Inﬂuence of spin-orbit coupling on the transport properties of ...Ludwig Maximilians-Universita¨t...

Ludwig

Maximilians-

Universität

München

Influence of spin-orbit coupling on the transportproperties of spintronics materials

1H. Ebert,1S. Lowitzer, V. Popescu,1D. Ködderitzsch,1J. Minar ,1S. Bornemann2P.H. Dederichs,2R. Zeller

3H. Akai, 3 M. Ogura

1Univ. München, Germany Germany2IFF J̈ulich, Germany

3 Osaka University

funded by theDFG within the programme

SFB 689Spinpḧanomene in reduzierten Dimensionen

Univ. Regensburg

.

JST-DFG workshop Kyoto 2009 – p.1/22

Ludwig

Maximilians-

Universität

MünchenOUTLINE

Introductionelectronic structure calculations

Transport in trilayer systemsTMR

in/out of plane anisotropy TAMR

in plane TAMR

Magnetotransport in bulkformalism

Residual resistivity tensor

spin transport

Summary JST-DFG workshop Kyoto 2009 – p.2/22

Ludwig

Maximilians-

Universität

MünchenGreen’s function in 2D

FM/SC/FM-trilayers with perfect matching

Green’s function (GF)

G±(~r, ~r ′; E) = limǫ→0

∑

λ

φλ(~r) φ×λ (~r)

E − Eλ ± iǫ= G±(~ri + ~Ri, ~r

′j + ~Rj , E)

tight-binding version of KKR-method

G+(~r, ~r ′; E) =1

ASBZ

∫

SBZ

d2k‖ei~k‖(~ρν − ~ρν′ )

×

[∑

ΛΛ′

RνΛ(~r, E) Gνν′

ΛΛ′(~k‖, E) R

ν′

Λ′(~r′, E) − i p δνν′

∑

Λ

RνΛ(~r, E)

]


Ludwig

Maximilians-

Universität

München

Accounting for Spin-Orbit Coupling (SOC)

Dirac Hamiltonian within LSDA(Local Spin Density Approximation)

ĤD = c~α~p + βmc2 + V (~r) + βB(~r)σz , ~α =

(0 ~σ~σ 0

), β =

(I2 00 −I2

)

=

[iγ5σrc

(∂

∂r+

1

r

(1 − βK̂

))+ V (r) + βσzB(r) + (β − 1)

c2

2

]

K̂ = σ̂ L̂ + 1 , γ5 =(

0 −I2−I2 0

), σr =

1

r~r · ~σ

four component Dirac formalism

accounts for SOC and spin-polarisationon same level


Ludwig

Maximilians-

Universität

München

Fe/GaAs/Fe – SOC-induced anisotropy

Bloch spectral functionsA(~k‖, EF) – SC interface layer

As-termination~M ‖ (001)

~M ‖ (110)

Ga-termination~M ‖ (001)

~M ‖ (110)


Ludwig

Maximilians-

Universität

München

Symmetry breaking at the FM/SC-interface

~M ‖ (001) − C2v

~M ‖ (110) − Cs

Fe/9(GaAs)/Fe -µorb and MAE

0.00

0.05

µ orb (

µ B)

bulkFe

FeFe

FeAs

GaAs

GaAs

GaAs

GaAs

FeFe

FeFe

bulk

x 10

M || (001)

M || (110)

crystallographic direction

0.0

1.0

2.0

3.0

E(φ

) -

E(1

10)

(mR

y)

(110) (1--10) (

--1

--10)(010) (

--100)

see also: Sjöstedt et al. 2002, PRL 89, 267203

Košuth et al. 2005, EPL 72, 816JST-DFG workshop Kyoto 2009 – p.6/22

Ludwig

Maximilians-

Universität

München

Transport properties

Conductance- Landauer-Büttiker formalism

G =e2

h

∑

~k‖

g(~k‖)

with ~k‖-resolved conductance:

g(~k‖) =

∫

ALWS

d2r

∫

ARWS

d2r′ jz(~r) G(~r, ~r′;~k‖; EF) jz(~r

′) G∗(~r, ~r ′;~k‖; EF)

jz(~r) =c

EF + mc2

(−i~∇z +

V

c~αz − i

B

cβ(~α × ~az)z

)

”pessimistic” MR ratio

T = GP−GAP

GP

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j

��

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j

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P / AP : parallel / antiparallel orientation of magnetisation KKR:Mavropoulos et al. (2004) JST-DFG workshop Kyoto 2009 – p.7/22

Ludwig

Maximilians-

Universität

München

Transport properties of Fe/ 13(AsGa)/Fe

~k‖-resolved conductance – AP orientation

effect of the spin-orbit coupling (SOC)

correct SOC

GAP = 0.0076 e2/h

SOC switched off

GAP = 0.0044 e2/h


Ludwig

Maximilians-

Universität

München

Tunneling Conductance for Fe/ n(GaAs)/Fe

Conductance

21 29 37 45 53 6110

-10

10-8

10-6

10-4

10-2

g (e

2 /h

/sur

face

ato

m)

PAP

TMR

21 29 37 45 53 61

0.7

0.8

0.9

1.0

TM

R=

(gP-g

AP)/

gP

number of GaAs layers

Symbols: SOC suppressed for inner GaAs layers

dashed lines: full SOC ⇒ SOC increases gAPJST-DFG workshop Kyoto 2009 – p.9/22

Ludwig

Maximilians-

Universität

München

Dependence on orientation of ~M

~M along (110)

AP orientation

~M along (001)

AP orientation

k‖-resolved conductance

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j

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j

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Ludwig

Maximilians-

Universität

München

Dependence on orientation of ~M

Conductance for ~M ‖ (110) and (001)

conductance for P/AP alignment

21 25 29 33 37 41 45 49 53 number of GaAs layers

10-8

10-6

10-4

g (e

2 /h

/ato

m)

PAP

TMR = (GP − GAP)/GP

21 25 29 33 37 41 45 49 53 number of GaAs layers

60

70

80

90

100

TM

R (

%)

j

M

j

M

j

M

j

M

symbols for ~M ‖ (110)

lines for ~M ‖ (001)

difference of G largest for AP configurationJST-DFG workshop Kyoto 2009 – p.11/22

Ludwig

Maximilians-

Universität

MünchenFe/GaAs/Fe – TAMR calculations

Fe/n(GaAs)/Fe tunnelling junction – Ga-termination

conductance

21 29 37 45 53 number of GaAs layers

10-5

10-4

g (e

2 /h

/sur

face

ato

m)

M || (001)M || (110)

TAMR

21 29 37 45 53 number of GaAs layers

-5.0

0.0

5.0

10.0

15.0

20.0

TA

MR

=(g

(00

1) -g

(11

0) )/

g(0

01

) (%

)

exact SOCSOC off I

j

M

j

M

strong TAMR effect for thin spacersuppressing SOC at the interface reduces the TAMR by ≈50%TAMR ≈ 6 × larger than for As termination JST-DFG workshop Kyoto 2009 – p.12/22

Ludwig

Maximilians-

Universität

München

Tunneling Anisotropic Magneto-Resistance

Gould et al. 2004, PRL 93, 117203Rüster et al. 2005, PRL 94, 027203

spin-valve behaviour with only one (ortwo coupled) magnetic layer

depending on φ width and sign change(unlike TMR)

at saturation, the sample is a sensor ofthe absolute directionof ~H

modelling conductance gg ∝ ’tunnelling’ DOS∆DOS≡DOS(M‖x) − DOS(M‖y)

spin-orbit coupling (SOC) inducedanisotropy


Ludwig

Maximilians-

Universität

München

Fe/29(GaAs)/Au – in-plane TAMR

theory

0.0

0.1

0.2

0.3

0.4

(R(φ

)-R

min)/

Rm

in

0.1

0.2

0.3

0.4

[110] [--110]

Ga termAs term x 20

experiment

0.996

0.997

0.998

0.999

1.000

R(Ω

)

0.5 Tesla5 Tesla10 Tesla

Fe(epitaxial)/GaAs(8nm)/Aumeasured at 4.2 K and -90 mV

[110

]

[1-1

0]

Moser et al., Univ. of Regensburg

Phys. Rev. Lett. 99 056601 (2007)

qualitative (but not quantitative) agreement between theory and experiment

φ

j

M


Ludwig

Maximilians-

Universität

München

Full conductivity tensor σ

Str̆eda, Smirc̆ka, JPF (1975,1977)

σµν = −~

2πV

∫ +∞

−∞dE f(E) Tr

〈ĵµ

dG+

dEĵν(G

+−G−)−ĵµ(G+−G−)ĵν

dG−

dE

〉

Crépieux, Bruno, PRB (2001), (T = 0)

σµν =~

4πVTr

〈

ĵµ(G+ − G−)ĵνG

− − ĵµG+ĵν(G

+ − G−)

〉

+e

4iπVTr

〈(G+ − G−)(rµĵν − rν ĵµ)

〉


Ludwig

Maximilians-

Universität

München

Residual resistivity of non-magnetic alloys

0 0.2 0.4 0.6 0.8 1X

0

10

20

30

40

ρ (1

0-6 O

hm c

m)

GuenaultCPA no VC CPA VC

AgxPd

1-x

0 0.2 0.4 0.6 0.8 1X

0

2

4

6

ρ (1

0-6 O

hm c

m)

no SROSROEXP ordered

CuxZn

1-x

Experiment:

Guénault, Phil. Mag. 30, 641, 1974 W. Webb, Phys. Rev. 55, 297, 1938


Ludwig

Maximilians-

Universität

München

(Ga1−xMnx)As

A

B

O

C

As

Ga/Mn0 0.05 0.1 0.15 0.2

x0

1000

2000

3000

4000

5000

ρ (1

0-6 O

hm c

m)

Turek et al. (TB-LMTO), J. Phys.: Cond. Mat. 16 (2004) Ogura, Akai (KKR), unpublishedpresent workEdmonds, Appl. Phys. Let. 81 3010 (2002), as-grownEdmonds et al., Appl. Phys. Let. 81 3010 (2002), annealedChoi et al., Appl. Phys. Let. 89 102503 (2006), annealedChun et al., Phys. Rev. Let. 98 26601 (2007), as-grown

Residual resistivitycomparison with experiment


Ludwig

Maximilians-

Universität

München

(Ga1−x+z−yMnx−zAs y)AsMn iz

O

A

C

B As

Mn

Ga/Mn/As

0 0.05 0.1 0.15 0.2x

0

1000

2000

3000

4000

5000

ρ (1

0-6 O

hm c

m)

Ga1-x

MnxAs

(Ga1-x-y

MnxAs

y)As, y=0.01

(Ga1-x+z

Mnx-z

)AsMni

z, z=0.01

(Ga1-x+z-y

Mnx-z

Asy)AsMn

i

z, y=0.01, z=0.01

Edmonds, Appl. Phys. Let. 81 3010 (2002), as-grownEdmonds et al., Appl. Phys. Let. 81 3010 (2002), annealedChoi et al., Appl. Phys. Let. 89 102503 (2006), annealedChun et al., Phys. Rev. Let. 98 26601 (2007), as-grown

Residual Resistivityinfluence of antisite and interstitial occupation


Ludwig

Maximilians-

Universität

München

Galvano-magnetic effects

Cubic ferromagnet ~M ‖ ~z

ρ =

ρxx ρxy 0

−ρxy ρxx 0

0 0 ρzz

Isotropic resistance ρ̄ = 13(2ρ‖ + ρ⊥)

Anomalous magneto-resistance AMR =ρ‖−ρ⊥

ρ̄

spontaneous Hall effect ρxy


Ludwig

Maximilians-

Universität

München

Resistivity tensor of Co xPd1−x and Co xPt1−x

Isotropic resistivity ρ

0.0 20.0 40.0 60.0 80.0 100.0at. % Co

0.0

10.0

20.0

30.0

40.0

ρ(µ

Ω.c

m)

CoPd TheoryCoPd Exp.CoPt TheoryCoPt Exp.

CoPt

CoPd

Anomalous Magneto-Resistivity ∆ρρ

0.0 20.0 40.0 60.0 80.0 100.0at. % Co

0.0

2.0

4.0

6.0

8.0

10.0

∆ρ/

ρ (%

)

CoPd

CoPt


Ludwig

Maximilians-

Universität

München

Spin conductivity tensor

~Σµν = −~

2πV

∫ +∞

−∞dE f(E) Tr

〈~Jµ

dG+

dEĵν(G

+−G−)− ~Jµ(G+−G−)ĵν

dG−

dE

〉

~Jµ spin current operator for direction µjν electrical current operator for direction ν (~j = ec~α)

Naive definition Jzµ = σz jµ/eIntroduce auxilary current that fulfills continuity equation

Jzµ =d

dt(σz rµ) = σz jµ/e +

(d

dtσz

)rµ

d

dtσz spin-flip events due to SOC or non-collinear magnetism

Semi-relativistic: Niu (2005) and several others

Relativistic: Gyorffy, Szunyogh (2008)JST-DFG workshop Kyoto 2009 – p.21/22

Ludwig

Maximilians-

Universität

München

Summary

Fully relativistiv description of transport withinLandauer-Büttiker

Kubo-Greenwood (and extensions)

Role of SOCinfluence on conductance G

in/out of plane anisotropy of TMR

in plane anisotropy of TMR-TAMR

source for bulk galvano-magnetotransport

Formal basis for corresponding spin transportcalculations


OUTLINEGreen's function in 2DAccounting for Spin-Orbit Coupling (SOC)Fe/GaAs/Fe -- SOC-induced anisotropySymmetry breaking at the FM/SC-interfaceTransport propertiesTransport properties of Fe/BFM {13}(AsGa)/Fe Tunneling Conductance for Fe/$n$(GaAs)/Fe Dependence on orientation of �m {vec M}Dependence on orientation of �m {vec M}Fe/GaAs/Fe -- TAMR calculationsTunneling Anisotropic Magneto-ResistanceFe/�m {29}(GaAs)/Au -- in-plane TAMRFull conductivity tensor $sigma $Residual resistivity of non-magnetic alloys(Ga$_{1-x}$Mn$_{x}$)As (Ga$_{1-x+z-y}$Mn$_{x-z}$As$_y$)AsMn$^{{m i}}_z$ Galvano-magnetic effectsResistivity tensor of Co$_x$Pd$_{1-x}$ and Co$_x$Pt$_{1-x}$ Spin conductivity tensor Summary

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