Chemistry Reference
In-Depth Information
We start with a 2-dimensional Hubbard model for graphene, which cap-
tures the physics of low energy spin dynamics. The Hamiltonian is:
c
i,σ
c
j,σ
+
H
=
−t
(
h.c
)+
U
n
i↑
n
i↓
(1)
i,j,σ
i
is the nearest neighbor hopping matrix element. While
the bare atomic U is more than 10
Here
t ∼
2
.
5
eV
eV
, the effective renormalized U can
be of the order of 3
−
4
eV
. We will keep U as a parameter to be fixed by
experiments.
The dispersion relation for the
π
∗
and
π
bands are:
1+4cos
√
3
k
x
a
2
cos
k
y
2
+4cos
2
k
y
2
ε
k
=
±t
(2)
with vanishing gaps at the two
points in the BZ (Fig. 2). The particle-
hole continuum of excitations is shown in Fig. 3. The 'Dirac cone single
particle spectrum' at the Γ and
K
points makes the particle hole continuum
very different from that of a free Fermi gas, or systems with extended Fermi
surface. In contrast to Fig. 4, the particle-hole spectrum of a 2d Fermi
liquid, our spectrum has a 'window'.
The 'window' is characteristic of a 1d
particle-hole spectrum
. In the Hubbard model two particles with opposite
spins at a given site repel with an energy U. This means an attraction
between an up spin particle and a down spin hole; or an attraction in the
spin triplet channel between a particle and a hole. A spin triplet particle-
hole pair could form a bound state, provided there is sucient phase space
for the attractive scattering. We find one spin-1 bound state for every
center of mass momentum of the particle-hole pair. In particular an effective
1d character of phase space also makes the collective mode energy vanish
linearly with momenta around the three points: Γ and
K
's.
The collective mode that we are after are obtained as the poles of the
particle-hole response function in the spin triplet channel. We will focus
on the zero temperature case. The magnetic response function within the
RPA (particle-hole ladder summation) is given by:
K
χ
0
(
q
,ω
)
χ
(
q
,ω
)=
(3)
χ
0
(
q
1
− υ
(
q
)
,ω
)
For Hubbard type on site repulsion,
υ
(
q
)=
U
and the
free particle
susceptibility is :
)=
1
N
f
k
+
q
− f
k
ω −
χ
0
(
q
,ω
(4)
(
ε
k
+
q
− ε
k
)
k
