Abstract
Observations of robust superconductivity in some of the iron based superconductors in the vicinity of a Lifshitz point where a spin density wave instability is suppressed as the hole band drops below the Fermi energy raise questions for spinfluctuation theories. Here we discuss spinfluctuation pairing for a bilayer Hubbard model, which goes through such a Lifshitz transition. We find s_{±} pairing with a transition temperature that peaks beyond the Lifshitz point and a gap function that has essentially the same magnitude but opposite sign on the incipient hole band as it does on the electron band that has a Fermi surface.
Introduction
The microscopic mechanism of pairing that gives rise to superconductivity in the iron based superconductors remains an unsettled issue^{1}. Spinfluctuation mediated pairing^{2,3,4}, in which electrons form pairs by exchanging virtual S = 1 particle hole excitations, is a leading candidate mechanism, since superconductivity appears near the onset of a magnetic phase. However, this picture relies on the nesting properties of the electronic band structure with both hole and electron Fermi surface pockets present and the absence of hole pockets in some of the iron based superconductors^{5,6,7,8,9,10,11} has challenged these theories. In these systems, the hole like band drops below the Fermi energy after a Lifshitz transition^{12,13}. Nevertheless, pairing remains strong, as evidenced e.g. by the high Tc superconductivity reported in monolayer FeSe films grown on SrTiO_{3}^{9,10,11,14}. Scanning tunneling microscopy experiments^{12} as well as ARPES measurements^{11,13} on these FeSe monolayers find that there are no hole pockets. Furthermore, the ARPES measurements of the variation of the gap magnitude around the electron pockets^{13} makes the possibility of dwave pairing, arising from pair scattering between the electron pockets, unlikely. However, these experiments also report the existence of an incipient hole band laying 50 to 100 meV below the Fermi energy, implying that the system is just beyond a Lifshitz transition^{15} where the hole Fermi surface has disappeared. In addition, photoemission measurements find evidence that superconductivity occurs in the monolayer FeSe film, when SDW order is suppressed by electron doping^{11} and density functional theory calculations^{16} predict that in the absence of electron doping, the ground state of the monolayer FeSe film would have SDW order. Thus, it appears that superconductivity is induced in the FeSe monolayer when the SDW order is suppressed by a Lifshitz transition arising from electron doping or strain^{11}. Motivated by these results, we have investigated the suppression of SDW order and the onset of superconductivity near a Lifshitz transition in a twolayer Hubbard model. This model was previously shown to have both s_{±} and dwave pairing depending upon the strength of the interlayer hopping^{17}. Here using this model, we show that spinfluctuation scattering of pairs between an electron and an incipient hole band can lead to s_{±} pairing for a system that has undergone a Lifshitz transition.
Results
The Hamiltonian for the two layer Hubbard model that we study is
Here creates/annihilates a fermion with spin σ on the n^{th} layer (n = 1 or 2). The intralayer hoping is t, the interlayer hoping is t_{⊥} and μ is the chemical potential. The band structure for this model is,
with t_{⊥}/t = 3.5 and μ set so that the site filling 〈n〉 = 1.05 is shown in Fig. 1(a). If the filling is kept constant as t_{⊥}/t is increased, the system has a Lifshitz transition such that for t_{⊥} > 3.67 the hole Fermi surface at the Γ point disappears as illustrated in Fig. 1(b). We are interested in studying the pairing for parameters such that the spin density wave (SDW) instability is suppressed by this Lifshitz transition.
In a random phase approximation (RPA) the spin susceptibility is given by
with
Here T is the temperature, G_{0}(k, ω_{n}) = (iω_{n} − ξ_{k})^{−1} and ω_{n} = (2n + 1)πT and Ω_{m} = 2mπT are the usual fermionic and bosonic Matsubara frequencies. For a fixed filling, as t_{⊥}/t is increased and the Lifshitz transition is approached, χ_{0} which peaks near wavevector (π, π, π), decreases. For 〈n〉 = 1.05, we take U = 2.4t so that the SDW instability determined from Eq. (3) is suppressed by the Lifshitz transition as shown in Fig. 2. With this suppression of the SDW order, one can imagine that superconductivity may appear following the usual paradigm. However, the Lifshitz transition that has suppressed the SDW instability can also lead to a suppression of the s_{±} pairing associated with the scattering of pairs between the electron Fermi surface and the incipient hole band. For a fixed pairing strength, T_{c} decreases as the hole band moves below the Fermi energy^{18}.
To explore this, we solve the BetheSalpeter equation
and determine T_{c} from the temperature at which the leading eigenvalue of Eq. (5) goes to 1. Here we use a spinfluctuation mediated interaction^{19},
Here the first term in the effective interaction is the bare interaction U which is momentum independent. The effect of this term is small due to the sign change of the gap between the two bands. In Eq. (5) we set G(k, ω_{n}) = [iω_{n} − ξ_{k} − Σ(k, ω_{n})]^{−1} with
Note that we keep the Fermi surface unchanged in the dressed Green’s function. For 〈n〉 = 1.05 and U = 2.4t, the resulting value of T_{c}, interpolated from the temperature at which λ crosses 1, is plotted in Fig. 2 as a function of t_{⊥}/t. As shown in this figure, after the SDW instability is suppressed by the Lifshitz transition, a pairing transition occurs at a T_{c} which peaks as t_{⊥}/t increases and then falls off as the hole band is pushed further below the Fermi energy.
The momentum dependence of the superconducting gap function Δ(k, ω = πT) ≡ Φ(k, πT)/Z(k, πT) is shown in Fig. 3. This is an A_{1g} (s_{±}) state in which the sign of Δ changes between k_{z} = 0 (bonding) and k_{z} = π (antibonding) bands. One can see that the magnitudes of the two gaps Δ(k_{x}, k_{y}, k_{z} = 0) and Δ(k_{x}, k_{y}, k_{z} = π) are comparable even though the hole band is below the Fermi energy.
In order to understand the peak in T_{c} that occurs as the hole band drops below the Fermi energy, it is useful to separately examine the dependence of T_{c} on the changes in and that occur as the T_{c} at which the eigenvalue of Eq. (5) goes to 1 as a functional of χ and the pair propagator GG. We can calculate the variation in T_{c} due to the change in χ with GG unchanged when t_{⊥} increases by Δt_{⊥},
and the variation due to the change in pair propagator GG when χ is unchanged and t_{⊥} increases by Δt_{⊥},
We set Δt_{⊥} = 0.01t. The results of the calculation are shown in Fig. 4. Here one sees that the initial increase in T_{c} arises from both the changes in χ and GG. The latter effect is associated with an increase in the quasiparticle spectral weight Z^{−1}(k, ω) on the electron Fermi surface that occurs as the hole band drops below the Fermi energy. This increase in the quasiparticle spectral weight initially ameliorates the decrease in T_{c} resulting from the submergence of the hole band. The initial positive contribution associated with the variation in χ reflects the change in the frequency structure of the spinfluctuations. As the hole band drops below the Fermi energy, a gap opens in the low energy q_{z} = π spin fluctuation spectrum and spectral weight is transfered to higher energies as shown in Fig. 5, which leads to stronger pairing^{20}. The ultimate decrease in T_{c} is due to the decrease of the pair propagator as t_{⊥} increases and the hole band drops further below the Fermi energy, as well as the decreasing strength of the spinfluctuations.
Discussion
To conclude, we have studied a twolayer Hubbard model with parameters chosen so that a SDW instability is suppressed by a Lifshitz transition in which the hole band at the Γ point drops below the Fermi energy. Here, we have kept the site filling fixed and varied the interlayer hopping to tune the system through the Lifshitz point. For a physical system this might by obtained via strain^{11}. Following the suppression of the SDW order, we find the onset of an s_{±} superconducting state whose transition temperature T_{c} initially increases as the system is pushed beyond the Lifshitz point by further increasing t_{⊥}/t. We find that this increase in T_{c} is associated with both an increase in the quasiparticle spectral weight and an increase in the strength of the pairing interaction, which are related to the incipient hole band and the resulting change in the spectral distribution of the spinfluctuations. We find that the gap function on the incipient hole band is similar in magnitude but of the opposite sign to that on the electron band which crosses the Fermi surface.
Additional Information
How to cite this article: Mishra, V. et al. s_{±} pairing near a Lifshitz transition. Sci. Rep. 6, 32078; doi: 10.1038/srep32078 (2016).
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Acknowledgements
Research sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UTBattelle, LLC, for the U. S. Department of Energy. DJS and TAM acknowledge the support of the Center for Nanophase Materials Sciences, a US DOE Office of Science User Facility. We acknowledge the Valinor cluster for computational resources. We thank A. Linscheid, S. Maiti, P. Hirschfeld for useful discussion.
Author information
Author notes
 Vivek Mishra
 , Douglas J. Scalapino
 & Thomas A. Maier
These authors contributed equally to this work.
Affiliations
Joint Institute of Computational Sciences, University of Tennessee, Knoxville, TN37996, USA
 Vivek Mishra
Department of Physics, University of California, Santa Barbara, CA93106, USA
 Douglas J. Scalapino
Computer Science and Mathematics Division & Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN37831, USA
 Thomas A. Maier
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Contributions
V.M., D.J.S. and T.A.M. contributed equally.
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The authors declare no competing financial interests.
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Correspondence to Vivek Mishra.
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