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SECTION I

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].

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, $\rho_{1}=\rho_{2}$ [6], or $h_{1}\to 0$ [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 $\rho_{1}=0$, when the thin film thickness $h_{2}$ 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 $\rho_{1}$ and $\rho_{2}$, and the geometric dimensions $a$, $b$, $h_{1}$, and $h_{2}$ 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 $h_{1}$ small and $\rho_{1}$ 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

Since the geometry [Fig. 1(a)] is symmetrical about the vertical $z$-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 $I=V_{0}/R$, where $R$ is the resistance between these two terminals, which we find to be 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 $W$ 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, $R_{c}$, and is expressed as $R_{c}={{\rho_{2}}\over{4\pi W}}\overline{R}_{c}$ for the Cartesian case. The normalized $\overline{R}_{c}$ depends on the aspect ratios $a/b$, $h_{1}/a$, and $a/h_{2}$, and on the resistivity ratio $\rho_{1}/\rho_{2}$, as explicitly shown in (1). The exact expression for $\overline{R}_{c}$ is derived in Appendix A (A7).

If we use the exact expression (A7) in the numerical evaluation of $\overline{R}_{c}$, we call the result the “exact theory”. [In the infinite sum in (A7), we use $10^{4}$ terms; we also use $10^{4}$ terms in the infinite sum in (A3b) to solve for $B_{n}$, $n=1, 2,..,10^{4}$.] From the vast amount of data that we collected from the exact theory at various combinations of $h_{1}$, $h_{2}$, $\rho_{1}$, $\rho_{2}$, $a$, and $b$, we attempted to construct simple fitting formulas for $\overline{R}_{c}$ so that the values of $\overline{R}_{c}$, or the bounds of $\overline{R}_{c}$, may be easily obtained without solving the infinite matrix (A3b). The bounds on $\overline{R}_{c}$ are identical to the corresponding asymptotic limits of $\overline{R}_{c}$, 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].

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

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

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 $h_{1}/a\rightarrow\infty$ limit, 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 $a$, $b(>a)$, $h_{2}$, $\rho_{1}$, and $\rho_{2}$, in the limit of $h_{1}/a\rightarrow\infty$.

In (3), $\overline{R}_{c0}\left({b/a}\right)$, $f(b/a)$, and $g(b/a)$ are derived by Lau and Tang [21], and by Zhang and Lau [(5) and (6) of [22]],^{1} $p(a/h_{2})$ is derived by Hall [(45) of [6], assuming $\rho_{1}/\rho_{2}=1$ and $h_{1}/a\rightarrow\infty$], $q(b/a)$ is derived by both Hall [(42) of [6], assuming $\rho_{1}/\rho_{2}=1$, $h_{1}/a\rightarrow\infty$, and $h_{2}/a\rightarrow\infty$] and Smythe [24]. The breakpoint in (2), ${{a}\over{h_{2}}}=\tan\left({{\pi a}\over{2b}}\right)$, was also stated by Hall [(46) of [6]]. At ${{a}\over{h_{2}}}=\tan\left({{\pi a}\over{2b}}\right)$, 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 $\overline{R}_{c}$ [6].

Fig. 4(a) shows the exact theory for $\overline{R}_{c}$ (A7) as a function of $\rho_{1}/\rho_{2}$, for various $a/h_{2}$ and $h_{1}/a$. Fig. 4(b) shows the exact theory for $\overline{R}_{c}$ (A7) as a function of $h_{1}/a$, for various $a/h_{2}$ and $\rho_{1}/\rho_{2}$. In both Fig. 4(a) and (b), we fixed $b/a=30$. In general, as either $\rho_{1}/\rho_{2}$ or $h_{1}/a$ increases, $\overline{R}_{c}$ increases. It is important to recognize from Figs. 2 –4 that dependence of $\overline{R}_{c}$ on $h_{1}/a$ and on $\rho_{1}/\rho_{2}$ is not significant, and that the major dependence of $\overline{R}_{c}$ is on $a/h_{2}$ and on $b/a$. Thus, for a given $a/h_{2}$ and $b/a$ 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 $h_{1}/a$ and $\rho_{1}/\rho_{2}$.

The field line equation, $y=y(z)$, may be numerically integrated from the first order ordinary differential equation $dy/dz=E_{y}/E_{z}=(\partial\Phi/\partial y)/(\partial\Phi/\partial z)$ where $\Phi$ 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 $\rho_{1}/\rho_{2}=1$, and $h_{1}/a=0.01$ with various $a/h_{2}$. We set $b/a=30$ in all calculations in Fig. 5. Note that the variation of $a/h_{2}$ in Fig. 5 may be interpreted this way: $a$, $b$, and $h_{1}$ are held fixed, $h_{2}$ decreases, i.e., $h_{2}=5a$, $a$, and 0.2 $a$, from Fig. 5(a)–(c). (Similar interpretation applies to other figures.) It is clear that as $a/h_{2}$ increases, the spreading of the field lines (also the current flow lines) in Region II becomes less significant. This explains the decrease in $\overline{R}_{c}$ as $a/h_{2}$ increases, as shown in Fig. 3. In the limit of $h_{2}\rightarrow 0$, 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 $h_{1}/a$ or $\rho_{1}/\rho_{2}$ increases, the spreading of the field lines in Region II slightly increases (not shown). The field line distribution is relatively insensitive to $h_{1}/a$ or $\rho_{1}/\rho_{2}$, as compared to the effect of $a/h_{2}$.

SECTION III

For the circular thin film vertical contact [Fig. 1(a)], the z-axis is the axis of rotation. The total current is $I=V_{0}/R$, where $R$ is the resistance between the two terminals AB and EF, given by 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, $R_{c}$, and is expressed as $R_{c}={{\rho_{2}}\over{4a}}\overline{R}_{c}$ for the circular case. The normalized $\overline{R}_{c}$ depends on the aspect ratios $a/b$, $h_{1}/a$ and $a/h_{2}$, and on the resistivity ratio $\rho_{1}/\rho_{2}$, as explicitly shown in (4). The exact expression for $\overline{R}_{c}$ is derived in Appendix B (B6).

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

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

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 $h_{1}/a\rightarrow\infty$ limit, given by (5) and (6) at the bottom of this page, 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 $\Delta=32/3\pi^{2}-1=0.08076$. Equation (5) is also plotted in Fig. 7, showing excellent agreement with the exact theory (B6) for arbitrary value of $a$, $b(>a)$, $h_{2}$, $\rho_{1}$, and $\rho_{2}$, in the limit of $h_{1}/a\rightarrow\infty$.

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

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

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 $h_{2}\rightarrow 0$, 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

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 $h_{1}$, $h_{2}$, $\rho_{1}$, $\rho_{2}$, $a$, and $b(>a)$ [Fig. 1(a)].

We found that the normalized constriction resistance $\overline{R}_{c}$ depended predominantly on $b/a$ and on $h_{2}/a$, i.e., on the geometry of Region II; but was relatively insensitive to $h_{1}/a$, and to $\rho_{1}/\rho_{2}$, 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 $(h_{2}\rightarrow 0)$, 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 $h_{2}$($h_{2}\rightarrow 0$ [17]) of the two edges G and D in Fig. 1(b), suggesting severe local heating there for the horizontal thin film contact.

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 $+V_{0}$. The solutions to Laplace's equation are, 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 $\Phi_{\rm{+}}$ and $\Phi_{\rm{-}}$ are the electrical potential in the regions I and II respectively, and $A_{\rm{n}}$ and $B_{\rm{n}}$ are the coefficients that need to be determined.

At the interface $z=0$, from the continuity of electrical potential and current density, we have the following boundary conditions 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 $A_{\rm{n}}$ in favor of $B_{\rm{n}}$, we have 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 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 $aA_{0}=+1$. The infinite matrix in (A3b) can be solved directly for $B_{n}$ with convergence guaranteed [22], from which $A_{n}$ follows in (A3a).

The total current from AB to EF is, [Fig. 1(a)] 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 $aA_{0}=+1$, and $W$ is the width in the third, ignorable dimension that is perpendicular to the paper. The terminal voltage $V_{0}$ may be expressed in terms of $B_{\rm{n}}$ as 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 $B_{0}=(\rho_{2}/\rho_{1})/b$, after taking $\partial\Phi_{-}/\partial z$ in (A1) and using (A2b) and 2c) in the resultant Fourier series.

The constriction resistance, $R_{c}$, is defined as the difference between the resistance from AB to EF, $R=V_{0}/I,$ and the bulk resistance, $R_{u}=\rho_{1}h_{1}/2aW+\rho_{2}h_{2}/2bW$, 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 $a$, $b(b>a)$, $h_{1}$, $h_{2}$ and $\rho_{1}/\rho_{2}$. In (A7), $B_{\rm n}$ is solved from (A3b). Equation (A7) appears in (1) of the main text.

The solutions to Laplace's equation in cylindrical geometry are [15], [20], [22], 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 $\Phi_{+}$ and $\Phi_{-}$ are the electrical potential in the regions I and II, respectively, $\alpha_{\rm{n}}$ and $\beta_{\rm{n}}$ satisfy $J_{1}(\alpha_{\rm{n}}a)=J_{1}(\beta_{\rm{n}}b)=0$, $J_{0}(x)$ and $J_{1}(x)$ are the Bessel functions of order zero and one respectively, and $A_{\rm{n}}$ and $B_{\rm n}$ are the coefficients that need to be determined.

At the interface $z=0$, from the continuity of electrical potential and current density, we have the following boundary conditions: 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 $A_{\rm{n}}$ is expressed in terms of $B_{\rm{n}}$, 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 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 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 $g_{nl}$ and $g_{ml}$ is in the form of the last part in (B3b). In deriving (B4), we have set $aA_{0}=1$ for simplicity.

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

The constriction resistance, $R_{c}$, is the difference between the total resistance $R$ and the bulk resistance $R_{u}=\rho_{1}h_{1}/\pi a^{2}+\rho_{2}h_{2}/\pi b^{2}$. We find 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 $a$, $b(b>a)$, $h_{1}$, $h_{2}$ and $\rho_{1}/\rho_{2}$. In (B6), $B_{n}$ is solved from (B4). Equation (B6) appears in (4) of the main text.

We thank Derek Hung for help in data collection.

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.

^{1}An anonymous referee suggested alternate, elegant fitting formulas, $f(x)=3/[(2x-1)(2+x)]$, $g(x)=(4/x^{2})(x^{2}-1)^{2}/(4x^{2}-3)$ where $x=b/a(>1)$. While these fitting formulas for $f(x)$ and $g(x)$ 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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