Wire Resistance and Size Effects

1. Theory

• Definition: The resistance of an interconnect line is $\rho L/(wh)$, but in lines whose width or thickness approaches the electron mean free path, the resistivity rises above its bulk value because electrons scatter at the surfaces, interfaces and grain boundaries; this rise is the resistivity size effect (Chawla et al.; Gall).
• Mean free path: Bulk Cu has a room-temperature mean free path of about 39 $\mathrm{nm}$ and a resistivity of 1.71 $\mu \Omega \mathrm{cm}$, so lines below about 100 $\mathrm{nm}$ already show a clear rise, and 10 $\mathrm{nm}$ lines are about an order of magnitude above bulk (Chawla et al.; Gall).
• Two mechanisms: Surface scattering is described by the Fuchs-Sondheimer model with a specularity parameter $p$, and grain-boundary scattering by the Mayadas-Shatzkes model with a reflection coefficient $R_{gb}$ (Chawla et al.).
• Barrier and liner penalty: Cu needs a diffusion barrier and liner of a few nanometres, which take a growing share of the trench and raise the line resistance about three times at 10 $\mathrm{nm}$ half-pitch (Gall).
• Alternative metals: Metals with a small product ${\rho }_0\lambda$, such as Rh, Ir and Ru, suffer less from the size effect and may not need a barrier, so they can beat Cu in the narrowest lines despite a higher bulk resistivity (Gall; Moors et al.).

2. Working Principle

• Surface scattering: An electron that reaches a surface is either reflected specularly, keeping its momentum along the wire, or scattered diffusely; diffuse scattering at Cu-Ta, Cu-air and Cu-MgO interfaces ($p=0$) shortens the effective mean free path by about $3\lambda /8$ divided by the dimension (Chawla et al.).
• Grain-boundary scattering: Each grain boundary reflects a fraction $R_{gb}$ of the electrons; since the grain size in damascene lines scales with the line dimension, the number of boundaries crossed per mean free path grows as the line shrinks (Chawla et al.; Gall).
• Additivity: The bulk phonon, surface and grain-boundary contributions add approximately as in Matthiessen’s rule, so the size effect is nearly independent of temperature and lowers the temperature coefficient of narrow wires (Chawla et al.; Huang et al.).
• Why ${\rho }_0\lambda$ matters: Since the size-effect increase scales with ${\rho }_0\lambda /w$, a metal with a short mean free path loses less conductivity at small widths even if its bulk resistivity is higher (Gall; background knowledge for the scaling).

3. Applications

• Line and via resistance estimates: Resistance per length of local and intermediate wires for RC delay and IR-drop calculations at advanced nodes (Gall; background knowledge).
• Metallization choices: Co local interconnects at Intel’s 10 $\mathrm{nm}$ node, and Ru, Mo and other barrierless metals evaluated for the tightest pitches (Gall; Moors et al.).
• Process optimization: Grain growth, twin-boundary formation and thinner barriers and liners to lower Cu line resistance (Chawla et al.; Gall).
• Thermometry and test structures: The lowered temperature coefficient of narrow wires matters for on-chip resistance thermometers and electromigration test structures, where temperature is inferred from resistance (Huang et al.; Metal and Interconnect page).

Resistance and Size-Effect Models

Name	Formula	Related Formulae	Variables/Constants
Line resistance and sheet resistance	$R=\frac{\rho L}{wh},R_s=\frac{\rho }{h}$	$\frac{R}{L}=\frac{{\rho }_{eff}}{wh},{\rho }_{eff}={\rho }_{metal}\frac{wh}{A_{metal}}$	$R:lineresistance$ $\rho :resistivity$ $L:length$ $w:width$ $h:thickness$ $\begin{array}{r}{R}_s:sheetresistance,R/L:resistanceper\\ length\end{array}$ $\begin{array}{r}{\rho }_{eff}:effectiveresistivityoverthefull\\ trenchcross-section\end{array}$ ${\rho }_{metal}:resistivityoftheconductoritself$ $\begin{array}{r}{A}_{metal}:conductorcross-sectionleftafterthe\\ barrierandliner(backgroundknowledge\\ fortheforms);Gall:fora10\mathrm{nm}\\ half-pitchCulinetheseed,barrierand\\ linerlayersraisethelineresistanceby\\ roughlyafactorof3\end{array}$ $\begin{array}{r}(computed:a10by20\mathrm{nm}trenchwith3\mathrm{nm}\\ ofbarrierandlineronthesidesand\\ bottomkeeps4by17\mathrm{nm}ofCu,2.9times\\ lessarea,matchingGal{l}^{\prime }sfactorof3;\\ with2\mathrm{nm}thefactoris1.85)\end{array}$
Surface scattering: Fuchs-Sondheimer and Dingle approximations	$\frac{{\rho }_f}{{\rho }_0}\approx 1+\frac{3\lambda }{8d}\left(1-\frac{p_1+p_2}{2}\right)(d\gg \lambda \text{asymptote, used down to}d\sim \lambda )$	$\frac{{\rho }_{wire}}{{\rho }_0}\approx 1+\frac{3}{8}\lambda \left(\frac{1}{w}+\frac{1}{h}\right)\left(1-p\right)$	${\rho }_f:filmresistivity$ ${\rho }_0:bulkresistivity$ $\lambda :bulkmeanfreepath$ $d:filmthickness$ $\begin{array}{r}{p}_1,p_2:specularityofthetwosurfaces(0\\ diffuse,1specular)\end{array}$ ${\rho }_{wire}:wireresistivity$ $w,h:widthandthickness$ $\begin{array}{r}p:specularityofallfoursurfaces;\\ Chawlaetal.:theapproximateFSform\\ validfor\kappa =d/\lambda muchgreaterthan1and\\ theDingleformforsingle-crystalwires\\ withwandhabove\lambda ,with\lambda =39\mathrm{nm}and{\rho }_0\\ =1.71\mu \Omega \mathrm{cm}forCuatroomtemperature\\ andp=0atCu-air,Cu-TaandCu-MgO\\ interfaces;Moorsetal.:aband-structure\\ generalizationoftheFSconductivityfor\\ anisotropicmetals\end{array}$ $\begin{array}{r}(computed:a10by20\mathrm{nm}Cuwirewithp=\\ 0has\rho /{\rho }_0=3.2fromsurfacesalone)\end{array}$
Grain-boundary scattering: Mayadas-Shatzkes	$\frac{{\rho }_g}{{\rho }_0}={\left[1-\frac{3}{2}\alpha +3{\alpha }^2-3{\alpha }^3\ln \left(1+\frac{1}{\alpha }\right)\right]}^{-1},\alpha =\frac{\lambda }{D}\frac{R_{gb}}{1-R_{gb}}$	${\rho }_{total}\approx {\rho }_0\left[1+\frac{3}{8\kappa }\left(1-\frac{p_1+p_2}{2}\right)+\frac{3}{2}\alpha \right](\kappa \gg 1,\alpha \ll 1),D=A{d}^{\xi }$	${\rho }_g:resistivitywithgrainboundaries$ $\alpha :grain-boundaryparameter$ $D:averagelateralgrainsize$ $\begin{array}{r}{R}_{gb}:reflectioncoefficientofagrain\\ boundary(0to1)\end{array}$ $\begin{array}{r}{\rho }_{total}:combinedapproximateresistivity,\kappa =\\ d/\lambda ,A\end{array}$ $\begin{array}{r}\xi :grain-sizefitconstants;Chawlaet\\ al.:R_{gb}of0.25plusorminus0.05for\\ filmsand0.30plusorminus0.05for\\ wires,D=22.4d^{0.43}(Danddinnm),and\\ theapproximateexpressionsunderestimate\\ theresistivityby6.5percentat50\mathrm{nm}\\ risingto17percentat20\mathrm{nm}forfilms\\ and3to16percentfrom350to50\mathrm{nm}for\\ wires;Huangetal.:bestfit{R}_{gb}=0.34\\ andp=0.50with\lambda =40\mathrm{nm}and25\mathrm{nm}\\ grains;Gall:R_{gb}of0.25to0.43forCu\\ andabout0.016for\Sigma 3twinboundaries,\\ andnegligiblegrain-boundaryeffectsfor\\ grainsabove200\mathrm{nm}ortentimestheline\\ width\end{array}$ $\begin{array}{r}(computed:\alpha =0.1,0.5and1.3give{\rho }_g/{\rho }_0\\ =1.15,1.70and2.78;withD=w=10\mathrm{nm}\\ and{R}_{gb}=0.25,\alpha =1.3)\end{array}$

Measured Copper Behaviour

Name	Formula	Related Formulae	Variables/Constants
Measured resistivity of Cu films and wires	$\rho \left(d\right)\text{and}\rho \left(w\right)\text{at}298\mathrm{K}$	$\frac{\Delta \rho }{\rho }/\frac{\Delta D}{D}$	$\begin{array}{r}\rho (d),\rho (w):resistivityagainstthickness\\ andwidth,\Delta \rho /\rho :relativeresistivity\\ changeforarelativegrain-sizechange\Delta \\ D/D;Chawlaetal.:filmsof40\mathrm{nm}at\\ 2.50,2.65and2.89\mu \Omega \mathrm{cm}\\ (single-crystal,large-grain,small-grain)\\ and26\mathrm{nm}at2.81,3.55and3.45\mu \Omega \mathrm{cm};\\ 45\mathrm{nm}thickwiresfrom2.95\mu \Omega \mathrm{cm}at340\\ \mathrm{nm}to3.62\mu \Omega \mathrm{cm}at68\mathrm{nm}width(small\\ grain)and2.70to3.38\mu \Omega \mathrm{cm}(large\\ grain);surfacescatteringdominatesin\\ thesefilms,grain-boundaryscattering\\ dominatesonceD=d;growinggrains1.8\\ timeslowerstheresistivityby10to15\\ percentfor30to50\mathrm{nm}filmsand7to9\\ percentforwires,andsinglecrystals\\ are18to22percentbelowsmall-grain\\ layers;Matthiesse{n}^{\prime }sruleholdswithin\\ about4percent;Huangetal.:50\mathrm{nm}\\ thickwiresrisefromabout3to7\times 10^{-8}\\ \Omega \mathrm{m}aswidthfallsfromabout350to100\\ \mathrm{nm}(readfromafigure)\end{array}$ $\begin{array}{r}(computed:theapproximateFSplusMS\\ expressionwithp=0,R_{gb}=0.25andD=\\ 22.4d^{0.43}gives2.64and3.04\mu \Omega \mathrm{cm}for40\\ and26\mathrm{nm},9and12percentbelowthe\\ measuredsmall-grainvalues,consistent\\ withthestatedunderestimate)\end{array}$
Temperature coefficient of narrow wires	$\rho \left(T\right)={\rho }_r+\frac{d\rho }{dT}T,TCR=\frac{1}{\rho }\frac{d\rho }{dT}$	$\rho ={\rho }_{phonon}\left(T\right)+{\rho }_{size}(\text{Matthiessen’s rule})$	$\begin{array}{r}{\rho }_r:temperature-independentresidualpart,\\ d\rho /dT:temperatureslope\end{array}$ $TCR:temperaturecoefficientofresistance$ ${\rho }_{phonon}:phonon-limitedbulkpart$ $\begin{array}{r}{\rho }_{size}:temperature-independentsize-effect\\ part(Matthiesse{n}^{\prime }srule,background\\ knowledge);Huangetal.:fora140\mathrm{nm}\\ widewire\rho =3.38\times 10^{-8}+8.05\times 10^{-11}T\Omega \mathrm{m}(T\\ inK,correlation0.997),aTCRof\\ 1.45\times 10^{-3}{\mathrm{K}}^{-1}atroomtemperatureagainst\\ 4.1\times 10^{-3}{\mathrm{K}}^{-1}forbulkCu,andaTCRfalling\\ withwidthover80to360\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:thefitgives5.77\times 10^{-8}\Omega \mathrm{m}and\\ aTCRof1.40\times 10^{-3}{\mathrm{K}}^{-1}at297\mathrm{K};itsslopeof\\ 8.05\times 10^{-11}\Omega m/Kisabout20percentabove\\ thebulkslopeofabout6.7\times 10^{-11}\Omega m/K\\ (backgroundknowledgeforbulk),a\\ deviationfromMatthiesse{n}^{\prime }srulelarger\\ thanChawlaetal{.}^{\prime }s4percent)\end{array}$

Alternative Metals

Name	Formula	Related Formulae	Variables/Constants
Figure of merit for alternative metals	${\rho }_0=\frac{m^{\ast }{v}_F}{n{e}^2\lambda }\Rightarrow {\rho }_0\lambda =\frac{m^{\ast }{v}_F}{n{e}^2}=\frac{\hslash k_F}{n{e}^2}(\text{free-electron form})$	${\rho }_{wire}-{\rho }_0\propto \frac{{\rho }_0\lambda }{w}$	$\begin{array}{r}{\rho }_0\lambda :productofbulkresistivityandmean\\ freepath,independentofthescattering\\ time\end{array}$ $m^{\ast }:effectivemass$ $v_F:Fermivelocity$ $n:carrierdensity$ $e:elementarycharge$ $\tau :scatteringtime$ $\hslash :reducedPlanckconstant$ $\begin{array}{r}{k}_F:Fermiwavevector(free-electronform,\\ backgroundknowledge);Gall\\ (first-principlesvalues,1\times 10^{-16}\Omega {\mathrm{m}}^2):Rh\\ 3.2,Pt3.4,Ir3.7,Nb3.9,Ru5.1and\\ 3.8(perpendicularandparalleltothec\\ axis),Ni4.1,Ta4.2,Os6.4and4.3,Al\\ 5.0,Fe5.6,Mo6.0andCu6.7;Irjudged\\ themostpromisingelementalmetal,\\ followedbyRuandRh;measuredepitaxial\\ datagivelargerproductsformanymetals,\\ suggestingclassicalmodelsfailbelow\\ about10\mathrm{nm};Moorsetal.:Ruinall\\ directions,V2AlCinplaneandMoPtalong\\ xandymayoutperformCu,bytradinga\\ lowerFermivelocityforahighercarrier\\ density\end{array}$ $\begin{array}{r}(computed:thefree-electronformwithn=\\ 8.47\times 10^{28}{\mathrm{m}}^{-3}(Moorsetal{.}^{\prime }sCudensity)\\ gives6.6\times 10^{-16}\Omega {\mathrm{m}}^2,closetoGal{l}^{\prime }s\\ 6.7\times 10^{-16};6.7\times 10^{-16}\Omega {\mathrm{m}}^{2}with1.71\mu \Omega \mathrm{cm}\\ gives\lambda =39\mathrm{nm}forCu;withanassumed\\ bulk7.1\mu \Omega \mathrm{cm}forRu,5.1\times 10^{-16}\Omega {\mathrm{m}}^{2}gives\\ \lambda =7.2\mathrm{nm})\end{array}$
Barrierless metals against Cu at narrow widths	${\rho }_{eff,Cu}={\rho }_{0,Cu}\left[1+\frac{3{\lambda }_{Cu}}{8}\left(\frac{1}{w}+\frac{1}{h}\right)\right]\frac{wh}{\left(w-2t_b\right)\left(h-t_b\right)},{\rho }_{eff,X}={\rho }_{0,X}\left[1+\frac{3{\lambda }_X}{8}\left(\frac{1}{w}+\frac{1}{h}\right)\right]$	${\rho }_{eff,X}<{\rho }_{eff,Cu}\text{when}w\text{is small}$	$\begin{array}{r}{\rho }_{eff,Cu}:effectiveresistivityofaCuline\\ withabarrierandlineroftotal\\ thickness{t}_{b}onthesidesandbottom\end{array}$ $\begin{array}{r}{\rho }_{eff,X}:effectiveresistivityofabarrierless\\ metalX,othersymbolsasabove\\ (surface-scattering-onlycomparisonwithp\\ =0,myreading;grainboundaries\\ neglected);Gall:metalsthatavoidliners\\ andbarriersgainconductance;Inte{l}^{\prime }s10\\ \mathrm{nm}-nodeColocalinterconnectsat20\mathrm{nm}\\ half-pitchorlessaccepteda1.7times\\ higherlineresistanceforbetter\\ electromigrationresistanceandlower\\ contactresistancewithTi-basedbarriers;\\ electromigrationontheMetaland\\ Interconnectpage\end{array}$ $\begin{array}{r}(computed:fora10by20\mathrm{nm}linewith3\\ \mathrm{nm}ofbarrier,Cureaches5.5\mu \Omega \mathrm{cm}in\\ themetaland16\mu \Omega \mathrm{cm}effective,while\\ Ruwith\lambda =7.2\mathrm{nm}andnobarrierreaches\\ about10\mu \Omega \mathrm{cm})\end{array}$

Important Questions

Ques: Is a 10 nm wide copper line about five or about ten times more resistive than bulk copper?

The classical models give the lower value and Gall the higher. For an assumed 10 by 20 $\mathrm{nm}$ Cu line with fully diffuse surfaces, the Dingle form gives $\rho /{\rho }_0=3.2$, and the Mayadas-Shatzkes term with 10 $\mathrm{nm}$ grains and $R_{gb}=0.25$ adds about 1.8 more, about 5 times bulk or 8.5 $\mu \Omega \mathrm{cm}$ (computed). Gall states that 10 $\mathrm{nm}$ Cu lines are approximately an order of magnitude above bulk, and that classical models underestimate the resistivity below about 10 $\mathrm{nm}$. Chawla et al. also found that the approximate expressions underestimate their measurements, by up to 17 per cent at 20 $\mathrm{nm}$, and the error grows as the dimension falls. The order-of-magnitude value is therefore the more likely one for real 10 $\mathrm{nm}$ lines (my reading). Reference: D. Gall, ‘The search for the most conductive metal for narrow interconnect lines’, Journal of Applied Physics 127(5), 050901 (2020), doi:10.1063/1.5133671 (fetch-tool summary; Table II truncated); J. S. Chawla, F. Gstrein, K. P. O’Brien, J. S. Clarke and D. Gall, ‘Electron scattering at surfaces and grain boundaries in Cu thin films and wires’, Physical Review B 84, 235423 (2011), doi:10.1103/PhysRevB.84.235423 (accepted manuscript; equations reconstructed) (fetch-tool summary).

Ques: Is surface scattering in copper wires diffuse (p = 0) or partly specular (p = 0.5)?

Chawla et al. found completely diffuse scattering, $p=0$, at Cu-air, Cu-Ta and Cu-MgO interfaces, fitting single-crystal and polycrystalline films and wires with a 39 $\mathrm{nm}$ mean free path and a grain-boundary reflection coefficient of 0.25 to 0.30. Huang et al. fitted their 50 $\mathrm{nm}$ thick wires with $p=0.50$ and $R_{gb}=0.34$. The two parameters trade against each other in a fit: a higher $p$ with a higher $R_{gb}$ can reproduce the same total resistivity. Chawla et al. separated the two by comparing single-crystal and polycrystalline samples, which Huang et al. did not (my reading), so $p=0$ is the more likely value for Cu with Ta-based barriers. Gall notes that specular scattering needs a smooth, epitaxial or weakly bonded interface with a low density of states in the liner. Reference: J. S. Chawla, F. Gstrein, K. P. O’Brien, J. S. Clarke and D. Gall, ‘Electron scattering at surfaces and grain boundaries in Cu thin films and wires’, Physical Review B 84, 235423 (2011), doi:10.1103/PhysRevB.84.235423 (accepted manuscript; equations reconstructed) (fetch-tool summary); Q. Huang, C. M. Lilley, M. Bode and R. S. Divan, ‘Electrical Properties of Cu Nanowires’, Proceedings of the 2008 IEEE International Conference on Nanotechnology (DTIC copy; Eq. 1 garbled) (fetch-tool summary); D. Gall, ‘The search for the most conductive metal for narrow interconnect lines’, Journal of Applied Physics 127(5), 050901 (2020), doi:10.1063/1.5133671 (fetch-tool summary; Table II truncated).

Ques: How much do the barrier and liner add to the resistance of a narrow copper line?

Gall estimates that the seed, barrier and liner raise the resistance of a 10 $\mathrm{nm}$ half-pitch Cu line by roughly a factor of 3. A simple area count reproduces this: an assumed 10 by 20 $\mathrm{nm}$ trench with 3 $\mathrm{nm}$ of barrier and liner on the sides and bottom leaves 4 by 17 $\mathrm{nm}$ of Cu, 2.9 times less area, while 2 $\mathrm{nm}$ would give 1.85 times (computed). The penalty compounds the size effect, because the remaining Cu is narrower and scatters more: with fully diffuse surfaces the 4 by 17 $\mathrm{nm}$ core alone would rise to about 5.5 times bulk (computed). Since the barrier does not scale with the line, the penalty grows quickly below about 20 $\mathrm{nm}$. This is why barrierless metals and thinner barriers are pursued. Reference: D. Gall, ‘The search for the most conductive metal for narrow interconnect lines’, Journal of Applied Physics 127(5), 050901 (2020), doi:10.1063/1.5133671 (fetch-tool summary; Table II truncated).

Ques: Why can ruthenium or rhodium beat copper in very narrow lines?

Because the resistivity rise from the size effect scales with ${\rho }_0\lambda$ divided by the line dimension, not with ${\rho }_0$ alone. Gall’s first-principles values are $6.7\times 10^{-16}$ $\Omega {\mathrm{m}}^2$ for Cu but 3.2 for Rh, 3.7 for Ir and 5.1 or 3.8 for Ru, depending on direction. Cu’s 39 $\mathrm{nm}$ mean free path follows from its product and 1.71 $\mu \Omega \mathrm{cm}$, while Ru, with an assumed bulk 7.1 $\mu \Omega \mathrm{cm}$, has only about 7.2 $\mathrm{nm}$ (computed). For an assumed 10 by 20 $\mathrm{nm}$ line, Cu with 3 $\mathrm{nm}$ of barrier reaches an effective 16 $\mu \Omega \mathrm{cm}$, and barrierless Ru only about 10 $\mu \Omega \mathrm{cm}$ (computed, surface scattering only). Gall ranks Ir first, then Ru and Rh, and Moors et al. also identify Ru, along with V2AlC and MoPt, as able to outperform Cu. Reference: D. Gall, ‘The search for the most conductive metal for narrow interconnect lines’, Journal of Applied Physics 127(5), 050901 (2020), doi:10.1063/1.5133671 (fetch-tool summary; Table II truncated); K. Moors, K. Sankaran, G. Pourtois and C. Adelmann, ‘First-principles-based screening method for resistivity scaling of anisotropic metals’, arXiv:2111.11121 (2022) (fetch-tool summary).

Ques: Does growing larger grains still lower the resistance of narrow copper lines?

Yes, but by a limited amount. Chawla et al. found that growing grains 1.8 times lowered the resistivity by 10 to 15 per cent for 30 to 50 $\mathrm{nm}$ films and 7 to 9 per cent for 70 to 350 $\mathrm{nm}$ wide, 45 $\mathrm{nm}$ thick wires, and that single-crystal layers were 18 to 22 per cent below small-grain layers, the most that grain growth can give. In their films surface scattering dominated, and grain-boundary scattering took over only when the grain size fell to the film thickness. Gall adds that grain-boundary effects become negligible for grains above 200 $\mathrm{nm}$ or ten times the line width, and that Sigma-3 twin boundaries reflect only about 0.016 of electrons against 0.25 to 0.43 for ordinary boundaries, so twin-rich microstructures are a further route. Reference: J. S. Chawla, F. Gstrein, K. P. O’Brien, J. S. Clarke and D. Gall, ‘Electron scattering at surfaces and grain boundaries in Cu thin films and wires’, Physical Review B 84, 235423 (2011), doi:10.1103/PhysRevB.84.235423 (accepted manuscript; equations reconstructed) (fetch-tool summary); D. Gall, ‘The search for the most conductive metal for narrow interconnect lines’, Journal of Applied Physics 127(5), 050901 (2020), doi:10.1063/1.5133671 (fetch-tool summary; Table II truncated).

Ques: Why do narrow wires have a lower temperature coefficient of resistance than bulk copper?

Because the size-effect part of the resistivity barely depends on temperature, while the phonon part does. Huang et al. measured $\rho =3.38\times 10^{-8}+8.05\times 10^{-11}T$ $\Omega \mathrm{m}$ for a 140 $\mathrm{nm}$ wide Cu wire, about $5.77\times 10^{-8}$ $\Omega \mathrm{m}$ at 297 $\mathrm{K}$, so the coefficient is $1.40\times 10^{-3}$ ${\mathrm{K}}^{-1}$ (computed) against their stated $1.45\times 10^{-3}$ ${\mathrm{K}}^{-1}$ and about $4.1\times 10^{-3}$ ${\mathrm{K}}^{-1}$ for bulk Cu. A fixed slope added to a larger total gives a smaller relative coefficient. The fitted slope is about 20 per cent above the bulk slope of about $6.7\times 10^{-11}$ $\Omega m/K$ (computed, background knowledge for bulk), more than the 4 per cent deviation from Matthiessen’s rule that Chawla et al. found. Resistance thermometry on narrow lines must therefore be calibrated on the line itself. Reference: Q. Huang, C. M. Lilley, M. Bode and R. S. Divan, ‘Electrical Properties of Cu Nanowires’, Proceedings of the 2008 IEEE International Conference on Nanotechnology (DTIC copy; Eq. 1 garbled) (fetch-tool summary); J. S. Chawla, F. Gstrein, K. P. O’Brien, J. S. Clarke and D. Gall, ‘Electron scattering at surfaces and grain boundaries in Cu thin films and wires’, Physical Review B 84, 235423 (2011), doi:10.1103/PhysRevB.84.235423 (accepted manuscript; equations reconstructed) (fetch-tool summary).

Related pages in this vault: Electron Backscatter Diffraction (EBSD), Metal and Interconnect, Interconnect Joule Heating, Contacts and Vias, Chip-to-Package Interconnects, RC Delay and Repeater Insertion, Wire Capacitance and Low-k Dielectrics, Thin-Film Metal Resistors (TaN, NiCr and SiCr), Four-Point Probe, Picosecond Ultrasonic Film Metrology, CMP Polisher.