### University of Groningen

### Applied Physics

### Bachelor thesis

### Qualitative analysis of the Drude and Plasma model for calculating

### the Casimir Force in a variety of material configurations

### Author:

### Steven van der Veeke

### Supervisors:

### Prof. dr. George.

### Palasantzas Prof. dr. ir. Bart J. Kooi

### September 24, 2014

Abstract

The Casimir force for configurations involving Gold, SiC and both phases of the phase change material AIST have been modeled using the Drude, Plasma and no model for the extrapolation of the dielectric function in the experimental inaccessible low range. Computations in the range 1nm - 200nm have been done by using the Lifshitz theory at room temperature and using ellipsiometrically obtained data. The theory for extrapolation using the models for all the materi- als are discussed and computed. The results of the force calculations have been put in perspective to each other. As a result we conclude that computations that involve gold approximate the theoretical Casimir force best, followed by the phase change material. It is clear that an extrapolation model has to be used to calculate the Casimir force and the Plasma models results in a signif- icant higher force compared to the Drude model. It is concluded from spring system simulations that extrapolation models used for the dielectric function to calculate the Casimir force have to be tested and developed to approximate the actual Casimir force with great precision before the usage of this force in MEMS and NEMS can be exploited.

## Contents

1 Introduction 2

2 Theory 4

2.1 Casimir Force . . . 4

2.2 Lifshitz theory . . . 5

2.3 Dielectric function . . . 7

2.4 Ellipsometry[1] . . . 8

2.5 Extrapolation models . . . 9

2.5.1 Drude model . . . 10

2.5.2 Plasma model . . . 11

3 Materials 12 3.1 Gold [2] . . . 12

3.2 Silicon-Carbide [3] . . . 14

3.3 Phase Change Materials . . . 15

3.3.1 Amorphous phase . . . 16

3.3.2 Crystalline phase . . . 16

4 Results 18 4.1 Dielectric functions . . . 18

4.1.1 Dielectric function for Gold . . . 18

4.1.2 Dielectric function for SiC . . . 19

4.1.3 Dielectric function for PCM (Amorphous) . . . 19

4.1.4 Dielectric function for PCM (Crystalline) . . . 20

4.1.5 Summary . . . 20

4.2 Force simulations . . . 22

4.2.1 Force per Material . . . 22

4.2.2 Comparison of the models . . . 23

4.2.3 Special configurations . . . 24

4.2.4 Force Summary . . . 25

4.3 Further advances . . . 28

5 Conclusions 32

6 Acknowledgments 34

### Chapter 1

## Introduction

In 1948 the Dutch scientist Hendrik Casimir published a paper that predicted a force between two perfectly reflecting plates due to perturbation of quantum fluctuations in the electromagnetic field. Casimir used the derivation of the van der Waals force as a starting point but shifted the emphasis from action at a distance between molecules to local action of fields. While the van der Waals force was described using fluctuating dipoles, Casimir considered it using the fluctuating electric fields as a basis. This change in viewpoint opened a new array of phenomena, which today we refer to as the ’Casimir effect’. [4]

The result that Casimir found was very small and it is generally overruled by the electrostatic repulsion between the plates. From this it is concluded that the Casimir effect only has a significant contribution to the force between plates below the µm regime.

However, today Micro Electo Mechanical Systems (MEMS) have become com- mon in many technological applications (accelerometers, biochips for detection of biological agents and DNA amplifications, actuators in ultrasonic an optical imaging varactors and many more). The ongoing reduction in size lead to Nano Electro Mechanical Systems (NEMS). Both systems are characterized by having compliant moving parts such as cantilevers, gears, ratchets etc. Typically these devices are actuated by electrostatic forces because these are easy to control by inducing potential differences.

These devices are becoming so small that forces such as capillary, van der Waals and Casimir Forces can play a role in the operation of these devices. The precise influence of these forces is determined by several factors, for example the scale of the device, the humidity or the temperature [5]. The unknown interaction of these factors onto micro-devices puts a limit on the use of these devices. The design of a MEMS or NEMS should be designed in such a way that they take into account or even better make use of the above mentioned forces.

Those forces cause a positional instability of the moving parts of a MEM- S/NEMS, known as pull-in or jump to contact [6]. This failure occurs when compliant surfaces adhere to each other when an elastic restoring force is over- come by an interfacial force between the surfaces. There have been many ex-

periments done to determine the effects of the forces on the MEMS and NEMS and it was shown by Chan [7] that the Casimir force is indeed strong enough to actuate MEMS and can even drive them in the nonlinear regime which lead to a failure of control. [8]

The last few years there have been published many papers that deal with Casimir forces in MEMS and NEMS. A significant amount of these papers study the control of the force and the application possibilities. The expectancy for the coming years is that the understanding as well as the application of the Casimir force will increase significantly and we can expect the exploitation of this force in future technological devices. [9]

### Chapter 2

## Theory

### 2.1 Casimir Force

In the article that Casimir published he considered two parallel perfectly re- flecting plates separated by a distance a [10].

Figure 2.1: Schematic drawing of the Casimir force [11].

The average electric field is given by: hEi = 0 if there is no charge on the plates.

However this does not have to be the same for the square of the electromagnetic fields:

hE^{2}i 6= 0, hB^{2}i 6= 0 (2.1)

and because of this the expectation value of the energy is not zero either:

H = 1 2

Z

(dr)(E^{2}(r) + B^{2}(r)) 6= 0 (2.2)
Because the expectation value of the energy is not zero this causes a measurable
force on the plates. Casimir did the derivation of this force and came to the
conclusion that in the case of two perfectly reflecting parallel plates this is an
attractive force. The result of Casimir’s derivation for the energy per unit area
and the force per unit area are:

E = − π^{2}

720a^{3}~c, F = − d

daE = − π^{2}

240a^{4}~c (2.3)

That equation 2.3 has a dependence on a is completely determined by dimen- sional considerations. If we fill in the values in the above formula we get the following result for the Casimir Force:

F = −8.11MeV fm a^{−4} = −1.30 × 10^{−27}N m^{2}a^{−4} (2.4)
This result is very small and it is mentioned before that it is generally overruled
by the electrostatic repulsion between the plates. It is overruled if each plate
has an excess electron density n greater than 1/a^{2}. In the µm regime and below,
the Casimir Force can not be neglected.

### 2.2 Lifshitz theory

Real materials are not perfect reflectors and those material properties have be taken in account. The widely used theory of dispersive forces between macro- scopic bodies was developed by Lifshitz in 1955. It makes use of the theory of fluctuating electromagnetic fields [12]. Dispersive forces can not be calculated when starting from microscopic interactions between separate atoms due to the presence of other atoms and their associated fields. Electrons in a field do not keep their position and therefore it is not possible to calculate the individual distortion in the atoms in a lattice. When using the macroscopic point of view where the lattice is seen as a continuous media one can give a description of the dissipation by using the fluctuation dissipation theorem. These fluctuating currents find their origin in fluctuations in the electric polarization.

The Fluctuation Dissipation Theorem (FDT) [13, 14, 15] states that in thermal equilibrium the correlation of fluctuating currents are related with the dissipa- tion in the medium according to the following formula:

hJα(ω, r)J_{β}^{∗}(ω^{0}, r^{0})i = ωε^{00}(ω)

~ω

2 + ~ω

e^{~ω/kT} − 1

×δ(ω−ω^{0})δ(r−r^{0})δαβ (2.5)
with α, β = x, y, z and ε^{00}(ω) = Im(ε(ω)). From this one can see that the
dissipation is proportional to the imaginary part of the dielectric function. This
relation also states that the dispersive force is closely related to the dissipation
in the materials of interacting bodies. In equation 2.5 the contributions of zero-
point energy and the thermal fluctuations are explicitly separated within the
large brackets. The fluctuating currents are the source of electromagnetic field,
which in turn obeys the Maxwell equations. Solutions of the Maxwell equations
can be expressed through the Green Functions, with Gαβas the Green tensor:

E_{α}(ω, r) = i
ω

Z

dr^{0}G_{αβ}(ω, r, r^{0})J_{β}(ω, r^{0}) (2.6)
Using the Green’s relation 2.6 together with the fluctuating dissipation theorem
from equation 2.5 and exploit some of the general properties of the Green tensor,
we end up with the correlations function for components of the electric field
expressed by the Green functions:

hE_{α}(ω, r)E^{∗}_{β}(ω^{0}, r^{0})i = 2π~ coth

~ω 2kT

Im (G_{αβ}(ω, r, r^{0})δ(ω − ω^{0})) (2.7)

Using the Maxwell tensor to give an expression for the forces acting on two bodies interacting via the long wave fluctuations that are placed in vacuum:

Tαβ(t, r) = ε0M_{αβ}^{E}(t, r) + M_{αβ}^{H}(t, r) (2.8)
where M_{αβ}^{E} is defined as:

M_{αβ}^{E} (t, r) = 1
4π

Eα(t, r)Eβ(t, r) −δαβ

2 E^{2}(t, r)

(2.9)
and M_{αβ}^{H} is similar defined with the change E → H Then if we want to find
the force due to the fluctuations, the tensor in equation 2.8 must be averaged
over the state of the electromagnetic field. Averaged quadratic combinations of
the field can be expressed via the correlation functions. When using this the
averaged stress tensor can be represented as:

hTαβi =~Λαβρσ

16π^{2}

∞

Z

−∞

dω coth ~ω

2kTIm[Gρσ(ω, r, r^{0})]|r=r^{0} (2.10)
where the tensor Λαβρσ has the following form:

Λαβρσ =

δαγδβδ−1 2δαβδγδ

×

ε0δγρδδσ+ c^{2}

ω^{2}γµρδνσ∂µ∂_{ν}^{0}

(2.11)
From inspecting equation 2.10 one can conclude that the Green function of the
systems has to be known to find the dispersive forces acting on the bodies. We
can find the explicit form of the Green function for two parallel plates inter-
acting via the long wavelength fluctuations. The most general configuration
that can be solved consists of two slabs separated by a gap d and each slab is
composed of different material with dielectric function ε_{1}(ω) and ε_{2}(ω) . The
gap is filled with a material that possesses the dielectric function ε_{0}(ω).

The force (pressure) acting on the opposite body is calculated as hTzzi from
equation 2.10 taken at z = z^{0} → d/2. The limit is taken from inside the gap
where the stress tensor is well defined. The result of this is:

F (T, d) = ~
2π^{2}

∞

Z

0

dω coth

~ω 2kT

× Re

∞

Z

0

dQ Q k0g(Q, ω)

(2.12)

Where the z-component in the gap is k0 = pε0ω^{2}/c^{2}− Q^{2} and the function
g(Q, ω) is defined as:

g(Q, ω) = X

ν=s,p

∞

X

m=1

(r_{1}^{ν}r^{ν}_{2})^{n}e^{2imk}^{0}^{d} (2.13)
The r^{ν}_{1,2} are the reflection coefficients from the inner surfaces of the plates.

The subscript ν = s, p stands for the polarization components for Transverse
Electric (TE, ν = s) or for Transverse Magnetic (TM, ν = p). The reflection
coefficients enter the Lifshitz formula as the Fresnel reflection coefficients. In
the described configurations these reflection coefficients are described in the gap
via the z-components for the i^{th}body k_{i} as:

r_{i}^{s}= k0− ki

k0+ ki

, r_{i}^{p}=εik0− ε0ki

εik0+ ε0ki

(2.14) Where

k0= r

ε0(ω)ω^{2}

c^{2} − Q^{2}, ki=
r

εi(ω)ω^{2}

c^{2} − Q^{2} (2.15)
There are two main things we can conclude from equation 2.12. First the
frequency dependent factor (~ω/2kT ) clearly originates from the Fluctuation-
Dissipation Theorem and secondly, equation 2.12 represents the Lifshitz formula
as the integral over real frequencies [14]. However this is not convenient in many
applications of the formula since the integrand contains one fast oscillating fac-
tor e^{ik}^{0}^{d}.

By taking the contour rotation in the frequency complex plane we can avoid the problem of the fast oscillating factor in the integral [16]. Using this technique the force can be expresses as the integral over imaginary frequencies. Contribution to the integral gives only the poles of coth(~ω/2kT ) located at:

ωn = iζn = i2πkT

~

n, n = 0, 1, 2, ... (2.16) The imaginary frequency is represented by ζ. When included the Lifshitz for- mula takes the following form:

F (T, d) = kT π

∞

X

n=0

’

∞

Z

0

dQ Q |k0| g(Q, iζ), (2.17)

Where the prime at the sum means that n = 0 has to be taken in account with
weight 1/2. The transformation causes |k0| =pε0ζ_{n}^{2}/c^{2}+ Q^{2}and the function
g(Q, iζ_{n}) is not oscillating any more. The same transformation has to be applied
to the reflection coefficients. They become:

k_{0}= i
r

ε_{0}(iζ)ζ^{2}

c^{2} − Q^{2}, k_{i}= i
r

ε_{i}(iζ)ζ^{2}

c^{2} − Q^{2} (2.18)
Any wave decays inside a material and thus the general sign is defined by the
condition that Im(k0) > 0.

### 2.3 Dielectric function

The reflection coefficients that have been defined in the previous section will depend on the dielectric function, or more specific: on the dielectric function at imaginary frequencies ε(iζ). The dielectric function describes the absorption and polarization properties of a material versus the wavelength, frequency or energy. Real materials are imperfect conductors and thus imperfect dielectrics [17]. The dielectric properties of a material strongly influence the Casimir force.

This is especially interesting at very small separations where the Casimir force is, according to the theory, at its largest. For precise calculations of the Casimir force, the frequency dependent dielectric functions have to be known.

Theoretically it is possible to calculate the optical properties of perfect crystals using quantum physics [18]. The semiconductor industry is capable of producing single crystal samples with a very low rate of defects and impurities [19], but today’s technology does not allow the existence of perfect crystals at macro- scopic length scales. The typical samples used for Casimir force experiments are fabricated using vacuum deposition techniques [2, 11, 17] and have local trenches up to 5 nm deep and the amount of defects in the samples can still be very large [2].

It has been shown that the variation in the sample that leads to a variation in the frequency dielectric function can lead to an uncertainty in the Casimir force calculation up to 8% by using the Lifshitz theory.

The optical properties of a material are characterizes by two measurable quan- tities: the index of refraction n(λ) and the extinction coefficient k(λ). Both depend on the wavelength of the electromagnetic radiation. The complex index of refraction is defined by:

ñ = n(λ) + ik(λ) (2.19)

In this equation the real part is related to the phase velocity inside a medium by ν = c/n where c is the speed of light inside that medium. The imaginary part of the equation is related to the adsorbed part of the electromagnetic radiation that travels through the medium. The real and imaginary part of the frequency dependent dielectric function are determined with the following equations[1]:

ε^{0}= n^{2}− k^{2} and ε^{00}= 2nk (2.20)
where ε^{0} represents the real part and ε^{00} represents the imaginary part of the
dielectric function. The index of refraction, the extinction coefficient, the real
part of the dielectric function and the imaginary part of the dielectric function
obey the Kramers-Kroning relation [2, 17, 20]. And as a result of this relation
the dielectric function can be expressed at imaginary frequencies by [14]:

ε(iζ) = 1 + 2 π

∞

Z

0

dωωε^{00}(ω)

ω^{2}+ ζ^{2} (2.21)

From the previous section it is clear that equation 2.21 is sufficient to calculate the Casimir force. Note that the frequency dependent dielectric function here has to be known to calculate imaginary dielectric function. This is largely acquired by using a technique called ellipsometry.

### 2.4 Ellipsometry[1]

Ellipsometry is a nonperturbing optical technique to determine surface charac- teristics of samples of interest. Ellipsometry exploits the polarization charac- teristics of electromagnetic radiation. It measures the change in the state of polarization of the light after it has been reflected of a surface, interface, thin film or nanostructured material.

An ellipsometry measurement is the description of the change in polarization.

From a ellipsometric measurement you typically get two results. As the polar- ized electromagnetic radiation is reflected from the sample it will we expressed as two parameters for each wavelength-angle combination: Psi (Ψ) and delta (∆). Where Ψ represents the change in amplitude of the reflected light and ∆ represents the phase change of the reflected light. Above values can be related to the ratio of the complex Fresnel reflection coefficients rsand rpfor the s- and p- polarized light, respectively:

ρ = rp

rs

= tan(Ψ)e^{i∆} (2.22)

The dielectric function for a simple homogeneous surface can be written as an expression of the ellipsometric variables as:

ε or hεi = ε^{0}+ iε^{00}= sin^{2}ϕ

1 + tan^{2}ϕ 1 − ρ
1 + ρ

(2.23) When we separate the real and imaginary part of the last part of the above formula we get:

ε^{0}= sin^{2}ϕ

1 + (cos^{2}(2Ψ) − sin^{2}(2Ψ)sin^{2}(∆))tan^{2}ϕ
(1 + sin(2Ψ)cos(∆))^{2}

(2.24) and

ε^{00}=−2cos(2Ψ)sin^{2}(2Ψ)sin(∆)sin^{2}(ϕ)tan^{2}(ϕ)

(1 + sin(2Ψ)cos(∆))^{2} (2.25)

It is clear from the above formula that ε^{00}can be computed from the ellipsomet-
rical data. This can be used to compute the dielectric function at imaginary
frequencies as the equation 2.21 prescribes.

Limitations

The ellipsometrical technique to find the imaginary dielectric function has some limitations. The most conventional ellipsometric have a measurement range for the wavelength of the electromagnetic radiation from about ∼130 nm up to

∼30 µm [2]. Ellipsometry in the range for a wavelength greater than ∼30 µm is difficult because it needs very intense sources that are still under development [21]. On the other hand is it also difficult for dielectric data in the far UV regime (below ∼130 nm) because this needs high energetic photons that have to be produced by a synchrotron[22]. Because there is limited available data for the imaginary part of the dielectric function it needs to be extrapolated outside the measured regions. The height of the contribution to the dielectric function in those regimes depends on the material and will be discussed in the following sections.

### 2.5 Extrapolation models

Ellipsometry can measure the dielectric function in a wide range of frequencies, but not in the whole range. Here the dielectric function has to be extrapolated.

Especially the low frequencies, down to ω = 0 are of interest because there the dielectric function can play an important role for the force calculations,

depending on the material that is studied. There are two widely accepted models to extrapolate the dielectric function at low frequencies: The Drude model and the Plasma model.

### 2.5.1 Drude model

Paul Drude designed in 1900 a model for the dielectric function using Maxwell equations:

∆ × H = 4π c j − iω

cD, j = σE, D = ε0E (2.26)

This leads to:

∆ × H = −iω c

ε0+ i4πσ ω

E (2.27)

and from this we can extract:

ε(ω) = ε0+ i4πσ

ω , ω → 0 (2.28)

Approximating the dielectric function in the low range:

ε^{00}(ω) → 4πσ

ω 1 when ω → 0 (2.29)

From this we can conclude that indeed the dielectric function at low frequencies is important for the force calculation. The dielectric function at low frequencies can be described by the following function [23]:

ε(ω) = 1 − ω_{p}^{2}

ω(ω + iω_{τ}) (2.30)

This function is defined by two parameters; the plasma frequency ωp and the relaxation frequency ωτ. Both frequencies are determined by using the relations [24]:

ω^{2}_{p}= N e^{2}

ε0m^{∗}_{e}, σ = ω_{p}^{2}
4πωτ

(2.31)
where N is the number of conduction electrons per unit volume, e is the charge,
and m^{∗}_{e} is the effective mass of electron.

Separating the real and imaginary part of the Drude equation 2.30 results in:

ε^{0}(ω) = 1 − ω_{p}^{2}

ω^{2}+ ω^{2}_{τ}, ε^{00}(ω) = ω^{2}_{p}ωτ

ω(ω^{2}+ ω_{τ}^{2}) (2.32)
The Drude model is in agreement with the Fluctuating Dissipation Theorem
and has a dissipative term in the result.

### 2.5.2 Plasma model

The plasma model ignores the FDT and is referred to as the non-dissipative model to extrapolate the dielectric function for low frequencies. It does not have a rigid theoretic basis other than that it fits the actual measurements nicely. The Plasma model has the following form [25]:

ε(ω) = 1 −ω_{p}^{2}

ω^{2} (2.33)

And this result to a contribution to the dielectric function of:

ε(iζ) = ω^{2}_{p}

ζ^{2} (2.34)

Where the usability of the plasma model is determined by the plasma wavelength given by: λp = (2πc)/(ωp). Above equation has no dissipative term and thus does not utilize the relaxation frequency.

### Chapter 3

## Materials

Casimir predicted a force between two perfectly reflecting plates in vacuum.

Lifshitz was the one who made the transition to the theorem that utilizes the dielectric function of real materials to characterize the Casimir force between the surfaces. [26]. Lifshitz theory predicts the same outcome as Casimir found for large distances (depending on the reflective properties of the material) as stated in formula 2.3. But lifshitz also found a reduction in the force for small distances where the material has pour reflection properties for low frequencies.

The conclusion of his findings were the dependence of the dielectric function for the Casimir force. In the following chapter we will explore the extrapola- tion possibilities for different materials of interest and their implications for the dielectric function.

### 3.1 Gold [2]

Today metals are widely used in all electrical devices and it will continue to find it’s applications in these devices. In the design of MEMS and NEMS there are often metals present and therefore it is a field of interest for Casimir physicists.

The general properties of metals are similar, however as stated in section 2.4 caution should be taken when trying to formulate general magnitudes of vari- ables using specific samples. To make use of the Casimir force in MEMS and NEMS one would need to know the underlaying working principles of the force.

In our calculations we make use of data from a gold sample. A second reason why metals are a field of interest is because it has been confirmed that metals give the maximum Casimir force due to their high absorption of conduction electrons in the low frequency ranges (which is in the far infrared)[3].

Equation 2.21 states how one could find the imaginary part of the dielectric function. However, with ellipsometry it is only possible to get data in a certain range which has as result that there is no experimental data available for the high- and low-frequency range. And thus it needs to be extrapolated to be taken into account for the Casimir force.

At low frequencies this can be done by the Drude or Plasma model. Because the transition to higher frequencies is not sharp it can practically be applied for

wavelengths up to λ > 2µm (approximately 0.62 eV). In that range the equation
can be compared with the measured data and the parameters ω_{p} and ω_{τ} can be
extracted from this. The value for gold is found to be ωp = 9.0 eV, and this is
largely adopted by the Casimir physics community [2, 27, 28, 29].

Using formula 2.21 and splitting it in the following parts:

ε(iζ) = 1 + εcut(iζ) + εexp(iζ) + εhigh(iζ), (3.1) where εexp stands for the part where there is experimental data is available, which is between ωcut and ω2. This term is constructed as:

εexp(iζ) = 2 π

ω_{2}

Z

ω_{cut}

dωωε^{00}(ω)

ζ^{2}+ ω^{2} (3.2)

εcut is constructed by using the the Drude model or the Plasma model. And εhigh should also be in the form of equation 2.21. However it has been found that the dielectric function for metals does not have a contribution in the high frequency regime when you have sufficiently high ellipsometric data, and thus can be neglected. It is sufficient to construct the dielectric function by using the data from the measurable regime and extrapolate the function in the low regime. In our case we have a high upper limit of our measurements (9eV) which makes the term εhigh negligible.

Drude dielectric function

The dielectric function for gold and other metals can be constructed using the Drude model. The experimental data is interpolated and used in formula 2.21.

Where the low regime is extrapolated according to the right part of equation 2.32. When using this result, plugging it in equation 2.21 and working it out analytically we get:

εcut(iζ) = 2 π

ω^{2}_{p}
(ζ^{2}− ω^{2}_{τ})

tan^{−1} ωcut

ω_{τ}

−ωτ

ζ tan^{−1} ωcut

ζ

(3.3) The full equation for the imaginary part of the dielectric function using the Drude model is given by:

ε(iζ) = 1 +2 π

ω^{2}_{p}
ζ^{2}− ω_{τ}^{2}

tan^{−1} ω_{cut}
ωτ

−ωτ

ζ tan^{−1} ω_{cut}
ζ

+2 π

ω_{2}

Z

ωcut

dωωε^{00}(ω)
ζ^{2}+ ω^{2}
(3.4)
In above result we neglected the contribution of the extrapolation above the
frequency ω2 since it is mentioned before that we can neglect this term.

Plasma dielectric function

Doing the same for the dielectric function but now making use of the plasma model gives us as a final result for the dielectric function:

ε(iζ) = 1 +ω^{2}_{p}
ζ^{2} +2

π

ω2

Z

ω_{cut}

dωωε^{00}(ω)

ζ^{2}+ ω^{2} (3.5)

Where again the contribution above ω2 can be neglected. The second term in above equation is the result of the extrapolation done according to the Plasma model.

### 3.2 Silicon-Carbide [3]

Metals are the materials that cause the highest possible resulting Casimir force, but for some applications metals are not suitable. For insatance if the device has the have high durability combined with high stiffness and low thermal expan- sion. An material that does have these properties is Silicon-Carbide (SiC). This material is currently used in precise instrumentation frames and mirrors and it is expected to have future applications in macro- and nanoassembly technologies via direct (optical) bonding concepts [30]. Besides that it is also expected to gain applications in MEMS, NEMS, automotive and space applications [31, 32].

The reason why SiC is considered as an replacement for conventional metals (such as Si) is because it exhibits strong polytypism where all polytypes have identical planar arrangements. The difference lies in the stacking of planes. Dis- order in the stacking of planes results in different polytypes. Polytypism causes a resemblance of the properties of metals such as Si [33] .

SiC has a relatively low residual stress level in the layers, high stiffness and good etch-stop properties that allows the fabrication of micro-structures using standard micro-machining techniques [31, 32]. The excellent properties of SiC such as high hardness, chemical inertness and the ability to survive operation at high temperatures and harsh environments makes it a very interesting material for future applications.

To calculate the Casimir force for SiC we need to construct the imaginary part of the dielectric function as stated in equation 2.21. Similar as with the met- als there is a limited range where the imaginary part of the dielectric function can be measured, and the ranges that lie outside this measurement has to be extrapolated.

Unlike with the metals, the dielectric function for the high frequencies can not be neglected. For the extrapolation of the high frequency regime we use [3]:

at ω > 9.34eV: ε^{00}(ω) = A

ω^{3} (3.6)

Where A is determined by the continuity of ε^{00}(ω) at ω = ω2 and ω2= 9.34eV .
When above extrapolations are filled in in equation 2.21 and then worked out
analytically we get the contribution the the imaginary part of the dielectric
function for the the high frequency regime:

∆Hε(iζ) = 2ω_{2}^{3}ε^{00}(ω2)
πζ^{2}

1
ω_{2} −

π

2 − tan^{−1}(ω1/ζ
ζ

(3.7)

Drude model

The Drude model for the low frequency regime is the same as we used for metals (but the parameters differ for SiC):

at ω < 0.03eV: ε^{00}(ω) = ω_{p}^{2}ωτ

ω(ω^{2}+ ω^{2}_{τ}) (3.8)
Working this out analytically we get the exact same result as equation 3.3:

∆_{L(Dr)}ε(iζ) = 2ω_{p}^{2}ωτ

π(ζ^{2}− ω^{2}_{τ})

tan^{−1}(ω_{1}/ω_{τ})
ωτ

−tan−1(ω_{1}/ζ)
ζ

, (3.9)

In total this construct a dielectric function in the form of:

ε(iζ) = 1 + 2 π

ω2

Z

ω_{1}

dωωε^{00}_{exp}(ω)

ζ^{2}+ ω^{2} + ∆_{L(Dr)}ε(iζ) + ∆_{H}ε(iζ) (3.10)

Plasma Model

When using the plasma model for the extrapolation in the low regime we get:

∆L(P l)ε(iζ) = ω^{2}_{p}

ω^{2} (3.11)

And this has as a result for the dielectric function:

ε(iζ) = 1 + 2 π

ω_{2}

Z

ω_{1}

dωωε^{00}_{exp}(ω)

ζ^{2}+ ω^{2} + ∆L(P l)ε(iζ) + ∆Hε(iζ) (3.12)

### 3.3 Phase Change Materials

In the previous two section we have discussed two material with static result- ing Casimir Force, however there is a particular interesting application for the Casimir force if one would succeed in producing a Casimir force that is switch- able between high and low force states. For this it is necessarily that we use a material which optical properties can be changed. This means that the dielectric function of the particular material has to change drastically to induce a large change in the Casimir force. The modification of the dielectric function has been studied for changing carrier densities in semiconductors [34, 35]. However there is group of materials that posses the above properties and that are renowned for their rapid switch between configurations: Phase Change Materials (PCM).

Phase change materials have two rapidly reversible states: Amorphous and Crys- talline phase [36, 37, 38]. This technique has been exploited for constructing rewritable optical data storage [36, 37, 38] where the optical contrast between both phases is used to store information. By using an intense focused laser

beam the PCM is heated above their melting temperature. Rapid colling of the molten material produces a glass-like amorphous state.

The amorphous phase has a lower reflectivity and thus can be distinguished from the crystalline phase. To reverse the transition, the amorphous phase has to be gradually heated to revert to the crystalline state. This principle is used in three generations of optical data storage devices (CD-RW, DVD-RW and BD-RW). These material properties and good reversibility of both states makes them a point of interest for the Casimir physics department. The material that was used to get the experimental data was AIST (Ag5In5Sb60T e30).

### 3.3.1 Amorphous phase

Again we make use of equation 2.21 to determine the imaginary dielectric func- tion. In the range between ω > 0.04eV, ω < 8.9eV we can make use of el- lipsometric measurements. The amorphous phase is characterized by its trans- parency in the infrared (ω < 0.04eV) range. Due to this property the amorphous phase of the PCM has no contribution to the dielectric function in the low range.

The imaginary part of the dielectric function at high frequencies was extrapo- lated as:

ε(iζ)^{00}= A

ω^{3} f or ω > 8.9eV (3.13)

This extrapolation is justified by a good Kramers-Kroning consistency for amor-
phous films and is in good agreement with the previously found permittivities
[39, 40]. And A is again determined by the continuity of ε^{00}(ω) at ω = 8.9eV.

It is important to notice that the extrapolated function in this regime is in the form of 1/ω and thus will decrease rapidly and have a very small contribution to the dielectric function.

### 3.3.2 Crystalline phase

The crystalline phase is not transparent in the low frequency range, in fact it has a high contribution to the imaginary part of the dielectric function in this range. In the crystalline state PCM is characterized by a large number of free carriers [41]. The drude model that has been used to fit the optical data for the crystalline state. In the regime where we want to extrapolate the date we use the simplified model where.

ε(ω) ' C + i ω^{2}_{p}

ωω_{τ} (3.14)

This one looks very similar to the equation for the metallic state (equation 2.30)
and is widely used in the study of the Casimir force [2, 7, 8, 27, 35, 42]. Above
equation is used to extrapolate to the low regime. The approximation is justi-
fied because ω << ω_{τ} here.

But since the material does not show a clear plasma edge, it is impossible to derive both ωp and ωτ, only the ratio that is state in the above formula. This ratio is related to the optical conductivity which is given by:

σ = ε_{0}ω_{p}^{2}
ωτ

(3.15)
where ε0 is the permitivity of vacuum. By fitting above formula to the known
data for ω < 0.07eV we can obtain C = 51 and (ω_{p}^{2}/ωτ) = 10.1eV. Considering
that the extrapolation in the high regime is done by the same extrapolation as
the amorphous state (equation 3.13). By taking the imaginary part of equation
3.14, combining it with equation 3.13 and equation 2.21 we can determine the
imaginary part of the dielectric function for the crystalline phase of the phase
change material.

### Chapter 4

## Results

### 4.1 Dielectric functions

In Matlab we have scripted the formula’s that have been stated in the pre- vious section to construct the dielectric function that belongs to the different materials. In this section we have summarized the results of these inter and extrapolation to construct the dielectric function for the materials of interest.

For all the materials there is a range for which data was obtained by using ellipsometric measurements. In some cases this data is used to construct the extrapolations. This this data is specific for the samples used to do the mea- surement and the results should be carefully considered before extending these result to the general case.

At the end of this section the parameters for all samples have been listed. These parameters are also obtained from these specific samples.

### 4.1.1 Dielectric function for Gold

The dielectric function for gold has an extrapolated function in the low regime.

From figure 4.1a and figure 4.1b it can been seen that this is extrapolated part in the low regime has a significant contribution to the dielectric function. Also it can be seen from the data in figure 4.1a that indeed we do not need to extrapolate in the high regime since it is sufficiently small and does not contribute to the dielectric function.

(a) ε^{00}for Gold (Au) (b) ε(iζ) for Gold (Au)
Figure 4.1: Data for the gold dielectric function

### 4.1.2 Dielectric function for SiC

The dielectric function for SiC has an extrapolated part in the high and in the low regime. The measured part consists of two different data sets (UV-regime and Infrared regime) that have been combined and interpolated to cover the mid-range. In figure 4.2b the separate terms formula 3.10 and 3.12 are shown.

Here the εexp contains both the data sets (UV and Infrared). εhigh is mostly significant in the low energy regime and the εlow has a very small contribution to the dielectric function for SiC and could be neglected as well.

(a) ε^{00}for SiC (b) ε(iζ) for SiC

Figure 4.2: Data for the SiC dielectric function

### 4.1.3 Dielectric function for PCM (Amorphous)

The PCM (Amorphous) has only an extrapolated part in the high regime since it is optically transparent in the low regime. From figure 4.3b it is clear that the extrapolation in the high regime has a small contribution to the dielectric function.

(a) ε^{00}for PCM(Amorphous) (b) ε(iζ) for PCM(Amorphous)
Figure 4.3: Data for the PCM(Amorphous) dielectric function

### 4.1.4 Dielectric function for PCM (Crystalline)

The PCM (Crystalline) has two extrapolated parts. The high regime part uses the same method that has been used for the PCM(Amorphous) phase. Besides from a parameter for continuity it is the same and thus as we concluded in the previous subsection, it does not have a significant contribution to the dielectric function. The crystalline phase is not transparent in the low regime and has an extrapolated part there. As seen from figure 4.4b this has a high contribution to the dielectric function and changes the resulting dielectric function drastically compared to the Amorphous phase.

(a) ε^{00}for PCM(Crystalline) (b) ε(iζ) for PCM(Crystalline)
Figure 4.4: Data for the PCM(Crystalline) dielectric function

### 4.1.5 Summary

Dielectric functions

To put the above data in perspective all dielectric function are shown in the figure below. From this it is clear that there is a large difference between the dielectric functions of the materials and models that we studied. Obviously that will result in a difference in the Casimir force.

Figure 4.5: Dielectric function for several materials and extrapolation models

Parameters

To construct the sample dependent dielectric functions for the above materials the following parameters have been used:

Material ω_{τ}(meV) ω_{p} (eV) ω_{down} (eV) ω_{up} (eV)

Gold (Drude and Plasma) 48.8 7.79 0.04 8.9

SiC (Drude and Plasma) 74.0 0.173 0.03 9.34

PCM (Crystalline) C C 0.04 8.9

Table 4.1: Summary of the parameters used to construct the dielectric functions
In the above table the parameters for PCM (Amorphous) is not included because
it does not use any parameters and C is defined as the ratio: ω^{2}_{p}/ω_{τ} and has the
value of 10.1eV.

### 4.2 Force simulations

The results from the previous section do not lead to a straightforward result for the resulting forces. In this section we show the results for the force simulations for several configurations. These configurations consist of Gold (Au), Silicon Carbide (SiC), the previously discussed phase change material in the amorphous phase (PCM(A)) and the crystalline phase (PCM(Cr)). To express the force we use to convenient scale of the force in comparison with the theoretical force, called the reduction factor:

η(z) = F (z)

FC(z), F_{C}(z) = −π^{2}~c

240a^{4} (4.1)

Where the second part is defined before as equation 2.3 and represents the theoretical force described by Hendrik Casimir. We simulated the force for the different materials and models in the range of 1nm - 200 nm.

### 4.2.1 Force per Material

In this section the Forces for the above four described materials and their con- figurations with each other are shown.

(a) η for Gold - * (b) η for SiC - *

(c) η for PCM(A) - * (d) η for PCM(Cr) - *

Figure 4.6: η for Au, SiC, PCM(A) and PCM(Cr) in comparison with their respective counter plates

### 4.2.2 Comparison of the models

The Plasma model is only defined for the Gold and SiC samples. Here we compare the force of both these models. The calculations have also been done without an extrapolation in the low regime. The First comparison is for the gold-gold configuration, the second comparison for the SiC-SiC configuration and the third and last comparison is for the gold-SiC configuration:

(a) η for three different extrapolation models for gold

(b) Difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.7: Difference in resulting force for different models for Gold

(a) η for three different extrapolation models for SiC

(b) Difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.8: Difference in resulting force for different models for SiC

(a) η for three different extrapolation models for gold-SiC

(b) Difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.9: Difference in resulting force for different models for gold-SiC

### 4.2.3 Special configurations

To complete the comparison, the calculations of a configuration using two mod- els have been done as well. In the graphs below the resulting reduction factor is shown when we compare two plates that each have a different model that extrapolates the dielectric function in the low regime. The same setup as in the previous subsection has been used. In the first figure the gold and SiC config- urations are shown, in the second figure the SiC-SiC configuration and in the third figure the gold-SiC configuration is shown :

Again first for the Gold model:

(a) η for Gold models

(b) Difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.10: Difference in resulting force for different models for Gold: Mixing the models

(a) η for SiC models

(b) difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.11: Difference in resulting force for different models for SiC: Mixing the models

(a) η for Gold-SiC models

(b) difference of resulting force compared to the Drude model expressed as a percent- age

Figure 4.12: Difference in resulting force for different models for gold-SiC:

Mixing the models

### 4.2.4 Force Summary

Below a summary of the calculated forces. The histograms have been made for the plate-plate separation of 1 nm, 100 nm and 200 nm.

Figure 4.13: Collection of calculated forces with a plate plate separation of 1 nm

Figure 4.14: Collection of calculated forces with a plate plate separation of 100 nm

Figure 4.15: Collection of calculated forces with a plate plate separation of 200 nm

### 4.3 Further advances

In this thesis we considered the plate-plate configuration for calculating the Casimir force. However when moving down to the nanoscale you have to deal with serious parallelism problems. To solve this problems in actual measure- ments the plate-sphere geometry is widely used [2, 11, 39, 43, 44]. Assuming the sphere radius is indicated with R and the distance between the plate and the sphere is much smaller than the radius of the sphere (z R) we can find an expression for the force between the plate and the sphere:

F^{ps}(z) = ~cR
16πz^{3}

X

ν=s,p

∞

Z

0

dxx^{2}ln(1 − r^{ν}_{1}r_{2}^{ν}e^{−x}) (4.2)

Above expression is valid for T = 0 or for short separations (z < 300 nm) at room temperature. The thermal fluctuations have to be in the range where they have negligible contribution, at room temperature the thermal wavelength is in the order of λT ∼ 3µm and thus can be neglected. The subscript ν = s, p

stands for the previously defined polarization components (equation 2.14 and the reflection coefficients have been previously defined in formula 2.15.

This force can also be expressed with the reduction factor:

ηsp(z) = F^{ps}(z)

F_{C}^{ps}(z), where F_{C}^{ps}(z) = π^{3}R~c

360z^{3} (4.3)

To use the Casimir force in MEMS or NEMS we have to consider the case where
the sphere (or plate) is moving. We assume that the sphere is moving by being
connected to a spring. Having assumed this the Casimir Force F^{ps} is opposed
the elastic restoring force: F (z) = −K(L_{0}− z) where K is the spring constant.

The equation of motion then is given by:

Md^{2}z

dt^{2} + M ω
Q

dz dt

= −K(L0− z) + F^{ps}(z) (4.4)

Figure 4.16: A schematic overview of the described configuration

In above equation it is assumed that the system had an initial im- pulse to trigger the actuation and the spring is not stretched (K

∂Fc/∂z|z=l_{0}). M is the mass of
the sphere and the second term
(M ω/Q)(dz/dt) is the intrinsic en-
ergy dissipation of the moving sys-
tem. Q stands for the qual-
ity factor, a large Q means a
lower rate of energy loss relative
to the stored energy of the sys-
tem.

The sphere and plate can be coated with a material of interest (Au, SiC or AIST) and then used in MEMS or NEMS.

To study the behavior of this configuration in these devices we introduce a bifurcation parameter [45, 46, 47, 48]:

λ = F^{ps}(L0)
KL0

(4.5)
This parameters represents the ratio of the minimum Casimir force and the
maximum elastic force. If we then set the total force zero at z = z^{∗}, which
is represented by the right part of equation 4.4 we can calculate the locus of
equilibrium points (z^{∗}):

λ = F^{ps}(L0)
F^{ps}(z^{∗})

1 − ^{z}_{L}^{∗}

0

(4.6)

If the condition dF/dz^{∗}= K + dFc/dz^{∗}is satisfied the above equation also gives
the (stationary) critical equilibrium points.

Figure 4.17 shows the bifurcation parameter λ for different material configura- tions. A comparison of the Drude and Plasma model have been made in figure 4.17b. Because we can approximate λ ∼ 1/K we can state that if λ < λmax

then there are two equilibriums. The equilibrium on the left side of the peak is
the unstable point that will lead to stiction while the equilibrium at the right
side of the peak represents a stable solution around which a periodic solution
exists. So the locus point where z^{∗}> z_{max}^{∗} is a stable actuation.

When the spring constant K is lower so that λ ≥ λmax the motion is unstable and will likely lead to stiction. From figure 4.17b it is clear that the Plasma model is less stable due to the higher force it causes. As a result we can conclude that the materials that have weaker resulting Casimir forces are less favorably to stiction because of their wider range of stable operation.

(a) (b)

Figure 4.17: (a) Bifurcation diagrams for all material combination Au, SiC, and PCMs using the Drude model for the Casimir force calculations (b) Bi- furcation diagrams for Au-Au and SiC-SiC using both the Drude and Plasma models for the sphere and plate configuration.

The phase portraits of the materials are shown in figure 4.18. These plots show the velocity of the actuating element (dz/dt) versus the displacement [43].

Figure 4.18b shows the difference when taking different samples or even using different models. In both figures a closed orbit represents a periodic movement around a stable equilibrium where the elastic force is strong enough to counter- balance the Casimir Force.

In figure 4.18b there are phase portraits for the same sample but using a different extrapolation model. It is clear from this figure that there is notably significant difference in the result. This difference is especially large at the position where the distance between the sphere and the plate is smallest and thus where the force is at it’s largest.

(a) (b)

Figure 4.18: (a)Phase portraits for Au, SiC and PCM (C) systems using the Drude model. (b) Phase portraits when the plate is coated with three different Au films from [11] using the Drude model, while for Au (3) we considered for the plate the plasma model and for the sphere the Drude model. The spring constant in all cases was K=0.00015 N/m.

Finally we consider the phase portrait in figure 4.19 we can see that in this particular case the drude model predicts a stable motion where the Plasma model does not. The later models predicts stiction and thus a breakdown of the device.

Figure 4.19: Phase portrait dz/dt vs. z for the Au-Au and the Au-SiC sys- tems with respect to the Drude and plasma models. The spring constant was K=0.00012 N/m.

### Chapter 5

## Conclusions

In this thesis we analyzed the properties for several materials and several models to calculate the Casimir force. This analysis has been done for Gold, Silicon Carbide, and the two different phases of the phase change material AIST (amor- phous phase and crystalline phase). The optical responses of those materials were measured ellipsometrically in a wide range. The unaccessible wavelengths have been extrapolated using the Drude or the Plasma model. Using this the di- electric function have been constructed by making use of this Drude and Plasma model. A third configuration was also considered. This configuration neglected the contribution of the dielectric function in the low regime.

It has been found that there is a significant difference in the resulting forces between the materials. It is determined that the dependence on the extrap- olation model is more important for the metallic model than for SiC model.

The widely accepted result which states that metallic material exhibit a larger resulting force was confirmed. It was also confirmed that the Plasma model, which neglects dissipation, results in a larger force.

To calculate the Casimir force, the dielectric function at imaginary frequencies,
ε(iζ) has to be known. This dielectric function is expressed through the mea-
surable function ε^{00}(ω) by the dispersion relation (equation 2.21). Metals and
SiC show a strong contribution in the low regime. The contribution in the low
regime for the phase change material strongly depended on the phase of the
material. The amorphous phase had no contribution in the low regime whereas
the crystalline phase did have a strong contribution in the low regime.

Dielectric function

The above mentioned low regions of interest are not accessible by ellipsiometric
experiments. Ellipsometry has a limit in the low frequency range caused by
energetic limitations which makes measurements practically impossible. For all
the materials considered in this thesis the low regime is extrapolated by using
the Drude model. Gold an SiC have also been extrapolated in the low regime
by using the Plasma model. These models make use of the parameters ω_{p} and
ω_{τ} that were extracted from the measured data.

The possible resulting dielectric functions are shown in figure 4.5. From this collection of results it is clear that there is not only a significant difference in the magnitude of dielectric function between materials, but also a significant difference in dielectric function between the extrapolated Plasma and Drude model. This difference is largest in the low energy regime.

Force simulations

The dielectric functions for the different models have been used to calculated the Casimir force by using the Lifshitz’s formula stated in equation 2.17. All possible configurations for the materials of interest have been calculated and compared. Using the Drude model to compare the other models, it is found that using a different model can give a difference in the resulting force up to 15

% for the (metallic) gold sample. The difference in resulting force for metallic material gradually descends at larger separations, where the difference in result- ing force for the SiC model is only around 1 % for the calculated separations, but this difference increases when the separation is increased.

The difference in models results for metals in a significant difference, where the difference is relatively largest at low separations and absolutely largest at larger separations. The main conclusion that can be drawn from this analysis is the non-negligible difference between the Plasma and the Drude model. It is clear that neglecting the low regime is not acceptable for the resulting force calculation. This difference is mainly true for metals. For materials such as SiC this difference is very small, especially at low separations and could there be neglected when not dealing with precise measurements. Because metals have the largest contribution to the resulting Casimir force the future developments in theory should be focused on the metals.

The application of both models has to be tested thoroughly in real experimental setups to determine which of these models approximates the measurable Casimir force best. Possible new models should be developed when both models do not fit the actual force. For the usage of the Casimir force in applications such as MEMS and NEMS, the force has to be known very precisely. When using the current models in designing these devices there is still a difference that is not negligible. We have considered a spring system for the usage of the Casimir force in a device. From this simulation we have to conclude that the Plasma model can drive the spring in the breakdown regime where the Drude model does not. Therefore more research has to be done to determine which model should be used for these designs. We can conclude that for the usage of the Casimir force in those devices the current models differ too much to be of use.

Measured data should be compared extensively with these models. With these future developments the Casimir force has some very promising applications in devices.

### Chapter 6

## Acknowledgments

In the fantastic process of writing this bachelor thesis I have had many setback which were mostly cause by inconclusive errors in the Matlab simulations. But there was a person who I could consult when dealing with these exhaustive, pain in the ass problems. Therefore I would like to give special thanks Mehdi Sedigh for helping me with these difficult and exhausting problems. Also I would like to thank him and George Palasantzas for having discussion about the topic which made me find my way in this research.

Besides the special thanks mentioned above I would like to thank everybody who has contributed in a way to this bachelor thesis. This has been done by providing useful viewpoint as a result of a discussion, by making nice cups of coffee that I drank excessively when writing this thesis or by supporting me in doing the research at moments when I was exhausted by setbacks.

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