Lombardi Surface Mobility

This example demonstrates how to use the Lombardi surface scattering model for the electron mobility in a simple MOSFET. The additional nonlinearity introduced by the field dependence of the mobility is readily overcome by the autogenerated solver sequence upon detection of such mobility models by the Semiconductor physics interface. The current density profile and total current flowing into the terminal is compared with the constant mobility case.

Introduction

Surface acoustic phonons and surface roughness have an important effect on the carrier mobility, especially in the thin inversion layer under the gate in MOSFETs. The Lombardi surface mobility model adds surface scattering resulting from these effects to an existing mobility model using Matthiessen’s rule. The mobility model is based on the equations presented in Ref. 1.

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

where T is the lattice temperature, μin,n and μin,p are the electron and hole input mobilities, Na- is the ionized acceptor concentration, Nd+ is the ionized donor concentration, E⊥,n is the component of the electric field perpendicular to the electron current and E⊥,p is the component of the electric field perpendicular to the hole current. All other parameters in the model are material properties (note that δn and δp have units of V/s). The material properties for silicon are also obtained from Ref. 1.

Model Definition

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

The model sweeps the drain voltage form 0 V to 1 V with an applied gate voltage of 1 V. We expect a significant perpendicular (to the currents) electric field underneath the gate contact. In order to ensure convergence of the nonlinear model, the Semiconductor physics interface creates a Segregated solver to first solve the main dependent variables with the electron and hole perpendicular components of the field (E⊥,n and E⊥,p) fixed, and then update the electric field variables separately.

Figure 1: Schematic of the modeled device.

Results and Discussion

Figure 2 shows the effect of the Lombardi surface mobility model on the solution. The comparison of the constant mobility (the perpendicular fields are multiplied by 0) and the Lombardi surface mobility (E⊥,n and E⊥,p multiplied by 1) models shows a more pronounced saturation effect for the Lombardi surface mobility model than for the constant mobility model (without field dependent parameters).

Figure 2: Plot of terminal current for the constant mobility and surface mobility cases. The current is reduced when the surface mobility model is active.

Figure 3 shows that the electron current density varies substantially in the inversion layer depending on the used mobility model. The Lombardi surface mobility model (top) shows a lower current density in comparison to the one observed with the constant mobility model (bottom).

Figure 4 shows that the electron mobility also varies along the device. As a consequence of the surface scattering effect, the electron mobility is smaller in the vicinity of the gate contact.

Figure 3: Plot of the electron current density on the same scale with the Lombardi surface mobility model active (top) and without (bottom).

Figure 4: Plot of the electron mobility for a drain-source voltage of 1 volt with the Lombardi surface mobility model active.

Reference

1. C.C. Lombardi, S. Manzini, A. Saporito, and M. Vanzi, “A physically based mobility model for numerical simulation of nonplanar devices,” IEEE Transactions on Computer-Aided Design, vol. 7, no. 11, pp. 1164–117, 1988.

Semiconductor_Module/Transistors/lombardi_surface_mobility

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select .

Click .

Click

In the tree, select .

Click

Enter model parameters.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Vds	0[V]	0 V	Drain-source voltage
Vgs	1[V]	1 V	Gate-source voltage
cp	0	0	Continuation parameter

Create model geometry representing a simplified MOSFET.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type 0.5.

Locate the section. In the text field, type -0.5.

In the text field, type -0.5.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (µm)
Layer 1	0.01

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Line Segment 1 (ls1)

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Line Segment 2 (ls2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the text field, type -0.18.

Locate the section. In the text field, type -0.5.

Line Segment 3 (ls3)

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and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the text field, type 0.18.

Locate the section. In the text field, type 0.5.

Click

Mesh Control Edges 1 (mce1)

In the toolbar, click

and choose .

On the object , select Boundary 4 only.

It might be easier to select the correct boundary by using the window. To open this window, in the toolbar click and choose . (If you are running the cross-platform desktop, you find in the main menu.)