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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.
𝑋 𝑟 : normalized 2𝑝𝑧 wavefunction for
an isolated atom.
1 conduction electron for each C atom in
the 2𝑝𝑧 state.
Unit cell (WXYZ) contains 2 atoms
(𝐴 and 𝐵).
𝑎1 = 𝑎2 = 𝑎 = 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
𝜓𝑘 𝑟 = 𝜑1 𝑟 + 𝜆𝜑2 𝑟
𝜑1 𝑟 =
𝜑2 𝑟 =
𝐴
𝐵
𝑒 2𝜋𝑖𝒌∙𝒓𝑨 𝑋 𝒓 − 𝒓𝑨
𝑒 2𝜋𝑖𝒌∙𝒓𝑩 𝑋 𝒓 − 𝒓𝑩
Extended wave function
THE BAND THEORY OF GRAPHENE
Variational principle to obtain the best
value of 𝐸, by substituting the wavefunction
in the Schroedinger equation:
𝜓𝑘 𝑟 = 𝜑1 𝑟 + 𝜆𝜑2 𝑟
𝐻 𝜑1 + 𝜆𝜑2 = 𝐸 𝜑1 + 𝜆𝜑2
By pre-multiplication by
𝜑1 ∗ or 𝜑2 ∗ and integration
we have:
𝐻11
𝐻21
𝐻11 + 𝜆𝐻12 = 𝐸𝑆
𝐻21 + 𝜆𝐻22 = 𝜆𝐸𝑆
𝐻𝑖𝑗 =
𝑆=
𝜑𝑖 ∗ 𝑟 𝐻𝜑𝑗 𝑟 𝑑𝑟
𝜑𝑖 ∗ 𝑟 𝜑𝑖 𝑟 𝑑𝑟 = 𝑁
Number of
unit cells
𝐻12
1
1
∙
= 𝐸𝑆 ∙ 𝐼2 ∙
𝐻22
𝜆
𝜆
𝐸
=
1
𝐻 /𝑁 + 𝐻22 /𝑁
2 11
We obtain:
𝐸 = 𝐻11 ′ ± 𝐻12 ′
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene
𝐻𝑖𝑗 ′ = 𝐻𝑖𝑗 /N
𝐸: interaction between an 𝐴 or 𝐵 atom with
itself
𝛾0 ′ : interaction between first neighbors of the
same type (𝐴 or 𝐵)
𝛾0 : interaction between first neighbors of
opposite type (𝐴 and 𝐵)
𝐸 = 𝐻11 ′ ± 𝐻12 ′
𝐻11 ′
=𝐸
− 2𝛾0 ′ cos 2𝜋𝑘𝑦 𝑎
Energy levels as function of ky (kx=0)
E
𝐸 − 𝐸 ≈ ±𝛾0 𝑠𝑞𝑟𝑡 1 + 4 cos2 𝜋𝑘𝑦 𝑎
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:
𝑁 𝐸
1
𝜖
=
𝐸 − 𝐸𝑐 =
𝑁𝑎
𝜋 3𝛾0 2
𝜋 3𝛾0 2
∞
2
0
𝑁 𝐸
𝜋 𝑘𝐵 𝑇
𝑓 𝐸 𝑑𝐸 =
𝑁𝑎
6 3 𝛾0
𝑁𝑎 : number of atoms in the lattice
2
𝑓 𝐸 : Fermi distribution
At room temperature (𝑘𝐵 𝑇 = 0.025 eV)
the effective number of free electrons
(𝑛𝑒𝑓𝑓 ), per atom, is 𝑛𝑒𝑓𝑓 = 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:
𝜒0 = 𝑛𝑒𝑓𝑓 𝜇𝐵 2 /𝑘𝐵 𝑇 ∝ 𝑇
G. Wagoner, Phys. Rev., 118, 647 (1960).
A. Barbon, Corso di Magnetochimica. A.A. 2013-14. Graphene