Magnetic and Superconducting Ordering at LaAlO{}_{3}/SrTiO{}_{3} Interfaces

Magnetic and Superconducting Ordering at LaAlO/SrTiO Interfaces

Abstract

We formulate a model for magnetic and superconducting ordering at LaAlO/SrTiO interfaces containing both localized magnetic moments and itinerant electrons. Though these both originate in Ti 3d orbitals, the former may be due to electrons more tightly-bound to the interface while the latter are extended over several layers. Only the latter contribute significantly to metallic conduction and superconductivity. In our model, the interplay between the two types of electrons, which is argued to be ferromagnetic, combined with strong spin-orbit coupling of the itinerant electrons, leads to magnetic ordering. Furthermore, we propose a model for interfacial superconductivity, consisting of random superconducting grains in the bulk STO driven, via coupling to the interface conduction band, towards long-ranged or quasi-long-ranged order. Most interestingly, the magnetic order and strong spin orbit coupling can lead in this manner to unconventional interfacial superconductivity, yielding a possible realization of Majorana physics.

I Introduction

It was recently discovered that, although LaAlO and SrTiO are both insulators, the interface between them is metallicOhtomo and Hwang (2004). Furthermore, the electrons at this interface have shown a variety of remarkable properties, including magnetismBrinkman et al. (2007); Ariando et al. (2011); Li et al. (2011); Bert et al. (2011) and superconductivityReyren et al. (2007). Magnetism and superconductivity often appear in the phase diagram of strongly-correlated materials, where they compete. However, it is very unusual for them to occur simultaneously, which appears to be the case at the LAO/STO interface Li et al. (2011); Bert et al. (2011); Mannhart and Schlom (2010). A basic question, then, is whether the same electrons are exhibiting both superconductivity and magnetism or if, instead, there is a precise sense in which there are two species of electrons – two different electronic bands, for instance – one of which is superconducting and the other of which is magnetic. In the former case, the superconductivity must be exotic, perhaps -wave superconductivity or a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state Fulde and Ferrell (1964); Larkin and Ovchinnikov (1965). This would contradict the conventional wisdom that the interface electrons are simply exhibiting the superconductivity of doped SrTiO, which is presumed to be a phonon-mediated -wave superconductor.

The experiments that reveal interesting magnetic behavior fall into two classes, those which deduce magnetism from transport and those which attempt to measure it more directly. The former include experiments that observe hysteresis in the electrical resistance as a function of magnetic field Brinkman et al. (2007); Ariando et al. (2011), which show that there is a magnetic field-driven first-order phase transition which has a large effect on the resistance. A magnetic transition is the most natural hypothesis. These signatures are found up to temperatures in excess of K. The latter include a torque magnetometry measurement Li et al. (2011) that shows that a field as small as a few milliTesla leads to a large magnetization, approximately per interface unit cell. This implies that the system has ferromagnetic domains which become aligned by even a very small field. This experiment shows that the magnetic moment, which points in the plane, has an onset temperature that is at least as high as K and persists below the superconducting . Finally, scanning SQUID magnetometry Bert et al. (2011) finds micron sized ferromagnetic domains in a paramagnetic background. From their estimates, most of the interfacial electrons which are predicted by polar catastrophe arguments Ohtomo and Hwang (2004) are paramagnetic. An order of magnitude smaller number of electrons are in ferromagnetic regions, and a two orders of magnitude smaller number of electrons are in superconducting regions. There does not appear to be any correlation between the magnetic and superconducting regions (unless there is spatial segregation in the direction), which implies that magnetism in a region does not prevent superconductivity from occurring in that region, counter to conventional wisdom. Taken together, these experiments constitute strong evidence that there are ferromagnetic domains at the LAO/STO interface that strongly affect normal state transport and are also present in the superconducting state.

Since neither LAO nor STO is magnetic, there is clearly a puzzle here: what is the cause of (at least local) ferromagnetism at their interface? It has been suggested that there is a narrow band at the interface Pentcheva and Pickett (2006) which gives rise to itinerant electron ferromagnetism. Alternatively, the magnetism may be due to localized electrons which don’t participate in the metallic (or superconducting) behavior. To make matters even less clear, there is evidence Caviglia et al. (2010); Fête et al. (2012) for strong spin-orbit coupling due to the broken inversion symmetry of the interface Ben Shalom et al. (2010) (Rashba spin-orbit coupling), which would ordinarily be antithetical to a uniform ferromagnetic moment.

An equally vexing problem is how ferromagnetism can coexist with superconductivity Dikin et al. (2011). Even if there were a sense in which different electronic bands were becoming ferromagnetic and superconducting, one would expect the magnetic moments of the former electrons to destroy superconductivity in the latter. One possibility is that the system is in an inhomogeneous superconducting state, e.g. the FFLO state, as suggested in Ref. Michaeli et al., 2012. Or, one could imagine larger scale inhomogeneity, so that the system breaks up into domains, some of which are superconducting while others are ferromagnetic. But the ferromagnetic moment would be anti-correlated with superconductivity in either type of inhomogeneous state. For instance, in the FFLO state, the system forms superconducting stripes, separated by magnetic ones. This disagrees with the experimental finding of Ref. Bert et al., 2011. Therefore, it is natural to consider, instead, a -wave superconducting state, but this begs question of what the superconducting mechanism is. Presumably, -wave superconductivity must be due to electron-electron interactions, rather than the electron-phonon coupling. Thus, superconductivity at the LAO/STO interface is a puzzle. It is generally assumed that it is related to superconductivity in doped STO, but this assumption does not provide any clues to how it can coexist with ferromagnetism.

In this paper, we present a physical picture for magnetism and superconductivity at the LAO/STO interface. Density functional theory calculations show that there are bands at the interface, corresponding to Ti orbitals, with and symmetry respectively Pentcheva and Pickett (2006); Popovic et al. (2008). We hypothesize that the band forms a band of localized electrons which accounts for most of the charge required by the ‘polar catastrophe’ Okamoto and Millis (2004). According to our picture, the bands form metallic bands of itinerant electrons at the interface. Coulomb interactions between localized and itinerant electrons generates a ferromagnetic interaction between them, thereby leading to a ferromagnetic Kondo model – but one in which the itinerant electrons have significant Rashba spin-orbit coupling. By analyzing a spin-orbit-coupled ferromagnetic Kondo lattice, we argue that the localized electrons develop magnetic order. Thus, in our picture, the magnetic moment of the system is due primarily to localized electrons.

Our picture for superconductivity is the following. We suppose that there are droplets of local superconductivity in the STO substrate. If the STO were doped, then these droplets would grow and percolate across the system, giving rise to superconductivity. In the absence of doping, this cannot occur, and the STO substrate is insulating. However, these droplets can interact with the itinerant electrons at the interface. Through the proximity effect, superconducting droplets in the STO substrate can induce a gap in the itinerant electrons at the interface. Thus seeded, the itinerant electrons at the interface can develop long-ranged or quasi-long-ranged superconducting order. However, these itinerant electrons must move in the magnetic background created by the localized electrons. Naively, the magnetism should destroy the superconductivity. However, the strong spin-orbit coupling of the interface electrons allows these two competing phenomena to coexist peacefully Gor’kov and Rashba (2001). Spin-orbit coupling mixes -wave and -wave superconductivity, so that the -wave superconductivity which is present in the droplets in STO can induce a mixture of -wave and -wave superconductivity at the interface. This mixture can tolerate a magnetic moment, unlike pure -wave superconductivity. This is very similar to the situation in proposals of topological insulators in contact with -wave superconductors Fu and Kane (2008, 2009); Cook and Franz (2011), superconductor-semiconductor heterostructures Sau et al. (2010); Alicea (2010), and spin-orbit-coupled quantum wires Lutchyn et al. (2010); Oreg et al. (2010). Therefore, according to our theory, even though superconductivity is -wave in STO, interfacial superconductivity is unconventional, as a result of magnetism and spin-orbit coupling.

An especially exciting consequence of the unconventional nature of the interfacial superconductivity is the possibility of realizing Majorana fermion physics. Indeed, the fabrication of narrow quasi-one dimensional conducting channels on an otherwise insulating LAO/STO interface Cen et al. (2008, 2010); Bi et al. (2010) by ‘writing’ them with an atomic-force microscope (AFM) tip suggests a natural implementation for the proposals of Refs. Lutchyn et al., 2010; Oreg et al., 2010. In the following sections we show that our model naturally generates a ‘helical’ interfacial band structure that can be driven into a topological phase when in proximity to ordinary s-wave superconductivity. As shown by Kitaev [Kitaev, 2001], 1D spinless superconductor with p-wave (or effectively p-wave) pairing supports Majorana zero-energy modes at the ends. Although our droplet model can at best lead to quasi-long ranged superconducting order, signatures of Majorana physics still remain, as shown in Refs. [Fidkowski et al., 2011; Sau et al., 2011; Cheng and Tu, 2011].

In Section II, we set up and justify a model of spin-orbit-coupled itinerant electrons interacting with Kondo spins. In Section III, we give a saddle-point analysis of a large- limit of this model. In Section IV, we solve this model numerically by the density-matrix renormalization group (DMRG) in the one-dimensional limit. In Section V, we analyze the superconducting proximity effect in the presence of a helical wire due to the presence of ferromagnetic Kondo interactions and spin-orbit coupling. We show that such an interfacial superconducting state might support Majorana zero-energy modes. We conclude in Sec. VI with the discussion of our results and proposing a schematic phase diagram for LAO/STO interface.

Ii Spin-Orbit-Coupled Ferromagnetic Kondo Model

We will now argue for a model of spin-orbit coupled electrons interacting with localized spins to describe the LAO/STO interface. The emerging consensus is that the electrons active at the LAO/STO interface come from the bands of Ti orbitals in the STO Ariando et al. (2011); Joshua et al. (2011); Breitschaft et al. (2010); Popovic et al. (2008); Pavlenko et al. (2012); Delugas et al. (2011); Salluzzo et al. (2009); Pentcheva and Pickett (2006); Mannhart et al. (2008). Furthermore, band structure calculations Joshua et al. (2011) and density functional theory Delugas et al. (2011) suggest a picture of successive Ti sub-bands near the interface being occupied as LAO thickness is increased (or gate voltage decreased).

Due to the inversion symmetry breaking at the interface, the lowest sub-band is predicted to be . It contains most of the charge required by the polarization catastrophe, but these electrons are thought to be localized, as seen from the low mobile carrier densities extracted from Hall transport measurements. We thus model these electrons as localized spins. Further indirect evidence for a picture involving localized spins in LAO/STO comes from transport experiments on a related system consisting of pure STO, with the doping effect of the LAO simulated by an polarized gel overlayer Lee et al. (2011). The longitudonal resistance in this system exhibits a Kondo minimum as a function of temperature, indicative of the presence of localized impurity spins.

The lowest extended interface states are thought to have and symmetry. Due to the geometry of their orbitals, their Fermi surfaces are highly anisotropic, with heavy and light carrier directions. Spin-orbit coupling also plays an important role in determining the electronic band-structure, with the authors of Ref. Joshua et al., 2011 arguing for an atomic spin orbit (ASO) effect of about . Furthermore, there is a Rashba contribution arising from the broken inversion symmetry of the interface and gating. Its magnitude is expected to be dependent on the gate voltage and the details of the sample, but in Ref. Caviglia et al., 2010 a value of is obtained through a fit to a weak anti-localization measurement. (Larger values close to are obtained for larger gate voltages; smaller values, for smaller gate voltages.) The Rashba nature of the coupling was deduced from the dependence of the spin relaxation time on the elastic scattering time. The electronic band-structure for the two dimensional interface depends on the precise ratio of Rashba and ASO coupling.

Motivated by recent experiments fabricating one dimensional conducting channels on otherwise insulating LAO/STO interfaces Cen et al. (2010); Bi et al. (2010), as well as by our desire to realize Majorana physics, we find it useful to examine one-dimensional channels at the LAO/STO interface. In a 1D channel, the anisotropy of the orbitals suggest that for a very narrow channel along the direction, transport should primarily be through the states Fête et al. (2012). Therefore, the neglect of the band is justified in this situation. In Ref. Cen et al., 2010; Bi et al., 2010, conducting channels of thickness are constructed. For an effective mass of order the electron mass, such channels correspond to a transverse confining energy of , and hence in principle we need to consider many subbands at the energy scales in which we are interested. However, if narrower conducting wires could be constructed (e.g. , corresponding to ), a one dimensional model would be more readily applicable.

We are thus led to a analyze a minimal model of a single spinful band interacting with a large density of localized spins. The localized spins, which give the dominant contribution to the magnetic moment come from the band. However, their tendency to order is driven primarily by their interaction with electrons. Although the experiment of Ref. Lee et al., 2011 suggests an antiferromagnetic coupling between conduction and impurity spins, we believe that it is more natural, according to Hund’s rule, take the interaction between electrons and electrons to be ferromagnetic. In either case, the Hamiltonian takes the form

(1)

where is the Pauli matrix acting on a spin degree of freedom, corresponds to classical spins due to localized magnetic impurities, and we have suppressed spin indices in the notation. In a lattice version of this Hamiltonian the kinetic term is modeled by a nearest neighbor hopping , whose magnitude is determined by band structure calculations Popovic et al. (2008); Pavlenko et al. (2012) to be about . From magnetoconductance experiments Caviglia et al. (2010); Fête et al. (2012) we take , which translates to a lattice spin orbit coupling or even as large as (see below). The final term is the interaction between the itinerant electrons and localized spins. Although we believe that it is more natural, according to Hund’s rule, to have a ferromagnetic coupling, , we will also, for completeness, consider the case of antiferromagnetic coupling, , with a magnitude of roughly one third of the bandwidth, as obtained in Ref. Lee et al., 2011.

Figure 1: (a) Sample one dimensional electronic band-structure. We expect a magnetic instability at ordering wave-vector , where a gap can open in the spectrum at the crossing point between the two bands. When this occurs, the band structure takes the form in (b)

Motivated by recent SQUID Bert et al. (2011) and torque magnetometry Li et al. (2011) experiments, which find at least local ferromagnetism, we are interested in examining the magnetic instabilities of the Hamiltonian (III). The one dimensional spin-orbit coupled electronic band-structure in figure 1 shows that it is natural to expect ordering at wave-vector , since it is there that a gap can be opened and the electronic energy lowered. Of course, ordering could also occur at the wavevectors that connect various pairs of Fermi points. In the next section we will study a large- limit of this model and find a tendency toward in-plane ferromagnetism, consistent with [Bert et al., 2011; Li et al., 2011]. In the following section we will reach a similar conclusion with DMRG.

Iii Large- Analysis of the Model

We now analyze magnetic instabilities of a generalization of the Hamiltonian (III) to species of fermions , with :

(2)

where the matrix is given by

(3)

We now integrate out fermions in Eq.(III):

(4)
(5)

where is the Berry phase term for the localized spins. In the large- limit, the functional integral

(6)

is equal to its saddle-point value. The saddle-point equations are given by:

(7)

The term is and is much smaller than the second term which is , so it can be neglected. Since the temperature of interest is much lower than the energy scales associated with the couplings in (III), we can effectively set it to zero and convert Matsubara sum to an integral. Thus, we arrive at the following mean-field equations:

(8)

where are the two eigenvalues of :

(9)
(10)

The explicit evaluation of derivatives yields

(11)

One can notice that we only obtain a non-zero contribution if and have opposing signs, so that the saddle point equation reduces to

(12)

Here is the unit step function ( for ) and

(13)

The -dependent contribution to the energy of a particular background spin configuration at one loop reads

(14)

To diagonalize, we take nearest neighbor hopping with amplitude normalized to , resulting in a kinetic term

(15)

For convenience we also absorb into : , and drop it from the following discussion. We then compute the one loop energy as a function of , and , and determine the propensity for magnetic ordering in various directions. From (9) we see that the and directions become equivalent at one loop, so it suffices to set and work with nonzero and . Symmetry considerations show that is an extremum of the one loop energy. In fact, we empirically see that it is a global maximum, and that the energy is unbounded from below, becoming more negative with increasing . This makes sense since the saddle-point Hamiltonian treats as a classical field with no dynamics, and larger leads to lower electronic energy. Physically, we expect that ultimately saturates. For the purposes of the saddle-point approximation we pick an appropriate value for and evaluate the energy difference between ordering in the and directions:

(16)

In figure 2 we plot at a specific value of , and note that it is everywhere negative. Although the percentage difference in energies is small, it is robustly negative over a very large range of physical values of , and . We conclude that in saddle-point approximation the spins prefer to develop magnetic order in the (in-plane) direction.

Figure 2: , where is defined in (III), plotted at as a function of chemical potential and spin-orbit coupling for a one dimensional conducting wire aligned in the direction sitting on the dimensional plane. We note that this quantity is everywhere negative, implying a propesity towards magnetic ordering in plane, perpendicular to the wire i.e. along the direction. Other values of give similar results.

We can perform a similar analysis for a two dimensional version of (III), obtaining a similar result: the spins prefer to order in-plane in the large- limit.

One very interesting part of our analysis is that we find ferromagnetic order developing at weak coupling. The underlying reason for this is that the band structure in the presence of Rashba spin-orbit coupling has a crossing at ; a small magnetic moment opens a gap there. This occurs even at arbitrarily weak coupling, if the chemical potential passes through this crossing. This is very similar to the case of other Fermi surface instabilities, such as the BCS instability or density-wave ordering for nested Fermi surfaces. If the chemical potential does not pass through the crossing, then a small minimum coupling must be exceeded, as in the case of small detuning away from a nested Fermi surface. This scenario stands in stark contrast to the usual case of the Stoner instability: ordinarily, ferromagnetism does not open a gap at the Fermi surface, and only occurs when the coupling exceeds the inverse of the density of states.

Figure 3: (color online) Ground state phase diagram of the model Hamiltonian in Eq.(17) at filling , determined by accurate DMRG simulations with system size up to sites. Changing coupling parameters and , three different phases are found, including the fully polarized phase, the partially polarized phase, as well as the easy-plane partially polarized phase. Here and .

Iv DMRG Solution of the 1D Limit

We now consider a 1D lattice Hamiltonian describing itinerant electrons coupled to localized impurity spins:

(17)

Here () is the electron creation (annihilation) operator with spin index at site ; is the spin operator, representing the localized magnetic moment; denotes the nearest neighbor (NN) tunneling matrix element (henceforth we set ) and is the lattice constant. is the Kondo coupling between localized magnetic moments and itinerant electrons, is the ferromagnetic exchange coupling between NN localized magnetic moments, and is the spin-orbit coupling. is the on-site Hubbard repulsion for the itinerant electrons, and we have also included Zeeman terms for both the localized spin and itinerant electron spin along the and directions. In the Hamiltonian (17), impurity spins have their own dynamics and the ground-state of the is determined by taking into account both electron and spin degrees of freedom on equal footing. If we use mean-field approximation for impurity spins, i.e. , and neglect electron-electron interaction, we obtain the Hamiltonian (III) considered in the previous section. As we show below, our conclusions regarding preferred magnetization persist in the strongly interacting limit .

Figure 4: (color online) Relative polarization as functions of and , along direction () and direction (), for the system in Eq.(17) at filling and system size sites. Relative polarization for in (a), in (b), in (c), and in (d). Here , , and is the saturated magnetization. Note that the polarization is induced by applying a small magnetic field along direction or along direction separately.

In our picture, most of the electrons required by the polarization catastrophe argument become localized spins, with a much lower density of itinerant electrons. Therefore, we focus on the low density case, taking for the sake of concreteness in most of our calculations. The coupling parameters are taken to have values and . (However, our results are not very sensitive to the value of .) We then map out the phase diagram as a function of and . By the arguments given in Sec. II, we expect that and . However, given that there is some uncertainty in these parameters, it behooves us at this stage to see how the physics of our Hamiltonian changes as we vary them. We have included small magnetic fields in the - and directions. These fields are necessary to break time-reversal symmetry and rotational symmetry about the -axis; otherwise, we would necessarily find . In an infinite system, but in the absence of these symmetry-breaking fields, the system can spontaneously choose to order along or or it could spontaneously pick a direction in the plane (if it orders at all). Employing the unbiased density matrix renormalization groupWhite (1992) method, we determine the ground-state phase diagram of the system (17).

We find the following phase diagram, depicted in Fig.3. For large and negative (i.e. a ferromagnetic coupling between localized and itinerant spins) and large, the system is partially-polarized and the moment points in the -direction (i.e. in plane, but perpendicular to the nanowire, and corresponding to the light blue region in the upper left corner of Fig.3). For but lying below the blue phase boundary in Fig. 3, we have a fully-polarized phase in which the spins can point equally-well in any direction (the grey region near the middle of Fig.3). For , there is a partially-polarized phase at small or small (narrow purple region). In this region, the spins can point can point equally-well in any direction. Finally, if is positive and either or is large, then the system is partially-polarized and the spins point in the -direction (light blue region in upper right of Fig. 3).

Figure 5: (color online) The Fermi surface of our 1D model at three points in the phase diagram. (a) The occupation number , which shows the region of momentum space occupied by the filled Fermi sea. (b) The equal-time density-density correlation function, which has singularities at .

We now examine the magnetically-ordered state in more detail. The occupation number clearly shows a filled Fermi sea with approximately one electron in each occupied state, with , as illustrated in Fig. 5a, at the parameter values listed in Fig. 5b. The Fermi wavevector is consistent with , in agreement with Luttinger’s theorem. The structure factor shown in in Fig. 5b has cusps at , from which we can more precisely extract the Fermi wavevector . The occupation numbers and Luttinger volume are consistent with the chemical potential depicted in Figure 1. There are two Fermi points and there is a single state at each Fermi point because the spin is locked to the momentum. Such a 1D electron gas is often called a “helical wire”. (Note that our system is not simply fully spin-polarized; we have checked that, at these parameter values, neither nor is ever equal to one, so the spins are not polarized in the -direction; they are also not equal to each other, so the spins are not polarized in the - or - directions.) Therefore, an odd number of bands (in fact, just one) crosses the Fermi surface; as we discuss in Section V, this means that the system is primed for the development of topological superconductivity.

V Proximity-Induced Superconductivity

We propose the following picture for superconductivity at the LAO/STO interface. Our starting point is superconductivity in STO, which occurs when insulating STO is doped. We assume that insulating STO has small islands or droplets of local superconductivity, which are too far apart and too weakly-coupled to develop long-ranged superconducting order. We suppose that these droplets are caused by unintentional local defects in STO. When STO is doped, the islands of local superconductivity grow in size and become more strongly coupled, until long-ranged superconducting order sets in. However, the presence of an interface with LAO changes matters. Itinerant electrons at the interface can mediate a coupling between superconducting droplets in STO that are close to the interface. As we will show, this can enable superconductivity to develop even when the droplets are too weakly-coupled to percolate across STO on their own.

We show how this can occur with a calculation in a simplified model. We suppose that there are some superconducting droplets in STO that are near the LAO/STO interface. In each droplet, a single-particle gap is assumed to be well-developed, but the interactions between the droplets are assumed to be too weak for superconducting order to set in. A 1D channel at the interface couples to a subset of these droplets, which form a linear array. The 1D channel induces interactions between the droplets, so that the linear array can be modeled as a 1D spin-gapped electron system, which we assume to be just slightly on the disordered side of the Kosterlitz-Thouless transition. (Since doped STO superconducts, this is a reasonable approximation.) The coupling between this system and a 1D channel at the LAO/STO interface can nudge the system into the basin of attraction of the quasi-long-range ordered superconducting phase on the other side of the Kosterlitz-Thouless transition. We thereby see how proximity to a metallic interface can stabilize long-ranged or quasi-long-ranged superconducting order.

Once superconductivity is established in STO, it is induced at the interface by the proximity effect. This mechanism of establishing superconducting order works for generic conducting 1D channels. However, the interesting scenario for us occurs when, due to Rashba spin-orbit coupling and ferromagnetism, the 1D channel realizes the helical band structure in Fig 1, and the induced superconductivity is topological. Hence, we begin with a bosonized helical 1D channel at the interface, coupled to an array of isolated, i.e. effectively zero dimensional, droplets:

(18)

Here, is the Fermi velocity in the 1D channel and is its Luttinger parameter. The 1D channel is assumed to have repulsive interactions, so . The factor of in the exponent in the third line has been inserted for later convenience so that it agrees with the convention for the charge boson of a spin-gapped electron system (see, for instance, Ref. Fidkowski et al., 2011). The droplets are assumed to have an average spacing , an average Josephson coupling to the 1D channel, and an average charging energy . We will neglect random variations in the spacing between droplets, in the Josephson couplings, and in the charging energies, and simply set , , and . Random variations in these parameters are certainly important in the physical system but are an unnecessary complication for this calculation. Note that if , then the droplets are completely decoupled from each other and there is no quasi-long-ranged superconducting order.

We now integrate out fluctuations of at length scales shorter than . This generates a coupling between droplets. At length scales much longer than , we can take the continuum limit for the array of droplets, thereby leading to the following effective action:

(19)

The droplets are now effectively described by a 1D wire with a spin gap; is the charge boson for such a wire. Since we assume that there is, initially, a weak interaction between the droplets and the 1D channel, we assume that . In other words, although the wire has a spin gap, we do not assume that it can superconduct without further mediation on the part of the 1D channel at the interface. Indeed, the parameters and can be related to the effective superfluid stiffness and compressibility of the array: and with being the cross-sectional area. We assume here that the superfluid stiffness is such that .

If the two velocities were equal, , one could analyze the model by forming the combinations . In terms of new variables the action reads

(20)

Here, we have rescaled the time coordinate by and have introduced the dimensionless parameter . We can now see, at a heuristic level, how the coupling between the droplets and the 1D channel can stabilize quasi-long-ranged superconducting order. Let us suppose, for a moment, that the coupling is relevant. Then is pinned, and we can ignore its fluctuations. Then we are left with , which exhibits algebraically-decaying superconducting order. This order is stable if weak impurity-backscattering or, equivalently, vortex tunneling is irrelevant. Because the wire is helical, we can only tunnel vortices Fidkowski et al. (2011), i.e. . Since is pinned, this means that must also wind . Consequently, in such a process, . The operator that accomplishes this is , where . This operator is irrelevant if , and when this inequality is satisfied, the system exhibits quasi-long-ranged superconducting order. Note that this can be satisfied even if and . So two systems, neither of which could sustain superconductivity on their own, can develop superconductivity when in proximity to each other. The key to this is the topological nature of the superconductivity: since only vortices can tunnel through a helical wire, the stability condition is less strict than for an ordinary superconductor Giamarchi (2004).

To support the aforementioned scenario, we need to show that that Cooper-pair tunneling term is relevant. The corresponding RG equation for is:

(21)

where . If , will grow from the initial small value to at which point gets pinned. Given that also flows under RG, we need to compute its flow equation and complete the system of RG equations for this model. To do that we rewrite Eq. (20) in the following form:

(22)

where, initially, , . The reason that we have introduced three couplings when there are, seemingly, only two couplings and is that the RG flow for this theory will carry the system away from the initial point . A real-space RG calculation yields the following equations for :

(23)

The coupling can be expressed in terms of , and (or equivalently in terms of and the initial parameters and since and do not flow):

(24)

Since monotonically depends , the growth of under the RG flow results in

(25)

Thus, the -coupling becomes more and more relevant, and eventually pins as assumed above.

We now consider more general case of unequal velocities . Proceeding as before, we find that Eq. (20) can be written as

(26)

where the coupling constants are defined as

(27)
(28)
(29)
(30)

The parameters and are initially equal to and , respectively, but they flow under RG as explained above.

We now sketch the real-space RG procedure. We start out by integrating out short-distance modes, but allowing arbitrarily short times. Thus, we have an effective action in which there are modes with and , where is the momentum cutoff. Following a real-space RG approach Giamarchi (2004), we integrate shells while keeping time integrals unconstrained and eventually rescale , . Here . Our RG procedure involves calculating the correlation function . To do it safely one has to normal order the exponent:

(31)

where , and the average is computed with respect to the bare action () defined in Eq. (26). Initially, when and , the correlation function in Eq. 31 can be easily calculated

(32)

However, in general it is a complicated function of the coupling constants (27) as well as and . One can show that at the tree-level the RG equation for becomes

(33)

where, again, .

We now compute the RG equations for this model at one-loop level. Using Eqs.(31) and (32), we find

(34)
(35)

where and the dimensionless function

(36)

As follows from Eqs.(34), the RG equations for and are given by

(37)
(38)

Thus, according to these Kosterlitz-Thouless-type RG equations, Eqs. (33),(37) and 38, we see that grows. The parameter has a complicated dependence on and which follows from Eq. (32). However, at small initial velocity mismatch , one finds

(39)

One can see that the -correction is quickly decaying with and thus do not change qualitatively our results obtained for the case, cf. with Eq. (24). In general, we find that is a monotonically increasing function of , see Fig.6. Thus, the growth of implies the growth of . Therefore, once relevant, will grow to strong coupling and pin .

Figure 6: Dependence of on the flow parameters and . Here we used , and . The function is a monotonically increasing function of .

In this case, drops out from Eq. (26) and the effective action now reads:

(40)

Following the argument that we used for , we observe that quasi-long-ranged order is stable so long as flux vortex tunneling is irrelevant, i.e. when is irrelevant. This occurs when . Note that this can be satisfied even if and . As in the equal velocity case, two systems, neither of which could sustain superconductivity on their own, can develop superconduct when in proximity as a result of the helical nature of one of the systems.

We thereby arrive at the model of Ref. Fidkowski et al., 2011: a 1D channel that is proximity-coupled to a quasi-long-range-order superconducting wire. As shown there, such a wire supports Majorana zero modes. We also expect our results discussed in this section to apply to multichannel nanowires with an odd number of occupied subbands coupled to superconducting droplets, see, e.g., Refs. Wimmer et al., 2010; Potter and Lee, 2010; Lutchyn et al., 2011.

Vi Discussion

In this paper, we have adopted the point of view that SrTiO has the seeds of both magnetism and superconductivity. However, these local tendencies only come to fruition when brought into contact with a metallic layer or 1D channel. We have focussed on the latter case, for reasons of tractability as well as potential relevance to the experiments of Refs. Cen et al., 2010; Bi et al., 2010, but we believe that our general mechanism works in 2D as well. We have shown that local moments in SrTiO that are near the LAO/STO interface can order ferromagnetically, as a result of their interaction with mobile electrons at the interface. We have also shown that droplets of local superconductivity in STO – which would interact too weakly to develop superconducting order if left to their own devices – can develop superconducting order as a result of their interaction with mobile electrons at the interface. Finally, we have noted that the interface electrons can form a topological superconducting state as a result of their proximity to ferromagnetic and superconducting order.

We have shown that our model leads to ferromagnetism by two different calculations: a large- calculation and a DMRG calculation. Both calculations find a ferromagnetic state with spins pointing in the plane, i.e. along -axis with being the direction along the wire. (In the DMRG calculation, the polarization may be either partial or full, depending on the strength of the spin-orbit coupling.) Interestingly, our large- calculation finds a ferromagnetic state even at weak coupling, which is a feature of the band structure in the presence of Rashba spin-orbit coupling. A 1D wire with sufficiently strong spin-orbit coupling and Zeeman field will form a helical wire. Our calculations – both large- and DMRG – show that our model gives rise to a helical wire. Remarkably, recent transport measurements can be interpreted as evidence that 1D channels at the LAO/STO interface are helical wires Cheng et al. (2012).

In a helical wire, it is possible for -wave superconductivity to coexist with a magnetic moment. As a result our model allows for a proximity coupling between -wave superconducting droplets in STO and ferromagnetic electrons at the LAO/STO interface. We have analyzed our model for superconductivity by mapping it to the theory of a single 1D boson in the vicinity of the Kosterlitz-Thouless transition. We find that such a model could be on the disordered side of the Kosterlitz-Thouless transition for very weak coupling between the droplets and a 1D channel at the interface but it could be on the ordered side of the transition if the coupling is sufficiently strong.

Our results on magnetism and superconductivity imply that the superconducting state of a 1D channel at the LAO/STO interface is in a topological superconducting state. This actually stabilizes the system against quantum phase slips: phase slips are forbidden, and only phase slips, which are less relevant in the RG sense, could disrupt the superconductivity. Furthermore, this topological superconducting state supports Majorana fermion zero modes, whose presence would lead to a -periodic ac Josephson effect Kitaev (2001); Lutchyn et al. (2010).

The main goal of our work has been to show why magnetism occurs, why superconductivity occurs, and why they can coexist. However, magnetism is probably not found at all carrier concentrations above the metal-insulator transition Thiel et al. (2006). Neither is superconductivity. A plausible schematic phase diagram is given in Fig. 7, based on the phase diagram in Refs. Moetakef et al., 2012; Levy, 2012. Therefore, it is also important to understand when and why they do not occur. If the magnetic order is too strong, so that the spins are fully polarized, then our mechanism does not work. This may explain why superconductivity is suppressed at large carrier concentration. However, a more detailed understanding of the phase diagram is definitely an important target for further investigation.

Figure 7: A schematic phase diagram for the LAO/STO interface as a function of carrier concentration.

We note that our calculations rely heavily on simplifying features of one dimensional systems – the applicability of DMRG calculations to the magnetic ordering of the system and the applicability of bosonization to the superconducting ordering of the system. It would be interesting to give a fully two-dimensional analysis of a model similar to ours. Furthermore, it would be useful to investigate other related materials exhibiting the same phenomena. In particular, recent work on epitaxially grown GdTi-SrTi interfaces Moetakef et al. (2012) indicates that ferromagnetism and superconductivity can also coexist in such systems. Being much cleaner than LAO/STO from a materials point of view, these interfaces might provide an attractive environment for the investigation of the ideas proposed in this paper.

Acknowledgements.
We would like to thank Guanglei Cheng, Harold Hwang, Jeremy Levy, Susanne Stemmer, and Joshua Veazey for discussions. H.C.J. is partially supported by the the KITP NSF grant PHY05-51164 and the NSF MRSEC Program under Award No. DMR 1121053. C.N. is supported by the DARPA QuEST program and the AFOSR under grant FA9550-10-1-0524. We thank the Aspen Center for Physics for hospitality and support under NSF grant #1066293.

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