The ATK-SE Package


The ATK-DFT package	External Calculators

Table of Contents

Introduction

ATK-SemiEmpirical (ATK-SE) can model the electronic properties of molecules, crystals and devices using both self-consistent and non-self-consistent tight-binding models. In this chapter is presented the implemented tight-binding models based on the Slater-Koster model and the extended Hückel model.

The Slater-Koster tight-binding model follows closely the DFTB formalism described in Ref. [4], and it is recommended that this paper is cited in publications using the SlaterKosterCalculator and DeviceSlaterKosterCalculator in ATK.

The extended Hückel model in ATK-SE is described in Ref. [3], and it is recommended that this paper is cited in publications using the HuckelCalculator and DeviceHuckelCalculator in ATK.

In ATK-SE, the non-self-consistent part of the tight-binding Hamiltonian is parametrized using a two-center approximation, i.e. the matrix elements only depend on the distance between two atoms and is independent of the position of the other atoms. In the extended Hückel model, the matrix elements are described in terms of overlaps between Slater orbitals on each site. In this way, the matrix elements can be defined by very few parameters. In the Slater-Koster model, the distance-dependence of the matrix elements is given as a numerical function; this give higher flexibility, but also makes the fitting procedure more difficult.

The self-consistent part of the calculation is identical for both SE models. The density matrix is calculated from the Hamiltonian using non-equilibrium Green's functions for device systems, while for molecules and crystals it is calculated by diagonalization. The density matrix defines the real-space electron density, and consequently the Hartree potential can be obtained by solving the Poisson equation. The following describes the details of the mathematical formalism behind the implementation.

For a list of the built-in parameter sets in ATK-SE, see the section called “Built-in parameter sets in ATK-SE”. In addition to these, it is possible to directly use parameter sets downloaded from the DFTB website, or define your own Slater-Koster table or extended Huckel basis set in the ATK input scripts.

	Note
	To use semi-empirical models in ATK it is necessary to have the `ATKSEMaster` feature in the license file (additionally, `ATKSESlave` features are needed to run SE calculations in parallel). This constitutes a separate component compared to the DFT module, and can be purchased separately.

Background information

Non-self-consistent Hamiltonian

The Hamiltonian is expanded in a basis of local atomic orbitals (an LCAO expansion)

$\displaystyle \phi_{nlm}({\bf r}) = R_{nl}(r) Y_{lm}(\hat{r}),$

where $Y_{lm}$ is a spherical harmonic and $R_{nl}$ is a radial function. Typically, the atomic orbitals used in the LCAO expansion has a close resemblance to the atomic eigenfunctions.

Onsite terms

With this form of the basis set, the onsite elements are given by

$\displaystyle S_{ij}^\mathrm{onsite} = \delta_{ij},$

$\displaystyle H_{ij}^\mathrm{onsite} = E_i \delta_{ij},$

where $E_i$ is an adjustable parameter, which often is close to the atomic eigenenergy.

Offsite terms in the extended Hückel model

The central object in the extended Hückel model is the overlap matrix,

$\displaystyle S_{ij} = \int_V \phi_i({\bf r} -{\bf R}_i) \phi_j({\bf r}- {\bf R}_j) \; \mathrm{d}{\bf r}.$

To calculate this integral the form of the basis functions must be specified. In the extended Hückel model the basis functions are parametrized by Slater orbitals

$\displaystyle R_{nl}(r) = \frac{r^{n-1}}{\sqrt{(2n)!}} \left[C_1 (2 \eta_1)^{n+\frac{1}{2}} e^{-\eta_1 \, r}+C_2 (2 \eta_2)^{n+\frac{1}{2}} e^{-\eta_2 \, r} \right].$

The LCAO basis is described by the adjustable parameters $\eta_1, \eta_2, C_1$ , and $C_2$ . These parameters must be defined for each angular shell of valence orbitals, for each element.

The overlap matrix defines the Hamiltonian

$\displaystyle H_{ij}= \frac{1}{4} (\beta_i+\beta_j) (E_i+E_j) S_{ij},$

where $E_i$ is the onsite orbital energy and $\beta_i$ is a Hückel fitting parameter (often chosen to be 1.75).

Weighting schemes

The are two variants of the weighting schemes of the orbital energies of the offsite Hamiltonian. The scheme used above

$\displaystyle \frac{1}{2} \beta (E_i + E_j) S_{ij},$

where $\beta = \frac{1}{2} (\beta_i+\beta_j)$ is due to Wolfsberg [14], while Hoffmann [15], [16] uses

$\displaystyle \frac{1}{2} (\beta +\alpha^2+(1-\beta) \alpha^4) (E_i + E_j),$

where $\alpha = (E_i - E_j)/(E_i + E_j)$ .

Both variants are available in ATK-SE through the parameters WolfsbergWeighting and HoffmannWeighting which can be given to the HuckelCalculator and the DeviceHuckelCalculator classes.

Offsite Hamiltonian in the Slater-Koster model

The overlap matrix is given by pairwise integrals between the different basis functions. These integrals can be precalculated for all relevant distances and different orbital combinations, and stored in so-called Slater-Koster tables. The Slater-Koster table stores the distance-dependent parameters $s(d , Z_1, Z_2, l_1, l_2, m)$ , where $d$ is the distance, $Z_1$ , $Z_2$ the element types, $l_1$ , $l_2$ the angular momentum of the two orbitals, and the index $m \le \min(l_1, l_2)$ .

From the Slater-Koster tables, the overlap matrix elements are given by

$\displaystyle S_{ij} = \sum_{m \le \min(l_i, l_j)} \alpha_{l_i, m_i, l_j, m_j, m}(\hat{R}_{ij}) s(d_{ij}, Z_i, Z_j, l_i, l_j, m),$

where $\alpha$ are the Slater-Koster expansion coefficients.

In the Slater-Koster model it is assumed that also the Hamiltonian has a pairwise form, and a Slater-Koster table is generated for the Hamiltonian matrix elements. This table may be generated by calculating Hamiltonian matrix elements for a set of dimer distances or by simply fitting matrix elements to the band structure for different lattice constants.

In ATK-SE, the Slater-Koster table is constructed either by providing the path to a directory containing compatible Slater-Koster files (see DFTBDirectory and HotbitDirectory), or directly using the SlaterKosterTable class.

Note that the extended Hückel model is a Slater-Koster model too, with a special fitting procedure for the Hamiltonian matrix elements.

Self-consistent Hamiltonian

In the self-consistent semi-empirical models in ATK, the electron density is computed using the tight-binding model as described above. Th density gives rise to a Hartree potential $V_H$ . The calculation of the Hartree potential is described in detail in the section called “The Hartree Potential”

The Hartree potential is included through an additional term in the Hamiltonian

$\displaystyle H_{ij}^{SCF} = \frac{1}{2}(V_H({\bf R}_i)+V_H({\bf R}_j)) S_{ij}.$

Electron density

The electron density is given by the occupied eigenfunctions

$\displaystyle n({\bf r}) = \sum_{\alpha}|\psi_\alpha({\bf r})|^2 f\left(\frac{\varepsilon_\alpha-\varepsilon_F}{k T}\right),$

where $f(x)=1/(1+e^x)$ is the Fermi function, $\varepsilon_F$ the Fermi energy, $T$ the electron temperature, and $\varepsilon_\alpha$ the energy of eigenstate $\psi_\alpha$ .

Next write the eigenstates in the Slater orbital basis as

$\displaystyle \psi_\alpha = \sum_i c_{\alpha i} \phi_i,$

and see that the total number of electrons, $N=\int_V n({\bf r}) \; \mathrm{d}{\bf r}$ is given by

$\displaystyle N = \sum_{i j} D_{ij} S_{ij},$

where $D_{ij} = \sum_{\alpha}c_{\alpha i}^* c_{\alpha j} f\left(\frac{\varepsilon_\alpha-\varepsilon_F}{k T}\right)$ is the density matrix.

An approximate atom-based electron density

In practice, a simple approximation is used for the electron density. To this end, introduce the Mulliken population

$\displaystyle m_\mu = \sum_{i \in \mu} \sum_j D_{ij} S_{ij},$

of atom number $\mu$ ,and write the total number of electrons as a sum of atomic contributions, $N=\sum_\mu m_\mu$ . The radial dependence of each atomic-like density is represented by a Gaussian function, and the total induced charge in the system is approximated by

$\displaystyle \delta n({\bf r})=\sum_\mu \delta m_\mu \sqrt{ \frac{\alpha_{\mu}}{\pi}} e^{-\alpha_\mu |{\bf r}-{\bf R}_\mu|^2},$

where $\delta m_\mu = m_\mu - Z_\mu$ is the total charge of atom $\mu$ , i.e. the sum of the valence electron charge $m_\mu$ and the ionic charge $-Z_\mu$ .

To see the significance of the width $\alpha_\mu$ of the Gaussian orbital, calculate the electrostatic potential from a single Gaussian density at position ${\bf R}_\mu$

$\displaystyle V_{H\mu}({\bf r}) = (m_\mu-Z_\mu) \frac{\text{Erf}(\sqrt{\alpha_\mu} | {\bf r}-{\bf R}_\mu|) }{ | {\bf r}-{\bf R}_\mu| }.$

The onsite value of the Hartree potential is $V_{H\mu}({\bf R}_\mu)= (m_\mu-Z_\mu) U_\mu$ , where $U_\mu= 2 \sqrt{ \frac{\alpha_{\mu}}{\pi}}$ is the onsite Hartree shift. In ATK-SE, it is the value of $U_\mu$ which is specified, and this value is used to determine the width $\alpha_\mu$ of the Gaussian using the above relation.

Onsite Hartree shift parameters

The shell-dependent onsite Hartree shift ( $U_l$ ) can be obtained from an atomic calculation.

$U_l$ is related to the linear shift of the eigenenergy $\varepsilon_l$ , of shell $l$ , as function of the shell occupation $q_l$ :

$\displaystyle U_l = \frac{d \varepsilon_l}{d q_l}.$

Thus, $U_l$ can be obtained by performing atomic calculations with different values of $q_l$ .

ATK provides a database for $U_l$ calculated using the DFT GGA.PBE functional. Access to the data is through the function ATK_U.

A table showing the values of the electrostatic parameter $U$ used in ATK-SE is shown here.

Spin polarization

The inclusion of spin in the tight-binding Hamiltonian follows the scheme in Ref. [5]. The following spin dependent term is added to the Hamiltonian

$\displaystyle H_{ij}^\sigma= \pm \frac{1}{2} S_{ij} \left(dE_{l_i} + dE_{l_j} \right),$

where the sign in the equation depends on the spin.

The spin splitting $dE_{l}$ of shell $l$ is calculated from the spin-dependent Mulliken populations $\mu_l$ of each shell at the local site as

$\displaystyle dE_{l} = \sum_{l' \in \mu_l} W_{l l'} \, (m_{l'\uparrow}- m_{l'\downarrow}).$

Onsite spin-split parameters

The shell-dependent spin splitting strength $W_{ll'}$ can be obtained from a spin-polarized atomic calculation, using [5]

$\displaystyle W_{l l'} = \frac{1}{2}\left(\frac{d \varepsilon_{l\uparrow}}{d m_{l' \uparrow}}-\frac{d \varepsilon_{l\uparrow}}{d m_{l'\downarrow}} \right).$

Since $W_{l l'}$ enters symmetrically in the Hamiltonian, it is convenient to symmetrize it

$\displaystyle \bar{W}_{l l'} = \frac{1}{2} (W_{l l'} + W_{l' l}).$

ATK provides a database for $\bar{W}_{l l'}$ . Access to the data is through the function ATK_W.

Tight-binding total energy

The calculation of the total energy follows [21] and [4]. The total energy has five terms:

$\displaystyle E=E_{\rm H^0}+E_{\rm\delta H}+E_{\rm ext}+E_{\rm spin} +E_{\rm pp}.$

The terms in the equation are

$E_{H^0}$ is the one-electron energy of the non-self-consistent Hamiltonian, given by

$\displaystyle E_{\rm H^0} = \sum_{ij} D_{ij} H^0_{ij}$
$E_{\rm\delta H}$ is the electrostatic difference energy,

$\displaystyle E_{\rm\delta H} = \frac{1}{2}\int V^H_0({\bf r}) \delta n({\bf r})d{\bf r}$
$E_{\rm ext}$ is the electrostatic interaction between the electrons and an external field.

$\displaystyle E_{\rm ext} = \int V^{ext}({\bf r}) \delta n({\bf r})d{\bf r}$
$E_{\rm spin}$ is the spin polarization energy [5]

$\displaystyle E_{\rm spin} = -\frac{1}{2} \sum_\mu \sum_{l \in \mu} \sum_{l' \in \mu} W(Z_{\mu}, l,l') m_l m_{l'}$
$E_{\rm pp}$ is the repulsive energy from a pair-potential between each atom pair, $V^{\rm pp}(Z_{\mu}, Z_{\mu'},R_{\mu, \mu'})$ .

$\displaystyle E_{\rm pp} = \sum_{\mu<\mu'} V^{\rm pp}(Z_{\mu}, Z_{\mu'},R_{\mu, \mu'})$

It is optional to add this term to the tight-binding model, it does not affect the electronic structure. The tight-binding model will, however, not give sensible geometries without a repulsive pair-potential.

	Note
	In version ATK 11.8 only the DFTB and Hotbit parameter sets contain a repulsive pair potential term, and only these methods can be used for geometry optimizations.

Parameters

Parameters for the Slater-Koster method

The Slater-Koster parameters can be provided either through the SlaterKosterTable class or through various 3. party formats. The supported 3. party formats are the slater-koster files from the DFTB consortium and the slater-koster files from the Hotbit consortium.

To use parameters in the DFTB or Hotbit format, put the parameter files in a single directory, and setup the basis set and pair potentials using the functions DFTBDirectory or HotbitDirectory.

Build in DFTB and Hotbit Parameters

The current version of ATK-SE is shipped with a number of DFTB style parameters from the CP2K consortium and the hotbit consortium. It is recommended that these home pages are cited if the parameters are used in publications. It is most easy to setup these basis sets using the Virtual NanoLab (VNL).

Build in Slater-Koster Table

A number of orthogonal tight-binding parameters are provided. The parameters are from Vogl et. al.[37] and Jancu et. al.[38] and it is recommended that these papers are cited if the parameters are used for publications. It is most easy to setup these basis sets using the Virtual NanoLab (VNL).

Parameters for the extended Hückel method

The parameters $\eta_1,\eta_2 , C_1, C_2$ , and $E$ must be defined for each valence orbital, while $\beta$ and $U$ only depend on the element type. Different parameter sets are provided with ATK-SE, but it is also possible to provide user-defined parameters in the input file using the HuckelBasisParameters class.

The tables below provide a mapping between the symbols in the equations and the corresponding keywords.

Table 1: HuckelBasisParameters

Symbol	HuckelBasisParameters
$E_i$	ionization_potential
$\beta$	wolfsberg_helmholtz_constant
$U$	onsite_hartree_shift
$W$	onsite_spin_split
$E^{\mathrm{VAC}}$	vacuum_level

Table 2: SlaterOrbital parameters

Symbol	SlaterOrbital parameters
$n$	principal_quantum_number
$l$	angular_momentum
$\eta$	slater_coefficients
$C$	weights

The current version of ATK comes with built-in Hoffmann and Müller parameter sets, which are appropriate for organic molecules. For crystalline structures, both metals and organic materials like graphene, parameters from J. Cerda are provided. When using these parameters, Ref. [9] should be referenced. The parameter sets are available via keywords as listed in the tables below.

To combine parameters from different sources, it is important to make sure they use the same energy zero level, in order to obtain correct charge transfers. This can be obtained by ensuring that the crystals have the correct work function and molecules the correct ionisation energies. For this purpose, an additional parameter $E^{\mathrm{VAC}}$ is introduced, which shifts the energy of the vacuum level. I.e. if a calculation with $E^{\mathrm{VAC}}=0$ eV has a work function of 6.5 eV, then by setting $E^{\mathrm{VAC}}=-1.5$ eV all bands shift rigidly upwards by 1.5 eV and the work function becomes 5.0 eV.

	Note
	The Hückel parameters have been fitted for non-self-consistent calculations. To use the parameters in self-consistent calculations, the self-consistent onsite shifts must be compensated by a reverse shift of the vacuum_levels.

Tables showing the values used for the built-in extended Hückel basis set parameters can be found in the atomic data appendix.