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2009, 113, 6883–6886 Published on Web 04/06/2009
Piezoelectric Properties of AlN, ZnO, and HgxZn1-xO Nanowires by First-Principles Calculations Alexander Mitrushchenkov, Roberto Linguerri,* and Gilberte Chambaud UniVersite´ Paris-Est, Laboratoire Mode´lisation et Simulation Multi Echelle, MSME FRE3160 CNRS, 5 bd Descartes, 77454 Marne-la-Valle´e, France ReceiVed: January 23, 2009; ReVised Manuscript ReceiVed: March 20, 2009
The piezoelectric constants of aluminum nitride and zinc oxide wurtzite-type nanowires have been calculated by first-principles approach and compared with the data obtained for bulk structures. The methods adopted here include the Hartree-Fock and density functional theory procedures in their periodic formulation. The piezoelectric response is seen to be higher in nanowires than in bulk. In zinc oxide wires, it is found that the piezoelectric constant is significantly enhanced by partial substitution of zinc by mercury. During recent years ionic nanowires have been the subject of numerous studies due to their unique physical and chemical properties, particularly in comparison with bulk structures, arising from surface effects and quantum confinement.1 Ionic nanowires are of interest in many technological applications, such as the creation of nanogenerators, nanosensors/ nanoactuators, or the manufacturing of very hard composite coatings and films. In many respects, aluminum nitride and zinc oxide are among the most attractive materials in nanowire research field. The III-V semiconductor aluminum nitride has several outstanding physical properties. Due to its wide band gap of 6.2 eV, excellent thermal conductivity, hardness, and resistance to oxidation, together with a high melting point, it is largely employed in microelectronics.2 Zinc oxide is a promising semiconductor for nanooptics and nanoelectronics applications: its wide band gap of 3.37 eV and high excitonic binding energy of 60 meV at room temperature make it suitable for use in the blue and ultraviolet spectral region.3 A variety of experimental techniques have been set up to grow arrays of AlN and ZnO nanowires with diameters ranging from a few to several hundred nanometers.4-9 Well-aligned ZnO nanorods with hexagonal sections have been synthesized by vapor transport deposition8 and guided vapor-liquid-solid growth.9 Unlike the bulk materials, for which a vast literature exists (see refs 10 and 11 for instance), only a limited number of theoretical works deal with structural and electronic parameters of zinc oxide and aluminum nitride nanowires.12-16 The work of Xiang et al.12 is the only one to focus on ab initio calculations of piezoelectric properties in nanowires, to date. In the present paper state-of-the-art first-principles calculations are performed to obtain optimized structures and elastic and piezoelectric parameters for several ZnO and AlN nanowires of different size. The improvement of electronic properties in semiconductors by transition-metal doping is a research field * To whom correspondence should be addressed. E-mail: linguerr@ univ-mlv.fr.
10.1021/jp9007015 CCC: $40.75
that attracted considerable interest in recent years (see ref 17 for instance). In this perspective, we studied the change in the piezoelectric response of ZnO nanowires produced by Hgdoping. The approaches adopted here include the density functional theory (DFT), in the local density and generalized gradient approximations (LDA and GGA), B3LYP hybrid functional approach, and the Hartree-Fock self-consistent-field method (HF-SCF). To test the accuracy of our results, the calculated lattice parameters and the piezoelectric tensor components are evaluated for the bulk crystals as well and compared to known experimental data. The information inferred from this preliminary study on small nanowires may serve as a guiding tool in deriving reliable models for larger one-dimensional nanostructures and to inspect the dependence of bulk-surface effects on the system size. In our calculations, we considered wurtzite-type nanowires A, B, C, D, and E with 12, 48, 108, 192, and 300 atoms per unit cell, respectively. These nanowires have diameters ranging from 0.4 to 2.8 nm. Piezomechanical properties have been evaluated for nanowires A, B, C, and D of AlN and A and B of ZnO. For AlN nanowire D, we found that the calculations of the piezoelectric constant may suffer from numerical inaccuracies, as discussed below, and require more detailed study. Hence, for E aluminum nitride nanowire, only geometrical parameters have been optimized. Wires A-D are shown in Figure 1. For HgxZn1-xO, four hypothetical nanowires of different composition, corresponding to x ) 1/4, 1/2, 3/4, and 1, have been studied. Pure HgO (x ) 1) has been included only for completeness, since under atmospheric pressure the most stable structure of mercury(II) oxide is not wurtzite, but cinnabar or orthorhombic. The strong piezoelectric effect along the [0001] direction in wurtzite structure is due to the asymmetric position of the anion with respect to cation. This is the direction in which the nanowires normally grow.4-9 Contrary to wurtzite, in centrosymmetric one-dimensional structures, such as cinnabartype nanowires, no along-the-wire piezoelectric effect is possible. The first-principles calculation of piezoelectric constants is done within the framework of the Berry phase theory of 2009 American Chemical Society
6884 J. Phys. Chem. C, Vol. 113, No. 17, 2009
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Figure 1. Structures of several ZnO (first row), AlN (second row), and HgxZn1-xO (last row) nanowires of different size. Different colors are used for zinc (yellow), aluminum (orange), mercury (gray), oxygen (red), and nitrogen (blue).
polarization.18 Since the conventional piezoelectric constants for bulk materials are not appropriate for nanowires, the effective piezoelectric constants eaik ) eikV/Nion are used instead,12 where Nion is the number of atoms in the unit cell and V is the unit cell volume. In the study of wurtzite-like nanowires grown in the [0001] direction, the only meaningful effective piezoelectric constant a , whose expression reduces to is e33
ea33 )
|e| ∂φ3 c 2πNion ∂ε3
(1)
where c is the lattice parameter, φ3 is the geometric phase along z, and ε3 is the strain tensor component in Voigt’s notation. Most of the computations presented here have been done with the CRYSTAL code,19 using the PWGGA, PBE, LDA, B3LYP, and HF methods for AlN (see Table 1), and PWGGA and LDA for ZnO nanowires (see Table 2). In the latter case, we could compare the quality of the obtained results with those by Xiang et al. (see ref 12). In the case of AlN, we also used the SIESTA code20 with the PBE functional for comparison. In the calculation of the piezoelectric response, geometry optimization of the lattice parameters for the bulk and the nanowires, including full relaxation of the atoms in the primitive cell, has been carried out for each selected method. The following Gaussian basis sets have been used with CRYSTAL: 86-21G* for Al,21 86-411d31G for Zn,22 the basis set from Weihrich et al. in ref 23 for Hg, 8-411G for O24 and 6-31d1G for N.24 In the PBE computations with SIESTA we used for Al and N the nonrelativistic GGA pseudopotentials, generated by the Troullier-Martins method,26 available in ref 27. According to eq 1, the piezoelectric constants can be obtained by evaluating the geometric phase as a function of deformation of the lattice parameter c. To evaluate φ3 with CRYSTAL, we used the localized crystalline orbitals approach,19 as we found that the Berry phase approach as implemented in CRYSTAL06, can only be used with 3D (bulk) systems. Both methods lead
to identical results for the bulk. At each deformation of the unit cell, before the actual computation of the geometric phase, internal relaxation of the atoms is necessary. The piezoelectric constant is then evaluated from a fit of φ3 vs the strain. In the case of D AlN nanowire, φ3 vs deformation along z does not perfectly fit to a smooth curve. The use of polynomials of different degree in the fit of the calculated points leads to slightly different values of the piezoelectric constant, as shown a in Table 1. In this case, the first and second value of e33 correspond to polynomials of degree 2 and 4, respectively. These numbers are thus only indicative and additional study will be performed. In the determination of the structural parameters, all the selected methods agree reasonably well with experiment (see Tables 1 and 2). HF underestimates the piezoelectric constant of bulk AlN and, compared with DFT, leads to smaller a for nanowires A and B (see Table 1). values of e33 A strong increase of the piezoelectric constant ea33 in nanowb is observed in all cases. The ires as compared to bulk e33 behavior is quite similar for AlN and ZnO: the enhancement gets larger for decreasing size of the nanowires. For AlN we observed the same irregular behavior of the piezoelectric constants as a function of the size of nanowire as found for a for wires C, ZnO by Xiang et al.,12 who calculated a larger e33 D, and E than for wire B. For AlN, we found that wire D has a than wire C. a slightly larger e33 a when the size As was shown in ref 12, the limit value of e33 of nanowire increases, can be expressed as follows: a e33,lim ) eb33 - 2νeb31 b e33
(2)
b e31
where and are the effective piezoelectric constants of the bulk and ν is the Poisson ratio
ν)-
∂ε1 c ∂a )∂ε3 a ∂c
(3)
where the derivative is taken by varying the lattice constant c and allowing a and internal atom positions to fully relax. The following relation links the Poisson ratio to the elastic tensor components:
ν)-
c13 c11 + c12
(4)
Equations 2-4 were used to estimate the experimental limits a b ’s given in Tables 1 and 2. Since ν is positive and e31 for the e33 negative, it follows that the effective piezoelectric constants are larger in nanowires than in bulk. In addition to this, surface effects in nanowires might contribute as well. From the analysis of data in Tables 1 and 2 for bulk AlN and ZnO, it is found that the piezoelectric constant is overestimated by the LDA approach and underestimated by HF and that the PWGGA or the PBE methods are the best choice for the calculation of the piezoelectric constants. For a given functional, the results obtained with CRYSTAL compare more favorably with the experimental data for bulk than those obtained with SIESTA. In Tables 1 and 2, we also present the energy per number of molecular units (Nmol) of the nanowires with respect to the bulk crystal. This energy is found to decrease as the inverse of the diameter of the nanowire, as it should be for a property that depends on the surface over volume ratio. We have found that in larger nanowires, for a given value of c, the atoms close to the central axis relax in exactly the same way as they do in the bulk. Work is in progress to calculate the piezomechanical response of wurtzite (101j0) surface and to
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J. Phys. Chem. C, Vol. 113, No. 17, 2009 6885
TABLE 1: Geometric Parameters (a, c, u) of the Wurtzite Structure, Effective Piezoelectric Constants and Energy for Several AlN Nanowires of Different Sizea PWGGA CRYSTAL PBE CRYSTAL PBE SIESTA LDA CRYSTAL B3LYP CRYSTAL HF CRYSTAL bulk
wire A, wire B, wire C, wire D, wire E,
a c u eb33 a e33,lim c ea33 ∆(E/Nmol) c ea33 ∆(E/Nmol) c ea33 ∆(E/Nmol) c ea33 ∆(E/Nmol) c ea33 ∆(E/Nmol)
3.125 5.007 0.3820 1470 1756 4.873 3574 2.314 5.045 2153 1.115 5.031 1922 0.760 5.024 2051/1941c 0.578 5.020
3.128 5.010 0.3821 1488 1777 4.870 3626 2.292 5.048 2166 1.105
3.171 5.047 0.3831 1567
3.088 4.942 0.3822 1635 1952 fails
4.916 3653 2.148 5.107 1980 0.993
3.119 5.002 0.3816 1327 1520 4.945 2970 2.310 5.051 1936 1.103 5.031
4.973 2361 1.244
3.107 4.971 0.3819 1396 1599 4.945 3142 2.512 5.027 1911 1.201
exp. 3.11029 4.98029 0.382030 161631 1830b
0.754 5.022 0.573 5.017
0.466
0.463
a
-16
ea33
b
Lattice parameters a and c in Å, in 10 µC Å/ion, and energy in eV/molecule. Estimated according to eqs 2-4 and using experimental data for lattice constants from ref 29, elastic constants from ref 32, and piezoelectric constants from ref 31. c The first and second values are obtained from fits of the geometric phase vs the deformation along z using polynomials of degree 2 and 4 respectively.
TABLE 2: Geometric Parameters (a, c, u) of the Wurtzite Structure, Effective Piezoelectric Constants and Energy for Several ZnO Nanowires of Different Sizesa PWGGA CRYSTAL bulk,
wire A, wire B,
a c u eb33 a e33,lim c ea33 ∆(E/Nmol) c ea33 ∆(E/Nmol)
LDA CRYSTAL
LDA SIESTA12
LDA33
exp.
3.181 5.159 0.377 1316 1815 5.414 1908 1.571 5.310 1944 0.774
3.17 5.18 0.374 1453
3.197 5.166 0.380 1463
3.24734 5.20334 0.38134 114335 1676b
3.264 5.270 0.380 1055 1675 5.488 2002 1.289 5.430 1704 0.625
5.335 2025 5.302 1837
Lattice parameters a and c in Å, ea33 in 10-16 µC Å/ion, energy in eV/molecule. b Estimated according to eqs 2-4 and using experimental data for lattice constants from ref 34, elastic constants from ref 36, and piezoelectric constants from ref 35. a
TABLE 3: Geometric Parameters and Effective Piezoelectric Constants of Several HgxZn1-x O Type-B Nanowires (See Figure 1) Calculated at the PWGGA (CRYSTAL) Levela ZnO Hg1/4Zn3/4O Hg1/2Zn1/2O Hg3/4Zn1/4O HgOb
c
ea33
5.430 5.564 5.844 5.888 6.431
1704 2536 3055 3563 3808
a Lattice parameter c in Å and ea33 in 10-16 µC Å/ion. b HgO does not exist in wurtzite form as a natural product.
understand its influence on the properties of nanowires. In the near future, the comprehension of the surface contribution to the piezoelectric effect will provide the tools for calculations at a larger scale. As predicted by theoretical analysis,28 the addition of mercury in B-type zinc oxide wires strongly enhances the piezoelectric properties (see Table 3). Moreover, a is found to increase with increasing mercury the calculated e33 to zinc ratio. The same behavior is expected for larger wires and/or non regular (chaotic) inclusions.
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