Physik-DepartmentLehrstuhl E23

Magneto-Transport Based

Determination of the Magnetic

Anisotropy in Materials for

Spintronics

Diplomarbeit von

Sebastian Werner Schink

Technische Universitat Munchen

ii

Contents

1 Introduction 1

2 Fundamentals 32.1 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Zeeman energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Shape anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Crystalline anisotropy . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Ferromagnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Determination of the anisotropy parameters by FMR . . . . . . 8

2.4 Theoretical description of magneto-transport phenomena . . . . . . . . 102.4.1 Anisotropic magneto resistance . . . . . . . . . . . . . . . . . . 102.4.2 Magneto resistance in a single crystal . . . . . . . . . . . . . . . 112.4.3 The Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.4 The anomalous Hall effect . . . . . . . . . . . . . . . . . . . . . 14

2.5 Magnetic anisotropy and switching fields . . . . . . . . . . . . . . . . . 162.5.1 In-plane magnetic anisotropy . . . . . . . . . . . . . . . . . . . 162.5.2 The switching fields . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.3 Switching for dominantly biaxial magnetic anisotropy . . . . . . 222.5.4 Determination of the anisotropy type from the switching fields . 242.5.5 Predominantly uniaxial magnetic anisotropy . . . . . . . . . . . 262.5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 In-plane magneto-transport experiment . . . . . . . . . . . . . . . . . . 282.6.1 Measured quantities . . . . . . . . . . . . . . . . . . . . . . . . 292.6.2 Comment on ρxy . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7.1 CrO2 thin film samples . . . . . . . . . . . . . . . . . . . . . . . 312.7.2 The (Ga,Mn)As sample . . . . . . . . . . . . . . . . . . . . . . 342.7.3 The Fe3O4 reference sample . . . . . . . . . . . . . . . . . . . . 34

3 Magneto-transport in (Ga,Mn)As 373.1 Magneto-transport with applied field in the film plane . . . . . . . . . . 37

3.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iii

3.1.2 Evaluation of the magnetic easy axes . . . . . . . . . . . . . . . 433.1.3 The effect of temperature on the magnetic anisotropy in

Ga0.96Mn0.04As . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.4 AMR at high fields . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Magneto-transport with H out of plane . . . . . . . . . . . . . . . . . . 513.2.1 Magneto-resistance with H out of plane . . . . . . . . . . . . . 513.2.2 Hall and Anomalous Hall effect . . . . . . . . . . . . . . . . . . 52

4 Magnetotransport in CrO2 594.1 In-plane magnetotransport in (100) oriented CrO2 thin films . . . . . . 59

4.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.2 Determination of the switching fields and the magnetic easy axes 654.1.3 Quantitative analysis of the changes MTHx and MTHy at mag-

netization switches . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.4 AMR at high fields . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.5 Outlook on fully quantitative magneto-transport . . . . . . . . . 74

4.2 Magneto-transport in CrO2 with H out of plane . . . . . . . . . . . . . 764.2.1 Longitudinal magneto-resistivity . . . . . . . . . . . . . . . . . . 764.2.2 Hall effect and anomalous Hall effect in CrO2 . . . . . . . . . . 774.2.3 Temperature dependence of ρxx and ρAHE . . . . . . . . . . . . . 804.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Electrically detected ferromagnetic resonance 855.1 The ferromagnetic resonance setup . . . . . . . . . . . . . . . . . . . . 865.2 Ferromagnetic resonance in CrO2 . . . . . . . . . . . . . . . . . . . . . 875.3 The setup for electrical detection of FMR . . . . . . . . . . . . . . . . . 875.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.1 Time constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 EDFMR in the AHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Conclusions and Outlook 1076.1 Switching fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2 Fully quantitative description of the magneto-resistance . . . . . . . . . 1096.3 Electrically detected ferromagnetic resonance . . . . . . . . . . . . . . . 1106.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Acknowledgements 115

Bibliography 117

iv

Chapter 1

Introduction

Whenever magnetic materials are applied in devices, the magnetic anisotropy (MA) isof interest, as it determines the possible orientations of the magnetization M at low orzero applied field.

For example, perpendicular recording materials [LVM+93] are distinguished bytheir MA which favors the magnetization to orient perpendicular to the film plane.Magnetic nanostructures are very interesting for fundamental research, for example inthe context of quantum effects, e.g. triplet superconductivity [KGK+06], or quantuminformation processing. The drive towards higher integration densities in applicationsalso requires magnetic nanostructures.Quantitative measurement of the MA, as well as the theoretical description thus isimportant. However, most techniques to determine the MA, e.g. direct magnetizationmeasurements by SQUID1 or vibrating sample magnetometry, ferromagnetic resonance(FMR), etc., cannot be applied to magnetic micro- and nanostructures. This is dueto the fact, that in all these techniques, the signal intensity is proportional to thetotal magnetic moment and hence to the volume of the sample. The noise floor ofthe measurement apparatus limits the sensitivity, so that a minimal total magneticmoment is required. This minimal moment typically is larger that the magnetizationof a nanometer size sample. Therefore, it is important to develop scalable MAmeasurement methods for which the signal intensity is independent of the samplevolume.

In this thesis, we explore magneto-resistance (MR) based measurements as meansto measure the MA. The MR is scalable, the signal to noise ratio is independent of thesample volume, and small structures can be investigated provided electric contacts canbe attached. The MR in many material systems is well established and understood,and rich features such as anisotropic MR (AMR), planar Hall effect (PHE) andanomalous Hall effect (AHE) have been observed.

In this thesis, we study two (nearly) half metallic ferromagnets (HMF), CrO2 and

1SQUID = Superconducting QUantum Interference Device

1

Ga0.96Mn0.04As . HMF are ferromagnets with unique properties. While the densityof states on the Fermi energy is finite for one spin orientation, it exhibits a gap forthe other. This results in a 100% spin polarization. CrO2 is the best establishedhalf-metal, with spin polarizations of up to 98% [LAS97] observed experimentally.(Ga, Mn) As is one of the best understood ferromagnetic semiconductors [SJC04]. Forthe injection of holes from (Ga,Mn)As, spin polarizations as high as 80% have beenobserved [vDLvR+04]. The samples investigated are single crystalline thin films ofthe two materials. They consist of single ferromagnetic domains [GRM+05, SSG+00].These unique properties make them promising candidates for the application inspintronics.

Many aspects of MA can be inferred from the low field anisotropic magneto-resistance. A first series of experiments in this direction already was performed atTU-Delft [vD05, GKS+06, GRM+05]. At small applied fields, M is strictly orientedalong the easy axes (EA), which are determined by the MA. When the applied fieldamplitude or orientation is varied, M switches abruptly from one EA to another.These switches are reflected in the MR and allow to determine the MA. The mag-nitude of the applied magnetic field at which those switches occur (switching fields)as a function of the orientation of the applied field yield a characteristic pattern.From basic symmetry considerations, we find, that the superposition of first andsecond order uniaxial and first order cubic MA can be exhaustively described by 3different constellations. These constellations yield net uniaxial MA, net biaxial MAwith orthogonal or non-orthogonal easy axes. These 3 cases can be unambiguouslydetermined from the switching field pattern. For all 3, there is one experimentalexample in this thesis. The switches are determined from low field magneto-resistancemeasurements. However they could be measured with a number of other methods,such as MOKE2 as well. We find that the switches agree nicely with simulations forthe respective MA.

Ferromagnetic resonance (FMR), is one of the most sensitive techniques. Conven-tional, microwave absorption-detected FMR is not scalable. However, FMR can bedetected via the magneto-resistance [EJ63, Tod70, KKS78, GHMH05]. This techniqueshould be scalable, as the MR is. In this thesis, the full spectroscopic equivalencyof microwave absorption detected FMR and electrically detected FMR (EDFMR) isproven by simultaneous measurements. We fully understand the EDFMR in terms ofJoule heating. Furthermore, we observe resonant changes of the anomalous Hall effect(AHE), which to the best of our knowledge has not been reported before. The relativesignal size of the EDFMR in the AHE and in the longitudinal resistivity can be un-derstood quantitatively, and allows to access the temperature dependence of both theresistivity and the magnetization. This opens the way for the investigation of the MA,the MR and the magnetization dynamics in micro- and nanostructures.

2Magneto-Optical-Kerr-Effect

2

Chapter 2

Fundamentals

2.1 Ferromagnetism

Magnetism, and especially ferromagnetism in solids, shall be briefly introduced. Let Hbe the applied magnetic field, M the magnetization and B = µ0(H+M) the magneticinduction. µ0 = 4π × 10−7 Vs

Amis the vacuum permeability. For small H, the first

order effect on the magnetization is a good approximation, and M = χH. In generalthe susceptibility χ is a tensor, for isotropic materials it becomes a scalar. All solidshave a small negative contribution to χ, the diamagnetism. If there are free magneticmoments (spins Si), they will align parallel to the applied field, which yields a positivecontribution to χ, the paramagnetism. The Hamiltonian for the free magnetic momentsin an applied field reads

Hpara = −gµ0µB

∑i

Si ·H. (2.1)

From this can be derived, that the magnetization M(H, T ) is determined by theBrillouin function. For high temperatures T and low fields H, the susceptibility canbe expressed as χ ∝ T−1 (Curie-Law).Long range magnetic ordering, and hence ferromagnetism, requires exchange betweenthe single magnetic moments. This can be direct or indirect exchange such as superexchange, double exchange or RKKY exchange. [Blu01, L00, Mor01] In a simple pic-ture, the exchange energy between two magnetic moments (spins Si,j) is equal for allmoments. Then the Hamiltonian reads

H = Hpara − J∑

i,j(i<j)

Si · Sj. (2.2)

The strength of the coupling is then parameterized into J. Below the Curie-temperatureTC , a spontaneous magnetization is possible (M(H = 0) 6= 0). This also implies, thatM depends not only on H and T , but also on the history of the ferromagnet (hys-teresis). In a large ferromagnet, the magnetization is not necessarily homogenouslyoriented. Instead spatially separated parts of differently oriented magnetization, so

3

called domains, may be energetically more favorable.Above the Curie-temperature, the coupling is not strong enough to achieve a spon-taneous magnetization. Then, the magnetic moments behave similar to those in aparamagnetic system. Instead of the Curie law, the Curie Weiss-law

χ ∝ (T − Tc)−1 (2.3)

then describes the temperature dependence of the susceptibility at low H.In many ferromagnets, the shape of the M(H) depends on the orientation of the appliedfield H. This is due to the magnetic anisotropy, described in the following.

2.2 The magnetic anisotropy

In single domain ferromagnetic samples with finite dimensions or crystalline structure,the magnetization M will not necessarily align parallel to the applied field H. Insteadthere are energetically favorable (easy axes, EA) and unfavorable (hard axes) direc-tions. This anisotropy stems from various breaks of symmetry, due to the crystallinestructure, strain and shape of the ferromagnet. In general, the magnetic anisotropycan be expressed as contributions to the free energy. In general, M orients such, thatthe free energy is minimal. At low magnetic fields, this implies that M will only pointalong EA.Thus the knowledge of the magnetic anisotropy is crucial, for any application of mag-netic materials, e.g. for spin injection, magnetic recording etc.Following the discussion by Refs. [Mec97, Goe03], the free energy per unit volume Ftot

is given by:Ftot = Fstat + Fdemag + Faniso (2.4)

with the Zeeman energy Fstat, the shape anisotropy or demagnetization contributionFdemag and the crystalline anisotropy energy Faniso. These terms will be described indetail in the following paragraphs.For this description, a coordinate system shall be introduced. Since we will investigatethin films in the experiments, we introduce polar coordinates. θ and Φ are the anglesbetween the projections of M and H, respectively into the film plane and an arbitrarilychosen reference axis in the film plane. Ψ, ψ are the angles with respect to the filmnormal.

M = M(M, θ, Ψ) (2.5)

H = H(H, Φ, ψ) (2.6)

2.2.1 Zeeman energy

The Zeeman term, the magnetostatic energy of the magnetization in the externallyapplied field H, is given by

4

Fstat = µ0M ·H (2.7)

= −µ0MH(cos θ sin Ψ cos Φ sin ψ + sin θ sin Ψ sin Φ sin ψ + cos ψ cos Ψ)

2.2.2 Shape anisotropy

The demagnetization term Fdemag represents the energy in the stray field. In ferromag-nets with a large total magnetic moment, such as CrO2 the demagnetization is large.If the shape of such a ferromagnet is extremely anisotropic, e.g. a thin film, the Morientation with the least energy in the stray field is favored, e.g. M in the plane of thethin film. In ferromagnetic systems with relatively small total magnetic moment, suchas the diluted magnetic semiconductor (Ga, Mn) As (Ga,Mn)As, Fdemag is negligible,and the crystalline terms dominate.

Fdemag can be formalized into the demagnetization tensor←→N , and

Fdemag =µ0

2M←→N M (2.8)

In general tr←→N = 1, and for a thin (001) oriented film, only Nzz 6= 0, thus

Fdemag =µ0

2M2 sin2 Ψ cos2 θ (2.9)

For other extreme ratios between the length, width and height, like in nanowires, onecan approximate the sample as an ellipsoid, and then find analytical solutions for thetensor elements[SHCS05].

2.2.3 Crystalline anisotropy

Let for simplicity the local moments in the single domain ferromagnet be parallel tothe total magnetization M, and mi be the direction cosines.The symmetry of the environment of the magnetic moments M is crucial for the mag-netic anisotropy. Consider a single crystal with cubic symmetry, then the crystallinecontribution to the free energy reads as:

Fcubic = Kc1(m2xm

2y + m2

ym2z + m2

xm2z) + Kc2(m

2xm

2ym

2z) (2.10)

with the cubic anisotropy constants Kc1 and Kc2. In the polar coordinates used beforeit reads:

Fcubic =Kc1

4(sin2(2Ψ) + sin4 Ψ sin2(2θ)) +

Kc2

16(sin2 Ψ sin2(2Ψ) sin2(2θ)) (2.11)

5

Every breaking of the cubic symmetry will introduce an uniaxial contribution tothe free energy. Such symmetry breaking can arise e.g. from uniaxial strain via amagneto-elastic effect. For uniaxial strain perpendicular to the film plane, the effectivefree energy contribution writes as

Fu = Ku,strain(1

2− sin2 Ψ cos2 θ) (2.12)

The surface also breaks the symmetry of the free energy, and the phenomenologicalfirst order contribution can be written as:

Fsurface =Ks1

d(1

2− sin2 Ψ cos2 θ) (2.13)

with the film thickness d and the surface anisotropy constant Ks1. The surface contri-bution is very small in the samples used here.

All uniaxial contributions have the same angular dependence, thus cannot be distin-guished. Hence, they can be summarized, introducing one effective uniaxial anisotropyconstant.

Ku = Ku,strain +Ks1

d. (2.14)

For Ftot = constant, the total free energy surface is spherical. Exemplary plotsof the free energy surface for first order cubic, first order uniaxial, a combination ofboth, and combinations of two uniaxial anisotropy contributions along different axesare shown in Fig. 2.1.

In CrO2, the demagnetization contribution, and in Ga0.96Mn0.04As on GaAs theuniaxial out-of-plane crystalline anisotropy always dominate. Thus all other, smallereffects are only visible when M is rotated in the film plane, and these large anisotropycontributions therefore vanish. This cut through the free energy surface yields maximaand minima that are not necessarily extremal values for the 3D surface. The minimawill be the easy axes (EA), the maxima the hard axes. If there is only one EA in thefilm plane, e.g. minima are located at (100) and (-100), the 2D free energy surface willhave elliptical shape. This kind of system is called uniaxial.If there are two EA in the film plane, we call it biaxial. The two EA do neither neces-sarily enclose an angle of 90 nor do they always have the same Ftot. These asymmetricsituations occur from superposition of uniaxial and cubic contributions. As Fig. 2.1(d,eand f) shows, a superpositions of two uniaxial anisotropy contributions along differentaxes, cannot yield biaxial behavior. In the example shown in Fig. 2.1(f), the anisotropyalong x is twice as strong as along z. In the x-z-plane, there is no minimum in thefree energy between the x and the z axis, rather the z-axis itself is the easy axis. Thisresults from the free energy depending on the direction cosines squared.In Fig. 2.1(d,e), the free energy surface for uniaxial anisotropy contributions ofequal size along the x-axis and along the z-axis is shown. With the trivial identitym2

x + m2y + m2

z = 1 this can be transformed to a single uniaxial anisotropy along they-axis, as corroborated by Fig. 2.1(d,e).The first order of the cubic term (2.10) scales with m4

i . This yields two easy directions

6

Figure 2.1: Surfaces of constant free energy, calculated for cubic anisotropy (a), for uniaxialanisotropy with a hard direction along x (b), superposition of both (c), superposition ofuniaxial anisotropy along x and along z of equal strength (d,e), and of factor 2 differentstrength (f).[Bra06a]

within a plane, e.g. the cut along the x-y-plane through the free energy surface dis-played in Fig. 2.1(a or c).The total crystalline anisotropy can be summarized as

Faniso = Fu + Fcubic. (2.15)

Further discussion of the magnetic anisotropy is performed in context of the fundamen-tals of ferromagnetic resonance (Section 2.3) and of magnetic switching (Section 2.5).

7

2.3 Ferromagnetic resonance

In analogy to electron paramagnetic resonance, that allows to investigate paramagneticcenters in solids, ferromagnetic resonance applies to exchange coupled systems, that wecall ferromagnetic. The magnetic moments can absorb microwave photons resonantly.The resonance condition is given by the classical equation of motion

M = γM × Btot (2.16)

with the total magnetic field Btot = B0 + BFM, consisting of both the externallyapplied dc magnetic field B0 and the internal field in the ferromagnet BFM. In generalM is not parallel to B0 due to magnetic anisotropy.From Eq. (2.16) one can derive

Msat

γΨ sin Ψ =

∂Ftot

∂θ(2.17)

−Msat

γθ sin Ψ =

∂Ftot

∂Ψ(2.18)

with the saturation magnetization Msat, which shall only deviate slightly from theequilibrium orientation (Ψ0, θ0).The deviation shall be periodical in time (∆Ψ, ∆θ ∝ exp(iωt)).

The equilibrium orientation of M is given by

∂θFtot|θ=θ0 = ∂ΨFtot|Ψ=Ψ0 (2.19)

and then the power expansion of the free energy around the equilibrium condition isgiven by

Ftot = Ftot,0 +1

2∂2

θFtot(∆θ)2 + (∂θ∂ΨFtot)∆θ∆Ψ +1

2∂2

ΨFtot(∆Ψ)2 (2.20)

Still at the equilibrium condition, Eqs. (2.17) and (2.18) are only solvable, if

(ω

γ

)2

=1

M2sat sin2 Ψ

(∂2

θ

). (2.21)

The energy formulation of the FMR (2.21) and the two equations for the equilibriumcondition (2.19) describe the FMR.

2.3.1 Determination of the anisotropy parameters by FMR

For orientations with high symmetry of the free energy surface Eq. (2.19) yields M ‖H.These are the easy axes, and for sufficiently high H the intermediate hard and hardaxes. Consider a thin (001) oriented film, and Φ measured with respect to the (010)axis. Then Eq. (2.21) can be simplified. For H ⊥ film (ψ = 0) this yields:

8

(ω

γ

)=

(Bres,⊥ − µ0M + 2

Kc1

M+ 2

Ku

M

)(2.22)

and for H ‖ film (ψ = 90) and e.g. Φ = 0, H ‖ (010) this yields:

(ω

γ

)=

(Bres,‖(100) + µ0M + 2

Kc1

M− 2

Ku

M

)(Bres,‖(100) + 2

Kc1

M

)(2.23)

For the other in plane orientation with high symmetry (110), (ψ = 90, Φ = 45),or equivalent, one can rewrite

(ω

γ

)=

(Bres,‖(110) + µ0M + 2

Kc1

M− 2

Ku

M

)(Bres,‖(110) − 2

Kc1

M

)(2.24)

These equations allow to determine the cubic anisotropy constant quantitatively bymeasuring at different in plane orientations. For the uniaxial contributions the angulardependence is equal to that of the demagnetization term. Thus, the only and effectiveuniaxial anisotropy constant

Keff,u = Ku − µ0M2

2(2.25)

can be measured. To distinguish between those two contributions, one has to vary themagnetization, and this means that the microwave frequency and the magnetic fieldhave to be varied. Since every resonator is good only for one certain frequency, thisusually means that another setup has to be used. Still FMR is the most precise wayto quantify the anisotropy parameters, and detailed results have been published onvarious material systems. [RYK+06, vvS+98, LSF03, HWUS03, Ber02, Kre02]

9

2.4 Theoretical description of magneto-transport

phenomena

Here the theory necessary to describe the phenomena that occur in the magne-totransport experiments shall be briefly introduced. These are the anisotropicmagneto resistance (AMR), the (ordinary) Hall effect (OHE) and the anomalous Halleffect(AHE). In the following, the coordinate system from Section 2.2 will be used(Eqs. (2.5),(2.6)). For simplicity, angles θ and Φ of the in-plane projections of M andH are measured with respect to the current density j.

2.4.1 Anisotropic magneto resistance

In ferromagnetic materials, generally speaking, the resistivities ρ‖ parallel and ρ⊥perpendicular to the in-plane projection of the magnetization M, are not equal[Jan57, MP75]. This anisotropy, termed anisotropic magneto resistance (AMR) stemsfrom spin-orbit-coupling. Spin-orbit coupling is inherently dependent on the angle be-tween spin orientation, given by the magnetization, and current direction. We considerthin films, and assume for simplicity, that M is oriented in the film plane, and hence noHall effect like phenomena are superimposed onto the AMR. Considering Ohm’s lawE = ρj, with the electric field E. In this anisotropic case, the resistivity ρ is a tensor.With E‖, j‖ the components parallel to the magnetization, and E⊥, j⊥ perpendicularto it, Ohm’s law reads:

(E‖E⊥

)=

(ρ‖ 00 ρ⊥

)(j‖j⊥

). (2.26)

If ρ⊥ 6= ρ‖, then ρ is diagonal only in this coordinate system. When the coordinatesystem is rotated by choosing another current direction than parallel or perpendicularto the magnetization, the resistivity tensor is not diagonal.

(Ex

Ey

)=

(ρxx −ρxy

ρxy ρyy

) (jx

jy

). (2.27)

The resulting off-diagonal elements ρxy represent a transverse electric field, calledplanar Hall effect (PHE). This name is somewhat misleading, since the PHE is not aHall like effect, the PHE is symmetric with respect to H and M inversion, while the Halleffect is antisymmetric. In other words, the PHE is 180 periodic in M. For arbitraryangles θ between the current density j and in-plane M, the longitudinal resistivity ρxx

and the transverse resistivity ρxy are given by[O’H00]:

ρxx = ρ⊥ + (ρ‖ − ρ⊥) cos2(θ) (2.28)

ρxy = (ρ‖ − ρ⊥)1

2sin(2θ) (2.29)

10

This simple model is strictly speaking valid only for polycrystalline samples (cf.Section 2.4.2).

In elementary ferromagnetic metals like Co or Ni, the ratio (ρ‖ − ρ⊥) / ρ‖ is verysmall and thus quantitative comparison of ρxx and ρxy is difficult. In (Ga,Mn)As thisratio is much larger, and consequently the PHE is large[TKAR03]. In CrO2 the PHEcan also be well resolved[GKS+06].

Note that the assumption that M is strictly in-plane, requires to observe the purePHE, without any Hall components, in the transverse voltage. This is well achievablein experiment, by choosing the film plane such, that it is also an easy plane for themagnetization, and applying H in the film plane.

2.4.2 Magneto resistance in a single crystal

Until now, the relative orientation of j and M was important, however their orientationwith respect to the crystal did not influence the result. This seems plausible for amor-phous or polycrystalline materials, while in single crystals, the symmetry is reduced.Thus one expects anisotropic transport properties with respect to the crystalline axis,especially in single crystalline samples of high quality, such as the (Ga,Mn)As and theCrO2 samples investigated in this work. A completely different approach to the mag-neto resistance in general has been taken [MFH+05] and recently worked out in detailfor (Ga,Mn)As with specific film and current orientations by Limmer et al.[LGD+06].We follow Limmer’s approach. Let m be the unit vector along M, along j, n perpen-dicular to the film plane, and t := × n the in-plane orthogonal to j. Furthermore,mi are the direction cosines of m with respect to the crystalline axes and so forth.Now, using the Einstein summation convention, the resistivity tensor can be written,in the basis of the crystalline axes, as a power series expansion of mi:

ρij = aij + akijmk + aklijmkml + ... (2.30)

For cubic symmetry (crystal class m3m), as in (Ga,Mn)As, but not in CrO2, theexpansion up to the second order, can be rewritten as:

ρm3m = A

1 0 00 1 00 0 1

+ B

m2x 0 0

0 m2y 0

0 0 m2z

(2.31)

+ C

0 mxmy mxmz

mxmy 0 mymz

mxmz mymz 0

+ D

0 mz −my

−mz 0 mx

my −mx 0

with the resistivity coefficients

A = a11 + a1122, B = a1111 − a1122, C = a2323, D = a123 (2.32)

11

While Limmer [LGD+06] allows the coefficient A, which describes the resistance in thenonmagnetic case, to depend on H to fit his data quantitatively, the other coefficientsB, C, D are constants with the dimension of resistivity. This description works well, aslong as the magnitude of the magnetization, and hence the magnitude of the magneto-resistive effects are constant. However, as various temperatures are investigated, likein this thesis, the magnitude of the MR effects varies. Then B, C, D have to dependon the magnitude of the magnetization. Otherwise the model could not cope with thedecrease of magneto resistance effects, as the Curie temperature is approached. The Band C terms represent the AMR. They are symmetric with respect to H inversion, anddepend only on the M component in the film plane(, t). The D term is antisymmetricwith respect to H inversion and depends only on the out of plane M component. Itrepresents the AHE. Let us now further simplify this description by looking at (001)oriented samples, like the one used in the experiment later on, and taking the specificcurrent directions [110] (like the Hall bar used), and [100]. First for n = (001) andj ‖ [110],

ρxx = A +1

2(B − C) + C ( · m)2 − 1

2(B − C)(n · m)2 (2.33)

ρxy = B( · m)(t · m)−D(n · m) (2.34)

The last term in ρxx and in ρxy vanishes if M is oriented parallel to the film plane.With the angle θ between the current density j and in-plane M, Eqs. (2.33),(2.34) thenread as

ρxx = A +1

2(B − C) + C cos2 θ (2.35)

ρxy = B1

2sin(2θ) (2.36)

The main difference between Eqs. (2.28)(2.29) and Eqs. (2.35),(2.36) is that the pref-actor of the cos2 θ and 1

2sin(2θ) terms are two independent parameters in the latter

formulation. This view gets more interesting when the same formalism is applied toj ‖ [100]:

ρxx = A + B( · m)2 (2.37)

ρxy = C ( · m)(t · m) − D(n · m) (2.38)

For completely in plane oriented M, this yields

ρxx = A + B cos2 θ (2.39)

ρxy = C1

2sin(2θ) (2.40)

12

x

y

z

Ex

jBz

+ + + + + + +

- - - - - - - - - - - Ey

x

y

z x

y

z

Ex

jBz

+ + + + + + +

- - - - - - - Ey

Figure 2.2: Sketch of the Hall geometry. The sign of the resulting transverse electric fieldis correct for positive charges.

Now the prefactors of the cos2 θ and 12sin(2θ) terms in ρxx and ρxy are exchanged.

It has been shown, that for polycrystalline samples the prefactors of both, the cos2 θand 1

2sin(2θ) term, become equal (2

5B + 3

5C) by averaging over all crystal orientations,

equally for transverse and longitudinal resistance.The extension of this theory to the tetragonal CrO2 lattice, as performed by Limmeret al. will be discussed in section 4.1.4.

2.4.3 The Hall effect

The Hall effect (OHE) is a standard method to measure the carrier density. In socalled Hall bar samples, like those used here (see Section 2.7), on a geometrically welldefined structure, ohmic contacts are applied such that current can be driven, and fourpoint longitudinal and transverse (Hall) voltage can be measured simultaneously. Letus consider a cuboid sample with thickness d, length l and width w [Kop89, Kit71].An electric field E shall be applied along the x-axis, and a magnetic field B along thez-axis. The equation of motion of charge carriers is given by the Lorentz equation:

md2r

dt2+

m

τ· dr

dt= +q(E + (v × B

c)) (2.41)

With the speed of light c, a phenomenological scattering time τ , the electron mass m,that is assumed isotropic and the charge q = −e for electron- and q = +e for holetransport. In the equilibrium case, a drift velocity vd := v − vth can be introduced,where vth is the thermal velocity

m

τ· vd = +q(E + (vd × B

c)) (2.42)

Further the current density j = qnvd, with n the electron or hole concentration, andthe cyclotron frequency ωc = eB

mcis defined. To compensate the Lorentz-force and allow

j ‖ x-axis, a transverse electric field is present in equilibrium:

Ey =q

e· ωcτjxρxx (2.43)

13

with the longitudinal resistance ρxx = mne2 τ . Following

jxρxx = Ex (2.44)

And the Hall constant can be expressed as

RH =Ey

jxBz

(2.45)

RH =ωcτρxx

Bz

=1

qn(2.46)

Consequently the sign of the Hall effect depends on the type of charge carriers, andthe magnitude of the Hall effect is small when n is high.

2.4.4 The anomalous Hall effect

In ferromagnetic, and strongly paramagnetic materials, the Hall effect is temperaturedependent, and the Hall coefficient is different at low and high fields. These effectscan be understood as a Hall like effect caused by the magnetization. The empiricformula ρxy = R0B + RSM separates the Hall effect (OHE) as described before,from the anomalous Hall effect (AHE), which is proportional to the magnetizationM . The AHE stems from spin orbit coupling, but there are two different mechanismsyielding different phenomenology. The skew-scattering mechanism [Cha74] demandsRS ∝ ρxx, while the side-jump mechanism [Ber70] and the Berry phase [CB01] yieldRS ∝ ρ2

xx.The skew scattering mechanism is a classical effect. Spin-orbit-coupling yields anasymmetric potential in which the incoming plain wave is scattered, which equalsa deviation by a small angle. The corresponding probability is proportional to thespin-orientation (M) and to the frequency of scattering events (ρxx). If the spins areevenly distributed, no asymmetry occurs in the scattering, and thus no AHE. For(Ga,Mn)As the skew-scattering process dominates [VVD+97, MOSS98, Fra05]

The side-jump scattering is a quantum mechanical process. For the elastic scatter-ing, energy should be conserved. Since the potential energy is changed in the scatteringprozess, the impact parameter has to be changed. This causes a sideward jump, usu-ally in the order of 10−14 to 10−10 nm. Strongly simplified one can state, that boththe width of the jump and the probability are proportional to ρxx, while of course Malso has to be involved to have an asymmetry in the scattering. The side-jump yieldsρxy ≈ RSM ∝ Mρ2

xx. Watts et al.[WWM+00] find this proportionality in CrO2. How-ever, our experiments (Section 4.2.3) clearly indicate, that only the side-jump mech-anism cannot explain the temperature dependence of the AHE. Instead, coincidingside-jump and skew-scattering mechanism are required for the quantitative descriptionof the AHE in CrO2.Another mechanism that yields the same proportionality as the side-jump mechanism

14

is the Berry phase theory [YS02, CB01, YKM+99]. Here topological spin defects areintroduced as scatterers. The interaction is again spin-orbit coupling. At low tempera-tures it is predicted that RS increases exponentially, then at intermediate temperatureswith the well known ρxy ≈ RSM ∝ Mρ2

xx law.

15

2.5 Magnetic anisotropy and switching fields

In this paragraph we discuss how the orientation of the magnetic easy axes can bederived from magneto-resistance measurements. We follow the approach outlined inthe paper by Cowburn et al. [CGF+95]. In the spirit of recent work at TU Delft[GK05], we generalize the approach to the different combinations of uniaxial and biaxialanisotropy.

2.5.1 In-plane magnetic anisotropy

The idea is simple: Consider a magnetic film which is a single domain. During themagnetization reversal process, the changes in the orientation of the magnetizationM in this film will result in abrupt changes of slope or switches of the magneto-resistance. These switches occur at characteristic magnetic fields Hc1 and Hc2, whichdepend sensitively on the shape and crystalline magnetic anisotropy. An analysis ofthese characteristic fields thus yields information about the magnetization orientationand the magnetic anisotropy.Different measurement techniques can be used to this end, for example Kerr-microscopyas in Cowburn’s work or magneto-transport as discussed here.

Consider the external magnetic field H applied in the film plane. Assume further-more that the magnetization also always lies within the film plane, and forms a singledomain, so that magnitude of the magnetization M = |M| is identical to the saturationmagnetization Msat. The magnetic free energy is then given by

E(θ, Φ, H) = −MH cos(θ − Φ) + Ku sin2(θ − λ) +Kb

4sin2(2(θ − λ− β)), (2.47)

with the effective first-order uniaxial anisotropy constant Ku and the effective first-order cubic anisotropy constant Kb[CGF+95]. The anisotropy terms are termed ‘ef-fective‘ because the shape anisotropy contributions can also be written in the form ofEq. (2.47), so that both shape and crystalline anisotropy are lumped into Ku and Kb.Since only the free energy landscape within the film plane is investigated, one cannotdistinguish between second order uniaxial and cubic anisotropy. We thus introduceKb, the biaxial anisotropy term, taking account of these two contributions. Note alsothat higher-order magnetic anisotropy terms have been omitted, as they typically aresmall with respect to Ku and Kb.In the experimental configuration, the direction of the applied field H is known onlywith respect to the crystalline axes. In other words, the orientation of the magneticeasy axes still has to be determined. The angles Φ, θ, λ and β are defined as illustratedin Fig. 2.3. The angle Φ is enclosed by H and a crystalline axis, we chose (010) inCrO2 and (110) in Ga0.96Mn0.04As , respectively. M encloses the angle θ with this axis.The easy axis of the uniaxial contribution is oriented at λ with respect to the sameaxis, and encloses the angle β with the easy axis of the biaxial contribution.

In general, the easy axes of the biaxial anisotropy contribution can enclosean arbitrary angle (β)with the easy axis of the uniaxial anisotropy. However the

16

λ Φ

H

(010)

(001)

β

uniaxial

e.abiaxial

e.a

θ

M

λ Φ

H

(010)

(001)

β

uniaxial

e.abiaxial

e.a

θ

M

Figure 2.3: The experiment yields the angle Φ enclosed by H and a crystalline axis, here(010). M encloses the angle θ with this axis. The easy axis of the uniaxial contributionis oriented at λ with respect to the same axis, and encloses the angle β with the easy axisof the biaxial contribution. The labels for the crystalline axes are valid for CrO2.

(100) plane in (Ga, Mn) As and the (001) plane in CrO2, that are investigated inthis thesis, are planes of high symmetry. They are axially symmetric with respect totwo orthogonal directions. The main idea is, that each contribution to the magneticanisotropy in these planes, has to be of the same symmetry. The easy directions of thebiaxial and the uniaxial contribution hence can only be oriented along those symmetryaxes, or along their bisectors. This yields only two different possible arrangements, βis either 0 or 45. The differentiation of these two cases however, can be lumped intothe sign of Kb. θ = 0 either is a hard axis for the biaxial contribution (Kb < 0) or aneasy axis (Kb > 0). Therefore we will use β = 0 in all following discussions, withoutloss of generality.The easy axis of the uniaxial contribution has to fulfill the same symmetry argument.This leaves only two possible orientations, along the two symmetry axes. In thefollowing, Ku > 0 without loss of generality, since a negative Ku can be transformedto a positive one by rotating the coordinate system by 90.To emphasize again, for surfaces with low symmetry, such as (113) GaAs, theseconsiderations are no longer true, and β 6= 0, 45 may occur. In this case anotherassumption, crucial for our argumentation, would also fail: the surface of (113) GaAsis not an easy plane, thus M is not always oriented in plane, even for H in plane.Still most of the single crystalline ferromagnetic samples studied, like ours, fulfill thesymmetry criterion discussed here. Hence we provide a powerful tool, to determinethe magnetic anisotropy.

17

According to the Stoner-Wohlfarth model of coherent magnetization rotation, Mwill always point along the direction θ corresponding to a local minimum in E(θ, Φ, H).In not too large magnetic fields (more exactly for fields H for which the magneticanisotropy terms in Eq. (2.47) are much larger than the Zeeman term), the influenceof H on the magnetization orientation can be neglected. The magnetization will thenpoint along the directions θ given by (∂E/∂θ)|H=0 = 0 ∧ (∂2E/∂θ2)|H=0 > 0. Theseare the net magnetic easy axes, yielded by the superposition of the easy axis of the uni-axial and biaxial contributions. The superposition of uniaxial and biaxial anisotropyis exhaustively described by 4 possible situations, as illustrated in Fig. 2.4. For domi-nating biaxial anisotropy, the net easy axes are either orthogonal to each other (i) ortilted towards each other (ii), depending on the sign of Kb. For dominating uniaxialanisotropy, the net magnetic easy axis is always parallel to the easy axis of the uniaxialcontribution. (iii) and (iv).

We now explicitly determine the orientation of the net easy axes (e.a.) for the 4different cases. Equation (2.47) yields

∂E(θ, Φ, H)

∂θ

∣∣∣∣H=0

= Ku sin(2(θ − λ)) +Kb

2sin(4(θ − λ)) = 0, (2.48)

∂2E(θ, Φ, H = 0)

∂θ2

∣∣∣∣H=0

= 2Ku cos(2(θ − λ)) + 2Kb cos(4(θ − λ)) > 0. (2.49)

Equation (2.48) is equivalent to 0 = sin(2(θ − λ)) Ku + Kb cos(2(θ − λ)), with thesolutions

sin(2(θ − λ)) = 0 ⇔ θ =

θa = λθb = λ + 90

θ−a = λ + 180

θ−b = λ + 270

, (2.50)

or

Ku + Kb cos(2(θ − λ)) = 0 ⇔ θ =

θc = λ + αθd = λ + 180 − αθ−c = λ + 180 + α

θ−d = λ− α

, (2.51)

with

α =1

2cos−1(−Ku

Kb

). (2.52)

For 0 ≤ (−Ku

Kb) ≤ 1, α = 1

2cos−1(a) yields 0 ≤ α ≤ 45.

Which of the solutions (2.50) and (2.51) corresponds to an energy minimum(easy axis) and which to an energy maximum (hard axis) depends on the valuesof the constants Ku and Kb. As outlined above, there are only 4 qualitativelydifferent possibilities: (i) Kb > Ku > 0,(ii) −Kb > Ku > 0, (iii) Ku > Kb > 0, and(iv) Ku > −Kb > 0. The corresponding energies E, as well as the first and second

18

0 Ku-Ku

(ii) (i)(iii)(iv)

0 Ku-Ku

K1

(ii) (i)(iii)(iv)

i)iii)iv)ii)

0 < Ku < -K10 < -K1 < Ku 0 < K1 < Ku

0 < Ku < K1

Figure 2.4: The superposition of uniaxial and biaxial anisotropy is exhaustively describedby 4 possible situations. The easy axes of the biaxial contribution are symbolized by twoorthogonal black lines, the easy axis of the uniaxial contribution by a single black line.The lines representing the dominating anisotropy are fat, and the resulting anisotropy isshown by the net easy axes in blue. In total biaxial (i,ii) or uniaxial (iii,iv) anisotropyresult.

derivatives, are plotted in Fig. 2.5. As obvious from the periodicity of E, one hasuniaxial behavior for |Kb| < |Ku|, with one hard and one easy axis. The Kb-termthen only changes the slope of the energy landscape, but not the angles at which theenergy minimum occurs. This is due to the fact that Ku is positive in both cases,so that Eq. (2.51) can never be satisfied. Only the angles given by Eq. (2.50) thuscorrespond to energy extrema, alternating between minimum and maximum, butalways at multiples of 90.

The biaxial, often also called cubic, case |Kb| > |Ku| is much more rich, with twohard and thus also two easy directions. For (i) (Kb > Ku > 0) Eq. (2.50) yieldsthe energy minima and thus the easy axes. As obvious from the Fig. 2.5, the two

19

(i): +Kb>Ku>0

(iv): Ku>-Kb>0

(iii): Ku>+Kb>0(ii): -Kb>Ku>0

Figure 2.5: E(θ, λ = 0) together with its first and second derivative, for the cases (i)..(iv)shown in Fig. 2.4.

easy directions are not equivalent in energy in this case, one being energetically morefavorable than the other. The two hard axes, however, are energetically identical.Note also that the slope of the energy landscape on both sides of the energy maximais different, depending on whether one goes towards the easy-easy direction (θ = 0 inFig. 2.5) or the semi-easy direction (θ = 90). In other words, the energy landscape isaxially symmetric around the energy minima, but asymmetric around the maxima inthis case.

For the other biaxial case, (ii) (−Kb > Ku > 0) the easy directions are given byEq. (2.51). The energy is now axially symmetric around the maxima, and asymmetricaround the minima. Moreover, the angle between the two easy directions, the possibleorientations of M in the sample, now differs from 90. This can be thought of as ofthe two easy axes being rotated one towards the other, as in a scissor closing.

2.5.2 The switching fields

Having established the orientation of the easy directions in the previous paragraph,we now address the magnetization reversal process. For simplicity, we depart from theStoner-Wolfarth model and assume that M will always point along one of the easyaxes, also for finite magnetic fields. Because the magnetization reorientation typically

20

e.a.

H

M(1)

M(2)M(3)

M(2’)

e.a.

H

M(1)

M(2)M(3)

M(2’) e.a.

HM

-M

(i), (ii): Biaxial Anisotropy (iii), (iv): Uniaxial Anisotropy

Figure 2.6: In the case of biaxial magnetic anisotropy ((i),(ii)) two paths for a 180 re-orientation are possible. From the initial magnetization state M(1) to the finial stateM(3), M can either rotate clockwise, via the intermediate state M(2) or anticlockwisevia M(2′). The path with the energetically more favorable first switch is realized. Foruniaxial anisotropy ((iii),(iv)) the magnetization reorients directly from −M to M. Themagnetization is strictly oriented along easy axes (e.a.).

occurs at small fields, this is a reasonable approximation.

To include the effect of domain walls, or M pinning, we adopt the approachproposed by Cowburn et al.[CGF+95]. Assuming that the propagation (and notthe formation) of domain walls through the sample is limiting the magnetizationreorientation, the whole magnetization reversal process can be parameterized by onesingle energy ε. The energy ε can be interpreted as the maximum domain wall pinningenergy, so that a domain wall will nucleate and propagate through the entire sampleas soon as ε is provided. ε can also be thought of as the maximum energy barrier thedomain wall encounters during its propagation. The magnetization thus will undergoabrupt switches from one easy direction to the next, if the difference ∆E between themagnetic free energy of the initial and the final state is larger than ε. The switchingfields Hc1 and Hc2 can thus be obtained from the condition ∆E = ε.

Depending on the symmetry of the free energy, M occurs either via one switch orvia two switches. In an uniaxial landscape (iii,iv) with only one easy axis, M reversesby one single switch of 180. In a biaxial landscape, there are two easy axes, and thereversal occurs via an intermediate state. Hence two subsequent switches are observed.

In the biaxial case, for a reorientation by 180, two sequences of subsequent switchesare possible, the magnetization can rotate either clockwise or counterclockwise (seeFig. 2.6). It will always switch in the direction with the maximal energy gain. This

21

determines the sequence unambiguously.

2.5.3 Switching for dominantly biaxial magnetic anisotropy

Let us consider |Kb| > |Ku|, i.e. the cases (i) or (ii) of Fig. 2.4, with two easy directionsin the film plane. First we will discuss case (i), with collinear uniaxial and cubic energycontributions. For Kb > Ku > 0, the cubic and the uniaxial term in Eq. (2.47) becomeminimal for the same θ. The local minima of the magnetic free energy then occur atthe angles given by Eq. (2.50). The two easy magnetic axes are orthogonal, but oneis easier than the other, as the uniaxial contributions to the energy are different. Thevalues of E, for M along the respective easy directions read:

E((θ − λ) = 0) = E0 = −MH cos(Φ− λ), (2.53)

E((θ − λ) = 90) = E90 = −MH sin(Φ− λ) + Ku, (2.54)

E((θ − λ) = 180) = E180 = MH cos(Φ− λ), (2.55)

E((θ − λ) = 270) = E270 = MH sin(Φ− λ) + Ku. (2.56)

The resulting energy differences for the first and second switch are (see Fig.2.6(left)):

∆E1 = E180 − E90 = MH cos(Φ− λ)− (−MH sin(Φ− λ) + Ku), (2.57)

∆E2 = E90 − E0 = −MH sin(Φ− λ) + Ku − (−MH cos(Φ− λ)), (2.58)

∆E1 = E270 − E0 = MH sin(Φ− λ) + Ku − (−MH cos(Φ− λ)), (2.59)

∆E2 = E0 − E90 = −MH cos(Φ− λ)− (−MH sin(Φ− λ) + Ku), (2.60)

∆E1 = E270 − E180 = MH sin(Φ− λ) + Ku − (MH cos(Φ− λ)), (2.61)

∆E2 = E180 − E90 = MH cos(Φ− λ)− (−MH sin(Φ− λ) + Ku), (2.62)

∆E1 = E0 − E90 = −MH cos(Φ− λ)− (−MH sin(Φ− λ) + Ku), (2.63)

∆E2 = E90 − E180 = −MH sin(Φ− λ) + Ku − (MH cos(Φ− λ)). (2.64)

This yields the switching fields

180 ⇒ 90 : Hc1 =(ε + Ku)

M(cos(Φ− λ) + sin(Φ− λ))= HA, (2.65)

90 ⇒ 0 : Hc2 =(ε−Ku)

M(cos(Φ− λ)− sin(Φ− λ))= HB, (2.66)

270 ⇒ 0 : Hc1 =(ε−Ku)

M(cos(Φ− λ) + sin(Φ− λ))= HC , (2.67)

0 ⇒ 90 : Hc2 =(ε + Ku)

M(− cos(Φ− λ) + sin(Φ− λ))= HD, (2.68)

22

270 ⇒ 180 : H =(ε−Ku)

M(− cos(Φ− λ) + sin(Φ− λ))= −HB, (2.69)

180 ⇒ 90 : H =(ε + Ku)

M(cos(Φ− λ) + sin(Φ− λ))= HA, (2.70)

0 ⇒ 90 : H =(ε + Ku)

M(− cos(Φ− λ) + sin(Φ− λ))= HD, (2.71)

90 ⇒ 180 : H =(ε−Ku)

M(− cos(Φ− λ)− sin(Φ− λ))= −HC . (2.72)

We will now address the other possible case (ii) with net biaxial magnetic anisotropy.For −Kb > Ku > 0, the uniaxial and biaxial term in Eq. (2.47) become minimal for θand θ + 45, respectively. In this case a hard axis of the biaxial contribution coincideswith the easy uniaxial direction (Fig. 2.4(ii)). The local energy minima in the magneticfree energy then occur at the angles given by Eq. (2.51). The two easy magnetic axesthus are tilted one towards the other, and they are equivalently easy. One then has

E(θ − λ = α) = Eα = −MH cos(α− (Φ− λ)) + Ku sin2(α) +1

4Kb sin2(2α),

E(θ − λ = 180 − α) = E180−α = MH cos(α + (Φ− λ)) + Ku sin2(α) +1

4Kb sin2(2α),

E(θ − λ = 180 + α) = E180+α = MH cos(α− (Φ− λ)) + Ku sin2(α) +1

4Kb sin2(2α),

E(θ − λ = 360 − α) = E360−α = −MH cos(α + (Φ− λ)) + Ku sin2(α) +1

4Kb sin2(2α).

The resulting energy differences for the first and second switch are:

∆E1 = E180+α − E360−α = MH cos(α− (Φ− λ)) + MH cos(α + (Φ− λ)), (2.73)

∆E2 = E360−α − Eα = −MH cos(α + (Φ− λ)) + MH cos(α− (Φ− λ)), (2.74)

∆E1 = E180+α − E180−α = MH cos(α− (Φ− λ))−MH cos(α + (Φ− λ)), (2.75)

∆E2 = E180−α − Eα = MH cos(α + (Φ− λ)) + MH cos(α− (Φ− λ)), (2.76)

∆E1 = E360−α − Eα = −MH cos(α + (Φ− λ)) + MH cos(α− (Φ− λ)), (2.77)

∆E2 = Eα − E180−α = −MH cos(α− (Φ− λ))−MH cos(α + (Φ− λ)), (2.78)

∆E1 = E360−α − E180+α = −MH cos(α + (Φ− λ))−MH cos(α− (Φ− λ)), (2.79)

∆E2 = E180+α − E180−α = MH cos(α− (Φ− λ))−MH cos(α + (Φ− λ)).(2.80)

23

The condition ∆E = ε again yields the switching fields for a given switching sequence:

180 + α ⇒ 360 − α ⇒ α :

Hc1 =ε/M

(cos(α− (Φ− λ)) + cos(α + (Φ− λ)))= HI , (2.81)

Hc2 =ε/M

(cos(α− (Φ− λ))− cos(α + (Φ− λ)))= HII , (2.82)

180 + α ⇒ 180 − α ⇒ α :

Hc1 =ε/M

(cos(α− (Φ− λ))− cos(α + (Φ− λ)))= HII , (2.83)

Hc2 =ε/M

(cos(α− (Φ− λ)) + cos(α + (Φ− λ)))= HI , (2.84)

360 − α ⇒ α ⇒ 180 − α :

Hc1 =ε/M

(cos(α− (Φ− λ))− cos(α + (Φ− λ)))= HII , (2.85)

Hc2 = − ε/M

(cos(α− (Φ− λ)) + cos(α + (Φ− λ)))= −HI , (2.86)

360 − α ⇒ 180 + α ⇒ 180 − α :

Hc1 = − ε/M

(cos(α− (Φ− λ)) + cos(α + (Φ− λ)))= −HI , (2.87)

Hc2 =ε/M

(cos(α− (Φ− λ))− cos(α + (Φ− λ)))= HII . (2.88)

2.5.4 Determination of the anisotropy type from the switchingfields

In the experiment, the switching fields Hc1 and Hc2 are determined as a functionof the magnetic field orientation Φ. As they reflect the magnetic symmetry of thesystem, the cases (i)-(iv) can be identified from this dependence. In the cases withnet biaxial anisotropy, (i) and (ii), the switching fields converge for 4 orientationsΦ1−4: Hc1(Φ1−4) = Hc2(Φ1−4). In case (i) Eqs. (2.65)-(2.72) yield for 180 ⇒ 90 ⇒0 : Φ1 = tan−1

(Ku

ε

), and for 270 ⇒ 0 ⇒ 90 : Φ2 = tan−1

(−εKu

)and so forth.

Hence Φ1 = 90+Φ2. In other words, the case (i) can be identified, when the switchingfield coincide at directions orthogonal to each other. The switching fields according toEqs. (2.65)-(2.72) for this case are shown in Fig. 2.7(i), with the ratio Ku

ε= 0.1 and

λ = 0 chosen arbitrarily. The larger Ku

εis, the more asymmetric are the switching fields

with respect to the hard axes. At the hard axes, Hc2 diverges, and Hc1 is discontinuous

24

(i) (ii)

Figure 2.7: Simulation of Hc1 and Hc2 as a function of the orientation Φ() of the magneticfield, in case of dominantly biaxial magnetic anisotropy. For case (i) easy axes of the inplane uniaxial and biaxial contributions coincide. The easy axes (e.a.) are the bisectors ofthe hard axes (h.a.). For the case (ii) the hard axes of the in plane uniaxial and the biaxialcontribution coincide. The easy axes (dark green) are orthogonal to those orientationswhere the switching fields coincide (red).

with the one-sided limits H<c1 and H>

c1. Their relation gives information about theuniaxial anisotropy relative to the energy barrier:

|H<c1 −H>

c1|H<

c1 + H>c1

=Ku

ε. (2.89)

The easy axes are the bisectors of the hard axes, and not, as one might think, the Φ1−4

discussed above.The symmetry is different in case (ii). The hard axes are, as for case (i) orthogo-

nal, and occur at the orientations where Hc2 diverges. In contrast to case (i), Hc1 iscontinuous there. Let us emphasize again, that the easy axes are neither the bisectorsof the hard axes, nor those orientations Φ0 for which HI = HII = H0. Instead the easyaxes are given by Eq. (2.51), with the parameter α given by

HI = HII ⇔ cos(α + (Φ0 − λ)) = − cos(α + (Φ0 − λ))

⇔ cos(α + (Φ0 − λ)) = 0 ⇔ α = 90 − (Φ0 − λ). (2.90)

Thus, α = 90 − (Φ0 − λ), the magnetic easy axes are orthogonal to the directionsof coinciding Hc1 and Hc2. That these directions are not orthogonal to each other,identifies case (i) from experimental Hci(Φ) data. All these symmetry properties canbe seen from the simulation, where α = 33 and λ = 90, in Fig. (2.7(ii)). The valuesof the ratio ε/M can also be obtained from Φ0: ε/M = H0 sin(2(Φ0 − λ)).The value of α allows to determine the ratio of cubic to uniaxial anisotropy in thesample from a measurement of the switching fields. From (2.52), Ku

Kb= − cos(2α). As

25

only the ratios Ku

Kband ε

Mare relevant for the switching fields, the absolute values of

Ku and Kb unfortunately cannot be determined.

2.5.5 Predominantly uniaxial magnetic anisotropy

The symmetry of the free-energy contour with respect to magnetization orientationchanges drastically when |Kb| < |Ku|, i.e. for uniaxial magnetic anisotropy (cases (iii)and (iv)). Collinear easy (iii) or hard (iv) axes of the uniaxial and biaxial contributionsdo yield a different free-energy. However there is only one net magnetic easy axis, whichis equal to the uniaxial easy axis. In both cases θ = 0 and θ = 180 give the twopossible orientations of M and the energy reads:

E(θ = 0, iiiiv) = Eα = −MH cos(Φ) + 0

14

Kb (2.91)

E(θ = 180, iiiiv) = E180 = MH cos(Φ) + 0

14

Kb. (2.92)

The energy difference, however for both (iii) and (iv) is given by:

∆E = ±2MH cos(Φ). (2.93)

M reorients by 180 in one single step. Hence there is only one switching field, Hc,which is minimal for H applied along the easy axis, and diverges for H along themagnetic hard axis.

Hc =ε

2M cos(Φ)(2.94)

A simulation of Hc is shown in Fig. 2.8 as a function of the orientation Φ of the magneticfield H.

2.5.6 Summary

In conclusion, we have shown that for E of the form (2.47) the different possible mag-netic anisotropy symmetries (i)-(iv) can be determined from measurements of Hc1(Φ)and Hc2(Φ). For dominantly biaxial magnetic anisotropy, Hc2(Φ) diverges 2 times per180. If Hc1 = Hc2 occurs every 90 the symmetry is of case (i) type. In this caseHc1 can be discontinuous at the angles Φ where Hc2 diverges. If Hc1 = Hc2 is fulfilledfor angles Φ that are not orthogonal to each other, the symmetry is of type (ii). Foruniaxial anisotropy, Hc2 diverges only once per 180.

26

0.1

1

10

0

30

6090

120

150

180

210

240270

300

330

0.1

1

10

Hc (

/2M)

hard axis

easyaxis

Hc

Figure 2.8: Simulation of the switching field as a function of the orientation Φ() of themagnetic field, in case of dominantly uniaxial magnetic anisotropy. The smaller biaxialcontribution does not influence the minima of the free energy. It therefore also does notaffect the switching fields, and thus cannot be measured by means of magnetic switching.

27

|µ0H| range [mT] µ0H increment [mT]

<5 0.1

5-30 0.2

30-100 1

100-200 5

Table 2.1: The applied field is varied stepwise according to the same algorithm for all in-plane measurements (Sections 3.1 and 4.1). The step size (increment) is decreased closeto H = 0.

2.6 In-plane magneto-transport experiment

The experiments are performed in a 10 T superconducting magnet system. Thetemperature is controlled with a variable temperature insert. The sample is mountedon a rotatable stage on the dipstick. This stage is designed such, that the appliedfield is always parallel to the film plane, but its orientation with respect to the currentdensity is variable.

The magnetic field control was programmed to rapidly access specific H values,and then keep H constant (non-persistent mode), while the measurement is performed.To allow for some averaging, the waiting time, for which H was kept constant, waschosen to 1 s. The H increment is decreased for small H, to better resolve the effectsof magnetic switching that occur here, and increased for larger H to accelerate themeasurement. Table 2.1 displays the algorithm according to which the increment waschosen in all measurements in sections 3.1 and 4.1. The magnetic field is accuratewithin an error of ≈ 1 mT, which stems from trapped flux in the superconductingcoils.

The variable temperature insert contains a resistive heater located on the bottomof the sample space. Here 4He gas flow can be applied by means of a needle valve.Since the sample space is pumped, with typical pressures in the order of mBar,temperatures down to 1.7 K can be reached. For temperatures above 100 K, no 4Hegas flow is applied, and the maximal temperature is 300 K.

To allow for rotation of the magnetic field with respect to the current density,the sample is mounted on a rotatable sample holder. For 5 K and 25 K, 2

steps, for 50 K 5 steps, and for 60 K 20 steps are chosen. Φ is in the followingused as the angle between the direction of the applied field H and current pathalong the crystalline (110) axis for the experiments on Ga0.96Mn0.04As and along(010) for the experiments on CrO2 , respectively. The automatically rotatablesample holder allows to access Φ values from 0 to 200 in the definition and thegeometry used for Ga0.96Mn0.04As , and −97 to +95 in the case of CrO2 , respectively.

28

All measurements are performed in dc mode. Current is driven by dint of a KeithleyK2400 Source Meter in the current source and current measuring mode. A limit is setfor the 2 point voltage, so that the current will be reduced if the 2 point voltage exceeds21 V. During the measurements the current is recorded to verify that this limit is notexceeded. On both current paths of the sample, along the (110) and the (110) axis forGa0.96Mn0.04As , and along the (010) and the (001) axis for CrO2 (sample geometriessee Figs. 2.13 and 2.12), the longitudinal Vxx and the transverse voltage drop Vxy arerecorded, all four simultaneously, using Keithley K2010 and K2000 multimeters. Theyare set to the ’slow’ mode and 5 point moving average, which yields an effective timeconstant of 0.5 s. The sensitivities were set to 1 V for the longitudinal and 0.1 V forthe transverse voltages. The I-V curves are ohmic up to the applied current density.It has been shown that the common practice of switching the current between positiveand negative values at every data point introduces noise and would extend the necessarytime for each data point by a factor of 10. Therefore this method was not applied here.

2.6.1 Measured quantities

The physical quantities directly measured in the magneto-transport experiments arethe angle Φ and the magnitude H of the applied magnetic field, the temperature T ,the current I and the voltages Vxx and Vxy. To allow for quantitative comparisonwe will, as described in the following, convert the latter into the resistivities ρxx andρxy. For B442, the Ga0.96Mn0.04As sample on which extensive measurements have beenperformed, the current level is of I = 10 µA. The Hall bar width is w = 50 µm. Thedistance between the longitudinal voltage probes is l = 175 µm. The film thickness isd = 20 nm. The resistances are Rxx = Vxx

Iand Rxy = Vxy

I. It is common practice to

define the resistivities as follows:

ρxx =Ex

jx

= Rxxwd

l(2.95)

ρxy =Ey

jx

= Rxywd

w(2.96)

For Data, the CrO2 sample used in the in-plane magneto-transport measurements,the Hall-bar geometry (w, l) equals that of the Ga0.96Mn0.04As sample, and the filmthickness is d = 100 nm. The current applied is I = 5 mA. For Bonomi, theCrO2 sample used in ED-FMR and out of plane magnetotransport the film thicknessis d = 100 nm and the width of the Hall bar is w = 80 µm. In the magnetotransportexperiments, the current applied is I = 1 mA, and the longitudinal voltage probes areseparated by l = 600 µm. The situation in the ED-FMR setup will be discussed inchapter 5. Table 2.2 shall give an impression of the characteristic order of magnitudeof the voltages, resistances and resistivities discussed here. In the discussion of theexperiments we focus only on the resistivities.

29

CrO2 Bonomi CrO2 Data Ga0.96Mn0.04As B442

Vxx (V) 10−2 10−2 10−1

∆Vxx (V) 10−6 10−4

Vxy (V) 10−5 10−4 10−3

∆Vxy (V) 10−6 10−3

Rxx (Ω) 10+1 10+1 10+4

∆Rxx (Ω) 10−3 10+1

Rxy (Ω) 10−2 10−1 10+2

∆Rxy (Ω) 10−3 10+2

ρxx (µΩcm) 10+1 10+1 10+3

∆ρxx (µΩcm) 10−3 10+1

ρxy (µΩcm) 10−1 10−1 10+2

∆ρxy (µΩcm) 10−3 10+2

Table 2.2: The orders of magnitude of the longitudinal and transverse voltage, resistanceand resistivity in the measurements described in this thesis. The absolute values (V , R,ρ), as well as the variations with M orientation (∆V , ∆R, ∆ρ) are shown.

2.6.2 Comment on ρxy

The transverse resistivity has been defined as ρxy = Ey

jxin analogy to the definition

of ρxx from Ohm’s law. While ρxx has to be always positive, to satisfy basic rules ofthermodynamics, the sign of ρxy is not a priori given. It depends on the coordinatesystem chosen, and on characteristic properties of the sample. Furthermore the off-diagonal elements of the resistivity tensor are connected via ρxy = −ρyx, and thus oneof them is always negative. This does not violate any fundamental laws like energyconservation, since Ey and jx are components in orthogonal directions, the contributionfrom ρxy to the power density is zero. However most authors avoid negative resistivitycomponents, and so do we, as long as it makes sense. In the in-plane magnetotransportin (Ga, Mn) As and in the AHE experiments, the transverse signal is positive for some Hvalues, negative for others. As discussed in Section 3.1, we understand the underlyingphysics well. ρxy is calculated as described in Eq. (2.96) and we allow, against commonpractice, positive and negative values.

30

100% spinpolarisation predicted

energy

spin-

spin-

DO

S (

a.u

.)

0

EF

100% spinpolarisation predicted100% spinpolarisation predicted100% spinpolarisation predicted

energy

spin-

spin-

DO

S (

a.u

.)

0

EF

a

b

c

Chromium

Oxygen

(a) (b)

Figure 2.9: (a): Rutile type crystalline structure of CrO2, with a = b = 0.4419 nm andc = 0.2912 nm. (b): The band structure calculations [LAS97] yield 100% spin polarizationat the Fermi energy.

2.7 Samples

Four samples are investigated in this thesis: the CrO2 samples Data and Bonomi, themagnetite sample M18a and the Ga0.96Mn0.04As sample B442. All four samples havebeen fabricated by collaborators [vD05, Bra06b, Fra05]. We keep their naming.

2.7.1 CrO2 thin film samples

The single-crystalline (100) CrO2 films were deposited epitaxially on (100)-orientedTiO2 substrates by chemical vapor deposition in the group of A. Gupta atUA Tuscaloosa. For details about the film growth and the structural and magneticproperties, we refer to Refs. [LGX99, MXG05]. The CrO2 lattice is tetragonal, witha = b = 0.4419 nm and c = 0.2912 nm. The films are strained on the tetragonal TiO2

substrate, with a lattice mismatch of 3.8% along b and 1.6% along the c-axis. Fig-ure 2.9(a) shows the rutile type crystalline structure of CrO2, with a = b = 0.4419 nmand c = 0.2912 nm. The CrO2 films grow such, that the b− and c− axes are orientedin the film plane.

CrO2 is the best established half-metallic ferromagnet known today, [dGMvEB83,LAS97, HTC+03, JSY+01, PWIX02] with an experimental spin-polarization of up to98% [SBO+98]. A sketch of the band structure is shown in Fig. 2.9(b).The magnetization of a 2 mm2 × 100 nm sample of CrO2, is investigated by SQUIDmagnetometry. For details on the SQUID magnetometer confer to Ref. [Gep04]. Fig-ure 2.10(a) and (b) shows the hysteresis loops with H applied parallel and perpendicularto the film plane, for 300 K (a) and 5 K (b), respectively. The saturation magnetizationis reduced upon increasing temperature. The applied field at which the magnetization

31

Figure 2.10: SQUID magnetization measurements on a 2 mm2 × 100 nm sample of CrO2,with H applied parallel and perpendicular to the film plane, for 300 K (a) and 5 K (b).

saturates, is much smaller for H in the film plane than for H perpendicular to it.

At TU-Delft, Hall-bar structures were patterned into the CrO2, using optical lithog-raphy and wet chemical etching. The CrO2 is covered by an insulating, antiferromag-netic native Cr2O3 layer of ≈ 2 nm thickness, which is very robust with respect toetching. Hence, Ohmic contacts were realized by depositing Au immediately aftersputter-cleaning the film surface in-situ in an Ar plasma [KGK+06]. Investigations ofthe magneto-resistance of different CrO2 Hall bars have been published recently, basedon work at TU-Delft [KGvD+05] and the measurements introduced in Section 4.1[GKS+06].The sample Bonomi is a 80 µm wide bar, width three pairs of contacts, each 300 µmfrom the next pair, etched into a 100 nm thick CrO2 film. Such a ”large” Hall barstructure contains enough CrO2 material to enable sensitive FMR experiments. It hasbeen investigated in sections 4.2 and 5, and at TU-Delft [vD05]. An image of Bonomiis displayed in Fig. 2.11.

The sample Data is a 50 µm wide, 90 angled bar. Its two branches are orientedalong the crystalline (001)- and (010)-axes. This allows to investigate the magne-totransport with both current orientations simultaneously. As Bonomi, it has beenpatterned at TU-Delft[vD05]. An optical microscopy image is displayed in Fig. 2.12.The angles Φ and θ, of the applied field and the magnetization, respectively, are mea-sured with respect to the crystalline (010)-axis. This convention is used, independentof whether the magnetotransport along (001) or (010) is investigated.

32

Figure 2.11: Optical microscopy image [vD05] of the CrO2 sample Bonomi. The Hall baris 80 µm wide, and the distance between sequent voltage probes is 300 µm.

(001)

(010)

HM

θ Φ

(001)

(010)

HM

θ Φ

Figure 2.12: Optical microscopy image of the CrO2 sample Data. The angles Φ and θ, ofthe applied field and the magnetization, respectively, are measured with respect to thecrystalline (010)-axis.

33

Φ

θ

(110)

(110)

H

M

Figure 2.13: Optical microscopy image of the Ga0.96Mn0.04As sample B442. The angles Φand θ, of the applied field and the magnetization, respectively, are measured with respectto the crystalline (110)-axis.

2.7.2 The (Ga,Mn)As sample

Single crystalline (Ga, Mn) As thin films were grown by molecular beam epitaxyon (001) GaAs wafers at Universitat Ulm [KFS+03, Koe01, Fra05]. We consider aMn concentration of 4.2%, a film thickness of 20 nm a Curie temperature of TC ≈ 70 K.

The sample (B442) investigated in chapter 3 has been patterned by photolithog-raphy and wet chemical etching, using the same mask as for Data. An optical mi-croscopy image of the Ga0.96Mn0.04As sample B442 is displayed in Fig. 2.13. In thein-plane magneto-transport measurements (Section 3.1), the angles Φ and θ, of theapplied field and the magnetization, respectively, are measured with respect to thecrystalline (110)-axis, independent of whether the magnetotransport along (110) or(110) is investigated.

2.7.3 The Fe3O4 reference sample

The magnetite sample M18a is investigated in chapter 5 to verify the bolometric na-ture of the electrically detected ferromagnetic resonance. This work was performedin cooperation with A. Brandlmaier, who gives a more detailed description on thesample fabrication and properties in his thesis [Bra06b]. The magnetite (Fe3O4) lay-

34

ers were grown on (100) MgO substrates using pulsed laser deposition.[RBK+03] Wehere consider a 32 nm thick, coherently strained single crystalline film. Ohmic con-tacts are attached to a 2 × 2 mm2-piece of this sample by wedge bonding Al wires inVan-der-Pauw-geometry.

35

36

Chapter 3

Magneto-transport in (Ga,Mn)As

Magneto-transport measurements allow to access many different aspects of the mag-netic properties of the ferromagnetic semiconductor (Ga,Mn)As. With an externalmagnetic field H applied in the film plane, the magnetic anisotropy (MA), namelythe orientation of the magnetic easy axes and the coefficients B and C in Eq. (2.31),describing the anisotropic magneto-resistance (AMR), can be quantified. With H per-pendicular to the film plane, the Curie temperature TC can be determined from thelongitudinal resistivity, from the magneto-resistance and from the anomalous Hall re-sistivity at high temperatures. In the same geometry the origin of the anomalous Halleffect and the magnetization as a function of temperature and applied field will be dis-cussed. The evaluation of the carrier density, several 1020 cm−3 is troublesome, sincethe (ordinary) Hall effect (OHE) is very small. However a rough estimation will begiven.

3.1 Magneto-transport with applied field in the

film plane

In this section, we discuss the magneto-transport in a Ga0.96Mn0.04As sample, withapplied field H, magnetization M and current density j all in the film plane. In thisgeometry the Hall effect and the anomalous Hall effect vanish and resistance is domi-nated by the AMR. The magnetic anisotropy can be mapped by monitoring character-istic magneto-transport features, such as switching fields, or maxima in the hystereticcontribution to the magneto-transport, as a function of H orientation. The sample isbiaxial at 5 K, with the magnetic easy axis along the (100) and (010) crystalline axes.At 25 K it is uniaxial with the easy axis along (110).

3.1.1 Results

Both ρxx(H) and ρxy(H) exhibit clear switches, enclosing a range of hystereticbehavior. ρxy(H) switches between levels nearly symmetric to zero. In ρxx(H) the

37

VxxH

j

−20°VxxVxxH

j

−20° VxxVxxVxx

H

j+20°

H

j+20°

H

j

−20°

Vxy

H

j

−20°

VxyVxy

VxyVxyVxy H

j+20°

H

j+20°

Figure 3.1: Longitudinal ρxx and transverse resistivity ρxy for the Ga0.96Mn0.04As sampleat 5 K. H is applied in the film plane, at ± 20 with respect to the current direction,corresponding to Φ = 70, 110. Sweeps from negative to positive H are red, vice versablack. Clear switches can be observed. ρxy shows the symmetry expected for a planarHall effect.

switching events occur at the same H values, but reside on top of a large non hystereticmagneto-resistance.

Figure 3.1 shows typical data, for Φ = 70 and Φ = 110, corresponding to Horiented at ± 20 with respect to the current direction along (110).

ρxx and ρxy are symmetric with respect to field reversal. For ρxx the switches areof the order of 10 µΩ cm on a background of 8 × 103 µΩ cm. ρxy switches betweentwo values nearly symmetric to ρxy = 0. Data recorded with applied magnetic fieldswept from negative to positive H values (up sweep) will be denoted ρup sweep

xx (H)and ρup sweep

xy (H) respectively. Those with applied magnetic field swept from positiveto negative H values (down sweep) will be denoted ρdown sweep

xx (H) and ρdown sweepxy (H)

respectively. Up sweep and down sweep only differ within ± 20 mT. In this interval,the magnetization is hysteretic. At one specific H value, M is different for the up andthe down sweep. Furthermore for the two angles in Fig. 3.1 the respective orientationsof H are symmetric with respect to the current direction, which also is a hard axis ofthe crystal as we will see later. Therefore the orientations of M in the two geometriesshould be symmetric with respect to the same axis.

38

The spectra for ρxx are, within the experimental error the same for both angles.For ρxy a sign change for the two orientations is observed. Considering Eqs. (2.35),we expect the planar Hall (PHE) signal to be proportional to 1

2sin (2(Φ− 90)), which

yields antisymmetric values for 70 and 110. For the longitudinal resistance the AMRpredicts proportionality to cos2 ((Φ− 90)). This would lead to the same value for70 and 110. Observing this symmetry proves that the signal observed in ρxy is areal planar Hall signal. This check is important, since imperfectly aligned transversecontacts could impose a spurious contribution proportional to ρxx on ρxy.

To further investigate the magnetic switching behavior, it seems desirable to sepa-rate the hysteretic parts from the non-hysteretic magneto-resistance. To this end weintroduce the magneto-transport hysteresis (MTH) as

MTHx(H) = ρdown sweepxx (H)− ρup sweep

xx (H) (3.1)

MTHy(H) = ρdown sweepxy (H)− ρup sweep

xy (H) (3.2)

As evident from Fig. 3.2, the MTH contains only hysteretic features. A non hys-teretic offset on the signal, as it would be caused by galvanic or thermal voltages, willvanish in the MTH. The symmetry properties of the planar Hall effect can again clearlybe seen.

Note also that the signal in MTHy is more than one order of magnitude larger thanthe signal in MTHx and less noisy. Thus for the mapping of the magnetic anisotropy,MTHy will be chosen.The evolution of the planar Hall effect with magnetic field orientation, is shown inFig. 3.3 (a,b).

First it is remarkable that the data for the current direction along (110) and (110)hardly differ. Then the 90 periodicity is nearly perfect, the symmetry axes at Φ = 0,90 and 180 are the magnetic hard axes of the sample. Those at Φ = 45 and 135

are the magnetic easy axes. The magnetization orients along those for low fields. Atthe abrupt changes (switches) in the PHE signal, the magnetization reorients discon-tinuously from along one easy axis to another. We term the corresponding H valuesswitching fields. For every up or down sweep at 5 K, there are two switches. First theorientation of M is changed by 90 to an intermediate state. According to Eqs. 2.35the planar Hall signal should change the sign ( sin(2 · 45) = − sin(2 · (45− 90)) ). Atthe second switch the orientation of M is changed again by 90, and the planar Hallsignal comes back to its original value.

If H is applied exactly along an easy axis, the intermediate state is skipped andM reorients directly by 180. In this case the planar Hall signal does not show abruptjumps. Close to this geometry, the intermediate state appears only in a narrow fieldrange, because in this state M is nearly perpendicular to H. Therefore the easy axesare those orientations of H, where the two switching fields converge.

If H is applied parallel to a magnetic hard axis, the first switch in magneti-zation occurs at lower fields, than for any other orientation. The Zeeman energyterm (see Eq. (2.7)) changes for this switch from M · H = MH sin(−45) to

39

-0.05 0.00 0.05-300

-200

-100

0

100

200

300

MTHxx

110°

MTHxy

70°

MTHxy

110°

MTH

xy, M

THxx

( c

m)

0H (T)

Ga0.96Mn0.04As 5K

MTHxx

70°

Figure 3.2: MTHρxx and MTHρxy for the Ga0.96Mn0.04As sample at 5 K. H is appliedin the film plane, at ± 20 with respect to the current direction, corresponding to Φ =70, 110.

M ·H = MH sin(+45), which is the maximal energy gain for a 90 reorientation. Forthe second switch to M ·H = MH sin(+135) the energy gain is zero, and thus thesecond switching field diverges.

Simultaneously to the two PHE voltages also the longitudinal voltage drops for cur-rent along (110) and (110) have been recorded, the corresponding figures are 3.3(c,d).

At the first glance the graphs for the longitudinal resistance look very similarto those of the PHE. The magnetic easy and hard axes of the sample can be welldiscerned again. The switching field values are reproduced. However symmetryproperties are different. Comparing two H orientations symmetric to a magnetic hardaxis, e.g. Φ = 88 and Φ = 92, MTHx is identical, while MTHy has opposite sign.Analogously to the discussion of Fig. 3.1, this is due to the symmetry properties ofthe planar Hall effect ( sin(2 · 88) = − sin(2 · 92) ), and the AMR in the longitudinalresistance ( cos2 88 = + cos2 92 ).

In a detailed comparison of the MTHx spectra with current along (110) and (110)

40

a) b)

c) d)

Figure 3.3: Magneto-transport hysteresis in the planar Hall resistivity (a,b) and in thelongitudinal resistivity (c,d) for the Ga0.96Mn0.04As sample at 5 K along both axes of theHall bar. 90 periodicity can be clearly seen. No difference for the two axes is observed.

opposite sign is found (see Figs. 3.3(c,d), 3.4(c,d)). To understand this we recall,that the ordinates Φ are defined as the angle of H with respect to the crystalline(110) axis. The features characteristic for the magnetic anisotropy depend on theangle between H and the easy and hard magnetic axes, and thus appear at thesame Φ for both current paths. However the AMR depends on the angle betweencurrent density and magnetization M. This angle differs by 90 for the two currentpaths. Let at one specific H point, M be oriented at θ with respect to the (110)axis for the down sweep and at θ∗ for the up sweep. The MTHx will then beproportional to cos2(θ) − cos2(θ∗). For the current path along (110), the respectiveangles are (θ − 90) and (θ∗ − 90) and the MTHx will then be proportional tocos2(θ − 90) − cos2(θ∗ − 90) = −[cos2(θ) − cos2(θ∗)]. Thus the opposite sign can beunderstood, independent from the magnetic anisotropy. Note that the sign change isunequal to a shift by 90 in Φ, and will also be observed for not 90 periodic situations.Analogously opposite sign in MTHy for both current orientations can be explained.sin 2(θ − 90) − sin 2(θ∗ − 90) = −[sin 2(θ) − sin 2(θ∗)] The expected sign changeis observed in MTHy (see Figs. 3.3(a,b), 3.4(a,b)). The polarity of the transversecontacts is chosen such, that ’hi’ is on the right hand side of the current path for

41

both current orientations. Without this definition the sign of MTHy would becomearbitrary.The data in Figs. 3.3, 3.4 within the range ± 30 mT appear noisier due to a smallerH step size.

42

3.1.2 Evaluation of the magnetic easy axes

While the MTH plots contain the complete set of measured data, to discuss the mag-netic anisotropy quantitatively it is desirable to extract the most relevant features.Those are the magnetic fields at which a switch in the magnetization occurs (switchingfields), and a measure of the magnitude of the MTH, and finally by means of those,the magnetic easy and hard axes of the sample. We first discuss the experimentaldetermination of the switching fields from the MTH traces. Both MTHy and MTHxshould ideally be antisymmetric with respect to H inversion, as well seen in Figs. 3.2and 3.3. Neither the positive nor the negative H axis is discerned for this evaluation.Considering the antisymmetric MTH, noise can further be reduced by the appropriateaveraging:

MTHxanti(H) =MTHx(H)−MTHx(−H)

2

MTHyanti(H) =MTHy(H)−MTHy(−H)

2

As mentioned above, MTHy is much better in signal to noise ratio than the corre-sponding MTHx.

Thus MTHyanti will be used for the evaluation of the switching fields. Even thoughthe switches are very steep, the slope dMTHy

dHis not infinite. This slope can be un-

derstood as the finite speed, at which the magnetization reorients. Defining an uni-versal criterion to determine the corresponding H values for each switch, Hc1 andHc2 is a trade of between several errors. One way would be to calculate the inflec-tion points of MTHy(H), but the second derivative is too noisy to allow for a pre-cise evaluation. Alternatively a fixed limit is chosen. The smallest positive H valuewith |MTHyanti(H)| > limit defines Hc1 and the smallest value H > Hc1 with|MTHyanti| < limit defines Hc2. Exemplary for one orientation (Φ = 194) and cur-rent parallel to (110), this evaluation in shown in Fig. 3.6 (e). The switching fields Hc1

and Hc2 are the intersections with the constant MTHyanti = 12µΩ cm.

We define one limit value, that is then constant for all orientations. Around theeasy axes, the total values of MTHyanti(H) in the intermediate magnetization state aresmall, thus the evaluation will fail for the more orientations, the higher the limit valueis chosen. If the value is chosen very low, dMTHy

dHis small at the intersection. This yields

significantly higher values for Hc2, and smaller values for Hc1, than those where theslope is steepest. A trade of between these two problems, led to the choice of 12µΩ cmas the universal limit value. However one should keep in mind, that especially thesecond switch is not completely abrupt, and the value for Hc2 can differ by values upto 10% for different, ”reasonable” limits.

For apparative reasons, only the range 0 < Φ < 200 can be covered. The an-tisymmetric MTH implies that switching fields are the same if H is inverted. H in-version changes Φ by 180. Thus, to obtain the whole 360, the data are mirrored

43

Hc1,2(Φ + 180) = Hc1,2(Φ) and displayed as open symbols, while original data arerepresented by full symbols.

The resulting switching fields as a function of angle Φ are shown in Fig. 3.6 (a) for5 K.

The biaxial magnetic anisotropy, represented by the 90 periodicity, is clearly re-solved. The easy axes are the intersections of Hc1 and Hc2 at Φ = 45 and Φ = 135.Here µ0Hc1 and µ0Hc2 converge to ≈ 12 mT. At the magnetic hard axes, at Φ = 0

and Φ = 90, µ0Hc2 reaches its maximum of ≈ 100 mT, while µ0Hc1 has a minimumaround the hard axes of 5 mT ≤ µ0Hc1 ≤ 8 mT.

Exactly at the hard axes Hc1 has a local maximum. This effect can also be seen inFig. 3.3(a,b).To allow for the evaluation of the magnetic anisotropy even from noisier MTH data,with less pronounced hysteresis, a simpler, alternative algorithm shall be introduced.Each MTHyanti(H, H > 0) curve has a well defined extremum. As a function of Φ theseextrema’s abscissae (H) and their ordinates both tell about the magnetic anisotropyof the sample. The abscissae yield a figure very similar to that of the switching fields.To go beyond this point, and get information on the magnitudes of the MTH, we willin the following focus on the ordinates, so to say the maximal values of MTHyanti(H).From Fig. 3.3(a,b) the main features are already clear. Within several degrees aroundthe easy axes the hysteresis vanishes. Around the hard axis, there is a small range withless pronounced hysteresis, most likely due to multi domain formation. In between easyand hard axes, the maximal amplitude of MTHyanti(H) is more or less constant over awide range of orientations.

The extremal values of MTHyanti(H) as a function of angle Φ are shown inFig. 3.6(b) for 5 K. The highest value for MTHyanti(H) far away from easy axesis 300 µΩ cm. However close to the easy axes the highest value for MTHyanti(H) isbelow 12 µΩ cm, and thus for these Φ the switching field evaluation failed. The biaxialmagnetic anisotropy is well resolved.As in all previous discussions, no major difference is observed for current along (110)and (110), respectively.

3.1.3 The effect of temperature on the magnetic anisotropyin Ga0.96Mn0.04As

We now address the effect of temperature. As described for 5 K above, magneto-transport measurements have been performed at 25 K, 50 K and 60 K. The magneticanisotropy changes from biaxial at 5 K to uniaxial at higher temperatures.

MTHy at 25 K is shown in Fig. 3.4 (a,b) and MTHx in (c,d). The symmetryproperties for the PHE and the longitudinal resistance are identical as discussed forthe 5 K data. In contrast to 5 K, at 25 K for almost all angles Φ only one abruptstep in MTHy and MTHx is observed. Additionally a relatively steep slope in bothMTH spectra is observed around zero applied field. Here the magnetization rotatescontinuously, yielding a gradual change in the AMR. As we still assume the sample to

44

a) b)

c) d)

Figure 3.4: Magneto-transport hysteresis in the planar Hall (a,b), and the longitudinalresistivity (c,d) for the Ga0.96Mn0.04As sample at 25 K. Current flows along (110) (a,c)and (110) (b,d) respectively. Uniaxial magnetic anisotropy is observed.

a) b)

Figure 3.5: Magneto-transport hysteresis in the planar Hall (a) and the longitudinal re-sistivity (b) for the Ga0.96Mn0.04As sample at 50 K with current flow along (110). Weakuniaxial magnetic anisotropy is observed.

45

represent a single magnetic domain, this domain’s orientation must rotate coherently.The change in the magnetic anisotropy, evaluated from the singularities (hard axes)and the minima (easy axes) in the switching fields is striking. While Φ = 90, (110) isstill a hard axis, Φ = 0 (110) and Φ = 180 (110) respectively, are easy axes at 25 K.The easy axes at 5 K, Φ = 45 (100) and Φ = 135 (010) are not discerned at 25 K. Asthere is only one easy axis left, it makes well sense, that there is only one magnetizationswitch, observed. The values for the switching fields are reduced at 25 K with respectto 5 K, as well as the magnitudes of the MTHx and MTHy. We will come back tothis point in the next paragraph. MTHy at 50 K is shown in Fig. 3.5(a), and thecorresponding MTHx in (b). For briefness only data measured with current parallel to(110) crystalline axis are shown. Both the magnitude of the MTH and the switchingfields are further reduced. However in ρxy there is still enough hysteresis to visualize themagnetic hard axis at Φ = 90 with MTHy. At 60 K hysteresis is under the noise level,therefore the evaluation of the MTH does no longer make sense. Analogously to 5 K,the switching fields have been evaluated at 25 K, and are displayed in Fig. 3.6 (c). Theuniaxial anisotropy is well resolved. Within ± 2 around the easy axis at Φ = 0 theswitching fields cannot be determined by the algorithm used. In the regime of uniaxialanisotropy, there should be only one switch at most. Furthermore, if M reorients byexactly 180, both longitudinal and transverse resistance remain constant in the simpleAMR picture. Coherent rotation of the magnetization at relatively low fields cannotonly explain the slope in MTHx and MTHy, but also that the switching field is stillobserved. The switch is then from an intermediate magnetization orientation to theeasy axis, which is an angle smaller then 180. Still, within ± 20 around the easy axisvalues larger than 10−4 T for µ0Hc1 can be determined. This may stem from coherentrotation setting in later, the more favorable the direction of the external field is for themagnetization. Another explanation would be that a small biaxial contribution stillexists. In that case Fig. 3.6 (c) gives an upper limit for the angle the two easy axescould enclose by the intersections of Hc1 and Hc2 of 6. On the other hand, Fig. 3.6 (d)clearly shows only one easy axis, with an angular resolution of 2. After all, coherentrotation can explain the finite slope in MTH(H) around H = 0, and the observationof magnetization reorientation in the regime of uniaxial magnetic anisotropy, throughabrupt jumps in MTHx and MTHy. It has to be discussed, whether the magnitudeof the slope around H = 0, can be understood quantitatively with a microscopicsimulation.As it comes to 50 K (MTH in Fig. 3.5), hysteresis is only observed within 30 < Φ <150. The magnitudes of the MTH is reduced, and the signal to noise ratio does notallow a reasonable evaluation of the switching fields. Instead we will directly come tothe maximal values of MTHyanti(H). They are, as a function of Φ, shown in Fig. 3.6(f).Uniaxial anisotropy is clearly seen, with the easy and hard axes oriented as at 25 K.The absolute values for MTHyanti are drastically reduced with respect to 5 K and 25 K,at most 60 µΩ cm are reached. As already seen in Fig. 3.5, the magnitude of MTHyanti

is, in contrast to 5 K and 25 K, not constant far from the easy axis, but exhibits apronounced maximum near the hard axis.

46

hard axis

easy axis

hard axis

easy axis

easy axis

hard axis

hard axis

hard axis

easy axiseasy axis

hard axis

hard axis

easy axiseasy axis

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

j || (110)

a) b)

c) d)

e) f )

Figure 3.6: Evaluation of the magnetic anisotropy of the Ga0.96Mn0.04As sample. Thedefinition of the switching fields Hc1 and Hc2 is shown in (e). The switching fields Hc1

and Hc2 as a function of the magnetic field orientation Φ() at 5 K and 25 K are plottedin (a) and (c), respectively. The maximal values of MTHyanti at 5 K, 25 K and 50 K aredisplayed in (b), (d) and (f) as a function of the magnetic field orientation Φ().

47

H || hard axis

H || hard axis

H || hard axis

H || easy axis

H || easy axis

H || easy axis

Figure 3.7: Angular dependence of ρxx(+0.2 T, Φ) in red and of ρxy(+0.2 T, Φ) in blackfor the Ga0.96Mn0.04As Hall bar at 5 K, 25 K, 50 K and 60 K.

3.1.4 AMR at high fields

So far we have only discussed the AMR obtained by varying the magnitude of themagnetic field while the field orientation was kept fixed (field sweep). We now considerthe AMR for constant field magnitude |H| as a function of Φ (angular sweep). Inthe field sweeps discussed in the last section, for each Φ, H was tuned in steps fromµ0H = +0.2 T to µ0H = −0.2 T and back. At the maximal applied field valuesµ0H = +0.2 T, M should be saturated parallel to H, thus Φ = θ. These data pointsat µ0H = +0.2 T were extracted, and plotted as a function of Φ. The resultingsets of data, ρxx(µ0H = +0.2 T, Φ) and ρxy(µ0H = +0.2 T, Φ), should accordingto Eqs. (2.28),(2.29), obey a cos2 Φ and sin 2Φ dependence respectively. At 50 Kand above, these dependencies are found, supporting Φ = θ. At lower temperatureshowever, deviation from the dependencies, expected according to Eqs. (2.28),(2.29),are observed, suggesting that θ is deviating by a small angle from Φ. For |H| À Hc2

no hysteresis is observed in ρxy and ρxx. Thus the magnetization reorientation is fullyreversible. This suggests that the magnetization rotates coherently.

In Fig. 3.7, ρxx and ρxy obtained at µ0H = +0.2 T are plotted as a function ofΦ for 5 K, 25 K, 50 K and 60 K. For 50 K and 60 K the cos2 Φ and sin 2Φ behaviorrespectively is nearly perfect. At these temperatures the magnetization thus follows Hcompletely. In other words: at µ0H = +0.2 T the orientation of M is not influencedby magnetic anisotropy. In contrast, at 25 K the curves deviate from the dependence

48

according to Eqs. (2.28),(2.29). This deviation is particularly pronounced around themagnetic hard axis (Φ = 90) and around the magnetic easy axis (Φ = 0). Themagnetic anisotropy apparently is significant at µ0H = +0.2 T and 25 K. Therefore itis energetically favorable for M to enclose a bigger angle with the magnetic hard axisthan H does. Around the magnetic hard axis, θ then must turn faster than aroundΦ = 90, and thus the curves appear jolted. Around the magnetic easy axis the curvesappear stretched. Since a magnetization orientation in a smaller angle to the easyaxis is more favorable than in a larger angle, θ is turning slower than Φ = 90 whenthe easy axis is passed.

At 5 K the deviation is even stronger than at 25 K. However the physics is thesame. As discussed before, the hard axes are at this temperature at Φ = 0 andΦ = 90, the easy axes at Φ = 45 and Φ = 135.

The magnetic anisotropy becomes the stronger the lower the temperatures, and onecan well understand that the magnetization at |H| = 0.2 T is not completely saturatedat 5 K, while it is at 50 K.

In their paper, Limmer and others [LGD+06], show ρxx(Φ) and ρxy(Φ) in(Ga, Mn) As only at 4.2 K at various fields. |H| was kept constant while Φ was swept.The cos2 Φ and sin 2Φ are well reproduced at 0.7 T, suggesting Φ = θ. At 0.1 T theyobserve a deviation, that can be understood as the persistent magnetic anisotropy.Even though in this paper the applied field magnitude is varied, while we vary thetemperature, both experiments agree in the order of magnitude for both, where themagnetic anisotropy becomes relevant.

We now turn to a more quantitative analysis of the data. The absolute differencebetween the maximal and minimal values of ρxx and ρxy, are termed ∆ρxy and∆ρxx. According to the AMR model Eqs. (2.28),(2.29), ∆ρxy and ∆ρxx should beof the same order of magnitude. This model is designed for a single ferromagneticdomain without crystalline anisotropy, or a polycrystalline single domain sample.In all previous discussion of the magnetotransport data, only ρxy was evaluatedquantitatively. In this section, for the first time longitudinal and transverse resistivitychanges are compared quantitatively. However we find that ∆ρxy

∆ρxxis 1.13 at 50 K and

1.38 at 5 K, the ratio is plotted in Fig. 3.8 for all available temperatures. Limmerand others [LGD+06] reported ∆ρxy

∆ρxx= 1.35 at 4.2 K for a sample with 5 % Mn

from the same MBE. Taking into account the crystalline anisotropy, as discussedin section 2.4.2, and especially in Eq. (2.35), ∆ρxy and ∆ρxx become independentparameters termed B and C. In this model, the orientation of M and currentwith respect to the crystalline axis is important. The deviation of ∆ρxy from ∆ρxx

that we observe motivates taking into account the crystalline anisotropy, and thusintroducing an additional free parameter. Up to now, we need this additional degreeof freedom to describe our data, but there is no quantitative test of the model.Having Hall bars processed from the same film with current along (110) and (100),

49

25 50 6050

100

150

200

250

300350

B, C

[cm

]

T (K)

B

C

5

Figure 3.8: Resistivity coefficients B (squares) and C (circles) for Ga0.96Mn0.04As as afunction of temperature.

one could perform such a test. Then the ratio ∆ρxy / ∆ρxx should be inverted for both.

Finally we note a clear decrease in both ∆ρxy and ∆ρxx for increasing temperature.This can be understood, with the temperature dependence of the magnetization, lead-ing to reduced AMR near the Curie temperature. In the paper of Limmer, differenttemperatures were not addressed, and they evaluate constant values for B and C. Tofit our data however, B and C have to depend on temperature.

50

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

8000

9000

10000

11000

25K

18K

5K

120K

100K

78K

48K

xx (

cm

)

0H (T)

63K33K

12K8K

2K

Ga0.96

Mn0.04

AsH film plane

Figure 3.9: Longitudinal resistivity for the Ga0.96Mn0.04As sample with M applied per-pendicular to the film plane. The magneto resistance is particularly large close to theCurie temperature. For temperatures up to 18 K hysteretic features can be resolved.

3.2 Magneto-transport with H out of plane

We now turn to magneto-transport measurements with H applied perpendicular tothe film plane. The (ordinary) Hall effect (OHE) and the anomalous Hall effect (AHE)contribute to the transverse voltage. The temperature and H dependence of both theAHE and the longitudinal resistance ρxx are compared, and the microscopic origin ofthe AHE in (Ga, Mn) As is discussed.

3.2.1 Magneto-resistance with H out of plane

We now address the longitudinal resistance ρxx, for the Ga0.96Mn0.04As sample, asplotted in Fig. 3.9 as a function of H applied perpendicular to the film plane, forvarying temperatures. The resistance values at zero field agree with those measuredwith H in plane. At 18 K and below hysteretic features at small fields are resolved. Upto 33 K a local maximum in the resistance at a finite H value is observed, this suggeststhat for H below this maximum M is not fully oriented perpendicular to the film plane.Conventionally that is attributed to a strong uniaxial magneto crystalline anisotropy

51

perpendicular to the film plane as observed early by Dietl and others [DHd97], thatforces the magnetization towards in-plane orientation. For larger H or for temperaturesabove 33 K, the magneto-resistance is negative. It gets maximal close to TC , with

ρxx(µ0H = 13 T)− ρxx(µ0H = 0)

ρxx(µ0H = 0)= −25% (3.3)

at 63 K. This is conventionally attributed to the interplay of spin order and disorder.At 0 K spin order is perfect due to ferromagnetic coupling, even without appliedfield. In contrast, at the Curie temperature the spontaneous order is completely lost.However an applied field can still easily force the spins to align, and thereby theapplied field most effectively alters the spin configuration here. As the temperatureis further increased, the energy scale and thus the necessary fields for spin alignmentincrease. The better the spin alignment, the lower the spin-flip scattering rate of themobile charge carriers. And hence the resistance is the smaller the better the spinsare aligned. Similarly the absolute value of ρxx(µ0H = 0) has a maximum near theCurie temperature (TC,transp.).

3.2.2 Hall and Anomalous Hall effect

Next we come to the transverse resistance, arising from the OHE and the AHE.ρxy (H) for various temperatures is shown in Fig. 3.10 (a). Except for the enlargedpart in (c), the curves are perfectly antisymmetric. From the symmetry of the PHE inthe previous section, it was already clear that the transverse voltage is free of spuriouscontributions from sample geometry. Now, besides confirming that, the antisymmetryproves that only Hall like signals contribute and that the signal is free of AMRcontributions, except for the domain effects discussed later on. The Hall effect (OHE)is small compared to the AHE. At low temperatures the AHE exhibits the expectedbehavior. As the magnetization orients to perpendicular to the film plane the AHEincreases rapidly, and then saturates as the magnetization saturates. In this saturatedregime, ρxy (H) still is not linear (enlarged in Fig. 3.10 (b)), as one would expect fora constant AHE contribution plus a small OHE.ρxy (H) for −0.5 T < µ0H < +0.5 T is shown in Fig. 3.10 (c) for 5 K, 12 K and18 K. A narrow hysteresis can be observed. Just before H changes sign, sudden jumpsoccur in ρxy. These features are symmetric with respect to H = 0. Accordingly theycan be a Hall effect, but rather stem from AMR. Domains are formed, in which themagnetization orients into the film plane. This is energetically favorable due to thestrong uniaxial magneto crystalline anisotropy. The in-plane oriented M then yieldsan AMR contribution to the transverse voltage. This effect is limited to small fields|H| < 0.5 T, and temperatures below 25 K.

Near and above TC the curves get more and more Brillouin function like. Still thevalues for ρxy(µ0H = 13 T) are not significantly reduced. Similar results have been

52

a)

b) c)

Figure 3.10: (a): Transverse resistance ρxy(H) with H applied perpendicular to the filmplane. The anomalous Hall effect dominates. For temperatures above TC, a Brillouinfunction like signal is observed in ρxy. The magnitude of ρxy(H) at high fields is similarat all temperatures. In the regime of saturated AHE, enlarged in (b), ρxy is not linear,since magneto-resistance effects from ρxx dominate the ordinary Hall effect. At smallfields, enlarged in (c) and low temperatures hysteretic features stemming from domainformation are observed.

53

reported earlier [MOSS98, VVD+97], and will be discussed below.

We will now briefly discuss the Hall effect (OHE). Measuring the OHE and hencethe carrier density in (Ga, Mn) As has intensely been discussed in literature [EWC+02,Ohn99, RSB+04] and is found to be troublesome as the AHE is at least one order ofmagnitude larger than the OHE. There remains an uncertainty of a factor of 2 on allmethods up to now. The most common approach is to go to very low temperatures andhigh applied fields. In this sense we will estimate the Hall slope and then the carrierdensity from the 2 K data. From the slope in the ρxy-data at high fields and at 2 K(Fig. 3.13(b)) a Hall coefficient

RH = 1.75µΩcm

T(3.4)

can be evaluated. The corresponding carrier density is n = 3.6× 1020 cm−3. Since ρxx

does not show Arrhenius-like temperature dependence, the carrier density should notchange significantly over temperature. The order of magnitude does well make sense,and a similar value is reported for a sample of the same growth run (B442) by theUlm group [Fra05]. However this picture is a bit naive. ρxy(H) in Fig. 3.13(b) is notlinear in the regime, where a saturated magnetization is reasonable. To understandthis, we will now turn to the anomalous Hall effect (AHE) before we come back to analternative way of evaluation of the ordinary Hall effect.

To address the microscopic origin of the AHE, both ρxx(T) and ρxy(T) have alsobeen measured at fixed applied fields µ0H = 0, +13 T and −13 T from 1.5 K to300K (R(T)). ρxy(T) is corrected by the constant Hall contribution from Eq. (3.4) forall temperatures. Furthermore the symmetry properties with respect to H inversion ofthe AHE (antisymmetric) and ρxx (symmetric) are used to separate them from possiblespurious signals.

antisymmetrized : ρxy(±13 T, T ) = ρxy(+13 T, T )− ρxy(−13 T, T ) (3.5)

symmetrized : ρxx(±13 T, T ) = ρxx(+13 T, T ) + ρxx(−13 T, T ) (3.6)

The results are shown in Fig. 3.11 (lines) together with data extracted from Fig. 3.9and Fig. 3.10 (symbols, termed R(H)).

The sets of data as a function of temperature and those as a function of H agreevery well. The dashed curve for ρxx(T, µ0H = 0 T ) is also shown in Fig. 3.12(b).From it, TC,transp. = 72 K can be evaluated. ρxx(T, µ0H = 13 T ) has its maximumat higher temperatures. Magneto resistance, corresponding to the difference betweenboth curves, has its maximum just below TC,transp.. Now the focus shall be on the setsof data at 13 T.

To decide whether side-jump or skew-scattering mechanisms dominate, one shouldcompare ρxx(T ) and ρxy(T ) at the same H, here 13 T. In case we foundρxy(T ) ∝ ρxx

1(T ), the skew scattering mechanism would be the main contribution tothe AHE. In case we found ρxy(T ) ∝ ρxx

2(T ) it would be the side jump mechanism.

54

0 50 100 150 200 250 3000

100

200

300

400

500

600

8000

9000

10000

11000

xy

(cm

)

T (K)

Symmetrized xx

@ 0H= 13T

from R(H) from R(T)

xx @

0H=0T

from R(H) from R(T)

Antisymmetrized xy

@ 0H= 13T

from R(H) from R(T)

xx

(cm

)

Figure 3.11: ρxx(T ) and ρxy(T ) for the Ga0.96Mn0.04As sample with H applied perpendic-ular to the film plane. The symbols represent data extracted from Figs. 3.9 and 3.10.

The key message is, a simple relation cannot be found. Both curves have extremal val-ues at different temperatures. This can be understood as the magnetization is stronglytemperature dependent, because TC is comparable to the temperature range appliedhere. Thus the temperature dependence of both the magnetization and the longitudi-nal resistance impact on the temperature dependence of the AHE. In the following wewill, based on literature [Fra05] assume the skew-scattering mechanism as dominant.Since the AHE is an effect of M, one could assume that it vanishes above TC . As inFig. 3.11 displayed, an AHE like ρxy is observed up to 300 K. Still 20% of the valueat low temperatures are found at 300 K. The values for ρxy(13 T ) are just above TC

not rapidly decreasing. Van Esch discussed the appearance of the AHE in the para-magnetic regime in a very early paper [VVD+97], terming it the extraordinary Halleffect. However, the physics in both the ferromagnetic and the paramagnetic regimeis identical. The AHE arises from spin polarization, which is proportional to M. M isabove TC still much larger than zero, because the applied field of 13 T is large.The Hall resistivity was introduced in section 2.4.4 as

ρxy = R0B + RSM (3.7)

55

a) b)

Figure 3.12: (a): The longitudinal resistivity divided by the transverse resistivity forGa0.96Mn0.04As . The paramagnetic signature yields TC =71 K. (b): The longitudinalresistivity at zero applied field has a maximum at TC =72 K.

The empiric constants R0 and RS both are not of the dimension of resistance. Thesmall OHE contribution R0 is corrected of the data according to Eq. (3.4), leavingthe dominant anomalous component RS. The magnetization far from saturation canbe written as M = χH. Above TC, in the paramagnetic regime, the Curie-Weiss law(Eq. (2.3)) determines the susceptibility for low applied fields: χ ∝ (T − TC)−1.This yields a hyperbolic temperature dependence of the magnetization above TC .Assuming the skew-scattering mechanism as the origin of the AHE, RS becomes directlyproportional to ρxx. Consequently for T À TC we can write:

ρxy ∝ ρxx · M ∝ ρxx

T − TC

(3.8)

ρxx

ρxy

∝ (T − TC). (3.9)

This is corroborated in Fig. 3.12(a). The resulting TC =71 K fits very well withTC evaluated from squid magnetization measurements and with TC evaluated fromthe maximum in ρxx (Fig. 3.12(b)). The fact, that in the paramagnetic regime inFig. 3.12(a) a linear dependence is found, supports that the assumption of the skewscattering mechanism is reasonable.

Starting again from the approach in section 2.4.4 and assuming skew-scattering,the magnetization can be derived as

M ∝ ρxy − µ0H ·RH

ρxx

. (3.10)

56

a)

c)

b)

d)

Figure 3.13: (a) and enlarged in (b): ρxy(H) at 2 K. From the high field data, an estimateof the OHE can be taken. (c) and enlarged in (d): Magnetization in arbitrary units,under the assumption of the side-jump mechanism for the AHE, evaluated from transportdata at 2 K. The Hall coefficient is chosen such, that the magnetization converges to aconstant at high fields.

The AHE is as discussed before not constant at high fields, and thus dominatesthe slope in ρxy(H) even at high fields. However the magnetization should convergeto a constant at high fields. This gives an additional approach to determine the Hallcoefficient RH.

The result for 2 K is shown in Fig. 3.13(c,d). The Hall coefficient chosen is

RH = 3.0µΩcm

T(3.11)

and the resulting carrier concentration n = 2.1 · 1020 cm−3. This estimate differsnearly by a factor of two from the carrier concentration estimated before from Fig.3.13(a,b). The deviation arises only from the influence of the magneto-resistance in ρxx.

57

Furthermore at 2 K, where this evaluation is made, the effective mass approximationno longer holds[CW80] since hopping conduction is involved. Then strictly speaking,the evaluation of the carrier density is no longer valid.After all, we find that the estimation of the carrier density is extremely troublesome.Even though the mechanism of the AHE cannot be determined via a simple relationbetween ρxx and ρxy, the skew-scattering mechanism is not in opposition to our findings.Several independent methods consistently allow to determine TC from our transportdata.

58

Chapter 4

Magnetotransport in CrO2

In analogy to the magneto-transport experiments described in chapter 3 we have alsomeasured the MR of two 100 nm thick (100) oriented CrO2 thin films. We show thatthe magnetic anisotropy of CrO2 can be deduced from measurements with H appliedin the film plane, while experiments with the field applied out of plane reveal themicroscopic origin of the anomalous Hall effect.

4.1 In-plane magnetotransport in (100) oriented

CrO2 thin films

In this section, we discuss magnetotransport experiments with H applied in the filmplane of (100) CrO2 thin films, in analogy to those described in Section 3.1 on(Ga,Mn)As. After refining the algorithm for the switching field evaluation, we showthat the in-plane magnetic easy axes are at ± 34 with respect to the crystalline (010)-axis for T ≤ 25 K.The resistivity ρ(T ≤ 25 K) = 3.7 µΩcm of CrO2 is about three orders of magni-tude smaller than the resistivity of the Ga0.96Mn0.04As sample. To achieve similarvoltage levels, a larger current, I = 5 mA, was applied in this series of experiments.Besides that, the experimental setup is completely analogous to the one described inSection 2.6. Note however, that the angle Φ in the following represents the angle be-tween the crystalline (010)-axis and the applied in-plane magnetic field H. The sampleinvestigated, Data, is described in detail in Section 2.7.

4.1.1 Results

Typical ρxx(|H|) and ρxy(|H|) traces are shown in Fig. 4.1. In this measurement, thecurrent path is parallel to the crystalline (010)-axis. The magnetic field sweeps fromnegative to positive values (up sweeps) are displayed in red, while those from positiveto negative values (down sweeps) are displayed in black. Two orientations of H arecompared in the figure. They both enclose an angle of 30 with j, as schematically also

59

shown in the figure. For both H orientations clear switches and a hysteretic region forµ0|H| ≤ 50 mT can be clearly discerned. In analogy to Section 3.1, we attribute theseeffects to the abrupt reorientation of a single ferromagnetic domain.

However, the transverse resistance ρxy does not change sign in CrO2 (Fig. 4.1, incontrast to the situation in Ga0.96Mn0.04As (Fig. 3.1). In CrO2, a large backgroundsignal, about 100 times larger than the change in ρxy with magnetization orientationseems to be superimposed. At present, we can only speculate about the origin ofthis spurious background. It could be due to imperfectly aligned transverse contacts,which are not exactly orthogonal with respect to the current flow. This would resultin a crosstalk of a fraction of the longitudinal voltage drop into the transverse signal.The misalignment required to obtain the signal magnitude observed is ≈ 1 µm. Whilethis might appear large, it is not unreasonable. As CrO2 (and the native Cr2O3 on topof the samples) is very hard to etch, our wet etching process leads to a significant edgeroughness of ≤ 1 µm. Other explanations could be an offset voltage in the voltmetersor crosstalk of the longitudinal resistance via ground loops.

To ascertain that the switches and hysteretic features in ρxy indeed are the trueplanar Hall signals, and not e.g. crosstalk from ρxx, it is instructive to compare Fig. 4.1(a) with (b). As expected from the cos2(θ) dependence in Eq. (2.28), the longitudinalresistance does not differ significantly for the two field orientations. One finds that themagnitude of the background remains the same, and does not change sign, while thepolarity of the hysteretic features is reversed. This is exactly what is expected fromthe sin(2θ) dependence in Eq. (2.29), and thus is a proof that this signal indeed is aPHE. To address the hysteretic features only, and to separate them from non-hystereticmagneto-resistance effects, the magneto-transport hysteresis in the longitudinal andtransverse resistivities, MTHx and MTHy given by Eqs. (3.1), again are very useful.

The MTH’s, calculated from the resistivity data of Fig. 4.1, are displayed in Fig. 4.2.The two orientations of H, Φ = +30 (blue) and Φ = −30 (brown), yield identicalMTHx, while the sign in MTHy is reversed, as mentioned for the resistivity data, thischaracteristic for the planar Hall effect.

The longitudinal and transverse resistivities have been measured from Φ = −97

to Φ = +95, in two degree steps. Both current along the crystalline (010) and alongthe (001) axis could be recorded simultaneously due to the shape of the Hall bar (seeFig. 2.12). For simplicity, for current along both (010) and (001), Φ is defined as theangle between the applied field and the (010) axis. We will not explicitly discuss theresistivity data, but instead directly analyze the MTH.

The evolution of MTHy and MTHx with Φ is shown in Fig. 4.3(a),(c) for j ‖ (010)and in Fig. 4.3(b),(d) for j ‖ (100). Comparing the MTHx and the MTHy signal forthe same Φ for the two current directions, one sees that both change sign. This makessense assuming that the whole Hall-bar consists of a single ferromagnetic domain. Thetwo current directions (j ‖ (010), j ‖ (100)) enclose an angle of 90. Therefore the anglebetween current and magnetization also changes by 90 when one considers j ‖ (100)

60

30°−

30°−

(a) (b)

(c) (d)

CrO

j||(010)

5K

Figure 4.1: Transverse resistance ρxy(H) of the CrO2 sample Data at Φ = −30 (a) andΦ = +30 (b) and the longitudinal resistance ρxx(H) (c) and (d) respectively. The sweepsfrom negative to positive H values (up sweeps, red) and vice versa (down sweeps, black)differ in a hysteretic region, limited by abrupt changes in both ρxy(H) and ρxx(H). Thecurrent path is parallel to the crystalline (010)-axis. While ρxx(H) is identical withinexperimental error for both orientations, the signature on ρxy(H) is inverted. This provesthat the switches in ρxy(H) indeed are due to a ”planar Hall effect”.

instead of j ‖ (010). Thus, the transverse resistances, according to Eq. (2.29), read:

j ‖ (010) : ρ(010)xy = (ρ‖ − ρ⊥)

1

2sin(2θ) (4.1)

j ‖ (100) : ρ(100)xy = (ρ‖ − ρ⊥)

1

2sin(2(θ − 90)) (4.2)

⇒ ρ(010)xy = −ρ(100)

xy . (4.3)

61

(a)

(b)

Φ=-30°

Φ=+30°

Φ=-30°

Φ=+30°

CrOj||(010)5K

CrOj||(010)5K

Figure 4.2: MTHy (a) and MTHx (b) for the CrO2 sample, with the current path parallelto the crystalline (010) direction. The external field H is applied in plane, at Φ = +30

(blue) and Φ = −30 (brown) with respect to the current direction.

Analogously the longitudinal resistances according to Eq. (2.28) read:

j ‖ (010) : ρ(010)xx = ρ⊥ + (ρ‖ − ρ⊥) cos2(θ) (4.4)

j ‖ (100) : ρ(100)xx = ρ⊥ + (ρ‖ − ρ⊥) cos2(θ − 90). (4.5)

According to the definitions in Eqs. (3.1), MTHx and MTHy are the differences

62

(a) (b)

(c) (d)5 Kj || (010)MTHx

5 Kj || (001)

MTHx

5 Kj || (001)

MTHx

5 Kj || (010)MTHy

Figure 4.3: The MTHy for the CrO2 sample at 5 K is displayed in (a) for current flowingalong the crystalline (010)-axis and along the (001)-axis in (b). The MTHx for bothcurrent directions are displayed in (c) and (d) respectively.

of the up sweep and the down sweep of ρxx and ρxy, respectively. We term the angleenclosed by the magnetization and the (010)-axis (b-axis) at one specific H value θdown

and θup, in the down sweep and in the up sweep, respectively. MTHx and MTHy thenread:

j ‖ (010) : MTHy(010) = sin 2θdown − sin 2θup (4.6)

j ‖ (100) : MTHy(100) = sin 2(θdown − 90)− sin 2(θup − 90) (4.7)

j ‖ (010) : MTHx(010) = cos2 θdown − cos2 θup (4.8)

j ‖ (100) : MTHx(100) = cos2(θdown − 90)− cos2(θup − 90). (4.9)

Hence MTHx and MTHy are of opposite sign for specific H and Φ, and current parallel

63

to (010) and (100), respectively:

MTHy(010) = −MTHy(100) (4.10)

MTHx(010) = −MTHx(100). (4.11)

In Fig. 4.3, the MTH signals for −30 mT < µ0H < 30 mT and for−5 mT < µ0H < 5 mT seem noisier. In fact, the noise level does not change,and the appearance simply originates from the point density.However, the signal to noise ratio is much better for MTHx compared to MTHy.Nevertheless, the data shown in Fig. 4.3(a),(b) are the best resolved planar Halleffect in CrO2 to our knowledge[KGvD+05]. The MTHx gives a clear picture of themagnetic anisotropy, that will be discussed in the following. For each orientation Φ,MTHx(H) is antisymmetric within experimental error. Hence we will without loss ofgenerality, focus on H > 0 in the following discussion. In this partition, two jumpsoccur in MTHx. There the magnetization M reorients abruptly. It will be provedlater on that the sample is indeed biaxial at 5 K and at 25 K, but the angle enclosedby the easy axes is not 90 as in Ga0.96Mn0.04As at 5 K, but 68 and 112. At the firstreorientation, M, represented by at single ferromagnetic domain, is displaced fromalong one easy axis to along the other. At the second one, M is again displaced fromalong one easy axis to along the other, such that it is oriented at 180 with respect tothe original direction. To clarify again: For the up sweep, these jumps occur both atH > 0. For the down sweep both at H < 0. The MTH contains both features of theup and the down sweep. The H values at which the first and second reorientationsoccur are Hc1 and Hc2 respectively. Equivalent to the argumentation in section 3.1.1,Hc1(Φ) and Hc2(Φ) are determined by the gain in Zeeman energy (see Eq. (2.7)) inthe reorientation. If H is applied along a magnetic hard axis, the energy gain in thefirst reorientation is maximal, and thus Hc1 is minimal. The second reorientationdoes, if H is applied exactly along a magnetic hard axis, yield zero energy gain, thusHc2 diverges. The evaluation of the magnetic easy axis requires a deeper discussion ofthe Zeeman energy, that is given in section 2.5. It is found that the switching fieldsHc1 and Hc2 coincide, if H is applied orthogonal to the magnetic easy axes. Theorientation of coinciding switching fields is not, as one might suspect, an easy directionof the magnetization. In Fig. 4.3, Hc1 and Hc2 coincide at Φ = ±56. Hence the easyaxes are at Φ = ±34. Hc1(Φ) = Hc2(Φ) is an important symmetry axis of the spectra.MTHx(H) on opposite sides of this axis, correspond to identical initial and final Morientations, however the reorientation path of M is clockwise and counterclockwiserespectively. Therefore the MTHx(H) are of reversed sign. The initial and final Morientations change, when a hard axis is crossed. Here the sign of MTHx(H) remainsunchanged, as the direction of the reorientation path is equal on both sides of a hardaxis.

64

4.1.2 Determination of the switching fields and the magneticeasy axes

To address the angular dependence of magneto-transport in more detail, we will in thefollowing refine the algorithm used in Section 3.1.2 to extract the switching fields fromcharacteristic features in the magneto-transport data. Based on these fields, we thendetermine the orientation of the magnetic easy axes. The most relevant features arethe divergence of Hc2 which yields the hard axis, and intersection of Hc1 and Hc2.As evident from Fig. 4.2, MTHx is much better in terms of the signal to noise ratiothan MTHy. Therefore we will concentrate on MTHx to determine the switching fieldsHc1 and Hc2. The algorithm used to determine has to be robust, so that the noiseon the MTHx(H) data does not yield artefacts, and that the switching fields can beautomatically determined by a computer program. We use the following 3 criterion,shown in Fig. 4.4(a) for Φ = −12. For all Φ, MTHx(H) is antisymmetric, hence wecan focus on H > 0. Hc1 is defined as the first zero in MTHx forµ0H > 2 mT. Atµ0H = 2 mT MTHx is significantly above noise level, hence the noise cannot introducefalse Hc1 values, as it would do if the criterion was µ0H > 0 mT. Furthermore wealways find µ0Hc1 ≥ 4 mT, hence the restriction µ0H > 2 mT is safe. As the switchshortly after H = 0 is very abrupt, the error in Hc1 is small.

The features in MTHx for H > Hc1 are much less abrupt, especially for H appliedclose to a hard axis, like in Fig. 4.4(a). We therefore use two alternative definitionsfor the second switching field. Hc2 is taken as the field, where MTHx(H > Hc1)has its extremum. Hc3 is finally defined as the field at which the hysteresis closesagain, i.e. MTHx ≈ 0. To deal with the noise floor, we use a threshold criterion of|MTHx| < 86× 10−12 Ωcm, which corresponds to |Vxx| < 1µV at the level I = 5 mA.

Fig. 4.4(b) shows Hc1 in black, Hc2 represented in red, and Hc3 in blue symbols.Note that data was taken only for −92 < Φ < 100 to visualize the symmetry, thedata are shown again rotated by 180 as open symbols. Within 70 < Φ < 110, Hc2

and Hc3 nearly coincide, because the switch is abrupt, and the hysteresis closes rapidly.For −50 < Φ < 50, Hc3 is a factor of two larger than Hc2. The hysteresis closes slowlyhere. Quantitatively, a larger threshold value for Hc3 reduces this discrepancy, but thenthe evaluation fails over a wider interval around Φ = ±60. This is problematic, sincethe accuracy of the evaluation of the orientation of the easy axes is limited by missingswitching field values in this range of Φ. Another disadvantage of Hc3 is, that thevalues strongly depend on the threshold value chosen, which introduces arbitrariness.

In comparison, the extremum of MTHx for H > Hc1 gives the most reliableand reasonable results. Hence, Hc2 is used as the second switching field from nowon. Within 10 around the orientation where Hc1 and Hc2 converge, all attempts toevaluate the switching fields by an automated procedure fail. The switches becomevery small and ill defined, and the MTHx(H) traces are dominated by noise. This isthe main limitation for the determination of the easy axes. We attempted to increasethis accuracy, by manually evaluating the switching fields around those orientations.However, the results scatter significantly, and hence do not yield a better resolution.

65

(a)

(b)

Figure 4.4: (a): The switching fields are determined as follows: Hc1 is the field for µ0H >2 mT for which MTHx=0. Hc2 is the field at which MTHx has an extremum, and Hc3 thefield at which the hysteresis closes (MTHx=0). More precisely, Hc3 is taken as the fieldfor which |MTHx| < 86× 10−12 Ωcm. The switching fields determined according to thosecriterion are displayed in (b) as a function of Φ(): Hc1 (black), Hc2 (red), Hc3 (blue) aredisplayed as full symbols. To represent the whole symmetry, the data are rotated by 180

and displayed as open symbols again.

66

-90 -60 -30 0 30 60 90

5

10

15

20

2530

CrO2

5Kj||(010)

56°

0Hc1

, 0H

c2 (m

T)

(°)

-56°

Figure 4.5: The switching fields Hc1 (full circles) and Hc2 (open circles) in CrO2 at 5 K withcurrent parallel to the (010)-axis. The black lines represent fits according to Eqs. (2.81)-(2.88). Hc1 = Hc2 is found for Φ = ±56, and according to Eq. (2.90), the easy axes areoriented at Φ = ±34 = ±α.

Figure 4.5 shows the switching fields Hc1 (full circles) and Hc2 (open circles) in theCrO2 sample Data at 5 K with current parallel to the (010)-axis, corresponding toFig. 4.3(c) and Fig. 4.6(a). Fits according to Eqs. (2.81)-(2.88) are plotted as blacklines. The fits of the two switching fields intersect (Hc1 = Hc2) at Φ = ±56. Thisyields according to Eq. (2.90), that the easy axes are oriented at Φ = ±34 = ±α.With the error of ±5, this clearly excludes orthogonal easy axes, and hence is a proofof type (ii) magnetic anisotropy according to Fig. 2.4.

The fit describes the switching fields well, except for a narrow range around thecrystalline (001) axis (Φ = ±90). Note that Eqs. (2.81)-(2.88) allow only three freeparameters in the fit: the orientation of the hard axes (here: λ = 0) and the minimalvalues of Hc1 at the hard axes, µ0Hc1(Φ = 0) = 3.4 mT and µ0Hc1(Φ = 90) = 5.0 mT.The orientation of the easy axes is therewith determined as

α = tan−1 Hc1(Φ = 0)Hc1(Φ = 90)

= tan−1 0.68 = 34 (4.12)

The switching fields Hc1 and Hc2, evaluated from MTHx for the current both alongthe crystalline (010) and along the (001) axes are shown in Fig. 4.6, for T = 5 Kand for T = 25 K. The symmetry properties of the data, and thus the position ofthe easy axes, are very similar for all 4 cases. The effect of temperature is negligible.

67

j||(010)

H

φ

(a) (b)

(c) (d)

EA

EA

EA

EA

EA

EA

EA

EA

j||(010)

H

φ

H

φ

H

φ

(001)

(001)

Figure 4.6: The switching fields Hc1 and Hc2 in CrO2 at 5 K (a),(b) and 25 K (c),(d),evaluated from magneto-transport with current along the crystalline (010) and (001) axes.The magnetic easy axes (black) are orthogonal to the orientations with Hc1 = Hc2 (red).The magnetic hard axes are the easy axes’ bisectors at Φ = 0, 90. Note, that themagnetic field orientation Φ() is always measured with respect to the CrO2 (010) axis(cf. Fig. 2.12)

Unfortunately it not possible to access higher temperatures with our setup, since smalltemperature fluctuations introduce large noise on MTHx. This effect is less severe inGa0.96Mn0.04As , since there the resistivity is less affected by temperature. Furthermore,the Curie temperature in CrO2 is 4 times higher than in Ga0.96Mn0.04As , and hence weare not able to investigate possible variations in the magnetic anisotropy in CrO2 withtemperature.

However, the different current paths do yield slightly different absolute values ofthe switching fields, especially for current along (001) (see Fig. 4.6(b,d)) and H close

68

to (001) the evaluation of Hc2 fails over a larger range than it does for current along(010). Besides that, also the different current path does not have a significant effect onthe evaluated switching fields. This may indicate that the sample is no longer a singleferromagnetic domain, when H is applied closely to the hard axis along (001).The quantitative discussion is based on the ideas about magnetic switching more ex-tensively discussed in Section 2.5. The orientations for which Hc2 diverges, yields thenet in-plane hard magnetic axes. They are with good accuracy oriented along Φ = 0

and along Φ = 90, corresponding to the CrO2 (010) and (001) axes. Next we deter-mine the angles Φ, for which both switching fields converge (Hc1 = Hc2), representedby two red lines in Fig. 4.6. They are not orthogonal, but tilted towards Φ = 90.Thus the magnetic anisotropy is of type (ii) (cf. Fig 2.4), with a dominant biaxial anda superimposed in-plane uniaxial contribution. The magnetic easy axes of the biaxialcontribution are oriented at Φ = ±45. The net magnetic easy axes, are tilted dueto the in-plane uniaxial contribution to the magnetic anisotropy. They are not thetwo red lines, but their perpendiculars, represented by a pair of black lines in Fig. 4.6.The main result is the orientation of the net easy axes, along ±34 with respect to theCrO2 (010) axis.

Φeasy = ±34 (4.13)

The easy axis of the uniaxial contribution is oriented at Φ = 0, thus λ = 0. Theangle between the easy axis of the uniaxial contribution and the net easy axis thus isα = 34. From this we can calculate the ratio of the uniaxial and biaxial anisotropycontributions:

Ku

Kb

= − cos(2α) = 0.5 (4.14)

The minimal value of Hc1 is ≈ 4 mT at Φ = 0 and ≈ 8 mT at Φ = 90. As only theratio Ku

Kbdetermines the orientation of the easy axes, it is not possible to determine the

values of Ku and Kb separately. According to Eq. (2.81), we can relate the ratio of theminimal values to α:

Hc1(Φ = 0)Hc1(Φ = 90)

= tan α (4.15)

The values of Hc1 have a minimum close to each hard axis. However, exactly ata hard axis the values are higher, or the evaluation fails. Thus the error on Hc1(Φ =0)/Hc1(Φ = 90) is relatively large, and we cannot use Eq. (4.15) for a quantitativeevaluation of α. However it is suitable for a rough check of consistency of the model,agreeing within the error on Hc1(Φ = 0)/Hc1(Φ = 90).

Finally the ratio ε/M can be determined from Eqs. (2.81)-(2.88):

ε

M= Hc1(Φ = 0) · 2 cos α = µ0 · 7 mT (4.16)

ε and M cannot be calculated separately, however a separate measurement of M wouldallow to determine the energy barrier for domain formation ε.

69

Microscopically, we can determine the relative orientation of the uniaxial and biaxialcontributions to the magnetic anisotropy and their quantitative ratio from the switch-ing fields.The main result however is that CrO2 is biaxial below 25 K, with the easy magneticaxes along ±34 with respect to the (010) axis. In contrast, uniaxial in-plane magneticanisotropy at room temperature in equally grown CrO2 thin films has been reported[RYK+06, RGM+04]. According to these reports, the easy axis is oriented along (001)for 100 nm thick films like the one investigated here, and along (010) for film thickness≤ 30 nm. We expect a tremendous change in the magnetic anisotropy between 25 Kand 300 K, since we found both axes to be the hard axes at low temperatures. Themost straight forward experiment to do next, is to investigate the conventional FMRon the films used here for 25 K and 300 K, and then for intermediate temperatures, totrack the change in magnetic anisotropy. An extension of these experiments would beto investigate the thickness dependence of this change in the magnetic anisotropy aswell. To this end, temperature dependent FMR on a ≈ 30 nm thick CrO2 film shouldbe investigated as well.The investigation of magnetic switching via the longitudinal or transverse resistivityrequires that the changes in AMR with magnetization orientation are large comparedto spurious variations of the resistance, e.g. by temperature fluctuations. The tem-perature derivative of the resistivity of CrO2 is several orders of magnitude smaller forT ≤ 25 K than for T = 300 K. Hence, to perform AMR measurements in CrO2 atT = 300 K , is challenging with respect to the stability of the sample temperature.Conventional FMR experiments, as used by Rameev et al., are much more promising,since they work equally well for low and high temperatures.

4.1.3 Quantitative analysis of the changes MTHx and MTHyat magnetization switches

In the following, we will quantitatively discuss the magnitudes of the MTHx and MTHyshown in Fig. 4.3. The current paths are oriented at Φ = 0 and Φ = 90, being thecrystalline (010) and (001) axes respectively. Those are exactly the hard axes, thebisectors of the easy axes. The longitudinal and transverse resistivities, with currentpath at Φ = 0, for M oriented along one (EA1, −34) or the other (EA2, +34) easyaxis, read as:

ρxx,EA1 = ρ⊥ + ∆ρ cos2(−34) (4.17)

ρxy,EA1 = ∆ρ1

2sin(−56) (4.18)

ρxx,EA2 = ρ⊥ + ∆ρ cos2(+34) (4.19)

ρxy,EA2 = ∆ρ1

2sin(+56) (4.20)

70

according to Eq. (2.28) and with ∆ρ = ρ‖ − ρ⊥. If the orientation of M is changedby 180, both ρxx and ρxy remain unchanged. For a single ferromagnetic domain, andM strictly oriented along easy axes, the MTHx and MTHy for a specific H and Φ areeither zero, when M is equal for up and down sweep, or M is oriented along differenteasy axes for up and down sweep. The latter case, occurring in the hysteretic region,the MTHx and MTHy can be calculated using the orientations of the easy axes (EA1,EA2):

MTHx = ±∆ρ(cos2(−34)− cos2(+34)) = 0 (4.21)

MTHy = ±1

2∆ρ(sin(−56)− sin(+56)) = ±∆ρ(sin(+56)) (4.22)

This finding predicts no switches in MTHx, while there should be switches observed inMTHy. In contrast, in our experiments, we observe clear switches in MTHx, with evenbetter signal to noise ratio than in MTHy. The approach in Section 2.4.2, introducingcrystalline anisotropy, does not relieve this problem, as long as the coefficients B and C,that replace ∆ρ are constants. Goennenwein et.al. [GRM+05] discussed on (Ga,Mn)As,whether an induction dependent ∆ρ allows to describe the magnetotransport dataquantitatively. They find good agreement of ∆ρ ∝ |H + M| with experiment. Ananalogous, fully quantitative study of our magnetotransport experiments failed, due tothe noise level, and because the M switches are much less abrupt than in the modelsystem (Ga,Mn)As. Qualitatively, ∆ρ ∝ |H + M| yields jumps in MTHx for abruptM reorientation, whenever H is not applied along the hard axis, bisecting the Morientation before and afterwards. Furthermore the slope in MTHx around H = 0,can be understood in this assumption, without leaving the picture of a single domain,orientated strictly along an easy axis.Independent of how the M switches affect the MTHx or MTHy, and independent oftheir magnitudes, the H values, at which the switches occur provide information onthe magnetic anisotropy.

4.1.4 AMR at high fields

So far we have only discussed the hysteretic features in the AMR obtained by varyingthe magnitude of the magnetic field for fixed field orientations. We now address theAMR for constant field magnitude |H| as a function of Φ (angular sweep). Experi-mentally this is realized analogously to the angular sweeps in Ga0.96Mn0.04As describedin Section 3.1.4. The resistance values at µ0H = +0.2 T are extracted from the fieldsweeps. One may ask, whether the magnetization state after a field sweep is identicalto that before. Note, that all hysteretic effects observed occur within µ0|H| ≤ 70 mT,hence we can assume that the magnetization state in angular sweeps at µ0|H| = 200 mTis fully reversible and thus is equal to that in a ’real’ angular sweep, with constant ap-plied field magnitude. In the case of large enough applied field, M ‖ H and θ = Φ.

71

Thus, Eqs. (2.28) and (2.29) read:

ρxx = ρ⊥ + (ρ‖ − ρ⊥) cos2(Φ) (4.23)

ρxy = (ρ‖ − ρ⊥)1

2sin(2Φ). (4.24)

The angular sweeps allow to directly measure ∆ρ = ρ‖ − ρ⊥.

Figure 4.7 shows the longitudinal and transverse resistance in the CrO2 sample as afunction of Φ at 5 K, extracted from field sweeps at µ0H = +0.2 T. As observed alreadyin Fig. 4.1, ρxy does not change sign, as in Ga0.96Mn0.04As , but the variations withΦ are dominated by a large background. We attribute this to geometric misalignmentof the transverse contacts, yielding a superposition of a fraction of the longitudinalvoltage drop onto the real planar Hall voltage. We find this fraction is different for thetwo branches of the Hall bar. However for both current orientations, the offset voltagecan be understood by a misalignment of the order of 1µm.We find a clear cos2 Φ dependence for ρxx(Φ), and a sin(2Φ) dependence for ρxy(Φ), inagreement with the simple model, Eqs. (2.28),(2.29). The values of the longitudinalresistivity is a factor of 2.5 larger for the current oriented along the (010)-axis thanalong the (001)-axis. Note that the Hall-bar geometry, namely width and length, isidentical for both current paths. The amplitudes of the cos2 Φ oscillation differs aswell, being larger for current along (001). That cannot be explained by misalignmentor other experimental problems, especially since the absolute value of ρxx along (001)is smaller, but the variations are larger than along (010). Obviously a more advanceddescription than in Eqs. (2.28),(2.29) is required . As in Section 3.1.4, we will applythe model from Section 2.4.2, taking into account the crystalline structure. We followagain the approach by Limmer et al. [LGD+06]. As long as (Ga, Mn) As was concerned,the symmetry of the crystal is cubic (m3m), and the power expansion for a (100)-surface yielded equivalent results (Eq. (2.37)) for current path along (010) or (001). InCrO2 however, additional terms in the power expansion have to be taken into account,since the symmetry is reduced and (010) and (001) are no longer equivalent directions.Starting from ρm3m given by Eq. (2.31), we get ρtetrag = ρm3m+δρt for lattice distortionalong (001), as in CrO2, with

δρt =

b1mz dmz + cmxmy 0−dmz + cmxmy b1m

2z 0

0 0 a + b2m2z

(4.25)

and the additional resistivity parameters a, b1, b2, c and d. Until here this modelused only the symmetry properties of a tetragonal bulk crystal. For any surface andcurrent orientation, the longitudinal and transverse resistivities can be related to theparameters in this model. For a (100) plane, and M strictly in plane, ρxx and ρxy for

72

(a)

(b)

Figure 4.7: ρxx(µ0H = +0.2 T, Φ) (a) and ρxy(µ0H = +0.2 T, Φ) (b) at 5K, with currentpath along the crystalline (010)-axis (filled squares, right scales) and along the (001)-axis(open circles, left scales).

73

current oriented along (010) and (001) are given by:

j ‖ (010) : ρxx = A + (B − b1) cos θ2 (4.26)

j ‖ (010) : ρxy =C

2sin(2θ) (4.27)

j ‖ (001) : ρxx = A + a + (B + b2) cos θ2 (4.28)

j ‖ (001) : ρxy =C

2sin(2θ) (4.29)

Thus upon going from cubic to tetragonal symmetry, additional degrees of freedomare introduced. The offset of the transverse resistances, that are not described bythis theory, but rather by misalignment are not equal for the two current directions.From the magnitude of ρxx for both current directions we obtain A = +1.463µΩcmand a = −0.874µΩcm. Furthermore the amplitude of the cos2 Φ oscillation is givenby independent parameters for j ‖ (010) and j ‖ (001) in the case of tetragonalsymmetry. This yields (B − b1) = +1.8 nΩcm and (B + b2) = +2.8 nΩcm. Howeverthe parameters B, b1 and b2 cannot be determined separately, since only two relationsfor three degrees of freedom are found.However, according to this model, the amplitude of the 1

2sin(2θ) oscillation in the

transverse resistance is strictly equal for j ‖ (010) and j ‖ (001), termed C. Thisis exactly what is found in experiment. For both current directions, we obtainC = −0.6 nΩcm within experimental error. Note again, that only the symmetryproperties of the crystal have been used, to come to this power expansion. Thisnicely corroborates the model proposed by Limmer et al. as a very useful tool toquantitatively describe magneto-transport.To summarize, we find, that both absolute value and variation of ρxx are differentfor current orientation along (010) and (001), as predicted by Eqs. (2.31),(4.25). Incontrast, the oscillation amplitude in the transverse signal is equal for both currentorientations, as predicted by Eqs. (2.31),(4.25).

4.1.5 Outlook on fully quantitative magneto-transport

In order to determine all free parameters in Eqs. (2.31),(4.25), more experiments indifferent geometries have to be performed. The Hall-like parameters D, d can only befound in experiments with M not in the film plane. In the Hall bar investigated here,only the parameter D could be determined in this way. Based on (100) CrO2 thinfilms, a Hall-bar with current orientation along (011) would allow to determine B, b1

and b2 separately:

j ‖ (011) : ρxx = A +

(B − b1 + b2

2

)cos θ2 (4.30)

j ‖ (011) : ρxy = B +

(b1 + b2

2

)sin(2θ)

2(4.31)

74

Current along (011) yields equal resistivities as along (011). Therefore it would bemore useful to fabricate an angled Hall-bar with branches parallel to (011) and (001),rather than parallel to (011) and (011).The parameters d and c however, can only be determined by magnetotransport in filmsoriented other than (100) or (010). (110) films are promising in order to fully determineall parameters describing the CrO2 magnetotransport. However, exotic current orien-tations such as (111) and (112), have to be chosen in order to achieve low symmetrywith respect to the tetragonal distortion along (001).Once magneto-transport in any arbitrary orientation of M and j is described quanti-tatively, it is a powerful tool to determine the magnetic anisotropy of nano structures.

75

4.2 Magneto-transport in CrO2 with H out of plane

We now turn to magneto-transport measurements with H applied perpendicular to thefilm plane. In this configuration, the ordinary Hall effect (OHE) and the anomalousHall effect (AHE) contribute to the transverse voltage. We present a comparison of theAHE and the longitudinal resistance ρxx as a function of temperature and magneticfield |H|. From these measurements, the side-jump mechanism can be identified as themain, but not the only cause of the AHE.In contrast to Section 4.1, the experiments presented in the following were performedon the sample Bonomi, a d = 100 nm thick and w = 80 µm wide Hall bar, describedin Section 2.7. Bonomi was also used in the electrically detected FMR experiments(Section 5), which require a quantitative comparison with the magneto-transport. Thecurrent applied is I = 1 mA in all experiments described in the following section.

4.2.1 Longitudinal magneto-resistivity

First we address the longitudinal resistance. Figure. 4.8(a) shows a typical I-V-curve,taken at 300 K and zero field, which is ohmic, as comparison to the straight dashedline proves.Its temperature dependence at µ0H = 0 mT is shown in Fig. 4.8(b). ρ decreases withdecreasing T , showing that the sample is metallic, indicating high quality CrO2. Below40 K, the resistance is constant, and for higher temperatures increasing rapidly. Thehigh residual resistivity ratio

RRR =ρxx(300 K)

ρxx(5 K)= 45 (4.32)

is characteristic for a good metal. At 2 K and µ0H = 0 mT, ρ = 3.8µΩcm are observed.The RRR and the magnitude of ρ observed, are in agreement with those published forhigh quality single crystalline CrO2 thin films [CV02, YOS+03, LGM+99, WWM+00].Poly crystalline films exhibit higher low temperature resistivity and smaller RRR.Between 100 K and 300 K, ρ obeys the power law ρ ∝ T 2.5. On the basis of our data,we can exclude a pure ρ ∝ T 2 dependence. The data agree with results publishedby Watts et al. [WWM+00]. These authors argue, that the temperature dependencecan be understood as electron-magnon interaction, however an additional factorexp(−∆/T ) for a gap ∆ in the magnon spectrum has to be introduced. However, ourdata, and those published by Watts et al. exclude a pure ρ ∝ T 2 dependence.

The magneto-resistance (MR) with H applied perpendicular to the film plane is dis-played in Fig. 4.9 for various temperatures. For temperatures above 135 K, a negativeMR of up to

MR =R(13 T)−R(0)

R(0)=

R(13 T)

R(0)− 1 = −5.8% (4.33)

76

3 10 30 100 3003.5

10

30

100

-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9-120

-80

-40

0

40

80

120

xx(

cm)

T (K)

CrO2

300K0H=0T

xx T2.5(b)

Vxx

(mV

)

I (mA)

(a)

Figure 4.8: (a): The I-V-curve in the CrO2 sample Bonomi at 300 K and zero field. Thedashed line is a guide to the eye. (b) shows the temperature dependence of the resistivityat zero field. The metallic behavior and the RRR = 45 characteristic of high-qualityCrO2 is evident. Above 100 K, ρxx ∝ T 2.5 is observed.

at 13 T and 300 K is found. At 50 K and below, a positive MR of up to +23.4% at 13 Tand 5 K is observed. The negative MR at higher temperatures can be understood interms of suppression of spin disorder by the applied field. Since spin disorder increasesthe scattering rate and thus the resistivity, the applied field reduces the resistivity.These findings agree with measurements published in [LGM+99, WWM+00].

4.2.2 Hall effect and anomalous Hall effect in CrO2

Figure 4.10(a) displays the transverse resistivity ρxy = Vxy·dI

, where d = 100 nm is thefilm thickness. While an antisymmetric Hall contribution is clearly present in the data,a spurious signal is superimposed on the Hall voltage, as ρxy(H = 0) 6= 0. Moreover thefield up- and down-sweeps show an offset with respect to each other. The latter effectis due to the measurement procedure: in contrast to all other experiments describedin this thesis, the field was swept continuously. During the sweep, the voltmeterscontinuously record data in the moving average mode, with an effective time constant

77

-10 -5 0 5 10

0.0

0.1

0.2

-1.0 -0.5 0.0 0.5 1.0

0.000

0.002

0.004

(xx

/xx

(0T))

-1

0H (T)

300K 200K 50K 270K 150K 25K 250K 135K 5K

(a)

300K 200K 50K 270K 150K 25K 250K 135K 5K

0H (T)

(b)

Figure 4.9: (a): Magneto-resistance (MR) with H applied perpendicular to the film plane.For temperatures above 135 K, a negative MR of up to −5.8% at 13 T and 300 K is found.At 50 K and below, a positive MR of up to +23.4% at 5 K is observed. (b) shows a blow-upof the MR at low fields.

τ = 1 s. This results in an artificial shift of the R(H). To eliminate this hystereticcontribution as good as possible, we have averaged the up and down sweeps.

To extract the antisymmetric Hall contributions we consider the anti-symmetrizedtransverse resistance:

ρantixy (H) =

1

4

(ρup

xy(H) + ρdownxy (H)− ρup

xy(−H)− ρdownxy (−H)

). (4.34)

In the following, the transverse resistance for magnetic field sweeps from negativeto positive H (up-sweep) and vice versa (down sweep) are termed ρup

xy and ρdownxy , re-

spectively. ρantixy (H) consists of the AHE and the OHE contributions only.

ρantixy is shown in Fig. 4.10(b) for temperatures from 5 K to 300 K. The Hall signal

consists of both, an ordinary and an anomalous Hall contribution (see Section 2.4.4).Below 115 K, the ordinary Hall effect dominates, i.e. ρanti

xy ∝ H over the entire rangeof applied field. The Hall coefficient RH is given by the slope of ρanti

xy (H) at high fields,when the AHE is saturated, and RH = 1

nq(see Eq. (2.46)).

78

(b)(a)

Figure 4.10: (a): ρxy(H) for the CrO2 sample Bonomi at various temperatures, with Happlied perpendicular to the film plane. A spurious signal is superimposed on the Hallsignal. (b): In the antisymmetrized ρanti

xy (H), only Hall signals are retained. The steepnegative slope at small fields is due to the AHE, as M reorients. Below 115 K, the AHEvanishes. The small positive slope at higher fields is due to OHE, it is constant for alltemperatures above 75 K, but increases below.

The AHE is dominant for −1 T < µ0H < 1 T, with a very steep negative slopein ρanti

xy , as the magnetization reorients in this field range. For µ0|H| > 1 T, the AHEsaturates for all temperatures and its magnitude increases strongly as temperature isincreased.

Below 50 K a significant increase of RH is observed.

In contrast to our findings in Ga0.96Mn0.04As (Section 3.2), ρantixy (H) is linear in a

good approximation at high fields. This is due to the small MR (see Fig. 4.9) in thetemperature range of large AHE. This allows for a quantitative determination of theordinary Hall effect in CrO2, even though the charge carrier concentration is one orderof magnitude larger, and hence the OHE is one order of magnitude smaller than inGa0.96Mn0.04As .

To discuss the Hall coefficient quantitatively, the derivative of ρantixy (H) is calcu-

lated. RH = ∂∂H

ρantixy (H) is shown in Fig. 4.11(b). Around zero field, the AHE is

dominating, and its temperature dependence can again be clearly seen. At higherfields, RH converges to a constant value. This for the Hall effect relevant range isshown in the inset in Fig. 4.11(a). For T ≥ 75 K, RH = (20± 3)× 10−9Ω cm is nearlyindependent of temperature. However, RH becomes larger at low temperatures, withRH(T ≤ 25 K) = (30±2)×10−9Ω cm. According to Eq. (2.46), the hole concentration

79

is given by p = 1eRH

, with e the elementary charge. The sign of the slope in ρantixy (H)

determines the charge carriers as holes. This corresponds to a hole concentration of

p(T ≥ 75 K) = (3.2± 0.5)× 1022 cm−3 = (0.91± 0.14)(f.u.)−1 (4.35)

p(T ≤ 25 K) = (2.1± 0.1)× 1022 cm−3 = (0.61± 0.04)(f.u.)−1. (4.36)

The hole concentration of 0.9 per formula unit (f.u.) is about 3 times larger, than thereported in literature [LGM+99, WWM+00]. However, we tacitly assumed unipolarcharge transport so far, while two-band transport is required to fit experimental dataquantitatively [WvMJ02, WWM+00, MSAD99, YS02].

A two band model fits the change in hole concentration at low temperatures weobserve. However, Watts et al. [WWM+00] report a sign change in the Hall effect withtemperature. Furthermore these authors report that less mobile electrons outnumberthe highly mobile holes by a factor of 500. This is in contrast to our observations,since we observe hole like transport at all temperatures. To quantitatively understandthe sign change in the anomalous Hall effect, the models in Section 2.4.4 have to beextended to a two-band model.

4.2.3 Temperature dependence of ρxx and ρAHE

The main goal of this paragraph is to determine the microscopic origin of the AHE.As discussed in Section 2.4.4, a proportionality ρAHE ∝ ρxx

α is considered as strongevidence that the AHE is caused by the skew-scattering mechanism for α = 1, whileα = 2 is attributed to the side-jump mechanism.In the measurements described until now, H was swept at various fixed temperaturesT (field sweep). We now consider temperature sweeps from 2 K to 300 K at fixedapplied fields. Such sweeps have been performed at 0 T and ±13 T.

In this way, antisymmetrized ρantixy = ρxy(+13 T)−ρxy(−13 T) can straightforwardly

be calculated and compared to the results of field sweeps at the same T . As evidentfrom Fig. 4.12, the results of temperature sweep and field sweeps agree very well.

Note that the Hall coefficient evaluated from the field sweeps has been interpolatedand than subtracted off the antisymmetrized temperature sweeps ρanti

xy , in order toobtain ρAHE(T, µ0H = ±13 T).

The result is displayed as the blue line in Fig. 4.12. At 70 K ρAHE changes sign.Except for the 100 K data point, ρAHE from field and from temperature sweeps agreewell. Between 100 K and 300 K, ρAHE depends on the temperature as ρAHE ∝ T 3.5

to ρAHE ∝ T 4 in good approximation. In this range, ρAHE increases two orders ofmagnitude. The data published by Watts et al. are shown for comparison as filledsymbols in Fig. 4.12 on an arbitrary scale. The temperature dependence of both setsof data agree well.

ρxx(T, µ0H = 13 T) differs from ρxx(T, µ0H = 0 T) only in the low temper-ature regime. This is the magneto-resistance, discussed in Fig. 4.9. According to

80

Figure 4.11: (a),(b): The field derivative of Fig. 4.10(b), ∂∂H ρanti

xy (H). It switches betweentwo levels. The AHE yields the larger, negative level around zero field, while the smallpositive level corresponds to the ordinary Hall coefficient RH at high fields. At 50 K andbelow, RH is significantly higher than from 75 K upwards. Around zero field, a positiveAHE is resolved for 50 K and below. (c): After subtraction of the OHE from ρanti

xy (H),only the AHE is left (ρAHE(H)). (d) shows the low temperature curves. For 100 K theAHE is 100 times smaller than at 300 K, but of the same sign. At lower temperatures,the AHE sign changes from positive to negative.

section 2.4.4, for a quantitative comparison of ρAHE and ρxx, in order to obtain α inρAHE ∝ ρxx

α, both ρAHE and ρxx have to be taken at the same applied field. SinceρAHE is determined from antisymmetrizing temperature sweeps at ±13 T we have touse ρxx(T, µ0H = 13 T). Using ρxx(T, µ0H = 0 T) would introduce the magneto-resistance as an error. For 100 K to 300 K, we find ρxx(T, µ0H = 13 T) ∝ T 2.5.Within the accuracy of the experiment, we can exclude a pure T 2 dependence, asdiscussed in Section 4.2.1.

81

For comparison, ρAHE(T ) has been extracted from the field sweeps as follows:ρAHE(T ) = ρAHE(T, µ0H = −13 T) − ρAHE(T, µ0H = +13 T) and is displayed asblue squares in Fig. 4.12.

A comparison of ρAHE(T ) and ρxx(T ) thus yields the exponent

1.4 =3.5

2.5< α <

4

2.5= 1.6

The uncertainty stems from ρAHE(T ), that depends on temperature as ρAHE ∝ T 3.5

to ρAHE ∝ T 4

In other words, ρAHE ∝ ρxx1.4...1.6. Within the accuracy of our measurement, we can

exclude a pure ρxx2 dependence, in contrast to the results by Watts et al. [WWM+00].

This makes clear, that a quantitative comparison between longitudinal resistanceand the anomalous Hall effect has to be performed with extreme carefulness. Watts etal. assumed ρxx ∝ T 2 instead of ρxx ∝ T 2.5 that would fit the data better, resultingin a completely different conclusion in the physics of the AHE. While these authorsconclude pure side-jump mechanism, we find that only a coexistence of side-jump andskew-scattering can be the origin of the AHE in CrO2.

We now attempt to address the influence of M(T ) on ρAHE(T ), according to

ρAHE(T ) ∝ M(T )ραxx(T ). (4.37)

The temperature range of concern is 100 K < T < 300 K, and the Curietemperature of the CrO2 films is ≈ 380 K. Magnetization measurements havebeen performed in a SQUID magnetometer.We find M(300 K) = 0.70 × M(100 K).This decrease is negligible with respect to the evaluation of α, and especially it isinsufficient, to agree with ρAHE ∝ Mρxx

2. However, the curvature in ρAHE at 300 Kcan be understood in terms of decreasing magnetization.

4.2.4 Summary

Our CrO2 thin films show the characteristics of a good metal with a residual resistivityratio of 45 and ρxx(300 K) = 170µΩcm. The resistivity is proportional to T 2.5 above100 K. The magneto-resistance (MR) between 75 K and 300 K is negative, and canreach for several percent at 13 T. Below 25 K however, a MR of +20% is found.The Hall effect indicates a hole concentration in the 1022 cm−3 range. Below 50 K,the Hall coefficient increases. This regime has been discussed controversially in theliterature, and a two band model with competing hole and electron like transport,with significantly different mobilities has been invoked. In the simple approximationof unipolar transport, our measurements indicate nearly one hole per Cr atomcontributing to the transport at high temperatures, and two thirds of this value below25 K. The carrier concentration in CrO2 has been discussed controversially. While

82

35 70 100 140 210 280350100

101

102

103

104

xx ( 0H = 0T)

xx ( 0H = 13T)

T3.5

AHE from R(T)

AHE from R(H)

AHE from Watts et al. (arb. units)

T2.5

xx (

cm),

AHE (1

0-9 c

m)

T (K)

T4

Figure 4.12: Both ρxx and ρxy significantly change with temperature. The lines representthe temperature sweeps ρAHE(T ) (blue), ρxx(T, µ0H = 0 T) (black) and ρxx(T, µ0H =13 T) (red). The open squares correspond to the ρAHE data points from field sweeps. Thefull pentagons correspond to the data by Watts et al. [WWM+00], on an arbitrary scale.ρAHE(T ) is well fitted by a T 4 or a T 3.5 dependence (solid lines), and ρxx(T ) ∝ T 2.5 (dashdotted line) above 100 K.

Watts et al. [WWM+00] report 0.4 electrons per Cr, Li et al. reported 0.3 holes perCr [LGM+99]. The anomalous Hall effect is negative and proportional to T 4 above100 K. It changes sign at 70 K, possibly due to an electronic contribution to thetransport at lower temperatures. A tremendous change in the transport propertieshas been observed by Watts et al., well agreeing with our observations, as temperatureis increased through T = 80 K. The MR becomes negative, the resistance increasesfaster than linear with temperature, and the AHE increases rapidly. These authorsrelate all these effects to the occurrence of spin-flip scattering events due to local spinfluctuations that give a finite, effective density of minority spin states at the Fermienergy.From the temperature dependence of the AHE and longitudinal resistivity, we cannotidentify the side-jump mechanism as the origin of the AHE, in contrast to Watts etal. Instead, the AHE is best described as ρAHE ∝ ρxx

α, with 1.4 < α < 1.6.

83

While the anomalous Hall effect therewith is described well in the temperature rangeabove 100 K, further understanding is required for the low temperature mechanism.This question is closely related to the enduring discussion of the carrier type anddensity. Therefore the Hall effect has to be addressed in terms of a two band model.Furthermore samples with different film orientations should be investigated, providingthat Li et al. used (100) CrO2, as in our experiments, while Watts et al. used (110)CrO2.To further access the charge carrier types, the determination of the heat transportcoefficients, i.e. the Seebeck- and the Nernst-coefficient, is challenging, but wouldreveal tempting insights to the transport-mechanism in CrO2.

84

Chapter 5

Electrically detected ferromagneticresonance

Ferromagnetic resonance (FMR) is one of the most sensitive methods to investigatethe magnetic anisotropy. However, conventional FMR cannot be used to investigatemagnetic nanostructures. This is due to the fact that the absorption of microwaveis used to detect FMR. The absorption signal intensity is proportional to the samplevolume. Beyond a certain sample size, the FMR then becomes smaller than the noisefloor, and cannot be detected.This sensitivity limitation can be overcome by detecting the FMR via other physicalquantities. The occurrence of FMR affects the temperature[MLKM00], and the otherquasi-static properties of a magnetic material, such as the magneto-resistance[EJ63,Tod70, KKS78, GHMH05], the magneto-impedance[BMM+00]. These effects offer scal-able techniques to detect FMR in magnetic microstructures.[MLKM00, GHMH05] Thisis an attractive perspective for a sensitive investigation of magnetic anisotropy beyondthe resolution limit of conventional FMR.

To fully exploit the potential of these unconventional FMR detection methods,their equivalence with the well established, conventional, cavity-based FMR must bedemonstrated.In a complementary approach, resonant experiments have been used to selectivelyinvestigate magneto-resistive phenomena, such as anisotropic magneto-resistance orthe anomalous Hall effect (AHE)[EJ63, Tod70, CS72, KKS78, GHMH05].

In this chapter [SBB+06], we report how the magneto-resistance of thin CrO2 andFe3O4 films changes upon microwave irradiation. We analyze the resonant changes ∆ρin the longitudinal resistance ρ and show that this electrically-detected ferromagneticresonance (EDFMR) signal is spectroscopically equivalent to conventional ferromag-netic resonance, measured simultaneously.

Furthermore, we also observe resonant changes ∆Vtrans of the transverse voltage inCrO2. The sign and the magnitude of the resonant changes ∆ρ/ρ and ∆Vtrans/Vtrans canbe consistently described in terms of a Joule heating effect, if one takes into account theresults from conventional magneto-transport experiments (Section 4.2). This demon-

85

Lock-InDetector

Circulator

Microwave

Source

Magnet

ESRBias

Sample,Cavity,Cryostat

Lock-InDetector

Circulator

Microwave

Source

Lock-InDetector

Circulator

Microwave

Bridge

(a) (b)

Figure 5.1: (a): Sketch of the setup used for the FMR measurements. (b) Mode pattern ina TE 102 cavity[Mec97]. At the sample position, the microwave magnetic field is maximaland perpendicular to the dc magnetic field. The electric field has a node at the sampleposition.

strates that the quantitative measurement of EDFMR in the longitudinal resistanceand in the Hall signals allows to separate the resistivity- and magnetization-dependentterms in the anomalous Hall voltage V AHE = cραMzI/d (see Eq. 4.37), and thus is analternative tool for the study of the microscopic origin of magneto-resistive phenomena.The CrO2 sample used, Bonomi is described in Section 2.7. Its dc transport proper-ties have been investigated in Section 4.2. The Fe3O4 sample is described elsewhere[Bra06b, RMO+04]. It is contacted by wedge bonding. The current path implies anuncertainty of a factor of 2 on the absolute value of the resistivity. However, the relativechanges in the resistivity are precise.

5.1 The ferromagnetic resonance setup

The theoretical background of the FMR has been discussed in Section 2.3. For the con-ventional, microwave absorption-detected FMR experiments, the samples are insertedinto the TE 102 cavity of a commercial Bruker electron spin resonance setup operatingat 9.3 GHz. The FMR spectra are recorded at room temperature, using magnetic fieldmodulation at 100 kHz. The setup used for the experiments described in Section 5.5operates at 9.8 GHz, also with a TE 102 cavity.

The setup is sketched in Fig. 5.1(a). Linearly polarized microwave radiation isgenerated in the microwave bridge by means of a Gunn diode array and a klystron at9.3 GHz and at 9.8 GHz, respectively. It is then split into two parts of equal power. Onepart (bias) is phase shifted, variably attenuated and then applied to the detection diode.This keep the diode in the working point, the region of its characteristic, where thesignal changes most for small changes in applied power. The other part goes through a

86

circulator to the resonator, in which the sample is located. The circulator is a one waydevice, i.e. it is transparent only radiation coming from the source to the resonator andfrom the resonator to the detector, and blocks the reverse paths. The coupling of theresonator is adjusted via an iris, such that it is critical. This means, that no power isreflected from the resonator. When the FMR occurs, the quality factor of the resonatoris changed, thus this critical coupling is disturbed, and then microwave is reflected fromthe resonator via the circulator to the detection diode, where it constructively interfereswith the bias. The detection diode converts the microwave power to a voltage, whichis fed to the lock-in.

The resonator supports a standing wave pattern, a TE 102 mode, as sketched inFig. 5.1(b) [Mec97]. The sample is positioned such, that the microwave H field ismaximal, and the microwave E field is minimal.

The dc magnetic field is applied via normal-conducting coils, with µ0|H| ≤ 1 T inthe 9.3 GHz setup, and µ0|H| ≤ 1.4 T in the 9.8 GHz setup, respectively. To allow forlock-in technique, an ac magnetic field (ν = 100 kHz) is superimposed parallel to thedc field.

The coils generating this modulation field are separated by a gold layer of several10 µm thickness from the cavity, which is reflecting the microwave, but is transparentfor the modulation field.

In the following, we term the signal detected by the lock-in as a function of ap-plied dc magnetic field the FMR amplitude (for short: FMR). Due to the magneticfield modulation, the FMR signal is proportional to the derivative of the absorption(Eq. (5.1)).

5.2 Ferromagnetic resonance in CrO2

The FMR spectra for H in the film plane, and nearly perpendicular to it, are shown inFig. 5.2. The FMR line position changes drastically with field orientation due to shapeanisotropy. Since the line width in-plane is much larger, the FMR signal amplitude issmaller, even though the absorption intensity is comparable for both H orientations.For H oriented exactly perpendicular to the film plane, the resonance field exceeds 1 T,and thus cannot be resolved with the setup used. Since the resonance field in this rangeis very sensitive to the field orientation, a deviation of about 2 from H perpendicularto the film plane is sufficient to shift the resonance into the accessible field range.

5.3 The setup for electrical detection of FMR

Simultaneously to the FMR, the magneto-resistance of the sample is measured in four-point geometry. To this end, an ac current I of frequency νI is driven through thesample, and the resulting voltage drop V is detected with differential preamplifiers.We carefully checked that there is negligible crosstalk between the magnetic field mod-ulation (at ν = 100 kHz) and the ac current modulation for the frequencies νI ≤ 1.1 kHz

87

H

H

x5

ϕ=0°

ϕ=88°

Figure 5.2: FMR signal of the CrO2 Hall bar with H applied in the film plane (ϕ = 0)and perpendicular to it (ϕ = 88). The FMR line position changes drastically with ϕ dueto shape anisotropy. The line width in-plane is much larger, and thus the signal amplitudeappears smaller than out of plane even though the absorption intensity is comparable forboth H orientations.

used in the measurements.The relative resistance changes in resonance are small, and thus the digitalizationdepth of the lock-in amplifiers (18 Bit ≈ 105) becomes relevant. When the resistanceis recorded directly by the differential input of the lock-in, this digitalization depth islimiting the resolution. This limitation can be overcome, by a compensation circuit,in which a constant voltage of the same magnitude as the four-point voltage over thesample is subtracted from this four-point voltage. The small difference signal is thendetected by the lock-in.

Figure 5.3 shows the compensation scheme used for the resistance measurements.As a voltage source, the oscillator output of the lock-in (or an external voltage source)is used. This ac voltage is converted to a current by the resistance Rcurrent = 10 kΩ

88

+

-

+

-

Vosc.outA-B

Lock-in Amp.

Rcurrent

RHBC1

RHBC2

A1

A2

OUT

+

-

+

-

+

-

+

-

Vosc.outA-B

Lock-in Amp.

Rcurrent

RHBC1

RHBC2

HBC

A1

A2

OUT

SR560

SR560

sample

Figure 5.3: The compensation circuit (The initiators of this setup at TU-Delft, obviouslymembers of the star-track-generation, termed it the Heisenberg-compensator (HBC). How-ever, it is neither violating quantum mechanics nor difficult to realize.) used to measuresmall changes in resistance is based on an additional resistor of the same size as the four-point resistance of the sample (Hall-bar in green, substrate in yellow). The voltage dropover RHBC1 and the four-point voltage drop over the Hall-bar are recorded with differen-tial preamplifiers and finally subtracted by the differential input of the lock-in amplifier.This compensator circuit overcomes the digitalization depth of the lock-in.

which is much larger than the resistance of the sample. Hence Rcurrent determines thetotal resistance of the bridge circuit and thus also the current level. The current isthen fed through the first resistor RHBC1, which consists of a switchable resistor forthe coarse, in series with a smaller potentiometer for the fine adjustment. The voltagedrop over RHBC1 is detected by a differential preamplifier (Stanford Research SR560).After passing RHBC2, which allows for a simultaneous compensation of the transverseresistance (not shown in Fig 5.3), the current is applied to the sample and after thatflows to ground. The four-point voltage drop V = IR over the sample is measuredwith a differential preamplifier (SR560). Both signals are applied to the differentialinput of the lock-in-amplifier. RHBC1 is adjusted such, that the difference signal at thelock-in is around zero. The input sensitivity of the lock-in can thus be increased byup to two orders of magnitude. So that the digitalization depth in increased by thesame amount. When the output of the preamplifier detecting the four-point voltage

89

drop over the Hall-bar is applied to the direct input of the lock-in, the absolute valueof V can be measured. With exception of the cables that go to the sample inside theresonator (≈ 20 cm) all cables used are coaxial cables.In our experiments, we found that optimal signal to noise ratio is obtained with νI ≈1 kHz. Upon further increment, the resistance signal detected by the lock-in is shiftedin phase to ≈ 10 at 10 kHz. Hence we used νI = 1.023 kHz in the experiments. Wechecked that the magneto-resistance is not changed when the magnetic field modulation(ν = 100 kHz) is turned on or off.The current level I is a crucial parameter for the measurements. The absolute changes∆R of the resistance in resonance are small, typically ∆R ≤ 1mΩ. To obtain sizablevoltage drops across the Hall bar, we applied 1.75 mA. For lower current levels, thesignal to noise got significantly worse, since the absolute noise level is quasi independenton I. Although the high current level leads to a sample temperature increase of ≈ 1 K,this effect is much smaller than the non-resonant microwave heating.

5.4 Results

Figure 5.4(a) shows how the longitudinal resistivity ρ = VI· wd

lof the CrO2 Hall

bar changes upon microwave irradiation. The data were taken with the externalmagnetic field µ0H applied in the film plane, perpendicular to the current flow inthe Hall bar. When the microwave source is ”off” (attenuated to an output powerlevel of 10 µW), one observes the typical negative low-field magnetoresistivity ofCrO2. The resistivity ρ ≈ 165 Ω for µ0H = 0 mT is in good agreement with otherhigh-quality thin films [WWM+00, ST98] and the results of Section 4.2. When themicrowave source is turned ”on” to an output power level of 200 mW, the resistivityof the film increases by about 30% (see Fig. 5.4(a)), and a broad, peaked structureappears around µ0Hres = 163 mT. Without microwave irradiation, the temperatureof the sample TS equals that of the cavity TCAV. When microwaves are applied, thesample is heated up due to Eddy-currents (non-resonant heating). The increase of theresistivity, independent of H, by 30% can be understood as an increase in TS due tonon-resonant heating. In the experiments described in this section, TS is estimatedfrom this increase in ρ, using ρ(T ) from conventional magneto-transport measurements(Section 4.2.1)[WWM+00, LGM+99].

The peaked structure in Fig. 5.4(a) is the signature of FMR in the electrical re-sistance. In Fig. 5.4(b), the magneto-resistivity of the sample is compared to theconventional FMR signal, recorded simultaneously. Because we use magnetic fieldmodulation, the FMR signal is the first derivative ∂/∂H of the resonant absorptioncurve:

FMR =∂Pabs

∂H. (5.1)

Pabs is the absorbed microwave power. To make a direct comparison possible, we define

90

H

Figure 5.4: (a) The resistivity ρ of the CrO2 Hall bar (full black line) increases uponmicrowave irradiation (blue symbols), and additional, peaked features appear aroundµ0Hres = 163 mT. The data were taken at room temperature, with the externally appliedmagnetic field in the film plane, but perpendicular to the current flow. (b) Upon plottingthe difference quotient ∆ρ/∆H, the close correspondence between the microwave-inducedchanges in ρ (electrically detected ferromagnetic resonance, EDFMR) and the conventionalFMR signal becomes apparent.

the EDFMR analogously:

EDFMR =∆ρ

∆H, (5.2)

with ∆ρ = ρ (H + ∆H/2) − ρ (H −∆H/2). µ0∆H = 3.2 mT is chosen, correspond-

91

ing to the magnetic field modulation amplitude used in FMR. So we can avoid todeform the spectrum additionally to the deformation caused by H modulation, butstill achieve a reasonably smooth derivative. The resistivity data of Fig. 5.4(a) cor-respond to the EDFMR in Fig. 5.4(b). In this representation, the peaked structurearound µ0Hres = 163 mT in the magneto-resistivity can be unambiguously identified asan electrically-detected ferromagnetic resonance (EDFMR) [SS66, NKK79, GHMH05].All FMR modes in the conventional FMR signal are reproduced in the EDFMR trace(cf. Fig. 5.4(b)). The slight discrepancies in signal shape and intensity between EDFMRand FMR are due to the bolometric nature of the EDFMR signal, as discussed in moredetail below. At present, we can only speculate about the microscopic origin of theFMR modes. They could be spin wave modes [RYA+03], or stem from the differ-ent parts of our microstructured sample (main body of the Hall bar, small voltageleads, CrO2 beneath the metalized contact pads, etc.). While further experimentswill be necessary to clarify the origin of these modes, we would like to emphasizethat the resonance field µ0Hres = 163 mT of the strongest resonance quantitativelyagrees with the results of more detailed, conventional FMR investigations of CrO2 thinfilms.[RGM+04, RYK+06]

The microwave-induced changes in the magneto-resistivity can be consistently un-derstood as a bolometric effect. As discussed e.g. by Neppl et al.,[NKK79] the bolo-metric resistivity change upon microwave irradiation can be written as

∆ρ =∂ρ

∂T∆T =

∂ρ

∂T

Pabs τ

C, (5.3)

with the temperature T of the sample, the increase in sample temperature ∆T uponirradiation, the absorbed microwave power Pabs, the thermal relaxation time constantτ between sample and environment, and the heat capacity C of the sample. Becausethe EDFMR signal is proportional to ∆ρ, a purely bolometric EDFMR signal shouldobey EDFMR ∝ (∂ρ/∂T ) Pabs according to Eq. (5.3).

We first analyze the influence of ∂ρ/∂T on the EDFMR signal. CrO2 is a goodmetal, with ∂ρ/∂T > 0 around room temperature (Fig. 5.5(b))[LGM+99]. A tem-perature increase ∆T > 0 due to the absorption of microwave photons (in either aresonant or a non-resonant process) should thus lead to ∆ρ > 0. Both the global andthe resonant increase of ρ upon microwave irradiation of the CrO2 sample shown inFig. 5.4(a) are thus straightforwardly explained. To further test EDFMR ∝ ∂ρ/∂T ,we have also measured the magneto-resistivity of a Fe3O4 thin film under microwaveirradiation. Magnetite has ∂ρ/∂T < 0 (Fig. 5.5(b)), so that both non-resonant andresonant microwave absorption should result in a resistance decrease. This is indeedthe case, as shown in Fig. 5.5(a): ρ globally decreases when the microwave source isswitched on, with an additional, dip-like decrease around µ0H = 170 mT. The latter isthe EDFMR, which again closely reproduces the conventional magnetite FMR signalrecorded simultaneously (not shown). Thus, both in CrO2 and in Fe3O4, the sign ofthe microwave-induced changes in the resistance follows ∂ρ/∂T , as expected for a bolo-metric effect. In the following, we discuss the characteristic changes of the FMR and

92

0 200 400

7240

72607440

7460

0 100 200 30010-2

10-1

100

101

TCAV 300K

T (K)0H (mT)

(b)(a)

CrO2

Fe3O4MW off

MW on

Fe3O4

/ (300K)

(cm

)

Figure 5.5: (a) In contrast to CrO2, the resistivity ρ of the magnetite film decreasesupon microwave irradiation, with an additional resistance dip at the FMR resonance fieldµ0H = 170 mT. (b) The resistivity of CrO2 decreases with decreasing temperature, whilethe ρ of Fe3O4 increases. This accounts for the sign of the microwave-induced resistancechanges in the two materials (Fig. 5.4 and panel (a)), demonstrating the bolometric natureof the microwave effect.

the EDFMR signal intensity SFMR and SEDFMR as a function of the applied microwavepower. For conventional FMR the signal intensity is given by the total area under theabsorption curve:

SFMR ∝∫

PabsdH. (5.4)

Recorded with magnetic field modulation, the signal intensity for a given microwavepower is given by

SFMR ∝ App (∆Hpp)2 (5.5)

[RRL+04, Poo86]. App and ∆Hpp are the corresponding peak-to-peak amplitudeand line width, respectively.

Analogously, the EDFMR signal intensity is given by:

SEDFMR ∝ App (∆Hpp)2 , (5.6)

93

(a) (b)

Figure 5.6: (a): FMR spectra of CrO2 at 300 K with applied field in the film plane formicrowave powers (PMW) ranging from 5 mW to 200 mW. To allow for comparison, thespectra have been divided by the square root of the microwave power. Then they all havethe same amplitude, as expected from Fig. 5.7. The line shape is independent of PMW,while the line position at higher PMW shifts to higher fields (closer to the paramagneticlimit) due to microwave heating. (b): The FMR signal magnitude intensity SFMR (blue,left scale) is proportional to the square root of PMW. The dashed lines is a guide to theeye. The line width ∆Hpp (red) remains constant within experimental error for all appliedPMW. In contrast, the line position Hres (green) clearly increases above 100 mW, while itconverges to a constant at low powers.

where App and ∆Hpp are the corresponding peak-to-peak amplitude and line width, ofthe EDFMR spectra.

We first investigate the influence of the applied microwave power on the line width∆Hpp, the line position Hres and the line shape. Figure 5.6(a) shows FMR spectraof CrO2 at 300 K with applied field in the film plane for microwave power (PMW)ranging from 5 mW to 200 mW, divided by the square root of the microwave power.The line shape is conserved, over the whole range of applied powers. In contrast, theline position at higher PMW deviates from the constant position at low powers dueto a change in the magnetic anisotropy with temperature. In Fig. 5.6(b) the FMRsignal magnitude SFMR is displayed on the left (blue) scale. Clear proportionality tothe square root of PMW is observed. The line width ∆Hpp (green) and the resonancefield Hres (red) are displayed on the right scale. While ∆Hpp remains constant withinexperimental error for all applied PMW, the increase in Hres is obvious in this picture.For all microwave power levels 10 µW ≤ PMW ≤ 200 mW accessible in our setup,SFMR is directly proportional to App, since ∆Hpp remains constant within experimentalerror. This is also true for SEDFMR. In Fig. 5.7, the FMR and EDFMR intensities ofthe strong resonance at µ0H = 163 mT (see Fig. 5.4(a)) are shown in comparison. We

94

10-1

100

101

102

103

10-1

100

101

102

103

FMR

EDFMR

signal a

mplit

ude (arb

. units

)

PMW

(mW)

Figure 5.7: The EDFMR signal amplitude (full circles) scales linearly with the appliedmicrowave power PMW (full line), as expected for a bolometric effect. In contrast, theFMR signal amplitude (open squares), which is proportional to the signal intensity obeysP

1/2MW (dashed line) characteristic of conventional magnetic resonance.

again find SFMR ∝ √PMW, as expected for conventional FMR below saturation. In

contrast, SEDFMR ∝ PMW is in good agreement with Eq. (5.3). Thus, the microwavepower dependence shown in Fig. 5.7 corroborates the notion that the EDFMR signalis bolometric.

5.4.1 Time constants

To further test the bolometric nature of EDFMR, time constants of the thermal re-laxation of the CrO2 sample has been investigated. Two approaches, via the resonantand via the non-resonant process have been chosen. First we address the non-resonantheating process. To this end, the applied microwave power was switched between twolevels, 200 mW and 20 mW, and the time evolution of the resistivity was recorded. The

95

Figure 5.8: Resistivity of the CrO2 Hall Bar sample at TCAV ≈ 300 K. H is appliedperpendicular to the film plane, with µ0|H| = 0.55 T far from resonance. When themicrowave power is switched from 20 mW to 200 mW, the sample temperature TS raisesdue to non-resonant microwave absorption, and the resistance increases. Time constantsin the order of tens of seconds are observed. Equivalent characteristics are observed forthe cooling caused by a reduction in applied power.

resistivity of the CrO2 Hall Bar sample at TCAV ≈ 300 K with H applied perpendicularto the film plane and µ0|H| = 0.55 T far from resonance, is shown in Fig. 5.8. Whenthe microwave power is switched from 20 mW to 200 mW, the sample is heated up dueto non-resonant microwave absorption, and the resistance increases. Time constantsin the order of tens of seconds are observed. Equivalent characteristics are observedfor the cooling caused by a reduction in applied power. The appearance of such longtime constants suggests, that all microscopic contributions to the heat capacity (elec-tronic, lattice, magnetic..) are in equilibrium on the time scale of our measurement.Accordingly, the heat capacity and the time constant τ in Eq. (5.3) are those of thewhole sample, not those of the electrons alone in contrast to the conjecture by Gui etal.[GHMH05].

Analogously to the non-resonant heating, the resonant microwave-induced resistiv-

96

Figure 5.9: Resonance field of the main EDFMR mode (filled circles) and FMR mode (opensquares) as a function of the magnetic field sweep speed. Positive speed corresponds tosweeps from small to large H. While the FMR is independent of the sweep speed withinexperimental error, the EDFMR significantly depends on the sweep speed, indicating atime constant of ≈ 1 s. The lines are guides to the eye.

ity changes respond slowly to changes. We find that the (almost) quantitative agree-ment between FMR and EDFMR spectra shown in Fig.5.4(b) is only obtained if themagnetic field is swept at a rate of 1 mT/s or slower. In faster sweeps, the shape ofthe EDFMR signal is strongly distorted. This distortion can be best quantified by theposition Hres of the main resonance line. It is plotted as a function of the magnetic fieldsweep speed in Fig. 5.9. Positive speed corresponds to sweeps from small to large H.The FMR (open squares) is independent of the sweep speed within experimental error.This proves, that delays caused by the experimental setup, such as the integration timeof the lock-in amplifiers, are not dominating. In contrast to FMR, the EDFMR (filledcircles) depends significantly on the sweep speed. Quantitatively, this picture yields atime constant of ≈ 1 s for the resonant process. This again indicates, that the heatcapacity of the whole sample is involved.

97

5.5 EDFMR in the AHE

In addition to EDFMR in the longitudinal resistivity ρ, we also have observed aresonant change in the Hall signal of the CrO2 Hall bar sample. The externalmagnetic field was perpendicular to the CrO2 film plane in these experiments. Werecorded the transverse voltage Vtrans in four-point geometry with an ac current bias,as described above, but the microwave setup is changed. When the transverse voltageis measured in the unipolar ESR setup used until now, only data corresponding tothe H > 0 branch of the blue curve in Fig. 5.10 can be measured. At zero field, alarge offset is found, which we attribute to crosstalk of the longitudinal resistancedue to misalignment of the transverse contacts. Without anti-symmetrizing (seeEq. (4.34)), the crosstalk cannot be clearly separated from the AHE, and the featureat the resonance field µ0H = 1 T cannot be unambiguously identified. However, toanti-symmetrize it is necessary to acquire data with H > 0 and H < 0.

To overcome this issue, the Hall experiments were performed in another setup. Here,the conventional magnet can access a field range |H| < 1.4 T. The microwave frequencyis 9.8 GHz. The polarity of the magnet can be reversed by manually exchanging thepower connectors. The larger field range also makes it possible to measure the FMRwith H applied exactly perpendicular to the film plane.

Furthermore the heating due to non-resonant microwave absorption was muchstronger in this other setup. The intensity of the resonant effect (EDFMR) scales with|HMW|2, with the amplitude HMW of the microwave magnetic field at the sample posi-tion. The non-resonant heating stems from Eddy-currents and hence is proportional to|EMW|2, with the amplitude EMW of the microwave electric field at the sample position.We estimate, that the ratio |HMW|2 / |EMW|2 is a constant in both resonators, and isone order of magnitude smaller in the resonator used in the second setup.For the Hall-EDFMR, data were taken for alternating field polarity, to acquire datasets for the anti-symmetrizing procedure, with the least possible drift in between them.

Figure 5.10 shows Vtrans as a function of applied field with and without microwavepower applied. Each spectrum is acquired in two sequent sweeps, one with H > 0 andone with H < 0. In contrast to EDFMR in the longitudinal resistivity, we used nocompensation circuit for the measurement of Vtrans. Vtrans depends strongly on H andhence it is not possible to measure with smaller sensitivity limits on the lock-in, whensubtracting a constant from Vtrans.

It becomes obvious from Fig. 5.10, that the use of both magnetic field polaritiesis crucial. Although Hall-like (odd) symmetry with respect to H is clearly evident, alarge offset is observed. Moreover, the resonant feature at µ0H = ±1 T seems to havethe same sign for H > 0 and H < 0. However, we will see shortly that this is not quitetrue.

To eliminate spurious resistivity contributions in the Hall signal, Vtrans measured

98

Figure 5.10: Transverse voltage of the CrO2 Hall-bar at TCAV ≈ 300 K, with (blue) andwithout (black) microwave irradiation. The measurements are taken in the bipolar setupat 9.8 GHz. Under microwave irradiation, TS increases to (325± 15) K. Both curves havea large offset to the origin, in the order of magnitude of the AHE. The resonant featureat µ0H = ±1 T has the same sign for H > 0 and H < 0, but is of different size.

for both magnetic field polarities, is anti-symmetrized to obtain

V Hall(H) =1

2Vtrans(H)− Vtrans(−H) . (5.7)

Turning the argument around, we can extract the spurious contributions, whichshould be even in H like ρ, by a symmetrizing procedure:

V symtrans(H) =

1

2Vtrans(H) + Vtrans(−H) . (5.8)

As a consistency check, we furthermore will use the difference spectrum of twosweeps with the same polarity (e.g. H > 0)

V ++trans(H) =

1

2Vtrans1(H)− Vtrans2(H) , (5.9)

99

⊥

µ

µ

µ

µ

Figure 5.11: The Hall voltage V Hall(H) of the CrO2 Hall bar without (full black line) andwith microwave irradiation (open blue circles) at TCAV ≈ 300 K. Under microwave irradi-ation, TS increases to (325±15) K. Analogous to the longitudinal resistance (cf. Fig. 5.4),V Hall(H) increases under microwaves and shows an additional resonant increase aroundthe ferromagnetic resonance at µ0Hres = 998 mT. The inset shows the resonant changesin V Hall(H) in more detail.

where Vtrans1(H) and Vtrans2(H) are two subsequent sweeps, both with H > 0.

Upon antisymmetrization, only the Hall-signal remains (Eq. (5.7)). As for the lon-gitudinal resistance in CrO2, we observe a clear increase in V Hall upon microwave irra-diation (Fig. 5.11). Additionally, a peaked feature appears around µ0Hres = 998 mT,i.e. at the field at which conventional FMR occurs for this magnetic field orientation.To ascertain that this feature is not an artefact, careful checks are mandatory. Inparticular, one must address the question whether the resonant feature is due to e.g.a temperature drift between two subsequent measurements. Then, some kind of fea-

100

ture will occur upon antisymmetrization, even if there was no resonant effect in theAHE. To critically test this issue, a series of measurements has been performed withalternating H polarity, and spectra with different as well as equal polarity have beenantisymmetrized, according to Eq. (5.7) and Eq. (5.9), respectively. A comparison ofthe difference spectra is shown in Fig. 5.12. In the top panel, the Hall voltage V Hall(H)(see Eq. (5.7)) in the CrO2 sample at TS = 325 K is shown for typical measurements.The average (orange), is calculated from 6 anti-symmetrized pairs of measurements(12 single measurements) and is shifted up by 1 µV for clarity. A double peak atµ0Hres = 1 T is well resolved. The line shape, however is different from that observedin Fig. 5.11. In the experiments of Fig. 5.11, the H sweep rate was 0.45 mT/s, whileonly 0.1 mT/s in Fig. 5.12. Apparently, the time constants involved are long enough,that this difference in sweep speed yields such a great difference in the line shape. Inthe lower part of Fig. 5.12, V ++

trans(H) (the antisymmetrized of two spectra with thesame polarity) is shown. It does not exhibit a peak or dip around µ0Hres = 1 T. Thisproves that the resonant feature (top) indeed stems from AHE, and is not an artefactdue to temperature or other drifts between subsequent measurements.

We thus observe an EDFMR signal in the Hall voltage, which has not been reportedbefore to the best of our knowledge.To address the mechanism leading to the Hall-EDFMR signal, recall that the Hallvoltage V Hall = (R0µ0H + RAMz)I/d in a ferromagnetic film of thickness d comprisesboth the ordinary and the anomalous Hall effect.[O’H00] The ordinary Hall coefficientR0 is inversely proportional to the carrier density. The anomalous Hall voltage V AHE =RAMzI/d depends on the magnetization component Mz perpendicular to the sample,and the anomalous Hall coefficient RA = cαρα scales with the resistivity in manycases.[O’H00] Conventional magneto-transport experiments (see Section 4.2.3) in ourHall bar samples corroborate this phenomenological notation, with RAMz À R0µ0Hand 1.4 < α < 1.6 for temperatures T > 100 K, and exclude the α ≈ 2 reported byothers. [YS02, WWM+00]

The Hall voltage without microwave irradiation (cf. the full black line in Fig. 5.11)thus is given by V Hall ≈ cαραMzI/d in good approximation. Using Eq. (5.3), therelative increase in the Hall signal upon a change ∆T in temperature due to microwaveirradiation then reads

∆V Hall

V Hall

=(∂V Hall/∂T ) ∆T

V Hall

= α∆R

R+

∆Mz

Mz

, (5.10)

with ∆Mz = (∂Mz/∂T ) ∆T . If the relative resonant change in magnetization ∆Mz/Mz

is not too large, one thus expects ∆V Hall/V Hall ≈ α∆R/R according to Eq. (5.10).However, in CrO2 at room temperature, ∆Mz/Mz cannot be neglected. SQUID

magnetometry experiments on a larger piece of the same CrO2 sample give(∂Mz/∂T ) /Mz as shown in Fig. 5.13 for µ0H = 1 T. Apparently the sample tem-perature is crucial, as (∂Mz/∂T ) /Mz is very large near TC . Figure 5.13 then yields

− 6× 10−3 K−1 <∂Mz

∂T

1

Mz

< −2× 10−3 K−1 (5.11)

101

Figure 5.12: (Top): The Hall voltage V Hall(H) = Vtrans(H)− Vtrans(−H) /2 in theCrO2 sample at TCAV ≈ 300 K with H ⊥ film plane, for several typical pairs of measure-ments. The average (orange) is shifted up by 1 µV for clarity. A peak at µ0Hres = 1 T iswell resolved. The effective sample temperature due to non-resonant heating, estimatedfrom the resistance is TS ≈ 325 K. (Bottom): V ++

trans(H), the difference of two spectrataken at the same polarity does not exhibit a peak or dip around µ0Hres = 1 T. Thisproves that the resonant feature stems from AHE.

in the relevant temperature and magnetic field range, which is comparable in magnitudeto (∂T ρ) /ρ = (8± 1)× 10−3 K−1 determined from the transport measurements.Using these values and α = 1.5 in Eq. (5.10), one obtains a Hall-EDFMR signal ∆V Hall

V Hall=

(8± 4)× 10−3 ∆TK

, which of the same order of magnitude as ∆ρ/ρ = (8± 1)× 10−3 ∆TK

.These values agree well with experiment.

The Hall-EDFMR traces shown in Figs. 5.11 and 5.12 yield 1% ≤ ∆V Hall

V Hall≤ 2%,

while 2% ≤ ∆ρρ≤ 4% for the magnetic field both perpendicular to the film plane and

parallel to it (Fig. 5.4). The spread in the values reflects the uncertainties in the signal

102

Figure 5.13: The normalized temperature derivative of the perpendicular magnetizationcomponent at µ0H = 1 T, (∂Mz/∂T ) /Mz as a function of temperature, measured in ahomogenous 2 mm2 × 100 nm CrO2 sample.

amplitude and line shape.

Furthermore we investigated the symmetric contribution in the transverse voltage

V symtrans(H) =

1

2Vtrans(H) + Vtrans(−H) , (5.12)

which can be considered as a direct measure of the resistivity. The comparison be-tween V sym

trans and V Hall thus allows a direct comparison of ∆ρρ

and ∆V Hall

V Hallacquired

simultaneously. Figure 5.14 shows the relative deviations of the symmetric and theantisymmetric components of the transverse voltage, with respect to their values atµ0H = 0.9 T, ∆V sym

trans/Vsymtrans and ∆V Hall/V Hall. Surprisingly, the line shape is differ-

ent. In the antisymmetric contribution two separated peaks are found, while in thesymmetric contribution only a shoulder and a clear peak can be discerned. We in-terpret this as evidence for different time constants involved in the resistance and themagnetization related processes in resonance. The relative magnitude of the resonantsignal is 1% for V Hall, and 4% for V sym

trans. However, the total line width is equal for the

103

Figure 5.14: The relative deviations of the symmetric (V symtrans, open circles, left scale) and

the antisymmetric (V Hall, filled squares, right scale) components of the transverse voltage,with respect to their values at µ0H = 0.9 T. The line shape is drastically different, andthe relative magnitude of the resonant peak is 1% for V Hall, and 4% for V sym

trans.

symmetric and anti-symmetric components. We thus find

1

4

∆ρ

ρ≤ ∆V Hall

V Hall

≤ 1

2

∆ρ

ρ(5.13)

in experiment. Finally, we would like to note that |(∂Mz/∂T ) /Mz| increases signif-icantly if the sample temperature approaches the Curie temperature TC ≈ 390 K ofCrO2, while (∂ρ/∂T ) /ρ does not change much. According to Eq. (5.10), ∆V Hall

V Hallshould

thus decrease, and eventually change sign, if the sample temperature is increased. Forthe highest temperature TS = (370 ± 20) K accessible in our FMR setup, we indeedobserve ∆V Hall

V Hall≈ 0, while the anomalous Hall effect is even more pronounced at this

temperature. We thus seem to be close to the temperature at which the sign of ∆V Hall

V Hall

changes, but unfortunately could not go further.

104

5.6 Summary

In conclusion, we have investigated the magneto-resistance properties of thin ferromag-netic CrO2 and Fe3O4 films under microwave irradiation. Both the resistivity ρ andthe transverse voltage Vtrans characteristically change when a ferromagnetic resonance(FMR) occurs in the film. The electrically-detected ferromagnetic resonance (EDFMR)signals closely match the conventional FMR, measured simultaneously, in both reso-nance fields and line shapes. EDFMR thus is spectroscopically equivalent to FMR. Weshow that the sign and the magnitude of the resonant changes ∆R/R and ∆Vtrans/Vtrans

can be consistently described in terms of a Joule heating effect. This demonstratesthat the quantitative measurement of EDFMR in the longitudinal resistivity and inthe Hall signals allows to separate the resistivity- and magnetization-dependent termsin the anomalous Hall voltage V AHE = cραMzI/d. Bolometric EDFMR thus can givevaluable insights into the microscopic origin of magneto-resistive phenomena. Fur-thermore, EDFMR should also allow to investigate the magnetic anisotropy in fer-romagnetic nanostructures, which cannot be studied with conventional FMR due tosensitivity limitations.

105

106

Chapter 6

Conclusions and Outlook

The magnetic anisotropy (MA) is a fundamental material property of ferromagnets,as it determines the possible orientations of the magnetization (M) at small appliedfields. Before a ferromagnet can be used in commercial devices or for fundamentalresearch, the MA first has to be determined. We have investigated the MA in singlecrystalline (100) oriented CrO2 and (001) oriented Ga0.96Mn0.04As thin films. Bothmaterials are ferromagnets with unique properties. CrO2 is a half-metal, with spinpolarizations of up to 98% [LAS97] observed experimentally. (Ga, Mn) As is one of thebest understood ferromagnetic semiconductors [SJC04]. Spin polarizations as high as80%, for holes injected from (Ga,Mn)As, has been observed [vDLvR+04].

Mesoscopic thin film samples of both materials form single domains[GRM+05, SSG+00].One one hand, many well established methods to determine the magnetic anisotropy,such as SQUID magnetometry or FMR are not scalable, and can thus not be appliedto nanostructures. On the other hand, nano structures are very interesting for funda-mental research, since one expects new quantum effects, e.g. triplet superconductivity[KGK+06], as well as for commercial applications in order to achieve higher integrationdensities. Hence, scalable methods to determine the MA open a promising area of newphysics.

In this thesis three magneto-transport based methods to determine the magneticanisotropy have been discussed:

1. At low applied fields, the magnetization switches abruptly between orientationsthat are determined by the MA, the easy axes (EA). We detect these switchesvia abrupt changes in the anisotropic magneto-resistance. (Sections 3.1 and 4.1,and [GKS+06]) The magnetic fields at which these switches occur, the switchingfields, allow to determine the orientation of the EA, and the MA.

2. A quantitative analysis of the magneto-resistance (Section 2.5, [LGD+06,CGF+95]), taking into account the crystalline symmetry, allows to unambigu-ously relate specific values of the longitudinal and transverse resistance to a spe-

107

cific magnetization orientation. This yields quantitative knowledge of the MA.

3. Resistance measurements performed simultaneously to ferromagnetic resonance(FMR), allow to detect the FMR electrically (EDFMR). The mechanism can befully understood in terms of Joule heating (Chapter 5, [NKK79, SS66, GHMH05]).As EDFMR requires only resistivity measurements, EDFMR should, in contrastto conventional FMR, be applicable to any nanoscale sample with electric con-tacts.

As all three methods are based on magneto-resistance measurements, they are suit-able for the investigation of nano structures, provided they can be contacted electrically.

6.1 Switching fields

At low applied fields, the magnetization of a single domain ferromagnet switchesabruptly between orientations determined by the MA, the easy axes (EA). The ap-plied field at which those switches occur, yield a pattern, characteristic for the MA ifthey are plotted as a function of the orientation of the applied field . This patternis, as is the MA, three dimensional. However, in the thin (100) CrO2 films on TiO2

and thin (001) (Ga, Mn) As films on GaAs, investigated in this thesis, the out-of-planeuniaxial anisotropy is particularly pronounced due to shape anisotropy [LGX99] andmagneto-crystalline anisotropy [LLD+06], respectively and the film plane is an easyplane. Since we apply H in the film plane, M is strictly in the film plane. Hence onlythe in-plane MA is investigated in our experiments.We model the in-plane switching field pattern, taking into account first order in-planeuniaxial and biaxial magnetic anisotropy, the Zeeman energy, and an energy barrierfor domain wall propagation. We find that the superposition of first order in-planeuniaxial and biaxial MA is exhaustively described by 3 different situations:

(i) biaxial magnetic anisotropy, with orthogonal easy axes

(ii) biaxial magnetic anisotropy, with non-orthogonal easy axes

(iii) uniaxial magnetic anisotropy

The calculations and the simulation of the switching field pattern for the 3 situationsis discussed in Section 2.5. Each situation can be unambiguously related to one patternof switching fields. From such an experimental pattern, the ratio of the anisotropyconstants, and the geometric arrangement of the uniaxial and biaxial contributionscan be determined. However, in contrast to FMR or SQUID, the anisotropy constantscannot be determined separately from such magneto-resistance measurements.We find a realization of all 3 situations in our magneto-resistance experiments:

(i) Ga0.96Mn0.04As at T = 5 K (Section 3.1)

108

(ii) CrO2 at T ≤ 25 K (Section 4.1)

(iii) Ga0.96Mn0.04As at T = 25 K (Section 3.1)

In Ga0.96Mn0.04As we observe the transition from biaxial MA at T = 5 K to uniaxialMA at T = 25 K in . This agrees with earlier findings [WVVL+03, HTK+03, LGD+06,LLD+06].In CrO2, we observe biaxial magnetic anisotropy at low temperatures, with the easyaxes oriented at ±34 with respect to the crystalline (010) axis. This agrees with recentfindings by Keizer et al. [KGK+06], to within a rotation by 90. However, other groupsobserved uniaxial magnetic anisotropy at T = 300 K, [RGM+04, RYK+06, FSS+06,MXG05, Rod66]. We will come back to this contradiction in the outlook.Taken together, the switching fields are a powerful tool to determine the MA in singledomain ferromagnets. This method may be used for micro- and nano- structures,provided they can be contacted electrically. Even though it is not possible to measurethe anisotropy constants separately, their ratio and the orientation of the easy andhard axes can be determined. We also would like to mention that the switching fieldscan be inferred from MOKE or other magnetization measurements.

6.2 Fully quantitative description of the magneto-

resistance

The switching fields are only a specific feature of the magneto-resistance (MR). Clearly,a complete quantitative description of the MR requires (and thus yields) the MA. Theclassical description of the AMR ([Jan57], Eqs. (2.28) and (2.29)), neglecting the crys-talline symmetry is insufficient to describe the complex resistance behavior we observein CrO2. Starting from symmetry considerations proposed long ago [Bir66, CGF+95]Limmer et al.[LGD+06], developed a description for magneto-resistance in single crys-tals, taking into account not only the relative orientation of the magnetization and thecurrent, but also their orientation with respect to the crystal. We apply this modelto CrO2 (Eqs. (2.31) and (4.25)), an find an excellent description of our data. In par-ticular, for current along (010) and (001), the longitudinal resistance, should be thesame according to the classical description (Eq. (2.28)) but should differ according toEqs. (4.28) and (4.26). In experiment we find a deviation by a factor of 2.5. Similarlythe magnitude of (ρ‖ − ρ⊥) in Eq. (2.28) should be equal for both current orientationsaccording to the classical description, but differs, as corroborated experimentally, ac-cording to Eqs. (4.28) and (4.26). Finally the variation of the transverse resistivity withM orientation in predicted to be equal for current along (010) and (001), according toboth models. This is corroborated in experiments Fig. 4.7.Hence our experiments strongly indicate that Limmer’s formulation of the magneto-resistance in single crystals can be extended to CrO2. For future discussions of themagneto-resistance in single crystalline ferromagnets, this model is indispensable.

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6.3 Electrically detected ferromagnetic resonance

We have investigated the magneto-resistance properties of thin ferromagneticCrO2 films under microwave irradiation (Chapter 5). Both the longitudinal resistanceR and the Hall voltage VHall characteristically change when a ferromagnetic reso-nance (FMR) occurs in the film. The electrically-detected ferromagnetic resonance(EDFMR) signals closely match the conventional FMR, measured simultaneously, inboth resonance fields and line shapes. EDFMR thus is spectroscopically equivalent toFMR. The electrical detection of FMR has been demonstrated early [SS66, NKK79]and investigated again recently [GHMH05].

We show that the sign and the magnitude of the resonant changes ∆R/R and∆VHall/VHall can be consistently described in terms of a Joule heating effect. Thisdemonstrates that the quantitative measurement of EDFMR in the longitudinal andtransverse signals allows to separate the resistivity- and magnetization-dependentterms in the anomalous Hall voltage V AHE = cραMzI/d. Bolometric EDFMR thuscan give valuable insights into the microscopic origin of magneto-resistive phenomena.

The contribution of this thesis is the proof of full spectroscopic equivalency ofEDFMR and FMR by simultaneous measurement (Fig. 5.4), and the detection of thesignature of FMR in the Hall voltage (Fig. 5.11), which has not been reported beforeto the best of our knowledge.

6.4 Outlook

In (Ga,Mn)As, the magnetic anisotropy is well understood [WVVL+03, HTK+03,LGD+06, LLD+06], and our findings agree with earlier reports. In (100) CrO2 thinfilms epitaxially grown on TiO2 we observe biaxial MA in magneto-transport ex-periments at low temperatures. This agrees with other MR experiments in similarfilms [KGK+06]. However, other groups report uniaxial MA at room temperature[RGM+04, RYK+06, FSS+06, MXG05], and at low temperature [SSG+00].

It is therefore crucial, to check whether a transition from biaxial to uniaxialanisotropy occurs with temperature in our CrO2 samples. The best method wouldbe investigate the sample, on which the magneto-transport measurements have beenperformed, by conventional FMR at T ≤ 25 K and at T = 300 K. At T ≤ 25 K weexpect to find biaxial MA and uniaxial MA at T = 300 K.An extension of these experiments would be to investigate the effect of CrO2 film thick-ness as well. Rameev et al. [RGM+04, RYK+06] reported on a change of the uniaxialeasy axis from (001) for thicker samples to (010) below 30 nm film thickness. Hence itwould be most promising, to investigate FMR for various temperatures on the availablesample of 100 nm thickness an on a sample thinner than about 30 nm.The investigation of magnetic switching via the longitudinal or transverse resistivity

110

requires that the changes in AMR with magnetization orientation are large comparedto spurious variations of the resistance, e.g. by temperature fluctuations. The tem-perature derivative of the resistivity of CrO2 is several orders of magnitude smaller forT ≤ 25 K than for T = 300 K. Hence, to perform AMR measurements in CrO2 atT = 300 K , is challenging with respect to the stability of the sample temperature.Conventional FMR experiments, as used by Rameev et al., are much more promising,since they work equally well for low and high temperatures.The next question on magnetic switching is how the process works microscopically,where and which domain walls nucleate and how they progress through the sample.Two methods to image the domain structure of a thin ferromagnetic film shall be pro-posed to be performed simultaneously to magneto-transport. The domain structurecan then be compared with the magneto-transport.

• Bolometric laser scanning microscopy.

• Magneto-optical Kerr microscopy (MOKE)

Bolometric laser raster microscopy is described in detail in [WEN+06]. The main ideais that the sample is scanned with focussed laser pulses and the global change in theresistance is recorded. Each pulse causes local Joule heating. The resistance change isdifferent, when domain wall or the central part of a larger domain is hit. By scanningthe sample, an image of the domains can be acquired.MOKE allows to spatially image the magnetization [HAR90], and thus to monitor thenucleation and motion of domain walls. Simultaneous magneto-transport experimentsthen allow to relate the switches in the resistance to a specific microscopic process.To this end, MOKE optics should be added to a conventional transport cryostat,either through windows or via optical fibers. Furthermore it has not been investigateduntil now, how domain formation affects the planar Hall effect (PHE) and the AMR.While our (Ga, Mn) As and CrO2 thin film samples consist of single ferromagneticdomains [GRM+05, SSG+00], for other materials like Fe3O4 [KN03], we always expectmulti-domain behavior. A setup at which the domain structure can be imaged eitherby MOKE or by bolometric laser raster microscopy, and magneto-transport can bemeasured simultaneously, would allow to quantitatively relate the domain structure tofeatures in the PHE and the AMR in those materials.Another drive for in-situ magnetization measurements with magneto-transportexperiments is the investigation of the microscopic origin of the anomalous Hall effect(AHE). In the measurements discussed in this thesis, the out-of-plane magnetizationcomponent Mz was assumed constant (Fig. 4.12) or estimated from separate SQUIDmeasurements (Chapter 5). It would be desirable to measure ρ, AHE and Mz

simultaneously. This could be performed in the same MOKE-transport-cryostatsystem suggested before.Another possibility to measure ρ, AHE and Mz simultaneously would be to performmagneto-resistance measurements in a SQUID magnetometer. To this end, wireswould have to be introduced into the sample space of the SQUID magnetometer to

111

contact the sample electrically. This is not straight forward, as those wires introducemagnetic material into the sample space. This spurious signal superimposed ontothe magnetization of the thin film investigated. Since the magnetization is larger inCrO2 than in (Ga,Mn)As, this spurious contribution is not as likely to dominate theSQUID signal when CrO2 is investigated than it is with (Ga,Mn)As. Therefore anexperiment of this type should preferable be performed with CrO2.

Limmer’s model [LGD+06] leads towards a fully quantitative description of themagneto-resistance in mono-domain single crystals. For crystals with tetragonalsymmetry, such as CrO2, 8 parameters (Eqs. (2.31),(4.25)) have to be determined inorder to describe the magneto-resistance with arbitrary orientations of current andmagnetization. To this end, experiments in several different geometries have to beperformed. The applied field has to be rotated in-plane and in-plane to out-of-plane,to determine the Hall-like parameters as well. On the available (100) CrO2 thin filmsit would be desirable to structure Hall-bars with current oriented along (011),(010)and (001). To fully determine all parameters of the magneto-resistance, (100) orientedfilms alone are not sufficient due to their high symmetry. Additional experiments on(110) films with exotic current orientations such as (111) and (112) then allow for adetermination of all remaining parameters due to low symmetry with respect to thetetragonal distortion along (001).

Once all parameters are determined from measurements at high fields (H ‖ M),the longitudinal and transverse resistance allow to unambiguously determine theorientation of M independent of H. Therewith the magnetic anisotropy can bedetermined, and changes in the magnetic anisotropy, e.g. due to temperature orinduced strain, can be detected.

When it comes to the investigation of the MA in a specific nano structure, it islikely, that the resistivity parameters differ from their bulk values, e.g. size dependentresistivity in metal wires with less than 100 nm width has been reported [SSS+05].Therefore, it is crucial to determine the resistivity parameters on every investigatedstructure individually.As discussed before, a single Hall-bar is insufficient to determine all resistivityparameters. Nevertheless, those parameters that can be determined on a specificHall-bar are sufficient to determine the MA of this specific Hall-bar.The idea is simple. In an individual Hall-bar, there is a one-to-one relation betweenlongitudinal and transverse resistivity on the one side, and magnetization on the other.At high fields M ‖ H, and thus this relation can be determined for all M orientations.It is described by the resistivity parameters. In the next step, measurements at lowapplied fields are performed, and now the M orientation is calculated back from theresistivities. Eventually this yields M(H), and this allows to calculate the MA.The findings on the magnetic anisotropy from this work, then have to be comparedin detail with angular resolved ferromagnetic resonance (FMR). The FMR still is a

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very powerful tool measure the MA, and is the reference method to be compared withevery new technique.

With the proof of spectroscopic equivalency of EDFMR and FMR, the FMRcan be detected via magneto-transport. Magneto-transport is scalable, and henceEDFMR should be, in contrast to FMR, also scalable. However, a quantitativeinvestigation of the scaling behavior of EDFMR still has to be performed. To this end,structures down to the limit of optical lithography shall be quantitatively investigatedby EDFMR and FMR.Besides the change of resistance, the occurrence of FMR also affects the magneto-impedance[BMM+00], the temperature[MLKM00] and causes a dc voltage dropwithout even without current applied [EJ63]. The latter effect also allows to detectthe FMR and is scalable. These interesting physical effects certainly deserve futureattention.

As the discussions in Section 5.5 showed, conventional FMR setups with unipolarpower supplies are not satisfying when it comes to combinations of FMR andmagneto-transport. Magnetic field and temperature control is very precise in thetransport cryostat system used in chapters 3 and 4. It would be very attractiveto introduce a state of the art EPR spectrometer into a conventional transportcryostat. Then one could measure FMR, EDFMR in the longitudinal and in the Hallresistance simultaneously, with the resolution required to make more quantitativestatements about the magnetization dynamics and the microscopic origin of theAHE. Furthermore, for investigation of the dc voltage [EJ63], it is crucial to acquiredata at various orientations of the applied field, hence it is tempting to performthese measurements in such a EPR spectrometer in a transport cryostat system as well.

Overall, magneto-transport is a very powerful tool to investigate the magneticanisotropy and the magnetization of ferromagnetic thin films. All magneto-transportbased techniques are scalable, and may thus but used to investigate the magneticanisotropy in micro and nano structures. This opens a wide field of interesting physics,for devices as well as for the fundamental understanding of magnetism.

113

114

Acknowledgements

I would like to thank everybody who contributed to this thesis, supported me during mywork and made the last year at the Walther-Meissner-Institut and the Walter SchottkyInstitut a great time. In particular, I would like to thank

• Prof. Dr. Rudolf Gross for allowing me to work at the Walther-Meissner-Institutand to visit the DPG-conference in Dresden, and for interesting discussions.

• Dr. Sebastian T. B. Goennenwein, the best boss ever, for his unlimited support,all the time and energy he spent on me in the lab and in very fruitful discussions;for always being one step ahead and for his demanding way that saved me fromcomplacency; and for not despairing of semi-Russian manuscripts.

• Priv. Doz. Dr. Martin S. Brandt for fruitful discussions, doubting my”schmutzige Taschenspielertricks”, and especially for the opportunity to workin his ESR-lab and his support, creativity and ambition there.

• Hans Hubl for his unlimited help in the ESR-lab and for showing me all the dirtytricks and secrets.

• Andreas Brandlmaier for all the joint work on EDFMR, for sharing worries andsuccess and for the chocolate keeping me awake.

• Christoph Bihler for his help in the ESR-lab, and for generously assigning timefor measurements.

• Ruurd Keizer for fruitful discussions and good cooperation with TU-Delft.

• Arunava Gupta for providing CrO2 samples.

• Robert Muller for helping me incredibly fast with smaller and larger workshopjobs.

• Matthias Opel for keeping me calm in the last days and more than carefullyreading this thesis.

• again the Schottkies, especially Martin, Hans and Christoph, for treating me likeone of them, and constructive cooperation on the EDFMR-manuscript.

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• Petra Majewski for sharing her special knowledge as ”Transportdrachen”.

• All the other Magnetiker, especially Sebastian Bauer and Wolfgang Kaiser, forthe good time.

• Thatsawan for her understanding and support.

• of course my parents for their unlimited support.

116

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