QuantumPhysics.dvi

Here,meis the electron mass andvits velocity. The last term account for the dissipative processes
which lead to Ohm’s law, and we have defined the relevant dissipation constant such thatσ 0 is the
conductivity in the absence of magnetic fields. In the stationary regime, we have a flow of charge
carriers with constant velocity, so thatdv/dt= 0, and we thus obtain,

nev=j=σ 0 (E+v×B) (11.27)

Eliminatingvin favor ofjgives,

j=σ 0

(

E+

1

ne

j×B

)

(11.28)

Choosing thez-axis alongBwithEin thexy-plane, the equation becomes,

jx = σ 0 Ex+

σ 0 B

ne

jy

jy = σ 0 Ey−

σ 0 B

ne

jx (11.29)

Inverting this relation to get the resistivity as a functionof the magnetic field,
(
Ex
Ey

)

=

(

ρxx ρxy

ρyx ρyy

)(

jx

jy

)

RL=ρxx=ρyy =

1

σ 0

RH=ρxy=−ρyx =

B

ne

(11.30)

Remarkably, the off-diagonal resistivity is independent of the normal conductivity and involves only
the density of carriers, besides fundamental constants andthe magnetic field itself.

11.5 The quantum Hall effect

Although the above picture of the Hall effect is applicable in the regime at room temperature and
for a large class of materials, there are special experimental settings where the Hall resistivityρxy
is not just a linear function ofB, but exhibits remarkable structure.^9
The experimental conditions for the Quantum Hall Effect (QHE)are as follows,

• A layer of electrons is trapped at the interface of two semi-conductors (a hetero-junction);


• The system is cooled to temperatures in the milli-Kelvin range;


• The system is very pure.

(^9) There is extensive literature on both the integer and fractional quantum Hall effects; a good source
with both experimental and theoretical contributions is inThe Quantum Hall Effect, R.E. Prange and S.M.
Girvin, Eds, Springer 1990.