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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

Chapter 4 Modeling of Charge Transfer Reactions at Defects in MOS devices

While the previous chapters discuss the relevance of BTI and TAT, the experimental characterization of the degradation mechanisms and the microscopic structure of defects considered to be responsible for the observed phenomena, in this chapter models to describe the underlying physical processes, i.e. charge transfer reactions, will be presented. Previous works have revealed that inelastic tunneling processes are essential to describe many experimentally observed features of BTI, RTN and TAT. However, the amorphous nature of the deployed insulating layers results in broad distributions of structural defect properties, and hence also the charge transfer kinetics of defects extends over many orders of magnitude in time. The extraction of physically meaningful defect properties from detailed models employing a number of parameters, e.g. the 4-state NMP model, becomes arbitrarily cumbersome, especially in large area devices, where only the macroscopic response of a defect ensemble is experimentally accessible. This becomes even more challenging when dealing with materials which can host a plethora of distinct defect types as is the case for SiC based power MOSFETs. Therefore, the modeling approaches presented in this work target the description of physical charge transfer reactions based on defect parameters that can be linked to experimentally inferred parameters as well as theoretical ab-intio calculations of particular defect candidates. In particular the methods presented here for simulation and defect parameter extraction have been developed with a focus on computational efficiency, in order to allow for the explicit treatment of a large number of defects across a wide parameter space.

4.1 From State Diagrams to the Master Equation

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Figure 4.1: A detailed description of the charge transfer kinetics requires to include meta-stable defect states for many defect structures leading to a 4-state Markov chain (left). However, often a reduction to an effective 2-state diagram (right) can be justified, e.g. in case the meta-stable states are experimentally not observable in large-area devices or the transition rates \( k_{ij} \) between stable and meta-stables states (black), stemming from purely thermally activated reactions, are much faster than the charge transfer transitions (blue).

As already discussed in Chapter 3, within the 4-state defect model a point-defect can dwell in one of two stable (1, 2) or two meta-stable states (1 \( ^\prime \), 2 \( ^\prime \)). As indicated in Figure 4.1 (left) for an electron trap, the states (1, 1 \( ^\prime \)) denote the neutral and (2, 2 \( ^\prime \)) the negative charge states. Thermal transitions within the same charge state are linked to structural relaxation of the defect without the need for an additional charge transfer to occur, hence such transitions are not directly detectable experimentally. However, certain observations in single-defect measurements like switching traps [75] or anomalous/temporal RTN [76] prove the existence of additional meta-stable states. When dealing with a large ensemble of defects as is typically the case for large area devices, these details do not play a significant role and a reduction of the four state model to an effective two state model shown in Figure 4.1 (right) is justified [203]. This holds also true when modeling defects like polarons, which are not expected to exhibit meta-stable states. Additionally, the purely thermally activated transitions from meta-stable to stable states are often much faster than the charge transfer transitions, e.g. considering typical bias conditions, where the barriers for charge transfer are higher than thermal barriers of about 0.25 eV to 0.5 eV in the case of HB or HE \( ^\prime \)-center defects in SiO2 [161].

In general, the temporal evolution of the charge state occupation of a defect is of particular interest. In order to compute the probability of a state to be occupied or change its occupation within a certain time, the basic assumption is that the transition rates \( k_{ij} \) from state \( i \) to state \( j \) are independent of the defect’s occupation history and only depend on the state in which the defect currently dwells. This is the key assumption for modeling the state transitions within a Markov chain and is justified as the time-scale for reaching the equilibrium configuration after a charge transfer (typically within a few ps) to and from a defect is much faster than typical transition rates ranging from µs to ks and above. With this assumption of memory-less transitions, the defect’s charge state can be described using a simple state machine, which can be mathematically described by a time-continuous discrete Markov process with the probability \( P \) of changing the state \( X \) from i to j within the time \( \Delta t \) [204]

(4.1) \{begin}{align} P\left \{ X_{t+\Delta t} = j | X_{t} = i \right \} = k_{ij} \Delta t + \mathcal {O} \big ( \Delta t^2 \big ). \label {equ:probab1} \{end}{align}

On the other hand, the probability of remaining in state \( i \) is given by

(4.2) \{begin}{align} P\left \{ X_{t+\Delta t} = i | X_{t} = i \right \} = 1 - P\left \{ X_{t+\Delta t} \neq i | X_{t} = i \right \} = 1 - \sum _{i\neq j} k_{ij} \Delta t + \mathcal {O}\big (\Delta t^2\big ). \label {equ:probab2} \{end}{align}

By using the law of conditional probabilities and taking the limit to infinitesimal time steps, the evolution of the probability to be in state \( i \) is given by the following differential equation:

(4.3) \{begin}{align} \frac {\partial P_i}{\partial t} = \lim _{\Delta t \to 0} \frac {P_i\left (t+\Delta t\right ) - P_i\left (t\right )}{\Delta t} = \sum _{j \neq i} P_j k_{ji} - P_i k_{ij}. \label {equ:master_equ} \{end}{align}

The entire system is then described by the so-called Master equations, a set of coupled linear differential equations, given by

(4.4) \{begin}{align} \dot {\vec {P}} &= \bm {K}\vec {P} \label {equ:master_equ_sys} \{end}{align}

with the coefficients of the rate matrix \( \bm {K} \)

(4.5) \{begin}{align} K_{ij} = \begin{cases} k_{ji}&, i \neq j \\ - \sum _{l \neq i} k_{il} &, i = j. \end {cases} \{end}{align}

Note that at all times the total occupation has to be conserved, i.e.

(4.6) \{begin}{align} \sum _i P_i = 1 \label {equ:sum_occ} \{end}{align}

has to hold. This condition is preserved from the initial state \( \vec {P}\left (0\right ) \) within the Master equation [205] and by solving (4.4) the occupation probabilities at all times \( t \) can be computed as

(4.7) \{begin}{align} \vec {P}\left (t\right ) = \exp \left (\bm {K}t\right ) \vec {P}\left (0\right ). \label {equ:masterequation_final} \{end}{align}

For Markov processes with a ring topology as constituted within the 4-state model (or reduced state models) the rate matrix \( \bm {K} \) is sparse with only two off-diagonal elements, which significantly reduces numerical computation efforts for solving (4.7). For the simple case of a two-state defect model the steady state solution \( \left ( t \to \infty \right ) \) for the occupation probability of state 1 reads

(4.8) \{begin}{align} P_1\left ( \infty \right ) = \frac {k_{21}}{k_{12}+k_{21}} \label {equ:2state_occupation_steady} \{end}{align}

and the temporal evolution of the occupation probability is given by an exponential decay function, i.e.

(4.9) \{begin}{align} P_1\left ( t \right ) = P_1\left ( \infty \right ) + \left [ P_1\left ( 0 \right ) - P_1\left ( \infty \right ) \right ] \exp { - t \left ( k_{12} + k_{21} \right )}. \label {equ:2state_occupation_trans} \{end}{align}

For an ensemble of \( N \) non-interacting defects their occupations can be simply computed individually by using (4.7), if their initial occupations and transition rates are known.

To include the possibility of a charge transfer between individual defects, the transition rates in (4.3) have to include the rates from and to each other defect and therefore depend on the occupation of all other defects. By assuming a two-state Markov process and denoting \( f_i = P_{1,i} \) with \( i \) running over \( N \) defects, a non-linear system of coupled Master equations with components

(4.10) \{begin}{align} \frac {\partial f_i}{\partial t} = k_{\mathrm {in},i} \big (\vec {f}\big ) \left ( 1 - f_i \right ) + k_{\mathrm {out},i} \big (\vec {f}\big ) f_i \label {equ:coupled_master_equation} \{end}{align}

results. The in- and out-rates at each defect are then given by

(4.11–4.12) \{begin}{align} k_{\mathrm {in},i} \big (\vec {f}\big ) &= \sum _{j \neq i}^N k_{ji} f_j \\ k_{\mathrm {out},i} \big (\vec {f}\big ) &= \sum _{j \neq i}^N k_{ij} \left ( 1 - f_j \right ). \label {equ:coupled_rates} \{end}{align}

Note that (4.10) is fundamentally different compared to the previous case of a single defect treated in (4.3) and (4.4), as it refers to a defect occupation probability in a state system in which \( \sum _i^N f_i \neq 1 \). An efficient algorithm for solving the non-linear system of ordinary differential equations (ODE) as required for calculating charge hopping currents within a two-state NMP model will be presented in Section 4.5. In the following sections, physical charge transitions rates for charge transfer between defects and carrier reservoirs, as well as between defects, are derived.