Article pubs.acs.org/crystal
Atomistic Simulations of CdS Morphologies Shafqat H. Shah,*,†,‡ Abdullah Azam,‡ and Muhammad A. Rafiq‡ †
Theoretical Physics Division, PINSTECH, Nilore, Islamabad, Pakistan Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, Pakistan
‡
S Supporting Information *
ABSTRACT: Atomistic simulations based on the static lattice model are performed to calculate the equilibrium and growth morphologies of CdS polymorphs. Morphologically important surfaces are optimized to calculate their structural and energetical properties such as surface and attachment energies. A common feature of all the nonpolar CdS surfaces is the outward movement of their anions and the inward movement of their cations. The relaxation of surfaces is critically important as it changes the surface and attachment energies significantly. The {112̅0} surface has the lowest surface energy (0.58 J/m2) for the wurtzite phase of CdS, whereas {110} surface has the lowest surface energy (0.62 J/m2) for the zincblend phase of CdS. The {101̅0}, {123̅0}, and {11̅00} surfaces of wurtzite CdS all have the same surface energy value (0.60 J/m2), which is very close to that of {112̅0} surface. Therefore, all these surfaces appear in the equilibrium morphology of the wurtzite CdS. The equilibrium morphology of the zincblend CdS is completely dominated by the {110} surface. The growth morphology of the wurtzite CdS consists of {101̅0}, {11̅00}, {0001}, and {0001̅} surfaces. The growth morphology of the zincblend CdS is found to be identical to its equilibrium morphology and, therefore, includes only the {110} surface.
1. INTRODUCTION Crystal morphology is a vital material design parameter like size and composition at the nanoscale. It is very important in many industries such as semiconductors, chemical, pharmaceutical, petrochemical, food, and cement, where routine processes of crystallization, material dissolution, flow and blending could be affected by crystal morphology. It is a complex phenomenon that depends on the crystallization process and its environment.1 Many internal (such as crystal symmetry) and external (such as temperature, additives, solvents, and defects) factors can influence the morphology of a crystal.2 The end-use of several materials depends on their morphology. Therefore, it is imperative to control the morphology of a nanocrystal for a particular application. For instance, in catalysis, different surfaces exhibit different reactivities. As a result certain surfaces become more favorable for a catalytic reaction than the others.3 Similarly, optoelectronic properties of II−VI semiconductors such as CdS are greatly affected by their morphologies.4 Therefore, in this study we calculate the equilibrium and growth morphologies of CdS. CdS is an important member of the II−VI semiconductor group with a wide range of applications such as cathodoluminescence, lasers, light-emitting diodes, waveguides, field emitters, logic circuits, memory devices, photodetectors, gas sensors, photovoltaics, and photochemistry.5−8 It has two polymorphs; at low temperature it exists in the wurtzite structure (WZ), whereas at high temperature it exists in the sphalerite or zincblend structure (ZB). It has several interesting properties such as direct bandgap, low work function, high electronic mobility, excellent thermal and chemical stability, © 2015 American Chemical Society
good transport properties, high refractive index, good piezoelectric coefficients, and transport properties.9−12 Only a few theoretical studies have been performed for the calculation of CdS morphology. Barnard and Xu13 used density functional theory (DFT) to calculate the equilibrium morphology of the wurtzite CdS. They also developed a thermodynamic model to predict the size, shape, and aspect ratio of CdS nanorods by using surface stresses. No growth morphology of CdS(WZ) has been predicted so far. Several other morphological studies of the related materials such as ZnS14 and CdSe15 have been performed using DFT and the static lattice model. In the present study, the static lattice model is used for the first time to calculate the equilibrium and growth morphologies of CdS. For several low index surfaces of CdS, surface and attachment energies are calculated. We also discuss the role of relaxation on the surface structures. To the best of our knowledge, no morphological study of CdS in wurtzite (WZ) and zincblend (ZB) phases is done using the static lattice model based on interatomic potentials. The static lattice model is an accurate, powerful, and highly efficient tool to calculate a wide range of material properties.16 The whole paper is arranged as follows. Section 2 gives the computational details. In Section 3 we discuss in detail all the structural properties, energetics of various surface terminations, and the equilibrium Received: December 19, 2014 Revised: March 4, 2015 Published: March 11, 2015 1792
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overall dipole of the surface is zero. Type 3 surfaces consist of charged planes and cannot be cleaved in such a manner that their dipoles become zero and thus always exhibit a nonzero dipole. In nature such surfaces can only exist if their dipole is suppressed by either geometrical reconstruction or their oxidation/reduction. In the present study, polar CdS surfaces are stabilized by geometrical reconstruction which involves removal of some ions from the top surface layer and their addition in a new layer at the bottom of the surface model (see Figure S1 in Supporting Information). In nature it is tantamount to creating point defects at the surface and has been used widely in many morphological predictive studies.14,21,22 Harding23 suggested a formula to estimate the amount of charge to be removed from the top layer of the surface model in order to stabilize a polar surface. According to Harding, a surface model is a stack of a number of surface unit cells. Each surface unit cell of size (ao) may comprise of a number of charged atomic planes (Qj) at positions (rj). Then α factor is defined as
and growth morphologies of both phases of CdS. Section 4 summarizes the main results of this study.
2. COMPUTATIONAL DETAILS We used the GULP code16 based on the static lattice model to calculate the structures and energies of various morphologically important CdS surfaces. An open source package, Wulffman,17 is used to construct the Wulff plots for the equilibrium and growth morphologies of CdS. The GULP code uses the Born model of solids in which total energy of a solid is described by two types of interactions, i.e., long-range interactions and short-range interactions. The long-range interactions are specified by Coulombic interactions of all the ions of the solid. On the other hand, short-range interactions are defined by interatomic potentials such as two-body, three-body, and higher-order potentials. For instance, in the present study, Buckingham potential (a two-body potential) and exponentialharmonic potential (a three-body potential) are used. The Buckingham potential consists of a repulsive exponential term and an attractive dispersion term,
UijBuck
⎛ r ⎞ C ij ij = Aij exp⎜⎜− ⎟⎟ − 6 rij ⎝ ρij ⎠
p
α=1+
j=1
⎛ r ⎞ ⎛ rij ⎞ 1 k b(θjik − θo)2 exp⎜⎜ − ⎟⎟ exp⎜⎜ − ik ⎟⎟ 2 ⎝ ρ2 ⎠ ⎝ ρ1 ⎠
(2)
In addition to long- and short-range interactions, the dipolar polariziability is introduced by the shell model of Dick and Overhauser.18 According to the shell model, ions are divided into a massless shell representing the valence electrons and a core representing the nucleus and the core electrons. The overall charge of the ion is split between the core and the shell of the ion. Uijcore − shell =
1 2 Krij 2
Q jrj Q 1ao
(4)
where p is the number of atomic planes or layers in a surface unit cell. By using the α factor, we can estimate the amount of charge, (1 − α) Q, to be removed from the top layer of the surface unit cell in order to stabilize the surface model. For equilibrium morphology, surface energies of the pertinent faces need to be calculated. Surface energy, Es(hkl), determines the relative thermodynamic stability of a surface (hkl). It is amount of energy required to cleave a surface from a bulk crystal and is given by the formula,
(1)
Since CdS is semicovalent in nature, an exponential-harmonic threebody potential is introduced due to the directionality of Cd−S−Cd bonds. The functional form of the exponential-harmonic potential is b U jik =
∑
Es(hkl) =
Eregion1(hkl) − nE bulk A(hkl)
(5)
where Eregion1(hkl) and Ebulk are the energies of the region 1 and bulk crystal, respectively. Here n is the number of formula units of CdS present in region 1, and A is the area of the surface under consideration. Surface energy values are positive, thus, implying an endothermic process. The smaller the value of surface energy, higher will be its stability. Once the surface energies of all the pertinent faces are calculated, the equilibrium morphology of a crystal can be constructed with the help of Gibbs criteria.24 According to Gibbs criteria the minimum of total surface energies for a given volume defines the equilibrium morphology of a crystal. Mathematically it is written as
(3)
In this study we used the shell model only for the S ion as it has a much higher ionic polarizibility than Cd ion. The potential parameters derived by Wright and Gale19 for CdS are used in the model potentials given by eqs 1−3. They used these model potentials to calculate the structures and the relative stability of two polymorphs of CdS. In addition to this they also calculated the elastic and dynamical properties of both phases of CdS and found them in close agreement with the experimental data. We employed the two region model of GULP code to simulate the surfaces of CdS. According to this model, the whole surface model is divided into region 1 and region 2. Region 1 includes the desired surface and the near-surface atoms which are explicitly relaxed, whereas region 2 represents the bulk material and simulates the effect of bulk on the surface. All the atoms in region 2 are fixed to their bulk relaxed positions. These regions have to satisfy two conditions, namely, region 1 should be large enough that the calculated surface energy converges with respect to its size. On the other hand size of region 2 should be such that the atoms at the bottom of region 1 have minimum interaction with the atoms at the bottom of region 2. In the present study region 1 and region 2 consists of 16 and 20 surface unit cells, respectively. The sizes of surface models are 1 × 1 (approximately 144 no of atoms) for nonpolar surfaces and 2 × 2 (approximately 296−576 no of atoms) for polar surfaces of CdS. The dimensions of the surface vectors U and V defining the surface unit cells are in the range of 4−19 Å and 4−17 Å, respectively. The surfaces of ionic materials are either polar or nonpolar in nature. Polar surfaces are unstable; therefore, their energies diverge due to their normal dipole. Tasker20 classified surfaces into three types. Type 1 surfaces consist of charged neutral planes and thus exhibit no dipole, whereas Type 2 surfaces are cleaved in such a way that the
Estotal =
∑ Es(hkl)A(hkl) hkl
(6)
For growth morphology prediction, attachment energies of all the relevant surfaces are required. Attachment energy, Ea(hkl), determines the growth rate of a particular crystal face. It is the amount of energy released when a stoichiometric growth slice attaches to a surface and is given by the formula
Ea(hkl) = Ecrystal − Egrowth slice
(7)
In GULP code attachment energy is calculated by the interaction energy between the growth slice and the rest of the material. Its value is negative and, thus, represents an exothermic process. Since it is closely associated with the growth rate, Ghkl, of a surface, we can write
GhklαEa(hkl)
(8)
It means that a particular surface will grow faster if the absolute value of its attachment energy is large. As a first step, we calculated the equilibrium structures of two phases of CdS using interatomic potentials derived by Wright and Gale.19 The structure optimization is performed using the Newton− Raphson procedure, and the Broyden−Fletcher−Goldfarb−Shanno (BFGS) method is used for updating the Hessian.25 These optimized structures are then used to create all the surface models. The 1793
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Table 1. Details of CdS(WZ) Surface Structures and Their Properties Including Terminations, Shifts, Interplanar Displacements (dhkl), Sizes of Surface Unit Cells, Number of Planes Per Surface Unit Cell, Composition of Each Plane and Total Charge of Each Planea surfaces (hkl)
shift
dhkl (Å)
(112̅0) (101̅0) (123̅0) (11̅00) (1012̅ )Cd (0001)Cd (101̅2)S (0001̅)S (0001)̅ Cd (0001)S
0 0 0 0 0 0 0.098 0.117 0 0.883
2.097 3.632 1.373 3.632 2.454 6.657 2.454 6.657 6.657 6.657
surface unit cell
no. of planes per surface unit cell
composition of planes
charge per plane
α factor
ΔC
Δcomposition
× × × × × × × × × ×
1 2 2 2 4 4 4 4 4 4
2Cd2S CdS,CdS CdS,CdS CdS,CdS 4Cd,4S,4Cd,4S 4Cd,4S,4Cd,4S 4S,4Cd,4S,4Cd 4S,4Cd,4S,4Cd 4Cd,4S,4Cd,4S 4S,4Cd,4S,4Cd
0 0 0,0 0,0 8+, 8−, 8+, 8−, 8−, 8+, 8−, 8+, 8+, 8−, 8−, 8+,
0 0 0 0 0.53 0.76 0.47 0.77 0.23 0.23
0 0 0 0 4+ 2+ 4− 2− 6+ 6−
0 0 0 0 −2Cd −Cd −2S −S −3Cd −3S
1 1 1 1 2 2 2 2 2 2
1 1 1 1 2 2 2 2 2 2
8+, 8− 8+, 8− 8−, 8+ 8−, 8+ 8+, 8− 8−, 8+
The factor (α) determines how much charge must be removed to quench the surface dipole moment. The parameters ΔC and Δcomposition define the amount of the charge to be removed and the consequent change in the composition of the top surface layer, respectively, for the stabilization of a polar surface. The negative sign in the Δcomposition column indicates the removal of ions from the top surface layer. a
Table 2. Details of CdS(ZB) Surface Structures and Their Properties Including Terminations, Shifts, Interplanar Displacements (dhkl), Sizes of Surface Unit Cells, Number of Planes Per Surface Unit Cell, Composition of Each Plane, and Total Charge of Each Planea surfaces (hkl)
shift
dhkl (Å)
(110) (431) (111)Cd (321) (311)S (11̅ 1̅ )̅ S (310) (211) (100)Cd (311)Cd (100)S (1̅1̅1̅)Cd (111)S
0 0 0 0 0.75 0.25 0 0 0 0 0.25 0 0.75
4.15 1.15 3.39 1.57 1.77 3.39 1.86 2.39 5.87 1.77 5.87 3.39 3.39
surface unit cell no. of planes per surface unit cell composition of planes 1 1 2 1 2 2 1 1 2 2 2 2 2
× × × × × × × × × × × × ×
1 1 2 1 2 2 1 1 2 2 2 2 2
2 2 2 2 2 2 2 2 4 2 4 2 2
CdS,CdS CdS,CdS 4Cd,4S CdS,CdS 4S,4Cd 4S,4Cd CdS,CdS CdS,CdS 4Cd,4S,4Cd,4S 4Cd,4S 4S,4Cd,4S,4Cd 4Cd,4S 4S,4Cd
charge per plane
α factor
ΔC
Δcomposition
0 0 +8, −8 0, 0 −8, +8 −8, +8 0, 0 0, 0 +8, −8, +8, −8 +8, −8 −8, +8, −8, +8 +8, −8 −8, +8
0 0 0.25 0 0.25 0.75 0.75 0 0 0.5 0.5 0.75 0.25
0 0 −2 0 −2 −2 0 0 −4 −6 −4 0 −6
0 0 −Cd 0 S -S 0 0 −2Cd −3Cd −2S 0 −3Cd
a The factor (α) determines how much charge must be removed to quench the surface dipole moment. The parameters ΔC and Δcomposition define the amount of the charge to be removed and the consequent change in the composition of the top surface layer, respectively, for the stabilization of a polar surface. The negative sign in the Δcomposition column indicates the removal of ions from the top surface layer.
convergence is achieved when the gradient norm is less than 0.001 for the optimized bulk CdS and for its various surfaces.
morphologically important surfaces. In order to describe and compare structural properties of different surfaces, we defined an atomic displacement parameter, i.e., Dlmn (where l = 1, m = 2, 3, 4..., and n = x, y, z). A surface unit cell consists of different atomic planes or layers. The atoms in top layer are always labeled as 1, and atoms in subsequent layers are labeled as 2, 3, 4... and so on. The top atomic layer is always undercoordinated, whereas all other layers may or may not be undercoordinated. The relative displacement of atoms in layers 2, 3, 4... with respect to atoms in top atomic layer (labeled as 1) along the xor y- or z-axes is indicated by Dlmn, therefore, l = 1, m = 2, 3, 4..., and n = x, y, z. It is important to note that Dlmn defines the position of atoms in layers (2, 3, 4...) with respect to atoms in top layer (1). We can also estimate the magnitude of atomic relaxation in different layers and directions by comparing the Dlmn values in the optimized and unoptimized surface models. The atomic displacements of all the nonpolar and polar CdS(WZ) surfaces are given in Tables S1 and S2 of the Supporting Information, respectively, and compared with the other theoretical studies. No relevant experimental data are found for comparison. Four most stable CdS(WZ) surfaces are
3. RESULTS AND DISCUSSION The optimized bulk structures and several other physical properties of wurtzite (WZ) and zincblend (ZB) phases of CdS could be found in ref 19. It can be seen that all the calculated values are in good agreement with the available experimental data. This confirms the accuracy of the interatomic potentials derived by Wright and Gale19 for modeling bulk CdS polymorphs. We used these potentials for modeling CdS surfaces. In order to calculate the morphologies of CdS, we have to build and optimize different surface models. In the case of polar surfaces, it is important to reconstruct the surface models using α factor given by eq 4. All the details of a surface model creation and its reconstruction, where it is necessary, are given in Tables 1 and 2. 3.1. Structural Properties. First step to calculate the morphologies of CdS in wurtzite (WZ) and zincblend (ZB) phases is to determine the equilibrium structures of 1794
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Figure 1. Relaxed surface structures of CdS (WZ phase). Viewed along the [100] direction. (a) (112̅0), (b) (101̅0), (c) (123̅0), (d) (101̅2) Cdterminated. The numbers on atoms are used to define their atomic displacement parameters (Dlmn). The relaxed atomic coordinates of these atoms are also shown. The S atoms are colored yellow, whereas Cd atoms are colored green.
Figure 2. Relaxed surface structures of CdS(ZB phase). Viewed along the [001] direction. (a) (110), (b) (431), (c) (111) Cd-terminated, (d) (321). The numbers on atoms are used to define their atomic displacement parameters (Dlmn). The relaxed atomic coordinates of these atoms are also shown. The S atoms are colored yellow, whereas Cd atoms are colored green.
tors.13,14,21 Now we highlight the important structural features of different CdS surfaces. For the (112̅0) CdS(WZ) surface, our calculated atomic positions in different layers are found comparable with other theoretical studies13,26 (see Table S1 of the Supporting
shown in Figure 1. All the relaxed nonpolar CdS (both WZ and ZB) surfaces exhibit a common feature that the anions (i.e., S atoms) move away from the bulk, whereas the cations (i.e., Cd atoms) move toward the bulk. This pattern of atomic movements is also observed in other II−VI semiconduc1795
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Table 3. Calculated Unrelaxed and Relaxed Surface, Es (J/m2), and Attachment, Ea (eV/unit), Energies of Various CdS(WZ) Surfacesa surfaces (hkl)
unrelaxed Es (J/m2)
relaxed Es (J/m2)
others
%ΔEs
unrelaxed (Ea) (eV/unit)
relaxed (Ea) (eV/unit)
%ΔEa
(112̅0) (101̅0) (123̅0) (11̅00) (101̅2)Cd (0001)Cd2 (101̅2)S (0001̅)S2 (0001̅)Cd1 (0001)S1
0.95 1.00 0.99 1.00 2.02 2.90 2.02 2.90 2.90 2.90
0.58 0.60 0.60 0.60 0.95 1.06 1.14 1.19 1.74 1.84
0.43b, 0.51c, 0.55d, 0.29e 0.41b, 0.48c, 0.52d, 0.28e
−38.95 −40.00 −33.33 −34.00 −52.97 −63.45 −43.56 −58.97 −40.00 −36.55
−1.42 −0.87 −2.57 −0.87 −14.39 −4.25 −18.57 −4.25 −4.25 −4.25
−1.39 −0.86 −2.51 −0.86 −14.70 −4.28 −18.27 −4.12 −4.09 −4.25
−2.11 −1.15 −2.33 −1.15 +2.15 +0.7 −1.62 −3.06 −3.76 0
a Comparison with other theoretical results, when available, is also presented. bRef 31 using DFT GGA-PW91. cRef 31 using DFT LDA-CAPZ. dRef 31 broken-bond model. eRef 13 using DFT GGA-PBE.
Table 4. Calculated Unrelaxed and Relaxed Surface, Es (J/m2), and Attachment, Ea (eV/unit), Energies of Various CdS(ZB) Surfacesa surfaces (hkl)
Es (unrelaxed)
Es (relaxed)
others
% ΔEs
Ea (unrelaxed)
(110) (431) (111)Cd (321) (311)S (11̅ 1̅ )̅ S (310) (211) (100)Cd (311)Cd (100)S (1̅1̅1̅)Cd (111)S
1.06 1.57 2.38 1.74 2.26 2.38 2.44 2.18 3.08 2.26 3.08 2.38 2.38
0.62 0.89 0.97 0.99 1.06 1.08 1.28 1.29 1.35 1.35 1.63 1.66 1.75
0.35b, 0.53c, 0.54d
−41.51 −43.31 −59.24 −48.28 −53.10 −54.62 −47.54 −40.83 −56.17 −40.27 −47.10 −30.25 −26.47
−0.81 −5.40 −4.35 −3.57 −9.00 −4.35 −3.80 −2.90 −6.50 −9.41 −6.50 −4.66 −4.66
Ea(relaxed) −0.81 −5.07 −4.38 −3.41 −6.59 −4.29 −3.95 −2.95 −6.64 −9.38 −6.59 −4.70 −4.21
%ΔEa 0 −6.11 −0.69 −4.48 −26.78 −1.38 +3.95 +1.72 +2.15 −0.31 +1.38 +0.88 −9.66
a Comparison with other theoretical results, when available, is also presented. bRef 31 using DFT GGA-PW91. cRef 31 using DFT LDA-CAPZ. dRef 31 broken-bond model.
corresponding values in the unoptimized (0001)Cd2. All the calculated atomic displacements for these surfaces are given in Table S2 of the Supporting Information. It can be seen that the calculated atomic displacements show good agreement with available results.13 However, the displacement of S atoms in layer 2 of (0001)Cd2 differs approximately by 86% and 63% along the y- and z- axes respectively, when compared to those of ref 13. All the nonpolar and polar relaxed surface structures of CdS(ZB) are given in Tables S3 and S4 of the Supporting Information, respectively. No experimental or theoretical data are available for comparison. Among all the studied CdS(ZB) surfaces, four energetically most stable surfaces are shown in Figure 2. For CdS(ZB), we compare the atomic displacements in relaxed and unrelaxed surfaces. This shows how atoms in different layers and directions relax with respect to atoms in the top layer. It is interesting to note again the typical atomic displacement pattern for CdS(ZB) as well in which anion move away from the bulk and cations move toward the bulk. When the relaxation of nonpolar and polar CdS(ZB) surfaces is compared, we note that the overall relaxation for polar surfaces is much larger than the nonpolar surfaces. Among all the nonpolar surfaces studied here, (310) and (110) are the highest and the lowest relaxed surfaces, respectively. For instance, when the corresponding atomic displacements in relaxed and unrelaxed (310) surfaces are compared, the atoms in layers 2,
Information). For instance, the atomic positions of second layer atoms are in close agreement with the available first-principles data.13 However, the displacements of atoms in third (D13x, D13z) and fourth (D14x, D14y, D14z) layers differ as much as 10− 50% from the corresponding atomic displacements reported in refs 13 and 26. The (101̅0) CdS(WZ) surface is perhaps the most studied surface. It is interesting to note that values of all the reported atomic displacements are very scattered (see Table S1 of the Supporting Information). Our calculated displacements are in close agreement with those of ref 13. However, one of our calculated displacements (D12z) is at least 36% underestimated compared to all other reported values.13,26−28 In the case of (0001) and (0001̅) CdS(WZ) polar surfaces, two terminations for each are possible, namely, one with Cd and other with S on the top layer. This results in four surfaces, i.e., Cd terminated surfaces such as (0001)Cd and (0001̅)Cd and S terminated surfaces such as (0001)S and (0001̅)S. Since these surfaces are polar, we employed geometric reconstruction to quench their dipole moment. Consequently some surfaces have one dangling bond, and others have three dangling bonds. For instance, (0001)S1 and (0001)̅ Cd1 surfaces have only one dangling bond, and (0001)Cd2 and (0001̅)S2 surfaces have three dangling bonds. A common feature of all these surfaces is the large atomic relaxation along the z-axis. For example, atoms in layers 2, 3, and 4 of (0001)Cd2 surface show approximately 6%, 17%, 10%, and 9% relaxation compared to the 1796
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Figure 3. Morphological plots of CdS nanocrystals (a) equilibrium morphology of CdS(WZ), (b) equilibrium morphology of CdS(ZB), (c) growth morphology of CdS(WZ), (d) growth morphology of CdS(ZB).
surface and attachment energies. It is also noteworthy that surface energies always decrease with relaxation, whereas the attachment energies may or may not decrease after relaxation. These findings show the importance of surface relaxation as mentioned in earlier theoretical studies.21,29,30 The largest change in surface energy after relaxation is found to be 63% for (0001)Cd2 surface and 59% for (111)Cd surface in wurtzite and zincblend phases of CdS respectively. In the case of CdS(WZ), the most stable surface is found to be the (112̅0) surface with a surface energy value of 0.58 J/m2. It is different from other first-principles studies13,31 which found (101̅0) to be the most stable surface. According to our calculations, the (101̅0) surface is the second most stable surface along with the (123̅0) and (11̅00) surfaces. The calculated surface energies of the (112̅0), (101̅0), (123̅0), and (110̅ 0) surfaces are almost equal (≈ 0.60 J/m2), which is consistent with the other first principle studies.13,31 For CdS(ZB), the most stable surface is (110) with a surface energy of 0.62 J/m2, which is in close agreement with other DFT calculations.31 The smallest absolute value of attachment energy is found to be (0.86 eV/unit) for (101̅0) and (11̅00) surfaces of CdS(WZ), whereas for CdS(ZB) the lowest value is (0.81 eV/unit) for (110) surface. Since the growth rate is directly proportional to absolute value of the attachment energy, small absolute values of the attachment energies are important because they represent relatively slow growing facets which finally appear on the morphology of a nanocrystal. 3.3. Morphology. The simplest method for calculating the morphology of a crystal is BFDH method.32 In this method, morphology of a crystal depends on the interplanar distances (dhkl). The crystal faces with large values of dhkl are morphologically more important than faces with smaller values
3, 4, 5, and 6 relax on the average in any one direction approximately by 0.72, 0.74, 0.71, 0.68, and 1.04 Å respectively. On the other hand for the (110) surface, the atoms in layers 2, 3, 4, 5, and 6 relax on the average in any one direction by 0.16, 0.09, 0.11, 0.12, and 0.11 Å, respectively. Similarly, for polar surfaces (311)S and (111)S are the most relaxed and the least relaxed surfaces, respectively. For example, when the corresponding atomic displacements in relaxed and unrelaxed (311)S surfaces are compared, the atoms in layers 2, 3, 4, 5, and 6 relax on the average in any one direction by 1.11, 1.27, 1.35, 1.20, and 1.25 Å respectively. On the contrary, for (111)S surface atoms in layers 2, 3, 4, 5, and 6 relax only by 0.03, 0.1, 0.05, 0.05, and 0.04 Å, respectively. The (311)S surface shows an unusual surface relaxation in which some of Cd atoms in layer 2 move above the top layer of S atoms. In the case of the (11̅ 1̅ )̅ Cd surface atoms in different layers relax substantially along the z-axis. For instance, atoms in layers 2, 3, 4, 5, and 6 relax along the z-axis approximately by 1.00, 0.14, 0.87, 0.66, and 0.80 Å, respectively, when compared to their unoptimized surface. 3.2. Surface Energetics. Next step to calculate the morphology of CdS is to ascertain the morphological importance of various surfaces. The morphologically important (MI) surfaces are those which have the slowest growth rates. Facets with high growth rates quickly disappear from the surface of a faceted nanocrystal. The growth rate of a crystal facet can be estimated by either its interplanar spacing (dhkl) or its surface energy (Es) or its attachment energy (Ea). We can calculate the surface and attachment energies of a facet by using eq 5 and eq 7 respectively. Tables 3 and 4 give the calculated surface and attachment energies of CdS surfaces in wurtzite and zincblend phases, respectively. It is evident from Tables 3 and 4 that surface relaxation could significantly affect the values of 1797
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Crystal Growth & Design
(≤200 °C) favors the formation of high aspect ratio multiarmed CdS nanorods. The CdS nanorods were synthesized at 120, 200, and 260 °C using a monomer concentration of 0.3 g for a reaction time of 3 h. It is found that at 120 °C, a mixture of four-armed, three-armed, and two-armed nanorods are formed with an aspect ratio of approximately 7. At 200 °C, three-armed and two-armed nanorods of CdS are found with an aspect ratio of about 4. At high temperature (260 °C), singlearmed nanorods with an aspect ratio of approximately 2.3 are formed. Similarly in the case of the CdS(ZB) phase, depending on the temperature and concentration of reagents, different shapes of nanocrystals (such as spherical, rod, and prismatic shapes) could be formed.38,41 At low temperature (70−140 °C) and concentration (1.3−31.6 mM of cadmium acetate), spherical particles of predominantly hexagonal phase are formed, whereas at high temperature (140−250 °C) different shapes of CdS(ZB) nanocrystals are formed depending on the concentration of reagents. For example, at high temperature and concentration (140 °C and 31.6 mM cadmium acetate) CdS(ZB) nanorods are formed. At even higher temperature 250 °C, the spherical CdS(ZB) nanocrystals are synthesized when the concentration is low (1.3 mM of cadmium acetate), and prism-shaped nanocrystals are formed when the concentration is high (31.6 mM of cadmium acetate). The calculated equilibrium and growth morphologies of CdS(WZ) are different, whereas they are identical in the case of CdS(ZB), as shown in Figure 3. One possible reason could be the nature of the mechanism which controls the final morphologies of CdS in two phases. For instance, only the most stable facet, i.e., (110), appears in the equilibrium and growth morphologies of CdS(ZB), which means that its shape is thermodynamically controlled. On the other hand, the growth morphology of CdS(WZ) does not include all the most stable facets, for example, {112̅0}. This is because the most stable facets do not necessarily have the slowest growth rates as given by their attachment energies; therefore, they quickly disappear from the final morphology. This means that the final shape of CdS(WZ) is controlled by kinetics rather than by thermodynamics. The intrinsic crystal structures of the two phases also promote particular shapes. The crystal structure of CdS(WZ) is hexagonal which favors the anisotropic shape. On the other hand CdS(ZB) has a cubic crystal structure which promotes a more isotropic shape. However, anisotropic shapes such as nanorods of CdS(ZB) could also be formed.38 Formation of anisotropic shapes from a highly isotropic cubic CdS(ZB) seems unexpected, but the presence of a small amount of hexagonal phase in the CdS(ZB) nanocrystal favors the growth of rodlike structures.38,41 Different shapes of CdS polymorphs (e.g., rods and rhombic dodecahedrons) could be potentially useful for different applications. For instance, nanorods are excellent candidates for photovoltaic cells,42 linearly polarized emission,43 and lasing in the visible range.44 Furthermore, different shape parameters such as length, diameter, and aspect ratio of the nanorods could be modulated to tune different properties such as optical properties.45,46 Similarly, the rhombic dodecahedral shape of nanocrystals potentially could be exploited for photocatalytic properties.47 The equilibrium morphology of CdS(WZ) shows a range of facets, whereas only a subset of these facets appear in its growth morphology. Different facets have different properties depending on the type of surface atoms, their relative arrangement and
of dhkl. It is a geometric method which primarily depends on the crystal symmetry and its structure. The relatively more accurate method is the equilibrium morphology based on Gibbs criteria.24,33 According to Gibbs, the minimum of total surface energy of a crystal of a fixed volume determines its equilibrium morphology. Mathematically it is given by eq 6. This is important when the size of the crystal is small. It depends on a thermodynamic quantity, namely, surface energy which is given by eq 5. The equilibrium morphologies of CdS are shown in Figure 3a,b for the wurtzite and zincblend phases, respectively. The morphology of CdS(WZ) is a cylindrical shape with a flat bottom and hexagon like top which is corroborated by a combined DFT and experimental study of Barnard and Xu.13 The sides of the cylinder have different sizes. For example, larger sides have either {112̅0} or {101̅0} or {11̅00} surfaces, and smaller sides have only {123̅0} surface. All these surfaces that make up the sides of the cylinder have almost equal surface energy (≈ 0.60 J/m2). The difference between the surface energy of {112̅0} surface and those of {101̅0}, {123̅0}, and {11̅00} surfaces is only 3%; therefore, all these surfaces appear on the equilibrium morphology of CdS(WZ). It is different in the case of CdS(ZB) where the relaxed equilibrium morphology is completely dominated by {110} surface and forms a rhombic dodecahedral shape. For the CdS(ZB) phase, the {110} surface has the lowest surface energy (0.62 J/m2), which is far lower than those of all the other surfaces studied in this paper. The second most stable surface of CdS(ZB) phase is {431}, which is almost 30% larger than the lowest {110} surface. Therefore, the equilibrium morphology of CdS(ZB) nanocrystal includes only the {110} surface. The growth morphology34−36 at best can be classified as quasi kinetic as it does not include any growth rate coefficients. Here growth rate is estimated by a thermodynamic quantity, i.e., attachment energy given by eq 7. It gives the steady state morphology of a given crystal. Figure 3c,d gives the growth morphologies of CdS in the wurtzite and zincblend phases, respectively. The growth morphology of CdS(WZ) is a faceted rod shape in which top and bottom are closed by {0001} and {0001}̅ surfaces and the sides comprise either {1010̅ } or {11̅00} surfaces. In the case of CdS(ZB), the growth morphology is similar to its equilibrium morphology (i.e., rhombic dodecahedral shape) in which each facet is of rhombus shape with the {110} surface. The next lowest attachment energy is (2.95 eV/unit) for the {211} surface, which is almost 70% larger than that of {110} surface. This explains why the {110} surface dominates the growth morphology of the CdS(ZB) nanocrystal. It is noteworthy that the growth conditions such as temperature, reaction time, nature of solvents, concentration of surfactants, and metal precursors play a critical role in determining the final shape of CdS nanocrystals.37−39 The predicted morphologies of CdS polymorphs, as shown in Figure 3, are calculated using the static lattice model which is valid at 0 K. Experimental studies37,39,40 have shown that temperature is pivotal in developing the intrinsic shape of a nanocrystal by determining the crystalline phase of the initial seed at the nucleation stage. It also controls the growth rate of the nanocrystal and provides requisite thermal energy for the synthesis. The anisotropic growth of nanocrystals such as nanorods requires high temperature and is kinetically controlled. For instance, Li et al.37 found that high temperature (∼260 °C) favors the formation of low aspect ratio singlearmed CdS(WZ) nanorods, whereas the low temperature 1798
DOI: 10.1021/cg5018449 Cryst. Growth Des. 2015, 15, 1792−1800
Crystal Growth & Design
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bonding. Different applications exploit these properties. For instance, two different products could be obtained depending on the exposed platinum facet which acts as a catalyst in a reaction between benzene and hydrogen. If the {100} facet is available, only a saturated hydrocarbon cyclohexane will be produced. If {111} platinum facet is available, an unsaturated analogue of cyclohexane, called cyclohexene, is also produced.48 For a particular application, a specific facet could be stabilized using different chemical species such as surfactants, polymers, and small gas molecules. These species favor directiondependent crystal growth. For example, the (0001) facet is common both in the equilibrium and growth morphologies of CdS(WZ). Recently it has been shown that this facet could be potentially important in the hydrogen production from water through photocatalysis in the visible light range.49 This facet also changes the bandgap of CdS from 2.24 to 2.39 eV in its nanoleaves morphology. Therefore, the shapes of nanocrystals as well as the facets that constitute them are crucial from an application viewpoint.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +92(0)3365166551. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are thankful to Dr. Masood-ul-Hasan and Dr. Sikander Majid Mirza for the fruitful discussions on this project. We are also grateful to the anonymous reviewers for their useful criticism.
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REFERENCES
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4. CONCLUSION We performed atomistic simulations based on the static lattice model to calculate the structural, energetical, and morphological properties of CdS polymorphs. The force fields used in this study successfully reproduce the bulk and surface properties of different phases of CdS. The calculated surface structures of the wurtzite and zincblend phases of CdS are found in good agreement with the available theoretical data. A common feature of all the relaxed surfaces of CdS is the typical relaxation behavior in which S atoms relax away from the bulk, whereas the Cd atoms relax toward the bulk. This behavior is corroborated by other theoretical studies as well. It is noted that the relaxation of surface structures is imperative as it could considerably reduce their surface energies and, hence, potentially influence the crystal morphology. The {1120̅ } and {110} surfaces are the most stable surfaces in the wurtzite and zincblend phases of CdS respectively. The equilibrium morphology of CdS(ZB) consists of the {110} surface only. Because of the almost equal surface energy of {1120̅ }, {1010̅ }, {123̅0}, and {11̅00} surfaces, they appear as the sides of the cylindrical equilibrium morphology of CdS(WZ), whereas the {0001} and {0001̅} surfaces cap the top and bottom of the cylinder. The lowest absolute value of attachment energy is found to be 0.86 eV/unit for the {101̅0} and {11̅00} surfaces of the CdS(WZ) phase. These two surfaces make up a larger part of the rod-shaped growth morphology of CdS(WZ) along with polar {0001} and {0001}̅ surfaces which close the top and bottom of the shape. The calculated rhombic dodecahedral growth morphology of CdS(ZB) is identical to its equilibrium morphology and is completely dominated by the {110} surface. This study helps us to understand why different surfaces appear in the morphology of CdS nanocrystals. More sophisticated morphological methods are needed to account for the role of external factors such as solvents, additives, and defects on the CdS morphology.
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Article
ASSOCIATED CONTENT
S Supporting Information *
Tables of atomic displacement parameter (Dlmn) for polar and nonpolar surfaces of CdS(WZ) and CdS(ZB) phases and a figure of geometric reconstruction for a polar surface. This material is available free of charge via the Internet at http:// pubs.acs.org. 1799
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