First-principles calculations of the superconducting properties in Li-decorated monolayer graphene within the anisotropic Migdal-Eliashberg formalism

TL;DR

This study uses first-principles anisotropic Migdal-Eliashberg calculations to analyze the superconducting mechanism in lithium-decorated monolayer graphene, confirming phonon-mediated pairing with a predicted Tc of 5.1-7.6 K.

Contribution

It applies an ab initio anisotropic Migdal-Eliashberg approach to elucidate the pairing mechanism and gap structure in Li-decorated monolayer graphene, aligning theoretical predictions with experimental data.

Findings

Superconductivity in Li-decorated graphene is phonon-mediated.

Predicted Tc range matches experimental observations.

Superconducting gap is anisotropic and single in nature.

Abstract

The ab initio anisotropic Migdal-Eliashberg formalism has been used to examine the pairing mechanism and the nature of the superconducting gap in the recently discovered lithium-decorated monolayer graphene superconductor. Our results provide evidence that the superconducting transition in Li-decorated monolayer graphene can be explained within a standard phonon-mediated mechanism. We predict a single anisotropic superconducting gap and a critical temperature Tc = 5.1-7.6 K, in very good agreement with the experimental results.

Equations

$${\alpha }^{2}F(\omega )=\frac{1}{N_{\mathrm{F}}{N}_{\mathbf{k}}{N}_{\mathbf{q}}}\sum _{\mathbf{k},{\mathbf{k}}^{\prime },\nu }|g_{\mathbf{k}{\mathbf{k}}^{\prime }}^{\nu }{|}^2\delta ({ϵ}_{\mathbf{k}})\delta ({ϵ}_{{\mathbf{k}}^{\prime }})\delta (\omega -{\omega }_{\mathbf{q}\nu }),$$

$$\lambda (\omega )=2{\int }_0^{\omega }d{\omega }^{\prime }{\alpha }^{2}F({\omega }^{\prime })/{\omega }^{\prime },\text{vspace}-0.1cm$$

$${\lambda }_{\mathbf{k}}=\sum _{{\mathbf{k}}^{\prime },\nu }\delta ({ϵ}_{{\mathbf{k}}^{\prime }})|g_{\mathbf{k}{\mathbf{k}}^{\prime }}^{\nu }{|}^2/{\omega }_{\mathbf{q}\nu }.\text{vspace}-0.1cm$$

$$Z(\mathbf{k},i{\omega }_n)=1+\frac{\pi T}{N_{\mathrm{F}}{\omega }_n}\sum _{{\mathbf{k}}^{\prime }{n}^{\prime }}\frac{{\omega }_{n^{\prime }}}{\sqrt{{{\omega }_{n^{\prime }}}^2+{\Delta }^2({\mathbf{k}}^{\prime },i{\omega }_{n^{\prime }})}}$$

$$\times \delta ({ϵ}_{{\mathbf{k}}^{\prime }})\lambda (\mathbf{k},{\mathbf{k}}^{\prime },n-n^{\prime }),$$

$$Z(\mathbf{k},i{\omega }_n)\Delta (\mathbf{k},i{\omega }_n)=\frac{\pi T}{N_{\mathrm{F}}}\sum _{{\mathbf{k}}^{\prime }{n}^{\prime }}\frac{\Delta ({\mathbf{k}}^{\prime },i{\omega }_{n^{\prime }})}{\sqrt{{{\omega }_{n^{\prime }}}^2+{\Delta }^2({\mathbf{k}}^{\prime },i{\omega }_{n^{\prime }})}}$$

$$\times \delta ({ϵ}_{{\mathbf{k}}^{\prime }})\left[\lambda (\mathbf{k},{\mathbf{k}}^{\prime },n-n^{\prime })-{\mu }_c^{\ast }\right].$$

$$\lambda (\mathbf{k},{\mathbf{k}}^{\prime },n-n^{\prime })=N_{\mathrm{F}}\sum _{\nu }\frac{2{\omega }_{\mathbf{q}\nu }}{({\omega }_n-{\omega }_{n^{\prime }}{)}^2+{\omega }_{\mathbf{q}\nu }^2}|g_{\mathbf{k}{\mathbf{k}}^{\prime }}^{\nu }{|}^2.$$

$$\frac{S_{es}(t)}{{\gamma }_n{T}_c}=-\frac{6\alpha }{{\pi }^2}{\int }_0^{\infty }[f\ln (f)+(1-f)\ln (1-f)]d\tilde{\epsilon },$$

$$\frac{C_{es}(t)}{{\gamma }_n{T}_c}=t\frac{d(S_{es}/{\gamma }_n{T}_c)}{dt},$$

Full text

I INTRODUCTION

During the past decade graphene has revolutionized many areas of nanotechnology from organic electronics to photovoltaics, plasmonics, photonics, and energy storage Novoselov_NAT12 . One notable application that was missing from this list was superconductivity, despite numerous theoretical predictions of either a conventional or an unconventional pairing mechanism einenkel ; profeta ; Margine2014 ; uchoa ; nandkishore ; kiesel . Very recently, a high-resolution angle-resolved photoemission spectroscopy (ARPES) study has presented evidence supporting the appearance of a superconducting phase in Li-decorated monolayer graphene (LiC6) around 5.9 K Ludbrook , within the standard phonon-mediated coupling mechanism. This work has been followed by two more studies that reported the observation of superconductivity in Ca-intercalated bilayer graphene Hasegawa and in Ca-intercalated graphene laminates Chapman .

In this article, we investigate from first principles the nature of the superconducting gap in LiC6. To this end, we solve the fully anisotropic Migdal-Eliashberg equations allen_mitrovic ; Margine2013 to obtain the superconducting transition temperature ($T_{c}$) and the variation of the superconducting energy gap on the Fermi surface. While previous ab initio calculations have shown that the electron-phonon coupling is sufficient to yield a critical temperature in 6.7-10.3 K range using the Allen-Dynes formula profeta ; Guzman ; kaloni or the isotropic Eliashberg formalism Szcz2014 , the nature of the superconducting gap has not yet been addressed. We find that, similar to bulk CaC6 Sanna_PRB07 ; Margine2016 ; Szcz2014 ; Szcz2015 , Li-decorated monolayer graphene exhibits a single anisotropic gap in agreement with the experimental work Ludbrook .

II METHODOLOGY

The calculations are performed within the local density approximation to density-functional theory lda1 ; lda2 using planewaves and norm-conserving pseudopotentials nc1 ; nc2 , as implemented in the Quantum-ESPRESSO package QE . The planewaves kinetic energy cutoff is 100 Ry and the structural optimization is performed until the forces on atoms are less than 10 meV/Å . Li-decorated monolayer graphene is described in the $\sqrt{3}\times\sqrt{3}R$30∘ graphene supercell with one lithium atom per unit cell. The optimized lattice constant and the adatom-graphene distance are $a=4.24$ Å  and $h=1.78$ Å. A Brillouin-zone (BZ) $\Gamma-$centered $\mathbf{k}$-point mesh of 24$\times$24 and a Methfessel-Paxton smearing mp of 0.02 Ry are adopted for the electronic charge density calculations. The phonon modes are computed within density-functional perturbation theory baroni2001 on a 6$\times$6 $\mathbf{q}$-mesh. We employ the EPW code Giustino2007 ; Giustino2016 ; EPW ; Margine2013 ; Ponce to obtain the superconducting gap. The calculation of the electronic wavefunctions required for the Wannier-Fourier interpolation Marzari ; Mostofi in EPW is performed on a uniform unshifted BZ $\mathbf{k}$-point grid of size 12$\times$12. For the anisotropic Eliashberg equations, we use 120$\times$120 and 60$\times$60 $\mathbf{k}$- and $\mathbf{q}$-point grids. The Matsubara frequency cutoff is set to five times the largest phonon frequency ($5\times 200$ meV), and the Dirac delta functions are replaced by Lorentzians of widths 100 meV and 0.5 meV for electrons and phonons, respectively.

III ELECTRONIC AND VIBRATIONAL PROPERTIES

In Figs. 1 (a)-(c), we show the crystal structure of Li-decorated monolayer graphene along with the corresponding decomposed electronic band structure and Fermi surface. Three bands cross the Fermi level around the $\Gamma$ point, in agreement with previous reports profeta ; Guzman . The inner and outer C $\pi^{*}$ bands, labeled as $\alpha$ and $\beta$ (blue dots), are obtained by folding the $\pi^{*}$ states of graphene from $K$ to $\Gamma$, following the superstructure induced by Li adsorption. The Li-derived band, labeled as IL (red dots), displays a nearly-free electron like dispersion upwards from about 0.56 eV below the Fermi energy. Similar weakly bound free-electron states have been observed in other layered materials profeta ; Calandra ; Boeri ; Kolmogorov and nanotubes Margine_NFE , and their rapid downshift under doping is due to the combined effects of quantum confinement and electrostatic response profeta ; Boeri ; Margine_NFE . The corresponding Fermi surface of LiC6 can be divided into two concentric regions centered around the $\Gamma$ point. The inner region is characterized by a snowflake-like electron pocket intersecting a hexagonal electron pocket, which arises from the mixing of the inner C $\pi^{*}$ states with the Li $s$ states. These Fermi sheets resemble the $\Gamma$-centered Fermi surface observed in Ca and Li intercalated bilayer graphene Margine2016 ; Shimizu_PRL14 . The outer region also has a snowflake-like shape and originates on the outer C $\pi^{*}$ states and the Li $s$ states.

We now focus on the vibrational properties and the electron-phonon coupling (EPC) in LiC6. Similar to bulk CaC6 Calandra and bilayer C6CaC6 Margine2016 , one can clearly identify in Fig. 2(a) three regions in the phonon dispersion associated to (i) the Li-related modes (up to 50 meV, where above 37 meV are Liz modes mixed with carbon out-of-plane Cz modes), (ii) the carbon out-of-plane Cz vibrations (50-100 meV), and (iii) the carbon in-plane Cxy modes (above 100 meV). The size of the symbols in Fig. 2(a) is proportional to the atomic displacements corresponding to Li and C in-plane and out-of-plane contributions.

The isotropic Eliashberg spectral function $\alpha^{2}F(\omega)$

[TABLE]

and the cumulative electron-phonon coupling strength $\lambda(\omega)$

[TABLE]

are shown in Figs. 2(b)-(c). In these expressions ${N_{\rm F}}$ represents the density of electronic states per spin at the Fermi level, $N_{\bf k}$ and $N_{\bf q}$ are the total numbers of ${\bf k}$ and ${\bf q}$ points, ${\epsilon_{\bf k}}$ is the Kohn-Sham eigenvalue with respect to the Fermi level, and ${g_{\bf k\bf{k^{\prime}}}^{\nu}}$ is the screened electron-phonon matrix element for the scattering between the electronic states ${\bf k}$ and ${\bf{k^{\prime}}}$ through a phonon with wave vector ${\bf q}\!=\!{\bf{k^{\prime}}}\!-{\bf k}$, frequency ${\omega_{\bf q\nu}}$ and branch index $\nu$. Here ${\bf k}$ and ${\bf{k^{\prime}}}$ indicate both the electron wavevector and the band index. We find that the low-energy phonons are key to achieving a high electron-phonon coupling in LiC6 as they account for 0.28 (51%) of the total EPC ($\lambda$ = 0.55). On the other hand, the electron-phonon coupling strengths associated with the out-of-plane Cz and in-plane Cxy modes are 0.12 (22%) and 0.15 (27%), respectively. Such behavior has also been found in bilayer C6CaC6, where the most significant contribution to the EPC comes from the low-energy phonon modes Margine2016 . Overall, our calculated EPC $\lambda=0.55$ is in good agreement with the experimental value $0.58\pm 0.05$ observed at the highest Li coverage Ludbrook and the values reported in previous theoretical studies profeta ; Guzman .

To quantify the anisotropy in the electron-phonon coupling, we further evaluate the momentum-resolved EPC $\lambda_{\mathbf{k}}$ Margine2013 , defined as:

[TABLE]

The calculated $\lambda_{\mathbf{k}}$ displays a significant anisotropy with a distribution in the 0.42-0.78 range as shown in Fig. 3. This is in line with experimental ARPES measurements where a marked anisotropy in the electron-phonon coupling has been observed in the case of decorated graphene Ludbrook ; Fedorov_NatCom14 , intercalated bilayer graphene Kleeman_JPSJ14 , and intercalated graphite Gruneis_PRB09 ; Valla_PRL09 . An alternative way to look at the EPC anisotropy is presented in the inset of Fig. 3, where the variation of $\lambda_{\mathbf{k}}$ on the Fermi surface is shown. When compared with the Fermi surface plot in Fig. 1(c), one can clearly see that the largest value of $\lambda_{\mathbf{k}}$ is attained on the portions of the Fermi surface dominated by the Li states. Notably, in bulk CaC6, $\lambda_{\mathbf{k}}$ was also found to be larger for the states with Ca dominant orbital character on the Fermi surface Sanna_PRB07 . An important implication of this finding is that the IL state and its associated interaction play a critical role in the superconducting paring of LiC6. A recent ARPES study has provided compelling evidence regarding the importance of the IL band in the pairing mechanism of bulk CaC6 Yang . Furthermore, the lack of any sign of superconductivity down to 3.5 K in few-layer graphene under large charge doping induced by electrochemical gating Gonnelli provides additional proof of the vital role of dopant atoms.

IV SUPERCONDUCTING PROPERTIES

The superconducting properties of LiC6 are obtained by solving self-consistently the fully anisotropic Migdal-Eliashberg equations along the imaginary axis at the fermion Matsubara frequencies ${\omega_{n}}=(2n+1)\pi T$ (with $n$ an integer) for each temperature $T$  Margine2013 ; Margine2014 ; allen_mitrovic ; Choi :

[TABLE]

$Z({\bf k},i{\omega_{n}})$ is the mass renormalization function, $\Delta({\bf k},i{\omega_{n}})$ is the superconducting gap function, $\lambda({\bf k},{\bf{k^{\prime}}},\!n-\!n^{\prime})$ is the momentum- and energy-dependent EPC, and $\mu_{c}^{*}$ is the semiempirical Coulomb parameter. The anisotropic $\lambda({\bf k},{\bf{k^{\prime}}},\!n-\!n^{\prime})$ to be used in the Eliashberg equations is given by:

[TABLE]

Figure 4(a) shows the superconducting energy gap $\Delta_{\mathbf{k}}$ as a function of temperature, calculated for a screened Coulomb parameter $\mu^{*}=0.14$, together with the average value of the gap (red squares). The superconducting gap $\Delta_{\mathbf{k}}$ on different parts of the Fermi surface at 0.5 K is shown in Fig. 4(b). We find that monolayer LiC6 displays a single anisotropic gap with an average value $\Delta_{0}=0.89$ meV in the $T=0$ K limit, in very good agreement with the ARPES result of $0.9\pm 0.2$ meV, measured at 3.5 K Ludbrook . This situation is similar to bulk CaC6 where the multiple-sheet Fermi surface gives rise to a single gap structure with a sizable anisotropy Sanna_PRB07 ; Margine2016 , but unlike bilayer C6CaC6 for which a two gap structure has been recently predicted Margine2016 .

The superconducting $T_{c}$ is identified as the highest temperature at which the gap vanishes. From Fig. 4(a) we find $T_{c}=5.9$ K and a ratio $2\Delta_{0}/k_{B}T_{c}=3.50$, very close to the ideal BCS value of 3.53 BCS . The predicted superconducting critical temperature is in excellent agreement with the experimental estimation of 5.9 K based on measurements of the size of the superconducting gap Ludbrook . The temperature dependence of the superconducting gap can be well fitted with a BCS model, as obtained by solving numerically the BCS gap equation Johnston with $\Delta_{0}$ and $T_{c}$ from our first-principles calculations. This is shown by the blue dashed line in Fig. 4(a). These results provide support for a conventional phonon-mediated mechanism as the superconducting origin in Li-decorated graphene. For completeness, we also explore the sensitivity of the calculated superconducting energy gap and critical temperature to the choice of the Coulomb parameter $\mu^{*}$, as shown in Figs. 4(c)-(d). For $\mu^{*}$ = 0.12 and 0.16, we obtain $\Delta_{0}$ = 1.10 meV and 0.69 meV and $T_{c}$ = 7.6 and 5.1 K, respectively.

Finally, using the $\alpha$–model Johnston ; Padamsee , we obtain the temperature dependence of the reduced electronic specific heat in the superconducting state. Within this model, the ratio $\alpha$ = $\Delta_{0}/k_{B}T_{c}$ is an adjustable parameter and the normalized superconducting state electronic entropy $S_{es}$ and heat capacity $C_{es}$ are expressed in terms of $\gamma_{n}T_{c}$ as:

[TABLE]

[TABLE]

where $f=\left[\exp(\alpha\tilde{E}/t)+1\right]^{-1}$ is the Fermi-Dirac distribution function, $t=T/T_{c}$ is the reduced temperature, and $\gamma_{n}=(2/3)\pi^{2}k_{B}^{2}{N_{\rm F}}$ is the Sommerfeld coefficient. The reduced quasi-particle energy is defined as $\tilde{E}=\sqrt{\tilde{\varepsilon}^{2}+\delta^{2}(t)}$, where $\tilde{\varepsilon}=\varepsilon/\Delta_{0}$ is the reduced normal state single-particle energy relative to the Fermi level and $\delta(t)=\Delta(T)/\Delta_{0}$ is the reduced gap function. The upper limit in the integral in Eq. (7) is set to $500\gg 1$.

Figure 4(e) shows the calculated $C_{es}/\gamma_{n}T_{c}$ for $\alpha=1.75$. The temperature dependence of the normalized gap $\delta(t)$ is assumed to be the same as in the BCS theory Johnston ; Padamsee . We checked the numerical results by comparing the data for $\alpha_{BCS}=1.764$ with Tables II–IV in Ref. [Johnston, ] and by verifying that the entropy at the critical temperature is equal to that of the normal state. The shape of the calculated specific heat curve is consistent with a one-gap BCS model and undergoes a discontinuous jump at the critical temperature. The specific heat jump at $T_{c}$ is found to be $\Delta C_{e}(T_{c})/\gamma_{n}T_{c}=1.385$ [shown in the inset of Fig. 4(e)], close to the weak limit BCS value of 1.426 BCS . Furthermore, this result is comparable to the experimetal and theoretical values reported for the normalized specific heat jump in bulk CaC6 Kim ; Sanna_PRB07 .

V CONCLUSIONS

In conclusion, we have studied the superconducting properties in Li-decorated monolayer graphene within the ab initio anisotropic Migdal-Eliashberg theory. Our results provide support for a standard phonon-mediated mechanism at the origin of the superconducting transition. Most of the electron-phonon coupling originates from the low-energy modes dominated by the motion of Li atoms similar to bilayer C6CaC6. We find a sizable anisotropy in the electron-phonon coupling which yields a single anisotropic gap over the Fermi surface. Further enhancement in the critical temperature of LiC6 is expected in the presence of a substrate kaloni or under applied strain Pesic2014 .

VI ACKNOWLEDGMENTS

J-J Zheng acknowledges the support from the China Scholarship Council (Grant No. 201508140043).