A Gate-Defined Quantum Point Contact in an InAs Two-Dimensional Electron Gas

TL;DR

This paper reports on experimental observations of quantized conductance and spin-resolved subbands in an InAs two-dimensional electron gas using a gate-defined quantum point contact, revealing detailed magnetic field effects.

Contribution

It provides new experimental insights into spin-resolved conductance quantization and subband structure in InAs quantum point contacts under magnetic fields.

Findings

Observation of conductance plateaus at integer multiples of e^2/h

Visualization of spin-resolved magnetoelectric subbands

Measurement of subband spacings and g-factor under magnetic fields

Abstract

We experimentally study quantized conductance in an electrostatically defined constriction in a high-mobility InAs two-dimensional electron gas. A parallel magnetic field lifts the spin degeneracy and allows for the observation of plateaus in integer multiples of $e^2/h$. Upon the application of a perpendicular magnetic field, spin-resolved magnetoelectric subbands are visible. Through finite bias spectroscopy we measure the subband spacings in both parallel and perpendicular direction of the magnetic field and determine the $g$-factor.

Equations

$$\frac{m^{\ast }}{2}{\omega }_0^2\cdot L_n^2=\hslash {\omega }_0(n-\frac{1}{2}),$$

$$E_n=E_0+(n-\frac{1}{2})\cdot \hslash \sqrt{{\omega }_0^2+{\omega }_c^2}+g{\mu }_{\mathrm{B}}{B}_{\perp },$$

Full text

When a ballistic two-dimensional charge carrier system is confined to a narrow constriction, a striking quantum mechanical phenomenon can be observed: the conductance is quantized in integer multiples of $2e^{2}/h$ van Wees et al. (1988a); Wharam et al. (1988). Such a structure, called a quantum point contact (QPC), has since its inception been demonstrated in a plethora of different materials. InAs is a material system featuring a low effective mass and a high spin-orbit interaction, and can be contacted with ease with superconductors due to Fermi level pinning at the surface. This provoked a substantial amount of recent interest in this material for the purposes of topological quantum computation Kitaev (2001); Oreg et al. (2010); Lutchyn et al. (2010); Kjaergaard et al. (2016). An architecture based on a two-dimensional material system would be beneficial for scaling and integration, which is why full control and understanding of nanostructures in InAs quantum wells has profound implications for future research. However, experiments on QPCs in InAs have been scarce, performed in trench-etched structuresDebray et al. (2009); Martin et al. (2010); Lehmann et al. (2014); Matsuo et al. (2017), and were limited by parallel conduction Koester et al. (1996) due to surface charge carrier accumulation inducing trivial edge conduction Noguchi et al. (1991); Olsson et al. (1996). In the following we study a completely electrostatically defined QPC in InAs without any background conductance in order to provide a detailed understanding of its energy levels, $g$-factor and magnetoelectric subband structure.

The device was fabricated on a heterostructure grown by molecular beam epitaxy on an undoped GaSb substrate in (100) crystal orientation, as described in Ref. Thomas et al., 2018. The schematic cross-section in Fig. 1 shows the layer sequence, which in growth direction consists of a $25\text{\,}\mathrm{n}\mathrm{m}$ GaSb layer, a $800\text{\,}\mathrm{n}\mathrm{m}$ Al0.8Ga0.2Sb0.93As0.07 quaternary buffer layer, two $20\text{\,}\mathrm{n}\mathrm{m}$ Al0.8Ga0.2Sb barriers below and above the $24\text{\,}\mathrm{n}\mathrm{m}$ wide InAs quantum well, and a $1.5\text{\,}\mathrm{n}\mathrm{m}$ InAs cap layer. The mobility of the two-dimensional electron gas in the quantum well is $\mu=$1.4\text{\times}{10}^{6}\text{,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s} at an electron density $n=$5.1\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2} as measured in a Hall bar geometry at $T=$1.5\text{,}\mathrm{K}. We determine an elastic mean free path $l_{e}=$16.5\text{\,}\mu\mathrm{m}, which is much larger than the size of the constriction we investigate in the following.

In order to avoid the trivial edge conduction inherent to InAs Noguchi et al. (1991); Olsson et al. (1996) we refrain from any etching steps and employ a fully gate-defined sample geometry as introduced in Ref. Mittag et al., 2018 and shown in Fig. 1. We begin by depositing Ge/Au/Ni/Au Ohmic contacts on the sample surface, followed by atomic layer deposition (ALD) of a $30\text{\,}\mathrm{n}\mathrm{m}$ dielectric layer of $\mathrm{Al}_{\mathrm{2}}\mathrm{O}_{\mathrm{3}}$ at a temperature of $150\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Subsequently, a rectangular ($10\text{/}70\text{\,}\mathrm{n}\mathrm{m}$) Ti/Au frame gate is deposited on the surface, followed by another $30\text{\,}\mathrm{n}\mathrm{m}$ of $\mathrm{Al}_{\mathrm{2}}\mathrm{O}_{\mathrm{3}}$ grown by ALD. Finally, ($5\text{/}25\text{\,}\mathrm{n}\mathrm{m}$) Ti/Au fine gates are deposited on top of the second dielectric layer forming the nanostructure whose shape can be seen in the scanning electron micrograph in the inset of Fig. 1. It consists of 5 pairs of opposing gates with separations of $100\text{\,}\mathrm{n}\mathrm{m}$ and $150\text{\,}\mathrm{n}\mathrm{m}$. The two gates colored in red are vertically offset by $150\text{\,}\mathrm{n}\mathrm{m}$ and horizontally offset by $100\text{\,}\mathrm{n}\mathrm{m}$ with respect to the plane of the image and were used to define the QPC investigated in this study. We refer to them as gates QPC1 and QPC2 and they are labelled accordingly in the inset of Fig. 1. This configuration showed the best transport data out of all combinations of gates, likely due to a locally favorable potential landscape. All subsequent measurements were performed in a dilution refrigerator with a base temperature of $T=$60\text{,}\mathrm{m}\mathrm{K}$$ using low-frequency AC lock-in techniques.

We bias the frame gate to the voltage $V_{\mathrm{FG}}=$-0.85\text{,}\mathrm{V} which is sufficient to deplete the electron gas underneath, thus isolating the inner part of the electron gas from the surrounding electron gas and the sample edges. A constriction can now be defined in the central region using the two fine gates colored in the inset of Fig. [1](#S0.F1) which create the QPC. We apply a fixed AC voltage bias of $\mathrm{d}V_{\mathrm{AC}}=$10\text{\,}\mathrm{\SIUnitSymbolMicro V} between the source (S) and drain (D) contact, measure the current $\mathrm{d}I_{\mathrm{AC}}$ and thereby determine the two-terminal differential conductance $G=\mathrm{d}I_{\mathrm{AC}}/\mathrm{d}V_{\mathrm{AC}}$. Fig. 2 shows $G$ as a function of the voltage applied to the QPC gates $V_{\mathrm{QPC}}$, displaying three plateaus of conductance in steps of $2e^{2}/h$. The two QPC gates are biased slightly asymmetrically, such that $V_{\mathrm{QPC1}}=V_{\mathrm{QPC}}$ and $V_{\mathrm{QPC2}}=V_{\mathrm{QPC}}+$300\text{,}\mathrm{mV}$$. The visibility of three steps is compatible with an estimate of the number of modes observable by comparing the lithographic width of the QPC to half the Fermi wavelength. The QPC pinches off the electron gas completely and no subtraction of a background conductance was needed, in contrast to previous experiments Koester et al. (1996); Debray et al. (2009). The resistance at pinch-off is at least $10^{8}\,\Omega$ and the curves showed small hysteresis of the order of $10\text{\,}\mathrm{mV}$ between up and down sweep direction. We note that, despite the high quality of the material at hand, we did not observe a feature corresponding to an 0.7 anomaly Thomas et al. (1996).

In order to map out the energy levels of the QPC, we perform finite bias spectroscopy by applying an additional DC bias voltage between source and drain leading to a bias voltage $V_{\mathrm{DC}}$ across the QPC. Fig. 3 depicts the numerical derivative with respect to $V_{\mathrm{QPC}}$ of the differential conductance $\mathrm{d}G/\mathrm{d}V_{\mathrm{QPC}}$, as a function of $V_{\mathrm{DC}}$ and $V_{\mathrm{QPC}}$. The data have been corrected to account for the voltage drop across the Ohmic contacts and then rescaled onto the resulting bias axis. For this, an I-V characteristic of the contacts alone without operating any fine gates has been recorded and the voltage drop at the contacts as a function of the total current through the sample has been determined. This voltage is then subtracted from the applied DC bias voltage to determine $V_{\mathrm{DC}}$, the actual bias voltage across the QPC.

The dark areas correspond to regions where the conductance does not change as a function of $V_{\mathrm{QPC}}$, i.e. conductance plateaus, and the bright regions mark transitions between them. The resulting diamond-shaped structure visualizes the magnitude of the mode spacings in energy, and the slope which has been highlighted by dashed white lines delivers a calibration from $V_{\mathrm{QPC}}$ to energy. The extent of the diamonds in $V_{\mathrm{DC}}$ direction decreases for increasing mode index $n$, corresponding to decreasing energy spacings of the modes. These were determined to be $\Delta E_{12}=$2.73\text{,}\mathrm{meV}, $\Delta E_{23}=$2.31\text{\,}\mathrm{meV}, and $\Delta E_{34}=$2.1\text{,}\mathrm{meV}$$. To assess the validity of these energies, we calculate the real space extent of the modes $L_{n}$ assuming an harmonic oscillator potential of the frequency $\omega_{0}$, according to

[TABLE]

with the electron effective mass $m^{*}$. Using $\hbar\omega_{0}=\Delta E_{nm}$ for the energy spacings of the respective modes determined from our previous finite bias measurements, we find the length scales $L_{1}=$32\text{,}\mathrm{nm}, $L_{2}=$53\text{\,}\mathrm{nm}, and $L_{3}=$72\text{,}\mathrm{nm}$$. These length scales are within expectation considering the geometric dimensions of the constriction used in this experiment.

As a next step, we apply a magnetic field $B_{\parallel}$ in parallel to the plane of the two-dimensional electron gas and along the transport direction of the QPC as depicted in the inset of Fig. 1 and investigate the resulting spin splitting of the modes. Fig. 4 shows the transconductance $\mathrm{d}G/\mathrm{d}V_{\mathrm{QPC}}$ as a function of $V_{\mathrm{QPC}}$ and $B_{\parallel}$. The three diamond-shaped regions centered along a line through $B_{\parallel}=$0\text{,}\mathrm{T}$$ correspond to the spin-degenerate conductance plateaus of the first three modes of the QPC. When increasing $B_{\parallel}$ these modes split and spin-polarized plateaus emerge as additional dark, diamond-shaped regions arranged in a regular pattern around the spin-degenerate plateaus. From the extent of the diamonds in $B_{\parallel}$ and the magnitude of their energy spacings extracted from the finite bias measurements shown in Fig. 3 we can determine the $g$-factor leading to the Zeeman splitting $\Delta E=g\mu_{\mathrm{B}}\Delta B$. We extract a value of $|g|=12.6$, closer to the bulk value $g=-15$ of InAs Konopka (1967); Pidgeon et al. (1967) than previous measurements on InAs nanowire quantum dots Fasth et al. (2007) or etched QPCs Matsuo et al. (2017). Using an expression obtained from $\vec{k}\cdot\vec{p}$ theory Lassnig (1985), we can calculate the effective $g$-factor $g^{*}$ while taking into account the band-edge parameters of InAs and the quantization energies arising due to the finite quantum well width. This yields a value of $g^{*}=-13.6$, which corroborates the experimental results as the quantum well investigated in this study is much wider than the ones used in previous experiments Matsuo et al. (2017); Shabani et al. (2014) and thus shows a larger $g$-factor.

Now we investigate the effect of a magnetic field perpendicular to the plane of electron gas, $B_{\perp}$, in order to demonstrate the magnetic depopulation of the quantized one-dimensional subbands. In Fig. 5 (a) the transconductance $\mathrm{d}G/\mathrm{d}V_{\mathrm{QPC}}$ of the QPC is depicted as a function of $V_{\mathrm{QPC}}$ and $B_{\perp}$. The three bright regions along a line through $B_{\perp}=$0\text{,}\mathrm{T} mark the transitions between the first three, spin-degenerate modes of the QPC. For increasing $B_{\perp}$ they curve towards higher energy in a parabolic fashion and then transition into a linear slope as they merge with the Landau levels forming in high field. We can observe two parabolas emerging from a single step at $B_{\perp}=$0\text{\,}\mathrm{T}, which are the magnetoelectric subbands van Wees et al. (1988b). A fit of these following a model by Beenakker and van Houten Beenakker and van Houten (1991)

[TABLE]

where $\omega_{c}$ is the cyclotron frequency, and an energy offset $E_{0}$ is superimposed on the data as white dashed lines. The white diamonds mark transitions between plateaus and were extracted from line cuts of $G$ as a function of $V_{\mathrm{QPC}}$. They serve as a guide to the eye in the regions where $B_{\perp}$ is large and the magnitude of the transconductance is small. In order to better visualize the spin splitting occurring in perpendicular field, we show in Fig. 5 (b) line traces of the differential conductance as a function of $V_{\mathrm{QPC}}$ at three different values of $B_{\perp}$ depicted by the colored dashed lines in Fig. 5 (a). At $B_{\perp}=$0.82\text{,}\mathrm{T}, the plateau corresponding to the first magnetoelectric subband is not yet visible due to the broad transition to the first mode of the QPC, and the plateau at $G=3e^{2}/h$ is a narrow shoulder. For $B_{\perp}=$1.2\text{\,}\mathrm{T}, clear steps in units of $e^{2}/h$ are visible, and for $B_{\perp}=$1.42\text{,}\mathrm{T}$$ and higher fields these plateaus become wider and more pronounced.

The vertical black lines in Fig. 5 (a) correspond to regions where no signal was measured due to a finite spacing between the Ohmic contacts and the edge of the frame gate. Therefore, at certain values of $B_{\perp}$ where the bulk enters a fully developed quantum Hall state, the contacts become decoupled from the rest of the sample Faist et al. (1991) and the signal vanishes periodically in $1/B_{\perp}$.

In order to further characterize the QPC, we laterally shift it by individually changing the voltages applied to the gates forming the constriction. The result can be seen in Fig. 6 where we plot the absolute value of the gradient of the differential conductance $|\nabla G|$ with respect to the voltages applied to the two QPC gates, $V_{\mathrm{QPC1}}$ and $V_{\mathrm{QPC2}}$ as a function of these two voltages. Semicircular bright features correspond to transitions between conductance plateaus and show that they are influenced evenly by both gates. The bright horizontal and vertical line, which only depend on $V_{\mathrm{QPC1}}$ or $V_{\mathrm{QPC2}}$ correspond to the pinch-off underneath the respective gate. Impurities in the QPC can be seen as resonances in this type of plot Williamson et al. (1990), and their slopes correspond to their capacitances to the different QPC gates. Hints of noise are visible on the transition to the first plateau. For all measurements in this paper we chose to sweep the gates along the path in gate voltage space marked by the white dashed line in Fig. 6. We note that compared to GaAs QPCs with similar scattering times van Wees et al. (1988a); Wharam et al. (1988) the quantization of the plateaus is worse and noise is more pronounced, presumably caused by a background disorder potential. Similar conclusions have been drawn from measurements on quantum Hall states in InAs quantum wells Ma et al. (2017). The reason for this remains an open question and will need to be the subject of future investigation.

In etched InAs QPCs, plateaus quantized at odd integer multiples of $e^{2}/h$ at zero external magnetic field have been reported Debray et al. (2009); Lehmann et al. (2014); Matsuo et al. (2017). We use the tunability of our structure by lateral gates to look for possible origins of such quantization, and to shine some light on the deviations of some of our plateaus from quantization at precise integer multiples of $2e^{2}/h$. By tuning the voltage on nearby gates surrounding the QPC we change the potential landscape around the QPC and thereby influence the coupling of electron waves from the contact regions into the channel. More specifically, we change the voltage $V_{\mathrm{sur}}$ applied simultaneously to the three neighboring gates of the QPC (colored yellow in the inset of Fig. 1) and $V_{\mathrm{QPC}}$ and plot the resulting transconductance of the QPC at zero external magnetic field in Fig. 7 (a). For decreasing $V_{\mathrm{sur}}$, the transitions between plateaus shift to more positive $V_{\mathrm{QPC}}$ due to cross-capacitance, as expected. Furthermore, the onset to the first plateau splits at $V_{\mathrm{sur}}=$-0.15\text{,}\mathrm{V}$$ and a new conductance plateau arises. Other smooth changes of the conductance as a function of $V_{\mathrm{QPC}}$ are evident in Fig. 7 (a).

As examples, line cuts of the conductance at three values of $V_{\mathrm{sur}}$ corresponding to the colored dashed lines in Fig. 7 (a) are displayed in Fig. 7 (b). They show how the conductance traces are continuously deformed as $V_{\mathrm{sur}}$ changes. For example, the red trace ($V_{\mathrm{sur}}=$-0.2\text{,}\mathrm{V}) shows a new conductance plateau seemingly quantized at $1.6e^{2}/h$, and the green trace ($V_{\mathrm{sur}}=$-0.3\text{\,}\mathrm{V}) resembles a reentrant plateau feature at $V_{\mathrm{QPC}}=$-1.45\text{,}\mathrm{V}$$ quantized at about $e^{2}/h$. Also the precise values of the conductance on plateaus quantized near integer multiples of $2e^{2}/h$ changes slightly as a function of $V_{\mathrm{sur}}$.

These measurements exemplify how spurious plateau-like features at unconventional conductance values can be generated by suitably molding the potential landscape in the vicinity of the quantum point contact constriction. They show not only that adiabatic coupling of the QPC channel to the leads is of crucial importance, but also that this adiabaticity can be easily spoiled in our InAs samples. In comparison to fully etched constrictions, the degree of tunability provided by the additional gates around the QPC allow us to access these otherwise not attainable regimes. While adiabaticity of the coupling is related to the long-range variations of the potential, resonant structures in the energy-dependent transmission (see green curve in Fig. 7 (b)) can be caused by shorter range static disorder McEuen et al. (1990) in the vicinity of the constriction. Additionally, time-dependent fluctuators may lead to noise and broaden transitions between plateaus as observed for the first plateau in our sample. Our results shown in Fig. 7 may be relevant for the interpretation of previously observed plateaus quantized at odd integer multiples of $e^{2}/h$ at zero external magnetic field Debray et al. (2009); Lehmann et al. (2014); Matsuo et al. (2017).

In conclusion, we have presented measurements of quantized conductance in a fully gate-defined QPC in an InAs quantum well. In both parallel and perpendicular magnetic field, spin-resolved transport through the constriction could be observed, and the $g$-factor as well as the energy spacing of the modes could be determined. Further optimizing both sample design and growth with the goal of eliminating noise and fluctuations in these nanostructures is needed to pave the way towards reliable operation of more complex nanostructures such as single or double quantum dots. Realizing spin qubits in laterally defined InAs quantum dots will enable performing qubit operations via the spin-orbit interaction similar to nanowires Flindt et al. (2006); Nadj-Perge et al. (2010) while benefiting from easier integration due to their two dimensional nature.