Current-Induced Metallization and Valence Transition in Black SmS

Shin-ichi Kimura^1,2,3, , Hiroshi Watanabe^1,2 , Shingo Tatsukawa², Takuto Nakamura^1,2 , Keiichiro Imura^4, , Hiroyuki S. Suzuki⁵ , and Noriaki K. Sato^4,

Insulator-to-metal transition (IMT) of solids has been one of the critical topics of interest for a long time because of the drastic change in electrical resistivity that is useful for microscopic switching.^1,2) IMT has been observed in many transition-metal compounds and organic conductors, in which they accompany a Jahn–Teller effect,³⁾ a charge/orbital ordering,⁴⁾ and a lattice distortion due to the creation of charge/spin-density waves.⁵⁾ On the other hand, in rare-earth compounds, for instance, in SmB₆ and YbB₁₂ and others, metallic property at higher temperatures changes to a Kondo insulator at low temperatures with a tiny energy gap due to the hybridization between conduction (c) and f states (namely c–f hybridization) by the Kondo effect.⁶⁾ The nonmagnetic insulating property of EuO at high temperatures changes to a ferromagnetic metal due to the indirect exchange interaction.^7,8) These rare-earth compounds do not show a lattice distortion because of the weak crystal electric field of \(4f\) electrons. Samarium mono-sulfide (SmS), the title compound, also shows IMT by applying pressure without lattice distortion as other rare-earth compounds.

SmS with NaCl-type crystal structure has the pressure-induced first-order phase transition from the black-colored semiconductor (b-SmS) to the golden-yellow-colored semimetal (g-SmS) (black-to-golden phase transition: BGT).^9–14) BGT can also be produced by applying chemical pressure of the substituting trivalent yttrium (Y) ions with a smaller ionic radius to Sm³⁺ ions.^15–18) The BGT accompanies the valence transition from Sm²⁺ to Sm³⁺, and the lattice constant decreases by about 5% due to the shrinking of the ionic radius without change in the crystal structure.^19,20) At BGT, the electrons' contribution of Sm ions changes from Sm²⁺ \(4f^{6}\) in b-SmS to Sm³⁺ \(4f^{5}+c\) in g-SmS by the BGT, so the appearance of the carriers is considered the origin of the IMT. Since the valence change and the shrinking of the lattice constant coincidently occur, however, more than 50 years after its discovery, the origin of BGT of this material is still under debate.²¹⁾ One of the proposed ideas for the origin of BGT is the BEC–BCS transition of Sm \(4f\)-\(5d\) exciton.²²⁾ The origin of the BEC–BCS transition is the spontaneous emergence of electron–hole pairs, namely exciton, due to the decrease of the energy gap by applying pressure. If so, creating the \(4f\)-\(5d\) exciton due to other methods is expected to make BGT.

Recently, Ando et al. reported that the voltage (V) as a function of current (I) changes non-linearly from the high differential electrical resistance (\(dV/dI\)) at low I to the low \(dV/dI\) at high I via a negative \(dV/dI\) at lower temperatures than 100 K, namely a current-induced IMT (CIMT), which is discussed to be related to the BEC–BCS transition of the \(4f\)-\(5d\) excitons.²³⁾ A similar nonlinear I–V curve and an electrical oscillation by a constant external voltage have been reported, attributed to an electric-field-induced insulator-to-metal transition.²⁴⁾ The change of the \(dV/dI\) value reflects the difference in the parameters of the conduction electrons, the carrier density (N), and/or the mobility (μ), which is related to the electronic structure change. However, the origin of the CIMT has yet to be revealed.

In this paper, to investigate the origin of the CIMT of b-SmS from the view of the electronic structure, we measured reflectivity [\(R(\omega)\)] spectra in the broad energy region from the terahertz (THz) to the vacuum-ultraviolet and angle-integrated photoelectron (PE) spectra by applying electric currents. As a result, the carrier density rapidly increases with decreasing mobility by applying current in the negative resistance region. In addition, the energy gap size increases, and the mean valence of Sm-ions increases with increasing current. However, the carrier density and the mean valence after CIMT do not reach those of g-SmS, suggesting a different mechanism of BGT or a precursor of BGT. The obtained results suggest that the localized Sm \(4f\) electrons change to an itinerant character by applying current, and the itinerant \(4f\) electrons hybridize with the Sm \(5d\) conduction band. The observed CIMT of b-SmS is concluded as a novel type of valence transition.

Single crystalline samples of b-SmS were grown using the vertical Bridgman method with a high-frequency induction furnace: Pre-reacted materials starting 99.99% pure (4N) samarium and 6N powdered sulfur were sealed in a vacuum W crucible.¹⁴⁾ Cleaved samples with the size of about \(1\times 1\times 1\) mm³ were mounted on the sample folder with the current flow and cooled down to about 20 K for the \(R(\omega)\) measurement with a liquid-He-flow-type cryostat and to 30 K for the PE measurement with a closed-cycle cryostat. The direct current up to 1.6 A was applied while maintaining a temperature range of up to 5 K to avoid temperature rise due to Joule heating and to prevent sample degradation. To obtain the carriers' parameters of the effective carrier density (\(N_{\textit{eff}}\)) and the relaxation time (τ), the near-normal-incident \(R(\omega)\) spectra along the (001) plane in the THz region of 6–50 meV were measured, and fitted with the Drude–Lorentz function because the absolute reflectivities have not been obtained. The detail is explained in the Supplementary Material.²⁵⁾ To obtain optical conductivity [\(\sigma_{1}(\omega)\)] spectra via the Kramers–Kronig analysis, \(R(\omega)\) spectra along the (001) plane were acquired in a wide photon-energy range of 6 meV–10 eV. The measurements were performed at UVSOR-III Synchrotron, Institute for Molecular Science, Japan.^26–28) PE spectra have been accumulated using a He-II discharge lamp with a monochromator (VG-Scienta VUV 5000+5040), a hemispherical analyzer (MBScientific A-1), and an own-developed cold sample finger with a current-flow system. The base pressure for the PE measurement was less than \(1\times 10^{-8}\) Pa.

Firstly, the temperature dependence of the I–V behavior has been checked. Figure 1(a) shows the I–V characteristics of b-SmS measured with increasing electric current at temperatures from 20 to 250 K, where the I–V curves show a sizable nonlinear behavior at temperatures below 50 K, especially negative \(dV/dI\) appears in the current region from 30 to 120 mA below 40 K, which is qualitatively consistent with that previously reported.^23,24) On the other hand, at higher temperatures than 80 K, almost linear I–V curves are observed, which suggests that the electrical resistance (\(V/I\)) value is almost constant. A Joule heating effect is expected, as described elsewhere,²⁹⁾ because the applied power is as large as 1 W above 1 A (The current density \({\sim}10^{2}\) A/cm²). However, the \(V/I\) value must rapidly decrease if the sample temperature increases due to the semiconducting property. The \(V/I\) value at the current region of \(I\geq 1.0\) A at temperatures higher than 80 K is about 80% of those at \(I=0.1\) A. The small \(V/I\) value change suggests that the temperature increase is less than 20 K evaluated from the energy-gap size of about 1200 K.¹⁴⁾ According to the discussion, the Joule heating effect above 80 K can be considered insignificant.

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Figure 1. (Color online) (a) I–V curves of b-SmS at temperatures from 20 to 250 K. Some discontinuities appearing in the I–V curves originate from changing the connection between the sample and electric contacts. (b) Current-dependent reflectivity [\(R(\omega)\)] spectra of b-SmS in the THz region at \(T=40\) K. (c, d) Temperature and external-current dependence of (c) the effective carrier density (\(N_{\textit{eff}}\)) and (d) the relaxation time (τ) of b-SmS evaluated by the DL fitting of the THz \(R(\omega)\) spectra shown in (b) and at other temperatures. (e, f) Same as (c) and (d), but all of the DL fitting data and schematic curved surfaces are also plotted as guides for the eye.

At lower temperatures than 80 K, on the other hand, the negative \(dV/dI\) appears, suggesting the rapid decrease of the electrical resistivity. If the decreasing \(V/I\) is assumed to originate only from the Joule heating effect, the increasing temperature can be evaluated from the temperature-dependent electrical resistivity curve, where the decrease in electrical resistivity over many orders of magnitude appears with increasing temperature. Under this assumption, the increasing temperature at \(T=20\) K with \(I=40\) mA is evaluated as less than 5 K, consistent with the measurements with a diode temperature sensor close to the sample. We believe that such small temperature increases cannot make the insulator-to-metal transition observed in this work because no thermal phase transition occurs in SmS, i.e., the metallization and valence transition, as shown in the following, is considered to originate from the current-induced effect.

To obtain information on carriers at different currents, we derived the temperature and current dependences of the carrier density and the relaxation time from the Drude–Lorentz (DL) fitting of the \(R(\omega)\) spectra in the THz region.^25,30) As an example of the current-dependent THz \(R(\omega)\) spectra, spectra at \(T=40\) K are shown in Fig. 1(b). At \(I = 0\) A, the spectrum is a typical insulating one, with a phonon peak at ∼25 meV, and the background intensity is low, suggesting a low carrier density. With increasing current, the background intensity increases, accompanied by a rapid change in the phonon shape originating from the interference between the phonon and background absorptions. After analyzing the spectral change using the DL fitting, we obtained the strong temperature- and current-dependent \(N_{\textit{eff}}\) and τ as shown in Figs. 1(c) and 1(d). To show the temperature and current dependences more clearly, the three-dimensional plots are also shown in Figs. 1(e) and 1(f). On the other hand, obtained parameters for the phonon peak showed little change with applied current as shown in Fig. S2 in the Supplementary Material,²⁵⁾ suggesting the Joule heating effect is not large, consistent with the temperature kept within 5 K. Therefore, the current dependence of \(N_{\textit{eff}}\) and τ originates from the applied current.

As shown in Fig. 1(c), \(N_{\textit{eff}}\) increases with increasing current at all temperatures. At higher temperatures than 150 K, the rate of increase (\(dN_{\textit{eff}}/dI\)) is almost constant at different temperatures [\({\sim}1.2\times 10^{16}\) cm⁻³ A⁻¹, dotted lines in Fig. 1(c)], but the intercept at 0 A increases with increasing temperature. The \(N_{\textit{eff}}\) at 0 A originates from the thermally excited carriers because the smallest (indirect) energy gap size of the material is about 0.1 eV,³¹⁾ comparable to the thermal excitation energy of \(k_{\text{B}}T\). The similar \(dN_{\textit{eff}}/dI\) value is also observed at higher currents than 0.6 A of lower temperatures than 100 K, suggesting the electronic structure in the region is consistent with the high-temperature range.

At lower currents than 0.6 A and lower temperatures than 100 K, on the other hand, the \(dN_{\textit{eff}}/dI\) value is about \(3.2\times 10^{16}\) cm⁻³ A⁻¹ [the dashed line in Fig. 1(c)], which is about three times larger than that at the high-current region. τ also enhances in the same temperature and current ranges. Usually, τ, as well as the electron mobility μ, have a carrier-density dependence due to the electron–electron scattering, but, in the low-carrier-density region at around \(10^{16}\) cm⁻³, they do not usually change so much.^32,33) Therefore, the change of τ at low temperature and low current region suggests that the character of carriers and the electronic structure are modulated from the region of the low temperature and low current to that of the high temperature and high current.

To clarify the electronic structure change by the current application, the current-dependent \(\sigma_{1}(\omega)\) spectra at \(T = 50\) K are shown in Fig. 2. At the low-current region of \(I=0.1\) A, a peak appears at about 0.56 eV originating from the exciton absorption at the direct gap between the Sm \(4f\) and \(5d\) states at the X point,^30,31) as evidenced in the inset of Fig. 2. By applying a current of ∼0.9 A, the peak at ∼0.64 eV shifts to the high-energy side. The onset of the reading edge also increases by about 40 meV from about 0.28 to 0.32 eV with increasing current. These results indicate the enlargement of the energy gap size. The obtained \(\sigma_{1}(\omega)\) spectrum at high current is completely different from that of g-SmS, in which there is a Drude-like spectral shape with a plasma edge at about 3 eV.^11,12,35) Therefore, the electronic structure of the high-current phase is not the same as that of g-SmS. Instead, the high-current spectrum in the low-energy region below 0.2 eV is similar to the spectrum at the pressure of 0.6 GPa, just below the boundary of BGT.³⁰⁾ However, the exciton peak shifts to the low-energy side by a pressure application, which trend is the opposite behavior of the current application. The pressure-induced energy gap narrowing below 0.6 GPa indicates that the material is still in the black phase at the pressure; therefore, the thermally excited carrier appears. However, in the high-current spectrum, the energy gap size increases, even though the carrier density increases with increasing current. This fact suggests that the carriers in the high-current region do not originate from thermal excitation. One scenario for the energy-gap enlargement at high currents is the appearance of the bonding and anti-bonding states of the hybridization between the Sm \(4f\) valence band and the \(5d\) conduction band. In the low-current region, a direct gap is expected at the X point in the Brillouin zone.³¹⁾ Then, since the Sm \(4f\) states are isolated, the Sm \(5d\) conduction band, which orbital the carriers go through, is not hybridized. At the high-current region, the isolated \(4f\) electrons change to itinerant; the \(4f\) electrons can be hybridized with the carriers, i.e., the hybridization bands, namely bonding and anti-bonding states of the \(4f\) states and \(5d\) conduction bands, are created. The energy gap between the bonding and anti-bonding states is enlarged with increasing the hybridization intensity. In addition, the mean valence of Sm ions is expected to be changed from 2+ to \(2+\delta\) due to the \(4f\)-\(5d\) hybridization. Then, delocalized electrons released from the Sm \(4f\) states can increase with increasing the hybridization intensity.

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Figure 2. (Color online) Current-dependent optical conductivity [\(\sigma_{1}(\omega)\)] spectra of b-SmS in \(\hbar\omega\) below 0.7 eV at \(T=50\) K, derived from the Kramers–Kronig analysis (KKA) of the reflectivity [\(R(\omega)\)] spectra. In the low current of \(I=0.1\) A, the onset of the reading edge is about 0.28 eV, whereas it shifts to about 0.32 eV in the high current of 0.9 A. The inset shows that the absorption edges in the absorption [\(\alpha(\omega)\)] spectra obtained from the KKA obey the \(\exp(\hbar\omega)\)-law, suggesting evidence of a direct gap from the Urbuch rule of exciton peaks.³⁴⁾ The arrow indicates the indirect energy gap size.³¹⁾ A sharp peak at about 20 meV originates from a TO phonon that is the same as the peak shown in Fig. 1(b).

To check the hybridization between the Sm \(4f\) and \(5d\) states at high currents, we measured the current-dependent valence band PE spectra as shown in Fig. 3(a). The figure shows that the PE spectrum has strong current dependence, i.e., the intensity of the Sm²⁺ \(4f\) multiplet peaks in the binding energy range below 3 eV decreases, and in contrast, the Sm³⁺ peak at about 8 eV increases with increasing current. Similarly, as shown in Fig. 3(b), the intensity of the Sm²⁺ \(5p\) peaks decreases, and that of the Sm³⁺ \(5p\) peaks increases with the current application. By the fitting of two Gaussian pairs assumed as the mixture of the Sm²⁺ and Sm³⁺ components and the spin–orbit splitting of the \(5p\) states, the Sm mean valence increases from \(2.19\pm 0.03\) at \(I=0\) A to \(2.41\pm 0.04\) at 0.6 A. Therefore, the scenario of the appearance of the hybridization state between Sm \(4f\) and \(5d\) at the high-current region is plausible.

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Figure 3. (Color online) (a) Current-dependent angle-integrated PE spectra in the valence band energy region as a function of binding energy (\(E_{B}\)). The PE spectra are accumulated at about 30 K using the He-II line (\(h\nu=40.8\) eV) as a light source. (b) Current-dependent Sm \(5p\) core level spectra. The marks and lines represent the raw data (a background was subtracted) and fitted data (dotted and dot-dashed lines: Sm²⁺ and Sm³⁺ components, respectively; solid line: sum of the components). Each component was fitted with two Gaussian functions assuming a spin–orbit pair. (c) The evaluated mean valence from the fitting in (b).

To explain all of the experimental evidence, the lowering τ, the energy gap enlargement, and the valence increasing by current application, we propose a model shown in Fig. 4. At the ground state without current application, the Sm²⁺ \(4f\) states are localized, and the energy gap between the Sm \(4f\) and \(5d\) states opens.³¹⁾ Since the \(5d\) conduction band is a free-electron-like band and the Sm ions are localized and non-magnetic divalent, the carriers' relaxation time τ is long because of the lack of strong interactions. With applying current, the flowing electrons push the localized \(4f\) electrons out from their localized sites by an electron–electron interaction because the energy level of the localized \(4f\) electrons is very close to \(E_{\text{F}}\), and then the \(4f\) electrons become itinerant. The itinerant \(4f\) electrons can hybridize with carriers in the Sm \(5d\) states, so the \(4f\)-\(5d\) hybridization state appears. Carriers in the hybridization band have a shorter relaxation time because of the strong interaction.

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Figure 4. (Color online) The scenario of the current-induced metallization and valence transition of b-SmS. (a) Real space description of the current-induced metallization. Localized \(4f\) electrons are induced to be itinerant by applied currents. (b) Reciprocal lattice description of the current-induced metallization and valence transition. After changing to the itinerant \(4f\) electrons at high currents, they hybridize to carriers, namely \(4f\)-\(5d\) hybridization. Then the bottom of the conduction band becomes flat. The relaxation time τ, which corresponds to the mobility (\(\mu=e\tau/m\), where e is the electron charge and m the electron mass), becomes shorter than that in the low-current case. After the appearance of the hybridization, the Sm valence increases. Then the Fermi level (\(E_{\text{F}}\)) moves to the higher energy side.

Similar phenomena to the CIMT and valence transition on b-SmS have not been observed, to the best of our knowledge, even though current-induced phenomena, such as a Néel vector switching,³⁶⁾ phase transitions in cuprate,³⁷⁾ and a magnetic thin film,³⁸⁾ has recently been reported. Also, electric-field-induced metallization on Ca₂RuO₄^39,40) is similar to our observation, but the electric field in our study is not so high. Our observation of the current-induced metallization and valence transition completely differs from such works. This phenomenon may be related to other anomalous phenomena, such as pressure-induced BGT, alkali-metal-induced surface valence transition,⁴¹⁾ photo-induced band shift,⁴²⁾ electrical oscillation by a constant external voltage,²⁴⁾ and the BEC–BCS transition of Sm \(4f\)-\(5d\) exciton; therefore, a comprehensive understanding of these phenomena will help to elucidate the fruitful physical properties of SmS. Similar valence transitions due to external perturbations are expected to be observable in materials near the critical point of the phase transition.

To summarize, we have investigated the electronic structure change of insulating b-SmS by the current application. By analyzing the THz reflectivity spectra at temperatures below 100 K, the carrier density rapidly increased with current application up to 0.8 A and slowly after that, accompanied by the decreasing relaxation time, suggesting the changing of the carriers' character and the electronic structure at high currents. The current application also increased the direct energy gap size at the X point in the Brillouin zone and the increase of Sm ions' mean valence, suggesting that the applied current induces the Sm \(4f\)-\(5d\) hybridization. The observed current-induced metallization accompanying the valence transition in b-SmS is a novel phenomenon of insulator-to-metal transitions.

References

• 1 N. F. Mott, Rev. Mod. Phys. 40, 677 (1968). 10.1103/RevModPhys.40.677 Crossref, Google Scholar
• 2 N. F. Mott, Metal–Insulator Transitions (CRC Press, Boca Raton, FL, 2004). Crossref, Google Scholar
• 3 J. E. Han, E. Koch, and O. Gunnarsson, Phys. Rev. Lett. 84, 1276 (2000). 10.1103/PhysRevLett.84.1276 Crossref, Google Scholar
• 4 Y. Moritomo, H. Kuwahara, Y. Tomioka, and Y. Tokura, Phys. Rev. B 55, 7549 (1997). 10.1103/PhysRevB.55.7549 Crossref, Google Scholar
• 5 G. Grüner, Density Waves in Solids (CRC Press, Boca Raton, FL, 2018). Crossref, Google Scholar
• 6 T. Kasuya, Europhys. Lett. 26, 277 (1994). 10.1209/0295-5075/26/4/007 Crossref, Google Scholar
• 7 V.-C. Lee and L. Liu, Phys. Rev. B 30, 2026 (1984). 10.1103/PhysRevB.30.2026 Crossref, Google Scholar
• 8 H. Miyazaki, T. Ito, H. J. Im, S. Yagi, M. Kato, K. Soda, and S. Kimura, Phys. Rev. Lett. 102, 227203 (2009). 10.1103/PhysRevLett.102.227203 Crossref, Google Scholar
• 9 A. Jayaraman, V. Narayanamurti, E. Bucher, and R. G. Maines, Phys. Rev. Lett. 25, 1430 (1970). 10.1103/PhysRevLett.25.1430 Crossref, Google Scholar
• 10 B. Batlogg, J. Schoenes, and P. Wachter, Phys. Lett. A 49, 13 (1974). 10.1016/0375-9601(74)90647-1 Crossref, Google Scholar
• 11 B. Batlogg, E. Kaldis, A. Schlegel, and P. Wachter, Phys. Rev. B 14, 5503 (1976). 10.1103/PhysRevB.14.5503 Crossref, Google Scholar
• 12 G. Travaglini and P. Wachter, Phys. Rev. B 30, 5877 (1984). 10.1103/PhysRevB.30.5877 Crossref, Google Scholar
• 13 K. Matsubayashi, K. Imura, H. S. Suzuki, G. Chen, N. Mori, T. Nishioka, K. Deguchi, and N. K. Sato, J. Phys. Soc. Jpn. 76, 033602 (2007). 10.1143/JPSJ.76.033602 Link, Google Scholar
• 14 K. Matsubayashi, K. Imura, H. S. Suzuki, T. Mizuno, S. Kimura, T. Nishioka, K. Kodama, and N. K. Sato, J. Phys. Soc. Jpn. 76, 064601 (2007). 10.1143/JPSJ.76.064601 Link, Google Scholar
• 15 T. Penney and F. Holtzberg, Phys. Rev. Lett. 34, 322 (1975). 10.1103/PhysRevLett.34.322 Crossref, Google Scholar
• 16 G. Guntherodt and F. Holtzberg, Solid State Commun. 18, 181 (1976). 10.1016/0038-1098(76)91449-6 Crossref, Google Scholar
• 17 M. Kaneko, M. Saito, T. Ito, K. Imura, T. Hajiri, M. Matsunami, S. Kimura, H. S. Suzuki, and N. K. Sato, JPS Conf. Proc. 3, 011080 (2014). 10.7566/JPSCP.3.011080 Link, Google Scholar
• 18 Y. Yokoyama, H. Hasegawa, Y. Mizuno, D. Asai, Y. Okamoto, H. S. Suzuki, K. Takehana, Y. Imanaka, and K. Takenaka, Phys. Rev. B 100, 245143 (2019). 10.1103/PhysRevB.100.245143 Crossref, Google Scholar
• 19 I. Jarrige, H. Yamaoka, J.-P. Rueff, J.-F. Lin, M. Taguchi, N. Hiraoka, H. Ishii, K. D. Tsuei, K. Imura, T. Matsumura, A. Ochiai, H. S. Suzuki, and A. Kotani, Phys. Rev. B 87, 115107 (2013). 10.1103/PhysRevB.87.115107 Crossref, Google Scholar
• 20 K. Imura, Y. Suzuki, N. Kawade, K. Sakamoto, K. Deguchi, H. Yamaoka, Y. Yamamoto, J. Mizuki, N. Hiraoka, H. Ishii, H. S. Suzuki, and N. K. Sato, JPS Conf. Proc. 30, 011131 (2020). 10.7566/JPSCP.30.011131 Link, Google Scholar
• 21 S. Watanabe, J. Phys. Soc. Jpn. 90, 023706 (2021). 10.7566/JPSJ.90.023706 Link, Google Scholar
• 22 C. M. Varma, Rev. Mod. Phys. 48, 219 (1976). 10.1103/RevModPhys.48.219 Crossref, Google Scholar
• 23 H. Ando, K. Sakamoto, K. Imura, K. Deguchi, H. S. Suzuki, and N. K. Sato, JPS Conf. Proc. 30, 011132 (2020). 10.7566/JPSCP.30.011132 Link, Google Scholar
• 24 H. Takahashi, R. Okazaki, H. Taniguchi, I. Terasaki, M. Saito, K. Imura, K. Deguchi, N. K. Sato, and H. S. Suzuki, Phys. Rev. B 89, 195103 (2014). 10.1103/PhysRevB.89.195103 Crossref, Google Scholar
• 25 (Supplemental Material) The Drude–Lorentz fitting method and the fitting data of phonons are available online. Google Scholar
• 26 S. Kimura, E. Nakamura, T. Nishi, Y. Sakurai, K. Hayashi, J. Yamazaki, and M. Katoh, Infrared Phys. Technol. 49, 147 (2006). 10.1016/j.infrared.2006.01.008 Crossref, Google Scholar
• 27 S. Kimura, Y. Sakurai, E. Nakamura, and T. Mizuno, AIP Conf. Proc. 879, 595 (2007). 10.1063/1.2436131 Crossref, Google Scholar
• 28 K. Fukui, R. Ikematsu, Y. Imoto, M. Kitaura, K. Nakagawa, T. Ejima, E. Nakamura, M. Sakai, M. Hasumoto, and S. Kimura, J. Synchrotron Radiat. 21, 452 (2014). 10.1107/S1600577513032931 Crossref, Google Scholar
• 29 K. Kishida, J. Shibutani, T. Hatano, Y. Yokoyama, N. Katayama, Y. Okamoto, Y. Nakamura, H. Kishida, H. S. Suzuki, K. Takehana, Y. Imanaka, and K. Takenaka, Appl. Phys. Lett. 121, 121901 (2022). 10.1063/5.0108716 Crossref, Google Scholar
• 30 T. Mizuno, T. Iizuka, S. Kimura, K. Matsubayashi, K. Imura, H. S. Suzuki, and N. K. Sato, J. Phys. Soc. Jpn. 77, 113704 (2008). 10.1143/JPSJ.77.113704 Link, Google Scholar
• 31 S. Kimura, T. Mizuno, K. Matsubayashi, K. Imura, H. S. Suzuki, and N. K. Sato, Physica B 403, 805 (2008). 10.1016/j.physb.2007.10.095 Crossref, Google Scholar
• 32 G. Masetti, M. Severi, and S. Solmi, IEEE Trans. Electron Devices 30, 764 (1983). 10.1109/T-ED.1983.21207 Crossref, Google Scholar
• 33 S. Reggiani, M. Valdinoci, L. Colalongo, M. Rudan, and G. Baccarani, VLSI Des. 10, 467 (2000). 10.1155/2000/52147 Crossref, Google Scholar
• 34 F. Urbach, Phys. Rev. 92, 1324 (1953). 10.1103/PhysRev.92.1324 Crossref, Google Scholar
• 35 J. L. Kirk, K. Vedam, V. Narayanamurti, A. Jayaraman, and E. Bucher, Phys. Rev. B 6, 3023 (1972). 10.1103/PhysRevB.6.3023 Crossref, Google Scholar
• 36 H. Wu, H. Zhang, B. Wang, F. Groß, C.-Y. Yang, G. Li, C. Guo, H. He, K. Wong, D. Wu, X. Han, C.-H. Lai, J. Gräfe, R. Cheng, and K. L. Wang, Nat. Commun. 13, 1629 (2022). 10.1038/s41467-022-29170-2 Crossref, Google Scholar
• 37 T. Naito, T. Nishizaki, and N. Kobayashi, J. Appl. Phys. 129, 043901 (2021). 10.1063/5.0033000 Crossref, Google Scholar
• 38 D.-k. Kim, A. Domenicucci, and S. S. Iyer, J. Appl. Phys. 103, 073708 (2008). 10.1063/1.2899959 Crossref, Google Scholar
• 39 F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki, and Y. Maeno, Sci. Rep. 3, 2536 (2013). 10.1038/srep02536 Crossref, Google Scholar
• 40 J. Zhang, A. S. McLeod, Q. Han, X. Chen, H. A. Bechtel, Z. Yao, S. N. Gilbert Corder, T. Ciavatti, T. H. Tao, M. Aronson, G. L. Carr, M. C. Martin, C. Sow, S. Yonezawa, F. Nakamura, I. Terasaki, D. N. Basov, A. J. Millis, Y. Maeno, and M. Liu, Phys. Rev. X 9, 011032 (2019). 10.1103/PhysRevX.9.011032 Crossref, Google Scholar
• 41 T. Nakamura, T. Nakaya, Y. Ohtsubo, H. Sugihara, K. Tanaka, R. Yukawa, M. Kitamura, H. Kumigashira, K. Imura, H. S. Suzuki, N. K. Sato, and S. Kimura, Phys. Rev. B 107, L041102 (2023). 10.1103/PhysRevB.107.L041102 Crossref, Google Scholar
• 42 Y. Chen, T. Nakamura, H. Watanabe, T. Suzuki, Q. Ren, K. Liu, Y. Zhong, T. Kanai, J. Itatani, S. Shin, K. Okazaki, K. Imura, H. S. Suzuki, N. K. Sato, and S. Kimura, arXiv:2212.00983. Google Scholar