Multi-Orbital Superconductivity in SrTiO{}_{3}/LaAlO{}_{3} Interface and SrTiO{}_{3} Surface

Multi-Orbital Superconductivity in SrTiO/LaAlO Interface and SrTiO Surface


April 23, 2013; accepted May 28, 2013 \abst We investigate the superconductivity in two-dimensional electron systems formed in SrTiO nanostructures. Our theoretical analysis is based on the three-orbital model, which takes into account orbitals of Ti ions. Because of the interfacial breaking of mirror symmetry, a Rashba-type antisymmetric spin-orbit coupling arises from the cooperation of intersite and interorbital hybridyzation and atomic LS coupling. This model shows a characteristic spin texture and carrier density dependence of Rashba spin-orbit coupling through the orbital degree of freedom. Superconductivity is mainly caused by heavy quasiparticles consisting of d and d orbitals at high carrier densities. We find that the Rashba spin-orbit coupling stabilizes a quasi-one-dimensional superconducting phase caused by one of the d or d orbitals at high magnetic fields along interfaces. This quasi-one-dimensional superconducting phase is protected against paramagnetic depairing effects by the Rashba spin-orbit coupling and realizes a large upper critical field beyond the Pauli-Clogston-Chandrasekhar limit. This finding is consistent with an extraordinarily large upper critical field observed in SrTiO/LaAlO interfaces and its carrier density dependence. The possible coexistence of superconductivity and ferromagnetism in SrTiO/LaAlO interfaces may also be attributed to this quasi-one-dimensional superconducting phase. \kwordNon-centrosymmetric superconductivity, two-dimensional electron gas, multi-orbital model

Two-dimensional conducting electron systems formed on SrTiO heterostructures are attracting much attention. For instance, electron gases with a high carrier density on the order of have been realized in SrTiO/LaAlO (STO/LAO) interfaces [1], SrTiO/LaTiO interfaces [2], SrTiO (STO) surfaces [3], and -doped STO. [4] The discovery of superconductivity [5], ferromagnetism [6, 7, 8, 9, 10], and their coexistence [7, 8, 9, 10] shed light on innovating phenomena in these systems. These quantum condensed phases are controlled by a gate voltage through the change of carrier density [3, 11, 12, 13, 14]. One of the key issues is the role of Rashba-type antisymmetric spin-orbit coupling [15] arising from the interfacial breaking of mirror symmetry, which may realize an exotic quantum condensed phase, such as non-centrosymmetric superconductivity [16], chiral magnetism [17], and their coexistent phase. In this research, we theoretically study the non-centrosymmetric superconductivity realized in STO nanostructures from the microscopic point of view.

It has been shown that a two-dimensional electron gas is confined in a few TiO layers of the STO/LAO interface and STO surface in the high-carrier-density region [3, 18, 19, 20, 21, 23, 22]. The conduction bands mainly consist of three t orbitals of Ti ions [18, 19, 20, 21, 23, 22]. Although the degeneracy of t orbitals significantly affects the band structure of two-dimensional electron gases, a theory of superconductivity based on the multi-orbital model has not been conducted. Multiband models have been studied [24, 25], but the symmetry of t orbitals is taken into account in this study for the first time. We show that the synergy of broken inversion symmetry and orbital degeneracy stabilizes an intriguing superconducting phase in the two-dimensional electron gases.

Our study is based on a two-dimensional tight-binding model that reproduces the electronic structure of the STO/LAO interface indicated by first principles band structure calculations [20, 21, 22, 23, 26, 27, 28] and experiments [18, 19]. We here focus on the STO/LAO interface, which has been intensively investigated, but our main results are also valid for other STO heterostructures. The model is described as {align} H=H_0 + H_I + H_Z, where the single-particle Hamiltonian is


We denote (d, d, d) orbitals using the index , respectively. The first term describes the kinetic energy of each orbital and includes the chemical potential . is the intersite hybridization term of d and d orbitals. represents the crystal electric field of tetragonal systems. Because the mirror symmetry is broken near the interface/surface, hybridization is allowed between d and d/d orbitals, and is represented by the “odd parity hybridization term” . The atomic spin-orbit coupling term (LS coupling term) of Ti ions is taken into account in . We here adopt the tight-binding model reproducing first principles band structure calculations for STO heterostructures [26, 27, 28], , , , , , and . The same tight-binding model has been adopted for the study of surface spin-triplet superconductivity in SrRuO [29]. Recent studies have examined the Rashba-type antisymmetric spin-orbit coupling [27, 28, 30] and magnetotransport [30] in STO/LAO interfaces on the basis of this model.

In this paper, we focus on the role of Rashba-type antisymmetric spin-orbit coupling in the interface superconductivity. In the above model, the Rashba spin-orbit coupling is induced by the combination of the odd parity hybridization term, , and the LS coupling term, . The former arises from the parity mixing of local orbitals, which is a general source of antisymmetric spin-orbit coupling [31, 32]. For instance, the () term describes the mixing of d (d) and d orbitals of Ti ions, which mainly occurs through the parity mixing with the p orbital (p orbital) on oxygen ions.

We consider the -wave superconductivity as expected in the bulk STO [33]. Unconventional pairing due to the electron correlation has been studied, [34] however, we do not touch this possibility. Our reasonable assumption has been justified by the recent experiment on superfluid density [35]. For simplicity, we take into account the intraorbital attractive interaction and the interorbital attractive interaction in the spin-singlet channel; {align} H_I=&U∑_i∑_m n_i,m↑n_i,m↓+U’∑_i∑_m≠m’n_i,m↑n_i,m’↓. For the discussion of the superconducting state in the magnetic field, we consider the Zeeman coupling term {align} H_Z=&-∑_k∑_m∑_s,s’μ_BH⋅σ_ss’ c^†_k,   ms c_k,   ms’, in which is the Pauli matrix and is the Bohr magneton. The orbital depairing effect arising from the coupling of electron motion and vector potential is suppressed by the geometry when we consider the magnetic field parallel to the two-dimensional conducting plane, . The orbital polarization due to the magnetic field is also ignored since the orbital moment along the plane vanishes for the degenerate d/d orbitals.

Now, we formulate the linearized gap equation, by which we determine the instability to the superconducting phase. First, we diagonalize the noninteracting Hamiltonian () using the unitary matrix . Thereby, the basis changes as , where and . With the use of the operators of quasiparticles, and , the noninteracting Hamiltonian is described as,


where is a quasiparticle’s energy and for .

Next, we introduce Matsubara Green functions in the orbital basis, {align} G_m’s’,  ms(k,iω_l)&=∫_0^βdτe^iω_lτ⟨c_k,   m’s’(τ)c^†_k,   ms(0) ⟩,
=&∑^6_j=11iωl-Ej(k)u_m’s’, j(k)u^*_ms, j(k), where is the Matsubara frequency. The linearized gap equation is obtained by looking at the divergence of the T-matrix, , which is given by


The wave vector represents the total momentum of Cooper pairs. In our model, the matrix element of the irreducible T-matrix is obtained as


where is the temperature. When we represent the T-matrix using the basis , the interaction term is represented by the diagonal matrix, with for and for others. The superconducting transition occurs when the maximum eigenvalue of the matrix, , is unity. Then, an element of the eigenvector is proportional to the order parameter , where for and for . In what follows, we assume a zero total momentum of Cooper pairs, namely, . Although a helical superconducting state with is stabilized in non-centrosymmetric superconductors under the magnetic field, [16, 36] a finite momentum does not play any important role in the following results. This is because the paramagnetic depairing effect is suppressed by the orbital degree of freedom, as we show below.

Figure 1: (Color online) (a) Band structure of our model. We show the dispersion relation for . The origin is the chemical potential for a carrier density . (b) Carrier density dependence of spin-orbit coupling on the Fermi surface. We show (solid line) and (dashed line) with being the Fermi wave number of the -th band along the [100] axis. (c) and (d) show Fermi surfaces for and for , respectively. Other parameters are assumed as .

We choose the parameters


so as to reproduce the electronic structure of two-dimensional electron gases. [18, 19, 20, 21, 23, 28, 22, 26, 27] We choose the unit of energy as . Band structure calculations resulted in meV [26], giving rise to an anisotropic Fermi velocity, , for . For the parameters in eq. (15), the d orbital has a lower energy than the d/d orbitals, as expected in STO heterostructures [18, 19, 20, 21, 23, 28, 22, 26, 27]; the level splitting at the point is meV. The chemical potential is determined so that the mobile carrier density per Ti ion is . Although an enormous carrier density of corresponding to at the STO/LAO interface was predicted by the “polar catastrophe” mechanism, [1] recent experiments have shown a rather low density of mobile carriers [11, 12, 13, 14, 18, 19]. One of our purposes is to clarify the carrier density dependence of the superconducting state. The sources of Rashba spin-orbit coupling are assumed to be and unless mentioned otherwise explicitly. We here assume rather large values of and so that the amplitude of Rashba spin-orbit coupling is larger than the transition temperature of superconductivity. We assume attractive interactions so that the transition temperature at zero magnetic field is K. A large transition temperature compared with the experimental K is assumed for the accuracy of numerical calculation. Since we discuss the normalized , the following results are hardly altered by the magnitude of . As we show elsewhere, the superconducting phase is almost independent of the ratio .

Figure 1(a) shows the band structure of our model. We see the spin splitting caused by the Rashba spin-orbit coupling. Because the Rashba spin-orbit coupling is enhanced around the band crossing points [31], the magnitude of spin splitting shows a nonmonotonic carrier density dependence. Figure 1(b) shows the spin splitting in the lowest pair of bands [] and that in the second lowest pair of bands [] as a function of carrier density, where is the Fermi wave number of the -th band along the [100] axis. The nonmonotonic behavior of a spin splitting, , is consistent with experimental observations for STO/LAO interfaces. The seemingly contradictory carrier density dependence [13, 14] of Rashba spin-orbit coupling is probably caused by the peak of , as pointed out by Zhong et al. [27]. In our model, the Fermi level crosses the bottom of the second lowest pair of bands [] at approximately . The Fermi surfaces for and are shown in Figs. 1(c) and 1(d), respectively. The isotropic Fermi surfaces mainly consist of the d orbital for a low carrier density, , while large anisotropic Fermi surfaces mainly consist of the d/d orbitals for a large carrier density, .

First, we discuss the superconducting state at zero magnetic field. While the superconductivity is mainly caused by the d orbital at low carrier densities, , the intraorbital Cooper pairing of d and d orbitals is the main source of superconductivity at high carrier densities, . This crossover of the superconducting state coincides with the change of quasiparticles on the Fermi surfaces discussed for Figs. 1(c) and 1(d). When we assume the attractive interactions independent of carrier density, the transition temperature monotonically increases with increasing carrier density. The nonmonotonic carrier density dependence observed in experiments [11] is reproduced by assuming a decreasing function of against carrier density. In this study, We avoid such a phenomenological assumption and discuss the normalized values such as . Note that the odd-parity hybridization and LS coupling hardly affect the superconducting state at zero magnetic field.

Figure 2: (Color online) (a) Normalized upper critical field, , for the field parallel to the [100] axis. Solid, dashed, and dash-dotted lines show the results for high carrier densities, , , and , respectively. The dotted line is obtained in the crossover region, , while dash-two-dotted line assumes a low carrier density, . Fermi surfaces mainly consist of the d orbital (d/d orbitals) in the low (high) carrier density region. The other parameters are the same as those in Fig. 1. (b) Carrier density dependence of at the lowest temperature [circles]. We also show the results for [squares], [diamonds], and [pluses] for comparison.

On the other hand, the Rashba spin-orbit coupling arising from the combination of and leads to an intriguing superconducting phase in the magnetic field. Figure 2(a) shows the phase diagram against temperatures and magnetic fields for various carrier densities. We see an extraordinarily large normalized upper critical field, , beyond the Pauli-Clogston-Chandrasekar limit, [37] around . It has been shown that the upper critical field is enhanced by the Rashba spin-orbit coupling [38], but that the enhancement is minor in the canonical Rashba-type non-centrosymmetric superconductors as  [39]. We here find that the rather large enhancement of the upper critical field is caused by the synergy of the Rashba spin-orbit coupling and the orbital degree of freedom. Indeed, when we decrease the carrier density to , the orbital degree of freedom is quenched and the upper critical field is suddenly decreased.

As shown in Fig. 2(b), the normalized upper critical field shows a broad peak at approximately and decreases with increasing carrier density for except for a sharp enhancement at around . The decrease in is attributed to the decrease in Rashba spin-orbit coupling [see Fig. 1(b)]. A sharp peak at around is induced by the appearance of small Fermi surfaces around the point, that is, the Lifshitz transition. Because the g-factor of this band vanishes at ( point) in the presence of atomic LS coupling , Cooper pairing in the small Fermi surfaces is not disturbed by the magnetic field. Thus, a sharp enhancement of the normalized upper critical field, , is a signature of the Lifshitz transition. It will be interesting to look for this Lifshitz transition since the Class D topological superconducting phase is realized near the Lifshitz transition by applying a magnetic field [40]. Since the renormalization of the g-factor is not due to the broken inversion symmetry, a sharp peak of also appears for [diamonds in Fig. 2(b)], for which the Lifshitz transition occurs at approximately . Aside from this peak, a small upper critical field below the Pauli-Clogston-Chandrasekar limit is obtained when either or is zero, because the Rashba spin-orbit coupling vanishes. As expected, the normalized upper critical field increases as we increase or . For instance, we obtain for and , in agreement with the experimental result of STO/LAO interfaces [13].

Figure 3: (Color online) Magnetic field dependence of order parameters for . We show the amplitude at the transition temperature, which is proportional to the order parameters below . The main components are (thick solid line) and (thick dashed line). The other small components are shown by the thin lines, as described in Fig. 3(d). We assume (a) , (b) , (c) , and (d) . The other parameters are the same as those in Fig. 2.

In order to clarify the roles of the orbital degree of freedom, we show the magnetic field dependence of order parameters for a high carrier density, . When both odd parity hybridization, , and LS coupling, , are finite [Fig. 3(d)], the magnetic field along the x-axis substantially enhances the Cooper pairs of the d orbital represented by while those of the d orbital () are suppressed. This means that a quasi-one-dimensional superconducting state dominated by the d orbital is stabilized in the magnetic field. Since this high-field superconducting phase is robust against the paramagnetic depairing effect, a large upper critical field is obtained, as shown in Fig. 2. It should be stressed that the Rashba spin-orbit coupling plays an essential role in stabilizing the quasi-one-dimensional superconducting phase. Indeed, we obtain a nearly isotropic two-dimensional superconducting phase with when either the odd parity hybridization or the LS coupling is zero [Figs. 3(a)-3(c)].

Figure 4: (Color online) g-vector of the lowest band (), , which is defined in eq. (18). Arrows show the direction of the g-vector; the length of arrows is proportional to the amplitude of the g-vector. Solid lines show the Fermi surfaces for . The other parameters are the same as those in Fig. 1.

We here illustrate why the quasi-one-dimensional superconducting phase is protected against the paramagnetic depairing effect. For this purpose, we derive the Rashba spin-orbit coupling in the band basis as we have performed in ref. 29. We reduce the single-particle Hamiltonian to the three-band model as , where


and . The Rashba spin-orbit coupling of the -th band is represented by the g-vector


whose direction is obtained by calculating the average for the -th eigenstate. Figure 4 shows the g-vector and the Fermi surfaces for . It is shown that the momentum dependence of the g-vector is quite different from an often-assumed form, . This is the characteristic property of orbitally degenerate non-centrosymmetric systems [29]. In the case of STO heterostructures, the g-vector is nearly parallel to the y-axis for , while it is almost along the x-axis for . The quasiparticles mainly consist of the d orbital (d orbital) for the former (latter). Since the Cooper pairing is disturbed by the paramagnetic depairing effect when the g-vector is parallel to the magnetic field, the field along the -axis suppresses the Cooper pairs of the d orbital. On the other hand, the Cooper pairs formed by the d orbital are protected by the g-vector nearly perpendicular to the magnetic field. In this way, the quasi-one-dimensional superconducting phase is stabilized by the orbital degree of freedom so as to avoid the paramagnetic depairing effect. This is an intuitive explanation for the large upper critical field shown in Fig. 2.

Finally, we discuss experimental results of the superconducting phase in STO/LAO interfaces. The superconducting transition temperature shows a non-monotonic carrier density dependence [11] and its peak at around cm coincides with the crossover from d-orbital-dominated Fermi surfaces to d/d-orbital-dominated Fermi surfaces [19]. The Rashba spin-orbit coupling seems to have the maximum amplitude in the crossover region [13, 14], consistent with the three-orbital tight-binding model adopted in this study. Interestingly, a large upper critical field, , beyond the Pauli-Clogston-Chandrasekar limit has been reported for a high carrier density cm close to the crossover [13]. The decrease in the normalized upper critical field with increasing carrier density was also observed for cm [13]. These behaviors are consistent with our finding in Fig. 2, although the signature of Lifshitz transition has not been found. This agreement with experimental results indicates that the quasi-one-dimensional superconducting phase is realized in the STO/LAO interfaces with high carrier densities. In contrast to the theoretical proposal for a helical superconducting phase with a finite total momentum of Cooper pairs [15], a large upper critical field is attributed to the entanglement of orbitals and spins in our three-orbital model. Indeed, we confirmed that the finite total momentum of Cooper pairs, namely, the finite in the T-matrix, hardly changes our results. The coexistence of superconductivity and ferromagnetism [7, 8, 9, 10] may also be attributed to the quasi-one-dimensional superconducting phase protected against spin polarization. We would like to stress that such a spin-polarized superconducting state is hardly stabilized in the multiband models, [41, 42] which phenomenologically assume the Rashba spin-orbit coupling and neglect the orbital degree of freedom. Our proposal for the quasi-one-dimensional superconducting phase can be verified by experiments using a tilted magnetic field. For instance, a vortex lattice structure elongated along the [010] axis will be observed in the field slightly tilted from the [100] axis to the [001] axis. As for a quantitative discussion, the crossover between low and high carrier density regions occurs in our model at around , which corresponds to a carrier density of cm. This is in reasonable agreement with experimental carrier density of cm [19], and a discrepancy probably arises from our inexact choice of tight-binding parameters. Note that a large upper critical field has been observed in -doped STO thin films [43]. Although the global inversion symmetry is not broken in this system, surface Rashba spin-orbit couplings play a similar role to the spin-orbit coupling in this study, as demonstrated for locally non-centrosymmetric superconductors [44].

In summary, we studied the superconductivity in the two-dimensional electron systems formed at the STO/LAO interface and STO surface. We analyzed the three-orbital model taking into account orbitals of Ti ions, and found that an unconventional structure of Rashba spin-orbit coupling arises from the orbital degeneracy and protects the quasi-one-dimensional superconducting phase against the paramagnetic depairing effect. The orbital degree of freedom plays an essential role in the response to the magnetic field and leads to a large upper critical field. The peak of the upper critical field as a function of carrier density coincides with the crossover from d-orbital-dominated Fermi surfaces to d/d-orbital-dominated Fermi surfaces. These observations provide a systematic understanding of superconducting properties at the STO/LAO interface.

The authors are grateful to S. Fujimoto and T. Shishido for fruitful discussions. This work was supported by KAKENHI (Grant Nos. 25103711, 24740230, and 23102709), and by a Grant for the Promotion of Niigata University Research Projects.


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