Size-dependent Debye temperature

As mentioned earlier, a softening of an effective Θ_D can be ascribed to the increased number of phonon modes due to additional surface phonons that

5The as-obtained βBG’s are nearly not changed. Its variation with diameter is almost the same as that in Fig. 3.7.

Figure 3.11: Variation of β_int with diameter.

electrons can scatter from. For that to be valid, an electron mean free path l comparable to the characteristic dimension of the system is certainly required.

For example, in thin metallic films grown by ultrahigh vacuum molecular beam epitaxy [62], the Θ_D was found to start deviating from the bulk value in the films of thickness below ≈ l. A considerable reduction of ≈ 50 % in Θ_D was found in the thinnest film, which had a thickness approximately just one third of the l. However, in our case, the observed significant decrease in Θ_D with decreasing diameter apparently cannot be simply explained in terms of such surface scattering because of the l being much shorter than the diameters of the NWs. Even for the thinnest one, its diameter is still several tens times longer than its l.

Obviously, we used a very simple model to describe the temperature be-haviour of the resistivity in our NWs. In deriving the Bloch-Gr¨uneisen formula, the real but complicated and barely known phonon spectral function α²F (ω) is replaced by a more simpler one, in which the phonon density of states hav-ing a quadratic dependence with a cutoff frequency (i.e., the Debye frequency) ω_D = k_BΘ_D/~ is used. In this Debye model, the phonon dispersion relation (spectrum) follows the form ω = v_sq, where v_s is the sound speed of the ma-terial and q is the phonon wavevector. Although the approximation is rather crude, the Bloch-Gr¨uneisen formula has often been found to work remarkably well when compared with experiments. However, one might still conjecture that this simplification would make serious illegitimacy if the acoustic phonon dispersion relation of the NW is substantially modified from that of its bulk as the diameter of the NW decreases down to a certain scale. The change in Θ_D thus corresponds to an outcome that results from the application of an inappropriate and too simple model.

Indeed, spatial confinement of phonons by boundaries or interfaces of the system could render material properties like optical spectra, electrical and ther-mal conductivities which depend on their dynamics subject to change when one or more dimensions of the system are reduced down to the scale close to the characteristic length, such as the phonon mean free path (MFP) and the dominant phonon wavelength λ [63, 64]. Specifically, for acoustic phonons [63],

no significant modification in the phonon spectrum has been found in systems with the feature size W smaller than the phonon MFP. On the other hand, however, a pronounced effect on the phonon spectrum is found to occur in sys-tems with the W further down to the scale close to or smaller than the λ. In fact, it has been theoretically shown [65] that such an acoustic phonon confine-ment effect can manifest itself by introducing not only acoustic subbands but also non-linear and considerably changed phonon dispersion relation curves at q close to the Brillouin zone center. For a free-standing nanostructure, even more complex dispersion relation curves could occur at the free surfaces due to the wave coupling of different vibrational modes [66].

The confinement effect can be roughly understood by simple means. It is known that when fundamental excitations, like phonons and electrons, are confined by the boundaries of the system, they are only allowed to exist at certain energy levels due to their wave nature. In bulk materials at tempera-ture T , these energy levels are almost identical and basically not distinguish-able from each other by experiments. As the W of the system decreases and becomes comparable with the wavelength of the excitation, the energy sepa-ration between these levels can be raised to a value larger than the thermal energy of the lattice kBT ,⁶ and hence becomes non-negligible and causes the related material properties to change. (Clearly, instead of reducing the W , the confinement phenomenon can as well be observed by decreasing the temper-ature of the system.) It is this energy “gap” that alters the total number of phonons at different levels and accordingly plays a role in possibly affecting the electron-phonon scattering rate that leads to the observed ρ(T ).

From this energy point of view, we can estimate in our NWs the tem-perature below which the energy separation between different phonon modes

6We have assumed that at temperature T , every phonon in the system has the same energy kBT , which is in fact the exact energy of the phonon that has the occupancy of ≈ 0.6 if the classical Planck distribution function is obeyed. This is the same approximation often used in textbooks to obtain a qualitative explanation of the Debye T³law for the heat capacity of solids due to lattice vibrations.

becomes appreciable. Suppose that we use a sphere in momentum space to ap-proximate the phonon density of states,⁷ we end up with the simple expression for the Debye temperature, which is given by

ΘD= ~v_s

where ~ and kB have their usual meanings, N is the number of primitive unit cells in the sample, and V is the volume of the sample. In the above equation, the N/V can be approximated by 1/a²c, where a and c are the lattice constants for RuO₂ and equal ≈ 4.5 and 3.1 ˚A, respectively. Using the value of Θ_D= 400 K for bulk RuO₂, we first estimate the sound speed of RuO₂ to be v_s ≈ 5360 m/s.⁸ At temperature T , the dominant phonon in the system is approximated to have an energy k_BT . The phonon wavelength subject to the influence of changing the feature size W is λ = 2W/n, where n = 1. Therefore, the temperature below which the confinement effect becomes considerable is

T = ~πv_s

k_BW . (3.13)

For the case of our thinnest NW of diameter ≈ 37 nm, this leads to the occurrence of perceivable energy splittings at temperatures below ≈ 3.4 K.

This is apparently a much lower temperature range than our fit range. At temperatures higher than ≈ 3.4 K, many phonon modes are occupied and have almost no differences from each other. In this case, the acoustic phonon spectrum and density of states may be regarded to be Debye-like, and thus the phonon confinement effect seems very unlikely to occur in our NWs, and thus unable to account for the observed reduction in Θ_D.

As a matter of fact, to be more specific, we shall consider the influence of acoustic phonon dimensionality on the temperature dependent resistivity directly from the point of view of the Bloch-Gr¨uneisen formula. As we know, if there exists any influence, it would manifest itself when the W is close to

7This is the same approximation which we assume in the derivation of the Bloch-Gr¨uneisen formula.

8This value is comparable to the speed of longitudinal sound wave in Al.

the wavelength of the dominant acoustic phonons that electrons scatter with.

All we have to do is find out this dominant phonon wavelength. It should be noted that the dominant acoustic phonons here might not be the same ones which we concerned about in the previous paragraph. By our definition, the dominant acoustic phonons, which we refer to here, mean the acoustic phonons that contribute most to the temperature dependent resistivity, i.e., the ρ_BG(T ) [Eq. (3.4)]. We note that, for a specimen with a β_BG and a Θ_D, the largest contribution at temperature T to the integral in ρBG(T ) would occur around some phonon frequency ωd which makes the integrand f (x) in Eq. (3.4) has the largest value, where f (x) has the form:

f (x) = x⁵

(e^x− 1)(1 − e^−x) , (3.14) and the x is given by ~w/kBT . The phonons that have this vibrating frequency ωdcan thus be regarded as the most important phonons in the current electron-phonon scattering process.

We plot in Fig. 3.12 the f (x) as a function of x for a material with a Θ_D at several different temperatures. The shaded area denotes the integral in Eq (3.4). What particularly deserves our attention in Fig. 3.12 is that when we are at different temperatures, we have different upper limit of integration x_max, as indicated in the figure. Also notice that, for T larger than ≈ Θ_D/5, the f (x) always has the maximum value at x ≈ 5. From Fig. 3.12, it can be clearly seen that the dominant phonon frequency is temperature dependent.

For example, at T = ΘD/2, the maximum value of f (x) occurs right at the Debye frequency ωD (i.e., ωd = ωD), while at T = ΘD/10, the maximum value of f (x) occurs at the phonon frequency ωD/2 (i.e., ωd= ωD/2).

Evidently, we can categorize them into two different groups. The first group is when we are at temperatures above ≈ Θ_D/5, at which the dominant phonon wavelength is exactly the Debye wavelength. The second group is when we are at temperatures below ≈ Θ_D/5, at which the dominant phonon wavelength is larger than the Debye wavelength, and increases as the temperature decreases.

Figure 3.12: f (x) as a function of x for a material with a Θ_D at several different temperatures.

From the definition of the Debye model:

~ω_D= ~2π vs

λ_D ≡ k_BΘ_D . (3.15)

With the use of Eq. (3.12), the Debye wavelength λ_D can be written as

λ_D= µ4πV

3N

¶_1/3

. (3.16)

As we did previously, we can approximate the N/V by the lattice constants, and then we have, in RuO₂, the λ_D ≈ 6.4 ˚A. Therefore, for RuO₂ at higher temperatures (i.e., at temperatures above ≈ 80 K for the case of the first group), the dominant phonon wavelength λ_d (= λ_D ≈ 6.4 ˚A) is far below the diameter of our thinnest NW of ≈ 37 nm, and hence the phonon confinement effect is certainly not expected to occur. When the temperature decreases down to below ≈ 80 K (i.e., for the case of the second group), the dominant phonon wavelength λd starts to increase, and would, for example, have the value of ≈ 25.6 (128) ˚A at T ≈ 20 (4) K. Accordingly, to observe any possi-ble influence of the phonon confinement effect on the temperature dependent resistivity, we have to reduce the temperature down to at least below ≈ 4 K so that the λ_d would become comparable with the diameter of our thinnest NW of ≈ 37 nm. However, this is obviously not the temperature range that concerns us in our analysis.

So far, we have demonstrated that the observed significant reduction in the Debye temperature Θ_D in our RuO₂ NWs can not be satisfactorily accounted for by the quantization of the acoustic phonon spectrum that results from the phonon confinement effect, since for our experimental range of temperature, such an energy splitting, if any, can hardly make any difference to the electron-phonon scattering process in our NWs.

Other than the afore-discussed possibilities which could lead to a reduction in Θ_D, a reduced sound speed v_s would also bring about a reduction in Θ_D, as can be seen from Eq. (3.12) or (3.15). The sound speed is known to be a constant in a classical elastic continuum. In the Debye model for real solids, the v_s is also presumptively taken to be a constant for each polarization type,

and has the relationship:

where C is the effective force constant between nearest neighbouring lattice planes, and M is the equivalent mass of a lattice point. Undoubtedly, the equivalent mass of a lattice point would not change with the size of a partic-ular specimen. If the reduction in Θ_D is truly caused by a reduced v_s, it then implies that the effective force (characterized by C) between nearest neigh-bouring lattice planes might be weakened as the size of a specimen reduces down to a certain length scale. Actually, such a conjecture is not implausible at all since it has been known that the effective forces can be of quite long range, possibly up to several tens lattice planes in some materials [67]. When the size of the system reduces down to this scale, the C is accordingly prone to be affected. Indeed, the recently developed bond-order-length-strength cor-relation theory [68] has revealed that the broken bonds of surface atoms could make the remaining bonds between the undercoordinated atoms shorter and stronger. In a calculation [57] of Young’s modulus based on this theory, the relative change in Θ_Dwith decreasing size has been found, and it could be pos-itive or negative, depending on the bond nature and the testing temperature.⁹ The comparison has been made with the results obtained for nanoparticles. A general agreement is satisfied.

For the case of metallic NWs, a size and material dependent reduction in Θ_D has been previously reported [69] (by using the same experimental ap-proach as we use here), though the relative change in ΘD is quite small, as compared with that observed in our RuO2 NWs. However, it should be noted that in Ref. [69], the measured NWs were actually embedded in a porous medium, and the possible interface interaction with the surrounding medium might further complicate the situation.

9Theoretically, the ΘD could be temperature dependent. However, in most transport studies, it is assumed and taken to be a constant independent of temperature.

3.4 IrO

Figure 3.13 shows the measured resistivity ρ as a function of temperature from 300 K down to liquid-helium temperatures for two IrO2 NWs with similar lat-eral sizes defined as ≡ W²/4, where W is the hypotenuse as schematically depicted in Fig. 2.3(b). From inspection of the SEM images, we obtained W ≈ 180 ± 5 nm for both samples. In Fig. 3.13, the symbols are the ex-perimental data and the solid curves are the theoretical predictions (see be-low). Clearly, both samples reveal electrical-transport characteristic of a typi-cal metal, i.e., the resistivities decrease as the temperature decreases from the room temperature down. However, close inspection indicates that the resistiv-ity ρ(300 K) ≈ 270 ± 40 µΩ cm in our NWs is considerably higher than that (≈ 85 ± 35 µΩ cm) in bulk single crystals [19]. In addition, the resistivity ratio ρ(300K)/ρ0 ≈ 1.1−1.2 in our NWs is considerably lower, as compared with the corresponding bulk values (≈ 10−1000, depending on the crystal quality) [18, 19, 37], where ρ₀ is the residual resistivity.

Similar with that observed in RuO₂ NWs, such a low residual resistivity ratio suggests the presence of a high level of (point) defects in our IrO₂ NWs.

This observation is also in sharp contrast to the conclusion drawn from conven-tional materials characterization techniques such as XRD and high-resolution TEM [35], which often revealed seemingly high-quality atomic structure. In fact, this is exactly one of the great advantages of the electrical-transport mea-surements which are very sensitive to the microscopic motions of conduction electrons in the NWs.

As usual, our experimental data still can be explained in terms of the same theoretical description given in Section 3.2, despite the fact that the observed RRR in IrO2 NWs are comparatively lower than those in RuO2NWs. Since the level of disorder (or, the RRR) in our NWs is similar to that in the sputtered IrO₂ thick films previously studied [48], the adequate formula to account for our experimental data should be thus the one described by Eq. (3.11), which would be a more complete model than Eq (3.9) for this sample.

Figure 3.13: Resistivity as a function of temperature for two IrO₂ NWs measured by the four-probe method. The symbols are the experimental data and the solid lines are the theoretical predictions of equation (3.11). The inset shows the SEM image of one NW contacted by four Cr/Au electrodes.

According to the electron-phonon-impurity interference theory [59] lead-ing to Eq. (3.10), the prefactor β_int is a constant for a given material, being independent of the amount of disorder contained in the sample. Therefore, taking the value of β_int = 4.5 × 10⁻⁷ K⁻² from [48], the measured normalized resistivity, (ρ − ρ₀)/ρ₀, can then be fitted to Eq. (3.9) with ρ₀, β_BG, Θ_D, β_E and Θ_E as adjusting parameters. We find that Eq. (3.9) can well describe the resistance in the Ir-1 (Ir-2) NW between 80 and 300 K (25 and 300 K). The fitted values of the relevant parameters are listed in Table 3.3. It should be noted that the fitted values of ΘD and ΘE are nearly identical for these two samples, which in turn are close to those values obtained in the previous mea-surements on this material [19, 48]. However, the ratio of our fitted values of β_BG/β_E ≈ 3.1 ± 0.3, which determines the relative strength of the coupling of electrons with acoustic-mode and optical-mode phonons, is somewhat larger than the previous result (≈ 2) [19, 48]. The relative importance of the contri-bution from each term in Eq. (3.9) is more clearly illustrated in a log-log plot as shown in Fig. 3.14. We point out that, below about 50 K, the resistivity of the Ir-1 NW increases more drastically than that of the Ir-2 NW as the temper-ature decreases. This notable resistance rise with decreasing tempertemper-ature may originate from the weak-localization and electron-electron interaction effects [51] and two-level systems [52], as the sample Ir-1 possesses a higher value of resistivity than the sample Ir-2. These just mentioned disorder induced effects are not considered in Eq. (3.9).

Table 3.3: Values of the relevant parameters for the two IrO₂ NWs measured by the 4-p method. For both samples, W ≈ 180 ± 5 nm and L ≈ 0.83 ± 0.01 µm. The uncertainties in resistivities mainly arise from the uncertainties in the dimensions of the NWs.

ρ₃₀₀ ρ₃₀₀/ρ₀ Θ_D β_BG Θ_E β_E β_BG/β_E

(µΩ cm) (K) (µΩ cm/K) (K) (µΩ cm/K)

Ir-1 295 ± 20 1.09 320 0.204 840 0.071 2.87

Ir-2 220 ± 15 1.19 310 0.341 820 0.099 3.44

Figure 3.14: Log-log plot of the variation of the normalized resistivity

∆ρ/ρ₀ = (ρ − ρ₀)/ρ₀ with temperature for the two IrO₂ NWs studied in figure 3.13. Notice that the scales of the axes are different in (a) and (b). The contribution from the ρint/ρ0 term in the Ir-1 NW has the same magnitude as that (∼ 10⁻³) in the Ir-2 NW, and thus is outside the range displayed in (a).

Chapter 4

Electronic transport through metal nanowire contacts

4.1 Introduction

Nanoscale materials are important for fundamental researches and applications due to their promising potential for both new physics and technology. One ma-jor motivation for studies on self-assembled Q1D metallic NWs is their poten-tial use as interconnects in the future nanoelectronics. Probing the electronic transport properties of these nanostructures thus becomes the very key step for the realization of numerous novel applications. Usually, connections of the instruments to the individual nanodevices are accomplished by the nanofab-rication techniques such as the electron-beam lithography (which we adopted in this work) and the focused ion beam deposition method. However, a criti-cal hindrance is that the often obtained non-negligible temperature dependent contact resistances¹ R_c are prone to complicate the experiments and could seriously mislead the physical interpretation of the data. Therefore, in this context, the main theme of this section – uncovering the physics and mecha-nism of the electronic nanocontact resistances – could provide indispensable information and valuable solution for this problem.

From the experimental point of view, the lead resistances and the electronic contact resistances in electrical measurements must be small to minimize ther-mal noises. Usually, the R_c of a macroscopic metal-metal contact is on the

1The large contact resistance (≈ tens kΩ) may result from a thin, dirty insulating layer incidentally formed at the interface between the submicron electrodes and the NW. This insulating layer could be the lightly contaminated or oxidized metals introduced during the electrode evaporation, the amorphous coating resulting from the complex growth process, the vacancies caused by dramatic surface roughness near the contact region, or the breaking induced by tensile stress.

order of ∼ 1 Ω. As the area of electronic contact shrinks, the magnitude of R_c may increase considerably. Moreover, it has been pointed out that, as the nanoscale being approached, the detailed atomic structure of the contact could affect the R_c significantly in fairly different manners [70].

In our case, the electronic contact resistances formed between the EBL-patterned electrodes and the NWs normally fall between several tens and sev-eral hundreds Ω, and are not much dependent on temperature. However, highly resistive electronic contacts with room temperature resistances of order several kΩ or higher may also be obtained in many fabrications. Figure 4.1 shows three diverse temperature behaviours of the measured resistance R, as

In our case, the electronic contact resistances formed between the EBL-patterned electrodes and the NWs normally fall between several tens and sev-eral hundreds Ω, and are not much dependent on temperature. However, highly resistive electronic contacts with room temperature resistances of order several kΩ or higher may also be obtained in many fabrications. Figure 4.1 shows three diverse temperature behaviours of the measured resistance R, as