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By Kuramoto

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A low areal density of electrons is trapped in an inversion layer between the two different bulk semiconductors. Great effort goes into making these 2DEGs very clean, with long scattering lengths. Because the electron density is so low compared with the interatomic distance, the dispersion relation near the Fermi energy is almost perfectly quadratic, with an effective mass much lower than that of free electrons. The inversion layer is located quite close to the upper surface of the wafer (typically around 100 nm below it).

Although it may be permissable to ignore Coulomb interactions in the leads, this is not permissable in the quantum dot itself. A simple and standard approach is to add a term to the Hamiltonian of the form Q2 /(2C), where Q is the charge on the quantum dot and C is its capacitance. 103) where n ˆ is the number operator for electrons on the dot and n0 ∝ V . An important dimensionless parameter is t2 ν/U , where t is the tunneling amplitude between leads and dot. If t2 ν/U 1 then the charge on the dot is quite well defined and will generally stay close to n0 , with virtual fluctuations into higher-energy states with n = n0 ± 1.

To see this, it is very convenient to consider a lattice model, ∞ H = −t σ † (ψi† ψi+1 + ψi+1 ψi ) + Jψ0† ψ0 · S. 26) The strong-coupling limit corresponds to J t. It is quite easy to solve this limit exactly. One electron sits at site 0 and forms a spin singlet with the impurity, which I assume to have S = 1/2 for now. | ⇑↓ − | ⇓↑ . ) The other electrons can do anything they like, as long as they don’t go to site 0. Thus, we say the impurity spin is “screened,” or, more accurately, has formed a spin singlet.

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