Griffith Criterion for Nanoscale Stress Singularity in Brittle Silicon Takashi Sumigawa,*,§ Takahiro Shimada,*,§ Shuuhei Tanaka, Hiroki Unno, Naoki Ozaki, Shinsaku Ashida, and Takayuki Kitamura Department of Mechanical Engineering and Science, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan S Supporting Information *
ABSTRACT: Brittle materials such as silicon fail via the crack nucleation and propagation, which is characterized by the singular stress field formed near the crack tip according to Griffith or fracture mechanics theory. The applicability of these continuum-based theories is, however, uncertain and questionable in a nanoscale system due to its extremely small singular stress field of only a few nanometers. Here, we directly characterize the mechanical behavior of a nanocrack via the development of in situ nanomechanical testing using a transmission electron microscope and demonstrate that Griffith or fracture mechanics theory can be applied to even 4 nm stress singularity despite their continuum-based concept. We show that the fracture toughness in silicon nanocomponents is 0.95 ± 0.07 MPa√m and is independent of the dimension of materials and therefore inherent. Quantum mechanics/atomistic modeling explains and provides insight into these experimental results. This work therefore provides a fundamental understanding of fracture criterion and fracture properties in brittle nanomaterials. KEYWORDS: fracture mechanics, Griffith criterion, nanocracking, fracture toughness, silicon
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which the crack becomes unstable and starts to propagate using the intensity of a singular stress field formed near the crack tip, where the stress diverges with a singularity of 1/√r at the tip. Therefore, not a point stress, but the singular field of stress determines the fracture initiation. In nanoscale materials, however, the singular stress field is shrunk and confined to several nanometers, in which only a small number of atoms exists compared to macroscale materials. Such a situation contradicts with a continuum assumption basing fracture mechanics theory, i.e., a huge enough number of atoms exists at the near-tip region to regard the region as a continuum medium. This inconsistency prevents the fundamental understanding of fracture in nanoscale materials, and it is therefore of central interest and importance to reveal whether fracture mechanics or Griffith theory is still applicable at the nanoscale. In an extremely small dimension, however, direct observation of nanocracking and measurement of its fracture toughness have been a major challenge due to the unstable and rapid nature of a crack.28 For toughness measurement, one needs to control a pre-introduced sharp crack (i.e., ideal Griffith crack) well-defined to a nanometer scale, and numerous challenges for precise control of brittle cracking have been done for its path and velocity with a designed specimen/loading23−27 or for its
ilicon-based small-scale devices have become ubiquitous in many modern technological applications owing to an intriguing rich variety of exquisite functionalities including electrical,1 piezoelectric,2 thermoelectric,3,4 electromechanical,5 and -optical6,7 properties of silicon at nanometer scales. Despite such widespread applications, reliability issues related to uncertain mechanical failure of silicon nanocomponents still remain a major factor preventing further development and miniaturization of the nanodevices. Quantitative characterization of failure of silicon at the nanoscale is therefore of central importance, and even promising for stretchable electronics8 and strained-silicon technologies.5,7 Numerous efforts have been made to mechanically test silicon micro- and nanocomponents and measure their fracture strength,9−14 which is however not inherent but exhibits remarkable size/geometry dependence.9−17 Because of the brittle nature of silicon, the actual fracture strength is usually determined by its fracture toughness,18,19 which is a material resistance to fracture through propagation of a crack, and the toughness is therefore the most essential to understand failure of silicon. Nevertheless, knowledge of mechanical behavior of a crack in small-scale silicon and its fracture is limited to the micrometer scales,20−27 but is seriously underdeveloped for nanoscales. This is mainly due to a scientifically important and practically critical issue arising at the nanoscale: Can fracture mechanics or Griffith theory still be applicable at the nanoscale? Fracture mechanics was proposed on the basis of continuum mechanics and describes the critical mechanical condition at © 2017 American Chemical Society
Received: April 11, 2017 Accepted: May 26, 2017 Published: May 26, 2017 6271
DOI: 10.1021/acsnano.7b02493 ACS Nano 2017, 11, 6271−6276
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Figure 1. Precrack introduction to Si nanospecimens. (a) SEM image of specimen fabricated by the FIB processing. (b) TEM images and diffraction patterns before and after annealing. (c) TEM images for during precrack introduction for Specimen 1. The notch is gradually opened by inserting the tip, and the nanoscale precrack with a length of 966 nm is nucleated from the notch root (see also Supporting Movie 1). (d) Same as c, but for a smaller Specimen 2. The precrack length is controlled to 326 nm.
length by indentation tests;29,30 however, they are still limited to a micrometer scale. Fabrication using chemical etching or focused ion beam (FIB) is an alternative, however, introduces a notch31−33 instead of a sharp crack (ideal Griffith crack), and the latter in addition causes an amorphous layer to crystalline samples, inhibiting intrinsic fracture toughness of silicon single crystals. In situ observation is also essential during testing to detect any plastic phenomena at the crack tip including dislocation emission, which extrinsically affects fracture toughness. Intrinsic fracture toughness at the nanoscale is, therefore, not directly accessible with current techniques, which underscores the critical need for development of a new testing method. Here, we realized a natural sharp nanocrack (ideal Griffith crack) and directly observed mechanical behavior of the nanocrack with a controlled length and experimentally measured its fracture toughness via the development of in situ nanomechanical testing for a designed nanospecimen using a transmission electron microscope (TEM). We measured the fracture toughness to be 0.95 ± 0.07 MPa√m. The measured value is consistent with those in different lengths, indicating that the fracture toughness is inherent in silicon, contrary to the strength, which exhibits significant size dependence. We simultaneously showed that the classical Griffith and continuum-based fracture mechanics theories are still valid to the brittle fracture governed by only a 4 nm region of singular stress at the crack tip. Our quantum mechanics/atomistic modeling provides validation and explanation for experimental results. This work provides fundamental understanding of fracture criterion and fracture properties in brittle nanomaterials.
carved out from a single-crystal Si (100) plate so that the front side surface coincides with the (100) plane by means of FIB processing. The cleavage plane (011) is employed upright in the perpendicular direction, and the thickness direction is [100]. The central region of the specimen is thinned down to
