# Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

### 4 Reliability Modeling

First models for MOS transistors to calculate the potential across the devices were developed more than 50 years ago [195, 196]. In 1966 the Pao-Sah model using a double integral formulation for solving the Poisson equation [197] was published. However, this model was not feasible for integration in circuit simulations, because it was numerically inefficient [198]. Therefore, later in 1978 the Pao-Sah model was simplified by Brews employing the charge-sheet approximation [199] forming the basis for threshold-voltage based models like BSIMv3 or MM9 [200]. For several years threshold-voltage based models were mainly used in device simulations, however, due to discontinuities and inaccuracies in inversion, surface-potential based models like PSP [201] or MM11 and charge-based models like EKV [202], ACM [203] or BSIM-Bulk [204] became dominant.

The simulation framework used in this work is called Comphy (short for compact physics) [136] and is based on surface potential computation, such as PSP and MM11. Comphy has been developed for calculating the threshold voltage  changes for transient input parameters, that are the gate voltages  and temperatures . This is done by defining a gate stack by its geometry and fundamental material parameters and by using the electrostatics derived for a one-dimensional device. By sampling defects in the oxide, degradation curves, e.g. the drift of the threshold voltage  extracted from BTI measurements, can be simulated using a 2-state NMP model as discussed in Chapter 3.

#### 4.1 Electrostatics of MOS Structures

By combining the one-dimensional Poisson equation with the charge neutrality equation, a surface potential based expression for the space charge per unit area can be derived [205]

Here, the positive sign refers to while the negative sign is used for . The abbreviation is the Debye length for holes with being the relative permitivity of the channel. and are the electron and the hole concentration in thermal equilibrium which can be efficiently computed using the Joyce-Dixon approximation [206]. is the elementary charge, and is the permittivity of the channel. For computing the temperature dependence of the Si band gap the model proposed by Bludau [207] is used. For effective masses for the valence band and the conduction band the models proposed by Lang et. al. [208] and Green [209] are used, respectively. With this, effective densities of states, Fermi level, intrinsic Fermi level, carrier concentrations, etc. can be computed as shown in [205].

By using simple electrostatic considerations [180] the following equation can be derived

where is the intrinsic work function difference and is the Fermi level in the channel at thermal equilibrium. The oxide capacitance per area is given by

where is the relative permittivity of the oxide, and are channel width and length, and is the thickness of the oxide. Equation (4.2) can be solved by using an iterative scheme, e.g. a Newton solver to obtain . By considering the trapped charges in the oxide, (4.2) can be extended to (4.4) which then allows to be minimized for with

where stands for the voltage shift caused by the oxide charges. The calculation of is the central operation in surface potential based compact models and allows to compute the potential curve across the insulator which is approximated as being defect free. This enables computing the shift of trap levels and then the computation of charge capture and emission rates using 2-state NMP theory, which is discussed in detail in Section 3.2. Depending on the defect occupation and the defect density, a mean captured charge can be computed at every time step for an ensemble of defects. The sum of all charge states then leads to a shift of the threshold voltage. This finally results in a transient  characteristics which can be compared to experimental data.