Bulk Electron Mean Free Path Research Articles

Resistivity size effect in epitaxial VNi2 layers

Epitaxial VNi2 layers are deposited onto MgO(001) and their resistivity ρ measured as a function of layer thickness d = 10.5–138 nm to quantify the resistivity size effect. The layers exhibit resistivity minima at both stoichiometric V:Ni = 1:2 composition and 700 °C growth temperature, which is attributed to electron scattering at V–Ni antisite defects and is described using the ordering parameter within the framework by Rossiter. A cube-on-cube epitaxy of the fcc parent structure on MgO(001) leads to two possible layer orientations for orthorhombic VNi2(010) and VNi2(103), resulting in considerable atomic disorder at domain boundaries, consistent with relatively small x-ray coherence lengths of 8 and 14 nm in-plane and along the growth direction of a 33.5 nm thick layer. In situ ρ vs d measurements yield a bulk resistivity of ρo = 46 ± 2 μ Ω cm and a benchmark quantity of ρoλ = (138 ± 5) × 10−16 Ω m2, where λ is the bulk electron mean free path. Air exposure causes a minor resistivity increase due to 2 ± 1 nm thick surface oxide that perturbs the surface potential. Resistivities at 77 K are Δρ = 16 ± 3 μ Ω cm below those at room temperature. This Δρ is thickness independent and is close to the previously predicted 13.9 μ Ω cm bulk resistivity for VNi2 along [100]. However, the measured bulk resistivity is well above this prediction, which is attributed to electron scattering at domain boundaries/atomic disorder. Consequently, the theoretically predicted superior directional conduction cannot be experimentally confirmed in this study. The overall results indicate that VNi2 is only a promising compound for narrow interconnects if a synthesis scheme can be developed that results in a strong atomic order, a negligible domain boundary density, and a [100] crystalline orientation along the transport direction.

First-principles prediction of electron grain boundary scattering in fcc metals

The electron reflection probability r at symmetric twin boundaries Σ3, Σ5, Σ9, and Σ11 is predicted from first principles for the eight most conductive face-centered cubic (fcc) metals. r increases with decreasing interplanar distance of atomic planes parallel to the boundary. This provides the basis for an extrapolation scheme to estimate the reflection probability rr at random grain boundaries, which is relatively small, rr = 0.28–0.39, for Cu, Ag, and Au due to their nearly spherical Fermi surfaces, but approximately two times higher for Al, Ca, Ni, Rh, and Ir with a predicted rr = 0.61–0.72. The metal resistivity in the limit of small randomly oriented grains with fixed average size is expected to be proportional to the materials benchmark quantity ρoλ × rr/(1 − rr), where ρo and λ are the bulk resistivity and bulk electron mean free path, respectively. Cu has the lowest value for this quantity, indicating that all other fcc metals have a higher resistivity in the limit of small randomly oriented grains. Thus, the conductivity benefit of replacement metals for narrow Cu interconnect lines can only be realized if the grains are larger than the linewidth or exhibit symmetric orientation relationships where r < rr.

The search for the most conductive metal for narrow interconnect lines

A major challenge for the continued downscaling of integrated circuits is the resistivity increase of Cu interconnect lines with decreasing dimensions. Alternative metals have the potential to mitigate this resistivity bottleneck by either (a) facilitating specular electron interface scattering and negligible grain boundary reflection or (b) a low bulk mean free path that renders resistivity scaling negligible. Recent research suggests that specular electron scattering at the interface between the interconnect metal and the liner layer requires a low density of states at the interface and in the liner (i.e., an insulating liner) and either a smooth epitaxial metal-liner interface or only weak van der Waals bonding as typical for 2D liner materials. The grain boundary contribution to the room-temperature resistivity becomes negligible if the grain size is large (>200 nm or ten times the linewidth for wide or narrow conductors, respectively) or if the electron reflection coefficient is small due to low-energy boundaries and electronic state matching of neighboring grains. First-principles calculations provide a list of metals (Rh, Pt, Ir, Nb, Ru, Ni, etc.) with a small product of the bulk resistivity times the bulk electron mean free path ρo × λ, which is an indicator for suppressed resistivity scaling. However, resistivity measurements on epitaxial layers indicate considerably larger experimental ρo × λ values for many metals, indicating the breakdown of the classical transport models at small (<10 nm) dimensions and suggesting that Ir is the most promising elemental metal for narrow high-conductivity interconnects, followed by Ru and Rh.

The Resistivity Size Effect in Epitaxial Nb(001) and Nb(011) Layers

Epitaxial Nb(011) and Nb(001) layers are sputter deposited onto ${a}$ -plane and ${r}$ -plane sapphire substrates, respectively, and their resistivity $\rho $ measured in situ , ex situ , and at 77 K as a function of layer thickness ${d}= 4$ –400 nm. The resistivity increase with decreasing ${d}$ is independent of layer orientation and is described with the model by Fuchs and Sondheimer (FS), providing a value for the bulk electron mean free path $\lambda = {20} \pm {2}$ nm at room temperature. Exposure to air causes a 1.5-nm-thick surface oxide and an increase in $\rho $ by up to 74%, suggesting a decrease in the surface scattering specularity from ${p}_{{{1}}}= {0.9} \pm {0.1}$ at the Nb-vacuum interface to completely diffuse scattering ( ${p}_{{{1}}}= {0}$ ) at the oxidized Nb surface. Alternatively, this increase in resistance can be attributed to roughening during surface oxidation while retaining completely diffuse scattering, yielding a lower bound for the room-temperature $\lambda $ of 9.0±0.4 nm. The product of the bulk resistivity $\rho _{o}$ times $\lambda $ is temperature-independent and, depending on either choosing the roughness or the specularity interpretation, $\rho _{o}\,\,\lambda = {14}\times {10}^{{-{16}}}$ or ${30} \times {10}^{{-{16}}}\,\,\Omega \text{m}^{{{2}}}$ , respectively. These values are 3.9 and 8.5 times larger than $\rho _{o}~\lambda $ from a previous theoretical prediction, indicating a dramatic break down of the classical FS model for Nb and indicating that the resistivity size effect in Nb is considerably larger than predicted earlier. They are also larger than for W, Ru, and Co, making Nb not promising for high-conductivity narrow interconnect lines.

Resistivity size effect in epitaxial Ru(0001) layers

Epitaxial Ru(0001) layers are sputter deposited onto Al2O3(0001) substrates and their resistivity ρ measured both in situ and ex situ as a function of thickness d = 5–80 nm in order to quantify the resistivity scaling associated with electron-surface scattering. All layers have smooth surfaces with a root-mean-square roughness <0.4 nm, exhibit an epitaxial relationship with the substrate: Ru[0001]||Al2O3[0001] and Ru[101¯0]||Al2O3[112¯0], and show no resistance change upon air exposure, suggesting negligible resistivity contributions from geometric surface roughness and grain boundary scattering and negligible changes in the surface scattering specularity p upon oxygen exposure. The room temperature ρ vs d data are well described by the semiclassical Fuchs-Sondheimer (FS) model, indicating a bulk electron mean free path λ = 6.7 ± 0.3 nm. However, the measured ρo × λ product at 77 K is 43% lower than at 295 K, suggesting a breakdown of the FS model and/or a thickness-dependent electron-phonon coupling and/or a temperature- or environment-dependent p. Transport simulations employing the ruthenium electronic structure determined from first-principles and a constant relaxation time approximation indicate that ρ is strongly (by a factor of two) affected by both the transport direction and the terminating surfaces. This is quantified with a room temperature effective mean free path λ*, which is relatively small for transport along the hexagonal axis independent of layer orientation (λ* = 4.3 nm) and for (0001) terminating surfaces independent of transport direction (λ* = 4.5 nm), but increases, for example, to λ* = 8.8 nm for (112¯0) surfaces and transport along [11¯00]. Direct experiment-simulation comparisons show a 12% and 49% higher λ from experiment at 77 and 295 K, respectively, confirming the limitations of the semi-classical transport simulations despite correct accounting of Fermi surface and Fermi velocity anisotropies. The overall results demonstrate a low resistivity scaling for Ru, suggesting that 10 nm half-pitch Ru interconnect lines are approximately 2 times more conductive than comparable Cu lines.

Influence of surface and grain-boundary scattering on the resistivity of copper in reduced dimensions

We examine the influence of surface and grain-boundary scattering on the total electrical resistivity of copper as dimensions are reduced close to the bulk electron mean free path (39 nm). Through resistivity and grain size characterization on copper wires with sizes down to 95×130 nm2 in a temperature range of 4.2 to 293 K, it was found that the influence of surface scattering is less than previously speculated, while grain-boundary scattering is dominant. A reduction of the background scattering length due to small grains accounts for the observed behavior. The reflection coefficient varies as expected from impurity enrichment in the grain boundaries.

Size effects in thermal noise characteristics of electrons in polycrystalline metallic films

We present a theory of thermal electronic noise in a polycrystalline metallic film the thickness a of which is comparable to the intrinsic or bulk electron mean-free path l 0. It is assumed that the film consists of grains with an average size d larger than or equal to a. Within the model including background, grain-boundary and external-surface electron scattering we have calculated the autocorrelation function and spectral density of the thermal noise as a function of the ratio l 0/ a and the value of the grain-boundary scattering parameter α depending on l 0/ d and on the grain-boundary reflection coefficient R. The results of the calculation for purely diffuse surface scattering show a substantial deviation of thermal noise characteristics from those in bulk monocrystalline materials. We have found that with the increase of the parameters l 0/ a and/or α the autocorrelation function becomes non-exponential and the noise spectrum deviates from the standard Lorentzian shape. The following features of the fluctuation spectrum are predicted and can be verified experimentally: (i) the redistribution of noise power over the spectrum, namely, the suppression of the low-frequency fluctuations and an increase of the magnitude of noise in the high-frequency region; and (ii) a considerable ‘blue’ shift of the cut-off frequency in the noise spectrum.

Electrical resistivity of epitaxial molybdenum films grown by laser ablation deposition

The results of electrical resistivity measurements at room and low temperatures, and also of reflection high energy electron diffraction analysis of thin (less than 110 nm) Mo films, grown by laser ablation deposition are presented. The films deposited under ultrahigh vacuum conditions on sapphire (1̄012) substrates at temperatures of 20–750 °C are monocrystalline, with a [001] axis perpendicular to the substrate. It is shown that the ratio of room temperature to residual resistance which characterizes structural perfection, is in the range 12–32 for the films 70 nm thickness. It increases abruptly at film growth temperatures of 200–370 °C and changes weakly at further temperature increase. The analysis of data on the size (thickness) effect of the Mo films deposited at 750 °C revealed that the effective electron mean free path is in the range 0.1–1 μm for the films 15–110 nm thickness. Surface scattering was found to be the basic channel of electron scattering at helium temperatures, with the specular coefficient q∼0.3 and residual bulk electron mean free path ∼100 μm. At low temperatures T<25 K, the temperature dependent part of the resistance is approximated by ρ(T)−ρ0=ATn. The power degree smoothly varies from 3.2 in high quality films deposited at 750 °C, where the effective electron mean free path exceeds film thickness to 4.5 in the films deposited onto cold unannealed substrates, where the effective electron mean free path is comparable with film thickness.

Electrical properties of epitaxial tungsten films grown by laser ablation deposition

An investigation was made of electrical properties of tungsten films of thickness in the range of 30–140 nm, grown by laser ablation deposition in ultrahigh vacuum on [1̄012] sapphire substrate. From the data on the size effect and the temperature dependence of the resistivity r, supported with reflection high energy electron diffraction measurements, we find that the films, deposited onto clean substrates kept at temperatures higher than 500 °C, grow epitaxially with high quality crystalline structure. The effective electron mean free path changes from 300 to 1400 nm while the residual resistance ratio RRR=r (295 K)/r (4.2 K) changes from 7 to 35. At low temperatures we find the temperature dependent part ρ(T)≊ATn, with A=1.56 mΩ cm K−n, n=3–3.4 for the temperatures T<20 K. From the fit of the data to the equations of the classical size effect theory it was found that the main source of electron scattering at helium temperatures is interface scattering with a specular coefficient q≊0.3 and the bulk electron mean free path l∞ (4.2 K)∼200 μm, which is up to 103 times larger than the film thickness.

Anisotropy in surface resistivity due to electron-phonon scattering in aluminium

The electrical resistivity rho has been measured at temperatures from 1.5 to 75 K for two series of foils with (111) and (110) surfaces on zone-refined Al, under the same current direction of ((112)). It is found that rho is larger on the (111) foil than the (110) one at 1.5-60 K and their difference peaks at around 35 K. The analysis of the data with Fuchs-Sondheimer theory gives the following results. (i) The phonon-limited part of the bulk-electron mean free path, lb(T), is larger along about ((111)) (lb111(T)) than the orientations around (110) (lb110(T)), and agrees with the value from the electron-phonon scattering time calculated by Leung et al. at 20 K. (ii) lb111(T) varies as T-3.0, but lb110(T) varies as T-3.5 at 10-20 K; this is consistent with Meader and Lawrence's theoretical prediction. The surface resistivity rho s of the single crystals is about half of that of the polycrystal of pure Al. It is concluded that the enhancement of rho s due to anisotropy in lb(T) is the main cause of the discrepancies between the measured and predicted rho s in polycrystals.

The effects of surface scattering upon resistivity

It is shown how surface scattering may lead to substantial deviations from Matthiessen's rule. Using an angularly dependent surface scattering parameter as modelled by Soffer (1967), limit expressions for surface scattering in foils and wires are presented for both high and low kappa regions, where kappa is the ratio of the foil thickness, d, or wire radius, a, to the bulk electron mean free path, lambda infinity . From considerations for general values of kappa it is shown that surface scattering, at temperatures where the residual resistivity dominates, simply enhances the bulk temperature dependence. For higher temperatures for which the bulk resistivity varies as T5 the surface scattering can lead to an apparent T2 dependence. Further, it is shown that only for very thick samples or high temperatures where kappa >>1 is it possible to use a simplifying expression of the form rho - rho infinity =3/8 rho infinity lambda infinity /t, for the resistivity of foils or wires, where t is d or a. For most experiments the full theory is necessary to deduce rho infinity lambda infinity ; consequently virtually all previous determinations of this parameter are erroneous. Moreover, because of the influence of the surface scattering on the apparent temperature dependence of the resistivity, even of relatively thick material, several of the 'bulk' temperature dependences previously deduced for pure materials are almost certainly in error.

The electrical resistivity of thin metal films with unlike surfaces

The variation of the resistivity with thickness d and bulk electron mean free path lambda infinity of thin metal films with unlike surfaces is calculated using Soffer's angularly dependent surface scattering model (1967). Tabular values for the resistivity enhancement due to a range of surface roughness combinations are presented. It is shown that in the limit of one very smooth and one very rough surface, the effect, as expected, is simply an apparent doubling in the sample thickness. This contrasts with the much more likely situation of one relatively rough surface and one smooth surface. For this case it is shown that at high kappa values ( kappa =d/ lambda infinity ) the dependence of the resistivity on kappa varies approximately as the rougher surface limit, while at low kappa values the variation approaches the limit of a sample of double thickness having, on both surfaces, the higher roughness. The overall effect in this case is that the resistivity appears to vary in the manner expected from an intermediate value of surface roughness with a changed rho infinity lambda infinity value, rho infinity being the bulk resistivity. It is demonstrated that only in the limit of identical surfaces, or one very smooth and one very rough surface, with in either case no, or at least known and quantifiable, grain boundary scattering, will the determination of rho infinity lambda infinity values be meaningful. Consequently rho infinity lambda infinity values determined in the past from the measurement of the variation with thickness of the resistivity of thin films are often substantially in error.

Electrical conduction in metal foils

The most popular theory concerning the electrical resistivity of metal foils is that given by Fuchs, despite the presence of a physically more realistic theory developed by Soffer. It is shown that as a result of using Fuchs' theory experimentalists have been led to erroneous conclusions regarding both the surface-induced deviations from Mathiessen's rule (DMR) and the temperature dependence of the product of the bulk resistivity and the bulk electron mean free path rho infinity lambda infinity . It is shown that Soffer's theory may be well fitted to the available data on aluminium foils using a temperature-independent rho infinity lambda infinity whilst also satisfactorily explaining surface-produced DMR. Consequently surface scattering is shown to be better characterised by Scoffer's model, which involves an angle-dependent scattering parameter, than by Fuchs' model which assumes that surface scattering is independent of angle of incidence. The two theories are shown to be equivalent only when the sample thickness is greater than the bulk electron mean free path.

Measurements and interpretation of the electrical resistivity and Hall coefficient in polycrystalline gold films: Part I

By measuring the resistivity of gold films as a function of temperature, the phonon contribution can be separated from the intrinsic one. The temperature dependence of the phonon terms can be fitted to a Sondheimer plot with the bulk electron mean free path and the value of the scattering parameter p varying with temperature. The phonon term at room temperature is not very different from the bulk. The intrinsic term in the resistivity is important, and in very thin films even more important than the phonon contribution. Electron microscopy including automatic image processing has shown that the product of grain size and intrinsic resistivity is constant. This agrees with the mosaic theory fairly well.

The audio frequency size effect seen in the modulation of the radio frequency size effect

The modulation field used in the radio frequency size effect is screened by induced audio frequency currents. The incomplete penetration of the modulation field can be clearly seen in the phase lag of the Gantmakher resonance signal behind the signal due to surface bound states. A comparison of the phase lag as a function of modulation frequency with a simple theory leads to an estimate of the bulk electron mean free path. The influence of modulation field screening on measurements of the phonon limited mean free path and other applications are discussed.

Resistivity Studies of Single-Crystal and Polycrystal Films of Aluminum

The resistivities of condensed polycrystal and single-crystal aluminum films have been measured as a function of thickness (1600-36000 \AA{}) and temperature (4.2-300\ifmmode^\circ\else\textdegree\fi{}K). Both the temperature-independent residual resistivity and the temperature-dependent resistivity increase with decreasing film thickness. The residual resistivity agrees very well with the Fuchs-Sondheimer theory. Both groups of films have the same bulk electron mean free path but different reflection parameters. The increase in the temperature-dependent part of the resistivity is larger but similar to the deviations predicted by the Fuchs-Sondheimer theory.