J Mater Sci (2012) 47:7587–7593 DOI 10.1007/s10853-012-6351-0
FIRST PRINCIPLES COMPUTATIONS
Polarization, piezoelectric properties, and elastic coefficients of InxGa12xN solid solutions from first principles L. Dong • S. P. Alpay
Received: 16 January 2012 / Accepted: 16 February 2012 / Published online: 8 March 2012 Ó Springer Science+Business Media, LLC 2012
Abstract III-nitrides GaN and InN, and InxGa1-xN solid solutions are polarizable semiconductors that crystallize in the prototypical wurtzite (W) structure. We present here the results of a density functional theory study carried out to determine the spontaneous polarization, piezoelectric coefficients, and elastic coefficients of InxGa1-xN alloys as a function of In the concentration x. To calculate these properties, we construct three distinct hexagonal/orthorhombic equivalent InxGa1-xN supercells that are derived from the disordered W unit cell of GaN and InN. These include an ordered W lattice (P63mc/Pmc21) and orthorhombic O-16 and O-32 lattices with Pmn21/Pna21 or P21 symmetry, respectively. Depending on the crystal structure, spontaneous polarization as a function of the In concentration x shows a downward bowing (W), a linear interpolation (O-16), and an upward bowing (O-32) between -0.033 C/m2 and -0.043 C/m2, the spontaneous polarizations of the end components GaN and InN, respectively. The composition dependence of the effective basal plane and out of plane (along the [0001] direction) piezoelectric coefficients (e// and e33, respectively) in the W and O-16 structure is non-linear and varies between e// = -0.287 C/ m2 and e33 = 0.598 C/m2 for GaN, and e// = -0.455 C/m2 and e33 = 1.044 C/m2 for InN. While the bulk modulus of InxGa1-xN in the W and O-16 structures follows Vegard’s law from 170 GPa (x = 0) to 124 GPa (x = 1), in the O-32 L. Dong (&) S. P. Alpay Department of Physics, University of Connecticut, Storrs, CT 06269, USA e-mail:
[email protected] S. P. Alpay Department of Chemical Materials and Biomolecular Engineering and Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA
structure it shows a strong downward bowing for compositions 0 \ x \ 0.5.
Introduction III-nitrides such as GaN and InN and their solid solutions have been studied intensively because of their unique properties, including a tunable band gap energy, high breakdown voltage, and high electron mobility [1]. As such, these materials find applications in solid state lighting, solar cells, chemical sensors, and photocatalysis [2–4]. Most of the devices involving III-nitrides are derived from thin films, often as multiple quantum wells, to take advantage of the highly mobile electron gases at the interfaces introduced by the discontinuity of the large spontaneous and piezoelectric polarizations [5]. In recent years, there has been a growing interest in the synthesis of III-nitrides as nanoparticles and one-dimensional (1D) nanowires for such applications as white light emission diodes (LEDs), miniaturized lasers, and high-mobility nanoelectronics [6, 7]. More complex InGaN/ GaN nanowire heterostructures have also been produced via metal–organic chemical vapor deposition and molecular beam epitaxy [8, 9]. In such 1D heterostructures and thin films constructs including quantum well structures, the macroscopic polarization mismatch between layers is an important parameter that determines the sheet density of the electron gases at the interlayer interfaces. Polarization of nitride compounds originates from the dipole moments associated with Ga–N and In–N bonds in a non-centrosymmetric unit cell. In bulk, under stress-free conditions, these materials possess a strong spontaneous polarization (PS) along the [0001] (c-axis). In heteroepitaxial films grown on, e.g., a suitable (0001) substrate, there are in-plane misfit strains
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(and commensurate out-of-plane stress-free strains) due to the mismatch of the lattice parameters of the layer(s) and the substrate. We note that the strain and polarization in polarizable materials are invariably coupled as internal and external strains that change the bond length, and hence the strength of the dipole moments. As such, the total polarization P can be expressed very generally via
coefficients of InxGa1-xN are also strongly affected by the choice of the atomic configuration. This is particularly important for potential device applications that employ quantum wells and 1D nanostructures of III-nitride solid solutions which may have a crystal structure different than the bulk disordered W lattice.
P ¼ PS þ PPZ ;
Crystal structure and calculation methodology
ð1Þ
where PPZ is the piezoelectric polarization. PPZ may vary linearly with in-plane strains. The measurement of the spontaneous and piezoelectric polarizations of III-nitrides and their alloys is difficult, mainly due to the unavoidable screening by surface charges [10]. As such, there are large variations in the experimental polarization values in such materials systems [1]. Simulations of device properties usually assume that the elastic stiffness and piezoelectric coefficients of the ternary InxGa1-xN solid solution follow a linear approximation as a function the composition x [11]; while PS has a parabolic dependence with a bowing parameter bS [12]: PS ðInx Ga1x NÞ ¼ xPS ðInNÞ þ ð1 xÞPS ðGaNÞ bS xð1 xÞ:
ð2Þ
We note that the latter assumption is based on results obtained from a density functional theory (DFT) study [13]. DFT calculations have been applied to study the crystal structure, electronic properties, surface morphology, and growth kinetics of compound polarizable semiconductors such as GaN, InN, and ZnO and a number of ternary solid solutions [14–19]. Recent DFT studies on InxGa1-xN and crystallographically similar Zn1-xBexO alloys have shown that orthorhombic supercells obtained by breaking the in-plane hexagonal symmetry of the (0001) layers of the binary nitride unit cell might energetically be more favorable than an ordered wurtzite structure [20, 21]. In this study, we employ a first principles methodology based on DFT to examine the structure, polarization, piezoelectric, and elastic properties of InxGa1-xN alloys. For a given composition, we take into account the different atomic configurations in three symmetries: two orthorhombic supercells (O-16 and O-32, with 16 and 32 atoms, respectively), in addition to an ordered wurtzite (W) supercell with 16 atoms. While the (effective) lattice sizes are nearly identical for all three structures, our results show that the formation energies (Eform) of the O-16 and O-32 lattices are significantly lower than that of the W lattice. Furthermore, PS in O-16 and O-32 constructs is markedly larger than the PS of the W due to the variations in the interatomic separations between the Ga, In, and N ions in the O-16 and O-32 lattices compared to the W supercell. Our findings indicate that piezoelectric and elastic
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GaN and InN usually crystallize in the prototypical W lattice (P63mc) structure of their end components, with an ‘‘inplane’’ lattice parameter a0, an ‘‘out-of-plane’’ lattice parameter c0 along the [0001] direction (c-axis), and an internal lattice parameter u0, which measures the bonding length along the c-axis (Fig. 1a). For the W (ordered), O-16, and O32 supercells of InxGa1-xN used in the computations, we retain the out-of-plane periodicity of the W structure along the c-axis. However, the in-plane periodicities are expanded distinctly from a smaller 1 9 1 hexagonal base in GaN to a pffiffiffi larger 2 9 2 hexagonal (W), 2 9 3 rectangular (O-16), pffiffiffi and 4 9 3 rectangular (O-32) in the InxGa1-xN alloys (Figs. 1b–d). This way one can envision the P63mc/Pmc21 (W), Pmn21/Pna21 (O-16), and P21 (O-32) symmetries to represent the disordered W structure (Fig. 2). For a more meaningful comparison of different alloy supercells, we use
Fig. 1 (Color online) a A W unit cell of GaN, and the planar view along the c-axis of b the 2 9 2 hexagonal base of the W In0.5Ga0.5N supercell, pffiffiffi c the 2 9 3 rectangular base of the O-16 In0.5Ga0.5N supercell, and d pffiffiffi the 4 9 3 rectangular base of the O-32 In0.5Ga0.5N supercell
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their equivalent in-plane lattice parameter a0 in the hexagonal 1 9 1 format (see the basal plane of Fig. 1a). To keep a fixed In/Ga ratio in all the basal planes, the compositions considered in W and O-16 lattices are limited to x = 0.25, 0.5, and 0.75. But since there are eight cation sites in the basal plane of O-32, it becomes possible to model additional compositions with x = 0.125, 0.375, 0.625, and 0.875 in a similar fashion (not shown). In this analysis, we performed first principles calculations as implemented in the VASP code [22] with the PW91 generalized gradient approximation (GGA) [23], projectoraugmented wave pseudo-potentials [24], and a cutoff energy of 400 eV for the plane wave expansion of the wave functions. The Ga 3d and In 4d electrons were explicitly included in the valence states of the pseudopotentials. For pure GaN and InN, a 9 9 9 9 6 C-centered k-point mesh in the first Brillouin zone was found to yield well-converged results. For the alloy supercells, 5 9 5 9 6, 5 9 6 9 6, and 3 9 6 9 6 C-centered k-point meshes were employed for the W, O-16, and O-32 structures, respectively. The atomic configuration of each composition in each supercell was optimized to reach the lowest energy state and the atomic positions were relaxed until all components of the force ˚ . The vector on each atom were reduced below 0.02 eV/A elastic coefficients were derived from the linear stress–strain relationship by performing finite distortions of the lattices in a symmetry-general approach [25]. The polarizations of InxGa1-xN supercells were calculated using the Berry-phase approach where a reference structure with zero polarization is required [13]. This reference phase was taken to be the undistorted zinc-blende structure (F 43 m) because it is centrosymmetric.
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Atomic relaxations, formation energy, and polarization ˚, Our calculated lattice parameters for GaN (a0 = 3.223 A ˚ , and u = 0.3770) and InN (a0 = 3.582 A ˚, c0 = 5.239 A ˚ , and u = 0.3792) agree with previous c0 = 5.786 A experiments and theoretical simulations [14]. Despite the difference in the crystal symmetry, the W, O-16, and O-32 supercells have almost the same lattice parameters a0 and c0 that obey Vegard’s Law (Fig. 3a, b): a0 ðInx Ga1x NÞ ¼ xa0 ðInNÞ þ ð1 xÞa0 ðGaNÞ;
ð3Þ
c0 ðInx Ga1x NÞ ¼ xc0 ðInNÞ þ ð1 xÞc0 ðGaNÞ:
ð4Þ
This result is consistent with X-ray diffraction studies on both thin films and nanostructures [6, 26]. Due to the different sizes of Ga and In cations, atomic relaxations occur in a given alloy lattice, resulting in slight deviations of Ga(In)–N bond lengths and N–Ga(In)–N bond angles from the values of bulk GaN and InN. These variations require an additional energy, or formation energy (Eform), which is given as Eform ðInx Ga1x NÞ ¼ EðInx Ga1x N) xEðInNÞ ð1 xÞEðGaNÞ:
ð5Þ
where E(InxGa1-xN), E(GaN), and E(InN) denote the internal energies of InxGa1-xN, GaN, and InN, respectively. As seen in Fig. 4, Eform of the ordered W supercell is significantly higher than that of the O-16 and O-32 lattices for x = 0.25, 0.5, and 0.75. This can be explained by analyzing the average lengths of Ga–N and In–N bonds of the equilibrium supercells for a given composition for all the three crystal structures of InxGa1-xN. The bond lengths can be obtained from the positions of the Ga, In, and N
Fig. 2 (Color online) The ordered W, O-16, and O-32 supercells of InxGa1-xN for x = 0.25, 0.5, and 0.75. The space groups of each supercell are also shown
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Fig. 3 (Color online) Lattice parameters a a0, b c0, and c average u of the W, O-16, and O-32 supercells of InxGa1-xN as a function of In concentration x
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crystal structure. As shown in Table 1, the average contraction in the In–N bonds in the W lattice is at least twice as large as that in the other two constructs. One may argue that at x = 0.75, the variation in the Ga–N bonds in the W structure is significantly smaller than O-16 and O-32 lattices. However, the dominant contribution to Eform comes from the larger number of In–N bonds at this composition. This correlation can also explain the trend in Eform as a function of x in a given symmetry. For instance, the magnitude of percent variation in Ga–N bond lengths in O-32 lattices shows a monotonically rising trend from x = 0.125 to 0.875 and that of In–N is decreasing in the same range. As such, for Ga-rich alloys (0 \ x \ 0.5), Eform increases mainly from the contribution of the formation of Ga–N bonds, while for In-rich alloys (0.5 \ x \ 1), it decreases due to the smaller contribution from the formation of In–N bonds. Due to this interplay, there is a peak in Eform in O-32 alloys for x = 0.5 (Fig. 2). Furthermore, the bond lengths along the c-axis, as characterized by the internal lattice parameter u, have a significant effect on the spontaneous polarization. For example, previous studies have found that PS varies linearly with u in pure GaN and InN if the other two lattice parameters (a0 and c0) were kept constant [27]. Our results show that there is a strong correlation between the average value of u (Fig. 3c) and PS (Fig. 5) in the three supercells of InxGa1-xN as a function of x. Calculated PS values for GaN (-0.033 C/m2) and InN (-0.043 C/m2) are in good agreement with prior theoretical work [14]. PS of the W InxGa1-xN alloys displays parabolic composition dependence as described in Eq. 2, and the bowing parameter bS in our study is -0.026 C/m2, very close to the value given by another DFT calculation (-0.038 C/m2) [13]. However, as seen in Fig. 5, the magnitude of PS in O-16 and O-32 supercells is significantly larger than that of the W lattice for a given x. While PS follows Vegard’s Law in the O-16 structure, it is bowed in O-32 supercell with the bowing parameter bS = 0.053 C/m2. The largest difference in PS Table 1 Percent variation in the average Ga–N and In–N bonding lengths in the W, O-16, and O-32 supercells; the variation is positive (negative) if the Ga–N or In–N bond is stretched (compressed) compared to bulk GaN or InN x
W
O-16
O-32
Ga–N
In–N
Ga–N
In–N
Ga–N 0.16
-1.24
0.25 0.375
-0.20
-2.06
0.30
-0.82
0.21 0.28
-0.55 -0.37
0.5
0.84
-0.60
0.27
-0.20
0.49
-0.32
0.45
-0.13
0.55
-0.15
0.97
-0.07
0.125 Fig. 4 (Color online) The formation energy per cation of the W, O-16, and O-32 supercells as a function of In concentration x
ions. As such, it becomes possible to calculate average percent variations in the lengths of Ga–N (In–N) bonds in InxGa1-xN with respect to bulk GaN (InN). We consider here all three-dimensional atomic displacements in a given
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0.625 0.75 0.875
0.25
-0.82
0.51
-0.07
In–N
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Fig. 5 (Color online) Spontaneous polarization of the W, O-16, and O-32 supercells as a function of the In concentration x
occurs in the O-32 and W structures for x = 0.5. PS is -0.052 C/m2 in O-32 and is -0.031 C/m2 in W, corresponding to a 66% variation. These findings indicate that the magnitude of PS of AxB1-xN (A, B = Al, Ga, or In) solid solutions in 1D nanostructures, where only limited number of atoms are available, may deviate significantly from the linear or parabolic compositional dependence.
Fig. 6 (Color online) Piezoelectric coefficients a e// and b e33 in the W and O-16 supercells as a function of the In concentration x
Piezoelectric and elastic coefficients In a polarizable material, the elastic stiffness coefficients (at constant polarization), and the polarization are intimately connected. In Voigt notation, piezoelectric polarization along the c-axis is given as [28] PPZ ¼
6 X
e3j ej ;
ð6Þ
j¼1
where eij are the piezoelectric coefficients. For quantum well structures or 1D quantum wires grown along the c-axis, there usually exist an equibiaxial in-plane strain (e// = e1 = e2) and a commensurate out-of-plane strain (e3), so that PPZ in a W lattice is reduced to W W W W PW PZ ¼ e31 ðe1 þ e2 Þ þ e33 e3 ¼ 2e== e== þ e33 e3 ;
ð7Þ
whereas in the orthorhombic supercells, it is given as O O O O O PO PZ ¼ ðe31 þ e32 Þe== þ e33 e3 ¼ 2e== e== þ e33 e3 :
ð8Þ
O O Here, we employ eO == ¼ ðe31 þ e32 Þ=2 as an effective ‘‘in-plane’’ piezoelectric coefficient. This way, we can directly compare the piezoelectric coefficients in W, O-16, and O-32 supercells. As shown in Fig. 6, e// and e33 in the W and O-16 structures generally follow the linear approximation for x = 0.5. For x = 0.25, e// and e33 of the W lattice are significantly higher than the Vegard’s law predictions. Similar trends can be said of e// of the W lattice, and e// and e33 of the O-16 lattice for x = 0.75.
The calculated piezoelectric coefficients of the two end components (e// = -0.287 C/m2, e33 = 0.598 C/m2 for GaN, and e// = -0.455 C/m2, e33 = 1.044 C/m2 for InN) agree well with previous results [1, 14, 27]. For heteroepitaxial films and 1D nanostructures, the strains (e// and e3) for a given set boundary conditions are correlated in the elastic limit. Therefore, in addition to the piezoelectric coefficients, it is also important to know the elastic stiffness coefficients and the bulk modulus to determine PPZ. The elastic stress (ri) and the commensurate strain (ej) are related through the Hooke’s Law [28]. In the contracted (Voigt) notation, this relation for the W lattice is given by: 10 1 0 1 0 e1 C11 C12 C13 : r1 : : CB e 2 C B r2 C B C12 C11 C13 : : : CB C B C B CB e 3 C B r3 C B C13 C13 C33 : : : CB C; B C¼B CB e 4 C B r4 C B : : : C : : 44 CB C B C B A@ e 5 A @ r5 A @ : : : : C44 : : : : : : 0:5ðC11 C12 Þ r6 e6 ð9Þ where Cij are elastic stiffness coefficients at constant polarization. The bulk modulus (B) of the crystal in the W structure can be obtained by B¼
2 ðC11 þ C12 ÞC33 C13 : C11 þ C12 þ C33 4C13
ð10Þ
The calculated values of Cij and B of GaN and InN (Table 2) fall in the range given by experimental and other
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DFT studies [1]. On the other hand, for an alloy supercell in the orthorhombic structure, the elastic stiffness tensor CijO has a different format as 10 1 0 1 0 O O O e1 r1 C11 C12 C13 : : : CB e 2 C B r2 C B C O C O C O : : : 22 23 CB C B C B 12 B r3 C B C O C O C O B C : : : C 23 33 CB e 3 C; B C ¼ B 13 O CB e 4 C B r4 C B : : : C : : 44 CB C B C B O @ r5 A @ : : : : C55 : A@ e 5 A O r6 e6 : : : : : C66 ð11Þ where the superscript O corresponds to the orthorhombic supercell. For each of the O-16 and O-32 supercells O O considered in this study, our results show that C11 &C22 , O O O O C13 &C23 , and C44 &C55 with very small differences (typically \5%). Therefore, we construct an equivalent elastic stiffness tensor CijE in the hexagonal format for the O-16 and O-32 supercells in such a way that
Table 2 Elastic stiffness coefficients (Cij) and bulk modulus (B) of GaN and InN (unit: GPa) C11
C12
C13
C33
C44
B
GaN
329
116
76
344
90
170
InN
196
103
77
207
45
124
1 0 E r1 C11 B r2 C B C E B C B 12 B r3 C B C O B C ¼ B 13 B r4 C B : B C B @ r5 A @ : r6 : 0
E C12 E C11 E C13 : : :
E C13 E C13 E C33 : : :
: : : E C44 : :
: : : : E C55 :
10 1 e1 : B C : C CB e 2 C B C : C CB e 3 C; C C : CB B e4 C A A @ e : 5 E e C66 6 ð12Þ
E O O E O E O where C11 ¼ 0:5ðC11 þ C22 Þ, C12 ¼ C12 , C13 ¼ 0:5ðC13 þ O E O E O O E O C23 Þ, C33 ¼ C33 , C44 ¼ 0:5ðC44 þ C55 Þ, and C66 ¼ C66 . These new elastic coefficients establish the relations between the W (Eq. 9) and the reference orthorhombic structure (Eq. 11). We note that the special relation in a W lattice E E E C66 ¼ 0:5ðC11 C12 Þ
ð13Þ
is always satisfied in the orthorhombic lattices used in this study with high precision. As such, we simply use Cij to present our results of the elastic stiffness coefficients of the W, O-16, and O-32 structures. Cij and B of the alloys are plotted in Fig. 7 as a function of the Be composition x. C33 and C44 in all three supercells follow the linear approximation from C33 = 344 GPa and C44 = 90 GPa in GaN to C33 = 207 GPa and C44 = 45 GPa in InN. However, values of C11, C12, C13, and B in the O-32 structures are significantly lower than those in W and O-16 structures for x from 0 to 0.5. The largest difference in these elastic
Fig. 7 (Color online) Elastic coefficients a C11, b C12, c C13, d C33, e C44, and f bulk modulus B of the W, O-16, and O-32 supercells of InxGa1-xN as a function of In concentration x
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coefficients between the O-32 and W/O-16 supercells is for x = 0.25. At this composition, C11, C12, C13, and B are 19.8, 47.8, 43.5, and 27.0% smaller than the values from Vegard’s law, respectively. For compositions x [ 0.5, the elastic stiffness coefficients and the bulk modulus do not display any discernable dependence on the supercell symmetry. We note that these coefficients have not been experimentally measured in InN–GaN solid solutions. Previous studies on the critical thickness and dislocation of InGaN/GaN epitaxial layers usually take a linear approximation of the elastic coefficients between InN and GaN (e.g., Ref. [29]). This assumption is confirmed by our calculation in the W and O-16 lattices, and partly in the O-32 lattice for 0.5 \ x \ 1.
Conclusions We have carried out a theoretical study based on DFT calculations to understand the crystal structure, PS, piezoelectric coefficients, and elastic stiffness coefficients of InxGa1-xN solid solutions. Our results indicate that the formation energies of orthorhombic crystal symmetries with O-16 or O-32 supercells in such alloys may be lower than that of the prototypical W lattice. Since the spontaneous polarization is directly related to the atomic arrangements in a given crystal structure, there are significant variations in PS when different supercell structures are compared. A downward bowing (bS = -0.026 C/m2), a linear interpolation, and an upward bowing (bS = 0.053 C/m2) are found in W, O-16, and O-32 lattices, respectively. Our calculations show that the elastic coefficients follow linear approximations in W and O-16 lattices, but are substantially smaller in O-32 for 0 \ x \ 0.5. Acknowledgement The authors gratefully acknowledge discussions with Dr. J. V. Mantese (United Technologies Research Center—East Hartford, CT) and Prof. George A. Rossetti at UConn.
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