First-principles prediction of semiconductor heterojunction band offsets and interface charge at the kT accuracy needed for device design (Kroemer's 1983 lineup problem)

Herbert Kroemer's 1983 review 'Heterostructure devices: A device physicist looks at interfaces' (Surface Science 132, 543–576) closed with five unresolved interface-physics issues standing between heterostructure device concepts and quantitative device design: (1) no purely theoretical band-lineup rule approached the accuracy devices need, since thermionic currents change by a factor of $e$ per $1\,kT$ of barrier change while existing rules erred by about $0.2$ eV; (2) the origin of offset variations with growth sequence and crystallographic orientation was unknown; (3) the microscopic configurations fixing the built-in interface charge and dipole of real as-grown interfaces were unidentified; (4) it was an open problem whether compound-on-element (polar/nonpolar) interfaces could combine structural quality with device-grade electrical properties; and (5) deliberate small misorientation had been proposed as an interface-charge design tool but was untested. The surviving open core, uniting (1) and (3), is a purely theoretical, parameter-free capability that predicts the band offsets of an arbitrary semiconductor interface to $\pm 1\,kT$ (about $25$ meV at $300$ K) and identifies microscopically the residual interface charge and dipole fixed by its as-grown atomic structure. State-of-the-art first-principles methods (explicit interface supercells with hybrid functionals) currently agree with experiment at the $0.05$–$0.15$ eV level, several $kT$.

Background

A semiconductor heterostructure is a single crystal containing an abrupt junction between two different semiconductors, and its central electronic feature is the band offset: the discontinuity in the valence- and conduction-band edges across the interface. Because carrier confinement, injection, and tunneling in devices such as heterojunction bipolar transistors, quantum-well lasers, and superlattice detectors are governed by these steps, the offset acts as an additional design parameter beyond ordinary electrostatic band bending. A lineup rule is any theory that predicts the offsets of a given material pair from bulk properties alone; the offset is usually decomposed into a valence-band offset $\Delta E_v$ and a conduction-band offset $\Delta E_c$ whose sum equals the band-gap difference. By 1983 the available rules were the electron-affinity rule (offsets from the difference of the semiconductors' electron affinities, i.e., their vacuum-level positions), the Frensley–Kroemer pseudopotential-plus-ionic-dipole model, Harrison's LCAO (linear combination of atomic orbitals) tight-binding table of valence-band maxima, and the common-anion rule — the conjecture, descending from the McCaldin–McGill–Mead correlation of Au Schottky barrier heights with anion electronegativity, that two semiconductors sharing an anion have nearly zero valence-band offset. Herbert Kroemer's 1983 review assessed all of these as semi-quantitative at best, with typical errors around $0.2$ eV, while device currents depend on barrier heights at the scale of $kT$ ($kT \approx 25.9$ meV at $300$ K) and below. The review further stressed: that offsets are not material constants — they can vary with growth sequence and crystallographic orientation; that real interfaces carry built-in charge and dipole layers whose microscopic origin was unknown; that polar/nonpolar interfaces (a compound, polar semiconductor such as GaAs or GaP grown on an elemental, nonpolar substrate such as Ge or Si) face electrostatics that force a large residual interface charge on the $\{100\}$ orientation, plus antiphase disorder (the two sublattices of the zincblende compound nucleating in the wrong relative order) during compound-on-element growth; and that a small tilt $\theta$ away from the ideal orientation generates a built-in interface charge density $\sigma = (q\sqrt{3}/a^2)\sin\theta$, about $4.7\times10^{12}$ elementary charges per $\mathrm{cm}^2$ at $0.5^\circ$ for GaAs, which he proposed to exploit deliberately as an electrostatic design parameter. The subsequent theoretical literature replaced lineup rules with explicit first-principles (density-functional, ab initio) interface supercell calculations initiated by Van de Walle and Martin, charge-neutrality-level lineup theory by Tersoff, and d-orbital corrections to the common-anion rule by Wei and Zunger. The field's two standard reviews are those of Yu, McCaldin, and McGill (1992) and Franciosi and Van de Walle (1996), the latter reframing the agenda as 'band offset engineering' — controlling offsets by interface design where they could not be predicted. Modern practice computes offsets from explicit superlattices using semi-local density functional theory (DFT) or hybrid functionals such as HSE (Heyd–Scuseria–Ernzerhof, which mixes a fraction of exact exchange into the exchange-correlation functional), with typical agreement with experiment at the $0.05$–$0.15$ eV level — still several $kT$ short of the device-design target. Three of the five issues of the 1983 review have since been resolved with direct external evidence: for lattice-matched isovalent systems such as GaAs/AlAs the offsets are bulk-like, independent of orientation and growth sequence to within about $0.05$ eV; device-quality compound-on-element interfaces exist in production-class devices (GaInP/GaAs/Ge multijunction solar cells, quantum-dot lasers epitaxially grown on silicon); and substrate misorientation is a routine, reproducible epitaxial design parameter.

Problem Statement

Stated in the source's own terms, the question is whether the interface parameters a heterostructure device designer needs — the band offsets and the built-in interface charge and dipole — can be obtained from theory rather than from per-system experiment: thermionic currents change by a factor of $e$ per $1\,kT$ of barrier change, so quantitative design requires the offsets at the $\pm 1\,kT$ level (about $25$ meV at $300$ K) and a microscopic account of the built-in interface charge and dipole, while the best 1983 lineup rules erred by roughly $0.2$ eV. The open problem that remains, uniting the prediction-accuracy and microscopic-interface-charge components, is:

Can purely theoretical methods — first-principles or otherwise, with no parameter fitted to interface data of the system being predicted — deliver, for an arbitrary, atomically specified semiconductor heterojunction (including heterovalent, polar/nonpolar, and lattice-mismatched pairs), (i) the valence- and conduction-band offsets accurate to $\pm 1\,kT$, and (ii) the residual interface charge and dipole fixed by the atomic configuration actually formed during growth (abrupt versus intermixed, stoichiometry, defects, antiphase boundaries)?

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