Identifying a forward-scattering superconductor through pump-probe spectroscopy

The discovery of enhanced superconductivity in FeSe intercalates [1] and ultrathin films of FeSe grown on oxides substrates [2], has attracted considerable interest. In these systems, significant electron doping drives the Γ-centered hole bands below the Fermi level E_F [3]. At face value, this Fermi surface topology challenges the picture of $s_\pm$ pairing mediated by spin fluctuations enhanced by Fermi surface nesting [4]. Motivated by this, several alternative proposals have been advanced, including contributions from the so-called incipient hole band below E_F [5–7]; or from some other bosonic mode such as nematic fluctuations [8] or phonons [2,9–13]. In particular, a unique cross-interfacial electron-phonon (e-ph) coupling has been invoked to explain the higher T_c values observed in FeSe films grown on oxide substrates in comparison to the doped FeSe intercalates [10,11,14,15]; this interpretation, however, is controversial [16–21], and finding experimental probes that might provide new insight into this problem is imperative.

A common feature of the scenarios based on nematic fluctuations or cross-interface e-ph coupling is the importance of forward scattering, where the relevant electron-boson coupling constant is peaked strongly at small momentum transfers q. Such a momentum structure can have significant consequences for superconductivity [12,22–24], enhancing pre-existing superconductivity in unconventional pairing channels, or resulting in significant T_c values if the interaction is sufficiently peaked around q = 0 [10,12,22,24–26]. These results naturally raise the question: how can one uniquely identify superconductivity mediated by a forward-focused interaction?

We address this issue in this letter by studying the problem in the time domain. We consider superconductivity mediated by an optical phonon in the idealized perfect forward-scattering limit (i.e., a delta-function coupling) and contrast its behavior with the case in which superconductivity is mediated by an isotropic interaction. (While we adopt a phonon mode for simplicity, our results are also applicable to the pairing mediated by nematic fluctuations.) In both cases, we drive the system into a non-equilibrium state by an ultrafast pump pulse and study the properties of the transient non-equilibrium state. We find that the two different types of momentum structures —isotropic and forward focused— show significant contrasting behavior in spectroscopic measurements. First, the suppression of the superconducting order in the isotropic case shows a strong dependence on the pump fluence, while the forward-scattering superconducting state is comparatively robust against pump-induced suppression. The energy absorption is also much smaller in the forward focused case. Second, while the suppression of the superconducting gap in the isotropic case is uniform in momentum space, the forward scattering shows a specific momentum-dependent suppression of the gap. This particular contrasting behavior can be measured in momentum- and time-resolved measurements, such as trARPES, and can provide evidences to either validate or falsify the forward-scattering mechanism in FeSe monolayers.

The observed effects can be understood as a direct consequence of the restricted phase space available to scattering process in the forward-scattering case. Figure 1 shows schematic scattering processes at E_F, where electrons are driven by an ultrafast pump field A(t). In the case of an isotropic e-ph interaction, an electron can scatter to all energy (and momentum) conserving eigenstates. In contrast, in the forward focused case, the only final states close to the initial state are available to an electron, i.e., momentum transfer $\mathbf{q}\approx0$. This restriction on phase space hinders the energy absorption in forward scattering and enables a non-trivial momentum structure of the gap [27], resulting in the observed effects.

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We consider the Holstein Hamiltonian on a 2D lattice,

$$\begin{eqnarray} \mathcal{H} =\,& \sum_{\mathbf{k},\sigma}\xi(\mathbf{k}) c^\dagger_{\mathbf{k},\sigma} c^{\phantom\dagger}_{\mathbf{k}, \sigma} + \sum_{\mathbf{q},\nu} \Omega_\nu \left(b_{\mathbf{q},\nu}^\dagger b^{\phantom\dagger}_{\mathbf{q},\nu} + \frac{1}{2} \right) \nonumber\\ & {}+\frac{1}{\sqrt{N}}\sum_{\substack{\nu,\sigma\\&&\mathbf{k},\mathbf{q}}}g(\mathbf{k},\mathbf{q}) c_{\mathbf{k}+\mathbf{q}, \sigma}^\dagger c^{\phantom\dagger}_{\mathbf{k},\sigma}(b^{\phantom\dagger}_{\mathbf{q},\nu}+ b_{-\mathbf{q},\nu}^\dagger). \end{eqnarray} \tag{ 1 }$$

Here, $\xi(\mathbf{k})$ is the nearest-neighbor tight-binding energy dispersion measured relative to the chemical potential, $c^\dagger_{\mathbf{k}}, c^{\phantom\dagger}_{\mathbf{k}}\,(b^\dagger_{\mathbf{q}}, b_{\mathbf{q}})$ are the standard creation and annihilation operators for an electron (phonon), $g(\mathbf{k},\mathbf{q})$ is the momentum-dependent e-ph coupling constant, and $\Omega_\nu$ is the frequency for the Einstein phonon mode ν. Here we take an effective-minimal model to capture the distinctive features of the forward scattering. Although the interaction term in the Hamiltonian produces an effective attractive electron-electron interaction, a formally identical boson exchange model can be used to treat repulsive spin-fluctuation–mediated interactions [28]. We also note that while FeSe monolayers are quite complicated (e.g., in their multiband nature, spin and nematic fluctuations), we assume the forward-scattering mechanism to be dominant and ignore further complications. This is sufficient to test the forward-scattering mechanism.

The superconducting state is modeled by the standard two-particle Nambu spinors in the self-consistent Migdal-Eliashberg formalism and time evolution is done by solving the Gor'kov equations self-consistently on the Keldysh contour [29] with an ultrafast pump field $A(t) = A_{\mathrm{max}} \exp(-t^2/2\sigma_p^2)\sin(\omega_p t)$, included via Peierls' substitution (see Supplemental Material Supplementarymaterial.pdf (SM)). We consider two types of e-ph interactions: an isotropic coupling with $g(\mathbf{k},\mathbf{q}) = g\sqrt{N}$, a constant, and a forward-scattering interaction where $g(\mathbf{k},\mathbf{q})$ is taken in the limit of perfect forward scattering $g(\mathbf{k},\mathbf{q}) = g\sqrt{N} \delta_{\mathbf{q},\mathbf{0}}$ [27] to reduce the computational cost. The different coupling strengths are tuned to get the same approximate value of the superconducting gap $(\Delta \sim 51\ \text{meV})$ for both isotropic and forward focused interactions. For completeness, we include a solution in the d-wave channel as well. To create an s- and d-wave symmetric superconducting gap, a particular symmetry is imposed on the matrix elements of the interaction, i.e., coupling strength. For the isotropic interaction we choose $g(\mathbf{k},\mathbf{q}) = g_{s} + g_d d_{\mathbf{k}} d_{\mathbf{q}}$, where g_s (g_d) are constants for s-wave (d-wave) pairing, $d_{\mathbf{k}} = \frac{1}{\sqrt{2}} [\cos(k_x) - \cos(k_y)]$ is the form factor for the d-wave pairing [30,31]. For the forward focused interaction we choose a particular symmetry of the superconducting gap as the initial condition for the self-consistent procedure, that converges to the s- or d-wave paring. Note that the particular symmetry is imposed only on the anomalous part of the self-energy matrix.

Table 1:.  Parameters used in the simulation.

Parameters	Isotropic	Forward
Phonon frequency $(\Omega_{1,2})$	0.2, 0.001 eV	0.2, 0.05 eV
Phonon coupling $(g^2_{1,2})$	0.12, 0.001 eV	0.034, 0.01 eV
Band parameters	$V_{nn}=0.25\ \text{eV}$, $\mu = 0.0\ \text{eV}$
Temperature	$\approx 83\ \text{K}$
Pump pulse	$\omega_p = 1.5\ \text{eV}$, $\sigma_p = 8\ \text{(1/eV)}$
Probe pulse	$\sigma = 25\ \text{(1/eV)}$

The nature of the perfect forward scattering restricts the system's ability to absorb any energy via scattering processes. Realistically, the coupling has some finite width [10,24,32], and there are additional phonons to scatter from in the system. To account for these effects, we include a second isotropically coupled phonon in the system, whose coupling is weak enough that it can dissipate energy but does not significantly contribute to the superconductivity. Table 1 lists the parameters used in the simulation.

First, we compare the effect of the pump pulse on the superconductor through the single-particle electron spectral function. Figure 2 displays the calculated time-resolved and angle-resolved photoemission spectroscopy (tr-ARPES) spectra of an s-wave superconductor for both types of scattering. The tr-ARPES spectra are calculated using the lesser part $(G_{\tilde{\mathbf{k}}}^<)$ of the normal Green's function [33,34] (see SM). For isotropic scattering (the top panels) the equilibrium band structure in normal state (fig. 2(a)) shows a kink at the phonon energy $(\pm \Omega_1)$, which is common for strong coupling to a phonon in the system [35]. The normal-state band structure for the forward focused case (fig. 2(e)) exhibits different behavior [12]; most notably there is a replica band (indicated by the red markers) below the main band similar to the experimentally observed [10,15,36] features of the FeSe monolayer superconductors that have been explained by forward scattering from phonons. The band is flatter in the vicinity of E_F due to the perfect forward-scattering limit adopted here [27].

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The remaining panels in fig. 2 show the tr-ARPES spectra in the superconducting state, where a pump pulse of amplitude $A_{\mathrm{max}} = 1.2 (V/a_0)$ is used to drive the system along the (11) direction. This fluence is high enough to melt the superconducting order significantly for isotropic pairing [29], as shown in figs. 2(c), (d). The superconducting state in equilibrium is shown in fig. 2(b). The spectra show the characteristics of a strong-coupling superconductor: a decrease of the spectral weight near E_F indicating the opening of a gap, the appearance of a particle-hole symmetric Bogoliubov quasiparticle band, and a shift of the kink by the maximum gap value [37,38]. When the pulse hits the system at $t_{\mathrm{delay}}=0\ \text{(1/eV)}$ (fig. 2(c)), the spectral weight redistributes itself along (11) direction. The movement of the spectral weight towards E_F represents the melting of the superconducting order. After the pump (fig. 2(d)), we observe that the superconducting order has partially melted, as the spectral weight is enhanced near E_F. The spectrum suggests that the system has entered the normal state; however, superconductivity is still present [29], as we will show when we discuss the dynamics of the superconducting order parameter (fig. 3).

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Figure 2(f) shows the contrasting case of the superconducting state in the forward-scattering scenario. We observe a flattening of the band near E_F compared to the normal state (fig. 2(e)). Figures 2(g) and (h) show the spectrum when the same pump pulse as used in the isotropic scattering case drives the system. We observe a weaker redistribution of the spectral weight, even for times after the pump (fig. 2(h)), indicating that the superconducting order based on forward focused pairing is remarkably robust and persistent.

To gain further insight we examine the evolution of the spectral gap in the time domain, which is estimated from the retarded anomalous self-energy $\Sigma_F^{R}(\mathbf{k};t,t')$,

$$\begin{eqnarray} &&\Delta_{\mathbf{k}}(t_{\mathrm{delay}}, \omega_{\mathrm{rel}}=0) = \int \mathrm{d}t_{\mathrm{rel}} \Sigma_F^{R}(\mathbf{k};t_{\mathrm{delay}},t_{\mathrm{rel}}), \end{eqnarray} \tag{ 2 }$$

where a Wigner transformation $t_{\mathrm{delay}} = \frac{t+t'}{2}$, $t_{\mathrm{rel}} = t-t'$ is used to transform the reference frame to average and relative times. First we calculate the order parameter at the Fermi momentum $\Delta(t_{\mathrm{delay}}) \equiv \Delta_{\mathbf{k} = \mathbf{k}_{\mathrm{F}}}(t_{\mathrm{delay}})$ for both types of scattering; see fig. 3(a) which shows $\Delta(t_{\mathrm{delay}})$ for different pairing symmetries after being normalized by the equilibrium value $\Delta_0$. We observe that the superconducting gap in the forward-scattering case exhibits a smaller degree of melting in comparison to the isotropic case. To quantify the suppression we determine the minimum value of the transient order parameter and plot it as a function of fluence and incident energy density $U_{\mathrm{in}} = \int \mathrm{d}t |\mathbf{E}(t)|^2$ in fig. 3(b) (a calculation of the energy absorption may be found in the SM). Since less energy is absorbed when forward scattering is the dominant source of scattering, a superconducting order built from forward-scattering pairing will also be affected less. Also, we observe that after the pump Δ relaxes in an oscillatory fashion (fig. 3(a)). These amplitude mode, or Higgs mode oscillations have been proposed and seen before in the literature [29,30,39–41] for pumped superconductors. In forward scattering, we observe similar oscillations for low fluences.

Finally, we demonstrate that there is an additional fingerprint of the forward focused pairing interaction —a pump-induced momentum dependence of the order parameter $\Delta_{\mathbf{k}}(t_{\mathrm{delay}})$ that goes beyond the standard s-/d-wave forms. In equilibrium, isotropic scattering pairing and forward-scattering pairing result in momentum-independent and momentum-dependent s-/d-wave order parameters, respectively [12,27]. Isotropic pairing leads to a gap that is a constant across the Brillouin zone $(\Delta_{\mathbf{k}} = \Delta_0)$. Upon pumping, there is only an isotropic suppression of the gap, as was shown previously [29].

On the other hand, the forward scattering pairing (in equilibrium) gives a gap that is peaked around the Fermi surface in both the s- and d-wave channels. This is shown in fig. 4(a). Although the order parameter is of s-/d-wave $(A_{1g}/B_{1g})$ in symmetry, it has additional momentum dependence (it is not constant up to a symmetry factor across the Brillouin zone) due to the forward focus in the interaction [27].

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We study the evolution of the forward-scattering–driven momentum-dependent gap for s-wave and d-wave pairing. Figure 4(b) shows the change in the transient order parameter when the system is driven out of equilibrium by a pump of amplitude $A_{\mathrm{max}} = 1.5~(\mathrm{V}/a_0)$. For s-wave pairing, $\Delta_{\mathbf{k}}$ is suppressed the most at k_F (third snapshot in fig. 4(b) at $t_{\mathrm{delay}} = 5\sigma_p$). Although the overall change in the order parameter is negative because the system absorbs a finite amount of energy which leads to dissociation of the Cooper pairs, surprisingly, in some regions of the Brillouin zone, the order parameter increases when the pump is active (second snapshot in fig. 4(b) at $t_{\mathrm{delay}} = 3\sigma_p$). For d-wave pairing, the superconducting gap does not show any enhancement after the pump and is suppressed at all k-points (see bottom panels of fig. 4).

To analyze the dynamics further we show the order parameter for the forward-scattering pairing s-wave gap as a function of k along the (11) direction for different delay times in fig. 5(a). The same dynamics as in fig. 4 may be seen here —a suppression of the gap relative to $t_{\mathrm{delay}}=-\infty$, with a brief enhancement around $t_{\mathrm{delay}}=3\sigma_p$. Figure 5(b) displays the magnitude of the order parameter as a function of delay time for three representative k-points marked by the vertical lines in fig. 5(a), which highlights the non-uniform changes in $\Delta_{\mathbf{k}}$. Such order suppression is characteristically different for the isotropic interaction scenario, where the suppression is larger and momentum independent, as shown in fig. 3.

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In summary, we have studied the effect of e-ph scattering focused in the forward direction on the superconducting state in the time domain and compared our results with the isotropic scattering case. The forward-scattering superconductor shows a robust, persistent superconducting order against the pump-induced perturbations. We also observe a momentum structure of the superconducting order parameter in the forward scattering. The order parameter exhibits a non-uniform melting in momentum space in the non-equilibrium state. These characteristic features of the forward scattering are in direct contrast with the conventional BCS isotropic superconductors, where a reasonable fluence can melt the superconducting order and the suppression of the superconducting order shows no momentum dependence. In fact, even unconventional (cuprate) superconductors show a rapid suppression of the gap for a weak pump pulse [42]. We explain our results for the forward pairing superconducting order using the restriction of phase space available for the scattering process to occur, which results in less absorption of energy. We support our explanation by estimating the absorbed energy and studying the dynamics of the superconducting order parameter in the time domain. These distinctive features of forward scattering may be detectable using momentum-resolved experiments such as time-resolved ARPES.

Recent work has suggested a weak forward-scattering phonon as a source of the high T_C as an enhancement of an unconventional pairing mechanism. In this scenario, the time-resolved signatures discussed here would appear essentially identical to a standard BCS superconductor since the lack of phase space does not apply, and a quantitative comparison would have to be made with respect to some reference material; an FeSe intercalate which has no interfacial phonons may be used for an absolute comparison to see if there is an increase in energy absorption commensurate with the enhanced pairing.

We acknowledge fruitful discussion with Doug Scalapino, Yan Wang, Louk Rademaker, Avinash Rustagi and Omadillo Abdurazakov. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.