4 Coulomb blockade

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86 4 Coulomb blockade J [arb. u.] V 0 2 4 6 8 V 4 2 0 -2 Fig. 4.11. Coulomb staircase. -4 -1 0 1 VG 2 3 Fig. 4.12. Contour plot of the differential conductance. above the Fermi-level (the electron charge is negative, and a negative potential means a positive energy shift). Nevertheless, existing stability patterns are of the same origin and form the same structure. The qualitatively new feature is additional lines correspondent to the additional discrete-level steps in the voltage-current curves.In general, the current and conductance of quantum dots demonstrate all typical features of discrete-level systems: current steps, conductance peaks. Without Coulomb interaction the usual picture of resonant tunneling is reproduced. In the limit of dense energy spectrum ∆ɛ → 0 the sharp single-level steps are merged into the smooth Coulomb staircase. 0.7 0
4.5 Cotunneling 4.5 Cotunneling 87 Until now we have considered only the sequential tunneling regime, which is described by the first-order perturbation in tunneling. At low temperatures the sequential tunneling is suppressed by the Coulomb blockade (excepting the degeneracy points), and the conductance of the system become exponentially small. In this case the second-order processes start to be important. These processes include the events, when two electrons with different energy participate in tunneling simultaneously (so-called inelastic cotunneling), or one electron tunnels coherently twice (elastic cotunneling), which is equivalent, however, to the simultaneous tunneling of two electrons with the same energy. Note, that both terms ”inelastic” and ”elastic” have no relation to the presence or absence of the additional inelastic interactions inside the system. The standard second-order perturbation theory yields the transition rate from some initial state |i〉 to some final state |f〉 Γ (2) 2π i→f = ¯h f ˆHT λ λ ˆHT 2 i Eλ − Ei δ(Ei − Ef ). (4.62) λ Theintermediatestates|λ〉 are virtual states with Eλ >Ei. Inthispointwe already assume that the system is in the Coulomb blockade regime with fixed number of electrons n, and the temperature is low T ≪ ∆E + n , because in the opposite case thermally excited quasiparticles in the leads can overcome the energy barrier, and the sequential tunneling through high-energy states will determine the current. Fig. 4.13. Cotunneling processes, elastic (a,c) and inelastic (b,d). (from Aleiner et al.).
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4 Coulomb blockade

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