Superconducting properties of doped blue phosphorene Eects of non-adiabatic approach Mohammad Alidoosti1 Davoud Nasr Esfahani12and Reza

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Superconducting properties of doped blue

phosphorene: Eﬀects of non-adiabatic approach

Mohammad Alidoosti1, Davoud Nasr Esfahani1,2and Reza

Asgari3,4

1Pasargad Institute for Advanced Innovative Solutions (PIAIS), Tehran

19916-33361, Iran

2Department of converging technologies, Khatam University, Tehran 19916-33357,

Iran

3School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran

19395-5531, Iran

4School of Physics, University of New South Wales, Kensington, NSW 2052,

Australia

E-mail: r.asgari@unsw.edu.au, d.nasr@khatam.ac.ir

5 October 2022

Abstract.

We study the eﬀects of Kohn anomalies on the superconducting properties in

electron- and hole-doped cases of monolayer blue phosphorene, considering both

adiabatic and non-adiabatic phonon dispersions using ﬁrst-principles calculations. We

show that the topology of the Fermi surface is crucial for the formation of Kohn

anomalies of doped blue phosphorene. By using the anisotropic Eliashberg formalism,

we further carefully consider the temperature dependence of the non-adiabatic phonon

dispersions. In cases of low hole densities, strong electron-phonon coupling leads to a

maximum critical temperature of Tc= 97 K for superconductivity. In electron-doped

regimes, on the other hand, a maximum superconducting critical temperature of Tc=

38 K is reached at a doping level that includes a Lifshitz transition point. Furthermore,

our results indicate that the most prominent component of electron-phonon coupling

arises from the coupling between an in-plane (out-of-plane) deformation and in-plane

(out-of-plane) electronic states of the electron (hole) type doping.

Keywords: superconductivity, Kohn anomaly, charge density wave, adiabatic and non-

adiabatic phonons

Submitted to: 2D Mater.

arXiv:2210.01151v1 [cond-mat.supr-con] 3 Oct 2022

Superconducting properties of doped blue phosphorene: Eﬀects of non-adiabatic approach 2

1. Introduction

Recent advances in the fabrication of two-dimensional (2D) materials reveal many

unprecedented phenomena for further investigation of the various properties of these

structures due to excellent electronic and phononic band energies [1–8]. One of

the distinctive features is the superconducting properties of atomically thin 2D

systems [9–18]. According to the BCS theory [19] the higher superconducting transition

temperature (Tc) can be achieved by having a remarkable electronic density of states

at the Fermi level, N(εF), with a large proportion of the Debye frequency, ωD. On the

other hand, to get a signiﬁcant Tc, one has to reach a tangible electron-phonon coupling

(EPC) parameter λ= 2N(εF)hg2i/ωD. Nevertheless, a signiﬁcant λarising from a large

value of N(εF) and hg2i, an average of EPC matrix elements over the ﬁrst Brillouin zone

(1BZ), is indeed more favorable than a value of λarising from a small ωD. The main

reason for this is that a small ωDgives rise to a smaller ωlog, characteristic phonon

energy, and subsequently suppresses the Tc.

Based on these analyses, one can look for light materials with a large λto achieve

reasonable Tc. In the context of 2D superconductivity, there are several materials with

a sharp peak in their N(ε) spectrum near the valence band maximum. Examples of

such systems are 2D buckled structures based on nitrogen group elements such as blue

phosphorene (BLP), which is the lightest in this material group [20,21], and 2D M2X2

with (M= In, Ga and X= S and Se) [22–27]. For systems with large N(εF), there is

often the possibility that a large nesting, ξ, occurs at some speciﬁc phonon wave vectors

(qs) leading to a considerable reduction in the bare charge susceptibility, χ0, when the

temperature reduces. This character leads to an intense softening at speciﬁc qs in some

branches of the phonon spectrum, known as the Kohn anomaly [28,29], and in some

cases, a charge density wave (CDW) instability will appear in the form of the imaginary

modes for some particular phonon wave vectors (qCDW ) when the temperature is below

TCDW . Accordingly, access to such a high Tccan be prohibited by this instability [22,30].

Therefore, a more uniform distribution of λin terms of phonon wave vectors over the

1BZ is more favorable to gain large Tc, while, the formation of the CDW phase becomes

less likely.

For systems where the Kohn anomaly exists, one can further include the eﬀects

of phonon spectrum normalization on temperature variations which are applied to the

Eliashberg function (α2F). This can lead to a temperature-dependent λ, so the presence

of the Kohn anomaly can alter λand ωlog simultaneously. Precisely, it is also important

to consider the non-adiabatic renormalization of the phonon spectrum, leading to a more

accurate determination of the formation of either the superconducting or the CDW

phase at low temperatures. The non-adiabatic phonon dispersions can be naturally

obtained by diagonalizing the phonon dynamical matrix related to non-adiabatic non-

self-consistent force constants at a physical temperature. Therefore, a self-consistent

solution should be employed to achieve a better Tc[31].

In this paper, based on ﬁrst-principles calculations, we investigate the

Superconducting properties of doped blue phosphorene: Eﬀects of non-adiabatic approach 3

superconducting properties of BLP for some electron (N)- and hole (P)-type dopings by

utilizing both adiabatic (A) and non-adiabatic (NA) approaches having been exploited in

three approaches; modiﬁed isotropic Allen-Dynes [32], isotropic Migdal-Eliashberg (ME)

and anisotropic ME theory [33–35]. First of all, the topology of the Fermi surface (FS)

is investigated in terms of various charge dopings. Then we focus on phonon dispersion

and CDW instability along with the origin of such instability at a given temperature.

Furthermore, we determine that the out-of-plane phonons are strongly coupled to the

electrons in P-doped cases, while, in-plane deformations have the largest contribution

to the electron-phonon coupling for N-doped cases. Ultimately, we calculate the Tcand

CDW phase transition temperature, TCDW , in both adiabatic and non-adiabatic regimes

for diﬀerent charge carrier densities. In this case, to better align the isotropic Migdal-

Eliashberg and Allen-Dynes mechanisms, we adopted the computational implementation

of potential µ∗

Nrather than µ∗

cas explained in supplementary materials. Moreover, the

direction-dependent α2Fwas extracted to reveal which types of deformations have the

greatest contribution to the charge carrier-phonon coupling (for both the electron and

hole cases) and subsequently for Tc. Our results show that the presence of non-adiabatic

eﬀects is an obstacle to entering into the unstable CDW phase for some P-doped levels.

The maximum Tcs are calculated at around 97 and 91 K for P-doping +0.03 and +0.02

in units of electron per formula unit (e/f.u.), respectively.

It should be noted that literature has primarily discussed intercalated 2D

systems [36–40] which require numerous atoms, however, our result demonstrates that

we are able to obtain a large critical temperature at a doping level that is easily accessible

in the experiment.

we would like to emphasize not only does our study provide a platform for

understanding the fundamental physics underlying the superconductivity beyond the

standard adiabatic model, but it also, and perhaps more importantly, oﬀers an

experimental route for investigating the hole doping in blue phosphorene at accessible

densities. The nature of symmetry in 2D materials is what sets blue phosphorene

apart from other well-known 2D materials (like graphene), as it signiﬁcantly enforces in-

plane electron-phonon coupling in those materials [41]. In contrast, blue phosphorene’s

buckling structure (lack of σhsymmetry) makes spectacular results stemming from out-

of-plane deformations. Additionally, such buckling results in electron-phonon interaction

strengths that are far higher than those of other 2D materials, for example, compared

to monolayer InSe, graphene and TMDs. Furthermore, it should be noted that non-

adiabatic eﬀects are applied to the entire Brillouin Zone to explore the competition

between the superconducting and CDW phases.

The paper is organized as follows. In Sec. 2we present, brieﬂy, utilized methods

and computational details. In Sec. 3we describe our results about CDW formation,

Kohn anomaly, and superconducting properties. Ultimately, we present our conclusions

in Sec. 4.

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Superconductingpropertiesofdopedbluephosphorene:Eectsofnon-adiabaticapproachMohammadAlidoosti1,DavoudNasrEsfahani1;2andRezaAsgari3;41PasargadInstituteforAdvancedInnovativeSolutions(PIAIS),Tehran19916-33361,Iran2Departmentofconvergingtechnologies,KhatamUniversity,Tehran19916-33357,Iran3SchoolofPhysi...

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Superconducting properties of doped blue phosphorene Eects of non-adiabatic approach Mohammad Alidoosti1 Davoud Nasr Esfahani12and Reza

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