3D Density-Gradient Simulation of a Nanowire MOSFET

This 3D model of a nanowire MOSFET employs the density-gradient theory to add the effect of quantum confinement to the conventional drift-diffusion formulation, without requiring excessively high computational costs. The oxide layer is simulated explicitly with geometric domains, and quantum confinement at the silicon-oxide interface is accounted for via a dedicated boundary condition. The density-gradient effective mass is anisotropic. Various selection utilities are used to simplify the assignment of physics settings and plot selections. The result matches well with the Id-Vg curves and electron density profiles published in the reference paper.

Introduction

As the semiconductor fabrication technology progresses to produce smaller and smaller devices, the effect of quantum confinement becomes more and more important. This tutorial follows the approach described in Ref. 1 to compute the electron density distribution and drain current as a function of the gate voltage of a Si nanowire MOSFET using the density-gradient theory (as reviewed in Ref. 2).

Model Definition

The model computes the DC characteristics and the electron density profile along the transverse and longitudinal centerlines of a Si nanowire MOSFET, to be compared with Fig. 2, 4, and 5 in Ref. 1. The effects of changing the value of the longitudinal effective mass are shown in those figures.

Not all simulation details are given in the paper. Nevertheless, the approach described in the paper is followed as closely as possible. In particular, the treatment of the potential barrier at the silicon-oxide interface follows the one discussed in Ref. 3. When a parameter is not listed, a typical value found in the literature is used in the model. The silicon material data built in the Semiconductor Module are used, with constant mobility. The gate metal work function and the drain voltage are selected to best match the Id-Vg curves in the reference paper. The section Modeling Instructions lists all the parameters used in the model.

The channel of the simulated structure is formed by a Si nanowire with a 3.2 nm square cross section, surrounded by an oxide layer of thickness 0.8 nm. The length of the channel is 4 nm. (See Fig. 1 in Ref. 1.) The source and drain lengths are arbitrarily set at 15 nm, long enough to provide the comparison with the figures in the reference paper.

The oxide layer is modeled explicitly as domains using the domain condition. The quantum confinement effect for the silicon-oxide interface is added by selecting the option for the boundary condition, based on Ref. 3. The temperature is assumed to be 300 K. Maxwell-Boltzmann statistics is used, as indicated by Eq. (2) in Ref. 1.

The density-gradient effective mass is anisotropic in this model. This is done by selecting the option under the section in the settings window for the domain condition.

For the ease of meshing and plotting along the centerlines, some extra layers are created in the 3D geometry. Various functionalities under both the and the nodes are then used to group the multiple geometric entities into different regions for convenient assignment of domain conditions, boundary conditions, and plot selections.

To reduce the computation time and file size, a relatively coarse mesh is used in this tutorial. Interested users are encouraged to perform mesh refinement studies by parameterizing the settings of the nodes used in the meshing sequence.

See the comments in the section Modeling Instructions for more detailed discussions on the model construction, meshing techniques, solution processes, and result visualization.

Results and Discussion

Figure 1 shows the drain current as a function of the gate voltage for a set of longitudinal effective mass values, to be compared with Fig. 2 in Ref. 1. The general trend and subthreshold slopes compare well, with some minor differences in the magnitude of the current.

Figure 1: Drain current versus gate voltage curves for a few effective mass values.

Figure 2 and Figure 3 plot the electron concentration profiles along the longitudinal and transverse centerlines, which compares well with Fig. 4 and 5 in the reference paper.

Figure 2: Electron concentration profile along the longitudinal centerline.

Figure 3: Electron concentration profile along the transverse centerline.

References

1. A.R. Brown, A. Martinez, N. Seoane, and A. Asenov, “Comparison of Density Gradient and NEGF for 3D Simulation of a Nanowire MOSFET,” Proc. 2009 Spanish Conf. Elec. Dev., p. 140, Feb. 11–13, 2009.

2. M.G. Ancona, “Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices,” J. Comput. Electron., vol. 10, p. 65, 2011.

3. S. Jin, Y.J. Park, and H.S. Min, “Simulation of Quantum Effects in the Nano-scale Semiconductor Device,” J. Semicond. Tech. Sci., vol. 4, no. 1, p. 32, 2004.

Semiconductor_Module/Transistors/nanowire_density_gradient_3d

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

The study step is suitable either to solve for systems known to be in thermal equilibrium, or, in this case, to provide a good initial condition for subsequent gate voltage sweeps.

In the tree, select .

Click

Geometry 1

The Model Wizard starts the COMSOL Desktop at the node. Use this chance to select a convenient length unit. Then enter some dimensional parameters for building the geometry.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1: Geometry

In the window, under click .

In the window for , type Parameters 1: Geometry in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
Lgate	4[nm]	4E-9 m	Gate length
Wchannel	3.2[nm]	3.2E-9 m	Channel width
dOx	0.8[nm]	8E-10 m	Oxide thickness
Lsource	15[nm]	1.5E-8 m	Source length
Ldrain	15[nm]	1.5E-8 m	Drain length
Ltot	Lsource+Lgate+Ldrain	3.4E-8 m	Total length
Wtot	Wchannel+dOx*2	4.8E-9 m	Total width

Start building the geometry. Add some extra layers and lines for meshing and plotting purposes. Use the various functionalities under both the and the nodes to group the multiple geometric entities into different regions for convenient assignment of domain conditions, boundary conditions, and plot selections.

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type -Lsource-Lgate/2.

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type Wtot.

Locate the section. From the list, choose .

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


Layer name	Thickness (nm)
Layer 1	Wtot/2

Select the check box.

Work Plane 1 (wp1)>Square 2 (sq2)

In the toolbar, click

In the window for , locate the section.

In the text field, type Wchannel.

Locate the section. From the list, choose .

Extrude 1: Source

In the window, right-click and choose .

In the window for , type Extrude 1: Source in the text field.

Locate the section. From the list, choose .

On the object , select Boundaries 3, 4, 6, and 7 only.

From the list, choose .

Locate the section. In the table, enter the following settings:


Distances (nm)
Lsource

Locate the section. Select the check box.

From the list, choose .

Find the subsection. Click .

In the dialog box, type Si in the text field.

Click .

In the window for , click

Extrude 2: Channel

Right-click and choose .

In the window for , type Extrude 2: Channel in the text field.

Locate the section. Click

On the object , select Boundaries 17–20 only.

Locate the section. Click

In the table, enter the following settings:


Distances (nm)
Lgate/2
Lgate

Locate the section. From the list, choose .

Click

Extrude 3: Drain

Right-click and choose .

In the window for , type Extrude 3: Drain in the text field.

Locate the section. Click

On the object , select Boundaries 33–36 only.

Locate the section. Click

In the table, enter the following settings:


Distances (nm)
Ldrain

Locate the section. From the list, choose .

Click

Extrude 4: Source oxide

In the window, under right-click and choose .

In the window for , type Extrude 4: Source oxide in the text field.

Locate the section. Click

On the object , select Boundaries 1, 2, 5, and 8 only.

Locate the section. From the list, choose .

Find the subsection. Click .

In the dialog box, type Oxide in the text field.

Click .

In the window for , click

Extrude 5: Gate oxide

In the window, under right-click and choose .

In the window for , type Extrude 5: Gate oxide in the text field.

Locate the section. Click

On the object , select Boundaries 25–28 only.

Locate the section. Find the subsection. From the list, choose .

Click

Extrude 6: Drain oxide

In the window, under right-click and choose .

In the window for , type Extrude 6: Drain oxide in the text field.

Locate the section. Click

On the object , select Boundaries 49–52 only.

Locate the section. From the list, choose .

Find the subsection. From the list, choose .

Click

Definitions

Adjacent 1

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

Adjacent 2

Right-click and choose .

In the window for , locate the section.

In the list, select .

Under , click

In the dialog box, select in the list.

Click .

Gate

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. Under , click

In the dialog box, select in the list.

Click .

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

In the window for , type Gate in the text field.

Source+Drain

In the toolbar, click

In the window for , type Source+Drain in the text field.

Locate the section. Under , click

In the dialog box, select in the list.

Click .

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

Box 1: x axis

In the toolbar, click

In the window for , type Box 1: x axis in the text field.

Locate the section. From the list, choose .

Locate the section. In the text field, type -0.1.

In the text field, type 0.1.

In the text field, type -0.1.

In the text field, type 0.1.

Locate the section. From the list, choose .

Box 2: z axis

Right-click and choose .

In the window for , type Box 2: z axis in the text field.

Locate the section. In the text field, type -0.1.

In the text field, type 0.1.

In the text field, type -inf.

In the text field, type inf.

Use the built-in silicon material for the silicon domains.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Si - Silicon (mat1)

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 2: Physics

In the toolbar, click

and choose .

Configure physics settings (first enter some physics parameters). Select the density-gradient formulation from the section of the main physics settings window. Specify the anisotropic density-gradient effective mass in the settings window for the domain feature.

In the window for , type Parameters 2: Physics in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
T0	300[K]	300 K	Temperature
fmx	0.8	0.8	Longitudinal DG effective mass
fmy	0.12	0.12	Transverse DG effective mass
mx	fmx*me_const	7.2875E-31 kg	Longitudinal DG effective mass
my	fmy*me_const	1.0931E-31 kg	Transverse DG effective mass
mz	my	1.0931E-31 kg	Transverse DG effective mass
epsrOx	3.9	3.9	Oxide dielectric constant
mox	0.5*me_const	4.5547E-31 kg	Oxide DG effective mass
moxstar	0.22*me_const	2.0041E-31 kg	Oxide DG effective mass
PhiBox	3.15[V]	3.15 V	Oxide potential barrier
Nd	1e20[cm^-3]	1E26 1/m³	Doping
cp	1	1	Continuation parameter for doping
Vd	0.05[V]	0.05 V	Drain voltage
Phig	4.5[V]	4.5 V	Gate metal work function
Vg	0.8[V]	0.8 V	Gate voltage

Semiconductor (semi)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

Semiconductor Material Model 1

In the window, under click .

In the window for , locate the section.

In the T text field, type T0.

Locate the section. From the list, choose .

In the meDG table, enter the following settings:


mx	0	0
0	my	0
0	0	mz

Apply the domain condition to model the oxide layer explicitly. Use the option to include quantum confinement effects at the silicon-oxide interfaces.

Charge Conservation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the εr list, choose . In the associated text field, type epsrOx.

Insulator Interface 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the meOx text field, type mox.

In the meOx* text field, type moxstar.

In the ΦnOx text field, type PhiBox.

Add doping and metal contacts.

Analytic Doping Model 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the ND0 text field, type Nd*cp.

Metal Contact 1: Source

In the toolbar, click

and choose .

In the window for , type Metal Contact 1: Source in the text field.

Locate the section. From the list, choose .

Metal Contact 2: Drain

In the toolbar, click

and choose .

In the window for , type Metal Contact 2: Drain in the text field.

Locate the section. From the list, choose .

Locate the section. In the V0 text field, type Vd.

Terminal 1: Gate

In the toolbar, click

and choose .

In the window for , type Terminal 1: Gate in the text field.

Locate the section. From the list, choose .

Locate the section. In the Φc text field, type Phig.

From the list, choose .

In the V0 text field, type Vg.

Create the mesh. Use the feature to arrange an efficient mesh that minimizes the number of mesh elements while still resolving large gradients in the solution fields. To reduce the computation time and file size, a relatively coarse mesh is used in this tutorial. Interested users are encouraged to perform mesh refinement studies by parameterizing the settings of the nodes.

Mesh 1

Mapped 1

In the toolbar, click

and choose .

Select Boundaries 1, 4, 8, 11, 15, 18, 21, and 24 only.

In the window for , click to expand the section.

Select the check box.

Distribution 1

Right-click and choose .

Select Edges 9, 20, 21, 24, 27, and 31 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 3.

In the text field, type 2.

Distribution 2

Right-click and choose .

In the window for , locate the section.

Click

Select Edges 10, 12, 13, 15, 23, and 33 only.

Locate the section. Select the check box.

Distribution 3

In the window, right-click and choose .

Select Edges 5, 17, 26, and 34 only.

In the window for , locate the section.

In the text field, type 1.

Swept 1

In the toolbar, click

Distribution 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type 3.

Distribution 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type 10.

In the text field, type 3.

Select the check box.

Click

The study step provides a good initial condition for subsequent gate voltage sweeps. Use the feature to ramp up the doping concentrations from small values in order to achieve convergence. Add a study step to sweep the gate voltage for a set of longitudinal DG effective mass values.

Study 1

Step 1: Semiconductor Equilibrium