User talk:RadityaUtama

Dimerization in Polyacetylene

Polyacetylene is an organic compound that consists of 1D polymer chain of carbon and hydrogen atoms as the lattice points and branches. The structure is basically repetitions of acetylene molecule $(C_{2}H_{2})$ and produces a high electrical conductivity. The interesting part of it is its alternating bonds between single and double ones and will produce a small energy gap within its spectrum band. The energy band makes it behave as a good semi-conductor material which we can practically use as electronic components.

(The structure of Polyacetylene); http://upload.wikimedia.org/wikipedia/commons/thumb/5/58/Trans-(CH)n.png/400px-Trans-(CH)n.png

Now let's trace how polyacetylene can create such energy gap. First, we assume the chain as 1D lattice chain that contains dimers constructed by alternating bond types. This means that the electron jumps with different probabilities between the sites. The double bond jump is more probable than the single one since it needs lower energy to apply. In addition to that, we also consider the "stifness" and kinetic energy factor. By taking all of these relevant compositions, the Hamiltonian of the chain becomes,

${\mathcal {H}}=\sum _{j}{\frac {m}{2}}{{\dot {u}}_{j}}^{2}+\sum _{j}{\frac {\kappa }{2}}(u_{j+1}-u_{j})^{2}-\sum _{js}(t_{j+1,j}{c_{j+1,s}}^{\dagger }c_{j,s}+h.c)$

$t_{n+1,n}=t_{o}-\alpha (u_{n+1}-u_{n})$

Some simplifications can be made by getting rid of the kinetic term cause the mass is relatively too big for the detail energy dynamics. Then we can assume 2 interdependent fixed energies of hopping term, means the single bond expands with a certain number from equilibrium while the double bond contracts with the same number

$u_{j}=\pm (-1)^{n}u_{o}$

${\begin{aligned}t_{j+1,j}&\Rightarrow t_{o}-t_{1}......single\\&\Rightarrow t_{o}+t_{1}......double\end{aligned}}$

This difference creates dimers model and so we can separate the lattice into 2 which are the odd and even parts with half filling for each. Then we can rewrite the Hamiltonian as,

${\mathcal {H}}=\sum _{j}{\frac {\kappa }{2}}(u_{j+1}-u_{j})^{2}-(t_{o}-t_{1})\sum _{js}^{N/2}({c_{2j-1,s}}^{\dagger }c_{2j,s}+h.c)-(t_{o}+t_{1})\sum _{js}^{N/2}({c_{2j+1,s}}^{\dagger }c_{2j,s}+h.c)$

Now, we want to couple all separate sites as few modes. That's why we have to Fourier transform the creation and annihilation operators form lattice space to momentum space.

$a_{2j-1,s}={\sqrt {\frac {2}{N}}}\sum _{k=1}^{N/2}e^{i{\frac {2\pi }{N}}k(2j-1)}a_{s}(k)$

$b_{2j,s}={\sqrt {\frac {2}{N}}}\sum _{k=1}^{N/2}e^{i{\frac {2\pi }{N}}k(2j)}b_{s}(k)$

These new operators already commute to each other. Let's replace both inside the Hamiltonian as,

${\begin{aligned}{\mathcal {H}}_{hop}&=-(t_{o}-t_{1})\sum _{s,k=1}^{N/2}(e^{i{\frac {2\pi }{N}}k}{a_{s}(k)}^{\dagger }b_{s}(k)+h.c)-(t_{o}+t_{1})\sum _{s,k=1}^{N/2}(e^{-i{\frac {2\pi }{N}}k}{a_{s}(k)}^{\dagger }b_{s}(k)+h.c)\\&=-\sum _{s,k=1}^{N/2}[2t_{o}cos({\frac {2\pi }{N}}k)-2t_{1}isin({\frac {2\pi }{N}}k)]{a_{s}(k)}^{\dagger }b_{s}(k)-\sum _{s,k=1}^{N/2}[2t_{o}cos({\frac {2\pi }{N}}k)+2t_{1}isin({\frac {2\pi }{N}}k)]{b_{s}(k)}^{\dagger }a_{s}(k)\\\end{aligned}}$

By doing matrix rearrangement and diagonalize it, we get 2 modes of single particle excitation spectrum which is,

$\epsilon (k)=\pm 2t_{o}{\sqrt {{cos}^{2}({\frac {2\pi }{N}}k)+({\frac {t_{1}}{t_{o}}})^{2}{sin}^{2}({\frac {2\pi }{N}}k)}}$

If we plot this spectrum we'll get the maximum width between 2 modes and their gap which are equal to $4t_{o}\approx 10eV$ and $4t_{1}\approx 1.4eV$. Finally we need to discuss some structure possibilities of polyacetylene. It can have cis or trans molecule which are mirrored with each other or could even be a mix of them. If it is so, then there will be a boundary in between which we call it domain wall, behaving like a localized, stable, and dynamic soliton. A neutral soliton is proposed to be one bond length unpaired spin wall that makes it harder to move to other sites. Nowadays, the focus of polyacetylene research is the wide-range control possibility of the doped one. This doping might come from charged domain walls since it shows some interesting properties like anomalous magnetic susceptibility, infrared absorbtion, and power from temperature.