Chapter 6 Calculations of electronic structure and density of states in the wurtzite
7.3 Surface states of nanowire with identical lateral facets
We have analyzed the surface spectrum of the semiinfinite wurtzite crystal. Now we want to understand how the spectrum will change on the surface of a nanowire. Let us
consider a wurtzite nanowire surrounded by the identical lateral facets [1-100]. First of all, we note that the surface spectrum and the wave function of each equivalent facet
should be identical. The cross section of the nanowire under investigation is hexagonal, we can assume that the cation-anion dimers wrap the nanowire along the lateral surface
of a hexagonal prism, which is encircled by a cylinder of radius R equal to that of the nanowire as shown in Fig. 7-4.
rz R
r⊥ [0001]
rz R
r⊥ [0001]
Fig. 7-4 Wurtzite nanowire with identical lateral facets wrapped by a cylinder of radius R equal to the
radius of nanowire.
By definition (7-3), the wave function of the surface state is a Bloch function
)
) (
( ) ,
(r⊥ rz =uk r eik⊥r⊥+kzrz
Ψ r (7-15)
What is important is that this wave function depends on the surface vector rr=rr⊥+rrz
with the components rr (perpendicular) and ⊥ rr (parallel) to the nanowire axis as z
shown in Fig. 7-4. Accordingly, the momentum kr kr krz +
= ⊥ should be decomposed
into the components kr⊥ ||ΓX
and krz ||ΓK
see inset in Fig. 7-3. The periodic function uk(rr is defined by )
)]
In the case of the semi-infinite crystal, the lattice vector Rr
runs over N sites of the one
[1-100] surface, while in the case of the nanowire, Rr
runs over N sites of the six identical [1-100] surfaces. On the surface of a nanowire with hexagonal cross section, the wave function (7-15) must satisfy the cyclic boundary conditions
)
Taking into account the periodicity of the function uk(rr , we obtain that this boundary ) condition can be satisfied only at discrete values of the wave vector perpendicular to the axis of the nanowire:
3 π R n
k⊥ = for any n=1,2,3… (7-18)
As a result the dispersion equation (7-13) will be reduced to
, 0
Fig. 7-5 demonstrates the dispersion of quantized levels of the surface bands through the
Brillouin zone of the nanowire, krz =[0,π /c]
, with the vector kz being parallel to the Γ
K direction of the surface Brillouin zone. We present here the data for the two surface bands close to the conduction (Figs. 7-5 (a) and 7-5 (b)) and valence bands (Figs. 7-5 (c) and 7-5 (d)) for ZnO nanowires with radius equal to 20a (~80 nm) and 10a (~40 nm) in
Figs. 7-5 (a) and 7-5 (c) and in Figs. 7-5 (b) and 7-5 (d), respectively.
Fig. 7-5 (Color online) Surface spectrum of GaN nanowire surrounded by identical [1-100] lateral facets
with radius R=20 u.c. (a), (c) and 10 u.c. (b), (d). The two surface bands close to the conduction (a), (b) and
valence bands (c), (d) are shown.
(a) (b)
(c) (d)
We note that, at room temperature, because of temperature broadening, the quantized surface spectrum can be distinguished only for the surface band close to the
conduction band (Figs. 7-5 (a) and 7-5 (b)). As we see from Figs. 7-5 (c) and 7-5 (d), the quantized levels of surface band close to the valence band can be distinguished only
at low temperature. At room temperature the spectrum of this band can be considered as quasicontinuous.
7.4 Band structures for polar and nonpolar surface layers
In this section, we solved eigenvalue problem of Eq. (7-2) and Eq. (7-4) to obtain electronic behavior for ZnO finite well structures for different direction. In Figs. 7-6(a)
and 7-6(b), we overlaid the band structures (solid curves) of the five-layers [0001] and [1-100] ZnO slabs, respectively, on the projected band structures (shaded regions) of bulk
crystal using the approach described in above. The extrema of the conduction and the valence bands of the bulk ZnO are found to be Ec = 3.36 eV and Ev = 0 eV, respectively.
In the above calculation, we used the transfer matrix method to prove that the dangling bonds of the cations (Zn) result in a surface band as a solid curve close to the bottom of
the conduction band in Fig. 7-6(a), while the dangling bonds of the anions (O) give rise to the surface bands near the middle of the band gap (around 1.3 eV). This is in agreement
with the origin of the conduction (valence) band mostly from the cation (anion) atomic orbitals [22, 24]. We also perceived that there are energy dips at the midst of band gap for the oxide-settled surface bands at theΓ point. The depth of energy dip is function
of slab thickness that may be caused by coupling between the O- and Zn-terminated
surfaces. Further study to clarify this problem is required and is under the way. In Fig.
7-6 (b), we can see that the band gap of the slab is formed by four bands, which include
two bands slightly above the bottom of conduction band and other two bands in the middle of the band gap at the Γ point. Particularly, we observed the splitting of the
these bands around the neighborhood of the Γ point and the difference between the
near-conduction bands and the middle-gap bands decrease with increase in the slab
thickness. On the other hand, there is strong coupling between the surface states with the allowed bands especially at the Γ point. In order to eliminate these coupling effects,
we calculated the wave functions for the parallel (to the layers of the well) wave vector, k//, away from the Γ point for ensuring accurately characterizing the properties of surface
states.
Fig. 7-6 The band structures of the ZnO slabs with five layers along (a) polar and (b) nonpolar surface
layers. The solid curves are the surface states and the shaded regions show the projected band structure of
the bulk.
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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Chapter 8 Conclusions and Prospective
8.1 Conclusions
ZnO QDs were synthesized successfully via a simple sol-gel method. The average
size of QDs can be tailored using well-controlled concentration of zinc precursor. The size of QDs was estimated by X-ray diffraction consistent with the results of
high-resolution transmission electron microscope (HRTEM) image. Furthermore, the measured Raman spectral shift and asymmetry for the E2 (high) mode caused by
localization of optical phonons agree well with that calculated by using the modified spatial correlation model. From the resonant Raman scattering, the coupling strength
between electron and longitudinal optical phonon, deduced from the ratio of the second- to the first-order Raman scattering intensity, diminishes with reducing the ZnO QD
diameter. The size dependence of electron-phonon coupling is principally a result of the Fröhlich interaction.
We also analyzed the exciton’s behavior in ZnO QDs. Size-dependent blue shifts of PL and absorption spectra revealed the quantum confinement effect. The band gap
enlargement is in agreement with the theoretical calculation based on the effective mass
model. Moreover, the exciton longitudinal-optical-phonon (LO-phonon) interaction was observed to decrease with reducing ZnO particle size to its exciton Bohr radius (aB).
The unapparent LO-phonon replicas of free exciton (FX) emission and the smaller FX energy difference between 13 and 300 K reveal decreasing weighting of exciton-LO
phonon coupling strength. The diminished Fröhlich interaction mainly results from the reducing aB with size due to the quantum confinement effect that makes the exciton less
polar.
In theoretical study, we first calculated the electronic structure and the density of
states in the wurtzite structure of Zn1-xMgxO (ZMO) alloys using sp3 semi-empirical tight-binding model, we observed increases of both band gap and electron effective mass
that agree with the experimental results as increasing Mg composition up to x = 0.3.
From the calculated total density of states, the increasing electron effective mass is a
result of less orbital overlap of cation sites due to extra density of modes coming from Mg3s and Mg3p orbitals as introducing more Mg composition. Additionally, reducing
electronegative characteristic of oxygen was caused by the O2p was less localized around the oxygen atom. In additionally, we theoretically studied the surface states in wurtzite
semiconductor nanowires with identical lateral facets. Using the transfer-matrix technique, we first calculated the surface states for surfaces [0001] and [1-100] of the
semi-infinite wurtzite semiconductor. We then used the cyclic boundary condition for the surface wave function in order to find the quantized spectrum for a nanowire.
Electronic band structures and surface states were investigated for ZnO finite wells or slabs grown along <0001> and <1-100> directions using tight binding representation.
The dangling bonds on two end-surfaces caused surface bands for different directions grown slabs, of which the wavefunctions tend to localize at the end surfaces. The
increasing splitting of the degenerate surface bands at the Γ point was observed decreasing with the thickness of the nonpolar [1-100] slab. And, the quantum
confinement effect is distinctively enhanced by the extra electron-field induced in the
<0001> grown finite well with the polar end-surfaces.
8.2 Prospective
In the next-generation optoelectronic devices, the nanometer-scale materials promise to be important due to numerous unique properties expected in the
low-dimensional system. Low-dimensional ZnO nanostructures, such as QDs, nanoparticles (NPs), nanobelts, nanowires, and quantum wells, have been widely
investigated for the feasible requirement. In particular, ZnO QDs and NPs are of great interest because of the three-dimensional confinement of carrier and phonon leads not
only continuous tuning of the optoelectronic properties but also improvement in device performance. Nevertheless, the surface of QDs is usually composed of uncoordinated
atoms, which make the QDs highly active and quench the PL emission. Since, the modification of surface of ZnO QDs becomes imperative issue for next generation of
optoelectronic devices.
In experiment, in order to combine the advantages of QDs and Mg-doped ZnO, ZnO-MgO core-shell structure is the most interesting topic for our works. That can overcome the effect of surface defects in ZnO, especially in QDs system which has large
surface-volume ratio. Because the band gap energy of MgO is much larger than that of ZnO, the ZnO-MgO core shell QDs should have better emission efficiency than ZnO
QDs.
We will suggest to also use simple sol-gel method to add Mg(OAc)2·4H2O in ZnO QDs solution then stir at 40°C. When the prepared new solution drops on Si then take to be annealed for 300°C to 700°C. The primary results of the low temperature PL spectra
of samples annealed at various temperatures that we observed a broaden emission from MgZnO alloy and two sharp emission from ZnO. We also found that the defect emission has been diminished from sample 300°C to 700°C. To further investigate more
characteristic of ZnO-MgO core-shell QDs, the temperature dependent PL and
time-resolved PL should proceed.
In other way, according to above experiment data, the effective-mass approximation
apparently gives a good understanding of the blue shift of the optical absorption threshold.
However, this approach fails for the smallest crystallite sizes because of the
oversimplified description of the crystal potential as a spherical well of infinite depth. A better description of the band structure can be obtained from a tight-binding (TB)
framework. Since the atomic structure is implicitly considered, this method is more adequate for small crystallites. In the future, I will calculate the electronic structure and
optical properties of ZnO QDs using TB approximation. Once the tight-binding parameters are known, we can calculate the eigenvalues of Hamiltonian H. This matrix
is formed by 4×4 block matrices describing the interactions on the same atom (intra-atomic) or between two first-nearest neighbors (interatomic). If N is the number
of atoms in the crystallite, the dimension of H is 4N and a direct diagonalization becomes impossible for several hundred atoms. To circumvent this problem we will use the
recursion method or symmetry basis method. Such large matrices can be diagonalized with the help of group theory; partial diagonalization is effected by using the projection
operators of the point group to form-basis states. Thus, computation time is reduced based on symmetry-TB method.
林國峰簡歷 (Vita)
基本資料
姓名: 林 國 峰 (Kuo-Feng Lin) 性別: 男
出生年月日: 1977 年 10 月 18 日 籍貫: 花蓮縣
永久通訊處: 花蓮縣豐濱鄉大港口村 9 鄰 50 號 Email: xia117.eo92g@nctu.edu.tw
xia117.eo94g@nctu.edu.tw 學歷:
2001.9 - 2003.6 國立台北科技大學光電科技系 學士 2003.9 – 2005.6 國立交通大學光電工程研究所 碩士 2005.9 – 2008.10 國立交通大學光電工程研究所 博士 博士論文題目:
實驗及理論探討奈米結構之氧化鋅光學性質研究
Experimental and theoretical study on the influence of finite crystallize optical properties in ZnO
nanostructures
Publication list
I. Refereed Journal Publications:
1. Kuo-Feng Lin, Hsin-Ming Cheng, Hsu-Cheng Hsu, Li-Jiaun Lin, Wen-Feng Hsieh, “Band gap variation of size-controlled ZnO quantum dots synthesized by sol–gel method,” Chemical Physics Letters 409, 208 (2005).
2. Hsin-Ming Cheng, Kuo-Feng Lin, Hsu-Cheng Hsu, Chih-Jen Lin, Li-Jiaun Lin, and Wen-Feng Hsieh, “Enhanced Resonant Raman Scattering and Electron-Phonon Coupling from Self-Assembled Secondary ZnO Nanoparticles,”
Journal of Physical Chemistry B 109, 18385 (2005).
3. Kuo-Feng Lin, Hsin-Ming Cheng, Hsu-Cheng Hsu, and Wen-Feng Hsieh,
“Band gap engineering and spatial confinement of optical phonon in ZnO quantum dots,” Appl. Phys. Lett. 88, 263117 (2006).
4. Hsin-Ming Cheng, Kuo-Feng Lin, Hsu-Cheng Hsu, and Wen-Feng Hsieh, “Size dependence of photoluminescence and resonant Raman scattering from ZnO quantum dots,” Appl. Phys. Lett. 88, 261909 (2006).
5. Ching-Ju Pan, Kuo-Feng Lin, Wei-Tse Hsu and Wen-Feng Hsieh, “Raman study on alloy potential fluctuations in MgxZn1-xO nanopowders,” J. Phys.:
Condens. Matter 19, 186201 (2007).
6. Ching-Ju Pan, Kuo-Feng Lin and Wen-Feng Hsieh, “Acoustic and optical phonon assisted formation of biexcitons,” Appl. Phys. Lett. 91, 111907 (2007).
7. Ching-Ju Pan, Kuo-Feng Lin, Wei-Tse Hsu and Wen-Feng Hsieh, “Reducing exciton-LO phonon coupling with increasing Mg incorporation in MgZnO powders,” J. Appl. Phys. 102, 123504 (2007).
8. Wei-Tse Hsu, Kuo-Feng Lin, and Wen-Feng Hsieh, “Reducing exciton-longitudinal-optical phonon interaction with shrinking ZnO quantum dots ,“Appl. Phys. Lett. 91, 181913 (2007).
9. S. C. Ray, Y. Low, H. M. Tsai, C. W. Pao, J. W. Chiou, S. C. Yang, F. Z. Chien and W. F. Pong, K. F. Lin, H. M. Cheng and W. F. Hsieh, “Size dependence of the electronic structures and electron-phonon coupling in ZnO quantum dots,”
Appl. Phys. Lett. 91, 262101 (2007).
10. Kuo-Feng Lin, Ching-Ju Pan, and Wen-Feng Hsieh, “Calculations of electronic structure and density of states in the wurtzite structure of Zn1-xMgxO alloys using sp3 semi-empirical tight-binding model,” Appl. Phys. A. (2008).
11. Kuo-Feng Lin and Wen-Feng Hsieh, “Electronic band structures and surface
11. Kuo-Feng Lin and Wen-Feng Hsieh, “Electronic band structures and surface