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Organic Light Emitting Diodes

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disorder (which tends to localize the wavefunctions) and by the Coulomb interaction, which binds<br />

electrons when transferred to a nearby repeat unit to the positive charge left behind (a hole).<br />

The construction of the remarkably successful SSH Hamiltonian is based on two assumptions [2]:<br />

(a) The π-electronic structure can be treated in the tight-bonding approximation with a transfer integral<br />

t ≈ 2.5 eV, and (b) The chain of carbon atoms is coupled to the local electron density through the<br />

length of the chemical bonds.<br />

tn, n+<br />

1 = to<br />

+ α ( un<br />

+ 1 − un<br />

)<br />

(1)<br />

where tn , n+<br />

1 is the bond-length dependent hopping integral from site n to n +1 and n is the<br />

displacement from equilibrium of the carbon atom. The first assumption defines the lowest order<br />

hopping integral, , in the tight-binding term that forms the basis of the Hamiltonian (Eqn. 2). The<br />

second assumption provides the first-order correction to the hopping integral. This term couples the<br />

electronic states to the molecular geometry, giving the electron-phonon (el-ph) interaction where α is<br />

the el-ph coupling constant. The precise form of Eqn. (1), in which the dependence of the hopping<br />

integral on the C-C distance is linearized for small deviations about , is the first term in a Taylor<br />

expansion.The resulting SSH Hamiltonian is then written as the sum of three terms:<br />

u<br />

th<br />

n<br />

to<br />

to<br />

H<br />

∑<br />

2<br />

+<br />

+<br />

pn<br />

1<br />

2<br />

[ −to<br />

+ α ( un+<br />

1 − un<br />

)]( cn+<br />

1,<br />

σ cn,<br />

σ + cn,<br />

σ cn+<br />

1.<br />

σ ) + ∑ + K∑(<br />

un<br />

1 − un<br />

) (2)<br />

2m<br />

2<br />

SSH = +<br />

n,<br />

σ<br />

n n<br />

where pn are the nuclear momenta, n are the displacements from equilibrium, m is the carbon mass,<br />

and K is an effective spring constant. The and are the fermion creation and annihilation<br />

operators for site n and spin σ. The last two term are, respectively, a harmonic »spring constant« term<br />

which represents the increase in potential energy that results from displacement from the uniform<br />

bonds lenghts in (CH)x and a kinetic energy term for the nuclear motion.<br />

u<br />

+<br />

cn, σ cn,<br />

σ<br />

Figure 6: Electronic structure of semiconducting PA; left - Band structure, right - Density of<br />

states. The energy opens at k = π/2a as a result of Peierls distortion<br />

The spontaneous symmetry breaking due to the Peierls instability implies that for the ground state<br />

of a pristine chain, the total energy is minimized for u n > 0 .Thus to describe the bond alternation in<br />

the ground state, we use:<br />

u )<br />

n<br />

n → un<br />

= (−1<br />

uo<br />

(3)<br />

With this mean-field approximation, the value uo<br />

which minimizes the energy of the system can be<br />

calculated as a function of the other parameters in the Hamiltonian. Qualitatively, however, one sees<br />

6

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