Heat generation in spatially confined solids through electronic light scattering

1.1 Theoretical backgrounds

Optical heating of an opaque homogeneous medium is mostly contributed by direct absorption of light when resonant (band-to-band) pumping, labelled as ‘1’ in Figure 1a. Nobusada et al. demonstrated the enhanced absorption due to indirect optical transitions in a spatially-confined structure [14]. For the sake of simplicity, let us consider a heterogeneous medium composed of only two sub-wavelength structures with the sizes of r ₁ and r ₂ (r ₁ < r ₂ ≪ λ). Such a system absorbs and scatters light poorly since the cross-section of absorption, C _a , and scattering, C _s , scales with the size as C _a ∼ r ³ and C _s ∼ r ⁶ [24], rapidly disappearing as r → 0. On the other hand, these optical inhomogeneities generate near-field photons, which allow changing the momentum of an electron by Δk ₁ and Δk ₂ (Δk ₁ > Δk ₂) during its transition from the valence band (VB) to the conduction band (CB) (Figure 1a). According to Fermi’s golden rule, the strength of optical transitions is driven by not only the interaction Hamiltonian but also the electronic density of states that is maximum at the energy band edge [14]. This means that only the indirect transition is realized during which the momentum of the near-field photon, Δk ₁, is transferred to the electron (labeled as ‘2’ in Figure 1a). Another indirect transition associated with the transfer of momentum Δk ₂ is unlikely to be done. Further thermalization of electrons to the bottom of the CB leads to optical heating (the shaded area on the right panel in Figure 1a). Obviously, direct/indirect absorption mechanisms are convoluted and a tunable sub-band laser probe is needed to distinguish them. Direct absorption is the dominant mechanism when resonant laser pumping, whereas indirect absorption is governed by both energy and momentum matching. Because the first limitation is dropped for electronic light scattering, as shown in Figure 1b, all indirect optical transitions become accessible, transferring both Δk ₁ and Δk ₂ to the electron. More importantly, inelastic broadband emission is an inherent signature of heterogeneous systems with strong spatial dispersion. These include ceramics and high-entropy crystals [25], amorphous and porous solids [26], perovskites [27], liquid crystals [25], highly-associated liquids [28], intercellular water [29], hydrogels [30], peptides and proteins [31], to name a few. This diversity underscores the fundamental importance of spatial dispersion in getting into insights of complex systems.

The increased charge population in the CB inevitably leads to optical heating through electron-phonon interaction, as seen on the right panel of Figure 1b. As a result of indirect scattering, an amplified electrical current is generated at the junction of glass and crystal [3]. At last, the most fundamental consequence of indirect scattering is an increase in the real refractive index of heterogeneous systems that can exceed the fundamental limit caused by temporal dispersion [10]. For optically transparent (κ ≈ 0) semiconductors the spatially varying refractive index reads as

where Ω _cv is a vibronic frequency corresponding to optical transitions between the Bloch electronic states v and c (Ω _cv = 0 for metals), m _e is the effective mass of an electron, ϰ is a light polarization direction. The oscillator strength f c v ϰ k is modified as follows

Here ℏ is the Planck’s constant, d c v ϰ k = c | e k r ϰ r ∂ / ∂ r | v is the transient electrical dipole moment that considers spatial dispersion. In homogeneous media, a change in the refractive index is possible at resonance only (temporal dispersion) mainly due to direct absorption (Figure 1a). In Eq. (2), integration runs over the entire Brillouin zone, and, thus, indirect transitions can contribute to the real refractive index significantly.

Depending on the energy band landscape, this process can in addition lead to optical heating of a light-scattering medium through electron-phonon interaction (Figure 1b). This process is determined by the slope of the energy band dE/dk and the population of electrons governed by Fermi–Dirac statistics. Thus, the change in overall temperature without heat transfer to the environment can be estimated using the following formula:

Eq. (4) indicates the fact that solids with the flat energy band valley are not heated under cw sub-band pumping.

The k-dependent ELS intensity is determined by the photonic density of states ℱ k and the population of electrons, driven by Fermi–Dirac statistics f F D k , namely:

where C is a constant proportional the scattering cross-section, E k = ℏ ω 0 − E k is the scattered photon energy. In the simplest case when ℱ k − k ′ = I 0 δ k − k ′ (I ₀ is the incident intensity, δ k − k ′ is the Dirac-delta function), one gets

where k _B is the Boltzmann constant, T is a temperature. This formula serves as a good approximation provided that high spatial homogeneity ℱ k ∼ δ k and E k − E p ≫ k B T . In Eq. (6), we introduced an additional parameter E _p analogous to Penn energy [32]. This seminal concept specifies the energy at which the electronic density of states is maximum throughout the entire Brillouin zone [33]. In our model, this parameter E p = ℏ ω 0 − E k p denotes the frequency shift corresponding to the statistical mode of near-field photon momentum k _p . This shift tends to zero for electrons trapped near the conduction band edge. Such transitions basically contribute to the appearance of the low-energy ELS peak. This is the reason why the central peak is dominant in spatially confined metals where indirect transitions near the Fermi level occur.

2.1 A silicon atomic-force-microscopy tip

Figure 2a schematically shows a TERS setup for measuring the optical heating of spatially confined crystalline silicon (c-Si), earlier suggested in Ref. [13] In our experiment, a c-Si AFM cantilever (VIT_P, NT-MDT) top-illuminated by a focused 633 nm laser light (x100, N.A. = 0.7), oscillates in semicontact mode over a borosilicate glass substrate covered with a 50 nm Au film (TED PELLA, Inc.). We first scan the film surface with the AFM cantilever to visualize its roughness. The size of surface irregularities ranges from 1 to 2 nm (Section III, the SI). Next, the tip apex is positioned over a protrusion with the size of choice: the smaller, the stronger photon confinement is achieved [13]. Upon selecting a 1 nm high protrusion, we then record Raman spectra (exposure 1 s) and the AFM cantilever phase kinetics with a gradual increase in the pumping intensity to 5.7 MW/cm² for 140 s. The optical heating is recognized by three observations: i) a temperature-dependent blueshift of the first-order vibrational Raman scattering (VRS) peak of c-Si at 521 cm⁻¹ (Figure 2b), ii) high-energy ELS (Figure 2d), exhibiting a nonlinear pump-dependent intensity behavior (Figure 2c), and iii) a change in the AFM cantilever phase above the pumping intensity of 1.5 MW/cm² (Figure 2e). The pump-dependent temperature of the tip apex is determined by a Raman-shift-based probe with an accuracy of 50 K (1800 grooves per mm grating) using Eq. S4 (Section IV, the SI). A calibration of the temperature probe was prior performed by temperature-dependent far-field Raman measurements in the range from 25 to 600 °C using a heating stage (Linkam Scientific Model THMS600).

Figure 2:

Photoheating of a silicon AFM tip apex. (a) Sketch of a TERS setup (upright configuration). (b) A Raman map of the AFM tip apex exposed to a focused laser light with different pumping intensity (the color indicates Raman intensity). The bottom inset displays SEM images of the AFM cantilever tips before and after laser impact with the intensity of 5 MW/cm². (c) A plot of the VRS intensity versus the pumping intensity. (d) Raman spectra taken at the intensity of 1 MW/cm² (gray) and 5 MW/cm² (red) in Figure 2 (b). The dashed curve denotes a fit by Eq. (6). (e) A kinetics of the AFM cantilever phase (red) and a dependence of Raman-shift-calculated temperature versus the pumping intensity (blue). (f) The energy-momentum diagram for c-Si.

Upon turning the level of 4.2 MW/cm² (Figure 2b and e), the temperature steeply climbs above 2,000 K, making the tip apex melt. A close inspection of scanning electron microscopy (SEM) images of the tip apex (the inset in Figure 2b), visualized before and after laser impact, confirms its destruction within 900 nm extent. However, the AFM cantilever phase starts to respond at 1.5 MW/cm² (red curve, Figure 2e). The phase shift Δ φ ≈ Q K ∂ F ∂ z (where K is the spring constant, Q is the quality factor) is mainly driven by the normal force gradient ∂F/∂z between the tip apex and the sample, that is greatly sensitive to temperature [34], [35]. It is important to note that the temperature of the entire cantilever remains constant during our experiment, except the tip apex. It follows from the fact that the resonant frequency of the AFM cantilever, equal to 300 kHz, is fixed until the tip apex is damaged, the event marked with the arrow in Figure 2b. In this figure, we highlight two regions labeled as ‘1’ and ‘2’ where giant temperature bursts are seen (Figure 2e), accompanied by ELS (Figure 2d). Thermal blinking is solely associated with tip damage. In a sense, thermal bursts in Figure 2e represent a non-stationary melting process.

This observation is in good agreement with an increase in the Raman intensity at this level (Figure 2c) that occurs due to ELS, as follows from Figure 2d. Large temperature fluctuations above 4 MW/cm² (blue curve, Figure 2e) are directly related to melting and a change in the tip apex morphology. We conclude that the ELS mechanism holds promise for probing a local temperature at the hot spot.

Anomalous optical heating of spatially confined silicon is provided by near-field photons mainly generated by a rough Au film (Figure 2a). Clearly, the film itself should get heated as well. Due to the high thermal conductivity of the Au film, equal to 320 W/mK, the film is rapidly thermalized and, therefore, it is not heated.

In c-Si, that is an indirect bandgap semiconductor, the optical absorption is driven by three-body photon-phonon-electron interaction. Alternatively, this process can be activated using a near-field photon carrying not only energy, but also momentum sufficient for indirect transitions [13]. Since the electronic density of states is greatest near the band edge [14], absorption-based indirect transitions are limited because of energy band k-dispersion, shown in Figure 2f. In general, one should consider optical transitions to those electronic states that correspond to energy E ₀. Despite the fact that indirect transitions supported with large momenta are allowed, their contributions to absorption are negligible. In contrast, light scattering allows indirect transitions to all accessible states below energy E ₀ near the conduction band edge. This leads to broadband inelastic light scattering with a frequency shift: ω k = ω 0 − Ω k (ω ₀ is the pumping frequency, Ω k is a vibronic frequency associated with the energy band), directly observed in Figure 2b and d. Eq. (6) can be used for fitting the high-energy ELS peak at 5 MW/cm² (Figure 2d), by using the regularized least squares method that yields the Penn energy E _p = 136 meV (1,100 cm⁻¹). The net temperature rise of the tip apex is mainly determined by thermal conductivity of the tip shaft served as a heatsink, whereas heat exchange with the surroundings through its surface (Kapitza resistance [36]) is negligible. It is important to notice that the mechanism of inhomogeneous broadening of a central peak at ω k ≈ ω 0 is related to low-energy ELS from a rough gold film that exists even at modest pump (Figure 2d) [10], [21].

2.2 A gold shear-force-microscopy tip

Let us now consider the spatial confinement effect for metals. Figure 3a shows a bottom-illumination TERS setup in which a rough gold tip glued to a quartz tuning fork oscillates in a plane parallel to the surface of bare coverslip with a frequency of 32 kHz. The inset in Figure 3b displays the gold tip apex of 25 nm in curvature radius. Using the same protocol as for c-Si, we reveal optical melting of the Au tip apex within 800 nm extent. This paradoxical result is explained by the fact that the melting points of Si (1,687 K) and Au (1,337 K) differ by 30 %, whereas their thermal conductivities (150 W/mK and 320 W/mK, respectively) are roughly two-fold distinct. In addition, melting points of spatially confined media can be reduced due to the size effect. In this case, a near-field photon is directly generated at the Au tip apex that is first heated and then destroyed. This effect indicates the fundamental role of expanded near-field photon momentum, enhancing light–matter interaction significantly.

Figure 3:

Photoheating of a gold tip apex. (a) Sketch of a TERS setup (inverted configuration). SEM images of a rough Au tips before (b) and after (c) laser illumination with the intensity of 5 MW/cm². (d) A Raman map of the Au tip upon optical heating. The inset shows a dependence of Raman intensity versus pumping intensity for different wavenumbers. (e) A kinetics of the Au tip phase (red) and a dependence of temperature (blue) versus pumping intensity when illuminated by a 633 nm laser light. (f) Schematic illustration of anti-Stokes/Stokes ELS with varying electron momentum Δk _e and interband absorption in gold. The up-left inset shows the Brillouin zone for gold bulk.

We detect not only a low-energy ELS signal often found in spatially confined metals, but also a broadband high-energy ELS band (Figure 3d). The optical melting occurs upon the intensity of 3.2 MW/cm². Like Si, the ELS intensity increases nonlinearly above this threshold, as seen in the inset of Figure 3d. Since the Q-factor of the tuning fork is significantly worse than that of the AFM cantilever, the shear-force-microscopy phase is less sensitive to temperature (red curve in Figure 3e). Following Sheldon et al., we used a combination of both Bose–Einstein and Fermi–Dirac statistics as a fitting function for estimating a temperature inside the Au tip apex by using the regularized least squares method [37]:

where χ is a relative contribution of both to the distribution. The melting temperature is in good agreement with the intensity threshold value at which the ELS intensity starts to increase in Figure 3d. Multiple temperature spikes are caused by dynamical destruction of the tip apex upon cw illumination impact (blue curve, Figure 3e). In contrast to semiconductors, indirect optical transitions in metals occur near the Fermi level, contributing to both anti-Stokes and Stokes signals. The balance between indirect transitions and interband absorption supports charge density oscillations within spatially confined metals, known as localized plasmon resonance.

The temperature, calculated by using Eq. (7), was found from the anti-Stokes wing, displayed in Fig. S5 (Section V, the SI). Interband absorption at 2.4 eV and above reduces the intensity of anti-Stokes scattering and, therefore, this method should be exploited with caution as temperature estimates can be biased. It is important to note that estimating temperature uncertainty is a challenging task because it requires a priori information on the energy band diagram, as implied by Eq. (4). In our case, the pump-dependent temperature of the gold tip apex is determined by an anti-Stokes probe with the accuracy of ca. 150 K (600 grooves per mm). Ultimately, the optical heating is determined by thermalization of charge carries and depends on the slope of the sp-conduction band (Figure 3f).

4.1 Supplementary information

Details on electronic polarization in homogeneous and heterogeneous media, expanded momentum of near-field photon, atomic force microscopy measurements, Raman thermometry, electronic light scattering spectra, methods and materials.