Disorder-limited quantized conductance in long InAs/AlSb split-gate ballistic constrictions: surface-donor fluctuation statistics, the subband-spacing criterion, and charge-trapping effects

The source paper (Koester, Bolognesi, Hu, Kroemer, Rooks, Phys. Rev. B 49, 8514, 1994) reported quantized conductance in a 1.0 μm split-gate ballistic constriction in an InAs/AlSb quantum well and attributed the unusually good length performance to one-dimensional (1D) subband spacings ($\Delta E_{1D}\approx 7-10$ meV) exceeding the amplitude of the potential fluctuations induced by the GaSb-cap surface donors (~4 meV, estimated by analogy with GaAs/AlGaAs simulations). This attribution, the spatial statistics of the surface donors (assumed random; ordering was entertained only as a speculation for which the authors found 'no evidence at present'), the maximum channel length for clean quantization, and the mechanism of the +2 V premeasurement gate-bias effects were all left open or untested by the source work. The authors' own follow-ups directly measured $\Delta E_{1D}$ (up to 10.2 meV) and extended the demonstrated length to 2.0 μm (1996), but the fluctuation amplitude itself remained only model-estimated with an assumed random donor distribution, the donor statistics remained untested, the precise length limit was explicitly left undetermined, and the promised dedicated study of the bias-history mechanism never appeared.

Background

Conductance quantization. When a two-dimensional electron gas (2DEG — a thin sheet of conduction electrons confined at a semiconductor interface) is forced through a short, narrow constriction, its two-terminal conductance is quantized in steps of $2e^2/h$ per occupied one-dimensional (1D) mode, $G=2ne^2/h$ with $n$ the number of occupied 1D subbands (van Wees et al. 1988; Wharam et al. 1988). The constriction — a quantum point contact (QPC) — is usually defined electrostatically by a split gate: two negatively biased surface electrodes that deplete the 2DEG beneath them and leave a narrow channel between. Clean steps require essentially ballistic transport (electrons crossing without scattering) and transmission close to unity within each mode.

The length problem. In modulation-doped GaAs/Al$_x$Ga$_{1-x}$As structures the electrons are supplied by intentionally introduced Si donors placed in the barrier layer away from the channel; a randomly distributed donor plane produces a slowly varying fluctuating potential in the 2DEG. Numerical simulations with realistic random-donor potentials (Nixon, Davies, Baranger 1991) found a fluctuation correlation length near 0.2 μm and predicted serious degradation of the quantization for constrictions longer than this, because screening in the carrier-depleted 1D channel is much weaker than in the 2D reservoirs. Experiments nevertheless disagreed: deep-etched GaAs-based wires maintained quantization to 2 μm (Ismail, Washburn, Lee 1991), while split-gate constrictions degraded beyond roughly 0.6 μm even on wafers with mean free paths of several micrometers (both reports are summarized and cited in the source paper). The theory was refined by first-order small-angle scattering analyses (Glazman, Jonson 1991) and modal analyses with realistic potentials (Laughton et al. 1991), which identified indirect backscattering via quasilocalized states — disorder-induced, quasi-bound states that mediate reflection of propagating modes and become more probable with higher mode occupancy. A companion analysis of soft (long-range) disorder (Zagoskin et al. 1995; Rashkeev et al. 1996) showed that the density of such states decreases exponentially with the ratio $\Delta E_{1D}/\sigma_{imp}$ of the 1D subband spacing to the standard deviation of the impurity potential.

The InAs/AlSb platform. InAs quantum wells clad by AlSb barriers differ fundamentally in their doping: without intentional dopants, the electrons come from donors located at the surface of the GaSb capping layer (GaSb passivates the air-reactive AlSb surface). These surface donors dominate the electron sheet concentration of nominally undoped wells — Nguyen, Brar, Kroemer, and English (1992) found the Fermi level at the surface pinned about 850±50 meV below the conduction-band edge of the AlSb top layer in their wells — and the resulting electron accumulation at the InAs well was treated theoretically by Chadi (1993). In the GaSb-capped structure of the source work the surface Fermi level is pinned about 450±50 meV below the GaSb conduction-band edge; beyond this, little is known about the donors themselves. Because the 'doping plane' is the surface, it can be kept comparatively far from the channel (40 nm in the source work), while the small InAs effective mass ($0.023m_0$, about one-third that of GaAs) makes the 1D subband spacing — the energy gap between the transverse quantization levels of the constriction — larger for the same gate geometry. The residual disorder in this system is dominated by interface roughness (atomic-scale steps at the InAs/AlSb well interfaces; Bolognesi, Kroemer, English 1992) and residual impurities rather than by a nearby doping sheet.

The source paper and its proposal. After an initial report of quantized conductance in shorter InAs/AlSb constrictions (Koester et al. 1993), the source work (Koester et al. 1994) observed eight conductance steps, all within about 20% of their ideal ballistic values, in a 1.0 μm-long split-gate constriction — at the time the longest split-gate channel showing evidence of ballistic transport over its full length — with little degradation relative to a 0.2 μm control device on the same mesa. The authors proposed that the good length performance follows because $\Delta E_{1D}$ exceeds the amplitude of the donor-induced potential fluctuations, so that elastic backscattering channels are suppressed; this idea had been speculated for large-spacing GaAs devices in earlier work quoted in the source paper (where constrictions engineered for large subband spacing operated up to 30 K, Snider et al. 1991). The companion measurement paper (Koester et al. 1994, Phys. Rev. B 50, 5710) determined $\Delta E_{1D}$ directly by magnetic depopulation (squeezing the 1D modes with a perpendicular magnetic field and fitting the mode-index versus inverse-field curve), obtaining spacings up to 10.2 meV. The authors also reported that a positive premeasurement gate bias of +2 V applied for 1–2 min was required to obtain reproducible conductance traces, deferring a full explanation to a future publication.

Status after the follow-up. The authors' systematic length study (Koester et al. 1996) extended the demonstrated length to 2.0 μm (step heights within about 10% of ideal, over 50% of the 3.8 μm transport mean free path), measured $\Delta E_{1D}=5.8-9.7$ meV in the 2.0 μm device by thermal-smearing analysis, and estimated the fluctuation amplitude $\sigma_{imp}\approx 3-4$ meV from electrostatics-based modeling with Thomas–Fermi screening (the simplest self-consistent screening model), under assumed inputs: deep donors at $10^{16}$ cm$^{-3}$ randomly distributed in the AlSb barrier layers (contributing about 2.4 meV) plus at most 2 meV of InAs/AlSb interface nonuniformity. This made the spacing-criterion picture quantitatively consistent with the quasilocalized-state theory of Zagoskin et al., but left the fluctuation amplitude itself unmeasured, the donor statistics assumed, and the breakdown length undetermined. Longer devices showed markedly larger device-to-device variability and stronger sensitivity to the premeasurement bias and to thermal cycling, attributed qualitatively to charging and discharging of impurity or surface states under the gates. What remained open — and remains open — is stated in the problem below.

Problem Statement

Establish quantitatively the disorder and charge-trapping physics that limits how long an InAs/AlSb split-gate ballistic constriction can be while retaining clean quantized conductance. The limiting potential fluctuations arise from the remote donor distribution — GaSb-cap surface donors and deep donors in the AlSb barrier layers — together with InAs/AlSb interface nonuniformity; the relative weights of these contributions are not settled by the source work. The following are separately open issues identified by the source papers — the static disorder landscape, its dynamical charge-state counterpart, the resulting scattering, and the length limit they impose — each of which is independently open. They are posed for the InAs/AlSb quantum-well platform of the source work (2DEG roughly 40 nm below the GaSb-capped surface, electrons supplied by remote donors — GaSb-cap surface donors and deep donors in the AlSb barrier layers — and 1D subband spacings as reported by the source studies: about 6–10 meV per the 1996 summary, and up to 10.2 meV measured by magnetic depopulation, with about 12 meV only extrapolated):

(1) Determine the amplitude and spatial correlation length of the potential fluctuations induced in the InAs channel by the remote donor distribution and InAs/AlSb interface nonuniformity — quantities that the 1996 follow-up could only estimate ($\sigma_{imp}\approx 3$–$4$ meV) from a Thomas–Fermi model and that it described as 'not an easily determined quantity'. Prior work assumed the donors (GaSb surface donors and deep donors in the AlSb barriers) to be randomly distributed — the 1994 paper noted possible donor ordering only to record that there was 'no evidence at present' — so this assumption should be tested or justified, and the separation of the surface-donor contribution from that of deep donors in the AlSb barriers and of InAs/AlSb interface roughness remains open (the 1994 estimate invoked randomly distributed GaSb surface donors, while the 1996 model invoked barrier deep donors plus interface nonuniformity).

(2) Establish whether the criterion $\Delta E_{1D}>\sigma_{imp}$ (one-dimensional subband spacing exceeding the fluctuation amplitude) is sufficient to suppress impurity backscattering and thereby permit clean quantization over long channels.

(3) Determine the precise length at which the quantized conductance can no longer be observed in InAs/AlSb split-gate constrictions. The criterion for 'clean quantization' should be stated explicitly (the source studies judged conductance steps within about 20% (1994) and about 10% (1996) of their ideal $2ne^2/h$ values to be adequately quantized).

(4) Identify the microscopic origin of the gate-bias-history effects: which impurity or surface states charge and discharge under the premeasurement gate bias (the +2 V, 1–2 min conditioning of the source work), with what energies, spatial locations, and trapping dynamics, and how they shift threshold voltages, cause hysteresis, and modify the potential landscape and quantization of the 1D channel.

References

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