Anomalous electrical transport behavior in

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NANOTECHNOLOGY

Nanotechnology 19 (2008) 185711 (5pp)

doi:10.1088/0957-4484/19/18/185711

Anomalous electrical transport behavior in nanocrystalline nickel G S Okram, Ajay Soni and R Rawat UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452017, MP, India

Received 11 February 2008 Published 2 April 2008 Online at stacks.iop.org/Nano/19/185711 Abstract We have prepared nanocrystalline Ni (n-Ni) samples of grain sizes 40–100 nm using a polyol method and investigated the electrical transport on their compacted pellets in the temperature range 3–300 K. The resistivity, ρ , decreases nearly linearly with increase in compaction pressure but without a change in its slope, dρ/dT . ρ is anomalously large, and is strongly temperature and grain-size dependent. The resistivity at room temperature, ρ300 K , is in the range ∼40–759 μ cm but with a positive coefficient of resistivity α (metallic). This is associated with the significantly enhanced dρ/dT with increase in residual resistivity ρ0 . These characteristics are attributed to the disorder in the grain boundaries that represents effectively a series resistor network.

prepared powders of average grain sizes D = 40–100 nm have been compacted into pellets of ∼8 mm diameter and 0.3 mm thick at various pressures (0.5–2 GPa). The pellets are then cut into rectangular shapes, loaded to the sample holder and put into evacuation for resistivity measurements. This is done as soon as possible to avoid excessive oxidation from the atmosphere. The mass density of the compacted n-Ni pellets is 85% of the bulk value, representing n-Ni bulk samples [11], and the density values for all the samples are approximately the same within the error (±2%). For references, arc-melted 99.99% bulk Ni, and one followed by annealing at 900 ◦ C for 24 h are used. These compacted n-Ni samples have exceptionally large resistivities and anomalously high temperature dependences of their resistivities.

1. Introduction The manner in which mobile electrons move freely throughout the material determines the electrical conduction of a metal. In good (crystalline) metals such as Ni, Fe or Co, according to the conventional Boltzmann transport theory that relies on weak scattering, kFl 1 even at temperatures T Tc (the Curie temperature), where kF is the Fermi wavevector and l is the mean free path. In real life, disorder in any material is omnipresent in varying degree. This ranges from a few impurities or interstitials in an otherwise perfect crystal to the strongly disordered limit of alloys or glassy materials. The disorder can also be introduced purposefully. This leads to significant changes in the electrical transport properties and hence to the discoveries of several classes of materials such as the so-called bad metals including (high Tc ) superconductors [1–3] and itinerant ferromagnets [4]. Bad metals exhibit anomalously high resistivity, ρ , implying that l is shorter than the physically reasonable length scales. When l is shorter than the interatomic spacing or kF−1 , the Boltzmann theory of transport is not self-consistent [5]. This self-consistency boundary is drawn from different criteria such as the Ioffe–Regel limit kFl > 1 [6] or the Mooij–Tsuei limit ρ ∼ 30–400 μ cm [7, 8]. In the latter limit, near room temperature, the sign and size of α(ρ) = ρ −1 dρ/dT tend to correlate with the magnitude of ρ in many disordered metallic systems. Good metals deviate from a metallic temperature dependence of the resistivity at or near this boundary [9, 10]. We have studied nanocrystals of a good metal, nickel (n-Ni), which can be prepared to mimic a bad metal. The 0957-4484/08/185711+05$30.00

2. Experimental details The n-Ni powder was prepared tuning it systematically with the variation of nickel acetate (Ni(ac)2 ·4H2 O) to ethylene glycol (CH2 OHCH2 OH) mole ratio. The light green solution of nickel acetate in ethylene glycol at pH ∼ 6.4 was refluxed for 10 h at ∼185 ◦ C. Once the solution becomes supersaturated with Ni ions, a black precipitate is formed by homogeneous nucleation. At the end of the reaction, the solution becomes light brown. On excess dilution with acetone, precipitates are settled at the bottom of beaker; these are further washed with acetone several times with intermediate sonication and centrifugation. The precipitate was vacuum dried at 60 ◦ C in 1 h. The samples thus prepared for the mole ratios 0.003:1, 1

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Nanotechnology 19 (2008) 185711

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ion beam. The C 1s line from either adventitious carbon or intentionally added graphite powder on the surface has been widely used for charge referencing [13, 14]. For this study, the adventitious carbon was used as a reference and the binding energy (BE) of the reference C 1s line was set as 284.6 eV. For each sample, a calibration factor was calculated from the difference between the measured C 1s BE and the reference value 284.6 eV [15]. The original BE data were corrected according to the calibration factor.

3. Results and discussion The resistivity at 300 K, ρ300 K , of the 100 nm n-Ni samples (figure 1) turns out to be ∼161, 117, 79 and 43 μ cm respectively for the compaction pressure 0.49, 0.98, 1.47 and 1.96 GPa; they have mass density of about 59, 68, 76 and 85% of the bulk value (8.46 g cm−3 ), respectively. The ρ(T ) feature down to 3 K remains the same. This means that by increasing the compaction pressure, only the magnitude of resistivity decreases without affecting the temperature dependence. The pellet with applied pressure 1.96 GPa showed drastically reduced porosity (figure 2(a)) and good connectivity among the nanograins compared to the pellet with 0.49 GPa pressure (figure 2(b)). Their morphology is seen to be significantly different from those of arc-melted 99.99% bulk Ni ingot having density 8.02 g cm−3 (figure 2(c)) and a similar Ni ingot, sintered at 900 ◦ C for 24 h having density 8.46 g cm−3 (figure 2(d)). Granularity is not clear in the simply arc-melted ingot. Here, there is not enough time to form grains while quenching and hence more disorder in the sample. Very clear granularity of grain sizes ∼75 μm is seen in the annealed ingot (figure 2(d)). The compaction pressure for the results presented in this study was therefore chosen as 1.96 GPa.

Figure 1. Temperature dependence of resistivity ρ of 100 nm n-Ni sample at various pressures as indicated. Inset: ρ at 300 K against compacted pressure. The line is a guide to the eyes.

0.004:1, 0.006:1 and 0.008:1 are well characterized and are of single-phase good quality n-Ni [12]. The grain size determined from the atomic force microscopy data is D ∼ 40, 50, 70 and 100 nm respectively; from now onward, the samples will be denoted by their D values. No further annealing was done, to avoid additional grain agglomeration. The standard four-probe resistivity measurements were performed in the temperature range 3–300 K; the four electrodes were collinear and the estimated error in the measurements is less than 3%. Surface chemistry of the samples was studied using VSW (UK) x-ray photoelectron spectroscopy (XPS) at a vacuum of 10−9 Torr. A monochromatic Al Kα x-ray source (hν = 1487 eV) was used for the analysis. The compacted n-Ni was etched for 30 min using a 10 μA and 500 eV argon

a

b

c

d

Figure 2. The SEM image of (a) n-Ni pellet with 1.96 GPa pressure, (b) n-Ni pellet with 0.49 GPa pressure, and (c) arc-melted Ni ingot and (d) ingot (c) annealed at 900 ◦ C for 24 h showing a tri-granular junction.

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Figure 4. Ni 2p core level x-ray photoelectron spectrum of argon-etched compacted nickel nanoparticles for the 50 nm sample as a representative spectrum.

Figure 3. Temperature dependence of the resistivity, ρ , of various samples as indicated. Inset: ρ at 5 K (ρ5 K ) and 300 K (ρ300 K ) against D . The lines are guides to the eyes.

The plot of resistivity against temperature of the 40, 50, 70 and 100 nm samples along with unsintered bulk Ni is shown in figure 3. It is clear that ρ300 K increases with the decrease in D . For the unsintered arc-melted bulk Ni ingot ρ300 K is ∼27 μ cm, showing disorder. On sintering (ordering), this comes down to ∼7.2 μ cm. Indeed, the comparable value of the resistivity ρ300 K in 100 nm n-Ni (∼43 μ cm) and unsintered bulk Ni (∼27 μ cm) samples suggests that the observed resistivity is an intrinsic property of these n-Ni samples. The value of ρ300 K is ∼40–759 μ cm for the n-Ni samples and it decreases logarithmically as D increases (figure 3, inset). The particle size distribution effect on the resistivity has been accounted for from this plot. By extrapolation, for example for the 40 ± 3 nm sample, ρ300 K varies from 630 μ cm (for 37 nm) to 950 μ cm (for 43 nm) through its measured value 759 μ cm (for 40 nm). This smallest value, namely 630 μ cm, does not reach the maximum of 530 μ cm (for 45 nm) for the 50 ± 5 nm sample, and so on. Therefore, the observed high resistivity is believed to be an intrinsic property of this nanocrystalline material. Similarly, the resistivity at 5 K (ρ5 K ) also decreases logarithmically as D increases (figure 3, inset). This trend is similar to those of Ag thin films [16], but rather distinct compared with those of Au, Al and Cu thin films wherein the resistivity increase is comparatively smaller [17]. The increase in ρ with reduction in D is therefore worth consideration. It is not due to the presence of NiC and NiO impurities either incorporated into the Ni lattice or from thin impurity shells (e.g., some amorphous carbon or oxide layer) on the surface of the n-Ni particles, which may escape the ordinary XRD study [12]. If the large ρ is due to C in the Ni bulk, it should have been about 17 μ cm for 3.4 μ cm/at.% C [18, 19], at ∼5 at.% C found in the 40 nm sample, for example. Also, if a NiO separation layer were to exist there, ρ300 K ∼ 10−3 –102 cm is expected and the resistivity should exhibit semiconducting behavior [20, 21]. However, the resistivity is metallic for all the samples. This is supported by our XPS data for the samples collected for the Ni 2p core level (figure 4), which indicates the presence of the metallic Ni (2p3/2 peak at ∼852 eV), not its carbide or

oxide peaks. Thus, XPS and XRD [12] data therefore clearly show the absence of carbide or oxide of nickel in this n-Ni. These findings intuitively provide convincing evidence for the observed metallic resistivity behavior for all the samples. The low value of ρ300 K ∼ 14.4 μ cm of electrodeposited nNi of grain sizes ∼50 nm [22] is apparently consistent with its probably very high density of ∼99.4% of the theoretical density; the high density indicates no vacancy-cluster-like volumes or missing crystallite pores [23]. This means that compacted n-Ni samples have a much higher number of grain boundary defects and hence less density compared with thin films, in particular the electrodeposited ones. This is expected because, in electrodeposited thin films, an external electric potential is applied in addition to its homogeneous nucleation potential in a polyol method of sample preparation. The ρ300 K value can be considered to be of the order of 27 μ cm, the value we obtain for arc-melted unsintered bulk Ni. It therefore can be grossly ascribed to the Ni samples of varying degree. This, however, is not so in compacted n-Ni samples, in which a coalescence regime may prevail. This in turn is very sensitive to the details of the growth conditions of the material [16]. The value of α calculated near 300 K suddenly drops (figure 5(a)) from 10.36 × 10−3 K−1 (bulk Ni) to 3.25 × 10−3 K−1 , beyond which it remains almost constant with the decrease in D . Therefore, the decreasing trend of α does not appear to cross over to a negative value with further decrease in D , maintaining its strong metallic behavior (positive α value) even with very high resistance value. The slope of resistivity dρ/dT versus residual resistivity ρ0 (= ρ5 K ) shows nearly linear behavior (figure 5(b)), which is significant relative to the bulk, i.e., ρ0 is directly proportional to dρ/dT . In fact, dρ/dT would have been constant (Mathiessen’s rule), if it were due to the residual resistivity of impurities as in alloys [24]. Both ρ0 and the temperature dependence ρ(T ) as suggested by an approximately linear dρ/dT are significantly enhanced relative to the bulk behavior. This proportionality indicates that there is a common source for the renormalization that gives rise to the anomalously large resistivity value. This discrepancy arises as the current has to flow through the grain boundaries, which are overall disordered compared to the bulk. 3


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The volume fraction of nanograin boundary is appreciably large compared to the total volume. While the structure of the grain boundaries is controversial [27, 28], they are largely disordered compared to the bulk of the sample [11, 12, 29, 30]. The n-Ni exhibits lattice expansion with grain size reduction [12]. The cause for this is explained quantitatively [12] based on the linear elasticity model [31]. It is found that the grain boundary width in these n-Ni samples is ∼0.5 nm and is the contributory factor to the lattice expansion [12]. This is associated with the estimated lattice strain of 0.18–0.25%. These grain boundaries are therefore disordered and make up a disordered shell in each nanograin [11, 12, 29, 30]. When such disordered shelled grains are compacted, they form a bulk three-dimensional array of a disordered shelled granular pellet (cf figure 2). The pellet therefore constitutes a disordered system representing effectively a series resistor network, which may explain the large resistivity observed. The mean free path obtained using a relaxation approach increases with increase in grain size. The 50, 70 and 100 nm samples follow the Ioffe–Regel limit, kFl > 1 [6]. This limit, however, breaks down for 40 nm sample with kFl ∼ 0.91 (ρ300 K ∼ 759 μ cm). Thus, a compacted n-Ni pellet perhaps mimics a bad metal of the type similar to Ag thin film [16]. Indeed, for such disordered n-Ni, it may not be sufficient to define a mean free path l for the whole polycrystal and a relaxation time approach may not describe grain boundary scattering accurately [32, 33].

4. Conclusions In summary, we have investigated the electrical transport of compacted 40–100 nm nanocrystalline nickel (n-Ni) samples prepared by a polyol method. A systematic decrease in the resistivity ρ with compaction pressure is found; its slope dρ/dT , however, remains the same. The resistivity at room temperature increases from ∼40 to 759 μ cm as the grain size decreases. This is associated with the significantly enhanced dρ/dT with increase in residual resistivity ρ0 . The highly resistive nature of the compacted n-Ni pellets is attributed to the disorder in the grain boundaries, which effectively represent a series resistor network.

Figure 5. (a) Plot of α against ρ300 K . The connecting line is a guide to the eyes. (b) Plot of dρ/dT versus ρ0 . The solid straight line is the linear fit to the data points, which are identified by the respective D values. (c) ρ0 and dρ/dT versus D . The solid curves are exponential fits.

The proportional enhancement of ρ0 and dρ/dT is believed to be a signature of distorted conduction paths and extremely thin links enhancing the resistivity [16]. Further, ρ0 , and dρ/dT versus D (figure 5(c)) display first-order exponential decay with increase in D . These trends therefore clearly show the dominant role of grain boundaries in enhancing the overall ρ value with reduction in D . Remarkably, the influences of these grain boundaries do not, however, lead to electron localization as the positive values of α suggest. The value of ρ300 K ∼ 759 μ cm for the 40 nm sample is large, well beyond the Mooij–Tsuei limit (ρ ∼ 30– 400 μ cm) [7, 8]. However, the ρ300 K values of our other n-Ni samples are within this limit. In the case of ordinary disordered metals (such as (Ag0.5 Cu0.5 )100−x Mgx ), α changes sign in this limit [25]. The values of α ∼ 2–10 × 10−3 K−1 in these n-Ni samples are significantly larger than those (−4 to +4 × 10−4 K−1 ) of the usual bad metals [8] but comparable to those of rhodium thin films [26].

Acknowledgments The authors acknowledge with thanks the encouragements of Dr P Chaddah and Professor A Gupta, Dr T Shripathi for the XPS data, and Dr D M Phase for the SEM data.

References [1] Emery V J and Kivelson S A 1995 Phys. Rev. Lett. 74 3253 [2] Hebard A F, Palstra T T M, Haddon R C and Fleming R M 1993 Phys. Rev. B 48 R9945 [3] Gurvitch M and Fiory A T 1987 Novel Superconductivity: Proc. Int. Workshop on Novel Mech. Superconductivity ed S A Wolf and V Z Krezin (Berkeley, CA: Plenum) p 663

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[20] Chen H L, Lu Y-M and Hwang W S 2005 Surf. Coat. Technol. 198 138 [21] Sasi B and Gopchandran K G 2007 Nanotechnology 18 115613 [22] Bakonyi I, Toth-Kadar E, Pogany L, Cziraki A, Gerocs I, Varga-Josepovits K, Arnold B and Wetzig K 1996 Surf. Coat. Technol. 78 124 [23] Lu L, Sui M L and Lu K 2000 Science 287 1463 [24] Gerritsen A N and Linde J O 1952 Physica 18 877 [25] Mizutani U and Yoshino K J 1984 J. Phys. F: Met. Phys. 14 1179 [26] Koshy J 1980 J. Phys. D: Appl. Phys. 13 1339 [27] Thomas G J, Siegel R W and Eastman J A 1990 Scr. Metall. Mater. 24 201 [28] Stern E A, Siegel R W, Newville M, Sanders P and Haskel D 1995 Phys. Rev. Lett. 75 3874 and references therein [29] Haubold T, Birringer R, Lengeler B and Gleiter H 1989 Phys. Lett. A 135 461 [30] Beloborodov I S, Lopatin A V, Vinokur V M and Efetov K B 2007 Rev. Mod. Phys. 79 469 [31] Qin W, Chen Z H, Huang P Y and Zhuang Y H 1999 J. Alloys Compounds 292 230 [32] Landauer R 1957 IBM J. Res. Dev. 1 223 [33] Vancea J, Riess G and Hoffmann H 1987 Phys. Rev. B 35 6435

[4] Klein L, Dodge J S, Ahn C H, Reiner J W, Mieville L, Geballe T H, Beasley M R and Kapitulnik A 1996 J. Phys.: Condens. Matter 8 10111 [5] Allen P B, Beaulac T P, Khan F S, Butler W H, Pinski F J and Swihart J C 1986 Phys. Rev. B 34 4331 [6] Ioffe A F and Regel A R 1960 Prog. Semicond. 4 237 [7] Mooij J H 1973 Phys. Status Solidi a 17 521 [8] Tsuei C C 1986 Phys. Rev. Lett. 57 1943 [9] Woodard D W and Cody G D 1964 Phys. Rev. 136 A166 [10] Marchenko V A 1973 Sov. Phys.—Solid State 15 1261 [11] Tschope A and Birringer R 1993 Acta Metall. Mater. 41 2791 [12] Okram G S, Devi K N, Sanatombi H, Soni A, Ganesan V and Phase D M 2008 J. Nanosci. Nanotechnol. at press [13] Moulder J F, Sticker W F, Sobol P E and Bomben K D 1992 Hand Book of X-ray Photoelectron Spectroscopy (Eden Prairie, MN: Perkin Elmer) [14] Stec W J, Morgan W E, Albridge R G and Van Wazer J R 1972 Inorg. Chem. 11 219 [15] Barr T L and Seal S 1995 J. Vac. Sci. Technol. A 13 1239 [16] Arnason S B, Herschfield S P and Hebard A F 1998 Phys. Rev. Lett. 81 3936 [17] Camacho J M and Oliva A I 2005 Microelectron. J. 36 555 [18] Cadeville M C, Lerner C and Friedt J M 1977 Physica B 86–88 432 [19] Schwerer F C 1969 J. Appl. Phys. 40 2705

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REVIEW OF SCIENTIFIC INSTRUMENTS 79, 125103 共2008兲

Resistivity and thermopower measurement setups in the temperature range of 5–325 K Ajay Soni and G. S. Okram UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road Indore, Madhya Pradesh 452017, India

共Received 6 August 2008; accepted 24 November 2008; published online 17 December 2008兲 Automated precision measurement setups for electrical resistivity of eight metallic samples simultaneously and thermoelectric power of different types of samples in the temperature range of 5–325 K have been developed. The details of the setups and their capabilities have been described. Usually each run takes nearly 5 h and typical error is within 3% and 4%, respectively. The results of high purity Nb and Pt samples are used as examples to demonstrate this. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3048545兴

I. INTRODUCTION

Resistivity 共␳兲 and thermoelectric power 共TEP or S兲 at low temperatures are two fundamental properties used for studying various materials including metals.1 These aspects require reliable and accurate measurements with reasonable speed. However, they are generally complex and need expertise. Toward these goals, there are several reports2–8 in literature on measurements of S wherein attaining the desired thermal stability to obtain an accurate datum at a set temperature usually requires about 5–15 min. This amounts to somewhat 24–74 h duration to record data points for 5–300 K at 1 K intervals. The main reasons for this long stability time for temperature and experimental data collection stem from the basic cryostat design and the accompanying control or detection system. For a usual earlier cryostat or insert 共e.g., Refs. 2 and 3兲, the vacuum that can be achieved is generally rather poor. It is because the vacuum chamber is quite often constituted with, say, 4.6 cm internal diameter 共ID兲 ⫻ 18.5 cm length 共L兲 ⫻ 0.2 cm thick 共T兲 brass can, fitted at the bottom of a brass flange without a set of nuts and bolts but with the help of just its vacuum inside; evacuation is done through a long and narrow stainless steel 共SS兲 tube of, say, 1.0 cm ID ⫻ 118 cm L ⫻ 0.07 cm T fitted on the top of the flange.2,3 The main disadvantage for such cryostat is that the actual vacuum that can be attained at the sample housing region is quite low, sometimes as low as ⬃10−3 mbar. Due to this, good temperature control is usually difficult. In addition, because of the fitting of the vacuum can with its vacuum, associated with its significant own weight, the system quite often faces problem of vacuum maintainability at temperatures below, say, 60 K or so. Once the cryostat part is settled as is done here, the next important aspect particularly for the thermopower measurements is the sample mounting arrangement for good thermal and electrical conduction or isolation to enable reliable data collection. In this direction, spring loaded system is one of the best and most popular methods.2–4,8–10 Following this method, fabrication of an efficient and thermally easily con0034-6748/2008/79共12兲/125103/4/$23.00

trollable cryostat supported by an automatic electronic control and detection unit using high precision equipments 共all Keithley or Lakeshore make兲 and a personal computer with programs written in Visual Basic is described here. The system is capable of recording the resistivity of eight metallic samples simultaneously and separately the TEP of one sample of various types in the temperature range of 5–325 K. Remarkably, it takes nearly 5 h for one run of either resistivity data set of eight samples or TEP of one sample, for either ramping or control at the usual rate of 1 K/min for 5–300 K range. This achievement is quite commendable as this leads to, among others, saving of liquid helium and electricity. Such rapid and efficient throughput is desirable due to the large number of samples requiring to be investigated. II. DESCRIPTION OF MEASUREMENT SETUPS A. General-purpose cryostat

The heart in the present setup is the fabricated generalpurpose vacuum insert 共cryostat兲 that can be fitted inside a commercially available 60 l liquid helium or nitrogen Dewar. The cryostat is essentially fabricated using an SS tube 共93 cm L ⫻ 5 cm ID⫻ 0.05 cm T兲 brazed with 共10 cm L ⫻ 5.1 cm ID⫻ 0.2 cm T兲 oxygen-free highly conducting copper 共OFHC兲 solid rod made as cup at the bottom, a ball valve, and a vacuum port near the neck with the top-end fitted with an appropriate sample holder 关Fig. 1共a兲兴. The cryostat with a vacuum maintained inside better than 10−6 mbar houses the sample holder assembly, either resistivity or thermopower, up to its neck. The advantage for the ball valve and a vacuum port near the neck assembly is that they allow us to load and unload the sample or its holder with ease, yet fast, without disturbing the vacuum inside the cryostat. B. Thermopower measurement sample holder

The thermopower 共S兲 measurement system is configured for differential dc method with OFHC as reference.1,2 The type of samples is a small size ceramic, metallic, or semiconductor material in the form of single crystal, ribbon, wire,

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Rev. Sci. Instrum. 79, 125103 共2008兲

FIG. 1. 共Color online兲共a兲 Schematic diagram of the general-purpose vacuum insert. 共b兲 Photograph of the bottom portion of sample holder and the radiation shield for TEP measurements. 共c兲 Schematic diagram of thermoelectric power measurement setup connected to computer-interfaced system.

or small bar. Typically, a pellet of about 5 mm diameter having 0.5–2 mm thickness is ideal as is usually suitable for scientific investigations. The sample with freshly clean surfaces is secured between two cylindrical OFHC blocks using an SS spring and an axial brass screw 关Figs. 1共b兲 and 1共c兲兴. This arrangement makes the sample mounting very convenient, which is done after cleaning the top 共bottom兲 of lower 共upper兲 OFHC block surface using a Scotch Brite scrubber, followed by a fine polish paper. The two junctions of the chromel-Au–Fe共0.07%兲 thermocouple are anchored thermally to the two OFHC blocks to which electrical leads are attached that enable good electrical contacts for the thermally generated potential difference 共⌬V兲 measurements. A temperature controller is used to maintain the temperature difference 共⌬T兲 between the two OFHC blocks using the chromel-Au–Fe共0.07%兲 thermocouple and a heater wire wound on the lower OFHC block 关Fig. 1共c兲兴. The ⌬T can be set to a desired value, say, 2 K. A silicon diode and another heater anchored to the upper OFHC block are connected to another temperature controller to sustain a desired controlled sample temperature 共⫾10 mK兲. The nanovoltmeter coupled through a nanovolt scanner card measures the ⌬V and the ⌬T via the chromelAu–Fe共0.07%兲 thermocouple. The sample holder is attached to a heat sink via three SS rods and a hylum disk 关Fig. 2共b兲兴. A removable cylindrical OFHC jacket fits at the bottom of the brass flange as a radiation shield. The sample holder with wiring assembly is loaded in the cryostat. The TEP data Smeas 共=⌬V / ⌬T兲 are recorded through a personal computer for a desired temperature step and range. The Seebeck coefficient of the copper reference 共SCu兲 is subtracted from the measured Smeas such that the sample S is given1,2 by S = Smeas − SCu. The reliability of the SCu data used for subtraction from the measured Smeas has been satisfactorily proven as the measured

data on the standard samples match the published data as shown below. Indeed, taking into account of all the component detection system times especially ⬃30 ms for the lowest voltage noise region of the nanovoltmeter, the possible recording speed is well within 100 ms/data point. However, to obtain the better and stable data, each data point is recorded within 8 s measuring ⌬V and ⌬T thrice and taking their averages.

FIG. 2. Schematic diagram of the four-probe resistivity setup of eight samples data recording simultaneously connected to computer-interfaced system.


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FIG. 3. 共Color online兲 Thermopower data 共full circle兲 of a 99.999% pure platinum wire and such data 共open star兲 of Huebner et al. 共Ref. 13兲 for comparison. Inset: 500 thermopower data points collected at an interval of 10 s at 300 K under a vacuum of 10−6 mbar. The variation in T is ⫾10 mK and in S is ⫾0.22 ␮V / K at 300 K.

C. Resistivity sample holder

Measured in four-probe configurations of different types,11 the schematic diagram for the computer-interfaced resistivity 共␳兲 measurement setup is shown in Fig. 2. The sample holder for resistivity measurements is parallelepiped OFHC block 12⫻ 12⫻ 54 mm3 attached to a brass flange to which the SS feedthrough tube for various electrical wires is connected on the upper side. A removable cylindrical OFHC jacket fits at the bottom of the brass flange as a radiation shield. On the two opposite faces of the copper block, a cigarette paper is fixed on each face using GE 7031 varnish to electrically isolate the samples from the copper block sample holder. Printed circuit board strips are fixed for electrical terminals on the other remaining opposite faces. From these terminals, four uninsulated copper wires 共two each for current and voltage leads兲 for each sample are taken for electrical contacts, which are done using a highly conducting silver paste. Proper grounding and electrical shielding are made wherever required. A calibrated silicon diode, used for temperature 共T兲 measurement 共⫾10 mK兲, and a cartridge heater anchored to the sample holder are connected to a temperature controller. At a selected temperature, a desired current 共forward and reverse兲 is passed through the outer two terminals of a sample from a subfemtoampere sourcemeter, and the resulting voltages 共forward and reverse兲 in the inner two terminals are recorded by a sensitive nanovoltmeter. A current switching card 共having 1 pA offset current兲 and a nanovolt scanner card 共with ⬍30 nV contact potential兲 both coupled to a switching system enable the data acquisition. The measurement proceeds for the next sample until the record for all of the samples is over. Then, the next temperature is set and repeats the above scan steps similarly for the desired range of temperatures. Using dimensions of the sample, the resistivity ␳ is calculated. The typical error estimated in the measurement, especially due to the sample 共pellet兲 dimensional factor, is about 3%. This value, a little higher for smaller sample sizes, matches that reported by Okram et al.12 but is better than earlier report of 5% error or higher.8

FIG. 4. Resistivity data of a 99.999% niobium wire. Inset: d␳ / dT vs T curve showing a peak at Tc = 9.4 K 共left兲 and superconducting transition region of the resistivity 共right兲.

III. RESULTS AND SUMMARY

As representative results, the TEP data of a 160 ␮m diameter 99.999% pure platinum wire are shown in Fig. 3 along with the literature data.13 Interestingly, despite our fast rate of data collection, the data agree well with those reported in literature.6,10,13,14 The thermopower data that were obtained in our Pt sample match the published data on Pt to within ⬃0.5 ␮V / K all throughout the measured temperature range.13 This is strikingly close, showing good reproducibility. The inset shows the variation in data points at 300 K with time, which indicates concentration of the data points near this temperature. These data points provide the deviation in T and S within ⫾10 mK and ⫾0.22 ␮V / K, respectively. They exemplify the stable data, and hence the capability of the present system. From these data 共a deviation of ⫾0.22 ␮V / K from, say, ⬃5.85 ␮V / K at 300 K兲, an error of typically less than 4% 共3.76% at actual兲 is estimated. This is better than the earlier reports of ⬃5% error.7,15 Similarly, Fig. 4 shows the resistivity of a 99.999% niobium wire that exhibits a superconducting transition temperature of Tc ⬃ 9.4 K 共inset兲. These data match quite well the published data.16 It may be noted from the inset that the data points are recorded at an interval of 100 mK, displaying very abrupt transition to superconductivity 共Fig. 4, bottom inset兲. This has been further confirmed from the derivative of resistivity curve 共Fig. 4, top inset兲 showing quite symmetric peak corresponding to the Tc 共⬃9.4 K兲. Remarkably, the data have been collected several times, particularly near the transition region, and found that they are highly reproducible. Similarly, we have collected such reproducible data for several other samples. These data convincingly indicate the reliability of these setups. Thus, the present experimental setups can measure four-probe resistivity of eight metallic samples simultaneously and thermopower of one sample of various types in the form of single crystal, ribbon, wire, or small bar of ceramic, metallic, or semiconductor material in about 5 h, each with precision. ACKNOWLEDGMENTS

The authors acknowledge with thanks Dr. P. Chaddah, Professor V. N. Bhoraskar, Dr. B. A. Dassannacharya, Pro-


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fessor A. Gupta, and Dr. V. Ganesan for their support and encouragement, and the staff of workshop and cryogenics for their assistance. 1



F. J. Blatt, P. A. Schroeder, C. L. Foiles, and D. Greig, Thermoelectric Power of Metals 共Plenum, New York, 1976兲; R. D. Barnard, Thermoelectricity in Metals and Alloys 共Taylor & Francis, London, 1972兲; J. M. Ziman, Electrons and Phonons 共Oxford University Press, New York, 1960兲. 2 G. S. Okram, M. Muralidhar, M. Jirsa, and M. Murakami, Physica C 402, 94 共2004兲. 3 M. K. Marhas, K. Balakrishnan, V. Ganesan, and R. Srinivasan, Rev. Sci. Instrum. 67, 2867 共1996兲. 4 K. M. Sivakumar, R. K. Singh, N. K. Gaur, and V. Ganesan, Proceedings of the 21st International Conference on Thermoelectrics, 2002 共unpublished兲, p. 337. 5 G. R. Caskey, D. J. Sellmyer, and L. G. Rubin, Rev. Sci. Instrum. 40, 1280 共1969兲. 6 A. L. Pope, R. T. Littleton IV, and T. M. Tritt, Rev. Sci. Instrum. 72, 3129

共2001兲. O. Boffoue, A. Jacquot, A. Dauscher, B. Lenoir, and M. Stolzer, Rev. Sci. Instrum. 76, 053907 共2005兲. 8 V. Ponnambalam, S. Lindsey, N. S. Hickman, and T. M. Tritt, Rev. Sci. Instrum. 77, 073904 共2006兲. 9 A. T. Burkov, A. Henrich, P. P. Konstantinov, T. Nakama, and K. Yagasaki, Meas. Sci. Technol. 12, 264 共2001兲. 10 L. S. S. Chandra, A. Lakhani, D. Jain, S. Pandya, P. N. Vishwakarma, M. Gangrade, and V. Ganesan, Rev. Sci. Instrum. 79, 103907 共2008兲. 11 L. J. van der Pauw, Philips Res. Rep. 13, 1 共1958兲; F. M. Smits, Bell Syst. Tech. J. 37, 711 共1958兲; H. C. Montgomery, J. Appl. Phys. 42, 2971 共1971兲. 12 G. S. Okram, A. Soni, and R. Rawat, Nanotechnology 19, 185711 共2008兲. 13 R. P. Huebener, Phys. Rev. 140, A1834 共1965兲. 14 J. P. Moore and R. S. Graves, J. Appl. Phys. 44, 1174 共1973兲. 15 N. E. Cusack and P. W. Kendall, Proc. Phys. Soc. London 72, 898 共1958兲. 16 N. Morton, B. W. James, G. H. Wostenholm, and R. J. Nichols, J. Phys. F: Met. Phys. 5, 85 共1975兲; C. Kittel, Solid State Physics, 7th ed. 共Wiley, New York, 1996兲, p. 336. 7


APPLIED PHYSICS LETTERS 95, 013101 共2009兲

Size-dependent thermopower in nanocrystalline nickel Ajay Soni and Gunadhor S. Okrama兲 UGC-DAE Consortium for Scientific Research, Indore 452001 MP, India

共Received 5 May 2009; accepted 11 June 2009; published online 6 July 2009兲 Thermopower 共Sn兲 of compacted nanocrystalline Ni of average particle size D = 38, 32, and 25 nm at low temperatures 共5–300 K兲 is reported here. Sn exhibits striking deviations from bulk thermopower Sbulk behavior revealing evolution of Sn with change in D. As D decreases, there is significant decrease in Sn, gradual disappearance of phonon drag minimum and sign change to positive below 38 K compared to usual negative Sbulk. They have been attributed to electron and phonon scattering at grain boundaries and phonon confinement. Small minima observed near 25 K in thermopower difference ⌬S are correlated with superparamagnetic transitions. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3167302兴 Miniaturization of devices into nanoscale has given rise to several issues in fundamental research and nanotechnology. From fundamental point of view, integrated circuits with nanoscale sizes and interconnects in semiconductor fabrication, for example, are approaching electron mean free path l, at which distinct deviation from bulk electrical transport properties in the diffusive regime takes place. To elucidate these intriguing physics, extensive, experimental, and theoretical works on variety of thin films and nanowires or nanoparticles, including single nanowire and compacted nanoparticles, have been carried out.1–7 Theoretical predictions have opened up several related aspects in terms of size dependence of scattering mechanism of electrons and phonons relevant to thermoelectric properties of nanoscale metals.8,9 While their experimental verifications are reported scarcely for single wires limited to above 100 nm,3 and for 30 nm wire,1 evolution of transport process with change of size below 50 nm is unavailable despite the vital importance for searching more efficient thermoelectric nanostructures.10–12 Fascinating aspects of nanostructures in this regard are spatial confinement of carriers and corresponding change in carrier density of states. With nanoparticles as quantum wells, phonon dispersion and group velocities are changed due to spatial confinement induced by boundaries.13 Phonon confinement affects all phonon relaxation rates and should affect thermoelectric transport as in electrical transport, wherein grain boundary 共GB兲 disorder prevails.2,14 We have investigated thermopower Sn of compacted 38, 32, and 25 nm nanocrystalline nickel 共n-Ni兲 as one can obtain oxygen-free compacted pellets.2 As D decreases, there is a considerable decrease in 兩Sn兩, gradual disappearance of phonon drag minimum 共PDM兲, and anomalous sign changes to positive at low temperature T 共below 38 K兲 compared to Sbulk. We have discussed size effects on Sn of this ferromagnet at nanoscale, role of electron and phonon scattering at GB, and confinement of phonons. Three samples of 38, 32, and 25 nm were prepared using a chemical reduction method; details reported in Ref. 2. Further, compacted pellet of commercial Ni powder of a micron size 共com-Ni兲 and annealed polycrystalline Ni ingot 共bulkNi兲共both 99.99%兲 were used for comparison. Thermopower

共S兲 measurements down to liquid helium temperature 共5–300 K兲 were carried out using compacted pellets sandwiched between two Cu blocks, with reference to which absolute S was measured.15 From now onward, n-Ni samples are denoted by their D values or n. For total thermopower, Sx and S are used interchangeably with x = bulk-, com-, n- 共25, 32, and 38 nm兲 Ni. Zero-field cooled 共ZFC兲 and field-cooled 共FC兲 magnetization measurements were done in 50 G using a sensitive superconducting quantum interference device magnetometer. A significant drop in 兩S兩 and phonon drag thermopower Sph at low T with reduction in D is observed 共Fig. 1兲. Note that there is a shift in linear region to lower T of Sx of all samples compared with that of Sbulk with negligible feature at a certain T, at which its slope changes. This is ⬃180 K for com-Ni sample. Linearity below 180 or 280 K of Sx extends down to ⬃80, 100, 95, and 100 K, respectively for com-, 38, 32, and 25 nm Ni samples. While other features are nearly similar, 兩S兩 in overall systematically drops signifying effectively reduction in overall diffusion Se and Sph as D diminishes. This gradual decrease in 兩Sn兩, with decreasing D, finally leads to crossing of its sign from negative to positive. This clear change over the sign of S from negative to positive value has not been reported so far in alloys of Ni;16 there is such a tendency only in 1.0 at. % Pd in Ni.17 Thus, anomalous positive signs of Sn are observed below 15, 25, and 38 K for 38, 32, and 25 nm Ni samples.

a兲

FIG. 1. 共Color online兲 Thermopower of various Ni samples, as indicated. Inset: Low temperature regime.

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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FIG. 2. 共Color online兲 Phonon drag thermopower of various Ni samples, as indicated, after subtraction of linear part due to electron diffusion.

Furthermore, with the exception of Scom laying near 300 K below Sbulk, all Sn for n-Ni samples lay above Sbulk. This is more systematically increased at lower T 共Fig. 1, inset兲. Indeed, these trends of Sn共T兲 and their magnitudes match well than reported for 30 nm Ni single wire.1 Such a good comparison with Sn共T兲 of compacted 38 nm sample18 is attributed to polycrystalline nature in both samples.1 Remarkably, with decreasing D, depths of PDM in Sn are smaller than that of Sbulk. This is in contrast to depths of PDM in most of Ni alloys investigated greater than that in pure Ni16,19 except in, say, 1 at. % Pd in Ni.17 These data were explained in terms of spin mixing related to spin up and spin down conductivities.20 Spin mixing process has important consequences for electrical conductivity ␴共T兲 variation and for Sph. This is applicable in S data 共Fig. 1兲. Magnon drag Sm similar to Sph in Ni alloys is quite peculiar as alloying can alter Sph either attenuating or enhancing PDM.16 Hence, present observation might also somehow be possible by simply alloying it but observation of clear positive sign of Sn at low T is reported here. To assess Sph contribution more precisely, linear part of Sx共T兲 of each sample is considered using a linear fit to generate and subtract Sex from Sx共T兲共Fig. 2兲. It is evident that PDM values of Sn gradually increase with decrease in D, except in Scom. This shows dominant role of nanostructure. Further, T of PDMs, TPDM, is found near 35 共25 nm Ni兲, 37 共32, 38 nm, and com-Ni兲, and 50 K 共bulk-Ni兲 with the last, consistent with Blatt et al.21 Considering that Debye temperature, ␪D共Ni兲 = 375 K, and TPDM normally laying at low T, say ␪D / 10− ␪D / 5,16 TPDM fall in this range. This is in contrast to additional scattering effects in this material as a consequence of prevailing GB disorder.2,22 Moreover, as low temperature Sph or Se of metals is proportional to corresponding heat capacity total16 S = aT + bT3

or

S/T = a + bT2 .

FIG. 3. 共Color online兲 S / T vs T2 plots of various Ni samples as indicated. Inset: Plots of coefficients a and b in Eq. 共1兲 as a function of D; 100 nm is chosen for bulk-Ni.

at GB and particle size, difference in S, ⌬S = Sn − Sbulk are plotted 共Fig. 4兲, where ⌬S = ⌬Se + ⌬Sph.8 Clearly, magnitude of ⌬S共T兲 increases in overall as D decreases from com-Ni sample down to 38, 32 and 25 nm Ni samples. These results clearly indicate dominant role of GB disorder, which influence scattering of phonons and electrons in a significant way. Interestingly, this is contrary to the way in which resistivity becomes anomalously large at smallest D sample yet with no saturation even when its value crosses Mooij–Tsuei limit.2 Notably, ⌬S exhibit more prominent and broader peaks with a marginal shift in their positions to higher T as D decreases. Size effect on electrons and phonons is in particular attributed to their l being reduced when D is comparable to or smaller than l.2,8 In such situations, flow of phonons is restricted, i.e., localized continually and transport of phonons is limited thereby decreasing Sph as D decreases.10 This can be explained by considering influence of phonon confinement in nanoparticles at low T on the probability for electron-phonon scattering Pe−p,14 given by Pe−p ⬃

bulk Pe−p

e共ប␻min/kBT兲 − 1

,

共2兲

where ␻min is the minimum nonzero phonon frequency due to confinement i.e., ␻min = ␲␷s / D, with ␷s as the sound velocbulk is the scattering probability in bulk. For n-Ni ity, and Pe−p

共1兲

Competing size effect in Sx is estimated from S / T versus T2 plot 共Fig. 3兲. Note that there is a linear portion of Sx / T in low T region below ⬃20 K, which is less than ␪D / 10. Coefficient a 共b兲 in Eq. 共1兲 increases 共decreases兲 gradually as D decreases 共Fig. 3, inset兲. Briefly, as D decreases, Sph x diminishes at the cost of Sex. Highly disordered GB in nanocrystalline metals introduce numerous interface scattering for electrons and phonons in compacted n-Ni.2,8 To get extra contribution of phonon drag 共⌬Sph兲 and diffusion 共⌬Se兲 thermopower due to disorder

FIG. 4. 共Color online兲 Difference thermopower, ⌬S, of various Ni samples as indicated. Inset: 共a兲 ⌬S at low temperatures. 共b兲 FC and ZFC magnetization in a field of 50 G for n-Ni as indicated.


013101-3

samples at 10 K, exponent exceeds unity significantly resulting in a considerable decrease in scattering probability and hence, Sph n decrement. From 75 to 100 K, for thermopower Sn of 38, 32, and 25 nm Ni samples, Pe−p becomes comparable bulk in Eq. 共2兲, where diffusion behavior dominates. Such to Pe−p probability is indeed revealed in our data. Interestingly, there are small minima 关Fig. 4, inset 共a兲兴 in ⌬S near 25 K of n-Ni samples that can be matched with superparamagnetic 共SPM兲 transition 关Fig. 4, inset 共b兲兴. Therefore, small minima are attributed to SPM transitions observed in magnetization data. Sharpness in transition to SPM changes with D. In fact, S of magnetic materials, particularly alloys of Ni, have attracted the scientific community for many decades because of their possible extra contribution arising from d-electrons.16 However, such extra contribution due to, say, magnons Sm has been proven to be negligible compared to those of Se and Sph.16,17 In conclusion, thermopower 共Sn兲 of compacted n-Ni at low T 共5–300 K兲 has been investigated. Results provide clear evolution of Sn with drop of D showing many interesting features. There is a significant decrease in 兩Sn兩, gradual attenuation of PDM and sign change to positive value at low T 共below 38 K兲 as D decreases. These are attributed to electron and phonon scattering at GB and phonon confinement. In addition, thermopower differences ⌬S exhibit small minima near 25 K, which are correlated with SPM transition. Authors thank Dr. D. T. Adroja, RAL, DIDCOT, U.K. for magnetic data. 1

Appl. Phys. Lett. 95, 013101 共2009兲


E. Shapira, A. Tsukernik, and Y. Selzer, Nanotechnology 18, 485703 共2007兲.

G. S. Okram, A. Soni, and R. Rawat, Nanotechnology 19, 185711 共2008兲. C. Strunk, M. Henny, C. Schönenberger, G. Neuttiens, and C. Van Haesendonck, Phys. Rev. Lett. 81, 2982 共1998兲. 4 G. Kästle, H.-G. Boyen, A. Schröder, A. Plettl, and P. Ziemann, Phys. Rev. B 70, 165414 共2004兲. 5 G. Reiss, J. Vancea, and H. Hoffmann, Phys. Rev. Lett. 56, 2100 共1986兲. 6 O. Ujsaghy, L. Szunyogh, and A. Zawadowski, Phys. Rev. B 75, 064425 共2007兲. 7 B. M. Askerov, S. R. Figarova, and V. R. Figarova, Nanotechnology 18, 424024 共2007兲. 8 R. P. Huebener, Phys. Rev. 140, A1834 共1965兲; R. F. Moreland and R. R. Bourassa, Phys. Rev. B 12, 3991 共1975兲; W. F. Leonard and S. F. Lin, J. Appl. Phys. 41, 1868 共1970兲. 9 G. D. Mahan, in Solid State Physics, edited by H. Ehrenreich and F. Spaepen 共Academic, USA, 1998兲, Vol. 51, p. 81. 10 R. Venkatasubramanian, E. Silvola, T. Colpitts, and B. O’Quinn, Nature 共London兲 413, 597 共2001兲. 11 L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, 12727 共1993兲; Y.-M. Lin and M. S. Dresselhaus, ibid. 68, 075304 共2003兲. 12 T. E. Humphrey and H. Linke, Phys. Rev. Lett. 94, 096601 共2005兲. 13 A. Balandin and K. L. Wang, J. Appl. Phys. 84, 6149 共1998兲. 14 A. Malikyan and H. Minassian, Chem. Phys. Lett. 331, 115 共2000兲. 15 A. Soni and G. S. Okram, Rev. Sci. Instrum. 79, 125103 共2008兲. 16 F. J. Blatt, P. A. Schroeder, C. L. Foiles, and D. Greig, Thermoelectric Power of Metals 共Plenum, New York, 1976兲. 17 T. Farrell and D. Greig, J. Phys. C 3, 138 共1970兲. 18 Based on another sample of 25nm 共XRD兲 and ⬃14 nm 共TEM兲, present sample size of 38nm 共XRD兲 might well be ⬃30 nm 共TEM兲 that in turn matches that in Ref. 1. TEM data normally show lower value because of wide distribution of particle size in which number of bigger particles is small; yet their volume fraction is large 共⬃82%兲, which is measured by XRD. 19 T. Farrell and D. Greig, J. Phys. C 1, 1359 共1968兲. 20 A. Campbell, A. Fert, and A. R. Pomeroy, Philos. Mag. 15, 977 共1967兲. 21 F. J. Blatt, D. J. Flood, V. Rowe, P. A. Schroeder, and J. E. Cox, Phys. Rev. Lett. 18, 395 共1967兲. 22 F. Napoli and D. Sherrington, J. Phys. F: Met. Phys. 1, L53 共1971兲. 2 3


JOURNAL OF APPLIED PHYSICS 110, 023713 (2011)

Size-dependent resistivity and thermopower of nanocrystalline copper Gunadhor S. Okram1 and Netram Kaurav2,a) 1

UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452001, MP, India 2 Department of Physics, Government Holkar Science College, A. B. Road, Indore 452001, MP, India

(Received 18 April 2011; accepted 7 June 2011; published online 29 July 2011) Nanocrystalline copper (NC-Cu) of average particle size (D) ranging from 29 to 55 nm was prepared using the polyol method. The compacted pellets of these nanoparticles were investigated using electrical resistivity (qn) and thermopower (Sn) measurements in the temperature range from 5 to 300 K. The observed electrical resistivity and thermopower data for all the samples are typical of a good metal and the qn(T) data are analyzed in the framework of the Bloch-Gru¨neisen theory. Our analysis indicates systematic departure from the bulk property for NC-Cu samples, decreasing effective Debye temperature, exponential decay of both the residual resistivity ratio (RRR) and the temperature coefficient of resistivity [a ¼ (1/q)dq/dT] as D decreases, yet the Boltzmann theory of electron transport still holds true (kFl 1). Further, the validity of the Nordheim-Gorter rule is also discussed. The temperature dependence of Sn is found to be quite sensitive as compared to bulk thermopower SBulk behavior, revealing the evolution of Sn and exhibiting a significant enhancement of the phonon drag peak as D decreases. The present findings overall suggest the C 2011 significant influence of the grain boundaries, surface atoms, and phonon confinement. V American Institute of Physics. [doi:10.1063/1.3610791]

I. INTRODUCTION

In recent years, nanocrystalline materials (NC-Ms) have attracted attention from the scientific community because of the rich physics involved as well as their potential use in device applications.1–6 This special interest in nanosystems comes from the fact that at the nanometer level the physical properties of these objects not only depend on their structure, but also on their size. An example is the enhancement of the thermoelectric figure of merit that further opens the possibility of novel efficient thermoelectric materials.7 In many applications of nanomaterials, the sensitivity or efficiency is proportional to the surface area and size. The presence of a large amount of grain boundaries (GBs) and/or the broad distribution of interatomic spacing in the GBs gives rise to the unusual properties of NC-Ms when compared to conventional polycrystals or single crystals with the same chemical composition.8 To elucidate these intriguing properties, extensive experimental and theoretical works on a variety of thin films and nanowires or nanoparticles, including single nanowires and compacted nanoparticles, have been carried out in the recent past.8–15 Theoretical predictions have opened up several related aspects in terms of size dependence of the scattering mechanism of electrons and phonons, relevant to the thermoelectric properties of nanoscale metals.16–18 In the regime where the width of the wire is a few tens of nanometers or less, it has been established adequately that the electrical resistivity (q) is not determined by the material alone but by its size as well.19–24 Fascinating aspects of nanostructures in this regard are spatial confinement of cara)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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riers and the corresponding change in carrier density of states.8,25–28 Detailed calculations reveal that on decreasing the film thickness, the relaxation rate of electrons will either increase or decrease depending on the character of the process—interband, intraband, and so on.26,27 With nanoparticles as quantum wells, phonon dispersion and group velocities change due to spatial confinement induced by the GBs.25 Phonon confinement affects all phonon relaxation rates and should affect thermoelectric transport as in electrical transport, wherein GB disorder prevails.8,10,28 This, however, is a positive point for reducing the thermal conductivity and enhancing the thermoelectric figure of merit of the material.29 The importance of resistivity studies in such NCMs or nanowires has been stressed because this is likely to be the dimension of metallic interconnects in potential electronic devices in the immediate future. In this size regime the concept of Boltzmann transport approaches its limit. A proper understanding of q in this regime is desirable because it allows one to get a quantitative estimate of q of the wire from its dimensions without really measuring it. For NC-Ms of these dimensions, the mean free path is comparable to or even larger than the particle size (D) (particularly for clean nanoparticles) and one would expect the size effect to be active. In this range q typically increases significantly as the diameter of the particle is decreased. This is a serious issue in interconnects as an increase in the resistance of the wire increases the propagation delay time constant of the system, which is directly translated to slowing down the speed of the device.24 The scattering due to small size and more GBs shows up as residual resistivity at T 20 K in nearly all metallic solids. If the NC-M is comparatively more disordered30 or has a proximity effect due to its substrate or matrix,31 the

110, 023713-1

C 2011 American Institute of Physics V

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023713-2

G. S. Okram and N. Kaurav

resistivity can show localization instead of causing a constant residual value. In addition, a substantial part of the resistivity should arise from the temperature-dependent part, which in a non-magnetic metal arises from the electronphonon interaction.32 Knowledge of size and GB effects of a NC-M vis-a`-vis the contribution of electron-phonon interaction will thus give us a control on the residual q and S of the NC-M. While a good estimate of predicting the resistivity contribution from the electron-phonon interaction is desirable for interconnects, such a situation will not arise in the case of thermoelectric material for which disorder at GBs and enhanced interfaces are desirable. This is the case in compacted NC-Ms. In this paper the primary focus is on the particular case of the contribution of electron-phonon scattering to the resistivity of metallic NC-Cu and we establish by experiments to what extent such established theories as the Bloch-Gru¨neisen theory, Mathiessen’s rule for resistivity, and Nordheim-Gorter rule for resistivity-thermopower correlation in metals and alloys are applicable or not as the NC-M diameter goes down to as small as 29 nm. Both theoretical and experimental studies on the temperature dependence of the resistivity and thermopower of NCMs have been reported.8–10,33–37 A residual resistivity at low temperature associated with metallic q ! T for T 100 K in elemental metal and alloy NC-Ms of size 15 nm has established experimentally unless the NC-Ms are not disordered.33 In the case of NC-Ms with size less than 10 nm, the resistivity may show a negative temperature coefficient ((1/q)@q/@T). This has been found in NC-Ms of Au,33 AuPd alloy,34 Zn,35 Cu,22,24,31 Ni,10 and Sn (above its superconducting transition temperature Tc 3.7 K).36 Thermopower has been reported in Cux(SiO2)1x nanogranular thin films,31 compacted NC-Ni,8 and NC-Ni wire.9 In the report on the resistivity of Cu nanowires by Bid et al.,24 the temperature coefficients were positive for all the size of wires down to 15 nm, showing the highly conducting nature. Similar results are reported by Wu et al.22 However, in Cux(SiO2)1x nanogranular thin films, the temperature coefficient of resistivity is negative, probably due to the proximity effect of SiO2. This latter assignment may not be true as the magnitude of thermopower is metallic and that too nearly linear down 50 K without a clear phonon drag peak.31 However, these closely-related transport properties, having contrasting responses, need further investigation with samples free from proximity effect of the matrix. There has not been any experimental study to the best of the authors’ knowledge that specifically addresses these issues together for which we use well-characterized compacted NC-Cu over an extensive range of temperature, particle size (D) in the range of 29-55 nm and analyze the data quantitatively. With these objectives we have studied the resistance and thermopower of nanocrystalline metal (copper) as a function of D in the temperature range 5–300 K. The systematic investigation allows us to analyze the data in the framework of the Bloch-Gru¨neisen theory. The electron-phonon coupling parameter enhances as D drops, showing clear departure from the bulk property. This gave us the decreasing effective Debye temperature as D decreases. Further, a clear evolution of the residual resistivity ratio (RRR) and temperature coefficient of resistivity (a ¼ (1/q)dq/dT)

J. Appl. Phys. 110, 023713 (2011)

allowed us to establish their exponential decay (i.e., departure from Mathiessen’s rule) with the reduction in size of the nanoparticles. While the Boltzmann transport theory of weak scattering still holds true (kFl 1), there is systematic evolution of the thermopower associated with significant enhancement of the phonon drag peak as D decreases. These findings overall suggest the significant influence of the disordered GBs and surface atoms that lead to phonon confinement processes, causing modifications in the electronphonon scattering. II. EXPERIMENT

To enable the present investigations, nancrystalline copper (NC-Cu) samples were prepared by refluxing a solution of copper acetate (Cu(ac)2 H2O) in ethylene glycol (EG, H2OH-CH2OH). A series of NC-Cu samples were prepared with variations in the mole ratios of the metal ion (Cu2þ) with respect to that of EG for obtaining particles of different sizes. The mole ratio was adjusted as 0.10:1, 0.07:1, 0.05:1 and 0.03:1 (Cu2þ: EG). Typically, 1.00 g of the copper acetate was dispersed in 100 ml of ethylene glycol at 192 C over one and a half hours. A condenser column with continuous flow of cold water is attached at the top of the flask. As the reflux was in progress, after about half an hour, the initially dark greenish color solution turned a milky brown color and slowly turned dark brown as Cu particles were precipitated. The reaction product was allowed to cool to room temperature and excess acetone was added. The diluted product was consequently centrifuged to collect the precipitate. This step was repeated several times. The precipitate was sonicated and dried in an oven at 60 C for half an hour. The powder was compacted into pellets near its bulk density immediately to avoid bulk oxidation.10 The characterization of particle size and the structural and crystallographic nature of the nanoparticles forms an essential part in the analysis of the data. The NC-Cu used in this investigation is single crystalline in nature. The determination of the average particle sizes has been established from x-ray diffraction (XRD) data and the nanostructure was investigated using atomic force microscopy (AFM). Electrical resistivity and thermopower measurements down to liquid helium temperature (5–300 K) were carried out using our homemade setups.38 Correction for the reference copper in thermopower measurements was done, which is extremely crucial for accurate determination of the measured thermopower. III. CHARACTERIZATION A. X-ray diffraction

Figure 1 shows the XRD patterns of the samples with different mole ratios. The diffraction peaks observed are matched exactly with standard (JCPDS) data for the face centered cubic copper, without any extra peaks due to impurities. The lattice parameters calculated using the indexed peaks are 0.3611(1), 0.3613(3), 0.3612(1), and 0.3613(1) nm, respectively, which although relatively smaller are very close to that of the bulk copper (0.36147 nm). A close


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mole ratios 0.05:1, 0.07:1, and 0.10:1, respectively. These D values are consistent (within the error) with those determined from XRD. It is now clear that as the mole ratio decreases the average particle size decreases. From now onward, we shall designate the samples by the particle size found with AFM. III. RESULTS AND DISCUSSION A. Electrical resistivity

FIG. 1. (Color online) XRD patterns of different mole ratio nanocrystalline copper samples, 0.03 (D ¼ 29 nm), 0.05 (D ¼ 37 nm), 0.07 (D ¼ 45 nm), and 0.10 (D ¼ 55 nm), as indicated.

observation indicates that the peaks broadened marginally with decreasing mole ratios, implying a decrease in the average particle size (D). The D was calculated from full width at half maximum (FWHM) of the most intense diffraction peak (111) using the Debye-Scherrer relation. The estimated values of D were 27, 37, 44, and 56 (62) nm for the mole ratios 0.03:1, 0.05:1, 0.07:1, and 0.10:1, respectively, showing the influence of the mole ratio on particle size. B. Atomic Force Microscopy

To understand the particle size and microstructure better, we have taken the AFM image of samples of different mole ratios. For this, the representative AFM image of the lowest mole ratio (0.03:1), for which the particle is expected to be the smallest, is presented in Fig. 2(a). The size D was estimated from the image from more than 100 particles by noting down the frequency of particles found for an interval of 5 nm was 29 6 2 nm. For this, a bar chart was plotted and a Gaussian fitting was performed (Fig. 2(b)). It was found that the size distribution is reasonably narrow, with a standard deviation of 3.4. The same procedure was followed to deduce the D values of 37 6 3, 45 6 3, and 55 6 4 nm for

Figure 3 shows the temperature dependence of the electrical resistivity qn(T) of the compacted NC-Cu samples with D 29, 37, 45, and 55 nm in the temperature range from 5 to 300 K. The insets show the same q(T) plotted with a logarithmic scale for the temperature axis to show more clearly the low temperature behavior of the resistivity. It can be seen that there is a significant increase in the resistivity at 300 K and in the residual resistivity at 5 K as D decreases. Compared to the bulk value, it comes out to be of the order of tenfold for 29 nm sample. However, this enhancement in the resistivity is significantly smaller compared to those of other NC-Ms such as nickel10 but relatively higher compared to those of the copper nanowires.24 Therefore, the observed electrical resistivity data for all samples are typical of a good metal that shows a fairly linear T-dependence of resistivity down to about 100 K. The residual resistance ratios RRR ¼ q300K/q5K for 29, 37, 45, and 55 nm samples are 3.1, 2.9, 4.6, and 9.9, respectively. This shows an exponential decay of RRR, i.e., RRR(D) ¼ 2.77 þ 2.260 103 eD/6.829 as D decreases (Fig. 4, right axis). This scenario implies that the disordered surface and GB atoms act as virtual impurities in NC-Cu, as is usually observed in such materials.10 However, it is inferred from the inset of Fig. 3 that q(T) does not show any upturn at low temperature, thus ruling out any drastic disorder in the system or quantum confinement that can give rise to effects such as localization.31,39 The temperature coefficient of resistivity (a) (Fig. 4) shows an exponential drop yet again, i.e., a(D) ¼ 0.00235 þ 1.62913 eD/8.817 as D decreases (Fig. 4, left axis). Therefore, Fig. 4 exemplifies the fact that RRR and a decay as D decreases, implying that the enhancement of resistivity in nanocrystalline copper does not follow the Mathiessen rule.32 This means that the displaced GB or surface atoms are not simply acting as

FIG. 2. (Color online) AFM image (left) and particle size distribution (right) of 0.03 mole ratio sample (D ¼ 29 nm).


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processes based on the phonon confinement, which might be one of the important factors responsible for the observed anomalous nature of thermopower presented in the later part of this paper. Now, we turn to a quantitative viewpoint of the electrical transport data analyzed theoretically, more comprehensively using Matthiessen’s rule32 qðT Þ ¼ q0 þ qeph ðT Þ:

(1)

Here, q0 is the residual resistivity, and the second term corresponds to resistivity due to the electron-phonon interaction, which is generally expressed as

qeph ðT; hD Þ ¼ Aeph ðT=hD Þ FIG. 3. (Color online) Variation of electrical resistivity with temperature for compacted NC-Cu samples for various particle sizes. The solid line represents the theoretical fit using the Bloch-Gru¨neisen relation [equation (2)]. The inset shows the electrical resistivity with the temperature axis in a logarithmic scale in order to show more clearly the low-temperature behavior of the resistivity.

impurities but rather in a complicated way in these NC-Cu samples. This observation is distinct from impurities present at different concentrations in alloys40 but is similar to several other alloys.32 The relatively more resistive nature of compacted NC-Cu pellets is attributed to the disorder in the GBs and enhancing the number of atoms at the surface that consequently lead to enhanced scattering as D decreases.10 This trend clearly indicates the significant role of smaller grains and, hence, their GBs effects on the smooth flow of charge carriers. The other possibility comes from confinement of phonons, which is also enhanced due to the GBs present in the system.8,10 Previously, it has been shown that the phonon confinement affects all phonon relaxation rates and should affect thermal transport, wherein GB disorder prevails.8 The present NC-Cu provides an opportunity to understand these

FIG. 4. (Color online) The variation of residual resistance ratio (right axis) and temperature coefficient of resistivity (a) (left axis) with the particle size (D). The plot of [q(T)q(4.2 K)]/q(hD) as a function of T/hD is shown in the inset.

5

hDð=T

x5 ðex 1Þ1 ð1 ex Þ1 dx:

0

(2) Equation (2) represents the Bloch-Gru¨neisen function of the temperature-dependent resistivity contribution of an acoustic phonon.32 Here, x ¼ hx/kBT and Ae-ph is a constant of proportionality. It is natural to choose a model phonon spectrum consisting of an acoustic Debye branch characterized by the Debye temperature hD. In equation (2), when we divide both sides by the resistivity q at hD, Ae-ph becomes the electronphonon interaction AD (4.223) at hD.41 It is instructive to mention here that the presence of additional low-energy optical vibrational states and the boson peak in the present NC-Cu samples would not have considerable roles as the temperature range is rather low to excite these processes. As pointed out earlier,24 the lower limit of the integral in equation (2) will no longer be zero as we approach nanometer scale due to the phonon confinement effect. To account for the phonon confinement effects we have set the lower limit of the integral as hmin/T. These are found to be pronounced in thermopower, where thermal gradient is involved as the only driving force. Equation (1) incorporated with equation (2) is fitted to the experimental data on the temperature-dependent resistivity, as shown in Fig. 3. It is inferred from the fits that the qn(T) for all sizes could be fitted to the above function over the entire temperature range of the investigation reasonably well. The values of coefficients (Ae-ph, AD, and hD) obtained for different particle sizes, resulting from the electron-phonon temperature-dependent contribution of resistivity, are listed in Table I. The electron-phonon interaction constant Ae-ph is found to be systematically increasing while the resistive Debye temperature hD decreases with decreasing D. The decrease in hD (615 K) with particle size is consistent with those observed from the specific data of NC-Ni.42 These trends are interpreted as a the direct consequence of the enhancement of electrical resistivity with D and are attributed to the increase in electron-phonon scattering as we move toward smaller D, wherein the surface scattering induced by particle size reduction and GBs dominate. In the nanocrystalline solids, wherein the mean free path of the electrons is comparable to the particle size, the bulk assumption will no longer be valid as the surface provides extra


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TABLE I. Parameters such as room temperature resistivity (q300K), electron mean free path (l), product of Fermi wave vector (kF) and l, Debye temperature (hD), electron phonon interaction constant (Ae-ph), and electron phonon interaction at hD (AD) obtained from the resistivity and fitting of experimental data using equation (2) for different particle sizes. Diameter (nm) q300K (lXcm) l (nm) kFl 29 37 45 55 Bulk

14.77 11.95 9.88 5.34 1.75

4.4 5.5 6.6 12.2 37.5

60 77 90 167 510

hD (K) Ae-ph (lX cm) 175 180 210 270 310

24.0 19.9 19.4 16.1 7.4

AD 2.288 2.183 2.880 3.455 4.225

scatterers to the charge carriers.8,10,34,41–44 Therefore, these magnitudes suggest that surface atoms and GBs in the present NC-Cu samples are quite usual, as in such NCMs8,10,34,41–44 but are different from other results,24 wherein the nanowires were suggestively significantly aligned. This situation is in line with the deviation of AD (see Table I) from its bulk value (4.22) and consistent with the nanoscale nature of the compacted NC-Cu, which however was not the case in the NC-Cu wires.24 Further, to give a quantitative view on the applicability of the Bloch-Gru¨neisen theorem and the electron-phonon interaction, we have plotted [q(T)q(4.2 K)]/q(hD) as a function of T/hD (Fig. 4, inset). It can be seen that the resistivity curves overlap at low temperature. At higher temperature, low D (29 nm and 37 nm) samples still maintain almost perfect overlapping that might be related to the almost identical hD and, hence, the same q(hD). However, in the case of larger D (45 nm and 55 nm) samples, the resistivity curves do not overlap significantly as particle size increases; this is related to the considerable difference in hD values and, hence, the corresponding difference in q(hD). The other factor that may be more dominant is the exponential variation in the coefficient of resistivity (Fig. 4), due to which there is departure from overlapping. This observation in turn signifies additional evidence of the change in AD value in the case of NC-Cu particles, as distinct from those of NC-Cu wires.24 Thus, there is a change in electron-phonon interaction with particle size in the present samples. We attribute this to the considerable disorder in the GBs and the enhanced number of atoms at the surface as a possible consequence of random orientation of the NC-Cu particles, leading to an effective phonon confinement. To test the applicability of Boltzmann transport theory based on weak scattering (kFl 1, where kF is the Fermi wave vector and l is the mean free path) in this disordered system, we estimated these parameters (Table I). It is seen that kFl values are all above 60 indicating that the condition kFl 1 applies well in these NC-Cu samples. In summary, the resistivity of the present NC-Cu samples shows an enhancement in its value, a clear deviation from Mathiessen’s rule, an increase in the electron-phonon scattering parameter, and a decrease in resistive Debye temperature as the particle size decreases while the Boltzmann transport theory of weak scattering (kFl 1) is still applicable. These are rather intriguing as there are deviations in Bloch-Gru¨neisen parameters from that of bulk, which is understood as enhanced electron-phonon scattering due to the possible confinement of phonons and

electrons in these nanoscale materials. To understand this better, we studied their thermopower. B. Thermopower

Figure 5 displays the temperature dependence of the thermopower Sn(T) of the NC-Cu samples in the temperature range from 5 to 300 K. The S(T) data of Cu bulk (99.99%) is also included for comparison, which shows a positive sign, a broad phonon drag peak around 55 K, and values within 2 lV K1. However, as it is seen, these features have drastically changed in NC-Cu samples. As the temperature increases, 29, 37, 45, and 55 nm samples display a positive sign up to 35 K (with slight variation in particle size), turning negative above this temperature. Their values fall sharply beyond this temperature and a consequent rise renders S to a negative phonon drag peak at 50 K. Then, the rise stops at the onset point of the phonon drag peak and the diffusion contribution takes dominance over the phonon drag, causing a broad hump around 120 K. The |S| value still lies within 2 lV K1. On close observations, it is seen that these features evolve gradually as D decreases. The hump gradually turns out to be more prominent and the negative phonon drag peak sharper with an associated gradual shift of the peak value (to 40 K for 29 nm sample) as the particle size drops. As we decrease the temperature, this evolution is dramatically significant in the 29 nm sample, for which the thermopower increases from a negative to a positive value at 210 K, manifesting the hump with a consequent fast drop from positive to negative values at 55 K. This in turn generates the sharp negative phonon drag peak and consequent crossing from negative to positive values at 20 K as the temperature further drops. Thus, the room temperature absolute S (S300K) values for studied samples are found to be less than 2 lV K1 and to increase with increasing D. It is therefore inferred from Fig. 5 that the behavior of thermopower of NC-Cu samples is significantly different from Cu-bulk. These features are

FIG. 5. (Color online) Variation of thermopower of compacted NC-Cu samples with temperature. The thermopower of copper standard is also shown.


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striking and might indicate a substantial change in the band structure or the conduction mechanism at the nanoscale; and they are distinct from those of the thermopower of a variety of copper samples45,46 (more details are to be found in Ref. 47) although the dips are to some extent similar, yet at higher temperatures in the present nanoparticles. The flipping of sign of the thermopower in these NC-Cu samples indicates, in principle, involvement of both the electrons and holes in the electrical transport, which changes systematically with size. From 120 to 300 K, |S| in general increases systematically as the temperature rises, indicating that the diffusion thermoelectric transport prevails in the high temperature range but suggesting electrons as the majority charge carriers. Therefore, these results clearly indicate the distinct behavior of the NC-Cu samples, associated with the sharp onset of a negative phonon drag peak. This is intriguing, as it is distinct from that of the bulk (hole-like), because thermopower in general is usually dependent on the topology of the Fermi surface and consequently closely related to the energy (E) and phonon wave vector (k) dependence of the relaxation time, s(E,k). In the low temperature limit, the carrier relaxation time is limited by impurity scattering. From the extrapolation of the electrical resistivity of Cu-nanoparticles (Fig. 3) to zero temperature, it turns out that the present samples correspond to a distinct value of resistivity at zero temperature. This means that there is an essential presence of equivalent impurity scattering in the present samples. Thus for low temperatures, where phonons have not yet started to play a significant role, the electrical resistivity is proportional to the inverse of relaxation time. For a three-dimensional system, the electron diffusion contribution to the thermopower is distinguished by its simple linear temperature dependence. The low as well as high temperature phonon drag and diffusion mechanisms have been adequately discussed for metals.47,48 In the low temperature regime, thermopower follows the relation S ¼ AT þ BT 3 :

(3)

Here the first term in the isotropic relaxation time approximation represents the contribution from the diffusion component (Sd), and is linear in T. The second term corresponds to the phonon drag contribution (Sph) at low temperature, analogous to the specific heat that follows the Debye T3 law. On the other hand, at higher temperatures, the conventional diffusion and phonon drag components to S are written as S ¼ aT þ b=T:

(4)

Here, A (or a) and B (or b) are the diffusion coefficient and phonon drag coefficient at low (high) temperatures, respectively. In order to elucidate these mechanisms and to extract the diffusion coefficient (a or A) and phonon drag coefficient (b or B) from the observed thermopower of NC-Cu samples, we have plotted S/T versus T2 and S/T versus T2 for the low and high temperature regions in Figs. 6 and 7, respectively. We have plotted in turn the resultant A and B (Fig. 6, inset) and a and b (Fig. 7, inset) coefficients versus D. It is seen

FIG. 6. (Color online) Diffusion and phonon drag contributions of various NC-Cu samples. S/T is found to be linear with T2 below 50 K. The inset shows the plots of coefficients A and B in Eq. (3) against particle size.

that the magnitude of coefficient A in the low temperature regime decreases gradually as D decreases but suddenly jumps up for the 29 nm sample. Conversely, |B| increases gradually as D decreases but suddenly jumps down for 29 nm sample. This means that these (diffusion and phonon drag contribution) coefficients operate oppositely to each other, as expected. For the high temperature regime, the coefficient a behaves very similarly with that observed for A in the low temperature regime. For b, it behaves similar to that of the low temperature case, yet interestingly the 29 nm sample follows the general trend. This means that the phonon contribution or confinement is prominent in this sample compared with other bigger nanoparticle samples. Because the coefficient a is inversely proportional to EF the decreasing nature of it with particle size eventually would enhance the Fermi energy.47,48 These variations of electronic band parameters near the Fermi energy should also affect the bare Seebeck coefficient as particle size decreases.49 Interestingly, this is found so, as the calculated Fermi energy EF from the coefficient a is 4.5 eV, 4.2 eV, and 3.8 eV for the 37 nm, 45 nm, and 55 nm samples, respectively, while the 29 nm sample shows an exceptionally low value of 2.6 eV. The last value, however, is in line with other observations so far on this sample and is nearly consistent with a spectroscopically determined value of 2.19 eV.50 These values of EF are, however, relatively much smaller than that of the bulk (7.0 eV) estimated from Sommerfeld’s model.49 We interpret this as related to modifications in the materials we investigated at nanoscale, at which the carrier density of states is expected to change. In the free electron model, Fermi energy is determined from bulk of the metal which is considered infinitely extended. This means that in a macroscopic crystal of 1024 atoms, typically of 108 atoms on a side, only about one in 108 reside near the surface. This situation, however, is no longer valid in the case of nanocrystalline metal, wherein the volume fraction of the surface atoms could be nearly comparable to those in the bulk. Taking these factors into account and considering the difficulties one experiences in understanding the thermopower of even the simple bulk


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FIG. 7. (Color online) Diffusion and phonon drag contributions of various NC-Cu samples. S/T is found to be linear with 1/T2 above 150 K. The inset shows the plots of coefficients A and B in Eq. (4) against particle size.

metals,47,48 the present observations do not seem to be surprising. To further elucidate speculation on the phonon confinement effect in the present NC-Cu samples, we recall that the effect of electron-phonon confinement would manifest itself as a broad hump in S at low temperatures for highly disordered GBs scattering.8,9 In the present case the low temperature limited resistivity for all measured NC-Cu samples is less than 6 lX cm (Fig. 3), suggesting that the phonon confinement effect in thermopower is expected to be noticeable in these compacted nanoparticles. Significantly disordered GBs in nanocrystalline metals introduce numerous interfaces leading to extra scattering centers due to the size effect for electrons and phonons.8–10 To get the extra contribution of phonon drag and diffusion thermopower due to disorder at grain boundaries and in particle size, the difference in S, DS ¼ Sn Sbulk is plotted in Fig. 8. It is seen that the absolute value of DS is significantly larger for all the samples compared to S, i.e., there is significant enhancement in S in NC-Cu samples with a systematic variation of DS as a function of temperature. We interpret this to the random nature of the GBs and surface atoms. As the GBs and surface atoms increase with decreasing particle size, |DS| systematically increases and consequently the diffusion and phonon-drag contributions become relatively more prominent than those in |S|. In order to appreciate better the phonon drag contribution, we subtracted the diffusion contribution of the thermopower SD from the total thermopower S, S SD. The plot for S SD as a function of temperature (Fig. 9) clearly shows that the phonon-drag contribution is relatively the same for the 37, 45, and 55 nm samples, while the 29 nm sample shows a very prominent phonon-drag peak. This indicates that in the 29 nm sample the contribution from the phonondrag is very significant, which is attributed to the phonon confinement caused by the disordered GBs and surface atoms. We thus argue that the origin of low temperature phonon-drag peaks presented here originates from the phonon confinement effect. It has been well established that if the

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FIG. 8. (Color online) Difference in thermopower, DS, of compacted NC-Cu samples.

medium surrounding a nanoparticle does not support the vibrational wavenumbers of a material, the optical and acoustic phonons get confined within the grain of the nanostructured material.51–54 This can be explained by considering the influence of phonon confinement in nanoparticles at low T on the probability for electron-phonon scattering Pep, given by28 Pep

Pbulk ep eðhxmn =kB T Þ 1

;

(5)

where xmn is the minimum nonzero phonon frequency due to confinement, i.e., xmin ¼ pts/D, with ts as the sound velocity and Pbulk ep is the scattering probability in bulk. The observed behaviors of Ae-ph (or AD), A, and a parameters and the thermopower difference DS(T) clearly demonstrate the dominant role of GB disorder, which influences scattering of phonons and electrons in a significant way. It is evident from the definition of xmn and equation (5), for a compacted

FIG. 9. (Color online) The plot for S SD as a function of temperature, showing evidence of a large phonon-drag peak for the 29 nm sample.


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NC-Cu sample of lowest size we studied and temperature of 10 K, the exponent will exceed from unity by several orders of magnitude, which will considerably decrease the scattering probability as compared with that of the bulk. On the other hand, at room temperature, the exponent will be much less than unity for the same set of parameters. We also establish our conclusion to elaborate diffusion thermopower in the framework of two band conduction mechanism, in line with the observations of negative and positive thermopower as the temperature varies. For this purpose, it is convenient to explicitly separate contributions to the thermopower from scattering identified with the electron diffusion process (SD) and the other due to the presence of significantly disordered GBs, surface atoms, and the phonon confinement processes (SI). The relation then will have the well-known Nordheim-Gorter (N-G) expression55 S ¼ SD þ ðqI =qÞðSI SD Þ;

(6)

where qI and q are, respectively, resistivity due to disorder and to the NC-Cu sample as whole. The significance of Eq. (6) is that, for fixed T, the thermopower S is linear in the extra resistance qI, if (SI SD) is constant and not affected by the GBs or surface atoms. In Fig. 10, we display the N-G plot (S versus q1) for different NC-Cu samples at 8 and 300 K. As can be seen in Fig. 10, the data exhibit a nearly linear behavior, suggesting that NC-Cu follow N-G relation at 300 K. However, it is found to deviate from expected linearity at low temperatures (8 K). We stress here that the differences in the anisotropy of relaxation times for phonon and impurity scattering can lead to breakdown of the N-G rule because of the deviations from Mathiessen’s rule as we have discussed above, which however is consistent with some typical alloys.56 Moreover, for the temperature regime at which the plot is made, the phonon confinement becomes more prominent and hence deviates from the validity of the N-G relation. For the validity of the N-G rule among other things, an invariant band structure is required on addition of the impurities,47 which, however, is not so, as discussed above.

FIG. 10. Nordheim-Gorter plot (S vs q1) for different NC-Cu samples at 8 K and 300 K.

In the present context, impurities we are concerned with are due to the presence of significantly disordered GBs, surface atoms, and phonon confinement processes. IV. CONCLUSIONS

We have investigated electrical resistivity (qn) and thermopower (Sn) of compacted Cu nanoparticles of average particle size (D) ranging from 29 to 55 nm in the temperature range 5–300 K. The observed electrical resistivity qn(T) data for all samples are typical of a good metal and have a fairly linear temperature dependence of resistivity down to about 100 K; the temperature dependence of qn can be fitted using the Bloch-Gru¨neisen relation. Our analysis strongly indicates clear signatures of departures of the physical properties of these NC-Cu samples from their bulk counterpart even when electrical transport obeys the Boltzmann transport theory reasonably well (kFl 1). The temperature dependence of Sn was found to be significantly different from SBulk behavior, which is attributed to the effect of the size of nanoparticles and resultant significant modifications of band structure due to the electron/phonon scattering at grain boundaries, surface atoms, and phonon confinement processes. As a consequence, the Fermi energy changes systematically with size, showing an exception in the 29 nm sample. ACKNOWLEDGMENTS

The authors gratefully acknowledge the assistance rendered by M. Babu of Aligarh Muslim University, UP, India and A. K. Parchur of HSG University, Sagar, MP, India in sample preparation; and M. Gupta and V. Ganesan, CSR, Indore, India for XRD data and AFM data collection, respectively. 1

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