Wavepacket dynamics for Massive Dirac electron C.P. Chuu Q. Niu Dept. of Physics Ming-Che Chang

Semiclassical electron dynamics in solid (Ashcroft and Mermin, Chap 12) Lattice effect hidden in E(k) Derivation is non-trivial Explains oscillatory motion of an electron in a DC field (Bloch oscillation, quantized energy levels are known as Wannier-Stark ladders) cyclotron motion in magnetic field (quantized orbits relate to de Haas - van Alphen effect) … Limits of validity Negligible inter-band transition (one-band approximation) “never close to being violated in a metal”

Semiclassical dynamics - wavepacket approach 1. Construct a wavepacket that is localized in both the r and the k spaces. 2. Using the time-dependent variational principle to get the effective Lagrangian Berry connection Magnetization energy of the wavepacket Wavepacket energy Self-rotating angular momentum

3. Using the Leff to get the equations of motion Three quantities required to know your Bloch electron: Bloch energy Berry curvature (1983), as an effective B field in k-space Anomalous velocity due to the Berry curvature Angular momentum (in the Rammal-Wilkinson form) time-reversal symmetry lattice inversion symmetry (assuming there is no SO coupling) Ω(k) and L(k) are zero when there are both

Single band Multiple bands Basic quantities Basics quantities Dynamics Magnetization Dynamics Dynamics Covariant derivative SO interaction Chang and Niu, PRL 1995, PRB 1996 Sundaram and Niu, PRB 1999 Culcer, Yao, and Niu PRB 2005 Shindou and Imura, Nucl. Phys. B 2005

Relativistic electron (as a trial case) Semiconductor carrier

Construction of a Dirac wave packet Plane-wave solution 2mC2 Center of mass This wave packet has a minimal size Classical electron radius

r Angular momentum of the wave packet Energy of the wave packet Ref: K. Huang, Am. J. Phys. 479 (1952). Energy of the wave packet r The self-rotation gives the correct magnetic energy with g=2 ! Gauge structure (gauge potential and gauge field, or Berry connection and Berry curvature) SU(2) gauge potential SU(2) gauge field Ref: Bliokh, Europhys. Lett. 72, 7 (2005)

Semiclassical dynamics of Dirac electron Precession of spin (Bargmann, Michel, and Telegdi, PRL 1959) L Center-of-mass motion + + + + + + + + + + L To liner fields > - - - - - - - - - - For v<<c Spin-dependent transverse velocity Or, “hidden momentum”

Shockley-James paradox (Shockley and James, PRLs 1967) A simpler version (Vaidman, Am. J. Phys. 1990) A charge and a solenoid: E B S q

Resolution of the paradox Penfield and Haus, Electrodynamics of Moving Media, 1967 S. Coleman and van Vleck, PR 1968 A stationary current loop in an E field m Gain energy Lose energy Larger m Smaller m E Power flow and momentum flow Force on a magnetic dipole magnetic charge model current loop model (Jackson, Classical Electrodynamics, the 3rd ed.)

Energy of the wave packet Where is the spin-orbit coupling energy?

Re-quantizing the semiclassical theory: (Chuu, Chang, and Niu, to be published. Also see Duvar, Horvath, and Horvath, Int J Mod Phys 2001) Re-quantizing the semiclassical theory: Effective Lagrangian (general) (Non-canonical variables) Standard form (canonical var.) Conversely, one can write (correct to linear field) new “canonical” variables, (generalized Peierls substitution) This is the SO interaction with the correct Thomas factor! For Dirac electron, to linear order in fields (Ref: Shankar and Mathur, PRL 1994)

Relativistic Pauli equation Pair production Dirac Hamiltonian (4-component) Foldy-Wouthuysen transformation Silenko, J. Math. Phys. 44, 2952 (2003) generalized Peierls substitution Semiclassical energy Pauli Hamiltonian (2-component) correct to first order in fields, exact to all orders of v/c! Ref: Silenko, J. Math. Phys. 44, 1952 (2003)

Anomalous magnetic moment (Cf: eq. 5.64 of Brown and Gabrielse, RMP 1986) for muon, a=0.001165923. choose magic = 29.3 to eliminate the effect of confinement E field when E=0, a is velocity-independent

Newton-Wigner and Foldy-Wouthuysen Pryce, Proc. Roy. Soc. London 1948 Newton and Wigner, RMP 1949 Silagadze, SLAC-PUB-5754, 1993 Blount, PR 1962 NW’s position operator (whose eigenstate is a localized function) = FW’s mean position operator  PrP = rs+R Foldy and Wouthuysen, PR 1950

Why heating a cold pizza? advantages of the wave packet approach A coherent framework for A heuristic model of the electron spin Dynamics of electron spin precession (BMT) Trajectory of relativistic electron (Newton-Wigner, FW ) Gauge structure of the Dirac theory, SO coupling (Mathur + Shankar) Canonical structure, requantization (Bliokh) 2-component representation of the Dirac equation (FW, Silenko) Also possible: Dirac+gravity, K-G eq, Maxwell eq… Pair production Relevant fields Relativistic beam dynamics Relativistic plasma dynamics Relativistic optics …

a PRA editor a PRL editor

Relativistic electron (as a trial case) Semiconductor carrier

Skew scattering ( Mott scattering) (Ref: Takahashi and Maekawa, PRL, 2002, Landau and Lifshitz, QM) Transition rate: (for  impurities, up to 2nd order Born approx.)

Hall effect (E.H. Hall, 1879) (Extrinsic) Spin Hall effect (J.E. Hirsch, PRL 1999, Dyakonov and Perel, JETP 1971.) (Extrinsic) Spin Hall effect skew scattering by spinless impurities no magnetic field required

Intrinsic spin Hall effect in p-type semiconductor (Murakami, Nagaosa and Zhang, Science 2003; PRB 2004) Luttinger Hamiltonian (1956) (for j=3/2 valence bands) Valence band of GaAs: (Non-Abelian) gauge potential Berry curvature, due to monopole field in k-space

Emergence of curvature by projection Non-Abelian Free Dirac electron Curvature for the whole space Curvature for a subspace 4-band Luttinger model (j=3/2) x y z v u Analogy in geometry Ref: J.E. Avron, Les Houches 1994

QW with structure inversion asymmetry: Rashba coupling (Sov. Phys QW with structure inversion asymmetry: Rashba coupling (Sov. Phys. Solid State, 1960) Datta-Das current modulator (aka spin FET, APL 1990) (Initial spin eigenstate is not energy eigenstate) spin-orbit coupling (current) tunable by gate voltage spin manipulation without using magnetic field not realized yet due to spin injection problem

Berry curvature in conduction band? 8-band Kane model Rashba system (in asymm QW) Is there any curvature simply by projection? There is no curvature anywhere except at the degenerate point

8-band Kane model Efros and Rosen, Ann. Rev. Mater. Sci. 2000

Gauge structure in conduction band Gauge potential, correct to k1 Angular momentum, correct to k0 Gauge structures and angular momenta in other subspaces Chang et al, to be published

Re-quantizing the semiclassical theory: generalized Peierls substitution: Effective Hamiltonian Ref: Roth, J. Phys. Chem. Solids 1962; Blount, PR 1962 vanishes near band edge higher order in k Spin-orbit coupling for conduction electron Ref: R. Winkler, SO coupling effect in 2D electron and hole systems, Sec. 5.2 Same form as Rashba In the absence of BIA/SIA

Effective Hamiltonian for semiconductor carrier Spin part orbital part Yu and Cardona, Fundamentals of semiconductors, Prob. 9.16 Effective H’s agree with Winkler’s obtained using LÖwdin partition

Position and velocity for a carrier (for B=0) Projected theory (eg. Pauli in Dirac) Unprojected (small) theory (eg. Pauli itself) Position Hamiltonian Gauge field Velocity is missing Neglecting dR/dt underestimates the transverse velocities by a factor of 2 (to leading order of k). Same factor of 2 exists for Dirac vs Pauli as well.

Projected theory: dependence on parent theories B C (= Roth, PR 1960) Revisiting the spin Hall effect in p-type semiconductor

Observation of non-Abelian Berry phase? Energy splitting in nuclear quadruple resonance Conductance oscillation for holes in valence bands (Zee PRB, 1988; Zwanziger PRA 1990) (Arovas and Lyanda-Geller, PRB 1998)

Forward jump and “side jump” Covered in this talk: Wave packet dynamics in multiple bands Relativistic electron Spin Hall effect Wave packet dynamics in single band Anomalous Hall effect Quantum Hall effect (Anomalous) Nernst effect Not covered Forward jump and “side jump” Berger and Bergmann, in The Hall effect and its applications, by Chien and Westgate (1980) optical Hall effect (Picht 1929+Goos and Hanchen1947, Fedorov 1955+Imbert 1968, Onoda, Murakami, and Nagaosa, PRL 2004; Bliokh PRL 2006) wave packet in BEC (Niu’s group: Demircan, Diener, Dudarev, Zhang… etc ) Not related: thermal Hall effect phonon Hall effect (Leduc-Righi effect, 1887) (Strohm, Rikken, and Wyder, PRL 2005, L. Sheng, D.N. Sheng, and Ting, PRL 2006)

Thank you !