COMSOL 6.4 - Heterojunction 1D

Heterojunction 1D

This benchmark model simulates three different heterojunction configurations under forward and reverse bias. It shows the difference in using the continuous quasi-Fermi level formulation versus the thermionic emission formulation for the charge transfer across the heterojunction. The simulated energy levels are compared between each configuration in order to illustrate the origin of the charge transfer, that is, whether it is primarily from holes in the valence band or from electrons in the conduction band. The computed I–V curves for each configuration are compared with results from the literature. Several methods for better convergence are demonstrated in the setup of the various study steps.

Introduction

Semiconductor heterojunctions are interfaces where two different semiconducting materials are in contact. There are two types of heterojunctions:

•

Isotypes: same type of majority carrier on both side of the junction, for example, n–n or p–p.

•

Anisotypes: the majority carrier are electrons on one side and hole on the other, for example, n–p or p–n.

Depending on the configuration, heterojunctions show different behaviors depending on the material band-gap difference and doping type.

Table 1: Heterojunction configurations.
Doping material1	Doping material 2	Band-gap difference Eg2-Eg1	Anode contact side	Cathode contact side	Current mostly carried by
n-doped	n-doped	>0	material 1	material 2	electrons
p-doped	p-doped	>0	material 2	material 1	holes
n-doped	n-doped	<0	material 2	material 1	electrons
p-doped	p-doped	<0	material 1	material 2	holes
p-doped	n-doped	>0	material 1	material 2	electrons
n-doped	p-doped	>0	material 2	material 1	holes
p-doped	n-doped	<0	material 1	material 2	holes
n-doped	p-doped	<0	material 2	material 1	electrons

Heterojunctions are characterized by a discontinuous conduction (and/or valence) band at the interface. In the Semiconductor Module, one can choose between two models to define the heterojunction. The continuous quasi-Fermi level model and the thermionic emission model.

The continuous quasi-Fermi level model defines the current continuity at the interface by setting the quasi-Fermi level to be continuous on each side of the junction.

The thermionic emission model defines the current continuity through a thermionic current density generated at the potential barrier.

Model Definition

This model simulates the behavior of an heterojunction under reverse and forward bias for three different configurations: isotype n–n, anisotype p–n, and anisotype n–p. The modeled junction is composed of two different materials, GaAs and Al0.25Ga0.75As, respectively located on the left and right side of the junction. As displayed on Figure 1, the junction has a length of 10 μm and a net doping concentration of: 1·1015 cm−3 for the GaAs and 1·1016 cm−3 for the Al0.25Ga0.75As. A Shockley–Read–Hall recombination feature is also added to the model for the sake of comparison with the reference model reported in Ref. 1.

Figure 1: Schematic of the geometry. The GaAs and Al0.25Ga0.75As are, respectively, located on the left and on the right side of the junction (center). Depending on the configuration (n–n, p–n, or n–p), the doping might be of donor type (Nd) or acceptor type (Na) for both materials. Note that the impurity concentration stays the same for each material for all three configurations, that is, 1·1015 cm-3 for GaAs and 1·1016 cm-3 for Al0.25Ga0.75As.


	The equation system for heterojunctions is highly nonlinear and numerically challenging. In this tutorial we show a few different ways to achieve better convergence. See the section Modeling Instructions for details.

Results and Discussion

Figure 2 shows a comparison between the continuous quasi-Fermi level model and the thermionic emission model for the n–n isotype junction under forward and reverse bias. The figure also compares our results with the ones obtained in Ref. 1. Figure 2 shows that the thermionic current model is in good agreement with the reference. The continuous quasi-Fermi level model is a little larger than the reference model in the reverse case.

Figure 2: Comparison of the current densities obtained using the continuous quasi-Fermi level model (green) and the thermionic emission model (blue) with the results obtained in Ref. 1 for the n–n isotype junction under forward and reverse bias.

Figure 3: Comparison of the current densities obtained using the thermionic emission model (blue) with the results obtained in Ref. 1 for the p–n anisotype junction under forward and reverse bias.

Figure 3 shows a comparison between the thermionic emission current density obtained with the model and the reference under forward and reverse bias for the p–n junction. A very good agreement is observed between the two models. Figure 4 shows the same kind of agreement for the n–p junction.

Figure 4: Comparison of the current densities obtained using the thermionic emission model (blue) with the results obtained in Ref. 1 for the n–p anisotype junction under forward and reverse bias

The thermionic current at the junction can come from the electrons in the conduction band or the holes in the valence band. Depending on the configuration, the current is primarily due to one of the two carriers. This is the result of the existence (or nonexistence) of a potential barrier generated by the bending of one of the bands (conduction or valence). Figure 5 displays the energy diagram for the n–n junction under forward bias at Va = 0.05 V. The figure shows that, for this configuration, it is the conduction band that forms the barrier (spike) over which electrons can transit. The current for the isotype heterojunction is then mainly due to electrons. A similar observation can be made on Figure 6 and Figure 7. The latter figures show that the current is mainly produced by electrons for the p–n heterojunction and by holes for the n–p heterojunction. Note that for the n–p junction, the forward bias appears when the applied potential is negative.

Figure 5: Energy diagram for the n–n heterojunction using the thermionic emission model.

Figure 6: Energy diagram for the p–n heterojunction using the thermionic emission model.

Figure 7: Energy diagram for the n–p heterojunction using the thermionic emission model.

Reference

1. K. Horio and H. Yanai, “Numerical Modeling of Heterojunctions Including the Thermionic Emission Mechanism at the Heterojunction Interface,” IEEE Transaction on Electron Devices, vol. 37, no. 4, pp. 1093–1098, 1990.

Semiconductor_Module/Verification_Examples/heterojunction_1d

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
sign	1	1	Sign of applied voltage
Va	0[V]	0 V	Voltage
ramp	1	1	Continuation parameter
L	5[um]	5E-6 m	Length
T0	300[K]	300 K	Temperature

Create the geometry, the junction is located at x = 0.

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 (m)
-L
0

Interval 2 (i2)

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (m)
0
L

Choose um as the default length units.

In the window, click .

In the window for , locate the section.

From the list, choose .

Load the material properties for GaAs and Al_0.25Ga_0.75As. The properties will be respectively assigned to the left and right side of the junction.

Definitions

GaAs

In the toolbar, click

In the window for , type GaAs in the text field.

Locate the section. From the list, choose .

Select Domain 1 only.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file heterojunction_1d_gaas.txt.

Al_0.25Ga_0.75As

In the toolbar, click

In the window for , type Al_0.25Ga_0.75As in the text field.

Locate the section. From the list, choose .

Select Domain 2 only.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file heterojunction_1d_algaas.txt.

Set the reference temperature to T0.

Click the

button in the toolbar.

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

Click .

Semiconductor (semi)

In the window, under click .

In the window for , click to expand the section.

In the T0 text field, type T0.

Set the lattice temperature to T0.

Semiconductor Material Model 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type T0.

Add a Shockley-Read-Hall recombination feature to all domains.

Trap-Assisted Recombination 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The voltage is applied on the left side of the junction (GaAs). The sign variable is used by the study steps to sweep the applied voltages from reverse to forward.

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 sign*Va.

The right side of the junction (Al_0.25Ga_0.75As) is grounded.

Metal Contact 2

In the toolbar, click

and choose .

Select Boundary 3 only.

The doping concentration stays constant for each material and for any modeled configuration. Accordingly, the variable Doping, defined in the domain variables GaAs and Al_0.25Ga_0.75As, is used for all the added doping feature. Depending on the chosen configuration, the doping type changes on each side of the junction. For this reason, n- and p-type features are created for each side of the junction. To define the desired configurations, doping features are enabled/disabled in their respective study steps.

Donor doping (left)

In the toolbar, click

and choose .

In the window for , type Donor doping (left) in the text field.

Select Domain 1 only.

Locate the section. From the list, choose .

In the ND0 text field, type Doping.

Donor doping (right)

In the toolbar, click

and choose .

In the window for , type Donor doping (right) in the text field.

Select Domain 2 only.

Locate the section. From the list, choose .

In the ND0 text field, type Doping.

Acceptor doping (left)

In the toolbar, click

and choose .

In the window for , type Acceptor doping (left) in the text field.

Select Domain 1 only.

Locate the section. In the NA0 text field, type Doping.

Acceptor doping (right)

In the toolbar, click

and choose .

In the window for , type Acceptor doping (right) in the text field.

Select Domain 2 only.

Locate the section. In the NA0 text field, type Doping.

In order to observe the effect of the different heterojunction models, create an additional feature and change the default continuous quasi-Fermi model for the thermionic emission model.

Continuity/Heterojunction 2

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

Add a material to the model and enter the material properties defined in the domain variables GaAs and Al_0.25Ga_0.75As. This discontinuous material is used to define both sides of the junction.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electron mobility	mun	mun0	m²/(V·s)	Semiconductor material
Hole mobility	mup	mup0	m²/(V·s)	Semiconductor material
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	er	1	Basic
Band gap	Eg0	Eg	V	Semiconductor material
Electron affinity	chi0	chi	V	Semiconductor material
Effective density of states, conduction band	Nc	Nc0	1/m³	Semiconductor material
Effective density of states, valence band	Nv	Nv0	1/m³	Semiconductor material
Electron lifetime, SRH	taun	taun0	s	Shockley-Read-Hall recombination
Hole lifetime, SRH	taup	taup0	s	Shockley-Read-Hall recombination