Calculated strain energy of hexagonal epitaxial thin films

Transcription

1 Journal of Crystal Growth 240 (2002) 6 13 Calculated strain energy of hexagonal epitaxial thin films Jianyun Shen a,b, *, Steven Johnston a, Shunli Shang b, Timothy Anderson a a University of Florida, Department of Chemical Engineering, Gainesville, FL , USA b Research Institute for Non-ferrous Metals of Beijing, Beijing, China Received 21 June 2001; accepted 21 November 2001 Communicated by C.R. Abernathy Abstract Generalized formulae for the strain energy of hexagonal thin films on both hexagonal and rhombohedral substrates have been developed. These formulae require knowledge of the elastic stiffness coefficients and the lattice parameters of the film and only the lattice parameters of the substrate. Example calculations of the strain energy present in the strained film-substrate material combinations GaN/Al 2 O 3 and GaN/LiGaO 2 are presented for different film crystallographic directions and rotation with respect to the substrate. Finally, phase equilibrium calculations are performed for the Ga N binary system which show the substantial influence of strain energy on equilibrium in the system. r 2002 Elsevier Science B.V. All rights reserved. PACS: Hc; s; 64.; Wt Keywords: A1. Phase diagrams; B1. Gallium compounds; B1. Nitrides 1. Introduction The increased interest in heteroepitaxy of semiconductor films has motivated the development of a reliable method to calculate the strain energy produced by the mismatch between the epitaxial growth layer and the underlying substrate. Lattice mismatch as well as the subsequent strain can have significant influence on the extent of dislocation generation, the morphological, electrical, and optical properties, and the phase equilibrium behavior of the growth layer. While *Corresponding author. Department of Chemical Engineering, University of Florida, Gainesville, FL , USA. Tel.: ; fax: address: jshen@che.ufl.edu (J. Shen). cubic film/cubic substrate combinations have been analyzed previously [1], systems involving hexagonally oriented material as either the film or substrate have not been thoroughly investigated to date. Examples of important semiconductor materials that exist in the hexagonal crystal structure include the wide band gap compound semiconductors GaN, SiC, BN and many II VI semiconductors such as CdS. These materials are promising candidates for use in optoelectronic applications including visible and ultraviolet emitters, high power high temperature electronics, and in the case of cbn as an abrasive [2,3]. The lack of large area bulk crystals for most of these hexagonal compounds has necessitated the use of alternative substrates with different lattice parameters and thermal expansion coefficients /02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 J. Shen et al. / Journal of Crystal Growth 240 (2002) Furthermore, device applications often require alloying that introduces additional stress in the film. As an example, alloying GaN with InN to lower the band gap energy increases the value of the lattice parameter. The strain energy stored in a film grown on a mismatched substrate increases the Gibbs energy of the material. This strain energy contribution modifies such equilibrium quantities as melting temperature, vapor pressure, and miscibility limits. While the growth of hexagonal materials has been extensively studied experimentally, quantitative calculation of the inherent strain energy has not been fully performed [4]. Furthermore, the effect of the strain energy on the resulting equilibrium has not been addressed. In this work elastic compliance equations are developed and their relationship to the overall strain energy of a hexagonally oriented film and substrate are presented. These general relations are then applied to the growth of GaN on different substrates. 2. Analysis The development of the general equations necessary to calculate the strain energy of a hexagonal film on either a hexagonal or rhombohedral substrate is presented in this section. The general form of strain energy for an elastic body can be written as U ¼ 1 2 ð s xe x þ s y e y þ s z e z þ t xy g xy þ t yz g yz þ t zx g zx Þ; ð1þ where U is the strain energy per unit volume, s x ; s y and s z are the normal stresses, t xy t yz and t zx are the shear stresses, e x ; e y and e z are the normal strains, and g xy ; g yz and g zx are the shear strains. Generally only the normal strain and stress in two perpendicular axes are considered to be produced as a result of lattice mismatch in epitaxial growth. Hook s law can be written as e x ¼ s 0 11 s x þ s 0 12 s y; ð2þ e y ¼ s 0 22 s y þ s 0 12 s x; ð3þ where s 0 ij are the elastic compliances of the plane of interest. These values can be obtained from the fourth-rank transform of the s ij values in the principle plane. The fourth-rank transform of the s ij values in the principle plane is s 0 ijkl ¼ X3 X 3 X 3 X 3 a im a jn a kp a lq s mnpq ; ð4þ m¼1 n¼1 p¼1 q¼1 where a im ; a jn ; a kp ; and a lq are the direction cosines of the axes of the plane studied with respect to those of the principle plane. It should be noted that s ij is actually the reduced representation of the fourth tensor s ijkl according to Voigt s notation [5]. In the case where the elastic compliances are unknown but the elastic stiffness coefficients, c ij ; are available, the s ij values can be calculated from c ij since the matrix of s ij and that c ij are mutually inverted. The matrix of s ij for the hexagonal crystal lattice is given by Lekhnitskii [5] as: 2 3 s 11 s 12 s s 12 s 11 s s 13 s 13 s s s ðs 11 s 12 Þ For the hexagonal lattice, the elastic compliances in the plane of interest can be written as s 0 11 ¼ð1 l2 3 Þ2 s 11 þ l3 4 s 33 þ l3 2 ð1 l2 3 Þð2s 13 þ s 44 Þ; ð5þ s 0 12 ¼ s 11ðl 1 m 1 l 2 m 2 Þ 2 þ s 12 ðl 1 m 2 l 2 m 1 Þ 2 þ s 13 ½ð1 l 2 3 Þm2 3 þ l2 3 ð1 m2 3 ÞŠ þ s 33l 2 3 m2 3 þ s 44 l 3 m 3 ðl 2 m 2 l 1 m 1 Þ: s 0 22 ¼ð1 m2 3 Þ2 s 11 þ m 4 3 s 33 ð6þ þ m 2 3 ð1 m2 3 Þð2s 13 þ s 44 Þ; ð7þ where l 1 ; l 2 and l 3 are the direction cosines of the x 0 axis of the plane studied with respect to the x; y; and z axes of the principle plane. Similarly m 1 ; m 2 ; and m 3 are that of y 0 axis. The relationship between s ij and c ij in the principle plane can be written as c 11 c 33 c 2 13 s 11 ¼ ðc 11 c 12 Þðc 11 c 33 þ c 12 c 33 2c 2 13 Þ; ð8þ

3 8 J. Shen et al. / Journal of Crystal Growth 240 (2002) 6 13 c 2 13 s 12 ¼ c 12c 33 ðc 11 c 12 Þðc 11 c 33 þ c 12 c 33 2c 2 13 Þ; c 13 ð9þ s 13 ¼ c 11 c 33 þ c 12 c 33 2c 2 ; ð10þ 13 c 11 þ c 12 s 33 ¼ c 11 c 33 þ c 12 c 33 2c 2 ; ð11þ 13 s 44 ¼ 1 : ð12þ c 44 Once these relationships are known calculation of a meaningful elastic energy term can made. Two different planes of film growth were considered, the ( ) basal plane and the (1 0 %1 0) prism plane (see Fig. 1). Since the ( ) basal plane of a hexagonal crystal is isotropic, the x and y axes can be chosen coincident with those of the principle plane. Therefore, e x ¼ s 11 s x þ s 12 s y ; ð13þ e y ¼ s 22 s y þ s 12 s x : ð14þ For a growth on the ( ) basal plane there are two possible film/substrate crystallographic relationships. The first one occurs when the planes of film growth and the underlying substrate are both hexagonal. The second one is when either the film growth plane or the substrate is not hexagonal (e.g., zincblende, rhombohedral). Each case was analyzed. (0001) -a 2 [1210] Å a Å [1010] [0001] -a 1 a 1 -a 3 (1010) a 2 [1210] Fig. 1. Basal plane ( ) and prism plane (1 0 %1 0) of a hexagonal crystal. The [% ] direction is parallel to the lower basal plane. When both the film and substrate are hexagonal and the basal plane of each is used in deposition, the strain in the x and y directions are equal as required by symmetry and can be written as e x ¼ e y ¼ a a 0 ; ð15þ a 0 where a and a 0 are the lattice parameters of the growth layer and the substrate in the basal plane, respectively. Using Eqs. (13) and (14), the normal stresses to the film/substrate interface can then be written as 1 a a 0 s x ¼ s y ¼ : ð16þ s 11 þ s 12 a 0 Thus from Eq. (1) the strain energy per unit volume is simply 2 : ð17þ 1 a a 0 U ¼ s 11 þ s 12 a 0 In the second case, when either the growth layer or substrate is not hexagonal, the strain in the x and y directions will not be equal (e x ae y )asa result of the departure from symmetry. From Eq. (13) and (14), the following relations are evident: r x ¼ s 11e x s 12 e y s 2 11 ; ð18þ s2 12 r y ¼ s 11e y s 12 e x s 2 11 s2 12 with a resulting strain energy expression of ð19þ U ¼ s 11ðe 2 x þ e2 y Þ 2e xe y s 12 2ðs 2 11 s2 12 Þ : ð20þ Deposition on the [1 0 %1 0] prism plane in the [%12%1 0] and [ ] directions (defined as x 0 and y 0 direction of the plane studied as shown in Fig. 1) was also considered. On this plane the direction cosines of the x 0 and y 0 axes are l 1 ¼ 0; l 2 ¼ 1; l 3 ¼ 0; m 1 ¼ 0; m 2 ¼ 0; and m 3 ¼ 1: Therefore the elastic compliances can be transformed from the plane of interest to the principle plane using s 0 11 ¼ s 11; ð21þ s 0 12 ¼ s 13; s 0 22 ¼ s 33; ð22þ ð23þ

4 J. Shen et al. / Journal of Crystal Growth 240 (2002) The strains in the [%12%1 0] and [ ] directions are thus e x 0 ¼ s x 0s 0 11 þ s y 0s0 12 ¼ s x 0s 11 þ s y 0s 13 ; ð24þ e y 0 ¼ s y 0s 0 22 þ s x 0s0 12 ¼ s y 0s 33 þ s x 0s 13 ð25þ and the normal stresses are r x 0 ¼ e x 0s 33 e y 0s 13 s 11 s 33 s 2 ; ð26þ 13 r y 0 ¼ e y 0s 11 e x 0s 13 s 11 s 33 s 2 : ð27þ 13 It directly follows as before that the strain energy per unit volume is: U ¼ e2 x 0s 33 þ e 2 y 0s 11 2e x 0e y 0s 13 2ðs 11 s 33 s 2 13 Þ : ð28þ If a hexagonal lattice structured film with lattice parameters a and c is grown with its prism plane on a rectangular face of a substrate with lattice parameters a 0 and c 0 ; then the system s strain is e x 0 ¼ða a 0 Þ=a 0 ; e y 0 ¼ðc c 0 Þ=c 0 ; and its resulting strain energy is U ¼ ða a 0=a 0 Þ 2 s 33 þðc c 0 =c 0 Þ 2 s 11 2ðs 11 s 33 s 2 13 Þ 2ða a 0=a 0 Þðc c 0 =c 0 Þs 13 2ðs 11 s 33 s 2 13 Þ : ð29þ Eq. (29) is general and can thus be used for all the other prism planes as well as internal planes such as (1 %2 1 0) because of the isotropic nature of the basal plane. 3. Lattice mismatch and strain energy calculation Three independent sets of calculations were performed to demonstrate the strain energy calculation of a GaN film grown on Al 2 O 3 and LiGaO 2 substrates, using the fundamental data provided in Tables 1 and 2 [6]. Table 3 summarizes the strain energy calculated for three different film/ substrate orientations and their corresponding lattice mismatches. From both Eqs. (17), (20) and (29) and the data in Table 3, it is clear that an increase in the lattice mismatch increases the system s strain energy. Table 1 Lattice constant ( ( A) of GaN, Al 2 O 3, and LiGaO 2 Material a b c GaN (wurtzite) F Al 2 O 3 (wurtzite) F LiGaO 2 (orthorhombic) Table 2 Elastic stiffness coefficients c ij and compliances s ij of GaN c ij (Gpa) c 11 c 12 c 13 c 33 c s ij (1/GPa) s 11 s 12 s 13 s 33 s The minimum lattice mismatch in the GaN/ Al 2 O 3 system occurs when the GaN film is rotated 301 with respect to the sapphire substrate so that the ( )8( ), [%1010]8[1 %2 1 0] GaN/Al 2 O 3 interface is formed and a 3:1 film/substrate unit cell ratio is used. For instance, the lattice mismatch for no rotation and 301 rotation with a 3:1 GaN/ Al 2 O 3 unit cell ratio are 33% and 16.1%, respectively. Rotation of GaN films on Al 2 O 3 has been observed experimentally [7] and is illustrated in Fig. 2. The strain energy of the 16.1% mismatched system is GPa or kj/mol. Of course these values of lattice mismatch exceeds the limits to maintain elastic deformation in the growth layer. Therefore, experimentally, a relaxation of the strain energy by large-scale dislocation formation would be inevitable without the use of a buffer layer (typically GaN or AlN). Consistent with experimental results and a recent set of calculations we have performed it is energetically favorable for the majority of the stress in the GaN/Al 2 O 3 system to be compensated by dislocation formation with approximately a 2% residual strain remaining at the interface [8]. Deposition of the ( )8(0 0 1), [%12%10]8[0 1 0] GaN/LiGaO 2 system (Fig. 3) is an attractive alternative due to its lower mismatch [9]. LiGaO 2 has an orthorhombic structure consisting of an alternate stacking of a two-dimensional array

5 10 J. Shen et al. / Journal of Crystal Growth 240 (2002) 6 13 Table 3 Strain energy produced by lattice mismatch Growth layer/substrate Growth orientation Lattice mismatch (%) Strain energy e x e y GPa kj/mol w-gan / w-al 2 O 3 ( )8( ) [%1010]8[1 %2 1 0] 16.1 F w-gan /o-ligao 2 ( )8(0 0 1), [%12%10]8[0 1 0] w-gan /o-ligao 2 (1 %210)8(0 1 0) [1 0 %10]8[1 0 0] Al 2 O 3 GaN [1010] of GaN [1210] of Al 2 O 3 consisting of oxygen tetrahedra centered Ga and Li ions. Therefore the material has a wurtzitic superstructure with a small departure from hexagonal symmetry as a result of the need to accommodate metallic atoms of different size. When GaN is grown on LiGaO 2 in this orienta- pffiffi tion, the normal strains are e x ¼ð 3 3:189 5:402Þ=5:402 ¼ 2:25% and e y ¼ð23:189 6:372Þ=6:372 ¼ 0:09%: Thus the strain energy is reduced by Eq. (20) to GPa or 1.29 kj/mol. Another possible film/substrate orientation for prism plane growth is the (1 %2 10)8(0 1 0), [1 0 %10]8[1 0 0] GaN/LiGaO 2 system (Fig. 4). The normal strains pffiffi for this material combination are then e x ¼ð 3 3:189 5:402Þ=5:402 ¼ 2:25% and e y ¼ð5:185 5:007Þ=5:007 ¼ 3:56%: The strain energy for this orientation is then calculated using Eq. (29) to be GPa or 4.94 kj/mol. Fig. 2. Scheme for the deposition of the ( )8( ), [%1010]8[1 %2 1 0] GaN/Al 2 O 3 system Å Å Å [1210] of GaN [010] of LiGaO Å GaN LiGaO Å Interfacial bond of Ga (or Li)-N; Ga and Li from LiGaO 2, N from GaN after deformation before deformation Fig. 3. Scheme for the deposition of the ( )8(0 0 1), [%12%10]8[0 1 0] GaN/LiGaO 2 system. 4. Results and discussion Thermodynamic equilibrium calculations were carried out on the strained Ga N system to demonstrate the importance of strain energy on a system s final equilibrium state. Thermochemical data for all species except for GaN(s) were taken from the ThermoCalc version K sub94 database [10]. The data for GaN(s) were reassessed by Davydov and Anderson [11]. Their results were incorporated. A strain energy term calculated from Eqs. (16), (21) or (29) was added to the value of G H ser for GaN(s). The effect of strain energy on the strained Ga N equilibrium binary system is shown in Figs. 5 and 6. Fig. 5 depicts equilibrium in the unstrained Ga N heterogeneous system at 0.1 MPa and will be used as a reference for the strained GaN/LiGaO 2 and GaN/Al 2 O 3 equilibrium systems. Fig. 6 illustrates the effect of strain energy on the resulting Ga N equilibrium. A decrease in the GaN thermodynamic stability (i.e., its melting point temperature) results from the addition of a positive term to the Gibbs energy expression of

6 J. Shen et al. / Journal of Crystal Growth 240 (2002) (1210) GaN / (010) LiGaO 2 interface GaN (1210) Å Å a 3 -a 1 -a 2 a 2 LiGaO 2 a 1 [1010] -a 3 Fig. 4. Scheme for the deposition of the (1 %210)8(0 1 0), [1 0 %10]8[1 0 0] GaN/LiGaO 2 system Gas Gas Temperature (K) Liquid + Gas 1055 K 500 GaN + Liquid GaN + Gas 303K Ga + GaN N Mole Fraction Fig. 5. T x diagram of the unstrained heterogeneous system Ga N as a function of temperature and N mole fraction at 0.1 MPa. GaN. As Fig. 6 shows, at a system pressure of 0.1 MPa the melting temperature of GaN decreases by 11 and 43 K with the addition of the strain energies associated with the ( )8(0 0 1), [%12%10]8[0 1 0] GaN/LiGaO 2 and (1 %210)8(0 1 0), [1 0 %10]8[1 0 0] GaN/LiGaO 2 growth systems, respectively. The smaller change in the melting temperature corresponds to less strained GaN basal plane deposition (Fig. 3) while the larger melting temperature is represents GaN prism plane growth (Fig. 4). Due to the decrease in the GaN melting point temperature the liquid+gas Temperature (K) (a) Temperature (K) (b) Liquid + Gas 1055K 500 GaN + Liquid GaN + Gas 303K Ga + GaN N Mole Fraction K 1044K GaN + Liquid Liquid + Gas 1012K No Strain U = 1.29 kj/mol U = 4.94 kj/mol GaN + Gas N Mole Fraction Fig. 6. T x diagram of the strained heterogeneous system Ga N on LiGaO 2 at 0.1 MPa system pressure. (a) Overall diagram; (b) Zoomed view of change of melting point.

7 12 J. Shen et al. / Journal of Crystal Growth 240 (2002) 6 13 stability domain will increase as the film/substrate strain energy is increased. However, since the strain energy only affects solid GaN there is no change in the dew point line. Furthermore, the GaN+liquid/GaN+gas as well as the GaN+ liquid/ga+gan equilibrium lines remain unchanged because the strain energy affects the GaN in both phases equally. Thus its impact on each phase cancels. It is desirable in most semiconductor deposition techniques including CVD, MOCVD, and MBE to have only solid and gas phases present. Thus, from a thermodynamic viewpoint, deposition of GaN on LiGaO 2 at near atmospheric conditions must occur below 1012 or 1044 K (1055 K in the unstrained case) and Ga:N molar ratios less than unity. Fig. 7 shows the thermodynamic equilibrium deposition of GaN on Al 2 O 3. A striking difference between Figs. 5 and 7 can be seen immediately. The strain energy of the GaN film deposited on Al 2 O 3, kj/mol, is so large that it prevents GaN from being a stable solid equilibrium phase at any temperature or mole fraction. Other calculations revealed that the maximum allowable strain energy that would still result in a region of GaN stability is E89 kj/mol. Hence the strain energy present when GaN is deposited on sapphire is over 80% too large for solid GaN stability. The phase diagrams shown in Fig. 7 are experimentally inaccurate because a high dislocation density would occur at this large of a lattice misfit which would significantly relax the strained growth layer. Therefore the phase diagram should be calculated by combining the strain and the dislocation energies with the Gibbs energy of GaN. This calculation will be published in our next paper. A general relationship between the decrease of the GaN melting point temperature and the lattice mismatch at 0.1 MPa system pressure is presented in Fig. 8. As shown in Fig. 5, the maximum possible change in the melting temperature, and thus the limit of GaN stability, is 752 K ( K). Therefore, the maximum symmetric strain the Ga N system can possess and still have solid GaN stability is when e x and e y equal Elastic incorporation of this much lattice mismatch induced strain energy to enable epitaxial growth is unreasonable and would experimentally result in dislocation formation and strain relief. Changes in the GaN melting point temperature are evident even at lower film strain energy values characteristic of high quality films Gas Temperature (K) Liquid + Gas K Ga + Gas N Mole Fraction Fig. 7. T x diagram of the strained heterogeneous system Ga N on Al 2 O 3 at 0.1 MPa system pressure. Fig. 8. Generalized T x diagram of the strained heterogeneous system Ga N at 0.1 MPa system pressure.

8 J. Shen et al. / Journal of Crystal Growth 240 (2002) Conclusions A series of generalized equations have been developed from fundamental stress/strain principles to calculate the lattice mismatch induced strain energy of a hexagonal film on hexagonal and rhombohedral substrates. Addition of strain energies to the thermodynamic equilibrium calculations of the Ga N phase diagram shows dramatic differences between the strained and unstrained cases. At 0.1 MPa system pressure the GaN melting point decreased 11 and 43 K depending on the film/substrate orientation when GaN was deposited on LiGaO 2. Even more interesting was the disappearance of a thermodynamically stable GaN region when the strain energy arising from its deposition on Al 2 O 3 was included. This contradicts what is observed experimentally. The dew point line and Ga melting point temperature lines were unchanged due to the canceling effect of the GaN in each adjacent deposition region. At equilibrium N 2 partial pressures, deposition on LiGaO 2 again decreased the GaN melting point temperature though a large GaN+liquid phase remained. The profound effect of strain energy in a lattice mismatched system indicates that the predicted equilibrium behavior of epitaxial thin films may differ greatly from that of unstrained thermodynamic systems. These calculations suggest that strain energy has significant influence on both the experimental and equilibrium deposition processes of semiconductor materials. Further refinement of this model by the inclusion of dislocation and surface energy effects will be presented in a future paper. References [1] W.A. Brantly, J. Appl. Phys. 44 (1973) 534. [2] H. Amano, T. Asahi, I. Akasaki, Jpn. J. App. Phys. Lett. 28 (1990) L150. [3] S.N. Mohammad, A.A. Salvador, H. Morkoc, Proc. IEEE 83 (1995) [4] R. Thokala, J. Chaudhuri, Thin Solid Films 266 (1995) 189. [5] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, CA, [6] Y. Takagi, M. Ahart, T. Azuhata, T. Sota, K. Suzuki, S. Nakamura, Physica B (1996) 547. [7] I. Akasaki, H. Amano, Y. Koide, K. Hiramatsu, N. Sawaki, J. Crystal Growth 98 (1989) 209. [8] O. Ambacher, J. Phys. D: Appl. Phys. 31 (1998) [9] Y. Tazou, T. Ishii, S. Miyazawa, Jpn. J. Appl. Phys. 36 (1997) L746. [10] B. Sundman, B. Jansson, J.O. Andersson, CALPHAD 9 (1985) 153. [11] A.V. Davydov, T.J. Anderson, Electrochem. Soc. Proc (1998) 38.