Single dislocation density

Germanium single crystals intended for electronic apphcations are usuaHy specified according to conductivity type, dopant, resistivity, orientation, and maximum dislocation density. They may be specified to be lineage-free unless the specified resistivity is below about 0.05 H-cm. Minority carrier lifetime and majority carrier mobHity are occasionaHy specified. 

Calculations of this type are carried out for fee, bcc, rock salt, and hep crystal structures and applied to precursor decay in single-crystal copper, tungsten, NaCl, and LiF [17]. The calculations show that the initial mobile dislocation densities necessary to obtain the measured rapid precursor decay in all cases are two or three orders of magnitude greater than initially present in the crystals. Herrmann et al. [18] show how dislocation multiplication combined with nonlinear elastic response can give some explanation for this effect. 

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

Kumar and Clifton [31] have shock loaded <100)-oriented LiF single crystals of high purity. The peak longitudinal stress is approximately 0.3 GPa. Estimates of dislocation velocity are in agreement with those of Flinn et al. [30] when extrapolated to the appropriate shear stress. From measurement of precursor decay, inferred dislocation densities are found to be two to three times larger than the dislocation densities in the recovered samples. 

To answer questions regarding dislocation multiplication in Mg-doped LiF single crystals, Vorthman and Duvall [19] describe soft-recovery experiments on <100)-oriented crystals shock loaded above the critical shear stress necessary for rapid precursor decay. Postshock analysis of the samples indicate that the dislocation density in recovered samples is not significantly greater than the preshock value. The predicted dislocation density (using precursor-decay analysis) is not observed. It is found, however, that the critical shear stress, above which the precursor amplitude decays rapidly, corresponds to the shear stress required to disturb grown-in dislocations which make up subgrain boundaries. 

N. Miyazaki, S. Okuyama. Development of finite element computer program for dislocation density analysis of bulk semiconductor single crystals during Czochralski growth. J Cryst Growth 183 S, 1998. 

Figure 5.16 Resolved shear stress as a function of dislocation density for copper. Data are for polycrystalline copper O single-crystal copper with one slip system operative 0 single-crystal copper with two slip systems operative and A single-crystal copper with six slip systems operative. From K. M. Rails, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress is a function of the dislocation density, Pd -... 

GaN crystals of both low and high electric conductivity can be grown under high pressure of nitrogen. Routinely grown GaN single crystals in a form of hexagonal platelets with surface area 60 - 100 cm2 and dislocation densities lower than 105 cm 2 can be used as substrates for homoepitaxy. 

More recently Mackwell, Kohlstedt, and Paterson (1985) studied the deformation of single crystals of San Carlos (Arizona) olivine deformed under hydrous conditions at 1,300 C, 300 MPa confining pressure, and 10 s strain-rate and found they were a factor of 1.5-2 weaker than those deformed in an anhydrous environment. TEM observations showed that specimens deformed under dry conditions, in an orientation such that the slip systems (001)[100] and (100)[001] would be activated, were characterized by a microstructure of generally curved dislocations and dislocation loops, but no organization into walls. The dislocation density was 10 -10 cm compared with an initial value of < 10 cm . Most of the dislocations and the loops lie approximately in the (010) plane because they are in contrast for g = 004, they probably have b = [001] dislocations with b = [010] and [100] would be out-of-contrast for this reflection. However, the slip system (010) [001] is not expected to be active. It is not clear, therefore, if these dislocations are actually involved in the deformation. The general geometry of the dislocation microstructure is not inconsistent with some climb mobility in fact, on the basis of the observations of Phakey et al. (1972), climb is certainly expected at 1,300°C. 

The dislocation method of stress analysis is also useful for determining craze stress fields in anisotropic (e.g., oriented) polymers . All one needs here is the stress field of a single dislocation in a single crystal with the same symmetry as the oriented polymer (the text by Hirth and Lothe provides a number of simple cases plus copious references to more complete treatments in the literature) the craze stress field can be generated by superposition of the stress fields of an array of these dislocations of density a(x). Dislocations may also be used to represent the self-stress fields of curvilinear crazes (produced by craze growth in a non-homogeneous stress field for example). Such a method has been developed by Mills... 

The existence of a supersaturation or overvoltage threshold is a characteristic feature of nucleation-induced processes as, e.g., electrochemical phase formation. Based on this phenomenon, several experimental techniques for electrociystallization studies have been developed (cf. Section 4.2). Before going into further details, however, let us discuss some technical skills that can lead to the preparation of well-developed low dislocation density single crystal faces. 

A major breakthrough in the understanding of basic concepts of crystal growth has been achieved in model systems with single crystal substrate faces, either free of screw dislocations (quasi-perfect) or with a low screw dislocation density. 

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