Problem: Crossover to Quasi-One-Dimensional Metal

A quasi-one-dimensional electron band is described by

 

where , , and are the components of the electron momentum and .    

  
Figure 4.2: Density of states for quasi-one-dimensional energy bands, as defined by Eq. 4.2.

Figure 4.2 shows the DOS at several values of A and B. At low band fillings, when the Fermi energy is close to the bottom of the band, the difference between the truly one-dimensional (B=0) and quasi-one-dimensional DOS is significant.

(a) Estimate the value of the electron density when the 1D 3D crossover happens.

(b) Compare the asymptotic behavior of the 1D band and the 3D band at low energies.