Gauge structure and effective dynamics in semiconductor energy bands 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 harder than expected Limits of validity Negligible inter-band transition (one band approximation) “never close to being violated in a metal” 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) …

Semiclassical dynamics - wavepacket approach 1. Construct a wavepacket |W that is localized in both the r space and the k space. 2. Using the time-dependent variational principle to get the effective Lagrangian 3. Using the Leff to get the equations of motion Anomalous velocity due to the Berry curvature Self-rotating angular momentum

Three quantities required to know your Bloch electron: Bloch energy Berry curvature (1983), as an effective B field in k-space 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

Semiclassical dynamics (valid to the first order of the fields only) spinor Single band Multiple bands Basic quantities (scalar) Basics quantities (matrix, boldface) Non-Abelian Dynamics Dynamics Covariant derivative Anomalous velocity 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

Semiclassical Dirac electron as a trial case (Free particle in external fields, positive-energy projection) We’ll call this “energy band” (in the sense of “empty-lattice”) 2mC2 2-fold degeneracy  SU(2) gauge field Pair production

Semiclassical dynamics of Dirac electron (Based on Culcer’s formalism) Precession of spin (reproduces the BMT eq.) L L + + + + + + + + + + - - - - - - - - - - Center-of-mass motion (for v<<c) Spin-dependent transverse velocity To linear fields > “hidden” momentum (Jackson, Classical ED, the 3rd ed.) Or, Transverse remains conserved (= 0) ! Particle momentum hidden momentum

Emergence of curvature by projection Ref: J.E. Avron, Les Houches 1994 Non-Abelian Free Dirac electron Curvature for a complete space Curvature for a subspace 4-band Luttinger model (j=3/2) x y z v u Analogy in geometry (Murakami, Nagaosa and Zhang, Science 2003; PRB 2004)

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 R. Winkler, SO coupling effect in 2D electron and hole systems, App. C

Gauge structure and angular momentum in conduction band Gauge potential, correct to k1 Angular momentum, correct to k0 Energy of the wave packet Where is the spin-orbit coupling energy? (see below)

Canonical variables the semiclassical theory: generalized Peierls substitution old variables, new “canonical” variables, (correct to linear field only) Hamiltonian in canonical variables Roth, J. Phys. Chem. Solids 1962; Blount, PR 1962 vanishes near band edge, higher order in k. will be neglected Hamiltonian in old variables SO coupling

Effective Hamiltonian for conduction electron (-e not shown) Spin part orbital part Same form as the Rashba coupling But in the absence of BIA/SIA Yu and Cardona, Fundamentals of semiconductors, Prob. 9.16 R. Winkler, SO coupling effect in 2D electron and hole systems, Sec. 5.2

Gauge structures and angular momenta for other subspaces Effective Hamiltonian for valence bands consistent with those obtained using LÖwdin partition R. Winkler, SO coupling effect in 2D electron and hole systems Chang et al, to be published

summary gauge structure (gauge potential and gauge field) in semiconductor effective quantum Hamiltonian via generalized Peierls substitution (agrees with unitary rotation) semiclassical dynamics Advantage of the semiclassical theory :-) kinetics (effective Hamiltonian) + dynamics in one framework, plus a very clear physical picture. Disadvantage of the semiclassical theory :-( it’s semiclassical.