- Email: [email protected]
GaN (0001) superlattice
Physics Letters A 379 (2015) 2384–2387
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
First-principles simulations of two dimensional electron gas near the interface of ZnO/GaN (0001) superlattice Jie Lei a,b , Da-Peng Zhu a , Ming-Chun Xu a , Shu-Jun Hu a,∗ a b
School of Physics and National Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong, 250100, PR China School of Science, Qilu University of Technology, Jinan, Shandong, 250353, PR China
a r t i c l e
i n f o
Article history: Received 16 April 2015 Received in revised form 11 July 2015 Accepted 22 July 2015 Available online 29 July 2015 Communicated by R. Wu Keywords: First-principles calculation Two dimensional gas Superlattice
a b s t r a c t By applying the on-site Coulomb interaction corrections on the anion:2p and the cation:3d electrons, we find that the GGA + U approach can completely compensate the underestimation of band gap of ZnO and GaN, two wide band gap semiconductors. Based on such approach, we investigate the electronic structure of ZnO/GaN (0001) heterostructure, particularly for the two dimensional electron gas formed near the polar interface. The polarization difference between ZnO and GaN induces the surface charge, resulting in the accumulation of band electrons on the N-polar interface. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Electronic states at the oxide interfaces exhibit various interesting properties, such as the magnetism, the high mobility, the quantum Hall effect, the superconductivity, etc. [1]. The novel electronic states are induced by the charge transfer effect [2] or the electrostatic boundary conditions [3], etc. Near the interface of an epitaxial heterostructure, the strong mismatch in electric polarization may induce a two dimensional electron gas (2DEG) [4], which is often accompanied by the observation of quantum Hall effect [5]. The heterostructures which possess the 2DEG have potential application in the high mobility transistor. ZnO/GaN heterostructures are promising candidates of optoelectronic devices [6]. Both ZnO and GaN hosts exhibit spontaneous polarization and piezoelectric properties [7]. The polarization mismatch between ZnO and GaN may induce a 2DEG at the interface of heterostructures [4]. The ZnO/GaN heterostructure is easy to achieve by utilizing the epitaxial technique, due to the small lattice mismatch (∼1.8%) between ZnO and GaN [6]. Determination of the electronic structures of ZnO/GaN interface is crucial to understand its electric and optical properties, and to further design novel optoelectronic devices. To our knowledge ZnO/GaN heterostructures have seldom been investigated by the first-principles calculations [8]. The primary obstacle is the underestimation of the band gaps of semiconductors by the theoretical
*
Corresponding author. E-mail address: [email protected] (S.-J. Hu).
http://dx.doi.org/10.1016/j.physleta.2015.07.033 0375-9601/© 2015 Elsevier B.V. All rights reserved.
method based on the density functional theory with the local density approximations (LDA) or generalized gradient approximations (GGA). Taking ZnO for instance, the first-principles calculations with GGA predicted a rather small band gap of 0.8 eV, in contrast to the experimentally detected 3.4 eV. Since the band offset of ZnO/GaN heterostructures is about 1 eV [9], the first-principles calculations can hardly be used to study the electronic properties of ZnO/GaN heterostructures without precise band gap predictions. In this Letter, we performed the first-principles calculations with GGA + U approximation to study the ZnO/GaN heterostructures. The on-site Coulomb interaction correction in terms of the Hubbard U has usually been restricted to the d shell electrons [10]. In the current study, we extend the correction to both the p and the d shells [11]. We find that the predicted band gap (as well as the predicted lattice parameters) strongly depends on the Hubbard U, thus it is possible to achieve the experimental band gaps by carefully selecting the proper values for Hubbard U. Based on such approach, we reveal the origin of 2DEG at the interface of ZnO/GaN heterostructure. The calculated results show the polarization discontinuity at the ZnO/GaN interfaces, which induces the charge transfer between two types of interfaces to compensate the surface charge. 2. Calculation details The first-principles calculations were performed by the Quantum-Espresso package [12] with a plane-wave basis set (energy cutoff of 60 Ryd) and ultra-soft pseudo-potentials [13] in GGA + U approximation for the exchange-correlation interactions [14]. The
J. Lei et al. / Physics Letters A 379 (2015) 2384–2387
Fig. 1. (Color online.) (a) Relation between the Hubbard U and the band gap (in eV) of ZnO. Black line indicates the values of U corresponding to the experimental band gap of 3.4 eV. (b) and (c) Band gap of ZnO predicted by the GGA + U (UO:2p = 7.4 eV and UZn:3d = 8.5 eV) and the GGA method, respectively. The energy of the valence band maximum is set to zero.
Monkhorst and Pack scheme of k-point sampling was used for integration over the Brillouin zone. The grids for the k-point sampling are 6 × 6 × 3 for primitive cell and 6 × 6 × 1 for ZnO/GaN superlattice. Structure optimization was performed until the stress of the supercell is smaller than 1 kbar and the forces of the ions is smaller than 2 × 10−3 Ryd/Bohr. The spontaneous polarization and the piezoelectric properties of ZnO and GaN were calculated based on the Berry-phase technique [15]. 3. Results First of all, we discuss the criteria of determining UO:2p and UZn:3d for the calculations of ZnO band structures. We find that by adjusting UO:2p and UZn:3d , GGA + U calculation can yield the experimental band gap, thus a series of UO:2p and UZn:3d are obtained. It is well known that the first-principles calculations always yield a small error (∼1%) for the lattice parameter predictions [16]. If the experimental lattice parameters are used to perform the first-principles calculations without any correction, the ZnO cell suffers from an undesired tension, which may induce errors for the calculation of the piezoelectric properties. We find that by altering the values of UO:2p and UZn:3d , the lattice parameters predicted by the GGA + U method approach to the experimental values (error less than 0.2%). Therefore, by combining the above two criteria, including the band gap and the lattice parameters, the values of UO:2p and UZn:3d can uniquely be determined. Fig. 1(a) shows the dependence of band gap of ZnO on the Hubbard U. The band gap is monotonically increased as increasing the values of UO:2p and UZn:3d . More significant influence of UO:2p is observed since the O:2p states dominate the valence band maximum (VBM). Based on the results, a series of UO:2p and UZn:3d which can yield the correct band gap is obtained, as shown by the black line in Fig. 1(a). Fig. 1(b) shows the band structure of ZnO calculated by the GGA + U method, where UO:2p = 7.4 eV and UZn:3d = 8.5 eV, which is compared by the GGA results shown in
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Fig. 2. (Color online.) (a) Relation between the Hubbard U and the stress (in kbar) of ZnO. Positive stress means the overestimation of lattice parameters, and vice verse. (b) Possible values of Hubbard U which can predict the correct lattice parameters (dashed line) and the band gap (solid line), derived from Fig. 1(a) and Fig. 2(a).
Fig. 1(c). In additional to the correct band gap, the incorporation of a large UZn:3d induces the deeper Zn:3d states (7 eV below the VBM), in good agreement with experiments [17], while the Zn:3d states are always overestimated in energy by the GGA calculations. To demonstrate the influence of Hubbard U on the lattice parameters, we calculate the in-plane stress of a ZnO primitive cell with the lattice parameters fixed at the experimental value, as shown in Fig. 2(a). The positive stress indicates the overestimation of the lattice parameters, and vice verse. At UO:2p = 0 eV and UZn:3d = 0 eV, the GGA method overestimates the lattice parameters of ZnO. We find that the UZn:3d has slight influence on the lattice parameters [18], while more precise results are obtained by increasing the value of UO:2p . Since the O:2p states dominate the bonding interaction between Zn and O atoms, the incorporation of UO:2p suppresses such interaction, and consequently leads smaller but more precise lattice parameters. Given the relation between the Hubbard U and two intrinsic properties of ZnO, the lattice parameters and the band gap, the values of UO:2p and UZn:3d are uniquely determined by the intersection of two lines shown in Fig. 2(b): UO:2p = 7.4 eV, and UZn:3d = 8.5 eV. Using the same approach we deduce the values of UN:2p and UGa:3d which are equal to 4.7 eV and 10.0 eV [19], respectively. In the following we study the electronic structures of (0001) oriented ZnO/GaN heterostructure, which is simulated by a superlattice composed of six ZnO bilayers and six GaN bilayers in each periodic unit along c direction. We focus on the heterostructure where GaN is grown on the (0001) oriented ZnO buffer layers. In the ZnO/GaN superlattice with the anion/cation ordering reserved, there are two types of interfaces: · · ·[–O–Zn–N–Ga–]· · · named by interface-I, and · · ·[–N–Ga–O–Zn–]· · · named by interface-II. The band structures of ZnO/GaN superlattice are shown in Fig. 3(a) with resolved projection of band on the corresponding bilayers (fat-band) as shown in Fig. 3(b). We find that the band projection along the (0001) direction shows a monotonic evolution in energy, which indicates the existence of a built-in electric field and the accumulation of charges at the interfaces. We calculate
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J. Lei et al. / Physics Letters A 379 (2015) 2384–2387
Fig. 4. Schematic illustration of (a) structure of ZnO/GaN superlattice; (b) accumulated charge at the interface, polarization ( P ) and built-in electric field (E); (c) band evolution induced by the built-in electric field.
Fig. 3. (Color online.) (a) Band structure of ZnO/GaN superlattice. Red dots indicate the band projections weighted by the contribution of each bilayer shown on the right. The vertical lines at zero energy are the Fermi level. (b) Crystal structure of ZnO/GaN superlattice and the integrated local density of states from E F − 1.0 eV to E F + 1.0 eV. Isovalue = 0.005 e/Bohr3 . Large pink balls stand for the zinc atoms and small red balls for the oxygen atoms. (c) Lowdin charge of each cation-anion bilayer. The values have been subtracted from the background of bulk values. The solid line are guide for the eyes. Table 1 Spontaneous polarization P SP , piezoelectric constants e 31 and e 33 of the bulk ZnO and GaN. Both the GGA and the GGA + U results are given. P SP
Approximation ZnO GaN
−0.022 −0.030 −0.022 −0.015
GGA GGA + U GGA GGA + U
e 31
e 33
−0.91 −0.95 −0.59 −0.78
1.17 1.22 0.78 1.02
the Lowdin charge of each bilayer, which is then subtracted from the background of bulk values, and finally the net charge is shown in Fig. 3(c). We note that the GaN bilayer close to the interface-I possesses additional 0.2 electron, which is mainly derived from the ZnO bilayer close to the interface-II. The integrated local density of states from E F − 1.0 eV to E F + 1.0 eV (E F represents the Fermi level) are shown in Fig. 3(b), indicating that the spatial distribution of the accumulated charges is confined to the region near the interface-I, in good agreement with the experiments [6]. We also find that the distribution of the accumulated charges slightly extends to the GaN side. The charge transfer between two types of interfaces is induced by the polarization ( P ) discontinuity of two constituents [4]. As SP shown in Fig. 4(b), the spontaneous polarization ( P ZnO ) of ZnO layer associated with the Zn-polarity points to the (0001) direction. The in-plane lattice parameter of bulk GaN is smaller than that of ZnO. Therefore the GaN layers in the heterostructure are under a tensile strain. The total polarization of GaN layer ( P GaN ) SP is the sum of the spontaneous polarizations ( P GaN ) and the piezoPE PE electric ( P GaN ) contributions. P GaN is given by: PE P GaN =2
a − a0 a0
e 31 − e 33
C 13 C 33
(1)
where e 31 and e 33 are two independent piezoelectric constants, a0 and a are the equilibrium and strained in-plane lattice parameters, C 13 and C 33 are the elastic constants [20], respectively. For the case of ZnO, there is only spontaneous polarization contribution. To evaluate the polarization difference between ZnO and GaN in the heterostructure, we calculated the spontaneous polarization and the piezoelectric constants e 31 and e 33 for the bulk ZnO and GaN. The results are given in Table 1. For the ZnO, the difference between the GGA and the GGA + U results is small, while the difference is more significant for the case of GaN. Since the GGA + U method predicts the precise band gaps and the lattice parameters, the following discussion is based on the GGA + U results. Based on equation (1) and the calculated piezoelectric constants, the total polarization of the strained GaN layer is calculated to be −0.070 C/m2 , which is much larger than that of ZnO. Such polarization discontinuity leads to a sheet charge at the interface-I. The sheet charge density is given by
ρ P = −∇ P
(2)
which predicts positive charges. The valence electrons thus tend to compensate the induced charge, resulting in the charge transfer between two interfaces, negative charges on interface-I and positive charges on interface-II, as shown in Fig. 4. The accumulated charge induce the build-in electric field, resulting in the reverse band evolution for ZnO and GaN. The determination of band offset of heterostructures is crucial to the understanding of their electric properties. Based on the band projection on two interfaces (Fig. 5), we deduce the valence band offset at the interface-I (interface-II) to be 1.58 (1.72) eV, in agreement with the result (1.3 eV) given by the transition level alignment of hydrogen [9]. The slight difference of band offset of two interfaces may be induced by the opposite charge accumulation. The experimentally determined band offset of ZnO/GaN heterostructure by the X-ray photoelectron spectroscopy is 1.0 eV [21]. The little difference may be due to the large Hubbard U corrections for Zn:3d and O:2p in our calculations. Resorting to an optimized pseudo potential may improve the results. For the conduction band offset, however, it is quite hard to determine the position of the conduction band minimum (CBM) due to the delocalized nature of 4s electrons of cation which dominates the conduction band. For the ZnO bilayer close to interface-II (see Fig. 2(a)), we obtain a band gap of 5.4 eV, which is much larger than the bulk value. The gap opening associated with the valence band narrowing is induced by the quantum confinement of superlattice [22]. To further verify the applicability of the current parameterized GGA + U approach, the band offsets between the bulk ZnO
J. Lei et al. / Physics Letters A 379 (2015) 2384–2387
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by calculating the spontaneous polarization and the piezoelectric constants. The band offset is deduced by the band projections on the bilayers near the interface. Our calculation results show agreement with the experiments but without significantly increasing the computational time compared with the hybrid functional method. Therefore our theoretical approach used in the current studies is promising in elucidating the electronic structures of wide band gap semiconductors and the heterostructures. Acknowledgements
Fig. 5. (Color online.) Valence band maximum (indicated by the arrows) of GaN and ZnO near the interfaces highlighted by the band projection on the corresponding bilayers. (a) Interface-I; (b) interface-II. Red dots: ZnO, green dots: GaN.
and GaN are calculated, and are compared with the experimental and theoretical results given by the VASP code with PBE0 hybrid functional [23]. The calculation details is briefly described in the following [24]. Firstly a slab model with the non-polar (11–20) surface orientation is constructed, including nine ZnO (or GaN) layers. Then we calculate the averaged electrostatic potential difference between the inside of the slab and vacuum region. Next, the average electrostatic potential between the inside of the slab and the bulk is aligned to determine the vacuum level of the bulk. Finally, the valence band maximum (VBM) and CBM with respect to the vacuum level is determined, respectively. The calculated band offset of CBM between ZnO and GaN is 1.16 eV and 0.55 eV based on the current GGA + U approach and PBE0 method, respectively. The experimental result of band offset can be deduced from the electron affinity of ZnO (4.5 eV) [25] and GaN (3.3 eV) [26]. The GGA + U results show good agreement with the experimental value (∼1.2 eV) as well as the reported theoretical results given by the DFT calculations with GW (∼1.2 eV) and HSE hybrid functional (∼1.1 eV) [27], while the PBE0 functional is hard to yield the coincident result. 4. Conclusions In conclusion, by applying the on-site Coulomb interaction corrections to the anion:2p and the cation:3d electrons, the firstprinciples calculations predict the precise band gaps and the lattice parameters for ZnO and GaN. Using such theoretical approach, we study the electronic structure of ZnO/GaN heterostructures, and characterize the 2DEG formed at the interface and the band offset between ZnO and GaN. The polarization discontinuity is revealed
This work was supported by National Basic Research Program of China (2013CB922303), Shandong Provincial Natural Science Foundation (ZR2013AM024) and National Natural Science Foundation of China (21273132). References [1] H.Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, Y. Tokura, Nat. Mater. 11 (2012) 103. [2] K.S. Takahashi, M. Kawasaki, Y. Tokura, Appl. Phys. Lett. 79 (2001) 1324. [3] A. Ohtomo, H.Y. Hwang, Nature (London) 427 (2004) 423. [4] O. Ambacher, J. Smart, J.R. Shealy, N.G. Weimann, K. Chu, M. Murphy, W.J. Schaff, L.F. Eastman, J. Appl. Phys. 85 (1999) 3222. [5] A. Tsukazaki, A. Ohtomo, T. Kita, Y. Ohno, H. Ohno, M. Kawasaki, Science 315 (2007) 1388. [6] D.C. Oh, T. Suzuki, J.J. Kim, H. Makino, T. Hanada, T. Yao, H.J. Ko, Appl. Phys. Lett. 87 (2005) 162104. [7] F. Bernardini, V. Fiorentini, D. Vanderbilt, Phys. Rev. B 56 (1997) R10024. [8] T. Nakayama, M.J. Murayama, J. Cryst. Growth 214 (2000) 299. [9] C.G. Van de Walle, J. Neugebauer, Nature (London) 423 (2003) 626. [10] B. Rakshit, P. Mahadevan, Phys. Rev. Lett. 107 (2011) 085508. [11] A. Walsh, J.L.F. Da Silva, S.-H. Wei, Phys. Rev. Lett. 100 (2008) 256401. [12] P. Paolo Giannozzi, et al., J. Phys. Condens. Matter 21 (2009) 395502. [13] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [14] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [15] R.D. King-Smith, D. Vanderbilt, Phys. Rev. B 47 (1993) 1651. [16] A. Janotti, D. Segev, C.G. Van de Walle, Phys. Rev. B 74 (2006) 045202. [17] C.L. Dong, C. Persson, L. Vayssieres, A. Augustsson, T. Schmitt, M. Mattesini, R. Ahuja, C.L. Chang, J.H. Guo, Phys. Rev. B 70 (2004) 195325. [18] X. Ma, Y. Wu, Y. Lv, Y. Zhu, J. Phys. Chem. C 117 (2013) 26029. [19] M. Magnuson, M. Mattesini, C. Höglund, J. Birch, L. Hultman, Phys. Rev. B 81 (2010) 085125. [20] A. Polian, M. Grimsditch, I. Grzegory, J. Appl. Phys. 79 (1996) 3343. [21] S.K. Hong, T. Hanada, H. Makino, Y. Chen, H.J. Ko, T. Yao, A. Tanaka, H. Sasaki, S. Sato, Appl. Phys. Lett. 78 (2001) 3349. [22] M.N. Huda, Y. Yan, S.-H. Wei, M.M. Al-Jassim, Phys. Rev. B 78 (2008) 195204. [23] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [24] V. Stevanovic, S. Lany, D.S. Ginley, W. Tumas, A. Zunger, PCCP, Phys. Chem. Chem. Phys. 16 (2014) 3706. [25] K. Jacobi, G. Zwicker, A. Gutmann, Surf. Sci. 141 (1984) 109. [26] C.I. Wu, A. Kahn, J. Appl. Phys. 86 (1999) 3209. [27] Y. Hinuma, A. Grüneis, G. Kresse, F. Oba, Phys. Rev. B 90 (2014) 155405.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
First-principles simulations of two dimensional electron gas near the interface of ZnO/GaN (0001) superlattice Jie Lei a,b , Da-Peng Zhu a , Ming-Chun Xu a , Shu-Jun Hu a,∗ a b
School of Physics and National Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong, 250100, PR China School of Science, Qilu University of Technology, Jinan, Shandong, 250353, PR China
a r t i c l e
i n f o
Article history: Received 16 April 2015 Received in revised form 11 July 2015 Accepted 22 July 2015 Available online 29 July 2015 Communicated by R. Wu Keywords: First-principles calculation Two dimensional gas Superlattice
a b s t r a c t By applying the on-site Coulomb interaction corrections on the anion:2p and the cation:3d electrons, we find that the GGA + U approach can completely compensate the underestimation of band gap of ZnO and GaN, two wide band gap semiconductors. Based on such approach, we investigate the electronic structure of ZnO/GaN (0001) heterostructure, particularly for the two dimensional electron gas formed near the polar interface. The polarization difference between ZnO and GaN induces the surface charge, resulting in the accumulation of band electrons on the N-polar interface. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Electronic states at the oxide interfaces exhibit various interesting properties, such as the magnetism, the high mobility, the quantum Hall effect, the superconductivity, etc. [1]. The novel electronic states are induced by the charge transfer effect [2] or the electrostatic boundary conditions [3], etc. Near the interface of an epitaxial heterostructure, the strong mismatch in electric polarization may induce a two dimensional electron gas (2DEG) [4], which is often accompanied by the observation of quantum Hall effect [5]. The heterostructures which possess the 2DEG have potential application in the high mobility transistor. ZnO/GaN heterostructures are promising candidates of optoelectronic devices [6]. Both ZnO and GaN hosts exhibit spontaneous polarization and piezoelectric properties [7]. The polarization mismatch between ZnO and GaN may induce a 2DEG at the interface of heterostructures [4]. The ZnO/GaN heterostructure is easy to achieve by utilizing the epitaxial technique, due to the small lattice mismatch (∼1.8%) between ZnO and GaN [6]. Determination of the electronic structures of ZnO/GaN interface is crucial to understand its electric and optical properties, and to further design novel optoelectronic devices. To our knowledge ZnO/GaN heterostructures have seldom been investigated by the first-principles calculations [8]. The primary obstacle is the underestimation of the band gaps of semiconductors by the theoretical
*
Corresponding author. E-mail address: [email protected] (S.-J. Hu).
http://dx.doi.org/10.1016/j.physleta.2015.07.033 0375-9601/© 2015 Elsevier B.V. All rights reserved.
method based on the density functional theory with the local density approximations (LDA) or generalized gradient approximations (GGA). Taking ZnO for instance, the first-principles calculations with GGA predicted a rather small band gap of 0.8 eV, in contrast to the experimentally detected 3.4 eV. Since the band offset of ZnO/GaN heterostructures is about 1 eV [9], the first-principles calculations can hardly be used to study the electronic properties of ZnO/GaN heterostructures without precise band gap predictions. In this Letter, we performed the first-principles calculations with GGA + U approximation to study the ZnO/GaN heterostructures. The on-site Coulomb interaction correction in terms of the Hubbard U has usually been restricted to the d shell electrons [10]. In the current study, we extend the correction to both the p and the d shells [11]. We find that the predicted band gap (as well as the predicted lattice parameters) strongly depends on the Hubbard U, thus it is possible to achieve the experimental band gaps by carefully selecting the proper values for Hubbard U. Based on such approach, we reveal the origin of 2DEG at the interface of ZnO/GaN heterostructure. The calculated results show the polarization discontinuity at the ZnO/GaN interfaces, which induces the charge transfer between two types of interfaces to compensate the surface charge. 2. Calculation details The first-principles calculations were performed by the Quantum-Espresso package [12] with a plane-wave basis set (energy cutoff of 60 Ryd) and ultra-soft pseudo-potentials [13] in GGA + U approximation for the exchange-correlation interactions [14]. The
J. Lei et al. / Physics Letters A 379 (2015) 2384–2387
Fig. 1. (Color online.) (a) Relation between the Hubbard U and the band gap (in eV) of ZnO. Black line indicates the values of U corresponding to the experimental band gap of 3.4 eV. (b) and (c) Band gap of ZnO predicted by the GGA + U (UO:2p = 7.4 eV and UZn:3d = 8.5 eV) and the GGA method, respectively. The energy of the valence band maximum is set to zero.
Monkhorst and Pack scheme of k-point sampling was used for integration over the Brillouin zone. The grids for the k-point sampling are 6 × 6 × 3 for primitive cell and 6 × 6 × 1 for ZnO/GaN superlattice. Structure optimization was performed until the stress of the supercell is smaller than 1 kbar and the forces of the ions is smaller than 2 × 10−3 Ryd/Bohr. The spontaneous polarization and the piezoelectric properties of ZnO and GaN were calculated based on the Berry-phase technique [15]. 3. Results First of all, we discuss the criteria of determining UO:2p and UZn:3d for the calculations of ZnO band structures. We find that by adjusting UO:2p and UZn:3d , GGA + U calculation can yield the experimental band gap, thus a series of UO:2p and UZn:3d are obtained. It is well known that the first-principles calculations always yield a small error (∼1%) for the lattice parameter predictions [16]. If the experimental lattice parameters are used to perform the first-principles calculations without any correction, the ZnO cell suffers from an undesired tension, which may induce errors for the calculation of the piezoelectric properties. We find that by altering the values of UO:2p and UZn:3d , the lattice parameters predicted by the GGA + U method approach to the experimental values (error less than 0.2%). Therefore, by combining the above two criteria, including the band gap and the lattice parameters, the values of UO:2p and UZn:3d can uniquely be determined. Fig. 1(a) shows the dependence of band gap of ZnO on the Hubbard U. The band gap is monotonically increased as increasing the values of UO:2p and UZn:3d . More significant influence of UO:2p is observed since the O:2p states dominate the valence band maximum (VBM). Based on the results, a series of UO:2p and UZn:3d which can yield the correct band gap is obtained, as shown by the black line in Fig. 1(a). Fig. 1(b) shows the band structure of ZnO calculated by the GGA + U method, where UO:2p = 7.4 eV and UZn:3d = 8.5 eV, which is compared by the GGA results shown in
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Fig. 2. (Color online.) (a) Relation between the Hubbard U and the stress (in kbar) of ZnO. Positive stress means the overestimation of lattice parameters, and vice verse. (b) Possible values of Hubbard U which can predict the correct lattice parameters (dashed line) and the band gap (solid line), derived from Fig. 1(a) and Fig. 2(a).
Fig. 1(c). In additional to the correct band gap, the incorporation of a large UZn:3d induces the deeper Zn:3d states (7 eV below the VBM), in good agreement with experiments [17], while the Zn:3d states are always overestimated in energy by the GGA calculations. To demonstrate the influence of Hubbard U on the lattice parameters, we calculate the in-plane stress of a ZnO primitive cell with the lattice parameters fixed at the experimental value, as shown in Fig. 2(a). The positive stress indicates the overestimation of the lattice parameters, and vice verse. At UO:2p = 0 eV and UZn:3d = 0 eV, the GGA method overestimates the lattice parameters of ZnO. We find that the UZn:3d has slight influence on the lattice parameters [18], while more precise results are obtained by increasing the value of UO:2p . Since the O:2p states dominate the bonding interaction between Zn and O atoms, the incorporation of UO:2p suppresses such interaction, and consequently leads smaller but more precise lattice parameters. Given the relation between the Hubbard U and two intrinsic properties of ZnO, the lattice parameters and the band gap, the values of UO:2p and UZn:3d are uniquely determined by the intersection of two lines shown in Fig. 2(b): UO:2p = 7.4 eV, and UZn:3d = 8.5 eV. Using the same approach we deduce the values of UN:2p and UGa:3d which are equal to 4.7 eV and 10.0 eV [19], respectively. In the following we study the electronic structures of (0001) oriented ZnO/GaN heterostructure, which is simulated by a superlattice composed of six ZnO bilayers and six GaN bilayers in each periodic unit along c direction. We focus on the heterostructure where GaN is grown on the (0001) oriented ZnO buffer layers. In the ZnO/GaN superlattice with the anion/cation ordering reserved, there are two types of interfaces: · · ·[–O–Zn–N–Ga–]· · · named by interface-I, and · · ·[–N–Ga–O–Zn–]· · · named by interface-II. The band structures of ZnO/GaN superlattice are shown in Fig. 3(a) with resolved projection of band on the corresponding bilayers (fat-band) as shown in Fig. 3(b). We find that the band projection along the (0001) direction shows a monotonic evolution in energy, which indicates the existence of a built-in electric field and the accumulation of charges at the interfaces. We calculate
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J. Lei et al. / Physics Letters A 379 (2015) 2384–2387
Fig. 4. Schematic illustration of (a) structure of ZnO/GaN superlattice; (b) accumulated charge at the interface, polarization ( P ) and built-in electric field (E); (c) band evolution induced by the built-in electric field.
Fig. 3. (Color online.) (a) Band structure of ZnO/GaN superlattice. Red dots indicate the band projections weighted by the contribution of each bilayer shown on the right. The vertical lines at zero energy are the Fermi level. (b) Crystal structure of ZnO/GaN superlattice and the integrated local density of states from E F − 1.0 eV to E F + 1.0 eV. Isovalue = 0.005 e/Bohr3 . Large pink balls stand for the zinc atoms and small red balls for the oxygen atoms. (c) Lowdin charge of each cation-anion bilayer. The values have been subtracted from the background of bulk values. The solid line are guide for the eyes. Table 1 Spontaneous polarization P SP , piezoelectric constants e 31 and e 33 of the bulk ZnO and GaN. Both the GGA and the GGA + U results are given. P SP
Approximation ZnO GaN
−0.022 −0.030 −0.022 −0.015
GGA GGA + U GGA GGA + U
e 31
e 33
−0.91 −0.95 −0.59 −0.78
1.17 1.22 0.78 1.02
the Lowdin charge of each bilayer, which is then subtracted from the background of bulk values, and finally the net charge is shown in Fig. 3(c). We note that the GaN bilayer close to the interface-I possesses additional 0.2 electron, which is mainly derived from the ZnO bilayer close to the interface-II. The integrated local density of states from E F − 1.0 eV to E F + 1.0 eV (E F represents the Fermi level) are shown in Fig. 3(b), indicating that the spatial distribution of the accumulated charges is confined to the region near the interface-I, in good agreement with the experiments [6]. We also find that the distribution of the accumulated charges slightly extends to the GaN side. The charge transfer between two types of interfaces is induced by the polarization ( P ) discontinuity of two constituents [4]. As SP shown in Fig. 4(b), the spontaneous polarization ( P ZnO ) of ZnO layer associated with the Zn-polarity points to the (0001) direction. The in-plane lattice parameter of bulk GaN is smaller than that of ZnO. Therefore the GaN layers in the heterostructure are under a tensile strain. The total polarization of GaN layer ( P GaN ) SP is the sum of the spontaneous polarizations ( P GaN ) and the piezoPE PE electric ( P GaN ) contributions. P GaN is given by: PE P GaN =2
a − a0 a0
e 31 − e 33
C 13 C 33
(1)
where e 31 and e 33 are two independent piezoelectric constants, a0 and a are the equilibrium and strained in-plane lattice parameters, C 13 and C 33 are the elastic constants [20], respectively. For the case of ZnO, there is only spontaneous polarization contribution. To evaluate the polarization difference between ZnO and GaN in the heterostructure, we calculated the spontaneous polarization and the piezoelectric constants e 31 and e 33 for the bulk ZnO and GaN. The results are given in Table 1. For the ZnO, the difference between the GGA and the GGA + U results is small, while the difference is more significant for the case of GaN. Since the GGA + U method predicts the precise band gaps and the lattice parameters, the following discussion is based on the GGA + U results. Based on equation (1) and the calculated piezoelectric constants, the total polarization of the strained GaN layer is calculated to be −0.070 C/m2 , which is much larger than that of ZnO. Such polarization discontinuity leads to a sheet charge at the interface-I. The sheet charge density is given by
ρ P = −∇ P
(2)
which predicts positive charges. The valence electrons thus tend to compensate the induced charge, resulting in the charge transfer between two interfaces, negative charges on interface-I and positive charges on interface-II, as shown in Fig. 4. The accumulated charge induce the build-in electric field, resulting in the reverse band evolution for ZnO and GaN. The determination of band offset of heterostructures is crucial to the understanding of their electric properties. Based on the band projection on two interfaces (Fig. 5), we deduce the valence band offset at the interface-I (interface-II) to be 1.58 (1.72) eV, in agreement with the result (1.3 eV) given by the transition level alignment of hydrogen [9]. The slight difference of band offset of two interfaces may be induced by the opposite charge accumulation. The experimentally determined band offset of ZnO/GaN heterostructure by the X-ray photoelectron spectroscopy is 1.0 eV [21]. The little difference may be due to the large Hubbard U corrections for Zn:3d and O:2p in our calculations. Resorting to an optimized pseudo potential may improve the results. For the conduction band offset, however, it is quite hard to determine the position of the conduction band minimum (CBM) due to the delocalized nature of 4s electrons of cation which dominates the conduction band. For the ZnO bilayer close to interface-II (see Fig. 2(a)), we obtain a band gap of 5.4 eV, which is much larger than the bulk value. The gap opening associated with the valence band narrowing is induced by the quantum confinement of superlattice [22]. To further verify the applicability of the current parameterized GGA + U approach, the band offsets between the bulk ZnO
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by calculating the spontaneous polarization and the piezoelectric constants. The band offset is deduced by the band projections on the bilayers near the interface. Our calculation results show agreement with the experiments but without significantly increasing the computational time compared with the hybrid functional method. Therefore our theoretical approach used in the current studies is promising in elucidating the electronic structures of wide band gap semiconductors and the heterostructures. Acknowledgements
Fig. 5. (Color online.) Valence band maximum (indicated by the arrows) of GaN and ZnO near the interfaces highlighted by the band projection on the corresponding bilayers. (a) Interface-I; (b) interface-II. Red dots: ZnO, green dots: GaN.
and GaN are calculated, and are compared with the experimental and theoretical results given by the VASP code with PBE0 hybrid functional [23]. The calculation details is briefly described in the following [24]. Firstly a slab model with the non-polar (11–20) surface orientation is constructed, including nine ZnO (or GaN) layers. Then we calculate the averaged electrostatic potential difference between the inside of the slab and vacuum region. Next, the average electrostatic potential between the inside of the slab and the bulk is aligned to determine the vacuum level of the bulk. Finally, the valence band maximum (VBM) and CBM with respect to the vacuum level is determined, respectively. The calculated band offset of CBM between ZnO and GaN is 1.16 eV and 0.55 eV based on the current GGA + U approach and PBE0 method, respectively. The experimental result of band offset can be deduced from the electron affinity of ZnO (4.5 eV) [25] and GaN (3.3 eV) [26]. The GGA + U results show good agreement with the experimental value (∼1.2 eV) as well as the reported theoretical results given by the DFT calculations with GW (∼1.2 eV) and HSE hybrid functional (∼1.1 eV) [27], while the PBE0 functional is hard to yield the coincident result. 4. Conclusions In conclusion, by applying the on-site Coulomb interaction corrections to the anion:2p and the cation:3d electrons, the firstprinciples calculations predict the precise band gaps and the lattice parameters for ZnO and GaN. Using such theoretical approach, we study the electronic structure of ZnO/GaN heterostructures, and characterize the 2DEG formed at the interface and the band offset between ZnO and GaN. The polarization discontinuity is revealed
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