Magnetic Oscillations in a Normal State of Organic Conductors: Many-Body Approach

Magnetic Oscillations in a Normal State of Organic

Conductors: Many-Body Approach

A. Lebed

To cite this version:

A. Lebed. Magnetic Oscillations in a Normal State of Organic Conductors: Many-Body Approach. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.1819-1936.<10.1051/jp1:1996190>. <jpa-00247283>

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J. Phys. I France 6 (1996) 1819-1836 DECEMBER 1996, PAGE 1819

Magnetic Oscillations in a Normal State of Organic Conductors:

Many-Body Approach

A.G. Lebed (*)

Institute for Materials Research, Tohoku University, Sendai 980-77, Japan

and

L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow,

2 Kosygina St., Russia

(Received 26 April 1996, received in final form 27 June 1996, accepted 4 Jul,v 1996)

PACS.74.70.Kn Organic superconductorsPACS.71.27.+a Strongly correlated electron systems; heavy fermions

Abstract. We argue that magnetic and angular oscillations observed in quasi-one-dimen-

sional (QID) orgamc conductors represent a new type of many-body phenomena. The physical

nature of such effects as"magic angles" in

(TMTSF)2Cl04, (TMTSF)2PF6, and (DMET-TSeF)2

AuC12 aswell

as"rapid magnetic oscillations" in (TMTSF)2Cl04

isshown to be beyond the

standard theory of metals. Below wediscuss an

explanation of these phenomena which utilizes

unusual many-body effect achange of an

effective dimensionality of electron-electron ("e-e"

interactions with changing botha

magnitude anda

direction of a magnetic field. We show that

some exotic transport properties ofa

metallic state caribe interpreted in terms of these dimen-

sional crossovers. We also demonstrate that magnetic field dependence of "e-e" interactions has

to break Fermi liquid description of quasiparticles at high magnetic fields, H > 25 30 T. This

leads to the appearance of strong forbidden oscillations of magnetic susceptibility, ôx/x0+~

1-10,

and magnetic moment, ôM/Mo+4

0.1. Ail of the above mentioned unique properties ofa

metalbc

phase in (TMTSF)2X and (DMET-TSeF)AuC12 allow us to call it ananomalous metallic phase.

1. Introduction

Numerous families of organic conductors which were synthesized during last two decades

demonstrate a wide variety of properties in a magnetic field (for a review, seeiii). Some

of them, quasi-two-dimensional (Q2D) compounds (ET)2X IX =13, IBr2), (ET)2MHg(SCN)4,

(ET)2MHg(SeCN)4 (M=

NH4, K, Rb, Tl), and (ET)2Cu(NCS)2, contain closed quasiparticle

orbits in their electron spectra. These organic metals exhibit well known Shubnikov-de Haas

(SdH) oscillations, de Haas-van Alphen (dHvA) oscillations, and magneto-breakdown oscilla-

tions of resistivity. Trie basic properties of SdH, dHvA, and magneto-breakdown oscillations in

(ET)2X, (ET)2Cu(NCS)2, and (ET)2MHg(SCN)4 compounds are found to be in a good agree-

ment with standard Lifshits-Kosevich (LK) formula inmoderate magnetic fields [2]. At higher

magnetic fields, H ct 25 30 T, experimental data are becoming more complicated. Some

measurements on(ET)2MHg(SeCN)4 materials indicate that an effective mass is becoming

(* e-mail: lebed©vostok.imr.tohoku.ac. jp

© Les Éditions de Physique 1996

1820 JOURNAL DE PHYSIQUE I N°12

magnetic field dependent at high fields [3], but at trie moment it is not clear if this correspondsto some new physics or not.

From our point of view, trie major physical question regarding trie magnetic oscillations

in Q2D materials could be trie following: "Which deviations from trie LK formulaare there

appear at high fields when cyclotronic frequency, ~dc, is of trie order of electron bandwidth intrie direction perpendicular to plane, ~dc m~

ti [4, 5], or of trie order of Fermi energy?" Trie latter

seems to be possible in (ET)2NH4Hg(SCN)4 material under pressure where small pockets ofquasiparticles are expected to exist [6].

In a tilted magnetic field Q2D compounds demonstrate nontrivial angular resistivity oscil-lations which are well described in terms of standard theory of anisotropic Q2D metals in trie

case of (ET)2X (X=

13,IBr2) materials I?i. As to angular peaks and dips of resistivity re-

cently discovered in so-called "reconstructed" phase of (ET)2MHg(SCN)4 compounds [8], theyare believed to be understood in terms of Yamaji's and Osada's effects [î, 9] within some more

complicated variant of "fermiology" (see, for example, [loi ).Trie physical origin of magnetic and angular oscillations observed in quasi-one-dimensional

(QID) organic conductors (TMTSF)2X (X=

Cl04, PF6, etc.) and (DMET-TSeF)2AuCl~is not so simple. In a metallic state these compounds (whic( bave only open Fermi surfaces(FS's) demonstrate such unusual phenomena as "magic angles" (MA'S) Ii e., nontrivial angulardependences of both transport [11-18] and thermodynamic [15] properties), "rapid magneticoscillations" (RMO) of resistivity [19-26], Danner-Chaikin's (DC) angular resonance [27], and

Osada-Kagoshima's (OK) angular effect [28j. By trie present moment there bas been done alot of attempts to understand trie origin of MA, RMO, DC, and OK phenomena. All existingtheories of magnetic properties in a metallic state of QID conductors can be subdivided intothree groups:

A) "Fermiological" theories (FL) [9, 27-32j Ii.e., refined variants of trie standard theory ofmetals).

B) Perturbative many-body (MB) theories [33-42j (these theories consider trie changes ofmany-body effects

in a magnetic field, particularly, trie dependence of "e-e" interactions onboth a magnitude and a direction of a magnetic field).

C) "Luttinger liquid" (LL) approach (trie authors of Refs. [43, 44] suppose that Tomonaga-Luttinger liquid is created in large enough magnetic fields and daim that magnetic propertiescannot be understood within trie framework of perturbative MB theories).

Let us discussin brief trie relationship between these different groups of theories. Important

methods of an investigation of open FS'S, DC and OK angular resonances, were suggested[27, 28j and explained [2î,32] within trie standard "fermiology". These angular oscillations are

observed in a magnetic field which is almost parallel to trie chains of QID compounds. Forsuch directions of a magnetic field, as it is shown in references [2î, 28, 32], FL approach workswell.

It is important that, unhke DC and OI< effects, MA and RMO phenomenaare observed

in magnetic fields with large enough projection perpendicular to trie chains, Hi > 1- 2 T.From papers on spin-density-wave (SDW) formation in (TMTSF)2X compounds [45-50], it

is known that perpendicular projection of a magnetic field, Hi, increases "e-e" interactionsdue to trie phenomenon of "one-dimensionahzation" of an electron spectrum [45, 46, 50]. Atlow temperatures this effect results in trie appearance of a cascade of field-induced SDW sub-phases [19. 51]. A variant of perturbative MB approach to magnetic phenomena developedby us and other authors [35-42] is based on trie similar idea but it daims more. We arguethat perpendicular component of a magnetic field changes

an effective dimensionality of "e-e"interactions in QID metals and that these changes of trie dimensionahty

are responsible forexotic magnetic properties. From this point of view, a dramatic increase of "e-e" interactions

N°12 MAGNETIC OSCILLATIONS: MANY-BODY APPROACH 1821

in trie presence of large enough perpendicular magnetic field [33-36,45-47]seems to be trie

main reason why FL theories [9, 29, 30] meet with serious difficulties [31] while describing MA

and RMO phenomena.

Below, we present our subjective perturbative MB view of trie physical nature of magneticand angular oscillations in QID metals. In 1986, thermodynamic MA phenomena were pre-dicted to exist in field-induced SDW state of (TMTSF)2X conductors [50]. A few years later,

trie similar effect was suggested for a resistivity in a metallic state [35] and then it was ex-

perimentally discovered [11-18]. Although at first there were serious discrepancies between

original prediction [35] and experimental data [11-18], all of trie qualitative discrepancies bave

been ehminated in a different variant of this model [36]. To meet experimental data [11-18],

a crossover between ID "e-e" interactions (at arbitrary directions of a magnetic field) and 2D

"e-e" interactions (at MA directions of a magnetic field) was suggested in reference [36] (foralternative point of view, see Ref. [37]). Several years ago perturbative MB theories [39, 40]

succeeded in explanation of trie thermodynamical MA phenomenon observed in a torque in

(TMTSF)2Cl04 (15]. And very recently, there was appeared perturbative MB theory [38] that

describes RMO observed in a metallic state of (TMTSF)2Cl04 (19-26j. At trie end of trie

review, we discuss a possibility of a break of trie Fermi hquid description at high magneticfields within a framework of trie perturbative MB approach [39-42j. On trie contrary to Fermi

liquid theory, we predict [42j trie existence of strong forbidden thermodynamic oscillations of

magnetic susceptibility and magnetic moment in (TMTSF)2Cl04 conductor which bas only

open FS'S.

To summarize, trie major qualitative features of MA and RMO phenomena (at least in

(TMTSF)2Cl04 and (DMET-TSeF)2AuC12) seem to be understood within perturbative MB

theories [35-41j, although new experiments on (TMTSF)2PF6 (44j demonstrate that a real

physical picture of magnetic phenomena is more complicated.

Among experiments which are still far from trie understanding are:

A) Nonmonotonous temperature dependence of a resistivity and unusual temperature depen-

dence of a magnetic susceptibility inmoderate magnetic fields [52j.

B) Non-analytical dependence of resistivity on a magnetic field in (TMTSF)2PF6 (44j.C) Disappearance of a quasiclassical magnetoresistance background and FS-effects

in(TMT

SF)2PF6 when perpendicular magnetic field is applied [44j.From our opinion, trie nonmonotonous temperature dependence of resistivity [52j might be pre-

scribed to a modification of "e-e" interactions intrie vicinity of SDW transition [53] or to some

localization effects in a field [52, 54]. As to trie other phenomena mentioned above, it seems that

they represent some new non-Fermi-liquid physics. Note that non-analytica( magnetic field de-

pendence of resistivity [32] observed in experiments on OK angular resonance [28] bas been

explained by FL approach [32]. Nevertheless, trie non-analytical magnetic field dependence of

resistivity under more general conditions of experiment [44] as well as trie disappearance of

FS effects at high enough perpendicular magnetic fields [44] still bave no explanation within

both FL and perturbative MB approaches. Trie authors of reference [43] speculate that these

phenomena are a manifestation of Luttinger liquid formation.

2. Experimentson

"Magic Angles" and "Rapid Magnetic Oscillations"

In this section we present a brief review to some experimental aspects of MA and RMO

phenomena in a metallic state of QID materials. MA phenomena are usually observed in a

magnetic field H=

(o,Hsinô,Hcosô) which is rotated in (b,c)-plane perpendicular to trie

direction of trie chains, a.If magnetic field is applied along one of trie possible vectors of a


83

b

c

Fig. l. "Magic" directions ofa magnetic field

aredefined by vectors of trie crystalline lattice

m

(b, c) plane. One of the main MA'S is shown byarrow

(see Eq. (I)).

B=5T

T/K

Îi 0.iÙéa iii

~ [ij1 0.4

~ ÎÎÎI, il 1-o

~ iii1

=8654 /~ 2 /~ i

-90 -80 -70 -50 -50 -40

6 (degree]

Fig. 2. A number of themain MA

areclear visible

onthe experimental

curved~pa /d0~ [15].

crystalline lattice:

~ ~*tan(ôk,m)" m

Il)ill~ C

(see Fig. 1), there appear nontrivial pecuharities on angular dependences of resistivity and

magnetic torque [11-18]. (Here, à is trie angle between H andc axis in an orthorombic model

of a unit cell; b* and c* are trie lattice constants; k and m are integers).A typical experimental plot of trie second derivative of a magnetoresistance with respect

to angle à is presented in Figure 2 where a number of maxima is observed. As it follows

from Figure 2, trie most common MA'S are trie integer ones which correspond to m =1 in

equation (1). Note that two major MA'S are observed at ôa~,1 "~/2 and ôo,i

"o when

magnetic field is applied parallel to b and c axes, correspondingly. Trie third MA with on"

arctan(b* /c*) is usually also pronounced whereas trie higher order MA'S correspond to local

minima (which bave much smaller magnitudes) and so they are visible only on a plot of trie

second derivative of magnetoresistance (see Fig. 2).


Pb

' Pa~ '

# J~#

ù'

7T

b

Fig. 3. A typical QID electron spectrum.

We point out that MA phenomena are very common in alow-dimensional conductors. MA'S

are observed in experiments on transport properties [11-18] as well as in experiments on ther-

modynamic ones [15,16] in a number of QID compounds in both a metallic and a SDW phases.

Trie other unusual magnetic phenomenon, RMO, is very common in a SDW state [20-26, 55-58].

As to metallic state, RMO bave been revealed only in (TMTSF)2Cl04 compound by trie present

moment [19-26]. A typical experimental geometry of an observation of trie resistive RMO cor-

responds to Hjj c

and Ijj a.

3. "Fermiological" Approach to "Magic Angles" and "Rapid Magnetic Oscilla-

tions" Difliculties

Electron spectrum of (TMTSF)2X (X=

Cl04,PF6,AsF6, etc.) and (DMET-TSeF)2AuC12

compounds corresponds to open slightly deformed sheets of trie FS:

e(p)=

+~F(Pa ~ FF) + 2tb cos(pbb*) + 2tc cos(pcc*) +~j 2tk,m cos(kpbb* + mp~c*) (2)

k>0,m=1

(see Fig. 3).

In equation (2) trie first term represents a free motion of electrons along trie chains on trie

right (+) and left (-) sheets of trie FS, with pF and VF being trie Fermi momentum and

trie Fermi velocity, correspondingly (pFvF t 2.5 x103 K). Trie second and trie third terms

correspond to trie hopping of electrons in trie perpendicular directions, b and c (tb t 200 K,

tc ct 5 K). Trie summation over k and m in equation (2) represents a small effective hopping

of electrons in(b, c) plane. Note that, due to trie extremely small overlapping of trie wave

functions along c axis, we retain only trie terms with m =1. It is known that in (TMTSF)2PF6

and (DMET-TSeF)AuC12 compounds ti,1 » t2,1 » t3,1 and so we may retain only a few terms

(say, with k=

and k=

2) in trie summation in equation (2). On trie contrary, in trie case of

(TMTSF)2Cl04 we bave to consider all higher harmonics, tk,1+~

K (k=

1, 2, 3,. ). due to a

corrugation of FS resulted from an anion ordering (AO) gap, D [ii (see Fig. 4).

In this section we show that in QID conductors with electron spectrum (2) negligible mag-

netoresistance effects areexpected within trie standard "fermiology" if inchain current, I j a,


Pb

~

l 2~

b*

i'a

X

')ioEiM-+

Fig. 4. Due to the AO phenomena,an electron spectrum in (TMTSF)2Cl04 corresponds to four

open sheets of FS.

is measured in a magnetic field perpendicular to trie chains, H 1 a. Indeed, let us consider an

electron motion along open sheets of trie FS in perpendicular magnetic field (see Figs. 3, 4).Due to small values of trie parameters tb,tc, and D, longitudinal component of an electronvelocity, va, is almost independent

on trie position ofan electron on trie FS, va(p)

ct +~F.This means that magnetic field does not disturb an electron motion along trie chains and thus

no any magnetic effects are expected.

Trie same is directly seen from Boltzmann kinetic equation:

E~~()~~+ ~)) Iv x HI ~~

()~~ = ~)) lflP) folP)1, 13)

where trie Lorentz force, F=

je /c)[v x Hi ct (e/c)~FH(b/b*),is almost perpendicular to trie

electric field, F 1 E(a);e is trie electron charge and c is trie velocity of light.

A simple analysis of equation (3) shows that all magnetic effectsin longitudinal resistivity

bave to be of trie order of ôpa(H)/pa(o)m~

(tb/PF~F)~+~

10~~ 10~3 This conclusion oftrie standard theory of metals is in a sharp contradiction with experimental data, since trieexperimental relative magnitudes of MA Ill,13,14,16-18j and RMO [19-26] phenomena

are oftrie order of 1 and 10~~, correspondingly.

To avoid trie above mentioned contradiction, an interesting phenomenological "bot spots"model was recently proposed [30]. This model suggests trie existence of some spots on trie QID

FS where electron relaxation time is much less than one on trie rest part of FS. Althougha

subsequent "microscopic" theory [31] confirms trie existence of "bot lines" on trie FS, neverthe-less, it is not able to explain trie appearance of MA'S

in a magnetic field. There exist also twoother FL theories of MA'S [9, 29], but both of them are inconsistent with large experimentalmagnitudes of MA effects and their main experimental features.

To summarize, we conclude that an observation of MA'S and RMO in a QID conductors isin a dramatic contradiction with trie predictions of "fermiology" based on trie standard theoryof metals.

N°i2 MAGNETIC OSCILLATIONS: MANY-BODY APPROACH 1825

4. "One-Dimensionalization" ofan

Electron Spectrum ina

Tilted Magnetic Field

[50]

Before proceeding to trie formulation of a perturbative MB approach to trie magnetic prop-

erties in QID metals, let us consider a phenomenon of "one-dimensionalization" of a QID

electron spectrum in a tilted magnetic field [soi. In this section we ignore a small last term in

equation (2) and discuss a quasiclassical equations of electron motion intrie case of a simplified

electron spectrum:

e(p)=

+vF(pa ~ pF) + 2tb cos(pbb*) + 2t~ cos(p~c*) (4)

in a magnetic field H=

(o, Hsinô, Hcosô).Due to small corrugations of trie FS (4), there are basically two projections of trie Lorentz force

which lie in 16, c) plane:

dpb /dt= e

~~H cos à, dp~ /dt

= -e

~~H sin à (5

c c

Using well known quasiclassical equations [59]

?'b =dfiP)/dPb, ~c =

dfiP)/dPc, 16)

it is easy to obtain electron trajectories in a magnetic field:

xi(t)=

~~b~*cosj~abtj, xj(t)

=

-~~~~*cosj~a~tj, j7)

ldb lac

where xf and xfare electron coordinates in (b, c) plane, t is a time: ~db "

eH~Fb* cosà/c, ~d~ =

eH~F c* sin à /c.From equations (7), it is followed that in trie plane perpendicular to trie chains electron

motion is a quasiperiodic and localized within a tube which cross section area, Si(H)=

(2tbb* /~db) x (2t~c*/~dc), is decreasing with increasing of a magnetic field. Using a quantum

mechanical language we can say that electron wave functions are locahzed in 16, c) plane and

are extended only along a axis.

Indeed, let us consider a quantum mechanical problem. In a tilted magnetic field in trie

Landau gouge, A=

(o,Hcos Hz, -Hsin Hz), trie Schrodinger equations are given by Peierls

substitution p - p (e/c)A [59]:

~~~~~ ~ ~~~ ~~~~~~~~ ~Î~~ ~ ~~~ ~~~~~~~~ ~j~~ÎIfi~D

IX, Pb, Pc)"

fÙ(~ IX, pb, p~) (8)

From equation (8), it is easy to derive electron wave functions in (x, pb, pc) representation:

~filD lx, Pb, pc)= exp(+1)) expl+1j sinlPbb* )) )1expl~i

jsin(pcc* +

)))1

(9)

and to convert them into Wannier representation:

ifi) p~(x, y =

ib*,z =

nc*= exp

+1~~ ~~~

~~~~ ~ ~" ~~~~~x)

,~F

x J+jL-i) (Àb/2)J~jN-~j(À~ /2), (10)

where Àb "4tb/~db, À~ =

4t~/~d~.


Using properties of Bessel functions, Jn(x) [60], it is possible to make sure that wave functions(10) exponentially decay in (b,c) plane on trie scales xf

m~Àbb* and xf

m~À~c*. On trie

contrary, wave functions are extended along trie chains and thus energy dependson momentum

pa and two quantum numbers:

AP)"

+VF(Pa ~ PF) ~ i'ldb + 7l'Lac (III

To summarize, in this section we demonstrated that at arbitrary direction of a magneticfield electron motion is becoming effectively ID if we take a simphfied electron spectrum (4).

5. A Change of trie Dimensionality ofan Electron Spectrum at "Magic" Directions

ofa Magnetic Field [36j

Below, we consider a more realistic electron spectrum (2) to describe an important for furtherdevelopment effect of "two-dimensionalization" of an electron spectrum at MA directions of a

magnetic field [36j.Since electron hoppings, tk,i, are small comparable with tb and t~, in trie vicinity of each MA

only one term is important in trie summation in equation (2). Suppose we retain trie term tk,iin equation (2). Let us see what is happened if a magnetic field is applied strictly along trieMA Ùk,i (see Eq. (1)). Due to a conservation of momentum projection along a magnetic field,electrons can move free along trie direction (k,1) if tk,1 ~ o. This motion obviously destroys

trie "one-dimensionalization" of electron spectrum (9)-(11) and results in trie appearance of 2Dproperties since a free electron motion is now allowed along trie chains and along trie direction

of a magnetic field.

Let us demonstrate trie above mentioned crossover between ID and 2D electron motion byusing a quantum mechanical language. In trie case of electron spectrum (2) trie Schrodingerequations can be rewritten as follows:

[~m~~

+ 2tb cos(pbb* "~~) + 2t~ cos(p c* + ~~~)dz

~F~

~F

+ 2tk,1COS(kpbb* + pcc* ~~'~~)jÎÎ~D(X, pb,Pc)"

f~~D(X,Pb,Pc) (~~)~F

They bave trie following solutions:

ÎÎ(~ lx, pb, Pc "~~PÎ~~~~'~ ~~~~ ~ÎÎ~ ~~~~~~~~~ ~ ~~~~ ~~~ ~~~~ ~~'~~ ~~~ ~~~~

where ~l(~ (z, pb, pc) are defined by equation (9); ~dk,i " k~db -~dc =e(~F/c)Hc*

cos (tan Ùk,itan Ù), Àk,1 "

4tk,1/~°k,1.A change of an effective dimensionahty at MA'S is directly followed from equations (13).

Indeed, if trie direction of a magnetic field tends to trie angle= Ùk,i then ~dk,i -

o andÀk,i sin(w)

-2tk,ix/~F. As a result, at

= Ùk,i we bave essentially 2D electron wavefunctions: ~

lYj~(x, pb, Pc "exp[~12tk,ix cos(kpbb* + p~c*)/~~] ifi(~(x, pb, p~ (14)

and 2D electron spectrum:

f(P)"

~~F(Pa ~ FF) + 2tk,1COS(kpbb* + pcc*) ~ (Î' R'k)ùJb (15)

Unhke ID spectrum (11), electron energy at "magic" directions of a magnetic field (15)depends

on two momenta: momentum along trie chains, pa, and momentum along a magneticfield, pH "

kpb + pc.


To summarize this part, we can make a remarkable conclusion. At arbitrary irrational

directions of a magnetic field electron spectrum is effectively ID (see Eq. (Il)), whereas at

rational directions (1) (1.e., at MA'S) electron spectrum is becoming 2D (15). Since tk,i is

quickly decreasing with trie number k, in practice only a few MA'S are important. Therefore,

if a magnetic field is far enough from trie main MA'S we cari regard electrons as eoEectively ID.

6. Two Kinds of "Magic Angles" [36] (The Secondary "Magic Angles" [35, 50])

Let us call MA'S which correspond to negligible values of tk,m in equation (2) trie secondary

MA'S to distinguish them from trie main MA'S. As it is shown in trie next section, trie secondary

MA'S can be defined by a simple inequality tk,m < T. From equation (15), it follows that for

negligible values of tk,m electron spectrum is again becoming effectively ID. It is important

that at "magic" directions of a magnetic field kuJb= mure in equation (11) and thus at trie

secondary "magic" directions of a field ID electron spectrum (11, 15) depends only onoile

quantum number (1'-n'k). This degeneration means that cyclotronic frequencies of an electron

motion along b and c axes in a real space are commensurate and that electron trajectories

in (b,c) plane are periodic (see Eq. (7)). This crossover from a quasiperiodic motion (atarbitrary directions of a field) to a periodic one

(at trie secondary MA directions of a field

[35, 50]) is particularly important in a SDW phase where trie secondary MA'S, Ùi.2 and Hi,3, are

sometimes more pronounced than trie main MA'S [12,16]. As to metalhc phase, trie secondary

MA are visible only as a local maxima of resistivity on a plot of trie second derivative of

magnetoresistance (see Fig. 2).

7. Model

In trie previous sections we described unique phenomena of a change of trie effective dimen-

sionality of electrons in a tilted magnetic field. It is known that "e-e" interactions are dra-

matically increasing with trie decreasing of trie dimensionality. In ID case "e-e" interactions

are becoming singular that leads to Peierls instability at low temperatures and high magnetic

fields [19,45-51].In this review we discuss properties of a metallic phase well above SDW phase transition. Our

message is that trie changes of "e-e" interactions with changing of an effective dimensionahty

define some magnetic properties in a metallic phase. We show that such nontrivial experimental

properties as MA'S and RMO can be prescribed to trie dependence of many-body effects on

effective dimensionality in a magnetic field.

From experiments on SDW formation in a magnetic field [19, 51,61,62], it is known that

basic properties of trie field-induced SDW state are understood within trie framework of a weak

coupling model [45-50]. As it is shown in trie previous sections, trie effective dimensionality of

electrons is decreasing in a magnetic field, therefore anapplicability of trie perturbative MB

approach is becoming not clear. Nevertheless, even in strong magnetic fields, H cf 20 50 T,

there exists an important large parameter which is trie area of trie localization of electron wave

functions in 16, c) plane expressed in terms of trie area of unit cell: Si (H)=

Si (H)/(b*c*)ce

10 20. It is possible to make sure that trie existence of a small parameter 1/S [(H) < 10~~

allows us to consider "e-e" interactions within trie perturbative approach.

8. Mary-Body Theory of "Magic Angles" [36]

It is known that in trie absence of a magnetic field longitudinal resistivity obeys aT~ law in

(TMTSF)~X materials in a broad temperature region. This indicates that "e-e" scattering is


likely a dominant mechanism of resistivity in these QID materials. Moreover, (TMTSF)2Xconductors can be considered as 1:2 commensurate ones that provides trie existence of an effec-

tive mechanism of resistivity trie so-called Umkiapp "e-e" scattering with nonconservation

of trie total momentum: pi + p(=

p( + p( + 4pF Ill. Below, we suggest that Umkiapp scat-

tering is a major mechanism of reiistivity and consider trie influence of changes of an effective

dimensionality in a magnetic field on trie probability of Umkiapp scattering.Let us take a realistic electron spectrum (2) and calculate a longitudinal resistivity in a

magnetic field by means of variational principle for "e-e" scattering [63]. According to [63], to

estimate a longitudinal resistivity it is necessary to average a probability of Umkiapp scattering,U(pi, P21P3, P4),

over electron momenta using Fermi factors, n(e):

Pa ~

l/T"

/~~ dpidpidpidpi U(pi, P2, P3, P416(Pi + p2 P3 P4)

X ôifiPi +filli f(P3 f(P4))n(f(Pi )i"(f(P21) il "(f(P3))1il "(f(P4))1, (16)

where pa m~

1/Tin a QID case; 1/T is an inverse "e-e" relaxation time.

Substituting electron wave functions (13) into equation (16) we get trie following expressionfor resistivity [36]

~ j+" 2~T/~F 2~Tjxj /~F 2~Tjxj /~F j~~~~'~~~ ~~ ~

-~J

~~sinh~(2~Tx/vF) exp(4~T(x(/~F) ~

2~~~~~'

(17)

A=

exp[-2iÀb sin( "~~cos a(sin ai + sin a2 )1, (18)

2

B=

exp(-2iÀ~ sin("~~cos fl[sin(fli koi

~~+ sin(fl2 ka2

~~)l) (19)

2 2 2 '

C=

exp[-2iÀk 1sin( "~'~~ cos(ok + fl)(sin pi + sin fl2 )1, (20)' 2

where gu is a dimensionless constant of Umkiapp "e-e" scattering, .) means trie averagingover all variables o, fl, ai, and flj (1 =

1, 2). (Note that Zeeman splitting is not important whilecalculating a probability of Umkiapp scattering in a magnetic field [35, 36], therefore, we omitelectron spins everywhere).

In an experimentally interesting case, when ~db, ~d~ » T, it is possible to derive some analyt-ical expressions both for resistivity at trie main MA,

= Ùk,i (tk,i W T):

1 jn2~~~~'~~'~~

~ T~~'~~'~~ ~~ÀbÀctk1 ~~~~~~~~ ~~ ~ ~~~

i jn2Pa(H,~k>i)

+~ p(H,~k>1) = 91~~~~~

in~(Àb), ~ùc » tc, (21)

and for magnetoresistance background (when trie direction of a magnetic field is far enoughfrom trie main MA, 69

= jÙ Ùk,i » (T/~d~) < 1):

P~IH'ÙI'~ ÎlH'Ù)"

gl(i~~lÀb) Slù(26j,

Ldc < tc,

Pa(lf,~)~

)(H, Ù) "9~) In~(Àb) Cos Ù, ~d~ » t~. (22)


Pa

,

,

,

'

'

'

,

Ùo,1 #0 Ùii Ùz,1 Ù3,1 Ùi,o ~

7r/2

Fig. 5. A few main MA'Sare

shownm

the plot of the calculated resistive MA phenomenon (seeEq. (17)).

Here, we also present trie expressions for resistivity in trie vicinity of MA, ôÙ < (T/~d~) < 1, [36]

in trie case when Ùk,i is trie main MA:

a(~'~i~ jji '~k>i~=

Ai(ôôj2 ~~°(~~,t~,i » T; (23j

and in trie case when Ùk,i is trie secondary MA:

~~~~Î(HÎiÎI)~~~~

" ~~~~~~~~12~l)~' ~~'~ ~ ~' ~~~~

where Ai, A2 > o are dimensionless parameters; Ai, A2+~

1.

From equations (21) and (22), it follows that in trie case tk,i > T an effective "twc-

dimensionalization" of an electron spectrum (see Eqs. (14, 15)) leads to trie appearance of

2D (T~) temperature dependence of resistivity at trie main "magic" directions of a magneticfield whereas at trie directions of a field which are far enough from trie main MA'S resistivity

possess ID (T~) temperature dependence. This phenomenon results in trie appearance of an-

gular minima of resistivity at trie main MA'S (see Eq. (23) and Fig. 5). In trie opposite case,

T > tk,i, a secondary MA phenomenon (see Sect. 6) leads to trie appearance of maxima of

resistivity (24). Trie prediction of trie main and trie secondary MA'S [36j is in a qualitative

agreement with experimental data on(TMTSF)2Cl04 (12,15j (see Fig. 2 in Ref. [36j)

Below, we also present trie expressions for resistivity at two major "magic" directions of a

magnetic field, Ùa~,1 "~/2 and Ùo,i "

o, that are shghtly dioEerent from trie common equa-

tion (21):~

p(H,Ùa~1)+~

(H, Ùa~,1) "

91~ In~(tb/T) (25)' T tb


Note that at Ù = ~/2 agnetoresistance is egligible inaccordance with trie

data 11-18j,whereas

9. Mary-Body Theory of "Rapid Magnetic Oscillations" [38j

In this section we extend perturbative MB approach to describe another exotic phenomenonRMO of resistivity experimentally observed in (TMTSF)2Cl04. As it is mentioned in

Section 3, RMO of longitudinal resistivity, pa, are completely unexpected within a framework

of standard theory of~metals. Below, we prescribe trie existence of RMO in (TMTSF)~Cl04 to

unusual many-body effect trie oscillations of a probability of "e-e" Umkiapp scattering in a

magnetic field.

For further development it is important that, unlike (TMTSF)2PF6, (TMTSF)2Cl04 bas

AO gap, D, in its electron spectrum (see Fig. 4). By present moment, it bas been established

that when AO gap is suppressed strong RMO of resistivity disappear [24, 64j. This means that

trie appearance of RMO in (TMTSF)2Cl04 is closely related to trie existence of AO gap. Thus

to describe RMO we need to finda solution of quantum mechanical problem in a magnetic

field for some more complicated electron spectrum.Trie AO introduces a periodic potential, D(y)

=D cos(~y/b*), which leads to a doubling of

trie crystalline lattice in y(b) direction. Therefore, in trie presence of trie AO gap there are

four components of an electron wave function which obey trie following Schrodinger equations:

1+~flpa ~ pF) + 2tb coslpbb*)1~flpb) + °~f lpb + ~/b*)=

e~fif (pu ), 127)

1+vF(pa ~ pF) 2tb coslpbb*)li'flpb + ~/b*) + nç~flpb)=

e~flpb + ~/b" (28)

As a result, electron spectrum sphts into four open QID branches:

fn(P) ~VF(P~ ~ FF + (~Î)"~/Î2tb C°S(Pbb*)Î~ + C~, (29)

where n =1, 2 (see Fig. 4).

(All small parameters, t~ ct 5 K and tk,m+~

K, are omitted in equations (27-29)since we

study RMO in a high field metallic state (H > 15 30 T) which is stable at temperatures~T > 17 K > 2t~, 2tk,m).

In a magnetic field H jc

trie Schrodinger equations in trie Landau gauge, A=

(o, Hz, o),take a form [38]

i~i~~j + 2t~ cas(p~b* jj~ii~bt(x,p~) + n~t(xp~ +j)

=eqfit ix, p~i, 130)

~~~~~X

~~~~°~(Pbb* ~~~)j~j+~~ ~

~'~~ ~ ~~~~~ ~ °i~f(x, Pb)=

e~ôj(x, p~ + ~ /~~ ~ ~~~~

where trie amplitudes ifi) lx, pb) and ifi) lx, pb + ~/b*) are related to electron Bloch functions,

lYf~~(X, y)1

~lpb(X, VI "~XP(~Pb#)[lfiÎ(~,Pb) + eXP(~~Y/~~)lfi~(X,Pb + ~/b*)j. (32)

(Note that equations (30-32) demonstrate an effective "one-dimensionahzation" of electrons

which bave QID electron spectrum (29), since electron energy, e, does not depend on momenta

perpendicular to trie chains, pb and p~).


From experimental works on RMO [21-26], it is known that magnetic breakdown phe-

nomenon occurs between two open sheets of FS denoted byn =

1 and n =2 in equation

(29) (see also Fig. 4). Experimental estimation of magnetic breakdown field [23] gives trie

following value: HMB Ù 15 T. If we take a theoretical expression for HMB [65] calculated for

electron spectrum (29):HMB

"

~cU~/2e~Ftbb*, (33)

we can conclude that D m 50 70 K < 2tb ~Î 400 K.

From general theory [59], it is known that perturbative approach to Schrodinger equations

(30, 31) is valid if H > HMB. In this case trie zeroth-order wave functions are a symmetrical

(S) and an antisymmetrical (A) combinations of two solutions of equations (30, 31) in trie

absence of trie AO gap [38,66, 67]:

(~~ '~~~~ j exp

( 1~~

+

2~F

The S and

spectrum (38, 66,6îj:

fÎ~P)"

~~F(Pa

~ + C~,

where

U*m

Ufi@cos(4tbclevFHb*) (38)

is a difference between trie energies of S and A wave functions. Trie absolute value of this

energy sphtting, jD*(H)j, rapidly oscillates with inverse magnetic field:

~ l= ~~~~~~(39)

H 4tbc

At this point we are ready to calculate inverse "e-e" relaxation time by substituting known

electron wave functions (34. 35) and electron spectrum (36, 37) into equation (16). As a result,

we bave trie following formula for a longitudinal resistivity:

~~ ~~~~

~~~ ~~ ~ÎÎ

~~ sinh~ÎÎÎÎ/~F)exp(Î~Îj~Îj~ÎÎÎ)

1

~~~~~~Î~~ ~~

~~ ~~~~~ ~~~~ ÎÎÎ ~~~~~~ ~ ~°~~

Î/~' ~~~~

where we bave used inequality Àb » 1.

Since equation (40) contains trie oscillatory function, D*(H), longitudinal resistivity bave

to possess a strong magnetic oscillations. We note that frequency of trie resistive oscillations

is given by equation (39) and corresponds to area between two open orbits (see Fig. 4). Trie

calculated magnitude of these oscillations possess unusual magnetic field and temperature


dependences. It is important that at high enough temperatures resistive oscillations (39, 40)should demonstrate a power law temperature decay [38] due to their many-body nature:

~~~~~ =El

~Î~~cos(8tbc/~e~fb*) + 82

~Î~~~cos(16tbc/~e~fb* (41)pa( ~Ttb ~Tt~

where El, 82+~

1.

A power law temperature decay (41) clear distinguish trie many-body oscillations (39-41)from all known kinds of one-body oscillations Ii. e., SdH oscillations, so-called, Stark oscillations,and magneto-breakdown ones). Indeed, as it is known [59], SdH and magneto-breakdown

oscillations decay exponentially with temperature whereas Stark oscillations do not dependon temperature per se. We would like to point out that Stark oscillations and many-body

oscillations suggested in this section can coexist in metals with open FS'S such as (TMTSF)2Xconductors. Therefore, we stress that, besides different temperature dependences, there exists

another important difference between these two kinds of oscillations. It is possible to make

sure that equations (40) and (41) are qualitatively applicable at arbitrary direction of an

electron current, I. This means that many-body oscillations (39-41) bave to exist at arbitrarydirections of a current, while Stark oscillations bave to be observed only when I

jjb (see, for

example, [59]). Trie above mentioned features of many-body RMO are in accordance withpreliminary experimental data [68].

In conclusion of this section we note that equation (40) exhibits a weak oscillations, ôpa(H) /pa (Hi < 10~~,

even in trie absence of trie AO gap Ii. e., when D*=

o). This type of oscillations

was early proposed in reference [33]. Recent experiments [24] show that strong RMO (39-41)may coexist with weak oscillations [33] in a metallic state of (TMTSF)~Cl04.

la. Forbidden Magnetic Oscillations:a Break of Fermi Liquid Description [40, 42]

In this section we show that perturbative MB approach to magnetic properties of a QID metalspredicts some phenomenon which is in disagreement with Fermi liquid theory. It is well knownthat Fermi liquid picture does not allow any thermodynamic oscillations to exist in metalswhich bave only open quasiparticle orbits [59]. Below, we demonstrate that a magnetic fielddependence of "e-e" interactions in Peieris channel dramatically changes this statement at

high magnetic fields, H > 20 30 T. We show that, due to trie oscillatory nature of magneto-breakdown electron spectrum (see Eqs. (36, 37) and Fig. 6), a magnetic field dependence of

Peierls interaction resultsin trie appearance of a strong forbidden oscillations of magnetic

susceptibility and magnetic moment [42].A few years ago it was shown [39, 40] that magnetic field dependence of trie contribution to

free energy due to "e-e" interactions in a metallic state con explain thermodynamic MA phe-nomenon [15]. In addition,

in reference [40] a new mechanism of thermodynamical magneticoscillations (which is not related to trie presence of any closed orbits)

was suggested. Unfor-tunately, trie calculated magnitudes of trie oscillations were occured too small to be observed,ôM/Mo

+~

10~3 -10~~ [40].In this section we discuss another mechanism of thermodynamic magnetic oscillations in

QID metals [42] which results in trie appearance of strong forbidden oscillations. According toreference [40], we consider a second order correction to free energy due to "e-e" interactions in a

magnetic field H jc

(see Fig. 7). Unhke reference [40], we study oscillationsin

(TMTSF)2Cl04compound which bas a remarkable oscillatory magneto-breakdown spectrum (36-38).

Below, we describea method of trie calculations of reference [42] and then proceed to trie

discussion of trie final results. Once wave functions and energy spectrum are known (see


e

es

eA

Pa

-kF ÉF

Fig. 6. Electron spectrum of (TMTSF)2Cl04in a magnetic field consists of four lD branches of

FS (see Eqs. (36-38 )) with energy splitting, a*, beingan

oscillatory functions of 1/H.

x x

Pb'Pb ~ F Ù3 ~'~'~'~ ~b*

i

Pb,Pb +) Éà

Pb>Pb +)

r grPb,Pb+j~gPb>Pb+j

i

Pb, pb +) ~p

p~, p~ +)

Fig. 7. A second order correction to free energy due to "e-e" interactions; arrowsstand for electron

spms.

Eqs. (34-38)), one con derive Green functions by using a standard procedure [69]:

jqj/ (x, p~)j*ifi/ IX', Pb)(42)G~(X, X'l

Pb, Pb) ~?~dn

f/(P)e(~~~

Ai


~(ifi/ IX, Pb)Î~lfil(X"Pb ~

j~3)G~ IX, X'( Pb, Pb + p) ~?~dn El (P)

f~I=sAi

where ~dn =2~T(n +1/2) is a fermion Matsubara frequency.

To estimate a second order correction to free energy it is necessary to make a summation

of trie Feynman diagrams corresponding to all possible combinations of G+ lx, x'; pb,pb) andG+ lx, x'; pb, pb + ~/b*) functions which conserve a total electron momentum (see Fig. 7). As a

result of this procedure, we get trie following correction to trie free energy, AF(H), calculated

per one electron [42]:

AF(H)=

~~~~~~~ /~ dp /~dz ~°~)~~~~~~~ Jj [2Àb sin( ~~~ sin(p)] cos~

~~~

,PF~F o dsinh (2~Tx/~F) 2~F ~F

(44)where we bave used inequality Àb » 1 and bave retained only term which does not depend on

Zeeman splitting (see Fig. 7); geR is an effective constant of "e-e" interactions.

Although integral (44) is divergent at small x, all measurable physical quantities such as

magnetic moment, AM=

-d(AF)/dH, and magnetic susceptibility, AK=

d(AM)/dH,possess only a weak logarithmic divergency. This allows us to estimate these quantities with

logarithmic accuracy in trie experimentally interesting case when ~db » T.

Below, we present trie final expressions for relative magnitudes of trie forbidden oscillations:

1'~

ÎÎ ~ÎÎi) ~

~~Î~ ~~~ ~ÎÎÎb~~~ ~~maxÎÎ,

D*j~~~ ~eÎÎb*) ~~~~

ôx 2g)~y ~dbtb HMB~

8~~tb~db 8tbc

Xo ~~ (ILBH)2 H~

~db

~~maxjT,D*j ~°~

e~FHb*'~~~~

where Mo and ~o stand for magnetic moment and magnetic susceptibility of noninteractingelectrons.

Note that constants of "e-e" interactions in a Q ID conductor, gi iii,are strongly renormalized

above SDW(CDW) phase transition (see, for example, [53] and jg~~r ct g~ < 1 at temperaturesT > TsDw(cDw). If we take experimental values of trie parameters ~db, tb, and HMB (~db Ù

3~BH, tb t 200 K, HMB t 15 T),we can estimate trie magnitudes of trie forbidden oscillations

at T ct 6 10 K and H ct 25 35 T as follows:

~~~

m~o-1

,

~~m~

10 (4î)Mo Xo

As it is seen from equation (4î), trie magnitudes of trie forbidden oscillations are unexpectedlylarge. A period of trie forbidden oscillations is given by equation (39). It corresponds to an area

between open sheets of FS (see Fig. 4) that is in a sharp contradiction with Fermi liquid picture.We suggest a high field measurements of magnetic susceptibility and magnetic moment in a

metallic state of (TMTSF)2Cl04 to reveal this possible deviation from Fermi hquid behavior.

11. Conclusion: an Anomalous Metallic State

In trie previous sections we showed that effective dimensionahty of an electron gas is decreasingat trie main "magic" directions of a magnetic field in QID conductors. We demonstrated that"e-e" interactions are dramatically changing with changing of trie effective dimensionahty.Trie explanations of such phenomena as MA'S and RMO

were discussed in terms of these


unusual dimensional crossovers. Following similar ideas, we predicted trie appearance of a

strong forbidden magnetic oscillations. On trie basis of trie above mentioned unique properties

of QID compounds, we formulate a concept of "anomalous metal" a metal which magnetic

properties are defined by magnetic field dependent many-body correlations.

Acknowledgments

I am thankful to Profs. N-N- Bagmet, E-V- Brusse, S-B- Brusse, J-S- Brooks, P-M- Chaikin,

L.P. Gor'kov, D. Jerome, S. Kagoshima, M.V. Kartsovnik, V.M. Laukhin, M. Motokawa, N.

Nagaosa, M.J. Naughton, T. Osada. T. Sasaki. J-R- Schrieffer, T. Toyota, V.M. Yakovenko,

and K. Yamaji for useful discussions. I also wish to thank Prof. M. Motokawa for trie kind

hospitality

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Magnetic Oscillations in a Normal State of Organic Conductors: Many-Body Approach

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