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Phys. Status Solidi RRL 6, No. 3, 99–101 (2012) / DOI 10.1002/pssr.201105541 pss Strength of metals at the Fermi length scale www.pss-rapid.com Jason N. Armstrong, Susan Z. Hua, and Harsh Deep Chopra * Laboratory for Quantum Devices, Materials Program, Mechanical and Aerospace Engineering Department, The State University of New York at Buffalo, Buffalo, NY 14260, USA Received 21 November 2011, revised 8 December 2011, accepted 12 December 2011 Published online 14 December 2011 Keywords yield strength, Fermi length scale, Sharvin length scale, surface energy, atomic force microscope * Corresponding author: e-mail hchopra@buffalo.edu, Phone: +1 716 645 1415, Fax: +1 716 645 2883 Using silver and gold, we have measured the size-dependence of the yield strength of atomic-sized samples as small as a single-atom bridge, with pico-level resolution in the applied force and displacement. The strength approaches theoretical values as the diameter of the sample becomes comparable to the Fermi wavelength of electrons (~0.5 nm); in the limit of a single-atom bridge, the strength is over four orders of magnitude higher than in bulk single crystals. Results provide direct evidence for Pauling’s prediction of bond stiffening with reduced atomic coordination. Beginning with a single-atom bridge, strength evolves in a staircase manner in Ag, instead of the intuitively assumed continuous approach to a saturating bulk value. Yield Strength (GPa) 150 100 50 t 0 0 1 2 3 Cross-section area (nm 2 ) Measured strength approaching theoretical (ideal) values at the Fermi length scale, corresponding to a sample made of a single-atom Au bridge. t 2a t t Strain e ≅a/2a;Idealstrength t= e*E a © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The use of scanning probes to study atomic-sized samples has led to basic insight into mechanical forces, quantum properties, and their interplay [1–8]. For example, isomorphous Ag and Au have nearly identical bond length and lattice constant. Being monovalent, conductance across a single-atom Au or Ag bridge becomes indistinguishable, equal to one quantum of conductance, G 0 = 2e 2 /h. Recent studies reveal new ‘markers’ for their chemical identity at atomic level based on differences in the transition from tunneling to contact [8]. As another example, we have reported a modulus enhancement in Au at the Fermi length scale [7]. In addition to the modulus, strength is another property of interest (stress at which material yields). Here, for the first time, we measure the strength of metals at the Fermi and Sharvin length scales. 2 Experimental details A modified atomic force microscope (AFM) was used to simultaneously measure force–deformation and conductance traces across atomicsized samples; the experimental setup is described in detail elsewhere [6, 7]. Measurements were made at room temperature in inert atmosphere. The AFM assembly consists of a dual piezo [6, 7]. With this configuration, the noise band is 5 pm (peak-to-peak), and its center line can be shifted by a minimum step of 4 pm. Conductance for Ag and Au was recorded at 100 mV and 250 mV, respectively. Increased instability was seen in Ag at higher voltages, possibly due to electro-migration away from ballistic contacts. Hence a lower voltage for Ag is used. Piezo was retracted at a rate of 5 nm/s. Silver and gold films (200 nm thick) were magnetron sputtered (30 W) on Si substrates and cantilevers in Ar atmosphere with partial pressure of 3 mTorr in a UHV chamber with base pressure of ~10 –8 – 10 –9 Torr. The sputtering targets were 99.999% pure. 3 Results and discussion Figure 1 shows an example of simultaneously measured force and conductance across atomic-sized Ag samples, where piezo retraction © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
physica status solidi rrl 100 J. N. Armstrong et al.: Strength of metals at the Fermi length scale 200 0 200 Conductance (2e 2 /h) 150 100 50 0 G v u v w u v w F v Yield 8 9 10 11 12 Piezo Retraction (nm) Figure 1 (online colour at: www.pss-rapid.com) Simultaneously measured force and conductance across atomic-sized Ag bridges. causes an initially large diameter bridge to be progressively broken down. Samples from single-atom bridges to hundred of atoms in diameter were formed. Similar traces were obtained for Au. Note that while Ref. [9] proves chain formation using selective examples, over a large experimental dataset at room temperature, formation of single-atom bridges instead of chains dominates [6, 7]; chains become prevalent at low temperatures. For a given atomic configuration, say, ‘v’ in Fig. 1, force increases (more negative values) until a critical force F Yield is reached that causes a new configuration ‘w’ to form abruptly; negative force in Fig. 1 denotes experiments in retraction. In Fig. 1, successive atomic configurations (such as the ones labeled ‘u’, ‘v’, ‘w’, etc.) are separated by a stepwise change in force and conductance. From hundreds of such traces, F Yield for different sized atomic configurations was measured; F Yield is the maximum numerical value of force to break a given configuration. Our analysis used experimental conductance values to calculate the cross-section area applying the Sharvin formula, as described previously [7]. Only in the limit of a single-atom bridge Sharvin analysis deviates, and atomic diameters can be used (also estimated but not shown). However, this difference is small (~5%). Thus all the data was base-lined using Sharvin estimate. In this manner, the size dependence of strength (force per unit area to rupture) can be determined. The size dependence of strength of Au and Ag is plotted in Fig. 2(a) and (b), respectively. The strength rises sharply in the limit of a single-atom bridge, and approaches values of ~125 GPa for Au and ~14–25 GPa for Ag. These values are comparable to theoretical values, and over four orders of magnitude (10,000 times) higher than the yield strength of 99.999% annealed single crystals; for example, the strength of Ag single crystals is ~0.5 MPa [10]. As an estimate of ideal strength itself, recently, we have shown that at Fermi length scale and beyond (up to ~1.45 nm diameter in the Sharvin regime), deformation occurs by homogeneous shear instead of defect mediated deformation; in addition, a large modulus E enhancement of -10 -20 -30 -40 Force (nN) Yield Strength (GPa) Yield Strength (GPa) 150 100 50 1-atom bridge Au (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 20 30 1-atom 1-atom bridge 25 15 10 5 (b) 20 15 10 5 0 Ag 2-atom 4-6 atoms 7-13 atoms 13-19 atoms 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 Cross-section area (nm 2 ) Figure 2 (online colour at: www.pss-rapid.com) Size dependence of the strength of (a) Au and (b) Ag. Inset in (b) shows a zoom-in view of strength at the Fermi length scale with data from additional experiments. ~210–500 GPa was seen (shear modulus of bulk Au being 27 GPa) [7]. As illustrated in the Abstract figure, the ideal shear strain ε is ~0.5. Taking into account modulus enhancement, the theoretical shear strength (εE) is ~100– 240 GPa, which is comparable to the experimental value of ~125 GPa in Fig. 2(a). Similarly, using 30 GPa as shear modulus for bulk Ag, its ideal strength is ~15 GPa. Recently, we have also measured the modulus of Ag at Fermi length scale; data shows a smaller modulus enhancement (~1.5–2 times the bulk value) [11]. Thus the ideal strength of Ag is ~22–30 GPa, which is the same order of magnitude as in Fig. 2(b). As the coordination number of atoms decreases, the radius of atoms shrinks [12, 13]. Reduction in bond length is associated with bond stiffening [14], and qualitatively explains the observed strengthening. Higher strength for Au versus Ag can also be qualitatively explained on the basis of former’s higher surface energy versus Ag. At these length scales, surface effects dominate. When the surface energy is high, more energy is required to form a given surface. Therefore, higher forces are required to deform the higher surface energy Au. The Bond–Order–Length– © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-rapid.com
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