Thin Solid Films 434(2003) 316–322

0040-6090/03/$ - see front matter� 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0040-6090(03)00504-2

Piezoresistance and electrical resistivity of Pd, Au, and Cu films

S.U. Jen*, C.C. Yu, C.H. Liu, G.Y. Lee

Institute of Physics, Academia Sinica, Taipei 11529, Taiwan, ROC

Received 19 February 2003; received in revised form 3 March 2003; accepted 6 March 2003

Abstract

Electrical resistivity and piezoresistance of some metal films, such as the Pd, Au, and Cu films, were measured. Surfacefeatures of a few characteristic films, used in this study, were revealed by using an atomic force microscope. The electricalresistivity r is plotted as a function of the film thicknesst, and the strain gauge factorg is plotted as a function of the sheetresistivity R . Two parameters are found useful: i.e.hyl and 2hyt, where h is the amplitude of undulations of the surfacesq

roughness andl is the electron mean free path at room temperature. By taking the Cu film as an example, the area-distributionfunction of island-like humps on the specimen surface is also analyzed. Two models, namely the surface roughness and theelectron tunneling models, are employed to explain the electrical resistivity and piezoresistance data observed. It is found that(1)if hyl(0.3 and 2hyt-0.5, the films are continuous, and the former model is effective, and(2) if 0.5-2hyt-1, the films arecritically coalesced, and the latter model is important. Finally, from the area-distribution plot, we can show indirectly that if thefilm is at the coalescence stage, the average separationDd between two neighboring humps is wider than if the film is at thecontinuous stage.� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Piezoresistance; Resistivity; Surface roughness; Tunneling

1. Introduction

Piezoresistance effect means a change in electricalresistance of a specimen, when it is subjected to aphysical strain. For example, the use of a strain gaugeemploys this phenomenon. The strain gauge factorg isthen defined asw1x,

1 DR 1 Drg' s(1q2n)q , (1)

R D´ r D´0

whereDRsRyR , R is the resistance of the specimen,0

when stressed by a strainD´, R the unstrained resis-0

tance,n the Poisson’s ratio, andDr/(rD´) the straincoefficient of electrical resistivity.

For a metal or an alloy specimen, theoretical studiesw1–3x have shown that when it is in a bulk form thegauge factorg s2(1qn)q2G(1y2n), where G isb

Gruneisen’s constant, and when it is in a thin-film form¨gs1q3n. Therefore, if the metal film is continuous,g

*Corresponding author. Fax:q886-2-2783-4187.E-mail address: physjen@gate.sinica.edu.tw(S.U. Jen).

should increase from the thin-film valueg to the bulkvalueg as the film thicknesst is increased. This meansb

that the free-electron-scattering model is good in thebulk as well as in the thin-film case. Note, in the lattercase, the conditiont)l, wherel is the mean free path(mfp) of the free electrons, has been assumed. Hence,as the film becomes thinner, i.e.t(l or t-l, g shouldbehave differently from the value 1q3n. For example,from this study and Ref.w2x, whent(l, the value ofgis in general lower than 1q3n. Moreover, when thefilm is thin enough to be at the coalescence region(i.e.t-l), g is known to increase sharplyw2,4,5x. As alreadyknown w2,5x, g depends much on the growing conditionof the film. As a result, the surface morphology of athin-film specimen may affectg in two ways: one isthe surface roughness effect and the other is the electrontunneling effect. In this article, we shall discuss thesetwo effects in details by taking the Pd, Au, and Cu filmsas examples. In particular, sinceg is closely related tothe electrical resistivityr according to Eq.(1), thethicknesst dependence ofr of the Pd, Au, and Cu filmis also studied.

317S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

Fig. 1. The electrical resistivityr vs. the film thicknesst plot for thePd, Au, and Cu films.

Table 1t is the film thickness, 2h is the mean height of roughness of thespecimen’s surface,l is the electron mfp for the various metal filmsw9x

Specimen t 2h l hyl 2hyt tyl r r`

(A)˚ (A)˚ (A)˚ (mV cm) (mV cm)

Cu-300 300 143 240 0.30 0.48 1.25 9.3 1.8Cu-250 250 217 240 0.45 0.87 1.04 4.4=102 –Au-320 320 149 310 0.24 0.47 1.03 10.4 2.2Au-275 275 194 310 0.31 0.71 0.89 2.8=104 –Pd-200 200 9.5 120 0.04 0.05 1.67 47.1 10.5Pd-43 43 7.1 120 0.03 0.17 0.36 1.0=107 –

Note, becauser of the Pd film is approximately equal tor of` `

the Pt film, we assumel sl .Pd Pt

2. Experiments

Both the film-making procedure and the piezoresist-ance measurement have been described in detail else-wherew6x. Here, we only outline a few points which areimportant to discussion in this study.

In regard to the film-making, the following conditionswere used. All the films were made by the evaporationmethod in an 1=10 Torr vacuum. The substrate wasy6

a piece of 0211 Corning glass. The substrate temperatureduring evaporation wasT s180 8C. The film thicknesss

t was measured by a step profiler, Dektak 3, and theaverage thicknesst of substrate was calculated from thes

density method.In regard to the piezoresistance measurement, the

following techniques were employed. The strainD´ ona thin-film specimen was calculated by applying aknown force on the free end of the glass-substratecantileverw6x. The electrical resistance of the specimenwas recorded for the two conditions: ifR of the0

specimen was higher than 1 kV, the two-wire methodwas used, and ifR was lower, the four-wire method0

was usedw7x. In order to examine the effect of contactresistance of the silver-paint pads, we did the followingtests. First, a film specimen withR , known to have a0

value approximately 1 kV was chosen. Then, we meas-ured its R by both the two-wire and the four-wire0

methods. The two results were consistent. This showedthat no significant contact effects were present. It alsoshowed that after the silver-paint has been set overnight,the electrical contacts were quite reliable. Secondly,when there was no current in the specimen, the voltageleads connected to the digital multimeter(Keithley2000) gave the valueV . Next, a maximum strain was0

applied(but still with no current) on the specimen, andthe voltage reading wasV. The difference betweenV0

and V was at most 0.1mV. In comparison, in a typicalexperimental run, when the probe current was 1 mA,the corresponding piezoresistance signals were in therange 1–4mV. In general, theDR vs. D´ plot is quitelinear. From a fit using Eq.(1), we find the value ofg.As to the estimation ofr it is calculated based on theformula rsR (twyL), where w and L are width and0

voltage-probe length of the film specimen.In addition, the surface morphology of a few charac-

teristic films, used in the piezoresistance or resistivitymeasurement, was examined in contact mode by anatomic force microscope(AFM), the Autoprobe LSsystem made by Veeco Instruments Inc. The tip usedwas the etched ultrasharp Si tip(UL06B). The softwareprovided can measure the mean height 2h of the surfaceroughness within a selected(or scanned) enclosure onthe specimen. In other words, if for simplicity theundulations of the cross-section of a film surface can beconsidered as a sinusoidal wave, as described in Ref.w8x, the amplitude of the undulations by definition isequal toh.

3. Results and discussion

Fig. 1 shows the electrical resistivityr vs. filmthicknesst plot for the Pd, Au, and Cu films. Table 1also shows the electron mfpl of these metal films atroom temperaturew9x. From Fig. 1 and Table 1, it iseasy to illustrate that if the films are thick enough, i.e.l<t the Sondheimer’s approximation,

brsr q , (2)` t

wherer is the electrical resistivity atts`, and bs`

(3y8)r l. However, when the films become thinner,`

e.g. t s50 A-l , t s280 Afl , and t s260Pd Pd Au Au Cu˚ ˚

Afl , r increases rapidly. This may indicate twoCu˚things: first, according to Ref.w8x, the surface roughnesseffect plays a significant role, and secondly, according

318 S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

Fig. 2. The strain gauge factorg vs. the sheet resistivityR for thesq

Pd, Au, and Cu films.

to Ref. w10,11x, the film is about to coalesce. Beforegoing into detailed discussion, we mention that thesubstrate temperatureT is the main cause for the shifts

of the critical thickness at which the coalescence isestablishedw10,12x. In our case withT s180 8C, thes

ability to grow a film continuously on the glass substrateis much better for the Pd metal than for the Au or Cumetal. This is due to the different wetting situation ofeach metal on the glass substratew11x. Hence, the surfacemorphology of the Pd film must be very different fromthat of the Au or Cu film. More discussion in thisrespect will come later.

In order to distinguish the two resistivity-incrementeffects, i.e. the surface roughness effect and the electrontunneling effect, we introduce two parameters as shownin Table 1: the ratioshyl and 2hyt. From the data inFigs. 1 and 2, we can summarize the following twocriteria. First, if hyl(0.3 and 2hyt-0.5, the film iscontinuous and the former(roughness) mechanism iseffective. Secondly, if 0.5-2hyt-1, the film is criticallycoalesced and the latter(tunneling) mechanism becomesimportant. Note, by definition, 2h can not be larger thant.

From Ref.w8x and using the surface roughness modeltherein,r can be expressed as,

t r(tylqhyl sinkx)Ÿrs dx, (3)|lŸ tylqhyl sinkx0

whereŸ is the roughness wavelength andks2pyŸ. Byassuming uniform roughness with long wavelength(Ÿ)h) and the conditionl(t, Eq. (3) is approximated asw13x,

w z1 brs r q qF(tyl) , (4)x |` 22 w xt 1y(hyt)y y ~1y(hyt)

whereF(tyl) is a complex function of(tyl), and canbe approximated as,

B E1 y tyl( )C FF(tyl)s e r , (for tyl(1) (5)`D G4

Comparing Eqs.(2) and (4), we see how the ratios(hyt) and (lyt) affect r, respectively. For example, forthe Au-320 and Cu-300 films in Table 1 we find thatthe first criterion has been met. In other words, thefollowing parameters in Table 1,r (2.2 mV cm, 2hy`

ts0.47, andtyls1.03, have been taken for the Au-320film, and r s1.8 mV cm, 2hyts0.48, andtyls1.25`

for the Cu-300 film. Then, based on Eqs.(4) and (5),the calculated ratio for the Au-320 film isryr (1.5`

and for the Cu-300 film isryr (1.4. In comparison,`

from Table 1 the experimental value ofryr for the`

Au-320 film is ryr s4.7, while for the Cu-300 film,`

ryr s5.2. Thus, considering the approximation made`

in Eqs.(4) and (5), the agreement above is reasonablygood.

Next, because the first criterion is not met for theAu-275 and Cu-250 films, the surface roughness mech-anism cannot be used to explain the correspondingresistivity data listed in Table 1. For example, for Au-275 or Cu-250 itshyl is approximately 0.3–0.4, whichis slightly larger than the critical value 0.3. If one stillsubstitutes the above values ofhyl into Eqs. (4) and(5), he will find thatryr s2–3. But, the experimental`

data in Table 1 show thatryr for either the Au-275`

or the Cu-250 film is already of the order of 10 –10 .2 4

Apparently, the surface roughness model cannot explainthe rapid increase of resistivity, if the metal film is atthe coalescence stage. The electron tunneling model,which is to be discussed later, should be raised. Further,we consider the experimental data of the Pd-200 film.Clearly, the first criterion is also satisfied by this film.Therefore, for the Pd-200 film, the agreement betweentheory and experiment is also good: from Eqs.(4) and(5), ryr s1.27, and from Table 1,ryr s4.5. How-` `

ever, for the Pd-43 film, even if the first criterion,hyls0.03<0.3, is met, Eqs.(4) and (5) fail completelyto predict any satisfactory value forryr . From Table`

1, the experimental value ofryr of the Pd-43 film is`

extremely large, approximately 10 .6

Fig. 2 shows the strain gauge factorg of the Pd, Au,and Cu films as a function of the sheet resistivityR ,sq

which is defined asR sR (wyL). In agreement withsq 0

Ref. w2x, all the following three features aboutg areobserved here in Table 2. First, for lowR (thickersq

films) the g value tends to approach the experimental

319S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

Table 2n is the Poisson’s ratio,g is the experimental value of the strain gauge factorg for a bulk specimen.g is the theoretical value ofg by usingB b

the equation in this table

Film n gB g s2(1qn)q2G(1y2n)b gmin gs1q3n gs1q2n

Pd 0.39 4.06 3.76 1.89 2.17 1.78Au 0.42 4.48 3.81 0.81 2.26 1.84Cu 0.35 3.30 3.88 0.93 2.05 1.70

G is Gruneisen’s constant.g is the minimumg found in Fig. 2.gs1q3n means thatg is calculated according to Ref.w2x, andgs1q2nmin¨means that according to Eq.(6) in the text.

bulk valueg , e.g.g s4.06 for Pd,g s4.48 for Au,B B B

and g s3.30 for Cu. Secondly, there is a minimum inB

g (i.e. g ) for the R value in the range 10 –10Vy2 3min sq

h. Thirdly, in the case of very highR (i.e. the thinnestsq

films), g increases greatly. What is more interesting isthat theR dependence ofg for the three kinds of metalsq

films in Fig. 2 is roughly the same. Hence, the plots inFig. 2 imply that the mechanism for theg behavior foreach kind of metal film must be universal and consistent.

In Fig. 2, for the Pd-200, Au-320, and Cu-300 films,their g)g and R -R (min)f10 Vyh. In addi-3

min sq sq

tion, from Table 1, their(2hyt) values are smaller than0.5. Therefore, we may consider all the three films tobe continuous. From the free-electron-scattering model,it states that from the bulk to the film form the valueof g should decrease fromg s2(1qn)q2G(1yn) tob

gs1q3n. However, by comparingg and gs1q3nmin

in Table 2, obviously, the former is smaller than thelatter for all the films. We are going to show that thisdiscrepancy can be reduced, if the correction, due to thesurface roughness effect, is considered. From Eqs.(1)and(4), if the film is continuous, the surface roughnessh will lead to the result,

2gs1q2nqn(hyt) . (6)

Since for the Pd-200, Au-320, and Cu-300 the(hyt)value is negligible,g is approximated as 1q2n. In short,when the surface roughness effect is taken into account,both Dr and r in Eq. (1) will increase. However, theincrease ofr is more than that ofDr. As a result, thevalue of g from Eq. (1) decreases from 1q3n to 1q2n. Indeed, from Table 2, we find that after the rough-ness correction has been made, the value ofgs1q2n

agrees withg better.min

Next, for the Pd-43, Au-275, and Cu-250 films, theR values fall in the range 10Vyh-R -10 Vyh.3 7

sq sq

From Table 1, the(2hyt) values of Au-275 and Cu-250are equal to 0.8. Because 0.8 is close to 1, the secondcriterion asserts that the two films should be criticallycoalesced. According to Refs.w2,14x, the electron tun-neling mechanism must be considered, and the relationbetweeng andR is expressed as,sq

gAŸn R ADd, (7)Ž .sq

where Dd is the average distance between two neigh-boring island-like humps. Clearly, in Fig. 2 if 10Vy3

h-R -10 Vyh, the logarithmic dependence ofR ,7sq sq

i.e. the first part of Eq.(7), is in general followed.Hence, the argument above proves that if the secondcriterion is satisfied for a film at the coalescence stage,the electron tunneling mechanism is important. Later,we can show that qualitatively speakingR increasessq

dramatically, asDd increases slightly. In other words,the tunneling model predicts an exponential increase ofR or r, once t is near the critical thickness ofsq

coalescence. This kind of rapid increment inr agreeswith the most rising part of ther vs. t plot for the Pd,Au, or Cu film in Fig. 1. However, from the surfaceroughness model, as discussed previously, it can onlyexplain the initial rising part.

Still, there is one problem with the Pd-43 film. Asstated before, it violates our theory(or criteria), becausePd-43 is so smooth(i.e. hyt<1), and yet ryr is`

extremely large. It is likely that the top surface of Pd-43 is covered with a thin layer of oxide. However, thissituation does not change our belief that the bare(insitu) surface of Pd-43 is smooth. The reason is givenbelow. Because thez-resolution of the scanner of ourAFM is down to 0.1 A, 2h detected on Pd-43, though˚as small as 7.1 A, is reliable. Further, Ref.w2x has˚shown that the Pd film does not change significantly inresistanceR on exposure to air. Finally, since the oxide0

layer is thin, the morphology on top layer should besimilar to what is underneath. Nevertheless, the resolu-tion limit of the step profiler is approximately 10 A.˚The thicknesst of the Pd-43 film should be less accuratethan that of the others. Hence, instead of consideringthe absolute value of(2hyt), we may consider the ratioof increase in(2hyt) with decreasingt; e.g. the ratio isapproximately 3.4 for the Pd films and 1.8 for the Cufilms. Since the two ratios are of the same order ofmagnitude, it may indicate that the situation of electrontunneling is the same for the Au-275, Cu-250, as wellas for the Pd-43 film.

In Figs. 3–5, the surface morphology of a fewselected specimens, such as the Pd-43, Pd-200, Au-275,Au-320, Cu-250, and Cu-300 films, scanned by ourAFM, are shown. For each set of figures, part(a)belongs to a thinner specimen, while part(b) belongs

320 S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

Fig. 3. The AFM images for(a) the Pd-43 and(b) the Pd-200 films.

Fig. 4. The AFM images for(a) the Au-275 and(b) the Au-320 films.

to a thicker specimen. Specifically speaking, the posi-tions of the part(a) specimens and that of the part(b)specimens are indicated by arrows in Figs. 1 and 2,respectively. As mentioned previously, the surface fea-tures of the Pd films in Fig. 3 are less definable.However, together with the data in Table 1, we concludethat the surface of the Pd film is smoother than that ofthe Cu or Au film. For the Au and Cu films in Figs. 4and 5, they are full of surface features. Table 1 alsomarks the increasing degree of surface roughness dis-tinctively (i.e. 2hyt increases), as each kind of the filmgrows thinner.

As shown in Figs. 4 and 5, there are many island-like humps in each AFM photograph. The size of eachhumpD is approximately 0.1mm in general. One thingH

about D should be made clear here. From the peak-H

broadening X-ray analysis of the(1 1 1) diffraction lineof these metal films, we find that the average grain sizeD is only approximately 100 A. Therefore,D is quiteH

˚different from D. In fact, from this study it is believedthat a typical island-like hump may contain several(e.g.10) crystalline grains.

Further, to be more specific about hump distribution,we have used the gray levels of each AFM image todefine the boundary of a designated hump, and thenestimated its area. Then, in principle, we can plot thearea(A) distribution of the island-like humps in eachpicture. However, it is found out that only the imagesof Fig. 5 are clear enough for a meaningful comparison.Hence, the area distribution for the Cu-250 film isplotted as in Fig. 6a, and that for the Cu-300 film is inFig. 6b. Note, the normalizationNyN means that the0

total enclosure area for the number counting in Fig. 6aand b is the same. By employing the Gaussian fits forthe plots in Fig. 6, we find that for the Cu-300 film themean area of a humpNAMs16=10 mm , and for they3 2

Cu-250 film NAMs14=10 mm . Therefore, the abovey3 2

results ofNAM indicate that the average separationDdbecomes slightly larger, as the Cu film becomes thinnerfrom the continuous stage(e.g. Cu-300) to the coales-cence stage(e.g. Cu-250). From Fig. 1, it is known thatthe value ofr of the Cu-250 film is much larger than

321S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

Fig. 5. The AFM images for(a) the Cu-250 and(b) the Cu-300 films.

Fig. 6. The normalized area distribution of the island-like humps inFig. 5: (a) for Cu-250 and(b) for Cu-300. A stands for the area ofhumps.

that of the Cu-300 film. Hence, we have shown quali-tatively that the second relation of Eq.(7), i.e.rAR Ae , is confirmed by the AFM data of the CuDd

sq

films. This conclusion gives another support for theelectron tunneling model for the thinner metal films,especially if they are at the coalescence or the discon-tinuous stage.

4. Conclusion

We made the Pd, Au, and Cu films in differentthicknesst, and measured the electrical resistivity andthe piezoresistance of the film specimens. Then, theexperimental data are analyzed by plottingr as afunction of t, and by plotting the strain gauge factorg

as a function of the sheet resistivityR . Two character-sq

istic groups of specimens were chosen to reveal theirsurface morphology by AFM. One involves the charac-teristics of the thicker and continuous films, such as thePd-200, Au-320, and Cu-300 films. The other involvesthe characteristics of the thinner and coalesced films,such as the Au-275 and Cu-250 films. However, the Pd-43 film is an exception from this study. In addition, theundulation amplitudeh of the surface roughness of thesesix specimens was measured.

Two theoretical models, i.e. the surface roughness andthe electron tunneling models, are employed to explainthe rapid rise of ther and theg data in ther vs. t andthe g vs. R plots, respectively. Parameters, likehylsq

and 2hyt, are useful in the model discussion. Two criteriaare set for the validity of the two models: first, ifhyl(0.3 and 2hyt-0.5, the roughness model is effective,

322 S.U. Jen et al. / Thin Solid Films 434 (2003) 316–322

and secondly, if 0.5-2hyt-1 the tunneling model isdominant. The former model explains the initial rise ofr or the initial drop ofg, when the film is continuous.The latter model explains the rapid rise of bothr andg when the film is critically coalesced. By consideringthe surface roughness effect,g of the continuous film isequal to 1q2n, which agrees with theg value better.min

It also explains why there is an initial drop ofg (i.e.from 1q3n to 1q2n). Further, the AFM images of Cufilms near the coalescence region confirm the tworelations, i.e.gAln(R ) andR Ae , which are derivedDd

sq sq

from the electron tunneling model.Finally, because the thinnest Pd film(e.g. Pd-43) has

posed some difficulties to the analysis in this study. Inthe future, more work is needed on this interesting thin-film system.

Acknowledgments

This work is supported by the National ScienceCouncil.

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