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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

4.2 The Shockley-Read-Hall Model

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Figure 4.2: The classic Shockley-Read-Hall (SRH) model was derived for the calculation of recombination (generation) rates of electrons and holes within a non-degenerate bulk semiconductor (left). Later it was ex- tended to account for charge trapping in interface and oxide defects (right) by simply adding a tunneling probability factor to the transition rates [61].

In their original work, Shockley and Read, as well as Hall, derived rates for non-radiative recombination of excess electrons and holes [62, 206] in bulk semiconductors, introduced either by light or carrier injection via defects, as shown in Figure 4.2 (left). For non-degenerated semiconductors the total rate for electron capture was derived as [62]

(4.13) \{begin}{align} k_{\mathrm {c},n} = \left [ 1 - \mathrm {exp} \left ( \frac {E_\mathrm {T} - E_{\mathrm {F},n}}{k_\mathrm {B} T} \right ) \right ] f_{p,\mathrm {T}} n C_n. \label {equ:srh_capture} \{end}{align}

Thereby \( n \) denotes the electron density, \( f_\mathrm {p,T} \) the hole occupation of the trap ( \( f_{p,\mathrm {T}} = 1 - f_{n,\mathrm {T}} \)) and \( C_n = v_\mathrm {T} \sigma _n \), \( v_\mathrm {T} \) stands for the thermal carrier velocity and \( \sigma _n \) is the capture cross section. This factor describes the probability of electron capture per unit time, which is equivalent to the influx for a certain defect volume. The rates for electron emission \( k_\mathrm {e,n} \), hole capture \( k_{\mathrm {c},p} \) and hole emission \( k_{\mathrm {e},p} \) can be analogously derived. Under steady-state conditions, the capture and emission rates are equal, leading to the well known SRH recombination rate

(4.14) \{begin}{align} R_\mathrm {SRH} = \frac {C_n C_p \left (p n - p_1 n_1\right )}{C_n \left (n+n_1\right ) + C_p \left (p+p_1\right )}. \{end}{align}

This model can be extended to describe electron trapping at interface defects from the semiconductor conduction band. Using Boltzmann statistics, the transition rate reads

(4.15) \{begin}{align} k_{\mathrm {c},n} = k_0 \mathrm {exp} \left ( \frac {E_\mathrm {T}-E_\mathrm {c}}{k_\mathrm {B} T} \right ), \label {equ:srh_interface} \{end}{align}

with \( k_0 = \sigma _n v_\mathrm {th} N_\mathrm {c} \), where \( N_\mathrm {c} \) denotes the effective density of states in the semiconductor conduction band. This model is still widely used for calculating interface state densities from different device characterization methods [143], but also to calculate transient (math image) shifts caused by charge capture and emission at interface and oxide defects [207]. In order to treat charge transfer rates to oxide defects, the energetic barrier (see Figure 4.2 right) at the oxide interface between the defect and the semiconductor has to be taken into account. It is usually included in (4.15) by simply introducing a tunneling probability \( \vartheta \) to the prefactor [61]. However, it should be noted that this modeling approach only accounts for elastic tunneling processes, thereby fully neglecting any energetic exchange of the charge carrier with the surrounding phonon bath upon the charge transfer reaction. The reason for the success of the SRH model for the extraction of interface state densities was recently demonstrated by Ruch et. al [208] by comparing it to NMP theory for interface defects (Pb,0-centers), with small relaxation energies. As will be derived in the next section, these relaxation energies \( E_\mathrm {R} \) determine the strength of electron-phonon coupling, i.e. small \( E_\mathrm {R} \) typically is a consequence of weak electron-phonon coupling. As a result, the obtained interface trap density spectra of both modeling approaches as extracted via CP experiments are very similar, in contrast to such calculated for defects with larger \( E_\mathrm {R} \) [208].