Path integral approach to the density of states between landau levels in a two-dimensional electron gas

Abstract

The density of states (DOS) of an electron confined in two dimensions in the presence of a transverse magnetic field and interacting with disorder is derived, using path-integral method, we find that the DOS of the electron can be written in an integral form N(E) = ((s))/лɦ ∫_͚^͚▒dt (m/2лiɦT)(ΩT/(2sin⁡(ΩT/2)))exp [ iET/C-ϚLT/〖2ɦ〗^( ^2 ) ∫_0^T▒〖dy[G(T,〖y]〗^(-1) 〗] where G(T,y) = 1+(8 sin⁡〖[Ω(T〗-y)/2]sin⁡[Ωy/2])/(xsin[ΩT/2])

In the special case of consideration, for the states at low energies, we make a large-T approximation, it is found that the DOS becomes an analytical form of a sum of Gaussians centered at each Landau energy En, N(E) = no ɦΩ(2π｢2)-1/2 ∑_(n=0)^͚▒exp-[(E-E_n )2/(2｢)], Where ｢2 =ϚLx/(4+x ) ,x=ɦΩ/EL , EL = ɦ^2/〖2mL〗^2 and ϚL is the magnitude of Gaussian variance. From the analytical form of the DOS, for disorder having a finite correlation length L ~ 100 Å we find broad Landau Levels which leads to a large DOS between Landau levels, in agreement with experiments on heterojunctions. Furthermore, from the integral form of the DOS, which can be rewritten as n(E) = no (2/π) ∑_(n=0)^∞▒∫_0^∞▒〖dtRee^(2i(v/x-(n+1/2)t+｢(t)) 〗 where ｢(t) = -tsint/2ixϚL ∫_0^t▒〖dy 1/([(x/4i)sint-cost+cosy)〗

using numerical calculation with the correlation length L~100 Å, we find our numerical results agree well with the experiments of Kukushkin and Timofeev on MOS inversion layer. The essential new feature, which provides a DOS between Landau Levels, is recognizing the role of correlation length.