Spontaneous appearance of nonzero-momentum Cooper pairing: Possible application to the iron-pnictides

We suggest that an inhomogeneous (nonzero-momentum) paired phase can appear in the absence of an external magnetic field in a system with a predominant interband pairing and with separate Fermi-surface sheets. The Fermi wave vector mismatch which appears in such a situation can be compensated by nonzero center-of-mass momentum of the Cooper pairs, which can lead to a spontaneous appearance of the Fulde-Ferrell type of superconducting state. The idea is examined using a tight-binding model which emulates the holelike and the electronlike bands of iron-based superconductors. The state can appear for the cases of both spin-singlet and spin-triplet pairing channels.

Figure 1

Electronic structure (a) and Fermi surface sheets (b) for $n=1.94$. The black solid line in (b) marks the boundaries of the reduced Brillouin zone. The $\Delta k_1\approx 0.236$ and $\Delta k_2\approx 0.210$ parameters represent the Fermi wave vector mismatch between the hole- and electronlike bands, respectively.

Figure 2

The $\Delta k_1$ parameter (a) and the chemical potential (b), both vs the total band filling $n$ for the limited case of two holelike bands in the model.

Figure 3

Free energy of the paired state as a function of the Cooper-pair total momentum for $n=1.75$ (a) and $n=1.79$ (b), for the limited case of two holelike bands only. The pairing strength is set to $U_0=0.39$. Note the four distinct free-energy minima in (b) which signal the FF phase appearance.

Figure 4

Phase diagram on the $(T,n)$ plane for $U_0=0.39$ for the limited case of two holelike bands only. The labels of the phases are explained in the main text. The transition from homogeneous (SC) to inhomogeneous (FF) spin-singlet phase is discontinuous. The solid line marks the border between the SC and NS phases in the case of absence of the FF phase.

Figure 5

Gap amplitudes ${\Delta }_{1\mathbf{Q}}^{\left(0\right)}$ and ${\Delta }_{1\mathbf{Q}}^{\left(1\right)}$ as functions of both $T$ and $n$, for $U_0=0.39$ for the limited case of two holelike bands only. Note the discontinuity in the gap parameter at the $\text{SC}\to \text{FF}$ transition, while the $\text{FF}\to \text{NS}$ transition is continuous.

Figure 6

The center-of-mass momentum component $Q_x$ of the Cooper pairs (for $Q_y=0)$ which corresponds to a stable FF phase as a function of both the reduced temperature and the band filling with the parameter $U_0=0.39$. These results correspond to the limited case of two holelike bands only.

Figure 7

The phase diagram on the $(n,U_0)$ plane for $U_1=U_0/5$ and for the limited case of two holelike bands. The labels representing the phases are explained in the main text.

Figure 8

Free energy of the system in the paired state as a function of the Cooper-pair total momentum for $n=1.72$ (a) and $n=1.795$ (b) for the model with two electron- and two holelike bands. The pairing strength is set to $U_0=0.33$. In (c) and (d) we show the $Q_x$ dependence of the free energy for $Q_y=0$. As one can see from (b), for a proper value of band filling, minima of the free energy appear for nonzero values of the Cooper-pair momentum.

Figure 9

Quasiparticle energies from Eq. (9) along the trajectory in the reduced Brillouin zone marked by the dashed line in Fig. 10 for the value of the $Q_x$ component which corresponds to the free-energy minimum shown in Fig. 8. In (b)–(e) the quasiparti-cle energies are shown zoomed around four points in the Brillouin zone marked by squares in (a) from left to right, respectively. As shown in (d) and (e) the quasiparticle energies intersect the Fermi energy at certain points in the Brillouin zone which leads to the depairing region presented in Fig. 10.

Figure 10

Depairing region in reciprocal space which corresponds to the free-energy minimium shown in Fig. 8.

Figure 11

Temperature dependences of the superconducting gaps for two values of the band filling $n=1.72$ (a),(c) and $n=1.795$ (b),(d) for the model with two electron- and two holelike bands. The pairing strength is set to $U_0=0.33$.

Figure 12

Phase diagram on the $(T,n)$ plane for $U_0=0.33$ for the case of the four-band Hamiltonian. The labels of the phases are explained in the main text. Note that the FF phase appears only in a narrow range of the band filling.

Figure 13

Band filling dependence of the gap components: $s$-wave (a) and extended $s$-wave (b) for $U_0=0.33$.