DC Characteristics of a MOS Transistor

DC Characteristics of a MOS Transistor (MOSFET) (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 electro de. 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. Hellström 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 at low 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 (R on on). In digital devices the transistor operates as a switch, making R on on an important parameter influencing the transistor’s power loss and driving abili ty. Its ability to drive an output is of special importance. This characte ristic is described by b y the fanout, the number of outputs that a device is capable of driving. In high-speed s ystems, parasitic effects make the situation more complicated. Model Definition DEVICE GEOMETRY The structure in Figure 12-7 is 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 n-t ype material. This process, called inversion, creates a conducting channel between the hi ghly 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 μm, a gate oxide

thickness of 5 nm, and a source-drain junction depth of 0.1 μm.

Figure 12-7: The model geometry. The simulation replaces the contacts with boundary conditions. DOMAIN EQUATIONS This example models a MOS transistor using the standard drift-diffusion approximation coupled with Poisson’s equation. With the aid of some generally accepted simplifications —  such as neglecting magnetic fields, assuming a constant densit y of states in both the valence and the conductance bands, and assuming a Boltzmann distribution of the carriers — it is  possible to derive these equations from Maxwell’s equations and Boltzmann transport theory. The model in this example formulates the problem using three dependent variables: ψ (the electrostatic potential), n, and p. The three basic semiconductor equations are

Where p and n are the hole and electron concentrati ons, respectively, and N represents the fixed charge associated with ionized donors. You can express the electron and hole current densities, Jn and J p, with a drift term and a diffusion term

where μn and μ p are the carrier mobilities, and Dn and D p are the carrier diffusivities. Another term, R SRH, represents the Shockley-Read-Hall recombination, which is a general recombination process using traps constituting energy levels in a semiconductor’s forbidden  band gap. In the recombination equation

ni is the intrinsic carrier concentration, τn and τ p are the carrier lifetimes, and n1 and p1 are  parameters related to the trap energy level. If that level is located in the middle of the band gap (which this model assumes), then n 1 and p1 equal the intrinsic carrier concentration, ni. BOUNDARY CONDITIONS For boundaries in contact with an insulator or far awa y from the active device area, you can use the symmetry or zero charge (flux) boundary condition for the electrostati c (diffusion)  problem. At boundaries in contact with a metal, the electrostatic potential is fixed. Assuming infinite recombination velocity at the contact, the mass action law

is valid. Using this law along with the assumption that there is no charge at the contact, it is  possible to calculate the carrier concentrations. The applied voltage equals the Fermi level in the semiconductor at the contact, so the electrostatic potential at the contact is the applied

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 12-8 shows 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 align the 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

1.46⋅1016 m-3 (T = 300K)

εr (Si)

epsilonr_Si

11.8

εr (SiO2)

epsilonr_Si

4.2

μn(Si)

mun

0.10 m 2/Vs (low concentration)

μ p(Si)

mup

0.05 m 2/Vs (low concentration)

-4

2

Dn = μn(Si) kT/q

Dn

20.7⋅10 m /s

D p = μ p (Si)kT/q

Dp

5.17⋅10-4 m2/s

τn

taun

~0.1 µs

τ p

taup

~0.1 µs

EG(Si)

Eg_Si

1.08 eV

χ Si

X_Si

4.0 eV

χ SiO2

X_SiO2

0.3 eV

χ 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 Poisson’s 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 Discussion The system’s 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

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

Figure 12-9: Drain current as a function of drain voltage for V GS = 0.8 V. Note that R onis 1/ (slope of the curve) at the point where V DS = 0, which falls at the far left of the curve. A surface plot is useful to view the shape of the solution variables. Figure 12-10 shows 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.

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. Enter phi0 in the Dependent variables edit field, and enter init in the Application mode 4name edit field. 5Click Add. 6Enter phi in the Dependent variables edit field. 7Click Add. Select the COMSOL Multiphysics>Diffusion>Convection and Diffusion>Steady-state 8analysis application mode. Enter nc in the Dependent variables edit field, and enter cde in the Application mode 9 name edit field. 10Click Add Click Application mode properties, select Conservative in the Equation form list and 11Ideal in the Weak Constraints list. Click OK . Enter pc in the Dependent variables edit field, and enter cdh in the Application mode 12name edit field. 13Click Add. Click Application mode properties, select Conservative in the Equation form list and 14Ideal in the Weak Constraints list. Click OK . 15Click OK . OPTIONS AND SETTINGS 1From the Options menu, choose Constants.

In the Constants dialog box, define the following constants with names, expressions, and 2descriptions (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 (m 2/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)

x2

2e-7

Help coordinate (m)

ch

yj/sqrt(log( Characteristic length of   NDimpl/NAsub)) 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)

3Click OK . 4From the Options menu, choose Functions. 5In the Functions dialog box, click the New button. In the New Function dialog box, type flguass in the Function name edit field. Click OK to 6create the new function. Type x, y, x1, y1, ch in the Arguments edit field, and type the following in the Expression 7edit field. (exp(-((y-y1)/ch)^2)*(y=y1))*(exp(-((x-x1)/ch)^2)*(x>x1)+(xScalar Expressions. In the Scalar Expressions dialog box, define the following variables with names and 2expressions, 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)

Electron concentration in thermal equilibrium

 p0

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

Hole concentration in thermal equilibrium

R_srh

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

Shockley-ReedHall recombination term

 phi_init

Vt*(-log(p_init/ni)* (Ndoping=0))-X_si -0.5*Eg_si

Initial guess for   phi0

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

3Click OK . Boundary Variables 1From the Options menu, choose Expressions>Boundary Expressions. In the Boundary Expressions dialog box, define the following variables with names and 2expressions: BOUNDARY 6, 19

Jn

q*lm3

Jp

q*lm4

ALL OTHER 

3Click OK . 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 . In the Boundary Integration Variables dialog box, select boundary 19 and then type Id in 2the 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 

εr 

epsilonr

epsilonr 

Ρ

q*(p0n0+Ndoping)

0

4

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

SUBDOMAIN ALL

 phi0(t0)

phi_init

5Click OK . 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 

εr 

epsilonr

epsilonr 

Ρ

q*(pcnc+Ndoping)

0

8

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

SUBDOMAIN ALL

 phi(t0)

phi0

9 Click OK . 10On the Multiphysics menu, select the Convection and Diffusion (cde) application mode. Open the Subdomain Settings dialog box, select subdomains 3, 4, 5, 7, and 11 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

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

13Click OK . 14On the Multiphysics menu, select the Convection and Diffusion (cdh) application mode. 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 15 following settings for the other subdomains SUBDOMAIN 1, 2, 6

D isotropic

Dp

U

mup*phix

V

mup*phiy

R

-R_srh

16

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

SUBDOMAIN 1, 2, 6

 pc(t0) 17Click OK .

p0

Boundary Conditions 1On the Multiphysics menu, select the Electrostatics (init) application mode. From the Physics menu, select Boundary Settings. Select the Interior boundaries check  2 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

Boundary condition

Electric  potential

Electric  potential

Zero charge/Symmetry

V0

phi_init

Vg-X_poly

4Click OK . 5On the Multiphysics menu, select the Electrostatics (es) application mode. Open the Boundary Settings dialog box, select the Interior boundaries 6 check box, and enter the following settings: BOUNDARY 2, 6

19

Boundary condition

Electric Electric  potential  potential

V0

phi_init

11, 12, 17

1, 3, 5, 7, 18, 21 – 23

Electric  potential

Zero charge/ Symmetry

phi_init+Vd VgX_poly

7Click OK . 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

10Click OK . 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

Boundary

ALL OTHER 

Concentration Insulation/Symmetry

condition  pc0

p_init

13Click OK . 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 click  OK . COMPUTING THE SOLUTION 1From the Solve menu, choose Solver Manager 2In the Solver Manager dialog box, click on the Script tab. Click the Solve using a script check box, and enter the following script in the large text 3area. 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;

Click OK and then click the Solve button to start the simulation. Note that the script 4overrides all other settings in the Solver Parameters and Solver Manager dialog boxes. POSTPROCESSING AND VISUALIZATION 1Open the Plot Parameters dialog box. Enter the expression phi as both Surface expression and Height expression on the Surface 2 page of the Plot Parameters dialog box. Remember to select the Height data check box. Click OK to get the plot in the figure below.

3 To plot the drain characteristics open the Domain Plot Parameters dialog box from the 4Postprocessing menu. Choose the Point plot type and check that all parameters are selected in the Parameter 5value list. 6On the Point tab, type Id in the Point expression edit field. 7Select vertex 1 and click  OK . In a separate window you should see the plot below.