Electric Circuit Model Analogy for Equilibrium Lattice Relaxation in Semiconductor Heterostructures

Transcription

1 Journal of ELECTRONIC MATERIALS, Vol. 47, No. 1, 2018 DOI: /s z Ó 2017 The Minerals, Metals & Materials Society Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures TEDI KUJOFSA 1,2 and JOHN E. AYERS 1 1. Electrical and Computer Engineering Department, University of Connecticut, 371 Fairfield Way, Unit 4157, Storrs, CT , USA Tedi.Kujofsa@gmail.com The design and analysis of semiconductor strained-layer device structures require an understanding of the equilirium profiles of strain and dislocations associated with mismatched epitaxy. Although it has een shown that the equilirium configuration for a general semiconductor strained-layer structure may e found numerically y energy minimization using an appropriate partitioning of the structure into sulayers, such an approach is computationally intense and non-intuitive. We have therefore developed a simple electric circuit model approach for the equilirium analysis of these structures. In it, each sulayer of an epitaxial stack may e represented y an analogous circuit configuration involving an independent current source, a resistor, an independent voltage source, and an ideal diode. A multilayered structure may e uilt up y the connection of the appropriate numer of these uilding locks, and the node voltages in the analogous electric circuit correspond to the equilirium strains in the original epitaxial structure. This enales analysis using widely accessile circuit simulators, and an intuitive understanding of electric circuits can easily e extended to the relaxation of strained-layer structures. Furthermore, the electrical circuit model may e extended to continuously-graded epitaxial layers y considering the limit as the individual sulayer thicknesses are diminished to zero. In this paper, we descrie the mathematical foundation of the electrical circuit model, demonstrate its application to several representative structures involving InxGa 1x As strained layers on GaAs (001) sustrates, and develop its extension to continuously-graded layers. This extension allows the development of analytical expressions for the strain, misfit dislocation density, critical layer thickness and widths of misfit dislocation free zones for a continuously-graded layer having an aritrary compositional profile. It is similar to the transition from circuit theory, using lumped circuit elements, to electromagnetics, using distriuted electrical quantities. We show this development using first principles, ut, in a more general sense, Maxwell s equations of electromagnetics could e applied. Key words: Electrical circuit model, relaxation, equilirium in-plane strain, multilayers, compositionally-graded, InGaAs, GaAs INTRODUCTION The understanding of the equilirium lattice relaxation has important implications in the (Received March 16, 2017; accepted August 16, 2017; pulished online Septemer 13, 2017) determination of the staility criteria for electronic and optical devices. 1 6 Furthermore, the equilirium configuration serves as the starting point for kinetically-limited lattice relaxation calculations and is critical in determining the effective stress and therefore the driving force for dislocation flow. Several models have een developed for the 173

2 174 Kujofsa and Ayers determination of the equilirium configuration 7 15 and, although it has een shown that the equilirium configuration for a general semiconductor strained-layer structure may e determined numerically y energy minimization using an appropriate partitioning of the structure into sulayers, 7 11 such an approach uses specialized code, is computationally intense, and does not lend itself to an intuitive understanding necessary for innovative structure design. To avoid these limitations, and to enale the development of analytical solutions for compositionally-graded heterostructures, we propose the use of an electrical circuit model analogy. Several mechanical electrical analogs have een developed and used, particularly for load eam analysis. The most common of these are the so-called force-current and force-voltage analogs, ut others have also een developed. It is possile to use any of these to provide a physically correct description of ehavior in a mechanical system; however, some are etter suited to certain applications. For example, our work relates to the static ehavior of a semiconductor heterostructure in equilirium, and there is no need to include electrical components such as capacitors and inductors, which may e included for transient (time-dependent) modeling. Among the previously pulished work on mechanical electrical analogies, a report of particular interest is that y Olsson and Bath, 20 which descries two particular choices of analogies for application to prolems of geophysics. Their second transcriptive system considers electrical current to e analogous to mechanical stress, electrical voltage to e analogous to mechanical strain, and electrical resistance to e analogous to the reciprocal of an elastic modulus. Olsson and Bath point out that an advantage of this transcriptive system is that it facilitates simple electrical analogies, and lends itself to series or parallel connections, which correspond to each other in the mechanical and electrical domains. The purpose for developing an analogy etween one field of physics and another is to take advantage of theoretical framework, which exists in one field ut not the other. In a recent short pulication, 21 we have shown that an aritrary semiconductor heterostructure, approximated y a stack of uniform composition layers, may e modeled using an electric circuit analogy similar to Olsson and Bath s, in which electrical voltage is considered analogous to mechanical strain. In this model, each sulayer of a general strained-layer device may e represented y an analogous electrical circuit configuration involving an independent current source, a resistor, an independent voltage source, and an ideal diode. A multilayered structure may e uilt up y the connection of the appropriate numer of these asic uilding locks, after which the node voltages in the electric circuit correspond to the equilirium strains in the original epitaxial structure. If any sulayer in the structure is grown coherently on the sulayer elow, the difference in strain in these sulayers is given y the lattice mismatch difference. This is modeled y introducing an independent voltage source etween the nodes representing the two sulayers, and the resulting strains are analogous to the node voltages, which may e found using supernode theory. The theoretical framework of supernodes exists in electrical circuits ut not in mechanical systems, and represents an important motivation for using the electric circuit analogy in this case. Another significant point is that the voltage source, which is related to the coherency state of the sulayer, is connected through an ideal diode, which conducts only in the case of coherent growth. Use of this device in the electric circuit analogy allows a physically correct description of the mechanical ehavior, though no such device exists in mechanical systems. This represents a second key motivation for the application of the electric circuit model analogy to a strained semiconductor heterostructure. Whereas, in the short paper, 21 we only considered structures which were approximated y a stack of uniform layers, here, in the full-length paper, we show the application of the electrical circuit model approach to continuously-graded structures, wherein the strain is a continuous function of distance from the interface. This enales the development of a closed-form expression for the strain as a function of distance in a continuously-graded layer with any aritrary compositional profile, including linear, exponential, power-law, or complementary error function profiles. This in turn allows an analytical solution for the critical layer thickness, widths of misfit dislocation free zones, and misfit dislocation density in these graded structures. In the present paper, we further develop the mathematical framework for the electrical circuit model approach, starting with a single strained layer and then generalizing to a multilayer structure. We relate the physical quantities in the epitaxial stack to those in an analogous electrical circuit. The electrical circuit model developed in this work is applicale for the determination of the equilirium in-plane strain in any epitaxial material system. However, for application to crystals outside the diamond and zinc lende classes of materials, it will e necessary to modify the geometric factors related to the elastic properties and dislocation geometry. Here, we confine our discussion to zinc lende materials, and demonstrate the equilirium analysis of a numer of In x Ga 1x As/GaAs (001) epitaxial structures, including a single strained layer, threelayer stacks, step-graded layers, linearly-graded layers and S-graded layers. We show that the strain results of the circuit model calculations are in agreement with the theory of Matthews and Blakeslee 12 for the single strained layer. We also develop exact results for the case of a linearly graded layer, whereas previously only an

3 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures 175 approximate solution had een developed y Tersoff. 13,14 In the approximate solution of Tersoff, it was assumed that the strain was completely relieved in the region containing dislocations, and that the dislocation line energy was independent of distance from the surface. We have not relied on these simplifying assumptions when applying the circuit model, and therefore provide exact in-plane strain results for the linearly-graded case. Furthermore, we show the extension of the circuit model to any continuously-graded semiconductor layer y taking the limit as the thickness of the individual sulayers approaches zero. This enales the development of analytical expressions for the strain, misfit dislocation density, critical layer thickness and widths of misfit dislocation free zones (MDFZ) in a continuously-graded epitaxial layer having any compositional grading profile, including linear, 22 exponential, 23 power law, and S-graded 9,24 profiles. The extension from a finite numer of sulayers to the continuously-graded case is analogous to the transition from circuit theory, using lumped circuit elements, to electromagnetics, wherein the electrical quantities are distriuted. We show the development of these expressions ased on first principles, ut, in a more general sense, Maxwell s equations of electromagnetics could e applied to continuously-graded strained layers. PHYSICAL MODEL FOR EQUILIBRIUM STRAIN RELAXATION Matthews and Blakeslee 12 developed a forcealance model for the equilirium strain in a single layer with uniform composition. In it, the equilirium strain was considered to e the value at which the glide force on a grown-in dislocation is equal to the opposing line tension. It has een shown that an equivalent result may e found y minimizing the sum of the strain energy and the line energy of misfit dislocations. 15 In a partly-relaxed and compositionally uniform epitaxial layer (as shown in Fig. 1a) which contains misfit dislocations, the inplane strain e is given y e ¼ f f q sin a sin /; jf j ð1þ where f is the lattice mismatch, f ða s a e Þ=a e, a s and a e are the relaxed lattice constants of the sustrate and epitaxial crystal (cm), respectively, the term f = jf j accounts for the sign of the mismatch, q is the linear misfit dislocation density in at the mismatched interface (cm 1 ), is the length of the Burgers vector (cm), a is the angle etween the Burgers vector and dislocation line vector, and / is the angle etween the glide plane and interface. We will define the lattice mismatch of the sustrate (layer 0) as f 0 0 to simplify the mathematical descriptions that will follow. The areal strain energy (erg/cm 2 ) associated with a partially-relaxed epitaxial layer with thickness h and an in-plane strain e,is E e ¼ Yhe 2 ; ð2þ where Y is the iaxial modulus (dyn/cm 2 ), Y ¼ C 11 þ C 12 2C 2 12 =C 11 ¼ 2Gð1 þ mþ= ð1 mþ, G is the shear modulus (dyn/cm 2 ), C 11 and C 12 are the elastic stiffness constants (dyn/cm 2 ), and v is the Poisson ratio (unitless). The line energy of dislocations per unit area (erg/cm 2 ), assuming two orthogonal networks with equal cross-sectional density, is E d ¼ q G2 1 m cos 2 a ln h 2pð1 mþ ¼ ðf eþ f G 1 m cos 2 a jf j2pð1 mþsin a sin u ln h ; ð3þ Here, the dislocation energy is determined using a mean-field approach without including dislocation dislocation interactions, and this is the main approximation used in this work. The equilirium condition is found y minimizing the sum of the dislocation line energy and the strain energy, E ¼ E d þ E e. Differentiating the energy, E, and setting the partial derivative to zero ð E e þ E d ¼ 2Yhe f jf j G 1 mcos 2 h 2pð1 mþsina sinu ln h ¼ 0: ð4þ The solution for equilirium in-plane strain, accounting for the possiility of pseudomorphic growth, is ( eðhþ ¼ f ; h h c f ð1m cos 2 aþ jf j8phð1þmþsin a sin u ln h ; ; h>hc ð5þ where h c is the critical layer thickness at which it ecomes energetically favorale to introduce misfit dislocations. Below h c, the in-plane strain is equal to the coherency strain (lattice mismatch). If we consider an epitaxial structure involving a stack of three disparate layers as shown in Fig. 1, the strain in each layer may e related to the misfit dislocation densities for that layer and those elow it. The in-plane strain e n in the nth sulayer of a general structure is given y e n ¼ f n f n f n1 q jf n f n1 j j j sin a sin u: ð6þ j¼1 For the case of three layers illustrated in Fig. 1, the in-plane strains are X n

4 176 Kujofsa and Ayers Fig. 1. Schematic representation of (a) a single compositionally uniform epitaxial layer with lattice mismatch f and thickness h, () an epitaxial structure comprised of three compositionally uniform layers with varying thickness and compositional mismatches and (c) a generalized epitaxial structure divided into N sulayers with varying thickness and mismatch. e 3 ¼ f 3 q 3 3 sin a sin / q 2 2 sin a sin / q 1 1 sin a sin / ð7þ e 2 ¼ f 2 q 2 2 sin a sin / q 1 1 sin a sin / e 1 ¼ f 1 q 1 1 sin a sin /: By rearranging the equations aove, the linear misfit dislocation densities for the three mismatched interfaces are given as q 3 ¼ f 3 f 2 ðf 3 e 3 Þðf 2 e 2 Þ jf 3 f 2 j 3 sin a sin / q 2 ¼ f 2 f 1 ðf 2 e 2 Þðf 1 e 1 Þ ð8þ jf 2 f 1 j 2 sin a sin / q 1 ¼ f 1 f 0 f 1 e 1 jf 1 f 0 j 1 sin a sin / The sum of the strain and dislocation line energy per unit area may e found y adding the contriutions of the three sulayers: E ¼ E d;1 þ E e;1 þ E d;2 þ E e;2 þ E d;3 þ E e;3 2 h i 3 G 1 q 2 1ð 1m 1 cos 2 aþ 1 2pð1m 1 Þ ln h 1þh 2 þh 3 1 þ 2Y 1 h 1 e 1 h i G ¼ 2 þq 2 2ð 1m 2 cos 2 aþ 2 2pð1m 2 Þ ln h 2þh 3 2 þ 2Y 2 h 2 e 2 : h i 5 G 3 þq 2 3ð 1m 3 cos 2 aþ 3 ð Þ ln h 3 þ 2Y 3 h 3 e 3 2p 1m 3 3 ð9þ To determine the equilirium strain of the threesulayer system shown in Fig. 1, we must differentiate the energy with respect to the in-plane strain at each sulayer and set each partial derivative equal to zero: h i 0 ¼ 2Y 3 e 3 f3f2 G33ð1m3 cos 2 aþ h3 jf3f2j2pð1m3þsin a sin / ln 3 h i 0 ¼ 2Y 2 e 2 f2f1 G22ð1m2 cos 2 aþ h2þh3 jf2f1j2pð1m2þsin a sin / ln 2 þ f 3 f 2 G m 3 cos 2 a jf 3 f 2 j2pð1 m 3 Þsin a sin / ln h ¼ 2Y 1 h 1 e 1 f 1 f 0 G m 1 cos 2 1 jf 1 f 0 j2pð1 m 1 Þsin a sin / ln h 1 þ h 2 þ h 3 1 þ f 2 f 1 G m 2 cos 2 a jf 2 f 1 j2pð1 m 2 Þsin a sin / ln h 2 þ h 3 : 2 ð10þ Concurrent solution of the three equations aove yields the equilirium in-plane strains e 1, e 2,and e 3. A similar analysis may e extended to any multilayered and compositionally-graded structure with N sulayers (as shown in Fig. 1c) and, in the general case, we consider the sum of the strain and dislocation line energy, E ¼ P N j¼1 E d;j þ E e;j. The equilirium in-plane strains are found y setting each partial derivative to n ¼ 0 and solving the resulting system of N equations:

5 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures 0 ¼ 2Y N h N e N f N f N1 G N N 1 m N cos 2 N jf N f N1 j2pð1 m N Þsin a sin / ln h N ; n ¼ N N 0 2Y n h n e n f "! # 1 n f n1 G n n 1 m n cos 2 a XN h j ln 0 jf n f n1 j2pð1 m n Þsin a sin / j¼n n n þ f "! # ; 1 n<n: B nþ1 f n G nþ1 nþ1 1 m nþ1 cos 2 a ln XN h j A jf nþ1 f n j 2pð1 m nþ1 Þsin a sin / nþ1 j¼nþ1 177 ð11þ Essentially, the analysis provided aove is the asis for the development of the Bertoli et al. 7 model, which utilizes an ad hoc numerical approach to minimize the sum of the strain energy and dislocation line energy for an aritrary multilayered heterostructure. The approach is generally applicale, and compositionallygraded layers may e represented y staircase profiles with aritrary precision. ELECTRICAL CIRCUIT MODEL FOR EQUILIBRIUM STRAIN RELAXATION The development of an electric circuit model stems from the fact that Eq. 4 resemles the node Fig. 2. (a) A simple resistive circuit comprising a resistor and an independent current source; () the equivalent electrical circuit to determine the equilirium lattice relaxation of a single compositionally uniform epitaxial layer with either compressive and tensile cases; (c) the equivalent circuit for an epitaxial layer consisting of three sulayers with varying compositional mismatch and thickness; and (d) the equivalent circuit for an epitaxial layer roken down into N sulayers.

6 178 Kujofsa and Ayers voltage expression for the top node of the simple electrical circuit shown in Fig. 2a: 0 ¼ 2Yhe f G 1 m cos 2 h jf j2pð1 mþsin a sin / ln h $ V R I ¼ 0: ð12þ The node voltage is determined from Kirchhoff s current law, which states that the algeraic sum of the currents flowing away from a node must equal zero.* In Eq. 12, the symol $ implies that quantities or relationships on either side of the arrow are analogous though they generally possess different units. The derivative of the strain energy with respect to the in-plane strain is analogous to the current flowing through the resistor e =@e $ I R, whereas the change in the dislocation energy with respect to the in-plane strain is analogous to the value of the independent current source D =@e $ I. Given that the partial derivatives are analogous to electrical currents, comparison of the two forms of Eq. 12 reveals that the equilirium strain is analogous to the node voltage**: e $ V; ð13þ the factor multiplying the strain is analogous to a conductance (reciprocal of resistance): 2Yh $ 1 R ; ð14þ and the sutracted term is analogous to an independent current source entering the top node of the circuit (see Fig. 2a): f Gð1m cos 2 hþ jf j2pð1mþsin a sin / ln h $ I : ð15þ To account for the possiility of pseudomorphic epitaxy, we can include an independent voltage source and an ideal diode in the circuit, which together form a clipping circuit as shown in Fig. 2. The ideal diode acts as a switch that is conductive only when an epitaxial layer is coherently grown. The numerical value of the voltage source V S is equal to the coherency strain in the layer: f $ V S : ð16þ To properly account for the sign of the lattice mismatch (tensile or compressive), the ideal diode must always face toward the true positive terminal of the independent voltage source (Fig. 2 illustrates oth cases). Therefore, in terms of the electrical circuit model, the two analogous forms of the solution for the node voltage (or the equivalent equilirium strain) are given as: V ¼ V S; h h c $ eðhþ I R; h>h c ( f ; h h ð17þ c ¼ ð1m cos 2 aþ : ; h>hc f jf j 8phð1þmÞsin a sin / ln h During pseudomorphic epitaxy, the in-plane strain of the heterostructure is determined y the lattice mismatch, and, in the electrical circuit model, this is analogous to the conduction of an ideal diode connected in series with an independent voltage source. An ideal diode acts as a perfect conductor without any internal resistance which results in the formation of a supernode in the circuit shown on Fig. 2; in other words, the separation of two essential nodes y an independent voltage source. We can extend the electrical circuit model descried aove to the three-layer structure shown in Fig. 1. By a similar approach, Eq. 10 resemles the node voltage expressions for three essential nodes à and therefore we can consider the consecutive stacking of the electrical circuit given in Fig. 2 to otain an equivalent circuit that descries a three-layered heterostructure (Fig. 2c). The appropriate connections of the electrical circuit lock are done in such a way that the separation of two essential nodes consist on one end the comination of the independent voltage source and the ideal diode and in the other end the independent current source. Furthermore, in each uilding lock, the resistor shares one of its terminals with the essential node and the other with the ground connection. Thus, in the analogous electrical circuit model, the node voltages for three essential nodes are given y *In this analysis, an electrical current which enters the node is considered negative. **It should e noted that the choice to represent the partial derivatives y analogous current sources is not unique, ut was made for convenience. The analogous circuit could e defined differently and still yield the correct results, as long as a consistent set of analogs was used. An ideal diode acts as a switch which allows the flow of a forward ias current with any magnitude ut which locks current in the reverse ias direction. à An essential node is defined as a node connected to more than two circuit elements. Therefore, the numer of essential nodes in the analogous circuit corresponds to the minimum numer of equations which must e utilized to solve the circuit.

7 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures ¼ 2Y 3 h 3 e 3 f 3f 2 ð h Þ G 3 3 1m 3 cos 2 a jf 3 f 2 j2pð1m 3 Þsin a sin / ln h 3 h 3 G 2 2 ð1m 2 cos 2 aþ jf 2 f 1 j2pð1m 2 Þsin a sin / ln h 2þh ¼ 2Y 2 h 2 e 2 f 2f 1 h i þ f 3f 2 G 3 3 ð1m 3 cos 2 aþ jf 3 f 2 j2pð1m 3 Þsin a sin / ln h 3 3 h 0 ¼ 2Y 1 h 1 e 1 f 1f 0 G 1 1 ð1m 1 cos 2 aþ jf 1 f 0 j2p 1m h 1 i þ f 2f 1 G 2 2 ð1m 2 cos 2 aþ jf 2 f 1 j2pð1m 2 Þsin a sin / ln h 2þh 3 2 i i ð Þsin a sin / ln h 1þh 2 þh 3 1 i $ V 3 R 3 I 3 ¼ 0 $ V 2 R 2 I 2 þ I 3 ¼ 0 : ð18þ $ V 1 R 1 I 1 þ I 2 ¼ 0 It can e shown that the numerical value of the voltage at each node is equivalent to the equilirium strain of that sulayer, e 1 $ V 1, e 2 $ V 2 and e 3 $ V 3. In the three-layer system, the diode-connected independent voltage sources are determined y the difference in the lattice mismatch of the two adjacent layers where: V S3 $ f 3 f 2 V S2 $ f 2 f 1 : ð19þ V S1 $ f 1 f 0 For the case in which all sulayers are coherently grown, the diodes all operate in the forward conduction mode, and therefore the voltage at each essential node is determined y accounting for the sum of all the independent voltage sources up to and including the layer in consideration: V 3 ¼ V S3 þ V S2 þ V S1 $ e 3 ¼ f 3 V 2 ¼ V S2 þ V S1 $ e 2 ¼ f 2 : ð20þ V 1 ¼ V S1 $ e 1 ¼ f 1 Upon the growth of strained material in which misfit dislocation networks are present at each mismatched interface, the diodes are all in the reverse locking mode (non-conductive), and the node voltages (layer strains) may e found y solution of the node voltage equations without inclusion of the independent voltage sources: V 3 ¼ R 3 I 3 $ e 3 V 2 ¼ R 2 ði 2 I 3 Þ $ e 2 : ð21þ V 1 ¼ R 1 ði 1 I 2 Þ $ e 1 In some cases, a coherent epitaxial layer may e grown on top of a metamorphic uffer. In such a case, the coherent interface, free from misfit dislocations, corresponds to a conducting diode in the electrical circuit. The presence of one or more interfaces free from misfit dislocations can e descried y the existence of a misfit dislocation free zone. The conduction of the diode results in the connection of an independent voltage source directly etween essential nodes, which in circuit theory can e considered to form a supernode. In other words, the presence of a MDFZ may e likened to the formation of a supernode in electrical circuit theory. A supernode in electrical circuit theory refers to the case where two essential nodes are separated y an independent voltage source. The existence of the supernode modifies the node voltage equations for the nodes involved, and therefore the resulting node voltages, which will e descried in more detail elow when considering the general treatment of an aritrary heterostructure. In the most general case, where we can consider an aritrary heterostructure consisting of multiple and/or compositionally-graded epitaxial layers as shown in Fig. 1c, we can extend the aove analysis y dividing the epitaxial layer into N disparate sulayers (Fig. 2d). The partial derivatives shown in Eq. 11 are given in terms of their analogous electrical circuit components as h i 0 ¼ 2YNhNeN f Nf N1 GNNð1mN cos 2 aþ hn jfnfn1j 2pð1mN Þsin a sin / ln N 0 "! # 1 fn fn1 Gnn 1 mn cos 2 a XN hj 2Ynhnen ln jfn fn1j2pð1 mnþsin a sin / j¼n n 0 ¼ "! # B fnþ1 fn Gnþ1nþ1 1 mnþ1 cos 2 a þ ln XN hj A jfnþ1 fnj 2pð1 mnþ1þsin a sin / j¼nþ1 nþ1 $ VN RN $ Vn IN ¼ 0; n ¼ N Rn In þ Inþ1 ¼ 0; 1 n<n : ð22þ In this extended analogy, the nth sulayer may e modeled y an electrical sucircuit in which: R n $ 1 ; 1 n N; ð23þ 2Y n h n I n $ fnfn1 and G n n 1m n cos 2 h jf nf n1 j2pð1m n Þsin a sin / ð "! # Þ PN h ln j ; 1 n N; n j¼n ð24þ V Sn $ f n f n1 ; 1 n N ; ð25þ The ideal diode in each sulayer is always facing the true positive terminal of the independent voltage source at that sulayer; when the diode conducts, the independent voltage source is dissipative. In the case in which each sulayer contains misfit dislocations, none of the diodes conduct, and the in-plane strain (node voltage) at the nth sulayer is determined y e n $ V n ¼ R n ði n I nþ1 Þ; 1 n<n : ð26þ R n I n ; n ¼ N The linear misfit dislocation density at each sulayer may then e determined y

8 180 Kujofsa and Ayers Tale I. Material properties for InAs, GaAs and the alloy In x Ga 12x As 25 where a is the lattice constant, C 11 and C 12 are the elastic stiffness constants and x is the indium mole fraction Parameter Material a (nm) C 11 (GPa) C 12 (GPa) InAs GaAs In x Ga 1x As x(0.0405) x(35.1) 53.7 x(8.4) Linear interpolation following Vegard s law was used to determine the material properties of the In x Ga 1x As alloy. ðf n e n Þðf n1 e n1 Þ q n ¼ f nf n1 jf n f n1 j n sin a sin / ; 1 n N : ð27þ From a farication point of view, the growth of mismatched and compositionally-graded epitaxial layers yields metamorphic heterostructures which may contain misfit dislocation free zones. In this situation, where the epitaxial structure as a whole is incoherent ut some of the sulayers are coherently grown with respect to the ones elow, the presence of a misfit dislocation free zone is equivalent to the formation of a supernode in the analogous electrical circuit model. Therefore, the node voltage (or the equivalent in-plane strain) in the ottom layer of the supernode is determined y accounting for the equivalent resistance of all the layers included in the supernode. If the supernode is ounded inclusively y sulayers r and x, then the equilirium strain (node voltage) in the ottom layer of the supernode is given y " P P # e r $ V r ¼ ði r I xþ1 Þ Px j V r i¼1 Si V i¼1 Si R j R SN ; j¼r ð28þ where the equivalent parallel resistance of the supernode (R SN ) is defined as the equivalent resistance for a series of resistors in parallel, R SN ¼ R r jjjjr x, and is given y! 1 R SN ¼ Xx 1 : ð29þ R j¼r j The in-plane strain (node voltage) at each sulayer of the supernode is then determined y adding the appropriate sum of independent voltage sources to the voltage at the ottom of the supernode. In other words, the node voltage (or the equivalent in-plane strain) of each sulayer of the supernode is determined from e i ¼ e r þ Xi j¼rþ1 f j f j1 $ V i ¼ V r þ Xi j¼rþ1 V Sj ; r <i x; ð30þ RESULTS AND DISCUSSION In applying the electrical circuit model to multilayered heterostructures and comparing its results to the numerical minimum energy calculations, we considered the following cases (Tale I). Uniform Composition Layers The simplest structure is one composed of three pseudomorphic layers in which the whole structure is coherently grown on the sustrate. Figure 3a shows a multilayered In x Ga 1x As/GaAs (001) heterostructure (Sample A) consisting of three uniform and pseudomorphic layers with indium compositions of x 1 ¼ 1%, x 2 ¼ 3% and x 3 ¼ 5%, respectively, and thicknesses of h 1 ¼ 10 nm, h 2 ¼ 5 nm and h 3 ¼ 10 nm, respectively. Figure 3 illustrates the equivalent electrical circuit model for the entire structure; the numerical values for the electrical components used here are summarized in Tale II. Given that the thickness of each layer is elow the critical thickness for misfit dislocation formation, the in-plane strain at each sulayer is equal to the coherency strain and therefore its lattice mismatch. In the equivalent electrical circuit model, the ideal diode at each node is conductive and therefore the respective current sources play no role in determining the voltage (i.e., in-plane strain) at each node. Furthermore, the voltage at each node is determined from the sum of all voltage sources up to and including the layer under consideration. The numerical value of the voltage in node one is equal to the lattice mismatch of sulayer one, the voltage at node two is equal to the sum of the voltage sources for sulayers one and two, and the voltage at node three is equal to the sum of all three voltage sources. Therefore, the numerical value of the voltage at each node is equal to the lattice mismatch of that layer as expressed y e n ¼ f n $ V n ¼ V Sn.As shown in Tale III, the results otained for Sample A using the circuit model agree with results from the numerical energy minimization approach. The growth of mismatched epilayers which are eyond the critical layer thickness requires the formation of a misfit dislocation network to relax

9 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures 181 Fig. 3. (a) Schematic representation of a coherently-grown multilayer heterostructure with three compositionally uniform epitaxial layers consisting of 10 nm In 0.05 Ga 0.95 As/5 nm In 0.03 Ga 0.97 As/10 nm In 0.01 Ga 0.99 As on a GaAs (001) sustrate. () The equivalent electrical circuit for the multilayer heterostructure consisting of the series connection (stacking) of the electrical circuit locks where the material properties of each sulayer are modeled using the equivalent electrical components (resistor, ideal diode, independent current and independent voltage sources). Tale II. The numerical values of the electrical components for the electrical circuits Sample Resistance (nx) Voltage source (mv) Current source (ka) A (Fig. 3) R 1 = V S1 = I 1 = R 2 = V S2 = I 1 = R 3 = V S3 = I 1 = B (Fig. 4) R 1 = V S1 = 7.1 I 1 = R 2 = V S2 = 13.9 I 1 = R 3 = V S3 = 10.4 I 1 = C (Fig. 5) R 1 = V S1 = 7.1 I 1 = R 2 = V S2 =0 I 1 =0 R 3 = V S3 =0 I 1 =0 D (Fig. 6) R 1 = V S1 = I 1 = R 2 = V S2 = I 2 = R 3 = V S3 = I 3 = R 4 = V S4 = I 4 = R 5 = V S5 = I 5 = R 6 = V S6 = I 6 = R 7 = V S7 = I 7 = R 8 = V S8 = I 8 = R 9 = V S9 = I 9 = R 10 = V S10 = I 10 = Tale III. The in-plane strains of various multilayered heterostructures and comparisons etween numerical minimum energy results and the electrical circuit model Sample Numerical minimum energy Electrical circuit model A (Fig. 3a) e 3 ¼0:3569% e 3 ¼0:3569% e 2 ¼0:2144% e 2 ¼0:2144% e 1 ¼0:0716% e 1 ¼0:0716% B (Fig. 4a) e 3 ¼þ0:1035% e 3 ¼þ0:1035% e 2 ¼0:3267% e 2 ¼0:3267% e 1 ¼0:0228% e 1 ¼0:0228% C (Fig. 5a) e 3 ¼0:0499% e 3 ¼0:0498% e 2 ¼0:0499% e 2 ¼0:0498% e 1 ¼0:0499% e 1 ¼0:0498%

10 182 Kujofsa and Ayers Fig. 4. (a) Schematic representation of an incoherently-grown multilayer heterostructure with three compositionally uniform epitaxial layers consisting of 150 nm In 0.15 Ga 0.85 As/100 nm In 0.3 Ga 0.7 As/75 nm In 0.1 Ga 0.9 As on a GaAs (001) sustrate. () The equivalent electrical circuit for the multilayer heterostructure consisting of the series connection (stacking) of the electrical circuit locks where the material properties of each sulayer are modeled using the equivalent electrical components (resistor, ideal diode, independent current and independent voltage sources). some portion of the lattice mismatch. In uniform layers, misfit dislocations are introduced at the mismatched interfaces and they can e modeled using the Dirac delta function. The heterostructure shown in Fig. 4a includes three incoherently-grown and mismatched sulayers with lattice mismatch f 1 ¼0:71%, f 2 ¼2:1% and f 3 ¼1:06% and thicknesses h 1 ¼ 75 nm, h 2 ¼ 100 nm and h 3 ¼ 150 nm. The analogous electrical circuit is shown in Fig. 4. The numerical minimum energy results shown in Tale III for Sample B suggest that all three sulayers are partly relaxed and all interfaces contain misfit dislocation networks. It is interesting to note that sulayers one and two exhiit compressive strain, similar to the lattice mismatch, whereas sulayer three exhiits tensile strain which is opposite to the lattice mismatch; the tensile strain present in the third sulayer suggests that misfit dislocations with Burgers vectors opposite in sense to those at the other two interfaces have een introduced. Standard electrical circuit simulator (SECS) modeling results of the electrical circuit shown in Fig. 4 are in excellent agreement with minimum energy calculations. Furthermore, diodes at each node are non-conductive and therefore the node voltage is determined y the sum of the currents flowing through that node and the resistor attached to it. Thus, the voltage (in-plane strain) at each node is given y e 3 ¼þ0:1035% $ V 3 ¼ R 3 I 3 ¼þ1:035mV e 2 ¼0:3267% $ V 2 ¼ R 2 ði 2 I 3 Þ¼3:267mV : e 1 ¼0:0228% $ V 1 ¼ R 1 ði 1 I 2 Þ¼0:228mV ð31þ electrical circuit model, we divided the epitaxial layer into three sulayers with varying thicknesses (h 1 ¼ 50 nm, h 2 ¼ 200 nm and h 3 ¼ 100 nm), ut each sulayer contained the same composition (and therefore the same lattice mismatch with respect to the sustrate). The in-plane strain of the heterostructure considered in Fig. 5a could e easily calculated using the well-known Matthews and Blakeslee s model (Eq. 4) y considering the total epitaxial thickness. The equilirium in-plane strain determined for Sample C (Tale III) from all three models are in excellent agreement. In addition, SECS modeling of the electrical circuit shown in Fig. 5c indicates that the ideal diode at sulayer one is non-conductive, which suggests the presence of a misfit dislocation network at the sustrate interface. However, at the second and third nodes, the ideal diodes are conductive; diode conduction at nodes two and three indicates the presence of a misfit dislocation free zone in the epitaxial sulayers two and three and implies that a three-fold supernode will e formed. As a consequence of the supernode formation, the current source at node one (I 1 ) flows through resistors R 1, R 2 and R 3 arranged in parallel (Fig. 5c); the voltage at each node is determined from the product of the current source I 1 and the equivalent parallel resistance R 1 jjr 2 jjr 3. Thus, the in-plane strain of each sulayer is given as: e 1 ¼ e 2 ¼ e 3 ¼0:0498% $ V 1 ¼ V 2 ¼ V 3 ¼ 1 1 I 1 ¼0:498 mv: R 1 R 2 R 3 ð32þ For the third heterostructure, we considered the important case of a single uniform and incoherentlygrown epitaxial layer as shown in Fig. 5a with a total thickness of h ¼ 350 nm and a lattice mismatch of f ¼0:71%. For the purpose of illustration, and also to provide a stringent test of the Step-Graded Epitaxial Layers The electrical circuit model could e extended to any aritrary multilayered heterostructure employing compositional-grading. Figure 6a illustrates a step-graded uffer comprising ten uniform layers

11 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures 183 Fig. 5. (a) Schematic representation of an incoherently-grown multilayer heterostructure with three sulayers where the top two are coherentlygrown. Physically, the heterostructure can e considered as a single compositionally uniform 350-nm-thick In 0.1 Ga 0.9 As layer deposited on a GaAs (001) sustrate; however, for the purpose of illustration, it is roken down to 100-nm In 0.1 Ga 0.9 As/200-nm In 0.1 Ga 0.9 As/50-nm In 0.1 Ga 0.9 As. () The equivalent electrical circuit for the multilayer heterostructure consisting of the series connection (stacking) of three electrical circuit locks where the material properties of each sulayer are modeled using the equivalent electrical components (resistor, ideal diode, independent current and independent voltage sources). each with 10 nm thickness, and in which there are equal compositional changes from one layer to the next (linear step-grading). For the structure studied here, the indium composition is varied from 1% at the sulayer closest to the sustrate interface to a final surface indium composition of 10% corresponding to a lattice mismatch of 0.71%; the lattice mismatch profile as a function of the distance from the sustrate interface z is shown in Fig. 6c (dashed line). The corresponding electrical circuit is illustrated in Fig. 6. For the case of a step-graded layer consisting of N sulayers, the maximum numer of misfit dislocation peaks is equal to the numer of mismatched interfaces (10 in the current case). Kujofsa and Ayers 8,9 have shown that, in general, continuously-graded layers contain misfit dislocation free zones adjacent to the sustrate interface and surface, provided that there is zero interfacial mismatch and/or the grading coefficient is sufficiently small. However, in step-graded layers, the thicknesses of the interfacial and surface misfit dislocation free zones are constrained to e related to multiples of the step-layer thickness, which in this case is one tenth of the total uffer layer thickness (h T ¼ 100 nm). 10 In In x Ga 1x As/GaAs (001) step-graded layers, the presence of an interfacial MDFZ is evident only when a very small grading coefficient is used (approximately 2% indium per micron). In addition, for a low ending composition, and therefore ending lattice mismatch, the surface misfit dislocation free zone may extend eyond the top step layer. Figure 6c depicts the equilirium in-plane strain distriution determined y numerical minimum energy calculations (solid lines) and the electrical circuit (circle symols) as a function of the distance from the interface. The in-plane strain profile comprises a series of step functions with discontinuities at the mismatched interfaces. The first five interfaces contain misfit dislocations which relax most of the mismatch strain and result in small values of the residual strain. The asence of misfit dislocation networks at the top five interfaces results in high uilt-in strains. These results from the circuit analysis are in excellent agreement with numerical energy minimization calculations, as shown in Fig. 6c. Linearly-Graded Epitaxial Layers The electrical circuit model could e applied to any heterostructure with an aritrary numer of sulayers. Figure 7 illustrates the lattice mismatch profile and equilirium strain for a linearly-graded layer of In x Ga 1-x As grown epitaxially on a GaAs (001) sustrate. Here, the dashed line shows the profile of lattice mismatch, which varies linearly from zero (corresponding to zero indium mole fraction) to f h ¼0:71% at the surface (corresponding to 10% indium mole fraction). Figure 7 also shows the equilirium strain profile for the linearly-

12 184 Kujofsa and Ayers Fig. 6. (a) Schematic representation of 100-nm-thick step-graded In 0.1 Ga 0.9 As epitaxial layer with 10 sulayers where the indium composition is varied in steps of 1% starting from 1% at the sulayer nearest the interface and ending at 10% in the sulayer at the surface. () The equivalent electrical circuit for the step-graded heterostructure consisting of the series connection (stacking) of 10 electrical circuit locks where the material properties of each sulayer are modeled using the equivalent electrical components (resistor, ideal diode, independent current and independent voltage sources). (c) Lattice mismatch (dashed) and in-plane strain (solid, circle) as a function of the distance from the interface for the stepgraded epilayer. The in-plane strain is determined using the numerical minimum energy (solid line) and electrical circuit (open circles) models. graded layer, as determined y numerical minimum energy calculations (solid line) and the electric circuit model (open circles). These two results are in excellent agreement, and oth show the existence of an interfacial misfit dislocation free zone as well as a surface misfit dislocation free zone, as expected on the asis of previous studies. 7,9 11 The misfit dislocation free zones exhiit residual strain profiles with the same slope as the lattice mismatch, and exhiit widths of 5 nm (at the interface) and 46 nm (at the surface). Between these, there is a dislocated region of thickness 49 nm in which the strain is nearly constant. In our work, the fundamental equations for the strain energy and dislocation line energy are essentially the same as those used y Matthews, 15 and later y Tersoff 13,14 Although the equilirium configuration in the linearly-graded epitaxial layer has een explored in great detail y Tersoff, 13,14 Fitzgerald et al. 26 Dunstan 27 and Romanato et al., 28 these models assume that graded material can relax completely in the presence of misfit dislocations. This is a simplifying assumption which does not strictly hold in either equilirium or kineticallylimited relaxation. More specifically, there are two key assumptions emedded in these models: first,

13 Electric Circuit Model Analogy for Equilirium Lattice Relaxation in Semiconductor Heterostructures 185 Fig. 7. Lattice mismatch (dashed) and in-plane strain (solid, circles) as a function of the distance from the interface for a 100-nm-thick linearly-graded In x Ga 1x As epitaxial layer where the indium composition is varied linearly from 0 at the sustrate interface to 10% at the surface. The in-plane strain is determined using the numerical minimum energy (solid line) and electrical circuit (open circles) models. there is zero strain in the dislocated region; and second, they neglect the thickness dependence of the line energies for dislocations. As a consequence of these simplifying assumptions, the interfacial misfit dislocation-free zone is not seen and there is zero strain in the dislocated region. Therefore, the in-plane strain characteristic is descried y these models as, eðzþ ¼ 0 z z c; ð33þ C f ðz z c Þ z c <z h: where z c is the edge of the dislocated region near the surface and C f is the grading coefficient. The numerical and the equivalent electrical circuit models do not make such simplifying assumptions, and therefore the residual strain characteristics is slightly different from the previously developed models. If the edges of the interfacial and surface MDFZs are located at distances of z 1 and z 2 from the sustrate interface, as illustrated in Fig. 7, and therefore the misfit dislocation density is concentrated in the middle region (z 1 z z 2 ), then the residual strain can e analytically modeled as follows: in the interfacial MDFZ, the asence of misfit dislocations indicates that the residual strain is equal to the lattice mismatch profile and therefore: eðzþ ¼C f z; z z 1 : ð34þ In the dislocated region (z 1 z z 2 ), the residual strain is modeled y the electrical circuit model as e n $ V n ¼ R n ði n I nþ1 Þ: ð35þ An important application of the electric circuit model analogy is to the case of a continuously-graded layer. This can e considered y approximating the continuously-graded material y a stack of uniform composition sulayers and then taking the limit as the thickness of the individual sulayers approaches zero. This development is similar to the transition from electric circuit theory using lumped circuit elements to electromagnetic theory using distriuted electrical quantities. This enales the development of analytical expressions for the strain, misfit dislocation density, critical layer thickness and widths of misfit dislocation free zones in a continuously-graded layer having any aritrary compositional profile. Previously, only the linear 22 and exponential 23 grading cases had een considered theoretically, ut the circuit model analogy allows the analysis of any continuously-graded layer, including those with power law, S-graded, or some type of non-linear compositional profiles. This development will e descried elow. Using the electric circuit analogy and assuming that the physical constants, m, G and Y are slowly varying functions of n, the in-plane strain in the n th sulayer is given y e n 8 f n f n1 ¼ 1 >< jf n f n1 j 2Y n h n f nþ1 f n >: jf nþ1 f n j G 1 m cos 2 a 2pð1 mþsin a sin / G 1 m cos 2 a 2pð1 mþsin a sin / "! # XN h j ln j¼n "! # XN h j ln j¼nþ1 9 >= : ð36þ As a specific example for a monotonic continuouslygraded layer f nf n1 j j ¼ f nþ1f n j j. Thus f n f n1 f nþ1 f n e n ¼ 1 f nþ1 f n G 1 mcos 2 a 2Yh n jf nþ1 f n j2pð1 mþsinasin/ ln Equation 37 can e simplified to e n ¼ 1 f nþ1 f n 2Y jf nþ1 f n j G 1 m cos 2 a 2pð1 mþsin a sin / P N h j j¼n P N h j j¼nþ1 >;! : ð37þ ln h nþhz h n ; h n ð38þ where h is the epitaxial layer thickness and z is the distance from the sustrate interface to the top of layer n. For simplicity let A ¼ 1 f nþ1 f n G 1 m cos 2 a 2Y jf nþ1 f n j2pð1 mþsin a sin / : ð39þ The limiting case of a continuously-graded layer, in which h n! 0, may e understood using L Hopital s rule, giving 8 lim f h n! 0 e ng ¼ lim < h n! 0 : ¼ lim A h n! 0 h n þ h n ¼ h ln h nþhz n ½h n A h z ¼ eðzþ: Š i9 = ; ð40þ

14 186 Kujofsa and Ayers 8 < C f z z z 1 ; eðzþ¼ A z hhz ð Þ þ C f z 1 z 1 <z z 2 ; and : C f ðz z 2 ÞþA z 2 hhz ð 2 Þ þ C f z 1 z 2 <z h: ð43þ The equation for the surface MDFZ oundary z 2 is given y Fig. 8. Lattice mismatch (dashed) and in-plane strain (solid, circles) as a function of the distance from the interface for a 100-nm-thick S- graded epilayer In x Ga 1x As epitaxial layer where the indium composition is varied linearly from 0 at the sustrate interface to 10% at the surface. The mean parameter was fixed at half of the epitaxial layer thickness l ¼ 50 nm, and the standard deviation parameter was r ¼ 10 nm. The in-plane strain is determined using the numerical minimum energy (solid line) and electrical circuit (open circles) models. Therefore, in the dislocated region of any continuously-graded layer, the in-plane strain will have this dependence on distance from the interface, apart from small compositional variations in, m, G and Y. This contrasts with simple models previously developed in which it was assumed that the dislocated region of a graded layer would e unstrained. The characteristic of Eq. 40 descries the residual strain in structures where the misfit dislocation region extends all the way to the sustrate interface, however, in linearlygraded epitaxial layers, the presence of the interfacial MDFZ leads to the adjustment of the strain profile. In addition, the strain characteristics descried here accounts for the variation of the residual strain in the dislocated region. Thus, the in-plane strain in the dislocated region is modeled y eðzþ ¼ A hz A h þ C f z 1 ; z 1 <z z 2 : ð41þ The second and third terms in the equation aove represent adjustments to account for the strain at the top of the interfacial MDFZ. By a similar analysis, in the surface MDFZ, the asence of misfit dislocations implies that the residual strain is proportional to the lattice mismatch and therefore the strain in this region is given y A eðzþ ¼C f ðz z 2 Þþhz 2 A h þ C f z 1 ; z 2 <z h : ð42þ Therefore, according to this model, the equilirium strain profile in the linearly-graded layer is given y Z h Z h eðzþ dz ¼ E d z Y ¼ C f ðz z 2 Þþeðz 2 Þ dz z 2 ¼ A ln h z ð44þ 2 : Solving the expression aove and recognizing that the width of the surface MDFZ is W MDFZ ¼ h z 2, yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A W MDFZ ¼ ln W MDFZ þ 2 Az 2 C f h C f z 1 : ð45þ Rearrangement of the equation aove results in the surface in-plane strain characteristic to e accurately modeled y e S ¼ A ln W MDFZ þ A W MDFZ h C f z 1 : ð46þ The sum of the second and third terms yields a small contriution to the equation, since the oundary for the interfacial misfit dislocation free zone z 1 is very small in these structures, however, its value could e found y a similar approach where Z z1 0 eðzþ dz ¼ E d 2 0 Y ¼ Z z1 0 C f z dz ¼ A ln h z 1 : ð47þ Solving, the expression aove results in transcendental expression similar to the Matthews and Blakeslee critical layer thickness equation as is applicale to linearly-graded layers, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A z 1 ¼ ln h z 1 : ð48þ C f Therefore, Eq. 46 is modified accordingly to e S ¼ A sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln W MDFZ þ Ah W MDFZ 2AC f ln h z 1 : ð49þ If we make the exact simplifying assumptions as the previous models, the expressions given in Eqs. 45 and 46 reduce to the ones provided y Tersoff.