Electric Circuit Model Analogy for Equilibrium Lattice Relaxation in Semiconductor Heterostructures

Journal of Electronic Materials, Sep 2017

The design and analysis of semiconductor strained-layer device structures require an understanding of the equilibrium profiles of strain and dislocations associated with mismatched epitaxy. Although it has been shown that the equilibrium configuration for a general semiconductor strained-layer structure may be found numerically by energy minimization using an appropriate partitioning of the structure into sublayers, such an approach is computationally intense and non-intuitive. We have therefore developed a simple electric circuit model approach for the equilibrium analysis of these structures. In it, each sublayer of an epitaxial stack may be represented by an analogous circuit configuration involving an independent current source, a resistor, an independent voltage source, and an ideal diode. A multilayered structure may be built up by the connection of the appropriate number of these building blocks, and the node voltages in the analogous electric circuit correspond to the equilibrium strains in the original epitaxial structure. This enables analysis using widely accessible circuit simulators, and an intuitive understanding of electric circuits can easily be extended to the relaxation of strained-layer structures. Furthermore, the electrical circuit model may be extended to continuously-graded epitaxial layers by considering the limit as the individual sublayer thicknesses are diminished to zero. In this paper, we describe the mathematical foundation of the electrical circuit model, demonstrate its application to several representative structures involving In x Ga1−x As strained layers on GaAs (001) substrates, 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 arbitrary compositional profile. It is similar to the transition from circuit theory, using lumped circuit elements, to electromagnetics, using distributed electrical quantities. We show this development using first principles, but, in a more general sense, Maxwell’s equations of electromagnetics could be applied.

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Electric Circuit Model Analogy for Equilibrium Lattice Relaxation in Semiconductor Heterostructures

Journal of ELECTRONIC MATERIALS Electric Circuit Model Analogy for Equilibrium Lattice Relaxation in Semiconductor Heterostructures JOHN E. AYERS 0 0 1.-Electrical and Computer Engineering Department, University of Connecticut , 371 Fairfield Way, Unit 4157, Storrs, CT 06269-4157, USA. 2.- , USA The design and analysis of semiconductor strained-layer device structures require an understanding of the equilibrium profiles of strain and dislocations associated with mismatched epitaxy. Although it has been shown that the equilibrium configuration for a general semiconductor strained-layer structure may be found numerically by energy minimization using an appropriate partitioning of the structure into sublayers, such an approach is computationally intense and non-intuitive. We have therefore developed a simple electric circuit model approach for the equilibrium analysis of these structures. In it, each sublayer of an epitaxial stack may be represented by an analogous circuit configuration involving an independent current source, a resistor, an independent voltage source, and an ideal diode. A multilayered structure may be built up by the connection of the appropriate number of these building blocks, and the node voltages in the analogous electric circuit correspond to the equilibrium strains in the original epitaxial structure. This enables analysis using widely accessible circuit simulators, and an intuitive understanding of electric circuits can easily be extended to the relaxation of strained-layer structures. Furthermore, the electrical circuit model may be extended to continuously-graded epitaxial layers by considering the limit as the individual sublayer thicknesses are diminished to zero. In this paper, we describe the mathematical foundation of the electrical circuit model, demonstrate its application to several representative structures involving InxGa1 xAs strained layers on GaAs (001) substrates, 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 arbitrary compositional profile. It is similar to the transition from circuit theory, using lumped circuit elements, to electromagnetics, using distributed electrical quantities. We show this development using first principles, but, in a more general sense, Maxwell's equations of electromagnetics could be applied. Electrical circuit model; relaxation; equilibrium in-plane strain; multilayers; compositionally-graded; InGaAs; GaAs TEDI KUJOFSA INTRODUCTION The understanding of the equilibrium lattice relaxation has important implications in the determination of the stability criteria for electronic and optical devices.1–6 Furthermore, the equilibrium 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 been developed for the determination of the equilibrium configuration7–15 and, although it has been shown that the equilibrium configuration for a general semiconductor strained-layer structure may be determined numerically by energy minimization using an appropriate partitioning of the structure into sublayers,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 enable the development of analytical solutions for compositionally-graded heterostructures, we propose the use of an electrical circuit model analogy. Several mechanical–electrical analogs have been developed and used, particularly for load beam analysis. The most common of these are the so-called ‘‘force-current’’ and ‘‘force-voltage’’ analogs,16–20 but others have also been developed. It is possible to use any of these to provide a physically correct description of behavior in a mechanical system; however, some are better suited to certain applications. For example, our work relates to the static behavior of a semiconductor heterostructure in equilibrium, and there is no need to include electrical components such as capacitors and inductors, which may be included for transient (time-dependent) modeling. Among the previously published work on mechanical–electrical analogies, a report of particular interest is that by Olsson and Bath,20 which describes two particular choices of analogies for application to problems of geophysics. Their second transcriptive system considers electrical current to be analogous to mechanical stress, electrical voltage to be analogous to mechanical strain, and electrical resistance to be analogous to the reciprocal of an elastic modulus. Olsson and Bath point out that an advantage of this transcriptive system is that it facilita (...truncated)


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Tedi Kujofsa, John E. Ayers. Electric Circuit Model Analogy for Equilibrium Lattice Relaxation in Semiconductor Heterostructures, Journal of Electronic Materials, 2017, pp. 173-187, Volume 47, Issue 1, DOI: 10.1007/s11664-017-5750-z