Nanostructured graphene for energy harvesting

Engineered nonlinearities have been shown to play an important role in increasing the efficiency of energy harvesting devices. Macroscopic prototypes using this approach have been demonstrated recently [F. Cottone, H. Vocca, and L. Gammaitoni, Phys. Rev. Lett. 102, 080601 (2009).] Here, in order to implement such a scheme at the nanoscale, we propose a simple device which is based on strained nanostructured graphene and discuss how it can respond to many energy sources that, although having a low intensity, are freely available, such as ambient vibrations or thermal noise. We discuss in some detail the case of thermal fluctuations harvesting in the steady-state nonequilibrium regime and of ambient vibrations.

Figure 1

Buckled ground-state configuration of a graphene sheet under compressive strain. The atoms of the primitive cell explicitly introduced in the calculation within periodic boundary conditions are shown in light yellow (light gray). Atoms of the computational cell that need to be constrained to sample transition states (at the apex and at the clamped ends) are displayed with a darker color (gray). The potential profile is obtained by performing a series of calculations of sinusoidal deformations with increasing amplitudes $h_i$, as shown schematically in the inset.

Figure 2

Potential as a function of the out-of-plane coordinate $h$ for compressions $\epsilon$ ranging from 0% to 0.1%. Finite values of $\epsilon$ favor buckling of the graphene sheet, with two symmetric minima at $h\ne 0$. The inset displays the separation between the minima and the transition barrier as a function of $\epsilon$, together with fits to ${\epsilon }^2$ and $\sqrt{\epsilon }$, respectively.

Figure 3

Evolution of the position $x$ in the three possible regimes: (i) At low applied compressions the system oscillates in a single-well potential at approximately $x=0$ (black circles); (ii) at the optimal compression a bistable behavior is clearly observed, with long swings from one well to the other (red diamonds); (iii) at large compressions the buckled graphene gets trapped in one of the minima (blue squares). The left-hand panel shows the time evolution $x=x\left(t\right)$; the right-hand panel is the attractor diagram $x=x\left(v\right)$.

Figure 4

Root mean square (rms) of the position vector $x$ (left-hand side) and mechanical power (right-hand side) as a function of the compressive strain $\epsilon$. The optimal compression that maximizes $x_{\text{rms}}$ is $\epsilon \sim 0.17$. This value of $\epsilon$ also maximizes the piezoelectrically generated voltage in the transduction circuit described in the text. The dashed line in the right-hand panel gives an estimation of the mechanical power accumulated by a linear oscillator of comparable size to the one discussed.

Figure 5

Spectral response in the three operating regimes. Zero or too large compressive strains yields rather selective frequency responses (black circles and blue squares, respectively). Around the optimal compression, on the other hand, the spectral response is much broader and extends significantly to the low-frequency domain (red diamonds). The spectral response is defined as the Fourier transform of the position vector as follows: $X\left(\omega \right)=\mathcal{F}\left\{x\left(t\right)\right\}=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{+\infty }x\left(t\right)e^{-i\omega t}dt$.