Band hybridization induced odd-frequency pairing in multiband superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Band Hybridization Induced Odd-Frequency Pairing in Multiband Superconductors
L. Komendov´a, A. V. Balatsky,
2, 3 and A. M. Black-Schaffer Department of Physics and Astronomy, Uppsala University, Box 530, SE-751 21 Uppsala NORDITA, Center for Quantum Materials, KTH Royal Institute of Technology,and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, NM 87545, United States (Dated: September 18, 2018)We investigate how hybridization (single-quasiparticle scattering) between two superconductingbands induces odd-frequency superconductivity in a multiband superconductor. An explicit deriva-tion of the odd-frequency pairing correlation and its full frequency dependence is given. We findthat the density of states is modified, from the sum of the two BCS spectra, at higher energies byadditional hybridization gaps with strong coherence peaks when odd-frequency pairing is present.
Many materials have multiple bands close to the Fermilevel. It is then not surprising that there also exist manysuperconductors with more than one superconductingband. Well-known multiband superconductors are MgB [1–3], which hosts two distinct superconducting gaps, andthe iron-based superconductors [4–6], where the orderparameter changes sign between different bands [7, 8].Multiband superconductivity has also been suggested tobe important in simple metals [9, 10], heavy fermion com-pounds [11–15], different carbides [16, 17], and Chevrelphases [18], as well as for engineering time-reversal in-variant topological superconducting states [19, 20].Multiple superconducting bands allow for unusual cou-pling effects. Early, and much current, theoretical focushas been oriented towards studying the effects of Joseph-son coupling between different superconducting bands,i.e. the exchange of whole Cooper pairs between bands[21–26]. However, conceptually much simpler is single-quasiparticle scattering, or tunneling, between the su-perconducting bands. The origin of interband quasipar-ticle scattering can be impurity scattering, but a com-mon intrinsic source is superconductivity associated withspecific orbitals, which subsequently hybridize to formthe low-energy bands around the Fermi surface. Tun-neling spectroscopy has found significant effects of inter-band quasiparticle scattering in silicon clathrate Ba Si [27, 28], iron-based superconductors [29], and MgB [30].Recent ARPES data on MgB have also shown that in-terband scattering due to disorder can be important [31].Theoretically, interband single-quasiparticle scatteringresults in both band hybridization and interband pairing,where the two electrons forming a Cooper pair belongto different bands. Straightforward effects of band hy-bridization were studied already quite early on [32–34].More recently, band hybridization has also been proposedto influence the nodal structure of the superconductinggap in iron-based superconductors [35–38]. On the otherhand, consequences of the induced interband pairing havebeen much less discussed. Pure interband pairing in coldatoms and quantum chromodynamics systems has beenproposed to give a “breached” regime containing both asuperfluid and a normal liquid [39, 40], however, in band- hybridized superconductors the interband pairing is alsoaccompanied by (usually large) conventional intrabandpairings. Still, it was recently pointed out, using symme-try arguments and simple mean-field BCS calculations,that odd-frequency, odd-interband pairing can be ubiq-uitous in multiband superconductors [41].The fermionic nature of the superconducting wavefunction usually renders a division into spin-singlet even-parity (i.e. s -, d -wave) or spin-triplet odd-parity ( p -wave) superconductors, but it can also be even or oddunder time, or equivalently frequency [42–44]. Exam-ples of this are odd-frequency spin-triplet s -wave super-conductivity giving rise to long-scale proximity effectsin superconductor-ferromagnet systems [45–47] or odd-frequency spin-singlet p -wave pairing in non-magneticjunctions [48–50]. In multiband superconductors theband index offers the additional possibility of spin-singlet s -wave pairing which is odd in both band and frequency,without translation or spin rotation symmetry breaking.In this work we derive the exact Green’s functionsfor a generic multiband superconductor with single-quasiparticle scattering. We thus obtain the full fre-quency dependence of the interband pairing and find thatthe odd-interband pairing has odd frequency dependence,while the even-interband pairing is even in frequency. Bystudying the density of states (DOS) we also discover thatfinite band hybridization gives rise to an extra gap lo-cated beyond the original gap edges. This hybridization-induced gap is not fully depleted, but has very pro-nounced BCS-like coherence peaks. Moreover, the hy-bridization gap disappears whenever the odd-frequencyinterband pairing is zero and is thus a directly measur-able signal of odd-frequency superconductivity.To model a generic multiband superconductor we con-sider two bands, band 1 and 2, with dispersions ǫ k and ǫ k , where ǫ k = ǫ − k . For simplicity, we assume thatthe two bands independently develop conventional spin-singlet s -wave superconducting order parameters ∆ and∆ , respectively, but one of them can also be zero. Fi-nally, we add a small single-quasiparticle hybridization,or scattering, term proportional to Γ between the twobands, resulting in the Hamiltonian: H = X kσ ǫ k a † kσ a kσ + ǫ k b † kσ b kσ + X kσ Γ( k ) a † kσ b kσ + H . c . + X k ∆ ( k ) a † k ↑ a †− k ↓ + ∆ ( k ) b † k ↑ b †− k ↓ + H . c ., (1)where a ( b ) is the annihilation operator in band 1 (2).Alternatively, the band hybridization can be interpretedas a coupling process in real space, with the a - and b -electrons living to the left and the right of a junction orin different layers [32, 51]. In this picture the Josephsoncoupling would instead be a two-particle tunneling term(not included here).We start by calculating the spin-singlet s -wave inter-band anomalous Green’s functions F and F , whichexpress the pairing correlations of two electrons belong-ing to different bands. Assuming Γ to be a small pa-rameter, we can use standard perturbation theory [52].The first order contributions are then represented by theschematic diagrams in the inset of Fig. 1(a) and give: F (1)12 = F Γ G − ←− G Γ F , where ←− G is the hole propaga-tor, i.e., ←− G = − G ( − k, − ω ) . The minus sign before thesecond term is due to scattering of hole propagators (left-going arrows). The normal and anomalous propagatorswithout the hybridization are as usual [52]: G j F j F † j ←− G j ! = 1( iω ) − E j (cid:18) iω + ǫ kj ∆ j ∆ ∗ j iω − ǫ kj (cid:19) , (2)where E j = E kj = ǫ kj + | ∆ j | and ω = ω n = π (2 n +1) k B T are the fermionic Matsubara frequencies. Usingthese expressions we arrive at F (1)12 = Γ[ iω (∆ − ∆ ) +∆ ǫ k +∆ ǫ k ] / [( ω + E )( ω + E )]. The next to leadingorder terms for F are cubic in Γ. Further organizingthe perturbation expansion of the interband pairing of agiven order n in a systematic way, we find several recur-sion relationships [61]. These can be compactly writtenin a matrix form as: ←− G ( n )12 F ( n )12 ! = g (cid:18) e f − f ∗ e ∗ (cid:19) ←− G ( n − F ( n − ! , (3)where g = Γ / [( ω + E k )( ω + E k )], e = ( iω − ǫ )( iω − ǫ ) − ∆ ∆ ∗ and f = − iω (∆ ∗ − ∆ ∗ ) + ∆ ∗ ǫ + ∆ ∗ ǫ . Forthe starting point of the recursion we use the first or-der anomalous interband Green’s function F (1)12 and thecorresponding normal propagator: ←− G (1)12 = Γ[∆ ∆ ∗ − ( iω − ǫ )( iω − ǫ )] / [( ω + E )( ω + E )]. From Eq. (3)we recognize that the Green’s functions to infinite or-der can be written as a geometric series, where the quo-tient is a two-by-two matrix. The formal criterion fora matrix geometric series to be convergent is that thenorm (i.e. the largest singular value) of the coefficientmatrix is <
1. From this we arrive at the condition Γ . ( ω + E ) / ( ω + E ) / , which translates into:Γ . p | ∆ || ∆ | , (4)meaning that the series is always convergent for suffi-ciently small Γ, provided that ∆ and ∆ are both finite.This allows us to sum the infinite series and we arriveat F = Γ[ iω (∆ − ∆ ) + ∆ ǫ + ∆ ǫ ] /D , where D =( ω + E )( ω + E ) − Γ [2( ǫ ǫ − ω ) − ∆ ∗ ∆ − ∆ ∗ ∆ ]+Γ .The expression for F is obtained by exchanging theband indices and we can also form the odd and evencombinations of F and F with respect to the bandindex: F odd12 ( k , iω ) = F − F iω Γ(∆ − ∆ ) /D (5) F even12 ( k , iω ) = F + F ǫ k + ∆ ǫ k ) /D. (6)The odd-band combination is directly seen to be odd infrequency, whereas the even-band combination has a con-ventional even-frequency dependence. This is fully con-sistent with Fermi-Dirac statistics for spin-singlet s -wavesuperconducting pairing. Furthermore, we see that in-terband pairing always requires a finite band hybridiza-tion and that the odd-interband pairing also requires∆ = ∆ . Equations (5)-(6) can be Fourier transformedto real space and then evaluated numerically, with theresult shown in Fig. 1. By definition, the odd-frequencypairing amplitude must be zero at ω = 0. Close to ω = 0,we find that the leading term in F odd is linear in ω . How-ever, the slope can initially be large and then abruptlychange sign to asymptotically go to zero for large ω , re-sulting in an approximate 1 /ω -dependence, which hasalso been found for some odd-frequency states [53, 54].By forming an analogue of Eq. (3) for the intrabandanomalous Green’s function in band 1 we find F = { ∆ [( iω ) − E ] − Γ ∆ } /D and the corresponding nor-mal Green’s function: G = { ( iω + ǫ )[( iω ) − E ] − Γ ( iω − ǫ ) } /D . The expressions for F and G are ob-tained by mutually exchanging band indices. The ex-pressions for the normal Green’s functions allow us tocompute the DOS using the standard formula: N ( E ) = − π Im Tr G ( E + iδ ) , δ → + . (7)Here the trace involves a sum over band and spin indicesand an integral over k -space. We use similar expressions,but unsummed over the band index, to define the partialDOS N and N .The total and partial DOS offer a direct connectionto experimental measurements on multiband supercon-ductors. In Figs. 2 - 4 we show numerically obtainedresults for the DOS, explicitly exploring the effect of in-terband pairing. To most clearly illustrate the effect ofinterband pairing we use two generic parabolic bands: ǫ j = ~ k / m j − µ j , with effective masses m = 20 m e , !" Figure 1: Odd- (a) and even- (b) interband pairing amplitudesin meV per nm when ∆ = 2 . = 2.5, 2.8, 4.5 meV(blue, orange, green), and Γ = 3 meV, with the band structurespecified in the main text. Inset: first order hybridizationcontributions to the interband pairing; F Γ G (top), −←− G Γ F (bottom). Solid (dashed) line represents the propagator inband 1 (2) and × the hybridization. m = 22 m e and distances from the bottom of the bandsto the Fermi level µ = 100 meV and µ = 105 meV. Forthis and other band structures we studied, the interbandeffects are most clearly visible when Γ is comparable to | ǫ − ǫ | for fixed k ≈ k F . The contributions to the DOSare obtained by numerical integration in the range includ-ing both Fermi surfaces and we use a smearing parameter δ = 0 .
01 meV. We have also independently verified theDOS results by exact diagonalization of the Hamiltonianin Eq. (1). In fact, we find a perfect agreement evenfar beyond the theoretical condition for convergence inEq. (4).We here especially showcase that unusual features inthe DOS are only seen when the conditions Γ = 0 and∆ = ∆ are both satisfied, which are exactly the twokey criteria for odd-frequency pairing, see Eq. (5). First,the DOS without any band hybridization, as shown inFig. 2(a), is just a sum of two BCS spectra with energygaps E g = ∆ and E g = ∆ , respectively, as expected.However, when we turn on hybridization, see Figs. 2(b-d), we see extra, very notable, dips in the DOS locatedbeyond the original gap edges E g , . These dips, symmet-ric around zero energy, clearly resemble superconductinggaps with their pronounced BCS-like coherence peaks.However, the DOS in the gap regions are not zero, butinstead equal to the partial DOS at these energies. Still,we refer to these features as hybridization-induced gaps. D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.5 meV, ∆ = 1 meV, Γ = 0 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.5 meV, ∆ = 1 meV, Γ = 0.5 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.5 meV, ∆ = 1 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.5 meV, ∆ = 1 meV, Γ = 6 meVNN N (a) (b)(c) (d) Figure 2: Total ( N ) and partial densities of states ( N , N )when ∆ = 2 . = 1 meV for different values ofΓ = 0 , . , , The hybridization-induced gaps grow in size and moveto higher energies for larger band hybridizations, andwe associate these features with interband superconduct-ing pairing. We note that beyond these hybridization-induced gaps, we see no other distinctive features as-sociated with interband pairing. Notably, there are nozero-energy or subgap states, otherwise often associatedwith odd-frequency pairing [43, 49, 50, 55–58]. Recentworks have pointed out that zero-energy states do notalways accompany odd-frequency pairing [41, 54, 59, 60],and odd-frequency, odd-interband pairing provides an-other example when this is not the case.In Fig. 3 we instead fix Γ = 3 meV, and ∆ = 2 . . Distinct hybridization-induced gaps arepresent at energies beyond the original gap edges, in-dependent on the relative size of the two original gaps.The only exception is exactly when ∆ = ∆ , thenthe hybridization-induced gaps completely disappear, de-spite the finite band hybridization, see Fig. 3(c). Detun-ing the value of ∆ slightly from that of ∆ results insmall, but noticeable, dips in the DOS, at the positionswhere the full gaps develop for increasing differences be-tween ∆ and ∆ . We thus find that the hybridization-induced gaps are only present in the DOS when bothΓ = 0 and ∆ = ∆ . These are exactly the two keycriteria for odd-frequency pairing, as seen in Eq. (5).In fact, only the odd-frequency interband pairing dis-appears at ∆ = ∆ , the even-frequency part is in gen-eral non-zero as soon as Γ = 0. Specifically, the even-and odd-frequency interband pairing amplitudes corre-sponding to the parameters in Fig. 3(c)-(e) are plottedin Fig. 1. From there it is clear that the even-frequencyinterband pairing is large and not changing significantlyaround ω = 0, while the odd-frequency part changesfrom identically zero in Fig. 3(c) to a notable non-zeroderivative at ω = 0 for Figs. 3(d)-(e). We can thus con- D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 0.5 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.5 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 2.8 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 4.5 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 7.5 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 10 meV, ∆ = 2.5 meV, Γ = 3 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 15 meV, ∆ = 2.5 meV, Γ = 3 meVNN N (a) (b)(c) (d) (e) (f)(g) (h) Figure 3: Total ( N ) and partial densities of states ( N , N )when ∆ = 2 . = 0 . , , . , . , . , . , ,
15 meV (a-h), with the bandstructure specified in the main text. clude that odd-frequency interband pairing is necessaryfor producing the hybridization-induced gaps. Detect-ing gaps beyond the original two superconducting gapsin multiband superconductors is therefore a clear sign ofthe presence of odd-frequency pairing. Intriguingly, ad-ditional gap features have already been reported in themultiband superconductor Ba Si [27].Finally, we also show that only one band has to benatively superconducting for the hybridization-inducedgaps to be present. In Fig. 4 we display how thehybridization-induced gap grows with increasing Γ when∆ = 0. Finite single-particle hybridization Γ results ina proximity-induced gap also in the second band, whichis manifested as a gap around zero energy, although al-ways smaller than E g = ∆ . For all parameter choicesin Fig. 4 the spectrum always has a gap at zero energy,i.e. there are no zero-energy states. This is not clearlyseen in Fig. 4(a) due to the finite smearing parameter,but it is clearly visible in exact diagonalization resultswhich do not suffer from the same problem. In addition,a finite Γ results in finite odd-frequency interband pair-ing, and we consequently also see hybridization-induced D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 3 meV, ∆ = 0 meV, Γ = 1 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 3 meV, ∆ = 0 meV, Γ = 2 meVNN N D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 3 meV, ∆ = 0 meV, Γ = 3 meVNN N (cid:2)(cid:3)(cid:4) D O S [ s t a t e s / e V / n m ] E - E F [meV] ∆ = 3 meV, ∆ = 0 meV, Γ = 5 meVNN N (a) (b)(c) (d) Figure 4: Total ( N ) and partial densities of states ( N , N )when ∆ = 3 meV and ∆ = 0, i.e. only one band with nativesuperconductivity, for different values of Γ = 1 , , , gaps beyond the ∆ gap, which also grow with Γ. Infact, these gaps are much more pronounced than theproximity-induced gap in band 2 at zero energy.In summary, we have studied the effect of single-quasiparticle hybridization or scattering in a two-bandsuperconductor. By performing perturbation theoryto infinite order in the hybridization term, we haveobtained the exact, fully frequency dependent, ex-pression for the interband pairing, which can be di-vided up into odd-frequency odd-interband and even-frequency even-interband pairing. The conditions for fi-nite odd-frequency interband pairing are (a) finite single-quasiparticle hybridization and (b) a non-zero differencebetween the original superconducting gaps; no appliedmagnetic field, inhomogeneity, or interface is required.Furthermore, we have shown that the DOS develops non-trivial gaps features with distinct coherence peaks be-yond the original gap edges only if the conditions forodd-frequency pairing are satisfied, otherwise the spec-trum just a sum of two BCS spectra. Detecting suchadditional gaps thus provides experimental evidence ofodd-frequency pairing in multiband superconductors.We would like to thank to K. Bj¨ornson, D. Kuz-manovski, T. L¨othman, and S. Nakosai for valuable dis-cussions. We acknowledge funding from the Wenner-Gren Foundations, the Swedish Research Council (Veten-skapsr˚adet), the G¨oran Gustafsson Foundation, and theSwedish Foundation for Strategic Research (SSF) (LKand ABS), and the European Research Council (ERC)DM-321031 (AVB). Work at Los Alamos was supportedby the US DOE Basic Sciences E 304 for the NationalNuclear Security Administration of the US Departmentof Energy under Contract No. DE-AC52-06NA25396. [1] J. Nagamatsu N. Nakagawa, T. Muranaka, Y. Zenitani,and J. Akimitsu, Superconductivity at 39 K in magne-sium diboride , Nature , 63 (2001).[2] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S.G. Louie,
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Band Hybridization Induced Odd-Frequency Pairing in Multiband Superconductors:Supplementary material
In this Supplementary Material we derive in detail theexact result (within perturbation theory) of the inter-band pairing amplitude in a two-band superconductorwith hybridized bands.The Hamiltonian for a generic multiband superconduc-tor can be written as (same as Eq. (1) in the main text): H = X kσ ǫ k a † kσ a kσ + ǫ k b † kσ b kσ + X kσ Γ( k ) a † kσ b kσ + H . c . + X k ∆ ( k ) a † k ↑ a †− k ↓ + ∆ ( k ) b † k ↑ b †− k ↓ + H . c .. (S1)It describes two superconducting bands coupled by a hy-bridization, or scattering, term Γ( k ), which we treat as aperturbation. We thus start with two copies of the singleband superconducting Green’s functions (same as Eq. (2)in the main text): G j F j F † j ←− G j ! = 1( iω ) − E j (cid:18) iω + ǫ kj ∆ j ∆ ∗ j iω − ǫ kj (cid:19) , (S2)which we use together with the hybridization term tobuild up the full Green’s functions of the system. Here G is the normal electron propagator, which we schemat-ically denote by , and ←− G is the hole propaga-tor denoted by . Moreover, F is the anoma-lous propagator, denoted by , and F † is itsHermitian conjugate . We will furthermoreuse solid lines to denote the propagators in band 1 anddashed lines for band 2. The hybridization term is rep-resented by × , which always connects two propagatorsfrom different bands with the same direction of the ar-row (due to momentum conservation).Now, we want to calculate the interband pairing am-plitude F , which corresponds to the sum of all pro-cesses of the type . The simplest processes,here indicated by the superscript (1), include just onescattering event × : F (1)12 = − = F Γ G − ←− G Γ F . The signof a particular process is simply given by ( − l , where l is the number of hole scattering events, i.e. scatteringconnecting two left-going arrows.Since F of any given order has to end with a right-pointing arrow in band two, there are only two possibili-ties, either it ends with or with .Similarly, it needs to start with a left-going solid arrow,which also limits the options. By drawing all possibleprocesses and translating them into formulas, we canwrite: F ( n )12 = −←− G ( n − + F ( n − . (S3) Carrying this procedure one step further, we write ←− G ( n − and F ( n − using F ( n − and ←− G ( n − : ←− G ( n − = F ( n − − ←− G ( n − (S4) F ( n − = F ( n − − ←− G ( n − . (S5)Plugging these into Eq. (S3) we arrive at: F ( n )12 = ←− G ( n − [ − ]+ F ( n − [ − ] . (S6)By a similar procedure we get: ←− G ( n )12 = ←− G ( n − [ − ]+ F ( n − [ − ] . (S7)Thus, we get a closed set of equations if we con-sider F and ←− G together, which is easiest done ina matrix formalism. Note that all arrow diagramscan be directly translated to specific formulas usingEq. (2), e.g. = − Γ ←− G ( − Γ) F =( − Γ) ( iω − ǫ )∆ / { [( iω ) − E ][( iω ) − E ] } .Summarizing, we arrive at the matrix recursion rela-tion in Eq. (3) in the main text: ←− G ( n )12 F ( n )12 ! = g (cid:18) e f − f ∗ e ∗ (cid:19) ←− G ( n − F ( n − ! , (S8)where g = Γ / [( ω + E k )( ω + E k )], e = ( iω − ǫ )( iω − ǫ ) − ∆ ∆ ∗ and f = − iω (∆ ∗ − ∆ ∗ ) + ∆ ∗ ǫ + ∆ ∗ ǫ .This is a matrix geometric series and is thus easilysummed giving: (cid:18) ←− G F (cid:19) = q (cid:18) − ge ∗ gf − gf ∗ − ge (cid:19) ←− G (1)12 F (1)12 ! , (S9)where q = [(1 − ge )(1 − ge ∗ ) + g | f | ] − , ←− G (1)12 = − =Γ[∆ ∆ ∗ − ( iω − ǫ )( iω − ǫ )] / [( ω + E )( ω + E )].Some terms cancel and we get F = qF (1)12 . Finally F = Γ[ iω (∆ − ∆ ) + ∆ ǫ + ∆ ǫ ] /D , where D =( ω + E )( ω + E ) − Γ [2( ǫ ǫ − ω ) − ∆ ∗ ∆ − ∆ ∗ ∆ ]+Γ .For the normal propagator G , needed for calculatingthe DOS, we create a matrix equation together with F †1