Magneto-transport study of top- and back-gated LaAlO$_3$/SrTiO$_3$ heterostructures

We report a detailed analysis of magneto-transport properties of top- and back-gated LaAlO$_3$/SrTiO$_3$ heterostructures. Efficient modulation in magneto-resistance, carrier density, and mobility of the two-dimensional electron liquid present at the interface is achieved by sweeping top and back gate voltages. Analyzing those changes with respect to the carrier density tuning, we observe that the back gate strongly modifies the electron mobility while the top gate mainly varies the carrier density. The evolution of the spin-orbit interaction is also followed as a function of top and back gating.

The two-dimensional electron liquid (2DEL) present at the interface between the insulating oxides LaAlO 3 (LAO) and SrTiO 3 (STO) exhibits several fascinating properties, including superconductivity and a large spin-orbit coupling. [1] It has also attracted much attention in the context of device applications following the realization of field-effect transistors and devices with nanoscale dimensions. [2][3][4][5][6] The large electric permittivity of the STO substrate, especially at low temperatures, facilitates tuning of the 2DEL properties by an electric field effect in back gate transistors. [2] This back gate configuration allows considerable control of multiple parameters of the 2DEL such as the superconducting critical temperature, spin-orbit interaction, carrier density, and mobility. [7][8][9][10][11][12] Additionally, it might simultaneously change the confinement of the 2DEL. [8,13,14] This confinement modifies the bulk STO electronic structure and leads to an orbital ordering with light d xy and heavy d xz /d yz sub-bands. [15,16] Recently, there has been growing interest in employing top electrodes in gating experiments of LAO/STO devices, [17][18][19][20][21][22] as LAO has a large band gap and reasonably large dielectric constant. [23] The transistor operation of a top-gated LAO/STO device was demonstrated with an on-off switch of the conductivity using less than 1 V. [3] However, the responses of the 2DEL to top and back gate can be different: a first comparison revealed, indeed, a different modulation of the mobility by the two approaches. [19] In this paper, we present a systematic study of transport properties of top-and backgated LAO/STO heterostructures in the presence of a perpendicular magnetic field. Electric field effect tuning is achieved by using the LAO film and STO substrate as top and back gate dielectrics respectively. Large tunability in resistance, carrier density and magnetoresistance were observed in both configurations. The top-gate approach shows reversible tuning with voltage sweeping, contrary to back gate. [7,14] Top and back gates affect the mobility differently, back gate being more effective to boost it. The evolution of the weaklocalization/weak-antilocalization behavior is also extracted from the magneto-transport data. This allows us to study inelastic and spin-orbit magnetic fields for constant carrier densities achieved by a combined use of top and bottom gates. We try to link the differences in the response of the 2DEL to the confining potential shape and to the multi-band conduction.
The Hall bars are defined by pre-depositing an amorphous STO layer as a hard mask. [5] Structures down to 25 µm in width are realized by optical lithography. Subsequently, epitax-ial LAO films are deposited on TiO 2 -terminated STO (001)-oriented single crystals by pulsed laser deposition in an oxygen pressure of 8 × 10 −5 Torr at 800 • C or 830 • C and annealed in 0.2 bar of oxygen at 520 • C for an hour as documented elsewhere. [7,24] After growth, ex-situ sputtered Pt is used for the top gate and Au films are sputtered on the backside of the substrate for the back gate electrodes. A schematic of a device with a top and back gate is displayed in Figure 1(a). Figure 1 Capacitance characterization with an AC voltage of 10 mV at 200 Hz demonstrates good capacitor behavior (small loss tangent and negligible leakage current) of top-gated devices prepared in this fashion. [25] We consistently observe an effective dielectric constant ( ef f ) for the Pt/LAO/2DEL capacitor smaller than the bulk value, which is possibly due to interfacial effects. [26] Moreover, this interfacial effect might be the reason for the weak dependence of ef f on temperature as shown in Figure 1(c) (bulk LAO has a temperature independent dielectric constant). ef f has little dependence on the top gate voltage V T G as shown in Figure 1(d). We also do not observe in the investigated V BG range substantial changes in the capacitance when we modulate the 2DEL carrier density using back gate. [27] Figure 2 compares the low temperature resistance behavior sweeping the gate voltages from the top (panel (a)) and from the back (panel (b)).
[28] For top gate voltages V T G , the tuning of the sheet resistance R S is reversible. We note the modulation of R S is about 40%, the V T G range being set to avoid leakage across the LAO layer. For back gate voltages V BG , a large history dependent behavior manifests in R S when V BG is first swept upwards and then downwards. In this geometry, from the as-grown state, R S first decreases and then saturates for a wide range of V BG (red curve in panel (b)). If we stop increasing V BG (at V max BG = 50 V as in panel (b)) and reverse the sweeping direction, we observe that the resistance increases strongly (blue curve), deviating from the upward sweep. Then, as long as V BG is kept below V max BG , the sweeps are reversible (the resistance moves along the blue curve). In this configuration, extremely high resistance states can be achieved. If V BG exceeds V max BG , the resistance remains constant in the upward sweep (green curve); however, the new reversible behavior is shifted to higher voltages (purple curve). This "forming process" of the resistance state suggests that many added electrons spill out of the quantum well and go into trapped states. [14] To build a more comprehensive understanding of the different effects of V T G and V BG , we investigate the magneto-transport properties of the 2DEL using both gates. The data are displayed in Figure 3: in the left panels, the measurements are performed at various V T G while V BG is maintained fixed, and vice versa for the right panels. We observe that the changes in R S induced by sweeping V T G or V BG are quite comparable, but on a completely different voltage range. Figures 3(c) -3(f) display the Hall resistance R xy as a function of magnetic field for different values of V T G and V BG : we observe an evolution from linear to non-linear behavior pointing to a transition from single to multiple band conduction. [11,29,30] Since correctly extracting the total carrier density n 2D from a non-linear R xy is generally difficult, the modulation of the total carrier density δn 2D is calculated using a parallel plate capacitor model. For the top gate, using ef f = 12, we obtain a change in n 2D that agrees with the estimation from the variation of the Hall coefficient with V T G when R xy is linear in field (see Figure 3(c)). For the back gate, δn 2D is obtained by the capacitor measurement as a function of V BG . [7] These δn 2D modulations are plotted as top axes in panels (a) and (b). Now comparing the change in R S versus δn 2D for the two gates, we see that the same R S modulation is achieved by top gate with a variation in n 2D that is roughly twice that obtained with the back gate.
This different response of R S to the top and back gates, as well as the evolution of the Hall effect, may suggest distinct effects of the two gates on the confining potential and hence on the sub-band structure. Self-consistent calculations using a Schrödinger-Poisson approach show that a pure increase in carrier density results in a stronger confinement. [31,32] Modeling of back gate transistors [8] and analysis of back gate voltage sweeps [14] indicate that positive voltages increase the confinement width (weaker confinement) of the 2DEL; as a consequence, the energy splitting between the lower energy d xy -and lower mobility -and higher energy d xz /d yz -and higher mobility -bands should be reduced.
In order to highlight the difference between the two gate approaches, we plot the mobility µ as a function of the carrier density in Figure 4 for top (panel (a)) and back (panel (b)) gate voltage sweeps. n 2D is estimated from the linear Hall effect and the carrier change induced by the capacitor effect, and µ is defined as σ(0)/n 2D e, where e is the electron charge and σ(0) is the zero-field conductance; µ calculated in this fashion is therefore an effective mobility. In the same graphs, we use a color code to indicate the degree of non-linearity observed in the Hall resistance ∆R xy /R xy (0) = |R xy (7.5 T) − R xy (0 T)|/R xy (0 T), R xy being the derivative ∂R/∂B at a given field. For single band conduction, ∆R xy /R xy (0) is zero (represented as blue) while for multi-band conduction, it becomes non-zero (represented as yellow/red). The mobility curves in Figure 4(a) show that top gate sweeps result in a strong modulation of n 2D and a smaller change in µ. When we change the back gate voltage, we notice a sharp increase in the carrier mobility: this is more evident in panel (b) where mobility is strongly enhanced by back gate voltages for a small variation in the carrier density. We also observe that, concomitant with the increase in µ, the Hall resistance becomes non-linear, suggesting that a second band with high mobility comes into play. [11,15,16,29,30] A possible scenario to understand the response of the transport properties to the fieldeffect tuning relates the effect of the two gates to the confining potential. As discussed above, a deconfinement of the 2DEL would reduce the d xy versus d xz /d yz band splitting, this energy becoming negative in bulk STO. [33] Along with this effect, carriers could be transferred to the more mobile band, resulting in higher mobility and non-linear Hall resistance. On the other hand, if electrons are more confined, the energy splitting should increase, and a shift in the Fermi level would mainly induce an increase in n 2D with little change in µ.
We finally explore the effect of top and back gates on the spin-orbit interaction of the system. We fit the magneto-conductance curves using the Maekawa-Fukuyama formula [34] and extract the inelastic field B i and the spin-orbit field B so for different sets of V T G and V BG . [35] Figure 5 shows the evolution of B i and B so as a function of n 2D . We observe that for each V T G the back-gate tuning results in an effective modulation of B so , in agreement with previous studies on the spin-orbit interaction. [9,30] While B i seems to be determined mainly by n 2D , B so should be more sensitive to the modification of the confining potential.
To get further insight into the evolution of B so upon gate tuning, in the inset of Figure 5, we plot B so versus V BG for the three carrier densities indicated by the shaded areas in the main panel of the figure. For the plot, V BG is chosen as the "tuning parameter" as it should affect the electron confining potential most and hence the strength of the spin-orbit interaction, although probably in a complex way. [11,[36][37][38] The data plotted in the inset show that the spin-orbit coupling can be tuned at constant n 2D and suggest that V BG is indeed modifying the confinement of the 2DEL.
To conclude, we present a systematic study of the effects of top and back gate on LAO/STO heterostructures. The detailed characterization reveals differences between the evolution of R S , R xy and magneto-conductance using the two approaches.

I. SUPPLEMENTARY MATERIAL
In this supplementary information, we fit the magneto-conductance curves using the Maekawa-Fukuyama (MF) formula: [9,34] ∆σ(B)/σ 0 = Ψ B B i + B so + 1 where Ψ(x) = ln(x) + ψ( 1 2 + 1 x ) with ψ(x) the digamma function, and γ = gµ B B/4eDB so . Here σ(B) = R S (B) R 2 S (B)+R 2 xy (B) and σ 0 = e 2 /πh is a universal value of conductance. Figure 6 displays magneto-conductance curves. In each panel, the top gate is fixed and the back gate is varied. For low doping samples, the data seem to reflect the evolution of the weak-localization/weak-antilocalization behavior as reported in references 9 and 10. At high dopings and high magnetic fields, data deviate from weak anti-localization fittings, making