Length Scales Characterizing Mesoscopic Systems

One important length scale characterizing mesoscopic systems is the Fermi wavelength λ _F = 2π/k _F , where k _F is the Fermi wave number. At zero temperature, electrons occupy states specified by the wave vector k with |k| ≤ k _F . The Fermi wave vector is related to the electron density n through 1.2.1$$ n = \left\{ {\begin{array}{*{20}{c}} {\frac{2}{{{{(2\pi )}^3}}}\frac{{4\pi }}{3}k_F^3 (d = 3)} \\ {\frac{2}{{{{(2\pi )}^2}}}\pi k_F^2 (d = 2)} \\ {\frac{2}{{(2\pi )}}{k_F} (d = 1)} \end{array}} \right. $$ where d is the system dimension (0D for d = 0, 1D for d = 1, 2D for d = 2, and 3D for d = 3) and the factor 2 comes from the electron spin. In typical metals such as Cu and Ag, the Fermi wavelength is of the order of a few angstrom and in semiconductors such as 2D systems realized in GaAs/AlGaAs heterostructures we have λ _F ~400 Å for the electron concentration n~3×1011 cm−2.