Tilting the balance towards d-wave in iron-based superconductors

TL;DR

This paper investigates how magnetic fields and temperature influence the competition between s-wave and d-wave superconducting states in iron-based superconductors, revealing a universal phase boundary shape that favors d-wave pairing.

Contribution

It provides a theoretical analysis of the phase boundary between nodal and nodeless superconducting states under magnetic field and temperature, specifically for s_ ext{±}-wave and d-wave pairing in iron-based superconductors.

Findings

Universal cubic line shape of phase boundary near critical point

Magnetic field and temperature favor d-wave state

Quantitative model for competing pairing symmetries

Abstract

The intricate interplay of interactions and Fermiology can give rise to a close competition between nodeless (e.g. s-wave) and nodal (e.g. d-wave) order in electronically driven unconventional superconductors. We analyze how such a scenario is affected by a Zeeman magnetic field $H_{\text{Z}}$ and temperature $T$. In the neighborhood of a zero temperature first order critical point separating a nodal from a nodeless phase, the phase boundary at low $H_{\text{Z}}$ and/or low $T$ has a universal line shape cubic in $H_{\text{Z}}$ or $T$, such that the nodal state is stabilized at the expense of the nodeless. We calculate this line shape for a model of competing s_\pm-wave and d-wave pairing in iron-based superconductors.

Equations

$$H=\sum _{\mathbf{k},\sigma ,{\sigma }^{\prime }}\left(\epsilon (\mathbf{k})-{\mu }_{\text{B}}{H}_{\text{Z}}{\sigma }_{\sigma ,{\sigma }^{\prime }}^z\right)c_{\mathbf{k},\sigma }^{†}{c}_{\mathbf{k},{\sigma }^{\prime }}^{}$$

$$+\frac{1}{2}\sum _{\mathbf{k},\sigma ,{\sigma }^{\prime }}\left({\Delta }_{\sigma ,{\sigma }^{\prime }}(\mathbf{k})c_{\mathbf{k},\sigma }^{†}{c}_{-\mathbf{k},{\sigma }^{\prime }}^{†}+h.c.\right)\text{,}$$

$$\mathcal{H}=\frac{1}{2}\sum _{\mathbf{k}}{\vec{C}}_{\mathbf{k}}^{†}\left(\begin{array}{cc}\epsilon (\mathbf{k})-{\mu }_{\text{B}}{H}_{\text{Z}}{\sigma }^z& i\Delta (\mathbf{k}){\sigma }^y\\ -i{\Delta }^{\ast }(\mathbf{k}){\sigma }^y& -\epsilon (\mathbf{k})+{\mu }_{\text{B}}{H}_{\text{Z}}{\sigma }^z\end{array}\right){\vec{C}}_{\mathbf{k}}\text{.}$$

$$E^{\pm ,\pm }(\mathbf{k})=\pm {\mu }_{\text{B}}{H}_{\text{Z}}\pm \sqrt{\epsilon (\mathbf{k}{)}^2+\text{absolutevalue}{\Delta (\mathbf{k})}^2}\text{,}$$

$${\psi }_{\mathbf{k}}^{†,(-,\pm )}$$

$${\psi }_{\mathbf{k}}^{†,(+,\pm )}$$

$$\displaystyle v_{\text{F}}\equiv\frac{\partial E({\bf k})}{\partial k_{\perp}}\bigg{|}_{{\bf k}={\bf k}^{*}_{\text{F}}}\quad\text{,}\quad v_{\Delta}\equiv\frac{\partial E({\bf k})}{\partial k_{\parallel}}\bigg{|}_{{\bf k}={\bf k}^{*}_{\text{F}}}\quad\text{,}$$

$$\displaystyle E({\bf k})\big{|}_{{\bf k}\approx{\bf k}^{*}_{\text{F}}}=v_{\text{F}}k_{\perp}+v_{\Delta}k_{\parallel}\quad\text{.}$$

$${\left(k_{\perp }-k_{F,\perp }^{\ast }\right)}^2{v}_{\text{F}}^2+{\left(k_{\parallel }-k_{F,\parallel }^{\ast }\right)}^2{v}_{\Delta }^2={\mu }_{\text{B}}^2{H}_{\text{Z}}^2\text{.}$$

$$N_{\uparrow }$$

$$m={\mu }_{\text{B}}\left(n_{\uparrow }-n_{\downarrow }\right)$$

$$e_d-e_s=-m{H}_{\text{Z}}=-N_n\frac{{\left({\mu }_{\text{B}}{H}_{\text{Z}}\right)}^3}{2\pi v_{\text{F}}{v}_{\Delta }}\text{,}$$

$$\mathcal{Z}=\sum _{\{|\Psi ⟩\}}{e}^{-\beta E_{|\Psi ⟩}}=\prod _{\mathbf{k}\sigma }\left(1+e^{-\beta E^{\sigma ,+}(\mathbf{k})}\right)\text{,}$$

$$f=-k_{\text{B}}T\sum _{\sigma }{\int }_{-\infty }^{+\infty }{\rho }_{\sigma ,+}(E)\ln \left(1+e^{-\beta E^{\sigma ,+}}\right)\mathrm{d}E.$$

$$f_d-f_s\stackrel{\beta \gg 1}{\approx }-\sum _{\sigma }{\int }_0^{E_{\text{max}}}\frac{{{\rm Y}}_{\sigma ,+}(E)}{e^{\beta E^{\sigma ,+}}+1}\mathrm{d}E\text{,}$$

$$\frac{\alpha }{2}{\int }_0^{E_{\text{max}}}\frac{E^2}{e^{\beta E}+1}\mathrm{d}E=\frac{\alpha }{2}\frac{1}{{\beta }^3}{\int }_0^{\beta E_{\text{max}}}\frac{x^2}{e^x+1}\mathrm{d}x\text{.}$$

$$\Xi (x)=\frac{x^3}{3}-x^2\ln \left(1+e^x\right)-2x{\mathrm{Li}}_2(-e^x)+2{\mathrm{Li}}_3(-e^x)\text{,}$$

$$\Xi (\beta E_{\text{max}})-\Xi (0)\approx -2{\mathrm{Li}}_3(-1)=2\eta (3)\text{,}$$

$$f_d-f_s\stackrel{\beta \gg 1}{\approx }-\frac{3}{2}\alpha \zeta (3){\left(k_{\text{B}}T\right)}^3\text{,}$$

Full text

Introduction. Superconductivity is a quantum many-body state of matter which is a phase-coherent superposition of paired electrons schrieffer1983theory . In order to estimate whether such a state is energetically preferable, we have to consider both the aspects of energy gain due to the formation of the pair condensate as well as the conditions of pairing, i.e., the challenge to overcome the Coulomb repulsion between the electrons. In phonon-mediated superconductors, the interplay of electrons and lattice distortions imply an effective attraction between electrons at appropriate retardation scales PhysRev.108.1175 ; the generically preferred superconducting phase then is an $s$-wave state, which optimizes the condensation energy and, at the same time, is least susceptible to disorder. For electronically driven superconductors, the initial challenge is to avoid Coulomb repulsion, which, despite the screening of the long range part of the interactions due to the electron fluid, is still present at small distances. In the cuprates, this is accomplished by $d$-wave superconductivity, i.e., through establishing a condensate with finite relative angular momentum of the Cooper pairs. Depending on an itinerant or localized moment point of view, such a condensate is either preferred due to an enhanced $(\pi,\pi)$ channel of particle hole fluctuations PhysRevLett.15.524 or due to doping an antiferromagnetic Mott insulator ANDERSON1196 ; RevModPhys.78.17 . Condensation energy dictates which of the possible $d$-wave solutions is preferred: among the in-plane polarized states, the $d_{x^{2}-y^{2}}$-wave solution is chosen over $d_{xy}$-wave, as the Fermi level density of states of the cuprate band structure is smallest along the nodal lines of $d_{x^{2}-y^{2}}$, which guarantees a minimal loss of condensation energy by the presence of the nodes. Due to the momentum dependence of pairing, $d$-wave is more susceptible to disorder than $s$-wave andi-dis .

Iron-based superconductors doi:10.1021/ja800073m are located such in parameter space that maximizing condensation energy and minimizing Coulomb repulsion are of similar importance. Despite of electron-electron interactions as the apparent dominant role in pairing PhysRevLett.101.026403 , the interaction scales are weaker than in the cuprates, and multiple Fermi surfaces as well as multiple orbitals that contribute to the pairing significantly complicate the picture 0034-4885-74-12-124508 ; doi:10.1146/annurev-conmatphys-020911-125055 . Different theoretical approaches from an itinerant PhysRevLett.101.087004 ; PhysRevB.80.180505 ; 2009NJPh…11b5016G and localized moment PhysRevLett.101.076401 ; PhysRevLett.101.206404 picture have found $s$-wave and $d$-wave to be in close competition to each other for iron-based superconductors. The majority of experimental evidence suggests $s$-wave superconductivity for most pnictide compounds RevModPhys.83.1589 , however, this observation might change as the crystal quality will improve in upcoming waves of refined material synthesis. Some indication along these lines has been recently obtained in \BPChemKFe_2As_2 as the most strongly hole-doped limit of \BPChemK_xBa_1-xFe_2As_2, where the crystal quality is significantly enhanced through the absence of any Ba content. There, as predicted theoretically PhysRevLett.107.117001 , thermal conductivity measurements 2012PhRvL.109h7001R ; PhysRevB.89.064510 have found strong indication for $d$-wave, while the overall situation is still far from settled (see e.g. Ref. PSSB:PSSB201600350, and references therein). Recently, Raman scattering has found $d$-wave pairing to be of nearly competitive propensity to $s$-wave in \BPChemK_xBa_1-xFe_2As_2 crystals of low to intermediate hole doping level $0.22<x<0.7$ tommy . The specific details of this subleading $d$-wave state fit the superconducting form factor predicted in PhysRevB.80.180505 .

In this Letter, we elaborate on how the competition between an $s$-wave and a $d$-wave state is affected perturbatively by a Zeeman field $H_{\text{Z}}$ or temperature $T$. As a phenomenological starting point, we place ourselves at a first order zero temperature critical point where we assume a gapped $s$-wave and a nodal $d$-wave state to be of equal energy density. As partly elaborated on above, such a situation might occur in a multiple Fermi surface scenario with a balanced energetic significance of pairing formation and condensation energy. When a weak Zeeman field is turned on, the $s$-wave state will not respond to it since all quasiparticles are gapped. The $d$-wave state, however, possesses gapless quasiparticles at the nodes, which are polarized due to the Zeeman field. This gain in magnetic polarization energy makes the $d$-wave state preferable, and thus serves as a parameter to tune a phase transition from $s$-wave to $d$-wave. This similarly applies for finite temperature, where the $d$-wave state, in contrast to the $s$-wave state, gains free energy through the generation of entropy in the nodal regime. It warrants similarity to the Pomeranchuk effect in He3 where the crystal phase is stabilized at finite $T$ pome ; rich .

Model. At zero temperature, a singlet superconductor subject to a Zeeman field is described by the Hamiltonian

[TABLE]

where $\varepsilon({\bf k})$ is the kinetic term of the electrons, $\Delta_{\sigma,\sigma^{\prime}}({\bf k})=i\Delta({\bf k})\sigma^{y}_{\sigma,\sigma^{\prime}}$, and $\sigma^{y}$, $\sigma^{z}$ denote the Pauli matrices. The orbital magnetic field contribution be weak in comparison to the Zeeman term, e.g. as accomplished by an in-plane field PhysRevB.57.8566 in a quasi two-dimensional crystal 111Orbital magnetic field effects of a nodal superconductor generically dominate over the Zeeman term at weak field strength for a field orientation perpendicular to the 2D plane. The Volovik effect volo then predicts a $\propto\sqrt{H}$ scaling of the induced density of states, as opposed to the $\propto H$ scaling for the Zeeman term.. For $H_{\text{Z}}=0$ and $T=0$, our phenomenological ansatz assumes energy densities $e_{s}=e_{d}$ of an $s$-wave and a $d$-wave state.

As a specific competing candidate state in materials such as the pnictides, we consider extended s-wave (s*±*) $\Delta^{s_{\pm}}({\bf k})=\Delta_{s}\left(\cos(k_{x})\cos(k_{y})\right)$ PhysRevLett.101.057003 ; PhysRevLett.102.047005 and extended $d$-wave $\Delta^{d\pm}({\bf k})=\Delta_{d}\left(\cos(2k_{x})-\cos(2k_{y})\right)$ PhysRevB.80.180505 (fig. 1). This choice is motived by recent Raman scattering experiments on K-doped Ba-122 where both the leading and subleading superconducting order can be detected, rendering the $s_{\pm}$-wave and $d_{\pm}$-wave states close competitors tommy . In our argument to follow, it is irrelevant which type of $d$-wave or $s$-wave is realized. The only assumption is that the anisotropy in the $s$-wave state be small enough to provide a minimal quasiparticle gap $\Delta_{s}>\mu_{\text{B}}H_{\text{Z}}$ (the reduction of $s^{\pm}$-wave gap anisotropy, which by itself can be vital to explaining the experimental data in iron pnictides PhysRevLett.106.187003 , is often caused by disorder PhysRevB.79.094512 ), and that the $d$-wave state is nodal. The latter is a property protected by symmetry, as long as the nodal lines intersect with at least one Fermi pocket.

Bogoliubov quasiparticles. To obtain the SC quasiparticle spectrum, we represent (1) by its Bogoliubov-de Gennes (BdG) form, i.e. we the employ Nambu spinor $\vec{C}^{\dagger}_{{\bf k}}=\left(c^{\dagger}_{{\bf k},\uparrow},c^{\dagger}_{{\bf k},\downarrow},c_{-{\bf k},\uparrow}^{\phantom{\dagger}},c_{-{\bf k}^{\phantom{\dagger}},\downarrow}\right)$ notation and obtain

[TABLE]

The eigenvalue spectrum of (2) is given by

[TABLE]

where the two binary indices label the up/down spin (later denoted $\sigma$) and Bogoliubov particle/hole character, respectively. The eigenstates are given by

[TABLE]

Quasiparticle magnetization. Assume $k_{\text{B}}T\approx 0$ to be the smallest of all energy scales, and consider the spectrum (3) in the case of $\Delta({\bf k})=\Delta^{s_{\pm},d_{\pm}}({\bf k})$ for one representative Fermi pocket (fig. 2): for the case of $s_{\pm}$-wave in fig. 2a, $\Delta({\bf k})\approx\Delta_{s}$. For $\mu_{\text{B}}H_{\text{Z}}<\Delta_{s}$, the Zeeman field has a negligible effect on the spectrum because the binding energy of the singlet pair is larger than the applied field. For $d_{\pm}$-wave, however, while gapped by $\Delta({\bf k})\approx\Delta_{d}$ in the anti-nodal regime, there is a nodal regime $\Delta({\bf k})\approx 0$ illustrated in fig. 3b where the quasiparticles will respond to the Zeeman field PhysRevB.57.8566 . We expand the quasiparticle dispersion (3) around the nodal regime in momentum space at ${\bf k}^{*}_{\text{F}}$. Define

[TABLE]

where $k_{\perp}$ and $k_{\parallel}$ are the momenta perpendicular and tangential to the Fermi surface of $\varepsilon({\bf k})$ (fig. 3a). The dispersion takes the form of an elliptic Dirac cone

[TABLE]

Due to $H_{\text{Z}}$, the Fermi levels of spin $\uparrow$ and $\downarrow$ quasiparticles shift against each other, and a Fermi pocket emerges, with its surface given by the elliptic equation

[TABLE]

Computing the magnetization from there, without loss of generality, we constrain ourselves to the spin $\uparrow$ species. The number of particles in a given volume $\mathcal{V}$ is given by the area enclosed by the Fermi surface (7)

[TABLE]

From the density $n_{\uparrow}(E)=\frac{1}{4\pi}\frac{E^{2}}{v_{\text{F}}v_{\Delta}}$, we obtain the energy density of states $\rho(E)=\frac{\partial n_{\uparrow}}{\partial E}=\frac{1}{2\pi}\frac{E}{v_{\text{F}}v_{\Delta}}$. Considering both spin $\uparrow$ and $\downarrow$ contributions, we eventually obtain the quasiparticle magnetization density

[TABLE]

Assuming we had started from an equal energy density for $s$-wave and $d$-wave, the Zeeman field hence yields a preference for $d$-wave according to

[TABLE]

where $N_{n}$ denotes the total number of nodes induced by the $d$-wave state on the multi-pocket Fermi surface.

Quasiparticle entropy. Assume $H_{\text{Z}}\approx 0$ and finite temperature $k_{\text{B}}T<\Delta_{s}$. The free energy density is given by $f=u-Ts=-k_{\text{B}}T/N\ln\mathcal{Z}$, where $u$ denotes the inner energy density, $T$ the temperature, $s$ the entropy per particle, and $\mathcal{Z}$ the (grand) canonical partition function. The latter be denoted by

[TABLE]

with $\frac{1}{\beta}=k_{\text{B}}T$, spin index $\sigma=\pm$, many-particle state $\ket{\Psi}$, and the corresponding energy $E_{\ket{\Psi}}$ constructed from the single-particle states (LABEL:eqn:bogoliubov_quasiparticle_eigenstates) and the quasiparticle energy $E^{\sigma,+}({\bf k})$ given by (3), as we only consider the particle-type Bogoliubov branch. We set $e_{d}=e_{s}$ such that for the relative change in free energy density $f_{d}-f_{s}$, the inner energy does not enter. The nodal regime of the $d$-wave state then is the only relevant contribution to the change of $f_{d}-f_{s}$ at low temperatures. Since $f$ depends only on the momentum ${\bf k}$ through the energy $E({\bf k})$, we express it by means of the density of states $\rho_{\sigma,+}(E({\bf k}))$ to find

[TABLE]

Due to the discontinuity of $\rho_{\sigma,+}(E)$ at the band edge, we have to include the boundary terms upon partial integration. The boundary terms, however, may be neglected for $\beta\gg 1$, which, together with $f_{s}\approx 0$ in this limit, leaves us with

[TABLE]

where $\rho_{\sigma,+}(E)=\frac{\mathrm{d}\Upsilon_{\sigma,+}(E)}{\mathrm{d}E}$ and $E_{\text{max}}=\limsup_{{\bf k}\in\text{BZ}}E^{\sigma,+}({\bf k})$. We are interested in the low-temperature behavior of $f_{d}$, which is why we are allowed to model the density of states by a linear function along with (6), i.e. $\rho_{\sigma,+}(E)=\alpha E_{\sigma,+}$ such that $\Upsilon_{\sigma,+}(E)=\frac{\alpha}{2}E_{\sigma,+}^{2}$. This reduces our task to solving the Fermi-Dirac integral

[TABLE]

It already shows that the change in free energy density has the low-temperature behavior $f_{d}-f_{s}\propto-T^{3}$. The integral is solved by the integrand’s primitive 1995JMP….36.1217L ; 2009arXiv0909.3653M

[TABLE]

where $\operatorname{Li}_{i}(x)$ denotes the polylogarithm abramowitz1972handbook . The result may be easily checked using $\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{Li}_{s}(x)=\frac{1}{x}\operatorname{Li}_{s-1}(x)$ and $\operatorname{Li}_{1}(x)=-\ln(1-x)$. The integral in (14) gives

[TABLE]

where we neglect $\Xi(\beta E_{\text{max}})$ for small temperatures since $\lim_{x\rightarrow\infty}\Xi(x)=0$. We express the polylogarithm in terms of the Dirichlet $\eta$-function, which in turn is written in terms of the Riemann $\zeta$-function $\eta(s)=(1-2^{1-s})\zeta(s)$. Having thus fixed the exact prefactor, we find

[TABLE]

with $\zeta(3)\approx 1.20206$ (Apéry’s constant), and the universal cubic phase boundary shape depicted in Fig. 4.

This finding is calculated for one single node, and would have to be multiplied by $N_{n}$ to account for the total number of nodes. In (17), $\alpha$ is the non-universal coefficient which is sensitive to the details of the microscopic system. To calculate an explicit example, we take the one-band Hubbard model $\epsilon({\bf k})=-2t(\cos(k_{x})+\cos(k_{y}))-\mu$ with a nearly circular Fermi pocket for $t=1.0$, $\mu=-1.9$. Its density of states at the nodes (which may be derived by means of $v_{\text{F}}$ and $v_{\Delta}$ in Eq. 5) gives $\alpha\approx 1.58$.

Conclusion. At the latest since the discovery of the iron pnictides, materials with competitive unconventional nodal and nodeless superconducting pairing tendencies have established an intricate problem to be further studied experimentally and theoretically. We have shown that the perturbative Zeeman-magnetic and entropic response of nodal quasiparticles pave the way to tilt the thermodynamic balance in favor of the nodal, in our case $d$-wave, superconducting state by a term which is cubic in the perturbation parameter, i.e. either $H_{\text{Z}}$ or $T$. In the neighborhood of a zero temperature first order critical point between the nodal and the nodeless state, this gives rise to a universal cubic phase transition line shape in favor of the nodal state. Whether this contribution can yield an observable magnetically or entropically induced superconducting phase transition crucially relies on a careful material design as well as the detailed tunability of system parameters towards such a critical point. From a broader perspective, iron-based superconductors are not the only class of materials displaying comparable superconducting pairing tendencies towards a nodal and a nodeless state. For instance, the nodeless chiral $p$-wave state predicted for strontium ruthenate 0953-8984-7-47-002 is challenged by a nodal $d$-wave state, where unfortunately the possibly strongly anisotropic gap profile makes it hard to discriminate between both superconducting states 0295-5075-104-1-17013 ; elena . It is likely, however, that the $d$-wave state eventually has to win beyond a critical amount of strain Hicks283 , rendering the latter an ideal experimental tuning parameter towards the critical point regime between a nodal and nodeless superconducting state we have envisioned in this work.