Role of Electromagnetic Fluctuations in Organic Electronics

Vladimir M.Mostepanenko|Elena N.Velichko|Maksim Aleksandrovich Baranov

Abstract—Thin organic films find expanding applications in electronic and optoelectronic devices,biotechnology,food packing,and for many other purposes.Among other factors,the stability of films with a thickness below a micrometer is determined by the zero-point and thermal fluctuations of the electromagnetic field.These fluctuations result in the van der Waals and Casimir free energy and forces between a film and a substrate.The fluctuationinduced force may be both attractive and repulsive making the film either more or less stable,respectively.Here,we review recently obtained results for the Casimir free energy of both freestanding and deposited on the metallic and dielectric substrates peptide films.We also perform computations for the free energy of the peptide films deposited on a silica glass substrate in the region of parameters where this free energy vanishes.Possible applications of the obtained results are discussed.

1.Introduction

During the last few years,thin organic films were used in field-effect transistors,light-emitting diodes,solar cells,biomarkers,and many other devices and technologies[1]-[10].This progress has been made possible due to the beneficial mechanical,electrical,and optical properties of these films[11]-[17].In doing so,some applications of protein and peptide films are related to organic microdevices[5],[18]where the thickness of a film may be below a micrometer.In this case,one should take into account the roles of zero-point and thermal fluctuations of the electromagnetic field,which contribute to the free energy of a film and respective pressure.It is well known that electromagnetic fluctuations may play both a positive and negative role in respect to the film stability[19].

It is common knowledge that electromagnetic fluctuations result in the van der Waals and Casimir free energy and forces[20].This effect arises not only for two closely spaced bodies but also for freestanding in vacuum thin films or films deposited on a substrate.Recently it was investigated for thin films made of various inorganic materials including nonmagnetic and magnetic metals and dielectrics[21]-[26].Although the van der Waals and Casimir forces between the organic and,specifically,peptide films have been investigated over a period of years[27]-[30],the effects of electromagnetic fluctuations on the free energy and pressure of organic films became the subject of study only recently[31],[32].In these papers,the fluctuation-induced free energy was computed at room temperature for the modeled peptide films,both freestanding and deposited on dielectric(SiO2)or metallic(Au)substrates.The free energy was investigated as a function of the film thickness and the fraction of water contained in the film.It was shown that for a freestanding film the fluctuation-induced(Casimir)free energy is negative which corresponds to the attractive pressure favorable for the film stability.For the peptide films deposited on an Au substrate,both the free energy and respective pressure turned out to be positive,which corresponds to Casimir repulsion.This makes the peptide coating less stable.The free energy of the peptide films deposited on a dielectric substrate(SiO2)was found to be a non-monotonous function of the film thickness,but the region of the thickness where it vanishes and changes its sign was not investigated.

In this paper,we investigate the free energy of the modeled peptide films with different fractions of water deposited on a SiO2substrate in the region of the film thickness where the Casimir free energy per unit area of the film vanishes or takes the minimum value.In the latter region,the Casimir pressure changes its sign from positive to negative with increasing the film thickness,which makes the film more stable.Possible applications of this result are discussed.

The paper is organized as follows.In Section 2,the Lifshitz formula for the free energy of a film deposited on a substrate is presented,as well as the dielectric permittivities of the used materials along the imaginary frequency axis.Section 3 contains our computational results.In Section 4,several conclusions are formulated.

2.Lifshitz Formula for the Free Energy of a Film Deposited on a Substrate

The fundamental theory of the van der Waals and Casimir forces was developed by Lifshitz[33].In the framework of this theory,the free energy and pressure caused by the electromagnetic fluctuations are expressed via the frequency-dependent dielectric permittivities of interacting bodies.Here we consider a peptide film with the thickness ofabelow a micrometer deposited on a thick dielectric substrate plate which can be treated as a semispace in our computations.It has been shown[34]that,in the computations of the fluctuation-induced free energy and forces,this assumption is valid if the thickness of a substrate exceeds 2 μm.The dielectric permittivities of a peptide film and of a substrate plate are denoted asεf(ω) andεs(ω),respectively.

According to the Lifshitz theory,the fluctuation-induced free energy of a film is expressed via the reflection coefficients of the boundary surfaces between the film and vacuum,and between the film and substrate plate,These reflection coefficients are different for two independent polarizations of the electromagnetic field,transverse magnetic(α=TM)and transverse electric(α=TE).They depend on the magnitude of the projection of the wave vector on the plane of a film,k⊥,and,through the dielectric permittivities,on the pure imaginary Matsubara frequencies(,wherekBis the Boltzmann constant,Tis the temperature,l=0,1,2,…,andis the Planck constant divided by 2π).In fact,these reflection coefficients are the well-known Fresnel coefficients with the only difference that they are calculated along the imaginary frequency axis.The explicit expressions for them are shown as the followings:

where

Another pair of reflection coefficients is given by

where

With these notations,the Lifshitz formula for the Casimir free energy per unit area of a film is written as[20],[24],[33]

where the prime on the summation sign inlmeans that the term withl=0 should be divided by 2.

To calculate the Casimir free energy in(5),one needs to know the dielectric permittivity of a peptide filmεfand that of a substrate plateεsover a wide range along the imaginary frequency axis.For specific materials used in computations,these permittivities are presented in Section 3.Here we only note that the peptide films usually contain some volume fractions of waterΦ.Because of this,the dielectric permittivity of a film,εf,should be obtained as a combination of the dielectric permittivity of water,εw,and of the peptide itself,εp.Taking into account that the peptide molecules are randomly distributed in water and have an irregular shape,the dielectric permittivity of the peptide film can be found from the following combination law[35]:

In the next section,(1)to(6)are used in numerical computations of the fluctuation-induced Casimir free energy of the peptide films.

3.Computational Results for the Casimir Free Energy of Peptide Films

We perform numerical computations for the model peptide which combines the dielectric properties of 18-residue zinc finger peptide found for the frequencies up to the microwave frequency region[36]with that of cyclic tripeptide RGD-4C calculated in the ultraviolet frequency region[37].Using these data,the dielectric permittivity of the model peptide along the imaginary frequency axisεpwas found in[32].It is shown by the bottom line in Fig.1 as a function of the imaginary frequency normalized to the first Matsubara frequency.

In the same figure,the dielectric permittivity of water,which has been much studied in the literature,is shown by the top line[38].At the zero frequency,the values of the dielectric permittivities are equal toεp(0)=15.0 andεw(0)=81.2,respectively.

Fig.1.Dielectric permittivities of water and peptide as functions of the imaginary frequency normalized to the first Matsubara frequency.

Fig.2.Dielectric permittivities of silica glass(the gray line)and peptide films with 0.10,0.25,and 0.40 volume fractions of water (the black lines plotted from bottom to top,respectively)as functions of the imaginary frequency normalized to the first Matsubara frequency.

Fig.3.Casimir free energy per unit area of the peptide films with Φ=0.10,0.25,and 0.40 volume fractions of water as a function of the film thickness by the lines 1,2,and 3,respectively.

Using the dielectric permittivity of peptideεpand that of waterεw,the dielectric permittivities of peptide filmsεfcontaining different volume fractionsΦof water are calculated with the help of(6).The obtained permittivity values for the films withΦ=0.10,0.25,and 0.40 are shown in Fig.2,as the functions of the imaginary frequency normalized to the first Matsubara frequency by three black lines,plotted from bottom to top,respectively.

The gray line in Fig.2 shows the well-known dielectric permittivityεsof a SiO2substrate plate[35].At the zero Matsubara frequency,one hasεs(0)=3.801.

The dielectric permittivitiesεfandεshave been substituted in (1) to (5) in order to compute the fluctuation-induced Casimir free energy per unit area of the peptide films deposited on a SiO2plate as functions of the film thickness.Computations have been performed at room temperatureT=300 K for the peptide films containingΦ=0.10,0.25,and 0.40 volume fractions of water.The computational results are shown in Fig.3 as functions of the film thickness by the lines 1,2,and 3 plotted for the peptide films withΦ=0.10,0.25,and 0.40 volume fractions of water,respectively.

As shown in Fig.3,the Casimir free energy of the peptide films deposited on a SiO2plate is nonmonotonous and changes its sign from positive to negative with increasing the film thickness.For the films with the volume fractions of waterΦ=0.10,0.25,and 0.40,the Casimir free energy takes the zero value for the film thickness equal toa=87.4 nm,84.1 nm,and 75.7 nm,respectively.With increasing the film thickness,the Casimir free energy reaches its minimum value.For the films withΦ=0.10,0.25,and 0.40,this happens for the thickness ofa=135.0 nm,127.5 nm,and 115.0 nm,respectively.

With further increasing the film thickness,the Casimir free energy approaches to zero remaining negative.At the points of minimum free energy,the Casimir pressure,

changes its sign from positive to negative.This means that for the films thinner than 135.0 nm,127.5 nm,and 115.0 nm,the Casimir pressure is repulsive and,thus,makes these films less stable,whereas for thicker films the Casimir pressure is attractive and contributes to their stability.

4.Conclusions

In the foregoing,we have calculated the Casimir free energy of thin peptide films with different volume fractions of water deposited on a silica glass plate.This was done in the framework of fundamental Lifshitz theory describing the van der Waals and Casimir forces.Taking into account that the Casimir free energy may contribute from 5% to 20% of cohesive energy of the thin peptide films[32],the problem of its calculation is not of entirely academic character,but is important for creating new electronic microdevices exploiting thin organic films.

According to our results,the Casimir free energy of the peptide film deposited on a SiO2substrate decreases,remaining positive with increasing the film thickness,takes zero value,reaches some minimum(negative)values,and then increases to zero with further increasing the film thickness.The value of the film thickness,at which the free energy vanishes and reaches its minimum value,depends on the volume fraction of water contained in a peptide film.For the films with minimum Casimir free energy,the Casimir pressure is equal to zero.Thus,for thinner films,electromagnetic fluctuations make these films less stable and contribute to their stability for thicker films.This effect may be useful to ensure the film stability in the next generation of organic-based electronic microdevices with further decreased dimensions.