Lattice Dynamics

Part II Theory

2.6 Lattice Dynamics

andΦ₀is the static potential energy of the crystal. Because the force on any atom must vanish in the equilibrium configuration,

Φ_αlκ =0. (2.47)

As an approximation that we will revisit in the next section, we neglect terms of order three and higher in the displacements in Eq. 2.44. The Hamiltonian can be written as

Hn =X

lκ

p²_lκ 2mκ

+Φ0+ 1 2

X

αlκ

X

α⁰l⁰κ⁰

Φ_αα⁰_lκl⁰_κ⁰uαlκuα⁰l⁰κ⁰. (2.48)

This is the harmonic approximation of lattice dynamics.

It is possbile to rewrite the Hamiltonian in matrix form

Hn=X

lκ

p²_lκ

2m_κ +Φ0+ 1 2

X

lκ

X

l⁰κ⁰

u^T_lκ Φ_lκl⁰_κ⁰ u_l⁰_κ⁰, (2.49)

where Φ_lκl⁰_κ⁰ is a 3×3 force-constant matrix defined for each atom pairlκ and l⁰κ⁰ ((l, κ)≠(l⁰, κ⁰)):

Φ_lκl⁰_κ⁰ =[Φ_αα⁰_lκl⁰_κ⁰] . (2.50)

This is the famous Born-von Kármán model. [8,26] If(l, κ)=(l⁰, κ⁰),Φ_αα⁰_lκlκ is a

“self-force constant”, determined by the requirement of no overall translation of the crystal

Φ_lκlκ = − X

(l⁰,κ⁰)≠(l,κ)

Φ_lκl⁰_κ⁰. (2.51)

Because equal and opposite forces act between each atom of a pair, the matrix

Φ_lκl⁰_κ⁰ must be real and symetric.

Φ_lκl⁰_κ⁰ =













a b c b d e c e f













(2.52)

To satisfy the translational symmetry, the force constant matrices must also have the following property

Φ_lκl⁰_κ⁰=Φ_0κ(l⁰_−l)κ⁰ =Φ_(l−l⁰_)κ0κ⁰. (2.53)

It should be noted that the interatomic force constants, such asΦ_αα⁰_lκl⁰_κ⁰, are in fact not constant but functions of temperature and lattice parameters.

2.6.2 Equations of Motion

Using the the harmonic approximation and the potential above, the equations of motion for nuclei are

mκu_lκ(t)= − X

l⁰,κ⁰

Φ_lκl⁰_κ⁰u_l⁰_κ⁰(t). (2.54)

There are 3× R ×N^cell equations of motion for a finite crystal containingN^cell unit cells, each containingRatoms. [8]

It is easily calculated that for any crystal of reasonable size (on the order of 10²³ atoms) the atoms on the surface account for only a very small fraction of all atoms. Typical atom in that crystal has many, many atoms in any direction extending beyond the distance that interatomic forces can reach. As a result, it is convenient to apply the approximation that displacements separated by a certain number of cells are equal. Imposing these periodic boundary conditions on the crystal, the solutions to the equation of motions can be written in the form of

plane waves of wavevectork, angular frequencyωkj, and “polarization”e_κj(k)

u_lκkj(t) =

s 2~

N m_κω_kj e_κj(k)e^{i(k·rl−ωkjt)}, (2.55)

= ~ v u

t2n(ε_kj, T )+1 N mκεkj

e_κj(k)e^{i(k·rl−ωkjt)}, (2.56)

where we take the real part to obtain physical displacements. The phase factor, e^ik·rl, provides all the long-range spatial modulation, while the dependence onκ, a short-range basis vector index, is placed in the complex constante_κj(k). It is convenient for this constant and the exponential to have modulus unity. j is the

“branch index” discussed below and there are 3Rdifferent such branches from symmetry.

2.6.3 The Eigenvalue Problem of the Phonon Modes

The polarization vectors,e_κj(k), contain all information on the excursion of each atom κ in the unit cell for the phonon mode k, j, including the displacement direction of the atom and its relative phase with respect to the other atoms. These

“polarization” vectors and their associated angular frequenciesω_kj(normal modes) can be calculated by diagonalizing the “dynamical matrix” D(k). The detailed descriptions can be found elsewhere. [8,23–25]

The dynamical matrix can be obtained by substituting Eq. 2.56into2.54. It has the dimensions 3N×3N and is constructed from 3×3 submatrices D_κκ⁰(k)

D(k)=













D11(k) . . . D1N(k) ... . .. ... DN1(k) · · · DNN(k)













, (2.57)

where each sub-matrix D_κκ⁰(k) is the Fourier transform of the force-constant

matrixΦ_lκl⁰_κ⁰

D_κκ⁰(k)= 1

√m_κm_κ⁰ X

l⁰

Φ_0κl⁰_κ⁰e^{ik·(rl0−r0)}. (2.58)

To simplify, the equations of motions (Eq. 2.54) with the plane wave solutions (Eq. 2.56) can be represented as an eigenvalue problem:

D(k)e_j(k)=ω²_kje_j(k), (2.59)

where

e_j(k)=

































ex1j(k) e_y1j(k) ez1j(k) e_x2j(k)

... e_zNj(k)

































(2.60)

is the eigenvector(s).

It can be shown that the dynamical matrix D(k)is hermitian for any value of k. As a result it is thus fully diagonalizable and the eigenvaluesω²_kj are real. The eigenvectors and eigenvalues of the dynamical matrix evaluated at a particular wavevector k correspond to the 3R eigenmodes of vibration of the crystal for that wavevector. It should be noted that the angular frequency (ωkj) dependence on the wavevector (k), called a dispersion relation, can be quite complicated. The speed of propagation of phonons in solids (speed of sound) is given by the group velocity as the slope of this relation

vsound =vg= ∂ω_kj

∂k , (2.61)

instead of the phase velocity

vp= ωkj

k . (2.62)

2.6.4 The Phonon Density of States

With the phonon vibrations calculated above, it is straightforward to calculate the phonon density of states (DOS) of a crystal. This is done by diagonalizing the dynamic matrixD(k)at a large number ofkpoints in the first Brillouin zone, and then binning them into the DOS histogram.

Similarly, a phonon partial DOSg_d(ε), which gives the spectral distribution of motions by one atom speciesdin the unit cell, can be obtained through weighting the vibrational contribution

gd(ε)=X

k|

X

ακj

δdκ|eακj(k)|²g(ε). (2.63)

Because the eigenvalues of the dynamical matrix are normalized for eachkas

X

ακj

|e_ακj(k)|²=1, (2.64)

the total DOS is the sum of the partial DOSs of all atoms in the unit cell,

g(ε)=X

d

gd(ε). (2.65)

This seems simple but some software for phonon calculations does not implement it correctly. As a result, the partial DOSs do not add up to the total DOS, and the total DOS may also have an incorrect weighting on the different atom species.

2.6.5 Group Theory

With group theory, it is possible to find eigenvectors and eigenvalues without diagonalizing the dynamical matrix. [8] Maradudin and Vosko’s review showed how group theory can be used to understand the normal modes of crystal vi- brations and classify them. [27] In their work, the degeneracies of the different normal modes were addressed rigorously, and their association with the symmetry elements of the crystal were used to label them. In the same issue of the journal, Warren discussed the point group of the bond, [28] which refers to the real space symmetries of the interatomic interactions, i.e., how the force constants transform under the point group operations at a central atom.