Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance
Chapter 3 Computational Methods
This chapter outlines the computational methods used in this thesis to investigate the properties of atomic defects in semiconductors and insulators. In particular, density functional theory (DFT), molecular dynamics (MD) and machine learning interatomic potential (MLIP) are discussed.
3.1 Density functional theory (DFT)
The quantum mechanical description of electrons in a material is governed by the Schrödinger equation. Solving this equation for a solid state material system is analytically impossible and computationally infeasible already for a system size of a few atoms. Every electron and ion in the system interacts with each other and different quantum mechanical effects have to be considered, leading to coupled high-dimensional partial equations. Therefore, such a problem has to be solved approximately. An established approach that is computationally efficient while still giving an accurate description of the electronic structure is DFT. This modeling method has been proven for decades to reliably reproduce and predict material properties based on first-principles calculations. The theorems and approximations that led to this theory are outlined in the following.
3.1.1 Hohenberg-Kohn theorems
A significant step towards the development of DFT was made by Hohenberg and Kohn in 1964 with the formulation of two groundbreaking theorems [169], that
demonstrated that the ground state properties of an atomic system are uniquely defined by the electron density
where
The first Hohenberg and Kohn theorem states that the external potential
A functional of a certain parameter will be denoted with square brackets in the following. Because the total energy of the system in the ground state is a functional of
Determining
The exact form of the functional
where the first term corresponds to the Hartree energy and the non-classical many-body correlation effects such as self-interaction, exchange and Coulomb correlation are all packed in the non-classical
The second Hohenberg and Kohn theorem states that only for the true ground state density
The Hohenberg-Kohn theorems give the basic principle of DFT and motivate the search for
3.1.2 Kohn-Sham equations
Due to the electron-electron interactions in
By construction,
As a result, also the full functional
The resulting Schrödinger-like equations for the
These equations include all interaction functionals in an effective potential and are given for the non-interacting particles with wave functions
During the derivation, every component that can not be solved directly was packed into the unknown
If this potential where known, the Kohn-Sham equation would lead to the correct ground state energy and electron density of the Schrödinger equation [172]. Since this is not the case, significant efforts have been made to reasonably approximate the exchange-correlation effects, which will be outlined in the next section.
3.1.3 Exchange-correlation functionals
The foundation of many approximations for the exchange-correlation energy is based on the treatment of the electron density as a locally homogeneous electron gas. This concept was already introduced in the original work of Kohn and Sham [171] and is nowadays known as the local density approximation (LDA). The corresponding functional is given by
where
However, LDA cannot account for rapidly varying electron densities, which description is crucial for analyzing localized charges at defect sites – one of the key focuses of this thesis. Additionally, LDA is known to overestimate
binding energies and underestimate electronic band gaps. A significant improvement in the treatment of
Such a treatment is known as the generalized gradient approximation (GGA) and is generally given by
where
This functional uses only fundamental constants as parameters and the dependency on the gradient of
While GGA tends to provide a more accurate description of the total energies, energy barriers and structural energy differences when compared to LDA [175], it still lacks an accurate description of the electronic properties of a solid state system. In particular, GGA is known to considerably underestimate the electronic band gap of insulators and semiconductors [176]. Because predicting the band gap as accurately as possible is crucial for analyzing the charge trapping properties of defects in electronic devices as described in Section 2.1.4, more advanced models are needed.
A significant improvement of the description of electronic properties can be achieved by the employment of hybrid functionals, which incorporate calculations of the exact Hartree-Fock exchange
where
The idea behind hybrid functionals is to obtain a partially interacting system, where both
with a standard mixing parameter
The mixing parameter
PBE0_TC_LRC is presented in Appendix D.
3.1.4 Solving the equation
With all approximations applied and all functionals defined, the task is now to solve Eq. (3.7) as accurately and as efficiently as possible. Since the potentials depend on the
electron density, which itself is directly related to the Kohn-Sham orbitals by Eq. (3.4), the equation has to be solved in a self-consistent manner. Starting with an educated
initial guess for
Over the last decades, several software packages have been developed to tackle this problem in a numerically stable way. Although each of these packages offers its own approach for this challenge, the obtained values and properties of most recent codes only vary slightly [182]. A main difference between the packages lies in their approach to expand the Kohn-Sham orbitals in an appropriate basis set
which is necessary to allow for efficient numerical calculations.
Two established examples for possible basis sets are plane waves and Gaussian type orbitals (GTOs) [183]. Plane waves satisfy the periodic boundary conditions of bulk materials, offer an orthonormal and complete basis set and are well-behaved in reciprocal space by construction. The size of the basis set only depends on the system size and the cutoff energy. However, since plane waves are delocalized, they are not well suited to describe localized orbitals such as trapped charges at defect sites. Consequently, a large large basis set is required for such investigations which makes the calculations expensive. Additionally, the description of core electrons with plane waves is computationally expensive, which is why the fast varying wave functions near the core region are often replaced with pseudopotentials [184, 185].
GTOs, on the other hand, are centered at the nucleus and provide a compact description of localized orbitals. This is particularly helpful for calculations in amorphous systems, which lack any internal periodicity. Furthermore, GTOs considerably speed up the evaluation of most integrals as their simple mathematical form often allows for an analytical solution. On the downside, GTOs are not orthogonal and can thus suffer from numerical issues such as over-completeness or linear dependency.
The code utilized for this thesis is CP2K [186], which attempts to utilize the benefits of both approaches by combining them into a hybrid basis set called the Gaussian plane wave (GPW) method. The hybrid basis set consists of plane waves and Gaussian functions [187] and is employed in the Quickstep code [188] as implemented in CP2K. In this approach, the GTOs are used for expansion of the wave functions, while the electron density is modeled in a plane wave basis set.
For all calculations in this thesis, core electrons are not explicitly treated but replaced with pseudopotentials to considerably speed up the calculations [185].
Furthermore, to accelerate the computation of
In summary, DFT is an established tool to model the electronic density of the ground state of many-body atomic systems, capturing both classical and quantum-mechanical effects. When combined with sophisticated models for approximating the exchange and correlation effects, DFT mostly gives accurate predictions of structural and electronic properties. However, even though various approximations are applied, solving the Schrödinger equation remains complex and computationally demanding and other methods may be required for specific problems.