Electron-phonon coupling and spin uctuations in the Ising superconductor NbSe 2 S. Das1 2H. Paudyal3E. R. Margine3D. F. Agterberg4and I. I. Mazin1 2 1Department of Physics and Astronomy George Mason University Fairfax VA 22030

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Electron-phonon coupling and spin ﬂuctuations in the Ising superconductor NbSe2

S. Das,1, 2 H. Paudyal,3E. R. Margine,3D. F. Agterberg,4and I. I. Mazin1, 2

1Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030

2Quantum Science and Engineering Center, George Mason University, Fairfax, VA 22030

3Department of Physics, Applied Physics, and Astronomy,

Binghamton University-SUNY, Binghamton, New York 13902, USA

4Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53201, USA

Ising superconductivity, observed experimentally in NbSe2and similar materials, has generated

tremendous interest. Recently, attention was called to the possible role that spin ﬂuctuations (SF)

play in this phenomenon, in addition to the dominant electron-phonon coupling (EPC); the possi-

bility of a predominantly-triplet state was discussed and led to a conjecture of viable singlet-triplet

Leggett oscillations. However, these hypotheses have not been put to a quantitative test. In this

paper, we report ﬁrst principle calculations of the EPC and also estimate coupling with SF, in-

cluding full momentum dependence. We ﬁnd that: (1) EPC is strongly anisotropic, largely coming

from the K-K’ scattering, and therefore excludes triplet symmetry even as an excited state; (2)

superconductivity is substantially weakened by SF, but anisotropy remains as above; and, (3) we

do ﬁnd the possibility of a Leggett mode, not in a singlet-triplet but in an s++ –s±channel.

I. INTRODUCTION

Revolutionary progress in the growth and exfoliation

of single atomic layers over the last two decades has led to

a new era of scientiﬁc discoveries and technological inno-

vation. Following graphene, transition metal dichalco-

genides (TMDs) have taken the spotlight, as treasure

trove for a plethora of novel quantum phenomena. One

of the signiﬁcant discoveries in recent years was the phe-

nomenon of the so-called Ising superconductivity, driven

by spin-orbit (SO) coupling combined with absence of

the inversion symmetry [1–10]. Proximity eﬀects and in-

terfaces of Ising superconductors with other monolayer

TMDs, such as doped TaS2and TaSe2[8,11], or with

two-dimensional (2D) magnetic layered materials, such

as CrI3[12,13] and VI3[14], could lead to interesting

device applications for quantum information storage and

spintronics.

The combination of broken Kramer’s degeneracy due

to the lack of inversion symmetry and SO coupling in

monolayers of 2H-NbSe2leads to splitting of the elec-

tronic bands near the Kpoint, and its corresponding

inversion counterpart, K0=−K, in the Brillouin zone

(BZ). The magnitude of this splitting due to spin-orbit

eﬀects is considerably larger than the superconducting

order parameter [11,15]. Because of this splitting, the

formally s-wave singlet superconducting state well known

in the bulk NbSe2, splits into two mixed states: singlet

(S) and triplet (T) states combine to form an S + T

state on one SO partner and an S - T state on the other.

The same is true about the inversion-related partners,

e.g., the outer Fermi contours around Kand K0[15].

The emerging phenomenon was duly dubbed “Ising su-

perconductivity”(IS). While in most experimental probes

the two IS partners combine to form a (nearly) pure S

state, the incipient triplet component manifests itself in

many notable ways, most famously in the formally inﬁ-

nite thermodynamic critical ﬁeld along the ab layer plane.

Recent ﬁrst principles calculations, combined with

some limited experimental data, strongly suggest that

bulk NbSe2is close to a magnetic instability, and the

undistorted monolayers are even closer [15–17] (and also

likely for similar TMD superconductors). This fact led

to speculations that triplet pairing, even if not a lead-

ing instability, may play an important role in Ising su-

perconductivity in NbSe2[15]. Recent observation of

a low-temperature tunneling mode in NbSe2monolayers

was tentatively interpreted as a singlet-triplet Leggett

mode [18].

Recently, we investigated the full momentum-

dependent spin susceptibility [19] in NbSe2monolay-

ers [17], and found that it is rather strongly peaked

at a particular wave vector, close to q= (0.2,0) in

the 2D Brillouin zone. At the same time, experimen-

tal and density-functional theory (DFT) calculations

of charge density waves [20–24] and superconductiv-

ity [4,10,18,24,25] for some bulk [15,20,26,27] and

2D TMDs [15,28–31] have been reported. A subsequent

ﬁrst-principles study claimed [15] that density functional

calculations overestimate the superconducting transition

temperature in monolayer NbSe2. Together with the in-

dications of strong spin ﬂuctuations (SF) in this class

of materials, it strongly suggests that a proper quanti-

tative analysis of the pairing state in NbSe2, and likely

in other Ising superconductors, is not possible without

the simultaneous accounting of the anisotropic electron-

phonon coupling (EPC) and SF-induced interaction.

In this paper, we present such an analysis and ﬁnd sev-

eral expected and some rather unexpected results. First,

in agreement with existing calculations of bulk and 2D

TMDs, the standard DFT calculations of EPC strongly

overestimate the transition temperature in monolayer

NbSe2(far beyond typical inaccuracies of the method).

Second, including on the same footing SF-induced inter-

action (using the previously calculated SF spectrum [17])

brings the calculations in agreement with experiment (in-

cluding a proper frequency cutoﬀ for SF is essential).

Third, the calculated EPC is exceptionally anisotropic,

arXiv:2210.00745v1 [cond-mat.supr-con] 3 Oct 2022

with the lion’s share of the coupling coming from the

same-spin K−K0scattering. The calculated gap distri-

bution, formally speaking, should be visible in tunneling

experiments, and it has not been observed so far. We

discuss possible reasons for why the small gap on the Γ

Fermi surface pocket has so far eluded detection.

II. BACKGROUND LANDSCAPE

A. Tunneling

Tunneling experiments are an indispensable tool for

the discernment of the quantitative as well as qualitative

nature of superconducting order parameter in unconven-

tional superconductivity [4]. In Ref. [18] it was pointed

out that the diﬀerent character of the dominant Nb or-

bitals on the Γ and KFermi surface pockets suggests that

their tunneling probability through vacuum or insulating

barrier should be diﬀerent. The fact that the calculated

superconducting gap is rather diﬀerent at the two sets of

pockets suggest that this issue deserves a closer look.

One possible explanation for the lack of observation

of a smaller gap is that, due to impurity scattering, the

gap averages to one uniform value. We do not ﬁnd this

likely. Indeed, the observed 2∆/Tcratio is noticeably

larger than the weak-coupling value of 3.54, and our cal-

culations are far from the strong coupling regime where

such an enhancement would be possible. Rather, our

larger (K) gap agrees consistently with the experiment.

This calls into question, why the second, smaller gap is

not seen in the experiment? We do not have an answer

yet, but we can add to the body of known facts, our cal-

culations of the partial character of Se pzat the Fermi

level. Indeed, in STM experiments it is rather clear that

the main signal comes from Se atoms, and this orbital is

the most extended along the out-of-plane direction, so it

is expected to dominate the STM spectra. We show this

character as the faux map in Fig. 2.

Interestingly, while on average the Γ pocket has a

larger content of this character, there are hot spots along

the K−Mdirection that are expected to have the largest

tunneling probability; taking the calculated value of the

superconducting gap at this point yields a rather good

agreement with the experiment. On the other hand,

while the diﬀerence between the tunneling current from

pzorbitals is exponentially higher than that from the px,y

ones, the dependence on the pzweight is just linear, so, in

principle, one would expect to see subgap features corre-

sponding, ﬁrst of all, to the Γ pocket gap approximately

twice smaller than the maximal gap.

In order to address the nature of superconducting gap,

scanning tunneling measurements were performed and re-

ported on few-layer NbSe2[28]. The superconducting gap

as well as the critical temperature (Tc) have been found

to decrease with the number of layers. In particular, the

gap values measured at 0.3 K exhibited a reduction by

more than a factor of 2 from 1.3 meV in the bulk to

0.6 meV in the bilayer. Unfortunately, no tunneling cur-

rent was detectable in the monolayer devices, most likely

due to the diﬃculty of obtaining a clean NbSe2-hBN in-

terface. The decrease in the Tchas been found to be well

described by a linear dependence with the inverse thick-

ness, with the temperature dropping from 7.0 K in bulk

to 4.7 - 4.8 K and 2.0 - 2.5 K in bilayer and monolayer,

respectively. This drastic decrease in both the measured

superconducting gap and critical temperature has been

assigned to the surface energy contribution imposed by

the boundary condition upon the electronic wave func-

tion. Further, it has been conjectured that while for up

to 5 layers or higher, the gap is considerably anisotropic,

the anisotropy disappears and the gap obeys the isotropic

Bardeen Cooper-Schrieﬀer (BCS) gap equations for the

bilayer [28]. The hypothesis that the incommensurate

charge density wave is enhanced by the simultaneous ex-

istence of superconductivity in monolayer NbSe2has also

been proposed. [25]

B. Experimental results vs magnetic and

electron-phonon coupling calculations

Superconductivity in bulk NbSe2has been studied ex-

tensively both experimentally and theoretically, and the

superconducting transition temperature Tchas been ex-

perimentally identiﬁed as ∼7 K [32]. Compared to bulk,

Tcof monolayer NbSe2is about half, up to ∼3.5 K in best

samples (it is often as low as 1 K) [3,8]. It was argued

that that is due to the pair-breaking eﬀect of magnetic

moments associated with Se vacancies [7].

State-of-the-art ﬁrst-principles calculations that usu-

ally deliver accurate outcomes for superconductors where

the pairing is entirely due to EPC overestimate the Tcin

bulk NbSe2[20] and isostructural NbS2[24]. In the lat-

ter case, calculations using Eliashberg theory yield a Tc

and a zero-temperature gap a factor of ∼3 and 4 larger

than experiment, respectively [24]. At the same time, the

experimentally measured spin susceptibility, χs, in bulk

NbSe2was reported to be χs∼3×10−4emu/mole [33],

which signiﬁcantly exceeds the bare bulk Pauli suscep-

tibility χ0∼0.87 ×10−4emu/mole. DFT calculations

render χs= 4.2×10−4emu/mole [15,17], 40% larger

that in the experiment – a common overestimation in

itinerant systems, indicating that SF are strong in the

system.

Recently, we have calculated the static q-dependent

DFT susceptibility in NbSe2monolayer [17], and rescaled

it to account for the ﬂuctuational reduction; the latter

was deduced from the known experimental data for the

bulk compound. Together with the standard formalism

for calculating EPC, this forms the basis for addressing

superconductivity in monolayer NbSe2from ﬁrst princi-

ples.

C. Role of Charge Density Waves

The role played by charge density waves in either as-

sisting or opposing superconductivity has been a matter

of active debate in the ﬁeld of unconventional supercon-

ductivity. Several recent papers [29,30] ascribe the noto-

rious overestimation of the superconducting temperature

and order parameter to the charge density wave (CDW)

eﬀects. We do not believe that CDW alone provides a

comprehensive explanation, if at all, for the following rea-

sons:

•First of all, overestimation takes place both in the

bulk and in single layer calculations of NbSe2[20,

27,30]. Yet in NbSe2suppression of the CDW by

pressure or disorder has only a minor eﬀect on the

Tc[20,34].

•NbS2does not exhibit a CDW phase, yet the prob-

lem of overestimation there is as severe, if not more

so [24].

•It was shown that in bulk NbSe2anharmonic-

ity strongly suppresses the tendency to form the

CDW [20], hence it is likely that standard DFT

calculations overestimate the CDW amplitude and

leads to the partial gapping of the Fermi surface.

Bulk calculations for NbSe2, accounting for anhar-

monicity to suppress CDW at elevated pressure,

extrapolate to Tc≈12.3 K and λ≈1.4 at zero

pressure, a considerable overestimate [20].

•Overestimation of the Tcwas also recently at-

tributed to the empirical treatment of the Coulomb

interaction in the Eliashberg formalism compared

to the superconducting density functional the-

ory [27]. Assuming a value of the Coulomb

pseudopotential µ∗=0.11 yielded a superconduct-

ing Tc=16 K, whereas a signiﬁcantly higher value

of µ∗=0.28 was necessary to replicate the exper-

imental outcome. Note that, while resorting to

an unusually high value of µ∗reproduces the ex-

perimental gap, such Coulomb interactions are not

physical even for low density metals, since the value

of µ∗(as opposed to µ) is set by log (EF/ωph), and

not by the bare Coulomb coupling.

•The resistivity in the normal state shows absolutely

no detectable feature at the CDW temperature [21].

If, as suggested in Ref. [30], CDW reduces the EPC

constant by a factor of seven, the eﬀect on the nor-

mal transport would have been dramatic.

•In recent experiments [18], suppressing CDW in

single layer NbSe2by disorder (such as Mo dop-

ing) led to Tcsimultaneously suppressed.

For these reasons, we believe that the eﬀect of CDW on

superconductivity in previous works was overestimated

and CDW plays at best a small role in suppressing su-

perconductivity. Instead, in this paper we put emphasis

on the pair-breaking eﬀect of magnetic interactions.

III. RESULTS

A. Theoretical basis

The recipe for calculating electron-phonon interactions

from ﬁrst principles is well established [35,36]. How-

ever, the incorporation of the eﬀects of spin-ﬂuctuation

warrants reevaluation of the hitherto established proto-

col. A formalism incorporating spin-ﬂuctuation eﬀects

alongside electron-phonon coupling would set the stage

to delineate the concomitant landscapes of conventional

and unconventional superconductivity. The momentum-

dependent Eliashberg spectral function is given by:

α2Fep(k,k0, ω)=NFX

ν|gν

k,k0|2δ(ω−ωqν),(1)

where NFis the density of states per spin at the Fermi

level, gν

k,k0are the screened electron-phonon matrix ele-

ments, and ωqνare the phonon frequencies for a phonon

with wavevector q=k−k0and branch index ν.

A systematic incorporation of spin ﬂuctuations is less

well established, even though the problem goes back to

the 1960s [37]. The simplest recipe was summarized

by D. Scalapino [38], and stipulates that the eﬀective

pairing interaction in the singlet channel is given by the

Eliashberg function α2Fsf (k,k0, ω), deﬁned through the

dynamical spin susceptibility χk−k0(ω) and (in the mod-

ern DFT parlance) the Stoner factor I:

α2Fsf (k,k0, ω) = −3

2πNFI2Im[χk−k0(ω)].(2)

In the triplet channel the sign is positive (attraction) and

the spin-rotation factor 3 is replaced by 1. In practice,

the static integrated version of Eq. (2), calculated as the

Fermi surface average, is universally used:

λsf =−3

2NFhI2Reχqiq.(3)

More elaborate versions, taking into account ladder

diagrams in addition to polarization bubbles, have also

been put forward in the following years, most notably

by Fay and Appel [39], but in proximity to a magnetic

instability the only resonant term is the one given by

Scalapino [38,40]. The non-resonant part is usually as-

sumed to be incorporated in the Coulomb pseudopoten-

tial.

Equation (3) has one serious problem however: it com-

pletely neglects retardation eﬀects, implicitly assuming

that the characteristic time scale for the spin ﬂuctua-

tions is the same as for phonons, which is rarely the case.

Because of this, practical applications of this formalism

are plagued by overestimating the SF eﬀect compared

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Electron-phononcouplingandspinuctuationsintheIsingsuperconductorNbSe2S.Das,1,2H.Paudyal,3E.R.Margine,3D.F.Agterberg,4andI.I.Mazin1,21DepartmentofPhysicsandAstronomy,GeorgeMasonUniversity,Fairfax,VA220302QuantumScienceandEngineeringCenter,GeorgeMasonUniversity,Fairfax,VA220303DepartmentofPhysics,Appl...

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Electron-phonon coupling and spin uctuations in the Ising superconductor NbSe 2 S. Das1 2H. Paudyal3E. R. Margine3D. F. Agterberg4and I. I. Mazin1 2 1Department of Physics and Astronomy George Mason University Fairfax VA 22030.pdf

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Electron-phonon coupling and spin uctuations in the Ising superconductor NbSe 2 S. Das1 2H. Paudyal3E. R. Margine3D. F. Agterberg4and I. I. Mazin1 2 1Department of Physics and Astronomy George Mason University Fairfax VA 22030

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