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Dynamical Cooper Pairing in Non-Equilibrium Electron-Phonon Systems

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Dynamical Cooper Pairing in Non-Equilibrium Electron-Phonon Systems Dynamical Cooper pairing in non-equilibrium electron-phonon systems Michael Knap,1 Mehrtash Babadi,2 Gil Refael,2 Ivar Martin,3 and Eugene Demler4 1Department of Physics, Walter Schottky Institute, and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany 2Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA 3Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA 4Department of Physics, Harvard University, Cambridge MA 02138, USA (Dated: December 16, 2016) We analyze Cooper pairing instabilities in strongly driven electron-phonon systems. The light-induced non- equilibrium state of phonons results in a simultaneous increase of the superconducting coupling constant and the electron scattering. We demonstrate that the competition between these effects leads to an enhanced supercon- ducting transition temperature in a broad range of parameters. Our results may explain the observed transient enhancement of superconductivity in several classes of materials upon irradiation with high intensity pulses of terahertz light, and may pave new ways for engineering high-temperature light-induced superconducting states. The application of a strong electromagnetic drive has phonons into squeezed quantum states, which can have sig- emerged as a powerful new way to manipulate material prop- nificantly enhanced coupling to electrons, and, finally, the pe- erties [1,2]. Long-range charge density wave order has been riodic modulation of system parameters leading to Floquet melted by light [3–6], insulators have been destroyed [7– states. Of these effects, we find that phonon squeezing uni- 9], and the breaking of superconducting pairs has been ob- versally leads to an enhancement of Tc, by potentially a large served [10–14]. Even more remarkably, long range order can factor [29]. The periodic modulation of the system parame- be not only destroyed but also created by exciting samples ters (Floquet) can also enhance Cooper pairing, via a super- with light. For instance, the dynamic emergence of transient conducting proximity effect in time rather than space. By spin-density wave [15], charge-density wave [16], as well as contrast, the static renormalization of the system parameters superconducting order [17–21] has been demonstrated. These can, depending on material-specific details, either enhance or observations lead to the very fundamental theoretical ques- suppress Tc. tions of the origin of the light induced order, including: What Driving phonons into a highly excited state, unfortunately, is the mechanism for the emergence of transient collective also leads to an increased electron-phonon scattering rate, behavior? What determines the lifetime of transient ordered which weakens Cooper pairing. We analyze the competition states? How robust and universal are the observed phenom- between the enhanced Cooper pair formation and Cooper pair ena? Answers to these questions may hold the key to a breaking and show that the enhancement of pairing can dom- novel route for achieving ordered many-body states by peri- inate in a broad parameter range resulting in signatures of su- odic driving as opposed to cooling; a subject that has attracted perconductivity that appear at higher temperatures compared considerable theoretical attention recently [22–28]. to equilibrium. We note that the predicted enhancement of We propose a general mechanism for making a normal Tc should be understood as a transient phenomenon, since in- conducting metal unstable toward Cooper-pair formation by elastic scattering of electrons with excited phonons will even- irradiation with light. The key ingredient is the nonlinear tually heat the system and destroy the superconducting order. coupling between optically active infrared phonons and the Even though our study is motivated by specific experiments, Raman phonons that mediate electron-electron attraction, re- the minimal models that we introduce and study are intended sponsible for superconductivity (Fig.1). Depending on the to elucidate the qualitative origins of these effects, rather than form of the nonlinearity (Tab.I), several effects can arise. to provide detailed material-specific predictions. These include the parameter renormalization of the time- Furthermore, the proposed mechanism can be readily ap- averaged Hamiltonian, the dynamic excitation of the Raman plied to many other types of long-range order such as charge- density and spin-density waves, by performing the instability analysis in the appropriate channels. TABLE I. Types of the phonon nonlinearities and their static and arXiv:1511.07874v3 [cond-mat.supr-con] 15 Dec 2016 non-equilibrium effects on the system. The renormalization of the I. ROLE OF THE PHONON NONLINEARITY parameters leads to an effectively static Hamiltonian, while phonon squeezing and the periodic Floquet modulation of system parameters are purely dynamical. We now describe the consequences of different types of static renorm. dynamical periodic optically-accessible phonon nonlinearities. The pump pulse of parameters squeezing Floquet directly couples to infrared-active phonon modes, which have IR 2 R a finite dipole moment. Since the photon momentum is negli- I: (Q0 ) Q0 X × X IR 2 R R gible compared to the reciprocal lattice vector, the drive cre- II: (Q0 ) QkQ−k XXX ates a coherent phonon state at zero momentum, QIR (t) = IR R R q=0 III: Q0 QkQ−k × XX E cos Ωt, where Ω is the drive frequency and E is proportional 2 5 enhancement of Tc >100% pump pulse 1 80% parametric coherently driven resonance infrared phonons 2 60% Raman modes excited 40% by nonlinear coupling 3 to infrared phonons 20% pump frequency enhancement of SC temporal coupling 4 proximity effect 0% < -10% t pump time driving amplitude A FIG. 1. Dynamical enhancement of the superconducting transition temperature. A schematic representation of the physical processes leading to light-induced superconductivity: (1) The pump pulse couples coherently to (2) an infrared-active phonon mode which in turn (3) via nonlinear interactions drives Raman phonons that are responsible for superconducting pairing. The non-equilibrium occupation of the Raman phonons (4) universally enhances the superconducting coupling strength γ, which is a product of the density of states at the Fermi level NF and the induced attractive interaction between the electrons U, and hence (5) increases the transition temperature Tc of the superconducting state. We calculate the relative enhancement of Tc compared to equilibrium (Tc − Tc;eq)=Tc;eq by taking into account the competition between dynamical Cooper pair-formation and Cooper pair-breaking processes, as a function of the pump frequency Ω=!¯ and the driving amplitude A. The data is evaluated for linearly dispersing phonons with mean frequency !¯, relative spread ∆!=!¯ = 0:2, and negative quartic couplings of type II between the Raman and infrared-active modes; Tab.I. Moreover, the electron-phonon interaction strength is chosen to give an equilibrium effective attractive interaction U=W = 1=8 that is weak compared to the bare electronic bandwidth W . The static renormalization of Raman modes leads to the uniform increase of Tc with increasing driving amplitude A, the squeezed phonon state manifests in the strong enhancement near parametric resonance Ω ∼ !¯, and the temporal proximity effect dominates near Ω=!¯ ∼ 0. to the drive amplitude. In the presence of phonon nonlinear- II. A MINIMAL ELECTRON-PHONON MODEL ities, the driven infrared-active phonon mode couples to Ra- R man modes Qq of the crystal [18, 22, 30, 31], which in turn We illustrate the aforementioned effects by using a model can couple to the conduction electrons. with Frohlich-type¨ electron-phonon interactions that couple with strength gk to the displacement of the Raman phonons R R Qk (Pk is the conjugate momentum) to the local electron den- sity niσ: There are three leading types of phonon nonlinearities X X H^ = −J cy c + (P RP R + !2QRQR ) which can have static and dynamic effects (Tab.I). A static el-ph 0 iσ jσ k −k k k −k renormalization of the Hamiltonian parameters arises from hiji,σ k IR r phonon nonlinearities that involve even powers of Q , since X 2!k 0 + g eikri QR n : (1) these terms have finite time averages. For phonon nonlin- V k k iσ earities of type I (Tab.I), this leads to a static displacement ikσ of the zero-momentum Raman phonon while for nonlineari- We consider dispersive optical phonons !k with mean fre- ties of type II the frequency of the Raman phonons is modi- quency !¯ and spread ∆!. Here, J0 is the bare electron hop- fied, cf. App.A. Both affect the superconducting instability, ping matrix element and V the volume of the system. Further, causing either enhancement or suppression; the analysis can we assume a quartic nonlinearitiy of type II. The phonon drive be done with a standard equilibrium Migdal-Eliashberg for- term, introduced for t > 0, reads malism [32]. Here we will focus, however, primarily on the dynamical effects. There are two types of dynamical effects ^ X 2 R R Hdrv(t) = − !kAk(1 + cos 2Ωt)Qk Q−k; (2) that can be distinguished: First, a simple modulation of sys- k tem parameters, which makes system instantaneously more 2 2 or less superconducting, and, second, dynamical squeezing of where Ak = −ΛkE =2!k and E is the amplitude of the driven phonons, which is an explicitly quantum effect. Notably, both infrared-active phonon. dynamical effects lead to an increased superconducting insta- The effect of the drive is twofold. First, there is a static bility temperature. contribution, which renormalizes Raman phonon frequencies. 3 For negative nonlinearities, Λk < 0, phonons are softened c,eq (a) dyn. with pair breaking 2 dyn. \wo pair breaking by !k(1 − Ak). The mode softening suppresses the electron tunneling, increases the density of states and thus the over- equilibrium all pairing strength. Second, the time dependent part of the drive (2) realizes a parametrically driven oscillator which dy- namically generates phonon squeezing correlations and can strongly suppress the electron tunneling matrix element [33].
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