Theoretical Calculations of Characters and Stability of Glide Dislocations in Zinc SulfideTheoretical Calculations of Characters and Stability of Glide Dislocations in Zinc Sulfide
Masaya Ukita1,+1,+2, Ryota Nagahara1,+2, Yu Oshima1,+2, Atsutomo Nakamura1, Tatsuya Yokoi1 and Katsuyuki Matsunaga1,2
1Department of Materials Physics, Nagoya University, Nagoya 464-8603, Japan 2Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan
Generalized stacking fault energies were calculated to understand dislocation characters and stability in zinc sulfide (ZnS) by using the density functional theory calculations. Peierls stresses and dislocation self energies were estimated for perfect and dissociated dislocations on glide-set and shuffle-set planes in ZnS in a framework of the Peierls­Nabarro model. It was found that Peierls stresses of the shuffle-set dislocations are smaller than those of the glide-set dislocations whereas dislocation self energies of the shuffle set are larger. It is experimentally known that the dissociated glide-set dislocations can be more easily formed and multiplied during plastic deformation in darkness at room temperature. It is suggested that the glide-set dislocations can be primarily activated due to their lower self energy, in spite of their higher Peierls stress. [doi:10.2320/matertrans.M2018253]
(Received July 31, 2018; Accepted October 24, 2018; Published December 25, 2018)
Keywords: plasticity of crystals, zinc sulfide, first-principles calculations, generalized stacking fault energy
1. Introduction
Zinc sulfide (ZnS) is one of II-VI inorganic semi- conductors, and is also known as luminescent materials,1)
an infrared optical material2) and as a photo-catalyst3) due to its superior electric and optical properties. Electrons and holes are produced via interband transition by photons, and contribute to its electrical conductivity and luminescent properties. Moreover, it was reported that light illumination to ZnS and some II-VI inorganic semiconductors can increase flow stresses during mechanical tests, which is called the photoplastic effect.4­6) This indicates that electrons or holes excited by photons can strongly affect mechanical properties of inorganic semiconductors. Furthermore, our research group recently found out that ZnS crystals show typical brittle fracture at room temperature under white or ultraviolet light environments whereas they can plastically deform up to a deformation strain more than 40% in complete darkness.7) It can be said, therefore, that ZnS is not necessarily brittle but has large plastic deformation ability without light exposure.
In order to reveal an origin of the intrinsic large plastic deformation ability of ZnS without light exposure, it is essential to understand atomic and electronic structures of dislocations in ZnS. ZnS has the zinc blend crystal structure and the easiest slip system of the h110i direction on the {111} plane, which correspond to the closest packed atomic plane and direction.6) In the tetrahedrally coordinated crystals involving the diamond and zinc blend structure, it is thought that glide dislocations tend to have straight dislocation lines lying along the h110i.8) This may be due to the Peierls potential valleys on the {111} plane, so that the perfect dislocations with a Burgers vector b = 1/2h110i are either pure screw dislocations or 60° dislocations (inclined at 60° to the dislocation lines). This fact was experimentally observed
in Si.9) In addition, a stacking sequence of {111} plane in the diamond and zinc blend structures can be represented by repeated AbBcCa (see Fig. 1), and thus there are two possible glide planes: the shuffle set between the widely spaced a-A (or b-B, c-C) planes, and the glide set between the closely spaced A-b (or B-c, C-a) planes.6,8,10)
It is known that shuffle-set dislocations cannot dissociate into partial dislocations and thus they are considered to be perfect ones. This is because a high-energy intrinsic stacking fault would be produced between partial dislocations.8) On the other hand, glide-set dislocations can be separated by two Shockley partial dislocations of b = 1/6h211i with an intrinsic stacking fault on {111} in between. From inclination angles of the Burgers vectors to the partial dislocation lines, the dissociated form of the screw dislocation (the 60° dislocation) in the glide set is a pair of 30° partial dislocations (a 30° and a 90° partial dislocation).
It is likely that which type of dislocations can be introduced depends on materials. There is still a controversy about this issue even for Si.6,11) GaAs and InP with the zinc blend structure seemingly show glide dislocations in the
glide shuffle
Zn S
[01−1]
Fig. 1 Schematic illustration showing the atomic arrangement of ZnS with the zinc blende structure. Locations of glide-set and shuffle-set planes are indicated by dotted lines.
+1Graduate Student, Nagoya University. Corresponding author, E-mail: ukita.masaya@nagoya-u.jp
+2Graduate Student, Nagoya University
Materials Transactions, Vol. 60, No. 1 (2019) pp. 99 to 104 ©2018 The Japan Institute of Metals and Materials
shuffle set at room temperature.12) TEM observations showed dissociated Shockley partial dislocations of b = 1/6h211i in ZnS,7,13) indicating slip on the glide-set plane. However, it is still unknown why dislocations on the easiest {111}h110i slip system in ZnS can move in the glide set, unlike the other III-V semiconductors.
In order to investigate a physical origin of the dissociated dislocations in ZnS, Peierls stresses and dislocation self energies on the {111} plane in ZnS were evaluated with density functional theory (DFT) calculations followed by the Peierls­Nabarro (P­N) model. The P­N model is a simple dislocation model where only two lattice planes facing a glide plane of a dislocation are treated as a discrete lattice and other regions above and below the planes are approximated as elastic continuum.14,15) The P­N model is widely used to describe dislocation cores and Peierls stresses in semi- conducting materials.16) Generalized stacking fault (GSF) energies of glide-set and shuffle-set planes were obtained from DFT calculations, and were used in the P­N model analyses. The experimentally observed dislocation characters in ZnS were discussed in terms of Peierls stresses and dislocation self energies obtained theoretically.
2. Computational Method
The P­N model assumes that a misfit region of inelastic displacement is restricted to a slip plane, whereas linear elasticity applies far from the slip plane. In this manner, a dislocation is treated as a continuous distribution of displacement u(x) of an upper half of the crystal with respect to the lower half at a distance x from the dislocation line. Then the P­N equation is as follows,14,15)
K
dx0 dx0 ¼ FðuÞ; ð1Þ
where F(u) is a restoring force acting between atoms on either side of the interface, and K is the energy factor, which depends on dislocation types. For a dislocation in an isotropic elastic medium, K can be expressed as follows,16)
K ¼ ® sin2 ª
; ð2Þ
where ª is the angle between the dislocation line and its Burgers vector, ® and ¯ are the shear modulus and the Poisson’s ratio. F(u) in eq. (1) is given by a gradient of the GSF energy £(u),
FðuÞ ¼ @£
@u : ð3Þ
GSF energies were obtained from DFT calculations, as will be explained later. The displacement u(x) is determined by solving the P­N equation with trial fitting functions as follows,17,18)
uðxÞ ¼ b
cn þ b
2 ; ð4Þ
where ¡n, xn, and cn are variational constants and b is the size of the Burgers vector. In this study, it was found that N = 10 provides a good convergence of fitting. Substituting eqs. (3), (4) into eq. (1), the variational constants (¡n, xn, and cn) are
fitted from the least squares minimization of the difference between both sides of the P­N equation.
A misfit energy W can be considered as a sum of misfit energies for individual atomic pairs across the slip plane. When a dislocation line locates at xd, W can be written as
WðxdÞ ¼ Xþ1
m¼1 £ðuðma0 xdÞÞa0; ð5Þ
where aA is the periodicity of W, taken as the shortest unit cell parameter in the direction of the dislocation’s displacement. A Peierls stress can be obtained from a maximum slope of W as follows,
·p ¼ max 1
du
: ð6Þ
In addition to Peierls stresses, dislocation self energies (Edis) were evaluated based on the P­N model. According to the isotropic elastic theory, an elastic energy Eel of a dislocation per unit dislocation length can be express as follows,14)
Eel ¼ b2K
4³ ln
rc ; ð7Þ
where rc and R are the core radius and the range of elastic field centered at the dislocation core, respectively. Since the observed dislocation density in ZnS after deformation up to 25% without light exposure was around 5 © 108 cm¹2,7) the value of R = 0.22 µm was employed in this study, which corresponds to a half of the average distance of dislocations in two dimensions. In addition, rc was set to be a half of a dislocation core width defined in the P­N model as ¦.16) Here the value of 2¦ corresponds to a distance over which u(x) changes from b/4 to 3b/4 (see eq. (4)).
When a dislocation dissociates into two partials i and j, a dislocation self energy Edis involves elastic energies from the partial dislocations plus a stacking fault energy £SF in between. Namely, Edis is described as
Edis ¼ Eel i þ Eel
j þ £SFd: ð8Þ In this formula, d is a stacking fault width, which can be determined by force balance of a repulsive elastic force fij between the two partials and an attractive force due to an energy cost of the stacking fault extension. An elastic force fij acting between two parallel dislocations i and j can be described by the Peach-Koehler’s equation19) as follows,
fij ¼ ®
2³
1 d ; ð9Þ
where the be and bs are the magnitude of the edge and screw components of the partial dislocations.
GSF energies were calculated by imposing rigid shear displacements of atoms on a particular slip plane and along a particular slip direction.20) More details can also be seen in Refs. 7, 21 and 22. Supercells used in the present study were containing 36 atoms, and atomic positions were optimized only in the direction perpendicular to the slip plane of {111} until residual force on atoms became less than 0.1 eV/nm.
DFT calculations in this study were based on the projector augmented wave (PAW) method as implemented in the VASP code.23,24) In the PAW potentials, Zn 3d4s and S 3s3p
M. Ukita et al.100
electrons were treated as valence electrons. The generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof was used for the exchange-correlation terms,25) and wavefunctions were expanded by plane waves up to a cut-off energy of 400 eV. Since it is known that normal PBE-GGA calculations have a limitation describing localized d states due to the self-interaction error, onsite Coulomb repulsion was considered with effective parameters of U = 9.0 eV and J = 1.0 eV for Zn 3d.26) Values for the Poisson’s ratio ¯ and the shear modulus ® were taken to be 0.26 and 33.8GPa, which were obtained in the present DFT calculations. Lengths of burgers vector b and energy factors K for dislocations considered in this study (see eq. (1)) are listed in Table 1.
3. Results and Discussion
Figure 2 shows calculated GSF energies for perfect and partial slip on the {111} plane of ZnS. In the case of the perfect slip of b = 1/2h110i (Fig. 2(b)), the two GSF energy profiles for the glide set and shuffle set have maximum values at u/b = 0.5, and yet the shuffle set has a much smaller GSF energy than the glide set. As can also be seen in Table 2, the shuffle-set perfect dislocations have smaller Peierls stresses, as compared to the glide-set perfect dislocations. It can be said, therefore, that the shuffle-set dislocations can move more easily than the glide-set dislocations in the perfect form.
As mentioned before, the glide-set perfect dislocations of b = 1/2½110 can be dissociated into two Shockley partials, which is described by the reaction of 1/2½110 ¼ 1/6½211 + 1/6½121 (see Fig. 2(a)). GSF energies for the partial slip along the glide set was displayed in Fig. 2(c). Additionally, the characters “(i)”³“(iv)” and “(iv)A” in this figure correspond to individual positions denoted in Fig. 2(a). It can be seen that the GSF energy along the displacement route of (i)-(ii)-(iv)A for the glide set is much smaller than the one for the perfect slip in Fig. 2(b). The resultant Peierls stresses of the glide-set partials are also much smaller than those of the glide-set perfect dislocations (see Table 2). Moreover, the GSF energy profile for the partial slip exhibits the deep local minimum at “(ii)”. This indicates rather stable formation of the intrinsic stacking fault between the two partials. The corresponding stacking fault energy £SF was found to be 0.05 eV/nm2 at “(ii)”, which is in good agreement with the reported value27) of 0.03 eV/nm2. It is also noted that the stacking fault widths d estimated from £SF and eq. (9) are about 41 nm for a pair of 30° partials and
about 26 nm for a set of 30° and 90° partials. The similar amount of d was also observed experimentally.7)
It is generally considered that the shuffle-set perfect dislocation cannot dissociate into Shockley partials. In order to confirm that this is also true in ZnS, GSF energies for displacement route from “(i)” to “(iv)A” through “(ii)” along the shuffle set was displayed in Fig. 2(c). In contrast to the glide-set partial, the GSF energy profile has a single saddle point and no local minima at “(ii)”. Moreover, the stacking
Table 1 Sizes of Burgers vectors b and energy factors K used in this study.
b = 1 2
G SF
e ne
rg y,
γ / e
V· nm
(c) <−211> direction
displacement, u/b (b = 1/2<−110>)
G SF
e ne
rg y,
γ / e
V· nm
(b) <−110> direction
Fig. 2 (a) Atomic configuration of ZnS viewed normal to the ©111ª direction. (b) GSF energies as a function of shear displacement for the h110i direction (perfect slip). (c) GSF energies for the h211i direction (partial slip).
Theoretical Calculations of Characters and Stability of Glide Dislocations in Zinc Sulfide 101
fault energy £SF at “(ii)” (5.35 eV/nm2) was much larger than that of glide set. Assuming that the shuffle-set perfect dislocation was dissociated into Shockley partials with this stacking fault, the estimated staking fault width of d from eq. (9) were about 0.39 nm for a pair of 30° partials and about 0.24 nm for a set of 30° and 90° partials. Since these widths are as large as the original Burgers vector length of b = 1/2½110 (0.38 nm), the two partial dislocations are almost overlapping each other and can be regarded as perfect dislocations. Therefore, it is thought that the shuffle-set perfect dislocation cannot dissociate into two partials in ZnS.
Although the glide-set partial dislocations have smaller Peierls stresses than those for the glide-set perfect ones, their values are still larger than those for the shuffle-set perfect dislocations (see Table 2). The Peierls stress for the glide-set 90° partial (4.3GPa) is almost the same with that for the shuffle-set screw dislocation (4.3GPa). Since the two glide- set partial dislocations should move simultaneously under applied external stresses, however, their total motion should be rate-controlled by the 30° partial having the larger Peierls stress (5.9GPa), indicating that the shuffle-set dislocations can move more easily in ZnS. Therefore, formation and multiplication of the glide-set partial dislocations in ZnS experimentally reported7) cannot be explained by the Peierls stresses alone.
A Peierls stress is a measure of dislocation mobility once the dislocation is formed. This does not mention anything about energy cost for dislocation formation itself. In order to investigate excess formation energies of dislocations, dislocation self energies were evaluated from eqs. (7) and (8). Figure 3 shows calculated dislocation self energies Edis per unit length for the glide-set partials and the shuffle-set perfect dislocations. For the glide-set partials (a pair of two 30° partials and one of 30° and 90° partials), the stacking fault energies with the calculated widths of d (described above) were also taken into account. It is clear that the glide-set partial dislocations have smaller self energies than the shuffle-set dislocations. Since major parts of Edis are determined by elastic energies that are proportional to b2
(see eq. (7)), the smaller self energies of the glide-set dislocations arise from the smaller Burgers vectors of b = 1/6h211i than those of the shuffle set (b = 1/2h110i, see also Table 1). It is also worth mentioning here that contributions of the stacking fault energies are quite small for the glide-set dislocations. This can be understood from the extremely small £SF value of 0.05 eV/nm2 (see also
Fig. 2(c)). These results indicate that the glide-set partials are more easily nucleated and multiplied by applied external stresses after yielding. Although the Peierls stresses for the glide-set partials are slightly larger than those of the shuffle- set perfect dislocations, it is likely that more energetically favorable formation of the glide-set partials can contribute to plastic deformation ability of ZnS.
It was experimentally reported that ZnS single crystals are initially colorless but turn orange after deformation in darkness.7) This fact indicates that dislocations introduced by plastic deformation in ZnS may have characteristic electronic structures that are different from that in the perfect crystal. In order to address this issue, changes in electronic densities of states (DOSs) for the glide-set partial slip and the shuffle-set perfect slip were analyzed based on GSF-energy calculations. Local DOSs across the slip planes at u/b = 0.5 as b = 1/6h211i (see Fig. 2) are displayed in Fig. 4, because the dislocation cores seem to undergo shear strains corresponding to that u/b value (a half Burgers vector size). In this case, the DOS profile of the perfect crystal was displayed together. For comparison, local DOSs across the slip planes were aligned so as to match average potentials between in the perfect crystal and in the bulk-like region of the deformed supercells.28) The top of the valence band in the perfect crystal was set at 0 eV. It can be seen that a band gap value Eg of the perfect crystal was 2.72 eV. Although the theoretical value underestimates the experimental value (3.52 eV), such underestimation of Eg was often observed in standard DFT calculations. As compared to this, Eg values across the slip planes (1.88 eV for the glide-set partial slip and 2.11 eV for the shuffle-set perfect slip) tend to be smaller. Such a tendency is consistent with the color change observed experimentally.7) It can be said that the dislocation core regions in ZnS have characteristic electronic structures with smaller band gaps, irrespective of the dislocation types.
It is finally noted that the local DOS profiles for the glide- set partial slip and the shuffle-set perfect slip show different features around the valence-band (VB) and conduction-band (CB) edges. In the case of the glide-set partial slip (Fig. 4(a)), the VB top was significantly modified and shifted to the band gap. This may be due to changes in atomic coordination across the slip plane. As can be seen in Figs. 1 and 2(a), the glide-set partial slip takes place between the closest packed
Table 2 Dislocation core widths 2¦ and Peierls stresses ·p obtained from the P­N model.
0 5
(a) (b) (c) (d)
elastic energy SF energy
Fig. 3 Calculated dislocation self energies Edis of (a) a pair of two 30° glide-set partials with a stacking fault (SF), (b) a set of 30° and 90° glide- set partials with a SF, (c) shuffle-set screw, and (d) shuffle-set 60° dislocations.
M. Ukita et al.102
atomic Zn and S planes along the b = 1/6h211i. In this case, the coordination number of Zn and S decreases from four to three at u/b = 0.5 and takes the planer configuration parallel to the ©111ª direction (see Figs. 5(a) and 5(b)). The missing coordination number of S with Zn in the severely deformed atomic configuration induces the pronounced acceptor-like electronic states just above the VB.
In the same manner as the glide-set partial slip, acceptor- like and donor-like states above the VB and below the CB
appear, and yet the energy of the acceptor-like states is not so deep as that for the glide set (Fig. 4(a)). Also, the significant DOS profile change around the CB edge can be observed. This may be due to atomic coordinations of Zn and S ions around the shuffle-set plane at u/b = 0.5, where three of four Zn­S bonds remain even by the shuffle set slip, keeping the triangular pyramidal configurations (Figs. 5(c) and 5(d)). Although Zn and S ions are originally fourfold coordinated, they also tend to have quasi-fivefold coordinations across the slip plane (Fig. 6), which is a quite contrast to the smaller coordination numbers of ions in the case of the glide-set partial slip (Figs. 5(a) and 5(b)).
As stated above, activation of the glide-set partial dislocations in ZnS can be ascribed to their lower dislocation self energy. It was also found that the dislocation cores have a smaller band gap locally, which should be closely related to dislocation mobility of ZnS under varying light condition. These GSF-energy calculations followed by the P­N model analyses are useful to investigate the observed slip deformation behavior of ZnS. It should be mentioned, however, that tetrahedrally coordinated semiconductors are considered to have dangling bonds at dislocation cores. Since dangling bonds at dislocation cores are quite unstable in covalent materials, bond reconstruction between dangling bonds at dislocation cores can take place.29) An energy gain due to the bond reconstruction provides an additional resistance to dislocation motion, because such reconstructed bonds must be broken during dislocation slip motion. This is true for IV semiconductors such as Si, and yet it is still unknown whether other III-V and II-VI semiconductors also undergo bond reconstruction at their dislocation cores. For instance, Justo et al. reported that bond reconstruction at dislocation cores of III-V semiconductors is less stable than that of IV semicondoctors.30) In the previous report by Kweon et al.,31) it was found that a 90° partial dislocation in CdTe (II-VI semiconductor) tends to have an unreconstructed core energetically more favorably than a reconstructed core. It seems that chemical bonding states of the host crystals may affect detailed atomic structures at their dislocation cores. For more detailed discussion, therefore, it is desirable to treat detailed atomic structures of the dislocation cores, which will be done in the future work.
4. Conclusions
In this study, the Peierls stresses and self energies of glide-
0.0
0.5
1.0
1.5
2.0
u/b=0.0
u/b=0.5
u/b=0.0
u/b=0.5
1.88 eV
Fig. 4 Local DOS profiles for atoms across (a) the glide-set and (b) the shuffle-set planes at a shear displacement u/b = 0.5. The top of the valence band in the perfect crystal (u/b = 0.0) was set at 0 eV. The grey areas indicate the calculated band gap of the perfect crystal. The band gap values at u/b = 0.5 were also denoted.
(b)
[−110]
shuffle
Fig. 5 Atomic configurations of ZnS for the glide-set partial slip ((a) and (b)) and the shuffle-set perfect slip ((c) and (d)) at a shear displacement of u/b = 0.5. In (b) and (d), open circles indicate atomic positions before slip (at u/b = 0.0).
shuffle-set plane[1
11 ]
Zn
S
Fig. 6 Atomic coordination at u/b = 0.5 for the shuffle-set perfect slip. The original atomic positions of S that form ZnS4 tetrahedra in the perfect crystal are drawn by the dotted circles.
Theoretical Calculations of Characters and Stability of Glide Dislocations in Zinc Sulfide 103
set and shuffle-set dislocations were evaluated based on DFT calculations and the P­N model to investigate dislocation characters in ZnS. The calculated Peierls stresses of the shuffle-set perfect dislocations were smaller than those of the glide-set ones, indicating that the dislocations can move along the shuffle-set plane during plastic deformation. According to the experimentally reported multiplication of dissociated dislocations by plastic deformation in darkness, the plastic deformation ability of ZnS cannot be fully explained by the Peierls stresses alone. From dislocation self energies, it was found that the glide-set partials are energetically more stable than the shuffle-set perfect dislocations. This suggests that the glide-set partials can be more easily nucleated and multiplied, which may lead to the remarkable plastic deformation ability of ZnS in darkness. It was also found that the band gaps of the dislocation cores estimated from LDOS profiles become smaller than that in the perfect crystal, which is in agreement with experimental results.
Acknowledgement
This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grant numbers JP18H03838.
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