Pair symmetry conversion in driven multiband superconductors

It was recently shown that odd-frequency superconducting pair amplitudes can be induced in conventional superconductors subjected to a spatially nonuniform time-dependent drive. It has also been shown that, in the presence of interband scattering, multiband superconductors will possess bulk odd-frequency superconducting pair amplitudes. In this work we build on these previous results to demonstrate that by subjecting a multiband superconductor with interband scattering to a time-dependent drive, even-frequency pair amplitudes can be converted to odd-frequency pair amplitudes and vice versa. We will discuss the physical conditions under which these pair symmetry conversions can be achieved and possible experimental signatures of their presence.

Figure 1

In the left (right) column we plot the even-$\omega$ (odd-$\omega$) terms of the real part of the Wigner transform [defined in Eq. (27)] of the anomalous part of the Green's function in Eq. (14), $\langle {\widehat{F}}^{\text{R}}(\omega ,T=0)\rangle$, in black (solid), where we have taken the average value of ${\widehat{F}}^{\text{R}}(\mathbf{k};\omega ,T=0)$ at $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(a\right)}$ and $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(b\right)}$. In each case we have also plotted the parity-preserving terms, Eqs. (22), in green (dashed) and the parity-reversing terms, Eqs. (23), in red (dash-dotted). (a) the diagonal component for band-$a$, (b) the diagonal component for band-$b$, (c) the interband component. (d) The components of the drive from Eq. (26) plotted in the time domain over a full period, the green vertical line denotes the time, $T=0$, at which all plots in this figure are evaluated. The parameters used to describe the driven multiband superconductor in this case are: effective masses, $m_a=0.5{\AA }^{-2}\mathrm{eV}$ and $m_b=1{\AA }^{-2}\mathrm{eV}$; chemical potentials, ${\mu }_a={\mu }_b=2\mathrm{eV}$; $s$-wave gaps, ${\mathrm{\Delta }}_{aa}=2\mathrm{meV}, {\mathrm{\Delta }}_{bb}=7\mathrm{meV}, {\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_{ba}=0$, consistent with ${\mathrm{MgB}}_2$ [44]; interband scattering, $\mathrm{\Gamma }=10\mathrm{meV};$ dissipation described by $\eta =1\mathrm{meV};$ and a drive given by Eq. (26) with $U_0=10\mathrm{meV}$, and ${\mathrm{\Omega }}_0=1\mathrm{meV}$ (242 GHz).

Figure 2

In the left (right) column we plot the even-$\omega$ (odd-$\omega$) terms of the real part of the Wigner transform [defined in Eq. (27)] of the anomalous part of the Green's function in Eq. (14), $\langle {\widehat{F}}^{\text{R}}(\omega ,T=\pi /2{\mathrm{\Omega }}_0)\rangle$, in black (solid), where we have taken the average value of ${\widehat{F}}^{\text{R}}(\mathbf{k};\omega ,T=\pi /2{\mathrm{\Omega }}_0)$ at $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(a\right)}$ and $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(b\right)}$. In each case we have also plotted the parity-preserving terms, Eqs. (22), in green (dashed) and the parity-reversing terms, Eqs. (23), in red (dash-dotted). (a) the diagonal component for band-$a$, (b) the diagonal component for band $b$, (c) the interband component. (d) The components of the drive from Eq. (26) plotted in the time domain over a full period, the green vertical line denotes the time, $T=\pi /2{\mathrm{\Omega }}_0$, at which all plots in this figure are evaluated. The parameters used to describe the driven multiband superconductor in this case are: effective masses, $m_a=0.5{\AA }^{-2}\mathrm{eV}$ and $m_b=1{\AA }^{-2}\mathrm{eV}$; chemical potentials, ${\mu }_a={\mu }_b=2\mathrm{eV}$; $s$-wave gaps, ${\mathrm{\Delta }}_{aa}=2\mathrm{meV}, {\mathrm{\Delta }}_{bb}=7\mathrm{meV}, {\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_{ba}=0$, consistent with ${\mathrm{MgB}}_2$ [44]; interband scattering, $\mathrm{\Gamma }=10\mathrm{meV}$; dissipation described by $\eta =1\mathrm{meV}$; and a drive given by Eq. (26) with $U_0=10\mathrm{meV}$, and ${\mathrm{\Omega }}_0=1\mathrm{meV}$ (242 GHz).

Figure 3

In (a) and (b), the 2D DOS computed using: effective masses, $m_a=0.5{\AA }^{-2}\mathrm{eV}$ and $m_b=1{\AA }^{-2}\mathrm{eV}$; chemical potentials, ${\mu }_a={\mu }_b=2\mathrm{eV}$; $s$-wave gaps, ${\mathrm{\Delta }}_{aa}=2\mathrm{meV}, {\mathrm{\Delta }}_{bb}=7\mathrm{meV}, {\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_{ba}=0$, consistent with ${\mathrm{MgB}}_2$ [44]; interband scattering, $\mathrm{\Gamma }=10\mathrm{meV}$; dissipation described by $\eta =0.1\mathrm{meV}$; and a drive given by Eq. (26) with $U_0=10\mathrm{meV}$, and ${\mathrm{\Omega }}_0=1\mathrm{meV}$ (242 GHz). In both panels we show the case for no drive in black (solid), and the cases with the drive at times $T=0$ and $T=\pi /2{\mathrm{\Omega }}_0$ in green (dashed) and red (dash-dotted), respectively. In (a) we focus on the states near the Fermi surface, in (b) we focus on the range of energies near the crossing of the two bands at which we find the driven DOS at $T=0$ possesses two peaks shifted from the avoided crossing at $E_0$ by, $\pm {\mathrm{\Omega }}_0/2$. In (c) and (d), the 3D DOS plotted for the same parameters as in (a) and (b). Notice that the main difference is that in 3D the driven DOS at $T=0$ is slightly suppressed relative to the undriven DOS (see inset). In (e) we plot the spectrum of the two band superconductor given by ${\epsilon }_{\pm }\left(\mathbf{k}\right)$ in Eq. (11). The horizontal gray line denotes the avoided crossing (see inset) at $E_0$, Eq. (29), due to the finite interband scattering, $\mathrm{\Gamma }$. In (f) we show the drive from Eq. (26) plotted in the time domain over a full period, the green vertical line denotes the beginning of the period at $T=0$ where the drive has maximum amplitude, while the red line denotes $T=\pi /2{\mathrm{\Omega }}_0$ where the drive amplitude is zero.

Figure 4

In the left (right) column we plot the even-$\omega$ (odd-$\omega$) terms of the real part of the Wigner transform [defined in Eq. (27)] of the anomalous part of the Green's function in Eq. (14), $\langle {\widehat{F}}^{\text{R}}(\omega ,T)\rangle$, where we have taken the average value of ${\widehat{F}}^{\text{R}}(\mathbf{k};\omega ,T)$ at $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(a\right)}$ and $\left|\mathbf{k}\right|=k_{\text{F}}^{\left(b\right)}$. In each panel, $\langle {\widehat{F}}^{\text{R}}(\omega ,T)\rangle$ is calculated self-consistently using Eq. (30) for the parameters discussed in the text at times $T=0$, green (dashed), and $T=\pi /2{\mathrm{\Omega }}_0$, red (dash-dotted). Additionally, we note that the self-consistent results for these same parameters but with $U_0=0$ appears as a black curve, which overlaps almost exactly with the $T=0$ results. (a) the diagonal component for band $a$, (b) the diagonal component for band $b$, (c) the interband component. (d) The drive from Eq. (26) plotted in the time domain over a full period, the green vertical line denotes the time $T=0$ while the red line denotes $T=\pi /2{\mathrm{\Omega }}_0$.

Figure 5

In (a) and (b) we repeat the plots of the even-$\omega$ intraband pair amplitudes appearing in Figs. 4 and 4, which were calculated self-consistently using Eq. (30) for the parameters discussed in the text at times $T=0$, green (dashed), $T=\pi /2{\mathrm{\Omega }}_0$, red (dash-dotted), and without a drive black (solid). Plotted over this range, the three curves are essentially indistinguishable; however, there a slight differences which we highlight in (c)–(f). In (c)–(f) we show the symmetry preserving (green/dashed) and symmetry reversing (red/solid) contributions to the plots appearing in (a) and (b) calculated using Eqs. (22) and (23). In (c) and (e) we show the corrections to the even-$\omega$ intraband pairing in band-$a$ at times $T=0$ and $T=\pi /2{\mathrm{\Omega }}_0$, respectively; in (d) and (f) we show the corrections to the even-$\omega$ intraband pairing in band-$b$ at times $T=0$ and $T=\pi /2{\mathrm{\Omega }}_0$, respectively.