Structure de bande du graphène

Physique des Solides
Gilles Montambaux
le 11 octobre 2005
Petite Classe 4
Structure de bande du graphène
Le graphite est un cristal lamellaire dans lequel les atomes de carbone sont, pour une couche
donnée, distribués aux sommets d’hexagones réguliers s’emboı̂tant les uns dans les autres pour
former une structure en nid d’abeille. On se propose d’étudier les propriétés électroniques d’une
seule couche, appelée aussi graphène. Le graphène a été découvert expérimentalement en 2004.
Après le graphite, le diamant, les fullerènes et les nanotubes, il constitue une nouvelle variété
allotropique du carbone, dont les propriétés sont fascinantes.
a2
B
A
a1
On appelle a la distance interatomique. La maille élémentaire est un losange et comporte
deux types de sites que l’on notera A et B. On note ~a1 et ~a2 les vecteurs de base.
1 - Caractériser cet édifice par son réseau de Bravais et son motif.
2 - Représenter graphiquement le réseau réciproque.
3 - Calculer le facteur de structure S(h, k).
4 - La bande de conduction du graphite, constituée par le recouvrement des orbitales pz ,
est à moitié remplie. On se propose d’étudier la structure de cette bande, dans l’approximation
des liaisons fortes. On suppose que l’hamiltonien atomique n’admet qu’une valeur propre ²0
B
non dégénérée , associée à un état |ϕi. On note |ϕA
j i [resp. |ϕj i] l’état centré sur le site (j, A)
[resp. (j, B)], c’est-à-dire le site A [resp. B] de la maille élémentaire j. Ainsi:
h~r |ϕi = ϕ(~r)
~ j)
r−R
h~r |ϕA
j i = ϕ(~
~
~ j − d)
r−R
h~r |ϕB
j i = ϕ(~
où d~ est le vecteur qui relie A et B.
On cherche des solutions de l’hamiltonien total sous la forme de combinaisons linéaires
d’états atomiques:
1 X
~j
B
i~kR
|ψ~k i = √
( λA |ϕA
j i + λB |ϕj i ) e
N j
On suppose que les états centrés sur des atomes différents ont un recouvrement très faible, de
sorte que:
A
B B
hϕA
j |ϕj 0 i = hϕj |ϕj 0 i = δj,j 0
B
hϕA
j |ϕj 0 i = 0
Enfin, on suppose que les seuls éléments de matrice non nuls de l’hamiltonien H sont les éléments
diagonaux:
B
B
A
hϕA
j |H|ϕj i = hϕj |H|ϕj i = ²
ainsi que les intégrales de transfert entre sites premiers voisins:
B
hϕA
j |H|ϕj 0 i = −t
pour (j, A) et (j 0 , B) premiers voisins.
• Projeter l’équation de Schrödinger:
H|ψk i = E|ψk i
B
sur deux états atomiques |ϕA
l i et |ϕl i. En déduire deux équations couplées pour les coefficients
λA et λB :
~
~
~
~
²λA − tλB (1 + e−ik.~a1 + e−ik.~a2 ) = EλA
²λB − tλA (1 + e+ik.~a1 + e+ik.~a2 ) = EλB
5 - En déduire que la relation de dispersion s’écrit:
E = ² ± t [3 + 2 Σ(~k)]1/2
avec
Σ(~k) = cos ~k.~a1 + cos ~k.~a2 + cos ~k.(~a1 − ~a2 ).
Dans la suite, on prendra l’origine des énergies en ².
6 - On repère le vecteur ~k par ses coordonnées α et β dans l’espace réciproque : ~k =
∗
α~a1 + βa∗2 . Montrer que α = ~k.~a1 /2π et β = ~k.~a2 /2π et que, pour tout vecteur ~k,
9
(~k.~a1 )2 + (~k.~a2 )2 − (~k.~a1 ) (~k.~a2 ) = k 2 a2
4
où k = |~k|.
7 - Lorsque k → 0, montrer que
9
Σ(~k) → 3 − k 2 a2
4
et en déduire que l’énergie E varie comme
3
E = −3t + ta2 k 2
4
,
k → 0.
8 - Montrer que l’énergie E s’annule en un nombre fini de points dont on donnera les
coordonnées ~kP .
9 - Montrer qu’au voisinage de ces points, l’énergie varie linéairement avec le vecteur d’onde:
E=±
3
t a |~k − ~kP |
2
10 - Calculer la densité d’états au voisinage de E = 0.
11 - Comment varie la chaleur spécifique à basse température, dans le cas du demi-remplissage?
———————–
Sur le site de BBC News, 22 octobre 2004 :
Radical fabric is one atom thick
A new class of material,
which brings computer
chips made from a single
molecule a step closer, has
been discovered by
scientists.
Called graphene, it is a twodimensional, giant, flat
molecule which is still only the
thickness of an atom.
dimensional, they have limitations. Graphene is a plane
transistor - flat sheets."
Professor Andre Geim, who leads the research team,
explained that the material they had discovered could be
thought of as millions of unrolled carbon nanotubes which
had been stuck together to make an infinitely large sheet, an
atom thick.
The new class of material is much more
stable than others
The nanofabric's remarkable electronic properties mean that
an ultra-fast and stable transistor could be made.
The physicists from the University of Manchester and
Chernogolovka, Russia, published their research in Science.
"In my opinion, this is one of the most exciting things to
have happened in solid state physics in a decade," Professor
Laurence Eaves, semi-conductor expert from the University
of Nottingham, UK, told the BBC News website.
Graphene is part of the family of famous fullerene molecules,
discovered in the last 20 years, which include buckyballs and
nanotubes.
Their unusual electronic, mechanical and chemical properties
at the molecular scale promise ultra-fast transistors for
electronics, as well as incredibly strong, flexible and stable
materials.
Ballistic promise
Scientists have been trying to exploit this for computing
because smaller transistors mean the distances electrons
have to travel become shorter, meaning faster speeds.
Conventional transistors rely
on the semi-conducting
characteristics of silicon which
provide the switches that
change the flow of current in
computers and other
electronics.
"All the recent progress has
been on nanotubes for
Graphene is like millions of unrolled
transistors. These are sheets nanotubes stuck together
of graphite molecules wrapped in a cylinder - like a chocolate
cylinder you stick in your ice cream," explained Professor
Laurence Eaves.
"Although these are interesting, because they are one-
They showed that electrons could travel sub-micron distances
without being scattered, which means fast-switching
transistors.
"At the moment, the research
is in early stages," Professor
Geim said.
"The applications are too early
to say, but the material is
incredible. We have studied its
electronic properties and no
other material displays this."
He added: "People have been How nanotechnology is building the
future from the bottom up
trying to make transistors
faster and smaller. There is a In pictures
Holy Grail of electronics that
engineers call ballistic
transistors - ultimately faster than anything."
A ballistic transistor is one in which electrons can shoot
through without collisions, like a bullet. In other words, it has
what is called a long mean free path - the distance a
molecule travels without colliding into another.
Greater distances with nothing to collide with means faster
speeds. Fewer collisions means less energy is lost, too.
Although they have not demonstrated a ballistic transistor
yet, the latest experiments have shown that the new
material could, in theory, produce one.
The team is currently experimenting with relatively small
sheets of the graphene nanofabric, 10s of microns (millionths
of a metre) across, but the sheets are still "large" in
molecular terms.
The nanofabric would have to be produced in much larger
wafers, a few centimetres in area, before electronics
manufacturers could start using it.
But, said Professor Geim, judging by how quickly carbon
nanotubes developed, graphene could be ready for industrial
application in about 10 years.