General overview of the subjects plus a brief descriptions of my contributions during the phd thesis.

1. Role of the vacuum fluctuation forces in
microscopic systems
Colloquia Doctoralia XXI ciclo, Modena 9/2/09
Candidate
Andrea Benassi
Supervisor
Prof. Carlo Calandra
Buonaura

2. § Introduction to the vacuum fluctuation forces
i) Macroscopic and microscopic forces
ii) The sign of the force
§ New Developments
i) Quantum size effects
ii) Surface effects
§ Applications
i) Stability of thin films
ii) A vacuum force based device
Outline

3. Macroscopic and Microscopic Forces
Van
der
Waals-­‐London
force
MICRO
MACRO
NON
RELATIVISTIC
(small
d)
RELATIVISTIC
(large
d)
Casimir-­‐Polder
force
F(d) = −
18
πd7
∞
0
α1(ω)α2(ω)dω
α1 α2
d
α1 α2
d
Casimir
force
F(d) = −
cπ2
240d4
L2
Hamaker
force
d
L
F(d) = −
H(1(ω), 2(ω))
d3
L2
1 2
d
L
(0) → ∞
H.G.B. Casimir Proc.Ned.Akad.Wet. (1948) S.K. Lamoreaux Rep.Prog.Phys. (1997)

4. Macroscopic and Microscopic Forces
Van
der
Waals-­‐London
force
MICRO
MACRO
NON
RELATIVISTIC
(small
d)
RELATIVISTIC
(large
d)
Casimir-­‐Polder
force
F(d) = −
18
πd7
∞
0
α1(ω)α2(ω)dω
α1 α2
d
α1 α2
d
Lifshitz
theory
F = F(d, 1(ω), 2(ω), 3(ω))
•
Extension
to
other
materials
•
Extension
to
other
geometries
•
Extendion
to
finite
temerature
E.M. Lifshitz Sov.Phys.JEPT (1956) S.K. Lamoreaux Physics Today (2007)

5. The sign of the force
d d
AIracJve
BETWEEN
ONTO
ISOLATED
INTERACTING
Repulsive
if
Stretching
if
Squeezing
dd
3
1 2 1 2
33 1
1 3 2
1 3
J.N. Munday and F. Capasso Nature (2009) Benassi and Calandra J.Phys.A. 40,13453

6. Electron Confinement (EC)
When
the
dimension
of
the
interacJng
objects
is
comparable
to
the
electron
wavelength
the
quantum
nature
of
the
electrons
cannot
be
neglected,
this
is
the
case
of
nanometric
thickness
films:
The
film
is
considered
as
a
quantum
well:
•
The
electron
spill
out
is
simulated
arJficially
in
the
case
of
an
infinite
well,
it
comes
out
naturally
in
the
case
of
a
finite
well
•
The
discreJzaJon
of
the
energy
levels
gives
rise
to
kinks
in
the
electron
density
or
in
the
Fermi
energy
Wood and Ashcroft Phys.Rev.B (1982)

7. The
RPA
dielectric
tensor
of
the
nanometric
film
can
be
calculated…
•
The
system
is
anisotropic
•
The
dielectric
tensor
components
depends
on
the
film
(well)
thickness
•
The
zz
component
becomes
semiconducJng
while
the
parallel
ones
remains
metallic
•
The
dielectric
funcJon
depends
on
the
electron
density
so
kinks
are
also
present
in
the
dielectric
funcJon
Quantum Models for the film
Benassi and Calandra Europhys.Lett. 82, 61002

8. The interaction between films
Some
plot
of
the
relaJve
percent
difference
of
the
force
with
and
without
the
EC:
•
Including
the
EC
the
force
strength
decreases,
along
the
z
direcJon
the
metallic
film
becomes
transparent,
trapping
less
modes
inside
the
cavity
•
Kinks
appear
each
Jme
that
a
new
level
fall
below
the
Fermi
energy
•
For
large
film
thickness
and
large
electron
density
the
EC
is
negligible
and
the
relaJve
difference
goes
to
zero
δP =
Fbulk − FEC
Fbulk
d
d
Ωp = 1014
rad/s = 10nm
Ωp = 1014
rad/s = 50nm
Ωp = 5 · 1014
rad/s = 50nm
Benassi and Calandra J.Phys.Conf.Series (2009)

9. Conclusions I
The
inclusion
of
EC
gives
correcJons
depending
on
the
film
density,
the
film
separaJon
and
the
film
thickness,
these
correcJon
can
vary
between
few
percent
up
to
50%
The
EC
correc*ons
can
be
improved
in
order
to:
•
include
atomisJc
descripJon
of
the
film
la]ce
•
include
surface
effects
We
believe
that
for
all
these
purposes
an
ab-­‐iniJo
approach
can
be
suitable
!
The
final
goal:
•
treat
arbitrary
shape
objects
(designing
MEMS)

10. A Silicon Surface: (111)2x1
Surface
properJes
start
to
be
relevant
when
the
system
of
interest
is
strongly
confined.
A
semiconducJng
surface
introduces
some
new
features:
•
Presence
of
surface
states
inside
the
bulk
gap
•
Surface
reconstrucJon
Benassi and Calandra in preparation

11. Modification of Dielectric Properties
An
RPA
dielectric
funcJon
can
be
calculated
both
for
the
bulk
and
the
slab:
•
The
off-­‐diagonal
components
are
negligible
•
A
strong
anisotropy
is
present
in
the
slab:
the
Lifshitz
theory
must
be
extended
to
treat
anisotropy
•
For
symmetry
reasons
the
surface
states
affect
only
the
yy
component
Slab
real
slab
imaginary
yy
xx
zz
bulk
imaginary
RPA
exp
Benassi and Calandra in preparation

12. Modification of Vacuum Interaction
Using
the
Lifshitz
formalism
the
force
between
films
can
be
calculated
using
RPA
bulk
and
the
RPA
slab
dielectric
funcJons.
Their
difference
tells
us
how
important
can
be
the
surface
effects:
d = 2nm
•
enlarging
the
film
thickness
the
surface
effects
become
negligible
•
the
relaJve
percent
difference
is
large
for
large
separaJons,
where
only
the
staJc
value
is
important
Benassi and Calandra in preparation

13. Conclusions II
The
calculaJon
of
the
vacuum
force
between
thin
films
gives
very
different
results
when
performed
using
an
ab-­‐iniJo
calculated
dielectric
funcJon
instead
of
the
ordinary
bulk
one.
This
is
mainly
due
to:
•
the
appearance
of
surface
states
inside
the
bulk
band
gap
that
gives
rise
to
a
strong
absorpJon
at
low
frequency
•
the
presence
of
the
surface
reconstrucJon
that
brings
to
a
highly
anisotropic
dielectric
response
To
improve
the
calculaJons
one
has
to
go
beyond
the
RPA
approximaJon
of
the
dielectric
tensor
including
many-­‐body
effects
A
separaJon
of
the
confinement
contribuJon
from
the
surface
one
must
be
done:
hydrogen
passivaJon
prevent
surface
reconstrucJon.

14. A Model for the Film Stability
The
vacuum
force
can
cause
a
change
in
the
surface
morphology
of
a
thin
film.
In
an
epitaxial
film
3
main
contribuJons
can
be
considered…
La]ce
mismatch
stress
Surface
stress
Vacuum
stress
The
first
two
contribuJon
are
several
order
of
magnitude
larger
that
the
Vacuum
force
however,
close
to
the
equilibrium
they
cancel
out
and
the
system
becomes
sensible
to
the
vacuum
force
R. Asaro and W. Tiller. Met.Trans (1971)

15. A Model for the Film Stability
With
a
slight
(
)
sinusoidal
corrugaJon
we
have:
q d λ
The
change
in
energy
is:
CriJcal
thickness
and
wavelength
exist
if
the
vacuum
force
is
repulsive:
∆E = −
1 − ν2
E
σ2
πq2
+ γ
π2
q2
λ
−
q2
λH
8πd4
Benassi and Calandra J.Phys.A 41,175401

16. Film Stability Diagrams
(ω) = 1 −
Ω2
1
ω2
(ω) = 1 −
Ω2
3
ω2
Corrugated
interface
Plasma
model
for
simple
metals:
the
plasma
frequency
is
proporJonal
to
the
electron
density
of
the
metal
Benassi and Calandra J.Phys.A 41,175401

17. Conclusions I
We
have
shown
how
the
morphology
of
a
thin
deposited
film
can
be
affected
by
the
presence
of
vacuum
forces,
its
behaviour
depending
on
the
dielectric
properJes
of
both
the
film
and
the
substrate.
This
phenomenon
can
be
used:
•
to
modify
the
film
properJes
during/acer
their
growth
•
to
measure
the
Casimir
force
on
a
single
object
(not
yet
measured)
However,
to
compare
our
result
with
realisJc
situaJons
some
improvements
are
needed:
•
void
defects
and
dislocaJon
must
be
included
in
the
elasJc
model
of
the
film
•
the
relaxaJon
of
the
substrate
la]ce
must
be
included

18. A Vacuum Force Based MEMS
Understanding
vacuum
forces
properJes
is
crucial
in
the
world
of
micro
and
nano
mechanics:
•
to
prevent
micro
and
nano-­‐devices
from
sJcJon,
adhesion
and
breaking
•
to
actuate
micro
and
nano-­‐devices
using
vacuum
forces
Vacuum
forces
depends
on
the
dielectric
properJes
of
the
interacJng
media,
tuning
the
dielectric
properJes
we
can
control
the
force
tailoring
the
mechanical
moJon.
•
The
force
depends
on
the
integral
over
frequencies
of
the
dielectric
funcJon,
a
drasJc
change
in
the
dielectric
properJes
is
needed
to
modify
a
rather
insensible
force
•
This
change
must
be
reversible
in
order
to
be
able
both
to
increase
and
decrease
the
vacuum
force
H.B.Chen et al.Science (2001) Buks and Rouckes Europhys.Lett. (2001)

19. The candidate: GeTe
A
good
candidate
is
the
Germanium
Telluride,
which
undergoes
a
fast
and
reversible
crystalline-­‐amorphous
transiJon…
In
the
transiJon
the
dielectric
properJes
change
strongly
moving
from
a
metal
to
an
insulator
with
1
eV
gap.
CalculaJng
the
vacuum
force
we
found
we
found
large
differences
in
the
interacJon
between
two
amorphous
plates
FAA
and
two
crystalline
plates
FCC
Benassi and Calandra Europhys.Lett. 84,11002

20. with
the
equilibrium
condiJon:
the
soluJon
is
a
bi-­‐stable
potenJal:
•
the
distance
between
the
maximum
and
the
minimum
gives
the
mechanical
excursion
of
the
device
•
The
height
of
the
barrier
gives
an
idea
of
the
stability
•
The
posiJon
of
the
barrier
gives
the
s*c*on
point
The Model Device
We
can
model
a
mechanical
device
by
a
fixed
plate
interacJng
with
a
mobile
plate
with
a
given
elasJcity
represented
by
a
spring.
The
two
plates
are
covered
with
a
thick
GeTe
film
whose
phase
can
be
switched
using
a
laser
pulse
or
a
current
pulse.
F(x) = Fres. − Fdisp. = k(x − x0) − ΣF(x − x0)
(x − x0) −
Σ
k
F(x − x0) = 0
Σ
k
Batra et al. Europhys.Lett. (2007)

21. Tailoring the Device motion
What
happen
if
we
change
the
GeTe
phase?
•
The
mechanical
excursion
can
be
modified
by
10
%
•
The
stability
of
the
device
can
be
changed
up
to
80%
•
The
sJcJon
point
can
be
moved
by
10%
x/x0
CC
AC
AA
Benassi and Calandra Europhys.Lett. 84,11002

22. Conclusions IV
We
have
shown
the
feasibility
of
a
vacuum
forces
based
device
that
exploit
a
metal-­‐
insulator
phase
transiJon
to
modify
the
mechanical
properJes
of
a
micro
oscillator.
The
device
model
is
sJll
rough
and
can
be
improved
in
a
number
of
ways
before
it
can
be
used
to
design
a
real
device:
•
Surface
roughness
effects
are
known
to
be
relevant
in
the
vacuum
interacJon
between
plates,
they
must
be
included
in
our
calculaJons
•
The
finiteness
of
the
plates
introduces
edge
effects
whose
importance
increase
decreasing
the
device
dimension
•
In
our
model
we
considered
a
global
phase
change
of
the
plates
media,
but
each
phase
change
process
has
its
own
penetraJon
depth
inside
the
GeTe
film
•
Finite
Jme
required
by
the
phase
change

23. Acknowledgments
I
would
like
to
express
my
special
thanks
to
professor
Carlo
Calandra
Buonaura
for
his
in-­‐
valuable
guidance,
help
and
support
over
these
years.
Another
special
thanks
to
professor
Elisa
Molinari
who
hosted
me
in
S3
na*onal
research
center
giving
me
the
opportunity
to
aIend
schools,
workshop
and
seminars
all
over
the
world.
I
want
to
acknowledge
CINECA
consorzio
interuniversitario
for
funding
my
Ph.D.
fellowship.

24. Field and charge quantization
•
QuanJzed
charges
(perturbaJon
theory)
•
ConJnuous
fields
MICRO
MACRO
•
ConJnuos
media
(linear
response
theory)
•
QuanJzed
fields
α1
d
d
1 2
Two
approximaJons
of
the
same
theory,
the
QED,
in
which
both
the
fields
and
the
charges
are
quanJzed
The
field-­‐charge
interacJon
is
hidden
inside
the
response
funcJon

25. Temperature: a way to eliminate the kinks
The
kinks
issue
can
be
solved
introducing
a
finite
value
for
the
temperature,
the
energy
levels
becomes
conJnuously
populated,
the
kinks
disappear
from
the
energy
and
the
force:
T = 0◦
K
T = 1◦
K
T = 2◦
K
T = 30◦
K