CONTACT RESISTANCE WITH DISSIMILAR MATERIALS: BULK ...
CONTACT RESISTANCE WITH DISSIMILAR MATERIALS: BULK CONTACTS AND THIN FILM CONTACTS Peng Zhang1, Y. Y. Lau1, W. Tang2, M. R. Gomez3, D. M. French2, J. C. Zier4, and R. M. Gilgenbach1 1Department
of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, USA 2AFRL, Kirtland AFB, USA 3 Sandia National Lab, Albuquerque, USA 4NRL, Washington DC, USA
Exact Solution
INTRODUCTION
b ρ1 ρ1L1 ρ 2 L2 ρ 2 R= 2 + R c , + 2 πb 4a a ρ 2 πa
Our interest in contact resistance was stimulated by the recognition of its importance in our ongoing studies of the Z-pinch, high power microwave generation, triple point junctions, field emitters, and heating phenomenology. In learning the subject, we were always referred to the classical reference of Holm [1]. Holm’s a-spot theory gives the electrical contact resistance of a circular constriction between two contacting surfaces. Implicit in the theory of Holm are several assumptions: (A) the a-spot has a zero thickness, i.e., zero axial length in the direction of current flow, (B) the current channel is made of the same material, e.g., the effects of contaminants have been ignored, and (C) the contact members are bulk conductors, whose dimensions transverse to the current flow are infinite. Here, we present a vast generalization of the conventional theory of bulk contacts [2] and thin film contacts [3], by relaxing assumptions (A), (B) and (C) mentioned above.
Bulk
Interface
Scaling laws b ρ1 ∆ 2ρ1 b b R c , ≅ R c0 + × ×g a Timsit 2 ρ1 +ρ2 a a ρ2 R c0 ( b/a )
Timsit
=1-1.41581(a/b)+0.06322(a/b)2 +0.15261(a/b)3 +0.19998(a/b)4 ,
g(b/a)=1-0.3243(a/b)2 -0.6124(a/b)4 -1.3594(a/b)6 +1.2961(a/b)8 , ∆=32/3π2 -1=0.08076
Total Resistance of Composite Channel R Cartesian
c ρ1 ρ3L3 b ρ1 ρ1 × 2h ρ3 ρ2 L2 ρ2 = + Rc , + + Rc , + 2b × W 4πW a ρ 2 2a × W 4πW a ρ3 2c × W Bulk
R cylindrical
ρ ρ R= + 4a 4a
h=0 Radius a
Bulk
Interface
Bulk
Interface
Bulk
c ρ1 ρ3L3 b ρ1 ρ1 × 2h ρ3 ρ2 L2 ρ2 = + Rc , + + Rc , + 2 2 2 πb 4a a ρ 2 πa 4a a ρ3 πc Bulk
Interface
Bulk
Interface
Bulk
Symbols: Exact theory Solid lines: Scaling laws
THIN FILM CONTACTS h > 0,
A. Cartesian thin film contact
Radius a≠ b ≠ c,
100
(a)
Exact Solution
Resistivity ρ1≠ρ2≠ρ3
100
a a ρ1 ρ1L1 ρ2L2 ρ2 R= Rc , , + + 2h×W 4πW b h ρ2 2a ×W
Cylindrical or Cartesian
RII
ρ1/ρ2 =
Contact resistance Rc
Rc
10 1
10
RI
0.1 0.01
Scaling laws
BULK CONTACTS
R c0 ( a/h ) = R c ( a/h ) ρ / ρ 1
A. Cartesian semi-infinite channel Laplace’s equation nπz a
nπz + b
Φ + =Φ - , z=0, y ∈ (0,a);
10
100
a/h a/h = 10
20 15
5 0.1
10
0.3
5
1
Cartesian or Cylindrical
z0,y ∈ (0,a),
-E + ∞ z,
⇒ Potential profile & interface resistance for arbitrary a, b, ρ1, ρ2
LTZ
1
(b)
β ( a/h ) = −0.0003 ( a/h ) + 0.1649 ( a/h ) + 0.6727, 0.03 ≤ a/h ≤ 30.
Boundary conditions
R c0 ( b/a )
0.1
25
(b)
a/h =
10
1.4 1.2
Rc
nπy Φ + (y,z)= ∑ A n cos e a n=0
Bulk
1 0.01
2
∞
nπy Φ - (y,z)= ∑ Bn cos e b n=1
= 2πa / h − 4 ln sinh ( πa/2h ) ,
0.5346 ( a/h )2 + 0.0127 ( a/h ) + 0.4548, 0.03 ≤ a/h ≤ 1; ∆ ( a/h ) = 0.0147x 6 − 0.0355x5 + 0.1479x 4 + 0.4193x 3 +1.1163x 2 + 0.9970x + 1, x = ln(a/h),1 < a/h ≤ 30,
Interface Resistance with Dissimilar Materials
∞
2 →0
2
2ρ1 × , ρ1 +β ( a/h ) ρ 2
Rc
R c ( a/h, ρ1 / ρ 2 ) ≅ R c0 ( a/h ) +
∆ ( a/h )
1
5
0.8
0.1
0.6
0.3
0.4
1
0.2 0 0.01
0.1
1
10
100
ρ1/ρ2 Symbols: Exact theory Solid lines: Scaling laws
0.001 ≤ a/h < 10.
∞
Φ+ (r,z) = A0 + ∑AnJ0 ( αnr) e-αnz − E+∞z,
z > 0,r ∈(0,a);
n=1 ∞
Φ− (r,z) = ∑BnJ0 ( βnr ) e+βnz − E−∞z,
z < 0,r ∈(0,b).
J1 (α n a)=J1 (β n b)=0
Symbols: Exact theory Solid lines: Scaling laws
CONCLUSIONS
• Simple, accurate analytical scaling laws of contact resistance with dissimilar materials were constructed for Boundary conditions both bulk and thin film contacts. They were validated against known limiting cases, experiments, and 1 ∂Φ + 1 ∂Φ numerical simulations ∂Φ = , z=0, r ∈ (0,a); Φ + =Φ - , z=0, r ∈ (0,a); =0, z=0, r ∈ (a,b) ρ1 ∂z ρ 2 ∂z • Interface resistance of bulk contacts depends mainly on the electrical resistivity of the main channel (ρ2, ρ3); ∂z ⇒ Potential profile & interface resistance for arbitrary a, b, ρ1, ρ2 it is insensitive to the resistivity of the contact region (ρ1) • For fixed ρ1/ρ2, thin film contact resistance primarily depends on a/h, as long as either L2 >> a or L2 >> h • The minimum thin film contact resistance occurs at a/h ~ 1, regardless of ρ1 and ρ2 [1] Holm, Electric Contacts: Theory and Application, Springer-Verlag, NY (1967); A. M. Rosenfeld and R. S. Timsit, Quart Appl. Math., vol. 39, p. 405, 1981. [2] Y. Y. Lau and W. Tang, J. Appl. Phys.105, 124902 (2009); M. R. Gomez et al., Appl. Phys. Lett. 95, 072103 (2009); P. Zhang et al., J. Appl. Phys. 108, 044914 (2010). This work was supported by AFOSR, AFRL, L-3 Communications, and Northrop-Grumman Corporation [3] P. Zhang et al., Appl. Phys. Lett. 97, 204103 (2010); J. Appl. Phys. 109, 124910 (2011); Proc. of the 57th IEEE Holm Conf. on Electrical Contacts (2011). n=1
of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, USA 2AFRL, Kirtland AFB, USA 3 Sandia National Lab, Albuquerque, USA 4NRL, Washington DC, USA
Exact Solution
INTRODUCTION
b ρ1 ρ1L1 ρ 2 L2 ρ 2 R= 2 + R c , + 2 πb 4a a ρ 2 πa
Our interest in contact resistance was stimulated by the recognition of its importance in our ongoing studies of the Z-pinch, high power microwave generation, triple point junctions, field emitters, and heating phenomenology. In learning the subject, we were always referred to the classical reference of Holm [1]. Holm’s a-spot theory gives the electrical contact resistance of a circular constriction between two contacting surfaces. Implicit in the theory of Holm are several assumptions: (A) the a-spot has a zero thickness, i.e., zero axial length in the direction of current flow, (B) the current channel is made of the same material, e.g., the effects of contaminants have been ignored, and (C) the contact members are bulk conductors, whose dimensions transverse to the current flow are infinite. Here, we present a vast generalization of the conventional theory of bulk contacts [2] and thin film contacts [3], by relaxing assumptions (A), (B) and (C) mentioned above.
Bulk
Interface
Scaling laws b ρ1 ∆ 2ρ1 b b R c , ≅ R c0 + × ×g a Timsit 2 ρ1 +ρ2 a a ρ2 R c0 ( b/a )
Timsit
=1-1.41581(a/b)+0.06322(a/b)2 +0.15261(a/b)3 +0.19998(a/b)4 ,
g(b/a)=1-0.3243(a/b)2 -0.6124(a/b)4 -1.3594(a/b)6 +1.2961(a/b)8 , ∆=32/3π2 -1=0.08076
Total Resistance of Composite Channel R Cartesian
c ρ1 ρ3L3 b ρ1 ρ1 × 2h ρ3 ρ2 L2 ρ2 = + Rc , + + Rc , + 2b × W 4πW a ρ 2 2a × W 4πW a ρ3 2c × W Bulk
R cylindrical
ρ ρ R= + 4a 4a
h=0 Radius a
Bulk
Interface
Bulk
Interface
Bulk
c ρ1 ρ3L3 b ρ1 ρ1 × 2h ρ3 ρ2 L2 ρ2 = + Rc , + + Rc , + 2 2 2 πb 4a a ρ 2 πa 4a a ρ3 πc Bulk
Interface
Bulk
Interface
Bulk
Symbols: Exact theory Solid lines: Scaling laws
THIN FILM CONTACTS h > 0,
A. Cartesian thin film contact
Radius a≠ b ≠ c,
100
(a)
Exact Solution
Resistivity ρ1≠ρ2≠ρ3
100
a a ρ1 ρ1L1 ρ2L2 ρ2 R= Rc , , + + 2h×W 4πW b h ρ2 2a ×W
Cylindrical or Cartesian
RII
ρ1/ρ2 =
Contact resistance Rc
Rc
10 1
10
RI
0.1 0.01
Scaling laws
BULK CONTACTS
R c0 ( a/h ) = R c ( a/h ) ρ / ρ 1
A. Cartesian semi-infinite channel Laplace’s equation nπz a
nπz + b
Φ + =Φ - , z=0, y ∈ (0,a);
10
100
a/h a/h = 10
20 15
5 0.1
10
0.3
5
1
Cartesian or Cylindrical
z0,y ∈ (0,a),
-E + ∞ z,
⇒ Potential profile & interface resistance for arbitrary a, b, ρ1, ρ2
LTZ
1
(b)
β ( a/h ) = −0.0003 ( a/h ) + 0.1649 ( a/h ) + 0.6727, 0.03 ≤ a/h ≤ 30.
Boundary conditions
R c0 ( b/a )
0.1
25
(b)
a/h =
10
1.4 1.2
Rc
nπy Φ + (y,z)= ∑ A n cos e a n=0
Bulk
1 0.01
2
∞
nπy Φ - (y,z)= ∑ Bn cos e b n=1
= 2πa / h − 4 ln sinh ( πa/2h ) ,
0.5346 ( a/h )2 + 0.0127 ( a/h ) + 0.4548, 0.03 ≤ a/h ≤ 1; ∆ ( a/h ) = 0.0147x 6 − 0.0355x5 + 0.1479x 4 + 0.4193x 3 +1.1163x 2 + 0.9970x + 1, x = ln(a/h),1 < a/h ≤ 30,
Interface Resistance with Dissimilar Materials
∞
2 →0
2
2ρ1 × , ρ1 +β ( a/h ) ρ 2
Rc
R c ( a/h, ρ1 / ρ 2 ) ≅ R c0 ( a/h ) +
∆ ( a/h )
1
5
0.8
0.1
0.6
0.3
0.4
1
0.2 0 0.01
0.1
1
10
100
ρ1/ρ2 Symbols: Exact theory Solid lines: Scaling laws
0.001 ≤ a/h < 10.
∞
Φ+ (r,z) = A0 + ∑AnJ0 ( αnr) e-αnz − E+∞z,
z > 0,r ∈(0,a);
n=1 ∞
Φ− (r,z) = ∑BnJ0 ( βnr ) e+βnz − E−∞z,
z < 0,r ∈(0,b).
J1 (α n a)=J1 (β n b)=0
Symbols: Exact theory Solid lines: Scaling laws
CONCLUSIONS
• Simple, accurate analytical scaling laws of contact resistance with dissimilar materials were constructed for Boundary conditions both bulk and thin film contacts. They were validated against known limiting cases, experiments, and 1 ∂Φ + 1 ∂Φ numerical simulations ∂Φ = , z=0, r ∈ (0,a); Φ + =Φ - , z=0, r ∈ (0,a); =0, z=0, r ∈ (a,b) ρ1 ∂z ρ 2 ∂z • Interface resistance of bulk contacts depends mainly on the electrical resistivity of the main channel (ρ2, ρ3); ∂z ⇒ Potential profile & interface resistance for arbitrary a, b, ρ1, ρ2 it is insensitive to the resistivity of the contact region (ρ1) • For fixed ρ1/ρ2, thin film contact resistance primarily depends on a/h, as long as either L2 >> a or L2 >> h • The minimum thin film contact resistance occurs at a/h ~ 1, regardless of ρ1 and ρ2 [1] Holm, Electric Contacts: Theory and Application, Springer-Verlag, NY (1967); A. M. Rosenfeld and R. S. Timsit, Quart Appl. Math., vol. 39, p. 405, 1981. [2] Y. Y. Lau and W. Tang, J. Appl. Phys.105, 124902 (2009); M. R. Gomez et al., Appl. Phys. Lett. 95, 072103 (2009); P. Zhang et al., J. Appl. Phys. 108, 044914 (2010). This work was supported by AFOSR, AFRL, L-3 Communications, and Northrop-Grumman Corporation [3] P. Zhang et al., Appl. Phys. Lett. 97, 204103 (2010); J. Appl. Phys. 109, 124910 (2011); Proc. of the 57th IEEE Holm Conf. on Electrical Contacts (2011). n=1
