In-plane gate induced transition asymmetry of spin-resolved Landau levels in InAs-based quantum wells Open Access

Two-dimensional electron gases (2DEGs) may exhibit a topological behavior, leading to the observation of the fractional and integral quantum Hall (QH) effects in perpendicular high-magnetic fields at low temperatures.^1–3 The formation of Landau levels (LLs) results in quantum Hall edge channels (QHECs) as chiral 1D states and the inter-edge channel interactions raise continuing research interest as Tomonaga–Luttinger liquids⁴ or many-body effects as quantized charge fractionalization.⁵ In general, for such studies, QHECs may be transmitted or reflected at potential barriers created by top- and split-gates.^3,6–8 Recent experiments on counterflow edge transport in InAs quantum wells suggest that even in the integer QH regime, the microscopic structure of the edge states can differ from that of macroscopic transport experiments and require careful consideration.⁹ To date, transversal in-plane electric fields are rarely employed to control the formation of QHEC or the currents in the transition regimes between LLs. However, for InAs-based heterostructures, spin–orbit coupling may come into play^10–18 and in-plane fields may become a useful tool to act on spin-dependent transport properties. In general, InAs- and InSb-based heterostructures have evoked renewed interest regarding 2D topological insulator phenomena^19–21 and in combination with superconductor and topological material research.^22–24 

Here, we investigate transport from zero- up to high-magnetic fields of the QH regime with only one spin-polarized QHEC (filling factor ν = 1). We study the crossover from diffusive electron transport in a wide Hall-bar to diffusive but few-channel quasi-one-dimensional transport in a Hall-bar with a micro-constriction. This constriction can be electrostatically depleted to pinch-off by symmetric voltages applied to in-plane gates, which allows us to control the transmission and reflection of QHEC and their interaction in InAs-based heterostructures. In-plane electric fields, which are transversal with respect to the current or edge currents along the constriction, are applied by asymmetric in-plane gate voltages. Previous theoretical work on quantum point contacts (QPCs) indicated spin filtering effects in zero magnetic fields,^25,26 and the effect of spin–orbit interaction on the electron flow in QPCs was studied with scanning gate microscopy.^27,28

Hall-bars were fabricated by micro-laser photolithography and wet-chemical etching in a shallow inverted In_0.75Al_0.25As/In_0.75Ga_0.25As quantum well (InGaAs/InAlAs QW) with an inserted InAs channel. The wafer was grown by molecular-beam epitaxy and consists of an In_0.75Al_0.25As/In_0.75Ga_0.25As QW, with a strained, 4 nm thick InAs layer at about 53 nm below the surface.²⁹ In the following, we shorten In_0.75Al_0.25As and In_0.75Ga_0.25As to InGaAs and InAlAs, respectively. A sketch of the layer sequence is shown in Fig. 1, together with the calculated conduction band edge profile and the probability density of the two lowest-energy states in the QW. The 2DEG is localized in the narrow InAs QW, with penetration into the InGaAs QW. The characterization of the wafer at T_bath = 0.25 K in the dark yields an electron density of n_s = 4.1 × 10¹¹ cm⁻² and a mobility of μ = 1.2 × 10⁵ cm²/Vs. After continuous exposure with an infrared light emitting diode (LED), with wavelength 880 nm, for 30 s, the density increased to n_s = 6.8 × 10¹¹ cm⁻² and the mobility increased to μ = 1.8 × 10⁵ cm²/Vs. The constrictions were fabricated in a second step with high-resolution micro-laser photolithography and wet-chemical etching of two trenches in a V-like shape, electrically isolated from the 2DEG, as shown in Fig. 3(a). The width and length of the resulting gap are both ∼4 µm. The contacts to the 2DEG and the in-plane gates were made by sputter-deposition of an approximately 5 nm layer of titanium beneath an ∼50 nm layer of gold, without annealing. The samples were mounted on chip-carriers and contacted by wedge-bonding with aluminum wire. The surfaces of the chip-carriers on which the samples are mounted are metallic and can be used as back-gates.

Electric transport measurements were carried out in an Oxford Instruments Triton dilution refrigerator and in an Oxford Instruments HelioxVL ³He-cryostat. The chip carriers were mounted perpendicular to the magnetic field, before inserting the sample holder into the cryostats for cooldown. Stanford Research Systems SR830 or Signal Recovery 7265 lock-in amplifiers were used to measure the longitudinal and transversal voltages simultaneously. In the Hall-bar without constriction [see the inset of Fig. 2(a)], the current I_x was flowing between contacts 1 and 4, the longitudinal voltage V_x was measured between contacts 2 and 3, and the transversal (Hall) voltage V_y was measured between contacts 2 and 6. In the Hall-bar with constriction, the current and voltages are as depicted in Fig. 3(a). The gate voltages were generated by high-precision source-meter units by Keithley, followed by low pass filters with a cutoff frequency of 1 Hz to prevent noise from high-frequency signals. All the transport measurements were performed after exposure with the infrared LED and waiting for the conductivity to be stable.

The band edge profile and the probability density of electron wavefunctions were calculated by using the self-consistent 1D Poisson–Schrödinger solver by Snider et al.³⁰ This method does not account for the non-parabolicity of the bands. The band parameters for the calculation were obtained from the software provided with Ref. 31, and the Schottky barrier heights are taken from Ref. 32. The estimates of the effect of strain, under the assumption that the InAs layer has the same lattice parameter as InGaAs, are based on data from Ref. 33, using a linear interpolation of valence- and conduction band edges between InAs and GaAs well materials on the GaAs substrate. The parameters used for the calculation are summarized in the Appendix.

The results of band-profile calculations of the heterostructure, with the InGaAs quantum well (QW) and the asymmetrically inserted InAs channel as depicted in Fig. 1(a), are discussed in the following. The band profiles and the probability density of the electronic wavefunctions show that a two-dimensional electron gas (2DEG) localized in the quantum well (QW) is formed. While the wavefunction is centered in the narrow 4 nm-InAs channel, it also penetrates significantly into the InGaAs barrier [see Fig. 1(b)]; therefore, the transport properties of the electrons are not determined exclusively by the InAs channel. The calculations by the 1D Poisson–Schrödinger solver with unstrained InAs show that 46% of the probability density is in the InAs channel, 47% is in the InGaAs quantum well, and 6.4% is in the InAlAs spacer. The calculations with strained InAs show that the wavefunction of the lowest-energy state is still localized within the InAs channel but with a larger penetration in the InGaAs QW [see Fig. 1(c)]: 39% of the probability density is in the InAs channel, 53% is in the InGaAs quantum well, and 7.9% is in the InAlAs spacer. Applying a back-gate voltage shifts slightly the QW and the InAs channel in energy relative to the Fermi energy but does not significantly change the probability density shape (see Fig. 5 of the Appendix). Based on Ref. 10, the Rashba coefficient is estimated to be α = 99.5 × 10⁻²⁰ em². The calculations yield an average electric field in the growth direction of ⟨E_z⟩ ≈ 4 × 10³V/cm and a Rashba parameter of α_R = α⟨E_z⟩ ≈ 4 × 10⁻¹³ eVm.

First, we discuss the magnetotransport measurements of the Hall-bar without constriction, which confirm that the QW yields a high-mobility 2DEG. Application of a back-gate voltage changes the carrier density in a range of 10%.

The observed negative magnetoresistance at B < 0.6 T, see Fig. 2(b), is consistent with the occurrence of weak localization as a quantum correction due to backscattering, which indicates diffusive transport at low fields, and is not affected by the back-gate voltage. The absence of a weak anti-localization in the magnetoresistance at low fields and of a beating pattern in the SdH oscillations at intermediate fields indicates that the heterostructure and the perpendicular electric field from the back-gate do not induce observable zero-field spin splitting due to spin–orbit coupling, which confirms previous studies.¹¹ Zeeman splitting can be observed for low magnetic fields as B > 2 T because of the relatively large g*-factor.

The existence of topological QHEC transport is evident from the Hall resistance for which the expected QH plateaus occur [see Fig. 2(a)], and the slope of $RxyB$ and the SdH frequencies yield the same sheet carrier density [see Figs. 2(b) and 2(c)]. From the change in R_xx(0) and the shift of the SdH minima with varying back-gate voltage V_bg, the back-gate voltage changes only the sheet carrier density and not the mobility of μ_H ≈ 2.4 × 10⁵ cm²/Vs [see Figs. 2(b) and 2(c)]. The change in carrier density with back-gate voltage is consistent with the calculation of the band profiles [see Fig. 1(b)].

The quality of the high-mobility 2DEG is supported by the observation that the magnetoresistance peak of the Zeeman-split LLs at higher magnetic fields (lower energy) is systematically lower than the peak at lower fields (higher energy); see Figs. 2(b) and 2(c). The lower-field peak corresponds to the chemical potential lying in the higher energy Zeeman-split level, spin-up; the higher-field peak corresponds to spin-down. The reduction in the lower-energy peak is the result of a non-equilibrium population of electrons between the highest-occupied, but partially filled LLs and lowest full LLs along the Hall bar. An equilibrium distribution can be established by inter-edge state coupling only over length scales much larger than the typical Hall-bars.^34,35 In Figs. 2(b) and 2(c), the higher-field peak is significantly reduced for V_bg = −200 V, which indicates that the application of a negative back-gate voltages minimizes potential fluctuations.

The Zeeman-split resistance peaks were also used to determine the absolute value of the effective Landé g*-factor. Both the magnetic-field dependence of the Zeeman-splitting in energy and the coincidence method³⁶ yield the same result of $g*=16$⁠.

Forming the constriction in the Hall-bar [see Fig. 3(a)] leaves the positions of SdH oscillations and the edge channel transmission in the QH regime unchanged. The magnetotransport measurements of the Hall-bar with constriction after illumination with an infrared light-emitting diode are shown in Figs. 3(b) and 3(c) (blue curves). The Hall resistance R_xy shows that the Hall-bar with constriction has the same electron density as the Hall-bar without constriction, n_s ≈ 6.9 × 10¹¹ cm⁻². However, as expected for a constriction with a width W ≈ 4 µm of the same order as the mean free path l_e ≈ 2.5 µm and much larger than the Fermi wavelength λ_F ≈ 30 nm, we observe an increase in the zero-field resistance and the SdH peak heights.

Applying a voltage to the in-plane gates in the QH regime allows us to switch from an undisturbed transmission of edge channels to a reflection at the constriction, resulting in a filling factor mismatch. The constriction can be electrostatically depleted until a pinch-off occurs at −15 V. Cycling the in-plane gate voltages V_rg = V_lg between −15 and +20 V, while the back-gate is grounded, shows at first a hysteretic behavior, until stability is reached with a reduced carrier density, n_s ≈ 6.6 × 10¹¹ cm⁻² [see Figs. 3(b) and 3(c), red curves]. In Fig. 3(b), all the QH resistance plateaus show the same quantization at integer fractions of h/e². The zero-field resistance R_xx(0) is determined by the constriction itself: Here, the constriction is quasi-ballistic, because its length L ≈ 4 µm is of the same order as the mean free path l_e. Hence, the resistance increase can be associated with the reduced number of subbands in the constriction and is commonly denoted as Sharvin resistance.^37,38 From a zero-field resistance increase of $ΔRxx0≈90Ω$⁠, we estimate that the number of spin-degenerate subbands in the constriction is $Nc=12h/e2ΔRxx0≈143$⁠, which yields an effective width $Weff=πNckF≈2.3μ$m with k_F ≈ 2 × 10⁸ m⁻¹. From this, we obtain a depletion length after etching of ∼0.9 µm. In Fig. 3(c), the SdH oscillations of the Hall-bar with constriction are very similar to the SdH oscillations in the Hall-bar without constriction. However, the peaks are larger due to increased scattering in the constriction. Unlike the SdH oscillations in Fig. 2(b), the peaks corresponding to lower-energy Zeeman-split level are not reduced, because the equilibration length is smaller at higher bath temperatures.³⁹ 

As can be seen from Fig. 3(c), the longitudinal resistance R_xx is more sensitive to changes due to cycling of the in-plane gate voltages. R_xx(0)is higher and yields $ΔRxx′0≈390Ω$⁠, $Nc′≈33$⁠, and $Weff′≈520nm$⁠, with a doubling of the depletion length. The SdH oscillations are shifted due to the 4% decrease in the electron density in the 2DEG. The decrease is larger in the constriction, because it acts as a barrier potential, increasing the conduction-band edge in the constriction. This results in a filling factor mismatch in the quantum Hall regime and the reflection of one or more edge channels at the constriction.^40,41 For low magnetic fields, the absence of weak localization signals the strong reduction in backscattering in the quasi-ballistic constriction.

For the electrostatic control of the transmission and reflection of the QH edge channels, we use symmetric in-plane gate voltages V_sym = V_rg = V_lg, which change the effective width of the constriction. In Fig. 4(a), the constriction transmits only a few 1D subbands at zero magnetic field for two different scenarios. First, the transmission of edge channels remains largely unchanged for a widened constriction at positive symmetric in-plane gate voltages (see V_sym = 0 V, gray, and +10 V, blue curve), showing only a small change in LL spacing due to a change in electron density. In this case, the depletion of the channel is due to the symmetric in-plane gate voltage and leads to about ten subbands being transmitted at zero magnetic field. Second, an edge channel mismatch can be induced by narrowing the constriction due to depletion by a negative symmetric in-plane gate voltage (see V_sym = −5 V, blue dashed curve). In this case, only up to 2 subbands are transmitted in zero magnetic field. Thereby, it is possible to separate the ν = 1 and ν = 2 edge channels, and the fully spin-polarized current of the ν = 1 edge channel is transmitted through the constriction, at 6 T instead of 10 T. Between B ≈ 7 T and B ≈ 9 T, all three curves show fluctuations in R_xx, and the corresponding fluctuations can be observed in R_xy measured simultaneously. This is consistent with an increased probability of resonant tunneling events between edge states when the chemical potential in the wide 2DEG is shifting from ν₀ = 2 to ν₀ = 1.⁴² 

To apply a transversal in-plane electric field to the spin-polarized edge channels in the constriction, we use an asymmetric in-plane gate voltage, ΔV_as = V_rg − V_lg. The effect is an electrostatic distortion in the constriction, which can be approximated by a saddle-point potential. When ΔV_as = 0, the transversal in-plane electric field is zero and the states are centered on the minimum of the saddle-point potential.

The dependence of the longitudinal resistance on the transversal in-plane electric field applied by the asymmetric gate voltage ΔV_as is observed in the transition between Zeeman-split LLs (QH plateaus), as can be seen in Fig. 4(b) for the case with a filling factor mismatch for the transition from ν = 1 to ν = 2 and, in Fig. 4(c), for the case without mismatch between the wide Hall bar and the constriction in the transition from ν = 1 to ν = 2 and from ν = 2 to ν = 3. The following observations on the orientation dependence of the transversal in-plane electric field on the longitudinal resistance do not find an explanation in a purely electrostatic response of charges.

For the case of the transmission without a mismatch, the shift of the longitudinal resistance features most prominently occurs at the magnetic field of about 5 T, which is marked by (ii) in Fig. 4(c). This shows that the transversal in-plane electric field affects the magnitude and the magnetic field position of the resistance increase depending on the spin state. When the chemical potential is in the spin-up Zeeman-split LL (ii), the backscattering is increased (green curve) or suppressed (red curve) depending on the transversal in-plane field orientation. This indicates that the transversal in-plane electric field changes the density of states of the spin-split LLs.

Together with the peak position for the spin-down Zeeman split LL, marked by (i) in Fig. 4(c), the magnetic field shift can be measured to be ΔB ≈ 0.1 T, which is consistent with an effective magnetic field of B_eff = (v_FE_transv)/c² ≈ (6 × 10⁷ cm s⁻¹ × 2 × 10⁴ V/cm)/c² = 0.13 T for ΔV_as = 20 V, normal to the 2DEG plane, where v_F is the Fermi velocity and E_transv is the transversal in-plane electric field. From this, we determine the in-plane spin–orbit coupling parameter α_so(ΔV_as = 20 V) = |g*|μ_BΔB/2 k_F ≈ 2 × 10⁻¹³ eVm and the corresponding in-plane coefficient for the Hall bar with etched constriction. This value is smaller by a factor 10 compared to the calculations for this heterostructure in Sec. II A. The difference can partly be accounted for by the wavefunction penetration into InGaAs, possibly by the strain in InAs and perhaps by the etched constriction, which were all not included in the calculations.

A similar resistance increase and decrease depending on the orientation of the transversal in-plane field can be seen for the transition from ν = 2 to ν = 1 at 8 T marked by (iii) and (iv) in Fig. 4(c). Here, increased backscattering sets in at lower magnetic fields (green curve), while the backscattering is strongly suppressed (red curve) at the resistance maximum. Because in the transition to ν = 1, only one spin state remains available for the final state after scattering processes, the effect of the orientation of the transversal in-plane electric field is not symmetric with respect to the two spin states. This situation is similarly in the ν = 2 to ν = 1 transition for the case of a mismatch, see Fig. 4(b), marked with (ii).

From the above considerations, we conclude that, although spin–orbit coupling does not lead to observable spin-splitting of subbands at zero magnetic field as discussed in Sec. II B, it needs to be taken into account when transversal in-plane fields are applied in the QH regime for 2DEGs, which are subject to lateral confinement in the order of the mean free path and below. Due to spin–orbit coupling, the orientation of transversal in-plane electric fields with respect to the current flow can affect the spin-dependent scattering between QHECs situated in the wide Hall bar and the constriction. This becomes most evident when increased backscattering is possible as in the transition between the LLs (QH plateaus).

These experiments show that the transport in the QH regime under application of micro- or nanopatterned gates and/or constrictions requires a thorough investigation of the undisturbed 2DEG (wide Hall-bars) as a reference system. Furthermore, in particular for 2DEGs with spin–orbit interaction, such as InAs-based heterostructures, the application of in-plane gates may serve as a powerful tool to explore spin-dependent phenomena by transversal electric fields.

For a high-mobility 2DEG with a large absolute value of the effective Landé g*-factor of 16 in an InAs-based quantum well, it was demonstrated that for an etched micro-constriction with in-plane gates, the QH regime with only one spin-polarized QHEC (filling factor ν = 1) can be reached at a moderate magnetic field of 9 T (no filling factor mismatch) or lower, at 6 T (for filling factor mismatch). The electrostatic control by symmetric voltages applied to the in-plane gates allows us to tune the transmission and reflection of QH edge channels. The orientation of a transverse in-plane electric field by asymmetric voltages applied to the in-plane gates affects the magnetic field position and the magnitude of the longitudinal resistivity. This strongly indicates that the spin-dependent backscattering between spin-polarized QH edge channels in the transition between Landau levels (QH plateaus) is affected by spin–orbit coupling.

We thank Christian Riha for scientific and technical support. O.C., J.B., and S.F.F. acknowledge the support from the Deutsche Forschungsgemeinschaft under Grant No. INST 276/709-1, and the JAMA Lab.

The authors have no conflicts to disclose.

Olivio Chiatti: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Johannes Boy: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (supporting). Christian Heyn: Resources (equal); Writing – review & editing (supporting). Wolfgang Hansen: Methodology (supporting); Resources (equal); Writing – review & editing (supporting). Saskia F. Fischer: Conceptualization (equal); Funding acquisition (lead); Methodology (equal); Project administration (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Table I shows the layer sequence and the parameters used for the band-profile calculations with the 1D Poisson–Schrödinger solver.³⁰ The bandgap, the conduction band offset and the effective mass were obtained from the software provided with Ref. 31 and the Schottky barrier heights are from Ref. 32. The effect of strain (values in bold) was estimated using the assumption that the InAs layer has the same lattice parameter as InGaAs and is based on data from Ref. 33. A linear interpolation of valence- and conduction band edges between InAs and GaAs well material on GaAs substrate was used. Fig. 5 shows the effect on the band-profile of a back-gate voltage.