Synthetic Metals, 17(1987) 1 5 ]

THE Q U A N T U M HALL EFFECT: R E L A T I O N T O Q U A S I O N E

D I M E N S I O N A L C O N D U C T O R S

ROBERT SCHRIEFFER

Inst i tu te for Theoret ical Physics, Universi ty of California, Santa Barbara , CA (U.S.A.)

A B S T R A C T

Many features of quasi one dimensional conductors, such as (CH)x and the fractional

quan tum Hall effect in two dimensional layers, are shared in common. These include quanti-

zat ion of the electron density, i.e., for certain densities, incompressibil i ty of the electron gas,

a gap in the quasi part icle spec t rum and fractionally charged quasi particles. The underlying

physics relat ing these apparent ly distinct systems is discussed.

I N T R O D U C T I O N

The anomolous propert ies of a two dimensional electron gas in the presence of a strong

magnet ic field B0 along z normal to the plane have been studied for more than a decade [1,2].

Exper iments on M O S F E T s and heterojunct ions exhibit striking deviations from semiclassical

t ranspor t theory. For example, the transverse or Hall conduct ivi ty azy, defined as the ratio

of current density Jz and the transverse Hall field Ey

axy = j z / E y (1)

is given by semiclassical theory as

o~y = . (2)

where u = pAo is the mean number of electrons per flux quan tum area

A0 =- Co/B0 = hc/eBo (3)

wi th p being the two dimensional electron density. For B0 ~ 100 kilogauss, A0 is of order

(100/~) 2. While eq. (2) predicts tha t ~xy should vary linearly wi th B o 1, yon Klitzing, Dorda

and Pepper [3] observed tha t axy exhibits plateaus as a function of u, the plateaus occurring

for u = 1, 2, 3 . . . Fur thermore , o-xy is accurately given by ie2/h, where i is the corresponding

integer value of u. In addit ion, the longitudinal resistance pxx has sharp min ima at these

integer u values, exhibit ing activated resistance as the tempera ture is varied.

Elsevier Sequoia/Printed in The Netherlands

This integer quantum Hall effect (IQHE) is basically a one body problem. In essence,

the continuum of two dimensional electronic states in the absence of B0 collapses to a ladder

of equally spaced Landau levels of energy En = nhwc, n -- 1, 2, 3 . . . , where we -- eBo/m*c is

the cyclotron frequency. Each Landau level is A/Ao-fold degenerate, where A is the sample

area. When defects are present, localized states are split off from each Landau level. Since

electrons in these levels carry no current, axy exhibits plateaus as the localized levels are

filled. Ando [4] proved that the reduction in the number of continuum (current carrying)

states is precisely compensated by an increased current carrying capacity per state, such

that axy remains an integer mulitple of c2/h at each step.

FRACTIONAL QUANTUM HALL EFFECT (FQHE)

In high mobility heterojunctions, Tsui, StSrmer and Gossard [5] observed plateaus of axy

similar to those discovered by von Klitzing et al., but at fractional values of ~ = n / m , where

n is an integer and m is an odd integer. As opposed to the integer effect, the fractional Hall

effect depends fundamentally on the existence of Coulomb interactions between electrons

involved in the Hall transport. Basically, the plateaus of axy are due to interelectronic

correlation effects leading to an energy gap ~ 2A in the quasi particle spectrum for fractional

densities ~ = n / m . As in the IQHE, quasi particles introduced by deviations of ~ from n / m

are pinned by defects, causing axy to be independent of ~ while the localized states are

being filled. The question is what is the nature of the ground state of the system such

that it exhibits gaps in the excitation spectrum for rational values of v i -- n / m with odd

denominators m? Alternatively, why does the system's energy E(v) have downward cusps

at vi?

Restricting the discussion to densities 0 < v < 1, and very large B0, only the lowest

energy Landau level is involved and only spin up electrons are present. The Hamiltonian is

,4,

where P projects onto the lowest Landau level. The kinetic energy (p Jr eA/c)2/2rn * is

absorbed in the definition of the lowest Landau level. The natural length ~0 in the problem

is defined by the flux quantum area

~t0 2 = ¢ 0 / B 0 (s)

The condition ~, = n / m corresponds to the requirement that the mean area per electron

is an integer multiple of the flux quantum area. Alternatively, the magnetic and electronic

lattices are commensurate for v = n / m .

The first attempt to account for this density or area quantization was in Hartree-Fock

calculations [6]. Unfortunately, E(v) was found to be smooth. Thus, in contrast to quasi 1D

3

conductors, a charge density wave ground state is not found within the mean field approxi-

mation.

Subsequently, Laughlin [7] proposed a trial ground state wave function of the Jastrow

pair form

~O(Zl""' ZN) = 1-[ f iJ e - ~ e Ize]2/4e~ (6) i<j

where z l = x l + iye is the complex coordinate of electron £. Laughlin showed that as a

consequence of the restrictions imposed by Fermi statistics, lowest Landau levels only, and

angular momentum conservation, if k9 0 describes the ground state then f i j must be of the

form

f i j : (zi - z j ) m (7)

where m = 1 ,3 ,5 . . . is an odd integer. From eq. (6) it follows that ~0 describes an electron

fluid of density u =--lm, with m an odd integer. However, the plateaus in axy require that

there be a gap 2A for adding extra quasi particles to the state with u - n / rn . Laughlin

proposed describing a quasi hole by blowing a small bubble in the 2D fluid, centered about

any point z 0. The above symmetry constraints force the quasi hole state to be of the form

• zo : H ( z k - z0) 0 (8) k

It is readily seen that the product factor raises the angular momentum of each particle about

z0 by one unit, removing the charge - u e from the vicinity of z0. Thus, the quasi hole has

charge Q : +ue.

There is a striking analogy between this result and that obtained for soliton excita-

tions in quasi 1D conductors. Namely, for r -band filling factor u (1 for polyacetylene) the

charge of the quasi particle (or soliton) is re. This result applies only if the CDW period is

commensurate with the crystal lattice period, i.e., v : n / rn , where n and m are integers.

What is the physics underlying this analogous behavior of two apparently unrelated

systems. Firstly, commensurability of the mean electron spacing and the crystal or magnetic

lattice spacing lowers the electronic energy, due to the electron-phonon interaction in the

quasi 1D conductor case and due to cooperative ring exchange in the FQHE, as discussed

below. Secondly, the commensurability requirement renders the electron gas (as opposed to

the electron plus ion system) incompressible. That is, the total energy has a cusp at electron

density v : v i : n / m such that the chemical potential # = OE/cOn jumps discontinuously at

v i. Hence charge can be added or subtracted only as quasi particles with a non zero energy

gap. Therefore, if charge is pushed out of a given region, it cannot be broadly redistributed

over space as a nearly uniform compression of the electron gas, but rather it must be clumped

into one or more quasi particles. Otherwise, the system energy will increase without bound.

Lastly, what quantizes the quasi particle charge? For quasi 1D conductors it is the dis-

cretely degenerate ground state which localizes a fixed fractional charge on domain bound-

aries separating regions having different ground state. This charge cannot be screened by

4

the r-electrons because of their incompressibility. The situation is similar for the FQHE;

as Laughlin argued, locally only integer multiples of charge ±t/e can be created because of

the constraint of remaining in the lowest Landau level for large B0. This locally quantized

charge is also globally quantized because of the incompressibility of the electron system. In

the recently proposed cooperative ring exchange theory of the FQHE [8] gives the alterna-

tive explanation that only for charge equal to an integer multiple of ± r e will ring exchanges

encircling the quasi particle add coherently to form a low energy state for large B0.

Below we summarize the relation between quasi 1D conductors and the FQHE.

Property 1D Conductor FQHE

Commensurability electron/crystal lattice electron/magnetic lattice

quantized filling ~, n / m n / m (m odd)

Incompressibility electrons for fixed ions electrons

quasi particle gap 2A 2A

quasi particle charge per spin ±t/e =t=ve

fractional statistics yes yes

degenerate vacuum yes ?

CONCLUSIONS

While there are many similarities, there are also substantial differences in these systems

in that 1D conductors contain both electrons and ions which can move, leading to low lying

phonon modes. On the other hand, the Hall effect exhibits dissipationless flow pxz --~ 0 as

T --+ 0, while 1D undoped conductors become insulators as T ~ O, pzz ~ c~.

The essential similarity between these 1 and 2D systems is that their ground state is

incompressible for certain stable fractional densities. This stability is a consequence of a

commensurability between the mean spacing between electrons and the lattice spacing (1D)

or magnetic length (FQHE). While the physics behind the commensuration energy is quite

clear in the 1D case, it is considerably more subtle in the fractional Hall effect. In essence, a

non analytic energy lowering occurs due to addition of spontaneously fluctuating ring currents

which add in phase only when commensuration is achieved, as shown recently by Kivelson,

Kallin, Arovas and the author [8]. This approach is complimentary but not contradictory to

that of Laughlin who emphasizes the short range part of the wavefunction and the importance

of restrictions such as Fermi statistics and lowest Landau level only.

While many predictions of the fractional charge theory have been confirmed in quasi 1D

conductors, such as reverse spin charge relations [9], the direct observation of fractional steps

in the FQHE is a highly significant achievement. Hopefully, the direct observation of the

fractional quasi particle charge will also be possible.

ACKNOWLEDGEMENTS

This work was supported in part by NSF Grant No. DMR82-16582 and the Japan Society

for the Promotion of Science.

REFERENCES

1

2

3

4

5

6

7

8

9

T. Ando, J. Phys. Soc. Japan, 36(1974) 1167;37 (1975)622; 3__7_7 (1975) 1233.

S.M. Girvin, in The Quantum Hall Effect, Springer, New York, 1986.

K.V. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett., 45 (1980) 494.

T. Ando, J. Phys. Soc. Japan, 36 (1974) 1167; 37 (1975) 622:3"7 (1975) 1233.

D.C. Tsui, H.L. StSrmer and A.C. Gossard, Phys. Rev. Lett., 48(1982) 1559.

H. Fukuyama and P.A. Lee, Phys. Rev. B, 18 (1978) 6245.

R.B. Laughlin, Phys. Rev. Lett., 50(1983)1395.

S. Kivelson, C. Kallin, D.P. Arovas and J.R. Sehrieffer, Phys. Rev. Lett., 56(1986) 873.

A.J. Heeger, in this volume. Also, A.J. Heeger, S. Kivelson, J.R. Schrieffer and W.P. Su,

Rev. Mod. Phys., to be published.