Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance

Chapter 2 Theoretical Background

The quantum mechanical treatment of charge transitions at a defect site is governed by time-dependent perturbation theory, which was pioneered by Dirac already in 1927 [103]. Dirac introduced the interaction picture to quantum mechanics, a combination of the Heisenberg picture, where the operators evolve in time, and the Schrödinger picture, where the states $|\psi (t)⟩$ are time-dependent. This treatment later lead to the famous Fermi’s golden rule, which gives the time-independent transition rate between two quantum states in a first-order approximation. The main considerations that lead to its formulation are outlined below.

2.1 Charge transition model

2.1.1 Fermi’s golden rule

Consider a quantum mechanical system in equilibrium, such as a solid state material containing an atomic defect, where the physical interactions are described by the Hamiltonian $H_0$. The time-independent Schrödinger equation with wave function $|\psi (\mathit{r},\mathit{R})⟩$ and energy $E_0$ is given by

$$\begin{array}{}\text{(2.1)}& H_0|\psi (\mathit{r},\mathit{R})⟩=E_0|\psi (\mathit{r},\mathit{R})⟩.\end{array}$$

Here, $\mathit{r}$ and $\mathit{R}$ are vectors that describe the positions of all $N$ electrons and $M$ nuclei in the system, respectively. A time-dependent perturbation $H^{\prime }(t)$, such as an applied electric field or interactions with photons or phonons, can drive the system out of its equilibrium and alter the initial state ${\psi }_i$. The total time-dependent Hamiltonian is then written as $H(t)=H_0+H^{\prime }(t)$. For small perturbations, it can be shown that the probability for the system to transition from an initial state ${\psi }_i$ to a final state ${\psi }_f$ is given in a first-order description by Fermi’s golden rule [104]

$$\begin{array}{}\text{(2.2)}& {\mathrm{\Gamma }}_{if}=\frac{2\pi }{\hslash }{\left|⟨{\psi }_i|H^{\prime }|{\psi }_f⟩\right|}^2\delta (E_f-E_i),\end{array}$$

where the Dirac delta function $\delta (E_f-E_i)$ ensures energy conservation.

This result demonstrates that the transition rate solely depends on the perturbation matrix element $M_{i,f}=⟨{\psi }_i|H^{\prime }|{\psi }_f⟩$. To arrive at this simple expression, several approximations and assumptions had to be applied [105]. The time-dependent state was expanded in a Taylor series in powers of a unit-free perturbation parameter. By considering only small perturbations, which is generally true for e.g. a photon interacting with a bulk semiconductor, it is justified to only include the first order of the expansion. Furthermore, the transition is assumed to occur from a discrete initial state, such as a localized electron at a defect site, to a continuum of final states such as the conduction band. The transition probability is thus integrated over a continuum of final states, which becomes sharply peaked in the asymptotic limit of infinite observation times. Consequently, the $\delta$-function in Eq. (2.2) is only accurate for sufficiently long times with respect to the uncertainty principle. Otherwise, transitions with $E_f\ne E_i$ are also relevant, even though unlikely when the energy difference becomes too large.

2.1.2 Charge transition at a defect site

While the mathematical expression of Fermi’s golden rule appears straightforward, solving the equation analytically for a system with several electrons and nuclei is impossible due to the high-dimensionality of the problem. Moreover, certain approximations and concepts are necessary to even make the numerical solution for solid state systems computationally feasible. A widely used approach to reduce the complexity of such calculations in a first step is to employ the Born-Oppenheimer approximation [106].

Potential energy surface

According to the Born-Oppenheimer approximation [106], the electrons and nuclei in an atomic system can be treated separately due to the immense difference of their masses and, hence, their velocities. Electrons move considerably faster than the nuclei and it is thereby justified to assume that electrons instantaneously react to small motions of the ions. Consequently, when solving the Schrödinger equation for the electrons within this approximation, the nuclei are locked in position $\mathit{R}$, thereby defining the properties of the electrons at position $\mathit{r}$. Following this approximation, both the wave function $\psi$ and the Hamiltonian $H$ can be separated into a vibrational and an electronic term. The total wave function can thus be written as

$$\begin{array}{}\text{(2.3)}& |\psi (\mathit{r},\mathit{R})⟩=|{\varphi }_k(\mathit{r},\mathit{R})⟩|{\chi }_{k\alpha }(\mathit{R})⟩,\end{array}$$

where ${\varphi }_k(\mathit{r},\mathit{R})$ is the electronic state $k$ and ${\chi }_{k\alpha }(\mathit{R})$ the corresponding vibrational state $\alpha$ [107]. The defining equations for both terms are given by

$$\begin{array}{}\text{(2.4)}& H_{\mathrm{el}}|{\varphi }_k(\mathit{r},\mathit{R})⟩=E_k|{\varphi }_k(\mathit{r},\mathit{R})⟩H_{\mathrm{vib}}|{\chi }_{k\alpha }(\mathit{R})⟩=E_{k\alpha }|{\chi }_{k\alpha }(\mathit{R})⟩.\end{array}$$

Consequently, the electronic part of the equation can be solved for a fixed position of the nuclei $\mathit{R}$. For each minimum energy eigenstate of the electron, there are multiple phonon modes $\alpha$ which represent vibrations around the equilibrium positions. By slightly varying the atomic positions, the electronic energy as a function of $\mathit{R}$ can be obtained, thereby defining an energy landscape $E_k(\mathit{R})$, which is commonly referred to as the potential energy surface (PES) of the electronic state. The PES provides a multi-dimensional representation of how the electronic energy of the system changes as a function of the position of the nuclei. Stable or meta-stable configurations of the nuclei correspond to local minima of the PES. The concept of the PES is supported by the adiabatic theorem [108], stating that a physical system remains in its instantaneous eigenstate if a perturbation is applied slowly enough.

Charge transition rates

A charge capture or emissions process at a defect site corresponds to a transition between two different PESs of the defect system. To arrive at an expression for the according transition rate within the Born-Oppenheimer approximation, the perturbation operator is split into an electronic and vibrational part.

$$\begin{array}{}\text{(2.5)}& H^{\prime }=H_{\mathrm{el}}^{\prime }+H_{\mathrm{vib}}^{\prime }.\end{array}$$

By inserting this separation together with Eq. (2.3) into Eq. (2.2), the matrix element can by written as

$$\begin{array}{}\text{(2.6)}& M_{i\alpha f\beta }=⟨{\chi }_{i\alpha }{\varphi }_i|H_{\mathrm{el}}^{\prime }|{\chi }_{f\beta }{\varphi }_f⟩=\int {\chi }_{i\alpha }(\mathit{R}){\chi }_{f\beta }(\mathit{R})\int {\varphi }_i(\mathit{r},\mathit{R})H_{\mathrm{el}}^{\prime }{\varphi }_f(\mathit{r},\mathit{R})\mathrm{d}\mathit{r}\mathrm{d}\mathit{R},\end{array}$$

when assuming orthogonality of the electronic wave functions of the initial and final state $⟨{\varphi }_i|{\varphi }_f⟩=0$ [109]. This equation can further be simplified by applying the Condon approximation, stating that the electronic transition moment does not vary significantly with the nuclear coordinates [110, 111, 112, 107]. This approximation is again justified by the large difference in the masses of electrons and nuclei. Consequently, the electronic matrix element can be taken out of the integral over the nuclear coordinates

$$\begin{array}{}\text{(2.7)}& M_{i\alpha f\beta }=⟨{\chi }_{i\alpha }|{\chi }_{f\beta }⟩⟨{\varphi }_i|H_{\mathrm{el}}^{\prime }|{\varphi }_f⟩.\end{array}$$

Because in reality the electronic states do change with varying nuclei coordinates, the electronic matrix element is evaluated for the equilibrium positions of the nuclei to give a qualitatively good description in the ground state. The validity of this approximation is still under discussion and other approaches such as the static approximation [113, 114] have been proposed for the treatment of $M_{i\alpha f\beta }$.

The total transition rate according to Eq. (2.2) with the description of $M_{i\alpha f\beta }$ according to Eq. (2.7) is given by

$$\begin{array}{}\text{(2.8)}& {\mathrm{\Gamma }}_{if}=\frac{2\pi }{\hslash }{\left|⟨{\chi }_{i\alpha }|{\chi }_{f\beta }⟩\right|}^2{\left|⟨{\varphi }_i|H_{\mathrm{el}}^{\prime }|{\varphi }_f⟩\right|}^2\delta (E_{f\beta }-E_{i\alpha }).\end{array}$$

This simplified relationship is known as the Franck-Condon principle [111, 112, 107] which has important implications for the treatment of charge transitions at defect sites. The matrix element $M_{i\alpha f\beta }$ and consequently the transition rate of Eq. (2.2) only depend on the overlap integral of the initial and final vibrational wave functions and the matrix element of the electronic transition. Thus, Eq. (2.8) can be separated into an electronic part $A_{if}$ including the electronic matrix element and the spectral function $f_{if}$, which corresponds to the vibrational part. To capture the full probability for a charge transition at a defect site, the sum over all final states and the thermal occupation of the initial states has to be considered [115, 114]. The complete transition rate can then be written as

$$\begin{array}{}\text{(2.9a)}& {\mathrm{\Gamma }}_{if}& =A_{if}{f}_{if}\text{(2.9b)}& A_{if}& =\frac{2\pi }{\hslash }{\left|⟨{\varphi }_i|H_{\mathrm{el}}^{\prime }|{\varphi }_f⟩\right|}^2\text{(2.9c)}& f_{if}& =\sum _{\alpha ,\beta }{w}_{\alpha }{\left|⟨{\chi }_{i\alpha }|{\chi }_{f\beta }⟩\right|}^2\delta (E_{f\beta }-E_{i\alpha }),\end{array}$$ where $w_{\alpha }$ is the thermal occupation factor of the vibrational state $\alpha$ [114] and $E_{f\beta }-E_{i\alpha }$ the energy difference between the initial and final state.

The computation of the electronic matrix element in Eq. (2.9b) for charge transfers in heterostructures, as would be relevant for investigations in this thesis, is challenging. For example, when studying the transition of a charge carrier from a substrate of a MOS device to a defect site in the oxide, the matrix element between a localized defect wave function and a de-localized, exponentially decaying bulk wave function has to be obtained. The evaluation of the required interface structures in different charge states through first-principle methods is computationally demanding, particularly when a statistical analysis is needed to account for the randomness of amorphous structures. Consequently, these elements are typically approximated to estimate the full transitions rate, e.g. by a classical capture cross section modified by a Wentzel–Kramers–Brillouin (WKB) based tunneling factor [32, 14, 3, 1, 2]. An approach to calculate this matrix element for charge transfer within a semiconductor can be found in [114].

Given the reasons above, the exact transition rates will not be investigated in this thesis and the $A_{if}$ will only be considered to modify the transition rate by an additional factor. Instead, the focus will lie on analyzing the probability of charge transitions at specific defects sites according to $f_{if}$. The parameters to evaluate Eq. (2.9c), and consequently to obtain transitions barriers for non-radiative charge transitions and vibrational broadening of optical transitions, will be modeled by first-principles methods. In many cases, the PES can effectively be reduced from the multi-dimensional description to one dimension to evaluate the vibrational overlap and to analyze charge trapping phenomena, which drastically reduces the computational costs [116]. Although this one-dimensional (1D) approximation may appear overly simplistic, it can accurately capture the key features of charge trapping dynamics with strong electron-phonon coupling [102, 110]. This will be further discussed in the following section and Appendix A.

Effective phonon mode

Based on the derivations above, it is clear that for an accurate description of the defect dynamics upon charge transitions, both the vibrational and electronic properties, along with their interactions through electron-phonon coupling must be considered. When a charge carrier localizes at a defect site, the attractive and repulsive forces within the system change, leading to the relaxation to a new minimum energy configuration. Since the vibrational wave functions themselves depend on the electronic state, the electronic and phononic interactions are also inherently coupled.

The strength of electron-phonon coupling at a defect site is quantified by the Huang-Rhys factor $S$, which gives the average number of emitted phonons after a charge transition as will be further discussed in Section 2.1.3. A full description of the charge transition involves addressing a multi-dimensional vibrational problem. However, when $S$ is sufficiently high ($S\gg 1$), such a transition can efficiently be described by a one-dimensional CCD diagram [116]. More details about the validity of the 1D representation are given in Appendix A.

In the 1D harmonic approximation, the potential energy surface near the defect site is described by a parabolic function corresponding to the effective phonon mode

$$\begin{array}{}\text{(2.10)}& E(Q)=\frac{1}{2}{\mathrm{\Omega }}^2{Q}^2.\end{array}$$

Here, $\mathrm{\Delta }{Q}^2=\sum _{\alpha ,i}{m}_{\alpha }\mathrm{\Delta }{R}_{\alpha i}^2$ gives the mass-weighted configuration coordinate change between the equilibrium positions of the initial and final states. Typically, $m$ is hereby expressed in atomic mass units and $R$ in angstrom. $\alpha$ is the index of the respective atom and $\mathrm{\Delta }{R}_{\alpha i}=R_{e;\alpha i}-R_{g;\alpha i}$ the distortion vector with $R_{e;\alpha i}$ and $R_{g;\alpha i}$ being the positions of the excited and ground state given in Cartesian coordinates $i=x,y,z$, respectively. Accordingly, a modal mass $M$ can be defined by $M=\mathrm{\Delta }{Q}^2/\mathrm{\Delta }{R}^2$ [102]. The effective modes of the transition can be obtained by calculating the total energies of linearly interpolated images between the stable defect configurations in two different charge states. Here, the term image refers to a specific atomic configuration. When $E$ is known as a function of $Q$, the effective frequency $\mathrm{\Omega }$ can be obtained by a fit to Eq. (2.10). The diagrams that captures the relaxations upon charge transitions with one effective phonon mode are called 1D configuration coordinate diagrams (CCDs). As the multi-dimensional PES is thereby effectively mapped to 1D, $E(Q)$ will be referred to as potential energy curve (PEC) in the following. This concept can be applied to describe both radiative and non-radiative transitions, as will be discussed in the next sections.

2.1.3 Radiative transitions

The 1D CCD of a radiative transition at a defect site is shown for the $-3/-2$ transition of the aluminum split vacancy in $\alpha$-Al${}_2$O${}_3$ in Fig. 2.1. This transition will be discussed in more detail in Chapter 6. Hereby, an electron localized at the vacancy is excited to the conduction band minimum (CBM). The effective modes of the transition correspond to the PEC in charge state $-3$ and $-2$. The total energies of the linearly interpolated images between the initial and final defect configurations are drawn as circles. $E_{\mathrm{ZPL}}$ is the energy of the zero phonon line, corresponding to the transition where no phonons are involved.

$\mathrm{\Delta }Q$ gives the total configuration coordinate change as described previously. The position and shape of the PECs are determined by the thermodynamic CTL and the relaxation energies $E_{\mathrm{Relax}}^{q_1/q_2}$, which will be described in the following.

Charge transition level (CTL)

The CTL gives the thermodynamic trap level of a charge transition at a defect site and can be evaluated from the formation energy formalism. The formation energy $E_{\mathrm{form}}^q$ of a defect in charge state $q$ as a function of the Fermi energy is given by [117]

$$\begin{array}{}\text{(2.11)}& E_{\mathrm{form}}^q=E_{\mathrm{tot}}^q-E_{\mathrm{tot}}^{\mathrm{bulk}}+\sum _i{\mu }_i{n}_i+q{E}_{\mathrm{F}}+E^{\mathrm{corr}},\end{array}$$

where $E_{\mathrm{tot}}^q$ is the total energy of the defective bulk in charge state $q$, $E_{\mathrm{tot}}^{\mathrm{bulk}}$ is the total energy of the pristine bulk, $n_i$ is the number of atoms of type $i$ removed from or added to the system to create the defect and ${\mu }_i$ is the corresponding chemical potential. The Fermi energy $E_{\mathrm{F}}$ is given with respect to the VBM of the pristine bulk $E_{\mathrm{F}}=E_{\mathrm{VBM}}+{ϵ}_{\mathrm{F}}$, where $E_{\mathrm{VBM}}$ can be approximated by the highest occupied Kohn-Sham orbital of the defect-free bulk system [118] (see Section 3.1.2).
$E^{\mathrm{corr}}$ is a correction term accounting for the spurious interactions of a charged supercell with its periodic image as will be described in Section 2.2.1.

The CTL corresponds to the Fermi energy where the formation energies of a defect in two different charge states are equal and a transition in the lowest energy charge state occurs. CTLs determine the electrical activity of a given defect and can be observed in electrical or optical measurements. The CTL is given by

$$\begin{array}{}\text{(2.12)}& \epsilon (q/q^{\prime })=\frac{E_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}^q(E_{\mathrm{F}}=0)-E_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}^{q^{\prime }}(E_{\mathrm{F}}=0)}{q^{\prime }-q},\end{array}$$

where the formation energy is evaluated at the VBM ($E_{\mathrm{F}}=0$).

Relaxation energy

Relaxation energies can be calculated by evaluating the energy of a system in charge state $q_2$ in the equilibrium configuration $Q_1$ of charge state $q_1$ as illustrated by $E_{\mathrm{Relax}}^{q_1/q_2}$ in Fig. 2.1 and Fig. 2.3. $E_{\mathrm{Relax}}^{q_1/q_2}$ corresponds to the energy dissipated to the phonon bath due to structural relaxations after a spontaneous charge transfer from charge state $q_1$ to $q_2$. The relaxation energy for a charging process from $q_1$ to $q_2$ is given by

$$\begin{array}{}\text{(2.13)}& E_{\mathrm{Relax}}^{q_1/q_2}=E_{\mathrm{tot}}^{q_2}(Q_1)-E_{\mathrm{tot}}^{q_2}(Q_2),\end{array}$$

where $Q_2$ is the stable configuration in charge state $q_2$.

Within the 1D approximation, the Huang-Rhys factor of a charge transition is given by:

$$\begin{array}{}\text{(2.14)}& S_{\mathrm{g},\mathrm{e}}=\frac{E_{\mathrm{Relax}}}{\hslash {\mathrm{\Omega }}_{\mathrm{g},\mathrm{e}}},\end{array}$$

where ${\mathrm{\Omega }}_{\mathrm{g},\mathrm{e}}$ is the vibrational frequency in the ground (g) or excited (e) state. In the case of Fig. 2.1, $E_{\mathrm{Relax}}^{-2/-3}$ would be the ground state relaxation while $E_{\mathrm{Relax}}^{-3/-2}$ is the relaxation energy for the excited state. Only if $S$ is sufficiently large ($S\gg 1$), the CCD gives an accurate representation of the vibrational properties (see also Appendix A). The sum of the relaxation energies is sometimes referred to as the Franck-Condon shift ($d_{\mathrm{FC}}=E_{\mathrm{Relax}}^{q_2/q_1}+E_{\mathrm{Relax}}^{q_1/q_2})$.

Vibrational broadening

Radiative (or optical) transitions involve the interaction of a photon with an electron localized at the defect site. Two main optical processes can be distinguished. First, absorption, where the electron can be promoted to an excited state by absorbing a photon with frequency ${\omega }_a$. Second, emission, where the electron relaxes from the excited state and is recaptured at the defect site, which is often accompanied by the emission of photon with frequency ${\omega }_e$. In the classical Franck-Condon approximation [107, 115], it is assumed that the atomic coordinates do not change during an optical transition at a defect site. Such a process corresponds to a vertical transition in the CCD as shown for absorption and emission in Fig. 2.1. The transition probability depends on the the overlap of different vibrational modes of the ground and excited state as given by Eq. (2.9c).

Following the general theory of luminescence [107, 110], the line shape function or normalized luminescence intensity of a dipole-allowed optical transition is given by

$$\begin{array}{}\text{(2.15)}& L_{\mathrm{em}}(\hslash {\omega }_{\mathrm{e}})=C{\omega }^{3}f(\hslash {\omega }_{\mathrm{e}}),\end{array}$$

with $C^{-1}=\int f(\hslash {\omega }_{\mathrm{e}}){\omega }_{\mathrm{e}}^3\mathrm{d}(\hslash {\omega }_{\mathrm{e}})$, while $f(\hslash {\omega }_{\mathrm{e}})$ is the spectral function of Eq. (2.9c). However, for the optical transition to occur, the energy of the emitted photon $E_p=\hslash {\omega }_{\mathrm{e}}$ has to match the energy difference between the initial and the final state inside the $\delta$-function.

$$\begin{array}{}\text{(2.16)}& f(\hslash {\omega }_{\mathrm{e}})=\sum _{m,n}{w}_m(T)|⟨{\chi }_{\mathrm{e}m}|{\chi }_{\mathrm{g}n}⟩{|}^2\cdot \delta (E_{\mathrm{ZPL}}+\hslash {\mathrm{\Omega }}_{\mathrm{e}m}-\hslash {\mathrm{\Omega }}_{\mathrm{g}n}-\hslash {\omega }_{\mathrm{e}}).\end{array}$$

Here, $n$ and $m$ give the vibrational levels of the ground (g) and excited (e) states, respectively. The vibrational wave functions ${\chi }_{\mathrm{g}n}$ and ${\chi }_{\mathrm{e}m}$ are solutions of the quantum-mechanical oscillator with parameters ($Q,n,\mathrm{\Omega }$) as defined by the CCD:

$$\begin{array}{}\text{(2.17)}& {\chi }_{\mathrm{g}n}(Q)=N_0{\mathrm{e}}^{-\frac{\mathrm{\Omega }{Q}^2}{2\hslash }}{H}_n\left(\sqrt{\frac{\mathrm{\Omega }}{\hslash }}Q\right).\end{array}$$

$H_n$ are the Hermite polynomials in the physicists definition and $N_0$ is a normalization factor. The energy level of the excited state $n$ is given by $E_n=(n+1/2)\hslash \mathrm{\Omega }$.

For $S\gg 1$, quantities derived from the CCD give surprisingly good predictions of optical broadening as detected with luminescence and absorption spectroscopy measurements [116]. Following [119], the full width at half maximum (FWHM) of the broadened emission spectrum as a function of the temperature $T$ can be calculated with parameters extracted from the CCD by

$$\begin{array}{}\text{(2.18)}& \begin{array}{rl}\mathrm{FWHM}(T)& =W_0\sqrt{\coth (\frac{\hslash {\mathrm{\Omega }}_{\mathrm{e}}}{k_{\mathrm{B}}T})}\\ & =\sqrt{8\ln 2}\frac{S_{\mathrm{g}}\hslash {\mathrm{\Omega }}_{\mathrm{g}}}{\sqrt{S_{\mathrm{e}}}}\sqrt{\coth (\frac{\hslash {\mathrm{\Omega }}_{\mathrm{e}}}{k_{\mathrm{B}}T})}.\end{array}\end{array}$$

For calculations of the normalized absorption intensity, only minor changes to Eq. (2.16) have to be applied. The normalized absorption line shape can be expressed by [107, 120]

$$\begin{array}{}\text{(2.19)}& L_{\mathrm{ab}}(\hslash {\omega }_{\mathrm{a}})=D{\omega }_{\mathrm{a}}A(\hslash {\omega }_{\mathrm{a}}),\end{array}$$

with $D^{-1}=\int A(\hslash {\omega }_{\mathrm{a}}){\omega }_{\mathrm{a}}d(\hslash {\omega }_{\mathrm{a}})$. From Fig. 2.1 it seems obvious, that for defects with large $S$, the contribution of the ZPL to the total line shape is very small as the vibrational overlap for $n=m=0$ vanishes.

2.1.4 Non-radiative multi-phonon transitions

  In addition to optical transitions, the charge state of a defect can also change following a non-radiative multi-phonon (NMP) process [110, 1, 121, 114]. The CCD of such charge transfer reactions is schematically shown in Fig. 2.2.

While for the optical transition the atomic structure was assumed to stay the same during the absorption and emission processes, for the non-radiative transition the system has to be driven out of its equilibrium configuration for a charge transfer to occur. The corresponding effective phonon modes can be modeled in the same way as for the optical transitions. An increase in temperature allows for the population of higher vibrational modes for the same electronic state [122]. Consequently, the atomic configuration changes due to these vibrations. The probability for a transition is then again proportional to the vibrational overlap between two states according to Eq. (2.9c). When the energy is elevated towards the crossing point of the two PECs, the transition probability for a single transition between the vibrational states $n$ and $m$ increases, because the distance between two wave functions decreases and the overlap becomes larger.

Classical limit

At room temperature, the energy spacing between consecutive vibrational levels for defects in solids is in the range of the thermal energy [102] and the vibrational modes can be assumed to form a continuum. The main portion of the overlap integral to the spectral function is given at the crossing point (CP) of the two PECs [123, 14, 124, 114]. It is therefore justified in the classical limit to neglect quantum-mechanical tunneling and to approximate that the NMP transition solely occurs at the CP of the two PECs [14, 1]. The classical energy barrier for electron capture at the defect site is simply given by $\mathrm{\Delta }{E}_{i,f}=\mathrm{\Delta }{E}_{0,-1}=\mathrm{CP}-\min (V^0(Q))$ and vice versa $\mathrm{\Delta }{E}_{f,i}=\mathrm{\Delta }{E}_{-1,0}=\mathrm{CP}-\min (V^{-1}(Q))$. As emphasized in Appendix B, the CP of the two PECs depends on the energies $E_{\mathrm{Relax}}^{q_1/q_2}$, $E_{\mathrm{Relax}}^{q_2/q_1}$ and the CTL, but is independent of the configurational change as encoded in $\mathrm{\Delta }Q$. This approximation provides good agreement with the full transition probability at room temperature, but fails at cryogenic temperatures, where quantum-mechanical tunneling can not be neglected anymore [125, 32, 2]. For the qualitative analysis of charge transitions between the substrate and defects in insulating layers of e.g. MOSFET and CTF devices, which are commonly operated at room temperature, it is considered reasonably accurate in this thesis. The treatment of such transitions in the classical limit which will be discussed in the following.

Interaction with a charge reservoir

To capture all possible charge transitions between an oxide defect and the substrate, the interaction with the entire range of electronic states in the valence and conduction bands has to be considered. However, as the mobile charge carriers in semiconductors are concentrated near the band edges, a practical approximation is applied in this thesis [126, 3, 32], where it is assumed that the electron (hole) transfer occurs only with the conduction (valence) band edge of the substrate.

A schematic band diagram of a heterostructure in the example of a Si${}_3$N${}_4$/Si system with the $\epsilon (0/-1)$ CTL of a charge trapping site in Si${}_3$N${}_4$ as calculated by Eq. (2.12) is shown in Fig. 2.3.

In this case, the CBM of the Si substrate serves as an electron reservoir. The Fermi level of the device and thus the energy of the exchanged charge carrier lies at the CBM of the substrate $E_{\mathrm{F}}=E_{\mathrm{c},\mathrm{Si}}$. Additionally, the electron trap level $E_{\mathrm{T}}^{0/-1}$ is introduced, which is given for a charge reservoir at the CBM by

$$\begin{array}{}\text{(2.20)}& E_{\mathrm{T}}^{0/-1}=\epsilon (0/-1)-E_{\mathrm{c},\mathrm{Si}}.\end{array}$$

$E_{\mathrm{T}}$ also corresponds to the trap level in the more simplistic SRH-model of charge trapping [127], where only the electronic energies, but not the structural relaxations are considered to describe the charge transition. The band alignment can be obtained by theoretical calculations [11] or from experiment, e.g. photoemission spectroscopy [128]. Without an applied voltage, the CTL of the sample defect in Fig. 2.3, as denoted by $\epsilon (0/-1)$, lies above the CBM of the substrate. This is because the formation energy of the defect, as calculated with Eq. (2.11), is higher in the negative charge state than in neutral charge state (without the captured electron) for this Fermi level. Consequently, the defect is stable in its neutral state with the electron still at the CBM of the substrate, as illustrated by the black parabola. However, when the system is driven out its equilibrium due to lattice vibrations, the defect can capture the electron from the reservoir, given that the energy is elevated up to the CB. Upon charge trapping, the system relaxes to a minimum energy configuration in the negative charge state, corresponding to a different PEC as illustrated by the solid blue parabola. For the backward transition, the energy barrier is smaller and the emission of the electron to the substrate consequently has a higher probability to occur. Under these conditions, it is thus unlikely that an electron will remain localized at the defect site.

Applied electric field

When a gate voltage $V_{\mathrm{G}}$ is applied on a device like a MOSFET or a CTF, the energetic position of the defect level shifts with respect to the substrate bands due to band bending near the interface and the electric field $F_{\mathrm{g}}$ in the oxide. Consequently, also the CBs for NMP charge transitions change [129, 130, 1] as illustrated in Fig. 2.3. The resulting trap level shift is given in a non-self-consistent first-order estimation by

$$\begin{array}{}\text{(2.21)}& {\mathrm{\Delta }}_{\mathrm{S}}=q{x}_{\mathrm{d}}{F}_{\mathrm{g}},\end{array}$$

where $q$ is the charge of the electron and $x_{\mathrm{d}}$ is the position of the defect in the nitride with respect to the interface. Here, it is assumed that the field does not interact with the defect, that dipole interactions can be neglected and the electric field does not interact with the electrons in the reservoir. The resulting energetic shift and the band bending of the nitride are schematically depicted by the dashed lines.

If the trap level shift $\mathrm{\Delta }$ is sufficiently large, the PEC in the negative charge states lies below the Fermi level and the system becomes stable with the localized electron at the defect site. Consequently, the transition barriers for the charge capture and release processes are altered, making it less likely for the electron to be emitted back to the substrate. Within the description of Eq. (2.11), $E_{\mathrm{form}}^q$ for $q=-1$ is now lower than for $q=0$, because the Fermi level of the system is shifted with respect to its energy reference, the VBM of the nitride. If the CTL is equal to the energy of the charge reservoir, the minima of both parabolas are at the same level. Consequently, the charge transfer frequency and thus the power of the random telegraph noise (RTN) signal, as measured for example in time dependent defect spectroscopy [131, 18], is at its maximum.

2.1.5 Thermal transitions

  So far, only transitions between different PES have been discussed. However, transitions to different minimum energy configurations on the same PES can also occur, a process that is governed by the general transition state theory [132]. The reaction from one state to another within this theory occurs via a transition state (TST), which is a saddle point on the PES. The thermal energy barrier between one minimum energy state and the TST is called activation energy $E_{\mathrm{a}}$.

The corresponding energy profile in 1D for a thermally activated transition between two (meta)stable defect states is schematically shown in Fig. 2.4. Here, the PECs for different charge states of a single defect have two local minima.

The TST between the two minimum energy configurations is denoted as a diamond. In thermal equilibrium, the corresponding transition rate is given by an Arrhenius law

$$\begin{array}{}\text{(2.22)}& k\propto {\nu }_0{\mathrm{e}}^{\frac{-E_{\mathrm{a}}}{k_{\mathrm{B}}T}},\end{array}$$

where ${\nu }_0$ is an attempt frequency and $E_{\mathrm{a}}$ is the activation energy, given by the difference between the initial minimum energy and the TST.

In general, the transition can occur along any possible path, but is dominated by the smallest barrier due to the exponential dependence of the rate on the energy barrier. This path is therefore called the minimum energy path (MEP) and the TST along this path corresponds to a first-order saddle point at the PES. In amorphous systems, for example, a defect can be metastable and stable in two different configurations. Prominent examples of such defect systems are the puckered and unpuckered defect configurations in $a$-SiO₂ [26, 4] as shown for the hydrogen bridge defect in Fig. 2.5.

When the Si moves through the plane of its adjacent oxygen atoms, it relaxes to a different stable configuration, corresponding to another local minimum on the same PES. Such defect configurations will be further discussed in Section 4.1.

The occurrence of different metastable configurations of a single oxide defect can have significant consequences on the charge trapping processes. When a defect relaxes to a different configuration on the same PEC after charge trapping, this naturally also affects the CTL and consequently the classical energy barrier for a charge transfer at this specific defect site. Consequently, a single defect site can exhibit completely different charge transition rates. This motivated the formulation of the 4-state NMP model to explain anomalous charge trapping events such as electrically disappearing defects or anomalous RTN in $a$-SiO${}_2$ [1, 26, 133, 4]. The computational method to find the TST and the MEP utilized for this thesis will be described briefly below.

Nudged elastic band

The nudged elastic band (NEB) method [134] is a powerful computational tool to obtain the MEP between two stable states in a chemical reaction or a phase transition. With the NEB method, the TST of the reaction can be found, which determines the transition rate between two configurations according to Eq. (2.22).

To find the MEP, the first step is to create equally spaced intermediate configurations between the initial and final state, which are called images or replicas. These replicas are connected with an applied spring force ${\mathit{F}}^s$. The spring force which acts on image $i$ with a spring constant $k$ is given by

$$\begin{array}{}\text{(2.23)}& {\mathit{F}}_i^s=k({\mathit{R}}_{i+1}-{\mathit{R}}_i)-k({\mathit{R}}_i-{\mathit{R}}_{i-1}).\end{array}$$

The total force acting on image $i$ of the chain is then given by

$$\begin{array}{}\text{(2.24)}& {\mathit{F}}_i=-\mathrm{\nabla }V({\mathit{R}}_i)+{\mathit{F}}_i^s,\end{array}$$

where $V({\mathit{R}}_i)$ is the total potential energy of the system as a function of all atomic coordinates ${\mathit{R}}_i$. Trying to find the MEP by optimizing the total energy of the chain using Eq. (2.24) results in two problems [135]. First, the images tend to slide down the MEP due to the component of the true force along the path. Second, the true MEP is often not found because of cutting corners on the PEC and consequently missing the true saddle point, which is caused by component of the spring force perpendicular to the path. The NEB method overcomes these issues by projecting out the problematic components.

To obtain these components, the path direction is defined by the tangent vector ${\mathit{\tau }}_{\parallel }$. The component of a force along this path is obtained by projection according to ${\mathit{F}}_{\parallel }=(\mathit{F}\cdot {\mathit{\tau }}_{\parallel }){\mathit{\tau }}_{\parallel }$. Thus, the parallel component of the true force is given by $\mathrm{\nabla }V({\mathit{R}}_i){|}_{\parallel }=\mathrm{\nabla }V({\mathit{R}}_i){\mathit{\tau }}_{\parallel }{\mathit{\tau }}_{\parallel }$ and the component of a force perpendicular to the path can be calculated with ${\mathit{F}}_{\mathrm{\perp }}^s={\mathit{F}}^s-{\mathit{F}}_{\parallel }^s$. The total force on image $i$ of the NEB method becomes

$$\begin{array}{}\text{(2.25)}& {\mathit{F}}_i^{\mathrm{NEB}}=-\mathrm{\nabla }V({\mathit{R}}_i{)}_{\mathrm{\perp }}+{\mathit{F}}_{i\parallel }^s.\end{array}$$

As a result, the spring forces do not prevent the elastic band from converging to the MEP and the images remain equally spaced [136]. The perpendicular component of the spring force can occasionally be turned on again when the parallel component is too large and the elastic band starts forming kinks [137].

While this method provides a discrete representation of the MEP, it often underestimates the true TST, requiring interpolation between the highest points for an approximate solution. This issue is solved by the climbing image nudged elastic band (CI-NEB) [136] method. In this approach, the image with the highest energy after a few optimization iterations is allowed to climb up the MEP to converge to a saddle point. This is achieved by a slightly modifying Eq. (2.25) for the highest energy image $i_{\max }$

$$\begin{array}{}\text{(2.26)}& {\mathit{F}}_{i_{\max }}^{\mathrm{CI}-\mathrm{NEB}}=-\mathrm{\nabla }V({\mathit{R}}_{i_{\max }})+2(\mathrm{\nabla }V({\mathit{R}}_{i_{\max }}{)}_{\parallel }.\end{array}$$

The image $i_{\max }$ is influenced by the full force generated by the potential, with the component along the path inverted and without any impact from the spring force. CI-NEB is an established method to obtain the saddle point of a thermal transition and the corresponding MEP and thus used in this thesis for all TST calculations.