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Theory of Laser-Driven Nonequilibrium Superconducfvity Michael Sentef

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Theory of Laser-Driven Nonequilibrium Superconducfvity Michael Sentef Theory of laser-driven nonequilibrium superconduc7vity PRB 92, 224517 (2015) Theory collaborators: A. F. Kemper, B. Moritz, J. K. Freericks, T. P. Devereaux, PRB 93, 144506 (2016) A. Georges, C. Kollath Michael Sentef New3SC Bled, September 14, 2016 Max Planck Institute for the Structure and Dynamics of Matter Pump-probe spectroscopy (1887) • stroboscopic inves7gaons of dynamic phenomena Muybridge 1887 Max Planck Institute for the Structure and Dynamics of Matter 2 Pump-probe spectroscopy (today) • stroboscopic inves7gaons of dynamic phenomena TbTe3 CDW metal J. Sobota et al., PRL 108, 117403 (2012) F. SchmiA et al., Science 321, 1649 (2008) Image courtesy: J. Sobota / F. SchmiA Max Planck Institute for the Structure and Dynamics of Matter 3 Ultrafast Materials Science Understanding the nature of QuasiparRcles § Relaxaon dynamics PRL 111, 077401 (2013) PRX 3, 041033 (2013) PRB 87, 235139 (2013) PRB 90, 075126 (2014) arXiv:1505.07055 arXiv:1607.02314 Understanding ordered phases § Collec7ve oscillaons § Compe7ng order parameters PRB 92, 224517 (2015) PRB 93, 144506 (2016) CreaRng new states of maAer § Photo-induced phase transi7ons § Floquet topological states Image courtesy: Nature Commun. 6, 7047 (2015) D. Basov arXiv:1604.03399 Max Planck Institute for the Structure and Dynamics of Matter 4 Outline • Keldysh Green funcons • Ordered states: Driven superconductors – Higgs amplitude mode oscillaons for op7cal pumping (1.5 eV laser) PRB 92, 224517 (2015) – light-enhanced superconduc7vity via hopping control PRB 93, 144506 (2016) Max Planck Institute for the Structure and Dynamics of Matter 5 2 been successfully used to describe the melting of a charge non-dispersive optical phonon with energy ⇥. As our density-wave state24 and the time evolution of electrons starting point, we use the Holstein model which couples in correlated metals25,26 via tr-ARPES and tr-reflectivity. a band of electrons to a single species of optical phonon: We connect the microscopic interaction parameters of 1 the electron-phonon interaction directly to the observed = (k)c† c + ⇥ b† b + H k k q q 2 timescales, and discuss the extension of the results ob- k q ⇧ ⌃ tained here to more general models. + ck† +qck bq + b† q − k,q,i ⇤ ⌅ II. METHOD We include the electron-phonon interactions in the Migdal limit, which is appropriate for weak coupling. We describe the time-evolution of the system via the Furthermore, since we are primarily interested in the re- non-equilibrium Keldysh formalism. All Green’s func- sponse of the electronic system, we will limit the discus- tions have two time arguments, where each time is lo- sion to just the e⇤ects of the phonons on the electrons, cated on the Keldysh contour (see Fig. 1). The sys- and will neglect the feedback of the electronic system on tems starts in equilibrium at a temperature T and time the phonons. t = tmin, and evolves until t = tfinal. The time-ordered, For a non-dispersive optical phonon, the electronic self- energy is tmin tmax 2 0 0 Σ(t, t⇥)=ig D (t, t⇥)Gk(t, t⇥) k where g is the electron-phonon coupling strength. The 0 bare phonon Green’s function D (t, t⇥)is tmin iβ 0 − D (t, t⇥)= i [n (⇥)+⇥ (t, t⇥)] exp (i⇥(t t⇥)) − B c − i [n (⇥)+1 ⇥ (t, t⇥)] exp ( i⇥(t t⇥)) FIG. 1: Keldysh contour used in the description of the double- − B − c − − time Green’s functions and self-energies. where nB(⇥) is the Bose function at temperature T : nB(⌅)=1/(exp(⌅/T ) 1). In the following, we will use anti-time-ordered, lesser and greater Green’s functions various values of the electron-phonon− coupling strength are formed by selecting t and t⇥ on appropriate parts of Non-Equilibriumg and the optical phonon Keldysh frequency Formalism⇥. the contour (see Refs. 1–3,7,8). With the self-energy above, we solve the Dyson equa- The driving fields are included directly into the propa- Beyond BCS tion gators via the Peierls substitution in the standard double- time formalism, which leads to the bare non-equilibrium 0 0 00 GGk(t, t(⇥!)=)=Gk(t,G t⇥)+(!)+dt1dtG2Gk((!t,) t1⌃)Σ((!t1),tG2)G(k(!t2),t⇥) Green’s function k k k k self-energy Σ: 0 This can be done by casting the Dysonelectron-electron equation scattering as G (t, t⇥)=i [n ((k)) ⇥ (t, t⇥)] k F − c a matrix equation. However, for the caseelectron of-phonon electron- scattering t ... exp i dt¯ (k A(t¯)) phonon coupling, better numerical stability can be ob- ⇥ − − tained by expanding the integral through Langreth rules ⌥ t0 and solving the equations of motion for the retarded, real- where nF is the Fermi function at temperature T : imaginary, and lesser Green’s functions.4,5 This leads to nF (⌅)=1/(exp(⌅/T ) + 1), t and t⇥ lie on the Keldysh a set of Volterra integrodi⇤erential equationsInclude thatthe effects can of be driving contour, ⇥c is the contour-ordered Heaviside function, solved via…and about its history standard numerical integration.field6 throughWe find time- that (k) is the single-particle dispersion, which we choose the solution of the Dyson equation bydependent integrating electronic the dispersion to be a square lattice tight-binding model with nearest- Volterra equations leads to more stable and inherently neighbor and next-nearest-neighbor hoppings Vnn =0.25 System knows about its thermal ini7al causal algorithms. Some details about numberε(k) → ofε time(k,t) eV and V =0.075 eV, and a chemical potential state… nnn points go here.Max Planck Institute for the Structure and Dynamics of Matter 6 µ = 0.255 eV. − The pulse that is of direct interest to pump-probe ex- periments is, by nature, a propagating light pulse; this (k)= 2Vnn (cos kx + cos ky)+4Vnnn cos kx cos ky µ − − implies an oscillating field without a zero-frequency com- A(t) is the vector potential at time t, which is related ponent. We model the pump via an oscillating vector po- to the electric field via A(t)= E(t)dt. Here, we use tential along the (11) direction with a Gaussian profile, the convention thath ¯ = c = e −=´ 1, and we work in the where we assume that the field is slowly varying spatially Hamiltonian gauge, i.e. the scalar potential is set to zero. and thus neglect the spatial dependence: Energies and frequencies are measured in units of eV. 2 To illustrate the momentum-dependent quasiparticle Fmax (t t0) A(t)=(ˆx + ˆy ) sin(⌅pt)exp − 2 relaxation rates we introduce coupling of electrons to a ⌅p − 2⇤ ⇥ 2 Holstein-Hamiltonian. Our model contains both the re- Holstein Hamiltonian2 (el-ph model) tarded pairing interaction mediated by phonons as well as dissipation of heat into the lattice.Model The and change Method in = ✏(k,t)c† c + ⌦γ b† b Holstein-Hamiltonian. Our model contains both the re- Holstein Hamiltonian (el-ph model) H kσ kσ q,γ q,γ the lattice distortion is modelled by a slow change of kσ q,γ tarded pairing interaction mediated by phonons as well X X the electronic hopping amplitude J to a smaller value 0 † † as dissipation of heat into the lattice. The change in = ✏(k,t)ck† σckσ + ⌦γ bq†,γ bq,γ gγ ck+qσckσ bq,γ + b q,γ (1) Jf . Due to this change the electronicH density of states − − the lattice distortion is modelled by a slow change of kσ q,γ q,γ,σ close to the Fermi-surface is enhance,X which results in X X ⇣ ⌘ the electronic hopping amplitude J to a smaller value 0 • electrons (2D square la.)† + spectrum† of phonons + el-ph coupling (Holstein) an increase equilibrium order parameter ∆gfγ(seeck+q Fig.σckσ1).bq,γ +withb q, fermionicγ creation(1) operators c† for dimension- J . Due to this change the electronic density of states − − kσ f These equilibrium considerations lead• Migdal-Eliashberg (1st Born)q one,γ,σ to expect that + phonon heat bath approximaon close to the Fermi-surface is enhance, which results in X ⇣ less momentum⌘ k =(kx,ky) and spin σ = , on a for adiabatic changes of the hopping, the order param- two-dimensional square lattice with dispersion"✏(#k,t)= an increase equilibrium order parameter ∆f (see Fig. 1). eter should increase, followingwith fermionic thesuperconductor: equilibrium creation phase operators di- ck† σ for dimension- These equilibrium considerations lead one to expect that 2J(t)(cos kx + cos ky). The explicit time dependence agram (Fig. 1(b)). In materialsless momentum [15], typicalk =( timekx,k scalesy) and spin− σ = , on a for adiabatic changes of the hopping, the order param- of the electronic" # hopping amplitude J(t)mimicsade- for changes of the latticetwo-dimensional throughnormal nonlinear square phonon lattice cou- with dispersion ✏(k,t)= eter should increase, following the equilibrium phase di- formation of the ionic lattice induced via a nonlinear pling are fractions of a picosecond,2J(t)(cos k whichx + cos isky a). nonadia- The explicit time dependence agram (Fig. 1(b)). In materials [15], typical time scales − anomalous coupling to an IR active optical phonon driven by the batic change in particularof for the the electronic slow collective hopping dynam- amplitude J(t)mimicsade- for changes of the lattice through nonlinear phonon cou- THz light pulse [15]. We assume for t<⌧ a linear ics of the superconductingformation condensate. of the Hence, ionic lattice it is an induced via a nonlinear t pling are fractions of a picosecond, which is a nonadia- ramp J(t)=J0 +(Jf J0) ⌧ and for t>⌧ the con- important open questioncoupling on which to time an IR scales active the super-optical phonon driven by the − batic change in particular for the slow collective dynam- stant Jorder(t)= parameterJf with Δ initial, hoppingel-ph single- amplitudepar7cle J0 = 0.25 THz light pulse [15].
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