# A Two Dimensional Tunneling Resistance Transmission Line Model for Nanoscale Parallel Electrical Contacts

## Abstract

Contact resistance and current crowding are important to nanoscale electrical contacts. In this paper, we present a self-consistent model to characterize partially overlapped parallel contacts with varying specific contact resistivity along the contact length. For parallel tunneling contacts formed between contacting members separated by a thin insulating gap, we examine the local voltage-dependent variation of potential barrier height and tunneling current along the contact length, by solving the lumped circuit transmission line model (TLM) equations coupled with the tunneling current self consistently. The current and voltage distribution along the parallel tunneling contacts and their overall contact resistance are analyzed in detail, for various input voltage, electrical contact dimension, and material properties (i.e. work function, sheet resistance of the contact members, and permittivity of the insulating layer). It is found the existing one-dimensional (1D) tunneling junction models become less reliable when the tunneling layer thickness becomes smaller or the applied voltage becomes larger. In these regimes, the proposed self-consistent model may provide a more accurate evaluation of the parallel tunneling contacts. For the special case of constant ohmic specific contact resistivity along the contact length, our theory has been spot-checked with finite element method (FEM) based numerical simulations. This work provides insights on the design, and potential engineering, of nanoscale electrical contacts with controlled current distribution and contact resistance via engineered spatially varying contact layer properties and geometry.

## Introduction

Nanoscale electrical contacts have attracted substantial attention due to the advancements in nanotechnology, material sciences and growing demands for miniaturization of electronic devices and high packing density. Contact resistance and their electro-thermal effects have become one of the most critical concerns of very large scale integration (VLSI) circuit designers, because of the excessive amount of Joule heating being deposited at the contact region1,2,3,4,5,6. The electrical contact properties have been extensively studied in metal-semiconductor7,8,9, metal-insulator-semiconductor and metal-insulator-metal10,11,12,13 junctions. The growing popularity of novel electronic circuits based on graphene, carbon nanotubes (CNTs) and other new materials has made contact engineering crucial. CNT based devices, in particular, experience significant challenges because of the inter-tube connections. On macroscopic level, the exceptional intrinsic electrical properties14,15 of CNTs become elusive3,14,16. Contact resistances between CNTs profoundly affect the electron transport and reduce the electrical conductivity of carbon nanofiber (CNF)15,16,17, and greatly limit the performance of CNT thin film based Field Effect Transistors (FETs)18,19,20,21. One can naturally expect these issues also arising from other novel two-dimensional materials (boron nitride, molybedenum sulfide, black phosphorus, etc) as well as new nano-composites. While the work presented here is generalizable to these other material systems, here we choose carbon materials as examples.

Tunneling type of electrical contacts11,22,23,24,25 are commonly found for CNT-CNT16,22,26,27,28,29,30, CNT-Metal31,32,33 and CNT-graphene34,35 contacts, where the contacting members are separated by very thin insulating layers. Tunneling effects in contact junctions significantly lower the electrical conductivity of the CNT/polymer composite thin films23. It is also found that tunneling resistance plays a dominant role in the electrical conductivity of CNT-based polymeric or ceramic composites27.

For decades, the basic models of tunneling current between electrodes separated by thin insulating films have been those of Simmons36,37,38 in 1960s. Simmon's formula have since been used for evaluating tunneling current in tunneling junctions24,29,39. Though there have been attempts to extend Simmons’ models to the field emission and space-charge-limited regimes11,40,41, it is always assumed that the tunneling junctions are one-dimensional (1D), i.e. there is no variations on the voltages drops along the length of the tunneling junction and the insulating film thickness is uniform. Thus, these existing models of tunneling junctions give no hint on the variation of tunneling current along the contact length and the importance of current crowding near the contact area when the two contacting members are partially overlapping (cf. Fig. 1). On the other hand, the widely used transmission line models (TLM) for electrical contacts typically assume the contact resistivity of the interface layers are constant10,42,43,44. It is questionable to apply these models to study the tunneling contacts, as the tunneling resistance depends on the junction voltage that varies spatially along the contact length.

In this paper, we propose a two-dimensional (2D) transmission line model for partially overlapped parallel contacts with spatially varying specific contact resistivity. Spatial dependence of specific contact resistivity of the contact interface may be introduced by many factors, such as nonuniform distribution of the resistive contaminants, oxides, or foreign objects at the contact interface, formation of contact interfaces with spatially varying thickness, or the presence of tunneling contacts between contact members. In the latter case, because of the nonlinear current-voltage characteristics of the tunneling junctions11,36, the specific resistivity along the contact length will become spatially dependent, even for a tunneling layer with uniform thickness (Fig. 1). For the tunneling-type contacts, the model considers the variation of potential barrier height and tunneling current along the contact length, by solving the TLM equations coupled with the tunneling current self consistently. We provide comprehensive analysis of the effects of contact geometry (i.e. dimension of the contact, and distance between the contact electrodes), and material properties (i.e. work function, sheet resistance of the contact members, and permittivity of the insulating layer) on the spatial distributions of currents and voltages across these contacts, and the overall contact resistance of parallel contacts.

The formulation of our 2D contact resistance TLM model is given in the next section. We would like to point out that, albeit an application of the standard transmission line theory based on the Kirchhoff's laws, the TLM has been used extensively with great success to characterize mesoscale and nanoscale electrical contacts10,30,42,43. Here we further extend the TLM model with the effects of spatially dependent contact resistivity. In the results and Discussion section, we consider three cases of parallel contacts: (1) constant specific contact resistivity, (2) linearly varying specific contact resistivity, and (3) tunneling contact resistivity depending on local junction voltages along the contact length. The first case of uniform specific contact resistivity along the contact length has been verified with COMSOL45 2D simulations. For the last case, for simplicity, we use the Simmons’ model36,37,38 to determine the local current-voltage characteristics across the tunneling junction. Though full scale quantum mechanical calculations may have to be used to accurately evaluate the nanoscale circuits, our model based on Simmons formula reveals the fundamental scalings and parametric dependence of current and voltage profiles, as well as electric contact resistance of tunneling contacts. Summary and suggestions for future research are given in the last section.

Note that, although this paper is focused on the normal Schrödinger tunneling type electrical contacts, the proposed TLM with spatially varying contact resistivity can be used for many other types of electric contacts, such as nanoscale Schottky contacts based on 2D materials heterostructure46,47, and Klein tunneling junctions48.

## Formulation

Consider a parallel contact formed between two nanowires or nanotubes or between two conducting thin films or layers, as shown in Fig. 1(a,b), respectively. The distance between the two contact members is D, and the contact length is L. A thin resistive interface layer is sandwiched between them. Both contacts in Fig. 1(a,b) can be described by a two-dimensional (2D) model, as shown in Fig. 1(c). Note that the proposed formulation is generally applicable to parallel Cartesian nanojunctions with different shape of the electrodes, for example, electrical contact between a nanowire and a thin film. In the 2D model, the effects of the transverse dimension (perpendicular to the paper) can be included in the effective sheet resistances Rsh1 and Rsh2 for conductor 1 and 2, respectively, such that there is no variation along the width w in the transverse dimension. The spatial dependent specific interfacial resistivity (also termed specific contact resistivity) is ρc(x), which is either predefined, or calculated from the local tunneling current in case of insulating tunneling layer36,37,38. We use the DC equivalent lump circuit transmission line model (TLM)10,42,43,44, as shown in Fig. 1(d), to model the 2D parallel contact in Fig. 1(c).

In the contact region PQNM in Fig. 1(c,d), using Kirchoff's laws for current and voltage, we get the following equations,

$${I}_{1}(x)-{I}_{1}(x+\Delta x)=\frac{{V}_{1}(x)-{V}_{2}(x)}{{\rho }_{c}(x)}\,\Delta x\,w,$$
(1a)
$${V}_{1}(x)-{V}_{1}(x+\Delta x)={I}_{1}(x)\,{R}_{sh1}\Delta x/w,$$
(1b)
$${I}_{2}(x+\Delta x)-{I}_{2}(x)=\frac{{V}_{1}(x)-{V}_{2}(x)}{{\rho }_{c}(x)}\,\Delta x\,w,$$
(1c)
$${V}_{2}(x)-{V}_{2}(x+\Delta x)={I}_{2}(x)\,{R}_{sh2}\Delta x/w,$$
(1d)

where I1(x) and I2(x) represent the current flowing at x through the lower contact member, MN and the upper contact member, PQ respectively, and V1(x) and V2(x) the local voltage at x along MN and PQ, respectively, and w is the effective transverse dimension of the contacts. When Δx → 0, Equation (1) becomes,

$$\frac{\partial {I}_{1}(x)}{\partial x}=-\,w{J}_{c}(x),$$
(2a)
$$\frac{\partial {V}_{1}(x)}{\partial x}=-\frac{{I}_{1}(x){R}_{sh1}}{w},$$
(2b)
$$\frac{\partial {I}_{2}(x)}{\partial x}=w{J}_{c}(x),$$
(2c)
$$\frac{\partial {V}_{2}(x)}{\partial x}=-\frac{{I}_{2}(x){R}_{sh2}}{w},$$
(2d)

where Jc(x) = Vg(x)/ρc(x) and Vg(x) = V1(x) − V2(x) are the local current density and the local voltage drop across the contact interface at x, respectively.

Note that, from Eqs (2a) and (2c), I1(x) + I2(x) = Itot = constant, where Itot is the total current in the circuit, to be determined from the boundary conditions. The boundary conditions for Eq. (2) are,

$${V}_{1}(x=0)={V}_{o},$$
(3a)
$${I}_{2}(x=0)=0,$$
(3b)
$${I}_{1}(x=L)=0,$$
(3c)
$${V}_{2}(x=L)=0,$$
(3d)

where, without loss of generality, we assume the voltage of the upper contact member at x = L is 0, and the externally applied voltage at x = 0 of the lower contact member is V0. Note that I1(x = 0) = Itot. From Eqs (2) and (3), it is easy to show $${V}_{1}^{^{\prime} }(x=0)=-\,{I}_{tot}{R}_{sh1}/w$$, $${V}_{1}^{\text{'}}(x=L)=0$$, $${V}_{2}^{^{\prime} }(x=0)=\,0$$, $${V}_{2}^{^{\prime} }(x=L)\,=$$ $$-{I}_{tot}{R}_{sh2}/w$$, where a prime denotes a derivative with respect to x. For the contact model in Fig. 1(d), the contact resistance is defined as,

$${R}_{c}=\frac{{V}_{1}(0)-{V}_{2}(L)}{{I}_{tot}}=\frac{{V}_{o}}{{I}_{tot}}.$$
(4)

It is convenient to introduce non-dimensional quantities, $$\bar{x}=x/L$$, $${\bar{\rho }}_{c}(\bar{x})={\rho }_{c}(x)/{\rho }_{c0}$$, $${\bar{R}}_{sh2}={R}_{sh2}/{R}_{sh1}$$, $${\bar{I}}_{1}(\bar{x})={I}_{1}(x)/{I}_{o}$$, $${\bar{I}}_{2}(\bar{x})={I}_{2}(x)/{I}_{o}$$, $${\bar{J}}_{c}(\bar{x})={J}_{c}(x)LW/{I}_{o}$$, $${\bar{V}}_{1}(\bar{x})={V}_{1}(x)/{V}_{o}$$, $${\bar{V}}_{2}(\bar{x})={V}_{2}(x)/{V}_{o}$$, $${\bar{V}}_{g}(\bar{x})={V}_{g}(x)/{V}_{o}$$, and $${\bar{R}}_{c}={R}_{c}/{R}_{c0}$$, where we define Io = wV0/Rsh1L, ρc0 = V0wL/Io, and Rc0 = Rsh1L/w. In normalized forms, Eq. (2) can be recast into the following second order differential equations,

$$\frac{{\partial }^{2}\overline{{V}_{1}}(\bar{x})}{\partial {\bar{x}}^{2}}={\bar{J}}_{c}(\bar{x}),$$
(5a)
$$\frac{{\partial }^{2}{\bar{V}}_{g}(\bar{x})}{\partial {\bar{x}}^{2}}=(1+{\bar{R}}_{sh2}){\bar{J}}_{c}(\bar{x}),$$
(5b)
$${\bar{\rho }}_{c}(\bar{x})\frac{{\partial }^{2}\overline{{I}_{1}}(\bar{x})}{\partial {\bar{x}}^{2}}+\frac{\partial {\bar{\rho }}_{c}(\bar{x})}{\partial \bar{x}}\frac{\partial \overline{{I}_{1}}(\bar{x})}{\partial \bar{x}}-(1+{\bar{R}}_{sh2})\,\overline{{I}_{1}}(\bar{x})+\alpha {\bar{R}}_{sh2}=0,$$
(5c)

where $${\bar{J}}_{c}(\bar{x})={\bar{V}}_{g}(\bar{x})/{\bar{\rho }}_{c}(\bar{x})$$, and $${\bar{V}}_{g}(\bar{x})={\bar{V}}_{1}(\bar{x})-{\bar{V}}_{2}(\bar{x})$$. The corresponding boundary conditions to Eq. 5(ac) are respectively,

$${\bar{V}}_{1}(\bar{x}=0)=1,\,{{\bar{V}}^{{\rm{^{\prime} }}}}_{1}(\bar{x}=0)=-\alpha \,{\rm{a}}{\rm{n}}{\rm{d}}\,{\bar{V}}_{1}(\bar{x}=1)={\bar{V}}_{g}(\bar{x}=1),$$
(6a)
$${\bar{V}^{\prime} }_{g}(\bar{x}=0)=-\,\alpha ,\,{\bar{V}^{\prime} }_{g}(\bar{x}=1)=\alpha {\bar{R}}_{sh2},$$
(6b)
$${\bar{I}}_{1}(\bar{x}=0)=\alpha ,{\bar{I}}_{1}(\bar{x}=1)=0,$$
(6c)

where the unknown constant α = Itot/Io is the normalized total current in the circuit, and prime denotes a derivative with respect to $$\bar{x}$$. Note that integrating Eq. (5b) subject to Eq. (6b) gives $${\int }_{0}^{1}\,{\bar{J}}_{c}(\bar{x})d\bar{x}\,=\alpha$$, which means that the total current is conserved across the contact interface.

Equations (5) and (6) are solved to give the voltage and current distribution along and across the contact interface as well as the total contact resistance, for a given electrical contact (Fig. 1) with spatially dependent interface specific contact resistivity $${\bar{\rho }}_{c}(\bar{x})$$. An example of the procedure to solve Eqs (5) and (6) numerically is as follows. For an initially guess on α, Eq. (5b) is solved using the shooting method, subject to Eq. (6b). Next, Eq. (5a) is solved with the initial values of $${\bar{V}}_{1}(0)$$ and $${\bar{V}}_{1}^{^{\prime} }(0)$$ from Eq. (6a). It is then checked whether $${\bar{V}}_{1}(1)$$ is equal to $${\bar{V}}_{g}(1)$$, as in Eq. (6a). The above-mentioned process repeats for different input α until the condition $${\bar{V}}_{1}(1)={\bar{V}}_{g}(1)$$ is satisfied. Finally, Eq. (5c) is solved to get $$\overline{{I}_{1}}$$ (and $$\overline{{I}_{2}}$$).

In principle, Eqs (5) and (6) can be solved numerically for arbitrary spatial dependence of specific contact resistivity $${\bar{\rho }}_{c}(\bar{x})$$. Here, we focus on a few special cases of practical importance. We first consider the case of constant $${\bar{\rho }}_{c}$$, where analytical solutions can be obtained, which also serve to validate our numerical approach. We then consider the effects of spatially dependent $${\bar{\rho }}_{c}(\bar{x})$$ on the parallel electrical contacts. We focus on two situations: linearly varying specific contact resistivity along x, and thin tunneling junction with uniform thickness, where analytical solutions to the TLM current and voltage equations are no longer available, and Eqs (5) and (6) are solved numerically.

## Results and Discussion

### Case 1: Constant specific contact resistivity ρc along the contact length L

For the special case of constant specific contact resistivity ρc, the TLM equations, Eqs (5) and (6), can be solved analytically to give,

$${\bar{I}}_{1}(\bar{x})=\frac{q}{K}\,[\,\sinh \,q(1-\bar{x})+{\bar{R}}_{sh2}(\sinh \,q-\,\sinh \,q\bar{x})],$$
(7a)
$${\bar{I}}_{2}(\bar{x})=\frac{q}{K}\,[\,\sinh \,q(\bar{x}-1)+{\bar{R}}_{sh2}\,\sinh \,q\bar{x}+\,\sinh \,q],$$
(7b)
$${\bar{J}}_{c}(\bar{x})=\frac{{q}^{2}}{K}\,[\,\cosh \,q(1-\bar{x})+{\bar{R}}_{sh2}\,\cosh \,q\bar{x}],$$
(7c)
$${\bar{V}}_{1}(\bar{x})=\frac{1}{K}[\,\cosh \,q(1-\bar{x})+{\bar{R}}_{sh2}M+{\bar{R}}_{sh2}q(1-\bar{x})\sinh \,q],$$
(7d)
$${\bar{V}}_{2}(\bar{x})={\bar{V}}_{1}(\bar{x})-{\bar{\rho }}_{c}{\bar{J}}_{c}(\bar{x}),$$
(7e)
$$\,{\bar{R}}_{c}=\frac{(1+{{\bar{R}}_{sh2}}^{2})\cosh \,q+{\bar{R}}_{sh2}(2+q\,\sinh \,q)}{(1+{\bar{R}}_{sh2})q\,\sinh \,q},$$
(8)

where $$q=\frac{L}{{\lambda }_{0}}=\sqrt{\frac{1+{\bar{R}}_{sh2}}{{\bar{\rho }}_{c}}}$$, $$K=(1+{{\bar{R}}_{sh2}}^{2})\cosh \,q+{\bar{R}}_{sh2}(2+q\,\sinh \,q)$$ and $$M=\,\cosh \,q\bar{x}+1+{\bar{R}}_{sh2}\,\cosh \,q$$.

Figure 2 shows the current and voltage distributions along the contact length and across the contact interface for various specific contact resistivity $${\bar{\rho }}_{c}$$, for a parallel contact formed between similar contact members, $${\bar{R}}_{sh2}={R}_{sh2}/{R}_{sh1}=1$$. The voltage along both contact members $${\bar{V}}_{1}$$ and $${\bar{V}}_{2}$$ decrease with $$\bar{x}$$, as shown in Fig. 2(b,c), respectively. The current $${\bar{I}}_{1}$$ in contact member 1 decreases with $$\bar{x}$$ (Fig. 2(e)), whereas $${\bar{I}}_{2}$$ in contact member 2 increases with $$\bar{x}$$ (Fig. 2(f)), with the total current $$\overline{{I}_{1}}(\bar{x})+\overline{{I}_{2}}(\bar{x})$$ being kept a constant along $$\bar{x}$$. The profiles of both normalized voltage drop $${\bar{V}}_{g}(\bar{x})$$ and current density $$\bar{{J}_{c}}(\bar{x})$$ across the interface layer, are symmetric along the contact length, with the minimum at the center of the contact structure $$\bar{x}$$ = 0.5 and the maximum at the contact edges, as shown in Fig. 2(a,d), respectively. The current crowding effects near the contact edges are well-known phenomena42,49, as the current density is distributed to follow the least resistive path (i.e. minimum overall resistance). It is important to note that as the specific contact resistivity $${\bar{\rho }}_{c}$$ decreases, the interface current density $$\bar{{J}_{c}}$$ becomes more crowded towards the contact edges, as shown in Fig. 2(d). In other words, the less resistive the contact interface layer, the more severe of the current crowding effects, which is in agreement with previous studies using both TLM42,44 and field theory10,49,50.

Figure 3 shows the current and voltage distributions along the contact length and across the contact interface for various parallel contacts formed between dissimilar materials, $${\bar{R}}_{sh2}$$ = Rsh2/Rsh1, with fixed specific contact resistivity $${\bar{\rho }}_{c}$$ = 1. The voltage $$\overline{{V}_{1,2}}$$ and the current $$\overline{{I}_{1,2}}$$ along the two contact members show similar behaviors as those in Fig. 2. However, the voltage drop across the interface layer $$\overline{{V}_{g}}(\bar{x})$$ and the contact current density $$\overline{{J}_{C}}(\bar{x})$$ are no longer symmetric, as shown in Fig. 3(a,d), respectively. When $${\bar{R}}_{sh2}$$ < 1, the maximum of $$\overline{{V}_{g}}(\bar{x})$$ and $$\overline{{J}_{c}}(\bar{x})$$ occurs at $$\bar{x}$$ = 0; when $${\bar{R}}_{sh2}$$ > 1, the maximum of $$\overline{{V}_{g}}(\bar{x})$$ and $$\overline{{J}_{c}}(\bar{x})$$ occurs at $$\bar{x}$$ = 1. This current crowding effect can again be explained by the fact that current flows are self-arranged to take the least resistive path in the circuit by adjusting the current distribution according to the local resistance.

The normalized contact resistance, $$\overline{{R}_{c}}$$ calculated from Eq. (8) is plotted in Fig. 4 for various $${\bar{\rho }}_{c}$$ and $${\bar{R}}_{sh2}$$. It is clear that $$\overline{{R}_{c}}$$ increases with both $${\bar{\rho }}_{c}$$ and $${\bar{R}}_{sh2}$$. In general, the contact resistance $$\overline{{R}_{c}}$$ depends more strongly on the the specific contact resistivity of the interfacial layer $${\bar{\rho }}_{c}$$ than on the sheet resistance ratio of the contact members $${\bar{R}}_{sh2}$$. For the special case of $${\bar{R}}_{sh2}$$ = 0, Eq. (8) becomes,

$${\bar{R}}_{c}=\frac{\coth \,q}{q},$$
(9)

with $$q=L/{\lambda }_{0}\,=1/\sqrt{{\bar{\rho }}_{c}}\,$$, which is also plotted in Fig. 4. Note that Eq. (9) is identical to the expression typically used for metal-semiconductor contact10,42.

To verify the results obtained from our analytical solution, we performed numerical simulations using the COMSOL multiphysics software45, for various combinations of $${\bar{R}}_{sh2}$$ and $${\bar{\rho }}_{c}$$ on the geometry shown in Fig. 1(c). The finite-element-method (FEM) based COMSOL 2D simulation results are included in Fig. 4 (cross symbols), showing excellent agreement with our theory. The convergence iteration error in the simulation was less than 10−9 for each point.

### Case 2: Specific resistivity ρc varies linearly along the contact length L

We assume the specific resistivity varies linearly along the contact length (Fig. 1) as $${\bar{\rho }}_{c}(\bar{x})=1+A\bar{x}$$. By solving Eqs (5) and (6) numerically, we obtain the current and voltage distributions along the contact interface, as shown in Fig. 5. As A increases, the overall contact interface becomes more resistive, therefore, the voltage drop $$\overline{{V}_{g}}(\bar{x})$$ across the interface layer increases (Fig. 5a), whereas the current density $$\overline{{J}_{C}}(\bar{x})$$ across the interface layer decreases in general (Fig. 5d). The maximum $$\overline{{V}_{g}}$$ occurs at the contact edge with the highest specific resistivity $${\bar{\rho }}_{c}$$ (i.e., at $$\bar{x}$$ = 0 when A < 0, and at $$\bar{x}$$ = 1 when A > 0), while the maximum interface current $$\overline{{J}_{C}}$$ occurs at the contact edge with the lowest $${\bar{\rho }}_{c}$$ (i.e., at $$\bar{x}$$ = 1 when A < 0, and at $$\bar{x}$$ = 0 when A > 0). The effects of A on the voltage $$\overline{{V}_{1,2}}$$ and the current $$\overline{{I}_{1,2}}$$ along the two contact members are also shown in Fig. 5(b,c,e,f), respectively.

The normalized contact resistance, $$\overline{{R}_{c}}$$ calculated from Eq. (4) for linear specific contact resistivity $${\bar{\rho }}_{c}(\bar{x})=1+A\bar{x}$$ is plotted in Fig. 6. As A increases, $$\overline{{R}_{c}}$$ increases, since the contact interface becomes more resistive. As $${\bar{R}}_{sh2}$$ increases, the contact resistance $$\overline{{R}_{c}}$$ depends more strongly on the linear constant A.

### Case 3: Tunneling contact resistance

Here, we assume the parallel contacts are formed through a tunneling interface layer between the two contact members. For simplicity, we have made the following assumptions: (1) the thickness of interfacial insulating film in the contact area is uniform, and (2) the insulating film is sufficiently thin (in the nano- or subnano-meter scale) so that charge trappings are ignored51,52.

For dissimilar contact members, the (normalized) current density at any location along the contact from contact member 1 to contact member 2 is calculated using Simmons’ formula37,

$${\bar{J}}_{c}(\bar{x})=B[{\bar{\phi }}_{I}{e}^{-A\Delta \bar{y}\sqrt{{\bar{\phi }}_{I}}}-({\bar{\phi }}_{I}+{\bar{V}}_{g}(\bar{x}))\,{e}^{-A\Delta \bar{y}\sqrt{{\bar{\phi }}_{I}+{\bar{V}}_{g}(\bar{x})}}]$$
(10)

where $${\bar{V}}_{g}(\bar{x})={\bar{V}}_{1}(\bar{x})-{\bar{V}}_{2}(\bar{x})$$ is the local voltage drop across the contact interface at $$\bar{x}$$, $$A=1.025\sqrt{e{V}_{0}\,[{\rm{eV}}]}D[{\rm{\AA }}]$$, $$B=615\frac{{L}^{2}[\mu m]{R}_{sh1}[\Omega /\square \,]}{{D}^{2}[\AA ]{(\Delta \bar{y})}^{2}}$$ and $$\Delta \bar{y}={\bar{y}}_{2}-{\bar{y}}_{1}$$. Definitions of $${\bar{\phi }}_{I}$$, $${\bar{y}}_{1}$$ and $${\bar{y}}_{2}$$ for forward bias (when lower work function contacting member is given positive bias) are given below, $${\bar{\phi }}_{I}={\bar{\phi }}_{2}-(\overline{{V}_{g}}(\bar{x})+\Delta \bar{\phi })\frac{{\bar{y}}_{1}+{\bar{y}}_{2}}{2}-\frac{1.15\bar{\lambda }}{{\bar{y}}_{2}-{\bar{y}}_{1}}\,\mathrm{ln}(\frac{{\bar{y}}_{2}(1-{\bar{y}}_{1})}{{\bar{y}}_{1}(1-{\bar{y}}_{2})})$$, where $$\Delta \bar{\phi }={\bar{\phi }}_{2}-{\bar{\phi }}_{1}$$, $${\bar{\phi }}_{1}=\frac{{\phi }_{1}}{e{V}_{0}}$$, $${\bar{\phi }}_{2}=\frac{{\phi }_{2}}{e{V}_{0}}$$, φ1 = W1 − χ and φ2 = W2 − χ. W1 and W2 are the work functions of contacting member 1 and 2 respectively, χ is the electron affinity of the insulating layer, which is 0 for vacuum. For $$\,\overline{{V}_{g}}(\bar{x})\le {\bar{\phi }}_{1}$$, $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}}$$, $${\bar{y}}_{2}=1-\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{2}+4\bar{\lambda }-2(\overline{{V}_{g}}(\bar{x})+\Delta \bar{\phi })}+{\bar{y}}_{1}$$; and for $$\overline{{V}_{g}}(\bar{x}) > \,{\bar{\phi }}_{1}$$, $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}}$$, $${\bar{y}}_{2}=\frac{{\bar{\phi }}_{2}-5.6\bar{\lambda }}{(\overline{{V}_{g}}(\bar{x})+\Delta \bar{\phi })}$$, where $$\bar{\lambda }=\frac{2.49}{{\varepsilon }_{r}D[{\rm{\AA }}]e{V}_{0}[{\rm{eV}}]}$$.

On the other hand, the definitions of $${\bar{\phi }}_{I}$$, $${\bar{y}}_{1}$$ and $${\bar{y}}_{2}$$ for reverse bias (when higher work function contacting member is given positive bias) are, $${\bar{\phi }}_{I}={\bar{\phi }}_{1}+(\Delta \bar{\phi }-\overline{{V}_{g}}(\bar{x}))\frac{{\bar{y}}_{1}+{\bar{y}}_{2}}{2}-\frac{1.15\bar{\lambda }}{{\bar{y}}_{2}-{\bar{y}}_{1}}\,\mathrm{ln}(\frac{{\bar{y}}_{2}(1-{\bar{y}}_{1})}{{\bar{y}}_{1}(1-{\bar{y}}_{2})})$$; for $$0 < \overline{{V}_{g}}(\bar{x})\le \Delta \bar{\phi }$$, $${\bar{y}}_{1}=\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{1}+4\bar{\lambda }-(\overline{{V}_{g}}(\bar{x})-\Delta \bar{\phi })}-\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}-\overline{{V}_{g}}(\bar{x})}$$, $${\bar{y}}_{2}=1-\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}-\overline{{V}_{g}}(\bar{x})}$$; for $$\Delta \bar{\phi } < \overline{{V}_{g}}(\bar{x})\le {\bar{\phi }}_{2}$$, $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{1}}$$, $${\bar{y}}_{2}=1-\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{1}+4\bar{\lambda }-2(\overline{{V}_{g}}(\bar{x})-\Delta \bar{\phi })}+{\bar{y}}_{1}$$; and for $$\overline{{V}_{g}}(\bar{x}) > {\bar{\phi }}_{2}$$, $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{1}}$$, $${\bar{y}}_{2}=\frac{{\bar{\phi }}_{1}-5.6\bar{\lambda }}{(\overline{{V}_{g}}(\bar{x})-\Delta \bar{\phi })}$$.

For the special case of the same material for contact members 1 and 236, in Eq. (10), $${\bar{\phi }}_{I}={\bar{\phi }}_{0}-\overline{{V}_{g}}(\bar{x})\frac{{\bar{y}}_{1}+{\bar{y}}_{2}}{2}-$$ $$\frac{1.15\bar{\lambda }}{{\bar{y}}_{2}-{\bar{y}}_{1}}\,\mathrm{ln}(\frac{{\bar{y}}_{2}(1-{\bar{y}}_{1})}{{\bar{y}}_{1}(1-{\bar{y}}_{2})})$$, where $${\bar{\phi }}_{0}=\,\frac{{\phi }_{0}}{e{V}_{0}}$$, φ0 = Wχ, W is the work function of contacting member 1 and 2, $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{0}}$$, $${\bar{y}}_{2}=1-\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{0}+4\bar{\lambda }-2\overline{{V}_{g}}(\bar{x})}+{\bar{y}}_{1}$$ for $${\bar{V}}_{g}(\bar{x})\le {\bar{\phi }}_{0}$$, and $${\bar{y}}_{2}=\frac{{\bar{\phi }}_{0}-5.6\bar{\lambda }}{\overline{{V}_{g}}(\bar{x})}$$ for $${\bar{V}}_{g}(\bar{x}) > {\bar{\phi }}_{0}$$. Note that we use Simmon's formula, Eq. (10) here for simplicity, which is reliable only when the barrier height is relative high and the gap voltage is low in the direct tunneling regime11. More accurate results for the tunneling current may be calculated using quantum models by solving the coupled Schrodinger equation and Poisson equation with the inclusion of space charge and exchange-correlation effects11,25.

We keep the normalization consistent with our previous calculations in the Formulation section. For a given parallel tunneling contact (Fig. 1), the inputs of our model are the applied voltage V0, sheet resistance (Rsh1, Rsh2) and work function (W1, W2) of contacting members 1 and 2, permittivity (εr), thickness (D), and electron affinity (χ) of the interfacial insulator layer, and the contact length L. Using Eq. (10), the specific contact resistivity is obtained from $${\bar{\rho }}_{c}(\bar{x})=\overline{{V}_{g}}(\bar{x})/\overline{{J}_{c}}(\bar{x})$$, which is inserted into the TLM equations, Eqs (5) and (6), to give a self-consistent solution to the voltage and current profiles, as well as the contact resistance for the parallel tunneling contact.

We consider CNT-vacuum-CNT parallel contact as an example. Both contact members are made of the same single-walled CNTs. Using the typical value of linear resistivity of single-walled CNT ρL = 20 kΩ/μm53,54, and diameter (or the width w) of 3 nm, an equivalent sheet resistance for both CNT contact members are estimated as $${R}_{sh1}={R}_{sh2}={\rho }_{L}w=60\,\Omega /\square$$, where the unit of the sheet resistance $$\Omega /\square$$ means “ohm per square”10,42,49. The work function of CNTs is W1 = W2 = 4.5 eV55. The interfacial layer is assumed to be vacuum (relative permittivity εr = 1.0, and electron affinity χ = 0). The voltage drop Vg(x) across and the tunneling current density Jc(x) through the contact interface are shown in Fig. 7 for various contact length L, vacuum gap distance D, and applied voltage V0. The profiles of both Vg(x) and Jc(x) are symmetric about the center of the contact, as expected for similar contact members (similar to Fig. 2a,d above). As the contact length L increases, the local voltage drop Vg(x) across the contact interface decreases, so does the tunneling current density Jc(x), as shown in Fig. 7a,b. However, the total current in the contact structure, $${I}_{tot}={\int }_{0}^{L}\,{J}_{c}(x)dx$$ increases with L, since the total contact resistance of the tunneling junction decreases as the contact length increases (cf. Fig. 8a below). As shown in Fig. 7c,d, when the gap distance D increases, the voltage drop Vg(x) increases, but the current density Jc(x) decreases, which is because the tunneling junction becomes more resistive11,36. Figure 7e,f shows both voltage drop Vg(x) and current density Jc(x) increase when the applied voltage Vo increases. More importantly, both Vg(x) and Jc(x) exhibit a stronger spatial dependence as V0 increases. This strong voltage dependence of electrical properties of the tunneling junction is in sharp contrast with those of ohmic contacts, where the shape of the profiles of Vg(x) and Jc(x), and the total contact resistance are independent of the applied voltage, and the current density scales linearly with the voltage drop, as discussed in Cases 1 and 2 above.

Also plotted in Fig. 7 are the analytical results from Eq. (7), by assuming constant tunneling contact resistivity across the contact length L (i.e. the typically assumed one-dimensional tunneling junction24), by (a), setting Vg = V0 and using Eq. (10) (dashed lines) and (b), using ohmic approximations for the tunneling junction, in the limit of Vg → 0 (dotted lines)36,37. In the latter case, the tunneling current density is a linear function of Vg.

$${\bar{J}}_{c}(\bar{x})=B\sqrt{{\bar{\phi }}_{I}}{e}^{-A\Delta \bar{y}\sqrt{{\bar{\phi }}_{I}}}\,{\bar{V}}_{g}(\bar{x}),\,{V}_{g}\to 0,$$
(11)

where $$B=\,315.60\,\sqrt{{V}_{0}}\frac{{L}^{2}[\mu m]{R}_{sh1}[\Omega /\square ]}{D[\AA ]\Delta y}$$. A and $$\Delta \bar{y}$$ are the same as for Eq. (10). $${\bar{\phi }}_{I}$$ is calculated from the same expression for Eq. (10) by setting Vg = 0. $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}}$$, $${\bar{y}}_{2}=1-\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{2}+4\bar{\lambda }-2\Delta \bar{\phi }}+{\bar{y}}_{1}$$ for forward bias; $${\bar{y}}_{1}=\frac{9.2\bar{\lambda }}{3{\bar{\phi }}_{1}+4\bar{\lambda }+\Delta \bar{\phi }}-\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}}$$, $${\bar{y}}_{2}=1-\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{2}}$$ for reverse bias; and $${\bar{y}}_{1}=\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{0}}$$, $${\bar{y}}_{2}=1-\frac{1.2\,\bar{\lambda }}{{\bar{\phi }}_{0}}$$ for similar contacting members. $$\bar{\lambda }$$, $$\Delta \bar{\phi }$$, $${\bar{\phi }}_{2}$$, $${\bar{\phi }}_{1}$$, $${\bar{\phi }}_{0}$$ have the same definition as in Eq. (10). It is found that both assumptions of constant contact resistivity are not sufficiently reliable, especially when the tunneling thickness D decreases or the applied voltage Vo increases. As the tunneling junction resistance becomes nonlinear in these cases, it is necessary to use the coupled TLM equations, Eqs (5) and (6), and the localized tunneling equation, Eq. (10), to provide more accurate predictions.

The total contact resistance Rc of the CNT-vacuum-CNT parallel contact is shown in Fig. 8, as functions of contact length L, vacuum gap distance D, and applied voltage V0. The total contact resistance Rc increases very rapidly with increasing insulating layer thickness, D, and decreases with contact length, L. For the low applied voltage regime (V0 < 0.3 V), Rc is almost independent of V0, as shown in Fig. 8c,d. When the applied voltage V0 > 0.3 V, Rc decreases sharply with V0. This is because the junction is no longer ohmic and the tunneling resistivity ρc decreases nonlinearly with the junction voltage, as a function of position along the contact length. Ohmic approximations (Eqs 8, 11) fail to give accurate results in the latter case and it is necessary to use the self-consistent numerical model. As L increases, the dependence of contact resistance on L becomes less significant. Similar profiles of contact resistance with L were observed in other experimental and theoretical works10,22,31,42,44. The contact resistance lies between 5 kΩ to 10 MΩ for the cases shown in Fig. 8, which agrees with previously reported experimental and theoretical works24,26,29. The existing 1D models give an inaccurate estimation of the contact resistance because they do not consider the variation of tunneling current density along the contact length.

Next, we extend our calculations for contacts of CNT with different metals – calcium (Ca), aluminum (Al), copper (Cu) and gold (Au). The work functions of Ca, Al, Cu and Au are taken as 2.9, 4.08, 4.7 and 5.1 eV respectively56. The work functions and dimensions of the CNT are kept same as before. In addition, the dimensions of the CNT and contacting-metal-2 are assumed to be same (width of 3 nm, thickness of 3 nm) for the simplicity of calculations. The resistivity of Ca, Al, Cu and Au are known to be 3.36 × 10−8 Ωm, 2.7 × 10−8 Ωm, 1.68 × 10−8 Ωm and 2.2 × 10−8 Ωm respectively56,57.

Figure 9 shows the effects of the work function of contacting member 2 (W2) and the permittivity of the thin insulating layer (εr), on the current and voltage characteristics in CNT-insulator-metal contacts. As the two contact members are different, the voltage drop Vg(x) and the tunneling current density Jc(x) are no longer symmetric along the contact length L. Figure 9(a,b) show that the voltage drop increases and the tunneling current density decreases with increasing W2. Figure 9(c,d) show that the voltage drop increases and the tunneling current density reduces significantly when the relative permittivity of the insulating layer increases from 1 to 3.9. Analytical solutions obtained by assuming constant tunneling resistivity along the contact length are also included, similar to the previous cases of Fig. 7. In general, for the chosen value of D = 0.5 nm, the ohmic approximations using Eq. 11 do not yield accurate results. The constant tunneling resistivity approximation using Eq. 10 by setting Vg = V0 could be a good approximation for the self-consistent TLM model (Eqs 5, 6 and 10), for tunneling layers with higher permittivity εr.

Figure 10 shows the contact resistance (in Ω) for various contact metals and tunneling films for CNT-insulator-metal contacts. Contact resistance increases with insulating layer thickness D, insulating layer permittivity εr and work function of contacting member W2. It decreases with contact length L, as in the similar contacts in Fig. 8. The potential barrier in the insulating layer increases with the increase of work function of the contact metal, resulting in lower tunneling current and higher contact resistance.

## Summary

In this paper, we proposed a self-consistent model to characterize partially overlapped parallel contacts. Our model considers the spatial variation of contact resistivity along the contact structure. We solved the TLM equations for three cases: (1) constant specific contact resistivity, (2) linearly varying specific contact resistivity, and (3) spatial dependent specific contact resistivity along the contact length due to current tunneling. The analytical solutions for the first case have been verified with FEM based numerical simulations. Our study provides a thorough understanding of the contact tunneling resistance, current and voltage distributions across nano and sub-nano scale metal-insulator-metal (MIM) junctions in parallel electrical contacts. The effects of contact geometry (i.e. dimension of the contact, and distance between the contact electrodes), and material properties (i.e. work function, sheet resistance of the contact members, and permittivity of the insulating layer) on the spatial distributions of currents and voltages across these contacts, and the overall contact resistance are studied in detail. While predominately classical in nature, the inclusion of tunneling current starts to address quantum effects in these small scale objects.

It is found that in general the ohmic approximation of tunneling junctions (Eq. 11) is not reliable for predicting the contact resistance of parallel tunneling contacts. The one-dimensional (1D) tunneling junction models (Eq. 10 with constant voltage across the whole junction) are good approximations of the parallel contacts only when the thickness or the permittivity εr of the tunneling film is relatively large, or the applied voltage across the contact V0 is relatively small. When the 1D models become unreliable for small D or εr, or large V0, the self-consistent TLM equations coupled with the tunneling current (Eqs 5, 6 and 10) need to be used to accurately characterize the parallel tunneling contacts.

The parallel tunneling contact in this work may be considered as the basic building block to better understand the macroscopic electrical conductivity of CNT fibers, which contains a very large number of such parallel contacts between individual CNTs. Furthermore, our study elucidates key parameters for parallel electrical contacts over a wide range of spatially dependent contact resistivity, which paves the way to strategically design of contact structures with controlled current distribution profiles and contact resistance, by spatially varying the contact layer properties and geometry.

In this formulation, we have ignored the effects of space charge and exchange-correlation inside the tunneling gap11,40. We have also ignored possible charge trapping inside contact junctions. The model is assumed two-dimensional, where the effects of the transverse dimension are neglected. These issues will be the subjects of future studies. It is important to note that the transmission line model (TLM) is only a simplified approximation of the 2D electrical contacts, where the current crowding and the fringing fields near the contact corners cannot be fully accounted for10,49. In order to accurately evaluate these effects as well as the impact of finite thickness in the contact members and the contact junction, and the possible parallel component of current flows in the interface layer, field solution methods need to be used10,49,50,58. Future studies may also include the effects of various contact geometry, insulator layer non-uniformities and AC response on the electrical properties of tunneling type contacts. An additional feature might include the role of capacitance and inductance in nano- and micro-scale structures, especially when a large contact resistance coupled with these reactive effects might introduce new time scales into time-dependent dynamic problems.

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## Acknowledgements

The work is supported by the Air Force Office of Scientific Research (AFOSR) YIP Award No. FA9550-18-1-0061.

## Author information

Zhang and Luginsland conceived the problem. Zhang and Banerjee did the theoretical formulation. Banerjee did the numerical computation and data collection. All authors jointly wrote the paper.

Correspondence to Peng Zhang.

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### Competing Interests

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