Band-edge diagrams for strained III–V semiconductor quantum wells, wires, and dots

We have calculated band-edge energies for most combinations of zinc blende AlN, GaN, InN, GaP, GaAs, InP, InAs, GaSb, and InSb in which one material is strained to the other. Calculations were done for three different geometries (quantum wells, wires, and dots) and mean effective masses were computed in order to estimate confinement energies. For quantum wells, we have also calculated band-edges for ternary alloys. Energy gaps, including confinement, may be easily and accurately estimated using band energies and a simple effective mass approximation, yielding excellent agreement with experimental results. By calculating all material combinations we have identified interesting material combinations, such as artificial donors, that have not been experimentally realized. The calculations were perfomed using strain-dependent $\mathbf{k}\bullet \mathbf{p}$ theory and provide a comprehensive overview of band structures for strained heterostructures.

Figure 1

(Color online) (a) The lens-shaped geometry used for all quantum dot calculations. The dot height is $\frac{1}{4}$ the diameter of sphere out of which the lens is cut. The $\mathit{z}$-axis is along [001]. (b) Band structure as a function of position along an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ dot’s axis of symmetry. The energies are calculated for $k=0$ using the local value of the strain. (c) Histogram of the conduction and valence band edges including each point in the dot. (d) Histogram as a shaded density, as shown in Figs. 3, 4, 5, 6, 7, 8, 9, 10.

Figure 2

(Color online) Confinement energy as a function of the barrier height $V_0$, the effective mass $m^{*}$ (in the well and barrier material), and the dimension of the structure, $L$. For $V_0=1\mathrm{eV}$, $m^{*}=m_e$, and $L=1\mathrm{nm}$, $U=26.247$. There is always a bound state for the 1D and 2D cases, but for 3D a minimum potential strength is required.

Figure 3

(Color online) Band energies vs composition for dot, wire, and well structures on an AlN substrate. (All nitrides are zinc blende.) The two long lines spanning the graph indicate the conduction and valence band energies for the unstrained substrate material. The lines of medium length indicate the unstrained valence and conduction energies for different well materials. The short lines are histograms of the valence and conduction energies for dots, wires, and wells of the indicated composition with strain effects included. The shading of the short lines is proportional to the volume of material with the indicated energy (see Fig. 1). Energies are calculated for $k=0$ using the local value of the strain, and the histograms include all points in the well material.

Figure 4

(Color online) Band edge diagram of strained dots, wires, and wells on GaN. All nitrides are zinc blende (see Fig. 3 for details).

Figure 5

(Color online) Band edge diagram of strained dots, wires, and wells on InN (see Fig. 3 for details).

Figure 6

(Color online) Band edge diagram of strained dots, wires, and wells on GaAs (see Fig. 3 for details).

Figure 7

(Color online) Band edge diagram of strained dots, wires, and wells on InP (see Fig. 3 for details).

Figure 8

(Color online) Band edge diagram of strained dots, wires, and wells on InAs (see Fig. 3 for details).

Figure 9

(Color online) Band edge diagram of strained dots, wires, and wells on GaSb (see Fig. 3 for details).

Figure 10

(Color online) Band edge diagram of strained dots, wires, and wells on InSb (see Fig. 3 for details). For all well materials the gaps are negative, with the state containing mostly valence character being higher in energy.

Figure 11

Band diagram of an $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ dot along the [001] direction through the center of the dot. Electrons are strongly confined in the InAs dot, while holes see a strain induced potential well in the InSb barrier adjacent to the dot.

Figure 12

(Color online) Band edge diagram of alloyed strained wells on GaP. The $x$ axis is the lattice constant of the ternary alloy comprising the quantum well. Dotted lines are the valence band energies, solid lines are the conduction band energies, and the lighter lines are the energies for unstrained materials.

Figure 13

(Color online) Band edge diagram of alloyed strained wells on GaAs.

Figure 14

(Color online) Band edge diagram of alloyed strained wells on InP.

Figure 15

(Color online) Band edge diagram of alloyed strained wells on InAs.

Figure 16

(Color online) Band edge diagram of alloyed strained wells on GaSb.

Figure 17

(Color online) Band edge diagram of alloyed strained wells on InSb.