Unquantified polarization and doping-gradient corrections in capacitance-voltage profiling of heterojunction band offsets

Capacitance-voltage (C-V) profiling through nonabrupt (compositionally graded) isotype heterojunctions was shown by Kroemer in 1985 to yield the abrupt-limit band offsets, because the electrostatic potential drop across the junction, an integral of the free-carrier minus donor charge imbalance, is invariant under compositional grading when the asymptotic materials are fixed. The analysis left two systematic corrections undetermined: a possible composition-gradient-induced polarization, parameterized as $P=\alpha\epsilon_{0}\,dc/dx$ with an unknown coefficient $\alpha$ with the dimension of an electric potential, which would shift the extracted offsets by $\alpha\epsilon_{0}\Delta c/\epsilon$, where $\epsilon$ is the permittivity of the graded material; and the ambiguity that arises when the donor distribution $N_d(x)$ is itself graded and not independently known. Later literature quantified composition-induced polarization only where it is large — wurtzite nitride alloys (first-principles calculation plus C-V measurement itself), SiC polytype junctions, and spontaneously ordered (In,Ga)P — and turned C-V profiling into a tool for measuring polarization charge, but left the original disordered zincblende graded-alloy case unquantified and produced no general neglect criterion, while the unknown-doping-gradient inversion is still handled by ad hoc fitting, orientation choice, and consistency checks. The problem asks for quantitative determination of both corrections with derived offset errors and a stated neglect criterion, and for a validated procedure for detecting or eliminating the doping-gradient bias, so that the accuracy of C-V band-offset extraction on nonabrupt isotype heterojunctions can be certified.

Background

A semiconductor heterojunction (HJ) is an interface between two different semiconductors; its key electronic parameter is the band offset, that is, the discontinuities $\Delta E_C$ and $\Delta E_V$ of the conduction- and valence-band edges across the interface, which together account for the band-gap difference. An isotype HJ (written $n$-N or $p$-P, with the lower-case letter denoting the narrow-gap side) has the same carrier type on both sides, so the band bending is governed by electrostatics alone rather than by a p-n junction.

Capacitance-voltage (C-V) profiling is a standard characterization technique: a reverse-biased Schottky barrier (a rectifying metal-semiconductor contact) is placed on the surface, and the measured capacitance as a function of bias is converted into an apparent free-carrier concentration versus depth. Near an isotype HJ the band offset produces a characteristic electron accumulation/depletion pair; the apparent profile is smeared over a Debye length (the electrostatic screening length), but the dipole moment of the pair is preserved by the averaging. Kroemer, Chien, Harris, and Edwall (1980) showed that this dipole moment determines the electrostatic potential drop across the junction and hence the conduction-band offset, and the method became one of the most widely used sources of band-offset data (Kroemer, 2001).

Most modern junctions are grown abruptly (for example by molecular beam epitaxy), but much early data, in particular on liquid phase epitaxy (LPE) grown junctions, involved compositionally graded (nonabrupt) interfaces. Kroemer (1985) proved that C-V profiling is grading independent: the potential drop can be written as an integral of the charge imbalance between free electrons and donors,

$\Delta\phi=\frac{q}{\epsilon}\int_{-\infty}^{\infty}\left[n(x)-N_{d}(x)\right]\left(x-x_{i}\right)\,dx$

(with $q$ the elementary charge, $\epsilon$ the permittivity, and $x_i$ an arbitrary reference that cancels by overall charge neutrality), and this integral is unchanged by compositional grading as long as the asymptotic materials on the two sides are fixed. C-V profiling on graded junctions therefore yields the abrupt-limit offsets, which explained the agreement between old graded-junction data and newer abrupt-junction data. A companion result (Babic and Kroemer, 1985) showed that nonuniform dielectric permittivity does not affect the extracted offsets.

Two systematic corrections were nevertheless left unquantified. First, the derivation assumes that the dielectric polarization is entirely field-induced ($P=\chi\epsilon_{0}E$, the usual linear-response form with susceptibility $\chi$). In a compositionally graded alloy the composition gradient itself may induce a polarization not proportional to the local field; parameterizing it as $P=\alpha\epsilon_{0}\,dc/dx$, with $c$ the alloy composition and $\alpha$ an unknown materials coefficient with the dimension of an electric potential, such a term would modify Poisson's equation and shift the C-V-extracted offsets by $\alpha\epsilon_{0}\Delta c/\epsilon=\alpha\Delta c/\epsilon_{r}$, where $\epsilon=\epsilon_{r}\epsilon_{0}$ is the permittivity of the graded material. No estimate of $\alpha$ existed, so it was not known for which material systems and grading extents the mechanism can be neglected. Second, the inversion from measured C-V data to the abrupt-limit offset requires the donor distribution $N_d(x)$; if the doping is itself graded and not independently known, the inversion is ambiguous, and the sign and magnitude of the resulting error cannot be bounded a priori. Kroemer (1985) offered only practical expedients: match the doping levels on the two sides, and measure both $n$-N and $p$-P junctions so that the extracted offsets can be checked against the sum rule $\Delta E_C+\Delta E_V=\Delta E_G$.

Later work transformed the polarization question in polar materials. In wurtzite III-V nitrides (wurtzite is the hexagonal, non-centrosymmetric crystal structure of GaN and AlN, whose polar symmetry axis permits a spontaneous polarization even in the unstrained bulk), first-principles calculations showed that the discontinuity of the macroscopic polarization (spontaneous plus strain-induced piezoelectric polarization) across an interface generates internal fields of order $10^9$ V/m and requires a nonstandard evaluation of band offsets, with forward-backward asymmetries of 0.2-0.85 eV (Bernardini and Fiorentini, 1998); the composition dependence of the polarization in ternary nitride alloys, including its nonlinear bowing (deviation from a linear-in-composition law), was then computed ab initio (Bernardini and Fiorentini, 2001). C-V profiling itself became a tool for measuring polarization charge (Jia et al., 2001; Zhou et al., 2002; Miller et al., 2002; Zhang et al., 2004), and composition grading is now deliberately exploited for polarization doping (Simon et al., 2010). Analogous composition- and structure-induced polarization charges were detected by C-V at spontaneously ordered (In,Ga)P/GaAs interfaces (Krispin et al., 2002) and at 3C-SiC/6H-SiC polytype junctions (Lu et al., 2009). For the disordered zincblende (the nonpolar cubic crystal structure of GaAs and of most III-V arsenide and phosphide alloys) [001] graded alloys, such as the LPE-grown AlGaAs-type junctions of the early C-V literature, however, no dedicated estimate of the gradient-induced polarization is available, and no general neglect criterion spanning material systems, crystal orientations, and grading extents has been established. The doping-gradient question likewise remains without a settled methodology: later analyses handle graded band gap and/or doping by numerical forward modeling and fitting (Gray and Lundstrom, 1985; Sundaram and Gossard, 1993; Kikuchi et al., 2000; Kleider et al., 2015), amend the extraction equations for identified error sources (Leu and Forrest, 1989), extend C-V analysis to p-n junctions (Nemirovsky et al., 1998), or implement the sum-rule consistency check (Rao et al., 1987), but none provides a standardized procedure that detects or eliminates an unknown graded donor distribution.

Problem Statement

Quantify, and where possible eliminate, the two systematic corrections that Kroemer's 1985 grading-independence analysis of capacitance-voltage (C-V) profiling left undetermined for band-offset extraction through nonabrupt (compositionally graded) isotype heterojunctions.

(1) Composition-gradient-induced polarization. Determine the polarization induced by a composition gradient in the compositionally graded isotype heterojunctions of interest here — LPE-grown disordered zincblende III-V graded alloys such as AlGaAs-type junctions. In the parameterization of Kroemer (1985) this means the composition-gradient coefficient $\alpha$ in $P=\alpha\epsilon_{0}\,dc/dx$ with $c$ the alloy composition — the response of the polarization to a composition gradient; the composition-dependent macroscopic polarization $P(c)$ and its bowing are context here, not a separate deliverable; derive the resulting error $\alpha\epsilon_{0}\Delta c/\epsilon=\alpha\Delta c/\epsilon_{r}$ in C-V-extracted abrupt-limit band offsets, and state a neglect criterion for this setting: for which grading extents the correction is negligible relative to typical band-offset experimental uncertainties (of order tens of meV), and for which it must be corrected. In systems where composition-induced polarization has already been quantified (wurtzite nitrides, SiC polytype junctions, spontaneously ordered (In,Ga)P), the remaining question is limited to any genuinely gradient-proportional term and its effect on C-V-extracted offsets; these systems, and other material systems and crystal orientations, provide context and optional extensions rather than completion conditions.

(2) Unknown graded doping. Develop and validate a robust experimental or analytical procedure to detect, quantify, or eliminate the bias in C-V-extracted abrupt-limit band offsets caused by a donor distribution $N_d(x)$ that is itself graded and not independently known — going beyond the practical expedients of matching the doping levels on the two sides of the junction and of cross-checking $n$-N and $p$-P measurements against the sum rule $\Delta E_C+\Delta E_V=\Delta E_G$. Detection alone establishes only that the bias is present; the deliverable is met when the procedure also bounds or removes it. Here validation means demonstrating the procedure's correctness and its quantitative accuracy.

Both corrections enter the same space-charge integral that the C-V technique inverts; the overarching objective is to certify the accuracy of C-V band-offset extraction on nonabrupt isotype heterojunctions.

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