Contact Resistance with Dissimilar Materials: Bulk Contacts and Thin Film Contacts

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1 Contct Resistnce with Dissimilr Mterils: Bulk Contcts nd Thin Film Contcts Peng Zhng Y. Y. Lu * W. Tng M. R. Gomez 3 D. M. French J. C. Zier 4 nd R. M. Gilgenbch Deprtment of Nucler Engineering nd Rdiologicl Sciences University of Michign Ann Arbor USA Air Force Reserch Lbortory Kirtlnd AFB USA 3 Sndi Ntionl Lbortories Albuquerque USA 4 Nvl Reserch Lbortory Wshington DC USA Abstrct Contct resistnce is importnt to integrted circuits nd thin film devices crbon nnotube bsed cthodes nd interconnects field emitters wire-rry z-pinches metlinsultor-vcuum junctions nd high power microwve sources etc. In other pplictions the electricl contcts re formed by thin film structures of few microns thickness such s in microelectromechnicl system (MEMS) relys nd microconnector systems. This pper summrizes the recent modeling efforts t the University of Michign ddressing the effect of dissimilr mterils nd of finite dimensions on the contct resistnce of both bulk contcts nd thin film contcts. The Crtesin nd cylindricl geometries re nlyzed. Accurte nlyticl scling lws re constructed for the contct resistnce of both bulk contcts nd thin film contcts over lrge rnge of spect rtios nd resistivity rtios. These were vlidted ginst known limiting cses nd spot-checks with numericl simultions. Keywords- contct resistnce; electricl contcts; contct potentil; thin films; dissimilr mterils; constriction resistnce; spreding resistnce ignored nd (C) the contct members re bulk conductors whose dimensions trnsverse to the current flow re infinite. Mny subsequent models hve been developed bsed on Holm s -spot theory dopting these three ssumptions. The -spot theory hs been extended by to include the effects of finite bulk rdius [7 8] thereby relxing ssumption (C). We hve recently further extended the Holm- theory of -spot by relxing ssumption (A) with the inclusion of finite thickness in the contct bridge [9]. The scling lw developed in [9] ws fvorbly tested ginst experiment [0]. We next included the effects of dissimilr mterils [] thereby simultneously relxing ssumptions (A) (B) nd (C) mentioned in the preceding prgrph. This generlized -spot is shown in Fig. where the resistivity nd 3 nd the dimensions b c nd h re rbitrry. In Section II we present the scling lws for the contct resistnce in Fig.. I. INTRODUCTION Our interest in contct resistnce ws stimulted by the recognition of its importnce in our ongoing studies of the Z- pinch [] high power microwve genertion [] triple point junctions [3] field emitters [4] nd heting phenomenology [5]. In lerning the subject we were lwys referred to the clssicl reference of Holm [6]. Holm s -spot theory gives the electricl contct resistnce of circulr constriction between two contcting surfces s [6] R = () where is electricl resistivity nd is the contct spot rdius. Implicit in the theory of Holm [6] re severl ssumptions: (A) the -spot hs zero thickness i.e. zero xil length in the direction of current flow (B) the current chnnel is mde of the sme mteril e. g. the effects of contminnts hve been *Electronic mil: yylu@umich.edu This work ws supported by n AFOSR grnt on the Bsic Physics of Distributed Plsm Dischrges L-3 Communictions Electron Device Division nd Northrop-Grummn Corportion. Two of us (PZ nd DMF) grtefully cknowledge fellowship from the University of Michign Institute for Plsm Science nd Engineering. Figure. Two current chnnels II nd III re mde in contct through the bridge region I in either Crtesin or cylindricl geometries. Holm s -spot corresponds to the cylindricl geometry with h = 0 << b << c. Current flows from left to right. For the thin film geometry shown in Fig. we hve lso extended the trditionl theory [ - 6] to include the effects of dissimilr mterils. This theoreticl study ws n dpttion //$ IEEE

of the techniques tht we used to tret the bulk mterils shown in Fig. [7 8]. In Section III we present the scling lws for the thin film contct resistnce for the model shown in Fig.

2 of the techniques tht we used to tret the bulk mterils shown in Fig. [7 8]. In Section III we present the scling lws for the thin film contct resistnce for the model shown in Fig.. Thus the results presented in this pper represent vst generliztion of the conventionl theory for bulk contcts nd thin film contcts. Figure. Thin film structures in either Crtesin or cylindricl geometries. Terminls E nd F re held t constnt voltge (V 0) reltive to terminl GH which is grounded. The z-xis is the xis of rottion for the cylindricl geometry. The resistivity rtio / in Regions I nd II is rbitrry. II. CONTACT RESISTANCE OF BULK CONTACTS Figure shows the geometry of generlized -spot Region I which hs finite xil length h joining two conducting current chnnels (II III). This figure shows Crtesin (cylindricl) composite current chnnel with hlf chnnel width (rdius) of b nd c ( b c) nd electricl resistivity nd 3. It is ssumed tht the xil extents of chnnels II nd III re so long tht the current flow in these chnnels is uniform fr from the contct region I. The contct resistnce ws derived from series expnsion of the solutions in different regions nd by mtching the boundry condition t the interfces. b b b g + + (3) 0 b/ = 4ln( b/ π) + 4ln( π /) f( b/ ) ( ) f( b/ ) = ( / b) ( / b) 3 4 (4) 0.489( / b) + 0.5( / b) 4 gb ( / ) =.8( / b) + 0.3( / b) ( / b) ( / b) where 0 ( x ) is the normlized contct resistnce of the Crtesin -spot derived by Lu Tng nd Zhng [] for the specil cse in Fig. : h = 0 b = c nd = 3. It is the nlog for the Crtesin chnnel [cf. Eq. (7)]. Note tht in Eq. (4) f() = f( ) = 0 g() = 0 g( ) = R c0 () = 0 nd d 0 ( x) / dx = 0 when x = b/ =. Thus from Eq. (3) the normlized interfce resistnce ( / ) = 0 s expected of Fig. in the limit b/ =. A. Generlized Crtesin -spot For the Crtesin chnnel of Fig. the scling lw for the totl electricl resistnce in Regions II I nd III including the interfces of these regions for rbitrry vlues of b c h nd 3 reds [] L b h R = + + b W 4πW W 3 c 3L3 + + (Crtesin) () 4πW 3 c W where W denotes the chnnel width in the third ignorble dimension tht is perpendiculr to the pper nd the rest of the symbols hve been defined in Fig.. Eqution () ws synthesized from vst mount of dt. In Eq. () the first third nd fifth term represents the bulk resistnce in Regions II I nd III respectively. The second nd the fourth term represent the interfce (contct) resistnces between Regions I nd II nd between Regions I nd III respectively. In Eq. () is pproximtely given by [] Figure 3. ( b/ / ) for semi-infinite Crtesin current chnnels I nd II s function of () spect rtio b/ nd (b) resistivity rtio / ; symbols for the exct theory solid lines for the scling lw Eq. (3) nd the dshed lines in (b) for the Crtesin -spot theory ( R 0( b/ ) Eq. (4)). c

3 In the cse of Crtesin semi-infinite chnnel (L >> b h >> ; Fig. ) the interfce resistnce t z = 0 ( b/ / ) hs n exct expression. This exct expression is well represented by the scling lw Eq. (3) essentilly for the entire rnge of 0 < / < nd b/ s shown in Fig. 3. It is cler from Fig. 3() tht R c increses s b/ increses for given /. However for very brod rnge of / from 0 - to 0 R c vries t the most by difference of for given spect rtio b/ s is evident in Fig. 3 (b). The vlidity of the scling lw of Eq. () for Fig. is further estblished by our demonstrtion tht these scling lws re indeed n excellent pproximtion in severl known limiting cses []. The scling lw of Eq. () introduces n error of t most 0 percent in the normlized contct resistnce ( R c ) in the worst cse h = 0 [] s the interfce resistnce R b/ / ws derived under the ssumption h >>. c ( ) B. Generlized Cylindricl -spot For the cylindricl chnnel of Fig. the scling lw for the totl electricl resistnce in Regions II I nd III including the interfces of these regions for rbitrry vlues of b c h nd 3 reds [] ( ) / = 0 s expected of the interfce resistnce from Fig. in the limit b/ =. In the cse of cylindricl semi-infinite chnnel (L >> b h >> ; Fig. ) the interfce resistnce t z = 0 ( b/ / ) hs n exct expression. This exct expression is well represented by the scling lw Eq. (6) essentilly for the entire rnge of 0 < / < nd b/ s shown in Fig. 4. It is cler from Fig. 4() tht R c increses s b/ increses for given /. Agin similr to the Crtesin cse for very brod rnge of / from 0 - to 0 R c vries only by difference of Δ for given spect rtio b/ s is evident in Fig. 4(b). The vlidity of the scling lw of Eq. (5) for Fig. is further estblished by our demonstrtion tht these scling lws re indeed n excellent pproximtion in severl known limiting cses []. Like the Crtesin chnnel the scling lw of Eq. (5) introduces n error of t most 0 percent in the normlized contct resistnce ( R c ) in the worst cse h = 0 [] s Eq. (5) ws derived for h >>. L b h R = + R c + πb 4 π (Cylindricl) (5) 3 c 3L πc where the symbols hve been defined in Fig.. Eqution (5) ws synthesized from vst mount of dt. In Eq. (5) the first third nd fifth term represents the bulk resistnce in Regions II I nd III respectively. The second nd the fourth term represent the interfce (contct) resistnces between Regions I nd II nd between Regions I nd III respectively. In Eq. (5) R c is pproximtely given by [] b b Δ b 0 g + + (6) ( / ) =.458( / ) ( / ) 0 b b b ( / b) ( / b) 4 gb ( / ) = 0.343( / b) 0.64( / b) ( / b) +.96( / b) where 3 / 3π x is the normlized contct resistnce of the -spot derived by nd Rosenfeld [7 8] for the specil cse in Fig. : h = 0 b = c Δ= = nd ( ) R x re monotoniclly incresing functions of x = b/ with g() = 0 g( ) = nd = 3. Both g(x) nd c0 ( ) R c0 () = 0 ( ) c0 (7) R = nd therefore Eq. (6) yields Figure 4. ( b/ / ) for semi-infinite cylindricl current chnnels I nd II s function of () spect rtio b/ nd (b) resistivity rtio / ; symbols for the exct theory solid lines for the scling lw Eq. (6) nd the dshed lines in (b) for the cylindricl -spot theory ( R 0( b/ ) Eq. (7)). c

4 III. CONTACT RESISTANCE OF THIN FILM CONTACTS The models of contct resistnce in Section II re inpplicble to the thin film contct geometry shown in Fig.. This is prticulrly the cse when the current is mostly confined to the immedite vicinity of the constriction nd flows prllel to the thin film boundry. Figure shows both Crtesin nd cylindricl geometries of the thin film. In Crtesin (cylindricl) geometry the current flows inside the bse thin film with width (thickness) h nd electricl resistivity converging towrds the center of the joint region nd feeds into the top chnnel with hlf-width (rdius) nd electricl resistivity. A. Crtesin Thin Film Contct For the Crtesin geometry of Fig. the totl resistnce R from EF to GH is found to be [8] L L R = + + (Crtesin) (8) h W 4πW b h W where W denotes the chnnel width in the third ignorble dimension tht is perpendiculr to the pper nd the rest of the symbols hve been defined in Fig.. In Eq. (8) the first term represents the bulk resistnce of the thin film bse from A to F nd from D to E where L = b. The third term represents the bulk resistnce of the top region from B to G. The second term represents the remining constriction (or contct) resistnce R c for the region ABCD [7 8]. The exct expression for R c is derived by using Fourier series representtion for the potentils in Region I nd Region II nd then mtching the boundry conditions t the interfce BC [8]. From the exct theory it is found tht R c becomes lmost constnt if either L / >> or L /h >> in which cse R c is determined only by the vlue of /h nd / independent of b [7 8]. The vst mount of dt collected from the exct clcultions llows us to synthesize simple scling lw for the normlized contct resistnce in Eq. (8) s [8] Δ h 0 + (9) + β ( ) ( ) π ( π ) 0 / h = / h = / h 4ln sinh / 0 / h (0) ( h) ( h) / / / h ; Δ ( / h) = 0.047x x x x +.63x x + x = ln( / h) < / h 30 β ( / h) = ( / h) ( / h) / h 30. () Figure 5. ( / h / ) for Crtesin thin film structures in Fig. s function of () spect rtio /h nd (b) resistivity rtio / ; symbols for the exct theory solid lines for the scling lw Eq. (9). The scling lw of Crtesin thin film contct resistnce Eq. (9) is shown in Fig. 5 which compres extremely well with the exct theory for the rnge of 0 < / < nd 0.03 /h 30. (We hve not found the scling lw for /h > 30 for generl vlues of / s dt for /h > 30 re not esy to generte from the exct theory [8].) Ech dt point (symbol) in Fig. 5 consists of mny combintions of b/ nd b/h with either L >> or L >> h. Agin R c is independent of b provided either L >> or L >> h. It is cler tht there is minimum vlue of R c in the region of /h ner unity for given /. This /h vlue for minimum R c decreses slightly s / increses. For the specil cse of / = the minimum R c = π - 4ln = occurs exctly t /h = [6 7] nd if /h devites from R c increses logrithmiclly s -4ln(/h) for /h << nd R c 4ln(/h) for /h >> [6 7]. In the regime /h < the rnge of vrition ( / ) for given /h is insignificnt (Fig. 5 ()); however in the regime of /h > ( / ) for given /h my chnge by n order of mgnitude or more. Once more we hve not estblished the symptotic dependence of R c s /h.

5 B. Cylindricl Thin Film Contct For the cylindricl geometry of Fig. the totl resistnce R from EF to GH is found to be [8] b L R ln = + +. (Cylindricl)() πh 4 b h π where the symbols hve been defined in Fig.. In Eq. () the first term represents the bulk resistnce of the thin film bse from A to F nd from D to E [ 7]. The third term represents the bulk resistnce of the top region from B to G. The second term represents the remining constriction (or contct) resistnce R c for the region ABCD [8]. The exct expression for R c is derived by using Fourier series representtion for the potentils in Region I nd Region II nd then mtching the boundry conditions t the interfce BC [8]. Similr to the Crtesin cse from the exct theory it is found tht R c becomes lmost constnt if either L / >> or L /h >> in which cse R c is determined only by the vlue of /h nd / independent of b. The vst mount of dt collected from the exct clcultions llows us to synthesize simple scling lw for the normlized contct resistnce in Eq. () s [8] Δ h R c0 + + β ( / ) = ( / ) R h R h c0 c / 0 ( h) ( h) ( h) 4 5 ( h) ( h) ( h ) ( h ) h h (3) / / / = / / 0.00 / ; / / < / < 0 (4) ( h) ( h) / / / h ; Δ ( / h) = x 0.05x x x x = ln( / h) < / h < 0 β / = / / ( h) ( h) ( h) 0.00 / h< 0. (5) The scling lw of cylindricl thin film contct resistnce Eq. (3) is shown in Fig. 6 which compres very well with the exct theory for the rnge of 0 < / < nd 0.00 /h < 0. (We hve not found the scling lw for /h > 0.) Ech dt point (symbol) in Fig. 6 consists of mny combintions of b/ nd b/h with either L >> or L >> h. Agin R c is independent of b provided either L >> or L >> h. For given /h R c increses s / increses similr to the Crtesin cse. It is cler tht there is minimum vlue of R c in the region of /h ner.5 for given /. The /h vlue for minimum R c decreses slightly s / increses. For the specil cse of / = the minimum R c 0.4 occurs t / h.6[7]. In the regime /h < the vrition ( / ) for given /h is insignificnt; however in the regime of /h > ( / ) for given /h chnges by fctor in the single digits up to n order of mgnitude s shown in Fig. 6 (). The cylindricl cse differs from the Crtesin cse in one spect nmely s / h 0 R c converges to constnt vlues rnging from bout to.08 essentilly for 0 < / <. The explntion follows. If / h 0 both the rdius nd thickness of the film region re much lrger thn the rdius of the top cylinder (Fig. ) s if two semi-infinite long cylinders re joining together with rdius rtio of b/. In this cse the -spot [6] theory gives vlue of R c in the rnge of to.08 for 0 < / < [c.f. Eq. () of Ref. ]. Once more we hve not estblished the symptotic dependence of R c s /h. Figure 6. ( / h / ) for cylindricl thin film structures in Fig. s function of () spect rtio /h nd (b) resistivity rtio / ; symbols for the exct theory solid lines for the scling lw Eq. (3).

6 IV. SUMMARY In this pper we presented simple ccurte nlyticl scling lws of contct resistnce with dissimilr mterils for both bulk contcts (Fig. ) nd thin film contcts (Fig. ) over very lrge rnge of geometries nd resistivities. Both Crtesin nd cylindricl geometries re nlyzed. The bulk contct (Fig. ) is substntil generliztion of Holm s -spot with finite contct region I nd with rbitrry vlues of b c h nd 3. The explicit scling lw presented here provides the building block for the contct resistnce of single sperity to llow sttisticl tretment for contcting rough surfce. Bsed on our clcultions we found tht if the electricl contct (Region I) is highly resistive ( >> >> 3 ) then the bulk resistnce (the third term on the RHS of Eqs. () nd (5)) domintes over the interfce resistnce (the second nd fourth term on the RHS of Eqs. () nd (5)) once the contct region s xil length (h) exceeds few times ( / ) nd ( 3 / ). Once the geometry (bch) is specified the interfce resistnce depends minly on the electricl resistivity of the min chnnel ( 3 ); it is insensitive to the resistivity of the contct region ( ). The scling lws of the contct resistnce of thin film contct (Fig. ) re lso estblished for the rnge 0 < / < nd 0.03 /h 30 for the Crtesin cse nd 0.00 /h < 0 for the cylindricl cse. It is found tht t given resistivity rtio the thin film contct resistnce primrily depends on the rtio of constriction size () to the film thickness (h) only s long s either L >> or L >> h. In the ltter cses the electrosttic fringe field is restricted to the constriction corner only nd becomes insensitive to the loction of terminls for the thin film region. If the constriction size () is smll compred with film thickness (h) the thin film contct resistnce is insensitive to the resistivity rtio. However if /h > the contct resistnce vries significntly with the resistivity rtio. Typiclly the minimum contct resistnce is relized with /h ~ for both Crtesin nd cylindricl thin films. Our nlysis here my redily be dpted to therml conduction in vrious bulk nd thin film structures under stedy stte. It could lso be pplicble to the mximiztion of chnnel flows t given pressure. REFERENCES [] M. R. Gomez J. C. Zier R. M. Gilgenbch D. M. French W. Tng nd Y. Y. Lu Effect of soft metl gsket contcts on contct resistnce energy deposition nd plsm expnsion profile in wire rry Z pinch Rev. Sci. Instrum. vol 79 p [] R. M. Gilgenbch Y. Y. Lu H. McDowell K. L. Crtwright nd T. A. Spencer "Crossed-Field Devices" in Modern Microwve nd Millimeter Wve Power Electronics Chp. 6 edited by R.J. Brker N.C. Luhmnn J.H. Booske nd G.S. Nusinovich IEEE Press Pisctwy NJ 004. [3] N. M. Jordn Y. Y. Lu Dvid M. French R. M. Gilgenbch nd P. Pengvnich Electric field nd electron orbits ner triple point J. Appl. Phys. vol 0 p [4] R. Miller Y. Y. Lu nd John H. Booske Electric field distribution on knife-edge field emitters Appl. Phys. Lett. vol 9 p [5] P. Zhng Y. Y. Lu nd R. M. Gilgenbch Anlysis of rdiofrequency bsorption nd electric nd mgnetic field enhncements due to surfce roughness J. Appl. Phys. vol 05 p [6] R. Holm Electric Contct Springer-Verlg Berlin ed [7] A. M. Rosenfeld nd R. S. The potentil distribution in constricted cylinder: An exct solution Qurt Appl. Mth. vol. 39 p [8] R. S. Electricl contct resistnce: properties of sttionry interfces IEEE Trns. Compon. Pckg. Technol. vol p [9] Y. Y. Lu nd Wilkin Tng A higher dimensionl theory of electricl contct resistnce J. Appl. Phys. vol 05 p [0] M. R. Gomez D. M. French W. Tng P. Zhng Y. Y. Lu nd R. M. Gilgenbch Experimentl vlidtion of higher dimensionl theory of electricl contct resistnce Appl. Phys. Lett. vol 95 p [] P. Zhng nd Y. Y. Lu Scling Lws for Electricl Contct Resistnce with Dissimilr Mterils J. Appl. Phys. vol. 08 p There ws typo in this pper. In Eq. (6) of this pper the term -.8(/b) in g(b/) should red -.8(/b). [] R. Constriction resistnce of thin-film contcts Electricl Contcts Processdings of the 54th IEEE Holm Conference on pp Oct [3] M. B. Red J. H. Lng A. H. Slocum nd R. Mrtens Contct Resistnce in Flt Thin Films Electricl Contcts Proceedings of the 55th IEEE Holm Conference on pp [4] G. Norberg S. Dejnovic H. Hesselbom Contct resistnce of thin metl film contcts Components nd Pckging Technologies IEEE Trnsctions on vol. 9 No. pp [5] P. M. Hll Resistnce clcultions for thin film ptterns Thin Solid Films vol p [6] P. M. Hll Resistnce clcultions for thin film rectngles Thin Solid Films vol 300 p [7] P. Zhng Y. Y. Lu nd R. M. Gilgenbch Minimiztion of thin film contct resistnce Appl. Phys. Lett. vol 97 p [8] P. Zhng Y. Y. Lu nd R. M. Gilgenbch Thin film contct resistnce with dissimilr mterils J. Appl. Phys. (in the press 0).

Thin film contact resistance with dissimilar materials

Thin film contact resistance with dissimilar materials JOURNAL OF APPLIED PHYSICS 109, 124910 (2011) Thin film contct resistnce with dissimilr mterils Peng Zhng, Y. Y. Lu, ) nd R. M. Gilgench Deprtment of Nucler Engineering nd Rdiologicl Sciences, University