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Breakdown in a MOSFET
This model shows how to use the time-dependent solver to model the breakdown of a MOSFET as a result of impact ionization. MOSFETs typically operate in three regimes depending on the drain-source voltage for a given gate voltage. Initially the current-voltage relation is linear; this is the ohmic region. As the drain-source voltage increases, the extracted current begins to saturate; this is the saturation region. As the drain-source voltage is further increased, the breakdown region is entered, where the current increases exponentially for a small increase in the applied voltage. This is due to impact ionization.
Introduction
Under high electric fields, charge carriers’ energies can become greater than the gap energy. Under this condition, additional electron hole pairs are generated by the energy released when collisions occur. This process, called impact ionization, is responsible for the phenomenon of avalanche breakdown in semiconductors.
Practically speaking, the carrier generation rate due to impact ionization is given by:
The Semiconductor Interface uses the model of Okuto and Crowell for αn and αp (Ref. 1) where:
And where E||,n and E||,p are the components of the electric field parallel to the electron and hole currents, respectively. Tref, an, ap, bn, bp, dn, and dp are material properties. Those values are defined in Ref. 1 for silicon, germanium, gallium arsenide, and gallium phosphate).
Model Definition
The geometry and operation of the device are discussed in the DC Characteristics of a MOS Transistor (MOSFET) model.
This model adds impact ionization generation to the device in which the drain voltage is swept under a constant gate voltage. The drain voltage is varied from 0 V to just over 11 V, which is the inception of avalanche breakdown in the 2D MOSFET.
After having loaded the original MOSFET model, a third study is added to the model in order to observe the effect of the impact ionization on the drain current. This study contains two study steps. The first step computes the stationary solution under a drain voltage and a gate voltage set to 0 V and 3 V, respectively. The second step is a transient step where the drain voltage is ramped from 0 V to 11.4 V for 1 ms under a gate voltage of 3 V. The solution obtained from the stationary step is used as the initial condition for the transient step.
Results and Discussion
Figure 1 shows the drain current as a function of the drain voltage under a gate voltage of 3V. The current initially starts to increase linearly before reaching saturation and eventually runaway as the carrier generation starts to break out.
Figure 2 shows the generation rate at Vds = 11.4 V. Note that the generation is localized mainly on a spot beneath the drain contact.
Figure 3 shows the logarithm of the generation rate at the same voltage (Vds = 11.4 V). Note that the generation is localized where the electrical field is more intense, that is, at the drain junction, and especially where the curvature of the junction is more pronounced.
Figure 1: Plot of the terminal current vs. drain source voltage. When impact ionization is included, the current increases exponentially with increasing drain-source voltage.
Figure 2: Plot of the impact ionization source which is largest beneath the drain contact.
Figure 3: Plot of the logarithm of the impact ionization source which is largest beneath the drain.
Reference
1. Y. Okuto and C.R. Crowell, “Threshold Energy Effect on Avalanche Breakdown in Semiconductor Junctions,” Solid-State Electronics, vol. 18, pp. 161–168, 1975.
Application Library path: Semiconductor_Module/Transistors/mosfet_breakdown
Modeling Instructions
Root
Load the MOSFET model
1
From the File menu, choose Open.
2
Browse to the model’s Application Libraries folder and double-click the file mosfet.mph.
Global Definitions
Configure the model’s parameters for the breakdown study. Change the drain voltage to 0 V, this value will be used to determine the model’s initial solution. Change the gate voltage to 3 V, which is the intermediate value from the previous MOSFET study. Also, add a value for the Breakdown voltage, Vbreak, to the model’s parameters.
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
Vd
0[V]
0 V
Drain voltage
Vg
3[V]
3 V
Gate voltage
Vbreak
11.4[V]
11.4 V
Breakdown voltage
Component 1 (comp1)
This study will sweep over Vd, in the previous MOSFET study this was achieved using an auxiliary sweep within a Stationary study step. For the breakdown model it is more appropriate to use a Time Dependent study step, as the stepping algorithm for this solver is better able to handle the highly nonlinear nature of the breakdown effects. In order to implement the same parameter sweep using a Time Dependent study step the drain voltage sweep is defined as a time dependent function, thus enabling the solver to perform the sweep as the solver steps through time. This is achieved by creating a new transient drain voltage variable, Vds, which is defined using a ramp function which increases as a function of time.
Definitions
Ramp 1 (rm1)
1
In the Model Builder window, expand the Component 1 (comp1) node.
2
Right-click Component 1 (comp1)>Definitions and choose Functions>Ramp.
First create the ramp function. The has a gradient of 1000, such that its value increases from 0 to 1 over an argument range of 1e^{-3}.
3
In the Settings window for Ramp, locate the Parameters section.
4
In the Slope text field, type 1000.
5
Select the Cutoff check box.
Variables 1
1
In the Home toolbar, click  Variables and choose Local Variables.
Now define the drain voltage sweep as a function of time using the ramp function. The time, in seconds, is used as the argument for the ramp, resulting in a function which increases from zero to one over a time interval of 1 ms. This is multiplied by the Vbreak to create a drain voltage that sweeps between zero and the breakdown voltage linearly over 1 ms.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Vds
Vbreak*(rm1(t[1/s]))
V
 
Semiconductor (semi)
Add a metal contact to the drain boundary. This extra boundary will be enabled to generate the initial solution in the first study step (stationary).
Metal Contact 4
1
In the Model Builder window, under Component 1 (comp1) right-click Semiconductor (semi) 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 Vd.
Change the value of the drain voltage to Vds, the transient expression defined earlier. This boundary condition will be enabled only for the second study step (transient).
Metal Contact 2
1
In the Model Builder window, click Metal Contact 2.
2
In the Settings window for Metal Contact, locate the Terminal section.
3
In the V0 text field, type Vds.
Add the impact ionization feature to all domains. This feature will only be enabled in the second study step (transient).
Impact Ionization Generation 1
1
In the Physics toolbar, click  Domains and choose Impact Ionization Generation.
2
In the Settings window for Impact Ionization Generation, locate the Domain Selection section.
3
From the Selection list, choose All domains.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Stationary.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3
Step 1: Stationary
The stationary step is used to create the initial solution to the problem. In this step the impact ionization generation and the transient drain voltage are disabled.
1
In the Settings window for Stationary, locate the Physics and Variables Selection section.
2
Select the Modify model configuration for study step check box.
3
In the Physics and variables selection tree, select Component 1 (comp1)>Semiconductor (semi)>Metal Contact 2.
4
Click  Disable.
5
In the Physics and variables selection tree, select Component 1 (comp1)>Semiconductor (semi)>Impact Ionization Generation 1.
6
Click  Disable.
Add a transient step. This step will compute the solution of our problem starting from the initial solution obtained from the first step. Disable the stationary drain voltage to enable the transient voltage at the drain boundary.
Time Dependent
1
In the Study toolbar, click  Study Steps and choose Time Dependent>Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,0.2/15,0.18)*1e-3 range(0.2,0.6/5,0.8)*1e-3 range(0.85,0.15/5,1)*1e-3.
4
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step check box.
5
In the Physics and variables selection tree, select Component 1 (comp1)>Semiconductor (semi)>Metal Contact 4.
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Click  Disable.
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node.
3
In the Model Builder window, click Time-Dependent Solver 1.
4
In the Settings window for Time-Dependent Solver, click to expand the Absolute Tolerance section.
5
From the Tolerance method list, choose Manual.
6
In the Absolute tolerance text field, type 1e-6.
Use a stop condition to cause the solver to stop when the current exceeds a certain value.
7
Right-click Time-Dependent Solver 1 and choose Stop Condition.
8
In the Settings window for Stop Condition, locate the Output at Stop section.
9
From the Add solution list, choose Steps before and after stop.
10
Locate the Stop Expressions section. Click  Add.
11
In the table, enter the following settings:
Stop expression
Stop if
Active
Description
root.comp1.semi.I0_2>200e-6
True (>=1)
√
Stop expression 1
12
In the Model Builder window, click Study 3.
13
In the Settings window for Study, locate the Study Settings section.
14
Clear the Generate default plots check box.
15
In the Study toolbar, click  Compute.
Results
Id vs. Vd with impact ionization
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 3/Solution 3 (sol3).
4
Locate the Plot Settings section. Select the x-axis label check box.
5
In the associated text field, type Drain Voltage, Vds (V).
6
Select the y-axis label check box.
7
In the associated text field, type Terminal current (uA).
8
Click to expand the Title section. From the Title type list, choose None.
9
Locate the Legend section. From the Position list, choose Upper left.
10
Right-click 1D Plot Group 7 and choose Rename.
11
In the Rename 1D Plot Group dialog box, type Id vs. Vd with impact ionization in the New label text field.
12
Click OK.
Global 1
1
Right-click Id vs. Vd with impact ionization 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 Parameter list, choose Expression.
4
In the Expression text field, type Vds.
5
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
semi.I0_2
uA
Terminal current
6
Click to expand the Legends section. From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
Vg = 3 V
8
Click to expand the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Diamond.
9
From the Positioning list, choose In data points.
10
In the Id vs. Vd with impact ionization toolbar, click  Plot.
Impact Ionization
1
In the Model Builder window, right-click Electric Potential (semi) and choose Duplicate.
2
Right-click Electric Potential (semi) 1 and choose Rename.
3
In the Rename 2D Plot Group dialog box, type Impact Ionization in the New label text field.
4
Click OK.
5
In the Settings window for 2D Plot Group, locate the Data section.
6
From the Dataset list, choose Study 3/Solution 3 (sol3).
7
Click to expand the Title section. From the Title type list, choose None.
Surface 1
1
In the Model Builder window, expand the Impact Ionization node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type semi.Gii.
Impact Ionization
1
In the Model Builder window, click Impact Ionization.
2
In the Impact Ionization toolbar, click  Plot.
Impact Ionization log scale
1
Right-click Impact Ionization and choose Duplicate.
2
Right-click Impact Ionization 1 and choose Rename.
3
In the Rename 2D Plot Group dialog box, type Impact Ionization log scale in the New label text field.
4
Click OK.
Surface 1
1
In the Model Builder window, expand the Impact Ionization log scale node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type log10(-semi.Gii+eps).
4
Click to expand the Range section. Select the Manual color range check box.
5
In the Maximum text field, type 36.
6
In the Impact Ionization log scale toolbar, click  Plot.