Band offset formation at semiconductor heterojunctions through density-based minimization of interface energy

It is well known that the magnitude of band offset (BO) at any semiconductor heterojunction is directly derivable from the distribution of charge at that interface and that the latter is decided by a minimization of total energy. However, the fact that BO formation is governed by energy minimization has not been explicitly used in theoretical BO models, likely because the equilibrium charge densities at heterojunction interfaces appear difficult to predict, except via explicit calculation. In this paper, electron densities at a large number of (100), (110), and (111) oriented heterojunctions between lattice-matched, isovalent semiconductors with the zinc blende (ZB) structure have been calculated by first-principles methods and analyzed in detail for possible common characteristics among energy-minimized densities. Remarkably, the heterojunction electron density was found to largely depend only on the immediate, local atomic arrangement. In fact, it is so much so that a juxtaposition of local electron-densities generated in oligo-cells (LEGOs) accurately reproduced the charge densities that minimize the energy for the heterojunctions. Furthermore, the charge distribution for each bulk semiconductor was found to display a striking separability of its electrostatic effect into two neutral parts, associated with the cation and the anion, which are approximately transferrable among semiconductors. These discoveries form the basis of a neutral polyhedra theory (NPT) that approximately predicts the equilibrium charge density and BO of relaxed heterojunctions from the energy minimization requirement. Well-known experimentally observed characteristics of heterojunctions, such as the insensitivity of BO to heterojunction orientation and the identity of interface bonds, the transitivity rule, etc., are all in good agreement with the NPT. Therefore, energy minimization, which essentially decides the electronic properties of all other solid and molecular systems, also governs the formation of the charge density at these heterojunction interfaces. In particular, the approach presented here eliminates the need to invoke mechanisms that are specific to semiconductor interfaces.

Figure 1

Potential energy and energy band diagram at a semiconductor heterojunction interface. The lateral ($-e{V}_{\mathrm{Coul}.}$) and macroscopic ($-e\overline{V_{\mathrm{Coul}.}}$) averages of the electrostatic potential energy (see text for definition) are shown as solid and dashed thick lines, respectively. The internal ionization potential, μ, is the energy difference between the VBM and the macroscopically averaged electrostatic potential energy. $e{\mathrm{\Delta }}_{\mathrm{ISR}}$ is the difference between the bulk values of $-e\overline{V_{\mathrm{Coul}.}}$. $E_g^{A/B}$ is the bandgap of semiconductor A or B, and ${\mathrm{\Phi }}_{\mathrm{VBO}}^{A-B}$ is the valence band offset between the two semiconductors.

Figure 2

Ball-and-stick models of structures used in the calculations. Larger and smaller spheres represent anions and cations, respectively. (a) (100) supercell, (b) (110) supercell, and (c) (111) supercell. The outlines of the repeating unit used for the calculation, which contains 12 anions and 12 cations for each interface orientation, are marked.

Figure 3

Ball-and-stick models of the zinc-blende lattice, with cations and anions displayed in blue and yellow colors, respectively. (a) Anion-centered Wigner-Seitz (AWS) cell, (b) cation-centered Wigner-Seitz (CWS) cell, (c) quadrant, and (d) neutral polyhedron on the cation site.

Figure 4

Lateral two-dimensional (2D) average and macroscopic three-dimensional (3D) average of two types of potential energy, plotted along the length of a 12-layer GaAs/AlAs(100) supercell. The local potential energy ($-e{V}_{\mathrm{loc}}$) contains pseudopotential contributions. The electrostatic potential energy (EPE) ($-e{V}_{\mathrm{Coul}.}$) is the electrostatic (Coulomb) potential energy calculated from the pseudocharge density output. In the GaAs region of the supercell (left), the VBM position is computed and shown as a thin solid line, using the average $-e\overline{V_{\mathrm{loc}}}$ in this region and the known difference VBM $-e\overline{V_{\mathrm{loc}}}$ for bulk GaAs. Independently, the VBM in the GaAs region is also determined and shown as a coarse dotted line, using the average EPE ($-e\overline{V_{\mathrm{Coul}.}}$) in this region and relative position established for bulk GaAs. Similarly, the VBM is determined for the AlAs region (right) of the supercell with both methods, using relative positions established for bulk AlAs. Both methods yield indistinguishable VBM positions.

Figure 5

(a) Electron density, per nm and per area of the two-dimensional (2D) unit cell ($0.16\mathrm{n}{\mathrm{m}}^2$), of a GaAs/AlAs(100) supercell. (i) Obtained by ab initio calculation; (ii) from stitching bulk GaAs and bulk AlAs densities at the interface As plane; (iii) from stitching bulk GaAs and AlAs densities terminated at CWS boundaries; and (iv) from stitching the bulk GaAs AWS density, the bulk AlAs AWS density, and the density of an AWS cell from a “1 1” $\mathrm{GaAlA}{\mathrm{s}}_2\left(100\right)$ cell (see text for details). (b) Difference between the equilibrium electron density calculated for a GaAs/AlAs(100) supercell and that constructed for the heterojunction by the various stitching methods. (c) Schematic illustration of two stitching methods for the simulation of charge density at the GaAs/AlAs(100) heterojunction. The atomic arrangement of the heterojunction is drawn in the center diagram, with As shown as large spheres, Al as small spheres, and Ga as midsized spheres. In the CWS-stitching method shown on the right, the volume of the heterojunction is divided into CWSs, and the charge density is stacked CWS-by-CWS with charge densities calculated for “a” bulk AlAs and “b” bulk GaAs. In the “1 1”-stitching method shown on the left, the volume of the heterojunction is divided into AWSs, and the charge density is stacked AWS-by-AWS with charge densities calculated for “c” bulk AlAs and “e” bulk GaAs, except for sites of the interface As common to both semiconductors. On these interface sites, AWSs are filled with charge density calculated for “d” an $\mathrm{AlGaA}{\mathrm{s}}_2$ oligo-cell.

Figure 6

Superimposed plots of charge density (per nm) of quadrants, integrated in the direction perpendicular to the center axis, $\langle 111\rangle$, of Ga-anion and Al-anion quadrants, as extracted from the (100), (110), and (111) common anion III-V heterojunction interfaces, as well as from the bulk semiconductor. At the interfaces, the cation in the quadrant is bonded with both Ga and Al. (a) Ga-P and Al-P quadrants, (b) Ga-As and Al-As quadrants, and (c) Ga-Sb and Al-Sb quadrants.

Figure 7

(a) Electron density, per nm and per area of the two-dimensional (2D) unit cell ($0.27\mathrm{n}{\mathrm{m}}^2$), of a GaAs/AlAs(110) supercell, obtained (i) by first-principles calculation; (ii) from stitching bulk GaAs and bulk AlAs densities on the midpoint between atomic planes; (iii) from stitching GaAs, AlAs, and “1 1” densities terminated on atomic planes; and (iv) from stitching GaAs, AlAs, “2 1” and “1 2” densities. (b) The difference (in e nm area) between the equilibrium electron density calculated for a GaAs/AlAs(110) supercell and that stitched together for this heterojunction with electron densities of bulk semiconductors and oligo-cells. (c) Schematic illustration of two stitching methods for the simulation of charge density at the AlAs/GaAs(110) heterojunction. The atomic arrangement of the heterojunction is drawn in the center diagram, with As, Al, and Ga shown, respectively, as large, small, and midsized spheres. In the CWS-stitching method shown on the right, the volume of the heterojunction is divided into CWSs, and the charge density is stacked CWS-by-CWS with charge densities calculated for “a” bulk AlAs and “b” bulk GaAs. In the “1 1”-stitching method shown on the left, the volume of the heterojunction is divided into three regions, by two parallel planes at locations of interface atoms. Charge densities calculated for “c” bulk AlAs and “e” bulk GaAs fill the outside regions, while the charge density from “d” an $\mathrm{AlGaA}{\mathrm{s}}_2$ oligo-cell fill the immediate interface region.

Figure 8

(a) The difference (e per nm and per area) between the equilibrium electron density calculated for the GaAs/AlAs(111) supercell and that stitched together for this heterojunction with electron densities of bulk semiconductors and oligo-cells. (b) Schematic illustration of stitching methods for the simulation of charge density at the GaAs/AlAs(111) heterojunction. The atomic arrangement of the heterojunction is drawn in the center diagram, with As, Al, and Ga shown as large, small, and midsized spheres, respectively. In the “1 1”-stitching method shown on the right, the volume of the heterojunction is divided into AWSs, and the charge density is stacked AWS-by-AWS with charge densities calculated for bulk GaAs “a” and bulk AlAs “c,” except for sites of the interface As common to both semiconductors. On those interface sites, AWSs are filled with charge density calculated for “b” the $\mathrm{AlGaA}{\mathrm{s}}_2$ oligo-cell. In the “2 2”-stitching method shown on the left, the volume of the heterojunction is divided into CWSs, and the charge density is stacked CWS-by-CWS with charge densities calculated for bulk GaAs “d” and bulk AlAs “f,” except for cation sites closest to the interface. On those interface sites, charge density calculated for the $\mathrm{A}{\mathrm{l}}_2\mathrm{G}{\mathrm{a}}_2\mathrm{A}{\mathrm{s}}_4$ oligo-cell “e” is inserted.

Figure 9

(a) Electron densities, per nm and per area of the two-dimensional (2D) unit cell ($0.185\mathrm{n}{\mathrm{m}}^2$), of bulk semiconductors and oligo-cells that are stitched together to simulate the charge density at the InAs/GaSb(100) heterojunction with As-Ga interface bonds. (b) The difference between the equilibrium charge density calculated for the unrelaxed InAs/GaSb(100) supercell with As-Ga interface bonds and the simulated LEGO-stitched charge density for this heterojunction, based on two stitching methods. (c) Schematic illustration of two stitching methods for the simulation of charge density at the (100) heterojunction between InAs (upper) and GaSb (lower), with As-Ga bonds. The atomic arrangement of the heterojunction is drawn in the center diagram, with anions shown as larger spheres and cations shown as smaller spheres. In the 1Q-stitching method shown on the right, the charge of the heterojunction is stacked together with “a” In-centered CWSs extracted from bulk InAs and “b” Sb-centered AWSs extracted from bulk GaSb. The gap at the interface is filled with “c” As-Ga quadrants extracted from a binary GaAs calculation. In the “11 + 11 − 1Q”-stitching method shown on the left, the charge density of the heterojunction, except at the immediate interface, is stacked together with “d” As-centered AWSs extracted from bulk InAs and “e” Ga-centered CWSs extracted from bulk GaSb. The interface region is filled with “f” As-centered AWSs from an $\mathrm{InGaA}{\mathrm{s}}_2$ “1 1” cell and “g” Ga-centered CWSs from a $\mathrm{G}{\mathrm{a}}_2\mathrm{InAs}$ “1 1” cell. As drawn, the “1 1 + 1 1” -stitched density has overlaps in the interface As-Ga bond region. The charge density of a layer of “c” As-Ga quadrants is therefore removed to complete the “1 1 + 1 1 − 1Q” stitching on left.

Figure 10

The difference (in e nm area) between the equilibrium charge density calculated for an unrelaxed BeTe/MgS(110) supercell and the simulated charge density obtained for the same heterojunction from various stitching methods described in the text.

Figure 11

(a) Laterally averaged electrostatic potential energy profile across a layer of quadrants of III-V semiconductors, calculated with a lattice constant of 0.609 nm. The quadrants are aligned in the $\langle 111\rangle$ direction and have an areal density that is given for the zinc blende crystal by ${(\sqrt{3}/4·a^2)}^{-1}\sim 6.23\mathrm{n}{\mathrm{m}}^{-2}$. The potential energy curves have been vertically shifted by amounts suggested in Table 5. (b) Same as (a), except the calculations are done with a lattice constant of 0.566 nm and a corresponding areal density of $\sim 7.21\mathrm{n}{\mathrm{m}}^{-2}$.

Figure 12

Difference between the equilibrium charge density (in e per nm per area) calculated for a supercell and that stitched together with bulk and oligo-cell densities for the BeTe/MgS(100) heterojunction with (a) relaxed Be-S interface bonds and (b) relaxed Te-Mg interface bonds. There is an overall contraction between the two semiconductors of $\sim 0.031\mathrm{nm}$ for the Be-S bonded interface and an expansion of $\sim 0.042\mathrm{nm}$ for the Te-Mg bonded interface. (c) Schematic illustration of the two stitching methods for a heterojunction between BeTe (upper) and MgS (lower), with Te-Mg bonds. The atomic arrangement of the heterojunction is drawn in the center diagram, with anions shown as larger spheres and cations shown as smaller spheres. In the “1.5 1.5”-stitching method shown on the right, the charge of the heterojunction is stacked together with Be-centered CWSs extracted from bulk BeTe (upper right) and S-centered AWS extracted from bulk MgS (lower right). The gap at the interface is filled with density extracted from a $\mathrm{BeT}{\mathrm{e}}_2\mathrm{M}{\mathrm{g}}_2\mathrm{S}$ calculation (middle right). In the “2.5 2.5”-stitching method shown on the left, the charge density of the heterojunction is stacked together with Te-centered AWS's extracted from bulk BeTe and an Mg-centered CWS extracted from bulk MgS. The interface region is filled with charge density from a $\mathrm{B}{\mathrm{e}}_2\mathrm{T}{\mathrm{e}}_3\mathrm{M}{\mathrm{g}}_3{\mathrm{S}}_2$ calculation.

Figure 13

Difference between the equilibrium charge density (in e per nm per area) calculated for supercell and that stitched together with bulk and oligo-cell densities for the relaxed BeTe/MgS(110) heterojunction. The locations of relaxed atoms are indicated. Additionally, atoms relax laterally, and there is a small overall expansion, $\sim 0.004\mathrm{nm}$, between the two bulk crystals. The relaxed atomic structure for the supercell is also employed for the “3 3” and “4 4” oligo-cells.

Figure 14

Dipole moment along the [111] (cation-to-anion) direction of single quadrant of a ZB semiconductor as a function of lattice parameter.

Figure 15

Accumulated dipole moment [see Eq. (2) of text for definition] of a Mg-S single quadrant, along the length of the quadrant, integrated from the S end (upper) and from the Mg end (lower) of the quadrant, for different lattice parameters. The upper curves have been vertically shifted by 0.15 e nm.

Figure 16

Dipole moment accumulated from the cation end of selected (a) III-V semiconductors and (b) II-VI semiconductor ZB quadrants, calculated at the equilibrium lattice parameter of each semiconductor. For clarity, curves have been vertically shifted.

Figure 17

Accumulated dipole moment curves of the ZB semiconductors from Fig. 16, redisplayed with the neutral point for each quadrant shifted to origin. For clarity, the end points of each curve are marked by dots. (a) III-V semiconductors; (b) II-VI semiconductors.