Collective modes in multiband superconductors: Raman scattering in iron selenides

We study Raman scattering in the superconducting state of alkali-intercalated iron selenide materials $A_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ ($A=\text{K},\text{Rb},\text{Cs}$) in which the Fermi surface has only electron pockets. Theory predicts that both $s$-wave and $d$-wave pairing channels are attractive in this material, and the gap can have either $s$-wave or $d$-wave symmetry, depending on the system parameters. ARPES data favor $s$-wave superconductivity. We present the theory of Raman scattering in $A_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ assuming that the ground state has s-wave symmetry but $d$ wave is a close second. We argue that Raman profile in $d$-wave $B_{2g}$ channel displays two collective modes. One is a particle-hole exciton, another is a Bardasis-Schrieffer-type mode associated with superconducting fluctuations in $d$-wave channel. At a finite damping, the two modes merge into one broad peak. We present Raman data for $A_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ and compare them with theoretical Raman profile.

Figure 1

Low-frequency Raman response from ${\mathrm{K}}_{0.75}{\mathrm{Fe}}_{1.75}{\mathrm{Se}}_2$ superconductor in right-left scattering polarization channel in the normal state, at $T=41\mathrm{K}$ (red), and in superconducting state at $T=3\mathrm{K}$ (blue), from Ref. [62]. The upper panel shows the response with contributions from phonons subtracted, as discussed in Ref. [62]. The in-gap modes are found in the energy interval 8 meV $\lesssim \omega \lesssim 11$ meV.

Figure 2

Phase diagram of $A{\mathrm{Fe}}_2{\mathrm{Se}}_2$ in $(\kappa ,T)$ plane. At high temperatures, the system is in the normal state. For $\kappa <{\kappa }^{*}$ ($\kappa >{\kappa }^{*}$) the transition occurs to the $d$-wave $s^{+-}$ state. The two normal to superconductor transitions merge at the tetra-critical point, $({\kappa }^{*},T^{*})$. For $T<T^{*}$, the system is in the $d$-wave state at $0<\kappa <{\kappa }_1^{*}\left(T\right)$, in $s^{+-}$ state for $\kappa >{\kappa }_2^{*}\left(T\right)$ and the intermediate $s+id$ phase with broken time-reversal symmetry is obtained at ${\kappa }_1^{*}\left(T\right)<\kappa <{\kappa }_2^{*}\left(T\right)$. The inset shows the energy of BS mode in the $s^{+-}$ state in units of the $s^{+-}$ OP, and the dashed line shows the minimal energy of quasiparticle excitations. The BS mode softens closer to the transition to the $s+id$ state.

Figure 3

(a) Diagramatic representation of the Raman susceptibility, ${\chi }^{\prime \prime }$. (b) p-h and (c) Cooper Raman vertices within the ladder approximation. Vertex renormalizations (often called final state interaction in Raman literature) represent multiple interactions in particle-hole channel and processes which convert particle-hole into particle-particle channel and include multiple scattering events in the particle-particle channel.

Figure 4

${\mathrm{B}}_{2g}$ collective modes (solid lines) of an $s^{+-}$ superconductor at $T=0$ (the superconducting gap changes sign between the two hybridized Fermi pockets). The energies of the collective excitations are plotted as functions of the dimensionless parameter $\kappa$ introduced in Eq. (4). The energies are in units of $2{\Delta }_s$, where ${\Delta }_s$ is the magnitude of $s^{+-}$ superconducting OP. The modes are shown in the interval $\kappa >{\kappa }_2\approx 0.611525$ (see Fig. 2). At the energies of the two modes the denominator in Eq. (30) vanishes, giving rise to the $\delta$-functional peak in the Raman intensity ${\chi }^{\prime \prime }$. We used Eqs. (32) for the polarization operators ${\Pi }_{ii}$ in Eq. (30) and used the approximate form for ${\Pi }_{23}$, (34). The parameters used in the calculation are $u_d=0.4$, $u_{\rho }=0.6$, ${\mu }_{+}/{\mu }_{-}=0.15$. The lower (red) and upper(blue) dashed lines are obtained in the limit ${\Pi }_{23}={\Pi }_{32}=0$, and represent the BS mode and a particle-hole exciton, respectively. The coupling between the two channels mixes the BS mode and p-h exciton. This coupling is, however, weakened for $s^{+-}$ gap superconductivity and scales with the degree of ellipticity of electron pockets. The shaded area is the quasiparticle continuum whose lower boundary is defined by $2{\Delta }_s(\kappa /\sqrt{{\kappa }^2+1})$. This boundary approaches $2{\Delta }_s$ for large hybridization, $\kappa \gg 1$.

Figure 5

Calculated Raman susceptibility, ${\chi }^{\prime \prime }$ as a function of the dimensionless frequency, $\omega /2{\Delta }_s$ for the fixed $\kappa =0.62>{\kappa }_2$. The two peaks represent in-gap modes corresponding to the two solid lines in Fig. 4. The parameters are the same as in Fig. 4. The continuum starts once the frequency enters the shaded area in Fig. 4. The small imaginary part $0.003\left(2{\Delta }_s\right)$ was added to the frequency for regularization.

Figure 6

Calculated Raman susceptibility, ${\chi }^{\prime \prime }$ as a function of the dimensionless frequency, $\omega /2{\Delta }_s$ for the fixed $\kappa =0.63>{\kappa }_2$. The two peaks represent in-gap modes corresponding to the two solid lines in Fig. 4. The parameters used in the calculation are $u_d=0.4$, $u_{\rho }=0.98$, ${\mu }_{+}/{\mu }_{-}=0.05$. The finite imaginary part $0.07\left(2{\Delta }_s\right)\ll 2{\Delta }_s$ is added to the frequency to simulate the effect of the disorder induced smearing.

Figure 7

Raman intensity, ${\chi }_{r,r}^{\prime \prime }$ as a function of $x=\omega /2{\Delta }_s$. Solid (black) line is obtained by substitution of Eqs. (C3b), (C5), and (C6) in Eq. (30) for $u_s=0.5$, $u_d=0.4$, and $u_{\rho }=0.2$. Dashed (red) line is calculated Raman intensity with only Cooper channel included, $u_{\rho }=0$. Dashed (blue) line is calculated Raman intensity with only particle-hole channel included, $u_d=0$. A small imaginary part is added to the frequency for illustration purposes, $x\to x+i\Gamma$, $\Gamma ={10}^{-5}$. The quasiparticle continuum is shown only for the full Raman susceptibility.