Abstract
The recent development in the fabrication of artificial oxide heterostructures opens new avenues in the field of quantum materials by enabling the manipulation of the charge, spin and orbital degrees of freedom. In this context, the discovery of twodimensional electron gases (2DEGs) at LaAlO_{3}/SrTiO_{3} interfaces, which exhibit both superconductivity and strong Rashba spinorbit coupling (SOC), represents a major breakthrough. Here, we report on the realisation of a fieldeffect LaAlO_{3}/SrTiO_{3} device, whose physical properties, including superconductivity and SOC, can be tuned over a wide range by a topgate voltage. We derive a phase diagram, which emphasises a fieldeffectinduced superconductortoinsulator quantum phase transition. Magnetotransport measurements show that the Rashba coupling constant increases linearly with the interfacial electric field. Our results pave the way for the realisation of mesoscopic devices, where these two properties can be manipulated on a local scale by means of topgates.
Introduction
The interplay between superconductivity and spinorbit coupling (SOC) is at the centre of intensive research efforts as it can generate a variety of unique phenomena such as the occurrence of triplet superconductivity, for instance^{1}. Recently, hybrid nanostructures involving a superconductor in proximity to a semiconducting nanowire with a strong SOC have been proposed as an ideal system to observe a topological superconducting phase, which accommodates pairs of Majorana fermions^{2,3}. Following this idea, the first signatures of Majorana Fermions were obtained in devices made with indium antimonide in contact with niobium titanium nitride^{4}. However, the realisation of such devices remains a challenge because (i) the intrinsic value of the SOC in semiconductors is weak and cannot be tuned (ii) it is difficult to control the spin state at the interface between very different materials. For this reason, the discovery of a twodimensional electron gas (2DEG) at the interface between two insulating oxides such as LaAlO_{3}/SrTiO_{3} or LaTiO_{3}/SrTiO_{3} raised a considerable interest^{5}. Indeed, this 2DEG displays both superconductivity^{6,7} and a strong SOC which is expected to be Rashbatype^{8,9}, a combination of properties which is rarely observed in the same material.
The 2DEG whose typical extension in the SrTiO_{3} substrate is of order ~10 nm^{10,12} is confined in an interfacial quantum well buried under an few unit cells thick insulating LaAlO_{3} layer. By adjusting the Fermi level with a gate voltage, the conductivity of the 2DEG can be modulated from insulating to superconducting^{11,12}. In addition, the Rashba SOC, which is dominated by the local electric field at the interface, can also be controlled with a gate voltage^{8}. The combination of these two effects enables the realisation of nanostructures, where the very same material can be turned into different states by applying a local electric fieldeffect. Thus far, controlling the superconductivity and SOC have been demonstrated almost exclusively with gates deposited at the back of thick SrTiO_{3} substrates. Because of the very high value of the SrTiO_{3} dielectric constant at low temperatures ()^{13}, the electric fieldeffect can significantly modulate the carrier density with gate voltages on the order of 100 V^{11,12,14}. However, in such geometry, it is not possible to control the properties of the 2DEG on a scale much smaller than the typical thickness of the substrate (500 μm), making it impossible to realise devices with dimensions comparable to lengths that are characteristic of quantum orders (such as the superconducting coherence length and the spin diffusion length). To overcome this problem, fieldeffect control of the superconductivity and Rashba SOC needs to be achieved by means of local topgates. Forg et al. fabricated fieldeffect transistors in a LaAlO_{3}/SrTiO_{3} heterostructures using the insulating LaAlO_{3} layer as the gate dielectric and the YBa_{2}Cu_{3}O_{7} layer as the topgate electrode^{15}. Hosoda et al. achieved topgate control of the normal state properties using a metallic gate directly deposited on the LaAlO_{3} layer^{16}. More recently, a first attempt to modulate the superconductivity with a topgate gave promising results^{17}, despite the leaky insulating LaAlO_{3} layer. In this article, the realisation of a topgated fieldeffect device is reported. The properties of the 2DEG could be tuned over a wide range, from a superconducting to an insulating state. In addition, the control of the Rashba SOC by means of a topgate is also demonstrated.
A tenμmwide superconducting Hall bar was first fabricated with an amorphous LaAlO_{3} template method and then covered by a Si_{3}N_{4} dielectric layer and a metallic top gate (see Fig. 1a,b)^{18}. More information on the fabrication processes is given in the Methods section. The sample was anchored to the mixing chamber of a dilution refrigerator with a base temperature of 16 mK. Figure 1c shows the superconducting transition of the device at the critical temperature , which is similar to an unprocessed LaAlO_{3}/SrTiO_{3} heterostructure. The currentvoltage (IV) characteristics of the device abruptly switches from the superconducting state (R = 0) to the resistive state (R ≠ 0) at the critical current I_{c} = 460 nA which corresponds to a critical current density of approximately 500 μA/cm.
Electrostatic Control of the Carrier Density
After the sample was cooled, the topgate voltage V_{TG} was first increased to +110 V, beyond the saturation threshold of the resistance. During this operation, electrons are added in the quantum well, increasing the Fermi energy to its maximum value (i.e., the top of the well)^{19}. In comparison with backgate experiments where the relationship between the carrier density (n) and the backgate voltage V_{BG} is not trivial owing to the electricfielddependent dielectric constant of SrTiO_{3}^{13}, here, the carrier density is expected to increase linearly with V_{TG}. Figure 2 shows the sheet carrier density , extracted from the Hall effect measurements performed up to B = 4 T as a function of the topgate voltage V_{TG}, for two different backgate voltages (V_{BG} = 0 V and V_{BG} = −15 V). For V_{BG} = 0 V, the linear increase in n is observed with V_{TG} only for negative V_{TG}. The nonphysical decrease in n with V_{TG} for positive gate voltages is caused by the incorrect determination of the carrier density at low magnetic fields. It was shown that at the LaAlO_{3}/SrTiO_{3} interface, the Hall voltage is no longer linear with the magnetic field for strong filling of the quantum well because of multiband transport^{12,20,21}. To reach a doping regime where the oneband approximation is valid, a negative backgate V_{BG} = −15 V was applied producing a depletion of the highest energy subbands that accommodate the highlymobile carriers, responsible of the decrease of the Hall number at positive V_{TG}. Figure 2 shows that in this case, the linear dependence of with V_{TG} can be recovered. The linear fit of slope is obtained from numerical simulations of the electric fieldeffect by a finite elements method assuming a dielectric constant for the Si_{3}N_{4} layer (see the inset in Fig. 2). Finally, the following relationship between the carrier density and topgate voltage is deduced: n = 5.0 × 10^{10} V_{TG} + 1.69 × 10^{13} e^{} .cm^{2}.
Superconductivity and Phase Diagram
In the following, the back gate voltage V_{BG} was always set to 0 V unless otherwise stated. Figure 3a shows the sheet resistance of the device as a function of temperature measured for different topgate voltages in the range [−110 V, +110 V], where the leakage gate current is negligible (<0.1 nA). The variation in V_{TG} induces a modulation in the normal state resistance by two orders of magnitude. Figure 3b summarises the variations of the normalised resistance R/R(T = 350 mK) as a function of temperature (T) and topgate voltage V_{TG} on a phase diagram. The corresponding n is also indicated on the top axis. The device displays a gatedependent superconducting transition, whose critical temperature T_{c} describes a partial dome as a function of V_{TG}, similar to that observed with a backgate^{11,12,14}. The maximum T_{c}, corresponding to optimal doping, is around 250 mK. In the underdoped region, a decrease in the gate voltage causes T_{c} to continuously decrease from its maximum value to zero. A superconductortoinsulator quantum phase transition takes place around V_{TG} = −90 V. The critical sheet resistance at the transition is , which is close to the quantum of resistance of bosons with 2e charges, . For large negative voltages, corresponding to low electron densities, the sheet resistance increases strongly when approaching the insulating state. In the overdoped region, the addition of electrons into the quantum well with the topgate produces a small decrease in T_{c} whose origin is currently under debate. Such behaviour has also been observed in doped bulk SrTiO_{3}^{22} and could be reinforced by the twodimensionality of the interface^{23}. The currentvoltage characteristics of the device for different topgate voltages are shown in Supplementary Material.
Rashba Spinorbit Coupling
In LaAlO_{3}/SrTiO_{3} heterostructures, the accumulation of electrons in the interfacial quantum well generates a strong local electric field E_{z} perpendicular to the motion of the electrons, which translates into a magnetic field in their rest frame. It is expected that the coupling of the electrons spin to this field gives rise to a Rashbatype SOC described by the Hamiltonian , where is the electron wave vector, is a unit vector perpendicular to the interface and σ are the Pauli matrices^{24}. The constant α represents the strength of the SOC and has to be directly proportional to the interfacial electric field E_{z}. In electronic transport measurements, the presence of a spinorbit coupling results in an additional spin relaxation mechanism characterised by the relaxation time τ_{SO}. Caviglia et al. reported a τ_{SO} roughly proportional to the inverse of the elastic scattering time τ_{e} in agreement with a D’YakonovPerel mechanism characteristic of a Rashba interaction^{8}. However, to confirm experimentally the Rashbatype SOC, it is also important to establish the linear dependance of α with E_{z} (α ∝ E_{z}) when the filling of the quantum well is varied by gating.
The weak localization corrections to the conductance of a twodimensionnal system at low temperatures are modified by the presence of an additional spin relaxation mechanism due to SOC^{25,26} whose strength can therefore be determined by properly analysing the magnetoconductance Δσ(B) = σ(B) − σ(0). Δσ(B) was measured in the normal state at different temperatures and topgate voltages. For negative V_{TG} a positive magnetoconductance was observed beyond 1 T. This is characteristic of a weak localization regime with small SOC (Fig. 4). As V_{TG} is increased, an inversion of the sign of the magnetoconductance is observed and at large positive gate voltages the magnetoconductance remains always negative. The experimental data in Fig. 4 were fitted with the MaekawaFukuyama formula in a diffusive regime that describes the change in the conductivity with magnetic field with negligible Zeeman splitting^{25},
where Ψ is the digamma function, is the quantum of conductance and the parameters B_{tr}, B_{Φ}, B_{SO} are the effective fields related to the elastic, inelastic and spinorbit relaxation times respectively. B_{Φ} and B_{SO}, which are measured here by a transport experiment, are related to the relaxation times τ_{Φ} and τ_{SO} by the expressions and respectively, where D is the diffusion constant^{25,26}. Finally, to account for the orbital magnetoconductance, we have added in Eq. (1) a B^{2} term with a Kohler coefficient A_{K} which increases quadratically with the mobility^{27,28}. Good agreement is obtained between the experimental data and the theory over the whole electrostatic doping range.
The evolution of the fitting parameters as a function of the topgate voltage and equivalent carrier density is shown in Fig. 4b. B_{ϕ} varies only weakly over the whole range of gate voltage, indicating that the number of inelastic collisions does not depend on the carrier density. In the framework of the weak localisation theory the temperature dependence of the inelastic scattering time is given by τ_{Φ} ∝ T^{−p} and therefore B_{Φ} ∝ T^{p}, where p depends on the inelastic mechanism. The same fitting procedure was performed at different temperatures, giving a linear relationship between B_{Φ} and T (Fig. 4b inset and Supplementary Material). This is consistent with p = 1, which indicates that the inelastic scattering is dominated by electronelectron interactions^{6,29}.
Spinsplitting Energy
The spinorbit term (B_{SO}) increases with topgate voltage and, correspondingly, with the carrier density. The analysis of this dependence can shed light on the origin of the SOC at the LaAlO_{3}/SrTiO_{3} interface. If we assume that the spin relaxation is dominated by the D’YakonovPerel mechanism, based on a Rashba spinorbit interaction, ^{26,30}. We then obtain the relationship between the coupling constant and the spinorbit effective field . Integrating the MaxwellGauss equation in the direction perpendicular to the interface gives the interfacial electric field where is the dielectric constant of Si_{3}N_{4} at the interface and n_{t} is the carrier density of nonmobile charges trapped in the SrTiO_{3} substrate. The coupling constant being proportional to E_{z}, it is therefore expected to vary with carrier density with the form α = an + b, which is well satisfied experimentally for a wide range of electrostatic doping (Fig. 5). This confirms experimentally that the D’YakonovPerel mechanism in the presence of Rashba spinorbit interaction is dominant in these 2DEGs.
Assuming a Fermi energy of 100 meV and an effective mass m = 0.7m_{0} for the d_{xy} light subbands mainly occupied, we can estimate the characteristic spinsplitting energy Δ_{SO} = 2k_{F}α where k_{F} is the electron wave vector at Fermi energy. The order of magnitude of a few meV, which is much larger than in most semiconductors, is in agreement with previous studies^{8,9}. Neglecting the small changes in k_{F} with doping, we can plot the variation of Δ_{SO} with V_{TG} and correspondingly, n (Fig. 5). Δ_{SO} is independent of the temperature below 10 K as the shape of the quantum well and E_{z} do not change in this temperature range (see the inset in Fig. 5 and Supplementary Material). The Kohler term (parameter A_{K}) is proportional to the square of the mobility. For positive gate voltages where the 2DEG has a rather large mobility, this term dominates the magnetoconductance and must be taken into account in Eq. (1). As shown in the Supplementary Figure S4, fitting the data without this term leads to an incorrect determination of the SOC in a large range of positive gating.
In summary, LaAlO_{3}/SrTiO_{3} based fieldeffect devices were fabricated using the amorphous LaAlO_{3} template method. The superconductivity can be electrostatically modulated over a wide range by a topgate voltage, without any leakage. A superconductortoinsulator quantum phase transition is induced when the quantum well is strongly depleted. By analysing the magnetotransport measurements, the presence of strong spinorbit coupling that could be controlled with the topgate voltage was demonstrated. The spinspliting energy on the order of a few meV was found to increase linearly with the interfacial electric field in agreement with the Rashba mechanism. These results represent an important step toward the realisation of new mesoscopic devices, where the interaction between superconductivity and the Rasba SOC could give rise to nonconventional electronic states.
Methods
Device fabrication
Starting with a TiO_{2} terminated oriented SrTiO_{3} commercial substrate (Crystec), the template of a Hall bar with contact pads was defined by evaporating an amorphous LaAlO_{3} layer through a resist patterned by optical lithography. After a liftoff process, a thin layer of crystalline LaAlO_{3} (8 u.c) was grown on the amorphous template by Pulse Laser Deposition, such that only the areas directly in contact with the substrate (Hall bar and contact pads) were crystalline. A KrF excimer (248 nm) laser was used to ablate the singlecrystalline LaAlO_{3} target at 1 Hz, with a fluence between 0.6 and 1.2 J/cm^{2} under an O_{2} pressure of 2 × 10^{−4} mbar^{31}. The substrate was typically kept at 650 °C during the growth of the film, monitored in realtime by reflection highenergy electron diffraction RHEED. As the growth occurs layerbylayer, the thickness can be controlled at the unit cell level. After the growth of the film, the sample was cooled down to 500 °C under a O_{2} pressure of 10^{−1} mbar, which was increased up to 400 mbar. To reduce the presence of oxygen vacancies (in both the substrate and the film), the sample was kept under these conditions for 30 minutes before it was cooled to room temperature. The 2DEG forms at the interface between the crystalline LaAlO_{3} layer and the SrTiO_{3} substrate. Such method has already been used to fabricate ungated 500 nm wide channels without noticeable alteration of the 2DEG properties^{18}. Once the channel is defined, a 500 nm thick Si_{3}N_{4} dielectric layer was deposited on the Hall bar by a liftoff process. After this step, a gold topgate layer was deposited and liftedoff forming and appropriate geometry to cover the Hall bar. A metallic back gate was added at the end of the process.
Additional Information
How to cite this article: Hurand, S. et al. Fieldeffect control of superconductivity and Rashba spinorbit coupling in topgated LaAlO_{3}\SrTiO_{3} devices. Sci. Rep. 5, 12751; doi: 10.1038/srep12751 (2015).
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Acknowledgements
This work was supported by the french ANR, the DGA the CNRS PICS program and the Région IledeFrance through CNano IdF and Sesame programs.
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Affiliations
Contributions
J.L. and N.B. supervised the project. E.L. and N.R. fabricated the LaAlO_{3}/SrTiO_{3} heterostructures by PLD under the supervision of A.B. and M.B. S.H. and A.J. made the topgate devices with the help of C.F.P., X.L, C.U. and M.P.L. S.H., A.J. and G.S performed and analysed the measurements. C.F.P. performed the Comsol simulations. All authors contributed to the interpretation of the results. S.H., J.L. and N.B. wrote the manuscripts with inputs of J.B., S.C. and M.G.
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Hurand, S., Jouan, A., FeuilletPalma, C. et al. Fieldeffect control of superconductivity and Rashba spinorbit coupling in topgated LaAlO_{3}/SrTiO_{3} devices. Sci Rep 5, 12751 (2015). https://doi.org/10.1038/srep12751
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