# TB2J

## Introduction

TB2J is an open-source Python package for the automatic computation of magnetic interactions (including exchange and Dzyaloshinskii-Moriya) between atoms of magnetic crystals from density functional Hamiltonians based on Wannier functions or linear combinations of atomic orbitals. The program is based on Green’s function method with the local rigid spin rotation treated as a perturbation. The ABACUS interface has been added since TB2J version 0.8.0.

## Installation

The most easy way to install TB2J is to use pip:

pip install TB2J


You can also download TB2J from the github page, and install with:

git clone https://github.com/mailhexu/TB2J.git
cd TB2J
python setup.py install


## The Heisenberg model

The Heisenberg Hamiltonian in TB2J contains three different parts, which are:

$$E = -\sum_{i \neq j} \left[ J_{\text{iso}}^{ij} \vec{S}_i \cdot \vec{S}_j + \vec{S}_i J_{\text{ani}}^{ij} \vec{S}_j + \vec{D}_{ij} \cdot (\vec{S}_i \times \vec{S}_j) \right],$$

where $$J_{\text{iso}}^{ij}$$ represents the isotropic exchange, $$J_{\text{ani}}^{ij}$$ represents the symmetric anisotropic exhcange which is a 3 $$\times$$ 3 tensor with $$J^{\text{ani}} = J^{\text{ani,T}}$$, $$\vec{D}_{ij}$$ represents the Dzyaloshinskii-Moriya interaction (DMI).

Note: Exchange parameters conventions for other Heisenberg Hamiltonian can be found in Conventions of Heisenberg Model.

## How to use

With the LCAO basis set, TB2J can directly take the output and compute the exchange parameters. For the PW and LCAO-in-PW basis set, the Wannier90 interace can be used instead. In this tutorial we will use LCAO.

### Collinear calculation without SOC

We take Fe as an example to illustrate how to use ABACUS to generate the input files required for TB2J.

#### 1. Perform ABACUS calculation.

INPUT file:

INPUT_PARAMETERS
#Parameters (1.General)
suffix                         Fe
stru_file                      STRU
kpoint_file                    KPT
pseudo_dir                     ./
orbital_dir                    ./
calculation                    scf
ntype                          1
nspin                          2
symmetry                       1
noncolin                       0
lspinorb                       0

#Parameters (2.PW)
ecutwfc                        100
scf_thr                        1.0e-6
init_chg                       atomic
out_mul                        1

#Parameters (4.Relaxation)
ks_solver                      genelpa
scf_nmax                       200
out_bandgap                    0

#Parameters (5.LCAO)
basis_type                     lcao
gamma_only                     0

#Parameters (6.Smearing)
smearing_method                gauss
smearing_sigma                 0.001

#Parameters (7.Charge Mixing)
mixing_type                    broyden
mixing_beta                    0.2

# Variables related to output information
out_mat_hs2                    1


STRU file:

ATOMIC_SPECIES
Fe  55.845         Fe_ONCV_PBE_FR-1.0.upf

NUMERICAL_ORBITAL
Fe_gga_8au_100Ry_4s2p2d1f.orb

LATTICE_CONSTANT
1.8897261258369282

LATTICE_VECTORS
2.8660000000      0.0000000000      0.0000000000
0.0000000000      2.8660000000      0.0000000000
0.0000000000      0.0000000000      2.8660000000

ATOMIC_POSITIONS
Direct

Fe
5.0000000000
2
0.0000000000 0.0000000000 0.0000000000 1 1 1 mag 2.5
0.5000000000 0.5000000000 0.5000000000 1 1 1 mag 2.5


and KPT file:

K_POINTS
0
Gamma
8 8 8 0 0 0


After the key parameter out_mat_hs2 is turned on, the Hamiltonian matrix $$H(R)$$ (in $$Ry$$) and overlap matrix $$S(R)$$ will be written into files in the directory OUT.\${suffix} . In the INPUT, the line:

suffix                         Fe


specifies the suffix of the output, in this calculation, we set the path to the directory of the DFT calculation, which is the current directory (“.”) and the suffix to Fe.

#### 2. Perform TB2J calculation:

abacus2J.py --path . --suffix Fe --elements Fe  --kmesh 7 7 7


This first read the atomic structures from th STRU file, then read the Hamiltonian and the overlap matrices stored in the files named starting from data-HR-* and data-SR-* files. It also read the fermi energy from the OUT.Fe/running_scf.log file.

With the command above, we can calculate the $$J$$ with a $$7 \times 7 \times 7$$ k-point grid. This allows for the calculation of exchange between spin pairs between $$7 \times 7 \times 7$$ supercell. Note: the kmesh is not dense enough for a practical calculation. For a very dense k-mesh, the --rcut option can be used to set the maximum distance of the magnetic interactions and thus reduce the computation cost. But be sure that the cutoff is not too small.

The description of the output files in TB2J_results can be found in the TB2J documentation. We can find the exchange parameters with Fe by :

grep "Fe" exchange.out


the following contents showing the first, second and third nearest neighbor exchange parameters as 18.6873, 9.9213 and 0.8963 meV, resoectively. More equivalent exchange parameters are also shown.

   Fe1   Fe2   ( -1,  -1,  -1) 18.6873   (-1.433, -1.433, -1.433)  2.482
Fe1   Fe2   ( -1,  -1,   0) 18.6867   (-1.433, -1.433,  1.433)  2.482
Fe1   Fe2   ( -1,   0,  -1) 18.6866   (-1.433,  1.433, -1.433)  2.482
Fe1   Fe2   ( -1,   0,   0) 18.6873   (-1.433,  1.433,  1.433)  2.482
Fe1   Fe2   (  0,  -1,  -1) 18.6873   ( 1.433, -1.433, -1.433)  2.482
Fe1   Fe2   (  0,  -1,   0) 18.6866   ( 1.433, -1.433,  1.433)  2.482
Fe1   Fe2   (  0,   0,  -1) 18.6867   ( 1.433,  1.433, -1.433)  2.482
Fe1   Fe2   (  0,   0,   0) 18.6873   ( 1.433,  1.433,  1.433)  2.482
Fe2   Fe1   (  0,   0,   0) 18.6873   (-1.433, -1.433, -1.433)  2.482
Fe2   Fe1   (  0,   0,   1) 18.6867   (-1.433, -1.433,  1.433)  2.482
Fe2   Fe1   (  0,   1,   0) 18.6866   (-1.433,  1.433, -1.433)  2.482
Fe2   Fe1   (  0,   1,   1) 18.6873   (-1.433,  1.433,  1.433)  2.482
Fe2   Fe1   (  1,   0,   0) 18.6873   ( 1.433, -1.433, -1.433)  2.482
Fe2   Fe1   (  1,   0,   1) 18.6866   ( 1.433, -1.433,  1.433)  2.482
Fe2   Fe1   (  1,   1,   0) 18.6867   ( 1.433,  1.433, -1.433)  2.482
Fe2   Fe1   (  1,   1,   1) 18.6873   ( 1.433,  1.433,  1.433)  2.482
Fe1   Fe1   ( -1,   0,   0)  9.9213   (-2.866,  0.000,  0.000)  2.866
Fe2   Fe2   ( -1,   0,   0)  9.9213   (-2.866,  0.000,  0.000)  2.866
Fe1   Fe1   (  0,  -1,   0)  9.9211   ( 0.000, -2.866,  0.000)  2.866
Fe2   Fe2   (  0,  -1,   0)  9.9211   ( 0.000, -2.866,  0.000)  2.866
Fe1   Fe1   (  0,   0,  -1)  9.9210   ( 0.000,  0.000, -2.866)  2.866
Fe2   Fe2   (  0,   0,  -1)  9.9210   ( 0.000,  0.000, -2.866)  2.866
Fe1   Fe1   (  0,   0,   1)  9.9210   ( 0.000,  0.000,  2.866)  2.866
Fe2   Fe2   (  0,   0,   1)  9.9210   ( 0.000,  0.000,  2.866)  2.866
Fe1   Fe1   (  0,   1,   0)  9.9211   ( 0.000,  2.866,  0.000)  2.866
Fe2   Fe2   (  0,   1,   0)  9.9211   ( 0.000,  2.866,  0.000)  2.866
Fe1   Fe1   (  1,   0,   0)  9.9213   ( 2.866,  0.000,  0.000)  2.866
Fe2   Fe2   (  1,   0,   0)  9.9213   ( 2.866,  0.000,  0.000)  2.866
Fe1   Fe1   ( -1,  -1,   0)  0.8963   (-2.866, -2.866,  0.000)  4.053
Fe2   Fe2   ( -1,  -1,   0)  0.8963   (-2.866, -2.866,  0.000)  4.053
Fe1   Fe1   ( -1,   0,  -1)  0.8970   (-2.866,  0.000, -2.866)  4.053
Fe2   Fe2   ( -1,   0,  -1)  0.8970   (-2.866,  0.000, -2.866)  4.053


Several other formats of the exchange parameters are also provided in the TB2J_results directory , which can be used in spin dynamics code, e.g. MULTIBINIT, Vampire.

### Non-collinear calculation with SOC

The DMI and anisotropic exchange are result of the SOC, therefore requires the DFT calculation to be done with SOC enabled. To get the full set of exchange parameters, a “rotate and merge” procedure is needed, in which several DFT calculations with either the structure or the spin rotated are needed. For each of the non-collinear calcualtion, we compute the exchange parameters from the DFT calculation with the same command as in the collienar case.

abacus2J.py --path . --suffix Fe --elements Fe  --kmesh 7 7 7


And then the “TB2J_merge.py” command can be used to get the final spin interaction parameters.

### Parameters of abacus2J.py

We can use the command

abacus2J.py --help


to view the parameters and the usage of them in abacus2J.py.

### Acknowledgments

We thanks to Xu He to provide critical interface support.