Giant Enhancement of Electron–Phonon Coupling in Dimensionality‐Controlled SrRuO3 Heterostructures

Abstract Electrons in crystals interact closely with quantized lattice degree of freedom, determining fundamental electrodynamic behaviors and versatile correlated functionalities. However, the strength of the electron–phonon interaction is so far determined as an intrinsic value of a given material, restricting the development of potential electronic and phononic applications employing the tunable coupling strength. Here, it is demonstrated that the electron–phonon coupling in SrRuO3 can be largely controlled by multiple intuitive tuning knobs available in synthetic crystals. The coupling strength of quasi‐2D SrRuO3 is enhanced by ≈300‐fold compared with that of bulk SrRuO3. This enormous enhancement is attributed to the non‐local nature of the electron–phonon coupling within the well‐defined synthetic atomic network, which becomes dominant in the limit of the 2D electronic state. These results provide valuable opportunities for engineering the electron–phonon coupling, leading to a deeper understanding of the strongly coupled charge and lattice dynamics in quantum materials.


S3. Parameters used for the two-temperature model (TTM) analysis
In solving the so-called rate equations introduced as Eq. 1 and 2 in the main text, several parameters should be pre-determined. Among them is the electron heat capacity e of the SrRuO3 (SRO) layer. e of the SrTiO3 (STO) layer is ignored because of its insulating nature, and that for SRO is taken into account to be varied depending on the SRO thickness. We perform the density functional theory (DFT) calculation for the SRO system, and estimate e from the density of states at the Fermi level for the given SRO thickness as shown in Fig.S3(a). Our DFT calculation indicates that the SRO layer becomes insulating at the 2 unit-cell (uc) thickness. However, the [2|6]10 SRO/STO superlattice (SL) exhibits a metallic character at room temperature 4 . We therefore take e for the 2 uc SRO layer from a linear extrapolation of the thickness-dependent e. By the way, we use the bulk value for the SRO films thicker than 10 uc.
For the thermal conductivity , although of the SRO layer is taken as a fit parameter in our analysis as demonstrated in Fig. 2c in the main text, we consider that of the STO layer to be pre-determined based on the Boltzmann's transport model given as κ SRO = κ bulk 1 + (2β l MFP )/x , where  is a thicknessdependent parameter. Note that this prediction works well in describing the thickness-dependent thermal conductivity of the SRO layer as shown in Fig. 2(c). For the STO layer, the bulk thermal conductivity ( bulk ) and phonon mean free path are 11 Wm -3 K -1 and 2.35 nm 5,6 , respectively, and Fig.   S3(b) shows an effective thermal conductivity ( eff ) of the STO layer which is taken as a pre-determined parameter in the simulation. In our previous research 4 , our DFT calculation within the GGA+U scheme explains well the metal-insulator transition in SRO/STO SL consistently with experimental observations. Therefore, we believe that our DFT calculation can provide reasonable values of the density of states or the electron specific heat for ruthenates, at least to some extent.
To solve the rate equations (1) and (2) introduced in the main text, we adopt a finite difference method.
As depicted in Fig. S4, Eq. S1 and S2 are discretized by using the Euler method and the box method as Here, a spatial grid size (a) is set to be 0.4 nm, and each grid node is indexed by a subscript i. A temporal grid size (t) is set to be 10 fs, and a temporal evolution is indicated by a superscript j. We consider also a thermal boundary conductance which plays an important role in a thermal diffusion in the superlattice system. At the interface between SRO and STO, a boundary condition is given by the Fourier's law where q is a heat flux and is a thermal boundary conductance. At the surface, we simply choose the Neumann boundary condition. Figure S4. Schematic of the finite difference method for the two-temperature model analysis. Heat flux (q) is calculated by two adjacent grid points. We also consider the thermal boundary conductance between different thermal media.

S4. Sensitivity analysis of fitting parameters used in the two-temperature model
A transient reflectivity change is described by three different processes, namely (i) electron-phonon thermalization, (ii) thermalization between SRO-STO superlattice layers (SL thermalization), and (iii) heat diffusion from superlattice to substrate ( Fig. S5(a)). Each of these three processes is related to the electron-phonon coupling constant (Gep), the thermal boundary conductance (TBC) and the thermal conductivity of SRO (κSRO), respectively. To figure out whether these parameters can be determined reliably from our experimental results, we perform the sensitivity analysis for each parameter 7 .
Sensitivity function S of a fitting parameter  is given by Here, Y is a fitted function which is the transient reflectivity in this work. And, S dictates how much the fitting function Y is influenced by the fitting parameter . Figure S5

S5. Role of the electron-phonon thermalization in the hot carrier cooling process
The transient reflectivity becomes relaxed faster as the SRO layer becomes thinner, and this is attributed to the faster cooling of hot carriers. In understanding such results, although we have considered the contribution of the electron-phonon thermalization process, one may think about a possibility of the contribution of the thermal diffusion which would work more efficiently in a thinner layer of metal 8 . Indeed, the transient reflectivity change in SrVO3/SrTiO3 SLs could be explained by considering only the thermal diffusion without the electron-phonon thermalization 9 .
In SRO/STO SLs, however, we confirm that the electron-phonon thermalization process is essential in understanding the hot electron cooling process. As shown in Fig.S6, if electron and lattice subsystems would be immediately thermalized (Gep = inf.) after the photoexcitation, although the response after 2 ps can be reasonably fitted with the contribution of the thermal diffusion, an initial fast decay after pumping cannot be explained. Indeed, RMS error which is a difference between experiment and fitting results is larger before 2 ps when we ignore the electron-phonon thermalization ( Fig. S4(b)).

S7. Pump-power-dependent hot carrier cooling time
The electron-phonon coupling constant Gep may be given differently when an initial electron temperature is extremely high 10 . In our two-temperature model analysis, however, we assume that the hot carrier temperature is not high enough to have such a nonlinear effect. In the two-temperature model, the relaxation time ep of the hot carrier temperature via the electron-phonon thermalization is given Here, Ti is an initial temperature. Therefore, provided that Gep remains the same, ep should be in proportion to the pump power as it linearly increases Te.

S8. State-filling effect
An interpretation of the reflectivity change right after the photo-excitation is complicated due to an existence of non-thermalized photo-carriers which can have contributions to polarization grating, statefilling, anisotropic distribution, and so on 12 . Among them, we figure out the state-filling effect explicitly and analyze our results by separating out such contribution.
To extract the state-filling effect, we designed two different situations having an energy relationship of pump and probe beams, namely, Epump>Eprobe and Epump<Eprobe as depicted in Fig. S9(a). When Epump>Eprobe, the probe beam can respond to a rapid relaxation of non-thermalized carriers into the band edge; as the optical transition is less allowed due to the Pauli exclusion principle, the reflectivity of probe beam shows a drastic change. In particular, such state-filling effect will disappear as nonthermalize carriers become thermalized carriers via an optical phonon scattering and a carrier-carrier scattering 13 . When Epump<Eprobe, this process will not appear.

S9. Substrate effect for electron-phonon coupling in SrRuO3 superlattices
Figure S10(a) shows transient reflectivity changes of the SRO/STO superlattices prepared on various substrates. Transient reflectivity increased right after pumping rapidly drops on a similar timescale for all the cases. After 5 ps, the slope of reflectivity change is slightly different depending on the substrate used, and it is attributed to different thermal conductivity values of substrates.

S10. Temperature-dependent electron-phonon coupling
In the electron-lattice non-equilibrium state, the heat equation is described by Here, Ce is an electron specific heat, and H is an electron-to-lattice energy transfer rate per unit volume which is given as a function of an electron temperature Te and a lattice temperature Tl. As in conventional metals, an interaction between electron and acoustic phonons is first considered for the energy transfer process. In the two-temperature model, the energy transfer rate is described in a more detail as 11 Here, ∞ is an intrinsic electron-phonon coupling constant, and is the Debye temperature.
Temperature-dependent electron-phonon coupling constant Gep(T) is simply given by the first derivative of energy change rate, U(T).
If the electron would be coupled with optical phonons, the modification should be made for Eq. (S9) about the energy change rate U(T) as Here, the optical phonon is assumed to have no dispersion near the zone center. In this case, unlike the interaction with acoustic phonons, the energy change rate exponentially decreases as temperature decreases. Figure S11 summarizes the electron-phonon coupling strength of SRO films investigated in this work together with various noble metals. For noble metals, Gep shows an increase in proportion to Ce/T. This general trend can be naturally understood as the higher electron density of states at the Fermi level gives rise to the larger el-ph coupling strength. For SRO films, however, the relationship between Gep and Ce/T are largely different; although the SRO films in the thin film limit seem to show the general trend of noble metals, the major Gep enhancement starting from the bulk SRO largely deviates from such trend. This implies that the large Gep variations observed in SRO films should be understood by considering not only the electron density of states but also the variation in the phonon contribution as discussed in the main text.