TEM measurement of strain in coherent quantum heterostructures
Introduction
Strain plays an important role in the microstructural evolution of many material systems, yet strain measurements are typically limited to average values over large areas. Measuring strain on a microscopic scale has remained difficult. In the 1960s, Ashby and Brown demonstrated that transmission electron microscopy (TEM) is a powerful tool for measuring strain in simple systems [1], [2]. Using two-beam diffraction contrast images of a strained copper matrix with coherent cobalt inclusions, comparisons to image contrast predicted by the numerically integrated two-beam Howie–Whelan equations led to good measurements of the strain. But Ashby and Brown's case benefited from an unusual advantage: the strain field they studied has an analytical solution [1], [2]. A coherent inclusion acts like an electrostatic point charge, with lattice plane bending analogous to the electric field. Today, we want to study different systems, such as quantum dots, where analytical strain models do not exist. For these systems contemporary TEM studies generally rely on finite element (FE) strain simulation results incorporated into a Howie–Whelan image simulation. Because FE strain simulations are highly constrained – one FE simulation must be produced for each image simulation – recent TEM experiments have involved only qualitative comparisons between experimental data and one or two image simulations using FE strain results [3], [4]. Recently, we reported a measurement of strain created by a coherent Ge island on an Si (0 0 1) substrate. The measurement was performed without FE modeling [5]. In this paper we describe in detail the TEM method used to measure strain in that system, a method which is applicable to any microscopic system for which no simple analytical strain model exists.
Section snippets
Germanium on silicon: a model system
Interest in Ge–Si heteroepitaxy arises from the system's potential technological importance and due to basic questions regarding the system’s morphological evolution. The observation that Ge forms uniformly sized, non-dislocated, well-ordered arrays of islands when deposited onto Si (0 0 1) [6], [7], [8] has led to the hope that this simple system will find applications as zero-dimensional quantum electronic devices and in permeable base transistors. Several aspects of this system's morphological
The abrupt displacement approximation
No simple analytical model describes the strain field due to isolated Ge islands on Si (0 0 1). Analytical models exist for other, similar systems, such as the “wavy” growth surface [26], [27], buried coherent inclusions [1], [2], [28], surface steps [29], or phase boundaries [30], and the models have been used in conjunction with TEM to measure strain in those systems, but none of these suffice for measuring strain due to isolated Ge islands.
Consequently, in order to simulate image contrast, we
Example measurement
We provide an example measurement using a full simulation to demonstrate the technique. Dark-field TEM image contrast for a 10 nm diameter strained hemispherical inclusion at the surface of a 700 nm thick specimen was simulated. The displacement field increases according to inside the object, then decreases proportionally to 1/r2, where r is a radial coordinate in a spherical coordinate system. This model describes the strain due to a coherent inclusion in an infinite solid [25] and is
Experimental details
We repeat the above procedure using an experimental image. Experiments were performed using Si (0 0 1)-oriented samples on which 11 monolayers (ML) of Ge were deposited by chemical vapor deposition at Hewlett Packard Laboratories, Palo Alto, California. After baking the Si wafer at 1150°C in an H2 ambient and depositing an Si buffer layer at 1080°C, Ge was deposited at 600°C and a pressure of 10 Torr using GeH4 in an H2 carrier gas. The amount of Ge deposited was equivalent to 11 ML (1 ML=6.27×1014
Conclusion
This approach to using TEM to measure strain in individual nano-scale features offers a number of advantages that will be useful for further studies of this type of system. We point out that this method relies on plan-view imaging which minimizes surface relaxation effects, image interpretation is straightforward, and the method can be used to measure the strain in individual islands, not an average strain over a macroscopic area. Future studies, then, may involve in situ studies in which
Acknowledgements
The authors would like to acknowledge the valuable insight and honest critiques provided by Ray Twesten. We also thank Dr. Ted Kamins at Hewlett Packard Laboratories for providing samples, and Prof. Y. Huang and P. Zhang of the University of Illinois Department of Mechanical and Industrial Engineering for providing finite element strain simulations. This work was supported by the National Science Foundation under Award No. DMR-9705440. This work was carried out in the Center for Microanalysis
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Current address: Materials Science Division, Argonne National Laboratories, Argonne, IL 60439, USA.