CONTROLLED SURFACE STRUCTURING
WITH NONLINEAR LASER LITHOGRAPHY
a thesis submitted to
the graduate school of engineering and science
of bilkent university
in partial fulfillment of the requirements for
the degree of
master of science
in
electrical and electronics engineering
By
¨
Ozg¨
un Yavuz
Controlled Surface Structuring with Nonlinear Laser Lithography By ¨Ozg¨un Yavuz
January 2018
We certify that we have read this thesis and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.
Fatih ¨Omer ˙Ilday(Advisor)
O˘guz G¨ulseren
Mehmet Emre Ta¸sgın
Approved for the Graduate School of Engineering and Science:
Ezhan Kara¸san
Director of the Graduate School ii
ABSTRACT
CONTROLLED SURFACE STRUCTURING WITH
NONLINEAR LASER LITHOGRAPHY
¨
Ozg¨un Yavuz
M.S. in Electrical and Electronics Engineering Advisor: Fatih ¨Omer ˙Ilday
January 2018
Self-organisation has always fascinated researchers from all branches of sciences and engineering. Despite its ubiquity, our present understanding of its core princi-ples and in particular how to control self-organised phenomena is at its infancy. A particularly rich case of self-organisation arises from the interactions of intensely powerful laser beams with material surfaces. As this phenomenon leads to forma-tion of sub-wavelength, thereby, nanoscale periodic structures through a simple, one-step process performed in ambient atmosphere, there has been tremendous interest in its use in applications, ranging from tribology to data storage. How-ever, there remains much to be desired in terms of our ability to control, regulate, dynamically modify the resulting structures. A technique recently demonstrated in our group, Nonlinear Laser Lithography (NLL), has made possible the creation of extremely uniform, virtually perfectly periodic self-organised nanostructures, which are in the form of parallel nanoscale lines. These nanostructures can be used to cover or tile indefinitely large areas without any apparent loss in quality or uniformity. Armed with this advance, we are now in a position to look beyond getting simply periodic structures and to develop conceptual tools and practical techniques for advanced control of the self-organisation process and to create a vast array of self-organised structures. In this thesis, we first develop a rigorous theoretical model for NLL, which we then show to possess excellent predictive power and can efficiently guide the experiments. We first reveal an interesting, self-organised effect, namely that the nanostructures respond to a tilting of the laser beam’s wavefront in a manner that is strongly analogous to the well-known Doppler effect. Further, building on the rigorous model developed in this the-sis, we propose and experimentally demonstrate that noise or modulations in the laser beam or defects on the surface can each steer the self-organised process. We further show that by deliberately introducing noise or defects, we can achieve patterns that are impossible to achieve otherwise. As an ultimate demonstration
iv
of this capability, we report on the creation of all the Bravais lattices possible for a surface. While the main results to be reported concern the NLL technique, the conceptual tools developed in this thesis rely on general properties of self-organisation through an interplay of positive and negative nonlinear feedback mechanisms. This defines a broad class of self-organising systems. As such, it is likely that the techniques we introduce can be appropriately adapted to achieve similar control over self-organised patterns forming in entirely different physical systems.
Keywords: Self-organisation, Nonlinear Laser Lithography, Doppler Effect, Non-linear, Far-from Equilibrium Systems, Bravais Lattices.
¨
OZET
DO ˘
GRUSAL OLMAYAN LAZER L˙ITOGRAF˙IS˙I ˙ILE
KONTROLL ¨
U Y ¨
UZEY ˙IS
¸LEMES˙I
¨
Ozg¨un Yavuz
Elektrik ve Elektronik M¨uhendisli˘gi, Y¨uksek Lisans Tez Danı¸smanı: Fatih ¨Omer ˙Ilday
Ocak 2018
Kendili˘ginden organize olma, t¨um bilim ve m¨uhendislik dallarındaki ara¸stırmacı-ları etkilemi¸stir. Her yerde varolmasına ra˘gmen, temel mekanizmaları ve ¨
ozellikle kontrol edilebilmeleri hakkındaki bilgimiz daha emekleme a¸samasındadır. Olduk¸ca zengin bir kendinden organize sistem, kuvvetli lazer ı¸sınlarının malzeme y¨uzeyine vurulmasıyla a¸cı˘ga ¸cıkar. Bu fenomen ile lazer dalga boyundan kısa, dolayısıyla nanometre mertebesinde, tek a¸samalı, g¨unl¨uk atmosfer altında, yapılar ¨
uretilebilmektedir. Bu yapılar, tribolojiden veri kayıt teknolojisine kadar geni¸s bir kullanım alanına sahiptir. Fakat, bu yapıları kontrol etmek, d¨uzenlemek, di-namik olarak de˘gi¸stirmek hala ba¸sarılamamı¸stır. Bizim grubumuzda bulunan yeni bir teknik olan Do˘grusal Olmayan Lazer Litografisi (NLL) ile muazzam kalitede, periyodik, kendinden organize, nanometre mertebesinde paralel ¸cizgiler halinde yapılar elde edilmi¸stir. Bu nano yapılar ile sınırsız boyutlarda bir alan, her-hangi bir kalite kaybı ya¸samadan kaplanabilmektedir. Bu geli¸sme sayesinde, ba-sit ¸cizgiler dı¸sında, kendinden organize sistemler ¨uzerinde pratik bir bilgiye sahip olup, olduk¸ca ¸ce¸sitli kendinden oranize ¸sekiller elde edilmi¸stir. Bu tezde, ilk olarak, NLL tekni˘ginin detaylı bir matematiksel modeli yapılmı¸stır. Geli¸stirilen modelin deneyleri m¨ukemmel ¨ong¨ord¨u˘g¨u ve y¨onlendirdi˘gi g¨osterilmi¸stir. Bu model sayesinde, nano yapıların tepkisi ¨uzerinde, lazerin e˘gilmesiyle olu¸san dalga cephesi de˘gi¸simine ba˘glı, kendinden organize bir etki g¨osterilmi¸stir. Bu etki, iyi bilinen Doppler etkisine kuvvetli benze¸sim g¨ostermektedir. Bu tezde geli¸stirilen matematiksel modele dayanarak, g¨ur¨ult¨u ve mod¨ulasyonun kendinden organize olma s¨urecini y¨onlendirdi˘gi, deneylerce g¨osterilmi¸stir. Dahası, kasten dahil edilen g¨ur¨ult¨u veya kusurlar ile ba¸ska t¨url¨u olu¸sturulamayacak motiflerin olu¸sması m¨umk¨un kılınmı¸stır. Bu kabiliyetin nihai bir g¨osterimi olarak, olası b¨ut¨un Bravais kafesi yapılarının malzeme y¨uzeyinde olu¸sturuldu˘gu rapor edilmi¸stir. G¨osterilen asıl sonu¸clar her ne kadar NLL tekni˘gi ile alakalı olsa da; bu tezde geli¸stirilen
vi
kavramsal ara¸clar, eksi ve artı, do˘grusal olmayan geri beslemenin etkile¸simi sayesinde, kendinden organize olmanın genel ¨ozelliklerine dayanmaktadır. Bu, ¸cok ¸ce¸sitli kendinden organize sistemleri tanımlamaktadır. Bu tezde uygulanan teknikler, motif ¨ureten, kendinden organize, tamamen farklı fiziksel sistemleri benzer ¸sekilde kontrol etmek i¸cin uyarlanabilir.
Anahtar s¨ozc¨ukler : Kendili˘ginden Organize Olma, Do˘grusal Olmayan Lazer Litografisi, Doppler Etkisi, Do˘grusal Olmayan, Dengeden Uzak Sistemler, Bravais Kafesleri.
Acknowledgement
I would like to thank my supervisor F. ¨Omer Ilday for his wise guidance and patience through my research. I appreciate the opportunity he has given to me to conduct this research, and the scientific environment that he has created in UFOLAB.
I would like to thank Onur Tokel for being such a great mentor to me alongside F. ¨Omer Ilday. They have thought almost everything I learned about academia and conducting research. This work would not be concluded without the discus-sions we have made. I appreciate the invaluable guidance and friendship that they have supplied to me.
I would like to thank all UFOLAB members for their support, friendship and enjoyable discussions.
I would like to thank my family ¨Umit and Sabiha Yavuz for their support throughout my life. They have not let me down whenever I needed them, includ-ing this research period.
Contents
1 Introduction 1
2 Nonlinear Laser Lithography 4 2.1 Theory . . . 4 2.2 Simulations . . . 8 2.3 Experimental Setup and Experiments . . . 11
3 Phase Control and Doppler Effect 14 3.1 Theory . . . 14 3.2 Simulations and Experiments . . . 16
4 Controlling Surface Patterns via Structured Noise 19 4.1 Structured Noise . . . 19 4.2 Pattern Selection . . . 21 4.3 Stochastic Resonance . . . 24
CONTENTS ix
List of Figures
2.1 Flowchart of Nonlinear Laser Lithography (NLL). . . 9 2.2 Interference pattern and emergence of the surface structures from
a single defect being placed at the centre of simulation domain. (a) Total intensity on the surface from vertically polarised laser beam. (b) Emergence of stripes from a single defect within 10,000 pulses. (c) Total intensity on the surface from circularly polarised laser beam. (d) Emergence of hexagonal pattern from three defects within 200,000 pulses. . . 10 2.3 Simulations to identify the effect of defect on NLL. Laser beam has
a beam width of 4 µm (1/e2) and scanned from left hand side to
right hand side. (a), (b) Scanning done with vertically polarised laser beam. (c), (d) Scanning done with horizontally polarised laser beam. Defect sizes in diameter are (a), (b), (c), (d), 500 nm, 100 µm, 500 nm, 100 µm respectively. . . 11 2.4 Experimental setup of NLL. . . 12 2.5 Simulation an experimental results of NLL. Laser beam is scanned
starting from left hand side to right hand side. (a), (b) Simula-tion results with vertically and horizontally polarised laser beam, respectively. (c), (d) Experimental results corresponding (a) and (b) respectively. . . 13
LIST OF FIGURES xi
3.1 Experimental configuration and laser total intensity on the ma-terial surface from a single defect. (a) Usual NLL beam-sample configuration. (b) Configuration to realise the Doppler effect. (c), (d) Simulation of the total intensity on the sample surface from a single defect, being placed at the centre of the simulation domain, with vertically polarised laser beam, using configurations (a) and (b) respectively. . . 16 3.2 Illustration of Doppler effect and simulation of total intensity on
the material surface. (a) Schematic of Doppler effect with for source, where ‘O’ is observer, ‘S’ is source and ‘θ’ is the angle between observer and source with respect to horizontal axis. (b) Simulation of total intensity on the surface with circularly polarised laser beam. . . 17 3.3 Comparison of Doppler effect and NLL simulation. Each pixel
in images corresponds to a period with respect to incidence and polarisation angle. (a) Calculations according to Doppler effect, Eqn. 3.10, where ϕ is polarisation angle and θ is incidence angle. (b) NLL simulation results. . . 17 3.4 Comparison of simulation and experimental results for Doppler
effect in NLL. . . 18
4.1 Feedback strength of each lattice with a given NLL feedback sys-tem configuration. In the natural mechanism, the structures tend to evolve in hexagonal pattern shown in upper part. Depending on the structured noise, the pattern can be guided to another pattern. (a) The natural way for pattern emergence with given initial con-ditions. (b) Structure guiding via structured noise from hexagonal to square pattern. . . 20
LIST OF FIGURES xii
4.2 Cartoon illustrating Bravais lattices. (a) Square lattice where a1 =a2 and ϕ = 90o. (b) Hexagonal lattice where a1 =a2 and
ϕ = 60o. (c) Oblique lattice where a
1 6=a2 and ϕ 6= 90o. (d)
Rect-angular lattice where a1 6=a2 and ϕ = 90o. (e) Centred rectangular
lattice where a1 6=a2, ϕ 6= 90o, and a1⊥2a2. . . 22
4.3 Transitions between hexagonal pattern to square pattern using structured noise in the simulation. The laser is scanned from left to right in both figures. (a) Line defect is added at the centre of the simulation domain. (b) Spatial modulation is applied to the laser beam at the centre of the scanning such that the modulation frequency matches with the feature size of square pattern (870 nm). 23 4.4 Bravais lattices realised with NLL using structured noise in
sim-ulations and experiments. (a)-(e) NLL simulations. Surface is preconditioned with horizontal lines again using NLL. Then, the laser beam scanned from left hand side to right hand side with dif-ferent polarisations. Initial defect to guide the patterns are shown with white solid lines. In (a) and (b) vertically polarised laser beam is used. In (c) polarisation angle is 45o with respect to the
vertical axis. In (d) and (e) Doppler effect is employed in order to obtain period difference with vertically polarised laser beam. (f)-(j) Experimental realisation of Bravais lattices. (f) Atomic force microscope image of hexagonal lattice. (g)-(i) Microscope images for realisation of Bravais Lattice. All scale bars are 5 µm, except (g), which is 3.5 µm. . . 24
LIST OF FIGURES xiii
4.5 Explanation of Stochastic Resonance (SR) phenomena in a com-munication system. (a) Transmitter device with maximum peak intensity of 1 arbitrary unit (A.U.) over time. (b) Plot of the trans-mitted signal through a noisy channel with a noise density of 0.5 A.U.. Detection threshold of receiver is marked with red line at 1.5 A.U.. (c) Received signal (blue), and transmitted signal (red) over time. (d) Transmitted signal through a noisy channel with a noise density of 1 A.U.. (e) Received signal (blue), and transmitted signal (red) over time. . . 25 4.6 Illustration of SR phenomena in NLL simulations. In (a) each
colour represents a lattice. Blue pixels represent square lattice, yellow pixels correspond to hexagonal lattice. Colours in between blue and yellow represent combination of hexagonal and square lattices. (b) Illustration of SR behaviour with 15 realisation of simulations and the average of them. Blue data points and red solid line represent the average value of the 15 realisations and the moving average of the simulation results respectively. Modulation amplitude is fixed to 2.5% in simulations, which corresponds the marked region in (a) with red box. . . 26
Chapter 1
Introduction
Just after the invention of the laser, it found its usage in various fields from laser cutting [1, 2], drilling [3, 4, 5, 6], optical trapping [7, 8, 9] to detection [10] and characterisation [11, 12] systems. Further, laser material interactions are extensively investigated depending on the laser parameters, either in atomic scale [13, 14, 15] or in terms of material processing [16, 17]. One of the most exciting phenomena is called Laser Induced Periodic Surface Structures (LIPSS) and discovered just five years after the invention of the laser in 1965 [18]. The rea-son of being such important phenomena is that beyond the application capability, it has unique pattern formation aspect.
LIPSS is first explained by the interference between incident electric field and Surface Plasmon Polariton waves (SPP) induced by the incident laser beam [19, 20, 21]. Firstly observed surface ripples were perpendicular to the laser polarisation [18]. Typical feature size of the ripples are in the order of laser wavelength in the free space. In the following years, the structures forming parallel to the laser polarisation has been shown [22]. Thus, merely because of historical reasons, the ripples being perpendicular to the laser polarisation are called “normal” LIPSS and the ripples being parallel to the laser polarisation are called “anomalous” LIPSS. However, the explanation of the anomalous LIPSS is not that straightforward. The problem is that while SPP waves fully satisfy the
boundary conditions of air material interface, the scattered waves being parallel to the laser polarisation simply cannot. The explanation was made through the “radiation remnant” or “retarded potential” approach in which the waves does not have to propagate on the surface [23]. Recently, novel technique of Nonlin-ear Laser Lithography [24] (NLL) is developed which overcomes the stubborn problems of LIPSS. In the first chapter of this thesis, the rigorous mathematical description of NLL is given. The main problem with the LIPSS was the quality of the structures which has been overcome with the NLL technique. The technique uses the coherent extension of the surface structures via scanning a small laser beam over the material surface. The newly formed surface structures get feed-back from the already existing surface structures. Thus, highly stable, periodic and coherent structures have been achieved using NLL.
In order to have total control over the surface structures, one must be able to tune the feature size. The feature size can be changed in two ways. First, the laser wavelength can be changed. However, the response of the target material can differ depending on the laser wavelength. Interactions may not be explained through either aforementioned techniques or first chapter of this thesis. Second way to alter the wavelength is changing the effective wavelength on the material surface. Effective laser wavelength can be adjusted by introducing linear phase or oblique incidence to the target material. The approach of oblique incidence is analysed previously [19]. However, polarisation of the laser beam is assumed to be fixed to a certain axis in the analysis. In this thesis, the approach is extended to NLL case and improved by considering variable polarisation angle. The results show that the effect of additional linear phase on structure period is identical to the Doppler effect in classical electromagnetics.
Beyond the highly stable, periodic processing capability, NLL is a very rich platform to investigate the nonlinear, far-from-equilibrium dynamics. Classical mechanics approaches and thermodynamics does not necessarily (almost never) work with the far-from equilibrium and nonlinear systems. While the linear and equilibrium systems are static in terms of energy transfer and pattern formation, the far-from equilibrium, nonlinear systems have rich capability of pattern for-mation. In nonlinear, far-from-equilibrium systems the entropy is not a useful
parameter [25], even if it is calculated it does not necessarily maximise itself. In fact, rate of entropy production might decrease locally while total entropy in the larger scale keeps increasing. Typical example for this scenario would be Maxwell’s Daemon [26]. Maxwell’s Daemon is a thought experiment introduced by Maxwell himself. In the experiment there are two containers, one of which is empty and other is filled by a gas, are connected through a lid. When a particle hits to the door, if it has enough speed (energy), the daemon lets the particle pass, otherwise the door is locked. At the end of the day, one of the containers is filled by the slow (less energetic) particles, the other is filled by fast (energetic) particles. Since the uniformity of the particles in both boxes is provided, total entropy decreases. The experiment shows that the entropy does not necessarily increase. However, here, the mistake was excluding the daemon himself from the total system. In following years it is proven that the speed measurement of the daemon generates entropy itself. When the entropy is calculated including the entropy generated by the daemon, it increases [27]. This thought experiment shows that the local entropy (entropy of boxes in Maxwell’s Daemon) can locally decrease. Albeit, the total entropy of the system keeps increasing. At the third chapter of this thesis, NLL is approached as nonlinear, far-from equilibrium sys-tem. The approach enables to gain extensive control capability, which is 2-D tiling of the material surface with all possible configurations, over the system.
Chapter 2
Nonlinear Laser Lithography
2.1
Theory
Nonlinear Laser Lithography is a recently introduced novel method to create structures using ultrafast laser on material surfaces varying from metals (Ti, W, Cu) to semiconductors (Si) [24]. The underlying mechanisms can be decoupled into three feedback loops. First, the laser beam is focused on the surface of the material. The pulse impinging to the material surface scatters the light and this scattered light interferes with the incident pulse. Second, in the points where the local intensity is higher than activation intensity (or ablation intensity) the structure formation occurs. Although the formation mechanism can change depending on the material, in the example of Ti, at the points where the intensity exceeds activation threshold, TiO2 starts to form on the surface. First two steps
constitute positive feedback and nonlinearity in the system. Finally, the structure height is limited by the exhaustion of the reactants near the activated site. Since the experiments are conducted mainly on 50 nm thick Ti sample, in the case of Ti, either the sample runs out of Ti or the ambient O2 cannot penetrate to the
activated area due to the formed structures. The last step introduces the negative feedback to the process.
The total electric field on the sample surface, which is sum of incident and scattered field, can be written as:
~ Etot(~r) = ~Einc(~r) + αk2 ε0 Z V d3V hG↔~r, ~r0E inc ~ r0i, (2.1)
where Einc(x, y) is incident electric field to the sample, α is polarisability, k is wave
number defined as 2π/λ, λ being wavelength of laser near the surface, ε0 is the
permittivity of free space,
↔
G~r, ~r0is the 3 dimensional Green’s Tensor defining
the response of a dipolar scatterer on the surface [28]. The overall response of the surface can be calculated by treating every infinitesimal point on the surface as dipole scatterer. Since the sample is thin, the phase accumulation between the two different heights can be neglected. Thus, the 3 dimensional Green’s Tensor can be truncated into 2 by 2 tensor as,
~ Etot(~r) = ~Einc(~r) + αk2 ε0 Z S dSh ↔ G~r, ~r0E inc ~ r0hr~0i, (2.2)
where h(x, y) is the surface profile. To solve Eqn. 2.2 the Dyadic Green’s function can be employed which is defined as,
↔ G(~r, ~r0) = [↔1 + 1 k2∇∇]G0(~r, ~r 0) , (2.3) where G0(~r, ~r0) = e ik|~r− ~r0|
4π|~r−~r0| is the scalar Green’s function. ~r = (x, y), ~r
0 = (x0, y0)
are the observation location and the source point, respectively. From Eqn. 2.3, we get the components of the dyadic Green’s function as,
Gxx = 1 4π{ cos2θ r + (3 sin 2θ − 1)( 1 k2r3 − i kr2)}e ikr (2.4) Gxy = Gyx = 1 4π{− sin θ cos θ r + 3 sin θ cos θ( 1 k2r3 − i kr2)}e ikr (2.5) Gyy = 1 4π{ sin2θ r + (3 cos 2θ − 1)( 1 k2r3 − i kr2)}e ikr, (2.6)
where cos θ = y−yr 0, sin θ = x−xr 0 and r2 = (x − x0)2 + (y − y0)2.
To gain intuition, decompose scattered electric field components to near and far field components as,
Es = Esf ar+ Esnear, (2.7)
where far field components are decaying with 1/r and near field components are decaying with 1/r2, 1/r3. When Eqn. 2.4 - 2.6 are substituted in 2.2,
Es,xf ar(x, y) = γRSdx0dy0{h(x0, y0)[sin2θE
x(x0, y0) − ...
sin θ cos θEy(x0, y0)]e
ikr
r }
(2.8)
Es,yf ar(x, y) = γRSdx0dy0{h(x0
, y0)[− sin θ cos θEx(x0, y0) + ...
cos2θEy(x0, y0)]e ikr r } (2.9) Enear s,x (x, y) = γ R Sdx 0dy0{h(x0, y0)[(3cos2θ − 1)E x(x0, y0) + ...
3 sin θ cos θEy(x0, y0)](k21r3 −
i
kr2)eikr}
(2.10) Es,ynear(x, y) = γRSdx0dy0{h(x0, y0)[3 sin θ cos θE
x(x0, y0) + ... (3sin2θ − 1)Ey(x0, y0)](k21r3 − i kr2)e ikr}, (2.11) where γ = αk2
4πε0. Similarly, Eqn. 2.8 - 2.11 can be calculated convoluting the dipole
radiation pattern with the surface profile. Dipole radiation can be calculated in fully vectorial approach as [29],
~ Es= α 4πε0 {k2(~n × ~Einc) × ~n eikr r + [3~n(~n ~Einc) − ~Einc]( 1 r3 − ik r2)e ikr}, (2.12) where ~n is ~r/r.
For an arbitrary polarisation, the scattering terms in Eqns. 2.8 - 2.11 can be arranged in the convolution form, K(x, y) ∗ h(x, y), where the kernel K(x, y) for linear polarisation can be written as,
K (x, y) = α 4πε0 k2sin2(θ) e ikr r +3cos 2(θ) − 1 1 r3 − ik r2 eikr . (2.13) This compact form allows to perform integration in the Fourier domain, signifi-cantly reducing the computation time.
Second step is calculating the number of activated atoms on the surface. In order to calculate it is assumed that all the laser power is absorbed at the skin depth of the material. The volume activated by a laser pulse can be calculated by,
dV (x, y) = dA Z
dz [f (x, y, z)], (2.14)
where dA is the surface element and f (x, y, z) is the activation probabil-ity function. The activation probability, f (x, y, z), can be approximated as H(In(x, y, z) − Ith), where H(t) is the Heaviside function and Ith is the ablation
threshold. Since the chemical reaction can also take place for In(x, y, z, ) < Ith,
with a low probability, H(t) can replaced with a smooth, differentiable function F(t) as,
f (x, y, z) = F (In(x, y, z) − Ith), (2.15)
where F(t) can be selected as a function of error function as, F (t) = 1/2 + erf (k1 (In(x, y, z) − Ith))/2 and erf (t) is the “error function”.
For a metal target, intensity of incident pulse over the material is,
In(x, y, z) = Isurf,n(x, y)e−z/δs, (2.16)
where Isurf,n(x, y) is the intensity distribution on the surface for the nth pulse
and δs is the skin depth of the metal. Thus, by performing the integration,
dV (x, y) = dAR dz[f (x, y, z)], we get the activated volume element as,
dV (x, y) = δs dA k2 (erf (k1 (Isurf,n(x, y) − Ith)) + 1)/2, (2.17)
where k1 is a reaction rate dependent factor, and k2 is a constant.
The ablation threshold can be calculated as, Ith = ρLδs/τp, where L is the
energy required to ablate a unit volume, ρ is material density, and τp is the laser
pulse duration. Pulse energies below the ablation threshold may be enough to overcome the activation energy.
Finally, in order to calculate the change in surface structure’s height increase per laser pulse, how many Ti atoms have been activated and the density of O2
molecules in the immediate vicinity available for the reaction must be considered. When the laser arrives on the surface, the number of Ti atoms activated per pulse in a small volume dV (x, y) is given by,
NTi(x, y) = dV (x, y) ρNa/MTi, (2.18)
where MTiis the molar mass, ρ is the mass density of Ti, Nais the Avogadro
Num-ber and dV (x, y) is calculated from Eqn. 2.17. The numNum-ber of oxygen molecules 7
available for reaction is assumed to be, NO2(x, y) = c1 e
−h(x,y)/hc, (2.19)
where h(x, y) is the height of the build up layer and hc is assumed to be the
critical height which determines the diffusion characteristics of O2 into TiO2.
At the end of each pulse, the change in the surface height is,
∆hn(x, y) = c2NTiO2, (2.20)
where c2 is a proportionality constant and NTiO2 is the number of newly formed
TiO2 molecules, given as:
NTiO2 = NTi, if NTi< NO2
NTiO2 = NO2, if NTi > NO2.
(2.21) This rate limiting factor imposes the negative feedback in NLL by saturating the surface structures. After finishing the aforamentioned calculations for the nth pulse, the procedure is reiterated for the (n + 1)th
pulse with updated height h (x, y), NTi(x, y) and NO2(x, y) values.
2.2
Simulations
Simulations are done using the explained theory at previous section. The coding is done in Matlab. The flowchart of the system can be seen in Fig. 2.1.
The simulation is done in two steps. First, the scattering from the surface and interference pattern is calculated (upper part of Fig. 2.1, shown in black). Note that Eqn. 2.8 - 2.11 can be written in convolution form which decreases the computation time significantly. Second, the amount of activated and surrounding material is calculated. According to the Eqn. 2.20 and 2.21 surface height and remaining reactants are updated (lower part of Fig. 2.1, shown in red and blue). Preceding two steps are reiterated with updated reactant amount and surface profile. Another important parameter is the linearisation parameter. Since doing
Figure 2.1: Flowchart of Nonlinear Laser Lithography (NLL).
the simulations pulse by pulse basis would require 1 million iterations to calculate 1 second in experiment using 1 MHz repetition rate laser. Thus, the simulation considers that N pulses are increasing the height N × ∆h and typical value of N is 5,000. Linearisation parameter N has found empirically, where the simulations still predicts the experimental data accurately. For some cases such as emergence of the patterns, needs to be simulated with very low or unity N value.
The emergence of patterns with NLL can be seen in Fig. 2.2. Fig. 2.2 (a) shows the interference pattern with a vertically polarised light from a single defect being placed at the centre of the simulation domain. Fig. 2.2 (b) show the struc-ture emergence form a single defect. It can be seen that the first few nanometres of the structures simply follows the parenthesis-like shape of interference pat-ten. However, after few thousands of pulses, since the newly formed structures are scattering the laser beam in a similar fashion, the pattern corrects itself to perfectly stripe pattern (Fig. 2.2 (a)). This property builds up a basis for self healing property of NLL. Under the influence of circularly polarised laser beam, the interference pattern from a single defect forms concentric circles as shown in Fig. 2.2(b). Since the pattern is circularly symmetric, symmetry must be broken in order to get periodic pattern. In experiments, the symmetry is broken due to the randomness (roughness) of the surface. Here, in simulations, the symmetry is broken by placing three defects at the edge of an equilateral triangle of 1 µm side length which is placed to the very centre of the simulation domain. The resultant pattern is hexagonal which can be seen in Fig. 2.2 (b).
Figure 2.2: Interference pattern and emergence of the surface structures from a single defect being placed at the centre of simulation domain. (a) Total intensity on the surface from vertically polarised laser beam. (b) Emergence of stripes from a single defect within 10,000 pulses. (c) Total intensity on the surface from circularly polarised laser beam. (d) Emergence of hexagonal pattern from three defects within 200,000 pulses.
Self healing is another vital property of NLL which is investigated with sim-ulations as well. Fig. 2.3 shows the simsim-ulations with different sizes of defect introduced at the centre of simulation domain. In all simulations, beam diameter is selected to be 4 µm and it scanned from left hand side to right hand side of the simulation domain with a speed corresponding 5 µm/s in experiments. While in Fig. 2.3 (a) and (b) laser polarisation is vertical, and in c and d it is horizon-tal. The message should be taken from Fig. 2.3 is that the self healing property works for small defects on the way of scanning of NLL. When the defect is too large that prevents the newly formed structures getting feedback from previously formed structures, the structures starts to reform after the defect.
Here, it can be seen that defects having large size breaks the coherence of 10
Figure 2.3: Simulations to identify the effect of defect on NLL. Laser beam has a beam width of 4 µm (1/e2) and scanned from left hand side to right hand side.
(a), (b) Scanning done with vertically polarised laser beam. (c), (d) Scanning done with horizontally polarised laser beam. Defect sizes in diameter are (a), (b), (c), (d), 500 nm, 100 µm, 500 nm, 100 µm respectively.
the structures. However, not all kinds defects disrupt the stability of the surface structures. This point of view will be discussed at the last chapter of this thesis. The comparison of experimental and simulation of NLL will be done in next section.
2.3
Experimental Setup and Experiments
Experiments are conducted with an ultrafast Ytterbium doped laser oscillator operating at 1030 nm central wavelength, cascaded with two pre amplifiers and a final power amplifier [30]. The oscillator is built to have 37 M Hz repetition
rate. After two stage of pre amplifiers, pulses are picked with an acusto-optic modulator down to 200 kHz driven by a field-progammable gate array. Having reduced the repetition rate, chirped pulse amplification technique [31] is applied in which pulses are stretched with 200 metres of stretch fibre. At the end of the final power amplifier, the pulses can go up to 6 µJ pulse energy and 200 kHz repetition rate. After pulse compression stage, pulse energy and pulse duration is decreased down to 3 µJ and 150 f s respectively. However, since the NLL setup works in sub-ablation threshold regime, the laser is operated in very low powers (as low as 50 nJ of pulse energy). The experimental setup is shown in Fig. 2.4.
Figure 2.4: Experimental setup of NLL.
After the compression process, the laser beam is guided to a transmission microscope setup. The laser beam, which is coupled with a blue illumination light via a dichroic mirror, is focused on a sample being placed on a 2 dimensional translational stage. The choice of blue light is merely for increasing the resolution of imaging due to diffraction limit. Finally, the illumination light is filtred from the laser beam and sent to the charge coupled device array through an high
magnification objective (typically 40 to 100x). The power of using microscope setup coupled to the laser beam is getting live feedback from the imaging system while sample is being processed.
Figure 2.5: Simulation an experimental results of NLL. Laser beam is scanned starting from left hand side to right hand side. (a), (b) Simulation results with vertically and horizontally polarised laser beam, respectively. (c), (d) Experi-mental results corresponding (a) and (b) respectively.
Fig. 2.5 illustrates the comparison of simulation and experimental results. In Fig. 2.5 (a) and (b) the laser scanned from left hand side to right hand side both in simulations and experiments respectively. The laser polarisation is selected to be vertical. On the other hand, in Fig. 2.5 (c) and (d), the laser polarisation is horizontal. The great agreement between simulations and experiments can be appreciated.
Chapter 3
Phase Control and Doppler Effect
3.1
Theory
There are two ways to control the structure period of LIPSS. First, laser wave-length can be changed. Second, the effective wavewave-length on the material surface can be altered by introducing oblique incidence of laser beam to the sample. It has been shown that changing the incidence angle of the laser beam on the sample changes the period of LIPSS as [19],
Λ = 1
1 ± sin θ, (3.1) where Λ is structure period, θ is the angle between laser propagation vector and normal of the surface. However, in Eqn. 3.1, the effect of laser polarisation is totally neglected. In this section, the structure period formula is generalised with the effect of the polarisation effect. Further, this effect is found to be identical to the Doppler effect.
Assuming that the angle between wave propagation vector and surface normal is ϕ on x-axis, electric field vector can be written as,
Einc(x, y) = E0eik sin ϕx. (3.2)
Neglecting the near field, scalar scattered electric can be rewritten (Using Eqn. 14
2.8 - 2.11):
Es(x, y) = γ
Z
dx0dy0Einc(x0, y0) h (x0, y0) cos2(θ − β) eikr, (3.3)
where r2 = (x − x0)2
+ (y − y0)2, θ = tan−1x−xy−y00
, k = 2πΛ and β is polarisation angle.
Assuming that a defect is modelled as Dirac delta function located at x = y = 0, total electric field will be:
Etot(x, y) = Einc+ γE0cos2(θ − β) eikr/r. (3.4)
Total intensity can be written as: Itot(x, y) = |E0| 2
+ |γE0cos2(θ − β) /r| 2
+ ... 2γE0cos2(θ − β) cos (kr + k sin ϕx) /r.
(3.5) In polar coordinates, sin ϕx = cos θ sin ϕr. Substituting this in Eqn. 3.5 yields,
Itot(x, y) = |E0|2+ |γE0cos2(θ − β) /r| 2
+ ... 2γE0cos2(θ − β) cos 2πΛr (1 + sin ϕ cos θ) /r.
(3.6) Assuming kr >> 1, we can neglect 1/r decay term in Eqn. 3.6 and it can be rewritten as,
Itot(x, y) = |E0|2+ |γE0cos2(θ − β)| 2
+ ... 2γE0cos2(θ − β) cos 2πΛr (1 + sin ϕ cos θ) .
(3.7) In order to find period of the structure period on the surface, derivative with respect to r must be equal to zero as,
∂rItot(r, θ) = 2γE0cos2(θ − β) × ...
sin 2πΛr (1 + sin ϕ cos θ)2πΛr (1 + sin ϕ cos θ) = 0. (3.8) Since it is a periodic function, difference between zero crossings of the function will give the period of the structures as,
r = nΛ
2 (1 + sin ϕ cos θ), (3.9) where n = 2m and m is an integer. The period of the final structures can be written as,
Λ0 = r3−1=
Λ
1 + sin ϕ cos θ. (3.10) 15
Formula for the period is same as Doppler effect with a substitution of v/c = sin ϕ effect which is:
Λ0 = Λ
1 + v/c cos θ (3.11) Similarly, maximising Eqn. 3.7 gives the orientation of the structures with respect to θ. Maxima of Eqn. 3.7 can be found at the places where θ = β + lπ and l is an integer.
3.2
Simulations and Experiments
Figure 3.1: Experimental configuration and laser total intensity on the material surface from a single defect. (a) Usual NLL beam-sample configuration. (b) Con-figuration to realise the Doppler effect. (c), (d) Simulation of the total intensity on the sample surface from a single defect, being placed at the centre of the simu-lation domain, with vertically polarised laser beam, using configurations (a) and (b) respectively.
Fig. 3.1 illustrates usual configuration being used for NLL (Fig. 3.1 (a)) and the altered configuration (Fig. 3.1 (b)) in order to have linear phase on the pulse. Fig. 3.1 (c) and (d) the interference pattern from a single defect on the
centre of the computation domain for normal and oblique incidence can be seen respectively.
Figure 3.2: Illustration of Doppler effect and simulation of total intensity on the material surface. (a) Schematic of Doppler effect with for source, where ‘O’ is observer, ‘S’ is source and ‘θ’ is the angle between observer and source with respect to horizontal axis. (b) Simulation of total intensity on the surface with circularly polarised laser beam.
Fig. 3.2 (a) and (b) illustrate the resemblance of Doppler effect to the NLL total intensity pattern with additional linear phase respectively. In Fig. 3.2 circularly polarised light is used in order to emphasise the similarity wavefront patterns.
Figure 3.3: Comparison of Doppler effect and NLL simulation. Each pixel in images corresponds to a period with respect to incidence and polarisation angle. (a) Calculations according to Doppler effect, Eqn. 3.10, where ϕ is polarisation angle and θ is incidence angle. (b) NLL simulation results.
Beyond the similarity, there is a theoretical basis in between Doppler effect 17
and linear phase modification in NLL which is explained in previous section. In addition to the theoretical basis, simulations have been conducted to prove that the same Doppler effect behaviour can be observed in the experiments. Fig. 3.3 (a) shows the Doppler effect equation given in 3.11 and Fig. 3.3 (b) the simulation results of scanning the laser perpendicular to the polarisation direction. Each point in the surface plot corresponds to a period value. The agreement between simulation and theory can be seen.
Figure 3.4: Comparison of simulation and experimental results for Doppler effect in NLL.
In order to verify the simulation results, the experiments are conducted with the setup explained in Chapter 2. Fig. 3.4 illustrates the experimental realisation of Doppler shift and comparison with the simulation results. Each plot on the right hand side is a cross section of the 2-D image which is shown in left hand side. Detection of the period with microscope microscope setup is cumbersome due to the diffraction limit. However, since the structures are in the form of stripe, simply counting the lines and average the period over several periods gives accurate results.
Chapter 4
Controlling Surface Patterns via
Structured Noise
4.1
Structured Noise
In nonlinear, far-from-equilibrium systems the entropy is not a meaningful pa-rameter [25], even if it is calculated it does not necessarily maximise itself in local manner. In fact, rate of entropy production might decrease locally while total en-tropy in the larger scale keeps increasing. Typical example for this scenario would be Maxwell’s Daemon [26], where the entropy of one of the boxes should decrease by simple thought. However, it is shown that measurement of the daemon gen-erates entropy which increases the total entropy of the system in turn [27]. Since NLL is nonlinear, far-from-equilibrium system there is no necessity to maximise the entropy. In the example of creation of lines with NLL, the local entropy de-creases if we define the entropy as the complexity of the surface topology. From the noise level (roughness of the surface), with the laser influence, regular struc-tures can be obtained which suggests decrease in local entropy. However, there is no theory identifying selection rules of nonlinear, far-from-equilibrium systems. Therefore, at least for the case of NLL, one should define a metric that explains the selection rule.
Figure 4.1: Feedback strength of each lattice with a given NLL feedback system configuration. In the natural mechanism, the structures tend to evolve in hexago-nal pattern shown in upper part. Depending on the structured noise, the pattern can be guided to another pattern. (a) The natural way for pattern emergence with given initial conditions. (b) Structure guiding via structured noise from hexagonal to square pattern.
NLL enables us to examine a system with well separated feedback blocks. In NLL, scattering kernel and interference defines the resultant lattice on the material surface. The metric can be defined as the gain seen by each probable lattices, resembling the self consistency check. Simply, the metric answers the question of how similar is the interference pattern to the desired pattern. From a single defect, linearly polarised light creates lines parallel to polarisation (see Chapter 2) which is favoured by the scattering kernel. When the surface is preconditioned by horizontal lines again done by NLL, scanning the laser over it with perpendicular polarisation leads to hexagonal pattern. The reasoning can
be done through the gain analysis over the possible lattices. Fig. 4.1 illustrates the schematic of the gain analysis process. In Fig. 4.1 (a) the negative gain over lattice constant and angle (between lattice vectors) space can be seen. The choice of negative gain is due to building up the connections with the potential energy surfaces in classical mechanics. According to the Fig. 4.1 (a), periodicity of Λ√3 of lattice constant and 30o between the lattice vectors is favoured. Not surprisingly, the favoured lattice corresponds to hexagonal lattice. However, in classical mechanics, altering the forces leads to different potential energy surface configurations. Using the intuition gained from classical mechanics, in NLL, resultant pattern can be controlled by changing either the surface profile of the material or the incident laser beam. Fig. 4.1 (b) illustrates the altered of potential energy surface after introducing carefully chosen defect. It can be seen that the lattice constant of Λ and 0o of angle between lattice vector is more favourable
after application of defect which corresponds to square lattice. This technique can be called as using structured noise in the system. The following section is dedicated to explain how the final pattern on surface can be guided by employing structured noise.
4.2
Pattern Selection
There are 5 distinct and unique lattices that can tile 2-D planar surface with isotropic unit cells [32] which are called Bravais Lattices and shown in Fig. 4.2. Thus, material surface can be tiled with at most 5 lattice types with isotropic features employing NLL.
Although the system dynamics favour a distinct pattern, the tiling patterns can be guided using structured noise. For instance, square pattern is less favourable (has less gain) compared to hexagonal for given initial conditions as discussed in previous section. Therefore, system naturally selects hexagonal pattern over all other patterns. There are two different way to guide patterns in NLL tech-nique. First, one can introduce a seed to the surface such that scanning over this seed results in desired pattern. Second, one can add spatial modulation on
Figure 4.2: Cartoon illustrating Bravais lattices. (a) Square lattice where a1 =a2
and ϕ = 90o. (b) Hexagonal lattice where a1 =a2 and ϕ = 60o. (c) Oblique lattice
where a1 6=a2 and ϕ 6= 90o. (d) Rectangular lattice where a1 6=a2 and ϕ = 90o.
(e) Centred rectangular lattice where a1 6=a2, ϕ 6= 90o, and a1⊥2a2.
the laser beam which matches with the period of the desired pattern. Note that the applied technique differs from interference lithography [33, 34] by requiring significantly less power (defect size compared to the resulting pattern area or applied modulation depth). Fig. 4.3 illustrates guiding from the hexagonal to square lattice employing both aforementioned techniques. The surface is precon-ditioned with horizontal lines using NLL, then the laser beam is scanned starting from left hand side to right hand side. In Fig. 4.3 (a), additional periodic line defects are introduced at the middle of the simulation domain (between x = 33 and 36 µm) to guide pattern from hexagonal to square pattern. In Fig. 4.3 (b) the modulation is applied after the half way of scanning (at x = 35 µm) with 5% modulation depth and a period of 870 nm which matches with the period of square lattice of NLL.
In addition to the hexagonal and square tilings, the resulting pattern can be guided into all 5 Bravais Lattices. However, realising other lattices requires change of initial conditions of laser and sample. For instance, rectangular lattice differs from hexagonal and square lattices by having two different lattice constant in different directions. In order to create different aspect ratio structures, spatial
Figure 4.3: Transitions between hexagonal pattern to square pattern using struc-tured noise in the simulation. The laser is scanned from left to right in both figures. (a) Line defect is added at the centre of the simulation domain. (b) Spa-tial modulation is applied to the laser beam at the centre of the scanning such that the modulation frequency matches with the feature size of square pattern (870 nm).
phase distribution must be altered (see Chapter 3).
With the help of Doppler effect which is explained in Chapter 3 it is possible create all 5 Bravais lattices. Fig. 4.4 illustrates realisation of all 5 Bravais Lattices on Ti samples. In Fig. 4.4 (a)-(e) simulation results are shown. In Fig. 4.4 (h)-(j) experimental results corresponding the simulations (a)-(e) are shown respectively. Similar to the previous experiment, all surfaces are preconditioned by horizontal lines and laser is started to scan from left hand side to right hand side. Formation of square lattice has already discussed. While formation of oblique lattice requires change in polarisation of incident laser, being different than vertical polarisation, and slanted initial line as defect, rectangular and centred rectangular lattices require change in resulting period. The period difference is induced by changing the incidence angle of the laser beam to the sample.
Figure 4.4: Bravais lattices realised with NLL using structured noise in simula-tions and experiments. (a)-(e) NLL simulasimula-tions. Surface is preconditioned with horizontal lines again using NLL. Then, the laser beam scanned from left hand side to right hand side with different polarisations. Initial defect to guide the patterns are shown with white solid lines. In (a) and (b) vertically polarised laser beam is used. In (c) polarisation angle is 45o with respect to the vertical axis. In
(d) and (e) Doppler effect is employed in order to obtain period difference with vertically polarised laser beam. (f)-(j) Experimental realisation of Bravais lat-tices. (f) Atomic force microscope image of hexagonal lattice. (g)-(i) Microscope images for realisation of Bravais Lattice. All scale bars are 5 µm, except (g), which is 3.5 µm.
4.3
Stochastic Resonance
Stochastic Resonance (SR) is a technique introduced in telecommunications. This technique is using the environmental noise level in favour of the detection prob-ability [35]. SR enables signal detection with low transmission power.
Fig. 4.5 illustrates the concept of stochastic resonance. Assume that there is a signal to be transmitted with 1 arbitrary unit (A.U.) peak power, off-set of 0.5 A.U., which is illustrated in Fig. 4.5 (a), and a receiver with a detection threshold
Figure 4.5: Explanation of Stochastic Resonance (SR) phenomena in a communi-cation system. (a) Transmitter device with maximum peak intensity of 1 arbitrary unit (A.U.) over time. (b) Plot of the transmitted signal through a noisy channel with a noise density of 0.5 A.U.. Detection threshold of receiver is marked with red line at 1.5 A.U.. (c) Received signal (blue), and transmitted signal (red) over time. (d) Transmitted signal through a noisy channel with a noise density of 1 A.U.. (e) Received signal (blue), and transmitted signal (red) over time.
of 1.5 A.U.. If there is white-gaussian noise in the environment with a root-mean-square (RMS) value of 0.5, decision of true-positive rate increases significantly (Fig. 4.5 (c)). When the RMS value of noise is increased to an RMS value of 1, detection of true-positive rate decreases (Fig. 4.5 (e)). In one extremum, which is absence of noise, the given receiver would be unable to detect anything (true-positive probability is zero). In the other extremum, which is environment with very high noise level, noise will scramble the detected signal (true-positive probability is zero again). However, there is a sweet spot in between the two extrema which enhances the true-positive probability. This behaviour might be the simplest explanation of SR.
In NLL, similar to the decision process of an receiver, there is a thresholding mechanism between switching from hexagonal to square lattice. In simulations,
Figure 4.6: Illustration of SR phenomena in NLL simulations. In (a) each colour represents a lattice. Blue pixels represent square lattice, yellow pixels correspond to hexagonal lattice. Colours in between blue and yellow represent combination of hexagonal and square lattices. (b) Illustration of SR behaviour with 15 realisation of simulations and the average of them. Blue data points and red solid line represent the average value of the 15 realisations and the moving average of the simulation results respectively. Modulation amplitude is fixed to 2.5% in simulations, which corresponds the marked region in (a) with red box.
the surface is preconditioned, as in previous section, with horizontal lines and vertically polarised laser light is scanned over the sample. On the laser light a modulation applied is matching with the square lattice period. However, the modulation can be as small as 3% of the laser intensity. In order to realise SR phenomena in NLL, white-gaussian noise is applied with varying RMS values. Fig. 4.6 illustrates the SR behaviour of NLL. In Fig. 4.6 (a) probability of ending up with square lattice and hexagonal lattice is shown with blue and yellow colours respectively. The colours fall between blue and yellow indicate that at some point of scanning process, the surface recovers itself to hexagonal lattice. Fig. 4.6 (b) illustrates the SR behaviour with 2.5% of spatial modulation depth.
Chapter 5
Conclusion and Further
Improvements
In this thesis the Nonlinear Laser Lithography (NLL) technique is analysed with rigorous mathematical approach using the dipole radiation of each scatterer on the material surface. The experimental results are confirmed with the simu-lations. Stability of the structures has been analysed with simulations and in which conditions the structure stability is destroyed has been found. The per-spective of feedback disruption opens the way to control the surface structures via structured noise. Moreover, improved simulations gave a capability of predicting experimental results before the experiments being
The next step of the simulations/experiments is adding spatial phase profile to the laser beam. The first and simplest thing is adding linear phase on the beam via introducing oblique incidence. As result, Doppler Effect has been demonstrated, to the best of my knowledge, for the first time in the material processing. Further improvement would be adding arbitrary spatial phase profiles and altering the total intensity on the material surface. Since the surface topology follows the interference pattern of the total intensity on the surface, adding different phase profiles would allow us to tile arbitrary shapes on the surface even aperiodic ones, such as quasicrystals.
Finally, it has shown that it is possible to a control nonlinear, far-from equi-librium system. Analogous to the potential energy surfaces in classical dynamics, the feedback strength is introduced to analyse the outcome of the system. The control capability has been gained by engineering the feedback strength via either changing the initial conditions or changing the input parameter which is spatial profile of the laser beam in NLL case. Since the system has switching mecha-nism in between the tiling states (hexagonal and square lattices) with the input parameter. Stochastic resonance, which is inherent such systems, has shown. The further improvement would be generalising the control capability over large variety of systems using the feedback strength approach.
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