## Nonradiative Multiphonon (NMP) Model

For accurate modeling of NBTI, GTS Framework R.2013 offers a reliability module which comes with an implementation of the latest non-radiative multiphonon (NMP) model [Grasser12].

### Trap States and Transition Rates

The model assumes four trap states as shown in Fig.1. It assumes two different types of transitions between its states 1, 1', 2, and 2'.

The transition rates of the relaxation processes (k_{1'1} , k_{11'}, k_{2'2} and k_{22'}) are bias-independent:

The barriers ε_{1'1} and ε_{2'2} are defined trap parameters, ν is the attempt frequency and β = 1 / (kB*T).

The transition rates where charge exchanges are involved (k_{12'} , k_{2'1}, k_{1'2} and k_{21'}) are bias-dependent and take NMP processes and tunneling into account:

Here, σ is the effective trap capture cross section and v_{th} is the thermal velocity. For the full set of rate-equations of the model, please check Tibor Grasser's publications on BTI (available at IμE) or consult the Minimos-NT user manual.

### Trap Generation, Charge

Traps can be generated randomly from a continuum distribution using the Grid option (see Fig.2) or loaded from file using the ReadIn option. The physical trap parameters can be either specified directly or given by a Gaussian distribution with mean value and standard deviation.

Depending on simulation setup, the equilibrium solution or the transient progress of the occupation will be calculated self-consistently. The charge of a trap, which affects the behavior of a device, depends on the occupation probabilities of the states 2 and 2'.

### Stress and Relaxation Cycles

The typical stress and relaxation cycles used in BTI measurements can be simulated using a transient simulation setup. The trap-state occupation over time is given by the initial trap-state and the transition rates which are bias-dependent and follow self-consistently from device simulation [Bina12].

Fig.3a and Fig.3b show the charge of each trap at the beginning and at the end of the stress cycle. This yields the threshold voltage shift ΔV_{th} during the BTI stress. Fig.4 shows ΔV_{th} for a period of stress at the left followed by relaxation at the right, both at temperatures of 50°C and 150°C, with V_{G} of -1.0V and -2.0V respectively.

### Inverse Modeling

The NMP model can be applied for inverse modeling of statistical parameters like trap density and distribution. Fig.5 illustrates a single trap optimization: Based on an initial guess, the capture and emission time constants of a non-radiative multiphonon trap are optimized to match measurement data. At the left, Fig.5 shows the score function and trap parameters. At the right, the measurements to be fitted and the actual implementations during the process can be seen.