BCS Theory of Driven Superconductivity

Eur. Phys. J. B (2016) 89: 244 DOI: 10.1140/epjb/e2016-70528-1 THE EUROPEAN PHYSICAL JOURNAL B Regular Article

Andreas Komnik1 and Michael Thorwart2,3,a 1 Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany 2 I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstr. 9, 20355 Hamburg, Germany 3 The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany

Received 9 September 2016 Published online 9 November 2016 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2016

Abstract. We study the impact of a time-dependent external driving of the lattice phonons in a minimal model of a BCS superconductor. Upon evaluating the driving-induced vertex corrections of the phonon- mediated electron-electron interaction, we show that parametric phonon driving can be used to elevate the critical temperature Tc, while a dipolar phonon drive has no eﬀect. We provide simple analytic expressions for the enhancement factor of Tc. Furthermore, a mean-ﬁeld analysis of a nonlinear phonon-phonon interaction also shows that phonon anharmonicities further amplify Tc. Our results hold universally for the large class of normal BCS superconductors.

1 Introduction reduction of the electronic hopping amplitude, the re- sulting increase of the density of states near the Fermi Quantum many-body systems which are driven far away edge has been shown to enhance superconductivity [24]. from thermal equilibrium represent an increasingly fas- Using the nonequilibrium dynamical mean-field theory cinating realm of condensed matter physics, since recent for a strongly coupled electron-phonon system, a strong progress in the experimental techniques has made it pos- electron-mediated phonon-phonon interaction has been re- sible to manipulate condensed matter quantum states by vealed [25]. These theoretical approaches are all very ad- strong external fields [1]. Light can strongly modify phases vanced and specialized to particular classes of systems and of correlated quantum many-body systems. For instance, are rather successful in explaining experimental data for strong time-dependent fields can induce transient super- specific materials. Yet, it is still desirable to establish and conducting phases in different material classes [2–8]. More- analyze minimal models to reveal the fundamental mech- over, electromagnetic irradiation can induce a collapse of anisms in terms of simple and elegant analytical results. long-range ordered charge-density wave phases [9–12], de- Very recently, such a minimal model of a strongly driven construct insulating phases [13–15], or break up Cooper electron-phonon Hamiltonian has been analyzed upon us- pair quasiparticles [16–19]. ing Floquet formalism [26]. A Floquet BCS gap equation Conceptual insight into the possible physical mecha- is derived which calls for a numerical solution and does nisms has been greatly advanced recently [20–26]. In the not permit closed analytic results. presence of strong lattice anharmonicities, the nonlinear In this work, we aim to obtain a rather general and coupling of a resonantly driven phonon to other Raman- explicit analytical result to illustrate the driving-induced active modes leads to a rectification of a directly excited elevation of the critical temperature of a normal super- infrared-active mode and to a net displacement of the conductor by extending the conventional BCS theory. We crystal along the coordinate of all anharmonically cou- go beyond the existing approaches, which usually consider pled modes [20,21]. Selective vibrational excitation can the modification of the distribution function of charge car- also drive high-TC cuprates into a transiently enhanced riers (see e.g. [27]). We consider the standard Fröhlich- superconducting state. Moreover, on the basis of the non- type electron-phonon Hamiltonian with linear phonons equilibrium Keldysh formalism, partial melting of the su- subject to a time-dependent external driving. We show perconducting phase by the pump field has been iden- that a simple dipolar coupling of the driving field to tified [22]. Furthermore, an advanced extension of the the phonon displacement coordinates only yields a scalar single-layer t-J-V model of cuprates to three dimensions phase shift and does not modify the electron-phonon in- has been used to show that an optical pump can suppress teraction vertex. In contrast to that, a parametric driv- the charge order and enhance superconductivity [23]. In ing of the phonon frequencies strongly modifies the re- an effective approach on the basis of a driving-induced tarded Green’s function, thereby changing the effective electron-electron attraction in a fundamental way. In or- a e-mail: [email protected] der to quantify these effects, we introduce an elevation Page 2 of 5 Eur. Phys. J. B (2016) 89: 244 factor η of the critical temperature which can be directly energies ω and the higher is the number of electrons, for calculated in our approach. In the limits of weak and which the mutual interaction becomes attractive. This is strong driving fields, we obtain simple expressions for η, accompanied by an increase of the critical temperature Tc, which reveal how the critical temperature can be enhanced at which the superconducting gap vanishes. even if the driving is nonresonant. Finally, we show that a The critical temperature in a BCS superconductor in parametric phonon drive combined with a phonon-phonon a simplest model of an attractive constant potential of interaction can induce an additional elevation of the crit- strength V0 is given by (kB =1) ical temperature. This is apparent already on the level of −1/[V0ρ(EF )] a mean-field treatment of the nonlinear phononics. Tc ωDe , (6)

where ρ(EF ) is the density of the electronic states at the Fermi edge and ωD is the Debye frequency which fixes the 2 Minimal model of a driven BCS characteristic energy scale for the phonon degrees of free- superconductor dom. The expression in equation (6) can be considered as generic if one interprets V0 as an effective parameter The canonical modeling of the superconducting materials which measures the strength of the (in general energy and isbasedontheFröhlich-type Hamiltonian ( =1) momentum dependent) attractive potential. There are ba- sically three different options to increase Tc by changing λ † † H0 = Hψ[ck]+HΩ + √ (a + a−q)c ck, (1) one of the above parameters. We shall consider two of V q k−q k,q them: (i) the enhancement of the effective attraction V0; and (ii) the increase of ωD. † where Hψ[ck]= k(k − μ)ckck is a Hamiltonian of the One way to modify the denominator of equation (5) electronic conductance band, is to drive the phonons by strong electromagnetic external THz fields. The driving can induce phonon excitations † HΩ = Ωqaqaq (2) in sequential steps, in which the phonons are directly ex- q cited by applied EM field pulses. Alternatively, infrared- active phonon modes with a finite dipole moment can be describes the phonon degrees of freedom with the disper- excited, and due to nonlinear phonon coupling, normal sion Ωq and λ is the electron-phonon interaction strength. phonon Raman modes of the crystal are excited [20,26]. V is the volume of the sample. The deflection field Qq = We choose not to concentrate on these intricacies as they † aq + a−q of the phonons is the Fourier transform of the are strongly material-dependent and thus nonuniversal phonon coordinate. The BCS theory is build upon the fact and consider the driving as acting directly on the relevant that Qq can be integrated over, such that an exact effec- phonon mode. There are essentially two qualitatively dif- tive action ferent possibilities, the dipolar (or linear) driving where the drive couples to the phonon deflection field, and the 2 † S = S0 + λ dtdt ck+q(t)ck(t)G(q,t− t ) parametric (or quadratic) driving where the drive modu- q,k,k lates the phonon frequencies. † × ck−q(t )ck (t )(3)

3 The effect of phonon driving results. Here, G(q,t− t ) is the Green’s function (GF) of Q the deflection field q. Its retarded component is canoni- The dipolar phonon driving by an explicitly time- cally defined as: dependent driving field Δq(t) does not influence the re- GR q, q,ω GR(q, q,t− t)=−iΘ(t − t) tarded GF ( ). This immediately follows when we replace equation (2)by ×Qq(t)Qq (t ) − Qq (t )Qq(t). (4) † † HΩ = Ωqaqaq + Δq(t)(aq + a−q). (7) It generates an approximative interaction vertex ampli- q tude V (q,ω) of an effective electron-electron interaction mediated by the phonons and is at the heart of BCS the- The electron-phonon coupling strength λ is quite weak in ory of superconductivity. To lowest order in λ,onethen most of the known superconducting materials. For this obtains for the interaction vertex reason, the leading behavior of the GR(q, q,ω)isdomi- 2 nated by the contribution of the phonon subsystem only. 2 R 2Ωqλ V q,ω λ G q, −q,ω , Solving the equations of motion for HΩ, one readily finds ( )= 0 ( )= 2 2 (5) ω − Ω −iΩqt q aq(t)=[aq(0) + f(t)]e ,where

t where ω is the energy transfer during the scattering of the 2 2 f t −i dtΔ t eiΩqt electron pair. Obviously, if ω <Ωq, the eﬀective inter- ( )= q( ) action is attractive, thus leading to the Cooper instabil- ity and superconducting ground state [28]. In general, the is a simple time-dependent scalar phase shift. As the re- larger the overall scale of Ωq is, the larger is the range of tarded GF is a commutator of ﬁelds, a mere shift of them Eur. Phys. J. B (2016) 89: 244 Page 3 of 5

⎧ ⎫ ny ny ⎨ ∞ (ω − Ωq)Jn (ω + Ωq)Jn − ⎬ R −i2Ωq ω−Ωq ω+Ωq G q, q ,ω iδ − i − , ( )= −q,q 2 2 2 2 2 2 (11) ⎩ ω − Ωq ω − Ωq − nΓq/ ω Ωq − nΓq/ ⎭ n=1 ( ) ( 2) ( + ) ( 2)

1 ny 1 xnΔq J¯n = dxJn (13) ω − Ωq 2 −1 ω − Ωq n 2 1 nΔq 1+n 3+n nΔq = 1F2 ; , 1+n; − (14) (n +1)! 2(ω − Ωq) 2 2 2(ω − Ωq)

does not affect the GF at all. Thus, we conclude that slightly smaller than the typical Debye frequency of su- within our approximation the linear driving does not af- perconducting materials. Hence, we may average over the fect the conventional BCS superconductivity picture. period of the external driving with respect to τ.Forthe The parametric driving enters via the Hamiltonian time-averaged Bessel functions, we then obtain † see equations (13) and (14) above HΩ = [Ωq + Δq(t)]aqaq. (8) q for even n and zero otherwise. Here, 1F2 denotes the hy- pergeometric function [29]. Its maximum is of the order of It is, e.g., realized indirectly by resonantly driving 1forn<2 for any argument and it decays exponentially infrared-active phonon modes with a finite dipole moment, for n>2. Hence, we may focus on the lowest order term which couple quadratically to normal Raman modes of the n = 2 only. The physical meaning is immediate. n denotes crystal [1,3,20,21] or by the quadrupole component of an the number of phonons which participate in the renormal- electromagnetic field. The trivial case is the static driving, ization of the GF by the vertex. The odd phonon numbers i.e., Δq(t)=Δq, which simply is an increase of phonon do not contribute for symmetry reasons. The larger n,the frequencies. As the Debye frequency rises as well, an in- more efficient is the mutual cancellation during averaging. crease of Tc is obvious. This effect is known and is ex- As a result, only the two-phonon process survives which perimentally detected in crystals subject to high pressure. is the parametric resonance. In the dynamical case, the solution for the time evolu- In order to quantify the enhancement of the interac- −iα(t) tion equation is obviously aq(t)=aq(0)e with the tion vertex around the Fermi edge, we define the enhance- t R R ment factor η = G (q, q , 0)/G0 (q, q , 0) as the ratio of phase α(t)=Ωqt + 0 dt Δq(t ). Then, the retarded GF of equation (4) follows as: the two retarded GF in the low-energy limit. Moreover, we may exploit the asymptotic behavior of the hyper- R n x G (q, q ,t,t)=−iδ−q,q Θ(t − t ) geometric function for =2for 1intheform 1 x2 F / / , −x2 x2/ O x4 ±i[α(t)−α(t)] 6 1 2[3 2; 5 2 3; ]= 6+ ( ) to assess the × (±1)e . (9) quantitative behavior of the GF in equation (4)inthe ± vicinity of the Fermi edge ω → 0. Hence, for weak driving Δq Ωq,weobtain Without restricting the generality, we henceforth assume periodic time-dependent driving in the form R 2 G (q, q , 0) Δq η = =1+ . (15) GR q, q, Ω2 − Γ 2 Δq(t)=Δq cos(Γqt), (10) 0 ( 0) 6( q q ) η>1 implies a relative enhancement of the attractive in- where Δq is the strength and Γq is the frequency of the driving. After a Fourier expansion with respect to the time teraction around the Fermi edge and thus an increase in T η difference t − t, we obtain a result in terms of the nth c,since enters in the expression for the critical temper- ordinary Bessel function [29]: ature as a factor renormalizing the electron-phonon coupling strength according to V0 → ηV0. This occurs for see equation (11) above subresonant driving Ωq >Γq, which is the most realistic where regime from the point of view of contemporary experi- (t + t) ments, and can, at least in principle, become quite large. y = Δq sin Γq (12) Ω <Γ 2 In the opposite case of superresonant driving q q, there is a decrease of Tc. This kind of transition should be explicitly depends on the evolution time τ =(t + t )/2. experimentally observable. We recover the zero-order contribution of equation (5)as In the limit of strong driving Δq Ωq,weobtainwith the first term of the r.h.s. of equation (11). Moreover, the 1 2 2 2 6 x 1F2[3/2; 5/2, 3; −x ]=1/(2|x|)+O(1/x )forx 1 multiphonon parametric resonances are apparent from the the enhancement factor denominators when 2˜ωq = nΓq. Ω3 To proceed, we exploit that the typical driving fre- η q . =1+ 2 2 (16) quency in the experiments is in the THz regime, which is 2Δq(Ωq − Γq ) Page 4 of 5 Eur. Phys. J. B (2016) 89: 244

It shows the similar dependence of Ωq and Γq. Although a strongly driven system, where the phonon population an estimate of the validity region of our approximation is Nq is determined by the irradiation field. In this case, more involved, we believe our results to hold for η ∼ 1–2. the phonon subsystem is stiffer and is characterized by an effectively enhanced Debye frequency ωD. In order to illustrate this feature, we consider the simplest case Ωq = vsq, 4 Nonlinear phononics where vs is the bare sound velocity of the crystal. Then, Ωeff(q) ≡ Ωq + χ(q)Nq ≈ (vs + χ1Nq)q. Hence, the crit- Next, we address the role of the phonon anharmonicity. ical temperature is renormalized according to Tc → ξTc On the microscopic level, it arises due to a nonlinear inter- with ξ =1+Nqχ1/vs and is thus increased. action between the phonons. Usually, one encounters two Hence, if a nonlinear superconducting material is ex- different kinds: three- and four-phonon interaction pro- posed to strong external parametric driving, the critical cesses. They are described by the Hamiltonians temperature can be increased by two effects, so that equation (6)ismodifiedto H3 = M3(q, k)QkQqQ−k−q, (17) −1/[ηV0ρ(EF )] q,k Tc ξωDe , (22) H M q, k, p Q Q Q Q , 4 = 4( ) k q −k−p −q+p (18) when η, ξ > 1. Overall, the theory is expected to hold q,k,p quantitatively up to Δq/Γq 1. It is important to realize that the enhancement factor enters in the exponent of Tc. where M3,4 are the corresponding interaction amplitudes. As a rule, they are small and the appropriate way to assess their influence is the perturbation theory. It turns out that the three-phonon self-energy vanishes exactly for 5 Discussion and conclusions homogeneous systems and is strongly suppressed in lat- tices with high symmetry groups. Hence, we focus on the By considering a minimal model of a Fröhlich-type BCS four-phonon process. We are interested in the effective Hamiltonian of a normal superconductor in presence of properties of one single phonon mode. Therefore, the most a time-dependent periodic electromagnetic driving of the important contribution is expected to be given by the non- phonons, we illustrate the basic physical mechanisms by diffractive scattering processes of the given phonon mode which the critical temperature Tc can be elevated. We p k q on itself, when =0and = . The underlying effective show that while a dipole (linear) driving cannot change Hamiltonian [30] can be inferred from the above one and Tc of the material, quadratic (parametric) driving can en- one finds hance the effective attractive phonon-mediated electron- † † † electron interaction and thus increase the critical temper- HΩ = Ωqa aq + χ(q)a a aqaq. (19) q q q ature. The effect in this minimal model is illustrated in q terms of the enhancement factor of the interaction ver- The anharmonicity coefficient χ(q) can be obtained from tex caused by the external driving. In the limits of weak and strong external phonon driving, we find simple an- M4(q, q, 0) and is expected to be small. Since phonons alytic results for the vertex enhancement. Furthermore, at rest do not exist we can write χ(q) ≈ χ1q,where q = |q|. Although in a superconducting material at low although an additional phonon anharmonicity does not † change Tc in BCS superconductors held at equilibrium, temperatures the phonon expectation value aqaq = Nq is strongly suppressed, this is not the case in presence of nonlinear phononics can provide an additional contribu- T an external drive. Invoking a mean field approximation, tion to the elevation of c, in that the effective Debye fre- the effective Hamiltonian is found to be quency is renormalized. Finally, we note that the external phonon drive also increases electron scattering, which in † † HΩ ≈ Ωqaqaq + χ(q)aqNqaq general suppresses Cooper pairing. Yet, it has been shown q recently [26] that the dynamic enhancement of the forma- tion of Cooper pairs addressed here dominates over the Ω χ q N a† a . = [ q + ( ) q] q q (20) increase of the scattering rate. These results in terms of q a minimal model shed new light on the essential ingredi- ents needed for manipulating the characteristics of a BCS In equilibrium and without external driving, Nq is deter- superconductor. A detailed analysis of the quasiparticle mined from the self-consistency condition decay processes and their interplay with enhanced inter- −1 action vertex is an obvious avenue for further research [32]. β(Ωq+χqNq) Nq = e − 1 , (21) Another source of Tc suppression might stem from thermal phonons, the effect of which can be taken into account and turns out to be smaller in comparison to the linear along the lines of references [33,34]. In future work, one system with χq = 0. For this reason, the impact of the an- also could allow for more realistic interactions, as is, for harmonic phonon subsystem on the electronic properties instance, shown in reference [35]. These effects are, how- is negligible and does not induce any appreciable change ever, to a larger degree material-dependent and thus less in Tc without driving [31]. This is completely different in universal. Eur. Phys. J. B (2016) 89: 244 Page 5 of 5

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