Organic Light Emitting Diodes

disorder (which tends to localize the wavefunctions) and by the Coulomb interaction, which binds
electrons when transferred to a nearby repeat unit to the positive charge left behind (a hole).
The construction of the remarkably successful SSH Hamiltonian is based on two assumptions [2]:
(a) The π-electronic structure can be treated in the tight-bonding approximation with a transfer integral
t ≈ 2.5 eV, and (b) The chain of carbon atoms is coupled to the local electron density through the
length of the chemical bonds.
tn, n+
1 = to
+ α ( un
+ 1 − un
)
(1)
where tn , n+
1 is the bond-length dependent hopping integral from site n to n +1 and n is the
displacement from equilibrium of the carbon atom. The first assumption defines the lowest order
hopping integral, , in the tight-binding term that forms the basis of the Hamiltonian (Eqn. 2). The
second assumption provides the first-order correction to the hopping integral. This term couples the
electronic states to the molecular geometry, giving the electron-phonon (el-ph) interaction where α is
the el-ph coupling constant. The precise form of Eqn. (1), in which the dependence of the hopping
integral on the C-C distance is linearized for small deviations about , is the first term in a Taylor
expansion.The resulting SSH Hamiltonian is then written as the sum of three terms:
u
th
n
to
to
H
∑
2
+
+
pn
1
2
[ −to
+ α ( un+
1 − un
)]( cn+
1,
σ cn,
σ + cn,
σ cn+
1.
σ ) + ∑ + K∑(
un
1 − un
) (2)
2m
2
SSH = +
n,
σ
n n
where pn are the nuclear momenta, n are the displacements from equilibrium, m is the carbon mass,
and K is an effective spring constant. The and are the fermion creation and annihilation
operators for site n and spin σ. The last two term are, respectively, a harmonic »spring constant« term
which represents the increase in potential energy that results from displacement from the uniform
bonds lenghts in (CH)x and a kinetic energy term for the nuclear motion.
u
+
cn, σ cn,
σ
Figure 6: Electronic structure of semiconducting PA; left - Band structure, right - Density of
states. The energy opens at k = π/2a as a result of Peierls distortion
The spontaneous symmetry breaking due to the Peierls instability implies that for the ground state
of a pristine chain, the total energy is minimized for u n > 0 .Thus to describe the bond alternation in
the ground state, we use:
u )
n
n → un
= (−1
uo
(3)
With this mean-field approximation, the value uo
which minimizes the energy of the system can be
calculated as a function of the other parameters in the Hamiltonian. Qualitatively, however, one sees
6