Synthetic Metals, 41-43 (1991) 3837-3840 3 8 3 7

QUASI-ONE-DIMENSIONAL CHARGE DENSITY WAVE IN THE MAGNETIC FIELD

A. BJELL9

Department of Theoretical Physics, Faculty of Sciences, P.O.B.162,

41001Zagreb, Croatia, Yugoslavia

K. MAKI

Department of Physics, thiversity of Southern California,

CA 90089-484, U.S.A.

ABSTR&CT

The orbital effects of magnetic field on the charge density wave with an

imperfect nesting are e%Amined within a perturbative approach appropriate to

the limit of small cyclotron frequency with respect to the critical

temperature. We calculate the eorrectioru~ to the order parameter and the

density of collective carriers, and make a qualitative comparison with

measurements on NbSe . 3

I NTRODUCT I ON

The three-dimensional charge and spin density wave [C(S)DW] ordering in

quasi-one-dimensional systems is the consequence of anomalous electron-hole

correlations on the Fermi surface consisting of two open sheets. The deviation

from. the nesting of these two sheets leads however to a finite transverse

dispersion with pockets in the electron spect ruml below the critical

temperature T . Evenmore, if strong enough, this deviation prevents the DW c

stabilization, and keeps the metallic state down to T=0, like in

three-dimensional systems. Gor'kov and Lebed [I] pointed out that the external

magnetic field B favors again, via the quantization of orbitals in pockets,

the DW stabilization, and may even induce the three-dimensional ordering in

systems which are metallic at B--0. This phenomenon of field induced SDW was

observed in some Bechgaard salts. The corresponding regime of cyclotron

frequencies ~¢ = vebB larger or of the order of the transition temperature was

intensively studied theoretically in few recent years.

The subject of the present work is the opposite regime, which is

appropriate for e.g. NbSe 3 in which the effects of the magnetic field on the

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3838

second CDW (with T ~- 59 K) were observed in the temperature range T -< 20 K and ¢ c

at B <- 20 T [2-5]. In contrast to the problem of field induced SDW, one has

here the small parameter (dc/A O = ~q << i, where A O is the gap at T=0, ~ is the

longitudinal correlation lenght and q = ~c/VF is the reciprocal magnetic

length. We develop the perturbative method which is the generalization of that

introduced by Montambaux at T>T . [6] The method gives the series expansion of c

the electron Green's function in terms of ~c/A and 6o, the parameter of

imperfect nesting in the electron spectru~n

~(p) = v(p,- Pr ) - 2tb c°s(bp2) - ~oc°s(2bp2) (i)

This enables in particular the determination of the lowest order (quadratic in

B) corrections to some thergzx~ie CDW quantities at low temperatures. We

exhibit here only the main conclusions of rather lengthy calculations

presented elsewhere [ 7 ] .

ORDER PAP~, CONDENSATE DENSITY

The leading correction to the order ~ter, expressed in a form

convenient for the low temperature range (T<<A) is given by

~o

I n / /~o (T) = - g 1 ~ T 2 ~ (-1) n 2 2T --- i +{K2 Ii (2)

0 n = l

where

in[Ao(T)/Aol =-2 ~. (-l)"+IKoIo ,

n=l

and A ° is the order parameter at T=O, (oc=0. Here K i and I i are the modified

Bessel functions with the respective arguments nA/T and n6 o/T.

The surprising outcome of the result (2) is that the order parameter

amplitude decreases at T=0, although it is expected that the DW order is

strengthened in the magnetic field. On the other hand the critical temperature

increases with increasing ~) , and is given by c

0o

n ~ O

3839

The decrease of h at T:0 is gradually compensated by increasing the

temperature, as is seen from the second term in the e..~pression (2),

The quantity associated with the transport proporties of sliding CDN is the

density of condensate carriers. The straightforward calculation leads to the

conclusion tb~t it increases by increasing magnetic field in both asymptotic

limits of static (~ < vFp ) and ~ i e ((,~ > vFp) response, The respective

e_xpressions are

2 ~o 6

f(o> + ¢ o ~, (_!).+i 3 = n K2I i (4) fl i 3T 3

n=O

and

(0 6 f(o:~ c o ~ 3T2 K/ I i (5) = + ( - I ) "+I + K + a 2 ,

fo o 3T 2 h j n = O

where f ( o ) and f ( o ) a r e known r e s u l t s f o r B=0 [8] , and 1 0

~o

1 J I ---~ = dz sech z e.x]? - ~ cosh ,

O

These e.x]pressions are simplified at low temperatures, T << h-6 ° ,

( f l 1 ) / ( h / ~ o ) 1 / 2 ( f o - 1 ) / ( T 2 / A 6 o ) l / 2 - ( h - ¢ o ) I T . . . . e (1 - cO:6o/6T2A ). (6)

The resul t (6) is valid only for 2 c < 6T2A/6o , since otherwise i t does not

obey the requirement that the n~mber of carr iers in the condensate should not

be larger than the total ntBnber of band electrons, Finally, it follows from

eq, (6) that the correction to the dynamic density is much weaker than that to

the static one, i.e.

f _ f ¢ o ) T 0 0

-~ << 1 . ( 7 ) f _ f (o ) A

I I

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CONCLUSIONS

The present results (4,5) for the condensate density are in a qualitative

agreement with the observed increase of the ohmic magnetoresistance in NbSe 3

at T ~ 10-20K [2,3] and with the increase of the ratio of current density to

the fundamental frequency of narrow band noise above the threshold field

[4,5]. However, before making a quantitative comparison it is necessary to

complete the present analysis by taking into account the Zeen~an splitting,

~nich is more effective in CDW than in SDW systems. 9~is is particularly so if

~o Ao, which j~t seems to be the case of Nl~e . The further work on this

problem is in progress.

ACKNOWLEDGEMENTS

The work is supported by NSF under grant No DMR86-11829 and DMR89-15285.

and by Yu-US Collaboration Program No DOE-JF-738.

REFERENCES

1 L. P. Gor'kov and A. G. Lebed, J. Physique Lettres 45 (1984) L 433.

2 R. V. Coleman, G. Eiserman, M. P. Everson, A. Johnson and L. M. Falieov,

Phys. Rev. Lett.55 (1985) 863.

3 J. Richard, P. Monceau and M. Renard, Phys. Rev. B35 (1987) 4533.

4 P. Monmeau, J. Richard and O. Laborde, Synth. Metal 19 (1987) 801.

5 T. M. Tritt, D. J. Gilespie, A. G. Ehrlich and G. X. Tessema,

Phys. Rev. Lett. 61 (1988) 1778.

6 G. MontambeLLx, Phys. Rev. B38 (1988) 4788.

7 A. Bjeli~ and K. M~i, Phys. Rev. B (1990), to be published.

8 K. Maki and A. Virosztek, Phys. Rev. B41 (1990) 557.