Advanced Functional Properties 6PC20

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Advanced Functional Properties 6PC20

• Lecture 7 graphene and 2 D band structure

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Alignment of transpolyacetylene
By applying mechanical force the polymer chains
can be oriented
chain dir.
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Absorption spectrum of oriented
transpolyacetylene
a
Leising Phys. Rev. B 38, 10313 (1988)
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Anisotropy
Isotropic a material is isotropic if its
properties are the same in all three
directions x,y and z.
Anisotropic a materials is isotropic if its
properties differ for at least one of the
directions x,y or z in comparison with the
others.
A quantity of a material can have the following
character
scalar
always independent of direction e.g. density,
melting temperature,
vector
dependent on one direction e.g. polarization,
deformation
dependent on two (or more directions) e.g.
dielectric tensor
tensor
Px e0 (erx,x - 1) Ex
Cause Electric field in e.g. x
direction (direction 2)
Response Polarization in e.g. x
direction (direction 1)
Dielectric tensor er
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What do the anisotropic optical constants mean ?
Classical Lorentz model light absoption comes
from oscillation of electrons that experience
friction
chain dir.
E // chain dir.
If the electric field is parallel to the chain
direction the p-electrons can move in the
direction of the field and along the chain
If the electric field is perpendicular to the
chain direction the motion of the electrons is
hampered. There is little space to move in this
direction
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Anisotropic dielectric properties of aligned
transpolyacetylene
e
Electric field E parallel to the direction of
the chains
e?
Electric field E perpendicular to the direction
of the chains
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Sidestep After doping with AsF5 metallic
behavior in direction
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Absorption spectrum of oriented
transpolyacetylene
Can we relate this onset energy to the
bandstucture calculations ?
Leising Phys. Rev. B 38, 10313 (1988)
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Last week Band diagram trans polyacetylene
Density of states diagram
E
Band diagram
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empty
0.8
Optical excitation ?
-0.8
filled
-6
r
If we take bL 3.4 eV and bS 2.6 eV then
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Photoexcitation of an electron across the gap
Electron on polyacetylene
Photon in vacuum
l (nm) 1240 / Ephoton (eV)
800 nm ? 1.5 eV
E
1.5 eV
0
k
k 2p/8000 Å 0.0008 Å-1
Photon momentum 1000 times smaller than that of
electron
L 2.5 Å
k2p/2.5Å 2.5 Å-1
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Photon absorption

• Absorption of a photon excites an electron to a
  higher energy level. This changes
• the energy of the electron E ? E
  Ephoton
• - the momentum the electron ?k ? ?k
  ?kphoton which implies k ? k kphoton

kphoton of a visible or infrared photon is very
small (kphoton 2p/l 10-3 Å-1 ) Photo
excitation results in a practically vertical
transition of the electron
Zoom in band diagram polyacetylene
Empty
Ek(eV)
Gap 1.5 eV
photon
Filled
p
k (Å-1)
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Conclusion optical measurement of the bandgap
Sofar in our calculations of the bandstructure,
we have neglected the interactions between
electrons on the polymer. From this independent
electron approximation, it follows that the
band gap of the polymer can be estimated from the
onset of the light absorption. This
approximation turns out the be not very accurate
for conjugated polymers. Yet it give a useful
insight into the behavior of the polymer. The
independt electron appoximation is much more
accurante for inorganic semicondcutors (Si, GaAs
etc.) (For a more complete account 3S350
Organic Electronics)
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Band gaps of selected conjugated polymers
These 1D polymers are insulating in their neutral
state
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Question
Why are these polymers such bad conductors in
their neutral state and why is graphite such a
good conductor ?
Graphite Layered compound Sheets of sp2
hybridized carbon atoms
Graphene A single sheet of sp2 carbon atoms
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Single layers of graphene can be prepared and
studied
Recently it has become possible to manipulate
single sheets of graphene
Photograph of graphene in transmitted light.
This one atom thick crystal can be seen with the
naked eye because it absorbs approximately 2.3
of white light
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Graphene how do the electron wave functions
look like ?
Calculate wavefunction analogously as we have
done for polyacetylene using 1) the tight
binding approximation (electons in orbitals on
C atom) 2) Hückel approximation ( pz
orbitals separately)
Solve Schrodinger equation in matrix form
Linear combination of pz orbitals with
coefficients c
Important to note that we neglect
electron-electron interaction and use the
independent electron approximation
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Structure of Graphene
Sheet of sp2 hybridized carbon 2 atoms in the
unit cell a and b. Planar structure. pz
orbitals form a 2D p-system Two unit vectors 1
and 2 span the lattice
Lattice constant a
2.456 Å
2.456 Å
For simplicity we set a 1
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Graphene wavefunctions a0,0 site
a and b site chemically equivalent
Bloch relations
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Graphene wavefunctions
b-1,0
a0,0
b0,0
b1
b0
b2
b0,-1

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Graphene wavefunctions b0,0 site
a0,1
2
b0,0
a0,0
1
a1,0
Bloch relations
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Graphene wavefunctions b sites
a0,1
b2
?0
a0,0
b0,0
b1
a1,0
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Results
x
ei?
drops out !
k1 and k2 are wave vectors but in which direction
?
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Graphene moving to cartesian coordinates
Quantum number k1 describes the wave motion of
the electrons in the u1 direction. This motion is
independent of the wavemotion in the u2
direction. Quantum number k2 describes the wave
motion of the electrons in the u2 direction.
We can now define a wave vector such that
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Result
Two surfaces above each other
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Top surface E(kx,ky)
K points (E0)
Side view
b0 b1 b2
Ek
3?
Empty
ky
0
K points
G point (k0)
Six fold symmetry reemerges !
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Band structure of graphene
3D view
Side view
Ek
3?
Ek
Empty
0
ky
0
Filled
-3?
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How stable are the K points against structural
deformation ?
b0 b1 b2 0.5 b1
Side view
Ek
3?
Empty
ky
0
Filled
-3?
K points do not go away !
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Transistor measurements
VSD
ISD
ISD
drain
source
insulator
graphene
graphene
gate
With the gate potential one can adjust the energy
up which the levels are filled with electrons
x
Vgate lt0
Vgate gt0
Vgate 0
Extra electrons
Fewer electrons
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Transistor results
Novoselov Nature 179, 438 (2005)
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Graphene mobility
m 104 cm2/Vs Controlled by disorder in the
insulator. Real mobility estimated at gt 105
cm2/Vs Mobility is among the highes values found
for all possible materials.
Intrinsic conductivity
Novoselov Nature 179, 438 (2005)
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Electron mobility graphene vs conjugated
polymers
Neutral polymers can also be put in a transistor
In this case one can also determine a value for
the mobility Here one finds m ? 1 cm2 / V s
depending on the purity of the polymers
Why do the electrons in graphene have such an
enormously high mobility ?
Remember that from our earlier investigations we
know that low mobility is the result of frequent
scattering at defects
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How fast does an electron travel ?
A wavefunction describing a localized electron
can be constructed as a superposition of free
waves with different momenta (wavepacket,
compare Fourier representation of the delta
function)
Distribution function
Localized wave function (wave packet)
Follow the time evolution of the wave packet
Time Dt
?2
x
Dx
speed Dx/Dt
Then
mobility speed /force
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Simple wave packet of just two waves
Assume that the difference between k and k is
small (kk)/2 k
w w
(w w)/2 w
Amplitude modulation
Carrier frequency
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Speed of the wave packet
Speed with which the amplitude modulation
moves
t0
Group velocity
t15
Dx
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Effective mass of an electron m
Electrons are waves
A localized electron can be descried by a
wavepacket . Speed with which the wave packet
moves is given by the group velocity vg
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Then acceleration
Suppose now that an electric field E is applied.
This exerts a force F eE on the electron
accelerates it. In a short time interval dt, this
will increase the (kinetic) energy of the
electron by an amount dE. dE is equal to the work
done by the electric field over the distance
dlvgdt
This implies
2
and
According to Newton
1
2
from which
with m the effective mass of the electron
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Effective mass of electrons in graphene
Ek
3?
The effective mass of the electrons diverges for
Ek0 What does this mean ? In order to change
the speed of the electrons one needs an infinite
amount of energy Thus the electrons travel at
constant speed
kx, ky
0
-3?
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Electrons in graphene are not scattered very
efficiently
Ping pong ball scatters back easily because it
weighs so little in comparison with the wall
Sledge hammer hardly scatters back because it
weighs so much
The electrons in graphene at Ek0 do not scatter
very strongly on defects because of their
diverging effective mass. Hence withing a graphen
sheet the electrons move with high mobility
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Graphite
c-axis
c-axis
          graphite's unit cell
Electrical conductivity  s0?c  1.0 ?105
ohm-1m-1 (perpendicular to c-axis or in
graphene plane)  s0//c   2.4 ?104 ohm-1m-1
(parallel to c-axis or perp. to graphene
plane)
Cu s0  6 ?106 ohm-1m-1
Powell Childs 1972
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Dielectric response function graphite
c-axis
Draine, B.T. Astrophys. J., 598, 1026 (2003)
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Optical properties graphite
Draine, B.T. Astrophys. J., 598, 1026 (2003)
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Real and reciprocal space
Real space
Reciprocal space k space
2p/L
0
unit cell
x
k
L
Polymer chain
Electron wave function with largest k vector
possible k2p/L
Electron wave function with smaller vector
Electron wave function with smallest k vector k??
0
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Real and reciprocal space
Reciprocal space k space
0
p/L
3p/L
5p/L
etc.
-p/L
-3p/L
-5p/L
k
Frist Brioullin zone Contains all necessary
information,
Second Brioullin zone Identical to the first.
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Reciprocal lattice of Graphene
Unit cell of the reciprocal lattice
G
G points
b2
b0 b1 b2
K
K
G
b1
By definition
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Unit vectors of the reciprocal lattice
Definitions
G
b2
Unit cell of the reciprocal lattice
K
K
Together with
G
b1
Give
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Reciprocal space the artist impression
M.C Escher
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What is the exact location of the K point ?
G
G
G
b2
K
K
K
0
0
0
G
b1
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Location of the K point ?
G
K
0
G
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Single Walled Carbon NanoTubes (SWCNT)
Graphene rolled up !
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A carbon nanotubes cutting, rolling and
glueing a sheets of graphene
Each type of tube can be classified with two
numbers (n,m)
n4
m2
c
Cutting vector
c
y
x
?
?
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Classification by inspection of the cutting line C
Each type of nanotube has a unique characterisic
set of numbers (n,m)
y
x
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Examples
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After rolling up not all k vectors from the
graphene are allowed anymore
Along the diameter of the tube the wavefunction
has to fit onto itself
Going around the perimeter the phase of the
wavefunction can change only in multiple of 2?
j integer
This can be related to the cutting line
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Resistrction on k vectors
Only k vectors that end on one of the dashed
lines Are allowed in the nanotubes
kx
Cutting vector
0
2p?2
2p
2p?3
ky
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Metallic nanotubes
If a k vector to a K point on the graphene
reciprocal lattice is allowed in a nanotube then
electrons can enter and leave the region in k
space where the filled (valence) and empty
(conduction) band have zero energy separation.
The nanotube should then be electrically
conducting. If the k-vector to K point is
forbidden then the tube must be insulating (or
synonymously semiconducting)
From analysis of graphene
Condition for metallicity
Definition cutting vector
Metallic SWCN tubes
j0,1,2,
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(n,m) can be determined from microscopy
STM images
Transistor from a single nanotube. Conductivity
can be measured
S. Iijima Nature 354 (1991) 56
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Experiments on Metallic tubes
Ohms Law
Ohms Law in differential form
Exp.
Conductivity related to the density of states r
that can transport electrons
Theor.
Theor.
Hu, Odom, Lieber Acc. Chem. Res., 32 , 435 (1999)
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Chemistry of the nanotubes is not under control
Tubes are grown from gasphase carbon precursor.
They grow on catalyst particles. This yields an
mixture or nanotubes (metallic insulating )
Electrical characteristics of individual tubes
from the same reaction mixture
Dekker et al. Nature 391, 59 (1998)