Calculation of the Change in the Madelung Energy of Doped ZnO

Transcription

1 CHINESE JOURNAL OF PHYSICS VOL. 48, NO. 6 DECEMBER 010 Calculation of the Change in the Madelung Energy of Doped ZnO Jing Wen, 1, Chunying Zuo, 1 and Cheng Zhong 1 College of Arts and Science, Heilongjiang Bayi Agricultural University, Daqing , People s Republic of China College of Chemistry and Molecular Sciences of Wuhan University, Wuhan 43007, People s Republic of China (Received April 16, 010) One of the main mechanisms leading to doping difficulty is the low dopant solubility in realizing p-type ZnO. The calculation of the impurity formation energy is widely accepted for investigating doping methods. Actually the calculation of the change in the Madelung energy of doped ZnO can also be taken as a guide for investigating the doping methods and impurity solubility, due to its larger proportion of ionic bond to the chemical bond. But we found that more than one value can be obtained during the calculation because a conditionally convergent series must be considered. Firstly, a symmetry rule is introduced especially in this paper to calculate the Madelung constant by considering the NaCl-type, CsCl-type, and ZnO-type crystal lattices. The advantage of this method is that the series can converge quickly using the symmetry rule without changing the form of the series. Then we calculate the difference in the Madelung energy between pure ZnO and ZnO doped by C and C-Ga, respectively, with the help of the method mentioned above. The results show that the C-Ca co-doping method is better than the C doping method from the stability point of view, which can effectively enhance the C impurity solubility. The method can be used to calculate this type of conditionally convergent series, whereby the arbitrary results which can come from different arrangements of the terms during the calculation can be avoided. PACS numbers: Ah, Lt, 61.7.uj I. INTRODUCTION Doping of compound semiconductors is the basis of all semiconductor heterostructures and all optoelectronic devices. No doubt it is important to know the properties of a semiconductor doped with a certain concentration of impurity, especially the stability of the lattice [1]. Generally the binding and stability of the lattice of a compound semiconductor are determined by the cohesive energy contributed by ionic bonds and covalent bonds. The Madelung energy [] coming from the contribution of the ionic bonds is an important part of the cohesive energy of a compound semiconductor. Therefore, the calculation of the change in the Madelung energy of the doped compound semiconductor can be considered as an important result for analyzing the stability of their structures. Recently, extensive research has been done on ZnO, because of its potential for various short-wavelength optoelectronic devices and many other applications [3 6]. ZnO is a group II-VI compound semiconductor with a wide direct band gap of 3.37 ev at room temperature, exciton binding energy of 60 mev, and a hexagonal wurtzite structure of space group P63mc. For the purpose of replacing GaN as a material for short wavelength light emitting devices and lasers, intensive attention has been focused on p-type ZnO [7, 8]. It is well c 010 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 VOL. 48 JING WEN AND CHENG ZHONG 845 known that the shallow acceptor level cannot be obtained easily for ZnO, because it is naturally an n-type semiconductor, due to a large number of native defects [9 11]. Therefore, many doping and co-doping methods have been proposed in theory and in experiments to achieve reliable p-type ZnO [1 14]. There are two main reasons for the difficulty in attaining p-type ZnO. One is low dopant solubility; the other is a high defect ionization energy. The calculation of the impurity formation energy is generally accepted for investigating the doping methods. But the calculation of the change in the Madelung energy of the doped ZnO can be also taken as guidance for investigating the doping methods, due to its larger proportion of ionic bond to the chemical bond [8, 15]. The Phillips ionicity of ZnO is 0.616, so the Madelung energy is the main contribution to its cohesive energy. An increase of the Madelung energy of doped ZnO can make the structure of the lattice unstable and decrease the dopant solubility. The Madelung energy comes from the ion-ion interaction and involves the summation of the Coulomb potentials between different ions. Because of the long-range force and the interactions between a large numbers of ions, its calculation presents two problems. One problem is the slow convergence of the sum over the repetitions of the central cell. The other problem is that a conditionally convergent series must be considered during the calculation. The Riemann series theorem [16] says that any conditionally convergent series may be summed to give any desired value by a suitable re-arrangement of the terms. Therefore, many methods have been introduced to evaluate the sum [17 19]. For example, the Ewald method [0, 1] or the Lekner [] method, which require a sum of many terms involving trigonometric functions and the complementary error function or Bessel functions instead of the direct summation, can make the sum converge quickly. Here we introduce a symmetry rule for calculating the Madelung constant by considering the NaCl-type, CsCltype, and ZnO-type crystal lattices specially. The advantage of this method is that the series can converge quickly by using the symmetry rule without transforming the form of the series. The results can be viewed as the principal values of the potential [, 3]. Then we attempt to realize p-type ZnO doped by C and C-Ga. Using the techniques described in Refs. [4 8], we present our computation results for the effective ionic charges of the doped ZnO. Making use of these results, we calculate the difference in the Madelung energy between pure and doped ZnO. The calculation involves another series, so we introduce a reasonable arrangement of the terms for getting a quickly convergent result with the help of the symmetry rule mentioned above. II. A METHOD FOR CALCULATING THE MADELUNG CONSTANT The Madelung constant is determined only by the structure of an ionic crystal. Its calculation is one of the fundamental problems in the theory of solids. In this calculation, the basic assumption is that the solid can be considered as a system of positive and negative ions. Here we provide a detailed calculation of the Madelung constants by considering the NaCl-type, CsCl-type, and ZnO-type crystal lattices specially.

3 846 CALCULATION OF THE CHANGE IN THE MADELUNG ENERGY... VOL. 48 FIG. 1: Schematic diagram of the affine coordinates and a minimum unit defined here for the NaCl-type crystal lattice. II-1. The Madelung constant of a NaCl-type crystal lattice We first consider the NaCl-type crystal lattice with N anions and N cations located at the lattice points. The ion-ion interaction contributions to the total energy of this type of lattice called the Madelung energy is U = 1 i j q i q j r i r j, (1) where q i, q j are ionic charges located at r i, r j. Here we divide the total energy calculated by two, as each ion will be counted twice. It is convenient to introduce a quantity α ij such that r i r j = α ij ρ, where ρ is the nearest-neighbor equilibrium distance between neighboring ions in the crystal. Then Eq. (1) becomes U = N q ρ j ± f ij, () where f ij = α 1 ij. The sum j ±f ij including all ions except i = j for avoiding an infinity is called the Madelung constant. Here the condition q i = q j = q has been used. We have supposed that all lattice points are equivalent. Fig. 1 shows the affine coordinates with a Na + at the origin. The unit vectors of the three axes are labeled as a 1, a, a 3, which are perpendicular to each other and satisfy the condition a 1 = a = a 3 = ρ. The contribution of the jth ion to the Madelung constant in these coordinates is ±f(n 1,n,n 3 ) = ±1 n 1 + n +, (3) n 3 where (n 1,n,n 3 ) are the jth ion s coordinates. Here we have set the subscript i of f ij as the origin and used the jth ion s coordinates instead of the subscript j. We consider a type of cell including 4 NaCl ion pairs shown in Fig. 1. The contribution of each ion pair of the

4 VOL. 48 JING WEN AND CHENG ZHONG 847 jth cell to the Madelung constant can be written as F 1 = f(n 1,n,n 3 ) + f(n 1,n,n 3 + 1), F = f(n 1 + 1,n + 1,n 3 ) + f(n 1 + 1,n + 1,n 3 + 1), F 3 = f(n 1 + 1,n,n 3 ) f(n 1 + 1,n,n 3 + 1), F 4 = f(n 1,n + 1,n 3 ) f(n 1,n + 1,n 3 + 1). (4) The order of summation is arranged as M c (N) = (F 1 + F + F 3 + F 4 ), (5) where F 1 means the summation of F 1 should exclude f(0,0,0), N is an integer. The ion pairs in a cell have a symmetry structure. The contributions to the sum M c coming from the anions and cations are proportional under this arrangement of the terms. The Madelung constant M c can quickly converge to the result after N = by the direct summation of Eq. (5). Some other arrangements of the terms may lead to the disproportion in contributions of the two types of ions, which can make the sum converge slowly and yield different results. II-. The Madelung constant of a CsCl-type crystal lattice We imitate the steps above by constructing the affine coordinates with a Cs + at the origin. The unit vectors are perpendicular to each other and satisfy the condition a 1 = a = a 3 = αρ, where α = 4 3. The contribution of the jth ion to the Madelung constant in these coordinates is ±f(n 1,n,n 3 ) = ±1 αn 1 + αn +. (6) αn 3 We take a type of cell including one CsCl ion pair as the minimum contributing unit to the Madelung constant. First we arrange the order of summation just in accordance with the sequence of the Cl shown in Fig. a. The sum can be written as M 1 c (N) = n + 1 n + 1 n + 1 ( [f (n 1,n,n 3 ) f n 1 + 1,n + 1,n )], (7) n 1 = N 1 n = N 1 n 3 = N 1 where f means the summation should exclude f(0,0,0). But this arrangement decreases the contributions of Cs +. Then we re-arrange the order of summation just in accordance with the sequence of the Cs + shown in Fig. b. The sum becomes M c (N) = [ ( f n 1 + 1,n + 1,n ) ] f (n 1,n,n 3 ). (8)

5 848 CALCULATION OF THE CHANGE IN THE MADELUNG ENERGY... VOL. 48 FIG. : D Schematic diagram of the minimum unit defined here for the CsCl-type crystal lattice. (a) The order of summation according to the sequence of Cl. (b) The order of summation according to the sequence of Cs +. FIG. 3: Schematic diagram of the affine coordinates and a unit for the ZnO-type crystal lattice. But this arrangement decreases the contributions of Cl. For the purpose of arranging the order of summation strictly in accordance with the sequence of the ions located at the lattice points, the two sums should be averaged to eliminate the disproportion in contributions of the two types of ions. The final expression is M c (N) = M1 c (N) + M c (N). (9) The Madelung constant M c can quickly converge to the result after N = 3. II-3. The Madelung constant of ZnO-type crystal lattice We imitate the steps above by constructing affine coordinates with a Zn ion at the origin, as shown in Fig. 3. The unit vectors satisfy the conditions a 1 = a = αρ and a 3 = c = αρ, where α = 8 3. The angle between a 1 and a is 10 and a 3 is normal to the plane of a 1 and a. The contribution of the jth ion to the Madelung constant in these coordinates is ±1 ±f(n 1,n,n 3 ) = αn 1 + αn + α n 3 αn. (10) 1n

6 VOL. 48 JING WEN AND CHENG ZHONG 849 We take the primitive cell of ZnO as the minimum contributing unit to the Madelung constant, which includes two ZnO ion pairs. Considering the factors mentioned above, this structure has four different arrangements of terms. The unit in Fig. 3 can be divided into four groups. They are respectively (134), (345) or (3456), (4567), and (5678), where the numbers correspond to the ions in Fig. 3. Each group as a minimum term of the sum can be arranged as follows. The first arrangement including (134) is F 11 = f(n 1 + ξ,n + η,n 3 + ζ 1 1), F 1 = f(n 1 + ξ,n + η,n 3 + ζ 1), F 13 = f(n 1,n,n 3 + ζ 3 ), F 14 = f(n 1,n,n 3 ), (11) where ξ = 3, η = 1 3, ζ 1 = 1 ρ c, ζ = 1, ζ 3 = ρ c. The sum of the terms can be written as M 1 c (N) = The second arrangement including (345) or (3456) is F 1 = f(n 1 + ξ,n + η,n 3 + ζ 1), F = f(n 1,n,n 3 + ζ 3 ), F 3 = f(n 1,n,n 3 ), (F 11 + F 1 + F 13 + F 14). (1) F 4 = f(n 1 + ξ,n + η,n 3 + ζ 1 ). (13) The sum of the terms can be expressed as n Mc (N) = The third arrangement including (4567) is F 31 = f(n 1,n,n 3 ), F 3 = f(n 1 + ξ,n + η,n 3 + ζ 1 ), F 33 = f(n 1 + ξ,n + η,n 3 + ζ ), (F 1 + F + F 3 + F 4). (14) F 34 = f(n 1,n,n 3 + ζ 3 + 1). (15) The sum of the terms is M 3 c (N) = (F 31 + F 3 + F 33 + F 34 ). (16)

7 850 CALCULATION OF THE CHANGE IN THE MADELUNG ENERGY... VOL. 48 The fourth arrangement including (5678) is F 41 = f(n 1 + ξ,n + η,n 3 + ζ 1 ), F 4 = f(n 1 + ξ,n + η,n 3 + ζ ), F 43 = f(n 1,n,n 3 + ζ 3 + 1), F 44 = f(n 1,n,n 3 + 1). (17) The sum of the terms can be written as M 4 c (N) = (F 41 + F 4 + F 43 + F 44). (18) According to the rule which has been used for calculating the Madelung constant of a CsCltype crystal lattice, M 1 c, M c, M 3 c, M 4 c are arranged respectively just in accordance with the sequence of the No. 1 ion, No. or 3 or 4 ion, No. 5 or 6 or 7 ion, No. 8 ion. So M c and M 3 c should be calculated three times respectively. Then all the sums should be averaged to eliminate the disproportion in contributions of the two types of ions. The final expression is M c (N) = M1 c (N) + 3M c (N) + 3M3 c (N) + M4 c (N). (19) 8 The Madelung constant M c can quickly converge to the result after N = 3. In conclusion, we introduced a symmetry-offset method to speed up the convergence of the lattice sum. The basic steps should be obeyed as follows. First, we choose a neutral cell as the minimum contributing unit cell for calculating the Madelung constant, and take one of these ions as the origin. Then the sum can be summed by expanding cubes in accordance with the sequence of ions in the crystal lattice. Each ion in a neutral cell has its own sequences after the origin has been chosen. Each sequence corresponds to an order of summation, so we must find out the different sequences decided by the ions in the minimum unit. Finally, All the sums calculated by the different orders of summation should be averaged to eliminate the disproportion in contributions of the two types of ions to the Madelung constant, then the quickly convergent results consistent with the results calculated by other methods can be obtained. III. CALCULATION OF THE CHANGE IN THE MADELUNG ENERGY OF THE DOPED ZNO The change in the Madelung energy of doped ZnO has been investigated widely. The conditionally convergent series should be considered during its calculation. In order to illustrate the calculation method, we built a 1 supercell of ZnO and computed the effective ionic charges of ZnO doped by C and C-Ca. The chemical formulas of the doped and co-doped ZnO can be written as Zn 8 O 7 C and Zn 7 CaO 7 C. The techniques for calculating the effective ionic charges are based on the methods described in Refs. [4 8],

8 VOL. 48 JING WEN AND CHENG ZHONG 851 TABLE I: The effective ionic charges of doped ZnO. The unit of charge is e. C(O 31 ) means that O 31 is replaced by C, where the subscripts ij correspond to the serial numbers in Fig. 4. The numbers in the parentheses are the coordinates of the atoms in Fig. 4. C(O 31 ) C(O 31 )Ga(Zn 1 ) C(O 31 ) C(O 31 )Ga(Zn 1 ) Zn 11 (0, 0, 0) O 31 (0, 0, ζ 3 + 1) Zn 1 (1, 0, 0) O 3 (1, 0, ζ 3 + 1) Zn 13 (0, 1, 0) O 33 (0, 1, ζ 3 + 1) Zn 14 (1, 1, 0) O 34 (1, 1, ζ 3 + 1) Zn 1 (ξ, η, ζ ) O 41 (ξ, η, ζ 1 ) Zn (ξ 1, η, ζ ) O 4 (ξ 1, η, ζ 1 ) Zn 3 (ξ, η + 1, ζ ) O 43 (ξ, η + 1, ζ 1 ) Zn 4 (ξ 1, η + 1, ζ ) O 44 (ξ 1, η + 1, ζ 1 ) FIG. 4: The structure of a supercell is defined here, and the atoms have been numbered. which are performed using the Castep codes. The structures of pure and C, and C-Ga doped wurtzite ZnO were optimized in order to determine the influence of the impurity on the stability of the structure. The effective charges were computed after the geometry optimization. The results have been listed in TABLE I. Fig. 4 shows the crystal structure of the supercell and the serial numbers of the atoms which correspond with the subscripts of the atoms in TABLE I. The effective ionic charges and lattice parameters of the doped ZnO are different from those of the pure ZnO. Both of them are the basic parameters for calculating the Madelung energy. Here we only consider the difference in the Madelung energy between the doped and pure ZnO resulting from the changes of the effective ionic charges, which is enough for us to investigate the doping methods. It is convenient to take the ideal hcp structure as the structure of ZnO. We take the supercell as a unit and consider the change in the Madelung energy of a unit for the doped ZnO. First, we consider the change in the potential energy of the ion Zn 11 within the range of a supercell. It is

9 85 CALCULATION OF THE CHANGE IN THE MADELUNG ENERGY... VOL. 48 m 11 = 1 ρ 4 (q 11 q ij q11q o ij) o f (n 1 + a ij1,n + a ij,n 3 + a ij3 ), (0) i,j=1 where q ij are the effective ionic charges of the doped ZnO, (a ij1,a ij,a ij3 ) are the coordinates of the atoms in a supercell corresponding to the coordinates in TABLE I, q o 11 = qo ij = 0.8 are the effective ionic charges of pure ZnO, the function f has been given in Eq. (10) and f means the summation should exclude f(0,0,0). Taking m 11 as the minimum unit for calculating the change in the potential energy of Zn 11, the sum of the terms is M 11 (N) = n 3 =N n 1 = N n = N n 3 = N m 11. (1) This is a conditionally convergent series and it should be calculated according to the method which has been used in calculating the Madelung constant of ZnO. So the final result should be the average of the different sums defined in Section II.3 and can be written as M f 11 (N) = M1 11 (N) + 3M 11 (N) + 3M3 11 (N) + M4 11 (N). () 8 Then the changes in the potential energy of other atoms in the supercell can be calculated by imitating the method used above for Zn 11. Finally, the change in the Madelung energy of a supercell can be expressed as 4 M sup = M f ij. i,j=1 (3) With increasing N, the M sup as a function of N are plotted in Fig. 5. Fig. 5a shows the M sup for C doped ZnO and Fig. 5b shows the M sup for C-Ca co-doped ZnO. We find M sup > 0 for C doped ZnO, which increases gradually with increasing N, and M sup < 0 for C-Ca co-doped ZnO, which converges quickly to ev after N = 8. It is obvious that the C-Ca co-doping method is better than the C doping method from the stability point of view, which can effectively enhance the C impurity solubility. IV. CONCLUSIONS Enhancing the dopant solubility is a common problem in doping methods. Generally, the calculation of the impurity formation energy is the main theoretical instruction for the science of doping. Here we make use of the change in the Madelung energy of the doped ZnO to investigate the doping methods and introduce a method to calculate this change. A conditionally convergent series should be considered during the calculation of the change in the Madelung energy of the doped ZnO. Its calculation involves the evaluations

10 VOL. 48 JING WEN AND CHENG ZHONG 853 FIG. 5: The change in the Madelung energy of a supercell for doped ZnO. (a) C doped ZnO. (b) C-Ca co-doped ZnO. of the Madelung constant and the change in potential energy of a single ion. We have introduced a method to calculate the Madelung constant. The arrangements of the terms for calculating the Madelung constant are based on the condition that the sum should converge quickly to the result which has been accepted widely. The arrangement of the terms for calculating the change in the potential energy of a single ion is based on the condition that a supercell should be taken as a minimum contributing unit for the terms, which can give a quickly convergent result for investigating the change in the Madelung energy of the doped compound semiconductor. Our results show that doping using C increases the Madelung energy, whereas co-doping using C and Ga decreases the Madelung energy. The method can be used as a theoretical instruction to investigate the change of the impurity solubility from the stability point of view. References Electronic address: wenj3008@16.com [1] E. F. Schubert, Doping in III-V Semiconductors, no. 1 in Cambridge Studies in Semiconductor Physics and Microelectronic Engineering, (Cambridge University Press, 1993). [] C. Kittel, Introduction to Solid State Physics, 7th Edition, (John Wiley & Sons, Inc., New York, 1996), Ch. 3, p [3] Z. K. Tang et al. Appl. Phys. Lett. 7, 370 (1998). [4] D. M. Bagnall et al., Appl. Phys. Lett. 70, 30 (1997). [5] T. Aoki, Y. Hatanaka, and D. C. Look, Appl. Phys. Lett. 76, 357 (000). [6] Y. Li, G. W. Meng, L. D. Zhang, and F. Phillipp, Appl. Phys. Lett. 76, 011 (000). [7] D. C. Look, Mater. Sci. Eng. B 80, 383 (001). [8] S. J. Pearton, D. P. Norton, K. Ip, Y. W. Heo, and T. Steiner, J. Vac. Sci. Technol. B, 93 (004). [9] W. Walukiewicz, Phys. Rev. B 50, 51 (1994). [10] D. C. Look, J. W. Hemsky, and J. R. Sizelove, Phys. Rev. Lett. 8, 55 (1999). [11] C. H. Park, S. B. Zhang, and S.-H. Wei, Phys. Rev. B 66, 0730 (00).

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