Examples

# Origins of Superconductivity in FeSe

A QS*GW*+DMFT description of superconductivity in Bulk, Intercalated, and Monolayer FeSe

More than a half century passed before phonon mediated superconductivity was finally explained by Bardeen, Cooper and Schrieffer. Then came the high-*T _{c}* superconductors in mid 1980’s, and now almost four decades have passed; yet we understand very little of how such high

*T*’s can emerge.

_{c}High *T _{c}* materials are complicated because many factors play an important role in contributing to superconductivity. Model descriptions are very good at identifying possible low-energy degrees of freedom which are the proximate cause for superconductivity. However they cannot take adequate account of the chemistry — the high-energy degrees of freedom that modify the low-energy ones. Equally important, driving forces that cannot be encapsulated in a one-particle description are exceedingly difficult for models to capture. Vertices in spin and charge channels that govern the two-particle properties provide an excellent instance of this difficulty.

In a series of three papers (Refs 1-3), Acharya and coworkers used Questaal’s new implementation joining Quasiparticle Self Consistent *GW* (QS*GW*) and Dynamical Mean Field Theory (DMFT) theory^{4} to answer the following questions about FeSe from an *ab initio* perspective:

What role does nemacity play in affecting the spin susceptibility and superconductivity

^{1}?Why does

*T*in FeSe increase so dramatically when intercalated with alkali elements_{c}^{2}?Why does

*T*of FeSe in monolayer form (on a SrTiO_{c}_{3}substrate) increase by roughly eight-fold compared to the bulk^{3}?

*Ab initio* theories have historically proved of limited use for unconventional superconductivity, because their fidelity is insufficient capture the subtle and competing many-body effects driving unconventional superconductivity, or other instabilities. The superior fidelity of QS*GW*+DMFT has radically changed the situation. Combined with a vertex from the two-particle Green’s function, the *q*- and *ω*-dependent spin and charge susceptibilities *χ*^{m}(*q*,*ω*) and *χ*^{d}(*q*,*ω*) can be calculated. As a final step, the particle-particle vertex generated from *χ*^{m}(*q*,*ω*) and *χ*^{d}(*q*,*ω*) via another BSE provides a means to construct an Eliashberg-like equation for instability to superconductivity. This four-tier method is described in some detail in Ref 5, with an application to Sr_{2}RuO_{4}, as well as in Refs 2,3. The primary challenge for such a theory is to reliably compute two-particle vertex functions in different channels and identify the primary mechanism for superconductivity.

**Nemacity and Superconductivity**

**FIG 1.**Two energy cuts of the imaginary part of the dynamic spin susceptibility Im

*χ*(

*q*,

*ω*) plotted along the line

*q*= (H=1, K, L=0), compared against an inelastic neutron scattering (INS) experiment

^{6}.

It is known that FeSe undergoes a transition from a tetragaonal to orthorhombic phase at 90 K. The transition is not large; the ratio of lattice parameters *b*/*a* changes from 1 to 0.98. Nevertheless this induces a striking asymmetry (nematicity) in a number of observed properties: for example *d _{xz}* and

*d*bands forming the electron pockets at M are no longer degenerate, as well as the spin susceptibility from neutron measurements. The bare fact of nematicity has led to a great deal of speculation first, about the mechanism responsible for it, and second, its effect on superconductivity. In particular, since the superconductivity is connected to the electron pockets at M (and the hole pockets at Γ), there is speculation that nematicity affects the superconducting state. (For a discussion, see Ref 1 and the references therein.)

_{yz}As to the first issue, our QS*GW*+DMFT finds nematicity, but it underestimated. The two likely reasons for this are spin fluctuations or the electron-phonon interaction (or something connected with nuclear displacements.) Whatever the cause, much of the driving force is likely to originate from a nonlocal vertex, absent in both QS*GW* and DMFT. To address the second issue we mimic the true nematicity by increasing *b*/*a* until the asymmetry in electron pocket matches quantum oscillation measurements.

The paper’s primary focus is the second issue, namely what is the effect of nematicity on superconductivity? Since *T _{c}* (9 K) is much lower than the transition to the orthorhombic phase (99 K), this is a difficult question to answer experimentally. With the computational machinery at hand, however, we can well answer this question, subject to the caveat that our proxy for obtaining the correct nematicity is also a sufficient proxy for the true effective potential.

**FIG 2. The superconducting instabilities**inside the orthorhombic phase are shown: Instabilities correspond to the leading (Δ

_{1}) and lagging (Δ

_{2}) eigenvalues of the solutions to the BCS gap equation. The evolution of the leading eigenvalue as a function of temperature is shown for tetragonal, orthorhombic, and ortho- enhanced phases in the bottom panel.

The spin susceptibility was used as a benchmark of the theory, in part because it has been well characterized, but also it is a primary mechanism driving superconductivity in FeSe. Fig. 1 compares calculations with and without nematicity to neutron data. Agreement is good; note in particular how the predicted enhancement in *χ*^{m}(*q*,*ω*) from nemacity correlates with the experimental data.

Fig 2 shows that even while *χ*^{m} is significantly affected by nemacity, superconductivity is little affected. This is because it is controlled mostly by the *d _{xy}* channel, which nemacity hardly effects. The theory predicts that leading instability (top left panel of Fig 2) is of the form Δ

_{1}∼ cos(

*k*) + cos(

_{x}*k*), consistent with the

_{y}*s*

^{+}−

*s*

^{−}form factor widely expected for FeSe. Note the theory also predicts a weaker (lagging) instability (Δ

_{2}), which is not seen because (Δ

_{1}) is stronger.

**Intercalated FeSe**

In the second paper^{2}, the two-particle origin of superconductivity is emphasized. This is important, because the standard BCS and Bose Einstein Condensate (BEC) theories widely invoked to explain superconductivity emphasize the role of one-particle properties, e.g. density-of-states at the Fermi level, *D*(*E _{F}*). In BCS theory

*T*has an exponential dependence on

_{c}*D*(

*E*), while in BEC theory it has a power law dependence. For electron-phonon mediated (conventional) superconductors,

_{F}*T*does typically increase with

_{c}*D*(

*E*), and it also does for some (unconventional) cuprate superconductors.

_{F}It has been known for some time that when FeSe is intercalated, *T _{c}* is enhanced by roughly 4-5 times over the bulk, for many kinds of intercalated variants. Yet DFT calculations show no significant change of

*D*(

*E*) on intercalation (and the result is similar when higher level of theory is used). Thus it seems clear that standard application of BCS and BEC theories do not explain the phenomenon, and what leads to such dramatic enhancement in

_{F}*T*remains unanswered. It is not totally unexpected since even in the BCS theory the constant λ determining

_{c}*T*is the product of

_{c}*D*(

*E*) and the effective inter-electron interaction, and the primary reason why common attention is focused on

_{F}*D*(

*E*) is that it is much easier to calculate. This simplification can miss out on the essential physics, and as we will show here, intercalated FeSe is a very clear example where all the essential changes happen in the effective interaction only. In a more formal language, this effective interaction is nothing but the vertex, which is an essentially

_{F}*two-particle*characteristic of correlated systems.

**FIG 3. Effect of intercalation**on

*T*divided into two steps, as noted in the text. Reduction in the Fe-Se-Fe bond angle enhances T

_{c}_{c}, and increasing the

*c*axis enhances it further.

The first task is to establish whether Questaal’s *ab initio* four-tier QS*GW*+DMFT theory yields the observed increase in *T _{c}*. In addition to doping the system (which as noted, seems not to be significant for FeSe), intercalation modifies the underlying FeSe lattice in two significant ways: it changes the Fe-Se-Fe bond angle (planar-distortion, which we denote as

*p-d*), and it also increases the

*c*axis, changing separation between Fe planes. In what follows,

*f-d*denotes what happens when both effects are taken into account. As Fig 3 shows, the theory predicts that both effects enhance

*T*. In both cases, it is driven by an enhancement correlations from the vertex, but for different reasons. In the

_{c}*p-d*only case, the smaller Fe-Se-Fe bond angle that reduces the Fe-Se-Fe hopping and makes Fe 3

*d*states more correlated. In the

*f-d*case, correlations are enhanced because the increased

*c*-axis reduces electronic screening.

**FIG 4. Failure of one-particle properties**from LDA, QS

*GW*and QS

*GW*+DMFT to explain the enhancement in T

_{c}}. The total density of states from different levels of the theory are shown, for intrinsic FeSe,

*p-d*, and

*f-d*.

When both effects are taken into account, the theory predicts a five-fold enhancement in *T _{c}*, dovetailing nicely with measurements on a number of systems with different intercalation elements. The reader is referred to Ref 2 for a detailed analysis for why this happens, but in brief the theory enables us to draw several important conclusions.

First, the one-body properties, e.g. *D*(*E _{F}*), do not account for the enhancement in

*T*(Fig 4).

_{c}**Shortcoming of the RPA theory for magnetic and superconducting susceptibilities:**

*χ*resolved in different intra-orbital channels as a function of

^{m}*q*. RPA theory predicts magnetic and superconducting susceptibilities to be maximal in the non-intercalated sample, while vertex corrections predict a nearly five-fold enhancement in magnetic instability in the intercalated sample.

Second, the vertex plays a central role (Fig 5). Without vertex corrections the momentum dependence of χ^{m}(*q*) is rather weak and there is no clear separation between χ^{m} at *q*=0 and *q*=(1/2,1/2). Experimentally, in bulk FeSe magnetic fluctuations at *q*=0 are rather weak compared to the main instability at *q*=(1/2,1/2) and it is the vertex that brings in the needed momentum dependent variation in χ^{m}.

Third, superconductivity is predominately driven by the *d _{xy}* channel in intercalated FeSe as it does in pristine FeSe

^{1}.

**Monolayer FeSe**

Iron-pnictogen and iron-chalcogen based superconductors are classed as Hund’s metals, meaning correlations mostly originate from the Hund’s exchange parameter *J*. In recent years a consensus has evolved that strong Hund’s correlations drive the ubiquitous bad metallicity observed in their phase diagrams. Such metals have a multiplicity of bands near the Fermi level *E _{F}*; in particular FeSe has all five Fe

*d*states active there. Correlations are observed to be highly orbital-selective (a signature of ``Hundness’’) with

*d*the most strongly correlated orbital. However, very little is known whether Hund’s correlation can generate glue for superconducting pairing and control

_{xy}*T*.

_{c}Also, rather remarkably, the bulk *T _{c}* (9 K), has been observed to increase to ~75 K when grown as a monolayer on SrTiO

*(M-FeSe/STO), and 109 K on doped SrTiO*

_{3}*. Thus while ``Hundness’’ has been found to be important in controlling the single- and two-particle spectral properties of bulk FeSe, the multiplicity of factors (orbital character, spin-orbit coupling, shape of Fermi surface and dispersion of states around it, differences in susceptibilities, nematicity, electron-phonon interaction) obfuscate to what extent Hundness, or other factors, drive superconductivity, and whether `Hundness’ can at all explain the jump in*

_{3}*T*going from bulk to M-FeSe/STO.

_{c}**FIG 6.**The QS

*GW*band structure and QS

*GW*+DMFT spectral functions

*A*(

*k,ω*) are shown on a section of the

*Γ*-M path for bulk-FeSe (top panel) and M-FeSe/STO (middle panel). The Fe-

*d*state calculated by QS

_{xy}*GW*is depicted in blue. Note the strongly marked incoherence, particularly in (

*d*). In all cases DMFT narrows the width of

*d*relative to QS

_{xy}*GW*as is typical of narrow-band

*d*systems, but incoherence is highly sensitive to the position of

*d*. In both cases

_{xy}*d*is proximate to

_{xy}*E*and a high degree of incoherence is present. When

_{F}*d*is pushed well below

_{xy}*E*(figures not shown, but see Fig 2 of Ref [3]) spectral functions become much more coherent, and χ

_{F}^{m}loses intensity (Fig 7). The bottom panel shows how the leading eigenvalue

*λ*of the linearized particle-particle ladder BSE equation evolves treating

*J*as a free parameter (blue circles). This is because the peak in χ

^{m}at the antiferromagnetic nesting vector is extremely sensitive to

*J*.

**FIG 7.**Energy and momentum resolved spin susceptibility Im

*χ*shown for (a) bulk FeSe (B-FeSe) (

^{m}(q,ω)*J*=0.6 eV), (b) bulk FeSe with increased Hund's correlation (

*J*=0.68 eV), (c) reduced Fe-Se height (

*h*=1.27 Å, (d) 0.15 electron doped bulk FeSe, (e) free standing monolayer of FeSe, M-FeSe (f) M-FeSe/STO. The q-path (H,K,L=0) chosen is along (0,0)-(1/2,0)-(1/2,1/2)-(0,0) in the Brillouin zone corresponding to the two Fe-atom unit cell. The intensity of the spin fluctuations at (1/2,1/2) is directly related to the presence of the Fe-d

_{Se}*state at Fermi energy and its incoherence. The more incoherent the A(*

_{xy}*k*,

*ω*) is, the more intense is Im

*χ*(

^{m}*q*=(1/2,1/2,0),

*ω*).

The third paper^{3} shows that Hundness does explain much of the origins of superconductivity in FeSe. We show explicitly the connections between correlations in *d _{xy}* and the superconducting critical temperature

*T*. Starting from the

_{c}*ab initio*results as a reference, we consider various kinds of excursions in parameter space around the reference to determine what controls

*T*. We show small excursions in

_{c}*J*can cause colossal changes in

*T*. Additionally we consider changes in hopping by varying the Fe-Se bond length in bulk, in the free standing monolayer M-FeSe, and M-FeSe on a SrTiO

_{c}_{3}substrate (M-FeSe/STO). The twin conditions of proximity of

*d*to

_{xy}*E*, and the strength of

_{F}*J*emerge as the primary criteria for incoherent spectral response and enhanced single- and two-particle scattering that in turn controls

*T*. Using constrained RPA, we show further that FeSe in monolayer form (M-FeSe) provides a natural mechanism to enhance

_{c}*J*. These factors thus explain why

*T*increases so dramatically for M-FeSe/STO, whereas an isolated monolayer, M-FeSe, is predicted to have a vanishingly small

_{c}*T*.

_{c}Fig. 7 highlights how the twin effects of *J* and proximity of the Fe *d _{xy}* state to

*E*work in concert to govern the intensity of Im

_{F}*χ*

^{m}in the vicinity of

*q*=(1/2,1/2,0) and

*ω*=0. Panels (a) and (b) differ only in that

*J*=0.60 eV in the former case (the value computed from constrained RPA) and

*J*artificially increased to 0.68 eV in the latter case:

*T*is predicted to be enhanced by roughly a factor of 10. Panels (a) and (c) use the

_{c}*ab initio*value for

*J*, but in (c) the Se height is reduced. The intensity of Im

*χ*

^{m}is softened, and the peak lifted well above

*ω*=0. Panel (e) shows Im

*χ*

^{m}for a free-standing monolayer. Once again Im

*χ*

^{m}is very weak and for panels (c) and (e)

*T*is very small (see bottom panel of Fig. 6). These are the two instances where

_{c}*d*is submerged well below

_{xy}*E*. Finally Im

_{F}*χ*

^{m}for

*m*-FeSe/SrTiO

_{3}is depicted in the bottom right panel (f). O

*p*orbitals in SrTiO

_{3}couple to the Fe

*d*and push it up, close to

_{xy}*E*. Also because screening is reduced,

_{F}*J*increases to 0.67 eV, inducing a strong peak in Im

*χ*

^{m}. These are the two most favorable conditions for high

*T*.

_{c}**References**

^{1} Swagata Acharya, Dimitar Pashov, Mark van Schilfgaarde, *Role of nematicity in controlling spin fluctuations and superconducting T c in bulk FeSe*, Phys. Rev. B 105, 144507 (2022).

^{2} Swagata Acharya, Mikhail I. Katsnelson, Mark van Schilfgaarde, *Vertex dominated superconductivity in intercalated FeSe*, Preprint https://doi.org/10.48550/arXiv.2205.14422.

^{3} Swagata Acharya, Dimitar Pashov, Francois Jamet, Mark van Schilfgaarde, *Electronic Origin of Tc in Bulk and Monolayer FeSe*, Symmetry 13, 169 (2021).

^{4} Dimitar Pashov, Swagata Acharya, Walter R. L. Lambrecht, Jerome Jackson, Kirill D. Belashchenko, Athanasios Chantis, Francois Jamet, Mark van Schilfgaarde, *Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique*, Section 7, Comp. Phys. Comm. 249, 107065 (2020).

^{5} Swagata Acharya, Dimitar Pashov, Elena Chachkarova, Mark van Schilfgaarde, and Cedric Weber, *Electronic structure correspondence of singlet-triplet scale separation in strained Sr _{2}RuO_{4}*, Appl. Sci. 11, 508 (2021). https://www.mdpi.com/2076-3417/11/2/508.

^{6} Q. Wang et al, Nat. Commun. 7, 12182 (2016).

PAPERS · MAGNETISM · SUPERCONDUCTIVITY