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transport in graphene was experimentally found to quantitatively follow the spin drift-
diffusion equation derived for semiconductors, which can be written in terms of spin-
dependent electrochemical potentials,
d2(p1 - p1) vE d(p - p1) _ pT- 0 (9.7)
dx2 D dx L2
where v is the carrier mobility, D is the diffusion constant, E is the applied electric field,
and L is the intrinsic spin diffusion length. For ferromagnetic electrodes, the second term
in the above equation can be neglected, and Eq. (9.7) becomes the conventional spin
diffusion equation.
When Io = 0, i.e., the linear regime of spin injection, the spin splitting in electrochem-
ical potential, pT(x) - p1(x), is proportional to the AC current I, and we define the spin
accumulation and the spin injection efficiency as
RSA(x) = , - - I 2=0. (9.8)
Figure 19 shows spin-polarized currents, Is, and spin accumulation, RSA, as a function of
x for different transfer lengths. We see that when the transfer lengths are much smaller
than the contact size, both the spin current and accumulation concentrate at the inner
edge of the circuit. When the transfer lengths are much greater than the contact size, the
spin current and accumulation spread across the entire contact.
The injected spin current into the graphene will flow out at the two edges of the elec-
trode. The relative weights of spin currents at the two edges are P = (IT - It)x=_,/(IT -
I4)z=o and P = -(IT- I)x=o/(IT- Iizo, where P +P, = 1. In Fig. 20, we depict PF, Pr
and the spin injection efficiency, a, as a function of the transfer length. We see that for
Ls/ao < 1, most of the spin current flows to the inner edge, where the charge current is
finite, and the spin injection efficiency is small. As the transfer length increases, the spin
currents are equally distributed between the two edges and the spin injection efficiency
increases. Since spin injection signals are measured in a voltage circuit next to the current
circuit, it is spin accumulation and current at the outer edge, RSA(-ao) and PF, that are
more important to the measurements. In literature, spin accumulation and spin current
are always implicitly assumed to be symmetric at the two edges, which, as shown in Figs.
19 and 20, is valid only when L,/ao 1.
9.3 Transfer length and spin detection
To understand spin detection and its dependence of transfer lengths, we consider explicitly
only two contacts, p with size co from the voltage circuit and 1 with size ao from the current40
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Yu, Zhi Gang. Spin Properties of Transition-Metallorganic Self-Assembled Molecules, report, June 30, 2010; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014357/m1/41/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.