Real MOS devices are affected by the generation, transformation and charging of oxide and interface defects. The complex interplay of these physical processes leads to a shift of the threshold voltage in response to changes of the gate bias which results in hysteresis and biastemperature instability (BTI). Experimentally, for larger stress times, one often observes a recoverable BTI component, caused by charging of preexisting oxide defects, and a quasipermanent BTI component with long recovery times. The latter component is typically ascribed to defect generation, although the underlying physical mechanisms are less understood (see the discussion in
It has been shown that the recoverable component can be accurately captured using a 2state nonradiative multi phonon (NMP) model. The quasipermanent component, on the other hand, can be approximately described using a simple doublewell (DW) model with a field dependent energy barrier, at least at the present level of understanding. In this section we will add both defect models to our device to lay the foundation for the subsequent BTI simulation.
In order to simulate the threshold voltage shift \(\Delta V_\mathrm{th} \) caused by a (partially) charged defect ensemble we have to specify the underlying distribution of the defect parameters. The choice most frequently found in the literature is a normal distribution for the energetic parameters \(E_\mathrm{T}\), \(E_\mathrm{R}\) and \(R\), while the spatial positions \( x_\mathrm{T} \) of the defects are usually assumed to be uniformly distributed, resulting in a constant defect density in a given area of the oxide. This choice is referred to as Gaussian defect band and has been used successfully to reproduce experimental data across a large variety of different devices. However, in exemplary studies the distribution of \(R\) was found to be narrow, hence, \(R\) is approximated as a constant in the Comphy framework. Consequently, each Gaussian NMP defect band can be characterized by the following 8 parameters:
parameter  Comphy notation  description  unit 

\(\langle E_\mathrm{T} \rangle \)  Et  mean energy level  eV 
\( \sigma_{E_\mathrm{T}} \)  Et_sigma  standard deviation of energy levels  eV 
\(\langle E_\mathrm{R} \rangle \)  Er  mean relaxation energy  eV 
\( \sigma_{E_\mathrm{R}} \)  Er_sigma  standard deviation of relaxation energies  eV 
\( x_\mathrm{T,min} \)  xt_min  lower bound of spatial defect distribution  m 
\( x_\mathrm{T,max} \)  xt_max  upper bound of spatial defect distribution  m 
\( R \)  R  curvature ratio of defects  1 
\( N_\mathrm{T} \)  Nt  defect concentration of defect band  m⁻³ 
defect band name  Et  Et_sigma  Er  Er_sigma  R  Nt 

SiO2_shallow_trap_band  3.48  0.15  3.82  1.36  0.407  1.5e25 
SiO2_deep_trap_band  5.65  0.20515  9.3438  2.5526  1.809  4.449e26 
HfO2_shallow_trap_band  3.18  0.25455  0.8684  0.4045  0.0018121  9.0534e25 
HfO2_deep_trap_band  5.23  0.23203  9.2285  1.4948  1.5287  4.1597e26 




























my_device.trap_bands['SiO2_shallow_trap_band'] = SiO2_shallow_trap_band
SiO2_deep_trap_band = TrapBand_2S_NMP(




















my_device.trap_bands['SiO2_deep_trap_band'] = SiO2_deep_trap_band
HfO2_shallow_trap_band = TrapBand_2S_NMP(




















my_device.trap_bands['HfO2_shallow_trap_band'] = HfO2_shallow_trap_band
HfO2_deep_trap_band = TrapBand_2S_NMP(




















my_device.trap_bands['HfO2_deep_trap_band'] = HfO2_deep_trap_band
 Generation and annealing of oxide defects
 Activation and passivation of dangling bonds
 Transformation of existing oxide defects (H relocation)
 Metastability of preexisting oxide defects
Similar to the Gaussian NMP trap band, we assume \(\epsilon_{12,0}\) and \(\epsilon_{21}\) to be normally distributed while the other parameters are approximated as constant. Consequently, a Gaussian DW defect band can be described in the Comphy framework by the following 7 parameters:
parameter  Comphy notation  description  unit 

\(\langle \epsilon_{12,0} \rangle \)  eps1  mean activation energy barrier  eV 
\( \sigma_{\epsilon_{12,0}} \)  eps1_sigma  standard deviation of activation energy barriers  eV 
\(\langle \epsilon_{21} \rangle \)  eps2  mean passivation energy barrier  eV 
\( \sigma_{\epsilon_{21}} \)  eps2_sigma  standard deviation of passivation energy barriers  eV 
\( k_0 \)  k0  transition rate for zero barrier  s⁻¹ 
\( \gamma \)  gamma  empirical field dependence of activation barrier  eV m V⁻¹ 
\( N_\mathrm{it} \)  Nit  interface defect concentration  m⁻² 
defect band name  eps1 / eV  eps1_sigma / eV  eps2 / eV  eps2_sigma / eV  k0 / s  gamma / eV m V⁻¹  Nit / m⁻² 

interface_traps  2.5297  0.92612  1.649  0.060114  1.2336E+13  1.7052E10  1.9096E+17 




























Vth_init = my_device.Vth
my_device.apply_gate_voltage(






print("dVth after {:.2e}s = {:.3f}V".format(1.0E5, my_device.Vth  Vth_init))