applied sciences Review
Dynamics of Carrier Transport in Nanoscale Materials: Origin of Non-Drude Behavior in the Terahertz Frequency Range Koichi Shimakawa 1, *,† and Safa Kasap 2,† 1 2
* †
Joint Laboratory of Solid State Chemistry, University of Pardubice, Pardubice 530 02, Czech Republic Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada;
[email protected] Correspondence:
[email protected]; Tel./Fax: +81-58-241-2185 The authors contributed equally to this work.
Academic Editor: Takayoshi Kobayashi Received: 18 November 2015; Accepted: 27 January 2016; Published: 14 February 2016
Abstract: It is known that deviation from the Drude law for free carriers is dramatic in most electronically conductive nanomaterials. We review recent studies of the conductivity of nanoscale materials at terahertz (THz) frequencies. We suggest that among a variety of theoretical formalisms, a model of series sequence of transport involving grains and grain boundaries provides a reasonable explanation of Lorentz-type resonance (non-Drude behavior) in nanomaterials. Of particular interest is why do free carriers exhibit a Lorentz-type resonance. Keywords: nanomaterial; THz spectroscopy; non-Drude transport
1. Introduction The interaction of THz radiation (0.1–10 THz; 0.4–40 MeV) with charge carriers provides important information on carrier transport in a wide range of materials, when the charge carrier scattering time lies around 1014 –1013 s [1,2]. In this frequency range, the most prominent change in the frequency-dependent complex conductivity is expected to occur. Free carriers follow the Drude law if the medium is homogeneous [3]. It is of interest to know what happen in inhomogeneous media. It is known that a deviation from Drude behavior is observed in most electronically conductive nanostructured materials, such as metals [4–6], semiconductors [7–16], and oxides [17–19]. The non-Drude behavior in nanomaterials is a kind of Lorentz resonance. Why the dynamics of free carriers is dominated by resonance but not relaxation-type? In the present review, we will answer this question and how to model the dynamics of free carriers in inhomogeneous nanostructured materials. 2. Dynamics of Free Carriers in Nanomaterials Before proceeding with discussion, we will show a difference between the Drude relaxation and the Lorentz resonance in terms of the complex (optical) conductivity, σ*(ω) = σR (ω) + iσI (ω) [3]. Note that the local electric field is taken to vary in time as exp(iωt). Solid and dotted lines, (a) and (b), show the real and imaginary parts of conductivity, σR (ω) and σI (ω), respectively, for the Drude relaxation, and lines (c) and (d) are for the Lorentz resonance. A frequency-independent real part of conductivity in curve (a) gives the dc conductivity. We hence expect that the low-frequency real part of conductivity (energy loss) corresponds to the dc conductivity (dc loss), when the transport is dominated by the Drude law. As already stated, a Lorentz-type behavior dominates THz conductivity in most of
Appl. Sci. 2016, 6, 50; doi:10.3390/app6020050
www.mdpi.com/journal/applsci
Appl. Sci. 2016, 6, 50
2 of 10
Appl. Sci. 2016, 6, 50
2 of 10
nanomaterials. We, therefore, discuss why do free carriers behave as a Lorentz-like resonance, since We, therefore, discuss why do free carriers behave as a Lorentz-like resonance, since free carriers should free carriers should follow the Drude law in usual understanding. follow the Drude law in usual understanding. A few models have been proposed for the origin of the Lorentz-type resonance behavior [1]. A few models have been proposed for the origin of the Lorentz-type resonance behavior [1]. We know that a quantum mechanical electronic (optical) transition between states is well described by We know that a quantum mechanical electronic (optical) transition between states is well described the Lorentz resonance. In the following, we state briefly the models so far proposed: by the Lorentz resonance. In the following, we state briefly the models so far proposed: (1) When a motion of free carriers involves a restoring force, i.e., the restoring force is added in (1) When a motion of free carriers involves a restoring force, i.e., the restoring force is added in the Drude equation, the equation of motion is described by a damped harmonic oscillator, which the Drude equation, the equation of motion is described by a damped harmonic oscillator, which should should leads to the Lorentz oscillator [3]. A surface depletion or accumulation field can be a leads to the Lorentz oscillator [3]. A surface depletion or accumulation field can be a source of the restoring source of the restoring force. This is called the plasmon model and was applied to semiconductor force. This is called the plasmon model and was applied to semiconductor nanoparticles [1,20]. nanoparticles [1,20]. (2) By assuming only one backscattering of free carriers, complex conductivity shows the (2) By assuming only one backscattering of free carriers, complex conductivity shows the Lorentz-type behavior, which is called the generalized Drude model or the Drude-Smith (DS) model Lorentz-type behavior, which is called the generalized Drude model or the Drude-Smith (DS) [21]. As the DS model has been most widely used to explain the non-Drude behavior [1], we summarize model [21]. As the DS model has been most widely used to explain the non-Drude behavior [1], here the DS approach. A complex conductivity given by Smith [21] is given as: we summarize here the DS approach. A complex conductivity given by Smith [21] is given as:
8 ¸ (0) cn σp0q =q σ pω 1+ 1 (1 −p1) 1 −1 iωτ iωτqn ∗ ()
(1)(1)
n 1
where σ(0) σ(0) is is the the Boltzmann Boltzmanndc dcconductivity conductivitygiven givenby byee2 2nnfτ/m*, τ/m*, nnf isisthe m* the the where the density density of of free free carrier, carrier, m* f f effective mass, mass,ττ the thescattering scatteringtime, time,and andωωthe theangular angularfrequency frequency external excitation, assumed to effective ofof external excitation, assumed to be be exp(−iωt). The coefficient c n represents the fraction of the carrier’s original velocity that is retained exp(iωt). The coefficient cn represents the fraction of the carrier’s original velocity that is retained after the nth scattering. Note that n = 0 produces just the Drude law. When we take only one scattering after the nth scattering. Note that n = 0 produces just the Drude law. When we take only one scattering (n = 1) and −1.0 ≤ c1 < 0 (backscattering), free carrier behavior dramatically changes from the Drude (n = 1) and 1.0 ¤ c1 < 0 (backscattering), free carrier behavior dramatically changes from the Drude relaxation to the Lorentz-type resonance (see Figure 1). A large issue in the DS model is that there is relaxation to the Lorentz-type resonance (see Figure 1). A large issue in the DS model is that there is no no proper physical basis, while good fitting to the experimental results is obtained. Note also that the proper physical basis, while good fitting to the experimental results is obtained. Note also that the DS DS model should basically be applied to homogeneous media. model should basically be applied to homogeneous media. (a) (c)
σ*(ω) (arb. unit)
0.8 0.4 (b)
0.0 -0.4 1012
(d)
1013
1014
1015
ω (rad/s) 1. Optical Lorentz resonance resonance Figure 1. Optical conductivity conductivity in in the the Drude Drude relaxation relaxation ((a) and (b)) and in the Lorentz ((c) and (d)).
(3) When When we we discuss discuss the the electronic electronic transport transport in in inhomogeneous inhomogeneous media, media, such such as as nanoparticles nanoparticles (3) with grain boundaries, charge transfer can occur from one constituent to another. Thus, the role role of of with grain boundaries, charge transfer can occur from one constituent to another. Thus, the interfaces (grain (grain boundaries) boundaries) may may be be important. important. A A schematic schematic view view for for such such carrier carrier transport transport is is shown shown interfaces in Figure 2. in Figure 2.
Appl. Sci. 2016, 6, 50 Appl. Sci. 2016, 6, 50
3 of 10 3 of 10
Figure 2. Schematic Schematic view of carrier carrier transport transport within grains grains (intragrain) (intragrain) and through through grain grain boundaries boundaries (intergrain) in nanomaterials (intergrain) in nanomaterials
We call call this this aa series series sequence sequence of of the the free We free (intragrain) (intragrain) and and tunneling tunneling (intergrain; (intergrain; grain grain boundary) boundary) carrier (SSFTC) transport model [2,22]. Let us review the SSFTC formalism. As stated the carrier (SSFTC) transport model [2,22]. Let us review the SSFTC formalism. As stated already, already, the frequency dependence the Drude Drude law law (n (n = = 00 in in frequency dependence of of complex complex conductivity conductivity for for free free carrier carrier is is given given by by the Equation (1)) as: Equation (1)) as: σp0q σ f pωq (2) 1 iωτ (0) ∗ () (2) = 1 − as an effective relaxation time that accounts Note that τ in this equation should be viewed for various scattering processes within grain, intragrain processes. the Note that τ in this equation should bethe viewed as so-called an effective relaxationscattering time that accounts for For various tunneling transport, the so-called expression, based on anprocesses. effective medium approach, which scattering processes within the grain,Dyre so-called intragrain scattering For the tunneling transport, should be equivalent to a parallel capacitor-resistor random connection, is adopted [23]. The tunneling the so-called Dyre expression, based on an effective medium approach, which should be equivalent to a conductivity is given as: random connection, is adopted [23]. The tunneling conductivity is given as: parallel capacitor-resistor iωτt σt pωq σt p0q (3) lnp1− iωτt q ∗ () = (0) (3) where σt (0) is the dc tunneling conductivity and is ln(1 given − by n)t (ert )2 /2kTτt . Here, nt is the density of tunneling carrier, rt the tunneling length (distance between grains), and τt the tunneling time. where σt(0) is the dc tunneling conductivity and is given by nt(ert)2/2kTτt. Here, nt is the density of The effective (overall) complex conductivity, in a simple one-dimensional approximation [23–25], is tunneling carrier, rt the tunneling length (distance between grains), and τt the tunneling time. given as [2,22]: The effective (overall) complex conductivity, in a simple one-dimensional approximation [23–25], is 1 f 1 f (4) given as [2,22]: σeff pωq σ f pωq σt pωq where f is the spectral weight of intragrain 1 transport. 1− = shows + ∗ As will be discussed in Section 3, σ *(ω) Lorentz-type resonance, as was found in (4) the ∗ ∗ eff () () () DS model. The negative imaginary conductivity, curve (d) in the Lorentz resonance, originates from where f is the spectral weight charge. of intragrain transport. capacitive nature of electronic It should be emphasized here that a series sequence of the Drude eff*(ω) shows Lorentz-type resonance, as was found in the DS As will be discussed in Section 3, σ and the Debye-type relaxations produces Lorentz-type resonance. model. negative medium imaginary conductivity, curve (d) in the Lorentz resonance, from (4) The An effective theory (EMT) [26] is expected also to apply to THz originates conductivity in capacitive nature of electronic charge. It should be emphasized here that a series sequence of the inhomogeneous media [1]. The EMT should be employed to model the composite materials, in Drude andwhen the Debye-type relaxations produces Lorentz-type resonance. particular charge transfer between composites has not occurred, e.g., relatively small fraction (4) An(or effective medium theory (EMT) in [26] is expected also to apply to THz conductivity in of metallic high conductive) component insulators. The Lorentz-like resonance is predicted inhomogeneous media [1]. The EMT should be employed to model the composite materials, in from the EMT calculation [1]. A loss-peak frequency (see curve (c) in Figure 1) highly depends on the particular when charge transfer between composites has not occurred, e.g., relatively small fraction of plasma frequency, i.e., depends on free carrier density, and, hence, significant loss-peak shift should be metallic (or component in insulators. The Lorentz-like resonance is predictedof from the observed inhigh suchconductive) media, similar to the plasmon case. In fact, in common with nanomaterials metals EMT calculation [1]. A loss-peak frequency (see curve (c) in Figure 1) highly depends on the plasma and semiconductors, the loss peak may lie in THz frequency range (no big shift in a loss peak). frequency, i.e.,conclude dependsthat on the freeSSFTC carrier density, and, loss-peak shifttransport should be We thus model is the besthence, modelsignificant to explain the electronic in observed in such media, similar to the plasmon case. In fact, in common with nanomaterials of metals nanomaterials. Typical examples are shown in the following section. and semiconductors, the loss peak may lie in THz frequency range (no big shift in a loss peak). We thus conclude that the SSFTC model is the best model to explain the electronic transport in nanomaterials. Typical examples are shown in the following section.
Appl. Sci. 2016, 6, 50
4 of 10
Appl. Sci. 2016, 6, 50
4 of 10
3. Typical Examples of THz Conductivity in Nanomaterials For the purposes of this focused review, we will consider and discuss only a few selected 3. Typical Examples of THz Conductivity in Nanomaterials examples of the THz conductivity, such as popular nano- semiconductors, oxides, and metals, rather the purposes ofand thisextensive focused review, considerreview and discuss onlycollection a few selected examples than For a comprehensive review.we Anwill extensive of a large of materials is of the THz conductivity, such as popular nanosemiconductors, oxides, and metals, rather than a beyond the scope of this focused review. Note that the following experimental data are all provided comprehensive andbased extensive review. An extensive review of a large collectionand of materials beyond by pulsed sources on terahertz time-domain spectroscopy (THz-TDS) measuredis at room the scope of this focused that the following experimental data are alltherefore, provided we by pulsed temperature. There are areview. numberNote of excellent reviews on this topics [1,27] and, do not sources basedtechniques on terahertz discuss THz in time-domain this article. spectroscopy (THz-TDS) and measured at room temperature. ThereOpen are acircles number excellent thisthe topics [1,27] and, therefore, doimaginary not discuss THz andoftriangles inreviews Figure 3on show experimental results of realwe and parts of techniques in this article. conductivity, respectively, in nanocrystalline Si films (average grain size is 15–25 nm) [8]. To extract the Openconductivity circles and triangles in Figure 3 showbetween the experimental results of real andand imaginary parts complex we use the relationship the dielectric permittivity conductivity of conductivity, respectively, in nanocrystalline Si films (average grain size is 15–25 nm) [8]. To extract given as: the complex conductivity we use the relationship between the dielectric permittivity and conductivity given as: (5) ∗ () = − ∗ () σ pωq iωε0 ε pωq (5) where ε*(ω) + ε∞ = εR(ω) + iεI(ω), ε0 is the absolute permittivity, εR(ω) and εI(ω) are the real and imaginary where + ε8 = εR (ω) respectively, + iεI (ω), ε0 isand theεabsolute permittivity, εR (ω) and εI (ω)Note are the realthat anda part of ε*(ω) dielectric constants, constant. again ∞ is the background dielectric imaginary part of dielectric constants, respectively, and ε is the background dielectric constant. Note 8 pumping spectroscopy has been used for time dependence of the form exp(−iωt) is used here. Optical again that a time dependence of the form exp( iωt) is used here. Optical pumping producing enough free carriers (electrons) at room temperature, and, hence, we dospectroscopy not need ε∞ has for been used for producing enough free carriers (electrons) at room temperature, and, hence, we do calculation as a physical parameter [1]. The negative value of the imaginary part of conductivity is not need ε for calculation as a physical parameter [1]. The negative value of the imaginary part of 8 obtained. conductivity is obtained. 100 80
σ*(ω) (Scm-1)
60 40 20 0 -20 -40 1012
1013
1014
1015
ω (rad/s) Figure 3. Complex Complex conductivity conductivity in in nanocrystalline nanocrystalline Si Si films. films. Open Open circles circles and and triangles triangles are are experimental Figure 3. experimental results of the real and imaginary parts of conductivity, respectively; data extracted from [8]. Solid lines results of the real and imaginary parts of conductivity, respectively; data extracted from [8]. Solid are conductivity predicted from the SSFTC model. lines are conductivity predicted from the SSFTC model.
Solid lines in Figure 3 show the best fit fit of of the the SSFTC SSFTC model model to to the the experimental experimental data. data. Adjusted physical parameters are all reasonable and are listed in Table 1. 1. We now recognize that the SSFTC model predicts predicts aasimilar similarbehavior behaviorofofthe the Lorentz resonance as shown in Figure 1 (curves (c) (d)). and Lorentz resonance as shown in Figure 1 (curves (c) and (d)). is important to emphasize experimental data onlyavailable availablewithin withinaa certain certain narrow It is It important to emphasize thatthat experimental data areareonly frequency range. In In the the present present case, case, aa peak peak is predicted by the model outside the experimental experimental frequency a peak outside thethe data range, butbut there is no frequency window. window.Fitting Fittingwithin withinthe thedata datarange rangeleads leadstoto a peak outside data range, there is way of validating this prediction without extending peak. no way of validating this prediction without extendingthe theexperimental experimentaldata datarange rangeto to cover the peak. However, as will be shown for nanogold particles (Figure 7), a peak is indeed found in the observed frequency comparison between thethe Lorentz resonance andand the frequency range. range. We, We,therefore, therefore,discuss discussa adetailed detailed comparison between Lorentz resonance SSFTC prediction in the partpart of this section. the SSFTC prediction in final the final of this section.
Appl. Sci. 2016, 6, 50
5 of 10
Appl. Sci. 2016, 6, 50
5 of 10
Table 1. 1. Physical Physicalparameters parametersused usedfor forcalculation calculationininthe theseries seriessequence sequenceofofthe thefree freeand andtunneling tunneling carrier carrier Table (SSFCT) transport model. σ is the measured dc conductivity. dc (SSFCT) transport model. σdc is the measured dc conductivity.
Quantity Quantity
Ge 2Sb 2Te5 Si Si Ge 2 Sb 2 Te 5 f 0.83 0.996 f 0.83 0.996 0.3 m* m* 0.30.3 0.3 3 ) −3 20 20 n (cm 5.35.3 10× 181018 1.51.5 ×1010 ) n (cm 14 14−14 τ (s)τ (s) 2.0 2.0 10×10 10 −14 2.42.4 × 10 nt (cm3 ) −3 2.0 1019 19 2.0 1019 19 2.0 ×10 2.0 × 10 nt (cm ) τt (s) 4.0 10 13 2.0 1011 −13 −11 t (s) 4.0 × 10 2.0 × τ ε8 320 10 320 ε∞1 ) 28.0 15.4 σeff (0) (S cm 1–10 20 σdcσ(S cm eff(0) (S1 )cm−1) 28.0 15.4 σdc (S cm−1) 1–10 20
ZnO ZnO 0.94 0.94 0.3 0.3 7.0 7.0×10 101919 14 4.0 10 4.0 × 1018−14 1.6 10 18 1.6 × 10 2.0 1014 2.0 14 × 10−14 14 4.0 4 4.0 4
SnO SnO2 2 0.97 0.97 0.3 0.3 1.0×10 102020 1.0 14 5.0 10 5.0 × 10−1418 1.6 10 1.6 × 101813 9.0 10 9.0 × 15 10−13 15 11.4 0.6 11.4 0.6
Au Au Cryst-ZnO Cryst-ZnO 0.999 1 1 0.999 1 1 0.24 0.24 221022 4.5 ×4.51017 1017 1.0 1.0 × 10 3.0 1014 2.7 ×2.710−141014 3.0 × 10−14 1.5 1021 1.5 × 1021 12 3.0 10 −12 3.0 × 101 - 155,000 - 142 55000142 46 46
Open circles and triangles in Figure 4 show the real and imaginary parts of conductivity Open circles and triangles in Figure 4 show the real and imaginary parts of conductivity in the in the crystalline phase of Ge2 Sb2 Te5 (from the Kadlec group at the Academy of Sciences of the crystalline phase of Ge2Sb2Te5 (from the Kadlec group at the Academy of Sciences of the Czech Czech Republic in Pargue), which is the most useful phase-change material, developed for digital Republic in Pargue), which is the most useful phase-change material, developed for digital versatile versatile disk (DVD) [28]. As the crystalline phase of phase-change materials is a degenerate disk (DVD) [28]. As the crystalline phase of phase-change materials is a degenerate semiconductor semiconductor (witha high density of free carriers) and, hence, the optical-pumping (photocarrier) ( witha high density of free carriers) and, hence, the optical-pumping (photocarrier) technique, as in technique, as in nanocrystalline Si (see Figure 3) is not required. In this case, background conductivity nanocrystalline Si (see Figure 3) is not required. In this case, background conductivity (imaginary), (imaginary), σ8 = ωε0 ε8 , is taken into account [1,2]. Solid lines show the real and imaginary parts of σ∞ = ωε0ε∞, is taken into account [1,2]. Solid lines show the real and imaginary parts of conductivity conductivity predicted by the SSFTC model, and the physical parameters from the fitting are listed in predicted by the SSFTC model, and the physical parameters from the fitting are listed in Table 1. Note Table 1. Note that features obtained here are very similar to those obtained for other phase-change that features obtained here are very similar to those obtained for other phase-change materials [14]. materials [14]. 600
σ*(ω) (Scm-1)
400 200 0 -200 -400 1012
1013
1014
ω (rad/s) Figure GeGe 2SbSb 2Te5 films. Open circles and triangles are Figure 4. 4. Complex Complexconductivity conductivityinincrystalline crystallinephase phaseofof 2 2 Te5 films. Open circles and triangles experimental results of the real and imaginary parts of conductivity, are experimental results of the real and imaginary parts of conductivity,respectively. respectively.Solid Solid lines lines are are conductivity conductivity predicted predictedfrom fromthe theSSFTC SSFTCmodel. model.
Let us examine the THz conductivity in so-called oxide metals and semiconductors. ZnO and SnO2, Let us examine the THz conductivity in so-called oxide metals and semiconductors. ZnO and for example, show both metallic and semiconducting natures [17,29]. Open symbols in Figures 5 and 6 SnO2 , for example, show both metallic and semiconducting natures [17,29]. Open symbols in Figures 5 show the complex conductivity in nanostructured ZnO [17] and SnO2 [29] films, respectively. and 6 show the complex conductivity in nanostructured ZnO [17] and SnO2 [29] films, respectively. Fitting of the SSFTC model to the experimental data shown by solid lines produces reasonable physical parameters and these are listed in Table 1. The both nanostructured oxides discussed here have enough free carriers and hence the optical pumping technique is not employed. Note that the EMT or hopping transport model has been applied to interpret the THz conductivity in Sb-doped SnO2 [29]. However, as already stated, the EMT or hopping transport mechanism does not work properly, when nanoparticles are closely packed. The measured σdc value (0.5–10 S cm1 ) [29] is closed to the effective conductivity deduced from the SSFTC model (see Table 1).
Appl. Sci. 2016, 6, 50
6 of 10
Appl. Appl. Sci. Sci. 2016, 2016, 6, 6, 50 50
66 of of 10 10
40 40 30 30 -1 (Scm-1) ) σσ*(*(ωω) )(Scm
20 20 10 10 00 -10 -10 -20 -20 -30 -30 12 10 1012
13 10 1013 ωω (rad/s) (rad/s)
14 10 1014
Figure 5. conductivity nanostructured Open circles and triangles, respectively, Figure 5.Complex Complex conductivity innanostructured nanostructuredZnO ZnO films. Open circles andand triangles, respectively, Figure 5. Complex conductivity inin ZnOfilms. films. Open circles triangles, respectively, are the experimental data extracted from [17]. Solid lines are conductivity predicted from are experimental the experimentaldata dataextracted extracted from SolidSolid lines are conductivity predictedpredicted from the the SSFTC SSFTC are the from[17]. [17]. lines are conductivity from the model. model. SSFTC model.
(Scm-1-1) ) σσ*(*(ωω) )(Scm
40 40 20 20 00 -20 -20 -40 -40 12 10 1012
13 10 1013 ωω (rad/s) (rad/s)
14 10 1014
Figure 6. Complex conductivity in nanostructured Sb-doped SnO2 ((βC500P-5%). Open circles and
Figure 6. Complex conductivity nanostructured Sb-doped Sb-doped SnO 500P-5%). Open circles and Figure 6. Complex conductivity in in nanostructured SnO2 2((βC ((βC 500 P-5%). Open circles and triangles triangles are are the the experimental experimental data data extracted extracted from from [29]. [29]. Solid Solid lines lines are are predicted predicted conductivity conductivity from from triangles are the experimental data extracted from [29]. Solid lines are predicted conductivity from the the SSFTC model. the SSFTC model. SSFTC model.
σ*(ω) (Scm-1)
Fitting Fitting of of the the SSFTC SSFTC model model to to the the experimental experimental data data shown shown by by solid solid lines lines produces produces reasonable reasonable physical parameters and these are listed in Table 1. The both nanostructured oxides discussed here Let us examine the THz conductivity in nanostructured gold films. Open circles and triangles physical parameters and these are listed in Table 1. The both nanostructured oxides discussed here in have enough free carriers and hence the optical pumping technique is not employed. Note that the Figure 7 show onefree of examples the real imaginary of conductivity in nanogold have enough carriers andofhence the and optical pumpingparts technique is not employed. Note thatfilms the [30]. EMT or hopping transport model has been applied to interpret the THz conductivity in Sb-doped hopping transport model hasinbeen interpret conductivity SolidEMT linesorare predicted conductivity the applied SSFTC to model. Asthe theTHz imaginary part in ofSb-doped conductivity SnO 2 [29]. However, as already stated, the EMT or hopping transport mechanism does not work SnO However, as already the EMT mechanism behavior. does not work shows the2 [29]. positive signature (curvestated, (b) in Figure 1),orit hopping seems totransport be the Drude-type Fitting to −1) [29] is closed properly, when nanoparticles are closely packed. The measured σσdcdc value (0.5–10 S·cm −1) [29] is closed properly, when nanoparticles are closely packed. The measured value (0.5–10 S·cm the experimental data shows, however, that grain boundaries dominate the transport and hence the to to the the effective effective conductivity conductivity deduced deduced from from the the SSFTC SSFTC model model (see (see Table Table 1). 1). SSFTC model isexamine working well inconductivity gold films in as nanostructured well. In fact, the dcfilms. conductivity of nanogold films [4] is Let us the THz gold in Let2016, us examine the THz conductivity in nanostructured gold films. Open Open circles circles and and triangles triangles Appl. Sci. 6, 50 7 of in 10 very Figure much 7smaller than that of conventional gold. Figure 7 show show one one of of examples examples of of the the real real and and imaginary imaginary parts parts of of conductivity conductivity in in nanogold nanogold films films [30]. [30]. Solid lines are predicted conductivity in the SSFTC model. As the imaginary part of conductivity Solid lines are predicted conductivity in the SSFTC model. As the imaginary part of conductivity shows 80000 (b) shows the the positive positive signature signature (curve (curve (b) in in Figure Figure 1), 1), itit seems seems to to be be the the Drude-type Drude-type behavior. behavior. Fitting Fitting to the experimental data shows, however, that grain boundaries dominate the transport to the experimental data shows, however, that grain boundaries dominate the transport and and hence hence 60000 the the SSFTC SSFTC model model is is working working well well in in gold gold films films as as well. well. In In fact, fact, the the dc dc conductivity conductivity of of nanogold nanogold films films [4] [4] is is very very much much smaller smaller than than that that of conventional conventional gold. gold. 40000of 20000 0 -20000
1012
1013
1014
1015
ω (rad/s)
Figure 7. Complex conductivityininnanostructured nanostructured Au circles andand triangles are the Figure 7. Complex conductivity Au films. films.Open Open circles triangles are the experimental extracted from [30].Solid Solidlines lines are are the predicted by the SSFTC model. experimental datadata extracted from [30]. theconductivity conductivity predicted by the SSFTC model.
Next, we will show what is observed in a single crystalline material. An example is shown for ZnO epitaxial films in Figure 8. Open circles and triangles show the real and imaginary parts of conductivity [31]. Solid lines show the predictions from the Drude law and physical parameters are listed in Table 1.
0 -20000
1012
1013
1014
1015
ω (rad/s)
Appl. Sci. 2016,Figure 6, 50 7. Complex conductivity in nanostructured Au films. Open circles and triangles are the experimental data extracted from [30]. Solid lines are the conductivity predicted by the SSFTC model.
7 of 10
Next, weshow will show what is observedin inaa single single crystalline material. An example is shown for Next, we will what is observed crystalline material. An example is shown for ZnO epitaxial films in Figure 8. Open circles and triangles show the real and imaginary parts ZnO epitaxial films in Figure 8. Open circles and triangles show the real and imaginaryofparts of conductivity [31]. Solid lines show the predictions from the Drude law and physical parameters are conductivity [31]. Solid lines show the predictions from the Drude law and physical parameters are listed in Table 1. listed in Table 1.
σ*(ω) (Scm-1)
30
20
10
0 1011
1012
1013
1014
1015
ω (rad/s)
Figure 8. Complex conductivity in in single epitaxial ZnO films. Open and triangles Figure 8. Complex conductivity singlecrystalline crystalline epitaxial ZnO films. Open circlescircles and triangles the experimental extracted from[31]. [31]. Solid thethe conductivity predicted from the are the are experimental datadata extracted from Solidlines linesareare conductivity predicted from the SSFTC model. SSFTC model. A good fit of the Drude law to the experimental results in epitaxial GaN has been also reported [32].
A good fit of stated, the Drude law to the experimental inshould epitaxial GaN has reported [32]. As already the Drude-Smith model [21], in results principle, be applied to been singlealso crystalline materials since theDrude-Smith disorder effect model is not important in the DS model; it isbe simply a generalization of As already stated, the [21], in principle, should applied to single crystalline the Drude formula. No such non-Drude behavior, which fits to the DS model, is reported. materials since the disorder effect is not important in the DS model; it is simply a generalization of the It was stated that the SSFTC mechanism produces the Lorentz-type resonance, as shown in Figure 3. Drude formula. No such non-Drude behavior, which fits to the DS model, is reported. In the Lorentz resonance, however, conductivity at a low frequency should be zero. Therefore, the SSFTC It was stated that the SSFTC mechanism produces the Lorentz-type resonance, as shown in model produces a similar trend to the Lorentz resonance. As shown in Figure 9, when we change Figure 3. In the parameters, Lorentz resonance, however, conductivity a low should be zero. Therefore, physical e.g., a decrease of f in Equation (4), in at which thefrequency tunneling contribution becomes the SSFTC model produces a similar trend to the Lorentz resonance. As shown in Figure 9, when we large, the resonant conductivity predicted by the SSFTC model (solid lines) becomes very close to that using the Lorentz resonance (dashed lines), described as [1,3]: change physical parameters, e.g., a decrease of f in Equation (4), in which the tunneling contribution becomes large, the resonant conductivity predicted by the SSFTC model (solid lines) becomes very close to that using the Lorentz resonance ∗ () =(dashed lines), described as [1,3]: ∗
σ p ωq Appl. Sci. 2016, 6, 50
(6)
− ⁄
1−
Ne2 τL m 1 iτ L pω ω p 2 {ωq
(6) 8 of 10
here, Nhere, is the number of Lorentz oscillator, dampingtime, time, ωp plasma the plasma frequency. N is the number of Lorentz oscillator,ττLL the the damping andand ωp the frequency. The physical parameters are are shown ininthe ofFigure Figure9. 9. The physical parameters shown thecaption caption of 100 80 σ*(ω) (Scm-1)
60 40 20
(a)
(b)
0 -20
(c)
-40 -60 1011
(d)
10
12
13
10
1014
1015
ω (rad/s) 9. Complex conductivity conductivity predicted from thefrom SSFTC,the solidSSFTC, lines (a) and (c), where 4.5 ×and 10 cm Figure Figure 9. Complex predicted solid linesn =(a) (c),, where (b) and (d), are 12 nt = 4.0 × 1019 cm−3, f = 0.70, τ = 2.0 × 10−14 s, and τt = 4.0 × 10−12 s are used. Dashed 14 s, lines, n = 4.5 1018 cm3 , nt = 4.0 1019 cm3 , f 18= 0.70, τ = 2.0 10 and τ = 4.0 10 s t from the Lorentz resonance, where N = 4.5 × 10 cm−3, ωp = 4.5 × 1013 s−1, and τL = 1.3 × 10−13 s. are used. Dashed lines, (b) and (d), are from the Lorentz resonance, where N = 4.5 1018 cm3 , 13 s1 , and τ = 1.3 1013 s. ωp = 4.5The 10 L be clearly understood in terms of equivalent electrical circuits. The Drude reason for this can 18
−3
mechanism, i.e., free carrier scattering (and loss), is equivalent to LRg series connection, where L is the inductance and Rg the resistance in grains. The tunneling mechanism, i.e., charge accumulation (and loss), is equivalent to CRgb parallel connection, where C is the capacitance and Rgb the resistance in grain boundaries. We know that a series connection of these elements produces the Lorentz-type resonance at a certain frequency [2]. Current flow (conductivity) is dominated by Rgb at lower frequencies and is dominated by L at a high frequency, i.e., zero conductivity at low and high frequencies when Rgb is large.
Appl. Sci. 2016, 6, 50
8 of 10
The reason for this can be clearly understood in terms of equivalent electrical circuits. The Drude mechanism, i.e., free carrier scattering (and loss), is equivalent to LRg series connection, where L is the inductance and Rg the resistance in grains. The tunneling mechanism, i.e., charge accumulation (and loss), is equivalent to CRgb parallel connection, where C is the capacitance and Rgb the resistance in grain boundaries. We know that a series connection of these elements produces the Lorentz-type resonance at a certain frequency [2]. Current flow (conductivity) is dominated by Rgb at lower frequencies and is dominated by L at a high frequency, i.e., zero conductivity at low and high frequencies when Rgb is large. We recognize, therefore, that a negative imaginary conductivity at a lower frequency is dominated by the grain boundary transport and a positive imaginary conductivity is dominated by the free carrier scattering within the grain. The SSFTC model can judge which factor is dominant in nanostructured materials. Recent reports on the THz conductivity in InP nanowires [33] and silicon nanocrystal superlattices [34] show similar behaviors, as shown in Figure 9, suggesting the importance of the tunneling contribution of carriers in these new materials. We should mention another important modeling technique that is based on computer simulations. It is recognized that Monte Carlo simulations can provide useful insight and information on carrier dynamics [12,35], which replicate well the non-Drude type behavior. This technique is useful when one discusses the carrier dynamics inside semiconductor nanoparticles. Carrier scattering processes at boundaries have been discussed in great detail [12]. If the carrier mean-free-path (MFP) is much smaller than grain size (or grain size is much larger than MFP), the carriers are regarded as moving in homogeneous media. In this case, the carrier dynamics can be well interpreted by the Drude-Smith model [21,36]. Finally, it should be mentioned that the interband optical transition (Lorentz oscillator), combined with the Drude contribution of free carriers, also shows a non-Drude behavior, which has been reported in carbon-based nanotubes [37]. Thus, the examination of the SSFTC model for other material systems will shed a great deal of light on the mechanism of optical conductivity in the terahertz range. 4. Conclusions Current understanding of the THz conductivity in nanostructured materials, through metals to semiconductors was reviewed. It was shown that a model of series sequence of free and tunneling carrier (SSFTC) transport had a fundamental physical basis and well-explained the observed non-Drude behavior in nanomaterials. The effective conductivity deduced from the SSFTC model is close to the measured dc conductivity, as listed in Table 1. Of particular interest is that the SSFTC transport mechanism produces a Lorentz-type resonance. Surprisingly, almost the same profile with the Lorentz resonance is predicted from the SSFTC model under certain conditions. Acknowledgments: Koichi Shimakawa would like to thank Vito Zima for supporting works in the Joint Laboratory of Solid State Chemistry at University of Pardubice. The authors thank Tomas Wagner and Miloslav Frumar for fruitful discussions. SK would like to thank NSERC Discovery Grants Program for financial support. We are most grateful to Filip Kadlec and Christelle Kadlec, Academy of Sciences of the Czech Republic, Prague, for providing the terahertz data on Ge2Sb2Te5 films before their publication. Author Contributions: Both authors were equally involved in the scientific formulation and development of the model and discussions related to the application of the model. Koichi Shimakawa was further involved in modifyting and fitting the model to the experimental data. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2.
Lloyd-Hughes, J.; Jeon, T.-I. A review of the terahertz conductivity of bulk and nano-materials. J. Infrared Milli Terahertz Waves 2012, 33, 871–927. [CrossRef] Shimakawa, K. Electrical properties of nanocrystalline media: Optical conductivity and non-Drude behavior in the terahertz frequency range. Can. J. Phys. 2014, 92, 1–4. [CrossRef]
Appl. Sci. 2016, 6, 50
3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14.
15.
16. 17. 18.
19.
20. 21. 22. 23. 24. 25. 26.
9 of 10
Wooten, F. Optical Properties of Solids; Academic Press: New York, NY, USA, 1972. Thoman, A.; Kern, A.; Helm, H.; Walther, M. Nanostructured gold films as broadband terahertz antireflection. Phys. Rev. B 2008, 77. [CrossRef] Lovrincic, R.; Pucci, A. Infrared optical properties of chrorium nanoscale films with a phase transition. Phys. Rev. B 2009, 80. [CrossRef] Takano, K.; Shibuya, K.; Akiyama, K.; Nagashima, T.; Miyamaru, F.; Hangyo, M. A metal-to-insulator transition in cut-wire-grid metamaterials in the terahertz region. J. Appl. Phys. 2010, 107. [CrossRef] Nienhuys, H.-K.; Sundstrom, V. Influence of plasmons on terahertz conductivity measurements. Appl. Phys. Lett. 2005, 87. [CrossRef] Cooke, D.G.; MacDounald, A.N.; Hryciw, A.; Wang, J.; Li, L.; Meldrum, A.; Hegmann, F.A. Transient terahertz conductivity in photoexcited nanocrystal films. Phys. Rev. B 2006, 73. [CrossRef] Cooke, D.G.; MacDonald, A.N.; Hryciw, A.; Meldrum, A.; Wang, J.; Frank, Q.L.; Hegmann, F.A. Ultrafast terahertz conductivity of photoexcited nanocrystalline silicon. J. Mater. Sci. Mater. Electron. 2007, 18, 447–452. [CrossRef] Ahn, H.; Ku, Y.-P.; Wang, Y.-C.; Chuang, C.-H. Terahertz spectroscopic study of vertically aligned InN nanorods. Appl. Phys. Lett. 2007, 91. [CrossRef] Fekete, L.; Kuzel, P.; Nemec, H.; Kadlec, F.; Dejneka, A.; Stuchlick, J.; Fejfar, A. Ultrafast carrier dynamics in microcrystalline silicon probed by time-resolved terahertz spectroscopy. Phys. Rev. B 2009, 79. [CrossRef] Nemec, H.; Kuzel, P.; Sandstrom, V. Far-infrared response of free charge carrier localized in semiconductor nanoparticles. Phys. Rev. B 2009, 79. [CrossRef] Kadlec, F.; Kadlec, C.; Kuzel, P. Contrast in terahertz conductivity of phase-change materials. Solid. State Commun. 2012, 152, 852–855. [CrossRef] Shimakawa, K.; Wagner, T.; Frumar, M.; Kadlec, F.; Kadlec, C.; Kasap, S. Terahertz and direct current losses and the origin of non-Drude terahertz conductivity in the crystalline states of phase change materials. J.Appl. Phys. 2013, 114. [CrossRef] Liu, H.; Lu, J.; Tang, S.H.; Zhang, X.; Sow, C.H. Composition-dependent ultra-high photoconductivity in ternary CdSx Se1x nanobelts as measured by optical pump-terahertz probe spectroscopy. Nano Res. 2013, 6, 808–821. [CrossRef] Liu, H.; Lu, J.; Tang, S.H.; Sow, C.H.; Zhang, X. Composition-dependent electron transport in CdSx Se1x nanobelts: A THz spectroscopy study. Opt. Lett. 2014, 39, 567–570. [CrossRef] [PubMed] Turner, G.M.; Beard, M.; Schmuttenmaer, C.A. Conductivity of ZnO nanowires, nanoparticles, and thin films using time-resolved THz spectroscopy. J. Phys. Chem. B 2006, 110, 25229–25239. Qazilbash, M.M.; Brehm, M.; Chae, B.-G.; Ho, P.-C.; Andreev, G.O.; Kim, B.-J.; Yun, S.J.; Baratsky, A.V.; Maple, M.B.; Kleilmann, F.; et al. Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging. Science 2007, 318, 1750–1753. [CrossRef] [PubMed] Coker, T.I.; Titova, I.V.; Foumax, S.; Bandulet, H.-C.; Brassard, D.; Kieffer, Q.L.; El Khakani, M.A.; Hegmann, F.A. Teahertz conductivity of the metal-insulator transition in a nanogranular VO2 films. Appl. Phys. Lett. 2010, 97. [CrossRef] Parkinson, P.; Lloyd-Hughes, J.; Gao, Q.; Tan, H.H.; Jagadish, C.; Johnston, M.B.; Herz, L.M. Transient terahertz conductivity of GaAs nanowires. Nano Lett. 2007, 7, 2162–2165. [CrossRef] Smith, N.V. Classical generalization of the Drude Formula for the optical conductivity. Phys. Rev. B 2001, 64. [CrossRef] Shimakawa, K.; Itoh, T.; Naito, H.; Kasap, S. The origin of non-Drude terahertz conductivity in nanomaterials. Appl. Phys. Lett. 2012, 100. doi:10.1063/1.3697404. Erratum, 2012, 100. [CrossRef] Dyre, J.C.; Schroder, T.W. Universality of ac conduction in disordered solids. Rev. Mod. Phys. 2000, 72, 873–894. [CrossRef] Shimakawa, K.; Ganjoo, A. AC photoconductivity of hydrogenated amorphous silicon: Influence of long-range potential fluctuations. Phys. Rev. B 2002, 65. [CrossRef] Kasap, S., Capper, P., Kouighia, C., Ruda, H., Eds.; Electrical Conduction in Metals and Semiconductors. In The Springer Handbook of Electronic and Photonic Materials; Springer: Heidelberg, Germany, 2007; pp. 19–44. Bruggeman, D.A.G. Berechnung verschiedener physikalisher konstanten substanzen I. Ann. Phys. 1935, 416, 636–664. [CrossRef]
Appl. Sci. 2016, 6, 50
27. 28. 29.
30. 31.
32. 33.
34. 35. 36.
37.
10 of 10
Ferguson, B.; Zhang, X.C. Materials for terahertz science and technology. Nat. Mater. 2002, 1, 26–33. [CrossRef] [PubMed] Wuttig, M.; Yamada, N. Phase-change materials for rewriteable data storage. Nat. Mater. 2007, 6, 824–832. [CrossRef] [PubMed] Skoromets, V.; Nemec, H.; Kopec, K.; Kuzel, P.; Peters, K.; Fattakhova-Rohlfing, D.; Vetushka, A.; Muller, M.; Ganzerova, K.; Fejfar, A. Conductivity mechanisms in Sb-doped SnO2 nanoparticles assebmblies: DC and Terahertz regime. J. Phys. Chem. C 2015, 119, 19485–19495. Walther, M.; Cooke, D.G.; Sherstan, C.; Hajar, M.; Freeman, M.R.; Hegmann, F.C. Terahertz conductivity of thin gold films at the metal-insulator percolation transition. Phys. Rev. B 2007, 76. [CrossRef] Zhang, X.H.; Guo, H.G.; Yong, A.M.; Ye, J.D.; Tan, S.T. Terahertz dielectric response and optical conductivity of n-type single crystal ZnO epilayers grown by metalorganic chemical vapor deposition. J. Appl. Phys. 2010, 107. [CrossRef] Zhang, W.; Azad, A.K.; Grischkowsky, D. Terahertz studies of carrier dynamics and dielectric response of n-type freestanding epitaxial GaN. Appl. Phys. Lett. 2003, 82, 2841–2843. [CrossRef] Ponseca, C.S.; Nemec, H.; Wallentin, J.; Auttu, N.; Beech, J.P.; Iqbal, A.; Borgstrom, M.; Pistol, M.-E.; Samurlson, L.; Yartsev, A. Bulk-like transverse electron mobility in an array of heavily n-doped InP nanowires probed by terahertz spectroscopy. Phys. Rev. B 2014, 90. [CrossRef] Nemec, H.; Zajac, V.; Kuzel, P.; Maly, P.; Gutsch, S.; Hiller, D.; Zachrias, M. Charge transport in silicon nanocrystal superlattices in the terahertz regime. Phys. Rev. B 2015, 91. [CrossRef] Willis, K.J.; Hagness, S.C.; Knezevic, I. Terahertz conductivity of doped silicon calculated using the ensemble Monte Carlo/finite-difference time-domain simulation technique. Appl. Phys. Lett. 2010, 96. [CrossRef] Yang, C.-S.; Lin, M.-H.; Chang, C.-H.; Yu, P.; Shieh, J.-M.; Shen, C.-H.; Wada, O.; Pan, C.-L. Non-Drude behavior in Indium-Tin-Oxide Nanowhisker and thin films investigated by transmission and reflection THz time-domain spectroscopy. IEEE JQE 2013, 49, 677–690. [CrossRef] Slepyan, G.Y.; Maksimenko, S.A.; Lakhatakia, A.; Yevtushenko, O.; Gusakov, A.V. Electrodynamics of carbon nanotubes: Dynamic conductivity, impedance boundary conditions, and surface wave propagation. Phys. Rev. B 1999, 60, 17136–17149. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
