semiconductor model library manual

5/24/2018 Semiconductor Model Library Manual

1/88

VERSION 4.3b

Semiconductor Module

Model Library Manual


2/88

C o n t a c t I n f o r m a t i o n

Visit the Contact Us page atwww.comsol.com/contactto submit general inquiries,

contact Technical Support, or search for an address and phone number. You can also visit

the Wordwide Sales Offices page atwww.comsol.com/contact/officesfor address and

contact information.

If you need to contact Support, an online request form is located at the COMSOL Access

page atwww.comsol.com/support/case.

Other useful links include:

Support Center:www.comsol.com/support

Download COMSOL:www.comsol.com/support/download

Product Updates:www.comsol.com/support/updates

COMSOL Community:www.comsol.com/community Events:www.comsol.com/events

COMSOL Video Center:www.comsol.com/video

Support Knowledge Base:www.comsol.com/support/knowledgebase

Part number: CM024102

S e m i c o n d u c t o r M o d u l e M o d e l L i b r a r y M a n u a l

19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

This Documentation and the Programs described herein are furnished under the COMSOL Software LicenseAgreement (www.comsol.com/sla) and may be used or copied only under the terms of the licenseagreement.

COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are eitherregistered trademarks or trademarks of COMSOL AB. All other trademarks are the property of theirrespective owners, and COMSOLAB and its subsidiaries and products are not affiliated with, endorsed by,sponsored by, or supported by those trademark owners. For a list of such trademark owners, see

www.comsol.com/tm.Version: May 2013 COMSOL 4.3b
http:///www.comsol.com/contact/http://www.comsol.com/contact/offices/http://www.comsol.com/support/case/http://www.comsol.com/support/http://www.comsol.com/support/download/http://www.comsol.com/support/updates/http://www.comsol.com/community/http://www.comsol.com/events/http://www.comsol.com/video/http://www.comsol.com/support/knowledgebase/http://www.comsol.com/sla/http://www.comsol.com/sla/http://www.comsol.com/tm/http://www.comsol.com/tm/http://www.comsol.com/support/knowledgebase/http://www.comsol.com/video/http://www.comsol.com/tm/http://www.comsol.com/sla/http://www.comsol.com/events/http://www.comsol.com/community/http://www.comsol.com/support/updates/http://www.comsol.com/support/download/http://www.comsol.com/support/http://www.comsol.com/support/case/http://www.comsol.com/contact/offices/http:///www.comsol.com/contact/


3/88

Solved with COMSOL Multiphysics 4.3b

2 0 1 3 C O M S O L 1 | B I P O L A R TR A N S I S T O R

B i po l a r T r a n s i s t o r

This model shows how to set up a simple n-p-n bipolar transistor model. For this

particular model, the current voltage characteristics in the static common-emitter

configuration are computed and the common emitter current gain is determined.

Introduction

The invention of the Bipolar Transistor at Bell labs in the late 1940s was arguably themost significant technological development of the 20th century. Bipolar transistors

were the workhorses of the first integrated circuits, and consequently have had an

enormous impact on the modern world. Although bipolar transistors have largely been

replaced by field effect devices in modern integrated circuits, they are still important,

particularly in high frequency applications.

The bipolar transistor is basically a three-terminal device whose resistance between two

terminals is controlled by the third. In this example, we are going to show the static

characteristics of a n-p-n bipolar transistor in common-emitter mode, that is, when the

emitter is grounded and the base and collector voltage are larger than zero. Figure 1

shows the biasing configuration of a n-p-n transistor in this configuration.

Figure 1: A n-p-n transistor in a common-emitter configuration.

In the common-emitter mode, the base terminal of the transistor serves as the input,

the collector is the output, and the emitter is common to both (in this case grounded).

N-p-n transistors in common-emitter configuration give an inverted output and can

have a very high gain which is a strong function of the base current.

VBE

VCB

IE

IB

IC

- -

+

+


4/88


2 | B I P O L A R TR A N S I S T O R 2 0 1 3 C O M S O L

The model presented here will show the influence of the base current and voltage on

the collector current for the common-emitter configuration.

To save computation time, the model solves only for two different base currents

(IB= 1 A and 10 A). Nevertheless, the method developed here can easily be

reproduce to get the collector current for other base currents.

Model Definition

The structure in used is half of the cross section of a simplified n-p-n bipolar transistor;

see Figure 2. The model uses the following dimensions: an emitter contact length of1.2/2 m, a base contact length of 0.3 m separated from the emitter contact by

0.35 m and a collector contact length of 2.5/2 m. The total depth of the transistor

is 1 m with a width of 2.5/2 m. The modeled doping profile is shown in Figure 2

and consists in four regions (n+, p, n and n+) described in details in the Modeling

Instructions.

Figure 2: On the bottom, the cross-section of the modeled transistor. The red dashed lineshows the axis of symmetry used to simplified the model geometry (only the right part ismeshed). On the top, the implemented doping profile taken on the axis of symmetry. Thelatter represents a typical profile used in silicon bipolar transistors.

EmitterBase Base

Collector

n+

p

n+

n

Netdopingconcentration(log(cm-

3))

Depth (um)

n+

n+

n

p

Depth(um)


5/88



The model also include an Arora mobility model to take into account the scattering of

the carriers by charged impurity ions as well as a Shockley-Reed-Hall recombination

model which simulates the main recombination mechanism observed in silicon underlow electrical field.

This model uses three stationary studies to show the effect of two base currents on the

common-emitter gain and typical output current-voltage characteristics. The first

study sweeps the base-emitter voltage (VBE) for a fixed collector-emitter voltage

(VCE= 0.5 V). The first study is used to obtained initial solutions for the other studies

providing a good enough guess for the non-linear solver. The initial conditions will,

in fact, be chosen using the solution that gives the closest base-current to the one to

be use in the next study.

Using the right initial condition, the last two study will sweep the collector-emitter

voltage (VCE) for a fixed base current (IB= 1 A for the second study and IB=10 A

for the third study). The results of the two last studies will then be used to plot the

common-emitter current-voltage output characteristics (ICas a function of VCE) as

well as the common-emitter current gain (hFE IC/IB).

Results and Discussion

Figure 3displays the current in each terminal as a function of the base-emitter voltage

(VBE) for a fixed collector-emitter voltage (VCE=0.5 V). Note that the figure shows

the terminal currents using the software signs convention, that is, outward current

(from the semiconductor material) are positive and inward currents negative. In

Figure 3, one can also see that the current is conserved as the magenta line (thecalculated base current, IB,calc) matches the base current, that is,

(1)

Figure 4, shows the current density field at a collector-emitter voltage of VCE=2V for

a base current ofIB= 1A. Note that the current is directed from the base and

collector towards the emitter.

Figure 5shows the collector current (IC) as a function of the collector-emitter voltage

(VCE) and base current (IB). Here we can clearly see a important current gain, i.e. a

small base current give rise to a large collector current. One can also note that the

collector current shows a saturation trend as the collector-emitter increases. Plotting

the ratio IC/IB as a function of the base current (IB)see Figure 6shows that the

gain reduces as the base current increase.

IB IE IC=


6/88



Figure 3: Terminal currents as a function of the base-emitter voltage (VBE) for a fixedcollector-emitter voltage (VCE =0.5 V).


7/88



Figure 4: Current density in the n-p-n bipolar transistor at a collector-emitter voltage ofVCE=2 V for a base current of IB=1 A. Note that the current is directed from the baseand collector to the emitter.


8/88



Figure 5: Common-emitter output current-voltage characteristics for IB= 1 A andIB=10 A. To save computation time, only two curves have been drawn and those fromVCE=0.05 V. Nevertheless, the method can easily be reproduce to populate the figure withother base currents.


9/88



Figure 6: Common-emitter gain (hFE IC/IB) as a function of the base current (IB).Larger base currents reduce the gain.

Model Library path: Semiconductor_Module/Device_Models/

bipolar_transistor

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizardwindow.2 Click the 2Dbutton.

3 Click Next.

4 In the Add physicstree, select Semiconductor>Semiconductor (semi).

5 Click Add Selected.

6 Click Next.


10/88



7 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.

8 Click Finish.

G L O B A L D E F I N I T I O N S

Parameters

1 In the Model Builderwindow, right-click Global Definitionsand choose Parameters.

2 In the Parameterssettings window, locate the Parameterssection.

3 Click Load from File.

4 Browse to the models Model Library folder and double-click the filebipolar_transistor_parameters.txt .

G E O M E T R Y 1

1 In the Model Builderwindow, under Model 1 (mod1)click Geometry 1.

2 In the Geometrysettings window, locate the Unitssection.

3 From the Length unitlist, choose m.

Rectangle 1 (r1)

1 Right-click Model 1 (mod1)>Geometry 1and choose Rectangle.

2 In the Rectanglesettings window, locate the Sizesection.

3 In the Widthedit field, type w_BJT/2.

4 In the Heightedit field, type d_BJT.

5 Locate the Positionsection. In the yedit field, type -d_BJT.

The doping profiles will be created in the Semiconductor user interface. However,

to get a finer mesh in the junction vicinities, it is wise to create geometry objects

defining the doping regions in the semiconducting material.

Rectangle 2 (r2)

1 In the Model Builderwindow, right-click Geometry 1and choose Rectangle.


3 In the Widthedit field, type w_E/2.

4 In the Heightedit field, type d_E.

5 Locate the Positionsection. In the yedit field, type -d_E.

Fillet 1 (fil1)

1 Right-click Geometry 1and choose Fillet.

2 On the object r2, select Point 2 only.


11/88



3 In the Filletsettings window, locate the Radiussection.

4 In the Radiusedit field, type d_E.

Rectangle 3 (r3)

1 Right-click Geometry 1and choose Rectangle.


3 In the Widthedit field, type w_BJT/2.

4 In the Heightedit field, type d_B.

5 Locate the Positionsection. In the yedit field, type -d_B.

Define the emitter ohmic contact

Point 1 (pt1)

1 Right-click Geometry 1and choose Point.

2 In the Pointsettings window, locate the Pointsection.

3 In the xedit field, type w_cE.

Define the base ohmic contact

Point 2 (pt2)



3 In the xedit field, type w_BJT/2-w_cB.

4 Click the Build Allbutton.

Load the semiconductor material properties for silicon.

M A T E R I A L S

Mater ial Browser

1 In the Model Builderwindow, under Model 1 (mod1)right-click Materialsand choose

Open Material Browser.

2 In the Material Browsersettings window, In the tree, select Semiconductors>Silicon.

3 In the Material_browserwindow, click Add Material to Model.

S E M I C O N D U C T O R ( S E M I )

1 In the Model Builderwindow, under Model 1 (mod1)click Semiconductor (semi).

2 In the Semiconductorsettings window, locate the Thicknesssection.

3 In the dedit field, type l_BJT.


12/88



M A T E R I A L S


Semiconductor Material Model 1

1 In the Model Builderwindow, under Model 1 (mod1)>Semiconductor (semi)click

Semiconductor Material Model 1.

2 In the Semiconductor Material Modelsettings window, locate the Model Inputssection.

3 In the Tedit field, type T0.

Add the Arora mobility model. The Arora mobility model is used to simulate thedoping-dependent mobility in silicon and takes into account the scattering of the

carriers by charged impurity ions, which leads to a reduction of the carrier mobility.


Arora Mobil ity Model (LI) 1

1 Right-click Semiconductor Material Model 1and choose Arora Mobility Model (LI).

Add the doping profiles beginning with the epitaxial layer background doping.

Semiconductor Doping Model 1

1 In the Model Builderwindow, right-click Semiconductor (semi)and choose

Semiconductor Doping Model.

2 In the Semiconductor Doping Modelsettings window, locate the Domain Selection

section.

3 From the Selectionlist, choose All domains.4 Locate the Doping Profilesection. In the ND0edit field, type N_epi.

Next, add the base doping profile.


1 Right-click Semiconductor (semi)and choose Semiconductor Doping Model.


section.3 From the Selectionlist, choose All domains.

4 Locate the Dopingsection. From the Dopant distributionlist, choose Gaussian.


13/88



5 Locate the Uniformly Doped Regionsection. In the r0table, enter the following

settings:

6 In the Wedit field, type w_BJT/2.

7 In the Hedit field, type d_E.

8 Locate the Doping Distributionsection. In the djedit field, type d_E.

9 In the NA0edit field, type N_B+N_epi.10 From the Nblist, choose Donor concentration (user defined) (semi/sdm1).

Proceed with the emitter doping profile.




section.

3 From the Selectionlist, choose All domains.


5 Locate the Uniformly Doped Regionsection. In the Wedit field, type w_E/2-d_E.

6 Locate the Doping Distributionsection. In the djedit field, type d_E.

7 From the Dopant typelist, choose Donor doping.

8 In the ND0edit field, type N_E+N_B.

9 In the Nbedit field, type N_B.

Finally, add the collector doping profile.




section.



0[um] x

-d_E y


14/88




settings:

6 In the Wedit field, type w_BJT/2.

7 In the Hedit field, type d_C.

8 Locate the Doping Distributionsection. In the djedit field, type 1.3*d_C.


10 In the ND0edit field, type N_C.

11 From the Nblist, choose Donor concentration (user defined) (semi/sdm1).

Now add recombination.

Shockley-Reed-Hall Recombination 1

1 Right-click Semiconductor (semi)and chooseGeneration-Recombination>Shockley-Reed-Hall Recombination.

2 In the Shockley-Reed-Hall Recombinationsettings window, locate the Domain Selection

section.


Ohmic Contact 1

1 Right-click Semiconductor (semi)and choose Ohmic Contact.

2 Select Boundary 7 only.

3 In the Ohmic Contactsettings window, locate the Terminalsection.

4 In the V0edit field, type V_E.

5 Right-click Model 1 (mod1)>Semiconductor (semi)>Ohmic Contact 1and choose

Rename.

6 Go to the Rename Ohmic Contactdialog box and type Emitter voltagein the New

nameedit field.

7 Click OK.

Ohmic Contact 2




0[um] x

-d_BJT-d_

Cy


15/88



4 In the V0edit field, type V_B.


Rename.

6 Go to the Rename Ohmic Contactdialog box and type Base voltagein the New

nameedit field.

7 Click OK.

Ohmic Contact 3




4 In the V0edit field, type V_C.


Rename.

6 Go to the Rename Ohmic Contactdialog box and type Collector voltagein the

New nameedit field.

7 Click OK.

M E S H 1

In the Model Builderwindow, under Model 1 (mod1)right-click Mesh 1and choose Free

Triangular.

Size

1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1click Size.

2 In the Sizesettings window, locate the Element Sizesection.

3 From the Predefinedlist, choose Finer.

Size 1

1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1right-click Free

Triangular 1and choose Size.

2 In the Sizesettings window, locate the Geometric Entity Selectionsection.

3 From the Geometric entity levellist, choose Boundary.

4 Select Boundaries 2, 4, 610, and 13 only.


6 Click the Custombutton.


16/88



7 Locate the Element Size Parameterssection. Select the Maximum element sizecheck

box.

8 In the associated edit field, type 0.01.


Remove the default plot option. Because you are going to do multiple studies, it is

better to define the quantities to plot manually. Doing this avoids COMSOL plotting

the default variables for each study, which may not be of interest for this example.

S T U D Y 1

1 In the Model Builderwindow, click Study 1.

2 In the Studysettings window, locate the Study Settingssection.

3 Clear the Generate default plotscheck box.

Get the initial solution to plot the doping profile and make sure it agrees with what

you want to simulate.

4 Right-click Study 1and choose Get Initial Value.

R E S U L T S

1D Plot Group 1

1 In the Model Builderwindow, right-click Resultsand choose 1D Plot Group.

2 In the 1D Plot Groupsettings window, locate the Datasection.

3 From the Time selectionlist, choose First.

4 Right-click Results>1D Plot Group 1and choose Line Graph.Select the center boundaries and plot the logarithm of the absolute value of the

signed dopant concentration. Note that the value -6 is added to display the units in

log10(1/cm^3).

5 Select Boundaries 1, 3, and 5 only.

6 In the Line Graphsettings window, locate the y-Axis Datasection.

7 In the Expressionedit field, type log10(abs(semi.Ndoping))-6.

8 From the Unitlist, choose log10(1/cm^3).

9 Select the Descriptioncheck box.

10 In the associated edit field, type log10(abs(semi.Ndoping)).

Plot the doping profile along yto display the profile from the center of the emitter to

the collector.


17/88



11 Locate the x-Axis Datasection. From the Parameterlist, choose Reversed arc length.

12 Click the Plotbutton.

13 In the Model Builderwindow, click 1D Plot Group 1.

14 In the 1D Plot Groupsettings window, click to expand the Titlesection.

15 From the Title typelist, choose None.

16 Locate the Plot Settingssection. Select the x-axis labelcheck box.

17 Select the y-axis labelcheck box.

18 In the x-axis labeledit field, type Reversed arc length (um).

19 In the y-axis labeledit field, type log10(abs( Ndoping (1/cm^3) )).

20 Click to expand the Axissection. Select the Manual axis limitscheck box.

21 In the x minimumedit field, type -0.01.

22 In the x maximumedit field, type 1.01.

23 In the y minimumedit field, type 11.

24 In the y maximumedit field, type 21.

25 Right-click 1D Plot Group 1and choose Rename.

26 Go to the Rename 1D Plot Groupdialog box and type Doping profile centerin

the New nameedit field.

27 Click OK.

28 Right-click 1D Plot Group 1and choose Plot.

Define the sweep for the base voltage (V_B).

S T U D Y 1

Step 1: Stationary

1 In the Model Builderwindow, under Study 1click Step 1: Stationary.

2 In the Stationarysettings window, click to expand the Study Extensionssection.

3 Select the Continuationcheck box.

4 Click Add.

5 In the table, enter the following settings:

6 In the Model Builderwindow, right-click Study 1and choose Rename.

Continuation parameter Parameter value list

V_B range(0,0.1,0.8),range(0.805,0.005,1)


18/88



7 Go to the Rename Studydialog box and type V_B sweep, V_CE=0.5 V, V_E=0 V

in the New nameedit field.

8 Click OK.

9 Right-click Study 1and choose Compute.

R E S U L T S

Doping profile center

Plot the current at each terminal as a function of the base voltage (V_B). Note that

outward currents (from the semiconductor material) are positive and inward currents

are negative.

1D Plot Group 2


2 Right-click 1D Plot Group 2and choose Global.

3 In the Globalsettings window, click Replace Expressionin the upper-right corner of

the y-Axis Datasection. From the menu, choose Semiconductor>Terminals

group>Terminal current (semi.I0_1).

4 Click Add Expressionin the upper-right corner of the y-Axis Datasection. From the

menu, choose Semiconductor>Terminals group>Terminal current (semi.I0_2).


menu, choose Semiconductor>Terminals group>Terminal current (semi.I0_3).

6 Locate the y-Axis Datasection. In the table, enter the following settings:


The current is conserved. To see this, you can plot the base current as I_B= I_E -

I_C. Note that to follow COMSOL's sign convention (outward currents arepositive), you need to write I_B = -I_E - I_C.

8 In the Model Builderwindow, right-click 1D Plot Group 2and choose Global.

9 In the Globalsettings window, locate the y-Axis Datasection.

Unit Description

mA I_E

mA I_B

mA I_C


19/88




11 Click to expand the Coloring and Stylesection. Find the Line stylesubsection. From

the Linelist, choose None.

12 From the Colorlist, choose Magenta.

13 In the Widthedit field, type 1.

14 Find the Line markerssubsection. From the Markerlist, choose Circle.

15 In the Numberedit field, type 30.


17 In the 1D Plot Groupsettings window, locate the Titlesection.



20Select the

y-axis labelcheck box.

21 In the x-axis labeledit field, type V_BE (V).

22 In the y-axis labeledit field, type I (mA).

23 In the 1D Plot Groupsettings window, click to expand the Legendsection.

24 From the Positionlist, choose Lower left.


26 Right-click 1D Plot Group 2and choose Rename.27 Go to the Rename 1D Plot Groupdialog box and type I_E, I_B and I_C as a

function of V_BEin the New nameedit field.

28 Click OK.

Use a Global Evaluation node to determine the approximate solution at which the base

current equals I_B = -1 uA. You will use this solution as the initial value for the

collector sweep at constant base current for the npn transitor in emitter-common

configuration.

Derived Values

1 In the Model Builderwindow, under Resultsright-click Derived Valuesand choose

Global Evaluation.

2 In the Global Evaluationsettings window, locate the Expressionsection.

3 Clear the Expressionedit field.

Expression Unit Description

-semi.I0_1-semi.I0_3 mA I_B calc


20/88



4 Click Replace Expressionin the upper-right corner of the Expressionsection. From

the menu, choose Terminal current (semi.I0_2).

5 Locate the Expressionsection. From the Unitlist, choose A.

6 Click the Evaluatebutton.

Inspecting the table shows that the solution V_B= 0.82 V corresponds to a good

initial value.

R O O T

In the Model Builderwindow, right-click the root node and choose Add Study.


1 Go to the Model Wizardwindow.


3 Click Finish.

S T U D Y 2

Step 1: Stationary




4 In the Model Builderwindow, click Step 1: Stationary.

5 In the Stationarysettings window, click to expand the Values of Dependent Variables

section.

6 Select the Initial values of variables solved forcheck box.

7 From the Methodlist, choose Solution.

8 From the Studylist, choose V_B sweep, V_CE=0.5 V, V_E=0 V, Stationary.

9 From the Parameter value (V_B)list, choose 0.82.

10 Locate the Study Extensionssection. Select the Continuationcheck box.

11 Click Add.


21/88




The Base Voltage boundary condition is overridden by the new Base Current one.

For the sake of clarity, explicitly disable the boundary condition base voltage as you

are not going to use it in the second study.


Base voltage 1

1 In the Model Builderwindow, under Model 1 (mod1)>Semiconductor (semi)right-click

Base voltageand choose Duplicate.

2 Right-click Base voltage 1and choose Rename.

3 Go to the Rename Ohmic Contactdialog box and type Base currentin the New

nameedit field.

4 Click OK.5 In the Ohmic Contactsettings window, locate the Terminalsection.

6 In the Terminal nameedit field, type 4.

7 From the Terminal typelist, choose Current.

8 In the I0edit field, type I_B.

Base voltage

In the Model Builderwindow, under Model 1 (mod1)>Semiconductor (semi)right-clickBase voltageand choose Disable.

S T U D Y 2

Step 1: Stationary


2 Go to the Rename Studydialog box and type V_CE sweep, V_E=0 V, I_B=1 uA


3 Click OK.

To get the solution for V_C = 0.05 V, the solver has to do multiple iterations from

the initial value at V_C = 0.5 V. To ensure rapid convergence in the calculation of

the first value, set the minimum damping factor to 0.1 and the recovery damping

factor to 0.75.


V_C range(0.05,0.05,2)


22/88



4 Right-click Study 2and choose Show Default Solver.

V _ C E S W E E P , V _ E = 0 V , I _ B = 1 U A

Solver 2

1 In the Model Builderwindow, expand the Solver 2node.

2 In the Model Builderwindow, expand the V_CE sweep, V_E=0 V, I_B=1 uA>Solver

Configurations>Solver 2>Stationary Solver 1node, then click Fully Coupled 1.

3 In the Fully Coupledsettings window, click to expand the Method and Termination

section.

4 In the Minimum damping factoredit field, type 0.1.

5 In the Recovery damping factoredit field, type 0.75.

6 Click the Computebutton.

Plot the current density field.

R E S U L T S

2D Plot Group 3



3 From the Data setlist, choose Solution 2.

4 Right-click Results>2D Plot Group 3and choose Arrow Surface.

5 In the Arrow Surfacesettings window, click Replace Expressionin the upper-right

corner of the Expressionsection. From the menu, choose Semiconductor>Currentsand charge>Electron current density (semi.Jnx,...,semi.Jny).

6 Locate the Coloring and Stylesection. From the Arrow lengthlist, choose Logarithmic.

7 In the Model Builderwindow, right-click 2D Plot Group 3and choose Plot.


9 Go to the Rename 2D Plot Groupdialog box and type Current densityin the New

nameedit field.

10 Click OK.

To obtain the common-emitter current gain (hFE), you need at least two base

currents. Accordingly, add another study to the model.

R O O T

In the Model Builderwindow, right-click the root node and choose Add Study.

S l d i h COMSOL M l i h i 4 3b


23/88






3 Click Finish.

Inspecting the table you can see that the solution V_B = 0.88 V corresponds to a

good initial value for a base current of 10 uA.

S T U D Y 3

Step 1: Stationary




4 In the Model Builderwindow, click Step 1: Stationary.

5 In the Stationarysettings window, locate the Values of Dependent Variablessection.

6 Select the Initial values of variables solved forcheck box.

7 From the Methodlist, choose Solution.

8 From the Studylist, choose V_B sweep, V_CE=0.5 V, V_E=0 V, Stationary.

9 From the Parameter value (V_B)list, choose 0.88.

10 Locate the Study Extensionssection. Select the Continuationcheck box.

11 Click Add.



Parameters

1 In the Model Builderwindow, under Global Definitionsclick Parameters.




V_C range(0.05,0.05,2)

Expression

-10[uA]

Solved with COMSOL Multiphysics 4 3b


24/88



S T U D Y 3


2 Go to the Rename Studydialog box and type V_CE sweep, V_E=0 V, I_B=10 uAin the New nameedit field.

3 Click OK.

To get the solution for V_C=0.05 V, the solver has to do multiple iterations from

the initial value at V_C=0.5 V. To ensure a rapid convergence in the calculation of

the first value, set the minimum damping factor to 0.01, the maximum number of

iterations to 150 and the recovery damping factor to 0.75.

V _ C E S W E E P , V _ E = 0 V , I _ B = 1 0 U A

Solver 3

1 In the Model Builderwindow, right-click V_CE sweep, V_E=0 V, I_B=10 uAand choose

Show Default Solver.

2 Expand the Solver 3node.

3 In the Model Builderwindow, expand the V_CE sweep, V_E=0 V, I_B=10 uA>SolverConfigurations>Solver 3>Stationary Solver 1node, then click Fully Coupled 1.

4 In the Fully Coupledsettings window, locate the Method and Terminationsection.

5 In the Minimum damping factoredit field, type 0.01.

6 In the Recovery damping factoredit field, type 0.75.

7 In the Maximum number of iterationsedit field, type 150.

8 Click the Computebutton.

R E S U L T S

1D Plot Group 4



3 From the Data setlist, choose None.

4 Right-click Results>1D Plot Group 4and choose Global.

5 In the Globalsettings window, locate the Datasection.


7 Click Replace Expressionin the upper-right corner of the y-Axis Datasection. From

the menu, choose Semiconductor>Terminals group>Terminal current (semi.I0_3).

Solved with COMSOL Multiphysics 4 3b


25/88




9 Click to expand the Legendssection. From the Legendslist, choose Manual.


11 Right-click Results>1D Plot Group 4>Global 1and choose Duplicate.



14 Locate the Legendssection. In the table, enter the following settings:


16 In the 1D Plot Groupsettings window, locate the Titlesection.



19 In the associated edit field, type V_CE (V).


21 Locate the Legendsection. From the Positionlist, choose Upper left.



24 Go to the Rename 1D Plot Groupdialog box and type I_C as a function of V_CE


25 Click OK.

1D Plot Group 5

1 Right-click Resultsand choose 1D Plot Group.



4 Locate the Titlesection. From the Title typelist, choose None.


-semi.I0_3 mA I_C

Legends

I_B=1 uA

Legends

I_B=10 uA



26/88

p y





8 From the Parameter selection (V_C)list, choose Last.

9 Click Replace Expressionin the upper-right corner of the y-Axis Datasection. Locate

the y-Axis Datasection. In the table, enter the following settings:

10 Locate the x-Axis Datasection. From the Parameterlist, choose Expression.

11 In the Expressionedit field, type -semi.I0_4.

12 From the Unitlist, choose A.

13 Locate the Coloring and Stylesection. Find the Line stylesubsection. From the Line

list, choose None.

14 In the Widthedit field, type 3.

15 Find the Line markerssubsection. From the Markerlist, choose Circle.

16 From the Positioninglist, choose In data points.

17 Find the Line stylesubsection. From the Colorlist, choose Black.

18 Locate the Legendssection. Clear the Show legendscheck box.

19 Right-click Results>1D Plot Group 5>Global 1and choose Duplicate.




23 In the 1D Plot Groupsettings window, locate the Plot Settingssection.

24 Select the x-axis labelcheck box.

25 In the associated edit field, type I_B (uA).


27 Locate the Axissection. Select the Manual axis limitscheck box.

28 In the x minimumedit field, type 0.

29 In the x maximumedit field, type 11.

30 In the y minimumedit field, type 120.

31 In the y maximumedit field, type 160.


semi.I0_3/semi.I0_4 1 I_C/I_B



27/88




34 Go to the Rename 1D Plot Groupdialog box and type I_C/I_B as a function ofI_B, V_C=2 Vin the New nameedit field.

35 Click OK.



28/88




29/88

2 0 1 3 C O M S O L 1 | D C C H A R A C T E R I S T I C S O F A M O S T R A N S I S T O R ( M O S F E T )

DC Cha r a c t e r i s t i c s o f a MOS

T r a n s i s t o r (MOSFET )This model calculates the DC characteristics of a MOS (metal-oxide semiconductor)

transistor using standard semiconductor physics. In normal operation, a system turns

on a MOS transistor by applying a voltage to the gate electrode. When the voltage on

the drain increases, the drain current also increases until it reaches saturation. The

saturation current depends on the gate voltage.

Introduction

The MOSFET (metal oxide semiconductor field-effect transistor) is by far the most

common semiconductor device, and the primary building block in all commercial

processors, memories, and digital integrated circuits. During the past decades this

device has experienced tremendous development, and today it is being manufactured

with feature sizes of 90 nm and smaller.

Figure 1: Cross-section TEM (transmission electron microscope) image of a 70-nmMOSFET fabricated in the clean room at the Royal Institute of Technology in Kista,Sweden (a project of P.-E. Hellstrm and others).

This model shows the basic functionality of a MOS transistor, where the gate-to-source

voltage controls the drain-to-source resistance and thus the drain current. At a certain

gate-to-source voltage (VGS), and at low drain-to-source voltages (VDS), the drain

current (ID) is almost linearly dependent on VDS. When VDSincreases, the drain

current saturates. The level of saturation depends on the gate-to-source voltage.



30/88

2 | D C C H A R A C T E R I S T I C S O F A M O S TR A N S I S T O R ( M O S F E T ) 2 0 1 3 C O M S O L

Model Definition

The structure in used is a cross section of a simplified MOS transistor; see Figure 2. In

the figure are displayed the location of the drain, base, gate (metal-oxide interface) and

source where the voltages are applied. All the applied voltages are referenced to the

source that is grounded. In this model, the base-to-source (VBS) as well as the

gate-to-source (VGS) voltages are fixed and respectively set to VBS= 0 V and

VGS= 2 V. The drain-to-source voltage (VDS) varies from 0 to 3 V.

In this model, one will observe the influence of the electric field developing across the

oxide layer on the doped p-type silicon. More specifically, for a gate voltage larger than

the threshold voltage (VGS >VT), a thin layer of it, close to the silicon-oxide surface,

turns into an n-type material. This process, called strong inversion, creates a

conducting channel between the highly doped n-type source and the drain regions; see

Ref. 1. With this channel present, a voltage across the source and the drain drives a

current (ID) that will eventually saturates as the electron concentration vanishes near

the drain region (pinch-off point) with increasing VDS. Note that the threshold

voltage is the value of VGS at which the charges in the inversion layer become

significant. The latter is defined by:

(1)

Where VFBis the flat-band voltage, Bis the difference between the equilibrium

Fermi level and the intrinsic level (in volts),ris the relative permittivity of the

semiconductor, 0,is the vacuum permittivity, qthe electron charge, NAthe acceptorconcentration and Cox, the oxide capacitance per unit area. For this model

VT1.09 V.

Figure 2: Cross-section of the modeled device. All the applied voltages are referenced to thesource pad that is grounded.

VT VFB 2B2r0qNA 2B

1

2---

Cox---------------------------------------------------+ +

source drain

n+ n+

p

gate

base

DSGS

VBS

insulator



31/88



Figure 3shows the energy diagram along the center of the device from the gate

contact for VGS= 2 V and VDS= 3 V. From this figure one can see the length of

inverted region starting from the surface (y = 0) to the crossing of the intrinsic energy

level with the electron quasi-Fermi level (y approx - 0.03 m). Figure 4displays the

logarithm of the norm of the current density in the device under the same conditions.

In the figure, one can notice the inverted region that allows the current to pass

between the n-doped regions (drain and source).

Figure 5shows the logarithm of the electron concentration along the inverted channel

for different drain-to-source voltages. The figure shows the electrons pinched-off as

the charge near the drain end is reduced by the channel potential, i.e. as the

drain-to-source voltage increases.

Looking at the space charge density in Figure 6shows a larger depletion width on the

drain-side of the MOSFET compared to the source side. This is a consequence of the

electrons drawn from the drain-side of the inverted channel.

Figure 7shows a typical IDvs VDS curve of the device where a the current rises linearlyat low drain-to-source voltage to slowly saturates as the electron concentration

vanishes at the pinch-off point.



32/88


Figure 3: Energy diagram for VGS=2 V and VDS=3 V along the center of the device fromthe gate contact. This figure shows the separation of the quasi-Fermi levels (size of thedepletion region) as well as the inversion region where the intrinsic energy level crosses theelectron quasi-Fermi level (Efn).



33/88


Figure 4: Logarithm of the norm of the current density for VGS=2 V and VDS=3 V. Herewe can clearly see the inverted channel between the two n-doped regions.



34/88


Figure 5: Logarithm of the electron concentration along the inverted channel for VGS=2V and VDSranging from 0 to 3 V. This figure shows a reduction of the electronconcentration near the drain creating the saturation of the drain current.



35/88


Figure 6: Surface charge density in the device for VGS=2 V and VDS=3 V. The figure showsthe inverted channel (in blue) as well as the depletion region (orange). The red regionsshow the close-to neutral areas. The drain-side of the device shows a larger depletion regionto compensate the vanishing space charge at the pinch-off point.



36/88


Figure 7: Drain current as a function of the drain-to-source voltage for VGS=2 V.

Reference

1. S. M. Sze and Kwok K. NG., Physics of Semiconductor Devices, third edition,John

Wiley & Sons, Hoboken, NJ, 2007.

Model Library path: Semiconductor_Module/Device_Models/mosfet




2 Click the 2Dbutton.

3 Click Next.





37/88


6 Click Next.


8 Click Finish.


Parameters




4 Browse to the models Model Library folder and double-click the file

mosfet_parameters.txt.

G E O M E T R Y 1




Rectangle 1 (r1)



3 In the Widthedit field, type L_mos.

4 In the Heightedit field, type h_mos.

5 Locate the Positionsection. In the xedit field, type -L_mos/2.6 In the yedit field, type -h_mos.

Bzier Polygon 1 (b1)

1 In the Model Builderwindow, right-click Geometry 1and choose Bzier Polygon.

2 In the Bzier Polygonsettings window, locate the Polygon Segmentssection.

3 Find the Added segmentssubsection. Click the Add Linearbutton.

4 Find the Control pointssubsection. In row 1, set xto -L_g/2.

5 In row 2, set xto L_g/2.


1 Right-click Geometry 1and choose Bzier Polygon.




38/88



4 Find the Control pointssubsection. In row 1, set xto -L_mos/2+L_s.

5 In row 2, set xto -L_mos/2.


1 Right-click Geometry 1and choose Bzier Polygon.



4 Find the Control pointssubsection. In row 1, set xto L_mos/2-L_d.

5 In row 2, set xto L_mos/2.

In order to have a finer mesh in the channel area, create a rectangle below the gate

contact.

Rectangle 2 (r2)

1 Right-click Geometry 1and choose Rectangle.


3 In the Widthedit field, type L_g.

4 In the Heightedit field, type h_mos/20.

5 Locate the Positionsection. In the xedit field, type -L_g/2.

6 In the yedit field, type -h_mos/20.

Create a line at the center of the geometry. This line will be mesh finely in order to

get a clean energy diagram.

Bzier Polygon 4 (b4)1 Right-click Geometry 1and choose Bzier Polygon.



4 Find the Control pointssubsection. In row 1, set yto -h_mos/20.

5 In row 2, set yto -h_mos.





39/88


M A T E R I A L S

Mater ial Browser

1 In the Model Builderwindow, under Model 1 (mod1)right-click Materialsand chooseOpen Material Browser.


3 Click Add Material to Model.




3 In the dedit field, typeW_mos.

Define the background doping.


1 Right-click Model 1 (mod1)>Semiconductor (semi)and choose Semiconductor Doping

Model.

2 In the Semiconductor Doping Modelsettings window, locate the Domain Selectionsection.


4 Locate the Doping Profilesection. From the Dopant typelist, choose Acceptor doping

(p-type).

5 In the NA0edit field, type Na.

Define the n-doped regions.


1 In the Model Builderwindow, right-click Semiconductor (semi)and choose

Semiconductor Doping Model.


section.





40/88



settings:

6 In the Wedit field, type L_mos/2.

7 In the Hedit field, type h_mos/10.

8 Locate the Doping Distributionsection. In the djedit field, type 0.4*L_mos/2-L_g/

2.


10 In the ND0edit field, type Nd.

11 From the Nblist, choose Acceptor concentration (user defined) (semi/sdm1).




section.




settings:

6 In the Wedit field, type L_mos/2.

7 In the Hedit field, type h_mos/10.

8 Locate the Doping Distributionsection. In the djedit field, type 0.4*L_mos/2-L_g/

2.


10 In the ND0edit field, type Nd.

11 From the Nblist, choose Acceptor concentration (user defined) (semi/sdm1).

Define the source pad with an ohmic contact.

Ohmic Contact 1


0.4*L_mos/2 x

0[um] y

-1.4*L_mos/2 x

0[um] y



41/88




Rename.

4 Go to the Rename Ohmic Contactdialog box and type Sourcein the New nameedit

field.

5 Click OK.

Define the drain pad with an ohmic contact.

Ohmic Contact 2




4 In the V0edit field, type Vds.


Rename.

6 Go to the Rename Ohmic Contactdialog box and type Drainin the New nameedit

field.

7 Click OK.

Define the base pad with an ohmic contact.

Ohmic Contact 3


2 Select Boundaries 2 and 9 only.


4 In the V0edit field, type Vbs.


Rename.

6 Go to the Rename Ohmic Contactdialog box and type Basein the New nameedit

field.

7 Click OK.

Define the gate with a thin insulator gate.

Thin Insulator Gate 1

1 Right-click Semiconductor (semi)and choose Thin Insulator Gate.




42/88


3 In the Thin Insulator Gatesettings window, locate the Gate Contactsection.

4 In the V0edit field, type Vgs.

5 In the insedit field, type eps_ins.6 In the dinsedit field, type d_ins.

7 In the edit field, type phim.

8 Right-click Model 1 (mod1)>Semiconductor (semi)>Thin Insulator Gate 1and choose

Rename.

9 Go to the Rename Thin Insulator Gatedialog box and type Gatein the New nameedit

field.

10 Click OK.

Define the mesh in the gate vicinity and in at the center of the device.

M E S H 1


Triangular.

Size1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1click Size.


3 From the Calibrate forlist, choose Fluid dynamics.

Free Triangular 1

1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1click Free Triangular 1.

2 In the Free Triangularsettings window, locate the Domain Selectionsection.

3 From the Geometric entity levellist, choose Domain.

4 Select Domain 2 only.

Size 1

1 Right-click Model 1 (mod1)>Mesh 1>Free Triangular 1and choose Size.




box.



F T l 2


43/88


Free Triangular 2

In the Model Builderwindow, right-click Mesh 1and choose Free Triangular.

Size 11 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1right-click Free








box.



S T U D Y 1

Step 1: Stationary

1 In the Model Builderwindow, under Study 1click Step 1: Stationary.

2 In the Stationarysettings window, click to expand the Study Extensionssection.

3 Select the Continuationcheck box.

4 Click Add.

5 Click Range.

6 Go to the Rangedialog box.

7 From the Entry methodlist, choose Number of values.

8 In the Startedit field, type 0.

9 In the Stopedit field, type 3.

10 In the Number of valuesedit field, type 21.

11 Click the Addbutton.

Right click the Study 1node and select Compute.

R E S U L T S

Data Sets

1 In the Model Builderwindow, under Results>Data Setsclick Cut Line 2D 1.


2 In the C t Li 2D settings indo locate the Li D t section


44/88


2 In the Cut Line 2Dsettings window, locate the Line Datasection.

3 In row Point 2, set xto 0.

4 In row Point 2, set yto -h_mos.

1D Plot Group 4



3 From the Data setlist, choose Cut Line 2D 1.

4 From the Parameter selection (Vds)list, choose Last.

5 Right-click Results>1D Plot Group 4and choose Line Graph.6 In the Line Graphsettings window, click Replace Expressionin the upper-right corner

of the y-Axis Datasection. From the menu, choose Semiconductor>Fermi levels and

band edges>Energies>Conduction band energy level (semi.Ec_e).

7 Locate the y-Axis Datasection. From the Unitlist, choose eV.

8 Locate the x-Axis Datasection. From the Parameterlist, choose Expression.

9 In the Expressionedit field, type -y.

10 Click to expand the Coloring and Stylesection. Find the Line stylesubsection. From

the Colorlist, choose Blue.

11 Click to expand the Legendssection. Select the Show legendscheck box.

12 From the Legendslist, choose Manual.


14 Right-click Results>1D Plot Group 4>Line Graph 1and choose Duplicate.

15 In the Line Graphsettings window, click Replace Expressionin the upper-right corner


band edges>Energies>Electron quasi-fermi energy level (semi.Efn_e).

16 Locate the Coloring and Stylesection. Find the Line stylesubsection. From the Linelist, choose Dashed.

17 From the Colorlist, choose Black.

Legends

Ec


18 Locate the Legends section In the table enter the following settings:


45/88






band edges>Energies>Intrinsic energy level (semi.Ei_e).


list, choose Solid.22 From the Colorlist, choose Magenta.





band edges>Energies>Hole quasi-fermi energy level (semi.Efp_e).


list, choose Dotted.

27 From the Colorlist, choose Black.





band edges>Energies>Valence band energy level (semi.Ev_e).


list, choose Solid.

32 From the Colorlist, choose Green.

Legends

Efn

Legends

Ei

Legends

Efp


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


46/88




35 In the 1D Plot Groupsettings window, click to expand the Titlesection.

36 From the Title typelist, choose Manual.

37 In the Titletext area, type Energy diagram.



40 In the associated edit field, type Energy (eV).

41 Click to expand the Legendsection. From the Positionlist, choose Lower right.


43 Go to the Rename 1D Plot Groupdialog box and type Energy diagramin the New

nameedit field.

44 Click OK.

2D Plot Group 5


2 In the Model Builderwindow, under Resultsright-click 2D Plot Group 5and choose

Surface.

3 In the Surfacesettings window, locate the Expressionsection.

4 In the Expressionedit field, type log10(semi.normJ).

5 Click to expand the Rangesection. Select the Manual color rangecheck box.

6 In the Minimumedit field, type 1.

7 In the Maximumedit field, type 10.

8 In the Model Builderwindow, right-click 2D Plot Group 5and choose Rename.

9 Go to the Rename 2D Plot Groupdialog box and type Current densityin the New

nameedit field.

10 Click OK.


1D Plot Group 6


Legends

Ev




47/88


, g p

Line Graph.



of the y-Axis Datasection. From the menu, choose Semiconductor>Carrier

concentrations>Electrons>Electron concentration (N).

5 Locate the Legendssection. Select the Show legendscheck box.

6 Click to expand the Qualitysection. From the Resolutionlist, choose No refinement.

7 Click the y-Axis Log Scalebutton on the Graphics toolbar.

8 In the Line Graphsettings window, locate the y-Axis Datasection.

9 From the Unitlist, choose 1/m^3.


11 In the 1D Plot Groupsettings window, locate the Legendsection.

12 From the Positionlist, choose Lower left.


14 Go to the Rename 1D Plot Groupdialog box and type Electron concentration

at the gatein the New nameedit field.

15 Click OK.


2D Plot Group 7



Surface.

3 In the Surfacesettings window, click Replace Expressionin the upper-right corner of

the Expressionsection. From the menu, choose Semiconductor>Currents and

charge>Space charge density (semi.rho).

4 Locate the Rangesection. Select the Manual color rangecheck box.

5 In the Minimumedit field, type -5e4.6 In the Maximumedit field, type 500.

7 Select the Manual data rangecheck box.

8 In the Minimumedit field, type -1e10.

9 In the Maximumedit field, type 500.




48/88


11 Go to the Rename 2D Plot Groupdialog box and type space charge densityin

the New nameedit field.

12 Click OK.


1D Plot Group 8



Global.

3 In the Globalsettings window, click Replace Expressionin the upper-right corner of

the y-Axis Datasection. From the menu, choose Semiconductor>Terminals

group>Terminal current (semi.I0_2).


5 Click to expand the Legendssection. Clear the Show legendscheck box.


7 Go to the Rename 1D Plot Groupdialog box and type Id vs Vdsin the New name

edit field.

8 Click OK.

Current density1 Right-click 1D Plot Group 8and choose Plot.

2 From the Filemenu, choose Save Model Thumbnail.


-semi.I0_2 mA Terminal current


PN -D i od e C i r c u i t


49/88

2 0 1 3 C O M S O L 1 | P N - D I O D E C I R C U I T

PN -D i od e C i r c u i t

This model compares a full device level simulation with a lumped circuit model to

simulate a half-wave rectifier.

Introduction

The pn-diode is of great importance in modern electronic applications. The latter is

mainly used as a rectifier to convert alternative currents (AC) to direct currents (DC)

by blocking either the positive or negative half of the AC wave. The present examplesimulates the transient behavior of a pn-diode used as the active component of a

half-wave rectifier circuit -see Figure 1

Figure 1: A basic half-wave rectifier circuit. A AC voltage source is connected to the anode

of a pn-diode. The resistor represents the load of the circuit.

In this example, a full level device simulation is made by connecting a 2D meshed

pn-junction to a circuit containing a sinusoidal source, a resistor and a ground (the

half-wave rectifier circuit displayed on Figure 1). In order to validate the obtained

results, the outputs of the full device simulation are compared to the circuit response

obtained using a large signal diode model (see the electric circuit).

Model Definition

Figure 2shows the modeled device cross-section and doping profile. The diode has a

width of 10 m and a depth of 7 m. The length of the diode has been set to 10 m

(not meshed). A Shockley-Reed-Hall recombination is also added to the model in

order to simulate the type of recombination usually observed in indirect band-gap


semiconductor such as silicon (used material). The meshed diode is connected to the


50/88

2 | P N - D I O D E C I R C U I T 2 0 1 3 C O M S O L

half wave circuit using two ohmic terminals where the applied voltage is determined

by the circuit source and current. For the large signal diode model, the saturation

current and ideality factor have been set to values fitting the modeled diode I-V curve.

Figure 2: On the bottom is displayed the cross-section of the simulated device. To savecomputation time, only half of the diode is meshed, i.e. the right side delimited by the axisof symmetry (red dashed line). The top figure shows the net doping concentration along thesymmetry line (center of the diode cross-section).


Figure 3shows the output voltages obtained from both the full level simulation and

large signal model. As expected from the reverse operation of a pn-diode, clipping

occurs on the negative half of the wave in the diode.

p

n

n+

p

n

n+



51/88


Figure 3: Output voltages obtain from both the full level simulation and large signalmodel.Voltages have been monitored at the source, diode and load ends.

Model Library path: Semiconductor_Module/Device_Models/

pn_diode_circuit




2 Click the 2Dbutton.3 Click Next.



6 In the Add physicstree, select AC/DC>Electrical Circuit (cir).




52/88


8 Click Next.

9Find the

Studiessubsection. In the tree, select

Preset Studies for Selected

Physics>Time Dependent.

10 Click Finish.


Parameters




4 Browse to the models Model Library folder and double-click the file

pn_diode_circuit_parameters.txt .

G E O M E T R Y 1




Rectangle 1 (r1)



3 In the Widthedit field, type w_diode/2.

4 In the Heightedit field, type d_diode.

5 Locate the Positionsection. In the yedit field, type -d_diode.

The doping profiles will be created in the semiconductor interface.However, in

order to have a finer mesh in the junction vicinities, it is wise to create geometry

objects defining the doping regions in the semiconducting material.

Rectangle 2 (r2)

1 In the Model Builderwindow, right-click Geometry 1and choose Rectangle.


3 In the Widthedit field, type w_anode/2+d_p.

4 In the Heightedit field, type d_p.

5 Locate the Positionsection. In the yedit field, type -d_p.


Fillet 1 (fil1)

1 Right-click Geometry 1 and choose Fillet


53/88


1 Right click Geometry 1and choose Fillet.

2 On the object r2, select Point 2 only.

3 In the Filletsettings window, locate the Radiussection.

4 In the Radiusedit field, type d_p.

Point 1 (pt1)



3 In the xedit field, type w_anode/2.



M A T E R I A L S

Mater ial Browser

1 In the Model Builderwindow, under Model 1 (mod1)right-click Materialsand choose

Open Material Browser.


3 Click Add Material to Model.




3 In the dedit field, type l_diode.

Definition of the background doping (n region).


1 Right-click Model 1 (mod1)>Semiconductor (semi)and choose Semiconductor Doping

Model.


section.


4 Locate the Doping Profilesection. In the ND0edit field, type Nd_back.


Definition of the n+ region.


54/88



1 In the Model Builderwindow, right-click Semiconductor (semi)and chooseSemiconductor Doping Model.


section.




settings:

6 In the Wedit field, type w_diode/2.

7 Locate the Doping Distributionsection. From the Dopant typelist, choose Donor

doping.8 In the ND0edit field, type Nd_max.


Definition of the p region.




section.



5 Locate the Uniformly Doped Regionsection. In the Wedit field, type w_anode/2.

6 In the Hedit field, type d_p.

7 Locate the Doping Distributionsection. In the NA0edit field, type Na_max.


0[um] x

-d_diode y


Shockley-Reed-Hall Recombination 1

1 Right-click Semiconductor (semi)and choose


55/88


g ( )

Generation-Recombination>Shockley-Reed-Hall Recombination.

2 In the Shockley-Reed-Hall Recombinationsettings window, locate the Domain Selection

section.


Ohmic Contact 1




Rename.

4 Go to the Rename Ohmic Contactdialog box and type anodein the New nameedit

field.

5 Click OK.

Ohmic Contact 2




Rename.

4 Go to the Rename Ohmic Contactdialog box and type cathodein the New nameedit

field.

5 Click OK.

E L E C T R I C A L C I R C U I T ( C I R )

Voltage Source 1

1 In the Model Builderwindow, under Model 1 (mod1)right-click Electrical Circuit (cir)

and choose Voltage Source.

2 In the Voltage Sourcesettings window, locate the Device Parameterssection.

3 From the Source typelist, choose Sine source.

4 In the Vsrcedit field, type Vac.

5 In the fedit field, type f.


To connect the numerical model to the circuit, we use the current flowing out of the

cathode as a source for the circuit model.


56/88


External U Vs. I 11 In the Model Builderwindow, right-click Electrical Circuit (cir)and choose External U

Vs. I.

2 In the External U Vs. Isettings window, locate the Node Connectionssection.


4 Locate the External Devicesection. In the Iedit field, type semi.I0_2.

Use a 100 kOhm load resistor to limit the current in the circuit.

Resistor 1

1 Right-click Electrical Circuit (cir)and choose Resistor.

2 In the Resistorsettings window, locate the Node Connectionssection.


4 Locate the Device Parameterssection. In the Redit field, type 100[kohm].

Create an additional degree of freedom for the cathode voltage. The latter is

basically the voltage across the resistor (one end referenced to ground) obtained

from the current flowing in the circuit. To set the new degree of freedom, use a

global equation, accessible from the advance physics option.

5 In the Model Builderwindows toolbar, click the Showbutton and select Advanced

Physics Optionsin the menu.


Global Equations 1

1 In the Model Builderwindow, under Model 1 (mod1)right-click Semiconductor (semi)

and choose Global>Global Equations.

Label Node names

n 2

Label Node names

p 2


2 In the Global Equationssettings window, locate the Global Equationssection.



57/88


, g g

Set the cathode voltage to be the voltage across the circuit resistance (referenced to

ground).

cathode


cathode.


3 In the V0edit field, type Vback.

Set the anode voltage to be the voltage across the circuit source (referenced to

ground).

anode


anode.


3 In the V0edit field, type cir.V1_v.

M E S H 1


Triangular.

Size

1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1click Size.


3 From the Predefinedlist, choose Extra fine.

Free Triangular 1

Set a finer mesh at the junction and electrodes boundaries.

Name f(u,ut,utt,t)

Vback cir.R1_v-Vback


Size 1

1 In the Model Builderwindow, under Model 1 (mod1)>Mesh 1right-click Free


58/88





4 Select Boundaries 2, 46, and 9 only.




box.



Set the simulation time to two cycles.

S T U D Y 1

Step 1: Time Dependent

1 In the Model Builderwindow, under Study 1click Step 1: Time Dependent.

2 In the Time Dependentsettings window, locate the Study Settingssection.

3 In the Timesedit field, type range(0,tmax/100,tmax).

For a better accuracy, use intermediate time stepping to force the solver to take at least

one step in each subinterval of the specified times. Also use a small initial step to get a

better resolution at the beginning of the simulation.

Solver 1

1 In the Model Builderwindow, right-click Study 1and choose Show Default Solver.

2 In the Model Builderwindow, expand the Solver 1node, then click Time-Dependent

Solver 1.3 In the Time-Dependent Solversettings window, click to expand the Time Stepping

section.

4 From the Steps taken by solverlist, choose Intermediate.

5 Select the Initial stepcheck box.


6 In the associated edit field, type 1e-12.

Right click the Study 1 nodeand select Compute.


59/88


R E S U L T S

1D Plot Group 4




4 Locate the Titlesection. From the Title typelist, choose None.


6 In the associated edit field, type Time (s).


8 In the associated edit field, type Voltage (V).

9 Right-click Results>1D Plot Group 4and choose Rename.

10 Go to the Rename 1D Plot Groupdialog box and type Voltage probesin the New

nameedit field.

11 Click OK.

Voltage probes




4 Click Replace Expressionin the upper-right corner of the y-Axis Datasection. From

the menu, choose Electrical Circuit>Voltage across device V1 (cir.V1_v).


menu, choose Electrical Circuit>Voltage across device UvsI1 (cir.UvsI1_v).


menu, choose Electrical Circuit>Voltage across device R1 (cir.R1_v).

7 Click to expand the Legendssection. From the Legendslist, choose Manual.


8 In the table, enter the f

semiconductor model library manual

Documents

trademarks of comsol

comsol multiphysics

c o n t

comsol desktop

download comsol

comsol community

n f o r

t i o nvisit