Crystal lattice has a resistivity against electric current since the conducting electrons are scattered by phonons. This phenomenon is referred to as intrinsic resistivity of crystalline materials (hereafter denoted as r_i). The intrinsic resistivity is the electrical resistivity of a defect-free and chemically pure material. With decreasing temperature (T) lattice vibrations diminish; therefore, intrinsic resistivity strongly decreases. For metallic materials the resistivity at high temperatures (e.g., for Cu between 100 K and the melting point) is a linear function of T [1,2]. As the temperature is reduced, the temperature dependence of resistivity follows. At lower temperatures the resistivity varies with ∼T⁵ (if scattering by phonons is the main contribution), and the resistivity becomes constant at 10–20 K. This residual resistivity is caused by the lattice defects and the impurities, i.e., it is not related to the intrinsic resistivity. As an example, Fig. 9.1 illustrates the temperature dependence of resistivity of Cu in double logarithmic scale.

where T is the temperature and index l stands for the different lattice defects (including impurities and alloying elements). The contributions of lattice defects are practically independent of the temperature but increase with increasing concentrations or densities of defects. For vacancies, dislocations, stacking faults, twin faults (twin boundaries), and high-angle grain boundaries (HAGBs) the resistivity contributions can be approximated as:

respectively, where η_v, η_d, η_sf, η_TB, and η_HAGB are the resistivity increments for unit vacancy concentration, dislocation density (given in m⁻²), stacking fault density (ρ_sf given in m⁻¹), twin fault density (ρ_TB given in m⁻¹), and HAGB density (ρ_HAGB given in m⁻¹). The quantity ρ_sf gives the area of stacking faults in a unit volume of material. ρ_TB and ρ_HAGB are defined similarly. It is noted that the planar fault or grain boundary density can be given as the reciprocal of the fault or boundary spacing, respectively. Table 9.1 lists the values of the specific resistivities η_v, η_d, η_sf, η_TB, and η_HAGB for Cu. It should be noted that the specific resistivity of interstitials is about three times larger than that for vacancies. In addition, the electrical resistivity of a divacancy is about 20% less than the electrical resistivity of two separated vacancies [3]. It was shown that the specific resistivity of a vacancy in Cu is approximately the same in the temperature range between 4 K and the melting point [4]. It should also be noted that the resistivity of HAGBs depends on the structure of HAGBs. For instance, in Cu the value of η_HAGB decreased from 5.5 × 10⁻¹⁶ to 2 × 10⁻¹⁶ Ωm² due to grain boundary relaxation during annealing [4].

where d₁₁₁ is the spacing between the neighboring {111} planes and β is given in percentage. Inserting Eq. (9.7) into Eq. (9.5), the resistivity caused by twin faults can be related to twin fault probability as:

where V_p is the volume fraction of pores. For V_p < 0.5, the porosity factor, f(V_p), can be expressed as (Wiedemann–Franz–Lorenz equation) [15]:

The electrical resistivity increment caused by monovacancies, vacancy clusters, and dislocations was determined in high purity ultrafine-grained (UFG) Ni and Cu [8]. The resistivity contribution of single and double vacancies in 99.998% purity Ni processed by two turns of high-pressure torsion (HPT) at RT was about 3 × 10⁻¹⁰ Ωm, which was caused by a vacancy concentration of ∼10⁻⁴. This value was measured at the periphery of the HPT-processed disk where the strain was the largest. The resistivity increment caused by vacancy agglomerates, dislocations, and HAGBs was ∼17 × 10⁻¹⁰ Ωm. It was found that the resistivity increment caused by the lattice defects formed in Ni during HPT saturated at a shear strain of ∼20, which can be achieved already at the periphery of a disk deformed by one-third turn of HPT. After one turn of HPT, both resistivity and strength in Ni change only slightly with increasing either the distance from the disk center or the number of turns [17]. It is worth to note that the contribution of lattice defects to resistivity in HPT-processed Ni (∼20 × 10⁻¹⁰ Ωm) is two orders of magnitude larger than the intrinsic resistivity measured at 4 K. At the same time, at RT the resistivity of lattice defects is only ∼10% of the intrinsic electrical resistivity of Ni. In UFG copper processed by five passes of equal-channel angular pressing (ECAP) at RT the lattice defects formed during severe plastic deformation (SPD) yielded a resistivity increment of ∼4 × 10⁻¹⁰ Ωm, irrespective of the applied ECAP route (A, B_C, and C, as defined in Chapter 3) [18]. The change of resistivity for ECAP-processed Cu is much smaller than that in Ni deformed by HPT, which can be explained by the much lower shear strain applied in the former method (∼5). The contribution of lattice defects to resistivity of ECAP-processed pure Cu at RT is only ∼2%. Similar results were obtained for 99.96% purity UFG copper produced by 16 passes of ECAP via route B_C at RT [19]. Although, the grain size was refined from 6 μm to 500 nm when the annealed Cu sample was processed by 16 ECAP passes, the conductivity changed only from 97% IACS to 95% IACS, i.e., the contribution of lattice defects (including vacancies, dislocations, and grain boundaries) to resistivity is only ∼2%. In Cu–3 wt.% Ag alloy 12 passes of ECAP at 343 K yielded reduction of conductivity from 94% to 88% IACS [20]. Generally, it can be concluded that in pure metals and equilibrium solid solutions, SPD results only in a slight increase of resistivity owing to the increase of amount of lattice defects. At the same time, the lattice defects result in a strong increase of mechanical strength. For instance, the yield strength in pure copper increased from ∼100 to ∼350 MPa owing to 16 ECAP passes [19]. Therefore, SPD methods usually significantly improve the strength-to-resistivity ratio. In the case of Cu processed by 16 passes of ECAP, the strength-to-resistivity ratio increased from 0.56 × 10¹⁰ to 1.9 × 10¹⁰ MPa/Ωm. At the same, for low carbon steel four passes of constrained groove pressing (CGP) led to only a 20% increment in strength-to-resistivity ratio [21]. In the CGP process the grain size of the steel was refined from 30 μm to 231 nm. Table 9.2 summarizes the grain size, the mechanical strength, the conductivity in IACS, the electrical resistivity, and the strength-to-resistivity ratio for different UFG and nanocrystalline materials. In the present case, the mechanical strength is characterized by the yield strength or the strength calculated as one-third of the hardness, depending on the availability of these data. It should be noted that the stress determined as one-third of hardness gives the flow stress corresponding to 8% plastic strain since hardness measurement itself yields an additional plastic deformation. For materials exhibiting strong strain hardening after the onset of yielding, one-third of hardness may be considerably larger than the yield strength. At the same time, for SPD-processed samples the strain hardening is usually moderate; therefore, there is only a slight difference between one-third of hardness and yield strength.

A combination of high strength and good conductivity in alloys can be obtained by applying moderate annealing after SPD processing. Large strength-to-resistivity ratio can be achieved by producing very fine grain structure with very low solute content inside the grains. In these microstructures, the majority of alloying elements are at the grain boundaries and/or in nanosized intermetallic compound particles. The SPD step in the thermomechanical treatment provides a UFG or nanocrystalline matrix that is stabilized by the nanoscaled precipitates formed in the annealing step. The good conductivity is guaranteed by a nearly complete purification of the matrix grain interiors from the solute atoms. The high strength is caused by the high amount of grain boundaries and the nanosized secondary phase particles instead of solute hardening in the grain interiors. The effectiveness of this processing strategy in the optimization of strength and electrical conductivity was illustrated for Cu and Al alloys [22,35–37]. For instance, the strength-to-resistivity ratio of Cu–9.85 wt.% Cr deformed by five turns of HPT at RT increased with about 20% when a heat treatment was applied at 773 K for 1 h [31]. In this material the secondary phase particles were Cr nanodispersoids with the size of 10–15 nm. It is noted that for similar Cr content without the application of this thermomechanical treatment the strength-to-resistivity ratio is five times smaller mainly due to the much lower conductivity [40]. Fig. 9.4 illustrates the effect of SPD processing and subsequent annealing on the strength, resistivity, and strength-to-resistivity ratio for pure copper and its alloys. It can be concluded that UFG microstructure with fine precipitates and purified grain interiors (from solutes) may yield copper materials with high strength and low electrical resistance, and the strength-to-resistivity ratio of this alloy may be better than that for pure Cu. It should be noted, however, that a very small Mg solute concentration in SPD-processed UFG Cu can also result in a good strength-to-resistivity ratio, similar to dispersion-strengthened UFG alloys with purified grain interiors [30]. In this case, most probably the large solute Mg atoms are segregated at the grain boundaries, thereby reducing their contribution to resistivity. Annealing after SPD processing for this alloy reduced the strength-to-resistivity ratio since the strength increased significantly without a considerable improvement of conductivity. The best strength-to-resistivity ratio (∼3.3 × 10¹⁰) was obtained for thermomechanically treated Cu–0.9 wt.% Hf and Cu–0.7 wt.% Cr–0.9 wt.% Hf alloys [32]. Both materials were first annealed at 1173 K for 1 h and quenched to RT. Then, they were deformed by five turns of HPT at RT. Finally, an aging heat treatment was applied for 1 h at 723 and 773 K for Cu–0.9 wt.% Hf and Cu–0.7 wt.% Cr–0.9 wt.% Hf alloys, respectively. During this aging, Cr and Cu₅Hf intermetallic compound nanoparticles with the size of 10–20 nm were formed, which hindered the grain growth effectively. Hence, the grain size increased only by about 20% despite the high temperature of aging. The formation of the very fine precipitates yielded an exceptional change of yield strength, as it increased by about 20% during heat treatment. In addition, a two times higher conductivity was detected after aging, i.e., both the strength and the conductivity were improved during the heat treatment.

In general, thin films usually have higher resistivity than bulk materials with similar chemical composition. For instance, a sputtered pure polycrystalline Cu film with the thickness of 1 μm exhibited a resistance of ∼2.2 × 10⁻⁸ Ωm at RT, which is about 20% larger than that for pure bulk Cu with similar grain size (∼1.8 × 10⁻⁸ Ωm) [41]. For Cr film with the thickness of 1 μm, the resistivity (∼116 × 10⁻⁸ Ωm) was one order of magnitude higher than that for bulk Cr material (∼13 × 10⁻⁸ Ωm). In addition, the resistivity of Cr film only slightly increased with increasing the temperature from 4K to RT. These observations can be explained by the very small diameter of the columnar grains (∼30–35 nm) and the high free volume at the grain boundaries in the film. The latter phenomenon is manifested by the higher interatomic spacing at grain boundaries and the lower density of pure Cr films than that observed for bulk material (the density of a Cr thin film was found to be ∼96%–97% of the value of bulk Cr). Therefore, the resistivity of sputtered Cr films is determined mainly by the lattice defects and not by the temperature-dependent electron–phonon scattering [41]. It was suggested that the large interatomic spacing at the grain boundaries in Cr films is caused by large internal stresses. In Cu films, these stresses can relax by twinning in the grain interiors; therefore, the free volume at the grain boundaries is smaller, resulting in only a slightly larger resistivity than in bulk Cu. With increasing film thickness, the resistivity of thin films decreases and asymptotically converges to the resistivity of bulk materials.

where d_twin,111 and η_TB,111 are the average twin fault spacing and specific resistivity for Σ3{111} twin interfaces, respectively. d_twin,112 and η_TB,112 are the average twin fault spacing and specific resistivity for Σ3{112} twin faults, respectively. It should be noted that experiments suggested the larger specific resistivity for Σ3{112} interfaces. As a consequence, the experimentally determined value of the average specific twin fault resistivity in sputtered Cu films increased from ∼1.5 × 10⁻¹⁷ to ∼3 × 10⁻¹⁷ Ωm² with increasing the deposition rate, which can be attributed to the contribution of semicoherent Σ3{112} twin interfaces developed at higher sputtering rates. The higher deposition rate also caused an increase in hardness from 2100 to 2800 MPa at RT. Therefore, the strength-to-resistivity ratio of sputtered Cu with high density of nanotwins reached a superior value of ∼4 × 10¹⁰ MPa/Ωm, which did not change considerably with the variation of deposition rate. Similar high strength-to-resistivity ratio (∼5 × 10¹⁰ MPa/Ωm) was obtained for nanotwinned 99.998% purity Cu film processed by pulsed electrodeposition (see Table 9.2), although the microstructure was very different from the epitaxially grown sputtered film [27]. The as-deposited Cu film was polycrystalline with a thickness of ∼100 μm and consisted of irregular-shaped grains with random orientations. The microstructure resembles to that depicted in Fig. 9.6. The grains had a wide distribution with an average size of 400 nm and contained a high density of twin faults with a mean twin-fault spacing of 15 nm. Similar grain size was achieved in 99.96% purity Cu by 16 passes of ECAP at RT but without large amount of twin faults in the grain interiors [19]. The twin faults inside the grains of the electrodeposited film increased the strength by a factor of 2.6 without significant reduction of conductivity.