Properties of Metallic Helimagnets

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Properties of Metallic HelimagnetsKwan-yuet HoInstitute for Physical Science and Technology & Department of PhysicsUniversity of Maryland

Apr 3rd, 2012

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OutlineHelimagnets (Introduction)Model and Energy ScalesPhase DiagramGoldstone ModesElectronic PropertiesColumnar Phase and SkyrmionsConclusion

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HelimagnetsHelimagnets: materials

exhibiting helimagnetism in one of the phases.

In helimagnetism, there is ferromagnetic order on each plane.

The direction of the spin rotates as one goes along the helix.

Low-temperature systemq-1 >> aExamples: MnSi, FeGe,

FexCo1-xSi

For MnSi,Lattice constant: 4.56ÅHelical wavelength: 180ÅOrdering temperature: 29.5KResistivity ~ 0.33μΩ cm (T=0)

(kFl~6000)

(Ishikawa, Tajima, Bloch, Roth 1976)(Pfleiderer, Julian, Lonzarich 2001)

(Bauer et al 2010)

q

(Uchida, Onose, Matsui, Tokura 2006)

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HelimagnetsHelimagnets

generally have richer phase diagrams than the other magnets.

Helimagnets are sensitive to a change of pressure. Helical order is destroyed at high pressures.

Universal quantum fluctuations lead to the tricritical point (TCP).

(Thessieu et al 1997)

(Pfleiderer, Julian, Lonzarich 2001)

(Kirkpatrick, Belitz, Vojta 1997)Experimental Phase Diagrams of MnSi

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HelimagnetsAn exotic columnar

phase (A phase) at T≈Tc and intermediate H.

Six-fold symmetry was found in neutron scattering.

It is a 2D hexagonal columnar lattice, confirmed by Lorentz TEM images.

Believed to be a Skyrmion lattice.

(Mühlbauer et al 2009)

MnSi

(Yu et al 2010)Fe0.5Co0.5Si(For MnSi, see Mühlbauer et al 2009; for FeGe, see Yu et al 2010; for Fe0.5Co0.5Si, see Yu et al 2010)

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HelimagnetsHelimagnets

generally have richer phase diagrams than the other magnets.

Helimagnets are sensitive to a change of pressure. Helical order is destroyed at higher pressures.

Universal quantum fluctuations lead to the tricritical point (TCP).



(Kirkpatrick, Belitz, Vojta 1997)Experimental Phase Diagrams of MnSi

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HelimagnetsThe disordered phase at p>pc has

a non-Fermi-liquid (NFL) transport properties that Δρ~T3/2.


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HelimagnetsThese properties are due to the huge

fluctuations.Like cholesteric liquid crystals, a pure

helimagnet has a Goldstone mode (called helimagnon) with “transverse” susceptibility χ┴

-1~k||2+ck┴

4.◦True helimagnetic long-range order cannot

exist in d=3.Like columnar phases in liquid crystals,

columnar phase in helimagnets have a fluctuation spectrum k┴

2+ckz4.

(Belitz, Kirkpatrick, Rosch 2006)

(Kirkpatrick & Belitz 2010)

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Model and Energy ScalesFerromagnets can be described

by the LGW functional.

€

SFM M[ ] = d3xr2M2 +

a2∇M( )2 +

u4M2( )

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥∫

(Heisenberg 1930s)

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Model and Energy ScalesTo stabilize a

helimagnet over a ferromagnet, Dzyaloshinski-Moriya (DM) interaction is needed.

Spin-orbit coupling constant: gso(dimensionless) c=akFgso

It exists in systems with no inversion symmetry.

€

S M[ ] = SFM M[ ] +c2

d3x M⋅ ∇ ×M( )∫

DM interaction(Dzyaloshinski 1958, Moriya 1960)

B20 cubic crystal, P213

gso ≈ 0.05 for MnSi

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Model and Energy ScalesLGW functional

€

SFM M[ ] = d3xr2M2 +

a2∇M( )2 +

u4M2( )

2−H⋅M

⎡ ⎣ ⎢

⎤ ⎦ ⎥∫

€

SDM M[ ] =c2

d3x M⋅ ∇ ×M( )∫

€

Scf M[ ] = d3xb2∂iM i[ ]

2+b1

2∂iM i+1[ ]

2+v4M i

4 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

i=1

3

∑∫€

S M[ ] = SFM M[ ] + SDM M[ ] + Scf M[ ]

c~gso b, b1 ~gso2 q=c/2a~gso

(Ho, Kirkpatrick, Sang, Belitz 2010)

gso <<1

Cubic anisotropyPinning

(Bak & Jenson 1980)Space group P213

(Belitz, Kirkpatrick, Rosch 2006)v ~gso

4

O(gSO0)

O(gSO2)

O(gSO4)

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Phase DiagramMean-field theoryH=0: pinned helical

phase, q in (1,1,1) for b<0, |b|~gso

2

M(x)=msp [cos(q.x)e1+sin(q.x)e2]

0<H<Hc1: q rotates from (1,1,1) to H

M acquires a homogeneous component along H

Elliptical conical phase near Hc1


(Ishikawa, Tajima, Bloch, Roth 1976)

MnSi

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Phase DiagramHc1 < H < Hc2: conical

phaseq aligns with HM(x)=msp

[cos(qz)x+sin(qz)y] + m//z

When H increases, msp decreases and m// increases.

msp vanishes at H=Hc2Columnar phase: 2D

hexagonal lattice of columns

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Phase Diagram


MnSi

(Ishimoto et al 1995)

Fe0.8Co0.2Si

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Goldstone ModesA pure helimagnet breaks the continuous

translational symmetry Goldstone mode unusual electronic properties through electron-Goldstone-mode coupling

M(x)=msp (cos[qz+ϕ(x)], sin[qz+ϕ(x)],0)Energy fluctuations ~ ∫d3x [∇ϕ(x)]2 wrong,

because free energy is independent of the direction of q. Perpendicular fluctuations should not cost extra energy.

The next available order of perpendicular fluctuation ~ ∫d3x [∇⊥

2ϕ(x)]2 Fluctuation energy ~ ∫d3x {[∂zu(x)]2 + c[∇⊥

2u(x)]2}, leading to Goldstone mode kz

2+ck┴4


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Goldstone ModesMagnetic field and small crystal

field effects (~gSO4) make the

Goldstone mode less soft.For conical phase, χ┴

-

1~kz2+H2k┴

2+ck┴4

For pinned helical phase, χ┴-

1~kz2+|b| k┴

2+ck┴4


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Goldstone ModesThere are two

Goldstone modes in columnar phase, as the translational symmetry breaking is on a plane.

By similar argument, the Goldstone modes have the spectrum χ┴

-

1~k┴2+c’kz

4

Disordered columnar phase has one Goldstone mode of the same spectrum.


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Electronic PropertiesSpecific heat:

Helimagnon: C(T) ~ T2

Columnar phase: C(T) ~ T5/2(Belitz, Kirkpatrick, Rosch 2006)

(Kirkpatrick & Belitz 2010)(Ho, Kirkpatrick, Sang,

Belitz 2010)

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Electronic PropertiesSingle-particle relaxation rate:

Helimagnons (k||~T, k┴~T1/2): τ-1~ T3/2 (clean); τ-1~ T (ballistic disorder)

Columnar phase (k||~T1/2, k┴~T): τ-1~ T2 (clean); τ-1~ T3/2 (ballistic disorder)





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Electronic PropertiesTransport relaxation rate:

Helimagnons (k||~T, k┴~T1/2): τ-1~ T5/2 (clean); τ-1~ T (ballistic disorder)

Columnar phase (k||~T1/2, k┴~T): τ-1~ T3 (clean); τ-1~ T3/2 (ballistic disorder)





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Electronic PropertiesIn ballistic disorder, both

relaxation rates of the columnar phase show T3/2-dependence.

The NFL phase that has electrical resistivity T3/2 might be a liquid of columns.

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Columnar Phase and SkyrmionsColumnar phase:

believed to be a Skyrmion lattice

Skyrmion: a topological object

Winding number: W=(1/4π)∫d2x (n.∂xn×∂yn), where n=M/|M|.

n = (-2yl, 2xl, (x2+y2-l2))/(x2+y2+l2), in σ model.

Algebraic decay at large distances

Size: believed to be q-1

W=-1(Pfleiderer & Rosch 2010)

(Abanov & Prokrovsky 1998; Belavin & Polyakov 1975)

(Skyrme 1961)

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Columnar Phase and Skyrmionsn(x)=-sinθ(ρ)φ

+cosθ(ρ) zCore size R, defined

by core behavior θ(ρ) = π (1-ρ/R)

Tail length lT, defined by the long-range tail exponential decay length exp(-ρ/lT).

Matching length L as the size. (Ho, Kirkpatrick, Belitz 2011)

lT

(Röβler, Leonov, Bogdanov 2011) θ: polar angle between the spin direction and the ferromagnet

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Columnar Phase and Skyrmions

The Skyrmion size is not always q-1, but it is the result of the competition of different length scales, e.g., correlation lengths, magnetic length, q-1

(Ho, Kirkpatrick, Belitz 2011)

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Columnar Phase and SkyrmionsT~Tc, intermediate H:

columnar phase (A phase)

Believed to be 2D hexagonal columnar Skyrmion lattice

With some efforts, it can be derived from LGW functional

Core-to-core distance in a Skyrmion lattice ~ qξp

2 ~ (c/a) (a/r) ~ c/r

(Ho, Kirkpatrick, Belitz 2011)(Mühlbauer et al 2009)

(Han, Zang, Yang, Park, Nagaosa, 2010)

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ConclusionHelimagnets are more favored than

ferromagnetism through DM interaction.Helimagnets have a richer phase diagram

than other magnets in general.Helimagnets have softer Goldstone

modes, resulting in huge fluctuations and special electronic properties.

The size of Skyrmions in helimagnets is the result of competition of various physical length scales.

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AcknowledgmentsTheodore Kirkpatrick (UMD)Dietrich Belitz (UOregon)Yan Sang (UOregon)

Properties of Metallic Helimagnets

Science

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