physica status solid.. - Mechanical and Aerospace Engineering

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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
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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
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physica
status
solidi
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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–
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Rapid
Research Letter
Phys. Status Solidi RRL 6, No. 3 (2012) 101
Strength framework in Refs. [14, 15] appears well suited to
analyze the size dependence of strength and modulus enhancement.
Finally, the inset in Fig. 2(b) shows a zoom-in view of
the strength of Ag, with data from additional experiments.
Remarkably, the data reveals discrete behavior for 1-atom,
and 2-atom bridges (standard deviation of discrete values
shown in blue). The discreteness is due to orientation dependent
variation of the strength because each time a
bridge is formed, forces are measured along different crystallographic
directions relative to the tip and the substrate.
This is known from our previous measurements of discrete
atomic displacements [7]. This is followed by plateaus. Although
discrete values were also observed for Au, plateaus
were not seen. This is because the surface energy of Ag
varies little with crystallographic orientation (~6%), in
contrast to ~33% variation for Au [16]. This causes steps
in Au to smear out. Although samples are constrained between
tip and substrate, the existence of plateaus is analogous
to the islands of stability in free clusters; see for example
Ref. [17] for the behavior of clusters with 3–13 atoms.
The position of steps can be qualitatively illustrated
by simple geometric configurations of atoms with increasing
diameter. As shown schematically in Fig. 3, the 1-, 2-,
and 3-atom configurations are unique and discrete. Once a
3-atom diameter sample is formed, it gives rise to three
equivalent sites, where atoms 4–6 sit (first plateau from
4 atoms to ~6–7 atoms).
1
1
2
1
3
4 5
6
2
1
3 2
4
4 1 5
7’ 3 2
7 6
6-7
5 6 7
4 1 7
3 2
8
13
13
9
12
13
10
11
14
19
19
15 16
4
19
5 6
Figure 3 (online colour at: www.pss-rapid.com) Schematic
showing the atomic configurations to illustrate the occurrence of
plateaus. Arrows and numbers indicate available sites; different
colors are used as aid to illustrate the successive build-up of contact
diameter.
3
2
7
18
17
Once the 1 st ring is complete, positions ‘8–13’
(marked by arrows) are filled to give the 2 nd plateau from
7–13 atoms. The 3 rd plateau from 13–19 atoms is associated
with the formation of complete rings around each
atom (marked 2–7 in the 1 st outer ring). Addition of atoms
14–19 results in completion of the 2 nd outer ring for a total
of 19 atoms. Formation of the 2 nd ring creates 12 new
equivalent positions around it, and the process continues.
Transitions at diameters with 7 atoms, 19 atoms, and
37 atoms have been seen in our recent study on the magnitude
of discrete atomic displacements in Au [7]. However,
a large variation in surface energy with orientation in Au
smears out the steps. Even in Ag, the plateaus are highly
susceptible to perturbations. However, Ag provides a new
distinguishing ‘marker’, in addition to the one recently discovered
in Ref. [8]. In summary, the results open the possibility
to impart vastly different strengths based on the
size. Position and height of steps can be changed by mixing
elements of different size and surface energy, and offers
a realistic approach to ‘materials by design’. The results
also provide evolutionary trace of an emergent intensive
property (strength) – variation with size retains the
inherent trait of intensive properties in form of sizeindependent
steps.
Acknowledgements This work was supported by the National
Science Foundation, Grant Nos. DMR-0706074 and DMR-
0964830, and this support is gratefully acknowledged.
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