1. Wavepacket dynamics of electrons in solid
Dept. of Physics Ming-Che Chang

2. Wave packet dynamics in a single band
Anomalous Hall effect
Quantum Hall effect
Nernst effect
Wave packet dynamics in multiple bands
Relativistic electron
Spin Hall effect

3. Classical Hall effect (Hall, 1879)Hz Jx Ey H H

4. Anomalous Hall effect in ferromagnet (Hall, 1880, 1881)
The usual Lorentz force term
saturation H
SO coupling
required
Anomalous term
Temperature dependence (after saturation)
Berger and Bergmann, in The Hall effect and its applications, by Chien and Westgate (1980)

5. Modern view of the KL theory:Jy
Anomalous velocity due to Berry curvature Ex
gives correct order of magnitude for Fe, also explains

6. Alternative mechanisms (extrinsic):
Smit: KL’s (intrinsic) mechanism vanishes in a lattice w/o disorder
Alternative mechanisms (extrinsic):
Smit’s skew scattering mechanism (1955)
Berger’s side jump mechanism (1970)
“The difference of opinion between Luttinger and Smit seems never to have been entirely resolved.”
CM Hurd, The Hall Effect in Metals and Alloys (1972)
“It is now accepted that two mechanisms are responsible for the AHE: the skew scattering… and the side-jump…” (Crepieux and Bruno, PRB 2001)
Actual situation is messy.
Relative importance differs between different materials.

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

8. Side jump Side jump in materials (Ref: Crepieux and Bruno, PRB 2001)
In real material, 2mc2 is replaced by band gap (104 enhancement)
Microscopically it’s due to the same anomalous velocity in Luttinger’s theory
Side jump in materials

9. Karplus-Luttinger mechanism:
Mired in controversy from the start, it simmered for a long time as an unsolved problem, but has now re-emerged as a topic with modern appeal. -- Ong

10. Science 2001
Science 2003
Recent Theory

11. Recent experiment
Science 2004

12. In the 70's when experimentalists actively investigated the 'Kondo problem', numerous Hall measurements were performed on nonmagnetic metals (e.g. Cu) with a dilute concentration of magnetic impurities, e.g. Mn. The weak AHE observed was consistent with xy being linear in  (skew scattering). These systems are paramagnetic rather than ferromagnetic (hence not really relevant). Nonetheless, the results fostered a collective, if unjustified, tilt towards the skew scattering mechanism and away from Luttinger's theory. It is fair to say that most condensed-matter physicists today have some nodding acquaintance with "skew scattering", but have a harder time recalling what the Luttinger anomalous velocity is Ong

13. Semiclassical dynamics - outline of derivation
1. Construct a wavepacket from one Bloch band 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

14. 3. Using the Leff to get the equations of motion
Anomalous Hall conductivity
KL’s intrinsic mechanism:
Anomalous velocity due to Berry curvature
Limits of validity
Negligible inter-band transition (one-band approximation)
“never close to being violated in a metal”

15. Wave packet dynamics in a single band
Anomalous Hall effect
Quantum Hall effect (Hofstadter spectrum)
Nernst effect
Wave packet dynamics in multiple bands
Relativistic electron
Spin Hall effect

16. Hofstadter spectrum
The band structure of an electron subjects to both a lattice potential V(x,y) and a magnetic field B
Can be studied using either the nearly free electron model or the tight-binding model (TBM)
Surprisingly complex spectrum!
Split of energy band depends on flux/plaquette
If plaq/0= p/q, where p, q are co-prime integers, then a Bloch band splits to q subbands (for TBM) B
The tricky part:
q=3  q=29 upon a small change of B!
Also, when B  0, q can be very large!

17. Hofstadter’s butterfly (Hofstadter, PRB 1976)
A fractal spectrum with self-similarity structure 1 2
Self-similarity
B0 near band button, evenly-spaced LLs
The total band width for an irrational q is of measure zero (as in a Cantor set)

18. Reptiles, by M.C. Escher, 1943
集異璧
著作者：Douglas R. Hofstadter
翻譯者：郭維德

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

20. Dispersion of band energy
p/q=1/3

21. Berry curvature and angular momentum
Chang and Niu, PRB 1996

22. Quantization of the hyper-orbits
Bohr-Sommerfeld quantization condition (Onsager, 1952)
22/67=1/[3+(1/22)]
Crucial for accuracy
Chang and Niu, PRB 1996

23. Wave packet dynamics in a single band
Anomalous Hall effect
Quantum Hall effect
Nernst effect (in FM material)
Wave packet dynamics in multiple bands
Relativistic electron
Spin Hall effect

24. The “Tao” of materials E-B: Hall effect, magneto-electric material
E-T: Thomson effect, Peltier/Seebeck effect
E-B-T: Nernst/Ettingshausen effect, Leduc-Righi effect
E-O, B-O: Kerr effect, Faraday effect, photovoltaic effect, photoelectric effect
E-M, B-M: piezoelectric effect/electrostriction, piezomagnetic effect/magnetostriction
M-O: photoelasticity
...
Electric (E,P)
Magnetic (B,M)
Thermal (T)
Mechanical (M)
Optical (O)
solid state sensor
solid state motor, artificial muscle
solid state refrigerator
Landau and Lifshitz, Electrodynamics of continuous media
Scheibner, 4 review articles in IRE Transations on component parts, 1961, 1962

25. Thermo-galvano-magnetic phenomenaJx
-yT Bz
Ohm Hall Thomson Nernst
Thomson Fourier Leduc-Righi
Ettings-hausen
(Thermal Hall effect)
Onsager relations

26. Berry curvature effect on DOS and magnetization
Xiao, Yao, and Niu, PRL 2005
Berry curvature effect on DOS and magnetization
Streda formula
Total energy
Magnetization
T=0:
T>0:
Gat and Avron, PRL 2003
Thonhauser, Ceresoli, Vanderbilt, and Resta, PRL 2005
Berry curvature correction

28. Anomalous thermoelectric transport
(Xiao et al, PRLs 2006)
Coarse-grain average
At low T, one finds
Mott relation (valid for FM materials as well)
WL Lee et al, PRL 2004
Macroscopic charge current density M0
Transport current density d
Transverse current

29. Wave packet dynamics in a single band
Anomalous Hall effect
Quantum Hall effect
Nernst effect
Wave packet dynamics in multiple bands
Relativistic electron (as a trial case)
Spin Hall effect

30. Single band Multiple bands Basic quantities Basics quantities Dynamics
Covariant derivative
Magnetization
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

31. Construction of a Dirac wave packet
Plane-wave solution
2mC2
Center of mass
This wave packet has a minimal size

32. r Angular momentum of the wave packet Energy of the wave packet
Ref: K. Huang, Am. J. Phys. 479 (1952).
(>< result from Einstein-de Haas effect?)
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)

33. 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  side jump x 2 !
Or,
“hidden momentum”
Trajectory is curved but transverse momentum is conserved!

34. 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

35. 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.)

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

37. 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)

38. 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)

39. 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

40. 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 …

41. a PRA editor
a PRL editor

42. Wave packet dynamics in a single band
Anomalous Hall effect
Quantum Hall effect
Nernst effect
Wave packet dynamics in multiple bands
Relativistic electron
Spin Hall effect

43. Hall effect (E.H. Hall, 1879) Spin Hall effect
(J.E. Hirsch, PRL 1999, Dyakonov and Perel, JETP 1971.)
Spin Hall effect
skew scattering by spinless impurities
no magnetic field required
From spin accumulation to charge accumulation
L< spin coherence length s s 130 m at 36 K for Al
(Johnson and Silsbee, PRL 1985)

44. Anomalous Hall effect in ferromagnet
spin-polarized incident current
charge-polarized outgoing current
Spin Hall effect in semiconductor
spin-unpolarized incident current
charge-unpolarized outgoing current
but spin-polarized outgoing current

45. 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 in valence band,

46. Emergence of curvature by projection
Ref: J.E. Avron, Les Houches 1994
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

47. 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

48. Berry curvature in conduction band
Rashba system (asymm QW)
8-band Kane model (bulk)
There is no curvature anywhere except at the degenerate point
Is there any curvature (hence Rashba-like effect) as a result of projection?

49. 8-band Kane model
Efros and Rosen, Ann. Rev. Mater. Sci. 2000

50. Gauge structure in conduction band, indeed!
Chang et al, to be published
Gauge potential, correct to k1
Angular momentum, correct to k0
Spin-orbit coupling
Same form as Rashba
In the absence of BIA/SIA
Ref: R. Winkler, SO coupling effect in 2D electron and hole systems, Sec. 5.2
Gauge structures and angular momenta in other bands

51. Re-quantizing the semiclassical theory:
generalized Peierls substitution:
Ref: Roth, J. Phys. Chem. Solids 1962; Blount, PR 1962
vanishes near band edge
higher order in k
Effective Hamiltonian
Agrees with those in Winkler’s (obtained using LÖwdin partition)

52. 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)

53. Forward jump and “side jump”
Covered in this talk:
Wave packet dynamics in single band
Anomalous Hall effect
Quantum Hall effect
(Anomalous) Nernst effect
Wave packet dynamics in multiple bands
Relativistic electron
Spin Hall effect
Forward jump and “side jump”
Berger and Bergmann, in The Hall effect and its applications, by Chien and Westgate (1980)
Not covered (among others):
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
(Leduc-Righi effect, 1887)
phonon Hall effect
(Strohm, Rikken, and Wyder, PRL 2005)

54. Thank you !