Photo-molecular high temperature superconductivity

Superconductivity in organic conductors is often tuned by the application of chemical or external pressure. With this type of tuning, orbital overlaps and electronic bandwidths are manipulated, whilst the properties of the molecular building blocks remain virtually unperturbed.Here, we show that the excitation of local molecular vibrations in the charge-transfer salt $\kappa-(BEDT-TTF)_2Cu[N(CN)_2]Br$ induces a colossal increase in carrier mobility and the opening of a superconducting-like optical gap. Both features track the density of quasi-particles of the equilibrium metal, and can be achieved up to a characteristic coherence temperature $T^* \approxeq 50 K$, far higher than the equilibrium transition temperature $T_C = 12.5 K$. Notably, the large optical gap achieved by photo-excitation is not observed in the equilibrium superconductor, pointing to a light induced state that is different from that obtained by cooling. First-principle calculations and model Hamiltonian dynamics predict a transient state with long-range pairing correlations, providing a possible physical scenario for photo-molecular superconductivity.

Organic conductors based on the BEDT-TTF (bisethylenedithio-tetrathiafulvalene) molecules display low dimensional electronic structures and unconventional high-TC superconductivity (1)(2)(3). In the crystals with κ-type arrangement, the (BEDT-TTF) +0.5 molecules (henceforth abbreviated as ET) form dimers, which are organized in planes on a triangular lattice (4). Due to dimerization, the quarter-filled band originating from the overlap of the molecular orbitals splits into pairs of half-filled bands. The ET planes are separated by layers of monovalent anions, which act as charge reservoirs ( Fig. 1(a)).
The low-energy physics of the κ-phase compounds is captured by a Hubbard model on a triangular lattice with weak and strong bonds (Fig 1(a), lower right panel) (5)(6)(7). The ground state can be tuned either by applying hydrostatic pressure or by chemical substitution with different anions (8). In both cases, the spacing between the dimers changes (Fig 1(a)), directly acting on the hopping integrals (', ' ( ), leaving the on-site wavefunction and electronic correlations essentially unperturbed. Here, we explore the possibility of dynamically manipulating the Hamiltonian parameters and the manybody wavefunction by resonant excitation of ET molecular vibrations ( Fig. 1(b)).
Equilibrium optical spectra were measured in the normal state of κ-Br single crystals using a Fourier transform spectrometer for temperatures between 15 K and 300 K (see Fig. 2 and Supplementary Material), in good agreement with those reported previously in the literature (9,10). In the equilibrium normal state (% > % & ), κ-Br is a quasi-twodimensional Fermi liquid, characterized by a narrow Drude peak in the in-plane optical conductivity, observed up to % * ≃ 50 K (11,12). For temperatures % > % * the quasiparticle response vanishes and the material exhibits the behavior of a so-called "bad metal" (see Fig. 2(b)). A large gap and a temperature-independent insulating optical conductivity are found along the perpendicular (interlayer) direction (13) (Fig. 2(a)).
Several narrow peaks are also evident in the mid-infrared spectral region, corresponding to infrared-active vibrational modes of the ET molecules (14)(15)(16).
Femtosecond mid-infrared pump pulses were polarized along the out-of-plane crystallographic axis and tuned to the spectral region where the molecular modes are found (* +,-+ ≃ 900 − 2000 cm 56 , 8 +,-+ ≃ 5 − 11 µm ). The peak pump electric field was varied between ∼500 kV/cm and ∼4 MV/cm. The changes in low-frequency reflectivity and complex optical conductivity induced by mid-infrared excitation were measured for frequencies between ∼0.8 and 7 THz with probe pulses polarized along the conducting layers. These were detected by electro-optical sampling after reflection from the sample at different pump-probe time delays. Figure 3 shows the most suggestive results discussed in this manuscript. Therein, optical properties (=(?), A 6 (?) + CA D (?)) are reported for four representative base temperatures between % = 15 K and % = 70 K, measured before (red) and 1 ps after photo-excitation (blue). Here, the pump pulses were tuned close to resonance with molecular vibrations corresponding to distortions of the C=C bonds on the ET dimers (see sketch in Fig. 2). Importantly, these pump pulses, polarized along the insulating direction, penetrated here deeper than the THz probe field (polarized along the metallic planes, see also Supplementary Material). Hence, from the "raw" changes in electric field reflectance, F 6 (?) + CF D (?), one could directly extract the complex optical conductivity (A 6 (?) + CA D (?)) without the need to account for an inhomogeneously excited medium (17)(18)(19)(20)(21)(22).
The transient optical properties measured at base temperature % = 15 K, that is immediately above % & = 12.5 K, exhibited the largest changes. The reflectivity saturated to = = 1 over a broad frequency range, reducing only above ∼ 120 cm -1 .
A qualitatively similar response was recorded for % = 30 K and % = 50 K, although with smaller gap and A D (?) divergence. For these three temperatures, the metallic optical spectra measured at negative time delays were fitted with a Drude-Lorentz model (red lines in Fig. 3) for normal conductors, whereas all the transient optical properties were fitted with an extension of the Mattis-Bardeen model for superconductors of variable purity (23,24) (blue lines in Fig. 3, see also Supplementary Material).
A qualitatively different response was measured at % = 70 K, for which A 6 (?) increased rather than decreasing to zero, and A D (?) remained characteristic of a metal without diverging toward low frequencies. These 70 K spectra were then fitted with the same Drude-Lorentz model used to reproduce the equilibrium response.
For each of the three measurements at % ≤ 50 K reported in Fig. 3, we extracted the optical gap, 2∆, from the Mattis-Bardeen fits. The temperature dependent gap size could be fitted by the function Δ(%) = Δ(0) tanh PQR S T 5S S U, where 2Δ(0) = 23 meV was the zero-temperature non-equilibrium gap, Q = 1.74, and % ( ≃ 52 K provided an effective "critical" temperature for the non-equilibrium state (see Fig. 4(a)). Note that % (~% * ≃ 50 K, which coincides with the temperature at which a coherent Drude peak was observed in the equilibrium metallic state of Fig. 2. The connection between the superconducting-like optical properties measured in the transient state and the Fermi-liquid behavior of the equilibrium normal state (12,25) is further underscored by the analysis reported in Fig. 4 Even though the absolute values on the two vertical axes of Fig. 4(b) are different, the temperature dependences are similar, with the same onset at % * ≃ 50 K.
The time dependence of these features is visualized in Figure 5, where three different quantities are displayed. Fig. 5 diverged to values larger than 10 Ñ Ω 56 cm 56 (corresponding to mean free paths of at least ~100 unit cells), limited here by the 0.8 THz low-frequency cutoff of our measurement, which in turn was set by the relaxation time of the state (~ 2 ps).
In Figure 6, we report the transient complex optical conductivity after photo-excitation at different driving frequencies, all measured by keeping the pump fluence fixed to ~2 mJ/cm 2 . A superconducting-like response could only be observed for excitation at 8 µm In searching for a microscopic mechanism for the observed photo-molecular response, we note that in a previous work on the one-dimensional Mott insulator ET-F2TCNQ, featuring the same ET molecular building block as κ-Br, selective driving of an IR-active vibration was shown to provide a dynamical modulation of the on-site Hubbard-à interaction, achieved by quadratic electron-phonon coupling (26,27).
In the κ-Br system studied here, ab-initio calculations in the frozen-phonon Starting from these estimates, we performed simulations of a driven Fermi-Hubbard model on a triangular ladder system (see Supplementary Material for the Hamiltonian and specific geometry of the system). In the driven state, correlations between pairs of doublons residing on different sites were quantified by the doublon correlation function 〈| o ä | ã 〉 = 〈ç o↑ ä ç o↓ ä ç ã↓ ç ã↑ 〉 (here, ç oê ä and ç oê are single-particle creation and annihilation operators, respectively). In the simulations, the system was initialized in its half-filled zero-temperature ground state, yielding doublon correlations which decay exponentially with distance (blue dashed line in Fig. 7(a)). When the driving was          Bruker Vertex 80v interferometer. The sample was mounted on the tip of a cone-shaped holder, with the in-plane crystal surface exposed to the beam. The holder was installed on the cold finger of a He-flow cryostat, thus enabling to collect broadband infrared spectra in reflection geometry at different temperatures. These spectra were then referenced against a thin gold film evaporated in-situ on the same sample surface.
The reflectivity curves obtained with this procedure, covering a range between ~25 -5000 cm -1 , were extrapolated to -→ 0 with Drude fits (see also Section S3), and extended to higher frequencies with literature data (3,4). This allowed us to perform Kramers-Kronig transformations and retrieve full sets of in-plane equilibrium response functions to be used as reference in our pump-probe experiment.
Examples of selected reflectivity and complex optical conductivity spectra are reported in Fig. S1. As discussed in the main text, the normal state equilibrium response is characterized by a metallic Drude peak (see Fig. S1b) for temperatures below ) * ≃ 50 K, while the spectra at higher temperatures are those of a "bad metal", characterized by a non-zero low-frequency conductivity, in absence of a Drude peak (see also Ref. 3).
We did not observe any change in the optical spectra across the superconducting transition temperature ) * = 12.5 K. The absence of a clear optical gap in the superconducting κ-(ET)2X compounds has already been reported in the past (5-8).  c.

S2. Determination of the transient optical properties
In a series of mid-infrared pump / THz probe experiments, we investigated different k- In order to minimize the effects on the pump-probe time resolution due to the finite duration of the THz probe pulse, we performed the experiment as described in Ref. 9.
The transient reflected field at each time delay 9 after excitation was obtained by keeping fixed the delay 9 between the pump pulse and the electro-optic sampling gate pulse, while scanning the delay t of the single-cycle THz probe pulse.
The stationary probe electric field : ; (=) and the differential electric field ∆: ; (=, 9) reflected from the sample were recorded simultaneously by feeding the electro-optic sampling signal into two lock-in amplifiers and mechanically chopping the pump and probe beams at different frequencies. : ; (=) and ∆: ; (=, 9) were then independently Fourier transformed to obtain the complex-valued, frequency-dependent : A ; (-) and ∆: A ; (-, 9). The photo-excited complex reflection coefficient r C (-, 9) was determined by ∆: A ; (-, 9) : where r C E (-) is the stationary reflection coefficient known from the equilibrium optical response (see Section S1).
As the penetration depth of the excitation pulses was typically larger than that of the THz probe pulses (with the only exception of the 6.8 µm pump experiment, see Figure   S2), the THz probe pulse sampled a homogeneously excited volume. The transient optical properties could then be extracted directly, without the need to consider any pump-probe penetration depth mismatch in the data analysis.  For the limited set of data that required a pump-probe penetration depth analysis (4 5675 = 6.8 µm, ) ≥ 50 K), we followed the procedure described in Refs. 11 & 12. We treated the photo-excited surface as a stack of thin layers with a homogeneous refractive index and described the excitation profile by an exponential decay. By numerically solving the coupled Fresnel equations of such multi-layer system, the refractive index at the surface could be retrieved, and from this the complex conductivity for a homogeneously transformed volume.
Importantly, this renormalization only affected the size of the response, whereas the qualitative changes in optical properties were independent of it and the specific model chosen (12).

S3. Fitting models
The in-plane equilibrium optical properties of k-(ET)2Cu[N(CN)2]Br were fitted at all measured ) > ) * with a Drude-Lorentz model, for which the complex optical conductivity is expressed as: Here, - 5 and a ] are the Drude plasma frequency and momentum relaxation rate, while Ω E,U , Ω 5,U , and Γ U are the peak frequency, plasma frequency, and damping coefficient of the i-th oscillator, respectively.
The same Drude-Lorentz model was also employed to fit the transient optical spectra after photo-excitation. Here, all parameters related to the oscillators at frequencies outside the measurement range were kept fixed to the values determined at equilibrium, while the lowest frequency oscillator and the Drude term were left free to vary. Importantly, for each data set, the same parameters were used to simultaneously fit the reflectivity, the real part of the optical conductivity, as well as its imaginary part.
As shown, for example, in Fig. 3 & Fig. 6 of the main text, this Drude-Lorentz model is able to fully reproduce the experimental data for the temperatures and time delays that show a metallic behavior and no signature of a superconducting-like response.
In addition, the transient superconducting-like response of k-(ET)2Cu[N(CN)2]Br could also be captured (see fits in Fig. 6 of main text) by the simple a ] → 0 limit of Eq. S3.1, Here h i , j, and k are the superfluid density, electron charge, and electron mass, respectively.
That said, the Drude-Lorentz model of Eq. S3.1 could be applied to all transient optical spectra reported in this work, without any assumption on the nature of the nonequilibrium state (see also Ref. 13). The zero-frequency limit R E = lim H→E R C ]^( -) extracted from these fits (see Fig. 5 & Fig. 6 of main text), which is a finite quantity in a metal and diverges in a "perfect conductor", was used as a discriminator for the presence of transient superconductivity (13).
Additional fitting of the superconducting-like optical properties was performed with an extension of the Mattis-Bardeen model for superconductors of variable purity (14,15), which is typically used to describe the response of superconductors at equilibrium, for finite frequencies and temperatures. Fit curves extracted with this procedure are reported, for example, in Fig. 3 of the main text. The corresponding values of the optical gap, 2∆()), extracted with this procedure are shown instead in Fig. 4a.

S4. Extended data sets
We report here a comparison of transient optical spectra taken under the same conditions (temperature, excitation wavelength and fluence, pump-probe time delay) on three different k-(ET)2Cu[N(CN)2]Br crystal coming from the same batch of samples.
These data, measured at ) = 30 K, are reported in Fig. S4. Therein, we show how, aside from small differences, the non-equilibrium response appears to be sample independent, and all signatures of transient, photo-induced superconductivity, i.e., a reflectivity equal to 1, a gap in R o (-), and a ~1/-divergence in R V (-), are fully preserved.

S5. Pump fluence dependence
In Figure S5 we show pump fluence dependent data. In analogy with Fig. 4 of the main text, we plot the non-equilibrium "superfluid density", determined from the low- The saturation seems to occur for a value of h stt uvwMx that approaches the equilibrium

S6. Calculation of the effective Hubbard parameters Geometry optimization
We performed a geometry optimization of the crystal structure for κ-Br in order to obtain relaxed atomic coordinates. This was achieved using the density functional theory code Quantum ESPRESSO (16,17) (18,19).

Phonon mode calculations
We computed the phonon modes of the isolated ET dimer and constructed the Bu modes by applying the corresponding symmetries (20). These Bu modes are the infrared-active modes with dipole along the out-of-plane b direction. We obtained the eigenvectors of the dynamical matrix from Quantum ESPRESSO, using the same parameters and cell as for the geometry optimization of the full κ-Br crystal.

Extraction of ground-state Hubbard model parameters
The effective electronic parameters of the weak-bond triangular lattice Hubbard model were obtained following the approach proposed in Ref. 21. To this end we computed the ground state and corresponding band structure using the Octopus code (22,23). We employed the same norm-conserving pseudopotentials, PBE functional, and k-point grid as for the geometry optimization. The real-space grid was sampled with a spacing of 0.3 Bohr. The resulting band structure ( Fig. S6.2) was fitted by a tight-binding model for 8 ET sites, leading to four parameters (= o − = S ), and fixing the electronic occupation to 3/4 filling (24). From these fits, we extracted the effective parameters Ñ, =, and = Ö of the single band Hubbard model (21,24) involving only the electronic band that crosses the Fermi level in Fig. S6

Extraction of Hubbard model parameters for displaced structures
Starting from the geometrically relaxed structure, we systematically displaced the ions according to the phonon mode coordinates obtained in the calculations. For each displaced structure, we computed the adiabatic electronic ground state and band structure in the frozen-phonon approximation using the Octopus code with the same technical parameters employed in the ground state calculation. We then followed exactly the same steps as in the ground state case to perform the tight-binding fits and finally extracted the effective Hubbard model parameters. Figure S6.3 shows the resulting modulations of Ñ, =, and the ratios Ñ = ⁄ and = Ö = ⁄ for displacements along the Bu mode coordinates.

S7. Driven Hubbard model
The geometry of the triangular Fermi-Hubbard model considered here is shown in Fig.   S7 where 9 denotes the time, = is the nearest neighbout hopping element and =′ is the hopping element in the vertical direction. Following the frozen phonon simulations reported in Section S6, we assumed the on-site interaction Ñ and the strong bond hopping elements = to be modulated by the phonon driving as follows: Here, the amplitudes ñ o and ñ V were extracted from the calculations in Section S6, the modulation frequency Ω was set to the phonon frequency, while the parameters 9 ú and 9 E control the duration and delay of the sinusoidal pulse. We assumed that the vertical hopping strength =' remains constant.
In Fig. S7.2 we report surface plots of the doublon correlations in distance and time for various system sizes. For sufficiently large systems we observe the emergence of uniform long-range doublon correlations which stabilise and persist over the simulation timescale. The increase in long-range correlation strength becomes more pronounced as the system size increases.
Finally, in Fig. 3 we show the full time-dynamics of the doublon correlations for various driving strengths and a fixed system size.
The matrix product calculations used to produce these results were performed using the Tensor Network Library (26). The system was initialised in its ground state using   (1). The four panels show calculations for four different system sizes û. The system is initialised in its ground state with Ñ E = 4.82 = E , =′ = 0.22= E and half-filling. It was then allowed to evolve under time-modulated interaction strengths (see Eq. S7.2) with driving parameters 9 E = 5.0= E , 9 ú = 5.0= E , ü = 2.27= E , ñ o = 0.155 and ñ V = 0.07. The four panels show calculations for different driving amplitudes ñ o , ñ V . The system was initialised in its ground state with Ñ E = 4.73= E , =′ = 0.24= E , at half filling. It was then allowed to evolve under time-modulated interaction strengths (see Eq. S7.2) with driving parameters 9 E = 5.0= E , 9 ú = 5.0= E , ü = 2.27= E , and constant ratio ñ o ñ V ⁄ = 2.21.