Effects of temperature dependence of electrical and thermal conductivities on the Joule heating of a one dimensional conductor

Abstract

We examine the effects of temperature dependence of the electrical and thermal conductivities on Joule heating of a one-dimensional conductor by solving the coupled non-linear steady state electrical and thermal conduction equations. The spatial temperature distribution and the maximum temperature and its location within the conductor are evaluated for four cases: (i) constant electrical conductivity and linear temperature dependence of thermal conductivity, (ii) linear temperature dependence of both electrical and thermal conductivities, (iii) the Wiedemann-Franz relation for metals, and (iv) polynomial fits to measured data for carbon nanotube fibers and for copper. For (i) and (ii), it is found that there are conditions under which no steady state solution exists, which may indicate the possibility of thermal runaway. For (i), analytical solutions are constructed, from which explicit expressions for the parameter bounds for the existence of steady state solutions are obtained. The shifting of these bounds due to the introduction of linear temperature dependence of electrical conductivity (case (ii)) is studied numerically. These results may provide guidance in the design of circuits and devices in which the effects of coupled thermal and electrical conduction are important.

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

“…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 ̅ ( ̅ ).…”

“…An example of the procedure to solve Eqs. (5) and (6) numerically is as follows. For an initially guess on , Eq (5b) is solved using the shooting method, subject to Eq.…”

“…III Case 3), where analytical solutions to the TLM current and voltage equations are no longer available, and Eqs. (5) and (6) are solved numerically.…”

“…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 region [1][2][3][4][5][6] . The electrical contact properties have been extensively studied in metal-semiconductor [7][8][9] , metal-insulatorsemiconductor and metal-insulator-metal [10][11][12][13] junctions.…”

Field emission from carbon nanotube fibers in varying anode-cathode gap with the consideration of contact resistance

“…This kind of sudden drop is attributed to the self-heating effects, which can lead to a sudden failure to emit for some of the CNTs forming the fiber. 13,23 For d = 1 mm, the fiber failed to emit entirely, shortly after applying the maximum voltage of 1000 V. where A = 1.54 × 10 6 /W, B = 6.83 × 10 9 W 3/2 , S eff is the effective emission area (m 2 ) of the fiber, W is the work function (in eV) of the emitting surface, β eff is the effective field enhancement factor, D = h + d is the gap distance (in m) between the anode and the base of the CNT fiber cathode, which is the sum of the fiber length h and the fiber tip-to-anode distance d, and V g is the gap voltage. In the FN plot, Equation (1) yields a straight line when plotted as ln(I/V 2 g ) vs 1/V g , with the slope of BD/ β eff .…”