DC Characteristics of a MOS Transistor

7/30/2019 DC Characteristics of a MOS Transistor

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DC Characteristics of a MOS Transistor (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.

Cross-section TEM (transmission electron microscope) image of a 70-nm MOSFET

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 voltage controls

the drain-source resistance and thus the drain current. At a certain gate voltage, VGS, and atlow drain voltages, the drain current is almost linearly dependent on the drain voltage. When

the drain voltage increases, the drain current saturates. The level of saturation depends on the

gate voltage.

The gate voltage also influences the coefficient describing the linear dependence between the

gate voltage and gate current at low drain voltages. This coefficient is generally known as the

On resistance (Ron). In digital devices the transistor operates as a switch, making Ron an

important parameter influencing the transistors power loss and driving ability. Its ability to

drive an output is of special importance. This characteristic is described by the fanout, the

number of outputs that a device is capable of driving. In high-speed systems, parasitic effects

make the situation more complicated.

Model Definition

DEVICE GEOMETRY

The structure inFigure 12-7is a cross section of a simplified MOS transistor. The electric

field in the gate influences the low-doped p-type silicon; specifically, at a certain gate voltage

a thin layer of it, close to the silicon-oxide surface, turns into an n-type material. This process,

called inversion, creates a conducting channel between the highly doped n-type source and the

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

drain current. This model uses the following dimensions: a gate length of 0.2 m, a gate oxide
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voltage plus the potential difference between the Fermi level and the electrostatic reference

level:

where Va is the applied voltage. In this model the reference potential is the vacuum level. This

is often a good choice when several materials with different affinities or work functions are

present. As a result, the electrostatic potential is continuous across all material interfaces. This

reference level also finds use in heterojunction device simulations. The band diagram in

Figure 12-8shows how this model represents the metal-oxide semiconductor material system.

Figure 12-8: A band diagram of the MOS material system. The necessary parameters to alignthe materials are the affinities of the oxide and semiconductor, plus the work function of the

polysilicon gate, which this model assumes has the same properties as aluminum.

The material parameters in the MOS transistor are:

NOTATION VARIABLE VALUE

ni(Si) ni_Si 1.461016 m-3(T = 300K)

r(Si) epsilonr_Si 11.8

r(SiO2) epsilonr_Si 4.2

n(Si) mun 0.10 m2/Vs (low

concentration)

p(Si) mup 0.05 m2/Vs (low

concentration)
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Dn = n(Si)kT/q

Dn 20.710-4

m2/s

Dp = p(Si)kT/q

Dp 5.1710-4 m2/s

n taun ~0.1 s

p taup ~0.1 s

EG(Si) Eg_Si 1.08 eV

Si X_Si 4.0 eV

SiO2 X_SiO2 0.3 eV

Poly X_poly 4.2 eV

INITIAL-VALUE CALCULATION

The rapid changes and large dynamics of the solution variables in the PDEs require a special

technique to calculate the initial value. This model solves an extra electrostatics application

mode separately to get the initial value for the other application modes. This extra application

mode also solves Poissons equation but with the difference that it replaces the carrier

concentrations with the formulas

This step produces the exact solution for the full system when all applied voltages are zero.

You perform the steps to get the initial condition with the solver scripting functionality in the

Solver Manager dialog box.

Results and DiscussionThe systems nonlinear behavior makes it necessary to use the parametric solver to reach the

desired voltage for each contact. This example first raises the gate voltage to 0.8 V and then

sweeps the drain voltage to 1 V. Including the first initial-value calculation, three solution

steps are necessary, and by employing solver scripting you can execute all of them with one


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click on the Solve button. The result from the last step appears inFigure 12-9.

Figure 12-9: Drain current as a function of drain voltage for VGS = 0.8 V. Note that Ronis 1/

(slope of the curve) at the point where VDS = 0, which falls at the far left of the curve.

A surface plot is useful to view the shape of the solution variables. Figure 12-10shows the

electrostatic potential at the final bias condition. The large negative shift for the potential is

due to the vacuum reference potential.

Figure 12-10: Surface plot of the electrostatic potential inside the MOSFET.
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The conducting channel at the oxide-semiconductor interface becomes visible in a logarithmic

surface plot of the electron concentration (Figure 12-11).

Figure 12-11: The channel close to the interface becomes visible in an electron-concentration

plot. Note the sharp drop in the channel concentration close to the drain contact, which is

responsible for the saturation of the drain current.

Model Library Path: COMSOL_Multiphysics/Semiconductor_Devices/MOS_transistor

Modeling Using the Graphical User Interface

MODEL NAVIGATOR

1In the Model Navigator, select 2D in the Space dimension list.

2Click the Multiphysics button.

3Select the COMSOL Multiphysics>Electromagnetics>Electrostatics application mode.

4

Enter phi0 in the Dependent variables edit field, and enter init in the Application mode

name edit field.

5ClickAdd.

6Enter phi in the Dependent variables edit field.

7ClickAdd.

8

Select the COMSOL Multiphysics>Diffusion>Convection and Diffusion>Steady-state

analysis application mode.

9Enter nc in the Dependent variables edit field, and enter cde in the Application modename edit field.

10ClickAdd

11

ClickApplication mode properties, select Conservative in the Equation form list and

Ideal in the Weak Constraints list. ClickOK.

12

Enter pc in the Dependent variables edit field, and enter cdh in the Application mode

name edit field.

13ClickAdd.

14

ClickApplication mode properties, select Conservative in the Equation form list and

Ideal in the Weak Constraints list. ClickOK.

15ClickOK.

OPTIONS AND SETTINGS

1From the Options menu, choose Constants.
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2

In the Constants dialog box, define the following constants with names, expressions, and

descriptions (the Description field is optional):

NAME EXPRESSION DESCRIPTION

q 1.602e-19 Elementary charge (C)

T0 300 Lattice temperature (K)

k 1.38e-23 Boltzmann constant (J/K)

epsilonr_si 11.8 Rel. permittivity for Si

epsilonr_sio2 4.2 Rel. permittivity for SiO2

ni 1.46e16 Intrinsic carrier

concentration for Si (m-3)

mun_si 0.1 Electron mobility for Si(m2/Vs)

mup_si 0.05 Hole mobility for Si (m2/Vs)

X_si 4.2 Electron affinity for Si (eV)

X_sio2 0.3 Electron affinity for SiO2

(eV)

X_poly 4.5 Work function for poly-Si

(eV)

Eg_si 1.08 Band gap of Si (eV)

taun 0.1e-6 Electron carrier life time (s)

taup 0.1e-6 Hole carrier life time (s)

Vt k*T0/q Thermal voltage (V)

x1 -1e-7 Help coordinate (m)


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x2 2e-7 Help coordinate (m)

ch yj/sqrt(log(

NDimpl/NAsub))

Characteristic length of

implanted region (m)

yj 1e-7 Junction depth (m)

NDimpl 1e25 Peak concentration of

implanted profile (m-3)

NAsub 5e23 Substrate doping (m-3)

Vg 0.8 Gate voltage (V)

Vd 0 Drain voltage (V)

3ClickOK.

4From the Options menu, choose Functions.

5In the Functions dialog box, click the New button.

6

In the New Function dialog box, type flguass in the Function name edit field. ClickOKto

create the new function.

7

Type x, y, x1, y1, ch in the Arguments edit field, and type the following in the Expression

edit field.

(exp(-((y-y1)/ch)^2)*(y=y1))*(exp(-((x-x1)/ch)^2)*(x>x1)+(x


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alter the properties of the curve segments with the dialog box Object Properties. Do the

following to draw the small spacer regions on the sides of the poly-gate:

5

Click the 2nd Degree Bezier Curve button and draw an arc from (-1e-7, 0), with control

point (-1e-7, 1e-7), ending it at (0, 1e-7).

6

Continue the arc with straight lines by clicking the Line button and then click the coordinate

sequence (0, 0) and (-1e-7, 0). Close to a solid by clicking the right mouse button.7From the Draw menu, choose Object Properties.

8

In the Object Properties dialog box, change the properties for each curve segment

according the table below.

CURVE

NUMBER

POINT

NUMBER

(ROW IN

TABLE)

X Y WEIGHT

1 1 -5e-

8

0 1

2 -1e-

8

0 1

2 1 -

1e-

8

0 1

2 -

1e-

8

5e-

8

1

3 1 -

5e-

8

0 1

2 -

5e-

8

5e-

8

1/sqrt(2)

3 -1e-

8

5e-8

1

9 Select the objects CO1, CO2, and R3, and click on the Mirror button.

10

In the Mirror dialog box, enter 1e-7 in the x edit field at the Point on line row. Leave all

other fields at their default values and clickOK. The geometry should look like the figure


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below.PHYSICS SETTINGS

Expression Variables

1From the Options menu, choose Expressions>Scalar Expressions.

2

In the Scalar Expressions dialog box, define the following variables with names and

expressions, and descriptions (the Description field is optional):

NAME EXPRESSION DESCRIPTION

Dn Vt*mun Electron diffusivity

Dp Vt*mup Hole diffusivity

n0 ni*exp((phi0+X_si+0.5*Eg_si)/Vt)

Electronconcentration in

thermal

equilibrium

p0 ni*exp(-(phi0+X_si+

0.5*Eg_si)/Vt)

Hole concentration

in thermalequilibrium

R_srh (nc*pc-ni^2)/(taun*(pc+ni)+

taup*(nc+ni))

Shockley-Reed-

Hall recombination

term

phi_init Vt*(-log(p_init/ni)*(Ndoping=0))-X_si

-0.5*Eg_si

Initial guess forphi0


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n_init (abs(Ndoping)/2+

sqrt(Ndoping^2/4+ni^2))*

(Ndoping>=0)+

ni^2/(abs(Ndoping)/2+

sqrt(Ndoping^2/4+ni^2))*(NdopingSubdomain Expressions.

2In the Subdomain Expressions dialog box, define the following variables:

SUBDOMAIN 1, 2, 6 4, 5, 7 ALL

OTHER

Epsilonr epsilonr_si epsilonr_sio2 1

Mun mun_si

Mup mup_si

3ClickOK.

Boundary Variables

1From the Options menu, choose Expressions>Boundary Expressions.

2

In the Boundary Expressions dialog box, define the following variables with names and

expressions:

BOUNDARY 6, 19 ALL

OTHER

Jn -

q*lm3

Jp q*lm4


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3ClickOK.

Integration Coupling Variables

The drain current Id is defined as an integral of the normal current density along the drain

contact. This can be implemented by using an integration coupling variable:

1From the Options menu, select Integration Coupling Variable>Boundary Variables.

2In the Boundary Integration Variables dialog box, select boundary 19 and then type Id inthe Name column, and type Jn+Jp in the Expression column.

Subdomain Settings

1On the Multiphysics menu, select the Electrostatics (init) application mode.

2From the Physics menu, select Subdomain settings.

3 In the Subdomain Settings dialog box, enter the following settings:

SUBDOMAIN 1, 2, 6 ALL

OTHER

r epsilonr epsilonr

q*(p0-n0+Ndoping)

0

4 Click the Init tab, and enter the following initial value:

SUBDOMAIN ALL

phi0(t0) phi_init

5ClickOK.

6On the Multiphysics menu, select the Electrostatics (es) application mode.

7 Open the Subdomain Settings dialog box and enter the following settings:

SUBDOMAIN 1, 2, 6 ALL

OTHER

r epsilonr epsilonr

q*(pc-nc+Ndoping) 0


SUBDOMAIN ALL

phi(t0) phi0

9 ClickOK.

10On the Multiphysics menu, select the Convection and Diffusion (cde) application mode.

11 Open the Subdomain Settings dialog box, select subdomains 3, 4, 5, 7, and8, and clear the Active in this subdomain check box. Then enter the


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following settings for the other subdomains

SUBDOMAIN 1, 2, 6

D isotropic Dn

u velocity mun*phix

v velocity mun*phiy

R -R_srh

12 Click the Init tab and enter the following initial value:

SUBDOMAIN 1,

2,

6

nc(t0) n0

13ClickOK.

14On the Multiphysics menu, select the Convection and Diffusion (cdh) application mode.

15

Open the Subdomain Settings dialog box, select subdomains 3, 4, 5, 7, and

8, and clear the Active in this subdomain check box. Then enter the

following settings for the other subdomains

SUBDOMAIN 1, 2, 6

D isotropic Dp

U -mup*phix

V -

mup*phiy

R -R_srh


SUBDOMAIN 1,

2,

6

pc(t0) p0

17ClickOK.


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Boundary Conditions

1On the Multiphysics menu, select the Electrostatics (init) application mode.

2

From the Physics menu, select Boundary Settings. Select the Interior boundaries check

box.

3 In the Boundary Settings dialog box, enter the following settings:

BOUNDARY 2, 6, 19 11, 12, 17 1, 3, 5, 7, 18, 21

23

Boundarycondition

Electricpotential

Electricpotential

Zerocharge/Symmetry

V0 phi_init Vg-X_poly

4ClickOK.

5On the Multiphysics menu, select the Electrostatics (es) application mode.

6

Open the Boundary Settings dialog box, select the Interior boundaries

check box, and enter the following settings:

BOUNDARY 2, 6 19 11, 12,

17

1, 3, 5, 7,

18, 2123

Boundary

condition

Electric

potential

Electric

potential

Electric

potential

Zero

charge/

Symmetry

V0 phi_init phi_init+Vd Vg-

X_poly

7ClickOK.

8On the Multiphysics menu, select the Convection and Diffusion (cde) application mode.

9 Open the Boundary Settings dialog box and enter the following settings:

BOUNDARY 2, 6, 19 ALL OTHER

Boundary

condition

Concentration Insulation/Symmetry

nc0 n_init

10ClickOK.

11On the Multiphysics menu, select the Convection and Diffusion (cdh) application mode.

12 Open the Boundary Settings dialog box and enter the following settings:

BOUNDARY 2, 6, 19 ALL OTHER

Boundary Concentration Insulation/Symmetry


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condition

pc0 p_init

13ClickOK.

MESH GENERATION

1From the Mesh menu, choose Free Mesh Parameters.

2Click the Custom mesh size button and enter 5e-8 in the Maximum element size edit field.

3Click the Subdomain tab.

4Select subdomains 2 and 6 and type 1-8 in the Maximum element size edit field.

5Click the Boundary tab.

6Select boundaries 4, 20, 24, and 26 and type 5e-9 in the Maximum element size edit field.

7Select boundary 13 and type 2e-9 in the Maximum element size edit field.

8Click the Remesh button and then clickOK.

COMPUTING THE SOLUTION

1From the Solve menu, choose Solver Manager

2In the Solver Manager dialog box, click on the Script tab.

3

Click the Solve using a script check box, and enter the following script in the large text

area.

init = asseminit(fem);

fem.sol=femstatic(fem, ...

'init',init, ...

'solcomp',{'phi0'}, ...

'outcomp',{'lm3','phi','phi0','pc','nc','lm4'});

fem0=fem;

fem.sol=femstatic(fem, ...

'init',fem0.sol, ...

'solcomp',{'lm3','phi','pc','nc','lm4'}, ...'outcomp',{'lm3','phi','phi0','pc','lm4','nc'}, ...

'pname','Vd', ...

'plist',[0:0.02:0.2 0.25:0.05:1]);

fem0=fem;


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4

ClickOKand then click the Solve button to start the simulation. Note that the script

overrides all other settings in the Solver Parameters and Solver Manager dialog boxes.

POSTPROCESSING AND VISUALIZATION

1Open the Plot Parameters dialog box.

2

Enter the expression phi as both Surface expression and Height expression on the Surface

page of the Plot Parameters dialog box. Remember to select the Height data check box.

3

ClickOKto get the plot in the figure below.

4

To plot the drain characteristics open the Domain Plot Parameters dialog box from the

Postprocessing menu.

5

Choose the Point plot type and check that all parameters are selected in the Parameter

value list.

6On the Point tab, type Id in the Point expression edit field.

7Select vertex 1 and clickOK. In a separate window you should see the plot below.


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DC Characteristics of a MOS Transistor