Master equations for quantum transport with Franck-Condon blockade

The Master equation framework for single electron transport is extended to include the Franck-Condon effect that can dominate the transport properties of nanoelectromechanical devices. The quantum nature of the resulting vibroelectronic states leaves a signature in the electronic shot noise, which is accounted for by careful treatment of the coherences. Two model systems are shown to demonstrate the relevance of such a formulation.

Figure 1

Schematic model of the considered system: a vibrating structure with a single electron quantum dot near a plane where an image charge effect occurs.

Figure 2

(Color online) Potential of the charged system for different parameters $\Omega ∊\left\{10,12,14,16,18,20\right\}$. Near the origin, the potential has a hydrogen-potential-like shape, while for larger $x$ values the shape of the harmonic potential dominates. With increasing $\Omega$, the harmonic character becomes more pronounced at small $x$ values, leading to a potential well, with minimum around $x=1$.

Figure 3

(Color online) Eigenvalue spectrum of the Hamiltonian (14) for $ϵ=12$ as a function of the oscillator parameter $\Omega$, showing the six lowest eigenvalues. For $\Omega \to 0$, the spectrum has the $1/n^2$ characteristic of the Hydrogen atom, while for large $\Omega$, the spectrum assumes the fanlike shape typical for harmonic spectra. At intermediate values of $\Omega$, we observe a level repulsion between the first and the second level.

Figure 4

(Color online) The first two Eigenfunctions of the Hamiltonian (14) for values of the oscillator parameter $\Omega$ before and after the level anticrossing. The two wave functions exchange their characteristics at this point, while keeping the properties of the first and second state.

Figure 5

(Color online) FC factors ${\tau }_{1,j}$ and ${\tau }_{2,j}$ as functions of the oscillator parameter $\Omega$, which modify the transition rates between the charged and the uncharged oscillator states. We note that they can assume negative values and even vanish for certain $\Omega$ values, thus blocking the corresponding transition.

Figure 6

(Color online) Frequency-dependent Fano factor $F\left(\omega \right)=S\left(\omega \right)/I$ for systems with finite bias $V_{\text{Bias}}={\mu }_L-{\mu }_R=40$ and asymmetric tunneling coupling ${\Gamma }_L=1/\left(2\pi \right)=2{\Gamma }_R$ and different oscillator parameters $\Omega =13$ (lowest line) to $\Omega =14.2$ (highest line), which is the interval in which the level anticrossing in the eigenvalue spectrum occurs $\left(\Omega \approx 13.6\right)$. The labels refer to the lowest line, all other lines have an offset of $0.05n$. The colorful lines are results obtained using the FC Master equation (Eq. (12)), while the black lines are the corresponding results from a calculation where all coherences are set to zero.

Figure 7

(Color online) Moduli of steady state solutions of the FC Master equation (12) for $ϵ=12$ and $\Omega =11$ (top) and $\Omega =15$ (bottom) with 20 different values of the damping parameter $\gamma$ in the interval (0,10]. The scale goes from blue (black) through red (gray) to yellow (light gray). The four indices of the matrix elements of the Master equation are mapped to the one index $k$ via $k={\delta }_{\nu ,1}\left[2\left(n-1\right)+m\right]+{\delta }_{\nu ,0}\left[4+10\left(n+1\right)+m\right]$, which means that the equally spaced peaks refer to the populations of the uncharged oscillator states and the peaks at $k=1$ and $k=4$ are the populations of the charged states. We observe that the coherences are all smaller than the populations and that with increasing damping the populations of the higher uncharged states vanish as well as their coherences. Some coherences, however, increase with increasing damping, leading to an increase in the coherences on average.

Figure 8

(Color online) The proposed model. The structure is suspended with contacts to a source (l) and drain (r), while it experiences, in its charged state $\left(-e\right)$, the images charges of two metallic STM tips $\left(e\right)$, which are symmetrically applied perpendicular to the transport direction $\left({\Gamma }_L,{\Gamma }_R\right)$.

Figure 9

(Color online) Potential of the Hamiltonian (22) for $\epsilon =16$ and various values for $\Omega$ between 1 and 2.5.

Figure 10

(Color online) Spectrum of the Hamiltonian (22) for the Coulomb parameter $\epsilon =16$. Low lying states are pairwise degenerate before they experience level anticrossing with the next higher lying state.

Figure 11

(Color online) The first two Eigenfunctions of the Hamiltonian (22) for values of the oscillator parameter $\Omega$ before and after the level anticrossing.

Figure 12

(Color online) FC factors ${\tau }_{|5,1⟩\leftrightarrow |1,0⟩}$, ${\tau }_{|6,1⟩\leftrightarrow |2,0⟩}$ and ${\tau }_{|7,1⟩\leftrightarrow |1,0⟩}$. Only these three FC factors are nonzero, because for the others the product of wave functions under the integral is antisymmetric and hence vanishes when integrated over a symmetric interval.

Figure 13

(Color online) Frequency-dependent Fano factors $F\left(\omega \right)$ for different values of the oscillator parameter $\Omega$. In the left panels, the transition $|1,0⟩\leftrightarrow |6,1⟩$ has been allowed by letting the corresponding FC factor ${\tau }_{|1,0⟩\leftrightarrow |6,1⟩}=0.1$. In the right panels, this transition is forbidden according to the correct treatment of the FC factors. The lines A and B indicate corresponding $\Omega$ values of the top and bottom panels, which can be seen as cuts through the two-dimensional plots.