COMSOL 6.4 - Heterojunction Tunneling

In the tree, select > .

Load some global parameters from a text file. These parameters are independent of space and time in a model.

Geometry 1

The model wizard brings us to the node in the Model Builder tree structure. We can use this opportunity to set the length unit to µm.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Create the geometry of three rectangular domains according to Fig. 2(b) in the reference paper.

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

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type d1+d2+d3.

In the text field, type d2.

Locate the section. In the text field, type -d1.

Click to expand the section. Clear the checkbox.

Select the checkbox.


Layer name	Thickness (µm)
Layer 1	d1
Layer 2	d2

Define local variables and operators. The variables can be spatially varying and can be grouped into different nodes for convenience.

First, use three separate nodes for the n doping and mole fraction of Al in each of the three domains.

Definitions

Variables for Domain 1

In the window, under right-click and choose .

In the window for , type Variables for Domain 1 in the text field.

Locate the section. From the list, choose .

Select Domain 1 only.

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


Name	Expression	Unit	Description
N_D	N_D1	1/m³	n doping
Al_frac	0		Mole fraction of Al

Variables for Domain 2

Right-click and choose .

In the window for , type Variables for Domain 2 in the text field.

In the table, change the definition of N_D as shown below, and leave Al_frac unchanged.

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


Name	Expression	Unit	Description
N_D	N_D2	1/m³	n doping
Al_frac	Al_frac0*(1-x/d2)		Mole fraction of Al

Variables for Domain 3

In the window, right-click and choose .

In the window for , type Variables for Domain 3 in the text field.

Select Domain 3 only.

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


Name	Expression	Unit	Description
N_D	N_D3	1/m³	n doping

Define material properties according to the reference paper. The electron effective mass will be needed when we define variables for the tunneling effect later.

Materials

Al(x)Ga(1-x)As (Yang et al 1993)

In the window, under right-click and choose .

In the window for , type Al(x)Ga(1-x)As (Yang et al 1993) in the text field.

First define local material variables for the mole fraction and the effective masses.

In the window, expand the > > node, then click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
x	Al_frac		Mole fraction of Al
me	(0.067+0.083def.x)me_const	kg	Electron DOS effective mass
mh	(0.48+0.31def.x)me_const	kg	Hole DOS effective mass

Note that these variables are accessed with the scope name def, for example def.x, because the tag of the node is def.

Then enter the material properties using formulas and descriptions given in the reference paper.

In the window, under > click .

In the window for , locate the section.

Before defining the variables for the tunneling effect, we need to prepare the coordinate systems.


Property	Variable	Value	Unit	Property group
Electron mobility	mun	7200[cm^2/V/s]/(1+5.51e-17[cm^3](N_D+N_A))^0.233(T0/T)^2.3(1-0.127def.x/0.1)	m²/(V·s)	Semiconductor material
Hole mobility	mup	380[cm^2/V/s]/(1+3.17e-17[cm^3](N_D+N_A))^0.266(T0/T)^2.7(1-0.067def.x/0.1)	m²/(V·s)	Semiconductor material
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	13.1-3*def.x	1	Basic
Band gap	Eg0	1.519[V]+1.247[V]def.x-5.405e-4[V/K]T^2/(T+204[K])	V	Semiconductor material
Electron affinity	chi0	4.07[V]-0.61.247[V]def.x	V	Semiconductor material
Effective density of states, conduction band	Nc	2((def.mek_B_constT)/(2pi*hbar_const^2))^1.5	1/m³	Semiconductor material
Effective density of states, valence band	Nv	2((def.mhk_B_constT)/(2pi*hbar_const^2))^1.5	1/m³	Semiconductor material

In this example, domain 2 happens to cover the region of interest for the line integration across the potential barrier. In general, the region of interest can be defined by different boundaries drawn in the geometry, which may or may not coincide with the material boundaries.

First, set up the coordinate along the heterojunction using the first mathematical interface.

Curvilinear Coordinates (cc)

In the window, under click .

In the toolbar, click

and choose .

Inlet 1

In the toolbar, click

and choose .

Select Boundary 6 only.

Curvilinear Coordinates 2 (cc2)

In the window, click .

Outlet 1

In the toolbar, click

and choose .

Select Boundary 5 only.

Next, set up the coordinate along the field lines using the second mathematical interface. Here we assume the field lines are perpendicular to the heterojunction, which should be a good approximation when the tunneling effect is significant.

In the window, under click .

In the toolbar, click

and choose .

Inlet 1

In the toolbar, click

and choose .

Select Boundary 7 only.

Give the coordinates convenient names for subsequent definitions.

In the window, click .

Outlet 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Definitions

Variables for tunneling

In the window, under right-click and choose .

In the window for , type Variables for tunneling in the text field.

Locate the section. From the list, choose .

Set up the device physics. Since the tunneling effect is highly sensitive to the shape of the potential barrier, we switch to the finite element quasi-Fermi level formulation. The default finite volume formulation would require much finer mesh, due to its discontinuous nature.

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


Name	Expression	Unit	Description
pbcc	cc.diff1.U	m	Normalized arc length along the edge of the potential barrier
flcc	cc2.diff1.U	m	Normalized arc length along the field line

Semiconductor (semi)

In the window, under click .

In the window for , locate the section.

In the d text field, type thickness.

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

Enable to unhide and set the reference temperature.

Click the

button in the toolbar.

In the dialog, in the tree, select the checkbox for the node > .

Click .

In the window for , click to expand the section.

In the T0 text field, type T0.

Set up the lattice temperature and incomplete ionization according to the reference paper. Note that the paper does not use a consistent formula for the donor activation energy for all examples, and here we take the values for the example in section 3.3 of the paper which we reproduce in this tutorial.

Semiconductor Material Model 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type T.

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

In the ΔEd ≡ Ec − Ed text field, type 0.005[V]+0.03[V]*material.def.x/Al_frac0.

Note the use of the prefix material in the expression so that the software will use the material properties assigned to each domain, in the same way that the option of works in many lists for material properties in the user interface.

In the ΔEa ≡ Ea − Ev text field, type 0.026[V].

Set the model of the heterojunction to thermionic emission. This heterojunction boundary condition will be used to compute the case of no tunneling, to be compared with the case that includes tunneling.

Continuity/Heterojunction 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Duplicate the heterojunction boundary condition to add extra current contribution from the tunneling effect at the abrupt junction (the left heterojunction). In this example, only electron tunneling is significant.

Continuity/Heterojunction 2

Right-click > > and choose .

In the window for , locate the section.

WKB Tunneling Model, Electrons 1

Select Boundary 4 only.

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

Add a subnode to set up the tunneling effect.

In the toolbar, click

and choose .

As mentioned earlier, domain 2 covers the region of interest for the line integration across the potential barrier.

In the window for , locate the section.

Click to select the

toggle button.

Select the boundary that is on the other side of domain 2 from the tunneling boundary.

Locate the section. Click to select the

toggle button.

Select Boundary 7 only.

The electron effective mass is defined in the material properties.

Locate the section. In the m text field, type material.def.me.

The field line and tunneling boundary coordinates have been defined earlier as variables in terms of the curvilinear coordinates.

In the flc text field, type flcc.

In the bndc text field, type pbcc.

Add doping and SRH recombination. The latter requires more material parameters.

Analytic Doping Model 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the ND0 text field, type N_D.

Trap-Assisted Recombination 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Materials

Al(x)Ga(1-x)As (Yang et al 1993) (mat1)

In the window, under > click .

In the window for , locate the section.

Adjust mesh to save computation time. Mesh refinement studies are always recommended.


Property	Variable	Value	Unit	Property group
Electron lifetime, SRH	taun	1[ns]	s	Shockley-Read-Hall recombination
Hole lifetime, SRH	taup	1[ns]	s	Shockley-Read-Hall recombination

Add metal contacts.

Semiconductor (semi)

Metal Contact 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

In the V0 text field, type Va.

Metal Contact 2

In the toolbar, click

and choose .

Select Boundary 10 only.

Mesh 1

In the window, under right-click and choose .

Size 1

In the window, under > click .

Select Boundary 4 only.

First solve the case of no tunneling effects. Sweep the applied voltage to obtain the I-V curve.

Study 1: no tunneling

In the window, click .

In the window for , type Study 1: no tunneling in the text field.

Step 1: Stationary

For this study we only solve for the physics, so leave it enabled and disable the two interfaces.

In the window, under click .

In the window for , locate the section.

In the column of the table, under , clear the checkboxes for and .

Select the checkbox.

In the tree, select > > .

Click to expand the section. Select the checkbox.


Parameter name	Parameter value list	Parameter unit
Va (Applied voltage)	range(-1.2,0.1,-0.01) -0.05 range(0.025,0.025,0.25)	V

In the toolbar, click

The model has only n-type dopant therefore, the p-type part of the default generated plot is not needed.

P-Type

In the window, expand the node.

Right-click and choose .

Plot the I-V curve for the no tunneling case. Set the scale and range to be the same as Fig. 12 in the reference paper for ease of comparison.

I-V at 300 K

In the toolbar, click

In the window for , type I-V at 300 K in the text field.

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

In the text area, type I-V at 300K.

Locate the section.

Select the checkbox. In the associated text field, type Current density (104A/cm2).

Locate the section. Select the checkbox.

In the text field, type -1.2.

In the text field, type 0.4.

In the text field, type 2.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

Solve the equations for the curvilinear systems before solving the case with the tunneling effect. In general, the curvilinear coordinates provide a good approximation to the field lines and equipotential contours where the tunneling effect is significant.


Expression	Unit	Description
semi.I0_1/d2/thickness/1e4[A/cm^2]	1	no tunneling

In the toolbar, click

Study 2: curvilinear coordinates

In the toolbar, click

to open the window.

Go to the window.

For this study we only solve for the two interfaces, so leave them enabled and disable the physics.

Find the subsection. In the table, clear the checkbox for .

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

In the window for , type Study 2: curvilinear coordinates in the text field.

In the toolbar, click

Plot the curvilinear coordinates, which in general provide a good approximation to the field lines and equipotential contours where the tunneling effect is significant.

Curvilinear coordinates

In the toolbar, click

In the window for , type Curvilinear coordinates in the text field.

Locate the section. From the list, choose .

Contour 1

Right-click and choose .

In the window for , locate the section.

In the text field, type pbcc.

Locate the section. In the text field, type 5.

Contour 2

Right-click and choose .

In the window for , locate the section.

In the text field, type flcc.

Locate the section. In the text field, type 8.

Contour 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Now solve the case with tunneling effects. Since we will only solve the semiconductor physics, not the curvilinear systems, we use the settings to provide a good initial condition for the semiconductor physics from Study 1, but we use the settings to provide the curvilinear coordinates from the solution of Study 2. Sweep the applied voltage to obtain the I-V curve.

In the toolbar, click

to open the window.

Go to the window.

For this study we only solve for the physics, so leave it enabled and disable the two interfaces.

Find the subsection. In the table, clear the checkboxes for and .

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 3: tunneling

In the window for , type Study 3: tunneling in the text field.

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Find the subsection. From the list, choose .

From the list, choose .

Find the subsection. From the list, choose .

From the list, choose .

Locate the section. Select the checkbox.


Parameter name	Parameter value list	Parameter unit
Va (Applied voltage)	range(-1.2,0.1,-0.01) -0.05 range(0.025,0.025,0.25)	V

In the toolbar, click

Net Dopant Concentration (semi) 1

In the window, under right-click and choose .

Add the tunneling result to the I-V curve plot. This matches well with Fig. 12 in the reference paper.

Global 2

In the window, under > right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
semi.I0_1/d2/thickness/1e4[A/cm^2]	1	tunneling

In the toolbar, click

Add a 1D plot group to plot the band diagram at two bias points -0.5 V and 0.15 V. This compares well with Fig. 13 in the reference paper.

Conduction band profiles

In the toolbar, click

In the window for , type Conduction band profiles in the text field.

Locate the section. From the list, choose .

From the list, choose .

To select the two bias points -0.5 V and 0.15 V, hold down the Control key to enable multiple selection by mouse clicking.

In the list, choose and .

Line Graph 1

Right-click and choose .

Select Boundaries 2, 5, and 8 only.

In the window for , locate the section.

In the text field, type semi.Ec.

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

From the list, choose .

Click to expand the section. Select the checkbox.

Find the subsection. Select the checkbox.

Line Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type semi.Efn.

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

From the list, choose .

Conduction band profiles

In the window, click .

In the window for , locate the section.

Select the checkbox.

In the text field, type 0.1.

In the text field, type 0.4.

Locate the section. From the list, choose .

In the toolbar, click

Now add two studies for the low-temperature cases, to compare with Fig. 14 in the reference paper. First create an empty study and copy over the study step for the nontunneling case from .

Study 4: no tunneling, lower Ts

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

In the window for , type Study 4: no tunneling, lower Ts in the text field.

Study 1: no tunneling

Step 1: Stationary

In the window, under right-click and choose .

Study 4: no tunneling, lower Ts

In the window, right-click and choose .

The very nonlinear equation system often needs to be parametrically swept from a favorable initial condition. In this case, we found it easier for convergence to start the sweep from the other end of the parameter range, with the corresponding solution from as the initial condition.

In the window for , locate the section.

Add a parametric sweep node for the two low-temperature cases, with the Al fraction adjusted for each case according to the reference paper.


Parameter name	Parameter value list	Parameter unit
Va (Applied voltage)	-0.05 range(-0.1,-0.1,-1.2)	V

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

From the list, choose .

Parametric Sweep

In the toolbar, click

In the window for , locate the section.


Parameter name	Parameter value list	Parameter unit
T (Lattice temperature)	200 180	K


Parameter name	Parameter value list	Parameter unit
Al_frac0 (Mole fraction of Al at heterojunction)	0.27 0.28

In the toolbar, click

Net Dopant Concentration (semi) 1

In the window, under right-click and choose .

Add a 1D plot group to plot the I-V curves at the three temperatures for the nontunneling cases computed so far.

I-V at different Ts

In the toolbar, click

In the window for , type I-V at different Ts in the text field.

Global 1

Right-click and choose .

In the window for , locate the section.


Expression	Unit	Description
-semi.I0_1/d2/thickness	A/cm^2	T=300 K, Al_frac0=0.30, no tunneling

Locate the section. From the list, choose .

In the text field, type -Va.

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

I-V at different Ts

In the window, click .

In the window for , locate the section.

From the list, choose .

In the list, choose , , , , , , , , , , , , and .

Locate the section. From the list, choose .

In the text area, type I-V at 300, 200, and 180 K.

Locate the section.

Select the checkbox. In the associated text field, type Current density (A/cm2).

Global 2

In the window, under > right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Expression	Unit	Description
-semi.I0_1/d2/thickness	A/cm^2	no tunneling

Click the

button in the toolbar.

In the toolbar, click

Create another empty study and copy the study steps from to modify for the tunneling case. Similar to Study 3, here we use the settings to provide a good initial condition for the semiconductor physics from Study 4, but we use the settings to provide the curvilinear coordinates from the solution of Study 2.