Self-consistent model of the two-subband magnetocapacitance of gated InAs/AlSb quantum wells and the origin of the anomalous shifts of capacitance minima to lower magnetic field

A 1992 magneto-transport study of a gate-controlled 15-nm InAs quantum well with AlSb barriers (Nguyen, Ensslin, and Kroemer) showed that the electron density follows a simple parallel-plate capacitor law from $3\times10^{11}$ to $3\times10^{12}\ \mathrm{cm^{-2}}$ and that above $\approx1.4\times10^{12}\ \mathrm{cm^{-2}}$ a second size-quantized subband is populated. In that two-subband regime the 4.2 K magnetocapacitance departs from the simple pattern: minima at integer Landau-level filling factors become non-equidistant in gate voltage, the density of each subband becomes a nonlinear function of gate voltage, and - anomalously - some minima shift to lower magnetic field as the gate voltage is increased. The study left three issues open: the missing self-consistent calculation (incorporating the InAs/AlSb band lineup and the nonparabolic dispersion of InAs), the unexplained minima shifts, and the unidentified deep levels in the AlSb barrier whose trapped charge shifts the effective gate bias after thermal cycling. Later literature substantially settled the third issue: the deep levels are native antisite-related defects (Al on Sb sites in AlSb; As on Al sites at AlAs-like interfaces), as supported by cyclotron-resonance defect spectroscopy and persistent-photoconductivity studies. The first two remain open: no published self-consistent model for gated InAs/AlSb reproduces the two-subband magnetocapacitance, and the anomalous minima shifts have no published explanation. The surviving problem is to construct such a model - incorporating the deep-well band lineup, the strongly nonparabolic InAs dispersion, and the barrier deep-level charge - and to determine quantitatively the physical origin of the minima shifts to lower magnetic field.

Background

InAs/AlSb quantum wells confine electrons in an unusually deep potential well: the conduction-band offset between InAs and the AlSb barrier is roughly $1.3$-$1.35\ \mathrm{eV}$, several times larger than in the GaAs/AlGaAs workhorse system, while InAs has a small direct band gap ($\approx0.42\ \mathrm{eV}$) and a small band-edge effective mass ($0.023\,m_0$, where $m_0$ is the free-electron mass). Electrons confined to the well form a two-dimensional electron gas (2DEG) whose motion perpendicular to the layers is quantized into discrete standing-wave states called subbands. The subband spacing is large, but still small enough that at high sheet densities ($N_s\gtrsim10^{12}\ \mathrm{cm^{-2}}$) a second subband becomes populated. A second peculiarity of InAs is strong conduction-band nonparabolicity: because the gap is small, the effective mass grows appreciably with electron energy, so the masses, densities of states, and Landau levels of the two occupied subbands differ from each other and from the band-edge value.

Nguyen, Ensslin, and Kroemer (1992) studied such a structure - a 15-nm InAs well with AlSb barriers (about a 100-nm top barrier), a GaSb cap, and a large-area top gate insulated by a deposited silicon dioxide or silicon nitride dielectric - using Hall measurements, Shubnikov-de Haas oscillations (periodic oscillations of the longitudinal resistance in magnetic field, whose periodicity counts the carrier density), and magnetocapacitance. Magnetocapacitance is the small-signal electrical capacitance between gate and 2DEG measured versus perpendicular magnetic field $B$ at 4.2 K: in a magnetic field the 2DEG density of states collapses into discrete Landau levels, and whenever the Fermi energy lies between Landau levels - at integer filling factor $\nu=N_s h/(eB)$ - the 2DEG is incompressible and the capacitance shows a minimum. With a single occupied subband these minima are equally spaced in gate voltage and the density follows a simple parallel-plate law. The 1992 work established this linear capacitive control over the range $3\times10^{11}$-$3\times10^{12}\ \mathrm{cm^{-2}}$ (gate voltage -7 to +10 V), located the second-subband onset near $1.4\times10^{12}\ \mathrm{cm^{-2}}$ (via a mobility dip attributed to inter-subband scattering and a non-self-consistent envelope calculation that included InAs nonparabolicity but not the electrostatics), and then recorded the two-subband anomalies that motivate the present problem: non-equidistant capacitance minima, a nonlinear dependence of each subband's density on gate voltage, and the counter-intuitive shift of some minima to lower $B$ with increasing gate voltage (at fixed filling factor, a minimum should move to higher $B$ as the density grows). In the GaAs/AlGaAs analog, self-consistent calculations - simultaneously solving the Schrödinger equation for the confined states and the Poisson equation for the electrostatic potential of the gate and the space charge - had already explained the corresponding structure, including the nonlinear population process of the upper subband (Ensslin, Heitmann, Gerhardts, and Ploog, 1989), but no equivalent calculation exists for InAs/AlSb, where the deep-well band lineup, the strong nonparabolicity, and charge trapped in deep barrier levels all differ.

The barrier deep levels form the third thread. Undoped InAs/AlSb wells carry a large ($\sim10^{12}\ \mathrm{cm^{-2}}$) electron density supplied not by deliberate doping but by native defects in the AlSb acting as deep donors (Tuttle, Kroemer, and English, 1989, 1990); Nguyen, Ensslin, and Kroemer additionally found that thermal cycling shifts the entire $N_s(V_g)$ characteristic to more positive gate voltage at unchanged slope, indicating deep traps that capture and release charge across the thermal cycle. The microscopic identity of these levels was later established: antisite defects - an Al atom on an Sb site in the AlSb barrier ($\mathrm{Al}_{Sb}$) and an As atom on an Al site at AlAs-like interfaces ($\mathrm{As}_{Al}$) - act as the donors, whereas Tamm states (localized interface states intrinsic to the abrupt change of chemical bonding at the interface) lie too low in energy to supply the observed carriers (Shen et al., 1993, 1995); cyclotron-resonance spectroscopy (resonant microwave absorption at the frequency at which electrons orbit in a magnetic field; oscillations of the resonance-line splitting and linewidth as a defect level crosses the Landau levels turn it into a defect spectroscopy) located the near-interface deep donor at $\approx67$-$72\ \mathrm{meV}$ above the InAs well bottom, consistent with the $\mathrm{Al}_{Sb}$ antisite (Vasilyev, Meltser, Ivanov, and Kop'ev, 2009). What remains missing is a quantitative account of the two-subband magnetocapacitance of the gated InAs/AlSb well itself.

Problem Statement

The problem left open by the 1992 study of a gate-controlled 15-nm InAs quantum well with AlSb barriers (about a 100-nm AlSb top barrier, GaSb cap, deposited dielectric, metal gate; measurements at 4.2 K, magnetic fields up to 8 T, gate voltage -7 to +10 V), restricted to the part that survives in the current literature, is:

(i) Construct a self-consistent microscopic model of the gated two-subband InAs/AlSb quantum well - a simultaneous Schrödinger-Poisson solution for the confined subbands and the electrostatic potential that incorporates the InAs/AlSb conduction-band lineup (offset $\approx1.3$-$1.35\ \mathrm{eV}$), the strongly nonparabolic InAs conduction band (band-edge mass $0.023\,m_0$, gap $\approx0.42\ \mathrm{eV}$), and the trapped charge in the deep levels of the AlSb barrier (identified since as antisite-related native defects) - and demonstrate that it quantitatively reproduces the measured magnetocapacitance of that structure across the single-subband to two-subband transition: the linear single-subband gate response, the second-subband onset at total density $\approx1.4\times10^{12}\ \mathrm{cm^{-2}}$, the non-equidistant spacing in gate voltage of the capacitance minima at integer filling factors once the second subband is populated, and the nonlinear dependence of each subband's density on gate voltage.

(ii) Determine, quantitatively, the physical origin of the observed shifts of some magnetocapacitance minima to lower magnetic field with increasing gate voltage in the two-subband regime - identifying a mechanism whose predicted shifts agree in sign and magnitude with the published minima positions, given that the trivial fixed-filling-factor expectation ($B=N_s h/(e\nu)$ for integer filling $\nu$) predicts motion to higher field as the density increases.

References

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