320
12 Carbon-Nanotube Reinforced Polymers
Fig. 12.4 Undulations and rippling of the graphene layer at 300 ◦ K, as reproduced by TBMD
simulations using the NRL Hamiltonian. The height of the ripples is ∼ 1Å. (From [70])
anomalies, given the complex conductivity and permeability, of the anomaly. We
will investigate the enhancements needed to analyze problems of the type shown in
these figures. The analyses described in [35, 109, 110, 112] will be appropriate here.
12.6 An Example of a Bianisotropic System
In [137, 138] and Fig. 12.6 we have an example of a bianisotropic system in which
the electrical characteristics of the system are partially controlled by a magnetic
field operating through magnetic spins. To be specific, this is a four-terminal carbon
nanotube sensor for magnetic field measurements that relies upon spin-coherent
electron transport in single-wall carbon nanotubes.
The Landauer formula for the conductance of the FM-SWCNT-FM magnetic
tunnel junction system is [137]
G =
2e 2
h
σ
T r
Im(Σ
r
L )G
r Im(Σ
r
R )G
a
σ σ
,
(12.4)
where σ (↑↓) is the spin index. The retarded (advanced) Green’s functions, G r(a) ,
are given by
G
r(a) (E) =
1
E − H tube − Σ
r(a)
L
− Σ
r(a)
R
,
(12.5)
where the self-energies are