Fluctuation-induced tunneling conduction

With the metallic nature of our individual RuO₂ NWs being established, we can already safely use these NWs to study the electronic contact resistances R_c(T ) in lithographic-contacting nanostructures, using the 2-probe method.

Figure 4.2 shows the typical temperature behavior of R_c for such a high-resistance contact from 300 K down to liquid-helium temperatures, as deter-mined from the 2-probe method on a RuO2 NW. In strong contrast to the 4-probe results previously studied, now the measured resistance reveals semi-conducting or insulating behavior, i.e., the resistance increases rapidly with decreasing temperature.

The inset to Fig. 4.2 shows the variation of logR with T⁻¹ for the same de-vice. This figure indicates that the simple thermally activated conduction (the straight solid line) is only responsible near room temperatures. As the temper-ature reduces from room tempertemper-ature, the resistance does not increase as fast as would be expected from the thermal activation process. At liquid-helium temperatures, the resistance appears roughly constant, i.e., independent of temperature. Such a temperature independent resistance at liquid-helium tem-peratures can signify a conduction mechanism characteristic of simple elastic tunneling. Indeed, quantitative analysis (see below) indicates that the overall temperature behavior of the contact resistance can be well interpreted in terms of a tunneling conduction model. Previously, in order to explain the electrical-transport properties in certain classes of granular metal-dielectric composites, Sheng and coworkers [71, 72] have proposed a thermally “fluctuation-induced tunneling” (FIT) model, where the thermal effects arising from the capacitance C of a small junction formed between two large metal grains was considered.

According to Sheng and coworkers [71, 72], the temperature dependent resistance for small applied electric fields across a single small junction can be expressed as

Figure 4.2: Resistance as a function of temperature for two high-resistance contacts in series, as determined from the 2-probe method on the NW device C3 (see Table 4.1). The inset shows a plot of logR versus T⁻¹ for the same NW device. The straight solid line indicates the thermal activation conduction.

where R₀ is parameter which depends only weakly on temperature, and T₁ and T₀ are characteristic temperatures defined as

T₁ = 8ε₀

where ε₀ is the vacuum permittivity, ~ is the Planck’s constant divided by 2π, and m is the electron mass. In Eq. (4.1), T₁ can be regarded as a measure of the energy required for an electron to cross the potential barrier between the two conducting regions, and T₀ is the temperature below which the fluctuation effects become insignificant, since, at T ¿ T₀, Eq. (4.1) is temperature inde-pendent and reduces to an expression for the expected simple elastic tunneling.

In the derivation of Eq. (4.1), the conduction was first modeled [71] as the tun-neling of electrons through a single potential barrier of width w, height V0, and junction area A. (A is the size at the point of the two large conducting regions’

closest approach.) If A is small enough, it was found [71, 72] that the potential barrier seen by the electrons could be effectively narrowed and lowered by the thermal voltage fluctuations (≈ p

k_BT /C, where k_B is the Boltzmann con-stant) across the insulating gap due to the small effective capacitance of the junction. Such a potential-barrier modulation effect greatly influences the tun-neling probability in the low temperature limit, and consequently introduces a characteristic temperature behavior to the normally temperature independent tunneling conductivity (i.e., the elastic tunneling regime).

In the case of granular composites of macroscopic sizes, it was then argued [72], via the effective-medium theory, that in a network of independently fluc-tuating tunnel junctions with different values of T1 and T0, the conductivity of the network could still be well described in terms of a single junction with a representative set of T₁ and T₀.

Figure 4.3 shows a plot of our experimental results in double logarithmic scale for four representative NW devices having high contact resistances (as determined from the 2-probe method). The symbols are the experimental data

and the solid curves are the least-squares fits to Eq. (4.1), with R₀, T₁ and T₀ as the adjusting parameters. Inspection of Fig. 4.3 clearly indicates that the Eq. (4.1) can well describe the overall temperature behavior for a wide range of temperature between 2 and 300 K. The values of T₁ and T₀can then be reliably extracted. Furthermore, by using SEM and/or AFM, we can directly measure the diameter and the length of our NWs, as well as the width of the relevant submicron Cr/Au electrodes overlying the NW. Therefore, the junction area A which appeared in Eqs. (4.2) and (4.3) is independently determined. (The junction area A is given by the product of the NW diameter and the width of the overlying submicron electrode.) With the values of T₁, T₀ and A being determined, the microscopic parameters characterizing the electronic contacts, i.e., the width w and height V₀ of the potential barrier, may then be inferred.

Our experimental values of the relevant parameters are listed in Table 4.1.

Notice that, in Table 4.1, the measured resistance R(300 K) for each NW device is at least an order of magnitude higher than the intrinsic resistance of the NW, R_s(300 K), justifying our approximation R(T ) ≈ 2R_c(T ). Moreover, the resistance ratio R(T )/R_s(T ) increases rapidly as the temperature decreases below room temperature.

It should be noted that, in the FIT model, because the two conducting regions remain large in size, the charging energy Ec needed to transfer an electron from one conducting region to the other is completely negligible, i.e., E_c ¿ k_BT . This situation is very different from that in the case of Coulomb blockade which involves fine metal grains or quantum dots, where the charging energy E_c (À k_BT ) rather than the thermal voltage fluctuations plays the crucial roles in controlling the electronic transport properties. In the present work, the volume of our “long” NW is relatively large as compared to the sizes of the fine metal grains (e.g., ∼ 10³nm³) used in Coulomb blockade studies.[73]

The typical volume of our NWs is ∼ 100 nm × 100 nm × 3 µm. Thus, our NW can be envisioned as a large conducting region separated by an insulating layer from another large conducting region (the submicron Cr/Au electrode) with a junction area A (∼ 100 × 500 nm²). This size of A is already small enough to render the aforementioned thermal voltage fluctuations important while large

Figure 4.3: Double logarithmic plot of the resistances versus temperature for four high-resistance NW devices, as determined from the 2-probe method.

The symbols are the experimental data and the solid curves are the theoretical fits to Eq. (4.1).

Table 4.1: Values of relevant parameters for four high-resistance NW devices, as determined from the 2-probe method. For each device, the NW resistance R_s(300 K) was estimated from the 4-probe method, while the junction area A was determined from the SEM image.

R(300 K) Rs(300 K) R0 T1 T0 A w V0

(kΩ) (kΩ) (kΩ) (K) (K) (µm²) (nm) (meV)

C1 241 0.67 112 316 91 0.023 6.6 4.2

C2 33 0.33 13.6 363 104 0.022 6.4 4.5

C3 10.3 0.31 8.08 87 37 0.040 7.0 1.7

C4 3 0.33 2.82 20 8.6 0.028 8.7 1.1

enough to make Coulomb blockade irrelevant. In effect, our NW devices in the 2-probe configuration mimic two similar tunnel junctions in series in the context of the FIT model.

Previously, the FIT model has been successfully applied to explain the tem-perature behavior of the resistances in, among others, carbon polyvinylchloride composites [71, 74, 75], polymer composites [76], and tin-doped indium oxide thin films [77]. In those “macroscopic” composite systems, a very large num-ber of tunnel junctions with barely known junction properties were involved.

On the contrary, the situation is greatly simplified and straightforward in our case, since in the 2-probe configuration we deal with only two electronic con-tacts characterized by similar junction parameters, as discussed. Moreover, our junction area A is known. Interestingly, our experimental values of w and V₀ listed in Table 4.1 are on the same orders of magnitude to those obtained in carbon polyvinylchloride composites [71, 74, 75]. This coincidence may be due to the fact that the sizes of our NWs are approximately the same to the mean size of the conducting chains found in those composites.

Finally, it is worth noting that, if in our case, the effective junction area is somewhat reduced from the maximum possible area A defined above, our values of w (V0) would be slightly decreased (increased) from those listed in Table 4.1.

4.3 IrO

4.3.1 Electron hopping conduction

In this subsection, we report the temperature behaviour of the highly resistive R_c(T ) for two electronic contacts measured on a representative IrO₂ NW (Ir-3) connected by three submicron electrodes. The sample parameters of the Ir-3 NW are given in the caption to table 4.2. As discussed in Section 2.2, the values of R_c(T ) can be extracted from the electrical measurements by employing either the 3-p [Fig. 2.5(c)] or the 2-p [Fig. 2.5(d)] configuration, provided that R_c À R_sand R_cÀ R_el. At 300 K, the resistances obtained from the 3-p and 2-p configurations for this 2-particular NW are 2.4 and 5.6 kΩ, res2-pectively. These

measured values are indeed significantly larger than the intrinsic resistance (≈

0.5 kΩ) of this NW, suggesting that the contact resistance(s) being dominating the measured resistances. More precisely, the measured R_3−p and R_2−p are largely determined by the electronic contact resistances R_c2and R_c3as denoted in Figs. 2.5(c) and (d). In the 3-p method, the measured resistance R = R_3−p ≈ R_c3; while in the 2-p method, the measured resistance R = R_2−p ≈ R_c2+ R_c3.

Figure 4.4(a) shows our experimental results for R3−p and R2−p as a func-tion of temperature. Inspecfunc-tion of Fig. 4.4 indicates that, in sharp contrast to the 4-p results discussed previously, now the measured resistances also re-veal semiconducting or insulating behaviour, namely, the resistance increases rapidly with decreasing temperature. Below about 50 K, a sharp resistance rise is found. Quantitatively, as the temperature reduces from room temperature to liquid-helium temperatures, the resistance ratio R_2−p/R_4−p increases from

≈ 20 to ≈ 500, ensuring the predominance of the electronic contact resistances on the measured resistances especially at intermediate and low temperatures.

The relevant parameters measured for the 3-p and 2-p configurations are listed in table 4.2.

A plot of logR as a function of 1/T is shown in the inset of Fig. 4.4(a), the nonlinear dependences of the 3-p and 2-p results suggest that the simple thermally activated conduction is not the responsible mechanism for our obser-vations. Instead, if we plot logR as a function of T^−1/2, linear dependences are clearly obeyed for a very wide temperature range from about 100 K down to liquid-helium temperatures, as depicted in Fig. 4.4(b). The difference between the two curves are simply the electronic contact resistance R_c2(T ). Such a logR ∝ T^−1/2 behaviour is frequently observed in materials like granular met-als [78] and disordered semiconductors [79]. The resistance can be expressed as

R(T ) = R_∞exp[(T₀/T )^1/2] , (4.4) where R_∞ and T₀ are material dependent parameters and are insensitive to temperature. Our fitted values of R_∞ and T₀ are listed in table 4.2. In

Figure 4.4: (a) Resistances as a function of temperature for the 2-probe and 3-probe measurement configurations as depicted in Figs. 2.5(c) and (d). The inset shows logR as a function of 1/T . (b) logR as a function of T^−1/2. Since R_2−p ≈ R_c2+ R_c3 and R_3−p ≈ R_c3, the difference between the two curves is simply the electronic contact resistance R_c2.

Table 4.2: Values of the relevant parameters for the three-probe and two-probe electrical measurement configurations implemented on the Ir-3 NW con-tacted by three submicron Cr/Au (10/90 nm) electrodes. The Ir-3 NW has a hypotenuse W ≈ 115 ± 5 nm, length L ≈ 0.7 µm, and the sample resistance R_s(300 K) ≈ 0.5 kΩ. R_∞ and T₀ are defined in Eq. (4.4).

R(300 K) R(100 K) R(10 K) R_∞ T₀

(kΩ) (kΩ) (kΩ) (kΩ) (K)

3-p 2.4 5.2 56.1 1.9 113

2-p 5.6 11.6 117.4 4.3 109

disordered semiconductors, the form of Eq. (4.4) can be given by the one-dimensional Mott [80] variable range hopping (VRH) between localized car-rier states near the Fermi level, or by the Efros-Shklovskii [81] VRH if the Coulomb interaction between carriers is taken into account. In granular met-als, the form of Eq. (4.4) arises from the conductivity model of Sheng and co-workers [78, 82], in which a structural effect is considered. In our case, since the 90-nm thick Au film (which formed the top layer of the submicron electrode) and the IrO2 NW are “good” metals, it is conjectured that our measured resistance of ∼ several tens kΩ (at a few tens K and lower) must be dominated by a resistance due to the nominally 10-nm thick granular Cr layer deposited between the thick Au film and the IrO₂NW. It can also be due to the amorphous coating and/or the noncrystalline structure of the outermost one or two atomic layers of the as-grown IrO₂ NW. The granular structure of the deposited Cr thin layer might have accidentally formed due to the breaking induced by tensile stress, the vacancies caused by dramatic surface roughness near the contact region, or the lightly contaminated metal grains during evap-oration. To check the structure, we have made several samples comprising a Cr layer with a nominal thickness of 10 nm on mica substrates using simi-lar deposition conditions as used for the submicron electrode fabrication, and analyzed the Cr layer surface profiles by atomic force microscopy (AFM). As expected, a granular pattern with a distribution of disk-shaped grains having radius of ∼ several tens nm and height of ≈ 2−6 nm has been observed in several cases (see Fig. 4.5), supporting the aforementioned conjecture. Indeed, it is known that thermal-evaporation deposited thin Cr films can easily form island-like granular structures rather than continuous layers [83].

Apparently, the one-dimensional Mott [80] VRH process is inappropriate for the explanation of our data, considering the geometrical structure around the contact region of our samples, namely, the size of the thin Cr layer in the transverse directions of electrical transport is more than an order of magnitude larger than that in the longitudinal direction. Although the Efros-Shklovskii [81] VRH theory is often used to fit the resistivity data in the studies of gran-ular metals where a logR ∝ T^−1/2 behaviour is observed, it has been shown

Figure 4.5: (a) AFM image of a thin Cr layer with a mean thickness of 10 nm prepared by thermal-evaporation deposition on a mica substrate. (b) Surface profile along the line indicated in (a), showing a distribution of disk-shaped grains having radius of ∼ several tens nm and height of ≈ 2−6 nm.

[84, 85] that serious inconsistencies exist in applying such transport model to those systems due to the unreasonable extracted values of the relevant param-eters. For instance, an optimum hop distance is found to be too short to allow the electrons to tunnel beyond neighboring metal grains, or, a tunnelling bar-rier is found to be smaller than k_BT (where k_Bis the Boltzmann constant), and thus made the comparison unrealistic. Therefore, we believe that the adequate mechanism to describe our data is the conductivity model proposed by Sheng and co-workers [78, 82], i.e., conduction electrons are thermally activated and hop through, in our case, the nanoscale Cr granules sandwiched between the thick Au layer and the IrO₂ NW.

In the Sheng’s model [78, 82], it is proposed that, for a granular metal-dielectric composite sample with a uniform relative volume fraction of metal and dielectric, the ratio s/d should have the same value everywhere throughout the sample, where s is the separation of neighboring metal grains, and d is the diameter of metal grains. Since the electrostatic charging energy E_c ∼ 1/d (which is required to create a positive-negative charged pair of grains), it thus follows that sE_c is a constant everywhere in the sample and can be written as

sE_c= k_BT₀

4χ , (4.5)

where T₀is the characteristic temperature parameter in Eq. (4.4), χ = (2mφ/~²)^1/2, m is the effective electron mass, φ is the effective barrier height, and ~ is Planck’s constant divided by 2π. With a second assumption that only hop-ping between nearest-neighbor grains which are equal or nearly equal in size is included, the model found that, at each temperature T , the maximum conduc-tivity occurs at a dominant separation of neighboring metal grains, s_m, given by

From Eqs. (4.5) and (4.6), it follows that, at high (low) temperatures, the conductivity is governed by the hopping events between small (large) grains separated by a short (long) distance. From our fitted value of T₀ ≈ 110 K, and assuming a free electron mass and a barrier height φ ≈ 0.1 eV, we

obtain E_c ≈ 16 (3) meV and s_m ≈ 1 (5) ˚A at T = 300 (10) K. (A value of φ ≈ 0.1 eV corresponds to a small barrier height which is about an order of magnitude larger than the thermal energy k_BT , and is just probable for tunnelling to occur.) These values are quite close to the values obtained in the previous works on granular metallic systems [78] and suggest that our thin Cr layers lie in the dielectric regime approaching the threshold for classical percolation conductivity. The smallness of the values of s_m results from a high volume fraction of metal (Cr), which in turn renders a low value of the effective tunnelling barrier due to considerable image forces [78, 82].

Chapter 5

Conclusion

By using different probe configurations in our measurements, we have mea-sured down to liquid-helium temperatures not only the intrinsic electronic transport properteis of individual single-crystalline RuO₂ and IrO₂ NWs but also the temperature dependent behaviours of high-resistance electronic con-tacts R_c on these two kinds of NWs.

For the intrinsic properties of the NWs, although the measured tempera-ture dependent resistivities can be well accounted for by an existing theoretical description based on the Boltzmann transport theory, we found that the De-bye temperature in RuO₂ NWs is largely reduced as the diameter of the NW decreases. (Comparable experiments on IrO2 NWs with diameters down to this scale have not been performed.) Possible mechanisms accounting for this observation have been discussed. It is concluded that the chemical binding in the NWs may be gradually weakened as the diameter decreases.

By employing the 3- and 2-probe method, we have quantitatively charac-terized the temperature behaviours of high resistance electronic contacts, R_c, formed at the interfaces between the submicron electrodes and the NWs. Two different behaviours have been observed. For RuO₂ NWs, we found that the temperature dependence of the R_c can be well attributed to the thermally fluctuation-induced tunneling conduction through a junction formed at the interface between the electrode and the NW. The junction parameters such as the barrier width and height have been determined. On the other hand, For IrO2 NWs, a temperature behaviour obeying the law logR ∝ T^−1/2 is observed over a wide temperature range below ≈ 100 K. This behaviour is satisfactorily ascribed to the hopping of electrons through nanoscale Cr gran-ules and/or amorphous coating incidentally formed at the interface between the Cr/Au submicron electrode and the NW. Less direct evidence supporting

this argument has been provided.

This work demonstrates that, by properly applying a combination of elec-trical measurement configurations, both the intrinsic property of a NW and the electronic contact on it could be quantitatively studied. Under certain condi-tions, the electronic contacts between an interconnect and a metal nanodevice could be further modeled.

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