Chapter 6 Calculations of electronic structure and density of states in the wurtzite
7.5 Wave function and quantum effect for ZnO finite well
The wave functions |Ψ| of the six bands for the [0001] slab of 5 layers close to the M point being labeled in Fig. 7-6(a) are shown in ascending energy order in Fig. 7-7(b).
And the wave functions with the same k// for the 2 and 10 layers slabs were also plotted in
Fig. 7-7(a) and 7-7(c), respectively. Analyzing the wave functions across the [0001]
slabs in Fig. 7-7(a)-(c), we can see that the bands (the first and second bands) near the
conduction band edge are mainly contributed by cation layers with high |Ψ| being located at the cation sites, and the upper valence bands (the fifth and sixth bands) are mainly
contributed by the anion layers with high |Ψ| being located at the anion sites.
Observably, the wave function (the third curve) of the closest (surface) band to the
conduction band edge is localized in the top cation-terminated layer and the surface band at the midgap (~ 1.3 eV) is localized in the bottom anion-terminated layer. Our
calculations are consistent with the results of Kresse et al. [27] which calculated Zn-terminated surface and O-terminated surface using the density-functional theory.
Fig. 7-7 The wave functions of the six bands closest to the middle of the band gap away from Γ point for
the slabs of two, five, and ten layers along [0001] (a)-(c) and along [1-100] (d)-(f).
Similarly, in order to eliminate the strong coupling between the surface states with the allowed bands close to the Γ point we plotted the wavefunctions of the [1-100] slabs
with thickness L = 2, 5, and 10 layers near the K point in Fig. 7-7(d)-(f). We can see that besides the wavefunctions of these two midgap bands (the second and the third) there
are other two bands (the fourth curves the fifth curves) show a tendency toward surface localization. Theoretically [28], these four bands should be considered as surface bands
induced by the dangling bonds, in contrast to the allowed bands induced by the atoms in the interior layers (the first and the sixth bands).
The surface state is induced by each dangling bond in a unit cell of the end-surface.
The coupling of the degenerate surface states on the periodic surface generates a surface
band in the surface Brillouin zone. Furthermore, the coupling between these two identical end-surfaces sandwiching the slab causes their degenerate bands to split into
symmetrical and antisymmetrical bands. The larger overlap of the wave functions of the degenerate bands will cause the larger energy-splitting. As a result, the splitting of the
degenerate surface bands increases with decrease in the thickness of the slab. However, the energy-splitting will not be perceived for the [0001] slabs, since the polar
end-surfaces are not identical. Another important consequence is that the splitting at the Γ point is the largest in comparison with that at the other Brillouin-zone boundaries. As
in Fig. 7-6 (b) of the five layers [1-100] slab, the two higher surface bands near the Γ point embedded themselves in the conduction band of the bulk. These bands as
expected shift toward the conduction band edge of the bulk with increasing L. In general, a localization length of surface state is inversely proportional to its energy
separation from the related allowed band [28]. Since the surface bands more closely approach the allowed conduction and valence bands near the Γ point, their localization
length must be longer at the Γ point than at the other k// in the Brillouin zone. The larger localization length at the Γ point results in the larger overlap between the degenerate
surface states, which explains the largest splitting of the bands at the Γ point.
Additionally, we also noted in Fig. 7-8 that the edges of conduction and valence bands
shift towards each other so as the badgap decreases with increasing L. As a matter of fact, this is a direct evidence of the quantum size effect. Obviously, the enhancing the
band gap have more effective carrier confinement in c-axis rather than in other direction.
And both the band structure of the slab and the surface bands become similar to those of
semi-infinite crystal with increase in the number of layers and the bands finally bundle to form the band spectrum of the bulk.
Fig. 7-8 Variations of the band gap as a function of the thickness for ZnO well along [0 0 0 1] (solid curve), [1−1 0 0] (dashed curve) directions and experiment data (hollow circle point).
7.6 Summary
In summary, using tight binding representation of the layered systems grown along
<0001> and <1-100> directions, we calculated the band structures and wave functions for
various ZnO slab layers with non-relaxed and non-reconstructed surfaces. We first calculate the surface spectrum of a semi-infinite crystal using the transfer-matrix
technique. Then, using cyclic boundary conditions, we calculate the quantized spectrum of the surface states in nanowires. We find that the spectrum of the nanowire surfaces
consists of a number of quantized levels inside the band gap. We show that dangling bonds on the two end-surfaces cause surface bands for different direction grown slabs.
Analyzing the wave functions across the layers, the surface states show a tendency toward surface localization. Particularly, the splitting of the degenerate surface bands
increase due to increasing overlap between their wave functions, which are localized on two nonpolar [1-100] end-surfaces, while it is not present for the [0001] finite well with
polar end-surfaces. Finally, we also found that the enhancing the band gap along [0001]
polar due to more effective carrier confinement in c-axis.
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