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SECTION I

INTRODUCTION

THIN film contacts, in which a thin conducting film constitutes at least one of the contacting members, are basic circuit structures in modern electronic devices. They are widely used in micro-electronic and micro-electromechanical systems, and in semiconductor material and device characterization [1], [2]. In miniaturization of electronics, current crowding and Joule heating are important issues [3], [4], [5].

Figure 1
Fig. 1. Two basic types of thin film contact: (a) vertical type, and (b) horizontal type, with dissimilar materials for both Cartesian and cylindrical geometries. For the cylindrical case, the Formula$z$-axis is the axis of rotation. This paper focuses mainly on the vertical type (a); the horizontal type (b) was solved in [20].

We consider two basic types of thin film contact: the vertical type [Fig. 1(a)] and the horizontal type [Fig. 1(b)]. In the vertical contact [Fig. 1(a)], the thin film base AB is an equipotential; the current flow is orthogonal to AB, but is tangential to the thin film edges BC and AH. In the horizontal contact [Fig. 1(b)], the thin film edges BC and AH are equipotentials. The current flow is orthogonal to BC and AH, but is tangential to the thin film base AB. As a result, the constriction resistance (also known as the spreading resistance), as well as current crowding at the edges G and D for these two types of thin film contacts are very different. We assume that there are only two contact members, denoted as Region I and Region II, between the electrodes [Fig. 1(a) and (b)]. The interface GD is perfectly smooth and does not contain a resistive sheet.

In this paper, we focus mainly on the vertical contact shown in Fig. 1(a), and provide a comprehensive study of the constriction resistance and the current flows in regions I and II. We point out the marked difference in current crowding at the corners (edges) D and G between vertical contact [Fig. 1(a)] and horizontal contact [Fig. 1(b)]. Without loss of generality, we assume that the top terminal EF is grounded..

The vertical thin film contact has been studied by several authors [6], [7], [8], [9], [10], [11], [12]. In particular, Hall [6] studied the Cartesian geometry using conformal mapping. Denhoff [12] studied the constriction resistance of a round thin film contact by solving Laplace equation using analytic, numerical, and finite element methods. These studies are restricted to the highly special cases of Fig. 1(a): assuming either equal resistivity, Formula$\rho_{1}=\rho_{2}$ [6], or Formula$h_{1}\to 0$ [7], [8], [9], [12]. We relax these two assumptions in this paper.

The horizontal thin film contact has also been studied extensively, in both Cartesian [1], [2], [13], [14], [15], [16], [17] and cylindrical geometry [15], [16], [17], [18], [19]. Most recently, we treated the 3-terminal horizontal thin film contact in great detail [20], with arbitrary resistivities and geometric dimensions in the individual contact members, as well as arbitrary (unequal) voltages at the terminals BC and AH [Fig. 1(b)]. We found severe current crowding at the edges D and G, for Formula$\rho_{1}=0$, when the thin film thickness Formula$h_{2}$ is small [17]. There is a great variety of current crowding in 3-terminal horizontal thin film contact [20].

In both Cartesian and cylindrical geometries [Fig. 1(a)], the resistivities Formula$\rho_{1}$ and Formula$\rho_{2}$, and the geometric dimensions Formula$a$, Formula$b$, Formula$h_{1}$, and Formula$h_{2}$ may assume arbitrary values. Thus, this paper is applicable to both thin and thick films, and Region I may represent a resistive thin film by letting Formula$h_{1}$ small and Formula$\rho_{1}$ large. Following the same procedure in studying the bulk contact resistance [21], [22] and the horizontal thin film contacts [15], [20], we analytically solve Laplace equation in Regions I and II of Fig. 1(a), then match the boundary conditions at the contact interface. The potential profile and the constriction resistance are calculated from this exact formulation.

In Section II, we consider the Cartesian vertical thin film contact. We present a formula for the constriction resistance and illustrate the current pattern. We compare our results to those obtained from the MAXWELL 2-D finite element code [23]. In Section III, we consider the cylindrical vertical thin film which is qualitatively similar to the Cartesian solution of Section II. Concluding remarks are given in Section IV. Only the major results will be presented in the main text. Their derivations are given in the appendices.

SECTION II

CARTESIAN THIN FILM VERTICAL CONTACT

Since the geometry [Fig. 1(a)] is symmetrical about the vertical Formula$z$-axis, so also are the current flow patterns and the field lines. The field lines are normal to the two terminals AB and EF. The total current is Formula$I=V_{0}/R$, where Formula$R$ is the resistance between these two terminals, which we find to be Formula TeX Source $$R=\rho_{1}{{h_{1}}\over{2a\times W}}+{{\rho_{2}}\over{4\pi W}}\overline{R}_{c}({{a}\over{b}},{{h_{1}}\over{a}},{{a}\over{h_{2}}},{{\rho_{1}}\over{\rho_{2}}})+\rho_{2}{{h_{2}}\over{2b\times W}},\eqno{\hbox{(1)}}$$ where Formula$W$ denotes the channel width in the third ignorable dimension that is perpendicular to the paper. In (1), the first term represents the bulk resistance of Region I. The third term represents the bulk resistance of Region II. The second term represents the remaining constriction (spreading) resistance, Formula$R_{c}$, and is expressed as Formula$R_{c}={{\rho_{2}}\over{4\pi W}}\overline{R}_{c}$ for the Cartesian case. The normalized Formula$\overline{R}_{c}$ depends on the aspect ratios Formula$a/b$, Formula$h_{1}/a$, and Formula$a/h_{2}$, and on the resistivity ratio Formula$\rho_{1}/\rho_{2}$, as explicitly shown in (1). The exact expression for Formula$\overline{R}_{c}$ is derived in Appendix A (A7).

If we use the exact expression (A7) in the numerical evaluation of Formula$\overline{R}_{c}$, we call the result the “exact theory”. [In the infinite sum in (A7), we use Formula$10^{4}$ terms; we also use Formula$10^{4}$ terms in the infinite sum in (A3b) to solve for Formula$B_{n}$, Formula$n=1, 2,..,10^{4}$.] From the vast amount of data that we collected from the exact theory at various combinations of Formula$h_{1}$, Formula$h_{2}$, Formula$\rho_{1}$, Formula$\rho_{2}$, Formula$a$, and Formula$b$, we attempted to construct simple fitting formulas for Formula$\overline{R}_{c}$ so that the values of Formula$\overline{R}_{c}$, or the bounds of Formula$\overline{R}_{c}$, may be easily obtained without solving the infinite matrix (A3b). The bounds on Formula$\overline{R}_{c}$ are identical to the corresponding asymptotic limits of Formula$\overline{R}_{c}$, which in some cases (but not all) were solved in the literature by other means. Thus, the numerical fitting formulas, given for instance in (2), (3), (5), and (6), are synthesized from a judicious combination of these asymptotic limits, and from the numerical data generated from the exact theory. We validated the exact theory and the synthesized fitting formulas with MAXWELL 2-D simulation codes in Figs. 2, 3, 4, 6, 7, and 8. Similar approaches were used in our recent treatment of the horizontal thin film contact [20].

Figure 2
Fig. 2. Formula$\overline{R}_{c}$ [see (A7)] for Cartesian case as a function of Formula$b/a$, for Formula$\rho_{1}/\rho_{2}=1$, Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the three curves are for Formula$h_{1}/a=10$, 0.1, and 0.001 (top to bottom). The symbols represent MAXWELL 2-D simulation.

The exact theory of Formula$\overline{R}_{c}$ (A7) is plotted in Fig. 2 as a function of Formula$b/a$, for Formula$\rho_{1}/\rho_{2}=1$ with various Formula$a/h_{2}$ and Formula$h_{1}/a$. For a given Formula$a/h_{2}$, Formula$\overline{R}_{c}$ increases as Formula$b/a$ increases. For Formula$a/h_{2}$ on the order of 1 or larger, Formula$\overline{R}_{c}$ approaches almost a constant as Formula$b/a$ becomes large. For a given Formula$b/a$, Formula$\overline{R}_{c}$ decreases as Formula$a/h_{2}$ increases. The effect of Formula$h_{1}/a$ on Formula$\overline{R}_{c}$ is minor. Formula$\overline{R}_{c}$ increases slightly as Formula$h_{1}/a$ increases. The effect of Formula$h_{1}/a$ becomes even less significant as Formula$a/h_{2}$ increases.

Since Formula$\overline{R}_{c}$ is relatively insensitive to Formula$h_{1}/a$, in Fig. 3 we plot Formula$\overline{R}_{c}$ as a function of Formula$a/h_{2}$ at various values of Formula$b/a$ and Formula$\rho_{1}/\rho_{2}$, in the limit of Formula$h_{1}/a\to\infty$. When Formula$a/h_{2}\ll 1$, Formula$\overline{R}_{c}$ approaches a constant value (independent of Formula$h_{2}$) for a given Formula$b/a$. This is due to the fact that if both Formula$h_{2}$ and Formula$h_{1}$ become much larger than Formula$a$, the structure in Fig. 1(a) will become a semi-infinite constriction channel, whose constriction resistance is independent of Formula$h_{1}$ and Formula$h_{2}$, which was studied in detail in [22]. When Formula$a/h_{2}>1$, Formula$\overline{R}_{c}$ decreases as Formula$a/h_{2}$ increases. As Formula$a/h_{2}\rightarrow\infty$, Formula$\overline{R}_{c}=2\pi\left({{{h_{2}}\over{a}}-{{h_{2}}\over{b}}}\right)\rightarrow 0$, which is in sharp contrast to the behavior of the horizontal Cartesian thin film contact studied in [15], [16], [17], [20], where the current flows parallel to the thin film bottom boundary, and Formula$\overline{R}_{c}$ (which has a different definition for the horizontal contact [17], [20]) approaches a finite constant of 2.77 as Formula$a/h_{2}\rightarrow\infty$.

Figure 3
Fig. 3. Formula$\bar R_{c}$ for Cartesian case as a function of Formula$a/h_{2}$, in the limit of Formula$h_{1}/a\to\infty$, for Formula$b/a=30$, 10, and 5. For each Formula$b/a$, the three solid curves are for Formula$\rho_{1}/\rho_{2}=100$, 1, and 0.01 (top to bottom), representing the results from exact calculations [(A7)]. The dotted lines represent (2), and the symbols represent MAXWELL 2-D simulation.

By comparing the data calculated analytically from the exact theory (A7), with the published scalings for some limiting cases [6], [21], [22], [24], we synthesized an accurate, analytical scaling law for the constriction resistance of a general vertical thin film contact, in the Formula$h_{1}/a\rightarrow\infty$ limit, Formula TeX Source $$\eqalignno{&\left.{\overline{R}_{c}\left({{{a}\over{b}},{{a}\over{h_{2}}}, {{\rho_{1}}\over{\rho_{2}}}}\right)}\right\vert_{h_{1}/a\to\infty} \cr&=\cases{\overline{R}_{c0}\left({{b}\over{a}}\right)+0.2274\times\left({{2\rho_{1}} \over{\rho_{1}+\rho_{2}}}\right)\times g\left({{b}\over{a}}\right),\matrix{&{{{a}\over{h_{2}}}<\tan \left({{{\pi}\over{2}}{{b}\over{a}}}\right)}}\hfill\cr p\left({{a}\over{h_{2}}}\right)\times\left({1+{{0.2274}\over{q(b/a)}} \times{{\rho_{1}-\rho_{2}}\over{\rho_{1}+\rho_{2}}}}\right),\matrix{&{{{a}\over{h_{2}}}>\tan\left({{{\pi}\over{2}}{{b}\over{a}}}\right)},}}&{\hbox{(2)}} \cr& \overline{R}_{c0}\left({b/a}\right)=4\ln(2b/\pi a)+4\ln (\pi/2)\times f(b/a),\cr & f(b/a)=0.03250(a/b)+1.06568(a/b)^{2}-0.24829(a/b)^{3}\cr &\quad+0.21511(a/b)^{4},\cr & g(b/a)=1-1.2281(a/b)^{2}+0.1223(a/b)^{4}-0.2711(a/b)^{6}\cr &\quad+0.3769(a/b)^{8},\cr & p(a/h_{2})=2\pi (h_{2}/a)-4(h_{2}/a)\tan^{-1}(h_{2}/a)\cr &\quad+2\ln\left[{(h_{2}/a)^{2}+1}\right]-2\pi(h_{2}/b),\cr & q(b/a)=2\left(\scriptstyle{{{b}\over{a}}+{{a}\over{b}}}\right)\ln\left(\scriptstyle{{1+a/b}\over{1-a/b}}\right)+4\ln\left(\scriptstyle{{b/a-a/b}\over{4}}\right).&{\hbox{(3a-3e)}}}$$

Equation (2) is also plotted in Fig. 3, showing excellent agreement with the exact theory (A7) for arbitrary value of Formula$a$, Formula$b(>a)$, Formula$h_{2}$, Formula$\rho_{1}$, and Formula$\rho_{2}$, in the limit of Formula$h_{1}/a\rightarrow\infty$.

In (3), Formula$\overline{R}_{c0}\left({b/a}\right)$, Formula$f(b/a)$, and Formula$g(b/a)$ are derived by Lau and Tang [21], and by Zhang and Lau [(5) and (6) of [22]],1 Formula$p(a/h_{2})$ is derived by Hall [(45) of [6], assuming Formula$\rho_{1}/\rho_{2}=1$ and Formula$h_{1}/a\rightarrow\infty$], Formula$q(b/a)$ is derived by both Hall [(42) of [6], assuming Formula$\rho_{1}/\rho_{2}=1$, Formula$h_{1}/a\rightarrow\infty$, and Formula$h_{2}/a\rightarrow\infty$] and Smythe [24]. The breakpoint in (2), Formula${{a}\over{h_{2}}}=\tan\left({{\pi a}\over{2b}}\right)$, was also stated by Hall [(46) of [6]]. At Formula${{a}\over{h_{2}}}=\tan\left({{\pi a}\over{2b}}\right)$, there is a discontinuity between the two expressions in (2). This discontinuity at the breakpoint is also seen in the dotted curves in Fig. 3 which plot (2). The size of this step discontinuity is always less than 2% of the exact value of Formula$\overline{R}_{c}$ [6].

Fig. 4(a) shows the exact theory for Formula$\overline{R}_{c}$ (A7) as a function of Formula$\rho_{1}/\rho_{2}$, for various Formula$a/h_{2}$ and Formula$h_{1}/a$. Fig. 4(b) shows the exact theory for Formula$\overline{R}_{c}$ (A7) as a function of Formula$h_{1}/a$, for various Formula$a/h_{2}$ and Formula$\rho_{1}/\rho_{2}$. In both Fig. 4(a) and (b), we fixed Formula$b/a=30$. In general, as either Formula$\rho_{1}/\rho_{2}$ or Formula$h_{1}/a$ increases, Formula$\overline{R}_{c}$ increases. It is important to recognize from Figs. 2 4 that dependence of Formula$\overline{R}_{c}$ on Formula$h_{1}/a$ and on Formula$\rho_{1}/\rho_{2}$ is not significant, and that the major dependence of Formula$\overline{R}_{c}$ is on Formula$a/h_{2}$ and on Formula$b/a$. Thus, for a given Formula$a/h_{2}$ and Formula$b/a$ in Figs. 3 and 4, the bounds of the curves are fairly accurately predicted by (2), which are plotted as dashed lines, for all values of Formula$h_{1}/a$ and Formula$\rho_{1}/\rho_{2}$.

Figure 4
Fig. 4. (a) Formula$\overline{R}_{c}$ for Cartesian case [see (A7)] as a function of Formula$\rho_{1}/\rho_{2}$, for Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the five solid curves are for Formula$h_{1}/a=10$, 1, 0.1, 0.01, and 0.001 (top to bottom), and (b) Formula$\overline{R}_{c}$ as a function of Formula$h_{1}/a$, for Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the five solid curves are for Formula$\rho_{1}/\rho_{2}=100$, 10, 1, 0.1, and 0.01 (top to bottom). We fixed Formula$b/a=30$ in all calculations. The dashed lines represent the bounds calculated from (2): for each Formula$a/h_{2}$ in (a) and (b), the upper dashed line is calculated from (2) by setting Formula$\rho_{1}/\rho_{2}\rightarrow\infty$; the lower dashed line is calculated from (2) by setting Formula$\rho_{1}/\rho_{2}\rightarrow 0.$ Note that Formula$\rho_{1}/\rho_{2}\rightarrow 0$ is equivalent to Formula$h_{1}/a\rightarrow 0$, because in these two limits, the top terminal EF in Fig. 1(a) is in effect placed directly at the interface DG. The symbols represent MAXWELL 2-D simulation.

The field line equation, Formula$y=y(z)$, may be numerically integrated from the first order ordinary differential equation Formula$dy/dz=E_{y}/E_{z}=(\partial\Phi/\partial y)/(\partial\Phi/\partial z)$ where Formula$\Phi$ is given by (A1). The field lines in the right half of the thin film structure [Fig. 1(a)] are shown in Fig. 5 for the special case of Formula$\rho_{1}/\rho_{2}=1$, and Formula$h_{1}/a=0.01$ with various Formula$a/h_{2}$. We set Formula$b/a=30$ in all calculations in Fig. 5. Note that the variation of Formula$a/h_{2}$ in Fig. 5 may be interpreted this way: Formula$a$, Formula$b$, and Formula$h_{1}$ are held fixed, Formula$h_{2}$ decreases, i.e., Formula$h_{2}=5a$, Formula$a$, and 0.2 Formula$a$, from Fig. 5(a)(c). (Similar interpretation applies to other figures.) It is clear that as Formula$a/h_{2}$ increases, the spreading of the field lines (also the current flow lines) in Region II becomes less significant. This explains the decrease in Formula$\overline{R}_{c}$ as Formula$a/h_{2}$ increases, as shown in Fig. 3. In the limit of Formula$h_{2}\rightarrow 0$, there will be little spreading of field lines (little current crowding) at the edge of the constriction, leading to zero constriction resistance (Fig. 3), in sharp contrast to the horizontal contact [15], [17], [20]. Note from Fig. 5 that the field lines in Region I are almost straight, and are fairly uniformly spaced across the interface in all the cases, implying minimal enhanced heating at the edges G and D in a vertical contact [Fig. 1(a)]. We also found that as either Formula$h_{1}/a$ or Formula$\rho_{1}/\rho_{2}$ increases, the spreading of the field lines in Region II slightly increases (not shown). The field line distribution is relatively insensitive to Formula$h_{1}/a$ or Formula$\rho_{1}/\rho_{2}$, as compared to the effect of Formula$a/h_{2}$.

Figure 5
Fig. 5. Field lines calculated from (A1) for the right half of Cartesian thin film contact (Fig. 1(a)), for the special case of Formula$\rho_{1}/\rho_{2}=1$ and Formula$h_{1}/a=0.01$ with various Formula$a/h_{2}$. We fixed Formula$b/a=30$. The field line distribution is relatively insensitive to Formula$h_{1}/a$ or Formula$\rho_{1}/\rho_{2}$ (not shown), as compared to the effect of Formula$a/h_{2}$.
SECTION III

CIRCULAR THIN FILM VERTICAL CONTACT

For the circular thin film vertical contact [Fig. 1(a)], the z-axis is the axis of rotation. The total current is Formula$I=V_{0}/R$, where Formula$R$ is the resistance between the two terminals AB and EF, given by Formula TeX Source $$R={{\rho_{1}h_{1}}\over{\pi a^{2}}}+{{\rho_{2}}\over{4a}}\overline{R}_{c}\left({{{a}\over{b}},{{h_{1}}\over{a}},{{a}\over{h_{2}}},{{\rho_{1}}\over{\rho_{2}}}}\right)+{{\rho_{2}h_{2}}\over{\pi b^{2}}}.\eqno{\hbox{(4)}}$$

In (4), the first and third terms represent the bulk resistance of Region I and II, respectively. The second term represents the remaining constriction resistance, Formula$R_{c}$, and is expressed as Formula$R_{c}={{\rho_{2}}\over{4a}}\overline{R}_{c}$ for the circular case. The normalized Formula$\overline{R}_{c}$ depends on the aspect ratios Formula$a/b$, Formula$h_{1}/a$ and Formula$a/h_{2}$, and on the resistivity ratio Formula$\rho_{1}/\rho_{2}$, as explicitly shown in (4). The exact expression for Formula$\overline{R}_{c}$ is derived in Appendix B (B6).

The exact theory of Formula$\overline{R}_{c}$ (B6) is plotted in Fig. 6 as a function of Formula$b/a$, for Formula$\rho_{1}/\rho_{2}=1$ with various Formula$h_{1}/a$ and Formula$a/h_{2}$. For a given Formula$a/h_{2}$, Formula$\overline{R}_{c}$ increases as Formula$b/a$ increases. However, Formula$\overline{R}_{c}$ becomes almost constant for large Formula$b/a$, independent of the value of Formula$a/h_{2}$, which is different from the Cartesian case in Fig. 2, where Formula$\overline{R}_{c}$ increases logarithmically with Formula$b/a$ if Formula$a/h_{2}$ is small [(2a) and (3a)]. For a given Formula$b/a$, Formula$\overline{R}_{c}$ decreases as Formula$a/h_{2}$ increases. Formula$\overline{R}_{c}$ increases only slightly as Formula$h_{1}/a$ increases. The effect of Formula$h_{1}/a$ becomes even less significant as Formula$a/h_{2}$ increases (Fig. 6).

Figure 6
Fig. 6. Formula$\overline{R}_{c}$ [see (B6)] for cylindrical case as a function of Formula$b/a$, for Formula$\rho_{1}/\rho_{2}=1$, Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the three curves are for Formula$h_{1}/a=10$, 0.1, and 0.001 (top to bottom). The symbols represent MAXWELL 2-D simulation.

In Fig. 7, we plot Formula$\overline{R}_{c}$ as a function of Formula$a/h_{2}$ for various values of Formula$\rho_{1}/\rho_{2}$, in the limit of Formula$h_{1}/a\to\infty$. As noted above, Formula$\overline{R}_{c}$ is independent of Formula$b$ for large Formula$b/a$; we set Formula$b/a=30$ for the calculation in Fig. 7. When Formula$a/h_{2}\ll 1$, Formula$\overline{R}_{c}$ approaches a constant value (independent of Formula$h_{2}$) for a given Formula$b/a$. This is due to the fact that if both Formula$h_{2}$ and Formula$h_{1}$ become much larger than Formula$a$, the structure in Fig. 1(a) will become a semi-infinite constriction channel, whose constriction resistance is independent of Formula$h_{1}$ and Formula$h_{2}$, which was studied in detail in [22]. When Formula$a/h_{2}>1$, Formula$\overline{R}_{c}$ decreases as Formula$a/h_{2}$ increases. As Formula$a/h_{2}\rightarrow\infty$, Formula$\overline{R}_{c}\sim h_{2}/a\rightarrow 0.$ This is in sharp contrast to the behavior of the horizontal cylindrical thin film contact studied in [15], [16], [17], [20], in which the current flows parallel to the thin film bottom boundary, and Formula$\overline{R}_{c}$ (which has a different definition for the horizontal contact [17], [20]) approaches a finite constant of 0.28 as Formula$a/h_{2}\rightarrow\infty$.

Figure 7
Fig. 7. Formula$\bar R_{c}$ for cylindrical case as a function of Formula$a/h_{2}$, in the limit of Formula$h_{1}/a\to\infty$, for Formula$b/a=30$, with Formula$\rho_{1}/\rho_{2}=100$, 1, and 0.01 (top to bottom). The solid lines represent the exact calculations [(B6)], symbols represent MAXWELL 2-D simulation, and the dashed lines represent (5).

By comparing the data calculated from the exact theory (B6) with the published scalings for some limiting cases [12], [22], [25], we synthesized an accurate, analytical scaling law for the normalized constriction resistance of a general cylindrical vertical thin film contact, in the Formula$h_{1}/a\rightarrow\infty$ limit, given by (5) and (6) at the bottom of this page, Formula TeX Source $$\eqalignno{&\left.{\overline{R}_{c}\left({{{b}\over{a}},{{a}\over{h_{2}}},{{\rho_{1}}\over{\rho_{2}}}}\right)}\right\vert_{h_{1}/a\to\infty}=\cases{\overline{R}_{c0}\left({{b}\over{a}}\right)+{{\Delta}\over{2}}\times\left({{2\rho_{1}}\over{\rho_{1}+\rho_{2}}}\right)\times g\left({{b}\over{a}}\right),\hfill\cr\matrix{&{{\rm when}\left[\matrix{a/h_{2}\leq 1.8a/b\quad{\rm if}\quad b/a\geq 4.37,{\rm or}\hfill\cr a/h_{2}\leq\left[{1-(a/b)^{2}}\right]^{-1}{\rm if}\quad b/a<4.37\hfill}\right];}}\hfill\cr\left[{p\left({{a}\over{h_{2}}}\right)-{{4}\over{\pi}}\left({{a}\over{b}}\right)^{2}{{h_{2}}\over{a}}}\right]\times\left({1+{{\Delta}\over{\bar R_{c0}\left({b/a}\right)}}\times{{\rho_{1}}\over{\rho_{1}+\rho_{2}}}}\right),\matrix{&{\rm otherwise,}}}&{\hbox{(5)}} \cr&\qquad\quad\bar R_{c0}\left({b/a}\right)=1-1.41581(a/b)+0.06322(a/b)^{2}+0.15261(a/b)^{3}+0.19998(a/b)^{4},\cr&\qquad\quad g(b/a)=1-0.3243(a/b)^{2}-0.6124(a/b)^{4}-1.3594(a/b)^{6}+1.2961(a/b)^{8},\cr&\qquad\quad p(a/h_{2})=\cases{[1+0.441271\left({a/h_{2}}\right)+0.194720\left({a/h_{2}}\right)^{2}-0.009732\left({a/h_{2}}\right)^{3}\hfill\cr-0.046505\left({a/h_{2}}\right)^{4}+0.002110\left({a/h_{2}}\right)^{5}+0.052204\left({a/h_{2}}\right)^{6}\hfill\cr-0.011044\left({a/h_{2}}\right)^{7}]^{-1}, 0<a/h_{2}\leq0.4;\hfill\cr 4\times [0+0.31338(h_{2}/a)-0.25134(h_{2}/a)^{2}+0.12512(h_{2}/a)^{3}\hfill\cr-0.03436(h_{2}/a)^{4}+0.003908(h_{2}/a)^{5}], 0.4<a/h_{2}<\infty,}&{\hbox{(6a-6c)}}}$$ where Formula$\Delta=32/3\pi^{2}-1=0.08076$. Equation (5) is also plotted in Fig. 7, showing excellent agreement with the exact theory (B6) for arbitrary value of Formula$a$, Formula$b(>a)$, Formula$h_{2}$, Formula$\rho_{1}$, and Formula$\rho_{2}$, in the limit of Formula$h_{1}/a\rightarrow\infty$.

In (6), Formula$\overline{R}_{c0}\left({b/a}\right)$ is synthesized by Timsit [25], Formula$g(b/a)$ is derived by us [(2) of [22]], and Formula$p(a/h_{2})$ is from Denhoff [(26) and (28) of [12]]. At the breakpoint, Formula${{a}\over{h_{2}}}=1.8{{a}\over{b}}$ or Formula${{a}\over{h_{2}}}={{1}\over{1-(a/b)^{2}}}$, there is a discontinuity between the two expressions in (5). The size of this step discontinuity is within 2% of the exact value of Formula$\overline{R}_{c}$ (B6) in the worst case.

Fig. 8(a) shows the exact theory for Formula$\overline{R}_{c}$ (B6) as a function of Formula$\rho_{1}/\rho_{2}$, for various Formula$a/h_{2}$ and Formula$h_{1}/a$. Fig. 8(b) shows the exact theory for Formula$\overline{R}_{c}$ (B6) as a function of Formula$h_{1}/a$, for various Formula$a/h_{2}$ and Formula$\rho_{1}/\rho_{2}$. Spot checks by MAXWELL 2-D code [23] are also shown in Fig. 8. In general, as either Formula$h_{1}/a$ or Formula$\rho_{1}/\rho_{2}$ increases, Formula$\overline{R}_{c}$ increases. It is important to recognize from Figs. 6 8 that the dependence of Formula$\overline{R}_{c}$ on Formula$h_{1}/a$ and Formula$\rho_{1}/\rho_{2}$ is not significant, and that the major dependence of Formula$\overline{R}_{c}$ is on Formula$b/a$ and on Formula$a/h_{2}$, similar to the Cartesian case in Section II. Thus, for a given Formula$a/h_{2}$ in Fig. 8, the bounds of the curves are fairly accurately predicted by (5), which are plotted as dashed lines.

Figure 8
Fig. 8. (a) Formula$\overline{R}_{c}$ for cylindrical case [see (B6)] as a function of Formula$\rho_{1}/\rho_{2}$, for Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the five solid curves are for Formula$h_{1}/a=10$, 1, 0.1, 0.01, and 0.001 (top to bottom), and (b) Formula$\overline{R}_{c}$ as a function of Formula$h_{1}/a$, for Formula$a/h_{2}=0.1$, 1, and 10. For each Formula$a/h_{2}$, the five solid curves are for Formula$\rho_{1}/\rho_{2}=100$, 10, 1, 0.1, and 0.01 (top to bottom). We fixed Formula$b/a=30$ in all calculations. The dashed lines represent the bounds calculated from (5): for each Formula$a/h_{2}$ in (a) and (b), the upper dashed line is calculated from (5) by setting Formula$\rho_{1}/\rho_{2}\rightarrow\infty$; the lower dashed line is calculated from (5) by setting Formula$\rho_{1}/\rho_{2}\rightarrow 0.$ Note that Formula$\rho_{1}/\rho_{2}\rightarrow 0$ is equivalent to Formula$h_{1}/a\rightarrow 0$, because in these two limits, the top terminal EF in Fig. 1(a) is in effect placed directly at the interface DG. The symbols represent MAXWELL 2-D simulation.

The field lines for the cylindrical case (not shown) are very similar to those in Fig. 5 for the Cartesian case. In the limit of Formula$h_{2}\rightarrow 0$, there will be little spreading of field lines (little current crowding) at the edge of the constriction, leading to zero constriction resistance (Fig. 7), in sharp contrast to the horizontal contact [17], [20].

SECTION IV

CONCLUDING REMARKS

This paper presented an exact solution for the constriction resistance in a vertical thin film contact with dissimilar materials, for both Cartesian and cylindrical geometries. The model assumed arbitrary geometric aspect ratios and arbitrary resistivities in the individual contact members. The constriction resistance was calculated analytically, and spot-checked against the MAXWELL 2-D code. The current flow patterns from the exact theory were displayed. Scaling laws for, and bounds on, the constriction resistance were presented for arbitrary values of Formula$h_{1}$, Formula$h_{2}$, Formula$\rho_{1}$, Formula$\rho_{2}$, Formula$a$, and Formula$b(>a)$ [Fig. 1(a)].

We found that the normalized constriction resistance Formula$\overline{R}_{c}$ depended predominantly on Formula$b/a$ and on Formula$h_{2}/a$, i.e., on the geometry of Region II; but was relatively insensitive to Formula$h_{1}/a$, and to Formula$\rho_{1}/\rho_{2}$, i.e., insensitive to the geometry or resistivity of Region I [Fig. 1(a)]. We also found that in the limit of small film thickness Formula$(h_{2}\rightarrow 0)$, there was hardly any current crowding in the vertical contact represented in Fig. 1(a). The current was distributed quite uniformly across the interface GD, implying minimal enhanced heating at the edges G and D in Fig. 1(a). This was in sharp contrast to a horizontal thin film contact [Fig. 1(b)], where the current that crosses the interface GD was highly concentrated near the edges G and D. In fact, at least half of the current flew within a distance of 0.44 Formula$h_{2}$(Formula$h_{2}\rightarrow 0$ [17]) of the two edges G and D in Fig. 1(b), suggesting severe local heating there for the horizontal thin film contact.

APPENDIX A

General Solution to the Cartesian Vertical Contact [Fig. 1(a)]

The formulation follows that of [15], [20], and [22]. Referring to Fig. 1(a), EF is grounded, and AB is biased with a voltage of Formula$+V_{0}$. The solutions to Laplace's equation are, Formula TeX Source $$\eqalignno{&\Phi_{+}(y,z)=A_{0}(z-h_{1})+\sum_{n=1}^{\infty}{A_{n}\cos\left({{n\pi y}\over{a}}\right)\sinh\left({n\pi{{z-h_{1}}\over{a}}}\right)},\cr &\qquad 0<z<h_{1},\left\vert y\right\vert\in (0,a),\cr &\Phi_{-}(y,z)=V_{0}+B_{0}(z+h_{2})+\sum_{n=1}^{\infty}{B_{n}\cos\left({{n\pi y}\over{b}}\right)\sinh\left({n\pi{{z+h_{2}}\over{b}}}\right)},\cr &\qquad-h_{2}<z<0,\left\vert y\right\vert\in (0,b),&{\hbox{(A1)}}}$$ where Formula$\Phi_{\rm{+}}$ and Formula$\Phi_{\rm{-}}$ are the electrical potential in the regions I and II respectively, and Formula$A_{\rm{n}}$ and Formula$B_{\rm{n}}$ are the coefficients that need to be determined.

At the interface Formula$z=0$, from the continuity of electrical potential and current density, we have the following boundary conditions Formula TeX Source $$\eqalignno{&\qquad\quad\Phi_{+}=\Phi_{-},\qquad z=0,\left\vert y\right\vert\in(0,a)&{\hbox{(A2a)}} \cr&{{1}\over{\rho_{1}}}{{\partial\Phi_{+}}\over{\partial z}}={{1}\over{\rho_{2}}} {{\partial\Phi_{-}}\over{\partial z}},\qquad z=0,\left\vert y\right\vert\in (0,a),&{\hbox{(A2b)}}\cr&\qquad\quad{{\partial\Phi_{-}}\over{\partial z}}=0,\matrix{&}z=0,\left\vert{y}\right\vert\in(a,b)&{\hbox{(A2c)}}}$$

From (A1) and (A2a), eliminating coefficient Formula$A_{\rm{n}}$ in favor of Formula$B_{\rm{n}}$, we have Formula TeX Source $$\eqalignno{&{-}A_{n}\sinh\left({n\pi{{h_{1}}\over{a}}}\right)=\sum_{m=1}^{\infty} {g_{mn}B_{m}}\sinh\left({m\pi{{h_{2}}\over{b}}}\right),&{\hbox{(A3a)}} \cr&{{\rho_{1}}\over{\rho_{2}}}nB_{n} \cosh\left(n\pi{{h_{2}}\over{b}}\right)+\sum_{m=1}^{\infty}{\gamma_{nm}B_{m}}\sinh\left({{m\pi h_{2}}\over{b}}\right)\cr&\qquad={{2}\over{\pi}}{{\sin\left({n\pi a/b}\right)}\over{n\pi a/b}},\matrix{&}n=1,2,3\ldots&{\hbox{(A3b)}}}$$ where Formula TeX Source $$\eqalignno{\gamma_{nm}=&\,\gamma_{mn}=\displaystyle\sum_{l=1}^{\infty}lg_{nl}g_{ml}\coth\left({{l\pi h_{1}}\over{a}}\right),\cr g_{mn}=&\,\displaystyle{{2}\over{a}}\int\limits_{0}^{a}{dy\cos\left({{m\pi y}\over{b}}\right)\cos\left({{n\pi y}\over{a}}\right).}&{\hbox{(A4)}}}$$

In deriving (A3b), we have assumed that Formula$aA_{0}=+1$. The infinite matrix in (A3b) can be solved directly for Formula$B_{n}$ with convergence guaranteed [22], from which Formula$A_{n}$ follows in (A3a).

The total current from AB to EF is, [Fig. 1(a)] Formula TeX Source $$I=2W\int_{0}^{a}{{{1}\over{\rho_{1}}}\left.{{\partial\Phi_{+}}\over{\partial z}}\right\vert}_{z=0}dy={{2W}\over{\rho_{1}}},\eqno{\hbox{(A5)}}$$ where we have used (A1) and Formula$aA_{0}=+1$, and Formula$W$ is the width in the third, ignorable dimension that is perpendicular to the paper. The terminal voltage Formula$V_{0}$ may be expressed in terms of Formula$B_{\rm{n}}$ as Formula TeX Source $$V_{0}=-{{h_{1}}\over{a}}-B_{0}h_{2}-{{1}\over{a}}\sum_{n=1}^{\infty}{B_{n}\sin h\left({{n\pi h_{2}}\over{b}}\right)}{{\sin\left({n\pi a/b}\right)}\over{n\pi a/b}}\eqno{\hbox{(A6)}}$$

We found Formula$B_{0}=(\rho_{2}/\rho_{1})/b$, after taking Formula$\partial\Phi_{-}/\partial z$ in (A1) and using (A2b) and 2c) in the resultant Fourier series.

The constriction resistance, Formula$R_{c}$, is defined as the difference between the resistance from AB to EF, Formula$R=V_{0}/I,$ and the bulk resistance, Formula$R_{u}=\rho_{1}h_{1}/2aW+\rho_{2}h_{2}/2bW$, Formula TeX Source $$\eqalignno{R_{c}\equiv&\,{{\rho_{2}}\over{4\pi W}}\overline{R}_{c}={{V_{0}}\over{I}}-R_{u},\cr{\bar R_{c}}=&\,\bar R_{c}\left({{{a}\over{b}},{{h_{1}}\over{a}},{{a}\over{h_{2}}},{{\rho_{1}}\over{\rho_{2}}}}\right)=2\pi{{\rho_{1}}\over{\rho_{2}}}\sum_{n=1}^{\infty}{B_{n}\sinh\left({{n\pi h_{2}}\over{b}}\right)}{{\sin\left({n\pi a/b}\right)}\over{n\pi a/b}}.\cr& &{\hbox{(A7)}}}$$

Equation (A7) is the exact expression for the constriction resistance of Cartesian vertical thin film contact [Fig. 1(a)] for arbitrary values of Formula$a$, Formula$b(b>a)$, Formula$h_{1}$, Formula$h_{2}$ and Formula$\rho_{1}/\rho_{2}$. In (A7), Formula$B_{\rm n}$ is solved from (A3b). Equation (A7) appears in (1) of the main text.

APPENDIX B

General Solution to the Circular Vertical Contact [Fig. 1(a)]

The solutions to Laplace's equation in cylindrical geometry are [15], [20], [22], Formula TeX Source $$\eqalignno{&\Phi_{+}(r,z)=A_{0}(z-h_{1})+\sum_{n=1}^{\infty}{A_{n}J_{0}\left({\alpha_{n}r}\right)\sinh\left[{\alpha_{n}(z-h_{1})}\right]},\cr&\qquad\sim\sim0<z<h_{1},r\in (0,a);\cr &\Phi_{-}(r,z)=V_{0}+B_{0}(z+h_{2})+\sum_{n=1}^{\infty}{B_{n}J_{0}\left({\beta_{n}r}\right)}\sinh\left[{\beta_{n}(z+h_{2})}\right],\cr &\qquad\sim\sim-h_{2}<z<0,r\in (0,b),&{\hbox{(B1)}}}$$ where Formula$\Phi_{+}$ and Formula$\Phi_{-}$ are the electrical potential in the regions I and II, respectively, Formula$\alpha_{\rm{n}}$ and Formula$\beta_{\rm{n}}$ satisfy Formula$J_{1}(\alpha_{\rm{n}}a)=J_{1}(\beta_{\rm{n}}b)=0$, Formula$J_{0}(x)$ and Formula$J_{1}(x)$ are the Bessel functions of order zero and one respectively, and Formula$A_{\rm{n}}$ and Formula$B_{\rm n}$ are the coefficients that need to be determined.

At the interface Formula$z=0$, from the continuity of electrical potential and current density, we have the following boundary conditions: Formula TeX Source $$\eqalignno{&\qquad\quad\Phi_{+}=\Phi_{-},\quad z=0,r\in(0,a),&{\hbox{(B2a)}}\cr& {{1}\over{\rho_{1}}}{{\partial\Phi_{+}}\over{\partial z}}={{1}\over{\rho_{2}}} {{\partial\Phi_{-}}\over{\partial z}},\matrix{&}z=0, r\in(0,a),& {\hbox{(B2b)}}\cr&\qquad\quad{{\partial\Phi_{-}}\over{\partial z}}=0,\matrix{&}z=0, r\in (a,b).&{\hbox{(B2c)}}}$$

From (B1) and (B2a), the coefficient Formula$A_{\rm{n}}$ is expressed in terms of Formula$B_{\rm{n}}$, Formula TeX Source $$\eqalignno{&-A_{0}h_{1}=\sum_{n=1}^{\infty}{B_{n}\sinh\left({\beta_{n}h_{2}}\right){{2J_{1}\left({\beta_{n}a}\right)}\over{\beta_{n}a}}+V_{0}} +B_{0}h_{2},&{\hbox{(B3a)}}\cr&\qquad-\sinh (\alpha_{n}h_{1})A_{n}=\sum_{m=1}^{\infty}{B_{m}\sinh\left({\beta_{m}h_{2}}\right)g_{mn}},\cr&\qquad\sim\sim g_{mn}={{2}\over{a^{2}J_{0}^{2}(\alpha_{n}a)}}\int\limits_{0}^{a} {rdrJ_{0}\left({\alpha_{n}r}\right)J_{0}\left({\beta_{m}r}\right), n\geq 1.}&\hbox{(B3b)}}$$ Combining (B2b) and (B2c) and (B3b), we obtain Formula TeX Source $$\eqalignno{&{{\rho_{1}}\over{\rho_{2}}}{{b}\over{a}}\beta_{n}bJ_{0}^{2}(\beta_{n}b)\cosh (\beta_{n}h_{2})B_{n}+\sum_{m=1}^{\infty}{\gamma_{nm}B_{m}}\sinh\left({\beta_{m}h_{2}}\right)\cr&={{2J_{1}\left({\beta_{n}a}\right)}\over{\beta_{n}a}},\matrix{&}n=1,2,3\ldots,&\hbox{(B4)}}$$ where Formula TeX Source $$\gamma_{nm}=\gamma_{mn}=\sum_{l=1}^{\infty}{g_{nl}g_{ml}\alpha_{l}aJ_{0}^{2}\left({\alpha_{l}a}\right)\coth\left({\alpha_{l}h_{1}}\right)},\eqno{\hbox{(B5)}}$$ and Formula$g_{nl}$ and Formula$g_{ml}$ is in the form of the last part in (B3b). In deriving (B4), we have set Formula$aA_{0}=1$ for simplicity.

The total resistance from Formula$AB$ to Formula$EF$ is Formula$R=V_{0}/I$, where Formula$I=\int\limits_{0}^{a}{\left({{{1}\over{\rho_{1}}}\left.{{\partial\Phi_{+}}\over{\partial z}}\right\vert_{z=0}}\right)2\pi rdr}=\pi a/\rho _{1}$ is the total current from AB to EF [Fig. 1(a)], and Formula$V_{0}$ can be found from (B3a) with Formula$B_{0}=\left({\rho_{2}/\rho_{1}}\right)a/b^{2}$. This expression for Formula$B_{0}$ is obtained after taking Formula$\partial\Phi_{-}/\partial z$ in (B1) and using (A2b) and (A2c) in the resultant Fourier series.

The constriction resistance, Formula$R_{c}$, is the difference between the total resistance Formula$R$ and the bulk resistance Formula$R_{u}=\rho_{1}h_{1}/\pi a^{2}+\rho_{2}h_{2}/\pi b^{2}$. We find Formula TeX Source $$\eqalignno{&\hskip-20ptR_{c}\equiv{{\rho_{2}}\over{4a}}\overline{R}_{c}={{V_{0}}\over{I}}-R_{u},\cr&\bar R_{c}\left({{{a}\over{b}},{{h_{1}}\over{a}},{{a}\over{h_{2}}},{{\rho_{1}}\over{\rho_{2}}}}\right)={{8}\over{\pi}}{{\rho_{1}}\over{\rho_{2}}}\sum_{n=1}^{\infty}{B_{n}\sinh\left({\beta_{n}h_{2}}\right){{J_{1}\left({\beta_{n}a}\right)}\over{\beta_{n}a}}},&{\hbox{(B6)}}}$$ which is the exact expression for the circular vertical thin film constriction resistance with dissimilar materials for arbitrary values of Formula$a$, Formula$b(b>a)$, Formula$h_{1}$, Formula$h_{2}$ and Formula$\rho_{1}/\rho_{2}$. In (B6), Formula$B_{n}$ is solved from (B4). Equation (B6) appears in (4) of the main text.

ACKNOWLEDGMENT

We thank Derek Hung for help in data collection.

Footnotes

This work was supported in part by an AFOSR Grant on the Basic Physics of Distributed Plasma Discharge, AFOSR Grant FA9550-09-1-0662, L-3 Communications Electron Devices Division. The review of this paper was arranged by editor C. C. McAndrew.

The authors are with the Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109-2104 USA (e-mail: umpeng@umich.edu; yylau@umich.edu).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

1An anonymous referee suggested alternate, elegant fitting formulas, Formula$f(x)=3/[(2x-1)(2+x)]$, Formula$g(x)=(4/x^{2})(x^{2}-1)^{2}/(4x^{2}-3)$ where Formula$x=b/a(>1)$. While these fitting formulas for Formula$f(x)$ and Formula$g(x)$ are not as accurate as (3b) and (3c), they may be used for most practical purposes. We wish to thank this referee for his/her careful reading of the manuscript.

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Authors

Peng Zhang

Peng Zhang

Peng Zhang (S'07–M'12) received the B.Eng. and M.Eng. degrees in electrical and electronic engineering from Nanyang Technological University, Singapore in 2006 and 2008, respectively, and the Ph.D. degree in nuclear engineering and radiological sciences from the University of Michigan at Ann Arbor, Michigan, USA, in 2012. He has authored or co-authored more than a dozen refereed journal articles on electrical contacts, surface roughness-induced heating, surface flashover and discharge, relativistic magnetron, classical and quantum diodes, magneto-Rayleigh-Taylor instability, and laser-solid interaction.

He was a recipient of the 2012 IEEE Nuclear and Plasma Sciences Graduate Scholarship Award.

Y. Y. Lau

Y. Y. Lau

Y. Y. Lau (M'98–SM'06–F'08) received the S.B., S.M., and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 1968, 1970, and 1973, respectively.

From 1973 to 1979, he had been an Instructor and then an Assistant Professor in applied mathematics at MIT. He was with Science Applications Inc., McLean, VA, USA, from 1980 to 1983, and with the Naval Research Laboratory (NRL), Washington, DC, USA, from 1983 to 1992, both as a Research Physicist. In 1992, he joined the University of Michigan, Ann Arbor, as a Professor in the Department of Nuclear engineering and Radiological Sciences, and in the Applied Physics Program. He has worked on electron beams, coherent radiation sources, plasmas and discharges. He has over 190 refereed publications and 10 patents.

He served three terms as an Associate Editor of the Physics of Plasmas, and was a Guest Editor of the IEEE TRANSACTIONS ON PLASMA SCIENCE Special Issue on high-power microwave generation. While at the NRL, he was a recipient of several Invention and Publication Awards and the 1989 Sigma-Xi Scientific Society Applied Science Award. He became a fellow of the American Physical Society in 1986, an IEEE Fellow in 2008 and was the recipient of the 1999 IEEE Plasma Science and Applications Award.

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