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Anisotropy and twinning in cubic zinc sulfide crystals
I Phys. Chm Solids Vol. 43, No. 6, pp. 497-500. 1982 Printed in Great Britain.
MC?-36!W82.lW/O Pcrgamon Press Ltd.
ANISOTROPY AND TWINNING IN CUBIC ZINC SULFIDE CRYSTALS WAYNEL. GARRETT? U.S. Army Armament Research and Development Command, Dover, NJ 07801,U.S.A. GERHARDRUBAN Institut fib Kristallographie, Freie Universitit Berlin, 1000Berlin 33, Germany and FERD WILLIAMS Physics Department, University of Delaware, Newark, DE 19711,U.S.A. (Received 22 September 1980;accepted in revised form 28 October 1981) AbstractThe structure of zinc sulfide single crystals grown from the vapor has been studied by a unique combination of X-ray crystallography and thermal conductivity measurements. Weissenberg patterns reveal the face centered cubic structure with rotational twinning in the [ill] direction, which could be ascribed to two overlapping lattices. These could be explained by normal stacking faults or by inverted twins. A strong anisotropy in thermal conductivity was measured and interpreted as favoring the inverted twin model.
1.
INTRODUCTION
The growth of synthetic ZnS crystals by sublimation in a static, self-sealing tube has yielded large crystals of
excellent quality[l]. The growth parameters in this system have been described [2] and were generally followed in this study. The ambient atmosphere in the growth vessel consisted of H2S with a 20% admixture of HCI. The difference between the temperature at the growth tip and the sublimation temperature at the ZnS charge (1300°C)was set at 10°C at the beginning of a growth run. The tip was then advanced through the furnace gradient (30”C/cm) at a rate of 3.0cm/day. As the tip advances, the temperature difference and thus the supersaturation increase until a critical value necessary to initiate crystal growth is reached. The crystals grew five to six cubic centimeters in about ten hours. This procedure yielded large single crystals about 80% of the time. The success of this growth procedure was attributed to an improvement in the process of initial nucleation. The crystals were examined microscopically and by X-ray diffraction techniques. The crystals were clear throughout and showed well developed faces. The basal plane [ll 11,when present, appeared smooth and glassy whereas the perpendicular side faces contained striations. Striations have been used to explain a layer-type growth [3] and have been correlated to stacking faults [4]. Photomicrographs taken of the basal plane surfaces revealed growth pyramids and well defined growth layers with step heights about one micron thick. These growth layers apparently start near the edge of the basal plane surface and advance in two directions, the junction of these directions joining to make a 120”angle.
tPresent address: Bell Telephone Laboratories, Holmdel, NJ 07733.U.S.A. IPCS Vol. 43. No. 6-A
X-Ray analysis of the crystals using a Weissenberg camera revealed a face centered cubic structure containing a rotational twinning fault along the [ll 11 direction. Zinc sulfide is known to crystallize in a cubic face centered (sphalerite) or hexagonal close packed (wurtzite) structure. In addition, each structure may contain various polytypes characterized by alterations in the c-axis dimensions [5]. The ultimate structure depends both on the growth temperature and on impurities. The transition between the sphalerite and hexagonal structures takes place at 1020°Cand is reversible 161. Either the fee or hexagonal structure can be built up by stacking together Zn& and SZn, tetrahedrons. The structure characterized by a repeat pattern ACAC.. . is hexagonal whereas the fee structure has an ABCABC.. . repeat pattern. The most common fault in the sphalerite structure is the rotational twin which alters the repeat pattern to ABCACBA.. . . Such a pattern might also be described as the inclusion of a hexagonal sequence into a predominantly cubic stacking sequence. 2. CRYSTALLOGRAPEY Reciprocal Weissenberg
lattice reflections were recorded with a camera, using Cu-K, radiation (1.542 A).
The crystals were rotated about the [ilO] direction and photographs taken of the plane containing the [ 11l] and [ii2j axes. The measured lattice spacings were: (i 10)= 3.83; (172) = 6.63 and (111) = 9.32 A. When the cell is transformed to the standard cubic axes, the calculated lattice constant becomes 5.43 A, which agrees with published values. Weissenberg photographs of the 0th and 1st layer lines revealed two main features: there were several sharp reflections and additional diffuse reflections which might be assigned to stacking faults[71. After evaluating the reflections it was found impossible to build up a unique reciprocal lattice corresponding to the 497
WAYNEL. GARRETT et al.
498
sphalerite structure. However, it was possible to construct two overlapping lattices resembling the case of twinned crystals. These two distinguishable crystallites we name “individuals” in accordance with previous work on twinning[8]. Figure 1 shows the reciprocal lattices overlapping. The observed reflections of one individual are marked by squares and connected by broken lines; those of the other individual are marked by dots and connected by full lines. Sharp reflections occur at the intersections of broken and solid lines with the remaining reflections diffuse. From the measured reciprocal lattice spacings, assignments can be made for the directions. In general, there are two different ways of describing the overlapping observed. In both cases, the reciprocal lattice of one individual is rotated to match the reciprocal lattice points of the other. In the first case, the [ll l] directions of both individuals are set parallel (referring to Fig. 1, they meet in the direction marked L). The resulting reciprocal lattices correspond to a coincidence of [liO] and [IlO] as rotational axes or [llz] and [ii2], respectively, as directions perpendicular to [ill] and the rotational axes. Referring to cubic reciprocal vectors (a’, b’, c+), the first individual can be expressed by orthogonal vectors (A,‘, BI+, C,‘) accordmg to the transformation: (Al+, Blf, C,‘) = ~1. (a’, b+, c’) where T; is the adjunct matrix of 7,:
u1= i
l/2 l/2 1
-l/2 l/2 1
-0 -1 1
(1)
. r
[ii21
?
The Miller indices transform using the inverse matrix as: (H,, K,, L,) = u,(h, k, 1), small letters always referring to the original cubic crystal. The second individual is described by a 180” rotation around C,+: (A*+, Bz+, C,‘) = 7;. (a’, b’, c+), where
The cubic indices of the second individual (/I, k, I)z are related to (Zf,, K2, L) by (h, k, I), = Q(&, K,, L2). For overlapping reflections (H,, K,, L,) = (Hz. K,, L,); we eventually get (II, k, 1)* = TV’ u, ( (h, k, I),, with the product opeator
being orthogonal and thus suitable to describe any relationships between the two lattices, also in crystal space. The most interesting part of this result is the coincidence of several lattice directions having different spacings, e.g. [ill] of one individual coinciding with [5ii] of the other individual, (333) and (5ii) being the corresponding reflections which overlap. Trying to align such a crystal by means of rotational photographs leads to some difficulties because of the identity period dsri= 28.118, instead of dill = 9.37 8. Furthermore the symmetry observed no longer retains the three-fold rotational axes, which should occur parallel [[111]] in the cubic ZnS structure. In all, three of the four tetrahedral directions are affected, namely [Till, [iii], and [iii], leaving only [ill] untouched. The physical meaning of the described 180”rotation of the reciprocal lattice is an identical rotation of the crystal lattice itself, around [ 1111.
[115]
Fig. 1. Reciprocallattice of cubic zinc sulfide. 0, for one individual; 0, for other individual; EI, superposition of both. L is (1111for both individuals; in K direction one individual corresponds to [ll?], other has ]li2]; numbers are indices: horizontal ones in L direction, vertical in K direction.
Anisotropyand hvinnii in cubic zinc suhidecrystals
Such a turnover is still consistent with a lattice being built up from ZnS, and SZn., tetrahedrons as well. Assuming [ill] as the preferred orientation with respect to the growth process, only the stacking sequence is altered, creating a stacking fault. This type of twinning was observed in diamond [9]. The second way of describing the observed overlapping of the reciprocal lattices from the two individuals leads to a similar 180”rotation. Starting again with (Al+, B I+, Cl+) = 71 *(a+, b+, c’) as above, the second individual is described by (As+, B3+,C,‘) = 7; - (a+, b+, c+), where
7; =
-1 0 0 01 0 0 0 -1
(4)
Analogously we gain a product operator: -213 7, . u1 = i l/3 -213
l/3 -213 -213
-213 -213 l/3
(5)
describing the interrelation between (h, k, 01 and (h, k, I),. the latter being different from (h, k, 1)*. The interpretation in terms of coincident lattice directions having different spacings is literally the same, yet the conclusions are different. Rotating ZnS4 tetrahedrons around directions perpendicular to [ill] is impossible without breaking bonds and is therefore no longer consistent with a unique arrangement of the crystal in sequential, though faulted, stacking order. Assuming such a case would rather imply the occurrence of really distinct crystal individuals which grow ___together in opposite directions, i.e. [ 11l]r parallel to [ 11lb. These individuals eventually meet at inner surfaces, where equivalent atoms such as sulfur atoms face each other. Such an inverted twin contains barriers. A decision between these two models cannot be made from the X-ray diffraction results. A joint description of the two cases applying to both of the individuals discussed is possible by defining a new trigonal unit cell, in hexagonal setting:
(A, B, C) = ~(a, b, c); u =
l/2 l/2 1
-1 l/2 1
l/2 -1 1
6) i.e. N=9 instead of 4 ZnS molecules per unit cell, as before. The lattice dimensions are A = B = 6.63 and C = 9.36 A, different from the wurtzite type. There are some advantages in favor of such a new type of structure. The atomic positions in the cubic structure, for instance, acquire odd values after a turnover occurs. A sulfur position in the unfaulted cubic structure with fractional parameters (x, y, z) = l/4, 3/4,3/4 would shift, in the turnover, to 11/12, 5/12, 5/12. Transformation to the trigonal lattice results in (X, Y 2) parameters -l/3,- l/3, 7/12 and l/3, l/3, 7/12, respectively. Actually any atomic position can be expressed by X = Y = 0 mod l/3, leaving uncertain, ac-
499
cording to stacking faults, only Z = 0 mod l/3 for Zn and Z= l/4 mod l/3 for S atoms.
3. TEERMAL CONDUCTMTY
In an effort to distinguish between the two explanations of the overlapping lattices derived from the crystallographic data, the one involving normal stacking faults and the other inverted twins, we considered other experimental techniques. The inverted twin is more like an interface than are the normal stacking faults and thus is expected to scatter phonons to a greater extent and thereby reduce thermal conductivity. The thermal conductivity of zinc blende is of course isotropic, however, the presence of either normal stacking faults or inverted twins in planes perpendicular to the [ill] direction was expected to yield anisotropic thermal conductivity with the greater thermal impedance in the [ 11I] direction. Crystals were grown and polished to appropriate dimensions and orientation, cubes 0.4mm on an edge with an axis in the [ill] direction, and arrangements were made for the measurements of thermal conductivity in the different cube directions. The results for measurements in all three directions on the best sample are given in Table 1. Other samples gave results within the error limits indicated in Table 1, indicating that the concentrations of defects are quite uniform from sample to sample. Copper-doped zinc sulfide was also measured and found to have similar anisotropy but with somewhat lower conductivities in both directions, specifically 0.125f 0.005 and 0.19 f 0.01 W/cm”C. These data show that concentration of either normal stacking faults or inverted twins is large, larger in fact than was expected from the crystallographic data. Unfortunately, a quantitative theory of the effect of each of these two types of laminar defects on thermal conductivity has not been developed. A first approximation to an analysis of these data can be based on assuming the crystal to be inhomogeneous and consisting of parallel slabs of regions with defects of volume fraction V and of regions of perfect zinc blende of volume fraction (1- V). We assume that the two components are separately isotropic, thus we are modelling the laminar defects which scatter phonons anisotropically by laminar regions which are themselves isotropic but which result in anisotropic thermal conduction of the total system because of the orientation of these occlusions. The twin boundaries or stacking faults are thus considered as perpendicular to the c-direction [ 11l] of the cubic structure, of finite thickness in the cdirection and with isotropic thermal conductivity different from that of the zinc blende. The well-known Table 1. Direction
Thermalconductivity
IIIlIt 111111 1’[1111
0.149+ 0.007W/cm”C 0.204f 0.010W/cm”C 0.2182 0.010W/cm”C
equations [lo] for the components of conductivity for this model are: 1 V (1-v) -_=-+KII Kd Kb
(7)
and K,=
VKd+(l-
v)&
(8)
where K,I is parallel to [111] and thus perpendicular to the planes of the slabs, Kl is perpendicular to 11111and thus parallel to the planes of the slabs, and Kd and Kb are the conductivities of the regions of laminar defects and of the perfect zinc blende, respectively. Equations are available for more complex composite structures, for example with either interconnecting component of high or low conductivities[lO], however, the plane parallel model gives the maximum anisotropy and is probably the best macroscopic model for these crystals. Microscopic analyses of phonon scattering by the two types of laminar defects are being investigated. The value of K for natural crystals of cubic ZnS was measured more than fifty years ago[ll]; much more recently K was measured over an extended temperature from 2 to 300K for pure synthetic crystals[l2]. The accepted value of K at room temperature for cubic ZnS is 0.27W/cm deg. Substitution of this value for & and from Table 1 for K,I and K, into eqns (7) and (8) yield Kd = 0.073 and V = 0.30, a surprisingly large value for the volume fraction of the region of laminar defects. There may be other scattering centers for phonons in our crystals, for example self-activated centers, i.e. zinc vacancy plus nearest neighbour donor dopant, which reduce Kb below the literature value, similar to the effect of copper doping. In order to reduce V to 0.1, Kb would have to be as low as 0.23 W/cm”C, which seems unlikely. The assumption of greater scattering of twin boundaries compared to that of stacking faults can be justified as follows: At the highest frequencies, i.e. at and near the Debye frequency oD, the scatterings are probably of comparable strength. However, the scattering of phonons by twin boundaries should be to a good approximation frequency independent while the mean free path for scattering by stacking faults should vary approximately as w-* below wD The ratio of mean scattering cross sections for the two types of defects depends on the phonon distribution as well as on a more complete analysis of the scattering, however, the above shows that the scattering by twin boundaries is surely greater than that by stacking faults. The difference in scattering by twin boundaries and stacking faults, just described for phonons, is also valid for photons. Visible light should be scattered almost
specularly by twin boundaries and hardly at all by stacking faults. The visible appearance of the crystals tends to support the presence of twin boundaries. 4.CONCLUSIONS
The Weissenberg patterns of zinc sulfide crystals grown from the vapour phase are in accordance with the zinc blende structure but with rotational twinning in the [111] direction. Analysis shows two overlapping lattices. The fitting together of these lattices can be explained by normal stacking faults or by inverted twins. From the thermal conductivity we conclude that a substantial fraction, approximately 30%, of the crystal volume consists of laminar defects with large cross sections for phonon scattering. The inverted twins have more nearly the proper characteristics, however, normal stacking faults cannot be eliminated as contributing, at least in part, to the anisotropy of the thermal conductivity and to the crystallographic results. The visible appearance of the crystals, in particular the oriented striations, tends to support the existence of twin boundaries. Other techniques can supplement the two reported in this paper in fully characterizing the structure of these crystals, for example, the Zeeman effect on the optical spectra of Ni*+ dopants at mixed cubic-hexagonal environment of ZnS has recently been reported [ 131. Acknowledgements-The authors are indebted to Dr. Donald Spitzerof the ResearchLaboratoryof the AmericanCyanamidCo for makingthe thermal conductivitymeasurementsand for helpful discussions. One of the authors (FW) thanks the Humboldt Foundation for support during the final preparation of the manuscript.This researchwas also supportedin part by a grant from the U.S. Army Research O&e-Durham to the University of Delaware. The authors thank the referee for helpful theoretical observations on the scattering of phonons by twin boundaries and by stacking faults. REFERENCES
I. Piper W. W. and Polich S. J., 1. Appl. Phys. 32, 1278(l%l). 2. Indradev. Mat. Res. Bull. 1. 173(1%6). 3. Stock L.‘W. and Brophy VI A., Am. Mineral 40,94 (1955).
4. Green L. C., Reynolds D. C. and Czyzak S. J. and Baker W. M., L Chem. Phys. 29, 1375(1958). 5. Allen E. T. and Crenshaw J. L., Am. L Sci 34,341 (1912). 6. Dev I., Br. J Appl. Phys. 17,761 (1%6). 1. Jagodzinski H., Acla Crysr. 2,201 (1949). 8. Hartshorne N. N. and Stuart A., Practical Optical Crystalbgraphy. Arnold, London (1%9). 9. BraPa Sir Lawrence and Clarinabull G. F.. Crvstal Structure of &erals. G. Bell, London (1%5). ’ 10. Parrott J. E. and Stuckes Audrey D., Thermal Canductiuities of Solids pp. 129-132.Pion, London (1975). 11. Eucken A. and Kuhn G., Z Physik, Gem. (Leipzig) 134,193 (1928). 12. Slack G. A., Phys. Reu. I%,3791 (1972). 13. Broser I., Germer R. and Schulz H.-J., 1 Magnetism Magnetic Materials 15-l&l,29 (1980).
MC?-36!W82.lW/O Pcrgamon Press Ltd.
ANISOTROPY AND TWINNING IN CUBIC ZINC SULFIDE CRYSTALS WAYNEL. GARRETT? U.S. Army Armament Research and Development Command, Dover, NJ 07801,U.S.A. GERHARDRUBAN Institut fib Kristallographie, Freie Universitit Berlin, 1000Berlin 33, Germany and FERD WILLIAMS Physics Department, University of Delaware, Newark, DE 19711,U.S.A. (Received 22 September 1980;accepted in revised form 28 October 1981) AbstractThe structure of zinc sulfide single crystals grown from the vapor has been studied by a unique combination of X-ray crystallography and thermal conductivity measurements. Weissenberg patterns reveal the face centered cubic structure with rotational twinning in the [ill] direction, which could be ascribed to two overlapping lattices. These could be explained by normal stacking faults or by inverted twins. A strong anisotropy in thermal conductivity was measured and interpreted as favoring the inverted twin model.
1.
INTRODUCTION
The growth of synthetic ZnS crystals by sublimation in a static, self-sealing tube has yielded large crystals of
excellent quality[l]. The growth parameters in this system have been described [2] and were generally followed in this study. The ambient atmosphere in the growth vessel consisted of H2S with a 20% admixture of HCI. The difference between the temperature at the growth tip and the sublimation temperature at the ZnS charge (1300°C)was set at 10°C at the beginning of a growth run. The tip was then advanced through the furnace gradient (30”C/cm) at a rate of 3.0cm/day. As the tip advances, the temperature difference and thus the supersaturation increase until a critical value necessary to initiate crystal growth is reached. The crystals grew five to six cubic centimeters in about ten hours. This procedure yielded large single crystals about 80% of the time. The success of this growth procedure was attributed to an improvement in the process of initial nucleation. The crystals were examined microscopically and by X-ray diffraction techniques. The crystals were clear throughout and showed well developed faces. The basal plane [ll 11,when present, appeared smooth and glassy whereas the perpendicular side faces contained striations. Striations have been used to explain a layer-type growth [3] and have been correlated to stacking faults [4]. Photomicrographs taken of the basal plane surfaces revealed growth pyramids and well defined growth layers with step heights about one micron thick. These growth layers apparently start near the edge of the basal plane surface and advance in two directions, the junction of these directions joining to make a 120”angle.
tPresent address: Bell Telephone Laboratories, Holmdel, NJ 07733.U.S.A. IPCS Vol. 43. No. 6-A
X-Ray analysis of the crystals using a Weissenberg camera revealed a face centered cubic structure containing a rotational twinning fault along the [ll 11 direction. Zinc sulfide is known to crystallize in a cubic face centered (sphalerite) or hexagonal close packed (wurtzite) structure. In addition, each structure may contain various polytypes characterized by alterations in the c-axis dimensions [5]. The ultimate structure depends both on the growth temperature and on impurities. The transition between the sphalerite and hexagonal structures takes place at 1020°Cand is reversible 161. Either the fee or hexagonal structure can be built up by stacking together Zn& and SZn, tetrahedrons. The structure characterized by a repeat pattern ACAC.. . is hexagonal whereas the fee structure has an ABCABC.. . repeat pattern. The most common fault in the sphalerite structure is the rotational twin which alters the repeat pattern to ABCACBA.. . . Such a pattern might also be described as the inclusion of a hexagonal sequence into a predominantly cubic stacking sequence. 2. CRYSTALLOGRAPEY Reciprocal Weissenberg
lattice reflections were recorded with a camera, using Cu-K, radiation (1.542 A).
The crystals were rotated about the [ilO] direction and photographs taken of the plane containing the [ 11l] and [ii2j axes. The measured lattice spacings were: (i 10)= 3.83; (172) = 6.63 and (111) = 9.32 A. When the cell is transformed to the standard cubic axes, the calculated lattice constant becomes 5.43 A, which agrees with published values. Weissenberg photographs of the 0th and 1st layer lines revealed two main features: there were several sharp reflections and additional diffuse reflections which might be assigned to stacking faults[71. After evaluating the reflections it was found impossible to build up a unique reciprocal lattice corresponding to the 497
WAYNEL. GARRETT et al.
498
sphalerite structure. However, it was possible to construct two overlapping lattices resembling the case of twinned crystals. These two distinguishable crystallites we name “individuals” in accordance with previous work on twinning[8]. Figure 1 shows the reciprocal lattices overlapping. The observed reflections of one individual are marked by squares and connected by broken lines; those of the other individual are marked by dots and connected by full lines. Sharp reflections occur at the intersections of broken and solid lines with the remaining reflections diffuse. From the measured reciprocal lattice spacings, assignments can be made for the directions. In general, there are two different ways of describing the overlapping observed. In both cases, the reciprocal lattice of one individual is rotated to match the reciprocal lattice points of the other. In the first case, the [ll l] directions of both individuals are set parallel (referring to Fig. 1, they meet in the direction marked L). The resulting reciprocal lattices correspond to a coincidence of [liO] and [IlO] as rotational axes or [llz] and [ii2], respectively, as directions perpendicular to [ill] and the rotational axes. Referring to cubic reciprocal vectors (a’, b’, c+), the first individual can be expressed by orthogonal vectors (A,‘, BI+, C,‘) accordmg to the transformation: (Al+, Blf, C,‘) = ~1. (a’, b+, c’) where T; is the adjunct matrix of 7,:
u1= i
l/2 l/2 1
-l/2 l/2 1
-0 -1 1
(1)
. r
[ii21
?
The Miller indices transform using the inverse matrix as: (H,, K,, L,) = u,(h, k, 1), small letters always referring to the original cubic crystal. The second individual is described by a 180” rotation around C,+: (A*+, Bz+, C,‘) = 7;. (a’, b’, c+), where
The cubic indices of the second individual (/I, k, I)z are related to (Zf,, K2, L) by (h, k, I), = Q(&, K,, L2). For overlapping reflections (H,, K,, L,) = (Hz. K,, L,); we eventually get (II, k, 1)* = TV’ u, ( (h, k, I),, with the product opeator
being orthogonal and thus suitable to describe any relationships between the two lattices, also in crystal space. The most interesting part of this result is the coincidence of several lattice directions having different spacings, e.g. [ill] of one individual coinciding with [5ii] of the other individual, (333) and (5ii) being the corresponding reflections which overlap. Trying to align such a crystal by means of rotational photographs leads to some difficulties because of the identity period dsri= 28.118, instead of dill = 9.37 8. Furthermore the symmetry observed no longer retains the three-fold rotational axes, which should occur parallel [[111]] in the cubic ZnS structure. In all, three of the four tetrahedral directions are affected, namely [Till, [iii], and [iii], leaving only [ill] untouched. The physical meaning of the described 180”rotation of the reciprocal lattice is an identical rotation of the crystal lattice itself, around [ 1111.
[115]
Fig. 1. Reciprocallattice of cubic zinc sulfide. 0, for one individual; 0, for other individual; EI, superposition of both. L is (1111for both individuals; in K direction one individual corresponds to [ll?], other has ]li2]; numbers are indices: horizontal ones in L direction, vertical in K direction.
Anisotropyand hvinnii in cubic zinc suhidecrystals
Such a turnover is still consistent with a lattice being built up from ZnS, and SZn., tetrahedrons as well. Assuming [ill] as the preferred orientation with respect to the growth process, only the stacking sequence is altered, creating a stacking fault. This type of twinning was observed in diamond [9]. The second way of describing the observed overlapping of the reciprocal lattices from the two individuals leads to a similar 180”rotation. Starting again with (Al+, B I+, Cl+) = 71 *(a+, b+, c’) as above, the second individual is described by (As+, B3+,C,‘) = 7; - (a+, b+, c+), where
7; =
-1 0 0 01 0 0 0 -1
(4)
Analogously we gain a product operator: -213 7, . u1 = i l/3 -213
l/3 -213 -213
-213 -213 l/3
(5)
describing the interrelation between (h, k, 01 and (h, k, I),. the latter being different from (h, k, 1)*. The interpretation in terms of coincident lattice directions having different spacings is literally the same, yet the conclusions are different. Rotating ZnS4 tetrahedrons around directions perpendicular to [ill] is impossible without breaking bonds and is therefore no longer consistent with a unique arrangement of the crystal in sequential, though faulted, stacking order. Assuming such a case would rather imply the occurrence of really distinct crystal individuals which grow ___together in opposite directions, i.e. [ 11l]r parallel to [ 11lb. These individuals eventually meet at inner surfaces, where equivalent atoms such as sulfur atoms face each other. Such an inverted twin contains barriers. A decision between these two models cannot be made from the X-ray diffraction results. A joint description of the two cases applying to both of the individuals discussed is possible by defining a new trigonal unit cell, in hexagonal setting:
(A, B, C) = ~(a, b, c); u =
l/2 l/2 1
-1 l/2 1
l/2 -1 1
6) i.e. N=9 instead of 4 ZnS molecules per unit cell, as before. The lattice dimensions are A = B = 6.63 and C = 9.36 A, different from the wurtzite type. There are some advantages in favor of such a new type of structure. The atomic positions in the cubic structure, for instance, acquire odd values after a turnover occurs. A sulfur position in the unfaulted cubic structure with fractional parameters (x, y, z) = l/4, 3/4,3/4 would shift, in the turnover, to 11/12, 5/12, 5/12. Transformation to the trigonal lattice results in (X, Y 2) parameters -l/3,- l/3, 7/12 and l/3, l/3, 7/12, respectively. Actually any atomic position can be expressed by X = Y = 0 mod l/3, leaving uncertain, ac-
499
cording to stacking faults, only Z = 0 mod l/3 for Zn and Z= l/4 mod l/3 for S atoms.
3. TEERMAL CONDUCTMTY
In an effort to distinguish between the two explanations of the overlapping lattices derived from the crystallographic data, the one involving normal stacking faults and the other inverted twins, we considered other experimental techniques. The inverted twin is more like an interface than are the normal stacking faults and thus is expected to scatter phonons to a greater extent and thereby reduce thermal conductivity. The thermal conductivity of zinc blende is of course isotropic, however, the presence of either normal stacking faults or inverted twins in planes perpendicular to the [ill] direction was expected to yield anisotropic thermal conductivity with the greater thermal impedance in the [ 11I] direction. Crystals were grown and polished to appropriate dimensions and orientation, cubes 0.4mm on an edge with an axis in the [ill] direction, and arrangements were made for the measurements of thermal conductivity in the different cube directions. The results for measurements in all three directions on the best sample are given in Table 1. Other samples gave results within the error limits indicated in Table 1, indicating that the concentrations of defects are quite uniform from sample to sample. Copper-doped zinc sulfide was also measured and found to have similar anisotropy but with somewhat lower conductivities in both directions, specifically 0.125f 0.005 and 0.19 f 0.01 W/cm”C. These data show that concentration of either normal stacking faults or inverted twins is large, larger in fact than was expected from the crystallographic data. Unfortunately, a quantitative theory of the effect of each of these two types of laminar defects on thermal conductivity has not been developed. A first approximation to an analysis of these data can be based on assuming the crystal to be inhomogeneous and consisting of parallel slabs of regions with defects of volume fraction V and of regions of perfect zinc blende of volume fraction (1- V). We assume that the two components are separately isotropic, thus we are modelling the laminar defects which scatter phonons anisotropically by laminar regions which are themselves isotropic but which result in anisotropic thermal conduction of the total system because of the orientation of these occlusions. The twin boundaries or stacking faults are thus considered as perpendicular to the c-direction [ 11l] of the cubic structure, of finite thickness in the cdirection and with isotropic thermal conductivity different from that of the zinc blende. The well-known Table 1. Direction
Thermalconductivity
IIIlIt 111111 1’[1111
0.149+ 0.007W/cm”C 0.204f 0.010W/cm”C 0.2182 0.010W/cm”C
equations [lo] for the components of conductivity for this model are: 1 V (1-v) -_=-+KII Kd Kb
(7)
and K,=
VKd+(l-
v)&
(8)
where K,I is parallel to [111] and thus perpendicular to the planes of the slabs, Kl is perpendicular to 11111and thus parallel to the planes of the slabs, and Kd and Kb are the conductivities of the regions of laminar defects and of the perfect zinc blende, respectively. Equations are available for more complex composite structures, for example with either interconnecting component of high or low conductivities[lO], however, the plane parallel model gives the maximum anisotropy and is probably the best macroscopic model for these crystals. Microscopic analyses of phonon scattering by the two types of laminar defects are being investigated. The value of K for natural crystals of cubic ZnS was measured more than fifty years ago[ll]; much more recently K was measured over an extended temperature from 2 to 300K for pure synthetic crystals[l2]. The accepted value of K at room temperature for cubic ZnS is 0.27W/cm deg. Substitution of this value for & and from Table 1 for K,I and K, into eqns (7) and (8) yield Kd = 0.073 and V = 0.30, a surprisingly large value for the volume fraction of the region of laminar defects. There may be other scattering centers for phonons in our crystals, for example self-activated centers, i.e. zinc vacancy plus nearest neighbour donor dopant, which reduce Kb below the literature value, similar to the effect of copper doping. In order to reduce V to 0.1, Kb would have to be as low as 0.23 W/cm”C, which seems unlikely. The assumption of greater scattering of twin boundaries compared to that of stacking faults can be justified as follows: At the highest frequencies, i.e. at and near the Debye frequency oD, the scatterings are probably of comparable strength. However, the scattering of phonons by twin boundaries should be to a good approximation frequency independent while the mean free path for scattering by stacking faults should vary approximately as w-* below wD The ratio of mean scattering cross sections for the two types of defects depends on the phonon distribution as well as on a more complete analysis of the scattering, however, the above shows that the scattering by twin boundaries is surely greater than that by stacking faults. The difference in scattering by twin boundaries and stacking faults, just described for phonons, is also valid for photons. Visible light should be scattered almost
specularly by twin boundaries and hardly at all by stacking faults. The visible appearance of the crystals tends to support the presence of twin boundaries. 4.CONCLUSIONS
The Weissenberg patterns of zinc sulfide crystals grown from the vapour phase are in accordance with the zinc blende structure but with rotational twinning in the [111] direction. Analysis shows two overlapping lattices. The fitting together of these lattices can be explained by normal stacking faults or by inverted twins. From the thermal conductivity we conclude that a substantial fraction, approximately 30%, of the crystal volume consists of laminar defects with large cross sections for phonon scattering. The inverted twins have more nearly the proper characteristics, however, normal stacking faults cannot be eliminated as contributing, at least in part, to the anisotropy of the thermal conductivity and to the crystallographic results. The visible appearance of the crystals, in particular the oriented striations, tends to support the existence of twin boundaries. Other techniques can supplement the two reported in this paper in fully characterizing the structure of these crystals, for example, the Zeeman effect on the optical spectra of Ni*+ dopants at mixed cubic-hexagonal environment of ZnS has recently been reported [ 131. Acknowledgements-The authors are indebted to Dr. Donald Spitzerof the ResearchLaboratoryof the AmericanCyanamidCo for makingthe thermal conductivitymeasurementsand for helpful discussions. One of the authors (FW) thanks the Humboldt Foundation for support during the final preparation of the manuscript.This researchwas also supportedin part by a grant from the U.S. Army Research O&e-Durham to the University of Delaware. The authors thank the referee for helpful theoretical observations on the scattering of phonons by twin boundaries and by stacking faults. REFERENCES
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