# TRACK: A python code for calculating the transport properties of correlated
electron systems using Kubo formalism

Antik Sihi1<EMAIL_ADDRESS>Sudhir K. Pandey2<EMAIL_ADDRESS>1School of Basic Sciences, Indian Institute of Technology Mandi, Kamand -
175075, India
2School of Engineering, Indian Institute of Technology Mandi, Kamand - 175075,
India

###### Abstract

Exploring the transport properties of different materials brings new avenue
for basic understanding of emergent phenomena and practical applications in
many different fields. Here, we report a program named as TRACK (TRAnsport
properties for Correlated materials using Kubo formalism) which is written in
Python 3 for calculating temperature dependent electrical conductivity,
electronic part of thermal conductivity, Seebeck coefficient and Lorenz
number. In this code, Kubo linear-response formalism is utilized for computing
these parameters using both interacting and non-interacting electronic
structure methods. The formula for transport coefficients is accordingly
modified to obtain the transport parameters under relaxation time
approximation using band-theory. The basic inputs of this program are the
structural information, dense k-points sampling in the irreducible part of the
Brillouin zone and the information of velocity matrix elements, which can be
calculated using third-party ab-initio package. TRACK is expected to calculate
the transport properties of different class of materials. The code has been
benchmarked by performing calculation on three different types of materials
namely Vanadium (V), FeSi and LaCoO3, which are metal, semiconductor and Mott
insulator, respectively. The temperature dependent behaviour of the transport
coefficients for these materials show fairly good agreement with the
corresponding experimental data.

##  I Introduction

The world of materials is broadly divided into metal, semiconductor and
insulator. Experimental study of different transport properties is one of the
procedures to verify the properties of these classes of materials. In addition
to this, when experiments of transport properties at low temperatures are
performed, then many exotic phenomena like Kondo effect stewart_nfl ;
singh_nfl , non-Fermi Liquid behaviour fisk_kondo , superconductivity
bednorz_sc etc. reveal their existence. Temperature dependent experimental
observation is necessary for understanding above mentioned properties. With
the change in temperature the bands of quasiparticles (e.g. electron, phonon,
defects/impurity etc.) may alter their behaviour. The small changes in bands’
behaviour will directly reflect on the transport properties of material. It
happens because of the direct dependence of transport coefficients on band
features. Therefore, proper estimation of temperature dependent band-structure
of any material is important for theoretical calculation of transport
properties. Along with the temperature effect, another bottleneck in
theoretical prediction is the presence of different scattering sources in
material. The intrinsic scattering mechanisms are electron-electron
interaction (EEI), electron-phonon interaction (EPI) and phonon-phonon
interaction (PPI). Further, electron-defects interaction (EDI) is also
prominent as an extrinsic scattering source in the experimentally synthesized
materials aschroft . The EEI plays an important role in estimating the
electronic structure of strongly and moderately correlated materials. Thus,
investigation of transport properties will help us to understand the new
emergent phenomena as well as the scattering mechanism involved in the
material.

Nowadays, exploring the transport properties of materials is an important
research area for basic understanding and application purposes. Moreover, when
the correlation effect and transport are merged then it opens a new area to
understand the exotic phenomena as well as the materials through transport
studies. For example, the Kondo metals show an upturn in resistivity ($\rho$)
at low temperature, for non-Fermi liquid system $\rho$ changes with
$T^{\alpha}$ ($\alpha\neq 2$) and also many strongly correlated materials show
anomalous behaviour in transport properties fisk_kondo ; imada_rev . In
present days, the applications of these materials have been explored in the
field of thermoelectric (TE), optoelectronic etc coulter_opto ; jtomczakRev .
The correlated thermoelectric materials (CTM) provide new avenue to overcome
the limitations of transport properties of the existing uncorrelated
semiconductors jtomczakRev . In order to understand the transport behaviour,
usually one needs to calculate electrical conductivity ($\sigma$), Seebeck
coefficient ($S$) and thermal conductivity ($\kappa$). The $\kappa$ contains
both electronic ($\kappa_{e}$) and phononic ($\kappa_{ph}$) contributions. To
compute all these quantities except $\kappa_{ph}$, any ab-initio electronic
structure methodology is needed for obtaining the eigenstates of
quasiparticle.

In light of this, the density functional theory (DFT) is one of the widely
used methodology, where the non-interacting single particle Kohn-Sham orbitals
are considered as the eigenstates of the material payne_dft . The semi-
classical Boltzmann transport equations (BTE) is usually chosen to calculate
the transport parameters by employing Kohn-Sham eigenstates. BoltzTraP
boltztrap and BOLTZWANN boltzwann are most commonly used codes, which follow
the above mentioned theory. Here, the effect of temperature is considered
through the Fermi-Dirac distribution function. But in this scenario, the
ground state electronic structure is used for the finite temperature
calculation, which may not be desirable for the finite temperature transport
properties calculation. In addition to this problem, the correlation effect
within the partially filled $d$/$f$ electrons is not sufficiently considered
in DFT. Therefore, it fails to provide proper electronic structure of strongly
or moderately correlated electron systems. At this point, dynamical mean-field
theory (DMFT) comes to tackle the Coulomb interactions of these correlated
electrons using many-body interaction theory, where the other electrons are
taken care by DFT in combined DFT+DMFT method kotliar_dmft ; haule_dmft . In
such type of correlated materials, Kubo linear-response formalism (details of
the methodology discussed in section II A) is usually employed for estimating
the transport properties using the spectral function
($A{(\mathbf{k},\omega)}$) on real $\omega$ axis jtomczakRev . The
$A{(\mathbf{k},\omega)}$ can be obtained from any Green-function based
electronic structure method like DFT+DMFT and $GW$ methods. Moreover,
computing the transport coefficients using the self-energy in imaginary axis
is possible, but this procedure needs lot of computational efforts for
calculating the higher-order transport coefficients haule_dmft . As per our
knowledge, TRIQS/DFTTools is the only open-source code for correlated
materials, where the Kubo formalism using $A{(\mathbf{k},\omega)}$ is utilized
for calculating the different transport coefficients aichhorn_triqs . This
code is interfaced with WIEN2k for calculating the transport parameters wien2k
. In TRIQS/DFTTools code, the projected Wannier functions are implemented as
the basis set for the correlated orbitals, which has followed the work of
Aichhorn et al aichhorn_imple . However, Haule et al. mentioned that this type
of projection generally provides incorrect spectral weight but preserves the
causal DMFT equations haule_dmft . The eDMFTF code is developed by considering
the new projection operator, which defines the connections between real-space
objects and the corresponding orbital’s counterpart. This new projection
operator satisfies the causality of DMFT equations and leads to correct
spectral weight. Thus, the eDMFTF code is the better choice to perform the
electronic structure calculation for correlated materials haule_dmft . In this
case, the continuous-time quantum Monte Carlo (CTQMC) method is implemented as
the impurity solver haule_ctqmc . But to the best of our knowledge, this open-
source code, which is interfaced with WIEN2k wien2k , has no facility to
compute the transport properties using Kubo formalism. Therefore, easily
accessible open-source code is necessary for calculating the transport
coefficient of correlated materials using the $A{(\mathbf{k},\omega)}$, which
is obtained from eDMFTF code. The Kubo formalism is also possible to be used
for any $GW$ based program, where $A{(\mathbf{k},\omega)}$ is computed.

Tomczak $et$ $al.$ showed that the Kubo formalism along with the consideration
of vanishing scattering rate has capability to calculate the transport
properties using the eigenstates of band-theory jtomczakRev . In case of
BoltzTraP boltztrap and BOLTZWANN boltzwann codes, different transport
properties are calculated based on the semi-classical BTE using constant
relaxation time approximation (CRTA). However, in general, the relaxation time
($\tau$) is function of direction (r) and energy ($\omega$). But, for
homogeneous system, direction independent $\tau$ is usually considered
aschroft . Therefore, implementation of $\omega$ dependent $\tau$ is desirable
for calculating the transport properties when eigenstates of DFT are used. In
BoltzTraP2 code, the relaxation time approximation (RTA) along with CRTA is
implemented for finding the electron-phonon scattering rate where $\tau$ is a
function of density of states BoltzTraP2 . Moreover, the choice of basis
function always plays important role in calculating the physical properties.
The Kohn-Sham orbitals are required in BoltzTraP boltztrap and BoltzTraP2
BoltzTraP2 , where the calculation to get Kohn-Sham orbitals for highly dense
k-mesh is time consuming. To reduce the computational time, the maximally-
localized Wannier function is utilized in BOLTZWANN boltzwann for making the
model Hamiltonian in low-energy region. However, proper choice of Wannier
functions for the Kohn-Sham orbitals is a tedious task and the all electron
eigenstates are always preferable than the eigenstates obtained from model
Hamiltonian for calculating any physical properties. Using the Kubo formalism,
the implementation of $\omega$ dependent $\tau$ is possible using Kohn-Sham
orbitals, which brings to explore more insight of any material from the
transport properties simply at level of DFT. As per our present knowledge, no
open-source code is available, where the above mentioned technique is used to
study the transport coefficients.

Here, a Python 3 code named as TRACK (formally called as TRAnsport properties
for Correlated materials using Kubo formalism), which is interfaced with
WIEN2k wien2k and eDMFTF haule_dmft codes, is reported to study $S$,
$\sigma/\tau$, $\kappa_{e}/\tau$ and Lorenz number ($L$), respectively. Kubo
linear-response formalism is used for exploring the basic insights of
transport parameters with considering non-interacting ($e.g.$ DFT) and
interacting ($e.g.$ DFT+DMFT) electronic structure methods. In order to
benchmark the code, we choose different type of non-magnetic correlated
materials such as Vanadium (V), FeSi and LaCoO3, which are prototype of well
known correlated metal, semiconductor and insulator, respectively. The
calculated values from our code are also compared with the results obtained
from BoltzTraP boltztrap calculations for all these materials.

(a)

(b)

(c)

Figure 1: Screenshot of input files when using the eigenvalues of (a) many-
body theory and (b) & (c) band-theory. Figure 2: The workflow of the TRACK
code. Table 1: The details of different input parameters.

Name |  | Unit |  | Meaning
---|---|---|---|---
case |  | - |  | Case file name
mu_value |  | eV |  | Zero point of the energy
cal_direction |  | - |  |  Direction: 0$\rightarrow$ $xx$; 1$\rightarrow$ $yy$; 2$\rightarrow$ $zz$
cal_type |  | - |  |  Type of calculation: 1$\rightarrow$ DFT+DMFT; 2$\rightarrow$DFT
beta_T |  | eV |  | Inverse of the temperature
mag_order |  | - |  |  1$\rightarrow$ non-magnetic; 2$\rightarrow$ magnetic
spin |  | - |  |  0$\rightarrow$ non-magnetic; 1$\rightarrow$ spin up; 2$\rightarrow$ spin down
omega_min |  | eV |  | Minimum energy value
omega_max |  | eV |  | Maximum energy value
omega_iter |  | - |  | Division factor for step size of energy value
tau_type |  | - |  |  1$\rightarrow$constant relaxation time; 2$\rightarrow$ relaxation time
tau |  | s |  |  constant value of $\tau$
omega_pwr |  | - |  |  Power factor for $\omega$ dependent function
|  |  |  |  related with $Im\,\Sigma(\omega)$
omega_cnst |  | - |  |  Proportionality constant for $Im\,\Sigma(\omega)$
|  |  |  |  and $\omega$ dependent function
dft_im_selfen |  | eV |  | Imaginary part of self-energy
start_temp |  | K |  | Value of the starting temperature
end_temp |  | K |  | Value of the end temperature
temp_step |  | - |  | Step size for the temperature difference
trans_cal |  | - |  |  F$\rightarrow$ Fresh start
|  |  |  |  R$\rightarrow$ Restart after formatting the energy file

##  II Theoretical background

###  II.1 Using many-body electronic structure theory

Here, the Kubo linear-response formalism is utilized for computing different
transport properties without considering the vertex correction tomczak ;
oudovenko . Under this formalism without vertex correction, $\sigma$, $S$ and
$\kappa_{e}$ along $\alpha\alpha^{\prime}$ direction are defined as,

$\sigma^{\alpha\alpha^{\prime}}=\beta e^{2}K_{0}^{\alpha\alpha^{\prime}}$ (1)

$S^{\alpha\alpha^{\prime}}=-\frac{k_{B}}{|e|}\frac{K_{1}^{\alpha\alpha^{\prime}}}{K_{0}^{\alpha\alpha^{\prime}}}$
(2)

$\kappa^{\alpha\alpha^{\prime}}_{e}=k_{B}\Bigg{(}K_{2}^{\alpha\alpha^{\prime}}-\frac{(K_{1}^{\alpha\alpha^{\prime}})^{2}}{K_{0}^{\alpha\alpha^{\prime}}}\Bigg{)}$
(3)

where, $\alpha(=x,y,z)$ is the direction, $\beta$ denotes the inverse of
temperature (T), $e$ represents the electronic charge and
$K_{0}^{\alpha\alpha^{\prime}}$ ($K_{1}^{\alpha\alpha^{\prime}}$) shows
current-current (current-heat) correlation function. In general, the
correlation function is called as kinetic coefficient which is given by,

$K_{n}^{\alpha\alpha^{\prime}}=N_{sp}\pi\hbar\int
d\omega(\beta\omega)^{n}f(\omega)f(-\omega)\mathit{\Gamma}^{\alpha\alpha^{\prime}}(\omega,\omega)$
(4)

where, $N_{sp}$ and $f(\omega)$ are the spin factor and Fermi function,
$\hbar$ denotes the reduced Planck’s constant. Along with these,
$\mathit{\Gamma}^{\alpha\alpha^{\prime}}$ represents the transport
distribution. In matrix notation after neglecting the vertex correction, this
quantity is expressed by,

$\mathit{\Gamma}^{\alpha\alpha^{\prime}}(\omega_{1},\omega_{2})=\frac{1}{V}{\sum\limits_{k}}Tr\bigg{(}v^{\alpha}(\mathbf{k})A(\mathbf{k},\omega_{1})v^{\alpha^{\prime}}(\mathbf{k})A(\mathbf{k},\omega_{2})\bigg{)}$
(5)

where $V$ is the unit-cell volume, $v^{\alpha}(\mathbf{k})$ is the group
velocity and $A(\mathbf{k},\omega)$ represents the spectral function. Here,
the momentum matrix method is used for finding the $v^{\alpha}(\mathbf{k})$,
which is given by optic ,

$\mathbf{\upsilon}_{n}^{\alpha}(\textbf{k})=\frac{1}{m_{e}}\langle\varPsi_{n\textbf{k}}|\hat{\textbf{p}}^{\alpha}|\varPsi_{n\textbf{k}}\rangle$
(6)

where, $m_{e}$ and $\hat{\textbf{p}}^{\alpha}$ are mass of free electron and
momentum operator along $\alpha$ direction, respectively.
$\varPsi_{n\textbf{k}}$ denotes the Kohn-Sham wave function, where n (k) is
band index (crystal momentum in reciprocal space). This method is chosen
because it avoids the well known “band-crossing problem” as mentioned in the
earlier literatures scheidemantel ; transopt . The similar implementation is
carried out in WIEN2k optic module for computing this quantity optic . In
addition to this, the spectral function in real $\omega$ is expressed by
imada_rev ,

$A{(\mathbf{k},\omega)}=-{\Huge\frac{1}{\pi}\frac{Im\,\Sigma(\omega)}{[\omega-\varepsilon^{0}_{\mathbf{k}}-Re\,\Sigma(\omega)]^{2}+[Im\,\Sigma(\omega)]^{2}}}$
(7)

where, $\varepsilon^{0}_{\mathbf{k}}$ is the energy of non-interacting single
electronic state with crystal momentum k, $Im\,\Sigma(\omega)$
($Re\,\Sigma(\omega)$) represents the imaginary (real) part of self-energy
($\Sigma(\omega)$) obtained from DFT+DMFT calculation. Furthermore, the Lorenz
number ($L$) is defined as,

$L=\frac{\kappa^{\alpha\alpha^{\prime}}_{e}}{T\sigma^{\alpha\alpha^{\prime}}}$
(8)

Table 2: Calculated values of Seebeck coefficients ($S$) for Vanadium (V) at
different temperatures by using TRACKDMFT, TRACKDFT and BoltzTraP along with
experimental data. (Here, $S$ is in $\mu V/K$).

Temperature |  | Experimental |  |  |  | Calculated |  |
---|---|---|---|---|---|---|---|---
(K) |  |  |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
200 |  |  0.41jung , 0.27okram , 0.86mackintosh , 0.60mackintosh |  | 6.28 |  | 2.50 |  | 1.68
250 |  |  -0.11jung , -0.39okram , 0.6mackintosh , 0.39mackintosh |  | 7.62 |  | 4.34 |  | 2.10
288 |  |  -0.36jung , -0.89okram ,0.83mackintosh ,0.55mackintosh |  | - |  | - |  | -
300 |  |  -1.06okram , 1.0mackintosh |  | 5.95 |  | 5.86 |  | 2.49
350 |  | - |  | 2.65 |  | 7.13 |  | 2.83
400 |  | - |  | 1.84 |  | 8.23 |  | 3.14
450 |  | - |  | -3.59 |  | 9.22 |  | 3.38

###  II.2 Using band-theory

In order to use the eigenvalues obtained from band-theory’s calculation, the
transport distribution ($i.e.$ equation 5) of Kubo formalism needs to be
modified accordingly. The $v^{\alpha}(\mathbf{k})$ is already obtained by
using single-particle Kohn-Sham orbitals. The two $A{(\mathbf{k},\omega)}$ are
seen in the equation of transport distribution, where the form of
$A{(\mathbf{k},\omega)}$ is usually known as Lorentzian like function of
$\omega$ for particular $\varepsilon^{0}_{\mathbf{k}}$. For non-interacting
theory, the value of $Re\,\Sigma(\omega)$ is equal to zero. Therefore, to
transform one of the $A{(\mathbf{k},\omega)}$ from Lorentzian type to Dirac
delta function of $\delta(\omega-\varepsilon^{0}_{\mathbf{k}})$, typically
very small value of $Im\,\Sigma(\omega)$ must be considered. This step is
needed because the value of $Im\,\Sigma(\omega)$ represents the half width at
the half maxima of the Lorentzian function and taking small value of this
quantity will transform it into Dirac delta function. Another
$A{(\mathbf{k},\omega)}$ is replaced by
$\big{(}-\frac{1}{Im\,\Sigma(\omega)}\big{)}$. Further, the
$Im\,\Sigma(\omega)$ is function of $\omega$, which denotes the strength of
EEI within material. In general, this quantity is equal to
$\big{(}-\frac{\tau(\omega)}{\hbar}\big{)}$. Therefore, if all the above
mentioned changes are implemented in equations (1-7), then the $\sigma$ for a
particular band will retrieve the form of $\sigma$ as obtained from
semiclassical theory aschroft_eq . Thus, with these approximations, the
transport coefficients within Kubo formalism can describe transport parameters
which are obtained from the semi-classical BTE with RTA by utilizing the
single-particle eigenvalues.

Further, the convergence of $S$ needs to be checked with varying the value of
$Im\,\Sigma(\omega)$ to see whether proper
$\delta(\omega-\varepsilon^{0}_{\mathbf{k}})$ is obtained or not. The present
implementation provides the opportunity to study the importance of $\omega$
dependent $\tau$ for calculating the transport properties. However till now,
BoltzTraP and BOLTZWANN don’t provide the results with considering
$\tau(\omega)$. These two codes are most commonly used for studying transport
properties of materials. All these procedures are also applicable for
computing the transport coefficients from the estimated eigenvalues of the
static mean-field DFT+$U$ method, where $U$ is called as Hubbard parameter.

##  III Workflow and Technical details

###  III.1 Workflow

In order to compute the transport properties using this code, all the input
variables need to be provided in single file, which is named as $track.input$.
The details of different input parameters are described in Table 1 and the
screenshot of $track.input$ files are shown in Figs. 1(a) and 1(b). Along with
this, additional files named as $case.struct$, $case.klist$ and $case.pmat$
are necessary to be kept in the same directory of input file for getting the
structural information, a list of k-points in the brillouin zone and the
information of velocity matrix elements between all bands in the desired
energy window for each k-point, respectively. The $case.struct$ and
$case.klist$ are two basic files for any WIEN2k calculation. The detailed step
to find the velocity matrix elements are explained in the userguide of WIEN2k
wien2k . In addition to these, the $eigvals.dat$ file is needed for the
transport properties calculations using the eigenvalues of DFT+DMFT. The file
contains the information of momentum resolved spectral functions as calculated
by eDMFTF code. In similar manner, for performing the transport calculations
using the eigenvalues of DFT, $case.energy$ file is needed from the self-
consistent WIEN2k calculation along with $track.input$, $case.struct$,
$case.klist$ and $case.pmat$ files. In case of DFT+$U$ within WIEN2k,
$case.energyup$ or (and) $case.energydn$ file is (files are) needed for
finding the transport parameters for non-magnetic (magnetic) material. The
workflow of the code can be more clearly seen in the flowchart shown in Fig.
2.

###  III.2 Technical details

TRACK is implemented in modern Python 3 language, which is incompatible with
Python 2. The Numpy and Scipy libraries are extensively utilized for
performing the several numerical operations numpy ; scipy . Along with these,
the Time module is used for estimating computation’s time. In the current
version of the code, it is only interfaced with the WIEN2k and eDMFTF codes.

Two different modes of calculations are implemented in TRACK code ($i.e.$
TRACKDMFT and TRACKDFT), which depends on the choice of eigenvalues from the
electronic structure methods. In such scenario, when one uses the eigenvalues
of eDMFTF (WIEN2k) code then $track\\_DMFT.py$ ($track\\_DFT.py$) code needs
to be run by the command - $python3$ $track\\_DMFT.py$ ($track\\_DFT.py$).
Dense k-mesh is strongly advised for obtaining the eigenvalues from the
electronic structure calculation with checking the convergence of transport
coefficients. Proper sampling of $\omega$ around the $E_{F}$ is needed for
transport calculation, because transport behaviours of any material highly
depend on low-energy window. It is important to note that when one uses the
$track\\_DFT.py$ code then the difference between two successive $\omega$ must
be equal to $\sim\frac{1}{10}$ times of $Im\,\Sigma(\omega)$. The easy
implementation makes this code more user friendly to compute different
transport properties.

##  IV Test cases

Here, we describe the capabilities of the TRACK code by performing
calculations on three different types of correlated materials: (i) Vanadium
(V), a well known correlated metal; (ii) FeSi, a famous semiconductor showing
many exotic phenomena; (iii) LaCoO3, a well-studied correlated system along
with metal-insulator transition and good TE properties.

It is known that the values of $\sigma$ and $\kappa_{e}$ vary based on the
estimation of $\tau$. Here, in case of DFT+DMFT methodology, only EEI is
considered for getting temperature dependent electronic structure properties
of any material. Therefore, in present scenario, the other sources of
electronic interactions are not considered for finding the value of
temperature dependent $\tau$, which may lead to the bad prediction of these
two quantities from the TRACKDMFT code. Thus, we calculate temperature
dependent $L$ parameter using this code, which is a independent quantity of
$\tau$ value like $S$.

Table 3: Calculated values of Seebeck coefficients ($S$) for FeSi at different
temperatures by using TRACKDMFT, TRACKDFT and BoltzTraP along with
experimental data. (Here, $S$ is in $\mu V/K$).

Temperature |  | Experimental |  |  |  | Calculated |  |
---|---|---|---|---|---|---|---|---
(K) |  |  |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
100 |  |  25.0buschinger_fesi , 41.3yang_fesi , 23.6wolfe_fesi , 20.2tomczak_fesi |  | - |  | 13.5 |  | -31.2
110 |  |  -1.8buschinger_fesi , -2.6yang_fesi , 40.0sales_fesi , 8.8wolfe_fesi , 11.8tomczak_fesi |  | 11.7 |  | - |  | -
180 |  |  -12.4buschinger_fesi , -17.1yang_fesi , -13.3sales_fesi , -8.1wolfe_fesi , -6.6tomczak_fesi |  | 3.6 |  | - |  | -
200 |  |  -10.0buschinger_fesi , -12.6yang_fesi , -10.4sales_fesi , -8.8wolfe_fesi , -7.0tomczak_fesi |  | - |  | 2.6 |  | -49.1
290 |  |  2.1buschinger_fesi , 0.6yang_fesi , 3.4wolfe_fesi , -6.2tomczak_fesi |  | -33.5 |  | - |  | -
300 |  |  0.1yang_fesi , -2.1sales_fesi , 3.5wolfe_fesi , -6.5tomczak_fesi |  | -23.2 |  | -5.6 |  | -35.0
380 |  |  2.9sales_fesi , -13.1tomczak_fesi |  | 10.9 |  | - |  | -
400 |  |  3.6sales_fesi |  | - |  | -8.6 |  | -24.4
600 |  |  11.3sales_fesi , 0.5tomczak_fesi |  | 11.3 |  | -9.2 |  | -12.7
700 |  |  14.7sales_fesi , 2.4tomczak_fesi |  | 6.0 |  | -8.3 |  | -8.8
800 |  |  4.96tomczak_fesi |  | 18.5 |  | -6.9 |  | -5.2
1000 |  |  12.5tomczak_fesi |  | 30.3 |  | -3.8 |  | 2.0

Table 4: Calculated values of Seebeck coefficients ($S$) for LaCoO3 at
different temperatures by using TRACKDMFT, TRACKDFT and BoltzTraP along with
experimental data. (Here, $S$ is in $\mu V/K$).

Temperature |  | Experimental |  |  |  | Calculated |  |
---|---|---|---|---|---|---|---|---
(K) |  |  |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
150 |  |  207.9cortes_lco |  | - |  | 15.78 |  | -413.25
180 |  |  431.9cortes_lco |  | -165.6 |  | - |  | -
200 |  |  558.3cortes_lco |  | - |  | 24.65 |  | -644.41
250 |  |  623.3cortes_lco |  | - |  | 58.53 |  | -579.46
290 |  |  578.5cortes_lco |  | 93.5 |  | - |  | -
300 |  |  -351.1he_lco , -323.7ohtani_lco , 249.3singh_mes_lco , 524.8cortes_lco , 615.6sk_high_lco |  | 138.0 |  | 159.8 |  | -529.15
350 |  |  -230fu_lco , -219.8he_lco , -69.5ohtani_lco , 287.1singh_mes_lco , 474.1sk_high_lco |  | - |  | 309.6 |  | -482.0
400 |  |  -121.5fu_lco , -15.96he_lco , 111.1ohtani_lco , 218.2singh_mes_lco , 275.3sk_high_lco |  | 241.14 |  | 418.45 |  | -438.35
450 |  |  -3.6fu_lco , 122.7he_lco , 156.8ohtani_lco , 121.5singh_mes_lco , 147.5sk_high_lco |  | - |  | 459.04 |  | -398.20
500 |  |  54.9fu_lco , 177.3he_lco , 70.5singh_mes_lco , 83.9sk_high_lco |  | 294.6 |  | 457.81 |  | -361.52
550 |  |  60.5fu_lco , 159.5he_lco , 49.7singh_mes_lco , 54.7sk_high_lco |  | - |  | 438.64 |  | -328.31
600 |  |  52.9fu_lco , 132.6he_lco , 34.7singh_mes_lco , 41.7sk_high_lco |  | 258.15 |  | 413.53 |  | -298.49
650 |  |  45.2fu_lco , 33.8sk_high_lco |  | - |  | 387.60 |  | -271.93
700 |  |  26.2sk_high_lco |  | 131.62 |  | 362.87 |  | -248.40
750 |  |  20.2sk_high_lco |  | - |  | 340.03 |  | -227.65
800 |  |  14.9sk_high_lco |  | 121.98 |  | 319.24 |  | -209.42

Figure 3: Temperature dependent Seebeck coefficient ($S$) of (a) Vanadium (V)
(b) FeSi and (c) LaCoO3 with different values of $Im\,\Sigma(\omega)$ for
checking the convergence. Table 5: Calculated values of temperature dependent
Lorenz numbers ($L$) by using TRACKDMFT, TRACKDFT and BoltzTraP for Vanadium.
(Here, $L$ is in $\times$10-8 $V^{2}/K^{2}$ )

Temperature |  |  |  | Calculated |  |
---|---|---|---|---|---|---
(K) |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
200 |  | 2.4 |  | 2.472 |  | 2.43
250 |  | 2.3 |  | 2.466 |  | 2.43
300 |  | 2.2 |  | 2.456 |  | 2.42
350 |  | 2.15 |  | 2.448 |  | 2.40
400 |  | 2.1 |  | 2.438 |  | 2.39
450 |  | 2.0 |  | 2.425 |  | 2.37

###  IV.1 Computational Details

The detailed structural informations of V, FeSi and LaCoO3 are taken from
Refs. 26, 27 and 28, respectively for carrying out the present calculations.
92$\times$92$\times$92, 74$\times$74$\times$74 and 53$\times$53$\times$53
k-mesh sizes are used for calculating the transport properties of V, FeSi and
LaCoO3, respectively. For DFT+DMFT calculation, the values of $U$=3.4
($J$=0.52) eV, $U$=4.4 ($J$=0.89) eV and $U$=6.9 ($J$=0.3) eV are set for V
3$d$, Fe 3$d$ and Co 3$d$ electrons of V, FeSi and LaCoO3, respectively. It is
noted that the semiconducting ground state of FeSi is found from the DFT
calculation, whereas DFT+DMFT provides the semi-metallic behaviour from the
temperature dependent studydutta_fesi . Here, $E_{F}$ is fixed at middle of
the bandgap for all calculations where bandgap is clearly observed from total
density of states.

In case of LaCoO3, the presence of octahedral rotation in the compound brings
important task to decrease the strength of crystal-field splittingchakra_prm .
Therefore, in order to overcome this with observing the metal-insulator
transition, the value of $J$=0.3 eV is chosen for DFT+DMFT. Moreover, it is
seen that the experimental data of transport properties are nicely described
for this value of $J$ with $U$=6.9 eV, which is the another reason to take
these values for the present study. To perform DFT+$U$ calculation for LaCoO3,
we followed the work of Singh et al.singh_lco_phlio . The eigenvalues of this
calculation are used for TRACKDFT and BoltzTraP codes to estimate the
transport parameters.

In order to compare with the computed values from the BoltzTraP, the constant
value of $\tau$=$10^{-14}$ s is considered for computing the transport
properties, when eigenvalues of DFT or DFT+$U$ are used. The convergence of
transport properties with varying the value of $Im\,\Sigma(\omega)$ is checked
and also advised for the users. All test cases are performed by considering
the direction $\alpha=x$ and $\alpha^{\prime}=x$. Here, all DFT+DMFT
calculations are performed using density-density type of Coulomb interaction
within eDMFTF code. To get $A{(\mathbf{k},\omega)}$ in real axis, maximum
entropy method is used for analytical continuation jarrell_maxent .

###  IV.2 Seebeck Coefficient ($S$)

The computed values of temperature dependent $S$ for V are tabulated in Table.
II along with the experimental data for comparison. The experimental values of
$S$ show decreasing behaviour with increasing temperature for high purity
sample, where the sign of this physical property changes at $\sim$250 K jung ;
okram . However, for polycrystalline sample, the values of this quantity is
first decreasing till $\sim$250 K and then increasing with rise in temperature
mackintosh ; vedernikov . Moreover, the computed values of $S$ for different
temperatures using TRACKDMFT decreases with rise in temperature as similar to
experimental data, where the change in the sign of this quantity is found at
$\sim$450 K. The estimated value of $S$ is $\sim$6.3 ($\sim$-3.6) $\mu$V/K at
$\sim$200 ($\sim$450) K, where the experimental values are $\sim$0.5
($\sim$-0.36) $\mu$V/K at $\sim$200 ($\sim$289) K.

For FeSi, Table III represents the calculated values of temperature dependent
$S$ with corresponding experimentally reported data. From the table, the
different experimental and theoretical results are showing the change of sign
of $S$ value in between 110-180 K (from positive to negative) and 290-380 K
(from negative to positive), respectively. This behaviour is nicely captured
by using the TRACKDMFT as evident from the table. The computed values of $S$
at 300 K using this code is found to be $\sim$-23.2 $\mu$V/K, where the
experimental range of $S$ at this temperature is obtained from $\sim$-2.1 to
$\sim$3.5 $\mu$V/K. Therefore, the overall trend of computed value using
TRACKDMFT is in good agreement with the experimental data for the studied
temperature range.

Numerous experimental works on the transport properties of LaCoO3 have been
reported. The huge range of $S$ value reveals from the different experimental
works for 300-600 K. It happens for employing different sintering temperature
to make the sample. The calculated values of $S$ as function of temperature
along with experimental data are provided in Table IV. In case of high
sintering temperature ($i.e.$ $\geq$1100 K), it is observed that the $S$
values show increasing behaviour with rise in temperature till $\sim$500 K and
after this temperature, the decreasing trend in $S$ is found for the studied
temperature range fu_lco ; he_lco ; ohtani_lco ; singh_mes_lco . The similar
behaviour is seen for polycrystalline sample of this compound, where the above
mentioned change in $S$ is found after 250 Kcortes_lco . Here, the TRACKDMFT
provides the computed value of $S$ = $\sim$-165.6 ($\sim$130) $\mu$V/K at 180
K (300 K). Moreover, the decreasing nature for $S$ value is obtained after 500
K. At 500 K, it is estimated to be $\sim$294.6 $\mu$V/K, where the
experimental range is found to be $\sim$54.9 to $\sim$177.3 $\mu$V/K.
Therefore, it is found that the overall temperature dependent behaviour of $S$
is nicely explained using the estimated values of TRACKDMFT.

At this point, it is noted that by taking the appropriate values of $U$, $J$
and changing the type of Coulomb interaction within DFT+DMFT, the predicted
values of TRACKDMFT for different class of materials may be tuned to the
values more closer to the experimentally reported value. The presence of off-
stoichiometry and impurity in the samples also affects the transport
properties. Further, it is mentioned that the estimation of bandgap using
DFT+DMFT method needs to get proper value for the material, which may be the
reason for getting deviation within the calculated and experimental data of
the transport parameter.

For TRACKDFT, the convergence of $S$ as function of temperature is checked for
different values of $Im\,\Sigma(\omega)$ for all three materials, which are
shown in Fig. 3(a), 3(b) and 3(c) for V, FeSi and LaCoO3, respectively. The
figures show for V and FeSi that $Im\,\Sigma(\omega)$= 0.5 meV provides good
convergence of the $S$ for studied temperature range. But, for LaCoO3, the
convergence is obtained, when the value of $Im\,\Sigma(\omega)$ is reduced
from 2.0 to 0.5 meV, for all the studied temperature except for 300-500 K.
Moreover, it is expected that the further reduction of $Im\,\Sigma(\omega)$
will provide good convergence for the mentioned temperature range. But, due to
our present limitations of computational resources, we describe the computed
values of TRACKDFT code with the fixed value of $Im\,\Sigma(\omega)$=0.5 meV
for all the materials.

Figure 4: Calculated values of temperature dependent electrical conductivity
($\sigma$) by using TRACKDMFT for (a) Vanadium (V) (b) FeSi and (c) LaCoO3
along with corresponding experimental data v_resistivity ; wolfe_fesi ;
lco_resistivity . Table 6: Calculated values of temperature dependent Lorenz
numbers ($L$) by using TRACKDMFT, TRACKDFT and BoltzTraP for FeSi. (Here, $L$
is in $\times$10-8 $V^{2}/K^{2}$)

Temperature |  |  |  | Calculated |  |
---|---|---|---|---|---|---
(K) |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
100 |  | 4.35 |  | 15.3 |  | 18.9
110 |  | 6.52 |  | - |  | -
180 |  | 3.83 |  | - |  | -
200 |  | 2.59 |  | 8.8 |  | 10.1
290 |  | 2.71 |  | - |  | -
300 |  | 3.72 |  | 5.6 |  | 7.6
380 |  | 3.27 |  | - |  | -
400 |  | 2.51 |  | 4.2 |  | 6.4
600 |  | 1.63 |  | 3.3 |  | 5.1
700 |  | 1.67 |  | 3.1 |  | 4.7
800 |  | 2.17 |  | 3.0 |  | 4.4
1000 |  | 1.28 |  | 2.7 |  | 3.8

Figure 5: Calculated values of temperature dependent electrical conductivity
per relaxation time ($\sigma/\tau$) by using TRACKDFT and BoltzTraP for (a)
Vanadium (V) (b) FeSi and (c) LaCoO3.

The computed values of $S$ using TRACKDFT code for V are tabulated in Table.
II along with the calculated values from BoltzTraP. These values are obtained
for the temperature range of 200-450 K. The table shows higher values for
TRACKDFT than the BoltzTraP for observed temperature window. The computed
values from both these codes represent the increasing trend with rise in
temperature, which are providing opposite behaviour than that of experimental
data for pure sample.

Similarly, Table III contains the calculated values of this transport
parameter using TRACKDFT for FeSi. In lower temperature region ($i.e.$
$\lesssim$ 300 K), the values are closely matched with the experimental data.
But, above this temperature, experimentally it is seen that the sign of $S$ is
changed from negative to positive, where the calculated values from TRACKDFT
are showing negative sign for 400-1000 K. However, the values are
monotonically increasing with rise in temperature, which shows similar fashion
as observed from the experimental data for temperature above 300 K. Moreover,
BoltzTraP gives negative sign of $S$ for 100-800 K, which does not follow the
experimental data. But, the increasing behaviour in $S$ is observed after 200
to 1000 K as similar to experiment. The trend of the $S$ value for TRACKDFT
shows quite good matching with the BoltzTraP as well as with the experimental
data of this semiconductor.

In case of LaCoO3, the estimated values of $S$ as function of temperature
using TRACKDFT are listed in Table IV. The table represents that the trend
wise good matching is seen in between the calculated values of TRACKDFT and
experimentally reported data. The value of $S$ at 300 K is found to be
$\sim$309.6 $\mu$V/K, where the experimental range at this temperature is
found to be $\sim$-230 to $\sim$474.1 $\mu$V/K. However, at the same value of
$E_{F}$, BoltzTraP fails to provide good description of the experimental data
as evident from the table. The decreasing nature is seen in the computed
values of TRACKDFT above 450 K, where this behaviour is nicely following the
experimental data. But, the estimated values of $S$ above 450 K are
sufficiently larger than experimental result.

Here, the transport properties of different class of materials with various
range of $S$ values are studied using TRACKDMFT and TRACKDFT codes. The above
discussion reveals that the trend wise nicely matching is observed with
experimental data from the calculated values of TRACKDMFT for all these
materials. Moreover, the comparative studies suggest the importance of
electronic correlation for studying the $S$ of the correlated materials.

Figure 6: Calculated values of temperature dependent electronic part of
thermal conductivity per relaxation time ($\kappa_{e}/\tau$) by using TRACKDFT
and BoltzTraP for (a) Vanadium (V) (b) FeSi and (c) LaCoO3. Table 7:
Calculated values of temperature dependent Lorenz numbers ($L$) by using
TRACKDMFT, TRACKDFT and BoltzTraP for Vanadium. (Here, $L$ is in $\times$10-8
$V^{2}/K^{2}$)

Temperature |  |  |  | Calculated |  |
---|---|---|---|---|---|---
(K) |  |  TRACKDMFT |  |  TRACKDFT |  | BoltzTraP
180 |  | 26.2 |  | - |  | -
200 |  | - |  | 2.96 |  | 159
290 |  | 15.02 |  | - |  | -
300 |  | 14.36 |  | 11.3 |  | 84.8
400 |  | 10.1 |  | 12.9 |  | 56.5
500 |  | 2.4 |  | 8.95 |  | 42.4
600 |  | 2.9 |  | 8.0 |  | 34.0
700 |  | 4.4 |  | 7.6 |  | 28.5
800 |  | 1.0 |  | 7.1 |  | 24.5

###  IV.3 Electrical conductivity ($\sigma$)

The experimental values of $\sigma$ for V, FeSi and LaCoO3 are shown in Figs.
4(a), 4(b) and 4(c), respectively, along with the computed values of
TRACKDMFT. The code provides large value of $\sigma$ due to only considering
the EEI. But, whether the trend of the temperature dependent experimental
$\sigma$ depends on the EEI or not, the computed value of $\sigma$ at 300 K is
divided by experimental $\sigma$ of similar temperature. In addition to this,
the factor obtained by this operation is used for rescaling the computed
values of all studied temperatures. The factor is found to be 296.7, 2883.6,
282146.3 for V, FeSi and LaCoO3, respectively. The figures corresponding to
these three materials represent that the shape of the temperature dependent
experimental $\sigma$ is nicely described by the computed values. Therefore,
the results suggest the importance of EEI for providing the trend of the
$\sigma$ for all the sample materials.

The $\sigma/\tau$ as function of temperatures, which are calculated by using
TRACKDFT and BoltzTraP, are shown in Fig. 5 for three materials. In case of V,
BoltzTraP code gives higher values of $\sigma/\tau$ than the TRACKDFT code,
which is also found from Fig 5(a). The computed values of $\sigma/\tau$ using
TRACKDFT (BoltzTraP) for this metal is $\sim$1.16 ($\sim$12.9) ×1020
$\Omega^{-1}m^{-1}s^{-1}$ for the studied temperature window. For FeSi, Fig.
5(b) illustrates the $\sigma/\tau$ as function of temperature, where TRACKDFT
estimates larger value than BoltzTraP for the studied temperature range. The
temperature dependent curve of $\sigma/\tau$ for TRACKDFT nicely follows
BoltzTraP for this semiconductor. The computed values of temperature dependent
$\sigma/\tau$ for LaCoO3 is plotted in Fig. 5(c), where TRACKDFT and BoltzTraP
codes provide the increasing nature with rise in temperature. The deviation
within the computed values of these two codes is found after 600 K. The
TRACKDFT gives higher value than BoltzTraP for studied temperature window.
Therefore, it is evident from the above discussion that these transport
properties can be explained by the TRACKDFT as similar to BoltzTraP for
different types of materials.

###  IV.4 Electronic part of thermal conductivity ($\kappa_{e}$) and Lorenz
number ($L$)

The calculated values of $\kappa_{e}/\tau$ by using TRACKDFT and BoltzTraP are
shown in Fig. 6 for sample materials. For V, the BoltzTraP provides higher
values than TRACKDFT for all studied temperature, which is evident from Fig. 6
(a). But, the increasing behaviour with the rise in temperature is observed
from the calculated values. The estimated values by utilizing TRACKDFT
(BoltzTraP) are increasing from $\sim$0.57 ($\sim$6.3) to $\sim$1.27
($\sim$13.7) ×1015 $Wm^{-1}K^{-1}s^{-1}$ on changing the temperature from 200
K to 450 K. Therefore for V, the order of this physical parameter is found to
be similar for both cases. Fig. 6(b) illustrates the $\kappa_{e}/\tau$ as
function of temperature for FeSi. The figure represents that the calculated
values from both the codes are almost same except for 200-600 K, where
TRACKDFT is showing slightly higher value than BoltzTraP. Therefore, it is
noted that the behaviour of the curve as obtained from TRACKDFT is nicely
matching with the curve of BoltzTraP for this transport coefficient. The
comparison between the calculated values of temperature dependent
$\kappa_{e}/\tau$ by using TRACKDFT and BoltzTraP codes is plotted in Fig.
6(c) for LaCoO3. These two codes provide the increasing nature with rise in
temperature for this physical quantity. The slope of the curve and the values
for the studied temperatures of $\kappa_{e}/\tau$ of TRACKDFT are lower than
BoltzTraP. For LaCoO3, the major deviation is found after 600 K for this
transport parameter. Therefore, overall features of $\kappa_{e}/\tau$ curve,
which are computed using TRACKDFT, for test materials are in good match with
the results of BoltzTraP.

Further, the calculated values of temperature dependent $L$ by using
TRACKDMFT, TRACKDFT and BoltzTraP codes for V, FeSi and LaCoO3 are shown in
Table V, Table VI and Table VII, respectively. The Table V shows decreasing
behaviour of $L$ with increasing temperature. The computed value of $L$ is
found to be $\sim$2.4 ($\sim$2.472) ×10-8 V2/K2 at 200 K from TRACKDMFT
(TRACKDFT) code, whereas BoltzTraP provides $\sim$2.43 ×10-8 V2/K2 at the same
temperature. The calculated values of $L$ for different temperatures obtained
from the present code are in good agreement with the experimental data jung of
V. Next for FeSi, it is seen from Table VI that the values obtained from
TRACKDMFT calculation are showing good agreement with the typical order of
metals and degenerate semiconductors. The value of $L$ at 300 K is estimated
to be $\sim$3.72 ×10-8 V2/K2 from TRACKDMFT. Moreover, both TRACKDFT and
BoltzTraP codes provide larger value than TRACKDMFT as seen from the Table VI.
The computed value is found to be $\sim$5.6 ($\sim$7.6) ×10-8 V2/K2 by using
TRACKDFT (BoltzTraP) at 300 K. In case of LaCoO3, the Table VII provides the
calculated values of $L$=1.0 - 4.4 ×10-8 V2/K2 for 500 - 800 K using
TRACKDMFT, which is typical order of this parameter for any metal. Similarly,
the TRACKDFT gives slightly higher value than TRACKDMFT for this temperature
range, whereas BoltzTraP predicts larger value than the typical value of $L$
for the metal. Moreover, from 180-400 K, the computed value of $L$ using
TRACKDMFT is found to decrease from 26.2 ×10-8 V2/K2 to 10.1 ×10-8 V2/K2. This
large value of $L$ may be obtained due to the underestimation of the
experimental bandgap within this temperature window, where insulating phase is
seen in DFT+DMFT calculation. However, BoltzTraP shows nonphysical large
values of $L$ for the studied temperature window. Thus, the result suggests
the better predictive power of TRACK code for computing the temperature
dependent $L$ value.

##  V Conclusions

We presented a Python 3 code TRACK for calculating the transport properties of
different class of materials. Kubo linear-response formalism is used for
calculating the transport coefficients. The benefit of using this formalism is
to access all types of correlated materials ($i.e.$ weak, moderate and strong)
with help of interacting and non-interacting electronic structure methodology.
By introducing appropriate approximations in the Kubo formalism the transport
expression under relaxation time is obtained. The approximated Kubo formalism
provides promising results when band-theory is utilized as ab-initio method.
Here, Vanadium (V), FeSi and LaCoO3 are chosen as test samples to benchmark
the TRACK code. The results obtained from TRACKDFT are also compared with the
calculated values of BoltzTraP code. The behaviour of different temperature
dependent transport parameters are showing good agreement with the trend of
corresponding experimental results for all three materials. Therefore, this
code is simultaneously useful to understand the basic physics of transport
properties for many emergent materials as well as predicting the values of
temperature dependent transport parameters for application purpose.

##  VI References

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