COMSOL 6.3 - Superlattice Band Gap Tool

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Superlattice Band Gap Tool

The Superlattice Band Gap Tool model helps the design of periodic structures made of two alternating semiconductor materials (superlattices). The model uses the effective mass Schrödinger equation to estimate the electron and hole ground state energy levels in a given superlattice structure. Device engineers can use the model to quickly compute the effective band gap for a given periodic structure and iterate the design parameters until they reach a desired band gap value.

A superlattice is constructed from multiple thin layers of alternating material compositions. It finds applications in HEMTs, LEDs, lasers, and photodetectors. (Ref. 1)

The superlattice in the model is composed of layers of GaAs and AlGaAs. The envelope function approximation (Chap. 1, Sec. 1.7 in Ref. 2) can be employed to justify the use of the single-particle Schrödinger equation. Note that not only the electron potential energy, but also the effective mass, is a function of material composition. To properly take care of the discontinuity in the effective mass, the physics interface applies the BenDaniel–Duke boundary condition (Chap. 2, Sec. 2.6 in Ref. 2) by default.

Model Definition

The band gap energies of the well and the barrier materials are 1.424 and 2.168 eV, respectively. The conduction band offset is 0.283 eV. The electron effective masses of the well and the barrier materials are 0.063 and 0.71, respectively. The hole effective masses of the well and the barrier materials are 0.51 and 0.76, respectively. The widths of the well and the barrier are both 5 nm. The modeling domain consists of one unit cell.

Two Schrödinger Equation physics interfaces are set up to solve for the electron and hole ground state wave functions and energy levels, in two separate studies, respectively. Periodic boundary condition of the Continuity type is used at the two ends of the unit cell. The solutions from the two studies are combined to compute the effective band gap and to generate the plot of electron and hole wave functions. The Array 1D dataset is used to extend the solutions from one to three unit cells for plotting.

Results and Discussion

Figure 1 summarizes the result by plotting the conduction and valence band edges, and the real and imaginary parts of the electron and hole ground state wave functions. The wave functions are shifted to the corresponding ground state energy levels.

Figure 1: Summary plot for the superlattice, including the conduction and valence band edges, and the real and imaginary parts of the electron and hole ground state wave functions.

The Global Evaluations show that the electron ground state energy level is 0.039 eV from the conduction band edge, and the one for the hole is 0.021 eV. This leads to the effective band gap of 1.484 eV for the superlattice structure.

1. S. M. Sze and K.K. Ng, Physics of Semiconductor Devices, 3rd ed., John Wiley & Sons, Inc., 2007.

2. P. Harrison, Quantum Wells, Wires and Dots, 3rd ed., John Wiley & Sons, Inc., 2009.

Semiconductor_Module/Quantum_Systems/superlattice_band_gap_tool

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select > .

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In the tree, select > .

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The Model Wizard exits and starts the COMSOL Desktop at the Geometry node. We can set the length scale here right away. Then import some model parameters from a text file.

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In the window, under click .

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In the window for , locate the section.

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From the list, choose .

Global Definitions

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In the window, under click .

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In the window for , locate the section.

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

Create the geometry for one unit cell of the superlattice.

Interval 1 (i1)

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In the window, under right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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In the text field, type -lw/2-lb/2.

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In the table, enter the following settings:


Lengths (nm)
lb/2
lw
lb/2

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Add physics for the electrons in the conduction band. Use two separate nodes for the different potential energy and effective mass values in the well and barrier regions. Since we are only interested in the ground state, use the default Continuity option for the Periodic Condition at the exterior boundaries of the unit cell.

Schrödinger Equation e-

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In the window, under click .

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In the window for , type Schrödinger Equation e- in the text field.

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In the text field, type schre.

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Click to expand the section. In the table, enter the following settings:


psie

Effective Mass 1

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In the window, under > click .

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In the window for , locate the section.

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In the meff,e,11 text field, type meb*me_const.

Electron Potential Energy 1

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In the window, click .

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In the window for , locate the section.

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From the Ve list, choose . In the associated text field, type Egb*e_const.

Effective Mass 2

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In the toolbar, click

and choose .

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Select Domain 2 only.

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In the window for , locate the section.

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In the meff,e,11 text field, type mew*me_const.

Electron Potential Energy 2

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In the toolbar, click

and choose .

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Select Domain 2 only.

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In the window for , locate the section.

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From the Ve list, choose . In the associated text field, type -CBO*e_const.

Periodic Condition 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

Add physics for the holes in the valence band in a similar way.

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In the toolbar, click

to open the window.

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Go to the window.

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In the tree, select > .

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Click the button in the window toolbar.

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In the toolbar, click

to close the window.

Schrödinger Equation hole

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In the window for , type Schrödinger Equation hole in the text field.

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In the text field, type schrh.

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Locate the section. From the list, choose .

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


psih

Effective Mass 1

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In the window, under > click .

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In the window for , locate the section.

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In the meff,h,11 text field, type mhb*me_const.

Electron Potential Energy 1

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In the window, click .

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In the window for , locate the section.

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From the Vh list, choose . In the associated text field, type 0.

Effective Mass 2

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In the toolbar, click

and choose .

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Select Domain 2 only.

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In the window for , locate the section.

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In the meff,h,11 text field, type mhw*me_const.

Electron Potential Energy 2

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In the toolbar, click

and choose .

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Select Domain 2 only.

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In the window for , locate the section.

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From the Vh list, choose . In the associated text field, type VBO*e_const.

Periodic Condition 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

Create the mesh.

In the toolbar, click

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In the window, click .

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In the window for , locate the section.

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Click the button.

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

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Step 1: Eigenvalue

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In the window, under click .

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In the window for , locate the section.

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In the text field, type 1.

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In the text field, type (Egb-CBO)[1/V].

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Locate the section. In the column of the table, under , clear the checkbox for .

Use two studies, one for each carrier type. Then combine the solutions to inspect the result.

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In the window, click .

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In the window for , locate the section.

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Clear the checkbox.

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In the toolbar, click

.

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In the toolbar, click

to open the window.

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Go to the window.

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Find the subsection. In the tree, select > .

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

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Click the button in the window toolbar.

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In the toolbar, click

to close the window.

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In the window for , locate the section.

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In the text field, type 1.

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In the text field, type -VBO[1/V].

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In the window, click .

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In the window for , locate the section.

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Clear the checkbox.

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In the toolbar, click

.

In the window, expand the node.

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In the window, expand the > node.

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Right-click > and choose .

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In the window for , locate the section.

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From the list, choose .

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From the list, choose .

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Locate the section. From the list, choose .

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From the list, choose .

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Locate the section. From the list, choose .

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. In the text field, type 3.

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Click to expand the section. Select the checkbox.

Global Evaluation 1

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In the toolbar, click

.

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In the window for , locate the section.

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From the list, choose .

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


Expression	Unit	Description
data1(schre.Ei)/e_const-(Egb-CBO)	V	Conduction band edge shift

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Global Evaluation 2

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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


Expression	Unit	Description
data2(schrh.Ei)/e_const+VBO	V	Valence band edge shift

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Global Evaluation 3

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In the toolbar, click

.

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In the window for , locate the section.

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From the list, choose .

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


Expression	Unit	Description
(data1(schre.Ei)+data2(schrh.Ei))/e_const	V	Effective band gap

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1D Plot Group 1

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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

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Locate the section.

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Select the checkbox. In the associated text field, type x (nm).

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Select the checkbox. In the associated text field, type Energy (eV).

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Locate the section. From the list, choose .

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Right-click and choose .

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

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Locate the section. In the text field, type data1(schre.V)/e_const.

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Select the checkbox. In the associated text field, type Conduction band edge.

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Locate the section. From the list, choose .

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In the text field, type arr1x.

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

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From the list, choose .

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

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From the list, choose .

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Click to expand the section. Select the checkbox.

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From the list, choose .

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In the table, enter the following settings:


Legends
Conduction band edge

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Right-click and choose .

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

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Locate the section. In the text field, type data2(-schrh.V)/e_const.

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In the text field, type Valence band edge.

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Locate the section. From the list, choose .

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


Legends
Valence band edge

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Right-click and choose .

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

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Locate the section. In the text field, type data1(schre.Psi*Egw/4/schre.plot_fac+schre.Ei/e_const).

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In the text field, type e- wave function.

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Locate the section. From the list, choose .

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


Legends
Electron wave function (real part)

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Right-click and choose .

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

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Locate the section. In the text field, type data2(-schrh.Psi*Egw/4/schrh.plot_fac-schrh.Ei/e_const).

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In the text field, type hole wave function.

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


Legends
Hole wave function (real part)

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In the window, right-click and choose .

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In the window for , type Im(psie) in the text field.

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Locate the section. In the text field, type data1(imag(schre.Psi)*Egw/4/schre.plot_fac+schre.Ei/e_const).

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Locate the section. Find the subsection. From the list, choose .

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From the list, choose .

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


Legends
Electron wave function (imag part)

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In the window, right-click and choose .

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In the window for , type Im(psih) in the text field.

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Locate the section. In the text field, type data2(-imag(schrh.Psi)*Egw/4/schrh.plot_fac-schrh.Ei/e_const).

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Locate the section. Find the subsection. From the list, choose .

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From the list, choose .

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


Legends
Hole wave function (imag part)

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In the toolbar, click

.