Wannier90 User Guide Wannier90:
User Manual:
Open the PDF directly: View PDF 
.
Page Count: 137 [warning: Documents this large are best viewed by clicking the View PDF Link!]
wannier90: User Guide
Version 2.0
14th October 2013
Contents
I Introduction 5
II wannier90.x 9
1 Methodology 11
2 Parameters 13
3 Projections 39
4 Code Overview 47
5wannier90 as a post-processing tool 49
6wannier90 as a library 57
7 Transport Calculations with wannier90 63
8 Files 67
9 Sample Input Files 83
III postw90.x 87
10 Parameters 89
11 Overview of the berry module 117
12 Electronic transport calculations with the BoltzWann module 121
13 Generic Band interpolation 125
3
Part I
Introduction
5
Introduction
Getting Help
The latest version of wannier90 and documentation can always be found at http://www.wannier.org.
There is a wannier90 mailing list for discussing issues in the development, theory, coding and algorithms
pertinent to MLWF. You can register for this mailing list by following the links at http://www.
wannier.org/forum.html
Finally, many frequently asked questions are answered in Appendix B.
Citation
We ask that you acknowledge the use of wannier90 in any publications arising from the use of this
code through the following reference
[ref] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt and N. Marzari,
wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions,
Comput. Phys. Commun. 178, 685 (2008)
It would also be appropriate to cite the original articles:
Maximally localized generalized Wannier functions for composite energy bands,
N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)
Maximally localized Wannier functions for entangled energy bands,
I. Souza, N. Marzari and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)
Credits
The present release of wannier90 was written by Arash A. Mostofi (Imperial College London, UK),
Giovanni Pizzi (EPFL, Switzerland), Ivo Souza (Universidad del Pais Vasco, Spain) and Jonathan R.
Yates (University of Oxford, UK). Contributors to the code include Young-Su Lee (KIST, S. Korea),
Matthew Shelley (Imperial College London, UK) and Nicolas Poilvert (Harvard University, USA).
wannier90 is based on the Fortran 77 codes written for isolated bands by Nicola Marzari and David
Vanderbilt, for entangled bands by Ivo Souza, Nicola Marzari, and David Vanderbilt, and for quantum
transport by Marco Nardelli.
7
8wannier90: User Guide
Acknowledgements: Stefano de Gironcoli (SISSA, Trieste, Italy) for the pwscf interface, Timo Thon-
hauser and Graham Lopez (Wake Forest, USA) extended this to add terms needed for orbital mag-
netisation ; Michel Posternak (EPFL, Switzerland) for the original plotting routines, Raffaello Bianco
(University of Trieste) for improvements to the k-slice plotting. Daniel Aberg (LLNL, USA) for povray
plotting routines, w90vdw is written by Lampros Andrinopoulos, Nicholas D. M. Hine and Arash A.
Mostofi at Imperial College London.
wannier90 c
2007-2013 Arash A. Mostofi, Jonathan R. Yates, Young-Su Lee, Giovanni Pizzi, Ivo
Souza, David Vanderbilt and Nicola Marzari
Licence
All the material in this distribution is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free Software Foundation; either
version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program; if not,
write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
USA.
Part II
wannier90.x
9
Chapter 1
Methodology
wannier90 computes maximally-localised Wannier functions (MLWF) following the method of Marzari
and Vanderbilt (MV) [1]. For entangled energy bands, the method of Souza, Marzari and Vanderbilt
(SMV) [2] is used. We introduce briefly the methods and key definitions here, but full details can be
found in the original papers and in Ref. [3].
First-principles codes typically solve the electronic structure of periodic materials in terms of Bloch
states, ψnk. These extended states are characterised by a band index nand crystal momentum k.
An alternative representation can be given in terms of spatially localised functions known as Wannier
functions (WF). The WF centred on a lattice site R,wnR(r), is written in terms of the set of Bloch
states as
wnR(r) = V
(2π)3ZBZ "X
m
U(k)
mnψmk(r)#e−ik.Rdk,(1.1)
where Vis the unit cell volume, the integral is over the Brillouin zone (BZ), and U(k)is a unitary
matrix that mixes the Bloch states at each k.U(k)is not uniquely defined and different choices will
lead to WF with varying spatial localisations. We define the spread Ωof the WF as
Ω = X
nhwn0(r)|r2|wn0(r)i − |hwn0(r)|r|wn0(r)i|2.(1.2)
The total spread can be decomposed into a gauge invariant term ΩIplus a term ˜
Ωthat is dependant
on the gauge choice U(k).˜
Ωcan be further divided into terms diagonal and off-diagonal in the WF
basis, ΩDand ΩOD,
Ω=ΩI+˜
Ω=ΩI+ ΩD+ ΩOD (1.3)
where
ΩI=X
n"hwn0(r)|r2|wn0(r)i − X
Rm|hwnR(r)|r|wn0(r)i|2#(1.4)
ΩD=X
nX
R6=0|hwnR(r)|r|wn0(r)i|2(1.5)
ΩOD =X
m6=nX
R|hwmR(r)|r|wn0(r)i|2(1.6)
The MV method minimises the gauge dependent spread ˜
Ωwith respect the set of U(k)to obtain
MLWF.
wannier90 requires two ingredients from an initial electronic structure calculation.
11
12 wannier90: User Guide
1. The overlaps between the cell periodic part of the Bloch states |unki
M(k,b)
mn =humk|unk+bi,(1.7)
where the vectors b, which connect a given k-point with its neighbours, are determined by
wannier90 according to the prescription outlined in Ref. [1].
2. As a starting guess the projection of the Bloch states |ψnkionto trial localised orbitals |gni
A(k)
mn =hψmk|gni,(1.8)
Note that M(k,b),A(k)and U(k)are all small, N×Nmatrices1that are independent of the basis set
used to obtain the original Bloch states.
To date, wannier90 has been used in combination with electronic codes based on plane-waves and
pseudopotentials (norm-conserving and ultrasoft [4]) as well as mixed basis set techniques such as
FLAPW [5].
1.1 Entangled Energy Bands
The above description is sufficient to obtain MLWF for an isolated set of bands, such as the valence
states in an insulator. In order to obtain MLWF for entangled energy bands we use the “disentangle-
ment” procedure introduced in Ref. [2].
We define an energy window (the “outer window”). At a given k-point k,N(k)
win states lie within this
energy window. We obtain a set of NBloch states by performing a unitary transformation amongst
the Bloch states which fall within the energy window at each k-point:
|uopt
nki=X
m∈N(k)
win
Udis(k)
mn |umki(1.9)
where Udis(k)is a rectangular N×N(k)
win matrix2. The set of Udis(k)are obtained by minimising the
gauge invariant spread ΩIwithin the outer energy window. The MV procedure can then be used to
minimise ˜
Ωand hence obtain MLWF for this optimal subspace.
It should be noted that the energy bands of this optimal subspace may not correspond to any of the
original energy bands (due to mixing between states). In order to preserve exactly the properties of a
system in a given energy range (e.g., around the Fermi level) we introduce a second energy window.
States lying within this inner, or “frozen”, energy window are included unchanged in the optimal
subspace.
1Technically, this is true for the case of an isolated group of Nbands from which we obtain NMLWF. When using
the disentanglement procedure of Ref. [2], A(k), for example, is a rectangular matrix. See Section 1.1.
2As Udis(k)is a rectangular matrix this is a unitary operation in the sense that (Udis(k))†Udis(k)=1.
Chapter 2
Parameters
2.1 Usage
wannier90.x [-pp] [seedname]
•seedname: If a seedname string is given the code will read its input from a file seedname.win.
The default value is wannier. One can also equivalently provide the string seedname.win instead
of seedname.
•-pp: This optional flag tells the code to generate a list of the required overlaps and then exit.
This information is written to the file seedname.nnkp.
2.2 seedname.win File
The wannier90 input file seedname.win has a flexible free-form structure.
The ordering of the keywords is not significant. Case is ignored (so num_bands is the same as
Num_Bands). Characters after !, or # are treated as comments. Most keywords have a default value
that is used unless the keyword is given in seedname.win. Keywords can be set in any of the following
ways
num_wann 4
num_wann = 4
num_wann : 4
A logical keyword can be set to true using any of the following strings: T,true,.true..
For further examples see Section 9.1 and the the wannier90 Tutorial.
2.3 Keyword List
13
14 wannier90: User Guide
Keyword Type Description
System Parameters
num_wann I Number of WF
num_bands I Number of bands passed to the code
unit_cell_cart P Unit cell vectors in Cartesian coor-
dinates
atoms_cart * P Positions of atoms in Cartesian co-
ordinates
atoms_frac * R Positions of atoms in fractional co-
ordinates with respect to the lattice
vectors
mp_grid I Dimensions of the Monkhorst-Pack
grid of k-points
kpoints R List of k-points in the Monkhorst-
Pack grid
gamma_only L Wavefunctions from underlying ab
initio calculation are manifestly real
spinors L WF are spinors
shell_list I Which shells to use in finite differ-
ence formula
search_shells I The number of shells to search when
determining finite difference formula
kmesh_tol R The tolerance to control if two
kpoint belong to the same shell
Table 2.1: seedname.win file keywords defining the system. Argument types are represented by, I for
a integer, R for a real number, P for a physical value, L for a logical value and S for a text string.
*atoms_cart and atoms_frac may not both be defined in the same input file.
wannier90: User Guide 15
Keyword Type Description
Job Control
postproc_setup L To output the seedname.nnkp file
exclude_bands I List of bands to exclude from the
calculation
restart S Restart from checkpoint file
iprint I Output verbosity level
length_unit S System of units to output lengths
wvfn_formatted L Read the wavefunctions from a
(un)formatted file
spin S Which spin channel to read
devel_flag S Flag for development use
timing_level I Determines amount of timing infor-
mation written to output
optimisation I Optimisation level
translate_home_cell L To translate final Wannier centres to
home unit cell when writing xyz file
write_xyz L To write atomic positions and final
centres in xyz file format
write_vdw_data L To write data for futher processing
by w90vdw utility
write_hr_diag L To write the diagonal elements of
the Hamiltonian in the Wannier ba-
sis to seedname.wout (in eV)
Table 2.2: seedname.win file keywords defining job control. Argument types are represented by, I for
a integer, R for a real number, P for a physical value, L for a logical value and S for a text string.
translate_home_cell only relevant if write_xyz is .true.
16 wannier90: User Guide
Keyword Type Description
Disentanglement Parameters
dis_win_min P Bottom of the outer energy window
dis_win_max P Top of the outer energy window
dis_froz_min P Bottom of the inner (frozen) energy
window
dis_froz_max P Top of the inner (frozen) energy win-
dow
dis_num_iter I Number of iterations for the minimi-
sation of ΩI
dis_mix_ratio R Mixing ratio during the minimisa-
tion of ΩI
dis_conv_tol R The convergence tolerance for find-
ing ΩI
dis_conv_window I The number of iterations over which
convergence of ΩIis assessed.
Table 2.3: seedname.win file keywords controlling the disentanglement. Argument types are repre-
sented by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a
text string.
wannier90: User Guide 17
Keyword Type Description
Wannierise Parameters
num_iter I Number of iterations for the minimi-
sation of Ω
num_cg_steps I During the minimisation of Ωthe
number of Conjugate Gradient steps
before resetting to Steepest Descents
conv_window I The number of iterations over which
convergence of Ωis assessed
conv_tol P The convergence tolerance for find-
ing Ω
conv_noise_amp R The amplitude of random noise ap-
plied towards end of minimisation
procedure
conv_noise_num I The number of times random noise
is applied
num_dump_cycles I Control frequency of check-pointing
num_print_cycles I Control frequency of printing
write_r2mn L Write matrix elements of r2between
WF to file
guiding_centres L Use guiding centres
num_guide_cycles I Frequency of guiding centres
num_no_guide_iter I The number of iterations after which
guiding centres are used
trial_step * R The trial step length for the
parabolic line search during the min-
imisation of Ω
fixed_step * R The fixed step length to take dur-
ing the minimisation of Ω, instead
of doing a parabolic line search
use_bloch_phases ** L To use phases for initial projections
Table 2.4: seedname.win file keywords controlling the wannierisation. Argument types are represented
by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a text
string. *fixed_step and trial_step may not both be defined in the same input file. **Cannot be used in
conjunction with disentanglement.
18 wannier90: User Guide
Keyword Type Description
Plot Parameters
wannier_plot L Plot the WF
wannier_plot_list I List of WF to plot
wannier_plot_supercell I Size of the supercell for plotting the
WF
wannier_plot_format S File format in which to plot the WF
wannier_plot_mode S Mode in which to plot the WF,
molecule or crystal
wannier_plot_radius R Cut-off radius of WF*
bands_plot L Plot interpolated band structure
kpoint_path P K-point path for the interpolated
band structure
bands_num_points I Number of points along the first sec-
tion of the k-point path
bands_plot_format S File format in which to plot the in-
terpolated bands
bands_plot_project I WF to project the band structure
onto
bands_plot_mode S Slater-Koster type interpolation or
Hamiltonian cut-off
bands_plot_dim I Dimension of the system
fermi_surface_plot L Plot the Fermi surface
fermi_surface_num_points I Number of points in the Fermi sur-
face plot
fermi_energy P The Fermi energy
fermi_energy_min P Lower limit of the Fermi energy
range
fermi_energy_max P Upper limit of the Fermi energy
range
fermi_energy_step R Step for increasing the Fermi energy
in the specified range
fermi_surface_plot_format S File format for the Fermi surface
plot
hr_plot L Write the Hamiltonian in the WF
basis
hr_cutoff P Cut-off for the absolute value of the
Hamiltonian
dist_cutoff P Cut-off for the distance between WF
dist_cutoff_mode S Dimension in which the distance be-
tween WF is calculated
translation_centre_frac R Centre of the unit cell to which final
WF are translated
Table 2.5: seedname.win file keywords controlling the plotting. Argument types are represented by, I
for a integer, R for a real number, P for a physical value, L for a logical value and S for a text string.
* Only applies when wannier_plot_format is cube.
wannier90: User Guide 19
Keyword Type Description
Transport Parameters
transport L Calculate quantum conductance and
density of states
transport_mode S Bulk or left-lead_conductor_right-
lead calculation
tran_win_min P Bottom of the energy window for
transport calculation
tran_win_max P Top of the energy window for trans-
port calculation
tran_energy_step R Sampling interval of the energy val-
ues
fermi_energy R The Fermi energy
tran_num_bb I Size of a bulk Hamiltonian
tran_num_ll I Size of a left-lead Hamiltonian
tran_num_rr I Size of a right-lead Hamiltonian
tran_num_cc I Size of a conductor Hamiltonian
tran_num_lc I Number of columns in a left-
lead_conductor Hamiltonian
tran_num_cr I Number of rows in a
conductor_right-lead Hamilto-
nian
tran_num_cell_ll I Number of unit cells in PL of left
lead
tran_num_cell_rr I Number of unit cells in PL of right
lead
tran_num_bandc I Half-bandwidth+1 of a band-
diagonal conductor Hamiltonian
tran_write_ht L Write the Hamiltonian for transport
calculation
tran_read_ht L Read the Hamiltonian for transport
calculation
tran_use_same_lead L Left and right leads are the same
tran_group_threshold R Distance that determines the group-
ing of WFs
hr_cutoff P Cut-off for the absolute value of the
Hamiltonian
dist_cutoff P Cut-off for the distance between WF
dist_cutoff_mode S Dimension in which the distance be-
tween WF is calculated
one_dim_axis S Extended direction for a one-
dimensional system
translation_centre_frac R Centre of the unit cell to which final
WF are translated
Table 2.6: seedname.win file keywords controlling transport. Argument types are represented by, I for
a integer, R for a real number, P for a physical value, L for a logical value and S for a text string.
20 wannier90: User Guide
2.4 System
2.4.1 integer :: num_wann
Number of WF to be found.
No default.
2.4.2 integer :: num_bands
Total number of bands passed to the code in the seedname.mmn file.
Default num_bands=num_wann
2.4.3 Cell Lattice Vectors
The cell lattice vectors should be specified in Cartesian coordinates.
begin unit_cell_cart
[units]
A1xA1yA1z
A2xA2yA2z
A3xA3yA3z
end unit_cell_cart
Here A1xis the x-component of the first lattice vector A1,A2yis the y-component of the second lattice
vector A2, etc.
[units] specifies the units in which the lattice vectors are defined: either Bohr or Ang.
The default value is Ang.
2.4.4 Ionic Positions
The ionic positions may be specified in fractional coordinates relative to the lattice vectors of the unit
cell, or in absolute Cartesian coordinates. Only one of atoms_cart and atoms_frac may be given in
the input file.
Cartesian coordinates
begin atoms_cart
[units]
P RP
xRP
yRP
z
Q RQ
xRQ
yRQ
z
.
.
.
end atoms_cart
wannier90: User Guide 21
The first entry on a line is the atomic symbol. The next three entries are the atom’s position R=
(Rx, Ry, Rz)in Cartesian coordinates. The first line of the block, [units], specifies the units in which
the coordinates are given and can be either bohr or ang. If not present, the default is ang.
Fractional coordinates
begin atoms_frac
P F P
1FP
2FP
3
Q F Q
1FQ
2FQ
3
.
.
.
end atoms_frac
The first entry on a line is the atomic symbol. The next three entries are the atom’s position in
fractional coordinates F=F1A1+F2A2+F3A3relative to the cell lattice vectors Ai,i∈[1,3].
2.4.5 integer, dimension :: mp_grid(3)
Dimensions of the regular (Monkhorst-Pack) k-point mesh. For example, for a 2×2×2grid:
mp_grid : 2 2 2
No default.
2.4.6 K-points
Each line gives the coordinate K=K1B1+K2B2+K3B3of a k-point in relative (crystallographic)
units, i.e., in fractional units with respect to the primitive reciprocal lattice vectors Bi,i∈[1,3]. The
position of each k-point in this list assigns its numbering; the first k-point is k-point 1, the second is
k-point 2, and so on.
begin kpoints
K1
1K1
2K1
3
K2
1K2
2K2
3
.
.
.
end kpoints
There is no default.
Note: There is an utility provided with wannier90, called kmesh.pl, which helps to generate the
explicit list of kpoints required by wannier90. See Sec. A.1.
2.4.7 logical :: gamma_only
If gamma_only=true, then wannier90 uses a branch of algorithms for disentanglement and localisation
that exploit the fact that the Bloch eigenstates obtained from the underlying ab initio calculation are
manifestly real. This can be the case when only the Γ-point is used to sample the Brillouin zone. The
localisation procedure that is used in the Γ-only branch is based on the method of Ref. [6].
22 wannier90: User Guide
The default value is false.
2.4.8 logical :: spinors
If spinors=true, then wannier90 assumes that the WF correspond to singularly occupied spinor states
and num_elec_per_state=1.
The default value is false.
2.4.9 Shells
The MV scheme requires a finite difference expression for ∇kdefined on a uniform Monkhorst-Pack
mesh of k-points. The vectors {b}connect each mesh-point kto its nearest neighbours. Nsh shells of
neighbours are included in the finite-difference formula, with Msvectors in the sth shell. For ∇kto be
correct to linear order, we require that the following equation is satisfied (Eq. B1 of Ref. [1]):
Nsh
X
s
ws
Ms
X
i
bi,s
αbi,s
β=δαβ ,(2.1)
where bi,s,i∈[1, Ms], is the ith vector belonging to the sth shell with associated weight ws, and α
and βrun over the three Cartesian indices.
2.4.10 integer :: shell_list(:)
shell_list is vector listing the shells to include in the finite difference expression. If this keyword is
absent, the shells are chosen automatically.
2.4.11 integer :: search_shells
Specifies the number of shells of neighbours over which to search in attempting to determine an
automatic solution to the B1 condition Eq. 2.1. Larger values than the default may be required in
special cases e.g. for very long thin unit cells.
The default value is 12.
2.4.12 real(kind=dp) :: kmesh_tol
Two kpoints belong to the same shell if the distance between them is less than kmesh_tol. Units are
Ang.
The default value is 0.000001 Ang.
2.5 Projection
The projections block defines a set of localised functions used to generate an initial guess for the unitary
transformations. This data will be written in the seedname.nnkp file to be used by a first-principles
code.
wannier90: User Guide 23
begin projections
.
.
end projections
If guiding_centres=true, then the projection centres are used as the guiding centres in the Wan-
nierisation routine.
For details see Section 3.1.
2.6 Job Control
2.6.1 logical :: postproc_setup
If postproc_setup=true, then the wannier code will write seedname.nnkp file and exit. If wannier90
is called with the option -pp, then postproc_setup is set to true, over-riding its value in the
seedname.win file.
The default value is false.
2.6.2 integer :: iprint
This indicates the level of verbosity of the output from 0, the bare minimum, to 3, which corresponds
to full debugging output.
The default value is 1.
2.6.3 integer :: optimisation
This indicates the level of optimisation used in the code. This is a trade between speed and memory. A
positive number indicates fastest execution time at the cost of more memory. Zero or negative numbers
indicates a smaller memory footprint - at increased execution time.
At the moment the only values that have an effect are optimisation<=0 (low memory) and optimisation>0
(fast)
The default value is 3.
2.6.4 character(len=20) :: length_unit
The length unit to be used for writing quantities in the output file seedname.wout.
The valid options for this parameter are:
–Ang (default)
–Bohr
24 wannier90: User Guide
2.6.5 character(len=50) :: devel_flag
Not a regular keyword. Its purpose is to allow a developer to pass a string into the code to be used
inside a new routine as it is developed.
No default.
2.6.6 integer :: exclude_bands(:)
A k-point independent list of states to excluded from the calculation of the overlap matrices; for example
to select only valence states, or ignore semi-core states. This keyword is passed to the first-principles
code via the seedname.nnkp file. For example, to exclude bands 2, 6, 7, 8 and 12:
exclude_bands : 2, 6-8, 12
2.6.7 character(len=20) :: restart
If restart is present the code will attempt to restart the calculation from the seedname.chk file.
The value of the parameter determines the position of the restart
The valid options for this parameter are:
–default. Restart from the point at which the check file seedname.chk was written
–wannierise. Restart from the beginning of the wannierise routine
–plot. Go directly to the plotting phase
–transport. Go directly to the transport routines
2.6.8 character(len=20) :: wvfn_formatted
If wvfn_formatted=true, then the wavefunctions will be read from disk as formatted (ie ASCII) files;
otherwise they will be read as unformatted files. Unformatted is generally preferable as the files will
take less disk space and I/O is significantly faster. However such files will not be transferable between
all machine architectures and formatted files should be used if transferability is required (i.e., for test
cases).
The default value of this parameter is false.
2.6.9 character(len=20) :: spin
For bands from a spin polarised calculation spin determines which set of bands to read in, either up
or down.
The default value of this parameter is up.
wannier90: User Guide 25
2.6.10 integer :: timing_level
Determines the amount of timing information regarding the calculation that will be written to the
output file. A value of 1 produces the least information.
The default value is 1.
2.6.11 logical :: translate_home_cell
Determines whether to translate the final Wannier centres to the home unit cell at the end of the
calculation. Mainly useful for molecular systems in which the molecule resides entirely within the
home unit cell and user wants to write an xyz file (write_xyz=.true.) for the WF centres to compare
with the structure.
The default value is false.
2.6.12 logical :: write_xyz
Determines whether to write the atomic positions and final Wannier centres to an xyzfile, seedname_centres.xyz,
for subsequent visualisation.
The default value is false.
2.6.13 logical :: write_vdw_data
Determines whether to write seedname.vdw for subsequent post-processing by the w90vdw utility (in
the utility/w90vdw/ directory of the distribution) for calculating van der Waals energies. Brillouin
zone sampling must be at the Gamma-point only.
The default value is false.
2.7 Disentanglement
These keywords control the disentanglement routine of Ref. [2], i.e., the iterative minimisation of ΩI.
This routine will be activated if num_wann <num_bands.
2.7.1 real(kind=dp) :: dis_win_min
The lower bound of the outer energy window for the disentanglement procedure. Units are eV.
The default is the lowest eigenvalue in the system.
2.7.2 real(kind=dp) :: dis_win_max
The upper bound of the outer energy window for the disentanglement procedure. Units are eV.
26 wannier90: User Guide
The default is the highest eigenvalue in the given states (i.e., all states are included in the disentan-
glement procedure).
2.7.3 real(kind=dp) :: dis_froz_min
The lower bound of the inner energy window for the disentanglement procedure. Units are eV.
If dis_froz_max is given, then the default for dis_froz_min is dis_win_min.
2.7.4 real(kind=dp) :: dis_froz_max
The upper bound of the inner (frozen) energy window for the disentanglement procedure. If dis_froz_max
is not specified, then there are no frozen states. Units are eV.
No default.
2.7.5 integer :: dis_num_iter
In the disentanglement procedure, the number of iterations used to extract the most connected sub-
space.
The default value is 200.
2.7.6 real(kind=dp) :: dis_mix_ratio
In the disentanglement procedure, the mixing parameter to use for convergence (see pages 4-5 of
Ref. [2]). A value of 0.5 is a ‘safe’ choice. Using 1.0 (i.e., no mixing) often gives faster convergence,
but may cause the minimisation of ΩIto be unstable in some cases.
Restriction: 0.0<dis_mix_ratio ≤1.0
The default value is 0.5
2.7.7 real(kind=dp) :: dis_conv_tol
In the disentanglement procedure, the minimisation of ΩIis said to be converged if the fractional
change in the gauge-invariant spread between successive iterations is less than dis_conv_tol for
dis_conv_window iterations. Units are Å2.
The default value is 1.0E-10
2.7.8 integer :: dis_conv_window
In the disentanglement procedure, the minimisation is said to be converged if the fractional change in
the spread between successive iterations is less than dis_conv_tol for dis_conv_window iterations.
The default value of this parameter is 3.
wannier90: User Guide 27
2.8 Wannierise
Iterative minimisation of e
Ω, the non-gauge-invariant part of the spread functional.
2.8.1 integer :: num_iter
Total number of iterations in the minimisation procedure. Set num_iter=0 if you wish to generate
projected WFs rather than maximally-localized WFs (see Example 8 in the Tutorial).
The default value is 100
2.8.2 integer :: num_cg_steps
Number of conjugate gradient steps to take before resetting to steepest descents.
The default value is 5
2.8.3 integer :: conv_window
If conv_window>1, then the minimisation is said to be converged if the change in Ωover conv_window
successive iterations is less than conv_tol. Otherwise, the minimisation proceeds for num_iter itera-
tions (default).
The default value is -1
2.8.4 real(kind=dp) :: conv_tol
If conv_window >1, then this is the convergence tolerance on Ω, otherwise not used. Units are Å2.
The default value is 1.0E-10
2.8.5 real(kind=dp) :: conv_noise_amp
If conv_noise_amp>0, once convergence (as defined above) is achieved, some random noise fis added
to the search direction, and the minimisation is continued until convergence is achieved once more. If
the same value of Ωas before is arrived at, then the calculation is considered to be converged. If not,
then random noise is added again and the procedure repeated up to a maximum of conv_noise_num
times. conv_noise_amp is the amplitude of the random noise fthat is added to the search direction:
0<|f|<conv_noise_amp. This functionality requires conv_window >1. If conv_window is not
specified, it is set to the value 5 by default.
If conv_noise_amp ≤0, then no noise is added (default).
The default value is -1.0
28 wannier90: User Guide
2.8.6 integer :: conv_noise_num
If conv_noise_amp >0, then this is the number of times in the minimisation that random noise is
added.
The default value is 3
2.8.7 integer :: num_dump_cycles
Write sufficient information to do a restart every num_dump_cycles iterations.
The default is 100
2.8.8 integer :: num_print_cycles
Write data to the master output file seedname.wout every num_print_cycles iterations.
The default is 1
2.8.9 logical :: write_r2mn
If write_r2mn =true, then the matrix elements hm|r2|ni(where mand nrefer to WF) are written
to file seedname.r2mn at the end of the Wannierisation procedure.
The default value of this parameter is false.
2.8.10 logical :: guiding_centres
Use guiding centres during the minimisation, in order to avoid local minima.
The default value is false.
2.8.11 integer :: num_guide_cycles
If guiding_centres is set to true, then the guiding centres are used only every num_guide_cycles.
The default value is 1.
2.8.12 integer :: num_no_guide_iter
If guiding_centres is set to true, then the guiding centres are used only after num_no_guide_iter
minimisation iterations have been completed.
The default value is 0.
wannier90: User Guide 29
2.8.13 real(kind=dp) :: trial_step
The value of the trial step for the parabolic fit in the line search minimisation used in the minimisation of
the spread function. Cannot be used in conjunction with fixed_step (see below). If the minimisation
procedure doesn’t converge, try decreasing the value of trial_step to give a more accurate line search.
The default value is 2.0
2.8.14 real(kind=dp) :: fixed_step
If this is given a value in the input file, then a fixed step of length fixed_step (instead of a parabolic
line search) is used at each iteration of the spread function minimisation. Cannot be used in conjunction
with trial_step. This can be useful in cases in which minimisation with a line search fails to converge.
There is no default value.
2.8.15 logical :: use_bloch_phases
Determines whether to use the Bloch functions as the initial guess for the projections. Can only be
used if disentanglement = false.
Th default value is false.
2.9 Post-Processing
Capabilities:
–Plot the WF
–Plot the interpolated band structure
–Plot the Fermi surface
–Output the Hamiltonian in the WF basis
–Transport calculation (quantum conductance and density of states)
2.9.1 logical :: wannier_plot
If wannier_plot =true, then the code will write out the Wannier functions in a super-cell whose size is
defined by the variable wannier_plot_supercell, and in a format specified by wannier_plot_format
The default value of this parameter is false.
2.9.2 integer :: wannier_plot_supercell
Dimension of the ‘super-unit-cell’ in which the WF are plotted. The super-unit-cell is wannier_plot_supercell
times the unit cell along all three linear dimensions (the ‘home’ unit cell is kept approximately in the
middle) if wannier_plot_supercell is provided as a single integer.
30 wannier90: User Guide
Otherwise, if three integers are provided, the super-unit-cell is wannier_plot_supercell(i) times the
unit cell along the i−th linear dimension.
The default value is 2×2×2.
2.9.3 character(len=20) :: wannier_plot_format
WF can be plotted in either XCrySDen (xsf) format or Gaussian cube format. The valid options for
this parameter are:
–xcrysden (default)
–cube
If wannier_plot_format=cube: Most visualisation programs (including XCrySDen) are only able to
handle cube files for systems with orthogonal lattice vectors.1wannier90 checks this on reading the
seedname.win and reports an error if cube format has been selected and the lattice vectors are not
mutually orthogonal.
2.9.4 integer :: wannier_plot_list(:)
A list of WF to plot. The WF numbered as per the seedname.wout file after the minimisation of the
spread.
The default behaviour is to plot all WF. For example, to plot WF 4, 5, 6 and 10:
wannier_plot_list : 4-6, 10
2.9.5 character(len=20) :: wannier_plot_mode
Choose the mode in which to plot the WF, either as a molecule or as a crystal. Only relevant if
wannier_plot_format=xcrysden.
The valid options for this parameter are:
–crystal (default)
–molecule
2.9.6 real(kind=dp) :: wannier_plot_radius
If wannier_plot_format is cube, then wannier_plot_radius determines the cut-off radius of the WF
for the purpose of plotting. wannier_plot_radius must be greater than 0. Units are Å.
The default value is 3.5.
1It’s worth noting that the visualisation program VMD (http://www.ks.uiuc.edu/Research/vmd), for example, is
able to deal with certain special cases of non-orthogonal lattice vectors. See http://www.ks.uiuc.edu/Research/vmd/
plugins/molfile/cubeplugin.html. At present wannier90 only supports orthogonal lattice vectors for cube output.
wannier90: User Guide 31
2.9.7 logical :: bands_plot
If bands_plot =true, then the code will calculate the band structure, through Wannier interpolation,
along the path in k-space defined by bands_kpath using bands_num_points along the first section of
the path and write out an output file in a format specified by bands_plot_format.
The default value is false.
2.9.8 kpoint_path
Defines the path in k-space along which to calculate the bandstructure. Each line gives the start and
end point (with labels) for a section of the path. Values are in fractional coordinates with respect to
the primitive reciprocal lattice vectors.
begin kpoint_path
G0.0 0.0 0.0L0.0 0.0 1.0
L0.0 0.0 1.0N0.0 1.0 1.0
.
.
.
end kpoint_path
There is no default
2.9.9 integer :: bands_num_points
If bands_plot =true, then the number of points along the first section of the bandstructure plot
given by kpoint_path. Other sections will have the same density of k-points.
The default value for bands_num_points is 100.
2.9.10 character(len=20) :: bands_plot_format
Format in which to plot the interpolated band structure. The valid options for this parameter are:
–gnuplot (default)
–xmgrace
Note: it is possible to request output in both formats eg bands_format =gnuplot xmgrace
2.9.11 integer :: bands_plot_project(:)
If present wannier90 will compute the contribution of this set of WF to the states at each point of the
interpolated band structure. The WF are numbered according to the seedname.wout file. The result is
written in the seedname_band.dat file, and a corresponding gnuplot script to seedname_band_proj.dat
.
For example, to project on to WFs 2, 6, 7, 8 and 12:
bands_plot_project : 2, 6-8, 12
32 wannier90: User Guide
2.9.12 character(len=20) :: bands_plot_mode
To interpolate the band structure along the k-point path, either use the Slater-Koster interpolation
scheme or truncate the Hamiltonian matrix in the WF basis. Truncation criteria are provided by
hr_cutoff and dist_cutoff.
The valid options for this parameter are:
–s-k (default)
–cut
2.9.13 integer :: bands_plot_dim
Dimension of the system. If bands_plot_dim <3 and bands_plot_mode =cut, lattice vector R=
N1A1+N2A2+N3A3, where Ni= 0 if Aiis parallel to any of the confined directions specified by
one_dim_axis, are exclusively used in the band structure interpolation.
The valid options for this parameter are:
–3 (default)
–2
–1
2.9.14 logical :: fermi_surface_plot
If fermi_surface_plot =true, then the code will calculate, through Wannier interpolation, the
eigenvalues on a regular grid with fermi_surface_num_points in each direction. The code will write
a file in bxsf format which can be read by XCrySDen in order to plot the Fermi surface.
The default value is false.
2.9.15 integer :: fermi_surface_num_points
If fermi_surface_plot =true, then the number of divisions in the regular k-point grid used to
calculate the Fermi surface.
The default value for fermi_surface_num_points is 50.
2.9.16 real(kind=dp) :: fermi_energy
The Fermi energy in eV. This parameter is written into the bxsf file. If fermi_energy is specified,
fermi_energy_min,fermi_energy_max, and fermi_energy_step should not be specified, and vice-
versa.
The default value is 0.0
wannier90: User Guide 33
2.9.17 real(kind=dp) :: fermi_energy_min
Instead of specifyfing a single Fermi energy, it is possible to scan the Fermi level over a range of values,
and recompute certain quantities for each εF.2This is the minimum value in the range (in eV).
There is no default value.
2.9.18 real(kind=dp) :: fermi_energy_max
The maximum value in the range of Fermi energies. Units are eV.
The default value is fermi_energy_min+1.0.
2.9.19 real(kind=dp) :: fermi_energy_step
Difference between consecutive values of the Fermi energy when scanning from fermi_energy_min to
fermi_energy_max. Units are eV.
The default value is 0.01.
2.9.20 character(len=20) :: fermi_surface_plot_format
Format in which to plot the Fermi surface. The valid options for this parameter are:
–xcrysden (default)
2.9.21 logical :: hr_plot
If hr_plot =true, then the Hamiltonian matrix in the WF basis will be written to a file seedname_hr.dat.
The default value is false.
2.9.22 logical :: transport
If transport =true, then the code will calculate quantum conductance and density of states of a
one-dimensional system. The results will be written to files seedname_qc.dat and seedname_dos.dat,
respectively. Since both quantities are a function of energy, they will be evaluated from tran_win_min
to tran_win_max with an interval of tran_energy_step.
The default value of this parameter is false.
2.9.23 character(len=20) :: transport_mode
If transport_mode =bulk, quantum conductance and density of states are calculated for a perfectly-
periodic one-dimensional system. In this case, the transport part can either use the Hamiltonian
2Scanning the Fermi level is currently supported only by the postw90 module berry, for berry_task=ahc,morb. For
all other functionalities that require a knowledge of εF, use fermi_energy instead.
34 wannier90: User Guide
matrix in the WF basis generated by wannier90 or a Hamiltonian matrix provided by the external file
seedname_htB.dat.
If transport_mode =lcr, quantum conductance and density of states are calculated for a system
where semi-infinite, left and right leads are connected through a central conductor region. In this
case, the transport part will work independently from the disentanglement and wannierise procedure.
Details of the method is described in Ref. [7].
If tran_read_ht =true then the Hamiltonian matrices must be provided by the five external files:
seedname_htL.dat, seedname_htLC.dat, seedname_htC.dat, seedname_htCR.dat, seedname_htR.dat.
If tran_read_ht =false then the Hamiltonian matrices are found automatically provided the super-
cell adheres to conditions outlined in Section 7.3.
The valid options for this parameter are:
–bulk (default)
–lcr
2.9.24 real(kind=dp) :: tran_win_min
The lower bound of the energy window for the transport calculation. Units are eV.
The default value is -3.0.
2.9.25 real(kind=dp) :: tran_win_max
The upper bound of the energy window for the transport calculation. Units are eV.
The default value is 3.0.
2.9.26 real(kind=dp) :: tran_energy_step
Sampling interval of the energy values from tran_win_min to tran_win_max. Units are eV.
The default value is 0.01.
2.9.27 real(kind=dp) :: fermi_energy
The Fermi energy in eV. The energy axis of the quantum conductance and density of states data will
be shifted rigidly by this amount.
The default value is 0.0
2.9.28 integer :: tran_num_bb
Size of a bulk Hamiltonian matrix. This number is equal to the number of WFs in one principal layer.
A one-dimensional system can be viewed as an array of principal layers which are defined in a way
that localized basis functions inside a certain principal layer only interact with those in the nearest
wannier90: User Guide 35
neighbor principal layer. In wannier90 a principal layer will be an integer multiple of a unit cell, and
the size is determined by hr_cutoff and/or dist_cutoff. The criterion is rather arbitrary when WFs
are adopted as a localized basis set, and it is up to a user’s choice.
The default value is 0.
2.9.29 integer :: tran_num_ll
Size of a left-lead Hamiltonian matrix. If transport_mode =lcr and tran_read_ht =false then
tran_num_ll is the number of Wannier functions in a principal layer.
The default value is 0.
2.9.30 integer :: tran_num_rr
Size of a right-lead Hamiltonian matrix.
The default value is 0.
2.9.31 integer :: tran_num_cc
Size of a conductor Hamiltonian matrix.
The default value is 0.
2.9.32 integer :: tran_num_lc
Number of columns in a left-lead_conductor Hamiltonian matrix. Number of rows must be equal to
tran_num_ll.
The default value is 0.
2.9.33 integer :: tran_num_cr
Number of rows in a conductor_right-lead Hamiltonian matrix. Number of columns must be equal to
tran_num_rr.
The default value is 0.
2.9.34 integer :: tran_num_cell_ll
Number of unit cells in one principal layer of left lead. Used if transport_mode =lcr and tran_read_ht =
false.
The default value is 0.
36 wannier90: User Guide
2.9.35 integer :: tran_num_cell_rr
Number of unit cells in one principal layer of right lead. Not used at present.
The default value is 0.
2.9.36 integer :: tran_num_bandc
Half-bandwidth+1 of a band-diagonal conductor Hamiltonian matrix.
The Hamiltonian matrix of a central conductor part, which is read from seedname_htC.dat, will
be diagonally dominant when tran_num_cc is very large. tran_num_bandc is used to construct a
compact matrix which contains the non-zero band-diagonal part of a full conductor Hamiltonian matrix.
Setting this parameter is only meaningful when tran_num_bandc is greater than tran_num_lc and
tran_num_cr.
The default value is 0.
2.9.37 logical :: tran_write_ht
If tran_write_ht =true, then the Hamiltonian matrix formatted for the transport calculation will
be written to a file seedname_htB.dat.
The default value is false.
2.9.38 logical :: tran_read_ht
If tran_write_ht =true, then the Hamiltonian matrix formatted for the transport calculation will
be read from a set of files described in the parameter transport_mode. Set tran_write_ht =false
to perform automated lcr calculations (see Section 7.3).
The default value is false.
2.9.39 logical :: tran_use_same_lead
If tran_use_same_lead =true, then the left and the right leads are the same. In this case, seedname_htR.dat
is not required.
The default value is true.
2.9.40 real(kind=dp) :: tran_group_threshold
Used to group and sort Wannier functions according to the positions of their centres. Wannier functions
in a group are within tran_group_threshold from one another in x,y and zdirections. Units are Å
The default is 0.15
wannier90: User Guide 37
2.9.41 real(kind=dp) :: translation_centre_frac(3)
Centre of the unit cell to which the final Wannier centers are translated. Numbers are in fractional
coordinates with respect to the lattice vectors.
The default value is (0.0,0.0,0.0).
2.9.42 real(kind=dp) :: hr_cutoff
The absolute value of the smallest matrix element of the Hamiltonian in the WF basis. If hmn(R)>
hr_cutoff, then the matrix element hmn(R)is retained and used in the band structure interpola-
tion (when bands_plot_mode =cut) or in the transport calculation. Otherwise it is deemed to be
insignificant and is discarded. Units are eV.
The default value is 0.0.
2.9.43 real(kind=dp) :: dist_cutoff
The largest distance between two WFs for which the Hamiltonian matrix element is retained and used
in the band interpolation (when bands_plot_mode =cut) or in the transport calculation. Units are
Å.
The default value is 1000.0.
2.9.44 character(len=20) :: dist_cutoff_mode
Dimension in which the distance between two WFs is calculated. The vector connecting two WFs may
be projected to a line (one_dim) or a plane (two_dim). The size of the projected vector is calculated,
and dist_cutoff is applied. When one_dim or two_dim is used, one_dim_axis must be given to
specify extended or confined direction.
The valid options for this parameter are:
–three_dim (default)
–two_dim
–one_dim
2.9.45 character(len=20) :: one_dim_axis
Extended direction for a one-dimensional system or confined direction for a two-dimensional system.
This direction must be parallel to one of the Cartesian axes.
The valid options for this parameter are:
–x
–y
38 wannier90: User Guide
–z
No default.
Chapter 3
Projections
3.1 Specification of projections in seedname.win
Here we describe the projection functions used to construct the initial guess A(k)
mn for the unitary
transformations.
Each projection is associated with a site and an angular momentum state defining the projection
function. Optionally, one may define, for each projection, the spatial orientation, the radial part, the
diffusivity, and the volume over which real-space overlaps Amn are calculated.
The code is able to
1. project onto s,p,d and f angular momentum states, plus the hybrids sp, sp2, sp3, sp3d, sp3d2.
2. control the radial part of the projection functions to allow higher angular momentum states, e.g.,
both 3s and 4s in silicon.
The atomic orbitals of the hydrogen atom provide a good basis to use for constructing the projec-
tion functions: analytical mathematical forms exist in terms of the good quantum numbers n,l
and m; hybrid orbitals (sp, sp2, sp3, sp3d etc.) can be constructed by simple linear combination
|φi=Pnlm Cnlm|nlmifor some coefficients Cnlm.
The angular functions that use as a basis for the projections are not the canonical spherical harmonics
Ylm of the hydrogenic Schrödinger equation but rather the real (in the sense of non-imaginary) states
Θlmr, obtained by a unitary transformation. For example, the canonical eigenstates associated with
l= 1,m={−1,0,1}are not the real px, pyand pzthat we want. See Section 3.4 for our mathematical
conventions regarding projection orbitals for different n,land mr.
We use the following format to specify projections in <seedname>.win:
Begin Projections
[units]
site:ang_mtm:zaxis:xaxis:radial:zona
.
.
.
End Projections
Notes:
39
40 wannier90: User Guide
units:
Optional. Either Ang or Bohr to specify whether the projection centres specified in this block (if given
in Cartesian co-ordinates) are in units of Angstrom or Bohr, respectively. The default value is Ang.
site:
C,Al, etc. applies to all atoms of that type
f=0,0.50,0 – centre on (0.0,0.5,0.0) in fractional coordinates (crystallographic units) relative to the
direct lattice vectors
c=0.0,0.805,0.0 – centre on (0.0,0.805,0.0) in Cartesian coordinates in units specified by the optional
string units in the first line of the projections block (see above).
ang_mtm:
Angular momentum states may be specified by land mr, or by the appropriate character string. See
Tables 3.1 and 3.2. Examples:
l=2,mr=1 or dz2 – a single projection with l= 2,mr= 1 (i.e., dz2)
l=2,mr=1,4 or dz2,dx2-y2 – two functions: dz2and dxz
l=-3 or sp3 – four sp3hybrids
Specific hybrid orbitals may be specified as follows:
l=-3,mr=1,3 or sp3-1,sp3-3 – two specific sp3hybrids
Multiple states may be specified by separating with ‘;’, e.g.,
sp3;l=0 or l=-3;l=0 – four sp3hybrids and one s orbital
zaxis (optional):
z=1,1,1 – set the z-axis to be in the (1,1,1) direction. Default is z=0,0,1
xaxis (optional):
x=1,1,1 – set the x-axis to be in the (1,1,1) direction. Default is x=1,0,0
radial (optional):
r=2 – use a radial function with one node (ie second highest pseudostate with that angular momentum).
Default is r=1. Radial functions associated with different values of rshould be orthogonal to each other.
zona (optional):
zona=2.0 – the value of Z
afor the radial part of the atomic orbital (controls the diffusivity of the radial
function). Units always in reciprocal Angstrom. Default is zona=1.0.
Examples
1. CuO, s,p and d on all Cu; sp3hybrids on O:
Cu:l=0;l=1;l=2
O:l=-3 or O:sp3
2. A single projection onto a pzorbital orientated in the (1,1,1) direction:
c=0,0,0:l=1,mr=1:z=1,1,1 or c=0,0,0:pz:z=1,1,1
3. Project onto s, p and d (with no radial nodes), and s and p (with one radial node) in silicon:
Si:l=0;l=1;l=2
Si:l=0;l=1:r=2
wannier90: User Guide 41
3.2 Spinor Projections
When spinors=.true. it is possible to select a set of localised functions to project onto ‘up’ states
and a set to project onto ‘down’ states where, for complete flexibility, it is also possible to set the local
spin quantisation axis.
Note, however, that this feature requires a recent version of the interface between the ab-initio code
and Wannier90 (i.e., written after the release of the 2.0 version, in October 2013) supporting spinor
projections.
Begin Projections
[units]
site:ang_mtm:zaxis:xaxis:radial:zona(spin)[quant_dir]
.
.
.
End Projections
spin (optional):
Choose projection onto ‘up’ or ‘down’ states
u– project onto ‘up’ states.
d– project onto ‘down’ states.
Default is u,d
quant_dir (optional):
1,0,0 – set the spin quantisation axis to be in the (1,0,0) direction. Default is 0,0,1
Examples
•18 projections on an iron site
Fe:sp3d2;dxy;dxx;dyz
•same as above
Fe:sp3d2;dxy;dxx;dyz(u,d)
•same as above
Fe:sp3d2;dxy;dxz;dyz(u,d)[0,0,1]
•same as above but quantisation axis is now x
Fe:sp3d2;dxy;dxz;dyz(u,d)[1,0,0]
•now only 9 projections onto up states
Fe:sp3d2;dxy;dxz;dyz(u)
•9 projections onto up-states and 3 on down
Fe:sp3d2;dxy;dxz;dyz(u)
Fe:dxy;dxz;dyz(d)
•projections onto alternate spin states for two lattice sites (Cr1, Cr2)
Cr1:d(u)
Cr2:d(d)
42 wannier90: User Guide
3.3 Short-Cuts
3.3.1 Random projections
It is possible to specify the projections, for example, as follows:
Begin Projections
random
C:sp3
End Projections
in which case wannier90 uses four sp3orbitals centred on each C atom and then chooses the appropriate
number of randomly-centred s-type Gaussian functions for the remaining projection functions. If the
block only consists of the string random and no specific projection centres are given, then all of the
projection centres are chosen randomly.
3.3.2 Bloch phases
Setting use_bloch_phases = true in the input file absolves the user of the need to specify explicit
projections. In this case, the Bloch wave-functions are used as the projection orbitals, namely A(k)
mn =
hψmk|ψnki=δmn.
3.4 Orbital Definitions
The angular functions Θlmr(θ, ϕ)associated with particular values of land mrare given in Tables 3.1
and 3.2.
The radial functions Rr(r)associated with different values of rshould be orthogonal. One choice would
be to take the set of solutions to the radial part of the hydrogenic Schrödinger equation for l= 0, i.e.,
the radial parts of the 1s, 2s, 3s. . . orbitals, which are given in Table 3.3.
wannier90: User Guide 43
l mrName Θlmr(θ, ϕ)
0 1 s1
√4π
1 1 pz q3
4πcos θ
1 2 px q3
4πsin θcos ϕ
1 3 py q3
4πsin θsin ϕ
2 1 dz2 q5
16π(3 cos2θ−1)
2 2 dxz q15
4πsin θcos θcos ϕ
2 3 dyz q15
4πsin θcos θsin ϕ
2 4 dx2-y2 q15
16πsin2θcos 2ϕ
2 5 dxy q15
16πsin2θsin 2ϕ
3 1 fz3 √7
4√π(5 cos3θ−3 cos θ)
3 2 fxz2 √21
4√2π(5 cos2θ−1) sin θcos ϕ
3 3 fyz2 √21
4√2π(5 cos2θ−1) sin θsin ϕ
3 4 fz(x2-y2) √105
4√πsin2θcos θcos 2ϕ
3 5 fxyz √105
4√πsin2θcos θsin 2ϕ
3 6 fx(x2-3y2) √35
4√2πsin3θ(cos2ϕ−3 sin2ϕ) cos ϕ
3 7 fy(3x2-y2) √35
4√2πsin3θ(3 cos2ϕ−sin2ϕ) sin ϕ
Table 3.1: Angular functions Θlmr(θ, ϕ)associated with particular values of land mrfor l≥0.
44 wannier90: User Guide
l mrName Θlmr(θ, ϕ)
−1 1 sp-1 1
√2s+1
√2px
−1 2 sp-2 1
√2s−1
√2px
−2 1 sp2-1 1
√3s−1
√6px +1
√2py
−2 2 sp2-2 1
√3s−1
√6px −1
√2py
−2 3 sp2-3 1
√3s+2
√6px
−3 1 sp3-1 1
2(s+px +py +pz)
−3 2 sp3-2 1
2(s+px −py −pz)
−3 3 sp3-3 1
2(s−px +py −pz)
−3 4 sp3-4 1
2(s−px −py +pz)
−4 1 sp3d-1 1
√3s−1
√6px +1
√2py
−4 2 sp3d-2 1
√3s−1
√6px −1
√2py
−4 3 sp3d-3 1
√3s+2
√6px
−4 4 sp3d-4 1
√2pz +1
√2dz2
−4 5 sp3d-5 −1
√2pz +1
√2dz2
−5 1 sp3d2-1 1
√6s−1
√2px −1
√12 dz2 +1
2dx2-y2
−5 2 sp3d2-2 1
√6s+1
√2px −1
√12 dz2 +1
2dx2-y2
−5 3 sp3d2-3 1
√6s−1
√2py −1
√12 dz2 −1
2dx2-y2
−5 4 sp3d2-4 1
√6s+1
√2py −1
√12 dz2 −1
2dx2-y2
−5 5 sp3d2-5 1
√6s−1
√2pz +1
√3dz2
−5 6 sp3d2-6 1
√6s+1
√2pz +1
√3dz2
Table 3.2: Angular functions Θlmr(θ, ϕ)associated with particular values of land mrfor l < 0, in
terms of the orbitals defined in Table 3.1.
wannier90: User Guide 45
r Rr(r)
12α3/2exp(−αr)
21
2√2α3/2(2 −αr) exp(−αr/2)
3q4
27 α3/2(1 −2αr/3+2α2r2/27) exp(−αr/3)
Table 3.3: One possible choice for the radial functions Rr(r)associated with different values of r:
the set of solutions to the radial part of the hydrogenic Schrödinger equation for l= 0, i.e., the radial
parts of the 1s, 2s, 3s. . . orbitals, where α=Z/a =zona.
Chapter 4
Code Overview
wannier90 can operate in two modes:
1. Post-processing mode: read in the overlaps and projections from file as computed inside a first-
principles code. We expect this to be the most common route to using wannier90, and is
described in Ch. 5;
2. Library mode: as a set of library routines to be called from within a first-principles code that
passes the overlaps and projections to the wannier90 library routines and in return gets the
unitary transformation corresponding to MLWF. This route should be used if the MLWF are
needed within the first-principles code, for example in post-LDA methods such as LDA+U or
SIC, and is described in Ch. 6.
47
48 wannier90: User Guide
Wannier_libWannier_prog
Wannerise PlotOverlap Disentangle Transport
Hamiltonian
Kmesh
Constants
Parameters
io
Utility
Figure 4.1: Schematic overview of the module structure of wannier90. Modules may only use data
and subroutines from lower modules.
Chapter 5
wannier90 as a post-processing tool
This is a description of how to use wannier90 as a post-processing tool.
The code must be run twice. On the first pass either the logical keyword postproc_setup must be set
to .true. in the input file seedname.win or the code must be run with the command line option -pp.
Running the code then generates the file seedname.nnkp which provides the information required to
construct the M(k,b)
mn overlaps (Ref. [1], Eq. (25)) and A(k)
mn (Ref. [1], Eq. (62); Ref. [2], Eq. (22)).
Once the overlaps and projection have been computed and written to files seedname.mmn and seedname.amn,
respectively, set postproc_setup to .false. and run the code. Output is written to the file seedname.wout.
5.1 seedname.nnkp file
OUTPUT, if postproc_setup =.true.
The file seedname.nnkp provides the information needed to determine the required overlap elements
M(k,b)
mn and projections A(k)
mn. It is written automatically when the code is invoked with the -pp
command-line option (or when postproc_setup=.true. in seedname.win. There should be no need
for the user to edit this file.
Much of the information in seedname.nnkp is arranged in blocks delimited by the strings begin block_name
. . . end block_name, as described below.
5.1.1 Keywords
The first line of the file is a user comment, e.g., the date and time:
File written on 12Feb2006 at 15:13:12
The only logical keyword is calc_only_A, eg,
calc_only_A : F
5.1.2 Real_lattice block
begin real_lattice
49
50 wannier90: User Guide
2.250000 0.000000 0.000000
0.000000 2.250000 0.000000
0.000000 0.000000 2.250000
end real_lattice
The real lattice vectors in units of Angstrom.
5.1.3 Recip_lattice block
begin recip_lattice
2.792527 0.000000 0.000000
0.000000 2.792527 0.000000
0.000000 0.000000 2.792527
end recip_lattice
The reciprocal lattice vectors in units of inverse Angstrom.
5.1.4 Kpoints block
begin kpoints
8
0.00000 0.00000 0.00000
0.00000 0.50000 0.00000
.
.
.
0.50000 0.50000 0.50000
end kpoints
The first line in the block is the total number of k-points num_kpts. The subsequent num_kpts lines
specify the k-points in crystallographic co-ordinates relative to the reciprocal lattice vectors.
5.1.5 Projections block
begin projections
n_proj
centre l mr r
z-axis x-axis zona
centre l mr r
z-axis x-axis zona
.
.
end projections
Notes:
n_proj: integer; the number of projection centres, equal to the number of MLWF num_wann.
wannier90: User Guide 51
centre: three real numbers; projection function centre in crystallographic co-ordinates relative to the
direct lattice vectors.
l mr r: three integers; land mrspecify the angular part Θlmr(θ, ϕ), and rspecifies the radial part
Rr(r)of the projection function (see Tables 3.1, 3.2 and 3.3).
z-axis: three real numbers; default is 0.0 0.0 1.0; defines the axis from which the polar angle θin
spherical polar coordinates is measured.
x-axis: three real numbers; must be orthogonal to z-axis; default is 1.0 0.0 0.0 or a vector per-
pendicular to z-axis if z-axis is given; defines the axis from with the azimuthal angle ϕin spherical
polar coordinates is measured.
zona: real number; the value of Z
aassociated with the radial part of the atomic orbital. Units are in
reciprocal Angstrom.
5.1.6 spinor_projections block
begin spinor_projections
n_proj
centre l mr r
z-axis x-axis zona
spin spn_quant
centre l mr r
z-axis x-axis zona
spin spn_quant
.
.
end spinor_projections
Notes: Only one of projections and spinor_projections should be defined. Variables are the same as
the projections block with the addition of spin and spn_quant.
spin: integer. ‘1’ or ‘-1’ to denote projection onto up or down states.
spn_quant: three real numbers. Defines the spin quantisation axis in Cartesian coordinates.
5.1.7 nnkpts block
begin nnkpts
10
1 2 0 0 0
.
.
end nnkpts
First line: nntot, the number of nearest neighbours belonging to each k-point of the Monkhorst-Pack
mesh
Subsequent lines: nntot×num_kpts lines, ie, nntot lines of data for each k-point of the mesh.
52 wannier90: User Guide
Each line of consists of 5 integers. The first is the k-point number nkp. The second to the fifth specify
it’s nearest neighbours k+b: the second integer points to the k-point that is the periodic image of the
k+bthat we want; the last three integers give the G-vector, in reciprocal lattice units, that brings
the k-point specified by the second integer (which is in the first BZ) to the actual k+bthat we need.
5.1.8 exclude_bands block
begin exclude_bands
8
1
2
.
.
end exclude_bands
To exclude bands (independent of k-point) from the calculation of the overlap and projection matrices,
for example to ignore shallow-core states. The first line is the number of states to exclude, the following
lines give the states for be excluded.
5.1.9 An example of projections
As a concrete example: one wishes to have a set of four sp3projection orbitals on, say, a carbon atom at
(0.5,0.5,0.5) in fractional co-ordinates relative to the direct lattice vectors. In this case seedname.win
will contain the following lines:
begin projections
C:l=-1
end projections
and seedname.nnkp, generated on the first pass of wannier90 (with postproc_setup=T), will contain:
begin projections
4
0.50000 0.50000 0.50000 -1 1 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.50000 0.50000 0.50000 -1 2 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.50000 0.50000 0.50000 -1 3 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.50000 0.50000 0.50000 -1 4 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
end projections
where the first line tells us that in total four projections are specified, and the subsquent lines provide
the projection centre, the angular and radial parts of the orbital (see Section 3.4 for definitions), the
zand xaxes, and the diffusivity and cut-off radius for the projection orbital.
pwscf, or any other ab initio electronic structure code, then reads seedname.nnkp file, calculates the
projections and writes them to seedname.amn.
wannier90: User Guide 53
5.2 seedname.mmn file
INPUT.
The file seedname.mmn contains the overlaps M(k,b)
mn .
First line: a user comment, e.g., the date and time
Second line: 3 integers: num_bands,num_kpts,nntot
Then: num_kpts ×nntot blocks of data:
First line of each block: 5 integers. The first specifies the k(i.e., gives the ordinal corresponding to
its position in the list of k-points in seedname.win). The 2nd to 5th integers specify k+b. The
2nd integer, in particular, points to the k-point on the list that is a periodic image of k+b, and in
particular is the image that is actually mentioned in the list. The last three integers specify the G
vector, in reciprocal lattice units, that brings the k-point specified by the second integer, and that thus
lives inside the first BZ zone, to the actual k+bthat we need.
Subsequent num_bands ×num_bands lines of each block: two real numbers per line. These are the real
and imaginary parts, respectively, of the actual scalar product M(k,b)
mn for m, n ∈[1,num_bands]. The
order of these elements is such that the first index mis fastest.
5.3 seedname.amn file
INPUT.
The file seedname.amn contains the projection A(k)
mn.
First line: a user comment, e.g., the date and time
Second line: 3 integers: num_bands,num_kpts,num_wann
Subsequently num_bands ×num_wann ×num_kpts lines: 3 integers and 2 real numbers on each line.
The first two integers are the band indices mand n. The third integer specifies the kby giving the
ordinal corresponding to its position in the list of k-points in seedname.win. The real numbers are the
real and imaginary parts, respectively, of the actual A(k)
mn.
5.4 seedname.eig file
INPUT.
Required if any of disentanglement,plot_bands,plot_fermi_surface or hr_plot are .true.
The file seedname.eig contains the Kohn-Sham eigenvalues εnk(in eV) at each point in the Monkhorst-
Pack mesh.
Each line consist of two integers and a real number. The first integer is the band index, the second
integer gives the ordinal corresponding to the k-point in the list of k-points in seedname.win, and the
real number is the eigenvalue.
E.g.,
54 wannier90: User Guide
1 1 -6.43858831271328
2 1 19.3977795287297
3 1 19.3977795287297
4 1 19.3977795287298
5.5 Interface with pwscf
Interfaces between wannier90 and many ab-initio codes as pwscf,abinit (http://www.abinit.org),
siesta (http://www.icmab.es/siesta/), fleur,VASP and Wien2k (http://www.wien2k.at) are
available. Here we describe the seamless interface between wannier90 and pwscf, a plane-wave DFT
code that comes as part of the Quantum ESPRESSO package (see http://www.quantum-espresso.
org). You will need to download and compile pwscf (i.e., the pw.x code) and the post-processing inter-
face pw2wannier90.x. Please refer to the documentation that comes with the Quantum ESPRESSO
distribution for instructions.
1. Run ‘scf’/‘nscf’ calculation(s) with pw
2. Run wannier90 with postproc_setup =.true. to generate seedname.nnkp
3. Run pw2wannier90. First it reads an input file, e.g., seedname.pw2wan, which defines prefix
and outdir for the underlying ‘scf’ calculation, as well as the name of the file seedname.nnkp, and
does a consistency check between the direct and reciprocal lattice vectors read from seedname.nnkp
and those defined in the files specified by prefix.pw2wannier90 generates seedname.mmn,
seedname.amn and seedname.eig
4. Run wannier90 with postproc_setup =.false. to disentangle bands (if required), localise
MLWF, and use MLWF for plotting, bandstructures, Fermi surfaces etc.
Examples of how the interface with pwscf works are given in the wannier90 Tutorial.
5.5.1 seedname.pw2wan
A number of keywords may be specified in the pw2wannier90 input file:
•outdir – Location to write output files. Default is ‘./’
•prefix – Prefix for the pwscf calculation. Default is ‘ ’
•seedname – Seedname for the wannier90 calculation. Default is ‘wannier’
•spin_component – Spin component. Takes values ‘up’,‘down’ or ‘none’ (default).
•wan_mode – Either ‘standalone’ (default) or ‘library’
•write_unk – Set to .true. to write the periodic part of the Bloch functions for plotting in
wannier90. Default is .false.
•reduce_unk – Set to .true. to reduce file-size (and resolution) of Bloch functions by a factor of
8. Default is .false. (only relevant if write_unk=.true.)1
1Note that there is a small bug with this feature in v3.2 (and subsequent patches) of quantum-espresso. Please use
a later version (if available) or the CVS version of pw2wannier90.f90, which has been fixed.
wannier90: User Guide 55
•wvfn_formatted – Set to .true. to write formatted wavefunctions. Default is .false. (only
relevant if write_unk=.true.)
•write_amn – Set to .false. if A(k)
mn not required. Default is .true.
•write_mmn – Set to .false. if M(k,b)
mn not required. Default is .true.
•write_spn – Set to .true. to write out the matrix elements of Sbetween Bloch states (non-
collinear spin calculation only). Default is .false.
•spn_formatted – Set to .true. to write spn data as a formatted file. Default is .false. (only
relevant if write_spn=.true.)
•write_uHu – Set to .true. to write out the matrix elements
hunk+b1|Hk|umk+b2i.
Default is .false.
•uHu_formatted – Set to .true. to write uHu data as a formatted file. Default is .false. (only
relevant if write_uHu=.true.)
•write_uIu – Set to .true. to write out the matrix elements of
hunk+b1|umk+b2i.
Default is .false.
•uIu_formatted – Set to .true. to write uIu data as a formatted file. Default is .false. (only
relevant if write_uIu=.true.)
•write_unkg – Set to .true. to write the first few Fourier components of the periodic parts of
the Bloch functions.
For examples of use, refer to the wannier90 Tutorial.
Chapter 6
wannier90 as a library
This is a description of the interface between any external program and the wannier code. There
are two subroutines: wannier_setup and wannier_run. Calling wannier_setup will return informa-
tion required to construct the M(k,b)
mn overlaps (Ref. [1], Eq. (25)) and A(k)
mn =hψmk|gniprojections
(Ref. [1], Eq. (62); Ref. [2], Eq. (22)). Once the overlaps and projection have been computed, calling
wannier_run activates the minimisation and plotting routines in wannier90.
6.1 Subroutines
6.1.1 wannier_setup
wannier_setup(seed_name,mp_grid,num_kpts,real_lattice,recip_lattice,
kpt_latt,num_bands_tot,num_atoms,atom_symbols,atoms_cart,
gamma_only,spinors,nntot,nnlist,nncell,num_bands,num_wann,proj_site,
proj_l,proj_m,proj_radial,proj_z,proj_x,proj_zona,
exclude_bands,proj_s,proj_s_qaxis)
•character(len=*), intent(in) :: seed_name
The seedname of the current calculation.
•integer, dimension(3), intent(in) :: mp_grid
The dimensions of the Monkhorst-Pack k-point grid.
•integer, intent(in) :: num_kpts
The number of k-points on the Monkhorst-Pack grid.
•real(kind=dp), dimension(3,3), intent(in) :: real_lattice
The lattice vectors in Cartesian co-ordinates in units of Angstrom.
•real(kind=dp), dimension(3,3), intent(in) :: recip_lattice
The reciprocal lattice vectors in Cartesian co-ordinates in units of reciprocal Angstrom.
•real(kind=dp), dimension(3,num_kpts), intent(in) :: kpt_latt
The positions of the k-points in fractional co-ordinates relative to the reciprocal lattice vectors.
•integer, intent(in) :: num_bands_tot
The total number of bands in the first-principles calculation (note: including semi-core states).
57
58 wannier90: User Guide
•integer, intent(in) :: num_atoms
The total number of atoms in the system.
•character(len=20), dimension(num_atoms), intent(in) :: atom_symbols
The elemental symbols of the atoms.
•real(kind=dp), dimension(3,num_atoms), intent(in) :: atoms_cart
The positions of the atoms in Cartesian co-ordinates in Angstrom.
•logical, intent(in) :: gamma_only
Set to .true. if the underlying electronic structure calculation has been performed with only
Γ-point sampling and, hence, if the Bloch eigenstates that are used to construct A(k)
mn and M(k,b)
mn
are real.
•logical, intent(in) :: spinors
Set to .true. if underlying electronic structure calculation has been performed with spinor
wavefunctions.
•integer, intent(out) :: nntot
The total number of nearest neighbours for each k-point.
•integer, dimension(num_kpts,num_nnmax), intent(out) :: nnlist
The list of nearest neighbours for each k-point.
•integer,dimension(3,num_kpts,num_nnmax), intent(out) :: nncell
The vector, in fractional reciprocal lattice co-ordinates, that brings the nnth nearest neighbour
of k-point nkp to its periodic image that is needed for computing the overlap M(k,b)
mn .
•integer, intent(out) :: num_bands
The number of bands in the first-principles calculation used to form the overlap matricies (note:
excluding eg. semi-core states).
•integer, intent(out) :: num_wann
The number of MLWF to be extracted.
•real(kind=dp), dimension(3,num_bands_tot), intent(out) :: proj_site
Projection function centre in crystallographic co-ordinates relative to the direct lattice vectors.
•integer, dimension(num_bands_tot), intent(out) :: proj_l
lspecifies the angular part Θlmr(θ, ϕ)of the projection function (see Tables 3.1, 3.2 and 3.3).
•integer, dimension(num_bands_tot), intent(out) :: proj_m
mrspecifies the angular part Θlmr(θ, ϕ), of the projection function (see Tables 3.1, 3.2 and 3.3).
•integer, dimension(num_bands_tot), intent(out) :: proj_radial
rspecifies the radial part Rr(r)of the projection function (see Tables 3.1, 3.2 and 3.3).
•real(kind=dp), dimension(3,num_bands_tot), intent(out) :: proj_z
Defines the axis from which the polar angle θin spherical polar coordinates is measured. Default
is 0.0 0.0 1.0.
•real(kind=dp), dimension(3,num_bands_tot), intent(out) :: proj_x
Must be orthogonal to z-axis; default is 1.0 0.0 0.0 or a vector perpendicular to proj_z if
proj_z is given; defines the axis from with the azimuthal angle ϕin spherical polar coordinates
is measured.
wannier90: User Guide 59
•real(kind=dp), dimension(num_bands_tot), intent(out) :: proj_zona
The value of Z
aassociated with the radial part of the atomic orbital. Units are in reciprocal
Angstrom.
•integer, dimension(num_bands_tot), intent(out) :: exclude_bands
Kpoints independant list of bands to exclude from the calculation of the MLWF (e.g., semi-core
states).
•integer, dimension(num_bands_tot), optional,intent(out) :: proj_s
’1’ or ’-1’ to denote projection onto up or down spin states
•real(kind=dp), dimension(3,num_bands_tot), intent(out) :: proj_s_qaxisx
Defines the spin quantisation axis in Cartesian coordinates.
Conditions:
?num_kpts =mp_grid(1) ×mp_grid(2) ×mp_grid(3).
?num_nnmax = 12
This subroutine returns the information required to determine the required overlap elements M(k,b)
mn
and projections A(k)
mn, i.e., M_matrix and A_matrix, described in Section 6.1.2.
For the avoidance of doubt, real_lattice(1,2) is the y−component of the first lattice vector A1, etc.
The list of nearest neighbours of a particular k-point nkp is given by nnlist(nkp,1:nntot).
Additionally, the parameter shell_list may be specified in the wannier90 input file.
6.1.2 wannier_run
wannier_run(seed_name,mp_grid,num_kpts,real_lattice,recip_lattice,
kpt_latt,num_bands,num_wann,nntot,num_atoms,atom_symbols,
atoms_cart,gamma_only,M_matrix_orig,A_matrix,eigenvalues,
U_matrix,U_matrix_opt,lwindow,wann_centres,wann_spreads,
spread)
•character(len=*), intent(in) :: seed_name
The seedname of the current calculation.
•integer, dimension(3), intent(in) :: mp_grid
The dimensions of the Monkhorst-Pack k-point grid.
•integer, intent(in) :: num_kpts
The number of k-points on the Monkhorst-Pack grid.
•real(kind=dp), dimension(3,3), intent(in) :: real_lattice
The lattice vectors in Cartesian co-ordinates in units of Angstrom.
•real(kind=dp), dimension(3,3), intent(in) :: recip_lattice
The reciprical lattice vectors in Cartesian co-ordinates in units of inverse Angstrom.
60 wannier90: User Guide
•real(kind=dp), dimension(3,num_kpts), intent(in) :: kpt_latt
The positions of the k-points in fractional co-ordinates relative to the reciprocal lattice vectors.
•integer, intent(in) :: num_bands
The total number of bands to be processed.
•integer, intent(in) :: num_wann
The number of MLWF to be extracted.
•integer, intent(in) :: nntot
The number of nearest neighbours for each k-point.
•integer, intent(in) :: num_atoms
The total number of atoms in the system.
•character(len=20), dimension(num_atoms), intent(in) :: atom_symbols
The elemental symbols of the atoms.
•real(kind=dp), dimension(3,num_atoms), intent(in) :: atoms_cart
The positions of the atoms in Cartesian co-ordinates in Angstrom.
•logical, intent(in) :: gamma_only
Set to .true. if the underlying electronic structure calculation has been performed with only
Γ-point sampling and, hence, if the Bloch eigenstates that are used to construct A(k)
mn and M(k,b)
mn
are real.
•complex(kind=dp), dimension(num_bands,num_bands,nntot,num_kpts),
intent(in) :: M_matrix
The matrices of overlaps between neighbouring periodic parts of the Bloch eigenstates at each
k-point, M((k,b))
mn (Ref. [1], Eq. (25)).
•complex(kind=dp), dimension(num_bands,num_wann,num_kpts),
intent(in) :: A_matrix
The matrices describing the projection of num_wann trial orbitals on num_bands Bloch states at
each k-point, A(k)
mn (Ref. [1], Eq. (62); Ref. [2], Eq. (22)).
•real(kind=dp), dimension(num_bands,num_kpts), intent(in) :: eigenvalues
The eigenvalues εnkcorresponding to the eigenstates, in eV.
•complex(kind=dp), dimension(num_wann,num_wann,num_kpts),
intent(out) :: U_matrix
The unitary matrices at each k-point (Ref. [1], Eq. (59))
•complex(kind=dp), dimension(num_bands,num_wann,num_kpts),
optional, intent(out) :: U_matrix_opt
The unitary matrices that describe the optimal sub-space at each k-point (see Ref. [2], Sec-
tion IIIa). The array is packed (see below)
•logical, dimension(num_bands,num_kpts), optional, intent(out) :: lwindow
The element lwindow(nband,nkpt) is .true. if the band nband lies within the outer energy
window at kpoint nkpt.
•real(kind=dp), dimension(3,num_wann), optional, intent(out) :: wann_centres
The centres of the MLWF in Cartesian co-ordinates in Angstrom.
wannier90: User Guide 61
•real(kind=dp), dimension(num_wann), optional, intent(out) :: wann_spreads
The spread of each MLWF in Å2.
•real(kind=dp), dimension(3), optional, intent(out) :: spread
The values of Ω,ΩIand ˜
Ω(Ref. [1], Eq. (13)).
Conditions:
?num_wann ≤num_bands
?num_kpts =mp_grid(1) ×mp_grid(2) ×mp_grid(3).
If num_bands =num_wann then U_matrix_opt is the identity matrix and lwindow=.true.
For the avoidance of doubt, real_lattice(1,2) is the y−component of the first lattice vector A1, etc.
M_matrix(m,n,nn,nkp) =humk|unk+bi
A_matrix(m,n,nkp) =hψmk|gni
eigenvalues(n,nkp) =εnk
where
k=kpt_latt(1:3,nkp)
k+b=kpt_latt(1:3,nnlist(nkp,nn)) +nncell(1:3,nkp,nn)
and {|gni} are a set of initial trial orbitals. These are typically atom or bond-centred Gaussians that
are modulated by appropriate spherical harmonics.
Additional parameters should be specified in the wannier90 input file.
Chapter 7
Transport Calculations with wannier90
By setting transport =TRUE,wannier90 will calculate the quantum conductance and density of states
of a one-dimensional system. The results will be written to files seedname_qc.dat and seedname_dos.dat,
respectively.
The system for which transport properties are calculated is determined by the keyword transport_mode.
7.1 transport_mode = bulk
Quantum conductance and density of states are calculated for a perfectly periodic one-dimensional
conductor. If tran_read_ht =FALSE the transport properties are calculated using the Hamiltonian in
the Wannier function basis of the system found by wannier90. Setting tran_read_ht =TRUE allows
the user to provide an external Hamiltonian matrix file seedname_htB.dat, from which the properties
are found. See Section 2.9 for more details of the keywords required for such calculations.
7.2 transport_mode = lcr
Quantum conductance and density of states are calculated for a system where semi-infinite, left and
right leads are connected through a central conductor region. This is known as the lcr system. Details
of the method is described in Ref. [7].
In wannier90 two options exist for performing such calculations:
•If tran_read_ht =TRUE the external Hamiltonian files seedname_htL.dat, seedname_htLC.dat,
seedname_htC.dat, seedname_htCR.dat, seedname_htR.dat are read and used to compute the
transport properties.
•If tran_read_ht =FALSE, then the transport calculation is performed automatically using the
Wannier functions as a basis and the 2c2 geometry described in Section 7.3.
63
64 wannier90: User Guide
7.3 Automated lcr Transport Calculations: The 2c2 Geometry
Calculations using the 2c2 geometry provide a method to calculate the transport properties of an lcr
system from a single wannier90 calculation. The Hamiltonian matrices which the five external files
provide in the tran_read_ht =TRUE case are instead built from the Wannier function basis directly.
As such, strict rules apply to the system geometry, which is shown in Figure 7.1. These rules are as
follows:
•Left and right leads must be identical and periodic.
•Supercell must contain two principal layers (PLs) of lead on the left, a central conductor region
and two principal layers of lead on the right.
•The conductor region must contain enough lead such that the disorder does not affect the principal
layers of lead either side.
•A single k-point (Gamma) must be used.
PL3 PL4PL1 ConductorPL2
h
H
LC
L
00
hCR
10
L
H , HR
01
C
H
PL1
Figure 7.1: Schematic illustration of the supercell required for 2c2 lcr calculations, showing where
each of the Hamiltonian matrices are derived from. Four principal layers (PLs) are required plus the
conductor region.
In order to build the Hamiltonians, Wannier functions are first sorted according to position and then
type if a number of Wannier functions exist with a similar centre (eg. d-orbital type Wannier functions
centred on a Cu atom). Next, consistent parities of Wannier function are enforced. To distingiush
between different types of Wannier function and assertain relative parities, a signature of each Wannier
function is computed. The signature is formed of 20 integrals which have different spatial dependence.
They are given by:
I=1
VZV
g(r)w(r)dr(7.1)
where Vis the volume of the cell, w(r)is the Wannier function and g(r)are the set of functions:
g(r) = n1,sin 2π(x−xc)
Lx,sin 2π(y−yc)
Ly,sin 2π(z−zc)
Lz,sin 2π(x−xc)
Lxsin 2π(y−yc)
Ly,
sin 2π(x−xc)
Lxsin 2π(z−zc)
Lz, ...o(7.2)
upto third order in powers of sines. Here, the supercell has dimension (Lx, Ly, Lz)and the Wannier
function has centre rc= (xc, yc, zc). Each of these integrals may be written as linear combinations of
the following sums:
wannier90: User Guide 65
Sn(G) = eiG.rcX
m
Umn ˜u∗
mΓ(G)(7.3)
where nand mare the Wannier function and band indexes, Gis a G-vector, Umn is the unitary matrix
that transforms from the Bloch reopresentation of the system to the maximally-localised Wannier
function basis and ˜u∗
mΓ(G)are the conjugates of the Fourier transforms of the periodic parts of the
Bloch states at the Γ-point. The complete set of ˜umk(G)are often outputted by plane-wave DFT
codes. However, to calculate the 20 signature integrals, only 32 specific ˜umk(G)are required. These
are found in an additional file (seedname.unkg) that should be provided by the interface between the
DFT code and wannier90 . A detailed description of this file may be found in Section 8.27.
Additionally, the following keywords are also required in the input file:
•tran_num_ll : The number of Wannier functions in a principal layer.
•tran_num_cell_ll : The number of unit cells in one principal layer of lead
A further parameter related to these calculations is tran_group_threshold.
Examples of how 2c2 calculations are preformed can be found in the wannier90 Tutorial.
Chapter 8
Files
8.1 seedname.win
INPUT. The master input file; contains the specification of the system and any parameters for the run.
For a description of input parameters, see Chapter 2; for examples, see Section 9.1 and the wannier90
Tutorial.
8.1.1 Units
The following are the dimensional quantities that are specified in the master input file:
•Direct lattice vectors
•Positions (of atomic or projection) centres in real space
•Energy windows
•Positions of k-points in reciprocal space
•Convergence thresholds for the minimisation of Ω
•zona (see Section 3.1)
•wannier_plot_cube: cut-off radius for plotting WF in Gaussian cube format
Notes:
•The units (either ang (default) or bohr) in which the lattice vectors, atomic positions or projection
centres are given can be set in the first line of the blocks unit_cell_cart,atoms_cart and
projections, respectively, in seedname.win.
•Energy is always in eV.
•Convergence thresholds are always in Å2
•Positions of k-points are always in crystallographic coordinates relative to the reciprocal lattice
vectors.
67
68 wannier90: User Guide
•zona is always in reciprocal Angstrom (Å−1)
•The keyword length_unit may be set to ang (default) or bohr, in order to set the units in which
the quantities in the output file seedname.wout are written.
•wannier_plot_radius is in Angstrom
The reciprocal lattice vectors {B1,B2,B3}are defined in terms of the direct lattice vectors {A1,A2,A3}
by the equation
B1=2π
ΩA2×A3etc., (8.1)
where the cell volume is V=A1·(A2×A3).
8.2 seedname.mmn
INPUT. Written by the underlying electronic structure code. See Chapter 5 for details.
8.3 seedname.amn
INPUT. Written by the underlying electronic structure code. See Chapter 5 for details.
8.4 seedname.eig
INPUT. Written by the underlying electronic structure code. See Chapter 5 for details.
8.5 seedname.nnkp
OUTPUT. Written by wannier90 when postproc_setup=.TRUE. (or, alternatively, when wannier90
is run with the -pp command-line option). See Chapter 5 for details.
8.6 seedname.wout
OUTPUT. The master output file. Here we give a description of the main features of the output. The
verbosity of the output is controlled by the input parameter iprint. The higher the value, the more
detail is given in the output file. The default value is 1, which prints minimal information.
8.6.1 Header
The header provides some basic information about wannier90, the authors, and the execution time of
the current run.
wannier90: User Guide 69
+---------------------------------------------------+
| |
| WANNIER90 |
| |
+---------------------------------------------------+
| |
| Welcome to the Maximally-Localized |
| Generalized Wannier Functions code |
| http://www.wannier.org |
| |
| Wannier90 v2.0 Authors: |
| Arash A. Mostofi (Imperial College London) |
| Giovanni Pizzi (EPFL) |
| Ivo Souza (Universidad del Pais Vasco) |
| Jonathan R. Yates (University of Oxford) |
| |
| Wannier90 Contributors: |
| Young-Su Lee (KIST, S. Korea) |
| Matthew Shelley (Imperial College London) |
| Nicolas Poilvert (Harvard) |
| |
| Wannier77 Authors: |
| Nicola Marzari (EPFL) |
| Ivo Souza (Universidad del Pais Vasco) |
| David Vanderbilt (Rutgers University) |
| |
.
.
| Copyright (c) 1996-2013 |
| A. A. Mostofi, J. R. Yates, Y.-S. Lee, |
| I. Souza, D. Vanderbilt and N. Marzari |
| |
| Release: 2.0 14th October 2013 |
.
.
| |
+---------------------------------------------------+
| Execution started on 8Oct2013 at 18:39:42 |
+---------------------------------------------------+
8.6.2 System information
This part of the output file presents information that wannier90 has read or inferred from the master
input file seedname.win. This includes real and reciprocal lattice vectors, atomic positions, k-points,
parameters for job control, disentanglement, localisation and plotting.
------
70 wannier90: User Guide
SYSTEM
------
Lattice Vectors (Ang)
a_1 3.938486 0.000000 0.000000
a_2 0.000000 3.938486 0.000000
a_3 0.000000 0.000000 3.938486
Unit Cell Volume: 61.09251 (Ang^3)
Reciprocal-Space Vectors (Ang^-1)
b_1 1.595330 0.000000 0.000000
b_2 0.000000 1.595330 0.000000
b_3 0.000000 0.000000 1.595330
*----------------------------------------------------------------------------*
| Site Fractional Coordinate Cartesian Coordinate (Ang) |
+----------------------------------------------------------------------------+
| Ba 1 0.00000 0.00000 0.00000 | 0.00000 0.00000 0.00000 |
| Ti 1 0.50000 0.50000 0.50000 | 1.96924 1.96924 1.96924 |
.
.
*----------------------------------------------------------------------------*
------------
K-POINT GRID
------------
Grid size = 4 x 4 x 4 Total points = 64
*---------------------------------- MAIN ------------------------------------*
| Number of Wannier Functions : 9 |
| Number of input Bloch states : 9 |
| Output verbosity (1=low, 5=high) : 1 |
| Length Unit : Ang |
| Post-processing setup (write *.nnkp) : F |
.
.
*----------------------------------------------------------------------------*
8.6.3 Nearest-neighbour k-points
This part of the output files provides information on the b-vectors and weights chosen to satisfy the
condition of Eq. 2.1.
*---------------------------------- K-MESH ----------------------------------*
+----------------------------------------------------------------------------+
| Distance to Nearest-Neighbour Shells |
| ------------------------------------ |
wannier90: User Guide 71
| Shell Distance (Ang^-1) Multiplicity |
| ----- ----------------- ------------ |
| 1 0.398833 6 |
| 2 0.564034 12 |
.
.
+----------------------------------------------------------------------------+
| The b-vectors are chosen automatically |
| The following shells are used: 1 |
+----------------------------------------------------------------------------+
| Shell # Nearest-Neighbours |
| ----- -------------------- |
| 1 6 |
+----------------------------------------------------------------------------+
| Completeness relation is fully satisfied [Eq. (B1), PRB 56, 12847 (1997)] |
+----------------------------------------------------------------------------+
8.6.4 Disentanglement
Then (if required) comes the part where ΩIis minimised to disentangle the optimally-connected sub-
space of states for the localisation procedure in the next step.
First, a summary of the energy windows that are being used is given:
*------------------------------- DISENTANGLE --------------------------------*
+----------------------------------------------------------------------------+
| Energy Windows |
| --------------- |
| Outer: 2.81739 to 38.00000 (eV) |
| Inner: 2.81739 to 13.00000 (eV) |
+----------------------------------------------------------------------------+
Then, each step of the iterative minimisation of ΩIis reported.
Extraction of optimally-connected subspace
------------------------------------------
+---------------------------------------------------------------------+<-- DIS
| Iter Omega_I(i-1) Omega_I(i) Delta (frac.) Time |<-- DIS
+---------------------------------------------------------------------+<-- DIS
1 3.82493590 3.66268867 4.430E-02 0.36 <-- DIS
2 3.66268867 3.66268867 6.911E-15 0.37 <-- DIS
.
.
<<< Delta < 1.000E-10 over 3 iterations >>>
<<< Disentanglement convergence criteria satisfied >>>
Final Omega_I 3.66268867 (Ang^2)
72 wannier90: User Guide
+----------------------------------------------------------------------------+
The first column gives the iteration number. For a description of the minimisation procedure and
expressions for Ω(i)
I, see the original paper [2]. The procedure is considered to be converged when
the fractional difference between Ω(i)
Iand Ω(i−1)
Iis less than dis_conv_tol over dis_conv_window
iterations. The final column gives a running account of the wall time (in seconds) so far. Note that
at the end of each line of output, there are the characters “<– DIS”. This enables fast searching of the
output using, for example, the Unix command grep:
my_shell> grep DIS wannier.wout | less
8.6.5 Wannierisation
The next part of the input file provides information on the minimisation of e
Ω. At each iteration, the
centre and spread of each WF is reported.
*------------------------------- WANNIERISE ---------------------------------*
+--------------------------------------------------------------------+<-- CONV
| Iter Delta Spread RMS Gradient Spread (Ang^2) Time |<-- CONV
+--------------------------------------------------------------------+<-- CONV
------------------------------------------------------------------------------
Initial State
WF centre and spread 1 ( 0.000000, 1.969243, 1.969243 ) 1.52435832
WF centre and spread 2 ( 0.000000, 1.969243, 1.969243 ) 1.16120620
.
.
0 0.126E+02 0.0000000000 12.6297685260 0.29 <-- CONV
O_D= 0.0000000 O_OD= 0.1491718 O_TOT= 12.6297685 <-- SPRD
------------------------------------------------------------------------------
Cycle: 1
WF centre and spread 1 ( 0.000000, 1.969243, 1.969243 ) 1.52414024
WF centre and spread 2 ( 0.000000, 1.969243, 1.969243 ) 1.16059775
.
.
Sum of centres and spreads ( 11.815458, 11.815458, 11.815458 ) 12.62663472
1 -0.313E-02 0.0697660962 12.6266347170 0.34 <-- CONV
O_D= 0.0000000 O_OD= 0.1460380 O_TOT= 12.6266347 <-- SPRD
Delta: O_D= -0.4530841E-18 O_OD= -0.3133809E-02 O_TOT= -0.3133809E-02 <-- DLTA
------------------------------------------------------------------------------
Cycle: 2
WF centre and spread 1 ( 0.000000, 1.969243, 1.969243 ) 1.52414866
WF centre and spread 2 ( 0.000000, 1.969243, 1.969243 ) 1.16052405
.
.
Sum of centres and spreads ( 11.815458, 11.815458, 11.815458 ) 12.62646411
2 -0.171E-03 0.0188848262 12.6264641055 0.38 <-- CONV
wannier90: User Guide 73
O_D= 0.0000000 O_OD= 0.1458674 O_TOT= 12.6264641 <-- SPRD
Delta: O_D= -0.2847260E-18 O_OD= -0.1706115E-03 O_TOT= -0.1706115E-03 <-- DLTA
------------------------------------------------------------------------------
.
.
------------------------------------------------------------------------------
Final State
WF centre and spread 1 ( 0.000000, 1.969243, 1.969243 ) 1.52416618
WF centre and spread 2 ( 0.000000, 1.969243, 1.969243 ) 1.16048545
.
.
Sum of centres and spreads ( 11.815458, 11.815458, 11.815458 ) 12.62645344
Spreads (Ang^2) Omega I = 12.480596753
================ Omega D = 0.000000000
Omega OD = 0.145856689
Final Spread (Ang^2) Omega Total = 12.626453441
------------------------------------------------------------------------------
It looks quite complicated, but things look more simple if one uses grep:
my_shell> grep CONV wannier.wout
gives
+--------------------------------------------------------------------+<-- CONV
| Iter Delta Spread RMS Gradient Spread (Ang^2) Time |<-- CONV
+--------------------------------------------------------------------+<-- CONV
0 0.126E+02 0.0000000000 12.6297685260 0.29 <-- CONV
1 -0.313E-02 0.0697660962 12.6266347170 0.34 <-- CONV
.
.
50 0.000E+00 0.0000000694 12.6264534413 2.14 <-- CONV
The first column is the iteration number, the second is the change in Ωfrom the previous iteration, the
third is the root-mean-squared gradient of Ωwith respect to variations in the unitary matrices U(k),
and the last is the time taken (in seconds). Depending on the input parameters used, the procedure
either runs for num_iter iterations, or a convergence criterion is applied on Ω. See Section 2.8 for
details.
Similarly, the command
my_shell> grep SPRD wannier.wout
gives
O_D= 0.0000000 O_OD= 0.1491718 O_TOT= 12.6297685 <-- SPRD
O_D= 0.0000000 O_OD= 0.1460380 O_TOT= 12.6266347 <-- SPRD
.
.
O_D= 0.0000000 O_OD= 0.1458567 O_TOT= 12.6264534 <-- SPRD
74 wannier90: User Guide
which, for each iteration, reports the value of the diagonal and off-diagonal parts of the non-gauge-
invariant spread, as well as the total spread, respectively. Recall from Section 1 that Ω=ΩI+ΩD+ΩOD.
8.6.6 Plotting
After WF have been localised, wannier90 enters its plotting routines (if required). For example, if you
have specified an interpolated bandstucture:
*---------------------------------------------------------------------------*
| PLOTTING |
*---------------------------------------------------------------------------*
Calculating interpolated band-structure
8.6.7 Summary timings
At the very end of the run, a summary of the time taken for various parts of the calculation is given.
The level of detail is controlled by the timing_level input parameter (set to 1 by default).
*===========================================================================*
| TIMING INFORMATION |
*===========================================================================*
| Tag Ncalls Time (s)|
|---------------------------------------------------------------------------|
|kmesh: get : 1 0.212|
|overlap: read : 1 0.060|
|wann: main : 1 1.860|
|plot: main : 1 0.168|
*---------------------------------------------------------------------------*
All done: wannier90 exiting
8.7 seedname.chk
INPUT/OUTPUT. Information required to restart the calculation or enter the plotting phase. If we
have used disentanglement this file also contains the rectangular matrices Udis(k).
8.8 seedname.r2mn
OUTPUT. Written if write_r2mn =true. The matrix elements hm|r2|ni(where mand nrefer to
MLWF)
wannier90: User Guide 75
8.9 seedname_band.dat
OUTPUT. Written if bands_plot=.TRUE.; The raw data for the interpolated band structure.
8.10 seedname_band.gnu
OUTPUT. Written if bands_plot=.TRUE. and bands_plot_format=gnuplot; A gnuplot script to plot
the interpolated band structure.
8.11 seedname_band.agr
OUTPUT. Written if bands_plot=.TRUE. and bands_plot_format=xmgrace; A grace file to plot the
interpolated band structure.
8.12 seedname_band.kpt
OUTPUT. Written if bands_plot=.TRUE.; The k-points used for the interpolated band structure, in
units of the reciprocal lattice vectors. This file can be used to generate a comparison band structure
from a first-principles code.
8.13 seedname.bxsf
OUTPUT. Written if fermi_surface_plot=.TRUE.; A Fermi surface plot file suitable for plotting with
XCrySDen.
8.14 seedname_w.xsf
OUTPUT. Written if wannier_plot=.TRUE. and wannier_plot_format=xcrysden. Contains the wth
WF in real space in a format suitable for plotting with XCrySDen or VMD, for example.
8.15 seedname_w.cube
OUTPUT. Written if wannier_plot=.TRUE. and wannier_plot_format=cube. Contains the wth WF
in real space in Gaussian cube format, suitable for plotting in XCrySDen, VMD, gopenmol etc.
8.16 UNKp.s
INPUT. Read if wannier_plot=.TRUE. and used to plot the MLWF. Read if transport_mode=lcr
and tran_read_ht=.FALSE. for use in automated lcr transport calculations.
76 wannier90: User Guide
The periodic part of the Bloch states represented on a regular real space grid, indexed by k-point p
(from 1 to num_kpts) and spin s(‘1’ for ‘up’, ‘2’ for ‘down’).
The name of the wavefunction file is assumed to have the form:
write(wfnname,200) p,spin
200 format (’UNK’,i5.5,’.’,i1)
The first line of each file should contain 5 integers: the number of grid points in each direction (ngx,
ngy and ngz), the k-point number ik and the total number of bands num_band in the file. The full file
will be read by wannier90 as:
read(file_unit) ngx,ngy,ngz,ik,nbnd
do loop_b=1,num_bands
read(file_unit) (r_wvfn(nx,loop_b),nx=1,ngx*ngy*ngz)
end do
The file can be in formatted or unformatted style, this is controlled by the logical keyword wvfn_formatted.
8.17 seedname_centres.xyz
OUTPUT. Written if write_xyz=.TRUE.; xyz format atomic structure file suitable for viewing with
your favourite visualiser (jmol,gopenmol,vmd, etc.).
8.18 seedname_hr.dat
OUTPUT. Written if hr_plot=.TRUE.. The first line gives the date and time at which the file was
created. The second line states the number of Wannier functions num_wann. The third line gives the
number of Wigner-Seitz grid-points nrpts. The next block of nrpts integers gives the degeneracy of
each Wigner-Seitz grid point, with 15 entries per line. Finally, the remaining num_wann2×nrpts lines
each contain, respectively, the components of the vector Rin terms of the lattice vectors {Ai}, the
indices mand n, and the real and imaginary parts of the Hamiltonian matrix element H(R)
mn in the WF
basis, e.g.,
Created on 24May2007 at 23:32:09
20
17
412141121461112
1 2
0 0 -2 1 1 -0.001013 0.000000
0 0 -2 2 1 0.000270 0.000000
0 0 -2 3 1 -0.000055 0.000000
0 0 -2 4 1 0.000093 0.000000
0 0 -2 5 1 -0.000055 0.000000
.
.
.
wannier90: User Guide 77
8.19 seedname_qc.dat
OUTPUT. Written if transport =.TRUE.. The first line gives the date and time at which the file was
created. In the subsequent lines, the energy value in units of eV is written in the left column, and the
quantum conductance in units of 2e2
h(e2
hfor a spin-polarized system) is written in the right column.
## written on 14Dec2007 at 11:30:17
-3.000000 8.999999
-2.990000 8.999999
-2.980000 8.999999
-2.970000 8.999999
.
.
.
8.20 seedname_dos.dat
OUTPUT. Written if transport =.TRUE.. The first line gives the date and time at which the file
was created. In the subsequent lines, the energy value in units of eV is written in the left column, and
the density of states in an arbitrary unit is written in the right column.
## written on 14Dec2007 at 11:30:17
-3.000000 6.801199
-2.990000 6.717692
-2.980000 6.640828
-2.970000 6.569910
.
.
.
8.21 seedname_htB.dat
INPUT/OUTPUT. Read if transport_mode =bulk and tran_read_ht =.TRUE.. Written if tran_write_ht =
.TRUE.. The first line gives the date and time at which the file was created. The second line gives
tran_num_bb. The subsequent lines contain tran_num_bb×tran_num_bb Hmn matrix, where the in-
dices mand nspan all tran_num_bb WFs located at 0th principal layer. Then tran_num_bb is recorded
again in the new line followed by Hmn, where mth WF is at 0th principal layer and nth at 1st principal
layer. The Hmn matrix is written in such a way that mis the fastest varying index.
written on 14Dec2007 at 11:30:17
150
-1.737841 -2.941054 0.052673 -0.032926 0.010738 -0.009515
0.011737 -0.016325 0.051863 -0.170897 -2.170467 0.202254
.
.
78 wannier90: User Guide
.
-0.057064 -0.571967 -0.691431 0.015155 -0.007859 0.000474
-0.000107 -0.001141 -0.002126 0.019188 -0.686423 -10.379876
150
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
.
.
.
0.000000 0.000000 0.000000 0.000000 0.000000 -0.001576
0.000255 -0.000143 -0.001264 0.002278 0.000000 0.000000
8.22 seedname_htL.dat
INPUT. Read if transport_mode =lcr and tran_read_ht =.TRUE.. The file must be written in
the same way as in seedname_htB.dat. The first line can be any comment you want. The second line
gives tran_num_ll.tran_num_ll in seedname_htL.dat must be equal to that in seedname.win. The
code will stop otherwise.
Created by a WANNIER user
105
0.316879 0.000000 -2.762434 0.048956 0.000000 -0.016639
0.000000 0.000000 0.000000 0.000000 0.000000 -2.809405
.
.
.
0.000000 0.078188 0.000000 0.000000 -2.086453 -0.001535
0.007878 -0.545485 -10.525435
105
0.000000 0.000000 0.000315 -0.000294 0.000000 0.000085
0.000000 0.000000 0.000000 0.000000 0.000000 0.000021
.
.
.
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 0.000000
8.23 seedname_htR.dat
INPUT. Read if transport_mode =lcr and tran_read_ht =.TRUE. and tran_use_same_lead =
.FALSE.. The file must be written in the same way as in seedname_htL.dat.tran_num_rr in
seedname_htR.dat must be equal to that in seedname.win.
wannier90: User Guide 79
8.24 seedname_htC.dat
INPUT. Read if transport_mode =lcr and tran_read_ht =.TRUE.. The first line can be any com-
ment you want. The second line gives tran_num_cc. The subsequent lines contain tran_num_cc×tran_num_cc
Hmn matrix, where the indices mand nspan all tran_num_cc WFs inside the central conductor region.
tran_num_cc in seedname_htC.dat must be equal to that in seedname.win.
Created by a WANNIER user
99
-10.499455 -0.541232 0.007684 -0.001624 -2.067078 -0.412188
0.003217 0.076965 0.000522 -0.000414 0.000419 -2.122184
.
.
.
-0.003438 0.078545 0.024426 0.757343 -2.004899 -0.001632
0.007807 -0.542983 -10.516896
8.25 seedname_htLC.dat
INPUT. Read if transport_mode =lcr and tran_read_ht =.TRUE.. The first line can be any
comment you want. The second line gives tran_num_ll and tran_num_lc in the given order. The
subsequent lines contain tran_num_ll×tran_num_lc Hmn matrix. The index mspans tran_num_ll
WFs in the surface principal layer of semi-infinite left lead which is in contact with the conductor
region. The index nspans tran_num_lc WFs in the conductor region which have a non-negligible
interaction with the WFs in the semi-infinite left lead. Note that tran_num_lc can be different from
tran_num_cc.
Created by a WANNIER user
105 99
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
.
.
.
-0.000003 0.000009 0.000290 0.000001 -0.000007 -0.000008
0.000053 -0.000077 -0.000069
8.26 seedname_htCR.dat
INPUT. Read if transport_mode =lcr and tran_read_ht =.TRUE.. The first line can be any
comment you want. The second line gives tran_num_cr and tran_num_rr in the given order. The
subsequent lines contain tran_num_cr×tran_num_rr Hmn matrix. The index mspans tran_num_cr
WFs in the conductor region which have a non-negligible interaction with the WFs in the semi-infinite
right lead. The index nspans tran_num_rr WFs in the surface principal layer of semi-infinite right
lead which is in contact with the conductor region. Note that tran_num_cr can be different from
tran_num_cc.
80 wannier90: User Guide
Created by a WANNIER user
99 105
-0.000180 0.000023 0.000133 -0.000001 0.000194 0.000008
-0.000879 -0.000028 0.000672 -0.000257 -0.000102 -0.000029
.
.
.
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.000000 0.000000 0.000000
8.27 seedname.unkg
INPUT. Read if transport_mode =lcr and tran_read_ht =.FALSE.. The first line is the number
of G-vectors at which the ˜umk(G)are subsequently printed. This number should always be 32 since
32 specific ˜umkare required. The following lines contain the following in this order: The band index
m, a counter on the number of G-vectors, the integer co-efficient of the G-vector components a, b, c
(where G=ab1+bb2+cb3), then the real and imaginary parts of the corresponding ˜umk(G)at the
Γ-point. We note that the ordering in which the G-vectors and ˜umk(G)are printed is not important,
but the specific G-vectors are critical. The following example displays for a single band, the complete
set of ˜umk(G)that are required. Note the G-vectors (a, b, c) needed.
32
1 1 0 0 0 0.4023306 0.0000000
1 2 0 0 1 -0.0000325 0.0000000
1 3 0 1 0 -0.3043665 0.0000000
1 4 1 0 0 -0.3043665 0.0000000
1 5 2 0 0 0.1447143 0.0000000
1 6 1 -1 0 0.2345179 0.0000000
1 7 1 1 0 0.2345179 0.0000000
1 8 1 0 -1 0.0000246 0.0000000
1 9 1 0 1 0.0000246 0.0000000
1 10 0 2 0 0.1447143 0.0000000
1 11 0 1 -1 0.0000246 0.0000000
1 12 0 1 1 0.0000246 0.0000000
1 13 0 0 2 0.0000338 0.0000000
1 14 3 0 0 -0.0482918 0.0000000
1 15 2 -1 0 -0.1152414 0.0000000
1 16 2 1 0 -0.1152414 0.0000000
1 17 2 0 -1 -0.0000117 0.0000000
1 18 2 0 1 -0.0000117 0.0000000
1 19 1 -2 0 -0.1152414 0.0000000
1 20 1 2 0 -0.1152414 0.0000000
1 21 1 -1 -1 -0.0000190 0.0000000
1 22 1 -1 1 -0.0000190 0.0000000
1 23 1 1 -1 -0.0000190 0.0000000
1 24 1 1 1 -0.0000190 0.0000000
1 25 1 0 -2 -0.0000257 0.0000000
1 26 1 0 2 -0.0000257 0.0000000
wannier90: User Guide 81
1 27 0 3 0 -0.0482918 0.0000000
1 28 0 2 -1 -0.0000117 0.0000000
1 29 0 2 1 -0.0000117 0.0000000
1 30 0 1 -2 -0.0000257 0.0000000
1 31 0 1 2 -0.0000257 0.0000000
1 32 0 0 3 0.0000187 0.0000000
2 1 0 0 0 -0.0000461 0.0000000
.
.
.
Chapter 9
Sample Input Files
9.1 Master input file: seedname.win
num_wann : 4
mp_grid : 4 4 4
num_iter : 100
postproc_setup : true
begin unit_cell_cart
ang
-1.61 0.00 1.61
0.00 1.61 1.61
-1.61 1.61 0.00
end unit_cell_cart
begin atoms_frac
C -0.125 -0.125 -0.125
C 0.125 0.125 0.125
end atoms_frac
bands_plot : true
bands_num_points : 100
bands_plot_format : gnuplot
begin kpoint_path
L 0.50000 0.50000 0.50000 G 0.00000 0.00000 0.00000
G 0.00000 0.00000 0.00000 X 0.50000 0.00000 0.50000
X 0.50000 0.00000 0.50000 K 0.62500 0.25000 0.62500
end kpoint_path
begin projections
C:l=0,l=1
end projections
begin kpoints
83
84 wannier90: User Guide
0.00 0.00 0.00
0.00 0.00 0.25
0.00 0.50 0.50
.
.
.
0.75 0.75 0.50
0.75 0.75 0.75
end kpoints
9.2 seedname.nnkp
Running wannier90 on the above input file would generate the following nnkp file:
File written on 9Feb2006 at 15:13: 9
calc_only_A : F
begin real_lattice
-1.612340 0.000000 1.612340
0.000000 1.612340 1.612340
-1.612340 1.612340 0.000000
end real_lattice
begin recip_lattice
-1.951300 -1.951300 1.951300
1.951300 1.951300 1.951300
-1.951300 1.951300 -1.951300
end recip_lattice
begin kpoints
64
0.00000 0.00000 0.00000
0.00000 0.25000 0.00000
0.00000 0.50000 0.00000
0.00000 0.75000 0.00000
0.25000 0.00000 0.00000
.
.
.
0.50000 0.75000 0.75000
0.75000 0.00000 0.75000
0.75000 0.25000 0.75000
0.75000 0.50000 0.75000
0.75000 0.75000 0.75000
end kpoints
wannier90: User Guide 85
begin projections
8
-0.12500 -0.12500 -0.12500 0 1 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
-0.12500 -0.12500 -0.12500 1 1 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
-0.12500 -0.12500 -0.12500 1 2 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
-0.12500 -0.12500 -0.12500 1 3 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.12500 0.12500 0.12500 0 1 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.12500 0.12500 0.12500 1 1 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.12500 0.12500 0.12500 1 2 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
0.12500 0.12500 0.12500 1 3 1
0.000 0.000 1.000 1.000 0.000 0.000 2.00
end projections
begin nnkpts
8
1 2 0 0 0
1 4 0 -1 0
1 5 0 0 0
1 13 -1 0 0
1 17 0 0 0
1 22 0 0 0
1 49 0 0 -1
1 64 -1 -1 -1
2 1 0 0 0
2 3 0 0 0
2 6 0 0 0
2 14 -1 0 0
2 18 0 0 0
2 23 0 0 0
2 50 0 0 -1
2 61 -1 0 -1
.
.
.
64 1 1 1 1
64 16 0 0 1
64 43 0 0 0
64 48 0 0 0
64 52 1 0 0
64 60 0 0 0
64 61 0 1 0
64 63 0 0 0
86 wannier90: User Guide
end nnkpts
begin exclude_bands
4
1
2
3
4
end exclude_bands
Part III
postw90.x
87
Chapter 10
Parameters
10.1 Introduction
The wannier90.x code described in Part II calculates the maximally-localized Wannier functions. The
wannier90.x code is a serial executable (i.e., it cannot be executed in parallel on different CPUs).
For users of the previous wannier90 1.2 release, the wannier90.x executable has only a few minor
changes with respect to the 1.2 release. Note however that the checkpoint file format has changed from
the one used in version 1.2.
The postw90.x executable contains instead a series of modules that take the Wannier functions cal-
culated by wannier90.x and use them to calculate different properties. This executable is parallel
(by means of MPI libraries), so it can be run on multiple CPUs. The information on the calculated
Wannier functions is read from the checkpoint seedname.chk file. Note that this is written in an un-
formatted machine-dependent format. If you need to use this file on a different machine, or you want
to use a version of postw90.x compiled with a different compiler, refer to Sec. A.2 in the Appendices
for a description of how to export/import this file.
10.2 Usage
postw90.x can be run in parallel using MPI libraries to reduce the computation time.
For serial execution use: postw90.x [seedname]
•seedname: If a seedname string is given the code will read its input from a file seedname.win.
The default value is wannier. One can also equivalently provide the string seedname.win instead
of seedname.
For parallel execution use: mpirun -np NUMPROCS postw90.x [seedname]
•NUMPROCS: substitute with the number of processors that you want to use.
Note that the mpirun command and command-line flags may be different in your MPI implementation:
read your MPI manual or ask your computer administrator.
89
90 wannier90: User Guide
Note also that this requires that the postw90.x executable has been compiled in its parallel version
(follow the instructions in the file README.install in the main directory of the wannier90 distribution)
and that the MPI libraries and binaries are installed and correctly configured on your machine.
10.3 seedname.win File
The postw90.x uses the same seedname.win input file of wannier90.x. The input keywords of
postw90.x must thus be added to this file, using the same syntax described in Sec. 2.2.
Note that wannier90.x checks if the syntax of the input file is correct, but then ignores the value of
the flags that refer only to modules of postw90.x, so one can safely run wannier90.x on a file that
contains also postw90.x flags.
Similarly, postw90.x ignores flags that refer only to wannier90.x (as number of iterations, restart
flags, . . . ). However, some parts of the input file must be there, as for instance the number of Wannier
functions, etc.
The easiest thing to do is therefore to simply add the postw90 input keywords to the seedname.win
file that was used to obtain the Wannier functions.
10.4 List of available modules
The currently available modules in postw90.x are:
•dos: Calculation of the density of states (DOS), projected density of states (PDOS), net spin
etc.
•kpath: Calculation of k-space quantities such as energy bands and Berry curvature along a
piecewise linear path in the BZ (see examples 17 and 18 of the tutorial).
•kslice: Calculation of k-space quantities on a planar slice of the BZ (see examples 17 and 18 of
the tutorial).
•berry: Calculation of properties related to the BZ integral of the Berry curvature and Berry con-
nection, including anomalous Hall conductivity, orbital magnetisation, and optical conductivity
(see Chap. 11 and examples 18 and 19 of the tutorial).
•BoltzWann: Calculation of electronic transport properties for bulk materials using the semiclas-
sical Boltzmann transport equation (see Chap. 12 and example 16 of the tutorial).
•geninterp (Generic Band Interpolation): Calculation band energies (and band derivatives) on a
generic list of kpoints (see Chap. 13).
10.5 Keyword List
On the next pages the list of available postw90 input keywords is reported. In particular, Table 10.1
reports keywords that affect the generic behavior of all modules of postw90. Often, these are “global”
variables that can be overridden by module-specific keywords (as for instance the kmesh flag). The
subsequent tables describe the input parameters for each specific module.
92 wannier90: User Guide
Keyword Type Description
Global Parameters of postw90
kmesh I Dimensions of the uniform interpo-
lation k-mesh (one or three integers)
kmesh_spacing R Minimum spacing between kpoints
in Å−1
adpt_smr L Use adaptive smearing
adpt_smr_fac R Adaptive smearing prefactor
adpt_smr_max P Maximum allowed value for the
adaptive energy smearing (eV)
smr_type S Analytical form used for the broad-
ened delta function
smr_fixed_en_width P Energy smearing (if non-adaptive)
num_elec_per_state I Number of electrons per state
scissors_shift P Scissors shift applied to the conduc-
tion bands (eV)
num_valence_bands I Number of valence bands
spin_decomp L Decompose various properties into
up-spin, down-spin, and possibly
spin-flip parts
spin_axis_polar P Polar angle of the spin quantization
axis (deg)
spin_axis_azimuth P Azimuthal angle of the spin quanti-
zation axis (deg)
spin_moment∗L Determines whether to evaluate the
spin magnetic moment per cell
uHu_formatted L Read a formatted seedname.uHu file
spn_formatted L Read a formatted seedname.spn file
berry_curv_unit S Unit of Berry curvature
Table 10.1: seedname.win file keywords controlling the general behaviour of the modules in postw90.
Argument types are represented by, I for a integer, R for a real number, P for a physical value, L for
a logical value and S for a text string.
The keyword spin_moment does not affect the behavior of the modules in postw90, and does not really
belong to any of them. It is listed here for lack of a better place.
wannier90: User Guide 93
Keyword Type Description
dos Parameters
dos L Calculate the density of states and
related properties
dos_task S List of properties to compute
dos_energy_min P Lower limit of the energy range for
computing the DOS (eV)
dos_energy_max P Upper limit of the energy range for
computing the DOS (eV)
dos_energy_step R Step for increasing the energy in the
specified range (eV)
dos_project I List of WFs onto which the DOS is
projected
[dos_]kmesh I Dimensions of the uniform interpo-
lation k-mesh (one or three integers)
[dos_]kmesh_spacing R Minimum spacing between kpoints
in Å−1
[dos_]adpt_smr L Use adaptive smearing for the DOS
[dos_]adpt_smr_fac R Adaptive smearing prefactor
[dos_]adpt_smr_max P Maximum allowed value for the
adaptive energy smearing (eV)
[dos_]smr_fixed_en_width P Energy smearing (if non-adaptive)
for the DOS (eV)
[dos_]smr_type S Analytical form used for the broad-
ened delta function when computing
the DOS.
Table 10.2: seedname.win file keywords controlling the dos module. Argument types are represented
by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a text
string.
Keyword Type Description
kpath Parameters
kpath L Calculate properties along a piece-
wise linear path in the BZ
kpath_task L List of properties to evaluate
kpath_num_points I Number of points in the first kpath
segment
kpath_bands_colour S Property used to colour the energy
bands along the path
Table 10.3: seedname.win file keywords controlling the kpath module. Argument types are represented
by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a text
string.
94 wannier90: User Guide
Keyword Type Description
kslice Parameters
kslice L Calculate properties on a slice in the
BZ
kslice_task S List of properties to evaluate
kslice_corner R Position of the corner of the slice
kslice_b1 R First vector defining the slice
kslice_b2 R Second vector defining the slice
kslice_2dkmesh I Dimensions of the uniform interpo-
lation k-mesh on the slice (one or
two integers)
kslice_fermi_level P Energy level for plotting constant-
energy contour lines (eV)
kslice_fermi_lines_colour S Property used to colour the Fermi
lines
Table 10.4: seedname.win file keywords controlling the kslice module. Argument types are repre-
sented by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a
text string.
wannier90: User Guide 95
Keyword Type Description
berry Parameters
berry L Calculate Berry-type quantities
berry_task L List of properties to compute
[berry_]kmesh I Dimensions of the uniform interpo-
lation k-mesh (one or three integers)
[berry_]kmesh_spacing R Minimum spacing between kpoints
in Å−1
berry_curv_adpt_kmesh I Linear dimension of the adaptively
refined k-mesh used to compute the
anomalous Hall conductivity
berry_curv_adpt_kmesh_thresh P Threshold magnitude of the Berry
curvature for adaptive refinement
kubo_freq_min P Lower limit of the frequency range
for optical spectra and JDOS (eV)
kubo_freq_max P Upper limit of the frequency range
for optical spectra and JDOS (eV)
kubo_freq_step R Step for increasing the optical fre-
quency in the specified range
kubo_eigval_max P Maximum energy eigenvalue in-
cluded when evaluating the Kubo-
Greenwood conductivity and JDOS
[kubo_]adpt_smr L Use adaptive energy smearing for
the optical conductivity and JDOS
[kubo_]adpt_smr_fac R Adaptive smearing prefactor
[kubo_]adpt_smr_max P Maximum allowed value for the
adaptive energy smearing (eV)
[kubo_]smr_type S Analytical form used for the broad-
ened delta function when computing
the optical conductivity and JDOS
[kubo_]smr_fixed_en_width P Energy smearing (if non-adaptive)
for the optical conductivity and
JDOS (eV)
Table 10.5: seedname.win file keywords controlling the berry module. Argument types are represented
by, I for a integer, R for a real number, P for a physical value, L for a logical value and S for a text
string.
96 wannier90: User Guide
Keyword Type Description
BoltzWann Parameters
boltzwann L Calculate Boltzmann transport co-
efficients
[boltz_]kmesh I Dimensions of the uniform interpo-
lation k-mesh (one or three integers)
[boltz_]kmesh_spacing R Minimum spacing between kpoints
in Å−1
boltz_2d_dir S Non-periodic direction (for 2D sys-
tems only)
boltz_relax_time P Relaxation time in fs
boltz_mu_min P Minimum value of the chemical po-
tential µin eV
boltz_mu_max P Maximum value of the chemical po-
tential µin eV
boltz_mu_step R Step for µin eV
boltz_temp_min P Minimum value of the tempera-
ture Tin Kelvin
boltz_temp_max P Maximum value of the tempera-
ture Tin Kelvin
boltz_temp_step R Step for Tin Kelvin
boltz_tdf_energy_step R Energy step for the TDF (eV)
boltz_tdf_smr_fixed_en_width P Energy smearing for the TDF (eV)
boltz_tdf_smr_type S Smearing type for the TDF
boltz_calc_also_dos L Calculate also DOS while calculat-
ing the TDF
boltz_dos_energy_min P Minimum value of the energy for the
DOS in eV
boltz_dos_energy_max P Maximum value of the energy for the
DOS in eV
boltz_dos_energy_step R Step for the DOS in eV
[boltz_dos_]smr_type S Smearing type for the DOS
[boltz_dos_]adpt_smr L Use adaptive smearing for the DOS
[boltz_dos_]adpt_smr_fac R Adaptive smearing prefactor
[boltz_dos_]adpt_smr_max P Maximum allowed value for the
adaptive energy smearing (eV)
[boltz_dos_smr_]fixed_en_width P Energy smearing (if non-adaptive)
for the DOS (eV)
Table 10.6: seedname.win file keywords controlling the BoltzWann module (calculation of the Boltz-
mann transport coefficients in the Wannier basis). Argument types are represented by, I for a integer,
R for a real number, P for a physical value, L for a logical value and S for a text string.
wannier90: User Guide 97
Keyword Type Description
geninterp Parameters
geninterp L Calculate bands for given set of k
points
geninterp_alsofirstder L Calculate also first derivatives
geninterp_single_file L Write a single file or one for each
process
Table 10.7: seedname.win file keywords controlling the Generic Band Interpolation (geninterp) mod-
ule. Argument types are represented by, I for a integer, R for a real number, P for a physical value, L
for a logical value and S for a text string.
98 wannier90: User Guide
10.6 Global variables
10.6.1 integer :: kmesh(:)
Dimensions of the interpolation grid used in postw90.x.
Not to be confused with the mp_grid input flag, which instead specifies the Monkhorst–Pack grid used
in the ab-initio calculation!
If three integers l m n are given, the reciprocal-space cell subtended by the three primitive translations
is sampled on a uniform l×m×ngrid (including Γ). If only one integer mis given, an m×m×m
grid is used.
If you use a module which needs a k-mesh, either kmesh_spacing or kmesh must be defined.
10.6.2 real(kind=dp) :: kmesh_spacing
An alternative way of specifying the interpolation grid. This flag defines the minimum distance for
neighboring kpoints along each of the three directions in kspace.
The units are Å−1.
If you use a module which needs a k-mesh, either kmesh_spacing or kmesh must be defined.
10.6.3 logical :: adpt_smr
Determines whether to use an adaptive scheme for broadening the DOS and similar quantities defined
on the energy axis. If true, the values for the smearing widths are controlled by the flag adpt_smr_fac.
The default value is true.
10.6.4 real(kind=dp) :: adpt_smr_fac
The width ηnkof the broadened delta function used to determine the contribution to the spectral
property (DOS, ...) from band nat point kis calculated as
ηnk=α|∇kεnk|∆k,
where εnkis the energy eigenvalue and the dimensionless factor αis given by adpt_smr_fac.∆kis
taken to be the largest of the mesh spacings along the three reciprocal lattice vectors b1,b2, and b3.
If the calculated value of ηnkexceeds adpt_smr_max, the latter value is used.
The default value is √2.
10.6.5 real(kind=dp) :: adpt_smr_max
See description given immediately above.
The units are eV. The default value is 1.0.
wannier90: User Guide 99
10.6.6 character(len=120) :: smr_type
Defines the analytical form used for the broadened delta function in the computation of the DOS and
similar quantities defined on the energy axis.
•gauss: Gaussian smearing
•m-pN: derivative of the N-th order Methfessel-Paxton function (N≥0). Example: m-p2 for the
second-order Methfessel-Paxton function. If only m-p is provided, the first-order function is used,
i.e., it is equivalent to m-p1.
•m-v or cold: derivative of the Marzari–Vanderbilt cold-smearing function
•f-d: derivative of the Fermi-Dirac distribution function
The default value is gauss.
10.6.7 logical :: smr_fixed_en_width
Energy width for the smearing function for the DOS. Used only if adpt_smr is false.
The units are eV. The default value is 0 eV. Note that if the width is smaller than twice the energy
step (e.g. dos_energy_step for the dos module), the DOS will be unsmeared (thus the default is to
have an unsmeared properties when adpt_smr is set to false.).
10.6.8 integer :: num_elec_per_state
Number of electrons per state. It can only take the values one or two.
The default value is 1 if spinors=true, 2 otherwise.
10.6.9 real(kind=dp) :: scissors_shift
Scissors shift applied to the conduction bands.
The units are eV. The default value is 0 eV (i.e., no scissors shift applied).
10.6.10 integer :: num_valence_bands
Number of valence bands of the system. Used in different modules and for the scissors shift.
No default value.
10.6.11 logical :: spin_decomp
If true, extra columns are added to some output files (such as seedname-dos.dat for the dos module,
and analogously for the berry and BoltzWann modules).
100 wannier90: User Guide
For the dos and BoltzWann modules, two further columns are generated, which contain the decomposi-
tion of the required property (e.g., total or orbital-projected DOS) of a spinor calculation into up-spin
and down-spin parts (relative to the quantization axis defined by the input variables spin_axis_polar
and spin_axis_azimuth). For the berry module with berry_task = kubo, three extra columns are
added to seedname-jdos.dat, containing the decomposition of the JDOS into up →up, down →
down, and spin-flip transitions. In the same way, six extra columns are added to the data files
seedname-kubo*.dat where the complex optical conductivity is stored.
The file seedname.spn must be present at input. Furthermore, if this variable is set to true it requires
num_elec_per_state = 1.
The default value is false.
10.6.12 real(kind=dp) :: spin_axis_polar
Polar angle of the spin quantization axis.
The units are degrees. The default value is 0.
10.6.13 real(kind=dp) :: spin_axis_azimuth
Azimuthal angle of the spin quantization axis.
The units are degrees. The default value is 0.
10.6.14 logical :: spin_moment
Determines whether to evaluate the spin moment.
The default value is false.
10.6.15 logical :: uHu_formatted
If uHu_formatted=true, then the uHu matrix elements will be read from disk as formatted (ie ASCII)
files; otherwise they will be read as unformatted files.
The default value of this parameter is false.
10.6.16 logical :: spn_formatted
If spn_formatted=true, then the spin matrix elements will be read from disk as formatted (ie ASCII)
files; otherwise they will be read as unformatted files. Unformatted is generally preferable as the files
will take less disk space and I/O is significantly faster. However such files will not be transferable
between all machine architectures and formatted files should be used if transferability is required (i.e.,
for test cases).
The default value is false.
wannier90: User Guide 101
10.6.17 character(len=20) :: berry_curv_unit
Unit in which the Berry curvature is specified at input (in berry_curv_adpt_kmesh_thresh) or written
to file (when kpath_task=curv or kslice_task=curv).
•ang2: Angstrom2
•bohr2: Bohr2(atomic units)
The default value is ang2.
102 wannier90: User Guide
10.7 DOS
Note that the behavior of the dos module is also influenced by the value of some global flags (listed in
Table 10.1), as spin_decomp,spin_axis_polar,spin_axis_azimuth,scissors_shift, etc. Some of
the global flags can be possibly overridden by local flags of the DOS module, listed below, which have
the same name of the global flag but are prefixed by dos_.
10.7.1 logical :: dos
Determines whether to enter the DOS routines.
The default value is false.
10.7.2 character(len=20) :: dos_task
The quantity to compute when dos=true
The valid options for this parameter are:
–dos_plot Density of states. An output data file seedname-dos.dat is created, containing the
energy values in eV in the first column, and the total DOS per unit cell and unit energy range
(in eV−1) in the second. Two additional columns are present if spin_decomp=true
The default value is dos_plot.
10.7.3 real(kind=dp) :: dos_energy_min
Lower limit of the energy range for computing the DOS. Units are eV.
The default value is the minimum value of the energy eigenvalues stored in seedname.eig, minus
0.6667.
10.7.4 real(kind=dp) :: dos_energy_max
Upper limit of the energy range for computing the DOS. Units are eV.
If an inner energy window was specified, the default value is the upper bound of the innter energy win-
dow, plus 0.6667. Otherwise it is the maximum value of the energy eigenvalues stored in seedname.eig,
plus 0.6667.
10.7.5 real(kind=dp) :: dos_energy_step
Energy step for the grid of energies used to plot the dos. Units are eV.
The default value is 0.01 eV.
wannier90: User Guide 103
10.7.6 integer :: dos_project(:)
If present postw90 computes, instead of the total DOS, the partial DOS projected onto the WFs listed.
The WFs are numbered according to the file seedname.wout.
For example, to project onto WFs 2, 6, 7, 8, and 12:
dos_project : 2, 6-8, 12
The DOS projected onto a set Sof orbitals is calculated as
ρS(E) = 1
NkX
kX
nhψ(H)
nk|ˆ
Pk(S)|ψ(H)
nkiδ(εnk−E)(10.1)
ˆ
Pk(S) = X
m∈S |ψ(W)
nkihψ(W)
nk|,(10.2)
where Nkis the number of mesh points used to sample the BZ, and the superscript (H) and (W) refer
to Hamiltonian gauge and Wannier gauge [8].
10.7.7 integer :: dos_kmesh(:)
Overrides the kmesh global variable (see Sec. 10.6).
10.7.8 real(kind=dp) :: dos_kmesh_spacing
Overrides the kmesh_spacing global variable (see Sec. 10.6).
10.7.9 logical :: dos_adpt_smr
Overrides the adpt_smr global variable (see Sec. 10.6).
10.7.10 real(kind=dp) :: dos_adpt_smr_fac
Overrides the adpt_smr_fac global variable (see Sec. 10.6).
10.7.11 real(kind=dp) :: dos_adpt_smr_max
Overrides the adpt_smr_max global variable (see Sec. 10.6).
10.7.12 logical :: dos_smr_fixed_en_width
Overrides the smr_fixed_en_width global variable (see Sec. 10.6).
Note that if the width is smaller than twice the energy step dos_energy_step, the DOS will be
unsmeared (thus the default is to have an unsmeared DOS).
wannier90: User Guide 105
10.8 kpath
10.8.1 logical :: kpath
Determines whether to enter the kpath routines.
The default value is false.
10.8.2 character(len=20) :: kpath_task
The quantities to plot when kpath=true
The valid options for this parameter are:
–bands Energy bands, in eV. The following files are created:
·seedname-bands.dat (data file)
·seedname-bands.gnu (gnuplot script)
·seedname-bands.py (python script)
·seedname-path.kpt (list of k-points along the path, written in the pwscf format)
–curv Minus the Berry curvature given by Eq. (11.18) of Ch. 11, in units of berry_curv_unit.
The following files are created:
·seedname-curv.dat (data file)
·seedname-curv_{x,y,z}.gnu (gnuplot scripts)
·seedname-curv_{x,y,z}.py (python scripts)
–morb The integrand of the k-space orbital magnetization formula [Eq. (11.20) of Ch. 11] in eV·Å2.
Four output files are created:
·seedname-morb.dat (data file)
·seedname-morb_{x,y,z}.gnu (gnuplot scripts)
·seedname-morb_{x,y,z}.py (python scripts)
–Any combination of the above. The following combinations are of special interest
kpath_task = bands+curv
kpath_task = bands+morb
They generate the following files:
·seedname-bands.dat (data file)
·seedname-{curv,morb}.dat (data file)
·seedname-bands+{curv,morb}_{x,y,z}.py (python scripts)
Two-panel figures are produced, with the energy bands within ±0.65 eV of the Fermi level in the
top panel, and the Berry curvature (or k-space orbital magnetization) in the bottom panel.
The default value is bands.
106 wannier90: User Guide
10.8.3 integer :: kpath_num_points
If kpath =true, then the number of points along the first section of the bandstructure plot given by
kpoint_path. Other sections will have the same density of k-points.
The default value is 100.
10.8.4 character(len=20) :: kpath_bands_colour
When kpath_task=bands, colour code the energy bands according to the specified quantity.
The valid options for this parameter are:
–spin Spin projection (in units of ¯h/2) along the quantization axis defined by the variables
spin_axis_polar and spin_axis_azimuth, for a spinor calculation
–none no colour coding
The default value is none.
wannier90: User Guide 107
10.9 kslice
10.9.1 logical :: kslice
Determines whether to enter the kslice routines.
The default value is false.
10.9.2 character(len=20) :: kslice_task
The quantity to plot when kslice=true
The valid options for this parameter are:
–fermi_lines Lines of intersection between constant-energy surfaces and the slice. The energy
level is specified by the keyword kslice_fermi_level. Output files:
·seedname-kslice-fermi-spn.dat (data file when kslice_fermi_lines_colour = spin)
·seedname-bnd_n.dat (gnuplot data files when kslice_fermi_lines_colour = none)
·seedname-kslice-coord.dat (python data files when kslice_fermi_lines_colour = none)
·seedname-kslice-bands.dat (python data file when kslice_fermi_lines_colour = none)
·seedname-kslice-fermi_lines.gnu (gnuplot script)
·seedname-kslice-fermi_lines.py (python script)
–curv[+fermi_lines] Heatmap of the Berry curvature of the occupied states [together with the
constant-energy contours]. The unit of Berry curvature is berry_curv_unit.
Output files:
·seedname-kslice-coord.dat (data files)
·seedname-kslice-curv.dat (data file)
·[seedname-kslice-bands.dat] (data file)
·seedname-kslice-curv_{x,y,z}[+fermi_lines].py (python scripts)
–morb[+fermi_lines] Heatmap of the k-space orbital magnetization in eV·Å2[together with the
constant-energy contours]. Output files:
·seedname-kslice-coord.dat (data files)
·seedname-kslice-morb.dat (data file)
·[seedname-kslice-bands.dat] (data file)
·seedname-kslice-morb_{x,y,z}[+fermi_lines].py (python scripts)
The default value is fermi_lines.
Note: When kslice_fermi_lines_colour = none the gnuplot scripts draw the k-slices with a square
shape, even when kslice_b1 and kslice_b2 below are not at right angles, or do not have equal lengths.
(The python scripts draw the slices with the correct parallelogram shape.)
108 wannier90: User Guide
10.9.3 real(kind=dp) :: kslice_corner(3)
Reduced coordinates of the lower-left corner of the slice in k-space.
The default value is (0.0,0.0,0.0)
10.9.4 real(kind=dp) :: kslice_b1(3)
Reduced coordinates of the first reciprocal-space vector defining the slice.
The default value is (1.0,0.0,0.0).
10.9.5 real(kind=dp) :: kslice_b2(3)
Reduced coordinates of the second reciprocal-space vector defining the slice.
The default value is (0.0,1.0,0.0).
10.9.6 integer :: kslice_2dkmesh(2)
Dimensions of the k-point grid covering the slice. If two integers m n are given, the slice is sampled
on a uniform m×ngrid. If only one integer mis given, an m×mgrid is used.
The default value for kslice_kmesh is 50.
10.9.7 real(kind=dp) :: kslice_fermi_level
Energy level in eV of the constant-energy contours when kslice_task = fermi_lines.
The default value is fermi_energy, if defined. Otherwise, no default value is given.
10.9.8 character(len=20) :: kslice_fermi_lines_colour
When kslice_task=fermi_lines (but not when combined with curv or morb), colour code the Fermi
lines according to the specified quantity.
The valid options for this parameter are:
–spin Spin projection (in units of ¯h/2) along the quantization axis defined by the variables
spin_axis_polar and spin_axis_azimuth, for a spinor calculation
–none no colour coding
The default value is none.
wannier90: User Guide 109
10.10 berry
10.10.1 logical :: berry
Determines whether to enter the berry routines.
The default value is false.
10.10.2 character(len=120) :: berry_task
The quantity to compute when berry=true
The valid options for this parameter are:
–kubo Complex optical conductivity and joint density of states. Output files:
·seedname-kubo-S_{xx,yy,zz,xy,xz,yz}.dat (data files). First column: optical frequency
¯hω in eV. Second and third columns: real and imaginary parts of the symmetric conductivity
σS
αβ(¯hω) = σS
βα(¯hω)in S/cm. Six additional columns are present if spin_decomp = true.
·seedname-kubo-A_{yz,zx,xy}.dat (data files). First column: optical frequency ¯hω in eV.
Second and third columns: real and imaginary parts of the antisymmetric conductivity
σA
αβ(¯hω) = −σA
βα(¯hω)in S/cm. Six additional columns are present if spin_decomp = true.
·seedname-jdos.dat (data file). First column: energy difference ¯hω in eV between conduc-
tion (c) and valence (v) states with the same crystal momentum k. Second column: joint
density of states ρcv(¯hω)(number of states per unit cell per unit energy range, in eV−1).
Three additional columns are present if spin_decomp = true.
–ahc Anomalous Hall conductivity, in S/cm. The three independent components σx=σyz ,σy=
σzx, and σz=σxy are computed. Output files:
·seedname-ahc-fermiscan.dat (data file) . The first column contains the Fermi level εFin
eV, and the following three column the values of σx,y,z(εF). This file is written if a range of
Fermi energies is specified via fermi_energy_min and fermi_energy_max. If a single Fermi
energy is given, the AHC is printed in seedname.wpout only.
–morb Orbital magnetisation, in bohr magnetons per cell.
Output files:
·seedname-morb-fermiscan.dat (data file). The first column contains the Fermi level εFin
eV, and the following three column the values of Morb
x,y,z(εF). This file is written if a range of
Fermi energies is specified via fermi_energy_min and fermi_energy_max. If a single Fermi
energy is given, Morb is printed in seedname.wpout only.
There is no default value.
10.10.3 integer :: berry_kmesh(:)
Overrides the kmesh global variable (see Sec. 10.6).
110 wannier90: User Guide
10.10.4 real(kind=dp) :: berry_kmesh_spacing
Overrides the kmesh_spacing global variable (see Sec. 10.6).
10.10.5 integer :: berry_curv_adpt_kmesh
If a positive integer nis given and berry_task=ahc, an n×n×nmesh is placed around points on
the uniform mesh (defined by either berry_kmesh or berry_kmesh_spacing) where the magnitude of
the k-space Berry curvature exceeds the threshold value specified in berry_curv_adpt_kmesh_thresh.
This can be used to densify the BZ integration mesh around spikes in the Berry curvature.
The default value is 1.
10.10.6 real(kind=dp) :: berry_curv_adpt_kmesh_thresh
Magnitude of the Berry curvature (in units of berry_curv_unit) that triggers adaptive mesh refinement
when berry_task=ahc.
The default value is 100.0.
10.10.7 real(kind=dp) :: kubo_freq_min
Lower limit of the frequency range for computing the optical conductivity and JDOS. Units are eV.
The default value 0.0.
10.10.8 real(kind=dp) :: kubo_freq_max
Upper limit of the frequency range for computing the optical conductivity and JDOS. Units are eV.
If an inner energy window was specified, the default value is dis_froz_max-fermi_energy+0.6667.
Otherwise it is the difference between the maximum and the minimum energy eigenvalue stored in
seedname.eig, plus 0.6667.
10.10.9 real(kind=dp) :: kubo_freq_step
Difference between consecutive values of the optical frequency between kubo_freq_min and kubo_freq_max.
Units are eV.
The default value is 0.01.
10.10.10 real(kind=dp) :: kubo_eigval_max
Maximum energy eigenvalue of the eigenstates to be included in the evaluation of the optical conduc-
tivity and JDOS. Units are eV.
wannier90: User Guide 111
If an inner energy window was specified, the default value is the upper bound of the inner energy
window plus 0.6667. Otherwise it is the maximum energy eigenvalue stored in seedname.eig plus
0.6667.
10.10.11 logical :: kubo_adpt_smr
Overrides the adpt_smr global variable (see Sec. 10.6).
10.10.12 real(kind=dp) :: kubo_adpt_smr_fac
Overrides the adpt_smr_fac global variable (see Sec. 10.6).
10.10.13 real(kind=dp) :: kubo_adpt_smr_max
Overrides the adpt_smr_max global variable (see Sec. 10.6).
10.10.14 logical :: kubo_smr_fixed_en_width
Overrides the smr_fixed_en_width global variable (see Sec. 10.6).
10.10.15 character(len=120) :: kubo_smr_type
Overrides the smr_type global variable (see Sec. 10.6).
112 wannier90: User Guide
10.11 BoltzWann
10.11.1 logical :: boltzwann
Determines whether to enter the BoltzWann routines.
The default value is false.
10.11.2 integer :: boltz_kmesh(:)
It determines the interpolation kmesh used to calculate the TDF (from which the transport coefficient
are calculated). If boltz_calc_also_dos is true, the same kmesh is used also for the DOS. Overrides
the kmesh global variable (see Sec. 10.6).
10.11.3 real(kind=dp) :: boltz_kmesh_spacing
Overrides the kmesh_spacing global variable (see Sec. 10.6).
10.11.4 character(len=4) :: boltz_2d_dir
For two-dimensional systems, the direction along which the system is non-periodic. It can assume the
following values: xfor a 2D system on the yz plane, yfor a 2D system on the xz plane, zfor a 2D
system on the xy plane, or no for a 3D system with periodicity along all threee directions.
This value is used when calculating the Seebeck coefficient, where the electrical conductivity tensor
needs to be inverted. If the value is different from zero, only the relevant 2×2sub-block of the electrical
conductivity is inverted.
The default value is no.
10.11.5 real(kind=dp) :: boltz_relax_time
The relaxation time to be used for the calculation of the TDF and the transport coefficients.
The units are fs. The default value is 10 fs.
10.11.6 real(kind=dp) :: boltz_mu_min
Minimum value for the chemical potential µfor which we want to calculate the transport coefficients.
The units are eV. No default value.
10.11.7 real(kind=dp) :: boltz_mu_max
Maximum value for the chemical potential µfor which we want to calculate the transport coefficients.
The units are eV. No default value.
wannier90: User Guide 113
10.11.8 real(kind=dp) :: boltz_mu_step
Energy step for the grid of chemical potentials µfor which we want to calculate the transport coeffi-
cients.
The units are eV. No default value.
10.11.9 real(kind=dp) :: boltz_temp_min
Minimum value for the temperature Tfor which we want to calculate the transport coefficients.
The units are K. No default value.
10.11.10 real(kind=dp) :: boltz_temp_max
Maximum value for the temperature Tfor which we want to calculate the transport coefficients.
The units are K. No default value.
10.11.11 real(kind=dp) :: boltz_temp_step
Energy step for the grid of temperatures Tfor which we want to calculate the transport coefficients.
The units are K. No default value.
10.11.12 real(kind=dp) :: boltz_tdf_energy_step
Energy step for the grid of energies for the TDF.
The units are eV. The default value is 0.001 eV.
10.11.13 character(len=120) :: boltz_tdf_smr_type
The type of smearing function to be used for the TDF. The available strings are the same of the global
smr_type input flag.
The default value is the one given via the smr_type input flag (if defined).
10.11.14 real(kind=dp) :: boltz_tdf_smr_fixed_en_width
Energy width for the smearing function. Note that for the TDF, a standard (non-adaptive) smearing
scheme is used.
The units are eV. The default value is 0 eV. Note that if the width is smaller than twice the energy
step boltz_tdf_energy_step, the TDF will be unsmeared (thus the default is to have an unsmeared
TDF).
114 wannier90: User Guide
10.11.15 logical :: boltz_calc_also_dos
Whether to calculate also the DOS while calculating the TDF.
If one needs also the DOS, it is faster to calculate the DOS using this flag instead of using independently
the routines of the dos module, since in this way the interpolation on the kpoints will be performed
only once.
The default value is false.
10.11.16 real(kind=dp) :: boltz_dos_energy_min
The minimum value for the energy grid for the calculation of the DOS.
The units are eV. The default value is minval(eigval)-0.6667, where minval(eigval) i s the mini-
mum eigenvalue returned by the ab-initio code on the ab-initio qme sh.
10.11.17 real(kind=dp) :: boltz_dos_energy_max
The maximum value for the energy grid for the calculation of the DOS.
The units are eV. The default value is maxval(eigval)+0.6667, where maxval(eigval) i s the maxi-
mum eigenvalue returned by the ab-initio code on the ab-initio qme sh.
10.11.18 real(kind=dp) :: boltz_dos_energy_step
Energy step for the grid of energies for the DOS.
The units are eV. The default value is 0.001 eV.
10.11.19 character(len=120) :: boltz_dos_smr_type
Overrides the smr_type global variable (see Sec. 10.6).
10.11.20 logical :: boltz_dos_adpt_smr
Overrides the adpt_smr global variable (see Sec. 10.6).
10.11.21 real(kind=dp) :: boltz_dos_adpt_smr_fac
Overrides the adpt_smr_fac global variable (see Sec. 10.6).
10.11.22 real(kind=dp) :: boltz_dos_adpt_smr_max
Overrides the adpt_smr_max global variable (see Sec. 10.6).
wannier90: User Guide 115
10.11.23 logical :: boltz_dos_smr_fixed_en_width
Overrides the smr_fixed_en_width global variable (see Sec. 10.6).
10.12 Generic Band Interpolation
10.12.1 logical :: geninterp
Determines whether to enter the Generic Band Interpolation routines.
The default value is false.
10.12.2 logical :: geninterp_alsofirstder
Whether to calculate also the first derivatives of the bands at the given kpoints.
The default value is false.
10.12.3 logical :: geninterp_single_file
Whether to write a single seedname_geninterp.dat file (all I/O is done by the root node); or instead
multiple files (one for each node) with names seedname_geninterp_NNNNN.dat, where NNNNN is the
node number. See also the discussion in Sec. 13.1.2 on how to use this flag.
The default value is true.
Chapter 11
Overview of the berry module
The berry module of postw90 is called by setting berry = true and choosing one or more of the
available options for berry_task. The routines in the berry module which compute the k-space
Berry curvature and orbital magnetization are also called when kpath = true and kpath_task =
{curv,morb}, or when kslice = true and kslice_task = {curv,morb}.
11.1 Background: Berry connection and curvature
The Berry connection is defined in terms of the cell-periodic Bloch states |unki=e−ik·r|ψnkias
An(k) = hunk|i∇k|unki,(11.1)
and the Berry curvature is the curl of the connection,
Ωn(k) = ∇k×An(k) = −Imh∇kunk|×|∇kunki.(11.2)
These two quantities play a central role in the description of several electronic properties of crystals [9].
In the following we will work with a matrix generalization of the Berry connection,
Anm(k) = hunk|i∇k|umki=A∗
mn(k),(11.3)
and write the curvature as an antisymmetric tensor,
Ωn,αβ(k) = αβγΩn,γ (k) = −2Imh∇kαunk|∇kβunki.(11.4)
11.2 berry_task=kubo: optical conductivity and joint density of states
The Kubo-Greenwood formula for the optical conductivity of a crystal in the independent-particle
approximation reads
σαβ(¯hω) = ie2¯h
NkΩcX
kX
n,m
fmk−fnk
εmk−εnk
hψnk|vα|ψmkihψmk|vβ|ψnki
εmk−εnk−(¯hω +iη).(11.5)
Indices α, β denote Cartesian directions, Ωcis the cell volume, Nkis the number of k-points used for
sampling the Brillouin zone, and fnk=f(εnk)is the Fermi-Dirac distribution function. ¯hω is the
optical frequency, and η > 0is an adjustable smearing parameter with units of energy.
117
118 wannier90: User Guide
The off-diagonal velocity matrix elements can be expressed in terms of the connection matrix [10],
hψnk|v|ψmki=−i
¯h(εmk−εnk)Anm(k) (m6=n).(11.6)
The conductivity becomes
σαβ(¯hω) = 1
NkX
k
σk,αβ(¯hω)(11.7)
σk,αβ(¯hω) = ie2
¯hΩcX
n,m
(fmk−fnk)εmk−εnk
εmk−εnk−(¯hω +iη)Anm,α(k)Amn,β(k).(11.8)
Let us decompose it into Hermitian (dissipative) and anti-Hermitean (reactive) parts. Note that
δ(ε) = 1
πIm 1
ε−iη ,(11.9)
where δdenotes a “broadended” delta-function. Using this identity we find for the Hermitean part
σH
k,αβ(¯hω) = −πe2
¯hΩcX
n,m
(fmk−fnk)(εmk−εnk)Anm,α(k)Amn,β (k)δ(εmk−εnk−¯hω).(11.10)
Improved numerical accuracy can be achieved by replacing the Lorentzian (11.9) with a Gaussian, or
other shapes. The analytical form of δ(ε)is controlled by the keyword [kubo_]smr_type.
The anti-Hermitean part of Eq. (11.8) is given by
σAH
k,αβ(¯hω) = ie2
¯hΩcX
n,m
(fmk−fnk)Re εmk−εnk
εmk−εnk−(¯hω +iη)Anm,α(k)Amn,β(k).(11.11)
Finally the joint density of states is
ρcv(¯hω) = 1
NkX
kX
n,m
fnk(1 −fmk)δ(εmk−εnk−¯hω).(11.12)
Equations (11.9–11.12) contain the parameter η, whose value can be chosen using the keyword
[kubo_]smr_fixed_en_width. Better results can often be achieved by adjusting the value of ηfor each
pair of states, i.e., η→ηnmk. This is done as follows (see description of the keyword adpt_smr_fac)
ηnmk=α|∇k(εmk−εnk)|∆k. (11.13)
The energy eigenvalues εnk, band velocities ∇kεnk, and off-diagonal Berry connection Anm(k)entering
the previous four equations are evaluated over a k-point grid by Wannier interpolation, as described
in Refs. [8, 11]. After averaging over the Brillouin zone, the Hermitean and anti-Hermitean parts of
the conductivity are assembled into the symmetric and antisymmetric tensors
σS
αβ = ReσH
αβ +iImσAH
αβ (11.14)
σA
αβ = ReσAH
αβ +iImσH
αβ,(11.15)
whose independent components are written as a function of frequency onto nine separate files.
wannier90: User Guide 119
11.3 berry_task=ahc: anomalous Hall conductivity
The antisymmetric tensor σA
αβ is odd under time reversal, and therefore vanishes in non-magnetic
systems, while in ferromagnets with spin-orbit coupling it is generally nonzero. The imaginary part
ImσH
αβ describes magnetic circular dichroism, and vanishes as ω→0. The real part ReσAH
αβ describes
the anomalous Hall conductivity (AHC), and remains finite in the static limit.
The intrinsic dc AHC is obtained by setting η= 0 and ω= 0 in Eq. (11.11). The contribution from
point kin the Brillouin zone is
σAH
k,αβ(0) = 2e2
¯hΩcX
n,m
fnk(1 −fmk)Imh∇kαunk|umkihumk|∇kβunki,(11.16)
where we replaced fnk−fmkwith fnk(1 −fmk)−fmk(1 −fnk).
This expression is not the most convenient for ab initio calculations, as the sums run over the complete
set of occupied and empty states. In practice the sum over empty states can be truncated, but
a relatively large number should be retained to obtain accurate results. Using the resolution of the
identity 1 = Pm|umkihumk|and noting that the term Pn,m fnkfmk(. . .)vanishes identically, we arrive
at the celebrated formula for the intrinsic AHC in terms of the Berry curvature,
σAH
αβ (0) = e2
¯h
1
NkΩcX
k
(−1)Ωαβ(k),(11.17)
Ωαβ(k) = X
n
fnkΩn,αβ(k).(11.18)
Note that only occupied states enter this expression. Once we have a set of Wannier functions spanning
the valence bands (together with a few low-lying conduction bands, typically) Eq. (11.17) can be
evaluated by Wannier interpolation as described in Refs. [8, 12], with no truncation involved.
11.4 berry_task=morb: orbital magnetization
The ground-state orbital magnetization of a crystal is given by [9, 13]
Morb =e
¯h
1
NkΩcX
k
Morb(k),(11.19)
Morb(k) = X
n
1
2fnkIm h∇kunk| × (Hk+εnk−2εF)|∇kunki,(11.20)
where εFis the Fermi energy. The Wannier-interpolation calculation is described in Ref. [12]. Note
that the definition of Morb(k)used here differs by a factor of −1/2from the one in Eq. (97) and Fig. 2
of that work.
11.5 Needed matrix elements
All the quantities entering the formulas for the optical conductivity and AHC can be calculated by
Wannier interpolation once the Hamiltonian and position matrix elements h0n|H|Rmiand h0n|r|Rmi
are known [8, 11]. Those matrix elements are readily available at the end of a standard MLWF
120 wannier90: User Guide
calculation with wannier90. In particular, h0n|r|Rmican be calculated by Fourier transforming the
overlap matrices in Eq. (1.7),
hunk|umk+bi.
Further Wannier matrix elements are needed for the orbital magnetization [12]. In order to calculate
them using Fourier transforms, one more piece of information must be taken from the k-space ab-initio
calculation, namely, the matrices
hunk+b1|Hk|umk+b2i
over the ab-initio k-point mesh [12]. These are evaluated by pw2wannier90, the interface routine
between pwscf and wannier90, by adding to the input file seedname.pw2wan the line
write_uHu =.true.
Chapter 12
Electronic transport calculations with the
BoltzWann module
By setting boltzwann =TRUE,postw90 will call the BoltzWann routines to calculate some transport
coefficients using the Boltzmann transport equation in the relaxation time approximation.
In particular, the transport coefficients that are calculated are: the electrical conductivity σ, the
Seebeck coefficient Sand the coefficient K(defined below; it is the main ingredient of the thermal
conductivity).
The list of parameters of the BoltzWann module are summarized in Table 10.6. An example of a
Boltzmann transport calculation can be found in the wannier90 Tutorial.
Note: By default, the code assumes to be working with a 3D bulk material, with periodicity along
all three spatial directions. If you are interested in studying 2D systems, set the correct value for the
boltz_2d_dir variable (see Sec. 10.11.4 for the documentation). This is important for the evaluation
of the Seebeck coefficient.
Please cite the following paper [14] when publishing results obtained using the BoltzWann module:
G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari,
BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties
with a maximally-localized Wannier functions basis,
Comp. Phys. Comm. (2013), DOI:10.1016/j.cpc.2013.09.015 (arXiv:1305.1587).
12.1 Theory
The theory of the electronic transport using the Boltzmann transport equations can be found for
instance in Refs. [15–17]. Here we briefly summarize only the main results.
The current density Jand the heat current (or energy flux density) JQcan be written, respectively, as
J=σ(E−S∇T)(12.1)
JQ=TσSE −K∇T, (12.2)
where the electrical conductivity σ, the Seebeck coefficient Sand Kare 3×3tensors, in general.
121
122 wannier90: User Guide
Note: the thermal conductivity κ(actually, the electronic part of the thermal conductivity), which
is defined as the heat current per unit of temperature gradient in open-circuit experiments (i.e., with
J= 0) is not precisely K, but κ=K−SσST(see for instance Eq. (7.89) of Ref. [15] or Eq. (XI-57b)
of Ref. [16]). The thermal conductivity κcan be then calculated from the σ,Sand Ktensors output
by the code.
These quantities depend on the value of the chemical potential µand on the temperature T, and can
be calculated as follows:
[σ]ij(µ, T ) = e2Z+∞
−∞
dε −∂f(ε, µ, T )
∂ε Σij(ε),(12.3)
[σS]ij(µ, T ) = e
TZ+∞
−∞
dε −∂f(ε, µ, T )
∂ε (ε−µ)Σij(ε),(12.4)
[K]ij(µ, T ) = 1
TZ+∞
−∞
dε −∂f(ε, µ, T )
∂ε (ε−µ)2Σij(ε),(12.5)
where [σS]denotes the product of the two tensors σand S,f(ε, µ, T )is the usual Fermi–Dirac
distribution function
f(ε, µ, T ) = 1
e(ε−µ)/KBT+ 1
and Σij(ε)is the Transport Distribution Function (TDF) tensor, defined as
Σij(ε) = 1
VX
n,k
vi(n, k)vj(n, k)τ(n, k)δ(ε−En,k).
In the above formula, the sum is over all bands nand all states k(including spin, even if the spin index
is not explicitly written here). En,kis the energy of the n−th band at k,vi(n, k)is the i−th component
of the band velocity at (n, k),δis the Dirac’s delta function, Vis the cell volume, and finally τis the
relaxation time. In the relaxation-time approximation adopted here, τis assumed as a constant, i.e.,
it is independent of nand kand its value (in fs) is read from the input variable boltz_relax_time.
12.2 Files
12.2.1 seedname_boltzdos.dat
OUTPUT. Written by postw90 if boltz_calc_also_dos is true. Note that even if there are other
general routines in postw90 which specifically calculate the DOS, it may be convenient to use the
routines in BoltzWann setting boltz_calc_also_dos = true if one must also calculate the transport
coefficients. In this way, the (time-demanding) band interpolation on the kmesh is performed only
once, resulting in a much shorter execution time.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each energy εon the grid, containing a number of columns. The first column is the
energy ε. The following is the DOS at the given energy ε. The DOS can either be calculated using the
adaptive smearing scheme1if boltz_dos_adpt_smr is true, or using a “standard” fixed smearing, whose
type and value are defined by boltz_dos_smr_type and boltz_dos_smr_fixed_en_width, respectively.
1Note that in BoltzWann the adaptive (energy) smearing scheme also implements a simple adaptive k−mesh scheme:
if at any given kpoint one of the band gradients is zero, then that kpoint is replaced by 8 neighboring kpoints. Thus,
the final results for the DOS may be slightly different with respect to that given by the dos module.
wannier90: User Guide 123
If spin decomposition is required (input flag spin_decomp), further columns are printed, with the spin-
up projection of the DOS, followed by spin-down projection.
12.2.2 seedname_tdf.dat
OUTPUT. This file contains the Transport Distribution Function (TDF) tensor Σon a grid of energies.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each energy εon the grid, containing a number of columns. The first is the energy
ε, the followings are the components if Σ(ε)in the following order: Σxx,Σxy,Σyy,Σxz,Σyz ,Σzz . If
spin decomposition is required (input flag spin_decomp), 12 further columns are provided, with the 6
components of Σfor the spin up, followed by those for the spin down.
The energy εis in eV, while Σis in 1
¯h2·eV ·fs
Å.
12.2.3 seedname_elcond.dat
OUTPUT. This file contains the electrical conductivity tensor σon the grid of Tand µpoints.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each (µ, T )pair, containing 8 columns, which are respectively: µ,T,σxx,σxy,σyy,
σxz,σyz,σzz . (The tensor is symmetric).
The chemical potential is in eV, the temperature is in K, and the components of the electrical conduc-
tivity tensor ar in SI units, i.e. in 1/Ω/m.
12.2.4 seedname_sigmas.dat
OUTPUT. This file contains the tensor σS, i.e. the product of the electrical conductivity tensor and
of the Seebeck coefficient as defined by Eq. (12.4), on the grid of Tand µpoints.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each (µ, T )pair, containing 8 columns, which are respectively: µ,T,(σS)xx,(σS)xy,
(σS)yy,(σS)xz,(σS)yz,(σS)zz. (The tensor is symmetric).
The chemical potential is in eV, the temperature is in K, and the components of the tensor ar in SI
units, i.e. in A/m/K.
12.2.5 seedname_seebeck.dat
OUTPUT. This file contains the Seebeck tensor Son the grid of Tand µpoints.
Note that in the code the Seebeck coefficient is defined as zero when the determinant of the electrical
conductivity σis zero. If there is at least one (µ, T )pair for which det σ= 0, a warning is issued on
the output file.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each (µ, T )pair, containing 11 columns, which are respectively: µ,T,Sxx,Sxy,Sxz,
Syx,Syy,Syz,Szx,Szy,Szz.
124 wannier90: User Guide
NOTE: therefore, the format of the columns of this file is different from the other three files (elcond,
sigmas and kappa)!
The chemical potential is in eV, the temperature is in K, and the components of the Seebeck tensor ar
in SI units, i.e. in V/K.
12.2.6 seedname_kappa.dat
OUTPUT. This file contains the tensor Kdefined in Sec. 12.1 on the grid of Tand µpoints.
The first lines are comments (starting with # characters) which describe the content of the file. Then,
there is a line for each (µ, T )pair, containing 8 columns, which are respectively: µ,T,Kxx,Kxy,Kyy,
Kxz,Kyz,Kzz . (The tensor is symmetric).
The chemical potential is in eV, the temperature is in K, and the components of the Ktensor are the
SI units for the thermal conductivity, i.e. in W/m/K.
Chapter 13
Generic Band interpolation
By setting geninterp =TRUE,postw90 will calculate the band energies (and possibly the band deriva-
tives, if also geninterp_alsofirstder is set to TRUE) on a generic list of kpoints provided by the
user.
The list of parameters of the Generic Band Interpolation module are summarized in Table 10.7.
The list of input kpoints for which the band have to be calculated is read from the file named
seedname_geninterp.kpt. The format of this file is described below.
13.1 Files
13.1.1 seedname_geninterp.kpt
INPUT. Read by postw90 if geninterp is true.
The first line is a comment (its maximum allowed length is 500 characters).
The second line must contain crystal (or rel) if the k-point coordinates are given in relative (crys-
tallographic) units, i.e., in fractional units with respect to the primitive reciprocal lattice vectors.
Otherwise, it must contain frac (or abs) if instead the k−point coordinates are given in absolute
coordinates (in units of 2π/Å) along the kx,kyand kzaxes.
The third line must contain the number nof following kpoints.
The following nlines must contain the list of kpoints in the format
kpointidx k1 k2 k3
where kpointidx is an integer identifying the given kpoint, and k1,k2 and k3 are the three coordinates
of the kpoints in the chosen units.
13.1.2 seedname_geninterp.dat or seedname_geninterp_NNNNN.dat
OUTPUT. This file/these files contain the interpolated band energies (and also the band velocities if
the input flag geninterp_alsofirstder is true).
125
126 wannier90: User Guide
If the flag geninterp_single_file is true, then a single file seedname_geninterp.dat is written by
the code at the end of the calculation. If instead one sets geninterp_single_file to false, each pro-
cess writes its own output file, named seedname_geninterp_00000.dat,seedname_geninterp_00001.dat,
. . .
This flag is useful when one wants to parallelize the calculation on many nodes, and it should be used
especially for systems with a small number of Wannier functions, when one wants to compute the
bands on a large number of kpoints (if the flag geninterp_single_file is true, instead, all the I/O
is made by the root node, which is a significant bottleneck).
Important! The files are not deleted before the start of a calculation, but only the relevant files are
overwritten. Therefore, if one first performs a calculation and then a second one with a smaller number
of processors, care is needed to avoid to mix the results of the older calculations with those of the new
one. In case of doubt, either check the date stamp in the first line of the seedname_geninterp_*.dat
files, or simply delete the seedname_geninterp_*.dat files before starting the new calculation.
To join the files, on can simply use the following command:
cat seedname_geninterp_*.dat > seedname_geninterp.dat
or, if one wants to remove the comment lines:
rm seedname_geninterp.dat
for i in seedname_geninterp_*.dat ; do grep -v \# "$i" >> \
seedname_geninterp.dat ; done
The first few lines of each files are comments (starting with #), containing a datestamp, the comment
line as it is read from the input file, and a header. The following lines contain the band energies (and
derivatives) for each band and kpoint (the energy index runs faster than the k-point index). For each
of these lines, the first four columns contain the k-point index as provided in the input, and the k
coordinates (always in absolute coordinates, in units of 2π/Å). The fifth column contains the band
energy.
If geninterp_alsofirstder is true, three further columns are printed, containing the three first
derivatives of the bands along the kx,kyand kzdirections.
The kpoint coordinates are in units of 2π/Å, the band energy is in eV.
Part IV
Appendices
127
Appendix A
Utilities
The wannier90 code is shipped with a few utility programs that may be useful in some occasions. In
this chapter, we describe their use.
A.1 kmesh.pl
The wannier90 code requires the definition of a full Monkhorst–Pack grid of kpoints. In the input
file the size of this mesh is given by means of the mp_grid variable. E.g., setting
mp_grid = 4 4 4
tells wannier90 that we want to use a 4×4×4kgrid.
One has then to specify (inside the kpoints block in the the seedname.win file) the list of kpoints of
the grid. Here, the kmesh.pl Perl script becomes useful, being able to generate the required list.
The script can be be found in the utility directory of the wannier90 distribution. To use it, simply
type:
./kmesh.pl nx ny nz
where nx,ny and nz define the size of the Monkhorst–Pack grid that we want to use (for instance, in
the above example of the 4×4×4kgrid, nx=ny=nz=4).
This produces on output the list of kpoints in Quantum Espresso format, where (apart from a header)
the first three columns of each line are the kcoordinates, and the fourth column is the weight of each
kpoint. This list can be used to create the input file for the ab-initio nscf calculation.
If one wants instead to generate the list of the kcoordinates without the weight (in order to copy and
paste the output inside the seedname.win file), one simply has to provide a fourth argument on the
command line. For instance, for a 4×4×4kgrid, use
./kmesh.pl 4 4 4 wannier
and then copy the output inside the in the kpoints block in the seedname.win file.
129
130 wannier90: User Guide
We suggest to always use this utility to generate the kgrids. This allows to provide the kpoint
coordinates with the accuracy required by wannier90, and moreover it makes sure that the kgrid used
in the ab-initio code and in wannier90 are the same.
A.2 w90chk2chk.x
During the calculation of the Wannier functions, wannier90 produces a .chk file that contains some
information to restart the calculation.
This file is also required by the postw90 code. In particular, the postw90 code requires at least the
.chk file, the .win input file, and (almost always) the .eig file. Specific modules may require further
files: see the documentation of each module.
However, the .chk file is written in a machine-dependent format. If one wants to run wannier90 on a
machine, and then continue the calculation with postw90 on a different machine (or with postw90 com-
piled with a different compiler), the file has to be converted first in a machine-independent “formatted”
format on the first machine, and then converted back on the second machine.
To this aim, use the w90chk2chk.x executable. Note that this executable is not compiled by default:
you can obtain it by executing
make w90chk2chk
in the main wannier90 directory.
A typical use is the following:
1. Calculate the Wannier functions with wannier90
2. At the end of the calculation you will find a seedname.chk file. Run (in the folder with this file)
the command
w90chk2chk.x -export seedname
or equivalently
w90chk2chk.x -u2f seedname
(replacing seedname with the seedname of your calculation).
This command reads the seedname.chk file and creates a formatted file seedname.chk.fmt that
is safe to be transferred between different machines.
3. Copy the seedname.chk.fmt file (together with the seedname.win and seedname.eig files) on
the machine on which you want to run postw90.
4. On this second machine (after having compiled w90chk2chk.x) run
w90chk2chk.x -import seedname
or equivalently
wannier90: User Guide 131
w90chk2chk.x -f2u seedname
This command reads the seedname.chk.fmt file and creates an unformatted file seedname.chk
ready to be used by postw90.
5. Run the postw90 code.
A.3 PL_assessment
The function of this utility is to assess the length of a principal layer (in the context of a Landauer-
Buttiker quantum conductance calculation) of a periodic system using a calculation on a single unit
cell with a dense k-point mesh.
The utility requires the real-space Hamiltonian in the MLWF basis, seedname_hr.dat.
The seedname_hr.dat file should be copied to a directory containing executable for the utility. Within
that directory, run:
\$> ./PL_assess.x nk1 nk2 nk3 num_wann
where:
nk1 is the number of k-points in x-direction nk2 is the number of k-points in y-direction nk3 is the
number of k-points in z-direction num_wann is the number of wannier functions of your system
e.g.,
\$> ./PL_assess.x 1 1 20 16
Note that the current implementation only allows for a single k-point in the direction transverse to the
transport direction.
When prompted, enter the seedname.
The programme will return an output file seedname_pl.dat, containing four columns
1. Unit cell number, R
2. Average ’on-site’ matrix element between MLWFs in the home unit cell, and the unit cell R
lattice vectors away
3. Standard devaition of the quantity in (2)
4. Maximum absolute value in (2)
A.4 w90vdw
This utility provides an implementation of a method for calculating van der Waals energies based on
the idea of density decomposition via MLWFs.
132 wannier90: User Guide
For theoretical details, please see the following publication and references therein:
Lampros Andrinopoulos, Nicholas D. M. Hine and Arash A. Mostofi, “Calculating dispersion interac-
tions using maximally localized Wannier functions”, J. Chem. Phys. 135, 154105 (2011).
For further details of this program, please see the documentation in utility/w90vdw/doc/ and the
related examples in utility/w90vdw/examples/.
A.5 w90pov
An utility to create Pov files (to render the Wannier functions using the PovRay utility) is provided
inside utility/w90pov.
Please refer to the documentation inside utility/w90pov/doc for more information.
Appendix B
Frequently Asked Questions
B.1 General Questions
B.1.1 What is wannier90?
wannier90 is a computer package, written in Fortran90, for obtaining maximally-localised Wannier
functions, using them to calculate bandstructures, Fermi surfaces, dielectric properties, sparse Hamil-
tonians and many things besides.
B.1.2 Where can I get wannier90?
The most recent release of wannier90 is always available on our website http://www.wannier.org.
B.1.3 Where can I get the most recent information about wannier90?
The latest news about wannier90 can be followed on our website http://www.wannier.org.
B.1.4 Is wannier90 free?
Yes! wannier90 is available for use free-of-charge under the GNU General Public Licence. See the file
LICENCE that comes with the wannier90 distribution or the GNU hopepage at http://www.gnu.org.
B.2 Getting Help
B.2.1 Is there a Tutorial available for wannier90?
Yes! The examples directory of the wannier90 distribution contains input files for a number of tutorial
calculations. The doc directory contains the accompanying tutorial handout.
133
134 wannier90: User Guide
B.2.2 Where do I get support for wannier90?
There are a number of options:
1. The wannier90 User Guide, available in the doc directory of the distribution, and from the
webpage (http://www.wannier.org/user_guide.html)
2. The wannier90 webpage for the most recent announcements (http://www.wannier.org)
3. The wannier90 mailing list (see http://www.wannier.org/forum.html)
B.2.3 Is there a mailing list for wannier90?
Yes! You need to register: go to http://www.wannier.org/forum.html and follow the instructions.
B.3 Providing Help: Finding and Reporting Bugs
B.3.1 I think I found a bug. How do I report it?
•Check and double-check. Make sure it’s a bug.
•Check that it is a bug in wannier90 and not a bug in the software interfaced to wannier90.
•Check that you’re using the latest version of wannier90.
•Send us an email. Make sure to describe the problem and to attach all input and output files
relating to the problem that you have found.
B.3.2 I have got an idea! How do I report a wish?
We’re always happy to listen to suggestions. Email your idea to the wannier90 developers.
B.3.3 I want to help! How can I contribute to wannier90?
Great! There’s always plenty of functionality to add. Email us to let us know about the functionality
you’d like to contribute.
B.3.4 I like wannier90! Should I donate anything to its authors?
Our Swiss bank account number is... just kidding! There is no need to donate anything, please just
cite our paper in any publications that arise from your use of wannier90:
[ref]A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt and N. Marzari, wannier90: A Tool
for Obtaining Maximally-Localized Wannier Functions, Comput. Phys. Commun., 178, 685 (2008)
and http://arxiv.org/abs/0708.0650.
wannier90: User Guide 135
B.4 Installation
B.4.1 How do I install wannier90?
Follow the instructions in the file README.install in the main directory of the wannier90 distribution.
B.4.2 Are there wannier90 binaries available?
Not at present.
B.4.3 Is there anything else I need?
Yes. wannier90 works on top of an electronic structure calculation.
At the time of writing there are public, fully functioning, interfaces between wannier90 and pwscf,
abinit (http://www.abinit.org), siesta (http://www.icmab.es/siesta/), VASP (https://www.
vasp.at), Wien2k (http://www.wien2k.at), fleur (http://www.fleur.de).
To use wannier90 in combination with pwscf code (a plane-wave, pseudopotential, density-functional
theory code, which is part of the quantum-espresso package) you will need to download pwscf from
the webpage http://www.quantum-espresso.org. Then compile pwscf and the wannier90 interface
program pw2wannier90. For instructions, please refer to the documentation that comes with the
quantum-espresso distribution.
For examples of how to use pwscf and wannier90 in conjunction with each other, see the wannier90
Tutorial.
Bibliography
[1] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
[2] I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001).
[3] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys.
Commun. 178, 685 (2008).
[4] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
[5] M. Posternak, A. Baldereschi, S. Massidda, and N. Marzari, Phys. Rev. B 65, 184422 (2002).
[6] F. Gygi, J. L. Fattebert, and E. Schwegler, Comput. Phys. Commun. 155, 1 (2003).
[7] M. B. Nardelli, Phys. Rev. B 60, 7828 (1999).
[8] X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B 74, 195118 (2006).
[9] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
[10] E. I. Blount, Solid State Physics 13, 305 (1962).
[11] J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys. Rev. B 75, 195121 (2007).
[12] M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza, Phys. Rev. B 85, 014435 (2012).
[13] D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys. Rev. B 74, 024408 (2006).
[14] G. Pizzi, D. Volja, B. Kozinsky, M. Fornari, and N. Marzari, Comput. Phys. Commun. (2013),
doi:10.1016/j.cpc.2013.09.015 arXiv:1305.1587.
[15] J. Ziman, Principles of the Theory of Solids, 2nd ed. (Cambridge University Press, 1972).
[16] G. Grosso and G. P. Parravicini, Solid State Physics (Academic Press, 2000).
[17] G. D. Mahan, in Intern. Tables for Crystallography, Vol. D (2006) Chap. 1.8, p. 7828.
137
