# Overview

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`ShengBTE`

: a solver for the Boltzmann transport equation for phonons

## Authors:

- Wu Li wu.li.phys2011@gmail.com
- Jesús Carrete Montaña jcarrete@gmail.com
- Nebil A. Katcho nebil.katcho@gmail.com
- Natalio Mingo natalio.mingo@cea.fr

## How to download, compile and use `ShengBTE`

Development of `ShengBTE`

is hosted at Bitbucket. The latest version can be downloaded using the "download" link at https://bitbucket.org/sousaw/shengbte. Alternatively, it is possible to clone its GIT repository from the command line with

git clone git@bitbucket.org/sousaw/shengbte.git ShengBTE

or from one of the many graphical frontends available.

To compile the code it is enough to run `make`

in the `Src`

subdirectory of the distribution, but a suitable `arch.make`

must be present in that directory. An example is provided as `arch.make.example`

. As a minimum, `$MPIFC`

must contain a valid command to compile Fortran 90 code with MPI directives, while the combination of `$LDFLAGS`

and `$LIBS`

must contain any linker flags required in order to link against an implementation of `LAPACK`

and against Atsushi Togo's spglib. ShengBTE uses some Fortran 2003 extensions, most notably its new syntax for array initialization, and a recent Fortran compiler is required that supports them; `gfortran`

4.8.2 and `ifort`

12.0.0 are known to work.

After compilation succeeds, a `ShengBTE`

binary will be created in the root directory of the distribution. This executable takes no command-line options and accepts no input from the terminal. It can be invoked simply as

./ShengBTE

for serial mode, but most often it will be run using a command like

mpirun -n 32 ./ShengBTE 2>BTE.err >BTE.out

often as part of a script to be submitted to a batch system.

## Input files

Exactly three files are required for a `ShengBTE`

run: `CONTROL`

, one of `FORCE_CONSTANTS_2ND`

or `espresso.ifc2`

, and `FORCE_CONSTANTS_3RD`

. Their contents are detailed below; for a complete example, the reader is referred to the `Test-VASP`

and `Test-QE`

subdirectories of the distribution.

### The `CONTROL`

file

The contents of this file describe the system to be studied and specify a set of parameters and flags controlling execution. Its format is merely a sequence of four Fortran namelists, with a reasonably flexible syntax that should become apparent after looking at the example `Test-*/CONTROL`

for zincblence InAs and GaAs. Some parameters and flags are mandatory, whereas others are optional and take a default value when unspecified.

`&allocations`

namelist:`nelements`

(integer, mandatory): number of different elements in the compound`natoms`

(integer, mandatory): number of atoms in the unit cell`ngrid`

(integer, 3, mandatory): number of grid planes along each axis in reciprocal space`norientations`

(integer, default=0): number of orientations along which to study nanowires

`&crystal`

namelist:`lfactor`

(real, nm, default=1.0): unit of measurement for lattice vectors`lattvec`

(real, 3 x 3, mandatory): real-space lattice vectors, in units of`lfactor`

`types`

(integer,`natoms`

, mandatory): a vector of`natom`

integers, ranging from 1 to`nelements`

, assigning an element to each atom in the system`elements`

(string,`nelements`

, mandatory): a vector of element names`positions`

(real, 3 x`natoms`

, mandatory): atomic positions in lattice coordinates`masses`

(real,`nelements`

, g/mol, default=automatic): atomic masses corresponding to each element. If they are omitted and`autoisotopes`

is true and the element names are known, they are computed automatically.`gfactors`

(real,`nelements`

, default=automatic): g factors for isotopic scattering associated to each element. If they are omitted and`autoisotopes`

is true and the element names are known, they are computed automatically.`epsilon`

(real, 3 x 3, Ε<sub>0</sub>, default=1): dielectric tensor of the system in the Cartesian basis`born`

(real, 3 x 3 x`natoms`

, e, default=0): Born effective charge tensor of each atom in the Cartesian basis`scell`

(integer, 3, mandatory): supercell sizes along each crystal axis used for the 2nd-order force constant calculation`orientations`

(integer, 3 x`norientations`

, mandatory unless`norientations`

==0): terns of integer indices defining the crystallographic directions along which to study nanowires

`¶meters`

namelist:`T`

(real, K): temperature to be used in the case of single temperature calculation`T_min`

,`T_max`

,`T_step`

(real, K): the minimum temperature, the maximum temperature and the increment to be used for multiple-temperature calculation. T takes the priority if it is present.`omega_max`

(real, rad/ps, default=1.e100): the max angular frequency up to which the anharmonic scattering properties are calculated for limited purposes.`scalebroad`

(real, default=1.0): scale parameter for Gaussian smearing. The default is theoretically guaranteed to work, but significant speedups can sometimes be achieved by reducing it, with negligible loss of precision.`rmin`

(real, nm, default=5.0): minimum radius of nanowires whose thermal conductivity will be computed`rmax`

(real, nm, default=505.0): maximum radius of nanowires whose thermal conductivity will be computed`dr`

(real, nm, default=100.0): radius increment to be used when simulating nanowires from`rmin`

to`rmax`

`maxiter`

(integer, default=1000): maximum number of iterations allowed in the BTE convergence process`nticks`

(integer, default=100): number of different values of the mean free path at which to compute the cumulative thermal conductivity`eps`

(real, default=10<sup>-5</sup>): the iterative solver of the BTE will stop when the relative change in the thermal conductivity tensor is less than`eps`

. Such change between steps n-1 and n is measured as ||Κ<sub>n</sub>-Κ<sub>n-1</sub>||, where ||⋅|| denotes a matrix 2-norm.

`&flags`

namelist:`nonanalytic`

(logical, default=.true.): compute and use the nonanalytic part of the dynamical matrix`convergence`

(logical, default=.true.): if true, iterate the BTE solver until convergence is achieved. If false, compute thermal conductivities in the relaxation time approximation.`isotopes`

(logical, default=.true.): include isotopic scattering in the relaxation times`autoisotopes`

(logical, default=.true.): compute atomic masses and g factors automatically`nanowires`

(logical, default=.false.): study the thermal conductivity of nanowires in addition to that of the bulk`onlyharmonic`

(logical, default=.false.): stop the program after computing the specific heat and small-grain thermal conductivity`espresso`

(logical, default=.false.): read second-order force constants from`espresso.ifc2`

(Quantum Espresso format) instead of the default`FORCE_CONSTANTS_2ND`

(Phonopy format)

### The `FORCE_CONSTANTS_2ND`

file

This file contains the second derivatives of the system's energy with respect to the Cartesian coordinates of the nuclei, *i.e.* the interatomic force constant matrix. Its format is precisely that chosen in Phonopy for the `FORCE_CONSTANTS`

file, so that the result of a Phonopy calculation can be used directly. The first line of the file declares the total number of atoms in the supercell, `npairs`

, which must be equal to `scell(1)`

x `scell(2)`

x `scell(3)`

x `natoms`

, and is followed by `npairs`

blocks of four lines each. The first line of each of those blocks contains two integers with the 1-based indices of the atoms forming the pair; the remaining three lines contain the 3 x 3 matrix of second-order interatomic force constants linking those two atoms, in eV/Å<sup>2</sup>.

### The `espresso.ifc2`

file

The information contained in this file is equivalent to that in `FORCE_CONSTANTS_2ND`

, but the format is different. For details, consult the Quantum ESPRESSO documentation. Please note that although this file's header contains information about lattice vectors, atomic positions, Born effective charges and so forth, it is ignored by `ShengBTE`

. It is the user's responsibility to ensure that `espresso.ifc2`

and `CONTROL`

are compatible.

### The `FORCE_CONSTANTS_3RD`

file

Similarly, this file contains the third-order interatomic force constant matrix, but uses a sparse description to save space. All constants are implicitily refered to a central unit cell i taken as the origin of coordinates. The first line again contains a single integer, `nb`

, which is followed by `nb`

blocks with the following structure:

- A blank line
- A 1-based sequential index
- A line with the Cartesian coordinates of the second unit cell in Å
- A line with the Cartesian coordinates of the third unit cell in Å
- A line with the 1-based indices of the three atoms involved, each from 1 to
`natoms`

- 27 lines, each of which starts with a tern of integers specifying three Cartesian axes and is completed by a force constant in eV/Å<sup>3</sup>. The last element of the tern changes first.

The following is an example of one such block:

1 0.000 0.000 0.000 0.000 0.000 0.000 1 1 1 1 1 1 0.0000000000E+00 1 1 2 0.0000000000E+00 1 1 3 0.0000000000E+00 1 2 1 0.0000000000E+00 1 2 2 0.0000000000E+00 1 2 3 0.2346653425E+02 1 3 1 0.0000000000E+00 1 3 2 0.2346653425E+02 1 3 3 0.0000000000E+00 2 1 1 0.0000000000E+00 2 1 2 0.0000000000E+00 2 1 3 0.2346653425E+02 2 2 1 0.0000000000E+00 2 2 2 0.0000000000E+00 2 2 3 0.0000000000E+00 2 3 1 0.2346653425E+02 2 3 2 0.0000000000E+00 2 3 3 0.0000000000E+00 3 1 1 0.0000000000E+00 3 1 2 0.2346653425E+02 3 1 3 0.0000000000E+00 3 2 1 0.2346653425E+02 3 2 2 0.0000000000E+00 3 2 3 0.0000000000E+00 3 3 1 0.0000000000E+00 3 3 2 0.0000000000E+00 3 3 3 0.0000000000E+00

## Output files

Many files including temperature-dependent directories are created during a successful run of `ShengBTE`

. They contain not only the thermal conductivity and related quantities, but also a set of intermediate results that may be useful to diagnose problems. For some quantites, values only for the q points in the irreducible wedge are output, values for the rest can be recovered by looking into the equivilent points in the irreducible wedge. This section includes a brief description of their contents.

`BTE.ReciprocalLatticeVectors`

: three reciprocal lattice basis vectors b1, b2 and b3 in nm-1.`BTE.qpoints`

: This file gives q points in the irreducible wedge of Brillouin zone (BZ), of which the relative coordinates with respect to the reciprocal lattice vectors are shown in the last 3 columns. The 1st and 2nd columns correspond the indices of those q points numbered in the irreducible wedge and in the whole Brillouin zone, respectively. The 3rd column lists the corresponding degeneracies.`BTE.qpoints_full`

: this file lists all q points in`ngrid(1)`

x`ngrid(2)`

x`ngrid(3)`

Γ-centered regular grid. The 1st column is a sequentially increasing index, the 2nd column contains the index of the equivalent irreducible q point numbered in the irreducible wedge, and the 3 remaining columns are the relative coordinates with respect to the reciprocal lattice vectors for the q point.`BTE.omega`

: phonon angular frequencies of those q points in the irreducible wedge, in rad/ps.`BTE.v`

: group velocities of those modes (q index changes first, and then band index) in the irreducible wedge, in km/s (or nm THz)`BTE.v_full`

: group velocities of all modes (q index changes first, and then band index)for all points listed in`BTE.qpoints_full`

`BTE.w_boundary`

: boundary scattering rate (in ps<sup>-1</sup> , 2nd column) obtained for a characteristic length L=1 nm and a specularity parameter p=0 vs angular frequency (in rad/ps, 1st column) for those modes (q index changes first, and then band index) in the irreducible wedge.`BTE.w_isotopic`

: isotopic scattering rate (in ps<sup>-1</sup> , 2nd column) vs angular frequency (in rad/ps, 1st column) for those modes (q index changes first, and then band index) in the irreducible wedge.`BTE.dos`

: the phonon density of states (2nd column) vs the angular frequencies (1st column, in rad/ps)`BTE.pdos`

: the phonon density of states projected on each atom in the unit cell (from the 2nd column on) vs the angular frequencies (1st column, in rad/ps)`BTE.P3`

: volume in phase space available for three-phonon processes, for each irreducible q point and phonon band`BTE.P3_total`

: sum of all the contributions in`BTE.P3`

, total volume in phase space available for three-phonon processes`BTE.P3_plus*`

,`BTE.P3_minus*`

: equivalents of`BTE.P3`

and`BTE.P3_total`

, but including only contributions from emission (minus) or absorption (plus) processes`BTE.gruneisen`

: Grüneisen parameter for each irreducible q point and phonon band`BTE.cvVsT`

: specific heat of the system, in J/(m<sup>3</sup> K) as a function of T (1st column)`BTE.gruneisenVsT_total`

: total Grüneisen parameter obtained as a weighted sum of the mode contributions as a function of T (1st column)`BTE.KappaTensorVsT_sg`

: thermal conductivity tensor per unit of mean free path in the small-grain limit, in W/(m K nm) as a function of T (1st column).`BTE.KappaTensorVsT_RTA`

: total thermal conductivity tensor in unit of W/(m K) in the Relaxation Time Approximation (zero-order) as a function of T (1st column).`BTE.KappaTensorVsT_CONV`

: total CONVerged thermal conductivity tensor in unit of W/(m K) as a function of T (1st column). The last column gives the number of iterations reaching convergence.

Under temperature-dependent directories:

`BTE.cv`

: specific heat of the system, in J/(m<sup>3</sup> K)`BTE.kappa_sg`

: thermal conductivity per unit of mean free path in the small-grain limit, in W/(m K nm)`BTE.gruneisen_total`

: total Grüneisen parameter obtained as a weighted sum of the mode contributions`BTE.WP3`

: weighted phase space available for three-phonon processes (in ps<sup>4</sup>rad<sup>-4</sup> , 2nd column) vs angular frequency (in rad/ps, 1st column) for those modes (q index changes first, and then band index) in the irreducible wedge. See [Phys. Rev. B 91, 144304 (2015)] for definition of weighted phase space.`BTE.WP3_plus`

: WP3 contributed by phonon absorption processes alone`BTE.WP3_minus`

: WP3 contributed by phonon emission processes alone`BTE.w_anharmonic`

: contribution of three-phonon processes to the scattering rate, for each q point and each band, in ps<sup>-1</sup>`BTE.w`

: total zeroth-order scattering rate for each q point and each band, in ps<sup>-1</sup>`BTE.w_final`

: total converged scattering rate for each irreducible q point and each band, in ps<sup>-1</sup>`BTE.kappa`

: tensorial contribution to the thermal conductivity from each band, in W/(m K). The last line contains converged values, the rest show the convergence process.`BTE.kappa_tensor`

: total thermal conductivity, a 3 x 3 tensor expressed in W/(m K). The last line contains converged values, the rest show the convergence process.`BTE.kappa_scalar`

: average of diagonal elements of the thermal conductivity tensor, in W/(m K). The last line contains converged values, the rest show the convergence process.`BTE.kappa_nw_*`

: thermal conductivities of nanowires built along different directions of the bulk material, for different radii. The first column in each file is a diameter, the following 3 x`natoms`

contain the contributions of each band and the last column contains the total thermal conductivity. Diameters are expressed in nm and conductivities in W/(m K)`BTE.kappa_nw_*_lower`

: lower bounds to the thermal conductivities of nanowires built along different directions of the bulk material, for different radii. The first column in each file is a diameter, the following 3 x`natoms`

contain the contributions of each band and the last column contains the total thermal conductivity. Diameters are expressed in nm and conductivities in W/(m K). Each lower bound is estimated by using the set of zeroth-order bulk relaxation times.`BTE.cumulative_kappa_*`

: this set of files is analogous to`BTE.kappa*`

, except in that their 1st column specifies a cutoff mean free path (in nm) when calculating the total contribution.`BTE.cumulative_kappaVsOmega_tensor`

: this is analogous to`BTE.cumulative_kappa_tensor`

, except in that the 1st column specifies a cutoff angular frequency (in rad/ps) when calculating the total contribution.