Composition dependent band offsets of ZnO and its ternary alloys

We report the calculated fundamental band gaps of wurtzite ternary alloys Zn1−xMxO (M = Mg, Cd) and the band offsets of the ZnO/Zn1−xMxO heterojunctions, these II-VI materials are important for electronics and optoelectronics. Our calculation is based on density functional theory within the linear muffin-tin orbital (LMTO) approach where the modified Becke-Johnson (MBJ) semi-local exchange is used to accurately produce the band gaps, and the coherent potential approximation (CPA) is applied to deal with configurational average for the ternary alloys. The combined LMTO-MBJ-CPA approach allows one to simultaneously determine both the conduction band and valence band offsets of the heterojunctions. The calculated band gap data of the ZnO alloys scale as Eg = 3.35 + 2.33x and Eg = 3.36 − 2.33x + 1.77x2 for Zn1−xMgxO and Zn1−xCdxO, respectively, where x being the impurity concentration. These scaling as well as the composition dependent band offsets are quantitatively compared to the available experimental data. The capability of predicting the band parameters and band alignments of ZnO and its ternary alloys with the LMTO-CPA-MBJ approach indicate the promising application of this method in the design of emerging electronics and optoelectronics.

Zinc oxide (ZnO), having a direct band gap (E g ) of 3.37 eV and a high exciton binding energy of about 60 meV at room temperature 1,2 is a promising material for electronics and optoelectronics, including applications to solar cells 3 , light-emitting diodes 4,5 and ultraviolet micro-lasers 6 . In electronics technology, semiconductor heterojunctions are commonly used to provide potential barriers and/or quantum wells that tune and control carrier transport. Composition controlling techniques are the most widely applied method to alter the band gap of materials in the so called "band engineering" to build heterojunctions. For ZnO, composition doping of Mg and Cd atoms has been used to increase or decrease the band gap, respectively [7][8][9][10] .
It could be quite tedious and difficult to experimentally measure the band gaps of semiconductor alloys and band offsets of their heterojunctions over the entire doping concentration, therefore first principles theoretical calculations and predictions are of great fundamental interest as well as practical relevance. In particular, as nanoelectronic devices are reaching the sub-10 nm scale, atomistic first principles prediction of band parameters of semiconductor materials and heterojunctions is becoming very important in order to design and/or select new materials that have desired properties. However, it has been a serious challenge for existing first principles methods to accurately predict band information of semiconductor alloys and heterojunctions for several reasons. First of all, let's consider calculating the band offset of a heterojunction A 1−x B x /A where A is a semiconductor and x the doping concentration of some impurity specie B. If x is small as is usually the case, e.g. x = 0.1%, one has to calculate a system containing at least 1000 host atoms A in order to accommodate just a single impurity atom B. Namely the total number of atoms are very large when x is small which makes first principles analysis extremely difficult to do. Second, one has to perform a configurational average of the calculated results because the doped impurity atoms are randomly distributed in the sample, and the configurational average is much more computationally costly. Finally, it is well-known that the first principles methods of density functional theory (DFT) 11,12 with local-density approximation (LDA) [13][14][15] and generalized gradient approximation (GGA) 16,17 underestimate the band gaps of semiconductors and, without a correct band gap, the accuracy of the calculated band offset could be questionable. For these reasons, the composition dependent band offsets of many important semiconductors, including that of ZnO and its ternary alloys, have not been theoretically investigated to a satisfactory level.
During the past decades, considerable theoretical efforts have been devoted to solve the above band gap and configurational average issues. Among them the combined DFT approach of coherent potential approximation (CPA) 18 and modified Becke-Johnson (MBJ) semilocal exchange 19 under the linear muffin-tin orbital (LMTO) scheme 20,21 , has been successful in calculating the band information of many semiconductors, their alloy and heterojunctions. So far, the LMTO-CPA-MBJ approach has been applied to predict the band gaps of all group III-V semiconductors 22 , the alloy InGaN 23 , AlGaAs 24 , the group IV semiconductor alloys SiGeSn 25 , as well as the band offset of group III-V semiconductor heterojunctions GaAs/AlGaAs 24 . Quantitative comparison between the calculated results and measured data were very satisfactory. However, a more challenging and important semiconductor family -wurtzite group II-VI semiconductors, has not been investigated by the LMTO-CPA-MBJ approach yet.
In Together with the previously reported results on group III-V semiconductors [22][23][24] and group IV semiconductors 25 , we can confidently expect that the band parameters and band alignments of most conventional semiconductors can be well predicted from first principles by LMTO-CPA-MBJ approach.

Results and Discussions
Band structure of pure semiconductor ZnO. To carry out DFT calculations within the LMTO scheme, atomic sphere approximation (ASA) should be employed for space filling with placing atom and vacancy spheres at appropriate locations 20,21 . Different ASA scheme, i.e., different sphere positions and radii, will affect the calculated electronic structure. To verify our ASA scheme of the LMTO method, we calculated the band structure of pure ZnO crystal at the level of local LDA to compare with that obtained by the Vienna Ab initio Simulation Package (VASP) 26,27 . As shown in Fig. 1(a), the agreement is excellent -suggesting the appropriateness of our ASA scheme. Note that the calculated LDA band gap, though consistent with that reported in the literature 28,29 , is underestimated which is a well-known issue of LDA.
Having obtained the correct band structure of pure ZnO by LMTO at the LDA level, we further determine the c-values of the MBJ potential (See Methods Section) in order to produce the experimental E g , 3.37 eV, of pure ZnO. Figure 1(b) shows the MBJ result, and the opening of the E g as campared to our LDA result as well as the previously reported LDA E g of 0.74 eV 28,29 and GGA E g of 0.80 eV 30 , is evident. Till now, we have decided all the parameters for the LMTO-CPA-MBJ approach to produce the experimental band gap of pure ZnO, including the LMTO related ASA parameters for sphere positions and radii and the MBJ related parameter c-values for increasing the band gap of semiconductors. More complicated ASA scheme and more c-values for different spheres may produce much more accurate band parameters, we would note that our choice is a good compromise between simplicity and reasonable accuracy. Though a few parameters need to be pre-determined in our first principle calculations, it is acceptable since we are dealing a very challenging problem and these parameters can be easily obtained. In the further calculations of the band gap of semiconductor alloys and band offset of semiconductor heterojunctions, we will fix all the parameters obtained in the pure ZnO calculations.
Band gaps of ZnO ternary alloys. Having correctly determined the band gap of pure ZnO crystal, exactly the same ASA scheme and the MBJ semi-local exchange potential were used to calculate the electronic structures of ZnO alloys. Note that when ZnO crystal is doped with (substitutional) impurity atoms at random sites, translational symmetry is broken and momentum k is no-longer a good quantum number such that Bloch's theorem no longer exists. In our CPA calculations 18,31 , the spirit of the approach is to construct an effective medium by completing configurational average over the random disorder that restores the translational invariance, as such one can again speak of a "band structure" with bands that are broadened by impurity scattering. Figure 1(c,d) plots the calculated CPA band structure of ZnO alloy with 15% MgO and 15% CdO composition, respectively. From Fig. 1(c), we obtain E g = 3.70 eV that is 0.33 eV higher than that of pure ZnO. This is very close to the experimentally measured value of E g = 3.68 eV for Zn 0.85 Mg 0.15 O 10 . For Zn 0.85 Cd 0.15 O, the calculated band gap is E g = 3.07 eV, namely, 0.3 eV smaller than that of pure ZnO.
We have calculated E g of wurtzite ZnO alloys versus impurity concentration x up to x = 40%, for Zn 1−x M x O where M = Mg or Cd, as shown in Fig. 2. Beyond x = 40%, it is known experimentally 7,8 that ZnO alloy tends to form a cubic phase (which is not the focus of this work). For Zn 1−x Mg x O alloys, our calculated E g increases linearly with the MgO concentration as E g = 3.35 + 2.33x (the solid squares line in Fig. 2). This scaling is consistent with the experimental observations of E g = 3.4 + 2.3x in ref. 32, and E g = 3.37 + 2.51x in ref. 33. According to our calculations, E g of the wurtzite Zn 1−x Mg x O alloy can be tuned to 4.29 eV at x = 40%. Our fitted linear slope agrees outstandingly with previous calculated value of 2.03 by LDA 34 , though LDA gives underestimated band gaps. Moreover, if quadratic fitting is applied to our calculated data, the coefficient of the x 2 term gives the bowing parameter of 0.18 eV, which is close to the previous calculated bowing parameter of 0.44 35 . For the Zn 1−x Cd x O alloys, we found that E g changes with the Cd concentration as E g = 3.36 − 2.33x + 1.77x 2 which is nonlinear (the dots line line in Fig. 2), and the E g of the wurtzite Zn 1−x Cd x O alloy can be narrowed to 2.71 eV when x = 40%. Here, the coefficient of the x 2 term gives the bowing parameter of 1.77 eV, which agrees extremely well with the experimental value of 1.75 eV 36 . Note that previous calculated bowing parameter was considerably smaller, 1.21 eV in ref. 37  Band offsets of ZnO and its ternary alloys. Having obtained accurate E g for the ZnO alloys, a very important task is to predict the band offsets for the ZnO/Zn 1−x M x O heterojunctions. Since our method can accurately predict the band gaps of pure ZnO and its ternary alloys, both the conduction band offset (CBO) and the valence band offset (VBO) can be calculated in one shot. In comparison, for those calculations where E g was not correctly obtained (such as using LDA/GGA functional), the conduction band offset could not be predicted unless one amends E g after the valence band offset is calculated. Figure 3     Discussions. Based on our LMTO-CPA-MBJ calculations on ZnO and its alloys, a general procedure for predicting band gaps of semiconductor alloys and band offsets of heterojunctions is in order. (1) Choose a proper ASA scheme and determine the c-value of the MBJ semi-local exchange to obtain accurate band structures of the pure semiconductor; (2) calculate the band information of semiconductors alloys by CPA; (3) calculate the band offset of the semiconductor heterojunction with corrected determined band gaps, from which the valence and conduction band offset can be predicted in one shot. Since a major problem of semiconductor technology is to design materials having desired electronic structure, the theoretical procedure summarized here should be applicable to a wide range of research issues.
The combined LMTO-CPA-MBJ approach is firstly applied on the band gap calculation of group III-V semiconductor alloy In x Ga 1−x N, the results are in excellent agreement with the measured data for the entire range of 0 ≤ x ≤ 1 23 . Later on, the approach is implemented in NANODSIM software package 31    band gaps of group IV ternary alloys SiGeSn, in which three elements locate on the same site with certain probabilities and their average physical quantities were dealt with CPA. These previously reported results show that the LMTO-CPA-MBJ approach can accurately calculate the band information of group III-V and group IV semiconductors and their binary and ternary alloys and heterojunctions. In this work, we further applied this approach on a more challenging system -wurtzite group II-VI semiconductors, and the obtained results are quantitatively compared with the experimental measurements.
We used the effective medium CPA method under LMTO scheme to calculate the electronic structures of semiconductors and alloys, which does not capture any atomic relaxations. Considering that we are dealing a very challenging problem, which needs to solve the doping, configurational average, band gap underestimation and computational cost issues in one calculation, to this end, the combined LMTO-CPA-MBJ approach is a reasonably compromised method. Moreover, for many semiconductor alloys, their compositions have very close lattice parameters, making the structural change negligible. It could be expected that this approach can be used in the prediction of band parameters of wide range of semiconductors.

Summary
Using a state-of-the-art atomistic approach, we have calculated the band gap of wurtzite ZnO based group II-VI ternary alloys with Mg or Cd compositions, and the band offsets of ZnO and its ternary alloy heterojunctions. The calculated band gaps of the ZnO alloys scale as E g = 3.35 + 2.33x and E g = 3.36 − 2.33x + 1.77x 2 for Zn 1−x Mg x O and Zn 1−x Cd x O, respectively. Further calculations on the band offsets show that the pure ZnO formed type-I band alignment heterojunction with its Mg and Cd doped ternary alloys. The calculated band gaps of the alloys and band offset of the heterojunctions quantitatively agree with the experimental measurements for x < 40% where the ZnO and its Mg and Cd doped alloys are in the wurtzite phase. The success of the LMTO-CPA-MBJ on predicting the band gaps and band offsets of group II-VI semiconductor alloys and heterojunctions, together with its capability on calculating and predicting the band gaps and band offsets of group IV 25 and group III-V [22][23][24] semiconductors, make us can confidently conclude that the LMTO-CPA-MBJ approach can be widely used to predict the semiconductor band parameters and band alignments from first principles and will be very useful to the design of emerging electronics and optoelectronics.

Methods
Our calculations are based on the LMTO-CPA-MBJ self-consistent approach where DFT is carried out in the LMTO scheme with the atomic sphere approximation (ASA) 20,21 , as implemented in the NANODSIM software package 31 . For ASA, vacancy spheres were placed at appropriate locations for space filling [See Table 1 for details] 23 . In particular, the simulation cell is filled by slightly overlapped spheres within ASA, and the cell volume of the whole simulated system V cell should be equal to the total volume of the atom and vacancy spheres S spheres . With the sphere position of the wurtzite structure in Table 1, we optimized the sphere radii by reproducing the electronic band structures of pure ZnO crystals. Afterward, to deal with the impurity doped ZnO alloy, the statistical effective medium CPA theory 18 is applied which allows us to obtain the configurational averaged results without individually computing each atomic configuration.
We use the MBJ semi-local exchange that can accurately determine the band gap E g of the semiconductors 19 . Following the original paper 19 , the MBJ semi-local exchange potential has the following form, x MBJ x BR , , where subscript σ is spin index, ρ σ is the electron density for spin channel σ. The quantity t σ is the kinetic energy density and σ v r ( ) is the Becke-Roussel potential used in ref. 44. The above MBJ potential has two terms whose relative weight is given by a parameter c, which depends linearly on the square root of the average of ρ ρ ∇ / . According to many previous studies 19,[22][23][24][25]45,46 , the calculated E g increases monotonically with c. For simplicity, we only used different c-values for real atom spheres and vacancy spheres, respectively. While the c-values can be self-consistently calculated using the electron density, in our calculations we fixed it to be c = 1.75 and c = 1.13 for vacancy spheres and real atom spheres respectively, which gave an E g of ZnO in excellent agreement with the experimental value (3.37 eV). Having obtained the correct band gap of ZnO, the same values of the c-parameters were used in all the subsequent calculations of the alloys and heterojunctions.
The experimental lattice constants 47,48 of hexagonal wurtzite ZnO, a = 3.2495 Å 47 , c/a = 1.602 47 , and u = 0.382c 48 , were used in our calculations for pure ZnO and all the alloys. The primitive cell of the wurtzite structure was used to calculate the band structures and E g of ZnO and its alloys. To determine the band offset, (1120) supercell systems containing 8 unit cell of pure ZnO and 8 unit cell layers of ZnO alloy were used to calculate the potential profile along the heterojunction. A 10 × 10 × 10 k-mesh and a 6 × 6 × 3 k-mesh were used to sample the Brillouin Zone (BZ) for the primitive cells and the heterojunctions, respectively. The band gap values, as well as the energy position of the valence band maximum and conduction band minimum are read from the CPA band structure. The CPA band structure can intuitively trace the shape of the band structure with broadening due to impurity scattering, and provide more accurately and rich band information for the semiconductor and its alloys. The band offset values were calculated based on the electrostatistic potential profile of the heterojunction and the primitive cell, as more details in ref. 24.  Table 1. Positions of atomic spheres in the wurtzite structure. V T and V O denote the vacancy spheres at the tetrahedral center and octahedral center of the ZnO primitive cell, respectively. The optimized sphere radii of 1.1481 Å, 0.7654 Å, and 1.3023 Å are used for the real atom spheres, vacancies at the tetrahedral center and vacancies at the octahedral center, respectively.