W hile the preceeding section details one mechanism whereby chiral order can be introduced into a system where none previously existed, there is still th e strong suspicion th a t such a simple model may miss some im portant underlying features. In this section th e detailed atom ic stru ctu re of the underlying tu be is assumed to be im portant, and the electronic stru ctu re is no longer treated as an electron gas. Again, we consider the case of a coiling charged polymer, whose presence is accounted for by assuming th a t w hat lies in its neighbourhood becomes polarised. In this case, it polarises some of th e carbon atom s on the tube. The polarised atom s will follow th e same helical p ath as the charge distribution and thus acquire a different on-site energy to their non-polarised counterparts.
Under the initial assum ption th a t these bulky molecules cam iot resolve the atom ic stru ctu re of the underlying lattice [28, 84], th e tubes may then be exposed to two different helical pitches: the intrinsic chirality of the nanotube and th a t as sociated with the w rapping potential. For sim plicity’s sake, we tre a t the wrapping molecule as a continuous charge distribution of uniform w idth w rapped around the nanotube at a constant angle 6, as depicted in Fig. 4.5. We assume th a t th e induced ])olarisation represented by the perturbing potential in Eq. (3.2) affects all carbon atom s im m ediately below the charged stripe in an identical fashion. T he lateral dimension of th e w rapping molecule, together w ith the distance of the molecule from th e tube, determ ines the w idth W of this stripe. In principle, th e angle 9
may bear no direct relation to the chiral angle a of the underlying nanotube, and we trea t these two param eters as both being independent. It is our goal here to investigate th e effect th a t this com bination of chiralities may have on the nanotube band stru ctu re and, in particular, on its electronic density of states (DOS).
H am ilto n ian
//o = \j)l{j'\ , (4.31) hi'
w here |j) rep resen ts th e 7r-orbital cen tred a t ato m j , \j') is a nearest-neighbour o rb ita l c e n tred on a to m / , and
7
is th e nearest-n eig h b o u r electronic hopping, which we will choose to be our energy u n it. T h e to ta l H am ilto n ian is given as / i = IIq + V , w herel/ = ^ A | m ) ( r n | (4.32)
m
accounts for th e p o larisatio n of th e p e rtu rb e d atom s un d er th e stripe. Here, m labels those atom s affected by th e proxim ity to th e w rapping m olecule and A represents th e corresp o n d in g shift in th e ir on-site p o te n tia l due to th e induced polarisation.. A can in principle be d eterm ined in accordance w ith th e m easiued binding energy betw een th e n a n o tu b e and its w rap p in g m olecules. Also, th e fact th a t ionic doping agents can affect th e a m o u n t of charge carried on th e DNA backbone suggests th a t we m ay consider a range of values for A, which we will for th e m om ent regard as a free p a ra m e te r.
To assess how th e electronic s tru c tu re of a n a n o tu b e is influenced by th e w rap ping p o te n tia l, I have perform ed a sy ste m atic stu d y of how th e electronic DOS is affected as som e of these different p a ra m e te rs are varied. In p articu la r, we will take a d v a n ta g e of th e fact t h a t th e quasi-one-dim ensional n a n o tu b e s have densities of s ta te s c o n ta in in g several d istin ctiv e van Hove singularities (VHS). T hese are lo c a te d a t values of energy for w hich an a rb itra rily large nu m b er of electrons can l)e in s ta te s of th e sam e energy. C onsequently, allowed tra n s itio n s betw een van Hove sin g u larities is a key featu re of tlie optical a b so rp tio n sp ectru m . By locating these sin gularities a n d tra c k in g how th e y evolve as th e aforem entioned p a ra m ete rs are var ied, one can view , a t least from a q u a lita tiv e p o in t of view, how n a n o tu b e s res])ond to such p e rtu rb a tio n s . Such response could, in principle, be tracked by com paring th e o p tic a l ab so rp tio n sp e ctru m of co ated n a n o tu b e s w ith th a t of pristin e tu b es.
F ig u re 4.5: Schem atic rep re sen ta tio n of a hehcally w rapped n a n o tu b e . In (a) we display a polym eric m olecule represented by a continuous charge d istrib u tio n of c o n sta n t w id th 11' coiling a t an angle 0 a ro u n d th e surface of th e n a n o tu b e . In (b) we display th e co rresponding geom etry in th e im w rapped rep re sen ta tio n . T h e atom s to be p e rtu rb e d are in d ic a te d by th e d a rk (blue) balls, while th e u n p e rtiu 'b e d atom s are th e light (yellow) balls. D is defined in th e p ictu re as th e d ista n c e betw een ecjuivalent a to m s of neighbouring stripes. Also d ep icted is th e circum ference of th e n a n o tu b e {2-kR) w here /? is th e n a n o tu b e radius.
Tlie sequence adopted in this section is as follows: in the next section, we discuss the details of our calculations, as well as defining some useful concepts; in the following section, we present results for the dependence of a few physical observables on the inicrostructure param eters A,^ and W; and in the final section we present conclusions and discussion.