The physics and chemistry of the Schottky barrier height Free

The electrical current flowing across the interface between a metal and a semiconductor is usually non-linear against the applied bias voltage, as the result of a discontinuity on the energy scale of the electronic states responsible for conduction in the two materials. In Fig. 1, a band diagram of the interface is sketched to show this discontinuity. Delocalized electronic states around the Fermi level (FL) are responsible for electrical conduction in the metal, drawn on the left, but these states are not coupled to any delocalized electronic states in the semiconductor drawn on the right. The set of electronic states responsible for electrical conduction in the semiconductor depends on the doping type of the semiconductor. For n-type semiconductors, the electrons near its conduction band minimum (CBM) are primarily responsible for electrical conduction, and for p-type semiconductors, holes near the valence band maximum (VBM) carry most of the current. Because of the presence of the fundamental band gap, the lowest-lying states for n-type semiconductor that can communicate with electrons in the metal are now at an energy $ΦB,n0$ above the FL, as shown in the figure. For electronic transport across the metal-semiconductor (MS) interface, this energy offset, known as the n-type Schottky barrier height (SBH), manifests itself as a potential energy barrier that leads to rectifying behavior between the metal and the n-type semiconductor, i.e., the flow of electrons from the semiconductor to the metal is easier than conduction in the opposite direction. If the semiconductor in contact with the metal is doped p-type, the energy difference between the FL and the VBM, marked as $ΦB,p0$⁠, is now the energy barrier, i.e., the p-type SBH, that controls the transport of holes across this MS interface. Because the current flow across an MS interface depends exponentially on the magnitude of the SBH at common applied voltages, the SBH is clearly the most important property of an MS interface that decides its electrical characteristics. From a scientific perspective, the explanation of the alignment condition between the energy bands across an MS interface represents an interesting challenge, and there is a long history of extensive scientific investigations attempting to meet that challenge. As a matter of fact, the formation mechanism of the SBH has been an area of very active research in condensed matter physics and surface science, dating back at least sixty years. There were annual conferences dedicated to this topic and focused sessions at every large international meeting on electronic materials in the 1970s and 1980s to address the SBH “mystery.” This intense scientific interest in the SBH still paled in comparison with the pressure and demand within the semiconductor industry to solve the contact problem for various devices. In terms of applications, the magnitude of the SBH is of overriding importance for electronic devices, and “solving the contact problem” essentially means to be able to control, or “tune,” the SBH for the MS interfaces of any desirable device. Such a technology-related problem should not be difficult to solve if the formation mechanism of the SBH is thoroughly understood. With so much effort that has gone into SBH research, and given the astonishingly sophisticated state of today's semiconductor devices and fabrication technologies, it seems a sure bet that the formation mechanism of the SBH should be, by now, a well understood subject theoretically and the magnitude of the SBH is a quantity that can be skillfully controlled technologically. However, a glimpse into most recent literature on the SBH and ohmic contacts still finds a wide spectrum of conflicting opinions and models, with which SBH data are interpreted. Also, except for sporadic demonstrations of strategies capable of altering the SBH in isolated cases, techniques effective in achieving desired SBH for any semiconductor system in general are still lacking. It seems only natural to ask “why haven't we learned more about the SBH after all the scientific and technological investigations?”

The lack of a coherent explanation of a wide spectrum of experimentally observed SBH data and the lack of control over the magnitude of the SBH, despite decades of intensive investigations, speak volumes about the difficult and complex nature of the SBH problem itself. The main sticking point can be traced to a sharp dependence of the SBH on the atomic structure of the MS interface, which has slowly yet surely been established through careful experimental and theoretical investigations. Compounding that dependence by a lack of control and understanding of the atomic structure at common MS interfaces routinely fabricated in the laboratories, and the SBH researchers really had an uphill battle on their hands. Presently, short of full-fledged numerical calculations, there are still no comprehensive and quantitative theories on the formation of interface dipole to explicitly address this most crucial part, i.e., the structural dependence, of the SBH. On the other hand, simple models of SBH are the ones that are easy to understand, easy to use, and the only ones that have been treated in standard textbooks.^1–3 Even though it has been pointed out that the basic assumptions made in simple models of the SBH are at odds with what ab initio calculations revealed about real MS interfaces,⁴ researchers still cling to them out of convenience and are hesitant to look beyond them for a more complete and realistic view of the SBH, presumably one that's built on the rigor of quantum mechanics. This gap between the complexity of the problem and the simplicity of the models it has continued to be compared with has been the main obstacle to SBH research that really shouldn't have been let happen. Continued research and technology development efforts in the same mode will almost certainly leave us in the doldrums, which we cannot afford to do at a time when the continued advances in technologies and the searches for novel devices demand an immediate solution to the contact issue.

The main purpose of this review article is to point out, through an examination of experimental and theoretical results on atomically controlled MS interfaces and known fundamental principles, that a nearly complete quantum-mechanics-based picture of SBH formation can already be constructed. However, as is often the problem with quantum descriptions of any realistic and practical phenomena, the formation of the SBH turns out to be intricately dependent on the specifics of the MS interface under study. Therefore, quantum description of the SBH in general cannot be formulated into simple analytic equations that are applicable for all MS systems, especially since there are usually so many important details about any MS interface that are not experimentally known. However, if one is willing to look beyond the inconvenience of not having a short answer for every SBH situation waiting to be explained, there are actually several basic principles that transcend all interface specificities and it is the interplay of these overarching principles under different circumstances that lead MS interfaces to display a wide range of intriguing and sometimes puzzling SBH behavior. These basic rules, which traditionally belong to, and are discussed in, their respective disciplines of electrostatics, quantum theory of solids, chemical bond formation, quantum transport, etc., together describe the formation of the SBH and therefore they should also form the knowledge base with which we understand SBH data and tackle technological SBH problems. In Sec. III, we will introduce each of these principles in their traditional settings and discuss in detail the specific roles each plays in the SBH. As the prospects for ever finding one fixed rule for the entire field of MS interfaces look very dim, these concepts are the only faithful ones that we can fall back on and the ones that we would need to put together for specific MS interface, under the known circumstances surrounding its formation, in order to make sense of its particular SBH behavior. Putting together separate ideas from their varied origins is mainly what this article intends to do. We will go over the basic concepts that are important for the SBH, examine real examples to see what roles these concepts play in shaping the actual experimental observations. The concepts that are fundamentally important for a sound description of the SBH come from the realms of both “physics” and “chemistry.” Therefore, when an esteemed Editor of this journal suggested the current title for this article, which contains both of these buzzwords, the author gratefully agreed. This turned out to be far better than any title for this article the author had thought of on his own! The plan of this review article is as follows: We will summarize what has been discovered about the SBH through experiments and theoretical calculations, both historically and recently, in Sec. II, where we will also go over the philosophy and circumstances behind some simple models of the SBH. Since most of the discussions here are on topics that have already been covered in some detail in prior reviews, Sec. II will be brief in places but will serve the function of directing interested readers to original work for further reading. Section III will be devoted to in-depth discussions of classical and quantum concepts that are essential and necessary in order to properly analyze the formation of interface dipole for specific MS system. Section IV will examine recent experimental and theoretical work on SBH adjustment at technologically important interfaces, from the perspective of the physics and chemistry expected at these MS interfaces.

In-depth discussions and explanations of the magnitude of the SBH usually begin with a consideration of the Schottky-Mott Rule,^5,6 which is an excellent practice because the Schottky-Mott Rule is a stalwart reminder of the critical role played by electrostatics in all energy-level-alignment types of problem, including the SBH. When a crystal is in isolation, the positions of its internal energy bands can be referenced to the “vacuum level” outside, as illustrated in Fig. 2(a), which shows that the energy difference between the vacuum level and the Fermi level of an isolated metal is the metal work function, $ϕM$⁠. Hidden in a band diagram like this are the facts that a surface of the solid has been created and that the surface has specific orientation, atomic structure(s), and electronic structures that likely influence the work function. What should also be pointed out is that the relevant “vacuum level” here is the rest energy of an electron placed just outside the crystal surface, and that only in cases where the long-range electric field vanishes identically in vacuum can this level be equated with the vacuum level at infinity.^4,7 In Fig. 2(a), the bands of an isolated semiconductor are also sketched, with the CBM of the semiconductor drawn at an energy $χSC$ (semiconductor electron affinity) below the vacuum level. Again, we should keep mind that $χSC$ depends on the specifics of the semiconductor surface. The Schottky-Mott Rule prescribes that the alignment condition for the energy bands when the two crystals are each in their isolated state should also prevail over the intimate MS interface between these two crystals. From Fig. 2(a), before the two crystals make contact with each other, the CBM of the semiconductor is at an energy $ϕM−χSC$ above the metal FL. If, after the metal and the semiconductor make contact and form an interface, the CBM is still found to be so positioned with respect to the FL, then we say that the Schottky-Mott Rule,

is observed for this MS interface. Or, equivalently, the p-type SBH is said to obey the Schottky-Mott Rule, if it is found that

The underlying idea behind the Schottky-Mott Rule, and the related Anderson Rule for heterojunction band offset,⁸ is the “superposition principle” of electrostatic potential. In practice, these theories seem as simple as “lining up the vacuum levels outside the two surfaces.” Schematically shown in Fig. 2(a) is the distribution of the electrostatic potential energy near the free surfaces of the two crystals. In the interior of each crystal, the electrostatic potential is periodic, arising from the periodic distribution of electronic and nuclear charge. The average electrostatic potential energy per unit cell for the metal, when referenced to the vacuum level, is assumed to be $−eVM¯$⁠. In a bulk crystal, all the energy levels and bands have definite and fixed relationship with respect to the electrostatic potential of the crystal. We may identify the Fermi energy of the metal relative to its average potential energy as its internal chemical potential, $μM(i)$⁠, i.e.,

Note that $μM(i)$ is calculable with quantum mechanics and is a pure bulk quantity that does not depend on the structure of the crystal surface, while the other two terms in Eq. (3) depend on the specifics of the metal surface. Also note the electrostatic potential energy arises only from Coulomb interaction, i.e., the average electrostatic energy increases by $qVM¯$ when a test charge q of any kind enters from vacuum into the metallic crystal. Surface scientists have preferred to break down this drop in potential energy from the exterior upon entry into the metal crystal, $−eVM¯$⁠, into a surface dipole contribution that depends on the structure of the surface and a bulk contribution which does not.^9,10 However, a surface dipole cannot be uniquely defined, mainly because the “bulk contribution” cannot be uniquely defined,¹¹ as discussed later. From the distribution of the electrostatic potential energy near the surface of a semiconductor shown in Fig. 2(a), we can also define an average electrostatic potential energy, $−eVSC¯$⁠, for the interior of the semiconductor. Some clarification should be made about the electrostatic potential inside a semiconductor, which wasn't necessary for metallic crystals. Due to incomplete screening, long-range electric field can be present inside a semiconductor, or any insulating material, making it possible for the electric potential energy to vary with position. The $−eVSC¯$ shown in Fig. 2(a) refers to the average potential energy in the “near-surface” region of the semiconductor. The VBM of the semiconductor, being the highest occupied state in the semiconductor, is the sum of the average electrostatic potential energy of the crystal and its internal chemical potential,

While the metal and semiconductor are separated, as shown in Fig. 2(a), the difference between the average electrostatic potential energy of the metal and that of the semiconductor is $−eVM¯+eVSC¯$⁠. If the charge distribution of both the metal and the semiconductor is held fixed, and the two crystals are brought together, the potential energy difference will remain the same, as graphically demonstrated in Fig. 2(b). This is necessarily the case because the electrostatic potential energy arises only from charge distribution and obeys superposition principle. Therefore, under the condition that charge rearrangement can be strictly avoided during MS interface formation, the band alignment condition for the isolated crystals carries over to the final interface, i.e., the Schottky-Mott Rule will hold. One thus deduces that the Schottky-Mott Rule describes the SBH of the metaphysical, non-interacting MS interfaces, put together without charge or atomic relaxation. Because of such, the Schottky-Mott Rule has very limited relevance for real MS interfaces and is not expected to be able to account for experimentally observed SBH. However, this Rule is a worthwhile indicator of the electrostatics in the system before interface chemistry takes over.

Ever since Schottky diodes became available for experimental investigation decades ago, it was clear that the idealistic, non-interacting picture of the MS interface suggested by the Schottky-Mott Rule was in error. Shown in Fig. 3 are FL positions in the band gap found for various metals evaporated in ultrahigh vacuum (UHV) on n- and p-type GaAs(100) surface, plotted against the work function of the metal.¹² In this particular case, there is no clear dependence of the SBH on the metal work function. The absence of a strong dependence of the SBH on the metal work function was an unexpected and sensational phenomenon that was observed on various covalent semiconductors. This phenomenon was given the name “Fermi level pinning” and captured much of researchers' attention and imagination for the last quarter of the 20th century. The reason that the Schottky-Mott theory fails is obvious: the charge distribution at real MS interfaces is significantly different from a simple superposition of the charge distributions on the original surfaces. In reality, during the formation of an MS interface, the metal and the semiconductor come to within close range of each other. From quantum mechanics, we know that when an atom is brought so close to another atom such that some orbitals from the atoms significantly overlapped, these isolated orbitals are no longer eigenstates of the system. Instead, molecular orbitals would form which could lead to chemical bonds. The chemistry at MS interfaces should not be any different, because metals and semiconductors are made of atoms. Electronic states that used to belong to only the semiconductor or only the metal have to be modified, to some degree, so that they can exist in the combined, larger quantum mechanical system. States that are related to the surfaces of the two crystals will not survive the formation of the MS interface, at least not without serious modifications. New “interface states” that are localized in the interface region and are dependent on the interface atomic structure may form. Because of chemistry, the actual charge distribution at an MS interface will be quite different from a linear superposition of the charge distributions on the two starting surfaces, as depicted in Fig. 2(b). One can attribute the net change in the potential energy as a result of charge rearrangement at the interface to the formation of an additional “interface dipole” $eDint$⁠, namely,

where $ΦB,no$ is the n-type SBH actually observed from an MS interface. We caution that there are many definitions of interface dipole and that the use of this term without a prior definition would be meaningless. The interface dipole can be largely viewed as the result of a transfer of charge between the metal and the semiconductor. While the direction of the charge transfer is usually self-evident, the exact amount of charge transferred, on the other hand, cannot be determined unambiguously. This is because at an MS interface there is no rigorous way to define where the metal begins and the semiconductor ends! It is only by artificial means that a charge transfer can be determined or defined. We shall adopt the convention that a “positive” interface dipole arises when an amount of electron is transferred from the metal to the semiconductor. Since the Schottky-Mott relationship is almost never observed in experiment, it can be concluded that the interface dipole does not vanish and is actually quite significant for the vast majority of MS interfaces. The formation of dipole at MS interfaces thus becomes the main focus of research and development work in SBH. It is not much of an exaggeration to say that “the formation of interface dipole is the entire topic of the SBH.” Casting the formation mechanism of the interface dipole in the light of basic physical and chemical principles is the main focus of Sec. III.

There is an element of timing in all simple models of the SBH that cannot be ignored. These models were all proposed at a time in the history of SBH research when the dominant experimental finding was the FL pinning phenomenon and the sole motivation for SBH theories seemed to be the explanation of this effect. The investigation of the “pinning strength” at a semiconductor (surface) was popular at the time, as this was believed to be a parameter that revealed the FL pinning mechanism. SBH data of various metals measured on one semiconductor were customarily plotted against the work function of the metal, such as shown in Figs. 3 and 4, the latter shows experimentally observed SBH for metals and metal silicides on n-type Si.^13,14 The data presented in plots like these were typically fitted linearly to deduce a slope of the n-type SBH against the metal work function. This slope is called the “interface behavior parameter,” or the “S parameter” of the semiconductor

which is regarded as a measure of the ability of a semiconductor surface to screen out external (metallic) influences. If the Schottky-Mott Rule were observed for a semiconductor surface (⁠$Dint=0$⁠), i.e., no pinning, an S-parameter of 1 would be deduced. As can be seen from Fig. 4, the SBH on Si shows some correlation with the metal work function, but there is much scatter in the data and the overall trend of the data is far from the dashed line with the slope of unity drawn on the same figure to represent the Schottky-Mott prediction. A linear fit of the set of the experimental data shown^13–15 would render an S parameter of ∼0.2 for Si, the small value for which would imply that the FL is “strongly pinned” for Si interfaces. Other covalent semiconductors, such as Ge, GaAs, InP, etc., are also marked by small S-parameters or weak dependencies of the SBH on the metal work function, whereas compound semiconductors with larger ionicity tend to have larger S-parameters, although there are exceptions. How the S-parameters found for different semiconductors correlate with specific properties, such as the ionicity,¹⁶ the polarizability,¹⁷ or the dielectric constant,¹⁸ of the semiconductors also became the subject of discussions and debates in the literature. These research activities of attempting to experimentally establish a strength-of-pinning from the SBH dependence on the metal work function and then, from that, empirically deducing the SBH formation mechanism are unwittingly at the mercy of many other factors at the MS interface to “remain the same” to be meaningful. For example, although the data in Fig. 3 seem to suggest that the dependence of the SBH on the metal work function is weak, the SBH of the various metals nevertheless varies over a range of >0.2 eV in a manner unrelated to the work function. It is also known that SBH measurement of supposedly the same MS interface often exhibited very significant variations in the magnitude of the SBH and/or the details of the junction characteristics, as a result of differences in surface preparation, treatment, and measurement technique. Theoretical and experimental investigations in the last decade of the 20th century would reveal that the electrical characteristics found for most of the (polycrystalline) MS interfaces involved in these empirical studies actually exhibited clear evidence for SBH inhomogeneity.^19,20 In other words, the FL was not uniquely positioned at most of these MS interfaces but rather varied locally, and the measured, supposedly singular, SBH for the entire MS interface was actually just an average among a significant range in the SBH for that interface. The widespread observation of SBH inhomogeneity indicates that the FL is “unpinned” within most MS interfaces, making the dedicated efforts at the time to find the “pinning” mechanism ill-advised. Given these additional bits of information, it only makes sense today for us to look at plots such as Figs. 3 and 4 with some caution. First of all, since each “point” on these plots actually represents a range of SBH, there is at least one other parameter, in addition to the metal work function, that controls the SBH, i.e., $Dint$ in Eq. (5) cannot be a single-valued function of $ϕM$⁠. As $ΦB,no$ depends on multiple variables, the partial derivative in Eq. (6), mathematically speaking, should be taken with the other variable(s) held fixed, and the value of $SΦ$ would depend on the value of the other parameter, as well as $ϕM$⁠. It is not surprising then that quite different $SΦ$ values were deduced from analysis of experimental data obtained by different laboratories or under different processing conditions. The message one gets from plots such as Fig. 4 should not be that the S-parameter is such and such for this semiconductor, but rather something that inspires the question “beside the work function and the electron affinity, what other factors are important for the SBH that are making the local SBH changing?” Without a knowledge of all the parameters that control the SBH, the value of the S-parameter itself has little meaning. It seems that the investigation of the reason(s) for the SBH to be locally varying would more directly reveal the microscopic formation mechanism of the SBH than making much of the slopes in plots such as Figs. 3 and 4.

At the time when researchers attempted to solve the mystery of FL-pinning at MS interfaces, a different phenomenon, unfortunately by the same nickname, was also a topic of much interest and investigation. The pinning of the FL in the band gap at the free and adsorbate-covered surfaces of semiconductors was extensively investigated and, by then, well understood to be driven mainly by the overall neutrality of the semiconductor surface.²¹ When the periodic structure of a crystal lattice is terminated at a surface, electronic states particular to the surface are created. The FL for the semiconductor surface should always be positioned such that, just when all electronic states below the FL are occupied, the surface is electrically neutral. The reason for the neutrality requirement is obvious: excess charge on the surface costs energy, in the absence of external free charge. For the FL of semiconductor surface therefore, the energetic position of electronic states in the surface region is of primary importance. Surface electronic structure depends on the atomic structure of the surface, and conversely, the atomic structure of the surface is dictated by surface energy, to which the distribution of surface electronic states make a major contribution. Electronic states found at the surface can be true surface states, which exist only in the gaps of the bulk energy band structure and decay in amplitude away from the surface, both toward vacuum and toward the bulk crystal. They can also be tails of the Bloch states of the semiconductor crystal, which generally decay toward the vacuum but could have enhanced intensity in the surface region, in which case they are known as surface resonance states. These aspects of surface electronic structure are similar to what will be described for the MS interfaces in Sec. III B. Uniform semiconductor surfaces typically feature dangling bond states from the surface atoms, which broaden into two-dimensional (2D) surface state bands in the fundamental gap of the semiconductor. The charge neutrality level (CNL) of electronic states in the surface region is then the FL for the surface. We should stress here that the CNL is an energy level that is defined by the charge neutrality condition for all electronic states in the entire surface region and not a level that can be revealed through analysis of only the charge distribution of electronic states in the gap. There is no particular neutrality requirement that needs to be satisfied by the “gap states” alone. Most frequently for semiconductor surfaces, the CNL is indeed found to lie in the gap of the bulk band structure, thus effectively “pinning” the surface FL inside the band gap. There are exceptions, the most prominent of which are the cleaved, non-polar {110} surfaces of some III-V compound semiconductors,^22,23 which feature no surface state bands in the fundamental band gap. In principle, therefore, neutrality at these cleaved surfaces is satisfied if the FL is anywhere within the band gap.²⁴ 

The dominant role played by the CNL in pinning the FL position for semiconductor surfaces did not go unnoticed to researchers looking for an explanation for FL pinning at MS interfaces. The first of the proposals to involve the same mechanism in the explanation of both “FL pinning” effects came from Bardeen,²⁵ who suggested that the work function difference between a metal and a semiconductor with surface states could be compensated by an exchange of charge between the metal and the semiconductor surface states. This idea was developed into a quantitative model for the SBH by placing the metal not in direct physical contact with the semiconductor, but at a small gap distance away from it.²⁶ The schematic shown in Fig. 5 is the “fixed-separation model” of interface dipole formation,^4,26 which has been used not only for surface states but also for other types of interface states, including metal-induced gap states, defect states, etc. The small gap is imagined to be an (uncharged) oxide or dielectric layer that does not affect the surface states of either the semiconductor or the metal. Under these conditions, the n-type SBH can be deduced to be²⁶ 

where $ϕCNL$ is the energy of the CNL, measured with respect to the VBM, $SGS$ is a constant in this theoretical model, given by

which could be identified with the S-parameter experimentally deduced for the semiconductor, and $DGS$⁠, $δgap$⁠, $εint$ are, respectively, the density of surface states, the thickness of the gap, and the dielectric constant of the gap material. The slight separation between the metal and the semiconductor, ∼1–2 nm, avoids the problem that the electronic structure of both the metal and the semiconductor should be significantly modified upon contact and, at the same time, allow some chemistry expected at MS interface, i.e., the charge exchange between the metal and the semiconductor, to take place. As can be deduced from the form of Eq. (7), the dependence of the SBH on the metal work function in this model is reduced by a factor of $SGS−1$⁠, which could be a large number for a significant density of surface states, $DGS$⁠. Therefore, with a high density of surface states, the interface FL is expected to be strongly pinned, near the position of the CNL, as shown in Fig. 6. With two free parameters, $DGS$ and $ϕCNL$⁠, to play with, Eq. (7) can explain any linear relationship found in SBH vs. work function plots such as Fig. 4. The artificial insertion of a dielectric layer at the MS interface might seem reasonable in the early days of SBH research when the cleanliness of semiconductor surface was suspect, but it makes no sense today as high resolution characterization techniques routinely reveal that MS interfaces fabricated with proper care are invariably intimate. One should note that even in the case that a thin dielectric layer truly separates the metal and the semiconductor as assumed in Fig. 5, the distribution of states at either the semiconductor-dielectric or the metal-dielectric interfaces cannot remain unchanged from the surface states on, respectively, the semiconductor and the metal. As explained in Sec. III B, the electronic states at these interfaces would depend on the band structure of the dielectric layer as well as the atomic structure of the two dielectric interfaces! Furthermore, there could be significant dipoles associated with both the semiconductor-dielectric and the metal-dielectric interfaces which would render the baseline condition for the band diagram, i.e., the use of $ϕM$ and $χS$ in Fig. 5, inappropriate (cf. Fig. 47). A convenience afforded by the fixed separation models, which is also presumably the most attractive motivation for these models, is that there is no mistaking which part of the interface belongs to the metal, which part belongs to the semiconductor, and the amount of net charge on either side. At intimate MS interfaces, such unambiguous distinction would not be possible, which is a fact that makes the requirement of charge neutrality a meaningful concept only for the entire MS interface, and a meaningless one for only one (the semiconductor) part of the interface. We also note that the derivation of Eq. (7) presumes that the metal has nearly perfect screening and allows no penetration of the electric field inside,²⁷ i.e., the density of states at the metal-dielectric interface is assumed to be infinite. This is another feature common to all simple SBH models that is in disagreement with result from ab initio calculations, discussed below. While there is little doubt that a band diagram such as Fig. 5 can explain the weak dependence of the SBH on metal work function, the question has always been whether such a simple picture of the charge distribution at MS interfaces is realistic enough to be worthwhile considering.

Surface states in the band gap of the isolated semiconductor cannot remain at intimate MS interfaces,²⁸ because Bloch states from the metal are present at these energies. Inside the band gap of the semiconductor, metal wave functions cannot propagate into, but rather have to decay toward, the semiconductor. The tails of these wave functions penetrate a small distance into the semiconductor and are known as metal-induced gap states (MIGS). As a result of MIGS, there is a spillage of negative charge from the metal into the semiconductor side of the interface in this energy range. This gives the appearance of charge transfer, albeit uni-directional, between states near the FL of the metal and the MIGS in the semiconductor. Since the MIGS can be rationalized as what evolves from the surface states on the original semiconductor, there is the expectation that the CNL concept is still applicable to the distribution of MIGSs at MS interfaces.^29,30 To make use of the CNL concept for MIGS in simple models, however, it is necessary to justify that the distribution of the MIGS is independent of the metal. The idea was put forward that, even though a band gap exists for a bulk semiconductor, movement of the Fermi level inside the band gap at the interface of a semiconductor still leads to local charging.³¹ The distribution of induced gap states is then argued to be an intrinsic property of the semiconductor and, thus, is independent of the nature of the interface, being the same for MS interfaces and for semiconductor heterojunctions.^27,32 Under these assumptions, the exchange of charge between the metal and the MIGS is envisioned to provide the dipole which screens the work function difference between the metal and the semiconductor. A distribution of MIGS, with a characteristic CNL, is assumed to be placed at a fixed distance, e.g., the decay length of typical MIGS, away from the metal to accommodate the charge transfer. Therefore, the band diagram by which an MIGS-based model is used to analyze the SBH is exactly as that already shown in Fig. 5 for surface states, and the specific predictions in an MIGS model for the SBH and the pinning strength are exactly those already given in Eqs. (7) and (8). There are many proposals on what the CNL should be for MIGS formed in the gap of a semiconductor. Tersoff argued that the CNL should be the energy level of the evanescent state with equal conduction and valence band character.³⁰ In one dimension, this is the branch point of the complex band structure. Arguments were also given as to why this level roughly corresponds to the midgap level of the indirect band gap.³² Cardonna and Christensen proposed that the midgap state at the Penn gap, called the “dielectric midgap energy” (DME), is the intrinsic CNL of the semiconductor.³³ Harrison and Tersoff drew from the tight-binding concept to propose that the average hybrid energy in the semiconductor, i.e., the dangling bond orbital energy, should act as the CNL.³⁴ There is also an empirical rule on the CNL from a compilation of experimental data.³⁵ From the perspective of quantum chemistry, the electronic wave functions in a region occupied by a material can always be expanded in terms of atomic orbitals of that material, because the orbitals form a complete set. The wave functions in the semiconductor material at the immediate MS interface can also be rigorously expanded in wave functions for the bulk semiconductor crystal, because the bulk wave functions, of all energies including unbound states, also form a complete set. The expectation is thus understandable that the semiconductor region at the MS interface will become neutral when certain energy is reached. However, this argument falls apart if the contributions from higher lying states are appreciable, which is almost inevitable when metal with orbitals of very different symmetries becomes the neighbor of the semiconductor.

Many aspects of the MIGS model are vague. For example, “the exchange of charge between the metal and the MIGS” has often been stated as the mechanism for interface dipole formation. This has two obvious problems: (1) The semiconductor does not have a density of electrons with energy in the bandgap. Therefore, there cannot be a transfer of electrons from the semiconductor to the metal around the FL. Through the formation of MIGS, the transfer of charge around the FL can only be uni-directional. This is why early investigations of the role played by MIGS in SBH were under the banner of the “negative charge model.”^36–38 (2) Because MIGSs are not isolated states but extensions of Bloch states in the metal, it is not possible to add or remove MIGS from the semiconductor side without adding or removing states on the metal side. They are parts of the same state. There is nothing to exchange! As explained below, the charge transfer between the metal and the semiconductor actually takes place through the entire energy range(s) of occupied states in the metal and the semiconductor, of which states inside the band gap represent only a small portion. How the MIGS theory is used to generate interface dipole is a carbon copy of how the fixed-separation model works, as shown in Fig. 5, with the exception that the gap states in the semiconductor are MIGS rather than surface states. Therefore, the shortcomings of the fixed separation model discussed above are also inherited by the MIGS model. Like the former, the MIGS model is unable to account for the large scatter in SBH data and the inhomogeneity of MS interfaces. In addition, some of the most basic assumptions of the MIGS models, such as the assumption on the independence of the MIGS distribution on the metal and the assumption that the CNL is an innate property of the semiconductor, are in disagreement with the result of ab initio calculations.^4,39 It thus seems that the MIGS models have little connection with processes at real MS interfaces. Notwithstanding these known problems, pinning by MIGS has become, by far, the most frequently invoked model when SBH data are discussed in the literature. A browse into these publications often finds that it is the ability to use the linear dependence of the fixed-separation model, Fig. 6 and Eq. (7), which attracted the comparison with the MIGS model. In such comparisons, what the “MIGS model” represents is only the phenomenological aspect of its theoretical prediction and has little to do with the hypotheses within the model that led to the prediction.

In order to fabricate an epitaxial MS interface, the crystal structures of the metal and the semiconductor must be similar and the lattice parameters of the two must be closely matched. That does not happen very often in nature, and therefore high quality epitaxial MS interfaces that have been experimentally produced are very few in number. However, once such single crystal interfaces are produced, the SBHs measured at these MS interfaces offer the most direct clues to the formation mechanism of the SBH. This is because the atomic structure is homogeneous at these MS interfaces and thus is clearly and solely responsible for the observed SBH. Most notable among epitaxial MS interfaces are the type A and type B NiSi₂/Si(111) interfaces, both of which can be fabricated with very high structural perfection^40,41 and their atomic structures have been carefully determined by high-resolution electron microscopy and other technique to be 7-fold coordinated.^42–46 These two interface atomic structures are similar in immediate bonding arrangement and differ only in distances for second-nearest-neighbor and beyond,⁴³ yet experiments showed that there was a difference of 0.14 eV in the SBHs of these two interfaces,⁴⁷ after an early experimental mistake⁴⁸ was identified.^49–52 The NiSi₂/Si(100) interface was another epitaxial MS interface that could be fabricated with near perfection in its crystalline order^53,54 and its interface atomic structure was determined by high resolution electron microscopy.^43,55 The n-type SBH of this NiSi₂ interface was found to be about 0.40 eV lower than that at the type B NiSi₂/Si(111) interface,⁵⁶ as shown in Fig. 7. The experimental discovery of this huge variation in the SBH with orientation between a single phase of the silicide, NiSi₂, and Si was in stark contrast to the opinion prevailing at the time that the SBHs for silicides were independent of the substrate orientation, crystalline phase, and stoichiometry.^57,58 Ab initio calculations,^59–61 based on the experimentally determined structure, yielded SBH results which were in excellent agreement with the trend experimentally observed for the two NiSi₂/Si(111) interfaces.⁶⁰ One should note that even though the SBH difference was reproduced, the absolute values of the SBH from these calculations were not accurate, because of the problem with the local density approximation (LDA) employed in these calculations. The A- and B-type comparison filtered out systematic errors in the calculations, leaving the essence of the SBH formation mechanism bare to the view of researchers who study their interface electronic structures. The projected density of states for each layer for the two epitaxial NiSi₂/Si(111) interfaces are shown in Fig. 8. Although the densities of MIGS were high at both interfaces,⁶⁰ calculations found no easily identifiable feature in the energetic distribution of these states to suggest a “pinning” of the FL. Actually, the role played by MIGS in determining the FL position could not be established in these calculations.^59,60 The interface dipole arises from charge distribution of all electronic states below the Fermi level, of which the charge due to MIGS is only a small part. There are significant differences in the distribution of MIGS at these two interfaces, reflecting an expected dependence of the MIGS characteristics on the interface structure. The potential due to all charges at the two interfaces in Fig. 9 (Ref. 60) reveals that the local charge varies rapidly at the MS interface and extends a small distance into both the Si and the NiSi₂. Two somewhat surprising features in Fig. 9 for both the A- and the B-type interfaces should be noted:⁶⁰ (1) The potential does not simply decay into the metal (NiSi₂), but rather goes through some rapid changes inside the metal before decaying. This very significant chemical shift inside the metal can also be clearly observed in Fig. 8 by comparing the main features of the dashed curve (bulk NiSi₂ DOS) with the solid curve (interface NiSi₂ DOS) in the third panel from the top. Such a behavior is inconsistent with the expectation that the metal simply screens out the disturbance due to the presence of the semiconductor. (2) The total depth of potential variation is longer into the metal (∼0.9 nm) than it is into Si (∼0.5 nm), which is inconsistent with the suggestion from the MIGS model that the formation of the interface dipole is one-sided (longer into the semiconductor), because of the long tails of MIGS.³⁶ 

CoSi₂ is technologically more attractive^62,63 because of its lower resistivity than NiSi₂. Single-crystal CoSi₂/Si(111) interfaces could be fabricated, albeit only with the type B orientation.^64–66 Experimental and theoretical studies have indicated that the 8-fold model is most likely the structure for well-annealed, type B CoSi₂/Si(111) interface.⁶⁴ However, it is also known that the interfacial structure of type B CoSi₂/Si(111) may vary according to preparation. Evidence for 7-fold coordinated structure has been obtained from layers that have only been annealed to lower temperatures. The SBH of well-annealed type B CoSi₂ layers is in the range of 0.65–0.70 eV on n-type Si, which is usually attributed to the 8-fold structure. On the other hand, CoSi₂ layers grown at lower temperatures show considerable variation in their SBH, including an n-type SBH as low as 0.27 eV.^67,68 Electronic structure calculations of the CoSi₂/Si(111) interfaces showed that the SBH could vary by as much as ∼0.6 eV with the assumed atomic structure and orientation of the interface. In particular, the n-type SBH for the 7-fold B-type interface was calculated to be lower than the 8-fold B-type interface, by ∼0.1–0.4 eV, depending on computation details, in agreement with the experimentally observed trend. The SBH of epitaxial CoSi₂ /Si(100) interface⁶⁹ is usually found not to differ much from that observed for polycrystalline CoSi₂, ∼0.7 eV. A 2 $×$ 1 reconstruction is often observed at the interfaces of epitaxial CoSi₂, although its structure has not been definitively determined.^69–71 The existence of more than one stable structures has been suggested by HREM studies of the CoSi₂/Si(100) interface.^72,73 In particular, the SBHs of the two interfaces (upper and lower) of buried, epitaxial Si/ CoSi₂/Si(100) structures were found to be different,⁷³ a dependence which was attributed to the difference in the observed interface structures.⁶⁸ Theoretical calculations showed that the SBH of the CoSi₂/Si(100) interface depends on the assumed interface structure (cf. Fig. 29 below).^74–76 However, the n-type SBH for 6-fold coordinated interface was calculated to be about ∼0.3 eV larger than that for the 8-fold coordinated interface,⁷⁶ opposite to the trend experimentally deduced.⁷³ 

On compound semiconductor surfaces, epitaxial metallic compounds that have been successfully fabricated included elemental metals such as Al and Fe,^77,78 and two groups of cubic materials with the CsCl and the rock salt structures.^79–81 The interfaces between epitaxial Al and GaAs(100) were studied by a number of techniques.^82,83 MBE prepared GaAs(100) surfaces may have a variety of reconstructions which are associated with different surface stoichiometry.⁸⁴ Some studies found that the SBH of epitaxial metal was a function of the original surface reconstruction/stoichiometry of the GaAs(100)^85,86 and the ZnSe(100),⁸⁷ which was suggestive of a possible correlation of the interface structure and SBH. However, this correlation was not always found.^88–90 Possible interface structures and electronic structures of the Al/GaAs system have been theoretically investigated.^39,91,92 Needs et al. found the SBH at epitaxial Al/GaAs(110) interface to vary by as much as 0.7 eV with a change in the interface structure.⁹¹ A much smaller variation, ∼0.1 eV, was found by Bardi et al. who noticed the importance of interface relaxation.⁹² In addition, experimentally observed dependence of the SBH of epitaxial Al/Al_xGa_1-xAs(100) interface on the alloy composition was reproduced by a linear response theory calculation.⁹² For the epitaxial Al/GaAs(100) interface, Dandrea and Duke provided detailed analysis of the distribution of MIGS's as a function of interface structure.³⁹ The density of MIGS was calculated to be above the $1×1014$ cm⁻² eV⁻¹ level, usually considered high enough to pin the Fermi level. However, the SBH was found to vary by 0.23 eV for the Al-Ga and Al-As bonded interfaces. The MIGS were incapable of pinning the Fermi level because the electronic charge density in other energy range changed by the order of $6×1014$ cm⁻² eV⁻¹ over the interface plane and overwhelmed MIGS screening. These resulted show a lack of direct connection between the Fermi level in the gap and the neutrality of the semiconductor. The interfaces between GaAs(AlAs) and a number of epitaxial intermetallic compounds have been examined by HREM.^93,94 However, the atomic structures of these interfaces have not been firmly established. Evidence for a reconstruction has been observed at the interface between NiAl and overgrown GaAs. The SBH of various structures involving epitaxial metallic compounds has been studied in detail. It was found that buried NiAl has a different SBH with respect to the overgrown (AlAs)GaAs than that on an (AlAs)GaAs substrate.⁹⁵ The observed SBH also seems to depend on the thickness of the NiAl and CoAl layers.⁹⁶ Furthermore, evidence for inhomogeneous SBH was found for these interfaces. The most spectacular result from these epitaxial metallic compound interfaces is the observation of a dependence of the SBH on epitaxial orientation, observed at the lattice-matched Sc_1-xEr_xAs/GaAs systems,^97,98 and shown in Fig. 10. It shows that annealing may lead to significant changes in the observed SBH. The SBH of the ternary compound Sc_0.32Er_0.68As varies by as much as 0.4 eV when the n-type GaAs orientation is changed from (100) to (⁠$11¯1¯$⁠).⁹⁸ The electronic structure at ErAs/GaAs interfaces has been investigated in detail.^99,100 There are two possible 1 × 1 structures for the ErAs/GaAs(100) interface, the Ga-terminated (the “chain model”), which is in agreement with experiment, and the As-terminated (the “shadow model”). Straight calculations for these two un-reconstructed interfaces showed that the FL was pinned at close to the CBM and VBM, respectively. The buckling of the two kinds of atoms in the mixed (Er + As) plane was responsible for a very large adjustment (>1 eV) in the interface dipole.¹⁰⁰ The p-type SBH calculated for a 2 × 2 reconstructed, and yet energetically more favorable, (100) interface structure was lower than that calculated for the ErAs/GaAs(110) interface by about 0.2 eV, in agreement with experimental result shown in Fig. 10.

If either the metal or the semiconductor forming an MS interface is a compound phase, there is always a choice on whether the interface is terminated on anions or cations, except for non-polar interfaces. The type of silicide interfaces, e.g., 8-fold or 6-fold, described above is a subtle example of this termination issue. For every compound semiconductor studied by numerical calculations, termination at polar interfaces has been a prominent parameter that leads to different SBHs, e.g., for GaAs(100),³⁹ GaN(100),¹⁰¹ and ZnSe(100)⁸⁷ interfaces. General considerations would suggest that, everything else being equal, the anion-terminated interface should have a smaller n-type SBH than the one terminated on cations. Furthermore, the SBH difference between the cation- and anion-terminated interfaces should increase with semiconductor ionicity. A systematic study of these issues was conducted by Berthod et al.,¹⁰² who calculated the electronic structures at the unrelaxed interfaces of Al with different terminations of the lattice-matched, (100)-oriented, Ge, GaAs, AlAs, and ZnSe crystals. The calculated SBHs were shown to be in reasonable agreement with experiment. Furthermore, the possibility was explored, of explaining the calculated SBH within a model^103,104 assuming “linear response” of the interface charge distribution to “external disturbance” in the form of changes in the nuclear charge (or, more precisely, the pseudo-potential) of atoms at the interface. To determine the systematics of dielectric response from the electronic distribution at the MS interface, a “test charge sheet,” in the form of a dilute mixture in the pseudo-potential of the atoms on a selected plane, was first placed at different depth on the semiconductor side to induce a change in the potential drop across the MS interface. The result of this clever “numerical experiment,”^102,105 shown in Fig. 11, represented the linear response with which the difference in the cation- and anion-terminated SBH was compared. Excellent agreement was found for the Al/GaAs(100) and the Al/ZnSe(100) systems, while for the Al/AlAs(100) interface the trend in the calculated SBH was predicted in the right order of magnitude by the model. In general, this work suggested that the concept of image-charge screening, classically only valid at macroscopic dimensions, could be extended to distances shorter than the decay length of the MIGS. This is an excellent work that is a great assistance to the understanding of the interplay between chemistry and electrostatics at the MS interface.^102,106 Note that even though the interface charge distribution seems to respond linearly to small changes in the nuclear (pseudo-)potential, this is largely an electrostatic effect. The baseline (the non-polar version) of these MS systems, on which the numerical experiments were conducted, already had interface dipoles, via quantum mechanical calculations, that satisfied the chemical (or bond-forming, energy-minimizing) aspect of the interface interaction.¹⁰² It would not be right to conclude from the image-force picture that the act of bond formation between the metal and the semiconductor could also be treated by a linear response model. One wouldn't know where to begin, in that case.

A technologically important group of semiconductors whose SBH has been discovered to depend on the surface termination is silicon carbide.^107–112 The SBH of two polytypes of the SiC family, 3C-SiC and 6H-SiC, have been studied in detail. Experimentally, the dependence on the starting surface termination for the individual metal has been strong, although a systematic correlation has been lacking.^107–109 It is not known the extent to which the original surface termination has survived at the final MS interfaces. These MS interfaces typically do not have good lattice matches and are not expected to accommodate epitaxial growth. However, theoretical calculations have been carried out by artificially placing metal atoms on strained lattice sites, which typically found that the C-terminated surfaces were more chemically reactive, and the dependence on metal work function stronger, than the Si-terminated surfaces.^113–115 As shown by the summary in Fig. 12, the n-type SBH on C-terminated SiC is systematically larger than that found on Si-terminated SiC. Oxide semiconductors/insulators with the perovskite structure (ABO₃) are of great interest for a wide range of current and future applications.^116,117 Along the normal (100) direction, a perovskite crystal can end on an AO surface plane, where A represents a 2+ cations such as Sr and Ca, or a BO₂ plane, where B is a 4+ cation such as Ti or Si. Because of the ionicity of the compound, these two surface terminations could offer different band alignment conditions.¹¹⁸ Theoretical calculations of the electronic structure at the interface between SrTiO₃(100) (STO) band various metals showed that the SBH has a wider range of variation (∼0.7 eV) on SrO-terminated surface than the range (0.3 eV) observed on TiO₂-terminated surface.¹¹⁹ For each individual metal, however, the effect of the surface termination on the SBH is present but not very significant (⁠$≤$0.3 eV). An interesting insight into the formation mechanism of the SBH was offered in these calculations through a break-down of the final SBH into stages. The alignment condition between metals and STO was calculated as the difference at the free surface stage ("stage 0"), at a stage when the surfaces of the STO and the Pt were individually constrained to their eventual, relaxed, atomic arrangement at the interface (“stage 1”), and then at the final equilibrated relaxed interface structure (“stage 2”). This analysis showed that even though the two different surface terminations led to very significant differences (>1 eV) in the alignment conditions at stages 0 and 1, the rearrangement in the spatial distribution electronic density when the metal and the semiconductor made contact in stage 2 would dominate and render the SBHs on the two surface terminations similar.¹¹⁹ The rationale for this stage analysis was to separate out the effect due to atomic rearrangement during the formation of the MS interface from that due to the relaxation of the electronic density. Both of these relaxations are results of chemical bond formation. Another interesting aspect of the formation of SBH that was demonstrated in these calculations concerned the spatial extent of charge rearrangement at the MS interface. As shown in Fig. 13, the modification of charge density during bond formation extended roughly equal distances into the metal and the STO for both surface terminations. Actually, if anything, the extension of the disturbance of the interface into Pt was a little longer than that into STO. This roughly equal participation in the depth of the MS interface from the metal and the semiconductor is similar to that found for silicide interfaces⁶⁰ shown in Fig. 9, and also that for other oxides (see Fig. 14(b)). Experimentally, it is known that the observed SBH on STO can be affected by the oxidation condition of the interface,¹²⁰ which is a result also found at many other oxide interfaces. As implied by the more than one dashed vertical lines in Fig. 13 representing atomic positions, crystal planes (with mixed atomic species) that are flat in a bulk crystal could have different perpendicular shifts at the interface due to reduced symmetry in the bonding geometry. Such corrugations, or “rumpling,” within the atomic planes of some binary alkaline earth crystalline oxides has been shown to have a significant effect on the magnitude of the SBH.^121–123 Electronic structure calculation of approximately lattice-matched interfaces of BaO, CaO, and SrO with different metals and metal “intralayers” failed to produce a clear dependence of the SBH on the metal work function.¹²¹ However, a dependence of the SBH on the rumpling parameter of the fully relaxed MS interface was noticed, as shown in Fig. 14(a). The interface rumpling here is defined as half the difference in the interplanar distance (between the 1st and the 2nd oxide planes at the interface) for oxygen atoms and that for the alkaline earth metal atoms, normalized by the bulk interplanar distance. Notice that a positive rumpling parameter indicates a shift of the (oxygen) anions toward the metal and/or a shift of the (alkaline earth metal) cations toward the oxide, creating an additional dipole with a sign that is in agreement with the observed trend in the p-type SBH shown in Fig. 14(a). The rumpling parameter seemed to be an indicator of the strength of the interaction between the metal and the oxide. To further explore the role played by rumpling, this parameter was deliberately changed at the interface (to non-optimized atomic structure) leading to approximately linear change in the calculated SBH.¹²¹ As shown in Fig. 14(b), the depth/length of the charge transfer remained little changed, while the magnitude and even the polarity of the charge transfer could be adjusted.

Very consistent results have been seen from all the first principles calculations on the SBH of epitaxial MS interfaces. They can be summarized as follows: (1) The SBH depends on the atomic structure of the interface. (2) The distribution of MIGS and the location of the CNL are intrinsic properties of the semiconductor, but depend on the interface structure. (3) The distribution of all charges, including MIGS, states below the VBM, and states in the metal, determines the interface dipole. The sharp dependence of the SBH on the interface structure, experimentally and theoretically found for epitaxial MS interfaces, clearly put the explanation of the microscopic mechanism of SBH formation out of the reach of simple models.

The vast majority of MS interfaces studied experimentally are non-epitaxial and therefore have atomic structure that varies from place to place. The interface dipole and the SBH at these interfaces are expected to vary locally. Evidence for the presence of inhomogeneity in the SBH's was recognized and reported only sporadically before the 1990s,^124–126 and serious attention was not paid to the issue of SBH inhomogeneity. The development of the ballistic electron emission microscopy (BEEM) technique^127,128 provided the spatial resolution needed to examine the distribution of local SBH underneath ultrathin metal layers. Occasionally, large-scale variations (0.7–1.1 eV) of the SBH were observed at compound semiconductor.¹²⁹ The amplitude of SBH modulations was found to depend on the doping level of the semiconductor,¹³⁰ being apparently more uniform on lightly doped semiconductors^128,130,131 than on heavily doped semiconductors. The reason for the dependence of the apparent SBH on the doping level is potential “pinch-off,”¹³² taking place not at the immediate MS interface but in the space charge region some distance in front of the MS interface. This effect involves no special physics and is just a consequence of the band-bending (electrostatic potential) near an inhomogeneous SBH needing to satisfy the Poisson's equation. When a region with locally lower SBH is small in size compared with the depletion width, electronic transport across this region of the interface is not determined by the local SBH, as for uniform MS interfaces, but rather by the transport of carriers across the (inhomogeneous) space charge region. Illustrated in Fig. 15 are potentials with different doping levels, in front of MS interfaces with identical inhomogeneous interface SBH profile. The key parameter that controls the electron transport at an inhomogeneous MS interface is the “saddle-point potential” energy, which decreases with the doping level, as illustrated in Fig. 15, thus accounting for the doping level dependence of BEEM.¹³⁰ Direct evidence for the potential pinch-off effect has been observed by BEEM in artificially generated inhomogeneous SBs^133,134 and also at semiconductor/liquid interfaces.^135,136 An analytic theory of the potential distribution and the electron transport at inhomogeneous SBH was developed^20,132 and was shown to accurately describe the results of experiments with known inhomogeneous SBHs⁵⁶ and the results of numerical simulations.¹⁹ Potential pinch-off can mask the rapid fluctuations of local SBH and was a major reason why inhomogeneous SBH could escape experimental detection for so long and why this issue was largely ignored historically. It was only after all the ramifications of potential pinch-off were understood that it became clear how severe and widespread the SBH inhomogeneity problem really was in polycrystalline Schottky diodes.¹⁹ 

The most intriguing dependence of the saddle-point potential for a low-SBH region is the one on the applied bias. As shown in Fig. 16, with an applied forward (reverse) bias, the saddle-point potential energy increases (decreases). In reverse bias, the effective SBH decreases with voltage, leading to unsaturated reverse current. Enhanced current at large reverse leakage current has often been observed experimentally. As pointed out,²⁰ the traditional explanation based on negative charge of the MIGS tails³⁷ was incompatible with the entirety of available experimental data. Rather, the much varied behavior of reverse current at similarly processed MS interfaces was in better agreement with the voltage dependence of the saddle-point potential for inhomogeneous SBH.^19,20 At forward applied bias, the rise in the saddle-point potential leads to an increase in the effective SBH that controls the current transport with the applied voltage. This is manifested in a slower-than-normal rate of increase in the current with forward bias, i.e., an ideality factor that exceeds unity, which is a phenomenon commonly observed in the studies of SBH. Before the bias-dependence of the saddle-point potential was pointed out, there had been many explanations of the non-ideal I-V characteristics, including image force lowering,¹³⁷ generation-recombination, interface states (negative charge),^138,139 and tunneling,^140,141 all of which could explain a greater-than-unity ideality factor. However, the facts that experimentally observed I-V traces often contained kinks and shoulders, the measured I-V characteristics could be significantly different for diodes similarly fabricated, and the measured ideality factor displayed various dependencies on temperature made clear that it would be impossible to explain the wide range of junction current behavior with uniform SBH. SBH inhomogeneity is the most likely cause for experimentally observed greater-than-unity ideality factors.

When the interface SBH is non-uniform, the current transport across the MS interface is spatially inhomogeneous. As just explained, the inhomogeneous distribution of the electrostatic potential in the space charge region leads to the formation of saddle-points that exert actual control over electronic transport across the entire junction. The analytic expressions derived for the locations and the potential energies of saddle points, as a function of interface SBH profile, doping level, temperature and the applied bias, pertains to a space charge region with artificially homogenized distribution of dopants. In other words, the charges due to exposed dopants are assumed to be smeared into a density of charge that is homogeneous in the space charge region. In reality, however, charges due to exposed dopants are approximately point charges fixed at discrete lattice sites. The discrete nature of dopant charges (or other charged defects) can have a very profound effect on the saddle-point potential and on the current transport at real MS interfaces. Particularly, how far the discrete dopant atom(s) are from the location of a saddle point can have a very significant influence on the actual energy barrier that controls the electronic transport with this low-SBH region. As illustrated in Fig. 17, the presence of a charged defect near the saddle-point significantly modifies the potential distribution. Therefore, it can be concluded that a non-uniform MS interface will appear leakier on more heavily doped semiconductor not only because of an increased average electric field near the MS interface, but also because of additional SBH lowering from dopant ions being accidentally placed near the critical saddle points. With a current flowing at an inhomogeneous MS interface, locations near the saddle-points become “hot spots” with current density often orders of magnitude greater than that averaged over the entire MS interface. There are a few interesting consequences of such a localized “current crowding” phenomenon, one of which concerns the “spreading” series resistance and is an issue that has already been addressed theoretically.^19,20 When the electronic transport at an inhomogeneous MS interface is dominated by the current flow in small and dispersed low-SBH patches, the total junction current can be written as a linear sum of the current to each patch^19,20

where $Va$ and $A**$ are the applied voltage and the Richardson constant, respectively. $Φi$⁠, $Ai$⁠, $Ii$⁠, and $Ri$ are, respectively, the SBH, the area, the current, and the series resistance of the i-th conducting patch. The series resistance associated with an independent patch, assumed circular in shape, has been found empirically to be¹⁴² 

where $ρ$ is the electrical resistivity, and $tlayer$ the layer thickness, of the semiconductor. Note that the series resistance for a patch can vary from essentially uniform in nature (⁠$=ρtlayer/Ai$⁠) when $Ai>tlayer$⁠, to essentially spreading in nature (⁠$=ρ(16Ai/π)−1/2$⁠), as for a point contact, when $Ai≪tlayer$⁠.^143,144 Calculation with the analytic form of the series resistance, Eq. (10), has been shown to reproduce experimental I-V characteristics and also results in close agreement with full-fledged semiconductor numerical simulation package.^19,20 The current crowding may also cause the density of carriers in the vicinity of the saddle-point to far exceed that corresponding to the equilibrium majority-carrier quasi-Fermi-level of the space charge region. In the simulation of the current flow at an inhomogeneous SBH, this represents a small complication as the quasi-Fermi-level is likely laterally inhomogeneous. However, there are also real and observable consequences of current crowding. As discussed earlier, electron transport is greatly facilitated by the accidental placement of dopant ion(s) near the saddle-point, which is the location of current crowding. What this implies is that at forward bias the capture rate of majority carriers by the exposed dopant near the saddle-point is greatly enhanced. One consequence of carrier capture near the saddle point is the negation of the additional SBH lowering due to discrete dopants and an increase in the effective SBH. In other words, the ideality factor of this particular current component may be very large statistically, due to the dependence of the statistically averaged charge state of such defects on the applied bias. Another consequence of the capture and emission of carriers near the saddle-points is “noisy” junction current in the time domain.

When the MS interface is non-uniform, even the simple task of reporting the SBH deduced by experiments becomes non-trivial. Frequently, the SBH determined experimentally depends on the technique of measurement, e.g., SBH measured by C-V often exceeds that measured for the same diode by photo-response or I-V techniques.^145–147 That this is a necessary consequence of SBH inhomogeneity has long been realized.^148–150 For certain types of work, only the result of one type of SBH measurement is important, e.g., I-V measurement for ohmic contact development. For other types of applications and investigations, the entire profile of the SBH distribution at an MS interface is of interest, in which case the experimental techniques and conditions that are needed to provide the desired, complete information should be carefully thought out. When the semiconductor is uniformly doped, C-V measurement usually renders the arithmetic average of the distribution of the SBH profile at the MS interface, which is a good starting point. To help with the reduction of the in-phase component of the current in C-V measurement, a common trick is to conduct the measurement at low temperatures. There are also other pitfalls with C-V measurements that should be avoided.^151,152 Because of the exponential dependence of the magnitude of junction current on the SBH, $exp[−ΦB/(kBT)]$⁠, I-V measurement is dominated by the portions of the MS interface with lower-than-average SBH; and the lower the measurement temperature, the more pronounced the dominance of the low SBH patches. The upper-half of the SBH distribution of an MS interface is not accessible experimentally by I-V measurements, except by special techniques with spatial resolution such as BEEM or electron-beam induced conduction (EBIC). However, since the upper-half for one type of doped semiconductor, e.g., n-type, is the lower-half in SBH distribution for the other type of doping, e.g., p-type. In routine studies, the I-V characteristics on both types of semiconductor form a complete picture of the full SBH distribution. In addition, the pinch-off effect at inhomogeneous MS interfaces can be reduced by the use of more heavily doped semiconductor. To circumvent the uncertainties in both the effective area responsible for current transport at inhomogeneous MS interfaces and the appropriate Richardson constant to use, study of junction current as a function of temperature, i.e., activation-energy analysis, is an effective technique. For certain distribution in the inhomogeneous SBH, the Richardson plot of a diode may be curved, as shown in Fig. 18, in which case the slope deduced at the low measuring temperatures represents an upper limit to the minimum in the distribution of the interface SBH. An additional uncertainty one can avoid in I-V measurement of inhomogeneous interfaces concerns the contribution due to series resistance (cf. Eqs. (9) and (10)), which vanishes identically at zero current (zero-bias). Instead of plotting the logarithmic of the forward-bias junction current against voltage, which renders the current at small biases range unusable for analysis, a logarithmic plot of the quantity $I/{1−exp[−eVa/(kBT)]}$ against the applied bias⁸⁹ allows the current in the entire bias range to become usable as shown in Fig. 19.¹⁴³ Therefore, activation energy study of junction current obtained at a fixed (non-zero) bias should be avoided, in favor of the analysis of the logarithmic of the quantity $I/{1−exp[−eVa/(kBT)]}$ interpolated to zero-bias, as done for Fig. 18. Taking into consideration of various known factors, the most revealing investigation of the full range of SBH distribution of an MS interface, assuming that the same MS structure can be reproduced on different types of semiconductor, consists of studying I-V results, obtained at low applied bias voltage, on both types of moderately doped semiconductor. Reporting of SBH measured for inhomogeneous MS interfaces should consist of results obtained by various techniques under different conditions. It is generally inadequate to report a single SBH value for any inhomogeneous interface.

The SBH problem, which can be roughly stated as “How the bands of a semiconductor are aligned with those of a metal at an interface?”, constitutes a small, albeit very important, part of a much broader area of scientific research on the “alignment of energy levels” between different materials. Included in this broader area are topics that are also of interest to materials scientists and device engineers, such as the alignment of energy levels at semiconductor-semiconductor, metal-insulator, semiconductor-oxide, metal-polymer, metal-electrolyte, etc., interfaces. The physics and chemistry that govern band alignment at these solid or condensed matter interfaces are expected to be similar. But also related to this general area of research are topics that aren't usually associated with the SBH, such as “the electron transfer rate between donor ions and acceptor ions in an electrochemical solution” or “ the rate of a (bio-)chemical reaction.” The common thread that connects all of these fields of research is the critical importance of the relative position of the energy levels in one material to the energy levels in the second material in deciding the current flow or charge transfer rate between the two materials. In these problems, of interest is the relative position/shift between two sets of energy levels, belonging to two different substances physically located very close or right next to each other. Despite their spatial proximity, the two sets of energy levels must maintain their separate individuality for a study of their relative alignment condition to make sense. For example, asking the question of “how do the valence atomic levels on a carbon atom compare with those on an oxygen atom” would not make sense if the two atoms together formed a CO molecule, because in that case the molecular orbitals can no longer be regarded or associated exclusively with only one atom. Looking at Fig. 1, we notice that the energy bands for either the metal or the semiconductor are not drawn inside the interface specific region (ISR), even though the ISR for an abrupt MS interface may contain physically nothing more than the metal crystal lattice and the semiconductor crystal lattice all the way up to the plane that separates the two. The reason that the energy bands for the individual crystals should be drawn only outside of the ISR is obvious: the electronic states inside the ISR are characteristic of the coupling between the metal and the semiconductor and cannot be exclusively identified with either side. Therefore, as far as the SBH is concerned, the material inside the ISR can be thought of as a third substance (neither the metal nor the semiconductor) whose levels on the energy scale per se are not directly linked to the SBH. Rather, the relevant aspect of the electronic states in the ISR is their “spatial” distribution, which causes shifts in the relevant energy levels elsewhere, in the two regions just right and just left of the ISR. From Fig. 1, the p-type SBH can be written in terms of the average potential energy at two specific places

where $−eVM−I¯$ is the average potential energy near the metal-ISR boundary, $−eVS−I¯$ is the average potential energy near the semiconductor-ISR boundary, and the $μ(i)$'s, as before, are the internal chemical potential for the two bulk solids. As vividly demonstrated by the form of Eq. (11), the electrostatic potential of the crystal is of paramount importance to any energy alignment problem. It is thus worthwhile to delve deeper into the issue of electrostatics of crystalline materials.

The potential energy due to all the charges in a solid, from ions and electrons, as well as charges external to the solid, can be written as

where $ψj$'s are single electron wave functions, $R→j$'s are the positions of the positive ions in the crystal, and $Vext$ is the electrostatic potential due to external sources. For a large and perfect crystal, the potential in the interior portion of the crystal is expected to be periodic. The average potential energy at any point $r→$ inside the crystal can be defined as that spatially averaged over the volume of a unit cell

where $Ωcell$ is the volume of the unit cell and the integration is carried over a unit cell centered about $r→$⁠. A convenient shape for the integration is the Wigner-Seitz cell, although any acceptable choice for the unit cell will give the same answer as long as $V¯(r→)$ is a slowly varying function in a crystal. There are other weighting strategies that may be used to smooth out the periodic oscillations and define the average potential.¹⁰⁵ One may wonder, in the absence of external charges (⁠$Vext=0$ in Eq. (12)), what would be the average potential energy $−eV¯(r→)$ due to the (infinite) crystal itself. This seems like a reasonable question to ask and is one that has been asked many times in the literature, as the answer seems to provide some “baseline” condition for the crystal potential. After a highly publicized debate in the literature,^153–155 it is now clear that the average potential of an infinite crystal has no physical meaning,^156,157 because the structure of the surface assumed for the crystal, as its dimensions approach infinity, actually determines the average potential of the “infinite” crystal. In other words, the surface contribution to the crystal potential energy does not vanish but becomes a constant, as the crystal approaches infinity. The easiest way to describe and understand the surface effect is to look at well-known results concerning “model solids.”^10,158 A model solid is an artificial object with a charge distribution which mimics that of a real solid, except that its charge distribution can be arbitrarily specified.^159,160 Such an object is not a real crystal because its charge distribution, especially that near the edge of the model solid, is not governed by Schrödinger equation. Nevertheless, the electrostatic potential of a model solid, being solution to the Poisson's equation, is invaluable for the analysis of electrostatics of solids. To reproduce the crystal potential inside a real solid, we can construct a model solid by attaching identical cells of (fixed) charge distribution to every three dimensional crystal lattice point, as schematically shown in Fig. 20. For simplicity, we shall choose a unit cell with vanishing net charge, net dipole, and net quadrupole moments, and also satisfies the requirement that the placement of one cell at every lattice point seamlessly recreates the charge distribution of a real solid. For a crystal of high symmetry, such as cubic or hcp, a unit cell with these properties can always be found and, in fact, there are always many such choices. The advantage of choosing a cell with these properties is that the electrostatic potential of each cell is very short-ranged, falling off more rapidly than $r−5$⁠.¹⁶¹ This means that cells a few lattice spacings away will have little contribution to the local electric potential. The average potential inside a large model solid so constructed is a constant and is independent of the shape (or size) of the model solid. In fact, the average potential energy of a model solid is simply and uniquely related to the spatial distribution of charge within the unit cell (model-solid cell) used to construct the model solid.

After a particular charge distribution for the model-solid cell, $ρMSCell$⁠, is chosen, the electrostatic potential in the model solid is given by

where $R→i$'s are points of the Bravais lattice. Note that the integration in $r→′$ is only over the volume of one model-solid cell centered about $r→′=0$⁠, and that $ρMSCell$ contains not only the dispersed negative density due to electrons but also positive delta-function(s) due to the nuclei. To calculate the average electrostatic potential of a model solid, one integrates Eq. (14) over a unit cell in $r→$ coordinates and divides by the volume. Because of the superposition principle of electrostatic potential, the average electrostatic potential in the interior of a model solid is simply

It can be shown¹⁶² that the above integral renders

where the spherapole (trace of the second order density matrix) of the unit cell, $ΘMSCell$⁠, is given by

One cautions that the average electrostatic potential, per cell, is a constant and given by Eq. (16), only if the lower order moments of the chosen $ρMSCell(r→)$ vanish. Note also that a different choice of the cell density would produce a different model solid potential via Eq. (17). Any difference in the model solid potential is obviously a result of different surface termination, as can be verified by using “Wigner solids” as examples. A Wigner solid, as schematically shown in Fig. 21, is made up of a periodic lattice of (positive) point charges with a neutralizing uniform background of (negative) charge.^153,156 Assume that we have a proton placed on every lattice site of a simple cube with a lattice constant of the Bohr radius, $a0$⁠, and a uniform, compensating density of electrons throughout the lattice. The obvious choice for a unit cell is the Wigner-Seitz cell about the point charge, which consists of a proton (⁠$+e$⁠) at the origin (its center) and is in the shape of a cube of $a0$ on each side, with a uniform charge density of $−ea0−3$⁠. This choice of the unit cell would render an average model solid potential energy of $−π/6$ a.u. (atomic unit, Hartree = 27.2 eV) according to Eqs. (16) and (17). Another conceivable choice for the unit cell of the Wigner solid is a solid cube of uniform negative charge density with eight point charges of $+e/8$⁠, one on each corner of the cube. Obviously, the net charge, dipole, and quadrupole moments also all vanish for such a Wigner-Seitz cell centered about the body center of the cube, for which the average potential energy is calculated to be $+π/3$ a.u. This difference in average potential energy can be easily demonstrated to be identical to the potential energy drop across the “difference layer” between the two Wigner solid surfaces. For example, the potential difference along the cubic (001) direction across one-half the length of the unit cell can be calculated to be $π/2$ a.u. It should be noted that even though the two cells chosen above are the only two available non-overlapping cells with vanishing lower multi-pole moments, there are actually many (infinite) other choices of “overlapping” unit cells that are also valid. For example, a cell with the positive point charge $+e$ at the origin and a linearly graded distribution of negative charge density

seamlessly recreates the Wigner-solid and has vanishing lower moments. A model solid generated with such a cell has an average potential energy of $−π/3$ a.u., even more negative than non-overlapping cells. These examples illustrate the facts that the potential of a model solid is completely arbitrary and that the potential of an infinite solid cannot be defined without a knowledge of the surface structure.¹⁵⁶ 

If the surfaces of a real solid have charge distribution exactly the same way as a model solid, then obviously the electronic states inside can be referenced directly to the true vacuum level at infinity. For example if a metal were terminated exactly the same way a model solid built from $ρMSCell(metal)$ cells is, its Fermi level would be

with respect to the vacuum level.¹⁰ And for a semiconductor model solid built from cells with density $ρMSCell(SC)(r→)$⁠, its VBM would be

with respect to the vacuum level at infinity. The model-solid result, Eq. (16), is very helpful in the analysis of the average potential energy at points near the MS interface, cf. Eq. (11). We know that if the sign of a certain charge distribution is reversed, the potential also reverses its sign. Therefore, if in the interior of a model solid a (small) number of unit cells of charge is removed, the average potential in the “hollow” or “void” left inside the model solid vanishes because of the superposition principle of electrostatics. In other words, if an integer number of cells of $ρMSCell(r→)$ were removed around a location inside a (model) solid, while keeping the charge distribution elsewhere frozen, the local average potential decreases by $VModSol¯$⁠. We shall call the potential inside the void created by removing a few unit cells of $ρMSCell$ from a (real) charge distribution the local “empty potential” $VMT(r→)$⁠. In other words,

The empty potential naturally depends on the charge distribution of $ρMSCell$⁠. The few cells of charge that are removed from the crystal, with its periodic potential inside, can be viewed as the “host” of the electronic states of the crystal, with its energy levels referenced to the vacuum level via Eq. (19) or Eq. (20). This scenario is schematically demonstrated in Fig. 22. Empty potential is the electrostatic potential at any point in a crystal solid that is NOT due to the local periodic density of charge of the crystal itself, but due to other sources such as the surfaces, the interface, defects, free charge, external charge, etc. When the “host” is put back to the “hollow” in the crystal that it came from, these energy levels are simply shifted by the slow-varying empty-potential in the empty space, from superposition principle. For example, the Fermi level in a metal crystal is

which should be a constant inside a metal (⁠$VMT(r→)=VMT$⁠), and the VBM of a semiconductor crystal is

which may be position-dependent. As usual, the energy level positions in Eqs. (22) and (23) are referenced to the zero in the electrostatic potential energy. We stress that the writing of Eqs. (22) and (23) amounts to nothing more than a re-definition of the energy levels in terms of the empty potential. However, empty potential turns out to be a friendlier and much more transparent reference to use for the analysis of energy level alignment problems than the average of the full periodic potential of a crystal. Before moving on, an issue that should be mentioned is that, as the form of Eq. (15) suggests, the above analysis, strictly speaking, pertains only to the full potential and the full charge distribution of a solid, e.g., from an all-electrons calculation. If numerical computations of bulk solids made use of pseudo-potentials, which is a very common practice at the present, results discussed above would however remain valid, provided some modifications, detailed elsewhere,¹⁶³ were included.

Consider the case when a model solid of metal is placed near a semiconductor model solid, as shown in Fig. 23(a). Because the electric field from symmetric cells is short-ranged and can be regarded as vanishing identically outside the model solid, the empty potential inside either model solid vanishes independent of the relative position or orientation of the two model solids. For this arrangement, the p-type SBH is simply

When it comes to the analysis of the distribution of electrostatic potential and the SBH near a real MS interface, Eq. (24) now provides a definitive reference. From the charge distribution at an MS interface, Fig. 23(b), it is clear that Eq. (24) would have held if not for the possible dipole due to the presence of the charge distribution in the un-shaded region (ISR). Note that charge distribution far away from the interface region can shift the empty potentials in both shaded regions, but cannot lead to a difference in the two. We further note that any part of the charge distribution inside the ISR that is overall neutral and locally symmetric, such as a layer of unit cells with undistorted (bulk) charge distribution, cannot lead to a difference in the empty potentials either. Therefore, it is only the change/difference in the charge distribution from the bulk charge distribution of either crystal that can lead to a shift in the SBH from Eq. (24). In other words,

where the potential drop across the ISR, $(ΔV)ISR$⁠, is given by

where $δρ(z)$ is the difference between the ISR planar-average charge distribution $ρ(z)$ and reference bulk charge distribution

We remind ourselves that the choices of the cells to construct the two model solids now also define the boundary of the ISR. To make $(ΔV)ISR$ easier to handle by chemical principles and ideas, the cell for the metal and that for the semiconductor should be chosen consistently. For elemental metals with a monatomic unit cell, the use of the Wigner-Seitz cell about the nucleus is an obvious and reasonable choice, as this ends the model solid on the same boundaries that separate the “atoms” in the crystal. Common elemental semiconductors, Si, Ge, and $α$-Sn, have the diamond lattice, with a basis for the unit cell. A model solid can be constructed with a Wigner-Seitz cell (for the fcc lattice) centered about the mid-point between nearest neighbors. However, more convenience is afforded if cells centered about the nucleus are used. Because of symmetry, the two atoms in a unit cell of the diamond structure are topologically inequivalent, but electrically equivalent. The diamond lattice can be partitioned into spaces in closest proximity to each atom, treating the two atomic sites in a unit cell on equal footing. These two types of “proximity cells,” each with the volume of 1/8 of the conventional fcc unit cell, can be used to stack together a model solid. The shapes of these two proximity cells, which are related to each other through inversion, are shown in Figs. 24(a) and 24(c). The relationship of a proximity cell to the atomic positions is roughly sketched in Fig. 24(b). It can be verified that the model solid would have an average electrostatic potential given by (16) and (17), where the integration is taken over one proximity cell only and the cell volume is taken to be that of one proximity cell. The simplicity in this result is the consequence of a property of the spherapole, Eq. (17): When the charge distribution of a model-solid-cell can be partitioned into several sub-cells, each with vanishing lower moments, the spherapole of the model-solid-cell is a simple sum of the spherapoles of the sub-cells. This is another way of saying that the value of the spherapole calculated in Eq. (17) does not depend on the choice of the origin. Model solids constructed using both types of proximity cells would put the representation of the (elemental) semiconductor crystal on par with the metal model solid. The preference for terminating/building both crystals with cells centered about nucleus is obviously an attempt to preserve the notion of “atoms” in a solid, about which most of the chemical concepts evolve.

There are many more compound semiconductors/insulators than there are elemental semiconductors. Common crystal structures for these compounds include zinc blende, perovskite, hexagonal, and fluorite. Because of the ionic nature of these crystal structures, the best choice for the model solid is not definable per se but is dependent on the kind of MS interface structure to be analyzed. Below we shall briefly describe how several model solids might be constructed each for the zinc blende and perovskite crystals to be friendly with different stacking sequences at the MS interface. For the zinc blende structure, the most straightforward choice for the construction of a model solid would be the Wigner-Seitz cell centered about either the cation or the anion, although these choices do not appear particularly chemistry-friendly, as nuclei are exposed on the surface (corner) of such a cell. A more chemistry-friendly choice would be to first construct proximity cells for both the cation and anion, a step which leads to cells identical to those already shown in Figs. 24(a) and 24(c). The sum of one each of the cation-centered proximity cell (CPC) and the anion-centered proximity cell (APC) would constitute one whole unit cell. However, to make the lower-order moments of the model-solid-cell vanish, one may place a CPC at the origin and place four APC's, each with ¼ of its full density, at the nearest neighbor positions, as shown in Fig. 25(a). This is obviously an arrangement that will generate a model solid with the periodic charge distribution of the bulk crystal and, at the same time, have rapidly decaying potential outside the cells. Let the net charge and the spherapole of the stand-alone CPC be $qcpc$ (typically positive) and $Θcpc$⁠, respectively; and for the APC: $qapc(=−qcpc)$ and $Θapc$⁠. The spherapole of a cation-centered model-solid-cell is

where $3a/4$ is the distance between neighboring cation-anion pair. From Eq. (16), the average potential for a model solid constructed with cation(Ga)-centered model-solid cells (CCMSC) is

Alternatively, one could have placed an APC at the center of the MSC and put four quarter-strength CPCs at nearest neighbor position, as schematically shown in Fig. 25(b). A moment's consideration tells us that the spherapole for such an anion-centered MSC (ACMSC) is

which can give its model solid potential through an equation analogous to Eq. (29). The average model-solid potentials of common compound semiconductors are shown in Table I.¹⁶³ As expected the model solids constructed with proximity cells have lower potential energies than that constructed with Wigner-Seitz cells. Also, the cation-centered model solids have lower potential energies than the anion-centered model solids, consistent with Eq. (28). We should remember that proximity cell is simple but not the only way to represent the chemical identity of an “atom” in a crystal. Defining atoms in a molecule is a well-studied topic in molecular chemistry, for which many proposals exist.^164–168 These prescriptions in quantum chemistry may provide more chemical flavor in the construction of model solids. However, one notes that these methods are more involved, sometimes yield very different definitions of atoms,¹⁶⁹ and are not based on strict rigor. On the other hand, the proximity cells for cations and anions are equal in size by definition, the size for the anion will typically exceeds that for the cation in the same compound when one of these other methods is used to partition the solid.^164–168 With a specific definition, the anions and cations can be used to construct model solid according to the same procedures. For example, a cation defined by Bader's atoms-in-molecules (AIM) theory¹⁶⁵ is placed at the center and four anions, each with ¼ of the charge density, are placed at the corners, to form a CCMSC with the AIM method. Model-solid potentials calculated with the AIM method are also shown in Table I. Note that for the same semiconductor, the number of valence electrons on the AIM-defined cation, as a result of its smaller size, is significantly smaller than that in the cation proximity cell. Therefore, the difference between the potentials for cation-centered and anion-centered model-solids of a compound is much larger for AIM construction than proximity cell construction.

Except for the non-polar (110) surface, model solids of the zinc blende crystal structure constructed from any of the methods described above cannot lead to a surface terminated on a full layer of either cations or anions. Instead, a fraction of a monolayer of ions (or proximity cells) has to terminate any polar surface of the zinc blende model-solid. For example, zinc blende model solid built from CCMSCs ends with one-half monolayer of anions on {100} type surfaces and three-quarters monolayer of anions on {$1¯1¯1¯$}. This is a necessary consequence dictated by electrostatics: because the ion (or proximity cell) for an ionic compound has net charge, beginning the surface with a full layer of either cations or anions necessarily leads to long-range potential in the solid and a situation unsuitable for meaningful analyses.¹⁷⁰ Shown in Fig. 26 are schematic diagrams of several possible cation-terminations of the zinc-blende model-solids on the (100) surface. As can be verified, the net potential drop across any of these slabs is zero, reflecting the suitability of these model-solids as reference for interface analysis. Which of these “constructions” should be used in actual SBH analysis depends on the way the interface chemistry is handled, as explained in detail in Sec. III E. Here, we only examine the “canonical” (without interface chemistry) energy level alignment condition for polar interfaces. If an MS interface is known to be fabricated with metal deposition over an anion-terminated (100) surface, e.g., As-stabilized GaAs(100), then the use of CCMSC would lead to the situation depicted in Fig. 27(a), where we have assumed the metal to have a monatomic unit cell. Because no atoms (proximity cells) are distorted from their bulk state, Fig. 27(a) represents a MS interface without interaction or charge transfer. Under this condition, the empty potential is the same for the two crystals and the p-type SBH is given by

If it bothers to see missing atoms at the interface, one may imagine these voids filled with neutral atoms to recreate the arrangement of atoms expected at this MS interface, as shown in Fig. 27(b). The symmetry in the charge distribution of neutral atoms guarantees that the electrostatic effect of these atoms fades rapidly. Therefore, electrostatics in the solids is the same with (Fig. 27(b)) or without them (Fig. 27(a)). If one is interested in analyzing the SBH at an interface fabricated by metal deposition over a cation-terminated semiconductor (100) surface, a reasonable starting configuration pits the model solid constructed with ACMSC's against the metal model solid, for which the non-interacting SBH is

Using similar arguments as just demonstrated for the zinc blende (001) surface, one can deduce that the baseline condition for a metal interface with the anion-terminated {$1¯1¯1¯$}, also known as the {111}B surface, is that given in Eq. (32), while the baseline SBH with the {111} is given by Eq. (31). Therefore, without MS interaction, the p-type SBH fabricated on anion-terminated surfaces would exceed that on cation-terminated surfaces by

The non-polar (110) surface of the zinc blende crystal can be represented by a model solid potential of

As expected, this non-polar surface has a potential which is the average of the cation-terminated surface and anion-terminated surface.

The perovskite structure and its derivatives are important structures occupied by many technologically important materials. The SBH to perovskite oxide is of great interest to a wide range of possible applications. In an ABO₃ cubic unit cell, there is a nominally 2+ (A) ion, a 4+ (B) ion, and three 2- oxygen (O) ions, about which three different types of proximity cells can be sketched out. A simple cube with the same orientation as the unit cell but with a/2 on each side, is found to be the proximity cell centered on B-atom. The proximity cell centered on A-atom, shown in Fig. 28(a), also has the full point group symmetry of the cube, as it is of the exact shape and size of the Wigner-Seitz cell for an f.c.c. lattice with the same lattice parameter. Shown in Fig. 28(b) is an oxygen-centered proximity cell, which has the same mid-section as Fig. 28(a) but with both the top and the bottom chopped off. There are three different orientations of the oxygen proximity cell, with the square facets oriented to mate with B-centered proximity cells and the diamond-shaped facets face the A-centered proximity cells. There are two convenient choices for the model-solid-cell, centered on A and B, respectively. To construct an “A”-centered model solid cell, we put an A proximity cell at the origin, eight B-centered proximity cells, at 1/8 of the full density, at the eight body-centered positions, and twelve ¼ -strength oxygen proximity cells at midpoints between B-cells. The spherapole of this “A”MSC is

where $ΘApc$⁠, $ΘBpc$⁠, $ΘOpc$ are the spherapoles of A-cell, B-cell, and oxygen-cell, respectively, and $qApc$⁠, $qBpc$ are the net charge of A-cell and B-cell, respectively. Similarly, for the “B”-centered MSC, which includes a B-cell at the origin, eight 1/8-strength “A” cells at the body-centered positions, and six ½-strength O-cells at the ⟨1/2,0,0⟩ type positions, the spherapole is

The most common orientation employed for perovskite oxides is the {100}, for which there are two possible terminations for a flat surface: a layer of AO or a layer of BO₂. Surfaces with either termination may be prepared by modern epitaxial techniques and used to fabricate Schottky barriers.¹¹⁹ To analyze interfaces with an AO layer at the interface, the model solid constructed from B-centered MSC's is used, which creates a surface with ½ of the AO proximity cells removed. As in the case of zinc blende (100), the absence of a fraction of the proximity cells at the perovskite surface preserves surface neutrality. And, to analyze an interface with a BO₂ termination, the potential of an “A”-centered model solid serves as a useful reference for the non-interactive interface. It is interesting to notice that, in the absence of interface chemistry, the p-type SBH for the AO-terminated surface differs from that for the BO₂-terminated surface by

The chemistry between the metal and the oxide termination layer usually reduce this difference in SBH at the actual interfaces.

Generally speaking, the philosophy of establishing a baseline alignment condition for the bands of the metal and the semiconductor based on atoms in their bulk-like state seems more intuitive than say the philosophy behind the Schottky-Mott Rule. With the discussion so far in this section, we are actually in position to analyze what the Schottky-Mott Rule really implies about the magnitude of the interface dipole. We first notice that while a metal crystal is in isolation, its measured work function, $ϕM$ (defined positive), is the difference between the vacuum level just outside the metal and the E_F of the metal. From Eq. (22), we know that inside this isolated metal crystal, the empty potential energy is a constant

which is a positive quantity (⁠$VMT(MetalSurf)<0$⁠). In the above equation, it is implied that a particular cell has been chosen to construct the metallic model solid. If we define $−EF(ModSol)$ as the bulk contribution to the work function, the empty potential energy, $−eVMT(MetalSurf)$⁠, is then identified as the dipole due to the formation of the metal surface, i.e., the surface dipole. Likewise, for an isolated semiconductor crystal with an experimentally measured ionization potential of $I(SC_surf)$⁠, the empty potential at the near surface region of the semiconductor crystal is (cf. Eq. (23))

which can be regarded as the surface dipole of the free surface of the semiconductor. When the Schottky-Mott relationship holds, we write

Plugging in Eqs. (25), (38), and (39) into Eq. (40), one finds that

In other words, the Schottky-Mott Rule holds when the interface dipole is the difference between the dipoles of the two free surfaces. Note that this conclusion is reached, independent of the way the model solids are defined for the metal and the semiconductor.

The distribution of charge at real MS interfaces is different from the abrupt termination depicted in Fig. 27(b). Generally speaking, the distribution of electronic charge is the sum of the individual densities of all the occupied electronic states at the interface, while the characteristics of the available electronic states depend on the arrangement of the positive ions at the interface. Conversely, the positions of the ions relax to configurations favored by the total energy associated with the electronic structure. This complex, self-consistent, process depends on every detail of the interface and can be quite individualistic and unfit for generalization. In this section, we shall be concerned with only one small aspect of this process, namely, the formation of electronic states from known and fixed atomic positions at the interface. For simplicity, we shall also initially restrict ourselves to epitaxial interfaces. Our discussion shall pertain to essentially the very last step in a real electronic structure calculation, i.e., when the electronic states calculated actually self-consistently generate the charge distribution assumed for the calculation (from the previous iteration)! Within either the Hartree-Fock formalism or the Density Functional Theory (Kohn-Sham), this step involves finding solutions to the single-electron Schrödinger equation,

where the potential $V(r→)$ contains the Coulomb interaction with all charges in the system and also exchange-correlation interactions. Even though the entire process of electronic structure calculation is routinely carried out in all modern electronic structure software packages for any given arrangement of atoms, a discussion of the physics of this process as it specifically applies to MS interfaces has not appeared in print, to the best of our knowledge. Although there are similarities with how the electronic structure at solid surfaces is calculated, on which detailed accounts are available.^21,171 An understanding of the requirements on the interface states from quantum mechanics is crucial in distinguishing what may be assumed of the electronic states at an MS interface and what may not.¹⁷² 

Assume that we have a semi-infinite MS interface with metal atoms occupying the space $z<0$ and semiconductor atoms occupying $z>0$⁠. Since we assume that all the atomic positions and the electrostatic potential of the entire interface are known, we can choose an ISR as the region $zM−ISR≤z≤zSC−ISR$ such that the potential/charge distribution at $z<zM−ISR$ is identical to that in the bulk metal and the potential/charge distribution at $zSC−ISR<z$ is identical to that in bulk semiconductor. $|zM−ISR|$ and $zSC−ISR$ should be chosen to have the above described properties, but also as small as possible for simplicity, and typically they are <1 nm for physically abrupt interfaces. The common 2D unit cell for both the semiconductor and the metal, which may be viewed as rectangular in shape without loss of generality, has an area of $|a→1×a→2|$ $=a1a2$⁠. We apply the Born-von Karman periodic boundary condition (PBC) in 2D with a box of area $(N1a1)⋅(N2a2)$ to define a mesh of $N2D (=N1⋅N2)$ distinct $k→||$'s that are allowed in a 2D Brillouin zone (2DBZ), which has a total area of $4π2/(a1a2)$ in reciprocal space. As the entire epitaxial MS interface is assumed to be periodic in x-y directions, the known potential at any z can be expanded in plane waves with the periodicity of the 2D unit cell,

where $K→||$'s are the 2D reciprocal lattice vectors, including 0, and note that the Fourier coefficients, $UK→||(z)$'s, have explicit z-dependence. Inside either of the bulk regions, $UK→||(z)$ has the z periodicity of the respective crystal. However, there is no particular symmetry for $UK→||(z)$ inside the ISR. Because of the in-plane periodicity, Bloch theorem holds true in x and y directions. Inside the ISR, wave function with a specific $k→||$ can be written as

where band and spin indices have been suppressed. Equations (43) and (44) can be plugged into Eq. (42) to yield coupled equations for the $wk→||,K→||ISR$ coefficients,

In theory, the sums over the 2D reciprocal lattice vectors in Eqs. (43)–(45) have infinite number of terms. However, finer spatial details are unimportant especially when pseudopotentials are used, justifying the termination of these summations at some appropriate cut-off energy in actual computations. Therefore, these sums have, in fact, a fixed number (N) of terms. Since Eq. (45) is a second-order differential equation, the complete solution for the ISR region can be written down if a total of 2N independent boundary values are supplied from the M-ISR and the SC-ISR interfaces. $φk→||$ in the ISR thus serves as the link between state(s) in the semiconductor and those in the metal, with the same energy and $k→||$⁠, through matching of the value and slope of the wave function at $zM−ISR$ and $zSC−ISR$⁠. How the numerical solutions to all the $wk→||,K→||ISR(z)$ in Eq. (45) may be obtained is perhaps best visualized by imagining setting up a grid in z for a finite-element analysis, e.g., all points with values $zm+1=zm+δz$ are on the grid. In this analysis, the second derivative in Eq. (45) becomes simply

One thus sees that if all the $wk→||,K→||ISR$'s on two consecutive grid points, e.g., $zm$ and $zm−1$⁠, are known (for a total of 2N boundary values), then the solutions to Eq. (45) can be generated for points $zm+1$ and $zm−2$⁠; and from there on, throughout the entire grid for the ISR.

In the metal region, $z≤zM−ISR$⁠, the potential is 3D periodic and identical to that in a bulk crystal, except for a possible rigid shift due to the dipole of ISR. It is clear that any bulk Bloch function $ψk→jM_Bloch$ with the same energy (after the rigid shift) and $k→||$ would satisfy the Schrödinger equation in this region. A general form for the full MS wave function in this region is, therefore,

where the sum is over all the Bloch states with the same energy and $k→||$ (⁠$k→j=k→||+k→⊥,j$⁠). Because a 3D energy band of the bulk crystal is periodic in the repeated-zone representation, a straight line through the (3D) Brillouin zone typically intersects with an equi-energy surface at two points or a few pairs of points. Since the band index is suppressed in Eq. (47), we should generalize and include all such intersections with all the bands. Only in rare cases are an odd number of points encountered. This happens when the straight line is tangent to part of the equi-energy surface. Such cases are unimportant for MS interface electronics, as the lone state either is static or involves an electron traveling parallel to the interface (with its group velocity in the plane of the interface). What is important and happens quite frequently is the case when the $k→||=const$ straight line intersects with no part of the equi-energy surface of interest in any of the bands. Obviously, this happens when the chosen energy $εi$ is in a gap between bands of the metal, or the entire $k→||=const$ line is inside a pocket of the Brillouin zone, i.e., a “mini-gap,” missed by the relevant equi-energy surface, which resides elsewhere in the zone. As can be seen from Fig. 29(a), which shows the bulk energy bands of the CoSi₂ projected along the {100} direction,⁷⁶ there are many “pockets” of these mini-gaps at specific energy and location in the 2DBZ. Inside these mini-gaps, there is no solution to the Schrödinger equation for the metal that satisfies the PBC. Other types of solutions, i.e., non-Bloch, may still be found. This should be obvious from Eq. (45), the general form of which can also be used for bulk metal or semiconductor. As before, a supply of 2N boundary values will allow a solution to be generated. What are the forms of these non-Bloch solutions? It is difficult to generalize because there are different reasons for the presence of (mini) band gaps. Below, we examine the forms of solutions in just one type of band gap, for nearly free electrons, and surmise that similar conclusions can be drawn for other types of band gap. In the nearly free electron theory, the potential is assumed to be so weak that it does not influence either the plane wave nature of the free electron wave functions, $ψk→jM_Bloch(r→)∝eik→j⋅r→$⁠, or its energy dispersion, $εj=ℏ2kj22m$⁠, EXCEPT when $k→j$ is near a Brillouin zone boundary.¹⁶¹ Close to a Bragg plane, $k→j⋅K→≈12K2$⁠, two related, unperturbed plane wave states are nearly degenerate, $|k→j|2 ≈ |k→j−K→|2$⁠. Therefore, the coupling between these two plane wave components cannot be ignored, even if the coupling potential energy between them is weak $|eUK→| ≪ ℏ2kj2/(2 m)$⁠. Writing $ψk→jM_Bloch(r→)$ as a linear combination of the two plane waves and plugging into the Schrödinger equation, one gets

as the eigenvalues. Indeed, the two-fold degeneracy is removed and a gap of $|2eUK→|$ is opened up at the Bragg plane. On the Bragg plane, the two solutions are known to be^161,171

The sign of the coupling energy, $−eUK→$⁠, determines which of the two solutions corresponds to which root in Eq. (48). Our interest in this exercise has been to examine the kind of non-Bloch solutions that may survive near an interface. For that purpose, we let $K→$ be a primary lattice vector for the 2D reciprocal lattice, $K→=K→||$⁠, and ignore all other Fourier components of the potential except $−eUK→||$⁠. We assume the Fourier energy term to be positive, i.e., $|UK→|||=−UK→||$⁠, thus identifying $cos(K→||/2)⋅r→$ with the higher of the eigenvalues in (48). We now look for a solution with a (forbidden) energy just below the top of the band gap (⁠$εgap_top$⁠)