Two-dimensional electron gas (2DEG) formed at the interface between two band insulators LaAlO₃ (LAO) and SrTiO₃ (STO) exhibits many fascinating properties, such as magnetism,^[1] superconductivity,^[2,3] and their coexistence.^[4–6] Besides, the observation of the spin–orbit coupling (SOC) in this system provides possibilities to control the orbital motion of electrons by acting on their spins.^[7–9] Initially, the vast majority of the investigations were specific only to the LAO/STO(001) interfaces.^[1–10] However, recent findings have shown that 2DEGs can be formed at interfaces with different crystal orientations, such as LAO/STO(110).^[11,12] and LAO/STO(111),^[12] opening fresh perspectives to understand the fundamental physics and to offer new functionalities.

Via x-ray linear dichroism spectroscopy, Pesquera et al. recently demonstrated an intrinsic distinctive electronic structure of the 2DEG at the LAO/STO(110) interface due to the different orbital symmetry and hierarchy compared to those of the (001)-oriented interface.^[13] Therefore, the physical properties, such as superconductivity and SOC,^[14–17] could be influenced by the distinctive orbital symmetry and hierarchy. Latest studies on the LAO/STO(110) interfaces revealed distinct superconductivity and Rashba SOC,^[16] and the SOC is insensitive to the gate voltage owing to the stronger polarizations of d_xy, d_yz/d_zx orbitals along the confinement direction. Moreover, for the LAO/STO(110) interface, there are two different crystal directions, i.e., [001] and , in the plane of the interface. Recent angle-resolved photoemission (ARPES) studies on SrTiO₃(110) surface showed a high electronic in-plane anisotropy along the two specific directions, and the anisotropy is pronounced at lower carrier density when only d_yz/d_zx-derived ellipsoids are occupied.^[18] Hence, the in-plane crystallographic directions of LAO/STO(110) may offer an extra degree of freedom to tune the properties of 2DEG. A few of the latest works mentioned the existence of the in-plane anisotropy,^[11,17] however a systematic study of the in-plane anisotropy in the (110) system is still required. Here, we present a careful and systematic research of the influence of the crystallographic direction on both normal state and superconducting properties at the LAO/STO(110) interfaces.

The samples with ten unit cells of LAO were grown at two different oxygen partial pressures (P_O2), 10⁻⁴ Torr and 10⁻³ Torr, on treated (110)-oriented SrTiO₃ substrates by pulsed laser deposition. The details of the growth procedure have been given in Ref. [15]. After preparation, two Hall bar structures were patterned in the same sample by using a standard photolithography technique, the LAO layers and a few STO layers were removed by low energy argon ion milling. For the directional dependence magnetotransport measurements, the Hall bar structures were along [001] and crystallographic directions, respectively. An etched STO surface was insulating, indicating that the conducting channels were at the interfaces. Ohmic contacts between electrode pad and sample interface were made by wire bonding. Magnetotransport properties were measured at temperature T = 2 K and with a magnetic field up to 9 T by a commercial Physical Properties Measurement System (Quantum Design, PPMS) equipped with a rotating sample holder. The gate was applied to the backside of the SrTiO₃ substrate by a voltage source and no leakage current (< 1 nA) was detected for ±100 V gate voltages (V_G).

The anisotropic out-of-plane magnetoresistance (MR) along two directions, and [001], is shown in Fig. 1. The MR is defined as ΔR/R₀ = [R(B) − R(B = 0)]/R(B = 0), where R(B) is the sheet resistance in the magnetic field B. As we can see from Fig. 1(a), for the sample prepared at P_O2 ∼ 10⁻⁴ Torr, the MR linearly increases with increasing magnetic field for both directions. The anisotropy is not obvious; but for the interface prepared at P_O2 ∼ 10⁻³ Torr, the anisotropy of the MR becomes obvious, as shown in Fig. 1(b). Along the direction, the MR firstly increases and then decreases with increasing magnetic field. These results indicate that the anisotropy along the two directions is pronounced at lower carrier density, which is consistent with the ARPES results on the STO(110) surface, attributed to the occupation of orbitals.^[18] In order to effectively exploit the anisotropy of the 2DEG at the LAO/STO(110) interface, we now focus on the sample prepared at high P_O2 (10⁻³ Torr) with a relatively low carrier density.

Fig. 1.

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	Fig. 1. The magnetoresistance measured along (black) and [001] (red) directions of LAO/STO(110) for samples prepared under oxygen partial pressures of (a) 10−4 Torr and (b) 10−3 Torr.

Anisotropic MR was investigated for different angle θ between the direction of the interface normal and the magnetic field. For the investigation of angular MR, the magnetic field was rotated from out-of-plane (θ = 0°, i.e., perpendicular to the surface) to in-plane (θ = 90°, parallel to the current) at 2 K. As shown in Fig. 2(a), along the direction, as the magnetic field increases, the MR gradually increases in the low field regime and then starts to decrease in the higher field. There is a local MR maximum, a positive slope in the low magnetic field regime but a negative slope in the high magnetic field regime. Besides, a negative MR appears for θ > 60° and the negative MR regime widens out as θ increases. Moreover, with increasing θ, the maximum of the MR slightly shifts to low magnetic field and the value becomes smaller. Along the [001] direction, the MR decreases as θ increases from 0° to 90°; and the local MR maximum appears at large angles, as shown in Fig. 2(b). The behavior of the in-plane MR is similar, while an obvious difference of MR along two directions can be observed at θ = 0° (perpendicular magnetic field), indicating that the anisotropy is more pronounced in the plane of the interface.

Fig. 2.

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	Fig. 2. The magnetoresistance for different angles θ along (a) and (b) [001] directions.

The positive out-of-plane MR is attributed to the orbital effect, while the in-plane MR rules out this effect. The difference of out-of-plane MR between [001] and directions suggests that crystallographic anisotropy plays a critical role in magnetotransport properties. The negative MR appears and becomes visible as the parallel magnetic field increases, as shown in Fig. 2. The negative MR can be counteracted by the positive contribution for the out-of-plane field case. When applying the parallel magnetic field, the suppression of positive MR indicates that the confinement zone of 2DEG is within one carrier mean free path. We assume that only one subband is occupied, then the mean-free path can be naively calculated by using formula ,^[19] where l is the mean-free path, h is the Planck constant, e is the electron charge, R_s is the sheet resistance, is the Fermi wave number, and n_s is the carrier density. Then at T = 2 K, the mean-free path is found to be about 22 nm along the direction, while it is about 34 nm along the [001] direction, the same order of magnitude compared with that reported by Ben Shalom et al. for (001)-orientation LAO/STO interfaces. It is noted that the mean-free path is shorter along the direction than that of [001], suggesting that scattering becomes stronger along the direction.

The remarkable angular dependence of MR and the appearance of negative MR indicate that the 2D weak localization effect should be considered. The magnetic field dependence of MR provides an important insight into the coupling between spin dynamics and transport. The existence of a local maximum MR and the appearance of a negative MR were also observed in other two-dimensional disordered systems,^[20–22] which is associated with the weak localization related to the presence of spin–orbit interactions (weak antilocalization).^[23–26] In LAO/STO heterostructures, the local electric field is enhanced by the accumulation of electrons in the interfacial quantum well, giving rise to the SOC. A strong spin–orbit field has been demonstrated at both (001) and (110)-oriented LAO/STO interfaces.^[7,8,16,17]

For the investigation of directional dependence of SOC, we measured the MR as a function of V_G, as shown in Fig. 3. Due to the high dielectric constant of SrTiO₃,^[27] the properties of 2DEG can be effectively tuned by a backgate voltage.^[7,8] As shown in Fig. 3(a), the sheet resistances along two directions vary as V_G changes. Also the sheet resistance along the direction is more sensitive to the applied electric field and is larger than that along the [001] direction. Figures 3(b) and 3(c) show the gate-tunable MR at different V_G along two directions. Along the direction, the MR decreases with increasing V_G, the local MR maximum shifts to low magnetic field, while along the [001] direction, the local MR maximum appears only for negative V_G.

Fig. 3.

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	Fig. 3. (a) The sheet resistance as a function of VG along two directions. The magnetoresistance at different VG along (b) and (c) [001] directions.

The conductance correction due to the weak localization is modified by the SOC. The influence of the SOC can therefore be assessed by properly analyzing the magnetoconductance in the diffusion regime. The magnetic field dependence of the conductance correction normalized by the quantum of conductance as a function of V_G is investigated, as shown in Fig. 4. We fit the experimental data by using the Maekawa–Fukuyama (MF) form^[8,28]

Fig. 4.

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Fig. 4. The out-of-plane magnetoconductance (normalized by the quantum of conductance) at different VG along (a) and (b) [001] directions. The dark grey dashed lines are the fitting lines by using Eq. (1). (c) The obtained fitting parameters BSO (filled square) and Bϕ (open circle) as a function of VG along (black) and [001] (red) directions. (d) The τSO as a function of VG along two directions, the inset shows the τi versus VG. Comparing τSO with τi at different VG along (e) and (f) [001] directions. Spin relaxation time vs. elastic scattering rate along two directions is shown in the insets of panels (e) and (f). The lines are a guide to the eye.

The evolution of the fitting parameters as a function of V_G is shown in Fig. 4(c). Along both directions, the inelastic field B_ϕ decreases as V_G increases. When increasing V_G, the sheet resistance decreases and the carrier density increases, so the inelastic scattering weakens. Along the direction, with increasing V_G, the spin–orbit field B_SO decreases slightly, while along the [001] direction, the spin–orbit field increases as V_G increases. For both directions, the B_SO changes slightly with varying V_G. This insensitive gate tunable SOC may be attributed to the strong atomic orbital polarizations of d_xy and d_xz/d_yz orbitals along the confinement direction.^[16] The corresponding scattering time as a function of V_G is shown in Fig. 4(d). Along both directions, τ_i decreases with increasing V_G, and τ_i is smaller than τ_SO (Figs. 4(e) and 4(f)), suggesting that the effect of the SOC is stronger compared with the orbital effect of the magnetic field. Moreover, a significant difference of spin–orbit term B_SO is observed. The τ_i is shorter along the direction, suggesting that the weak localization is stronger than that of the [001] direction, which is consistent with the difference of sheet resistivity (Fig. 3(a)). For spin relaxation processes, there are mainly two typical scenarios: the Elliott–Yafet (EY) mechanism and the D’yakonov–Perel’ (DP) mechanism.^[30–32] In the latter mechanism, the spin relaxation time τ_SO is expected to be inversely proportional to the elastic scattering time τ. As shown in the insets of Figs. 4(e) and 4(f), τ_SO is proportional to the inverse of τ over a wide range of V_G, characterizing the Rashba spin–orbit interactions. Besides, we note a deviation for V_G > 50 V along the [001] direction, which may be due to the same order of τ and τ_SO. For intuitive comparison, all transport parameters for the sample prepared under the high oxygen pressure are listed in Tables 1 and 2.

Table 1.

Table 1.

Table 1. Transport parameters along [001] direction. . VG/V Rsheet/(kΩ/□) n/1013 cm−2 D/cm2·s−1 BSO/T Bϕ/T τSO/10−12 s τϕ/10−12 s τ/10−13 s − 100 9.30 2.21 0.54 1.30 0.54 2.37 5.73 0.52 − 75 6.84 2.36 0.73 1.38 0.43 1.63 5.30 0.66 − 50 5.00 2.51 1.00 1.44 0.35 1.15 4.73 0.85 − 25 3.51 2.73 1.42 1.62 0.29 0.72 4.07 1.11 0 2.25 2.80 2.21 1.84 0.23 0.40 3.18 1.69 25 2.13 2.85 2.34 1.71 0.28 0.41 2.47 1.75 50 199 2.77 2.50 1.95 0.23 0.34 2.88 1.93 75 1.66 2.73 2.99 2.27 0.23 0.24 2.37 2.34 100 1.48 2.67 3.36 2.27 0.22 0.22 2.21 2.68

Table 1. Transport parameters along [001] direction. .

Table 2.

Table 2.

Table 2. Transport parameters along direction. . VG/V Rsheet/(kΩ/□) n/1013 cm−2 D/cm2·s−1 BSO/T Bϕ/T τSO/10−12 s τϕ/10−12 s τ/10−13 s −100 24.82 3.73 0.20 1.17 0.85 7.03 9.65 0.11 − 75 17.37 3.57 0.29 1.10 0.71 5.20 8.14 0.17 − 50 10.79 3.22 0.46 1.03 0.58 3.45 6.13 0.31 − 25 7.11 2.88 0.70 0.99 0.50 2.39 4.71 0.52 0 5.19 2.81 0.96 0.98 0.47 1.74 3.62 0.73 25 4.92 2.87 1.01 0.95 0.53 1.70 3.09 0.75 50 4.33 2.83 1.15 0.94 0.47 1.53 3.08 0.87 75 3.80 2.80 1.311.49 0.88 0.43 1.42 2.92 1.00 100 3.35 2.58 0.83 0.40 1.33 2.74 1.23

Table 2. Transport parameters along direction. .

Moreover, for the P_O2 = 10⁻⁴ Torr sample, down to mK magnitude of temperature, the superconductivity emerges, while for the P_O2 = 10⁻³ Torr sample, the superconductivity is absent. To our surprise, the superconductivity is anisotropic along the two directions, as shown in Fig. 5. The superconducting transition temperature (T_c) is lower but the out-of-plane critical field (H_c2,⊥) is larger along the direction than that along [001] (inset of Fig. 5). The temperature dependence of H_c2,⊥ is extracted from the MR curves measured at different temperatures, where H_c2,⊥ is defined as the perpendicular magnetic field in which the sheet resistance of interfaces reaches to half of the normal state resistance taken at a given temperature. Generally, H_c2,⊥ should be proportional to T_c. We speculate that the unconventional larger H_c2,⊥ along the direction is due to the effect of disorder.^[33,34] The above MR results suggest the strong localization along the direction. The deviation at low temperatures indicates that the effect of disorder overtakes the thermal fluctuation.^[35] The mechanism of in-plane anisotropy of superconductivity needs further investigation and will be discussed elsewhere.

Fig. 5.

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	Fig. 5. Temperature dependence of the sheet resistance along two directions. The inset shows temperature dependence of the out-of-plane critical field, taken at 50% of normal resistance.

The anisotropic behavior along two directions and [001] at the LAO/STO(110) interfaces may be due to the difference of Ti t_2g (i.e., d_xy, d_yz, and d_zx) orbitals. For LAO/STO(110), contrary to (001), the lower energy states have d_yz/d_zx character.^[13] The ARPES results performed on SrTiO₃(110) surfaces reveal that the d_yz/d_zx-derived Fermi surface is anisotropic.^[18] For the interface prepared at low P_O2 (e.g., 10⁻⁴ Torr), the carrier density is high (in our case, ∼ 6 × 10¹³ cm⁻²), and then both d_xy and d_yz/d_zx subbands are filled, the anisotropy weakens. However, for the interface prepared at high P_O2 (e.g., 10⁻³ Torr), the carrier density is low (∼ 2.2 × 10¹³ cm⁻²), only d_yz/d_zx derived ellipsoids appear, the anisotropy is obvious. Along the direction, the d_yz/d_zx like subbands dispersing is weak and the effective electron mass is semiheavy, while the d_yz/d_zx like subbands dispersing is strong and the effective electron mass is light along the [001] direction.^[17,18] The difference of d_yz/d_zx like subbands along the two directions may influence the properties of 2DEGs at the LAO/STO(110) interfaces.

We demonstrate the in-plane anisotropic behavior of 2DEG at LAO/STO(110) interfaces by performing magnetotransport and superconducting measurements. The anisotropy becomes more obvious for samples of low carrier density. We find that the SOC is anisotropic and insensitive to the gate voltage, while the spin relaxation mechanism is the DP mechanism along both directions. The effect of localization is stronger along the direction. Moreover, the anisotropic superconductivity is also observed, lower T_c but larger H_c2,⊥ is shown along the direction. Such crystallographic tunable properties expand the possibilities for electronic engineering of two-dimensional electron gas at the LaAlO₃/SrTiO₃ interface.