VOLUME62, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 20 FEBRUARY 1989

"Fractional" Quantized Hall Effect in a Quasi-One-Dimensional Conductor

G. Montambaux(a) and P. B. Littlewood AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974

(Received 21 March 1988)

When the magnetic field is not aligned along one of the crystallographic axes of a quasi-ID conductor, the electronic motion is periodic only for special values of the field orientation. For these values, a cas­cade of spin-density-wave subphases is expected as the field increases. In each subphase the Hall resis­tance per layer is pxy =qh/pe2. q depends on the orientation of the field and p on its amplitude. The electron motion involves large orbits which spread over q unit cells. The integer quantization of these or­bits implies a fractional quantization of the density of carriers.

PACS numbers: 72.15.Gd, 71.30.4-h, 75.10.Lp

The integer quantized Hall effect (QHE) has been ob­served in two different physical systems. The first is the two-dimensional isotropic electron gas.l The second are the spin-density-wave (SDW) phases of the organic con­ductors of the Bechgaard salts family.2 In these quasi-1D compounds, the metallic phase in a magnetic field can be unstable versus the formation of a SDW ordered phase. The qualitative reason is that in a field, the elec­tronic motion is bounded and periodic in real space. The effective dimensionality of the electronic motion is thus reduced, enhancing the instability of the metallic phase characteristic of a ID band structure.3 At the same time, since the field adds a new periodicity in the elec­tronic motion, the spectrum in the ordered SDW phase, instead of exhibiting one gap as in zero field, shows a series of gaps located at quantized values of the wave vector.4"7 These gaps separate Landau bands. Since it is energetically favorable for the SDW formation that the Fermi level lies in one of these gaps, the SDW wave vector varies with the field to fulfill this condition.8 As a result, the Landau bands are either completely filled or completely empty. Integer quantization of the Hall effect is thus expected since the number of itinerant car­riers between the SDW gap and the Fermi level (n bands) varies linearly with the field: N = neH/h.Sf9

The fractional quantized Hall effect in the 2D electron gas is produced by correlation effects in a fluid state10

and is not expected in a crystalline or density-wave state where the Hall conductance remains integrally quan­tized. {l Here, we show that under certain conditions, the possibility of field-induced periodicities in a quasi-ID conductor leads to a fractional quantization of the Hall resistance per layer (the quantity of interest in a 3D compound), without the effect of the electronic correla­tions. We point out that, in general, the Hall resistance is not the inverse of the Hall conductance and we will see that in general only the Hall resistance obeys a simple ratio. We show first that such a fractional resistance is allowed by general arguments and then discuss a general model for quasi-ID conductors in which such behavior is found. Some physical arguments are given to interpret

these results, and we conclude by discussing the relevance to real materials.

Starting from the Kubo-Greenwood formula, Halpe-rin12 has derived the 3D quantized Hall conductivity for noninteracting electrons in a periodic potential, as a gen­eralization of the result of Thouless et al.l x for the 2D case. If the Fermi level lies in a gap, the Hall conduc­tivity in a phase perpendicular to the field is written, from Ref. 12,

a ± = ( e 2 / 2 ^ ) G - k , (1)

where G is a reciprocal-lattice vector and k is a unit vec­tor along the field. This formula allows for a noninteger resistance per layer. The above SDW mechanism in a quasi-lD conductor corresponds to G = « Q A : , e.g., an in­teger resistance. In this Letter, we present a mechanism for SDW ordering in a tilted magnetic field, a physical situation leading to such a noninteger resistance per lay­er pxy. When the Fermi level is pinned in a gap, the electronic motion is periodic, leading to the general re­sult from Eq. (1) that pxy=qh/pe2. Of course this re­sult comes from a generalization in 3D of the integer case and has nothing to do with the "usual" fractional QHE produced by correlations. As we will show at the end of this Letter, the flux enclosed by electronic orbits remains an integer number of flux quanta. The fraction­al quantization comes from the fact that the periodic motion can involve large orbits, extending over several Brillouin zones. The integer quantization of these orbits leads to the fractional quantization of the carrier densi­ty, when referred to the original unit cell.

The physics leading to the integer quantization in a quasi-ID conductor has been described starting from a model orthorhombic dispersion relation for the metallic phase:

E-v(\kx\ -kF) + ty[kyb-(eHb/h)x]. (2)

This dispersion has been linearized along the direction of highest conductivity around the Fermi level. ty and tz

are 2^-periodic functions which describe the warping of the open Fermi surface. Within the Landau gauge

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A = ( 0 , / / J C , 0 ) , a field H applied along the c direction in­troduces a new periodicity in real space, characterized by the wave vector G =eHb/ ft, which is the origin of the series of gaps in the energy spectrum.

The resulting quantization of the Hall conductance in the ordered phase has, of, course a totally different origin than the "usual" QHE. l Here, it is expected in a "pure" case, in the absence of impurities. The role of "reser­voir" is played by the electrons located below the SDW gap which do not contribute to the Hall conductivity.9

As the field varies, a series of SDW subphases is expect­ed, each subphase being characterized by the quantum number n which labels the Hall conductivity.8 The SDW vector in each subphase varies as Qx=2kF

+ neHb/h. Experimentally, in the quasi-ID organic conductor (TMTSF) 2C10 4 (TMTSF denotes tetra-methyltetraselenafulvalene) of the Bechgaard salts fami­ly, such plateaus have been observed, separated by first-order transitions.2 However, these plateaus are not so well formed as in usual QHE. Moreover, the longitudi­nal resistance does not go to zero, as expected. It has been proposed that the impurities induced some extend­ed states in the gaps, leading to an imperfect QHE. 1 3

Finally, that the Hall effect is expected to be quantized only if the dispersion along the c direction does not affect the quantization condition.8 This is believed to be the case in (TMTSF)2C104 .

In this Letter, we explore the role of another possible periodicity in the electron motion. Lebed has stressed that, in an orthorhombic quasi-ID conductor, if the field is not aligned along the c direction, it introduces two periodicities in the electron motion.14 He analyzed the instability of tlhe metallic phase which in principle can be very dependent on the field orientation (depending on the parameters of the dispersion relation and on the SDW vector). The corresponding structure of the ordered phase has been studied recently.15 Because of the new periodicity, the electronic spectrum exhibits new gaps and the Landau bands are thus split into subbands. We shall now argue that this may lead to a fractional quanti­zation of the Hall resistance pxy.

Let us first come to the metallic phase dispersion rela­tion. With a tilted field H = (0 , / / , , /7 z ) =(O, / / s in0 , HcosO), it is written, using the Landau gauge,

E=v(\kx\ - kF) + ty[kyb - (eHzb/h)x] . (3)

The field introduces two periodicities, with wave vectors

G\ =eHzb/h, Gi^eHyclh . (4)

When these periodicities are commensurate, the elec­tronic motion is again bounded and periodic14 and the metallic phase is unstable. Lebed showed that the sus­ceptibility exhibits logarithmic divergences for quantized values of the longitudinal components of the wave vector Qx =s2kF + mG\+nG2, m and n being integers. As a re­sult, the metallic phase is unstable toward the formation

of a SDW with one of these wave vectors. Let us assume the two periodicities are commensurate

so that

G2/Gl=cHy/bHz=(c/b)t<inO = r/q , (5)

where r and q are mutually prime. Gaps are expected at wave vectors

± T & = ± T 2kF + m-eHzb

h + 1 \2kF+2L^A\

q n (6)

The amplitude of these gaps can be related to the param­eters of the dispersion relation and to the SDW vector.15

The largest gap, sitting at the Fermi level, determines the SDW wave vector

eHzb Qx=2kF + R

h (7)

The effect of a finite Hy is thus to split each Landau band into q subbands of equal weight (see Fig. 1). As a result, we expect, when the amplitude of the field varies, a succession of SDW subphases characterized by different values of the quantum number p, q being fixed for a given orientation of the field. The different values of p are not necessarily successive and can be positive or negative, depending on the field. In the subphase labeled by p, there are p subbands between the SDW gap and the Fermi level. Each subband contains Ny = ( | e | / qc)Hz/h carriers per unit volume, p is positive if the SDW gap is above the Fermi level (carriers are holes) and negative in the other case. The Hall conductivity in the plane perpendicular to the field is thus expected to vary as

<J±=-Nv\e\

H = £ _ ! _

q h cos#

(8)

We have added the contribution of the p subbands. This formula can be proved directly by using the Streda for-

ASDW„-« —

/ •

FIG. 1. Structure of the spectrum when Gi/G\=cHy/bHz

= r/q = j . Each subband contains eHz/3hc =eHy/2hb states per unit volume. The largest gap at the Fermi level determines the SDW subphase. Here p—4 so that pxy = \hle1 and Pxz=2e2/h.

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mula for the calculation of cr± (Refs. 9 and 16): <J± = — | e | dNt/dH \ v. This formula involves the deriva­tive of the Nt, the total number of electrons per volume unit with respect to H, at fixed external potential which in our case means at fixed value of the SDW vector Q. Nt is simply given by Nt=2kF/27rbc so that dNt/ dH\Q= —pdNy/dH, which demonstrates Eq. (8). It can be checked easily that cr± has the form of the Halpe-rin formula (1).

In the plane perpendicular to the field, the Hall resis­tivity p± is given by p± = \/<J±. We now want to calcu­late the Hall coefficients in the x-y plane of the measure­ment. They are simply given by a rotation of angle Q. One easily deduces the Hall conductance per layer axy

and the Hall resistance per layer pxy,

P e ? <JXV =c(jJLcosO = J1- -—cos z 0,

q h (9)

P_L_COS0 a h

c p e2

It is again worth stressing that in this plane the Hall

resistivity is not the inverse of the Hall conductivity. Only the Hall resistivity is given by a simple fraction in­dependent of the cell parameters. One finds easily that the Hall resistance pxz is similarly given by

p±sin0 r_h_

P e2 (10)

J

The numerator is different while the denominator is the same as in pxy. Indeed, the gap hierarchy can be seen as well as a series of Landau bands with degeneracy eHz/hc per unit volume split into q subbands or a series of Landau bands with degeneracy eHy/hb split into r subbands. The two measurements of pxy and pyz reflect this duality.

The structure of the electronic spectrum and the sub­sequent fractional quantization of the Hall conductance do not necessarily require a tilted field. It happens whenever the magnetic field is not aligned along one of the crystallographic axes of the sample. For example, a field perpendicular to the a-b plane of a monoclinic crys­tal leads to the following structure of the dispersion ( c o s a = b * c / | b\ \c\):

E =v(\kx\ — kf) + ty[kyb — (eHb / h) x] + tz[kyc cosa + kzc since — (eHccosa/h)x] . (11)

The two periodicities G\ =eHb/ h and G2==eHccosa/hi if in a simple ratio, will induce a series of SDW phases with fractional quantization of the Hall effect. This geometry is simple in the sense that pxy = \/oxy.

This fractional quantization can be simply understood in terms of flux quantization. Although the crystal unit cell has an area —a2 (a being a typical interatomic distance), the SDW ordering leaves a very small pocket of itinerant car­riers. The number of these carriers is given by an area defined in a plane of the reciprocal space perpendicular to the magnetic field. This area is the zero-field area between one sheet of the Fermi surface and the other one translated by the nesting vector Q = (Qx,Qy,Qz) (Fig. 2). For the above case of monoclinic symmetry, we have (v = 1):

J + T

[Qx — 2kF — ty(kyb + Qyb)[(ky + Qy)ccosa4-(kz + Qz)csina] — tz(kyccosa + kzcsina)]dky , (12)

where T is the periodicity of the electronic motion. This motion is periodic only if G2/G1 =ccosa/b is a rational number r/q. In this case

A =(QX -2kF)lKq/b . (13)

orbits leads to a fractional quantization of the carrier density, when referred to the original unit cell.

Using the Halperin formula, one gets in this case

The fact that this area does not depend on kz reflects the fact that there is no dispersion along the field. In a mag­netic field, this fundamental area has to be quantized, leading to the quantization of the wave vector Qx. Since this area involves q original unit cells [Fig. 2(b)] , an in­teger quantization of this area implies that the elementa­ry area defined in the first Brillouin zone is fractionally quantized. A unit cell can be defined in real space, with area S,

2

&xy —- n +m ccosa

S = eH

A = hq(Qx~2kF)

H2e2b (14)

When the wave vector is fractionally quantized, the flux through this area remains an integer number of flux quanta hie. In conclusion, the fractional quantization comes from the fact that the periodic motion of electrons involves larger orbits. The integer quantization of these

A priori a SDW forms only if ccosa/b is commensurate so that now oxy is in general a fractional multiple of e2/h. However, if ccosa/b is incommensurate or com­mensurate with high order, it is believed that a SDW is still stabilized with the simplest ratio r/q close to c cosa/b.

If the topological structure of the spectrum does not depend on the energies of the problem (as soon as we have open orbits in the metallic phase), the observation of a fractionally quantized Hall conductance depends on the amplitude of the gaps. The gap Ap at the Fermi level has the structure

A P2 ( G , , G 2 ) O C X ' A 2 ( G 1 ) A 2 ( G 2 ) , (15)

m,n

where the sum is restricted to the numbers m and n such

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V O L U M E 62, N U M B E R 8 PHYSICAL REVIEW LETTERS 20 F E B R U A R Y 1989

S-rr

47T

Zrr

1 1 (a)

1

/ 1

Q j H 0 _ ^ -

1 1

r A

¥ *

6-rr

47T

Zir

FIG. 2. (a) When there is only one periodicity in the elec­tronic motion (tz =0) , the SDW vector varies with the field so that the number of itinerant carriers in the SDW phase is quantized. This number is proportional to the area included between one sheet of the Fermi surface and the other one translated by Q(/ / ) . In this case, the periodicity of the motion is the periodicity of the lattice. Integer quantization of the area A leads to integer quantization of the carrier density, (b) When the angle (H,c) is finite, the periodicity of the motion is larger than the periodicity of the lattice. Here the motion has periodicity 27zq=6n. The area A spreads over three Brillouin zones. Integer quantization of this area leads to fractional quantization of the carrier density.

that mq + nr —p. Am and An are the gaps corresponding to the periodicities G\ and G2 if they were alone. These gaps A are functions of the ratios G\/D\ or G2/D2 where Di is the deviation from perfect nesting (expressed in wave-number units) along the corresponding direction. The A's are large if the ratios G7/A are of order unity. For fractional quantization to be observable, the two hierarchies of gaps must have comparable amplitude. This will lead to a succession of values of p which can be

very complex, even with changes in sign. The thermo­dynamic stability of these different phases will depend on the energies D\ and D2. Phases with fractional piq may only be stable if these parameters are comparable. It would be tempting to ascribe the recently claimed v = y plateau found in the Bechgaard salt (TMTSF)2C1C>4 (Ref. 17) to the mechanism described in this Letter. However, in our present understanding, Bechgaard salts have, indeed, a quasi-2D electronic structure, as confirmed recently from experiments in a tilted field,18 so that Z>2<£>i. As a result, An(G2) — S(n) SO that only one periodicity is relevant and integer quantization is ex­pected. However, it is not excluded that interesting effects happen for some field orientations so that G2 can be of order of Z>2-

Although there is yet no experimental evidence of this fractional quantized Hall effect in quasi-ID conductors, we think that the possibility of such an effect without electron correlations reflects a new aspect of the extraor­dinary rich physics of low-dimensional conductors.

We acknowledge very useful discussions with C. Kal-lin, R. N. Bhatt, and B. I. Halperin.

^Also at Laboratoire de Physique des Solides associe au CNRS, Universite Paris Sud, 91405 Orsay, France.

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

2See review articles by P. Chaikin and M. Ribault, in Low Dimensional Conductors and Superconductors, edited by D. Jerome and L. Caron, NATO Advanced Study Institute, Ser. B., Vol. 155 (Plenum, New York, 1987).

3L. P. Gor'kov and A. G. Lebed, J. Phys. (Paris), Lett. 45, L433 (1984).

4D. Poilblanc, M. Heritier, G. Montambaux, and P. Lederer, J. Phys. C 19, L321 (1986); G. Montambaux and D. Poilblanc, Phys. Rev. B 37, 1913 (1988).

5K. Yamaji, Synth. Met. 13, 29 (1986). 6A. Viroztek, L. Chen, and K. Maki, Phys. Rev. B 34, 3371

(1986). 7M. Ya. Azbel, P. Bak, and P. Chaikin, Phys. Rev. A 39,

1392 (1986). 8M. Heritier, G. Montambaux, and P. Lederer, J. Phys.

(Paris), Lett. 45, L943 (1984); G. Montambaux, M. Heritier, and P. Lederer, Phys. Rev. Lett. 55, 2078 (1985).

9D. Poilblanc, G. Montambaux, M. Heritier, and P. Lederer, Phys. Rev. Lett. 58, 2701 (1987).

10R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). n D . J. Thouless, M. Kohmoto, M. P. Nightingale, and M.

den Nijs, Phys. Rev. Lett. 49, 405 (1982). 12B. I. Halperin, Jpn. J. Appl. Phys. 26, Suppl. 3, 1913

(1987). 13M. Ya Azbel and P. M. Chaikin, Phys. Rev. Lett. 59, 582

(1987). 14A. G. Lebed, Pis'ma Zh. Eksp. Teor. Fiz. 43, 137 (1986)

[JETP Lett. 43, 174 (1986)]. 15G. Montambaux (to be published). 16P. Streda, J. Phys. C 15, L717 (1982). 17R. V. Chamberlin et al.9 Phys. Rev. Lett. 60, 1189 (1988). 18X. Yan et al. (to be published).

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