L-7

Goal : This is the first lecture about the localization. Localization is a wave phenomenon induced by disorder.

The Gaussian packet of free particles: the ballistic behaviour

BCE and 1D localization

File:Localization1D.png

BCE and 1D localization

The conductance and the diffusive behaviour

Ohm's laws characterize electric transport of (good or bad) conductors:

• First law:

${\frac {V}{I}}=R,\quad {\text{or}}\quad {\frac {I}{V}}=G$

Here $R$ is the resistence of the sample and $G$ is its conductance.\

• Second law

$R=\rho {\frac {L}{S}}\sim \rho L^{2-d},\quad {\text{or}}\quad G=\sigma {\frac {S}{L}}\sim \sigma L^{d-2}$

Here $\rho ,\sigma$ are the resistivity and the conductivity. These are material properties, independent of the geometry of the sample These phenomenological laws are a macroscpic manifestation of the diffusive motion of the electrons. From the Drude model we know that disorder is the crucial ingredient to justify the presence of diffusion. ]