The band offset (BO) at semiconductor heterojunctions and the Schottky barrier height (SBH) at metal–semiconductor interfaces are important device parameters that are directly related to the charge distribution at the interface. Recently, an approach based on the neutral polyhedra theory (NPT) was developed that allowed interface charge density to be modeled and the BO/SBH at epitaxial interfaces to be quantitatively explained and predicted. The present work shows that the band alignment conditions for a large number of practical interfaces, the majority of which are polycrystalline, can also be explained by modeling the charge distribution at the interface with densities of bulk crystals. Two types of interfaces are distinguished: those between crystals with similar chemical bonds and those with dissimilar bonds. The majority of interfaces presently studied belong to the first category, with their experimentally measured BO/SBHs in good agreement with the structure-independent predictions from NPT. The similarity of bonds at the interface and in bulk crystals makes it unnecessary to make adjustments for the interface bonds and is argued to be the reason behind “bulk-like” behavior in band alignment conditions at such interfaces. The effect of interface bonds that cannot be ignored at some interfaces with perovskite oxides is successfully treated by model solids constructed with the atoms-in-molecules theory. The validity and the wide applicability of density-based theories in the explanation and prediction of band alignment at solid interfaces are demonstrated.
I. INTRODUCTION
The alignment of the electronic energy bands of one crystal relative to those of another across a solid interface is of vital technological importance as it controls the transport properties across these interfaces in a device.1,2 The search for a scientific explanation of the band offset (BO) for semiconductor/semiconductor heterojunctions and the Schottky barrier height (SBH) at metal/semiconductor (MS) interfaces has driven extensive scientific research for decades; yet, the formation mechanisms of SBH and BO are still under debate. The alignment of bands at solid interfaces is directly connected to the distribution of electric charges at the interface. Thus, the latter offers the most direct explanation of the BO/SBH. Indeed, it has been based on these distributions that theoretical calculations were able to yield BO/SBH values for many interfaces, which agree with the experiment.3,4 Because the charge distribution in matter is governed by chemical laws, it seems plausible that the BO/SBH may be predictable through chemical modeling. However, until recently, it was not possible to directly use the charge distribution to explain the BO/SBHs of intimate interfaces due to two obstacles.
The main obstacle has been the nearly exclusive use of surface quantities to represent the solids in band alignment analysis. At the time the Schottky–Mott theory (SMT)5,6 and the electron affinity rule (EAR)7 were proposed, the work function ( ) and the ionization energy ( ) were used to represent the metals and the semiconductors in BO/SBH analysis, because there were no alternatives. Errors associated with the significant surface dipole contributions,8,9 inherent in and , prevented density-based SMT and EAR theories from explaining the experimentally observed band alignment at intimate interfaces. A second obstacle, especially for MS interfaces, is the prevalence of poly-crystallinity and a lack of information on the atomic structure of the interfaces. Structural non-uniformity is known to lead to local inhomogeneity in the band alignment, as has been widely observed experimentally.10,11 Therefore, the BO/SBH values that are measured experimentally for polycrystalline interfaces are statistical averages and, thus, difficult to model through a charge distribution. Without a viable approach to model the charge distribution at interfaces, explanation of experimental SBH/BO had to rely on empirical models based on the charge neutrality level (CNL) assumption,12–16 which could not account for the richness of experimentally observed dependencies.
Recently, the main obstacle for the traditional charge density-based approach to explain the BO/SBH was removed by using bulk references without surface dipoles, established with neutral polyhedra theory (NPT). As a result, the BO values at interfaces between zincblende semiconductors,17,18 those between perovskite oxides,19 and the SBH values at epitaxial interfaces between elemental metals and zincblende semiconductors20 were all shown to be in excellent agreement with predictions through modeled charge distributions of the various interfaces. The dependence of the BO/SBH on an interface atomic structure was also reproduced. These recent successes obtained from epitaxial interfaces of known atomic structures confirmed the direct connection between the BO/SBH and the charge distribution at the interface and demonstrated the relative ease with which the latter may be modeled. They also raised the possibility that the BO/SBH of a polycrystalline interface with an unknown atomic structure may be explained through charge modeling. Here, that possibility is put to the test. The experimentally measured BO/SBH values for technologically important interfaces, including those found in newer types of photovoltaic solar cells, are compared to those modeled and predicted from charge distributions and chemistry. The expected charge distributions are found to adequately explain the BO/SBH values of the vast majority of interfaces presently investigated, many of which are polycrystalline in nature. The present work makes use of the causal relation between an interface charge distribution and band alignment and discusses different strategies that may be used to model the BO/SBH when the interface structure is unknown and when it may be reasonably deduced.
II. METHODOLOGY ADOPTED PREVIOUSLY FOR EPITAXIAL INTERFACES
To estimate the BO/SBH of an interface through charge density modeling, one needs to produce the density of a solid interface as closely as possible without actually conducting an interface calculation. A basic tenet of the density functional theory (DFT) is that the ground-state electron density of a solid results from a universal response to the “external potential” of the nuclei.21 Because the electron density at any specific location depends predominantly on the atomic structure in its close proximity,22 an opportunity arises for the charge distribution at an interface to be modeled. What this “nearsightedness principle” of electron density22 prescribes is that the local charge distribution in the immediate vicinity of a particular atomic structure does not vary with the larger structure far from this location. The local charge distribution is, thus, approximately transferable or interchangeable between different materials with the same local structure.
Taken at its face value, this nearsightedness suggests that the charge distribution at any location of a solid might be dominated by only one of the atoms closest to this location. Thus, the charge distribution of any solid might be assembled with charge densities that habitually reside with each of the atoms that make up the structure. If proven feasible, such a scheme would allow the unknown charge distribution of an interface to be approximately assembled with the known atomic densities of bulk crystals and the band alignment conditions to be predicted. A prerequisite for such an approach to succeed is the partition of the bulk charge density into “atoms” that are essentially transferable. This idea was tested in recent years, beginning with the heterojunction interfaces between semiconductors with zincblende structures.17 For these covalently bonded compounds, NPT was used to define the atoms in a bulk crystal that were used for interface modeling. Excellent agreements were found between the NPT predictions and the valence band offsets (VBOs) experimentally or theoretically observed at the heterojunction interfaces.17,18 These results lent credence to the general approach of simulating the interface charge distribution with stacked atomic densities obtained from the analysis of bulk materials. Later on, NPT was also found to be effective in predicting/explaining BO/SBH of other epitaxial interfaces.19
To see the possible reasons NPT was successful for epitaxial interfaces, we note that the neutral polyhedra in zincblende crystals have the following properties:
Between nearest neighbors, the partition is at a distance from each nucleus that approximately corresponds to the covalent radius of that atom.
The charge distribution of the neutral polyhedron for a cation (or an anion), defined from one semiconductor, is approximately interchangeable with that defined from a different bulk semiconductor containing the same cation (anion).
The average potential energy of an NPT model solid is independent of its surface structure.
Among all model solids with abruptly terminated charge density, the NPT model solid's orientation-independent band position is at the highest position with respect to the vacuum level.19
With their sizes conforming to their covalent radii, the neutral polyhedra on opposing surfaces, when stitched together at interfaces, are already at optimal bond lengths from each other. Therefore, lattice and charge relaxations may be expected to be small. The minimized total energy of the charge distribution of each NPT model solid19 further suggests that the effect due to charge relaxation may be small.
To model properly, it is important to identify what this term represents explicitly. The final charge distribution of a relaxed interface with a particular atomic structure may be envisioned to take shape in two steps. In step one, two model solids with carefully chosen surface terminations are stacked together to bring all the atoms needed for the formation of a neutral interface with the specified atomic structure. Still in their unrelaxed, bulk positions, some atoms in the ISR have not settled in their final relaxed positions. However, all atoms should be present, without any missing or extra ones. Essentially, step one involves the stacking of bulk atomic densities to create a temporary charge distribution of the interface. After step one, there is no , and the VBO is simply the difference between the two values. In step two, the charge density and atomic positions in the ISR are relaxed to reach equilibrium, at which point the interface may again be partitioned as in Fig. 1 and the term emerges. This conceptual exercise shows clearly that the term is the net effect of charge and lattice relaxations from stacked bulk densities to the equilibrium distribution. Because the first two terms of Eq. (1) depend on the method used to partition the bulk charge density, the value of and how this term can be modeled also depend on the type of model solid employed in step one. For it to be easily estimated from general principles, this term should be a minor correction. In other words, the partition method used should be well suited for the type of bonds at the interface such that the stacked densities of step one already resemble the final distribution.
From an artificial distribution assembled with bulk atomic densities, charge needs to relax to reach equilibrium. Because each atomic density is already energy minimized for a (bulk) atomic structure that is similar to that at the interface, the rearrangement of charge density within each “atom” is expected to be small and may be ignored. However, the average charge densities on the face of the two model solids are usually different. Obviously, there should be some transfer of electrons from the model solid with higher face density to the one with lower density to smooth out the charge distribution. Electrostatic analysis showed that a potential energy difference of eV Å would be generated if a density difference of were assumed to smooth out linearly over a distance of .19 Previously, the density mismatch was assumed to dissipate over one-half of the interface bond length,19 although density equilibration should be a universal phenomenon unrelated to bond formation. Another correction to the term becomes necessary when the bonds in the two bulk crystals are dissimilar. When that happens, the new bonds at the interface are not part of either of the bulk crystals, and the local environments for the atoms at the interface are significantly different from bulk. Then, considerable adjustments need to be made to the stacked atomic densities at the interface to make it resemble the charge distribution at the real interface. The main correction on the band alignment due to the formation of new interface bonds is to account for the polarization of the bonds. Previously, the correction due to bond formation was assumed to be proportional to the difference in the electronegativities of the atoms, the bond length, and the areal density of the bonds at MS interfaces.20 This particular correction, which obviously requires knowledge on the interface atomic structure, is generally unnecessary for interfaces between materials with similar bonds.
With only these two corrections included, the terms for various epitaxial interfaces were estimated adequately, leading to successful explanation of experimental and theoretical band alignment conditions.3,4,23 These investigations confirmed the charge distribution as the foundation of SBH/BO formation and that the nearsightedness of the electron density could be relied on to reasonably model the interface charge distribution with bulk densities. The same methodology is here applied to a large number of practical polycrystalline interfaces of covalent and ionic materials, with largely unknown atomic level interface structures, to look for common rules and practical limitations of charge density modeling at such interfaces.
III. CALCULATION OF CRYSTAL CHARGE DENSITY
A wide selection of crystalline solids has been chosen for the present investigation. These include materials that are presently used or being developed for solar energy applications as absorbers, electron and hole extraction materials, or contacts. Some interfaces employed in the fabrication of detectors, sensors, and other applications are also studied. Crystal structures and lattice parameters available from the literature and from the Materials Project database24 are used to compute the energy bands and the charge density. All DFT calculations are performed using a plane wave basis, along with projector augmented wave treatment of core-electrons, as implemented in the Vienna Ab initio Simulation Package (VASP).25–27 The screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE)28,29 is used for improved accuracy in the calculated magnitudes of the bandgap. For a few important practical solids that take the amorphous form under conventional device processing conditions, such as SiO2 and Al2O3, calculations are performed in their known crystalline form(s) of the same chemical formula. As a significant difference in the atomic density is usually found between an amorphous solid and its crystalline phase(s) and the average potential of the NPT model solid depends on the packing density, results of these calculations are mainly for reference only. On the same note, it should be kept in mind that the actual structure of a solid material, whose interface BO/SBH has been experimentally measured, is not always known precisely due to variations in the processing conditions in different experiments. For semiconducting materials that are frequently used as heavily doped degenerate contacts, only the insulating (undoped) versions of the band structure and the electron density are computed. A few alloy compounds frequently employed in applications are usually fabricated with their composition tuned for a specific lattice parameter, such as PbZr0.52Ti0.48O3 and Ga0.47In0.53As. They are studied here, albeit only approximately, with ordered alloy structures of a simpler composition, e.g., PbZr0.5Ti0.5O3 and Ga0.5In0.5As. Results for semiconductors with the zincblende/diamond structures and perovskite-structured oxides were previously published17,19 and are not repeated here. The present work only covers crystalline materials that have been used prominently and investigated extensively in the literature for a range of applications. Analyses, such as those demonstrated here, should be readily applicable to material interfaces omitted in this work.
IV. CURRENT METHODS AND PROCEDURES
A. Neutral polyhedra definition for a unit cell of any structure
Material . | Crystal structure . | Lattice parameter (Å) . | NPT (eV) . | Density (Å−3) . | Bulk reference (eV) . | AIM (eV) . |
---|---|---|---|---|---|---|
Al2O3 | rhomb. | a = 4.82, c = 13.12 | −4.47 | 0.315 | −5.66 | −3.73 |
AlN | hex. | a = 3.13, c = 5.02 | −3.43 | 0.284 | −4.50 | −3.84 |
Bi2O3 | monoc. | 4.22 × 3.67 × 5.58 | −4.07 | 0.175 | −4.73 | |
BiFeO3 | rhomb. | a = 5.59, c = 13.91 | −3.81 | 0.230 | −4.68 | −2.16 |
CdS | wurtz. | a = 4.17, c = 6.78 | −4.81 | 0.087 | −5.14 | −4.85 |
CH3NH3PbBr3 | orthorh. | 8.07 × 8.78 × 12.08 | −5.02 | 0.113 | −5.45 | |
CH3NH3SnI3 | orthorh. | 8.51 × 9.15 × 13.16 | −4.39 | 0.099 | −4.77 | |
CIGS | tetra. | a = 5.7, c = 11.4 | −3.47 | 0.098 | −3.84 | −3.58 |
CsPbBr3 | cubic | a = 5.949 | −4.76 | 0.054 | −4.96 | |
CsPbI3 | cubic | a = 6.275 | −4.53 | 0.026 | −4.63 | |
CsSnI3 | cubic | a = 6.286 | −3.88 | 0.047 | −4.05 | |
Cu2O | cubic | a = 4.25 | −3.06 | 0.218 | −3.88 | |
Cu2S | cubic | a = 5.55 | −1.52 | 0.152 | −2.09 | |
Cu2S | monoc. | 9.11 × 5.93 × 5.63 | −4.72 | 0.097 | −5.09 | |
Cu2Se | monoc. | 13.81 × 12.63 × 15.65 | −2.10 | 0.170 | −2.74 | |
Cu2Te_224 | cubic | a = 5.65 | −5.22 | 0.082 | −5.53 | |
Cu2Te | hex. | a = 4.3, c = 8.52 | −2.50 | 0.129 | −2.98 | |
Cu2Te | monoc. | 7.693 × 3.803 × 7.248 | −1.76 | 0.156 | −2.35 | |
Cu2Te_225 | cubic | a = 6.15 | −1.63 | 0.129 | −2.12 | |
CuGaSe2 | tetra. | a = 5.6, c = 11.2 | −3.36 | 0.101 | −3.74 | −4.92 |
CuInSe2 | tetra. | a = 5.81, c = 11.62 | −3.57 | 0.090 | −3.91 | −5.00 |
Fe2O3 α | rhomb. | a = 5.09, c = 13.77 | −4.14 | 0.256 | −5.10 | |
Ga2O3 β | monoc. | 12.28 × 3.05 × 5.82 | −5.01 | 0.245 | −5.93 | |
GaN | hex. | a = 3.24, c = 5.29 | −3.65 | 0.236 | −4.54 | −3.31 |
Gd2O3 | monoc. | 14.14 × 3.56 × 8.76 | −4.82 | 0.219 | −5.64 | |
HfO2 | monoc. | 5.09 × 5.15 × 5.3 | −4.31 | 0.281 | −5.37 | −2.14 |
HfSiO4 | tetra. | a = 6.55, c = 5.96 | −5.01 | 0.303 | −6.15 | −3.56 |
HgCdTe | tetra. | a = 4.58, c = 6.48 | −3.84 | 0.070 | −4.10 | |
In2O3 | cubic | a = 10.18 | −5.28 | 0.201 | −6.03 | −3.91 |
InGaAs | tetra. | a = 4.14, c = 5.87 | −3.05 | 0.093 | −3.40 | |
InN | hex. | a = 3.55, c = 5.73 | −3.43 | 0.176 | −4.09 | −2.74 |
InSe β | hex. | a = 4.04, c = 1.743 | −3.75 | 0.081 | −4.06 | |
Lu2O3 | cubic | a = 10.26 | −4.93 | 0.220 | −5.75 | |
MgO | cubic | a = 4.194 | −4.12 | 0.296 | −5.23 | −3.55 |
RuO2 | tetra. | a = 4.48, c = 3.11 | −2.94 | 0.269 | −3.95 | −1.76 |
Si3N4 α | trig. | a = 7.75, c = 5.62 | −3.02 | 0.293 | −4.12 | |
SiC 4H | hex. | a = 3.08, c = 10.07 | −1.61 | 0.293 | −2.72 | −2.56 |
SiC 6H | hex. | a = 3.08, c = 15.1 | −1.57 | 0.300 | −2.70 | |
SiC β | cubic | a = 4.35 | −1.68 | 0.274 | −2.71 | −2.16 |
SiO2 α | trig. | a = 4.92, c = 5.43 | −6.72 | 0.250 | −7.66 | −6.30 |
SnO2 | tetra. | a = 4.76, c = 3.21 | −5.05 | 0.254 | −6.00 | −3.24 |
Ta2O5 | tetra. | a = 3.88, c = 26.27 | −5.72 | 0.210 | −6.51 | −4.19 |
TiO2_ana | tetra. | a = 3.78, c = 9.62 | −5.35 | 0.259 | −6.33 | −4.08 |
TiO2_rut | tetra. | a = 4.6, c = 2.96 | −4.51 | 0.281 | −5.57 | −3.10 |
ZnO | hex. | a = 3.24, c = 5.22 | −4.84 | 0.207 | −5.62 | −4.35 |
ZrO2 | monoc. | 5.15 × 5.23 × 5.33 | −4.52 | 0.268 | −5.53 | −2.36 |
ZrSiO4 | tetra. | a = 6.62, c = 6.00 | −5.15 | 0.298 | −6.27 |
Material . | Crystal structure . | Lattice parameter (Å) . | NPT (eV) . | Density (Å−3) . | Bulk reference (eV) . | AIM (eV) . |
---|---|---|---|---|---|---|
Al2O3 | rhomb. | a = 4.82, c = 13.12 | −4.47 | 0.315 | −5.66 | −3.73 |
AlN | hex. | a = 3.13, c = 5.02 | −3.43 | 0.284 | −4.50 | −3.84 |
Bi2O3 | monoc. | 4.22 × 3.67 × 5.58 | −4.07 | 0.175 | −4.73 | |
BiFeO3 | rhomb. | a = 5.59, c = 13.91 | −3.81 | 0.230 | −4.68 | −2.16 |
CdS | wurtz. | a = 4.17, c = 6.78 | −4.81 | 0.087 | −5.14 | −4.85 |
CH3NH3PbBr3 | orthorh. | 8.07 × 8.78 × 12.08 | −5.02 | 0.113 | −5.45 | |
CH3NH3SnI3 | orthorh. | 8.51 × 9.15 × 13.16 | −4.39 | 0.099 | −4.77 | |
CIGS | tetra. | a = 5.7, c = 11.4 | −3.47 | 0.098 | −3.84 | −3.58 |
CsPbBr3 | cubic | a = 5.949 | −4.76 | 0.054 | −4.96 | |
CsPbI3 | cubic | a = 6.275 | −4.53 | 0.026 | −4.63 | |
CsSnI3 | cubic | a = 6.286 | −3.88 | 0.047 | −4.05 | |
Cu2O | cubic | a = 4.25 | −3.06 | 0.218 | −3.88 | |
Cu2S | cubic | a = 5.55 | −1.52 | 0.152 | −2.09 | |
Cu2S | monoc. | 9.11 × 5.93 × 5.63 | −4.72 | 0.097 | −5.09 | |
Cu2Se | monoc. | 13.81 × 12.63 × 15.65 | −2.10 | 0.170 | −2.74 | |
Cu2Te_224 | cubic | a = 5.65 | −5.22 | 0.082 | −5.53 | |
Cu2Te | hex. | a = 4.3, c = 8.52 | −2.50 | 0.129 | −2.98 | |
Cu2Te | monoc. | 7.693 × 3.803 × 7.248 | −1.76 | 0.156 | −2.35 | |
Cu2Te_225 | cubic | a = 6.15 | −1.63 | 0.129 | −2.12 | |
CuGaSe2 | tetra. | a = 5.6, c = 11.2 | −3.36 | 0.101 | −3.74 | −4.92 |
CuInSe2 | tetra. | a = 5.81, c = 11.62 | −3.57 | 0.090 | −3.91 | −5.00 |
Fe2O3 α | rhomb. | a = 5.09, c = 13.77 | −4.14 | 0.256 | −5.10 | |
Ga2O3 β | monoc. | 12.28 × 3.05 × 5.82 | −5.01 | 0.245 | −5.93 | |
GaN | hex. | a = 3.24, c = 5.29 | −3.65 | 0.236 | −4.54 | −3.31 |
Gd2O3 | monoc. | 14.14 × 3.56 × 8.76 | −4.82 | 0.219 | −5.64 | |
HfO2 | monoc. | 5.09 × 5.15 × 5.3 | −4.31 | 0.281 | −5.37 | −2.14 |
HfSiO4 | tetra. | a = 6.55, c = 5.96 | −5.01 | 0.303 | −6.15 | −3.56 |
HgCdTe | tetra. | a = 4.58, c = 6.48 | −3.84 | 0.070 | −4.10 | |
In2O3 | cubic | a = 10.18 | −5.28 | 0.201 | −6.03 | −3.91 |
InGaAs | tetra. | a = 4.14, c = 5.87 | −3.05 | 0.093 | −3.40 | |
InN | hex. | a = 3.55, c = 5.73 | −3.43 | 0.176 | −4.09 | −2.74 |
InSe β | hex. | a = 4.04, c = 1.743 | −3.75 | 0.081 | −4.06 | |
Lu2O3 | cubic | a = 10.26 | −4.93 | 0.220 | −5.75 | |
MgO | cubic | a = 4.194 | −4.12 | 0.296 | −5.23 | −3.55 |
RuO2 | tetra. | a = 4.48, c = 3.11 | −2.94 | 0.269 | −3.95 | −1.76 |
Si3N4 α | trig. | a = 7.75, c = 5.62 | −3.02 | 0.293 | −4.12 | |
SiC 4H | hex. | a = 3.08, c = 10.07 | −1.61 | 0.293 | −2.72 | −2.56 |
SiC 6H | hex. | a = 3.08, c = 15.1 | −1.57 | 0.300 | −2.70 | |
SiC β | cubic | a = 4.35 | −1.68 | 0.274 | −2.71 | −2.16 |
SiO2 α | trig. | a = 4.92, c = 5.43 | −6.72 | 0.250 | −7.66 | −6.30 |
SnO2 | tetra. | a = 4.76, c = 3.21 | −5.05 | 0.254 | −6.00 | −3.24 |
Ta2O5 | tetra. | a = 3.88, c = 26.27 | −5.72 | 0.210 | −6.51 | −4.19 |
TiO2_ana | tetra. | a = 3.78, c = 9.62 | −5.35 | 0.259 | −6.33 | −4.08 |
TiO2_rut | tetra. | a = 4.6, c = 2.96 | −4.51 | 0.281 | −5.57 | −3.10 |
ZnO | hex. | a = 3.24, c = 5.22 | −4.84 | 0.207 | −5.62 | −4.35 |
ZrO2 | monoc. | 5.15 × 5.23 × 5.33 | −4.52 | 0.268 | −5.53 | −2.36 |
ZrSiO4 | tetra. | a = 6.62, c = 6.00 | −5.15 | 0.298 | −6.27 |
Oxide . | Crystal structure . | Lattice parameter (Å) . | NPT (eV) . | Density (Å−3) . | Bulk reference (eV) . | AIM (eV) . |
---|---|---|---|---|---|---|
(Ba,Sr)TiO3 | tetra. | a = 3.98, c = 7.96 | −4.44 | 0.222 | −5.28 | −4.28 |
BaSnO3 | cubic | a = 4.18 | −5.04 | 0.211 | −5.83 | −4.21 |
BaTiO3 | cubic | a = 4.02 | −4.63 | 0.140 | −5.16 | −4.35 |
BaZrO3 | cubic | a = 4.22 | −4.84 | 0.124 | −5.30 | −3.78 |
CaHfO3 | cubic | a = 4.115 | −4.76 | 0.248 | −5.69 | −4.26 |
CaSnO3 | cubic | a = 4.07 | −4.82 | 0.227 | −5.68 | −4.35 |
CaZrO3 | cubic | a = 4.145 | −4.78 | 0.114 | −5.21 | −4.26 |
(K,Na)NbO3 | tetra. | a = 4.04, c = 8.08 | −5.06 | 0.191 | −5.78 | 1.91 |
KNbO3 | cubic | a = 4.07 | −5.17 | 0.085 | −5.49 | −1.96 |
LaAlO3 | cubic | a = 3.80 | −3.77 | 0.201 | −4.52 | −5.20 |
(Na,Bi)TiO3 | tetra. | a = 5.52, c = 3.91 | −4.46 | 0.221 | −5.29 | −1.84 |
Pb(Zr,Ti)O3 | tetra. | a = 5.69, c = 25.22 | −4.44 | 0.228 | −5.29 | −3.41 |
PbTiO3 | cubic | a = 3.97 | −4.07 | 0.156 | −4.66 | −3.57 |
PbZrO3 | cubic | a = 4.18 | −4.88 | 0.117 | −5.32 | −3.34 |
SrHfO3 | cubic | a = 4.145 | −4.72 | 0.121 | −5.17 | −4.12 |
SrSnO3 | cubic | a = 4.115 | −4.89 | 0.142 | −5.42 | −4.31 |
SrTiO3 | cubic | a = 3.94 | −4.38 | 0.149 | −4.94 | −4.56 |
SrZrO3 | cubic | a = 4.18 | −4.77 | 0.219 | −5.59 | −4.08 |
Oxide . | Crystal structure . | Lattice parameter (Å) . | NPT (eV) . | Density (Å−3) . | Bulk reference (eV) . | AIM (eV) . |
---|---|---|---|---|---|---|
(Ba,Sr)TiO3 | tetra. | a = 3.98, c = 7.96 | −4.44 | 0.222 | −5.28 | −4.28 |
BaSnO3 | cubic | a = 4.18 | −5.04 | 0.211 | −5.83 | −4.21 |
BaTiO3 | cubic | a = 4.02 | −4.63 | 0.140 | −5.16 | −4.35 |
BaZrO3 | cubic | a = 4.22 | −4.84 | 0.124 | −5.30 | −3.78 |
CaHfO3 | cubic | a = 4.115 | −4.76 | 0.248 | −5.69 | −4.26 |
CaSnO3 | cubic | a = 4.07 | −4.82 | 0.227 | −5.68 | −4.35 |
CaZrO3 | cubic | a = 4.145 | −4.78 | 0.114 | −5.21 | −4.26 |
(K,Na)NbO3 | tetra. | a = 4.04, c = 8.08 | −5.06 | 0.191 | −5.78 | 1.91 |
KNbO3 | cubic | a = 4.07 | −5.17 | 0.085 | −5.49 | −1.96 |
LaAlO3 | cubic | a = 3.80 | −3.77 | 0.201 | −4.52 | −5.20 |
(Na,Bi)TiO3 | tetra. | a = 5.52, c = 3.91 | −4.46 | 0.221 | −5.29 | −1.84 |
Pb(Zr,Ti)O3 | tetra. | a = 5.69, c = 25.22 | −4.44 | 0.228 | −5.29 | −3.41 |
PbTiO3 | cubic | a = 3.97 | −4.07 | 0.156 | −4.66 | −3.57 |
PbZrO3 | cubic | a = 4.18 | −4.88 | 0.117 | −5.32 | −3.34 |
SrHfO3 | cubic | a = 4.145 | −4.72 | 0.121 | −5.17 | −4.12 |
SrSnO3 | cubic | a = 4.115 | −4.89 | 0.142 | −5.42 | −4.31 |
SrTiO3 | cubic | a = 3.94 | −4.38 | 0.149 | −4.94 | −4.56 |
SrZrO3 | cubic | a = 4.18 | −4.77 | 0.219 | −5.59 | −4.08 |
B. Model solids created with AIM
The charge distribution in an ionic crystal is more conveniently described by placing charged cations and anions at sites in a unit cell, rather than neutral atoms. Because the majority of materials presently studied are at least partially ionic compounds, we explore the use of Bader's quantum theory of atoms in molecules (AIM)33,34 as a method to produce stacked atomic densities for the simulation of an interface charge distribution. When the interface bonds cannot be ignored, the properties of the Bader atoms offer a way to account for their effect. With available programs,33,34 Bader atoms may be defined on the boundaries of zero flux in the electron density of a solid or molecule, and they are typically non-neutral. The presence of net charge limits the way an AIM model solid can be defined, because all surfaces of a model solid have to be neutral and electrically equivalent. Thus, constructing model solids for compounds with differently charged cations and/or anions requires particular attention.
Another group of ionic compounds that have wide technological applications and are being investigated for band alignment modeling are the binary oxides, with MaOb composition. Even though the chemical formula may appear simple, the structure of the unit cell of some binary oxides may be too complicated for the construction of a model solid with desired surface termination. Because of the difficulty with finding suitable symmetric cells with vanishing dipole moment, Bader model solids with a specific surface structure, such as ones that terminate on a partial oxygen layer, are not available in general. However, there is always a “generic” model solid that can be constructed without considering the detailed structure for each binary oxide that provides a useful reference. The average potential energy of the generic Bader model solid is obtained by substituting the sum of spherapoles of the Bader atoms, , into Eq. (2). We now examine what surface termination(s) can be expected from a model solid with such an average potential energy. The use of only indicates that that is the spherapole of the entire symmetric cell. Ordinarily for any symmetric cell constructed with Bader atoms, the terms from all the cations and from the Bader oxygen atoms should be added to to complete the . Note that these two sums are of opposite signs, because . Their absence shows that these two sums cancel each other for the generic AIM model solids, which requires the cations and the oxygen anions to be distributed, on average per charge, at equal (squared) distances from the center of the symmetric cell. The exposed surfaces of such a model solid would have one-half layer of cations and oxygen Bader atoms in the composition of the binary oxide. In other words, the AIM references listed in Table I correspond to model solids with “non-polar” surfaces.
C. Choosing model solids best suited for specific interfaces
The present work aims to simulate the charge distribution for a variety of interfaces with charge densities of model solids. There are two types of model solids, NPT and AIM, to choose from, and the choice may depend on the interface. It is, thus, helpful to first compare generally how transferable the atoms, defined by these two methods, are for different classes of materials. We note that both methods are well in line with the nearsightedness principle, as the electron density at a location closest to or is dominated by a particular atom in a solid is usually partitioned to reside with that atom by either method. There are several properties of a partitioned atom that may potentially be used to determine the transferability of the atom when it is put in different compounds. Here, the spherapole of an atom or ion is chosen, because it is that property of a charge distribution that directly affects the average potential energy. In Table III, the spherapoles of the neutral polyhedron and the Bader atom for each cation and anion are compiled for zincblende semiconductors. Using the arsenide (As) anion in zincblende semiconductors as an example, we see that its NPT spherapoles are 6.48, 6.59, and 6.74 eÅ2, respectively, in the three semiconductors, AlAs, GaAs, and InAs. The similarity of the neutral polyhedron of an atom among zincblende compounds containing that atom as a common element was already emphasized when NPT was first discovered.17 In comparison, the Bader atoms in Table III are generally not interchangeable among different semiconductors. Thus, NPT is clearly favored over AIM for the analysis of interfaces with these covalent semiconductors. From the data for perovskite oxides in Table IV, both the NPT and the AIM seem to define individual atoms that are approximately transferable among different oxides, although AIM should be given a slight preference as the scatter in the Bader spherapoles is smaller than NPT. Based entirely on density transferability, NPT seems to be the preferred technique for all interfaces, except for those between some ionic compounds that may be slightly better handled by AIM.
III–V cation | Al NPT | Al Bader | Ga NPT | Ga Bader | In NPT | In Bader |
With P | 4.90 | 0.87 | 4.59 | 3.00 | 5.87 | 3.91 |
With As | 5.06 | 1.06 | 4.71 | 3.35 | 6.02 | 4.25 |
With Sb | 5.22 | 1.51 | 4.82 | 4.20 | 6.11 | 5.04 |
III–V anion | P NPT | P Bader | As NPT | As Bader | Sb NPT | Sb Bader |
With Al | 5.76 | 12.52 | 6.48 | 13.42 | 8.59 | 15.92 |
With Ga | 5.87 | 8.49 | 6.59 | 9.08 | 8.74 | 10.66 |
With In | 5.99 | 8.78 | 6.74 | 9.46 | 8.88 | 11.21 |
II–VI cation | Mg NPT | Mg Bader | Zn NPT | Zn Bader | Cd NPT | Cd Bader |
With S | 3.95 | 0.39 | 6.76 | 4.83 | 9.68 | 7.46 |
With Se | 4.04 | 0.44 | 6.81 | 5.08 | 9.76 | 7.76 |
With Te | 4.44 | 0.55 | 6.93 | 5.54 | 9.86 | 8.28 |
II–VI anion | S NPT | S Bader | Se NPT | Se Bader | Te NPT | Te Bader |
With Mg | 6.01 | 11.35 | 7.15 | 13.13 | 9.80 | 17.35 |
With Zn | 6.04 | 8.94 | 7.19 | 10.12 | 9.74 | 12.72 |
With Cd | 6.12 | 8.95 | 7.30 | 10.17 | 9.87 | 12.77 |
III–V cation | Al NPT | Al Bader | Ga NPT | Ga Bader | In NPT | In Bader |
With P | 4.90 | 0.87 | 4.59 | 3.00 | 5.87 | 3.91 |
With As | 5.06 | 1.06 | 4.71 | 3.35 | 6.02 | 4.25 |
With Sb | 5.22 | 1.51 | 4.82 | 4.20 | 6.11 | 5.04 |
III–V anion | P NPT | P Bader | As NPT | As Bader | Sb NPT | Sb Bader |
With Al | 5.76 | 12.52 | 6.48 | 13.42 | 8.59 | 15.92 |
With Ga | 5.87 | 8.49 | 6.59 | 9.08 | 8.74 | 10.66 |
With In | 5.99 | 8.78 | 6.74 | 9.46 | 8.88 | 11.21 |
II–VI cation | Mg NPT | Mg Bader | Zn NPT | Zn Bader | Cd NPT | Cd Bader |
With S | 3.95 | 0.39 | 6.76 | 4.83 | 9.68 | 7.46 |
With Se | 4.04 | 0.44 | 6.81 | 5.08 | 9.76 | 7.76 |
With Te | 4.44 | 0.55 | 6.93 | 5.54 | 9.86 | 8.28 |
II–VI anion | S NPT | S Bader | Se NPT | Se Bader | Te NPT | Te Bader |
With Mg | 6.01 | 11.35 | 7.15 | 13.13 | 9.80 | 17.35 |
With Zn | 6.04 | 8.94 | 7.19 | 10.12 | 9.74 | 12.72 |
With Cd | 6.12 | 8.95 | 7.30 | 10.17 | 9.87 | 12.77 |
A-site cation . | Ca NPT . | Ca Bader . | Sr NPT . | Sr Bader . | Pb NPT . | Pb Bader . | Ba NPT . | Ba Bader . |
---|---|---|---|---|---|---|---|---|
With Ti | 9.08 | 4.26 | 11.08 | 5.95 | 13.84 | 9.07 | 14.27 | 8.74 |
With Sn | 9.69 | 4.29 | 11.67 | 6.01 | 14.86 | 8.96 | ||
With Hf | 9.81 | 4.29 | 11.74 | 6.06 | ||||
With Zr | 9.92 | 4.31 | 11.87 | 6.05 | 14.46 | 9.46 | 14.97 | 9.05 |
With Ru* | 11.24 | 5.98 | 14.33 | 8.91 | ||||
With Nb* | 11.27 | 6.03 | 14.68 | 8.95 | ||||
With V* | 10.75 | 5.98 | 13.99 | 8.86 | ||||
With Ir* | 14.52 | 8.85 | ||||||
B-site cation | Ti NPT | Ti Bader | Sn NPT | Sn Bader | Hf NPT | Hf Bader | Zr NPT | Zr Bader |
With Ca | 7.75 | 4.01 | 10.42 | 5.71 | 10.35 | 5.28 | 10.91 | 5.70 |
With Sr | 7.88 | 4.00 | 10.52 | 5.63 | 10.41 | 5.27 | 11.01 | 5.67 |
With Pb | 7.98 | 3.94 | 11.04 | 5.85 | ||||
With Ba | 8.05 | 3.96 | 10.67 | 5.78 | 11.11 | 5.70 |
A-site cation . | Ca NPT . | Ca Bader . | Sr NPT . | Sr Bader . | Pb NPT . | Pb Bader . | Ba NPT . | Ba Bader . |
---|---|---|---|---|---|---|---|---|
With Ti | 9.08 | 4.26 | 11.08 | 5.95 | 13.84 | 9.07 | 14.27 | 8.74 |
With Sn | 9.69 | 4.29 | 11.67 | 6.01 | 14.86 | 8.96 | ||
With Hf | 9.81 | 4.29 | 11.74 | 6.06 | ||||
With Zr | 9.92 | 4.31 | 11.87 | 6.05 | 14.46 | 9.46 | 14.97 | 9.05 |
With Ru* | 11.24 | 5.98 | 14.33 | 8.91 | ||||
With Nb* | 11.27 | 6.03 | 14.68 | 8.95 | ||||
With V* | 10.75 | 5.98 | 13.99 | 8.86 | ||||
With Ir* | 14.52 | 8.85 | ||||||
B-site cation | Ti NPT | Ti Bader | Sn NPT | Sn Bader | Hf NPT | Hf Bader | Zr NPT | Zr Bader |
With Ca | 7.75 | 4.01 | 10.42 | 5.71 | 10.35 | 5.28 | 10.91 | 5.70 |
With Sr | 7.88 | 4.00 | 10.52 | 5.63 | 10.41 | 5.27 | 11.01 | 5.67 |
With Pb | 7.98 | 3.94 | 11.04 | 5.85 | ||||
With Ba | 8.05 | 3.96 | 10.67 | 5.78 | 11.11 | 5.70 |
A factor that also affects which of the two techniques should be used for the analysis of an interface is whether the interface structure is completely unknown, as may be assumed to be the case for most polycrystalline interfaces, or whether it is either known or may be reasonably inferred. Because of the charged nature of Bader atoms, the average potential energy of an AIM model solid depends sharply on its surface termination. Therefore, the appropriate AIM model solids to use for a particular interface can only be chosen when the interface structure is known. Even for an interface of a known atomic structure, it is not always possible to create a trial charge distribution that is overall neutral with stacked Bader atoms, again because of their net charges. Therefore, there are practical limitations with the use of AIM model solids, even though the AIM technique is preferred for ionic interfaces. Discussions above identified two different AIM model solids with well-defined surface terminations that can generate interface charge distributions that are overall neutral. Therefore, only ionic interfaces with structures that can be handled by these two AIM model solids are analyzed by the Bader method presently. NPT will be the method to model the BO/SBH for most of the studied interfaces, in the first place because the same NPT model solid is applicable to all (including unknown) interface structures, and in the second place because the interatomic distance at the interface is already approximately optimized, which minimizes the need for further lattice relaxation.
D. Relaxation of a charge density difference
From the stacked atomic densities, the difference in the average electron densities on the surfaces of the two model solids needs to equilibrate, leading to a correction to the modeled band alignment. This correction is known to be proportional to the density difference if the latter is assumed to smooth out linearly over a finite distance. As this is a universal phenomenon, a single formula, ,19 is used to account for this effect for all the present interfaces, regardless of the model solid used or whether the interface structure is known. These corrections are of comparable magnitudes as those used in previous studies and correspond to the density difference being dissipated linearly over a fixed distance of ∼0.07 nm. This formula is arrived at from an empirical fit to all the definitive SBH/BO values (with small or no quoted error bars) experimentally measured from interfaces between binary oxides of Table V. It, thus, provides an estimate on the distance over which charge density differences are dissipated in solids in general. A point of reference is the Thomas–Fermi screening length, which is ∼0.05 nm for some metals. The average charge densities on the surfaces of the NPT and the AIM model solids can be found in Tables I and II for all materials presently investigated. In addition to density differences, the formation of new bonds at the interface of dissimilar materials, notably between covalent and ionic crystals, requires charge rearrangement and a possible modification to the modeled band alignment condition. However, knowledge on the interface atomic structure is needed for such corrections to be made. For the present polycrystalline interfaces, no corrections are taken for bonds, as the atomic structure is unknown. On account of polycrystallinity and the randomness in the interface structure, it is assumed that the effect of bond polarization averages out to an insignificant amount. Note that such corrections are unnecessary for entirely ionic or entirely covalent interfaces.
Oxide no. 1 . | Oxide no. 2 . | VBO/p-SBH (eV) . | NPT (eV) . | Reference . |
---|---|---|---|---|
rut-TiO2 | ana-TiO2 | 0.7 | 0.76 | 35 and 36 |
HfO2 | SiO2 | 1.1–2.2 | 2.29 | 37 and 38 |
Cu2O | Al2O3 | 2.6 | 1.78 | 39 |
Cu2O | Bi2O3 | 0.9 | 0.84 | 39 |
Cu2O | Sn:In2O3 | 2.1 | 2.15 | 40 |
Cu2O | ZnO | 1.5–2.9 | 1.74 | 41–43 |
HfO2 | HfxSi1−xO2 | 0.8 | 0.78 | 44 |
HfO2 | β Ga2O3 | 0.5 | 0.56 | 45 |
ZrO2 | β Ga2O3 | 0.3 | 0.40 | 45 |
Al2O3 | Sn:In2O3 | 0.2 | −0.32 | 46 |
SnO2 | In2O3 | 0 | 0.03 | 15 |
SnO2 | ZnO | 0 | −0.38 | 14 |
Al2O3 | ZnO:Al | 0 | −0.04 | 14 |
ZnO | ana-TiO2 | 0.57 | 0.71 | 47 |
RuO2 | Ta2O5 | 2.38 | 2.55 | 14 |
RuO2 | ZnO | 2.2 | 1.67 | 14 |
RuO2 | ana-TiO2 | 2.35 | 2.37 | 14 |
RuO2 | rut-TiO2 | 1.65 | 1.62 | 14 |
RuO2 | Fe2O3 | 0.85 | 1.15 | 48 |
RuO2 | Bi2O3 | 0.95 | 0.77 | 48 |
Oxide no. 1 . | Oxide no. 2 . | VBO/p-SBH (eV) . | NPT (eV) . | Reference . |
---|---|---|---|---|
rut-TiO2 | ana-TiO2 | 0.7 | 0.76 | 35 and 36 |
HfO2 | SiO2 | 1.1–2.2 | 2.29 | 37 and 38 |
Cu2O | Al2O3 | 2.6 | 1.78 | 39 |
Cu2O | Bi2O3 | 0.9 | 0.84 | 39 |
Cu2O | Sn:In2O3 | 2.1 | 2.15 | 40 |
Cu2O | ZnO | 1.5–2.9 | 1.74 | 41–43 |
HfO2 | HfxSi1−xO2 | 0.8 | 0.78 | 44 |
HfO2 | β Ga2O3 | 0.5 | 0.56 | 45 |
ZrO2 | β Ga2O3 | 0.3 | 0.40 | 45 |
Al2O3 | Sn:In2O3 | 0.2 | −0.32 | 46 |
SnO2 | In2O3 | 0 | 0.03 | 15 |
SnO2 | ZnO | 0 | −0.38 | 14 |
Al2O3 | ZnO:Al | 0 | −0.04 | 14 |
ZnO | ana-TiO2 | 0.57 | 0.71 | 47 |
RuO2 | Ta2O5 | 2.38 | 2.55 | 14 |
RuO2 | ZnO | 2.2 | 1.67 | 14 |
RuO2 | ana-TiO2 | 2.35 | 2.37 | 14 |
RuO2 | rut-TiO2 | 1.65 | 1.62 | 14 |
RuO2 | Fe2O3 | 0.85 | 1.15 | 48 |
RuO2 | Bi2O3 | 0.95 | 0.77 | 48 |
V. MODELING EXPERIMENTALLY DETERMINED VBO/SBHs
The majority of interfaces presently investigated are polycrystalline and, therefore, may contain areas with different atomic structures and band alignment conditions. Polycrystallinity, thus, seems to make it difficult for the interface charge distribution to be modeled. However, this obstacle does not necessarily prohibit a density-based approach from accurately predicting the BO/SBH for certain polycrystalline interfaces, because of some common rules that all interfaces should follow, irrespective of its actual orientation or atomic structure. In any crystal with polar bonds, it is energetically favorable for the cations to stay away from each other and be surrounded by anions and vice versa for the anions. Following any path on the skeletal bond diagram of a bulk crystal, one expects to find cations and anions to alternate. This general rule, observed for all bulk compounds, makes a strong case for the network of alternating polarized bonds to continue across any interface, irrespective of the interface orientation. Some consistency in the arrangement of atoms may, therefore, be expected simply based on energetics, even at polycrystalline interfaces of an unknown atomic structure.
Between some materials, e.g., binary oxides, this rule of alternating arrangement of cations and anions alone is enough to allow the BO/SBH of the interface to be adequately modeled. For other interfaces, variations in the BO/SBH may arise depending on which of the several available cations and which of the several anions are actually at the interface. This concern is particularly valid for compounds with cations or anions of different valences, such as some perovskite oxides. For these interfaces, additional assumptions could be made about the interface structure to make more realistic predictions. Then, there are still other interfaces, e.g., those between covalent and ionic compounds, where the polarization of the interface bonds and, thus, the BO/SBH should depend on the interface structure. Because of competition, the structures of these interfaces are not always predictable from simple energetic arguments. As suggested above, successful modeling of band alignment conditions at these interfaces becomes possible only if the effect of variations in bond polarization is averaged out from polycrystallinity and the randomness in the interface structure. These different types of interfaces are discussed separately.
A. Interfaces of binary oxides
Each binary oxide has one type of metal cation and one type of anion (oxygen). Because oxygen is the common anion throughout, the bonding sequence for an interface between any two binary oxides is essentially predictable. For example, between HfO2 and SiO2, one expects to find –Hf–O–Si– at the interface, regardless of the orientation. Here, the band alignment conditions for these interfaces are modeled by NPT and compared with those experimentally measured. The good agreement between experiment and theory is shown clearly in Fig. 2, with the data for the individual interfaces listed in Table V. This agreement suggests that, under the condition of a continuous network of bonds, the effect of lattice and charge relaxation from stacked bulk densities is small and arises mainly from the smoothing out of charge density differences. There is no need to account for new interface bonds when the bulk network of polar bonds continues across these interfaces.
Because binary oxide interfaces are clearly ionic, one may wonder whether the generic Bader model solids also work here. Surprisingly, the answer is no. The non-polar Bader model solid for SiO2 is terminated on a ½ monolayer of SiO2, whereas that for HfO2 is terminated on a ½ monolayer of HfO2. When these two model solids are stacked together, a non-polar (SiO2)/2 + (HfO2)/2 layer is created at the interface. Such a finely juxtaposed structure costs energy and is not expected to be important for a real interface. Simply from the required continuation of a polarized bond network, a polar interface structure (–Hf–O–Si–) is expected, which cannot be reached by non-polar model solids. Non-polar AIM model solids generally do not produce reasonable band alignment conditions (not shown). The importance of using suitable model solids should be noted whenever charged AIM densities are employed in the analysis. On the other hand, the expected polar interface structure here may be recreated with NPT model solids, with each model solid contributing ½ of the interface oxygen layer and without modifications to the model solid potentials. For these reasons, the NPT technique is chosen for the analysis, even though these interfaces are clearly ionic. With Fig. 2 and Table V, one of the questions the present work aims to address is answered: band alignment prediction through atomic densities can be accurate even for polycrystalline interfaces.
B. Interfaces with perovskite oxides
Each perovskite oxide has two cations, usually of different formal valences. A simple application of the rule of an alternating cation and anion does not lead to a unique choice of the interface structure. Which of the two cations of the perovskite is actually at the interface with another material may affect the band alignment condition significantly. Therefore, to accurately model the BO/SBH for perovskite interfaces, the effect of polarized bonds should not be ignored, if possible. As the average potential energy of an NPT model solid is independent of its surface structure, it is difficult to correct for interface bonds with the NPT approach. An NPT prediction for a perovskite interface, thus, amounts to an interface average, in the absence of a dominant atomic structure. NPT-modeled band offsets of all the present perovskite interfaces, summarized in Table VI, show varied degrees of agreement with those experimentally measured.
Material no. 1 . | Material no. 2 . | VBO (eV) . | NPT (eV) . | Bader (eV) . | Reference . |
---|---|---|---|---|---|
BaTiO3 | SrTiO3 | 0 | −0.22 | 0.24 | 14 |
PbTiO3 | SrTiO3 | 1.1 | 0.28 | 0.97 | 51 |
LaAlO3 | SrTiO3 | 0.15 | 0.42 | 52 | |
SrTiO3 | BaSnO3 | 0.28 | 0.89 | −0.11 | 53 |
LaAlO3 | BaSnO3 | 0.5 | 1.31 | 53 | |
SrTiO3 | SrZrO3 | 0.5 | 0.66 | 54 | |
(Ba,Sr)TiO3 | a-TiO2 | 0 | 1.05 | −0.06 | 14 |
(Ba,Sr)TiO3 | Ta2O5 | 0 | 1.23 | −0.15 | 14 |
LaAlO3 | SiO2 | 2.5 | 3.14 | 55 | |
SrTiO3 | SnO2 | 0.85 | 1.06 | 56 | |
ana-TiO2 | SrTiO3 | 0.2 | −1.39 | 0.06 | 57 |
SrTiO3 | MgO | 0 | 0.30 | 58 | |
BiFeO3 | SrTiO3 | 1 | 0.26 | 1.24 | 59 |
GaAs | SrTiO3 | 2.5 | 1.57 | 60; see the text | |
Si | SrTiO3 | 2.1 | 2.32 | 57 and 61 | |
Si | BaTiO3 | 2.4–2.7 | 2.54 | 62 | |
Ge | SrHfO3 | 3.27 | 2.82 | 63 | |
Sn:In2O3 | (Ba,Sr)TiO3 | 0.1 | −0.75 | 0.46 | 64 |
Pb(Zr,Ti)O3 | Sn:In2O3 | 1.1 | 0.74 | 1.02 | 65 |
Al2O3 | (Ba,Sr)TiO3 | 0.15 | −0.38 | 0.20 | 66 |
RuO2 | BiFeO3 | 0.72 | 0.72 | 48 | |
RuO2 | (Bi,Na)TiO3 | 1.05 | 1.34 | 0.94 | 48 |
RuO2 | (K,Na)NbO3 | 2.15 | 1.81 | 14 | |
RuO2 | PbZrO3 | 0.7–0.8 | 1.36 | 0.64 | 14 |
RuO2 | (Ba,Sr)TiO3 | 2.35 | 1.32 | 2.35 | 67 |
RuO2 | Pb(Zr,Ti)O3 | 1 | 1.34 | 65 |
Material no. 1 . | Material no. 2 . | VBO (eV) . | NPT (eV) . | Bader (eV) . | Reference . |
---|---|---|---|---|---|
BaTiO3 | SrTiO3 | 0 | −0.22 | 0.24 | 14 |
PbTiO3 | SrTiO3 | 1.1 | 0.28 | 0.97 | 51 |
LaAlO3 | SrTiO3 | 0.15 | 0.42 | 52 | |
SrTiO3 | BaSnO3 | 0.28 | 0.89 | −0.11 | 53 |
LaAlO3 | BaSnO3 | 0.5 | 1.31 | 53 | |
SrTiO3 | SrZrO3 | 0.5 | 0.66 | 54 | |
(Ba,Sr)TiO3 | a-TiO2 | 0 | 1.05 | −0.06 | 14 |
(Ba,Sr)TiO3 | Ta2O5 | 0 | 1.23 | −0.15 | 14 |
LaAlO3 | SiO2 | 2.5 | 3.14 | 55 | |
SrTiO3 | SnO2 | 0.85 | 1.06 | 56 | |
ana-TiO2 | SrTiO3 | 0.2 | −1.39 | 0.06 | 57 |
SrTiO3 | MgO | 0 | 0.30 | 58 | |
BiFeO3 | SrTiO3 | 1 | 0.26 | 1.24 | 59 |
GaAs | SrTiO3 | 2.5 | 1.57 | 60; see the text | |
Si | SrTiO3 | 2.1 | 2.32 | 57 and 61 | |
Si | BaTiO3 | 2.4–2.7 | 2.54 | 62 | |
Ge | SrHfO3 | 3.27 | 2.82 | 63 | |
Sn:In2O3 | (Ba,Sr)TiO3 | 0.1 | −0.75 | 0.46 | 64 |
Pb(Zr,Ti)O3 | Sn:In2O3 | 1.1 | 0.74 | 1.02 | 65 |
Al2O3 | (Ba,Sr)TiO3 | 0.15 | −0.38 | 0.20 | 66 |
RuO2 | BiFeO3 | 0.72 | 0.72 | 48 | |
RuO2 | (Bi,Na)TiO3 | 1.05 | 1.34 | 0.94 | 48 |
RuO2 | (K,Na)NbO3 | 2.15 | 1.81 | 14 | |
RuO2 | PbZrO3 | 0.7–0.8 | 1.36 | 0.64 | 14 |
RuO2 | (Ba,Sr)TiO3 | 2.35 | 1.32 | 2.35 | 67 |
RuO2 | Pb(Zr,Ti)O3 | 1 | 1.34 | 65 |
As atoms defined by the Bader method have better transferability among perovskite oxides than NPT, their use, when appropriate, is expected to give more accurate predictions of the band offsets. For some of the interfaces shown in Table VI, the interface structure may actually be reasonably deduced. For example, at the BaTiO3/SrTiO3 interface, one expects the two crystals to be joined at a common TiOx layer. We could use group II-centered (A-centered) Bader model solids, listed in Table II, to represent both crystals of this interface. This strategy recreates the “shared” TiO2 layer that still remains overall neutral. Much improved VBO prediction over the NPT model is seen for the Bader model for this interface. Other interfaces that have been modeled with A-centered Bader model solids are PbTiO3/SrTiO3 and SrTiO3/BaSnO3. In the case of the latter, the common layer created at the interface is a mixed Sn0.5Ti0.5Ox layer. Because Sn and Ti are of the same formal valence, there is no issue with an interface charge. The modeled band offset may be viewed as an average result between an interface with an –SrOx–TiOx–BaOx– structure and that with the –SrOx–SnOx–BaOx– structure. The AIM projections for these interfaces are much closer to the experiment than those modeled with NPT.
Some interfaces between perovskite oxides and binary oxides have structures that are either deducible or may be narrowed down to a few choices with similar band offset consequences. These include all the yet-undiscussed interfaces in Table VI that are marked with predictions from the Bader method. The interface between anatase-TiO2 and SrTiO3 best demonstrates the suitable Bader strategy for these interfaces. The structure is expected to change locally from a TiOx environment to an SrOx environment immediately across the TiO2/SrTiO3 interface. Such an interface structure can be reproduced by using the “generic” Bader model solid, of which the VBM position is listed in Table I, to represent the anatase-TiO2 and the model solid centered about the Sr Bader atom to represent the SrTiO3. These two Bader model solids each contribute one-half monolayer of TiO2 to the interface. The expected atomic structure of the interface is, thus, recreated in this Bader approach without inflicting a net charge for the interface, and indeed, the experimental VBO is very well described by the Bader method, as illustrated in Table VI. Other interfaces in this group, such as the In2O3/(Ba,Sr)TiO3, do not have common cations between the two compounds. With the AIM procedures just described, a mixed InxTiyOz layer is formed at the interface. Such an analysis would produce a result that is an average of interface regions with the –InOx–TiOx–(Ba,Sr)Ox– and the –InOx–(Ba,Sr)Ox–TiOx– structures. Given that the valences of Ti and In are close, these structures are likely more favorable than other stacking sequences at the interface. The AIM model, thus, describes a more probable structure of this interface than the completely random structure assumed in the NPT analysis. Significantly more accurate band alignment conditions are found for the Bader method as a result. It should be noted that the BiFeO3 compound has been handled as a binary oxide because of the similarity in Bader charges for Bi and Fe.
Experimental band alignment results from Table VI are plotted in Fig. 3 against either the predictions of AIM analysis when available or the NPT projections. Very good agreement is found for most of the interfaces, thereby confirming the validity of density-based analysis of polycrystalline interfaces. The only real disagreement is seen for the GaAs/SrTiO3 system, where the experimental data were obtained from epitaxial (100) interfaces fabricated with an As–TiO2 structure at the interface.60,68 This is a structure with a disrupted network of bonds and one that cannot be simulated by placing Bader atoms at all positions of the atom of the interface, as doing so would lead to a net charge for the interface. The NPT value calculated without considering the bond polarization at the interface, 1.57 eV, is still of some use. The correction for bond polarization is not easily estimated here because there are three atomic species involved, although it seems almost certain to significantly increase the NPT value on account of a much larger electronegativity of oxygen than arsenic. Such a correction would significantly improve the agreement between the NPT prediction and the experimental VBO.
C. Interfaces between oxides and zincblende/wurtzite semiconductors
The experimental VBOs between oxides and zincblende/wurtzite/diamond semiconductors are compared with the NPT model in Table VII and plotted in Fig. 4. As more than one cation and anion are available for every interface, the VBO is expected to depend on the atomic structure for most of these interfaces. With a few exceptions, there is no strong energetic argument to favor a particular interface structure over another without actual calculation. Any agreement of the experimental VBO with the NPT model here is, thus, an indication that the effect of polar bonds at the interface is small or has been averaged to be insignificant due to polycrystallinity. An examination of Table VII shows that the agreement between the NPT and experiment is quite reasonable for most of these interfaces. There are a few interfaces with very significant disagreements that should be examined more closely. Of all interfaces, the largest errors, ∼2 eV, are found for all forms of the SiC semiconductors. Even though both of its constituents are group IV elements, all forms of silicon carbide have huge differences in the total number of electrons contained in the Bader basins of C (∼6.8 e) and Si (∼1.2 e). SiC is typically fabricated at high temperatures, and investigations have found signatures of C–C bonds and evidence for a graphitic structure at SiC interfaces.69–71 Based on these reports, it seems reasonable to assume the presence of an Si–C dipole, associated with the defects, at the interface with the positive pole pointing away from the interface and toward the silicon carbide, independent of other details of the interface. The large magnitude of this dipole would significantly lower the VBM for the SiC bulk crystal and move the NPT values in the direction of the experimental VBO for all SiC interfaces studied.
Material no. 1 . | Material no. 2 . | VBO (eV) . | NPT (eV) . | Reference . |
---|---|---|---|---|
GaAs | Ga2O3 | 2.6 | 2.56 | 72 |
GaAs | Gd2O3 | 1.9 | 2.54 | 73 |
GaAs | HfO2 | 2.3 | 2.00 | 50 |
GaAs | ZrO2 | 2.2 | 2.16 | 50 |
GaAs | SiO2 | 3.4 | 4.29 | 73 (see the text) |
GaAs | In2O3 | 2.7 | 2.66 | 74 |
GaAs | ZnO | 2.3 | 2.25 | 75 |
InGaAs | HfO2 | 3.37 | 1.97 | 76 |
InGaAs | InP | 0.345 | 0.36 | 77 |
InSb | Al2O3 | 3.6–3.8 | 2.33 | 78 and 79 |
GaN | HfO2 | 0.3 | 0.83 | 80 |
GaN | SiO2 | 2 | 3.12 | 81 (see the text) |
GaN | Gd2O3 | 1 | 1.10 | 82 |
GaN | Si3N4 | −0.5, 1 | −0.41 | 83 and 84 |
GaN | ZnO | 0.7–1.4 | 1.08 | 85–87 |
AlN | ZnO | 0.43 | 1.12 | 87 |
InN | ZnO | 1.95 | 1.53 | 87 |
Ge | Lu2O3 | 2.9–3.0 | 3.40 | 88 |
Ge | ZrO2 | 2.8–3.4 | 3.18 | 50 and 89 |
Ge | HfO2 | 3 | 3.01 | 90 and 91 |
Ge | Gd2O3 | 2.9 | 3.29 | 90 |
Si | HfO2 | 2.5–3.5 | 2.75 | 38, 50, and 82 |
Si | HfSiO4 | 2.69 | 3.53 | 92 |
Si | SiO2 | 4.75 | 5.04 | 37 |
Si | HfxSi1−xO2 | 3.8 | 3.53 | 44 |
Si | Al2O3 | 3.25 | 3.04 | 93 |
Si | ZrO2 | 1.9–3.3 | 2.91 | 89, 93, and 94 |
Si | ZnO | 2.8 | 3.00 | 95 |
Si | SiC beta3C | 1.7 | 0.09 | 96 (see the text) |
4H SiC | SiO2 | 2.9 | 4.94 | 97 (see the text) |
4H SiC | HfO2 | 1.74 | 2.65 | 98 (see the text) |
4H SiC | ZrO2 | 0.52 | 2.81 | 99 (see the text) |
6H SiC | SiO2 | 2.9 | 4.96 | 97 (see the text) |
CdS | TiO2 | 1.07 | 1.59 | 100 |
CdS | ZnO | 0.96–1.2 | 0.88 | 101 and 102 |
CdS | Sn:In2O3 | 1.25 | 1.29 | 103 |
CdS | SnO2 | 1.2–1.5 | 1.26 | 103 and 104 |
CdTe | SnO2 | 2 | 1.70 | 104 |
CdTe | In2O3 | 2.1 | 1.73 | 105 |
CdTe | CdS | 0.65–0.96 | 0.83 | 106 and 107 |
ZnTe | ZnO | 2.4 | 1.60 | 108 (see the text) |
InSe | In2O3 | 2.05 | 1.98 | 109 |
CIGS | ZnS | 0.8–1.4 | 1.05 | 110 and 111 |
CIGS | ZnO | 2.26 | 1.78 | 111 |
Cu2S | TiO2 | 2.9 | 3.48 | 112 |
CuInSe2 | CdS | 0.8–0.9 | 0.84 | 113 |
Material no. 1 . | Material no. 2 . | VBO (eV) . | NPT (eV) . | Reference . |
---|---|---|---|---|
GaAs | Ga2O3 | 2.6 | 2.56 | 72 |
GaAs | Gd2O3 | 1.9 | 2.54 | 73 |
GaAs | HfO2 | 2.3 | 2.00 | 50 |
GaAs | ZrO2 | 2.2 | 2.16 | 50 |
GaAs | SiO2 | 3.4 | 4.29 | 73 (see the text) |
GaAs | In2O3 | 2.7 | 2.66 | 74 |
GaAs | ZnO | 2.3 | 2.25 | 75 |
InGaAs | HfO2 | 3.37 | 1.97 | 76 |
InGaAs | InP | 0.345 | 0.36 | 77 |
InSb | Al2O3 | 3.6–3.8 | 2.33 | 78 and 79 |
GaN | HfO2 | 0.3 | 0.83 | 80 |
GaN | SiO2 | 2 | 3.12 | 81 (see the text) |
GaN | Gd2O3 | 1 | 1.10 | 82 |
GaN | Si3N4 | −0.5, 1 | −0.41 | 83 and 84 |
GaN | ZnO | 0.7–1.4 | 1.08 | 85–87 |
AlN | ZnO | 0.43 | 1.12 | 87 |
InN | ZnO | 1.95 | 1.53 | 87 |
Ge | Lu2O3 | 2.9–3.0 | 3.40 | 88 |
Ge | ZrO2 | 2.8–3.4 | 3.18 | 50 and 89 |
Ge | HfO2 | 3 | 3.01 | 90 and 91 |
Ge | Gd2O3 | 2.9 | 3.29 | 90 |
Si | HfO2 | 2.5–3.5 | 2.75 | 38, 50, and 82 |
Si | HfSiO4 | 2.69 | 3.53 | 92 |
Si | SiO2 | 4.75 | 5.04 | 37 |
Si | HfxSi1−xO2 | 3.8 | 3.53 | 44 |
Si | Al2O3 | 3.25 | 3.04 | 93 |
Si | ZrO2 | 1.9–3.3 | 2.91 | 89, 93, and 94 |
Si | ZnO | 2.8 | 3.00 | 95 |
Si | SiC beta3C | 1.7 | 0.09 | 96 (see the text) |
4H SiC | SiO2 | 2.9 | 4.94 | 97 (see the text) |
4H SiC | HfO2 | 1.74 | 2.65 | 98 (see the text) |
4H SiC | ZrO2 | 0.52 | 2.81 | 99 (see the text) |
6H SiC | SiO2 | 2.9 | 4.96 | 97 (see the text) |
CdS | TiO2 | 1.07 | 1.59 | 100 |
CdS | ZnO | 0.96–1.2 | 0.88 | 101 and 102 |
CdS | Sn:In2O3 | 1.25 | 1.29 | 103 |
CdS | SnO2 | 1.2–1.5 | 1.26 | 103 and 104 |
CdTe | SnO2 | 2 | 1.70 | 104 |
CdTe | In2O3 | 2.1 | 1.73 | 105 |
CdTe | CdS | 0.65–0.96 | 0.83 | 106 and 107 |
ZnTe | ZnO | 2.4 | 1.60 | 108 (see the text) |
InSe | In2O3 | 2.05 | 1.98 | 109 |
CIGS | ZnS | 0.8–1.4 | 1.05 | 110 and 111 |
CIGS | ZnO | 2.26 | 1.78 | 111 |
Cu2S | TiO2 | 2.9 | 3.48 | 112 |
CuInSe2 | CdS | 0.8–0.9 | 0.84 | 113 |
On a separate issue, it may be noticed that the VBOs predicted by NPT for SiO2 tend to be too large; i.e., the NPT VBM position with respect to the vacuum level is low. We are reminded that the structure used to calculate the NPT VBM is the crystalline α-glass and not the amorphous SiO2, used in the actual devices studied experimentally. As the atomic density for the crystalline material is much larger than that of the amorphous phase, the average electrostatic potential energy of the NPT model solid for amorphous SiO2 should be higher (less negative) than that of the α-glass. This tends to push the NPT VBM position for amorphous SiO2 higher, although the actual VBM position can only be calculated from amorphous SiO2. Another specific case concerns the ZnTe/ZnO interface, where because of the uninterrupted network of bonds, the interface can be inferred to contain the –Te–Zn–O– bond sequence. Because of the large difference in electronegativities of O and Te, with Bader charges of −0.51 e and −1.29 e, respectively, in ZnO and ZnTe, the effect of bond polarization should be large enough to increase the NPT-predicted VBO to significantly above the experimentally measured value. However, the experimental VBO value is only 2.4 eV, because the VBM of ZnTe cannot move above the conduction-band minimum (CBM) of ZnO, which has a bandgap of ∼2.4 eV.
VI. DISCUSSIONS AND CONCLUSIONS
The alignment of energy bands at solid interfaces is controlled by the distribution of charge at the interface, and the distribution of charge in all materials is governed by chemistry. A comprehensive understanding of the formation of interface band alignment has to start from the charge distribution. Yet, for many decades, traditional density-based theories, plagued by extraneous surface dipoles contained therein, could not explain the BO/SBH. It was only after a recent demonstration of how the surface dipole might be removed consistently for all crystals that the charge density of the interface could be accurately modeled. With the help of NPT, the BO/SBH values at various epitaxial interfaces were quantitatively explained.114,115 Additionally, the presence of metallic bonds at zincblende semiconductor interfaces was shown to be responsible for the (in)famous Fermi level pinning phenomenon.114,115 The main reason that the interface charge distribution is predictable is the similarity of the local density of the interface to that found in bulk crystals. With the nearsightedness principle of the electron density in effect, each interface charge distribution is approximately constructed with that harvested from the bulk crystals, leading to a reasonable deduction of the BO/SBH for that interface. These recent developments rightfully restore “the distribution of charge” back to the pivotal position in the explanation of all issues related to the alignment of energy bands at interfaces.
The novelty of the present work lies in showing how and when the band alignment conditions at polycrystalline interfaces with largely unknown structures may be predicted with the density-based approach. Polycrystalline interfaces are important because they are crucial for various devices, including solar cells. A wide range of materials with different structures are analyzed in the present work, which is made possible by expanding the NPT technique to efficiently partition unit cells with essentially an unlimited number of atomic species and/or inequivalent sites. To model the interface charge distribution as closely as possible, a study of the transferability of atomic densities is undertaken and it shows that atoms defined by NPT are generally transferable and are, thus, suitable for the analysis of all interfaces, while Bader atoms defined by AIM may have a slight edge for ionic compounds. Therefore, two types of model solids are also constructed with the AIM technique to help with the analysis of some ionic interfaces that are amenable.
Because the network of polar bonds for bulk compounds is expected to continue through an interface, the allowed stacking sequence(s) of atomic species at some interfaces are predictable despite polycrystallinity. It is particularly obvious for the interfaces between binary oxides, as only oxygen is available as the anion for the entire interface. Based on this analysis, the charge distribution at such an interface is expected, and indeed has been found, to be similar to that constructed from NPT model solids of the two bulk crystals, without the need to consider the polarization associated new bonds at the interface. The excellent agreement between experiment and NPT projections for interfaces between binary oxides not only demonstrates the accuracy for density-based theory to explain band alignment conditions at polycrystalline interfaces but also identifies “bond similarity” as the underlying reason for the occasional observation of bulk-like behavior in the experimental studies of interfaces among a group of compounds.
For perovskite oxide interfaces, the band alignment usually depends on which of the two cations of the perovskite is actually at the interface and, hence, is modeled by the structure-independent NPT technique. However, at interfaces with other oxides, oxygen becomes the only anion at the interface and the interface structure may, thus, be deduced or narrowed down to few choices. For these interfaces, the atomic densities defined by the AIM technique are preferred for better transferability. Because of the limited availability of AIM model solids, only some of the interfaces with perovskite oxides can be analyzed. Better agreement with the experimental band alignment conditions is indeed found for AIM predictions than by the NPT for these interfaces. Relying on the technique available and best suited for each interface, theoretical predictions are in good agreement with the experimental results.
Interfaces between covalent and ionic semiconductors have more than one cation or anion that may reside at the interface, with different consequences for the band alignment. In the absence of information on their structures, the band alignment conditions of these (polycrystalline) interfaces can only be modeled “generically,” without concerns for the specific bonds at the interface, with NPT model solids. The general philosophy here is that the randomness in the range of polycrystalline interface structures may partially average out the effect due to the variety of polarized bonds at the interface. This turns out to be a reasonable approximation for the majority of the actual polycrystalline interfaces, as the experimental band alignment conditions agree with the NPT predictions. The significant discrepancies between experimental results and NPT predictions for other polycrystalline interfaces, including those with SiC, are discussed and shown to be attributable to some specifics of the interfaces observed in the experiment. All in all, the NPT approach offers reasonable explanations of the band alignment at these interfaces, which is consistent with the general absence of large effects due to interface bonds at polycrystalline interfaces.
From an examination of Figs. 3–5, which span a wide range of materials with different types of bonds, the consistent picture that emerges is that the band alignment condition, even for polycrystalline interfaces, can be well understood in terms of the distribution of charge. In addition to simple chemistry, there is no evidence for other factors, such as those assumed in empirical models, that control the charge distribution and, thus, the band alignment. Unfortunately, there is no single approach to model the charge distribution that is right for all the interfaces. Attention should be paid to what chemistry tells us about the bonds at each interface. At interfaces where the bulk network of polar bonds is expected to continue without interruption, there is no need to consider the polarization of interface bonds. Both the NPT and the AIM, if proper model solids may be constructed, should be accurate. At interfaces between dissimilar materials, knowledge of the interface structure is required to accurately predict the effect of the interface bonds. With the atomic structure known, AIM is the preferred technique to model the effect of interface bonds. Without a known structure, one resorts to the NPT approach and relies on the randomness in a polycrystalline structure to average out the effect of interface bond polarization. Plenty of examples are shown in the present work. These recipes for band alignment prediction are built on the sound scientific basis of a charge distribution. As such, they serve as guidelines for band alignment prediction for yet unexplored or novel interfaces.
ACKNOWLEDGMENTS
The author thanks Leeor Kronik and David Cahen for scientific discussions and critical reading of the manuscript. He is grateful to Dahvyd Wing and Guy Ohad for technical assistance.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
Raymond T Tung: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.
APPENDIX: NEUTRAL POLYHEDRA DEFINITION
In NPT, the charge density of a unit cell is partitioned to define neutral volumes about each atom. As shown previously through the variational principle,19 the total spherapole of the unit cell is minimized when the boundary separating any two atoms i and j is planar and perpendicular to the displacement between the two nuclei, . Each neutral volume for the model solid with maximized average potential energy is, thus, in the shape of a polyhedron, giving this partition method its name. The mere fact that each neutral volume was in the shape of a polyhedron was enough for NPT analysis to be performed previously on crystals with a single element, such as metals; with two elements, such as zincblende semiconductors; and with three elements, such as perovskite oxides. To extend the NPT analysis to all technologically important solids, which include crystals of lower symmetry and with large unit cells, a definitive and efficient approach to search for the neutral polyhedra that minimizes the sum of the spherapoles is needed.