Capacitance–voltage (C–V) profiling from an adjacent Schottky barrier measures carrier depth profiles in semiconductors, but when the profiling depletion region crosses an isotype n–n heterojunction the apparent profile is broadened and asymmetric: a small voltage increment depletes charge both at the outer depletion edge and inside the conduction-band notch at the interface. Kroemer and Chien (1979 conference abstract; journal version 1980, the KCHE method) showed that this broadening conserves the total charge and the dipole moment of the space-charge distribution, which yields a two-integral inversion for the interface sheet charge and the conduction-band offset ΔEc\Delta E_c — provided the ionized-donor profile is known independently, Debye-length smearing is neglected, and the probing barrier is on the favorable side. The research objective is to remove these three restrictions. Debye averaging was quantified by the same authors in 1981 (RMS averaging distance 2LD\sqrt{2}L_D; moment-conservation theorem); compositional grading and nonuniform permittivity were proven in 1985 not to bias the extraction; and the probe-geometry asymmetry has been studied by numerical simulation and structure-design work. The requirement of an independently known donor profile, however, remains: the method's later applications, including the 2012–2014 III-nitride heterojunction work by its originators, still take the doping profile as input and list its uncertainty among unresolved limitations. The open core is a joint inversion of C–V data through an isotype heterojunction that simultaneously constrains the donor profile, interface charge, and band offset — or a proof that this is impossible — with quantified validity bounds including Debye-length and probe-geometry effects.

Background

Capacitance–voltage (C–V) profiling is a standard electrical method for measuring the depth profile of free carriers in a semiconductor. A rectifying contact — most often a Schottky barrier, i.e. a metal–semiconductor contact whose space-charge region behaves as a voltage-dependent capacitor — is reverse biased, and the carrier concentration is inferred from how the differential capacitance C(V)C(V) grows as the depletion region (the region swept free of mobile charge) widens. In the depletion approximation, the probed depth is w=ϵ/Cw=\epsilon/C and the apparent carrier concentration is n^(w)=[qϵ2d(1/C2)dV]−1\hat{n}(w)=\left[\frac{q\epsilon}{2}\frac{d(1/C^{2})}{dV}\right]^{-1}, with ϵ\epsilon the permittivity and qq the elementary charge; in a homogeneously doped region n^\hat{n} equals the true carrier density, so plotting −log⁡n^-\log \hat{n} maps the conduction-band edge.

An isotype n–n heterojunction is a junction between two different n-type semiconductors, for example GaAs and (Al,Ga)As grown by liquid-phase epitaxy (LPE) or molecular-beam epitaxy (MBE). The band-gap difference partitions into a conduction-band offset ΔEc\Delta E_c and a valence-band offset; these offsets, together with any fixed interface charge σi\sigma_i (from defects, or, in polar III–V nitrides, from spontaneous and piezoelectric polarization), control carrier confinement in high-electron-mobility transistors (HEMTs), quantum wells, and lasers, so their measurement is a central metrology problem. At such a junction the conduction band forms a notch (a local dip that accumulates electrons) on one side and a spike on the other.

When the profiling depletion region crosses such a junction, the simple interpretation fails: a small voltage increment removes mobile charge not only at the outer depletion edge but also inside the notch, so the measured n^\hat{n} is a weighted average of two spatially separated depletion events. Apparent profiles through n–n heterojunctions therefore look as if the junction were compositionally graded over implausibly large distances, and the distortion is asymmetric, most pronounced when the probing barrier is placed on the notch side. In a 1979 conference abstract (published in IEEE Trans. Electron Devices) and its journal version (Kroemer, Chien, Harris, and Edwall, 1980 — the method now called KCHE), Kroemer and Chien showed that this broadening nevertheless conserves two quantities: the total charge and the first moment (dipole) of the total space-charge distribution, up to a smearing over the Debye length LD=(ϵkT/q2n)1/2L_D=(\epsilon kT/q^{2}n)^{1/2}, the natural screening length of a semiconductor in thermal equilibrium. These two conserved integrals, evaluated from the measured n^(w)\hat{n}(w) and the ionized-donor profile ND(x)N_D(x), suffice to extract both the interface sheet charge and ΔEc\Delta E_c for an abrupt junction; the 1980 paper obtained ΔEc=0.248\Delta E_c=0.248 eV (about 0.66 ΔEg0.66\,\Delta E_g) for an LPE n-N GaAs/Al0.3_{0.3}Ga0.7_{0.7}As junction.

The 1979 analysis left three linked restrictions: (i) the inversion is determinate only if ND(x)N_D(x) is known from independent measurements — without it, the extraction of interface charge and ΔEc\Delta E_c is underdetermined; (ii) the Debye-length smearing is asserted to be minor but is not quantified; and (iii) the dependence of the asymmetric broadening on probe placement — which side of the junction the Schottky barrier, p–n junction, or electrolyte (electrochemical-profiling) barrier sits on — is uncharacterized. Later work by the same group quantified the Debye averaging (Kroemer and Chien, 1981: a moment-conservation theorem and an RMS averaging distance 2LD\sqrt{2}L_D), proved that compositional grading (Kroemer, 1985) and nonuniform dielectric permittivity (Babić and Kroemer, 1985) do not bias the extracted offsets, and, with others, developed numerical Poisson and Poisson–Schrödinger simulations of apparent profiles and structure-design guidelines. No later work supplies the removal of restriction (i): an inversion that does not need an independently measured donor profile. Even the 2012–2014 applications of the method by its originators to III-nitride heterojunctions still take the doping profile as input and list its uncertainty, together with interface-position metrology, among the unresolved limitations.

Problem Statement

How can the conserved-integral C–V inversion through isotype n–n heterojunctions be made quantitatively reliable and broadly applicable? The load-bearing unresolved core is an inversion that removes the requirement of an independently known donor profile. Given C–V profiling data through an isotype n–n heterojunction (and at most a stated-minimal set of auxiliary measurements), determine the interface sheet charge σi\sigma_i and the conduction-band offset ΔEc\Delta E_c without requiring ND(x)N_D(x) from separate metrology; recover the donor profile ND(x)N_D(x) in addition when the data support it, and state when they do not. The target is such an inversion, with its identifiability conditions and error bounds, or a rigorous demonstration that the joint problem is underdetermined — that distinct configurations (ND(x),σi,ΔEc)(N_D(x),\sigma_i,\Delta E_c) can produce indistinguishable C–V responses.

Validity conditions any proposed method or impossibility argument must account for: (i) quantified Debye-length validity bounds — the magnitude of, and the conditions under which, Debye-length smearing (and, in quantum-confined systems, wave-function-extent averaging) leaves the conserved-integral relations accurate to stated tolerances; and (ii) characterized probe-geometry dependence — the systematic dependence of the apparent broadening and its notch-side asymmetry on the placement and type of the probing rectifying contact (Schottky barrier, p–n junction, or electrolyte barrier, on either side of the junction), and the limits of validity of the inversion with respect to that geometry, including interface states that exchange charge during the measurement cycle. Both have been substantially addressed by the later literature (Debye smearing: Kroemer and Chien, 1981; probe-geometry asymmetry: Blood, 1986; Panda and Biswas, 2013) and enter as quantified validity conditions rather than fresh derivations. Any claimed method must state its auxiliary inputs and their required accuracy: the originators' 2012 work found extraction possible only provided the doping profile and the exact hetero-interface position are known, and a 1 nm interface misplacement shifted the extracted offset by roughly 80 meV in polar structures.

References

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