Electronic structure of free-standing InP and InAs nanowires
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M. Willatzen Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark
R. Melnik Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
L.C. Lew Yan Voon Department of Physics, Wright State University, Dayton, Ohio 45435 (Received 10 April 2006; accepted 16 August 2006)
An eight-band k⭈p theory that does not suffer from the spurious solution problem is demonstrated. It is applied to studying the electronic properties of InP and InAs free-standing nanowires. Band gaps and effective masses are reported as a function of size, shape, and orientation of the nanowires. We compare our results with experimental work and with other calculations. I. INTRODUCTION
In the last few years there has been a resurgence of interest in quantum wires, or nanowires as they are now more commonly called. Those with shorter longitudinal (transverse) aspect ratios are known as quantum rods or nanorods (nanoribbons). The interest is in part driven by new bottom-up growth methods such as laser-assisted catalytic growth1,2 and colloidal synthesis,3,4 and by the demonstration of lasing in ZnO,5 GaN,6 and CdS nanowires.7 The nanowires are now mostly of the free-standing type; typical dimensions are tens of nanometers in lateral linear size and microns in length, although nanowires as small as 3 nm in lateral size have been reported.1,2 In addition to the cubic semiconductors, nanowires of wirtzite structure have been synthesized. For the most part, they can be assumed to have low defect concentrations and close to bulk crystal structure, although there have been a few reports of strained nanowires.8 As expected, a number of studies have already been made. Experimentally, examples are polarized photoluminescence9,10 and Raman scattering.11 Surprisingly, the nature of the valence-band states and the calculation of the band gap is still a disputed problem. There were early calculations of the band structures of embedded nanowires using the k⭈p method. In 1989, Citrin and Chang12 applied the six-band Luttinger-Kohn k⭈p to rectangular GaAs/Al0.2Ga0.8As nanowires. One result of the latter is that a crossing behavior of the first two valence states in a square quantum wire (because of ⌫6 and ⌫7 symmetries of C4) becomes anti-crossing in a a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2006.0358 J. Mater. Res., Vol. 21, No. 11, Nov 2006
rectangular quantum wire (because of all states being ⌫5 of C2). Another result is the mixing of heavy-hole (HH) and light-hole (LH) states even at k ⳱ 0. Typical crosssections were 10 × 10 nm. Sercel and Vahala13–15 studied cylindrical GaAs/Al0.3Ga0.7As nanowires with infinite and finite barriers by using a four-band model within the axial approximation. They went down to 1 nm radius in their calculations, although one might question the applicability of k⭈p at that point. In kz-space, their results for the R ⳱ 2.5- and 5.0-nm nanowires show that the Fz ⳱ 1/2 state is the lowest in energy at kz ⳱ 0, but the Fz ⳱ 3/2 is lower at finite kz;
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