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Anomalous optical conductivity in disordered condensed matter
Journal of Non-Crystalline Solids 358 (2012) 2373–2376
Contents lists available at SciVerse ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Anomalous optical conductivity in disordered condensed matter K. Shimakawa a, b,⁎, T. Itoh a, H. Naito c, S.O. Kasap d a
Center of Innovative Photovoltaic Systems, Gifu University, Japan Nagoya Industrial Science Research Institute, Nagoya, Japan c Department of Physical Electronics, Osaka Prefecture University, Sakai, Japan d Department of Electrical Engineering, Saskatchewan University, Saskatoon, Canada b
a r t i c l e
i n f o
Article history: Received 2 August 2011 Received in revised form 20 September 2011 Available online 27 October 2011 Keywords: Free carrier absorption; Optical conductivity; GST chalcogenides; Nano-crystalline silicon; Metallic oxides
a b s t r a c t The real and imaginary optical conductivities in disordered semiconductors (and metals) do not follow classical Drude's law in the near infrared region. We discuss an origin of the anomalous free carrier absorption in metallic (and near metallic) states of chalcogenide materials (phase-change GST), dye-sensitized nanocrystalline TiO2 and nanocrystalline silicon films. A series sequence of “localized” and “extended” state type behavior of carriers produces a Lorentz-type resonance which is caused by “bound” (localized) electrons. The analytical expressions were fit to the experimental data and produced various important physical parameters such as the numbers of free and localized carriers, the scattering time of free carriers, tunneling time of localized carriers, and the Drude dc conductivity in the system. We also discuss whether the present model has generated reasonable physical parameters. © 2011 Elsevier B.V. All rights reserved.
1. Introduction It is well known that deviations from the Drude behavior of free carriers, near THz to infrared (or visible energy) range, occur in disordered materials such as metallic polymers [1,2] and nanomaterials [3–7]. Disorder-induced carrier localization [1] or backscattering of free carriers [4,8] in homogeneous media may induce such anomalies in optical conductivity. In nanomaterials, however, due to its random structural nature, percolative transport with random networks of connecting pathways are predicted [2] and hence the dynamics of free carrier transport in nanomaterials cannot be simple. Understanding the nature of electronic transport in such media is not only of fundamental scientific interest but also of practical importance in many optoelectronic applications. The treatment of a material as homogeneous or inhomogeneous strongly depends on the frequency ω of external excitation. At higher frequencies, due to the short duration of acceleration time of free carriers, the free carriers reside inside the grains and hence free carriers see homogeneous media. At lower frequencies, carriers should pass through grain boundaries, and hence carriers see an inhomogeneous medium. Study of optical conductivity in the THz energy range may help to understand the free carrier dynamics of nanomaterials. In the present study, we propose a new model (formulation) for the complex optical conductivities in nanometer-scale inhomogeneous media and then discuss the experimental data in a phase-
⁎ Corresponding author. Tel./fax + 81 58 241 2185. E-mail address: [email protected] (K. Shimakawa). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.09.042
change material of chalcogenide Ge2Sb2Te5 (GST), dye-sensitized nanocrystalline TiO2, and nanocrystalline-Si as case examples.
2. Optical conductivity in inhomogeneous media In metal-like nanomaterials, transport channel can be 3D-like transport through quantum tunneling among metallic grains, i.e. a series sequence of electronic transport inside metal-like domains and quantum tunneling through grain boundaries. The basic idea for this has been applied to explain the optical conductivity in metallic polymers [2], and we briefly introduce the basic formulation. The frequency dependence of the complex conductivity σopt(ω) (optical conductivity at an angular frequency ω) for free carriers follows the Drude free-electron model [9]:
σoptðfcÞ ðωÞ ¼
σ ð0Þ ≡ σf R ðωÞ − iσfI ðωÞ; 1 þ iωτ
ð1Þ
where σ(0) is Drude's dc conductivity (or the Boltzmann conductivity) given by
2
σ ð0Þ ¼
ne τ ; m
ð2Þ
in which n is the concentration of free electrons, e the electronic charge, m* the effective mass, and τ the relaxation time.
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K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376
For the tunneling conductivity σopt(t)(ω), we use the so-called Dyre expression in random media [10]: σoptðtÞ ðωÞ ¼ σt ð0Þ
iωτt ¼ σtR ðωÞ þ iσtI ðωÞ; lnð1 þ iωτt Þ
ð3Þ
where σt(0) is the dc hopping conductivity given by σt ð0Þ ¼
nt ðert Þ2 : 6kTτt
ð4Þ
τt here is the tunneling time, rt the tunneling distance (distance between grains), and nt the number of tunneling carriers. Complex conductivity due to a series sequence of free and hopping carries is given as 1 1 1 ¼ þ : σopt ðωÞ σoptðfcÞ ðωÞ σoptðtÞ ðωÞ
ð5Þ
To calculate the overall imaginary part of the conductivity Im σopt(ω), the term, ωε0ε∞, should be added into Eq. (5), where ε∞ is the background dielectric constant (ε∞ = 1 for conventional metals)
Fig. 1. A model calculation for σ(0) = 84 S cm− 1 (and n = nt): (a) Real part of σ(ω) as a function of tunneling time τt = 1 × 10− 11 (A), 1 × 10− 12 (B), 1 × 10− 13 s (C), and a Lorentz resonance curve with proper parameters (D), (b) Imaginary part of σ(ω) as a function of tunneling time τt = 1 × 10− 11 (A), 1 × 10− 12 (B), 1 × 10− 13 s (C), and a Lorentz resonance curve with proper parameters (D).
[2]. σR(0), the real part of σopt (0), deduced from Eq. (5), should be close to the measured dc conductivity σdc. Before fitting Eq. (5) to the experimental data, we first predict the gross features of the present model: Figs. 1 and 2 show the predicted real and imaginary parts of optical conductivities σopt (ω) in THz range, respectively. Here we set σ(0) = 85 S cm− 1 (τ = 6.0 × 10− 15 s), and the hopping time τt = 1.0× 10− 11 s (line A), 1.0 × 10− 12 s (line B), 1.0 × 10− 13 s (line C). Interestingly, the real part of the conductivity has a peak, which is completely different from the Drude behavior. When the tunneling time through grain boundaries increases (due to charge accumulation near grain boundaries), overall conductivity decreases but exhibits a more pronounced peak. Usually “a peaked nature” in optical conductivity is attributed to a Lorentz-type resonance [9] and hence the present result is both interesting and potentially useful: A series sequence of the Drude and the Debye-like mechanisms produces a Lorentz-type resonance behavior. It is of interest to compare the present peaked nature with that from the Lorentz resonance. The solid lines (D) in Figs. 1 and 2 show the optical conductivity arising from the Lorentz resonance. Recall that electrons are completely localized at atomic sites in the Lorentz model. Here we set, for example, the resonance frequency ω0 = 7.0 × 10 13 Hz, and the damping time τL = 5 × 10 − 15 s [9]. When we only consider the real part of conductivity, it is not easy to distinguish the dominating mechanism. As shown in Fig. 2, the behavior of the imaginary part of conductivity however is distinctly different for the present model and the Lorentz resonance. Thus, we need to analyze both the real and imaginary parts of conductivity to obtain a proper understanding of the dominating mechanism in the optical conductivity behavior. T = 300 K was set in the present calculations.
Fig. 2. (a) Real part of σopt(ω) in GST, (b) Imaginary part of σopt(ω) in GST. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376 Table 1 Physical parameters used for the model calculations. m* is taken from literature (not a deduced parameter).
n (cm− 3) τ (s) m* ntrt2(cm− 1) τt (s) ε∞ σR(0) (S cm− 1)
GST
TiO2
Si
3.0 × 1020 1.5 × 10− 15 0.3 4.5 × 107 3.0 × 10− 14 50 330
6.0 × 1019 2.5 × 10− 15 10 2.5 × 106 8.0 × 10− 11 2 0.03
1.5 × 1019 3.0 × 10− 14 0.3 7.0 × 106 4.0 × 10− 13 15 17
3. Case examples We analyze representative experimental data taken at room temperature (THz- or infrared-spectroscopy) on a selection of “disordered” crystalline materials selected from three major classes of materials, i.e. chalcogenides [11], oxides [4,12], and silicon [5], as case examples. First we consider Ge2Sb2Te5 (GST)-chalcogenides, which is known as “phase-change materials”. Open circles in Fig. 2(a) and (b) show the real and imaginary parts of infrared conductivity, σopt, in hexagonal GST (crystalline phase), respectively. Note that the original experimental data (open circles), with no information of the crystalline grain size, have been obtained from the refractive index n and extinction coefficient k in the literature [11]. The fit of Eq. (5) (solid line) to the experimental data is good, although the experimental data are presented in a limited frequency range. Further, the latter fit has produced the important physical parameters of the material system, which are tabulated in Table 1. It is important to examine both the real and imaginary part of the conductivity, simultaneously. Separate considerations of the real and
Fig.3. (a) Real part of σopt(ω) in mesoporous TiO2, (b) Imaginary part of σopt(ω) in mesoporous TiO2. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
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imaginary parts of the conductivity may lead to serious errors, since there are cases in which certain physical parameter may dominate either the real or imaginary part. Open circles in Fig. 3(a) and (b) show the real and imaginary parts of THz conductivity, σopt, in dye-sensitized nanocrystalline TiO2 films, respectively. The model calculations given by solid lines fit well to the experimental results. The imaginary part of conductivity takes a negative value, which is different from that of the GST. This can be due to the fact that the real part of conductivity has been measured a lower frequency from the peak (Re σopt in Fig. 3(a)). Finally we consider a case example of nanocrystalline Si (nc-Si). Open circles in Fig. 4(a) and (b) show the real and imaginary parts of THz conductivity, σopt, in nc-Si films (annealing of SiO films) with grain size of 15–25 nm, respectively [5]. The model calculations given by solid lines fit well with the experimentally obtained imaginary part of conductivity, while the fit is not as good for the real part of conductivity. The imaginary part of conductivity takes a negative value, similar to that of TiO2. Details of the physical parameters deduced for the present selection of materials will be discussed in the following section. 4. Discussion First, we discuss the physical parameters in the GST. m* = 0.3 is taken to be the same as in the literature [11]. The number of free carrier n is very close to the reported values (5 × 10 20 cm − 3 [13,14] and 2.7 × 10 20 cm − 3 [15]) in hexagonal crystalline phase of GST (degenerate p-type semiconductor). The scattering time τ is in the range of 10 − 15 s [11], which produces a free hole mobility
Fig. 4. (a) Real part of σopt(ω) in nanocrystalline Si, (b) Imaginary part of σopt(ω) in nanocrystalline Si. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
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K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376
μ (= eτ/m*) = 9 cm 2V − 1 s − 1. This is not far different than those reported values (30 cm 2V − 1 s − 1 [13], 20 cm 2V − 1 s − 1 [14]) for hexagonal GST. In the tunneling term (Eq. (4)), as it is not easy to deduce nt and rt separately, we take ntrt2 as a parameter. If we assume n = nt, then rt is estimated to be 3.9 nm which may be a reasonable value for a path length through grain boundaries. Relatively large value of ε∞ here is also close to 60, which has been reported for the GST [11]. The deduced σR(0) = 330 S cm − 1 is smaller than the measured dc conductivity (~ 1000 S cm − 1) [13–15], though the difference is not sufficiently large to cause concern. As already shown in Fig. 1, the tunneling time τt is an important factor for the overall features of optical conductivity and is larger than the scattering time τ. This feature was found in TiO2 and nc-Si (see Table 1). Such a larger tunneling time is expected in a resonant tunneling mechanism, i.e. τt ~ τ exp(rt/ξ), where ξ is the tunneling factor [16], although detailed tunneling mechanisms is not clear. The present TiO2 is a mesoporous nanocrystalline film and is an n-type semiconductor as well as the present nc-Si. The effective mass m* for TiO2 was taken to be 10 [4] and 0.3 for nanocrystalline Si [5]. The number of free carriers in TiO2 and nanocrystalline Si, respectively, is 5–10 times larger than that of published reports [4,5], while τ is comparable to the reports of ~ 10 − 14 s [4,5]. ε∞ = 2 seems to be small for TiO2, which may attributed to the porous nature of the present TiO2 films. We note that negative values of the imaginary part of conductivity (Im σopt in Figs. 3(b) and 4(b)) in TiO2 and Si, while it takes positive values for the GST (see Fig. 2(b)). The imaginary part of conductivity takes negative values when the real part of conductivity appears at the lower frequency side of the peak (see Figs. 3(a) and 4(a)). Finally, it should be mentioned that the optical conductivities in the present TiO2 and nc-Si have been interpreted well by the modified Drude model (the Drude-Smith model [8,17]) [4,5]. It consists of the basic Drude model, plus additional terms that account backscattering. The Drude–Smith model has been applied to the optical conductivity of liquid Hg and Te, and quasicrystals [17], which basically requires homogeneous media. When the mean-free-path within a periodic time in THZ frequency exceeds the size of grains, the Drude or Drude–Smith model cannot be applied. Our model
is very simple and can be applied to macroscopic inhomogeneous systems. 5. Conclusions Anomalous optical conductivity in near infrared or THz frequencies observed in nanomaterials was interpreted by a series sequence of the Drude and Debye-like mechanisms. A Lorentz-type resonance peak in the optical conductivity can be reproduced well by the present model, which can be applied to macroscopically inhomogeneous media. The present model was applied to three case studies comprising of Ge2Sb2Te5 (GST), TiO2, and nanocrystalline-Si films. For all three materials, the model produced reasonable physical parameters. Acknowledgement T. I. and K. S. would like to thank Prof. S. Nonomura for discussion. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
A.J. Heeger, Curr. Appl. Phys. 1 (2001) 35201. K. Shimakawa, H. Naito, S.O. Kasap, Philos. Mag. Lett. 89 (2009) 673. A. Ugawa, A.G. Rinzler, D.B. Tanner, Phys. Rev. B 60 (1999) R11305. G.M. Turner, M.C. Beard, C.A. Schmuttenmaer, J. Phys. Chem. B 106 (2002) 11716. D.G. Cooke, A.N. Mac Donald, A. Hryciw, Meldrum Al, J. Wang, Q. Li, F.A. Hegmann, J. Mater. Sci. - Mater. Electron. 18 (2007) S447. A. Thoman, A. Kern, H. Helm, M. Walther, Phys. Rev. B 77 (2008) 195405. T.L. Coker, L.V. Titova, S. Fourmaux, H.-C. Bandulet, D. Brassard, J.-C. Kieffer, M.A. Al Khakani, F.A. Hegmann, Appl. Phys. Lett. 97 (2010) 221905. N.V. Smith, Phys. Rev. B 64 (2001) 155106. F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972. J.C. Dyre, T.B. Schroder, Rev. Mod. Phys. 72 (2000) 873. A. Mendoza-Galvan, J. Gonzalez-Hernandez, J. Appl. Phys. 87 (2000) 760. T.L. Cocker, L.V. Titova, S. Fourmaux, H.-C. Bandulet, D. Brassard, J.-C. Kieffer, M.A. El Khakani, F.A. Hegmann, Appl. Phys. Lett. 97 (2010) 221905. J.M. Yanez-Limon, J. Gonzalez-Hernandez, J.J. Alvarado-Gil, I. Delgadillo, H. Vargas, Phys. Rev. B 52 (1995) 16321. T. Kato, Ke. Tanaka, Jpn. J. Appl. Phys. 44 (2005) 7340. B. Lee, J.R. Abelson, S.G. Bishop, D. Kang, B. Cheong, K. Kim, J. Appl. Phys. 97 (2005) 093509. V.N. Prigodin, A.J. Epstein, Physica B 338 (2003) 310. S. Kasap, H. Ruda, Y. Boucher, Cambridge Illustrated Handbook of Optoelectronics and Photonics, Cambridge University Press, Cambridge, 2009, p. 125.
Contents lists available at SciVerse ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Anomalous optical conductivity in disordered condensed matter K. Shimakawa a, b,⁎, T. Itoh a, H. Naito c, S.O. Kasap d a
Center of Innovative Photovoltaic Systems, Gifu University, Japan Nagoya Industrial Science Research Institute, Nagoya, Japan c Department of Physical Electronics, Osaka Prefecture University, Sakai, Japan d Department of Electrical Engineering, Saskatchewan University, Saskatoon, Canada b
a r t i c l e
i n f o
Article history: Received 2 August 2011 Received in revised form 20 September 2011 Available online 27 October 2011 Keywords: Free carrier absorption; Optical conductivity; GST chalcogenides; Nano-crystalline silicon; Metallic oxides
a b s t r a c t The real and imaginary optical conductivities in disordered semiconductors (and metals) do not follow classical Drude's law in the near infrared region. We discuss an origin of the anomalous free carrier absorption in metallic (and near metallic) states of chalcogenide materials (phase-change GST), dye-sensitized nanocrystalline TiO2 and nanocrystalline silicon films. A series sequence of “localized” and “extended” state type behavior of carriers produces a Lorentz-type resonance which is caused by “bound” (localized) electrons. The analytical expressions were fit to the experimental data and produced various important physical parameters such as the numbers of free and localized carriers, the scattering time of free carriers, tunneling time of localized carriers, and the Drude dc conductivity in the system. We also discuss whether the present model has generated reasonable physical parameters. © 2011 Elsevier B.V. All rights reserved.
1. Introduction It is well known that deviations from the Drude behavior of free carriers, near THz to infrared (or visible energy) range, occur in disordered materials such as metallic polymers [1,2] and nanomaterials [3–7]. Disorder-induced carrier localization [1] or backscattering of free carriers [4,8] in homogeneous media may induce such anomalies in optical conductivity. In nanomaterials, however, due to its random structural nature, percolative transport with random networks of connecting pathways are predicted [2] and hence the dynamics of free carrier transport in nanomaterials cannot be simple. Understanding the nature of electronic transport in such media is not only of fundamental scientific interest but also of practical importance in many optoelectronic applications. The treatment of a material as homogeneous or inhomogeneous strongly depends on the frequency ω of external excitation. At higher frequencies, due to the short duration of acceleration time of free carriers, the free carriers reside inside the grains and hence free carriers see homogeneous media. At lower frequencies, carriers should pass through grain boundaries, and hence carriers see an inhomogeneous medium. Study of optical conductivity in the THz energy range may help to understand the free carrier dynamics of nanomaterials. In the present study, we propose a new model (formulation) for the complex optical conductivities in nanometer-scale inhomogeneous media and then discuss the experimental data in a phase-
⁎ Corresponding author. Tel./fax + 81 58 241 2185. E-mail address: [email protected] (K. Shimakawa). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.09.042
change material of chalcogenide Ge2Sb2Te5 (GST), dye-sensitized nanocrystalline TiO2, and nanocrystalline-Si as case examples.
2. Optical conductivity in inhomogeneous media In metal-like nanomaterials, transport channel can be 3D-like transport through quantum tunneling among metallic grains, i.e. a series sequence of electronic transport inside metal-like domains and quantum tunneling through grain boundaries. The basic idea for this has been applied to explain the optical conductivity in metallic polymers [2], and we briefly introduce the basic formulation. The frequency dependence of the complex conductivity σopt(ω) (optical conductivity at an angular frequency ω) for free carriers follows the Drude free-electron model [9]:
σoptðfcÞ ðωÞ ¼
σ ð0Þ ≡ σf R ðωÞ − iσfI ðωÞ; 1 þ iωτ
ð1Þ
where σ(0) is Drude's dc conductivity (or the Boltzmann conductivity) given by
2
σ ð0Þ ¼
ne τ ; m
ð2Þ
in which n is the concentration of free electrons, e the electronic charge, m* the effective mass, and τ the relaxation time.
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K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376
For the tunneling conductivity σopt(t)(ω), we use the so-called Dyre expression in random media [10]: σoptðtÞ ðωÞ ¼ σt ð0Þ
iωτt ¼ σtR ðωÞ þ iσtI ðωÞ; lnð1 þ iωτt Þ
ð3Þ
where σt(0) is the dc hopping conductivity given by σt ð0Þ ¼
nt ðert Þ2 : 6kTτt
ð4Þ
τt here is the tunneling time, rt the tunneling distance (distance between grains), and nt the number of tunneling carriers. Complex conductivity due to a series sequence of free and hopping carries is given as 1 1 1 ¼ þ : σopt ðωÞ σoptðfcÞ ðωÞ σoptðtÞ ðωÞ
ð5Þ
To calculate the overall imaginary part of the conductivity Im σopt(ω), the term, ωε0ε∞, should be added into Eq. (5), where ε∞ is the background dielectric constant (ε∞ = 1 for conventional metals)
Fig. 1. A model calculation for σ(0) = 84 S cm− 1 (and n = nt): (a) Real part of σ(ω) as a function of tunneling time τt = 1 × 10− 11 (A), 1 × 10− 12 (B), 1 × 10− 13 s (C), and a Lorentz resonance curve with proper parameters (D), (b) Imaginary part of σ(ω) as a function of tunneling time τt = 1 × 10− 11 (A), 1 × 10− 12 (B), 1 × 10− 13 s (C), and a Lorentz resonance curve with proper parameters (D).
[2]. σR(0), the real part of σopt (0), deduced from Eq. (5), should be close to the measured dc conductivity σdc. Before fitting Eq. (5) to the experimental data, we first predict the gross features of the present model: Figs. 1 and 2 show the predicted real and imaginary parts of optical conductivities σopt (ω) in THz range, respectively. Here we set σ(0) = 85 S cm− 1 (τ = 6.0 × 10− 15 s), and the hopping time τt = 1.0× 10− 11 s (line A), 1.0 × 10− 12 s (line B), 1.0 × 10− 13 s (line C). Interestingly, the real part of the conductivity has a peak, which is completely different from the Drude behavior. When the tunneling time through grain boundaries increases (due to charge accumulation near grain boundaries), overall conductivity decreases but exhibits a more pronounced peak. Usually “a peaked nature” in optical conductivity is attributed to a Lorentz-type resonance [9] and hence the present result is both interesting and potentially useful: A series sequence of the Drude and the Debye-like mechanisms produces a Lorentz-type resonance behavior. It is of interest to compare the present peaked nature with that from the Lorentz resonance. The solid lines (D) in Figs. 1 and 2 show the optical conductivity arising from the Lorentz resonance. Recall that electrons are completely localized at atomic sites in the Lorentz model. Here we set, for example, the resonance frequency ω0 = 7.0 × 10 13 Hz, and the damping time τL = 5 × 10 − 15 s [9]. When we only consider the real part of conductivity, it is not easy to distinguish the dominating mechanism. As shown in Fig. 2, the behavior of the imaginary part of conductivity however is distinctly different for the present model and the Lorentz resonance. Thus, we need to analyze both the real and imaginary parts of conductivity to obtain a proper understanding of the dominating mechanism in the optical conductivity behavior. T = 300 K was set in the present calculations.
Fig. 2. (a) Real part of σopt(ω) in GST, (b) Imaginary part of σopt(ω) in GST. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376 Table 1 Physical parameters used for the model calculations. m* is taken from literature (not a deduced parameter).
n (cm− 3) τ (s) m* ntrt2(cm− 1) τt (s) ε∞ σR(0) (S cm− 1)
GST
TiO2
Si
3.0 × 1020 1.5 × 10− 15 0.3 4.5 × 107 3.0 × 10− 14 50 330
6.0 × 1019 2.5 × 10− 15 10 2.5 × 106 8.0 × 10− 11 2 0.03
1.5 × 1019 3.0 × 10− 14 0.3 7.0 × 106 4.0 × 10− 13 15 17
3. Case examples We analyze representative experimental data taken at room temperature (THz- or infrared-spectroscopy) on a selection of “disordered” crystalline materials selected from three major classes of materials, i.e. chalcogenides [11], oxides [4,12], and silicon [5], as case examples. First we consider Ge2Sb2Te5 (GST)-chalcogenides, which is known as “phase-change materials”. Open circles in Fig. 2(a) and (b) show the real and imaginary parts of infrared conductivity, σopt, in hexagonal GST (crystalline phase), respectively. Note that the original experimental data (open circles), with no information of the crystalline grain size, have been obtained from the refractive index n and extinction coefficient k in the literature [11]. The fit of Eq. (5) (solid line) to the experimental data is good, although the experimental data are presented in a limited frequency range. Further, the latter fit has produced the important physical parameters of the material system, which are tabulated in Table 1. It is important to examine both the real and imaginary part of the conductivity, simultaneously. Separate considerations of the real and
Fig.3. (a) Real part of σopt(ω) in mesoporous TiO2, (b) Imaginary part of σopt(ω) in mesoporous TiO2. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
2375
imaginary parts of the conductivity may lead to serious errors, since there are cases in which certain physical parameter may dominate either the real or imaginary part. Open circles in Fig. 3(a) and (b) show the real and imaginary parts of THz conductivity, σopt, in dye-sensitized nanocrystalline TiO2 films, respectively. The model calculations given by solid lines fit well to the experimental results. The imaginary part of conductivity takes a negative value, which is different from that of the GST. This can be due to the fact that the real part of conductivity has been measured a lower frequency from the peak (Re σopt in Fig. 3(a)). Finally we consider a case example of nanocrystalline Si (nc-Si). Open circles in Fig. 4(a) and (b) show the real and imaginary parts of THz conductivity, σopt, in nc-Si films (annealing of SiO films) with grain size of 15–25 nm, respectively [5]. The model calculations given by solid lines fit well with the experimentally obtained imaginary part of conductivity, while the fit is not as good for the real part of conductivity. The imaginary part of conductivity takes a negative value, similar to that of TiO2. Details of the physical parameters deduced for the present selection of materials will be discussed in the following section. 4. Discussion First, we discuss the physical parameters in the GST. m* = 0.3 is taken to be the same as in the literature [11]. The number of free carrier n is very close to the reported values (5 × 10 20 cm − 3 [13,14] and 2.7 × 10 20 cm − 3 [15]) in hexagonal crystalline phase of GST (degenerate p-type semiconductor). The scattering time τ is in the range of 10 − 15 s [11], which produces a free hole mobility
Fig. 4. (a) Real part of σopt(ω) in nanocrystalline Si, (b) Imaginary part of σopt(ω) in nanocrystalline Si. Solid line is model calculation with proper physical parameters tabulated in Table 1 and open circles are experimental data.
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K. Shimakawa et al. / Journal of Non-Crystalline Solids 358 (2012) 2373–2376
μ (= eτ/m*) = 9 cm 2V − 1 s − 1. This is not far different than those reported values (30 cm 2V − 1 s − 1 [13], 20 cm 2V − 1 s − 1 [14]) for hexagonal GST. In the tunneling term (Eq. (4)), as it is not easy to deduce nt and rt separately, we take ntrt2 as a parameter. If we assume n = nt, then rt is estimated to be 3.9 nm which may be a reasonable value for a path length through grain boundaries. Relatively large value of ε∞ here is also close to 60, which has been reported for the GST [11]. The deduced σR(0) = 330 S cm − 1 is smaller than the measured dc conductivity (~ 1000 S cm − 1) [13–15], though the difference is not sufficiently large to cause concern. As already shown in Fig. 1, the tunneling time τt is an important factor for the overall features of optical conductivity and is larger than the scattering time τ. This feature was found in TiO2 and nc-Si (see Table 1). Such a larger tunneling time is expected in a resonant tunneling mechanism, i.e. τt ~ τ exp(rt/ξ), where ξ is the tunneling factor [16], although detailed tunneling mechanisms is not clear. The present TiO2 is a mesoporous nanocrystalline film and is an n-type semiconductor as well as the present nc-Si. The effective mass m* for TiO2 was taken to be 10 [4] and 0.3 for nanocrystalline Si [5]. The number of free carriers in TiO2 and nanocrystalline Si, respectively, is 5–10 times larger than that of published reports [4,5], while τ is comparable to the reports of ~ 10 − 14 s [4,5]. ε∞ = 2 seems to be small for TiO2, which may attributed to the porous nature of the present TiO2 films. We note that negative values of the imaginary part of conductivity (Im σopt in Figs. 3(b) and 4(b)) in TiO2 and Si, while it takes positive values for the GST (see Fig. 2(b)). The imaginary part of conductivity takes negative values when the real part of conductivity appears at the lower frequency side of the peak (see Figs. 3(a) and 4(a)). Finally, it should be mentioned that the optical conductivities in the present TiO2 and nc-Si have been interpreted well by the modified Drude model (the Drude-Smith model [8,17]) [4,5]. It consists of the basic Drude model, plus additional terms that account backscattering. The Drude–Smith model has been applied to the optical conductivity of liquid Hg and Te, and quasicrystals [17], which basically requires homogeneous media. When the mean-free-path within a periodic time in THZ frequency exceeds the size of grains, the Drude or Drude–Smith model cannot be applied. Our model
is very simple and can be applied to macroscopic inhomogeneous systems. 5. Conclusions Anomalous optical conductivity in near infrared or THz frequencies observed in nanomaterials was interpreted by a series sequence of the Drude and Debye-like mechanisms. A Lorentz-type resonance peak in the optical conductivity can be reproduced well by the present model, which can be applied to macroscopically inhomogeneous media. The present model was applied to three case studies comprising of Ge2Sb2Te5 (GST), TiO2, and nanocrystalline-Si films. For all three materials, the model produced reasonable physical parameters. Acknowledgement T. I. and K. S. would like to thank Prof. S. Nonomura for discussion. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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