JOURNAL OF APPLIED PHYSICS 101, 063501 共2007兲
Strain relaxation in heteroepitaxial films by misfit twinning. I. Critical thickness Lilin Liu, Yousheng Zhang, and Tong-Yi Zhanga兲 Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
共Received 20 April 2006; accepted 28 November 2006; published online 16 March 2007兲 Based on the dislocation theory of twinning, an analytical solution is given to determine the critical thickness of the misfit twin formation in an epilayer with different elastic constants from its substrate. For a given epilayer thickness and lattice mismatch strain, the twin formation energy should reach its minimum to determine the twin width and a zero minimum formation energy determines the critical thickness for misfit twinning. The effect of elastic mismatch between the epilayer and the substrate on the critical thickness is studied comprehensively, revealing that an elastically soft epilayer has a large critical thickness. Moreover, a misfit-twin-and-perfectdislocation predominance chart is constructed to predict the predominant regions of misfit twinning and perfect dislocation in the mismatch strain and the specific twin-boundary energy domain. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2433368兴 I. INTRODUCTION
Thin films on substrates are of special interest due to their wide applications in microelectronic devices, microelectromechanical systems, oxidation/corrosion protection, etc.1 An important issue in thin film research is residual stresses that are usually induced in films during fabrication and play a significant role in the microstructural evolution of films. Residual stresses may damage a microelectronic or microelectromechanical device during fabrication and/or reduce its service life. The reliability of device performance depends, to a large extent, on the residual stresses that remain in thin films. Therefore, it is of both academic merit and practical significance to understand the behavior of residual stresses in thin films. Although a large elastic mismatch strain can be tolerated in very thin films deposited on substrates, beyond a critical thickness, the film/substrate system will react to relieve the mismatch strain usually by producing misfit dislocations, misfit twins, forming cracks, and/or causing surface instabilities. The residual stress relaxation through twinning mechanism has been frequently observed experimentally. Wu and Weatherly2 comprehensively studied stress relaxation in 2% tensile strained In1−xGaxAs1−yPy epilayers grown at a temperature of 480 ° C on InP 共100兲 substrates. Their results show that twinning is the only stress-relief mechanism in quaternary epilayers. Two orientations of twins were observed, which were nucleated at the rough surface by generating 90° partial dislocations. The twins may terminate themselves at the interface between the epilayer and the substrate or penetrate into the substrate frequently because the shear modulus of the epilayer is higher than that of the substrate. Their results indicate clearly that there is a critical epilayer thickness for twinning, below which no misfit twins are formed.2 Li et al.3 studied the strain relaxation in GaAsN and Author to whom correspondence should be addressed; FAX: 共852兲23581543; electronic mail:
[email protected]
a兲
0021-8979/2007/101共6兲/063501/12/$23.00
GaP films grown on 共100兲 GaAs by molecular beam epitaxy. They found that the relaxation of films with rough surfaces occurred by twinning or by formation of perfect dislocations, while in epilayers with smooth surfaces, film cracking was the predominant stress-relief mechanism. In strained Si/ Ge superlattice epilayers deposited on 共100兲 Ge, Wegscheider et al.4 found that twinning is the only stress-relief mechanism. Through experimental observations with a transmission electron microscope 共TEM兲 and theoretical calculations based on a half-loop dislocation nucleation model, Wegscheider and Cerva5 explored the effects of compressive and tensile mismatch strains on the formation of misfit defects in Si/ Ge superlattice epilayers on 共100兲 Ge substrates and SixGe1−x epilayers on 共100兲 Si substrates. Their results show that when a tensile mismatch strain is larger than a critical value, 90° partial dislocations will be predominantly generated, thereby leading to misfit twins. In contrast, if the tensile mismatch strain is lower than the critical value, generating 60° perfect dislocations will be the predominant stress-relief mechanism. With compressive mismatch strains, however, generating 60° perfect dislocations is always energetically more favorable than generating 30° partial dislocations, which yields a larger critical thickness of an epilayer with a compressive mismatch strain than that of an epilayer with the same magnitude of a tensile mismatch strain as the compressive mismatch strain. Using a TEM, Neethling and Alberts6 studied primary and multiple twins in GaAs epilayers grown on Si 共100兲 and Si 共111兲 substrates. Their results indicated that microtwins, originating from the uneven GaAs/ Si interface and propagating through the GaAs epilayer, are the main defect type and the surface morphology of the Si plays an important role in the formation of these microtwins. Dynna et al.7 experimentally and theoretically studied Au–Ni and Au–Cu epilayers deposited on Au 共001兲 substrates. Their results demonstrated that twinning, resulting from the glide of 90°partial dislocations on successive 兵111其 planes, is the major stress-relief mechanism. Halley et al.8 demonstrated
101, 063501-1
© 2007 American Institute of Physics
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Liu, Zhang, and Zhang
that the strain relaxation in ordered FePd films grown on Pd 共001兲 substrates took place mainly through misfit twining. The lattice misfit between the ordered FePd and Pd共001兲 lattices was about 1%. The degree of the relaxation was found to be approximately linear to the thickness of the FePd film. They observed that a misfit twin was composed of a group of partial dislocations sliding on successive 兵111其 planes. The partial dislocations did not terminate themselves all at the interface between the film and the substrate. The sliding distance of a partial dislocation could not be the same for all partial dislocations inside a misfit twin such that the end of the misfit twin was not flat. These experimental observations confirm again the dislocation mechanism of twinning. The relaxation mechanism of misfit strain was later found to depend on the chemical ordering in the FePd thin films.9 In disordered or weakly ordered FePd layers, the relaxation was dominated by the activity of perfect misfit dislocations.9 Using a high-resolution TEM, Liu et al.10 observed that misfit twinning and misfit dislocations were both concurrently active in the relaxation of compressive mismatch strain in electroplated Pd thin films on Ni 共001兲 substrates. In addition to semiconductor and metal epilayers, twinning is comprehensively studied for superconducting and ferroelectric epilayers,11–17 which are not discussed in detail here. In highly symmetrical crystals, such as face-centeredcubic 共fcc兲, zinc blende, and diamond crystal structures, a twin is formed by gliding of a group of partial dislocations along parallel 兵111其 planes with the Burgers vector of a / 6具112典,18 where a denotes the lattice spacing. For example, a misfit twin in an epilayer deposited on a 共100兲 substrate with the zinc blende crystal structure is formed by a group of misfit 90° partial dislocations if the mismatch strain in the epilayer is tensile or by a group of misfit 30° partial dislocations if the mismatch strain is compressive.5,10,19–22 This is because the interaction between a formed misfit twin 共a group of misfit partial dislocations兲 and the mismatch stress must be negative in order to reduce the total energy of the epilayer/substrate system. From the dislocation mechanism of twinning, the stress fields produced by a misfit twin are the superposition of the stress fields produced by individual misfit partial dislocations that form the misfit twin. Therefore, the theoretical approach used to describe misfit dislocations can be directly applied to the study of misfit twins. However, misfit twins have some unique features. First, a misfit twin is composed of m misfit partial dislocations with each misfit dislocation gliding over an atomic crystalline plane inside the twin. Second, there is a twin boundary between the twin and the matrix. The specific twinboundary energy is about half of the specific stacking fault energy.23 Based on the dislocation mechanism of twinning, Dynna and Marty21 systematically studied misfit twins in epilayers with the same elastic constants as their corresponding substrates. In the present work, we study misfit twins in epilayers with different elastic constants from their corresponding substrates based on the dislocation mechanism of twinning. Considering the elastic mismatch between an epilayer and its substrate, Zhang22,24 developed a hybrid method by combining superposition and Fourier transformation to cal-
J. Appl. Phys. 101, 063501 共2007兲
culate the stress field of misfit dislocations by extending the classical Fourier transformation method.25,26 The advantage of the hybrid method is that the formula determining the critical thickness can be approximately expressed in a closed analytic form. The hybrid method22,24 is used in the present work to explore the role of the elastic mismatch in misfit twinning. For misfit dislocations, the critical thickness of an epilayer is usually determined from the zero value formation energy of a misfit dislocation, which implies that forming the misfit dislocation does not change the total energy of the epilayer/substrate system at the critical thickness.27 When the epilayer thickness is larger or smaller than the critical thickness, forming a misfit dislocation will decrease or increase the total energy of the epilayer/substrate system. However, a misfit twin may change its width by changing the number, m, of misfit partial dislocations that form the twin and thus change the total energy of the epilayer/substrate system. Therefore, another condition, in addition to zero formation energy, is required to determine the critical thickness for misfit twinning. When studying the critical thickness for film cracking, Zhang and Zhao28 proposed that the formation energy for cracking should be minimized to determine the crack depth and then when the minimum formation energy is equal to zero, the critical thickness for film cracking is found. The same approach has been employed in the determination of the critical thickness for domain formation in strained ferroelectric and ferroelastic films,29 with the criterion that the minimum formation energy is equal to zero. In the present work, we follow the criterion to study the critical thickness of misfit twinning. Based on the dislocation mechanism of twinning, twodimensional isotropic elasticity analysis is carried out in the present work to calculate the change in energy, the critical thickness for misfit twinning, and the equilibrium configuration of misfit twins. In two-dimensional calculations, all energies are given in per unit length and hereafter this notation is used for simplicity unless otherwise noted. The critical thickness for misfit twinning is described in this paper, while an array of misfit twins is elucidated in Part II.30 The physical models, major equations and results are given in the text, and simplified formulas are emphasized so that readers may easily adopt them to estimate a critical thickness. The details of the derivations are given in the Appendix based on isotropic elasticity. Interested readers may find these details useful in further studies.
II. FORMATION ENERGY OF A MISFIT TWIN
Figure 1 shows a heteroepitaxial film with a thickness of h deposited on an infinitely thick substrate. The interface between the epitaxial layer and its substrate is chosen to be the x1 axis. An individual misfit twin with m partial misfit dislocations is located in the interface. For simplicity, the width of the misfit twin is noted as m in the following. The formation energy of a misfit twin, E f , is defined as the energy change due to forming the twin, which is given by
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Liu, Zhang, and Zhang *
*
*a = 共1 兲 + 共2 兲,
FIG. 1. An individual misfit twin of m partial misfit dislocations located at the interface between an epilayer and its substrate. m = 3 is used here as an example.
Ef =
1 2 −
冕冕 冕冕 兺
1 2
共T + m兲共T + m兲dx1dx2
兺
1 Es,twin = 2
共1a兲
冕冕
Eint,twin/misfit =
兺
TTdx1dx2 ,
冕冕
兺
mTdx1dx2 ,
共1b兲
共1c兲
where ⌫t is the total twin-boundary energy between the misfit twin and the matrix of the epilayer; Es,twin and Eint,twin/misfit denote, respectively, the self-energy of the twin and the interaction energy between the twin and the mismatch strain; T and T are, respectively, the strain and stress fields produced by the twin; and m and m are, respectively, the strain and stress fields produced by the mismatch. The integration domain, 兺, encloses the entire solid including the epilayer and the substrate, but it may exclude the partial dislocation cores. Based on the dislocation mechanism of twinning, the twin strain and stress fields are composed from m
m
i=1
i=1
共2兲
where a,i and a,i denote, respectively, the strain and stress fields produced by the ith misfit partial dislocation. Using the hybrid superposition and the Fourier transformation method, we divide the dislocation stress field, a, and the dislocation displacement field, ua, into three parts:
a = 共1兲 + 共2兲 + 共3兲,
ua = u共1兲 + u共2兲 + u共3兲
in the epilayer and into two parts:
共3b兲
in the substrate, where the parameters in the epilayer and the substrate are denoted without and with the superscript “ *”, respectively. As described by Zhang,31 the first part of the stress and displacement represents the fields produced by a partial dislocation in the interface between two semi-infinite media. The second stress field in the epilayer is determined by letting the first two stress fields satisfy the traction-free condition along the epilayer surface. The third stress field in the epilayer and the second stress field in the substrate are calculated using Fourier transformation. The details are described in the Appendix. For a biaxial mismatch stress field in an epilayer, the mismatch stress field is determined by the lattice mismatch between the epilayer and its substrate, which generates a mismatch strain, f, f=
a* − a , a
共4兲
m,11 = m,33 =
共3a兲
2共1 + 兲 f and other m,ij = 0, 1−
共5兲
where and are, respectively, the shear modulus and Poisson’s ratio of the epilayer. Equation 共5兲 indicates that the mismatch stresses are related only to the elastic constants of the epilayer when the substrate is treated as being infinitely thick. Since the mismatch stress field is analytic at the dislocation core, the interaction energy can be calculated without excluding the dislocation core in the integration. Thus, completing the integration of Eq. 共1c兲 gives Eint,twin/misfit =
2共1 + 兲 fmb1h. 1−
共6兲
As can be seen in Eq. 共6兲, the product of the b1 component of the Burgers vector and the mismatch strain must be negative to ensure negative interaction energy, which is necessary in order to form the misfit twin. The total twin-boundary energy is calculated by ⌫t =
T = 兺 a,i and T = 兺 a,i ,
*兲
where a and a* denote the lattice constants of the film and the substrate, respectively. The biaxial mismatch stress field is explicitly expressed by25
mmdx1dx2 + ⌫t = Es,twin + Eint,twin/misfit
+ ⌫t ,
*
u*a = u共1 兲 + u共2
2 ␥ Th , sin
共7兲
where ␥T is the specific twin-boundary energy between the twin and the matrix and denotes the orientation of the misfit twin, as shown in Fig. 1. If m 艌 2, there are two twin boundaries and that is why 2 appears in Eq. 共7兲. Usually, ␥T is approximately taken a half value of the specific stacking fault energy. If m = 1, there is only a stacking fault and Eq. 共7兲 holds also. The detailed derivation of the self-energy is given in the Appendix. The hybrid method yields the self-energy in an analytic form,
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Es,twin = m
冋
册
eb23 共J1 + J2兲共b21 + b22兲 2h + ln 4 2 r0
+m
J2共b21 − b22兲 − 2J1b22 2
冋
m−1
+
eb23 4共J1 + J2兲共b21 − b22兲 共m − k兲h2 + 兺 2 2 2 4 k=1 k d + 4h
+
共J1 + J2兲共b21 + b22兲 2
册兺
m−1
共m − k兲ln
k=1
k2d2 + 4h2 k2d2 + r20
m−1
4h2J1共b21 + b22兲 共m − k兲共k2d2 − 4h2兲 + + E0,3e 兺 共k2d2 + 4h2兲2 k=1 + E0,3s − +
再 冋
J 2共 *J 2 + J 1兲 m 共b21 − b22兲 4 *
冉
J 1共 J 1 + J 2兲 J1 J2 − 4J1 − 共J1 − J2兲 − *
− 2共b21 + b22兲
冉
J21 J22 + *
冊冎
冊册
,
共8a兲
where r0 is the core radius of the dislocation; b1, b2, and b3 are the components of the dislocation Burgers vector in the coordinator system, i.e., b = 共b1 , b2 , b3兲; e is the effective shear modulus; and J1 and J2 are bimaterial constants introduced by Zhang and Li,32 J1 =
* , + *
e =
2* , * +
J2 =
* , + * *
= 3 − 4;
共8b兲
d denotes the distance along the x1 axis between two adjacent misfit partial dislocations; E0,3e and E0,3s are induced by the third stress field of the edge and screw dislocation components, respectively, and are calculated via Fourier transformation. The energy contribution of E0,3e and E0,3s is attributed to the elastic mismatch in the elastic constants between the epilayer and the substrate. If the epilayer has the same elastic constants as the substrate, the self-energy and the formation energy will be reduced to the corresponding results in Ref. 21 III. CRITICAL THICKNESS FOR MISFIT TWINNING
Figures are plotted here to demonstrate the formation energy of various values of the epilayer thickness. For simplicity, we consider an epilayer deposited on a 共100兲 substrate. We set up the coordinate system such that the x1, x2, ¯ 兴 directions, and x3 axes are along the 关011兴, 关100兴, and 关011 respectively, and the dislocation lines are parallel to the ¯ 兴 direction in the two-dimensional calculations. In this 关011 case, the Burgers vector of misfit 90° partial dislocations ¯ 11兲 plane and in the coordinate syslying on the 共111兲 or 共1 冑6 冑6 1 冑2 tem is 6 a关− 冑3 , 3 , 0兴 or 6 a关− 冑13 , −冑 32 , 0兴, which will form a misfit twin if the mismatch strain is tensile. If the mismatch
FIG. 2. The formation energy of a misfit twin as a function of the twin width, m, for lattice mismatch strains: 共a兲 f = 0.02 and 共b兲 f = −0.02.
strain is compressive, a misfit twin is composed of a group of misfit 30° partial dislocations with a Burgers vector of 冑6 关 1 1 冑3 兴 or 冑66 a关 21冑3 , 冑16 , ⫿ 冑23 兴 in the coordinate 6 a 2冑3 , − 冑6 , ± 2 system. Due to the symmetry, the two types of 90° or 30° partial dislocations have the same Burgers vector component, 冑2 冑2 b1, which is − 6 a for the 90° and 12 a for the 30° partial dislocations, showing that the b1 magnitude of the 90° partial dislocations is double the b1 magnitude of the 30° partial dislocations. The Burgers vector of a 60° perfect dislocation ¯ 11兲 plane in the coordinate system is lying on the 共111兲 or 共1 冑2 冑2 also given here for comparison. It is 2 a关 21 , − 2 , ± 21 兴 or 冑2 冑 关1 2 1兴 2a 2, 2 ,⫿2 . We have conducted experiments on Pd epilayers and will report the results in Part II of this series. The specific twinboundary energy of Pd is ␥T = 0.013b0. For most semiconductor crystals, the dislocation core size is regarded as r0 = b0 / 4 with b0 denoting the magnitude of the Burgers vectors of the dislocation. The value of the specific twin-boundary energy of ␥T = 0.013b0 and the dislocation core size of r0 = b0 / 4 are used hereafter in most numerical calculations and may not be mentioned again. Assuming that the epilayer has the same elastic constants as the substrate, Fig. 2 shows the
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formation energy as a function of the twin width for various values of epilayer thickness for two mismatch strains of f = 0.02 and f = −0.02, where the used parameters are Poisson’s ratio of = 0.3 and the dislocation core size of r0 = b0 / 4 with b0 = 冑16 a denoting the magnitude of the Burgers vectors of the 90° and 30° partial dislocations. For a small epilayer thickness, the formation energy is always larger than zero and may always increase, as illustrated in Fig. 2共a兲 for h = 20.07b0, or first decrease and then increase, as illustrated in Fig. 2共b兲 for h = 100b0, when the width of the misfit twin increases. If the epilayer thickness reaches a critical thickness, hc, the formation energy has a minimum value of zero with a definite width of the misfit twin, namely, the critical misfit twin width. The critical thicknesses shown in Figs. 2共a兲 and 2共b兲 are hc = 34.323b0 and hc = 128.5b0, respectively, and the corresponding critical misfit twin widths are mc = 2 and mc = 3. When the epilayer thickness is larger than the critical thickness, hc, the formation energy decreases through zero, reaches a minimum, and then increases through zero again when the twin width increases, as shown for h = 47.5b0 in Fig. 2共a兲 and h = 150b0 in Fig. 2共b兲. Since the formation energy is calculated with integer m, the formation energy minimum or zero formation energy may not occur exactly at a given m. If the misfit twin width, m, is treated as a continuous variable, the critical thickness for misfit twinning should be determined by simultaneously satisfying
E f 共h,m兲 =0 m
共9a兲
E f 共h,m兲 = 0.
共9b兲
This is because, for a given epilayer/substrate system, the formation energy of a misfit twin is a function of the epilayer thickness and the misfit twin width, with the mismatch strain and the stacking fault energy as parameters. Simultaneously satisfying Eqs. 共9a兲 and 共9b兲 determines the critical thickness for misfit twinning and the critical misfit twin width, whereas Eq. 共9a兲 alone gives only the equilibrium misfit twin width, with which the formation energy reaches its minimum. Because m is an integer, caution should be used in numerical calculation when approximately satisfying Eqs. 共9a兲 and 共9b兲 simultaneously. The critical thickness for misfit twinning and the critical thickness for forming a misfit perfect dislocation are shown in Figs. 3共a兲 and 3共b兲 for tensile and compressive mismatch strains, respectively, as a function of the magnitude of the lattice mismatch strain for the shear modulus ratio of / * = 0.5, 1, and 2, wherein a misfit partial dislocation is treated as a special misfit twin with m = 1. It should be noted that the magnitude of the Burgers vector of a 60° perfect dislocation, 冑2 冑3 times that of a 90° or 30° partial 2 a, is as large as dislocation and thus the value of 冑3b0 is used as the core size of the 60° perfect dislocation. As expected, both critical thicknesses decrease monotonically with increasing mismatch strain. The elastic mismatch between the epilayer and substrate plays an important role in the critical thickness, especially with small mismatch strains, as indicated in Figs.
FIG. 3. The critical thickness for misfit twinning and the critical thickness for misfit perfect dislocations vs 共a兲 tensile and 共b兲 compressive mismatch strains when the shear modulus ratio, / *, is equal to 0.5, 1, and 2. The inset figures show the corresponding critical misfit twin width.
3共a兲 and 3共b兲, thereby illustrating that a softer epilayer gives a larger critical thickness for both misfit twinning and misfit dislocations. If a tensile mismatch strain is lower than the critical value of 0.038, the critical thickness for misfit perfect dislocations is smaller than the critical thickness for misfit twinning, indicating that a perfect misfit dislocation rather than a misfit twin may be easily formed at the interface. This is because a misfit twin has two twin boundaries that possess the twin-boundary energy against the formation of a misfit twin. The total twin-boundary energy could be much larger than the strain energy of the twin when the epilayer thickness is shallow. On the other hand, if a tensile mismatch strain is higher than the critical value, the critical thickness for misfit perfect dislocations is larger than the critical thickness for misfit twinning, thereby indicating that a misfit twin rather than a misfit perfect dislocation may be easily formed at the interface. For compressive mismatch strains, however, the critical thickness for misfit perfect dislocations is smaller than the critical thickness for misfit twinning for the used specific twin-boundary energy value. It should be noted that the critical value of the mismatch strain depends on the specific twin-boundary energy. Since the twin-boundary energy and the interaction energy are both
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epilayer/substrate systems, where misfit twinning is found to be the only strain-relief mechanism.2,3 Since the twinboundary energy used here is in units of b0, we tabulate values of the dimensionless twin-boundary energy for various crystals in Table I. For compressive mismatch strains, however, the critical thickness for misfit twinning with twinboundary energy of 0.0004b0 is still larger than that for misfit perfect dislocations. Considering that the lowest twinboundary energy reported in the literature is around 0.001b0, we do not plot the misfit-twin-and-perfectdislocation predominance chart for compressive mismatch strains, although the approach described above can be applied to compressive mismatch strains. IV. SIMPLIFIED FORMULAS FIG. 4. The dependence of critical mismatch strain on specific twinboundary energy yields a misfit-twin-and-perfect-dislocation predominance chart.
directly proportional to the epilayer thickness, we may define the effective driving force, G, for misfit twinning, which is given by G = ⌫t + Eint,twin/misfit = 2h
冉
冊
␥ T/ 1 + + fmb1 . sin 1 −
共10兲
If 共␥T / 兲 / sin + 关共1 + 兲 / 共1 − 兲兴fmb1 艌 0, clearly, there is no driving force for misfit twinning, so no misfit twin will be generated. Equation 共10兲 indicates that increasing the twin width enhances the effective driving force for misfit twinning. The numerical results confirm this expectation. The critical twin width corresponding to the critical thickness for misfit twinning shown in Figs. 3共a兲 and 3共b兲 is illustrated in the insets of Figs. 3共a兲 and 3共b兲. In general, the critical twin width is large when the mismatch strain is small. Take / * = 1 as an example. When the tensile lattice mismatch strain is 0.01, the critical twin width is m = 4, while a tensile lattice mismatch strain of 0.016 corresponds to a critical twin width of m = 2. When the tensile mismatch strain is higher than 0.034, the effective driving force is high enough for the given specific twin-boundary energy and thus the misfit twin is actually a single partial misfit dislocation. As mentioned above, the critical mismatch strain, at which the critical thickness for misfit perfect dislocations equals the critical thickness for misfit twinning, depends on the twin-boundary energy. Figure 4 shows the critical mismatch strain as a function of the specific twin-boundary energy for tensile mismatch strains. The critical mismatch strain varies almost linearly with the twin-boundary energy. The curve of the critical mismatch strain versus the twin-boundary energy separates the domain of the mismatch strain versus the twinboundary energy into two regions. In the upper region above the curve, the critical thickness for misfit twinning is smaller than the critical thickness for misfit perfect dislocation and thus may be called the predominant misfit twinning region. The predominant misfit perfect dislocation region is the lower region below the curve, where the critical thickness for misfit twinning is larger than the critical thickness for misfit perfect dislocation. The four symbols in Fig. 4 represent four
The exact solution of the self-energy of a misfit twin, i.e., Eq. 共8兲, might be too complex for practical applications. Therefore, we propose two simplified versions of the selfenergy solution as follows: Es,twin = m
冋
册
eb23 共J1 + J2兲共b21 + b22兲 2h + ln 4 2 r0
+m
J2共b21 − b22兲 − 2J1b22 2
冋
m−1
eb23 4共J1 + J2兲共b21 − b22兲 共m − k兲h2 + 兺 2 2 2 + 4 k=1 k d + 4h 共J1 + J2兲共b21 + b22兲 + 2
册兺
m−1
共m − k兲ln
k=1
k2d2 + 4h2 k 2d 2
m−1
+
4h2J1共b21 + b22兲 共m − k兲共k2d2 − 4h2兲 兺 共k2d2 + 4h2兲2 k=1
−
J 2共 *J 2 + J 1兲 J 1共 J 1 + J 2兲 m 共b21 − b22兲 + 4 *
再 冋
− 4J1 − 共J1 − J2兲 − 2共b21 + b22兲
Es,twin ⬇
冋
冉
冉
J1 J2 − *
J21 J22 + *
冊冎
冊册
eb23 共J1 + J2兲共b21 + b22兲 + 4 2
− k兲ln
册
k2d2 + 4h2 . k 2d 2
共11a兲
,
册冋
m ln
2h + r0
m−1
共m 兺 k=1 共11b兲
In the first simplified solution, i.e., Eq. 共11a兲, we ignore the terms with r0, because r0 ⬍ h, and E0,3e and E0,3s, because they have to be calculated by Fourier transformation. E0,3e and E0,3s are induced by the mismatch in the elastic constants between the epilayer and the substrate, whose effect on the estimation of the critical thickness will be negligible if the elastic mismatch is small. The second simplified self-energy solution takes only the logarithmic term that makes the major contribution to the self-energy. The two simplified self-
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J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang TABLE I. The dimensionless twin-boundary energy for various crystals. Crystal Ag Al Au Cu Ni Pd Pt Si Ge In0.25Ga0.75As In0.5Ga0.5As0.5P0.5 In0.72Ga0.28P InP GaP Ge1−xSix
␥T 共mJ/ m2兲 a
8 75a 15a 24a 43a 90k 161a 25.5c 306e 24.5f 18f 12f 9f 20.5i 30− 4.5xj
共GPa兲 b
30.48 25.99b 27.58b 48.32b 76.06b 43.59b 60.95b 51d 40.15d 29.1g 28.3a 27.1g 22.3g 34i 40.15 +10.85xd
␥ T / 共 b 0兲
a 共nm兲 b
0.408 53 0.404 96b 0.407 82b 0.361 49b 0.352 4b 0.389 07b 0.392 36b 0.543 06b 0.565 74b 0.575 4h 0.575 4h 0.575 4h 0.586 87h 0.545i 0.565 74 −0.022 68xd
0.0016 0.0175 0.0033 0.0034 0.0039 0.0130 0.0165 0.007 0.0019 0.0036 0.0027 0.0019 0.0017 0.0027 共6 − 0.9x兲冑6 共803+ 217x兲共5.6574− 0.2268x兲
a
Reference 33. b Reference 39. c Reference 35. d Reference 40. e References 36 and 37. f Reference 38. g Reference 41. h Reference 2. i Reference 3. j Reference 42. k Reference 34.
energy solutions are used to calculate the critical thickness as a function of the shear modulus ratio, / *, for given mismatch strains of 0.01 and 0.015, and the results are plotted in Fig. 5, where the Poisson’s ratio for both epilayer and substrate is taken to be 0.3. For comparison, the exact solution of the self-energy is also used here to plot the critical thickness in Fig. 5 with the same parameters. Figure5 shows that for a given mismatch strain, the critical thickness monotonically decreases as the ratio of / * increases. For a given value of / *, the smaller the mismatch strain, the larger the critical thickness. In the exact solution, the critical thick-
nesses for = * are 157.58b0 and 65.832b0 for the mismatch strains of 0.01 and 0.015, respectively. They increase to 272.61b0 and 111.38b0 when the ratio of / * equals 0.1, i.e., the epilayer is ten times as soft as the substrate. On the other hand, the critical thickness decreases to 49.829b0 and 13.563b0 when the ratio of / * approaches 10, i.e., the epilayer is ten times as hard as the substrate. The critical thicknesses estimated from the two simplified solutions are very similar to thicknesses calculated from the exact solution. When / * = 1, as expected, the first simplified solution leads to almost the same critical thickness as that determined by the exact solution, whereas the second simplified solution overestimates the critical thickness by about 16%. When / * = 0.1, the first and second simplified solutions lead to the critical thicknesses of 359.5b0 and 401.21b0, and 151.08b0 and 167.66b0 for the mismatch strains of 0.01 and 0.015, respectively, indicating that the simplified solutions overestimate the critical thickness. When / * = 10, however, the first and second simplified solutions lead to the critical thicknesses of 13.239b0 and 22.126b0, and 3.5612b0 and 8.4239b0 for the mismatch strains of 0.01 and 0.015, respectively, showing that the simplified solutions underestimate the critical thickness. Nevertheless, within a range of the shear modulus ratio from 0.5 to 2, the relative difference is within 16% of the critical thickness evaluated by the exact and simplified solutions. V. COMPARISON WITH EXPERIMENTAL OBSERVATIONS
FIG. 5. The critical thickness for misfit twinning as a function of the shear modulus ratios for the mismatch strains of f = 0.01 and f = 0.015 calculated from the exact solutions and the two approximations.
As mentioned in the Introduction, Wu and Weatherly2 made TEM observations of misfit twins in In0.25Ga0.75As/ InP
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063501-8
J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang
TABLE II. Material constants of the In0.25Ga0.75As/ InP 共100兲 system. Material constants 共units兲 Component
In0.25Ga0.75As InP 共100兲
a 共nm兲
共GPa兲
␥T 共mJ/ m2兲
0.5754a 0.58687a
29.1b 22.3b
0.322a 0.36a
24.5c 9c
a
Reference 2. b Reference 40. c Reference 38.
共100兲 system, for which the material constants are listed in Table II. From the lattice constants, we determine the mismatch strain to be f = 0.02 in the In0.25Ga0.75As/ InP 共100兲 system. The theoretical prediction of the critical thickness for twinning is hc = 4.2 nm with a partial dislocation core size of b0 / 4. The experimental results2 reveal that the epilayers with thicknesses less than 5 nm are misfit-twin-free, while the 10 nm thick epilayer has twins. These results are more or less consistent with our theoretical predictions. In Part II of this series, we report on good comparisons of twin morphologies in theoretical predictions and experimental observations. VI. CONCLUDING REMARKS
In this paper, a single misfit twin in a heteroepitaxial layer is studied based on the dislocation theory of twinning by using the hybrid superposition and Fourier transformation approach. Analytical solutions are derived for the calculation of various energies involved in the misfit twinning process. The critical thickness for misfit twinning is determined by letting the minimum formation energy equal zero, which simultaneously determines the critical thickness and the critical twin width for a given epilayer/substrate system. The results indicate that a softer substrate leads to a smaller critical thickness for misfit twinning. Furthermore, a misfit-twinand-perfect-dislocation predominance chart is constructed for tensile mismatch strains to predict the predominant stress-relief mechanism. The theoretical predictions agree with experimental observations. The continuum-based dislocation theory of twinning may not hold when the critical thickness is at the same order as the magnitude of the Burgers vector. In such cases, atomistic calculation may be necessary. ACKNOWLEDGMENT
This work was supported by an RGC grant from the Hong Kong Research Grants Council, HKSAR, China. APPENDIX: MATHEMATICAL DERIVATIONS
The mathematical derivations for the stress and displacement fields of a misfit dislocation are described in detail in Ref. 31. For convenience, we briefly present some of them here and then derive the solution for a misfit twin. As described in the text, the third part of the stress field in the epilayer should satisfy the traction-free condition along the epilayer surface, i.e.,
共3兲 i2 =0
共i = 1,2,3兲
共A1兲
at x2 = h.
The second part of the stress and displacement fields in the substrate and the second and the third parts of the stress and displacements fields in the epilayer should meet the displacement-continuity and the traction-continuity boundary conditions along the interface, i.e., u共2*兲 = u共2兲 + u共3兲, = 1,2,3兲
共2*兲 共2兲 共3兲 i2 = i2 + i2
共i
at x2 = 0.
共A2兲
In isotropic elasticity, in-plane 共for edge components兲 and antiplane 共for screw components兲 deformations can be treated separately.
1. Screw component
For antiplane deformation, the displacement, u3, should satisfy the governing equation,
冉
冊
2 2 + 2 u3共x1,x2兲 = 0. 2 x1 x2
共A3兲
Equation 共A3兲 is satisfied when the displacement u3 is the imaginary part of an analytic function, g共z兲, with z = x1 + ix2, i.e., u3 =
1 Im关g共z兲兴,
共A4兲
where Im denotes the imaginary part of a complex function. Then, the stress components, 31 and 32, are calculated from the analytic function,
32 + i31 = g⬘共z兲,
共A5兲
where the prime stands for the differential with respect to z. The complex potentials of the first and second parts for a misfit twin are m−1
g共1兲 t 共z兲
eb 3 = 兺 ln共z − xd − jd兲, 2 j=0
g共2兲 t 共z兲
eb 3 =− 兺 ln共z − xd − jd − i2h兲, 2 j=0
m−1
共A6兲
where the subscript t means twin, xd denotes the first partial dislocation located at the left end of the twin, and xd = 0 in the present work. Thus, the two parts of the stress field in the epilayer and the contribution to the self-energy from the two parts of the stress field are given by m−1
共1兲 31 =−
eb 3 x2 , 兺 2 j=0 共x1 − jd兲2 + x22 m−1
共1兲 32 =
x1 − jd eb 3 , 兺 2 j=0 共x1 − jd兲2 + x22
共2兲 31 =
x2 − 2h eb 3 , 兺 2 j=0 共x1 − jd兲2 + 共x2 − 2h兲2
共A7a兲
m−1
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063501-9
J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang m−1
共2兲 32
s E0,1
x1 − jd eb 3 =− , 兺 2 j=0 共x1 − jd兲2 + 共x2 − 2h兲2 1 =− 2
冕
h 共m−1兲d
兺 x =0
冋
r0
1 =− 2 =
冕
冋
兺 x =0
0
eb23 4
m−1
k 2d 2 + h 2
共m − k兲ln 2 2 2 兺 k d +r k=1 0
册
共A7c兲
,
册
1
m−1
共m − k兲ln 兺 k=1
k2d2 + 4h2 . 共A7d兲 k 2d 2 + h 2
Using Fourier transformation, ˜f 共,x 兲 = 1 2 冑2
冕
⬁
e−兩兩x2
m−1
eijd .
共A14兲
j=0
f共x1,x2兲eix1dx1 ,
where
冊
共A9兲
冑2
冋冕
−⬁
˜u3共,x2兲 = 0. 2x 2
1
1 E0,3s = − Re 2
we convert the governing equation, 共A3兲, into the space as − 2 +
冊兺
冉
冕
⬁
˜f 共,x 兲e−ix1dx , 2 1
共A15兲
−⬁
共3兲 we have the stress component, 31 , in the real space. Finally, the self-energy associated with the third part of the stress field is calculated from
=
冉
e兩兩x2 a21
f共x1,x2兲 =
共A8兲
2
2冑2关共⌫ − 1兲 − 共⌫ + 1兲a21兴
Using the inverse Fourier transformation,
共2兲 31 b3dx2
m ln 2 +
eb3共⌫ − 1兲
+
1
h 共m−1兲d
共3兲 ˜31 =
共1兲 31 b3dx2
eb23 h = m ln + 4 r0 s E0,2
共A7b兲
eb23 4
S t共 兲 =
冕
h
0
⬁
1
冑2
冕
⬁ m−1
兺 兩˜31共3兲e−ix 兩共x =kd兲db3dx2 1
1
−⬁ k=0
St共兲d ,
共A16a兲
0
再
共1 − ⌫兲共1 − 1/e2h兲 关共⌫ − 1兲 − 共⌫ + 1兲e2h兴
冎冋
册
1 − cos共md兲 . 1 − cos共d兲 共A16b兲
Thus, the third part of the displacement in the epilayer satisfying the traction-free boundary condition, Eq. 共A1兲, is given by 兩兩x2 ˜u共3兲 + A2e−兩兩x2 , 3 共,x2兲 = A1e
with A1 =
共A10a兲 共A10b兲
A2/a21 ,
where a1 = ea2 and a2 = 兩兩h. The second part of the displacement and stress fields in the substrate has to meet the zero remote stress condition at infinity and, therefore, it takes the form * 兩兩x2 ˜u共2*兲 . 3 共,x2兲 = A1e
共A11兲
Then, using Eq. 共A2兲 gives A*1 = A1 + A2 +
1 ˜u共2兲 3 , − i x1
A*1 = ⌫共A1 − A2兲 +
共2兲 ˜32 , 共*兩兩兲
共A12a兲 where
eb3 i兩兩 ijd , 2兺 e 2冑2 a1 j=0
⌫=
Solving Eq. 共A12兲 leads to A2 =
eb 3
2冑2关共⌫ − 1兲 − 共⌫ +
For in-plane deformation, the equilibrium equations can be expressed by the displacements as 2共1 − 兲
2u 1 2u 2 2u 1 + + 共1 − 2 兲 = 0, x21 x1x2 x22
2u 2 2u 1 2u 2 2共1 − 兲 2 + + 共1 − 2兲 2 = 0. x 2 x 1 x 2 x1
1兲a21兴
冉
. *
共A12b兲
冊
共A17兲
Two complex potentials, 共z兲 and 共z兲, are used to represent the displacement and stress fields,
11 + 22 = 2关⬘共z兲 + ⬘共z兲兴,
u共z兲 = u1 + iu2 =
m−1
共2兲 ˜32 =−
2. Edge component
22 − i21 = ⬘共z兲 + ⬘共z兲 + 共z − ¯兲 z ⬙共z兲,
m−1
˜u共2兲 eb 3 1 3 ijd =− , 2兺 e 冑 x1 2 2 a1 j=0
册
共A18兲
1 关共z兲 − 共z − ¯兲 z ⬘共z兲 − 共z兲兴. 2
The complex potentials of the first and second parts for a misfit twin are m−1
m−1
1 i 1 − 兺 eijd . * j=0 共A13兲
共3兲 , in the space is given by The stress component, ˜31
共1兲 t
J 1b e = 兺 ln共z − jd兲, i j=0
共A19a兲
m−1
共1兲 t =
J 2b e 兺 ln共z − jd兲, − i j=0
共A19b兲
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063501-10
J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang m−1
共2兲 t
共3兲 11 = 22A1e兩兩x2 + A2e−兩兩x2 − A3共兩兩x2 + 2兲e兩兩x2
m−1
z − jd J 2b e J 1b e =− ln共z − jd − i2h兲 + , 兺 兺 i j=0 i j=0 z − jd − i2h
+ A4共兩兩x2 − 2兲e−兩兩x2,
共A19c兲 m−1
共2兲 t
m−1
+ 2兲e兩兩x2 − A4共兩兩x2 + 1 − 2兲e−兩兩x2兴,
2hJ b 1 J 1b e = 兺 ln共z − jd − i2h兲 − 2 e 兺 i j=0 z − jd − i2h j=0 +
4h J1be 1 , 兺 i j=0 共z − jd − i2h兲2
e = E0,1
共J1 +
J2兲共b21
+
2 − k兲ln
冋 册
b22兲
k 2d 2 + h 2 k2d2 + r20
m ln
+
h + r0
+
4h2J1共b21
+
兺 k=1
= 兩兩关− A*1e兩兩x2 + A*3共兩兩x2 − *兲e兩兩x2兴, u共2*兲 2
−
m−1 b22兲
兺 k=1
共2*兲 11 = 2*2A*1e兩兩x2 − A*3共兩兩x2 + 2*兲e兩兩x2, 共2*兲 12 = 2*i兩兩关A*1e兩兩x2 − A*3共兩兩x2 − 1 + 2*兲e兩兩x2兴,
共A25兲
m−1
共m 兺 k=1
共2*兲 22 = 2*2关− A*1e兩兩x2 + A*3共兩兩x2 − 2 + 2*兲e兩兩x2兴.
The traction-free condition, Eq. 共A1兲, requires 共a2 + 1 − 2兲 A2 − A3共a2 − 1 + 2兲a1 − A4 = 0, a1 a1 共A26兲 共a2 + 2 − 2兲 A2 − A3共a2 − 2 + 2兲a1 + A4 = 0. A 1a 1 + a1 a1
A 1a 1 −
共m − k兲共k2d2 − 4h2兲 共k2d2 + 4h2兲2
The boundary condition, Eq. 共A2兲, gives
共m − k兲h k 2d 2 + h 2
2
A*1 + 共1 − 2*兲A*3 = ⌫关A1 − A2 + 共1 − 2兲共A3 − A4兲兴 +
m−1
+
4共J1 + J2兲共b21 − b22兲 共m − k兲h2 兺 2 2 2. k=1 k d + 4h
共A20b兲
2共1 − 兲
2˜u1 ˜u2 + 共1 − 2兲 2 = 0, x2 x2
u2 ˜u1 ˜ 2 = 0. − 2共1 − 2兲u 2 + 共− i兲 x2 x2 2˜
−
共A21兲
A*1 = A1 + A2 +
+ A4兩兩x2e−兩兩x2,
and
共2兲 22 , 2 2 *
共A22兲
1 u共2兲 1 , 2 x1
− A*1 − *A*3 = − A1 + A2 − 共A3 − A4兲 +
兩兩x2 u共3兲 + A2e−兩兩x2 − A3兩兩x2e兩兩x2 1 = iA1e
+ A4共兩兩x2 + 兲e−兩兩x2兴,
共A27兲
and
From Eq. 共A21兲, the third part of the displacements and stresses of the film in the space is given by
兩兩x2 + A2e−兩兩x2 + A3共兩兩x2 − 兲e兩兩x2 u共3兲 2 = 兩兩关− A1e
共2兲 12 , i2兩兩*
A*1 + 共2 − 2*兲A*3 = ⌫关A1 + A2 + 共2 − 2兲共A3 + A4兲兴
By using Fourier transformation, we convert the equilibrium equations 共A17兲 into the space, ˜ 1 + 共− i兲 − 22共1 − 兲u
共A24兲
and
共A20a兲
J2共b21 − b22兲 − 2J1b22 2
2J1共b22
= iA*1e兩兩x2 − A*3兩兩x2e兩兩x2, u共2*兲 1
共m 兺 k=1 m−1
冋 册
m−1 b22兲
+m
−
k2d2 + 4h2 k 2d 2 + h 2
Similarly, the second part of the displacements and stresses of the substrate in the space takes the form
m−1
2J1共b21 − b22兲 共m − k兲h2 兺 2 2 2, k=1 k d + h
共J1 + J2兲共b21 + b22兲 = m ln 2 + 2 − k兲ln
+ 2兲e兩兩x2 − A4共兩兩x2 + 2 − 2兲e−兩兩x2兴.
共A19d兲
where be = b1 + ib2. Thus, the energies in the film associated with the two parts are
共A23兲
共3兲 22 = 22关− A1e兩兩x2 − A2e−兩兩x2 + A3共兩兩x2 − 2
m−1
2
e E0,2
共3兲 12 = 2i兩兩关A1e兩兩x2 − A2e−兩兩x2 − A3共兩兩x2 − 1
共A28兲
1 u共2兲 2 , − i兩兩 x1
for traction continuity and displacement continuity, respectively. Solving 共A26兲–共A28兲 leads to A3 =
L1 f 1 + L2 f 2 , H
A4 =
a21共L3 f 1 + L4 f 2兲 , H
共A29a兲
where
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063501-11
J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang
L1 = 共⌫ − 1兲 − 共⌫ + 兲a21,
再 冋
u共2兲 −1 2,jd = 2hb2 共J1 + J2兲 − 共J1 x1 2共x2 + 4h2兲
L2 = − 2共⌫* + 1兲a2a21 , 共A29b兲
L3 = 2共⌫ − 1兲a2 , L4 = 共⌫* + 1兲a21 − 共⌫* − 兲,
+ J 2兲
H = L 1L 4 − L 2L 3 ,
− J 2兲 +
共A29c兲
f1 =
冉
冊
冊
1 x21 + 4h2
From 共A18兲, 共A19c兲, and 共A19d兲, the second part of the stresses and displacements in the film along the interface is
⇒
4h2 − 3x21 共x21 + 4h2兲3
共2兲 兺 22,jd, j=0
u共2兲 1 = x1
m−1
兺 j=0
u共2兲 1,jd , x1
m−1 共2兲 12 =
共2兲 兺 12,jd , j=0
u共2兲 2 = x1
m−1
兺 j=0
,
Using the following Fourier transformations:
共A29d兲
共2兲 22 =
册冎
共A30f兲
x = x1 − jd.
共2兲 共2兲 12 1 u共2兲 1 u共2兲 1 22 2 − − . f2 = − 2 1 + x1 − i兩兩 x1 2* 2 i兩兩
m−1
8共J1 − J2兲h2 8J1h2共x2 − 12h2兲 − x2 + 4h2 共x2 + 4h2兲2
共A30e兲
共2兲 共2兲 12 1 u共2兲 1 u共2兲 * 22 2 1 + − + , 2 x1 − i兩兩 x1 2* 2 i兩兩
冉
册 冋
x2 − 4h2 8J1h2共4h2 − 3x2兲 + + b1x 共J1 x2 + 4h2 共x2 + 4h2兲2
x1 ⇒ 2 x1 + 4h2
共A30a兲
u共2兲 2,jd , x1
x31 − 12hx1 共x21 + 4h2兲3
冑 ⇒
冑
冑 ⇒
1 , 2 2ha21
共x21 + 4h2兲2
⇒
冑
兩兩 , 2 a21
2 , 2 4ha21
i兩兩 , 2 a21
冑
4h2 − x21
冑
x1 ⇒ 2 共x1 + 4h2兲2
共A31a兲 i , 2 4ha21
− i兩兩 , 2 2a21
where
再 冋
and
−1 x2 − 4h2 共2兲 = 2hb 共J − J 兲 − 共J − J 兲 1 2 1 2 1 2 22,jd 共x2 + 4h2兲 x + 4h2
共2兲 12,jd =
册 冋
−
8J1h2共4h2 − 3x2兲 + b2x 共J1 + J2兲 共x2 + 4h2兲2
+
8共J1 + J2兲h2 8J1h2共x2 − 12h2兲 − x2 + 4h2 共x2 + 4h2兲2
册冎
,
共A30b兲 f e1
再 冋
8J1h2共4h2 − 3x2兲 + + b1x 共J1 + J2兲 共x2 + 4h2兲2 8J1h 共x − 12h 兲 8共J1 + J2兲h − x2 + 4h2 共x2 + 4h2兲2 2
−
2
2
2
再 冋
册冎
f e2
=
=
+ J 2兲
,
2
2
2
冑2a21 b1
冑2a21
冉
J2 2hJ1 − 2 兩兩
冉
J1 2hJ2 − 4h2J1 − 2+ 兩兩
共A30c兲 f o1
册 冋
x − 4h 8J1h 共4h − 3x 兲 + − b2x 共J1 x2 + 4h2 共x2 + 4h2兲2 2
b1
冊冉
* − *
冊兺
m−1
eijd ,
共A32a兲
j=0
冊冉
1 1 * −
冊兺
m−1
eijd ,
j=0
共A32b兲
=
u共2兲 −1 1,jd = 2hb1 共J1 + J2兲 + 共J1 x1 2共x2 + 4h2兲 2
共A31b兲
we calculate f 1 and f 2, which even and odd parts are expressed as
−1 x2 − 4h2 2hb 共J − J 兲 + 共J − J 兲 2 1 2 1 2 2 共x2 + 4h2兲 x + 4h2
册 冋
f共x1 − jd兲 ⇒ F共兲eijd ,
8共J1 − J2兲h2 8J1h2共x2 − 12h2兲 − J 2兲 − − x2 + 4h2 共x2 + 4h2兲2
册冎
f o2
=
ib2
冑2a21 ib2
冑2a21
冉
J2 + 2hJ1 兩兩
冊冉
冉
J1 + 2hJ2 + 4h2J1兩兩 兩兩
* − *
冊兺
m−1
eijd ,
共A32c兲
j=0
冊冉
1 1 * −
冊兺
m−1
eijd .
j=0
共A32d兲 ,
共A30d兲
The energy associated with the third part can be expressed in terms of as
Downloaded 16 Mar 2007 to 143.89.145.133. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
063501-12
J. Appl. Phys. 101, 063501 共2007兲
Liu, Zhang, and Zhang
冋冕
1 E0,3e = − Re 2
h
0
1
冑2
冕
⬁ m−1
兺
−⬁ k=0
共3兲 + ˜21 b2兲e−ix1兩x1=kdddx2
=−
2冑2
冕
⬁
−⬁
4
共3兲 兩共˜11 b1
册
m−1
1 兺 兵兩兩b1关共L1B1 + L3B2兲f 1 + 共L2B1 H k=0
+ L4B2兲f 2兴 + ib2关共L1B3 − L3B4兲f 1 + 共L2B3 − L4B4兲f 2兴其兩e−ix1兩x1=kdd .
共A33兲
Finally, the energy due to the third part can be given by E0,3e =
Eb1 =
2 共b Eb1 + b22Eb2兲, 2 1
冕
⬁
W1,t共兲d,
Eb2 =
0
冕
⬁
共A34a兲 W2,t共兲d ,
0
where W1,t共兲 =
−1 Ha21
关C1共J2 − 2J1a2兲共L1B1 + L3B2兲 − C2共J1
− 2J2a2 + 4J1a22兲共L4B2 + L2B1兲兴 W2,t共兲 =
1 Ha21
1 − cos共md兲 , 1 − cos共d兲
关C1共J2 + 2J1a2兲共L1B3 − L3B4兲 + C2共J1
+ 2J2a2 + 4J1a22兲共L2B3 − L4B4兲兴
1 − cos共md兲 , 1 − cos共d兲 共A34b兲
B1 = 1 − a21 − 2a2,
B2 = 1 − 2a2a21 − a21 ,
B3 = a21 − 2a2 − 1,
B4 = a21 − 2a2a21 − 1,
C1 = 1
* − , *
C2 =
1 1 − . *
共A34c兲
共A34d兲
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