Nanowires with diameters ' 100 nm

Despite the disorder nature of our NWs, we found that for the temperature behaviour above T_min our experimental data still can be explained within the framework of the Boltzmann transport theory for typical metals, as given in Section 3.2.

Figure 3.4 shows the log-log plot of the variation of the measured normal-ized resistivity, (ρ − ρ₀)/ρ₀, as a function of temperature for two NWs with diameters ≈ 100 nm. The solid line is the least-squares fit to Eq. (3.9). As mentioned, we merely took ρ0, ΘD, βBG, and ΘE as adjusting parameters in our fitting. The fitted values of these parameters for all NWs are listed in

Figure 3.3: Variation of the resistivity at T = 300 K with diameter of the NW for the NWs measured in the present work.

40 60 80 100 200 400

Figure 3.4: log-log plot of the variation of the measured normalized resistiv-ity, (ρ − ρ₀)/ρ₀, as a function of temperature for two NWs with diameters ≈

Table 3.2. Figures 3.5–3.8 show the variations of Θ_D, Θ_E, β_BG/ρ₀,³ and ρ₀ with diameter, respectively. In this section, only NWs with diameters ' 100 nm would be discussed. The results for NWs with diameters < 100 nm would be discussed later.

We first notice that the fitted values of Θ_D and Θ_E are nearly identical for the NWs with diameters ' 100 nm. However, in comparison with the corresponding bulk values (Θ_D ≈ 400–410 K and Θ_E ≈ 790–810 K) in single crystals [19, 43], we find that the ΘD has a value of ≈ 400 ± 25 K, while the ΘE has a value of ≈ 700 ± 25 K. A reduction of ≈ 13 % in ΘE is observed.

If we additionally set Θ_D and Θ_E to their bulk values (i.e., 400 and 790 K, respectively) and left only ρ₀ and β_BG as free variables, we found that for all the NWs no satisfactory results could be obtained as expected. A typical best fit in this way is shown in Fig. 3.9 for the sample Ru-4.

At first glance, since we are dealing with NWs, one might conjecture that this reduction in the effective phonon characteristic temperatures results from the increased electron scattering rate with surface phonons, which are softer than bulk phonons due to the lowered coordination number of surface atoms leading to their enhanced vibrational amplitudes. Nevertheless, in our case, as mentioned above, this electron-surface scattering is less effectual, because in our NWs, the much shorter electron mean free path as compared with the diameter of the NWs makes this process much infrequent. As a matter of fact, a change in phonon related properties such as these characteristic temperatures could also be induced by some structural [53, 54], compositional [55, 56], and mechanical [57, 58] effects. For instance, in thin gold films the lowering of the effective Debye temperature is ascribed due to the electron scattering with additional surfaces created by grain boundaries [53]. However,

3In fact, the measured resistance R was fitted to the equation of the form:

R − R0

R0 =ρ − ρ0

ρ0 =βBG

ρ0 f (ΘD, ΘE) ,

where R0 is the residual resistance, and f is a function of ΘD and ΘE, obtained from Eqs. (3.4) and (3.6). Therefore, the actual adjusting parameters are R0, βBG/ρ0, ΘD, and ΘE.

Table 3.2: Values of the relevant parameters for the eight RuO₂ NWs mea-sured by the 4-p method.

Fit range ρ₀ Θ_D (β_BG/ρ₀) Θ_E ρ₃₀₀/ρ₀

(K) (µΩ cm) (K) (10⁻³/K) (K)

Ru-1 55–300 93 ± 27 426 ± 15 3.66 ± 0.04 726 ± 13 1.595 Ru-2 30–300 420 ± 303 365 ± 4 4.30 ± 0.01 689 ± 4 1.725 Ru-3 40–300 194 ± 134 372 ± 5 4.42 ± 0.02 687 ± 4 1.746 Ru-4 50–300 166 ± 74 377 ± 22 4.07 ± 0.06 674 ± 16 1.695 Ru-5 55-300 117 ± 39 373 ± 19 3.39 ± 0.04 690 ± 12 1.571 Ru-6 70–300 569 ± 375 393 ± 22 2.66 ± 0.03 672 ± 8 1.449 Ru-7 35–300 300 ± 225 324 ± 7 3.58 ± 0.02 692 ± 5 1.611 Ru-8a 30–300 119 ± 62 213 ± 13 3.07 ± 0.01 682 ± 2 1.532 Ru-8b 30–300 100 ± 52 234 ± 14 3.21 ± 0.02 617 ± 2 1.585 Ru-8ab 30–300 92 ± 29 229 ± 15 3.11 ± 0.01 660 ± 1 1.542

Figure 3.5: Variation of Θ_D with diameter.

Figure 3.6: Variation of Θ_E with diameter.

Figure 3.7: Variation of β_BG/ρ₀ with diameter.

Figure 3.8: Variation of ρ₀ with diameter.

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Figure 3.9: Best fit for the sample Ru-4 by intentionally setting Θ_D and Θ_E to their bulk values (i.e., 400 and 790 K, respectively) and leaving only ρ and

apart from being caused by these effects, we found that in the present work another possibility could also account for the observed reduction in Θ_E owing to the simple fact that, in our moderately disordered NWs, we interpreted the resistivity variations as arising solely from the “pure” electron-phonon scattering process that usually dominate in clean metals.

Experimentally, it has been known for a long time that deviations from Matthiessen’s rule exist in many real metals. That is, the additivity and in-dependence which bring about Eq. (3.8) are no longer valid as the level of disorder in a metal increases. Recently, it becomes clear that such a devia-tion in an impure metal can be readily accounted for by merely adding an additional term into Eq. (3.8). This additional contribution was theoretically proposed and calculated by Reizer and Sergeev [59] as a consequence of the quantum interference mechanism between the elastic electron scattering and the electron-phonon scattering. It can be written as

ρ_int(T ) = β_intρ₀T² where β_int is a material dependent constant, being independent of the amount of disorder contained in the sample. Such an “electron-phonon-impurity in-terference” has been experimentally demonstrated by many groups in many different material systems [48, 54, 60, 61]. It is shown that this effect can dominate the overall behaviour of ρ(T ) at lower temperatures, especially at temperatures below ≈ 0.1Θ_D. However, as temperature increases, the increase of this contribution becomes less temperature dependent and finally saturates to a constant value at T ≈ Θ_D. In principle, at these higher temperatures Eq. (3.10) contributes just a small amount to the total ρ(T ) (as compared with that contributed by the electron-phonon scattering) and thus can some-times be neglected in the first approximation.

Figure 3.10 shows the result obtained by fitting the experimental data of the same NW presented in Fig. 3.9 to the equation of the form:

ρ(T ) = ρ₀+ ρ_BG(T ) + ρ_E(T ) + ρ_int(T ) , (3.11) in which we intentionally set Θ_D and Θ_E to their bulk values, and then left

only ρ₀, β_BG, and β_int as adjusting parameters. In comparison with Fig. 3.9, it is clear to see that the experimental data turns out to agree quite well with the theoretical description including this additional correction.

This correction is ≈ 6 % of the total resistivity at T ≈ 400 K. It thence slightly decreases as temperature decreases, and eventually becomes more im-portant at temperatures approaching ≈ 40 K.⁴ In fact, it is not surprising at all to have reduced values of Θ_D and Θ_E if we regard the measured ρ(T ) as exclusively originating from the electron-phonon scattering mechanisms de-scribed by Eqs. (3.4) and (3.6). Phenomenologically, in order to compensate for lack of including Eq. (3.10), more collisions of electrons with optical and acoustic phonons accordingly had to be produced to take this accountability.

At higher temperatures, the dominant behaviour of ρ(T ) in this range is con-trolled by Eq. (3.6), in which Θ_E corresponds to the unique optical phonon energy in the Einstein approximation. To offer the needed extra resisting force against the motion of electrons, the Θ_Ehad to lower itself so that more optical phonons could be more easily excited. The environment consequently became more chaotic as seen by electrons. At last, the resistivity was compensated.

Likewise, at lower temperatures, the optical phonons would quickly disappear, and the acoustic phonons thus became in this range the dominant source that the electrons could collide with. As described above, Eq. (3.4) has a tem-perature behaviour which crosses over from a T dependence at temtem-peratures above ≈ Θ_D to a T⁵ dependence at temperatures approaching zero. A lower Θ_Dmeans that such a crossover would occur at relatively lower temperatures.

Therefore, the T dependence would prevail over a wider range of temperature to make the needed compensation for not including Eq. (3.10), which has a weaker T² dependence at lower temperatures.

4Due to the low temperature upturn in our NWs, any contribution originating from Eqs. (3.4), (3.6), and (3.10) would be masked. Without this upturn, Eq. (3.10) would dominate the temperature behaviour down to the lowest temperature until it gets masked again by the residual resistivity, as can be seen in Ref. [61].

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Figure 3.10: Best fit for the sample Ru-4 to the Eq. (3.10) by intentionally setting Θ_D and Θ_E to their bulk values (i.e., 400 and 790 K, respectively) and leaving only ρ0, βBG, and βint as free variables.