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Crystal Lattice Vibrations Phonons

- Introduction to Solid State Physics

http//www.physics.udel.edu/bnikolic/teaching/phy

s624/phys624.html

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.

Vibrations of small amplitude 1D chain

Classical Theory Normal Modes

2

3

1

4

Quantum Theory Linear Harmonic Oscillator for

each Normal Mode

Normal modes of 4-atom chain in pictures

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

Adiabatic formalism Two Schrödinger equations

(for electrons and ions)

The non-adiabatic term can be

neglected at Tlt100K!

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

Properties of quasielastic force coefficients

Solving equations of motion Fourier Series

Example 1D chain with 2 atoms per unit cell

1D Example Eigenfrequencies of chain

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!

2D Example Normal modes of chain in 2D space

- Constant force model (analog of TBH) bond

stretching and bond bending

3D Example Normal modes of Silicon

L longitudinal T transverse O optical A

acoustic

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.

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)

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!

Quantum theory of small amplitude lattice

vibrations First quantization of LHO

?First Quantization

Second quantization representation Fock-Dirac

formalism

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

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!

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).

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

Classical theory of neutron scattering

Bragg or Laue conditions for elastic scattering!

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

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