Single-molecule devices: materials, structures and characteristics

Abstract

This review article provides a brief survey of materials, structures and current state-of-the-art techniques used to measure the charge conduction characteristics of single molecules. Single molecules have been found to exhibit several unique functionalities including rectification, negative differential resistance and electrical bistable switching, all of which are necessary building blocks for the development and configuration of molecular devices into circuits. Conjugated organic molecules have received considerable interest for their low fabrication cost, three dimensional stacking and mechanical flexibility. Furthermore, the ability of molecules to self-assemble into well-defined structures is imperative for the fabrication of molecule based circuits. The theoretical formalisms are presented for studying single-molecule Coulomb blockade effects, ballistic transport in a molecular chain and electromagnetic coupling between a surface-plasmon field and a single molecule. Moreover, the experimental current–voltage results are discussed using basic principles of carrier transport mechanisms.

Appendices

Appendix 1: Coupling of a quantum dot to contacts

If we include the coupling $\Gamma _c$ of a quantum dot to two electrodes, Eq. (1) will be modified to [112]

$$\begin{aligned} \mu _{N+1}=E_{N+1}+\frac{(N+1/2)e^2}{C}+\Gamma _c\,{\mathcal N}_L(\mu _0)\ . \end{aligned}$$

(10)

Here, C represents the capacitance of a quantum dot and ${\mathcal N}_L(\mu _0)$ is the average linear density of electrons in one-dimensional (1D) electrodes. In addition, $\{\mu _{N+1}\}$ in Eq. (10) is the resonant levels, which connect to $\mu _0$ (electrode chemical potential) only through $\mu _0$ dependence in ${\mathcal N}_L(\mu _0)$. Consequently, from Eq. (10) we find

$$ \frac{\partial \mu _{N+1}(\mu _0)}{\partial \mu_0}=\Gamma_c \, \frac{\partial{\mathcal{N}}_L(\mu_0)}{\partial \mu_0} \equiv \Gamma_c\,{\mathcal{G}}_L(\mu _0), $$

(11)

where ${\mathcal G}_L(\mu _0)$ is the density of states of electrons in 1D electrodes. After we take the electron subband edge as an zero-energy point, the gate potential $eV_g$ becomes just the difference $\delta \mu _0$ between the fixed substrate chemical potential and $\mu _0$. In this way, we are able to simply write down $\delta \mu _0=\gamma e\delta V_g$ with $\gamma <1$ being a ratio parameter.

Denoting $\nu _N$ as the resonant electrode chemical potential with the quantum-dot energy level $\mu _N$, we acquire the self-consistent equation $\nu _N=\mu _N(\nu _N)$. As a result, the period of $\mu _0$ in the conductance oscillations can simply be written as

$$\begin{aligned} \delta \mu _{0} & = \nu _{{N + 1}} - \nu _{N} = \Delta E + \frac{{e^{2} }}{C} + \Gamma _{{\text{c}}} \left[ {{\mathcal{N}}_{{\text{L}}} (\nu _{{N + 1}} ) - {\mathcal{N}}_{{\text{L}}} (\nu _{N} )} \right] \\ & = \Delta E + \frac{{e^{2} }}{C} + \Gamma _{c} {\mkern 1mu} {\mathcal{G}}_{L} (\mu _{0} )\delta \mu _{0} . \\ \end{aligned}$$

(12)

Equation (12) can also be rewritten as

$$\delta \mu _{0} = \frac{{\Delta E + e^{2} /C}}{{1 - \Gamma _{c} {\mkern 1mu} {\mathcal{G}}_{L} (\mu _{0} )}} \equiv \frac{{\Delta E + e^{2} /C}}{{{\mathcal{K}}(\mu _{0} )}} = \gamma e\delta V_{g} .$$

(13)

Here, ${\mathcal K}(\mu _0)=1-\Gamma _c\,{\mathcal G}_L(\mu _0)$ plays the role of a screening factor to the charge addition energy due to coupling to 1D electrodes. Furthermore, $\Gamma _c$ is found to be

$$\begin{aligned} \Gamma _c\sim \frac{e^2}{4\pi \epsilon _0\epsilon _bL}\left[ \frac{3}{2}+\ln \left( \frac{L}{W}\right) \right] \equiv \frac{e^2}{C}\, \end{aligned}$$

(14)

where L (W) is the gate length (chain width), and $\epsilon _b$ is the quantum-dot dielectric constant. It is clear from Eq. (14) that C is the bare dot capacitance, which is roughly proportional to L and related to W weakly.

If we consider a non-interacting 1D electron gas, its density of states per unit length is

$$\begin{aligned} {\mathcal G}_L(\mu _0)=\frac{\sqrt{2m^*}}{\pi \hbar }\,\sum \limits _{n=1}^{\mathrm{occupied}}\,\frac{1}{\sqrt{\mu _0-\varepsilon _n}}\ , \end{aligned}$$

(15)

where $m^*$ is the electron effective mass and $\varepsilon _n$ denotes the nth subband edge. Due to the fact that $\delta {\mathcal G}_L(\mu _0)/{\mathcal G}_L(\mu _0)=\delta V_g/[2(V_g-V_0)]$, we know that ${\mathcal G}_L(\mu _0)$ becomes a constant for $\delta \mu _0\ll V_g-V_0$ with $V_0$ being the threshold gate voltage.

Appendix 2: Charge and heat currents for symmetric electron subbands

By assuming symmetric subbands $\varepsilon _{n,k}=\varepsilon _{n,-k}$ for electrons with respect to $k=0$, where n is the subband index and k is the wave number of electrons along a chain, then we obtain an anti-symmetric relation $v_{n,-k}=-v_{nk}$ for electron group velocity. In this case, Eq. (5) leads to

$$\begin{aligned} {\mathcal J}^{(j)}&= {} \frac{2(-e)^{1-j}}{\pi }\sum _{n}\int _0^\infty dk\,|v_{nk}|\nonumber \\&\times (\varepsilon _{nk}-{\bar{\mu }})^j \left[ f_0(\varepsilon _{nk}-\mu _L)-f_0(\varepsilon _{nk}-\mu _R)\right] \ , \end{aligned}$$

(16)

where $f_0(x)$ is the Fermi function for electrons in a thermal-equilibrium state. Because $\mu _L+eU_b=\mu _R\equiv \mu$, we have $f_0(\varepsilon _{nk}-\mu _L)-f_0(\varepsilon _{nk}-\mu _R)=eU_b[\partial f_0(\varepsilon _{nk}-\mu )/\partial \varepsilon _{nk}]$ in the limit of $U_b\rightarrow 0$, and find ${\bar{\mu }}\rightarrow \mu =\mu _L=\mu _R$ simultaneously. Here, $U_b$ represents an infinitesimal voltage difference between the left and the right electrodes connecting to a conduction chain in the middle and ${\bar{\mu }}$ is the chemical potential of the chain. For such a situation, we find from Eq. (16)

$$\begin{aligned} {\mathcal J}^{(j)}&= {} \frac{2eU_b(-e)^{1-j}}{\pi }\sum _{n}\left(\int _{\varepsilon _{n,k=0}}^{\varepsilon _{n,k_1}}+ \int _{\varepsilon _{n,k_1}}^{\varepsilon _{n,k_2}}+\cdots + \int _{\varepsilon _{n,k^*}}^\infty \right)\nonumber \\&\times \mathrm{sign}(v_{nk})(\varepsilon _{nk}-\mu )^j\, \left[ \frac{\partial f_0(\varepsilon _{nk}-\mu )}{\partial \varepsilon _{nk}}\right] \,d\varepsilon _{nk}\ . \end{aligned}$$

(17)

Here, the energy integration over the range $0<k<\infty$ in Eq. (17) has been chopped into the sum of a series of sub-integrations between the successive extremum points $\varepsilon _{n,k_m}$ to guarantee that $\varepsilon _{n,k}$ is a monotonic function of k in these sub-ranges. In addition, $\varepsilon _{n,k^*}$ in Eq. (17) represents the last extremum (minimum) point. Each integration can be calculated analytically for both cases with $j=0$ and $j=1$, respectively.

Appendix 3: Coupling of a quantum dot to surface plasmons

In the presence of a light illumination, the semiconductor Bloch equations [45] for photo-excited electrons in a quantum dot are written as

$$\begin{aligned} \frac{dn^\mathrm{e}_\ell }{dt}&= {} \frac{2}{\hbar }\,\sum \limits _j\,\mathrm{Im}\left[ \left( p_\ell ^j\right) ^*\left( {\mathcal D}^\mathrm{eh}_{\ell ,j}-p_\ell ^j\,U^\mathrm{eh}_{\ell ,j;j,\ell }\right) \right] \nonumber \\&+\left. \frac{\partial n ^\mathrm{e}_\ell }{\partial t}\right| _\mathrm{rel}-\delta _{\ell ,1}\,{\mathcal R}_\mathrm{sp}\,n_1^\mathrm{e}\,n_1^\mathrm{h}\ , \end{aligned}$$

(18)

where $\ell =1,\,2,\,\ldots$ labels the energy levels, $n^\mathrm{e}_\ell$ represents the level occupation, and ${\mathcal R}_\mathrm{sp}$ denotes the spontaneous emission rate. We have introduced in Eq. (18) the non-radiative energy relaxation [113] shown as the term marked ‘rel’. The notations for the other terms, $p_\ell ^j$, ${\mathcal D}^\mathrm{eh}_{\ell ,j}$ and $U^\mathrm{eh}_{\ell ,j;j,\ell }$, will be given bellow in this part. In a parallel way, the semiconductor Bloch equations for holes in a quantum dot are derived as

$$\begin{aligned} \frac{dn^\mathrm{h}_j}{dt}&= {} \frac{2}{\hbar }\,\sum \limits _\ell \,\mathrm{Im}\left[ \left( p_\ell ^j\right) ^*\left( {\mathcal D}^\mathrm{eh}_{\ell ,j}-p_\ell ^j\,U^\mathrm{eh}_{\ell ,j;j,\ell }\right) \right] \nonumber \\&+\left. \frac{\partial n^\mathrm{h}_j}{\partial t}\right| _\mathrm{rel}-\delta _{j,1}\,{\mathcal R}_\mathrm{sp}\,n_1^\mathrm{e}\,n_1^\mathrm{h}\ , \end{aligned}$$

(19)

where $j=1,\,2,\,\ldots$ labels the hole energy levels and $n^\mathrm{h}_j$ stands for the level occupation. Again, we have included the non-radiative energy relaxation in Eq. (19). The optical coherence introduced in Eqs. (18) and (19), on the other hand, is found to satisfy

$$\begin{aligned} i\hbar {\mkern 1mu} \frac{d}{{dt}}p_{\ell }^{j} & = {\text{ }}\left[ {\bar{\varepsilon }_{\ell }^{{\text{e}}} (\omega ) + \bar{\varepsilon }_{j}^{{\text{h}}} (\omega ) - \hbar (\omega + i\gamma _{0} )} \right]p_{\ell }^{j} \\ & \quad + \left( {1 - n_{\ell }^{{\text{e}}} - n_{j}^{{\text{h}}} } \right)\left( {\rm{\mathcal{D}}_{{\ell ,j}}^{{{\text{eh}}}} - p_{\ell }^{j} {\mkern 1mu} U_{{\ell ,j;j,\ell }}^{{{\text{eh}}}} } \right) \\ & \quad + p_{\ell }^{j} \left[ {\sum\limits_{{j_{1} }} {n_{{j_{1} }}^{{\text{h}}} {\mkern 1mu} \times \left( {U_{{j,j_{1} ;j_{1} ,j}}^{{{\text{hh}}}} - U_{{j,j_{1} ;j,j_{1} }}^{{{\text{hh}}}} } \right) - \sum\limits_{{\ell _{1} }} {\mkern 1mu} n_{{\ell _{1} }}^{{\text{e}}} {\mkern 1mu} U_{{\ell _{1} ,j;j,\ell _{1} }}^{{{\text{eh}}}} } } \right] \\ & \quad + p_{\ell }^{j} \left[ {\sum\limits_{{\ell _{1} }} {n_{{\ell _{1} }}^{{\text{e}}} {\mkern 1mu} \times \left( {U_{{\ell ,\ell _{1} ;\ell _{1} ,\ell }}^{{{\text{ee}}}} - U_{{\ell ,\ell _{1} ;\ell ,\ell _{1} }}^{{{\text{ee}}}} } \right) - \sum\limits_{{j_{1} }} {\mkern 1mu} n_{{j_{1} }}^{{\text{h}}} {\mkern 1mu} U_{{\ell ,j_{1} ;j_{1} ,\ell }}^{{{\text{eh}}}} } } \right], \\ \end{aligned}$$

(20)

where $\hbar \gamma _0=\hbar \gamma _\mathrm{eh}+\hbar \gamma _{ext}$ corresponds to the total energy-level broadening from finite carrier lifetime and loss of an external field with angular frequency $\omega$. Moreover, $\overline{\varepsilon }^\mathrm{e}_\ell (\omega )$ and $\overline{\varepsilon }^\mathrm{h}_j(\omega )$ in Eq. (20) denote the dressed-state kinetic energies for electrons and holes. [45] It can be seen from Eq. (20) that the diagonal dephasing ($\gamma _0$), the renormalization of interband Rabi coupling, the renormalization of electron and hole energies, as well as the exciton binding energy, are all included.

For a steady-state solution presented in Eq. (8), we calculated the renormalized interband energy-level separation as

$$\begin{aligned} {\mathcal W}^\mathrm{eh}_{\ell ,j}(\omega \vert t)&= {} \overline{\varepsilon }^\mathrm{e}_\ell (\omega \vert t)+\overline{\varepsilon }^\mathrm{h}_j(\omega \vert t)-U^{eh}_{\ell ,j;j,\ell }\nonumber \\&+\sum \limits _{\ell _1}\,n^\mathrm{e}_{\ell _1}(t)\left( U^\mathrm{ee}_{\ell ,\ell _1;\ell _1,\ell }-U^\mathrm{ee}_{\ell ,\ell _1;\ell ,\ell _1}\right) \nonumber \\&+\sum \limits _{j_1}\,n^\mathrm{h}_{j_1}(t)\left( U^\mathrm{hh}_{j,j_1;j_1,j}-U^\mathrm{hh}_{j,j_1;j,j_1}\right) \nonumber \\&-\sum \limits _{\ell _1\ne \ell }\,n^\mathrm{e}_{\ell _1}(t)\,U^\mathrm{eh}_{\ell _1,j;j,\ell _1}-\sum \limits _{j_1\ne j}\,n^\mathrm{h}_{j_1}(t)\,U^\mathrm{eh}_{\ell ,j_1;j_1,\ell }\ . \end{aligned}$$

(21)

Here, in Eqs. (18), (19), (20) and (21) we have introduced the Coulomb interaction matrix elements, defined by

$$\begin{aligned} U^\mathrm{ee}_{\ell _1,\ell _2;\ell _3,\ell _4}&= {} U_0\int \frac{d^2\mathbf{q}_\Vert }{q_\Vert }\,{\mathcal Q}^\mathrm{e}_{\ell _1,\ell _4}(\mathbf{q}_\Vert )\,{\mathcal Q}^\mathrm{e}_{\ell _2,\ell _3}(-\mathbf{q}_\Vert )\ , \end{aligned}$$

(22)

$$\begin{aligned} U^\mathrm{hh}_{j_1,j_2;j_3,j_4}&= {} U_0\int \frac{d^2\mathbf{q}_\Vert }{q_\Vert }\,{\mathcal Q}^\mathrm{h}_{j_1,j_4}(\mathbf{q}_\Vert )\,{\mathcal Q}^\mathrm{h}_{j_2,j_3}(-\mathbf{q}_\Vert )\ , \end{aligned}$$

(23)

$$\begin{aligned} U^\mathrm{eh}_{\ell ,j;j^\prime ,\ell ^\prime }&= {} U_0\int \frac{d^2\mathbf{q}_\Vert }{q_\Vert }\,{\mathcal Q}^\mathrm{e}_{\ell ,\ell ^\prime }(\mathbf{q}_\Vert )\,{\mathcal Q}^\mathrm{h}_{j,j^\prime }(-\mathbf{q}_\Vert )\ , \end{aligned}$$

(24)

where $U_0=e^2/8\pi ^2\epsilon _0\epsilon _b$, Furthermore, two dimensionless form factors, ${\mathcal Q}_{\ell ,\ell ^\prime }^\mathrm{e}(\mathbf{q_\Vert })$ and ${\mathcal Q}_{j,j^\prime }^\mathrm{h}(\mathbf{q}_\Vert )$ have been employed in Eqs. (22)–(24) due to confinement by a quantum dot. On the other hand, in Eqs. (18), (19) and (20), the Rabi coupling matrix elements to an electromagnetic field $\displaystyle {{\varvec{E}}}(\mathbf{r};\,t)=\Theta (t)\,{\varvec{E}}(\mathbf{r};\,\omega )\,e^{-i\omega t}$ are found to be

$$\begin{aligned} {\mathcal D}^\mathrm{eh}_{\ell ,j}(t)=-\delta _{\ell ,1}\,\delta _{j,1}\,\Theta (t)\,\left[ {\varvec{E}}^\mathrm{eh}_{\ell ,j}(\omega )\cdot \mathbf{d}_\mathrm{c,v}\right] \ , \end{aligned}$$

(25)

where $\Theta (x)$ is a unit-step function, $\mathbf{d}_\mathrm{c,v}$ is interband dipole moment, and the effective electric field in Eq. (25) is defined as

$$\begin{aligned} {{\varvec{E}}}^\mathrm{eh}_{\ell ,j}(\omega )=\int d^3\mathbf{r}\left[ \phi ^\mathrm{e}_\ell (\mathbf{r})\right] ^*{\varvec{E}}(\mathbf{r};\,\omega )\left[ \phi ^\mathrm{h}_j(\mathbf{r})\right] ^*\ . \end{aligned}$$

(26)

Here, $\phi ^\mathrm{e}_\ell (\mathbf{r})$ [$\phi ^\mathrm{h}_j(\mathbf{r})$] in (26) corresponds to the electron (hole) wave functions, respectively.

Under the steady-state condition, the interband optical polarization ${{\varvec{\mathcal P}}}^\mathrm{loc}(\mathbf{r};\,\omega )$, related to the optical coherence in the quantum dot, is derived as [114]

$$\begin{aligned} {\varvec{\mathcal{P}}}^{{{\text{loc}}}} ({\mathbf{r}};\omega ) & = 2\left| {{\mathcal{F}}_{0} ({\mathbf{r}})} \right|^{2} {\mathbf{d}}_{{{\text{c}},{\text{v}}}} \left\{ {\int {d^{3} } {\mathbf{r}}^{\prime } {\mkern 1mu} \phi _{1}^{{\text{e}}} ({\mathbf{r}}^{\prime } ){\mkern 1mu} \phi _{1}^{{\text{h}}} ({\mathbf{r}}^{\prime } )} \right\} \\ &\quad \times {\mkern 1mu} \frac{1}{\hbar }{\mkern 1mu} \mathop {\lim }\limits_{{t \to \infty }} \left[ {\frac{{1 - n_{1}^{{\text{e}}} (t) - n_{1}^{{\text{h}}} (t)}}{{\omega + i\gamma _{0} - {\mathcal{W}}_{{1,1}}^{{{\text{eh}}}} (\omega |t)}}} \right]{\mathcal{D}}_{{1,1}}^{{{\text{eh}}}} (t), \\ \end{aligned}$$

(27)

where the profile function $|{\mathcal F}_0(\mathbf{r})|^2$ comes from the confinement of a quantum dot. This optical polarization plays the role of a source term in Maxwell equations for self-consistent total field.