1. Wavepacket dynamics for Massive Dirac electron
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 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”

3. 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

4. 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

5. 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

6. Relativistic electron (as a trial case)
Semiconductor carrier

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

8. 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)

9. 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”

10. 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

11. 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.)

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

13. 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)

14. 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)

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

16. 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

17. 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 …

18. a PRA editor
a PRL editor

19. Relativistic electron (as a trial case)
Semiconductor carrier

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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 8-band Kane model
Efros and Rosen, Ann. Rev. Mater. Sci. 2000

27. 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

28. 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

29. 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

30. 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.

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

32. 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)

33. 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)

34. Thank you !