Angle-Dependent Magnetoresistance in Organic Metals

HAL Id: jpa-00247284

https://hal.archives-ouvertes.fr/jpa-00247284

Submitted on 1 Jan 1996

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Angle-Dependent Magnetoresistance in Organic Metals

Stephen Blundell, John Singleton

To cite this version:

Stephen Blundell, John Singleton. Angle-Dependent Magnetoresistance in Organic Metals. Journal

de Physique I, EDP Sciences, 1996, 6 (12), pp.1837-1847. �10.1051/jp1:1996191�. �jpa-00247284�

Angle.Dependent Magnetoresistance in Organic Metals Stephen

J. Blundell

(*)

and John

Singleton

Department

of

Physics,

University of Oxford, Clarendon

Laboratory.

Parks

Road,

Oxford OXI 3PU, UK

(Received

17

April

1996, revised ii

July1996, accepted

19

August 1996)

PACS.71.20.Rv Polymers and

organic compounds

PACS.71.18.-y

Fermi surface: calculations and measurements; ^effective mass, g factor

PACS.72.15.Gd

Galvanomagnetic

and other magnetotransport ^effects

Abstract. Recent

experimental

studies of trie

angle-dependent

magnetoresistance in vanous

organic

metals bave been remarkably successful in

elucidating

trie nature of trie low-temperature

ground

state and

providing

information about trie Fermi surface shape which is hard _or impos- sible to obtain

using

other

techniques.

We _{review vanous} theoretical

approaches

to describe

angle-dependent

magnetoresistance ^and a number of important

experimental

results which bave been obtained.

1. introduction

A

key question

in trie

physics

of _any of trie

organic

metals and

superconductors

^{which bave} been discovered in trie last

twenty

_years concerns trie nature of trie electronic bandstructure and trie

shape

of trie Fermi surface

(FS).

A vital issue to be resolved for each material _is trie extent to which _a Fermi

liquid description

is

applicable.

To this

end,

various

magnetoresistance (MR) experiments

^{bave been}

particularly

useful in

elucidating

trie nature of trie

low-temperature ground

state and. where _a Fermi

liquid approach

_appears

valid, determining

trie

shape

of trie FS.

Trie

experimental

use of MR

techniques

to determine trie FS of various metals bas _a

long

and successful

history il, _2].

Trie _presence of _open and closed orbits _can be

easily distinguished by

trie field

dependence

of trie MR while trie _area of trie FS

pockets

_can be measured

by

trie

frequency

of Shubnikov-de Haas

(SdH)

oscillations [2]. ^These

techniques

are now

routinely applied

to

organic

metals [3]. SdH oscillations _can be used to determine information

concerning

trie FS

geometry

^{because trie}

frequency

of each series of oscillations is

proportional

to trie _area of _an extremai orbit _on trie FS [2]- ^The orbits which _are not extremal do not contribute to this

oscillatory signal

but give use to _a

non-oscillatory background

MR-

However,

this

background

MR _can

depend quite dramatically

_on trie direction of trie

apphed magnetic field,

and in _some

cases very

large angle-dependent magnetoresistance

oscillations

(AMROS)

at constant field _can

be found.

Experimentally,

AMROS _are measured

by rotating

a

sample

in a fixed

magnetic

field while

monitoring

^trie

resistivity

of trie

sample.

AMROS _can be observed at much

higher temperatures (*)

Author for

correspondence je-mail: sjb@vax.ox.ac.uk)

©

Les

Éditions

_de

Physique

1996

and in much lower

applied

fields than SdH oscillations. This is because SdH oscillations _anse

from trie movement of Landau levels

through

trie Fermi energy

(EF)

and therefore _require

that trie temperature ^{is low}

enough

for trie FS to be

sharply defined;

this restriction does not

apply

_so

stringently

to AMROS since

they

do _not

originate

from trie motion of _energy levels

through

trie FS- The information obtained from AMROS _can therefore be

complementary

to SdH oscillations since trie effect is due to all electrons _on trie

FS,

not

just

those

performing

extremal orbits.

Trie

technique

of _measuring trie

dependence

of trie resistance _on trie direction of _an

applied magnetic

field bas

proved

useful in trie

study

of conventional metals

(see

Sect. 2

below)

and also in trie

study

of _various

ferromagnetic films,

due to trie

~anisotropic magnetoresistance'

effect

[4]. However,

it bas

recently produced striking

results in _a number of

organic

metals.

Organic

metals and

superconductors (for

_{reviews, see}

là-?]

_are

particularly

attractive

systems

to

study

because their bandstructures _are

extremely

well defined and

samples

_can often be of

exceptional

chemical

purity.

This article reviews AMROS in

organic

metals and is

arrangea

_as follows: _in Section 2 _we

briefly

_summarise trie

principles

behind AMROS _in

general

three-dimensional Fermi

surfaces;

in Section 3 these results _are

specialised

to trie _case when trie FS consists of

arbitrarily corrugated

sheets

(quasi-one-dimensional FS);

in Section 4 _we consider trie _case of

corrugated cylinders (quasi-two-dimensional FS).

As will be apparent from trie cited

references,

Professor

Schegolev

and bis group bave made many important contributions to this field-

2. 3D Fermi Surfaces

In order _to calculate

galvanomagnetic

effects in _a

metal,

_{a necessary}

preliminary

is to under- stand which electron orbits _are

possible

_across trie FS for _{a given} orientation of trie

magnetic

field [8].

Then,

trie

conductivity

_{a~j can} be calculated

using

trie Boltzmann

transport equation:

~2

ôf~(~j

⁰

'"

4~3

Îs ~~~

ôE(k)

~~~~'~~

Î_~ ~~~~'~~~~~~~~'

_~~~

where _T is trie relaxation

time, ~i(k, t)

_is trie i~~ component ^{of trie}

velocity

of _an electron with

wave vector k at time t, ^and

fo(k)

=

(e(~(~)~~F)/~B~'+1)~~

is trie Fermi function. This

is an

integral (over

all states at trie

FS)

of trie

velocity-velocity

correlation function for each FS orbit. This _can

change dramatically

_as trie direction of trie

magnetic

field is

changed,

because this alters trie

patins

of all trie FS orbits. It _can be

particularly

sensitive to whether trie orbits _are open or

closed,

_so that these two _cases may be

distinguished by differing

MR

behaviour

[8,9]-

For

multiply

connected

(or necked)

Fermi

surfaces,

_some directions of trie

magnetic

field result in closed orbits

(Fig. la)

whereas others lead to open orbits

(Fig. lb)-

The consequence of trie former is that trie

resistivity

either _increases

quadratically

with field

or saturates

(depending

_on whether trie metal _is

compensated

or

uncompensated)

whereas

trie latter result in _an anisotropic

resistivity:

such measurements

give

new information _not available from conventional

quantum-oscillation

measurements [9]. ^{This idea bas been used} to

great

^{effect in}

determining

trie FS of _e-g- tir

[loi

and copper

[11] by rotating single crystal samples

_{in an}

applied magnetic

field.

In

Figure

la _we show _a closed orbit around _a

cylinder

of FS which lies

along

trie _z direction.

Trie

velocity v(k)

is

always

normal to trie FS _so that _as trie

magnetic

field B

increases,

trie

velocity-velocity

correlation function decreases _as trie components of trie

velocity

_{are very}

effectively averaged.

At

high fields,

this falls _as

B~~

_so that in _a

compensated

metal

(where

a~y =

o)

like bismuth. _p~~, p~~ c~

B~ (this

_{occurs in magnesium} and _zinc

also,

but _{at very}

(ai (b)

~

v(k)

z

x~-y

Fig.

l-

(a)

A closed orbit

(b)

An open orbit-

high

fields in these materials trie

quadratic

increase is

interrupted by magnetic

^breakdown

Ill

In _an

uncompensated ^metal,

a~y c~

B~~

_at

high

^fields so that _p~~ and pyy ^{saturate at}

high

fields

[ii.

For _{an open} orbit

along

z

(see Fig. lb),

with B

along

trie _x

direction,

trie

component

of

velocity along

z is

similarly averaged (so

that _{pzz c~}

B~

but trie

component

^of

velocity along

y

quickly

reaches a non-zero average value _as B increases

(so

that p~~

quickly saturates) iii.

3. Fermi Sheets

In this section _we consider AMROS from

quasi-one-dimensional (quasi-1D)

^FSS i-e- a FS

consisting only

of

sheets;

our discussion follows that of reference

[12].

^We assume trie

following

energy

dispersion

^relation:

E(k)/à

m

~~(jk~j k~) 1(k~, k~). (2)

The FS

(defined by E(k)

₌

o)

consists of two sheets at k~ ₌

+kF

which _are

warped

_or

corrugated

in _a manner described

by

trie function

f (assume

that

f(k~, kz)j

<

kF~F

_so that trie two sheets are

only slightly warped

and do not

touch).

Trie

velocity v(k)

of each electron

as a function of momentum k can then be calculated

using

v(k)

₌

h~~(dE(kj/dk)

=

+~Fê~ (à ilôky)êy (ôilôkz)êz (3)

where

ê~, ê~

^and

êz

_are unit vectors in trie

k~,

_k~ and k~ directions

respectively.

^Trie

^velocity

will be

time-dependent

since trie electron's momentum k varies with time

according

to trie

equation

^{of motion}

hi

= -ev x B.

(4)

This

equation

^of motion

implies

that k _remains

perpendicular

to v; this condition ensures that trie electron remains _on trie FS. For this _reason, we need

only

to calculate trie

equation

^of

motion for

kjj

=

(o, ky, kz).

Therefore _we find that trie

equation

^{of motion} can be written as

Éjj ⁼

(e/h)V4l

x

ê~

where 4l

=

~F(Byky

+

Bzkz)

+

B~ f

_so that trie electron motion _is

along

contour lines of constant 4l. In trie

simple

_case where trie

magnetic

^{field lies} in trie

plane

of

la) _{16) lC)}

tan ₌

1/2

_tan

=

12/23

tan

=

K/6

~l

/2

Il

2

/23

~

K/6

~

l

Fig.

2. If trie

magnetic

field lies in the

plane

of the sheets all orbits become

straight

lines when projected _on to the

ky-kz

plane.

Depending

_on whether tan 0 is rational

(a,b)

_or irrational

(c)

the orbits _on the FS will be either

periodic

_or

ergodically

fill the Fermi sheet in the reduced Brillouin _zone

respectively.

trie sheets

(B~

=

o),

^electrons _appear to travel in

straight

lines _across trie Fermi sheet in _a direction

perpendicular

to B

=

(o, By, Bz)

when viewed

along k~ (see Fig. 2).

We shall _write trie

corrugation

function

f(ky, kz)

in Fourier components:

Elki

=

h~F('k~' kfi ~

_tmn

_COS(Rmn

kil

⁺

~mni (si

where

Rmn

_are

appropriately

defined

real-space

vectors which define trie

corrugations

and ç2mn are

phase

factors. We shall let trie sheets bave

arbitrary

orientation with

respect

to trie

crystallographic

_axes

jour

coordinates _are

aligned

with trie

sheets,

not trie

underlying crystal)

so trie

requirement

of translational

symmetry implies only

that trie _vectors

Rmn

should lie

on trie _most

general

2D

lattice,

trie

oblique

lattice. _Thus without loss of

generality,

_we set

Rmn

₌

(o,

mb ₊

nd, nc) _[12]

Furthermore _{we set} toc ₌ o _since this term

only produces

_a shift in trie Fermi energy.

When trie

magnetic

field lies in trie

plane

of trie sheets

(B~

= o _so that trie

magnetic

field B is given

by

B

=

(o, BsinÙ,

_{Bcos Hi}

),

all orbits become

straight

lines when

projected

on to trie

ky-kz plane,

all

lying

_{in a} direction

perpendicular

to trie

magnetic

field.

Then, depending

on whether _tan is rational

(equal

to

m/n

where

m and _{n are}

integers

and _n

~

o,

Figs-

2a and

b)

_or irrational

(as

will almost

always

be trie case,

Fig. 2c),

trie orbits _on trie FS will be either

periodic (with period ni

_or

ergodically

fill trie Fermi sheet _in trie reduced Brillouin _zone

respectively. Hence,

trie

type

^{of orbit}

depends

_upon trie value of in _an

extremely

sensitive

fashion!

If

by

some mechanism trie

conductivity depends

_{to some}

degree

_on this

topological _property

of electron

orbits,

trie presence of

conductivity

_{maxima or} minima is

expected

at

particular

'magic angles'.

Trie

question

^{of trie} mechanism

by

which trie

conductivity might couple

to trie

degree

of

periodicity

of trie electron orbits _is of _course trie heart of trie _matter-

(a) (b)

Fig.

3. If magnetic field lies _in trie

plane

of trie

quasi-iD

sheets ail orbits become

straight

fines when

projected

_on to the

ky-kz

plane. For _a

given

Fourier component of the

corrugation,

the

velocity

is more

effectively averaged

when electrons

(a)

_are not

travelling along

the _axis of the

corrugation

than

(b)

when

they

_are.

Applying

trie Boltzmann transport

equation (1)

to trie

equation

of trie Fermi sheets

là)

_we

find that

[12,13]

at T ₌ o

a~~ =

e~g(EF)~)T (6)

and a~y = ay~ ⁼ a~z = az~ = o but

layy

^ayz

g(EF )e~T ~ ^(mb

⁺

^{ndj2 (mb}

⁺

^{ndjnc t$~} _~~j

az~ azz h~

(mb

₊

nd)nc (nc)~

l +

(GmnT.uF)~

m>n

where

Gmn

=

~/ ((mb

+

nd)

_{cos nc} sin

Hi. (8)

The

dependence

is contained in

Gmn

and maxima in trie

conductivity

_are ^observed whenever

Gmn

₌

0,1-e-

tan ₌ ^~ x +

~. (9)

n c c

This _is trie

'magic angle'

condition described above

(trie

factor

d/c expressing

trie fact that trie Brillouin _{zone is} not

rectangular

but defined _{on an}

oblique

lattice [12]

).

Recall that trie electron motion in this

geometry

_is

along straight

^lines when viewed

along k~.

Then trie AMROS in this _case arise from trie fact that for any Fourier

component

of

corrugation,

trie

velocity

is _more

effectively averaged

when electrons _are not

travelling along

trie axis of trie

corrugation (Fig. 3a)

than when

they

_are

(Fig. 3b);

thus

sharp

resistance minima _are obtained when trie orbits _run

along

a Fourier

component

^{of trie}

corrugation.

Thus in this model it

is not _so much that these

"magic angles" correspond

to

periodic orbits,

but that trie

angle corresponds

to _a

particular

Fourier

component

^of trie FS

corrugation.

Trie contribution of _an orbit at _a

particular

value of is

governed by

tmn

(where

_m and _{n are} related to

by Eq. (9))

and will

produce

no resonance if tmn = 0. Thus trie

geometry

of trie Fermi

sheet, parameterized by

trie Fourier components tmn ^{of trie}

corrugation, entirely

controls trie AMROS.

A

special

_case of

equation (5)

is often studied in which trie

only

_non-zero Fourier components of trie

corrugation

_are tic _"

2tb

and

toi

_"

2tc (and

_we set all ç2mn =

0).

^{This model bas been}

extensively

used to

study

trie

(TMTSF)2X family

of

quasi-one-dimensional

_organic ^conductors

in which trie two sheets _are

only weakly

modulated

by

a

single

^Fourier component ^{in each} of

trie _y and _z directions

(here

TMTSF is

tetramethyltetraselenafulvalene

and X

=

Cl04, PF6).

Trie

amplitude

of trie modulation is determined

by

trie

tight binding

transfer

integrals

tb and tc, where in TMTSF salts their ratio is

typically given by tb/tc

m~ 30. In this _case

equation (7) implies

_some smooth

angle-dependence

in trie MR but _no AMRO features.

In fact it _seems that

quasi-ID

Fermi sheets in

organic

materials _are almost

always

insuf-

ficiently corrugated

to

give strong

AMRO features- This is because trie tmn are related to transfer

integrals

and these fall off _very

quickly

with distance _{as one} goes

beyond

trie nearest

neighbour

level

[12].

^It is sometimes the _case that

organic

metals with

quasi-ID

_open sections also contain

cylinders

of FS _or

quasi-two-dimensional (quasi-2D)

sections

(though

this _is not the _case with TMTSF

salts).

In this case, there may be observed AMROS associated with closed orbits around trie

quasi-2D

sections of FS

(see

Sect-

4),

trie

quasi-ID

sheets

contributing only

_a

slowly varying background-

In such _{cases, one can}

imagine

that trie

conductivity

is a

sum of

appropriately weighted

contributions from different sections of trie FS.

If _an

organic

material bas

only

_a

weakly warped

Fermi sheet

(as

in trie _case of trie TMTSF

softs),

AMROS _are

only expected

to be observed if trie

magnetic

field is rotated close to trie direction

perpendicular

to trie sheets

(1.e.

close to trie

highly conducting direction).

In this

case oscillations _may be observed which _are associated with _open orbits which _weave between islands of closed orbits around trie local -maxima and minima

(around

trie local hillocks and

valleys

on trie sheets

[14],

see also

il 2j)

These oscillations bave been

experimentally

observed

in

(TMTSF)2Cl04 by Danner, Wang

and Chaikin

by rotating

the

magnetic

field close to the

a-axis

[14j.

However,

even

though

the sheets in TMTSF salts _are believed _to be

relatively smooth,

and there _are believed to be _no closed

pockets

in the

FS, sharp

AMROS _are observed _even when the

magnetic

field is in the

plane

of the sheets

[15-17j.

If the effect is _a semiclassical _one, then either the sheets _are much _more

corrugated

than is

presently thought (this

is

probably unlikely

because it would

imply large

transfer

integrals corresponding

to interactions _{over a} much

greater

distance than

just nearest-neighbour)

_or trie

scattering

on trie FS is

strongly k-dependent,

1-e- there _exist FS

"bot-spots" _[18j. Assuming

trie latter

model,

at a magic

angle

trie

periodic

electron motion in trie reduced _zone scheme allows _some fraction of trie electrons _to miss trie

bot-spot;

at unmagic

angles

trie electron

trajectories

_are incommensurate and

ergodically

_sweep out trie whole FS, all electrons

scattering strongly

at trie

bot-spots _[18]

Another

explanation

of trie

sharp

AMRO

dips

bas been to

ignore bot-spots

and

keep only

nearest

neighbour

transfer

integrals

but to _{use a} non-linearized band model

[19]

^thus

higher

order

corrugations

in trie

shape

of

quasi-

^ID Fermi sheets _are induded

imphcitly.

Alternatively,

trie effect _may be outside trie _scope of _a semiclassical

explanation.

Lebed _ar-

gued

^first that at

magic angles

the effective

dimensionality

of the electron-electron interactions

is

reduced, thereby changing

trie

scattering [20]

^More

recently,

it bas been

proposed

that trie

magnetic

^field renormahzes trie coherent part of trie _c-axis

hopping

to _zero

[21];

^{in this latter}

model, hopping parallel

_to trie

magnetic

^{field is}

unaoEected,

_so that trie

dips

_are

predicted

when trie field

points along

_a real space lattice vector

(see

also

[22j

)-

However, although

trie

interpretation

of AMROS in trie TMTSF softs is rather

comphcated,

trie _case of

(ET)2MHg(SCN)4 (M

=

K, Tl, Rb)

is rather dioEerent-

(Here

ET is

bis(ethylene- dithio)tetrathiafulvalene,

known also _as

BEDT-TTF).

These salts consist of _a sandwich struc- ture of alternate

layers (along

_ac

planes)

of ET molecules and

MHg(SCN)j

_aurons. Trie ET

molecules _are linked to each other _in these

planes by overlap

of their molecular ~-orbitals and

they

stack

alongside

_one another.

They

_are

separated

in trie b* direction

by

sheets of trie anion

MHg(SCN)/,

to form _a two-dimensional

(2D)

conductive network. Trie resistance _is therefore niuch

greater

^with trie current _across trie

planes (parallel

_to trie b*

direction)

than with it in trie _ac

plane.

Trie most

extensively

studied salt in this

family

bas M

= K and in trie

following

k~

(a)

v 16)

Fig.

4. Candidate Fermi surfaces for

o-ET2KHg(SCN)4. la)

Calculated FS

(after _[30]) consisting

of _a 2D closed noie

pocket

and _a pair of iD

planar

FS sheets. This is believed to be

probably

valid above TN _or in fields greater ^than m~ 22 T.

(b)

Below

TN,

the FS is

thought

to be nested

by

_a SDW _as shown

(nesting

vector

Q) resulting

in the destruction of the iD

planar

FS sheets and the formation of new inclined open sheets and small closed

pockets

from the 2D closed pocket in the original FS.

(from [27]).

discussion _we will restrict _our attention to this

material, although

trie

arguments apply

at least

qualitatively

to trie M ₌

Tl,

Rb salts. In _a

transport experiment

on

(ET)2KHg(SCN)4,

resistance is measured with

voltage

contacts

placed

_on both _ac

(conducting plane) platelet

faces

[23]

^and very

large

AMRO effects bave been observed

[23-26].

This material is very

interesting

in trie _context of trie _current

review,

_as it _can exhibit either

quasi-iD

_or

quasi-2D AMRO, depending

on trie

experimental temperature

and field. Trie mechanism which allows trie M

= K Salt _to behave in this way was

originally proposed by

Kartsovnik et ai.

[27, 28];

it is based _on trie calculated Fermi surface of Mori et ai.

[29, 30]

(Fig. 4a),

which consists of _a

quasi-2D

hole

pocket

and _a

pair

of

quasi-ID

sheets

[31]

^{At low}

temperatures,

^below trie Néel

temperature TN,

_a

spin-density

_wave

(SDW)

is

thought

to nest trie FS _so that

only

_some small

pockets

and _a

quasi-

ID section of trie FS remains

[27] (Fig. 4b);

this

quasi-ID

section is

thought

to be inclined at _m~ 21° to trie

crystallographic

b*c

plane.

This reconstructed

ground

state can be

destroyed by temperature (above TN)

or

magnetic

^field

(above

about

Bkink m~22-24T,

at trie so-called 'kink' transition

[35])

so that trie FS reverts

back to trie

predicted

form. This

picture

^is

supported by

AMRO

experiments:

trie observed AMROS _are found to be

quasi-ID

in trie nested

region (T

<

TN

and B _<

Bkink)

and

quasi-2D

outside it

(T

_>

_{TN [36]}

_or B >

Bk;nk

_123,

24j).

An

example

of this is shown in

Figure

5 which shows AMROS in

o-ET~KHg(SCN)4

for different values of trie

magnetic

field. At trie lowest field shown there _are

only

AMRO

dips corresponding

to _a

highly corrugated quasi-ID

sheet.

As trie field increases

through Bk;nk,

these

dips disappear

and _are

replaced by Sharp

AMRO

peaks

characteristic of _a

quasi-2D

section of FS

(see

Sect.

4) _[23] (see

also

[33]).

At low

temperatures

and at fields below trie kink in

a-ET2KHg(SCN)4

trie

nesting

of trie

FS _removes trie

weakly corrugated ^quasi-ID

^{sections of FS and} "cuts up and

glues together"

pieces

of

quasi-2D

FS. Trie reconstructed FS then consists of _{a very}

highly corrugated

Fermi sheet _since it consists of _a

periodic assembly

of

cyhndrical sections;

trie

corrugation

thus bas

a very

high

harmonic content. In consequence, trie tmn should _not be

interpreted

here _as transfer

integrals

_so much _as Fourier components ^{of trie}

corrugation. Although

_some

quasi-2D pockets

_are also formed in trie reconstructed

FS,

these do _not contribute to trie AMROS. This

is

perhaps

because

they

themselves _are

irregularly corrugated;

trie

quasi-2D

AMRO eoEect relies

on

cylindrical

sections of FS with

relatively

weak and

reguiar corrugation-

The

obliqueness

of

T

T

~ ~

© É

Éfi

~

T

T T

5T

~90

6

(degàees)

Fig.

5. AMROS

in

o-ET2KHgjSCN)4

at i-à K for several values of the

applied magnetic

field

(after [33]).

^{Below Bk~nk the data} are characteristic of

a

quasi-iD

sheet, with

sharp

dips and _a broad

maximum near 0

= o

(see [12]).

As the field increases

through

Bk~nk these

dips disappear

and _are replaced by sharp AMRO peaks and the broad maximum

near 0

= o

disappears.

This _is characteristic of _a quasi-2D section of FS. SdH oscillations aise appear on the

higher

field data at (ô( ~ ^50°.

trie lattice _on which trie

corrugations

_are defined reflects trie structure of trie

quasi-iD

Fermi sheet which is at an

angle

_(m~

21°)

to trie

(triclinic) crystallographic

_axes in

(ET)2KHg(SCN)4 (specifically,

trie b*c

plane).

Trie information obtained from AMRO

experiments

^about trie

vectors

Rmn

concerns trie

geometry

of trie

corrugations

of trie Fermi sheet

produced by

trie

SDW

nesting.

We also note that it

possible

to describe trie

quasi-ID

AMROS in this salt

by considering only

a

quasi-2D

section of FS and

treating

trie effect of _a

periodic potential (due

_to trie assumed

SDW) _using

_a Landau level

description [37].

We

strongly prefer

_a semiclassical

description

since it

provides

not

only predicted

conductivities that _can be

directly compared

with

experiment [12],

^{but is valid} even at much

higher temperatures

where _a Landau level

description

_{is quite}

inappropriate

and where

reasonably strong

AMROS _are still observed in

experiments.

^One of trie attractions of AMRO studies is that trie effect _can be _{seen in}

samples

_in which SdH oscillations

(resulting directly

from distinct Landau levels _moving

through EF

_are unobservable

(because

of either trie

temperature being

too

high

_or trie

sample quality being

too

low).

4. Fermi

Cylinders

Consider _now trie AMROS due to a

warped cylindrical

FS

pocket lying along

trie

kz

direction.

This situation

apphes

to many salts of BEDT-TTF- The electron

dispersion

_{can in} this _case be written as

E(k)

=

£(kj 2tcos(kzd) (10)

where

kjj

=

(k~,k~)

and

kz

_are

respectively

trie

components

^{of trie} wave vector

parallel

and

perpendicular

to trie

conducting planes.

If trie

magnetic

field is

perpendicular

to trie

planes,

bath neck and

belly

orbits will _occur around trie FS- It _was realised

by Yamaji

that _at certain inclination

angles

of trie

magnetic

field

(for

trie _case

£(kjj)

=

(h~/2m)(kj

₊

_k))

this is

given by kfd

tan 9

=

~(n -1/4),

where

n is _an

integer)

all orbits will bave identical _area S which he

argued

could

produce

AMRO

peaks

because trie SdH oscillation

amplitude

would be

largest

at these

angles [38].

^This

explained

trie AMROS which had been

previously

observed in

fl- ET2IBr2 _{(39, 40]}

and

9-ET2I3 (41]. However,

^{since trie effect is} seen at

higher temperatures

than

those at which SdH oscillations _occur, trie concept ^of constant cross-sectional _area

maximising

trie SdH oscillation

amplitude

is _trot

primarily

relevant. Rather _{one cari use} trie Boltzmann

equation (Eq. (iii

to calculate trie

conductivity

of all orbits around trie FS for

arbitrary

field orientation

[42, 43j-

^The

equivalence

of trie two

approaches

_was shown in

[42j:

trie average

velocity perpendicular

to trie 2D

loyers

is

proportional

to

ôS/ôKz (where Àz

labels trie

kz position

^of trie

orbit)

and thus vanishes when S is net a function of

Kz-

In this way one con

show that

Yamaji's

result is nevertheless correct and trie AMRO

peaks

_are connected with trie

vanishing

^{of trie electronic} group

velocity perpendicular

to trie 2D

loyers-

The

angles 9n

at which trie _{maxima occur are}

given by kf~~d ^tan(9n)

=

gr(n +1/4)

+

A(çi),

where trie

signs

and ₊

correspond

to

positive

and

negative 9n respectively,

d is trie effective

interplane spacing,

kf~

is trie maximum Fermi _{wave vector}

projection

on trie

plane

of rotation of trie

field,

and

n =

+1, +2, [42j.

Here

positive

_n

correspond

to

9n

_> o and

negative

n to

9n

< o

[42j.

^Trie

gradient

^of a

plot

of tan

9n against

n may thus be used to find _one of trie dimensions of trie FS

and,

if trie process is

repeated

for several

planes

of rotation of trie

field,

trie

complete

FS _may be

mapped

out-

A(çi)

is determined

by

trie inclination of trie

plane

of

warping:

hence this may also be found

[42j.

^Numerical simulations of this kind of calculation bave been

presented [44].

At fields above trie

kink,

trie FS of

(ET)2KHg(SCN)4

is _no

longer

reconstructed and trie AMROS _{are very} different from trie

quasi-

^ID case discussed in trie

previous

section. Trie

high

field AMROS

(see Fig. 5)

consist of _a series of

peaks

rather thon

dips (see

also

[23, 24, 33]).

Trie open sections _are

insufficiently corrugated

to

produce

_any AMROS. Trie AMROS _are

then dominated

by

trie closed sections

[23].

^{2D AMROS bave also been observed in} a

graphite

intercalation

compound [45].

^{Trie effect bas also been} demonstrated in _a

GaAs/A[Gai-~As superlattice _[46].

5. Conclusion

The

study

^of AMROS in various

organic

metals bas been

remarkably

successful in

providing

information about FS

shapes.

In

particular,

trie

shape

of _a closed section of FS _{can now} be

easily

^determined and the _area derived from AMRO measurements

compared

with trie value

obtained from trie

frequency

of trie

corresponding

SdH oscillation.

To observe AMROS from _open sections of FS it is necessary ^to bave _a rather

highly corrugated FS;

this

requirement

_seems to be

only rarely satisfied,

trie

only

studied

example being

that of

(ET)2KHg(SCN)4

which bas _a

highly corrugated

_open section due to FS reconstruction

~ia _a

spin-density

_wave

(this argument

^{will of} course also

apply

to trie isostructural salts

(ET)2TlHg(SCN)4

and

(ET)2RbHg(SCN)4,

_see

[47-49]).

Trie AMROS observed in TMTSF salts _seem somewhat _more

mysterious

^and a consensus on their origin bas _trot yet

emerged, although

_a

particular

Mass of oscillations observed in

(TMTSF)2Cl04 by

Danner et ai- admits to a

straightforward

semiclassical

explanation [14. 22].

MR measurements con

yield

_{an enormous} amount of useful information and _a combination

of SdH and AMRO measurements

(field

_sweeps ^and

angle sweeps)

_can be

especially

fruitful.

This _was

particularly apparent

ⁱⁿ a recent

study

of

fl"-ET2AuBr2

where SdH oscillation data could

only

be

correctly interpreted

when it _was

reahsed,

because of trie field

dependence

of

AMRO measurements, ^that a field-induced transition altered trie FS _at loT

[50].

Acknowledgments

Data for

Figure

5

(From [33j)

_were ^obtained at trie National

High Magnetic

Field

Laboratory, Tallahassee,

which is

supported by

NSF

Co-operative Agreement

DMR-9016241 and

by

trie State of Florida. We _are

grateful

to trie

EPSRC,

trie

EU,

and to Merton

College,

^Oxford

(SJB).

References

iii Pippard A.B., Magnetoresistance

in Metals

(Cambridge University Press, 1989).

[2]

Shoenberg D-, Magnetic

oscillations in metals

(Cambridge University Press, 1984).

[3] ^For a recent review, see trie

Proceedings

of trie International Conference of

Synthetic

Metals, Seoul,

Korea

July 1994, Synth-

Met. 69-71

(1995).

[4] McGuire T-R. and Potter

R-I-,

IEEE Trans.

Magn.

Il

(1975)

1018-

[si

^{Jérome D. and Schulz}

H-J-,

Adv-

Phys.

31

(1982)

299.

[6] ^Jérome

D-,

Soiid. State Commun. 92

(1994)

89.

[7]

Ishiguro

T. and

Yamaji K-, Organic Superconductors (Springer-Velag, Berlin, 1990)-

[8] ^Lifshitz

I.M.,

Azbel M.I. and

Kaganov M.I.,

Sou.

Phys-

JETA 4

(195î)

41.

[9] Lifshitz I.M. and

Peschanskiiv-G.,

Sou.

Phys-

JETA 8

(1959) 875;

ibid. Il

(1960)

137.

[loi

Alekseevskii N-E- and Gaidukov

Yu.P-,

Sou.

Phys-

JETA 9

(1959) 311; AlekseevskiiN-E.,

Gaidukov

Yu-P-,

Lifshitz I.M. and

Peschanskiiv.G-,

So~.

Phys.

JETPI2

(1961)

837.

[1Ii

Klauder

J-R-,

Reed

W-A-,

Brennert G-F- and Kunzler

J-E-, Phys-

Re~. 141

(1966)

592.

[12j

^{Blundell S-J- and}

Singleton J-, Phys.

Re~- B 53

(1996)

5609-

[l3j

Osada

T-, Kagoshima

S. and Miura

N., Phys.

Re~- B 46

(1992)

1812.

[14j

^Danner

G-M-, Kang

W- and Chaikin P-M-,

Phys-

Rev- Lett. 72

(1994)

3714.

[15j Boebinger G-S-,

Montambaux

G-, Kaplan M.L.,

Haddon

R-C-,

Chichester S-V- and

Chiang L-Y-, Phys-

Rev- Lett. 64

(1990)

591.

[16]

^Osada

T-,

Kawasumi

A-, Kagoshima S-,

Miura N. and Saito

G-, Phys.

Rev- Lett. 66

(1991)

1525.

[17] Kang W-,

Hannahs S-T- and Chaikin

P-M-, Phys.

Rev- Lett. 69

(1992)

2827.

[18]

^Chaikin

P-M-, Phys.

Rev. Lett. 69

(1992)

2831.

[19]

^Maki

Il-, Phys-

Rev. B 45

(1992)

51ii

[20]

^{Gor'kov L.P. and Lebed}

A.G.,

J.

Phys-

Lett- France 45

(1984) 833;

Chaikin

P-M-, Phys.

Rev- B 31

(1985)

4770- Lebed

A.G.,

JETA Lett. 43

(1986) 174;

Lebed A.G. and Bak

P-, Phys.

Rev- Lett. 63

(1989) 1315;

Lebed

A.G.,

J.

Phys-

I France 4

(1994)

351. Lebed

A.G., Synth-

Met. 70

(1995)

993.

[21] Strong S-P-,

Clarke D.G. and Anderson

P-W-, Phys.

Rev- Lett. 73

(1994)

1007-

[22j

Danner G-M- and Chaikin

P-M-, Phys-

Rev- Lett. 75

(1995)

4690.

[23] Caulfield

J-,

Blundell

S-J-,

du Croo de

Jongh M.S.L-,

Hendriks

P-T-J-, Singleton J-,

Dc- porto

M-,

Pratt

F-L-,

House

A.,

Perenboom

J-A-A-J-, Hayes W-,

Kurmoo M- and

Day P-, Phys-

Rev. B 51

(1995)

8325.

[24j

^Sasaki T. and

Toyota N., Phys-

Rev. B 49

(1994)

10120.

[25]

^Caulfield

J-, Singleton J-,

Hendriks P-T

-J-,

Perenboom

J-A-A-J-,

Pratt

FL., Doporto M-,

Hayes

W-,

Kurmoo M- and

Day P-,

J.

Phys-

Condens. Matter 6

(1994)

L155.

j26j Iy/Y-, _{Yagi R-,}

_Hanasaki

_{N-, Kagoshima} S-,

Mon H.,

Fujimoto

H. and Saito

G-,

J-

Phys.

Soc-

Jpn

63

(1994)

674.

[2îj

Kartsovnik

M.V.,

Kovalev A.E. and Kushch

N-D-,

J-

Phys.

I France 3

(1993)

1187.

[28j

^{The Fermi surface} reconstruction was

proposed originally

for trie M

= Tl salt- Whilst trie details may ailler for trie _case of M

=

K,

_we believe that trie

explanation

^of trie

change

ⁱⁿ

AMRO

dimensionality

that trie model ooEers _is

qualitatively

correct

(see

also

[31]).

[29j

^Mori

H-,Tanaka S.,

Oshima

M-,

Saito

G.,

Mori

T-, Maruyama

Y- and Inkuchi

H.,

Buii- Chem- Soc.

Jpn

63

(1990)

2183.

[30]

^Oshima

M.,

Mon

H-,

Saito G. and Oshima

K.,

Chem. Lett. 1989

(1989)

1159.

[31j

^Recent theoretical calculations

[32j

^{and AMRO}

experiments [33j (for

trie

high

_pressure

FS _see

[34]) imply

^{that trie}

quasi-2D

_part of trie Fermi surface is in fact

elliptical

in

cross-section,

with trie

major

axis of trie

ellipse

tilted with

repect

to trie _a axis. This

will of _course affect trie nested Fermi surface. In

spite

of

this,

_we believe that trie correct

nested Fermi surface is

likely

to be

only

_a modification of trie

picture

of Kartsovnik et

ai.;

for

pedagogical

_{reasons we} include trie model _as

derived, recognising

its

important

contribution to _our

understanding

of trie

o-phase

ET salts.

[32]

^Seo

D.-K-, Whangbo M.H.,

Fravel B. and

Montgomery L.K.,

Sofia State Commun. 100

(1996)

191.

[33]

^House

A.A.,

Blundell

S-J.,

Honold

M.M., Singleton J.,

Perenboom

J.A.A.J., Hayes W-,

Kurmoo M- and

Day P-,

J-

Phys-:

Condens- Matter 8

(1996)

8829.

[34]

^Hanasaki

N-, Kagoshima S-,

Miura N- and Saito G-, J.

Phys.

Soc.

Jpn

65

(1996)

1010.

[35]

^Osada

T., Yagi R-, Kagoshima S.,

Miura

N-,

Oshirna M- and Saito

G., Phys.

Rev. B 41

(1990) 5428;

Sasaki T- and

Toyota N-,

Soiid- State Commun. 82

(1992) 447;

Brooks

J-S-, Agosta C.C., Klepper S-J-,

Tokomoto

M-,

Kinoshita

N.,

Anzai

H-, Uji S.,

Aoki

H-,

Perel

A.S.,

Athas G-J- and Howe

D.A., Phys.

Re~. Lett. 69

(1992)

156.

[36]

^Kovalev

A-E.,

Kartsovnik

M-V-,

^Shibaeva

R-P-, Rozenberg L-P., Schegolev

I.F. and Kushch

N-D-,

Soiid State Commun. 89

(1994)

575.

[37]

^Yoshioka

D.,

J-

Phys.

Soc.

Jpn

64

(1995)

3168.

[38] Yamaji K.,

J.

Phys-

Soc-

Jpn

58

(1989)1520.

[39]

^Kartsovnik

M.V.,

Kononovich

P-A-,

Laukhin

V.N.,

and

Schegolev I.F.,

JETA Lett. 48

(1988)

541

(Pis'ma

Zh-

Eksp-

Teor-

Fiz.)

48

(1988) 498).

[40] Schegolev

I.F., Kononovich

P-A-,

Laukhin V.N. and Kartsovnik

M.V., Phys-

Scr. T29

(1989)

46.

[41] Kajita K.,

Nishio

Y-,

Takahashi

T-,

Sasaki

W.,

Kato

R., Kobyashi H-, Kobyashi

A- and

Iye Y.,

Sofia. State Commun. 70

(1989)

1189.

[42]

^Kartsovnik

M.V.,

Laukhin

V.N.,

Pesotskii

S.I., Schegolev I.F.,

and Yakovenko

V.M.,

J-

Phys.

I France 2

(1993)

89.

[43] Peschansky V.G.,

Roldan

Lopez

J-A- and Yao

T.G.,

J-

Phys.

I France 1

(1991)

1469.

[44] Yagi R., Iye Y-,

Osada T- and

Kagoshima S.,

J.

Phys-

Soc.

Jpn

59

(1990) 3069; Yagi

^R.

and

Iye Y.,

Soiid State Commun. 89

(1994)

275.

[45] Iye Y.,

Baxendale M. and Mordkovich

V-Z.,

J.

Phys-

Soc.

Jpn

63

(1994)

1643.

[46] Yagi R-, Iye Y-,

Hashimoto

Y., Odagiri T-, Noguchi H.,

Sasaki H. and Ikoma

T-,

J-

Phys.

Soc-

Jpn

60

(1991)

3784.

[47]

^Kartsovnik

M.V.,

Ito

H., Ishiguro T.,

Mori

H.,

Mon

T-,

Saito G- and Tanaka

S-,

J-

Phys.:

Condens- Matter 6

(1994)

L479.

[48] Klepper S-J-,

Brooks

J.S.,

Athas

G.J.,

Cheii

X-,

Tokumoto M-, Kinoshita N. and Tanaka

Y-, Surf.

Sci. 305

(1994)

181.

[49]

^Kartsovnik

M.V.,

Kovalev

A.E.,

Laukhin V.N. and Pesotskii

S.I.,

J.

Phys.

^{I France} 2

(1992)

223.

[soi

House A.A., Harrison

N.,

Blundell

S-J-,

Deckers

I., Singleton J.,

^Herlach

F., Hayes W.,

Perenboom

J.A.A.J.,

Kurmoo M. and

Day P., Phys.

Re~. B 53

(1996)

9127.