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```BULK MAGNETIZATION OF GRAPHENE
Tight binding approximation: the mobile
electrons are always located in the
proximity of an atom, and then are
conveniently described by the pz atomic
orbital of the atoms it touches.
: normalized 2 wavefunction for
an isolated atom.
1 conduction electron for each C atom in
the 2 state.
Unit cell (WXYZ) contains 2 atoms
( and ).
1 = 2 =  = 2.46 Å
(foundamental lattice displacement).
The base functions are periodical functions
with the same periodicity as the (2D) lattice.
k is a wave vector. It defines a reciprocal
lattice and acts as a kind of quantum number.
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
= 1  + 2
1  =
2  =

2∙   −
2∙   −
Extended wave function
THE BAND THEORY OF GRAPHENE
Variational principle to obtain the best
value of , by substituting the wavefunction
in the Schroedinger equation:
= 1  + 2
1 + 2 =  1 + 2
By pre-multiplication by
1 ∗ or 2 ∗ and integration
we have:
11
21
11 + 12 =
21 + 22 =
=
=
∗
∗     =
Number of
unit cells
12
1
1
∙
=  ∙ 2 ∙
22

=
1
/ + 22 /
2 11
We obtain:
= 11 ′ ± 12 ′
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
′ =  /N
: interaction between an  or  atom with
itself
0 ′ : interaction between first neighbors of the
same type ( or )
0 : interaction between first neighbors of
opposite type ( and )
= 11 ′ ± 12 ′
11 ′
=
− 20 ′ cos 2
Energy levels as function of ky (kx=0)
E
−  ≈ ±0  1 + 4 cos2
kx=0
Zero
band-gap
ky
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
Calculation of the
density of states:
It is possible to show that the number of
electronic energy states per atom:
N° of free electrons plus positive holes per
atom:

1

=
−  =

30 2
30 2
∞
2
0

=

6 3 0
: number of atoms in the lattice
2
: Fermi distribution
At room temperature (  = 0.025 eV)
the effective number of free electrons
( ), per atom, is  = 2.3 ∙ 10−4
N(E)
4
3.5
E
f(E)
3
f (E) 

x 10
2.5
1
exp[( E  Ec) / kT ]  1
2
1.5
1
0.5
Ec
0
E
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
0
50
100
150
Temperature [K]
200
250
300
Magnetic susceptivity:
0 =   2 /  ∝
G. Wagoner, Phys. Rev., 118, 647 (1960).
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
```