Contact Resistance with Dissimilar Materials

Transcription

1 Contact Resistance with Dissimilar Materials Peng Zhang, and Y. Y. Lau Department of Nuclear Engineering and Radiological Sciences University of Michigan Ann Arbor, MI, USA, Michigan Institute for Plasma Science and Engineering 1st Annual Graduate Student Symposium September 29, 2010 This work was supported by AFOSR, L-3 Communications, Northrop-Grumman Corporation, and a MIPSE Graduate Fellowship.

2 Introduction Electrical contact is important to Thin film devices and integrated circuits Carbon nanotubes based cathodes and interconnects Field emitters Tribology Wire-array Z pinches Metal-insulator-vacuum junctions High power microwave (HPM) sources Faulty electrical contact caused the recent failure of the Large Hadron Collider (LHC), and threatens the International Thermonuclear Experimental Reactor (ITER) There is no prior theory of electrical contact with dissimilar materials 2

3 Current Metal 1 Current Current Metal 2 Current : True points of contact => High contact resistance Current flows only through the true points of contact 3 Contact resistance is highly random, affected by surface roughness, pressure, hardness, residing oxides and contaminates, etc 3

4 Holm s Model*: h = 0 Our Model: h > 0 2b 2b ρ 2 2a ρ 1 2a 2h ρ 3 ρ 3 h = 0 2c 4 *Holm, Electric Contacts: Theory and Application, Springer-Verlag, NY (1967). h > 0, a b c, ρ 1 ρ 2 ρ 3 4

5 Interface Resistance with Dissimilar Materials Cylindrical (Cartesian) Semi-infinite Channel 5 5

6 A. Cylindrical semi-infinite channel Boundary conditions Φ = Φ, z = 0, r (0, a) + 1 Φ+ 1 Φ =, z = 0, r (0, a) ρ z ρ z 1 2 Φ = 0, z = 0, r ( a, b) z 6 Laplace s equation αn z + 0 n 0 ( α n ) + n= 1 Φ ( r, z) = A + A J r e E z, z > 0, r (0, a); n= 1 0 ( β ) + βn z Φ ( r, z) = B J r e E z, z < 0, r (0, b). n n 6

7 A. Cylindrical semi-infinite channel , 4 c b L R a a a L R b ρ ρ ρ π π ρ ρ = + + Bulk Bulk Interface 2 4 c c R R a ρ = Contact Resistance: 7 7

8 A. Cylindrical semi-infinite channel Scaling law (Cylindrical): 8 Symbols: Exact theory Solid lines: Scaling law Dashed lines: R ( / ) 0 b a c Timsit 8

9 B. Cartesian semi-infinite channel Boundary conditions Φ = Φ, z = 0, y (0, a) + 1 Φ+ 1 Φ =, z = 0, y (0, a) ρ z ρ z 1 2 Φ = 0, z = 0, y ( a, b) z 9 Laplace s equation nπ z nπ y a Φ + ( y, z) = An cos e E+ z, n= 0 a z > 0, y (0, a), nπ z nπ y + b Φ ( y, z) = Bn cos e E z, n= 1 b z < 0, y (0, b), 9

10 B. Cartesian semi-infinite channel R ρ 2L ρ 2 2 b 1 1L1 Rc, ρ = + + ρ 2b W 4πW a ρ2 2a W Bulk Interface Bulk Contact Resistance: Rc = 4 ρ π 2 W R c 10 10

11 B. Cartesian semi-infinite channel Scaling law (Cartesian): 11 Symbols: Exact theory Solid lines: Scaling law Dashed lines: R ( / ) 0 b a c LTZ 11

12 Total Resistance of Composite Channel 12 Cylindrical or Cartesian 12

13 Cylindrical: Cartesian: Total Resistance of Composite Channel ρ L ρ b ρ ρ 2h ρ c ρ ρ L R = + R, + + R, + π ρ π ρ π c 2 c 2 b 4a a 2 a 4a a 3 c Bulk Interface Bulk Interface Bulk ρ L ρ b ρ ρ 2h ρ c ρ ρ L R = + R, + + R, + 2b W 4 W a 2a W 4 W a 2c W c c π ρ2 π ρ3 Bulk Interface Bulk Interface Bulk 13 13

14 Test of Scaling Laws Test A. h >> a Electrostatic fringe field at one interface has an exponentially small influence on the other interface Both interface resistance the same as in the semi-infinite channel 14 14

15 Test B. h 0 h = 0, ρ 2 = ρ 3 and b = c a-spot 1. Cylindrical 2. Cartesian R R ( b a) c0 / ( b a) c0 / Timsit LTZ ρ 1 0, Region I is perfectly conducting, contact resistance at each interface equivalent to ½ of the symmetrical a-spot b/a, c/a, but ρ 2 ρ 3, the cylindrical scaling laws differs by at most 8% from (ρ 2 + ρ 3 )/4a 15 15

16 Test C. ρ 1 = ρ 2 = ρ 3 All channels are made of the same material [2] Analyzed in great detail in [2] Verified by experiment in [3] 16 [2] Y. Y. Lau and Wilkin Tang, J. Appl. Phys. 105, (2009). [3] M. R. Gomez et al., Appl. Phys. Lett. 95, (2009) 16

17 Test D. Comparison of 3D Maxwell code ρ 2 = 0.038Ωm, ρ 3 = 0.001Ωm, a = 4mm, b = 8mm, c =10 mm, L 2 = L 3, 2h from 1.5 to 16 mm, L 2 +2h + L 3 = 80 mm, Excitation voltage = 10V Symbols: Maxwell 3D Solid lines: Scaling law 17 17

18 Gomez s Experimental Validation of Scaling Law (ρ 1 = ρ 2 = ρ 3, b = c) Experimental Configuration Cross Sectional View Copper Endcap Plastic Copper Sulfate Case: A B C D E Theoretical contact resistance geometry was mimicked by machining holes of varying diameter in a piece of plastic and filling it with copper sulfate. h 2b 2a 18 [3] M. R. Gomez et al., Appl. Phys. Lett. 95, (2009) 18

19 Gomez s Experimental Validation (ρ 1 = ρ 2 = ρ 3, b = c) A plot comparing the second set of experimental data and the theoretically predicted values [3] 19 [3] M. R. Gomez et al., Appl. Phys. Lett. 95, (2009) 19

20 Conclusion Simple scaling laws for contact resistance with dissimilar materials are constructed. They have been validated in various tests and simulations, and experiments. If the electrical contact is highly resistive (ρ 1 >> ρ 2, ρ 1 >> ρ 3 ), bulk resistance dominates over the interface resistance, if 2h exceeds a few times (ρ 2 /ρ 1 )a and (ρ 3 /ρ 1 )a. Once the geometry (a,b,c,h) specified, interface resistance depends mainly on resistivity of main channel (ρ 2, ρ 3 ); insensitive to that of contact region (ρ 1 ). This work vastly generalized Holm s classical theory (1967) of contact resistance to higher dimensions, with dissimilar materials. Peng Zhang, and Y. Y. Lau, "Scaling laws for electrical contact resistance with dissimilar materials", J. Appl. Phys 108, (2010)