ELSEVIER Physica B 201 (1994) 487-489

PHYSICA]

Field-induced spin density wave and quantum Hall effect in (TMTSF)2X

K. Machida a' *, Y. Hasegawa b, M. Kohmoto c, Y. Hori a, K. Kishigi a

aDepartment of Physics, Okayama University, Okayama 700, Japan bFaculty of Science, Himeji Institute of Technology, Akou-gun, Hyogo 678-12, Japan

¢Institute of Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan

Abstract

The sign reversal phenomenon of the quantized Hall conductance observed in (TMTSF)2 X is explained theoretically in terms of the mean field solution for the field-induced spin density wave (FISDW) characterized by many-order parameters (MOP). The many competing OPs can make the sign of the Hall constant change particularly near the subphase boundary. It is shown that the jump of the Hall constant is accompanied by the energy gap closing of the FISDW,

1. Introduction

The field-induced spin density wave (FISDW) phe- nomena in quasi-two dimensional organic conductors (TMTSF)2X (X = CIO4, PF6, ReO4) are associated with one-dimensionalization of highly anisotropic two-dimen- sional electron motions. The external magnetic field (H) applied perpendicular to the two-dimensional conduc- tion plane plays a non-trivial role, inevitably leading to a Peierls instability in spin or charge degrees of freedom for a system where in the absence of H no Peierls instabil- ity is expected to occur because of imperfect nesting under a given band structure.

Experiments (see for a review Ref. [1]) performed on (TMTSF)2X reveal several remarkable characteristics of FISDW phenomena; Among other things: (1) Recursive and hierachicai fine structures in the phase diagram in which each main integer subphase (N = 0, 1, 2 . . . . ' from high fields) is further subdivided into subsubphases as the temperature (T) is lowered and this process iteratively

continues ad infinitum. (2) The quantized Hall constants whose quantum numbers roughly coincide with the inte- ger N of the main integer subphases. However, the sign reversals of the quantized Hall number, corresponding to the so-called Rebault anomaly for X = C104, have been observed for certain H and T domains.

The purposes of this paper are to give a theoretical explanation to the above two characteristic phenomena associated with FISDW which has not been done before in a coherent manner.

2. Many-order parameter state

In order to understand the essence of the FISDW problem, we start with the following "generic" mean-field Hamiltonian [2] which describes a one-dimensional sys- tem with a tendency toward the DW formation under an external periodic potential:

*Corresponding author. ~ = ~rt°o + ~1 ,

0921-4526/94/$07,00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 1 18-F

(2A)

488 K. Machida et al./ Physica B 201 (1994) 487-489

9fro = ~ coskCICk -- v ~ (CICk+e + h.c.), (2.2) k k

• "~, = Z Z Au CI+2kF+u~Ck, (2.3) N k

with the self-consistent equations

Au = -- U ~ (C~Ck+ 2kv+Ua), (2.4) k

where U is the electron-phonon coupling constant for the CDW or the on-site Coulomb interaction for the SDW and the energy unit is normalized by half the bandwidth. We allow the OP's Au (N = 0, ___ 1, _+ 2, ... ) to appear. This generic model Hamiltonian is equivalent to that in the FISDW problem, namely, the incommen- surate potential arises from the external magnetic field in this case. The periodicity of the potential is controlled by the magnetic flux passing through the unit lattice, or the field strength at will.

In order to concisely illustrate how our situation dif- fers from that in the ordinary Peierls state, we take up the simplest example: we consider the quarter-filling case where 2kv = x/2 and the wave number of the potential 6 = ~/4 (the lattice constant of x-axis is taken as a = 1 from now on). The OPs to be taken account are A _ a, Ao, A ~ and A 2 in this case because the wave numbers; k + N6 (N = 0-7) in the first Brillouin zone are coupled to each other through the above Hamiltonian. The matrix ele- ments connecting to these wave numbers give rise to the OPs which have a chance to grow. Simple diagonaliza- tion of the 8 x 8 Hamiltonian matrix readily yields a self- consistent solution with all Au non-vanishing. For example, when v = 0.02 (v = 0.04), A-1 = - - 0 . 0 2 9 (-0.055) , Ao = 0.035 (0.037), A1 = 0.006 (0.014), and A2 = - 0.004 (0.001) for U = 1.0. The band splits into eight bands.

The many-order parameter (MOP) state can appear irrespective of the filling, 6, v and U. The above example clearly illustrates that the inevitable infinite chain of the coupling among k points occurs when 6- ~ is incommen- surate with the lattice constant. This ultimately leads to the MOP. The above also shows that the largest OP could be other than Ao in contrast with what is expected by the Peierls theorem that the filling uniquely deter- mines the nesting vector of Q = 2kv.

3. Sign reversal of quantized Hall conductance

By linearizing the dispersion near the Fermi wave numbers: + kF, (2.1) can be rewritten in x-representation a s

fdky f dx " x de= 3 ~ £

- ivvOx 2vcos(6x A 1~ , ky) (x) A*(x) iVvSx -- 2VCOS(6X -- ky) /

(3.1)

where the spatial dependence of the OP; A (x) is given by the Fourier transformation in terms of At as

A(x)= L Al eit~x (3.2) t = - o o

t g=(@. (x , ky),@_(x, ky)) r and we have explicitly re- stored the kr-dependence suitable for the FISDW prob- lem. A transformation from ~ to ~ through

~k+ (x, kr) = ~+ (x, ky)exp[ + i(2V/VF6)sin(6x -- ky)]

yields

= k "~7 [ - ivv~x A(x, ky)~ f d k y f d x ~ " ( x ,

x @(x, ky), (3.3)

where

z J ( X , k y ) = A ( x ) e -iotsin(6x-k')

= ~ At Jn(ct)eilkre i " - n)6x, (3.4) l,n

ot = 4V/Vv,5, 6 = ebH/hc (b: the lattice constant of y-axis) and J,(e) is the Bessel function with order n. It is seen that the energy gap at + kv is determined by 21 Z(kr)l where the complex function

Z(ky)= L At Jr( OOeitkr (3.5) l = --c~

has been introduced. According to Yakovenko's argument [3], the winding

number L of Z(kr) (0 ~< ky < 2x) in complex plane gives rise to the quantized Hall conductance a~y = - 2LeZ/h. When each FISDW subphase is described by only a single OP, say, Au, the complex function (3.5) is Z(k r) = As Ju(a)e i/vky. Then the Hall number is simply N. This is the case discussed by Poilblanc et al. [4]. How- ever, as shown above the FISDW phases are generally characterized by a sequence {At} of many OPs. Depend- ing on the sequence, the winding number or Hall number can take both signs.

An example for a mean field solution is shown in Fig. 1 where as a function ofe or H, the sign of Hall number is reversed (Fig. lib)) even when {Al} continuously evolves (Fig. l(a)).

In the case where a FISDW phase is continuously evolving upon varying H, the energy gap at the Fermi

K. Machida et al./ Physica B 201 (1994) 487-489 489

0.~

0

-0.I

A I

A9

-2

0.0,3

0,0~

0.01

I l l l i { I q ~

o ,

. , o . .

0.05 0.1 V

(a)

(b)

(c)

Fig. 1. A mean field solution of Eqs. (2.1)-(2.4) and sign reversal of the Hall constant as a function of v (U = 1.5, 3/2rt = 1/36). (a) The magnitudes of various order parameters AN; (b) the corres- ponding Hall numbers calculated by Eq. (3.5) and (c) the asso- ciated energy gap closings evaluated by 2[Z(ky)l.

level must close when the Hall conductance jumps be- cause Z(ky) function crosses the origin in the complex plane, meaning that the gap collapses at that point in the Brillouin zone. In Fig. l(c) we plot the energy gaps or the maxima of 2lZ(ky)l as a function of v. It is seen that the energy gap closes whenever the Hall constant jumps from one integer to another.

It is a necessary condition for the sign change to occur that the F I S D W phase must contain the OPs {A~} with I being both signs, otherwise the sign change never occurs even if a solution contains many OPs. Even within the Nth integer subphase region where AN is largest the Hall number is not necessarily N and could have the opposite sign with the subphase index N (Fig. 1). This is parti- cularly true near the phase boundary region where the magnitudes of AN, A-N, AN-1 and A_tN_ 1) all compete, the Hall number has a chance to change its sign when going from N to N - 1 upon decreasing H as is seen from Eq. (3.5).

4. Conclus ion

We have shown an origin why the quantized Hall constant changes its sign in the F I S D W problem. By using a mean field solution with M O P s for the simplified standard model we have demonstrated that the sign reversal of the Hall constant can occur as a function of H, a feature absent for the single-order parameter state. In the case where a F I S D W state continuously evolves against H the jumps of the Hall constant must be accom- panied by the gap closing.

References

[1] See for a review: P.M. Chaikin, W. Kang, S. Hannahs and R.C. Yu, Physica B 177 (1992) 353; also see F. Tsobnang, F. Pesty, P. Garoche and M. H6ritier, Synth. Met. 41 (1991) 1707.

[2] K. Machida, Y. Hori and M. Nakano, Phys. Rev. Lett. 70 (1993) 61; Y. Hori, K. Kishigi and K. Machida, J. Phys. Soc. Japan 62 (1993) 3598; also see K. Machida and M. Nakano, J. Phys. Soc. Japan 59 (1990) 4223; K. Machida, Y. Hori and K. Nakano, J. Phys. Soc. Japan 60 (1991) 1730; Y. Hori and K. Machida, J. Phys. Soc. Japan 61 (1992) 1246; K. Machida, Y. Hasegawa, M. Kohmoto, V.M. Yakovenko, Y. Hori and K. Kishigi, Phys. Rev. B 50 (1994).

[3] V.M. Yakovenko, Phys. Rev. B 43 (1991) 11353. [41 D. Poilblanc, G. Montambaux, H6ritier and P. Lederer,

Phys. Rev. Lett. 58 (1987) 270.