Spin Orbit Interaction in Semiconductors J

UNIVERSIDADE DE SÃO PAULO Instituto de Física de São Carlos

Spin orbit interaction in semiconductors J. Carlos Egues Instituto de Física de São Carlos Universidade de São Paulo [email protected]

International School of Physics Enrico Fermi: Quantum Spintronics and Related Phenomena Varenna, June 19 2012 Lecture 1 • Motivation • Spin & orbit: an early experiment

• Spin-orbit interaction: a simple picture

• Sommerfeld model + spin orbit: spin splitting

• New experiment: interband s-o coupling (2D)

• „Cross-dressed‟ atoms as cool spins (energy scales, Zitterbewegung) • Summary

Carlos Egues, Varenna 2012 Some (early) motivation 2012 Conventional electronics: charge plays a fundamental role in devices

[integrated circuits: resistors, transistors (billions), etc.] Varenna , ,

Vg Egues

Source Drain Carlos Carlos Field Effect Transistor (FET)

“spin: factor of 2” (http://www.pbs.org/transistor/science/)

„Rashba effect‟ Datta and Das (APL 56, 1990) Emmanuel Rashba

Supriyo.Datta

Gate

FM1 2DEG FM2 (courtesy of Th. Schäpers) Motivation 2012

Spin FET (Semiconductor spintronics, etc.)

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• Most popular proposed spintronic device (Datta and Das ’90); Egues

• Coherent electric control of magnetic degrees of freedom; Carlos Carlos • Low energy to rotate/flip spins; faster (?); (Hall & Flatté, APL/2006) • More control & various modes of operation. Logic gates with “dynamic” functions: reprogrammable on the fly; Multifunctional: logic, storage, processing, communication, etc, in a single chip? • Design flexibility (hybrid devices, geometry, etc. etc.) • Quantum effects: coherence? Quantum computation? [Awschalom, Flatte, and Samarth, Sci. Am., 2002 (also Sci. Am. Oct/2004); Awschalom & Flatté Nat. Phys. (2007)] • General: fertile ground for novel spin-dependent phenomena. Interplay of many-body effects (tunability) & quantum confinement; spin-flip mechanisms, spin injection and spin polarized transport, non-local quantum correlations (EPR), etc. Spin orbit: renewed interest

 Topological insulators ‘Two-faced’ solid: an insulator from the inside… a conductor from the outside… Spin-orbit interaction: define the ‘topology’ of the bands  Majorana fermions Exotic (half) fermion: ‘particle=antiparticle’ • Spin-orbit interaction + quantum wire + s-wave SC: proximity eff.  unconventional p-wave pairing in the wire • Zeeman (or exchange) gap: topological superconducting wire (Kitaev: spinless p-wave SC chain  Majorana end modes)  Synthetic gauge potentials: cold atoms in optical lattices

Carlos Egues, Varenna 2012 Spin-orbit: an early experiment Richarson (1908); Einstein & de Haas (1915)

Carlos Egues, Varenna 2012 Angular momentum  “spin orbit” 2012

Ltot=0  magnet rotates

conservation Varenna

, ,

Egues Carlos Carlos

Ltot=0

-L

Compass: difficulties • Ships (steel) • Submarines • Proximity to poles

L Patent office (Bern): • navigation system using gyroscopes Sci. Am., October/2004 Experiment: Einstein & de-Haas 2012

(1915) Basic idea: Varenna

Sci. Am., October/2004 ,

1) Orbital motion (loop) magnetic moment. Egues 2) Alignment of moments (via a field B)  change in angular momentum of bar. Carlos

string 3) Angular momentum conservation  Fe bar must rotate (“s-o coupling”).

4) The experiment worked beautifully! mirrors 5) However (factor of 2): the ferromagnetism “Laser” of Fe is due to the intrinsic electron spin Electromagnet (rather than the orbital one). More mathematically: Fe cylinder Lloops Lmech 0, loops loops bra bar MLbar  bar bar magnet loops mec ('since',  s )   Mbar / Lbar electron spins C. Kittel, Phys. Rev. 76, 743 (1949)

2012

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Solids: basics Sommerfeld electrons Sommerfeld model: free & non-interacting electrons in a box with periodic BC. („Fermi gas‟) 2 pˆ ikx x ik x() x L ik x L HH  , e = e e 1 2m0 bare mass 1 22k 2  kk,, exp(ik  r )  &   , knii V 2m0 L orbital spin (i=x,y,z) ni Z

Volume V, N electrons Fermi sea Parabolic band “Potential” N (degenerate gas – spin) n  k z V z SF  (k )  x 2012 (Fermi surface) Fermi level  F

y k Varenna k y , x x 3 2 k  3 n kx F Egues Plane waves: k space: Density of states: ()  

delocalized localized (DOS) Carlos Carlos Rashba spin orbit: basics (spin orbit interaction in two-dimensional electron gases) Electric manipulation of intrinsic magnetic degrees of freedom.

(coherent spin control)

Carlos Egues, Varenna 2012 Rashba spin-orbit interaction (“poor man’s derivation”) Atomic case: Heterojunction: 2012 v Potential profile: V(z) Note structural -e Nitta et al. PRL ‘97 inversion asymmetry r Engels et al. PRB ’97 Varenna (SIA) , , r -e +Ze +Ze Two-dimensional Egues Can control -v slope of V(z) electron gas (2DEG) In its rest frame, the electron feels: Carlos via gate! 1 B  v  E eff nucleus dV 2c „Thomas precession‟ In analogy with the atomic case:V  zˆ Spin-orbit interaction: (Kroemer, Am.J.Phys. 2004) dz ge Hence, the Rashba s-o term is: E  μ B  S(vE) so spin eff 2mc2 dV “Electric HR  zˆ ()σp  Field” Corresponding Hamiltonian: so dz ge σ H  ( pV ) Defining a s-o coupling constant  : so 2 2mc 2  Ze2  or, H   LS  so 2 2 3     2m c r  H R  (σp)   p  p  dV dV so  z  x y y x H so  rˆ(σp), V  rˆ Bernardes, et al. PRL 99, 076603 (2007) dr dr Calsaverini, el al. PRB 78, 155313 (2008) Datta-Das spin transistor

2012 S. Datta and B. Das, Appl. Phys. Lett. 56, 655 (1990).

polarizer analyzer

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z Egues y Awschalom, Flatté & Samarth Sci. Am. 2002

Carlos Carlos x L („Relativistic chip‟ (Prinz, Phys. Today, adapted) Sci. Am. 2004) Main ingredients: 2DEG + spin orbit + “FM” emitter/collector Hamiltonian: (lowest subband: simplicity)

1 2 2  1 2  H  p  p   p  p  H1DEG  px   y px 2DEG 2m x y  x y y x 2m  “Time evolution”: rotation about the y axis 1  cos( / 2)  U     R  cos( / 2) sin( / 2) R    ik y  / RR 0  sin(R / 2) UeR  sin(RR / 2) cos( / 2) 2   R  2mL /  ( basis) z I. 1 cosR L  vF  (kF / m) Sommerfeld 2D + Rashba: quasi-1D case 2012 Rashba Confinement

1 22 Varenna H px  py  i yx   x  y  V()y , 2m

Zeroth-order solution (no Rashba): quantum wire subbands Egues

2 2 Carlos Carlos kx nn, ()kx  , nb  a , ,... 01 2 2 2m Hp pxy  V()y 2m ixkx k,, n ey  n (),,    

Effective quasi-1D Rashba Hamiltonian: two lowest subbands a and b

kkx 

a, ()k i  k0  i  d d  a d / dy b i k () k i  d 0 a, (“interband mixing”) H   0 i d () k i  k Moroz & Barnes, „99 b, Mireles & Kirczenow, ‟01 i d0  i  k  () k Egues, Burkard, Loss,‟ 02 b, Sommerfeld 2D + Rashba: quasi-1D case 2012 Rashba Confinement

1 22 Varenna H px  py  i yx   x  y  V()y , 2m

Zeroth-order solution (no Rashba): quantum wire subbands Egues

2 2 Carlos Carlos kx nn, ()kx  , nb  a , ,... 01 2 2 2m Hp pxy  V()y 2m ixkx k,, n ey  n (),,    

Effective quasi-1D Rashba Hamiltonian: two lowest subbands a and b

kkx 

1 22 Varenna H px  py  i yx   x  y  V()y , 2m

Zeroth-order solution (no Rashba): quantum wire subbands Egues

2 2 Carlos Carlos kx nn, ()kx  , nb  a , ,... 01 2 2 2m Hp pxy  V()y 2m ixkx k,, n ey  n (),,    

Effective quasi-1D Rashba Hamiltonian: two lowest subbands a and b

kkx 

a, ()k i  k0  i  d d  a d / dy b i k () k i  d 00 a, (“interband mixing”) H   0 i d () k i  k Moroz & Barnes, „99 b, Mireles & Kirczenow, ‟01 i d0  i  k  () k Egues, Burkard, Loss,‟ 02 b, Strictly 1D Case: d=0

Neglects interband mixing a a ()k i k H  (lowest subband) 1D i k() k Eigenvalues: a Eigenvectors (spin): 2 2 2  2  kR 1  s (k)  k  skR   , s         i   2m 2m y 2

m  R 1 k       i   R 2 y 2

No Rashba Rashba (gate induced)

(k) ()k

b 2012 bs

Quasi 1D  

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 a  as Egues

k Carlos

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A recent experiment

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 42 meV Egues Carlos Carlos

31 meV

14 meV

Tunable  : # of ML

`12 , ´: const. in this range (12 ,  10 )

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 Egues Carlos Carlos

„Ref. [11]‟ 2012

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Intersubband coupling strength

„Ref. [12]‟

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Eigenvectors

12,  1 Eigenvalues

„Ref. [12]‟

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Crossings

Dirac cones Eigenvectors

12,  1 Eigenvalues

„Ref. [12]‟

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Anticrossings

Dirac cones Eigenvectors

12,  1 Eigenvalues

H Bentmann, PRL 108, 196801 (2012)

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(ab initio) Carlos

J. Fu, M. Hachiya, P. H. Penteado, & CE   0

214

12   0

Crossings

H Bentmann, PRL 108, 196801 (2012)

2012

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(ab initio) Carlos

J. Fu, M. Hachiya, P. H. Penteado, & CE

 01. 2 3 2 1 1 2 214

12  01. 2  x

H Bentmann, PRL 108, 196801 (2012)

2012

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(ab initio) Carlos

J. Fu, M. Hachiya, P. H. Penteado, & CE

 0. 25 1

21 0. 25

12 08.

 0. 05 2

 x

Important interplay of Rashba and Dresselhaus!

„Ref. [13]‟

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Spin-resolved bands

„kink‟

Two-photon photoemission from a Bi/Cu(111) surface alloy „First principles calculations‟

Quasi 1D channels Phys. Rev. Lett. 89, 176401 (2002) 2012

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Appl. Phys. Lett., 82, 2658 (2003)

2 d V  g G (e2 / h) dR  kc L 2mL 0 R    0 2 F R  2 2 d

2012

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Egues Carlos Carlos An interesting realization of the spin orbit interaction

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Egues Carlos Carlos „Cross-dressed‟ atoms as cool spins 2012

Optical traps & lattices

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• Lasers + magnetic fields trap & cool atoms Carlos Carlos

• Control interaction among atoms & BECs

• Simulate canonical condensed matter models

• Light induced gauge fields (Synthetic E’s & B’s) Atoms as two-level (spin) systems 2012

1) Hyperfine ground and excited states of alkali atoms, e.g., F=1 & 2 in 87Rb; Varenna

2) Dipole trap; ,

3) Intersecting Raman lasers detuned from the resonance 5S1/2 5P1/2 , 5P3/2;

4) Zeeman fields – linear and quadratic: state selection. Egues

e Carlos Carlos  Atom: Effective two-level system

 R

   F=1 multiplet: 0, ±1   R H    1   z R  Zeeman 0 

H        Quadratic Zeeman 1 RR      

Optical lattice

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Crossed lasers

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Egues Carlos Carlos Spin-orbit with ultra cold atoms • Main ingredients • Physical picture ‘Ultracold’ spin orbit: physics 2012

1) Crossed (perpendicular) Raman lasers (+ dipole trap) Varenna , , 2) Optical lattice (counterpropagating beams)

y Egues Carlos Carlos

‘Ultracold’ spin orbit: physics 2012 Varenna Crossed Raman lasers impart momentum to the atom while exciting it. ,

y Egues Carlos Carlos

x In ‘spinish’ 2012

(spin language) Varenna Crossed lasers impart momentum to the atom while flipping its (pseudo) spin! ,

y Egues Carlos Carlos

Δk k2 k1

 Momentum and spin are coupled! x

Real thing! 2012

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Equal Rashba & Dresselhaus couplings (1D)

Schliemann, Egues, and Loss, Phys. Rev. Lett. 90, 146801 (2003)

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Hamiltonian for the atoms: Egues

2 2    Carlos H   1  Fr  2m 2x2 Periodic spin-dependent potential:

 y x Fr   R sinG0 y y  R sinG0 x x  x y R cosG0 x R cosG0 y z

k space Schroedinger equation:     εk εk  F G F G  2 n      n k  k 1  z    2 Ck  0   n  2 G0 k  G    k  Lowest bands (k→0)

2 2 n  R n kC 1  z kkyxy x   0 kx k k  k  k ky x y 2 2EGR 0 Zeeman only Zeeman+Rashba Rashba Interaction + Zeeman 2012

2 Varenna  p    , 2    i p  p   2m 2     Egues H     p   p    H  2 2m 2 z  x y y x   p    i p     2m 2  Carlos

Eigenvalues & eigenvectors: H  

2k 2 1 ei sin( / 2) 2 2 2  (r)   eikr   (k)     4 k    2m 2  cos( / 2) 

2k 2 1  ei cos( / 2) 2 2 2  (r)   eikr   (k)     4 k    2m 2  sin( / 2) 

k 2 2 2 with: ei  i  and cos() 1/ 1 4 k /  k Rashba dispersions: 1D Case

ε(k) ε(k) 2k 2 1  (k)   2  4 2k 2  2m 2 -kR kR k ε(k) k Δ = 0 & α = 0 Δ = 0 & α ≠ 0 (no Zeeman; no Rashba) (no Zeeman; Rashba) m kR  2 Δ 

k 2012

Varenna , , No orbital effects included. Not an approximation!!! (cold atoms) Egues Δ ≠ 0 & α ≠ 0 (spin gap: Goldhaber-Gordon/2010) (Zeeman+Rashba) Carlos

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What can we do with this? Carlos

Relativistic QM on a chip: Zitterbewegung 2012 Zitterbewegung: basic idea

“Orthodox” zitterbewegung: Free Dirac equation (beating: E>0 & E<0 eigenstates)

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Semiconductor zitterbewegung: Rashba Hamiltonian (Kane model + V(z) + “FW”) Egues (Schliemann et al. PRL/2005, Bernardes et. al. PRL/2007)

(confinement) Carlos Carlos

Rashba spin-orbit: total angular momentum Jz = Lz + Sz is conserved.

[Jz,HR]=0  [Lz,HR]=-[Sz,HR]  dSz/dt = - dLz/dt

(Sz and Lz: constrained dynamics)

Sz precesses  Lz= xpy - ypx oscillates  jittery motion of x & y

y Spin-orbit coupling  zitterbewegung x Spin space

wave Real space Zitterbewegung packet motion

Summary – Lec. 1 2012 Varenna

• Early motivation: spin FET & coherent control , Egues

• Spin orbit: atomic physics  semiconductors Carlos Carlos • Intersubband s-o coupling: wells with 2 subbands (details in lecture 3)  generalized Rashba model: two Dirac cones (Bernardes et al. PRL/2007; Calsaverini et al. PRB/2008)  anticrossings due to intersubband coupling (recent experiment – Bentmann, PRL/2012)  unusual spin textures • Spin orbit in optical lattices: physical picture  „cross-dressed‟ atoms as cool spins