J. Phys. Chem. C 2009, 113, 18027–18031
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Temperature Dependence of Electrical Resistance in Films of Gold Nanoparticles Linked by Organic Molecules K.-H. Mu¨ller,* J. Herrmann,† G. Wei, B. Raguse, and L. Wieczorek CSIRO Materials Science and Engineering, P.O. Box 218, Lindfield, New South Wales 2070, Australia ReceiVed: April 8, 2009; ReVised Manuscript ReceiVed: August 31, 2009
The electrical resistances of films of gold nanoparticles linked by hexanedithiol and octanedithiol molecules were measured as a function of temperature from 4 up to 375 K. The electrical resistance of these films goes through a minimum at a temperature T*. The change of resistance with temperature is semiconductor-like below T* and metal-like above T*. We argue that this behavior of the electrical resistance might not be the signature of a Mott-Hubbard metal-to-insulator transition but instead the signature of thermally activated electron conduction with a temperature-dependent electron tunneling where the tunneling gap between nanoparticles increases when the films expand thermally. Measuring the electrical resistances of different films revealed how T* relates to the size of the nanoparticles and to the expansion coefficients of the substrates on which the films were deposited. Simple model calculations agree well with the experimental data. Water/ ice trapped within nanopores in the films leads to a hysteretic resistance versus temperature behavior between 220 and 300K. Introduction Just as atoms can be used to form traditional materials, nanoparticles can be assembled to form artificial materials with novel properties. Understanding the fundamental properties of these materials is of critical importance for the design of nanoparticle-based architectures with tunable optical and electronic properties.1 One of the properties that is of fundamental and practical interest is charge transport.2,3 It has been shown that the charge transport through these artificial materials depends strongly on the type and size of the nanoparticles and on the electronic coupling between nanoparticles.2,4 The coupling is usually achieved by bridging the gaps between the nanoparticles with organic linker molecules. The electrical resistance of such materials, in the case of metal nanoparticles, has revealed two interesting temperature regimes.5–9 At low temperatures, thermally activated hopping (tunneling) takes place, and the change in electrical resistance with temperature is semiconductor- (or insulator-) like, whereas at higher temperatures the increase in the resistance with increasing temperature resembles that of a metal. Both temperature regimes are separated by a minimum in the electrical resistance. In the literature, this temperature dependence has been qualitatively attributed to a Mott-Hubbard metal-to-insulator transition,5–9 though no model calculations have yet been presented that quantitatively describe the experimental data. In this article, we show that an alternative and quite simple explanation can be used to describe quantitatively the experimental observation of a minimum in the electrical resistance by taking into account the thermal expansion of the nanoparticle film. Experimental Section The fabrication of the gold nanoparticle films has been described previously.10 Briefly, the nanoparticle films are * To whom correspondence should be addressed. E-mail: karl.muller@ csiro.au. Tel: +6194137052. † National Measurement Institute, P.O. Box 264, Lindfield, New South Wales, 2070, Australia.
produced in a two-step procedure: First, a solution of linker molecules is added to an aqueous suspension of gold nanoparticles which causes the nanoparticles to cross-link and form aggregates. In a second step, these nanoparticle aggregates are vacuum-filtered through a nanoporous membrane substrate, resulting in a film of nanoparticles accumulating on top of the membrane substrate. Once the solvent has evaporated, the assembled nanoparticle films are quite robust, can be handled and flexed without delaminating from the substrate, and do not redisperse in water. The film thickness can be varied by simply adjusting the volume of cross-linked nanoparticle suspension filtered through the substrate. We used two different nanoporous substrates, that is, polycarbonate track-etched membranes11 and alumina membranes.12 These substrate materials have quite different coefficients of thermal expansion RS (polycarbonate: RS ) 10-4 K-1 at 300 K, alumina: RS ) 5 × 10-6 K-1 at 300 K),13 allowing us to study the influence of the thermal expansion of the substrate on the conduction properties of the gold nanoparticle films. For each type of substrate, we assembled two different gold nanoparticle films, (a) films with a gold nanoparticle size of D ) 26 ( 3 nm linked by octanedithiol molecules, that is H-S-(CH2)n-S-H with n ) 8, and (b) films with a nanoparticle size of D ) 16 ( 2 nm linked by hexanedithiol molecules, that is H-S-(CH2)n-S-H with n ) 6. Thus, in total, four different samples were prepared for electrical resistance versus temperature measurements. The thickness of each of the four nanoparticle films was approximately 300 nm. For electrical resistance measurements, films were cut into strips with a typical width of 1 mm and length of 5 mm, and gold wires were attached using conductive silver paint with the contacts spaced about 1 mm apart. The electrical resistances of these films at room temperature were between 0.2 and 2 MΩ. These resistance values indicate that the film thickness had crossed the percolation threshold and that the resistance can be described in terms of 3D percolation as has been discussed previously.14 We measured the temperature dependence of the resistance of these films in the temperature interval 4-375 K
10.1021/jp9032639 CCC: $40.75 2009 American Chemical Society Published on Web 09/28/2009
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Figure 1. Normalized electrical resistance R/R(150 K) versus absolute temperature T for films of 26 nm diameter gold nanoparticles linked by octanedithiol molecules on either a polycarbonate or an alumina membrane substrate. The full curves show fits to the experimental data using eqs 1 and 2.
(using a Quantum Design MPMS platform) by applying between the contacts a fixed bias voltage of 1 V. Instead of four-probe resistance measurements, we used simple two-probe measurements, which are justified because all of our films have large resistances that make effects from contact resistances negligible. Also, we repeated some measurements using Be-Cu springs to contact the films and measured very similar film resistances indicating the unimportance of contact resistances. Results and Discussion For an electrical current to flow through a nanoparticle assembly, the electrons have to tunnel across gaps of width L between adjacent nanoparticles. The presence of linker molecules in the tunnel gaps enhances the tunneling rate compared to the tunneling rate across vacuum gaps.14,15 To initiate a current flow, an electron first has to tunnel from a charge-neutral particle to another charge-neutral particle. When doing so, the system has to enhance its energy by at least EC - e V12, where EC is the Coulomb blockade energy, V12 the applied voltage drop between adjacent nanoparticles and e the electron charge. The energy EC - e V12 can be delivered by thermal fluctuations, that is kB T, where kB is the Boltzmann constant and T the absolute temperature. When we apply a bias voltage of 1 V to our samples, the voltage V12 is extremely small because of the large number of nanoparticles along a film of 1 mm length, and thus e V12 < < kB T even at the lowest temperatures. Therefore, the temperature dependence of the electrical resistance R(T) of nanoparticle assemblies can be approximated by the expression16
R(T) ∝ eβL(T)eEC/kBT
(1)
Here, β is a parameter that describes the electron tunneling across the gap filled with linker molecules. The linker molecules themselves have a very high electrical resistance compared to the gold nanoparticles.15 The parameter β has been determined
experimentally for alkanethiol molecules and found to be approximately β ) 13 nm-1.17 The gap L(T) between nanoparticles in eq 1 is taken as temperature dependent to account for the fact that the film expands thermally. We have shown previously4,18 that, at low temperatures, disorder in the nanoparticle films leads to a deviation of R(T) from the simple exp (EC/kBT) Arrhenius behavior assumed in eq 1. In our previous work, we proposed that this deviation is due to disorder in our nanoparticle assemblies, which results in a wide distribution of charging energies EC.18 At low temperatures, electrons percolate along paths dominated by small EC’s, which results in an R(T) ∝ exp (EC/kBT)γ behavior where γ < 1.4 Figure 1 shows the measured normalized electrical resistance R/R(150 K) versus absolute temperature T for films with D ) 26 nm sized gold nanoparticles linked by octanedithiol (n ) 8) molecules on either a polycarbonate or an alumina porous substrate. The measurements were performed by first cooling the samples down from T ) 300 to 4 K (arrows in Figure 1) and then by warming up the samples from T ) 4 to 300 K. Strong hysteretic behavior can be seen above T ) 210 K, whereas below this temperature no clear evidence of any hysteric behavior could be found. The full curves in Figure 1 show fits through the experimental data applying eq 1 where for simplicity a linear thermal expansion has been assumed for the tunneling gap such that
L(T) ) L0[1 + Reff(T - 300K)]
(2)
Here, L0 is the average width of tunneling gaps between adjacent nanoparticles at T ) 300 K and Reff is the effective thermal expansion coefficient, which characterizes the thermal expansion of the tunneling gaps. The gap width L0 is approximated by the length of the linker molecules sandwiched between two gold surfaces, which for octanedithiol molecules was taken as L0 ) 1.51 nm. While fitting the normalized R(T)/ R(150K)data with eqs 1 and 2, EC and Reff were varied. As can be seen from Figure 1, good fits were obtained for a Coulomb blockade energy of EC ) 3 meV and Reff ) 1.3 × 10-4 K-1 for the gold nanoparticle film on the polycarbonate substrate, and Reff ) 3 × 10-5 K-1 for the film on the alumina substrate. A summary of the parameters is shown in Table 1. As can be seen from Table 1, the value of Reff depends on the substrate and Reff is larger for the substrate with the larger expansion coefficient. The assumption of a linear thermal expansion (eq 2) over a wide temperature is certainly an oversimplification. Most materials have a temperature-dependent thermal expansion coefficient that increases with increasing temperature. Assuming a more realistic thermal expansion would result in better fits to the experimental data but would require more parameters. The effective thermal expansion coefficient Reff, which describes the thermal expansion of the gaps between nanoparticles, arises from the interplay between the thermal expansion of substrate, nanoparticles, and linker molecules, and depends on the Young’s moduli of both nanoparticles and linker molecules. We have derived a simple formula for the effective thermal expansion coefficient Reff in terms of the substrate thermal expansion coefficient, Rs, the thermal expansion coefficient of gold, RAu, and of the linker molecules, Rm, and their Young’s moduli, EAu and Em, respectively. First, the geometrical constraint of a thin nanoparticle film on a thick substrate leads to the relation
Electrical Resistance of Gold Nanoparticles
J. Phys. Chem. C, Vol. 113, No. 42, 2009 18029
(∆L)T + (∆L)P + (∆D)T + (∆D)P ) (D + L0)RS∆T
(3) where (∆L)T and (∆D)T are the changes of the separation gap and of the nanoparticle size induced by the temperature change ∆T, and (∆L)P and (∆D)P are the changes induced by the intrinsic stress in the film parallel to the substrate. Second, the balance between the intrinsic stress in the molecules and in the nanoparticles leads to the expression
Em
(∆L)P (∆D)P ) EAu L0 + (∆L)T D + (∆D)T
(4)
Because (∆L)T < < L0, (∆D)T < < D, ∆L ) (∆L)T + (∆L)P and Reff ) ∆L/( L0 ∆T ), one obtains from eqs 3 and 4 for the effective thermal expansion coefficient
Reff )
[
D + L0 1 D RS + γRm - RAu 1+γ L0 L0
]
(5)
where γ ) Em D/( EAu L0 ). We have taken the values Em ) 53 GPa,19 EAu ) 78 GPa,20 and RAu ) 14 × 10-6K-1 21 from the literature. We obtain the best match of the Reff calculated from eq 5 with the Reff fitting our experimental data for the 26 nm nanoparticles on an alumina substrate (first row in Table 1) by using for the thermal expansion of the linker molecules the value of Rm ) 45.5 × 10-6K-1. We then use this Rm value to calculate Reff for the remaining three cases shown in Table 1. We find reasonable agreement between the calculated and the experimental Reff as shown in Table 1. The exact value of Em is debated in the literature,19 and, thus, if instead of Em ) 53 GPa, we would use Em ) 73 GPa, and Rm ) 41 × 10-6K-1, we obtain calculated Reff values that agree with the experimentally determined Reff within 2% in all four cases shown in Table 1. It should be pointed out that eq 5 neglects disorder, voids, and water/ice, which in addition might affect Reff. The Coulomb charging energy EC in 3D nanoparticle assemblies can be estimated using the Abeles formula,22 that is
EC )
L0 e2 2πε0εr D(D + 2L0)
(6)
where ε0 is the vacuum permittivity and εr the relative permittivity of the molecules surrounding the gold nanoparticles. In the literature, a value of εr ) 2.2 is often used for the dielectric constant of a surrounding alkane medium.6,18 As shown in Table 1, the calculated EC agrees reasonably well with the fitted EC value. The hysteretic behavior of R(T) seen during the cooling and warming-up cycle is shown more clearly in Figure 2. Inevitably, during the film formation process, residual water will be trapped
Figure 2. Normalized electrical resistance R/ R (150 K) versus absolute temperature T for films of 26 nm diameter gold nanoparticles linked by octanedithiol molecules on either a polycarbonate or an alumina membrane substrate. The R(T) water/ice hysteresis part is shown in detail. The homogeneous nucleation temperature TH of supercooled water and the melting temperature TM of ice are indicated by dashed vertical lines.
inside the nanopores of the membrane substrate as well as in the nanopores of the disordered nanoparticle film, and the hysteresis in Figure 2 can be attributed to the contraction/ expansion of water/ice in these pores.23,24 Supercooled water seems to exist in these nanopores down to the homogeneous nucleation temperature of water, that is TH ) 235 K.25,26 From Figure 2, it is apparent that the pronounced features in the hysteresis loop coincide well with TH and with the melting temperature TM of ice, that is TM ) 273 K. For the film assembled on the polycarbonate substrate, one can see a minimum in R(T) at TH when the sample is cooled down and a maximum at TM when the sample is warmed up. The volume of ice is 9% larger than that of water at 273 K and therefore the formation of ice causes the nanoparticle assembly to expand, which explains why the warming up part of R(T) appears above the cooling down part in Figure 2. It is interesting to note that during the warm-up cycle shown in Figures 1 and 2, a small fraction of ice seems to convert into water already at TH, far below the ice melting temperature TM. It is unclear from our data whether the R(T) hysteresis is dominated by the water/ice in the pores of the film or by the water/ice in the pores of the substrate. Nevertheless, the hysteresis observed in R(T) provides indirect evidence for the plausibility of eq 1, namely that substrate expansion/contraction can strongly affect R(T).
TABLE 1: Parameters substrate
D (nm)
n
L0 (nm)
EC, fit (meV)
EC, eq 3 (meV)
Rs @300K (10-6 K-1)
Reff, fit (10-6 K-1)
Reff, eq 7 (10-6 K-1)
alumina polycarbonate alumina polycarbonate
26 26 16 16
8 8 6 6
1.51 1.51 1.26 1.26
3.0 3.0 7.0 7.0
2.6 2.6 5.6 5.6
5a 100a 5a 100a
30 130 30 130
30 166 29 164
a
Ref 13.
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Figure 3. Normalized electrical resistance R/ R (150 K) versus absolute temperature T for films of 16 nm diameter gold nanoparticles linked by hexanedithiol molecules on either a polycarbonate or an alumina membrane substrate. The full curves show fits to the experimental data using eqs 1 and 2.
Figure 3 shows the measured normalized electrical resistance R/ R (150 K) versus absolute temperature T for films of gold nanoparticles with size D ) 16 nm linked by hexanedithiol (n ) 6) molecules. Again, the substrate is either polycarbonate or alumina and the measurements were performed by first cooling the samples down from T ) 300 to 30 K and then by warming the samples up from T ) 30 to 300 K. As in Figure 1, hysteretic behavior can be seen above T ) 210 K. Parameters and fitted values for EC and Reff are shown in Table 1. The calculated EC agrees reasonably well with the fitted EC value. The same parameters for the effective thermal expansions Reff that were used before in Figure 1, again give good fits to the data. In contrast to Figure 1 where, for the polycarbonate substrate, the minimum in the resistance R(T) is at T* ≈ 135 K, the minimum in Figure 3 has shifted to the higher temperature of T* ≈ 190 K. Thus, the data show that an increase in EC leads to an increase in T*. Using eqs 1 and 2, one finds for the temperature T* where R(T) has a minimum
T* )
EC kBβL0Reff
Figure 4. Temperature T* where R(T) has a minimum versus the Coulomb charging energy EC. The full lines represent eq 7 using the values of Figure 1 for the effective thermal expansion Reff. The values for L0 correspond to the length of the octanedithiol (n ) 8) or hexanedithiol (n ) 6) linker molecules. The triangles show the experimental T* values obtained from the measurements shown in Figures 1 and 3 for the polycarbonate substrate.
Figure 5. Normalized electrical resistance R/ R (220 K) of nanoparticle films versus absolute temperature T for hexanedithiol-linked gold nanoparticles of 16 nm diameter on an alumina membrane substrate and for octanedithiol-linked gold nanoparticles of 26 nm diameter on a polycarbonate and on an alumina membrane substrate. The temperature was increased from 200 to 375 K and then decreased to 290 K.
(7)
Figure 4 displays T* versus the charging energy EC calculated using eq 7 where Reff ) 1.3 × 10-4 K-1 for the films on the polycarbonate substrates and Reff ) 3 × 10-5 K-1 for the films on the alumina substrates. For L0 the values for the lengths of hexanedithiol and octanedithiol linker molecules are used. The triangles in Figure 4 represent the experimental T* values taken from Figures 1 and 3. Good agreement between our simple model and the experimental data is obtained. As can be seen in Figure 4, in the case of films with D ) 26 nm on alumina substrates, a minimum in R(T) is expected at T* ≈ 250 K. Unfortunately, the predicted minimum in R(T) cannot be identified clearly in Figure 1 because of the presence of the water/ice-R(T) hysteresis at that temperature.
We also measured the electrical resistances of the nanoparticle films up to a temperature T ) 375 K. The time interval between two consecutive data points was 5 min. Figure 5 displays R/ R (220 K) versus absolute temperature for three different nanoparticle films. The data shows that at temperatures above 330 K the electrical resistance starts to drop. In the case of the film with D ) 26 nm and n ) 8 on an alumina substrate, the film resistance dropped by more than 2 orders of magnitude when the temperature was taken up to 375 K. Decreasing the temperature from 375 down to 293 K revealed irreversibility of R(T). We did not measure below 293 K to see if the films had become metal-like down to 4.2 K or still showed thermally activated behavior, neither did we inspect the films whether they had undergone morphological changes. Because of the large resistance drop seen in Figure 5, we suspect that partial sintering
Electrical Resistance of Gold Nanoparticles of gold nanoparticles might have occurred upon heating to 375 K and that a metallic network might have been formed.27 Despite of the large drop in resistance, the electrical resistivity of this nanoparticle film at 293 K was found to be still about 7000 times larger than that of bulk gold. It should be noted here that metals by definition do not have any barrier that needs to be overcome in order for charges to flow. It has been pointed out previously by others28,29 that films of gold nanoparticle films linked with hexanedithiols (n ) 6) and octanedithiols (n ) 8) do not behave like metals. It also has been observed by some researchers29 that, when linked with dithiols where n e 5, R(T) showed behavior down to 4.2 K that is typical for a metal, thus indicating a possible MottHubbard metal-to-insulator transition. We have previously investigated gold nanoparticle films linked with ethanedithiol (n ) 2) and butanedithol (n ) 4) and in contrast, in our films, we have not observed metallic but instead thermally activated behavior even for these short dithiol molecules.4,18 It is important to note that even for n ) 2, the Coulomb blockade energy that we have measured did correspond to the size of a single nanoparticle and not to a larger size of metal-like aggregates formed by several nanoparticles.4 In our case, the nanoparticle films were prepared by filtration of preaggregated dithiol-gold nanoparticles onto a porous substrate. Because during film formation there is an excess amount of dithiols present in the aqueous solution, the aggregates are thought to link with each other via dithiol linker molecules when landing on the substrate.10 The films made by this method are strongly disordered and have a large void content. Methods that have been used previously by others that show R(T) behavior with a minimum, similar to ours, used either the drop-casting method6,9 or a Langmuir method.7,8 In these cases, nanoparticles are not cross-linked but capped with thiols and weak van der Waals interactions keep the particles together, producing films that are quite well ordered with high packing densities. Despite differences in the film morphology between these films and our films, the physics of electrical conduction is that of electron tunnelling across gaps between nanoparticles and, independent of the morphology, the temperature dependence of the electrical resistance, R(T), can be described by eq 1. The effect of percolation of electrons through a disordered tunnel-resistor network seems to become only important at low temperatures (
