Transport in metallic multi-island Coulomb blockade systems: A systematic perturbative expansion in the junction transparency

We study electronic transport through metallic multi-island Coulomb-blockade systems. Based on a diagrammatic real-time approach, we develop a computer algorithm that generates and calculates all transport contributions up to second order in the tunnel-coupling strengths for arbitrary multi-island systems. This comprises sequential and cotunneling, as well as terms corresponding to a renormalization of charging energies and tunneling conductances. Multi-island cotunneling processes with energy transfer between different islands are taken into account. To illustrate our approach we analyze the current through an island in Coulomb blockade, that is electrostatically coupled to a second island through which a large current is flowing. In this regime both cotunneling processes involving one island only as well as multi-island processes are important. The latter can be understood as photon-assisted sequential tunneling in the blockaded island, where the photons are provided by potential fluctuations due to sequential tunneling in the second island. We compare results of our approach to a $P\left(E\right)$ theory for photon-assisted tunneling in the weak coupling limit.

Figure 1

An example of a multi-island geometry. The system consists of $M=5$ islands with $I=4$ island-island and $J=4$ island-lead junctions. Any junction, e.g., $i=1$, in between a pair ($m_1=1$, $m_1^{\prime }=2$) of islands is characterized by a transparency ${\alpha }_0^1$ [see Eq. (11) below for definition] and a capacitance $C_{m=1,m^{\prime }=2}$. Only some stray capacitances are shown in the sketch. The state of the system is given by the excess charges on the islands: $\mid \chi ⟩=\mid -3,1,2,-3,2⟩$.

Figure 2

Diagram describing the time evolution of the islands’ density matrix with sequential tunneling, cotunneling (two tunneling lines at a time), and higher order processes (from left to right). The diagram is reducible, i.e., it can be cut at a certain time without cutting a tunneling contraction line. Summing irreducible parts (shaded) yields the self-energy. In the left-most part a tunneling process through junction $i$ in between a pair of islands $(m_i,m_i^{\prime })$ changes the state of the island on the upper branch from $\mid {\chi }_0⟩=\mid n_1,n_2,\dots ,n_{m_i},\dots ,n_{m_i^{\prime }},\dots ,n_M⟩$ to $\mid {\chi }_1⟩=\mid n_1,n_2,\dots ,n_{m_i}-1,\dots ,n_{m_i^{\prime }}+1,\dots ,n_M⟩$.

Figure 3

Two capacitively coupled SETs. The upper one is the noise generator, and the lower the quantum noise detector.

Figure 4

Sketch of functions relevant to calculation of the photon-assisted tunneling rate [Eq. (22)]. Without energy exchange with the environment [$P\left(\epsilon \right)$ is $\delta$ peaked, dotted line] the photon-assisted rate is given by the rate function ${\alpha }^{-}$ (broken line) at the effective energy gap ${\Delta }_{\pm }$. Coupling to the noise generator SET changes $P\left(\epsilon \right)$ (solid line) and the rate is given by the integration in Eq. (26). The inset shows the asymmetric charge noise of an SET as calculated in Ref. 52.

Figure 5

Results of the diagrammatic technique for the generator/detector setup (solid lines) compared to standard cotunneling (dotted lines) and photon assisted current with slope as predicted from $P\left(E\right)$ theory (dashed lines, $x$ intercepts fitted). All curves at $T=25\mathrm{mK}$, $e{V}_D^{\mathrm{sd}}=0.5k_B\mathrm{K}$, and $n_{x,\mathrm{eff}}^G(n_D=0)=0.5$ with ${\alpha }_0^j=0.01$ for each junction; $E_G=10k_B\mathrm{K}$, $E_D=10k_B\mathrm{K}$ for $A–C$ and $5k_B\mathrm{K}$ for $D$, $E_{\mathrm{int}}=0.5k_B\mathrm{K}$ for $A–C$ and $1k_B\mathrm{K}$ for $D$. $n_x^D$ as follows: $A$, 0.3; $B$, 0.4; $C$, 0.45; $D$, 0.4.

Figure 6

Two topologically distinct types of diagrams are found. This yields four different integral terms, to be evaluated.

Figure 7

Sketch of ${\alpha }_L^{+}\left(\omega \right)$ (broken line) and $\mathrm{Im}\Pi \left(\omega \right)$ (solid line) in the integrand of Eq. (B4). Perturbative expansion works, if the rate ${\alpha }_L^{+}$ is approximately linear in the region given by the peak in $\mathrm{Im}\Pi \left(\omega \right)$, corresponding to $\mid \sigma \left(\Delta \right)\mid <k_{B}T$ on resonance (a) or to $\mid \Delta -{\mu }_L\mid >\mid \sigma \left(\Delta \right)\mid$ in (c) and (d). The expansion fails (b) for $\mathrm{Im}\Pi \left(\omega \right)$ incorporating the kink in ${\alpha }_L^{+}$.