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

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

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

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)

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

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.

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

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

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

Science 2001 Science 2003 Recent Theory

Recent experiment Science 2004

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

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

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”

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

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!

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)

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

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

Dispersion of band energy p/q=1/3

Berry curvature and angular momentum Chang and Niu, PRB 1996

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

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

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

Thermo-galvano-magnetic phenomena Jx -yT Bz Ohm Hall Thomson Nernst 1826 1879 1851 1886 Thomson Fourier Leduc-Righi Ettings-hausen 1851 1886 1807 1887 (Thermal Hall effect) Onsager relations

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

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

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

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

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

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)

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!

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

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

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

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)

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

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,

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

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 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?

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

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

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)

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 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)

Thank you !