Cobalt Interconnect Metallization

Resistivity scaling is the breakdown of bulk conduction in interconnects whose critical dimension approaches the electron mean free path. In copper at room temperature that path is

$$ \lambda_{\text{mfp}} \approx 39\ \text{nm} $$

so a 20 nm wide line confines carriers to roughly half a mean free path. Every additional scattering surface then contributes resistance that does not exist in bulk metal, and measured resistivity rises steeply and non-linearly as linewidth falls.

Fuchs-Sondheimer: external surface scattering

The Fuchs-Sondheimer model treats the film boundaries as partially specular reflectors. A fraction $p$ of electrons reflect specularly and retain their momentum along the wire; the remaining $(1-p)$ scatter diffusely and lose it entirely. For a film of thickness $d$ with $\kappa = d / \lambda_{\text{mfp}}$,

$$ \frac{\rho_0}{\rho_{\text{FS}}} = 1 - \frac{3(1-p)}{2\kappa}\int_{1}^{\infty}\!\left(\frac{1}{t^{3}}-\frac{1}{t^{5}}\right)\frac{1-e^{-\kappa t}}{1-p\,e^{-\kappa t}}\,dt $$

For the idealized film boundary conditions, the integral form is the reference model. Its thick-film asymptote is useful for intuition:

$$ \rho_{\text{FS}} \approx \rho_0\left[1 + \frac{3}{8}(1-p)\frac{\lambda_{\text{mfp}}}{d}\right] $$

The physical content is the $1/d$ dependence. The specularity $p$ is a fitted boundary parameter, not a transferable material constant. Roughness, chemistry, texture and the validity of the semiclassical model all influence its apparent value. Rectangular wires also expose multiple surfaces, so film asymptotes require geometry-aware correction before line prediction.

Mayadas-Shatzkes: internal grain-boundary scattering

Mayadas and Shatzkes model grain boundaries as a set of partially reflecting planes perpendicular to the current, spaced by the average grain diameter $g$, each reflecting carriers with probability $R$. Defining the reflection coefficient

$$ \alpha = \frac{\lambda_{\text{mfp}}}{g}\cdot\frac{R}{1-R} $$

the resistivity ratio is

$$ \frac{\rho_0}{\rho_{\text{MS}}} = 3\left[\frac{1}{3}-\frac{\alpha}{2}+\alpha^{2}-\alpha^{3}\ln\!\left(1+\frac{1}{\alpha}\right)\right] $$

In damascene copper, grain size and texture evolve with trench geometry and anneal rather than obeying a universal $g=d$ rule. As characteristic grain size falls, $\alpha$ grows and boundary scattering can become a major term; the actual contribution must be separated experimentally from surface and impurity effects.

Combined model

A common first-order decomposition is Matthiessen-like addition. Because the scattering channels and fitted parameters are coupled, this expression is an accounting approximation—not a guarantee of predictive accuracy:

$$ \rho_{\text{total}} \approx \rho_0 + \Delta\rho_{\text{FS}} + \Delta\rho_{\text{MS}} + \rho_{\text{imp}} $$
Interconnect resistance is a transport problem and a geometry problem surface scattering + grain boundaries raise resistivity; liners reduce the area that actually carries current A ELECTRON TRANSPORT IN A CONFINED POLYCRYSTAL grain boundarygrain boundary diffuse surface events: probability 1−p boundary reflection: probability R when width, height and grain size approach λ, momentum-relaxing events multiply B CONDUCTING AREA Cu core barrier + liner consume area candidatemetal linerless concept uses more area same drawn width; integration assumptions differ C SCHEMATIC SCALING TRADE SPACE drawn linewidth → decreasingline resistance / length Cu + barrier/liner linerless candidate crossover is stack-specific schematic trends — not a universal material ranking D MODEL → MEASUREMENT LOOP Fit the actual structure • width and height; all interfaces • grain-size distribution and texture • roughness, impurities, liner conductivity • temperature coefficient and variability p and R are fitted interface parameters, not universal constants.
Scaled-Line Resistivity Is an Extraction Problem electrical resistance becomes material physics only after contacts, geometry, temperature, and variability are resolved A · FOUR-TERMINAL LINE MEASUREMENT force I+force I− sense V+sense V− known sense length L R′ = (V+ − V−)/(I L) contact drops lie outside the voltage taps B · GEOMETRY SETS THE DENOMINATOR drawn / measured width height conducting core Acore ≠ Wdraw × Hdraw liner may conduct in parallel TEM/CD uncertainty propagates directly into ρeff = R′Acore C · FIT A FAMILY, NOT ONE LINE inverse conducting area 1/Acore → R per length → rough / fine-grain split smooth / coarse-grain split JOINT EXTRACTION INPUTS • multiple widths, heights, lengths, and temperatures • measured CD/TEM geometry and grain distribution • blanket-film controls and liner-only structures • wafer/location replicates and model residuals one resistance number cannot uniquely identify p, R, grain size, conducting area, and impurities report measured R′, geometry, extraction model, fitted covariance, temperature, and structure—not “bulk resistivity” alone

The measured quantity is resistance per length; scaled resistivity is inferred. Four-terminal structures suppress pad and contact drops, but the conversion $\rho_{\mathrm{eff}}=R'A_{\mathrm{core}}$ still inherits uncertainty in sidewall angle, line height, barrier thickness, liner continuity, and parallel conduction. A useful extraction therefore fits families of structures and temperatures with independent geometry and microstructure measurements, then tests whether the fitted parameters predict held-out structures.

Why alternative metals can win after integration

Bulk resistivity alone is insufficient for a scaled-line comparison. The product $\rho_0 \lambda_{\text{mfp}}$ is a useful semiclassical screening metric for surface-scattering sensitivity, but it does not by itself rank integrated conductors: grain structure, interfaces, liner area, aspect ratio, impurities, reliability, and patterning also enter. Copper combines low bulk resistivity with a comparatively long mean free path, so its size penalty can become important under confinement.

Metal$\rho_0$ (μΩ·cm)$\lambda_{\text{mfp}}$ (nm)$\rho_0\lambda$ (10⁻¹⁶ Ω·m²)Note
Cu1.7396.7needs barrier + liner
Co6.211.87.3barrierless, good EM
Ru7.86.75.1shortest λ, liner-free
Mo5.311.25.9emerging, low ρλ
W5.315.58.2high ρλ, legacy contact

A damascene copper line does not use its full drawn cross-section for copper conduction: barrier and liner thicknesses follow integration and reliability constraints and may scale more slowly than the trench, so the copper core can shrink faster than the drawn dimension. A qualified linerless or reduced-liner scheme can devote more of the drawn cross-section to conduction. That geometric benefit can outweigh higher bulk resistivity at tight dimensions, but the crossover is specific to deposition, dimensions, aspect ratio, interfaces, patterning and reliability requirements.

Practical consequences

Measured narrow-line copper resistivity can substantially exceed the bulk value once surface, grain-boundary and impurity scattering are included; barrier and liner displacement further raise line resistance by reducing conductive area. The resulting RC pressure motivates alternative integration schemes and metals, but does not by itself dictate one universal architecture. Modeling this requires fitting an identifiable parameter set to the actual process rather than adopting literature values. The apparent $p$ and $R$ summarize interface and microstructure behavior inside a chosen model; without independent geometry and grain information, different parameter combinations can explain the same resistance data.

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