Making capacitance-voltage profiling a reliable method for conduction-band-offset extraction in polar III-nitride heterostructures: sub-nanometer interface location and unexplained reconstruction discrepancies

At a polar (c-plane, Ga-face) III-nitride heterointerface, the spontaneous-plus-piezoelectric polarization sheet charge $\sigma_i$ (a charge areal density, with $|\sigma_i/e|\sim10^{12}$-$10^{13}\ \mathrm{cm}^{-2}$) amplifies any error $\xi$ in the assumed interface position into a conduction-band-offset error $\xi\,\sigma_i/\epsilon$ — roughly 80-90 meV per nanometer of misplacement for $|\sigma_i/e|\approx 4.7\times10^{12}\ \mathrm{cm}^{-2}$ (Hurni, Kroemer, Mishra, and Speck, 2012). Because self-consistent Poisson(-Schrödinger) reconstructions of the measured apparent carrier profiles show unexplained systematic mismatches, the C-V data alone cannot locate the interface with the sub-nanometer precision required, and the origin of the mismatches (interface grading, deep defects, spatially varying donor ionization, solver convergence) was not established. The open problem is to make C-V profiling a quantitatively reliable method for $\Delta E_c$ and $\sigma_i$ extraction in polar III-nitride heterostructures — reliably either by locating the interface position accurately enough that the misplacement bias $\xi\,\sigma_i/\epsilon$ falls below the tolerances, or by an analysis insensitive to the assumed interface position — and thereby demonstrate quantitatively reliable $\Delta E_c$ and $\sigma_i$ extraction. The requirement is met, for every polar (c-plane, Ga-face) $n$-$N$ GaN/Al$_x$Ga$_{1-x}$N profile, by either of two alternatives, satisfying either of which resolves it: (a) the interface position is supplied from independent metrology accurate enough that the misplacement bias $\xi\,\sigma_i/\epsilon$ contributes less than 30 meV to $\Delta E_c$, i.e. $\xi \le 30\ \mathrm{meV}\cdot\epsilon/\sigma_i$ — roughly 0.3 nm at $|\sigma_i/e|\approx4.7\times10^{12}\ \mathrm{cm}^{-2}$ and 0.15 nm at $10^{13}\ \mathrm{cm}^{-2}$ — and the extraction then returns $\Delta E_c$ within 30 meV and $\sigma_i$ within 10% of their true values; or (b) the extraction is position-free, neither fixing nor assuming an interface position internally, so that its output is unchanged as any assumed position is varied over the full interface-location uncertainty of at least 1 nm, and it recovers the true $\Delta E_c$ and $\sigma_i$ rather than a merely invariant output. Alternative (b) excludes freezing the interface position at a nominal value, which keeps the full $\xi\,\sigma_i/\epsilon$ bias while appearing invariant and does not qualify. The result must hold across the whole class rather than for a single favourable profile, and may be illustrated, but not replaced, by a demonstration on any measured or forward-modelled profile; and identify the physical origin of the reconstruction-measurement discrepancies.

Background

Capacitance-voltage (C-V) profiling is a standard electrical characterization technique: the small-signal capacitance $C(V)$ of a Schottky diode or of a one-sided $p^+$-$n$ junction (a junction whose $p$ side is so heavily doped that essentially the whole depletion region lies in the $n$ layer) is measured as a function of reverse bias, and from it one forms an apparent free-electron concentration profile

$\hat{n}(w)=\left[\frac{q\epsilon}{2}\frac{\mathrm{d}}{\mathrm{d}V}\left(\frac{1}{C^2}\right)\right]^{-1},\qquad w=\frac{\epsilon}{C},$

where $q$ is the elementary charge, $\epsilon$ the dielectric permittivity, and $w$ the depletion depth. Because mobile carriers are smeared over roughly a Debye length, $\hat{n}$ is an apparent, thermally broadened profile rather than the true carrier density — the "Debye averaging" analyzed by Kroemer and Chien (1981). Nevertheless, a reconstruction method initiated by Kroemer (1985) — with the effect of the dielectric-permittivity mismatch across the interface analyzed by Babic and Kroemer (1985) — recovers, from $\hat{n}(w)$ together with the donor doping profile $N_d(x)$ and the heterointerface position $x_i$, both the net charge $\sigma_i$ residing at the interface and the electrostatic potential difference between two field-free points, from which the conduction-band offset $\Delta E_c$ (the discontinuity of the conduction-band edge across the heterojunction) follows; Hurni et al. refer to this reconstruction as the Kroemer-Chien-Harris-Edwall (KCHE) analysis.

The wurtzite III-nitrides (GaN and its Al and In alloys) are polar crystals: spontaneous polarization plus strain-induced piezoelectric polarization jump abruptly at heterointerfaces, producing effective sheet charges of order $10^{12}$-$10^{13}\ \mathrm{cm}^{-2}$, one to two orders of magnitude larger than interface charges in conventional III-V heterostructures. Where the polarization sheet charge is positive it is compensated by a two-dimensional electron gas (2DEG), a quantum-confined, high-density electron layer. Miller et al. (2002) measured these polarization charges directly by C-V profiling, and modeling had shown that deep levels distort the C-V curves of AlGaN/GaN structures (Osvald, 2011).

Hurni, Kroemer, Mishra, and Speck ("Capacitance-voltage profiling on polar III-nitride heterostructures", J. Appl. Phys. 112, 083704, 2012) applied the KCHE analysis to Ga-face c-plane (polar-orientation) $n$-$N$ GaN/AlGaN heterostructures — isotype heterojunctions with n-type conduction on both sides, the more lightly doped layer written in lowercase — using one-sided $p^+$-$n$ junction diodes and a self-consistent numerical reconstruction — solving Poisson's equation for the electrostatic potential, coupled where needed to the Schrödinger equation for the quantized 2DEG states, iterated to self-consistency. They proved an exact linear error law: if the interface is assumed at $x_i+\xi$ while the true position is $x_i$, the reconstructed potential difference acquires the error $\xi\,\sigma_i/\epsilon$, so the extracted $\Delta E_c$ is proportionally biased; numerically, with $|\sigma_i/e|\approx 4.7\times10^{12}\ \mathrm{cm}^{-2}$, each nanometer of misplacement shifts the extracted $\Delta E_c$ by roughly 80-90 meV (on one sample, extracted $\Delta E_c$ was 260, 180, and 100 meV for $x_i$ = 236, 237, and 238 nm). They also settled one prior ambiguity: because the C-V displaced-charge response includes all depletable electrons, the integral relations must use the total donor concentration $N_d$ rather than the ionized concentration $N_d^+$ (relevant since Si donors in GaN and AlGaN are only partially ionized at room temperature), with only a small logarithmic correction term. However, their reconstructions systematically mismatched the measured apparent profiles — the measured $\hat{n}$ decreases earlier and to a higher minimum than the reconstruction, the reconstructed space-charge extent is shorter than measured, and one reconstruction near a 2DEG interface developed an unphysical peak attributed to solver convergence difficulties — so the C-V data plus the reconstruction could not themselves locate $x_i$ to the required precision. Candidate causes were left unidentified: finite interface grading (alloy interdiffusion over ~nm instead of a perfectly sharp interface), deep defect states, spatially varying donor ionization or trapping, and solver convergence or approximation limits, along with uncertainties in the true local doping profile $N_d(x)$ (day-to-day growth fluctuations of order 10% were observed between sister growths). The paper concluded that C-V profiling is not, as it stands, a reliable stand-alone method for $\Delta E_c$ extraction in polar III-nitrides.

Problem Statement

For polar (c-plane, Ga-face) III-nitride heterostructures such as $n$-$N$ (isotype, n-type on both sides) GaN/Al$_x$Ga$_{1-x}$N with interface charge $|\sigma_i/e|\sim10^{12}$-$10^{13}\ \mathrm{cm}^{-2}$, make capacitance-voltage profiling a quantitatively reliable method for extracting the conduction-band offset $\Delta E_c$ (and the interface charge $\sigma_i$). Concretely: (1) make the extracted $\Delta E_c$ and $\sigma_i$ quantitatively reliable — reliably either by locating the interface position accurately enough that the misplacement bias $\xi\,\sigma_i/\epsilon$ falls below the tolerances, or by an analysis insensitive to the assumed interface position — and thereby demonstrate quantitatively reliable $\Delta E_c$ and $\sigma_i$ extraction. The requirement is met, for every polar (c-plane, Ga-face) $n$-$N$ GaN/Al$_x$Ga$_{1-x}$N profile, by either of two alternatives, satisfying either of which resolves it: (a) the interface position is supplied from independent metrology accurate enough that the misplacement bias $\xi\,\sigma_i/\epsilon$ contributes less than 30 meV to $\Delta E_c$, i.e. $\xi \le 30\ \mathrm{meV}\cdot\epsilon/\sigma_i$ — roughly 0.3 nm at $|\sigma_i/e|\approx4.7\times10^{12}\ \mathrm{cm}^{-2}$ and 0.15 nm at $10^{13}\ \mathrm{cm}^{-2}$ — and the extraction then returns $\Delta E_c$ within 30 meV and $\sigma_i$ within 10% of their true values; or (b) the extraction is position-free, neither fixing nor assuming an interface position internally, so that its output is unchanged as any assumed position is varied over the full interface-location uncertainty of at least 1 nm, and it recovers the true $\Delta E_c$ and $\sigma_i$ rather than a merely invariant output. Alternative (b) excludes freezing the interface position at a nominal value, which keeps the full $\xi\,\sigma_i/\epsilon$ bias while appearing invariant and does not qualify. The result must hold across the whole class rather than for a single favourable profile, and may be illustrated, but not replaced, by a demonstration on any measured or forward-modelled profile; (2) identify the physical origin of the systematic discrepancies between self-consistent Poisson(-Schrödinger) reconstructions and measured apparent carrier profiles — the earlier decrease and higher minimum of the measured $\hat{n}$ and the extended space-charge extent (the isolated 2DEG-interface peak is not among these: it is identified below as a numerical artifact) — by determining which single mechanism or combination of mechanisms — among finite interface grading, deep defect states, spatially varying donor ionization or trapping, numerical solver convergence, and uncertainties in the true local doping profile $N_d(x)$ — is actually responsible for the observed discrepancies. The unphysical peak that appeared in one reconstruction near a 2DEG interface is, as described in the Background, a numerical artifact attributed to solver convergence difficulties: it is to be excluded as numerical through robust self-consistent convergence, not reproduced as a feature of the measured apparent profile. That exclusion applies only to the isolated 2DEG-interface peak; numerical solver convergence and approximation limits nevertheless remain a candidate origin of the remaining reconstruction-measurement discrepancies — the earlier decrease, higher minimum, and extended space-charge extent — where the reconstruction can be biased without producing a manifestly unphysical feature.

References

1. Christophe A. Hurni, Herbert Kroemer, Umesh K. Mishra, James S. Speck, "Capacitance-voltage profiling on polar III-nitride heterostructures", Journal of Applied Physics 112, 083704 (2012). DOI: 10.1063/1.4757940, https://doi.org/10.1063/1.4757940.
2. Christophe A. Hurni, Herbert Kroemer, Umesh K. Mishra, James S. Speck, "m-plane (101̄0) and (202̄1) GaN/AlxGa1–xN conduction band offsets measured by capacitance-voltage profiling", Applied Physics Letters 105, 232108 (2014). DOI: 10.1063/1.4903180, https://doi.org/10.1063/1.4903180.
3. Monir Rychetsky, Ingrid Koslow, Baran Avinc, Jens Rass, Tim Wernicke, Konrad Bellmann, Luca Sulmoni, Veit Hoffmann, Markus Weyers, Johannes Wild, Josef Zweck, Bernd Witzigmann, Michael Kneissl, "Determination of polarization fields in group III-nitride heterostructures by capacitance-voltage-measurements", Journal of Applied Physics 119, 095713 (2016). DOI: 10.1063/1.4943185, https://doi.org/10.1063/1.4943185.
4. Norman Susilo, Georgios G. Roumeliotis, Michael Narodovitch, Bernd Witzigmann, Monir Rychetsky, Silvio Neugebauer, Martin Guttmann, Johannes Enslin, Armin Dadgar, Tore Niermann, Tim Wernicke, André Strittmatter, Michael Lehmann, Dimitra N. Papadimitriou, Michael Kneissl, "Accurate determination of polarization fields in (0001) c-plane InAlN/GaN heterostructures with capacitance-voltage-measurements", Journal of Physics D: Applied Physics 51, 485103 (2018). DOI: 10.1088/1361-6463/aae464, https://doi.org/10.1088/1361-6463/aae464.
5. Norman Susilo, Marcel Schilling, Michael Narodovitch, Hsin-Hung Yao, Xiaohang Li, Bernd Witzigmann, Johannes Enslin, Martin Guttmann, Georgios G. Roumeliotis, Monir Rychetsky, Ingrid Koslow, Tim Wernicke, Tore Niermann, Michael Lehmann, Michael Kneissl, "Precise determination of polarization fields in c-plane GaN/AlxGa1-xN/GaN heterostructures with capacitance–voltage-measurements", Japanese Journal of Applied Physics 58, SCCB08 (2019). DOI: 10.7567/1347-4065/ab09dd, https://doi.org/10.7567/1347-4065/ab09dd.
6. Yinlu Gao, Dan Sun, Xue Jiang, Jijun Zhao, "Ab initio analytic calculation of point defects in AlGaN/GaN heterointerfaces", Journal of Physics: Condensed Matter 33, 035002 (2020). DOI: 10.1088/1361-648X/abbdbb, https://doi.org/10.1088/1361-648X/abbdbb.
7. Pramod Reddy, Isaac Bryan, Zachary Bryan, James Tweedie, Shun Washiyama, Ronny Kirste, Seiji Mita, Ramon Collazo, Zlatko Sitar, "Charge neutrality levels, barrier heights, and band offsets at polar AlGaN", Applied Physics Letters 107, 091603 (2015). DOI: 10.1063/1.4930026, https://doi.org/10.1063/1.4930026.
8. E. J. Miller, E. T. Yu, C. Poblenz, C. Elsass, J. S. Speck, "Direct measurement of the polarization charge in AlGaN/GaN heterostructures using capacitance–voltage carrier profiling", Applied Physics Letters 80, 3551 (2002). DOI: 10.1063/1.1477275, https://doi.org/10.1063/1.1477275.
9. J. Osvald, "Influence of deep levels on capacitance-voltage characteristics of AlGaN/GaN heterostructures", Journal of Applied Physics 110, 073702 (2011). DOI: 10.1063/1.3643000, https://doi.org/10.1063/1.3643000.
10. Herbert Kroemer, "Determination of heterojunction band offsets by capacitance-voltage profiling through nonabrupt isotype heterojunctions", Applied Physics Letters 46, 504 (1985). DOI: 10.1063/1.95572, https://doi.org/10.1063/1.95572.
11. Herbert Kroemer, Wu-Yi Chien, "On the theory of Debye averaging in the C-V profiling of semiconductors", Solid-State Electronics 24, 655 (1981). DOI: 10.1016/0038-1101(81)90195-7, https://doi.org/10.1016/0038-1101(81)90195-7.
12. Dubravko I. Babic, Herbert Kroemer, "The role of nonuniform dielectric permittivity in the determination of heterojunction band offsets by C–V profiling through isotype heterojunctions", Solid-State Electronics 28, 1015-1017 (1985). DOI: 10.1016/0038-1101(85)90032-2, https://doi.org/10.1016/0038-1101(85)90032-2.
13. Ziyi Zhang, Maki Kushimoto, Masahiro Horita, Naoharu Sugiyama, Leo J. Schowalter, Chiaki Sasaoka, Hiroshi Amano, "Space charge profile study of AlGaN-based p-type distributed polarization doped claddings without impurity doping for UV-C laser diodes", Applied Physics Letters 117, 152104 (2020). DOI: 10.1063/5.0027789, https://doi.org/10.1063/5.0027789.
14. Tetsuya Suemitsu, Hiroki Imabayashi, Kenji Shiojima, "Impact of Polarization Charges on Threshold Voltage and Band Offset in AlGaN/GaN Heterostructures", physica status solidi (a) 223, e202501001 (2026). DOI: 10.1002/pssa.202501001, https://doi.org/10.1002/pssa.202501001.