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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.
Application Library path: Semiconductor_Module/Transistors/lombardi_surface_mobility
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Semiconductor > Semiconductor (semi).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Enter model parameters.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
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
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose µm.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Height text field, type 0.5.
4
Locate the Position section. In the x text field, type -0.5.
5
In the y text field, type -0.5.
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
0.01
7
Clear the Layers on bottom checkbox.
8
Select the Layers on top checkbox.
9
Click  Build All Objects.
Line Segment 1 (ls1)
1
In the Geometry toolbar, click  More Primitives and choose Line Segment.
2
In the Settings window for Line Segment, locate the Starting Point section.
3
From the Specify list, choose Coordinates.
4
Locate the Endpoint section. From the Specify list, choose Coordinates.
5
Locate the Starting Point section. In the x text field, type -0.12.
6
Locate the Endpoint section. In the x text field, type 0.12.
Line Segment 2 (ls2)
1
In the Geometry toolbar, click  More Primitives and choose Line Segment.
2
In the Settings window for Line Segment, locate the Starting Point section.
3
From the Specify list, choose Coordinates.
4
Locate the Endpoint section. From the Specify list, choose Coordinates.
5
Locate the Starting Point section. In the x text field, type -0.18.
6
Locate the Endpoint section. In the x text field, type -0.5.
Line Segment 3 (ls3)
1
In the Geometry toolbar, click  More Primitives and choose Line Segment.
2
In the Settings window for Line Segment, locate the Starting Point section.
3
From the Specify list, choose Coordinates.
4
Locate the Endpoint section. From the Specify list, choose Coordinates.
5
Locate the Starting Point section. In the x text field, type 0.18.
6
Locate the Endpoint section. In the x text field, type 0.5.
7
Click  Build All Objects.
Mesh Control Edges 1 (mce1)
1
In the Geometry toolbar, click  Virtual Operations and choose Mesh Control Edges.
2
On the object fin, select Boundary 4 only.
It might be easier to select the correct boundary by using the Selection List window. To open this window, in the Home toolbar click Windows and choose Selection List. (If you are running the cross-platform desktop, you find Windows in the main menu.)
Add the built-in silicon material properties.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Semiconductors > Si - Silicon.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Enter physics settings.
Semiconductor (semi)
1
In the Settings window for Semiconductor, click to expand the Continuation Settings section.
2
In the Cp text field, type cp.
Metal Contact 1
1
In the Physics toolbar, click  Boundaries and choose Metal Contact.
2
Select Boundaries 2 and 3 only.
Metal Contact 2
1
In the Physics toolbar, click  Boundaries and choose Metal Contact.
2
Select Boundary 7 only.
3
In the Settings window for Metal Contact, locate the Terminal section.
4
In the V0 text field, type Vds.
Thin Insulator Gate 1
1
In the Physics toolbar, click  Boundaries and choose Thin Insulator Gate.
2
Select Boundary 5 only.
3
In the Settings window for Thin Insulator Gate, locate the Terminal section.
4
In the V0 text field, type Vgs.
5
Locate the Gate Contact section. In the εins text field, type 4.2.
6
In the dins text field, type 5[nm].
7
In the Φ text field, type 4.5[V].
Analytic Doping Model 1
1
In the Physics toolbar, click  Domains and choose Analytic Doping Model.
2
Select Domain 1 only.
3
In the Settings window for Analytic Doping Model, locate the Impurity section.
4
In the NA0 text field, type 5e17[1/cm^3].
Analytic Doping Model 2
1
In the Physics toolbar, click  Domains and choose Analytic Doping Model.
2
Select Domain 1 only.
3
In the Settings window for Analytic Doping Model, locate the Distribution section.
4
From the list, choose Box.
5
Locate the Uniform Region section. Specify the r0 vector as
0.2[um]
X
0[um]
Y
6
Locate the Impurity section. From the Impurity type list, choose Donor doping (n-type).
7
Locate the Uniform Region section. In the W text field, type 0.3[um].
8
In the D text field, type 0.2[um].
9
Locate the Impurity section. In the ND0 text field, type 1e19[1/cm^3].
10
Locate the Profile section. In the dj text field, type 0.1[um].
11
From the Nb list, choose Acceptor concentration (semi/adm1).
Analytic Doping Model 3
1
In the Physics toolbar, click  Domains and choose Analytic Doping Model.
2
Select Domain 1 only.
3
In the Settings window for Analytic Doping Model, locate the Distribution section.
4
From the list, choose Box.
5
Locate the Uniform Region section. Specify the r0 vector as
-0.5[um]
X
0[um]
Y
6
In the W text field, type 0.3[um].
7
In the D text field, type 0.2[um].
8
Locate the Impurity section. From the Impurity type list, choose Donor doping (n-type).
9
In the ND0 text field, type 1e19[1/cm^3].
10
Locate the Profile section. In the dj text field, type 0.1[um].
11
From the Nb list, choose Acceptor concentration (semi/adm1).
Trap-Assisted Recombination 1
1
In the Physics toolbar, click  Domains and choose Trap-Assisted Recombination.
2
Select Domain 1 only.
Semiconductor Material Model 1
In the Model Builder window, click Semiconductor Material Model 1.
Lombardi Surface Mobility Model (S) 1
1
In the Physics toolbar, click  Attributes and choose Lombardi Surface Mobility Model (S).
2
In the Settings window for Lombardi Surface Mobility Model (S), click to expand the Continuation Settings section.
3
From the Continuation type list, choose Use interface continuation parameter.
Do not forget to select the mobility model in the Semiconductor Material Model node, otherwise the mobility model subnode has no effect.
Semiconductor Material Model 1
1
In the Model Builder window, click Semiconductor Material Model 1.
2
In the Settings window for Semiconductor Material Model, locate the Mobility Model section.
3
From the μn list, choose Electron mobility, Lombardi (semi/smm1/mmls1).
Set up a user-defined mesh.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type 0.04.
5
In the Maximum element growth rate text field, type 1.04.
Size 1
In the Model Builder window, right-click Size 1 and choose Delete.
Size 2
Right-click Size 2 and choose Delete.
Free Triangular 1
Right-click Free Triangular 1 and choose Delete.
Edge 1
1
In the Mesh toolbar, click  More Generators and choose Edge.
2
Select Boundaries 3–7 only.
3
In the Settings window for Edge, click to expand the Control Entities section.
4
From the Smooth across removed control entities list, choose Off.
Size 1
1
Right-click Edge 1 and choose Size.
2
Select Boundaries 4–6 only.
3
In the Settings window for Size, locate the Element Size section.
4
From the Calibrate for list, choose Semiconductor.
5
Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.005.
Copy Edge 1
1
In the Model Builder window, right-click Mesh 1 and choose Copying Operations > Copy Edge.
2
In the Settings window for Copy Edge, locate the Source Boundaries section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 3-7 in the Selection text field.
5
Click OK.
6
In the Settings window for Copy Edge, locate the Destination Boundaries section.
7
Click to select the  Activate Selection toggle button.
8
Select Boundary 11 only.
9
Click to expand the Control Entities section. From the Smooth across removed control entities list, choose Off.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
5
Click to expand the Control Entities section. From the Smooth across removed control entities list, choose Off.
6
Click to expand the Reduce Element Skewness section. Select the Adjust edge mesh checkbox.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundary 9 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 6.
6
In the Element ratio text field, type 7.
7
From the Growth rate list, choose Exponential.
8
Select the Reverse direction checkbox.
Free Triangular 1
1
In the Mesh toolbar, click  Free Triangular.
2
In the Settings window for Free Triangular, click to expand the Control Entities section.
3
From the Smooth across removed control entities list, choose Off.
4
Click  Build All.
5
Click the  Zoom Extents button in the Graphics toolbar.
The user-defined mesh is shown in the image below. The mapped mesh with the specific distribution helps create layers of thin elements underneath the gate, where the large gradient of the carrier concentration should be resolved by the mesh.
Study 1
Step 1: Stationary
Set up an auxiliary continuation sweep for the continuation parameter cp, with the value 0 for constant mobility and 1 for mobility model.
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
Click  Add.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
Vds (Drain-source voltage)
range(0,0.2,1)
V
cp (Continuation parameter)
0 1
 
7
From the Sweep type list, choose All combinations.
8
From the Reuse solution from previous step list, choose Auto.
The field-dependent mobility models are very nonlinear and difficult to solve in a fully coupled manner. The Semiconductor 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 the default is sufficient. Take a look at the autogenerated solver sequence.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 node.
Observe that the default number of iterations for the Segregated solver is 3.
4
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 > Segregated 1 node, then click Segregated Step 1.
5
In the Settings window for Segregated Step, click to expand the Method and Termination section.
Observe that the main variables are solved in the first Segregated Step with the Automatic Newton solver.
Click on Segregated Step 2, and observe that the electric field variables are updated in the second Segregated Step with the Constant Newton solver.
6
In the Study toolbar, click  Compute.
Results
Electron Concentration (semi)
Click the  Zoom Extents button in the Graphics toolbar.
Add a 1D plot to compare the I-V curves with and without the mobility model.
Terminal current comparison
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Terminal current comparison in the Label text field.
Global 1
1
Right-click Terminal current comparison and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Semiconductor > Terminals > semi.I0_2 - Terminal current - A.
3
Locate the x-Axis Data section. From the Axis source data list, choose Vds.
4
In the Terminal current comparison toolbar, click  Plot.
5
Click to expand the Legends section. From the Legends list, choose Manual.
6
In the table, enter the following settings:
Legends
Constant mobility
Lombardi surface mobility
Terminal current comparison
1
In the Model Builder window, click Terminal current comparison.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Lower right.
Add a 2D plot to compare the current density profiles with and without the mobility model.
Electron current density comparison
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Electron current density comparison in the Label text field.
3
Locate the Data section. From the Parameter value (cp) list, choose 0.
For this plot group, we will zoom in to a region underneath the gate to look at the current density profile. In order not to interfere with the zoom settings of other plot groups, we use a different view setting for this plot group by selecting "New".
4
Locate the Plot Settings section. From the View list, choose New view.
5
Clear the Plot dataset edges checkbox.
Surface 1
1
Right-click Electron current density comparison and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Semiconductor > Currents and charge > Electron current > Electron current density - A/m² > semi.JnX - Electron current density, X-component.
Filter 1
1
Right-click Surface 1 and choose Filter.
2
In the Settings window for Filter, locate the Element Selection section.
3
In the Logical expression for inclusion text field, type (y>-0.05[um])&&(x>-0.1[um])&&(x<0.1[um]).
4
In the Electron current density comparison toolbar, click  Plot.
Surface 2
1
In the Model Builder window, under Results > Electron current density comparison right-click Surface 1 and choose Duplicate.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (sol1).
4
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Deformation 1
1
Right-click Surface 2 and choose Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the X-component text field, type 0.
4
In the Y-component text field, type 0.06.
5
Locate the Scale section.
6
Select the Scale factor checkbox. In the associated text field, type 1.
7
In the Electron current density comparison toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Add a 2D plot to show the electron mobility.
Electron mobility
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Electron mobility in the Label text field.
Surface 1
1
Right-click Electron mobility and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Semiconductor > Mobility > semi.mun_ls - Electron mobility, Lombardi - m²/(V·s).
3
In the Electron mobility toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.