The Drude-Smith Model for Conductivity: de novo Derivation and Interpretation
TThe Drude-Smith Model for Conductivity: de novo Derivation and Interpretation
Keno L. Krewer, ∗ Marco Ballabio, and Mischa Bonn Max Planck Institute for Polymer Research, 55128 Mainz, Germany (Dated: August 19, 2020)The Drude-Smith model successfully describes the frequency and phase-resolved electrical con-ductivity data for a surprisingly broad range of systems, especially in the terahertz region. Still,its interpretation is unclear since its original derivation is flawed. We use an intuitive physicalframework to derive the Drude-Smith formula for systems where microscopically free charges areaccumulated on a mesoscopic scale by localized scatterers. Within this framework, the model al-lows us to quantify the microscopic momentum relaxation time of the charges and the fraction ofmesoscopically localized charges in addition to the direct current limit of the conductivity. We showthat the Drude-Smith model is unique among different Drude-Lorentz models because the relaxationtime of the free carriers also determines the frequency and damping of the resonance of the boundcharges.
Characterising charge carrier dynamics by conductivityspectra:
Charge carriers in conductors and semiconductors formthe basis of several important technologies, includingcomputers, semiconductor lasers, and light-emitting de-vices. These technologies consist of increasingly smallstructures. There is, therefore, both a technological andfundamental interest in characterizing and understand-ing the properties of charge carriers in both bulk andnanostructured materials. A very suitable way of char-acterizing charge carriers dynamics in different materialsand material configurations is through their frequency-dependent conductivity, or, equivalently, dielectric re-sponse. The response of mobile carriers and polaronsis dictated by carrier-phonon interactions leading to ran-domization of the carrier momentum typically occurringon (sub-)picosecond time scales, giving rise to dispersionin the dielectric response on meV energy scales. Scatter-ing from defects also typically occurs on that time- andenergy scale. The dielectric response in the same energyrange is modified for carriers that undergo different typesof transport, such as hopping transport in non-crystallinesemiconductors. The ability to probe charge carriers inthe (sub-)meV energy or, equivalently, megahertz to ter-ahertz frequency range, therefore, allows their detailedcharacterization through the distinct spectral signaturesin this frequency range. Empirically, the Drude-Smith(DS) model describes such frequency-resolved electricalconductivity data measured in a wide variety of sys-tems, ranging from liquid metal, for which the modelwas first formulated, [1–3] over percolated metals [4–7]and other nano structures [8–15], amorphous metals [16],graphene[17], semiconductors [18–22], to organic conduc-tors [23–26]. The feature that all these systems have incommon is a restriction of charge carrier motion that istranslated into the observation of a reduced conductivityat low frequencies. The original derivation of the modelby Smith was based on microscopic arguments [1], butis flawed to the point that it contradicts itself. This has left the meaning of the model and its parameters un-clear. Observed Drude-Smith conductivities have, there-fore, sometimes been interpreted in terms of Smiths pro-posed preferential one-off backscattering, or alternativelyas a more mesoscopic charge confinement induced bygrain boundaries or as nanoscale disorder [27–30]. On oc-casion, the Drude-Smith model has just been consideredan empirical approximation, e.g. of hopping transport,without substantial physical interpretation [18]. Hence aclear understanding of the Drude-Smith type conduction,and an unambiguous physical assignment of the param-eters employed describing it, are desirable.Here, we derive the Drude-Smith equation based onmesoscopic arguments. This may explain why the for-mula can describe microscopically very different sys-tems. We start by explaining the problems and self-contradiction in Smiths original derivation.
Smiths problematic physical picture and contradictingassumptions:
Smith [1] motivated his model by considering the im-pulse response of n identical, non-interacting particlesper unit volume carrying a charge e whose scatteringprocess is governed by Poisson statistics, that means eachscattering event is independent from all preceding events.The current density j of such a system has the followingresponse in the time domain [1]: j ( t ) j (0) = exp (cid:18) − tτ c (cid:19) + ∞ (cid:88) s =1 β s s ! (cid:18) tτ c (cid:19) s exp (cid:18) − tτ c (cid:19) (1)where τ c is the expected time between collisions, s isthe number of collisions that a charge carrier has un-dergone up to time t . At time t = 0, an electric fieldhad accelerated the charge carrier to a certain veloc-ity, and β s reflects the expected fraction of this origi-nal velocity the charge carrier has retained after s col- a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug lisions. In case of classical trajectories, ” β s is the ex-pectation of cos( θ ) after s collisions.” [1] θ is the an-gle between the original and final velocity. Smith cor-rectly states that for independent collisions, β s = ( β ) s and eq. (1) simplifies to j ( t ) /j (0) = exp ( − t/τ ), where τ = τ c / (1 − β ) is the velocity relaxation time. This re-sponse is the well-known Drude result [1] equivalent to acomplex frequency-domain conductivity ˜ σ of˜ σ ( ω ) = ne τm − iωτ = σ − iωτ (2)where ω is the angular frequency and m the mass ofeach particle. We abbreviate the zero-frequency limit ofthe conductivity (i.e., DC conductivity) for this case ofentirely free charges as σ .Smith then made the assumption that β (cid:54) = 0 and β s =0 for all s ≥
2. This implies for the conductivity that [1]:˜ σ DS ( ω ) = σ − iωτ C (cid:18) β − iωτ C (cid:19) (3)Physically, this means each charge carrier scatters intoa preferential direction, defined by β , as a consequenceof its first scattering process after time 0, and thencompletely randomly forever after. The special spec-tral shape that Smith wanted to capture with his modelappears when β is negative, implying the moving elec-tron charge is preferentially scattered in the backwarddirection, but only on the first scattering event. Smithsmodel thus implies the very strange physical picture ofsome kind of counter attached to each charge carrier.All counters of all carriers would somehow be initializedat time 0. Each counter would then count every scat-tering event the carrier undergoes, directing it to startscattering randomly after the first event. To compoundthe oddity of this physical picture, we remember that t = 0 is an arbitrary choice. The odd physical picture isproblematic for Smiths model; a self-contradiction makesSmiths derivation of his formula mathematically wrong.Smith starts by assuming a Poisson distributed scatter-ing process, allowing the use of eq. (1). But by definition,events of a Poisson process must be statistically indepen-dent. Choosing any β s (cid:54) = ( β ) s assumes a correlationbetween scattering events, which therefore precludes theuse of eq. (1). A Poisson process must always lead to theDrude shape of eq. (2). Hence the model is not internallyconsistent. Deriving Smiths formula for a localized and a continuousscattering process
Here, we offer a self-consistent derivation of the Drude-Smith formula that is general enough to explain the ob- servation of Drude-Smith type conduction in a wide rangeof microscopically different materials. We start similarlyto Smith, and consider a material with free carriers whichrelax their velocity within the time τ due to scattering. Aconstant applied field E would lead to a current j . Afterthe field is switched off at time t = 0, the current decaysexponentially (fig 1a). E ( t ) = E Θ( − t ) j ( t ≥
0) = E σ exp ( − t/τ ) (4)We obtain the frequency domain current by Fouriertransformation: ˜ j ( ω ) = ˜ E ( ω ) σ − iωτ (5)This is the Drude response. Now we insert an obstaclein our material (fig. 1b)). When we induce a current j I inthe material, the obstacle will reflect a fraction C of theapplied current. After the initial current j I is switched offat t = 0, the reflected current j B will relax exponentiallywith the same characteristic relaxation time τ . j I ( t ) = j Θ( − t ) j B ( t ≥
0) = j · ( − C ) exp ( − t/τ )(6)This implies a frequency domain relationship:˜ j B ( ω ) = ˜ j I ( ω ) ( − C )1 − iωτ (7)Taking the initial current spectrum ˜ j I as the result ofan applied electric field ˜ E , we can easily derive ˜ j B in thefrequency domain (fig. 1c)).˜ j B ( ω ) = ˜ E ( ω ) σ − iωτ ( − C )1 − iωτ (8)The net current spectrum ˜ j as a result of the electricalfield at the obstacle is˜ j ( ω ) = ˜ j I + ˜ j B = ˜ E ( ω ) σ − iωτ (cid:18) − C − iωτ (cid:19) (9)This is Smiths conductivity formula. For now, it onlydescribes the current at the obstacle, not far away fromit.The reflection of the current caused by the mesoscopicobstacle induces a charge accumulation between the ob-stacle itself and any other position x in the material(Fig. 1d)). We can call on the continuity equation to en-sure that the formula is correct for all positions along thedirection x of current and field. In the frequency domain,the continuity equation reads: FIG. 1. Fig. 1 a): An electric field E is switched off at t = 0. In a material with free carriers, all with the same relaxationtime τ , the current j at t = 0 will be proportional to the dc-conductivity σ . After time 0, the current will decay exponentially,described by the relaxation time. The Drude model of complex conductivity follows from Fourier transformation.Fig. 1 b): A current j I is injected into the material. Some form of obstacle reflects a fraction C of this current. After theincoming current is switched off at time t = 0, the back-reflected current j B will again decay exponentially due to backgroundscattering characterized by the relaxation time. In the frequency domain, this exponential decay will again result in a (1 − iωτ )denominator.Fig. 1 c): Replacing the injected current in b) by a current launched by an electric field as in a) results in the back currentthat has been subject to scattering twice: before and after reflection. Combining the initial ˜ j I and the back current ˜ j B yieldsthe Drude-Smith conductivity for the total current ˜ j .Fig. 1 d): The back current will obviously also decay as a function of distance from the obstacle. However, this decay impliesan accumulation of charge (green circles). The displacement current ˜ j D (green) by this time-dependent charge accumulationwill compensate for the decrease in actual back current. − ∂ ˜ j∂x = − iω ˜ ρ ( x ) (10)with ˜ ρ the position-dependent charge density spec-trum. Integrating this equation, we obtain the differencebetween the reflected current at the position of the ob-stacle x = 0 and another position x . This change incurrent equals the displacement current ˜ j D generated bythe transient polarization ˜ P ,x of the charge that has toaccumulate between 0 and x :˜ j B (0) − ˜ j B ( x ) = − iω (cid:90) x ˜ ρ ( x ) dx = − iω ˜ P ,x = ˜ j D ( x )(11) The total current, including the displacement current,is therefore constant. This picture of spatial variations ofelectrical currents and fields due to localized scatteringcenters was introduced for the direct current by Landauer[31] more than ten years before Smith. Landauer explainsin detail how to derive the fraction C of the microscopi-cally free current dammed up by the obstacles based ontheir microscopic scattering cross-section and arrange-ment. For the macroscopic conductivity model, we donot need to know the microscopic details of the velocitydistribution of the charge carriers or any details aboutthe obstacles apart from the fact that they are localized.Localized means that the mesoscopic obstacles are farenough away from each other so that only a tiny fractionof back current is reflected again by another obstacle.Hence, the obstacles must be separated by distances ex-ceeding the mean free path of the continuous scatteringmechanism. In the limiting case of many obstacles andlittle background scattering, the build-up of back reflec-tions upon back reflections leads to a similar situationas that described by eq. (1), which will result again in aDrude model with the obstacle scattering dominating theresulting Drude relaxation time. At this point, we shouldmention that the reflection from the obstacles does notneed to be specular or particularly in the backward di-rection. Any interaction with the obstacle that leads toa change in current ∂j/∂x will cause a charge accumula-tion and, therefore, localization. The model parameter C does not directly relate to microscopic scattering (back-wards or otherwise). Rather, it is the fraction of the flowof microscopically free charges that is dammed up by thelocalized obstacles, causing charge accumulation. Distinguishing Drude-Smith from other conductivity models:
Now that we have derived the Drude-Smith formula,we investigate what makes it empirically successful andunique. We rewrite Smiths conductivity in the followingnotation, to show its place in the Drude-Lorentz formal-ism: ˜ σ DS ( ω ) = σ − iωτ (cid:18) − C − iωτ (cid:19) (12)= σ (1 − C )1 − iωτ − iω τ Cσ − iωτ − ( ωτ ) (13)We remind that the relaxation time τ has a positivevalue, conductivity σ is also a positive parameter and C is a parameter between 0 and 1. Note that we take C as positive, which is opposite to the current convention,as the convention is still based on the misidentificationof C with Smiths β . The product version in eq. (12) isclose to the original and conventional formulation. Thesum version eq. (13) allows more insightful comparisonswith other conduction models, since it shows that theDrude-Smith formula is a special case of a Drude-Lorentzconductivity ˜ σ DL . The general Drude-Lorentz formulafor a system with one species of free charge carriers anda single resonance of bound charges (e.g., a phonon modeor an exciton transition) is:˜ σ DL ( ω ) = σ DC − iωτ D − iω P − iωγ/ω − ( ω/ω ) (14)We can identify the DC-limit of the conductivity σ DC with σ DC = σ (1 − C ) in the Drude-Smith case. Thesteady-state limit of the polarization resonance P isgiven by the product τ Cσ of all three parameters of theDrude-Smith model. C is the parameter describing theimportance of the bound resonance relative to the freecarrier response, distinguishing Drude-Smith from plain Drude. When mapping the Drude-Lorentz perspectiveonto the Drude-Smith formula, the truly unique featureis the importance and universality of the relaxation time τ . It is not only equivalent to the Drude relaxation time τ D of the free carrier response, but it also determinesboth the damping rate γ = 2 /τ and the resonance fre-quency ω = 1 /τ of the Lorentz oscillator. Moreover, asthe direct current conductivity σ DC is also proportionalto the relaxation time, we conclude that each of the fiveparameters of the general Drude-Lorentz formalism de-pends on τ in the Drude-Smith case. The universalityof the relaxation time sets the Drude-Smith model apartfrom other Drude-Lorentz type conduction models; Thefact that the same relaxation time guides the behavior ofboth free and bound charges indicates that the chargesare the same kind of microscopically free charge carriers,undergoing the same microscopic dissipation (scattering)processes. In conclusion, the parameter C denotes thefraction of these microscopically free charges that is con-fined on a larger, mesoscopic scale; τ is the microscopicmomentum relaxation time that determines free and con-fined carrier relaxation as well as confined carrier reso-nance.Many other models, despite starting from different mi-croscopic hypotheses, arrive at versions of the macro-scopic Drude-Lorentz conductivity similar to Drude-Smith. Their adoption is, however, restricted to very spe-cific cases, compared to the apparent universal applica-bility of the Drude-Smith model. Here, we present someof those models in more detail, illustrating their similar-ities and points of contrast with the macroscopic Drude-Smith conduction and the challenge to resolve those dis-tinctions.As first, we consider the case of a localized surfaceplasmon. This is an example of a resonance of micro-scopically free charge carriers which are mesoscopicallyconfined in a conductor, surrounded by a dielectric. Nien-huys and Sundstrm [32] describe the complex conductiv-ity of a surface plasmon localized in a small conductiveparticle ˜ σ P ( ω ) by˜ σ P ( ω ) = (cid:15) ω P τ D − iωτ D ( − ( ω/ω P P ) ) . (15) (cid:15) ω P τ D is the formula for the DC conductivity σ DC ofthe Drude material of the plasmonic particle. The per-mittivity of vacuum (cid:15) is a constant, not a parameter. τ D is the Drude relaxation time, ω P the plasma frequency ofthe conductive material. The particle plasmon resonancefrequency ω P P is proportional to the plasma frequencyby factors depending on the geometry of the particle andthe surrounding dielectric, resulting in a resonance fre-quency that is lower, but of the same order of magnitudeas the plasma frequency ω P of the bulk material. Theplasma frequency hence influences both the magnitudeand resonance frequency. This is the key conceptual dif-ference to the Drude-Smith model, where the relaxationtime plays the most prominent role. The DC-limit of theplasmon conductivity is 0, since all carriers are meso-scopically localized in the particle. Therefore, only if thesurface plasmon frequency coincides with half the relax-ation rate, the plasmon model becomes equivalent to alimiting case of a Drude-Smith model with C = 1. Onemight consider that a combination of isolated particlesand percolation paths leads to an effective medium thatis a combination of free Drude conductivity and a plas-monic part. When interrogating the conductivity at fre-quencies low compared to the relaxation rate and plasmaresonance, this combination of plasmons and free carrierswill be hard to distinguish from the Drude-Smith model[33].We continue with the idea of an effective medium. Wecan construct a very crude effective circuit for a perco-lated medium (fig 2a)): a percolation path representedby resistor R and a discontinuous path where the con-ductive parts are summed up by resistor R and the gapsby capacitance C . The total complex conductance ˜Σ eff will be ˜Σ eff ( ω ) = Σ1 − iωτ D (cid:18) − b − iωτ RC (cid:19) . (16) τ D is the Drude relaxation time of the material in theresistor, Σ is the DC conductance of the resistor elements R and R in parallel, b is R / ( R + R )) and the RC-response time of the capacitive branch τ RC is R · C . Thesimilarities between this and Smiths model are apparent.For frequencies sufficiently lower than 1 / ( τ RC τ D ) the twomodels will again be hard to distinguish. The key differ-ence is the appearance of τ RC , a second time constantunrelated to τ D .Another model with two distinct time constants wasderived by Cocker et al. [34] for the case of micro-scopic, Maxwell-Boltzmann distributed charges confinedby reflecting walls (Fig 2b)). For totally reflecting walls,Cocker et al. derive a conductivity˜ σ Co ( ω ) = ne τ /m − iωτ (cid:18) − − iωτ (cid:19) (17)with τ = (cid:16) τ D + vL (cid:17) − and τ = (cid:16) vL vτ D L +2 vτ d (cid:17) − . L isthe distance between the reflecting walls, v the averagespeed of the electrons (the thermal velocity). The for-mula was derived for a fixed reflectance set to 1; there-fore, a distinct C parameter is lacking. Cocker et al.smore general formula for variable reflectance does con-tain terms which can be interpreted as the C parameterin the Drude-Smith formula [34]. The macroscopic con-ductivity is essentially the same as the effective circuit FIG. 2. Fig. 2 a): Equivalent circuit that may serve asa representation of an effective medium, i.e. for a conduc-tor perforated by holes/scratches. The resulting conductivitymodel appears similar to Smiths model, with the ratio be-tween resistances R and R replacing Smiths C parameter.However, the decay time of the confined charges is given bythe RC-time τ RC of the capacitive branch, not by the mo-mentum relaxation time τ D that determines the inductanceof R . Therefore, two time-constants matter, contrary to thesingle relaxation time in the Drude-Smith conductivityFig. 2 b): Reproduced following Cocker et al. [34], whoconsider (Maxwell-Boltzmann distributed) charge carriers ina box with reflecting walls. This microscopic model is also apossible scenario for the localization of carriers in very specificscenarios. However, just as the equivalent circuit, the relax-ation time of the confined portion of the carriers is differentfrom that of the free ones; here, it depends on the distance L between reflecting walls. Again, we have two time-constantsinstead of the single constant seen in the Drude-Smith model. eq. (16), both conductivities contain two different re-laxation times τ and τ instead of the one relaxationtime of the Drude-Smith model. Obviously, these two-relaxation times models will be hard to distinguish fromDrude-Smith when their two relaxation times are simi-lar to each other. Even for a larger difference betweenthe two relaxation times, the difference to the Drude-Smith result will only show up as a slightly wider peakof the two-relaxation time model and can only be re- C ondu c t i v i t y [ s ] w [1/t] Drude Smith, C=0.9 2 relaxation times Particle Plasmon
Re Im s P = 0.52 s ; w pp =1/ t ; t D = 0.5 ts t = 0.62 s ; C t = 0.84; t = 0.5 t; t = 2 t C ondu c t i v i t y [ s ] w [1/t] Drude Smith, C=0.9 Drude Lorenz
Re Im s D = 0.1 s ; t D = t;w =1.1 /tg = 0.48 /t P =1.04 s a) b) FIG. 3. Fig. 3 a): Comparison between a Drude-Smithmodel for a C parameter of 0.9 and other simple Drude-Lorentz models. The conductivity is displayed in units ofthe σ parameter of the Drude-Smith model, the frequencyin units of the relaxation rate. The line indicating the realconductivity is solid, the imaginary dashed. The Drude Smithcurve is black, a conductivity model with two relaxation timesdiffering by a factor of 4 is shown in red (as in the case ofCockers model or the effective circuit model), and a particleplasmon conductivity in blue. The two relaxation times modelcan only be distinguished from the Drude-Smith model by theslightly wider resonance. The particle plasmon is hardly dis-tinguishable above the resonance frequency.Fig. 3 b): Comparison between a Drude-Lorentz and a Drude-Smith model. The Drude-Lorentz may approximate the con-ductivity of an electron-hole plasma that has partially con-densed into excitons. The specific parameters of this modelapproximate photo-excited ZnO at 30 K. Below the resonancefrequency, Drude-Smith approximates any model comprisingfree and bound charges well. solved close to the resonance frequency ω = √ τ τ − .Similarly, a Drude-Smith response will look very similarto that predicted by the plasmon model above the res-onance frequency, even if a sizeable fraction of the flowof charges is not dammed. These two cases are displayed in fig. 3 a). Fig. 3 b) displays a Drude-Lorentz conduc-tivity, which approximates an electron-hole plasma thathas partially condensed into excitons. The relation be-tween the free carrier relaxation time τ D , the resonancefrequency ω and the damping rate γ was chosen to ap-proximate photo-excited ZnO at 30 K [35]. For ZnO[35], up to ca. 1.5 THz ( ωτ D ≈ . Conclusion:
The Drude Smith model arises when mesoscopically lo-calized obstacles dam the flow of a current of microscop-ically free carriers, which undergo continuous dissipationby microscopic scattering events, in analogy to what hasbeen formulated in the Drude model. This means one candraw the following conclusions when identifying Drude-Smith conductivity behavior:1. Microscopically free carriers exist in the system.2. At least two different classes of scattering processesexist.(a) A microscopic class, that can be treated as aspatially homogeneous background.(b) A mesoscopic class of obstacles localized onthe scale of the mean free path due to back-ground scattering.3. The fraction of charge carriers banked up by thelocalized scatterers in steady-state is representedby the confinement parameter C .4. τ is the velocity relaxation time due to the micro-scopic scattering process.5. σ is the DC-conductivity the system would have ifthe mesoscopic obstacles were removed.6. σ DC = σ (1 − C ) is the DC-limit of the conductiv-ity.Several other models for a combination of bound andfree carriers lead to similar conductivities over most ofthe frequency range. In order to identify the Drude-Smith behavior from experimental conductivity data andjustify the full mesoscopic interpretation, a comparativehypotheses test with those models should be performed.Despite being successfully adopted, thanks to its limitedamount of parameters and their apparent simple mean-ings, the Drude-Smith model had to be considered a merephenomenological expression up to this point. This work,on the other hand, provides an alternative to the originalderivation whose basic premise of preferential backscat-tering must lead to a simple Drude, not Drude-Smithconductivity. Our results show that the Drude-Smithmodel should be considered as the time-dependent ver-sion of Landauers idea of localized scatters damming theflow of microscopically free charge carriers. Acknowledgements
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