Cavity quantum-electrodynamical polaritonically enhanced superconductivity
Laser control of solids has so far mainly been discussed in the context of strong classical nonlinear light-matter coupling in a pump-probe framework. Here we propose a quantum-electrodynamical setting to address the coupling of a low-dimensional quantum material to quantized electromagnetic fields in nanocavities. Using a protoypical model system describing FeSe/SrTiO, we study how the formation of phonon polaritons at the 2D interface of the material modifies its superconducting properties in a Migdal-Eliashberg simulation. We find that through highly polarizable dipolar phonons, cavity-induced superconductivity is possible at temperatures above the bare critical temperature of the system. Our results demonstrate that quantum cavities enable the engineering of fundamental couplings in solids paving the way to unprecedented control of material properties.
Strong coupling and manipulation of matter with photons in quantum-electrodynamical (QED) environments is becoming a major research focus across many disciplines. Among the topics with large potential are the creation of exciton-polariton condensates byrnes_exciton-polariton_2014 , polaritonic chemistry galego_cavity-induced_2015 ; ebbesen_hybrid_2016 ; flick_kohnsham_2015 ; flick_atoms_2017 and transport hagenmuller_cavity-assisted_2018 , quantum nanoplasmonics tame_quantum_2013 , light-induced topology lindner_floquet_2011 ; claassen_all-optical_2016 ; hubener_creating_2017 and magnetism in 2D materials shin_phonon-driven_2018 , and novel spectroscopies ruggenthaler_quantum-electrodynamical_2018 . In condensed matter, the search for control knobs that allow to design properties of quantum materials is an ongoing broad research effort basov_towards_2017 . One possible route is to employ the nonequilibrium dynamics and coherent manipulation of quantum many-body systems with ultrashort laser pulses rini_control_2007 ; forst_nonlinear_2011 ; zhang_dynamics_2014 ; mitrano_possible_2016 ; sentef_theory_2017 ; pomarico_enhanced_2017 ; kennes_transient_2017 ; sentef_light-enhanced_2017 ; tancogne-dejean_ultrafast_2017 . However, in these cases “classical” light has been typically used. Here we propose a new route towards manipulating microscopic couplings in solids and inducing ordered phases especially at interfaces and in two-dimensional materials.
The discovery of enhanced superconductivity in monolayer FeSe on SrTiO liu_electronic_2012 ; qing-yan_interface-induced_2012 ; huang_monolayer_2017 and its possible relation to a cross-interfacial electron-phonon coupling lee_interfacial_2014 ; rademaker_enhanced_2016 ; zhang_role_2016 has stimulated considerable interest with an ongoing open debate zhou_dipolar_2017 ; wang_phonon_2017 ; song_phonon-enhanced_2017 ; jandke_unconventional_2017 ; lee_what_2015 ; kulic_electronphonon_2017 . Irrespective of the outcome of this debate, the interfacial phonon mode under consideration is of particular interest for light-control purposes as it has a dipole moment implying bilinear phonon-photon coupling, while at the same time the phonon also couples bilinearly to in-plane FeSe electrons, with a vertex that is strongly peaked for small momentum transfers known as forward scattering. This combination of features is due to the high degree of anisotropy owing to the interfacial structure. Here we employ a prototypical model system, related to FeSe/SrTiO, for such extreme forward scattering to investigate how photon-phonon coupling in cavities can affect electron-boson coupling and boson-mediated superconductivity.
ii.1 Setup: Two-dimensional material inside a cavity
In Figure 1A we show the setup for a two-dimensional material inside a QED cavity environment with perfectly reflecting mirrors. The mirrors confine the photon modes inside the cavity and can lead to strong light-matter coupling even when only vacuum of the electromagnetic field is considered todorov_ultrastrong_2010 ; todorov_few-electron_2014 . Specifically, we propose a layered structure of a 2D material (e.g., monolayer FeSe) on a dielectric substrate with large dielectric constant (e.g., SrTiO) that further helps confine the cavity photon modes of interest.
For the particular example of FeSe/SrTiO, the effect of the cavity is to couple the electromagnetic field of the photons polarized along the direction, perpendicular to the interfacial plane, to a cross-interfacial phonon mode. Importantly, here we go beyond the often-employed rotating-wave and dipole approximations for the light-matter interaction and use full minimal coupling including the and terms (see Section B of the Supplementary Materials), which makes the theory manifestly gauge-invariant and avoids unphysical divergences. The phonon has a dipole moment along that involves motion of the O and Ti ions in the topmost layer of SrTiO, spatially very close to the FeSe monolayer. Specifically, one quasi-dispersionless optical Fuchs-Kliewer phonon at 92 meV zhang_role_2016 has been identified as the most relevant phonon mode that strongly couples to the FeSe electrons both in angle-resolved photoemission lee_interfacial_2014 and high-resolution electron energy loss spectroscopies zhang_role_2016 . The influence of screening on this mode is not settled yet, in particular when it comes to phonon linewidths zhou_dipolar_2017 ; wang_phonon_2017 . However, the experimental evidence for its influence on electronic properties lee_interfacial_2014 ; zhang_role_2016 is definitely present suggesting to use this mode to build a simplified model Hamiltonian to address the impact of reaching strong light-matter coupling on the superconducting behavior of the material. We specifically use a single-band model for the electrons in two spatial dimensions in a partially filled band with filling per spin, as previously used to model the relevant electronic structure fitting angle-resolved photoemission data rademaker_enhanced_2016 . A bilinear electron-phonon scattering is introduced with a coupling vertex that is strongly peaked near momentum with a coupling range . The coupling strength is adjusted to keep a total dimensionless coupling strength independent of , where is determined from the effective mass renormalization in the metallic normal state above the superconducting critical temperature in absence of the cavity coupling. This conservative choice of is for instance below the value of 0.25 that was given in Ref. zhang_role_2016, .
We employ phonon-photon coupling and study phonon-polariton formation in this setting. In Figure 1B we show schematically the resulting polariton branches that stem from a gauge-invariant coupling involving both and terms, where is the current of phononic dipoles associated to an infrared-active phonon mode, and the electromagnetic gauge field of the photons. The relevant effective coupling strength between photons and phonons is given by the phononic plasma frequency , with the effective ionic mass of the phonon (see Section B of the Supplementary Materials). For the 2D system in the cavity the plasma frequency is controlled by the length of the vacuum inside the cavity in direction, , and the 2D unit cell area , with , the length and number of unit cells of the system in direction, respectively. The plasma frequency sets the splitting between the upper and lower polariton branches, reminiscent of the LO-TO splitting in bulk semiconductors. Obviously this splitting is only relevant at very small momenta since the photon energies become large compared to the phonon frequency quickly as increases due to the large magnitude of the speed of light.
The formation of phonon polaritons leads to a redistribution of the electron-boson coupling vertex to the two polariton branches. In Figure 1C we plot the squares of the coupling vertices between electrons and the respective polaritons as a function of , where is the Fermi momentum. A realistic value of the coupling range for FeSe/SrTiO was estimated as , as needed to create replica bands in angle-resolved photoemission that duplicate primary band features without significant momentum smearing lee_interfacial_2014 ; rademaker_enhanced_2016 . In a microscopic model, this value depends on the distance between the topmost TiO layer and the FeSe monolayer as well as the anisotropy of in-plane and perpendicular dielectric constants via , with realistic estimates and . This coupling range is larger than the momentum at which photon and phonon branches cross and mix most strongly in the polariton formation process. This implies that the modification of electron-boson coupling due to the cavity only happens at very small momenta typically smaller than . Thus to investigate how the degree of forward scattering influences the way in which cavity coupling is able to modify electronic properties, we employ different values for below, envisioning that cavity effects are enhanced when becomes smaller, which would in practice be achieved by making the dielectric-constant anisotropy ratio larger.
ii.2 Light-induced superconductivity
The critical question to answer here is how the redistribution of the coupling vertex to the upper and lower polariton branches affect the electronic properties. We investigate this by a diagrammatic approach employing Matsubara Green’s functions. We adopt the same approximations used in Ref. rademaker_enhanced_2016, and compute the self-consistent Migdal-Eliashberg diagram with dressed electronic Green’s function in Nambu space, allowing us to take into account superconducting order. The central quantity is the electronic self-energy written in terms of the Pauli matrices , the effective mass renormalization , the band dispersion renormalization , and the anomalous self-energy , which vanishes in the normal state.
Figure 2A shows the resulting temperature-dependent superconducting order gap evaluated at the smallest Matsubara frequency and at a Fermi momentum along the Brillouin zone diagonal for a coupling range representative of FeSe/SrTiO. Starting from a critical temperature K in the absence of the cavity (), we find a slight enhancement of superconductivity as the cavity is introduced and its extension in the direction perpendicular to the 2D material is reduced, resulting in a nonzero . For perhaps unrealistically large values (eV) and , a sizeable enhancement of of a few Kelvin is found in our simulations, which would likely require cavity sizes of a few lattice constants and might in practice be too small to achieve at the moment.
In order to investigate the effect of the forward-scattering coupling range, we look at the change of the superconducting order in the case of that is reduced by a factor of two from the realistic value described above, see Figure 2B. In this case the polaritonic redistribution of the coupling is expected to be more effective as there is a better match between the coupling range and the polariton avoided crossing. This is indeed observed in the superconducting order enhancement. Where a value of was needed in Figure 2A to obtain a visible enhancement of , here a smaller value is sufficient to enhance by K. Even larger lead to enhancements of order 10 . Finally if we decrease the range by another factor of two, , the enhancement is very strong and leads to more than a doubling of (Figure 2C).
ii.3 Cavity-modified electron-boson interaction
Having discussed the influence of the polariton formation in the cavity on superconducting order, we now turn to its effect on the effective electron-boson coupling itself. This is of interest independently of superconductivity, as the electron-phonon coupling affects also many other properties of materials, such as the conductivity or structural phase transitions. In particular it plays a pivotal role for THz-driven nonequilibrium phases of materials. In Figure 3 we show how cavity coupling modifies the temperature-dependent quasiparticle mass renormalization obtained from the normal self-energy for the different coupling ranges, realistic (Figure 3A), reduced (Figure 3B), and very small (Figure 3C). The first observation is that independent of the cavity shows a strong temperature dependence with a peak around , decreasing both towards higher temperatures and towards lower temperatures deep inside the ordered phase. The former is readily understood as a usual temperature effect when at high temperature the system becomes more and more classical and less correlated. The latter is understood by considering the fact that correlation effects are reduced deep in the ordered phase when quantum fluctuations lose their importance and a quasi-classical mean-field description can be adopted. Therefore the effect of the cavity coupling decreasing below is opposite to its effect above where it enhances . Again the cavity effects are more pronounced as increases for fixed , and as increases for fixed .
In summary we propose to employ polaritonic couplings and their tunability in QED cavity settings to control bosonically mediated effects in low-dimensional materials. In reality the size of the achieved effects will depend on the quality factor of the cavity, the degree to which our idealized boundary conditions are realized in practice, and on the required large coupling strengths that can actually be reached in real devices. Importantly, however, our above results are ground-state modifications that are still qualitatively valid even in dissipative systems liberato_virtual_2017 ; flick_atoms_2017 . Moreover for organic molecules in cavities the ultrastrong-coupling regime was even achieved in bad cavities with small quality factors bahsoun_electronic_2018 . Here we predict enhancements of in a few percent range for moderate few-percent changes of the electron-boson coupling . Furthermore, interface engineering, for example by replacing the dielectric substrates, is expected to have a strong influence on the physics in the cavity as well, as this will change both the forward-scattering coupling range as well as the polarizability of phononic dipole and thus the plasma frequency that determines the cavity coupling strength.
We note that a related idea of exciton-mediated superconductivity allender_model_1973 in 2D heterostructures has been introduced laussy_exciton-polariton_2010 and recently discussed in the context of transition-metal dichalcogenides cotlet_superconductivity_2016 . These proposals require exciton-polariton condensates to exist in the first place, which then mediate superconductivity in doped nearby layers via coupling of quasifree electrons to condensed exciton polaritons. By contrast, our present proposal does not rely on bosonic condensation but rather focusses on directly enhancing the electron-boson coupling relevant to superconductivity through polariton formation in a cavity.
Here we have provided a showcase example for light-induced equilibrium superconductivity as an extreme that is very different from optically stimulated short-lived superconducting-like states for strong laser pumping mitrano_possible_2016 . Conceptually, it will be very interesting to investigate the case of cavities pumped with external classical lasers, enabling researchers to explore the range of possibilities of light-induced effects between the QED limit and the strongly-driven classical limit, and also providing an additional knob for enhancing the above discussed cavity-mediated electron-boson coupling. For the example of FeSe/SrTiO, our proposal could help shed light on the above-mentioned debate about the role of the forward-scattering phonon for superconductivity. If the coupling of the phonon to electrons is unimportant, the polaritonic pairing idea will not work out, which could serve as a test for the influence of the phonon on the electronic properties. Similarly, it was recently suggested to use classical lasers in a pump-probe setting to study the forward-scattering nature of the phonon kumar_identifying_2017 . Ongoing work focuses on a realistic ab initio computation of cavity-enhanced couplings via dipolar phonons using the framework of quantum-electrodynamical density functional theory ruggenthaler_quantum_2014 .
Iv Materials and Methods
We employ a cavity quantum-electrodynamical setting with plane-wave mode expansion inside a cavity, with fixed-node boundary conditions for confined cavity photon modes along the direction, and periodic boundary conditions in the extended 2D plane (see Section A of the Supplementary Materials). Specifically, we use the Migdal-Eliashberg approximation to the electronic self-energy to a coupled electron-polariton model Hamiltonian involving electron-phonon forward scattering and dipolar phonon-photon coupling. The electron-polariton Hamiltonian has the form
with () the electron creation (annihilation) operators at wavevector and spin , the electronic band dispersion measured relative to the chemical potential which is adjusted to fix a band filling of 0.07 per spin. Furthermore is the number of points in the 2D Brillouin zone, and is the polariton-momentum -dependent electron-polariton coupling to branch with bosonic polariton creation (annihilation) operators () for the polaritons with energies
Here the underlying bare energies are given by the electronic hopping rademaker_enhanced_2016 , the phonon frequency zhang_role_2016 , the bare photon dispersion is with speed of light , and we use a variable effective phononic plasma frequency throughout the main text. Further details can be found in Sections B and C of the Supplementary Materials.
We apply the Migdal-Eliashberg approximation to the electronic Matsubara self-energy given by
with self-consistent electronic Nambu Green’s function , decomposed into Pauli matrices , unrenomalized polaritonic Green’s function , and fermionic Matsubara frequencies and bosonic Matsubara frequencies , , and inverse temperature . This amounts to the approximation that the bare phonon mode already contains the energy-shift renormalization due to electron-phonon coupling as the bare phonon frequency is taken from experimental data, and further renormalizations of the phonon polaritons due to electron-polariton coupling are small. The self-consistent computation of is initialized with a seed for the anomalous superconducting self-energy of eV and a convergence criterion of eV. Further details can be found in Section D of the Supplementary Materials.
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Discussions with H. Appel, S. Johnston, S. Latini, A. J. Millis, and L. Rademaker are gratefully acknowledged. M.A.S. acknowledges financial support by the DFG through the Emmy Noether programme (SE 2558/2-1). A. R. acknowledges financial support by the European Research Council (ERC-2015-AdG-694097), Grupos Consolidados (IT578-13), and European Union’s H2020 program under GA no. 676580 (NOMAD).
Vi Figures and tables
Vii Supplementary materials
Text (Sections A to D)
References mahan_many-particle_2000 .
vii.1 Relevant photon modes in cavity
In this work we consider a 2D material on a dielectric substrate in a nanocavity. We impose reflecting mirror boundary conditions with and for the magnetic and electric components of the photonic field, and the surface normal. The size of the cavity in direction is . If the dielectric substrate has a very high dielectric constant, such as for SrTiO at low temperature, it can be considered almost metallic and is reduced accordingly in our effective description.
Assuming periodic boundary conditions in the plane, we obtain for example for the vacuum electric field, obeying the wave equation with the speed of light,
with , and
We assume and to be large to obtain a fine momentum grid in the plane. By contrast is assumed to be small (), implying that for the photon energy is at least well above typical phonon energy scales and thus irrelevant to the problem of our interest. We retain only the , component that has constant mode amplitude along the direction. Thus we will use only one mode for each in-plane momentum with
vii.2 Phonon-photon Hamiltonian
We consider the generic Hamiltonian for phonon-photon coupling mahan_many-particle_2000 ,
Throughout we approximate the phonon dispersion relevant for FeSe/SrTiO with a dispersionless zhang_role_2016 . Here summations are over the first Brillouin zone in the 2D square lattice with lattice constant , implying a high-frequency cutoff to the photons, which is irrelevant to the electron-boson physics happening at much lower energy. For the photon, we take only the mode polarized along the direction parallel to the phonon dipoles, and restrict it to the lowest branch due to cavity confinement as discussed above, implying .
We write the phononic dipole current operator via bosonic operators
with polarization vector , and similarly for the relevant component of the photonic vector potential
assuming periodic boundary conditions inside the 2D plane. Here () creates (annihilates) a phonon with wavevector ; () creates (annihilates) a cavity photon with wavevector . is the number of unit cells, the system volume, the unit cell volume, and the ionic mass. In momentum space we have
Now we first diagonalize the bare photon plus terms of the Hamiltonian,
Here we have introduced canonical operators
We have also defined the phononic plasma frequency
which for the 2D system in the cavity is governed by the length of the vacuum inside the cavity in direction, , and the 2D unit cell area .
with renormalized bosonic operators
Importantly, the apparent infrared divergence due to as in stemming from terms containing the vector potential (Eq. (S16)) is cured by the plasma frequency giving the photons a finite mass. This will be undone by the bilinear coupling term discussed next, but overall a divergence-free theory emerges.
The bilinear coupling term is written as
Here we have used the important identity , which indeed shows that the term removes the infrared divergence from the term.
Diagonalization of the total phonon-photon Hamiltonian gives the polaritonic Hamiltonian
with new bosonic creation and annihilation operators and , , for the upper and lower phonon-polariton branches with dispersions that fulfil
vii.3 Electron-polariton Hamiltonian
The electron-polariton model Hamiltonian for FeSe/SrTiO inside the cavity reads
Here, () creates (annihilates) an electron with wavevector and spin ; is the electronic band dispersion measured relative to the chemical potential ; is the momentum dependent electron-phonon coupling. The direct electron-photon coupling of electrons in the FeSe plane to the photon branch of interest is neglected, which amounts to the assumption that the paramagnetic electronic current density inside the FeSe layer is perfectly two-dimensional, thus not coupling to the photonic vector potential which points perpendicular to the plane, implying .
Adopting the FeSe/SrTiO single-band model from Rademaker et al. rademaker_enhanced_2016 , we take an electronic band dispersion , where is the in-plane lattice constant. We set and use as an initial guess , which is adjusted during the self-consistent calculations (see below) to a fixed band filling for each spin. We neglect the fermion momentum dependence in the electron-phonon coupling , where is the momentum transfer, and use . Here, is adjusted to fix the total dimensionless coupling strength (see below) of the electron-phonon interaction in absence of the cavity coupling, and sets the range of the interaction in momentum space.
The full electron-polariton is then obtained as
where results from the projection of the phonon component in the polaritonic operators , by which the polaritons inherit part of the electron-phonon coupling through the unitary transformation .
vii.4 Migdal-Eliashberg simulations
The electronic self-energy in Migdal-Eliashberg theory on the Matsubara frequency axis employing Nambu notation reads rademaker_enhanced_2016
where are the Pauli matrices, and renormalize the electronic single-particle mass and band dispersion, respectively, and is the anomalous self-energy, which vanishes in the normal state. In Migdal-Eliashberg theory, the self-energy corresponding to the Hamiltonian (S36) is computed by self-consistently evaluating
where is the bare phonon propagator, is the dressed electron propagator, is number of momentum grid points, and is the inverse temperature.
Inside the cavity with , these well-known equations are modified to account for the Hamiltonian (S37) by using polariton branches instead of the phonon:
where is the bare polariton propagator,
In practice, we use an initial guess of eV for the anomalous self-energy and run the self-consistency until a convergence to better than eV is achieved. The 2D momentum grid to sample the Brillouin zone is chosen as and convergence checked by comparing against grids in selected cases. The momentum convolution in Equations (S39) and (S40) is performed by fast Fourier transforms to a real-space grid and products on the real-space grid. The Matsubara cutoff is eV for the frequency summations, and convergence in this cutoff also checked.