Interface Trapping Effects of a MOSCAP

This tutorial compares experimental data from the literature with a COMSOL model of a MOSCAP with interface traps (surface states). The Trap-Assisted Surface Recombination feature is used to simulate the effects of the trap charges and the processes of carrier capturing and emitting by the traps. The effect of the fixed charges in the gate oxide is also included. The computed values of the capacitance and conductance as functions of the gate voltage and frequency reproduce the qualitative behavior of the experimental data with comparable magnitudes. The model uses the quasi-Fermi level formulation and shows how to plot quantities such as the trap occupancy as a function of the energy.

Introduction

The metal-silicon-oxide (MOS) structure is the fundamental building block for many silicon planar devices. In a practical interface between the silicon and the oxide, defects and dangling bonds form interface traps (surface states) that can affect the charge transport by capturing and emitting carriers, and can affect the electrostatics via the net surface charge density of the traps. In addition, fixed charges in the oxide also affect the electrostatics. This tutorial constructs a simple 1D model of a MOS capacitor (MOSCAP) to explore these effects and compare with experimental observations.

Model Definition

The 1D MOSCAP model is based on the experimental device (the n-type sample) described in Fig. 14 of Ref. 1. The simulation parameters are taken from the nominal experimental values whenever possible, as detailed below.

The experimental samples were prepared with a 10 μm thick epitaxial layer grown on low-resistivity substrates to minimize the effect of the bulk series resistance. In the model the epilayer is assume to be the same thickness (10 μm), and the substrate is 2 μm thick, assuming the bulk series resistance can be ignored. The oxide thickness is assumed to be 60 nm, in the middle of the experimental range of 50~70 nm. The diameter of the gate is 3.8·10−2 cm as given in the figure caption.

The electron mobility is assumed to be constant 1450 cm2/V/s. The n-doping concentrations in the epilayer and the substrate are then computed from the experimental values of resistivity of 0.75 and 0.005 ohm-cm, respectively.

The oxide dielectric constant is assumed to be 3.9. The oxide capacitance is then computed from its dielectric constant, thickness, and the gate diameter.

The fixed oxide charge density is 9·1011 cm−2 as given by the reference paper.

The trap energy distribution is assumed to be rectangular, with a range of 0.2 eV, centered around the mid-gap. The height of the rectangle is assumed to be 2·1011 cm−2eV−1, as suggested by Fig. 15 of the reference paper. For the capture process, the thermal velocity is assumed to be 107 cm/s, and the cross sections are 1·10-15 cm2 and 2.2·10-16 cm2 for the electrons and holes, respectively, as given on the same page as Fig. 15 of the paper.

The metal work function of the gate is assumed to be 4.5 eV.

Results and Discussion

Figure 1 shows the computed terminal capacitance and the equivalent parallel conductance as functions of the gate voltage to compare with Fig. 23 in Ref. 1. The curves show the same qualitative behavior with comparable magnitudes.

Figure 1: Cm-V and Gp-V curves.

Figure 2 shows the computed terminal capacitance and the equivalent parallel conductance as functions of the small signal frequency. The qualitative behavior of the equivalent parallel conductance compares well with Fig. 25 in Ref. 1 (The paper did not include capacitance in the figure.)

Figure 2: Cm and Gp vs. frequency.

Figure 3 shows the rectangular density of states for the traps.

Figure 3: Density of trap states along the energy axis.

Figure 4 and Figure 5 shows the trap occupancy to gain some insight on the effect of the traps on the Cm-V and Gp-V curves. See the discussions in the section Modeling Instructions for more details.

Figure 4: Trap occupancy in the static case.

Figure 5: Small signal response of the trap occupancy.

Reference

1. E.H. Nicollian and A. Goetzberger, “The Si-SiO2 interface – electrical properties as determined by the metal-insulator-silicon conductance technique,” The Bell System Technical Journal, vol. 46, no. 6, 1967.

Semiconductor_Module/Device_Building_Blocks/moscap_1d_interface_traps

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

We will use the study to obtain the DC condition, and add a step later for the AC small signal analysis.

In the tree, select .

Click

Geometry 1

The Model Wizard exits and starts the COMSOL Desktop at the Geometry node. We can set the length scale to um here right away.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Load parameters and variables from files. The expressions for variables will be yellow-colored because the variable semi.iomega will not be available until the study step is solved.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file moscap_1d_interface_traps_parameters.txt.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file moscap_1d_interface_traps_variables.txt.

Note the use of the lindev operator in the variable definitions to evaluate the complex-valued small-signal amplitude of the terminal charge semi.Q0_2. Note that when plotting a variable defined with the lindev operator, the check box in the plot settings window should be cleared. We will show an example below.

Create line intervals in the geometry to represent the epilayer and the substrate, using the thickness parameters defined above in the node.

Geometry 1

Interval 1 (i1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (µm)
0
t_epi
t_epi+t_sub

Add the default silicon material to the model.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Enter the cross-section area to the main physics node, and select the option for the discretization.

Semiconductor (semi)

In the window, under click .

In the window for , locate the section.

In the A text field, type area_g.

Click to expand the section. From the list, choose .

Create doping for the substrate and the epilayer, using the concentration parameters defined above.

Analytic Doping Model 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

From the list, choose .

In the ND0 text field, type Nd_sub.

Analytic Doping Model 2

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the ND0 text field, type Nd_epi.

Ground the bottom surface of the substrate and add a boundary condition on the top surface of the epilayer, using parameters defined above for the gate properties.

Metal Contact 1

In the toolbar, click

and choose .

Select Boundary 3 only.

Thin Insulator Gate 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

In the V0 text field, type Vg.

Locate the section. In the εins text field, type epsr_ox.

In the dins text field, type t_ox.

In the Φ text field, type phiM.

Add a subnode to the gate for the small signal analysis.

Harmonic Perturbation 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the V0 text field, type Vac.

Use a boundary condition to take into account the fixed charges in the gate oxide.

Surface Charge Density 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

In the ρs text field, type rhos_ox.

Finally use a boundary condition to take into account the effect of the interface traps, which form a continuum of levels around the middle of the band gap.

Trap-Assisted Surface Recombination 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

The option requires one or more subnodes to specify the distribution of the trap energy level(s), in order for the boundary condition to take effect.

Continuous Energy Levels 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the Nt text field, type Nss*Ew0*e_const.

From the list, choose .

In the Ew text field, type Ew0.

In the ΔEtran text field, type Ew0/100.

We can shrink the range of discretization to be the same as the one for the rectangle distribution that we have just specified above, to make more efficient use of the extra dimension, which by default approximates the continuous energy distribution by 25 discrete levels distributed within the range. The number of discrete levels can be adjusted using the input field .

In the Et,min text field, type semi.tasr1.ctb1.Et0-Ew0/2.

In the Et,max text field, type semi.tasr1.ctb1.Et0+Ew0/2.

Locate the section. In the σ < n > text field, type sigma_n.

In the Vnth text field, type v_th.

In the Vpth text field, type v_th.

In the σ < p > text field, type sigma_p.

Set up a user-defined mesh for a better compromise between discretization error and roundoff error.

Mesh 1

Edge 1

In the toolbar, click

Distribution 1

Right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

In the text field, type 100.

Distribution 2

In the window, right-click and choose .

Select Domain 2 only.

In the window for , locate the section.

In the text field, type 20.

Click

In the first study, we fix the small signal analysis frequency at 50 Hz and sweep the gate voltage from accumulation to inversion.

Study 1 - Vg sweep at 50 Hz

In the window, click .

In the window for , type Study 1 - Vg sweep at 50 Hz in the text field.

Step 1: Semiconductor Equilibrium

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Vg (Gate voltage)	range(1,-0.2, -2) range(-2.1,-0.05,-4) range(-4.2,-0.2,-5)	V

Add a study step for the small signal analysis. Make sure to specify the same gate voltage swept values.

Frequency Domain Perturbation

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type f0.

Click to expand the section. Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Vg (Gate voltage)	range(1,-0.2, -2) range(-2.1,-0.05,-4) range(-4.2,-0.2,-5)	V

In the toolbar, click

Results

Energy Levels (semi)

By default for the study step, the plots show the small signal variation of the quantities being plotted. To look at the static solution, either change the drop-down menu in each of the plots, or just change the dataset for the plot group. We will do the latter for simplicity.

In the window for , locate the section.

From the list, choose .

Locate the section. Clear the check box.

In the toolbar, click

Carrier Concentrations (semi)

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. Clear the check box.

In the toolbar, click

The plots of the energy levels and carrier concentrations show that the voltage sweep indeed covers the range from accumulation to inversion.

Electric Potential (semi)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Now create a plot for the measured capacitance and the equivalent parallel conductance to compare with Fig. 23 in the reference paper. The two quantities can have different magnitudes. So we will use two separate Global plots and two y-axes; one for each quantity.

Cm and Gp vs. Vg

In the toolbar, click

and choose .

In the window for , type Cm and Gp vs. Vg in the text field.

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
Cm	pF	Cm

Global 2

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
Gp	nS	Gp

Cm and Gp vs. Vg

In the window, click .

In the window for , locate the section.

Select the check box.

In the table, select the check box for .

Locate the section. From the list, choose .

In the toolbar, click

The peaking of the equivalent parallel conductance and the distinct wiggle of the measured capacitance curve are clearly seen to be qualitatively similar to the figure in the reference paper with comparable magnitudes. As mentioned earlier, since the variables being plotted have been defined using the lindev operator to evaluate the complex-valued small-signal amplitude, the check box in the plot settings window should be cleared, as is the case here by default.

Now create a study to sweep the frequency while keeping the gate voltage constant.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Stationary

By default a step is included when adding the study. Here we can remove the step and set up the solver sequence to use the static solution already computed in the previous study for the small signal analysis. In this case select the solution for the gate voltage of -3 V.

Right-click and choose .

In the window, click .

In the window for , type Study 2 - Freq sweep at -3 V in the text field.

Step 1: Frequency Domain Perturbation

In the window, under click .

In the window for , locate the section.

In the text field, type f0 * 10^range(-2,0.2,3).

Locate the section. Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Vg (Gate voltage)	-3	V

Solution 3 (sol3)

In the toolbar, click

Step 1: Frequency Domain Perturbation

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

Create a plot for the equivalent parallel conductance to compare with Fig. 25 in the reference paper. The measured capacitance will also be plotted for the fun of it.

Results

Cm and Gp vs. Freq

In the window, right-click and choose .

In the window for , type Cm and Gp vs. Freq in the text field.

Locate the section. From the list, choose .

Global 2

In the window, expand the node, then click .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
Gp/(2pifreq)	pF	Gp/\omega

Locate the section. From the list, choose .

Global 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

In the toolbar, click

To gain some insight on the behavior of the capacitance and conductance shown in the plots above, it helps to plot the trap occupancy along the energy axis. The energy axis is added to the model by the physics interface using an extra dimension component. To plot any quantity along the energy axis, we need to create a dataset pointing to the extra dimension component where the quantity is defined. We will first plot the density of states of the traps as an example.

Study 1 - Vg sweep at 50 Hz/Solution 1 XD

In the window, expand the node.

Right-click and choose .

In the window for , type Study 1 - Vg sweep at 50 Hz/Solution 1 XD in the text field.

Locate the section. From the list, choose .

Density of trap states vs. Energy

In the toolbar, click

and choose .

In the window for , type Density of trap states vs. Energy in the text field.

Locate the section. From the list, choose .

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type atxd0(0[um],(semi.tasr1.ctb1.gt/(e_const*Ew0))).

In the field, type 1/(cm^2*eV).

Select the check box. In the associated text field, type Density of trap states.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type atxd0(0[um],semi.tasr1.ctb1.Vxd-semi.tasr1.ctb1.Et0)*e_const.

From the list, choose .

Select the check box. In the associated text field, type Energy.

From the list, choose .

In the toolbar, click

Note that the operator atxd0 is used with a "0" because the energy axis (extra dimension) is defined in a boundary condition, which has a dimension of 0 in the 1D model. The first argument for the operator is 0[um] since the boundary condition is applied on the boundary located at 0 um. The x-axis for the energy is centered around the center of the trap energy level distribution. The density of trap states shows a rectangular distribution of 2e11[cm^-2*eV^-1] as specified. The option of is selected for both the x- and y-axes.

Now we plot the trap occupancy along the energy axis and compare two cases: gate voltage = 1 V (accumulation) and gate voltage = -3 V (peak of equilibrium parallel conductance). First the static solution.

Trap occupancy, static