Interface superconductivity in LaAlO-SrTiO heterostructures
The interface superconductivity in LaAlO-SrTiO heterostructures reveals a non-monotonic behavior of the critical temperature as a function of the two-dimensional density of charge carriers. We develop a theoretical description of interface superconductivity in strongly polar heterostructures, based on the dielectric function formalism. The density dependence of the critical temperature is calculated accounting for all phonon branches including different types of optical (interface and half-space) and acoustic phonons. The LO- and acoustic-phonon-mediated electron-electron interaction is shown to be the dominating mechanism governing the superconducting phase transition in the heterostructure.
pacs:71.10.Ay, 71.38.Fp, 74.20.-z, 72.10.Bg
Recent progress in the development of multilayer structures based on complex oxides (1), provides the means to generate a two-dimensional electron gas (2DEG) at the oxide interfaces. The discovery of superconductivity at the LaAlO-SrTiO interface (2); (5); (3); (6); (4) has stimulated increasing interest in the experimental and theoretical study of these structures.
Because strontium titanate is a highly polar crystal, the electron-phonon mechanism of superconductivity seems to be the most promising for the explanation of the experimental data on superconductivity in the LaAlO-SrTiO heterostructures. The Migdal – Eliashberg theory of superconductivity (7); (8), as well the BCS theory, is valid when the phonon frequencies are much smaller than the electron Fermi energy. This is not the case for polar crystals with sufficiently high optical-phonon frequencies, like strontium titanate. To tackle such systems, non-adiabatic extensions of the theory of superconductivity have been developed. Pietronero et al. (9); (10); (11) generalized the Eliashberg equations to include non-adiabatic corrections beyond Migdal’s theorem. The method developed by Kirzhnits et al. (12) (see also Refs. (13); (14); (15)) is focused on the superconductivity caused by the Fröhlich electron-phonon interaction with polar optical phonons. It uses the total dielectric function of a polar crystal. The Pietronero and Kirzhnits approaches are complementary: the former is non-perturbative with respect to the coupling strength and perturbative with respect to the Debye energy, while the latter is weak-coupling but non-perturbative with respect to the optical-phonon energies.
Strontium titanate is a unique example of a polar medium in which superconductivity has been detected at very low carrier densities, so that the optical-phonon energies can be larger than the Fermi energy. Moreover, as found in Refs. (16); (17), the electron – LO-phonon coupling constant in SrTiO is not very large. Therefore for the investigation of superconductivity in a LaAlO-SrTiO heterostructure, the Kirzhnits method seems to be appropriate. Here, we apply the Kirzhnits method for a multilayer structure with several polar layers.
Ii Superconductivity in a multilayer polar structure
We consider the quasi 2D electron-phonon system described by the Hamiltonian:
Here are the phonon second quantization operators, is the 2D in-plane phonon wave vector, the index labels phonon branches in the LAO-STO structure, is the phonon frequency. Furthermore, and are, respectively, the creation and annihilation operators for electrons with spin , in-plane wave vector , band index and size-quantization quantum number . The energy corresponding to the single-particle state is . is the lateral size of the system, and is the matrix element of the electron-electron interaction potential,
The electron-phonon interaction amplitudes can be written as:
where is the matrix element of the amplitude that we will specify below. The index labels the phonon branches of the multilayer structure. In the calculations, we assume that in the LaAlO-SrTiO heterostructure under consideration, the electron gas is confined to a very thin layer 2 nm. Consequently, only the lowest energy subband () is filled, and transitions to higher subbands can be neglected.
The electron-electron interaction potential , the equations for the eigenfrequencies of the interface modes, and the amplitudes of the electron-phonon interaction in a multilayer structure are derived within the dielectric continuum approach accounting for the electrostatic boundary conditions in a similar way as in Refs. (19); (20). We use Feynman units: , , , where is an effective LO-phonon frequency (taken equal to the highest of the LO-phonon frequencies of SrTiO). The potential is
expressed using the dimensionless Coulomb coupling constant
The coefficient depends on the dielectric constants of the media constituting the heterostructure. For the system without an electrode at the LaAlO-vacuum interface, is
where is the width of the LaAlO layer. The index in the dielectric constant labels the layers: for the SrTiO substrate, for the LaAlO layer, and for the vacuum. For the system with an electrode, is obtained from (6) in the limit .
We take into account the following phonon branches: (1) the interface optical phonons, (2) the half-space optical phonons, and (3) the acoustic phonons. For the interface optical phonons, the eigenfrequencies are found from the equation
with the functions
The amplitudes of the electron-phonon interaction with these interface phonon modes are
where is the Heaviside step function, and the factor is
For the structure with an electrode, we set in the above formulae.
Because the 2DEG layer is positioned at the SrTiO side of the interface, the half-space phonons of strontium titanate can contribute to superconductivity. The frequencies of the half-space phonons are the same as for the bulk LO phonons. The amplitudes of the electron-phonon interaction for the half-space phonons differ from those for the bulk LO phonons only by the boundary condition of zero amplitude at the interface. Although the half-space phonons turn out to give a relatively very small contribution to the resulting phonon-mediated interaction potential, we take them into account for completeness. For the acoustic-phonon contribution, we use the frequencies and interaction amplitudes for the deformation potential from Ref. (21):
with the dimensionless coupling constant
where is the mass density of strontium titanate, is the deformation potential, and is the sound velocity.
with the kernel function
where , is the effective mass for the motion along the surface, and is the total effective electron-electron interaction potential. The energy is counted from the Fermi energy . The kernel function (17) is essentially energy-nonlocal, as distinct from the BCS and Migdal – Eliashberg approaches, since it is provided by a retarded effective electron-electron interaction , through the plasmon-phonon excitations. Consequently, the frequency dependence of the gap can differ from that within the BCS or Migdal-Eiashberg pictures.
The gap equation (16) with the effective interaction potential described above allows for the determination of the gap function and the critical temperature in a LaAlO-SrTiO heterostructure. In the low-temperature range, when the thermal energy is much lower than the Fermi energy of the charge carriers , the approximation method proposed by Zubarev (18) allows to find the normalized gap function as a numeric solution of the Fredholm equation,
The critical temperature is given by the expression,
where is the Euler constant, and the parameter is determined explicitly through the normalized gap parameter
In the present treatment, the effective electron-electron interaction includes contributions from both optical and acoustic phonons. Since the Kirzhnits theory assumes the weak-coupling regime, we suggest that the effective phonon-mediated interaction due to the acoustic phonons can be taken into account in an additive way with respect to the combined contribution of Coulomb interaction and optical phonons. The total effective interaction can be thus approximated by the sum:
where is the effective interaction described in terms of the total dielectric function, and is the effective interaction due to the acoustic phonons.
The effective potential in a quasi 2D system is determined following Ref. (15). Within RPA (22), the relation between the effective potential taking into account dynamic screening, , and the effective potential without screening, , is
where is the polarization function of a free 2DEG. Here, we use the RPA-polarization function (22); (13); (23). The non-screened potential is a sum of a Coulomb contribution and a contribution from the phonon mediated interaction between the two electrons:
where the Coulomb contribution is given by the expression (2), and the effective optical phonon mediated interaction is approximated by the Bardeen – Pines form (24); (25). For the acoustic-phonon contribution to the effective potential, we also apply the Bardeen-Pines approximation, as in Ref. (26):
There are indications from experiment (2); (5); (3); (6) that the superconducting phase transition in a LaAlO-SrTiO heterostructure is governed by the Berezinskii – Kosterlitz – Thouless (BKT) mechanism (27); (28); (29). It is shown in Refs. (3); (6) that the temperature dependence of the resistance just above is specific for a BKT phase transition, corresponding to the 2D nature of the superconducting system.
The critical temperature of the BKT phase transition is determined by the equation (28) which includes the pair superfluid density :
The superfluid density monotonously decreases with increasing temperature, and turns to zero at . Therefore, the critical temperature must be necessarily lower than . In the case when the BKT transition is present in the LaAlO-SrTiO heterostructure, can be interpreted as the pairing temperature at which the preformed pairs appear. In the LaAlO-SrTiO heterostructures, the superfluid density extracted from the BKT equation (25) is several orders of magnitude lower than the actual electron density: . This inequality can be satisfied only when the gap parameter is very small compared to its value at , and, consequently, when . Therefore, as already concluded in Ref. (2), and are extremely close to each other in the LaAlO-SrTiO heterostructures.
Iii Results and discussion
For the numerical calculations, we use the set of material parameters already used in earlier works (17); (26). The dielectric constants for SrTiO are and (calculated using the Lyddane-Sachs-Teller relation for the LO- and TO- phonon frequencies and the ratio ). The effective mass for the present calculation has been taken (17) (where is the electron mass in vacuum). The dielectric constants for LaAlO are used from Ref. (30): and . The only material parameter which is not yet well-determined, is the acoustic deformation potential in strontium titanate. It should be noted that the deformation potential responsible for the interaction of an electron with the acoustic phonons is the “absolute” rather than “relative” deformation potential (31); (32); (33). In the literature, we can find several different suggestions on the values of the deformation potential in strontium titanate. Koonce et al. (34) applied the value eV to fit the experimental data on in bulk strontium titanate. In Ref. (35) the deformation potential is estimated to be eV on the basis of the value of the Fermi energy of the electrons. In Ref. (36) the value eV is calculated on the basis of first principles density functional theory, that seems to be more reliable than two other values, because the many-valley band model of Ref. (34) is not confirmed by later studies, and the deformation potential of Ref. (35) is a rough estimation using the Fermi energy of the electrons. As we discuss below, the results from our theory compare favorably with values of (35); (36), but are incompatible with the large value used in (34).
Here we have calculated in the LaAlO-SrTiO heterostructure using several values of the deformation potential: , and . They seem to be physically reasonable, because they lie in the same range as the values used in Refs. (35); (36). For the comparison of the calculated critical temperatures with the known experimental data, we use in the numeric calculations the model of the LaAlO-SrTiO heterostructure accounting for the presence of an electrode at the oxide layer.
In Fig. 1, the kernel function is plotted for the set of parameters indicated above, choosing the deformation potential eV suggested in Ref. (36). This kernel function is qualitatively similar to the kernel function for a 2D electron gas from Ref. (14). There exists a distinction between the kernel functions for the 2D and 3D systems within the Kirzhnits – Takada method: for a 3D electron gas, the kernel function tends to zero when achieving a local maximum in the interval . On the contrary, for a 2D electron gas, is a monotonically decreasing function of and in the range of negative frequencies.
As explained in Refs. (13); (14); (15), superconductivity in an electron-phonon system can exist despite the fact that the kernel function is positive for all frequencies. The kernel for energies larger than is dominated by the Coulomb repulsion between two electrons whose spatial distance is small, while the behavior of the kernel near the Fermi surface is due to both the Coulomb interaction and the attraction mediated by the plasmon-phonon excitations, between two electrons whose distance is rather large. Consequently, when, for example, is much larger than , the two electrons can avoid the region of the Coulomb repulsion and form a Cooper pair. We can see that, although is not exactly equal to zero, achieves its minimum at , facilitating pairing.
The measured critical temperatures are taken from several sources (2); (3); (4). The experimental work by N. Reyren et al. (2) contains only two points: K for a 2D density of the electrons cm, and K for cm. The paper (3) represents the critical temperature as a function of the gate voltage, and the dependence of the modulation of the electron density on the gate voltage. The total electron density is related to the modulation as , where cm, according to Ref. (3). Using the experimental data for and represented in these figures, we obtain the dependence for the experiment (3). We also include recent experimental results on the superconductivity in the LaAlO-SrTiO heterostructure (4).
In Fig. 2 (a), the critical temperatures as a function of the 2D carrier density calculated in the present work are compared to the experimental data for discussed above. It is worth noting that there exists a substantial difference between experimental values of obtained in different experiments. However, they all are of the same order of magnitude and lie in the same range of the carrier densities.
The observed differences of the experimental results on the critical temperatures in the LaAlO-SrTiO heterostructures can be explained as follows. In different experiments (2); (3); (4), the LaAlO-SrTiO heterostructures have been grown independently. Therefore those heterostructures can at least slightly differ from each other. The thermal energy is extremely small compared to the characteristic energies involved in the superconducting phase transition: the Fermi energy of the electrons and the LO-phonon energies (which both are of order meV). Under these conditions, the critical temperatures can be very sensitive to relatively small difference of the internal properties of the fabricated heterostructures. Additional factors (e. g., disorder, local phonons, defects, etc.) can substantially influence . Moreover, also even the critical temperatures for bulk strontium titanate measured in different experiments (34); (37) differ substantially from each other. Consequently, the relatively large variation of the experimental results on the critical temperature in LaAlO-SrTiO heterostructure is not surprising. Despite that uncertainty, the measured critical temperatures in different experiments are of the same order.
We can see that for each , the curve is close to only few data points. However (i) experimental data are obtained with a substantial numeric inaccuracy, (ii) experimental data from different sources do not agree even with each other. It should be noted also that, since the thermal energy corresponding to the critical temperature in the LAO-STO structure is very small with respect to other energies participating in the superconductivity (the optical-phonon energies and the Fermi energy), even a small uncertainty of these parameters can then lead to a significant change of the critical temperature. Our calculation, performed without fit using parameter values known from literature, yields the critical temperatures within the same range as in the experiments. We can therefore conclude that, taking into account the uncertainty of the experimental results on the critical temperatures, the suggested theoretical explanation of the superconducting phase transition in the LaAlO-SrTiO heterostructures leads to tentative agreement with experiment.
We can see different regimes of as a function of density in Fig. 2. First, the critical temperature exhibits almost linear dependence at small up to . After reaching the maximum, as seen from Fig. 2 (b), the critical temperature falls down at high densities as with a positive constant depending on the parameters of the system. This density dependence of for low and high densities can be explained as follows.
In the limit of low densities, the effective interaction potential given by (22) tends to the non-screened effective interaction potential that does not depend on the density. Also the contribution to the total effective interaction due to the acoustic phonons is density-independent. Therefore at low densities the parameters tends to a constant depending on the material parameters of the system. According to (19), the critical temperature in the low-density limit is proportional to the carrier density.
where is the Debye frequency, is the density of states at the Fermi energy, and is the model BCS matrix element. The BCS theory describes the adiabatic regime when , and pairing occurs for the electrons whose energies lie in the layer of width near the Fermi energy. As a result, within the BCS picture. On the contrary, in strontium titanate and in the LaAlO-SrTiO heterostructure at low densities the anti-adiabatic regime is realized: , where is an optical-phonon frequency. In the anti-adiabatic regime, all electrons participate in the superconductivity. Therefore the factor in the adiabatic regime corresponds to the factor in the anti-adiabatic regime.
Because in the anti-adiabatic regime all electrons contribute to superconductivity, the parameter hardly can be interpreted as . In a non-adiabatic regime must be, in general, a functional of the density of states for all energies . However, the density of states for a 2D system and for a sufficiently low energy (where the band nonparabolicity is relatively small) is
so that (and hence also ) does not depend on the carrier concentration at low concentrations. Thus the aforesaid qualitative physical estimation leads to the the low-density behavior , in agreement with the result obtained in the present work.
In the opposite regime of high carrier densities, the plasma frequency can exceed both the Fermi energy and the optical phonon energies. In this regime, the plasmon mechanism of superconductivity (14) must dominate. According to Ref. (14), the critical temperature for an electron gas in 2D with the effective mass and the dielectric constant due to the plasmon mechanism can be modeled by an analytic expression,
with the averages
and the function
Here, is the Euler beta function, is the conduction band degeneracy, is the Fermi momentum, and is the Thomas-Fermi wave vector. For a 2D electron gas, does not depend on the carrier density. Here, the factor is equal to 1 because the conduction band in SrTiO is split due to the spin-orbit interaction (40).
where is given by:
The upper bound for the density when is determined by the equation
The parameter is proportional to , and for sufficiently high densities. Therefore in the high-density range, but for densities smaller than that determined by (33), the model critical temperature (28) due to the plasmon mechanism behaves approximately as
Since , the estimation (34) following Ref. (14) is in agreement with the critical temperature obtained in the present work, as seen from Fig. 2 (b). In the figure, the ratio is plotted in the logarithmic scale as a function of , focusing at the high-density range (larger than in the experiments (2); (3); (4)). We can see that at relatively small acoustic deformation potential , the dependence as a function of is almost linear for high densities. For larger , the acoustic-phonon mechanism stronger influences the density dependence of leading to deviations from the purely plasmon picture.
In conclusion, we have re-formulated the Kirzhnits method for a multilayer structure with several polar layers. The developed technique is capable to describe superconductivity in multilayer structures, where the electrostatic electron-electron interaction, the optical-phonon spectra, and the amplitudes of the electron-phonon interaction are modified compared to bulk. In the present treatment, all phonon branches existing in the multilayer structure are taken into account.
We have found that at low densities, the critical temperature is well described by a BCS-like expression with the Fermi energy instead of the Debye energy. This is a direct consequence of the anti-adiabatic regime, which occurs at low carrier densities. At high densities, the density dependence of the critical temperature shows the domination of the plasmon mechanism of superconductivity.
The obtained agreement of the calculated critical temperatures with experiment gives support to the hypothesis that the mechanism of superconductivity is provided by the electron – optical-phonon interaction (see, e. g., (38)), at least in the multilayer structure analyzed in the present work.
Acknowledgements.We thank H. Boschker for the experimental data on critical temperatures and for discussions. We are grateful to M. Cohen, A. Zunger and C. G. Van de Walle for data on the deformation potential resulting from their works. This work was supported by FWO-V projects G.0370.09N, G.0180.09N, G.0119.12N, G.0122.12N, the WOG WO.035.04N (Belgium), the SNSF through Grant No. 200020-140761 and the National Center of Competence in Research (NCCR) “Materials with Novel Electronic Properties-MaNEP”.
- thanks: On leave of absence from: Department of Theoretical Physics, State University of Moldova, str. A. Mateevici 60, MD-2009 Kishinev, Republic of Moldova.
- thanks: Also at Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA.
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