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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

4.4 Two-State NMP Model for Trap-Trap Interaction

Up to this point, charge transfer between a reservoir and a defect has been described, while defect to defect charge transfer has been neglected. However, there are strong indications  [219, CSJ2] that “charge hopping" between defects in dielectrica can lead to enhanced leakage currents in semiconductor devices.

4.4.1 Parameter Transformation

Therefore, with the goal of efficient computational models in mind, an approach to calculate defect to defect charge transfer will be presented. To minimize the parameters needed for a full-scale computation of both defect/reservoir and defect/defect NMP charge transfer reactions, as well as to contain the defect properties represented by the PECs for defect to reservoir charge reactions, as shown in Figure 4.3, a harmonic two-state PEC for defect/defect reaction will be derived from the corresponding defect/reservoir interaction.

The top two panels of Figure 4.4 represent the harmonic approximation for two-state PECs for two individual defects A and B, which can be denoted by repeating (4.21) and (4.22) with a negative state 1 and neutral state 2 by

(4.32–4.33) \{begin}{align} V\sub {1}^\text {A,B}\left (q\right ) &= V\sub {1,0}^\text {A,B} + c\sub {1}^\text {A,B} {q}^{2} \label {equ:pec_or1}\\ V\sub {2}^\text {A,B}\left (q\right ) &= V\sub {2,0}^\text {A,B} + c\sub {2}^\text {A,B} \left (q-\Delta q^\text {A,B}\right )^{2}. \label {equ:pec_or2} \{end}{align}

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Figure 4.4: The PECs of two individual defects (top) and (center) are shown for charge transfer from/to the channel conduction band. Superposition of the two PECs results in an effective PEC (bottom) to calculate barriers for charge transfer between the defects. Assuming negligible reservoir relaxation, the effective parameters \( R^\mathrm {eff} \) and \( E_\mathrm {R}^\mathrm {eff} \) are fully determined by the parameters of the individual defect/reservoir PECs. (taken from [CSJ2]))

Under the assumption of linearly independent defect/reservoir state PECs and negligible relaxation of the charge reservoir, together with the requirement of \( R^\mathrm {A,B} = 1 \), a superpostion of the individual PECs yields an effective PEC for defect/defect charge reaction (detailed discussion about limitations and requirements is given in Section 4.4.2). For deriving the defect/defect transition PECs, the minima \( V\sub {2,0}^\text {A,B} \) of the reservoirs in both reservoir/defect PECs are assumed to be the conduction band edge EC of the semiconductor channel substrate. When an electron is transferred from defect A to B, the charge state of A changes from negative to neutral, while B changes its state from neutral to negative simultaneously. The superpostion for PECs of the individual states of defect A and B, together with a transformation of the reaction coordinate accordingly, results in the effective defect/defect charge transfer PEC

(4.34–4.35) \{begin}{align} V\sub {1}^\text {eff}\left (q\right ) &= V\sub {1}^\text {B}\left (q\right ) + V\sub {2}^\text {A}\left (-q\right ) \label {equ:pec_eff1} \\ V\sub {2}^\text {eff}\left (q\right ) &= V\sub {1}^\text {A}\left (-q\right ) + V\sub {2}^\text {B}\left (q\right ) \label {equ:pec_eff2}. \{end}{align}

State 1 thereby represents the state in which an electron dwells at the site of defect A (negatively charged) while defect B resides in its neutral state. The effective state 2 represents the opposite charge states. By inserting (4.32) and (4.33) into (4.34) and (4.35) and subtracting \( V\sub {2,0}^\text {A,B} \) = EC from both equations, the effective states read

(4.36–4.37) \{begin}{align} V\sub {1}^\text {eff} \left (q\right ) &= V\sub {1,0}^\text {B} + c\sub {1}^{B} q^2 + c\sub {2}^\text {A} \big (\Delta q^{A} - q \big )^2 \label {equ:pec_eff_long} \\ V\sub {2}^\text {eff}\left (q\right ) &= V\sub {1,0}^\text {A} + c\sub {1}^{A} q^2 + c\sub {2}^\text {B} \big (q - \Delta q^{B} \big )^2. \label {equ:pec_eff_long2} \{end}{align}

Thereby the energy reference level is shifted by \( V\sub {2,0}^\text {A,B} \), which is allowed as only energetic differences are of interest. By applying a quadratic extension and shifting the reaction coordinate by \( \Delta q\sub {1}^\text {eff} = -\left (c_\mathrm {2}^\mathrm {A} \Delta q^\mathrm {A} \right ) / c_\mathrm {1}^\mathrm {eff} \) with \( c_\mathrm {1}^\mathrm {eff} = c\sub {1}^{B} + c\sub {2}^\text {A} \) the equation is brought to the form of (4.33):

(4.38–4.39) \{begin}{align} V\sub {1}^\text {eff}\left (q\right ) &= V\sub {1,0}^\text {B} - s\sub {1} + \big ( c\sub {1}^\text {B} + c\sub {2}^\text {A} \big ) q^{2} \label {equ:pec_eff_long4} \\ V\sub {2}^\text {eff}\left (q\right ) &= V\sub {1,0}^\text {A} - s\sub {2} + \big ( c\sub {1}^\text {A} + c\sub {2}^\text {B} \big ) \big (q+\Delta q^\text {eff}\big )^{2} \label {equ:pec_eff_long5} \{end}{align}

with \( \Delta q^\text {eff} = \Delta q_\mathrm {2}^\text {eff} - \Delta q_\mathrm {1}^\text {eff} \). Note that the shifts of the minima of the harmonic oscillators

(4.40–4.41) \{begin}{align} s\sub {1} &= c\sub {2}^\text {A} \big ( \Delta q^\text {A}\big )^2 - \frac {\left (c\sub {2}^\text {A} \Delta q^\text {A}\right )^2}{c\sub {1}^\text {eff}} = c\sub {2}^\text {A} \big ( \Delta q^\text {A}\big )^2 \big ( 1 - \frac {c\sub {2}^\text {A}}{c\sub {1}^\text {eff}}\big )\\ s\sub {2} &= c\sub {2}^\text {B} \big ( \Delta q^\text {B}\big )^2 - \frac {\left (c\sub {2}^\text {B} \Delta q^\text {B}\right )^2}{c\sub {2}^\text {eff}} = c\sub {2}^\text {B} \big ( \Delta q^\text {B}\big )^2 \big ( 1 - \frac {c\sub {2}^\text {B}}{c\sub {2}^\text {eff}}\big ) \{end}{align}

are a result of different curvatures in the original PECs and vanish for the case of identical curvature ratios. The parameters \( R^\text {eff} \) and \( E_\mathrm {R}^\text {eff} \), which uniquely define the effective PEC, are readily described by the parameters used in (4.33) for defect/reservoir interaction yielding

(4.42–4.43) \{begin}{align} {R^\text {eff}} &= \sqrt {\frac {c\sub {1}^\text {eff}}{c\sub {2}^\text {eff}}} = \sqrt {\frac {c\sub {1}^\text {B} + c\sub {2}^\text {A}}{c\sub {1}^\text {A} + c\sub {2}^\text {B}}} \label {equ:pec_eff_long1} , \\ E_\mathrm {R}^\text {eff} &= c\sub {1}^\text {eff} {\Delta q^\text {eff}}^{2} = c\sub {1}^\text {eff} \bigg ( \frac {c\sub {2}^\text {A} \Delta q^\text {A}}{c\sub {1}^\text {eff}} + \frac { c\sub {2}^{B} \Delta q^\text {B}}{c\sub {2}^\text {eff}} \bigg )^{2}. \label {equ:pec_eff_long12} \{end}{align}

The case of two identical defects with \( R^\mathrm {A} = R^\mathrm {B} \) and \( E_\mathrm {R}^\mathrm {A} = E_\mathrm {R}^\mathrm {B} \) results in \( R^\mathrm {eff} = 1 \) and \( E_\mathrm {R}^\mathrm {eff} = 2 E_\mathrm {R}^\mathrm {A,B} \), which intuitively states that twice the energy is exchanged with the thermal bath upon electron transfer from defect to defect (as both sites undergo structural relaxation), compared to the defect / reservoir case. It has to be noted that it is inherently assumed that the carrier reservoirs are not undergoing structural relaxation upon charge capture or emission. Thus, \( c_\mathrm {2}^\mathrm {A} = c_\mathrm {2}^\mathrm {B} \), which due to \( R^\mathrm {eff} = \sqrt {\left ( R^\mathrm {A} + 1 \right ) / \left (R^\mathrm {B} + 1 \right )} \) is only a function of the curvature ratios of the defect PECs and with the restriction of \( R^\mathrm {A} = R^\mathrm {B} = 1 \) the effective ratio is also \( R^\mathrm {eff} = 1 \) for non-identical defects with \( E_\mathrm {R}^\mathrm {A} \neq E_\mathrm {R}^\mathrm {B} \).

By calculating the barriers with the effective parameters according to (4.31) and following the derivations given in Section 4.3, using a discrete density of states at ET with the trap density NT , the analytical expressions for the rates for defect/defect charge transfer are

(4.44) \{begin}{align} k_{12,21}\left ( f_{1,2}, E_\mathrm {T;1,2} \right ) = N_\mathrm {T}\left ( E_\mathrm {T;1,2} \right ) f_{1,2} v_{\mathrm {th},n,\mathrm {ox}} \sigma \vartheta \exp \left ( -\varepsilon _{12,21} / \left ( k_\mathrm {B} T \right ) \right ). \label {equ:rates_defects} \{end}{align}

The fundamental advantage of the transformation of the PECs from the defect/reservoir to the defect/defect interaction case is the reduced parameter set, as only one reaction coordinate diagram needs to be parameterized. These parameters can be readily compared to such derived from DFT computations for suspected defect candidates [77]. However, the superposition of defect states can only be considered valid within the limits that are discussed in the following section.

4.4.2 Limitations

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Figure 4.5: Large defect densities and therefore low average distances between the defects lead to a strong coupling, resulting in a reduced relaxation energy and energetic barriers for defect to defect charge transfer, as calcu- lated by constrained-DFT for defects in HfO2 [220] and MgO [221(left). A schematic for one-dimensional Coulomb potentials with large (top, right) and small distance (bottom, right) between the defects illustrates the exponential barrier reduction (red) with decreasing distance (taken from [CSJ2])

As mentioned in the previous section, a superposition of defect/reservoir charge transfer PECs is strictly valid for linearly independent potentials only. For the exact quantum mechanical solution, a Coulomb interaction term needs to be considered in the electronic Hamiltonian of the many-body Schrödinger equation. By using constrained DFT including exact Hartree-Fock exchanges, the reduction of the relaxation energies of the potential energy surfaces for defect to defect electron transfer has been calculated as a function of their mutual distance for defects in MgO [221] and HfO2 [220]. The results of these calculations are reprinted in  Figure 4.5 together with an extrapolation to large distances using a Marcus-like functional of the form [221]

(4.45) \{begin}{align} E_\mathrm {R} = E_\mathrm {R,\infty } \left ( 1 - C / d_\mathrm {t} \right ). \label {equ:er_red} \{end}{align}

\( E_\mathrm {R,\infty } \) was thereby calculated for fully decoupled defects in the dilute limit. The fitting constant \( C \) has the physical meaning of a cavity (for ions in solvents in the original work of Marcus), however, this interpretation may not apply for electron transfer in solids [222]. Nonetheless, the general form of ER as a function of distance as given in (4.45) can still be considered valid. The results show that even at large defect densities of \( N_\mathrm {t} \) = 1 × 1020/cm3 corresponding to low average distances calculated when assuming close-packed spheres

(4.46) \{begin}{align} d_\mathrm {t} = \frac {1}{\sqrt {2}} \bigg ( \frac {4}{N_\mathrm {t}} \bigg )^{\frac {1}{3}} \approx \SI {2.3}{\nano \meter }, \label {equ:dist_dens} \{end}{align}

the relative errors due to \( E_\mathrm {R} \) reduction are about 10 %.

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Figure 4.6: A two dimensional representation of the PES, with harmonic approximations in both lateral reaction coordinates \( q_x \) and \( q_y \), illustrates a slight offset of the energetic minimum of the transition point (star) between the two state minima, compared to the intersection of the one-dimensional approximation (direct line), resulting in enlarged barriers.

Another restriction for the PECs superposition in Section 4.4.1 is given by the requirement of the curvature ratios satisfying \( R^\mathrm {A} = 1 / R^\mathrm {B} \). This seems quite restrictive, however, recent DFT investigations for defects in SiO2 show that the curvature ratios are close to 1 and hence fulfill this restriction. Besides this case study in SiO2, \( R = 1 \) is also well justified and a frequently used approximation for defects in other materials [223, 224, 110]. Additionally, if the requirement is not met, the minimum energy crossing point of both two-dimensional PESs does not lie on the direct line between the two minima of the states, as shown in Figure 4.6. A correct calculation of the intersection point would then require the calculation of the energetic minimum of the transition point of the two dimensional PES, which exceeds the acceptable computational effort for a device reliability study by far.

Additionally, the choice of \( R = 1 \) prevents a cross-correlated defect parameter search for \( R \) and \( E_\mathrm {R} \), when trying to fit experimental data, as will be outlined in Section 4.8. With the NMP model for defect to defect charge transfer outlined and keeping its limitations in mind, the next section will present its application for the calculation of trap-assisted leakage currents.