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arXiv:cond-mat/0411022 · Journal of Statistical Mechanics Theory and Experiment

Effective charging energy of the single-electron box

Philipp Werner, Matthias Troyer

Published Jan 20, 2005• 6 citations
arXivHTMLPDF

Abstract

We present numerical results on electron tunneling in a single-electron box at low temperature. The effective action of this device is equivalent to the Hamiltonian of a classical XY spin chain with long ranged interactions. Using an efficient cluster algorithm and a new transition matrix Monte Carlo approach, we are able to compute the effective charging energy $E_C^*$ in the limit of very small tunneling resistance. While previous Monte Carlo simulations were restricted to the weak and intermediate tunneling regimes, our method extends the range of $E_C^*$-values by more than 30 orders of magnitude. This allows us to clearly observe the exponential suppression of $E_C^*$ with increasing tunneling conductance $α$. For large, but fixed $α$, the correction to the leading exponential behavior exhibits a crossover from an intermediate temperature behavior at $βE_C^* \ll 1$, to zero temperature behavior at $βE_C^*\gg 1$. We determine this correction in both regimes and compare the numerical results to the numerous and controversial theoretical predictions for the strong tunneling limit.

DOI
10.1088/1742-5468/2005/01/p01003

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We present numerical results on electron tunneling in a single-electron box at low temperature. The effective action of this device is equivalent to the Hamiltonian of a classical XY spin chain with long ranged interactions. Using an efficient cluster algorithm and a new transition matrix Monte Carlo approach, we are able to compute the effective charging energy EC∗E_C^*EC∗​ in the limit of very small tunneling resistance. While previous Monte Carlo simulations were restricted to the weak and intermediate tunneling regimes, our method extends the range of EC∗E_C^*EC∗​-values by more than 30 orders of magnitude. This allows us to clearly observe the exponential suppression of EC∗E_C^*EC∗​ with increasing tunneling conductance ααα. For large, but fixed ααα, the correction to the leading exponential behavior exhibits a crossover from an intermediate temperature behavior at βEC∗≪1βE_C^* \ll 1βEC∗​≪1, to zero temperature behavior at βEC∗≫1βE_C^*\gg 1βEC∗​≫1. We determine this correction in both regimes and compare the numerical results to the numerous and controversial theoretical predictions for the strong tunneling limit.
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Tunnelling measurements on fractional quantum Hall systems are continuing to increase in popularity since they provide a method to probe the non-Fermi liquid behaviour of fractionally charged excitations occupying the edge states of a quantum Hall system. When considering tunnelling one must resort to an effective theory and typically a phenomenological tunnelling Hamiltonian is used analogous to that used for a conventional Luttinger liquid. It is the form of this tunnelling Hamiltonian that is investigated in this work by making a comparison to an exact microscopic calculation of the zero mode tunnelling matrix elements. The computation is performed using Monte Carlo and results were obtained for various system sizes for the ν=1/3ν=1/3ν=1/3 Laughlin state. Here we also present a solution to overcome the phase problem experienced in Monte Carlo calculations using Laughlin-type wavefunctions. Comparing the system size dependence of the microscopic and phenomenological calculations for the tunnelling matrix elements, it was found that only for a particular type of operator ordering in the tunnelling Hamiltonian was it possible to make a good match to the numerical calculations. From the Monte Carlo data it is also clear that for any system size the electron tunnelling is always less relevant than the quasiparticle tunnelling process, supporting the idea that when considering tunnelling at a weak barrier, the electron tunnelling process can be neglected.
We calculate the average number of electrons on a metallic single-electron box as a function of the gate voltage for arbitrary values of the tunneling conductance. In the vicinity of the plateaus the problem is equivalent to calculating the helicity modulus of a classical inverse square XY model in one dimension. By a combination of perturbation theory, a two-loop renormalization group calculation, and a Monte Carlo simulation in the intermediate regime we provide a complete description of the smearing of the Coulomb staircase at zero temperature with increasing conductance.
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The Cooper-pair tunneling in superconducting charge boxes in tunable electromagnetic environments is experimentally studied using a single-electron-transistor electrometer and via current-voltage characteristics measurement with small excitations. In the low-impedance regime, the environmental effect is quantified with an effective Josephson coupling energy, which is found to decrease as the environment impedance increases. In the high-impedance regime, the Cooper-pair tunneling is blockaded. For impedance above $250\phantom{\rule{0.3em}{0ex}}\mathrm{M}\ensuremath{\Omega}$, Coulomb oscillation of the zero-bias conductance suggests that Cooper-pair tunneling is completely suppressed and subgap quasiparticle tunneling becomes dominant.