Strong Electromechanical Coupling of an Atomic Force Microscope Cantilever to a Quantum Dot

We present theoretical and experimental results on the mechanical damping of an atomic force microscope cantilever strongly coupled to a self-assembled InAs quantum dot. When the cantilever oscillation amplitude is large, its motion dominates the charge dynamics of the dot which in turn leads to nonlinear, amplitude-dependent damping of the cantilever. We observe highly asymmetric line shapes of Coulomb blockade peaks in the damping that reflect the degeneracy of energy levels on the dot. Furthermore, we predict that excited state spectroscopy is possible by studying the damping versus oscillation amplitude, in analogy with varying the amplitude of an ac gate voltage.

Figure 1

(a) Schematic of the setup. (b) Equivalent circuit diagram, where $C_{\mathrm{tip}}$ depends on $x$.

Figure 2

(a) First damping peak calculated from simulation (dots) and semianalytic theory (solid lines). The green dashed line is from linear response. $\mathcal{N}$ is plotted in reverse for consistency with experiment. (b) Adiabatic approximation (dash-dotted), semianalytic theory (solid), and simulation (dots) for oscillation amplitude $a=1\mathrm{nm}$. (c)–(e) Average dot charge versus time (solid) for $a=1\mathrm{nm}$, at voltages marked in (a). Cantilever position is also shown (thin dashed) as a reference. We took $2E_C=31\mathrm{meV}$, ${\omega }_0/\Gamma =1$, and $A=10\mathrm{meV}/\mathrm{nm}$. Other parameters were taken from the experiment (see setup).

Figure 3

(a) Experiment (solid) and theory (dashed) for the first damping peak. We converted $e{V}_B$ to $E$ using $\alpha =0.04$ extracted at weak coupling. One fit value $A=7.8\mathrm{meV}/\mathrm{nm}$ produced the theory curves for all three amplitudes. (b) Second peak with $A=9.2\mathrm{meV}/\mathrm{nm}$. (c) Measured damping over the $p$ shell; theory shown in (d) with $A=11\mathrm{meV}/\mathrm{nm}$. Other parameters were taken from experiment as described in the text.

Figure 4

Differential damping versus applied voltage (converted to $E$), and oscillation amplitude $a$. For large amplitudes, a peak appears on the line $E=Aa-{\Delta }_{\mathrm{sp}}$ (white dashed line). Inset: cut along black dashed line. Parameters are the same as in Fig. 2.