Intersubband pairing induced Fulde-Ferrell phase in metallic nanofilms

We consider a freestanding metallic nanofilm with a predominant intersubband pairing which emerges as a result of the confinement in the growth direction. We show that the Fermi wave vector mismatch between the subbands, detrimental to the intersubband pairing, can be compensated by the nonzero center-of-mass momentum of the Cooper pairs. This leads to the spontaneous appearance of the intersubband Fulde-Ferrell (IFF) state, even in the absence of an external magnetic field. Our study of the intrasubband pairing channel on the stability of the IFF phase shows that the former strongly competes with the intersubband pairing, which prohibits the coexistence of the two superconducting phases. Interestingly, upon application of the magnetic field we find a transition to an exotic mixed spin-singlet subband-triplet and spin-triplet subband-singlet paired state. Finally, we discuss the possibility of existence of the IFF pairing in novel superconducting materials.

Figure 1

Schematic illustration of pairing in the two-band model (cross section along $k_y=0)$ for magnetic field (a) $H=0$ and (b) $H\ne 0$. Electronic states from opposite sides of the Fermi surface, forming Cooper pairs, are connected by arrows. The intersubband finite-momentum pairing is marked by the blue arrows. The horizontal dashed line denotes the Fermi level $\mu$.

Figure 2

(a) The phase diagram in the $(V_{12},\mathrm{\Delta }E)$ plane, where one can distinguish the regions of NS (normal state), IFF (intersubband Fulde-Ferrell phase), and ISP (intersubband paired phase). The dashed horizontal line denotes $\mathrm{\Delta }E=5$ meV, chosen for further analysis. (b) The Cooper pair momentum, which minimizes the energy of the system, as a function of both $\mathrm{\Delta }E$ and $V_{12}$. (c) and (d) The free energy of the system as a function of the Cooper pair momentum $\mathbf{Q}=(Q_x,Q_y)$ for two selected values of the interband pairing strength $V_{12}$, for which the IFF and ISP phases are stable, respectively.

Figure 3

(a) The superconducting gap parameter as a function of both inter- and intrasubband pairing strength with visible regions of stability of ISP (intersubband paired state), ESP (equal subband paired state), and NS (normal state). Results are for $\mathbf{Q}=0$ and a selected value of the energy separation $\mathrm{\Delta }E=5$ meV. (b) The region of stability of the Fulde-Ferrell intersubband state with values of the $\mathbf{Q}$ vector which minimize the free energy in the inset. Results are for $\mathbf{Q}\ne 0$ within the parameter range marked by the white rectangle in (a).

Figure 4

(a) The phase diagram in the $\left(V_{12},B\right)$ plane for the selected value of the intrasubband pairing $V=0.05$. (b) The spin-singlet, subband-triplet pairing amplitude and (c) the spin-triplet, subband-singlet pairing amplitude vs $V_{12}$ and $B$. In (d) we show ${\mathrm{\Delta }}_{ST}$ and ${\mathrm{\Delta }}_{TS}$ as a function of $V_{12}$ for a selected value of $B$. Additionally, in (e) the schematic representation of the spin-split subbands in the external magnetic field is provided.

Figure 5

The superconducting gaps corresponding to the nonzero momentum pairing (a) within the subbands and (b) between the subbands as functions of $V_{12}$ and $B$. (c) and (d) The values of the Cooper pair momentum which minimize the system energy.