Plastic deformation in InSb single crystals is governed by the motion of dislocations. Since InSb has a diamond cubic lattice, it possesses two sets of slip planes: a shuffle set and a glide set. Transmission electron microscopy analysis of deformed bulk single crystals shows that, at low temperatures (<20 °C), dislocations have narrow cores, while at higher temperatures, they have extended cores. However, it remains unclear to which slip plane set these dislocations belong. In this paper, by combining experiments and atomic-level calculations, we show that dislocations with narrow and extended cores, respectively, belong to the shuffle and glide sets. The conclusion is reached by calculating the generalized stacking fault energy curves and ideal shear stresses using density functional theory calculations and the intrinsic stacking fault width associated with dislocations using atomistic simulations. It is also found that while the shuffle set dislocations are easier to activate at lower temperatures, dislocations on the glide set become dominant at higher temperatures.

InSb, a narrow bandgap (∼0.2 eV) semiconductor material, is extensively used in infrared and thermal imaging cameras and many other optoelectronic devices. The changes in mechanical properties of semiconductor materials with temperatures pose a challenge to their applications as optoelectronic devices in today's advanced miniaturized technologies. InSb, like most semiconductor materials, is known to show a ductile-to-brittle transition (DTBT) with a decrease in the temperature around 60% of its melting temperature (∼0.6 Tm).1 

The deformation of InSb crystals is governed by the motion of dislocations on crystallographic planes. Mechanically, InSb deforms differently in three temperature regions (a) T > 150 °C, (b) 150 °C > T > 20 °C, and (c) T < 20 °C. It is ductile in the highest temperature range (T > 150 °C) and brittle below 150 °C. Experimental analyses suggest that the DTBT is associated with the change in the {111} slip plane on which the dislocations glide. The interplanar spacing of the {111} glide plane in the diamond cubic lattice of InSb is not uniform but alternates between closely and widely spaced planes. Perfect dislocations having Burgers vector a0/2 {111} 〈110〉, where a0 is the lattice parameter, and gliding between the two more closely spaced {111} planes are thought to be dissociated dislocations. This means that the dislocation prefers to dissociate into two Shockley partial dislocations with Burgers vector a0/6 {111} 〈112〉 with an intrinsic stacking fault (ISF) in-between. These are known as “glide set” dislocations.

In the lowest temperature range (T < 20 °C) where InSb is brittle, InSb only deforms by the motion of perfect non-dissociated dislocations that are believed to belong to the shuffle set.2,3 “Shuffle set” dislocations glide between the two more widely spaced {111} planes. In the intermediate temperature regime (150 °C > T > 20 °C) between room temperature (RT) and the DTBT temperature, both dissociated and non-dissociated dislocations have been reported. Within this regime, close to 150 °C, twins coexist with extended stacking faults, like in Si around the transition temperature,4,5 whereas close to RT (20 °C), perfect non-dissociated dislocations instead accompany the extended stacking faults.

Whether the non-dissociated dislocations observed at low temperatures belong to the shuffle set is still under debate in InSb, as well as in many other semiconductors. There are no experimental observations of the core structure for these dislocations. Only simulations on Si have unambiguously suggested the shuffle set character of these dislocations.6 Also, there is a debate about the early nature of dislocations at nucleation. Simulations in Si suggest that the dislocations nucleate in the shuffle set, but when thermal fluctuations allow, they evolve toward the glide set.7 

It has also been observed that InSb exhibits drastic changes in the mechanical properties with a reduction in the crystal size. At RT, it is brittle in the bulk or macroscopic samples but becomes ductile when the sample size becomes small (in order of micrometers).2,3,8,9 Macroscopic single-crystalline samples of InSb are brittle at RT. However, Thilly et al.8 and Wheeler et al.9 have shown that InSb single-crystalline samples with a very small size, e.g., micro-pillars with small diameter, can be plastically deformed, and the brittleness can be suppressed by an increase in the surface-to-volume ratio. In these studies, micro-compression tests at various temperatures and subsequent dislocation analyses have shown that the plastic deformation of the micro-pillars is mediated by dislocations of the same nature as the ones observed in macroscopic samples at the same temperature. It has been proposed that elastic strain accumulated during mechanical deformation is released as cracks in the bulk polycrystalline material, but it can instead be dissipated with the formation of dislocations in small crystals.10 

The energetics of the slip of both the shuffle set and glide set dislocations in InSb deserve further study and may provide insight into the DTBT either via a change in the temperature or a change in the crystal size. To understand the energetics of the dislocation glide between two lattice planes, the generalized stacking fault energy (GSFE) associated with the shear of the crystal is commonly calculated and analyzed. The GSFE is the change in the energy associated with the rigid displacement of one half of the crystal relative to the other half across a crystallographic plane along a crystallographic direction. From the GSFE profile, the lattice resistance for shearing by a given slip mode can be estimated. Also, for some dislocations with planar dissociated states, it is possible to identify from the GSFE curve a set of partial dislocations that can produce the same displacement as a full dislocation. From the derivative of GSFE curves, one can also estimate the ideal shear strength as well as a relative barrier to form a full or partial dislocation. Besides the GSFE curves, dislocation core structures can be directly obtained by atomistic simulations, provided that the interatomic potential is suitable for studying plasticity.

In this work, we first provide experimental evidence of the different nature of dislocations in the different temperature ranges by reporting transmission electron microscopy (TEM) observations on thin foils extracted from plastically deformed InSb bulk single-crystalline samples. We then use first-principles density functional theory (DFT) calculations to study the GSFE curves for the shuffle and glide sets in the zinc blende structure of InSb. By comparing the GSFE curves for {111}〈110〉 and {111}〈112〉 slip systems corresponding to the perfect and Shockley partial dislocation, we show that dissociation into two partial dislocations is favorable for the glide set but not the shuffle set. To accompany these results, we conduct atomistic simulations to analyze ISF widths of pure- and mixed-type dislocations for the glide set. The DFT results also identify that the shuffle set has the lower ideal shear strength and the lower energy barrier to form. The implication is that thermal energy is needed to activate the glide set dislocations, providing a rationale for the higher activity of dissociated dislocations at higher temperatures.

InSb single-crystalline samples (3 × 3 × 8 mm3) were plastically deformed in compression between 400 °C and RT, at a strain rate of 2.5 × 10−5 s−1, in the Paterson apparatus (uniaxial compression under a confining gaseous pressure of 300 MPa) along the [213] direction to favor a single slip on the a0/2 [011] (1¯11¯) slip system. Below RT, micro-indentation tests were performed on the InSb wafer inside a controlled-atmosphere chamber filled with argon to avoid ice formation at the sample surface upon cooling, while a liquid nitrogen circulation was installed under the sample holder to decrease the sample temperature from RT down to −176 °C. Several hundred Vickers indentations were performed at different temperatures under a maximum load of 50 g.8 We remark that the deformation mechanisms in InSb do not depend on the deformation technique (i.e., compression or indentation) but only on the temperature of the test.2 

For TEM observation, thin foils were extracted from the deformed samples polished down to a thickness of 10–15 μm with the Tripod technique at very low speed. To obtain electron transparency, the thin foils were further thinned down by ion bombardment using a PIPS (GATAN Precision Ion Polishing System) with a small incidence angle (4°), small energy (2.5 keV), and reduced duration (<20 min). With such procedures, the microstructure is preserved in the thin foils. Deformation microstructures were then characterized by diffraction contrast techniques in Bright Field (BF), Dark Field (DF), and Weak Beam Dark Field (WBDF) with a 200CX JEOL microscope operating at 200 kV. In conventional TEM, the dislocation Burgers vector b and stacking fault vector R are determined by contrast extinction in two-beam conditions: gb = 0 and gR = 0 or n (n is an integer), where g is the diffraction vector.9 

To study the structural parameters and the GSFE curves for the glide and shuffle sets in InSb, we perform DFT calculations using generalized gradient approximation (GGA) for the exchange correlation functional with the Perdew–Burke–Erzenhof (PBE) parameterization as implemented in the VASP code.11,12 The interaction between valence electrons and ionic cores is treated using Projector augmented-wave (PAW) potentials.13,14 The number of valence electrons in PAW potentials for In is 13 (4d10, 5s2, 5p1) and for Sb is 5 (5s2, 5p3). We used a plane wave energy cutoff of 520 eV and optimized the structure until the force on each atom is smaller than 0.01 eV/Å. We used a 9 × 9 × 9 gamma-centered Monkhorst–Pack k-point mesh15 to integrate the Brillouin zone of the cubic primitive unit cell to calculate the lattice constants and 9 × 5 × 1 gamma-centered Monkhorst–Pack k-point mesh for the supercell for GSFE calculations.

The dislocation core structures in InSb are too large to be directly simulated by DFT. Atomistic simulations are then used to simulate the core structure of dislocations in InSb under no stress. The accuracy of atomistic simulations significantly depends on the validity of the interatomic potential. To our best knowledge, there exist two interatomic potentials for InSb. The first one was developed by Costa et al.17 and the second one by Rino et al.17 The second potential was a result of the re-parameterization of the first potential, such that the structural stability and transformation of InSb can be better predicted. Hence, the second interatomic potential developed by Rino et al.17 is used in this work. In particular, the lattice parameter, elastic constants, and GSFE curves must be accurately predicted for studies involving dislocations. In Sec. V A, we will calculate these parameters using this interatomic potential and compare the results with DFT. It will be shown that the glide set GSFE curves are accurately predicted by this potential but not the shuffle set GSFE curves. Then in Sec. V B, ISF widths of the screw, 30°, 60°, and edge dislocations at different temperatures in the glide set of InSb will be presented. All atomistic simulations are carried out using LAMMPS.18 

Figure 1(a) is a BF TEM micrograph obtained with g = 022 in the [1¯11¯] axis and showing a typical microstructure observed in the 300 °C-deformed sample. It is observed that dislocation segments are aligned along 〈110〉 directions [see the projection of the Thomson tetrahedron in Fig. 1(b)] and interact to form small segments. Applying successively different BF and WBDF imaging conditions allows determining the Burgers vector of the different dislocation segments as observed in Fig. 1(b). It is found that all segments are perfect dislocations with b1 = ±(a0/2) [011], b2 = ±(a0/2) [101¯], and b3 = ±(a0/2) [110]; note that b3 = b1 + b2. These perfect dislocations have a strong screw character. A pair of dislocation nodes is presented at very high magnification in Figs. 1(d)1(f): the upper node is constricted (CN), while the lower node is extended (EN). EN consists of three Shockley partial dislocations that are separated by an ISF. In the TEM micrographs of Fig. 1, the EN is not always fully visible, but it can be easily reconstructed by combining different micrographs.

FIG. 1.

300 °C-deformed sample: (a) BF micrograph with g = 022, (b) BF micrograph with g = 022, (c) BF micrograph with g = 242, (d) WBDF micrograph (g, 3 g) with g = 022, (e) WBDF micrograph (g, 3 g) with g = 220, (f) WBDF micrograph (g, 3 g) with g=202¯, and (g) schematization of dislocation interactions in two nodes: the constricted node (CN) and the extended node (EN). (h) Geometric parameters describing the EN. The scale bar of (c)–(f) is the same as the one given in (c).

FIG. 1.

300 °C-deformed sample: (a) BF micrograph with g = 022, (b) BF micrograph with g = 022, (c) BF micrograph with g = 242, (d) WBDF micrograph (g, 3 g) with g = 022, (e) WBDF micrograph (g, 3 g) with g = 220, (f) WBDF micrograph (g, 3 g) with g=202¯, and (g) schematization of dislocation interactions in two nodes: the constricted node (CN) and the extended node (EN). (h) Geometric parameters describing the EN. The scale bar of (c)–(f) is the same as the one given in (c).

Close modal

Figure 1(g) provides a schematic of the interactions formed by dislocations of 1, 2, and 3 types. The EN node is formed by three dislocations (numbered 4, 5, and 6) belonging to the same slip plane as the perfect dislocations. In this case, the dislocations labeled 4, 5, and 6 are successively out of contrast with g = 220, 202¯, and 022. Their respective Burgers vector is, therefore, (a0/6) [1¯12], (a0/6) [121], and (a0/6) [211¯], which confirms that they are Shockley partial dislocations. This type of dissociated configuration was repeatedly observed, confirming the dissociated character of all the ENs resulting from dislocation interactions in the samples deformed at 300 °C.

Such a configuration is used to measure the ISF energy, γsf, considering that the EN is at an equilibrium state. This approach was proposed initially by Whelan19 and then improved by Brown and Thölén.20 In this theory, the equilibrium shape of a symmetric EN is calculated under isotropic elasticity conditions by considering the dislocation self-stress. Each branch of the node is divided into several straight segments that interact with each other, including the one with the ISF at the node center. Given the shape at the equilibrium state, γsf is calculated by measuring the inner node radius y, the curvature R, and the character angle α of the partial dislocations [Fig. 1(h)]. The relation between y, R, α, and γsf is given by the following equation:

γsf=μbp2y{0.552υ1υ0.6υ1υcos2α+0.182υ1υ+0.36υ1υcos2αlog10Rbp},
(1)

where μ is the shear modulus, υ is the Poisson ratio, and bp is the magnitude of the Burgers vector of a partial dislocation. Since the observation is performed here directly in the slip plane, there is no projection to be considered when measuring y, R, and α. Also, in the present case, the character of all partial dislocation segments is about 30° for all EN chosen randomly in the TEM micrograph [Fig. 1(a)]. For example, the inner node radius y measured at the EN visible in Figs. 1(d)1(f) is 9.5 nm ± 0.5 nm. The curvature of the partial dislocation R is more difficult to estimate and has a larger uncertainty. In the present case, R is 39 nm ± 5 nm. Combining the measured values of y, R, and α, we can estimate γsf to be 36 mJ/m² ± 3 mJ/m².

In the case of the RT deformed samples, Fig. 2(a) shows images, obtained in BF conditions with g = 022 in the [1¯11¯] axis, of regions in the microstructure where evidently complex interactions occur between dislocations. Numerous partial dislocations are seen. Moreover, the micrograph obtained in BF with g=311¯ in the [2¯33¯] axis [Fig. 2(b)] reveals contrasts that are characteristic of ISFs. We find that these ISF contrasts, however, impede obtaining unambiguous extinctions of dislocations in BF, DF, and WBDF conditions. Therefore, we employed another characterization technique, not based on extinctions but on interferences with defects, called large-angle convergent beam electron diffraction (LACBED).3 This characterization technique indicated that this microstructure is mainly formed by the interaction of 30° or 90° Shockley partial dislocations belonging to different slip systems.

FIG. 2.

RT deformed sample: (a) BF micrograph with g=02¯2¯ and (b) BF micrograph with g=311¯. The scale bar is the same in both subfigures.

FIG. 2.

RT deformed sample: (a) BF micrograph with g=02¯2¯ and (b) BF micrograph with g=311¯. The scale bar is the same in both subfigures.

Close modal

For the lowest temperature studied, −176 °C, Fig. 3(a) presents the typical deformation microstructure observed in the sample indented below RT with long dislocation segments aligned with the 〈110〉 directions. Figure 3(b) presents two dislocation families from the same region at 60° from each other observed in BF conditions with g = 022 in the [1¯11¯] axis. Successive “extinction” of each family with, respectively, g=4¯2¯2 and g=2¯4¯2¯ in the [1¯11¯] axis suggests that these dislocations are perfect screw dislocations with b1= ±(a0/2) [011] and b2= ±(a0/2) [101¯]. Careful inspection of Fig. 3(b) also shows that the dislocation segments present several cross-slip events, including a double cross-slip, a mechanism that is possible with non-dissociated screw dislocations. It is worth noting that such double cross-slip events are also visible at a macroscopic scale by inspection of the slip traces at the surface of the sample with optical microscopy, as reported in Ref. 2.

FIG. 3.

−176 °C-indented sample: (a) the deformation microstructure with long dislocation segments aligned with the 〈110〉 directions and (b) the enlargement of (a) observed in BF conditions with g = 022 in the [1¯11¯] axis. Reproduced with permission from Kedjar et al., Acta Mater. 58, 1426 (2010). Copyright 2010 Elsevier.

FIG. 3.

−176 °C-indented sample: (a) the deformation microstructure with long dislocation segments aligned with the 〈110〉 directions and (b) the enlargement of (a) observed in BF conditions with g = 022 in the [1¯11¯] axis. Reproduced with permission from Kedjar et al., Acta Mater. 58, 1426 (2010). Copyright 2010 Elsevier.

Close modal

First, DFT is used to calculate the lattice parameter, elastic constants, bulk/cohesive energies, and the phonon frequency for InSb in the zinc blende crystal structure (space group No. 216, F4¯3m), as shown in Fig. 4(a). Results are summarized in Table I. For comparison, the experimental values of the lattice parameter,21 elastic constants,22 and cohesive energies23 at RT are also shown. The DFT-based values are in good agreement with the experimentally measured ones. Note that DFT calculations based on PBE-GGA usually overestimate the lattice parameter of a crystal.24,25 We also calculated the structural properties of InSb in the rock salt crystal structure, which is another common crystal structure for binary semiconductors, as shown in Fig. 4(b). Between the bulk energies of the two crystal structures (each of which contains one In atom and one Sb atom), the lowest energy structure of InSb is zinc blende, which is consistent with prior experimental observations.

FIG. 4.

The zinc blende and rock salt crystal structures of InSb. The In atoms and Sb atoms are shown in blue and yellow, respectively.

FIG. 4.

The zinc blende and rock salt crystal structures of InSb. The In atoms and Sb atoms are shown in blue and yellow, respectively.

Close modal
TABLE I.

DFT-based structural parameters for InSb in the zinc blende and rock salt crystal structures. DFT found that the ground state structure is the zinc blende structure, for which experimental21–23 and MS17 results are also shown for comparison. Note that experimental measurements were conducted at RT and atmospheric pressure. Both the bulk and cohesive energies are for one In atom plus one Sb atom.

StructureZinc blendeRock salt
MethodDFTExperimentMSDFT
a0 (Å) 6.636 6.479 6.473 6.147 
C11 (GPa) 54.8 65.94 65.8 103.2 
C12 (GPa) 28.4 34.45 35.7 20.9 
C44 (GPa) 25.4 30.31 43.4 42.5 
Bulk energy (eV) −7.104  −5.6 −6.813 
Cohesive energy (eV) 5.3584 5.5255 5.6 5.0674 
Phonon frequency (cm−1165   97 
StructureZinc blendeRock salt
MethodDFTExperimentMSDFT
a0 (Å) 6.636 6.479 6.473 6.147 
C11 (GPa) 54.8 65.94 65.8 103.2 
C12 (GPa) 28.4 34.45 35.7 20.9 
C44 (GPa) 25.4 30.31 43.4 42.5 
Bulk energy (eV) −7.104  −5.6 −6.813 
Cohesive energy (eV) 5.3584 5.5255 5.6 5.0674 
Phonon frequency (cm−1165   97 

To study the GSFE in DFT for the (111) slip plane, we take a supercell consisting of 72 atoms (36 In and 36 Sb), as shown in Fig. 5. In this supercell, the 〈110〉 direction is along the x axis, the 〈112〉 direction is along the y axis, and the 〈111〉 direction is along the z axis. For the calculation of the GSFE curves, we use periodic boundary conditions along the x and y directions. To select the number of atoms in the supercell, we test the convergence of the energy with respect to the number of atoms in the system (or the number of atomic layers along the z direction). Based on this, we found that 18 layers along the z direction achieve convergence within 5 mJ/m2, and thus, this supercell size is selected for the calculations that follow.

FIG. 5.

(a) The crystallographic orientation of the supercell for calculation of GSFE for (111) 〈110〉 and (111) 〈112〉 slip systems in InSb. The In atoms and Sb atoms are shown in blue and yellow, respectively. The stacking sequence in InSb along the 〈111〉 direction is the AaBbCc type (where capital letters represent “In” atom planes and small letters represent “Sb” atom planes). The slip plane for the dislocations could be either between glide sets (between B and b atoms) or it could between shuffle sets (between a and B atoms). The inter-layer separation between the planes for the glide set is 0.96 Å and for the shuffle set is 3.83 Å. In panels (b) and (c), we show the in-plane atomic structures for the glide and shuffle sets of planes from the side and top views.

FIG. 5.

(a) The crystallographic orientation of the supercell for calculation of GSFE for (111) 〈110〉 and (111) 〈112〉 slip systems in InSb. The In atoms and Sb atoms are shown in blue and yellow, respectively. The stacking sequence in InSb along the 〈111〉 direction is the AaBbCc type (where capital letters represent “In” atom planes and small letters represent “Sb” atom planes). The slip plane for the dislocations could be either between glide sets (between B and b atoms) or it could between shuffle sets (between a and B atoms). The inter-layer separation between the planes for the glide set is 0.96 Å and for the shuffle set is 3.83 Å. In panels (b) and (c), we show the in-plane atomic structures for the glide and shuffle sets of planes from the side and top views.

Close modal

Achieving a minimum energy structure can involve local atomic shuffling. To isolate the role of atomic shuffling, we calculate GSFE values using two different approaches: (a) a standard approach and (b) a relaxed approach. In the standard approach, we shift the upper half of the crystal with respect to the lower half of the crystal along the glide direction.26 At each displacement, we minimize the energy of the system by fixing all the positions of atoms in both the upper and lower crystals in the x and y directions and allow all other atoms to relax along the z direction. In the relaxed approach, at each displacement, in addition to the relaxation along the z direction, we also allow atoms to relax along the direction normal to the glide direction.

To understand the dislocation glide between two closely spaced (111) planes in InSb, we first calculate the GSFE curve for the glide set (GS) along the 〈110〉 direction using both the standard and relaxed methods. The calculated profiles are shown by circle symbols in Fig. 6. We find a significant difference between the calculated profiles using standard and relaxed methods (open and filled circles in Fig. 6). The GSFE curve for the standard method does not show any local minima, whereas the GSFE curve for the relaxed method shows a very deep local minimum at a 0.5 normalized displacement. Analysis of the atomic structures obtained from the standard and relaxed methods shows that atomic shuffling is involved in accommodating the shear displacement and achieving the stacking fault configuration. As shown in Fig. 7, additional relaxation of atoms along the 〈112〉 direction (which is normal to the 〈110〉 direction) allows atomic shuffling that leads to the minima on the GSFE curve.

FIG. 6.

DFT-based GSFE curves on the (111) plane in InSb for the glide set (GS) and the shuffle set (SS) along 〈110〉 and 〈112〉 glide directions. The displacement on the horizontal axis for each curve is normalized by the Burgers vectors along that direction. The open symbols are GSFE curves obtained from the standard GSFE calculations and the filled symbols are GSFE curves obtained from the relaxed GSFE calculations. The standard and relaxed GSFE curves are the same for both glide and shuffle sets along the 〈112〉 glide direction; therefore, we only show the relaxed GSFE curves along the 〈112〉 glide direction.

FIG. 6.

DFT-based GSFE curves on the (111) plane in InSb for the glide set (GS) and the shuffle set (SS) along 〈110〉 and 〈112〉 glide directions. The displacement on the horizontal axis for each curve is normalized by the Burgers vectors along that direction. The open symbols are GSFE curves obtained from the standard GSFE calculations and the filled symbols are GSFE curves obtained from the relaxed GSFE calculations. The standard and relaxed GSFE curves are the same for both glide and shuffle sets along the 〈112〉 glide direction; therefore, we only show the relaxed GSFE curves along the 〈112〉 glide direction.

Close modal
FIG. 7.

Relaxed atomic structures in the planes above and below the slip plane for the glide set during GSFE calculation along the 〈110〉 direction using the (a) standard and (b) relaxed methods for normalized displacements of 0, 0.25, and 0.5.

FIG. 7.

Relaxed atomic structures in the planes above and below the slip plane for the glide set during GSFE calculation along the 〈110〉 direction using the (a) standard and (b) relaxed methods for normalized displacements of 0, 0.25, and 0.5.

Close modal

In the low temperature range, the experimentally observed slip plane for dislocations to glide lies between two widely spaced (111) atomic planes or shuffle sets. Therefore, we calculate the GSFE curve for the shuffle set (SS) along the 〈110〉 direction using both standard and relaxed methods. The GSFE profiles for the shuffle set along 〈110〉 are shown by square symbols in Fig. 6, and they do not show any local minima on the curves. In contrast to the glide set, for the shuffle set, the difference between the calculated GSFE profiles using standard and relaxed methods (open and filled squares in Fig. 6) is very small. Again, because of the additional relaxation, GSFE based on the relaxed method is lower than that based on the standard method, but we do not see atomic shuffling associated with additional relaxation on this plane.

To understand the observation of a partial dislocation, we also calculate the GSFE curves for the glide and shuffle sets along the 〈112〉 direction using both the standard and relaxed methods. We found that the two methods give the same GSFE along the 〈112〉 direction. This is expected because the atomic structure along the direction normal to 〈112〉 is symmetric. The calculated GSFEs for both shuffle and glide sets using the relaxed method are shown in Fig. 6. The glide set shows a local minimum at a 0.33 normalized displacement (which corresponds to the Burgers vector for Shockley partial a0/6 〈112〉), whereas the shuffle set shows a local minimum at a 0.44 normalized displacement. Burgers vectors of full and partial dislocations are illustrated in Fig. 8. For more detailed information on the full and partial dislocations in semiconductors, we refer the readers to Ref. 27.

FIG. 8.

An illustration of the 〈110〉 and 〈112〉 directions on a {111} plane in cubic diamond crystals. The angle β is 30°.

FIG. 8.

An illustration of the 〈110〉 and 〈112〉 directions on a {111} plane in cubic diamond crystals. The angle β is 30°.

Close modal

The lattice parameter and elastic constants predicted by the interatomic potential17 are in reasonable agreement with DFT and with experimental measurements, as shown in Table I. Here, we calculate the GSFE curves on both glide and shuffle sets using molecular static (MS) simulations, following the same procedures used in DFT calculations (Sec. IV B). It is found that the MS predictions on the glide set are in good agreement with DFT, as shown in Figs. 9(a) and 9(b). However, the GSFE curves on the shuffle set predicted by MS differ greatly from DFT [Fig. 9(c)]. Thus, only the dislocation core structures on the glide set are studied by MS and molecular dynamics (MD) simulations in Sec. V B.

FIG. 9.

MS-based GSFE curves on the (111) plane in InSb for the glide set (GS) and the shuffle set (SS) along the 〈110〉 and 〈112〉 glide directions. DFT results, with the same symbols in Fig. 6, are also shown for comparison. It is found that MS well represents the GSFE curves on (a) and (b) the glide set but not (c) the shuffle set.

FIG. 9.

MS-based GSFE curves on the (111) plane in InSb for the glide set (GS) and the shuffle set (SS) along the 〈110〉 and 〈112〉 glide directions. DFT results, with the same symbols in Fig. 6, are also shown for comparison. It is found that MS well represents the GSFE curves on (a) and (b) the glide set but not (c) the shuffle set.

Close modal

To study the core structures of glide set dislocations, we set up a 3D periodic simulation cell containing a dislocation dipole with two dislocations of the same type but with an opposite Burgers vector.28 Four dislocation character angles are considered: 0° (screw), 30°, 60°, and 90° (edge), where the angle is defined as the in-plane angle between the Burgers vector and the dislocation line. Each dislocation is built by applying the corresponding isotropic elastic displacement field to all atoms.29 It follows that equilibrated core structures of dislocations are calculated at four temperatures: −273.15 °C, −176 °C, 20 °C, and 300 °C. At the lowest temperature (absolute zero), MS simulations are conducted, for which an energy minimization via the conjugate gradient scheme is conducted, until the change in the energy between successive iterations divided by the most recent energy magnitude is less than or equal to 10−15. At the other three higher temperatures, which were used in our experiments, MD simulations are carried out, for which dynamic equilibrium is achieved by running dynamic simulations for 100 000 time steps with a time step size of 1 fs.

Figure 10 presents the ISF width, d, associated with the four types of dislocations, based on the disregistry fields.30,31 At the same temperature, d increases with the character angle, with the screw dislocation being nearly undissociated and the edge dislocation being the most dissociated. Similar trends are found for dislocations in pure FCC crystals.28,29 This is expected since elastic anisotropy in InSb is cubic as in pure FCC metals and the GSFE surface on the glide set plane is similar in topology to that in FCC. In addition, we find that, for the same character angle, the ISF width increases with the temperature, but only by about 10% from −273.15 °C to 300 °C. The finding that the glide set dislocations do not possess a narrow core regardless of the temperature is aligned with our experimental observation that dislocations with narrow cores belong to the shuffle set.

FIG. 10.

ISF width, d, predicted by MS (at −273.15 °C) and MD (other temperatures) simulations for dislocations with the edge, 30°, 60°, and screw characters, respectively.

FIG. 10.

ISF width, d, predicted by MS (at −273.15 °C) and MD (other temperatures) simulations for dislocations with the edge, 30°, 60°, and screw characters, respectively.

Close modal

In order to understand which dislocation modes could easily move under an applied shear stress in the system, we also calculated the ideal shear stresses (ISSs) using calculated DFT- and MS-based GSFE energy profiles. The ISS is defined as 1bγu, where γ is the GSFE energy profile. The calculated ISSs for the glide and shuffle sets are shown in Fig. 11. The maximum values corresponding to the first peak, which are also called ideal shear strengths (ISSth), are given in Table II. They suggest that the shuffle set in the 〈112〉 slip direction is the easiest to move and the shuffle set in the 〈110〉 slip direction is the next easiest. This comparison takes into account that the former dislocations are compact and the latter dislocations are partial dislocations belonging to the extended glide set 〈112〉 dislocation (see Table II).

FIG. 11.

ISS from the derivative of the relaxed GSFE curves on the glide set (GS) and the shuffle set (SS), based on (a) DFT and (b) MS. MS simulations are done only for the glide set because the MS-based GSFE curves for the shuffle set are not accurate [Fig. 9(c)].

FIG. 11.

ISS from the derivative of the relaxed GSFE curves on the glide set (GS) and the shuffle set (SS), based on (a) DFT and (b) MS. MS simulations are done only for the glide set because the MS-based GSFE curves for the shuffle set are not accurate [Fig. 9(c)].

Close modal
TABLE II.

Summary of DFT-based GSFE calculations in Fig. 6 and ISS calculations in Fig. 11(a) for the glide set and the shuffle set in InSb using the relaxed method. γusf represents the unstable stacking fault energy value corresponding to the first maximum and γsf is the ISF energy corresponding to the local minimum on the GSFE curves. ISSth is the ideal shear strength. The final column is an approximate estimate of volume b3 of the dislocation core.

Slip systemBurgers vector magnitude b (Å)γusf (mJ/m2)γsf (mJ/m2)ISSth (GPa)Volume (Å3)
PerfectPartial
(111) 〈110〉 Glide 4.69 2.346 798 at 0.25b 65 at 0.5b 8.72 12.91 (partial) 
Shuffle 4.69  578 at 0.5b 3.87 103.29 
(111) 〈112〉 Glide 8.13 2.709 789 at 0.167b 39 at 0.33b 5.92 19.88 (partial) 
Shuffle 8.13 3.576 637 at 0.33b 592 at 0.44b 3.42 45.729 (partial) 
Slip systemBurgers vector magnitude b (Å)γusf (mJ/m2)γsf (mJ/m2)ISSth (GPa)Volume (Å3)
PerfectPartial
(111) 〈110〉 Glide 4.69 2.346 798 at 0.25b 65 at 0.5b 8.72 12.91 (partial) 
Shuffle 4.69  578 at 0.5b 3.87 103.29 
(111) 〈112〉 Glide 8.13 2.709 789 at 0.167b 39 at 0.33b 5.92 19.88 (partial) 
Shuffle 8.13 3.576 637 at 0.33b 592 at 0.44b 3.42 45.729 (partial) 

Taken together, the GSFE curves and ISS calculations can lend insight into the selection of the dislocation type during glide. The relaxed glide set GSFE 〈110〉 curve suggests that the glide set would split into two with equal Burgers vector along the 〈110〉 direction with a fault in-between. The ISF energy, γsf, is low, suggesting wide stacking faults. The relaxation results in a lower energy barrier. It is interesting to note that the calculated value of γsf, 39 mJ/m², is in very good agreement with both the one measured on the EN (Sec. III A), 36 mJ/m² ± 3 mJ/m², and other literature data, 38 mJ/m² ± 4 mJ/m², as reported in Ref. 32. The glide set GSFE 〈112〉 curve would also dissociate into two Shockley partials, with differing Burgers vector directions but equal in value. Between these two glide set reactions, the 〈112〉 direction is the more likely to be preferred since the formation energetic barrier γusf is lower, the displacement jump associated with γusf is shorter, and the ISSth, indicating mobility, is lower than those for 〈110〉 dissociation.

Based on the unstable stacking fault energy (Fig. 6) and calculated ISSth at 0 K (Table II), it follows that shuffle set dislocations will nucleate first and will be observed in the very low temperature range consistent with suggestions from experimental observations below RT. At very low temperatures, glide set dislocations would need more load/applied stress as ISSth values are large compared to shuffle set dislocations. It is speculated that the glide set may only be activated with the help of thermal activation in the higher temperature range.

Below, we revisit the experimental observations within each distinct temperature regime in light of the atomic scale calculations.

The unstable stacking fault energy (γusf) for the shuffle set along the 〈110〉 direction is the lowest (578 mJ/m2) compared to other modes; therefore, it would be nucleated first in the low temperature range. Also, as shown in Fig. 6, the shuffle set along the 〈110〉 direction would not dissociate into partial dislocations, which is consistent with experimental observations of perfect non-dissociated dislocation in the low temperature brittle region.

Experimental results suggest that both dissociated glide set dislocations and non-dissociated shuffle set dislocations are possible in this regime, particularly near RT. On the one hand, the shuffle set 〈112〉 Shockley partial dislocations are easy to activate since the ISSth value is low, but based on their unstable stacking fault values and Burgers vector values, they are energetically less favorable compared to the shuffle set 〈110〉 full dislocations (see Table II). Both shuffle sets along the 〈112〉 and 〈110〉 glide directions are easier to activate than the 〈112〉 and 〈110〉 glide sets. On the other hand, Fig. 6 suggests that the glide set can dissociate along the 〈110〉 and 〈112〉 directions. The formation of both glide set dislocations has similar activation barriers to nucleate, but the 〈112〉 direction has lower ISSth, which implies that it is more mobile and favored. Therefore, it may be that when sufficient amounts of thermal energy are provided, glide set dislocations can form and subsequently dissociate, forming a dissociated dislocation possessing a wider core. Once it has a wider core, it is more mobile than an undissociated shuffle set dislocation. The probability of this reaction increases with the temperature. At and above 20 °C, thermal energy can facilitate the dissociation in some regions of the material but not all.

The ISSth values for the glide set dislocations are large compared to the shuffle set ones. Therefore, glide set dislocations may be activated in the higher temperature range and would be favored in the higher temperature range as a perfect dislocation would split into two partials. The magnitude of each partial is relatively small, making them more mobile than that of a perfect dislocation and more kinetically favorable.

Kaxiras and Duesbery33 have shown for Si that γusf is low for the shuffle set and dislocations in this set are nucleated first and are preferred in the low temperature range. This aspect is similar to InSb. Furthermore, for Si, it has been suggested that the glide set has higher entropy compared to the shuffle set.33,34 Assuming that the same applies to InSb, then with increasing temperature, the free energy, F =E(γusf) − TS + PV, where S is the entropy, P is the pressure, and V is the volume, associated with the glide set dislocation could be lower than that for the shuffle set, resulting in a shuffle to glide transition.

In this work, dislocations in InSb, a semiconductor material, are studied via atomic-level calculations and deformed bulk InSb single crystals via TEM over a range of temperatures. TEM analysis of the deformed bulk single crystals shows that at low temperatures (<20 °C), dislocations have narrow, non-dissociated cores, while at higher temperatures (>150 °C), they have dissociated cores. DFT calculations are conducted to obtain the GSFE curves for the shuffle and glide sets in InSb. Ideal shear strengths, which indicate the level of difficulty to activate dislocations, are derived from the GSFE curves. Atomistic simulations are performed to study the ISF widths associated with glide set dislocations of pure and mixed characters. Taken together, our atomic-level calculations indicate that the glide set dislocations dissociate into two partial dislocations, while the shuffle set dislocations would not. Yet, the shuffle set dislocations possess lower ideal shear strengths, indicating that it is easier to be activated than the glide set ones. Hence, it is proposed that at low temperatures, the motion of shuffle set dislocations, which have more compact cores, dominate plasticity. As the temperature increases, the motion of glide set dislocations, which have dissociated core structures, would become dominant.

Y. S. and I.J.B. acknowledge financial support from the National Science Foundation Designing Materials to Revolutionize and Engineer our Future (DMREF) program (NSF CMMI-1729887). Use was made of computational facilities purchased with funds from the National Science Foundation (NSF) (No. CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; No. NSF DMR 1720256) at UC Santa Barbara. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation under Grant No. ACI-1053575. This work partially pertains to the French Government program “Investissements d’Avenir” (LABEX INTERACTIFS, Reference No. ANR-11-LABX-0017-01).

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