Thermal dissipation and variability in electrical breakdown of carbon nanotube devices

We study high-field electrical breakdown and heat dissipation from carbon nanotube (CNT) devices on ${\text{SiO}}_2$ substrates. The thermal “footprint” of a CNT caused by van der Waals interactions with the substrate is revealed through molecular dynamics simulations. Experiments and modeling find the CNT-substrate thermal coupling scales proportionally with CNT diameter and inversely with ${\text{SiO}}_2$ surface roughness $\left(\sim d/\Delta \right)$. Comparison of diffuse mismatch modeling and data reveals the upper limit of thermal coupling $\sim 0.4\text{W}{\text{K}}^{-1}{\text{m}}^{-1}$ per unit CNT length at room temperature, ($130\text{MW}{\text{K}}^{-1}{\text{m}}^{-2}$ per unit area), and $\sim 0.7\text{W}{\text{K}}^{-1}{\text{m}}^{-1}$ at $600°\text{C}$ for the largest diameter ($\sim$3.2 nm) CNTs. We also find semiconducting CNTs can break down prematurely and display more variability due to dynamic shifts in threshold voltage, which metallic CNTs are immune to; this poses a fundamental challenge for selective electrical breakdowns in CNT electronics.

Figure 1

(Color online) (a) Schematic cross section of typical CNT device with diameter $d$ and thermal footprint $b_t$ (also see Fig. 5) on ${\text{SiO}}_2$ substrate with thickness $t_{ox}$ and surface roughness $\Delta$. The $p+\text{silicon}$ is used as a backgate. The device layout with source and drain terminals is shown in Fig. 2a. As current $\left(I_D\right)$ passes in the CNT, the generated Joule heat dissipates through the substrate. The equivalent thermal circuit includes the CNT-${\text{SiO}}_2$ interface thermal resistance $\left(1/g\right)$ and spreading resistance in the ${\text{SiO}}_2$ $\left(1/g_{ox}\right)$. (b) Typical electrical breakdowns of CNTs having similar dimensions shows higher breakdown power in vacuum $\left(\sim {10}^{-5}\text{torr}\right)$ than in ambient air. This illustrates the role of oxygen for CNT breakdown in air. (c) AFM images of CNTs broken in air (left) and vacuum (right). Due to higher power, breakdowns in vacuum can lead to ${\text{SiO}}_2$ surface damage, which is not observed for air breakdowns.

Figure 2

(Color online) (a) SEM image of CNT device showing breakdown location $\left(L_{BD}\right)$. (b) Histogram of breakdown location normalized by CNT length $\left(L_{BD}/L\right)$ indicating the majority of m-CNTs break near the middle and s-CNTs break closer to the grounded drain. (c) Computed temperature distribution at breakdown (maximum $\text{temperature}=T_{BD}$) along a $2\text{−}\mu \text{m}$-long CNT with Eq. (2) using $C_1=1$ and varying $C_2$. $C_2=0$ corresponds to m-CNTs (uniform heat dissipation) and $C_2>0$ corresponds to s-CNTs. For s-CNTs biased under hole conduction the heat generation and temperature profile are skewed toward the grounded drain terminal. Block arrows in (b) and (c) show direction of hole flow.

Figure 3

(Color online) The PDOS for a (10,10) nanotube from MD simulations (Ref. 16). The Bose-Einstein occupation $\left(f_{\text{BE}}\right)$ at room temperature is plotted in red against the right axis. Shaded in gray is the product of the PDOS with $f_{\text{BE}}$, showing diminished contribution from higher frequency phonon modes. The inset shows the PDOS of the CNT and that of ${\text{SiO}}_2$, the latter displaying a lower cutoff near 40 THz.

Figure 4

(Color online) van der Waals potential (blue solid line) interaction between CNT and ${\text{SiO}}_2$, as used in calculations to derive the thermal footprint (Fig. 5). The second derivative of the potential (red dashed line) with respect to distance from the surface $\left(z\right)$ represents a spring constant of the interatomic interaction. The square root of this spring constant is used to weigh the contribution of each atom to the effective thermal footprint $\left(b_t\right)$ of the CNT.

Figure 5

(Color online) (a) Nanotube height $H$ (solid red square), geometric footprint $b_g$ (△), and thermal footprint $b_t$ (▲) on the ${\text{SiO}}_2$ substrate as a function of CNT diameter from MD simulations. A fit to the thermal footprint is shown as a solid line from Eq. (13). (b) Calculations reveal two distinct regimes: in regime I $\left(d<2.1\text{nm}\right)$ the CNT shape is nearly circular, dominated by the curvature energy; in regime II $\left(d>2.1\text{nm}\right)$ the CNT shape becomes flattened with a stronger influence of the surface vdW interaction. Example CNTs from each regime are shown, (7,7) with $d=9.6\text{Å}$ for regime I, and (22,22) with $d=30.6\text{Å}$ for regime II. Small vertical arrows indicate the relative magnitude of vdW force coupling with the substrate at each atomic position. The ${\text{SiO}}_2$ surface is at $z=0$. The minimum CNT-${\text{SiO}}_2$ distance is $2.5\text{Å}$ at the middle of the (7,7) CNT and toward the sides of the (22,22) CNT. The middle of the (22,22) CNT is slightly “buckled” with $2.8\text{Å}$ separation from ${\text{SiO}}_2$ (also noted by the lower vdW force there).

Figure 6

(Color online) (a) Electrical breakdown power (in air) of CNTs vs diameter $d$, showing proportional scaling. (b) Extracted CNT-${\text{SiO}}_2$ thermal coupling $g$ vs $d$ (see text) for both m- and s-CNTs. Solid line is the DMM calculation and dashed-dotted lines are fitted to MD simulations with different vdW coupling strengths ($\chi =1$ and $\chi =2$, respectively, from Ref. 16). (c) CNT-${\text{SiO}}_2$ thermal coupling per unit area $h$ vs $d$, showing the DMM represents an upper-limit scenario of heat dissipation. The spread in the data and lower apparent thermal coupling in practice are attributed to ${\text{SiO}}_2$ surface roughness, and charge trapping near semiconducting CNTs (see text). (d) Calculated temperature dependence of the upper-limit thermal coupling per unit area. Thermal coupling at room temperature $\left(\sim 130\text{MW}{\text{K}}^{-1}{\text{m}}^{-2}\right)$ is $\sim 40\mathrm{\%}$ lower than at the breakdown temperature $\left(\sim 220\text{MW}{\text{K}}^{-1}{\text{m}}^{-2}\right)$.

Figure 7

(Color online) (a) CNT-${\text{SiO}}_2$ thermal coupling $g$ vs diameter $d$ $\left(\text{symbols}=\text{data}\right)$ and DMM simulations (lines) for perfect substrate contact (100%) and for 75% and 50% effective contact area due to ${\text{SiO}}_2$ surface roughness (also see Fig. 1). (b) Replot of same experimental data vs diameter scaled by rms surface roughness $\left(d/\Delta \right)$ measured by AFM near each CNT. This indicates the role of ${\text{SiO}}_2$ surface roughness for thermal dissipation from CNTs. Dashed lines are added to guide the eyes. (c) Breakdown power $P_{BD}$ for s-CNTs alone plotted with respect to their threshold voltage $\left(V_{TH}\right)$. The variance in $V_{TH}$ is also a contributing factor to the spread in extracted thermal coupling data for s-CNTs.