Thank you very much for this niceness to
the action and with afternoon to everyone

and welcome to this talk on carbon base in
a context for an electronic systems so.

Before I start talking about
these carbon based in a context.

It's good to recognize why
these in a connects are so

important and we need to look for.

New in a kind of technologies.

So if you look at the.

Power dissipation on a chip or.

In other wars if you look at the kept
the total capacitance on a chip.

This data from intil four

point one three Micron Technology North
shows that in a kind of accounts for

fifty want more than fifty one
percent of the capacitance on a chip.

So more than fifty percent of
the dynamic power on the chip at that

particular technology North was
dissipated by in economics and

there ist is for transistors and and

Mondo's in a kind of excuse
that you see that worth

short local in economics and
no one in a context long.

No one in a context are contributing
to power dissipation So

this data is presented by Intel at
that particular technology north and

the authors of the paper predicted within
five years the share of in a kind of

power dissipation really reach
sixty five to eighty percent.

And then if you look at
the contribution that in a kind of X.

make to to see it could
delay this charge from

Texas Instruments is quite revealing.

Because it shows that transistors
account for only forty three percent of.

Could delay and resistance and
capacitance of warthe

interest sail into comics and into in
a kind of accounts for the rest of the.

The layoffs circuits and
to see how the how

did delay contribution of in a kind of X.
evolves with technology this chart

again from Texas Instruments
shows that at one hundred thirty

nanometer technology north sixty four
percent of the delay was determined by.

By transistors and this is for
if Syria kit made out of

a few NAND gates and there aren't any
of the long in that kind it's in it.

They're all short in a kind of X..

Then in ninety nine to be surprise you
know that the share of chance this service

is reduced to fifty seven percent and
then in sixty five nanometer.

Technology you know it is reduced
to four to six percent and

obviously the share of in a kind of X.
has continuously.

Increased So this tells us that both
in terms of power dissipation and

in terms of super performance
in a kind of dominate.

In terms of the limitation
that the imposed.

Now once we get to sixty five and

forty five not only technology
nodes then something else becomes

even more problematic you read in
a kind of X. and that's the rise in.

Copper resistivity.

So whenever the cross-section
of dimensions of of wire become

comparable with the mean free path of
electrons in Baltimore to yield then.

The scatterings that electrons experience
at the surface of these wires become

important and therefore the effective
means the path becomes.

All and smaller as we scaled down the
dimensions of the worst just because of

the extra scattering Nick events
that happened at the wire surfaces.

The other problem with
copper resistivity is that

the size of the grains that form
these copper wires the size of

these grains is also proportional to the
cross-section of dimensions of the wires.

So when wires become narrower and
femur the grain sizes become

smaller which means that electrons have
to pass through more grain boundaries and

every time they go they pass
through green boundaries.

There is a.

There is a probability of being back
scatter so worth of these increase

the resistivity of copper wires
as we scale down the dimensions.

So this plot shows the resistivity of
copper wires versus lines with and

the solid line here shows
how resistivity increases.

Once we get two dimensions below
one hundred nanometer and so and

there is north to nor
known technology solution to this problem.

Now quite interesting.

Really if you go to the extreme form of.

Skid wires which are quantum way.

Then the mean free path of
electrons can be extremely large.

There isn't for this is that in a quantum
way electrons can move in one dimension

only and therefore the only way that they
can get scattered it's just is to get

scattered backward which requires
a large sudden change in momentum and

therefore in a high quality quanta wire.

Back scattering of electrons does

not happen as often as it does in a three
dimensional wire in which electrons can

go through a series of small angle
scattering and get back scatter.

So.

Because of this property of quantum waves
in a carbon nanotube beaches of good

high quality quantum wire.

You can have me free pass
on your order of Micron

whereas in a good MIT good
conductor like copper

the path of electrons is in
due order of forty nanometers.

So this brings us to
carbon based nanomaterials

graph an energy bins graphene is
a single layer of carbon atoms and

if you call it and pad there knew it you
would have a graph and then a ribbon.

And if you rolled this graph in the ribbon
you have a single wall in a tube and

if you have a set of course centric.

Cubes then you have a model
to all night it you.

So these can be quite
close to quantum wires

because the dimensions of these
materials can be quite small.

For example for

single attitudes damages as small
as one nanometer are quite common.

And.

Because of that the need for
path of electrons is quite large in these

materials the bonds between carbon
atoms is extremely strong and

therefore these materials
are mechanic you raise robust.

And then Grapheme has a very
peculiar band structure.

It's a zero gap semiconductor and
then when you pattern every been or.

Form a cube then you apply
quantum confinement and

depending on the geometry and how this
quantum confinement has been applied.

You may end up having a metallic
material or a semiconductor material.

Therefore the same material carbon
it using graphic evidence can be

can be either metallic or semiconductor.

And therefore there are potential
candidates for both in that context and

turns.

Sisters.

So today I retarding about
the electronic properties of these

materials and try to see how we can
potentially use them as in a kind of and

solve some of the problems that
we deal with copper in a context.

So I restarted carbon nanotubes and
instead of talking about carbon attitudes.

I would talk about the sick it models for
ideal Kuantan where

their carbon nanotubes are one
example of these quantum wires.

And then taken to a convent
nun ideologies and talked

about in a kind of applications of carbon
nanotubes then I'll talk about graphing

manage ribbons and the suit could model an
ideal conductance of graph or narrative.

And there in a crank up to cations And
finally the conclusion.

And to stop me if you have.

Questions.

So let's focus first on an ideal
quantum where the first

thing that is quite interesting about
quantum wires is this kinetic inductance.

Let's first take a look at the classical
explanation for Quantum for

kinetic inductance.

Conventional inductance is defined
as resistance to current change.

Due to fire there is law and.

Inductance represents the energy
stored in magnetic field.

And so you can define you can
integrate over the whole space and

find the total energy stored in magnetic
fields and equate that with half

squared and this for you can calculate
the inductance of these conductivity.

The mechanical counterpart for

inductance is mass because that's
the resistance to velocity change.

And the energy is.

Sure they did that is half and the script.

Now when you think of
current It's movement

of carriers which have none zero mass and
therefore.

They have some kinetic energy as.

In addition to the energy that
is stored in the magnetic field.

So if you want to be very accurate.

You need to if you want to know the.

The total energy of the system
you need to add the energy

stored in the magnetic field with
the total kinetic energy of electrons.

But in most conductors in three
dimensional conductors this terror is

many orders of magnitude
smaller than this.

And that's why we conventionally
ignore the second part.

But when you deal with quantum wires.

Current in classical terror
is the product of electric

electron charge carrier density and
electron velocity.

So for for
a given current since in a quantum where

carrier density is extremely low.

Then the lost city of carriers
has to be very large and

when velocity of electrons is large
then it means that the kinetic terror

here becomes quite substantial and
in quantum wires often

this term becomes much larger than this
and you can ignore the first part.

So quantum so
kinetic inductance can be defined

to represent this term the same day that
magnetic inductance is used to define.

Or to represent the energy
stored in the magnetic field.

So obviously this is
the classical explanation and

we know that as we increase the current.

Velocity of carious doesn't change
proportionally in quantum mechanical.

Theory.

So in quantum mechanical explanation.

You can.

For a quantum wire you
can solve the sure thing.

Your creation and apply the boundary
condition and find the energy states that

are allowed and these energy states
are plotted here versus of a vector.

And when you don't have any current
there are the same number of

electrons are moving from
left to right with the ones

that are moving in opposite directions.

So all electrons on this side
are moving from left to right

because their velocity is positive.

And the ones on this side have
in the negative the lost city in

order to have current You have to
convert some of the left move or

their turns to dry conversions and
because of the poly exclusion principle.

You cannot put more than two
electrons in each energy state so

to convert the electrons have to go
to higher available energy states and

in this way when you increase the current
energy of your system the kinetic

energy of your system increases and since
in Quantum was the density of states is

very low this increase in total
energy is quite substantial.

So for each quantum conduction.

Channel.

You can find the kinetic inductance to be
around for nine or energy per my Tron.

And it is more than four orders of
magnitude larger than the magnetic

inductance of a normal conductor.

Now.

Similarly you can explain
quantum resistance because

now the currents that
are moving from left to right.

These electrons you have to inject
them at a higher energy level compared

to the ones that are moving in opposite
direction because now you have you have

two different fermion energies for your
right movers and left mover electrons.

And therefore you need to apply a word to
each to give electrons enough energy so

that they can be injected
to the conduct to the.

Quantum wires.

And this want to is proportional
to the current that you get.

So you can define a quantum resistance
which is about twelve point nine killer or

as the minimum resistance that
a quantum where can have.

This is a fundamental limit.

If there is no scattering anywhere.

Neither at the contacts nor
along the length of the quantum wire then

the resistance of this quantum wire
would be this twelve point nine.

Q. or for each conduct channel and

in a single wall then it you if it is
metallic there are true quanta in.

Two conduction channels and
therefore you have six point five kill.

Now knowing the inductance and
capacitance and

resistance of use for
an ideal quantum wire

we can develop this simple
transmission line model for an ideal.

Q You have contact real
have quantum resistance

that has been split between the two ends
because this quantum resistance is length.

Independent.

And the rest of the components
are distributed along the list you have

magnetic inductance like any other mid to

any other conductor you have kinetic
inductance and you have capacitance.

In the transmission line.

You know that the velocity develop
vs propagation velocity is one over

school route of L C L is inductance
per unit and C. is capacitance period.

And in a normal transmission line.

We know that the very for appreciation
speed is equal to the speed of light.

So now that you have an extra chair here

this kinetic inductance which is more
than forward is of magnitude larger than

the magnetic inductance then
our propagation velocity is.

More than a hundred times smaller.

So every wave.

An electromagnetic wave in a quantum
wire propagates hundred times slower

compared to to the speed of light or
compared to a normal transmission line.

Now this would be perfect if you want
to make a delay line because then

you can have a shorter structure and
have the same delay but

if you want to make it in a kind
of this would be a problem.

So for example here this step response
of these two part with physical

in a car next is plotted versus time.

This is the voltage at
the end of the line venue

excite these two in a kind
of you to step functions.

For copper wire which is this black line.

You have this behavior.

Whereas for the ideal carbon nanotube.

In a kind of.

First you have a large travel time.

This is the time that it takes for the.

For the ray for the electromagnetic wave
to arrive at the end of the line and

then you have this R.C.

type behavior because of the quantum
resistance and the load capacitance.

So you can see that this ideal
Carbonite you is much slower

compared to this copper wire.

Just because of the quantum resistance and

kinetic inductance worth of
reach our fundamental limits.

So you can not do anything
about those things.

But we need to keep in mind
that kinetic inductance inside

is somewhat different
from magnetic inductance.

For magnetic inductance there
are two properties that.

Dictate the very last city they
vary from piggish in speed being

independent of the geometry.

First the distance to the return path.

The term is the magnetic inductance so
if you have.

With a transmission like this
you can increase the spacing

between the ground plane and
the wire and lure the capacitance but

the problem is that inductance
increases by the same factor and

the vapor peckish and speed doesn't
change the other thing is you have

mutual magnets you conductance So if you
connect two wires in parallel magnetic

inductance doesn't change much because of
the mutual inductance between the two.

So as you change the geometry.

Magnetic inductance and
capacitance are so.

So related such that the very for
up again velocity does not change but

for kinetic inductance it is
only a self inductance and

its value doesn't depend on
the distance to every chair passed so

by connecting nanotubes or
quantum words in parallel.

You can lure the kinetic inductance and
make your conductor faster.

And therefore this is something that
you can do in order to achieve a fast

in a connect you connect these and
these quantum wires in Palin.

And the inductance or the.

Kinetic inductance would
decrease linearly in

as you increase the number of
the nanotubes in this structure and

if you scale the dimensions proportionally
you can keep the capacitance constant and

therefore you can improve
the velocity by a square root of and.

And this is quite important
because this not only

deals with carbon nanotubes it also
applies to any other car kind of

quantum where that you want to use as
a transmission line or as in a kind of X..

So this way you can get around
this fundamental limits

of kinetic inductance and

also quantum resistance with scaled down
linear knew if you did number of Nann.

An attitude is that you can act in power.

So here this red line shows
the step response of this bundle

then you connect when you have
an area of ten by ten nano tubes and

now the travel time has become so
small that you can ignore it and

since this has a small of resistance
compared to do copper wire.

You see that now you can
outperform the copper of wire.

So it's quite important to have.

Bundles of quantum wires in order
to lure the kinetic inductance and

quantum resistance.

So up to here everything was about ideal.

Quantum wires but obviously we
never deal with ideal materials and

we need to take into account the extra
scatterings that electrons experience due

to for nons or due to do you fix or
add the contacts.

So this is the complete secret mother for
a carbon nanotube at both ends.

You have this quantum resistance
which is what you talked about in

addition to that you have
also contact resistance and

this is because some of the electrons
gets reflected right at the contacts and

the value of this contact resistance
me vary from a few hundred ohms

to hundreds of Q. or
depending on the quality of the contact.

Then the rest of the components
are distributed along the length.

You will have various electronic
form on scattering mechanisms and

the resistance components
that you see here.

Each represent one form of electron for
non scattering mechanism and.

If you are interested you can
look at these references for

the details of these
resistance components.

Then you have magnetic induction is
kinetic inductance electrostatic

capacitance and also quantum capacitance
but in most cases you can ignore discount

on capacitance and that's why I didn't
talk about these quantum capacitance but

again this quantum capacitance
is because of the small.

Density of states.

One thing that we can.

Conclude from this secured model is that
if you look at the effective mean free

path which is something that represents
all these resistance components.

It can be approximated by this simple
expression would be is damage here and

T.'s temperature and
Tina is just a constant hundred K..

So.

This shows that the effective.

Mean free path here is legally
proportional to dam meter and

qualitatively you can explain this
by noting that the electrons wave

packet in a car when attitude is like a do
not extended around the circumference.

And therefore the effective disorder that
an electron experiences gets average

around the circumference and for a larger
diameter tube the electron experiences

as smaller effect and the probability
of being scattered becomes smaller.

Now using the suit as mother lead
you just saw here conductivity has

been prompted to rest this length.

For a bundle of carbon nanotubes.

Temperatures assumed to be one
hundred degrees C. and the geometry

of these tubes has been assumed to
be random therefore statistically

only one third of the tubes are metallic
their wrists are semiconducting.

And there is a certain spacing between

the tubes which is dictated
by the random vol forces.

And contact resistance is assumed to be

relatively small compared
to the quantum resistance.

So.

Here this red care shows conductivity
of this bundle as a function of Lance.

And also the dashed lines showed
the conductivity of copper wires.

When you scaled down
the length conductivity.

Of this carbon nanotube one that decreases
because of the quantum resistance

that does not scale of it if you have
a quantum resistance that is fixed and

is independent of the live.

So when you make to the man its you.

Very short then this quantum
resistance limits your conductivity.

And also you see that copper wires.

RIDDIFORD cross-section of dimensions
have different conductive it is and

if you have a copper wire with fifty
nanometer cross-section of dimensions.

It would have about
the same conductivity as.

These carbon at your bundles.

So this carbon at your bundle
can outperform copper wires

below fifty nine meter dimensions.

And if you increase
the diameter of each cube.

You your overall
conductivity drops because

the number of cubes per cross-sectional
area decreases quadratically if you didn't

read the diameter there
as the number of cubes.

There as the new free pass
increases only lean years.

So ideally you want to have a very small

nanotubes nanotubes
with minimum damage or.

So this table that is now part of the eye
to receive the international technology

roadmap for semiconductors shows
that at each technology north.

What is the minimum density that
you need to meet in order to beat.

Minimum sized copper wires in
terms of conductivity because

in terms of fabrication probably
one of the most difficult thing for

fabricating these carbon attitude
bundles is to make them difference.

These bonds is typically are quite porous.

And in terms of technology it's
quite important to know what density

you need to achieve to become
competitive with copper and this tape.

The minimum density in each
Technology know it in order to beat.

Copper and the good thing is that as
technology advances this minimum density

becomes smaller which means that it
becomes easier to beat copper and that

simply because copper becomes Warse when
you go to smaller and smaller dimensions.

Now there are some properties
associated with this

these bundles of carbon
attitudes which are important.

If you want to make one of those of
carbon atoms for in a crack applications.

First the Internet a tube.

So the coupling between the attitudes
in a bundle is very weak and

this is because of these nanotubes
have random chiral it is and

therefore the eigenstates of electrons
in neighboring magnitudes are different.

Therefore there is very little coupling
between electrons in in neighboring

carbon nanotubes so
you can treat it is not its use as

isolated nanotubes in terms
of electronic properties.

This is good because it means that
metallic cubes remain metallic.

But it's bad because there is
no coupling between them and

it means that at the contacts you
need to make good or any connections

to all tubes within the bundle otherwise
only do those that are connected to

the contacts will be conducting and
the rest will be just wasted.

So this makes making connections
two bundles of carbon atoms

quite challenging especially for
horizontal.

In a context because if you
have a side contact like this

you are making connections
only through the outer.

Nanotubes.

And the other point is that all
the tubes within the bundle need to be

the same size and they need to go
from one contact to the other contact

if they are broken in the middle.

They do not.

And tribute much to conduction.

Now let's move on to graphene.

Manners had been so.

In terms of electronic properties
you can't think of graphene

as a carbon nanotube that is undrilled.

In terms of fabrication.

That is not the case and

you cannot only rule a carbon attitude
to obtain graphic ribbons and

usually you grow graphene either
on silicon carbide or exfoliating

from it from graphite and then using the
fog to feed to pad there in these reboots.

But in terms of electronic properties.

There are some similarities
between the two.

And this share many of
the good properties that.

That we talked about before they can
conduct very large current density is

the electron can be quite large and
they are mechanically robust.

There are some major differences on the
positive side for graphene you can use new

thought graphy to pattern them and
that is that can be a major

advantage compared to carbon nanotubes
that you have to grow them and you have to

control how they are grown and it's very
difficult to make sure that they are.

They are placed at the right place and
so on.

And the other advantage is that
since you can pattern them.

You may be able to pad them devices.

I mean transistors and in a crisis on the
same graph and shoot in a seamless fashion

and therefore you could potentially avoid
some contact resistances and also some.

Quantum resistances.

On the negative side you will see that
quantum resistance in graphic driven

is twice larger.

And also if the edges are rough.

Then there would be extra
scatterings at the edges and

that would limit the conductance
of natural ribbons.

Now this secret mother photograph.

Reuben he's shown here which is
quite similar to what you saw for

carbon nanotubes Harvard
the values are different.

You have again contact
resistance quantum resistance.

You have kinetic inductance and quantum
capacitance and then you have different

resistance come you know resistive
components here which are due to.

For non scatterings defects scatterings
and also scatterings at the edges.

Now most of these components are somehow
dependent on the number of conduction

channels that you have for example
quantum resistance is about thirteen Q.

or per conduction channel or

committee conductance is eight nine or
Henri pair Micron for each channel.

And also these resistance values depend
on the mean free path of electrons and

also on the number of conduction channels.

So this number of conduction channels
which determines the ideal conductance is

a very important parameter.

So the next few slides are I would be
just talking about these the number of

conduction chance in graphing ribbons of
different geometries and different rates.

So he if you have a ribbon rare along
the edges you see these arm chair pattern.

Then the ban structure depends
on the number of carbon atoms

that you see across the reef if the number
of carbon atoms across the reef is.

And multiply of three.

Plus two then the this
would be a metallic ribbon.

Otherwise it would be a semiconductor

this is based on simple
type binding approximation.

Later on I will talk about.

Some modification to this but.

To start we can look at this.

Type binding approximation and
we have this kind of.

Dependency of band structure on the number
of carbon atoms across the reef.

So here.

The number of conduction Challen
which you can also call it as

conductance divided by conductance
are one conduction chance.

This has been plotted versus reason for

this armchair ribbon for both metallic and
similar conduct in humans.

So the gap between the sub bands is
inversely proportional to writhe so

when you have very void
ribbons the gap is so small

that there is almost no difference between
metallic and semiconductors ribbons.

But when you come to smaller
dimensions then the gap increases

in semiconductors ribbons and therefore
conductance of metallic ribbons and

semiconductor ribbons
gives very substantially.

And we could also look at.

Conductance of carbon nanotubes
metallic carbon attitudes and

semiconductor tubes and plug that versus
our conference and compare the two.

And you see two differences first here for
carbon that you metallic Carbonite use you

see that there is a factor
of two difference year.

And also the difference between
conductance of metallic and semiconducting

ribbons is much smaller compared
to that for four carbon nanotubes.

And this goes back to two the band
structure of these two materials.

In in carbon nanotubes the back
the the boundary condition is that

the electron wave packet has to be
periodic around the circumference.

So both a sign and cosign are soon
to the surely going to creation.

Whereas in a ribbon the very
function has to vanish at the edges.

So only a sun function can be there.

So the sub sub bends in the.

Carbon match you are degenerate for
each energy level you have two subclans

on top of each other but
in a graph and ribbon.

That's not the case and therefore.

You see this this factor of
two difference over here.

Instead in graph and ribbons the gap
between the sub bands is half of what it

is for carbon nanotubes and that's what
the difference between the between.

Conductance is of metallic and
semiconductor ribbons

is much smaller compared to what it is for
carbon nanotubes.

Then the other point is that these two
plots that you see represent discrete

points because the riff of this ribbon
can change by a lattice constant.

And if for example you have
a drafting group interviewed.

Ten point two nanometer Reus
there would be eighty three

Adams across the wrist and
therefore this is metallic ribbon.

But if you are off by one
carbon atom either larger or

narrower then you would have
semiconductor ribbons and

therefore your conduct and we'd be
five times around five times smaller.

And if you wanted to make in that
context out of this material.

This would have been a major problem
because you cannot guarantee

that the roof of a ribbon is specific
value of an atomic resolution.

However this is the this
is a problem only for for

the case when the fermion
energy is equal to zero and

in graphene most of the time to
spare me energy is shifted because

some electrons get trapped
between the graph in earlier and

substrate because the work
functions are different and

also some molecules might
drop the graphene layer.

And then for me energy is
shifted the cares that you just.

Are shifted to the left.

So this is point one electron ward and
this is point two electrons or

does are the fermion edges for
these ribbons.

And now the.

Metallic and
semiconductor ribbons are conducting

equally down to the dimensions
that you are interested.

I mean even up to down to five or
three nanometers

you can have a smear tactic and some
are conducting ribbons that are equally

conducting So this is very good news
because it may tell you that you may not

have to control the Kyra lity of these
ribbons for in a kind of applications.

If the Fermi energies elevated
the other interesting point here is in

this region for the for the Fermi
energy zero point two electrons or

do you see that the semiconductor ribbon
is conducting even more than the metallic

ribbon and this sounds odd but
when you look at the bass searcher it's.

It's easy to on to to understand why

this is the band structure
of this metallic ribbon.

So you have fifty carbon atoms here.

Then for.

Similar conduct this
is the band structure.

So the gap between the sub
bands in the semiconductor

is about one third of
what it is over here.

So if your family unit is point two
electron void in the metallic ribbon you

have populated only one subband So
your quantum conductance is equal to

one quantum conductance and
here you have two sub bands populated.

So the number of conduction
channels here is larger.

So this is good because it tells you
that some are conducting ribbons can be

equally conductivity or
even better than metallic ribbons.

If the fermion A-G. is elevated.

For is exactly bins of simple tight
binding tells us that all ribbons should

be metallic and therefore if
you plot conductance versus vs.

For very narrow ribbons you have
only one conduction channel.

But when you increase the reef then
the gap between the sub and becomes so

small that some higher level
has also get populated

because of the thermal energy
of electrons and then.

Conductance.

Starts increasing and again it's been
compared to carbon nanotubes and

again you see that factor
of two difference.

And as you increase the fair manager of
the zigzag rebrand conduction channel

conductance increases especially at higher

at a knowledge of its because
then the bandgap is smaller.

Now up to here.

It was based on simple type binding.

There are some detail for example first
principle calculations show that.

Carbon atoms at the edges are closer
to each other by about three point five

percent and this opens a gap in the band
structure of this ribbon that was

that simple type binding showed that
this should be a metallic ribbon so.

Now if you take this into account
this changing lattice constant

There's a small band gap.

Even in the band structure
of this ribbon and

the value of this band gap is inversely
proportional to v If Which makes sense.

Qualitatively because if you
have a very vital ribbon

what happens the details of what happens
at the edges is not important but

if you have a very narrow ribbon
then this can be quite important.

And also this changing lattice
constant changes the bandgap in

semiconductor bins for three P.
ribbons it decreases the band gap.

And for three people us want
it increases the bandgap.

And there are some experimental
results which support this and.

Here is the biggest group of staff four

hundred eyes grew at Stanford they
have chemically derives some.

Graph and ribbons and they have measure.

The bandgap and they have compared the
results with the theoretical models and

there's a good match there.

So taking.

This changed a lot this constant into
account here conductance has been plotted

versus with and
if the Fermi energy is equal to zero for

these ribbons that very
supposed to be metallic

now you have this drop in conduction
because of that gap that their peers.

But when you elevate defer manager to.

Even a small firm in energy or
in zero point one electron ward.

Then you don't see much difference
the dash line is what we opted for

speed simple type binding.

Now this is when you take this
change in constant into account.

And again you see that for.

For this very energy there is no
difference between the conductance is of

these different kinds of ribbons.

If you are a balls eight nanometers and

four in a kind of applications we
are mainly interested in this region.

For our Also for zigzag ribbons first
principle calculations showed that

if you take into account the spin.

Degree of freedom a small
band gap opens in

the band structure of zigzag ribbons and
this is because and and

again this can lure the conduct
of zigzag treatments.

Especially if the federal
energy is equal to zero.

But if you elevate defend
the energy then this band gap is.

Is is not important because your feminine
energy is larger than disband gap

unless you go to very small dimensions.

So for in a crank up occasions we need
to know what is the resistance and

resistivity of these
materials to see whether or

not they can outperform copper wires.

The number of conduction channels
the other thing that we need to know

is the mean free path of electrons for
graph and ribbons.

The mean free pass.

Associated with scattering with acoustic
for nuns which are dominant at low bias

world it is predicted to be in
the range of tens of micro meters.

And.

Some experimental results on mobility
sure mobility is underage of two

hundred thousand since in descriptive
words second and therefore.

That corresponds to several.

Micro meter mean free paths.

So this corroborates
this prediction that the.

Intrinsic need for a path.

Dictated by electron for
interaction should be quite large.

So it is quite so it is generally expected
that the path be dictated by disorder.

By the substrate in use to disorder and
the effects and

also by edges roughness and
for the next few slides for

the sake of time and also to look
at the upper bound on performance.

I will assume that the edges
are Smoove and just assume

constant mean free path dictated by
the substrate induced disorders.

So here resistance period length has
been proud to address this with four

two fer me energies and different kinds of
graph and ribbons and the need for a path.

Dictated by.

Substrate induced disorders
is assumed to be one micron.

And you see that if the fermion edge is
point to electron walked down to four or

five nanometers these different
kinds of graphing treatments equally

conduct will and I can compare this
with him on a lead out single wall and

its use and you see that I'm on
a layer of single tubes that are.

Densely packed is.

Is better.

However the problem with carbon and.

It is that no one knows
how to pack them and

how to fabricate them in a straight and
aligned fashion.

And you can also compare
it with copper wires.

This is a couple of where everything
is quote to its thickness and

therefore this is not a fair comparison
we are comparing a thick copper wire

to the single layer of graphene
which is only one atom thick but

still at eight nanometer there's
a there is this break even point and

below that graphing Ruben's can offer
smaller resistances per unit limits.

Then you can plot sheets resistance or

physicists like to call these
two dimensional resistivity.

And this is plotted versus Raef for

different kinds of graph and ribbons for
me energies again point to electron vote.

And now if you assume a certain
thickness for this graph and

ribbon point three five nanometer which
is the spacing between graphite layers.

Then you can calculate the normal
three dimensional resistivity and

this is shown by the right axis and

then you can compare this resistivity
view to what copper wires offer and

what wonders of carbon nanotubes offer and
you see that.

For small dimensions the advantage that
these graphing ribbons can potentially

offer it is.

Enormous.

Of course this is assuming
that the edges are.

Are very smooth and
also fairly energy everywhere in this

stack of graphing layers is
point to electron forward.

So these are does requirements
that we need to meet.

So we need stacks of non interacting
graphing layers to take advantage of this

small three dimensional resistivity.

So again to repeat.

Visitors requirements that.

Graphing stacks have to meet.

We need stacks.

Not interacting graphing manners ribbons
and there are some examples of these

layers for example in a potassium
graphene it is shown that.

Different layers are decoupled
because of some retain

that each layer experiences compared
to the early years or upper layers.

And all the layers must have fairly
energies that are different than zero.

And the edges need to be smoothed and

we need to make good electrical
connections to all layers because

this is again like carbon nanotubes if
we don't make connection to all layers.

Only those that are connected will
be conducting So to conclude.

Result that the limits that
are imposed by in a kind of excess on

cheap power dissipation and
also performance are quite dominant and

they are increasing with
technology scaling.

A common base in a crisis showed great
potential because of being one in one or

two dimensional materials and therefore
having a Large Electron need free pass.

Hope I didn't find time to talk about
this but in some cases we can use.

Carbonite you in that kind of X.
that are very thin and

are in a kind of capacitance and
therefore offer much smaller.

Power dissipation and still be fast.

If you know cranks are short.

Then all graphs in.

And none of your bands have a band gap.

So there is no true
metallic graphing ribbon.

However this band get depends on reef and
country.

Ratatouille and if you're fairly
energy is slightly elevated

then there is at least one conduction
channel in all of these ribbons and

therefore they can be used as
inequalities and finally we need.

Backs of money interacting
graph in layers.

In order to outperform copper wires in
terms of resistivity four dimensions

below twenty nine in meters and
with that I would like to stop and

thank you very much for your attention and
please thank you thank you.

Just start.

Your slide where you talk about all.

Measurements for one year old boy

with your knowledge you can
actually write it short.

Again for the sake of time I didn't show
the analysis for Monta one at us and.

If you deal with it long lens for
long in a kind of X..

Large diameter monitor worn attitudes
can offer better performance compared to

bundles of seem to want to use and the
advantage that more to warm attitudes that

they have is that it is by nature a dense
structure and I mean many cases it is

even easier to group them onto one minute
you compared to seem to want to hit you.

But in terms of contact
the problem is still there

because the shells within a mile to one
night you are still non interacting.

So you need to make connections
to all she has been in the.

For the month you want to use Otherwise
only the outer shell would be conduct.

All the more the better.

So for any given damage you want to have

maximum number of layers and it seems that
the spacing between the layers is fixed.

It's dictated by the van to run forces.

So you want to have
a larger in a damage or

smaller in a dam meter to maximize enough.

Players should

thank you.