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1 H 1 --<-<-j + 2 Ho j + 1
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(11.85)
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For H in this interval,
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[N - (j
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(11.86)
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'The experimental effect was discovered by W. J. De Haas and P. M. Van Alphen, Leiden Cornrnun., 212 (1931). Our simplified model is that of R. E. Peierls, Z. Phys. 81, 186 (1933). For a more realistic treatment see 1. M. Luttinger, Phys. Rev. 121,1251 (1961).
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FERMI SYSTEMS
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I I I I
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Fig. 11.10 De Haas-Van Alphen elfeet.
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Introducing the parameter x = H/Ho we can summarize the results as follows: /lOHOX 1 ( N Eo B) = (11.87)
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/loHox[(2j + 3) - (j + 1)(j + 2)x]
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(11.88)
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0,1,2, ... )
The magnetization per unit volume and the magnetic susceptibility per unit volume are respectively given by
-/lon
Jt =
(x> 1)
/lon [2(j + 1)(j + 2)x - (2j + 3)] (
(x>l)
2 < x < j
~ 1' j
(11.89) 0,1,2, ... )
{ - ( j + 1)(j + 2) H
2/l on
(j ~ 2 < x < j ~ 1 ' j = 0,1,2, ... )
(11.90)
These are shown in Fig. 11.10
11.5 THE QUANTIZED HALL EFFECT
The Hall effect was discovered in the nineteenth century: When crossed magnetic and electric fields are applied to a metal, a voltage is induced in a direction orthogonal to the crossed fields, as evidenced by an induced current flowing in
STATISTICAL MECHANICS
that direction-the Hall current. This effect is easy to understand on the basis of the free electron theory of a metal, as follows. Crossed magnetic and electric fields, denoted, respectively, by Hand E, act as velocity filters to free charges, letting through only whose those velocity v is such that E + (vje)B = 0, or
(11.91)
For free charge carriers in a metal, the current density is
j = qnv
(11.92)
where q is the charge, and n the density. The Hall resistivity Pxy is defined as the ratio of the electric field (in the y direction) to the Hall current density (in the x direction): E j=(11.93)
Substituting this into (11.92) and then into (11.91), we obtain
(11.94)
Measurements of the Hall resistivity in various metals has yielded charge carrier densities and provided the first demonstrations that there are not only negative charge carriers (electrons), but also positive ones (holes). The two-dimensional electron system used as a model in the last section can now be created in the laboratory, thanks to developments in the transistor technology. It can be made by injecting electrons into the interface of an alloy sandwich, which confines the electrons in a thin film about 500 A thick. The Hall experiment has been performed on such two-dimensional electron systems at very low temperatures, and the direct resistivities Pxx and the Hall resistivities Pxy have been measured, as indicated in Fig. 11.11. The experimental results are quite dramatic, as shown in Fig. 11.12. As the magnetic field H increases the degeneracy of the Landau levels increases. Since the electron density does not depend on the field the filling fraction p of the lowest Landau level decreases: hen
(11.95)
The Hall resistivity exhibits plateaus at p = 1, ~, t, with values equal to 1jP, in units of hje 2 At the same time, the conventional resistivity Pxx drops to very low values. This indicates that in the neighborhood of these special filling fractions the two-dimensional electron fluid flows with almost no resistance. The value at p = 1, called the integer quantized Hall effect, was first observed in a MOSFIT (metal-oxide semiconductor field-effect transistor) at T = 1.5 K. The Hall resis-
FERMI SYSTEMS
Uniform magnetic field H
tttt
Applied voltage
Hall current
The Hall effect. A current I flows in a direction orthogonal to crossed electric and magnetic fields. The Hall resistivity is defined as Pxy = V/1. The conventional resistivity Pxx can be obtained by measuring the voltage drop along the direction of the current.
Fig. 11.11
tlVlty was found to be quantized with a preClSlon of one part in 10 5.* The fractional values were found soon after. t The integer effect is easy to understand on a naive basis. Since at p = 1 the lowest Landau level is completely filled, there is an energy gap above the Fermi level. Low-energy excitations are therefore impossible, and so the centers of the electron orbits flow like a free gas. Using (11.94) with n = eH/hc, the Landau degeneracy per unit area, we immediately obtain the desired result.