Spin current across a tunnel junction driven by ...

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Spin current across a tunnel junction driven by transverse electric fields Kjetil Børkje and Asle Sudbø Norwegian University of Science and Technology

Spin current across a tunnel junction driven by transverse electric fields – p.1/14

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Outline Motivation Dissipationless currents in heterostructures 2DEG Spin-orbit coupling Model 1 Results (Model 2)


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Motivation Spin-FET - S. Datta and B. Das, Appl. Phys. Lett. 56 665 (1990)

Gate voltage control of spin-orbit coupling


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Motivation Spin-FET - S. Datta and B. Das, Appl. Phys. Lett. 56 665 (1990)

Gate voltage control of spin-orbit coupling General problems/challenges in spintronics: Injection of spin polarised current into semiconductor Manipulation of spins by electric rather than magnetic fields Spin current across a tunnel junction driven by transverse electric fields – p.3/14

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Dissipationless currents in heterostructures Superconductors: IJ = I0 sin(θB − θA )

ΨΑ = |Ψ| eiθΑ

ΨΒ = |Ψ| eiθΒ

B. D. Josephson, Phys. Lett. 1 251 (1962)

Ferromagnets:

MA

Izspin = I0 sin θ

θ

MB

J. Slonczewski, Phys. Rev. B 39 6995 (1989)

y x


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2DEG

Two-dimensional electron gas (2DEG)

Asymmetry in confining electrostatic potential ⇒ Perpendicular effective electric field

Vconf

z


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Spin-orbit coupling Pauli equation + relativistic correction: Ψ↑ Ψ↑ =H i~∂t Ψ↓ Ψ↓ (p + eA)2 e H= − eφ + B · σ 2m m ~ (E × p) · σ + 2 2 4m c Atomic physics: E ∼ r ⇒ HSO ∼ L · S Spin current across a tunnel junction driven by transverse electric fields – p.6/14

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Spin-orbit coupling in 2DEG Rashba Hamiltonian: X † H= φk (εk + α(E × k) · σ)φk k

† φk

=

† (ck↑ ,

† ck↓ )

,

α material-dependent E z y


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Spin-orbit coupling in 2DEG Rashba Hamiltonian: X † H= φk(εk + α(E × k) · σ)φk | {z } k

† φk

=

† (ck↑ ,

Bk

† ck↓ )

,

α material-dependent E z

Bk k

y

Momentum-dependent “magnetic field”: B−k = −Bk Spin current across a tunnel junction driven by transverse electric fields – p.7/14

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Model 1 A x z

E

T

B

z y

E

A

B

Why different planes? The electrons will feel a change in “magnetic field” as they tunnel from A to B Spin current across a tunnel junction driven by transverse electric fields – p.8/14

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Model 1 A

k = (kx , 0, kz ) , EA = E A y ˆ A × k) = α(E BA k

A x z

E

T

B

z y

E

B

A

B

p = (0, py , pz ) , EB = E B x ˆ B BB p = α(E × p)


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Model 1 A


A x z

E

B

T

z y

E

B

A

B


A

H =

X

φ†k (εk + BA k · σ)φk ,

φ†k = (c†k↑ , c†k↓ )

X

φ†p (εp + BB p · σ)φp ,

φ†p = (d†p↑ , d†p↓ )

k

B

H =

p


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Model 1 A


A x z

E

B

T

z y

E

B

B

A


A

H =

X

φ†k (εk + BA k · σ)φk ,

φ†k = (c†k↑ , c†k↓ )

X

φ†p (εp + BB p · σ)φp ,

φ†p = (d†p↑ , d†p↓ )

k

B

H =

p

HT =

X kpσ

∗ † dpσ ckσ Tkp c†kσ dpσ + Tkp


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Results Izspin

=

P

A

˙ Ti σh N σ σ

N˙ σT = i[H T , NσA ]

x z

E

T

B

z y

E

A

B


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Results Izspin

=

P

A

˙ Ti σh N σ σ

x

E

B

T

z

z y


E

B

A

Linear response: Izspin = s

X kp

F |Tkp |2 kz pz q (kx2 + kz2 )(p2y + p2z )

s = −sgn(E A E B ) F = F (|k|, |p|, α, E A , E B , eV, T )

,

F (eV = 0) 6= 0


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Results Izspin

=

P

A

˙ Ti σh N σ σ

x

T

z

B

z y


E

B

A

Tunneling in z-direction: |Tkp | = 0 Izspin = 2s

E

X

k,kz >0 p,pz >0

if

sgn(kz ) 6= sgn(pz )

F |Tkp |2 kz pz q

(kx2 + kz2 )(p2y + p2z )

s = −sgn(E A E B ) F = F (|k|, |p|, α, E A , E B , eV, T )

,

F (eV = 0) 6= 0


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Why similar to ferromagnets? Bk

Bp k

x y

B−p p

−k

−p

B−k

z

Polar angle in xy-plane (θ ∈ (−π, π]): θ1A = 0

θ1B = − π2

θ2A = π

θ2B =

π 2


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Why similar to ferromagnets? Bk

Bp k

x y

B−p p

−k

−p

B−k

z

Polar angle in xy-plane (θ ∈ (−π, π]): θ1A = 0

θ1B = − π2

θ2A = π

θ2B =

π 2

Difference in polar angle: ∆θ1 = θ1B − θ1A = − π2

∆θ2 = θ2B − θ2A = − π2


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Model 2 Rashba spin-orbit coupling: H R = α(x)(kx σy − ky σx )

,

α|E| → α

Include another type of spin-orbit coupling present in materials lacking inversion symmetry: Linear Dresselhaus term: H D = β(kx σx − ky σy )

,

β constant


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Model 2 Rashba spin-orbit coupling: H R = α(x)(kx σy − ky σx )

,

α|E| → α

Include another type of spin-orbit coupling present in materials lacking inversion symmetry: Linear Dresselhaus term: H D = β(kx σx − ky σy )

A

αΑ

β

d=1

x=0

⇒

,

β constant αΒ

β

B x

Bk = kx (β, α, 0)


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Model 2

A

αΑ

β

αΒ

x=0

B

β

x

A ik± x hσz i(x) for scattering of ΨA (x) = χ : ±e in

hσz i(x) for scattering of ΨBin (x) = χB± e−ip± x :

Amplitude ∼ sin θ ,

θ = arctan

αB β

− arctan

αA β


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Summary Spin (polarised) current across a tunnel junction due to spin-orbit coupling - cond-mat/0506024 Origin of effect: Change in direction of momentum-dependent “magnetic field” No magnetic fields or magnetic materials Spin current in absence of charge current May control both direction and size Possibility for spin injection in semiconductors Spin current across a tunnel junction driven by transverse electric fields – p.14/14