On galvanomagnetic effects in layered conductors

(1)

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On galvanomagnetic effects in layered conductors

V. Peschansky, J.A. Roldan Lopez, Toyi Gnado Yao

To cite this version:

V. Peschansky, J.A. Roldan Lopez, Toyi Gnado Yao. On galvanomagnetic effects in layered con- ductors. Journal de Physique I, EDP Sciences, 1991, 1 (10), pp.1469-1479. �10.1051/jp1:1991220�.

�jpa-00246428�

(2)

J Phys. Ifmnce 1 (1991) 1469 -1479 O3OBRE1991, PAGE 1469

Classification

Physics

Absuacts 7215G 72.20M

Proofs not corrected

by

the authors

On galvanomagnetic effects in layered conductors

VG.

Peschansky,

J.A~ Roldan

Lopez

and

lbyi

Gnado

Yao(*

Kharkov State University, Institute for LOW

lbmperature Physics

and

Engineering

Ukr SSR Academy of Sciences, Kharkov, U.S.S.R.

(Received

24 June 1991,

accepted18

Ju~y1991)

Abstract. In the quasidassical

approximation

_we analyze the

dependences

of the resistivity _p and Hall field _on the magnetic field orientation and

intensity

in layered conductors with quasi-» energy spectrum ^of arbitrary form~ The Shubnikov-de Haas

amplitudes

are shown to drop abruptly and the smooth part ^of

magnetoresistance

to increase at certain orientation of H when the respective extreme

(Fermi surface)

^FS cross section is self-intersecting-

Anisotropy

^of _p, depending essentially on FS topology, manifests itself most when the current flows _across the layers.

In recent years, ^interest ⁱⁿ

physical properties

of

layer

conductors has grown

considerably.

A wide class of

superconductors NbSe2, lbS2 dichalcogenides

of transition

metals, cuprate

based metal oxide

compounds, organic

conductors such as salts of tetrathiafulvalen

~BEDT- TTF~, halogens

of tetraseleniumtetracen

~TSeT),

etc.

represents layered

structures with _a

sharp anisctropy

of electrical

conductivity

in the normal

(non-superconducting)

states.

They

have _an

anomalously

low electrical

conductivity along

the _n direction normal to a certain

plane.

The _ex-

perimentally

observed Shubnikov-de Haas oscillation of _ma

gnetoresistance

in

organic

conductors

(see,

for

example [1-10]

and Refs, there

in)

and metal

type

^of ^electrical

conductivity

in most of them make it

quite

reasonable _to believe that the well founded

concept

^of

charge

^carriers ⁱⁿ metals is also

applicable

for

describing

electron

properties

of the

layered

conductors. Nevertheless

the

only

_way to prove ^the correctness of

using

the

concept

^of

elementary excitations,

similar to conduction electrons in metals, is the solution of the inverse

problem

^of

reconstructing

the Fermi surface

(FS)

from

experimental

data, ln

particularly, topology

of

FS,

can be

reliably

determined

by studying galvanomagnetic phenomena [I ii.

lUet _us consider _a conductor with

quasi-two

dimensional electron energy

spectrum, placed

in d,c, uniform

magnetic

^field II. A small curvature in the direction normal to

layers

is characteristic of all FS

geometries

of

sharply anisotropic

conductors. Therefore it is

quite justified

to make _use in this _case of the strong

coupling approximation,

It _means that _a coefficient An in the

expression

(° l+esent address: High school in Sokode, Togo.

(3)

1470 JOURNAL DE PHYSIQUE I N°10

for the energy of

charge

carriers

£(P)

₌

i~

_An

_{(Pz, Pv)}

CDS

(nPz/hq) (1)

can be assumed to be

rapidly decreasing

^with _n and all

An

at n > I to be much less than Fermi energy SF. Here h is the Planck's constan~

q =

ja

x

bjla(b

_x

c), (2)

and a, b, c the basic vectors of the

single crystal, forming

the

primitive elementary cell,

with _a and b

lying

^{in the}

layer plane (z, y).

A weak

dependence

of the energy ^on ^the ^electron ^momentum

projection

_pz is due to the slow

charge

carriers motion across the

layers.

The

velocity

of this motion

Vz "

°£/°Pz

"

j§(~/h~) An (Pz Py)

^~'~

(~Pz /hg) (3)

b

considerably

smaller than the maximum drift

velocity

_vF of

charge

carriers in the

layer plane.

The condition

v~""

= vo = QVF < vF

(4)

will be assumed below _to be

arrays

satisfied.

The

isoenergetic

surface

s(p)

₌ _s,

depending

_on the

type

^{of the} ^function Ao (Pz>

Py)

can

be either _a system ^of

weakly corrugated cylinders (isolated

or interconnected

by

thin

links),

or

elongated

closed surfaces in the _momentum space, ^when s is _near to the energy band boundaries.

lb determine the electric current

density

11 =

/ d~P2evif(P)/(2~h)~

=

«;>E> (5)

it is necessary ^to ^solve ^the BoltzJnann

equation

for the distribution function of

charge

carders

f(p)

₌

fo(s) eE#

3

folds,

which within _a linear

approximation

~vith

respect

^to ^weak ^electrical

field E and _T

approximation

of collision

integral fi'( f)

=

f fo) /r

has _a rather

sbnple

form

o~/ot

₊

~/r

₌ _v.

(6)

Its solution

1lS =

/~ ^eXP((t' t)/r)vS(t')dt' (7)

makes it

possible

to

represent

^the

components

^of

conducthity

tensor in the

folloming

form

[I Ii

:

a,;

(H, q)

₌

(vii;

₌

)~

~~ / _dpH _/~ _dtvi(t) _/~ _dt'v;(t')

exp

~~'

~~

(8)

It _can also be done in the Fburier

representation

of electron velocities:

(4)

N° lo ON GALVANOMAGNEnC EFFECTS IN LAYERED CONDUCTORS 1471

Here _e, _r and

fo(s)

are the

charge,

the _mean free time and Fermi distribution of conduction electron,

respectively,

t and Q

=

eH/m*c-

the time of its motion and

gyrofrequency

in the magnetic field on the orbit

s(p)

₌ const, _pH ₌

PH/p

₌ coast, m*

(s, pH)

is the

cyclotron

effective mass, c- the

fight velocity

in vacuum, T

= 2~

IQ,

~

T

vj

=

T~~ /

v;

(t)

_exp

(-ikot)dt.

o

In the base of _a two-dimensional energy

spectrum

^of ^conduction ^electrons ^the

conductivity

tensor

components

a;z and az; go ^to ^zero, and Yap

(a, fl

₌ _z,

y) depends only

on the

magnetic

field

projection

Hz ₌ H cos @. In this case the electrical

resistivity

of the

sample along layers,

_p,

depends

on H and in a universal _manner,

namely

p =

p(H

_cos

@)

(10)

One _can

easily

_prove it

by making

_use of the

equations

of

charge

motion in the

magnetic

field

3p~ /0t

₌ _(vy cm vz sin

@)

eH/c; _0py /0t

₌

(v~

cm @)

eH/c;

3pz /3t

=

(v~

^sin _@)

eH/c.

^~~~~

The _z axis is normal to the vector H and _n. In the case of

weakly corrugated

FS the

angular dependence

of the

component

a~p in _a broad range has to be

only slightly

different from the

dependence aap(H

_cos _@) _as

long

as the parameter q remains small and

only

at certain

angles

=

[

the relative value of correction to the resistance of the

type (IQ)

can be of the order of

unity. However,

the

dependence

of _a,z and az; ^on will be

substantially

different because there are no zero order

approximations

in

parameter

q in those components ^{of the}

conductivity

tensor.

I. An

asymptotic expression

for _azz

«zz

(H> n)

₌

~~lill~q~~ (,

_n<

^~~~~

^°~ ^~

_dPH ^'~

_dt

_£~

_{dt' exP}

l~' r

An _(PH>

t)

_x

x

Aj (pH, t')

sin ^~

~

(pH

py

(pH, t)

sin

)lsin

(pH

_Py

(pH, t')

sin _@)

(hq

cos

(hq

_cos

(12)

at q < I _can

easily

be calculated if _one makes _use of the slow

dependence

_{of py}

(pH,t)

₌

py(t)

+

q/hpy

_(pH>

t)

_on _pH at small _q.

lbking

into account this slow

dependence

one gets

only

small corrections _to _azz, which _are of

higher

orders than q2.

Retaining only

terms

proportional

to

q2 one gets the

following asymptotic expression

for _azz

'~~

'

_~

ii3~

~

~~

' /

^~~" ~~~'~~'~~" _~'~~

j~ ^~~~~

^~~"~

^~~~~'~

^~

~ ~°~ (Xn

(i')

_Xn

(i"))

~

(~~)

Xn(l~)

_"

nPv(l~)~i

7 "

(~lr) 'I

_l~

"

iltl

l~' _" ill'

Although components

adz and _aza contain

only

terms linear in _vz

they

_are, nevertheless, of the 6rder of

q2,

since _v~ and pa

(at

_q _-+

0)

are

independent

_{of pH} _so that _at any

magnetic

field

(5)

1472 JOURNAL DE PHYSIQUE I N° lo

there are no terms

proportional

to q in the

expressions

2~2~

^2«aq - # 2« _~ cc

'°~ "

(2~h)3hq ~P~"'°7 ~~'/m

^~~" ^{~~P ~7}~~" ~'~~

~~n

_~PH>h"~ ~

~ ~

(~~ ~)

~j~ ^~ (PH Pv (PH>

i")

SlJ'

~) l,

° '

hq

cos ^'

~

(14)

'~°

2~hq ~~

^~ ^~

~~~'~'~

_~

~

'/

^~~"

^~'

^~~" ^~'~~ ^"°

~~~'~"~

_~

~ ~~~

~~~'~'~

~~~'

~~ i~'~

^~~~

^~~

(15) Therefore,

when the _current flows _across the

layers (j

_ii

n)

the

resistivity

_can be written _as

pit = «Iz~

(16)

to a sufficient accuracy. ^This

resistivity steadily

grows with

magnetic

field and reaches the saturation level at

sufficiently high

field intensities. Since

s(p)

is _an _even function one can reduce _azz in the limit of strong

magnetic

fields

(7

«

1)

to the

following

form:

«zz (CO, Q) =

4e~Tm* (2~h)~~(hq)

^~~ Cm

f (n f~

An_(§~) Cm

(Xn(§~)) §a)

^~

(17)

n=J °

azz

(H)

at ₌ 0is

equal

to its maximum value _azz

(0)

and at small it decreases with

increasing (/hazz

_~- _@2) ^since ^the

integrands

ⁱⁿ ^formula

(17)

are

alternating sign

functions each term in

(17)

can become

arbitrarily

small at > I and _even go to zero at some value of

@, which _are d#ierent for different _q.

However,

the

asymptotic

^value ^of _azz

(co)

at <

~/2

_cannot vanish for any form of the

energy-momentum relationship because,

_as its follows from

(9),

«zz(o)

>

«zz(H)

>

«zz(cc)

=

(olr)

@z = VI-

(18)

Therefore _even if

An

= 0 for _n > 2 the

asymptotic expression

for _azz in strong

magnetic

fields at some values

equal

to

[

is

proportional

not to

q2

but to some

higher

_power of _q. In the latter _case the

higher

terms of

expression

of _azz in ₇ become rather

substantial,

and as

long

as strong

inequality (An+i

<

An

= A#~x

(p~, py))

^is

hold,

the

resistivity

_p _even _{for 7} _« 1 is a

growing

function of

magnetic

field at

(n~

+

A]/A])

«

7~

« 1

(19)

and saturates at

higher magnetic

field than in the case

# [.

This

brings

to a substantial

anisotropy

of

magnetoresistance

when the

magnetic

^field ^devhtes ^from ^the ^normal to the

layer planes.

If the current flows in the

layer plane

ja

₌

&apEp; hap

= Yap

aazazplazz (20)

a considerable

growth

of the

resistivity

with

increasing

H is

possible

if there are thin links between

cylinders

with the thickness of order of q. Such links

give

rise to open ^FS section oriented

along

p~. In this _case ayy ^involves ^a ^term ^{of the} order of «off and the

resistivity

can be written

[iii

as

P " Po

(1

_{+ Q7} ^~

C°S~#) (21)

(6)

N°10 ON GALVANOMAGNEITIC EFFECTS IN LAYERED CONDUCTORS 1473

where

#

is _an

angle

between the

j

vector and the _z

axis,

ao =

ayy(0, q)

_{and po} ₌

ap~

lUet _us _assume that the difference between the numbers of electrons _and holes per ^unit

volume,

N,

is _nonzero.

Making

use of

equation

of

charge

motion

(11)

one can

easily

show that at 7 « 1 and in the absence of open ^electron

trajectories,

the

following

relations _are valid _:

(c~p(r)

_Nec

'~~

(eHT

_cos _@)2' ^'" ^~°~ ^~ ^'~~ ^~~~ ^~ H ^'

'"

~'~~

~ '~~ ^~~~ ~ _~ _'~~ _~'~~~~~~

(e~~~~)2'

^~~~~

«yz «zy "

Cpz(t) /~~ dt'eXp (- ^'()

_Vz

_(t'))

-m

T

and the

resistivity,

^for

example,

for the current

parallel

to z, has the

following asymptotic

form:

jH

^cos

⁾

^~

^(P$r)

^~ ^~ ^"~ ^~ ^~ ^x

P " Pzz ^~ 'YY

Nec

(Ne2r)~

^2~~~

(«~z «z~)~ («yz

+ «zv

~'zz

~~~ ~~~

(23)

X

«zz

Resistance

anisotropy

^is ^manifested

only

in small corrections

proportional

to q2 ^which

depend

rather

substantially

on the orientation of the

magnetic

field with

respect

to the

crystallographic

axes. If a~z az~ ^E£

27a(~,

^and

a(~

^b nonzero, then at _some values of ₌

[

and

magnetic

fields

satisfying

the condition

(19)

when

azz(H)

»

azz(co),

the

resistivity

is

growing

function of the

magnetic

field. In this case the

resistivity

saturation level is reached with

growing

H

only

in _a rather

high field,

when

(azz(H) azz(co))

«

azz(co),

that is when at least the condition

72

<

(q~

⁺

(A2/A,)~)

^b ^satbfied. ^A ^maximum ^{in the}

^angular ^dependence

^p(@) ^appears ^at ^the

same as those at which pjj ^is ^maximal. If a~z

(H)

₌ _az~

(H),

then at ₌ [ the

resistivity

_{p is}

minimum in the

magnetic

fields

satisfjing

the condition

(19).

In

sufficiently strong magnetic

fields the last term in

(23)

is

proportional

to

H~2

_and _thus _may _be _omitted.

In the weak

coupling approximation

in the

layer plane,

which is

widely

used in various calculations

(see

for

example [12-14]),

when

Ao

(p~,p~)

₌

p( /2m,

₊

p( /2m2> (24)

the

resisthity

in the

layer plane

^is

completely independent

^of ^the

magnetic

^field.

Making

use of the relation

(22)

and the

equation (I I)

one can

readily

show that all

components

^{of the} _ai; ^matrix

can be

expressed through

_ao and _azz

72'zy ,~

'"j " -a~y

~ ~'zz 72

(1

_{+ ~i}~~) ^~l ^ta~

~~~~~ ~~ ~ ~~~~~ ^~~ ^~

~z (~+ ^~~2)

_tan _f

~((

~~ ~ ~~~~~ ^~~~ ^~

'

(7)

tan ^~ «072

~_

C"~I

I =

1,

2_

(26)

'~Y ^~ '~~~~

l + 7172 + 7172 ^' ^~ e Hr C°S ~

Rather

simple

calculations result in

following expressions

of the

resistivity,

_p, and the Hall field EHan _:

p =

(ml ^sin~#

⁺ m2

cos~#) /Ne~r;

_EHau

=

(j

_x

H)/Nec (27)

If the electric current is normal _to

layers

then _even in _a rather

simplified

model of electron energy

spectrum

^of

type (24)

the

resisitivity

Pjj "

«£~

+

«i (tan ~)~/ (l

₊

7172) (28)

depends

_on the

magnetic

field

intensity

and

orientation,

and the Hall

components

are

E~ =

jH

sin

@/Nec; Ey

₌ ^0.

(29)

It should be noted that pjj ^is

only weakly

affected

by

the

shape

of the Ao

(Pz

_Py ^function ^and

the formula

(28),

which is valid for any

magnetic

field

intensity,

describes

correctly

at least

qual- itatively

the behavior of

magnetoresistance.

The fact that _p is

independent

of

magnetic

field _as indicated

by

the formula

(27)

is _a consequence ^both ^of ^the

parabolic

form of the momentum de-

pendence

of Ao

(PzPy)

^and of the

asumption

that other

charge

^carrier _groups can be

ignored.

Violation of

only

one of the above condition

gives

rise to _a

magnetoresistance, /hp.

At tan » I when electrons _move _on closed

sharply elongated

orbits the main contribution to the

integral

over ~g in the formula

(17)

comes from _narrow vicinities around the

steady phase points

where _V~ _rz 0, In the case of _a

weakly corrugated

FS there _are

only

two

tuming points

on

the orbit. Those _are

points

at _~g ₌ ~gi, 1~2 where py (~gi =

Pi"

and py (~g2) =

Pi~ Retaining

only

terms of order ofq2 one can

easily

obtain an

explicit dependence

of _azz

(m)

on for

arbitrary

dhpersion

law of

charge

carriers in _a

quasi-two-dimensional ^c6nductor.

The n'th term reaches its minimum _at _an

angle

₌ _@n ^which ^satbfies the condition _n tan @n =

tan

[

=

~)~

^M ^where ^M ^is an

integer.

It _means that the

asymptotic

value

az~(co)

p

is nonzero at any

angle

@, lf _one _assumes that An decreases with

growing

_n as

q~

then the terms

with _n > 2 in the formula

(30)

can be omitted.

Thus,

investigating

the

dependence

of the

resistivity

on the orientation of the

magnetic

field

one can determine the

shape

of the FS section

by

the

plane

_pH ₌ const and _some other details of the electron energy

spectrum.

If b -+ ~

/2

the condition 7 « cannot be satisfied _at any ^value of the

magnetic field,

since the effective

cyclotron

electron _mass m*

=

(2~

_cos

@)-I j

dp~ /v~

in thin _case

diverges.

At

=

r/2

the

resistivity

_p is _a

parabolic

function of H because of the carriers

drifting along

_open tra

jectories.

2. If _q tan > I or in the case, when An decreases with

growing

_n slower than

q~,

then also at G£

I,

the FS _can contain _a self-intersection

point

at

p°,

where

V~

(p°)

_cos _Vz

(p°)

sin

=

0,

V~

(p°)

=

0,

_s

(p°)

₌ _SF

(31)

(8)

N° lo ON GALVANOMAGNEITIC EFFECTS IN LAYERED CONDUCTORS 1475

The rotation

period

of the

charge

carders in _a small

layer

in the

neighbourhood

of the self

intersecting

^orbit _pH ₌

pl diverges logarithmically

when _pH

-+

p(

T ₌

2rQp~ln (pF/

(PH

P~)

_PF =

hq (32)

and the condition of the strong

magnetic

field

(Qr

»

I)

cannot be satbfied at all _pH values. Con- tribution of these conduction electrons to the

conductivity

tensor

components

does not

satisJy

the

Onsager principle

^of ^kinetic coefficient

symmetry

^and can

substantially

affect the _ma

gnetore-

sistance behaviour in

magnetic

fields that

satisJy

the condition

nor

_» I. Here

no

is _a rotation

frequency

^of an electron in

magnetic

^field on an orbit that is removed

sufficiently

far from the

self-intersecting

_one.

Averaging

in the formula

(9)

_{for a,;} over pH values that _are found _on FS in _a small

layer 6p

₌

pH PH resembles

averaging

the

polycrystal

resistance _over all

crystallite

orientations when FS is open ^and ^electron ^rotation

period

on

strongly elongated

orbits may also take

arbitrary

great

Values

[11,lsl.

A fraction of electron orbits

6p/pF

for which Qr < I

exponentially

decreases with the rise of

nor,

_as _can be _seen in the formula

(32). Therefore,

their contribution

Aa,;

to the

conductivity

tensor

components

can be

neglected

^if the

magnetic

field is

sufficiently strong. Nevertheless,

there exists _a

magnetic

field range, ^where

/ha;;

exceeds the contribution of all other electrons to ai;.

Let _us evaluate the contribution of electrons in a,;, for which Qr ^<

I, supposing

that the Fourier

components

^of the electron

velocity

_vj~~ in _a

region

of _not very

large

values of k decrease when k grows:

2x ~'~

vj~~ =

(2K)~~ /

v; (§2) eXp

(-ik§2)d§2

₌

v)

v a;

/k~;

_{q >}

I;

_a; < 1.

(33)

0

Summation _over k _at Qr _« I _can be

replaced by integration

and after _some

relatively simple

calculations _one

gets

at q = I the

following expression

^for

/ha;;

/ha,;

₌ ao

(c;;In£

⁺

D,;QorInln))

^~~ ⁼ ^aooor

^(c;;

⁺

^Di;Inoor) ⁽³⁴⁾

P P PF

Numerical factors ci; = c;, and

D;;

₌

D;,

are of order of

unity,

^ifI and

j

do not coincide with z, and

C,z

and

D,z

_are

proportional

to

q2.

The

resistivity

at

j

I H in _the _case of _a FS section _pH

=

pl _containing _only

_a

_single

self- intersection

poin~

has _a

folloming

form

P = Po

Ii

₊ _{(QOT)~ exP}

_(-Qor)) / Ii

₊

_(Qor)~

_exP

_(-2Qor)

In~

(Qor) (35)

It _can be

easily

seen that the

magnetoresistance

_can both grow ^and ^decrease ^with an increase of H when nor _< 5

p =

P°(~1°~)~

~~P

(~~1°~)

~~

(Q°~)(l'~ (Q°~)

< ~~P

(Q°~)

< (QOT)~

~~~

Po ^at

(nor)

_< _exp

(nor)

The _ma

gnetoresistance

of _some

organic

conductors behaves in _a similar _manner

(see

for _exam-

ple [6,10]).

Anomalous behaviour of the

magnetoresistance

_can also appear at

higher magnetic

field intensities if the

self-intersecting

electron orbit is close to the central FS section. On the FS section

(9)

1476 JOURNAL DE PIiYSIQUE I N° lo

pH = 0 there _can be

only

an even number of self-intersection

points

or none at all

(see Fig. I).

If the FS sections _pH ₌ 0 and _pH ₌

pl

lie close _to each other the electron has to pass ^at ^least

two

"dangerous" part

of its orbit

(in

the

vicinity

of

point

Oi and

02

in the

Fig. 2)

where it _moves

slowly

on the orbit in the

magnetic

field. In that case nor in the index of power ^{of the} exponent ⁱⁿ the formula

(36)

_{for p} should be

replaced by Qor/2,

which widens up ^to

nor

₌ 10 the

magnetic

field range ^where ^the

resistivity

can still be

appreciably

affected

by magnetic

^field ^variations

po

(nor)

^~'

^ln~~

(QOT) exp (QOT

/2)

at exp

(nor /2)

_< _(QOT)~ ln(QOT) < (QOT)~

p =

po(Qor)~

_exp

(-QOT/2)

at

(Qor)~In (nor)

< exp

(QOT/2)

<

(Qor)~

po ^at

(Qor)~

< exp

(Qor/2).

(37)

t~~f~t'

A'S' A'

8'

Q

,

O

j

Fig,

I. section _pH

= const. of Fermi surface corrugated

cylinder

type in the cases when pH = 0 (AA

and

BB')

and at 8

= 0

(CC').

l~° $

~ g

z j

4*° 1 )

( ^~

_g

~<° _~ (

$

i

Fig.

Z

3. Quasi-two-dimensional electron energy

spectrum

^favours an inclusion of _a great number of

charge

carriers into the formation of

quantum oscillatory

effects and therefore results in _a _con-

(10)

siderable increase of the

amplitude

of Shubnikov-de Haas and de Haas-van

AJphen oscillations,

in

comparison

with

isotropic

conductors. In metals the Shubnikov-de Haas effect is

mainly

man-

ifested

by

a

dependence

of the thermoelectric power ^on ^the inverse value of the

magnetic

field

strength

and Shubnikov-de Haas oscillations of the

magnetoresistance

were

experimentally

estab- lished

only

ⁱⁿ semimetals of the bkmuth

type.

The

specific

^feature of FS of

layered

conductors results not

only

ⁱⁿ an increase of the

quantum

oscillation

amplitude

but in their

peculiar

_angu-

lar

dependence

as

wel( espechlly

in the

angle

_range, where

self-intersecting

electron orbits _are certain to be

presenL

In the

quasiclassical approximation

the energy levels of

charge

carriers in the

magnetic

field

~vith closed orbits in the _momentum space can be determined

using

the

Lifshits-Onsager

relation

s

(g pH)

₌

2rheH(n

+

7(n)) /ci

(38)

n

=~o,

1,

2, 3,..

,

0 _<

7(n)

I ₁₁

where S is _an _area of _a section of the

isoenergetical

surface

by

the

plane

_pH ₌ consL We

neglect

here

spin splitting

of conduction electron energy

levels,

that

gives

^rise to Pauli

paramagnetism.

Quantization of the

charge

carrier energy En

(pH)

reduces the appearance ^of

singularities

in the

density

of states of the

charge

carriers

V(£)

_"

2Hl£1(2~h)~~/C f / _{dPH6 (£}

En

(PH))

_'~

L _(3£n _(PH) _/3PH)~~

(«,(pH)"«

(39)

which appear

periodically

with _a

step proportional

to I

/H,

and that is the _catwe of the

quantum

oscillation effects [16,17~. ^Since at n = const we have

05/3pH

+

35/3s

as

/3pH

= 0 it follows that the

singularities

in the

density

of states

v(s)

occur for

charge

^carriers ^with

35/3pH

₌ 0 _or

carriers with

35/3s

= co; those _are conduction electrons with _an extremal section

Se

= S

(pe)

^of

FS,

or electrons whose orbits contain _a self-intersection

point p°.

Far bom the section _pH

=

p(

electron energy

spectrum

ⁱⁿ the

quasiclassical approximation (n

»

I)

is _a sequence ^of

equidistant

levels with _an interval

hoo

between _two

neighbouring

states,

and with ₇

=

1/2.

However in the

neighbourhood

of

p(

_a

rearrangement

ôf êlectron ôrbits ând

changes

of their

connectivity

at pH -+

p~

lead to a considerable

complication

of the energy spectrum and

oscillatory dependence

^of ₇ upon ^H [18].

Azbel,

who has calculated the electron energy

spectrum

near the

saddle-point p°

_on _FS, indicated that their contribution into the quantum os-

cillation effect is

sF/hoc

smaller than that of electrons with the extremal section _area FS

[19].

Using quantum

kinetics

equations

_[17~ or Kubo's method [20] ^it can be

easily

shown that _am-

plitudes

of the Shubnikov-de Haas oscillations of

magnetoresistance,

^formed

by

the above said

electrons,

_are related in _a similar _manner. We don't

present

^here ^details of those calculations and limit ourselves

by

remarks of

general

nature.

Azbel's

theory

^of calculation of the electron energy

spectrum

near the

saddle-points

on FS _can be

readily generalized

on an

arbitrary

number of self-intersection

points

of the same classical orbit in the _momentum space.

The energy

spectrum

^{of the}

charge carrier,

whose orbit contains _two

"dangerous" _parts

situated

closely

to the

saddle-points Oi

and

02 (see Fig. 2),

is determined from condition of

matching

^of

(11)

Schrodinger equation

solutions at the

points

Oi and

02>

which has the

following

form:

cos

(~

^~~~

) _(~/ ^~~

+ _~g

(ki)

_i9

k2))

+ q2 ^cos

(~

^~~~

) _(~~

^~~~

+ _~g

(ki))

⁼

e e

~~ ^~°~

~

_~~~

2e~h

^~~~ ^~

^'~~~~~

~~~~ ~"

~~~~

2e~/ ~~~'

~

T ₊

ik;) ^(4o)

ia

(k;

₌

2k>In"

_{+ fin} _argtan

_(tan

h

(~k> ))

~ T

ik;)

q; =

(2

cos h

(2~k; _))~~~~

_exp

(-~k; _k;

= c

(m; mj (~~ (s

_so;

(pH)) /2eHh;

j

₌ 1, 2.

Here _ml and

m[

are

principal

values of the effective mass tensor in the

saddle-points Oi

and

02,

_so;

(pH)

^is an energy ^on pH

along

a coordinate

line, passing

the

points Oj

and

02

in the

direction

parallel

to the _pH axis.

Calculating

the

oscillating

terms of

thermodynamical

and kinetic characteristics _one has to take

into account that

352/3pH

₌

0,

but

0Si/3pH #

0 and

353/3pH #

0. If the sections _pH

= pe and

pH =

pl

_are close to each other but nevertheless

separated by

_an interval

/hpH

₌ _(Fe

PI

_>

pF

(hoc/sF)"~

,

the

charge

carriers in those sections

neighbourhood

make

independent

contribu- tions _to the

quantum

oscillation effect.

Following Azbel,

_one _can

readily

show that contribution of

electrons with the

spectrum satisfying

the condition

(40),

in the

quantum

oscillation Shubnikov- de Haas

effect,

as in the _case of _a

single "dangerous" section,

is

sF/hoo

times less than the _con-

tribution of electrons with the extremal FS section at infinite _mean free time _r of

charge

car-

rier. As

/£pH

declines the

Dingle

^factor

begins

to

play

a substantial par~

decreasing

_exp

(I /QT)

times the

amplitude

^of oscillations that _are due to electrons with

Se,

and

according

to the formula

(31)

the oscillation

amplitudes

in that _case _are

multiplied by

_a small factor

[/hp/pfl'~~°~

If

/hp

< pF

(hoo/sF)~~

_a substantial

rearrangement

^of energy ^levels ^of

charge

carriers _at the extremal FS section takes

place

and _at

p(

-+ pe their contribution to

quantum

oscillation effect becomes

considerably

smaller. Varhtions of the

magnetic

field orientation may reduce

Ap

to an

arbitrary

small

quantity

and a

sharp

fall of the Shubnikov-de Haas osdllation

amplitude

at variations of _seems to be due to appearance ^of self-intersections

points

on the extremal FS section.

Knowing

the

angle

at which the

quantum

^osdllation

amplitude abruptly

decreases _one can determine _q, I-e- the _extent of

corrugation

of the Fermi surface.

Acknowledgements.

We would like to thank I.F

Shchegolev,

^M.V Kartsovnik and VN. l~aukhin for

helpful

discussions of _our results and

kindly presenting

us information on their research of

galvanomagnetic

^effect ⁱⁿ

organic

conductors.

(12)

References

Ill

KAKWVNIK M-V, LAUKHIN VN., NtJANKOVSKII VI., IGNATIEV A~A~, transverse

magnetoresistance

and shubnikov de Haas oscillations in

organic superconductor

fl

(ET)2IBr2,

Pis'ma Zk

Ekp.

lkor Fiz 47

(1988)

302-3o5.

[2] KABTSOVNIC M.V, KONON0VICH PA., LAUKHIN VN., SHHEGCLEV I-E,

Anisotropy

of magnetoresis-

tance and shubnikov-de Haas oscilations in organic superconductor fl

(ET)2IBr2,

Pis'ma Zk

Eksp.

fiat Fiz All

(1988)

49&501.

[3] OSHIMA K., M0RIT, INOKUCHI H., URAYAMA H., YAMOCHI H., sAtTc C., shubnikov de Haas effect and Fermi surfase in

ambient-pressure organic superconductor [bistetrathiafulvalene] 2Cu(NCS)~,

Phys. ^Rev ^{B 38}

(1988)

938-941.

[4] PARKER I-D-, PIGRAM D-D-, FRIEND R-H-, KuRmo M., DAY P, Observation of de Haas shubnikov and de Haas van Alphen oscillations in fl

(BEDT TTF)~AuI2,

Synth. Met. 27

(1988)

A387-

A39~

[5l ^TOYOTA N., sAsAKJ T, MURXrA Xl, HONDA Y., TOKUMCTC M., BANDO H., KINOSHITA N., ANzAi H.,

IsHtGuRo T, MUTO Y.,

Cyclotron

_mass and

Dingle

temperature ^of ^conduction ^electrons

moving

in

layered

plane of

organic superconductors

fl

(BEDT TTF)2IBr2,

fl

(BDT TTF)213

and _m

(BDT TTF)2Cu(NCS)2,

J Phys. Sac.

Jpn

57

(1988)

2616-2619.

[6j ^KANG W, MONTAMBAUX G., COCI~ER J.R., JEROME D., BATAIL P, LENOIR C., Observation of

giant magnetoresistance

oscillations in the high-Tc phase Of the two-dimensional

organic

conductor fl

(BEDT TTF)213,

Phys. Rev Lea. 62

(1989)

2559-2562.

[7j ^KmmovNic M-V, KoNoNoV1cH PA., LAUKHIN VN., PESCTSKn S.I., SCHEGOLEV I-E,

Galvanomag-

netic

properties

and Fermi surface of the organic conductor fl

(ET)2IBr2,

Zk

Eksp.

fiat Fiz 97

(1990)

1305-1315.

[8] SCHEGOLEV I-E, KONON0VICH PA., KARrSOVNIC M-V, LAUKHIN VN., PESCTSKII S.1., ^HILTI B., MAYER C-W, Shubnikov de Haas oscillations and magnetoresistance in semi metallic state of

(lSeT)2Cl, Sj,ntk

Met 39

(1990)

357-363.

[9] TOKUMCTC M., SWANSON A-G-, BRmKS J.S., AGOSTA C-C-, HANNAHS S-T, RINOSHrrA N., ANzM H., ANDERSON J-R-, Direct observation of

spin

splitting of Shubnikov de Haas oscilations in a quasi-

two- dimensional organic conductor

(BEDT TTF)~KHg(SCN)~,

J Phys. Sm. Jpn ⁵⁹

(1990)

2324-2327.

[10] ^OSADA T, YAGI R., KAGCSHIMA S., MIURA N., OSHIMA M., SAtTC G.,

High-field

_ma gnetotransport ⁱⁿ

an

Organic

conductor

(BEDT TTF)2KHg(SNC)4, Praceding

of ISSP International Symposium

on the

Physics

and

Chemistry

of

Organic Superconductors (Springer-Verlag,

1990)p. 1-12.

[11] ^LIFSHITz I-M-, ^PESCHANSKY VG., Galvanomagnetic characteristics of metals with open ^Fermi ^Sur- faces, ^Zk

Eksp.

fiat Fiz 35

(1958)

^1251-1264.

[12] ^PESCHANSKY VG., DASSANAYAKE M., TSYBUUNA E-V,

Weakly damped

Reuter-Sondheimer _waves in

anisotropic

conductors, Fiz Nisk

lkmp.

II

(1985)

^297-304.

[13] ^YAMAGJI K., On the angle dependence ^of the magnetoresistance in

quasi-two-dimensional

_supercon- ductors, J Phys. Sm.

Jpn

58

(1989)

^1520-1523.

[14]^YAGI R., IYE Y., OSADA T, KAGOSHIMA S., Semiclassical Interpretaton of Angular

Dependent

Ossi-

latory Magnetoresistance in Quasi-Two-Dimensional

Systems,

J Phys. Sm. Jpn 59

(1990)

3069.

[15l ^ZIMAN J-M-,

Galvanomagnetic properties

of cylindrical Fermi surface, Phdbs.

Map

3

(1958)

1117- 1127.

[16j LANDAU L-D-, On the de Haas _van Alphen effect, l+m. Rt~z. Sm. ^A170

(1939)

363-364.

[17j ^LIFSHITz I.M., KosEV1cH A-M-, On the theory of Shubnikov- de Haas Effect, ZIL

Eksp,

lkor Fiz ₃₉

(1960)

1276-1285.

[18] AzBEL M.Ya., Quasidassical

quantization

near the

special

classical

trajectories,

Zk

Eksp,

7bor Fiz 39

(1960)

878-887.

[19] ^AzBEL M.Ya., Quantum oscillations Of thermodynamical

quantities

at the arbitrary Fermi surface, Zk

Eksp.

lkor Fiz 39

(1960)

878-887.

[20] ^KuBc R., HASEGAWA H., HASHITSUME N., Quantum theory ^Of galvanomagnetic phenomenon, J Phys.

Sm.

Jpn

14

(1959)

56-76.