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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.) Varenna , • 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) Solids: basics Carlos Egues, Varenna 2012 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(vE) so spin eff 2mc2 dV “Electric HR zˆ ()σp Field” Corresponding Hamiltonian: so dz ge σ H ( pV ) Defining a s-o coupling constant : so 2 2mc 2 Ze2 or, H LS so 2 2 3 2m c r H R (σp) p p dV dV so z x y y x H so rˆ(σp), 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 Varenna , 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 2mL / ( basis) z I. 1 cosR L vF (kF / m) Sommerfeld 2D + Rashba: quasi-1D case 2012 Rashba Confinement 1 22 Varenna H px py i yx 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 yx 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 yx 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 Varenna , a as Egues k Carlos A recent experiment Carlos Egues, Varenna 2012 2012 Varenna , 12, 42 meV Egues Carlos Carlos 31 meV 14 meV Tunable : # of ML `12 , ´: const. in this range (12 , 10 ) 12, Carlos Egues, Varenna 2012 Intersubband coupling strength coupling Intersubband „Ref. [11]‟ „Ref. Carlos Egues, Varenna 2012 „Ref. [12]‟ 2012 Varenna , Egues Carlos Carlos Eigenvectors 12, 1 Eigenvalues „Ref. [12]‟ 2012 Varenna , Egues Carlos Carlos Crossings Dirac cones Eigenvectors 12, 1 Eigenvalues „Ref. [12]‟ 2012 Varenna , Egues Carlos Carlos Anticrossings Dirac cones Eigenvectors 12, 1 Eigenvalues H Bentmann, PRL 108, 196801 (2012) 2012 Varenna , Egues (ab initio) Carlos J. Fu, M. Hachiya, P. H. Penteado, & CE 0 214 12 0 Crossings H Bentmann, PRL 108, 196801 (2012) 2012 Varenna , Egues (ab initio) Carlos J. Fu, M. Hachiya, P. H. Penteado, & CE 01. 2 3 2 1 1 2 214 12 01. 2 x 3 H Bentmann, PRL 108, 196801 (2012) 2012 Varenna , Egues (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]‟ 2012 Varenna , Egues Carlos Carlos 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 Varenna , Egues Carlos Carlos 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 An interesting realization of interesting realization An the spin orbit interaction the Carlos Egues, Varenna 2012 „Cross as coolspins - dressed‟ atoms dressed‟ Carlos Egues, Varenna 2012 2012 Optical traps & lattices Varenna , Egues • 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.
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