052182172X cambridge university press transport in laser microfabrication fundamentals and applications aug 2009

Thus, an electric field Einduces an electricdipole moment or polarization vector P, while the magnetic induction field B drives a magnetic dipole moment or magnetization vector M.. In ot

Transport in Laser Microfabrication

Emphasizing the fundamentals of transport phenomena, this book provides earchers and practitioners with the technical background they need to understand laser-induced microfabrication and materials processing at small scales It clarifies the lasermaterials coupling mechanisms, and discusses the nanoscale confined laser interactionsthat constitute powerful tools for top-down nanomanufacturing In addition to analyzingkey and emerging applications for modern technology, with particular emphasis on elec-tronics, advanced topics such as the use of lasers for nanoprocessing and nanomachining,the interaction with polymer materials, nanoparticles and clusters, and the processing ofthin films are also covered

res-Costas P Grigoropoulos is a Professor in the Department of Mechanical Engineering

at the University of California, Berkeley His research interests are in laser materialsprocessing, manufacturing of flexible electronics and energy devices, laser interactionswith biological materials, microscale and nanoscale fluidics, and energy transport

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-82172-8

ISBN-13 978-0-511-59515-8

© C Grigoropoulos 2009

2009

Information on this title: www.cambridge.org/9780521821728

This publication is in copyright Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any partmay take place without the written permission of Cambridge University Press

Cambridge University Press has no responsibility for the persistence or accuracy

of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate

Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.org

eBook (EBL)Hardback

To Mary, Vassiliki, and Alexandra

2.4 Definition of laser intensity and fluence variables 48

5.3 Modeling of ablation-plume propagation 116

6.3 Femtosecond-laser interaction with semiconductor

6.4 Phase transformations induced by femtosecond laser

6.7 Nonlinear absorption and breakdown in dielectric materials 176

7.1 Modeling of energy absorption and heat transfer in pulsed-laser

7.4 Nanosecond-laser-induced temperature fields in melting and

7.6 Lateral crystal growth induced by spatially modified irradiation 222

Contents ix

10.1 Rapid vaporization of liquids on a pulsed-laser-heated surface 282

10.3 Nonlinear interaction of short-pulsed lasers with dielectric liquids 304

Lasers are effective material-processing tools that offer distinct advantages, includingchoice of wavelength and pulse width to match the target material properties as well asone-step direct and locally confined structural modification Understanding the evolution

of the energy coupling with the target and the induced phase-change transformations iscritical for improving the quality of micromachining and microprocessing As currenttechnology is pushed to ever smaller dimensions, lasers become a truly enabling solu-tion, reducing thermomechanical damage and facilitating heterogeneous integration ofcomponents into functional devices This is especially important in cases where con-ventional thermo-chemo-mechanical treatment processes are ineffective Componentmicrofabrication with basic dimensions in the few-microns range via laser irradiationhas been implemented successfully in the industrial environment Beyond this, there

is an increasing need to advance the science and technology of laser processing to thenanoscale regime

The book focuses on examining the transport mechanisms involved in the laser–material interactions in the context of microfabrication The material was developed in

the graduate course on Laser Processing and Diagnostics I introduced and taught in

Berkeley over the years The text aims at providing scientists, engineers, and graduatestudents with a comprehensive review of progress and the state of the art in the field bylinking fundamental phenomena with modern applications

Samuel S Mao of the Lawrence Berkeley National Laboratory and the cal Engineering Department of UC Berkeley contributed major parts of Chapters 5,

Mechani-6, and 9 I wish to acknowledge the contributions of all my former and current dents throughout this text Hee K Park’s, David J Hwang’s, and Seung-Hwang Ko’sinput extended beyond their graduate studies to post-doctoral stints in my laboratory

stu-I am grateful to Gerald A Domoto of Xerox Co for introducing me to an ing laser topic that evolved into my doctoral thesis at Columbia University DimosPoulikakos of the ETH Z¨urich talked me into starting this book project when I was

interest-on sabbatical in Zurich in 2000 His cinterest-ontributiinterest-ons in collaborative work form a keypart of the text I thank Professor Jean M J Fr´echet of the UC Berkeley College ofChemistry for his contributions as well as Costas Fotakis of the IESL FORTH, Greece,and Dieter B¨auerle of Johannes Kepler University, Austria, for their support and input

I am indebted to the NSF, DOE, and DARPA for funding work this book benefited from.The expert help of Ms Ja Young Kim in preparing the artwork was key in completingthis book.

Costas P GrigoropoulosBerkeley, California, USA

1 Fundamentals of laser

energy absorption

1.1 Classical electromagnetic-theory concepts

1.1.1 Electric and magnetic properties of materials

Electric and magnetic fields can exert forces directly on atoms or molecules, resulting

in changes in the distribution of charges Thus, an electric field
Einduces an electricdipole moment or polarization vector
P, while the magnetic induction field
B drives

a magnetic dipole moment or magnetization vector
M It is convenient to define theelectric displacement vector
Dand the magnetic field
Hsuch that

where ε0 and µ0 are the electric permittivity and magnetic permeability, respectively,

in vacuum For isotropic electric materials the vectors
D,
E, and
P are collinear, whilecorrespondingly for isotropic magnetic materials the vectors
H,
B, and
Mare collinear

Introducing the electric susceptibility χ , the polarization vector is written as

where µ is the material’s magnetic permeability and µr the relative magnetic

per-meability In a medium where the charge density ρ moves with velocity
v, the free

current-density vector
J is defined as

The magnitude of this current,|
J|, represents the net amount of positive charge crossing

a unit area normal to the instantaneous direction of
v per unit time The current-density vector is related to the electric field vector via the electric conductivity, σ,

electromag-homogeneous (uniform) materials, for which ε and µ are constants independent of

position, the following relations hold:

1.1.2 Boundary conditions

Consider an interface i, separating two media (1) and (2) of different permittivities

ε1, ε2and permeabilities µ1, µ2 (Figure 1.1) According to Born and Wolf (1999) thesharp and distinct interface is replaced by an infinitesimally thin transition layer Within

this layer ε and µ are assumed to vary continuously Let
n12 be the local normal at theinterface pointing into the medium (2) An elementary cylinder of volume␦V and surface

area␦A is taken within the thin transition layer The cylinder faces and peripheral wall

are normal and parallel to vector
n12, respectively Since
Band its derivatives may beassumed continuous over this elementary control volume, the Gauss divergence theorem

1.1 Classical electromagnetic-theory concepts 3

elementary rectangle

1

2

C

interface δA

elementary surface 1

2

i S

Figure 1.1. Schematics of an elementary volume of height␦n and an elementary rectangular

contour of width␦s across the distinct interface separating media 1 and 2.

The second integral is taken over the surface of the cylinder In the limit, as the height

of the cylinder␦h → 0, contributions from the peripheral wall vanish and this integral

yields

(
B1·
n1+
B2·
n2)␦A = 0, (1.13)where
n1= −
n12and
n2=
n12 Consequently,

n12· (
B2−
B1)= 0. (1.14)The electric displacement vector
Dis treated in a similar manner by applying the Gausstheorem to Equation(1.10):

where B2n =
B2·
n, B1n=
B1·
n, D2n=
D2·
n, and D1n =
D1·
n In other words, the

normal components of the magnetic induction vector
Bare always continuous and thedifference between the normal components of the electric displacement
D is equal in

magnitude to the surface charge density σs

To examine the behavior of the tangential electric and magnetic field components,

a rectangular contour C with two long sides parallel to the surface of discontinuity isconsidered Stokes’ theorem is applied to Equation(1.8):

In the limit as the width of the rectangle␦h → 0, the last surface integral vanishes and

the contour integral of
Eis reduced to

E1·
t1+
E2·
t2= 0. (1.20)Considering the unit tangent vector
talong the interface,
t1= −
t = −
s ×
n12,
t2=
t =

s ×
n12, Equation(1.20)gives

If a similar procedure is applied to Equation(1.9), then

n × (
H2−
H1)=
K, (1.22)where
Kis the surface current density

The boundary conditions(1.21)and(1.22)are written in the following form:

The subscript t implies the tangential component of the field vector Thus, the tangentialcomponent of the electric field vector
Eis always continuous at the boundary surfaceand the difference between the tangential components of the magnetic vector
H is

equal to the line current density K, and in radiation problems where σs= 0, K = 0.

Consequently, the normal components of
Dand
Band the tangential components of
E

and
Hare continuous across interfaces separating media of different permittivities andpermeabilities

1.1.3 Energy density and energy flux

Light carries energy in the form of electromagnetic radiation For a single charge qe, therate of work done by an external electric field
E is qe
v ·
E, where
v is the velocity of

the charge If there exists a continuous distribution of charge and current, the total rate

of work per unit volume is
J·
E, since
J = ρ
v Utilizing(1.9),

J·
E=
E· (∇ ×
H)−
E·∂
D

1.1 Classical electromagnetic-theory concepts 5

The following identity is invoked:

∇ · (
E×
H)=
H· (∇ ×
E)−
E· (∇ ×
H ), (1.25)and applied to(1.24):

The scalar U represents the energy density of the electromagnetic field and in the SI

system has units of [J/m3] The vector
S is called the Poynting vector and has units

[W/m2] It is consistent to view |
S| as the power per unit area transported by the

electromagnetic field in the direction of
S Hence, the quantity ∇ ·
S quantifies the net

electromagnetic power flowing out of a unit control volume Equation(1.27a)states the

Poynting vector theorem.

For propagation in vacuum, ρ = 0, σ = 0, µ = µ0, and ε = ε0, and Equations (1.29a)and (1.29b) give

The above are wave equations indicating a speed of wave propagation c0= 1/√µ0ε0,

i.e the speed of light in vacuum For propagation in a perfect dielectric, ρ = 0, σ = 0,

and the following apply

Equations (1.30) and (1.31) can be satisfied by monochromatic plane-wave solutions

with a constant amplitude A and of the general form

where
r and
s are the position vector and the wavevector, respectively.

The angular frequency ω and the magnitude of the wavevector
s are related by

1.1 Classical electromagnetic-theory concepts 7

y x

z

wavefront

s r

Figure 1.2. A schematic diagram depicting a plane wave propagating normal to the direction
s.

s

u2

u1

H E

Figure 1.3. A schematic diagram depicting the instantaneous vectors
Eand
H that form aright-hand triad with the unit vector
s along the propagation direction.

In a homogeneous, charge-free medium,∇ ·
E= ∇ ·
H = 0 Hence,

time-averaged flux of energy

S = | E0|2

2ωµ
u3= E
∗·
E

H E

H

s E

1.1.5 Electromagnetic theory of absorptive materials

The optical properties of perfect dielectric media are completely characterized by thereal refractive index In such media, it is assumed that electromagnetic radiation interactswith the constituent atoms with no energy absorption In contrast, especially for metals,very little light penetrates to a depth beyond 1␮m at visible wavelengths Considerthen media with nonzero electric conductivity that absorb energy but do not redirect acollimated light beam Let
Eand
Hbe the real parts of periodic variations:

1.1 Classical electromagnetic-theory concepts 9

Combining the above with (1.43a) and (1.43b) gives

∇2E
c= iωµ(σ + iωε)
Ec, (1.46)or

is the complex dielectric constant

A complex velocity vcand a complex refractive index nccan then be defined:

Let nc= n − ik, where n is the real part of the complex refractive index and k the

imaginary part, the so-called attenuation index:

c =ω (n − ik)

the above can be written as

T, the first two periodic terms will contribute very little Hence, the averaged energy is

The above expressions indicate that the energy flux carried by a wave propagating in

an absorbing medium is proportional to the squared modulus of its complex amplitudeand to the real part of the complex refractive index of the medium The modulus of the

Poynting vector, i.e the monochromatic radiative intensity, I λ, is

In the above, λ0is the wavelength in vacuum As shown inFigure 1.5, the energy flux

drops to 1/e of I λ,0 after traveling a distance d, the so-called absorption penetration

depth:

1.1 Classical electromagnetic-theory concepts 11

Figure 1.5. A schematic diagram depicting the exponential decay of a monochromatic beam in a

medium of complex refractive index n – ik.

1.1.7 Refraction and reflection at a surface

Perfect dielectric media

Consider a plane light wave of electric field
E+

0 incident at the plane interface of

two perfect dielectric media, characterized by the real refractive indices n0 and n1(Figure 1.6) The electric field is decomposed to the two polarized components E0p+ and

0s, where p and s indicate polarizations parallel and normal to the plane of incidence

(xOz) The superscripts+ and – indicate forward and backward wave propagation Let

− 0p

E

− 0s

E

+ 0s

1

Figure 1.6. A plane wave is incident on the flat interface (z= 0) separating two semi-infinite

media of refractive indices n0 and n1 The electric field vector is decomposed to the parallel (p)and normal (s) polarizations

n0sin θ+= n0sin θ−= n1sin θt. (1.68)

Since sin θ+= sin θ−and cos θ+= −cos θ−, it is inferred that

Equation(1.68)and the statement that the directional vector
s−of the reflected wave

lies on the plane of incidence (xOy), as expressed by (1.67b), constitute the law of

1.1 Classical electromagnetic-theory concepts 13

The above equation, combined with the fact that the directional vector
st of therefracted wave lies on the plane of incidence as deduced from (1.67c), expresses Snell’s law of reflection The tangential components of the electric and magnetic vectors have

also to be continuous across the interface:

Figure 1.7. The angular dependence of the surface reflectivity for a bulk dielectric sample having

refractive index n1= 1.45 for radiation incident through a medium of refractive index

n0= 1 The curve labeled “reflp” is for parallel polarization and the curve marked “refls” is for

normal polarization The Brewster angle is indicated by “θB.”

For θ > 0, Equations (1.73), combined with (1.72) and Snell’s law of refraction, are

used to obtain the reflectivities and transmissivities Equations (1.72) can be simplified:

r F,1p= tan(θ − θt)

tr1p = 2 sin θtcos θ sin(θ + θt)cos(θ − θt), (1.75b)

r F,1s= −sin(θ − θt)

tr1s = 2 sin θtcos θ

The denominators are finite, except in the case of p-polarization, when tan(θ + θt)→ ∞,

for θ + θt= π/2 In this case,

At this angle, the polarizing or Brewster angle is ρλ,p= 0.Figure 1.7gives an example

of the angular reflectivity variation for a perfectly dielectric (transparent) bulk materialfor the parallel and normal polarizations It is noted that for unpolarized light, thereflectivity is

Reflection at the surface of an absorbing medium

The equations for light propagation in a transparent medium can be modified for thecase of an absorbing medium, by replacing the real refractive index by its complex

1.1 Classical electromagnetic-theory concepts 15

Figure 1.8. The angular dependence of the surface reflectivity for a bulk absorbing sample having

refractive index nc= 0.7 − 2i for radiation incident through a medium of refractive index n0=

1 The curve labeled “reflp” is for parallel polarization and the curve marked “refls” is for normalpolarization

counterpart:

In this case the angle of refraction is complex, and is identified by the generalized version

of Snell’s law of refraction:

“pseudo-Brewster” angle, although it does not vanish there

Consider light impinging on the surface of an absorbing medium at an oblique angle

of incidence The complex electric field in the medium (1) is

E1c(
r, t) =
E t,0c e−i[kc1 (
r·
s t )−ωt], (1.80)

where kc= ω(n1− ik1)/c0.

The complex unit vector of propagation in medium (1) is defined by

where q and δ are real By taking the squares and equating the real and imaginary parts

of (1.80c) and (1.81), it can be shown that

c0[x sin θ + zq(n1cos δ + k1sin δ) + izq(n1sin δ − k1cos δ)]. (1.84)

On examining(1.80)and(1.84), it is evident that the surfaces of constant amplitudeare given by

1.1 Classical electromagnetic-theory concepts 17

planes of constant phase

absorbing medium

air

planes of constant amplitude

1.1.8 Laser light absorption in multilayer structures

A laser beam is incident on a multilayer stack of films, z0≥ ≥ z j−1 ≥ z ≥

z j ≥ z N , which is stratified in the z-direction, as shown inFigure 1.10 The

laser-beam propagation axis is on the x–y plane A wave of unit strength, and wavelength,

λ0, is considered incident on the stratified structure at the angle θ0 The case of trary polarization can be treated as a superposition of TE (transverse electric) and TM(transverse magnetic) polarized waves For TE-polarized light

z N

laser beam

Figure 1.10. A schematic diagram of a laser beam incident on a multilayer structure

Using Maxwell’s equations, it is found that W is linearly dependent on U, and that the

solution can be expressed in the form of a characteristic transmission matrix, defined by

In the above expression, pc= nccos θc for a TE wave, and pc= cos θc/ncfor a TM

wave The angle θcis complex for absorbing films and is defined by the generalizedversion of Snell’s law of refraction:

The lumped structure reflectivity and transmissivity can be obtained The reflection and

transmission Fresnel coefficients, rFand tr, are

1.1 Classical electromagnetic-theory concepts 19

Absorptivity Reflectivity

Transmissivity 200

reflec-length λ = 0.5145 ␮m, upon a silicon layer of thickness dSi= 0.5 ␮m, deposited on a

glass substrate The variation with temperature of the complex refractive index of silicon

(Sun et al., 1997) generates distinct changes in the optical properties of the film due

to interference effects, even though the bulk-silicon normal-incidence reflectivity variesslowly with temperature When silicon melts, it exhibits a metallic behavior with anabrupt rise in reflectivity and drop in absorptivity

Returning to the stratified multilayer structure, the time-averaged power flow per unit

area that crosses the plane perpendicular to the z-axis is given by the magnitude of the

Poynting vector,

2Re[
E (z)×
H∗(z)]. (1.94)

A plane wave is assumed incident on the structure, with electric field amplitude Ea+ The

corresponding energy flow along the z-direction is

S = na

2µc |E+

The electric field amplitudes of the reflected and transmitted waves, E−a and E+ss, are

obtained using the above expressions The electric and magnetic fields in the mth layer,

Calculation of the amplitudes E m+and E m−starts from the first layer, for which d m−1=

0 Once the electric field has been determined, the power flow is evaluated everywhere

in the structure At a location z within the mth layer

1.2 Optical properties of materials 21

1.2 Optical properties of materials

1.2.1 Classical theories of optical constants

Consider a medium characterized by a complex refractive index nc= n − ik The

com-plex dielectric constant (or comcom-plex permittivity) is defined by

εc= ε0(1+ χ) − i σ

ω = ε0(ε− iε), (1.102)where

ε0 = 1 + Re(χ) = n2− k2 (1.103a)and

the free current density, while the part Im(χ ) is caused by the current density associated with bound charges For a nonmagnetic material (i.e µ = µ0) the components of thecomplex refractive index are derived from (1.103a) and (1.103b):

The Lorentz model for nonconductors

According to this model, polarizable matter is represented as a collection of identical,

independent, and isotropic harmonic oscillators of mass m and charge e, whereas

elec-trons are permanently bound to the core and immobile atoms In response to a drivingforce produced by the local (effective) field, the oscillators undergo a displacement fromequilibrium
x and are acted upon by a linear restoring force Ks
x, where Ksis the spring

stiffness, and a damping force b ˙
x, where b is the damping constant The equation of

motion is

m ¨
x + b ˙
x + Ks
x = e
Elocal. (1.105)The excitation is assumed of periodic form,

Of interest is the resulting periodic response:

x = (e/m)
Elocal

where the resonance frequency ω0=√Ks/m and ζ

ment and the electric field are not in phase Equation(1.108)is rewritten

x = (e/m)
E0eiωteiφ

The induced dipole moment of an oscillator is e
x If N is the number of oscillators

per unit volume, the polarization
P (dipole moment per unit volume), neglecting localeffects, is

2 p

full-width-at-half-maximum points are at ω = ω0± ζ/2.

In the ideal case of no absorption, i.e ζ = 0, the real refractive index goes to infinity,

n ω →ω±⇒ ±∞ The regime of anomalous dispersion is the only region in the radiation

1.2 Optical properties of materials 23

spectrum of decreasing n with increasing frequency Taking the limit of Equation (1.112) for high frequencies, ω  ω0:

where ω pj , ω 0j , and ζ jare the plasma frequency, the resonance frequency, and the

damp-ing constant assigned to the j th harmonic oscillator This multiple-oscillator model can

be used for fitting the radiation properties of materials over a broad spectral range The

most important resonance arises from interband transitions of valence-band electrons to

the conduction band To induce an interband transition, the photon energy has to exceed

the band-gap energy Eg Insulators have band-gap energies in the deep ultraviolet rangeand concentrations of free carriers (electrons and holes) are very small in the visiblerange The band-gap energy of semiconductors is in the visible or near-IR range It istherefore possible for free carriers to contribute to the optical response spectra in thevisible range In addition to electronic transitions, resonant coupling to high-frequencyoptical phonons usually occurs at near-IR frequencies

The Drude model for conductors

In conducting media, not all electrons are bound to atoms The optical response of

metals is dominated by free electrons in states close to the Fermi level If an external

field is applied, their motion will become more orderly Since there are no resonancefrequencies, the optical response of a collection of free electrons can be obtained by

15 20

15 10 5 0 1

Photon Energy (eV)

¨
x + ζ ˙
x = 0 ⇒
x =
x0−1

ζ
v0e−ζ t (1.119)The velocity ˙
x =
v =
v0e−ζ t = v0e−τ t

The characteristic decay time or relaxation time τ = 1/ζ The damping constant is

therefore related to the time between collisions due to impurities, imperfections, and

lattice vibrations Typical relaxation times are O(τ )≈ 10−13s.

The solution to Equation(1.118)for a time-harmonic field
E=
E0eiωtis

where m is the effective electron mass.

1.2 Optical properties of materials 25

Assuming a volumetric number density N of electrons, the current vector is

For frequencies in the far IR, ω  ζ, the electrical conductivity may be approximated

by the static, dc, value:

which is real and positive

In straightforward analogy with the Lorentz model, the complex dielectric functionis

Table 1.1 Plasma frequency and relaxation times for various metals (Prokhorovet al., 1990) The relaxation times

derived from ε(ω = ζ) = −ε(ω = ζ) ∼ = ω2τ2/2 were used to obtain the data shown in the last column

These relations are identical to the behavior of the Lorentz model at high frequencies

For frequencies in the range 1/ζ  ω < ωp, i.e in the near-IR and visible spectrum,the refractive index is nearly completely imaginary while the surface absorptivity andabsorption coefficient are constant:

At frequencies much higher than the plasma frequency, Equation (1.128) suggests that

ε→ 1 and ε→ 0, and, consequently, n → 1, k → 0 This is the so-called regime

of UV transparency Typical behavior of the Drude-model prediction is displayed in

Figure 1.13 The simple Drude theory is remarkably effective in the prediction of opticalproperties of metals such as aluminum However, it fails by itself to explain the opticalbehavior of many other metals For example, the reflectivity of bulk silver exhibits aprecipitous drop to near zero at the photon energy of 4 eV (Palik,1985) Yet, above thisplasma frequency it rises and falls again to low values at higher frequencies This trend