Graphene conductivity
A lot of effort has been devoted to the question of transport
in pure graphene due to the remarkable fact that the dc
conductivity is finite without any dissipation process
present.
Recall the Drude theory:
0
J (t ) ne 2
1
ne 2
 ( ) 


, where  0 

E (t )
m 1  i 1  i
m
1 4 ne 2  p
(in simple metals,  0 

)
4
m
4
4 ne 2
2
p 
where  p  plasma frequency.
m
In Graphene the density of states at the Fermi level vanishes, so n should be small.
However,  is not well known but surely very large.
So we can see that the problem of evaluating  0 is not trivial.
2
M. Lewkowicz and B. Rosenstein, PRL 102, 106802 (2009)
Dynamics of Particle-Hole Pair Creation in Graphene find:
4 e2
lim lim  1 
disorder 0  0
 h
e2
 1.27
h
 e2
e2
 1.57
h
lim lim    2 
 0 disorder 0
2 h
They support this value of the dc conductivity of pure graphene
Other authors find
e2
3  
h
There is no accepted theoretical value
Measurement of
conductivity
They report  =1.7 1
Previous experiments on Si substrates give 4 1
There is no accepted experimental value
We study the time-dependent transport through Graphene using
J mn (t )  Tr[ fJˆmn (t )]
f 
1
 ( H0   )
e
1
Jˆmn (t ) in Heisenberg picture (M.Cini 1980)
,
4
semiinfinite
semiinfinite
a  2.46 Angstroms
After sudden switching of voltage we calculate the current
1
J mn (t )  Tr[ fJˆmn (t )]
f   ( H0   )
, Jˆmn (t ) in Heisenberg picture.
e
1
J mn (t )
Conductivity  (t)=
V
e2
0 
h
 d 2 
at t  t * 
L
vF
J
|E  0
E
 d 1 
I
|V 0
V
6
first plateau  =
 e2 W
2h L
4e 2
second plateau  =
independent of geometry
h
Fullerenes
Sir Harold W. Kroto,
University od
Sussex,Nobel Prize for
Chemistry in 1996
Discovery September 4,1985
Was known initially as soccerene
Fullerenes consist of 20 hexagonal and 12
pentagonal rings as the basis of an icosohedral
symmetry closed cage structure.
20 hexagons  12 pentagons  F  32
20 hexagons  12 pentagons  20*6  12*5  180  2S  S  90
g  0 Euler : F  S  V  2  V  60
8
In theory, an infinite number of fullerenes can
exist, their structure based on pentagonal and
hexagonal rings, constructed according to rules
for making icosahedra.
Il fullerene non è molto reattivo data la
stabilità dei legami simili a quelli
della grafite ed è inoltre
ragionevolmenteinsolubile nella maggioranza
dei solventi. I ricercatori hanno potuto
aumentare la reattività fissando dei gruppi
attivi alla superficie del fullerene.
9
per produrre i fullereni: arco elettrico, a circa
5300°K, con una corrente elevata e bassa
tensione, utilizzando elettrodi in grafite in
atmosfera inerte (argon) a bassa pressione.
10
Endohedral compounds
They are fullerene cages with La or other metal atoms inside. Some have been
crystallized and found to superconduct
11
The art of hitting the goal with every shot
We have observed de Broglie wave interference of the buckminsterfullerene C60 with a
wavelength of about 3 pm through diffraction at a SiNx absorption grating with 100 nm
period. This molecule is the by far most complex object revealing wave behaviour so
far. The buckyball is the most stable fullerene with a mass of 720 atomic units, composed of
60 tightly bound carbon atoms.
http://www.univie.ac.at/qfp/research/matterwave/c60/index.html
12
Carbon Nanotubes
Fascinating electronic and mechanical
Properties:
1. Depending on their chiralities,
nanotubes can be metallic,
semimetallic or semiconducting
2. Remarkably high Young’s moduli
and tensile strength
“Imagine the possibilities: materials with ten times the strength
of steel and only a small fraction of the weight!”
------Former resident Bill Clinton
13
14
Multi-Walled NanoTube
(MWNT)
15
S. Iijima. "Helical microtubules of graphitic carbon." Nature 354 56 (1991)
16
Carbon Nanotubes
S. Iijima, Nature 354, 56 (1991)
17
Carbon Nanotubes: Lattice Structure
From Wikipedia
18
Carbon Nanotubes: Lattice Structure
S. Iijima, Nature 354, 56 (1991)
L≈1m
d≈nm
Graphene sheet
Nanotube
Single Wall ( m, n) CNT
with a , b primitive lattice vectors,
na  mb  roll-up-vector of ( n, m) CNT
( atom at xa  yb is identified to the one at ( x  n)a  ( y  m)b )
Depending on n,m the CNT can be metal or semiconductor
19
This is a possible choice of the basis which is often used:
a1
b | a1 || a2 | a 3
a2
(m, n) translation in Graphene Rmn  ma1  na2
Rmn identified with R00  (m, n) nanotube.
Then, (n,0) nanotubes are called zigzag nanotubes, and (n,n)
nanotubes are called armchair nanotubes. Otherwise, they are
called chiral.
20
axis of CNT
a1
(n,0) alias Zigzag CNT
a2
path
towards the
tip
(4,0) zigzag CNT
(15,0) zigzag CNT
path around the
belt: 2n atoms
21
armchair CNT
a1
a2
CNT axis
= y axis
(2, 2) armchair CNT
armchair CNT
path along
the y axis
All armchair nanotubes are metallic, as suggested by paths along axis
22
“Armchair” geometry
(n,m) with m=n, always metallic
“Zig-zag” geometry
(n,m) with m=0 e.g.
(5,0),(6,4),(9,1) are
semiconducting
• “Chiral”
geometry
•all the rest
23
The alternative basis which we used for the band structure of Graphene is also
in use for CNT
a1
a2
3 3
3
3
a1  a( , ), a2  a( ,  ), a1  a2  3a
2 2
2
2
3
3
In this basis, (n, m) translation Rnm  na1  ma2  a( (n  m),
(n  m))
2
2
(n, m) CNT : Rnm identified with R00 .
Since both conventions are used we must be ready to handle both of them.
Zigzag CNT
path around the
belt: 2n atoms
24
a1
Zigzag CNT using alternative basis
3 3
3
3
a1  a( , ), a2  a( ,  ), a1  a2  3a
2 2
2
2
a2
(4, 0) zigzag CNT becomes
(4, 4) in new basis
CNT axis
= x axis
a
2
3
a
a 3
The circumference is na 3
Indeed, the shift along the belt of CNT is obtained with m  n,
Rn, n  a(0, n 3) .
The wavevector k around belt is quantized: k na 3  2 ,
na 3
The CNT radius 
2
 integer
25
pz Electronic bands of (n,-n) zigzag CNT-tight-binding
approximation
Since the wavevector k  k y around belt is quantized:
k na 3  2 ,
 integer, just insert into Graphene
band structure
2
E (k )  E0  J 0
ky  k 

 3

 3

3
k y a  .
k y a   4 cos 
1  4 cos  k x a  cos 

2

 2

 2
2
 ,   0, 1, 2,
na 3

 3
  
k y a   cos 
 cos 
 and one finds, setting k x  q (along axis)
 n 

 2
band structure: E (q,  )  E0  J 0
  
  
3 
cos
4

1  4 cos  qa  cos 



n
n
2






2
26
Carbon Nanotubes as quasi 1D systems: one component of k quantized
• Band Structure of graphene
• NT: Compact transverse dimension
Discretization of
k
Subbands
correspond
to different
values of k
k|| is a
continuous
variable
27
k||
27
Note:
3 
  
  
band structure: E (q,  )  E0  J 0 1  4 cos  qa  cos 

4
cos



2
n
n






  integer quantum number, (n,n)=chirality index
2
3 
dependence along x: cos  qa  since the distance along x between two A atoms
2 
3
3
is a. Setting c= a
2
2
1d BZ


q
c
c
3
with c  a ,
2
Note also: E (q,  )  E (q,  )
28
  
  
E (q,  )  E0  J 0 1  4 cos  qc  cos 

4
cos



 n 
 n 


3
  
  
BZ
 q  , c  a. Since cos 

cos


  E (q,  )  E (q,  )
c
c
2
 n 
 n 
2
Note: the (4,-4) zigzag CNT
has 8 atoms around the
belt. Generally,
(n,-n) zigzag  2n atoms in
belt  2n bands
Examples: (n,-n) zigzag: for n  5, 10 bands:   0, 1, 2
each occurs twice because of double sign in E ;
no more bands since   3 yields cos(
3
2
)  cos( ).
5
5
for n  6,   0, 1, 2,3 each with E  12 bands ,

  0,3 nondegenerate since cos( 3)=0.
6
29
3
  
  
1  4 cos  qc  cos 

4
cos
,
c

a



2
 n 
 n 
2
The dispersion E (q,  )  E0  J 0
3 
implies two nondegenerate bands E ( q, 0)  E0  J 0 5  4 cos  qa  ,
2 
n
n
and for even n,   , two flat bands E ( q, )  E0  J 0
2
2
Zigzag CNT are Metals for n multiple of 3.

1
cos( )  . Therefore :
3
2
   1
cos 
  if n  3 and   1, if n  6 and   2,.... if n  3* |  | ; then,
 n  2
  
  
2
radicand  1  4 cos  qc  cos 
  4 cos 
  2(1  cos  qc ).
 n 
 n 
qc
   1  cos( )
Using cos   
, E  E0  2 J 0 | cos( ) |
2
2
2
at the border of the BZ, qc   ,  E  E0 for both bands. No gap!
2
30
From Mahan’s nutshell book : band structure of a (5,0) zigzag
nanotube. Labels indicate angular momentum  values
31
From Mahan’s nutshell book : band structure of a (6,0) zigzag
nanotube. Labels indicate angular momentum  values. If m-n is a
multiple of 3 the nanotube is metallic.
32
armchair CNT
a1
a2
CNT axis
= y axis
(2,2) armchair CNT
armchair CNT
path along
the axis
All armchair nanotubes are metallic, as suggested by paths along axis
33
Recall Primitive
vectors
1

1
cos( ) 
3
2
 Cartesian components are :
a2
a1a2 angle 

i

3
a1  a2  
2

3

2

3
3 3
3
3
a1  a( , ), a2  a ( ,  ),
2 2
2
2
too; indeed,
j
3
2
3

2

k

3
3

0   k *3
| a1  a2 || a1 || a2 |
| a1 || a2 | sin( )
2
2
3


0

34
Armchairs are (n,n) using basis
a1
a2
CNT axis
Nanotubes: the site at
= y axis
3
3
Rmn  ma1  na2  a ( (n  m),
(n  m))
2
2
is identified  R00
orthogonal vector
CNT axis
Tmn  ma1  na2  ((n  m),  3(n  m))
taking m=n, armchair.
Rn ,n  a (3n, 0),
(2,2) armchair CNT
Length of belt:6a
k  kx
(belt of CNT along x axis),
k is quantized
k 3an  2
q  k y along y axis = CNT axis
35
Graphene : E (k )  E0  J 0
 3

 3

3

1  4 cos  k x a  cos 
k y a   4 cos 
k y a 
2

 2

 2

for armchair CNT: set k y  q, k x  k 
2
2
 ,   0, 1, 2,
3na
 3

 3 
3
cos 
k y a   cos 
qa   cos(qc), c 
a
2
 2

 2

3

 3 2 
  
cos  k x a   cos 
a   cos 

2

 2 3na 
 n 
E (q,  )  E0  J 0
2
  
1  4 cos 
 cos  qc   4 cos  qc 
 n 
For   0,
E (q, 0)  E0  J 0 [1  2 cos  qc ] cross at qc  
2
2
, E ( 
, 0)  E0
3
3
 armchair CNT are always metals.
36
From Mahan’s nutshell book : band structure of a (5,5) armchair
nanotube. Labels indicate angular momentum  values. All
armchair nanotubes are metallic.
37
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