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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

4.3 Non-Radiative Multi-Phonon Theory

As discussed in the previous section, the SRH model fails to describe charge transfer to and from oxide defects due to its purely elastic tunneling character that results in a “tunneling front" from the gate channel into the oxide and underestimation of defect charge transition time constants in thin oxides [73]. The missing model component within the SRH picture is the structural transformation of the defect upon charge capture. In order to compute exact transition rates for both radiative and non-radiative transitions, a full quantum mechanical treatment of the defect and the surrounding phonon bath is required [209]. However, such an approach is computationally unfeasible when applied to a large ensemble of defects with a wide variation of structural properties, as is the case in large area power MOSFETs with amorphous insulators. In this section, the detailed derivation of classical NMP transition rates from the exact quantum mechanical solution as given by Waldhoer [210] will be iterated. In such a form, the NMP model allows for transient computations for a large defect ensemble as is required to study charge trapping on a device level. It should be noted that these classical approximations of the transfer rates have shown to be able to capture the main features of charge trapping related effects in a large number of Si-based [211], as-well as novel 2D [140] and wide-band gap devices [158, 212]. Unlike in radiative transitions, in which the excess energy to overcome an energetic barrier is induced by a photon, both the energetic activation of the carrier and the dissipation of the excess energy upon relaxation to the final state’s equilibrium configuration is due to interaction with multiple phonons. Therefore, for the derivation of the non-radiative multi-phonon transition rates, both electronic states \( \psi \) and vibrational states \( \eta \) as well as their mutual interaction have to be considered. By using the simplifications induced by the Born-Oppenheimer approximation [213], the transition rates from an initial vibronic state \( \ket {\psi _i \otimes \eta _\alpha } \) to a final state \( \ket {\psi _j \otimes \eta _\beta } \) can be calculated with Fermi’s Golden Rule [214]

(4.16) \{begin}{align} k_{i\alpha ,j\beta } = \frac {2\pi }{\hbar } \vert M_{i\alpha ,j\beta } \vert \delta \left (E_{j\beta } - E_{i\alpha }\right ) \label {equ:fermi_goldenrule}

with the state eigenenergies \( E_{i\alpha } \), \( E_{j\beta } \) and the matrix element

(4.17) \{begin}{align} M_{i\alpha ,j\beta } = \mel {\psi _i \otimes \eta _\alpha }{\hat {H}^\prime }{\psi _j \otimes \eta _\beta }. \label {equ:matrix_elem} \{end}{align}

Within the Born-Oppenheimer approximation, the first order perturbation Hamiltonian \( \hat {H}^\prime \) can be separated in an electronic \( \hat {H}^\prime _\mathrm {e} \) and vibrational \( \hat {H}^\prime _\mathrm {v} \) part. This approximation is justified due to the large difference in timescales, at which the electrons and nuclei respond to changes of their coordinates and momenta within the system and the matrix element becomes [210]

(4.18) \begin{equation} \begin{split} M_{i\alpha ,j\beta } &= \mel {\psi _i \otimes \eta _{i\alpha }}{\hat {H}^\prime _\mathrm {e}+\hat {H}^\prime _\mathrm {v}}{\psi _j \otimes
\eta _{j\beta }} \\ &= \bra {\eta _{i\alpha }}\ket {\eta _{j\beta }} \mel {\psi _i}{{\hat {H}}^\prime _\mathrm {e}}{\psi _j} + \bra {\psi _i}\ket {\psi _j} \mel {\eta _{i\alpha }}{\hat
{H}^\prime _\mathrm {v}}{\eta _{j\beta }} \\ &= \bra {\eta _{i\alpha }}\ket {\eta _{j\beta }} \mel {\psi _i}{{\hat {H}}^\prime _\mathrm {e}}{\psi _j}. \end {split} \label
{equ:matrix_elem_bo_approx} \end{equation}

Due to the orthogonality of the different electronic wave-functions, the second term in (4.18) vanishes and the matrix element is determined by the overlap integral of initial and final vibrational states multiplied by the electronic matrix element, as stated by the Franck-Condon principle [215, 216].

As for typical operating conditions a large number of different vibrational modes will contribute to the overall rate, the thermal average over all partial rates (4.16) needs to be considered. With the initial state assumed to be in thermal equilibrium this results in the expression of the rate by a product

(4.19) \{begin}{align} k_{ij} = A_{ij} f_{ij} \label {equ:rates_average} \{end}{align}

with \( A_{ij} \) being the electronic matrix element describing the electronic coupling between the defect wavefunction and the charge reservoir. The line-shape function \( f_{ij} \) describes the vibrational interactions in the classical high temperature limit, in which the difference between the vibrational energies \( \Delta E_{i\alpha } \) is much smaller then the thermal energy \( k_\mathrm {B}T \) and the classical barrier \( \varepsilon _{ij} \) defined by the crossing point between the Potential Energy Surfaces (PESs) of the initial and final state (c.f. Figure 4.3), i.e. \( \Delta E_{i\alpha } < k_\mathrm {B} T < \varepsilon _{ij} \), and can be approximated by  [210]

(4.20) \{begin}{align} f_{ij} = \gamma _{ij} \exp \left (-\frac {\varepsilon _{ij}}{k_\mathrm {B}T} \right ). \label {equ:lsf} \{end}{align}

The exponential term typically dominates \( f_{ij} \), whereas the prefactor \( \gamma _{ij} \), which depends on the particular PES shape, can be neglected [210].

Up to this point, the NMP formalism is fairly general. In order to reduce the complexity introduced by considering a full PES, the harmonic approximation of the one-dimensional Potential Energy Curve (PEC) of the neutral and negative state at their energetic minimum is used, as shown in Figure 4.3 (left). While, the PES shape is in general arbitrarily complex, the assumption of parabolic PECs has been shown to deliver a reasonably accurate approximation for important defect candidates in Si MOSFETs [210, 217]. For the calculation of charge transfer in MOSFETs, the rates (4.19) need to be calculated for the interaction with a carrier reservoir, which can be the channel conduction (valence) band or the gate contact. As the most relevant degradation mechanism for power switch applications is PBTI in a nMOSFET, the rates are derived for charge transfer between the channel conduction band and a defect, as shown in Figure 4.3 (right).


Figure 4.3: Harmonic approximations of the Potential Energy Curves (PECs) of two defect states along the dominant reaction coordinate \( q \) for charge transfer between the channel conduction band and the defect. In the classical high temperature limit the barriers \( \varepsilon _{-0} \) and \( \varepsilon _{0-} \) are determined by the intersection point. These barriers are uniquely defined by the relaxation energy \( E_\mathrm {R} \), the trap level \( E_\mathrm {T} \) relative to the conduction band edge \( E_\mathrm {c} \) and the curvature ratio \( R \). Note that the minima of the parabolas shift relative to each other with different field strength across the oxide (c.f dashed lines).

The system’s total energy in the neutral defect state is given by

(4.21) \{begin}{align} V_\mathrm {0} \left ( q, E_\mathrm {el} \right ) = V_\mathrm {0,cb}\left (E_\mathrm {el}\right ) + V_\mathrm {0,min} + c_\mathrm {0} \left ( q - \Delta q
\right )^2 \label {equ:PEC_individual_neutral} \{end}{align}

with the curvature \( c_0 \) of the neutral PEC and the reaction coordinate difference between the minima \( \Delta q = q_\mathrm {0} - q_\mathrm {-} \). The energy of the electron \( E_\mathrm {el} \) in the channel conduction band is included in the first term \( V_\mathrm {0,cb} \) and as indicated in Figure 4.3 each band state the electron can dwell in results in a separate PEC for the neutral state. When an electron with energy \( E_\mathrm {el} \) is captured from the conduction band, the defect becomes negatively charged and the total energy reads [210]

(4.22) \{begin}{align} V_\mathrm {-} \left ( q \right ) = V_\mathrm {0,cb} - E_\mathrm {el} + V_\mathrm {-,min} + c_\mathrm {-} \left ( q \right )^2. \label
{equ:PEC_individual_negative} \{end}{align}

As only energy differences are relevant for calculating the transition rates, \( V_\mathrm {0,cb} \) cancels out. With the additional definition of the thermodynamic trap level \( E_\mathrm {T} = E_\mathrm {c} - V_\mathrm {0,min} \) the transition rate (4.19) becomes a function of \( E_\mathrm {el} \) and ET as [210]

(4.23) \{begin}{align} k_{0-} \left ( E_\mathrm {el}, E_\mathrm {T} \right ) = A_{0-}\left ( E_\mathrm {el}, E_\mathrm {T} \right ) f_{0-}\left ( E_\mathrm {el}, E_\mathrm {T} \right
). \label {equ:rates_energydep} \{end}{align}

For a continuous energy state reservoir an integration over the energy space is required yielding [210]

(4.24) \{begin}{align} k_{0-} \left ( E_\mathrm {T} \right ) = \int _{E_\mathrm {c}}^{\infty } A_{0-} \left ( E, E_\mathrm {T} \right ) D_\mathrm {c} \left ( E \right ) f_n \left ( E
\right ) f_{0-}\left ( E, E_\mathrm {T} \right ) dE. \label {equ:rates_energydep_int} \{end}{align}

Thereby \( D_\mathrm {c} \) denotes the density of states in the conduction band and the electron (hole) occupation \( f_n \) ( \( f_p \)) of the reservoir needs to be weighed by Fermi-Dirac statistics [210]

(4.25) \{begin}{align} f_n \left ( E \right ) = 1 - f_p \left ( E \right ) = \frac {1}{1+\exp \left [ \left ( E - E_\mathrm {F} \right ) / \left ( k_\mathrm {B} T \right ) \right ]}.
\label {equ:fermi_dirac} \{end}{align}

Finally, to solve the integral (4.24) over the band states, further approximations are necessary, in order to reduce the numerical expense. First, by using the bandedge approximation, which assumes that the carriers within a semiconductor are predominantly located around the bandedges, the matrix elements and line-shape function can be factored out of the integral [210]

(4.26) \{begin}{align} k_{0-} \left ( E_\mathrm {T} \right ) = A_{0-} \left ( E_\mathrm {c}, E_\mathrm {T} \right ) f_{0-}\left ( E_\mathrm {c}, E_\mathrm {T} \right ) \int
_{E_\mathrm {c}}^{\infty } D_\mathrm {c} \left ( E \right ) f_\mathrm {n} \left ( E \right ) dE. \label {equ:rates_energydep_int_be} \{end}{align}

The left over terms in the integral simply yield the electron concentration

(4.27) \{begin}{align} n = \int _{E_\mathrm {c}}^{\infty } D_\mathrm {c} \left ( E \right ) f_n \left ( E \right ) dE. \{end}{align}

Furthermore, the electronic matrix element \( A_{ij} \) is typically approximated by a simple tunneling factor [74], which is reasonable given the strong localization of the defect wave functions [210]. This approximation yields [72]

(4.28) \{begin}{align} A_{0-} = v_{\mathrm {th},n} \sigma _n \vartheta , \{end}{align}

with the electron thermal velocity \( v_{\mathrm {th},n} \), the capture cross section \( \sigma _n \) and the tunneling probability, which is typically calculated by using a Wentzel-Kramers-Brillouin (WKB) approximation [218], which gives analytical expressions for trapezoidal and triangular barriers. Together with the line-shape function in the classical limit, we arrive at the analytical expression for the electron capture rate from the conduction band edge [210]

(4.29) \{begin}{align} k_{0-}^\mathrm {c} \left ( E_\mathrm {T} \right ) = n v_\mathrm {th,n} \sigma _\mathrm {n} \vartheta \exp \left ( - \varepsilon _{0-} / \left ( k_\mathrm {B} T
\right ) \right ) \label {equ:rates_analytical_cap} \{end}{align}

Analogously and by using the relation \( f_p \left ( E \right ) = f_n \exp \left ( \left ( E_\mathrm {F} - E \right ) / \left ( k_\mathrm {B} T \right ) \right ) \) the electron emission rate is given by [210]

(4.30) \{begin}{align} k_{-0}^\mathrm {c} \left ( E_\mathrm {T} \right ) = n v_{\mathrm {th},n} \sigma _n \vartheta \exp \left ( \left ( E_\mathrm {F} - E_\mathrm {T} - \varepsilon
_{-0} \right ) / \left ( k_\mathrm {B} T \right ) \right ). \label {equ:rates_analytical_em} \{end}{align}

In the same fashion analytical rates for electron capture (emission) can be calculated for the interaction with the channel valence band \( k_{0-}^\mathrm {v} \) ( \( k_{-0}^\mathrm {v} \)) and the gate contact \( k_{0-}^\mathrm {g} \) ( \( k_{-0}^\mathrm {g} \)). The total capture and emission rates for solving the Master equation (4.4) are then given by the sum of all capture and the sum of all emission partial rates. As can be seen from the equations (4.29) and (4.30), the rates are dominated by the barriers \( \varepsilon _{0-}, \varepsilon _{-0} \), which can be analytically calculated from the intersection point of the PECs in the harmonic approximation (c.f. Figure 4.3) by [73]

(4.31) \{begin}{align} \varepsilon _{-0} = E_\mathrm {R} \left (\frac { 1 - R\sqrt {1+\left (R^2-1\right ) E_\mathrm {0-} / E_\mathrm {R}}}{R^2-1}\right )^2 \label
{equ:barriers_parabolas} \{end}{align}

with the relaxation energy \( E_\mathrm {R} \) and the square root of curvature ratio \( R = \sqrt {c_\mathrm {0} / c_\mathrm {-}} \). The reverse barrier \( \varepsilon _{0-} \) can be calculated by \( \varepsilon _{0-} = \varepsilon _{-0} - E_\mathrm {0-} \) with the energetic difference between the PECs minima \( E_\mathrm {0-} = E_\mathrm {c} - E_\mathrm {T}-x_\mathrm {T}F_\mathrm {ox} \) and \( F_\mathrm {ox} \) the electric field strength within the oxide. The left panel of Figure 4.3 thereby illustrates how the minima of the PECs are shifted relative to each other by applying \( F_\mathrm {ox} \) across the oxide in the gate stack, which imposes the gate bias dependence of the energy barriers. Note, that this is the main difference of the Grasser two-stage NMP model, compared to the Kirton and Uren model [67], which extended the SRH model by a bias independent Boltzmann factor, which is only a good approximation in the case of interface defects [73].