Predicting band offset of lattice matched ZnO and BeCdO heterojunction from first principles

ABSTRACT Using first-principles approach, we calculated the band gaps of wurtzite Cd ternary alloy and the band offset of the lattice matched heterojunction, where the modified Becke–Johnson semi-local exchange was used to determine the band gap and the coherent potential approximation was applied to deal with doping effect in disordered alloys. The heterojunction was determined to have a type II band alignment, with valence and conduction band offset being 0.28 and 0.50 eV, respectively. The calculation approach and procedure demonstrated here can be used to predict the band offset of more lattice matched semiconductor heterojunctions. GRAPHICAL ABSTRACT IMPACT STATEMENT The band offsets of lattice matched heterojunction were predicted from first principles, where band gaps of semiconductors and configuration average of disorders were both accurately and efficiently calculated.


Introduction
For the design of electronic devices, the band alignment of heterostructure plays a vital role as the distribution of carriers determine the optical characteristics of the device [1,2]. If a semiconductor heterojunction A/B that formed by two different semiconductor compounds A and B has a type I band alignment, both electrons and holes are confined in the narrower band gap semiconductor and may recombine with each other more efficiently. Such heterojunctions are commonly used in LEDs or laser diodes. For instance, in ZnO-based semiconductor separation of the electron and hole is a key step in the generation of solar power. In nanometer architecture photovoltaic device, the charge separation is often facilitated by a heterojunction A/B with type II band alignment. For instance, in ZnO-based semiconductor technology, ZnO/ZnSe [13], ZnO/ZnS [14], ZnO/CdTe [15], ZnO/ZnFe 2 O 4 [16] heterojunctions are all determined to have type II band alignment and have been intensively investigated for solar cell applications. However, the lattice mismatch ratios in the above heterostructures are larger than 18%, and such a big lattice mismatch ratio, in principle, induces larger crystal mosaics [17,18], higher residual carrier concentration [19], and threading dislocations [20], which greatly hinder the optical performance of the photovoltaic devices. It is therefore very important to grow lattice matched heterostructures, and doping one part or both parts of the heterostructure is an effective way.
Considering that the lattice constant of a semiconductor can be simply adjusted by doping, it is clearly possible to form lattice matched heterojunction using pure ZnO and a ternary semiconductor alloy. Because the concentration of each composition is both sensitive to the lattice constant of the semiconductor alloy and the band alignment of the heterojunction, it is therefore very important and useful to develop a theoretical method which can predict the band alignment between semiconductor alloys, so that the composition of the semiconductor species and the corresponding concentration can be determined in advance to avoid costly and tedious experimental explorations. First-principles approaches are obviously the most promising method to make these predictions, whereas, the calculations of the band offset of semiconductor alloys from first principles have always been a serious challenge [21][22][23][24]. Firstly, the traditional first-principles methods usually underestimate the band gap of semiconductors within a reasonable calculation cost, resulting in the failure to predict the conduction band offset of the heterojunction. Some advanced methods such as GW [25] and hybrid functional [26] can yield correct band gap but require huge calculation cost which are hard to calculate heterojunctions involving around hundred atoms or more. Next, the calculations of semiconductor with impurities [27] also increase the calculation cost, especially when the concentration of one species is very small, because one needs to compute systems with a large number of host atoms to accommodate the impurities and to take configuration average over many calculations to obtain a more reliable physical result. Thirdly, transport property is crucial for heterojunction and related devices. One of the most widely used and standard methods to calculate the transport properties of mesoscopic systems from first principles is the so-called NEGF-DFT method which combines the non-equilibrium Green's function (NEGF) and density functional theory (DFT) to deal with the non-equilibrium statistics [28,29]. To calculate using NEGF-DFT method, only localized basis sets such as LMTO and linear combination of atomic orbitals (LCAO) are suitable, because plane-wave basis set will generate dense matrix which needs huge time to inverse when constructing the Green's function. To overcome the calculation challenges and further study the transport properties, we have proposed a theoretical method to calculate the composition dependent band offset of semiconductor heterojunctions [5,30], which uses modified Becke-Johnson (MBJ) semi-local exchange functional [31] to obtain accurate band gaps for semiconductors within reasonable calculation cost [32] and the coherent potential approximation (CPA) [33] to deal with the doping effect for semiconductor alloys and to overcome the prohibitively large computation required for performing configuration average. Previously, the method has been used to calculate the composition dependent band gaps and band offsets for group III-V [30,34], group IV [35], group II-VI [5] semiconductor alloys and their heterojunctions, and quantitative comparisons to the existing experimental data are made.
In this work, we carried out first-principles calculations within the CPA-MBJ approach to predict the band alignment of ZnO/Be 1−x Cd x O (x = 0.56) heterojunction, where the Be 1−x Cd x O alloy and ZnO have matched lattice constant. It is well accepted that the theoretical 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-principle prediction of band parameters of semiconductor heterostructures is becoming very important to design of new device architecture and select of new electronic materials that have desired properties for the next generation devices. The rest of the paper is organized as follows. In the next section, the calculation method and details are briefly discussed. Section 3 presents the results and Section 4 is a short summary.

Calculation methods and details
Our first-principles calculations are based on density functional theory (DFT), where the linear muffin-tin orbital (LMTO) is used with the atomic sphere approximation (ASA) [36], as implemented in Nanodsim software package [37]. In our calculations, the primitive cell of the wurtzite structure was used to calculate the band structures and band gaps (E g ) of the semiconductor compounds. The lattice constants of these pure hexagonal wurtzite structures are adopted from Refs. [38][39][40] and where x is the concentration Cd in Cd-doped wurtzite BeO. To calculate the band offset of ZnO and the ternary alloy, ZnO/BeCdO heterojunction with the lattice constant of pure ZnO is used to determine the potential profile across the interface. Clearly, the lattice constant of the Be 0.44 Cd 0.56 O ternary alloy equals to the lattice constant of the pure ZnO according to the Vegard law. Although CdO usually forms rock-salt structure, the wurtzite CdO is calculated to be an additional local minimum, and the total energy of both CdO structures are the same within the precision of the first-principles calculations [40]. Moreover, experiments also show that CdO can form wurtzite ternary alloys in a wide range of CdO concentration from 0 to 69% [42]. Therefore, in all the calculations of this work, we only consider wurtzite structure for the studied semiconductors. For ASA, empty spheres were placed at appropriate locations for space filling. The wurtzite structure can be made nearly close packed by adding two types of vacancy sites [ Figure 1(d)]. For technical details of the atomic sphere positions and radii, we refer interested readers to Ref. [34]. The MBJ semi-local exchange functional is used to accurately determine the band gap of semiconductors. Following the original paper [31], the MBJ semi-local exchange potential has the following form: where subscript σ denotes spin, and ρ σ the electron density for spin channel σ . The quantity t σ is the kinetic energy density, and v BR x,σ (r) the Becke-Roussel potential. The above MBJ potential has two terms whose relative weight is given by a parameter c . It was shown in Ref. [31] that the value of c depends linearly on the square root of the average of ∇ρ/ρ. The parameter c can be determined self-consistently as discussed in Refs. [31,43]. In our calculations the c values are 'local', namely, different c values are used for different real atom and vacancy spheres in ASA, and these values are pre-determined by fitting the accurate band gap of pure ZnO and BeO [44]. The c parameter of Cd is the same as that of Be. During the self-consistent iterations, Zn and Cd 3d electrons are treated as valence electrons. The band gap of semiconductor alloy can be obtained from the spectral function calculations. To more intuitively read the band gap information, we plot the spectral function as a function of both momentum k and energy E as where G r (E, k) is the disorder average of retarded Green's function, which can be carried out analytically in disordered bulk materials. In this way, the spectral function can trace like the band structure of the pure semiconductor as shown in the following section.

Results and discussions
The calculated band structures of wurtzite ZnO, BeO and CdO with band gap of 3.37, 10.60 and 2.30 eV, respectively, are shown in Figure 1. Compared with the band gaps from the local density approximation (LDA), and generalized gradient approximation (GGA), the opening of band gap is evident as listed in the Table 2, and our calculated MBJ band gaps are in good agreement with the existing experimental values. Furthermore, using the LMTO-CPA-MBJ method, we calculated band gap across the entire concentration range for wurtzite Be 1−x Cd x O alloy as plotted in Figure 2. Note that when BeO crystal is doped with impurity atoms  O-2p is enhanced and, therefore, the valence band maximum is risen. The energy of the Cd-5s is less than the energy of Be-2s, therefore, doping Cd in BeO will significantly reduce the conduction band minimum [51]. Moreover, from the inset of Figure 2, we can read that the band gap of the Be 0. 44 can be obtained with respect to the average of the electrostatic potential E es (A) or E es (B) in each bulk material. However, because the calculations on the two materials A and B are carried out independently, namely, the energies in A or B are not referred to the same reference point in energy, the energy relation between the two calculations are ill-defined. Therefore, the manipulation of adding or subtracting energies between A and B is of no any physical meaning.
To calculate the band offset of heterojunction A/B, we further carried out a calculation on the heterojunction A/B where each part A and B are long enough so that the electrostatic potential E es (A) or E es (B) in the center regions of both A and B are not affected by the interface. Because the energies in A or B in the heterojunction are referred to the same reference point in the calculation, we can obtain the valence band offset (VBO) from the difference between the valence band edge E v (A) and E v (B) inside the heterojunction. To calculate the VBO as defined by VBO ≡ E v (A) − E v (B) , we start from the following relationship: The physical meaning of the above equations is that the difference between the energy edge and the average electrostatic potential of a bulk material keeps unchanged. We can easily obtain from the above equations as follows: The difference of the first two square brackets correspond to the energy difference E v of the valence band edges for the two bulk semiconductors A and B that form the heterojunction, and the items in the third square brackets give the lineup of the potential V through the heterojunction. For the conduction band offset, the same calculation procedure can be applied. Therefore, the band offset can be calculated as [52][53][54] where Our calculated band offsets and the corresponding atomic structure of the heterojunction are shown in Figure 3, the band alignment of ZnO/Be 0.44 Cd 0.56 O heterojunction is type II with CBO = 0.50 eV and VBO = 0.28 eV. We note that the CPA does not include local relaxations so that the interface stays rigid in our calculations. How the microscopic details affect the band offset still needs a further study. To understand why type II band offset in ZnO/Be 0.44 Cd 0.56 O heterojunction was achieved, it must to recall the band alignment of ZnO/BeO heterojunction, in which the valence band maximum of BeO is of lower energy than that of ZnO, while the conduction band minimum of BeO is of higher energy than that of ZnO [55]. A type I energy alignment was formed. This is because the p-d (p orbital of O and d orbital of Zn) repulsion pushes up the valence band maximum, whereas Be have no active d electrons [51]. As Cd is doped in BeO, the valence band maxima of BeCdO shift upwards due to the repulsion effect between Cd-4d and O-2p, and with the increasing of Cd concentration in  Be 1−x Cd x O alloy, the repulsion effect between Cd-4d and O-2p is enhanced. Especially, for x = 0.56, in our model the valence band maximum of Be 0.44 Cd 0.56 O will higher than that of ZnO. As a result, a type II energy alignment was formed in ZnO/Be 0.44 Cd 0.56 O heterojunction. From the analysis, we can find that the band offsets of the heterojunction are strongly dependent on Cd composition, just as reported in the ZnO/ZnCdO heterojunction [5].
Besides using the above calculation method to accurately determine the band offset values, a more intuitive way to obtain the band alignment is to calculate the local projected density of states (LPDOS) of the heterojunction [56][57][58], through which we can obtain the band alignment of the heterojunction directly, and an intrinsic staggered band offset (type II band alignment) can be observed as shown in Figure 4. Clearly, the LPDOS indicated band offsets qualitatively agree well with our accurately determined band offset values.

Summary
We have predicted the band gaps across the entire concentration range for Be 1−x Cd x O and band offset of lattice matched ZnO/Be 0.44 Cd 0.56 O heterojunction using from first principles under LMTO scheme with CPA-MBJ approach. Our calculated results show that the heterojunction has a type II band alignment with the CBO and VBO being 0.50 and 0.28 eV, respectively. It can be expected that this approach can be used to predict more band information for semiconductor alloys and heterojunction.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by the National Natural Science Foundation of China (grant nos. 11504072, 11774217 and 11404273).