Introduction

Low-dimensional narrow bandgap InSb nanostructures, such as nanowires and quantum wells, have in recent years attracted great interests. Due to their small electron effective mass, strong spin–orbit interaction (SOI), and large Landé g-factor, these nanostructures have potential applications in high-speed electronics1, infrared optoelectronics2, spintronics3, quantum electronics4,5 and topological quantum computation6. The past decade has witnessed booming investigations of devices made from epitaxially grown InSb nanowires, including field-effect transistors7,8, single9,10,11 and double quantum dots12,13, and semiconductor–superconductor hybrid quantum devices14,15,16,17. Among the most influential, pioneer developments are the topological superconducting quantum devices made from InSb nanowires14,16, in which zero-energy modes, a signature of Majorana fermions18,19 in solid state, were detected and studied. However, to build a device in which braiding of topological quantum states, such as Majorana fermions, can be conveniently performed and thus topological quantum computations can be designed and realized, it could be inevitable to move from single-nanowire structures to multiple-nanowire20,21 and two-dimensional (2D) planar quantum structures22,23,24. Recently, high-quality InSb/InAlSb heterostructured quantum wells25,26 and free-standing InSb nanosheets27,28,29,30 have been achieved by epitaxial growth techniques. In comparison with InSb/InAlSb quantum well systems, the free-standing InSb nanosheets have advantages in direct contact by metals, including superconducting materials, in easy transfer to different substrates, and in convenient fabrication of dual-gate structures. With use of free-standing InSb nanosheets, lateral quantum devices, such as planar quantum dots31 and superconducting Josephson junctions32,33,34, have been successfully fabricated. A most intriguing perspective of these layered materials is to build topological superconducting structures from them, in which Majorana fermions and parafermions35,36 can be created and manipulated, enabling a different route of developments towards topological quantum computation technology. A desired ingredient in constructing topological superconducting states from such a semiconductor nanostructure is strong SOI (with a few 100 nm or shorter in spin–orbit length and about 100 µeV or larger in spin–orbit energy) possessed in the material37,38. Comprehensive studies of SOI have been carried out for InSb nanowires39,40 and quantum wells41. However, a desired study of SOI and, in particular, its controllability has not yet been carried out for free-standing InSb nanosheets, although it is highly anticipated that such a study would lead to great advancement in the developments of spintronics, quantum-dot based spin–orbit qubits, and topological quantum computation technology.

In this article, we report on magnetotransport measurements of an epitaxially grown, free-standing, zincblende InSb nanosheet and on employment of dual-gate technique to achieve tunable SOI in the nanosheet. Key electron transport characteristic lengths, such as the mean free path, phase coherence length, and SOI length, in the nanosheet are extracted from the measurements of the low-field magnetoconductance. We show that a strong SOI is present in the InSb nanosheet and is greatly tunable using a voltage applied over the dual gate. We also demonstrate, through band diagram simulation for the experimental structure setups, that the origin of an intrinsic SOI observed in the InSb nanosheet at zero dual-gate voltage comes from the build-in structure asymmetry in the dual-gate device. The advancement made in this work in understanding and controlling of strong SOI in the InSb nanosheet will greatly simplify the design and implementation technology for the construction of spintronic devices, spin–orbit qubits, and topological quantum devices.

Results and discussion

Dual-gate InSb nanosheet device

The dual-gate device studied in this work is made from a free-standing, single-crystalline, zincblende InSb nanosheet on an n-doped silicon (Si) substrate covered by a 300-nm-thick layer of silicon dioxide (SiO2) on top, using standard nanofabrication techniques (see “Methods”). Figure 1a shows a scanning electron microscope (SEM) image of the device and the measurement circuit setup. Figure 1b shows a schematic view of the layer structure of the device. The InSb nanosheet in the device is contacted by four stripes of Ti/Au (contact electrodes). The n-doped Si substrate (contacted by a thin gold film at the bottom) and the SiO2 layer are employed as the bottom gate and the gate dielectric. The top gate is made from a Ti/Au film with a layer of hafnium dioxide (HfO2) as the top-gate dielectric. The nanosheet has a width of ~550 nm and a thickness of ~30 nm (estimated based on the calibrated contrast in the SEM image). The separation between the two inner Ti/Au electrodes is 1.1 μm. Low-temperature transport measurements of the dual-gate device is carried out in a physical property measurement system (PPMS) cryostat, equipped with a uniaxial magnet, in a four-probe configuration, in which a 17-Hz AC excitation current (I) of 100 nA is supplied through the two outer electrodes and the voltage drop (V) between the two inner contact electrodes is recorded. The nanosheet channel conductance G is obtained from G = I/V. In measurements for the magetoconductance, ΔG = G(B) − G(B = 0), the magnetic field is applied perpendicular to the InSb nanosheet plane.

Fig. 1: Dual-gate InSb nanosheet device and its gate transfer characteristics.
figure 1

a False-colored SEM image of the device studied in this work and measurement circuit setup. The device is fabricated from an MBE-grown InSb nanosheet (transport channel) on an n-doped Si substrate (bottom gate) covered by a layer of SiO2 on top (bottom-gate dielectric). The InSb nanosheet is contacted by four stripes of Ti/Au (contact electrodes) and is then covered by depositing a layer of HfO2 (top-gate dielectric) and a metal bilayer of Ti/Au (top gate). The nanosheet has a width of ~550 nm and a thickness of ~30 nm. The separation between the two inner electrodes is ~1.1 μm. b Schematic view of the layer structure of the device. c Conductance G measured for the device as functions of top-gate voltage VTG and bottom-gate voltage VBG (gate transfer characteristics) at a temperature of T = 1.9 K. In the measurements, a 17-Hz AC current (I) of 100 nA is applied through the two outer electrodes, and the voltage drop (V) between the two inner electrodes is recorded and is then converted to the conductance through G = I/V. The red and yellow solid lines denote the constant conductance contours of ~9 and ~5e2/h, respectively. d G as a function of bottom-gate voltage VBG at VTG = 0 V. e G as a function of top-gate voltage VTG at VBG = 0 V.

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Figure 1c shows the measured conductance of the InSb nanosheet in the device as a function of voltages, VBG and VTG, applied to the bottom and top gates (transfer characteristics). Figure 1d shows a horizontal line cut of Fig. 1c (bottom-gate transfer characteristics) at VBG = 0 V, while Fig. 1e shows a vertical line cut of Fig. 1c (top-gate transfer characteristics) at VBG = 0 V. Conductance fluctuations superimposed on the transfer curves are observable. These fluctuations are reproducible and arise from universal conductance fluctuations42 (UCF). Overall, the top gate shows a strong coupling to the InSb nanosheet, while the bottom gate shows a relatively weak coupling to the nanosheet. The former is in accordance with the fact that a short distance between the top gate and the nanosheet and a high dielectric material (HfO2 in this case) are employed in the device. From Fig. 1b, one can infer that an electric field stretching perpendicularly through the InSb nanosheet can be present and can be tuned by a voltage applied over the two gates (dual-gate voltage).

The carrier density in the InSb nanosheet can be estimated from the measured transfer characteristics. Here, we extract the carrier density, at a fixed top-gate voltage of VTG = 0 V, from \(n = C_{{\mathrm{gs}}} \times \frac{{V_{{\mathrm{BG}}} - V_{{\mathrm{BG}}}^{{\mathrm{th}}}}}{e}\), where e denotes the elementary charge and \(C_{{\mathrm{gs}}} = \frac{{\varepsilon \varepsilon _0}}{d}\) is the unit area capacitance between the bottom gate and the nanosheet with ε0 being the vacuum permittivity, ε = 3.9 the dielectric constant of SiO2, and d = 300 nm the thickness of SiO2. In the above relation, \(V_{{\mathrm{BG}}}^{{\mathrm{th}}}\) is the threshold voltage at which the conductance G goes to zero. In our case, to extract the threshold, a line fit to the measured G − VBG curve in Fig. 1d is made (see Supplementary Fig. 1a). Then by extending the fitting line to intersect the horizontal axis, we obtain \(V_{{\mathrm{BG}}}^{{\mathrm{th}}}\). In this way, we have estimated out a carrier density of n = 7.2 × 1011 cm−2 at VBG = −5 V and VTG = 0 V, at which the measured conductance takes a value of G ~ 9e2/h. Note that along the red contour line in Fig. 1c, the measured conductance stays at the same value of G ~ 9e2/h and thus the carrier density in the nanosheet stays, to a good approximation, at the same value of n = 7.2 × 1011 cm−2. Similarly, the yellow contour line in Fig. 1c displays the measurements at a conductance of G ~ 5e2/h and a carrier density of n = 4.3 × 1011 cm−2 in the nanosheet. The electron mobility in the nanosheet is estimated from μ = σ/ne, where \(\sigma = \frac{{GL}}{W}\) is the sheet conductivity with L being the channel length (i.e., the distance between the two inner contact electrodes, 1.1 μm in this device) and W being the channel width (i.e., the width of the nanosheet, 550 nm in this device). Since the conductance is approximately a linear function of VBG and the same is for the electron density in the nanosheet, the same electron mobility of μ ~ 6000 cm2 V−1 s−1 in the nanosheet is extracted at both G ~ 9e2/h and G ~ 5e2/h. The electron mean free path in the nanosheet can be estimated from \(L_e = \frac{{\hbar \mu }}{e}\sqrt {2\pi n}\), where \(\hbar = \frac{h}{{2\pi }}\) with h being the Planck constant, giving Le ~ 84 nm at n = 7.2 × 1011 cm−2 (G ~ 9e2/h) and Le ~ 65 nm at n = 4.3 × 1011 cm−2 (G ~ 5e2/h). A larger value of Le obtained at the higher electron density could be due to screening of scattering by electrons in the nanosheet. For comparison, it is worthwhile to note that the Fermi wavelength can be estimated as \(\lambda _{\mathrm F} = \sqrt {2\pi /n} \sim\)30 nm at the carrier density of n = 72 × 1011 cm−2, which is close to the thickness of the nanosheet. Thus, only one or few 2D electron subbands in the InSb nanosheet are occupied and the InSb nanosheet is dominantly a 2D electron system. The same analysis based on the top-gate transfer characteristics should give the similar estimations for the carrier density and the mobility at the same setting of VBG and VTG. According to this, we have extracted a value of ε ~ 6.5 for the dielectric constant of the top-gate dielectric HfO2 using the G − VTG curve shown in Fig. 1e and the carrier densities extracted through the G − VBG curve (see Supplementary Note I for detail).

Quantum transport characteristics of the InSb nanosheet

In a quantum diffusive device, the electron transport can be characterized by a set of transport length scales, including phase coherence length (Lφ), SOI length (LSO), and mean free path (Le). In order to determine all these lengths in the InSb nanosheet, we have performed detailed magnetotransport measurements for the dual-gate InSb nanosheet device at low magnetic fields. Figure 2a shows the measured magnetoconductance, ΔG = G(B) − G(B = 0), at different VBG with top-gate voltage set at VTG = 0 V. Here, the magnetic field B is applied perpendicular to the nanosheet. It is seen that the measured magnetoconductance displays a peak in the vicinity of B = 0, i.e., the weak antilocalization (WAL) characteristics. The WAL arises from quantum interference in the presence of strong SOI and gives a positive quantum correction to the conductance at zero magnetic field. It is also seen that at VBG = 0 V, a well-defined WAL peak is observed, but the peak becomes less pronounced as VBG decreases.

Fig. 2: Magnetotransport measurements at various bottom-gate voltages.
figure 2

a Low-field magnetoconductance, \({\mathrm{{\Delta}}}G = G\left( B \right) - G(B = 0)\), measured for the device shown in Fig.1a at various bottom-gate voltages VBG at temperature T = 1.9 K. The bottom trace shows the measured magnetoconductance data at VBG = 0 V and all other measured magnetoconductance traces are successively vertically offset for clarity. Here the top-gate voltage is set at VTG = 0 V. The black solid lines are the theoretical fits of the experimental data to the HLN equation [Eq. (1)]. b Phase coherence length Lφ, spin–orbit length LSO, and mean free path Le, in the InSb nanosheet extracted from the fits as a function of VBG.

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For a 2D diffusive system, the low-field magnetoconductance is well described by the Hikami–Larkin–Nagaoka (HLN) quantum interference theory43. Assuming that the electron transport in the InSb nanosheet is in the 2D diffusion regime, the quantum correction to the low-field magnetoconductance is given by

$$\begin{array}{*{20}{l}} {{\Delta}G\left( B \right)} \hfill & = \hfill & { - \frac{{e^2}}{{\pi h}}\left[ {\frac{1}{2}{\Psi}\left( {\frac{{B_\varphi }}{B} + \frac{1}{2}} \right) + {\Psi}\left( {\frac{{B_{\mathrm{e}}}}{B} + \frac{1}{2}} \right)} \right.} \hfill \\ {\,} \hfill & {\,} \hfill & { - \frac{3}{2}{\mathrm{{\Psi}}}\left( {\frac{{\left( {4/3} \right)B_{{\mathrm{SO}}} + B_\varphi }}{B}\, + \,\frac{1}{2}} \right) - \frac{1}{2}\ln \left( {\frac{{B_{\upvarphi}}}{B}} \right)} \hfill \\ {\,} \hfill & {\,} \hfill & {\left. { - \ln \left( {\frac{{B_e}}{B}} \right) + \frac{3}{2}\ln \left( {\frac{{\left( {4/3} \right)B_{{\mathrm{SO}}} + B_\varphi }}{B}} \right)} \right]} \hfill \end{array}.$$
(1)

Here, Ψ(x) is the digamma function. Three subscripts, φ, SO, and e, in the above equation denote inelastic dephasing, spin–orbit scattering, and elastic scattering processes, respectively. Bφ,SO,e are the characteristic fields for the three scattering mechanisms and are given by \(B_{\varphi ,{\mathrm{SO}},{{e}}} = \hbar /(4eL_{\varphi ,{\mathrm{SO}},{{e}}}^2)\). The measured low-field magnetoconductance data at different VBG shown in Fig. 2a are fitted to Eq. (1) using Lφ, LSO, and Le as fitting parameters (see further detail in “Methods”). The black solid lines in Fig. 2a are the results of the fits.

Figure 2b shows the extracted Lφ, LSO, and Le in the InSb nanosheet from the fits at VTG = 0 V as a function of VBG. As shown in Fig. 2b, Lφ is strongly dependent on VBG, while LSO and Le show weak VBG dependences and stay at values of LSO ~ 130 nm and Le ~ 80 nm. Here, we note that the extracted Le ~ 80 nm is in good agreement with the values extracted above from the gate transfer characteristics. The weak VBG dependence of Le arises from the fact that at the low temperature we have considered, Le is primarily given by the distribution of scattering carriers, such as charged impurities and lattice defects, in the conduction InSb channel and the dielectric SiO2 layer, as well as at the InSb–SiO2 interface, and the distribution of scattering centers should be insensitive to a change in the gate voltage in the range we have considered. The LSO also shows a weak VBG dependence because it primarily depends on the perpendicular electric field penetrated through the InSb nanosheet, which is only weakly dependent on VBG when the InSb nanosheet is at open conduction state. At VBG = 0 V (a high carrier density case), the extracted Lφ reaches to ~530 nm. As VBG sweeps from 0 to −13 V, Lφ decreases rapidly to ~180 nm, indicating that the dephasing is stronger at a lower carrier density. The physical origin of this increase in Lφ with increasing carrier density is that, at this low temperature, the dephasing arises predominantly from electron–electron interaction with small energy transfers, in the form of electromagnetic field fluctuations generated by the motions of neighboring electrons (the Nyquist dephasing mechanism44), and such fluctuations get to be diminished at a higher carrier density and thus an increased bottom-gate voltage due to stronger charge screening. It is worthwhile to emphasized that Lφ is one order of magnitude larger than the thickness of the nanosheet. This, together with the fact that the typical Fermi wavelength λF ~ 30 nm is close to the thickness of the nanosheet, supports our assumption that the transport in the nanosheet is of a 2D nature. In addition, the extracted Le ~ 80 nm is one order of magnitude smaller than the distance between the two inner contact electrodes, indicating that the transport in the nanosheet is in the diffusion regime.

There are several possible mechanisms responsible for the spin relaxation process in the nanosheet. One is the Elliot–Yafet mechanim45,46, i.e., the spin randomization due to momentum scattering. In the Elliot–Yafet mechanism, the spin relaxation length can be estimated out as39,47 \(L_{{\mathrm{SO}},{\mathrm{EY}}} = \sqrt {\frac{3}{8}} \cdot \frac{{E_{\mathrm{G}}}}{{E_{\mathrm{F}}}} \cdot L_{{e}} \cdot \frac{{\left( {E_{\mathrm{G}} \, + \, {\Delta}_{{\mathrm{SO}}}} \right)\left( {3E_{\mathrm{G}} \, + \, 2{\Delta}_{{\mathrm{SO}}}} \right)}}{{{\Delta}_{{\mathrm{SO}}}\left( {2E_{\mathrm{G}} \, + \, {\Delta}_{{\mathrm{SO}}}} \right)}} \ge 500\,{\mathrm {nm}}\), using the bandgap EG = 0.23 eV, the Fermi energy \(E_{\mathrm F} = \frac{{\hbar ^2\pi n}}{{m^ \ast }} \le\) 50 meV (with n ≤ 7.2 × 1011 cm−2), bulk spin–orbit gap48 ΔSO ~ 0.8 eV, and the mean free path Le ~ 80 nm. The estimated LSO,EY is much larger than the experimentally extracted value of LSO ~ 130 nm. Therefore, the Elliot–Yafet mechanism does not play a key role in our system. Another one is the D’yakonov-Perel’ mechanism49, which considers the spin precession between scattering events. Since the InSb nanosheets used in our device is a zincblende crystal and the current flow would take along a <111> or a <110> crystallographic direction27, the Dresselhaus SOI50 would be either absent or negligible51. Based on the above analyses, we expect that the Rashba SOI52 is the primary cause of spin relaxation in the InSb nanosheet. This expectation is also consistent with our designed device structure with an enhanced structural asymmetry. Hence we can obtain a Rashba spin–orbit strength of αR ~ 0.42 eV Å according to \(L_{{\mathrm{SO}}} = \hbar ^2/m^ \ast \alpha _{\mathrm{R}}\), where \(m^ \ast = 0.014\,m_0\) denotes the effective mass of electrons in InSb with m0 being the free electron mass. The spin–orbit energy can be determined as \(E_{{\mathrm{so}}} = \frac{{m^ \ast \alpha _{\mathrm{R}}^2}}{{2\hbar ^2}}\) ~ 160 μeV in the InSb nanosheet. In comparison with most commonly employed III–V narrow bandgap semiconductor nanostructures with a strong SOI, the extracted spin–orbit strength of αR ~ 0.42 eV Å in our InSb nanosheet from the low-field magnetotransport measurements shown in Fig. 3 is smaller than but comparable to the values of 0.5–1 eV Å found in InSb nanowires39, but is significantly larger than the values of ~0.16 eV Å found in InAs nanowires53. In addition, our extracted spin–orbit strength in the InSb nanosheet is an order of magnitude larger than the values reported previously for InSb and InAs quantum wells41,54. Thus, the extracted αR ~ 0.42 eV Å in our InSb nanosheet corresponds to a strong SOI found in a III–V narrow bandgap semiconductor nanostructure.

Fig. 3: Magnetotransport measurements at various voltages applied over the dual gate at constant carrier densities in the InSb nanosheet.
figure 3

a Low-field magnetoconductance ΔG measured for the device at a constant conductance of ~9e2/h and temperature T = 1.9 K at various values of dual-gate voltage \(V_{\mathrm{D}} = V_{{\mathrm{TG}}} - V_{{\mathrm{BG}}}\). The bottom trace shows the measured magnetoconductance data at \(V_{\mathrm{D}} = - 2\) V and all other measured magnetoconductance traces are successively vertically offset for clarity. The black solid lines are the theoretical fits of the experimental data to the HLN equation [Eq. (1)]. b Phase coherence length Lφ, spin–orbit length LSO, and mean free path Le extracted from the fits in a as a function of VD. c The same as a but measured for the device at a constant conductance of ~5e2/h. Here, the bottom trace shows the measured magnetoconductance data at VD = −4.4 V and all other measured magnetoconductance traces are again successively vertically offset for clarity. d Phase coherence length Lφ, spin–orbit length LSO, and mean free path Le extracted from the fits in c as a function of VD.

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Tuning the SOI in the InSb nanosheet by dual-gate voltage

The SOI of the Rashba type is tunable by applying an electric field perpendicularly through the InSb nanosheet. Such an electric field can be achieved and tuned by a voltage VD applied over the dual gate. For example, with VTG being set at 0 V, we could sweep VBG to gradually change VD and thus the electric field through the nanosheet. However, as we showed above, sweeping VBG only also tunes the carrier density in the nanosheet. To demonstrate the manipulation of SOI solely via the vertical electric field in the nanosheet, the carrier density in the nanosheet ought to be fixed. In the present work, this is achieved by performing magnetotransport measurements along an equal conductance contour line, in which the carrier density in the nanosheet approximately stays at a constant value, but the dual-gate voltage, VD = VTG − VBG, is tuned continuously. Figure 3a shows magnetoconductance traces measured along a contour line of G ~ 9e2/h (the red contour line in Fig. 1c) at several values of VD. It is seen that all the measured magnetoconductance traces show the WAL characteristics. To extract the transport length scales as a function of VD, we fit these measured magnetoconductance traces to Eq. (1). The black solid lines in Fig. 3a show the results of the fits. Figure 3b displays the characteristic transport lengths Lφ, LSO, and Le extracted from the fits. It is shown that Lφ stays at a constant value of ~460 nm, independent of VD. This is in good agreement with the fact that Lφ is mainly influenced by carrier density and temperature, but not by an electric field applied perpendicular to the nanosheet. The same is also true for Le, which is found to stay at a value of ~85 nm. However, LSO shows a strong dependence on VD. As seen in Fig. 3b, LSO is monotonically increased from ~130 to ~390 nm as VD changes from −2 to 11 V, indicating that the SOI strength becomes weaker as VD moves towards more positive values. Figure 3c shows the magnetoconductance traces measured along a constant conductance contour line of G ~ 5e2/h (the yellow contour line in Fig. 1c) at varying VD from −4.4 to 10.7 V. Again, the WAL characteristics are observed in the measurements. The black solid lines in Fig. 3c show the fits of the measurements to Eq. (1) and Fig. 3d shows the transport lengths extracted from the fits. Again, it is seen that with varying VD, Lφ stays at a value of ~340 nm and Le stays at a value of ~78 nm. i.e., both are independent of VD. However, LSO is seen to increase from ~130 to ~270 nm as VD is tuned from −4.4 to 10.7 V. Our results presented in Fig. 3 clearly demonstrate that the SOI in the InSb nanosheet of our dual-gate device can be effectively tuned by applying a voltage over the dual gate without a change in the carrier density in the nanosheet. The achieved change in LSO from 130 to 390 nm corresponds to a change in the spin–orbit strength from 0.42 to 0.14 eV Å and a change in the spin–orbit energy from 160 to 18 μeV.

We have also performed the dual-gate voltage VD dependent measurements of the transport characteristics lengths Lφ, LSO, and Le in the InSb nanosheet along the constant conductance contour lines of ~2.6 and ~1.1e2/h, and an efficient tuning of SOI in the nanosheet by the dual-gate voltage VD is again observed (see Supplementary Note V). All the results presented in the present section (and in Supplementary Fig. 5) manifest that the SOI in the InSb nanosheet in a dual-gate structure can be efficiently tuned by a voltage applied to the dual gate at largely different but fixed carrier densities of the nanosheet.

Band diagram and intrinsic Rashba SOI in the InSb nanosheet

It is important to emphasize that the experimentally extracted Rashba spin–orbit length LSO is small, indicating a strong SOI, even at VD = 0 V. This seemly unexpected observation however reveals the presence of an intrinsic structural asymmetry even in the absence of a voltage difference between the top and bottom gates due to band offsets appeared in the HfO2–InSb–SiO2 heterostructure. To show this, the energy band diagram in the vertical direction is simulated using commercially available software COMSOL. The simulation is mainly based on Poisson’s equations and takes the material parameters of HfO2, InSb, and SiO2, including bandgaps, dielectric constants, electron effective masses, and electron affinities, as inputs (see Supplementary Table I for material parameters). Figure 4a displays the simulated energy band diagram of the HfO2–InSb–SiO2 structure at VD = 0 V (with VTG = VBG = −0.33 V) and carrier density n = 7.2 × 1011 cm−2 in the InSb nanolayer. The conduction band and the valence band edges exhibit a noticeable bending even at VD = 0 V, illustrating the presence of an intrinsic structure asymmetry in the InSb nanosheet. Figure 4b shows a zoom-in view of the simulated conduction band edge in the InSb nanosheet at three values of VD. The green, blue, and red solid lines represent the conduction band edges at VD = 0, −2 (with VTG = −0.46 V and VBG = 1.54 V), and 11 V (VTG = 0.4 V and VBG = −10.6 V), respectively, and the carrier density of n = 7.2 × 1011 cm−2 in the InSb nanosheet. With pushing the dual-gate voltage from VD = 0 V to VD = −2 V, we can see that the band bending becomes stronger, indicating an enhanced structural asymmetry and thus a stronger Rashba SOI. On the contrary, by moving from VD = 0 V to VD = 11 V, we compensate the band bending towards a nearly flat band, leading to a reduced asymmetry in the structure and a weaker Rashba SOI. These simulated results are fully in line with the experimental observations. Based on the simulations, the strength of the vertical, effective mean electric field in the InSb nanosheet can be estimated. It is found that the field strength gradually increases when changing from VD = 11 V to VD = −2 V (see Supplementary Fig. 3), providing a numerical evidence for tunable structural asymmetry and thus tunable SOI in the nanosheet. Furthermore, the material-specific, Fermi-level-dependent Rashba prefactor rR, defined via the Rashba coefficient αR = rRE with E being the strength of the perpendicular electric field in the InSb nanosheet48, can be determined (see Supplementary Note IV for details). The Rashba prefactors extracted for the InSb nanosheet are rR = 4.26e nm2 at carrier density n = 7.2 × 1011 cm−2 and rR = 3.48e nm2 at carrier density n = 4.3 × 1011 cm−2.

Fig. 4: Simulated band diagrams.
figure 4

a Simulated energy band diagram of the HfO2–InSb–SiO2 structure in the device at VD = 0 V (with VBG = VTG = −0.33 V). The conduction and valence band edges Ec and Ev, are marked by the dark blue and light blue solid lines, while the Fermi level EF is marked by the dark blue dashed line. b Zoom-in plots (red, blue, and green solid lines) showing the details of the conduction band edges inside the InSb nanosheet at three different values of VD. The red, blue, and green dashed lines mark the Fermi levels obtained in the simulations at the three different values of VD.

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Temperature effects

Figure 5a shows the measured low-field magnetoconductance of the device at VBG = 1.54 V and VTG = −0.46 V at temperatures of 1.9 to 20 K. At temperature T = 1.9 K, a sharp WAL peak is seen in the vicinity of zero magnetic field. As the temperature increases, both the height of the WAL peak and the fluctuation magnitude of the UCF patterns become gradually suppressed, although they still remain visible at T = 20 K. Again, we fit these measured magnetoconductance data to Eq. (1) and plot the results in black solid lines in Fig. 5a. Extracted Lφ, LSO, and Le from the fits are displayed in Fig. 5b. It can be found that both LSO and Le are weakly dependent on temperature, while Lφ shows a strong temperature dependence, decreasing rapidly from ~470 to ~210 nm with increasing temperature from 1.9 to 20 K. The temperature dependence of Lφ is found to follow a power law of Lφ ~ T−0.38 (see the solid line in Fig. 5b). The power index of −0.38 falls between the values of −1/2 and −1/3, corresponding to the Nyquist dephasing processes44 in a 2D (T−1/2) and a 1D system (T−1/3). Since Lφ is in the same order of length scale as the width of the conduction channel (~550 nm), the transport in the InSb nanosheet is more likely in an intermediate regime between the1D and 2D limits55.

Fig. 5: Temperature-dependent magnetotransport measurements.
figure 5

a Low-field magnetoconductance ΔG measured for the device at \(V_{{\mathrm{BG}}} = 1.54\,{\mathrm{V}}\) and \(V_{{\mathrm{TG}}} = - 0.46\,{\mathrm{V}}\), corresponding to a conductance value of ~9e2/h and a dual-gate voltage value of VD = −2 V, at different temperatures T. The bottom trace shows the measured magnetoconductance data at T = 1.9 K and all other measured magnetoconductance traces are successively vertically offset for clarity. b Phase coherence length Lφ, spin–orbit length LSO, and mean free path Le extracted by fitting the measured data to Eq. (1) as a function of temperature T. The black solid line is a power-law fit to the extracted phase coherence length Lφ, showing Lφ ~ T−0.38.

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In summary, a dual-gate planar device made from a single-crystalline zincblende InSb nanosheet is fabricated and the quantum transport properties of the InSb nanosheet in the device are studied by low-field magnetotransport measurements. Carrier density, mean free path, coherence length, and SOI strength in the InSb nanosheet are extracted. It is shown that the measured low-field magnetoconductance can be excellently described by the 2D diffusive HLN quantum transport theory and exhibits the WAL characteristics. The origin of the WAL characteristics is identified as the presence of strong SOI of the Rashba type in the InSb nanosheet. By performing the magnetoconductance measurements of the InSb nanoshhet at constant carrier densities, we demonstrate that the Rashba SOI strength can be efficiently tuned by a voltage applied over the duel gate. We also observe the presence of a strong SOI in the InSb nanosheet at zero dual-gate voltage. By simulations for the band diagrams of the device structure, we identify the origin of this intrinsic SOI in the InSb nanosheet as the presence of band bending in the nanosheet even at zero dual-gate voltage. The strong and tunable Rashba SOI in the InSb nanosheet, demonstrated in this work, lays the groundwork for employing this emerging layered material in the developments of spintronics, spin qubits, and topological quantum devices.

Methods

Material growth

High-quality, free-standing, single-crystalline, pure zincblende phase InSb nanosheets used in this work are grown by molecular-beam epitaxy (MBE) on top of InAs nanowires on a Si (111) substrate. The growth process starts by depositing a thin layer of Ag on the Si substrate in an MBE chamber. The film is subsequently annealed in situ to form Ag nanoparticles. Thin InAs nanowires are then grown with these Ag nanoparticles as seeds. The InSb nanosheets are grown on top of the InAs nanowires by abruptly switching the group-V source from As to Sb and with an increased Sb flux. High-resolution transmission electron microscopy and scanning electron microscopy analyses show that the as-grown InSb nanosheets are of high-quality, pure zincblende phase, single crystals and are up to several micrometers in sizes and down to ~10 nm in thickness. For further details about the growth process and structural properties of our MBE-grown InSb nanosheets, we refer to ref. 27.

Device fabrication

For device fabrications, the MBE-grown InSb nanosheets are mechanically transferred from the growth substrate onto an n-doped Si substrate covered with a 300-nm-thick layer of SiO2 on top. The Si and SiO2 layers are later used as a global bottom gate and its dielectric. After transferring, contact electrodes are fabricated on selected nanosheets with a thickness of t ~ 30 nm via a combined process of electron-beam lithography (EBL), electron-beam evaporation (EBE) of a Ti/Au (5/90 nm in thickness) metal bilayer, and lift off. We note that before the metal evaporation, the exposed areas on the InSb nanosheets are chemically etched in a de-ionized water-diluted (NH4)2Sx solution to remove the surface oxide and to subsequently passivate the fresh surface. After the contact electrode fabrication, a 20-nm-thick HfO2 dielectric layer is deposited on the sample by atomic layer deposition. Finally, a Ti/Au (5/90 nm in thickness) metal bilayer top gate is fabricated on each device by the combined process of EBL, EBE and lift off, again. Figure 1a shows a false-colored SEM image of a fabricated device measured for this work and the measurement circuit setup. In this device, four parallel contact electrodes are made on the InSb nanosheet and the distance between the two inner contact electrodes is ~1.1 μm. The top gate covers the entire InSb nanosheet as seen in Fig. 1a and as indicated in the schematic shown in Fig. 1b.

Gate transfer characteristics and magnetotransport measurements

Low-temperature transport measurements of the fabricated devices are carried out in a PPMS cryostat equipped with a uniaxial magnet. The InSb nanosheet conductance is measured in a four-probe configuration to eliminate the impact of the contact resistances using a lock-in technique, in which a 17-Hz AC excitation current I of 100 nA is supplied between the two outer electrodes and the voltage drop V between the two inner contact electrodes is recorded. The nanosheet channel conductance G is obtained numerically from G = I/V. For this work, the results of the measurements obtained from a representative device as shown in Fig. 1a are presented. The measurements are carried out with magnetic fields applied perpendicular to the nanosheet plane at temperatures of T = 1.9–20 K.

Fitting of the measured magnetoconductance data to the HLN formula

To extract the characteristic transport lengths of Lφ, LSO, and Le in the InSb nanosheet, the measured data are fitted to Eq. (1), based on the least-squares method, using both the “curve_fit” function in the SciPy package written in Python and the non-linear fit program in the Origin software for crosscheck. The two fitting procedures give almost the same results. The fitting bounds are set in order to make the corresponding length scales vary in a reasonable range. For example, we set the fitting bound of Le as Le ≤ 200 nm. The range of magnetic fields B is chosen to be |B| ≤ 20 mT in all the fittings presented in this work in order to make the low-field condition of Eq. (1) satisfied.

Band diagram simulation

To simulate the energy band diagrams of the HfO2–InSb–SiO2 heterostructure in the device, Poisson’s equations are solved using commercially available program COMSOL in compliance with the boundary conditions of the system. An effective one-dimensional model with three sections representing three different materials, HfO2, InSb, and SiO2, is considered. Material properties employed in the simulation, include bandgaps, dielectric constants, electron effective masses, and electron affinities, are listed in Supplementary Table I. The carrier density in the InSb nanosheet and the boundary conditions used in the simulation are acquired from the experiments. We first show the different degrees of band bending, i.e., the different degrees of asymmetry, when various voltages are applied to the top and bottom gates (Fig. 4 and Supplementary Fig. 3a). The carrier density distribution inside the InSb layer can also be calculated (see Supplementary Fig. 3c). It is seen that the carrier density is non-uniformly distributed, consistent with the conduction band bending profile obtained. The quantitative analysis of the asymmetry is carried out from the calculated effective electric field strength inside the InSb layer shown in Supplementary Fig. 3b.