The statistical theory of quantum dots

Abstract

A quantum dot is a sub-micron-scale conducting device containing up to several thousand electrons. Transport through a quantum dot at low temperatures is a quantum-coherent process. This review focuses on dots in which the electron's dynamics are chaotic or diffusive, giving rise to statistical properties that reflect the interplay between one-body chaos, quantum interference, and electronelectron interactions. The conductance through such dots displays mesoscopic fluctuations as a function of gate voltage, magnetic field, and shape deformation. The techniques used to describe these fluctuations include semiclassical methods, random-matrix theory, and the supersymmetric nonlinear σ model. In open dots, the approximation of noninteracting quasiparticles is justified, and electron-electron interactions contribute indirectly through their effect on the dephasing time at finite temperature. In almost-closed dots, where conductance occurs by tunneling, the charge on the dot is quantized, and electron-electron interactions play an important role. Transport is dominated by Coulomb blockade, leading to peaks in the conductance that at low temperatures provide information on the dot's ground-state properties. Several statistical signatures of electronelectron interactions have been identified, most notably in the dot's addition spectrum. The dot's spin, determined partly by exchange interactions, can also influence the fluctuation properties of the conductance. Other mesoscopic phenomena in quantum dots that are affected by the charging energy include the fluctuations of the cotunneling conductance and mesoscopic Coulomb blockade.[Published in Rev. Mod. Phys. 72, 895 (2000)] 2. Random-phase approximation in disordered dots 56 3. Parametric variation of the mean field 57 4. Anderson model with interactions 58 B. Spin effects and interactions 58 C. Peak-height statistics and interactions 61 D. Random interaction matrix model 62 VII. Charging Energy Effects in Quantum Dots 64 A. Mesoscopic fluctuations in elastic cotunneling 64 B. Mesoscopic Coulomb blockade 66 C. Mesoscopic fluctuations of the differential capacitance 68 VIII. Conclusion and Future Directions 69 IX. Acknowledgments 72 References 72

Disordered mesoscopic systems with interactions: Induced two-body ensembles and the Hartree-Fock approach

“…In such diffusive and/or chaotic open dots, the conductance exhibits "random" but reproducible fluctuations versus, e.g., gate voltage or magnetic field that can be successfully described by RMT. 2 The situation changes as the coupling of the dot to the leads is reduced and the dot becomes almost isolated. The charge on the dot becomes quantized, the conductance as a function of gate voltage displays sharp peaks (the Coulomb blockade resonances), and the Coulomb interaction between electrons becomes important.…”

Magnetic-field dependence of transport in normal and Andreev billiards: A classical interpretation of the averaged quantum behavior

“…The main signatures of classical integrability ͑or lack of it͒ on the statistics of energy levels and properties of the transport coefficients for closed and open systems, respectively, have been discussed in detail in various reviews. [1][2][3][4] Discussions on modifications owing to the possibility of Andreev reflection appear in more recent studies, 5,6,[9][10][11][12][13][14][15] mostly focusing on the features of the quantum mechanical level density.…”

“…Experimental realizations are based on exploiting the analogy between quantum and wave mechanics in either microwave and acoustic cavities or vibrating plates, 1 and on structured twodimensional electron gases in artificially tailored semiconductor heterostructures. [2][3][4] In the latter case, the particles are also charge carriers making these nanostructures relevant to applied electronics.…”

Spin effects in a quantum ring

“…Although many-electron systems are typically analyzed with statistical approaches [18], we were able to show previously that the level spectrum of high quality many-electron quantum ring structures in the Coulomb blockade regime can be understood in great detail [19]. Here we report experiments that exhibit the physics of the Hamiltonian (1), namely, the Zeeman effect and exchange interaction effects in this system.…”