PHYSICAL REVIEW B VOLUME 50, NUMBER 18 1 NOVEMBER 1994-II

Evidence of anomalous hopping and tunneling efFects on the conductivity of a fractal Pt-Slm system

Gao-xiang Ye, Jing-song Wang, Yu-qing Xu, Zheng-kuan Jiao, and Qi-rui ZhangDepartment ofPhysics, Zhejiang Unioersity, Hangzhou 3I0027, People's Repgblic of China

(Received 7 January 1994; revised manuscript received 1 June 1994}

The temperature dependence of the conductance and the nonlinear electrical response of a Pt-film per-colation system, deposited on fracture surfaces of a-A1203 ceramics, have been measured over three de-

cades of sheet resistance. We find that in the temperature interval of T=77—300 K, the resistance tem-

perature coefficient P=(1/R)dR/dT is not a constant, which is difFerent from that for fiat films. A dcI-V characteristic which strongly depends on the thickness of the film is found and it can be interpretedas a competition among the local Joule heating, hopping, and tunneling effects. The third-harmonic mea-

surement suggests that the critical exponent comes from 1/f noise, which obeys the power-law depen-

dences S& ~ (p —p, ) ", R ~ (p —p, ) ', and then S& ~ R with m =~/t, where S& is the mean square ofresistance fluctuations, p the surface coverage fraction, and p, its percolation critical value. We find thatw =0.45%0.06, which is lower than the flat-film exponent. This result indicates that the tunneling and

hopping effects in the fractal samples are much stronger than that of flat films.

I. INTRODUt. iiON

There is a very sensitive and complex dependence ofthin-film microstructure on growth conditions. The mi-crostructure at or below submicron length scales fre-quently has a profound efFect on all physical properties(magnetic, electrical, mechanical, optical, etc.) of thethin-film system. ' In fact, the self-affine fractal struc-tures of films are also essential to various multilayered5&~s, quantum wells, and superlattices, although itsphysical origins are less studied so far.

Usually, roughness thin films are made by the ion-bombardment method and the nonequilibrium growthtechnique. These fiims have been proven to have theself-affine fractal surfaces and can be well described byscaling theory. ' ' On the other hand, a type of rough-ness thin films, in which both the up and down surfacesare uneven, can be made by using the roughness sub-strates, and many interesting phenomena have beenfound. ' These films, particularly ultrathin films, alsohave the self-affine fractal structures. The experimentalresults indicate that, just as the fiat substrates, the self-aSne fractal substrates may also be very useful for bothfundamental and practical purposes.

In this paper, we report our measurements of the tem-perature dependence of resistivity and the nonlinear dcI-V behavior of the Pt-film percolation system depositedon fracture surfaces of a-A1203 ceramics. We find that,over three decades of sheet resistance, the films have apositive temperature coeScient of resistance in the inter-val T =77-300 K. The nonlinear electrical response ismuch different from that of the Sat-metal films. Thethird-harmonic generator B is found to scale as R +

with critical exponent w =0.45+0.06, which indicatesthat the hopping and tunneling effects in this system aremuch stronger than those of the Sat films.

II. SUMMARY OF THE THEORYAND ITS APPLICATION TO DC MEASUREMENT

where p is the surface coverage fraction and p, is its per-colation critical value. The film resistance obeys anotherpower law

R ~(p —p, )

Combining Eq. (3) with Eq. (4) yields

(4)

S„~R

where ur =x'It.Consider a resistor network carrying an ac current

I=Iocos(tot) with each element a carrying a currenti cos(cot) Assuming the .resistors have a positive tem-perature coeScient of resistance and the amplitude of the

The microgeometry of a percolative film can be probedby the 1/f noise measurement, in which the fourth mo-ment of the current distribution is measured. ' ' Thefihn system can be considered as a random resistor net-work with total sheet resistance R, carrying a current I,made up of elements of resistance r carrying currents i .By the conservation of energy, we have

I R =chroia .

Using Tellegen's theorem, ' the relative resistancenoise Sz is given by

S„=((5R ))/R = gi (5r ) gi,r, (2)

which probes the fourth-current distribution moment.Rammal et al. ' have introduced a nonuniversal ex-ponent x that describes the divergence of the noise power

(3)

0163-1829/94/50(18)/13163(5}/$06.00 50 13 163 1994 The American Physical Society

13 164 YE, WANG, XU, JIAO, AND ZHANG 50

local temperature oscillation is proportional to the ampli-tude of the Joule power generated, then Joule heatingwould cause the sample's resistance to oscillate at 2~

R =R0+bR cos(2cot +P),the amplitude of the resistance is proportional to thefourth moment of the current distribution

3.0

i~ I

, ~

hR =Ph(co)/(I )gi r

where P=(1/R )(dR /dT), h(co) is a function of frequencybut not of a, and P is the phase shift between the heatproduction and the local temperature. The voltageacross the sample is given by

V=IR =IORocos(cot)+( —,')IDLER cos(3cot+P) . (8)

The third-harmonic coeScient is

V3f Iob,R ~ ( 1 /Io )gi r

which is related to the fourth-current distribution mo-ment. Hence, using Eq. (2), the quantity Sii can be writ-ten

102 4 6 8 10 12 &4

1/T t, 10 K )

FIG. 1. Conductance vs reciprocal temperature for platinumfractal 61ms of different thicknesses.

Sii V3f /(Io R ) (10)

where, as usual, we take r =r for all a. ' Therefore, V3fshould scale as

8:—V3//Io 0- R +" .

Now, let co approach zero, then Eq. (8) becomes

R =Ro+M (12)

where Ro is the sheet resistance when I approaches zero.Then, the coefficient 8 can be obtained by the dc mea-surement. It has been proved that, if co~0, the value of8 obtained by the ac measurement will approach the dcresult and, furthermore, the critical exponent m is in-

dependent on frequency co although the coefficient 8 de-

pends on the frequency. '2' Therefore, using Eq. (12) tomeasure the critical exponent w is an ideal method for itssimplicity and accuracy. The third-harmonic-generationmethod described above can be used for measuring sam-

ples with lower sheet resistance. ' '

resistances (about 200-200000 0), a good linearity of thelog conductance vs 1/T plot is found, and the films havea positive temperature coefficient of resistance, indicatingthat, at low current (I =10 pA), the conductivities aremetallic. Such behavior is similar to that of ffat-metalfilms although the substrates are totally different. ' How-ever, the coefficient P is not a constant, which is contraryto that of the ffat system. ' The results of twelve sampleshave been fitted by cr =eoexp( —T /To ). WhereTo =y/k, k is the Boltzmann's constant, and p is the ac-tivation energy. The optimal values for To are listed inTable I. It is found that the samples could be dividedinto two groups. The 61ms belonging to the lower resis-tance group have larger values of To and those belongingto the higher group have smaller values of To. The ac-tivation energies of the higher resistance samples show notrend with the sheet resistance R, providing evidence for

III. EXPERIMENT

The sample preparation method has been described inour previous work. ' ' The fractal dimension of the sub-strates was Do =2.20+0.06. The 61ms were deposited bythe dc-magnetron-sputtering method among four gold-film electrodes. The size of each sample was 6.0X2.0mm . The dc sheet resistance R as a function of the tem-perature T and the dc current I was measured with thefour-probe method.

IV. RKSUI.TS AND DISCUSSION

A. The tempera jure dependence of the resistance

Figure 1 contains the conductance o. vs 1/T curves forfive Pt fractal Slms from 77 to 300 K. In a wide span of

Sample

1

2345

6

89

10

jz

272350

2 1682 6843 1983 35047534 9266072

1059531 110

188000

To (K}

373512141618131311}4IO

13

TABLE I. Parameter 10 obtained by sting the experimental

temperature dependence of the dc conductance0.( T)=o Oexp{ —To /T).

50 EVIDENCE OF ANOMALOUS HOPPING AND TUNNELING. . . 13 165

the importance of the hopping and tunneling conductivi-ties in this resistance span (see Table I). This is contraryto the results of flat-metal films. ' ' We propose that thesharp drop (compared with the flat-metal films) of To isbecause of the fractal structures in the films. For thelower resistance samples, i.e., R & 1 kQ, the films are rela-tively thicker and the fractal structures on the substratesare then submerged. However, when the film resistanceis beyond 1 kQ, the self-af5ne fractal microstructure will

begin to appear on the films and, will apparently changethe dependence of conductance on temperature.

For ultrathin fractal Slms, i.e., R ) 1 MQ, the resis-tances showed large fluctuations in time (the current usedis 10 pA). The amplitudes of the fluctuations are seem-

ingly not strongly dependent on the temperature. A de-tailed description of the phenomenon that occurs duringthe approach to zero conductance will be published sepa-rately.

8. I-V behavior measurements

The I-V characteristic was measured and three distinctregimes of the sheet resistance Ro were found. For thethick film, i.e., Ro (10kQ, the R-I behavior is similar tothat of the flat film, although their substrates are quitedifferent. ' ' As shown in Fig. 2, at low currents thefilm shows Ohmic behavior. At higher currents, howev-er, the resistance increases with the current, i.e.,dR /dI )0. This reversible R -I relation can be well fittedby Eq. (12). If the current is further increased andbeyond the breakdown current I„' then an irreversibleand discontinuous change will occur (in Fig. 2, I, =10mA). The nonlinear response described above has beenwell studied and is generally interpreted in terms of aheating and melting process of the hot spots (or links) dueto the local Joule heating. '

It is very reasonable that the thick fractal film has thequadratic R-I behavior, just as the flat film does. In fact,when the film thickness increases, the fractal structure of

the substrate will be covered and then the microstructurein the Slm will disappear gradually. Finally, the samplebecomes a thicker and smoother film. Thus, all thecharacteristics of the thick fractal films should be similarto those of the flat films. Our previous work has verifiedthis expectation. '

It has been proven that the power-law behaviors, exist-ing in the lattice-percolation and continuum-percolationsystems, would not be affected apparently by the fractalgeometries of the substrates. Fitting the experimentaldata in Fig. 2 by the Eq. (12), one finds B =2.0X10(V/A ). Using this method, the coeScient B of differentsamples with different sheet resistances are obtained andthe scaling of B as a function of the sheet resistance Ro isshown in Fig. 3. Again, a power law with the critical ex-ponent w =0.45+0.06 is well defined up to Ra=10 kQ.This value of w is much smaller than the flat-film ex-ponent' ' and the theoretical predictions in two-dimensional system. ' '

The R-I behavior of samples with resistances higherthan 10 kQ cannot be fitted by Eq. (12), as shown in Fig.4. At low current, the sheet resistance drops with the in-crease of I; when the current further increases andreaches a critical value, however, the resistance starts toincrease. From Fig. 4, one can see that R (as well as Ro)will be increased after the Srst R-I measurement. There-fore, this R-I relation corresponds to an irreversiblebehavior. The relative change of the resistancet=hR/R in Fig. 4 is about S%%uo.

For films with resistances higher than 30 kQ (see Fig.5), within our measurement range, the resistance will sim-

ply decrease with the current, i.e., dR/dI (0 and, for afixed current, the value of dR/dI (absolute value) wouldincrease when Ro becomes larger. This conclusion is alsotrue for the flat metallic films. ' However, in Fig. 5,dR/dI will approach zero and seems to be unchangedwith further increasing of the current, which is contraryto the situation of the flat films. ' Again, the R-Ibehavior described in Fig. 5 cannot be well repeated and

1249 1' ~ I ~ ~ ~ I I II ~ I ~ I I ~ ~ ~ ) ~ ~ I 1 ISIt

1245

1241C:

1233

10 p7

10 p6

gal

10

Slo

1229 e r ~

0.0 2.0 4.0 6.0 8.0 10.010 s-4

s

FIG. 2. R-I characteristic of one of the platinum fractalfilms. This behavior is similar to that of Sat-metal films and canbe interpreted in terms of the Joule-heating efFect. Dots are theexperimental data and the solid line represents the fitR =Ro+BI B =2.0X 10 (V/A ).

1010 100 1000

Ro (0)

10000

FIG. 3. The scaling of B for platinum fractal films,2+ w =2.45+0.06.

13 166 YE, WANG, XU, JIAO, AND ZHANG

26.0

25.0

24.0

23.0

c 0 0

16.8

16.6

16.4

16.2

16.0

15.8-

1.0 2.0 3.0 4.0

as various metal-insulator-metal (MIM) tunneling junc-tions, are constructed in the films. When a high currentpasses through a weak link, the local temperature changeis sufBcient to excite local hopping and breakdown of theMIN tunneling junction. Both the hopping and tunnel-ing effects mould reduce the current density of the linkand the sheet resistance. The higher the current is, themore the amount of the resistance will be reduced, whichcauses a weaker third-harmonic component [see Eq. (12)],and hence the critical exponent w becomes smaller. Ac-cording to this analysis, the measured value of m, whichis smaller than all the results of flat-film systems, ' indi-cates that the hopping and tunneling effects in fractalfilms are much stronger than those of the flat films. Thisconclusion is very reasonable since the weak links (or tun-neling junctions) in the self-afiine films are much moreplentiful than that in the Sat systems and the film micro-geometries are different from each other. We also pro-pose that the phenomenon of dR idI (0 (see Figs. 4 and5) is mainly caused by the short circuit due to the tunnel-

0.0 2 ' 0 4.0 6.0

FIG. 4. R-I characteristics of platinum fractal films withRo&10 kQ. Dots and plus signs represent the first and thesecond R-I measurement results, respectively; the solid lines area guide to the eye.

4000

$000

2000

the sheet resistance R will increase greatly after the firstR.-I measurement (see Fig. 6). The relative change of theresistance is much larger than that of the low resistancesamples (in Fig. 6, t is around 15%), which means that,for a certain current, the microstructure of the high resis-tance films changes much more than that of the low resis-tance samples.

That the nonlinear I-V characteristics in Figs. 4 and 5correspond to irreversible behavior indicates that thehot-spot melting process due to Joule heating occurs inthese cases. Therefore, the coeScient B cannot be ob-tained by the third-harmonic method since, in thesecases, the conductivity of the samples is not purely metal-lic due to the stronger hopping, tunneling, and the ir-reversible melting process, and the third-harmonic signalis weaker. In fact, such a phenomenon also exists in thesamples with resistance R o & 10 kQ although theinfluence on that R-I characteristic is relatively weakerand can be omitted (see Fig. 2). To the best of ourknowledge, the electrical breakdown behavior of thin-metal films with resistance Ro & 10 kQ has not been wellstudied for both smooth and roughness films. It is obvi-ous that the phenomenon described in Figs. 4 and 5 can-not be simply explained by the Joule-heating effect' andthe two-dimensional theoretical models. ' '

We proprose that the lower critical exponent m and theanomalous R -I behavior is caused by the rough surface ofthe substrate. Since the substrate has the self-afBne struc-ture, a huge number of defects and fractal structureswould exist in the film. Therefore, many weak links, just

1000

0.00 0.04 o.o8 0.12

200

180

160

140

100

0.00I I

o.20 0.40 0.60 0.80 1.00

48.o

46.0

44 F 0

42.0

40.0

38.0

0.0 0.5 1.5

FIG. 5. R-I characteristics of ultrathin platinum fractal films

due to hopping and tunneling sects. Dots represent the experi-mental results and the solid lines are a guide to the eye.

50 EVIDENCE OF ANOMALOUS HOPPING AND TUNNELING. . . 13 167

54.0

52.050.0

48.o46.044.o

42 ~ 0

40.038.0

0.0 0.5 2.0

FIG. 6. R-I characteristics of the Pt films with Ro & 30 kQ.Dots and plus signs represent the first and the second R-I mea-surement results, respectively; the solid lines are a guide to theeye.

ing and melting processes at the hot spots. This effectwill become more serious in measuring the samples withhigher sheet resistance. Therefore, the higher the sheetresistance, the bigger the value of dR/dI will be (withinour measurement span). When all the weaker MIM junc-tions have been broken down, then the value of dR/dIwill decrease, as shown in Figs. 4 and 5. This result isquite different from that of the flat films, in which thevalue of dR /dI will increase with the current. 's

V. CONCLUSION

In summary, we have measured systematically thedependence of conductance on temperature and the I-Vbehavior on a Pt-film percolation system deposited on thefractal surfaces of a-Alz03 ceramics. The measurementsprovide a wide span of resistance and current: three de-

cades of both Ro and I. Our measurement shows a goodlinearity of the log conductance vs 1/T plot atT =77-300 K and the films have a property of metallicconductivity. However, the resistance-temperaturecoefBcient P is not a constant (P~ T ), which indicatesthat, in this film-thickness interval, the temperaturedependence of the conductance could be affected distinct-ly by the self-aSne structure of the substrate. From theR-I behaviors, three distinct regimes of sheet resistanceare found (Figs. 2, 4, and 5). In diH'erent regimes, the I V-relation has different behaviors. By using a new dc tech-nique, a power law with critical exponent w =0.45+0.06is well defined up to R0=10 kQ; beyond this value, thetunneling and hopping processes would become so seri-ous that their influence on the R-I relation cannot beomitted. Therefore, the current interval, in which dR /dIis a negative value, appears obviously. The above resultscan be interpreted under the assumption that all the Jouleheating, hopping, and tunneling effects would make con-tributions to the R-I characteristics of the films simul-taneously. However, a flexible theoretical model, includ-ing all the aspects of Joule heating, hopping, and tunnel-ing, is still lacking.

The results above show us that the fractal substrateshave various influences on the electrical properties of themetal films. We have found that these influences arestrongly dependent on the film thickness. Furtherresearch on the physical origins of these effects is stillneeded.

ACKNOWLEDGMENTS

The authors would like to thank Professor X.J.Zhang,Dr. M. Q. Tan, X. M. Tao, and H. L. Wang for helpfuldiscussions and technical assistance. The financial sup-port from the Zhejiang Provincial Natural Science Foun-dation of China (Grant No. 193054) is gratefully ac-knowledged.

'Permanent address: Department of Physics, Hangzhou Uni-

versity, Hangzhou 310028, People's Republic of China.J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruyn-

seraede, Phys. Rev. Lett. 70, 57 (1993).~S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).J. Chevrier, V. Le Thanh, R. Buys, and J. Derrien, Europhys.

Lett. 16, 737 (1991).~G. X. Ye, Y. Q. Xu, J. S. Wang, Z. K. Jiao, and Q. R. Zhang,

Phys. Rev. B 49, 3020 (1994).5P. Meakin, Prog. Solid State Chem. 20, 135 (1990).F. Family, Physica (Amsterdam) 168A, 561 (1990).

7G. X. Ye, J. S. Wang, Y. Q. Xu, and Q. R. Zhang, Solid StateCommun. 88, 275 (1993).

sJ. S. Wang, G. X. Ye, Y. Q. Xu, and Q. R. Zhang, Chin. Phys.Lett. 11,43 (1994).

R. Kant, Phys. Rev. Lett. 70, 4094 (1993).R. H. Koch, R. B. Laibowitz, E. I. Alessandrini, and J. M.Viggiano, Phys. Rev. B 32, 6932 (1985).Y. Song, S. I. Lee, and J. R. Gaines, Phys. Rev. B 46, 14

(1992).M. A. Dubson, Y. C. Hui, M. B. Weissman, and J. C. Gar-land, Phys. Rev. B 39, 6807 (1989).R. Rammal, C. Tannous, P. Breton, and A.-M. S. Tremblay,Phys. Rev. Lett. 54, 1718 (1985); R. Rammal, C. Tannous,and A.-M. S.Tremblay, Phys. Rev. A 31, 2662 (1985).Y. Yagil and G. Deutscher, Phys. Rev. B 46, 16 115 (1992).Y. Yagil, G. Deutscher, and D. J. Bergman, Phys. Rev. Lett.69, 1423 (1992).

' M. Nishiura, S. Yoshida, and A. Kinbara, Thin Solid Films15, 133 (1973).C. A. Neugebauer and M. B. Webb, J. Appl. Phys. 33, 74(1962).L. de Arcangelis, S. Redner, and H. J. Herrmann, J. Phys.(Paris) Lett. 46, 585 (1985).C. J. Lobb, P. M. Hui, and D. Stroud, Phys. Rev. B 36, 1956(1987).G. A. Garfunkel and M. B. Weissman, Phys. Rev. Lett. 55,269 (1986).