1. Gauge structure and effective dynamics in semiconductor energy bands
C.P. Chuu Q. Niu
Dept. of Physics Ming-Che Chang

2. 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) …

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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)

9. 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

10. 8-band Kane model
R. Winkler, SO coupling effect in 2D electron and hole systems, App. C

11. 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)

12. 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

13. 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

14. 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

15. 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.