FIG. S4: Phase diagram for the driving period P and the thermal gradient | dT/dx | for (10,10)@(15,15) CNT-based oscillator with inner tube lengths of (a) 5 nm, (b) 10 nm, and (c) 20 nm (outer tube length of 100 nm). Blue region indicates the combination of period and thermal gradient which leads to a limited motion of the inner tube. Brown region indicates combinations where the inner tube is ejected from the outer tube. of Eq. (1) for different values of the thermal gradient and driving period P . These results are only estimations since we used parameters for the case of inner tube length of 5 nm and thermal gradient of 7 K/nm ( F$_{vdW}$ = 1 . 92 nN, γ = 1 . 48 pN ps $^{2}$/ ˚ A $^{2}$, and α = 8 . 08 pN nm/K) to all other thermal gradient cases. Figure S4 shows a phase diagram which indicates, for which combinations of the driving period and the thermal gradient | dT/dx | , the inner tube motion is limited. As one can see, the phase diagram exhibits a complicated structure. The longer the inner tube, the higher the thermal gradient necessary to completely eject the inner tube. The overall behavior of the oscillator will also depend upon the outer tube length. The inner tube can reach a steady velocity state if the outer tube is long enough. For example, for the thermal gradient of 7 K/nm and an inner tube length of 5 nm, steady velocity state was observed for an outer tube length of 1 µ m, as shown in Figure S5. For this case, the behavior of the device will be different as represented in the bifurcation diagram of Figure S6. The inner tube can now perform periodic motions only at one side of the outer tube. This is because the inner tube is not able to return to the middle of the outer tube for such large outer tube length. Furthermore, such large outer tube lengths may cause additional issues such as off-axis vibrations (if not clamped) which may make the device not viable. Modifying the thermal pulse (form, periods, etc) might provide a way to allow periodic 15