Abstract
Silicene has been synthesized recently, with experimental evidence showing possible superconductivity in the doped case. The noncoplanar lowbuckled structure of this material inspires us to study the pairing symmetry of the doped system under a perpendicular external electric field. Our study reveals that the electric field induces an interesting quantum phase transition from the singlet chiral d + id′wave superconducting phase to the triplet fwave one. The emergence of the fwave pairing results from the sublatticesymmetrybreaking caused by the electric field and the ferromagneticlike intrasublattice spin correlations at low dopings. Due to the enhanced density of states, the superconducting critical temperature of the system is enhanced by the electric field remarkably. Furthermore, we design a particular dc SQUID experiment to detect the quantum phase transition predicted here. Our results, if confirmed, will inject a new vitality to the familiar Sibased industry through adopting doped silicene as a tunable platform to study different types of exotic unconventional superconductivities.
Introduction
The past few decades have witnessed the glorious development of unconventional superconductivity (SC)^{1}. While most of the so far confirmed unconventional superconductors are singlet pairing including, say, the cuprates with dwave pairing and the ironpnictide with s ± pairing^{2,3}, no triplet superconductor has been confirmed except for the Sr_{2}RuO_{4} system which shows evidence for possible p + ip′ pairing^{4}. Recently, however, triplet pairing is catching more and more attentions partly due to its possible connection with topological SC^{5,6,7,8}, which has inspired great enthusiasm in searching for triplet superconductors. Here we predict a tunable quantum phase transition (QPT) from the singlet d + id′ superconducting state to the triplet fwave one in doped silicene via applying a perpendicular external electric field on the system. Although both pairing symmetries break the time reversal symmetry, they are quite different. While the former shows chiral complex gap structures which is a hot topic recently^{9,10,11,12,13,14,15}, the latter has a real pairing gap which changes sign with every 60° rotation. The fwave pairing has been proposed in the study of the quasi1D organic system^{16}, the triangular lattice system for Na_{x}CoO_{2}^{17}, and the spinless fermion system on the honeycomb optical lattice^{18}. However, unambiguously confirmed experimental evidence for it is still lack now. If the fwave SC predicted here is confirmed experimentally, it will be the first time to realize such an intriguing highangularmomentum pairing state.
Silicene, a single atomic layer of Si forming a 2D honeycomb lattice, can be regarded as the Sibased counterpart of graphene. It has attracted a lot of research interest since being successfully synthesized recently^{19,20,21,22,23}. Similar lattice structures between graphene and silicene bring them similar band structures with linear dispersion near the Fermi surface (FS), which further lead to similar physical properties between them. The most important structural difference between the two layered systems lies in that, while the graphene layer forms a regular flat plane, the silicene layer instead takes the form of noncoplanar lowbuckled (LB) structure, with the sublattices A and B forming two separate planes. The LB structure of silicene causes many fascinating consequences, such as the enhanced quantum spin Hall effect^{24,25}, the quantum anomalous Hall effect and valley polarized quantum Hall effect in external electric field^{26,27,28,29}, and possible d + id′ chiral SC in bilayer silicene^{30}. On the other hand, this LB structure provides the possibility to use gated silicene as a tunable source of the perfectly spinpolarized electrons^{31}. Interestingly, a recent tunnelling experiment reported electronic gap in doped silicene^{32}, probably caused by SC, which has attracted a lot of research interests^{33,34}.
In this paper, we report our study of the pairing symmetry of doped silicene under a perpendicular external electric field, which differentiates the onsite energies of the two sublattices of this LB system. Based on the random phase approximation (RPA) to the Hubbard model of the system, we reveal a tunable QPT from the singlet d + id′ chiral SC under weak field to the triplet fwave one under strong field, without the necessity of longrange Coulomb interaction^{35,36,37}. The physics behind this interesting QPT lies in that, under the strong sublatticesymmetrybreaking electric field, only one sublattice is left as the low energy subspace, and the ferromagneticlike intrasublattice spin correlations favor triplet fwave pairing in this subspace. Due to the enhanced density of states (DOS) near the Fermi level, the superconducting critical temperature of the system is enhanced by the applied electric field remarkably. To detect the QPT predicted here, we further design a particular experiment with a phasesensitive dc SQUID, which can distinguish between the d + id′ and fwave pairings. Our study will open up a new era to utilize the familiar Sibased material as a tunable platform to study the competition and QPT among different types of exotic unconventional SCs.
Results
Model
The LB honeycomb lattice of silicene is shown in Figs. 1(a) and 1(b). Since the two sublattices A and B lie in two parallel planes separated from each other, an applied perpendicular electric field induces an onsite energy difference Δ between them. We adopt the following Hubbard model as an appropriate start point to study the low energy physics of the system:Here the tterm (with t = 1.12 eV^{25}) describes the nearestneighbor (NN) hoppings, and for sublattice A (B). The Hubbard interaction strength U = 2 eV is adopted in the following, which is near the value obtained from the first principle calculations^{38}. The nextnearestneighbor (NNN) hoppings are not included here because they are qualitatively unimportant. The resulting particlehole symmetry enables us to focus the following discussions only on the electrondoped case.
The band structure of the system with the dispersion is shown in Fig. 1(c). Clearly, the applied field induces a band gap Δ near the Kpoint and the nearby low energy band structure is flattened, although it does not modify the shape of the FS for a given doping concentration. Note that the electric field breaks the sublattice symmetry of the system. In particular, the onsite energy of sublattice A, i.e., V_{A} = Δ/2, is closer to the Fermi level shown for the electrondoped case, and consequently this sublattice will dominate the low energy physics of the system. This effect turns out to be very important for our following discussions.
Susceptibilities
According to the standard RPA approach^{30,39,40,41}, we define the free susceptibility (U = 0) of the model aswhere p, q, s, t = 1, 2 is the sublattice index. The Hermitian static susceptibility matrix is defined as . The largest eigenvalue of this matrix represents the static susceptibility of the system in the strongest channel, and the corresponding eigenvector determines the pattern of the applied detecting field in that channel. In addition, the eigenvector also describes the pattern of the dominant intrinsic spin fluctuations in the system.
In Fig. 2(a)–2(c), we show the kspace distributions of the zero temperature static susceptibility of the system in the strongest channel mentioned above. From Fig. 2(a) to 2(c), one can clearly observe the doping evolution of the static susceptibility. In particular, when the doping increases gradually from zero to the Van Hove (VH) doping x = 1/4, the momenta of the maximum susceptibility evolves from the Γpoint (2(a)) first to a small circle around it (2(b)), and finally to the Mpoints (2(c)). Such an doping evolution of the susceptibility originates from the evolution of the FS which grows gradually from the Kpoints (the Diracpoints) first to small pockets around them, and finally to the connected large FS with perfect nesting at the VH singularity. From 2(a) to 2(c), one verifies that the dominant spin fluctuations on each sublattice of the system changes gradually with doping from ferromagneticlike to antiferromagneticlike. In Fig. 2(d), a typical pattern of the dominant spin fluctuations of the system is shown at 10% doping for Δ = 1 eV. Most prominently in Fig. 2(d), the magnetic moments are asymmetrically distributed between the two sublattices, with the magnitude of the moment on sublattice A obviously larger than that on sublattice B. This is consistent with the above band structure analysis which suggests that sublattice A will dominate the low energy physics of the system. Our calculation reveals that such asymmetry is enhanced by both electricfield and doping.
When the Hubbard interaction is turned on, the charge (c) or spin (s) susceptibility is given by RPA aswhere (U) is a 4 × 4 matrix, whose only two nonzero elements are ^{30}. Clearly, the repulsive Hubbard interaction suppresses χ^{(c)} and enhances χ^{(s)}. When the interaction strength U is greater than a critical value U_{c}, the spin susceptibility χ^{(s)} of the model diverges, which implies the instability towards the longrange spindensitywave (SDW) order. Below the critical interaction strength U_{c}, the spin fluctuations take the main role in mediating the Cooper pairing.
Pairing symmetries
Through exchanging the renormalized susceptibilities (3), one obtains the effective interaction vertex V^{αβ}(k, k′)^{41} between two Cooper pairs on the FS, which is then plugged into the following linearized gap equation near the superconducting critical temperature T_{c}^{41}:Here the integration is along various FS patches labelled by α or β, is the Fermi velocity and is the tangential component of k′ along the FS. Solving Eq. (4) as an eigenvalue problem, one obtains the largest eigenvalue λ, which determines T_{c} via T_{c} ≈ te^{−1/λ}, and the corresponding eigenvector Δ_{α}(k) as the leading gap form factor.
The phase diagram on the xΔ plane obtained by our RPA calculations is shown in Fig. 3(a). Clearly, except for the SDW phase near the VH doping x = 1/4, the singlet chiral and triplet nodeless fwave pairings beat other instabilities and serve as the leading instability in different regimes of the phase diagram. The gap function of the d_{xy} symmetry (for doping x = 0.15 and Δ = 0) shown on the FS in Fig. 3(b) is antisymmetric about the x and y axes, and that of its degenerate partner (not shown) is symmetric about these axes. Their mixing in the form of d + id′ minimizes the ground state energy. The main part of the d + id′ pairing in real space is distributed on NNbonds as shown in Fig. 3(d). The gap function of the fwave pairing (for doping x = 0.15 and Δ = 1 eV) is shown in Fig. 3(c). The main part of the fwave pairing in real space is distributed on NNNbonds as shown in Fig. 3(e). Clearly, the gap function of this timereversalbreaking fwave pairing changes sign with every 60° rotation either in kspace or in real space.
As shown in Fig. 3(a), the leading instability for Δ = 0 is d + id′ pairing at low dopings and SDW order near the VH doping. This result is consistent with previous calculations on the graphene system^{9,10,11,12,13,14}, which shares the same Hubbard model as here. In Fig. 3(f), we show the doping dependence of the SDW critical interaction U_{c}. Clearly, when the doping , U_{c} is less than U = 2 eV, which accounts for the emerging of the SDW state in Fig. 3(a). The most interesting and important discovery here is that the triplet fwave pairing, which is mediated by the NNNbond ferromagnetic spin fluctuations, wins over the d + id′ pairing and rises as the leading pairing symmetry when sufficiently strong electricfield is applied on the system at low dopings. Physically, such an electricfieldinduced QPT originates from the sublatticesymmetrybreaking effect mentioned before, which has selected the sublattice A as the low energy subspace of the system. All the relevant low energy physics, including the pattern of the spin fluctuations shown in Fig. 2(d), the effective pairing interaction mediated by these spin fluctuations, and the pairing itself take place mainly in this subspace in strong electric field. Consequently, at low dopings, the intersublattice pairing d + id′ symmetry shown in Fig. 3(d) has to give way to the intrasublattice pairing fwave symmetry shown in Fig. 3(e).
The superconducting critical temperature T_{c} ≈ te^{−1/λ} of the fwave pairing state is controllable, and can be remarkably enhanced by the applied electric field. As shown in Fig. 3(g) for doping x = 0.15, the eigenvalue λ of the fwave pairing can be enhanced to about 0.18 when Δ is tuned to 1 eV. Although the RPA approach tends to overestimate T_{c}, we still have space to enhance the T_{c} of the system to the experimentally accessible range by tuning the doping level closer to the VH doping or increasing the electric field. Physically, such enhancement is attributed to the increase of the DOS near the Fermi level (see the inset of Fig. 3(g)) caused by the flattening of the band structure under the electric field (see Fig. 1(c)). The increase of the DOS not only enhances the number of Cooper pairs near the FS, but also enhances the pairing interaction. The T_{c} of the d + id′ pairing can also be enhanced by sufficiently strong electric field due to this DOS enhancement, but it is lower than that of the fwave pairing as shown in Fig. 3(g). For a vanishingly small U, the fwave SC has been proposed even in the electricfieldfree system^{42}, but with extremely low T_{c}.
Detecting the QPT
The QPT from d + id′ to fwave pairings predicted here can be detected by the dc SQUID, a phasesensitive device which has been adopted in determining the pairing symmetries of such superconducting systems as cuprates^{43} and Sr_{2}RuO_{4}^{44}. Our basic scheme is shown in Figs. 4(a) and 4(d), where a slice of silicene is fabricated into a hexagonal shape, allowing the relative phase among different directions in the system to be detected. Two superconductornormal metalsuperconductor (SNS) Josephson tunneling junctions are formed on the opposite (4(a)) or adjacent (4(d)) edges of the hexagon, which are connected by a loop of a conventional swave superconductor, forming a bimetallic ring with a magnetic flux Φ threading through the loop.
As a result of the interference between the two branches of Josephson supercurrent, the maximum total supercurrent (the critical current) I_{c} in the circuit modulates with Φ according toHere I_{0} is the critical current of one Josephson junction, Φ_{0} = h/2e is the basic flux quantum, and δ_{ab} is the intrinsic phase shift inside the silicene system between pairs tunneling into the system in two directions a and b. In configuration 4(a), δ_{ab} is equal to 0 (π) for singlet pairing including the d + id′ one (triplet pairing including the fwave one), and in configuration 4(d), it is 2π/3 (π) for the d + id′ (f) symmetry. Consequently, the I_{c} ~ Φ curves for d + id′ and fwave pairings in configurations 4(a) and 4(d) show different patterns (Figs. 4(b), 4(c), 4(e), and 4(f)). Most prominently, the fwave pairing symmetry is characterized by the minima of I_{c} (which can be nonzero when the selfinductance of the loop is not negligible) at zero flux for both configurations 4(a) and 4(d). For the d + id′ symmetry at zero flux, while the critical current I_{c} is a maximum in configuration 4(a), it is not an extreme point in 4(d). Thus, by observing the modulation of the SQUID response vs applied magnetic flux, one can distinguish between the two pairing symmetries and hence detect the QPT.
Discussion
The nonmonotonic doping dependence of the critical electric field in Fig. 3(a) originates from the competition between two opposite effects. On the one hand, the sublatticesymmetrybreaking effect, which favors the fwave pairing, is enhanced by doping since the Fermi level sits in the middle of V_{A} and V_{B} at zero doping (see Fig. 1(c)). On the other hand, as shown in Fig. 2(a)–2(c), the intrasublattice spin correlations in the system evolves from ferromagneticlike at low dopings, which favors triplet fwave pairing, to antiferromagneticlike near the VH doping, which favors singlet d + id′ pairing.
Besides the nodeless fwave symmetry predicted here, there is also another nodal f′wave symmetry which is mainly composed of intrasublattice pairings with a bond length of three times of the lattice constant. This f′wave symmetry is not favored here, for its gap nodes along the ΓK lines do not avoid the FS. When a sufficiently strong KaneMele spinorbit coupling (SOC)^{45,46} is added into model (1), the Fermi pockets of the system would shift from near the Kpoints to near the Mpoints, and the f′wave symmetry would be favored, for its gap nodes avoid the FS. In a system with a medium SOC, the two fwave pairings could be nearly degenerate, and their mixing in the form of f + if′ might be realized to minimize the energy. As a result of its nontrivial topological property, this intriguing triplet chiral superconducting state can harbor the Majorana zeromode at its boundary^{6,47,48,49}, which is useful in the topological quantum computation. Although the SOC in the present silicene system is too small^{24,25} to favor the f + if′ pairing, we can expect the system with a LB honeycomb lattice similar to silicene and a medium SOC, such as BiH^{50,51}, could serve as the platform for this novel highangularmomentum chiral pairing state. We leave this subject for future studies.
As a result of the rapid development of modern experimental techniques, the electric field strong up to 0.3 V/Å has already been applied to the research works of materials^{52}, which has approached what we need here. Besides, the monolayer of silicene has been synthesized on top of the substrates of Ag, ZrB_{2}, Ir, etc^{19,20,21,22,23}. Due to the lowbuckled structure of silicene, the distance between the sublattice A and the substrate is different from that between the sublattice B and the substrate. As a result, the interaction between the silicene monolayer and the substrate provides an effective electric field, which causes the sublatticesymmetrybreaking and the difference between the onsite energies of the two sublattices^{53}. Since such an effective electric field is an internal electric field of chemical origin, it can generally be as strong as what we need here. Therefore, our proposal of the fwave pairing is feasible in practice.
In conclusion, we have systematically studied the pairing phases of doped silicene in a perpendicular external electric field. The results of our RPA study predict that with the enhancement of the electric field, the system will experience a QPT from singlet d + id′ superconducting state to triplet fwave one, and the superconducting critical temperature of the system will be enhanced due to the increase of the DOS. Our model needs neither longrange Coulomb interaction nor the situation of vanishingly small HubbardU, and thus is more realizable than other proposed ones.
Methods
Susceptibilities
According to the standard RPA approach^{30,39,40,41} adopted in our study, we first define the free susceptibility (U = 0) of the model (1) as in Eq. (2). Direct calculation yields the following explicit expression of the free susceptibility,where and are the αth eigenvalue and eigenvector of the single particle Hamiltonian of the system respectively, and n_{F} is the FermiDirac distribution function.
When the interaction is turned on, we define the charge (c) and spin (s) susceptibilities as
For U = 0, we have χ^{(c)} = χ^{(s)} = χ^{(0)}. In the RPA level, χ^{(c(s))} is given by Eq. (3).
Effective interaction and gap equation
Consider the scattering of a Cooper pair from the state (k′, −k′) in the βth (β = 1, 2) band to the state (k, −k) in the αth (α = 1, 2) band. This scattering process can be described by the following effective interaction,Here the projective interaction vertex V^{αβ}(k, k′) is given by the effective vertex throughand itself, in the singlet channel, readsand, in the triplet channel, readsThe meanfield decoupling of the effective interaction (9) in the Cooper channel gives rise to the selfconsistent gap equation. Near the critical temperature T_{c}, this equation is linearized as Eq. (4), solving which one obtains the leading pairing symmetries and the corresponding T_{c}.
If the leading pairing symmetries include two degenerate doublets such as and d_{xy}, we can express the gap function of the system asHere and represent the normalized gap functions of corresponding symmetries. Then the mixing coefficients C_{1}, C_{2}, and C_{3} are determined by the minimization of the total meanfield energy with the constraint of the average electron number in the superconducting state.
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Acknowledgements
We are grateful to Hong Yao, QiangHua Wang, Yi Zhou and YuZhong Zhang for stimulating discussions. The work is supported by the MOST Project of China (Grants Nos. 2014CB920903, 2011CBA00100), the NSF of China (Grant Nos. 11174337, 11225418, 11274041, 11334012), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grants Nos. 20121101110046, 20121101120046). F.Y. is supported by the NCET program under Grant No. NCET120038.
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School of Physics, Beijing Institute of Technology, Beijing 100081, China
 LiDa Zhang
 , Fan Yang
 & Yugui Yao
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Contributions
Y.Y. and F.Y. conceived the idea of searching the pairing symmetry in doped silicene subject to an applied electric field. L.D.Z. carried out the RPA calculations. All authors analyzed the data and wrote the manuscript.
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