Schottky-barrier formation and transport at inhomogeneous metal-semiconductor contacts: the surviving open problem descended from Mott's 1939 crystal-rectifier theory

Mott's 1939 theory of crystal rectifiers raised several open questions about the metal-semiconductor blocking layer. Mott's thermal-over-barrier theory of the metal-semiconductor blocking layer explained the rectification polarity and, qualitatively, the shape of the resistance-voltage curves of copper-oxide rectifiers, but left open the validity of its simplifying assumptions in real contacts, the reconciliation of the predicted forward slope (of order $e/kT$) with the smaller slopes extracted from experiment, and the quantitative role of thin low-barrier patches, space charge, the image force, and the crossover from over-barrier conduction to tunnelling. The later literature resolved the transport physics of the ideal homogeneous barrier (thermionic emission-diffusion theory, exact space-charge solutions, field and thermionic-field emission) and identified spatial barrier inhomogeneity as the semi-quantitative explanation of forward-slope anomalies; ballistic-electron-emission microscopy realized nanometer-scale mapping of local barrier heights. The surviving open core, supported by Tung's reviews (2001, 2014), Tung and Kronik (2021), and current contact-engineering literature, is the predictive problem: derive, from interface atomic structure and bulk semiconductor properties alone, the Schottky barrier height and its spatial distribution at real (inhomogeneous, defect-bearing) metal-semiconductor interfaces, and through them the full DC rectification characteristic including image-force lowering, space charge, and the thermionic-to-tunnelling crossover.

Background

A crystal rectifier is a contact between a metal and a semiconductor whose resistance depends strongly on the direction of current flow; the canonical 1930s example, analyzed by Mott (1939), is the copper-plate/cuprous-oxide (Cu$_2$O) rectifier. Rectification requires that a substantial fraction of the applied voltage drop across a high-resistance 'blocking layer' - a thin depleted or insulating region adjacent to the metal - rather than across the bulk semiconductor. Mott modeled this layer as a potential hill of height set by the contact potential difference $V_0=(\phi-\chi)/e$, where $\phi$ is the metal work function (the energy to remove an electron from the metal) and $\chi$ the semiconductor electron affinity; this identification of the barrier with the work-function difference is now called the Schottky-Mott rule (Schottky 1938, Schottky 1939). Because experimentally inferred blocking layers were far too thick (of order $10^{-4}$ to $10^{-3}$ cm) for quantum tunnelling, Mott took the current to be carried by electrons thermally excited over the barrier, obtaining a current law of the form $j\simeq(evn_1/d)(V_0-V)[1-\exp(eV/kT)]$, where $d$ is the layer thickness, $v$ the electron mobility in the layer, $n_1$ the electron density at the interface, $e$ the elementary charge, $k$ Boltzmann's constant, and $T$ the temperature. This gave the correct rectification polarity for hole (defect) conductors such as cuprous oxide, where earlier tunnelling-based theories predicted the wrong sign, and reproduced the qualitative shape of the resistance-voltage (R-V) curves, including the decrease of reverse resistance at volt-level biases caused by the image force - the electrostatic attraction between an electron and its mirror charge in the metal, which lowers the barrier by an amount growing with the field $F$ in the layer.

Mott's treatment left open: (i) the assumption $eV_0\gg E_0$, where $E_0$ is the energy needed to excite carriers from impurity levels into the band in the bulk semiconductor, so that only the linear portion of the potential hill matters; (ii) neglect of space charge generated by the current, i.e. a constant field $F$ in the layer; (iii) electron mean free path small compared with $d$; and (iv) essentially the whole potential drop occurring across the blocking layer. He also noted that the predicted initial slope of the R-V curve in the easy-flow direction, of order $e/kT$, is steeper than the slopes extracted from Schottky's measurements, and proposed - without quantitative demonstration - that microscopic inhomogeneities, small patches of anomalously thin blocking layer that carry most of the current at small bias, could reconcile the difference.

The subsequent history divides cleanly. The transport physics of the ideal, spatially uniform barrier was essentially completed between 1942 and 1966. Bethe's boundary-layer ('diode') theory treated the mean-free-path limit opposite to Mott's assumption (iii), in which the current is limited by thermionic emission over the barrier rather than by drift-diffusion through it; Macdonald (1962) obtained an accurate solution of the idealized one-carrier metal-semiconductor junction problem including space charge; Crowell and Sze (1966) combined thermionic emission with drift-diffusion, delimiting the validity of both limits; and Padovani and Stratton (1966) quantified field emission and thermionic-field emission - tunnelling through thin or heavily doped barriers - together with the crossover between over-barrier and through-barrier conduction, a crossover still computed quantitatively in modern device work (Hara et al. 2022). Billig (1951) had already extended image-force treatments to high reverse voltages, where earlier theories failed, and invoked barrier 'weak spots' to explain breakdown and hysteresis. What invalidates Mott's assumptions in real contacts, however, is deeper than transport: Bardeen (1947) showed that surface states can pin the Fermi level (the electron chemical potential) at the interface, so the barrier is not generally given by the work-function difference - the premise inside assumptions (i) and (iv). Measured barrier heights on polycrystalline contacts are largely insensitive to the metal work function ('Fermi-level pinning'), while epitaxial contacts (crystalline metal layers grown in a fixed orientation relation to the semiconductor lattice) show barrier heights that depend on the atomic structure of the interface. Tung (1991) showed that spatial barrier-height nonuniformities ('patches') analytically explain forward slopes smaller than $e/kT$ - ideality factors $n>1$ in the standard diode form $j\propto\exp(eV/nkT)$ - together with several other long-standing anomalies; Sullivan et al. (1991) confirmed numerically that low-barrier patches are electrostatically 'pinched off' by the surrounding depletion field, with saddle points controlling transport; Weiterling et al. (1996) measured broad, structure-dependent barrier-height distributions at epitaxial Ag/Si; and ballistic-electron-emission microscopy (BEEM, a scanning-tunnelling-microscopy technique that images the local barrier height of a buried metal-semiconductor interface with nanometer resolution; Kaiser and Bell 1988, Hecht et al. 1990) realized the spatially resolved measurement of barrier inhomogeneity. Yet the predictive problem has outlived these explanations: Tung (2001, 2014) concluded that existing barrier-height models are too simple to treat the chemistry of real interfaces and that no simple generally applicable equations exist, and Tung and Kronik (2021) achieved a quantitative, chemistry-based explanation of calculated barrier heights only for ideal epitaxial metal/zinc-blende-semiconductor interfaces (zinc blende being the diamond-like cubic crystal structure of Si, GaAs and related semiconductors); contact engineering in the current 2D-semiconductor literature still treats Fermi-level pinning and contact-resistance prediction as open problems (Liu et al. 2025; Xiang et al. 2025).

Problem Statement

Construct and validate a quantitative, predictive microscopic theory of the DC rectification characteristics of real metal-semiconductor contacts - contacts whose interfaces are spatially nonuniform and chemically/structurally imperfect, as in Mott's original copper-cuprous-oxide system and in modern polycrystalline contacts. The theory must derive, from the interface's atomic structure and composition and the semiconductor's bulk properties alone (no per-contact fitted barrier-height, patch-distribution, or pinning parameters): (1) the Schottky barrier height and its spatial distribution across the contact area - resolving the Schottky-Mott/Fermi-level-pinning problem, i.e. the deviation of measured barrier heights from the work-function difference $V_0$ and their dependence on interface structure and preparation; and (2) the full resistance-voltage (equivalently current-voltage) characteristic, including image-force lowering of the barrier, the space charge and non-constant internal field set by the current itself, the behavior as the applied potential approaches the barrier height (where the field can destroy the potential hill), and the crossover between thermal-over-barrier conduction in thick high-barrier regions and tunnelling through thin or low-barrier patches - such that the forward-direction initial slope (its deviation from $e/kT$, i.e. the ideality factor and its temperature dependence), the reverse-bias characteristics, and the measurement-technique dependence of extracted barrier heights are predicted rather than fitted. The two components are jointly required: transport theory through an inhomogeneous barrier is only as predictive as the barrier distribution that feeds it, and a barrier-formation theory is tested against rectification measurements only through such transport. Neither component has independent standing: a barrier-height distribution without the transport characteristic it feeds, or a rectification characteristic obtained by fitting the patch distribution or the pinning parameters, is a partial contribution to the problem, and both components are required together, free of per-contact fitted parameters, and validated against the independent measurements named below. The scientific test of the theory is agreement of its predictions with independent measurements - current-voltage/resistance-voltage characteristics, barrier heights from current-voltage and capacitance-voltage methods, and spatially resolved barrier maps such as ballistic-electron-emission microscopy - on real, spatially inhomogeneous contacts (agreement confined to ideal or purely calculated interfaces does not by itself meet this test). The theory is to be demonstrated on at least one of the two kinds of real contact named above - Mott's copper/cuprous-oxide rectifier or a modern polycrystalline metal-semiconductor contact - by comparing its predictions with independent measurements of that contact. For that contact, the theory's predictions are to be compared with the measured current-voltage/resistance-voltage characteristic, with barrier heights extracted by both the current-voltage and the capacitance-voltage methods, and, where spatially resolved barrier-map data (such as BEEM) exist for it, with that map as well. The predicted barrier heights are to agree with the measured values, as extracted by each comparable method, and the predicted ideality factor with its measured value, to within the measurement and extraction uncertainty, with the reverse-bias characteristics and the measurement-technique dependence of the extracted barrier heights reproduced rather than fitted. This is the surviving open core of the Schottky-barrier problem descended from Mott (1939), as delimited by the modern reviews cited above: the validity of assumptions (i)-(iv) of the thermal-over-barrier picture in real, possibly inhomogeneous contacts; the reconciliation of the predicted forward slope of order $e/kT$ with the smaller slopes extracted from experiment; the quantitative role of thin patches, space charge, and the image force; and the experimental determination of the spatial distribution of barrier-layer properties. The ideal-homogeneous-barrier portion of that 1939 program was resolved by later transport theory (Bethe 1942; Macdonald 1962; Crowell and Sze 1966; Padovani and Stratton 1966), and the forward-slope anomaly is semi-quantitatively explained by barrier inhomogeneity (Tung 1991; Sullivan et al. 1991); what remains open is the predictive first-principles treatment of real contacts stated in (1)-(2).

References

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