Does the delocalization of p-orbitals ensure conductivity?

For example, polyacetylene is a [conjugated system](https://en.wikipedia.org/wiki/Conjugated_system), i.e. with de-localized p-orbitals, but accroding to [Chiang et. al. (1977)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.39.1098) polyacetylene is semiconducting, and becomes conductive only when doped with Cl/Br/I/AsF5.


Isn't the conjugation alone enough to allow conductivity?


[![Electrical conductivity of trans-polyacetylene as a function of AsF5 dopant concentration](https://i.stack.imgur.com/oQ6K3.png)](https://i.stack.imgur.com/oQ6K3.png)([source](http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.39.1098))


UPDATE
------


One reason they mention in the paper is:



> 
> In the idealized situation of a uniform chain, the resulting
>  conduction band would give rise to metallic behavior. However, such a
>  system is unstable with respect to bond alternation, which causes the
>  formation of an energy gap in the electronic spectrum.
> 
> 
> 


But if this is true then (1) how the halogens improve the situation, and (2) why isn't there a similar phenomenon in all conjugated systems?




Delocalized $\pi$ orbitals **do** allow for mobile charges. The catch there is "allow." Just because these systems *can* have mobile charges does not make them *conductive*.


Conductivity can be defined as:
$\sigma = n e \mu$
where $e$ is the charge on an electron, $\mu$ is the charge mobility, and $n$ is the number of charges.


In the case of $doped$ conjugated polymers like you indicate above, the halogen vapors or other dopants *usually* oxidize the polymer and create more charges (i.e., $n$) and the conductivity increases. This ***does*** happen with pretty much every conjugated system I've seen: oxidize or reduce the neutral system and you'll have more charge carriers = greater conductivity. (NB: most conjugated organics are easier to oxidize than reduce, so you get + charges (holes) as the charge carrier, rather than electrons.)


That's the quick explanation as to why adding halogens increases the conductivity -- you get more charge carriers.


Incidentally, there are other ways to increase the charge carriers. Chemical or electrochemical doping usually makes a conductive system, but you can also use electrical bias (e.g., a [field-effect transistor](https://en.wikipedia.org/wiki/Field-effect_transistor)) or light (i.e., a [photovoltaic](https://en.wikipedia.org/wiki/Organic_solar_cell)) which will also increase carrier concentrations.


**As to why *trans*-polyacetylene isn't a metal in the first place?**


If we use a simple [particle-in-a-box](https://en.wikipedia.org/wiki/Particle_in_a_box) model, the difference in energy levels goes to zero as the box becomes very large. This would suggest *trans*-polyacetylene should be a metal (i.e., no band gap).


Consider also the [Hückel model](https://en.wikipedia.org/wiki/H%C3%BCckel_method) for polyacetylene. We can build up the nearly-infinite system by solving for ethylene, butadiene, etc.


[![enter image description here](https://i.stack.imgur.com/CgrgO.png)](https://i.stack.imgur.com/CgrgO.png)


Notice that the energy gap between levels becomes smaller and smaller. We can plot the HOMO-LUMO gap as a function of chain length:


[![enter image description here](https://i.stack.imgur.com/f1O0C.png)](https://i.stack.imgur.com/f1O0C.png)


The gap becomes smaller and smaller! 


[Wikipedia](https://en.wikipedia.org/wiki/H%C3%BCckel_method) even gives the general formulas:


$$E\_k = \alpha + 2\beta \cos \frac{k\pi}{(n+1)}$$
$$\Delta E = -4\beta \sin \frac{\pi}{2(n+1)}$$


So both particle in a box and Hückel predict no band gap when $n$ gets very large. Theory predicts a metal!


Why doesn't the *real* polyacetylene behave as a metal. For any one-dimensional system, there will be a [Peierls distortion](https://en.wikipedia.org/wiki/Peierls_transition) which lowers the total energy by distorting the geometry (long-short bond alternation) and increasing the band gap.1


**In short, a truly one-dimensional metal is impossible, but doped polymers are certainly very good electrical conductors. Record films of *trans*-polyacetlyene have conductivity close to conventional metals.**2


[![enter image description here](https://i.stack.imgur.com/P1pB2.png)](https://i.stack.imgur.com/P1pB2.png)




---



1 The long-short bond alternation is the Peierls distortion. It means some electrons are localized (shorter double bonds) and thus lower in energy. This creates a band gap (i.e., lower E for valence band).


2 The film isn't a 1D material, but the trans-polyacetylene chains are each 1D.



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