The fundamental unit of 2-dimensional resistivity may be readily evaluated, using ̄h= 1. 05457 ×
10 −^34 Jsande= 1. 60218 × 10 −^19 C, and we find,
2 π ̄h
e^2= 25, 812 .63 Ω (11.34)
The units work out as follows,Js/C^2 = (J/C)(s/C) =V/A= Ω. Note that, by comparison, the
expression for the fine structure constant is
α=
e^2
4 πǫ 0 ̄hc=
1
137. 04
(11.35)
Thus, we have, alternatively,
2 π ̄h
e^2=
1
2 ǫ 0 cα(11.36)
Measurement of the Hall resistivity now gives the most accurate determination of this combination
of the fundamental constantseand ̄h.
The actual experimental results are more complicated, and also even more interesting. In
Figure 12, it is the Hall resistivityρxywhich is shown as a function ofBdirectly. The resistivity is
expressed as a dimensionless coefficient 1/νtimes the fundamental unit of 2-dimensonal resistivity
2 π ̄h/e^2 , as follows,
ρxy=1
ν×
2 π ̄h
e^2(11.37)
In the vicinity of theν= 1 plateau, more plateaux are discovered. It turns out (as you can see
from Figure 12) thatν appears to always be a rational number whose denominator is an odd
integer. This is the Fractionally Quantum Hall Effect (FQHE).The dynamics responsible for the
FQHE is no longer that of independent electrons, but now results from collective effects, which are
fascinating, but into which we cannot go here.
It is nonetheless possible to make direct contact with the structure of Landau levels as follows.
We now consider the system to be governed byNelectrons, subject to the external magnetic field
B and to their mutual repulsive Coulomb interactions. It is impossible to solve such a system
exactly, but Robert Laughlin proposed an educated guess forthe wave function of such a system
ofN electrons. We denote the planar coordinates of each electron by the complex numberszi,
i= 1,···,N. First of all, all of these electrons are assumed to be in the lowest Landau level, so
their wave function must be of the form, (we takeeB >0),
f(z 1 ,···,zN) exp{
−
eB
2 ̄h∑N
i=1|zi|^2}
(11.38)wherefis a holomorphic function of all thezi. Now in a strong magnetic field, spins up and down of
the electron will be split, so the lowest energy states will have all the spins aligned. This makes the
states symmetric in the spin quantum numbers, and Fermi-Dirac statistics then requires that the