Phonon

Phonon propagating through a square lattice (atom displacements greatly exaggerated)

Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength λ is marked.

There is a minimum possible wavelength, given by twice the equilibrium separation a between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2a, due to the periodicity of the lattice. This can be thought as one consequence of Nyquist–Shannon sampling theorem, the lattice points are viewed as the "sampling points" of a continuous wave.

Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes do possess well-defined wavelengths and frequencies.

Animation showing the first 6 normal modes of a one-dimensional lattice: a linear chain of particles. The shortest wavelength is at top, with progressively longer wavelengths below. In the lowest lines the motion of the waves to the right can be seen.

In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.

The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step (adiabatic theorem):

n − 1   n   n + 1   ←   a   →

···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···

→→ → →→→

u_{n − 1} u_n u_{n + 1}

where n labels the nth atom out of a total of N, a is the distance between atoms when the chain is in equilibrium, and u_n the displacement of the nth atom from its equilibrium position.

If C is the elastic constant of the spring and m the mass of the atom, then the equation of motion of the nth atom is

${\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}$

This is a set of coupled equations.

Since the solutions are expected to be oscillatory, new coordinates are defined by a discrete Fourier transform, in order to decouple them.[5]

Put

${\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.}$

Here, na corresponds and devolves to the continuous variable x of scalar field theory. The Q_k are known as the normal coordinates, continuum field modes φ_k.

Substitution into the equation of motion produces the following decoupled equations (this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform,[6]

${\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}$

These are the equations for decoupled harmonic oscillators which have the solution

${\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.}$

Each normal coordinate Q_k represents an independent vibrational mode of the lattice with wavenumber k, which is known as a normal mode.

The second equation, for ω_k, is known as the dispersion relation between the angular frequency and the wavenumber.

In the continuum limit, a→0, N→∞, with Na held fixed, u_n → φ(x), a scalar field, and ${\displaystyle \omega (k)\propto ka}$. This amounts to classical free scalar field theory, an assembly of independent oscillators.

A one-dimensional quantum mechanical harmonic chain consists of N identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.

In some contrast to the previous section, the positions of the masses are not denoted by u_i, but, instead, by x₁, x₂…, as measured from their equilibrium positions (i.e. x_i = 0 if particle i is at its equilibrium position.) In two or more dimensions, the x_i are vector quantities. The Hamiltonian for this system is

${\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}$

where m is the mass of each atom (assuming it is equal for all), and x_i and p_i are the position and momentum operators, respectively, for the ith atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with waves in Fourier space which uses normal modes of the wavevector as variables instead coordinates of particles. The number of normal modes is same as the number of particles. However, the Fourier space is very useful given the periodicity of the system.

A set of N "normal coordinates" Q_k may be introduced, defined as the discrete Fourier transforms of the x_k and N "conjugate momenta" Π_k defined as the Fourier transforms of the p_k:

${\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}$

The quantity k_n turns out to be the wavenumber of the phonon, i.e. 2π divided by the wavelength.

This choice retains the desired commutation relations in either real space or wavevector space

${\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left[x_{l},p_{m}\right]\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0\end{aligned}}}$

From the general result

${\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}$

The potential energy term is

${\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}$

where

${\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}$

The Hamiltonian may be written in wavevector space as

${\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}$

The couplings between the position variables have been transformed away; if the Q and Π were Hermitian (which they are not), the transformed Hamiltonian would describe N uncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, periodic boundary conditions are imposed, defining the (N + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

${\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }$

The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode ω_k are:

${\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }$

The levels are evenly spaced at:

${\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }$

where 1/2ħω is the zero-point energy of a quantum harmonic oscillator.

An exact amount of energy ħω must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field is quantized, the quantum of vibrational energy is called a phonon.

All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later.[7]

This may be generalized to a three-dimensional lattice. The wavenumber k is replaced by a three-dimensional wavevector k. Furthermore, each k is now associated with three normal coordinates.

The new indices s = 1, 2, 3 label the polarization of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.

Dispersion curves in linear diatomic chain

Optical and acoustic vibrations in a linear diatomic chain.

Dispersion relation ω = ω(k) for some waves corresponding to lattice vibrations in GaAs.[8]

For a one-dimensional alternating array of two types of ion or atom of mass m₁, m₂ repeated periodically at a distance a, connected by springs of spring constant K, two modes of vibration result:[9]

${\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}$

where k is the wavevector of the vibration related to its wavelength by ${\displaystyle k={\tfrac {2\pi }{\lambda }}}$.

The connection between frequency and wavevector, ω = ω(k), is known as a dispersion relation. The plus sign results in the so-called optical mode, and the minus sign to the acoustic mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

The speed of propagation of an acoustic phonon, which is also the speed of sound in the lattice, is given by the slope of the acoustic dispersion relation, ∂ω_k/∂k (see group velocity.) At low values of k (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ωa, independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of k, i.e. short wavelengths, due to the microscopic details of the lattice.

For a crystal that has at least two atoms in its primitive cell, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the wavevector. The boundaries at −π/a and π/a are those of the first Brillouin zone.[9] A crystal with N ≥ 2 different atoms in the primitive cell exhibits three acoustic modes: one longitudinal acoustic mode and two transverse acoustic modes. The number of optical modes is 3N – 3. The lower figure shows the dispersion relations for several phonon modes in GaAs as a function of wavevector k in the principal directions of its Brillouin zone.[8]

Many phonon dispersion curves have been measured by inelastic neutron scattering.

The physics of sound in fluids differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support shear stresses (but see viscoelastic fluids, which only apply to high frequencies).

The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an operator, then it describes a quantum field theory of non-interacting bosons.[2] The second quantization technique, similar to the ladder operator method used for quantum harmonic oscillators, is a means of extracting energy eigenvalues without directly solving the differential equations. Given the Hamiltonian, ${\displaystyle {\mathcal {H}}}$, as well as the conjugate position, ${\displaystyle Q_{k}}$, and conjugate momentum ${\displaystyle \Pi _{k}}$ defined in the quantum treatment section above, we can define creation and annihilation operators:[10]

${\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)}$   and   ${\displaystyle {b_{k}}^{\dagger }={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{-k}-{\frac {i}{m\omega _{k}}}\Pi _{k}\right)}$

The following commutators can be easily obtained by substituting in the canonical commutation relation:

${\displaystyle \left[b_{k},{b_{k'}}^{\dagger }\right]=\delta _{k,k'},\quad {\Big [}b_{k},b_{k'}{\Big ]}=\left[{b_{k}}^{\dagger },{b_{k'}}^{\dagger }\right]=0}$

Using this, the operators b_k^† and b_k can be inverted to redefine the conjugate position and momentum as:

${\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)}$   and   ${\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)}$

Directly substituting these definitions for ${\displaystyle Q_{k}}$ and ${\displaystyle \Pi _{k}}$ into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:[2]

${\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)}$

This is known as the second quantization technique, also known as the occupation number formulation, where n_k = b_k^†b_k is the occupation number. This can be seen to be a sum of N independent oscillator Hamiltonians, each with a unique wave vector, and compatible with the methods used for the quantum harmonic oscillator (note that n_k is hermitian).[10] When a Hamiltonian can be written as a sum of commuting sub-Hamiltonians, the energy eigenstates will be given by the products of eigenstates of each of the separate sub-Hamiltonians. The corresponding energy spectrum is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.[10]

As with the quantum harmonic oscillator, one can show that b_k^† and b_k respectively create and destroy a single field excitation, a phonon, with an energy of ħω_k.[10][2]

Three important properties of phonons may be deduced from this technique. First, phonons are bosons, since any number of identical excitations can be created by repeated application of the creation operator b_k^†. Second, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contains sums over the position and momentum operators of every atom when written in position space (See position and momentum space).[10] Finally, using the position–position correlation function, it can be shown that phonons act as waves of lattice displacement.

This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:[10][2]

${\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}$

Which can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.[10]

Phonons have been shown to exhibit Quantum tunneling behavior (or phonon tunneling) where, across gaps up to a nanometer wide, heat can flow via phonons that "tunnel" between two materials.[15] This type of heat transfer works between distances too large for conduction to occur but too small for radiation to occur and therefore cannot be explained by classical heat transfer models.[15]