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Wavepacket dynamics for Massive Dirac electron

Wavepacket dynamics for Massive Dirac electron. 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 non-trivial. Explains.

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Wavepacket dynamics for Massive Dirac electron

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  1. Wavepacket dynamics for MassiveDirac electron C.P. Chuu Q. Niu Dept. of Physics Ming-Che Chang

  2. Semiclassicalelectron 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. Semiclassicaldynamics - 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. Ω(k) and L(k) are zero when there are both • time-reversal symmetry • lattice inversion symmetry • (assuming there is no SO coupling) 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)

  5. Single band Multiple bands Basic quantities Basics quantities 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. 2mC2 Construction of a Dirac wave packet Plane-wave solution Center of mass This wave packet has a minimal size Classical electron radius

  8. r r • Angular momentum of the wave packet Ref: K. Huang, Am. J. Phys. 479 (1952). • Energy of the wave packet 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 L Or, “hidden momentum” • Precession of spin (Bargmann, Michel, and Telegdi, PRL 1959) L • Center-of-mass motion + + + + + + + + + + To liner fields > - - - - - - - - - - For v<<c Spin-dependent transverse velocity

  10. A simpler version (Vaidman, Am. J. Phys. 1990) A charge and a solenoid: q S B E Shockley-James paradox (Shockley and James, PRLs 1967)

  11. Smaller m m Gain energy Lose energy Power flow and momentum flow E Larger m (Jackson, Classical Electrodynamics, the 3rd ed.) Force on a magnetic dipole • magnetic charge model • current loop model 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

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

  13. For Dirac electron, to linear order in fields This is the SO interaction with the correct Thomas factor! (Ref: Shankar and Mathur, PRL 1994) (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)

  14. Semiclassical energy generalized Peierls substitution Relativistic Pauli equation Pair production Dirac Hamiltonian (4-component) Foldy-Wouthuysen transformation Silenko, J. Math. Phys. 44, 2952 (2003) Pauli Hamiltonian (2-component) correct to first order in fields, exactto all orders of v/c! Ref: Silenko, J. Math. Phys. 44, 1952 (2003)

  15. Pair production 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… Relevant fields • Relativistic beam dynamics • Relativistic plasma dynamics • Relativistic optics • …

  16. Relativistic electron (as a trial case) • Semiconductor carrier

  17. Hall effect(E.H. Hall, 1879) (J.E. Hirsch, PRL 1999, Dyakonov and Perel, JETP 1971.) (Extrinsic) Spin Hall effect • skew scattering by spinless impurities • no magnetic field required

  18. 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

  19. z Analogy in geometry u v y x 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) Ref: J.E. Avron, Les Houches 1994

  20. 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

  21. 8-band Kane model Efros and Rosen, Ann. Rev. Mater. Sci. 2000

  22. Gauge structures and angular momenta in other subspaces Gauge structure in conduction band • Gauge potential, correct to k1 • Angular momentum, correct to k0 Chang et al, to be published

  23. Spin-orbit coupling for conduction electron • 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 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

  24. Effective Hamiltonianfor 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

  25. 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)

  26. Thank you !

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