11institutetext:  School of Basic Sciences, Indian Institute of Technology
Mandi, Kamand - 175075, India
School of Engineering, Indian Institute of Technology Mandi, Kamand - 175075,
India

# First-principles phonon calculations for lattice dynamics, thermal expansion
and lattice thermal conductivity of CoSi at high temperature region

Shamim Sk 1111E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Sudhir K. Pandey 2222E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This study presents the first-principles phonon calculations to understand the
experimental thermal expansion ($\alpha(T)$) and lattice thermal conductivity
($\kappa_{L}$) of CoSi at high temperature region. Phonon dispersion is
computed using finite displacement method and supercell approach by taking the
equilibrium crystal structures obtained from DFT. The calculation of
$\alpha(T)$ is done under quasi-harmonic approximation. The $\kappa_{L}$ is
calculated using first-principle anharmonic lattice dynamics calculations
under single-mode relaxation time approximation. Calculated $\alpha(T)$ in the
temperature range $0-1300$ K gives the good match with existing experimental
data. The calculated value of $\kappa_{L}$ ($\sim$8.0 W/m-K) at 300 K is found
to be in good agreement with the experimental value of $\sim$8.3 W/m-K. The
temperature dependent of phonon lifetime due to phonon-phonon interaction is
calculated to understand the behaviour of $\kappa_{L}$. Present study suggests
that ground state phonon dispersion obtained from DFT based methods gives
reasonably good explanation of experimental $\alpha(T)$ and $\kappa_{L}$.

##  1 Introduction

Theoretical understanding of the thermal transport properties by phonon is a
challenging task in condensed matter physics and materials science. The source
of scattering mechanisms for realizing the phonon transport properties are
phonon-phonon interaction, electron-phonon interaction, phonon-defect
interaction etc[1]. The capturing of these scattering mechanisms for any
system is really a challenging job computationally due to involvement of the
many-body interactions. At finite temperature, the study of these properties
becomes more difficult at the level of theory and computation due to the
presence of anharmonic effect. However, the recent advancement of high
performance computers improves the situation to some extent. For instance, to
study the lattice dynamics and dependent phonon properties at finite
temperature, an ab initio molecular dynamics simulations is recently used[2,
3, 4, 5]. But, this method is computationally costly and the implementation of
this method is not straightforward. In this context, the reliable and
relatively cheaper methods to understand the phononic properties are first-
principle density functional theory (DFT)[6] based techniques.

In addition to the electronic structure information, DFT also calculates the
force exerted on each atom of the compound. Once atom is displaced from its
equilibrium position, the forces of all atoms enhance. The systematic
displacement of atoms give a number of phonon frequencies. The method for
analysis of these phonon frequencies known as finite displacement method
(FDM)[7]. The another method to analysis the phonon frequencies is density
functional perturbation theory (DFPT)[8]. At the DFT level, it is the common
practice to use the ground state phonon dispersion to calculate the phonon
properties at finite temperature. But, the ground state of DFT itself faces
many challenges. For instance, many-particle wave function is approximated by
one-particle wave function in DFT. Apart from this, DFT approximates the
ground state results by various exchange-correlation (XC) functionals. On the
top of these challenges in ground state, it will be interesting to study up to
what extent DFT addresses the phonon properties at high temperature. Keeping
such a challenge in mind, we have chosen CoSi as a case example to study the
phonon related properties at high temperature region.

CoSi is recently marked as novel topological semimetal[9, 10]. This compound
with a B20 simple cubic structure is reported as a potential thermoelctric
(TE) material from last few decades[11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22]. The TE material is one which converts the waste heat into useful
electricity. The efficiency of TE materials is calculated using the
dimensionless parameter, called figure of merit[23], $ZT=S^{2}\sigma
T/\kappa$. Where, S is Seebeck coefficient, $\sigma$ is electrical
conductivity, $\kappa$ is thermal conductivity and T is absolute temperature
of the material. The $\kappa$ consists of two parts: electronic thermal
conductivity ($\kappa_{e}$) and lattice thermal conductivity ($\kappa_{L}$).
The efficient TE materials should have the value of ZT greater or equal to
unity[24]. Therefore, potential TE materials should possess high power factor
(PF = $S^{2}\sigma$) with low $\kappa$. The PF of CoSi is comparable with so
called state-of-the-art TE materials Bi2Te3[25] and PbTe[26]. But, ZT of CoSi
is diminished by it’s high $\kappa$. Actually, achieving high ZT is really a
difficult task, as S, $\sigma$ and $\kappa_{e}$ are strongly dependent to each
other through charge carrier[1, 27]. Minimizing of $\kappa_{e}$ without
affecting $\sigma$ is a difficult job as they are related linearly via
Wiedeman-Franz law: $\kappa_{e}=L\sigma T$, L is Lorenz number. Hence, the
brilliant way to maximize ZT of CoSi is to optimize $\kappa_{L}$. Therefore,
it is necessary to study the phonon properties in order to understand
$\kappa_{L}$.

In the context of thermoelectricity, phonon plays a crucial role to fully
evaluate the TE materials. The information of phonon must be considered for
understanding the lattice thermal conductivity, thermal expansion etc. Here,
the ground state phonon dispersion obtained from DFT based method can be used
to study the temperature dependent phononic transport properties under various
approximations. Within the harmonic and quasi-harmonic approximations the
basic phonon properties e.g. phonon dispersion, heat capacity at constant
volume (Cv), entropy, thermal expansion, heat capacity at constant pressure
(Cp) etc. can be calculated. But, calculating the lattice thermal conductivity
needs the anharmonic force constant which requires the many-body perturbation
theory and hence computational implementation becomes more challenging as well
as time consuming. In addition to the different approximations involved in
calculating the phonon properties, the different XC functionals, force
constant cutoff, supercell size, atomic displacement size etc. strictly affect
the ground state phonon dispersion[2]. Therefore, by taking this ground state
phonon dispersion, it will be a great challenge to understand the experimental
phonon related properties at high temperature.

In this work, the first-principle DFT based phonon calculations are carried
out to understand the experimental $\alpha(T)$ and $\kappa_{L}$ at high
temperature region. The finite displacement method and supercell approach are
used to calculate the phonon dispersion. The calculated thermal expansion in
the temperature range $0-1300$ K gives the nice match with the experimental
reported data. The anharmonic lattice dynamics has been introduced to
calculate $\kappa_{L}$. The computed value of $\kappa_{L}$ is $\sim$8.0 W/m-K
which is in consistent with the estimated experimental value of $\sim$8.3
W/m-K at room temperature. Temperature dependent phonon lifetime due to
phonon-phonon interaction is calculated in order to understand the
experimental feature of $\kappa_{L}$.

##  2 COMPUTATIONAL DETAILS

Phonon properties are calculated using PHONOPY code[28] based on finite
displacement method (FDM) and supercell approach[7]. A supercell of size 2
$\times$ 2 $\times$ 2 containing 64 atoms is used in order to calculate the
total forces on each atom in WIEN2k code[29]. The local density approximation
(LDA)[30] is used as XC functional. The k-mesh size of 4 $\times$ 4 $\times$ 4
is used in the full Brillouin zone for force calculation. The convergence
criteria for the calculations of forces is set as 0.1 mRy/Bohr. Using these
forces, second order force constants are extracted using PHONOPY code to
calculate the phonon dispersion. Thermal expansion of this compound is also
calculated under quasi-harmonic approximation (QHA) as implemented in PHONOPY
code.

The lattice thermal conductivity is calculated using PHONO3PY code[31] within
supercell approach. Same size of supercell is used here as used in PHONOPY.
The ABINIT software[32] is employed to calculate forces on each atom using
projector augmented wave (PAW) method under DFT. The PAW datasets are taken
from the work of Jollet et. al[33]. The LDA[30] is employed as XC functional
which was also used in the WIEN2k force calculation in the case of PHONOPY.
The self-consistency is achieved by setting the force convergence criteria of
5 $\times$ 10-8 Ha/Bohr. The k-mesh size of 4 $\times$ 4 $\times$ 4 is used in
the full Brillouin zone.

The output results of ABINIT containing forces on each atom are used in
PHONO3PY[31] to calculate second and third order force constants. These force
constants are used to calculate lattice thermal conductivity using single-mode
relaxation time approximation as implemented in PHONO3PY. A dense q-mesh size
of 21 $\times$ 21 $\times$ 21 is used for lattice thermal conductivity
calculation. A real-space cutoff distance of 5.16 Bohr is set to ensure the
three neighbour atoms interaction. This setting distance effectively reduces
the number of supercell calculations and hence minimizes the computational
cost.

##  3 RESULTS AND DISCUSSION

Phonon dispersion of CoSi is presented in fig. 1 along the high symmetry
directions in the first Brillouin zone. The positive phonon frequencies (or
energies) of all the branches signifying the mechanical stability of the
compound[28]. Phonon dispersion associates with twenty four branches as
primitive cell of CoSi contains eight atoms. Out of them three are acoustic
branches and twenty one are optical branches. The maximum phonon energy is
calculated as $\sim$52 meV, which closely matches with the experimental value
of 56 meV[34]. At $\sim$22 mev, all the three optical branches are nearly
degenerated with the five optical branches at $R$ point. The nature of
acoustic branches along $\Gamma-M$ and $\Gamma-R$ directions are almost
linear, which means that the group velocity is nearly close to phase velocity
at this region[1]. Here, it is important to note that the sound velocity can
be calculated from the slopes of acoustic branches[1]. This sound velocity is
associated with the lattice thermal conductivity in a solid[1, 35].

Figure 1: Phonon dispersion of CoSi.

A detailed study of the dependency of XC functionals on vibrational and
thermodynamic properties of CoSi has been carried out by Sk et al[36]. Though
the vibrational properties were reported as XC functional dependent, but
thermodynamic properties were almost insensitive of XC functional. However,
LDA was reported to give the better value of Debye temperature as compared to
other functionals. Here, we have chosen LDA as an XC functional for
calculating all the phonon related properties in this work.

In a pure harmonic solid, the phonon related properties like thermal expansion
and lattice thermal conductivity are not defined as phonons do not interact
under harmonic approximation[37]. In this context, quasi harmonic
approximation is found to be a reasonably good approximation to capture the
thermal expansion. The study of thermal expansion is much needful in those
areas where temperature dependent properties of the materials are taken into
account. The materials used for making modules of TEG experience temperature
gradient. Hence, study of thermal expansion of these materials are helpful
before putting these materials in TEG. Keeping this in mind, we have
calculated linear thermal expansion coefficient ($\alpha(T)$) of CoSi under
QHA. Here, QHA used for volume dependence of phonon properties. fig. 2(c)
displays the calculated $\alpha(T)$ in the temperature range $0-1300$ K. For
the better understanding of fig. 2(c), it is necessary to explain fig. 2(a)
and (b) first. Fig. 2(a) exhibits the change in total free energy as a
function of primitive cell volume at different temperatures starting from 0 K
to 1300 K with step size of 100 K. Here, the total free energy at given
temperature and volume is expressed as:
$F(T;V)=[U_{el}(V)-U_{el}(V_{0})]+F_{ph}(T;V)$. Where
$U_{el}(V)-U_{el}(V_{0})$ is the relative ground state electronic energy
obtained from first-principle calculation, $V_{0}$ is the equilibrium volume
at 0 K. $F_{ph}(T;V)$ is the phonon Helmholtz free energy. Fig. 2(a) shows
that for every temperature there are energy minima (denoted by solid square)
corresponding to equilibrium volume of primitive cell which are connected by
solid line (red). These equilibrium volumes are plotted as a function of
temperature as presented in fig. 2(b). The primitive cell volume increases
monotonically with increase in temperature after 100 K. The equilibrium volume
at 0 K is calculated as $\sim$83.6 Å3, whereas it reaches $\sim$86.4 Å3 at
1300 K. The volume is increased by $\sim$3.3$\%$ in the temperature interval
$0-1300$ K.

Figure 2: (a) Variation of total free energy $F$ with primitive cell volume. (b) Change in primitive cell volume with temperature. (c) Linear thermal expansion coefficient $\alpha(T)$ as a function of temperature for CoSi. Table 1: Calculated linear thermal expansion coefficients ($\alpha(T)$) at different temperatures compared with reported experimental values. |  |  |  |  $\alpha(T)$ [$\times 10^{-6}$K-1] |  |
---|---|---|---|---|---|---
Reported | 100 K | 300 K | 600 K | 900 K | 1200 K | 1300 K
This work | 3.1 | 7.9 | 9.3 | 10.0 | 10.6 | 10.9
Mandrus et al.[40] | 3.8 | 9.8 | — | — | — | —
Krentsis et al.[41] | 3.9 | 10.4 | 11.5 | 12.1 | — | —
Ruan et al.[42] | — | — | 12.4 | 13.4 | 15.2 | 15.8

Using the primitive cell volume at different temperatures (as shown in fig.
2(b)), we have calculated the volumetric thermal expansion coefficient as
follows: $\beta(T)=\frac{1}{V(T)}\frac{\partial V(T)}{\partial T}$. CoSi has
simple cubic structure, hence considering uniform expansion in all the three
directions, $\alpha(T)$ can be taken as one third of $\beta(T)$[1]. The
calculated $\alpha(T)$ is plotted in fig. 2(c). The same method to calculate
$\alpha(T)$ is used by earlier reported work[38, 39]. Fig. 2(c) shows the
rapid increment of $\alpha(T)$ up to $\sim$300 K, then increases slowly till
1300 K. The average increment rate of $\alpha(T)$ in the temperature range
$0-300$ K is calculated as $\sim$0.03/K, whereas this value is observed as
$\sim$0.004/K in the temperature range $300-1300$ K. The values of $\alpha(T)$
at 300 K and 1300 K are found to be $\sim$7.9$\times$10-6 K-1 and
$\sim$10.9$\times$10-6 K-1, respectively. The calculated values of $\alpha(T)$
are compared with experimental reported data (dilatometer measurement) as
displayed in Table 1. Table shows that at low temperature calculated
$\alpha(T)$ gives good match with experiment, but as the temperature increases
calculated $\alpha(T)$ deviates slightly from experiment with lower values.
However, the values of $\alpha(T)$ calculated under QHA are found in good
agreement with the reported data.

The study of $\alpha(T)$ gives the change in length of TE materials when
subjected to a heating or cooling cycle. In many cases, the TE materials need
to face the enough mechanical stress during a large number of heating and
cooling cycles, which is named as thermal fatigue. In such cases, study of
thermal fatigue is much helpful before putting the TE materials in TEG for
long time application of TEG. The product of elastic modulus and $\alpha(T)$
is a useful quantity to study the thermal fatigue[43, 44]. The Calculated
alpha(T) can be used in designing TEG. Thermal expansion is also responsible
for the microcracking and porosity in TE materials, which affect the
performance of TE materials. For instance, Zhang et al.[45] studied the effect
of microcracking on TE properties of skutterudite specimen in the temperature
range $300-800$ K. They showed that presence of microcracks affect the Seebeck
coefficient in minor scale but they observed drastic change in electrical
conductivity. Subsequently, power factor decreases and gives low ZT as they
said[45]. Therefore, before using of any TE material for making TEG, study of
$\alpha(T)$ is much helpful for long time application of TEG.

The lattice part of thermal conductivity ($\kappa_{L}$) is calculated using
first-principle anharmonic lattice dynamics calculations implemented in
PHONO3PY code[31]. Anharmonic force constants are computed from first-
principle calculation. First, phonon lifetime is calculated from the imaginary
part of phonon self-energy by considering the phonon-phonon interaction only.
The third order force constants contain anharmonicity are employed for
calculating the imaginary part of self-energy. The lifetime of the phonon mode
$\lambda$ is given by[46, 47]

$\tau_{\lambda}=\frac{1}{2\Gamma_{\lambda}(\omega_{\lambda})},$ (1)

where, $2\Gamma_{\lambda}(\omega_{\lambda})$ corresponds to the phonon
linewidth of the phonon mode $\lambda$ and $\omega_{\lambda}$ is the frequency
of a phonon mode. This phonon lifetime is related with $\kappa_{L}$. The
$\kappa_{L}$ is computed by solving the linearized phonon Boltzmann equation
(LBTE) under single-mode relaxation time (SMRT) method and given by[47, 48]

$\kappa_{L}=\frac{1}{NV_{0}}\sum_{\lambda}C_{\lambda}\textbf{v}_{\lambda}\otimes\textbf{v}_{\lambda}\tau_{\lambda}^{SMRT},$
(2)

where, N is the number of unit cells in the crystal, $V_{0}$ is volume of a
unit cell, $\textbf{v}_{\lambda}$ and $\tau_{\lambda}^{SMRT}$ are the group
velocity and single-mode relaxation time of phonon mode $\lambda$.
$C_{\lambda}$ is the mode dependent heat capacity. For calculating
$\kappa_{L}$, the single-mode relaxation time $\tau_{\lambda}^{SMRT}$ is
considered as the phonon lifetime $\tau_{\lambda}$, where $\tau_{\lambda}$ is
defined in eq. 1.

Figure 3: Lattice thermal conductivity as a function of temperature.

The calculated values of $\kappa_{L}$ for CoSi is shown in fig. 3 in the
temperature range $300-800$ K. The calculated $\kappa_{L}$ is compared with
the experiment in the same figure. Experimental $\kappa_{L}$ is taken from the
work of Sk et al.[49] by subtracting the experimental $\kappa_{e}$ (estimated
from Wiedeman-Franz law) from the total experimental $\kappa$. The
$\kappa_{L}$ at 300 K is calculated as $\sim$8.0 W/m-K which is closely
matching with the experimental value of $\sim$8.3 W/m-K at the same
temperature. As the temperature increases, the calculated values deviate from
the experimental value. At 800 K the calculated and experimental values are
found to be $\sim$2.9 W/m-K and $\sim$7.2 W/m-K, respectively. This deviation
may be due to the various factors. For instance, Lorenz number, L is taken as
constant for estimating $\kappa_{e}$, but in practical L is temperature
dependent quantity. For calculating temperature dependence of $\kappa_{L}$,
ground state phonon mode was used. But, thermal expansion of fig. 2(b) shows
that volume of primitive unitcell of CoSi is changed with temperature,
resulting a change in lattice parameter. Hence, phonon dispersion are expected
to be changed once we use temperature dependent lattice parameters. This
temperature dependent phonon dispersion may improve the $\kappa_{L}$ value.
Therefore, considering all these factors into account, one can expect the
better matching between calculated and experimental data at high temperature
region, which is beyond the scope of our present study.

In order to understand the temperature dependent behaviour of $\kappa_{L}$, we
have calculated the phonon lifetime. The lifetime of phonon in a solid is
decided by the various scattering mechanisms, e.g. phonon-phonon interaction
(PPI), electron-phonon interaction, phonon-defect interaction etc. Here, we
consider only PPI to calculate the phonon lifetime. The lifetime of each
phonon mode $\lambda$ is calculated using the imaginary part of phonon self-
energy as eq. 1. The imaginary part of the self-energy
$\Gamma_{\lambda}(\omega_{\lambda})$ is calculated within many-body
perturbation theory as[47]

$\displaystyle\Gamma_{\lambda}(\omega)=\frac{18\pi}{\hbar^{2}}\sum_{\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}|\Phi_{-\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}|^{2}\\{(n_{\lambda^{{}^{\prime}}}+n_{\lambda^{{}^{\prime\prime}}}+1)$
$\displaystyle\times\delta(\omega-\omega_{\lambda^{{}^{\prime}}}-\omega_{\lambda^{{}^{\prime\prime}}})+(n_{\lambda^{{}^{\prime}}}-n_{\lambda^{{}^{\prime\prime}}})$
$\displaystyle\times[\delta(\omega+\omega_{\lambda^{{}^{\prime}}}-\omega_{\lambda^{{}^{\prime\prime}}})-\delta(\omega-\omega_{\lambda^{{}^{\prime}}}-\omega_{\lambda^{{}^{\prime\prime}}})]\\},$
(3)

where, $\Phi_{-\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$
signifies the strength of interaction among three phonons $\lambda$,
$\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$ involved in the
scattering. $n_{\lambda}$ is the phonon occupation number at the equilibrium.

Fig. 4(a) shows the lifetime of twenty four branches (marked as b1 to b24) in
the temperature range $300-800$ K as calculated by taking the weight average
over the q-points in the Brilloun zone. The same method to calculate phonon
lifetime is employed earlier[50]. From fig. 4(a) it is seen that optical
branch b8 has the highest lifetime in the full temperature range signifies the
lesser scattering as compared to other phonon branches. Two optical branches
b23 and b24 (having almost same lifetime) show the lowest lifetime implying
the highest scattering. Finally, the acoustic, optical and total phonon
lifetimes are calculated by averaging the respective number of phonon branches
as shown in fig. 4(b). Figure shows that the acoustic branches have the higher
lifetime than optical branches. This suggests that the contribution of
lifetime in $\kappa_{L}$ from acoustic branches are higher than the optical
branches. The total lifetime of phonons due to PPI is calculated by averaging
all the branches. The total phonon lifetime is calculated as $\sim 2.40\times
10^{-12}$ s at 300 K, then as the temperature increases this value decreases
up to $\sim 0.85\times 10^{-12}$ s at 800 K. This decrement nature of phonon
lifetime with temperature is accordance with the temperature dependent trend
of $\kappa_{L}$ (see fig. 3). At this point it is important to note that for
calculating $\kappa_{L}$ (see eq. 2), group velocity is taken as temperature
independent quantity. In the work of Sk et al.[36] specific heat of CoSi has
been shown to increase with increasing temperature and almost constant at high
temperature (above $\sim$500 K). Therefore, the decrement nature of
$\kappa_{L}$ is mainly determined by the temperature dependent trend of phonon
lifetime. The phonon lifetime can be reduced (in order to maximize ZT) by
introducing the extra scattering centres via nanostructuring, alloying etc[24,
51]. In this way $\kappa_{L}$ can be minimized to get high ZT of CoSi.

Figure 4: (a) The phonon lifetime for twenty four phonon branches. (b) Total,
acoustic, optical phonon lifetime due to PPI as a function of temperature.

##  4 CONCLUSIONS

In summary, we have studied the first-principles DFT based phonon calculations
to understand the phonon properties of CoSi at high temperature region. The
finite displacement method has been employed to compute the phonon dispersion.
Then the ground state phonon dispersion is used to address the experimental
$\alpha(T)$) and $\kappa_{L}$. The calculated QHA based thermal expansion
gives quite good match with the existing experimental results. The
$\kappa_{L}$ is calculated using anharmonic force constant under many-body
perturbation theory. The calculated value of $\kappa_{L}$ ($\sim$8.0 W/m-K) is
in good agreement with the experimental value ($\sim$8.3 W/m-K) at room
temperature. The temperature dependent behaviour of $\kappa_{L}$ is seen to
decide by the trend of phonon lifetime. This study suggests that the phonon
band-structure obtained from DFT based method addresses the experimental
$\alpha(T)$) and $\kappa_{L}$ in reasonably good manner. However, for better
quantification of the studied phonon properties specially at high temperature,
one may require to go beyond the DFT based methods.

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