Electron mean free path in elemental metals

The electrical resistivity of metal wires increases as their width decreases.^1–4 This size effect represents a major challenge for nanoelectronics⁵ and for the further downscaling of integrated circuits^6,7 as the resistivity of, for example, 10-nm-wide Cu interconnect lines is approximately an order of magnitude larger than that of bulk Cu.^2,8 The primary reason for the resistivity increase is electron scattering at external surfaces and grain boundaries, which are most commonly described by the classical models by Fuchs and Sondheimer (FS)^9,10 and Mayadas and Shatzkes (MS),^11,12 respectively. Both models predict in their approximate form an additive resistivity contribution which is proportional to ρ_o × λ/d, where ρ_o and λ are the bulk resistivity and mean free path for electron phonon scattering, and d is the relevant length scale, that is, the wire width or the grain size for surface and grain boundary scattering, respectively. Therefore, in the limiting case of thin wires and/or small grain sizes, the wire resistivity becomes proportional to ρ_o × λ for any given fixed wire dimension and grain size distribution. Thus, the metal with the lowest product ρ_o × λ is expected to exhibit the highest conductivity in the limit of a small wire width. This argument neglects variations in the surface scattering specularity and grain boundary reflection coefficient which are expected to differ for different metals and enter the FS and MS models as phenomenological parameters that also depend on the surface^13–18 and grain boundary^3,19–22 structure and chemistry. Nevertheless, minimizing the product ρ_o × λ is a useful starting point in the focused search for metals that form high-conductivity narrow wires.

Values for the bulk resistivity of elemental metals are well established from measurements at room temperature where electron scattering is dominated by phonons and resistivity contributions due to impurities and crystalline defects are negligible for carefully prepared samples. In contrast, there is only limited information regarding the bulk electron mean free path. One approach is to calculate λ from the measured ρ_o using the free electron model. This assumes a constant Fermi velocity and a spherical Fermi surface, which is a reasonable approximation for alkali and group 11 metals. In fact, the free electron predictions for room temperature Cu and Ag of λ = 39 and 53 nm are in good agreement with values obtained from fitting the measured resistivity of epitaxial metal layers vs their thickness using the FS model.^14,23,24 In contrast, the limited reported results from metals with non-spherical Fermi surfaces suggest differences by up to an order of magnitude between the λ determined from the free electron model and from fitting resistivity vs thickness data.^25–29 These results are further complicated by anisotropy effects^30–33 and the possible breakdown of the FS model for narrow conductors.²⁵ 

In this paper, I present calculated values of the product ρ_o × λ based on the bulk electronic structure predicted from first-principles density functional calculations. Thus, these calculations correctly account for the anisotropy in the Fermi surface as well as the variation in the electron velocity v(k) as a function of wave vector k; that is, they go beyond the common effective mass approximation of the FS and MS models that are expected to cause inaccuracies for metals with anisotropic Fermi surfaces. Electron transport is described within a semi-classical Boltzmann transport formalism with individual scattering events that completely randomize the electron momentum, either within a constant mean free path or within a constant mean free time approximation. That is, the electron-phonon scattering probability is approximated by a k-independent average distance or average time, respectively. The tabulated results suggest that quite a few (5–10) metals have an effective resistivity that is lower than that of copper in the limit of narrow wires, despite that their bulk resistivity is up to a factor of five larger. Therefore, the decision regarding which of those metals will replace Cu for narrow interconnect lines may ultimately be dominated by secondary factors including processing feasibility and electromigration performance.

The electronic structure of the twenty most conductive elemental metals is obtained from density functional calculations using the Vienna ab initio simulation package (VASP), employing periodic boundary conditions, a plane wave basis set, the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) exchange correlation functional,³⁴ and the projector-augmented wave method.³⁵ Computational convergence with respect to the energy cut-off for the plane wave expansion, the k-point grid, and the unit cell size and shape is tested and adjusted such that all values reported in this letter have a computational uncertainty <1%. In addition to the outer-most electron shells, the following electrons are explicitly calculated; that is, they are not included in the core of the pseudo potentials: p for alkali, alkaline earth, and transition metals up to column VI, 4d for In, 2s and 2p for Al, and 1s for Be. Calculations are done using primitive unit cells containing one atom for fcc, bcc, and bct and two atoms for hcp structures, and lattice parameters are fixed at the established experimental room temperature values. The latter are typically slightly smaller (∼1%) than the values obtained when fitting calculated energy vs lattice parameters, as typical for the generalized gradient approximation.³⁶ Self-consistent calculations using a Γ-centered 40 × 40× 40 k-point grid are used to determine the charge distribution, which is subsequently used for non-self-consistent calculations with a finer 200 × 200 × 200 k-point mesh. Such a fine k-point sampling is required for good convergence of the numerical integration across the Fermi surface, which is very sensitive to band crossings. Careful convergence tests are performed for all reported elements to achieve the desired <1% numerical accuracy, requiring, for example, for Be an increase to 400 × 400 × 400 k-points.

The Fermi surface is determined from the calculated bands, i.e., the calculated electron energy vs k curves E_n(k) where n is the band index. This is done by dividing the Brillouin zone into irregular tetrahedra that are defined by four neighboring k-points. Linear interpolation between the corners yields the intercepts of the Fermi surface with the tetrahedron edges. Three or four intercepts per tetrahedron define one or two triangles of the Fermi surface, respectively,³⁷ with an electron velocity v_n(k) = $1ℏ∇k$ E_n(k) that is perpendicular to the triangle surface and is defined by the slope of the band. More elaborate interpolation schemes have been tested, including three-dimensional polynomial interpolation to create k-point submeshes, but do not consistently improve k-point convergence, which is attributed to these approaches being more sensitive to discontinuities at band-crossings.

I reiterate here that all calculations in this paper are done for bulk materials; that is, quantum size effects in narrow wires are not accounted for during the electronic structure calculations. This approximation is justified by the small screening length in metals, resulting in a thin film ballistic conductance that matches the bulk conductance for all but the narrowest layers. For example, a 2-nm-thick Cu layer has a calculated conductance of 0.58 × 10¹⁵ Ω⁻¹ m⁻² (Ref. 17), in perfect agreement with the bulk value.³⁸ In contrast, surface roughness in narrow conductors can cause a destruction of isotropic Fermi surface sheets which affects transport properties.¹⁷ 

Transport is simulated within a semiclassical approach, such that the bulk conductivity σ_o is given by

Here, the sum is over bands and the integration is over the Fermi surface $SFn$ of band n. The carrier relaxation time τ_n(k), the electron velocity along the transport direction v_t,n(k), and the electron velocity vector v_n(k) are functions of the wave vector k for each band with index n. The factor two in Eq. (1) is accounting for both spins for the case on non-magnetic metals, while this factor is removed for magnetic materials (Co and Ni in this letter) and the contribution from each spin is summed up separately, starting with a spin-polarized density functional calculation.

Using a constant relaxation time approximation, the relaxation time becomes independent of band, spin, and wave vector. That is, τ_n(k) = τ and thus

Correspondingly, using a constant mean free path approximation, the carrier relaxation time becomes τ_n(k) = λ/$|vn(k)|$ and therefore

I note here that for cubic symmetry, the integral in Eq. (2b) is exactly one third of the area of the Fermi surface A_f, such that the product λ × ρ_o is inversely proportional to the Fermi surface area, i.e., λρ_o = 12π³ℏ/(e²A_f) as used, for example, in Ref. 26. This fact is not explicitly used in the numerical code, primarily to keep modularity and flexibility of the computational approach, which first interpolates band energies in the reciprocal coordinate system of the primitive unit cell vectors, second defines the Fermi surface as a list of triangles with associated velocity vectors in a reciprocal Cartesian coordinate system, and third determines transport quantities by summation over all triangles according to Eqs. (2a) and (2b). Integration is done for all calculated metals using multiple transport directions corresponding to conventional [100], [010], [001], [110], [1 $1¯$ 0], and [111] directions. The results are identical within <1% deviation for all directions for the case of cubic fcc and bcc structures, and for directions within the basal plane for hcp and bct. This is expected based on the 3-dimensional isotropic resistivity for cubic materials and 2-dimensional isotropy within the basal plane of hexagonal and tetragonal symmetries. The values reported, in this letter, are the average from the [100] and [010] directions, while for hcp and bct the [001] values are separately listed.

Fig. 1 shows typical Fermi surfaces, in this case, for Cu, Mo, Ru, and Rh. The Brillouin zones are outlined by black lines, while the Fermi surface is colored according to the Fermi velocity v_f(k). The Fermi surface of Cu in Fig. 1(a) is nearly spherical, with eight necks that reach along ⟨111⟩ directions to touch the hexagonal faces of the zone and are due to the reduction of the band energy near the zone boundary.³⁹ The total Fermi surface area of 2.28 × 10²¹m⁻² almost perfectly matches ( <2% deviation) the free-electron prediction of 2.32 × 10²¹m⁻². In contrast, the average Fermi velocity of 1.11 × 10⁶m/s is 29% smaller than the free electron prediction of v_f = 1.57 × 10⁶m/s. The velocity is relatively constant over most of the surface, decreasing from a maximum v_f = 1.31 × 10⁶m/s along ⟨100⟩ to 1.0 × 10⁶m/s along ⟨110⟩, and drops to 0.7 × 10⁶m/s for the neck at the zone boundary. The Fermi surface for Mo in Fig. 1(b) is 2.55 × 10²¹m⁻² and much more complex than for Cu, as multiple 4d bands cross the Fermi level in Mo. The Fermi velocity varies by nearly an order of magnitude, from 0.2 to 1.5 × 10⁶m/s, with an average v_f = 0.92 × 10⁶m/s. More specifically, v_f is 0.8 × 10⁶m/s for the electron jack near the zone center and decreases continuously to a minimum v_f = 0.2 × 10⁶m/s in the electron knobs that extend along the ⟨100⟩ directions, v_f = 0.8–1.5 × 10⁶m/s for the hole octahedron and v_f = 0.9–1.3 × 10⁶m/s for the hole ellipsoid near the zone boundary along ⟨100⟩ and ⟨110⟩, respectively. The Brillouin zone for Ru in Fig. 1(c) is wider within the basal plane than along the hexagonal axis, reflecting the Ru lattice constants which are 2.706 and 4.282 Å perpendicular and parallel to the hexagonal axis, respectively. The Fermi surface area is 3.32 × 10²¹m⁻² and the average v_f = 0.72 × 10⁶m/s. The anisotropy is evident from both the shape and the velocity distribution, with the inner “electron star” exhibiting a relatively low v_f = 0.3 × 10⁶m/s along [0001] but a three times larger v_f = 0.9 × 10⁶m/s within the hexagonal plane while the “hole ring” has v_f = 0.3–0.8 × 10⁶m/s over most of its surface but also exhibits pockets with v_f = 0.9–1.2 × 10⁶m/s which is almost as high as v_f = 0.9–1.3 × 10⁶m/s for the additional separate pockets. The Fermi surface of Rh plotted in Fig. 1(d) is relatively large, 4.73 × 10²¹m⁻², and the average velocity small v_f = 0.67 × 10⁶m/s. It exhibits multiple nestled surfaces with both electron and hole character around the zone center and near the zone boundary, respectively. The large Fermi surface of Rh leads to a small product λ × ρ_o, which is the smallest of all investigated metals, as discussed below, such that Rh becomes a promising candidate for narrow high-conductivity metal wires.

Table I is a list of the calculated transport data for the twenty most conductive elemental metals. The list is sorted according to increasing bulk resistivity at room temperature ρ_o,rt, as reproduced from Ref. 40. The values for the average Fermi velocity v_f and the columns τ × ρ_o and λ × ρ_o are directly obtained from first principles calculations, using Eqs. 2(a) and 2(b) for the constant τ and λ approximations, respectively. These products are temperature-independent because ρ_o is proportional to the electron-phonon scattering, while τ and λ are inverse proportional to the electron-phonon scattering. The listed values for the room temperature τ_rt and λ_rt are determined from the τ × ρ_o and λ × ρ_o products by dividing by the known room temperature bulk resistivity. Two values are given for hexagonal and tetragonal metals, reflecting transport perpendicular and parallel to the hexagonal/tetragonal axes. That is, the first values are for conduction within the 2-dimensional basal planes and have double the weight in comparison to the second values for conduction in the 3rd dimension. I note here that, for purposes of data consistency, the listed bulk resistivity values are from large-grain randomly oriented polycrystalline samples, even for anisotropic (hcp and bct) metals. Thus, while the calculated data-pairs for τ × ρ_o and λ × ρ_o correctly account for the anisotropy in Fermi surface and velocity of hcp and bct metals, the columns for τ_rt and λ_rt contain two values, which is an artifact of this approach. For example, the calculated τ × ρ_o for Mg along the hexagonal axis is 15% smaller than the perpendicular value, indicating a resistivity anisotropy ratio ρ_||/ρ_⊥ = 0.85 for Mg. This is in excellent agreement with a reported measured anisotropy value of 0.84.⁴¹ The agreement suggests, since the calculations are done with a constant τ, that the electron-phonon scattering cross-section in Mg is isotropic. Correspondingly, the bulk resistivity could be listed in Table I as two values: ρ_⊥ = 4.62 μΩ cm and ρ_|| = 3.93 μΩ cm which, in turn, would result in single value for τ_rt of 1.86 fs. However, a similar argument within the constant mean free path approximation would yield an anisotropy of 0.90, and therefore slightly different ρ_⊥ and ρ_|| values, which would make data presentation in Table I confusing. Moreover, the resistivity anisotropy has not been reported previously for all metals, contains considerable uncertainty with deviations between different reports of up to 30%,^41,42 and also differs considerably (0%–25%) from our calculated anisotropy ratios.^41–44 

Cu has a calculated λ × ρ_o = 6.70 × 10⁻¹⁶ Ω m², which is within 2% of 6.59 × 10⁻¹⁶ Ω m² determined from the free electron approximation. Correspondingly, λ_rt = 39.9 nm matches the 39 nm reported for room temperature Cu based on the free electron model^2,45 which also matches resistivity-vs-thickness scaling data.^13–15 This agreement between the result from the calculated Fermi surface and a spherical free-electron surface is attributed to a combination of (i) the calculated surface area to be nearly identical to the free-electron area, as discussed above, and (ii) Eq. (2b) for cubic symmetry to be only dependent on the surface area, that is, independent of v(k). In contrast, the calculated τ × ρ_o of 6.04 × 10⁻²² Ω m s is 44% larger than the free-electron value of 4.19 × 10⁻²² Ω m s, reflecting the importance of correctly accounting for v(k).

The product λ × ρ_o listed in Table I is interesting for determining the most promising metals for narrow high-conductivity lines, because, based on the FS and MS models, the smallest λ × ρ_o value is expected to yield the lowest resistance in the limit of thin wires with fixed dimension, grain size distribution, and interface scattering cross sections, as discussed above. Various metals have a lower λ × ρ_o product than Cu, which is the metal currently used for interconnects in integrated circuits. Particularly, noteworthy are Rh, Ir, and Ni, with values that are 2.1, 1.8, and 1.6 times smaller than that for Cu, respectively. In addition, metals that have λ × ρ_o values comparable to that of Cu but exhibit either processing and/or electromigration advantages may become competitive as the wire cross-sectional area shrinks. Processing advantages include metals with an established atomic layer deposition process or metals that can be processed through annealing steps to form large grains, reducing the resistivity contribution from grain boundaries. Electromigration advantages primarily refer to metals that exhibit low electromigration such that adhesion/liner/barrier layers may become unnecessary, freeing up space which effectively increases the interconnect cross-sectional area and, in turn, lowers their effective resistivity. Promising metals in that regard are W, Mo, Co, and Ru, all exhibiting λ × ρ_o values within 30% of the value for Cu, despite their 3–5 times larger bulk resistivity.

The calculated product λ × ρ_o provides guidance in the search for metals that may be suitable for narrow interconnect lines, because the resistance of narrow lines is expected to be proportional to λ × ρ_o. The lowest λ × ρ_o is found for Rh, with a value that is 2.1 times smaller than for Cu, potentially yielding narrow lines that are twice as conductive as copper. A whole range of other metals, listed in Table I, also become competitive with Cu in regard to effective conductivity for the limiting case of narrow wires, suggesting that processing viability and electromigration performance will ultimately determine the best metal to replace Cu for integrated circuits.

This research was funded by the Semiconductor Research Corporation under Task 1292.094 and through the STARnet center FAME funded by MARCO, DARPA, and SRC. The author also acknowledges the NSF under Grant No. 1309490. Computational resources were provided by the Center for Computational Innovations at RPI.