A comparative study for colloidal quantum dot conduction bandstate calculations Page: 4 of 10
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bands uk,n (r)eik'r-
where k is the reciprocal lattice of the supercell and n is the bulk band index. Since the bulk
Bloch states are good basis functions of the quantum dot wavefunctions, one can truncate
this basis set significantly without introducing big errors compared to the original FSM
solved results .
Now, if only the first conduction bulk band (n=CB1) is used in Eq.(3), it will yield a QD
CBM state in Eq.(2), and the valence band states will be deflated in the effectiv Hamiltonian.
The LCBB results solved in this way are shown in Fig.1 as the open squares. They are all
higher than the original FSM/EPM CBM of Eq.(1). In the case of InAs and InP, the LCBB
result is very different from the EMA result, while for CdSe and Si, and especially for large
QDs, the LCBB and EMA results are virtually the same. The difference between LCBB
and EMA is due to their bulk band structures [item (i)]. The bulk band structures for InAs,
InP, and CdSe near the F point are shown in Fig.2(a), (b), and (c) respectively. In the
LCBB, since the exact bulk Bloch state is used in the basis set, the exact EPM conduction
band band structure is implicitly employed in the calculation. For a 50 A diameter QD, the
corresponding average reciprocal vector k is denoted by the vertical dotted lines in Fig.2(a)-
(c). As we can see, at this k, for InAs and InP, there is a big difference between the EMA
band structure and the EPM band structure, corroborating their big differences in QD CBM
energies. Note that, in a 8x8 k.p model [5, 20], the conduction band energy curve can be
much improved upon the single band EMA. But still, for small QD, there could be large
differences between the k.p band and the EPM band, especially when other band valleys
become important .
We now study the multi-bulk band coupling effect induced by the QD geometry. Here,
we have to distinguish two different multi-band coupling effects. One is the coupling in a k.p
like model, which exists even in bulk at the off F k-points. This coupling stems from the fact
that the off F k-point Bloch state un,keik.r is expanded with the F point Bloch state basis
un,reik.r in the k.p Hamiltonian. This coupling between Un,r is needed to get the correct
bulk band structure. However, here we are interested in another coupling, which is called
here multi-bulk band coupling. This is an coupling (mixing) between the bulk Bloch states
unkeik.r for different n's induced by the QD geometry. In the bulk, this coupling is zero. The
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Luo, Jun-Wei; Li, Shu-Shen; Xia, Jian-Bai & Wang, Lin-Wang. A comparative study for colloidal quantum dot conduction bandstate calculations, article, December 1, 2005; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc885234/m1/4/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.