Electron spins in nonmagnetic semiconductors

Transcription

1 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

2 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

3 The Zeeman Hamiltonian 3 Hamiltonian for an electron spin in a magnetic field : magnetic moment : magnetic field magnetic moment of an electron spin, ) : Landé g-factor (=2 for free electrons) : Bohr magneton 58 ev/t : electronic charge : Planck constant : free electron mass : spin operator : Pauli operator

4 The Zeeman Hamiltonian 4 Hamiltonian for an electron spin in a magnetic field : magnetic moment : magnetic field magnetic moment of an electron spin : Landé g-factor (=2 for free electrons) 2 2 : Bohr magneton 58 ev/t : electronic charge : Planck constant : free electron mass : spin operator : Pauli operator

5 The energy eigenstates 5 Without loss of generality, we can set the -axis to be the direction of, i.e.,,, Spin up Spin down 2 2 2

6 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 ex. 2 is equivalent to

7 Expectation values of the Pauli vector 7 Expectation values for the spin up state,, spin is pointing in the +z direction

8 Expectation values of the Pauli vector 8 One more example: Expectation values for a coherent superposition ,, 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

9 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 cos 2 sin 2 2 This is the most general state. 2 complex coefficients have 4 degrees of freedom, minus normalization condition and arbitrariness of the global phase. 2 θ ϕ 2 2

10 Time evolution of spin states How does a spin state that is perpendicular to magnetic field evolve with time? Initial state: 2. Obtain the time-evolution operator exp 2. Compute the state at time 3. Calculate the expectation value of

11 Time-evolution operator. Time-evolution operator is given by so exp where 2 where exp 2 exp 2 Ω is the Larmor frequency exp Ω The state at time is obtained by applying on the initial state exp Ω 2 4 GHz/T exp Ω 2 exp Ω 2 2 exp Ω 2 exp Ω 2

12 Larmor precession 2 3. Expectation value of 2 2 cosω 2 2 sinω spin precession in the plane at an angular frequency Ω 2

13 Heisenberg picture of Larmor precession 3 More intuitive semi-classical picture can be obtained,,, using commutation relations:,,,,, combined into one vector equation where Ω

14 Semi-classical equation of motion 4 Ω Ω rate of change for angular momentum in terms of magnetization and spin angular momentum torque exerted by magnetic field Ω (agrees with classical result!) Torque is perpendicular to both magnetic field and spin Parallel component of spin is conserved Perpendicular component precesses Spin precesses in a cone at frequency Ω

15 Spin relaxation and the Bloch equation 5 In a real world, interaction with the environment will eventually randomize spin. Longitudinal spin relaxation Requires energy relaxation 2. Transverse spin relaxation Phase relaxation is enough For an ensemble, Inhomogeneous dephasing Bloch equation

16 Larmor precession revisited 6 For spins initially pointing along in a magnetic field along, the solution for spin component along is given by 2 exp cos Ω Ω

17 Summary: physics of non-interacting spins 7 Zeeman Hamiltonian Bloch sphere Ω Larmor precession,, and Bloch equation

18 Optical spin injection and detection 8 In direct gap nonmagnetic semiconductors, Larmor precession can be directly observed in the time domain Time-resolved Kerr rotation data T = 5 K B = T Optical selection rules Time-resolved Kerr/Faraday rotation Hanle effect S x (a.u.) 2 Time (ns) B = T B = 3 T AlGaAs quantum well Electron spins are optically injected at t=

19 Optical selection rules 9 left circularly polarized photons carry angular momentum of + right circularly polarized photons carry angular momentum of - (in units of ħ) selection rules from angular momentum conservation conduction band c 6 E cb m j = 3/2 m j = /2 m j = +/2 m j = +3/2 8 v lh hh k valence band Zinc-blende: 5% polarization Wurtzite and quantum wells: % polarization

20 Time-resolved Faraday (Kerr) rotation 2. Circularly polarized pump creates spin population S B () pump ~fs pulses at 76MHz Ti:Sapphire Conduction electrons s x =±/2 s x = -/2 s x = /2 + Heavy and light holes j x = -3/2 j x = 3/2 j x = -/2 j x = /2 3 : : : 3

21 . Circularly polarized pump creates spin population 2. Linearly polarized probe measures S x S B (2) probe F S x s x = -/2 s x = /2 + j x = -3/2 j x = 3/2 K S x + absorption + + index of refraction photon energy

22 . Circularly polarized pump creates spin population 2. Linearly polarized probe measures S x 3. t is scanned and spin precession is mapped out S B (3) t F S x () pump (2) probe K S x F t F Ae t / T * 2 g B B / cos( t) S. Crooker et al., PRB 56, 7574 (997)

23 TRFR in bulk semiconductors 23 GaAs PRL 8, 433 (998) GaN PRB 63, 222 (2)

24 TRFR in quantum wells and quantum dots 24 ZnCdSe quantum well Science 277, 284 (997) CdSe quantum dots PRB 59, 42 (999)

25 Hanle effect 25 In DC measurements, magnetic field dependence of the spin polarization becomes Lorentzian. t exp cos el t 2 T2 dts * * T 2 S Optical Orientation (Elsevier, 984)

26 Imaging the spin accumulation due to spin Hall effect 26 Science 36, 9 (24) n s (a.u.) Reflectivity (a.u.) Spin accumulation at the edges are imaged by modulating the externally applied magnetic field and measuring the signal at the second harmonic frequency 5 B ext 5 Kerr rotation (rad) T = 3 K x = - 35 m x = + 35 m unstrained GaAs Magnetic field (mt) Position (m) Position (m) Position (m)

27 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

28 g-factor engineering 28 H = g B B S g: material dependent effective g-factor B: magnetic field Control the g-factor through material composition in semiconductor heterostructures g= -.44 in GaAs g= -4.8 in InAs g=.94 in GaN Change g in a static, global B! g-factor g-factor in Al x Ga -x As move electrons into different materials using electric fields Al concentration x Weisbuch et al., PRB 5, 86 (977)

29 Quasi-static electrical tuning of g-factor 29 Nature 44, 69 (2) T=5K, B=6T nm g-factor.v E= negative Kerr Rotation (a.u.) -.8V -.4V -2.V -3.V E> positive 5 Time Delay (ps) g-factor is electrically tuned

30 Frequency modulated spin precession 3 Time dependent voltage V(t)=V +V sin(2t) S B laser Blue: Kerr Rotation Red: microwave voltage Fits well to equation : Ae t cos t cos t Science 299, 2 (23) Spin precession is frequency modulated g-factor tuning at GHz frequency range

31 Spin-orbit interaction 3 H = ħ/(4m 2 c 2 ) (V(r) p) lab frame E e - electron s frame e - B k relativistic transformation Allows zero-magnetic field spin manipulation by electric field through control of k

32 Rashba and Dresselhaus effects 32 time-reversal symmetry (i.e., at zero magnetic field) inversion symmetry E E k, E k, k, E k, Kramers degeneracy Zero-magnetic-field spin splitting requires asymmetry Rashba effect: structural inversion asymmetry (SIA) of quantum wells Sov. Phys. Solid State 2, 9 (96) Ω, Dresselhaus effect: bulk inversion asymmetry (BIA) of zinc-blende crystal Phys. Rev., 58 (955) Ω 2,, In a () quantum well Ω 2, Strain-induced terms Ω,, Ω,,

33 Spatiotemporal evolution of spin packet in strained GaAs 33 Nature 427, 5 (24) pump e - V= d probe V=V E z Pump-probe distance (m) FR (a.u.) z E = V cm - E = 33 V cm - E 33 V cm - V cm - 66 V cm Time (ns) Pump-probe distance (m) T=5K FR (a.u.) 2 3 B=.T E = 66 V cm - E = V cm Time (ns) Spin precession at zero magnetic field!

34 z ( E ) B int k V cm - 5 V cm - V cm -

35 Coherent spin population excited with electrical pulses 35 [] contact to InGaAs InGaAs m probe m 6 m 9 m pump semi-insulating GaAs substrate B eff S B E contact to substrate F (arb. units) T = 5 K V pp = 2 V +44 mt -44 mt background 5 t (ns) Phys. Rev. Lett. 93, 766 (24) t (ns) F (au) B (mt) Precession at electron Larmor frequency of InGaAs Faraday rotation signal due to electrons The signal shows sign change with magnetic field spins excited in the plane of the sample (along the effective magnetic field) Electron spins are generated using picosecond electrical pulses

36 Hyperfine interaction 36 : nuclear spin : electron spin Nuclear spin polarization acts as an effective magnetic field for electron spins Dynamic nuclear polarization: spin injection causes nuclear spin polarization Lampel, PRL 2, 49 (968) PRL 56, 2677 (2)

37 AC Stark effect 37 Optical pulse below bandgap shifts the band gap (AC Stark effect) Circularly polarized pulse shifts one of the spin subbands, causing spin splitting + m j = 3/2 m j = +3/2 Science 292, 2458 (2)

38 Topics covered in this talk Zeeman Hamiltonian Bloch sphere Larmor precession,, and Bloch equation Optical selection rules Time-resolved Kerr/Faraday rotation Hanle effect g-factor engineering spin-orbit interaction hyperfine interaction AC Stark effect