Pressure tuning of light-induced superconductivity in K₃C₆₀

Abstract

Optical excitation at terahertz frequencies has emerged as an effective means to dynamically manipulate complex materials. In the molecular solid K₃C₆₀, short mid-infrared pulses transform the high-temperature metal into a non-equilibrium state with the optical properties of a superconductor. Here we tune this effect with hydrostatic pressure and find that the superconducting-like features gradually disappear at around 0.3 GPa. Reduction with pressure underscores the similarity with the equilibrium superconducting phase of K₃C₆₀, in which a larger electronic bandwidth induced by pressure is also detrimental for pairing. Crucially, our observation excludes alternative interpretations based on a high-mobility metallic phase. The pressure dependence also suggests that transient, incipient superconductivity occurs far above the 150 K hypothesised previously, and rather extends all the way to room temperature.

Resonant optical excitation of infrared-active phonon modes can drive the crystal lattice of solids nonlinearly¹, excite other orders coherently², switch lattice polarization³, drive insulator-to-metal⁴ or magnetic transitions⁵, and even induce transient superconductivity above equilibrium T_c^6,7. In the potassium-doped fulleride K₃C₆₀, a superconductor at temperatures below T_c = 20 K, excitation of local molecular vibrations was shown to induce superconducting-like optical properties in the high temperature metal (T > T_c)⁸. Key features of this state are a ∼12-meV-wide gap in the resistive optical conductivity σ₁(ω), twice as large as the equilibrium 6-meV-wide superconducting gap, and a divergent low-frequency imaginary conductivity σ₂(ω), indicative of large carrier mobility. This state was found to extend to at least 100 K, hence up to temperature scales far in excess of equilibrium T_c (20 K). For T > 100 K, a partially gapped state was reported. In this higher temperature regime, the light-induced state can be interpreted either as a high mobility metal “en route” to a transient superconducting state, or as an incipient superconductor, which is only partially coherent.

Many theoretical mechanisms have been invoked to explain these observations, ranging from a dynamical reduction of the electronic bandwidth⁹, to the parametric amplification of the pairing instability¹⁰ and to electron attraction¹¹ in vibrationally excited molecular sites^12,13. Recent experiments in bilayer graphene are consistent with some of these suggestions^9,10, as the optical excitation of a similar vibrational mode as that driven in K₃C₆₀ appears to increase the electron phonon interaction¹⁴. Finally, recent theoretical work has raised the possibility that photo-stimulation may involve optical excitation and cooling of above-gap thermal quasi-particles into a super-excitonic state with high electronic heat capacity¹⁵.

The face-centred cubic structure of doped fullerides A₃C₆₀ is shown in Fig. 1a. Three electrons are donated by the alkali atoms to each C₆₀ molecule, which then form three narrow, half-filled bands near the Fermi level. Figure 1c shows how the equilibrium superconducting transition appears in the steady-state optical properties of K₃C₆₀, measured above and below the critical temperature T_c = 20 K. When cooling metallic K₃C₆₀ (red curves) below T_c, one observes large changes in the optical properties: a saturation of the low-frequency reflectivity to R = 1, a 6 meV gap in the real part of the optical conductivity σ₁(ω), and a 1/ω divergence in the imaginary part σ₂(ω)^16,8 (blue curves).

Figure 1. Structure, equilibrium phase transition, and transient light-induced phase of K₃C₆₀.

a. The f.c.c. crystal structure of K₃C₆₀ (Ref. ²⁹). The C₆₀ molecules are represented by the green bonds connecting all the C atoms. The grey spheres are the K atoms, acting as spacers between neighbouring buckyballs. The equilibrium lattice constant is 14.26 Å at room temperature. b. C₆₀ molecular distortion (blue) along the T_1u(4) vibrational mode coordinates. The equilibrium structure is shown in red. c. Reflectivity (sample-diamond interface), real and imaginary optical conductivity - R(ω), σ₁ (ω) and σ₂(ω) - across the equilibrium superconducting transition in K₃C₆₀. The red curves are measured at 25 K, in the metallic phase. The blue curves refer to the equilibrium superconductor (10 K). d. Reflectivity (sample-diamond interface), real and imaginary optical conductivity - R(ω), σ₁(ω) and σ₂(ω) - of K₃C₆₀ at equilibrium (red) and 1 ps after photo-excitation (blue) at T = 100 K. The light-blue curves show the data reported in Ref. 8, measured with a fluence of 1 mJ/cm², while those in dark blue were measured with a broader probe spectrum and a higher pump fluence (3 mJ/cm²).

As superconductivity in K₃C₆₀ emerges from a combination of Jahn-Teller intramolecular distortions and electronic correlations^17,18,19, it is natural to explore the response of the material to direct excitation of optically accessible vibrational modes. In Fig. 1d we report the optical properties of polycrystalline powders of K₃C₆₀, 1 ps after the excitation tuned to “on ball” infrared active modes of T_1u symmetry at 170 meV energy (7.3 μm wavelength), whose atomic distortion is displayed in Fig. 1b. At this frequency, strongly correlated metallic carriers are also excited, as the radiation is also resonant with a broad absorption peak extending from ∼40 to 200 meV²⁰, whose precise origin is still unclear¹⁵. A broadband probe pulse was used to detect the light-induced changes in the optical reflectivity and complex optical conductivity between 1.6 and 7 THz (6.5 – 29 meV) using THz-time-domain-spectroscopy. Starting from the unperturbed metallic state at 100 K (red curves), we observed an increase in the reflectivity, which saturates to R = 1 for all probe photon energies below ∼12 meV, a gapped σ₁(ω) and a divergent σ₂(ω). These data confirm the results of Ref. 8, but were recorded with an improved apparatus, involving higher pump fluence and a broader probe bandwidth (see Supplementary Information S3).

In this paper, we study how the features reported in Fig. 1d change with the application of hydrostatic pressure. At equilibrium, the application of pressure reduces the superconducting transition temperature T_c, because of the increase in the electronic bandwidth when the intermolecular spacing is reduced^21,22. As shown in Fig. 2, the size of the optical gap (2Δ₀) and the critical temperature^23,24 decay linearly already at relatively modest pressures. Due to the low bulk modulus (28 GPa²⁴), a pressure of 3 GPa reduces the superconducting gap to less than half of the ambient-pressure value, as the electronic bandwidth increases by ∼25%²⁵.

Figure 2. Equilibrium pressure dependence of superconducting K₃C₆₀.

a. Schematic representation of a diamond anvil cell (DAC) acting on the K₃C₆₀ crystal structure. By applying external pressure, the inter-molecular distances get reduced. b. Superconducting transition temperature and calculated optical gap (2Δ₀/k_BT = 3.52²³), plotted as a function of lattice parameter and external pressure. Data adapted from Refs. ²⁴, ³⁰.

Figure 3 displays snapshots of the measured optical reflectivity R(ω) at the sample-diamond interface, along with complex conductivity spectra, σ₁(ω) and σ₂(ω), for different values of static pressure. The exact pressure was measured with calibrated ruby fluorescence (see Supplementary Information S3). In each panel, the red and blue curves trace the optical properties of the equilibrium metal and those of the non-equilibrium state induced by photo-excitation, respectively. For pressures up to 0.17 GPa (Fig. 3a-c) the transient optical response of K₃C₆₀ is similar to that observed at ambient pressure, with a reflectivity approaching R=1, a gapped σ₁(ω) and a divergent σ₂(ω) toward low frequencies. However, some spectral weight is also found in σ₁(ω) at low energies, indicative of reduced coherence.

Figure 3. Pressure dependence of the transient optical properties of K₃C₆₀ at T = 100 K.

Reflectivity (sample-diamond interface) and complex optical conductivity of K₃C₆₀ measured at equilibrium (red) and 1 ps after photoexcitation (blue) at T = 100 K, for different external hydrostatic pressures. All data were taken with the same pump fluence of 3 mJ/cm².

As the applied pressure increases, a stronger suppression of the light-induced changes in both the reflectivity and complex optical conductivity is observed (Fig. 3d-e). Above 0.3 GPa the enhancement in the reflectivity is clearly less pronounced, and a progressively broader Drude peak appears at low frequency in the σ₁(ω) spectrum.

The reduction in light-induced coherence observed as a function of pressure is clearly not compatible with what expected for a light-induced metallic state, as a lattice compression in a metal is typically associated with larger electronic bandwidth, smaller effective mass and larger mobility. This is for example evident when analysing the equilibrium metallic properties in the red curves of Fig. 3 (see also Supplementary Information S2), where one observes larger plasma frequencies ω_p with increasing pressure.

In Fig. 4a-c we report the fractional spectral weight loss for frequencies inside the gapped region of the spectrum, obtained by integrating σ₁(ω) between 6.5 and 12.9 meV for different pressures and base temperatures of 100 K, 200 K and 300 K (see Supplementary Information S9 for full data sets at 200 and 300 K). Shaded blue areas indicate the pressure-temperature ranges where the light-induced state is gapped. Overall, the light-induced gap fills already at moderate pressure values, becoming even smaller for increasing temperature. For $P\underset{˜}{>}0.3$ GPa, the pressure dependence of the light-induced effects is strongly reduced.

Figure 4. Scaling of the σ₁(ω) gap with external pressure.

Photo-induced reduction in σ₁(ω) spectral weight, integrated between 6.5 and 12.9 meV, normalised by the equilibrium value (integrated in the same range). The blue shaded areas identify the regions in which a photo-induced conductivity gapping is measured. All data were taken with the same pump fluence (3 mJ/cm²). Vertical error bars represent uncertainties determined from different sets of measurements and horizontal error bars show the calibration uncertainty of the ruby fluorescence measurements used to determine the pressure (see Supplementary Information S3).

Fits to the optical properties of Fig. 3 make the qualitative analysis above quantitatively significant (see Supplementary Information S10). By fitting the transient optical response of K₃C₆₀, we extrapolated the value of the low frequency optical conductivity σ₀ = lim_ω→0σ₁(ω). To compare both superconducting-like and metallic-like states in a consistent fashion, we used a Drude-Lorentz fit for the entire regime, in which σ₀ was allowed to float from finite (metal) to infinite values (perfect conductor), and a single lorentzian was used to capture the mid-infrared absorption band extending from 40 to 200 meV.

The results of this analysis are summarised in Fig. 5a-c, where we report the pressure dependence of σ₀ for three temperatures (100 K, 200 K, and 300 K). The red squares describe the pressure dependence for the optical properties of the equilibrium metal, while the blue diamonds that of the photoexcited state. As shown in these plots, the equilibrium metallic conductivity increases with applied pressure. On the contrary, two pressure regimes are found for the light-induced state, one in which σ₀ decreases for small pressures (dσ₀/dP < 0, blue shaded area) and one where it eventually increases slightly for higher pressures (dσ₀/dP < 0, yellow shaded area). Several indications can be extracted from these data.

Figure 5. Pressure dependence of the extrapolated conductivity.

Blue diamonds are extrapolated zero-frequency conductivities extracted from Drude-Lorentz fits of the transient optical spectra, as a function of pressure and for three different temperatures: 100 K (a), 200 K (b), and 300 K (c). Red squares are the corresponding zero-frequency conductivities determined at equilibrium. The blue areas identify the regions in which σ₀ is suppressed by pressure (dσ₀/dP < 0) while those in yellow indicate the regime in which dσ₀/dP > 0. All data were taken with the same pump fluence (3 mJ/cm²). The insets show a close-up of the low-pressure region. Vertical error bars reflect the fit uncertainty and horizontal error bars show the calibration uncertainty of the ruby fluorescence measurements used to determine the pressure (see Supplementary Information S3).

First, as mentioned above, from the optical properties alone reported in Ref. 8, one could not uniquely differentiate a superconductor from a perfect conductor, as optics only identifies the density of charge carriers and the scattering rate. The hydrostatic pressure dependence reported here adds crucial information. At low pressures, the photoexcited state has clear superconducting-like pressure dependence (dσ₀/dP < 0), whereas for higher pressures the response is clearly metal-like (dσ₀/dP > 0). Furthermore, at the high pressures σ₀ of the photo-excited state follows the same slope as that of the equilibrium metal.

In this context the results reported for high temperatures (T = 200 K and T = 300 K) are surprising. In that temperature range, a high mobility metallic state was proposed to interpret the data of Ref. 8. However, this interpretation was also not unique, as a superconducting-like state with progressively lower coherence could also have explained the data. Figure 5 suggests that in the low pressure regime the dσ₀/dP < 0 behaviour is retained all the way to 300 K, hinting that some incipient features of transient superconductivity may already be present up to room temperature.

These observations also provide guidance for a microscopic explanation of our results. Indeed, as summarised in Fig. 6, one finds a very strong dependence of the light-induced optical conductivity on pressure, and for the higher-pressure ranges (smaller lattice constants) the metallic phase (yellow) is stabilised. Our data sets an important benchmark for theories of photo-induced superconductivity^{9,10,11,15,27,28}, which should reproduce the observed pressure dependence. Figure 6 also indicates a clear path for future research in the broader context of A₃C₆₀ superconductivity, showing on the right-hand side the region of the phase diagram still to be accessed (panel b), with the interesting perspective of optimizing light-induced superconductivity further, for even larger lattice spacing.

Figure 6. Out-of-equilibrium phase diagram of f.c.c. K₃C₆₀.

a. K₃C₆₀ phase diagram (T vs room-temperature lattice constant and pressure). Green filled circles indicate the superconducting transition temperature (T_c) measured for different lattice parameters²⁴ at equilibrium. The light-blue filled area defines the equilibrium superconducting phase (SC). The dark blue shading stands for the Light-Induced Superconductor (LI-SC), which evolves into a High-Mobility Metal (HMM) under the application of pressure (red area). b. Extension of the f.c.c. A₃C₆₀ phase diagram for larger lattice constants. PI, PM, and SC refer to the equilibrium paramagnetic insulating, paramagnetic metallic, and superconducting phase, respectively. Based on Ref. 17,31,32.