Crystal Lattice Vibrations: Phonons

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Crystal Lattice Vibrations Phonons

• Introduction to Solid State Physics
  http//www.physics.udel.edu/bnikolic/teaching/phy
  s624/phys624.html

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Lattice dynamics above T0

• Crystal lattices at zero temperature posses long
  range order translational symmetry (e.g.,
  generates sharp diffraction pattern, Bloch
  states, ).
• At Tgt0 ions vibrate with an amplitude that
  depends on temperature because of lattice
  symmetries, thermal vibrations can be analyzed in
  terms of collective motion of ions which can be
  populated and excited just like electrons
  unlike electrons, phonons are bosons (no Pauli
  principle, phonon number is not conserved).
  Thermal lattice vibrations are responsible for

? Thermal conductivity of insulators
is due to dispersive lattice vibrations (e.g.,
thermal conductivity of diamond is 6 times larger
than that of metallic copper). ?
They reduce intensities of diffraction spots and
allow for inellastic scattering where the energy
of the scatter (e.g., neutron) changes due to
absorption or creation of a phonon in the
target. ? Electron-phonon
interactions renormalize the properties of
electrons (electrons become heavier).
? Superconductivity (conventional BCS) arises
from multiple electron-phonon scattering between
time-reversed electrons.
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Vibrations of small amplitude 1D chain
Classical Theory Normal Modes
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3
1
4

Quantum Theory Linear Harmonic Oscillator for
each Normal Mode
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Normal modes of 4-atom chain in pictures
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Adiabatic theory of thermal lattice vibrations

• Born-Oppenheimer adiabatic approximation
• Electrons react instantaneously to slow motion of
  lattice, while remaining in essentially
  electronic ground state ? small electron-phonon
  interaction can be treated as a perturbation with
  small parameter

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Adiabatic formalism Two Schrödinger equations
(for electrons and ions)
The non-adiabatic term can be
neglected at Tlt100K!
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Newton (classical) equations of motion

• Lattice vibrations involve small displacement
  from the equilibrium ion position 0.1Å and
  smaller ? harmonic (linear) approximation

• N unit cells, each with r atoms ? 3Nr Newtons
  equations of motion

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Properties of quasielastic force coefficients
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Solving equations of motion Fourier Series
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Example 1D chain with 2 atoms per unit cell
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1D Example Eigenfrequencies of chain
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1D Example Eigenmodes of chain at q0
Optical Mode These atoms, if oppositely charged,
would form an oscillating dipole which would
couple to optical fields with
Center of the unit cell is not moving!
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2D Example Normal modes of chain in 2D space

• Constant force model (analog of TBH) bond
  stretching and bond bending

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3D Example Normal modes of Silicon
L longitudinal T transverse O optical A
acoustic
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Symmetry constraints
?Relevant symmetries Translational invariance of
the lattice and its reciprocal lattice, Point
group symmetry of the lattice and its reciprocal
lattice, Time-reversal invariance.
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Acoustic vs. Optical crystal lattice normal modes
?All harmonic lattices, in which the energy is
invariant under a rigid translation of the entire
lattice, must have at least one acoustic mode
(sound waves)
?3 acoustic modes (in 3D crystal)
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Normal coordinates
?The most general solution for displacement is a
sum over the eigenvectors of the dynamical matrix

• In normal coordinates Newton equations describe
  dynamics of 3rN independent harmonic oscillators!

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Quantum theory of small amplitude lattice
vibrations First quantization of LHO
?First Quantization
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Second quantization representation Fock-Dirac
formalism
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Quantum theory of small amplitude lattice
vibrations Second quantization of LHO
?Second Quantization applied to system of Linear
Harmonic Oscillators
?Hamiltonian is a sum of 3rN independent LHO
each of which is a refered to as a phonon mode!
The number of phonons in state is
described by an operator
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Phonons Example of quantized collective
excitations
?Creating and destroying phonons
?Arbitrary number of phonons can be excited in
each mode ? phonons are bosons
?Lattice displacement expressed via phonon
excitations zero point motion!
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Quasiparticles in solids

• Electron Quasiparticle consisting of a real
  electron and the exchange-correlation hole (a
  cloud of effective charge of opposite sign due to
  exchange and correlation effects arising from
  interaction with all other electrons).
• Hole Quasiparticle like electron, but of
  opposite charge it corresponds to the absence of
  an electron from a single-particle state which
  lies just below the Fermi level. The notion of a
  hole is particularly convenient when the
  reference state consists of quasiparticle states
  that are fully occupied and are separated by an
  energy gap from the unoccupied states.
  Perturbations with respective to this reference
  state, such as missing electrons, are
  conveniently discussed in terms of holes (e.g.,
  p-doped semiconductor crystals).
• Polaron In polar crystals motion of negatively
  charged electron distorts the lattice of positive
  and negative ions around it. Electron
  Polarization cloud (electron excites longitudinal
  EM modes, while pushing the charges out of its
  way) Polaron (has different mass than
  electron).

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Collective excitation in solids
In contrast to quasiparticles, collective
excitations are bosons, and they bear no
resemblance to constituent particles of real
system. They involve collective (i.e., coherent)
motion of many physical particles.

• Phonon Corresponds to coherent motion of all the
  atoms in a solid quantized lattice vibrations
  with typical energy scale of
• Exciton Bound state of an electron and a hole
  with binding energy
• Plasmon Collective excitation of an entire
  electron gas relative to the lattice of ions its
  existence is a manifestation of the long-range
  nature of the Coulomb interaction. The energy
  scale of plasmons is
• Magnon Collective excitation of the spin degrees
  of freedom on the crystalline lattice. It
  corresponds to a spin wave, with an energy scale
  of

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Classical theory of neutron scattering
Bragg or Laue conditions for elastic scattering!
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Classical vs. quantum inelastic neutron
scattering in pictures

• Lattice vibrations are inherently quantum in
  nature ? quantum theory is needed to account for
  correct temperature dependence and zero-point
  motion effects.

Phonon absorption is allowed only at finite
temperatures where a real phonon be excited