DOI 10.1007/s00339-008-4721-x
Calculations of electronic structure and density of states
in the wurtzite structure of Zn
1
− x
Mg
x
O alloys using sp
3
semi-empirical tight-binding model
Kuo-Feng Lin· Ching-Ju Pan · Wen-Feng Hsieh
Received: 2 January 2008 / Accepted: 22 May 2008 / Published online: 21 June 2008 © Springer-Verlag 2008
Abstract Calculating the electronic structure and the
den-sity of states in the wurtzite structure of Zn1− xMgxO
(ZMO) alloys using sp3semi-empirical tight-binding model, we observed increases of both band gap and electron effec-tive mass that agree with the experimental results as increas-ing 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 that the O2p was less localized around the oxygen atom.
PACS 71.20.-b· 71.55.Gs · 74.25.Jb · 71.20.Nr · 87.15.Pc 1 Introduction
In the past decade, the advantageous technologies of light-emitting diodes (LED) and semiconductor lasers realize full-color display systems which have prompted the research for devices operating in the blue-ultraviolet (UV) [1–3]. Zinc-oxide (ZnO) materials have drawn considerable attention for the application in UV optoelectronics because of their exci-tonic transition energy (∼3.37 eV) and large exciton bind-ing energy (∼60 meV). The band gap becomes even larger if Zn atoms are substituted by magnesium (Mg) atoms, which have a similar ionic radius, allowing for quantum-well struc-tures and superlattices [4–6]. The change of band gap of the
Kuo-Feng Lin· Ching-Ju Pan · Wen-Feng Hsieh (
) Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Tahsueh Rd., Hsinchu 30050, Taiwane-mail:wfhsieh@mail.nctu.edu.tw
Zn1− xMgxO (ZMO) layers grown on sapphire [21] with
0.1 < x < 0.3 and powders [20] with 0 < x < 0.05 have been reported; and in the Al doped ZMO alloys with a preferential c-axis orientation, Lu et al. [23] experimentally showed that the electron effective masses increase with in-creasing Mg concentration. However, it is still lack of theo-retical confirmation of bandgap and electron effective mass as increasing Mg incorporation.
In the wurtzite Zn1−xMgxO (ZMO) alloys,
Malashe-vich et al. [28] investigated the polarization-related prop-erties based on density-functional-theory calculation under the local-density approximation and the Berry-phase ap-proach to calculate electric polarization. Li and Wei [25] used the first-principle band-structure method to show that if Mg atoms substitute Zn atoms to reduce the anion and cation kinetic p-d repulsion, the acceptor transition energy of NO–nZnZncan be reduced. Whereas, the application of the first-principles calculation to study electronic band struc-ture of disordered alloys and solid solutions generally re-quires using very large supercell in order to mimic the dis-tribution of local chemical environments. It is very compu-tationally demanded. On the other hand, the tight-binding (TB) theory is a versatile and simple method to calculate the electronic properties of solids. Additionally, due to the trans-ferability of the TB parameters, the method has been read-ily applied to systems with broken translational invariance such as low-dimensional structures, clusters, and alloys. Us-ing the TB approach, among the quantities that are suc-cessfully calculated, there are elastic constants [7], phonon spectra [8], vacancy-formation energies, and surface ener-gies [9,10], as well as cluster-formation [11] energies and magnetic moments. Particularly, the electronic structure of semiconductor compounds is well calculated in the com-plete Brillouin zone [12,13]. Additionally, the tight bind-ing method gives reliable results for bands originatbind-ing from
well-localized atomic states. ZnO and MgO are both ionic semiconductors; therefore, the tight binding model should be also suitable for studying ZMO compound semiconduc-tors. Nevertheless, there were few theoretically calculations of the electronic structure in the Zn1−xMgxO alloy system.
Particularly, for electronic devices, knowledge of the effec-tive masses is especially important for analyzing the device properties, especially the current transport characteristics.
In this letter, we present the electronic band structure and total density of states (DOS) of ZnO and ZMO al-loy crystallization using the nearest- and the next-nearest-neighbor semi-empirical tight-binding (SETB) approach sp3 model [14].
2 Theoretical details
We present the electronic band structure and total DOS of ZnO and ZMO alloy crystallization using the nearest-and the next-nearest-neighbor semi-empirical tight-binding (SETB) approach sp3model [14], which has proven its ca-pability in the case of bulk wurtzite crystals. The virtual-crystal approximation (VCA) [12] treats an alloy as a per-fectly periodic crystal by assuming the symmetry of the con-stituents being identical to the symmetry of pure crystal. In the compound Zn1− xMgxO semiconductor, a virtual atom,
whose wavefunctions of electrons are linear combinations of atomic orbitals of the constituent Zn and Mg atoms, is assumed to occupy the cation site in a unit cell, while anion site is occupied by O atom. Therefore, the effect of disor-der is absent in our results. However, in ordisor-der to ensure the virtual crystal approximation is valid, the ZMO alloy must keep with the wurtzite structure. The wurtzite ZMO alloy is stable with respect to the corresponding rocksalt alloy for
x <0.375 [17,18]. The VCA was used by taking weighted averages of the diagonal matrix elements for the band struc-tures of the alloys, namely, we write the SETB parameters of the binary compounds,
Eb(Zn1−xMgxO)= (1 − x)EbZnO+ xE MgO b ,
b= s, px, py, pz, (1)
where EbZnO,MgOare the binding energies of corresponding orbitals of Zn and Mg bound to O2p and O2s orbitals, and x is the incorporation concentration of Mg. The off-diagonal matrix elements, multiplied by the square of the bond length, are also averaged this way (Harrison’s rules) [15] with the bond length obtained from Vegard’s law [16],
d(Zn1−xMgxO)= (1 − x)dZnO+ xdMgO. (2) All the parameters are listed in Table1. The VCA treats an alloy as a perfectly periodic crystal, assuming its symme-try identical to the symmesymme-try of the constituents; therefore,
Table 1 Tight-binding parameters of ZnO and MgO alloys (in eV)
ZnO MgO E(s, a) −19.046 −20.02 E(p, a) 4.142 4.2 E(s, c) 1.805 1.753 E(p, c) 12.368 12.1986 V (s, s) −6.043 −6.89289 V (x, x) 7.157 8.16356 V (x, y) 10.578 12.0657 V (sa, pc) 4.703 5.36443 V (pa, sc) 8.634 9.84829 λa 0.032 0.032 λc 0.085 0.0074
it does not describe the different local atomic environment in inhomogeneous materials and has failed in unusual semi-conductors [29]. However, the VCA has the advantage of simplicity and computational efficiency, and this approach also has been successfully to solve problems about structure, thermodynamics, dielectric, and piezoelectric properties of some materials [30,31].
3 Results and discussion
3.1 Band structures of Zn1−xMgxO
The resultant band structures for ZnO and Zn0.7Mg0.3O are given in Fig.1. The deepest and lowest valence band of ZnO crystal (Fig.1a) near−20 eV corresponds to an atomic-like O2s state, and the upper valence bands are mainly com-posed of the O2p state and Zn4s and 4p states. The low-est conduction band is contributed by Zn4s state, and the uppermost conduction bands are primarily caused by Zn4p character. The ZMO is direct band gap semiconductor for all stable wurtzite structures and has band gap ranging from 3.429 eV for ZnO to 4.153 eV for Zn0.7Mg0.3O. In addition, we also estimate the p-doubly degeneracy in valence bands with renormalized atomic spin–orbital (SO) splitting of the anion and cation p states [19]. For ZnO we got 15 meV for SO energy (Δ0), offering a good agreement with experimen-tal results [20], while the modification of SO energy was not conspicuously observed in Zn1− xMgxO alloy. However, we
perceived that the band gap at Γ point increases as Mg con-centration increases.
3.2 Energy gaps and effective masses
In Fig.2we plotted the change of the band gaps at Γ points of ZMO alloys as a function of the Mg mole fraction x along with the experimental results from the ZMO layers grown
1− x x
Fig. 1 Calculated band structure of (a) ZnO and (b) Zn0.7Mg0.3O
us-ing semi-empirical tight-bindus-ing model
Fig. 2 The energy gaps of wurtzite Zn1− xMgxO alloys as a function
of Mg mole fraction x. The hollow circles and solid circles show ex-perimental data from Refs. [20,21]
on sapphire (solid circle point) [21] and powders (hollow circle point) [20] in the range 0 < x < 0.30. It shows very good agreement of our calculated curve with the
experimen-Fig. 3 Composition dependence of the electron effective masses. The dash line shows the electron effective mass parallel to the c-axis, and
the solid line is for perpendicular to the c-axis
tal results that the band gap increases with increasing Mg concentration. The curves can be fitted by the polynomials
EΓg = 3.429 + 2.377x + 0.1349x2(eV), (3) although the quadratic term is referred as the band-gap bow-ing factor bebow-ing 0.1349, which depends on the substrate on which the layer was grown [22].
The effective mass is given by the curvature of the disper-sion relation E(k) according to (m1∗)μν=12
d2E(k) dkμdkν, where
k is the wave vector of electron in the crystal, and μ and
ν are Cartesian coordinates. Figure3shows the calculated effective masses for the alloys ZMO as a function of Mg concentration x. We found that the electron effective mass increases when the Mg concentration increases. In ZnO wurtzite structure (x= 0), the electron effective masses of the lowest conduction band for parallel and perpendicular to the c-axis are me = 0.278m0 and m⊥e = 0.202m0, respec-tively. These values agree well with experimental data of Button et al. [26] determined from the cyclotron resonance measurements. As aforementioned, the lowest conduction band is mainly caused by Zn4s state; the electron effective masses are mainly determined by the overlapping wavefunc-tions of s-orbital among cation sites. Lu et al. [23] also ob-served the same results from the modified Burstein–Moss model in the Al doped ZMO alloys with a preferential c-axis orientation. They indicated that the electron effective masses increase with increasing Mg concentration, while the carrier mobility and electron concentration decrease. Furthermore, we note that variation of electron effective masses is more obvious for me than for m⊥e in the ZMO alloys. Since the
ZnO has wurtzite structure and the MgO is cubic, the lat-tice along c-axis changes more dramatically than that along a-axis as more Mg atoms substituting Zn atoms.
3.3 Density of states
We calculated the total DOS of ZMO alloys by summation over all bands using a Gaussian function with a broaden-ing parameter of 0.01 eV, e.g., due to temperature broad-ening around 77 K. As shown in Fig. 4, we obtained the similar DOS for the concentration ranging from 0 to 10%, while the DOS of upper valence bands were slightly broad-ened and decreased as increasing Mg incorporation. Imai et al. [24] used the first-principle pseudopotential method to show that the bond of Zn–O is more ionic than Zn–S due to the probability of electron lingering between Zn and O is much smaller than between Zn and S. Similar to those obtained by Li et al. [25] using the first-principle calcula-tion, the increasing probability of electron lingering between cation and anion with increasing Mg composition suggest that in the ZMO alloys the electrons from O2p orbital are less localized around the oxygen atom to cause reduction of electronegative characteristic of oxygen and ionization en-ergy of acceptors.
As shown in Figs. 4c and d, we found both the DOS of the upper valence band near −2.7 eV and the upper-most conduction bands near 19 eV increase as Mg com-position increases. Because the weighting of Mg’s atomic orbital increases with increasing Mg concentration in the VCA method, we believe that the Mg3s orbital mainly con-tributes extra density of modes to the upper valence band, and the Mg3p orbitals principally contribute extra density of modes to the uppermost conduction bands. Additionally, the DOS of the near lowest conduction band increases with more localized wavefunction as increasing Mg incorpora-tion. Since the orbital energy of Mg3s is lower than that of Zn4s [27], the overlapping integral among cation sites re-duces, and consequently, the less overlap of wavefunctions of the neighboring cation atoms results in narrowing the width of conduction band so that the larger effective mass. The wavefunction overlapping determines the rate of quan-tum tunneling of an electron from one ion to another so that
Fig. 4 Total density of states of various Mg concentrations in the
wurtzite Zn1− xMgxO alloys: (a) 0%, (b) 10%, (c) 20%, and (d) 30%
the electrons hop slower from one ion to an adjacent one in the lattice with increasing Mg concentration that shows the larger effective electron mass.
4 Conclusion
In conclusion, using sp3semi-empirical tight-binding model under virtual-crystal approximation, we have calculated the electronic band structure and the total density of states for the wurtzite structure of Zn1− xMgxO alloys
semiconduc-tor. We observed enlarging band gap and increasing electron effective mass that agree with the experimental results as increasing Mg composition up to x= 0.3. From the calcu-lated total density of states, due to extra density of modes coming from Mg3s and Mg3p orbitals as introducing more Mg composition, the increasing electron effective mass is a result of less orbital overlap of cation sites. In addition, the O2p is less localized around the oxygen atom to cause re-ducing electronegative characteristic of oxygen.
Acknowledgements This work was partially supported by the Na-tional Science Council and the Ministry of Economic of the Republic of China under Contract No. NSC-96-2628-M009-001-MY3.
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