Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of non-interacting spins Optical spin injection and detection Spin manipulation in nonmagnetic semiconductors Physics of non-interacting spins 2 In non-magnetic semiconductors such as GaAs and Si, spin interactions are weak; To first order approximation, they behave as non-interacting, independent spins. • Zeeman Hamiltonian • Bloch sphere • Larmor precession ∗ • , , and • Bloch equation The Zeeman Hamiltonian 3 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 58 eV/T , ) : electronic charge : Planck constant : free electron mass : spin operator : Pauli operator The Zeeman Hamiltonian 4 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 2 58 eV/T : electronic charge : Planck constant : free electron mass : spin operator 2 : Pauli operator The energy eigenstates 5 Without loss of generality, we can set the -axis to be the direction of , i.e., 0,0, 1 Spin “up” |↑ |↑ 2 1 2 2 1 Spin “down” |↓ |↓ 2 1 2 2 Spinor notation and the Pauli operators 6 Quantum states are represented by a normalized vector in a Hilbert space Spin states are represented by 2D vectors Spin operators are represented by 2x2 matrices. For example, the Pauli operators are 1 ex. |↑ |↑ is equivalent to 2 1 10 1 1 1 1 |↑ |↑ |↑ 2 2 01 0 2 0 2 Expectation values of the Pauli vector 7 Expectation values for the spin up state 01 1 0 ↑| |↑ 10 10 0 10 0 1 0 1 0 ↑| |↑ 10 10 0 00 10 1 1 ↑| |↑ 10 10 1 01 0 0 0,0,1 spin is pointing in the +z direction Expectation values of the Pauli vector 8 One more example: Expectation values for a coherent superposition 1 1 1 | |↑ |↓ 2 2 1 1 01 1 1 1 | | 11 11 1 2 10 1 2 1 1 0 1 1 | | 11 11 0 2 01 2 1 10 1 1 1 | | 11 11 0 2 01 1 2 1 1,0,0 spin is pointing in the +x direction spin can point in any direction, but when measured along a particular axis, it can only be “up” or “down” The Bloch sphere 9 In general, | |↑ |↓ where and are complex numbers. Any spin state (and any qubit) can be represented by a point on a sphere sin cos , sin sin , cos one-to-one correspondence 1 |↑ |↑ |↓ 2 | cos |↑ sin |↓ 2 2 θ 1 |↑ |↓ This is the most general state. 2 complex 2 coefficients have 4 degrees of freedom, 1 minus normalization condition and ϕ |↑ |↓ arbitrariness of the global phase. 2 1 |↑ |↓ |↓ 2 Time evolution of spin states 10 How does a spin state that is perpendicular to magnetic field evolve with time? 1 Initial state: | 0 |↑ |↓ 2 1. Obtain the time-evolution operator exp 2. Compute the state at time | | 0 3. Calculate the expectation value of Time-evolution operator 11 1. Time-evolution operator is given by 10 exp where 2 01 so Ω exp 0 exp 0 2 2 Ω 0exp 0exp 2 2 where Ω is the Larmor frequency 14 GHz/T 2 2. The state at time is obtained by applying on the initial state 1 1 1 | 0 |↑ |↓ 2 2 1 1 exp Ω⁄ 2 0 1 1 exp Ω⁄ 2 | | 0 2 0expΩ⁄ 2 1 2 exp Ω⁄ 2 Larmor precession 12 3. Expectation value of 1 01 || 2 10 1 2 cosΩ 1 0 || 2 0 2 spin precession in the plane at sinΩ an angular frequency Ω⁄ 1 10 || 110 2 01 Heisenberg picture of Larmor precession 13 More intuitive semi-classical picture can be obtained 1 , , , using commutation relations: , , , , , combined into one vector equation where Ω Semi-classical equation of motion 14 in terms of magnetization and spin angular momentum (agrees with classical result!) Ω rate of change for torque exerted by angular momentum magnetic field • Torque is perpendicular to both magnetic field Ω and spin • Parallel component of Ω spin is conserved • Perpendicular component precesses Spin precesses in a cone at frequency Ω Spin relaxation and the Bloch equation 15 In a real world, interaction with the environment will eventually randomize spin 1. Longitudinal spin relaxation Requires energy relaxation 2. Transverse spin relaxation Phase relaxation is enough ∗ For an ensemble, Inhomogeneous dephasing ∗ Bloch equation ∗ Larmor precession revisited 16 For spins initially pointing along in a magnetic field along , the solution for spin component along is given by Ω 1 exp ∗ cos Ω 2 Summary: physics of non-interacting spins 17 • Zeeman Hamiltonian • Bloch sphere Ω • Larmor precession ∗ • , , and • Bloch equation Optical spin injection and detection 18 Time-resolved Kerr rotation data In direct gap nonmagnetic semiconductors, T = 5 K Larmor precession can be directly observed B = 0 T in the time domain 0 0 B = 1 T • Optical selection rules 0 0 (a.u.) x • Time-resolved S B0 = 3 T Kerr/Faraday rotation 0 AlGaAs quantum well • Hanle effect 0 1 2 Time (ns) Electron spins are optically injected at t=0 Optical selection rules 19 left circularly polarized photons carry angular momentum of +1 right circularly polarized photons carry angular momentum of -1 (in units of ħ) selection rules from angular momentum conservation conduction band E c 6 cb 3113 + + v 8 hh lh mj = 3/2 mj = 1/2 mj = +1/2 mj = +3/2 0 k valence band Zinc-blende: 50% polarization Wurtzite and quantum wells: 100% polarization Time-resolved Faraday (Kerr) rotation 20 1. Circularly polarized pump creates spin population B S (1) ~100fs pulses at 76MHz pump Ti:Sapphire sx = -1/2 sx = 1/2 Conduction electrons sx=±1/2 + Heavy and j = -3/2 j = 3/2 light holes x x jx = -1/2 jx = 1/2 31: : 13: 1. Circularly polarized pump creates spin population 2. Linearly polarized probe measures Sx B S (2) probe F Sx + K Sx absorption sx = -1/2 sx = 1/2 + + + index of refraction j = -3/2 j = 3/2 x x photon energy 1. Circularly polarized pump creates spin population 2. Linearly polarized probe measures Sx 3. t is scanned and spin precession is mapped out B S (3) t (2) probe F Sx (1) pump K Sx * t /T2 F F Ae cos(t) gB B / t S. Crooker et al., PRB 56, 7574 (1997) TRFR in bulk semiconductors 23 GaN GaAs PRB 63, 121202 (2001) PRL 80, 4313 (1998) TRFR in quantum wells and quantum dots 24 ZnCdSe quantum well Science 277, 1284 (1997) CdSe quantum dots PRB 59, 10421 (1999) Hanle effect 25 In DC measurements, magnetic field dependence of the spin polarization becomes Lorentzian. t el dtS0 exp cost 0 T * * 2 2 T2 1 S0 “Optical Orientation” (Elsevier, 1984) Imaging the spin accumulation due to spin Hall effect 26 Science 306, 1910 (2004) ns (a.u.) Reflectivity (a.u.) -2-1 021 1 2 3 4 5 Spin accumulation at the edges are imaged 150 by modulating the externally applied magnetic field and measuring the signal at the second harmonic frequency 100 B ext 50 m) 2 T = 30 K 0 1 rad) 0 x = - 35 m Position ( -50 x = + 35 m 0 -1 unstrained -100 Kerr rotation ( GaAs -2 -40 -20 02040 -150 Magnetic field (mT) -40-20 02040-40-20 02040 Position (m) Position (m) Spin manipulation in nonmagnetic semiconductors 27 There are many interactions in nonmagnetic semiconductors that allow spin manipulation • g-factor engineering • spin-orbit interaction • hyperfine interaction • AC Stark effect g-factor engineering 28 H = gB B·S Control the g-factor through material composition in g: material dependent effective g-factor semiconductor heterostructures B: magnetic field g= -0.44 in GaAs g-factor in Al Ga As g= -14.8 in InAs x 1-x 0.6 g= 1.94 in GaN 0.4 0.2 Change g in a static, global B! 0.0 g-factor -0.2 -0.4 0.0 0.1 0.2 0.3 0.4 Al concentration x move electrons into different materials using electric fields Weisbuch et al., PRB 15, 816 (1977) Quasi-static electrical tuning of g-factor 29 Nature 414, 619 (2001) T=5K, B=6T 100 nm g-factor 0.0V E=0 negative 0 -0.8V 0 -1.4V 0 -2.0V Kerr Rotation (a.u.) Kerr Rotation 0 -3.0V 0 E>0 positive 0 500 1000 Time Delay (ps) g-factor is electrically tuned Frequency modulated spin precession 30 Time dependent voltage V(t)=V0+V1 sin(2t) S B laser Blue: Kerr Rotation Red: microwave voltage Fits well to equation : t Science 299, 1201 (2003) Ae 1 cos t 1 cos t Spin precession is frequency modulated 0 g-factor tuning at GHz frequency range Spin-orbit interaction 31 H = ħ/(4m2c2) (V(r) × p)· lab frame electron’s frame “E” “B” e- e- relativistic transformation k Allows zero-magnetic field spin manipulation by electric field through control of k Rashba and Dresselhaus effects 32 time-reversal symmetry Ek, E k, Kramers degeneracy (i.e., at zero magnetic field) inversion symmetry Ek, E k, Zero-magnetic-field spin splitting requires asymmetry Rashba effect: structural inversion asymmetry (SIA) of quantum wells Sov. Phys. Solid State 2, 1109 (1960) Ω , Dresselhaus effect: bulk inversion asymmetry (BIA) of zinc-blende crystal Phys. Rev. 100, 580 (1955) Ω 2 , , In a (001) quantum well Ω 2 , Strain-induced terms Ω ∝ , , Ω ∝ , , 33 Spatiotemporal evolution of spin packet in strained GaAs Nature 427, 50 (2004) -20 E = 0 V cm-1 E = 66 V cm-1 m) V=0 0 0 pump m) 20 FR (a.u.) 20 -1 0 132 E = 33 V cm-1 40 e- 0 E = 100 V cm-1 d 0 Pump-probe distance ( z 20 E 20 3 probe 2 z E Pump-probe distance ( 1 33 V cm-1 40 T=5K -1 0 -1 66 V cm B=0.0000T V=V FR (a.u.) 100 V cm 0 -1 0 1 2435 0 13524 Time (ns) Time (ns) Spin precession at zero magnetic field! z (“E”) k Bint 0 V cm-1 50 V cm-1 100 V cm-1 Coherent spin population excited with electrical pulses 35 Phys.