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CdTe heterojunctions
Journal of Crystal Growth 204 (1999) 447}452
Mis"t dislocations and stresses in Cd Hg Te/CdTe V \V heterojunctions I.V. Kurilo *, I.O. Rudyi , O.I. Vlasenko State University **Lviv Polytechnic++, 12 Bandera Str., 290646 Lviv 13, Ukraine Institute of Semiconductor Physics, NAS Ukraine, 45 Prospect Nauky, 252650 Kyiv 28, Ukraine Received 15 November 1998; accepted 24 March 1999 Communicated by M. Schieber
Abstract Elastic properties and characteristics of mis"t dislocations of Cd Hg Te/CdTe heterostructures were estimated: the V \V energy density of the "lm}substrate interface, the critical thickness for loss of coherency in a heterojunction, the elastic strain in the epitaxial system, the length of the cracks in the epilayer, the free edge displacement of a layer, the mis"t dislocations spacing, the magnitude of Burgers vector lying in di!erent mis"t planes, and the interface dangling bond density. 1999 Elsevier Science B.V. All rights reserved. PACS: 68.35.!p; 68.55.Ce; 68.60.Bs Keywords: Heterostructure; Interface; Mis"t dislocations; Elastic strain
1. Introduction Cd Hg Te is of interest since it is used in V \V infrared detectors for a wide range of applications. To obtain high-quality alloy layers of this material various epitaxial growth techniques have been developed. CdTe has a favourable lattice constant for Cd Hg Te epitaxial growth. This alloy V \V layer}substrate combination has been referred to as a `lattice matcheda system since lattice constants of HgTe and CdTe di!er only by 0.3%. The lattice mismatch between the Cd Hg Te epilayer (EL) V \V and the CdTe substrate may introduce strain-relieving mis"t dislocations (MD). * Corresponding author.
Dislocations in Cd Hg Te crystals have been V \V reported to degrade the performance of detectors made on these materials [1]. Furthermore, defects such as dislocations have been shown to have deleterious e!ects on optical and electrical properties of many semiconductors. Oldham and Milnes [2] have drawn attention to the fact that MD in heterojunction interfaces will provide high surface densities of dangling bonds which must be expected to act as acceptor and/or recombination centers. Thus, it is of interest to investigate heterojunction interfaces and the phenomena in the transition zone. The purpose of the present paper is to estimate some parameters of heterostructures of Cd Hg Te/ V \V CdTe important for practical applications.
0022-0248/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 9 0 - 6
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I.V. Kurilo et al. / Journal of Crystal Growth 204 (1999) 447}452
2. Experimental procedure Epilayers (EL), were grown by the isothermal vapour-phase epitaxy method in a closed evacuated ampoule on monocrystalline CdTe substrates. Substrates were prepared by mechano-chemical etching in a 5% bromine}methanol solution for 3 min to remove the polishing damage. The +1 0 0,, +1 1 0, and +1 1 1, crystallographic orientations of substrates were employed. As the material source, high-purity HgTe or Cd Hg Te powders were V \V used. The main parameters of the growth process were: the growth temperature in the range from 500 to 6003C, the growth time in the 12}72 h range, the substrate thickness in the 0.6}1.5 lm range and the EL thickness in the 20}200 lm range. EL with graded composition were obtained under such conditions. Moreover the EL surface was enriched by Hg, while the area adjacent to the substrate was enriched by Cd. The use of Cd Hg Te instead of V \V HgTe weakly in#uences the EL growing process, since a certain amount of CdTe is transported under isothermal conditions. However, there is another situation if the growth process occurs in conditions of thermal gradient between source and substrate and in the H #ow for vapour transport. So it is possible to get EL with high homogeneity of the composition over the whole thickness of EL. In this work calculations were carried out for the ultimate case of xP0 because the obtained values represent inequalities (unilateral restrictions).
3. Results and discussion To calculate heterostructure characteristics it is necessary to know elastic constants (C ), Poisson's GH ratio (l), Young's moduli (E) and shear moduli (G). It should be remarked that C values for A''B4' GH crystals taken from literature di!er from one another by 8}10%. In our calculations we used C values considered from Cottam and Saunders GH [3] for the HgTe crystal and from Greenoogh and Palmer [4] for the CdTe crystal: HgTe }C " 53.61, C "36.60, C "21.23 GPa; CdTe} C "53.80, C "37.40, C "20.18 GPa. Values of l we have calculated on the basis of C GH values [5].
It should be noted that calculation of E and G values on the basis of di!erent techniques leads to di!erent results. E values were calculated from Voigt and Reuss averagings [6]. Such averaged values of E are used for simpli"cation in most problems of continuum dislocation theory dealing with dislocations with di!erent Burgers vector orientation. These averagings allowed us to derive the elastic module of an isotropic polycrystal from the elastic constants of an anisotropic single crystal. This problem was investigated in detail in Ref. [5]. Results of the estimation of E values using other procedures are described in this paper. One of the useful parameters for the characterisation of the heterostructures is the energy density of the interface (a sum of both potential and elastic energies) [7]: Ga E " +1#b!(1#b ' 4p !b ln[2b((1#b)!2b],.
(1)
Here G is the interface shear modulus which may be obtained by measuring the interface hardness. We have assumed that G"(G #G )/2, where G and G are substrate and epilayer shear module, respectively (we have obtained G and G values of 1.539 and 1.614 GPa, respectively). The parameter b"2pa /(¸G) depends on the interface shear modulus, the meaning of K and the elastic properties of the bicrystal system. 1/K"(1!l )/G # (1!l )/G ; ¸!MD spacing. We assume the spac ing a and a of the lattices A (CdTe) and B (HgTe) to be such that Na "(N#1)a where N is an integer. We imagine A to be generated from C by a homogeneous extension which increases the distance Na of C to (N#0.5)a "Na in A. Similarly, B is generated by a homogeneous compression of the distance (N#1)C in C to (N#0.5)a "(N#1)a in B. The vernier period of registers is thus ¸"Na "(N#1)a "(N#0.5)a . (2) This equation de"nes a in terms of a and a . For a small lattice mis"t a /¸"(a !a )/a . Taking into account lattice parameters of a "0.6481 nm and a "0.6461 nm [8], we have obtained values of N"323, a "0.6479 nm, and ¸"0.2093 lm. The
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449
Ga value is considered as the `rigiditya degree of the EL}substrate bond. Substituting all the data into Eq. (1) we have obtained the value of E "2.461 lJ/m. One should note that expression ' (1) was obtained by assuming that in#uences normal to the interface stresses and layer thickness are neglected. This assumption is possible if the layer thickness exceeds half of the MD spacing. Therefore, it is possible to omit the layer thickness if it exceeds 0.0998 lm, which is realised in our case. Evaluation of ¸ by the formula
*N (1 1 1)"3.58;10 cm\. Thus, *N values depend on the crystallographic orientation of the plane, so both the easiness of epitaxial growth and electrical properties of heterojunctions must be expected to vary with the surface orientation. Such dependencies for di!erent pairs of contacting materials were revealed (see Ref. [9]). According to Jesser and Kuhlmann-Wilsdorf [10] an epitaxial layer remains MD free for thicknesses less than the so-called critical pseudomorphic growth thickness, given by
aa ¸" , (a !a )
4pG G (a !a ) 2G aa ln (G #G )G(1!v )(a #a ) . (4) h" p(a #a )D(1#2v )(G #G ) This is true under the condition of b;1 with b"0.023. Here G and v are averaged values for layers and source materials, D is the mis"t of the bicrystal system (D"0.0032). As a result we have obtained the value of h "0.036 lm. For h(h the elastic strain of contacting pairs achieves its maximum value:
(3)
which can now be estimated from relation (2) is not strict enough. According to Holt [9] in the relation of (2) it is necessary to use `modi"eda lattice parameters with some corrections for substrate orientation. In this case the calculated values of MD spacing are ¸ "¸ " ¸ "0.14 lm, ¸ "0.199 lm. Here the upper indices near ¸ represent line directions of dislocations occurring in the interface. One of the important parameters of MD is the Burgers vector magnitude. Using the data concerning the MD geometry of (1 0 0), (1 1 0) and (1 1 1) interfaces in sphalerite structures [9] we have obtained "b" ""b" " " b " " " b " " " b " " "b" "0.4575 nm,"b" "0.6470 nm. Here the "rst and the second upper indices represent the line and the Burgers vector direction of dislocations occurring in the (h k l) interface. MD leads to the formation of a high surface density of dangling bonds which act as pinning points and recombination centres. We have calculated the surface density of dangling bonds of *N "N !N , with N and N as dangling bond densities of contacting materials. Here formulae for *N (h k l) from Ref. [9] were used. It has been assumed that in (1 1 1) or A heterojunctions there may be only a-dangling bonds and in (1 1 1 ) or B heterojunctions } only b-dangling bonds, whereas in +1 0 0, and +1 1 0, planes the number of dangling bonds of a- and b-type is equal. Calculated values of *N (h k l) are *N (1 0 0)" 6.20;10 cm\, *N (1 1 0)"4.39;10 cm\,
(a !a ) . e "D"2
(a #a ) For h'h the elastic strain of the EL}substrate bicrystal system can be calculated using the relation a a ln[2b(1#b) !2b] . (5) e"! (1#G /G )(a #a )2p(1#v )h Here a "2a a /(a #a ) and b"2pG a / ¸(1!v )(1#G /G )G. One can easily see that relations (5) and (1) for b evaluation are identical. Let h"1 lm and ¸"0.14 lm, then we have deduced the value of e"5.62;10\. For h"200 lm the e value is equal to 2.81;10\. Then we have calculated the average elastic strain (e ) in small epitaxial islands for discontinu ous "lms: 3a a ln[2b(1#b) !2b] e "! . (6) (1#G /G )(a #a )2R Assuming that R"10\ lm island radius, we have obtained the value of e "6.32;10\. For
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the substrate average strain Gae e " . (7) 4a a We have got the value of e "1.66;10\. The critical radius of "nite epitaxial islands for loss of coherency in the interface plane is 3a (1#(G a/4G a)) ln((2b!b)/e) . (8) R "! 2D(1#G /G ) Here e is the natural logarithm base. After the substitution we have obtained the value of R "675.3 nm. In addition, using the relation suggested in Ref. [7] we have calculated the elastic energies ratio of E /E "0.84. One should note that E, h , e, e , e and R values obtained here refer to +1 1 0, interface planes which are often used for epitaxial growth. In such a case traditional and `modi"eda lattice parameters (see above) for [1 1 0] dislocation line and [0 0 1] direction of Burgers vector are equal. One of the most important types of defects in heterostructures are stresses which appear during the crystallisation process and the following cooling process down to room temperature. Heteroepitaxial stresses might be caused by the lattice constants di!erence (p ) as well as the ther ? mal expansion coe$cients di!erence (p ). Thermal ? stresses in heterostructures are determined by the generally used expression [11] E *a*¹ p " . (9) ? 1!l Here *a is the thermal expansion coe$cients di!erence, and *¹ the temperature range. Let *¹"500 K. Using a(HgTe)"4;10\ K\ and a(CdTe)"4.9;10\ K\ [8], the calculation gives the value of p "36 MPa. ? This value of p exceeds the p yield point of ? W HgTe crystals, which according to our measurements [12] for compression directions of 11 1 02 and 11 0 02 is in the 8}12 MPa range. It should be noted that the upper yield point for Cd Hg Te crystal is equal to 41.4, 25.3 and 28.6 MPa for 11 0 02, 11 1 02 and 11 1 12 compression axes, respectively. These stresses may result in bending of
heterostructures and sometimes in their fracturing or lifting from a substrate. In our experiments we did not observe the last two phenomena due to the high plasticity of HgTe and Cd Hg Te (HgTe is V \V semi-metal, for which the strain-hardening coe$cient according to our data in the 1}12% range of strain vary slightly; for the value of e"12% the value of stresses is equal to 40 MPa). Taking into account that the EL composition can signi"cantly change with the layer depth (besides the abovementioned case of EL formation with the constant composition of x+0.9 and 0.1 near the substrate and on the surface of EL accordingly) it may be one more reason for the stresses and the formation of cracks in heterostructures. Using the well-known Gri$th correlation we have estimated the possible crack length
p"
2Ep FIJ . p(1!l)l
(10)
Here l is the half-length of the crack, p the FIJ surface energy of the (h k l) plane, and E the Young's modulus. We have calculated p values using the procedFIJ ure proposed in Ref. [13]. For example, for +1 1 0, planes of HgTe crystals we have obtained p(1 1 0)"0.327 J/m value. Moreover, we have estimated values of p using the procedure [14], in FIJ which another approach of the surface energy calculation has been proposed. In this case we have obtained the value of p(1 1 0)"0.344 J/m. Both values di!er somewhat from the value of p(1 1 0)"0.077 J/m received by us from experiments of the brittle fracture under microindentation [12]. And, "nally, the estimation of the p value using the procedure proposed in Ref. FIJ [15] according to Marsh's works in the "eld of brittle fracture, gives the value of p(1 1 0) equal to 0.150 J/m. On the basis of this value and Young's modulus of E"45.8 GPa obtained by us using the Voigt's averaging procedure as well as the value of p "36 MPa we have obtained the crack length ? of about 8 lm. One should note that indentation of investigated crystals by the indenter load of 0.5 N leads to the crack length of about 20 lm. Under such conditions we have achieved higher stresses and more complicated nature of their distribution
I.V. Kurilo et al. / Journal of Crystal Growth 204 (1999) 447}452
451
compared to an axial compression, due to the indenter geometry. For other p values obtained by FIJ us the crack length was in the 4}18 lm range. Formation of such cracks is more probable in Cd Hg Te and CdTe crystals which have greater V \V brittleness [12]. Fig. 1 shows the crack (see arrow) which pierces the epitaxial layer and reaches its surface. The crack nucleates on the EL}substrate interface. But such cracks nucleate as a result of sudden cooling process of heterostructures down to room temperature. We note that the thermal expansion coe$cients di!erence of CdTe and HgTe is equal to 0.9;10 K\ (see above). Because of the high plasticity of HgTe, cracks are not formed and the relaxation of mechanical stresses in this crystal occurs without the fracture. Such relaxation was studied by us under conditions of uniaxial compression. Especially high degree of stress relaxation was observed in ternary alloys. Our calculated size of cracks may be considered as a critical thickness of EL, above which the crack formation in the epilayer can take place. Let us calculate stresses caused by the lattice constants di!erence. These stresses appear just during the EL growth process at the maximum temperature in a system: E (*a) p " . (11) ? [(1!l )a ] Taking into account all E values we have determined the average value of p "0.222 GPa. ? Therefore, these stresses exceed p stresses by sev ? eral times. However, at high temperatures there is a signi"cant relaxation of these stresses with simultaneous MD formation during a deposition process. Under the in#uence of stresses of di!erent types (e.g. thermal stress) EL bending is possible. Using the relation 3Kh p(1!l ) d" , (12) E h (1#E h )/(E h ) proposed in Ref. [16] we have calculated the displacement of the EL free edge using the known p value. Here K is the length of the specimen; h and h are thicknesses of the EL and substrate, respec tively. Let K"15 mm, h "200, 100 lm, h "500
Fig. 1. The heterostructure Cd Hg /CdTe with cracks: V \V (a) optical micrograph. A network of cracks is seen lying on the surface epilayer and a crack is seen perpendicular to the interface (arrow); (b) the enlarged marked region obtained by means of scanning electron microscope.
and 1000 lm, then we can get d value within the region of 2.3}5.0 lm depending on the E value obtained above. Using the known relationship Eh p" , 6(1!l )R h
(13)
we have obtained (for h "200 lm, h "1000 lm and p"36 MPa values) the radius of curvature of the heterostructure R "1.82 and 1.44 m for E values obtained using Voigt and Reuss aver agings, respectively.
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If a uniform elastic strain in EL takes place, then the interface energy density connected with this strain is
1#l h e. E "2G (14) C 1!l Let us calculate the E value for e"e that C
takes place if h (h . We have suggested h "0.01 lm and e "D"3.25;10\, then we
have obtained the value of E"7.11;10\. For C EL thicknesses of 1 and 200 lm (see above) we have obtained the values of E and E to be equal C C to 2.46;10\ and 1.22;10\ J/m, respectively. Taking into account the approximate nature of the formulae used as well as the signi"cant scattering of literature data concerning parameters used in these formulae, characteristics of heterostructures obtained here should be considered as approximate ones but quite proper in order to use them for technological purposes. 4. Conclusions We investigated the Cd Hg Te epitaxial layers V \V grown on monocrystalline CdTe substrates by the isothermal vapour-phase method. The lattice mismatch between EL and CdTe substrate may introduce strain-relieving mis"t dislocations (MD). We estimated some parameters of Cd Hg Te/CdTe V \V heterostructures important for practical application. Our calculations were carried out for the xP0 ultimate case because the obtained values represent inequalities (unilateral restrictions). The value of the interface energy density is equal to 2.461 lJ/m. Calculated values of the MD spacing for di!erent dislocation directions occurring in the interface are of 0.14 and 0.199 lm. Values of the surface density of dangling bonds corresponding to (1 0 0), (1 1 0) and (1 1 1) planes are of 6.20;10, 4.39;10 and 3.58;10 cm\, respectively. The
critical pseudomorphic growth thickness is equal to 0.036 lm. The elastic strain of the substrate}EL bicrystal system is equal to 2.8;10. The thermal stresses in heterostructures is of p "36 MPa. The ? p value that appears due to the di!erence in ? lattice constants is of p "0.222 GPa. These stres ? ses exceed the p stresses by several times. How ? ever, at high temperatures there is a signi"cant relaxation of such stresses during the deposition process with simultaneous MD formation. The results obtained here may be used in the EL deposition technology processes as well as for the devices production on the grounds of investigated materials.
References [1] H. Takigawa, M. Yoshikawa, T. Maekawa, J. Crystal Growth 86 (1988) 446. [2] W.G. Oldham, A.G. Milnes, Solid State Electron. 7 (1964) 153. [3] R.I. Cottam, G.A. Saunders, J. Phys. Chem. Solids 36 (1975) 187. [4] R.D. Greenough, S.B. Palmer, J. Phys. D 6 (1973) 587. [5] I.V. Kurilo, V.P. Alekhin, I.O. Rudyi, S.I. Bulychev, L.I. Osypyshin, Phys. Stat. Sol. (a) 163 (1997) 47. [6] J.R. Hirth, J. Lothe, Theory of Dislocations, McGraw-Hill, New York, 1968. [7] J.H. Van der Merwe, Single-crystal "lms. Conference, Pensilvania, 1963, New York, 1964, pp. 139}163. [8] O. Madelung (Ed.), Landolt-BoK rnstein, New Series, Vol. 17/b Semiconductors: Physics of II}VI and I}VII Compounds, Springer, Berlin, 1982. [9] D.B. Holt, J. Phys. Chem. Solids 27 (1966) 1053. [10] W.A. Jesser, D. Kuhlmann-Wilsdorf, Phys. Stat. Sol. 19 (1967) 95. [11] Y. Budo, J. Priest, Solid State Electron. 6 (1963) 159. [12] I.V. Kurilo, V.P. Alekhin, S.I. Bulychev, Preprint. Inst. Metall., Acad. Nauk SSSR, Moscow, 1982. [13] S.N. Zadumkin, Fiz. Tverd. Tela 2 (1960) 378. [14] B.F. Ormont, Dokl. Akad. Nauk SSSR 282 (1985) 320. [15] A. Evans, T.R. Wilschaw, Acta Metall. 24 (1976) 939. [16] A. Brenner, S.J. Sendero!, Ref. Nat. Bur. Stand. 42 (1949) 105.
Mis"t dislocations and stresses in Cd Hg Te/CdTe V \V heterojunctions I.V. Kurilo *, I.O. Rudyi , O.I. Vlasenko State University **Lviv Polytechnic++, 12 Bandera Str., 290646 Lviv 13, Ukraine Institute of Semiconductor Physics, NAS Ukraine, 45 Prospect Nauky, 252650 Kyiv 28, Ukraine Received 15 November 1998; accepted 24 March 1999 Communicated by M. Schieber
Abstract Elastic properties and characteristics of mis"t dislocations of Cd Hg Te/CdTe heterostructures were estimated: the V \V energy density of the "lm}substrate interface, the critical thickness for loss of coherency in a heterojunction, the elastic strain in the epitaxial system, the length of the cracks in the epilayer, the free edge displacement of a layer, the mis"t dislocations spacing, the magnitude of Burgers vector lying in di!erent mis"t planes, and the interface dangling bond density. 1999 Elsevier Science B.V. All rights reserved. PACS: 68.35.!p; 68.55.Ce; 68.60.Bs Keywords: Heterostructure; Interface; Mis"t dislocations; Elastic strain
1. Introduction Cd Hg Te is of interest since it is used in V \V infrared detectors for a wide range of applications. To obtain high-quality alloy layers of this material various epitaxial growth techniques have been developed. CdTe has a favourable lattice constant for Cd Hg Te epitaxial growth. This alloy V \V layer}substrate combination has been referred to as a `lattice matcheda system since lattice constants of HgTe and CdTe di!er only by 0.3%. The lattice mismatch between the Cd Hg Te epilayer (EL) V \V and the CdTe substrate may introduce strain-relieving mis"t dislocations (MD). * Corresponding author.
Dislocations in Cd Hg Te crystals have been V \V reported to degrade the performance of detectors made on these materials [1]. Furthermore, defects such as dislocations have been shown to have deleterious e!ects on optical and electrical properties of many semiconductors. Oldham and Milnes [2] have drawn attention to the fact that MD in heterojunction interfaces will provide high surface densities of dangling bonds which must be expected to act as acceptor and/or recombination centers. Thus, it is of interest to investigate heterojunction interfaces and the phenomena in the transition zone. The purpose of the present paper is to estimate some parameters of heterostructures of Cd Hg Te/ V \V CdTe important for practical applications.
0022-0248/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 9 0 - 6
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I.V. Kurilo et al. / Journal of Crystal Growth 204 (1999) 447}452
2. Experimental procedure Epilayers (EL), were grown by the isothermal vapour-phase epitaxy method in a closed evacuated ampoule on monocrystalline CdTe substrates. Substrates were prepared by mechano-chemical etching in a 5% bromine}methanol solution for 3 min to remove the polishing damage. The +1 0 0,, +1 1 0, and +1 1 1, crystallographic orientations of substrates were employed. As the material source, high-purity HgTe or Cd Hg Te powders were V \V used. The main parameters of the growth process were: the growth temperature in the range from 500 to 6003C, the growth time in the 12}72 h range, the substrate thickness in the 0.6}1.5 lm range and the EL thickness in the 20}200 lm range. EL with graded composition were obtained under such conditions. Moreover the EL surface was enriched by Hg, while the area adjacent to the substrate was enriched by Cd. The use of Cd Hg Te instead of V \V HgTe weakly in#uences the EL growing process, since a certain amount of CdTe is transported under isothermal conditions. However, there is another situation if the growth process occurs in conditions of thermal gradient between source and substrate and in the H #ow for vapour transport. So it is possible to get EL with high homogeneity of the composition over the whole thickness of EL. In this work calculations were carried out for the ultimate case of xP0 because the obtained values represent inequalities (unilateral restrictions).
3. Results and discussion To calculate heterostructure characteristics it is necessary to know elastic constants (C ), Poisson's GH ratio (l), Young's moduli (E) and shear moduli (G). It should be remarked that C values for A''B4' GH crystals taken from literature di!er from one another by 8}10%. In our calculations we used C values considered from Cottam and Saunders GH [3] for the HgTe crystal and from Greenoogh and Palmer [4] for the CdTe crystal: HgTe }C " 53.61, C "36.60, C "21.23 GPa; CdTe} C "53.80, C "37.40, C "20.18 GPa. Values of l we have calculated on the basis of C GH values [5].
It should be noted that calculation of E and G values on the basis of di!erent techniques leads to di!erent results. E values were calculated from Voigt and Reuss averagings [6]. Such averaged values of E are used for simpli"cation in most problems of continuum dislocation theory dealing with dislocations with di!erent Burgers vector orientation. These averagings allowed us to derive the elastic module of an isotropic polycrystal from the elastic constants of an anisotropic single crystal. This problem was investigated in detail in Ref. [5]. Results of the estimation of E values using other procedures are described in this paper. One of the useful parameters for the characterisation of the heterostructures is the energy density of the interface (a sum of both potential and elastic energies) [7]: Ga E " +1#b!(1#b ' 4p !b ln[2b((1#b)!2b],.
(1)
Here G is the interface shear modulus which may be obtained by measuring the interface hardness. We have assumed that G"(G #G )/2, where G and G are substrate and epilayer shear module, respectively (we have obtained G and G values of 1.539 and 1.614 GPa, respectively). The parameter b"2pa /(¸G) depends on the interface shear modulus, the meaning of K and the elastic properties of the bicrystal system. 1/K"(1!l )/G # (1!l )/G ; ¸!MD spacing. We assume the spac ing a and a of the lattices A (CdTe) and B (HgTe) to be such that Na "(N#1)a where N is an integer. We imagine A to be generated from C by a homogeneous extension which increases the distance Na of C to (N#0.5)a "Na in A. Similarly, B is generated by a homogeneous compression of the distance (N#1)C in C to (N#0.5)a "(N#1)a in B. The vernier period of registers is thus ¸"Na "(N#1)a "(N#0.5)a . (2) This equation de"nes a in terms of a and a . For a small lattice mis"t a /¸"(a !a )/a . Taking into account lattice parameters of a "0.6481 nm and a "0.6461 nm [8], we have obtained values of N"323, a "0.6479 nm, and ¸"0.2093 lm. The
I.V. Kurilo et al. / Journal of Crystal Growth 204 (1999) 447}452
449
Ga value is considered as the `rigiditya degree of the EL}substrate bond. Substituting all the data into Eq. (1) we have obtained the value of E "2.461 lJ/m. One should note that expression ' (1) was obtained by assuming that in#uences normal to the interface stresses and layer thickness are neglected. This assumption is possible if the layer thickness exceeds half of the MD spacing. Therefore, it is possible to omit the layer thickness if it exceeds 0.0998 lm, which is realised in our case. Evaluation of ¸ by the formula
*N (1 1 1)"3.58;10 cm\. Thus, *N values depend on the crystallographic orientation of the plane, so both the easiness of epitaxial growth and electrical properties of heterojunctions must be expected to vary with the surface orientation. Such dependencies for di!erent pairs of contacting materials were revealed (see Ref. [9]). According to Jesser and Kuhlmann-Wilsdorf [10] an epitaxial layer remains MD free for thicknesses less than the so-called critical pseudomorphic growth thickness, given by
aa ¸" , (a !a )
4pG G (a !a ) 2G aa ln (G #G )G(1!v )(a #a ) . (4) h" p(a #a )D(1#2v )(G #G ) This is true under the condition of b;1 with b"0.023. Here G and v are averaged values for layers and source materials, D is the mis"t of the bicrystal system (D"0.0032). As a result we have obtained the value of h "0.036 lm. For h(h the elastic strain of contacting pairs achieves its maximum value:
(3)
which can now be estimated from relation (2) is not strict enough. According to Holt [9] in the relation of (2) it is necessary to use `modi"eda lattice parameters with some corrections for substrate orientation. In this case the calculated values of MD spacing are ¸ "¸ " ¸ "0.14 lm, ¸ "0.199 lm. Here the upper indices near ¸ represent line directions of dislocations occurring in the interface. One of the important parameters of MD is the Burgers vector magnitude. Using the data concerning the MD geometry of (1 0 0), (1 1 0) and (1 1 1) interfaces in sphalerite structures [9] we have obtained "b" ""b" " " b " " " b " " " b " " "b" "0.4575 nm,"b" "0.6470 nm. Here the "rst and the second upper indices represent the line and the Burgers vector direction of dislocations occurring in the (h k l) interface. MD leads to the formation of a high surface density of dangling bonds which act as pinning points and recombination centres. We have calculated the surface density of dangling bonds of *N "N !N , with N and N as dangling bond densities of contacting materials. Here formulae for *N (h k l) from Ref. [9] were used. It has been assumed that in (1 1 1) or A heterojunctions there may be only a-dangling bonds and in (1 1 1 ) or B heterojunctions } only b-dangling bonds, whereas in +1 0 0, and +1 1 0, planes the number of dangling bonds of a- and b-type is equal. Calculated values of *N (h k l) are *N (1 0 0)" 6.20;10 cm\, *N (1 1 0)"4.39;10 cm\,
(a !a ) . e "D"2
(a #a ) For h'h the elastic strain of the EL}substrate bicrystal system can be calculated using the relation a a ln[2b(1#b) !2b] . (5) e"! (1#G /G )(a #a )2p(1#v )h Here a "2a a /(a #a ) and b"2pG a / ¸(1!v )(1#G /G )G. One can easily see that relations (5) and (1) for b evaluation are identical. Let h"1 lm and ¸"0.14 lm, then we have deduced the value of e"5.62;10\. For h"200 lm the e value is equal to 2.81;10\. Then we have calculated the average elastic strain (e ) in small epitaxial islands for discontinu ous "lms: 3a a ln[2b(1#b) !2b] e "! . (6) (1#G /G )(a #a )2R Assuming that R"10\ lm island radius, we have obtained the value of e "6.32;10\. For
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the substrate average strain Gae e " . (7) 4a a We have got the value of e "1.66;10\. The critical radius of "nite epitaxial islands for loss of coherency in the interface plane is 3a (1#(G a/4G a)) ln((2b!b)/e) . (8) R "! 2D(1#G /G ) Here e is the natural logarithm base. After the substitution we have obtained the value of R "675.3 nm. In addition, using the relation suggested in Ref. [7] we have calculated the elastic energies ratio of E /E "0.84. One should note that E, h , e, e , e and R values obtained here refer to +1 1 0, interface planes which are often used for epitaxial growth. In such a case traditional and `modi"eda lattice parameters (see above) for [1 1 0] dislocation line and [0 0 1] direction of Burgers vector are equal. One of the most important types of defects in heterostructures are stresses which appear during the crystallisation process and the following cooling process down to room temperature. Heteroepitaxial stresses might be caused by the lattice constants di!erence (p ) as well as the ther ? mal expansion coe$cients di!erence (p ). Thermal ? stresses in heterostructures are determined by the generally used expression [11] E *a*¹ p " . (9) ? 1!l Here *a is the thermal expansion coe$cients di!erence, and *¹ the temperature range. Let *¹"500 K. Using a(HgTe)"4;10\ K\ and a(CdTe)"4.9;10\ K\ [8], the calculation gives the value of p "36 MPa. ? This value of p exceeds the p yield point of ? W HgTe crystals, which according to our measurements [12] for compression directions of 11 1 02 and 11 0 02 is in the 8}12 MPa range. It should be noted that the upper yield point for Cd Hg Te crystal is equal to 41.4, 25.3 and 28.6 MPa for 11 0 02, 11 1 02 and 11 1 12 compression axes, respectively. These stresses may result in bending of
heterostructures and sometimes in their fracturing or lifting from a substrate. In our experiments we did not observe the last two phenomena due to the high plasticity of HgTe and Cd Hg Te (HgTe is V \V semi-metal, for which the strain-hardening coe$cient according to our data in the 1}12% range of strain vary slightly; for the value of e"12% the value of stresses is equal to 40 MPa). Taking into account that the EL composition can signi"cantly change with the layer depth (besides the abovementioned case of EL formation with the constant composition of x+0.9 and 0.1 near the substrate and on the surface of EL accordingly) it may be one more reason for the stresses and the formation of cracks in heterostructures. Using the well-known Gri$th correlation we have estimated the possible crack length
p"
2Ep FIJ . p(1!l)l
(10)
Here l is the half-length of the crack, p the FIJ surface energy of the (h k l) plane, and E the Young's modulus. We have calculated p values using the procedFIJ ure proposed in Ref. [13]. For example, for +1 1 0, planes of HgTe crystals we have obtained p(1 1 0)"0.327 J/m value. Moreover, we have estimated values of p using the procedure [14], in FIJ which another approach of the surface energy calculation has been proposed. In this case we have obtained the value of p(1 1 0)"0.344 J/m. Both values di!er somewhat from the value of p(1 1 0)"0.077 J/m received by us from experiments of the brittle fracture under microindentation [12]. And, "nally, the estimation of the p value using the procedure proposed in Ref. FIJ [15] according to Marsh's works in the "eld of brittle fracture, gives the value of p(1 1 0) equal to 0.150 J/m. On the basis of this value and Young's modulus of E"45.8 GPa obtained by us using the Voigt's averaging procedure as well as the value of p "36 MPa we have obtained the crack length ? of about 8 lm. One should note that indentation of investigated crystals by the indenter load of 0.5 N leads to the crack length of about 20 lm. Under such conditions we have achieved higher stresses and more complicated nature of their distribution
I.V. Kurilo et al. / Journal of Crystal Growth 204 (1999) 447}452
451
compared to an axial compression, due to the indenter geometry. For other p values obtained by FIJ us the crack length was in the 4}18 lm range. Formation of such cracks is more probable in Cd Hg Te and CdTe crystals which have greater V \V brittleness [12]. Fig. 1 shows the crack (see arrow) which pierces the epitaxial layer and reaches its surface. The crack nucleates on the EL}substrate interface. But such cracks nucleate as a result of sudden cooling process of heterostructures down to room temperature. We note that the thermal expansion coe$cients di!erence of CdTe and HgTe is equal to 0.9;10 K\ (see above). Because of the high plasticity of HgTe, cracks are not formed and the relaxation of mechanical stresses in this crystal occurs without the fracture. Such relaxation was studied by us under conditions of uniaxial compression. Especially high degree of stress relaxation was observed in ternary alloys. Our calculated size of cracks may be considered as a critical thickness of EL, above which the crack formation in the epilayer can take place. Let us calculate stresses caused by the lattice constants di!erence. These stresses appear just during the EL growth process at the maximum temperature in a system: E (*a) p " . (11) ? [(1!l )a ] Taking into account all E values we have determined the average value of p "0.222 GPa. ? Therefore, these stresses exceed p stresses by sev ? eral times. However, at high temperatures there is a signi"cant relaxation of these stresses with simultaneous MD formation during a deposition process. Under the in#uence of stresses of di!erent types (e.g. thermal stress) EL bending is possible. Using the relation 3Kh p(1!l ) d" , (12) E h (1#E h )/(E h ) proposed in Ref. [16] we have calculated the displacement of the EL free edge using the known p value. Here K is the length of the specimen; h and h are thicknesses of the EL and substrate, respec tively. Let K"15 mm, h "200, 100 lm, h "500
Fig. 1. The heterostructure Cd Hg /CdTe with cracks: V \V (a) optical micrograph. A network of cracks is seen lying on the surface epilayer and a crack is seen perpendicular to the interface (arrow); (b) the enlarged marked region obtained by means of scanning electron microscope.
and 1000 lm, then we can get d value within the region of 2.3}5.0 lm depending on the E value obtained above. Using the known relationship Eh p" , 6(1!l )R h
(13)
we have obtained (for h "200 lm, h "1000 lm and p"36 MPa values) the radius of curvature of the heterostructure R "1.82 and 1.44 m for E values obtained using Voigt and Reuss aver agings, respectively.
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If a uniform elastic strain in EL takes place, then the interface energy density connected with this strain is
1#l h e. E "2G (14) C 1!l Let us calculate the E value for e"e that C
takes place if h (h . We have suggested h "0.01 lm and e "D"3.25;10\, then we
have obtained the value of E"7.11;10\. For C EL thicknesses of 1 and 200 lm (see above) we have obtained the values of E and E to be equal C C to 2.46;10\ and 1.22;10\ J/m, respectively. Taking into account the approximate nature of the formulae used as well as the signi"cant scattering of literature data concerning parameters used in these formulae, characteristics of heterostructures obtained here should be considered as approximate ones but quite proper in order to use them for technological purposes. 4. Conclusions We investigated the Cd Hg Te epitaxial layers V \V grown on monocrystalline CdTe substrates by the isothermal vapour-phase method. The lattice mismatch between EL and CdTe substrate may introduce strain-relieving mis"t dislocations (MD). We estimated some parameters of Cd Hg Te/CdTe V \V heterostructures important for practical application. Our calculations were carried out for the xP0 ultimate case because the obtained values represent inequalities (unilateral restrictions). The value of the interface energy density is equal to 2.461 lJ/m. Calculated values of the MD spacing for di!erent dislocation directions occurring in the interface are of 0.14 and 0.199 lm. Values of the surface density of dangling bonds corresponding to (1 0 0), (1 1 0) and (1 1 1) planes are of 6.20;10, 4.39;10 and 3.58;10 cm\, respectively. The
critical pseudomorphic growth thickness is equal to 0.036 lm. The elastic strain of the substrate}EL bicrystal system is equal to 2.8;10. The thermal stresses in heterostructures is of p "36 MPa. The ? p value that appears due to the di!erence in ? lattice constants is of p "0.222 GPa. These stres ? ses exceed the p stresses by several times. How ? ever, at high temperatures there is a signi"cant relaxation of such stresses during the deposition process with simultaneous MD formation. The results obtained here may be used in the EL deposition technology processes as well as for the devices production on the grounds of investigated materials.
References [1] H. Takigawa, M. Yoshikawa, T. Maekawa, J. Crystal Growth 86 (1988) 446. [2] W.G. Oldham, A.G. Milnes, Solid State Electron. 7 (1964) 153. [3] R.I. Cottam, G.A. Saunders, J. Phys. Chem. Solids 36 (1975) 187. [4] R.D. Greenough, S.B. Palmer, J. Phys. D 6 (1973) 587. [5] I.V. Kurilo, V.P. Alekhin, I.O. Rudyi, S.I. Bulychev, L.I. Osypyshin, Phys. Stat. Sol. (a) 163 (1997) 47. [6] J.R. Hirth, J. Lothe, Theory of Dislocations, McGraw-Hill, New York, 1968. [7] J.H. Van der Merwe, Single-crystal "lms. Conference, Pensilvania, 1963, New York, 1964, pp. 139}163. [8] O. Madelung (Ed.), Landolt-BoK rnstein, New Series, Vol. 17/b Semiconductors: Physics of II}VI and I}VII Compounds, Springer, Berlin, 1982. [9] D.B. Holt, J. Phys. Chem. Solids 27 (1966) 1053. [10] W.A. Jesser, D. Kuhlmann-Wilsdorf, Phys. Stat. Sol. 19 (1967) 95. [11] Y. Budo, J. Priest, Solid State Electron. 6 (1963) 159. [12] I.V. Kurilo, V.P. Alekhin, S.I. Bulychev, Preprint. Inst. Metall., Acad. Nauk SSSR, Moscow, 1982. [13] S.N. Zadumkin, Fiz. Tverd. Tela 2 (1960) 378. [14] B.F. Ormont, Dokl. Akad. Nauk SSSR 282 (1985) 320. [15] A. Evans, T.R. Wilschaw, Acta Metall. 24 (1976) 939. [16] A. Brenner, S.J. Sendero!, Ref. Nat. Bur. Stand. 42 (1949) 105.