Phonon mechanism in the most dilute superconductor n-type SrTiO₃

Significance

Low-temperature superconductivity of SrTiO₃ gives insight into high-T_C (up to 110 K) superconductivity in atomic-thin FeSe layers on SrTiO₃. In low-doped SrTiO₃ long-range electron interactions with longitudinal (LO) polar modes provide the unique pairing mechanism. Frequencies of polar modes larger than the Fermi energy imply nonadiabatic pairing rather than the usual separate treatment of electron and phonon degrees of freedom. Transport experiments reveal the mobility edge. Random localized states control the interaction between electrons and LO phonons via the magnitude of the dielectric constant. T_C is proportional to the Fermi energy. Interaction of the FeSe electrons with surface LO phonons is governed by the dielectric constant at a diffuse surface. Larger 2D Fermi surface results in higher T_C for FeSe/SrTiO₃.

Abstract

Superconductivity of n-doped SrTiO₃, which remained enigmatic for half a century, is treated as a particular case of nonadiabatic phonon pairing. Motivated by experiment, we suggest the existence of the mobility edge at some dopant concentration. The itinerant part of the spectrum consists of three conduction bands filling by electrons successively. Each subband contributes to the superconducting instability and exhibits a gap in its energy spectrum at low temperatures. We argue that superconductivity of n-doped SrTiO₃ results from the interaction of electrons with several longitudinal (LO) optical phonons with frequencies much larger than the Fermi energy. Immobile charges under the mobility edge threshold increase the “optical” dielectric constant far above that in clean SrTiO₃ placing control on the electron–LO phonon interaction. T_C initially grows as density of states at the Fermi surface increases with doping, but the accumulating charges reduce the electrons–polar-phonon interaction by screening the longitudinal electric fields. The theory predicts maxima in the T_C-concentration dependence indeed observed experimentally. Having reached a maximum in the third band, the transition temperature finally decreases, rounding out the T_C (n) dome, the three maxima with accompanying superconducting gaps emerging consecutively as electrons fill successive bands. This arises from attributes of the LO optical phonon pairing of electrons. The mechanism of LO phonons opens the path to increasing superconducting transition temperature in bulk transition-metal oxides and other polar crystals, and in charged 2D layers at the LaAaO₃/SrTiO₃ interfaces and on the SrTiO₃ substrates.

In the rapidly growing field of transition-metal-oxide electronics (1) SrTiO₃ is a key material with the unique physical properties. A broadband insulator, SrTiO₃ is the rare case of a paraelectric with the ferroelectric transition aborted by the quantum fluctuations and is remarkable, in particular, for a large dielectric constant that at helium temperatures reaches the enormous value of 25,000 (2).

A number of features in the low-temperature behavior of SrTiO₃ are still poorly understood, primarily its electronic properties. Most challenging, perhaps, is the nature of superconductivity in doped SrTiO₃. Upon doping (via a chemical substitution or reducing the oxygen content), SrTiO₃ displays low-temperature superconductivity already at concentrations of doped electrons as low as $n_s\approx 5.5\times {10}^{17}{\text{cm}}^{-3}$; as such, doped SrTiO₃ was dubbed the “most dilute superconductor” (3). At LaAlO₃/SrTiO₃ interfaces the electrons form a metallic layer on the SrTiO₃ side that displays 2D ferromagnetism and superconductivity (4, 5). Further, a 1-unit-cell-thick layer of FeSe on the surface of SrTiO₃ becomes superconducting at the record temperatures $T_C\ge 100\hspace{0.5em}\text{K}$ (6). These and other examples suggest that the mechanism of superconductivity in SrTiO₃ might differ from that in the weak-coupling BCS (Bardeen, Cooper, Schrieffer) phonon model. Below we focus on superconductivity in bulk doped SrTiO₃ and argue that it owes its origin to interaction between electrons and longitudinal (LO) polar phonons with frequencies far exceeding the Fermi energy. Implications for other polar crystals and 2D superconductivity of the atomic-thick FeSe layer on substrates of SrTiO₃ are briefly mentioned.

Superconductivity in SrTiO₃, first discovered in 1964 (7), was supposed initially to be another case of the phonon-mediated Cooper pairing generalized to the case of doped multivalley semiconductors (8). The limit of a small number of carriers is known to present a special challenge for theory as at low doping the Fermi energy $E_F$ is small.

The dimensionless coupling parameter of the BCS theory ${\lambda }_{BCS}$ is proportional to the product of a matrix element of electron–phonon scattering $V_{e-ph}$ and density of states $v\left(E_F\right)$ at the Fermi surface: ${\lambda }_{BCS}\propto V_{e-ph}\times \nu \left(E_F\right)$. Whereas the former is of atomic scale, as in ordinary metals, $\nu \left(E_F\right)$ is small. It was therefore suggested in ref. 9 that in polar semiconductors the long-range interaction with an LO optical mode can compensate the smallness of density of states. After the analysis the authors, however, came to the conclusion that this mechanism may be effective only if the frequency ${\omega }_{LO}$ of such phonons is smaller than the Fermi energy.

Understanding of superconductivity in the doped insulator SrTiO₃ has met with additional difficulties (for a brief review see refs. 10, 11). Thus, the popular multivalley model (8) turned out not to be relevant as in the energy spectrum of the cubic SrTiO₃ there is only one minimum at the $\mathrm{\Gamma }$-point built on the titanium 3d$t_{2g}$ levels (11–13).

That the Debye temperature for strontium titanate is comparable with or, at low enough doping, even larger than $E_F$, was probably first stressed in ref. 10.

Recent experiments (3, 14, 15) demonstrated that upon doping SrTiO₃ passes several stages before at last reaching at high enough concentration a metallic ground state. We show below that taking this evolution into consideration is necessary for understanding the low-temperature physics in doped SrTiO₃. In what follows, we explore interactions between the electrons doped into the conduction band and LO phonons. One vital difference from ref. 9 is that there are four LO phonons at the $\mathrm{\Gamma }$-point in SrTiO₃ [in the cubic phase (16)]. The other one comes about from the fact that the presence of immobile charges embedded via the doping process significantly affects the dielectric characteristics of the material and, hence, the matrix element for the interactions of electrons with LO phonons. Most of the results below are obtained analytically. Among other things, the theory predicts the appearance of maxima in the dependence $T_C\left(n_s\right)$ on the number of carriers $n_s$, providing the explanation for one of the most intriguing experimental findings (14).

As is well known, in metals the weak-coupling BCS theory is generalized to the case of arbitrary strong electron–phonon interactions in the Migdal–Eliashberg equations (17, 18).

As the frequencies of the LO-polar phonon modes in bulk pure SrTiO₃ are known and are high (19), electron–phonon interactions in doped SrTiO₃ present the extreme case of the “antiadiabatic” limit ${\omega }_0\gg E_F$. We argue that apart from the self-evident significant interest in the realization of such extreme conditions specifically in the doped strontium titanate, the notion of superconductivity mediated by phonons with a frequency higher than or of the same order as the Fermi energy calls for more general exploration of peculiarities inherent in nonadiabatic pairing mechanisms.

One part of the problem concerns competition between the Coulomb repulsion and the phonon-mediated attraction. From the condition ${\omega }_0/E_F\ll 1$ in the adiabatic regime follows the “retardation” effect of the BCS theory: Two electrons of the pair evade each other, being a distance $d$ apart, on the order of $v_F/{\omega }_0$ larger by a factor $E_F/{\omega }_0\gg 1$ than the atomic scale $a\approx 1/p_F$. Effectively, this decreases the role of the Coulomb repulsion because the latter is screened on the atomic distances.

In the opposite limit of ${\omega }_0/E_F\gg 1$ electrons of the pair feel both the Coulomb repulsion and the potential of the local lattice distortion instantaneously. For the Cooper pair to be formed the strength of the phonon attraction must exceed the direct Coulomb repulsion. For the general theory of electron–phonon interaction in the nonadiabatic case the immediate difficulty is that the customary mathematical apparatus of the Migdal–Eliashberg equations is not applicable at ${\omega }_0/E_F\ge 1$; the case of doped SrTiO₃ is of particular value where the most significant peculiarities proper to a nonadiabatic superconductivity can be deduced in the logarithmic approximation.

Results

Excitations in Insulating SrTiO₃.

Consider at first two excitations in the conduction band of pure SrTiO₃ that scatter off each other via virtual exchange of LO phonons. In a polar semiconductor the scattering matrix element has the form (GFL is after names of the authors of ref. 9):

$${\mathrm{\Gamma }}_{GFL}(\overrightarrow{p},{\epsilon }_n|\overrightarrow{k},{\epsilon }_m)=\frac{4\pi e^2}{{\kappa }_{\infty }{q}^2}-\frac{4\pi e^2}{q^2}\left(\frac{1}{{\kappa }_{\infty }}-\frac{1}{{\kappa }_0}\right)\times \frac{{\omega }_{LO}^2}{{\omega }_{LO}^2+{({\epsilon }_n-{\epsilon }_m)}^2}.$$ [1]

We apply the thermodynamic technique (see ref. 20) that is more convenient for the analysis of the Cooper instability, compared with solving the linear integral equation for the gap parameter on the real frequency axis, as this technique circumvents the pole singularities in the kernel of the integral equation.

In Eq. 1 ${\omega }_{LO}^2$ is the square of the LO phonon frequency; $\overrightarrow{q}=\overrightarrow{p}-\overrightarrow{k}$ and ${\epsilon }_n-{\epsilon }_m$ are the momentum and frequency the electrons exchange upon scattering; ${\kappa }_0$ and ${\kappa }_{\infty }$ are the static and optical dielectric constants, respectively. If the frequency of LO phonon is large, ${\omega }_{LO}\gg E_F$, at low temperatures one can omit ${\epsilon }_n-{\epsilon }_m$ in the denominator of the phonon Green function. Doing so in Eq. 1, one obtains

$${\mathrm{\Gamma }}_{GFL}(\overrightarrow{p},{\epsilon }_n|\overrightarrow{k},{\epsilon }_m)={\mathrm{\Gamma }}_{GFL}\left(q\right)=\frac{4\pi e^2}{{\kappa }_0{q}^2}>0.$$ [2]

As mentioned above, in the case of SrTiO₃ there are four LO phonons at the $\mathrm{\Gamma }$-point (16). Strictly speaking, matrix elements of the interaction between electrons and LO phonons in a multimode polar crystal have a more complicated form than in Eq. 1. Characteristic of SrTiO₃, however, one phonon mode has a giant gap between the LO and the transverse optical phonons (16). For such a LO mode Eq. 1 is correct, at least approximately. As in ref. 9, its contribution practically compensates the direct Coulomb term as the static dielectric constant is large [${\kappa }_0\approx 2\times {10}^4$(2)]. Of the three LO phonons one mode is not infrared active (16). As a result, instead of [2], one has

$$\mathrm{\Gamma }\left(q\right)=\frac{4\pi e^2}{{\kappa }_0{q}^2}-{\alpha }_{eff}^2\frac{4\pi e^2}{q^2}\left(\frac{1}{{\kappa }_{\infty }}-\frac{1}{{\kappa }_0}\right)\approx -{\alpha }_{eff}^2\frac{4\pi e^2}{q^2{\kappa }_{\infty }}<0.$$ [3]

Here the factor ${\alpha }_{eff}^2$ comes about from the interaction of electrons with the two remaining LO-polar phonons. In clean strontium titanate their contribution in [3] should guarantee the attractive sign of the interaction between two excitations.

Doping SrTiO₃: Experimental Summary.

Recent experiments (3, 14, 15) revealed a number of new features concerning the microscopic properties of doped SrTiO₃. It is worth summarizing briefly some facts most essential for the discussion below.

• i)Pure SrTiO₃ is on the verge of a ferroelectric transition with its static dielectric constant ${\kappa }_0$ at low-temperature 24,000 being 4 orders of magnitude larger than that of the vacuum. The Bohr radius is very large, $a_B^{\ast }=0.53{\kappa }_0(m_e/m)\times {10}^{-8}\text{cm}$, giving $a_B^{\ast }\approx 5-7\times {10}^{-5}\text{cm}$ (2).
• ii)Correspondingly, the insulator-to-metal transition into the emerging “impurity band” occurs, according to the famous Mott criterion $n_s^{1/3}{a}_B^{\ast }>0.26$, already at low doping $n_s^{MT}<{10}^{12}{\text{cm}}^{-3}$. Experimentally, the cross-over from the regime of impurity band transport to the regime of conductivity of coherent carriers in the conduction band starts at $n_s^{\ast }\approx 2\times {10}^{16}{\text{cm}}^{-3}$ (15). In order of magnitude $n_s^{\ast }$ may be taken as an estimate for the mobility edge.

Shubnikov–de Haas (SdH) quantum oscillations (QO) in the magnetoresistance are first seen at $n_s\approx 4\times {10}^{17}{\text{cm}}^{-3}$; QO and superconductivity are observable in samples with somewhat higher carrier concentration, $n_s\approx 5.5\times {10}^{17}{\text{cm}}^{-3}$. At such concentrations the Fermi energy is extremely small ($E_F\approx 1.1-1.3\hspace{0.5em}\text{meV}$) (3, 14).

• iii)With $n_s$ varying between $n_s\approx 5.5\times {10}^{17}{\text{cm}}^{-3}$ and $n_s\approx 4\times {10}^{20}{\text{cm}}^{-3}$ the superconducting transition temperature $T_C$ in ref. 14 varies from 0.07 to 0.4 K.
• iv)In the experimental dependence of $T_C$ on the dopant concentration (14) several regimes are distinguishable. First, $T_C$ increases ($n_s$ varies from $n_s\approx 5.5\times {10}^{17}{\text{cm}}^{-3}$ to $n_s\approx 1.05\times {10}^{18}{\text{cm}}^{-3}$). After reaching a maximum at $n_{s\max }\approx 2\times {10}^{18}{\text{cm}}^{-3}$, $T_C$ then decreases before starting to grow again as electrons begin to fill the next band. Observation of a new SdH frequency at $n_{c1}=1.2\times {10}^{18}{\text{cm}}^{-3}$ signifies the concentration at which the chemical potential touches the bottom of the second band. Finally, another frequency of QO above $n_{s2}\approx 2\times {10}^{19}{\text{cm}}^{-3}$ signals that the chemical potential has reached the bottom of the third band (14).

From this behavior one can infer, in particular, that the threefold degeneracy of the spectrum of the cubic phase of SrTiO₃ at the $\mathrm{\Gamma }$-point is lifted.

Maxima in $T_C\left(n_s\right)$ are the most remarkable features among the findings (3, 14). It is argued here that such maxima are the signature of LO polar phonons that constitute the mechanism of the superconductivity of doped SrTiO₃.

Interaction of Electrons with LO Phonons in Doped SrTiO₃.

The Hamiltonian 3 describes the interaction between electrons via the exchange of LO phonons in clean SrTiO₃. One has to revisit the conjecture that the electron–lattice interaction in doped SrTiO₃ is long-range and governed by the electric fields inherent in LO optical modes. In the Fröhlich Hamiltonian,

$$\overrightarrow{P}=F_C\overrightarrow{u},$$ [4]

for the connection between the polarization $\overrightarrow{P}$ and the lattice displacement $\overrightarrow{u}$ one must however take into account the possibility of screening mechanisms.

Details of these mechanisms are not a priori clear in the case in hand. In regard to the Mott criterion $n_s^{1/3}{a}_B^{\ast }>0.26$ for the transition into a “metallic” impurity band, we note, together with ref. 15, that the binding energy $E_B=-13.6\hspace{0.5em}\text{eV}\times (m/m_e){\kappa }_0^{-2}$ of a hydrogen-like center at ${\kappa }_0=\mathrm{24,000}$ is so small that doping centers must be ionized at any temperature of interest. One obvious consequence of that is the absence of a distinct gap separating the impurity band from “the bottom” of the conduction band. Instead, there should be a threshold in the density of states equivalent, in effect, to the mobility edge. As mentioned above, the cross-over toward the coherent conduction band transport is observable above concentration $n_s^{\ast }\approx 2\times {10}^{16}{\text{cm}}^{-3}$ (15).

Lattice displacements in a polar crystal are accompanied by formation of electric dipoles also in the presence of disorder. Therefore, several optical phonons propagating in doped SrTiO₃ carrying longitudinal electric fields must exist. However, the metallic regime experimentally firmly establishes itself at a concentration more than a factor of 10 higher, $n_s^{\ast }\approx 2\times {10}^{16}{\text{cm}}^{-3}$.

We hypothesize that the carriers with their energy below the mobility edge contribute mainly by changing the value of the “optical” dielectric constant. The Fröhlich matrix element 4 for electrons in the conduction band will be written in the same canonical form (Eq. 1):

$$F_C={\left[4\pi e^2\frac{\hslash {\omega }_{LO}}{2}\left(\frac{1}{{\overline{\kappa }}_{\infty }}-\frac{1}{{\overline{\kappa }}_0}\right)\right]}^{1/2},$$ [5]

where ${\overline{\kappa }}_{\infty }$ and ${\overline{\kappa }}_0$ in [5] (${\overline{\kappa }}_0\gg {\overline{\kappa }}_{\infty }$) are the optical and the static dielectric constants in the presence of localized electrons in the background.

One hydrogen-like center possesses the dipole moment $(9/2)a_B^{\ast 3}$. As $a_B^{\ast }$ is large, reaching $a_B^{\ast }=5-7\times {10}^{-5}\text{cm}$, local centers can significantly contribute to the polarizability. At $n_s\approx n_s^{\ast }\approx 2\times {10}^{16}{\text{cm}}^{-3}$ this can result in a large dielectric constant ${\overline{\kappa }}_{\infty }$ on the order of ${10}^2-{10}^3$.

The long-range Coulomb potentials can also be screened by the mobile carriers in the conduction band. Combining the two mechanisms, instead of Eq. 3, one writes

$$\tilde{\mathrm{\Gamma }}(q,{\omega }_{mn})=\frac{4\pi e^2}{Q^2(q,{\omega }_{mn})}\times \left[\frac{1}{{\overline{\kappa }}_0}-{\alpha }_{eff}^2\left(\frac{1}{{\overline{\kappa }}_{\infty }}-\frac{1}{{\overline{\kappa }}_0}\right)\right]\approx -{\alpha }_{eff}^2\frac{4\pi e^2}{Q^2(q,{\omega }_{mn}){\overline{\kappa }}_{\infty }}<0.$$ [6]

Eq. 6 finalizes our choice of the model. Both ${\overline{\kappa }}_{\infty }$ and ${\alpha }_{eff}^2$ are model parameters that may depend on sample quality. It is convenient to redefine ${\overline{\kappa }}_{\infty }$ in [6] taking ${\alpha }_{eff}^2=2$. In a single band with parabolic energy spectrum $\epsilon \left(\overrightarrow{p}\right)=p^2/2m$ doped electrons fill all states up to the chemical potential $\mu \equiv E_F=p_F^2/2m$. (The case of several bands is considered later.)

In the so-called random-phase approximation (RPA) the general form of $Q^2(q,{\omega }_{mn})$ in [6] is

$$Q^2(q,{\omega }_{mn})=q^2+{\kappa }_{TF}^{2}S(q,{\omega }_{mn}).$$ [7]

Here

$$S(q,{\omega }_{mn})={\displaystyle \underset{0}{\overset{1}{\int }}\frac{{({\overrightarrow{v}}_F\cdot \overrightarrow{q})}^{2}d\mu }{{({\overrightarrow{v}}_F\cdot \overrightarrow{q})}^2+{\omega }_{mn}^2}},$$ [7a]

and ${\kappa }_{TF}^2$ is the square of the Thomas–Fermi radius:

$${\kappa }_{TF}^2=(4e^{2}m{p}_F/{\overline{\kappa }}_{\infty }\pi {\hslash }^3).$$ [7b]

It will be shown later in the calculations below that one can assume in [6] and [7a] that ${\omega }_{nm}=0$. That is, $S(q,0)=1$ (SI Appendix I).

Introduce the notation ${\overline{a}}_B={\overline{\kappa }}_{\infty }{\hslash }^2/e^{2}m$ for the optical Bohr radius and rewrite ${\kappa }_{TF}^2=(4e^{2}m{p}_F/{\overline{\kappa }}_{\infty }\pi {\hslash }^3)=4p_F^2/(\pi {\overline{a}}_B{p}_F/\hslash )$. In passing, note that from the formal point of view the regime of the large Mott parameter $n_s^{1/3}{a}_B^{\ast }\gg 1$ is analogous to the regime of the “degenerate plasma.” Owing to the mobility edge and the presence of immobile carriers, applicability of RPA in the conduction band is controlled by the value of parameter $\pi p_F{\overline{a}}_B/\hslash \gg 1$.

Cooper Instability in Logarithmic Approximation.

For ordinary superconductors the Migdal adiabatic provision ${\omega }_0/E_F\ll 1$ allows simplifying the diagrammatic expansion for the scattering amplitude by omitting all of the so-called “crossing diagrams.” The result is reduced to the closed system of Migdal–Eliashberg equations (17, 18). In the nonadiabatic case this theoretical tool is lost. Fortunately, the basic features of the superconductivity of doped SrTiO₃ can be explored taking advantage of the logarithmic approach. The latter is applicable at small $T_C\ll E_F$ and, as mentioned above, in the case at hand $T_C$ varies between $({10}^{-3}-{10}^{-2})E_F$.

Superconductivity manifests itself in the occurrence at $T=T_C$ of the pole in the scattering amplitude $\mathrm{\Gamma }(p,q-p|p\prime ,q-p\prime )$ at zero total momentum and frequency $q=0$ (20). The exact amplitude is the sum of all diagrams in the Cooper channel. Denoting $\mathrm{\Gamma }(p,q-p|p\prime ,q-p\prime ){|}_{q=o}\equiv \mathrm{\Gamma }\left(p\right|p\prime )$, the equation for $\mathrm{\Gamma }\left(p\right|p\prime )$ reads

$$\mathrm{\Gamma }(p,|p\prime )=\tilde{\mathrm{\Gamma }}(p,|p\prime )-\frac{T}{{\left(2\pi \right)}^3}{\sum }_{n\prime }\int d\overrightarrow{k}\tilde{\mathrm{\Gamma }}(p,|k)G\left(k\right)G(-k)\mathrm{\Gamma }(k,|p\prime ).$$ [8]

Usually, instead of calculating $\mathrm{\Gamma }\left(p\right|p\prime )$ from Eq. 8, the transition temperature is determined via the eigenvalue of the homogeneous equation for the gap function $\mathrm{\Psi }\left(p\right)$:

$$\mathrm{\Psi }\left(p\right)=-T{\sum }_m{\displaystyle \int \frac{d\overrightarrow{k}}{{\left(2\pi \right)}^d}}\tilde{\mathrm{\Gamma }}\left(p\right|k)\mathrm{\Pi }\left(k\right)\mathrm{\Psi }\left(k\right).$$ [9]

The Cooper instability owes its origin to the logarithmic divergence in the blocks $\mathrm{\Pi }\left(k\right)$ represented in Eq. 8 by the product of the two Green functions, $G\left(k\right)G(-k)$. In the thermodynamic formulation $G\left(k\right)={[i{\nu }_n-({\overrightarrow{k}}^2-p_F^2)/2m]}^{-1}$ and one has

$$\mathrm{\Pi }\left(k\right)=G\left(k\right)G(-k)=\frac{1}{{\nu }_m^2+{[({\overrightarrow{k}}^2-p_F^2)/2m]}^2}.$$ [10]

Solving the integral 9 exactly is expected to give for $T_C$ a BCS-like weak-coupling form:

$$T_C=W\exp (-1/\lambda ).$$ [11]

We however determine the exact dimensionless constant $\lambda$ in [11] without solving Eq. 9. As to the prefactor $W$, its value will be known only to the accuracy of a numerical factor of the order of the unity.

Substituting [10] in the right-hand side of Eq. 9 gives

$$-\frac{T}{{\left(2\pi \right)}^3}{\sum }_{n\prime }\int d\overrightarrow{k}\tilde{\mathrm{\Gamma }}\left(p\right|k)G\left(k\right)G(-k)\mathrm{\Psi }\left(k\right)$$

$$\Rightarrow -T{\sum }_{n\prime }\int \tilde{\mathrm{\Gamma }}\left(p\right|k)[m{p}_F\sin \theta d\theta )/{\left(2\pi \right)}^2]d\varsigma \frac{1}{{\nu }_m^2+{\varsigma }^2}\mathrm{\Psi }\left(k\right).$$ [12]

(In [12] $\varsigma =({\overrightarrow{k}}^2-p_F^2)/2m\approx v_F(p-p_F)$; $\theta$ is the angle between two vectors $\overrightarrow{p}$ and $\overrightarrow{k}$.)

Let the vectors $\overrightarrow{p}$ and $\overrightarrow{k}$ in [12] be on the Fermi surface. The expression for the vertex $\tilde{\mathrm{\Gamma }}\left(p\right|k)\equiv \tilde{\mathrm{\Gamma }}\left(\theta \right){|}_{FS}$ stands in front of the logarithmic singularity in Eq. 12:

$$\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\sin \theta d\theta }\tilde{\mathrm{\Gamma }}\left(\theta \right){|}_{FS}\right]\frac{m{p}_F}{{\left(2\pi \right)}^2}\times {\displaystyle \underset{0}{\overset{W}{\int }}\frac{d\varsigma }{\varsigma }}th\frac{\varsigma }{2T}\Rightarrow \lambda \ln \left(\frac{2W\gamma }{\pi T}\right).$$ [13]

Thereby, Eq. 13 gives for $\lambda$ the definition

$$\lambda =\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\sin \theta d\theta }\tilde{\mathrm{\Gamma }}\left(\theta \right){|}_{FS}\right]\frac{m{p}_F}{{\left(2\pi \right)}^2}.$$ [14]

In [13] $W$ is a characteristic scale of the dependence of $\tilde{\mathrm{\Gamma }}\left(p\right|k)$ on the energy variable $\mathit{\varsigma }$ that plays the role of an order-of-magnitude cutoff parameter in the integral over $\mathit{\varsigma }$ in Eq. 12.

In the extreme nonadiabatic limit the only relevant energy scale is the Fermi energy and hence $W$ must be on the order of $E_F$.

Expressions for the Transition Temperature.

In Eqs. 13 and 14 take for $\tilde{\mathrm{\Gamma }}\left(p\right|k)$ its expression $\tilde{\mathrm{\Gamma }}(q,{\omega }_{mn})$ from Eqs. 6 and 7:

$$\int \tilde{\mathrm{\Gamma }}\left(p\right|k)[m{p}_F\sin \theta d\theta )/{\left(2\pi \right)}^2=\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{e^{2}m\sin \theta d\theta }{\pi p_F[1-\cos \theta +2({\varsigma }_p+{\varsigma }_k)/E_F+{\kappa }_{TF}^2/2p_F^2]{\overline{\kappa }}_{\infty }}}\right],$$ [15]

substitute ${\kappa }_{TF}^2=(4e^{2}m{p}_F/{\overline{\kappa }}_{\infty }\pi {\hslash }^3)=4p_F^2/(\pi {\overline{a}}_B{p}_F/\hslash )$ and integrate over $\cos \theta$. (In [15] ${\varsigma }_k=v_F(k-p_F)$ and ${\varsigma }_p=v_F(p-p_F)$.)

At the Fermi surface ${\varsigma }_k$ and ${\varsigma }_p$ equal zero; from [15] follows:

$$\lambda =\frac{\hslash }{\pi p_F{\overline{a}}_B}\ln (1+\pi p_F{\overline{a}}_B/\hslash ).$$ [16]

The expression for $T_C$ (Eq. 11) can be presented in the convenient form:

$$T_C=const\times \frac{\gamma }{{\pi }^3}\times \frac{{\hslash }^2}{m{\overline{a}}_B^2}\left[{\overline{T}}_1(\pi p_F{\overline{a}}_B/\hslash )\right].$$ [17]

Denoting in [17] $x=\pi p_F{\overline{a}}_B/\hslash$, the function ${\overline{T}}_1\left(x\right)=x^2\exp \left[-x/\ln (1+x)\right]$ is plotted in Fig. 1.

Fig. 1.

Temperature of the superconducting transition $T_C$ as function of the concentration of electrons $n_s$ (in the first band). $T_C\left(n_s\right)$ is proportional to ${\overline{T}}_1\left(x\right)=x^2\exp \left[-x/\ln (1+x)\right]$. The relation between the variable $x$ and the concentration is $x=\pi p_F{\overline{a}}_B/\hslash =5.74\times {(n_s\times {10}^{-18})}^{1/3}$ (see text). The function ${\overline{T}}_1\left(x\right)$ has a maximum at $x=7.18$. At larger $x$, ${\overline{T}}_1\left(x\right)$ tends to zero and, as discussed in the text below, at higher concentrations superconductivity stems from filling the second band and then, at even higher $n_s$, from filling the third band. The emerging maximum in the $T_C\left(n_s\right)$ dependence has been observed experimentally (14).

To test the relevance of the theoretical expression [17] to the experiment, consider $T_C\left(n_s\right)$ in the vicinity of the maximum in Fig. 1B (14); the latter is reached at $n_{s\max }\approx 2\times {10}^{18}{\text{cm}}^{-3}$.

($T_{C\max }\approx 0.2\hspace{0.5em}\text{K}$). [Contributions into $T_C\left(n_s\right)$ from the second band at these concentrations supposedly remain small.] The maximum of ${\overline{T}}_1\left(x\right)$ in Fig. 1, ${\overline{T}}_1\left(x_{\max }\right)=1.69$, is at $x_{\max }=7.18$.

Substituting $p_F/\hslash \equiv {\left(n_s\right)}^{1/3}{\left(3{\pi }^2\right)}^{1/3}$ into ${(\pi p_F{\overline{a}}_B/\hslash )}_{\max }=7.18$, one finds ${\overline{a}}_B\approx 0.58\times {10}^{-6}\text{cm}$. According to figure 4A in ref. 14, the mass of the lower band is $m_1\approx 1.8m_e$. Eq. 17 gives for $T_{C\max }=const\times 1.1\hspace{0.5em}\text{K}$. Thereby $const\approx 1/5$. [${\overline{a}}_B=0.53{\overline{\kappa }}_{\infty }(m_e/m_1)\times {10}^{-8}\text{cm}=0.58\times {10}^{-6}\text{cm}$ allows the estimate for the dielectric constant ${\overline{\kappa }}_{\infty }\approx 2\times {10}^2$; for the pure bulk SrTiO₃ ${\kappa }_{\infty }=5.2$ (2).] Note the large $\pi p_F{\overline{a}}_B/\hslash \ge 7$, as it was assumed above.

Eq. 17 suggests the simple physical interpretation for the maximum of $T_C\left(n_s\right)$ in Fig. 1B (14). At the start of the doping $T_C$ in Fig. 1 increases, but at higher concentrations the RPA screening becomes important and $T_C$ decreases as the Thomas–Fermi radius (Eq. 7) $r_{TF}=1/{\kappa }_{TF}\propto n_s^{-1/3}$ gets shorter.

So far only one band with an isotropic energy spectrum has been considered. As already mentioned earlier, lattice deformations split the threefold degenerate minimum at the $\mathrm{\Gamma }$-point of the cubic SrTiO₃ into three bands (11, 12, 14, 21, 22). For a single band, from Eq. 17 and Fig. 1 it follows that at large $n_s$, $T_C\left(n_s\right)$ would tend to zero. Therefore, with an increase of dopant concentration superconductivity comes about with filling the second band and then, at even higher concentrations, the third one.

The maximum is inherent in the kernel [15] and in the expression for $T_C\left(n_s\right)$ (Eq. 17) and is the hallmark of the interaction of electrons with LO phonons.

The doping dependence of $T_C\left(n_s\right)$ below is studied analytically in the model of three parabolic bands with three different masses, as shown in Fig. 2 A and B.

Fig. 2.

Three-band model. (A) Shown schematically are three bands arising at low temperatures from splitting of the threefold degenerate minimum at the $\mathrm{\Gamma }$-point in cubic SrTiO₃. Here ${\mu }_{c1}$ and ${\mu }_{c2}$ mark the bottoms of the middle and of the upper bands at the concentrations $n_{c1}=1.2\times {10}^{18}{\text{cm}}^{-3}$ and $n_{c2}=2.5\times {10}^{19}{\text{cm}}^{-3}$, respectively (14). As depicted, the six points of the bands intersections are marked by small dashed ovals. The degeneracy at each such intersection is lifted; the two bands locally repulse each other (see in the larger oval on the right). (B) The resulting pattern of the three-band spectrum (schematic). With the chemical potential moving upward, one, two, and three different frequencies in the spectrum of quantum oscillations emerge in the consecutive order. The spectrum of oscillations becomes complex when the chemical potential position falls too close to one of the intersections not far from either ${\mu }_{c1}$ or ${\mu }_{c2}$.

The new frequency emerging in the SdH oscillations at the concentration $n_{c1}=1.2\times {10}^{18}{\text{cm}}^{-3}$ (15) signifies reaching the bottom of the second (the middle) band. The derivation (Eq. 16) can be repeated, this time for the chemical potential $\mu$ in a position above the bottom, ${\mu }_{c1}\equiv p_{c1}^2/2m_1$, of the second band. With the notations $m_2$ and $p_{F2}={(m_2/m_1)}^{1/2}{(p_{F1}^2-p_{c1}^2)}^{1/2}$ for the mass and the Fermi momentum in the second band, one writes $Q^2(q,0)=q^2+(4e^2/\pi {\kappa }_{\infty }{\hslash }^3)[m_1{p}_{F1}+m_2{p}_{F2}]$. Simple calculations give the factor ${\lambda }_2$ in the exponent of Eq. 13 for the temperature of the Cooper instability in the second (the middle) band (${\lambda }_1\to {\lambda }_2$):

$${\lambda }_2=e^2(m_2/\pi p_{F2}{\overline{\kappa }}_{\infty })\ln \{1+(p_{F2}^2\pi {\kappa }_{\infty }{\hslash }^3)/e^2[(m_1{p}_{F1}+m_2{p}_{F2}]\}.$$ [18]

The relation between concentration and position of the chemical potential $\mu =p_{F1}^2/2m_1$ in the second band is

$$[p_{F1}^3+p_{F2}^3](1/3{\pi }^2{\hslash }^3)=n_s.$$ [19]

At concentrations $n_s$ above $n_{c2}=2.5\times {10}^{19}{\text{cm}}^{-3}$, electrons begin filling the third (the upper) band (15). Accordingly,

$${\lambda }_3=e^2(m_3/\pi p_{F3}{\overline{\kappa }}_{\infty })\ln \{1+\left(p_{F3}^2\pi {\kappa }_{\infty }{\hslash }^3\right)/e^2[(m_1{p}_{F1}+m_2{p}_{F2}+m_3{p}_{F3}]\}.$$ [20]

Similarly to Eq. 19,

$$[p_{F1}^3+p_{F2}^3+p_{F3}^3](1/3{\pi }^2{\hslash }^3)=n_s.$$ [21]

Eqs. 16 and 18–21 present the exponents ${\lambda }_{\mathrm{1,2,3}}$ in the expressions for the onset temperatures of superconductivity in each of the three bands when the chemical potential moves from one band to the other with increasing dopants concentration.

SI Appendix I

In ref. 10 superconductivity of doped SrTiO₃ was ascribed to the plasmon-polar optical phonon mechanism. The RPA expressions above account for the presence of the plasmon pole; however, for the Hamiltonian in the form of Eq. 3 describing interaction of electrons with LO optical phonons, its contribution to the superconducting pairing is negligible. Indeed, return to Eqs. 13 and 14; see the main text. At ${\omega }_{nm}^2>{({\overrightarrow{v}}_F\cdot \overrightarrow{q})}^2$ in [14a]

$$\mathrm{\Gamma }(q,{\omega }_{mn})=-\frac{8\pi e^2}{q^2{\overline{\kappa }}_{\infty }}\times \frac{{\omega }_{mn}^2}{{\omega }_{mn}^2+{\omega }_{pl}^2}.$$ [AIa]

In [AIa] ${\omega }_{pl}^2=4\pi e^2{n}_s/m$ is the square of the plasma frequency. On the axis of the real frequencies ${\omega }_{nm}^2\to -{({\epsilon }_n-{\epsilon }_m)}^2$ and $\mathrm{\Gamma }(q,{\omega }_{mn})$ in Eq. AIa accept the form

$$\mathrm{\Gamma }(q,{\omega }_{mn})\Rightarrow -\frac{8\pi e^2}{q^2{\overline{\kappa }}_{\infty }}\times \frac{{({\epsilon }_n-{\epsilon }_m)}^2}{{\omega }_{pl}^2-{({\epsilon }_n-{\epsilon }_m)}^2}.$$ [AIb]

That is, the plasma frequency pole indeed emerges in [AIb]; however, for the pole to play any role in the Cooper pairing its residue should be ${\omega }_{pl}^2$, not ${({\epsilon }_n-{\epsilon }_m)}^2$. In the thermodynamic technique such pole is not singled out because with $\mathrm{\Gamma }(q,{\omega }_{mn})$ in the form [AIa] summation over ${\epsilon }_m$ does not contribute to the logarithmic singularity in Eqs. 9 and 10. Correspondingly, the dependence on ${\omega }_{nm}$ in Eqs. 14 and 14a can be omitted.

Discussion

The two expressions 18 and 20 are similar to Eq. 16 in that $T_{C2}\left(n_s\right)$ and $T_{C3}\left(n_s\right)$ both in the middle and the upper bands display maxima at some $n_{1\max }$ and $n_{2\max }$. This stems from the fact that ${\lambda }_2$ and ${\lambda }_3$ are small at small $p_{F2}$ and $p_{F3}$, and decrease again as the screening radius $r_{TF}=1/{\kappa }_{TF}\propto n_s^{-1/3}$ is getting shorter at larger concentrations. In Fig. 3A ${\lambda }_1\left(x\right)$ (Eq. 16), ${\lambda }_2\left(x\right)$ (Eq. 18), and ${\lambda }_3\left(x\right)$ (Eq. 20) in each band are plotted as functions of $x=(\pi p_F{\overline{a}}_B/\hslash )$; the larger is the value of $\lambda$, the higher is $T_C$. (For the explicit form of relations between $x$ and the dopants concentration $n_s$, see SI Appendix II.)

Fig. 3.

Low-temperature phase diagram of doped SrTiO₃ in the ($T,n_s$) plane in the three-band model (Fig. 2). (A) The concentration dependence of the $\lambda$-factors in exponents of the expression for the transition temperature $T_C=const\times W\exp (-1/\lambda )$ for each band. Relations between $x$ and $n_s$ in different bands are (i) the lower band ($0<x<x_{c1}$):$x=\pi p_F{\overline{a}}_B/\hslash =5.74{(n_s\times {10}^{-18})}^{1/3}$; (ii) in the middle band ($x_{c1}<x<x_{c2}$):$x^3+{(m_2/m_1)}^{3/2}{(x^2-x_{c1}^2)}^{3/2}=195(n_s\times {10}^{-18})$; (iii) in the upper band ($x>x_{c2}$):$x^3+{(m_2/m_1)}^{3/2}{(x^2-x_{c1}^2)}^{3/2}+{(m_3/m_1)}^{3/2}{(x^2-x_{c2}^2)}^{3/2}=195(n_s\times {10}^{-18})$. For masses, their experimental values $m_1=1.8m_e$, $m_2=3.5m_e$ (14); $m_3=6m_e$ (12, 14, 23) are taken. Here and below $x_{c1}\approx 6.1$ and $x_{c2}\approx 20$ correspond to the two concentrations $n_{c1}=1.2\times {10}^{18}{\text{cm}}^{-3}$ and $n_{c2}=2.5\times {10}^{19}{\text{cm}}^{-3}$(14). (B) Dimensionless temperature $\overline{T}\left(x\right)$ (for the definition, see Eq. 22 in the text) as a function of the concentration. Full lines depict dependence on the concentration of the temperature of superconducting transition across the whole $(T,x)$ plane. Dashed lines are lines of the pairing instability in preceding bands. In principle, at a given $x$ superconductivity sets in simultaneously for all bands owing to the proximity effects (nonzero interband matrix elements), but, as the effect of proximity is weak in the case of the pairing mechanisms under discussion, superconductivity in these bands manifests itself mainly below these dashed lines, thereby leading at low temperatures to the pattern of the well-distinguishable superconducting gaps. The three vertical arrows in B show that the number of gaps may vary from a single gap (A) to two gaps (B) and to three gaps (C).

Temperatures of the transition $T_{C;\mathrm{1,2,3}}$ in all three bands can be presented in the form

$$T_{C;\mathrm{1,2,3}}\left(n_s\right)={\left(const\right)}_{\mathrm{1,2,3}}\times \frac{\gamma }{{\pi }^3}\times \frac{{\hslash }^2}{m{\overline{a}}_B^2}\times {\overline{T}}_{\mathrm{1,2,3}}\left(x\right).$$ [22]

The dimensionless ${\overline{T}}_{\mathrm{1,2,3}}\left(x\right)$ are plotted in Fig. 3B. (For the analytical form of dependence of each of ${\lambda }_1\left(x\right)$, ${\lambda }_3\left(x\right)$, and ${\overline{T}}_{\mathrm{1,2,3}}\left(x\right)$ on x, see SI Appendix II.)

With no pretense to describing quantitatively the experiment (14), the three plots of ${\overline{T}}_{\mathrm{1,2,3}}\left(x\right)$ in Fig. 3 correctly reproduce the expected overall behavior of $T_C$ in the ($T_C$,$n_s$) phase diagram of doped SrTiO₃, including, in particular, the appearance of the three $T_C$ maxima. Two maxima are seen in Fig. 1B (14) at $\approx 0.2\hspace{0.5em}\text{K}$ and at $\approx 0.4\hspace{0.5em}\text{K}$(data at the intermediate concentrations in ref. 14 are absent).

Two interesting qualitative predictions follow. First, with electrons filling successively one band after another, tunneling experiments at low temperatures are expected to reveal one, two, and even three superconducting gaps developing in parallel with the increase in concentration, as shown schematically in Fig. 3B. [The two-gap structure in the $I\left(V\right)$ characteristics has been observed in ref. 24.]

Secondly, the three bands fully exhaust the electronic spectrum of SrTiO₃ at the $\mathrm{\Gamma }$-point. Therefore, the $T_C$ maximum observed after electrons began filling up the third band imposes the upper limit on the value of the temperature of the superconducting transition.

In the model of three bands shown in Fig. 2 the frequencies of QO in the each band would smoothly grow proportional to the chemical potential, except in the vicinity of the bands intersection.

Note that the $T_C\left(n_s\right)$ dependence, asymmetric with respect to the maximum at 0.2 K in Fig. 1B (14), is reproduced poorly in the single-band picture (Eq. 18) because of proximity to the two-band intersection.

SI Appendix II

(A) The dimensionlesss ${\overline{T}}_{\mathrm{1,2,3}}\left(x\right)$ in Fig. 3 A and B for the three bands are related to the corresponding temperatures of the Cooper instability as

$$T_{C;\mathrm{1,2,3}}\left(n_s\right)={\left(const\right)}_{\mathrm{1,2,3}}\times \frac{\gamma }{3{\pi }^3}\times \frac{{\hslash }^2}{m_i{\overline{a}}_B^2}{\overline{T}}_{\mathrm{1,2.3}}\left(x\right).$$

(B) Definitions of ${\lambda }_{\mathrm{1,2,3}}\left(x\right)$ are as follows:

(i) For the lower band, Eq. 17:

$${\overline{T}}_1\left(x\right)\equiv f_1\left(x\right)=x^2\exp \left[-x/\ln (1+x)\right],$$

$${\lambda }_1\left(x\right)=\frac{1}{x}\ln (1+x),$$

$$x=\pi p_F{\overline{a}}_B/\hslash =5.74{(n_s\times {10}^{-18})}^{1/3}.$$

(ii) In the middle band, Eq. 19:

$${\overline{T}}_2\left(x\right)=(x^2-x_{c1}^2)\exp \left\{-\frac{1}{{\lambda }_2\left(x\right)}\right\},$$

$${\lambda }_2\left(x\right)={\left(\frac{m_2}{m_1}\right)}^{1/2}\times \frac{\ln [1+F_2\left(x\right)]}{{(x^2-x_{c1}^2)}^{1/2}},$$

$$F_2\left(x\right)=\left(\frac{m_2}{m_1}\right)\times \frac{x^2-x_{c1}^2}{x+{\left(m_2/m_1\right)}^{3/2}{(x^2-x_{c1}^2)}^{1/2}}.$$

As a function of x, concentration of carriers in the second band $n_s={\left(3{\pi }^2{\hslash }^3\right)}^{-1}[p_{F1}^3+p_{F2}^3]$ is given by the equation

$$x^3+{(m_2/m_1)}^{3/2}{(x^2-x_{c1}^2)}^{3/2}=195\times (n_s\times {10}^{-18}),$$

$$x>x_{c1}=\mathrm{6.1.}$$

(iii) For the upper band, Eq. 21:

$${\overline{T}}_3\left(x\right)=(x^2-x_{c2}^2)\exp \left\{-\frac{1}{{\lambda }_3\left(x\right)}\right\},$$

$${\lambda }_3\left(x\right)={\left(\frac{m_3}{m_1}\right)}^{1/2}\times \frac{\ln [1+F_3\left(x\right)]}{{(x^2-x_{c2}^2)}^{1/2}},$$

$$F_3\left(x\right)=\left(\frac{m_3}{m_1}\right)\times \frac{x^2-x_{c2}^2}{x+{\left(m_2/m_1\right)}^{3/2}{(x^2-x_{c1}^2)}^{1/2}+{\left(m_3/m_1\right)}^{3/2}{(x^2-x_{c2}^2)}^{1/2}}.$$

The corresponding expression $n_s={\left(3{\pi }^2{\hslash }^3\right)}^{-1}[p_{F1}^3+p_{F2}^3+p_{F3}^3]$ for the concentration of carriers in the third band as a function of x is

$$x^3+{(m_2/m_1)}^{3/2}{(x^2-x_{c1}^2)}^{3/2}+{(m_3/m_1)}^{3/2}{(x^2-x_{c2}^2)}^{3/2}=195\times (n_s\times {10}^{-18}),$$

$$\left(x>x_{c2}=20\right).$$

Summary

The main results from the above can be summarized as follows:

• i)The introduced notion of the mobility edge is the fundamental concept underlying the low-temperature properties of n-doped strontium titanate. The electrons doped in SrTiO₃ fall into the two groups with different properties. The first group consists of the localized electrons. Owing to the large dielectric constant ${\kappa }_0\approx {10}^4$ in the insulator SrTiO₃, these electrons occupying states below the mobility edge are responsible for a large ${\overline{\kappa }}_{\infty }$ on the order from ${10}^2$ to ${10}^3$. Note that ${\overline{\kappa }}_{\infty }$ is the only free parameter in the theory. Its value, however, depends on the specific doping procedure and varies in different samples.
• ii)At concentrations exceeding the mobility edge, the superconducting pairing in the metallic bands is mediated by the interaction with LO polar phonons. Arguments are given that in SrTiO₃ such interaction is attractive. The value of the Migdal parameter ${\omega }_0/E_F$ is inverted.
• iii)$T_C\ll E_F$ in low-doped SrTiO₃. In this particular case calculations could be performed analytically in the logarithmic approximation. The superconducting transition temperature $T_C\left(n_s\right)$ and its qualitative dependence on the concentration $n_s$ was obtained within a numeric factor of order unity. Three maxima in $T_C\left(n_s\right)$ and three superconducting gaps emerging in succession, while electrons fill one band after another, are the hallmarks of the LO optical phonon pairing mechanism.
• iv)The value of ${\overline{\kappa }}_{\infty }$ depends on the sample quality and so does the overall pattern for superconductivity in the (T, x) plane, as is confirmed by the comparison in Fig. 1C (14) with the data (7, 25). Therefore, the concentration that corresponds to the mobility edge cannot be determined unambiguously from the experiments (15).
• v)Pairing in a nonadiabatic regime opens a new prospect for increasing the temperature of the superconducting transition in the transition-metal oxides and other polar crystals. Thus, in the framework of the logarithmic approximation $T_C$ is proportional to the Fermi energy: the larger is $E_F$, the higher is $T_C$ (at a given value of the dimensionless coupling constant).
• vi)The above approach was generalized in ref. 26 to 2D superconductivity of the single-unit-thick FeSe layer deposited on the SrTiO₃ substrate (6).

Finally, in the extreme nonadiabatic limit the only relevant energy scale is the Fermi energy and hence the prefactor $W$ must be on the order of $E_F$. For the prediction of the $T_C$ value, in practice knowing the numerical factor in the expression $T_C=const\times E_F\exp (-1/\lambda )$ would be of importance.

However, the issue remains basically unexplored, because in the nonadiabatic regime the analysis of Eq. 9 beyond the logarithmic approximation is complicated by the presence of contributions from the Born corrections to the matrix element (27).