We measure the gate voltage (Vg) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO3/SrTiO3 interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature Tc and the upper critical fields lying at the same Vg. The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as Tc. The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit, suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior.
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TY - JOUR
T1 - Link between the Superconducting Dome and Spin-Orbit Interaction in the (111) LaAlO3/SrTi O3 Interface
AU - Rout, P. K.
AU - Maniv, E.
AU - Dagan, Y.
N1 - Funding Information: LaAlO 3 / SrTi O 3 Interface Rout P. K. Maniv E. Dagan Y. * Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University , Tel Aviv 69978, Israel * firstname.lastname@example.org 4 December 2017 8 December 2017 119 23 237002 6 June 2017 © 2017 American Physical Society 2017 American Physical Society We measure the gate voltage ( V g ) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO 3 / SrTi O 3 interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature T c and the upper critical fields lying at the same V g . The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as T c . The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit, suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior. Israel Science Foundation http://dx.doi.org/10.13039/501100003977 ISF http://sws.geonames.org/294640/ 382/17 Ministry of Science and Technology, Israel http://dx.doi.org/10.13039/501100006245 Ministry of Science and Technology http://sws.geonames.org/294640/ 3-11875 United States-Israel Binational Science Foundation http://dx.doi.org/10.13039/501100001742 BSF 2014047 Oxide heterostructures provide unique platform where various degrees of freedom from the constituent materials can combine such that new collective phenomena emerge at the interfaces  . An interesting example is a two-dimensional (2D) electron liquid at the interface between (100)-oriented SrTiO 3 and LaAlO 3 that exhibits gate tunable superconductivity [2–4] and spin-orbit interaction [4–6] . Recent experiments on (111) LaAlO 3 / SrTi O 3 have shown 2D conduction [7–9] and superconductivity with a transition temperature ( T c ) of about 100 mK [10,11] . In a (111)-oriented LaAlO 3 / SrTi O 3 interface, the cubic lattice is projected onto the (111) plane of the interface, resulting in a 2D sixfold crystalline structure. Angle-resolved photoemission studies on the (111) SrTiO 3 surface reveal a sixfold symmetric electronic structure [12,13] . This 2D crystalline symmetry is also reflected in the magnetotransport properties  and has been predicted to host exotic electronic orders [14–17] . At low temperatures, this symmetry is lowered, since bulk SrTiO 3 undergoes multiple structural transitions. Below 105 K, a transition from a cubic to a tetragonal phase occurs  . The symmetry is further reduced to triclinic below ∼ 70 K , and polar domain walls where inversion symmetry is broken are created  . Such a domain wall can be pinned to the interface, resulting in unconventional superconductivity, which is linked to spin-orbit coupling. In a 2D superconductor, for a magnetic field applied perpendicular to the superconducting plane, superconductivity is broken when vortices become closely packed. By contrast, the parallel upper critical field ( H c ∥ ) is determined by the Chandrasekhar-Clogston limit [20,21] , which is set by comparing the Zeeman energy to the superconducting gap. In the presence of a spin-orbit interaction, this upper bound is relaxed [22,23] . In this Letter, we report a nonmonotonic (dome-shaped) dependence of T c with a gate voltage in the (111) SrTiO 3 / LaAl O 3 interfaces. From the gate dependence of T c and H c ∥ , we estimate the spin-orbit energy ( ϵ SO ), which follows the nonmonotonic behavior of T c . Remarkably, we found similar behavior for the spin-orbit field H SO extracted from weak antilocalization measurements. Epitaxial films of LaAlO 3 were deposited on an atomically flat SrTiO 3 (111) substrate using pulsed laser deposition. The details of the deposition procedure and substrate treatment are described in Ref.  . We control the layer-by-layer growth of 14 monolayers ( LaO 3 / Al layers) by reflection high-energy electron diffraction oscillations. The atomic force microscope images show the step and terrace morphology of the film with step heights of 0.22 nm. The electrical measurements with the current along the [ 11 2 ¯ ] direction were carried out in a Leiden Cryogenics custom-made dilution refrigerator. Figure 1(a) presents the temperature-dependent sheet resistance R S ( T ) at various gate voltages V g . A clear gate-dependent superconducting transition is observed. We define the critical temperature T c as the temperature at which R S reaches half of its value at 350 mK. The normal state resistance R S (350 mK) decreases monotonically with increasing V g [Fig. 1(b) ], which is consistent with previous reports [8,9] . The monotonic increase of R S is contrasted with the nonmonotonic dependence of T c on V g . A similar dome-shaped region in the carrier density-temperature phase diagram is seen in many unconventional superconductors and in the (100) LaAlO 3 / SrTi O 3 interface. 1 10.1103/PhysRevLett.119.237002.f1 FIG. 1. (a) Temperature dependence of sheet resistance R S ( T ) for various gate voltages. (b) T c and R S (350 mK) as a function of V g . (c) Gate dependence of the inverse Hall coefficient 1 / | e R H | at T = 5 K . In the (100) LaAlO 3 / SrTi O 3 interface, the Hall coefficient depends nonmonotonically on the gate voltage. Surprisingly, this nonmonotonic behavior is also seen in the gate dependence of the Shubnikov–de Haas oscillation (SdH) frequency. Both the SdH frequency and low field inverse Hall coefficient follow the gate dependence of T c for the (100) interface [3,24] , or the superconductivity starts appearing when the low field inverse Hall coefficient decreases from its maximum value  . By contrast, for the (111) interface the inverse Hall coefficient monotonically decreases with V g [Fig. 1(c) ] consistent with previous observations [8,9] . In the case of the (111) LaAlO 3 / SrTi O 3 interface, the titanium t 2 g bands are split into low and high spin states due to the atomic spin-orbit interaction [14,15] . We have shown that the lower spin state is first populated when accumulating electrons with increasing V g  . This two-band scenario complicates the interpretation of the Hall data. We have estimated the amount of carrier density modulation due to the electric field effect similar to Refs. [2,26] . Since the V g range used is relatively small, the nonlinearities in the dielectric constant ( ε ) can be neglected, and thus the corresponding modulation of electron density is ≃ 1.3 × 10 13 cm - 2 with ε ≃ 15 000 . This value is much smaller than the net change in 1 / | e R H | of ≃ 4.3 × 10 13 cm - 2 . Moreover, the electron density due to the field effect increases with V g in contrast to the observed behavior in Fig. 1(c) . All these observations indicate the presence of a hole band in addition to electron band(s) in the (111) interface. We have confirmed this scenario by analyzing the normal state transport data via a simplistic noninteracting two-band model with one hole and one electron band (see Ref.  for more details). Therefore, it is possible that the hole contribution to the electronic transport (and perhaps to superconductivity) becomes important in this V g range  . This is also consistent with the polar structure of the (111) interface  . The sheet resistance versus magnetic field at 90 mK for various gate voltages is plotted in Figs. 2(a) and 2(b) for perpendicular and parallel field configurations, where the sample is properly aligned with the field within an accuracy of 2°. We define the critical field ( H c ⊥ ) for the perpendicular magnetic field configuration such that R S ( H c ⊥ ) = R S ( 350 mK ) / 2 , and a similar criterion is followed for H c ∥  . In Fig. 2(c) , we plot H c ∥ and H c ⊥ as a function of V g , both exhibiting nonmonotonic behavior with the maximum at the same gate voltage as T c . H c ∥ > H c ⊥ for all gate voltages reaching a maximal ratio of ∼ 16 . Such strong anisotropy between two field orientations is evidence for 2D superconductivity in the (111) interface. Thus, it is expected that the superconducting layer thickness ( d ) should be smaller than the Ginzburg-Landau coherence length ( ξ GL ). To check this, we extract ξ GL from H c ⊥ using the relation ξ GL = Φ 0 / 2 π H c ⊥ . It is presented in Fig. 2(d) together with its extrapolation to zero temperature using H c ⊥ ( T ) = H c ⊥ ( 0 ) ( 1 - T / T c ) valid for a 2D superconductor. Since the parallel magnetic field fully penetrates a 2D ( d ≪ ξ ) superconductor, we can only estimate the upper limit for d denoted as d ¯ , which can be found from d ¯ = 3 Φ 0 / π ξ GL H c ∥ [see Fig. 2(d) ]. We note that, for all V g , d ¯ < ξ GL ( 0 ) , rendering superconductivity in the (111) SrTiO 3 / LaAl O 3 two dimensional. 2 10.1103/PhysRevLett.119.237002.f2 FIG. 2. Magnetoresistance R S ( H ) at T = 90 mK in (a) perpendicular ( H → perpendicular to the current and interface) and (b) longitudinal ( H → parallel to the current and interface) configurations for various V g . (c) H c ⊥ and H c ∥ at 90 mK as a function of V g along with the Chandrasekhar-Clogston limit H P . (d) Gate dependence of ξ GL (90 mK), ξ GL ( 0 ) , and d ¯ . For a parallel field configuration in a 2D superconductor, the orbital motion and vortices can be neglected, making the Zeeman energy the dominant pair-breaking effect. This leads to an upper (Chandrasekhar-Clogston) limit of H c ∥ given by H P = 3.5 k B T c / 2 g μ B ( μ B is the Bohr magneton) in the BCS weak coupling limit [20,21] . Assuming a gyromagnetic ratio of g ≃ 2 , we observe H c ∥ > H P for all gate voltages reaching a maximal ratio of ∼ 11 [Fig. 2(c) ]. In the presence of strong spin-orbit coupling, the Chandrasekhar-Clogston limit can be relaxed. Other reasons for breaking this limit could be strong coupling superconductivity, many-body effects, and an anisotropic pairing mechanism. To determine the spin-orbit interaction from H c ∥ , we use a somewhat oversimplified picture of spin-orbit scattering that suppresses spin orientation by the Zeeman field  . For a strong spin-orbit interaction, H c ∥ can be expressed in terms of the spin-orbit energy ( ϵ SO ) as H c ∥ = 0.602 ϵ SO / k B T c H P with ϵ SO = ℏ / τ SO , and τ SO is the spin-orbit scattering time. Remarkably, this analysis reveals a nonmonotonic dependence of ϵ SO on V g as shown in Fig. 3(b) . This is the main finding of our Letter. For (110) LaAlO 3 / SrTi O 3 , gate-independent spin-orbit coupling has been observed  , perhaps because of the nonpolar structure of this interface. The findings on the (110) interface are contrasted with our results of a strong and gate-tunable spin-orbit interaction for the (111) interface that follows the behavior of the superconducting dome. A weaker correlation between spin-orbit coupling and T c in the (100) interface can be deduced by combining Refs. [4–6] , where H c ∥ is smaller. 3 10.1103/PhysRevLett.119.237002.f3 FIG. 3. (a) The normalized perpendicular magnetoconductance Δ σ ( H ) / σ 0 for different V g at T = 1.3 K . The black solid lines are the fits according to Eq. (1) . (b) ϵ SO as a function of V g determined from H c ∥ (see the text for more details). (c) Gate dependence of H i and H SO extracted from the fitting of weak antilocalization. (d) Gate dependence of τ i , τ SO , and τ . The inset shows τ SO as a function of τ - 1 along with the solid line as a guide to the eye. To further confirm the presence of a spin-orbit interaction, we studied the perpendicular magnetoresistance well above T c at 1.3 K [Fig. 3(a) ]. For a 2D diffusive metallic system placed in a perpendicular magnetic field ( H ), the field-dependent quantum correction to conductivity Δ σ ( H ) normalized by quantum conductance ( σ 0 = 2 e 2 / h ) can be expressed as [5,30] Δ σ ( H ) σ 0 = Ψ ( H H i + H SO ) + 1 2 1 - γ 2 Ψ ( H H i + H SO ( 1 + 1 - γ 2 ) ) - 1 2 1 - γ 2 Ψ ( H H i + H SO ( 1 - 1 - γ 2 ) ) - A H 2 1 + C H 2 , (1) where Ψ ( x ) = ln ( x ) + ψ [ 1 2 + ( 1 / x ) ] [ ψ ( x ) is the digamma function] and γ = g μ B H / 4 e D H SO ( D is the diffusion coefficient). H i and H SO are the inelastic and spin-orbit fields, respectively. The classical orbital magnetoresistance contributes a Kohler term to Eq. (1) with the parameters A and C . Figure 3(c) shows H i and H SO for different V g (see Supplemental Material for the gate dependence of g , A , and C  ). Clearly, H SO > H i for all V g , suggesting that we are in the weak antilocalization regime [see Fig. 3(a) ]. H SO from weak antilocalization [Fig. 3(c) ] shows nonmonotonic behavior similar to ϵ SO inferred from superconductivity [Fig. 3(b) ], and, furthermore, they have a maximum value at the same gate voltage as T c . In general, the LaAlO 3 / SrTi O 3 interface has a complicated band structure involving multiple contributions from the titanium d bands [31,32] . Therefore, the extracted parameters from weak antilocalization do not correspond to an individual band; instead, an averaged value over all the bands should be considered  . We have extracted various averaged time scales, i.e., τ SO , τ i (inelastic time), and τ (elastic scattering time) [Fig. 3(d) ]. The τ SO ( i ) are related to H SO ( i ) determined from weak antilocalization as H SO ( i ) = ℏ / 4 e D τ SO ( i ) . The effective diffusion coefficient ( D ) and τ are calculated using a naïve Drude model for a 2D electron gas (see Ref.  ). Using this analysis, we find that τ SO depends linearly on τ - 1 for V g < - 25 V [see the inset in Fig. 3(d) ], while for V g > - 25 V both τ SO and τ increase with V g [Fig. 3(d) ]. The low V g regime ( V g < - 25 V ) is governed by a D’yakonov-Perel’-type spin-orbit relaxation mechanism for which τ SO ∝ τ - 1 . In this scenario, the electron precesses around the spin-orbit field, which is changing due to momentum scattering at a typical time τ  . The high V g regime, on the other hand, is characterized by τ SO ∝ τ , suggesting that the electron spin is coupled to the crystal momentum. Interestingly, these two regimes separated by the point where τ SO ≃ τ and the maximum of T c (and H c ∥ ) dome lies close to this V g . All these observations suggest the mixing of multiple bands in the presence of a strong spin-orbit interaction for higher V g . This scenario concurs with our recent report of crystalline sixfold anisotropic magnetoresistance in the (111) interfaces  , where the sixfold term appears as a result of another band with higher spin state J getting populated with increasing V g . It is therefore possible that the crystalline spin-orbit interaction becomes important close to this avoided band crossing region due to the orbital mixing [23,35] . This interaction becomes smaller as V g is further increased away from the band crossing regime, resulting in a dome in the spin-orbit energy versus V g . Such a multiband effect can also lead to dome-shaped superconductivity with maximum T c lying at this regime [as observed in Fig. 1(b) ] similar to the case for the (100) interface  . A more exotic mechanism of superconductivity in the LaAlO 3 / SrTi O 3 interface involves the formation of a Fulde-Ferrell-Larkin-Ovchinikov (FFLO) state due to large spin-orbit coupling  . This can somewhat explain the nonmonotonic gate dependence of H c ∥ and T c with the maxima lying at τ SO = τ . However, the H c ∥ for a quasi-2D superconductor in a FFLO state is estimated to be at most 2.5 times the Chandrasekhar-Clogston limit  , which is much lower than the observed values [see Fig. 2(c) ]. Therefore, a full theoretical understanding of the phenomenological link observed here between the superconducting dome and the spin-orbit energy is yet to be developed. Salje et al. have found that for SrTiO 3 below ∼ 70 K the tetragonal symmetry is lowered and the Sr atoms are displaced along the  direction, leading to the breaking of local inversion symmetry  . It is therefore possible that a (111) SrTiO 3 -based polar interface has such broken inversion symmetry in addition to conventional inversion symmetry breaking observed at polar oxide interfaces, which can result in an unconventional superconductivity. It has been recently suggested that dichalcogenide monolayers with hexagonal structure can be a realization of exotic Ising superconductivity where the spins are locked in an out-of-plane configuration due to the breaking of centrosymmetry [38–40] . We also note that the possibility for a nodeless time-reversal-symmetry-breaking superconducting order parameter has been proposed for (111) SrTiO 3 -based interfaces from symmetry considerations  . In summary, the superconducting transition temperature T c of the (111) LaAlO 3 / SrTi O 3 interface has a nonmonotonic dependence on the gate voltage. Maximum T c is found at the same gate voltage where maximal values of spin-orbit field H SO and spin-orbit energy ϵ SO are observed. H SO is extracted from weak antilocalization, while ϵ SO is estimated from the superconducting properties. The H c ∥ exceeds the Chandrasekhar-Clogston limit by more than an order of magnitude due to a strong spin-orbit interaction. 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PY - 2017/12/4
Y1 - 2017/12/4
N2 - We measure the gate voltage (Vg) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO3/SrTiO3 interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature Tc and the upper critical fields lying at the same Vg. The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as Tc. The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit, suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior.
AB - We measure the gate voltage (Vg) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO3/SrTiO3 interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature Tc and the upper critical fields lying at the same Vg. The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as Tc. The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit, suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior.
UR - http://www.scopus.com/inward/record.url?scp=85038379865&partnerID=8YFLogxK
U2 - 10.1103/PhysRevLett.119.237002
DO - 10.1103/PhysRevLett.119.237002
M3 - Article
C2 - 29286685
AN - SCOPUS:85038379865
VL - 119
JO - Physical Review Letters
JF - Physical Review Letters
SN - 0031-9007
IS - 23
M1 - 237002