Chemistry Reference
In-Depth Information
ω
qv
F
where
. Using the above expression we obtain the following disper-
sion relation for the collective mode:
z ≡
q
3
ω
=
qv
F
−
)
2
≡ qv
F
− E
B
(
q
)
π
2
v
F
(
U
−
k
c
4
πv
F
32
as
ω → q →
0. Here
E
B
(
q
)isthe
binding
energy of the particle-hole pair of
momentum
q
around the Γ point. The binding energy around the
K
points
is roughly half of this.
We mentioned earlier that our collective mode is a 'magnetic zero
sound'. While magnetic zero sound are dicult to get in normal met-
als, graphite manages to get it in the entire BZ because of the window in
the particle-hole spectrum (Fig. 3).
Having established the existence of a gapless spin-1 collective mode
branch within Hubbard model and the RPA approximation, we will dis-
cuss whether the semi-metallic screened interaction in the 3 dimensional
stacked layers will affect our result. As mentioned earlier, in tight binding
situation like ours, the spin physics is mostly captured by the short range
part of the repulsion among the electrons. We have numerically studied
the response function for a more realistic intra layer interaction namely
the screened coulomb interaction (including interlayer scattering between
layers separated by distance
)givenby
[
18
]
d
πe
2
0
q
)=
2
sinh(
qd
)
v
ω, q
[cosh(
˜
(
)+
2
πe
2
0
q
qd
sinh(
qd
)
χ
0
(
ω, q
)]
2
−
1
and find that the collective mode survives with small quantitative modifi-
cations.
Let us discuss life time effects, that is beyond RPA. A remarkable feature
of our collective modes is that it never enters the particle-hole continuum.
It does not suffer from Landau damping (resonant decay into particle-hole
pair excitations). To this extent our collective modes are sharp and pro-
tected; higher order processes will produce the usual life time broadening,
particularly at the high energy end. However, in real graphite there are
tiny electron and hole pockets in the BZ with a very small Fermi energy
∼
. This leads to 'Landau damping' of low energy collective
modes around the Γ and
10 to 20
meV
K
points, but only in a small momentum region
k
F
∼
50
π
a
k
F
is the mean Fermi momentum of the electron
and hole pockets. That is, only a few percent of the collective mode branch
in the entire BZ is Landau damped.
k ∼
∆
2
,where
