Molecular Model of a Quantum Dot Beyond the Constant Interaction Approximation

We present a physically intuitive model of molecular quantum dots beyond the constant interaction approximation. It accurately describes their charging behavior and allows the extraction of important molecular properties that are otherwise experimentally inaccessible. The model is applied to data recorded with a noncontact atomic force microscope on three different molecules that act as a quantum dot when attached to the microscope tip. The results are in excellent agreement with first-principles simulations.

Figure 1

Structure and charging of the single-molecule QD junction investigated in this work. (a) Equivalent circuit of the QD junction. $C_{\mathit{t}}$ and $C_{\mathit{s}}$ denote capacitances of the tip-molecule and molecule-surface junctions. (b) Experimental realization of the QD junction with $\mathrm{PTCDA}\text{−}{\mathrm{Cl}}_4$. Electron tunneling occurs through the tip-molecule junction, characterized by the distance $d$. $z$ is the total tip-surface distance. (c) $\mathrm{\Delta }f(V_b)$ spectrum recorded at $z=30\AA$. Two charging events are observed: At $V_{-}^{*}$, one electron is withdrawn from the QD, while at $V_{+}^{*}$, one electron is placed into it. $V_{\mathrm{cpd}}$ is the contact potential difference [12]. The inset shows examples of two experimentally measured $\mathrm{\Delta }f(z)$ curves used to calculate $\mathrm{\Delta }{F}_{-}(z=42.2\AA )$ (left curve, $V_b=-1.57\mathrm{V}$) and $\mathrm{\Delta }{F}_{+}(z=42.9\AA )$ (right curve, $V_b=4.51\mathrm{V}$) by integration (shaded area). (d) Energy diagram of the QD. Since only two charging voltages ($V_{\pm }^{*}$) are observed, two QD levels can be charged. These levels are shown in red as ${ϵ}_{\mathit{a}}$ and ${ϵ}_{\mathit{b}}$. When a bias voltage $V$ is applied to the junction, both levels are gated by $\alpha eV$ with respect to $E_{\mathit{F}}$ of the left electrode (tip). $V_{-}^{*}$ aligns ${ϵ}_{\mathit{a}}$ with $E_{\mathit{F}}$ of the tip, and one electron is transferred from ${ϵ}_{\mathit{a}}$ to the tip ($n_a=n_b=0$ for $V_b<V_{-}^{*}$). $V_{+}^{*}$ aligns ${ϵ}_{\mathit{b}}$ with $E_{\mathit{F}}$ of the tip, and one electron is transferred from the tip to ${ϵ}_{\mathit{b}}$ ($n_a=n_b=1$ for $V_b>V_{+}^{*}$). For $V_{-}^{*}<V_b<V_{+}^{*}$, as shown in the Figure, $n_a=1$, $n_b=0$.

Figure 2

Experimental characterization of QD junctions with PTCDA (black), $\mathrm{PTCDA}\text{−}{\mathrm{Cl}}_4$ (red), and TTCDA (green). (a) $z$ dependence of the charging voltages $V_{-}$ (circles) and $V_{+}$ (triangles). Chemical structures of 3,4,9,10-perylene tetracarboxylic acid dianhydride (PTCDA), 1,6,7,12-tetrachloroperylene-3,4,9,10-tetracarboxylic dianhydride ($\mathrm{PTCDA}\text{−}{\mathrm{Cl}}_4$), and 3,4,11,12- terrylene-tetracarboxylic dianhydride (TTCDA) are shown in the inset. (b) $z$ dependence of $\mathrm{\Delta }{F}_{-}$ (circles) and $\mathrm{\Delta }{F}_{+}$ (triangles).

Figure 3

Parameters of molecular QDs evaluated from the experiment. Color coding as in Fig. 2. (a) Gating efficiency $\alpha$. (b) ${ϵ}_a$ (circles) and ${ϵ}_b$ (squares). (c) Polarizability change $\mathrm{\Delta }\gamma$. The noise in the data arises from the numerical differentiation ${\partial }_z{V}_{\pm }$ in Eqs. (5)–(8).