Temperature-Dependent Optical Properties of Single Crystalline and

Apr 10, 2017 - The conclusion on the crystalline quality was made based on transmission electron microscope (TEM) and X-ray diffraction (XRD) measurem...
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Temperature-Dependent Optical Properties of Single Crystalline and Polycrystalline Silver Thin Films Harsha Reddy,† Urcan Guler,† Krishnakali Chaudhuri, Aveek Dutta, Alexander V. Kildishev, Vladimir M. Shalaev, and Alexandra Boltasseva* School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, United States S Supporting Information *

ABSTRACT: Silver holds a unique place in plasmonics compared to other noble metals owing to its low losses in the visible and near-IR wavelength ranges. With a growing interest in local heating and high temperature applications of plasmonics, it is becoming critical to characterize the dielectric function of nanometer-scale thin silver films at higher temperatures, especially near the breakdown temperature, which depends on the film thickness and crystallinity. So far, such a comprehensive study has been missing. Here we report the in situ high temperature ellipsometry measurements of ultrasmooth and epitaxial quality crystalline silver films, along with electron beam evaporated polycrystalline silver films at temperatures up to 700 °C, in the wavelength range of 330−2000 nm. Our findings show that the dielectric function of all the films changes remarkably at elevated temperatures with larger relative changes observed in polycrystalline films. In addition, low-loss epitaxial films were found to be thermally more stable at elevated temperatures. We demonstrate the importance of our findings for high temperature applications with a numerical simulation of field enhancement in a bow-tie nanoantenna, a near field transducer commonly used for heat-assisted magnetic recording. The simulated field profiles at elevated temperatures showed significant deviations compared to those at room temperature, clearly suggesting that the use of room temperature optical properties in modeling elevated temperature applications can be misleading due to the thermal deviations in the Ag dielectric function. We also provide causal analytical models describing the elevated temperature Ag dielectric functions. KEYWORDS: plasmonics, thin films, optical properties, thermal effects, ellipsometry

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(STPV), where the metal-based components are intentionally heated to approximately 1000 °C. With increasing temperatures, various physical processes, such as increasing electron− phonon interaction,17 reducing carrier density,18 and changes in the crystallinity, are expected to lead to significant deviations in the optical responses. However, numerical modeling of all these applications has so far been performed using tabulated room temperature optical properties,19 which is primarily due to the lack of a reliable resource for elevated temperature optical properties. The temperature evolution of optical properties in noble metals has been studied in the past.20,21 However, only the optical responses of bulk metals were probed in these studies, while their thinner nanoscale counterparts that are more relevant to nanoscience and nanotechnology were not investigated. For nanometer scale thick films, the enhancement in the plasmonic performance of Ag films at low temperatures22 and the elevated temperature optical properties of 200 nm thick silver films were previously reported.23,24 However, in these

ow losses in silver have made it one of the most widely used materials in plasmonic and optical metamaterial applications, such as near field superlens,1,2 negative index metamaterials,3 and far field hyperlens.4,5 More recently, single crystal silver films were also used for realizing hyperbolic metasurfaces in visible frequencies.6 All of the novel phenomena enabled by plasmonics and optical metamaterials are due to the excitation of surface plasmons, the collective oscillations of free electrons, which allow confinement and control of light at dimensions much smaller than the diffraction limit.7−9 On the other hand, all plasmonic materials suffer from ohmic losses, which lead to the dissipation of confined energy into the kinetic energy of free electrons, thereby increasing the temperature of the plasmonic components. For instance, in Heat Assisted Magnetic Recording (HAMR) the plasmonic near-field transducer is estimated to operate at temperatures close to 400−500 °C.10−12 Other self-heating applications imposing high temperatures include biosensors, photothermal therapy,13−15 and spasers.16 In addition to these self-heating applications, there has been a growing research interest in various high-temperature energy conversion applications, such as thermophotovoltaics (TPV) and solar thermophotovoltaics © 2017 American Chemical Society

Received: November 8, 2016 Published: April 10, 2017 1083

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studies, the maximum temperature was only 377 °C (650 K) and only 200 nm thick Ag films were considered. With the growing interest in high temperature applications, it has become critical to characterize the properties at even higher temperatures, close to the breakdown temperatures of nanometer scale thin films. Furthermore, a recent study on thin gold films showed that the temperature induced changes are quite sensitive to both the sample thickness and crystallinity.17 Such a study has so far been missing in silver films. Herein, we performed a comprehensive study of the temperature dependent optical properties of Ag films of varying crystallinities and thicknesses, up to 700 °C, in the spectral range 330−2000 nm using in situ high temperature ellipsometry. More specifically, we have investigated the temperature evolution of optical properties in 34, 200, and 450 nm thick single crystalline (SC) films as well as 40 and 100 nm thick electron beam (e-beam) evaporated films that were poly crystalline (PC) in nature. Our findings show that the optical properties change significantly with increasing temperature and the relative changes in the dielectric function are strongly dependent on both the sample thickness and crystallinity. Notably, the imaginary part of the complex dielectric function showed remarkable changes at longer wavelengths, increasing by nearly 2−4×, depending on the sample thickness and crystallinity. In addition, the singlecrystalline films were found to be more stable at elevated temperatures. We demonstrate the importance of our findings in self-heating applications by performing full-wave numerical simulations (using a finite element method (FEM) solver, COMSOL Multiphysics, Wave Optics Module) of the field profile in the gap between bowtie-nanoantennas, a plasmonic near field transducer in HAMR applications, using the extracted elevated temperature optical properties. Quite remarkably, the field enhancement reduces significantly at elevated temperatures, indicating that the temperature dependencies must be incorporated into the numerical models for the accurate description of plasmonic systems involving self-heating. We also provide experiment fitted causal analytical models to describe the temperature-dependent complex dielectric functions. These causal analytical models will be of critical assistance for rational designing and accurate modeling of Agbased nanophotonic components aimed at specific local and global heating applications.

The HRTEM image shows the regular arrangement of the atoms, clearly indicating the high degree of crystallinity and larger grain sizes in the sputtered sample. We therefore refer to the sputtered films as single crystalline and the evaporated films as polycrystalline in the rest of the discussion below. In addition, the surface morphology from AFM topographs on all the sputtered films showed subnm RMS roughness, while the PC films displayed larger roughness values (few nm). More details on growth conditions, TEM, XRD, and AFM characterizations are available in the Supporting Information.

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EXPERIMENTAL SECTION We performed the temperature-dependent studies using a variable angle spectroscopic ellipsometer (VASE) equipped with a heating stage (Linkam Scientific Model TS1500). The samples were mounted on a ceramic cup that could heat up the samples from ambient to 1500 °C with a rated temperature stability of ±2 °C. The heating stage was kept under high vacuum (∼10−6 Torr) to prevent oxidation and sulfidation of silver.25 As described in the schematic of the experimental setup in Figure S1, a quartz window provides optical access to the sample. This introduces fixed offsets into the ellipsometer measurements (tan ψ, Δ). Before each temperature-dependent study, we calibrated these offsets at room temperature and used them to correct the subsequent measurements (see Supporting Information for details on window offsets calibration). A major limiting factor for accurate measurements at high temperatures was the background thermal emission from the ceramic heating cup and the sample itself.17 With increasing temperature the background thermal emission becomes stronger, eventually saturating the VASE detector over 450 °C. We overcame this limitation by placing a pinhole in the reflected beam path. This suppresses the background thermal radiation while still allowing most of the reflected beam to reach the detector, hence enabling accurate measurements over 450 °C (see ref 17 for more details). The whole stage was water cooled, which kept the quartz window close to ambient temperature. The real and imaginary parts of the complex dielectric function (ε̂(ω) = ε1 + iε2) were then extracted by fitting the corrected VASE data with a Drude and one Lorentz oscillator model of the following form:

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ε(̂ ω) = ε∞ −

SAMPLE GROWTH Single crystalline Ag films were prepared by DC magnetron sputtering (PVD Products) of Ag (99.99%, Kurt J. Lesker) onto MgO substrates (500 μm thick) at 10−8 Torr. A thin TiN layer of 4 nm thickness was used as an underlayer to support crystalline growth (see Supporting Information). The polycrystalline Ag (99.99%, Kurt J. Lesker) films were also grown on MgO substrates held at room temperature in an e-beam evaporator (Leybold) at 10−6 Torr. No underlayer was used in this process. The conclusion on the crystalline quality was made based on transmission electron microscope (TEM) and X-ray diffraction (XRD) measurements. Figure S2 shows the XRD data on Ag films grown from both DC magnetron sputtering and e-beam evaporation. The thick sputtered films showed a strong peak corresponding to Ag (2 0 0), while the thick evaporated films had a weak Ag (1 1 1) peak in the XRD scans. No noticeable peak corresponding to Ag was observed in thin PC film. In Figure S3, we show high resolution transmission electron micrographs (HRTEM) of a 450 nm sputtered film.

ωp2 2

ω + i ΓDω

+

A1 E12

− (ℏω)2 − iγ1(ℏω) (1)

Here ε∞, ωp, and ΓD are the background dielectric constant, plasma frequency, and Drude damping, respectively; additionally, A1, E1, and γ1 are, respectively, the oscillator strength, oscillator energy, and oscillator damping. While retrieving the optical constants from the VASE data, the optically thick films (with thicknesses over a 100 nm) were considered to be semi-infinite. In thinner films (34 nm thick SC and 40 nm thick PC films), reflections from the interface between the Ag film and the substrate also contribute to VASE measurements. We model these films with multilayer models consisting of either Ag/TiN/MgO or Ag/MgO to account for additional reflections. We first collected data on a bare MgO substrate and 4 nm thick TiN film deposited on MgO, and extracted their dielectric functions by fitting with Cauchy and Drude-Lorentz oscillator models, respectively. These optical properties were subsequently used for MgO and TiN layers, 1084

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Figure 1. Temperature-dependent dielectric functions of 34 nm (a, b) and 450 nm (c, d) sputtered single crystalline Ag films. Various colors correspond to different temperatures as shown in the legend. (a, c) Real parts and (b, d) imaginary parts of the complex dielectric functions for 34 and 450 nm thick films, respectively. The imaginary part increases with increasing temperature, while the real part changes only marginally. Insets show the real and imaginary parts for selected wavelength ranges.

the real part becomes more negative, making the films more metallic. The increase in metallicity can be attributed to the decrease in the electron effective mass, which in turn increases the plasma frequency thereby making the films more metallic. A detailed description of the physics behind these temperatureinduced changes is given in the theory section. Subsequent optical images taken on the heat-treated samples revealed the formation of highly regular rectangular cracks (Figure S4). Depth profile analysis from atomic force microscope (AFM) measurements showed the depth of the cracks to be quite close to the thickness of the film (Figure S6). Such regular cracks are a good indication of a high degree of crystallinity in these films. In Figure 1c,d, we plot the temperature dependencies of ε1 and ε2 in a 450 nm thick SC Ag film from room temperature to 700 °C. Similar to the 34 nm thick film, the imaginary part (Figure 1d) increases, while the real part (Figure 1c) reduces marginally with increasing temperature. It should, however, be noted that the maximum increase in ε2 at longer wavelengths is only a factor of 2, whereas in the case of a thinner film, nearly a 4-fold increase was observed. These temperature dependencies can be attributed to an increase in the electron−phonon interaction and the reduced effective mass as described above. Figure S8 shows the temperature evolution of the dielectric function in a 200 nm SC film. The deviations in real part are marginal (Figure S8a), while the imaginary part increases with temperature, similar to 450 nm SC film. However, the relative changes in the imaginary part at 200 °C are smaller compared to 450 and 34 nm sputtered films, which is due to the experimental error, possibly due to fluctuations in the lamp intensity. Unlike the thin films, no cracks were found on the thick films after heat treatment. Note that the maximum temperature in case of thin SC films was 600 °C, whereas for

while retrieving the properties of Ag films. Mean square errors (MSEs) for all the fits were under 3, indicating that the fits were accurate. The extracted temperature-dependent Drude-Lorentz oscillator models for all Ag films are provided in the Supporting Information (see Tables S1−S5 in Supporting Information).

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RESULTS

Figure 1a,b show the temperature-dependent real and imaginary parts of the complex dielectric function of a 34 nm thick SC Ag film at selected temperatures, up to 600 °C. With increasing temperature we observe significant deviations in the complex dielectric function compared to those from room temperature measurements. These temperature-induced deviations are particularly evident at longer wavelengths (λ > 900 nm), where the interband transitions are insignificant, and only the Drude terms contribute. At these wavelengths, the imaginary part (ε2) increases with increasing temperature, and eventually at 600 °C it becomes nearly four times larger than that at room temperature (Figure 1b). The observed increase in ε2 is primarily due to the increasing phonon number. Qualitatively, as the temperature is raised the phonon number increases following Bose−Einstein statistics. This in turn increases the electron−phonon scattering, thereby elevating losses. As a result ε2, which is proportional to the scattering rates, also increases. The real part (ε1), on the other hand, only changes marginally with temperature (Figure 1a). The small deviations in the real part can be understood by considering the temperature dependencies at longer wavelengths where the interband transitions are insignificant (shown in the inset of Figure 1a). At these wavelengths, with increasing temperature 1085

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Figure 2. Temperature-dependent dielectric functions of 40 nm (a, b) and 100 nm (c, d) e-beam deposited polycrystalline Ag films. Different colors correspond to different temperatures (shown in legend). The real part is plotted in (a) and (c), while the imaginary part is shown in (b) and (d). In 40 nm thick films, both the real and imaginary parts change significantly at elevated temperatures. For 100 nm thick films, the real part changes only marginally with increasing temperature (c), while the imaginary part increases monotonically with increasing temperature (d). Insets show the real and imaginary parts for selected wavelength ranges.

the thick films we collected data up to 700 °C. This was because at temperatures over 600 °C the thin SC films showed severe structural deviation with the cracks occupying a significant fraction of the film area (see Figure S4 and AFM scan in Figure S6). As a result, it was no longer possible to reliably retrieve the dielectric function with a simple DrudeLorentz oscillator model beyond the breakdown temperature. Subsequently, we measured the temperature dependencies in polycrystalline films (Figure 2). Due to the smaller grain sizes in these films, the scattering rates and therefore the optical losses are expected to be larger than their SC counterparts. This is manifested in a larger imaginary part compared with singlecrystalline films. Figure 2a,b shows the extracted temperaturedependent dielectric functions of 40 nm thick PC films. Initially, when the temperature is increased to 100 °C, the imaginary part reduces a bit (inset of Figure 2b). This is known to be due to grain merging in polycrystalline films at slightly elevated temperatures, similar to what has been observed in Au.26 However, raising the temperature over 100 °C increases the imaginary part. Eventually, at 300 °C it increases by nearly four times compared to that for the room temperature. Significant deviations in ε1 were also observed in these films (Figure 2a). The observed temperature dependencies in ε1 are primarily due to the deviations in Drude broadening ΓD but not because of plasma frequency. This will be discussed later in the theory section. At even higher temperatures, we could no longer acquire reliable fits to our VASE measurements with a simple Drude-Lorentz oscillator model. Furthermore, optical images on the heat-treated film showed features with strong contrast (Figure S5). The irregularities in the fits and the contrast in the optical images led us to suspect that the film

may not be continuous upon heat treatment. This was subsequently verified from AFM topographs and depth profile analysis, which are shown in Figure S9. Several cracks could be seen in the film, revealing the structural degradation due to the heat treatment. In contrast to SC films, these cracks were completely irregular (see Supporting Information for more details). Analytical models incorporating the changes in the surface morphology would be required to retrieve the optical responses beyond the breakdown temperatures.27,28 These studies are beyond the scope of the current work. It should be noted that the temperature induced relative changes in the dielectric function are larger in PC films compared to their SC counterparts (Figure S10). We would like to reiterate that the breakdown temperature of thin polycrystalline films (around 300 °C) is significantly smaller than the breakdown temperature of thin single crystalline films of nearly the same thickness (600 °C). The temperature-induced deviations in a 100 nm thick PC film are shown in Figure 2c,d. The real part only shows marginal changes with temperature, similar to thick SC films. This is because of small deviations in the plasma frequency and the fact that the Drude broadening is significantly smaller than the photon energies considered in this study (Table S5). The imaginary part, on the other hand, increases by about two times upon heating the sample to 600 °C. At 700 °C, we could not get good fits, indicating that the film was no longer continuous. This was later confirmed from AFM topographs and optical images. The significant increase in the imaginary part at elevated temperatures greatly reduces the performance of plasmonic elements. We estimated the propagation lengths of surface 1086

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Figure 3. Computed SPP propagation lengths at Ag−air interface (a) and quality factor of localized surface plasmon resonance of a spherical particle (b) with optical constants taken from 34 nm thick sputtered SC film at various temperatures (shown in legend). Both the propagation lengths and QLSPR reduce with increasing temperature.

plasmon polaritons (SPPs) at the Ag-air interface and the quality factor of localized surface plasmon resonances (QLSPR) for a spherical particle using the extracted elevated temperature dielectric functions. The propagation length of SPPs29−31 is defined as the length over which the electric field drops down ω ε by a factor of e(LSPP = 1/(2Im( c ( 1 + ε )0.5 ))), while the quality factor of localized surface plasmon resonance is given by the ratio of enhanced local field to the incident field, and gives a measure of the strength of the enhanced local field.31 In the case of a spherical nanoparticle, under the quasi-static −ε assumption, it can be shown that Q LSPR = ε 1 (ref 31). The

Table 2. Computed Quality Factor of Localized Surface Plasmon Resonance of a Spherical Particle (QLSPR) at Different Temperatures and 820 nm Wavelength

2

computed SPP propagation lengths and QLSPR using the temperature-dependent dielectric functions of 34 nm thick SC films are shown in Figure 3a and b, respectively. Both the propagation lengths and QLSPR are reduced by nearly a factor of 3 to 4 at the maximum temperature. In Tables 1 and 2, we compare the SPP propagation lengths and QLSPR of all different films at various selected temperatures and a wavelength of 820 nm.

34 nm SC 200 nm SC 450 nm SC 40 nm PC 100 nm PC a

LSSP (μM) at 23 °C

LSSP (μM) at 400 °C

LSSP (μM) at 600 °C

LSSP (μM) at 700 °C

193

78

65

146

105

97

84

42.5

124

90

86

73

41.1

89

16a

85

53

15 21

18.5

70.6 45.6

21

19.6

16.5

45

4a 13

10

QLSPR at 400 °C

34 nm SC 200 nm SC 450 nm SC 40 nm PC 100 nm PC

51 34

18 23

30 22 21

QLSPR at 600 °C

81.8 52.4

Computed values at 300 °C.

the coercivity of the magnetic medium locally when the NFTs are brought into close proximity to the recording magnetic medium.10−12 The reduced coercivity then allows for writing/ recording information locally onto the magnetic medium using a nearby magnetic head. Although this is a promising approach, it suffers from a serious drawback. The strong absorption of laser light leads to significant heating of plasmonic nanoantennas. It has been estimated that the internal heating would lead to temperatures as high as 400 °C.12 However, nanostructured plasmonic elements made of conventional metals (Au and Ag) are unstable at such high temperatures. This has led the community to explore other plasmonic materials that are stable at such high temperatures.32 Here, we show that the temperature-induced changes to the dielectric function lead to significant modifications in the field profile, which should be accounted for an accurate estimation of the field enhancement. Furthermore, the larger breakdown temperatures of the single crystalline silver films make them better candidates for HAMR applications. We performed field profile simulations in the gap between the bowtie antennas on MgO (refractive index n = 1.73), a near-field transducer design, using a commercial FEM solver (COMSOL Multiphysics, see Supporting Information for more details). The extracted optical properties from the temperature dependent VASE measurements were used as the material properties for our simulations. Figure 4 shows the computed average electric field in a small volume (64 nm3) in the gap between the two nanoantennae at different temperatures computed using the optical properties of thinner SC (Figure 4a) and PC (Figure 4b) films. Insets show the incident field

% change at largest temperature 66.3

82 43

% change at largest temperature

QLSPR at 23 °C

a

Table 1. Computed SPP Propagation Lengths at Different Temperatures and 820 nm Wavelength sample type

QLSPR at 700 °C

sample type

49.4

Computed values at 300 °C.

We demonstrate the practical importance of our findings for an important high-temperature application by simulating the field profiles of an Ag nanoantenna that could be used as a heatassisted magnetic recording (HAMR) near-field transducer (NFT). In HAMR the nanoanatennas are excited with laser beams, close to their resonant frequency, resulting in intense localized electromagnetic fields, hotspots, in the near field of the plasmonic nanoantennas. It was proposed and later experimentally demonstrated that these hotspots could reduce 1087

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Figure 4. Computed field enhancement in the gap between the bow-tie nanoantennas at room temperature and elevated temperatures using the temperature-dependent dielectric function of 34 nm SC (a) and 40 nm PC (b) Ag films. Incident field polarization was along the bow-tie axis, as shown in the insets. Field enhancement reduces at elevated temperatures with larger relative changes observed in PC films.

Figure 5. Temperature-dependent plasma frequency (a) and Drude broadening (b) of 34 nm thick single crystalline Ag film. The plasma frequency increases with increasing temperature and saturates at temperatures over 400 °C. The Drude broadening also increases with increasing temperature. The model fit in (b) is based on eq 7.

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THEORY In this section, we discuss theoretically expected temperature dependencies in the plasma frequency and Drude damping and compare them with our experimental findings. Similar discussion along these lines can be found in an earlier paper.17 For the sake of completeness, we reiterate some of them here. At longer wavelengths, where the interband transitions are insignificant, the dielectric response can be described with just the Drude term in eq 1. The temperature dependencies of the two terms in the Drude model, plasma frequency, and Drude broadening are primarily due to (1) the reduced free electron density resulting from the volume thermal expansion, (2) the reduced electron effective mass, and (3) the increased electron−phonon interaction at elevated temperatures. The square of plasma frequency ωp is given by

polarization and the generated hot spots. It should be noted that the field enhancement is larger in the SC case due to lower losses. At 300 °C, the maximum field enhancement in SC and PC cases reduces by nearly 17% and 50%, respectively. These findings illustrate several key points: (1) Incorporating the temperature-dependent optical properties in modeling hightemperature applications leads to significantly different results versus those obtained with the room temperature data; (2) In the case of NFTs, as the field enhancement reduces at elevated temperatures, one would have to increase laser pumping to even higher intensities to achieve the same enhancement which would increase the temperature even further; (3) The relative changes in electric field enhancement in the SC case are smaller compared to the PC case. As a result, together with their larger thermal stability, SC films can play a major role in applications requiring moderate elevation of temperatures such as spasers, biosensors, along with NFTs. We would like to point out that several high temperature applications operate at atmospheric pressure. However, silver suffers from various contamination issues when exposed to ambient pressure, which would degrade its optical properties. To circumvent this problem, protective capping layers such as, SiN and HfO, that are chemically inert and at the same time sustaining high temperatures, can be used to prevent surface contamination. The temperature-dependent properties of such protective layers will be considered in a separate study.

ωp2 =

Ne 2 m*ε0

(2)

where N is the free electron carrier density and m* is the electron effective mass. Assuming no additional carriers are being generated at elevated temperatures, the temperature dependence of N can be described using the volume thermal expansion coefficient γ as N= 1088

N0 1 + γ(T − T0)

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1 1 ⎡2 T⎤ = ⎢ + ⎥ τeϕ τ0 ⎣ 5 θ⎦

where N0 is the carrier density at temperature T0. From eq 3 it is clear that with increasing temperature N reduces, thereby decreasing the plasma frequency. However, in metals, the electron effective mass m* has been reported to decrease with temperature33 which, in turn, increases ωp. The interplay between these two counteracting mechanisms determines the overall temperature dependence of ωp. The temperaturedependent ωp in thin (34 nm thick) SC film is plotted in Figure 5a. With increasing temperature, we observe an increase in the plasma frequency. Based on these results, we conclude that the decrease in the electron effective mass is the dominant mechanism. A similar increase in ωp was reported previously.24 We would like to emphasize that the conclusion on the decrease in m* was purely based on the fits from the experimental temperature dependent ellipsometer measurements. It is not possible to conclude which of the parameters, m*, N, and ΓD, has a dominant influence from just the temperature evolution of the real part. In order to systematically understand the role of each of the parameters we fitted the ellipsometer data with a Drude-Lorentz model. From the fits the temperature evolution of the two Drude terms ωp and ΓD were obtained. These temperature dependencies provided insights into which of the three parameters (m*, N, and ΓD) have a dominant influence on the evolution of the real part. The increase in ΓD and a decrease in N lead to a decrease in the magnitude of the real part. However, experimentally it was observed that the real part increases in magnitude, making the films more metallic. This can only be explained by assuming that the decrease in the effective mass is the dominant mechanism in the evolution of the real part. On the other hand, the Drude broadening ΓD is related to the scattering rates of free electrons 1 as

The electron−electron scattering, on the other hand, depends on both temperature and frequency, but the temperature-dependent term, (KBT)2, is significantly smaller than the frequency-dependent term,

ℏ τ

(4)

where ℏ is the reduced Planck constant. The dominant contributions to 1 come from electron−phonon ( 1 ) and τ

τeϕ

electron−electron ( 1 ) scatterings. The temperature and τee

frequency dependencies of these two terms are given by34−36 ⎛ T ⎞5 1 1 ⎡2 = ⎢ + 4⎜ ⎟ ⎝θ⎠ τeϕ τ0 ⎣ 5

∫0

θ/T

⎤ z4 dz ⎥ z e −1 ⎦

⎛ ℏω ⎞2 ⎤ 1 1 3 ⎛ 1 ⎞⎡ ⎟ ⎥ = π ΓΔ⎜ ⎟⎢(KBT )2 + ⎜ ⎝ 2π ⎠ ⎦ τee 12 ⎝ ℏE F ⎠⎣

2

( ℏ2ωπ ) ,

for all the

temperature and frequency ranges considered in this study. We note that the model with a frequency-independent Drude broadening and, hence, a frequency-independent electron− electron scattering is commonly used. Therefore, we ignore both the frequency and temperature dependencies in our model and treat the electron−electron scattering as a constant. In Figure 5b, we plot the temperature-dependent Drude damping extracted from VASE fits for thin SC film. The orange curve is the fit obtained using a linear temperature dependence as predicted by eq 7. Good agreement was found between theory and experiment. We observe similar deviations in ωp and ΓD in thick PC and SC films. The plasma frequency increases marginally with temperature, while the Drude broadening increases significantly (see temperature-dependent Drude-Lorentz models in Tables S1−S5). These temperature dependencies can be attributed to the reducing electron effective mass and the increasing electron−phonon interactions discussed above. We would like to point out that the underlying mechanisms leading to larger temperature induced deviations in ε2 in thin SC film are not entirely clear. Insights into the possible physical processes leading to these deviations can be gained from the simplified expression of electron−phonon coupling shown in eq 7. The material dependent constant τ0 in eq 7 depends on various band structure parameters. The most plausible explanation for the observed trend in thin SC samples is because of a smaller τ0, which in turn leads to larger deviations in ε2. Furthermore, the observed larger increase in the Drude damping in the evaporated films is also possibly because of a smaller τ0. However, quantitative estimates of deviations in τ0 would require detailed calculations of various material parameters, such as the Fermi-surface distortion parameter and the number of zone boundaries which intersect the Fermi surface (see ref 35 for a more detailed discussion on parameters that influence τ0). Such a study is beyond the scope of the current work. The evolution of the real part of the dielectric function in thin PC film (Figure 2a) can be understood from the temperature-dependent Drude-Lorentz oscillator models (shown in Table S4 in Supporting Information). Note that the plasma frequency varies only marginally with temperature (