3. 5.2 Graphene
Brillouin zone of the hexagonal lattice
lattice vectors:
a = a
!
1
2
ex −
√3
2
ey
"
, b = a
!
1
2
ex +
√3
2
ey
"
, c = cez .
reciprocal-lattice vectors:
Ga = 2π
b × c
a · (b × c)
=
4π
√3a2c
b × c =
4π
√3a
!√3
2
ex −
1
2
ey
"
,
Gb = 2π
c × a
a · (b × c)
=
4π
√3a2c
c × a =
4π
√3a
!√3
2
ex +
1
2
ey
"
,
Gc = 2π
a × b
a · (b × c)
=
4π
√3a2c
a × b =
2π
c
ez .
Γ: Center of the Brillouin zone
K: Middle of an edge joining two rectangular faces→ (Ga + Gb)/3
M: Center of a rectangular face → Ga/2
A: Center of a hexagonal face → Gc/2
H: Corner point → (Ga + Gb)/3 + Gc/2
L: Middle of an edge joining a hexagonal and a rectangular face → Ga/2 + Gc/2
10
5
0
−5
Energy relative to EF [eV]
−10
Γ KM Γ A H L Γ
Figure 6: Band structure of the normal state
of a superconductor, MgB2.
34
4. Unit cell and atomic structure of graphene
The computational cell for graphene is hexagonal one.
(a) (b)
Figure 7: The unit cell and the atomic struc-ture
of graphene used for first-principles calcu-lations:
(a) the side view and (b) the top view.
Input file of OpenMX to calculate electronic states of graphene
#
# OpenMX Inputfile
#
# File Name
#
System.Name graphene
DATA.PATH /home/gohda/openmx/DFT_DATA13
# Definition of Atomic Species
#
Species.Number 2
<Definition.of.Atomic.Species
C C6.0-s2p2d1 C_PBE13
E Rn13.0-s2p2d2f1 E
Definition.of.Atomic.Species>
# Atoms
#
Atoms.Number 3
Atoms.SpeciesAndCoordinates.Unit FRAC # Ang|AU
<Atoms.SpeciesAndCoordinates
1 C 0.33333333333333 0.66666666666667 0.50000000000000 2.0 2.0
2 C 0.66666666666667 0.33333333333333 0.50000000000000 2.0 2.0
3 E 0.00000000000000 0.00000000000000 0.50000000000000 0.0 0.0
Atoms.SpeciesAndCoordinates>
35
5. Atoms.UnitVectors.Unit Ang # Ang|AU
<Atoms.UnitVectors
1.23000000000000 -2.13042249330972 0.00000000000000
1.23000000000000 2.13042249330972 0.00000000000000
0.00000000000000 0.00000000000000 10.00000000000000
Atoms.UnitVectors>
# SCF or Electronic System
#
scf.XcType GGA-PBE # LDA|LSDA-CA|LSDA-PW|GGA-PBE
scf.SpinPolarization off # On|Off|NC
scf.energycutoff 300.0 # default=150 (Ry)
scf.maxIter 100 # default=40
scf.EigenvalueSolver band # DC|GDC|Cluster|Band
scf.Kgrid 11 11 1 # means n1 x n2 x n3
scf.Mixing.Type rmm-diisk # Simple|Rmm-Diis|Gr-Pulay|Kerker|Rmm-Diisk
scf.Init.Mixing.Weight 0.30 # default=0.30
scf.Min.Mixing.Weight 0.001 # default=0.001
scf.Max.Mixing.Weight 0.700 # default=0.40
scf.Mixing.History 10 # default=5
scf.Mixing.StartPulay 5 # default=6
scf.Mixing.EveryPulay 1 # default=5
scf.criterion 1.0e-6 # default=1.0e-6 (Hartree)
scf.lapack.dste dstevx # dstegr|dstedc|dstevx, default=dstevx
# DOS and PDOS
#
Dos.fileout on # on|off, default=off
Dos.Erange -20.0 10.0 # default = -20 20
Dos.Kgrid 48 48 1 # default = Kgrid1 Kgrid2 Kgrid3
# Band dispersion
#
Band.dispersion on # on|off, default=off
Band.Nkpath 4
<Band.kpath
200 0.0 0.0 0.0 0.33333333333333 0.33333333333333 0.0 G K
100 0.33333333333333 0.33333333333333 0.0 0.5 0.0 0.0 K M
173 0.5 0.0 0.0 0.0 0.0 0.0 M G
37 0.0 0.0 0.0 0.0 0.0 0.5 G A
Band.kpath>
# Others
#
scf.restart on
scf.fixed.grid 0.00000000000000 0.00000000000000 0.00000000000000
36
6. Band strucuture and density of states
You can excecute OpenMX by typing on a UNIX (MacOSX, Linux, etc.) terminal as follows:
$ openmx INCAR >log 2>&1 &
5
(a) (b)
0
−10
Γ K M Γ A
−5
Energy relative to EF [eV]
−15
−20
0
−20 −15 −10 −5 0 5 10
Energy relative to EF [eV]
Density of states [arb. unit]
w EA
w/o EA
w EA
w/o EA
(c)
Figure 8: (a) The energy band and (b) the de-sity
of states for graphene obtained by first-principles
calculations with enpty-atom basis
functions (red) and without them (black). (c)
Nearly-free-electron states that cannot be de-scribed
without the empty atom.
Comparison with the tight-binding model with the H¨uckel approximation
In the Huckel ¨approximation, only the |φpz ⟩ state is considered for the frontier orbitals of benzene
rings. The tight-binding Hamiltonian using the notation of |I⟩ = |φI
pz ⟩ is
H =
#
I
εI |I⟩⟨I|−
#
<IJ>
t|I⟩⟨J| = −
#
<IJ>
t|I⟩⟨J| , (23)
where t is the transfer integral and we define εI = 0. Using the Bloch sum
φB
i,k(r) =
1
√Nk
#
T
eik·T φpz (r − Ri − T ) (24)
with the approximation of ⟨I|J⟩ = δIJ (Thus Sij = δij), the matrix elements are
⟨φB
i,k|H|φB
i,k⟩ = 0 , and (25)
⟨φB
i,k|H|φB
j,k⟩ = ⟨φB
i,k⟩∗ = −t(eik·T0 + eik·T1 + eik·T2)
j,k|H|φB
= −t(1 + eik·a + e−ik·b) ≡ −tf . (26)
37
7. Thus the transformed Hamiltonian becomes
H = −t(f|φB1
⟩⟨φB2
| + f∗|φB2
⟩⟨φB1
|) . (27)
Diagonalizing H through ε2 − t2|f|2 = 0, we obtain
ε = ±t|f|
= ±t
$
(1 + cos k · a + cos k · b)2 + (sink · a − sin k · b)2
= ±t
$
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) . (28)
At the M point (k = Ga/2),
ε = ±t
$
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b)
= ±t
$
5 + 4 cosGa/2 · a
= ±t√5 + 4 cos π
= ±t . (29)
As for the Γ–K path, i.e. k = κ(Ga + Gb),
ε = ±t
$
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b)
= ±t
$
3 + 2 cos κGa · a + 2 cos κGb · b + 2 cos κ(Ga + Gb) · (a + b)
= ±t
$
3 + 4 cos (2πκ) + 2 cos(4πκ) (30)
Γ K
10
5
0
Energy relative to EF [eV]
−5
w EA
w/o EA
TB
Figure 9: Comparison of energy bands ob-tained
by first-principles calculations and the
tight-binding model with the H¨uckel approxi-mation.
The hopping integral was chosen as
t = 2.7 eV.
38
8. Γ K M
0 0
0 0
0
0
0
0 0
0
0
π 0
π
π
0 0
π π
0
0
π
π
π
π
π
π
B1
Figure 10: The phase θ = k · T appear in
the Bloch sum for the A sublattice (i = 1),
%
φk(r) = √1 ,Nk
T eik·T φpz (r − R1 − T ).
Let us take a closer look of the wave functions:
At the Γ point:
H = −3t(|φB1
⟩⟨φB2
| + |φB2
⟩⟨φB1
|) , ε= ±3t , (31)
|π⟩ =
1
√2
(|φB1
⟩ + |φB2
⟩) , |π∗⟩ =
1
√2
(|φB1
⟩ − |φB2
⟩) . (32)
In general:
H = −t(f|φB1
⟩⟨φB2
| + f∗|φB2
⟩⟨φB1
|) , ε= ±t|f| , (33)
|π⟩ =
1
√2
&
|φB1
⟩ +
f∗
|f||φB2
⟩
'
, |π∗⟩ =
1
√2
&
|φB1
⟩ −
f∗
|f||φB2
⟩
'
. (34)
At the K point:
H = 0 , ε= 0 , (35)
|π⟩ = cπ,1|φB1
⟩ + cπ,2|φB2
⟩ , |π∗⟩ = cπ∗,1|φB1 ⟩ + cπ∗,2|φB2
⟩ . (36)
Exercise
For k vectors around the K point, k = 13
(Ga + Gb) + δkxex + δkyey, derive the effective
Hamiltonian using the Talor expansion starting from the form
H =
⎛
⎝ 0 −tf
−tf ∗ 0
⎞
⎠ .
Use the Pauli matrices σ:
σx =
⎛
⎝ 0 1
1 0
⎞
⎠ , σy
=
⎛
⎝ 0 −i
i 0
⎞
⎠ , and σz =
⎛
⎝ 1 0
0 1
⎞
⎠ .
39
9. Answer:
Exercise
Calculate i) the eigenenergy and ii) the sign of the linear combination for π orbitals in a butadiene
molecule CH2CHCHCH2, a virtual fragment of graphene, using the H¨uckel approximation.
Hint: 2 < √5 < 3
Answer:
40