Single-gap superconductivity and dome of superfluid density in Nb-doped SrTiO3

TL;DR

This study investigates the superconducting properties of Nb-doped SrTiO$_3$ across its doping range, revealing a single-gap superconducting state and a dome-shaped superfluid density, with detailed microwave measurements down to very low temperatures.

Contribution

It provides the first detailed microwave optical response analysis of Nb-doped SrTiO$_3$ across its superconducting dome, demonstrating single-gap behavior and the dome-shaped evolution of key superconducting parameters.

Findings

Single-gap electrodynamics despite multiple bands.

Gap homogenization over the Fermi surface.

Superfluid density exhibits a dome shape with doping.

Abstract

SrTiO$_3$ exhibits a superconducting dome upon doping with Nb, with a maximum critical temperature \mbox{$T_\mathrm{c} \approx 0.4$~K}. Using microwave stripline resonators at frequencies from 2 to 23~GHz and temperatures down to 0.02~K, we probe the low-energy optical response of superconducting SrTiO$_3$ with charge carrier concentration from 0.3 to $2.2\times 10^{20}$~cm$^{-3}$, covering the majority of the superconducting dome. We find single-gap electrodynamics even though several electronic bands are superconducting. This is explained by a single energy gap $2\Delta$ due to gap homogenization over the Fermi surface consistent with the low level of defect scattering in Nb-doped SrTiO$_3$. Furthermore, we determine $T_\mathrm{c}$, $2\Delta$, and the superfluid density as a function of charge carrier concentration, and all three quantities exhibit the characteristic dome shape.

Full text

Amongst the numerous distinctive properties of SrTiO3 mueller-burkard_prb_1979 ; Szot2006 , its superconducting state is of particular interest Schooley1964 . Stoichiometric SrTiO3 is an insulating perovskite, but when charge carriers are introduced, e.g. by doping with Nb, SrTiO3 becomes metallic with Fermi-liquid properties Spinelli2010 ; Lin2015 ; marel_prb_2011 . Such doped SrTiO3 features several electronic bands at the Fermi level that can be filled consecutively, making SrTiO3 a model system for multi-band physics marel_prb_2011 ; behnia_prl_2014 . This includes superconductivity in SrTiO3 binnig_prl_1980 ; behnia_prl_2014 , which can reach a critical temperature $T_{\mathrm{c}}$ around 0.4 K, and is an important reference for the superconducting interface in SrTiO3/LaAlO3 heterostructures reyren-mannhart_science_2007 . Of particular interest is the evolution of superconductivity with doping: already a very small charge carrier density $n$ of $5.5\times 10^{17}$ cm*-3* suffices to induce superconductivity Lin2013 . Only surpassed by superconducting Bi Prakash_Science2017 , this charge carrier density places SrTiO3 in the non-adiabatic regime of superconductivity that is characterized by a Fermi velocity too low for conventional phononic coupling Behnia2017 ; Swartz_arxiv_2016 , thus leaving the mechanism for superconductivity in SrTiO3 a matter of ongoing discussion Edge2016 ; Stucky2016 . Particularly relevant for our study is the evolution of $T_{\mathrm{c}}$ as a function of Nb doping. With increasing charge carrier density, $T_{\mathrm{c}}$ first increases, but for densities larger than $1\times 10^{20}$ cm*-3* decreases again Lin2013 ; schooley-koonce_prl_1965 ; koonce-schooley_prl_1967 . Such behavior, a dome-shaped evolution of $T_{\mathrm{c}}$ as function of a tuning parameter, is found in the phase diagrams of numerous superconducting material classes. These include cuprate, heavy-fermion, organic, pnictide, granular, and interface superconductors Jerome1991 ; MathurNature1998 ; Lefebvre_PRL2002 ; BrounNaturePhys2008 ; Caviglia2008 ; RichterNature2013 ; ShibauchiRevCon2014 ; Chen_PRL2015 ; PrachtPRB2016 , and the nature of such superconducting domes remains in the focus of scientific activity. One aspect is the relation between $T_{\mathrm{c}}$ and other energy scales that are relevant for the superconducting state, such as the superconducting energy gap $2\Delta$ and the superfluid stiffness $J$ that is proportional to the superfluid density $\rho_{s}$. In particular, for various superconducting systems that exhibit domes of $T_{\mathrm{c}}$ and $\rho_{s}$, the causal relationship between these two quantities remains a highly controversial issue PrachtPRB2016 ; Uemura_PRL1989 ; Emery_Nature1995 ; Homes_Nature2004 ; Zuev_PRL2005 ; Broun_PRL2007 ; Luan_PRL2011 ; Bozovic_Nature2016 ; Lee-Hone_PRB2017 . Here superconducting SrTiO3 is an ideal model system to study the interplay between $T_{\mathrm{c}}$ and superfluid density: doping with Nb allows convenient control of electronic material properties including tuning through the superconducting dome. Furthermore, the role of disorder, which often complicates interpretation of composition-tuned superconducting domes, is well understood for SrTiO3. Finally, because of the small energy gap, all relevant parameters such as $2\Delta$ and $\rho_{s}$ can be obtained from a single (optical) experiment on the same specimen, as we show in this work.

Optical spectroscopy at THz and infrared frequencies is an established route for the investigation of superconductors with $T_{\mathrm{c}}$ of a few K and higher Basov2005 ; PrachtIEEE2013 , but for superconductivity in SrTiO3 one has to consider much lower relevant temperatures and frequencies. Consequently, we employ a microwave technique, namely superconducting stripline resonators suitable for the mK range SchefflerPSS2013 ; hafner_rsi_2014 . Since this spectroscopic technique probes the full complex electrodynamics of a superconductor, it directly indicates the superconducting gap $2\Delta$ and it quantifies the response of both the superfluid and the thermally excited quasiparticles. Combining our microwave investigation with normal-state transport measurements allows us to clearly elucidate the connections between pairing, phase stiffness, disorder, and multi-band electronic structure in Nb-doped SrTiO3.

In our stripline resonator configuration the Nb-doped SrTiO3 samples act as a ground plane hafner_rsi_2014 ; SchefflerActaIMEKO2015 , as shown schematically in the inset of Fig. 1(b). Using a cavity perturbation analysis, we obtain the surface impedance of the Nb-doped SrTiO3 for a wide frequency range of $\omega/2\pi=2$ to 23 GHz and at temperatures down to $T=0.02$ K, smoothly crossing from $\hbar\omega\ll k_{B}T$ to $\hbar\omega\gg k_{B}T$. In all cases the microwave penetration depth is greater than 1 $\mu$m, allowing the measurements to probe deeply into the superconducting bulk. Details on the samples, microwave measurements and analysis are discussed in the supplemental material SupplementalMaterial .

The microwave surface impedance gives direct access to the complex conductivity, $\sigma=\sigma_{1}+\mathrm{i}\sigma_{2}$, which in turn encodes the electrodynamic response of both quasiparticles and superfluid. At sub-gap frequencies, the only absorption mechanism contributing to $\sigma_{1}$ comes from thermally excited quasiparticles, and is exponentially small at low temperatures. Increasing the probing frequency further, pair-breaking sets in at $\hbar\omega=2\Delta$, leading to kinks in $\sigma_{1}(\omega)$ and $\sigma_{2}(\omega)$ at this frequency, as seen in Fig. 1, which displays conductivity spectra of the sample with $n_{\mathrm{Hall}}=2.0\times 10^{20}$ cm*-3* for several temperatures. Simultaneously fitting the Mattis–Bardeen equations mattis-bardeen_1958 to the real and imaginary parts of the conductivity spectra provides our first means of determining $2\Delta$, with results plotted as open symbols in Fig. 3(a). On a more fundamental level, the agreement of experimental spectra and Mattis–Bardeen fits suggests that the electrodynamic response of superconducting SrTiO3 is governed by a single superconducting gap. We also point out that the pronounced downturn in $\sigma_{2}(\omega)$ above the gap frequency (See Fig. 1b) is a subtle indication that disorder is relevant in this material SupplementalMaterial .

An independent way to extract $2\Delta(T)$ from our data, without relying on Mattis-Bardeen theory in detail, is by analyzing the temperature dependence of $\sigma_{1}$ for a set of frequencies, namely the resonator harmonics that we have available. For any given frequency $\omega$ that is smaller than the zero-temperature energy gap $2\Delta_{0}$, the temperature-dependent $\sigma_{1}(T)$ will indicate pronounced additional losses due to Cooper-pair breaking as the temperature is scanned through the point at which $2\Delta(T)$ falls below $\hbar\omega$. This is evident as kinks in $\sigma_{1}(T)$ in Fig. 2, which shows data for seven different resonator frequencies and $n_{\mathrm{Hall}}=2.0\times 10^{20}$ cm*-3*. The temperature dependence of the energy gap inferred by this method is plotted as the stars in the frequency-temperature plane of Fig. 2 and as solid symbols in Fig. 3(a).

The complete set of $2\Delta(T)$ is shown for all five samples in Fig. 3(a). These data are well described by the temperature dependence of $2\Delta$ predicted for a single-gap BCS superconductor TinkhamBook , where we treat the gap ratio $2\Delta_{0}/k_{\mathrm{B}}T_{\mathrm{c}}$ as an adjustable parameter. Fitted gap ratios are plotted in Fig. 3(b) and are in close accord with the BCS prediction for weak-coupling superconductivity, $2\Delta_{0}/k_{\mathrm{B}}T_{\mathrm{c}}=3.528$ TinkhamBook . This value is also in line with recent tunneling results for doped SrTiO3 thin films Swartz_arxiv_2016 . However, these findings are at odds with the early tunnelling data for doped SrTiO3, which appeared to resolve multiple gaps binnig_prl_1980 (as was observed for other multiband superconductors IavaronePRL2002 ; RubyPRL2015 ), but recently were interpreted in terms of surface superconductivity Eagles_JSNM2017 . Nevertheless this still leaves us with the question, why there is only one energy gap while there is strong evidence for multiple superconducting bands behnia_prb_2014 .

A resolution of this puzzle emerges from Anderson’s theorem Anderson1959 , which describes how conventional superconductivity is protected from nonmagnetic impurity scattering; disorder scattering does not suppress $s$-wave pairing, but instead homogenizes the energy gap over the Fermi surface. In the case of a multiband superconductor with a scattering rate $\Gamma$ larger than the superconducting gap(s), this homogenization causes a single value for the spectroscopic gap throughout the complete Fermi surface even if the different bands had well separated, distinct values in the absence of scattering. (See supplemental material for more details SupplementalMaterial .) To support this concept for our actual case, we have to consider the scattering rates that are present in our samples. Electronic scattering in doped SrTiO3 is typically quantified from the normal-state Hall mobility $\mu$ (see supplemental material SupplementalMaterial ), from which we extract a scattering rate via $\Gamma_{\mathrm{Hall}}\equiv e/(\mu\,m^{*})$ in a one-band interpretation and assuming several relevant values of the effective mass, $m^{*}=1.3,2$ and $4~{}m_{e}$ based on quantum oscillation measurements behnia_prl_2014 , with the heavier one expected to dominate in our dopant range. Furthermore, using the normal-state resistivity $\rho_{\mathrm{DC}}$ of our samples and the plasma frequency $\omega_{p}$ estimated from optical studies of Nb-doped SrTiO3 vanMechelen2008 , we can determine a transport scattering rate $\Gamma_{\rho}=\epsilon_{0}\omega_{p}^{2}\rho_{\mathrm{DC}}$ which in Fig. 4(c) is plotted together with comparable values $\Gamma_{\mathrm{IR}}$ for the samples of Refs. marel_prb_2011 ; vanMechelen2008 and $\Gamma_{\mathrm{Hall}}$. For all our samples, these scattering rates are larger than the respective superconducting energy gaps, thus validating our explanation of single-gap superconductivity caused by scattering in a multi-band system. Furthermore, the roughly linear increase of $\Gamma$ with $n_{\mathrm{Hall}}$ indicates that the dominant scattering mechanism is impurity scattering due to the Nb dopants.

It is important to place these ideas in the context of field-dependent measurements of thermal conductivity behnia_prb_2014 and surface impedance (see supplemental material SupplementalMaterial ), which indicate clearly that multiple bands contribute to superconductivity. These field-dependent probes reveal two distinct scales: the upper critical field, $B_{c2}=\Phi_{0}/2\pi\xi^{2}$, the point at which vortex spacing reaches the coherence length $\xi$ and superconductivity is globally destroyed; and a lower field scale, $B^{\ast}$, indicating an additional, longer superconducting coherence length. In a multiband system, band-specific coherence lengths $\xi_{i}\equiv\hbar v_{\mathrm{F},i}/\pi\Delta_{i}$ (index $i$ for different bands) TinkhamBook naturally arise from band-to-band variation in either the energy gap, $\Delta_{i}$, or Fermi velocity, $v_{\mathrm{F},i}$. In Nb-doped SrTiO3, quantum oscillation studies behnia_prl_2014 indicate enough variation in $v_{\mathrm{F},i}$ that we expect multiple field scales to emerge even in the presence of the homogeneous energy gap implied by the spectroscopic measurements.

Further insights into the structure of the superconducting gap come from the temperature dependence of superfluid density. In previous studies Kogan_PRB2009 , multi-band gap structure in $\rho_{s}(T)$ exhibited either as pronounced, mid-range upwards curvature, e.g., in V3Si Nefyodov_epl_2005 , or as the presence of small activation energies, e.g., in MgB2 Fletcher_prl_2005 and FeSe Teknowijoyo_PRB2016 ; Meng_njp_2016 . In our measurements, superfluid density is obtained from the low frequency limit ($\omega/2\pi\approx 2$ GHz) of the out-of-phase conductivity, $\rho_{\mathrm{s}}(T)\equiv\lambda^{-2}=\omega\mu_{0}\sigma_{2}(\omega,T)$, where $\lambda$ is the London penetration depth TinkhamBook . We note that for low $T_{\mathrm{c}}$ superconductors such as Nb-doped SrTiO3, electrodynamic measurements are an excellent means for determining absolute superfluid density, as the superconducting penetration depth is measured relative to the high-frequency skin-depth in the nearby normal state, providing a reliable reference point that can be calibrated in terms of dc resistivity measured on the same sample pippard . The temperature dependences of the absolute superfluid density of all five samples are shown in Fig. 4(a), along with fits to $\rho_{s}(T)$ from single-gap, weak-coupling BCS in the presence of nonmagnetic disorder (see supplemental material SupplementalMaterial ). These fits provide additional experimental evidence for single-gap behavior, and the values $\Gamma_{\mathrm{sf}}$ for the scattering rate that we obtain from the fits are consistent with the other estimates shown in Fig. 4(c) and thus confirm our interpretation of single-gap superconductivity in terms of Anderson’s theorem.

Furthermore, the zero-temperature limits $\rho_{s0}$ of the fits for $\rho_{s}$ allow us to study the evolution of the superfluid density as a function of normal-state charge carrier density. In Fig. 4(b) we plot $\rho_{s0}$ versus $n_{\mathrm{Hall}}$, and we find a dome shape rather similar to the domes of $T_{\mathrm{c}}$ and $2\Delta_{0}$. A dome of superfluid density in Nb-doped SrTiO3 has also recently been reported in Ref. Collignon2017, . As additional context, Fig. S5 shows that Nb-doped SrTiO3 follows the Homes’ law scaling between $\rho_{s}$ and $\sigma_{\mathrm{DC}}T_{\mathrm{c}}$.

Following the Ferrell-Glover-Tinkham sum rule, $\rho_{s0}$ is proportional to the spectral weight in $\sigma_{1}(\omega)$ that is lost in the superconducting state compared to the normal state TinkhamBook . For all our samples the scattering rate is larger than $2\Delta$, placing them in the dirty limit, and thus the transferred spectral weight is limited by $2\Delta$ (or $T_{\mathrm{c}}$). Consequently the $T_{\mathrm{c}}$ dome in SrTiO3 is not governed by $\rho_{s0}$. This result is particularly interesting in the context of the ongoing discussion concerning the superconducting domes in other material classes, most notably the cuprates. These domes of superconductivity are also often accompanied by $\rho_{s0}$ behavior that qualitatively tracks the rise and fall of $T_{\mathrm{c}}$. However, the causal relationship between $\rho_{s}$ and $T_{\mathrm{c}}$ remains a point of major controversy, in particular whether superfluid density places bounds on $T_{\mathrm{c}}$ or, instead, $T_{\mathrm{c}}$ controls the spectral weight available to form the superfluid Bozovic_Nature2016 ; Emery_Nature1995 ; Kogan_PRB2013 ; Lee-Hone_PRB2017 . Our findings demonstrate that for SrTiO3 the latter explanation holds and that the electronic scattering due to disorder has to be considered for a full understanding of the superfluid density. This scattering rate in superconducting SrTiO3, which is very small for a metallic system but unavoidable due to the required charge carrier doping, is also a crucial ingredient for our observation that Nb-doped SrTiO3 is a single-gap, multi-band superconductor.

We thank G. Untereiner for resonator and sample preparation and K. Behnia and H. Y. Hwang for helpful discussions. Financial support by Carl-Zeiss-Stiftung (M.T.), DFG, and DAAD is thankfully acknowledged. D.M.B. and N.R.L.-H. gratefully acknowledge financial support from the Natural Science and Engineering Research Council of Canada and the Canadian Institute for Advanced Research. M.T. and M.H.B. contributed equally to this work.