Casimir force calculations near the insulator-conductor transition in gold thin films
aa r X i v : . [ qu a n t - ph ] M a r Casimir force calculations near the insulator-conductor transitionin gold thin films
R. Esquivel-Sirvent ∗ Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20-364, D.F. 01000, M´exico (Dated: October 29, 2018)
Abstract
We present theoretical calculations of the Casimir force for Au thin films near the insulator-conductor transition that has been observed experimentally. The dielectric function of the Authin films is described by the Drude-Smith model. The parameters needed to model the dielectricfunction such as the relaxation time, plasma frequency and the backscattering constant dependon the thickness of the film. The Casimir force decreases as the film thickness decreases until itreaches a minimum after which the force increases again. The minimum of the force coincides withthe critical film thickness where a percolation conductor-insulator occurs. ∗ Corresponding author. Email:raul@fisica.unam.mx . INTRODUCTION The attractive force between two parallel neutral plates made of a perfect conductor isknown as the Casimir force. This force is explained in terms of the radiation pressure dueto quantum vacuum fluctuations of the quantized electromagnetic field [1] when boundariesare present. Casimir’s original derivation [2] substracted the vacuum energy between theplates, as obtained from the sum of allowed vacuum modes, from the vacuum energy whenthe plates were absent. The resulting force per unit area is given by, F = − ~ cπ L , (1)where L is the separation between the plates.In 1956 Lifshitz [3] generalized Casimir’s results to real materials characterized by their di-electric function. In this theory, the dissipative effects associated with the radiation reactionof the elementary atomic dipoles composing the dielectric is balanced by the fluctuating vac-uum field in accordance with the fluctuation-dissipation theorem. If we consider two plates i = 1 , F = ~ c π Z ∞ QdQ Z q> dk k q ( G s + G p ) , (2)where G s = ( r − s r − s exp (2 kL ) − − and G p = ( r − p r − p exp (2 kL ) − − . In these expres-sions, the factors r p,s are the reflectivities for p or s polarized light , Q is the wavevectorcomponent along the plates, q = ω/c and k = p q + Q .Experimentally the Casimir force has been measured by several groups [4, 5, 6, 7, 8, 9, 10,11] for separations between the plates as low as 20 nm [12], using different techniques such asforce balances, atomic force microscopes and micro mechanical balances. The experimentalresults verify the Lifshitz theory to a high precision. These experiments have also pointedout the importance that Casimir forces can have in micro and nano systems. In particularthe role it plays in stiction has been considered by several authors [13, 14, 15, 16, 17, 18, 19].The Lifshitz theory [3], requires the knowledge of the dielectric function of the materials.An important issue is determining which is the correct dielectric function that is consistent indescribing the optical properties of the materials and the measurements of the Casimir force.Although it may be thought that the problem is straight-forward, controversial results havebeen reported. The use of of the Drude model in Lifshitz theory has been argued to violate2ernst’s heat theorem, while the plasma model presents no problem at all [20]. This has beendisputed by several authors . For example, spatial dispersion in the calculation of the Casimirforce [21, 22] has been proposed to solve the problem or a non-vanishing damping constantin the Drude model at zero temperature . Recently, an interesting proposal suggestingJohnson noise as a possible finite-size effect to be included to solve this controversy wassuggested for a system made of two parallel wires [23]. Also, Ellingsen [24] has establishedthe criteria for the non violation of Nernst’s heat theorem working on the real axis ratherthan the imaginary axis. At this time the apparent violation of thermodynamics remainsan interesting issue under discussion in the literature [25, 26].Several experiments and theoretical studies have been done to study how the differentdielectric functions influence the value of the Casimir force. For example, an early attemptwas made by Iannuzzi that measured the Casimir force between hydrogen switchable mirrors(HSM) [27]. HSM’s change their dielectric function dramatically when immersed in anhydrogen rich environment. However, this has been shown to have little effect on the Casimirforce as the optical properties of the mirrors changed only in a narrow frequency range. Theseresults were refined recently, describing the conditions needed to observe changes in the forcebetween switchable mirrors [28]. Besides metals, semiconductors such as Si has been used[29, 30] and light modulation of the Casimir force has been measured [31, 32, 33]. In thiscase, light changes the carrier density of the semiconductor making it more or less metallic,thus changing the magnitude of the Casimir force. Changing the magnitude of the forceby a suitable choice of materials can be achieved using for example silicon based aerogels[34] that have the lowest index of refraction of any solid or with temperature changes ofthe dielectric function of materials such as V O that undergo a martensitic transition fromdielectric to conductor. [35, 36, 37]. Recently, the possibility of using metamaterials has alsobeen considered. The effect of having a negative index of refraction is to have a repulsiveCasimir force [38, 39].Up to now, gold is the most common metal to work with in Casimir force experiments.When comparing with theoretical calculations, tabulated data for the dielectric function isused [40]. The films are usually thick enough to be able to use the assumption that themeasured properties are close to those of bulk Au. At higher frequencies where quantumeffects come into play, the dielectric function of Au can be described by an analytic modelusing a phenomenological approach that adds to the Drude model a Lorentz type response3unction [41]. Another approach based on time dependent density functional theory hasbeen used to describe the intraband transitions of Au [42]. Tabulated dielectric data for Audepends on the conditions of the sample preparation [43] and the Drude parameters obtainedfrom the extrapolation to low frequencies gives different values for the plasma frequency anddamping parameter. This implies variations of up to 5% in the calculation of the Casimirforce.Another important parameter affecting the value of the Casimir force is the film thickness[12]. This was shown experimentally by Iannuzzi [44, 45] when it was demonstrated thatthe Casimir attraction between a metallic plate and a metal coated sphere depended on thethickness of the coating. Most recently, in reference [46] different Au samples were preparedunder similar conditions with thicknesses ranging from 120 nm to 400 nm. From measuredellipsometry data it was verified that the plasma frequency varies from 6.8 eV to 8.4 eV forthis set of particular samples, changing the Casimir force a few percent. Also, the influenceof slab thicknesses on the Casimir force was studied by Pirozhenko [37] for slabs of finitethickness for several materials such as V O and doped silicon taking into account differentcarrier concentrations.The reduction of size can significantly change the physical parameters of a system such asthe Debye temperature or the conductivity. The case of clusters is well known. For example,a metallic cluster can go from being an insulator to a conductor depending on the size of thecluster [47]. A similar phenomenon is observed in metallic thin films. As the thickness ofthe film approaches the mean free path, the Debye temperature and the conductivity showa sharp decrease in its values. This was shown experimentally by Kastle [48] with Au filmswhose thickness varied from 2 nm to 70 nm . Indeed, a conductor-insulator transition isobserved as a function of film thickness in Au [49].In this paper, we study the Casimir between Au thin films near the insulator-conductortransition. The dielectric function for the Au films is described by the Drude-Smith modeland the parameters for this model are obtained from reported experimental data. II. DIELECTRIC FUNCTION OF AU THIN FILMS
Thin films grown by thermal deposition, present anomalous behaviors in many of theirphysical properties, deviating from their bulk values with varying film thicknesses. In the4ase of the conductivity, as the film thickness decreases it becomes less conductive until itreaches a point where it becomes insulating. The reason for this change can be explainedas a percolation transition. During deposition, disordered gold islands or clusters startforming on the substrate confining the conduction electrons to these islands. As the depo-sition continues, the fraction of island covering the substrate increases making it possiblefor electrons to hop between islands and increasing the conductivity. Finally, more of theseclusters are connected until a percolation threshold is reached [49]. After this percolationtransition, the conductivity of the film will increase with increasing thickness until the bulkvalues are reached. Experimentally this has been measured by Walther [49] using terahertztime-domain spectroscopy.An analytic expression of the dielectric function that describes the Au films near thistransition is the Smith-Drude model. This is a classical correction to the Drude model,introduced by Smith [50] to describe the conductivity and dielectric properties of disorderedmetals, liquid metals and recently the metal-insulator transition of thin Au films [49]. Thismodel assumes that the electrons in a metal are scattered with a probability that followsa Poisson distribution. In this model, the current as a function of time after an impulseelectric field is applied is ~J ( t ) = ~J (0) exp ( − t/τ ) f ( n ), where f ( n ) is a function that has theinformation on the probability that an electron was scattered n times in the time interval[0 , t ]and τ is the relaxation time. For the Drude-Smith model we have that f ( n ) = 1 + ∞ X n =1 c n n ! (cid:18) tτ (cid:19) . (3)After a scattering event, the electron will retain only part of its initial velocity. This isrepresented by the value of the coefficients c n . The Drude model assumes that after eachcollision the electron has a velocity not related with the velocity before the collision [51]thus, c n = 0 for all n .For a material with a plasma frequency ω p and damping constant γ = 1 /τ , the conduc-tivity for the Drude-Smith model is σ ( ω ) = ω p π ( γ − iω ) " ∞ X n =1 γ n c n ( γ − iω ) n . (4)The behavior predicted by the Drude-Smith model differs from that of the original Drudemodel at low frequencies. Let σ D be the conductivity predicted by the Drude model. At5ero frequency the DC current for the Drude-Smith model is σ (0) = σ D (0)(1 + c c − ǫ ( ω ) = 1 − ω p ω ( ω + iγ ) " ∞ X n =1 i n γ n c n ( ω + iγ ) n . (5)Typically only the term n = 1 is enough to describe experimental results. If c = 0 theDrude result is obtained and the case where c = − c = − . c = − .
94 corresponds to the quasicrystal Al . Cu . F e [53].In the case of the Au thin films, that we consider in this work, we use Walther’s [49]measurements using teraHertz spectroscopy, that show the crossing from insulator to con-ductor as the film thickness increases. The samples were prepared by thermal evaporation ofAu and deposited on a Si substrate at room temperature and in vacuum. The conductivitymeasurements were best described by a Drude-Smith model. For films 20 nm thick the bulkvalues for the plasma frequency and damping parameter of Au were recovered. Decreasingthe film thickness also reduces the plasma frequency until a critical thickness of d c = 6 . nm when the conductor-insulator transition is observed. In table I we summarized the resultsof Walther [49] for some representative film thicknesses, as well as the value of the constant c used to describe the dielectric function. The damping parameter γ does not change exceptclose to the critical film thickness d c .In Figure 1 we present the plot of the Drude-Smith dielectric function ǫ ( iω ) as a functionof frequency for different Au film thicknesses. The vertical axis is normalized to the function ǫ ( iω ) D that is the Drude dielectric function ( c = 0) for bulk Au with the parameters givenby Walther [49]. We evaluated the dielectric function after rotating the frequency axis tothe complex plane ω → iω , that is used in the calculation of the Casimir force using Lifshitzformula [3]. The three top curves in Fig. 1 show that as the thickness decreases the dielectricfunction decreases since, as seen from Table 1, the plasma frequency is also decreasing.However, at the critical thickness d c = 6 . nm , the dielectric function is lower than thecorresponding curve for the d = 4 nm curve in this frequency range. The lower curve (opencircles) corresponds to the film thickness where the insulator-conductor transition occurs.For lower frequencies, we present in Figure 2 again the dielectric function for film thickness6f d = 6 . nm and d = 4 nm . For the latter the dielectric function decreases and goes to zeromeaning a zero DC conductivity since c = − ω = 10 s − .Now we consider the effect of the film thickness on the Casimir force, between a systemmade of a half space of Au and a Au thin film deposited on a Si substrate. The reductionfactor η defined from Eq. (1) and Eq. (2) as η = FF C is a convenient way to see the behaviorof the Casimir force. To use Eq. (2) we calculate the reflection coefficients for a Au thinfilm of thickness d on top of a substrate given by r p,s = r ps + r ps e − δ r ps r ps e − δ (6)where r i,jp,s are the Fresnel coefficients between material i and j where the subindex 0 standsfor vacuum, 1 is the Au thin film and 2 is the substrate. The optical length is defined as[37, 52] δ = dc p ω ( ǫ ( iω ) −
1) + c k . (7)In Figure 3, we plot the reduction factor as a function of separation using the parametersof the Au film from table 1. The dielectric function of the Si substrate is a Lorentz type os-cillator model [29]. As expected, the Casimir force decreases with decreasing film thickness.However, this trend changes after the critical film thickness of d = 6 . nm where the perco-lation transition occurs. At this critical distance the plasma frequency attains a minimumvalue, as seen in table 1. After the critical thickness the force increases again although thefilm thickness is decreasing. The behavior at the distance d c where we see a sudden increasein the relaxation time and a decrease of the plasma frequency is consistent with the singularbehavior in the dielectric constant in metal-dielectric percolation transitions [54, 55]. Thisbehavior is seen more clearly in Figure 3, where we show the reduction factor as a functionof film thicknesses when keeping the separation between the plates fixed at L = 400 nm .Following previous works by Piroshenko and Lambrecht [29, 37], we further analyze theeffect of the thin film using the optical length or phase factor δ of the Au thin film 7. Inparticular, the difference between using the Drude, Plasma or Drude-Smith model can beseen in the behavior of the optical length with frequency. For example at the critical distance d c we calculate the optical length assuming a simple plasma model δ p , the Drude model δ d and for the Drude-Smith model, and calculate the percent difference ∆ = 100 × | δ − δ p | /δ and ∆ = 100 × | δ − δ D | /δ . This is shown in Figure 5, where we plotted the percent difference7s a function of frequency. As expected, for high enough frequencies the plasma model isenough to describe the dielectric function of the Au thin film, if interband transitions areignored. However, at low frequencies the percent difference between the phase factor whenthe Drude or plasma model are used becomes larger in particular for the plasma model. Theoptical length was calculated for the particular value ck/ω = 1 as in reference ([37]). III. CONCLUSIONS
In this work we studied the effect that the film thickness has on the Casimir force forfilms near the critical thickness where a conductor-insulator transition occurs. To describethe dielectric function of the Au thin film we use the Drude-Smith model that describesthe experimental data available in the literature. The parameters used in our calculationsare best-fit parameters from experimental measurements. For thick films the force decreaseswith decreasing film thickness until a critical thickness is reached after which the Casimirforce increases even with decreasing film thickness. The minimum value of the Casimirforce is due to a decrease in the plasma frequency and an increase in the relaxation timewhen the metal-insulator transition occurs in the film. The use of Au thin films has theadvantage that the growing techniques are well known, and experimentalist working onCasimir force measurements use gold coated spheres and substrates routinely. Dependingon the film thickness the dielectric function deviates from the typical Drude and plasmamodels in accordance to the Drude-Smith model. As the thickness decreases the value ofthe backscattering constant c decreases changing the value of the DC conductivity. Thiscould lead to experimental settings that further explore the role of DC conductivities andfinite temperature controversies. Acknowledgments
The author wish to acknowledge the helpful comments of L. Reyes-Galindo. Partialsupport for this work comes from DGAPA-UNAM project 113208.8
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20 13.19 18 0 115 10.05 19 0 110 6.28 19 0 16.4 1.25 80 -0.7 0.34 1.88 20 -1 0TABLE I: Best fit parameters for the Drude-Smith model taken from Ref. ([49]). The last columnshows the ratio of the DC conductivity for the Drude-Smith model to the expected DC Drudemodel conductivity. −1 −2 −1 ω / ω ε / ε D d=20 nmd=15 nmd=10 nmd=6.4 nmd=4 nm FIG. 1: Dielectric function for thin films using the Drude-Smith model as a function of frequency.The parameters for each film thickness were taken from experimental data. The dielectric functionis compared to the bulk Drude dielectric function reported in the experimental data of Walther[49]. In all figures we take ω = 10 /s . ω / ω ε / ε D d=6.4 nmd=4 nm FIG. 2: In this figure we plot again the dielectric function for low frequencies of the two lowercurves of Fig.1. This is for d = 6 . nm and d = 4 nm . The dielectric function for the thinner filmdecreases We see a cross over since for d = 4 nm the backscattering constant is c = − ǫ ( iω ) →
100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.9 separation (nm) η d=20 nmd=15 nmd=10 nmd=6.4 nmd=5 nmd=4 nm FIG. 3: Reduction factor as a function of frequency for different Au film thicknesses. As the filmthickness decreases the Casimir force also decreases until it reaches the critical thickness d = 6 . nm ,after which the force increases even with decreasing film thickness. η L=400 nmd c FIG. 4: Reduction factor as a function of the Au film thickness. The reduction factor is calculatedassuming a separation between the plates of L = 400 nm and using the optical data for the Aufilms from Ref.( [49]). At the thickness of d c the percolation transition occurs, and the Casimirforce attains a minimum. −3 −2 −1 ω / ω ∆ δ % DrudePlasma
FIG. 5: Percent difference between the optical length for the Drude-Smith model and the Drudemodel (solid line ) and the plasma model (crosses). The curves are calculated for a Au film ofthickness d c . The low frequency behavior shows strong differences if the plasma or Drude modelare used. In this figure ω = 10 /s and ck/ω = 1.= 1.