4.2 Capacitance Calculation

Inverse modeling profiling techniques deal with the determination of the doping profile of a device from experimental capacitance measurements and the underlying physical equations that relate the capacitances values   to the doping variation. For semiconductor devices, the theoretical relationship takes the form of the basic semiconductor equations, namely Poisson's equation and the current continuity equations [91]. In the case of thermal equilibrium and negligible current flow, the solution of the continuity equations can be ignored. Hence, the space charge density within a device can   be calculated by solving Poisson's equation:

where: is the elementary charge, the semiconductor permitivity, the electrostatic potential, , the electron and hole concentrations, the donor and acceptor concentrations.

From the solution of Poisson's equation, the charges associated with the   device terminals are calculated by integrating the space charge density over a device region :

or by applying the law of Gauß to evaluate the line integral of the electric field on a closed loop:

The device capacitances are then approximated by numerically differentiating   the terminal charges. This procedure is inherently prone to numerical roundoff and integration errors. The following strategies are applied   to minimize their effects:

• Central differences are used for numerical differentiation:

  

  where and are the terminal charges at and respectively, , and . The size of the voltage step, , should be small enough to achieve a good linearized approximation of the derivative. However, a small change in the voltage yields a small change in the charge. This could result in numerical roundoff errors in calculating the quotient due to subtractive cancellation. For room temperature measurements, it was found that a 20mV step is a reasonable compromise between the two conflicting requirements.

• Discretization is a major source of errors in the   calculations. In principle one could increase the size of the grid to effectively eliminate numerical approximation errors. Whereas this is convenient and feasible for 1D problems, the computational demands in the 2D case prohibit such an approach. It is for that reason that the solver grid is carefully refined in regions where the space charge density varies rapidly.
• Use of the same grid at and in the solution reduces the integration errors as part of the errors in the calculation of the charges is canceled out by taking the difference.
• When applicable, the charge integration is limited to device charges that contribute to charge variation. This minimizes the loss of significant digits in the capacitance calculation. For example, in calculating the bulk charge, the ionized dopant atoms in the S/D diode region are not included in the summation because they remain constant when a small voltage perturbation is applied.

At a given voltage, the capacitance value is a measure of the incremental charge variation in response to a change of voltage. By measuring various capacitances under different bias conditions, one can ``probe'' various portions of the doping profile by depleting or accumulating that region of carriers. It follows that the device capacitance values can be represented as a nonlinear function of the doping profile. A nonlinear least-squares optimization can then be applied to determine the doping profile that minimizes the difference between measurements and calculated values using numerical device simulation. This is a continuous minimization problem. The target is to determine the complete functional variation of the profile. It is converted to a discrete problem by a proper parameterization of the profile as described next.