Nanopatterned Anchoring Layers for Liquid Crystals
by
Christopher S. Gear
MASSACHUSETTS WTNT~
OF TECHNOLOGY
B.S. Physics
United States Naval Academy, 2012
JUN 10 2014
IUBRARIES
SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MATERIALS SCIENCE AND ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2014
@
2014 Massachusetts Institute of Technology. All rights reserved.
Signature of Author:
Signature redacted
Department of Materials Science and Engineering
May 21, 2014
Certified by:
Signature redacted
Kenneth Diest
Technical Staff Member, MIT Lincoln Laboratory
Thesis Advisor
Signature redacted
Silvija Grade(ak
Professor of Materials Science and Engineering
Thesis Reader
Accepted by:
Signature redacted
Gerbrand Ceder
Professor of Materials Science and Engineering
Chair, Departmental Committee on Graduate Studies
Inhomogeneous Anchoring Layers for Liquid Crystals
Using Nanopatterned Grooves
by
Christopher S. Gear
Submitted to the Department of Materials Science and Engineering
on May 21, 2014 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Materials Science and Engineering
ABSTRACT
This thesis describes the theory and fabrication of inhomogeneous Liquid Crystal
anchoring layers. While chemical anchoring techniques have proved useful for many
applications, especially Liquid Crystal Displays, they have thus far been unable to
demonstrate the ability to provide anchoring energy that varies with high spatial
frequencies. This thesis describes the use of nano-grooves patterned with electron beam
lithography as a novel way to provide varied anchoring energies for Liquid Crystal devices.
A Liquid Crystal beam deflector is discussed and designed with computational simulations
as a possible application for varied anchoring layers. Anchoring grooves are patterned onto
fused silica substrates, then their anchoring energies are measured using optical methods.
It is shown that nanopatterned grooves are capable of producing anchoring energies which
can span an order of magnitude or more across a single substrate and vary across
extremely small regions (< 1 pm).
Thesis Advisor: Kenneth Diest
Title: Technical Staff Member, MIT Lincoln Laboratory
Thesis Ready: Silvia Gradeak
Title: Professor of Materials Science and Engineering
Contents
1
2
3
4
5
Introduction
4
1.1
Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Alignment Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Liquid Crystal Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Liquid Crystals and their Anchoring Layers
9
2.1
Liquid Crystal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Rubbed Polyimide Anchoring Layers . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Photoaligned Anchoring Layers
. . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
Mechanical Anchoring with Grooves . . . . . . . . . . . . . . . . . . . . . . .
15
Liquid Crystal Beam Deflector
19
3.1
Beam Deflector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2
Liquid Crystal Director Simulation . . . . . . . . . . . . . . . . . . . . . . .
22
3.3
Device Design and Optimization . . . . . . . . . . . . . . . . . . . . . . . . .
27
Mechanical Anchoring Energy Experiment
32
4.1
Experiment Objective
. . . . . . . . . . .
32
4.2
Twisted Nematic Cell Optical Model . . .
33
4.3
Patterning HSQ . . . . . . . . . . . . . . .
36
4.4
Cell Fabrication . . . . . . . . . . . . . . .
39
4.5
Experimental Setup . . . . . . . . . . . . .
40
4.6
R esults . . . . . . . . . . . . . . . . . . . .
41
4.7
A nalysis . . . . . . . . . . . . . . . . . . .
43
Discussion
45
3
1
Introduction
1.1
Liquid Crystals
Liquid Crystals (LCs) constitute a unique class of materials. They are able to flow as liquids,
but possess high degrees of short range order not unlike crystalline solids. They owe these
properties to their unusual rod-like molecular shape and interesting chemical properties. LCs
also derive their dielectric anisotropy from these properties. This makes them especially
susceptible to external electric fields and also imparts them with optical birefringence. Their
index of refraction is usually higher for electric fields oscillating along their long axis. Their
dielectric properties coupled with their ability to flow and reconfigure in-situ make them
uniquely suited for switchable optical devices. In recent years, LC devices have become
ubiquitous, particularly flat panel Liquid Crystal Displays (LCDs) [1].
1.2
Alignment Layers
Most LC devices require some form of alignment, or anchoring, layer to establish a preferred
orientation for molecules placed on the surface.
The strength of this anchoring layer is
referred to as its anchoring energy, which is the magnitude of the energy penalty faced by
the LC for aligning perpendicular to a preferred direction. These anchoring layers introduce
uniform alignment to the entire LC layer, and in their absence local defects within the LC
or on the substrate would dictate orientation. Broadly speaking, there are two categories of
anchoring energy to consider: polar anchoring energy, which is the energy penalty for the
LC aligning perpendicular to the plane of the substrate, and azimuthal anchoring energy,
which is the energy penalty for LC aligning perpendicular to a designated direction within
the plane of the substrate.
LC molecules can be anchored with two general techniques: chemical anchoring or mechanical anchoring.
Chemical anchoring mechanisms operate at the monolayer interface
between the substrate and the LC. A chemical is deposited onto the substrate and then its
isotropy is broken either by mechanical rubbing or radiation from polarized light. At this
4
point, the anchoring molecules (usually chain-like polymers) are arranged on the substrate
with some order such that one orientation is preferred over another.
The LC molecules,
which are deposited on top of said chemical layer, energetically favor anchoring parallel to
the molecules of the underlying substrate. The LC is thus anchored in that specific orientation [2].
LCs can also be anchored mechanically.
The governing mechanism of this anchoring
technique differs from chemical anchoring in that it does not rely on a monolayer interaction
but a long range elastic interaction within the LC itself. For mechanical anchoring, grooves
are patterned onto a substrate. Assuming the LC is anchored into the plane of the substrate,
in what is known as the low-pretilt angle regime, if a single LC molecule is introduced to this
substrate, it would not be more energetically favorable to align with or against the grooves;
however, by placing a film of LCs onto the substrate which completely covers the grooves,
it becomes energetically favorable for the LC molecules to align parallel with the grooves.
This is because if the LC molecules were aligned perpendicularly to the grooves, they would
constantly be changing direction as each individual LC molecule would lie locally tangent
to the groove. By aligning parallel to the grooves, each LC molecule may lie tangent to
the plane and maintain local alignment with neighboring molecules.
This arrangement is
energetically favorable and in fact becomes stronger with deeper groove depth and higher
spatial frequency of the grooves[3].
Most applications require a uniform anchoring layer, providing the same anchoring energy
and orientation across the entirety of a given substrate. However, many novel technologies
require patterned anchoring layers that vary in magnitude and orientation at high spatial
frequencies [4, 5]. Most modern technologies that require LC anchoring layers use rubbed
polyimiide layers. This technique is inherently coarse as it requires a mechanical brush [6].
Another promising group of materials are photo-aligned dyes. While development is ongoing,
both technologies fall short of allowing one to pattern anchoring layers with varying strengths
and directions at micrometer or nanometer resolutions - this will be discussed in more detail
in Sections 2.2 and 2.3.
5
This thesis discusses a novel approach to the problem in which one can produce variable
mechanical anchoring energy with grooves patterned by electron-beam lithography. Because
the anchoring energy produced by grooves is proportional to 1/A
3
[3], where A is the period
of the grooves, one can control the anchoring energy with a high degree of variability. With
this in mind, one can impose anchoring energies spanning multiple orders of magnitude on
a single substrate.
1.3
Liquid Crystal Devices
As stated earlier, because of their ability to reconfigure in-situ as well as their optical properties, LCs are natural candidates for switchable optical devices. The general public is probably
most familiar with the LCD. The LC layer in this device is arranged in what is known as
a "twisted-nematic" configuration. It is "twisted" in that the average orientation of the LC
director rotates 900 through the thickness of the LC layer. Thus, the director at the bottom
of the layer is perpendicular to the director at the top of the layer. The device operates in
what is known as the adiabatic approximation regime. This will be discussed in more detail
later, but basically if linearly polarized light is incident to the Twisted Nematic Cell with
its polarization direction parallel to the LC director at the input face of the cell, then the
light will maintain its linear polarization while traversing the cell and exit the cell with its
polarization direction rotated such that the new polarization direction is parallel with the
director of the LC at the exit face of the cell [7].
LCDs work with crossed polarizers placed immediately in front of and behind the LC
layer. Light going through this device becomes polarized then rotates polarization and exits
the device. The LCD is useful because one can "switch" the LC configuration. By applying
an electric field normal to the plane of the substrates, one can force the LC to become
aligned parallel to the electric field and thus break the twisted configuration.
With the
twist configuration broken, polarized light no longer rotates its polarization as it traverses
the cell and thus is blocked by the exit polarizer. Thus, with an electric potential one is
1Usually
the LC that is in the immediate vicinity of the anchoring layer remains aligned parallel to the anchoring direction,
as the anchoring energies used for LCDs are very high.
6
able to modulate the amount of light able to exit the LCD. Luckily for display engineers,
LCDs only require uniform anchoring layers with constant magnitude and direction of their
anchoring energies. This is not the case for novel technologies, such as beam steering devices
and beam deflectors, which require patterned anchoring layers. While LCDs are a mature
technology and are widely considered a solved problem, some are still attempting to develop
new techniques to expand the viewing angle and make further refinements.
Of course, LCs posses optical properties applicable to a bevy of optical devices, not just
displays. These include but are not limited to active lenses, non-mechanical beam steering
devices, imagers, and beam deflectors. This thesis will discuss beam deflectors in some detail.
1.4
Scope of the Thesis
Section 2 gives a description of LCs and how they are affected by various anchoring layers.
Section 2.1 describes LC properties and introduces some formalism which is required to
quantitatively describe the behavior of LC films. The following sections are devoted to LC
anchoring techniques. Section 2.2 describes LC anchoring using rubbed polymers, which is
probably the most common technique used in technology today, but as of right now can not
be used to pattern variable anchoring energies. Section 2.3 discusses photoaligned anchoring
layers.
This is a developing technology but holds promise for generating more intricate
anchoring layers by leveraging modern photolithography techniques.
Finally, section 2.4
discusses mechanical anchoring energy that is caused by grooves deposited on a surface. A
brief set of equations is shown that demonstrates the elastic nature of this form of anchoring
energy. The theory behind this form of anchoring is revisited in Section 4.
Section 3 describes the design of a LC beam deflecting device which uses a LC film in
what is known as a cycloidal configuration.
While devices like this have been reported
in the literature, this design is novel in that the device allows light to pass unimpeded
in its default state and then by applying a potential the device enters beam deflecting
rmode. Thus far in the literature only the opposite modality has been achieved. Section 3.1
discusses the Jones Calculus, explaining the operation of a cycloidal LC beam deflector.
7
Section 3.2 explains a computational simulation technique used in the design process of the
device. This simulation technique is used in an iterative design and optimization process
described in Section 3.3. Ultimately in order to generate a cycloidal LC configuration, it
was determined that spatially varying anchoring energies were necessary. This discovery
motivated the research into variable anchoring energies and the experiment described in
Section 4
Section 4.1 discusses the objective of the experiment and its motivation. It also touches on
the background theory behind the optical testing that was performed. Section 4.2 shows the
formalism for light passing through a Twisted Nematic LC cell. This knowledge allows one to
extract the twist angle of the cell using optical methods and in turn determine the anchoring
energy present in the cell. In Sections 4.3 and 4.4, the fabrication procedures, which span
patterning nanogrooves onto fused silica substrates and then bonding two substrates together
to yield a LC cell, are discussed. The experiment setup is described in Section 4.5, this
discusses the equipment and apparatus required to extract data. The results are shown in
Section 4.6 and analyzed in Section 4.7. Interesting trends are observed and a new model is
proposed to better explain the mechanisms at play. Finally, in Section 5, the implications
of the study are discussed and possible mechanisms that led to deviations of the data from
the theory are discussed.
8
2
Liquid Crystals and their Anchoring Layers
2.1
Liquid Crystal Properties
In order to discuss LCs and their applications, we should begin with a description of the
basic categories of LCs. There are three major classes of LCs: Lyotropic, Polymeric, and
Thermotropic. Lyotropic LCs are cases where molecules in a solution obtain local alignment
and even exhibit crystalline phases. The most common example of these LCs are amphiphilic
molecules dissolved in water.
These are molecules with one end that is hydrophilic and
another end that is hydrophobic - examples include soaps and lipids.
Polymeric Liquid
Crystals are polymers with LC molecules as monomers, they exhibit high viscosities and
thus are less susceptible to reconfiguration.
Finally Thermotropic Liquid Crystals, which
this thesis is concerned with, are rod shaped molecules with aromatic groups which prefer
uniform local alignment.
They are easily the most studied LCs and most employed in
technology. Their name is derived from the fact that they present a rich variety of properties
and phase transitions at or around room temperature making their variable properties easily
accessible
[1].
Within the Thermotropic LCs, there are several different classes of LC all of which have
an isotropic phase accessible at high temperatures, but at low temperatures exhibit different
ordered phases. These include Nematic LCs which prefer uniform local alignment and are the
focus of this thesis. Other examples include Cholesteric LCs, which contain a self induced
twist, and Smetic LCs, of which there are several subcategories (thus far nine have been
discovered) each of which has some form of translational order in addition to their directional
order [1].
Nematic LCs are easily the most well understood and commonly used form of LC. As
stated in Section 1.1, LCs exhibit high degrees of short range order despite their liquid
phase.
In order to more fully discuss Nematic LCs, we should define some of their basic
9
properties. Nematic LCs have an order parameter S defined as:
S
1
- < 3(k -h)(k - h) - 1 >
2
=-
1
< 3cos2
2
_ 1>
(1)
where k is the unit vector describing the orientation of any given LC molecule and ft is the
unit vector describing the average orientation for the LC at a given point. Thus, if all LC
molecules are perfectly aligned in a given region, S = 1. This definition is consistent for all
uniaxial LCs, which have infinite rotational symmetry around a single long axis. For biaxial
LCs (which have their rotational symmetry broken by the addition of another molecular
axis), the definition of S becomes increasingly complicated and requires a more general
tensor order parameter to be used [1]:
1
where i, j, and k are the unit vectors for each of the molecular axes. The diagonal components
of the tensor can be expressed by:
1
2
- < 3 sin 2 0 cos 2
Si
1
Sjj =
Skk =
< 3 sin 2 0 sin 2
I < 3 cos2
2
1
-1>
(3)
-_ >
(4)
- 1>
(5)
where 0 describes a molecule's polar axis deviation from the director and 0 describes a
molecule's azimuthal axis deviation from the director.
Thus, for biaxial molecules S is
always a traceless tensor. The LC used for the simulations and experiments in this work was
5CB, a common and readily available LC which is uniaxial, so S in this thesis will refer to
the scalar order parameter.
Other important properties of Nenatic LCs include their elastic constants. These define
the energy penalties for spatial director variations present in bulk LC. Using symmetry
arguments F.C. Frank dramatically simplified the expression for the total elastic energy
10
density contained within a bulk Nematic LC [1, 8]:
F =
IK(V . h)21 +
2
1v
x
-K 2 (h -V X
2
)2 + -K3(i X V X I)2
2
(6)
Each term represents a class of curvature that can exist within a bulk uniaxial material.
The first term is the "splay" deformation and K is likewise the Splay elastic constant. The
second term is the "twist" deformation and K 2 is the Twist elastic constant. Finally, the
third term is the "bend" deformation and K 3 is the Bend elastic constant.
Of course, other contributions to the total free energy of bulk LCs exist. The first of
which is the external field free energy. This is due to the dielectric anisotropy that exists
for LC molecules. For a uniaxial LC, there are three dielectric constants, two of which are
independent. They make up the dielectric tensor defined as:
0
0
E
0
0
Where E is the symbol for the dielectric tensor and cI is the dielectric constant for the LC
perpendicular to the long axis of the LC and E is the dielectric constant for the LC parallel
to the long axis of the LC. For almost all LCs El > EI, however exceptions exist. Also LC
dielectric constants have dispersion relations and some can have regions where Ell > EI and
others for which
E
<
EI
[1].
The second contribution to the LC free energy is from the surface interactions. As with any
dissimilar materials interface, there is some free energy associated with the interface between
LCs and whatever substrate on which they are deposited. For LC in the isotropic phase, the
surface energy is a constant and is only dependent on chemical properties. However, in an
ordered phase, the LC uniaxial symmetry dictates that the changing the orientation of the
molecules will cause changes in the surface energy. This surface energy can be broken down
into two components, polar energy and azimuthal energy. Polar anchoring energy refers
11
to the energy penalty for the LC alignment normal to the substrate plane. Positive polar
anchoring energies induce what is known as homogeneous or planar alignment where LC
molecules are aligned in the plane of the substrate. Negative polar anchoring energies cause
the LC to align perpendicular to the plane in what is referred to as homeotropic alignment.
Azimuthal anchoring energy refers to the energy penalty for aligning perpendicular to the
preferred orientation direction while remaining in the same plane as the substrate.
Anchoring layers are required for most devices as they provide uniform anchoring across
a substrate, such as is needed for LCDs. This allows for long range macroscopic order within
the bulk LC. Otherwise, defects would dictate local director orientations. One could make
the analogy of anchoring layers to seed crystals required for growing high quality crystals.
2.2
Rubbed Polyimide Anchoring Layers
The success of LCDs has led to the misconception that LC anchoring is a solved problem.
For the case of uniformly aligned substrates, that is largely true. However, there are few
if any viable technologies readily available which allow for the patterning of spatially varying anchoring energies. Currently, two technologies are used as anchoring layers - rubbed
polymers and photoaligned molecules. Both are still under development, but both present
difficult challenges in order to be implemented as a spatially varying anchoring layer.
Rubbed polymers, the most common being polyimide, are used in most LCDs and represent the most mature LC anchoring technology. Using rubbed polyimide seems almost too
simple as one only has to spin on a thin layer of the polymer, usually between 100 and 200
nm thick and then rub with a drum wrapped in velvet. It has been shown that by increasing the number of passes or the pressure while rubbing can increase the anchoring energy.
This process works by introducing anisotropy to the polymer layer. This is readily observed
in the increased hydrophilic tendencies exhibited only in the direction of the rubbing [6].
The anisotropy induces anchoring for the LC molecules deposited on the substrate. LC
molecules naturally favor the directions that exhibit the lowest surface roughness
[2].
The
strength of the anchoring however is not completely dictated by the roughness anisotropy,
12
but by the chemical interactions between the LC and the polymer layer. The actual chemical
mechanisms that cause the alignment were for a long time elusive, however it is now well
understood that wT-bonds between the polymers and LC are responsible [9, 10].
The chemical origin of the anchoring strength is both a blessing and a curse for modern
LC technology. It is favorable because one can consistently achieve very strong anchoring
energies which are uniform over a large substrate (in the case of LCDs, sometimes 100 inches
or more) with a simple fabrication process. Additionally, the anchoring layer is stable for
a wide range of chemical and temperature environments. Perhaps most importantly, very
large anchoring strengths are accessible with this technique.
Unfortunately, because the anchoring strength is dependent on the chemical properties of
the LC and polymer molecules, the technique leaves little room to engineer variable anchoring
strengths which might be useful for more nuanced applications. Anchoring energies are very
high for rubbed polyimide layers, on the order of 10- 4J/n
2,
this almost guarantees that LC
molecules deposited onto rubbed polyimide layers remain fixed in alignment parallel with
the rubbing direction. This rules out devices which depend on switching between parallel
and orthogonal alignments [11].
There have been attempts to modulate anchoring strengths for rubbed polyimide anchoring layers. The first being modulation of the rubbing strength, defined as the pressure
applied while rubbing the substrate.
It has been shown that by increasing the rubbing
strength, one is able to increase the amount of anisotropy in the alignment of the polymer
and ultimately increase the anchoring energy [11]. One can also modulate anchoring energy
by rubbing the polyimide multiple times [2, 6]. Another technique that could be used to
modulate the anchoring energy for a given anchoring layer is varying the concentration of
polyimide molecules within another non-anchoring polymer [12].
Another shortcoming of rubbed polyimide layers is the fact that rubbing is an inherently
coarse method. By its mechanical nature, bringing a brush into contact with a substrate and
then rubbing it some distance is going to be a coarse process. The ability to pattern very
small (on the order of a micron) regions with this method is extremely limited; however,
13
research is being done to utilize "micro-rubbing" techniques, such as scratching with an
AFM tip. This could be effective but will require additional development [13].
Even if these two technologies, variable pressure rubbing and micro-rubbing, can be developed and implemented, it will be difficult to combine them. This would be required if
an engineer wanted to design a truly arbitrary anchoring layer, one such that the anchoring
energy could be varied continuously at very small scales and the anchoring orientation could
be changed with high spatial frequency.
2.3
Photoaligned Anchoring Layers
Photoalignable materials can also be used to create patterned anchoring fields. This technique involves molecules that can be aligned by radiation of polarized light. This alignment
can be in turn transferred to LC molecules deposited on top of the anchoring layer. The
most researched chemicals for this approach are azo dyes [14, 15].
Azo dyes exist in two possible isomers, the cis and the trans states. Absorption of photons
can cause a transition from one isomer to the other. The probability of absorption increases
as the molecules are more closely aligned with the polarization direction of the incident light.
Upon absorption, the molecule will change isomers and realign in an orientation independent
of its original orientation. For this reason, prolonged exposure to polarized light can cause
the azo dye molecules to be aligned perpendicularly to the polarization direction of the
incident light. The mechanics which dictate the alignment of the LC in contact with the azo
dye layer are complex but are governed by the Van der Waals interactions. These make it
energetically favorable for the LC to align parallel to the direction in which a majority of
the azo dye molecules lie [16].
Azo dyes, like rubbed polyimide, generate very strong anchoring layers. Their strength
is again dictated by chemical interactions, but the topography of the substrate can also
contribute [16]. It has been shown that with prolonged exposure times, more ordering occurs
for the azo dye layer and the anchoring energy increases as a result [14]. It has also been
shown that increasing the thickness of the azo dye layer can increase the anchoring strength
14
[15]. These properties may be taken advantage of to utilize photoaligned layers to pattern
varying anchoring energies along a single substrate.
With the advancements made in modern photolithograpy processes, azo dyes present an
opportunity to generate patterned anchoring layers. With the use of multiple masked steps,
varying polarization directions, and variable exposure times it is conceivable that one could
generate arbitrary anchoring patterns. However, these techniques are still in development.
Photoaligned materials also allow for interference patterning. This method is used presently
with much success for various devices. By illuminating a substrate with two different light
sources, one can achieve a standing interference pattern which is deposited on the substrate.
This allows for varying anchor energies with high spatial frequencies [17, 18]. However, since
only a simple pattern may be repeated, interference lithography does not allow for truly
arbitrary anchoring layers.
2.4
Mechanical Anchoring with Grooves
This thesis will discuss the mechanical anchoring energy associated with grooved substrates
in detail. This is probably the least mature anchoring technology, but could be useful in
solving some of the problems associated with LCs today. The effect of grooves establishing
a preferred direction for LC alignment was first discussed by Berreman in 1972 [3].
Berreman hypothesized that LC molecules, given the opportunity, will align parallel to
grooves on a surface. In his paper, he derived the energy penalty incurred when LCs align
perpendicular to the grooves - this is the anchoring energy. As seen in Figure 1, aligning
parallel to the grooves allows the LC molecules to have uniform alignment; however, by
aligning perpendicularly the LC molecules are forced to constantly change orientation, as
shown in Figure 2. This ultimately increases the strain energy present in the LC, making
perpendicular alignment an unfavorable configuration.
The derivation is rather straightforward, first one starts with a sinusoidal pattern on a
surface, assume z = A sin(qx), then assume the LCs align themselves perpendicular to the
grooves in the x, z plane. The molecules will arrange themselves to produce the least amount
15
Figure 1: LC molecules deposited onto ridges and aligning parallel to the ridges. Seen here, the molecules
can all have uniform alignment while remaining tangent to the substrate.
Figure 2: LC molecules deposited onto ridges and aligning perpendicular to the ridges. In order to take on
this orientation and remain tangent to the substrate, LC molecules must constantly change their orientation.
of free energy which would satisfy the equation:
& 22 +
2
2
Tx
0
0
(7)
The solution to this equation is of the form:
$(x,z) = Aq cos(qx)e-qz
(8)
The free energy density for such a configuration is:
K
2
E
(02~
+ (,)2]
-(O2
K
Oz
(9)
(10)
2
16
where u is the volume free energy density and a single elastic constant approximation is used,
such that K is equal to the average of K + K 3 . To solve for the surface energy, integration
is required:
W, =
u(z) dz =
0
4
k( Aq)29q1
where Wa is the azimuthal anchoring energy. Let q = 27/A, where A is the period of pitch
of the pattern and one obtains:
Wa =
2wr3 KA 2
A3
(12)
This derivation requires that the molecules are aligned within the plane of the substrate,
meaning that the substrate must have very high polar anchoring energy (Wp). Such is often
the case, however when W is finite, this theory must be modified.
Work done by Faetti accounts for finite polar anchoring energies. It is shown that the
azimuthal anchoring energy for grooved surfaces is dependent on the LC remaining in the
plane of the substrate. This requires large W ard Wa decreases for small WP according to
[19]:
WA =
2
A3
1
[73K
7rK
1+
(13)
AWp.
This establishes two bounds for the strength of the anchoring layer the first being for high
W
in which case, Faetti's theory reduces to the original Berreman shown in Equation 12.
For low W, the term
Kbecomes large compared to one and the equation reduces to:
A2 H
27 2A2w
A2
Because of the anchoring energy's 1/A
3
(14)
dependence on the pitch of the grooves, a very wide
range of anchoring energy is accessible. Furthermore, with advances in the semiconductor
industry, photolithography, allows us to generate arbitrary patterns of very high resolution.
Using electron beam lithography as discussed in this thesis allows for even higher resolutions.
High resolution allows for very fine pitches and thus very high anchoring energies.
Addi-
tionally, electron beam lithography allows for completely arbitrary patterns of grooves to be
17
fabricated. Thus generating an anchoring energy that varies spatially in both magnitude
and direction becomes feasible. This is not currently afforded by any other technology.
18
3
3.1
Liquid Crystal Beam Deflector
Beam Deflector Model
While LCs are most commonly employed in LCDs, they also are good candidates for other
interesting devices, specifically switchable beam deflectors and non-mechanical beam steering devices. By configuring in a cycloidal pattern as shown in Figure 3.1, LCs can act as
a switchable diffraction grating, which eliminates Oth order transmitted light [20]. This has
been experimentally demonstrated by using interference lithography to pattern a photoalignment layer in a cycloidal pattern on a substrate [17, 21]. LC that is in contact with this
pattern assumes its configuration. Light that is incident on this LC film is decomposed into
left and right hand circularly polarized light, and no 0t order light is transmitted. When
an electric field is applied perpendicular to the substrate: the LC molecules align to it, the
cycloidal pattern is destroyed, and light passes through with no diffraction pattern.
I
I
I
-WOO
I-We
I
Figure 3: This is an example of LC molecules arranged in a cycloidal configuration.
In order to understand how a cycloidal configuration deflects light, Jones calculus is used.
The z-axis is defined as the optical axis and the LC film lies in the (x, y) plane, then to
19
model the the cycloidal LC film, let:
n(x) = (cos(qx), sin(qx), 0)
(15)
where n is the director field, and q = 27/A, A being the period of rotation for the cycloidal
pattern.
Consider a birefringent film which has a "slow-axis", that is a direction for which light
with a polarization direction parallel to experiences a higher index of refraction than for light
with a perpendicular polarization direction. For LCs, their slow axis is almost always their
long axis, largely because it has a higher dielectric constant. A cycloidal LC configuration
can be modeled as a birefringent film with a locally defined, constantly rotating with a trace
in the x direction, slow axis. With this in mind, a Jones transmission matrix for the cycloidal
LC film is defined:
cos(qx)
(sin(qx)
where , = 7r(n, -no)/A,
- sin(qx)
(zsq)
cos(qx)
0e
0
-iz')
cos(q))
sin(qx)
- sin(qx)
cos(qx)
(
ne is the LC extraordinary index of refraction, no is the LC ordinary
index of refraction, and A is the wavelength of the incident light.
Multiplication is then carried out to see the electric fields evolve as they transverse the
LC film:
(=
Ex (z)E(i1
T(x, y) ((O
Ex (0)
(17)
(7
Ey(z) )EY(O)
where z is the coordinate within the thickness of the LC Cycloidal film. From here one
obtains:
Ex(x, z) = Ex(x, 0) cos(zK) + (i) cos(2qx) sin(zK)1 + Ey(x, 0) [(i) sin(2qx) sin(z,)]
(18)
Ey (x, z) = Ey (x, 0) cos(zK) - (i) cos(2qx) sin(zrK) ]+
(19)
20
Ex(x, 0) [(i) sin(2qx) sin(zri)]
These equations can be rewritten in terms of circularly-polarized components. First recall:
R = Ex - f(EY)
(20)
L = Ex + i(Ey)
(21)
Then:
L(x, z)
L(x, 0)cos(zr) + R(x, 0)ie 2iqx sin(zri)
(22)
R(x, z)
R(x, 0)cos(zh) + L(x, O)ie- 2 1qx sin(zh)
(23)
Now these equations are further decomposed into 0 th and
1 st
diffraction orders:
L(x, z) = Lo(z) + L1(z)e21qx
(24)
R(x, z)= Ro(z) + R_1(z)e-
(25)
Finally, the transmitted light is expressed in terms of the light that was incident on the
surface (assuming low amounts of reflection and absorption):
Ro(x,z) = cos(zK,)Ro(x,0) + isin(zh)Lj(x,0)
R
1 (x,
z) = cos(zK)R- 1 (x, 0) + isinz)Lo(x, 0)
Lo(x, z)
L1(x, z)
cox(zi,)Lo(x, 0) + isin(zr.)R- 1 (x, 0)
=COX(ZK)L1(x,
0) + 'ISIn(zK) Ro (x,
0)
From here one can see that so long as the thickness of the cell fulfills the condition Li =
(26)
(27)
(28)
(29)
/2,
and only zeroth order light is incident on the LC film, then no zeroth order light is transmitted
[22]. The amount of diffraction is governed by the classical diffraction grating equation such
that:
sin o = A
A
21
(30)
where 0 is the diffraction angle [23]. This thesis will describe the design of a cycloidal grating
with A = 3 pm.
3.2
Liquid Crystal Director Simulation
Currently in the literature, several groups have demonstrated the ability to construct devices
that utilize a LC film in a cycloidal configuration to deflect incident light with photoaligned
anchoring layers such as the azo dyes discussed in Section 2.3. The cycloidal pattern can be
made permanent in the azo dyes with interference lithography [17, 21]. This anchoring layer
then transfers its configuration to the LC film. The device is switchable by applying an electric field perpendicular to the plane of the substrate; this breaks the cycloidal configuration
as the LC molecules become aligned parallel to the field. Conversely, it could be of interest
to design a device that operates under the opposite modality such that in the absence of an
electric field the device allows light to pass through undeflected, and then when an electric
field is applied, the device becomes a beam deflector.
In order for a beam deflector to operate with the opposite modality, such that the cycloidal
pattern exists only when a voltage is applied; a novel approach is required. This thesis
describes the design of new ways to achieve cycloidal patterns of LCs with the critical
and most novel step being patterning spatially varying anchoring energies. Electron beam
lithography is used to pattern grooves to provide alignment for the LCs such as described
in Section 2.4. The flexibility of electron beam lithography allows for arbitrarily patterned
anchoring layers which vary spatially, not only in anchoring energy, but also anchoring
direction.
Knowing that LC molecules in the nematic phase energetically prefer uniform alignment,
it is clear that achieving a cycloidal configuration will be non trivial. Furthermore, the device
must be bistable such that with no field applied, the LC layer is an optically transparent
layer, however with a field applied, the device must switch in order to allow for a cycloidal
configuration of the LC. Ultimately, one needs to consider the free energy present in the LC
layer at any given time such that the free energy is minimized for the desired configurations.
22
In order to design a cell that would produce a cycloidal LC configuration, computational
simulations were used.
Simulations calculated the LC configuration which contained the
minimum free energy for any given set of design parameters. The model used in this thesis
is based on the work by Mori and the equations that follow are taken from this paper [24].
In order to calculate the minimum free energy configuration, one must define and calculate
the different sources of free energy, starting with the elastic free energy present in a LC film:
(n o (V x n)) 2 + IK 3 (n x (V x n)) 2
fS = 2-K1(V . n) 2 + IK
2 2
2
where
f,
(31)
is the symbol for elastic free energy density, Ki is the splay elastic constant, K 2 is
the twist elastic constant, K 3 is the bend elastic constant, and n is the director field.
The external field free energy that is generated by the interaction of an applied electric
field and the LC molecules defined as:
f
1D-E
(32)
2
where D is the displacement vector which is in turn equal to
EoE
E, co being the permittivity
of vacuum and 6 being the dielectric tensor of the LC. Thus, the total electric field free
energy is equal to:
1
-
E
S2
-Ecoc
(33)
Finally, the anchoring energy is defined using the Rapini-Papolour approximation as [25]:
f =
2
W os 2(0)
(34)
Where W is the anchoring energy magnitude and 0 is the angle which the LC director
deviates from the preferred orientation of the anchoring layer.
In order to simulate the LC director field the model takes advantage of a Q-tensor description. The Q-tensor representation is useful as it incorporates the product of two directors,
thus canceling the sign that could be associated with orientation.
23
A vector approach is
unphysical as it breaks the symmetry inherent in LC molecules. The Q-tensor is defined as:
S
Qjk = 2 (3nJrnk - 6jk)
(35)
In order to reduce complexity of the model, a 2D version was implemented. This approximation is valid because for the cell thickness in question, twist behaviors would result in much
higher energies than the bend and splay behaviors present in a cycloidal configuration. This
leaves the Q-Tensor representation of the elastic free energy as:
fs
=
12G 2
G
2
G(3)
27(K3 - K1) G22 + 9_Ki SK
2 + 2(K3 - K1)
S633
K
2 ± 27 (K
S'
27
(36)
where
G
=Qjk,lQ3jk,l()
(37)
G2
= Qjk,kQjl,
(38)
QjkQIm,jQlm,k
(39)
G
In Q-Tensor representation the electric component becomes:
fe
1
(40)
+ *OEO EVJIVJ Qjk
where C is the average dielectric constant for the LC material such that C =
cL, and Vj =
Ell-
j,
I_being the dielectric constant for electric fields perpendicular to the
LC long axis and Ell being the dielectric constant for electric fields parallel to the LC long
axis.
The total free energy density for the system is equal to
fg
=
f,
- fe.
With the total
free energy defined, the LC layer can be meshed into cells and within any given mesh cell
the director is said to be uniform throughout the volume of the cell. The model operates
by calculating derivatives of free energy for small changes in director orientation and then
'relaxing' to the minimum free energy by iteratively rotating every mesh cell's director in
24
the direction which reduces the local free energy. By taking the derivative of free energy for
every given mesh cell with respect to its director orientation, one can arrive at the following
set of Euler-Lagrange equations:
0
(41)
[fg]i, + Anr
=
where i is an index which traverses every mesh cell in the simulation, A is the Lagrange
multiplier for the constraint that In= 1 and [fg]n, is defined by:
Ofg
On'
d
d
dy
Ofg>
dx (n,,
X)
Of(
On,,
Y)
d (Ofg
dz On., z
(42)
This can be expressed in terms of the Q-Tensor using the chain rule:
Kf91n
On',
=LfglQjk
&Qj-k
(43)
This, in turn, is written as a set of two Euler-Lagrange equations for the x and y components
of the director for every mesh cell in the simulation:
[fgln= 3nxS[fg]Q. + 3fnyS(fg]Qxy + [fg]Q,)
(44)
3nSVf
9 ]QY + 3iYStf 9 1QX + [fg]Qya:)
(45)
[fV91%
These equations contain the following terms:
[G 2 Q, = -
[G
]Q
2 Qgk,11
= -2Q'Ilk
[G6j]Qk = QIn,}QIm,k - Qlm,lQjkr - QIrnQjk,inl - Qhm,mQ3k,I - QImQjklri
(46)
(47)
(48)
All of these contribute to the dynamic equation for the director given by:
718tni -fgnj
25
+ An',
(49)
where -q is the rotational viscosity of the LC. One should note that in the steady state case
this equation reduces to the original Euler-Lagrange equations.
Now the anchoring energy contribution is included. Recall the expression for the anchoring
energy is:
1
fanc
2
W sin 2 (o)
(50)
This is expressed in terms of the director components such that:
Stan- 1
((nY
(51)
This simplification is possible because for the simulations used in this thesis all of the anchoring layers use the y-axis as the preferred direction. Now the derivatives are taken in
terms of nx and ny:
[fac]m, = W sin(#) cos(#) (
[fancnr
=
-W
ny
sin(q) cos(0)(nx) (12)
(52)
(53)
With all of these equations at hand, a relaxation algorithm can be implemented. Rather
than directly solving the Euler-Lagrange equations, the simulation iteratively steps through
small time steps constantly reducing the total free energy of the system. This eventually
yields the free energy minimum. In order to do this, the dynamic equation is discretized
such that:
Ani
71AmY
t
+ An.
YiAt
(54)
Now Thi is set to 1, ignoring the time scale for the simulation and the terms are redistributed:
An
-
A t ([fg]ni + Ani)
(55)
Now A, the Lagrange multiplier is ignored and the director is simply renormalized after
each time step, which yields the equations:
26
+
-
An[f, ]
(56)
~T+1
i+1-i(57)
The LC simulated was 5CB which is a common, readily available LC. Its constants are
given in the literature: Ac = 13.8, K 1 = 5.9 pN, K2 = 4.5 pN, and K3 = 9.9 pN [26]. The LC
film can be meshed into 10 nim by 10 nin squares and the model implemented using periodic
boundary conditions in the horizontal and vertical directions. The desired configuration is
a repeating cycloidal pattern in the horizontal direction with a 3 ptm period, so a unit cell
of the simulations is 3 pm wide but its vertical dimension is allowed to float with various
design parameters.
3.3
Device Design and Optimization
The generation of a standing cycloidal LC pattern was approached as an open ended design
problem. Several general designs were proposed and their design parameters were identified,
these included variable geometries, applied potentials, and variable anchoring energies. These
parameters were optimized using what is known as the NOMADS algorithm.
stands for Non-Linear Mesh Adaptive Direct Search Algorithm.
NOMADS
This is a powerful tool
developed in order to optimize non-linear "black box" functions of many variables [27, 28].
Optimally, the design must cause a thin LC film to arrange in a cycloidal configuration,
such that the director remains in the plane of the substrate but rotates spatially in the
horizontal, or x direction.
The configuration should have translational symmetry in the
vertical or y direction. An ideal LC configuration is shown in Figure 4.
In order to determine an optimal set of parameters, the NOMDADS algorithm requires
an objective function which it uses to rank the quality of any given set of parameters. For a
perfect end state, the objective function must return zero and for deviations, the objective
function must increase proportional to how 'imperfect' the end increases in value. It is up to
27
the user to define the objective function. In this work it was defined as the average angular
deviation of every mesh cell from a perfect cycloidal configuration such as is described by
Equation 15.
...
/
N
-
/
2.5
-
ll t\\.
/
/
1..525.
0.
g'm
Figure 4: This plot shows two periods of the Ideal LC configuration. Designs were evaluated based on their
ability to replicate this configuration.
Initially, the azimuthal anchoring energy was set to zero and the simulations relied on
electrostatics to attempt to achieve the desired configuration. Various patterns of electrodes
with variable geometries and potentials were simulated in hopes of generating an electrostatic
field which would generate a cycloidal pattern 2 . Through many iterations of simulations it
was determined that a good candidate for the electrode configuration would be the one
shown in Figure 5. Grounded bus bars ran vertically on the substrate with island electrodes
in between. Because of the high potentials in between every pair of bus bars, it was believed
that the horizontal fields present immediately in the vicinity of the bus bars would force the
local LCs to align horizontally. Additionally, the rows of island electrodes would alternate
between positive and negative potentials, causing vertical fields between them. The center
island electrodes had high potentials and those on the edges of the border unit cell had lower
potentials. This would generate a vertical electrostatic field that was strong in the center of
each unit cell and decayed toward the edges.
2
1n reality, the field would be driven at 1 KHz as strong DC applied electric fields can damage LC molecules.
28
____
()(b)
6
6
100
2
2
0.5
0
100
Figure 5: Image (a) shows the electrical potential (background color) with local electrical fields superimposed
with a vector plot.
Across the top and bottom are the negative potential island electrodes; the positive
run vertically along the left
potential island electrodes are in the middle of the image. Grounded bus bars
field lines indicate field
electric
The
Volts.
are
bar
color
the
on
displayed
and right sides of the image. Units
configuration.
LC
corresponding
the
shows
(b)
Image
strength.
direction only, not field
The fabrication challenges, while apparent, were for the moment set aside. The goal in
the modeling process was simply to demonstrate that electrostatic fields could be used to
create a free energy minimum for the LC layer in the cycloidal configuration. The NOMADS
optimization was run, allowing for unrealistically high potentials (voltages up to ± 100V).
The design seemed to gain traction for these potentials; however, the design could not
direction.
produce a cycloidal configuration with perfect translation symmetry in the vertical
Also, it was unclear how the fringe effect present in between every pair of island electrodes
would manifest physically. As seen in Figure 5, bands form, for which the cycloidal pattern
alternates between clockwise and counterclockwise rotations of the director field. This could
be acceptable for a beam deflecting device, but the objective function returned average
angular differences of about 50 per mesh cell, so other designs were investigated.
It seemed as though a variable electric fields discussed above would not be sufficient for
the LC to observe a cycloidal configuration so other design parameters were considered.
It was quickly discovered that with a spatially varying anchoring energy, a cycloidal LC
configuration could be achieved with a considerably more feasible fabrication procedure.
First, "binary" anchoring energy, that is anchoring layer that is present in some areas
but not others, was considered. The design parameters now included the strength of the
29
anchoring energy and its span on a give unit cell. The island electrodes were removed
and the vertical bus bars had alternating potentials applied to create a uniform horizontal
electric field throughout the LC film. This field is superimposed on a substrate with regions
of vertical anchoring energy at the borders of any given unit cell.
Thus at the borders
of a unit cell, the anchoring energy would force vertical alignment, toward the center of a
unit cell there would be no azimuthal anchoring energy and the electric field would dictate
that the LC molecules align horizontally. This had moderate success as the average angular
difference of the director versus the ideal configuration was 3.10, furthermore this device
posed no serious fabrication issues concerning the electrodes and the required potentials
were no longer unrealistic, the ideal design requiring only 1 Volt.
0
0
0.A
2
11.
2.5
3
Figure 6: This is the final design for the beam deflecting device. The green fields represent the anchoring energy fields with the darker green signifying stronger anchoring energy. The red arrows depict the electrostatic
field that can be applied in order to achieve the shown cycloidal pattern.
In attempts to improve upon this design, the "binary" anchoring energies were replaced
with regions of variable anchoring energy. Continuously varying anchoring energy would
probably be desirable, but its fabrication seemed unrealistic.
In order to simulate this
device, the anchoring energy regions were partitioned into discrete zones. Now the design
parameters included the electric field strength and the strength and width of the variable
anchoring layer. This design performed very well, with an average angular deviation being
just 1.1' from the ideal configuration as shown in Figure 6. The design parameters arrived
at the for the final design were as follows: W 1 = 6.1 x 10-5 J/m 2, W 2 = 1.2 x 10-
J/m 2,
and E = 0.2 V/pm, where W 1 is the anchoring strength of the dark green field, W2 is
the anchoring strength of the light green field, and E is the magnitude of the electrostatic
30
field present. This new design affords the ability to position as many pitches as necessary
between bus bars. For this reason, one can leave the potential and bus bar spacing as a
further design parameter. For example, if the fabricated device had four pitches (12 pn) in
between bus bars the potential difference from bus bar to bus bar is required to be 2.4 V,
thus the potential at each bus bar would be 1.2 V. While this design was very promising, it
was uncertain if the variable anchoring energies could be patterned, and this motivated the
investigation into LC anchoring layers and variable anchoring energies which is described as
follows.
31
4
4.1
Mechanical Anchoring Energy Experiment
Experiment Objective
As previously discussed, anchoring layers are common in everyday technology, the most
prevalent of which are rubbed polyimide layers. These layers provide extremely strong anchoring energies often times on the order of 10-4 J/m 2 . These strengths virtually guarantee
that the LC remain fixed in the preferred direction, which would be inappropriate for the
designs discussed in Section 3.3. Also, because the rubbing method is coarse, they could not
be patterned into discrete strong, moderate, and absent regions that span only hundreds of
nanometers as required by the designs. One possible alternative discussed in the literature
is the mechanical anchoring energy offered by patterned grooves. By using the fabrication
precision offered by electron beam lithography, the ridges could be patterned onto a surface
and offer anchoring energies according to the formula [19]:
WaWo=273A2K
=_
SA3
1
1+
7K
(8
(58)
Although the formula predicts very high anchoring energies, spatially varying anchoring
energies with the strengths and resolutions required by the designs arrived at in Section 3.3
have not been reported. It was believed that leveraging the technology of electron beam
lithography to pattern grooves of very narrow widths would allow access to high anchoring
energies patterned with high resolution. In order to confirm these anchoring energies could
be fabricated, twisted nematic LC (TNLC) cells were made and optically tested. A TNLC
cell such as in Figure 7, has two substrates with anchoring layers sandwiched together such
that the anchoring layers are in perpendicular directions. LC is then inserted into the cell.
For very strong anchoring layers the LC is aligned well with the anchoring layer on the
substrates; however, for moderate or weak anchoring layers, the LC finds a torque balance in
which the torque of the anchoring layers is balanced by the torque contained internally in the
LC, which exists because nematic LCs want to be uniformly aligned. This torque balance
generates a twist angle within the LC. By measuring the twist angle, one can directly observe
32
the magnitude of the anchoring energy, which follows:
Wa
12K20
d sin(Od - 0)
(59)
where W is the anchoring energy, K2 is the twist elastic constant of the LC molecules, 0 is
the twist angle, and Od is the "design" twist angle which is the rotation angle between the
anchoring directions on each substrate.
4.2
Twisted Nematic Cell Optical Model
Measuring the twist angle is a non trivial problem, but luckily all optical methods can be
used [7, 29, 30, 31]. This section describes the optical model used to solve for the twist angle
of the LC, which, as will be shown in the formalism that follows, exactly equals the rotation
of linearly polarized light's polarization angle at certain wavelengths.
Glass plate
LC molecules
Glass plate
Figure 7: Representation of a TNLC. The LC molecules twist between two substrates with perpendicular
anchoring layers, which are represented by the red lines.
In order to model light passing through a TNLC cell, Jones Calculus is used and the
33
derivation that follows is taken largely from Yariv's text and the work done by Moreno
[7, 32]. First, the LC is assumed to be uniformly aligned along any given slice of the x, y
plane. The cell is then modeled as a large number of thin birefringent slices. Each slice has
a Jones Matrix which is equal to M = R(-V)BoR(0) where R is the rotation matrix for any
given angle, V) is the orientation of the slow axis of birefringence, and B0 is the retardation
matrix which takes the form:
Bo
e
(60)
C-
0
where
4
e Ff/2
is the absolute phase shift and F is the retardation phase shift between the fast and
slow polarizations of light. At this point the cell is then "sliced" into N layers each with
Jones matrices defined by R(-mp)BoR(mp) where p is defined as 0/N. The overall Jones
Matrix for the cell is:
N
= r
M
R(-mp)BoR(mp)
(61)
m=1
Carrying out the multiplication leaves:
M = R(-0)[BoR(6/N)]N R(p)
(62)
For large N, p approaches 0, thus R(p) ~ I. Writing this product out gives:
[N
M = R(-0)
-!
sn )-iF/2N
N
N
-sin( )iF/ 2N COS(0)-iF/ 2 N
COS( 0)e-IF/2N
(63)
This rather unwieldy expression can be simplified quite nicely by use of Chebyshev's identity
which states that:
m
A
B
C DLj
Asin(mKA)-sin(m-1)KA
sin(KA)
[
Lsin(KA)
Csin(mKA)
34
Bsin(mKA)
sin(KA)
Dsin(mKA)-sin(m-)KA
sin(KA)
(64)
where
KA = cos-'[1/2(A + D)]
(65)
With this in mind, M takes the form:
Osin(y)
j sin(I)
Osin(-y)
cos(}) -
M = R(-)
_ Y Biy)
where
=dAn/A
T
and -y =
Q/2
1
(66)
c
) ^ sin(Y)j
cos(-) - ito' (7 _
+ 02, d being the thickness of the cell and An being the
difference in index of refraction for the ordinary and extraordinary axes of the system. From
here two salient regimes can be identified. The first is for cells for which 0 > 0, which occurs
for thick cells. In this case
~y- and the Jones Matrix reduces to:
[
M
= R(-0)
e--1
The second regime is for wavelengths for which
0
]
(67)
2w72
=/n
+ 02, and thus
= n7r.
For this
case, the Jones Matrix reduces to:
A =
1
R(-0)
L 0
(-
(68)
)"l
Both of these conditions are referred to as the adiabatic regime. The first condition, in which
cells are thick, is known as the adiabatic approximation and requires that incident light be
linearly polarized with its polarization direction parallel with the director of the LC at the
input face of the cell 3 . The second condition is known as a local adiabatic point as it is
satisfied only for a narrow band of wavelengths. It is valid for all linearly polarized light,
regardless of input polarization direction. For both cases, linearly polarized light passing
through the cells is able to maintain its linear polarization. This can be seen in the lack
of cross terms in the phase retardation matrices contained in Al above. In the adiabatic
regimes described, the matrix M reduces to effectively a simple rotation matrix as seen in
3
LCDs operate in this regime.
35
Equation 68, where the rotation angle is equal to 0, direct observation of the twist angle is
permitted.
4.3
Patterning HSQ
As stated earlier in Section 4.1, a grooved surface providing mechanical anchoring energy
was a possible candidate for the anchoring layers needed in the final design of the beam
deflecting device. In order to pattern the grooves, the electron-beam resist Hydrogen Silsequioxane (HSQ) was used. This is a rather complex spin on glass electron beam resist. It
contains hydrogen, silicon, and oxygen atoms and when exposed by electron beam radiation,
the silicon-hydrogen bonds are broken leaving only a silicon oxide compound; however, the
precise reaction chains are still topics of research [33, 34]. It is an inorganic resist capable
of extremely fine patterning and in fact possesses one of the best resolutions available for
electron beam resists [35].
Because the beam deflecting device being developed is to operate in the visible light
regime, transparent substrates are required for its fabrication. For this reason, fused silica
substrates are selected, these in turn present two challenges. First, the substrates are insulating, thus electron beam radiation leads to charging of the substrate which materially
changes the effective dose delivered. Also, the electron beam system used requires substrates
to have a reflective surface in order to measure local substrate height with the use of an interferometer. Local height must be precisely measured as small deviations in height, on the
order of 1 pm, can deteriorate beam's focus which has a spot size of approximately 10 nm.
With other resists, one is able to overcome said challenges by simply evaporating chromium
onto the top of the resist in order to dissipate the charging and provide a reflective surface
to accurately measure the height of the substrate. A 50 nm layer of chromium is used as
it is sufficient to provide an effectively opaque layer for reflection; however, when chromium
is applied directly to the HSQ it corrupts the resist with a mechanism that is not well understood. To determine if this effect was caused by the chromium itself or the substrate,
the HSQ with 50 nm layer of chromium deposited was written on a silicon substrate. It
36
was discovered that the same degradation pattern existed for HSQ topped with chromium
written on both silicon and fused silica substrates. A SEM image of lines and spaces written
in HSQ spun onto silcon with chromium evaporated directly on top of the HSQ is shown in
Figure 8.
Figure 8: SEM image of lines and spaces written on HSQ with 50 nm of Chromium on top on a
silicon
substrate. Clearly, the chromium corrupts the HSQ.
Chromium deposition ruins the resist entirely, so chromium was evaporated onto only
the outer ring of the substrate such that the area which was actually to be patterned was
uncovered. The outer ring was used to remove as much tilt as possible from the substrate
using an adjustable stage. By leveling the substrate, it was believed that beam focus would
not be an issue. The issue of charging was left unmitigated, the hope being that it would
not be severe enough to greatly affect the patterning. Unfortunately, as seen from Figure 9
the charging proved catastrophic, and while some interesting patterns were observed, these
were not useful.
After several rounds of tests, eventually it was discovered that by depositing a shielding
layer of polymer on top of the HSQ, one can mitigate the effects of chromium corruption and
write patterns onto the HSQ. It was discovered that a shielding layer of 360 nm of EL11, a
copolymer resist, was sufficient to protect the HSQ from the effects of the chromium. Atomic
37
Figure 9: SEM image of HSQ written on fused silica substrate with no charge dissipation layer.
Force Microscopy was used to take an image shown in Figure 10 on the patterned lines made
using the process described here.
25. 0 rn'
12.5 rwn
0
0.8
0.6
0.2
0.4
0.4
0.2
0.6
0.8.
Figure 10: AFM image of HSQ written on fused silica substrate with a shielding layer of 360 nm of EL11
and a 50 nm chromium charge dissipation layer. The z scale is in nanometers and the x and y scales are in
microns.
Even with these steps taken, the HSQ was not exposed perfectly and possessed significant
line edge and surface roughness.
However, because the anchoring properties of grooved
substrates are based on long range elastic interactions, it was determined that the grooves
patterned would prove effective at creating an anchoring layer.
38
4.4
Cell Fabrication
In order to test the anchoring energy of nanogrooves, several TNLC cells of differing groove
pitch and HSQ thickness were prepared.
the grooves onto the substrate.
fused silica.
First, the lithography was performed to pattern
Hexamethyldisilazane (HMDS) was first spun onto the
HMDS dehydrates the substrate surface, priming it for HSQ. An HSQ 2%
solids solution in Methyl Isobutyl Ketone (MIBK) was used at spin speeds of 2000, 3000,
and 4000 RPM, generating different HSQ thicknesses.
Following this step, the polymer
shielding layer mentioned in 4.3 was deposited. A copolymer resist was used which contained
Methylmetacrylate (MMA) and Methylacrylic Acid (MAA) in an 11% solid solution of ethyl
lactate. This is sold under the name EL11 by MicroChem. EL11 was spun on at 3000 RPM,
resulting in a layer that is 360 nm1
thick4 . Finally, the chromium was deposited.
The samples were then exposed in the electron beam lithography system using a dose of
525 pC/cn12 for HSQ spin speed of 2000 RPM, 575 piC/cm 2 for HSQ spin speed of 3000
RPM, and 600 pC/cn
2
for HSQ spin speed of 4000 RPM, these resulted in groove depths
of 26 nm, 22 nm, and 18 nm respectively.
These doses are much lower than required for
normal HSQ patterning on silicon, but are necessary due to the increased sensitivity when
patterning on an insulating substrate. The doses vary for each spin speed as was required to
ensure equal line and space width. After exposure the chromium was etched from the wafers
using chromium etch type 1020 which is a combination of Nitric Acid and Ceric Ammonium
Nitrate. The samples were then rinsed with acetone and IPA in order to remove the EL11
layer. The HSQ was then developed in Tetra-muethyl Ammonium Hydroxide, the developer
used was AZ 300MIF. The development was done at 50' C for two minutes. The development
was followed by an IPA rinse and an N 2 blow-dry.
The samples were then sealed together using SU8, a photoresist, which was patterned
to form a cavity for the LC cell. The SU8 acted as the spacer and bonding material for
the cell [36]. SU8 2001, provided by MicroChem, was spun on at 2000 RPM, then exposed
4
One should note that it is rather curious that such a thick layer is needed. Thinner shielding layers provided only slight
protection from the chromium. This suggests that the EL11 is not a simple chemical shield and the corruption mechanism
caused by chromium deposition is rather complex.
39
with ultra-violet radiation for 30 seconds; this primes the SU8 for subsequent crosslinking
which occurs at higher temperatures. A post exposure bake of 5 minutes was used at 950 C,
this lightly crosslinks the exposed resist so it may survive development. The SU8 was then
developed using SU8 developer for 30 seconds, removing the unexposed SU8. The samples
were then rinsed with IPA and blow-dried with N2 . After development the thickness of the
patterned SU8 layer is 1.1 pm. At this point the wafers were aligned 900 to each other and
pressed together at 1 psi and hardbaked at 150' C for at least 12 hours. The hardbake step
completes the crosslinking process and bonds the substrates together. This would decrease
the SU8 layer thickness slightly such that the total layer thickness after the cell was bonded
was 1.5 pm. The bonding introduces some beam blur but the thickness could be measured
by optical methods as will be described in Section 4.6.
A channel was patterned into the SU8 such that LC could be wicked into the cavity and
air could be let out from the opposite end of the channel. It is important to wick in LC
while it is in an isotropic state, thus the cells were raised to 400 C while wicking in the LC.
10 pL of LC was deposited onto the substrate next to the cavity channel. Once deposited,
surface tension naturally wicked in the LC until it filled the cavity. At this point, the cells
were left to cool to room temperature and once the LC was in the nematic state, the cells
were ready to be measured.
4.5
Experimental Setup
In order to accurately measure the twist angle, the cell must be probed with wavelengths of
light that fall on local adiabatic points. By using a white light source and a spectrometer,
many wavelengths could be probed simultaneously to ensure that the adiabatic condition
was satisfied. So long as -y = nr then linearly polarized light will have its polarization
twisted by the twist angle of the cell regardless of entrance orientation [7].
To test the
twist angle, each LC cell was placed in between two linear polarizers. For a given input
polarization, the analyzer is rotated through 1800 to find the maximum transmission. If the
adiabatic condition is satisfied, then the difference in angle of the two polarizers which yields
40
a maximum in transmission is constant as the polarizer changes orientation. By ensuring
that this is the case, one is able to verify that the light is in the adiabatic regime and measure
the cell twist angle as it is equal to the difference in the angle of the two polarizers.
Detector
Analyzer
White
light
source
Twisted liquid
crystal cell
Polarizer
Figure 11: Shown here is a schematic of a basic polarimeter.
The experiment utilizes a polarimeter. A schematic is shown in Figure 11. This entails a
white light source, two polarizers, the sample (a TNLC cell), and a spectrometer. The white
light source is fiber coupled and directed to the first polarizer which establishes linearly
polarized light.
The linearly polarized light enters the TNLC cell and undergoes some
rotation. The light then passes through the analyzer. From there the light enters a fiber
receiver which delivers the light to a spectrometer.
The experimental setup is shown in
Figure 12.
4.6
Results
The LC twist angles were observed for varying groove pitches and depths. Then, the anchoring energies were calculated by using the equation:
Wa =
2K 2 0
dsin(Od - 0)
where Wa is the azimuthal anchoring energy, K
2
is the twist elastic constant (4.5 pN) [26],
0 is the twist angle, and Od is the design twist angle.
41
(69)
LC Twist Cell
FiberAnalyzer
Polarizer
Fiber light
emitter
Fiber Coupled
Spectrometer
Fiber Coupled
White Light
Source
Figure 12: Photo of experimental apparatus.
100 nm pitch grating, height 18 nm
250
- - -
Adiabatic Point, X= 440 nm
Off Adiabatic Point, X= 460 nm
200
150
100
50
0
50
100
150
200
Polarizer Angle
Figure 13: This plot shows the analyzer angle required to achieve the maximum transmission vs the polarizer
angle exactly on and slightly off the local adiabatic point-intensities in transmitted light on and near the
adiabatic regime.
In order to confirm the cell thickness knowledge about the adiabatic regime was used.
As seen from in Figure 13, a very precise wavelength is required to observe the adiabatic
regime and when the regime is satisfied the analyzer angle tracks exactly with the polarizer
angle. Once the twist angle was obtained, the equation -y = r could be used to calculate
the thickness of the cell. Because -y2 -2
+ 32
42
and 3 is defined as: TAn(A)d/A, where An
is a function of A can be described by Cauchy equations for the extraordinary and ordinary
indices refraction [37]:
.00489
.0026
2
+ .
2
A4
A
no
1.676 +
Te
1.519 +
.0016
2
A
(70)
.0012
4
+
(71)
With a, An, and A known the thickness could be calculated. Once the thickness of the cells
was confirmed, the anchoring energies were calculated and are shown in Figure 14.
x 10'
0 18 nm depth
0 22 nm depth
0 26 nm depth
6
5
0D
W4
CD
3
2-
14
140
160
180
200
220
Groove Period (nm)
240
260
Figure 14: Azimuthal anchoring energies measured for varying groove depth and period.
The reader will notice the large error bars present for higher anchoring energies. This
makes sense as higher anchoring energies correspond to larger twist angles. Recall that
WaO( sin(6 -)'
thus as 0 approaches Od, a rather small uncertainty in 0 yields a very large
uncertainty in W. This is illustrated in the plot in Figure 15.
4.7
Analysis
Recall the model proposed by Faetti [19]:
Wae
27 3 A 2 K
A=_
A3
43
1
1+
7K
AWp_
(72)
---
er=
1 pm Thick Cell
5 pm Thick Cell
E 10
W 10'
10
106
10
20
10
60
40
Twist Angle (0)
80
Figure 15: Azimuthal anchoring energy as a function of measured twist for two cells of different thickness,
notice how they each provide better resolution for anchoring energies of different orders of magnitude.
where W, is the azimuthal anchoring energy, A is the groove depth, K is the average of the
elastic constants KI and K 3 ; for 5CB, this is 8 pN [26], A is the groove period, and W, is
the polar anchoring energy. The literature provides a range of values for the polar anchoring
energy of glass or fused silica substrates, so Wp is allowed to be a fitting parameter which
varies from 10-5 J/m
to 10-3 J/m 2 [38]. The results are shown in Figure 16, allowing the
2
polar anchoring energy to be fitted to the data using a standard least squares fit yielded WP
= 7.1 x 10-5. The results of this fit seem to suggest that the effect of narrowing the groove
pitch results in stronger anchoring energies than predicted by this model.
x 10
0
6
0
0
---
.5
18 nm depth experiment
22 nm depth experiment
26 nm depth experiment
18 nm depth theory
22 nm depth theory
- 26 nm depth theory
I4
0
2
E
I
?40
160
180
220
200
Groove Period (nm)
240
260
Figure 16: The anchoring energy data is shown with the energy predicted by Faetti's modification to the
Berreman model.
44
5
Discussion
While searching through the literature some interesting trends emerged that were not entirely understood or explained. It is only recently that technology has allowed for such fine
structures to be patterned on surfaces that researchers are encountering the limits of the
regimes for which the models of Berreman and Faetti may be applied. Other data sets in
literature deviate from theory for narrow groove pitch [39]. Two papers suggested that increased azimuthal anchoring energy led to an increase in the order parameter, one of which
examined grooved substrates [40], the other of which examined rubbed polymers [41]. Several papers also show that a linear increase in the order parameter led to a linear increase
in the polar anchoring energy [11, 41, 42]. Yet another paper observed a coupling between
the azimuthal anchoring energy and the polar anchoring energy such that increasing groove
depth led to an increased polar anchoring energy [43].
7
E
0 150 nm period
0 200 nm period
6
0
250 nm period
5S)4
E
2
2
900
400
600
500
700
2
(Groove Depth) (nm )
Figure 17: The anchoring energy is plotted versus the square of the groove depth. Each period is shown
with a fit line for which Wp was optimized.
In order to investigate these trends, the data is plotted versus the square of the groove
depth. According to the theory, this should yield straight lines. As shown in Figure 17, the
data seems to correlate well with the theory in this regard. For each groove period, a least
45
squares fit is used to calculate the polar anchoring energy that best fits the data. When this
technique is used, three separate polar anchoring energies are found: for the 150 inn groove
period, Wp = 1.9 x 10- 4 J/m 2 , for the 200 nm groove period, Wp = 3.9 x 10- 5 J/m 2 , and
for the 250 nm groove period, Wp = 1.8 x 10- 5 J/M 2 . This result agrees with the literature
that patterns that exert increases azimuthal anchoring energy increase the polar anchoring
energy present [43].
The mechanism that could couple the polar and azimuthal anchoring energies is unknown.
It is suspected that an increase in the azimuthal anchoring energy can lead to an increase
in the order parameter which in turn increases the polar anchoring energy. This increased
polar anchoring energy lowers the effect of Faetti's correction, which states that the azimuthal
anchoring energy is scaled by the term
1This
1+7rK/AWp--Ti
but
developed,
u
must
dvlpd
ul
ut beemrmore fully
this could provide insight into deviations from commonly held theories.
In any event, despite the deviation from theory of the results provided here, it was demonstrated that variable anchoring energies could be provided with the use of grooves patterned
by electron beam lithography. The energy ranges obtained from grooved patterns in this
study range from 1x 10-6 j/m2 to 60x 10-6 J/m 2 . This includes the range required by the
designs for the LC based beam deflector discussed in this paper and could also be used in a
variety of novel optical devices.
46
Acknowledgments
This work was sponsored by the U. S. Army Natick Soldier Research Development and
Engineering Center under Air Force Contract FA8721-05-C-0002. Opinions, interpretations,
recommendations and conclusions are those of the authors and are not necessarily endorsed
by the United States Government.
I have utmost gratitude for the US Navy for allowing me to pursue a Masters Degree,
the MIT Lincoln Laboratory for granting me a tuition award, and the MIT Department of
Materials Science and Technology for allowing me to conduct my research at the Lincoln
Laboratory. A special thanks to Professor Silvija Gradeeak for being my thesis reader.
Everyone at the Lincoln Lab has proven eager to help, especially Mike Marchant, who
evaporated more 50 nm layers of chromium than I care to count, Donna Lennon, who on
multiple occasions would drop everything to answer the slightest metrology question or help
with imaging, and everyone in Group 81 who showed interest in this work and volunteered
their time to help me work through whatever particular snag I had caught in my research
that day, especially those who attended my subgroup meetings - Michael Geis, Vladimir
Liberman, Lalitha Parameswaran, Rich Kingsborough, and Keith Krohn. Perhaps most
importantly, a thank you to Ken Diest, my thesis advisor - his patience is the stuff of legend.
47
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