10926 | J. Mater. Chem. C, 2019, 7, 10926--10932 This journal is©The Royal Society of Chemistry 2019

Cite this: J.Mater. Chem. C, 2019,

7, 10926

Emergence of superconductivity in a Diracnodal-line Cu2Si monolayer: ab initio calculations

Luo Yan,†abcd Peng-Fei Liu, †cde Tao Bo,cd Junrong Zhang,cd

Ming-Hua Tang, ab Yong-Guang Xiao*ab and Bao-Tian Wang *cde

Following the prediction given by first-principles simulations [J. Am. Chem. Soc. 137, 2757 (2015)], a

Dirac nodal-line Cu2Si monolayer has been successfully synthesized on a Cu(111) surface [Nat.

Commun. 8, 1007 (2017)] or on a Si(111) substrate [Phys. Rev. Mater. 3, 044004 (2019)]. However, its

superconducting properties have never been reported in experiments or theory. Here, through first-

principles calculations, we study its electron and phonon properties and electron–phonon coupling to

investigate the possibility of superconductivity for a metallic Cu2Si monolayer. The results show that it is

an intrinsic BCS-type superconductor, with the estimated superconducting temperature Tc being

B4.1 K. We further find that the Fermi surface nesting is partially responsible for its superconducting

character. Carrier doping as well as biaxial strains will suppress the Tc. Our results help to explain the

challenges to experimentally probe superconductivity in substrate-supported Cu2Si monolayers and provide

clues for further experiments.

1 Introduction

Superconductivity,1 one of the macroscopic quantum phenomena,is old but still a new field in condensed-matter physics. Thediscovery of graphene2 together with its unique properties3,4

immediately motivated both theoretical and experimental researchefforts, including regarding its potential superconductivity.Although pristine graphene is not a superconductor owing toits small density of states (DOS) around the Fermi level and itssymmetrical restriction in the electron–phonon coupling (EPC),superconductivity can be introduced into this system throughmetal decorating/intercalating,5–12 carrier doping,13–15 strain,14

and magic angle twisting,16 which opens the door for two-dimensional (2D) superconductivity. Beyond graphene, lots ofstudies have demonstrated that superconductivity can be intro-duced by the metallization of other monolayer materials, suchas strained silicene,17 Li-intercalated h-BN bilayers,18 strainedphosphorene,19 electron-doped arsenene,20 electron-doped phos-phorene,21 Li-intercalated bilayer phosphorene,22 electron-doped

single-layer MoS2,23,24 Li-intercalated bilayer MoS2,25 strainedMoX2 (X = S or Se),26 and metal-intercalated blue phosphorusbilayers.27 However, until now, intrinsic 2D superconductorsare still rather scarce in real planar monolayer systems.

Superconductors with one atomic thickness have been con-tinuously filled with numerous interests due to their interplaywith the quantum effects at the 2D limit,28–33 which areimportant for realizing next-generation quantum informationtechniques.34,35 However, to the best of our knowledge, few realplanar monolayer systems have been reported to be intrinsicsuperconductors.36–38 Several examples are as follows: singlelayer B2C, which is slightly corrugated with the boron andcarbon layers separated in the vertical direction by only 0.032 Å,has been predicted to be a 2D intrinsic Bardeen–Cooper–Schrieffer(BCS) superconductor;39 single-atomic-layer Cu-benzenehexathial(Cu-BHT), a 2D metal–organic framework, has been predicted byfirst-principles calculations40 and later experimentally verified41 asa BCS superconductor; w3 and b12 borophenes (w3-B and b12-B),fabricated on a Ag(111) surface, have been demonstrated to bephonon-mediated superconductors.42–44

In the present work, we focus on a Cu2Si monolayer (seeFig. 1) which was firstly predicted by a combination of first-principles calculations and the particle-swarm optimizationmethod45 and then was successfully synthesized by directlyevaporating atomic Si on a Cu(111) surface46 or Cu depositionon a Si(111) surface47 in an ultrahigh vacuum chamber. This 2Dsystem is a nonmagnetic metal with excellent stability45 andhas been identified as a 2D nodal-line semimetal.46 It haspotential for gas sensing,48 electrode materials49 and high-speed

a Key Laboratory of Key Film Materials & Application for Equipments

(Hunan Province), School of Material Sciences and Engineering,

Xiangtan University, Xiangtan, Hunan, 411105, China. E-mail: ygxiao@xtu.edu.cnb Hunan Provincial Key Laboratory of Thin Film Materials and Devices,

School of Material Sciences and Engineering, Xiangtan University, Chinac Institute of High Energy Physics, Chinese Academy of Sciences (CAS),

Beijing 100049, China. E-mail: wangbt@ihep.ac.cnd Spallation Neutron Source Science Center, Dongguan 523808, Chinae State Key Laboratory of Quantum Optics and Quantum Optics Devices,

Shanxi University, Taiyuan, 030006, China

† The first two authors contributed equally to this work.

Received 10th July 2019,Accepted 7th August 2019

DOI: 10.1039/c9tc03740c

rsc.li/materials-c

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low-dissipation devices,46 etc. Here, we investigate carefully thesuperconductivity and other exotic properties that may exist in thisreal planar monolayer system. On the basis of first-principlescalculations, we prove that the 2D Cu2Si is an intrinsic BCSsuperconductor with Tc = 4.1 K, and demonstrate that the partialmechanism of its superconductivity is from Fermi surface (FS)nesting. Moreover, we study the effects of carrier doping andbiaxial strain on its superconductivity. Unfortunately, the super-conductivity of 2D Cu2Si is sensitive to these external conditions,resulting in smaller Tc than its pristine form. This explains to someextent why this system has never been measured as a super-conductor, although it has been synthesized for years. Our resultsindicate that a high quality Cu2Si monolayer synthesized onsubstrates with negligible lattice mismatch is urgent for exploringits superconductivity in experiments.

2 Computational methods

The first-principles electronic structure calculations are performedusing the Perdew–Burke–Ernzerhof50 form of the generalizedgradient approximation (GGA),51,52 as implemented in the Viennaab initio simulation package (VASP).53,54 The cutoff energy of500 eV and a 32 � 32 � 1 Monkhorst–Pack k-point mesh are usedin the calculations. The length of the unit cell of 20 Å along the zdirection is used to get rid of the interaction between adjacentimages. All geometry structures are fully relaxed until the residualforces on each atom are less than 0.01 eV Å�1.

The EPC and superconductivity are calculated utilizing thedensity functional perturbation theory (DFPT)55 through theQuantum-ESPRESSO code.56,57 The PBEsol form of GGA fromthe Standard Solid State Pseudopotentials (SSSP) library58

is employed in the phonon calculations. The kinetic energycutoff and the charge density cutoff of the plane wave basisare chosen to be 80 and 800 Ry, respectively. Self-consistentelectron density is evaluated by employing 32 � 32 � 1 k mesh.Both phonon and EPC are calculated by using a 4 � 4 �1 q mesh.

Here, the magnitude of the EPC lqn is calculated accordingto the Migdal–Eliashberg theory,59,60 by

lqn ¼gqn

phNðEFÞoqn2; (1)

where gqn is the phonon linewidth, oqn is the phonon frequency,and N(EF) is the electronic DOS at the Fermi level. The gqn can beestimated by

gqn ¼2poqn

OBZ

Xk;n;m

gnkn;kþqm

��� ���2d ekn � eFð Þd ekþqm � eF� �

; (2)

where OBZ is the volume of the Brillouin zone (BZ), ekn and ek+qm

denote the Kohn–Sham energy, and g nkn,k+qm represents the EPCmatrix element. The g nkn,k+qm can be determined self-consistentlyby the linear response theory.61 The Eliashberg electron–phononspectral function a2F(o) and the cumulative frequency-dependentEPC l(o) can be calculated by

a2FðoÞ ¼ 1

2pNðEFÞX

qn

gqn

oqnd o� oqn� �

(3)

and

lðoÞ ¼ 2

ðo0

a2F ðoÞo

do; (4)

respectively.Using a typical value of the effective screened Coulomb

repulsion constant m* = 0.140,42,62–64 as well as the Eliashbergspectral function a2F(o) and l, one can calculate the logarithmicaverage frequency olog and the superconducting transitiontemperature Tc by

olog ¼ exp2

l

ð10

doo

a2F ðoÞlogo� �

(5)

and

Tc ¼olog

1:2exp � 1:04ð1þ lÞ

l� m�ð1þ 0:62lÞ

� �: (6)

According to eqn (2), we find that the phonon linewidth hastwo major contributions: the EPC matrix elements and the FSnesting factor x(q). This factor is in the form of

xðqÞ ¼Xk;n;m

d ekn � eFð Þd ekþqm � eF� �

: (7)

3 Results and discussion3.1 Atomic structure and electronic properties

The atomic structure of the freestanding form of a Cu2Simonolayer, along with the electron localization function, isshown in Fig. 1. The structure of the Cu2Si monolayer is truly aplanar structure without buckling, featuring planar hexacoordinatecopper and planar hexacoordinate silicon. The optimized latticeconstants along the a and b axes are both 4.10 Å. The calculatedCu–Si and Cu–Cu bond lengths are both 2.33 Å. Dominant ionicbonds form between Cu and Si atoms. Meanwhile, neighboringCu atoms are connected via metallic bonds.

The band structure and FS are plotted in Fig. 2. Clearly, theCu2Si monolayer is metallic with three bands crossing the Fermilevel. As illustrated by the FS, there are three bands contributingto the three closed contours around the G center: a hexagon,a hexagram, and a circle. These results accord well with the

Fig. 1 Electron localization function of a Cu2Si monolayer with anisovalue of 0.25 a.u. The unit cell is labeled by the black solid line. TheCu and Si atoms are indicated by blue and jacinth balls.

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previous studies.45–47,65 To measure the contributions of thethree bands to the intrinsic metallic feature for the Cu2Simonolayer, we calculate the charge density at each cross-overpoint of these bands and the Fermi level (see Fig. 3), corres-ponding to the a–f points in Fig. 2(a). In Fig. 3(b) and (f), we cansee that there are massive electron transfers at the b and fpoints, formed by the second band crossing the Fermi level. Asshown by Fig. 3(f), the electrons transfer in the line channels.While at the c and d points, there are few charges [see Fig. 3(c)and (d)], indicating weak electron transfer. As shown by theprojected electronic density of states (PDOSs) in Fig. 4, thereis apparent hybridization between the Si-3p and Cu-3d4pstates near the Fermi level. The dominant contributions to theelectronic states at the Fermi level are from dxy,x2�y2, dxz,yz andpx,y states of Cu, followed by some contributions from px,y andpz states of Si. Thus, the metallic nature of the Cu2Si monolayeris mainly controlled by its Cu-3d orbitals.

3.2 EPC and superconductivity

After analyzing the electronic structures, we now focus on thevibration properties and EPC. The phonon dispersions of 2DCu2Si are shown in Fig. 5. The real frequencies of all modes in theBZ indicate that this 2D crystal is dynamically stable, consistentwith previous results from first-principles calculations.45,65

To gain deep insights into the phonon vibrations, thephonon k�p theorem is used to sort the phonon branchesaccording to the continuity of their eigenvectors66–68

Xek;s1

�ð jÞ � ekþD;s2ð jÞ��� ��� ¼ ds1;s2 � 0ðDÞ

�� ��; (8)

where ek,s*( j) is the displacement of the atom j in the eigen-vector of (k,s) vibrational modes, and D is a small wave vector.As indicated in Fig. 5(a), three acoustic branches, including theout-of-plane (ZA), in-plane transverse (TA) and in-plane longitudinal(LA) modes, cross the two low-frequency optical branches. This kindof crossing indicates a strong optical-acoustic phonon coupling.

From the decomposition of the phonon spectrum withrespect to the vibrations of Cu and Si atoms [see Fig. 5(b)], wefind that the main contributions below 100 cm�1 are from out-of-plane modes of Cu-z, including the ZA and the lowest opticalbranch (LOB). The interactions between in-plane modes ofCu-xy and out-of-plane of Si-z contribute to the intermediate-frequency region from 100 to 350 cm�1. Meanwhile, thein-plane modes of Si-xy occupy the high frequencies above350 cm�1 mostly. The highest mode frequency, 421 cm�1, is muchsmaller than those of b12-B (1200 cm�1),42,43 w3-B (1290 cm�1)42,44

and B2C (1243 cm�1),39 indicating weak bonding (ionic andmetallic bonds) interactions between its component atoms. Fromthe projected phonon density of states (PhDOS), as shown inFig. 6(b), we can also draw these conclusions.

To explore the superconductivity of a Cu2Si monolayer, wecalculate the Eliashberg spectral function [eqn (3)], the EPCconstant l [eqn (4)], the logarithmic average frequency olog

[eqn (5)] and Tc [eqn (6)] and present them in Fig. 6. TheEliashberg spectral function exhibits three major peaks, at 35.8,62.1 and 153.7 cm�1, in the whole frequency region. It’s clear

Fig. 2 (a) Band structure of the Cu2Si monolayer. (b) Fermi contour ofCu2Si in the extended BZ scheme with the colour drawn proportional tothe magnitude of the Fermi velocity nF.

Fig. 3 (a–f) Charge density contours for a–f points in Fig. 2(a). The yellowand cyan areas represent electron gains and losses, respectively.

Fig. 4 PDOSs for a Cu2Si monolayer.

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that phonons in the low-frequency region (0–100 cm�1) con-tribute mainly to the EPC, as they account for 81.5% of the totalEPC (l = 0.81). The first and the second peaks of a2F(o) areresponsible for this part. We can also see from Fig. 5(c) that thelarge values of the lqn are around the M point in a frequencyrange of 19.7–69.6 cm�1, mainly on the softened ZA branch.They are responsible for the first peak of a2F(o). Besides, somemedium values of the lqn are on the flat LOB, which results inthe second peak. The middle-frequency region (100–350 cm�1),in which the TA and the second-lowest optical branches inter-act tightly, expand flat in BZ, and result in the third peak ofa2F(o), has partial contribution (14.8%) to the EPC. The high-frequency phonon modes (350–420 cm�1) contribute to theremaining weight (3.7%) of the EPC. Similar to the b12,42,43 w3

borophene42,44 and single layer Cu-BHT,40 the EPC induced byhigh-frequency phonons is almost negligible. With a typicalvalue of 0.1 for m*, we obtain the frequency-dependent super-conducting transition temperature Tc(o) [see Fig. 6(c)]. Itsderivative also exhibits three big peaks corresponding to thoseof a2F(o). The total Tc is calculated to be 4.1 K, which iscomparable to those of the Ca-intercalated bilayer grapheneC6CaC6 (4.0 K),6,69 2D tri-Mo2B2 (3.9 K)70 and 2D AlB6 (4.7 K).71

Overall, the Cu2Si monolayer is a weak BCS-type superconductorwith l = 0.81.

In Table 1, we list the superconductive parameters of m*,N(EF), olog and l for some realistic 2D superconductors with aflat geometry synthesized in experiments. These systems are/may beintrinsic 2D superconductors without external conditions, such ascarrier doping, high pressure, metal decorations/interalations and/or functional groups. Among them, the Cu-BHT was firstly predictedby the BCS theory40 as a superconductor and then has been verifiedby experiments.41 This fact clearly indicates that the combination ofthe first-principles calculations and the BCS theory is reliable andalso powerful in predicting 2D superconductors. Compared with thetwo forms of borophene (w3-B and b12-B),42 the Tc of Cu2Si isfundamentally small, although their values of l are comparable.This fact is mainly due to the small value of olog that Cu2Sipossesses. After all, the atomic masses of Cu and Si are largelyheavier than that of boron, constraining the phonon vibrations.

3.3 FS nesting

As discussed above, the large values of the EPC lqn are mainly inteh low-frequency region, in which the soft ZA phonon mode isalong the K–M and M–G directions. To clearly understand thephonon softening, we investigate the FS nesting in the Cu2Simonolayer by calculating the nesting function [eqn (7)]. Theresults of x(q) along the high-symmetry paths are shown in

Fig. 5 (a) Phonon spectra of the acoustic and optical branches along the high symmetry line for the Cu2Si monolayer. (b) Phonon frequency dispersionweighted by the motion modes of Cu and Si atoms. (c) Phonon spectrum with the size of the red dots drawn proportional to the magnitude of lqn.

Fig. 6 (a) Eliashberg spectral function a2F(o) and cumulative frequency-dependent EPC function l(o). (b) Projected phonon density of states forthe Cu2Si monolayer. (c) The frequency-dependent superconductingtransition temperature Tc and its derivative based on the modified McMil-lian equation.

Table 1 Summary of m*, N(EF) (in unit of states per spin per Ry per cell),olog (in K), l, and Tc (in K) for the realistic 2D superconductors with a flatgeometry synthesized in experiments. The value of Tc from an experimentis indicated by the datum within parentheses

Comp. m* N(EF) olog l Tc Ref.

w3-B 0.1 323.43 0.95 24.7 42b12-B 0.1 384.16 0.89 18.7 42Cu-BHT 0.1 51.8 1.16 4.43(3) 40 and 41Cu2Si 0.1 7.64 83.594 0.81 4.03 This work

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Fig. 7(a) and (b). It’s obvious that the contributions of the FSnesting are large around the G and M points. For the G point, itsx(q) only represents the entire FS nesting into itself, which hasno actual physical meaning.72 As for K points, we can believethat they are the superimpositions of M points. Consideringthese, we can conclude that the FS nesting is responsible for themajor contributions around the M point. In Fig. 7(b), we canalso see sharp peaks along K–M and M–G, demonstrating thatstrong FS nesting occurs along these directions. Not coincidentally,the regions of the phonon softening [see Fig. 5(c)] correspond tothe nesting areas, strongly indicating that the large lqn is partiallyinduced by the FS nesting. Besides, we display the distribution ofthe q-resolved EPC lqn in Fig. 7(c) and (d). It’s clear that the ZAbranch contributes mostly to the EPC lqn along the K–M andM–G directions, while the LOB has a small contribution. Thiscauses the emergence of six comet-shaped hot spots in the firstBZ, pointing from M to G.

3.4 Effects of doping and strain

5w?>Inspired by former successes in controlling the electronicproperties as well as the superconductivity of 2D systems bycarrier doing,64,73–76 here we also want to investigate its effects onthe Cu2Si monolayer. Within a compensating uniform chargebackground of opposite sign to maintain charge neutrality,64,77

the carrier doping is simulated by directly adding electrons intoor removing electrons from the system. For each doping concen-tration, we fully relax the plane lattice constants and atomiccoordinates. We apply both hole and electron dopings rangingfrom 0.2 h per cell to 0.2 e per cell. The changes in the super-conductive parameters of l, olog, N(EF) and Tc are shown inFig. 8(a)–(d), respectively. Increasing hole doping concentration,although l and olog change to some extent, N(EF) and Tc arealmost unchanged. As expected, N(EF) increases with the electrondoping concentration but the l and Tc decrease. Thus, we canconclude that the hole doping has almost no impact on thesuperconductivity of 2D Cu2Si, but electrons transferring fromthe Cu(111) substrate to the Cu2Si monolayer can suppress itssuperconductivity. Although we fail to modulate the Tc higher by

carrier doping, we provide theory clues for future experimentalinvestigations (Fig. 8).

Since the Cu2Si monolayer is synthesized on Cu(111) orSi(111) substrates,46,47 the estimation of a freestanding sampleis not enough to reflect real samples. Therefore, we studythe superconductivity of strained Cu2Si by biaxial strains (x)(�2% o x o 2%) to simulate the real samples grown onsubstrates with different lattice constants. The x is calculated

by x ¼ a� a0

a0� 100% (positive value means tensile strain while

negative one means compressive strain), atomic coordinatesare fully relaxed in each case. The calculated superconductiveparameters are presented in Fig. 9. Obviously, the l and Tc

decrease with applied both tensile and compressive strains,and even under tensile strains the olog and N(EF) have beenincreased. When compressing 2%, the l and Tc reach theminimum values about 0.39 and 0.2 K, respectively. Thus,biaxial strains can suppress the superconductivity of 2D Cu2Si.We further estimate the lattice mismatch between Cu2Si andCu(111)/Si(111) surface to reflect the practical situation ofexperimental synthesis. The result of lattice mismatch betweenthe Cu2Si and Cu(111) surface is�15.9% and the lattice mismatch

Fig. 7 (a) and (b) Fermi surface nesting function x(q) of Cu2Si. (c) and (d)The distribution of q-resolved EPC lqn.

Fig. 8 The changes of the superconductive parameters of (a) l, (b) olog,(c) N(EF) and (d) Tc under different doping concentrations.

Fig. 9 The related superconductive parameters of (a) l, (b) olog, (c) N(EF)and (d) Tc under different strains.

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between the Cu2Si and Si(111) surface is 6.1%. Based on ourcalculated results [see Fig. 9(d)], such large lattice mismatcheswould result in the absence of superconductivity in 2D Cu2Si.

As already mentioned above, the electron doping and biaxialstrains usually have negative effects on the superconductivity of2D Cu2Si. This supplies theoretical evidence for why the super-conductivity in the Cu2Si monolayer has never been reportedexperimentally, though it has been synthesized for years. Thisproblem has also been found in probing superconductivity insubstrate-supported b12-B.64,78 In order to preserve or evenenhance the Tc of 2D Cu2Si, according to our calculated results,it’s urgent to avoid interference from external conditions.

4 Conclusions

In conclusion, we systematically explore the electron structures,phonon vibrations and superconductivity of one recent experi-mentally synthesised real planar monolayer Cu2Si system,combining first-principles calculations and the BCS theory.Our results clearly indicate that 2D Cu2Si is not only a Diracnodal-line semimetal, but also a BCS superconductor withTc = 4.1 K. The FS nesting is partially responsible for itssuperconductivity. The Tc is found to be sensitive to externalconditions of doping or strain. Our work will stimulate more effortsin exploring superconductivity in newly synthesized or newlypredicted semimetals, such as ZrSiSe, ZrSiTe,79 b-PbO2,80

W2As3,81 2D SbAs82 and Be2P3N.83

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors would like to thank the financial support from theNational Natural Science Foundation of China under Grant No.11835008, and 51872250, the State Key Laboratory of IntensePulsed Radiation Simulation and Effect (Northwest Instituteof Nuclear Technology) under Grant No. SKLIPR1814, andKey Laboratory of Low Dimensional Materials & ApplicationTechnology of Ministry of Education (Xiangtan University) underGrant No. KF20180203. P. F. L. and B. T. W. also acknowledgethe PhD Start-up Fund of the Natural Science Foundation ofGuangdong Province of China (Grant No. 2018A0303100013)and the Program of State Key Laboratory of Quantum Optics andQuantum Optics Devices (No. KF201904). The calculations wereperformed at Supercomputer Centre at the China SpallationNeutron Source.

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