Strong and tunable spin-orbit interaction in a single crystalline InSb nanosheet

A dual-gate InSb nanosheet field-effect device is realized and is used to investigate the physical origin and the controllability of the spin-orbit interaction in a narrow bandgap semiconductor InSb nanosheet. We demonstrate that by applying a voltage over the dual gate, efficiently tuning of the spin-orbit interaction in the InSb nanosheet can be achieved. We also find the presence of an intrinsic spin-orbit interaction in the InSb nanosheet at zero dual-gate voltage and identify its physical origin as a build-in asymmetry in the device layer structure. Having a strong and controllable spin-orbit interaction in an InSb nanosheet could simplify the design and realization of spintronic deceives, spin-based quantum devices and topological quantum devices.


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 electronics 1 , infrared optoelectronics 2 , spintronics 3 , quantum electronics 4,5 and topological quantum computation 6 . The past decade has witnessed booming investigations of devices made from epitaxially grown InSb nanowires, including field-effect transistors 7,8 , single [9][10][11] and double quantum dots 12,13 , and semiconductor-superconductor hybrid quantum devices [14][15][16][17] . Among the most influential, pioneer developments are the topological superconducting quantum devices made from InSb nanowires 14,16 , in which zero-energy modes, a signature of Majorana fermions 18,19 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-nanowire 20,21 and two-dimensional (2D) planar quantum structures [22][23][24] . Recently, high-quality InSb/InAlSb heterostructured quantum wells 25,26 and free-standing InSb nanosheets [27][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 dots 31  Comprehensive studies of SOI have been carried out for InSb nanowires 39,40 and quantum wells 41 .
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 simply 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 ( ) of 100 nA is supplied through the two outer electrodes and the voltage drop ( ) between the two inner contact electrodes is recorded. The nanosheet channel conductance is obtained from = / . In measurements for the magetoconductance, = ( ) − ( = 0), the magnetic field is applied perpendicular to the InSb nanosheet plane.
Figure1c shows the measured conductance of the InSb nanosheet in the device as a function of voltages, BG and TG , applied to the bottom and top gates (transfer characteristics). Figure  1d shows a horizontal line cut of Fig. 1c Fig. 1a). Then by extending the fitting line to intersect the horizontal axis, we obtain BG th . In this way, we have estimated out a carrier density of n = 7.2 × 10 11 cm −2 at BG = −5 V and TG = 0 V, at which the measured conductance takes a value of G ~ 9 2 ℎ ⁄ .
Note that along the red contour line in Fig. 1c, the measured conductance stays at the same value of G ~ 9 2 ℎ ⁄ and thus the carrier density in the nanosheet stays, to a good approximation, at the same value of n = 7.2 × 10 11 cm −2 . Similarly, the yellow contour line in Fig. 1c displays the measurements at a conductance of G ~ 5 2 ℎ ⁄ and a carrier density of n = 4.3 × 10 11 cm −2 in the nanosheet. The electron mobility in the nanosheet is estimated from = / , where = 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 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 BG and the same is for the electron density in the nanosheet, the same electron mobility of ~6000 cm 2 · V −1 · s −1 in the nanosheet is extracted at both G~ 9 2 ℎ ⁄ and G~ 5 2 ℎ ⁄ . The electron mean free path in the nanosheet can be estimated from e = ℏ √2π , where ℏ = ℎ 2π with ℎ being the Planck constant, giving e~ 84 nm at n =7.2 × 10 11 cm −2 (G ~ 9 2 ℎ ⁄ ) and e~ 65 nm at n = 4.3 × 10 11 cm −2 (G ~ 5 2 ℎ ⁄ 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 ( φ ), SOI length ( SO ), and mean free path ( e ).
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. 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 BG = 0 V, a well-defined WAL peak is observed, but the peak becomes less pronounced as BG decreases.
For a 2D diffusive system, the low-field magnetoconductance is well described by the Hikami-Larkin-Nagaoka (HLN) quantum interference theory 43 . 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 Here, Ψ( ) 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.  Figure 2b shows the extracted φ , SO , and e in the InSb nanosheet from the fits at TG = 0 V as a function of BG . As shown in Fig. 2b, φ is strongly dependent on BG , while SO and e show weak BG dependences and stay at values of SO~1 30 nm and e~ 80 nm. Here, we note that the extracted e~8 0 nm is in good agreement with the values extracted above from the gate transfer characteristics. The weak BG dependence of e arises from the fact that at the low temperature we have considered, e 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 SO also shows a weak BG dependence because it primarily depends on the perpendicular electric field penetrated through the InSb nanosheet, which is only weakly dependent on BG when the InSb nanosheet is at open conduction state. At BG = 0 V (a high carrier density case), the extracted φ reaches to ~ 530 nm. As BG sweeps from 0 to −13 V, φ decreases rapidly to ~180 nm, indicating that the dephasing is stronger at a lower carrier density. The physical origin of this increase in φ 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 mechanism 44 ), 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 φ is one order of magnitude larger than the thickness of the nanosheet. This, together with the fact that the typical Fermi wavelength F~3 0 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 e ~ 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 mechanim 45,46 , i.e., the spin randomization due to momentum scattering. In the Elliot-Yafet mechanism, the spin relaxation length can be estimated out as 39,47 ≥ 500 nm, using the bandgap G = 0.23 eV, the Fermi energy F = ℏ 2 π * ≤50 meV (with ≤ 7.2 × 10 11 cm −2 ), bulk spin-orbit gap 48 ∆ SO ~ 0.8 eV, and the mean-free path e ~ 80 nm. The estimated SO,EY is much larger than the experimentally extracted value of SO ~ 130 nm. Therefore, the Elliot-Yafet mechanism does not play a key role in our system. Another one is the D'yakonov-Perel' mechanism 49 , 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 direction 27 , the Dresselhaus SOI 50 would be either absent or negligible 51 . Based on the above analyses, we expect that the Rashba SOI 52 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 , where * = 0.014 0 denotes the effective mass of electrons in InSb with 0 being the free electron mass. The spin-orbit energy can be determined as so = * R 2 2ℏ 2~1 60 μ V 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 R~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 nanowires 39 , but is significantly larger than the values of ~ 0.16 eV Å found in InAs nanowires 53 . 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 wells 41,54 . Thus, the extracted R ~ 0.42 eV Å in our InSb nanosheet corresponds to a strong SOI found in a III-V narrow bandgap semiconductor nanostructure.

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 D applied over the dual gate. For example, with TG being set at 0 V, we could sweep BG to gradually change D and thus the electric field through the nanosheet. However, as we showed above, sweeping BG 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, D = TG − BG , is tuned continuously. Figure 3a shows magnetoconductance traces measured along a contour line of G ~9 2 ℎ ⁄ (the red contour line in Fig. 1c) at several values of D . It is seen that all the measured magnetoconductance traces show the WAL characteristics. To extract the transport length scales as a function of D , 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 φ , SO , and e extracted from the fits. It is shown that φ stays at a constant value of ~460 nm, independent of D . This is in good agreement with the fact that φ 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 e , which is found to stay at a value of ~ 85 nm. However, SO shows a strong dependence on D . As seen in Fig. 3b, SO is monotonically increased from ~130 to ~390 nm as D changes from −2 to 11 V, indicating that the SOI strength becomes weaker as D moves towards more positive values. Figure  Temperature effects Figure 5a shows the measured low-field magnetoconductance of the device at BG = 1.54 V and TG = −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 φ , SO and e from the fits are displayed in Fig. 5b. It can be found that both SO and e are weakly dependent on temperature, while φ shows a strong temperature dependence, decreasing rapidly from ~470 nm to ~210 nm with increasing temperature from 1.9 to 20 K. The temperature dependence of φ is found to follow a power law of φ~− 0.38 (see the solid line in Fig. 5b).
The 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, the 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.

Material growth
High-quality, free-standing, single-crystalline, pure zincblende phase InSb nanosheets used in this  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.

Data Availability
The data supporting the findings of this study are available within the article and its     (1)]. b, Phase coherence length φ , spin-orbit length SO , and mean free path e in the InSb nanosheet extracted from the fits as a function of BG .          The carrier mobility in the InSb nanosheet can be obtained from = / , where = is the nanosheet channel conductivity with being the channel length (i.e., the distance between the two inner contacts, about 1.1 μm in this device) and the channel width (i.e., the width of the nanosheet, about 550 nm in this device). Here, we note that since both the nanosheet conductance and the carrier density in the nanosheet depend linearly on BG , the extracted carrier mobility from the transfer characteristic measurements (which is often called the field effect mobility) will be independent of BG . Thus, we can evaluate the carrier mobility µ by setting the back gate voltage value at, e.g., BG R = −5 V, at which the carrier density is n = 7.2 × 10 11 cm −2 and the nanosheet conductance is ~9 2 ℎ ⁄ . The obtained carrier mobility is then ~ 6000 cm 2 · V −1 · s −1 . The carrier mean free path in the nanosheet is given by e = ℏ √2π , where ℏ = ℎ 2π with ℎ being the Planck constant. From the measured back gate transfer characteristics shown in Fig. 1(a), we obtain e~ 84 nm at n = 7.2 × 10 11 cm −2 (and G ~ 9 2 ℎ ⁄ ) and e~ 65 nm at n = 4.3 × 10 11 cm −2 (and G ~ 5 2 ℎ ⁄ ).

Strong and tunable spin-orbit interaction in a single crystalline InSb
Using the top-gate transfer characteristics shown in Fig. 1 HfO2 layer in our device is in good amorphous phase, consistent with the fact that it was grown at a low temperature by atomic layer deposition.

Supplementary Note II. Comparison between the results obtained by analyses of magnetotransport measurements of the InSb nanosheet using the HLN and ILP theories
In the main article, the HLN model is utilized in the analysis of our magnetotransport data. This is suitable for a weak disordered system such as InSb nanosheet and other emerging 2D materials, where the electron elastic scattering length, or the mean free path, e is shorter than all other characteristic transport length scales, such as phase coherence length φ and spin-orbit length SO . However, in a clean 2D electron system with a ultrahigh mobility made from a semiconductor heterostructured quantum well, e can be exceedingly longer than SO . In this case, the HLN model may no longer be applicable and one might need to invoke the so-called ILP model, developed by Iordanskii, Lyanda-Geller and Pikus 1 , in analyses of the magnetotransport measurement data. Here, it is worthwhile to check whether the ILP model can be applied to the magnetotransport data obtained in our device. In the ILP model, the quantum conductance correction to the low-field magnetoconductance is given by Here, SO    Here we assume that each material layer is an infinite two-dimensional structure and we thus need to solve effectively only a one-dimensional Poisson's equation. Material parameters of InSb, SiO2 and HfO2 utilized in the simulations for the energy band diagrams are given in Table I. Poisson's equation used here to describe the electrostatics of the HfO2-InSb-SiO2 heterostructure has a form of where is the electric potential, r is the dieletric constant of the material, q is the

Supplementary Note IV. Analysis of the Rashba SOI in the InSb nanosheeet
In a semiconductor quantum structure, two predominant mechanisms that give rise to spin-orbit coupling and thus lift the spin degeneracy even in the absence of a magnetic field are the Dresselhaus and Rashba 2 SOI. The first one arises from an intrinsic bulk inversion asymmetry (BIA) of the underlying crystal structure, as described by Dresselhaus 3 , while the second one arises from a structural inversion asymmetry (SIA) induced by an electrical field = −∇ ( ) in the crystal, where ( ) is the electric potential, as described by Bychkov and Rashba 4 . The electric field could include both a built-in part in the structure and a tunable part created by, e.g., applying a gate voltage.
In the lowest-order approximation, the Rashba SOI Hamiltonian can be written as 5 where R is a material-specific, Fermi level dependent prefactor 6,7 and is the wave vector. To estimate the effect of the Rashba SOI in our dual-gate device structure, we assume that all conduction carriers experience a same electric field in the InSb nanolayer. We approximate this electric field by the mean electric field = (0, 0, ), As shown in Fig. 4, at both carrier densities, cm -2 and 0.41 V -2 at n = 4.3 × 10 11 cm -2 . This result supports our above assumption that the Rashba SOI induced spin precession process is the major cause for the observed gate voltage tunable WAL characteristics. The intercepts of the two fitting lines and the vertical axis are nearly the same and are very close to a value of 8 µm -2 , which gives 0 =8 µm -2 and represents all other field-independent contributions including the one from the Dresselhaus SOI. The Fermi level dependent prefactors R can be obtained from the extracted slopes κ using the relation R = ℏ 2 * ⋅ √ . The results are R = 4.26 e⋅nm 2 at n = 7.2 × 10 11 cm -2 and R = 3.48 e⋅nm 2 at n = 4.3 × 10 11 cm -2 . It is seen that similar dual-gate voltage dependences of the transport lengths φ , so , and e as observed in Fig. 3 of the main article are obtained. In particular, the spin-orbit length SO is seen to be efficiently controlled via the dual-gate voltage D at both constant conductance values. These data, together with those shown in Fig. 3 of the main article, demonstrate 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.