Influence of spin-orbit coupling on the transport properties of - - PowerPoint PPT Presentation

Ludwig Maximilians- Universit¨ at M¨ unchen

Influence of spin-orbit coupling on the transport properties of spintronics materials

• 1H. Ebert, 1S. Lowitzer, V. Popescu, 1D. K¨
• dderitzsch, 1J. Minar , 1S. Bornemann

2P.H. Dederichs, 2R. Zeller

• 3H. Akai, 3 M. Ogura
• 1Univ. M¨

unchen, Germany Germany

2IFF J¨

ulich, Germany

3 Osaka University

funded by the DFG within the programme SFB 689 Spinph¨ anomene in reduzierten Dimensionen

• Univ. Regensburg

.

JST-DFG workshop Kyoto 2009 – p.1/22

Ludwig Maximilians- Universit¨ at M¨ unchen

OUTLINE

Introduction electronic structure calculations Transport in trilayer systems TMR in/out of plane anisotropy TAMR in plane TAMR Magnetotransport in bulk formalism Residual resistivity tensor spin transport Summary

JST-DFG workshop Kyoto 2009 – p.2/22

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Green’s function in 2D

FM/SC/FM-trilayers with perfect matching Green’s function (GF)

G±( r, r ′; E) = lim

ǫ→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

d2kei

k( ρν − ρν′)

×

• ΛΛ′

Rν

Λ(

r, E) Gνν′

ΛΛ′(

k, E) Rν′

Λ′(

r ′, E) − i p δνν′

• Λ

Rν

Λ(

r<, E) Hν′

Λ (

r>, E)

• JST-DFG workshop Kyoto 2009 – p.3/22

Ludwig Maximilians- Universit¨ at M¨ unchen

Accounting for Spin-Orbit Coupling (SOC)

Dirac Hamiltonian within LSDA (Local Spin Density Approximation)

• HD

= c α p + βmc2 + V ( r) + βB( r)σz ,

• α =
• σ
• σ 0
• , β =
• I2

0 −I2

• =
• iγ5σrc

∂ ∂r + 1 r

• 1 − β

K

• + V (r) + βσzB(r) + (β − 1)c2

2

• K =

σ L + 1 , γ5 =

• −I2

−I2

• , σr = 1

r r · σ

four component Dirac formalism accounts for SOC and spin-polarisation

• n same level

JST-DFG workshop Kyoto 2009 – p.4/22

Ludwig Maximilians- Universit¨ at M¨ unchen

Fe/GaAs/Fe – SOC-induced anisotropy

Bloch spectral functions A( k, EF) – SC interface layer As-termination

• M (001)
• M (110)

Ga-termination

• M (001)
• M (110)

JST-DFG workshop Kyoto 2009 – p.5/22

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Symmetry breaking at the FM/SC-interface

• M (001) − C2v
• M (110) − Cs

Fe/9(GaAs)/Fe - µorb and MAE

0.00 0.05 µorb (µB) bulk Fe Fe Fe Fe As Ga As Ga As Ga As Ga As Fe Fe Fe Fe bulk

x 10

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

crystallographic direction 0.0 1.0 2.0 3.0 E(φ) - E(110) (mRy) (110) (1

• 10)

(

• 1
• 10)

(010) (

• 100)

see also: Sj¨

• stedt et al. 2002, PRL 89, 267203

Koˇ suth et al. 2005, EPL 72, 816

JST-DFG workshop Kyoto 2009 – p.6/22

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Transport properties

Conductance - Landauer-Büttiker formalism

G = e2 h

• k

g( k) with k-resolved conductance: g( k) =

• AL

WS

d2r

• AR

WS

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 − iB c β( α × az)z

• ”pessimistic” MR ratio

T = GP−GAP

GP

• j
• j
• P / AP : parallel / antiparallel orientation of magnetisation KKR:

Mavropoulos et al. (2004)

JST-DFG workshop Kyoto 2009 – p.7/22

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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

JST-DFG workshop Kyoto 2009 – p.8/22

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Tunneling Conductance for Fe/n(GaAs)/Fe

Conductance

21 29 37 45 53 61 10

• 10

10

• 8

10

• 6

10

• 4

10

• 2

g (e

2/h /surface atom)

P AP

TMR

21 29 37 45 53 61 0.7 0.8 0.9 1.0

TMR=(g

P-g AP)/g P

number of GaAs layers

Symbols: SOC suppressed for inner GaAs layers dashed lines: full SOC ⇒ SOC increases gAP

JST-DFG workshop Kyoto 2009 – p.9/22

Ludwig Maximilians- Universit¨ at M¨ unchen

Dependence on orientation of M

• M along (110)

AP orientation

• M along (001)

AP orientation

k-resolved conductance

• j
• j
• JST-DFG workshop Kyoto 2009 – p.10/22

Ludwig Maximilians- Universit¨ at M¨ unchen

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 /atom)

P AP

TMR = (GP − GAP)/GP

21 25 29 33 37 41 45 49 53 number of GaAs layers 60 70 80 90 100 TMR (%)

j M j M j M j M

symbols for

M (110)

lines for

M (001)

difference of G largest for AP configuration

JST-DFG workshop Kyoto 2009 – p.11/22

Ludwig Maximilians- Universit¨ at M¨ unchen

Fe/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 /surface atom)

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 TAMR=(g

(001)-g (110))/g (001) (%)

exact SOC SOC off I

j M j M

strong TAMR effect for thin spacer suppressing SOC at the interface reduces the TAMR by ≈50% TAMR ≈ 6 × larger than for As termination

JST-DFG workshop Kyoto 2009 – p.12/22

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Tunneling Anisotropic Magneto-Resistance

Gould et al. 2004, PRL 93, 117203 R¨ uster et al. 2005, PRL 94, 027203

spin-valve behaviour with only one (or two coupled) magnetic layer depending on φ width and sign change (unlike TMR) at saturation, the sample is a sensor of the absolute direction of H modelling conductance g g ∝ ’tunnelling’ DOS ∆DOS≡DOS(Mx) − DOS(My) spin-orbit coupling (SOC) induced anisotropy

JST-DFG workshop Kyoto 2009 – p.13/22

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Fe/29(GaAs)/Au – in-plane TAMR

theory

0.0 0.1 0.2 0.3 0.4 (R(φ)-Rmin)/Rmin 0.1 0.2 0.3 0.4 [110] [

• 110]

Ga term As term x 20

experiment

0.996 0.997 0.998 0.999 1.000 R(Ω) 0.5 Tesla 5 Tesla 10 Tesla

Fe(epitaxial)/GaAs(8nm)/Au

measured at 4.2 K and -90 mV [ 1 1 ] [ 1

• 1

]

Moser et al., Univ. of Regensburg

• Phys. Rev. Lett. 99 056601 (2007)

qualitative (but not quantitative) agreement between theory and experiment

φ

j M

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Full conductivity tensor σ

St˘ reda, Smir˘ cka, JPF (1975,1977)

σµν = − 2πV +∞

−∞

dE f(E) Tr

• ˆ

jµ dG+ dE ˆ jν(G+−G−)−ˆ jµ(G+−G−)ˆ jν dG− dE

• Crépieux, Bruno, PRB (2001), (T = 0)

σµν =

• 4πV Tr
• ˆ

jµ(G+ − G−)ˆ jνG− − ˆ jµG+ˆ jν(G+ − G−)

• +

e 4iπV Tr

• (G+ − G−)(rµˆ

jν − rνˆ jµ)

• JST-DFG workshop Kyoto 2009 – p.15/22

Ludwig Maximilians- Universit¨ at M¨ unchen

Residual resistivity of non-magnetic alloys

0.2 0.4 0.6 0.8 1 X 10 20 30 40 ρ (10

• 6Ohm cm)

Guenault CPA no VC CPA VC

AgxPd1-x

0.2 0.4 0.6 0.8 1 X 2 4 6 ρ (10

• 6Ohm cm)

no SRO SRO EXP ordered

CuxZn1-x

Experiment: Guénault, Phil. Mag. 30, 641, 1974

• W. Webb, Phys. Rev. 55, 297, 1938

JST-DFG workshop Kyoto 2009 – p.16/22

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(Ga1−xMnx)As

As Ga/Mn

0.05 0.1 0.15 0.2 x 1000 2000 3000 4000 5000 ρ (10

• 6Ohm cm)

Turek et al. (TB-LMTO), J. Phys.: Cond. Mat. 16 (2004) Ogura, Akai (KKR), unpublished present work Edmonds, Appl. Phys. Let. 81 3010 (2002), as-grown Edmonds et al., Appl. Phys. Let. 81 3010 (2002), annealed Choi et al., Appl. Phys. Let. 89 102503 (2006), annealed Chun et al., Phys. Rev. Let. 98 26601 (2007), as-grown

Residual resistivity

comparison with experiment

JST-DFG workshop Kyoto 2009 – p.17/22

Ludwig Maximilians- Universit¨ at M¨ unchen

(Ga1−x+z−yMnx−zAsy)AsMni

z

As Mn Ga/Mn/As

0.05 0.1 0.15 0.2 x 1000 2000 3000 4000 5000 ρ (10

• 6Ohm cm)

Ga1-xMnxAs (Ga1-x-yMnxAsy)As, y=0.01 (Ga1-x+zMnx-z)AsMn

i z, z=0.01

(Ga1-x+z-yMnx-zAsy)AsMn

i z, y=0.01, z=0.01

Edmonds, Appl. Phys. Let. 81 3010 (2002), as-grown Edmonds et al., Appl. Phys. Let. 81 3010 (2002), annealed Choi et al., Appl. Phys. Let. 89 102503 (2006), annealed Chun et al., Phys. Rev. Let. 98 26601 (2007), as-grown

Residual Resistivity

influence of antisite and interstitial occupation

JST-DFG workshop Kyoto 2009 – p.18/22

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Galvano-magnetic effects

Cubic ferromagnet

M z ρ =    ρxx ρxy −ρxy ρxx ρzz   

Isotropic resistance ¯

ρ = 1

3(2ρ + ρ⊥)

Anomalous magneto-resistance AMR =

ρ−ρ⊥ ¯ ρ

spontaneous Hall effect ρxy

JST-DFG workshop Kyoto 2009 – p.19/22

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Resistivity tensor of CoxPd1−x and CoxPt1−x

Isotropic resistivity ρ

0.0 20.0 40.0 60.0 80.0 100.0

• at. % Co

0.0 10.0 20.0 30.0 40.0 ρ(µΩ.cm) CoPd Theory CoPd Exp. CoPt Theory CoPt Exp.

CoPt CoPd Anomalous Magneto-Resistivity ∆ρ

ρ

0.0 20.0 40.0 60.0 80.0 100.0

• at. % Co

0.0 2.0 4.0 6.0 8.0 10.0 ∆ρ/ρ (%)

CoPd CoPt

JST-DFG workshop Kyoto 2009 – p.20/22

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Spin conductivity tensor

• Σµν = −

2πV +∞

−∞

dE f(E) Tr

• Jµ

dG+ dE ˆ jν(G+−G−)− Jµ(G+−G−)ˆ jν dG− dE

• Jµ

spin current operator for direction µ jν electrical current operator for direction ν ( j = ec α) Naive definition

Jz

µ = σz jµ/e

Introduce auxilary current that fulfills continuity equation

Jz

µ = d

dt (σz rµ) = σz jµ/e + d dt σz

• rµ

d dt σz 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

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Summary

Fully relativistiv description of transport within Landauer-Büttiker Kubo-Greenwood (and extensions) Role of SOC influence 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 transport calculations

JST-DFG workshop Kyoto 2009 – p.22/22