⇦Chinese Physics Letters, 2020, Vol. 37, No. 5, Article code 057101 Structural Transition from Ordered to Disordered of BeZnO$_2$ Alloy * Li-Xia Qin (秦丽霞)1†,**, Han-Pu Liang (梁汉普)1†, Rong-Li Jiang (蒋荣立)2 Affiliations 1School of Materials and Physics, China University of Mining and Technology, Xuzhou 221116 2School of Chemical Engineering, China University of Mining and Technology, Xuzhou 221116 Received 16 December 2019, online 25 April 2020 *Supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2017XKZD08).
†Li-Xia Qin and Han-Pu Liang contributed equally to this work.
**Corresponding author. Email: lixia@cumt.edu.cn Citation Text: Qin L X, Liang H P and Jiang R L 2020 Chin. Phys. Lett. 37 057101    Abstract Employing Monte Carlo simulations based on the cluster expansion, the special quasi-random structures and first-principles calculations, we systematically investigate the structure transition of BeZnO$_2$ alloys from the ordered to the disordered phase driven by the increased synthesis temperature, together with the solid-state phase diagram. It is found that by controlling the ordering parameter at the mixed sublattice, the band structure can vary continuously from a wide direct band gap of 4.61 eV for the fully ordered structure to a relatively narrow direct band gap of 3.60 eV for the fully disordered structure. Therefore, a better optical performance could be achieved simply by controlling the synthesis temperature, which determines the ordering parameters and thus the band gaps. DOI:10.1088/0256-307X/37/5/057101 PACS:71.23.-k, 78.30.Fs, 64.60.Cn © 2020 Chinese Physics Society Article Text Zinc oxide (ZnO) has become a widely used semiconductor for applications in ultraviolet (UV) optoelectronic devices because of its wide direct band gap (3.37 eV) and high exciton binding energy (60 meV).[1,2] ZnO-based devices can utilize quantum well structures to optimize device performance.[3] The UV laser diodes based on ZnO/BeZnO films and ZnO/BeZnO-based UV light emitting diodes (LEDs) with an active layer region composed of ZnO/BeZnO quantum wells (QWs) have been fabricated.[4] Ground-state BeO shares the same wurtzite $P6_3mc$ structure as ZnO and has a much wider band gap of 10.6 eV. Therefore, the band gap is expected to vary in a widely tunable range from 3.40 to 10.60 eV by alloying ZnO and BeO.[5] This suggests that the performance of ZnO-based devices, for example, the UV photodetectors, can be enhanced practicably by adjusting the band gap of Be$_x$Zn$_{1-x}$O alloys to satisfy desired technological applications. Meanwhile, although this alloy system has been experimentally synthesized, the solid-state phase diagram and the temperature dependence of optical absorption are still unknown. To effectively tune the band gap of Be$_x$Zn$_{1-x}$O alloys, conventional methods, including controlling alloying concentration, doping or strain engineering, have been widely reported.[6–14] However, the detailed effects of temperature on the band gap of this system have not been fully revealed so far. It is well-known that the ordering in semiconductor alloys would impose significant influences on their electronic band structures.[15–17] For example, the band gap of CuGaSe$_2$ can vary from 0.23 eV for the fully disordered sublattice to 1.64 eV for the fully ordered sublattice,[18] and that of Cs$_2$AgBiBr$_6$ from 0.44 to 1.93 eV.[19] As the synthesis temperature increases in experiments, the structure ordering would be destroyed and the disorder may be introduced into the ordered crystal structure subcutaneously. To the best of our knowledge, the ordered-to-disordered transition and their influence on the band structure have not been investigated thoroughly for Be$_x$Zn$_{1-x}$O alloys. Another aim is to point out the details of structure evolution and the quantitative relationship between disorder and temperature. In this work, we focus on the fully ordered, partially disordered and fully disordered alloys of BeZnO$_2$ based on the Monte Carlo (MC) simulations, which only requires a cluster expansion as an input, together with the special quasirandom structures (SQSs). The configurational disorder in alloys at specified temperatures is simulated by the MC methods with the energy evaluated using a cluster expansion effective Hamiltonian,[20–22] which was proposed to describe the order-disorder phenomena when the underlying structure is known. Our investigations reveal how the alloy energy and the ordering parameters vary quantitatively with respect to temperature, which reveals how the band structure evolves with the synthesis temperature in detail. The order-disorder transition of BeZnO$_2$ alloy was simulated using the cluster expansion approach as implemented in the $emc2$ utility of the Alloy Theoretic Automated Toolkit (ATAT) code.[23] The cluster expansion coefficients were fitted to total energies calculated using the PBEsol functional. The equilibrium structures of the solid solution at high temperatures were determined by the MC simulations using a supercell of 461464 atoms. The fully disordered solid solution was mimicked by the SQS in the wurtzite supercell of 64 atoms. The configuration was generated by the $mcsqs$ utility in the ATAT,[23,24] and the perfect matchings of atomic correlation functions were achieved. The partially disordered structures were constructed using the approach similar to that in generating SQS by finding a supercell structure with its relevant atomic correlation functions closest to the target partially disordered atomic correlation functions. The structural optimization, total energy, and electronic structure of the fully ordered, partially disordered and fully disordered structures are calculated using the density functional theory (DFT) within the local density approximation.[25,26] The projector augmented wave method[27,28] implemented in the VASP package[29–31] was employed. The plane wave energy cutoff is set to 500 eV and the structures are relaxed until the Hellmann–Feynman forces are less than 0.05 eV/Å. The $4\times4\times4$ and $2\times2\times2 \varGamma$-centered $K$-point meshes are adopted for the structure optimizations and the phonon and band-structure calculations, respectively. The electronic band structures were calculated using the standard Heyd–Scuseria–Ernzerhof hybrid functional (HSE06)[32] for the equilibrium lattice constants. The phonon-dispersion calculations were performed in the Phonopy package.[33]
cpl-37-5-057101-fig1.png
Fig. 1. Profiles of (a) the unit cell and (b) the $4\times4\times3$ supercell for the fully ordered BeZnO$_2$. The green, gray and red balls represent the Be, Zn and O atoms, respectively. The ground-state phonon dispersion curves in the whole Brillouin zone are plotted in panel (c).
cpl-37-5-057101-fig2.png
Fig. 2. HSE06 band structure and density of states of the fully ordered BeZnO$_2$. The Fermi level is set to 0.0, marked with red dashed lines.
The unit cell of the ordered structure of BeZnO$_2$ shares the hexagonal wurtzite symmetry, which contains two O, one Be and one Zn atoms. The structure details are shown in Fig. 1(a). In order to examine its structural stability, a $4\times4\times3$ supercell, as shown in Fig. 1(b), is constructed to calculate the phonon dispersion by finite difference methods. Figure 1(c) displays that there are no imaginary modes appearing in the calculated phonon dispersion curves in the whole Brillouin zone (BZ), thus this phase can remain dynamically stable. Figure 2 shows the HSE06 band structure of the ordered wurtzite BeZnO$_2$ along the high-symmetry points in the BZ, which remains a direct band gap with the valence band maximum (VBM) and the conduction band minimum (CBM) located at the $\varGamma$ point. The CBM and VBM states are mainly occupied by the O $2p$ orbital and the $3d$-orbital binding energy of Zn atom is $\sim $7 eV. The well-known $p$–$d$ coupling,[34–36] existing in ZnO instead in BeO, would push the VBM up and thus reduce the band gap, therefore the band gap of ZnO is much smaller than that of BeO. The increment of Be composition in the alloy would weaken such a repulsive effect and thus enhance the band gap. As a result, the band gap of 4.61 eV of BeZnO$_2$ is larger than that of ZnO, and still smaller than that of BeO. To investigate the ordered-to-disordered phase transition of BeZnO$_2$, it is essential to establish the relationship between the excess energy and the ordering parameter. The cluster expansion is a generalization of the well-known Ising Hamiltonian and parameterizes the energy per atom as a polynomial in the occupation variables,[22,37–39] $$ E(\sigma)=\!\sum_{\alpha}m_{\alpha}J_{\alpha}\Big\langle\mathop {\prod}\limits_{i\in{\alpha}'}{\sigma_i}\Big\rangle= E_{\rm R} +\! \sum_{km}J_{km}\overline{\varPi}_{km},~~ \tag {1} $$ where $m_{\alpha}$ indicates the number of clusters that are equivalent by symmetry to a cluster $\alpha$ consisting of a set of sites $i$. $J_{\alpha}$ ($J_{km}$) is the effective cluster interaction (ECI) coefficients, where the $k$ represents the number of atoms in the cluster such as pairs ($k=2$), triangles ($k=3$), tetragons ($k=4$), and $m$ represents the $m$th-neighbor distance. The averaged atomic correlation functions $\overline{\varPi}_{km}$ of a cluster ($k$, $m$) is the product of the spin-like occupation variable $\sigma_i$ of each site $i$, which, in the Ising lattice models, is +1 if the $i$ cation site is occupied by a Zn and $-1$ if by a Be atom. The $\overline{\varPi}_{21}=-1$ for a fully ordered phase and 0 for a fully disordered phase. $E_{\rm R}$ is the total energy of the fully random alloy. The cluster expansion presents an extremely concise and practical way to model the configurational dependence of an alloy's energy.[39] An accuracy that is sufficient for phase-diagram calculations can be achieved by keeping only clusters $\alpha$ that are relatively compact, e.g., short-ranged pairs or small triplets. The real advantage is that the unknown $J_{km}$ can be obtained by fitting them to the energy of a relatively small number of configurations through first-principles calculations, which is the key to determine the cluster expansion. Thereby, the framework can be extended to other configurations to make the energy of an alloy available even in the absence of experimental data. The fitting performed by the ATAT package reveals that $J_{21}= 821.3$ meV and $J_{22}=-17.6$ meV. This is consistent with the fact that the atomic interactions are relatively short ranged and reveals that the interactions of the nearest-neighbored pairs play a dominant role in the energy and phase transition. Therefore, the $\overline{\varPi}_{km}$ can be safely restricted to the pairs up to the second neighbors. Equation (1) is simplified to $$ E(\sigma) \approx E_{\rm R} + J_{21}\overline{\varPi}_{21} + J_{22}\overline{\varPi}_{22}.~~ \tag {2} $$ At the same time, the $\overline{\varPi}_{21}$, varying from 0 for a fully disordered phase to $-1$ for the wurtzite structure, can be treated as an ordering parameter to quantitatively describe the structure disorder induced by the increased synthesis temperature, which was previously observed in the cases of CuGaSe$_2$ and Cs$_2$AgBiBr$_6$ alloys.[18,19] Once the cluster expansion is known, the statistical mechanical techniques of MC simulations are used to simulate the solid-state phase diagrams by providing the free energy.
cpl-37-5-057101-fig3.png
Fig. 3. (a) Monte Carlo simulation of the excess energy as a function of temperature and (b) the corresponding averaged atomic correlation functions of pairs up to the second neighbor ($\overline{\varPi}_{21}$ and $\overline{\varPi}_{22}$).
A large supercell containing 461464 atoms was constructed for the Monte Carlo simulations at various temperatures to reveal the ordered-disordered phase transition in details. Figure 3 shows the excessive energy per site and the corresponding averaged atomic correlation functions of pairs up to the second neighbor ($\overline{\varPi}_{21}$ and $\overline{\varPi}_{22}$) as a function of temperature. As temperature increases from 0 to 400 K, the energy per cation site and $\overline{\varPi}_{2m}$ almost remain unchanged, thus the phase remains ordered under 400 K. Then the energy rises slowly as temperature further increases up to 800 K. However, in the range from 800 to 1500 K, the energy rises sharply, accompanied with the abrupt changes of $\overline{\varPi}_{2\,m}$, which suggests the ordered-to-disordered phase transition. As temperature further increases after the phase transition, the energy tends to reach the convergence and the $\overline{\varPi}_{21}$ and $\overline{\varPi}_{22}$ come to approach zero, the value of the fully disordered phase.
cpl-37-5-057101-fig4.png
Fig. 4. Structure models of (a) the partially disordered and (b) the fully disordered structures.
A partially disordered structure can be simulated by the SQS method through controlling the ordering parameter $\overline{\varPi}_{21}$ to satisfy a desired value. The order-disorder phase diagram suggests that the partially disordered structure can be obtained by quenching above 400 K and more disorder is introduced by higher temperature. We constructed a partially disordered supercell of 64 atoms sharing a wurtzite symmetry and focused on $\overline{\varPi}_{21}=-0.703$ with the corresponding temperature of 1100 K. We also adopted the SQS method to describe the fully disordered alloy, in which a small finite supercell is generated so that its averaged atomic correlation function $\overline{\varPi}_{km}$ is closest to the targeted correlation function $(2x -1)^k$, where $x$ is the alloy composition. Thus for the fully disordered BeZnO$_2$, the $\overline{\varPi}_{21}$ and $\overline{\varPi}_{22}$ are both equal to zero, whereas for the fully ordered structure, $\overline{\varPi}_{21}=-1$ and $\overline{\varPi}_{22}=1$. This is because of the different arrangements of atoms in the pairs. For example, in the ordered structure, the closest pair clusters are composed of different atomic pairs. The perfect matching is achieved for desired $\overline{\varPi}_{21}$ after several iterations for the partially disordered structure, meanwhile the $\overline{\varPi}_{22}$ and $\overline{\varPi}_{22}$ are as the same as those of the fully disordered solid solution. Figure 4 displays the structure models to simulate the partially and fully disordered BeZnO$_2$, respectively.
cpl-37-5-057101-fig5.png
Fig. 5. HSE06 band structures and density of states of (a) the partially disordered and (b) the fully disordered BeZnO$_2$. The Fermi level is set to 0.0, which is marked by the dashed lines.
cpl-37-5-057101-fig6.png
Fig. 6. Band alignments of the fully ordered, partially disordered, and fully disordered BeZnO$_2$.
To study the effects of disorder on the electronic properties, the band structures and also the density of states of the partially and fully disordered BeZnO$_2$ are plotted together in Fig. 5. The band structures remain direct with the VBM and CBM located at the $\varGamma$ point. It is noteworthy that when the disorder is enhanced by increasing temperature, the O $2p$ orbital contributes more to the VBM states compared to the fully ordered structure. This is consistent with the fact that as the $\overline{\varPi}_{21}$ increases from $-1$ to 0, the band gap is reduced to 3.60 eV of the fully disordered phase from 4.61 eV of the fully ordered phase. The partially disordered phase, related to the synthesis temperature of 1100 K, displays the similar band structure with the band gap of 4.04 eV to the fully ordered phase, sharing the similar electronic states near the VBM and CBM. To investigate the optical absorption of BeZnO$_2$ alloy at different synthesis temperatures in details, the band alignment of the fully disordered, partially disordered, and ordered phases is plotted in Fig. 6. The valence-band offset $\Delta{E_{\rm v}}$ is calculated following the same procedure as in the core-level photoemission measurement.[40,41] The conduction-band offsets $\Delta{E_{\rm c}}$ are described by adding the HSE06 band gaps to the valence-band offsets. When the degree of disorder increases from the fully ordered phase to the fully disordered phase, the VBM rises by 0.36 eV, while the CBM drops by 0.65 eV. For the partially disordered phase at $\sim $1100 K, the increment of VBM is 0.17 eV, the decrement of CBM is 0.40 eV, with respect to the fully ordered phase. Therefore, it is practically feasible to adjust the band gap according to a specified technological application by introducing disorder into the semiconductor alloy via precisely controlling the synthesis temperature. In summary, we have systematically investigated the solid-state phase diagram and the temperature dependence of optical absorption for BeZnO$_2$ alloy using the Monte Carlo simulations together with the special quasirandom structures, and the first-principles calculations. We show that by introducing disorder to the cation sites, the direct band gap can be tuned continuously from 4.61 eV of the fully ordered phase to 3.60 eV of the fully disordered phase by controlling the synthesis temperature, in order to facilitate light absorption and emission in the ultraviolet region. Our study points to new possibilities of band gap engineering in BeZnO$_2$ alloy for potential technological applications. We are grateful to the Advanced Analysis and Computation Center of CUMT for the award of CPU hours to accomplish this work. 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