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Superconductivity in the surface states of a Bi2X3 topological insulator: effects of a realistic model
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 255701 (http://iopscience.iop.org/0953-8984/27/25/255701) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 255701 (7pp)
doi:10.1088/0953-8984/27/25/255701
Superconductivity in the surface states of a Bi2X3 topological insulator: effects of a realistic model Lei Hao and Jun Wang Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China E-mail: [email protected] Received 9 March 2015, revised 27 April 2015 Accepted for publication 14 May 2015 Published 9 June 2015 Abstract
Superconductivity in the topological surface states is essential to both the surface spectrum of bulk superconducting state and the proximity-induced superconductivity of Bi2X3 (X is Se or Te) topological insulators. While previous theories were mostly based on simplified models for the bulk topological insulator and the surface states, the accumulating experiments stimulate us to make an analysis using realistic model for the normal state electronic structures, incorporating terms responsible for particle–hole asymmetry and hexagonal warping. An effective low-energy model for the topological surface states is derived first. Then we identify all the bulk timereversal-invariant superconducting pairings in the topological insulator that can open a gap in the topological surface states. Many more pairings are found to be able to gap the topological surface states as compared to conclusions based on simplified models. The number of proximityinduced pairing channels in the topological surface states increases by one as a result of the hexagonal warping term, but is not changed by the particle–hole asymmetry term. Keywords: topological surface states, superconducting topological insulators, proximity effect (Some figures may appear in colour only in the online journal)
1. Introduction
insulator (or, a semiconductor with large spin–orbital coupling), has been considered by many as a more promising way [36–55]. For example, high-temperature superconductivity up to 80 Kelvins was reported in Bi2Se 3 and Bi2Te 3 via proximity to Bi2Sr2CaCu2O8 + δ [53, 54]. Among the various topics concerning the superconductivity in three-dimensional topological insulators, the issue of pairings that can form in the topological surface states is very important to both the bulk and proximity-induced superconductivities of the topological insulators. For clarity, we use ‘topological surface states’ to indicate the surface states arising from the topological character of topological insulator, and ‘surface Andreev bound states’ to denote the surface modes appearing in the superconducting phase as a result of the topological character of the superconducting pairing. For the most studied superconducting topological insulator, CuxBi2Se 3, the topological surface states coexist and are well separated from the bulk conduction band around the chemical potential [2]. In this case, it was shown theoretically that the absence of gap opening in the topological
Searching for possible topological superconductors has attracted a large amount of experimental and theoretical attention [1–32]. One reason is that these new topological matters have Majorana fermions, which are promising candidates for realizing the topological quantum computations, as elementary boundary modes or zero modes pertaining to vortices and impurities [33]. Bulk topological superconductors, though intensively searched for, are scarce. While Sr2RuO4 appears to be a two-dimensional time-reversal-broken topological superconductor [34, 35], the possibility of time-reversal-invariant topological superconductivity in three-dimensional superconducting topological insulators including CuxBi2Se 3, Bi2Se 3 and Bi2Te 3 are presently under hot debate [1–7]. Besides, these bulk superconductors all have the drawback of having very low superconducting critical temperatures. Hence, the alternative approach of achieving a topological superconductor, by the proximity coupling of a superconductor with a topological 0953-8984/15/255701+7$33.00
1
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L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
surface states is intimately related to the possible existence of surface Andreev bound states [8]. In addition, a recent selfconsistent calculation by Mizushima et al shows that the topological surface states can induce interesting parity mixing effect and the enhancement of certain pairing components [56]. For the proximity-induced superconductivity in a topological insulator, the superconductivity in the topological surface states is even more important since it is possible in this case to tune the chemical potential of the topological insulator to the bulk energy gap, where only the topological surface states exist in the normal phase [57, 58]. Altogether, it is very important to study the superconductivity in the topological surface states. Previous theories mostly adopt some simplified model for the (normal state) electronic structures of three-dimensional topological insulators like Bi2Se 3, which ignore the terms relevant to particle–hole asymmetry and (or) hexagonal warping. A recent work of Fu studying a special odd-parity pairing by taking into account of the effect of hexagonal warping has shown some qualitative changes [59]. This finding together with the accumulating relevant experimental results [51–55], have made it increasingly urgent to study the possible pairing correlations in the topological surface states systematically, starting from a realistic microscopic model for the electronic structures of relevant topological insulators. It is the aim of the present work to fill up this gap. The intrinsic superconducting instabilities of the topological surface states, not referring to the bulk topological insulator or an external superconductor coupled by proximity effect, will however not be considered in the following [60]. In the remaining parts of this work, we first define in section 2 the model that would be used to produce a reasonable electronic structure for three-dimensional topological insulators like Bi2Se 3 and Bi2Te 3. Then in section 3 we derive the low-energy effective model for the topological surface states. In section 4, all the time-reversal-invariant pairings for the superconducting topological insulators that can gap the topological surface states are constructed, the number of which is shown considerably increased by the presence of the particlehole asymmetry and hexagonal warping terms. In section 5, the pairing channels that can appear in the topological surface states when the topological insulator is coupled to an external (spin-singlet or spin-triplet) superconductor are studied. A short summary is presented in section 6.
hexagonal BZ corresponding to an equivalent hexagonal lattice with two orbitals per unit cell [6–8, 64]. Taking the basis vector as ϕk† = [a1†k ↑, a1†k ↓, a 2†k ↑, a 2†k ↓], in which the subindices 1 and 2 indicate the two orbitals, the tight binding Hamiltonian matrix for wave vector k is written as [7] H0(k) = ϵ(k)I4 + M (k)Γ5 + B0cz(k)Γ4 + A0 [cy(k)Γ1 (1) −cx (k)Γ2] + R1d1(k)Γ3 + R2d2(k)Γ4,
in which ϵ(k) = C0 + 2C1[1 − cos(k ⋅ δ4 )] (2) 4 + C2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)], 3 M (k) = M0 + 2M1[1 − cos(k ⋅ δ4 )] (3) 4 + M2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)], 3 1 cx (k) = [sin(k ⋅ δ1) − sin(k ⋅ δ2 )], (4) 3 1 cy(k) = [sin(k ⋅ δ1) + sin(k ⋅ δ2 ) − 2 sin(k ⋅ δ3)], (5) 3 cz(k) = sin(k ⋅ δ4 ), (6) 8 d1(k) = − [sin(k ⋅ a1) + sin(k ⋅ a2) + sin(k ⋅ a3)], (7) 3 3 d2(k) = −8[sin(k ⋅ δ1) + sin(k ⋅ δ2 ) + sin(k ⋅ δ3)]. (8)
Here, the four nearest neighboring bond vectors of the effective hexagonal lattice are δ1 =
(
δ2 = −
3 2
a,
), δ
1 a, 0 2
3
(
3 2
1
)
a, 2 a, 0 ,
= (0, −a, 0), and δ4 = (0, 0, c ). a and
c are lattice parameters in the xy plane and along the z direction, respectively. The three in-plane second nearest neighboring bond vectors in d1(k) are a1 = δ1 − δ2, a2 = δ2 − δ3, and a3 = δ3 − δ1. The first term of H0(k) is responsible for the particle–hole asymmetry of the electronic structure. The last two terms of H0(k) induce hexagonal warping of the bulk Fermi surface and the topological surface states [62, 63]. Previous theoretical studies usually neglect the first and the last two terms of H0(k) [5, 6, 8–10, 61, 64], which will however be the focus of the present work. As was shown explicitly in a former work, the above model respects all the symmetry requirements of the D35d double group for Bi2X3 [7, 62]. The parameters contained in equation (1) are assumed to take values realistic for actual materials [7, 62]. While equation (1) is equivalent to the model introduced by [7], we would take in this work the parity basis for the two orbitals instead of the local basis adopted by [7]. The five Γ matrices as thus written as Γ1 = σ1 ⊗ s1, Γ2 = σ1 ⊗ s2, Γ3 = σ1 ⊗ s3, Γ4 = σ2 ⊗ s0, and Γ5 = σ3 ⊗ s0 [7, 62]. In this basis, the parity operator is written as P = σ3 ⊗ s0 [8]. In this work, σα (α = 1, 2, 3) and sα (α = 1, 2, 3) are defined as Pauli matrices
2. Model We consider superconducting pairings in the topological surface states of three dimensional topological insulators, including Bi2Se 3, Bi2Te 3, and Sb 2Te 3 [61]. We would concentrate on Bi2X3 (X is Se or Te) to analyze the related superconductors. These materials belong to the D35d space group and consist of Bi2X3 quintuple layers stacked along the outof-plane direction. They are very well described by an effective model defined close to the Brillouin zone (BZ) center (the Γ point) [61–63]. In this work we use a tight binding extension of the model. Instead of working in the original BZ for a D35d space group [61, 62], the k ⋅ p model is extended to a 2
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
in the orbital and spin subspaces, while σ0 and s0 are unit matrices in the orbital and spin subspaces. In studying proximity-induced pairings in the topological surface states, we consider an ideal interface between a superconductor and the top surface of the topological insulator. The possible pairing symmetries of the external superconductor that would be studied include the spin-singlet pairings (s-wave or d-wave), and a type of spin-triplet pairings with the p + ip′-wave state proposed for Sr2RuO4 as a special case.
For each λαβ, we can get two independent solutions for η as ηαβ1(E ) = [βB0λα, 0, L + − D+λα2 − E , 0]T , (18)
and ηαβ 2(E ) = [0, L− − D−λα2 − E , 0, −βB0λα ]T . (19)
The general solution for the surface modes for kx = k y = 0 is a linear combination of the above modes Ψ(z, E ) = ∑ cαβγ ηαβγ e βλαz. (20) αβγ
3. Topological surface states
Since the proper boundary condition for a sample occupying z < 0 is [8, 61, 62, 65]
Starting from the model defined by equation (1) for bulk topological insulators, we first derive the topological surface states for the upper surface of a sample occupying the lower half space z < 0. Expanding the kz-dependent terms of H0(k) in equation (1) to the leading order of kz, we have
Ψ(z = 0, E ) = Ψ(z = −∞, E ) = 0, (21)
we should retain only the four components of cαβγ with β = +. From Ψ(z = 0, E ) = 0, and requiring the existence of nontrivial solutions for Ψ(z, E ), we have the following constraint for the energy E
ϵ(k) ≃ C0 + C1kz2 + C2xy, (9)
det∣η1, +,1(E ), η1, +,2(E ), η2, +,1(E ), η2, +,2(E )∣ = 0. (22)
M (k) ≃ M0 + M1kz2 + M2xy, (10)
From this equation, we get the twofold degenerate eigenvalue for the surface modes as
B0cz(k) ≃ B0kz, (11)
C E = C0 − 1 M0. (23) M1
4
where C2xy = 3 C2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)] and M2xy = C2xy[C2 → M2]. The simultaneous presence of the particle–hole asymmetry and the hexagonal warping terms make the direct analytical calculation of the surface modes for a general surface momentum (kx, k y ) difficult. Instead, we make a perturbative calculation by first solving the surface modes for kx = k y = 0 and then taking the kx and k y dependent terms as perturbations. Since the Fermi momentum in the (kx, k y ) plane is small [57, 58], this should be a reasonable approximation. The eigenequation for the surface modes of kx = k y = 0 are written as
The above result is the same as the previous result of Shan et al for a slightly different yet unitarily equivalent model [62, 65]. Putting the above solution for E back to the boundary condition of equation (21), we get two degenerate solutions for {cα, +, γ } which give the two surface modes for kx = k y = 0 as Ψγ (z, E ) = N0η˜1, +, γ (e λ1z − e λ2z), (24)
where γ = 1, 2 labels the two modes. The number N0 is the normalization factor for the surface mode. η˜1, +, γ is defined as the original η1, +, γ in equations (18) and (19) multiplied by a positive normalization factor. Substituting equation (23) to equations (18) and (19), η˜1, +, γ can be simplified to
H0(kx = k y = 0, kz → −i∂z)Ψ(z, E ) = E Ψ(z, E ). (12)
The ansatz for the surface modes are taken as Ψ(z, E ) = ηe λz, (13)
1 η˜1, +,1 = [ −D− , 0, − D+ , 0]T , (25) 2M1
where η is a four component column vector. Substituting the ansatz to equation (12), the nontrivial solution satisfies
1 η˜1, +,2 = [0, −D− , 0, − D+ ]T . (26) 2M1
det∣H0(kx = k y = 0, −iλ ) − EI4∣ = 0, (14)
which gives
In obtaining the above simplification, we have assumed M1 > 0, D+ > 0 and D− < 0, which are consistent with the parameter sets for Bi2Se 3 and Bi2Te 3 derived from first- principles calculations [7, 62]. Ψγ (z, E ) (γ = 1, 2) are the exact eigen-basis for kx = k y = 0. To get the approximate eigenstates for nonzero k˜ , we begin by taking Ψγ (z, E ) (γ = 1, 2) as the starting basis, then take the terms in H0(k) that depend on kx or k y as perturbation, finally we get the approximate low-energy effective model of the surface states close to kx = k y = 0 as [8, 62, 65]
D+(L− − E ) + D−(L + − E ) − B02 + (−1)α − 1 F λα2 = , (15) 2D+D−
where α = 1, 2, D± = C1 ± M1, L ± = C0 ± M0, and F = [D+(E − L−) + D−(E − L +) + B02]2 (16) −4D+D−(E − L +)(E − L−).
Defining λα as the solutions with positive real parts, we get the four independent solutions for λ as
3
λαβ = βλα, (17)
h eff (k˜ ) = ∑ aα(k˜ )θα, (27)
where β = ±, α = 1, 2.
α=0
3
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
where k˜ = (kx, k y ) and the four coefficients are
where N± = 1/ 2 a12 + a22 + a32 ( a12 + a22 + a32 ± a3) is the normalization factor. The above two eigenvectors can be expressed in second quantized form in terms of equation (30) as
C C a 0 = C0 − 1 M0 + C2xy − 1 M2xy, (28) M1 M1
⎞ Nα ⎡ ⎛ ⎢c1α⎜ −D− a1†k˜ ↑ − D+ a 2†k˜ ↑⎟ 2M1 ⎣ ⎝ ⎠ (34) +c2α −D− a1†k˜ ↓ − D+ a 2†k˜ ↓ ⎤⎦,
M12 − C12 (29) [a1, a2, a3] = [−A0 cy, A0 cx, −R1d1]. M1 The dependencies on k˜ are made implicit in equations (28) and (29). Since equation (27) is the effective model for a general nonzero wave vector k˜ , its basis can be denoted as [d1†k˜ , d 2†k˜ ] ≡ φ k˜†. θα (α = 0, …, 3) in equation (27) are Pauli matrices acting in the subspace subtended by φ k˜†. Note that, though φ k˜† is formally the basis for nonzero k˜ , its dependence on the z coordinate inherits directly from Ψγ (z, E ) (γ = 1, 2) and thus does not vary with k˜ . This character, resulting from the perturbative nature of our approach, should be acceptable since the Fermi momentum in the (kx, k y ) plane is usually very small so that the relevant wave vectors are close to zero [57, 58]. According to the above explanations and recalling equations (24)–(26), we can express the basis for equation (27) as
fα†k˜ =
(
)
where we have defined c1α = α a12 + a22 + a32 + a3 and c2α = a1 + ia2, α = ±. To compare the above results with those for the simplified models, we can simply set C0 = C1 = C2 = 0 and (or) R1 = R2 = 0. For example, the eigenvalue of equation (23) would be E = 0 in the absence of the particle–hole asymmetry term, which is consistent with previous results for simplified models [8, 64]. 4. Bulk time-reversal-invariant pairings that gap the topological surface states
⎤ 1 ⎡ ⎢ −D− a1†k˜ σ − D+ a 2†k˜ σ ⎥, dγ†k˜ = (30) γ γ 2M1 ⎣ ⎦
Suppose the chemical potential crosses the Eα(k˜ ) (α = + or −) branch of the topological surface states and the topological surface states coexist but are well separated from the bulk band states. This assumption is consistent with the experimental observation of CuxBi2Se 3 [2]. Consider the possible time-reversal-invariant pairings that can form within the topological surface states. Since the conduction and valence surface bands defined by equations (32)–(34) are both nondegenerate, the pairing can only be of the form
where γ = 1, 2, σ1 =↑ and σ2 =↓. The four creation operators appearing on the right hand side of equation (30) are understood as linear combinations of the corresponding operators defined on the various quintuple layers of the sample. The amplitudes on the various layers follow an exponentially decaying behavior according to equation (24). Suppose the N0 factor defined in equation (24) has been determined by the normalization condition. Turning back from the continuous z dependence of equation (24) to the original discrete lattice representation, we can write approximately in the limit of very small c (the lattice parameter along the z direction) as
Δα (k˜ ) = g(k˜ )fα†k˜ fα†, −k˜ , (35)
where g(k˜ ) is an undetermined factor to ensure the timereversal-invariance of the pairing. By introducing the Nambu basis, this pairing term would be seen as the off-diagonal element in the corresponding Bogoliubov–de Gennes Hamiltonian [7, 8]. Recalling the transformation property of the timereversal operator T , that is T iT −1 = −i, Tc k ↑T −1 = c−k ↓ and Tc k ↓T −1 = −c−k ↑, a simple choice for g(k˜ ) is found to be
+∞
aγ†k˜ σ = N0 ∑ aγ†k˜ , −nz, σ (e−λ1nzc − e−λ2nzc) c , (31) nz = 1
where γ = 1, 2 for the two orbitals and σ =↑, ↓ for the two spins. Note that the operators in the summation of equation (31) have the same meaning as the operators in the definition of the bulk model, while the operator on the left side of equation (31) is for the surface modes. z = 0 in equation (21) of the boundary condition is identified with nz = 0 in the discrete lattice representation and can be understood as the first vacant layer outside of the sample. The factor c is added to ensure the (approximate) normalization of the discretized version of the surface modes. Now, we can diagonalize h eff (k˜ ) and get the full perturbative dispersion of the topological surface states as
a (k˜ ) − ia2(k˜ ) g(k˜ ) = 1 . (36) a12(k˜ ) + a22(k˜ ) g(k˜ ) can be different from the above form by an arbitrary real and even function of k˜ to keep the time-reversal-invariance of the pairing. Since for small wave vectors we have cx (k˜ ) ≃ kx and cy(k˜ ) ≃ k y, we see that the time-reversal-invariant pairing realizable within the topological surface states has the form of a spinless k y + ikx pairing [8, 36]. Recalling equations (34), (30) and (31) by which the f operators can be expressed in terms of the spin–orbital basis for the bulk topological insulator, we can expand the above time-reversal-invariant pairing in terms of this basis. The independent pairing terms appearing in this expansion are naturally considered as possible bulk superconducting
E± = a 0 ± a12 + a 22 + a32 . (32)
The corresponding eigenvectors in the basis of φ k˜† can be taken as ⎛ ⎞ 2 2 2 ⎜± a1 + a 2 + a3 + a3 ⎟, χ = N ± (33) ± ⎜ ⎟ a1 + ia2 ⎝ ⎠ 4
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
5. Proximity-induced pairing channels in the topological surface states
phases of the topological insulator. This expansion thus provides a complete list of all the bulk pairings that can open a gap within the topological surface states. This analysis had been used in previous studies of superconducting topological insulators where simplified model for the electronic structures were used [8, 66]. For the simplified model, very few bulk pairings can open a gap within the surface states, including [66]
The foregoing section studies the intrinsic bulk pairing of the topological insulator that can induce superconductivity in the surface states. Since the bulk superconducting transition temperature is only about 4 K, there have been increasing interest in realizing superconductivity in the topological surface states through proximity effect, where a sample with much higher superconducting transition temperature can be used. There have been plenty of theoretical and experimental studies in this direction [36–55]. Two experiments reported hightemperature superconductivity induced in Bi2Se 3 and Bi2Te 3 via proximity to Bi2Sr2CaCu2O8 + δ [53, 54]. The theoretical studies mostly start from some simplified model for the bulk topological insulator and the surface states [36–50]. In light of the studies of the last section, it is interesting to see how some important conclusions obtained in previous studies, such as the number and symmetry of the proximity-induced pairing channels, might be changed qualitatively. In the present study, we focus on the proximity-induced superconductivity within the topological surface states and disregard the proximity effect relevant to the bulk electronic states of the topological insulator. A perfect interface between the topological insulator and a two-dimensional superconductor is assumed. Defining ψ k˜† = [c k˜†↑, c k˜†↓] as the basis state vector for the single-band superconductor, its Hamiltonian is
Δ1(k) = ϕk†[cyσ0 ⊗ s3 + icxσ0 ⊗ s0](ϕ−†k )T , (37) Δ2(k) = ϕk†[cyσ1 ⊗ s3 + icxσ1 ⊗ s0](ϕ−†k )T , (38) Δ3(k) = ϕk†iσ0 ⊗ s2(ϕ−†k )T , (39) Δ4(k) = ϕk†iσ1 ⊗ s2(ϕ−†k )T . (40)
We have removed the tilde over k˜ to indicate the bulk pairing states. The nonzero C1, which means the presence of the particle–hole asymmetry term in equation (1), adds the following two bulk pairings to the list of being able to gap the topological surface states
Δ5(k) = ϕk†[cyσ3 ⊗ s3 + icxσ3 ⊗ s0](ϕ−†k )T , (41) Δ6(k) = ϕk†iσ3 ⊗ s2(ϕ−†k )T . (42)
The presence of the hexagonal warping term (R1 ≠ 0) add the following two pairings to the list of being able to gap the topological surface states
1 ∑ [ψk˜†Δ(k˜ )(ψ−†k˜ )T + H.c.], 2 k˜ k˜ (46) where ϵ k˜ is the energy dispersion, H.c. means taking the Hermite conjugation. Without losing generality, we suppose the superconductor is directly coupled only to the top quintuple layer of the topological insulator via HSC = ∑ ψ k˜†[ϵ k˜ − μ]s0ψ k˜ +
Δ7(k) = ϕk†d1(k)σ0 ⊗ s1(ϕ−†k )T , (43) Δ8(k) = ϕk†d1(k)σ1 ⊗ s1(ϕ−†k )T . (44)
Finally, the simultaneous presence of the particle–hole asymmetry term and the hexagonal warping term in equation (1) add to the above list an additional pairing
Ht = ∑ [t1c k˜†σ a1k˜ , −1σ + t2c k˜†σ a2k˜ , −1σ + H.c.]. (47)
Δ9(k) = ϕk†d1(k)σ3 ⊗ s1(ϕ−†k )T . (45)
k˜ σ
Note that the a operators in the above equation are defined the same as those on the right hand side of equation (31), and thus the ‘−1’ label indicates the degrees of freedom in the surface layer of the topological insulator adjacent to the superconductor. Recalling equations (30) and (31), we can get the coupling between the external superconductor and the topological surface states as
It is clear that for realistic model of the electronic structures incorporating all essential ingredients, the number of bulk pairing channels that can open a gap within the topological surface states is considerably increased. Since the gapping or not of the topological surface states changes the spectral property of the materials which is directly measurable by angle resolved photoemission spectroscopy or various tunneling spectroscopies [2–4, 6], it is essential in these cases to consider the full model to get experimentally relevant predictions. For example, Δ _ 6(k) was identified as a possible pairing symmetry for CuxBi2Se3 [5, 7]. While Δ _ 6(k) was found to keep the topological surface states intact for simplified model [8, 10, 66], it would according to the above conclusion gap the topological surface states when a more realistic model containing the particle–hole asymmetry term is considered. This conclusion has been verified by explicit numerical calculations of the surface spectral functions, the results of which will be reported in a later work.
HC = ∑ [tc k˜†σγ d γ k˜ + H.c.], (48) ˜ kγ
where γ = 1, 2, σ1 = ↑ and σ2 = ↓. This coupling Hamiltonian is the same as that used in previous works [40, 44]. The tunneling matrix element t is explicitly written as N0 t= (t1 −D− − t2 D+ )(e−λ1c − e−λ2c) c . (49) 2M1
For t = 0, there is no superconductivity in the topological surface states induced via proximity. For nonzero t, 5
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
the proximity-induced pairing in the topological surface states manifest through the nonzero correlation functions ⟨φ−k˜ φ k˜T⟩. These correlation functions can be calculated through the imaginary time anomalous Matsubara Greeen’s function
b″ = (ωn2 + ξ k˜2 + ∣dz∣2 )(ωn2 + a 02 − a12 − a 22 + a32 ) (58) + 2∣t∣2 (ωn2 − a 0ξ k˜ ) + ∣t∣4 , c″ = 2ia3[a 0(ωn2 + ξ k˜2 + ∣dz∣2 ) − ∣t∣2 ξ k˜ ], (59)
1 Fk˜ (τ ) = ⟨Tτ φ−k˜ (τ )φ k˜T(0)⟩ = e−iωmτ Fk˜ (iωm ), (50) β m
∑
d″ = 2a2ωn(ωn2 + ξ k˜2 + ∣dz∣2 + ∣t∣2 ) − 2a1a3(ωn2 + ξ k˜2 + ∣dz∣2 ), (60) Again, the four matrix elements of Fk˜ (iωm ) are in general all nonzero. Different from the spin-singlet case, the matrix elements of Fk˜ (iωm ) now have both even-frequency components and odd-frequency components, also consistent with the previous results of Black-Schaffer and Balatsky [44]. The particle–hole asymmetry term of equation (1) is again found not changing any qualitative results. The hexagonal warping term, on the other hand, is found to induce spin-singlet pairing correlation which would be absent if R1 = 0 such that c′′ = 0 [44]. As a conclusion to the present section, the particle–hole asymmetry term of equation (1) does not induce qualitative changes to previous results based on simplified models for the electronic structure of the topological insulators. The hexagonal warping term of equation (1), however, increases the number of the proximity-induced pairing channels by one. This is easy to understand because the proximity-induced pairing symmetry in the topological surface states other than the original symmetry of the external superconductor arises as a result of the spin–orbital coupling in the topological surface states, which can be seen from equations (51)–(60) by setting a1 = a2 = a3 = 0. Recalling equations (27)–(29), which shows the dependencies of ai (i = 0,1,2,3) on the parameters of equation (1), it is not the particle–hole asymmetry term but the hexagonal warping term that changes the character of the spin–orbital coupling of the topological surface states. In the above analysis, we have only considered the possible channels of proximity-induced pairing in the topological surface states. The amplitudes of the various proximity-induced pairing channels, which can be studied by integrating out the degrees of freedom of the external superconductor and work with the resulting effective low-energy model [40], is left to future works.
where β = 1/(kBT ) with kB the Boltzmann constant and T the temperature, ωm = (2m + 1)πkBT (m runs over all integers) are the fermionic Matsubara frequencies. Since the full Hamiltonian is of mean-field type with only bilinear terms, the anomalous Green’s functions can be calculated in terms of standard equation of motion method for an arbitrary pairing term Δ(k˜ ). For a general spin-singlet pairing Δ(k˜ ) = iΔ0(k˜ )s2 with Δ0(−k˜ ) = Δ0(k˜ ), Fk˜ (iωm ) is found to be
⎛ a′ − d′ −b′ + ic′⎞ i(t *)2Δ0(k) (51) ⎜ ⎟ F k(iWm ) = 2 a′ − b′ 2 − c′ 2 − d′ 2 ⎝ −b′ − ic′ a′ + d′ ⎠
where we have defined a′ = 2a2[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (52) b′ = 2ia3[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (53) c′ = −(ωn2 + ξ k˜2 + Δ20)(ωn2 + a 02 + a12 + a 22 + a32 ) (54) +2(a 0ξ k˜ − ωn2 )∣t∣2 −∣t∣4 , d′ = −2ia1[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (55)
in which ξ k˜ = ϵ k˜ − μ, t * means the complex conjugation of t, and ∣t∣2 = tt *. The four matrix elements of Fk˜ (iωm ) are generally nonzero. It is also clear that Fk˜ (iωm ) is an even function of frequency, consistent with the conclusion of a previous study by Black-Schaffer and Balatsky [44]. The presence of the particle–hole asymmetry term of equation (1) (i.e. a 0 ≠ 0) is found to bring about no qualitative changes. The hexagonal warping term, on the other hand, is found to induce unequalspin spin-triplet pairing correlations which would be absent if R1 = 0 such that b′ = 0 [44]. This last point is the only qualitative difference from previous works using simplified models for the electronic structures. We would also like to consider a type of spin-triplet pairings in the form of Δ(k˜ ) = (dz(k˜ )s3)is2. A special case defined by dz(k˜ ) = Δ0(kx ± ik y ) (Δ0 is a constant) breaks the time reversal symmetry and is considered as the most probable pairing of Sr2RuO4 [34, 35]. For this group of spin-triplet pairings, the anomalous Matsubara Green’s function is found to be
6. Summary We have studied the superconducting phases that can form in the topological surface states of a three-dimensional topological insulator, both in the case of bulk superconducting states of the topological insulator and for proximity-induced pairings. The particle–hole asymmetry term and the hexagonal warping term bring about some qualitative changes. In the case of bulk superconducting phase of the topological insulator itself, the number of pairings that can open a gap in the topological surface states is considerably increased by both the particle–hole asymmetry term and the hexagonal warping term. For the proximity-induced pairing, it is the hexagonal warping term that increases the number of pairing channels that form in the topological surface states by one. The present study illustrates the necessity of incorporating the particlehole asymmetry term and the hexagonal warping term in the
⎛ a″ − d″ −b″ + ic″⎞ (t *)2dz(k˜ ) (56) F ⎜ ⎟, k˜ (iωm ) = a″ 2 − b″ 2 − c″ 2 − d″ 2 ⎝−b″ − ic″ a″ + d″ ⎠
where we have defined a″ = −2ia1ωn(ωn2 + ξ k˜2 + ∣dz∣2 + ∣t∣2 ) (57) −2ia2a3(ωn2 + ξ k˜2 + ∣dz∣2 ) + 2a1ξ k˜ ∣t∣2 , 6
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
[29] Tanaka Y, Sato M and Nagaosa N 2012 J. Phys. Soc. Japan 81 011013 [30] Nakosai S, Tanaka Y and Nagaosa N 2012 Phys. Rev. Lett. 108 147003 [31] Nagai Y, Nakamura H and Machida M 2012 Phys. Rev. B 86 094507 [32] Chen L and Wan S 2013 J. Phys.: Condens. Matter. 25 215702 [33] Qi X-L and Zhang S-C 2011 Rev. Mod. Phys. 83 1057 [34] Luke G M et al 1998 Nature 394 558 [35] Xia J, Maeno Y, Beyersdorf P T, Fejer M M and Kapitulnik A 2006 Phys. Rev. Lett. 97 167002 [36] Fu L and Kane C L 2008 Phys. Rev. Lett. 100 096407 [37] Tanaka Y, Yokoyama T and Nagaosa N 2009 Phys. Rev. Lett. 103 107002 [38] Sau J D, Lutchyn R M, Tewari S and Das Sarma S 2010 Phys. Rev. Lett. 104 040502 [39] Alicea J 2010 Phys. Rev. B 81 125318 [40] Stanescu T D, Sau J D, Lutchyn R M and Das Sarma S 2010 Phys. Rev. B 81 241310 [41] Linder J, Tanaka Y, Yokoyama T, Sudbø A and Nagaosa N 2010 Phys. Rev. Lett. 104 067001 Linder J, Tanaka Y, Yokoyama T, Sudbø A and Nagaosa N 2010 Phys. Rev. B 81 184525 [42] Lababidi M and Zhao E 2011 Phys. Rev. B 83 184511 [43] Black-Schaffer A M 2011 Phys. Rev. B 83 060504 [44] Black-Schaffer A M and Balatsky A V 2012 Phys. Rev. B 86 144506 Black-Schaffer A M and Balatsky A V 2013 Phys. Rev. B 87 220506 [45] Zhang F, Kane C L and Mele E J 2013 Phys. Rev. Lett. 111 056402 [46] Chiu C-K, Gilbert M J and Hughes T L 2011 Phys. Rev. B 84 144507 [47] Lucignano P, Mezzacapo A, Tafuri F and Tagliacozzo A 2012 Phys. Rev. B 86 144513 [48] Alicea J 2012 Rep. Prog. Phys. 75 076501 [49] Lee K, Vaezi A, Fischer M H and Kim E-A 2014 Phys. Rev. B 90 214510 [50] Triola C, Rossi E and Balatsky A V 2014 Phys. Rev. B 89 165309 [51] Yang F et al 2012 Phys. Rev. B 85 104508 [52] Wang M-X et al 2012 Science 336 52 [53] Zareapour P et al 2012 Nat. Commun. 3 1056 [54] Wang E et al 2013 Nat. Phys. 9 621 [55] Xu J-P et al 2014 Phys. Rev. Lett. 112 217001 [56] Mizushima T, Yamakage A, Sato M and Tanaka Y 2014 Phys. Rev. B 90 184516 [57] Hsieh D 2009 Nature 460 1101 [58] Chen Y L 2009 Science 325 178 [59] Fu L 2014 Phys. Rev. B 90 100509 [60] Cheng M, Sun K, Galitski V and Das Sarma S 2010 Phys. Rev. B 81 024504 Santos L, Neupert T, Chamon C and Mudry C 2010 Phys. Rev. B 81 184502 Lu C-K and Herbut I F 2010 Phys. Rev. B 82 144505 [61] Zhang H, Liu C-X, Qi X-L, Dai X, Fang Z and Zhang S-C 2009 Nat. Phys. 5 438 [62] Liu C-X, Qi X-L, Zhang H, Dai X, Fang Z and Zhang S-C 2010 Phys. Rev. B 82 045122 [63] Fu L 2009 Phys. Rev. Lett. 103 266801 [64] Wang Q-H, Wang D and Zhang F-C 2010 Phys. Rev. B 81 035104 [65] Shan W-Y, Lu H-Z and Shen S-Q 2010 New J. Phys. 12 043048 Zhou B, Lu H-Z, Chu R-L, Shen S-Q and Niu Q 2008 Phys. Rev. Lett. 101 246807 [66] Hao L and Lee T-K 2015 J. Phys.: Condens. Matter. 27 105701 Hao L and Lee T-K 2015 arXiv:1407.3329v2
electronic structures of three-dimensional topological insulators like Bi2Se 3 and Bi2Te 3 to get theoretical predictions comparable to experiments related to superconductivity in the topological surface states. Acknowledgments We thank T-K Lee for enlightening discussions. LH is supported by NSFC.11204035 and SRFDP.20120092120040. JW is supported by NSFC.11274059, NSFC.11274260, and NSF of Jiangsu Province BK20131284. References [1] Hor Y S, Williams A J, Checkelsky J G, Roushan P, Seo J, Xu Q, Zandbergen H W, Yazdani A, Ong N P and Cava R J 2010 Phys. Rev. Lett. 104 057001 [2] Andrew Wray L, Xu S-Y, Xia Y, Hor Y S, Qian D, Fedorov A V, Lin H, Bansil A, Cava R J and Zahid Hasan M 2010 Nat. Phys. 6 855 [3] Levy N, Zhang T, Ha J, Sharifi F, Talin A A, Kuk Y and Stroscio J A 2013 Phys. Rev. Lett. 110 117001 [4] Peng H, De D, Lv B, Wei F and Chu C-W 2013 Phys. Rev. B 88 024515 [5] Fu L and Berg E 2010 Phys. Rev. Lett. 105 097001 [6] Sasaki S, Kriener M, Segawa K, Yada K, Tanaka Y, Sato M and Ando Y 2011 Phys. Rev. Lett. 107 217001 [7] Hao L, Wang G-L, Lee T-K, Wang J, Tsai W-F and Yang Y-H 2014 Phys. Rev. B 89 214505 [8] Hao L and Lee T K 2011 Phys. Rev. B 83 134516 [9] Hsieh T H and Fu L 2012 Phys. Rev. Lett. 108 107005 [10] Yamakage A, Yada K, Sato M and Tanaka Y 2012 Phys. Rev. B 85 180509 [11] Zhang J L 2011 Proc. Natl Acad. Sci. USA 108 24 [12] Zhang C, Sun L, Chen Z, Zhou X, Wu Q, Yi W, Guo J, Dong X and Zhao Z 2011 Phys. Rev. B 83 140504 [13] Sasaki S, Ren Z, Taskin A A, Segawa K, Fu L and Ando Y 2012 Phys. Rev. Lett. 109 217004 [14] Novak M, Sasaki S, Kriener M, Segawa K and Ando Y 2013 Phys. Rev. B 88 140502 [15] Arpino K E et al 2014 Phys. Rev. Lett. 112 017002 [16] Koren G, Kirzhner T, Lahoud E, Chashka K B and Kanigel A 2011 Phys. Rev. B 84 224521 [17] Kirzhner T, Lahoud E, Chaska K B, Salman Z and Kanigel A 2012 Phys. Rev. B 86 064517 [18] Kriener M, Segawa K, Ren Z, Sasaki S and Ando Y 2011 Phys. Rev. Lett. 106 127004 [19] Das P, Suzuki Y, Tachiki M and Kadowaki K 2011 Phys. Rev. B 83 220513 [20] Bay T V, Naka T, Huang Y K, Luigjes H, Golden M S and de Visser A 2012 Phys. Rev. Lett. 108 057001 [21] Schnyder A P, Ryu S, Furusaki A and Ludwig A W W 2008 Phys. Rev. B 78 195125 [22] Kitaev A 2009 AIP Conf. Proc. 1134 22 [23] Qi X L, Hughes T L and Zhang S C 2010 Phys. Rev. B 81 134508 [24] Qi X L, Hughes T L, Raghu S and Zhang S C 2009 Phys. Rev. Lett. 102 187001 [25] Roy R arXiv: 0803 .2868 [26] Sato M 2009 Phys. Rev. B 79 214526 [27] Sato M 2010 Phys. Rev. B 81 220504 [28] Sato M, Takahashi Y and Fujimoto S 2009 Phys. Rev. Lett. 103 020401 Sato M, Takahashi Y and Fujimoto S 2010 Phys. Rev. B 82 134521 7
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Superconductivity in the surface states of a Bi2X3 topological insulator: effects of a realistic model
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 255701 (7pp)
doi:10.1088/0953-8984/27/25/255701
Superconductivity in the surface states of a Bi2X3 topological insulator: effects of a realistic model Lei Hao and Jun Wang Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China E-mail: [email protected] Received 9 March 2015, revised 27 April 2015 Accepted for publication 14 May 2015 Published 9 June 2015 Abstract
Superconductivity in the topological surface states is essential to both the surface spectrum of bulk superconducting state and the proximity-induced superconductivity of Bi2X3 (X is Se or Te) topological insulators. While previous theories were mostly based on simplified models for the bulk topological insulator and the surface states, the accumulating experiments stimulate us to make an analysis using realistic model for the normal state electronic structures, incorporating terms responsible for particle–hole asymmetry and hexagonal warping. An effective low-energy model for the topological surface states is derived first. Then we identify all the bulk timereversal-invariant superconducting pairings in the topological insulator that can open a gap in the topological surface states. Many more pairings are found to be able to gap the topological surface states as compared to conclusions based on simplified models. The number of proximityinduced pairing channels in the topological surface states increases by one as a result of the hexagonal warping term, but is not changed by the particle–hole asymmetry term. Keywords: topological surface states, superconducting topological insulators, proximity effect (Some figures may appear in colour only in the online journal)
1. Introduction
insulator (or, a semiconductor with large spin–orbital coupling), has been considered by many as a more promising way [36–55]. For example, high-temperature superconductivity up to 80 Kelvins was reported in Bi2Se 3 and Bi2Te 3 via proximity to Bi2Sr2CaCu2O8 + δ [53, 54]. Among the various topics concerning the superconductivity in three-dimensional topological insulators, the issue of pairings that can form in the topological surface states is very important to both the bulk and proximity-induced superconductivities of the topological insulators. For clarity, we use ‘topological surface states’ to indicate the surface states arising from the topological character of topological insulator, and ‘surface Andreev bound states’ to denote the surface modes appearing in the superconducting phase as a result of the topological character of the superconducting pairing. For the most studied superconducting topological insulator, CuxBi2Se 3, the topological surface states coexist and are well separated from the bulk conduction band around the chemical potential [2]. In this case, it was shown theoretically that the absence of gap opening in the topological
Searching for possible topological superconductors has attracted a large amount of experimental and theoretical attention [1–32]. One reason is that these new topological matters have Majorana fermions, which are promising candidates for realizing the topological quantum computations, as elementary boundary modes or zero modes pertaining to vortices and impurities [33]. Bulk topological superconductors, though intensively searched for, are scarce. While Sr2RuO4 appears to be a two-dimensional time-reversal-broken topological superconductor [34, 35], the possibility of time-reversal-invariant topological superconductivity in three-dimensional superconducting topological insulators including CuxBi2Se 3, Bi2Se 3 and Bi2Te 3 are presently under hot debate [1–7]. Besides, these bulk superconductors all have the drawback of having very low superconducting critical temperatures. Hence, the alternative approach of achieving a topological superconductor, by the proximity coupling of a superconductor with a topological 0953-8984/15/255701+7$33.00
1
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L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
surface states is intimately related to the possible existence of surface Andreev bound states [8]. In addition, a recent selfconsistent calculation by Mizushima et al shows that the topological surface states can induce interesting parity mixing effect and the enhancement of certain pairing components [56]. For the proximity-induced superconductivity in a topological insulator, the superconductivity in the topological surface states is even more important since it is possible in this case to tune the chemical potential of the topological insulator to the bulk energy gap, where only the topological surface states exist in the normal phase [57, 58]. Altogether, it is very important to study the superconductivity in the topological surface states. Previous theories mostly adopt some simplified model for the (normal state) electronic structures of three-dimensional topological insulators like Bi2Se 3, which ignore the terms relevant to particle–hole asymmetry and (or) hexagonal warping. A recent work of Fu studying a special odd-parity pairing by taking into account of the effect of hexagonal warping has shown some qualitative changes [59]. This finding together with the accumulating relevant experimental results [51–55], have made it increasingly urgent to study the possible pairing correlations in the topological surface states systematically, starting from a realistic microscopic model for the electronic structures of relevant topological insulators. It is the aim of the present work to fill up this gap. The intrinsic superconducting instabilities of the topological surface states, not referring to the bulk topological insulator or an external superconductor coupled by proximity effect, will however not be considered in the following [60]. In the remaining parts of this work, we first define in section 2 the model that would be used to produce a reasonable electronic structure for three-dimensional topological insulators like Bi2Se 3 and Bi2Te 3. Then in section 3 we derive the low-energy effective model for the topological surface states. In section 4, all the time-reversal-invariant pairings for the superconducting topological insulators that can gap the topological surface states are constructed, the number of which is shown considerably increased by the presence of the particlehole asymmetry and hexagonal warping terms. In section 5, the pairing channels that can appear in the topological surface states when the topological insulator is coupled to an external (spin-singlet or spin-triplet) superconductor are studied. A short summary is presented in section 6.
hexagonal BZ corresponding to an equivalent hexagonal lattice with two orbitals per unit cell [6–8, 64]. Taking the basis vector as ϕk† = [a1†k ↑, a1†k ↓, a 2†k ↑, a 2†k ↓], in which the subindices 1 and 2 indicate the two orbitals, the tight binding Hamiltonian matrix for wave vector k is written as [7] H0(k) = ϵ(k)I4 + M (k)Γ5 + B0cz(k)Γ4 + A0 [cy(k)Γ1 (1) −cx (k)Γ2] + R1d1(k)Γ3 + R2d2(k)Γ4,
in which ϵ(k) = C0 + 2C1[1 − cos(k ⋅ δ4 )] (2) 4 + C2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)], 3 M (k) = M0 + 2M1[1 − cos(k ⋅ δ4 )] (3) 4 + M2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)], 3 1 cx (k) = [sin(k ⋅ δ1) − sin(k ⋅ δ2 )], (4) 3 1 cy(k) = [sin(k ⋅ δ1) + sin(k ⋅ δ2 ) − 2 sin(k ⋅ δ3)], (5) 3 cz(k) = sin(k ⋅ δ4 ), (6) 8 d1(k) = − [sin(k ⋅ a1) + sin(k ⋅ a2) + sin(k ⋅ a3)], (7) 3 3 d2(k) = −8[sin(k ⋅ δ1) + sin(k ⋅ δ2 ) + sin(k ⋅ δ3)]. (8)
Here, the four nearest neighboring bond vectors of the effective hexagonal lattice are δ1 =
(
δ2 = −
3 2
a,
), δ
1 a, 0 2
3
(
3 2
1
)
a, 2 a, 0 ,
= (0, −a, 0), and δ4 = (0, 0, c ). a and
c are lattice parameters in the xy plane and along the z direction, respectively. The three in-plane second nearest neighboring bond vectors in d1(k) are a1 = δ1 − δ2, a2 = δ2 − δ3, and a3 = δ3 − δ1. The first term of H0(k) is responsible for the particle–hole asymmetry of the electronic structure. The last two terms of H0(k) induce hexagonal warping of the bulk Fermi surface and the topological surface states [62, 63]. Previous theoretical studies usually neglect the first and the last two terms of H0(k) [5, 6, 8–10, 61, 64], which will however be the focus of the present work. As was shown explicitly in a former work, the above model respects all the symmetry requirements of the D35d double group for Bi2X3 [7, 62]. The parameters contained in equation (1) are assumed to take values realistic for actual materials [7, 62]. While equation (1) is equivalent to the model introduced by [7], we would take in this work the parity basis for the two orbitals instead of the local basis adopted by [7]. The five Γ matrices as thus written as Γ1 = σ1 ⊗ s1, Γ2 = σ1 ⊗ s2, Γ3 = σ1 ⊗ s3, Γ4 = σ2 ⊗ s0, and Γ5 = σ3 ⊗ s0 [7, 62]. In this basis, the parity operator is written as P = σ3 ⊗ s0 [8]. In this work, σα (α = 1, 2, 3) and sα (α = 1, 2, 3) are defined as Pauli matrices
2. Model We consider superconducting pairings in the topological surface states of three dimensional topological insulators, including Bi2Se 3, Bi2Te 3, and Sb 2Te 3 [61]. We would concentrate on Bi2X3 (X is Se or Te) to analyze the related superconductors. These materials belong to the D35d space group and consist of Bi2X3 quintuple layers stacked along the outof-plane direction. They are very well described by an effective model defined close to the Brillouin zone (BZ) center (the Γ point) [61–63]. In this work we use a tight binding extension of the model. Instead of working in the original BZ for a D35d space group [61, 62], the k ⋅ p model is extended to a 2
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
in the orbital and spin subspaces, while σ0 and s0 are unit matrices in the orbital and spin subspaces. In studying proximity-induced pairings in the topological surface states, we consider an ideal interface between a superconductor and the top surface of the topological insulator. The possible pairing symmetries of the external superconductor that would be studied include the spin-singlet pairings (s-wave or d-wave), and a type of spin-triplet pairings with the p + ip′-wave state proposed for Sr2RuO4 as a special case.
For each λαβ, we can get two independent solutions for η as ηαβ1(E ) = [βB0λα, 0, L + − D+λα2 − E , 0]T , (18)
and ηαβ 2(E ) = [0, L− − D−λα2 − E , 0, −βB0λα ]T . (19)
The general solution for the surface modes for kx = k y = 0 is a linear combination of the above modes Ψ(z, E ) = ∑ cαβγ ηαβγ e βλαz. (20) αβγ
3. Topological surface states
Since the proper boundary condition for a sample occupying z < 0 is [8, 61, 62, 65]
Starting from the model defined by equation (1) for bulk topological insulators, we first derive the topological surface states for the upper surface of a sample occupying the lower half space z < 0. Expanding the kz-dependent terms of H0(k) in equation (1) to the leading order of kz, we have
Ψ(z = 0, E ) = Ψ(z = −∞, E ) = 0, (21)
we should retain only the four components of cαβγ with β = +. From Ψ(z = 0, E ) = 0, and requiring the existence of nontrivial solutions for Ψ(z, E ), we have the following constraint for the energy E
ϵ(k) ≃ C0 + C1kz2 + C2xy, (9)
det∣η1, +,1(E ), η1, +,2(E ), η2, +,1(E ), η2, +,2(E )∣ = 0. (22)
M (k) ≃ M0 + M1kz2 + M2xy, (10)
From this equation, we get the twofold degenerate eigenvalue for the surface modes as
B0cz(k) ≃ B0kz, (11)
C E = C0 − 1 M0. (23) M1
4
where C2xy = 3 C2[3 − cos(k ⋅ δ1) − cos(k ⋅ δ2 ) − cos(k ⋅ δ3)] and M2xy = C2xy[C2 → M2]. The simultaneous presence of the particle–hole asymmetry and the hexagonal warping terms make the direct analytical calculation of the surface modes for a general surface momentum (kx, k y ) difficult. Instead, we make a perturbative calculation by first solving the surface modes for kx = k y = 0 and then taking the kx and k y dependent terms as perturbations. Since the Fermi momentum in the (kx, k y ) plane is small [57, 58], this should be a reasonable approximation. The eigenequation for the surface modes of kx = k y = 0 are written as
The above result is the same as the previous result of Shan et al for a slightly different yet unitarily equivalent model [62, 65]. Putting the above solution for E back to the boundary condition of equation (21), we get two degenerate solutions for {cα, +, γ } which give the two surface modes for kx = k y = 0 as Ψγ (z, E ) = N0η˜1, +, γ (e λ1z − e λ2z), (24)
where γ = 1, 2 labels the two modes. The number N0 is the normalization factor for the surface mode. η˜1, +, γ is defined as the original η1, +, γ in equations (18) and (19) multiplied by a positive normalization factor. Substituting equation (23) to equations (18) and (19), η˜1, +, γ can be simplified to
H0(kx = k y = 0, kz → −i∂z)Ψ(z, E ) = E Ψ(z, E ). (12)
The ansatz for the surface modes are taken as Ψ(z, E ) = ηe λz, (13)
1 η˜1, +,1 = [ −D− , 0, − D+ , 0]T , (25) 2M1
where η is a four component column vector. Substituting the ansatz to equation (12), the nontrivial solution satisfies
1 η˜1, +,2 = [0, −D− , 0, − D+ ]T . (26) 2M1
det∣H0(kx = k y = 0, −iλ ) − EI4∣ = 0, (14)
which gives
In obtaining the above simplification, we have assumed M1 > 0, D+ > 0 and D− < 0, which are consistent with the parameter sets for Bi2Se 3 and Bi2Te 3 derived from first- principles calculations [7, 62]. Ψγ (z, E ) (γ = 1, 2) are the exact eigen-basis for kx = k y = 0. To get the approximate eigenstates for nonzero k˜ , we begin by taking Ψγ (z, E ) (γ = 1, 2) as the starting basis, then take the terms in H0(k) that depend on kx or k y as perturbation, finally we get the approximate low-energy effective model of the surface states close to kx = k y = 0 as [8, 62, 65]
D+(L− − E ) + D−(L + − E ) − B02 + (−1)α − 1 F λα2 = , (15) 2D+D−
where α = 1, 2, D± = C1 ± M1, L ± = C0 ± M0, and F = [D+(E − L−) + D−(E − L +) + B02]2 (16) −4D+D−(E − L +)(E − L−).
Defining λα as the solutions with positive real parts, we get the four independent solutions for λ as
3
λαβ = βλα, (17)
h eff (k˜ ) = ∑ aα(k˜ )θα, (27)
where β = ±, α = 1, 2.
α=0
3
L Hao and J Wang
J. Phys.: Condens. Matter 27 (2015) 255701
where k˜ = (kx, k y ) and the four coefficients are
where N± = 1/ 2 a12 + a22 + a32 ( a12 + a22 + a32 ± a3) is the normalization factor. The above two eigenvectors can be expressed in second quantized form in terms of equation (30) as
C C a 0 = C0 − 1 M0 + C2xy − 1 M2xy, (28) M1 M1
⎞ Nα ⎡ ⎛ ⎢c1α⎜ −D− a1†k˜ ↑ − D+ a 2†k˜ ↑⎟ 2M1 ⎣ ⎝ ⎠ (34) +c2α −D− a1†k˜ ↓ − D+ a 2†k˜ ↓ ⎤⎦,
M12 − C12 (29) [a1, a2, a3] = [−A0 cy, A0 cx, −R1d1]. M1 The dependencies on k˜ are made implicit in equations (28) and (29). Since equation (27) is the effective model for a general nonzero wave vector k˜ , its basis can be denoted as [d1†k˜ , d 2†k˜ ] ≡ φ k˜†. θα (α = 0, …, 3) in equation (27) are Pauli matrices acting in the subspace subtended by φ k˜†. Note that, though φ k˜† is formally the basis for nonzero k˜ , its dependence on the z coordinate inherits directly from Ψγ (z, E ) (γ = 1, 2) and thus does not vary with k˜ . This character, resulting from the perturbative nature of our approach, should be acceptable since the Fermi momentum in the (kx, k y ) plane is usually very small so that the relevant wave vectors are close to zero [57, 58]. According to the above explanations and recalling equations (24)–(26), we can express the basis for equation (27) as
fα†k˜ =
(
)
where we have defined c1α = α a12 + a22 + a32 + a3 and c2α = a1 + ia2, α = ±. To compare the above results with those for the simplified models, we can simply set C0 = C1 = C2 = 0 and (or) R1 = R2 = 0. For example, the eigenvalue of equation (23) would be E = 0 in the absence of the particle–hole asymmetry term, which is consistent with previous results for simplified models [8, 64]. 4. Bulk time-reversal-invariant pairings that gap the topological surface states
⎤ 1 ⎡ ⎢ −D− a1†k˜ σ − D+ a 2†k˜ σ ⎥, dγ†k˜ = (30) γ γ 2M1 ⎣ ⎦
Suppose the chemical potential crosses the Eα(k˜ ) (α = + or −) branch of the topological surface states and the topological surface states coexist but are well separated from the bulk band states. This assumption is consistent with the experimental observation of CuxBi2Se 3 [2]. Consider the possible time-reversal-invariant pairings that can form within the topological surface states. Since the conduction and valence surface bands defined by equations (32)–(34) are both nondegenerate, the pairing can only be of the form
where γ = 1, 2, σ1 =↑ and σ2 =↓. The four creation operators appearing on the right hand side of equation (30) are understood as linear combinations of the corresponding operators defined on the various quintuple layers of the sample. The amplitudes on the various layers follow an exponentially decaying behavior according to equation (24). Suppose the N0 factor defined in equation (24) has been determined by the normalization condition. Turning back from the continuous z dependence of equation (24) to the original discrete lattice representation, we can write approximately in the limit of very small c (the lattice parameter along the z direction) as
Δα (k˜ ) = g(k˜ )fα†k˜ fα†, −k˜ , (35)
where g(k˜ ) is an undetermined factor to ensure the timereversal-invariance of the pairing. By introducing the Nambu basis, this pairing term would be seen as the off-diagonal element in the corresponding Bogoliubov–de Gennes Hamiltonian [7, 8]. Recalling the transformation property of the timereversal operator T , that is T iT −1 = −i, Tc k ↑T −1 = c−k ↓ and Tc k ↓T −1 = −c−k ↑, a simple choice for g(k˜ ) is found to be
+∞
aγ†k˜ σ = N0 ∑ aγ†k˜ , −nz, σ (e−λ1nzc − e−λ2nzc) c , (31) nz = 1
where γ = 1, 2 for the two orbitals and σ =↑, ↓ for the two spins. Note that the operators in the summation of equation (31) have the same meaning as the operators in the definition of the bulk model, while the operator on the left side of equation (31) is for the surface modes. z = 0 in equation (21) of the boundary condition is identified with nz = 0 in the discrete lattice representation and can be understood as the first vacant layer outside of the sample. The factor c is added to ensure the (approximate) normalization of the discretized version of the surface modes. Now, we can diagonalize h eff (k˜ ) and get the full perturbative dispersion of the topological surface states as
a (k˜ ) − ia2(k˜ ) g(k˜ ) = 1 . (36) a12(k˜ ) + a22(k˜ ) g(k˜ ) can be different from the above form by an arbitrary real and even function of k˜ to keep the time-reversal-invariance of the pairing. Since for small wave vectors we have cx (k˜ ) ≃ kx and cy(k˜ ) ≃ k y, we see that the time-reversal-invariant pairing realizable within the topological surface states has the form of a spinless k y + ikx pairing [8, 36]. Recalling equations (34), (30) and (31) by which the f operators can be expressed in terms of the spin–orbital basis for the bulk topological insulator, we can expand the above time-reversal-invariant pairing in terms of this basis. The independent pairing terms appearing in this expansion are naturally considered as possible bulk superconducting
E± = a 0 ± a12 + a 22 + a32 . (32)
The corresponding eigenvectors in the basis of φ k˜† can be taken as ⎛ ⎞ 2 2 2 ⎜± a1 + a 2 + a3 + a3 ⎟, χ = N ± (33) ± ⎜ ⎟ a1 + ia2 ⎝ ⎠ 4
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5. Proximity-induced pairing channels in the topological surface states
phases of the topological insulator. This expansion thus provides a complete list of all the bulk pairings that can open a gap within the topological surface states. This analysis had been used in previous studies of superconducting topological insulators where simplified model for the electronic structures were used [8, 66]. For the simplified model, very few bulk pairings can open a gap within the surface states, including [66]
The foregoing section studies the intrinsic bulk pairing of the topological insulator that can induce superconductivity in the surface states. Since the bulk superconducting transition temperature is only about 4 K, there have been increasing interest in realizing superconductivity in the topological surface states through proximity effect, where a sample with much higher superconducting transition temperature can be used. There have been plenty of theoretical and experimental studies in this direction [36–55]. Two experiments reported hightemperature superconductivity induced in Bi2Se 3 and Bi2Te 3 via proximity to Bi2Sr2CaCu2O8 + δ [53, 54]. The theoretical studies mostly start from some simplified model for the bulk topological insulator and the surface states [36–50]. In light of the studies of the last section, it is interesting to see how some important conclusions obtained in previous studies, such as the number and symmetry of the proximity-induced pairing channels, might be changed qualitatively. In the present study, we focus on the proximity-induced superconductivity within the topological surface states and disregard the proximity effect relevant to the bulk electronic states of the topological insulator. A perfect interface between the topological insulator and a two-dimensional superconductor is assumed. Defining ψ k˜† = [c k˜†↑, c k˜†↓] as the basis state vector for the single-band superconductor, its Hamiltonian is
Δ1(k) = ϕk†[cyσ0 ⊗ s3 + icxσ0 ⊗ s0](ϕ−†k )T , (37) Δ2(k) = ϕk†[cyσ1 ⊗ s3 + icxσ1 ⊗ s0](ϕ−†k )T , (38) Δ3(k) = ϕk†iσ0 ⊗ s2(ϕ−†k )T , (39) Δ4(k) = ϕk†iσ1 ⊗ s2(ϕ−†k )T . (40)
We have removed the tilde over k˜ to indicate the bulk pairing states. The nonzero C1, which means the presence of the particle–hole asymmetry term in equation (1), adds the following two bulk pairings to the list of being able to gap the topological surface states
Δ5(k) = ϕk†[cyσ3 ⊗ s3 + icxσ3 ⊗ s0](ϕ−†k )T , (41) Δ6(k) = ϕk†iσ3 ⊗ s2(ϕ−†k )T . (42)
The presence of the hexagonal warping term (R1 ≠ 0) add the following two pairings to the list of being able to gap the topological surface states
1 ∑ [ψk˜†Δ(k˜ )(ψ−†k˜ )T + H.c.], 2 k˜ k˜ (46) where ϵ k˜ is the energy dispersion, H.c. means taking the Hermite conjugation. Without losing generality, we suppose the superconductor is directly coupled only to the top quintuple layer of the topological insulator via HSC = ∑ ψ k˜†[ϵ k˜ − μ]s0ψ k˜ +
Δ7(k) = ϕk†d1(k)σ0 ⊗ s1(ϕ−†k )T , (43) Δ8(k) = ϕk†d1(k)σ1 ⊗ s1(ϕ−†k )T . (44)
Finally, the simultaneous presence of the particle–hole asymmetry term and the hexagonal warping term in equation (1) add to the above list an additional pairing
Ht = ∑ [t1c k˜†σ a1k˜ , −1σ + t2c k˜†σ a2k˜ , −1σ + H.c.]. (47)
Δ9(k) = ϕk†d1(k)σ3 ⊗ s1(ϕ−†k )T . (45)
k˜ σ
Note that the a operators in the above equation are defined the same as those on the right hand side of equation (31), and thus the ‘−1’ label indicates the degrees of freedom in the surface layer of the topological insulator adjacent to the superconductor. Recalling equations (30) and (31), we can get the coupling between the external superconductor and the topological surface states as
It is clear that for realistic model of the electronic structures incorporating all essential ingredients, the number of bulk pairing channels that can open a gap within the topological surface states is considerably increased. Since the gapping or not of the topological surface states changes the spectral property of the materials which is directly measurable by angle resolved photoemission spectroscopy or various tunneling spectroscopies [2–4, 6], it is essential in these cases to consider the full model to get experimentally relevant predictions. For example, Δ _ 6(k) was identified as a possible pairing symmetry for CuxBi2Se3 [5, 7]. While Δ _ 6(k) was found to keep the topological surface states intact for simplified model [8, 10, 66], it would according to the above conclusion gap the topological surface states when a more realistic model containing the particle–hole asymmetry term is considered. This conclusion has been verified by explicit numerical calculations of the surface spectral functions, the results of which will be reported in a later work.
HC = ∑ [tc k˜†σγ d γ k˜ + H.c.], (48) ˜ kγ
where γ = 1, 2, σ1 = ↑ and σ2 = ↓. This coupling Hamiltonian is the same as that used in previous works [40, 44]. The tunneling matrix element t is explicitly written as N0 t= (t1 −D− − t2 D+ )(e−λ1c − e−λ2c) c . (49) 2M1
For t = 0, there is no superconductivity in the topological surface states induced via proximity. For nonzero t, 5
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the proximity-induced pairing in the topological surface states manifest through the nonzero correlation functions ⟨φ−k˜ φ k˜T⟩. These correlation functions can be calculated through the imaginary time anomalous Matsubara Greeen’s function
b″ = (ωn2 + ξ k˜2 + ∣dz∣2 )(ωn2 + a 02 − a12 − a 22 + a32 ) (58) + 2∣t∣2 (ωn2 − a 0ξ k˜ ) + ∣t∣4 , c″ = 2ia3[a 0(ωn2 + ξ k˜2 + ∣dz∣2 ) − ∣t∣2 ξ k˜ ], (59)
1 Fk˜ (τ ) = ⟨Tτ φ−k˜ (τ )φ k˜T(0)⟩ = e−iωmτ Fk˜ (iωm ), (50) β m
∑
d″ = 2a2ωn(ωn2 + ξ k˜2 + ∣dz∣2 + ∣t∣2 ) − 2a1a3(ωn2 + ξ k˜2 + ∣dz∣2 ), (60) Again, the four matrix elements of Fk˜ (iωm ) are in general all nonzero. Different from the spin-singlet case, the matrix elements of Fk˜ (iωm ) now have both even-frequency components and odd-frequency components, also consistent with the previous results of Black-Schaffer and Balatsky [44]. The particle–hole asymmetry term of equation (1) is again found not changing any qualitative results. The hexagonal warping term, on the other hand, is found to induce spin-singlet pairing correlation which would be absent if R1 = 0 such that c′′ = 0 [44]. As a conclusion to the present section, the particle–hole asymmetry term of equation (1) does not induce qualitative changes to previous results based on simplified models for the electronic structure of the topological insulators. The hexagonal warping term of equation (1), however, increases the number of the proximity-induced pairing channels by one. This is easy to understand because the proximity-induced pairing symmetry in the topological surface states other than the original symmetry of the external superconductor arises as a result of the spin–orbital coupling in the topological surface states, which can be seen from equations (51)–(60) by setting a1 = a2 = a3 = 0. Recalling equations (27)–(29), which shows the dependencies of ai (i = 0,1,2,3) on the parameters of equation (1), it is not the particle–hole asymmetry term but the hexagonal warping term that changes the character of the spin–orbital coupling of the topological surface states. In the above analysis, we have only considered the possible channels of proximity-induced pairing in the topological surface states. The amplitudes of the various proximity-induced pairing channels, which can be studied by integrating out the degrees of freedom of the external superconductor and work with the resulting effective low-energy model [40], is left to future works.
where β = 1/(kBT ) with kB the Boltzmann constant and T the temperature, ωm = (2m + 1)πkBT (m runs over all integers) are the fermionic Matsubara frequencies. Since the full Hamiltonian is of mean-field type with only bilinear terms, the anomalous Green’s functions can be calculated in terms of standard equation of motion method for an arbitrary pairing term Δ(k˜ ). For a general spin-singlet pairing Δ(k˜ ) = iΔ0(k˜ )s2 with Δ0(−k˜ ) = Δ0(k˜ ), Fk˜ (iωm ) is found to be
⎛ a′ − d′ −b′ + ic′⎞ i(t *)2Δ0(k) (51) ⎜ ⎟ F k(iWm ) = 2 a′ − b′ 2 − c′ 2 − d′ 2 ⎝ −b′ − ic′ a′ + d′ ⎠
where we have defined a′ = 2a2[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (52) b′ = 2ia3[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (53) c′ = −(ωn2 + ξ k˜2 + Δ20)(ωn2 + a 02 + a12 + a 22 + a32 ) (54) +2(a 0ξ k˜ − ωn2 )∣t∣2 −∣t∣4 , d′ = −2ia1[a 0(ωn2 + ξ k˜2 + Δ20)−∣t∣2 ξ k˜ ], (55)
in which ξ k˜ = ϵ k˜ − μ, t * means the complex conjugation of t, and ∣t∣2 = tt *. The four matrix elements of Fk˜ (iωm ) are generally nonzero. It is also clear that Fk˜ (iωm ) is an even function of frequency, consistent with the conclusion of a previous study by Black-Schaffer and Balatsky [44]. The presence of the particle–hole asymmetry term of equation (1) (i.e. a 0 ≠ 0) is found to bring about no qualitative changes. The hexagonal warping term, on the other hand, is found to induce unequalspin spin-triplet pairing correlations which would be absent if R1 = 0 such that b′ = 0 [44]. This last point is the only qualitative difference from previous works using simplified models for the electronic structures. We would also like to consider a type of spin-triplet pairings in the form of Δ(k˜ ) = (dz(k˜ )s3)is2. A special case defined by dz(k˜ ) = Δ0(kx ± ik y ) (Δ0 is a constant) breaks the time reversal symmetry and is considered as the most probable pairing of Sr2RuO4 [34, 35]. For this group of spin-triplet pairings, the anomalous Matsubara Green’s function is found to be
6. Summary We have studied the superconducting phases that can form in the topological surface states of a three-dimensional topological insulator, both in the case of bulk superconducting states of the topological insulator and for proximity-induced pairings. The particle–hole asymmetry term and the hexagonal warping term bring about some qualitative changes. In the case of bulk superconducting phase of the topological insulator itself, the number of pairings that can open a gap in the topological surface states is considerably increased by both the particle–hole asymmetry term and the hexagonal warping term. For the proximity-induced pairing, it is the hexagonal warping term that increases the number of pairing channels that form in the topological surface states by one. The present study illustrates the necessity of incorporating the particlehole asymmetry term and the hexagonal warping term in the
⎛ a″ − d″ −b″ + ic″⎞ (t *)2dz(k˜ ) (56) F ⎜ ⎟, k˜ (iωm ) = a″ 2 − b″ 2 − c″ 2 − d″ 2 ⎝−b″ − ic″ a″ + d″ ⎠
where we have defined a″ = −2ia1ωn(ωn2 + ξ k˜2 + ∣dz∣2 + ∣t∣2 ) (57) −2ia2a3(ωn2 + ξ k˜2 + ∣dz∣2 ) + 2a1ξ k˜ ∣t∣2 , 6
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