Stress Relaxation in Copper CrystalsJ. M. Galligan and J. L. Davidson Citation: Journal of Applied Physics 38, 3420 (1967); doi: 10.1063/1.1710139 View online: http://dx.doi.org/10.1063/1.1710139 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurement of probability distributions for internal stresses in dislocated crystals Appl. Phys. Lett. 105, 181907 (2014); 10.1063/1.4901219 Isothermal stress relaxation in electroplated Cu films. II. Kinetic modeling J. Appl. Phys. 97, 103532 (2005); 10.1063/1.1904721 Effects of Passivation Layer on Stress Relaxation and Mass Transport in Electroplated Cu Films AIP Conf. Proc. 741, 256 (2004); 10.1063/1.1845857 Numerical Simulations of Stress Relaxation by Interface Diffusion in Patterned Copper Lines AIP Conf. Proc. 741, 62 (2004); 10.1063/1.1845837 Diffusional creep as a stress relaxation mechanism in electromigration J. Appl. Phys. 83, 100 (1998); 10.1063/1.366729

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3420 COMMUNICATIONS

in order to measure the change of the half-wave voltage. For the range of laser intensity tested, the damage reaches a steady state when 1 E. I,., 1 E. I. The electron density is estimated to be of the order of 3XI0'6 per cm3 when 1 E.I=3 kV/cm.

A similar distribution of inhomogeneity of refractive indices near a damaged region was observed for LiTaOs and LiNbOs, and more extensive study of it is being carried out.

I wish to thankL. G. Van Uitert and W. A. Bonner for supplying the samples. Many discussions with J. E. Geusic, R. T. Denton, M. DiDomenico, Jr., S. K. Kurtz, R. G. Smith and S. H. Wemple, and the assistance of R. P. Morris are gratefully acknowledged.

1 A. Ashkin. G. D. Boyd. J. M. Dziedzic. R. G. Smith. A. A. Ballman. H. J. Levinstein. and K. Nassan. App!. Phys. Letters 9,72 (1966).

2 F. S. Chen. J. E. Geusic. S. K. Kurtz. J. G. Skinner. and S. H. Wemple, J. App!. Phys. 37, 388 (1966).

• A. G. Chynoweth. Am. J. Phys. 20, 218 (1952).

Stress Relaxation in Copper Crystals*

J. M. GALLIGAN AND J. L. DAVIDSON

Columbia University. New York. New York

(Received 19 October 1966; in final form 27 March 1967)

Various studies of the work-hardened state of crystals have been made by changing the temperature of deformation or changing the strain rate, both as a function of the plastic strain.'·2 In these studies it is assumed that the dislocation structure-density of active dislocations as well as distribution-remains the same or at least similar throughout the change of temperature or the change in strain rate used. Also, it has recently been claimed that studies of stress relaxationS might be used to obtain the stress dependence of dislocation velocities, by using the phenomenological equation of dislocation motion given by Gilman and Johnston.' In the etch-pit method of Gilman and Johnston no assumption about

3000

E E 2000 ..... .. E l! '" .. .. ~ iii

1000

r Slop Loading

. -4' a 'x 10 Isec

b 3.6 x 10-"tsec

2s.ec.../.--.l Time (sec)

FIG. 1. Stress relaxation in a copper crystal after deformation at two different strain rates. Specimen deformed a~ 800 K.

N E E .... E ~ ~ .. .. ~ iii

2000

-4 I x 10

25 .. c-+-l Time (nc)

FIG. 2. The effect of strain-rate cycling upon the work-hardening rate and flow stress of crystals deformed at 80oK.

the constancy of the dislocation density is necessary since the motion of an individual dislocation is measured. The use of stress relaxation to determine the stress dependence of dislocation velocities, however, specifically requires a constant density of dislocations throughout the experiment in order for the experiment to correctly measure the stress dependence of the dislocation velocity. One of the basic parameters of such measurements is the strain rate used throughout the experiment. If, for example, so­called "m" values (see Gilman and Johnston for definition of m) are determined using the stress-relaxation method, but different rates of loading are used to reach the same total plastic strain in two crystals, the "m" values should be the same in both cases. If they are not the same, one must consider not only the total strain, but the path used to obtain the same total strain. In the case of strain-rate-change experiments in which the flow stress is meas­ured, such as those of Basinski,! and Thorton and Hirsch,2 if some unloading occurs then the subsequent measurement of the change in the flow stress (and work-hardening rate) is of very doubtful significance. We have carried out such experiments which distinctly show that "m" values, measured by stress­relaxation methods, depend upon the prior strain rate. In addition, in experiments on the same crystal, in which the crystal was de­formed at "slow" and "fast" strain rates, it is found that the work­hardening coefficient is sensitive to the prior strain rate.

The present experiment was performed on copper crystals grown by the standard Bridgman method, from American Smelting and Refining Company high-purity copper. Tensile tests were performed at 800 K in a laboratory tensile machine.6 The drive motor could be reversibly changed in rate by a factor of fifty with a measured reproducibility of better than 1%.

A typical experimental result used in the determination of "m" values is shown in Fig. 1, where the strain rate has been changed from lX1Q-4/sec to 3.6XIQ-4/sec in deforming a single­crystal of copper. The strain interval involved in this change of strain rate was kept as small as possible. The observed stress relaxation quite obviously depends upon the strain-rate history of the specimen. It might be mentioned, however, that the magni­tude of the stress relaxation increases with total strain, but the relative amount of relaxation is larger for higher strain rates.

This experimental result led us to examine the effect of strain­rate cycling upon the work-hardening rate of some copper crystals. Copper crystals were strained at a strain rate of 1 X 10-4/sec at 80oK, the crystals allowed to relax partially, the crystals again reloaded but at a very slow strain rate and then the crystals were reloaded at a higher strain rate. The results of this experiment are shown in Fig. 2, where it is clearIy·seen that the flow stress is reduced and the work hardening decreases, when the crystlll is given an intermediate deformation at a "low" strain rate,

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COMMUNICATIONS 3421

These experiments indicate that during deformation of copper at 78°K, there is some dynamic recovery occurring during defor­mation and this recovery depends upon the strain rate. This would be equivalent to saying that the mobile dislocation density is lower at lower strain rates, which should be taken into account in stress­relaxation measurements.

Similar experiments have been undertaken in polycrystalline copper, platinum, gold, aluminum, and nickel. A more complete account of this will be given elsewhere.

* Supported through the National Science Foundation. 1 Z. S. Basinski. Phil. Mag. 4, 393 (1959). 2 T. R. Thorton and P. B. Hirsch. Phil. Mag. 3, 738 (1958). 3 F. W. Noble and D. Hull. Acta Met. 12, 1089 (1964). 'J. J. Gilman and W. G. Johnston. J. App!. Phys. 30, 129 (1959). • J. L. Davidson and J. M. Galligan (unpublished results).

On the Resolution of the Field-Ion Microscope

V. G. WEIZER

Lewis Research Center. Cle.eland. Ohio

(Received 27 February 1967)

The resolution of an image spot on the screen of the field-ion microscope increases as the applied voltage is increased. There is, however, a voltage that produces maximum resolution, beyond which the resolution abruptly degenerates. A knowledge of the mechanism responsible for this sudden blurring is desirable not only because of the insight it would give to the detailed under­standing of the operation of the microscope, but also because it might indicate ways of controlling the blurring, and thus lead to increased resolution.

There are two possible sources for the ions causing the blurring effect, as indicated in the literature: (1) the gas phase above the tip surface! and (2) an accommodated or partially accommodated phase on the surface.2 To distinguish between these two sources,

. a measurement of the voltage required for blurring onset as a function of tip temperature was made. Here, lack of temperature dependence would indicate a gas-phase origin. Conversely, if there is a dependence, then at least partial accommodation has taken place before ionization.

The microscope used in these measurements3 permits varying the temperature of the tip over a range from 5° to 3000 K while the tip surroundings remain at 78°K. The background pressure is typically about 5XlO-9 Torr, and the screen is directly viewed continuously during the experiment.

Figure 1 shows a typical set of data for the (111) plane of tungsten with helium as the imaging gas. The results were found to be independent of gas pressure, which was varied from 0.2 to 0.8 p. (gauge readings, uncorrected for helium gas). The results were shown also to be independent of the temperature of the tip surroundings by warming the screen and its liquid nitrogen reser­voir to room temperature, exposing the tip to about a 2'1f-stera­dian solid angle of surfaces at 300oK.

The fact that the blurring threshold is dependent upon the tip temperature indicates that the blurring current originated from accommodated or partially accommodated atoms. The fact that there is a maximum at about 800 K leads one to suspect that there are two processes operating. The low-temperature process would seem rather easy to explain qualitatively using a transit-time theory for the atoms moving in the comparatively low-field regions between the imaging tungsten atoms. A decrease in tip temperature would cause lower helium atom velocities and longer transit times through these secondary ionization zones, which would, because the ionization rate is proportional to the transit time, increase the current from these areas. Therefore, a decrease

10.2

:: 10.0

i § 9.8 g' ;: 5 9.6 :s .e 9.4

o o

o Average experimental - Theoretical - equation 1

Pressure: 0.511 (uncorrected) Tip radIus: 260 A

40 80 120 160 200 240 280 320 360 Tip temperature. "K

FIG. 1. Comparison of Eq. (1) with experimental data. Scatter in voltage measurements about 2%. In Eq. (1) Fo has beert set equal to 4.5 V /A for an applied voltage of 9775 V.

in temperature would require a decrease in field to prevent between atom ionization. A lack of knowledge of field variation in these regions, however, makes calculation of the field-temperature relationship rather impractical.

The high-temperature region of the curve, as welJ as being unexpected, is a little more difficult to explain. One approach that seems to fit the experimental data consists of postulating the existence of two separate regions of ionization: (1) A region close to the surface where the equipotentials closely conform to the atomic structure of the surface, and where, if, under the proper con~ition~ of field and temperature, ionization does take place, the IOns diverge only slightly and form a well-resolved spot on the screen. (This is the source of ions during normal operation of the microscope.) (2) A second, broader region at some distance Xe2 from the surface, separated from the first region by a region of very low ionization probability. This second region, being further from the surface, would produce a more diffuse imaging beam on the screen.

Experimental evidence for the existence of such ionization bands is given by Jason et al.4 who have used mass spectrographic techniques to resolve structure in the energy distribution of ions produced near a tungsten tip. Their work was done on the hy­drogen-tungsten and the neon-tungsten systems, however, and it is assumed here that a similar structure exists for the helium­tungsten system.

It is postulated here that the tangential spreading of Jason's lowest energy deficit band, adjacent to the surface, with increasing field strength is the cause of the low-temperature blurring current. It is further postulated that the bands of higher energy deficit correspond to the ionization region, distant from the surface which is the source of the high-temperature blurring current:

To be more explicit, a simple calculation may be performed to check the general fit of the model. First, it is assumed that the group of Jason's ionization regions above the first, low energy deficit band is one broad ionization zone, and that the helium atoms at higher temperatures orbit into this zone. Then, if the probability of ionization in this zone is proportional to the time spent orbiting in it, it follows that, for small ionization rates, the probability of ionization (following Gomer's derivation5) is

c=2t2/r= 10!6 (2m) 112 (rt/aFo2) [k T-2aFo2 (Xc2/rt) J1I2

X exp- (0.6813/2/ Fo), (1)

where 2t2 is the time spent by a hopping atom in the region above the cutoff distance X c2,5 m is the gas atom mass, a is the gas atom polarizability, kT is in the initial gas atom kinetic energy, rt the tip radius, Fo the surface field, T the tip temperature, 1 the gas atom ionization potential, and T the ionization time, which is approximated by6

T=10-!6 expO.6813/2/Fo. (2)

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