Weak localization and electron-electron interaction in percolating nickel films

Solid State Communications, Vol. 102, No. 1, pp. 41-46, 1997 0 1997 Ekevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00

Pergamon

PII: SOO38-1098(96)00779-X

WEAK LOCALIZATION AND ELECTRON-ELECTRON

INTERACTION IN PERCOLATING NICKEL FILMS

M. Aprili,* J. Lesueur, L. Dumoulin and P. Nedellec Centre de Spectromitrie Nucliaire et de Spectrometrie de Masse, CNRS-IN2P3 91405 Orsay, France (Received

10 October 1996; accepted 19 December 1996 by J. Jofiin)

We have investigated the scaling effects on the electron interface in inhomogeneous Ni films near the percolation threshold. As a result of the high exchange interaction of the conduction electrons, the conductanceanomaly at low temperature is due to the enhancement of the Coulomb e-e interactions by disorder. However, the magnetoresistance analysis shows that a small contribution of weak electron localization remains, while the enhancement of the e-e interactions is insensitive to the applied magnetic field. The signature of the modified dimensionality introduced by the percolating structure appears at very low temperature (T < 4 K) when the phase coherence scale is of the same order as the percolation length. In this regime, the study of the magnetoresistance renormalization proves that localization and magnetism coexist in itinerant ferromagnetic materials. 0 1997 Elsevier Science Ltd. All rights reserved Keywords: A. magnetic films and multilayers, A. disordered systems, D. electronic transport, D. quantum localization.

disappears and the quantum correction to the transport can be described by the EEI only. However, an original It is well known that because of the weak localization of configuration should appear for a band ferromagnetism the conduction electrons (WEL) and the enhancement of [5]. In this case, the Zeeman splitting induced by the the Coulomb electron-electron interactions (EEI), a applied field is negligible compared to the split of the quantum correction to the Drude conductivity has to be conduction band, so the e-e MR is slight [6]. This poses taken into account in disordered metals [l]. A unified an interesting situation: the quantum corrections to the physical picture of these two phenomena has been conductivity as a function of temperature is dominated proposed by Bergmann in terms of a quantum interference by EEI effects, while the MR should be due only to of partial waves of one (WEL) or two (EEI) interacting the WEL. Because the magnetization value is much electrons making closed paths [2]. The phase coherence larger than the typical inelastic field (M % Wi) we is preserved on scales L r for EEI (L; = Dh&T) and L L expect a small effect from the applied field and no for the WEL (Lz-DTi) [l]. In a 2D system (d Q LT,L,), temperature-dependence [7]. both mechanisms provide the same logarithmic anomaly The presence of a morphological disorder enhances in the temperature dependence [3]. Fortunately the the quantum interferences by modifying the 2D behavior analysis of the magnetoresistance (MR) distinguishes either toward a fractal one (for scales smaller than the the localization contribution from the e-e one [4]. homogeneity scale), or by normalizing the coherence In the presence of a magnetic field, a phase shift effects by a geometrical factor dependent on the metallic results between backward partial waves leading to a coverage (p), at larger sale [8]. The study of the phase coherence breaking and thus a weakening of WEL [3]. coherence scaling in spin polarized materials as a function Then we expect that in magnetic materials the WEL of temperature, provides a sensitive way to investigate whether the small WEL MR, or the p-renormalization of * Present address: Department of Physics, UIUC, Urbana, EEI resistance anomaly dominates. We report here on percolating nickel films near the IL 61801, U.S.A. e-mail: [email protected] 1. INTRODUCTION

41

42

INTERACTION

IN PERCOLATING

percolation threshold and we study the percolation effects on low temperature transport mechanisms. The high spin polarization (P = 11%) of the electrons at the Fermi level [9] exhibits the interesting situation previously described. We find that two different scaling renormalizations corresponding to the 2D and the fractal dimensionality occur. In particular, from the analysis of the morphological disorder role on the magnetoconductance, we are able to show that a small WEL contribution is still present in itinerant ferromagnetic materials. 2. EXPERIMENTAL Percolating Ni films have been prepared by controlled coalescence through heating of homogeneous films, between 500°C and 550°C in ultra-pure He vapor. The homogeneous layers (cu. 6 nm thick) were previously evaporated by electron-gun in UHV (lop9 torr). The electrical resistance variations during heating were monitored together with sample temperature. Approaching the coalescence temperature, the morphological changes induce a fast resistance increase. Then the resistance measurement provides an accurate way to follow the evolution of the inhomogeneous structure. The annealing also allows a reduction of the amount of atomic size defects, leading to relatively large-scale topological disorder only. We have discussed this technique in detail elsewhere [lo]. We present in Fig. 1 a TEM image of a Ni film after annealing. The dark and clear areas correspond to metallic Ni grains and holes, respectively. We deduce the metallic coverage p = 0.56 & 0.02, the percolation length &, - 200 nm and the fractal dimensionality

NICKEL FILMS

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df = 1.8 2 0.1 which are in good agreement with the theoretical prediction for a 2D percolating square lattice df = 1.9 [ll]. Since the metallic volume is conserved during the coagulation process, the average physical thickness of the Ni clusters is about 11 nm, roughly twice the initial thickness. Furthermore, the study of the infinite conducting cluster mass density (mean metallic coverage), shows that for scales L < 30 nm (around the size o of a metallic channel) and L > .$, the system can be considered homogeneous. The change of resistivity at 4.2 K from 15 $-I-cm (virgin sample) to 1023 @cm (after annealing), is consistent with a 2D percolation law involving a reduction of about 50% of the metallic coverage. The magnetic properties of the percolating Ni films have been studied later by mean of the extraordinary Hall effect and has been discussed elsewhere [12]. No differences between a homogeneous and an inhomogeneous sample have been found. We will focus here on the resistance and MR measurements at low temperature (1.5-20 K). To avoid any damage of the weak metallic links, low current measurements (1 = 100 nA) have been carried out using an ordinary a.c.-bridge technique to detect very small signal variations (lo100 nV). As the temperature dependence is around 10’ times larger than the field dependence, the temperature was stabilized below the LHe temperature to within 1 mK by partial He pressure control during the MR measurements. 3. RESULTS AND DISCUSSION 3.1. Conductance Assuming that the diffusion coefficient D is around 5 X 1O-4 m2 s-l as found in Ni [13] and Fe [22] based transition metal alloys, in agreement with a large d-electron density at the Fermi level, the characteristic length L r is larger than the sample thickness for temperatures lower than about 30 K. As L, is larger than Lr, for T < 30 K a 2D behavior is expected. The appropriate parameter to study the WEL transport corrections to the diffusive transport is the square conductance (C, = dip) because it is independent of the disorder amount; we will adopt this representation later. In the limit of a weak spin-orbit (SO) scattering rate, the theoretical expression for &ZO~&) is [14] 2 ACOWEL(T)

Fig. 1. TEM image of an inhomogeneous Ni film near the percolating threshold. The black areas represent metallic clusters on top of the substrate (light areas).

=

a(T),

where CYis a coefficient dependent of the dephasing mechanism, i.e. inelastic scattering. At low temperature the number of excited phonons is negligible and the main inelastic mechanisms in Ni are the e-magnon and the s-d electron-electron interactions for which CY= 2, as

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studied by the analysis of the resistivity temperaturedependence (lS]. The EEI effects coming from a reduced screening factor F, also shows a loga~thmic behavior 1161

e2

GIIEEI(T) = m2(1-

F)ln(T),

(2)

where the factor 2(1- F) is the sum of the orbital [17] and spin [6] terms. Taking into account the WEL and EEI contributions, the conductance variation at low temperature is therefore given by the expression e2 ACn(0 = -[1 2h?r2

+ 2(1 -F)]ln(T)I

-&-$ 0

(3)

where the first term rises from the electron interference and the second one from the ordinary inelastic scattering, Kd being a coupling c~fficient amounting for the strength of the e-Macon or e-e ~tera~tion. We present in Fig. 2 the variation of the square conductance as a function of temperature for a homogeneous film and the percolating one shown in Fig. 1. The mean thickness of both samples is about 11 nm, the resistivities 15 and 1023 pa-cm, corresponding at a square resistance of 17 and 930 a, respectively. At low temperature the logarithmic decrease of AC0 is consistent with the 2D coherence effects described by equations (1) and (2). In contrast, at higher temperature the inelastic scattering destroys the electron interference and the usual T2 behavior is observed. For the

0 -s 1

2

3

456

2

3

456

10 T( Kelvin )

Fig, 2. Temperature-dependence of the square conductance for a homogeneous (H = 0 and H = 4 Tesla) and a percolating film of same thickness. The T = 4 K square resistances are 17 Q and 930 Q respectively. The lines enlighten the logarithmic dependence expected in a 2D regime.

43

homogeneous film the phase coherence is preserved for T < 4 K. When an external magnetic field (H = 4 Tesla) is applied pe~ndicular to the sample surface, no change on AQ(T) behavior appears, confirming that the WEL is mainly reduced by the internal magnetization and the EEI term is only responsible for the temperature dependence of the conductance. From a fit we obtained F = 0.55, in good agreement with the value for the screening factor F = 0.53 estimated using a simple model developed in [16]. We focus now on the percolating film. Although the conductance behaves as the homogeneous one, two important differences can be observed: a reduction of the prefactor in the logarithmic term together with an enhancement of the temperature range where the coherence effects occur. We discuss these differences separately. The slope decreasing might be associated with a geometrical factor taking into account the percolating st~cture. In fact, a possible e~~cement of the screening factor ~orres~nding to such a slope decrease can be ruled out because the F value so obtained is larger than the maximum screening value allowed (F = 1). Assuming that the screening factor F is the same for the homogeneous and percolating film, the slope ratio is 0.56 in agreement with the metallic coverage ratio. Therefore the EEI term seems to scale as p, pointing out that the square conductance has to be normalized by the effective surface. That can be understood because, for T > 4 K, Z,r is shorter or of the same order as the metallic path width w (w - 30 nm), while any signature of the percolating dimensionality can be seen only at the scale L between w and Fp This conclusion is supported also by the analysis of the conductance temperature-dependence that does not show any anomalous behavior ]8] as it should be in a fractal system. The weak AC~(~ deviation at low temperature from the logarithmic law is even consistent with this picture. In fact, at temperatures lower than 4 JS, Lr becomes larger than w and the EEI contribution will be sensitive to the fractal dimensionality. However a fractal behavior will be achieved only at very low temperature (T < 1 K) when L r is of the same order that .$. Moreover the assumption that the screening factor is the same for the homogeneous and the inhomogeneous samples seems reasonable because the disorder scale introduced through the percolating structure is much larger than the Thomas-Fermi screening length. The forward shift of the temperature value corresponding to the maximum conductance can be explain taking into account the reduction of the factor &/R& Using the ~-de~nden~e of the loga~thmic term, we are able to draw out from this shift the p-dependence for the Kd coefficient. We find that Kd must be normalized by the same geometrical factor 0, -P~)-‘.~ found for Ra,

44

INTERACTION IN PERCOLATING NICKEL FILMS

indicating that it follows also a 2D percolating law. Then we can summarize the effects of the percolating structure on the temperature-dependence of the square conductance in the temperature range we have investigated, saying that the normalization by the metallic coverage is different for the EEI term and the ordinary ones. This difference can be described by the expression

Cop(T)= @-Pc)‘~3Go + co(n) +pCOEE,(T),

(4)

where Cw and Co(T) are the conductance due to the elastic and inelastic scattering, respectively and ACUEs, is the coherent contribution due to the screening decrease of interacting electrons.

-2

r

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T=20K

q

-6 z ,x0 -8 -10 -12

3.2. Magnetoconductance In NI films the coherence effects on the MR are usually hard to see because the classical strong magnetic anisotropy due to the orientation of the magnetic domains dominates the low field-dependence [18]. However, increasing the disorder, allows the measure of weak contributions such as the WEL and EEL In fact, with ACU being independent of the disorder as previously indicated, A& is proportional to the square of Rn involving an important enhancement of MO when Rn increases one or two order of magnitude, as in the case of inhomogeneous films near the percolation threshold. Furthermore, the percolating structure changes the 2D geometry of the films providing a renormalization of the WEL and EEI terms. The analysis of the p-dependence on C&Z) will be developed below. The MR at 1.5 and 20 K for the percolating film of Fig. 1 is shown in Fig. 3. At low field, the magnetic domains flipping along the applied magnetic field direction provides the fast decrease of Rn. The saturation field H, = 0.2 Tesla is of the same order as that found in a homogeneous film of the same thickness (H$ - 0.3 Tesla) and any difference could be explained using a percolating demagnetized factor as indicated elsewhere [12]. At high field two different behaviors appear. For high temperatures (T > 20 K) where the quantum interference is negligible, as found by the study of the conductance temperature-dependence, a positive parabolic MR consistent with Kolher’s law [20] is observed. An interesting negative MR instead occurs at low temperature where the coherence effects are important. A negative MR has already been found in homogeneous high disordered thin Ni films [7]. Because the e-e MR term is positive, we rule out any particlehole channel (EEI) contribution [l] indicating that in Ni the particle-particle channel (WEL) only is affected by the external magnetic field. This result agrees with the intuitive idea announced above for which the Zeeman term is negligible in spin polarized electron gas. Consequently the field-dependence of Co is only due to the

0

1

2

3

4

5

6

H (Tesla)

Fig. 3. Resistance as a function of magnetic field for a percolating film. The field is oriented perpendicularly to the sample plan. The temperature-dependence distinguishes the quantum coherent effects from the classical magneto-transport. WEL. Obviously because of the spontaneous magnetization, the internal field will be higher than the applied one. In the case of weak SO scattering rate and for fields lower than Ha = li/4eDZYra the theoretical expression [21] for the AC~wnL(H) = CnwuL(H) -Cow&O) in a 2D system can be approximated by e2 ‘%wEL(H)=

m

ii&)] [( z&J-ln(

x 9 0.5 +

(5) where q is the diagamma function and ro, ri the elastic and inelastic relaxation time, respectively. We choose to study the WEL magnetoconductance by subtracting the classical contribution corresponding to the 20 K curve. We report the net WEL contribution in Fig. 4 for different temperatures. The logarithmic dependence predicted by eqn (5) is well followed in the high field limit. This behavior, characteristic of the quantum coherent effects, can not be dealt with any other classical mechanisms (spin-disorder scattering) [22]. We also note that any quantitative analysis at low fields is impossible because it is too sensitive to the background subtraction. However, two main differences with the theoretical model developed for a homogeneous system, occur: (i) the WEL magnetoconductance is strongly reduced (factor lo* at least) and (ii) a temperature dependence in the high field behavior appears. Two reasons can be

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INTERACTION IN PERCOLATING NICKEL FILMS

45

corresponding to the logarithmic term in Fig. 4 are displayed, as a function of one over the square root of temperature. Because CJv = 0.95 is almost one [24], we expect that KT depends on the temperature in the same way as the inelastic mechanisms (e-e, e-magnon). The experimental behavior in Fig. 4 (inset) shows that indeed KT scales as T2, following the temperature-dependence of inelastic relaxation time in Ni as known by the R(T) analysis. As a result, the normalization of the weak electron localization term in a percolating film is in good agreement with the MC scaling theory of Palevski and Deutscher [23]. 4. CONCLUSIONS 1 -I-I” H ( Tesla )

10

Fig. 4. Effect of the applied magnetic field on the electron interference in the fractal regime (T < 4 K). Note that the classical background has be removed by subtraction of the MR curve at 20 K. (inset) Dependence of the MR prefactor as function of one over the square root of temperature: analysis of the scaling renormalization due to the percolating structure. evoked for explaining the small effect of the applied magnetic field. First, the true magnetic field corresponds not to the external one, but to the magnetic induction. Second, a geometrical factor taking into account the percolating structure as in the R(T) analysis should be also introduced in the expression (5). But in this case, the localization length L, is larger than LT, therefore as for T < 4 K, LT > w, we conclude that L, 9 w (Le - 200 nm for rr - lo- l1 s [22]), then the scaling renormalization will be sensitive to the fractal dimensionality. We show now that the percolating geometry can explain the anomalous MC temperature dependence. In a 2D percolating network, where &, % (L,, w), two separated contributions give rise to the WEL magnetoconductance [23]. One part, on the scale of L,, consists of coherent loops of radius smaller than or of the order of L,; the second part is the rest of the network which does not contain loops at the L, scale. However the latter is negligible once L, P o. Then the expression (5) has to be multiplied by a scaling prefactor R(L,)= Ro

b1’

5” [

which takes into account the maximum size of coherent loops in a percolating system. As L, depends on the temperature, the scaling prefactor involves a temperature dependence in the high field magnetoconductance, as observed. In the inset of Fig. 4 the slopes Kr of the linear fits

The investigation of the quantum coherence effects (WBL and EEI) in Ni percolating films shows that while temperature-dependence of the conductance is dominated by the EEI term, a remnant of WEL magnetoresistance outlined by a disappearance of the EEI contribution, has been observed. We explain the negligible EEI MC, by inhibition of the Zeeman splitting in spin polarized electrons at the Fermi level. As the morphological disorder involves a fractal behavior at a scale larger than the metallic grain size, two different renormalizations due to the percolating structure affect the WEL (magnetoconductance) and the EEI (conductance) corrections at lower (T < 4 K) and higher (T > 4 K) temperatures, respectively. For T > 4 K with LT < w, the logarithmic anomaly in the temperature-dependence of the conductance has to be multiplied by the metallic coverage p, i.e. the effective surface of the sample: the scattering at the surfaces is equivalent to the atomic disorder. In contrast, when T < 4 K (i.e. L, > w), the amplitude of the WEL contribution varies as L, because the number of coherent loops is limited by the sample morphology. However, further investigations at lower temperatures (T C 1 K) should be carried out to explore the cross-over in the EEI term, from the 2D behavior to the fractal dimensionality one. Acknowledgements-We wish to acknowledge fruitful discussions with H. Bemas who has always encouraged this work. REFERENCES 1. Lee, P.A. and Ramakrishnan, T.V., Rev. Mod. Phys., 57, 1985, 287. 2. Bergmann, G., Phys. Rev., B35, 1987, 4205. 3. Bergmann, G., Phys. Rev., 107, 1984, 1. 4. Van Haesendonck, C. et al., Phys. Rev., B25,1982, 5090.

5. Connoly, J.W.D., Phys. Rev., 159, 1967, 415. 6. Lee, P.A. and Ramakrishnan, T.V., Phys. Rev., B26,1982,4009.

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10. 11. 12. 13. 14. 15. 16. 17.

INTERACTION IN PERCOLATING NICKEL FILMS Komori, F. et al., J. Msg. Msg. Mat., 35,1983,74. Gefen, Y. et al., Phys. Rev., B28, 1983, 6677. Tedrow, P.M. and Meservey, R., Phys. Rev., B7, 1973,318. Aprili, M., Thesis University of Paris Sud, 1994. Gefen, Y. et at., Phys. Rev. L&t., 47, 1981, 1771. Aprili, M. et al., ~~~Sta~ Coax., 9&1996,221. Raffy, H. et al., Phys. Rev., B36,1987,2158. Bergmann, G., Phys. Rev., BZ8,1983,515. CampbeII, LA and Fert, A, ~e~~~~~ate~~ (Edited by E.P. Worthy. Non-Homed, 1982. Febr, Y. et al., Phys. Rev., B33, 1986,663l. Fukuyama, H., J. Phys. Sot. Jpn., 50,1981,3407; 51,1982,1105.

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18. Wedler, G. and Schneck, H., Thin Solid Films, 47, 1977,137. 19. Bergmann, G. and Marquardt, P., Phys. Rev., Bl8, 1978,326. 20. Schwerer, F.C. and Silcox, .I., J. Appl. Phys., 39, 1968,2047. 21. Hikami, S. et al., Frog. Theor. Phys., 63,1980,707. 22. Trudeau, M.L. and Cocbrane, R.W., Phys. Rev., B38,1988,5353. 23. PaIevski, A. and Deutscher, G., Phys. Rev., B34, 1986,431. 24. Palevski, A. and Deutscher, G., J. Phys., Al7,1984, L895.