Wavepacket dynamics for Massive Dirac electron

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Wavepacket dynamics for Massive Dirac electron
Dept. of Physics Ming-Che Chang
C.P. Chuu Q. Niu
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Semiclassical electron dynamics in solid
(Ashcroft and Mermin, Chap 12)

• Lattice effect hidden in E(k)
• Derivation is non-trivial

• 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)

Explains
Limits of validity
Negligible inter-band transition (one-band
approximation) never close to being violated in
a metal
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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
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Three quantities required to know your Bloch
electron
3. Using the Leff to get the equations of motion

• 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)

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Single band
Multiple bands
Basic quantities
Basics quantities
Magnetization
Dynamics
Dynamics
Covariant derivative
SO interaction
Culcer, Yao, and Niu PRB 2005 Shindou and Imura,
Nucl. Phys. B 2005
Chang and Niu, PRL 1995, PRB 1996 Sundaram and
Niu, PRB 1999
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• Relativistic electron (as a trial case)
• Semiconductor carrier

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Construction of a Dirac wave packet
Plane-wave solution
Center of mass
This wave packet has a minimal size
Classical electron radius
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• 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 g2 !

• 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)
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Semiclassical dynamics of Dirac electron

• Precession of spin (Bargmann, Michel, and
  Telegdi, PRL 1959)

?L

• Center-of-mass motion

To liner fields gt
- - - - - - - - - -
For vltltc
Spin-dependent transverse velocity
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Shockley-James paradox (Shockley and James, PRLs
1967)
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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
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Energy of the wave packet
Where is the spin-orbit coupling energy?
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(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)
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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, exact to all
orders of v/c!
Ref Silenko, J. Math. Phys. 44, 1952 (2003)
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Anomalous magnetic moment
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Newton-Wigner and Foldy-Wouthuysen

• Pryce, Proc. Roy. Soc. London 1948
• Newton and Wigner, RMP 1949
• Silagadze, SLAC-PUB-5754, 1993
• Blount, PR 1962

• NWs position operator (whose eigenstate is a
  localized function) FWs mean position operator
  ? PrP rsR

Foldy and Wouthuysen, PR 1950
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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 Diracgravity, K-G eq, Maxwell eq

Relevant fields

• Relativistic beam dynamics
• Relativistic plasma dynamics
• Relativistic optics

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a PRA editor
a PRL editor
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• Relativistic electron (as a trial case)
• Semiconductor carrier

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

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Intrinsic spin Hall effect in p-type
semiconductor (Murakami, Nagaosa and Zhang,
Science 2003 PRB 2004)
Luttinger Hamiltonian (1956) (for j3/2 valence
bands)
Valence band of GaAs
(Non-Abelian) gauge potential
Berry curvature, due to monopole field
in k-space
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Emergence of curvature by projection
Non-Abelian

• Free Dirac electron

Curvature for the whole space
Curvature for a subspace

• 4-band Luttinger model (j3/2)

Ref J.E. Avron, Les Houches 1994
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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

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Berry curvature in conduction band?
8-band Kane model
Rashba system (in asymm QW)
There is no curvature anywhere except at the
degenerate point
Is there any curvature simply by projection?
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8-band Kane model
Efros and Rosen, Ann. Rev. Mater. Sci. 2000
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Gauge structure in conduction band

• Gauge potential, correct to k1

• Angular momentum, correct to k0

Chang et al, to be published
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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

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Effective Hamiltonian for semiconductor carrier
Spin part orbital part
Yu and Cardona, Fundamentals of semiconductors,
Prob. 9.16
Effective Hs agree with Winklers obtained using
LÖwdin partition
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Position and velocity for a carrier (for B0)
Projected theory (eg. Pauli in Dirac)
Unprojected (small) theory (eg. Pauli itself)
Position
Hamiltonian
Gauge field
Velocity
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Projected theory dependence on parent theories
( Roth, PR 1960)
Revisiting the spin Hall effect in p-type
semiconductor
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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)
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• 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

Covered in this talk
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 1929Goos and Hanchen1947, Fedorov
  1955Imbert 1968, Onoda, Murakami, and Nagaosa,
  PRL 2004 Bliokh PRL 2006)
• wave packet in BEC
• (Nius 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)
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Thank you !