Magnetism in diluted semiconductorsMagnetism in diluted semiconductors

I- Introduction.II- Theory for disordered Heisenberg models.III- Ab initio based studies for diluted magnetic semiconductors.IV- Magnetic excitations and phase diagram of the diluted III-V

compound GaMnAs.V- Model approach: the non perturbative V-Jpd model.VI- Conclusions..

Georges BouzerarInstitut Néel -MCBT

CNRS

CollaborationCollaboration

Josef Kudrnovsky (Institut of Physics-Prague)Richard Bouzerar (University of Amiens)Olivier Cepas (CNRS Paris/ Neel Institute CNRS)Lars Bergqvist (Jullich)

Magnetic semiconductors: • Early 60’s (non dilute magnetic materials): EuO (Tc=70K), EuS (17K) and the spinels as CdCr2S4 (84 K) or HgCr2Se4 (106 K)

•70’s and 80’s: Diluted Magnetic Semiconductors II-VI materials CdTe, ZnTe and ZnSe) II Mn2+ (isoelectric)

easy to incorporate Mn direct Mn-Mn AFM exchange interaction

PM, AFM, or SG (spin glass) behaviour codoping (holes) has led to FM for TC < 3K.

•Late 80’s: III-V materials MBE (In,Mn)As films on GaAs substrates FM with TC=35 K•Late 90’s: MBE (Ga,Mn)As films on GaAs substrates FM with TC=110 K

I- Introduction

Idea: combination of semiconductor technology (charge) with magnetism (spin) should give rise to new devices:

Long spin-coherence times (~ 100 ns) have been observed in semiconductorsUse the charge to Control the magnetism (Write bits through electric current) Quantum computing??

Need for materials with relatively high Curie temperature (beyond room temperature).

DMS based materials are promising candidates for Spintronic devices.

An exemple of spintronic application: Magnetic Random Access Memory (non volatile) based on GMR (nobel prices of Fert and Gruenberg)

•Role and effects of the disorder?

•Importance of thermal fluctuations?

•Nature of the magnetic couplings?

• Influence of the band structure of the material?

• Dependence with the carrier density and magnetic impurity concentration?

•Why carrier codoped II-VI compounds have much smaller TC than III-V materials?

• Effects of intrinsic defects (compensation)?

•Disagreement between different theoretical approaches?

Understand experimentally and theoretically which physical quantities control TC

In III-V materials the substitution of Ga3+ by Mn2+ introduces simultaneously a localized spin S=5/2 and an itinerant carrier (hole) in the valence/impurity band.

The exchange between impurities is of « generalized RKKY-type » (polarization of the gas of carrier + effects of resonances). Standard RKKY couplings are inappropriate to describe ferromagnetism in these materials

1- GaMnAs is one of the most studied diluted III-V material , it exhibits relatively high TC : xMn=5% , TC=110 K and xMn=7.5% , TC=175 K.

2- GaMnN is more controversial: some pretends to observe paramagnetism and other ferromagnetism with very high Curie temperature.

1- III-V compounds.

1- First source of disorder: the random position of the Mn impurities and possibly formation of inhomogeneities.

2- Second source of disorder: defects of compensation.

These defects appear during the MBE growth

of the samples. These defects have a drastic

effects on both TRANSPORT and MAGNETISM.

There are two types of such defects: As anti-sites which a priori only affect the density of carrier. Mn(I) affect both the hole density and the density of magnetically active Mn.

2- Disorder and Compensation defects.

Trapped holes

3- Some experimental resultsGaMnAs with Highest TC =172 K

T dependence of the Magnetization and inverse susceptibility (Wang et al. 2005)Curie temperature and hole concentration

(p) as a function of the impurity concentration. Full symbol (metallic samples) (Matsukura et al. PRB 1998).

Resistivity at zero field as a function of T Curie temperature (Matsukura et al. PRB

1998). Inset effect of H.

Resistivity at zero field as a function of temperature for various annealing time (Potashnik et al. 2001).

metal isolant

Resistivity after annealing:

Hayashi et al., APL 78, 1691 (2001)

Optical conductivity as a function of frequency for various concentration.The samples with x=0.052 are as grown and annealed (Burch et al. PRL 2006).

Magnetization and charge density profile in GaMnAs before and after annealing (Kirby et al. PRB 2004)

before

after

Peak in the conductivity is inconsistent with Valence Band picture

Mathieu et al., PRB 68, 184421 (2003)

Annealing dependence of magnetization curve

magnetization curves change straight/convex (upward curvature) → concave (downward curvature)

degradation for very long annealing (precipitates?)

Potashnik et al., APL 79, 1495 (2001)

4- Predictions based on Zener mean Field theory.

2 1/3C pd xpT J

Dietl et al. Science 2000 (cited 3000 times!!)

Is the Zener mean field theory really appropriate to describe ferromagnetism in diluted magnetic semiconductors??

0 pd i ii I

H H J S s

Includes a realistic bande structure for the host material

The theory should allow the study the magnetic properties : Curie temperature, magnetic excitations spectrum, local magnetizations, ...) of a system of Nimp interacting magnetic impurities with spin S (quantum or classical) randomly distributed on a given lattice.

i jj

ijijiH p Jp S S

The approach should be able to:

Treat disorder effects beyond a simple Virtual Crystal Approximation (VCA).

Treat properly thermal and quantum fluctuations beyond Mean Field (MFT).

II- Theory for disordered Heisenberg model.

pi=1 if the site is occupied otherwise 0

MFT has several shortcomings.MFT has several shortcomings. Underestimates both the effects of thermal fluctuations and Underestimates both the effects of thermal fluctuations and

disorder. disorder. Collective excitations are not included.Collective excitations are not included. Violate both theorems of Mermin-Wagner and Goldstone (finite Violate both theorems of Mermin-Wagner and Goldstone (finite

Curie temperature for 1D et 2D)Curie temperature for 1D et 2D)

= S z

i j

e ziM

ff ffi iF

ei jH avec h JSh

S ( )z

S

effi

B

hk TSB

013 ( 1)C jB

j

S S xT Jk

1- The Mean Field-VCA theory

1

( )1

23 ( 1) (q) (0) - ( ) , RPA

B C

qNq

S S J J qxk T

, .

1.44 , 1.39 , 2MC RPA MFB C B C B C

simple cubic lattice nearest neighbor exchange J

k T J k T J k T J

Includes collective excitations. Fulfils both theorems of Mermin-Wagner and Goldstone (in the absence of anisotropy TC=0 for1D and 2D)

Very good agreement with exactexact Monte Carlo calculations in the absence of disorder (x=1).

RPA : improvement of thermal and transverse fluctuations

2- The Random Phase Approximation -VCA theory

•RPA-CPA (Coherent Potential Approximation) (G.B & P. Bruno

PRB 2002) effective medium theory :

improvement of the disorder treatment and thermal fluctuations.

Does not include short range correlations, clusters,….

•Monte Carlo calculations: exact but too heavy et very costly in CPU, especially when the exchange couplings are long-ranged.

Generalization of RPA (accurate for clean systems): exact treatment of the disorder effects: the Self-Consistent Local RPA

3- Proper treatment of disorder and thermal fluctuations

Thermodynamic and dynamical properties within the real space SC-LRPA.

Choose a disorder configuration: positions of impurities.

For a given temperature T, after convergence one can calculate:

•The distribution of the local magnetizations mi (T) (Reflectrometry measurements).

•The spectral function A(q,w,T) and magnetic excitations (Inelastic Neutron Scattering exp..

•The spin-spin correlation functions < Si .Sj > and magnetic susceptibility.

•Etc. ………

convergence

( , )ijG T

( )i T( )i

TmDetermination of the thermodynamic and dynamical properties.

0

( ) ( ); (0)i j

i tijG S t Si e dt

2 1( 1)

3

B

imp

Ci i

k T S SN F

1

1

1

2,

m ( ( ))

( ) limN

imp

imp

iC

i

i i

z

i z

iNT T

i

i m S

EE

E

S

m

F dG

The Curie Temperature.Within SC-LRPA one can derive a semi-analytical expression for the Curie temperature in disordered ferromagnetic systems.

This expression is the generalisation of RPA to disordered ferromagnetic systems!

Fi depends on both the eigenvalues and eigenstates of the non symmetric effective Hamiltonian Heff

- eff z zij i ij ij l lj

l

H S J S J

Allow the determination of the possible good candidates (not enough to find the materials with the highest TC

Allow a direct quantitative comparison, we choose an approach with no adjustable parameters.

Includes in the most realisitic way the band structure and non perturbatively the effects of the substitution of an impurity (d orbitals and hybridization exactly treated).

Material specific: difficult to draw general conclusion.

Only valid for the Ground state properties (T= 0 K)

III- Ab initio based studies for diluted magnetic semiconductors

ZnO ZnS

ZnSe ZnTe

Which possible diluted ferromagnetic semiconductors ?

1- Some ab-initio results1- Some ab-initio results

II-VI materialsII-VI materials

GaN

GaP

GaAs GaSb

III-V materialsIII-V materials

Density of states in (Ga,Mn)As with 3.125% Mn

Wierzbowska et al., PRB 70, 235209 (2004)

LSDA: d-orbital weight at EF, VB top and CB bottom: absent in photoemission experiment and SIC calculations.

LSDA+U : phenomenological procedure to improve the tretament of elevtron-electron correlations

d orbitals

Mn d-orbital weight shifted away from EF, better agreement (especially for the feature at -4 eV)

similar results from other methods going beyond LSDA: GGA, SIC-LDA

LSDA

LSDA +U

Not RKKY couplings!

Zn1-xCrxTe

Ga0.95Mn0.05As

Couplings in diluted magnetic semiconductors (data from J.Kudrnovsky)

RKKY couplings

1- Determination of the Ground state properties and exchange integrals Jij from first principle methods (here LDA Tight Binding -Linear Muffin Tin Orbitals)

2- Proper treatment (Monte Carlo or SC-LRPA) of (i) thermal fluctuations and (ii) disorder in the treatment of the disordered effective Heisenberg Hamiltonian to calculate the magnetic properties.

The two step approach calculations (TSA):

How to proceed?

The Mean field VCA underestimate the fluctuations, and overestimate strongly the Curie temperature (for 5% Mn TC=300 K).

The «Ising» calculations: Effects of disorder included but not the transverse fluctuations lead to non realistic TC.

The SC-LRPA lead to much smaller values for the Curie temperature.

Good agreement with Monte Carlo: for 5% Mn and the same couplings (see next slide): 130 K [1] et 103 K [2].

[1] L. Bergqvist et al. Phys. Rev. Lett., 93 137202 (2004)[2] K. Sato et al. Phys. Rev. B, 70 201202 (2004)

2- Role of the disorder and thermal fluctuations.2- Role of the disorder and thermal fluctuations.

Disorder but no transverse fluctuations

Mean field + VCA

SC-LRPA

LSDA LSDA+U Courtesy to L. Bergqvist

3- Monte Carlo vs SC-LRPA.3- Monte Carlo vs SC-LRPA.

GaMnAs

We observe a very good agreement between experiment and theory for optimally annealed samples (calculations are done for non compensated systems).

« As-grown » samples exhibit much smaller Curie temperature: strong reduction due to the presence of compensating defects.

The calculations provide the maximum value of the Curie temperature for each concentration of Mn.

Below 1% no ferromagnetism (percolation threshold ).

Remark: (1) The TB-LMTO calculations to estimate the couplings and (2) the treatment of disorder and fluctuations within SC-LRPA are reliable.

4- Comparison between experiment and theory for GaMnAs.4- Comparison between experiment and theory for GaMnAs.

As grown samples

(Data from Edmonds et al. Nottingham).

Variation of TC for a fixed total density of Mn2+after different annealing treatment for Ga1-xMnxAs. The total Mn concentration is:

20 30.067 0.003 (15 1)10

Mncmx

nnhh (10 (1020 20 cm cm -3-3)) TTCC (K) (K) détailsdétails

4.1 +/- 0.504.1 +/- 0.50 6767 As grownAs grown

9.0 +/- 0.909.0 +/- 0.90 113113 6hrs@180 6hrs@180 ooCC

10.0 +/- 1.010.0 +/- 1.0 126126 15hrs@180 15hrs@180 ooCC

13.5 +/- 2.013.5 +/- 2.0 143143 140hrs@180 140hrs@180 ooCC

4.6 +/- 0.504.6 +/- 0.50 7676 As grownAs grown

Conclusion : TC varies strongly from 67 K to 143 K depending on the annealing conditions.

a- Experimental results.

5- Effects of compensating defects.5- Effects of compensating defects.

We assume that Mn(Ga) density is fixed x and we vary the density y of As anti-sites (Ga1-x-yMnx Asy )As . As on Ga sublattice is double donor, the density of holes is,

2h

n x y

Weak variation of TC with for

> 0.60

For < 0.55 the ferromagnetism becomes unstable, the super-exchange dominate in the nearest neighbour exchange (frustration??? See following).

Simple theory RKKY MF-VCA predicts that TC = x 4/3 1/3 inconsistent with our results and experiments

We can not explain the experimental results by assuming that As anti-sites is the dominating mechanism of compensation.

hn

x

b- Effects of As anti-sites.

We introduce the reduced variable

Instability of Ferro GS

??

It is energetically more favourable for Mn(I) to be located near Mn(Ga). The coupling is strongly antiferromagnetic: reduction of Mn(Ga) magnetically active.

Mn(I) is a double donor: reduction of the carrier density.

After annealing Mn(Ga)-Mn(I) pairs break and release carriers and increase active Mn(Ga).

The problem reduces to an effective model of interacting active Mnwith a hole density

2 ( )eff Mn Mnx x x I 3 ( )h Mn Mnn x x I

c- Effects of Mn interstitials (Mn(I)).

d- Comparison between experiment and theory.

For each sample to calculate TC we use the following expressions:

2 3

3 2Mn h h

eff effMn h

x n nx and

x n

Samples with highest TC are in very

good agreement with the calculations done for = 1.

For « as-grown » samples we also reproduce Tc by taking into account that < 1.

This study confirms that the dominating mechanism of compensation is due to Mn interstitials.

The theoretical curve ( = 1) allows to give a good estimate value for TC for each sample :

TC=649 ( xeff - 0.0088)1/2

effx

Experiment Theory

GaMnAs is the III-V material with the highest Curie temperature.

Instead of the theoretical predictions based on MF-VCA calculations (Dietl et al.) that TC = 600 K for xMn= 6% .GaMnN has a very small TC. In spite of a very strong nearest neighbour coupling (10 times that of GaMnAs)

InMnAs exhibits intermediate Curie temperatures

6- Comparison between different III-V materials.6- Comparison between different III-V materials.

For more details see G.B. ,EPL (2007).

c

, c cA q A q

i jiq(r -r ), ,j

ijimp

1q = e

Nc R c L cA i j

�������������������������� ��

IV- Magnetic excitations spectrum and phase diagram of GaMnAsThe magnetic spectral function is directly accessible via Inelastic Neutron Scattering experiment , it reads,

The index (c) denotes the configuration average!

- eff z zij i ij ij l lj

l

H S J S J

The eigenvalues (magnon mode) and respectively left and right eigenvectors are: ωα , |Ψ

Lα ˃ , |Ψ

Rα ˃ .

Excitations are well defined in a very small region around the Γ point

ω(q) = D q 2 in the limit of long wave length (Goldstone mode)

Second moment γ(q) = Cq : it does not correspond to the real width of the excitation.

The Mn density is

x=0.03

1, ( ) ( )A q m q q

The Mn density is

x=0.03

Both TC and the stiffness vanishes below the percolation threshold.

In contrast to non disordered systems the ratio D/Tc is not constant.

Values of D are as large as those reported in metallic manganites as LaCaMnO3.

The first moment of A(q, w) does not correspond to the magnetic excitation.

The ground state gets canted above a critical value of the SE coupling!

The unpaired spins remains almost uncanted.

In the limit of very large JAF , the pairs are

disconnected x---> xeff

Competition between extended ferro couplings and short range Superexchange.

G. B. R. Bouzerar and O. Cepas, PRB (R. C.) (2007).

Distribution of canting angle of the ground state for two different concentration of holes.

Phase diagram of GaMnAs.

This phase may explain inconsistency with magnetization measurements at low hole concentration

��������������

†

,

†

,ij i

ij i

ij ii

i ii ic S c cVsH Jct

Spins operators :si : carrier Si : magnetic impuritytij =t for (i,j) nearest neighbour

V controls the impurity band position.

Crucial term : source of scattering (Coulomb potential) due to the substitution of a cation by another one with a different charge: Vi=V if i is occupied by a magnetic impurity.

V- Model study: the V-Jpd model

R. Bouzerar et al. EPL 2007

2

( ) ( ) ( )4

pdij ij ji

JJ m f G G

Choice of a configuration of disorder and set of parameters V and Jpd, x

Calculation of the Mn-Mn magnetic couplings

Diagonalization of the itinerant carrier Hamiltonian in each spin sectors

Transport propertiesMagnetic properties

Procedure for the study of transport and magnetism

A.I. Liechtenstein et al., PRB (1995)

1- The damped RKKY model (V=0).

No finite size effects (the same results for both 800 or 2000 impurities dilute on the fcc lattice).

MF-VCA : strongly overestimates the true Curie temperature

SC-LRPA : -Variation of Tc is non monotonic-Even with the relatively large damping the tability region of ferromagnetim is very narrow (inconsistent with experimental results)

damping is introduced

R. Bouzerar et al. PRB 2006

2- The effects of the cutt-off.

The standard RKKY model is inappropriate to describe ferromagnetism in dilute magnetic semiconductors as GaMnAs or GaMnN.

An average over at least 200 configurations of disorder has been performed.

Pure RKKY regime: the region of stabilityfor the carrier is reduced to 10% of the magnetic impurity.

Even for a relatively strong damping the region of stability remains narrow..

3- The non perturbative treatment of the V-Jpd model

DOS as a function of V at fixed JpdS=5t

Magnetic couplings (similar to ab initio): (a) as a function of hole concentration (b) as a function of V at fixed JpdS=5t and fixed hole density

An average over at least 200 configurations of disorder has been performed.

The Jpd model captures the essential physics of the diluted magnetic semiconductors. Note the crucial role of the onsite potential!

Even at low carrier density we do not have:

We find the opposite TC reduces with incerasing Jpd!.

2 1/ 3C pdT J p

Significant increase of the stability region of the ferromagnetic phase.

4- Curie temperature (in units of t) at fixed V=-2.4t.

5- Comparison between TSA and full Monte Carlo treatment of the V-Jpd model .

(a)(b)

(a) and (b) hole density per defect is respectively nh/x=0.125 and 0.25

• Tc (SC-LRPA) exhibits a maximum in both cases • At small values of within Sc-LRPA • For ,Tc vanishes.• Disagreement with the full Monte Carlo simulations • (in the inset), for the amplitudes!

2C pdT J

2pdJ S W

for details see R. Bouzerar and GB, submitted (2009) condmat0902.4722

(c) MF-VCA vs SC-LRPA

• Tc within MF-VCA is much larger and saturates for JS→∞

• We observe only for small values of Jpd an agreement between both methods.

• For larger hole density the agreement for small values disappear (RKKY oscillations kills the ferromagnetic phase!)

(c)

6-Origin of the disagreement between full Monte Carlo and the two step approach?Finite size effects on TC (within SC-LRPRA): an average over at least 200 configurations of disorder for the largest systems and few thousands for the smallest systems is done.

See R. Bouzerar and GB, submitted (2009) condmat0902.4722

Magnetic couplings as a function of the distance r for different system sizes (huge effects near the typical distance between impurities)

Calculated distributions or the Curie temperatures (huge fluctuations for the small systems! )

The case of non dilute systems: the double exchange model in presence of on-site disorder?

see GB and O. Cepas ,PRB 2007

† †

, ,ij i i

ii j ii i

j i iiH ct J S c cVc s

��������������

1 and x J The on site potential Vi is chosen randomly in [-W/2,W/2] , W is the disorder strength

The TSA versus the full Monte Carlo calculations (a) no disorder W=0 and (b) the disordered case

In contrast to the dilute case, a very good agreement is obtained for the non dilute case even in the presence of disorder!

Thank you