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:
[email protected]
+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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