Flat bands as a route to high-temperature superconductivity in graphite
FFlat bands as a route to high-temperaturesuperconductivity in graphite
T.T. Heikkil¨a and G.E. Volovik
Abstract
Superconductivity is traditionally viewed as a low-temperature phenomenon.Within the BCS theory this is understood to result from the fact that the pairing ofelectrons takes place only close to the usually two-dimensional Fermi surface resid-ing at a finite chemical potential. Because of this, the critical temperature is expo-nentially suppressed compared to the microscopic energy scales. On the other hand,pairing electrons around a dispersionless (flat) energy band leads to very strong su-perconductivity, with a mean-field critical temperature linearly proportional to themicroscopic coupling constant. The prize to be paid is that flat bands can generallybe generated only on surfaces and interfaces, where high-temperature superconduc-tivity would show up. The flat-band character and the low dimensionality also meanthat despite the high critical temperature such a superconducting state would be sub-ject to strong fluctuations. Here we discuss the topological and non-topological flatbands discussed in different systems, and show that graphite is a good candidate forshowing high-temperature flat-band interface superconductivity.The purpose of this chapter is to propose a route to increasing the critical temper-ature of superconductivity by searching for special electronic dispersion that wouldpromote the superconducting strength. We first show that a huge increase in the(mean-field) critical temperature is possible, if a dispersionless energy spectrum, a flat band can be created in the system in the absence of the interaction leading to su-perconducting correlations. We then discuss a few known schemes to generate such(approximate or exact) flat bands.
T.T. Heikkil¨aUniversity of Jyvaskyla, Department of Physics and Nanoscience Center, P.O. Box 35, FI-40014University of Jyv¨askyl¨a, Finland e-mail:
G.E. VolovikAalto University, Department of Applied Physics and Low Temperature Laboratory, P.O. Box15100, FI-00076 AALTO, Finland and L. D. Landau Institute for Theoretical Physics, 117940Moscow, Russia e-mail: [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r T.T. Heikkil¨a and G.E. Volovik
Within the BCS mean-field theory, the occurrence of Cooper pairing at zero temper-ature can be studied via the free energy density for the pairing energy ∆ : F ∆ = − (cid:90) d d p ( π ¯ h ) d ( E p ( ∆ ) − E p ( ∆ = )) + ∆ | g | , (1)where d is the dimensionality, g < E p ( ∆ ) = (cid:113) ε p + ∆ is the quasiparticle excitation energy at momentum value p , evaluated ina system with the normal-state dispersion ε p . The first term in Eq. (1) demonstratesthat the formation of a gap ∆ decreases the energy of quasiparticles, which fill thenegative energy levels of Dirac vacuum. The second term is the cost of the formationof the gap, which perturbs the vacuum. For simplicity, we consider spinless fermionsand the gap that does not depend on momentum.Requiring ∆ to minimize F ∆ , we get the self-consistency relation ∆ = | g | (cid:90) d d p ( π ¯ h ) d ∆ E p ( ∆ ) , (2)or 1 = | g | (cid:90) d d p ( π ¯ h ) d E p ( ∆ ) . (3)Equation (3) dictates the behavior of ∆ at different dimensionalities d and for dif-ferent normal-state energy spectra ε p .For s -wave superconductivity in conventional metals with an isotropic Fermi sur-face and dispersion ε = v F ( p − p F ) expanded around the Fermi energy ε F = v F p F ,the integral in Eq. (3) is concentrated in the vicinity of the Fermi surface1 = | g | A d ( π ¯ h ) d v F (cid:90) ε uv d ε √ ε + ∆ ε uv (cid:29) ∆ ≈ | g | ν F ln ε uv ∆ . (4)Here ν F = A d ( π ¯ h ) d v F is the density of states in the normal metal; A d is the area ofthe d -dimensional Fermi surface; and ε uv (cid:28) ε F is the ultraviolet cut-off of the log-arithmically diverging integral, such as the Debye temperature. This leads to theexponentially suppressed gap: ∆ = ε uv exp (cid:18) − | g | ν F (cid:19) , (5)and correspondingly to the exponential suppression of the transition temperature T c .Situation drastically changes when the spectrum of the normal state has a flatband – a region in momentum p where ε p =
0. Since within the flat band E p ( ∆ ) = ∆ ,Eq. (3) becomes lat bands as a route to high-temperature superconductivity in graphite 3 = | g | V d ( π ¯ h ) d ∆ , (6)where V d is the volume of the flat band in momentum space. Hence instead of theusual exponentially suppressed behavior in Eq. (5) we have the gap ∆ that is linearlyproportional to the interaction strength: ∆ = | g | V d ( π ¯ h ) d , (7)Since the critical temperature of the superconductors is typically of the same orderof magnitude as the gap at zero temperature, the resulting superconductivity mayexist at high temperatures.It is instructive to consider the intermediate case when the quasiparticle spectrumis ε ( p ) = ε (cid:18) pp (cid:19) M . (8)Then for M > d , Eq. (3) gives the power law dependence of transition temperatureon the coupling constant: ∆ ∝ | g | MM − d . (9)In the limit of large M (cid:29)
1, the spectrum (8) transforms to the flat band concentratedat p < p , and the gap (9) asymptotically approaches the linear dependence on thecoupling g in Eq. (7). The case with d = M =
2, where ∆ ∝ g , has beenconsidered by Kopaev [1] and Kopaev-Rusinov [2].In the sections below, we consider different systems where the exact or approxi-mate flat bands could be realized. We start with the flat bands induced by interaction between the fermions. As wasfound by Khodel and Shaginyan [3], the interaction may lead to the merging ofdifferent fermionic energy levels, which results in the formation of a dispersionlessband, see also Refs. [4, 5]. The effect of the merging of discrete energy levels dueto interaction has been reported in a recent paper, Ref. [6].Their argument is based on the phenomenological Landau’s consideration on thederivation of the distribution function n p of fermions at T =
0. It is determined bythe energy functional E { n ( p ) } , whose variational derivative ε p is the quasiparticleenergy. The variation of this functional gives the equation for n p and ε p : δ E { n ( p ) } = (cid:90) d d p ( π ¯ h ) d ε p δ n p = . (10) T.T. Heikkil¨a and G.E. Volovik pp p flat bandFermi surface flat band two solutions: ε ( p ) = 0 or δ n ( p )=0 δ n ( p )=0 δ n ( p )=0 splitting of Fermi surfaceto Fermi ball(flat band) p ε ( p ) ε ( p ) n ( p ) n ( p ) p F δ n ( p )=0 δ n ( p )=0 ε ( p ) = 0 ε ( p ) = 0 δ E { n ( p ) } = ε ( p ) δ n ( p ) d d p = 0 Fig. 1
Illustration of the formation of the Khodel-Shaginyan flat band due to interaction betweenfermions. (
Top ): the Landau model of the Fermi liquid considers the energy E { n ( p ) } as the func-tional of the distribution function n ( p ) of quasiparticles. The variation of the functional gives twotypes of solutions: ε p = δ n p =
0. (
Left ): distribution of quasiparticles in the class of Fermiliquids. Two regions with solutions δ n p = ε p = Right ): Forstrong interaction (large interaction constant), the intermediate region with the solution ε p = Since the quasiparticle distribution function is constrained by the Pauli principle0 ≤ n p ≤
1, there are two classes of solutions of the variational problem. One classis ε p =
0, which is valid if 0 < n p <
1; another one is δ n p = n p = n p = Let us start with the Fermi gas – the system of free fermions with the spectrum ε p = p / m − µ , where µ >
0. The energy functional for free fermions is E { n ( p ) } = (cid:82) d d p ( π ¯ h ) d ε p n p , which gives the solution shown in Fig. 1 ( left ). The solution of theclass ε p = p F , where p F / m = µ . Outside ofthe Fermi surface the distribution function n p = const, with n p = p < p F and n p = p > p F . This corresponds to the class of solutions with δ n p = d = p = p F is the 1d circle. For that one has to consider the Green’s function at imaginaryfrequency: G ( ω , p ) = i ω − (cid:16) p m − µ (cid:17) . (11) lat bands as a route to high-temperature superconductivity in graphite 5 C p x p F p y ω Fig. 2
Illustration of the topological stability of the Fermi surface on an example case with dimen-sion d =
2, when the Fermi surface forms a closed loop. Green’s function has singularities on theline ω = p x + p y = p F in the three-dimensional space ( ω , p x , p y ) . Stability of the Fermi surfaceis protected by the invariant (12) which is represented by an integral over an arbitrary contour C around the Green’s function singularity. The Green’s function in Eq. (11) has singularities at ω = p belonging to theFermi surface. These points form a closed line in the three dimensional ( ω , p x , p y ) -space, see Fig. 2 for d =
2. This line has a topological winding number: the phase Φ of the Green’s function, G = | G | e Φ changes by 2 π along an arbitrary contour C around the line. In other words, the Fermi surface represents the p -space analog ofthe vortex lines in superfluids and superconductors, where the phase of the orderparameter changes by 2 π around the vortex. The 2 π winding of the phase Φ can-not change under small deformations of the parameters of the system, and thus isrobust to the interactions between the particles, if we do not consider the supercon-ducting, magnetic or other phase transition, which drastically (non-perturbatively)reconstructs the energy spectrum. This topological stability is the reason why inter-acting Fermi liquids preserve the Fermi surface.For more complicated cases, when the Green’s function has spin, band and otherindices, and for arbitrary dimension d the winding number N of the Fermi surfaceis expressed analytically in terms of the matrix Green’s function in the followingform: N = tr (cid:73) C dl π i G ( ω , p ) ∂ l G − ( ω , p ) . (12)Here the integral is taken over an arbitrary contour C around the Green’s functionsingularity in the d + N = When the interaction between the particles is strong enough, so that it starts dom-inating over the fermionic statistics, a more classical behavior of the distributionfunction may emerge, in which the natural solution of the variational problem corre-
T.T. Heikkil¨a and G.E. Volovik ππ p x p=p p=p p y ω Fig. 3
Illustration of the topological structure of the Khodel-Shaginyan flat band in a d = right ) the Fermi surface at p = p F spreads into a flat band concentrated inthe region p < p < p . Correspondingly the line of the Green’s function singularities in Fig. 2 –a vortex line – is spreading to an analog of a domain wall terminating on a pair of π -vortices at p = p and p = p . [4] sponds to a zero value of the variational derivative, δ E { n ( p ) } / δ n p =
0. Therefore,with an increasing interaction strength one may expect the topological quantumphase transition to the distribution in Fig. 1 ( right ), where the solution ε p = π winding transforms to the domain wall, at which the Green’sfunction has a jump, G ( ω = + , p x , p y ) − G ( ω = − , p x , p y ) (cid:54) =
0. This domain wallterminates on π vortices. Here we consider the formation of the Khodel-Shaginyan flat band in the vicinityof a saddle point in the d = E { n ( p ) } = ∑ p n p ε ( ) p + ∑ p , p (cid:48) f ( p , p (cid:48) ) n p n (cid:48) p . (13)We illustrate the flat band solution using an even simpler functional with contactinteraction: E { n ( p ) } = ∑ p (cid:34) ε ( ) p n p + U (cid:18) n p − (cid:19) (cid:35) , (14)where U >
0. This functional has always a flat band solution with 0 < n p < ε p = δ E δ n p = ε ( ) p + U (cid:18) n p − (cid:19) = , (15) lat bands as a route to high-temperature superconductivity in graphite 7 flat band ε ( p ) = 0 n ( p ) < 1 n ( p ) = 0 n ( p ) = 0 n ( p ) = 1 n ( p ) = 1 p x p y Fig. 4
Flat band emerging near a saddle point. (
Left ): from the simplified Landau-type theory inEqs. (14)-(16) with µ =
0. The flat band is concentrated in the black region. (
Right ): from the nu-merical solution of the Hubbard model [8] [D. Yudin, et al. , Phys. Rev. Lett. , 070403 (2014)],showing the spectral function within the reciprocal space of an interacting triangular lattice. Thelower left sextant corresponds to the noninteracting case U =
0. For large U the band flattening isclearly seen near the saddle points. n p = − ε ( ) p U , < n p < . (16)In the vicinity of the saddle point the non-perturbed spectrum (i.e. at U =
0) hasthe form ε ( ) p = p x p y m − µ . For µ (cid:54) =
0, there are two hyperbolic Fermi surfaces. Theyinterconnect at the Lifshitz transition, which takes place at µ =
0. When the inter-action U is switched on, the flat bands emerge. Figure 4 ( left ) demonstrates the flatband at µ =
0. The same shape of the region with the flat band has been obtainedfrom the numerical simulations of the Hubbard model [8], see Fig. 4 ( right ). In the previous section we discuss the fermion condensate – the flat band, whichmay emerge due to interactions in the vicinity of a singularity in the non-interactingspectrum. There are also other ways to generate flat bands or approximate flat bands.Historically the flat bands first appeared as Landau levels of charged particles ina magnetic field [9]. Here we discuss the flat bands that have purely topologicalorigin. They may exist without a magnetic field, and they are not very sensitive tointeractions. The flat band may emerge as the surface or interface state in topologicalsemimetals [10, 11, 12], which we discuss in this section. The flat band may alsoappear at the strained interfaces with misfit dislocations, which play the role ofeffective magnetic field. Such flat bands are discussed in Sec. 4.In this section we characterize semimetals that have an internal spin-like struc-ture. All the examples discussed here can be characterized via the topological in-variant of the form [13]
T.T. Heikkil¨a and G.E. Volovik N = tr (cid:73) C d l π i · [ Γ H − ( p ) ∂ l H ( p )] , (17)where H is the Hamiltonian in the momentum space; Γ is a matrix which commutesor anti-commutes with the Hamiltonian, such as the third Pauli matrix σ acting onthe spin-like degree of freedom; and C is a contour in momentum space, specifiedseparately for each semimetal.In semimetals, the flat bands are realized as a particular consequence of the bulk-boundary correspondence of topological media (see [14, 15, 16] or Sec. 22.1 in[17]). If for example a 3D bulk system contains Weyl points, then an interface witha topologically trivial material, or with a material having a different value of topo-logical invariants contains a line of zeroes – the Fermi arc [18]. The terminationpoints of the Fermi arc are given by projections of the Weyl points to the interface.In the same manner the Dirac lines in 3D bulk or Dirac points in 2D bulk give riseto the to nodes of higher dimension at the interface – the flat band [10, 11, 12]. Theboundaries of the flat band are determined by the projection of the Dirac line orDirac points to the interface. Let us consider an example semimetal characterized by an even number of 2-dimensional Dirac points, such as that found in graphene around the two val-leys. Close to the Dirac point the Hamiltonian can be written as a 2 × H = v F (cid:18) p x − ip y p x + ip y (cid:19) = v F p · σ , (18)where p = ( p x , p y ) and σ = ( σ x , σ y ) , where σ j are Pauli matrices. The eigenvaluesof H satisfy ε = v F | p | having a node ε ( p = ) in a single point in momentumspace. To illustrate the topological protection of this node, let us add a perturbationof the form V ( p ) = v ( p ) · σ , where v = ( v x , v y ) does not break the (pseudo)spin sym-metry. As a result, the dispersion becomes ε = ( v F p x + v x ) + ( v F p y + v y ) , whichagain has a single node at ( p x , p y ) = − ( v x , v y ) / v F . The only effect of the potentialis thus to shift the node, but not annihilate it. This property can be expressed via thepresence of the topological charge of the form (17), where Γ = σ z and the contour C goes around the Dirac point in the 2D momentum space (see Fig. 5). For the Hamil-tonian in Eq. (18), we get N =
1. In graphene, there are two Dirac points: the firstone has the form in Eq. (18), and the second is otherwise the same but p y (cid:55)→ − p y . Inthat case the second Dirac point has N = −
1. These topological charges stay invari-ant to perturbations of the form V ( p ) , as long as we shift the contour of integrationalong with the shift of the Dirac point, and as long as the two nodes do not mergedue to such a perturbation. lat bands as a route to high-temperature superconductivity in graphite 9 Fig. 5
2D Dirac/Fermi point in momentum space and the line of integration for the topologicalinvariant N . In 3D materials, the topological charge in Eq. (17) characterizes lines of nodes– the Dirac lines. The Dirac lines are readily obtained in superfluids and supercon-ductors, where the symmetry operator Γ contains the particle-hole symmetry. Inparticular, the Dirac line exists in the polar phase of superfluid He [19] and mayappear in superconductors without inversion symmetry [12]. The topologically sta-ble Dirac lines give rise to the topologically protected surface flat band. Accordingto the bulk-surface correspondence, the boundary of the flat band is determined bythe projection of the nodal line on to surfaces. In nonsuperconducting materials thecorresponding symmetry which enters Eq. (17) can be only approximate, being vio-lated by spin-orbit interaction, or by the higher order hopping elements. This leads toformation of approximately flat surface bands as it happens for example in graphite[20], graphene networks [21] and possibly in some other materials [22]. y a/ π x a/ π ǫ / γ K K KK ′ K ′ K ′ p p Fig. 6
Energy spectrum in the conduction and valence bands of graphene. Here a is the latticeconstant, and γ denotes the nearest-neighbour hopping parameter in the tight-binding lattice. Theunderlying contour plot shows the positions of the Dirac points. Only two of these points are non-equivalent, the others are connected via reciprocal lattice vectors.0 T.T. Heikkil¨a and G.E. Volovik We first consider the case of graphene, whose energy spectrum within the valenceand conduction bands is plotted in Fig. 6. This spectrum follows for example fromthe nearest-neighbour tight-binding model on a honeycomb lattice, see [23, 24].Around the specific points, marked K and K (cid:48) in the figure, the low-energy Hamilto-nian is of the form of Eq. (18). These points are described by the topological charge N , such that N ( K ) = + N ( K (cid:48) ) = − K and K (cid:48) points in the 2d momentum plane, andconsider the presence of an edge placed in the x -direction. Now, the edge marksa boundary between graphene, which is a nodal semimetal, and vacuum, which isa trivial insulator. By the bulk-boundary correspondence we may hence expect flatband states at the edge. Since we maintain translational invariance along the x di-rection, p x remains a good quantum number and it also parametrizes the edge statesand their dispersion ε e ( p x ) . According to the bulk-boundary correspondence, theprojections of the Dirac points to the boundary determine the end points of the flatband. From Fig. 7 it is clear that the termination points of the flat band must be lo-cated at p x values corresponding to the p x -component of the K and K (cid:48) points. When p x crosses these points, the normal to the interface run across either K or K (cid:48) pointsand the topological invariant in Eq. (17) along the normal changes. However, fromthe figure alone one cannot say which of the regions between the K and K (cid:48) pointscontain flat bands and which not. The solution is to construct the system by repeat-ing a set of infinite chains, and construct the topological invariant, in this case calledthe Zak phase [25] for these chains. As shown for example in [25, 10], the detailsof this procedure depend on the microscopic form of the edge, which is either of the“bearded” or the “zigzag” type. Figure 7 illustrates the resulting positions of the flatband dispersion in momentum space.If we were to place the interface in the y direction (in which case we would obtainthe “armchair” edge), there would be no flat band due to the fact that the normal tothe interface would run across both K and K (cid:48) points. In this case the change of thetopological charge across this normal line would be N ( K ) + N ( K (cid:48) ) = In graphite, the coupling between the graphene layers is much weaker than thatwithin the atoms producing the hexagonal lattice in individual layers. Because ofthis, we may assume that the individual layers remain to be described by the two-dimensional low-energy momentum-space Hamiltonian of the form in Eq. (18). Inthe (most stable) AB stacking, the pair of layers is arranged so that the graphenehexagons are rotated with respect to each other by 30 ◦ . As a result, one pair ofatoms (say, A on the bottom layer and B on the top layer) reside on top of each lat bands as a route to high-temperature superconductivity in graphite 11 Fig. 7
Formation of a flat band in the zigzag (lower part of the figure) or bearded (upper part)edges of graphene. other, whereas the other atom of the unit cell (B on the bottom and A on the top) areon the bottom/top of the other hexagon. Therefore, the interlayer coupling betweenthe first pair of atoms is much stronger than the coupling between the second pair.Taking into account only this strongest coupling then produces the Hamiltonian ofthe form H K = (cid:18) v F p · σ − γ σ ↓ − γ ∗ σ ↑ v F p · σ (cid:19) (19)around one of the graphene valleys ( K -point) for this pair of layers. Here γ quan-tifies the interlayer hopping. Around the other valley ( K (cid:48) point) the Hamiltonianis H K (cid:48) ( p x , p y ) = H K ( p x , − p y ) . It is straightforward to show that H K / K (cid:48) have fourbranches of eigenvectors, two gapped ones (with ε ( | p | = ) = ± γ ) and two witha quadratic dispersion around a Fermi node, ε = | γ | ( p / p FB ) , where p FB = | γ | / v F . The corresponding N = −
2) for H K ( H K (cid:48) ).Beyond the bilayer, the graphene layers can be stacked in two qualitatively differ-ent ways by respecting the AB stacking for each pair of layers. In Bernal stacking,the line of strongest interlayer coupling is straight, i.e., connects the same atoms ineach layer whereas for rhombohedral (ABC) stacking, it follows an armchair-typepattern, i.e., the strongly coupled atoms differ between neighbouring pairs of layers. The corresponding elements of the momentum space Hamiltonians coupling layers n and m for 2D momenta around the K point are H Bernal mn = v F p · σ δ mn − γ δ m , n + (cid:8) σ ↑ [( − ) m − ] + σ ↓ [( − ) m + ] (cid:9) / + h . c . (20) H rhombohedral mn = v F p · σ δ mn − γ δ m , n + σ ↑ + h . c . (21)In H Bernal the terms in square brackets take care of indexing the even and odd layersseparately. Note that since the choice of the A / B indices for the graphene sublatticeatoms is arbitrary, we could have also chosen to write the coupling in H rhombohedral in terms of σ ↑ instead of σ ↓ . Formally, these two choices correspond to differentsigns of the bulk topological invariants (see below), but they have a meaning onlyin the presence of (Bernal) stacking faults that change the coupling around someparticular interface. We first derive the bulk dispersion of Bernal graphite by including only the strongestinterlayer hopping term γ . We make the bulk Ansatz ψ Tn = (cid:16) α o β o (cid:17) e ip z na / ¯ h , n odd (cid:16) α e β e (cid:17) , n even (22)for the wave function on the n -th layer. Here p z is the momentum in the directionperpendicular to the layers and a is the distance between the layers. This is an eigen-function with energy ε provided that the coefficients α e / o , β e / o satisfy v F p − − γ cos ( p z a / ¯ h ) v F p + v F p − − γ ∗ cos ( p z a / ¯ h ) v F p + (cid:124) (cid:123)(cid:122) (cid:125) H B α o β o α e β e = ε α o β o α e β e , (23)where p ± = p x ± ip y ≡ p ⊥ e i φ , p ⊥ ≥
0. This has (four) zero-energy solutions at p x = p y =
0, regardless of the value of p z . Within this approximation, Bernalgraphite has therefore two Dirac lines, running through the K and K (cid:48) points of the2D graphene band structure, or between the H points in the 3D graphite band struc-ture, see Fig. 3.3. Including higher-order hopping terms in the Hamiltonian expandsthis line into electron and hole pockets [26]. However, let us first analyze the topol-ogy of H B . The topological charge in this case is of the form of Eq. (17), wherethe contour C runs around the Dirac lines as indicated in Fig. 8, and σ z should be lat bands as a route to high-temperature superconductivity in graphite 13 replaced by ( σ z ⊗ ) . This produces N = ±
2, where the factor 2 takes care of theadditional layer degree of freedom in Eq. (23).
Fig. 8
Fermi lines in Bernal graphite within the nearest-neighbour interlayer approximation. Thispicture expands the 2D momentum space ( p x / p y plane) of graphene to the 3D space where p z runsbetween − ¯ h π / a to ¯ h π / a ( a is the interlayer spacing). This is thus nothing but the generalization of the graphene topological charge tographite and it yields N = ± H − K − H / H (cid:48) − K (cid:48) − H [26], where the H ( (cid:48) ) point is shifted from the K ( (cid:48) ) point in the p z directionby ¯ h π / a . Based on the bulk-boundary correspondence we may therefore expect tohave surface states at the lateral boundaries of Bernal graphite, as extensions of theflat band states in zigzag graphene, but now the flat band extends throughout thefirst Brillouin zone in the p z direction.Let us now consider the effect of higher-order hopping elements. First they splitthe Dirac line with multiple charge N = N =
1. However, the situation ismore interesting, see Fig. 9 and Refs. [26, 27]. The N = N =
1, while the central line has N = −
1. The total topological invariant remains the same, N = + + − = Γ [20] underlying thetopological protection. Without the topological protection the four Dirac lines areexpanded into four pairs of electron and hole pockets, which touch each other atfour points. As a result the surface states of Bernal graphite cease to be flat on theenergy scale related to those higher-order hoppings. To obtain the bulk dispersion of rhombohedral graphite, it is enough to make theAnsatz holesN = 2electronselectrons Fig. 9
Graphite Fermi surface from Ref. [26]. It consists of the electron and hole pockets connectedat four points. Each point is described by topological invariant in Eq. (17): the point in the centerhas N = − N =
1. Such structure originates from the nodalline with N = Fig. 10
Fermi spirals in rhombohedral graphite. ψ n = (cid:18) αβ (cid:19) e ip z na / ¯ h . (24)This is an eigenfunction of H rhombohedral if α and β satisfy (cid:18) v F p − − γ exp ( ip z a / ¯ h ) v F p + − γ ∗ exp ( − ip z a / ¯ h ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) H RHG (cid:18) αβ (cid:19) = ε (cid:18) αβ (cid:19) (25) lat bands as a route to high-temperature superconductivity in graphite 15 for some energy ε . The zero-energy solutions are found along a spiral in the mo-mentum space, p z a = arctan (cid:18) p y p x (cid:19) = φ , v F p ⊥ = | γ | . (26)In this case the topological charge equals the spiral helicity, and can be defined byEq. (17) where the integration contour runs over the 1st Brillouin zone ( − π / a to π / a ) in the p z -direction. For p z running close to the line from the H point via Kback to H, N = p x ± ip y is inside the spiral,i.e., p ⊥ < | γ | / v F . For larger momenta away from the H − K − H line, N =
0. Onthe other hand, for the path H (cid:48) − K (cid:48) − H (cid:48) the spiral is defined by Eq. (26), where p y ↔ − p y , inverting the helicity and the sign of N to -1.As a result of the existence of the non-trivial topological charge, the surfacelayers of rhombohedral graphite contain flat bands, i.e., ε ( p ⊥ < | γ | / v F ) =
0. Thiscan be understood as follows. For each p ⊥ with p ⊥ (cid:54) = | γ | / v F , the Hamiltonian H p ⊥ ( p z ) describes a 1D insulator. For p ⊥ < | γ | / v F , this insulator is topological,since it has the topological charge N ( p ⊥ ) =
1. According to the bulk-boundarycorrespondence, each topological insulator has an edge state with zero energy. Thesestates form a flat band in the region p ⊥ < | γ | / v F . Recently another possible source of the topological flat band has been discussedin two materials: highly oriented pyrolytic Bernal graphite (HOPG) [28] and het-erostructures SnTe/PbTe, PbTe/PbS, PbTe/PbSe, and PbTe/YbS consisting of a topo-logical crystalline insulator and a trivial insulator [29]. In both cases the flat bandcomes from a misfit dislocation array, which is spontaneously formed at the inter-face between two crystals due to the lattice mismatch. In Ref. [28] the lattice ofscrew dislocations has been considered, which emerges at the interface betweentwo domains of HOPG with different orientations of crystal axes. In Ref. [29] themisfit dislocation array is formed at the interface between topological and trivialinsulators.The above two systems exhibit similar phenomenon. In both cases superconduc-tivity related with the interfaces has been found [30, 31]. The reported transitiontemperature essentially exceeds the typical transition temperature expected for thebulk materials. A possible origin of this phenomenon is the flat band at the inter-faces, where the transition temperature could be proportional to the coupling con-stant and the area of the flat band.The topological origin of the flat bands in these systems can be understood eitherin terms of the overlapping of the 1D flat bands formed within the dislocationsor using the following consideration. In case of the heterostructures, on one sideof the interface the insulator is topological, and thus the interface contains Diracfermions. The strain at the interface (with its dislocations) acts on Dirac fermionsas the effective magnetic field. Such emergent field is now extensively discussed for
Fig. 11
Misfit dislocation grid at the interface from [N. Ya. Fogel, Phys. Rev. B , 174513(2002)] [30]. strained graphene, see e.g. the most recent paper Ref. [32] and references therein.The effective magnetic field B produces the required flat band, since the first Landaulevel for massless Dirac fermions has zero energy. When the period decreases, thefield B increases, giving rise to enhanced density of states, which is proportional to B . This is equivalent to increase of the area of flat band in the scenario discussed inSec. 3. Such increase of transition temperature is discussed in Ref. [29].In particular, Ref. [29] considered the case where the massless Dirac Hamilto-nian of the topological insulator experience a periodic field. They arrived at theHamiltonian of the form H TF = − i ∂ x σ x + [ k y − A ( x )] σ y , (27)written in terms of (scaled) momentum k y along the dislocation, and a scaled strain-induced gauge field A ( x ) satisfying A ( x + d ) = A ( x ) and (cid:82) d A ( x ) dx =
0. In the caseof Ref. [29], A ( x ) = β cos ( π x / d ) , but the approach works for a more general pe-riodic vector potential as well. Due to the position dependent vector potential, thereis no translation symmetry in the x direction and the momentum in the x -directionis not a good quantum number. However, due to the periodicity of the potential wecan use Bloch’s theorem and define the pseudomomentum ˜ k x . This allows for calcu-lating the spectrum of Eq. (27), plotted in Fig. 12 and exhibiting (an approximate)flat band for k y < β / k x ∈ [ − π / d , π / d ] .The emergence of the approximate flat band in Eq. (27) can be understood by firstconsidering the case k y =
0. In that case the Hamiltonian has two (unnormalized)zero-energy solutions, lat bands as a route to high-temperature superconductivity in graphite 17 -5 0 5 x -3 -6-4-20246 -30 -20 -10 0 10 20 30-8-6-4-202468 Fig. 12
Spectrum of Eq. (27) with β =
30. Inset shows the (approximate) linear dispersion withlow values of k y , and with a speed as in Eq. (4). ψ + = (cid:18) (cid:19) exp [ (cid:90) x A ( x (cid:48) ) dx (cid:48) ] ψ − = (cid:18) (cid:19) exp [ − (cid:90) x A ( x (cid:48) ) dx (cid:48) ] . Let us include the term H = k y σ y as a perturbation. The second order secular equa-tion produces the 2 × H : (cid:32) H ( )+ − H ( ) − + (cid:33) = c (cid:18) k y k y (cid:19) with the spectrum E = ± ck y . The slope c of the spectrum is obtained from the matrixelements of the normalized eigenfunctions c = (cid:82) ∞ − ∞ dx (cid:82) ∞ − ∞ dx exp [ (cid:82) x A ( x (cid:48) ) dx (cid:48) ] = d (cid:82) d e (cid:82) x A ( x (cid:48) ) dx (cid:48) . For A ( x ) = β cos ( x ) we thus get c = / I ( β ) , where I ( x ) is the zeroth Bessellfunction of the first kind. When c (cid:28)
1, requiring β (cid:29)
1, the result is an approximateflat band.Alternatively, we may consider the case of a periodic line dislocation. There, thevector potential is proportional to the strain field, A ( x ) = α u xx ( x )+ α u yy ( x ) , where α and α are coupling constants and u xx ( x ) = bz π ( − ν ) x + z ( x + z ) u yy ( x ) = bz νπ ( − ν ) x + z , where b is the size of the Burgers vector of the dislocation, z is the distance from thedislocation plane, and ν is the Poisson ratio. We also assume that such dislocationsrepeat after each d . In this case we may estimate the slope c for d (cid:29) z by c ≈ exp (cid:20) − (cid:90) ∞ A ( x ) (cid:21) = exp (cid:20) − b ( α + να ) − ν (cid:21) . The flat band dispersion where c (cid:28) ν ≈ α i (cid:29) / b . Note that this result show that the estimates done in Ref. [29] areoveroptimistic, as there α b ≈ As discussed in Sec. 3, bulk neutral graphite and graphene are typically consideredto have a very low density of states. This is why the occurrence of superconductivityin these bulk systems would require a strong doping, shifting the Fermi energy to afinite value and thereby increasing the density of states. Let us consider the examplecase of a 2D Fermi line with the superconducting coupling g , but with a shiftedFermi level by a value µ . In this case Eq. (3) yields for µ = ∆ = ( a ε uv − / a ) / > , (28)where a = g / ( π ¯ h v F ) . This is non-zero only for a large coupling strength a > / ε uv .For weak coupling | a | < / ξ c , superconductivity is enhanced by doping, because fornon-zero µ the solution is ∆ = | µ | exp ( − / a − ξ c | µ | − ) .This strategy has been followed by some graphite experimentalists. For exampleRef. [34] demonstrates bulk superconductivity of two graphite intercalation com-pounds C Yb and C Ca with critical temperatures of 6.5 and 11.5 K, respectively.There one of the effects from the intercalant layers is the charge transfer, i.e., dopingthe graphite.The situation is opposite in flat band systems: doping does not aid supercon-ductivity, but rather decreases the critical temperature. Fith a non-zero chemicalpotential µ , the gap becomes ∆ = (cid:113) ∆ − µ , (29)where ∆ FB is the gap obtained for µ =
0, as in Eq. (7). lat bands as a route to high-temperature superconductivity in graphite 19
There are experimental indications for the presence of superconductivity atgraphite interfaces [31]. In this case, the interfaces form between differently ori-ented regions of Bernal graphite, so that the graphite c -axis ( p z in our work here)runs perpendicular to these surfaces. These observations hence cannot be explainedby the flat bands at the lateral interfaces (along p x or p y ) as one would expect inBernal graphite. There are also no direct indications of the presence of rhombo-hedral stacking in these systems, but the resolution of the imaging does not allowruling out some rhombohedral regions close to the interfaces exhibiting supercon-ducting properties. Note that rhombohedral stacking has been found in highly ori-ented pyrolytic graphite, see for example [35, 36]. The remaining scenario is basedon the formation of an array of dislocations (Sec. 4) at these surfaces as the surfacelayers try to adapt to the neighbouring graphite planes.The discussion in Sec. 1 concerns only the relation of the mean field order pa-rameter with normal-state electronic spectrum. The details of the flat band supercon-ducting state, such as the quasiparticle spectrum, supercurrent and collective modes,depend on the way the flat band is formed. The case of rhombohedral graphite hasbeen worked out in Refs. [37, 38, 39] and summarized in [40]. The density of statesin the superconducting state was computed in [41]. In that case the superconductingspectrum is not flat, but rather has an inverse parabolic shape and exhibits a minigapwhose value is inversely proportional to the distance between adjacent surfaces. Thesupercurrent is characterized by a large critical current that is linearly proportionalto ∆ . However, in this model the supercurrent does not only flow along the surfaces,but also between them. On the other hand it is clear that not all types of flat bandscan describe a state with a non-vanishing critical current. For example, an entirelyisolated region in momentum space having a flat band describes a set of completelylocalized electrons, which cannot be made to carry supercurrent.In this chapter we have concentrated on illustrating the results for mean-fieldmodels of superconductivity on flat bands. As discussed in [40], fluctuation effectsare much more important in flat band superconductors than in ordinary BCS super-conductors. The nature of the fluctuation response depends on the mechanism of theflat band formation, and the ensuing superconducting spectrum. The details of suchfluctuation effects we leave for further work.As we have discussed in this chapter, there are many ways of realizing (approxi-mate) flat bands. Some of these realizations may be relevant in interfaces of graphite,where indications of superconductivity with a very high critical temperature havebeen observed. It remains to be demonstrated whether these properties can be ex-plained within the flat band scenario of superconductivity.We acknowledge Pablo Esquinazi, Ville Kauppila and Timo Hyart for discus-sions. This work was supported by the Academy of Finland through its Center ofExcellence program, and by the European Research Council (Grant No. 240362-Heattronics). References
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