Role of the Electron–Phonon Interaction in the Superconductivity of the 2-Dimensional Sn/Si(111) Interface

Abstract

In order to elucidate the mechanism creating superconductivity in the 2-dimensional layer of a p-doped Sn/Si(111) surface, we have analyzed the many-body effects associated with the electron-phonon (e-ph) coupling and the electron–electron interaction. First, we have calculated the DFT surface band of the system and the coupling associated with the different interactions. In our calculations we find a mean field (DFT) electron bandwidth of 0.54 eV, an attractive coupling $U^{neg}=0.32$ eV associated with the e-ph coupling and an effective electron–electron Hubbard repulsion of $U=0.83$ eV. Then, we analyze the Hubbard Hamiltonian, neglecting in this step the e-ph coupling that is much smaller than the Hubbard coupling, by considering a p-doping in this Hamiltonian of 10%; by means of a Dynamical Mean Field (DMF) approach combined with an interpolative calculation for the self-energy, we deduce the local density of states (DOS) and show that the quasi-particle DOS induced by the doping is not large enough to induce magnetism in the Sn-monolayer. This leads us to analyze the possibility of having superconductivity by considering the attractive interaction induced by the e-ph coupling within an appropriate BCS-Hamiltonian. Our calculations show that the quasiparticle metallic system has a superconductivity critical temperature of ≈7–9 K, in good agreement with experiments.

1. Introduction

Low-dimensional solids [1] and surfaces [2] in particular show many different exotic phases due to the importance of fluctuations and the role played by electronic correlations in competition with other interactions like the electron-phonon coupling. These interactions create Mott insulators, magnetism or charge density waves, or superconductivity [3,4,5] as well as many other phases [6,7]. In recent papers, Weitering et al. [8,9] have reported the existence of superconductivity in the two-dimensional layer of a p-doped Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) structure. In this surface, Sn atoms, with a coverage of 1/3, are bonded to three Si atoms of the first Si-layer on T₄ sites; in this way, three dangling bonds of the ideal Si(111) surface are saturated by the interaction with each Sn adatom, leaving just the adatom dangling bond, which is occupied by one electron [10,11,12]. The undoped layer is a Mott insulator [12,13], but hole-doping is found to induce a superconducting phase with a critical temperature close to 5 K. It is not clear, however, which is the mechanism creating this new phase. On the other hand, a giant electron-phonon (e-ph) interaction [14] has been found in the related Sn/Ge(111)–($3\times 3$) interface, which appears between 120 and 150 K. In this $(3\times 3)$ reconstruction, one out of three of the Sn adatoms of the Sn/Ge(111)–($\sqrt{3}\times \sqrt{3}$) surface is displaced outwards $\sim 0.3$ Å relative to the other two; the dangling bond corresponding to the Sn atom displaced upwards is fully occupied, and the surface presents a half-filled electronic band around the Fermi energy [11,15,16,17]. High-resolution angle-resolved photoemission spectroscopy measurements [14] show that, around the Fermi energy, there appears an important electron mass enhancement, compatible with theoretical calculations, indicating the importance of the electron-phonon interaction. These results suggest that this giant e-ph interaction can play an important role in the formation of that new two-dimensional superconducting phase in the closely related Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) surface. In this paper, we have explored this possibility by analyzing in detail both the undoped and doped Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) surface. We first calculate for the ideal surface the e-ph and the electron–electron interactions, which turn out to have a very local character. Then, we discuss the Mott-phase and how a sharp density of states (DOS) around the Fermi energy is induced by the p-doping. Finally, using this DOS and the attractive interaction associated with the e-ph coupling, we analyze the possible superconducting state of the system by using a negative-U Hubbard model, after discarding the possibility of having a magnetic phase.

2. Sn/Si(111)–($\sqrt{\mathbf{3}}\times \sqrt{\mathbf{3}}$) Interface: Surface Geometry, Surface Bands, the Electron–Phonon Coupling and the Hubbard Interaction

Figure 1 shows the electronic surface band associated with the dangling bonds of the ideal Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) interface [18,19] as calculated using a DFT code [20] (see Appendix A). We next concentrate on the surface phonon states for the Sn atoms moving in the direction perpendicular to the surface: those displacements can be expected to induce strong re-hybridization of the Sn-p orbitals and an important coupling between the corresponding phonon and the Sn dangling bonds. The phonon surface band for those displacements in the Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) surface has been calculated in ref. [18] and is in good agreement with the calculations of ref. [21] if the phonons associated with the motion of the Sn atoms within the surface plane and the Si-atoms contributions to the phonon surface bands are discarded. That Sn-phonon band extends roughly between 10 and 6.2 meV (see also ref. [22]). Then, we approximate that phonon band by a flat surface band of 8.1 meV. Furthermore, we analyze the electron–phonon coupling by considering only one mode, the $q=0$ mode, in consistency with the flat surface band approximation. This is calculated by displacing all the Sn atoms a quantity $\Delta z$ and calculating the shift in energy, $\Delta E$, of the surface band [14] (see Appendix A). Then, the parameter g that defines the e-ph interaction can be calcuated by the equation [23]

$$g\sum _i({\widehat{b}}_i^{+}+{\widehat{b}}_i){\widehat{n}}_i,$$

(1)

where ${\widehat{b}}_i$ and ${\widehat{b}}_i^{+}$ are the bosonic operators for the (111)-surface phonon, ${\widehat{c}}_{i\sigma }^{+}$ and ${\widehat{c}}_{i\sigma }$ the fermionic operators for the i$\sigma$-electrons of the 2D-band, and ${\widehat{n}}_i={\sum }_{\sigma }{\widehat{n}}_{i\sigma }$, ${\widehat{n}}_{i\sigma }={\widehat{c}}_{i\sigma }^{+}{\widehat{c}}_{i\sigma }$ is the electron occupation number for the i-site given by [14]

$$g=|{\displaystyle \frac{\Delta E}{\Delta z}}|\sqrt{{\displaystyle \frac{\hslash }{2M{\omega }_0}}},$$

(2)

where M is the effective mass associated with the cluster formed by a Sn atom and the Si atoms coordinated to it; from reference [18] we take M = 127 m_p (m_p is the proton mass). With all these values we obtain the following e-ph coupling g = 36 meV and a value of $2g^2/{\omega }_0=0.32$ eV. Notice that this quantity can be considered at a low energy as an attractive electron–electron interaction induced by the e-ph coupling [24]: $U^{neg}=-2g^2/{\omega }_0$.

We should mention that Equation (1) follows from our approximation of a phonon flat band, since this implies having localized phonons ${\widehat{b}}_i$ and ${\widehat{b}}_i^{+}$ interacting with the dangling bond charge ${\widehat{n}}_i$.

We have also obtained the effective Hubbard interaction, U, using a restricted DFT method [25]; for this purpose, we consider a Sn/Si(111)–($2\sqrt{3}\times \sqrt{3}$) surface unit cell and perform restricted DFT calculations in which one electron is transferred between the two Sn-dangling bonds and determine the shift between the corresponding surface states. This shift is associated with the intra-atomic Hubbard interaction, $U_0$, as well as the electrostatic interaction between charges in the different dangling bonds; this long-range interaction between charges in different dangling bonds is evaluated using an electrostatic approach with $ϵ=12$ for the Si dielectric constant [26] (notice that the nearest neighbor adatom distance is ∼6 Å). This analysis yields $U_0=1.16$ eV (on-site electron–electron interaction); this interaction is screened by V, the nearest-neighbor inter-site electron–electron interaction (see, e.g., the discussion in Section 2.1 in ref. [27]). Thus, the effective Hubbard interaction is $U=U_{0}V=0.83$ eV.

3. Mott Phase for ${\mathit{n}}_{\mathit{i}\mathsf{\sigma }}=\mathbf{1}/\mathbf{2}$ and Surface States for p-Doping

We turn our attention to the possible phases that might appear in this 2-D system, assuming that the charge $n_i=n_{i\uparrow }+n_{i\downarrow }$ can be controlled by the amount of p-doping [8]. In the previous section we have found that $|U^{neg}|=0.32$ eV is much smaller than $U=0.83$ eV); thus, we take advantage of this result and analyze the system in two steps: (i) First, we neglect the e-ph coupling and analyze the following Hamiltonian [23]:

$$\widehat{H}=\sum _i{E}_0{\widehat{n}}_{i,\sigma }+\sum _{\sigma ,i,j}{t}_{i,j}{\widehat{c}}_{i\sigma }^{+}{\widehat{c}}_{j\sigma }+\sum _{i}U{\widehat{n}}_{i\uparrow }{\widehat{n}}_{i\downarrow },$$

(3)

where we introduce the effective intra-atomic U; the reference energy level corresponding to the Sn dangling bond states $E_0$; the hopping terms, $t_{i,j}$, fitted to the band calculated above (using first, $t_1$, and second, $t_2$, nearest-neighbor hopping interactions, $t_1=0.06$ eV and $t_2=-0.024$ eV); and look for the solution that corresponds to a given value of $n_i$. Note that $n_i=\langle {\widehat{n}}_i\rangle$ changes with the p-doping that induces a modification of the semiconductor boundary layer band-bending and consequently of the surface band occupancy [28].

Then, (ii) in a second step (see below), we introduce the effects of the e-ph coupling starting with the quasi-particle density of states provided by the solution of Hamiltonian (3). This procedure to find a solution in the superconducting gauge is much less involved than the one associated with the simultaneous introduction of the electron–electron and electron–phonon interactions and has the important added value that in this way we can disentangle the effects of the e-e and e-ph interactions in the behavior of this system, providing a valuable insight on the origin of superconductivity in this 2D system.

Regarding step (i), and in order to illustrate the important effect of doping in the electronic structure at the Fermi level, it is convenient to start discussing the solution of Hamiltonian (3) in the limit $t_{i,j}\to 0$. This can be achieved by introducing the appropriate local atomic self-energy given by [29,30,31]

$${\Sigma }_{i\uparrow }^{\infty }\left(\omega \right)={\displaystyle \frac{n_{i\downarrow }(1-n_{i\downarrow })U^2}{\omega -E_0-(1-n_{i\downarrow })U}}$$

(4)

then, Green’s function, $G_{i\uparrow }(\mathbf{k},\omega )$, in this limit is given by

$$G_{i\uparrow }(\mathbf{k},\omega )={\displaystyle \frac{1}{\omega -E_0-t_i\left(\mathbf{k}\right)-n_{i\downarrow }U-{\sum }_{i\uparrow }^{\infty }\left(\omega \right)}},$$

(5)

where

$$t_i\left(\mathbf{k}\right)=\sum _{j\ne i}{t}_{i,j}{e}^{i\mathbf{k}·({\mathbf{R}}_j-{\mathbf{R}}_i)}$$

(6)

independent of the local site i (${\mathbf{R}}_j$ is the position of site j). Equations (4) and (5) lead to

$$G_{i\uparrow }(\mathbf{k},\omega )={\displaystyle \frac{n_{i\downarrow }}{\omega -E_0-U-n_{i\downarrow }{t}_i\left(\mathbf{k}\right)}}+{\displaystyle \frac{1-n_{i\downarrow }}{\omega -E_0-(1-n_{i\downarrow })t_i\left(\mathbf{k}\right)}}.$$

(7)

From these equations we find the results shown in Figure 2 for U = 0.83 eV. Independent calculations [24,32], for a half-filled band $(n_{i\uparrow }=n_{i\downarrow }=1/2)$ indicate that the Mott transition, and the magnetic phase, appears for $U\langle \rho \rangle \ge 1.5$, where $\langle \rho \rangle$ is the average DOS of the initial band, which in this case (one electron in the band) is $\langle \rho \rangle =1/D=1.85$ eV⁻¹, where $D=0.54$ eV is the bandwidth; this suggests that the undoped surface should be marginally insulating, showing the Mott phase found experimentally [8]. Then we can expect the DOS shown in Figure 2b to give a fair approximation to the solution of Hamiltonian (3) for $n_{i\sigma }=1/2$.

Notice that for $n_{i\sigma }=1/2$, the DOS shown in Figure 2b presents the well-known empty Upper Hubbard Band (UHB) and the filled Lower Hubbard Band (LHB), both having a total number of states per site and spin equal to $1/2$. We also find that both bands are like the original one with a width reduced by a factor of 2; notice also that the two bands are split by ≈U. On the other hand, for $n_{i\sigma }<0.5$ (say 0.45, as suggested by the experimental evidence [8]), we find that the LHB is partially filled with a total number of 0.45 $\left(n_{i\sigma }\right)$ electrons, while the total number of states is 0.55 $(1-n_{i\sigma })$, in such a way that a metallic phase appears associated with the upper part of the LHB (Figure 2a). The empty part of this LHB coincides with the number of holes introduced by the p-doping. In this case, the different bandwidths of the Hubbard bands are modified by the corresponding total number of states each band has.

We can obtain an improved local self-energy, ${\Sigma }_{i\sigma }\left(\omega \right)$, by means of an interpolative approach [30,31], whereby we consider two limits, (a) $U\to 0$, and (b) $t_{i,j}\to 0$. For U small, we calculate the second order self-energy as

$$\begin{array}{c}\hfill {\Sigma }_{i\sigma }^{\left(2\right)}\left(\omega \right)=U^2{\int }_{-\infty }^{\infty }dxdydz{\displaystyle \frac{{\rho }_{-\sigma }\left(x\right){\rho }_{-\sigma }\left(y\right){\rho }_{\sigma }\left(z\right)}{\omega +y-x-z+i0}}\\ \hfill \left[f\left(x\right)f\left(z\right)(1-f(y\left)\right)-(1-f(x\left)\right)(1-f(z\left)\right)f\left(y\right)\right],\end{array}$$

(8)

where $f\left(s\right)$ is the Fermi distribution function; s is the corresponding energy, x, y or z; and ${\rho }_{\sigma }\left(s\right)$ is the local DOS at site i associated with the auxiliary Hamiltonian

$${\widehat{H}}^{\prime }=\sum _{i\sigma }{ϵ}_0{\widehat{n}}_{i\sigma }+\sum _{ij\sigma }{t}_{i,j}{\widehat{c}}_{i\sigma }^{+}{\widehat{c}}_{j\sigma }.$$

(9)

In this equation ${ϵ}_0$ is adjusted to obtain the same Fermi energy for Hamiltonians (3) and (9). On the other hand, for $t_{i,j}\to 0$, ${\Sigma }_{i\sigma }^{\infty }\left(\omega \right)$ is given by Equation (4). As ${\Sigma }_{i\sigma }^{\left(2\right)}\left(\omega \right)$ and ${\Sigma }_{i\sigma }^{\infty }\left(\omega \right)$ have the same limit for $\omega \to \infty$, ${\displaystyle \frac{U^2{n}_{i\downarrow }(1-n_{i\downarrow })}{\omega }}$, the following self-energy interpolates between those two cases:

$${\Sigma }_{i\uparrow }\left(\omega \right)={\displaystyle \frac{{\Sigma }_{i\uparrow }^{\left(2\right)}\left(\omega \right)}{1-\alpha {\Sigma }_{i\uparrow }^{\left(2\right)}\left(\omega \right)}},$$

(10)

where $\alpha$ is a constant independent of $\omega$, $\alpha ={\displaystyle \frac{(1-n_{i\downarrow })U+E_0-{ϵ}_0}{n_{i\downarrow }(1-n_{i\downarrow })U^2}}$. The crucial quantity to calculate ${\Sigma }_{i\sigma }^{\left(2\right)}\left(\omega \right)$ and ${\Sigma }_{i\sigma }\left(\omega \right)$ is the local DOS ${\rho }_{i\sigma }\left(\omega \right)={\rho }_{\sigma }\left(\omega \right)$ independent from i, which is taken as the DOS given by Equation (9). A better approximation for ${\rho }_{\sigma }\left(\omega \right)$ can be obtained by using a kind of Dynamical Mean Field (DMF) approach [33]; here, we calculate ${\rho }_{A\sigma }\left(\omega \right)$ embedding site A, considered as an impurity, in the lattice, in such a way that all the sites different from A have the same self-energy ${\Sigma }_{\sigma }\left(\omega \right)$; in other words, ${\rho }_{A\sigma }\left(\omega \right)$ is calculated from the following Green’s function ${\widehat{G}}_{ij\sigma }$

$${\widehat{G}}_{ij\sigma }={(\omega \widehat{I}-{\widehat{H}}^{imp})}_{ij\sigma },$$

(11)

where the effective operator, ${\widehat{H}}^{imp}$, is given by

$$\begin{array}{c}\hfill {\widehat{H}}^{imp}=(E_0+U{n}_{A\overline{\sigma }}){\widehat{n}}_{A\sigma }+\\ \hfill \sum _{i\ne A,\sigma }\left(E_0+U{n}_{i\overline{\sigma }}+{\Sigma }_{\sigma }\left(\omega \right)\right)n_{i\sigma }+\sum _{ij}{t}_{i,j}{\widehat{c}}_{i\sigma }^{+}{\widehat{c}}_{j\sigma }.\end{array}$$

(12)

($\overline{\sigma }=-\sigma$). In this equation, ${\Sigma }_{\sigma }\left(\omega \right)$ is absent from site A. Then, $G_{AA\sigma }$ (Equation (11)) yields [34] ${\rho }_{A\sigma }\left(\omega \right)$ in such a way that, on its turn, it defines the self-energy, ${\Sigma }_{A\sigma }\left(\omega \right)$, and is plugged into Equation (12) in the next step to obtain consistency in ${\rho }_{A\sigma }\left(\omega \right)$.

Figure 3 shows the 2-D DOS calculated for $n_{i\sigma }=0.45$ and U = 0.83 eV using this DMF approach. Our results show a narrow peak around the Fermi level and two sidebands reminiscent of the UHB and the LHB found for $t_{i,j}\to 0$. These results can be better understood by considering the $t_{i,j}\to 0$ limit presented in Figure 2a: in this case, $n_{i\sigma }=0.45$, we find two bands with different weights depending on the value of $n_{i\sigma }$ [30]. The crucial point to realize is that for p-doping, a sort of metallic band appears around a Fermi energy located in the LHB, in such a way that this quasiparticle DOS (QPDOS) is going to be responsible for the possible phase the system has, either magnetic, polaronic or superconductor [24,35,36,37]. As $|U^{neg}|$ is much smaller than U, we can discard the polaronic phase [24] and concentrate on the possibility of magnetism.

4. Magnetism or Superconductivity?

We analyze the possibility of having a magnetic phase in this 2-D system by realizing that the value of $\langle \rho \rangle$ for the quasiparticle DOS is less than 1.5 eV⁻¹ (see Figure 3), so that $U\langle \rho \rangle \approx 1.25<1.5$, indicating that it is unlikely that the doped surface has a magnetic phase [24].

We finally explore the possibility of having a superconductive phase due to the e-ph coupling. In our approach we consider the low-energy quasi-particles associated with the solution of Hamiltonian (3) and introduce Equation (1) describing the interaction of those quasi-particles, ${\widehat{c}}_{i\sigma }^{+}$, with the local phonons, ${\widehat{b}}_i^{+}$. Then, we integrate out the phonons, following Hewson et al. [24,38], to obtain the effective interaction $2g^2{\omega }_0/({\omega }^2-{\omega }_0^2)$ between quasi-particle states. The crucial point to realize is that, for quasi-particles of very low energy, much smaller than ${\omega }_0$, the effective interaction $2g^2{\omega }_0/({\omega }^2-{\omega }_0^2)\approx -2g^2/{\omega }_0$ is attractive and constant. Notice also that in our actual case the effect of g on the density of states is small [39,40]. Accordingly, we introduce for analyzing the possible superconducting gap the following Hamiltonian:

$${\widehat{H}}^{SC}=\sum _{i\sigma }{ϵ}_0^{\prime }{\widehat{n}}_{i\sigma }+\sum _{ij\sigma }{T}_{i,j}{\widehat{c}}_{i\sigma }^{+}{\widehat{c}}_{j\sigma }+\sum _i{U}^{neg}{\widehat{n}}_{i\uparrow }{\widehat{n}}_{i\downarrow }$$

(13)

where $U^{neg}=-2g^2/{\omega }_0=-0.32$ eV, while ${ϵ}_0^{\prime }$ and $T_{i,j}$ are associated with the QP states induced by the Hubbard interaction (${ϵ}_0$ and $t_{i,j}$ are associated with the DFT bands). We should stress that Hamiltonian (13) is only valid for quasi-particles close to the Fermi energy, with $|\omega -E_F|\ll {\omega }_0$, a condition satisfied by the Cooper quasi-particle pairs of the superconducting phase because the superconducting gap is much smaller than ${\omega }_0$.

We analyze the Hamiltonian (13) by means of a Nambu formalism within the possible superconductive gauge [41]. This yields the following superconducting gap, $\Delta$:

$$1=\left|U^{neg}\right|{\rho }_0{\int }_{-{\omega }_0}^{{\omega }_0}{\displaystyle \frac{d\omega }{\sqrt{{\omega }^2+{\Delta }^2}}},$$

(14)

where ${\rho }_0$ is the DOS associated with the quasi-particle peak at the Fermi level. From Figure 3 we see that ${\rho }_0=1.5-1.7$ eV⁻¹. Equation (14) yields

$$\Delta ={\omega }_0{e}^{-1/\left(\right|U^{neg}\left|{\rho }_0\right)}$$

(15)

so that, with the values calculated above, $|U^{neg}|=0.32$ eV, ${\rho }_0=1.5-1.7$ eV⁻¹, and ${\omega }_0=8.1$ meV, we find

$$\Delta =1.01–1.29\mathrm{meV}.$$

As $k_B$$T_C$$=2\Delta /3.53$ in BCS theory [41], we obtain a critical temperature $T_C$ of 6.9–8.8 K, in excellent agreement with the experimental value of 5 K [8]. We stress that this value for the critical temperature has been obtained in our calculations without any fitting to the experimental results. Obviously, a value closer to the experimental critical temperature could be obtained by, e.g., taking a slightly smaller value of $|U^{neg}|\sim$0.27 eV.

It might be argued that the electron–electron interaction would destroy the superconducting phase; however, as mentioned above, neither magnetism nor a polaronic phase is likely to appear in this doped surface. Furthermore, the compatibility of an e-ph-induced superconducting phase with a non-negligible U-interaction is also found in normal metals like Al (see also Appendix B). We also should remark that the crucial point in our analysis of this superconductivity phase is not how large the electron–phonon interactions are, but the disappearance of the Mott phase due to the p-doping.

5. Conclusions

In conclusion, we have analyzed the electronic properties of the 2-dimensional layer of a p-doped Sn/Si(111) surface, including the many-body effects associated with the e-ph and the electron–electron interactions. In a first step, we analyze the ideal surface, neglecting many-body effects, by means of a DFT calculation. Based on the DFT results, we calculate the main parameters defining the e-ph coupling and the electron–electron interaction: the e-ph coupling induces an attractive local interaction, $U^{neg}=-0.32$ eV, while a restricted DFT calculation yields an effective electron–electron interaction of $U=0.83$ eV. Then, we analyze how the Hubbard interaction U modifies the ideal DOS; for the undoped surface layer, we find that U is larger than the critical value defining the transition to the Mott phase; however, our results for a doped layer with $n_{i\sigma }$ = 0.45 indicate that the system is metallic with a DOS at the Fermi level too low for sustaining a magnetic phase. Discarding a magnetic phase in this system, we analyze the possibility of having a superconducting phase; this is achieved by introducing the attractive local interaction, $U^{neg}$, in the quasiparticle DOS induced by the electron–electron interaction. Our BCS analysis shows that, with these parameters, a superconducting phase can appear around 7–9 K, in good agreement with the experimental evidence [8]. We conclude that the superconductivity appearing in this system is due to the combined effect of the e-ph coupling acting on the quasiparticle DOS created by the electron–electron Hubbard interaction.

It is also interesting to mention that a similar phase might appear in the Sn/Ge(111)–($\sqrt{3}\times \sqrt{3}$) surface where the electron–phonon interaction is relevant [14]. It has been found [42] that this surface presents a Mott transition at very low temperature in similarity with the Sn/Si(111)–($\sqrt{3}\times \sqrt{3}$) case; it is an open question whether a p-doping of this surface might also induce a superconducting phase.