In this chapter, we discuss the collective vibrational motion of nuclei in a lattice at a specific frequency, so-called phonons (a detailed discussion can be found e.g. in the book byDove Ref. [77]). Many physical properties of materials, such as the heat capacity, conduction or thermal expansion, depend on phonons.
So far the nuclei were fixed at their nuclear positionsRI and decoupled from the motion of the electrons by the Born-Oppenheimer (BO) approximation, Sec.1.2. The many-body Hamiltonian (Eq.1.2) separates into an electronic and a nuclear part. In this sections, we are dealing with the Hamiltonian of the nuclear system RJ)is the interatomic potentialVnnand ˆTnis the kinetic energy of the nuclei
given in Eq.1.2. In a crystalline solid the instantaneous atomic positions RI =R0I +uI(t)is given by the atomic equilibrium positionsR0I and a time-dependent displacementuI(t). For small displacements from the atomic equilibrium positions, we can expandU(R)from Eq.3.1in a Taylor series
U(RI) = U(R0I) +
∑
up to the second order, this is known as theharmonic approximation. The first derivative is zero, because all atoms are in their equilibrium position.The second derivative with respect to the displacementuI(t) defines the interatomic force constant matrix ¯Cin real space. The Fourier transformed interatomic force constant matrix ˜CI J(q) is used to built the dynamical matrix ˜DI J(q)
D˜I J(q) = C˜I J(q) pMIMJ
. (3.3)
The full solution for all vibrational states is given by the 3Nat×3Nat determ-inant equation to be solved for any wave vectorqwithin the first Brillouin zone (BZ) with vibrational frequenciesωη,q
det
D¯˜(q)−ω21
=0, (3.4)
with the identity matrix1. The dependency of the vibrational frequencies ωη(q) on the wave vector q is known as the phonon dispersion and η is the band index. If there are Nat atoms per unit cell, Eq. 3.4gives 3Nat
eigenvalues – vibrational frequencies – per reciprocal lattice vectorq. These correspond to the 3Nat branches of the dispersion relation. In a one-atomic unit cell, there are only three phonon bands for which ωη(q = 0) = 0, they are called the acoustic modes. They split in one longitudinal and two transverse acoustic modes. In crystals with more than one atom per unit cell (Nat >1), in addition to the three acoustic modes are higher branches, so-called optical modes. For a crystal with more than one atom in the primitive cell, there are 3Nat−3 optical modes.
The harmonic oscillator energy levels or eigenvalues for the modeωqare given by The full knowledge of the detailed dispersion relationωη(q)is not always necessary. For expectation values of thermodynamic potentials or their derivatives, the information about the number of phonon modes per volume at a certain frequency or frequency range is sufficient, the so-called phonon density of states (phonon-DOS). The phonon-DOS, gphon(ω,η), of each branch η and the totalphonon-DOS, gphon(ω) are given by counting the respective number of phonons with a particular frequency ωη(q) within theBZ:
Phonons from first-principles: The finite difference method
The interatomic force constant matrix ¯C can be computed usingab initio quantum-mechanical techniques such as density-functional theory (DFT).
Two different methods are commonly used to compute phonons:
1. Thefinite difference method: This scheme is based on numerical differ-entiation of forces acting on the nuclei when atoms are displaced by a small amount from their equilibrium positions. In this work, we
use the phonopy software package to calculate phonons applying the finite difference method [327].
2. Thelinear response functionorGreen’s function method: Derivatives of the energy are calculated perturbatively up to third order. A detailed discussion is given for example in [8,111].
In the following, we focus on the discussion of thefinite difference method, as this is the methodology used in this work. In practice, a single atom in a supercell (SC) is displaced byuI,nand the induced forces on the displaced atom Iand all other atoms in theSC are calculated. The forces are
F(n,I) =−
∑
n0
C¯n,I;n0,J ·uJ,n=0 (3.7) where ¯Cn,I;n0,J is the interatomic force constant matrix relating atoms I in the unit cellnand Jin unit celln0(Eq.3.2). Then, the dynamical matrix in reciprocal space ˜¯D(q)(Eq.3.4) becomes additional phase factor, because the atomI is located in the primitive unit cell andJin then0unit cell of the supercell. Effects due to periodic boundary conditions are included by displacing only atoms in the central unit cell (n =0) and calculating the effect of the displacement on all the atoms in the SC. At discrete wave vectorsqthe finite difference method is exact, if the condition
exp(2πiq·L) =1 (3.9)
is fulfilled (Lis the lattice vector of theSC). Better accuracy of the phonon dispersion is achieved by choosing a largerSC.
The great advantage is that the calculation uses the same computational setup as other electronic structure calculations. To describe the phonon modes at a reciprocal-lattice vectorq, the linear dimension of theSCis of the order of k2πqk. Calculating a detailed phonon dispersion is demanding, because the cost of an interatomic-force constant calculation will scale as 3Nat×NUCwhere NUCis the number of unit cells in theSCand the factor 3 accounts for the three generally independent phonon polarisations. For-tunately, making use of symmetry relations in the crystal can reduce the computational effort and phonons are often well behaved in regular solids and can be interpolated reducing theSCsize needed [245].