COMSOL 6.4 - Caughey–Thomas Mobility

Caughey–Thomas Mobility

This example demonstrates how to use the Caughey–Thomas high-field saturation model for the electron and hole mobility. Field-dependent mobility makes a problem which is already highly nonlinear even more nonlinear. The Semiconductor physics interface automatically creates a suggested solver sequence for better convergence when a built-in field-dependent mobility model is used.

Introduction

With an increase in the parallel component of the applied field, carriers can gain energies above the ambient thermal energy and then transfer energy gained from the field to the lattice by optical phonon emission. This effect leads to a saturation of the carriers velocity.

The Caughey–Thomas mobility model adds high field velocity saturation to an existing mobility model (or to a constant input mobility), based on equations presented in Ref. 1.

The electron (μn,ct) and hole (μp,ct) mobilities are determined by the following equations:

Here T is the lattice temperature, μin,n and μin,p are the electron and hole input mobilities, respectively, and Fn and Fp are the driving forces for electrons and holes, respectively (several options are available, the default is Fn = E||,n and Fp = E||,p, where E||,n is the magnitude of the component of the electric field parallel to the electron current and E||,p is the magnitude of the component of the electric field parallel to the hole current). All other parameters in the model are material properties (note that v0,n and v0,p are the saturation velocities for electrons and holes at the reference temperature Tref and have units of m/s). The material properties for Silicon are also obtained from Ref. 1.

Model Definition

The model represents a simplified channel region of a 2D MOSFET, where the n-doped drain and source are located on the left and right side of the geometry, respectively; see Figure 1. The gate is positioned on top of the p-doped silicon section, which is located at the center of the device.

The model consists of sweeping the drain to source voltage form 0 V to 1 V under a gate to source voltage of 0 V. Sweeping the drain voltage creates a significant electric field at the left side of the gate at the p–n junction.

Figure 1: Schematic of the modeled device.

The field-dependent mobility models are very nonlinear and difficult to solve in a fully coupled manner. The Semiconductor physics interface automatically creates a suggested solver sequence when the built-in field-dependent mobility models are used. This suggested solver sequence alternately solves the main dependent variables with the electric field variables fixed and then updates the electric field variables afterward, using the Segregated solver. By default, 3 iterations are used; however, depending on the model, more iterations may be needed. For this model, 4 iterations are used for better accuracy.

Results and Discussion

Figure 2 shows the effect of the Caughey–Thomas mobility model on the solution. The comparison of the constant mobility (the driving forces Fn and Fp are multiplied by 0) and the Caughey–Thomas mobility model (Fn and Fp multiplied by 1) shows a more pronounced saturation effect for the Caughey–Thomas model.

Figure 3 shows that the electron mobility varies substantially along the device. The electron mobility is 3 order of magnitude lower in the vicinity of the drain–gate junction as a consequence of the high electric field generated by the applied potential on the drain contact.

Figure 4 and Figure 5 show the effect of the mobility model on the electron drift velocity. Figure 4 shows the electron drift velocity for the Caughey–Thomas mobility model (Fn and Fp multiplied by 1) and Figure 5 for the constant mobility model (Fn and Fp multiplied by 0). A comparison of the two figures clearly shows an important reduction of the electron drift velocity when the field effect is taken into account.

Figure 2: Comparison of the terminal current for the constant mobility and Caughey–Thomas mobility. The current is lower and levels off more rapidly due to the fact that the mobility decreases when the electric field is high.

Figure 3: Plot of the electron mobility at a drain–source voltage of 1 V. The mobility varies by 3 orders of magnitude over the modeling domain.

Figure 4: Plot of the electron drift velocity using Caughey–Thomas mobility. The drift velocity barely exceeds 1 x 105 m/s.

Figure 5: Plot of the electron drift velocity for the constant mobility case. Since the mobility does not decrease as the electric field increases, the drift velocity becomes very high.

Reference

1. C. Canali and others, “Electron and Hole Drift Velocity Measurements in Silicon and Their Empirical Relation to Electric Field and Temperature,” IEEE Transactions on Electron Devices, vol. 22, no. 11, pp. 1045–1047, 1975.

Note the correction in: G. Ottaviani, “Correction to ‘Electron and Hole Drift Velocity Measurements in Silicon and Their Empirical Relation to Electric Field and Temperature’,” IEEE Transactions on Electron Devices, vol. 23, no. 9, p. 1113, 1976.

Semiconductor_Module/Transistors/caughey_thomas_mobility

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Enter model parameters.

Global Definitions

Parameters 1

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In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Wfin	10[nm]	1E-8 m	Height
Lg	100[nm]	1E-7 m	Width
tox	1[nm]	1E-9 m	Gate oxide thickness
Vds	0[V]	0 V	Drain source voltage
Vgs	0[V]	0 V	Gate source voltage
ds	0	0	Continuation parameter