Thin film contact resistance with dissimilar materials

Transcription

1 JOURNAL OF APPLIED PHYSICS 109, (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 of Michign, Ann Aror, Michign , USA (Received 3 Ferury 2011; ccepted 30 April 2011; pulished online 28 June 2011) This pper presents results of thin film contct resistnce with dissimilr mterils. The model ssumes ritrry resistivity rtios nd spect rtios etween contct memers, for oth Crtesin nd cylindricl geometries. It is found tht the contct resistnce is insensitive to the resistivity rtio for /h < 1, ut is rther sensitive to the resistivity rtio for /h > 1 where is the constriction size nd h is film thickness. Vrious limiting cses re studied nd vlidted with known results. Accurte nlyticl scling lws re constructed for the contct resistnce over lrge rnge of spect rtios nd resistivity rtios. Typiclly the minimum contct resistnce is relized with /h 1, for oth Crtesin nd cylindricl cses. Electric field ptterns re presented, showing the crowding of the field lines in the contct region. VC 2011 Americn Institute of Physics. [doi: / ] I. INTRODUCTION Thin film contct is very importnt issue in mny res, such s integrted circuits, 1,2 thin film devices, 3,4 cron nnotue nd cron nnofier sed cthodes 5,6 nd interconnects, 5,7 field emitters, 6,8 nd thin film-to-ulk contcts, 9 etc. Even in the simplest form, the film resistor remins the most fundmentl component of vrious types of circuits. 3,4 Recently, it ecomes incresingly importnt in the minituriztion of electronic devices such s micro-electromechnicl system relys nd microconnector systems, where thin metl films of few microns re typiclly used to form electricl contcts. 9 In high energy density physics, the electricl contcts etween the electrode pltes nd in Z-pinch wire rrys re crucil for high current delivery. 10 For decdes, the fundmentl model of electricl contct hs een Holm s clssicl -spot theory, 11 which ssumes circulr contct region (of zero thickness) etween two ulk conductors. The -spot theory hs recently een extended to include the effects of finite ulk rdius, 12 of finite thickness of contct ridge, 13,14 nd of dissimilr mterils nd contminnts. 15 These prior works re inpplicle to the thin film geometry tht is studied in this pper (Figs. 1 3). 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 oundry. The two-dimensionl (2D) thin film resistnce hs een investigted for vrious ptterns in Crtesin geometry. 3 The spreding resistnce of three-dimensionl (3D) thin film disks is lso nlyzed. 9,16 These prior works ssume constnt nd uniform electricl resistivity in ll regions. In prticulr, Timsit 9 nlyticlly clculted the spreding resistnce of circulr thin conducting film of thickness h connected to ulk solid vi n -spot constriction of rdius, ut with the ssumption tht the current density distriution through the -spot of this film is the sme s the known current density ) Electronic mil: yylu@umich.edu. distriution through the -spot in semi-infinite ulk solid. 9,11,12 Timsit stted tht his model is relile only for 0 < /h As we shll see, in this pper, we re le to confirm Timsit s results for 0 < /h 0.5, nd t the sme time to extend his results for /h up to ten [cf., the lowest solid curve in Fig. 10]. Most recently, we developed simple nd ccurte nlyticl model for Figs. 1 3, under the sme ssumption of constnt nd uniform resistivity in ll regions. 17 We determined the condition which minimizes the thin film contct resistnce for oth Crtesin nd cylindricl geometries. Our scling lws were vlidted ginst MAXWELL 3D 18 simultion nd ginst conforml mpping results for the Crtesin geometry (Figs. 1 nd 2). In this pper, we gretly extend the nlytic theory of Ref. 17 y llowing the contct memers to hve n ritrry rtio in electricl resistivity. Figure 1 shows oth Crtesin nd cylindricl geometries of the thin film. The current flows inside the se thin film with width (thickness) h nd electricl resistivity, converging towrd the center of the joint region, nd feeds into the top chnnel with hlf-width FIG. 1. (Color online) 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 q 1 / in Regions I nd II is ritrry /2011/109(12)/124910/10/$ , VC 2011 Americn Institute of Physics Author complimentry copy. Redistriution suject to AIP license or copyright, see

2 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) FIG. 2. (Color online) Two cses of Crtesin thin film contct represented y Fig. 1: () thin film sheet resistor nd () hetsink geometry. (rdius) nd electricl resistivity q 1, in Crtesin (cylindricl) geometry. This configurtion is representtive to vrious pplictions. The Crtesin cse my represent thin film sheet resistor [Fig. 2()], 3 where the third dimension, which is perpendiculr to the plne of the pper, is smll. It my lso represent hetsink geometry [Fig. 2()], where this third dimension is lrge. The cylindricl cse (Fig. 3) my represent cron nnotue 5 8 or field emitter 6 setting on sustrte; or it my represent z-pinch wire connected to plte electrode. 10 It is ssumed tht the xil extent of the top chnnel (i.e., L 1 in Fig. 1) is so long tht the current flow in this region is uniform fr from the contct region. Our nlytic formultion (given in detil in the Appendices) ssume finite length L 2 in the se region (Fig. 1). Thus, we study the dependence of the contct or constriction resistnce on the geometries nd resistivities shown in Fig. 1, for ritrry vlues of,, h, q 1, nd (Figs. 4, 5, 9, nd 10). The potentil profiles re formulted exctly, from which the interfce contct resistnces re derived. Simple, ccurte FIG. 4. (Color online) for the Crtesin structure in Figs. 1 nd 2 is plotted s function of () L 2 / nd () L 2 /h for /h ¼ 0.1 nd 8.0, nd q 1 / ¼ 10, 1.0, nd 0.1 (top to ottom). scling lws for the thin film contct resistnce re synthesized (Figs. 6 nd 11). The ptterns of current flow re lso displyed. The conditions to minimize the contct resistnce re identified in vrious limits. Vlidtion of our theory ginst known results is indicted. Only the mjor results will e presented in the min text. Their derivtions re given in the ppendices. In Sec. II, FIG. 3. (Color online) Cylindricl cse of thin film contct represented y Fig. 1. FIG. 5. (Color online) s function of /h, for the Crtesin structure in Figs. 1 nd 2. The solid line represents the exct clcultions [Eq. (A8)], where ech curve consists of mny comintions of / nd /h, with either L 2 or L 2 h. The dshed lines represent the limiting cses of q 1 =!1[Eq. (2)] nd q 1 =! 0 [Eq. (3)]. Author complimentry copy. Redistriution suject to AIP license or copyright, see

3 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) FIG. 6. (Color online) for Crtesin thin film structures in Figs. 1 nd 2, s function of () spect rtio /h nd () resistivity rtio q 1 / ; symols for the exct theory, solid lines for the scling lw Eq. (4). the results for the Crtesin thin film contct resistnce (constriction resistnce) with dissimilr mterils re presented [Fig. 2]. In Sec. III, the results for the cylindricl thin film contct resistnce (constriction resistnce) with dissimilr mterils re presented [Fig. 3]. Concluding remrks re given in Sec. IV. II. CARTESIAN THIN FILM CONTACT WITH DISSIMILAR MATERIALS Let us first consider the 2D Crtesin T -shpe thin film pttern (Figs. 1 nd 2). The pttern is symmetricl out the verticl center xis. Current flows from the two terminls E, F to the top terminl GH (Fig. 1). We solve the Lplce s eqution for Regions I nd II, nd mtch the oundry conditions t the interfce BC, z ¼ 0. The detils of the clcultions re given in the Appendix A. The totl resistnce, R, from EF to GH is found to e R ¼ L 2 2h W þ 4pW ; h ; q 1 þ q 1L 1 2 W ; (1) where W denotes the chnnel width in the third, ignorle dimension tht is perpendiculr to the pper, nd the rest of the symols hve een defined in Fig. 1. In Eq. (1), the first term represents the ulk resistnce of the thin film se, from A to F, nd from D to E, where L 2 ¼. The third term represents the ulk resistnce of the top region from B to G. The second term represents the remining constriction (or contct) resistnce,, for the region ABCD. If we express the constriction (contct) resistnce s ¼ð =4pWÞ for the Crtesin cse, we find tht depends on the spect rtios /h nd /, nd on the resistivity rtio q 1 /, s explicitly shown in Eq. (1). The exct expression for is derived in Appendix A [cf., Eq. (A8)]. In Eq. (A8), the coefficient B n is solved numericlly in terms of q 1 /, /h, nd / [cf., Eq. (A6)]. These numericl vlues of B n then give from Eq. (A8). The exct theory of [cf., Eq. (A8)] is plotted in Fig. 4() s function of L 2 /, for vrious q 1 / nd /h. To explicitly exmine the dependence on the geometricl prmeters, in Fig. 4() is replotted s function of L 2 /h in Fig. 4(). It is seen from Fig. 4 tht ecomes lmost constnt if either L 2 / 1orL 2 /h 1, in which cse is determined only y the vlue of /h nd q 1 /, independent of. Mny other similr clcultions (not shown) led to the sme conclusion. This is due to the fct tht if L 2, the electrosttic fringe field t the corner B (Fig. 1) is restricted to distnce of t most few s, mking the flow field t the terminl F insensitive to. Likewise, if L 2 h, the electrosttic fringe field t the corner B is restricted to distnce of t most few h s, mking the flow field t the terminl F lso insensitive to. In Fig. 5, the exct theory of [cf., Eq. (A8)] is plotted s function of /h, for vrious q 1 /. Ech solid curve in Fig. 5 consists of mny comintions of / nd /h, with either L 2 or L 2 h. Agin, is independent of, provided either L 2 or L 2 h. For given /h, increses s q 1 / increses. It is cler tht there exists minimum of vlue of in the region of /h ner unity, for given q 1 /. This /h vlue for minimum decreses slightly s q 1 / increses. For the specil cse of q 1 / ¼ 1, the minimum ¼ 2p 4ln2¼ 3:5106 occurs exctly t /h ¼ 1, 3,17 nd if /h devites from 1, increses logrithmiclly s ffi 4lnð=hÞ 1:5452 for =h 1, nd ffi 4lnð=hÞ 1:5452 for =h 1. 3,17 In the regime /h < 1, the rnge of vrition ðq 1 = Þ for given /h is insignificnt (Fig. 5); however, in the regime of /h > 1, ðq 1 = Þ for given /h my chnge y n order of mgnitude or more. In the limit of q 1 /!1; is simplified s (cf., Eq. (A10) in Appendix A) j q1 =!1 ¼ 4 X1 coth½ðn 1=2Þph=Š n 1=2 sin 2 ½ðn 1=2Þp=Š ½ðn 1=2Þp=Š 2 2pð Þ=h; (2) which is lso plotted in Fig. 5. Note tht the exct theory for q 1 / ¼ 100 overlps with Eq. (2). In the limit of q 1 /!1; the minimum ffi 3.9 occurs t /h ¼ 0.85, s shown in Fig. 5. In the opposite limit, q 1 /! 0, the region BCHG (Fig. 1) cts s perfectly conducting mteril with respect to the se region BCEF. Thus, the whole constriction Author complimentry copy. Redistriution suject to AIP license or copyright, see

4 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) interfce BC is n equipotentil surfce, s if L 1 ¼ 0 nd the externl electrode is pplied directly to the interfce BC for the Crtesin geometry. This specil cse is nlyzed y Hll (cf., Fig. 2 nd Eq. (12) of Hll s 1967 pper 3 ), nd from which in the limit of q 1 /! 0 is given s j q1 =!0 ¼ 2p h h 4 ln sinh p i ; (3) 2 h which is lso plotted in Fig. 5. Note tht the exct theory for q 1 / ¼ 0.01 overlps with Eq. (3). This greement my e considered s one vlidtion of the nlytic theory presented in Appendix A.Inthelimitofq 1 /! 0; converges to constnt minimum vlue of 4ln2 ¼ 2.77 for /h > 2,sshown infig.5. As nother vlidtion, consider the specil cse q 1 / ¼ 1 nd L 2 ¼ 0 (Fig. 1). This cse hs n exct solution using conforml mpping. 3 The exct vlues of for /h ¼ 0.1 nd /h ¼ 8 otined from conforml mpping re, respectively, nd In comprison, our numericl vlues re, respectively, nd , s shown in the dt for L 2 ¼ 0 in Fig. 4. The vst mount of dt collected from the exct clcultions llows us to synthesize simple scling lw for the normlized contct resistnce in Eq. (1) nd Fig. 5 s (for L 2 or L 2 h) h ; q 1 ffi 0 þ D h h 2 2q 1 q 1 þ ; (4) h q2 0 ð=hþ ¼ ð=hþj q1 =!0¼ 2p=h 4 ln ½ sinh ð p=2h ÞŠ; (5) Dð=hÞ ¼ 8 >< >: 0:5346ð=hÞ 2 þ0:0127ð=hþþ0:4548; 0:03 =h 1; 0:0147x 6 0:0355x 5 þ 0:1479x 4 þ 0:4193x 3 þ 1:1163x 2 þ 0:9970x þ 1; x ¼ lnð=hþ; 1 < =h 30; ð=hþ ¼ 0:0003ð=hÞ 2 þ 0:1649ð=hÞþ0:6727; 0:03 =h 30: FIG. 7. (Color online) Field lines in the right hlf of Region II of the Crtesin geometry in Fig. 1 for q 1 / ¼ 1 with () /h ¼ 0.1, () zoom in view of () for 0 y/ 3, (c) /h ¼ 1, nd (d) /h ¼ 10. The results from series expnsion method [Eq. (A1)] (solid lines) re compred to those from conforml mpping (dshed lines). (6) This scling lw of Crtesin thin film contct resistnce, Eq. (4), is shown in Fig. 6, which compres extremely well with the exct theory, for the rnge of 0 < q 1 = < 1 nd 0.03 /h 30. (We hve not found the scling lw for /h > 30 for generl vlues of q 1 /,s dt for /h > 30 re not esy to generte from the exct theory, Eq. (A8).) The field line eqution, y ¼ y(z), my e numericlly integrted from the first order ordinry differentil eqution dy=dz ¼ E y =E z ¼ð@U =@yþ=ð@u =@zþ where U is given y Eq. (A1). Figure 7 shows the field lines in the right hlf of Region II (Fig. 1) for the specil cse of q 1 / ¼ 1, with vrious spect rtios /h. It is cler tht the field lines re most uniformly distriuted over the conduction region when / h ¼ 1, which is consistent with the minimum normlized contct resistnce t /h ¼ 1forq 1 / ¼ 1 (Fig. 5). The field lines re horizontlly crowded round the corner of the constriction when /h 1 [Fig. 7()], since in this limit most of the potentil vritions in the thin film (Region II in Fig. 1) re restricted to distnce of few s. The field lines ecome verticlly crowded round the corner of the constriction when /h 1 [Fig. 7 (d)], since in this limit most of the potentil vritions in the upper region (Region I in Fig. 1) re restricted to distnce of few h s. Both limits led to higher contct resistnce in generl (Figs. 5 nd 6). In Fig. 8, the field lines re shown for the specil cse of /h ¼ 1, with vrious resistivity rtios q 1 /.Asq 1 / increses, Region II ecomes more conductive reltive to Region I, the interfce etween Region I nd II (i.e., BC in Fig. 1) ecomes more nd more like n equipotentil, therefore, the field lines (nd the current density) t the interfce ecome more uniformly distriuted, s shown in Fig. 8(c). For q 1 / ¼ 1, the clculted field lines [from Eq. (A1)] re lso compred to those otined from conforml mpping, with excellent greement for ll clcultions, s shown in Figs. 7 nd 8(). This close greement of the field lines with the exct conforml mpping formultion is nother vlidtion of the series expnsion method. Author complimentry copy. Redistriution suject to AIP license or copyright, see

5 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) III. CYLINDRICAL THIN FILM CONTACT WITH DISSIMILAR MATERIALS We now consider the cylindricl configurtion of Fig. 1 using similr pproch. A long cylindricl rod of rdius with resistivity q 1, is stnding on the center of lrge thinfilm circulr disk of thickness h, nd rdius ¼ þ L 2 with resistivity. Current flows inside the thin-film disk from circulr rim E nd F to terminl GH (Figs. 1 nd 3). We solve the Lplce s eqution for Regions I nd II, nd mtch the oundry conditions t the interfce BC, z ¼ 0. The detils of the clcultions re given in the Appendix B. The totl resistnce, R, from EF to GH is found to e R ¼ q 2 2ph ln þ 4 ; h ; q 1 þ q 1L 1 p 2 : (7) In Eq. (7), the first term represents the ulk resistnce of the thin film in Region II, exterior to the constriction region ABCD. It is simply the resistnce of disk of inner rdius, outer rdius, nd thickness h. 9 The third term represents the ulk resistnce of the top cylinder, BCHG. The second term represents the remining constriction resistnce,, for the region ABCD. If we express the constriction (contct) resistnce s ¼ð =4Þ for the cylindricl cse, we find tht depends on the spect rtios /h nd /, nd on the FIG. 9. (Color online) for the cylindricl structure in Figs. 1 nd 3, is plotted s function of () L 2 /, nd () L 2 /h, for /h ¼ 0.1 nd 10.0, nd q 1 / ¼ 10, 1.0, nd 0.1 (top to ottom). FIG. 8. (Color online) Field lines in the right hlf of Region II of the Crtesin geometry in Fig. 1 for /h ¼ 1 with () q 1 / ¼ 0.1, () q 1 / ¼ 1, nd (c) q 1 / ¼ 10. For q 1 / ¼ 1, the results from series expnsion method [Eq. (A1)] (solid lines) re compred to those from conforml mpping (dshed lines). resistivity rtio q 1 /, s explicitly shown in Eq. (7). The exct expression for is derived in Appendix B [cf., Eq. (B8)]. In Eq. (B8), the coefficient B n is solved numericlly in terms of q 1 /, /h, nd / [cf., Eq. (B6)]. These numericl vlues of B n then give from Eq. (B8). The exct theory of [Eq. (B8)] is plotted in Fig. 9() s function of L 2 /, for vrious q 1 / nd /h,wherel 2 ¼ - (Fig. 1). To explicitly exmine the dependence on the geometricl prmeters, in Fig. 9() is replotted s function of L 2 /h in Fig. 9(). It is found tht ecomes constnt if either L 2 / 1orL 2 /h 1, in which cse is determined only y the vlue of /h nd q 1 /, independent of. Mny other similr clcultions (not shown) led to the sme conclusion. This is due to the fct tht if L 2, the electrosttic fringe field t the corner B (Fig. 1) is restricted to distnce of t most few s, mking the flow field t the terminl F insensitive to. Likewise, if L 2 h, the electrosttic fringe field t the corner B is restricted to distnce of t most few h s, mking the flow field t the terminl F lso insensitive to. In Fig. 10, the exct theory of [cf., Eq. (B8)] is plotted s function of /h, for vrious q 1 / nd /. Agin, is independent of, provided either L 2 or L 2 h. For given /h, increses s q 1 / increses, similr to the Crtesin cse. It is cler tht there is minimum of vlue of in the region of /h ner 1.5, for given q 1 /. The /h vlue for minimum decreses slightly s q 1 / increses. For the specil cse of q 1 / ¼ 1, the minimum ffi 0:42 occurs t =h ffi 1:6. 17 is fitted to the following formul for q 1 / ¼ 1: 17 Author complimentry copy. Redistriution suject to AIP license or copyright, see

6 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) which is lso plotted in Fig. 10. Note tht the exct theory for q 1 / ¼ 100 overlps with Eq. (9). In the limit of q 1 /!1; the minimum ffi 0.48 occurs t /h ¼ 1.3, s shown in Fig. 10. In the opposite limit, q 1 /! 0, the region BCHG (Fig. 1) cts s perfectly conducting mteril with respect to the se region BCEF. Thus, the whole constriction interfce BC is n equipotentil surfce, s if L 1 ¼ 0 nd the externl electrode is pplied directly to the interfce BC for the cylindricl geometry. This specil cse is nlyzed y Timsit (cf., Fig. 7 nd Eq. (18) of Ref. 9), whose in the limit of q 1 /! 0is FIG. 10. (Color online) s function of /h, for the cylindricl structure in Figs. 1 nd 3. The solid lines represent the exct clcultions [Eq. (B8)], where ech curve consists of mny comintions of / nd /h, with either L 2 or L 2 h. The dshed lines represent the limiting cses of q 1 =!1[Eq. (9)] nd q 1 =! 0 [Eq. (10)]. ffi 1:0404 2:2328x þ 5:0695x 2 7:5890x 3 þ 6:5898x 4 2:9466x 5 þ 0:5226x 6 ; x ¼ =h; =h 1:6; ffi 0:4571 0:1588y þ 0:1742y 2 0:0253y 3 þ 0:0015y 4 ; y ¼ lnð=hþ; 1:6 < =h < 100: (8) In the regime /h < 1, the vrition ðq 1 = Þ for given /h is insignificnt; however, in the regime of /h > 1, ðq 1 = Þ for given /h chnges y fctor in the single digits, up to n order of mgnitude s shown in Fig. 10. The cylindricl cse differs from the Crtesin cse in one spect, nmely, s =h! 0, our numericl clcultions show tht converges to constnt vlues, rnging from out 1 to 1.08, essentilly for 0 < q 1 = < 1. The explntion follows. If =h! 0, oth the rdius nd thickness of the film region re much lrger thn the rdius of the top cylinder, s if two semi-infinite long cylinders re joining together with rdius rtio of =!1. In this cse, the -spot theory 11 gives vlue of in the rnge of 1 to 1.08, for 0 < q 1 = < 1 [c.f., Eq. (2) of Ref. 15]. In the limit of q 1 /!1; is simplified s (cf., Eq. (B10) in Appendix B) R q1 c =!1 ¼ 16 X 1 p J 2 1 ðk n=þ k n = cothðk n h=þ k 2 n J2 1 ðk nþ 2 ph lnð=þ; (9) q1 =!0 ¼ 4 p X 1 cothðk n h=þ sinðk n=þ 2 k 2 n J2 1 ðk nþ ph lnð=þ: (10) Timsit cknowledges tht Eq. (10) is ccurte only for the rnge of 0 < =h 0:5, 9 eyond which the ssumption of equipotentil contct tht he introduces to derive Eq. (10) does not hold nd the result is not ccurte nymore. This insight of Timsit nd the ccurcy of his solution for / h < 0.5 re evident in Fig. 10, where Eq. (10) is plotted. Note tht the exct theory for q 1 / ¼ 0.01 overlps with Eq. (10) up to /h ¼ 0.5. For /h > 0.5, the exct clcultion of [cf., Eq. (B8)] is lso difficult in the limit of q 1 /! 0, since the determinnt of the mtrix for solving the coefficient B n in Eq. (B6) is close to zero. [This is the min reson why the scling lw given in Eq. (11) elow is vlid only for =h 10]. Nevertheless, our clcultions of for q 1 / ¼ 0.01 shown in Fig. 10 re ccurte up to =h 10, from the convergence of results s sufficiently lrge numer of terms in the infinite series of Eqs. (B6) nd (B8) re employed in our numericl clcultions. Thus, our greement with Timsit s clcultions for /h < 0.5 my e considered s vlidtion of our series expnsion method, nd we hve extended Timsit s clcultions 9 to /h ¼ 10 in Fig. 10. We lso spot checked our results ginst the MAXWELL 3D code for the cse q 1 / ¼ The vst mount of dt collected from the exct clcultions llows us to synthesize simple scling lw for the normlized contct resistnce in Eq. (7) nd Fig. 10 s (for L 2 or L 2 h) h ; q 1 ffi 0 þ D h h 2 2q 1 q 1 þ ; (11) h q2 8 < 1 2:2968ð=hÞþ4:9412ð=hÞ 2 6:1773ð=hÞ 3 0 ð=hþ ¼ ð=hþj q1 =!0 ¼ : þ3:811ð=hþ 4 0:8836ð=hÞ 5 ; 0:001 =h 1; 0:295 þ 0:037ðh=Þþ0:0595ðh=Þ 2 ; 1 < =h < 10; Dð=hÞ ¼ ( 0:0184 ð =h Þ2 þ0:0073ð=hþþ0:0808; 0:001 =h 1; 0:0409x 4 0:1015x 3 þ 0:265x 2 0:0405x þ 0:1065; x ¼ lnð=hþ; 1 < =h < 10; ð=hþ ¼ 0:0016ð=hÞ 2 þ0:0949ð=hþþ0:6983; 0:001 =h < 10: (12) (13) Author complimentry copy. Redistriution suject to AIP license or copyright, see

7 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) FIG. 11. (Color online) for cylindricl thin film structures in Figs. 1 nd 3, s function of () spect rtio /h, nd () resistivity rtio q 1 / ; symols for the exct theory, solid lines for the scling lw Eq. (11). This scling lw of cylindricl thin film contct resistnce, Eq. (11), is shown in Fig. 11, which compres very well with the exct theory, for the rnge of 0 < q 1 / < 1 nd /h < 10. (We hve not found the scling lw for /h > 10 for generl vlues of q 1 /, s explined in the preceding prgrph.) Similr to the Crtesin cse, the field lines in the thin film region re clculted from Eq. (B1), y numericlly solving the field line eqution dz=dr ¼ð@U =@zþ=ð@u =@rþ. Figure 12 shows the field lines in the right hlf of Region II (Fig. 1) for the specil cse of q 1 / ¼ 1, with vrious spect rtios /h. It is cler tht the field lines re most uniformly distriuted over the conduction region when /h ¼ 1, which is consistent with the smllest normlized contct resistnce ner /h ¼ 1 for q 1 / ¼ 1 (Figs. 10 nd 11). The field lines re horizontlly crowded round the corner of the constriction when /h 1 [Fig. 12()], nd ecome verticlly crowded round the corner when /h 1 [Fig. 12(d)], leding to higher contct resistnce in oth limits, in the sme mnner s lredy explined for the Crtesin cse. In Fig. 13, the field lines re shown for the specil cse of /h ¼ 1, with vrious resistivity rtios q 1 /. As q 1 / increses, Region II ecomes more conductive reltive to Region I, the interfce etween Regions I nd II (i.e., BC in Fig. 1) ecomes more nd more like equipotentil, therefore, the FIG. 12. Field lines in the right hlf of Region II of the cylindricl geometry in Fig. 1 for q 1 / ¼ 1 with () /h ¼ 0.1, () zoom in view of () for 0 r/ 3, (c) /h ¼ 1, nd (d) /h ¼ 10. field lines (nd the current density) t the interfce ecome more uniformly distriuted, s shown in Fig. 13(c). IV. CONCLUDING REMARKS This pper presents ccurte nlytic models which llow redy evlution of the contct resistnce or constriction resistnce of thin film contcts with dissimilr mterils over lrge rnge of prmeter spce. We show the lrge distortions of the field lines s result of film thickness. The models ssume ritrry spect rtios, nd ritrry resistivity rtios in the different regions for oth Crtesin nd cylindricl geometries. From the lrge prmeter spce surveyed, it is found tht, t given resistivity rtio, the thin film contct resistnce primrily depends only on the rtio of constriction size () to the film thickness (h), s long s either L 2 or L 2 h. In the ltter cses, the electrosttic fringe field is restricted to the constriction corner only, nd ecomes insensitive to the loction of terminls for the thin film region. The effects of dissimilr mterils re summrized s follows. If the constriction size () is smll compred to the film thickness (h), the thin film contct resistnce is insensitive to the resistivity rtio. However, if /h > 1, the contct resistnce vries significntly with the resistivity rtio. Author complimentry copy. Redistriution suject to AIP license or copyright, see

8 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) where U þ nd U - re the electricl potentil in the region BCHG nd BCEF, respectively, E þ1 is the uniform electric fields t the end GH, V 0 is the electricl potentil t the ends E nd F (y ¼ 6), nd A n nd B n re the coefficients tht need to e solved. Since the current flows prllel to the thin film oundry EF, ¼ 0; z ¼ h; jj2ð0; y Þ; (A2) which leds to ðn 1=2Þph C n ¼ B n coth : (A3) At the interfce z ¼ 0, from the continuity of electricl potentil nd current density, we hve the following oundry conditions: U þ ¼ U ; 1 q @z ¼ 0; z ¼ 0; jj2ð0; y Þ; z ¼ 0; jj2ð0; y Þ; (A4) z ¼ 0; jj2ð; y Þ: (A4c) (A4) FIG. 13. Field lines in the right hlf of Region II of the cylindricl geometry in Fig. 1 for /h ¼ 1 with () q 1 / ¼ 0.1, () q 1 / ¼ 1, nd (c) q 1 / ¼ 10. Typiclly the minimum contct resistnce is relized with /h 1, for oth Crtesin nd cylindricl cses. Vrious limiting cses re studied nd vlidted with known results. Accurte nlyticl scling lws re presented. Finlly, one my dpt the results in this pper to the stedy stte het flow in thermlly insulted thin film structures with dissimilr therml properties. This my e done with Fig. 1 y replcing the electricl conductivity (1/q j ) with the therml conductivity (j j ), j ¼ 1, 2, in the different regions, ssuming tht the j j s re independent of temperture. APPENDIX A: THE CONTACT RESISTANCE OF CARTESIAN THIN FILM Referring to Fig. 1, we ssume tht L 1, so tht the current flow is uniform t the end GH, fr from the joint region. For the two dimensionl Crtesin chnnel, the y- xis nd z-xis re in the plne of the pper. The solutions of Lplce s eqution re U þ ðy; zþ ¼A 0 þ X1 A n cos npy e ðnpz Þ Eþ1 z; z > 0; jj2ð0; y Þ; U ðy; zþ ¼V 0 þ X1 ðn 1=2Þpz B n sinh ðn 1=2Þpz ðn 1=2Þpy þ C n cosh cos ; z < 0; jj2ð0; y Þ; (A1) From Eqs. (A4) nd (A1), the coefficient A n is expressed in terms of B n A 0 ¼ X1 ðn 1=2Þph sin½ðn 1=2Þp=Š B n coth þ V 0 ; ðn 1=2Þp= (A5) A n ¼ X1 ðm 1=2Þph B m coth g mn ; m¼1 g mn ¼ 2 ð 0 cos npy ðm 1=2Þpy cos dy; n 1 (A5) Comining Eqs. (A3),(A4), (A4c), nd (A5), we otin 1 q B n þ 2 X 1 ðm 1=2Þph c n 1=2 q nm B m coth 1 m¼1 2 sin½ðn 1=2Þp=Š ¼ ; n ¼ 1; 2; 3::: ðn 1=2Þp q 1 ðn 1=2Þp= (A6) where c nm ¼ c mn ¼ X1 l¼1 lg nl g ml ; (A7) nd g nl nd g ml is in the form of the lst prt in Eq. (A5). Note tht in deriving Eq. (A6), we hve set E þ1 ¼ 1 for simplicity. It cn e shown from Eq. (A6) tht B n / 1=n 2 s n!1(c.f., Appendix B of Ref. 15). Thus, y writing Eq. Author complimentry copy. Redistriution suject to AIP license or copyright, see

9 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) (A6) in n infinite mtrix formt, B n cn e solved directly with gurnteed convergence. The totl resistnce from EF to GH is R ¼ (U F - U G )/ I ¼ V 0 /I, where I ¼ j2wðe þ1 =q 1 Þj ¼ 2W=q 1 is the totl current in the conducting chnnel, nd W is the chnnel width in the third, ignorle dimension tht is perpendiculr to the pper. The contct resistnce,, is the difference etween the totl resistnce R nd the ulk resistnce (exterior to ABCD) R u ¼ q 1 L 1 =2W þ L 2 =2hW. From Eq. (A1) nd (A5), we find ¼ j ¼ A 0 V 0 j I ; h ; q 1 ¼ 2p q 1 X 1 sin ðn 1=2Þp= ½ Š ðn 1=2Þp= L 2 2hW ¼ 4pW ; B n coth½ðn 1=2Þph=Š 2pð Þ ; (A8) h which is the exct expression for the contct resistnce of Crtesin thin film of dissimilr mterils (Fig. 1) for ritrry vlues of, ( > ), h, nd q 1 /. It ppers in Eq. (1) of the min text. Given the resistivity rtio q 1 / nd spect rtios /h nd /, the coefficient B n is solved numericlly from Eq. (A6), is then otined from Eq. (A8). In the limit of q 1 /!1; Eq. (A6) my e simplified to 2 q B n ¼ 2 sin½ðn 1=2Þp=Š ; n ¼ 1; 2; 3::: ðn 1=2Þp q 1 ðn 1=2Þp= (A9) Thus, from Eq. (A8), is found s ; ¼ 4 X1 h coth½ðn 1=2Þph=Š n 1=2 2pð Þ=h; q 1 =!1; which ppers s Eq. (2) in the min text. sin 2 ½ðn 1=2Þp=Š ½ðn 1=2Þp=Š 2 APPENDIX B: THE CONTACT RESISTANCE OF THIN FILM TO ROD GEOMETRY (A10) Referring to Fig. 1, similr to the Crtesin cse, we lso ssume tht L 1, so tht the current flow is uniform t the end GH, fr from the joint region. The solutions of Lplce s eqution in the cylindricl geometry re 9,15 U þ ðr;zþ¼a 0 þ X1 U ðr;zþ¼v 0 þ X1 J 0 k n r A n J 0 ð n rþe nz E þ1 z; z > 0;r 2ð0;Þ; B n sinh k nz þ C n cosh k nz ; z < 0;r 2ð0;Þ; (B1) where U þ nd U - re the electricl potentil in the region BCHG nd BCEF, respectively, E þ1 is the uniform electric fields t the end GH, V 0 is the electricl potentil t the thin film rim E nd F (r ¼ ), J 0 (x) is the zeroth order Bessel function of the first kind, n nd k n stisfy J 1 ( n ) ¼ J 0 (k n ) ¼ 0, nd A n nd B n re the coefficients tht need to e solved. Since the current flows prllel to the thin film oundry EF, we hve which ¼ 0; z ¼ h; r 2ð0; Þ; (B2) C n ¼ B n coth k nh : (B3) At the interfce z ¼ 0, from the continuity of electricl potentil nd current density, we hve the following oundry conditions: U þ ¼ U ; z ¼ 0; r 2ð0; Þ; (B4) þ q m¼1 ; z ¼ 0; r 2ð0; Þ; ¼ 0; z ¼ 0; r 2ð; Þ: From Eqs. (B1) nd (B4), the coefficient A n is expressed in terms of B n A 0 ¼ X1 B n coth k nh 2J1 ðk n =Þ þ V 0 ; (B5) k n = A n ¼ X1 B m coth k mh g mn ; 2 g mn ¼ 2 J0 2ð nþ ð 0 k m r rdrj 0 ð n rþj 0 ; n 1: (B5) Comining Eqs. (B3), (B4), (B4c), nd (B5), we otin B n þ 1 X 1 q 1 k n J1 2 ð k c nþ nm B m coth k mh m¼1 ¼ 2J 1 ðk n =Þ q 1 k 2 n J2 1 k ; n ¼ 1; 2; 3:::; (B6) ð nþ where c nm ¼ c mn ¼ X1 g nl g ml l J0 2 ð lþ; (B7) l¼1 nd g nl nd g ml is in the form of the lst prt in Eq. (B5). Note tht in deriving Eq. (B6), we hve set E þ1 ¼ 1 for simplicity. It cn e shown from Eq. (B6) tht B n / 1=k 2 n / 1=n 2 s n!1(c.f., Appendix A of Ref. 15). Thus, y writing Eq. (B6) in n infinite mtrix formt, B n cn e solved directly with gurnteed convergence. The totl resistnce from EF to GH is R ¼ (U F - U G )/ I ¼ V 0 /I, where I ¼ p 2 ðe þ1 =q 1 Þ ¼ p=q1 is the totl current in the conducting chnnel. The contct resistnce,,is the difference etween the totl resistnce R nd ulk resistnce (exterior to ABCD) R u ¼ q 1 L 1 =p 2 þð =2phÞ ln ð=þ. From Eq. (B1) nd (B5), we find Author complimentry copy. Redistriution suject to AIP license or copyright, see

10 Zhng, Lu, nd Gilgench J. Appl. Phys. 109, (2011) j ¼ A 0 V 0 j I ¼ 8 q 1 X 1 p 2 ph ln ; h ; q 1 q 2 2ph ln ¼ 4 ; B n cothðk n h=þ J 1ðk n =Þ k n = ; (B8) which is the exct expression for the contct resistnce etween thin film nd coxil rod of dissimilr mterils (Fig. 1) for ritrry vlues of, ( > ), h, ndq 1 /.It ppers in Eq. (7) of the min text. Given the resistivity rtio q 1 / nd spect rtios /h nd /, the coefficient B n is solved numericlly from Eq. (B6), is then otined from Eq. (B8). In the limit of q 1 /!1; Eq. (B6) my e simplified to B n ¼ 2J 1 ðk n =Þ q 1 k 2 n J2 1 k ; n ¼ 1; 2; 3::: (B9) ð nþ Thus, from Eq. (B8), is found s ; h ffi 16 p X 1 J1 2ðk n=þ cothðk n h=þ k n = k 2 n J2 1 ðk nþ 2 ph lnð=þ; q 1=!1; which ppers s Eq. (9) in the min text. ACKNOWLEDGMENTS (B10) This work ws supported y n AFOSR grnt on the Bsic Physics of Distriuted Plsm Dischrges, L-3 Communictions Electron Device Division, nd Northrop-Grummn Corportion. One of us (P.Z.) grtefully cknowledges fellowship from the University of Michign Institute for Plsm Science nd Engineering. 1 G. H. Gelinck, T. C. T. Geuns, nd D. M. de Leeuw, Appl. Phys. Lett. 77, 1487 (2000); W. J. Greig, Integrted Circuit Pckging, Assemly nd Interconnections (Springer, New York, 2007). 2 J. L. Cronero, G. Morin, nd B. Con, IEEE Trns. Microwve Theory Tech. 43, 2786 (1995). 3 P. M. Hll, Thin Solid Films 1, 277 (1967); iid. 300, 256 (1997). 4 H. Kluk, G. Schmid, W. Rdlik, W. Weer, L. Zhou, C. D. Sherw, J. A. Nichols, nd T. N. Jckson, Solid-Stte Electronics 47, 297 (2003). 5 R. H. Bughmn, A. A. Zkhidov, nd W. A. de Heer, Science 297, 787 (2002). 6 D. Shiffler, T. K. Sttum, T. W. Hussey, O. Zhou, nd P. Mrdhl, in Modern Microwve nd Millimeter Wve Power Electronics, edited y R. J. Brker, J. H. Booske, N. C. Luhmnn, nd G. S. Nusinovich (IEEE Press, Pisctwy, NJ, 2005), Chp. 13, p. 691; V. Vlhos, J. H. Booske, nd D. Morgn, Appl. Phys. Lett. 91, (2007). 7 W. Wu, S. Krishnn, T. Ymd, X. Sun, P. Wilhite, R. Wu, K. Li, nd C. Y. Yng, Appl. Phys. Lett. 94, (2009); Z. Yo, C. L. Kne, nd C. Dekker, Phys. Rev. Lett. 84, 2941 (2000); D. Mnn, A. Jvey, J. Kong, Q. Wng, nd H. Di, Nno Lett. 3, 1541 (2003). 8 R. Miller, Y. Y. Lu, nd J. H. Booske, Appl. Phys. Lett. 91, (2007). 9 R. Timsit, Proc. of the 54th IEEE Holm Conf. on Electricl Contcts, pp (2008); M. B. Red, J. H. Lng, A. H. Slocum, nd R. Mrtens, Proc. of the 55th IEEE Holm Conf. on Electricl Contcts, pp (2009); G. Norerg, S. Dejnovic, nd H. Hesselom, IEEE Trns. Compon. Pckg. Technol. 29, 371 (2006). 10 D. A. Chlenski, B. R. Kusse, nd J. B. Greenly, Phys. Plsms 16, (2009); M. R. Gomez, J. C. Zier, R. M. Gilgench, D. M. French, W. Tng, nd Y. Y. Lu, Rev. Sci. Instrum. 79, (2008). 11 R. Holm, Electric Contcts, 4th ed. (Springer-Verlg, Berlin, 1967). 12 R. S. Timsit, IEEE Trns. Compon. Pckg. Technol. 22, 85 (1999); A. M. Rosenfeld nd R. S. Timsit, Qurt. Appl. Mth. 39, 405 (1981). 13 Y. Y. Lu nd W. Tng, J. Appl. Phys. 105, (2009). 14 M. R. Gomez, D. M. French, W. Tng, P. Zhng, Y. Y. Lu, nd R. M. Gilgench, Appl. Phys. Lett. 95, (2009). 15 P. Zhng nd Y. Y. Lu, J. Appl. Phys. 108, (2010). There is typo in this pper. In Eq.(6) of this pper, the term (/) 2 in g(/) should red (/) M. W. Denhoff, J. Phys. D: Appl. Phys. 39, 1761 (2006). 17 P. Zhng, Y. Y. Lu, nd R. M. Gilgench, Appl. Phys. Lett. 97, (2010). 18 See for MAXWELL 3D softwre. Author complimentry copy. Redistriution suject to AIP license or copyright, see