Tests of nematic-mediated superconductivity applied to Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$

TL;DR

This paper proposes empirical tests, supported by calculations, to investigate whether nematic quantum fluctuations mediate superconductivity in Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$, emphasizing uniaxial strain experiments as a key probe.

Contribution

It introduces specific experimental tests and theoretical calculations to assess nematic-mediated pairing in a tunable superconductor system.

Findings

Predictions for uniaxial strain experiments to detect nematic fluctuations

Quantitative model supporting nematicity's role in pairing

Potential evidence for nematic-mediated superconductivity

Abstract

In many unconventional superconductors, nematic quantum fluctuations are strongest where the critical temperature is highest, inviting the conjecture that nematicity plays an important role in the pairing mechanism. Recently, Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$ has been identified as a tunable nematic system that provides an ideal testing ground for this proposition. We therefore propose several sharp empirical tests, supported by quantitative calculations in a simple model of Ba$_{1-x}$Sr$_x$Ni$_2$As$_2$. The most stringent predictions concern experiments under uniaxial strain, which has recently emerged as a powerful tuning parameter in the study of correlated materials. Since uniaxial strain so precisely targets nematic fluctuations, such experiments may provide compelling evidence for nematic-mediated pairing, analogous to the isotope effect in conventional superconductors.

Equations

$$V_{k,p}^{(ind)}\approx -h(k)\delta (k-p)$$

$$\delta \lambda =\left(\oint \frac{d{\mathbf{k}}_{\parallel }}{v_F^2({\mathbf{k}}_{\parallel })}h({\mathbf{k}}_{\parallel })\right){\left(\oint \frac{d{\mathbf{k}}_{\parallel }}{v_F({\mathbf{k}}_{\parallel })}\right)}^{-1},$$

$$\frac{T_c}{T_{c,0}}=$$

$$\approx$$

$$\frac{\Delta (k)}{{\Delta }_{max}}\approx 1+\left(\frac{1}{{\lambda }_0}\right)\left(\frac{h(k)}{v_F(k)}-{\left[\frac{h(k)}{v_F(k)}\right]}_{max}\right)$$

$$S_{int}=$$

$$D(q)=\frac{{\chi }_0}{1+{\xi }^2{q}^2}\Theta \left(\Lambda -|q|\right)$$

$$V_{k,p}\approx$$

$$\approx$$

$$\text{where }g\equiv$$

$$\frac{{\left(\delta T_c\right)}_{band}}{T_c}\propto \left(\frac{1}{{\lambda }_0}\right){\left(\frac{\alpha \kappa }{E_F}\right)}^2{\chi }_0^2{ϵ}^2$$

$$\frac{\delta {\chi }_0}{{\chi }_0}\approx -(b_{\chi }u{\kappa }^2){\chi }_0^{y_{ϵ}}{ϵ}^2,\frac{\delta \xi }{\xi }\approx -(b_{\xi }u{\kappa }^2){\chi }_0^{y_{ϵ}}{ϵ}^2$$

$$\frac{{\left(\delta T_c\right)}_{fluc}}{T_c}\propto +\left(\frac{[d-1]b_{\xi }-b_{\chi }}{{\lambda }_0}gu{\kappa }^2\right){\chi }_0^{y_{ϵ}}{ϵ}^2,$$

Full text

Introduction: Considerable debate remains regarding the pairing mechanisms of most unconventional superconductors. Many hypothesize that fluctuations of some order parameter(s), such as magnetism, mediate the bulk of the pairing interaction. These hypotheses are plausible, in part, because the superconducting region of the phase diagram is often close to various other forms of long range order. A ubiquitous form of such order is nematicity, which breaks the discrete point group symmetry of the crystal lattice. Theoretical studies have established, in the abstract, that nematic fluctuations promote superconductivityYamase and Zeyher (2013); Maier and Scalapino (2014); Raghu et al. (2015); Metlitski et al. (2015); Lederer et al. (2015); Kang and Fernandes (2016); Wang et al. (2016); Liu and Wang (2017); Lederer et al. (2017); Klein and Chubukov (2018); Klein et al. (2018); Abanov et al. (2018); Wu et al. (2019), but their relevance to the superconductivity of any given material is difficult to assess. There is thus a need for distinctive and testable predictions based on the hypothesis of nematic-mediated pairing.

In addition to concrete predictions, testing the hypothesis requires sufficiently simple material systems, preferably ones with tunable nematic fluctuations. For instance, while the Fe-based superconductors exhibit ubiquitousKuo et al. (2016) and tunable nematic fluctuations, these are (with rare exceptionsAbdel-Hafiez et al. (2015); Hosoi et al. (2016)) accompanied by the confounding factor of a nearby magnetic phase.

Happily, recent experiments have identified a tunable, nonmagnetic model system in the Ba1-xSrxNi2As2 seriesRonning et al. (2008); Sefat et al. (2009); Kothapalli et al. (2010); Eckberg et al. (2019). The nematic susceptibility in the B1g channel111We here use the crystallographic two-Ni unit cell. Note that the channel of strong nematic fluctuations in iron-based superconductors is $B_{2g}$ in this nomenclature (measured by elastoresistance) grows as the Sr concentration $x$ is reduced from 1.0 to 0.7, from below the noise floor to a large (dimensionless) value of nearly 20. Meanwhile, the superconducting critical temperature, $T_{c}$, rises dramatically, from $0.6K$ to 3.5 $K$. The typical determining factors of $T_{c}$ in BCS superconductors–the Debye frequency and the density of states–vary negligibly in this range of $x$, leading the authors of [Eckberg et al., 2019] to suggest that nematic fluctuations are responsible for the enhancement.

Here, we examine the influence of nematic fluctuations on various superconducting properties, based both on general considerations and on explicit calculations for a simple model of the Ba1-xSrxNi2As2 system, for $0.7<x<1.0$. One set of predictions follows from anisotropy (momentum dependence) of the superconducting gap, which nematic fluctuations promote by virtue of their anisotropic coupling to electrons. The other set concerns the effects of uniaxial strain, which explicitly breaks lattice rotation symmetry and reduces the strength of nematic fluctuations. Both sets of predictions are essentially general, but those regarding strain are sharper, may apply to a broader variety of materials, and are particularly timely in light of the increasing use of strain as an experimental tuning parameterHicks et al. (2014).

General considerations: The influence of nematic fluctuations on superconductivity can be comprehensively analyzed within a weak coupling framework except asymptotically close to a nematic quantum critical pointLederer et al. (2015). Since there is no indication of a zero temperature divergence of the nematic susceptibility in the $x-T$ phase diagram of Ba1-xSrxNi2As2Eckberg et al. (2019), the weak coupling approach may give qualitatively correct results in the entire doping range under consideration. For our purposes, the main results are that nematic fluctuations 1) enhance $T_{c}$, potentially by a large amount, as shown in Fig. 1; 2) promote a characteristic anisotropy of the superconducting gap, whose consequences are shown in Figs. 2 and 3; and 3) dramatically enhance the influence of uniaxial strain on superconductivity, as shown in Fig. 4. In this section, we summarize some key results of the weak coupling approach of [Lederer et al., 2015], to which we refer the reader for further details.

Consider, for simplicity, a weak coupling s-wave superconductor with density of states $\rho_{0}$ at the Fermi level. Assume the superconductivity is due to a momentum-independent attractive interaction of strength $V_{0}$ operating at frequency $\Omega_{0}$ (this would be the Debye frequency for conventional phonon-mediated attraction). Then the gap and $T_{c}$ are both of order $\Omega_{0}\exp(-1/\lambda_{0})$, where $\lambda_{0}=\rho_{0}V_{0}$ is the pairing eigenvalue.

Now weakly couple the system to a separate nematic mode with a long correlation length $\xi$. The nematic fluctuations mediate an attractive interaction $V^{(ind)}_{k,p}$, that is dominated by forward scattering, since the interaction is negligible for momentum transfer greater than $\xi^{-1}$. Accordingly, as $\xi$ increases, $V^{(ind)}_{k,p}$ becomes nearly diagonal in momentum, and can be approximated by a delta function:

[TABLE]

Here $h(k)\geq 0$ is a coupling function governed by the symmetry of the nematic order parameter, and we will take an explicit form in the next section. $h$ increases as the strength of nematic fluctuations increases. Because $V^{(ind)}_{k,p}$ is diagonal in momentum, it is simple to compute its pairing eigenvalues. The largest is $\lambda^{ind}=\text{max}[h(k)/v_{F}(k)]$, where $v_{F}(k)$ is the Fermi velocity and the maximum is computed for $k$ on the Fermi surface. We assume that $\lambda^{ind}\ll\lambda_{0}$, so that the “bare” attraction is still the dominant part of the pairing interaction222The opposite case $\lambda^{ind}\gg\lambda_{0}$ can also be analyzed straightforwardly. We ignore the effect of retardation, namely that $V^{ind}$ operates at a lower frequency scale than $V_{0}$. This effect can easily be incorporated, but in weak coupling is parametrically less important than the pairing eigenvalue effects considered here..

With these assumptions, the changes in $T_{c}$ and the gap function can be computed using perturbation theory in $V^{(ind)}$, applied to the linearized gap equation. The new pairing eigenvalue is $\lambda=\lambda_{0}+\delta\lambda$, where

[TABLE]

With the integrals over the Fermi surface. The pairing eigenvalue has been increased by a suitably weighted average of the coupling function, but by a small amount $\delta\lambda<\lambda^{(ind)}\ll\lambda_{0}$. $T_{c}$ has also been increased:

[TABLE]

Note that the enhancement of $T_{c}$ can be large despite the fact that $\delta\lambda/\lambda_{0}$ is small, provided $\delta\lambda\gtrsim\lambda_{0}^{2}$. The new gap function can also be computed in perturbation theory. To leading order,

[TABLE]

The gap function is now anisotropic, with the pattern of anisotropy determined by the band structure and $h(k)$, which is in turn governed by the symmetry of the nematic order parameter.

Model and numerical results: Here we consider an effective model for the physics of Ba1-xSrxNi2As2. In the absence of nematic fluctuations, the model comprises a fermion band structure and a weak attractive interaction, giving rise to conventional BCS superconductivity. We then couple the fermions to a nematic bosonic mode $\phi$ in a symmetry-appropriate way, and specify the “bare” fluctuation spectrum of $\phi$. Explicit calculations are performed using the perturbative renormalization group approach of [Lederer et al., 2015].

We model the low energy bandstructure (as measured in angle-resolved photoemissionZhou et al. (2011)) with four cylindrical sheets, two centered at the M point and of radius $k_{1}$ and one each at the X and Y points with radius $k_{2}<k_{1}$. We assume parabolic dispersion with masses $m_{1,2}$. Before coupling to nematic fluctuations, we take the interaction between fermions to be a momentum independent attraction of strength $V_{0}>0$, so that there is BCS superconductivity with $T_{c}$ and gap proportional to $\exp(-1/\lambda_{0})$, with $\lambda_{0}\propto(m_{1}+m_{2})V_{0}$ the unperturbed pairing eigenvalue. We couple the fermions to the nematic mode $\phi$ with the term

[TABLE]

where spin and band indices are implicit. The form factor $f(k)\equiv\sin(k_{x}a)\sin(k_{y}a)$ (with $a$ the in-plane lattice constant) specifies the the fermion bilinear to which $\phi$ couples, in this case an anisotropic next-nearest-neighbor hopping. The precise form of $f(k)$ depends on microscopic details, but must respect the $B_{1g}$ symmetry of the nematic fluctuations. Since $f(k)$ vanishes at the high symmetry points $X,Y,M$, it is natural that the smaller pockets at X and Y have overall weaker coupling than the larger pocket at M.

We assume a mean-field form of $D(q)$, the static correlation function of $\phi$:

[TABLE]

Here $\chi_{0}$ is the thermodynamic nematic susceptibility, $\xi$ is the correlation length, and we have introduced a hard momentum cutoff $\Lambda$ satisfying $\xi^{-1}\ll\Lambda\ll k_{1},k_{2}$. We will tune the strength of nematic fluctuations by tuning $\xi$ and $\chi_{0}$, which are related by $\chi_{0}\sim\xi^{2}$. Both $\xi$ and $\chi_{0}$ will increase as strontium concentration $x$ is decreased from $1.0$ to $0.7$ (i.e. as nematic correlations grow in strength). For explicitness we take $\chi_{0}$ to equal the (linearly-interpolated) nematic susceptibility measured by elastoresistanceEckberg et al. (2019).

Integrating out the boson, we obtain a four-fermion interaction $V^{(ind)}_{k,p}=-\alpha^{2}f^{2}([k+p]/2)D(k-p)/4$. For $\Lambda\xi\gg 1,$ $D(q)$ is sharply peaked at $q=0$, and so can be approximated by a delta function of weight $W\equiv\int d^{d-1}qD(q)$, where the integral is over the Fermi surface. $W$ depends on the dimensionality of the nematic fluctuations, so we treat both the two and three dimensional cases333Even in a system whose electronic structure is highly two dimensional, nematic fluctuations may be rendered three dimensional by coupling to the lattice. However, the coupling to acoustic phonons introduces long range interactions that qualitatively alter the low energy physicsZacharias et al. (2015); Labat and Paul (2017); Paul and Garst (2017), a complexity we ignore here for simplicity..

Making these approximations, the full pairing interaction is $V_{k,p}=-V_{0}+V^{(ind)}_{k,p}$, or:

[TABLE]

Here the dimensionless coupling constant $g$ sets the scale of fractional changes to the pairing eigenvalue. In $d=3$, $g$ also contains a factor $\log[1+(\Lambda\xi)^{2}]$.

We discretize the Fermi surface and numerically diagonalize the linearized gap equation, yielding the maximum pairing eigenvalue $\lambda$ and therefore $T_{c}$, both shown in Fig. 1. The corresponding eigenfunction is the pair wave function, which is increasingly anisotropic as nematic fluctuations grow with decreasing $x$, as illustrated by Eq. 4. This anisotropy can be experimentally identified by the splitting of peaks in the density of states, which occur at local maxima in the gap. The density of states for selected values of $x$ is shown in Fig. 2 and can be measured by tunneling spectroscopy in either planar or scanning geometries. The gap anisotropy is also indirectly measurable in the ratio of the zero temperature gap maximum to $T_{c}$, and in the fractional specific heat jump at $T_{c}$, as shown in Fig. 3. The gap to $T_{c}$ ratio increases, while the specific heat jump decreases.444Though not obvious from the figure, the change in specific heat is parametrically smaller (in the weak coupling limit) than the other effects of the induced gap anisotropy, going as $\delta\lambda^{2}$ instead of $\delta\lambda$, as discussed in the supplementary material

Effects of B1g strain**: When nematic fluctuations are strong, the system is highly susceptible to the influence of a symmetry breaking field such as uniaxial strain. The effect of such strain on $T_{c}$ becomes increasingly dramatic at large $\chi_{0}$, but whether strain increases or decreases $T_{c}$ depends on microscopic details. Strain also has two qualitatively distinct effects on $T_{c}$: 1) It alters the band structure, leading to changes in $T_{c}$, as in Eq. 9; and 2) It cuts off nematic fluctuations and alters the pairing interaction, as shown in Eqs. 10 and 11.

Let strain couple to $\phi$ with coupling constant $\kappa$. At weak strain, $\phi$ acquires an expectation value $\langle\phi\rangle\approx\chi_{0}(\kappa\epsilon)$, with $\chi_{0}$ the thermodynamic nematic susceptibility as previously defined. Therefore, any physical quantity with B1g symmetry takes on a value proportional to $\epsilon$. In particular the band structure acquires a (fractional) anisotropy of order $\alpha\chi_{0}\kappa\epsilon/E_{F}$. Such band structure effects alter any quantity that is nonzero at $\epsilon=0$, such as the density of states–and therefore $T_{c}$–by an amount proportional to the square of this anisotropy (linear variation is forbidden since such quantities are even under $\epsilon\to-\epsilon$)555The discussion in this section is largely general, but here we specialize to the case of a single-component order parameter. Multicomponent order parameters (for instance the degenerate $d_{xz}$ and $d_{yz}$ in a tetragonal system) permit a $T_{c}$ which varies as $|\epsilon|$.. In particular, this effect changes $T_{c}$ by

[TABLE]

where the proportionality constant is a dimensionless number of order one, whose magnitude and sign depend on microscopic details.

The application of strain alters not only the band structure, but the interactions. This is because strain cuts off nematic fluctuations, reducing the susceptibility $\chi_{0}$ and the correlation length $\xi$ by amounts proportional to $\epsilon^{2}$. As the (zero-strain) $\chi_{0}$ increases, the coefficient of $\epsilon^{2}$ diverges as $\chi_{0}^{y_{\epsilon}}$, where $y_{\epsilon}\equiv 2+2\beta/\gamma=\{2.26\text{ in }d=2,\ 3\text{ in }d=3\}$ and $\beta,\gamma$ are the order parameter and susceptibility exponents, respectively. See the Supplementary Material for a derivation. Note that $y_{\epsilon}>2$, so these effects are parametrically stronger than the band structure effects described in Eq. 9. Accordingly, the correlation length and susceptibility vary as

[TABLE]

Here $u>0$ is a dimensionful scale related to the self-interaction of $\phi$ (for instance the quartic term in Landau theory, as discussed in the Supplementary Material), and $b_{\chi,\xi}$ are dimensionless quantities. None of these quantities are specified by our model, but they are all positive since the symmetry breaking field $\epsilon$ cuts off nematic fluctuations.

Per (7), the relative strength $g$ of the nematic-mediated interaction varies as the ratio of $\chi_{0}$ to $\xi^{d-1}$. Accordingly, $g$ (and therefore $T_{c}$) can either increase or decrease, since both $\chi_{0}$ and $\xi$ decrease with $\epsilon$. Up to logs,

[TABLE]

so that this effect increases (decreases) $T_{c}$ for $[d-1]b_{\xi}>b_{\chi}$ ($[d-1]b_{\xi}<b_{\chi}$). We show in Fig. 4 the second derivative of $T_{c}$ with respect to $\epsilon$, normalized by its value at $\epsilon=0$. Though the sign depends on microscopic details, the magnitude grows dramatically with decreasing $x$, much more so than the pairing eigenvalue (Fig. 1) or the gap anisotropy (Figs. 2, 3).

Discussion: We have shown that even weak coupling of the electrons near the Fermi surface to nematic fluctuations can explain the observed increase in the $T_{c}$ of Ba1-xSrxNi2As2 as $x$ is reduced from 1.0 to 0.7. If this explanation is correct, there are a number of experimental consequences, as calculated above. In brief: 1) nematic fluctuations promote gap anisotropy, whose signatures are observable in tunneling conductance, specific heat, and the gap to $T_{c}$ ratio, among others; 2) uniaxial strain substantially affects $T_{c}$ and other superconducting properties, by altering both the band structure and the strength of the nematic-mediated interaction. The predictions regarding strain are asymptotically stronger effects than those regarding gap anisotropy, which could itself have explanations unrelated to nematic fluctuations. We therefore consider strain experiments to be the sharpest tests of the hypothesis of nematic-mediated pairing.

That said, it is intrinsically difficult to empirically establish a pairing mechanism based on the exchange of electronic fluctuations. In part, this is a matter of principle: Unlike the phonons of BCS theory, the electronic fluctuations considered are generally not well defined normal modes666Order parameter fluctuations are well defined near a quantum critical point at which they order. This is essentially the scenario considered here, since we assume large $\chi_{0}$., so it isn’t clear what it means for electrons to exchange them. But an even greater difficulty is that most forms of electronic fluctuations do not admit a simple tuning parameter to establish their relevance, as the isotope effect establishes the relevance of phonons. However, breaking the symmetry corresponding to a nearly ordered mode is essentially guaranteed to affect it much more than other aspects of the physics. As such, strain experiments to assess nematic-mediated pairing provide perhaps the most tightly controlled evaluations to date of (non-phonon) fluctuation-mediated pairing.

We close with a brief speculation regarding normal state properties. We have not computed these properties directly, but the symmetry of the nematic order parameter essentially guarantees substantial anisotropyDell’Anna and Metzner (2006) of single particle properties measurable in photoemission, such as the effective mass and the linewidth. Such anisotropy is indeed found in numerical simulations of simple models for nematic quantum criticalityLederer et al. (2017). The strength of this anisotropy should show substantial dependence on the nematic fluctuation strength $\chi_{0}$. We eagerly await further investigations of the normal and superconducting state of this exciting material.

Acknowledgements We gratefully acknowledge helpful discussions with Rafael Fernandes, Steve Kivelson, and Johnpierre Paglione, as well as data shared by JP and Christopher Eckberg. SL was supported in part by a Bethe/KIC fellowship at Cornell. EB was supported by the European Research Council (ERC) under grant HQMAT (grant no. 817799) and by the Israel-USA Binational Science Foundation (BSF). Theoretical studies by SL and E-AK were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-SC0018946.