Field Induced Spin Density Waves

(1)

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P. Chaikin

To cite this version:

P. Chaikin. Field Induced Spin Density Waves. Journal de Physique I, EDP Sciences, 1996, 6 (12),

pp.1875-7898. �10.1051/jp1:1996169�. �jpa-00247288�

(2)

J.

Phys.

l France 6

(1996)

1875-1898 DECEMBER 1996, PAGE 1875

Field Induced Spin Density Waves

P-M-

Chaikin (*)

Department

of

Physics,

Princeton

University, Princeton,

NJ 08544, USA and

Exxon Research and

Engineering Company,

Route 22 East,

Annandale,

NJ o8801, USA

(Received

20 June 1996, revised 20

August

1996, accepted 26

August 1996)

PACS.75.30.Fv

Spin-density

_waves

PACS.72.15.Gd

Galvanomagnetic

^and other magnetotransport elfects PACS.74.70.Kn

Organic superconductors

Abstract. Trie Field Induced

Spin Density

Waves

(FISDWS)

found in

organic

conductors

represent a unique serres of transitions which meld trie one-dimensional physics of trie Peierls

instability

with trie two-dimensional

physics

of the

Quantum

Hall Elfect. This _paper presents _a

pedagogical

introduction tu trie FISDW'S in the

Bechgaard softs, along

with _recent

experimental

results _on related

high magnetic

field

phenomena.

1, Introduction

The

Bechgaard salts, (TMTSF)2X (where

X

=

PF6, Cl04, Re04 etc.),

_are

probably

trie most

interestiug

electronic materials _ever discovered.

Dependiug

_ou

composition, temperature,

pressure and

magnetic

field

they

exhibit most of trie

ground

states aud

pheuomena

associated with

interacting

electrons in

vastly

different

systems.

^There are

competitions

between metallic and

insulating, magnetic

and

superconductiug, semicouducting

and semimetallic

phases.

Trie

Bechgaard

softs exhibit ail of the electronic transport ^mechanisms

yet discovered,

metalhc

conductivity, sliding density

_wave

conductivity, superconductivity

and the

quantum

^Hall ^effect.

What is _even _more remarkable is that ail of the above

properties

cau be observed in _oue

siugle crystal

of _one of trie

Bechgaard

softs

(TMTSF)2PF6

_as

temperature,

pressure, and

magnetic

field _are varied

iii.

Trie basis for

understanding

trie wide

variety

of behaviors is to be found in trie

strongly anisotropic

bandstructure

resultiug

from trie

quasi-one-dimensional

chainlike

crystal

structure.

Platelike TMTSF molecules stock face to face _lu _a _zigzag chaiu. Trie wavefunction

overlap

from _one molecule to trie next is

responsible'for

trie

large

^bandwidth

(1 eV)

in trie chair _a direction.

Neighboring

chairs _in trie b direction _are also

sulficiently

close for Se orbital

overlap

and

yield

_a bandwidth of 0.1 eV. Trie

coupled

chairs form two-dimensional

planes

in which trie electrons _are delocalized. Trie

planes

_are

separated

in trie third direction

by

_a sheet of anious.

Trie

overlaps

_are small aud trie baudwidth is dowu

by

_an additional factor of

r-

30 to 0.003 eV. There is _a fuit

charge

trausfer of _one electron _per unit cell to trie

anion, leaving

trie two TMTSF molecules with half _a noie _ou each. Were it _uot for _a

slight

dimerization of trie

zigzag

(* e-mail: chaikin@

pupgg.princeton.eau

©

Les

Éditions

_de

Physique

1996

(3)

lE+04

Ge lE+02

~Si

-

E+00

~_jgçj c~(Ncsj

~

)

lE~2

À~

g

lE~4

YBCO

à

lE-06

i~~~

COPPer

iE-io

i io ioo iooo

Tenperature (K)

Fig.

l.

Log-log plot

of trie resistivities of _some _common conductors. Circles indicate the _super-

conducting

transition temperatures. Note that

(TMTSF)2Cl04

looks most similar to Cu menai. Bonn

are very clean, with _mean free

patins

of many microns ai low temperature. Trie _main dilference is trie factor of

r- lo00 carrier

density

from the

larger

unit cell in the

organic.

chaiu,

_we would

expect

a quarter filled TMTSF baud. But trie dimerization

splits

trie TMTSF baud into two bauds and leaves _us with _a _one half filled _Upper baud _as

required by

trie

charge

transfer aud

stoichiometry.

Trie

highly anisotropic bandwidths, 4ta 4tb 4tc

re eV 0.1eV 0.003 eV

imply conductivity anisotropies

_in ^trie ratio (t~

/tj)~.

Trie measured conductivities _are: _aa _ab _i _a~ _m

10~

10(Q cm)

_[2]. The

temperature dependent

resistivities of several

preseutly interesting

materials is shown _in

Figure

1. The transition temperatures ^{of the}

supercouductors

are iudicated

by

the

large

circles

terminatiug

trie low eud of trie

resistivity

_curves. It is clear that trie

(TMTSF)2Cl04

sait is _a

good

metal with resistauce

decreasiug

about three orders of

magnitude

_ou

cooliug

from _room

temperature

to il K. It is very similar to

Copper,

trie _main difference

being

that trie _carrier

density

is much less. Trie

Bechgaard

salts bave about carrier per 1000 cubic

Angstrom

unit

cell,

while Cu bas _one carrier per 1 cubic

Angstrom

Cu atom.

This

essentially

accounts for trie thousandfold difference _in

conductivity.

Trie low temperature

phase diagram

for trie wunder material

(TMTSF)2PF6

is shown _in

Figure

2

[3,4].

Above 12 K trie

system

is metallic.

Upon coohng

at ambient pressure there _is _a ver»

weakly

first order transition to _a

Spin Density

Wave

(SDW) insulating phase

_as evidenced

by antiferromagnetic

_resonance, NMR and _muon _spm rotation [5]. Trie SDW transition _tem-

perature con be

suppressed by application

of moderate _pressure until ai _r- 6 kbar trie metallic

state is reestablished. Once trie SDW is

suppressed

trie

crystal

becomes

superconducting

at

1.2 K

[6j.

This material _was trie first _organic

superconductor

discovered. Further increasing trie pressure

gradually

reduces trie

superconducting

transition

temperature. Applying

a small

magnetic

field

along

trie least

conducting

direction

(to

mduce _screemng _currents _in trie

highly conducting plane)

kilts trie

superconductivity

above _a critical field of

r- 500 Gauss [7].

(4)

N°12 FIELD INDUCED SPIN DENSITY WAVES 1877

'~~,

, ^,

' ',

,~ ^i' ',,~

'~~Î~'~ ^~~~éÎ

^' _'

,

'

i

,

%$Î~

'

$~

~

~ ~

j~ ~

1~ _,

/~

fl

^'

,

' ^'

~

, ^,

s

_,

,

'

»j ~WÙ

Fig.

2. The low temperature

phase diagram

of

(TMTSF)2PF6 (3,4].

While trie

proximity

of trie SDW

or

antiferromagnetic

insulator

phase

to trie

superconduct,ing phase

is _common to many

sy~tems, trie field induced SDW cascade _seen above 6 kbar and

r- 5 testa is _su f~r

unique

tu trie

quasi-one-dimensional

_organic conductors [8]. ^Each separate

phase

_in trie cascade with

increasing

field

corresponds

to _a different quantum ^Hall state.

All of trie

phases

discussed _up to this

point

bave been observed in other materials. Trie

juxtaposition

of

superconducting

and SDW _or

aiitiferromagnetic phases

is _now often _seen _m

highly

correlated systems ^such as trie

high

temperature

superconductors.. However,

what is

new and _so far _unique to these materais [8] ^is ^trie

phase

which results from further

mcreasing

trie

magnetic

field while in trie low

temperature

metallic state

[9]. Imagine starting

at 10

kbar,

at 0.5 K and

applymg

_a field

(along

trie _c

direction).

500 Gauss kilts trie

superconductivity

and leaves _us with _a "normal" metal

(more

about that

later).

At

slightly

_more than 5 tes-

las _we _encounter another

phase boundary

where trie

resistivity sharply

_mcreases _as does trie

Hall coefficient. Further

increasing

field this transition is followed

by

_a cascade of at least 9 consecutive first order transitions to different semimetal states until at

r- 20 testas _we enter trie "final"

insulating phase.

Most remarkable of ail. each transition takes _us to _a state with

a well defined and

sequenced (..1/4,1/3,1/2,1) quantized

Hall resistance

[10j.

It is

particularly striking

since trie

Quantum

Hall Effect

(QHE)

is

intrinsically

trie

property

^of a two-dimensional

electron system and this materai _was trie first _to exhibit trie

QHE

in _a bulk three-dimensional

crystal.

This cascade of

phase

transitions is to Field Induced

Spin Density

Wave

(FISDW)

states and is trie main

topic

of this _paper.

2. One Dimensionalization in _a

Magnetic

Field

The Fermi surface which _we associate with this bandstructure _is cartooned in

Figure

3. It consists of two

non-intersecting slightly warped

sheets. Trie actual Fermi surface and Brillouin

(5)

k~

2x/b

Fig.

3. Cartoon of the idealized Fermi surface of the

Bechgaard

salis.

(Trie

real

crystal

structure is triclinic non

orthorhombic).

Trie Fermi surface consists of

nonintersecting

sheets

along

trie

highly

conducting

_a direction

warped by 4tbleF

and

4tcleF

in the b and _c directions.

Zone _are _more

complex il ii.

Trie

crystal

structure is

triclinic,

non orthorhombic _as in trie _car- toon and there is

overlap

between _one TMTSF molecule _on _a chair and several molecules _on _a

neighboring

chair. If trie softs _were

truly

one-dimensional

electronically

then trie Fermi surface would consist of two

parallel

sheets _at

+2kF.

Trie _warping is due to trie finite bandwidths _m trie b and _c direction

(trie

b

warping

is about trie correct size, but trie _c _axis is

exaggerated by

about _a factor of

30.)

A

good

deal of trie

phenomena

observed in the

Bechgaard

softs _are

directly

related to this Fermi surface which _at first

glance

_seems innocuous. There _are _no closed orbits and hence _no chance for Landau

quantization.

Trie

warping

_seems too

large

to allow the

one-dimensional instabilities associated with trie Peierls transition. For H (

c

(perpendicular

to trie

highly conducting plane)

_we expect small

saturating magnetoresistance along

trie _a axis and sizable

nonsaturating magnetoresistance along

b and _c

[12].

Let _us

forget

about trie least

conducting

_c direction for trie moment and concentrate _on the Fermi surface and bandstructure in trie a-b

plane.

Trie _essence of the

problem

is that of

large amsotropy

and

only

_open orbits of trie Fermi surface. A

dispersion

relation with these

properties

_con be _wntten _as

£(kx, ky

=

-2ta

_cos

k~a 2tb

_cos

kg

^b re

1~ ~) 2tb

_cas

kyb (2.1)

where _we

temporarily

choose _a free electron form for the

dispersion along

_x to avoid _com-

plications

due to

nesting

and

commensurability [13,14].

We want to _see the effect of _a field

perpendicular

to the

plane.

A Landau-Peierls substitution k _~ iv

eA/hc,

with the choice of _a Landau gauge A

=

(0, Hz)

^leads to:

£ $il

2tb Cos(ib) ~)~x)il

= fil

(2.2)

m x y c

Since _y

only

_appears _in trie

partial denvative,

trie wavefunction _con be factored _as

il(~, vi

=

e~~YY

<(-ri

and

defining

the

magnetic wavelength

as

2~/À

=

eHb/hc

_we bave:

~ÎÎÎ2~~ °~ Î ^~

^~~'~~

(6)

N°12

FJjLD

^INDUCED ^SPIN ^DENSITY ^WAVES ¹⁸⁷⁹

eHv~b

a~=-

4ic

k space

ÎÎÎ~ ~Î~

_~

~+~

r~al SPaC~

_~- ^~~

eHb

Fig.

4. Real space and momentum space

quasi-classical

trajectories of electrorts in the presertce of

a

magnetic

field

along

_c. In k space the motion

along

the open orbits

periodically

_crosses the Brillouin

zone with

frequency

_uJb. In real _space the motion

is Iocalized to _r~

4tô/~~b

_oc

1/H

chains

along

b.

The motion is extended along the chains but there _is _a

spatial

modulation ai the

magnetic Iength

=

hcleHb

= ~o

/Hb (where

~o _is the flux

quanta).

Since

ky

appears

only

in the

argument

^of ^the

cosine,

it

only

_serves to shift the

origin

of _x

(hence

the center of _mass of the

wavefunction,

x'

~ x

lbky/2~). Equation (3)

is then _a _one- dimensional

Schrodinger's equation

for _a

periodic potential (in

fact it is Mathieu's

equation) [15].

^In a

magnetic

field the

dispersion depends

_on

kz, ejkz

rather than trie _zero field

dispersion e(k~, ky)

^Thus ^the

magnetic

field makes trie

system electromcally

one-dimensional.

We _can understand this _one dimensionalization tram the

quasi-dassical

motion of the elec-

trons _on trie Fermi surface. Trie two-dimensional Fermi surface is

schematically

shown in

Fig-

ure 4. Trie electron

velocity

_vk

=

s7ke

is

perpendicular

to trie Fermi surface. In trie presence of _a

magnetic

^field ^electrons are constrained to move on constant energy surfaces. Electrons follow trie

equation

of motion

ltôk/ôt

₌ _evk x

B/c

~ trie real space

velocity

_vk _~

ôr/ôt

and k _space motion

ôk/ôt

_are

simply

rotated

by

90°. There is _a characteristic

frequency

uJb +

(ôkb/ôt)/(2~/b)

=

eufHb/hc

with which the electron _crosses trie Brillouin _zone _in trie b

(y)

direction.

(u~

^is

approxiniately

constant at tif on the Fermi surface. The

dispersion along

_x

is often taken in linearized form _as

ltufk~.)

The real space electron motion is shown _in

Figure

4 bottom. It is limited

along

_y and extended

along

_x, ^1-e- one-dimensional. Trie width of trie orbit

(which

is

actually

the extent of the

quantum

wavefunction

along

_y

(16]

^is

(4tb/ltuJb16

and there _is _a _new

periodicity,

G ₌ 2~

Il

=

eHb/ltc, given by

trie

magnetic length

1

along

_x

il _7j.

(The length

is such that

#o

" lbH with çio "

hcle

_a flux

quanta.

^Note that for _an

isotropic

two-dimensional electron system ^the

magnetic length lH

_is such that po "

1(H.

For 10 testas

(7)

Qa Q°

§Qt.K

.4

(Q"l~~~

Fig.

5.

Susceptibility

of trie electromc system to _a distortion of wavevector

Q. x(Q)

[29]. The FISDW wavevector is determined

by

the

Q

ai the _maximum of _x,

m ibis

case

Q". (SmaII changes

m bandstructure _can shift trie maximum to either of the Qo's

Ieading

to

changes

in the

sign

of the quantum Hall

resistance.)

trie wavefunction is

spread

_over about 60 airains in trie b direction and trie

periodicity

induced

on trie airains in trie _a direction is about 200 unit

cells.) _[18]

The effective one-dimensionalization of trie electrons

by

trie

magnetic

field _can and does bave

interesting

consequences. One-dimensional metals _are unstable

against

^trie ^formation ^of

charge

and

spin density

_waves

(Peierls transitions) _(19].

Trie effects of electron interactions

become _more

important

ⁱⁿ _one

dimension,

Fermi

liqmds

_give _way to

Luttinger liquids

and other _many

body ground

states

[20, 21].

^Disorder

plays

_a drastic rote and

always

leads to locahzation and

insulating

^behavior at least for

non-interacting

electrons. There is _no

long

range order and hence _no finite

temperature

transitions _in _one dimension. Trie absence of any closed current

loops

would

suggest

^that ^trie

superconducting

cntical field should tend to

mfinity

for one-dimensional

systems (22].

^Of ^this ^nch set of

proposed

consequences of field induced _one

dimensionahty,

trie

only

_one of which _we

presently

bave conclusive evidence is trie Field Induced

Spin Density

Wave

(FISDW)

transitions

[9,17, 23j.

3. Trie FISDW Phase

Diagram

The

instability

of trie metallic state _can be obtained

by calculating

trie

susceptibility

to _a

peri-

odic spm

density

modulation at trie wavevector q. This

susceptibility

_is shown _m

Figure

5. Trie one-dimensional character of trie electronic

dispersion

_in _a

magnetic

field _assures

divergences

in trie

susceptibility

_as temperature is lowered toward _zero.

Using

_a Stoner critena _we then

expect

a transition into an SDW _state

corresponding

to trie _wavevector at which trie

suscepti- bility

_is maximum. The wavevectors of trie _maxima shift with

magnetic

field

(17, 24, 25]. They

alwavs _occur at

2kF ~112~/À along ka

but trie kb value _varies _with bandstructure _and field in such _a _way _as to

yield quantized

k 9pace areas.

(8)

À

qualitative explanation

of the FISDW'S aise follows from

considering

the

possible ground

states of the

system (rather

than the

instability

_on

cooling

from the metallic

state).

À _Due-

dimensional

system

^is unstable

agamst

^the ^formation ^of a

density

wave because _a distortion of

wavevector

2kF exactly

nests trie two sides of trie Fermi surface. This

produces

_a _gap at the

Fermi energy. The total electronic _energy is lowered

by ^A~

In _eF

/~l

while trie distortion costs an

energy of order

KA~

where _K is _an elastic constant _or _an

exchange

interaction

(19).

^Because ^of the

log

term the electronic _energy

always

wins

(for

small

Al

and the

density

_wave is stable. For

a

quasi-one-dimensional dispersion,

the

nesting

of the Fermi surface is _net

perfect

as illustrated

schematically

in

Figure

6a. À distortion of

2kF Produces

_a _gap at

only

_a few

points

_on the Fermi surface. These _gaps close off electron and hale

pockets

and the

dispersion

is that of _a semi-metal. Trie electronic _energy

lowenng

is of order ti~ In _eF

IA (with

_m >

3),

net

enough

to _overcome the cost of

making

trie distortion. Trie metallic state of trie

quasi-one-dimensional

system

^is ^stable. ^This ^is ^the ^situation ^for

(TMTSF)2PF6

under pressure without _a

magnetic

field. It remains metallic to T

= 0.

Dimension ail _states at sf

,

£

coupled by q=2kf

,

~ £~ in gap

everywhere

c

distortion stable

£~ ² W

2n/b

(6aÎ)

-kf ka~ _kf

h _e

Imper§ect Nesting

^leaves pockets ^distodion

~

~b

_~f ^~°~~~~~~

(6a2)

k

~

a)

Fig.

6.

ai

Distortion with H

= 0.

ii

For

perfect nesting (as

m one

dimension)

_a

surgie

wavevector maps one side of the Fermi surface to the other opening a

complete

_gap ai the Fermi energy,

stabilizirtg

the distortion and the

msulating phase. 2) Imperfect

nesting Ieaves electron and

for

hole

pockets

_on

a

partially gapped

Fermi surface. There _is _no

longer

_a _gap ai _EF and ibis state is non lower energy thon trie menai without dïstortion.

b)

No distortion, finite H 1) ^For ^closed ^orbits area, ertergy artd Hall resistance _are

quantized. 2)

For the _open orbit

Bechgaard

salis there _are _no

interesting

elfects ai EF.

c)

Distortion and field. The _areas and energies m the electron and hole

pockets

_are

quantized,

the distortion wavevector adjusts _so that _EF lies between Landau Ievels. Since _EF _is

completely

in _a _gap trie distortion _is stabilized and

concurrently

_we bave

ouly

filled Landau Ievels and trie quantum ^Hall

eifect.

(9)

Closcd

°'bit landau

Lcvcls

QHE

g

(6b1) En

=

(n

+

la)

Jfo

c

Magnetic Field

Quasi-ID

^~

Opencfbit

No Landau

£j QllwtiZation

(6b2)

bj ka~

Distonion

₊

Field

£f ⁱⁿ Gap distonion

q ~f "~~

+

Landau Quantization QHE

~j (6ci

Fig.

6. Contmued.

Now lets consider the effect of _a

magnetic

^field _on an undistorted metal~

Figure

6b. If _we bave closed orbits then _we bave Landau

quantization,

discrete energy levels and trie

Quantum

Hall effect

(in

_a two-dimensional

system).

However. for _open orbits there _is _no

quantization

condition and trie energy

dispersion

_remains continuous with _a dassical

joften zero)

Hall effect

(26].

^Trie quasi-one dimensional metal _seems

umnteresting

bath in its

stability

and lack of

magnetic

field effects.

(10)

If _we bave bath _a distortion and _a

magnetic

^field _we

regain something interesting, Figure

6c

(27].

^Trie ^distortion ^leads ^to ^dosed ^orbits ⁱⁿ ^trie ^semi-metal

pockets.

These must be Landau

quantized

with

spacing huJ,

_uJ ₌

eH/m*c.

If trie Fermi energy lies between Landau

levels,

then

we bave _a

completely gapped system,

trie electronic _energy

gain again

beats eut trie distortion energy and trie

system

lowers its energy

by distorting, forming

an SDW. How _con _eF

always

lie in the

gap? Suppose

_we _were to

change

the

magnetic

field. The

spacing

of the Landau

levels shifts and _we

might expect

eF to enter a Landau level. Without _a gap the SDW would

collapse. However,

_eF remains in the gap as

long

_as _we have

only

filled Landau

levels,

1.e. the

area in the k space

pockets

must be _an

integral multiple

of

eH/hc (the

_area which _can

exactly

accommodate trie number of states in _a Landau

level).

Trie system can

adjust

trie _area of trie

pockets by changing

the wavevector of the SDW distortion.

Equivalently,

the distortion

wavevector dictates the

top

and bottom of the electron and hale

pockets.

As the field

changes

the _qsDw

changes

_SO _as to

keep

_eF in the _gap between Landau levels. Àt _some field it may be that trie _energy _is lowered

by

_qsDw

jumping

to _a different value _SO that _eF sits between

another set of Landau levels.

Trie situation where there _are

only completely

^filled landau levels is

precisely

trie condition which

gives

rise to the

Quantum

Hall effect

[28].

^In a conventional two-dimensional electron gas eF sits between Landau levels

only

_m the presence of disorder induced localized states. In the present case eF sits in the gap for intrinsic energy reasons. In fact _we could net have _a stable FISDW without the

QHE (and conversely).

As field is

changed

_we should have first order transitions between SD~V states with different quantum numbers. Trie

ordering

of trie

quantum occupations

_as field is mcreased

depends

_on details of trie bandstructure and trie

nesting [29].

^In ^ail cases trie

high

field state should bave _a wide

splitting

between the lowest energy electron Landau level and trie

highest

_energy hale Landau level. This

high

field limit is _an insulator. In trie

simplest picture

_as trie field is lowered there is _a

single pocket

and trie

quantum

numbers follow _n

=

0.1, 2, 3, 4,...

_as trie field is lowered.

Trie cartoon

description

_given above is culled from _a

great

^deal ^of theoretical work

by

_a

number of groups who bave

converged

on a "Standard model" _or

"quantized nesting

model"

for trie FISDW'S

[17, 24,25].

Trie

picture

which emerges from these detailed calculations _is illustrated in

Figure

7.

Starting

from

high

field _we should bave trie FISDW insulator state with _no filled Landau

levels,

N

=

0,

and

essentially

trie _same transition

temperature

as for trie

zero field SDW with

perfect nesting.

On

lowering

trie field _we enter trie N

=

phase

with _semi- met allic

integer

quantum Hall behavior. Further

lowering

trie field

yields

trie cascade of FISDW transitions to different

phases, separated by

^first order transitions and each associated with _a different Landau level

filling

and

quantum

^Hall

plateau.

Within each

phase

the wavevector for trie SDW

changes continuously

with field

(qfisDw

=

(2kF

~1~2~

Il,

_~

lb

+

e)

to maintain trie

complete filling

of trie Landau levels. Between

phases

trie wavevector

changes discontinuously

with the _a

component having

a different value of _n. In the

simplest

_case _i~

= N and the cascade results _m the

stepwise

decrease of _n _as

suggested

ⁱⁿ the

figure. However, depending

_on the

bandstructure,

bath the sequence of wavevectors i~ and their association with N

(trie

number of filled Landau

levels)

_may

change [29].

Àt pressures somewhat above the critical _pressure needed to suppress the ambient field

SDW,

bath trie

PF6

Salt and trie

ÀsF6

Salt show trie transitions _as

predicted by

the

simplest

form of trie

quantized nesting

model. Trie Hall resistance

(at

T

= 0.5

K)

and transition

temperature

are shown for _a

PF6 sample

at llkbar in

Figure

8, and Hall and

longitudinal

resistance _are shown in

Figure

9 for _an

ÀsF6 sample

at T ₌ 0.5 K and 10 kbar

[30].

We _see trie

stepwise

^decrease m trie Hall _resistance

following

_p~~ ₌ h

/2Ne~

_as

expected _[31].

The absolute value of the Hall resistance _m trie

plateaus

is

roughly

at trie

quantized

value given above

ii.

_e. 13 k Q

ID

_per

plane)

for

"good" samples (where good

_means

agreement

and

(11)

F-

1 n=0

1

Àlxy

f

2e~

o

H

Fig.

7.

According

to trie standard _or quantized

nesting

mortel there

is a cascade of FISDW _tran- sitions

as field is increased. Associated with each transition the Iow temperature Hall resistance should show _a quantized

plateau.

In trie

simplest

_case trie plateaus, _p~~

=

h/2Ne~,

mcrease as

N

= ..5, 4, 3,2. 1,0, with

increasing

field.

io

o

-

50 C

jzs

ce

o

0 5 10 15 z0 z5

H(T)

Fig.

8. Data for

(TMTSF)2PF6

_ai 11 kbar _pressure

showing

trie phase

diagram

and the Hall resistance ai 0.5 K

[3,4].

(12)

~~z

(TMTSF)2AsF6 to kbar oSK

6

1

~-z

DÎ~

ioo

6

50 1

DÎ

o

0 à 10 15 z0 25

H(T)

Fig.

9.

(TMTSF)2AsF6

shows similar behavior to PF6. Here the

longitudinal

resistance is _seen _to decrease with

increasing

field in the

regiou

of the Hall

plateaus

and to exhibit

peaks

_in the region of the transitions between

FISDW-QHE

states. Above 18 testa the

sample

enfers the N

= 0 _state which

is a SDW insulator

[3,4].

presumably

that the current

paths

_are

uniformly

distributed between the

conducting planes

that make _up trie

crystal).

Trie differences from

conventionalinteger quantum

Hall Effect in

single layer

two-dimensional electron

systems je-g- GaÀs)

_are the factor of 2 in the denominator

(from

the

spin degeneracy resulting

from the

spin pairing

in the SDW

state) [31],

the absence of _any linear

region

between the

plateaus (from

the first order nature of the transitions between the different N states, or

equivalently,

the fact that the

system

can

energetically

never find itself with _eF located in _a Landau

level),

and the tact that trie

plateaus

do _net sit _on _a hne p~y =

Hli~~e (with

n~ the fixed electron

density)

which

extrapolates

to zero at zero field

(from

trie fact that the effective carrier

concentration _is

changing

with field _as trie

nesting changes

and trie

pockets

shrink toward _zero at

high field.)

In trie conventionaltwo-dimensional electron

system (2DES) QHE

the

longitudinal

resistance p~~ goes toward _zero in the Hall

plateaus

and has local maxima between trie

plateaus.

À similar

behavior is _seen m

Figure

10. In fact _p~~ decreases with

decreasmg temperature

ⁱⁿ ^trie

plateaus (approximately exponentially)

and increases

exponentially

_in the N

= 0

phase _[32].

There _are several other

interesting

differences between this

system

^and ^the ^2DES. ^The ^dosed ^orbits which

quantize

into the Landau levels _are created

by

the SDW distortion and its

resulting

SDW _gap. The gap is

relatively

small and the

magnetic

fields present are of

comparable magnitude.

We _cari therefore bave

magnetic

breakdown

through

these gaps and the result is

tunneling

between Landau levels. Trie Landau levels become Landau bauds.

Depending

_on trie

particular

bandstructure and _qsDw _we bave _an intricate set of bauds and _gaps. An

example

_is

shown in

Figure

11 which shows trie calculated gaps about _eF _as trie field is vaned and different states m trie cascade _are present

[33].

(13)

io~

10°

§

~$

n"0

n=1

n-Z n~3 75

_

)

50

ç/

~~ 25

nm3

o 0

(K)

Fig.

10.

Temperature dependence

of the

longitudinal

and Hall resistance of

(TMTSF)2AsF6.

The

Hall resistance

monotonically

increases _to ils quantized value _as temperalure is Iowered

through

the

FISDW transition. For the N ₌ 0 SDW

insulating

state the resistance _mcreases due to the

opening

of the SDW gap and the absence of _any filled Landau levels. For N

#

0 states the resistance rises

just

below TsDw from the Ioss of _carriers but then decreases _as the

dissipationless

transport associated with the

QHE

takes

over ai Iower temperatures.

4. Tilted

Magnetic

Fields and Lebed Resonances

SO far trie FISDW bas been descnbed _as trie

mstability

of _a

quasi-one-dimensional

metal _m _a two-dimensional _space.

(Expenmentally

trie

phase diagram basically

scales

Orly

with the field

perpendicular

to the a-b

plane

when trie field is

tilted.)

Trie real electromc

system

is of _course three-dimensional and here _we consider _some of trie effects of trie

three-dimensionality.

If trie bandwidth in trie third direction _is

larger

than trie SDW gap, or trie

spacing

of trie Landau levels

(these

numbers _are

comparable)

then trie bands Will

overlap

and trie FISDW and trie associated

QHE

will

disappear.

Trie best measurements and calculations _agree that

4t,

_m 0.003 _eu

r- 30

K,

easily enough

to kilt trie two-dimensional effect. The clever solution that _nature bas found and

apphed

in these

softs,

is to choose _a wavevector of

~/c along

_c, 1-e- to alternate trie SDW in

neighbonng planes.

Just _as the i~>avevector

(2kF, ~/b)

leads to

perfect nesting

and

complete

_gapping ^{of the} Fermi surface for trie

dispersion

relation

e(k)

=

ltvfkz 2tb coskyb, (2kF, ~/b, ~/c)

does trie _same for

e(k)

=

ltvfkz 2tb

_Cos

kyb 2tb

_Cos

kzc.

If trie _x

dispersion

is

quadratic,

trie

nesting

is

imperfect

and what remains of trie

dispersion

is tbe~ ~

t)lef

and tc~~ ~w

t)lef

~w

10~~ev

~w o-1 K. For _a

strictly

two-dimensional

system

any finite

magnetic

field

(14)

uJ

o

' 6 à lu 20 3o

H 11)

Fig.

ii. The gap ^structure m the FISDW states _are shown _as _a functiou of

maguetic

field [33].

The dark

regions

_are the _gaps which separate the white Landau bands. Note that the

largest

_gap ^is

ceutered around _eF.

would

produce

_an FISDW if _we cooled to low

enough temperature [17].

^À limite tc~~ mandates

a threshold field below which _we carnet attain _an FISDW.

Experimentally

this bas been round to be when

huJcr,t

+~ tc~~

r-

o-1 K _as

expected

aise from consideration of trie

QHE

and Landau

quantization [34].

Lebed

[35] suggested

that there _was _a _way around trie three

dimensionality

and trie _zero

temperature

threshold field.

Up

to this

point

_we bave treated trie _case when trie

magnetic

field is

aligned along

trie _c _axis. If trie field is tilted in trie b-c

plane,

trie electronic orbits sweep

across trie Fermi surface _as illustrated in

Figure

12. Trie k space

equation

^of ^motion ^is

simply

hdk/dt

= eV x B _m _evf~ _x B _so that there _are _now two raturai

frequencies,

_Due for

crossing

trie Brillouin _zone in trie

direction, nô

"

Îô/(2~/b)

=

evfbB cos9/lt

and _Due for trie

(irection,

Q~ ₌

Îc/(2~/c)

=

evfcB

sin

9/h.

Àt certain

angles,

tan 9

=

pb/qc,

with p and q

integers,

these

frequencies

_are

rationally related, Qc /Qb

"

P/Q.

For these

particular angles,

trie electron orbits

retrace their motion and the

trajectory

_is a

hue,

for other

angles

trie electron orbits do trot

retrace and end _up _covenng trie entire Fermi surface. Lebed

argued

that the commensurate

motion at trie

"magic" angles couples

trie

dispersion

in trie two open orbit directions and reduces the three-dimensional

dispersion

to one-dimensional. Àt these

angles

trie threshold

(15)

2z/b

« »

a) arbibary angle j.space) c) p/q= ^In magic angle

/Î=1 /q=iQ

. . .

/~

. .

b) P'q=' ma#c anfle _d) mai space iatùce Î

Fig.

12.

a)

Motion of _an electron _across the Fermi surface in the presence of _a

magnetic

field

applied

in the b-c

plane (perpendicular

to

a)

ai _an

angle

à from the _c axis.

Umklapp scattering

at the Brillouin _zone boundaries

produce

_a

trajectory

which _covers the entire Fermi surface.

b), c)

For

a

specific angle

à

=

tan~~(pb/qc)

_the _k _space

trajectories

retrace _on the Fermi surface.

d)These "magic"

or Lebed

angles

correspond to the

magnetic

field

pointing

between molecular centers

(or along

real space translation

vectors).

field would be reduced to zero

(at

_zero

temperature).

Trie

analogous effect,

trie increase of trie transition

temperature

^of ^trie ^FISDW at trie Lebed

angles

bas been _seen

[36].

À few _years after Lebed's

prediction

of trie

commensurability

effect _on trie threshold

field,

came trie idea that trie

angular

_resonance _on trie Fenni surface should aise bave drastic effects _on trie

transport propeities _[37]. Expenmentally,

these _were first observed in trie

Cl04

Salt _a axis resistance

[38].

^Later

they

_were _seen

along

bath _a and _c directions _in bath trie

Cl04

and

PFô

salts when

they

_are _in trie metallic state at low

temperature [39].

^{Ii is}

fairly

_easy to understand

why _transport _properties

_are

changed

at trie Lebed

magic angles:

trie

repeating

orbits _cari avoid

high scattering

_regions of trie Fermi surface

and/or provide

different _averages _over trie velocities.

Trie _one _or two

dimensionality

of trie

dispersion

bas drastic effects _on trie _nature of bath

impufity

and electron-electron

scattefing.

To date there _are

r- 10 different

modelslexplanations

for trie magic

angle

resonances, none of which bas

satisfactorily explained

ail of trie data

[38,40,41].

This _is

partly

because trie effects differ from _one sali to trie orner. There _are

especially Sharp

and unusual structures found in trie

PF6

Salt under _pressure:

Figure

13.

Àlthough

the basic idea for trie Lebed _resonances _cames from k space, Fermi surface _argu- ments

Figure

12, trie faon that k space ^vectors are

perpendicular

to real space vectors~ and that trie

velocity

is

perpendicular

_to trie

field,

translates to trie fact thon trie

"magie" angles

bave

a real _space

interpretation. They

_are

simply

trie

angles

at which trie field is oriented in trie direction between trie actual molecules in different chains

(Fig. 12d).

This observation led _to

(16)

7

~ ^(Hcoso)~/~

6 _xx

5 Q

6~4

x

2

1

/

~~

~

zz

0.3 Ô~

0.

'~-ioo

d

Fig.

13. Angular

dependent magnetoresist~nce along

trie _a

(top)

and _c

(bottom)

_axes for

(TMTSF)2PF6

ai 11 kbar. 4 testa and 0.5K. trie Lebed

dips

ai

p/q

= 0, +1 _are evident. The "back-

ground"

magnetoresistance bas _an unusual form which _seems to

depend

_ou the componeut ^{of the} ^field

perpendicular

to the most

conducting plane

to _a power.

one of the most

intrigmng

ideas _ou the

ongin

of the

magic angle

effects, at least for the

PFU

Salt. The resistivities

along

the most and least

conducting

directions _are shown for this sali in

Figure

13. Trie

suggestion

was that trie Salt is

marginally

_a Fermi

hquid _[41].

It is

presumed

that the

planes

would be _m _a non-Fermi

liquid (NFL)

state if

they

_were

uncoupled.

The electronic

coupling along

the _c _axis, tc, _is

(presumably) just barely enough

to lead to coherent transport

along

_c and

consequently

trie destruction of trie 2 dimensional NFL state m favor of

a conventional 3 dimensional Fermi

liquid.

In this scenario _a small field

along

b reduces

taon

below trie critical value and leaves _a 2D NFL. In such _a 2D

system

ail

magnetotransport

must

depend only

on the field

perpendicular

to the 2D

plane

and

might

be _m trie form of _power

(17)

o-o

~

tlC

,oQ

1 ~

~~

~

^~l

~

~ m

~llÎ ^-l.5

O

log[Hcos8]

Fig.

14. Trie _power Îaw

dependence

_on field

perpendicular

to a-b plane

(H cos(à))

_is illustrated by

Iog-Iog

plot of _c axis

magnetoresistance

for

sample

rotations taken ai diifereut fields. Trie _same _curve

results when field _sweeps _are taken at fixed

angles.

Trie deviations _occur ai trie Lebed

angles.

Same parameters _as _in

Figure

13.

laws, (H

_Cos

6)~.

At the magie

angles tc~~

is net as reduced and trie _resistance

draps.

Measure- ments show that aside from trie

magie angles

the

magnetoresistances

_vary _p~~ _cç

(H

_Cos

6)°.~

and _pzz _cç

(Hcos6)~

^~ ~° ~.~ when either H is varied _at _constant 6 _or 6 is vaned at constant H

(42, 43], Figure

14. Fiom this point ^of ^view ^trie "normal" metal state of

(TMTSF)2PF6

_is _a

non Fermi

liqmd

in trie _presence of _a small field

along

b.

Trie

deceptively simple

Fermi

surface,

which _ai first

glance yields

_no hint of

interesting

behav-

ior in _a

magnetic field,

also

produces large angular dependent magnetoresistance

^for rotations in trie _a-c

plane [44]

^and ^most

recently

in trie a-b

plane [45j.

^{SO for} these _resonances bave been

readily explainable,

at least in trie

Cl04

sali in terms of conventional Boltzniann transport, with _a coherent Fermi surface

[46]. They

bave been able to

provide

useful information about trie bandstructure parameters much _as bas been done

previously

with closed orbit menais.

5. Trie Anomalous behavior of

(TMTSF)zCl04

Àlthough

the

PF6

sait of TMTSF _seems _an ideal

example

of trie

simplest

form of trie standard model for trie

FISDW'S,

trie

Cl04

sait presents a much _more

complex

behavior. To date trie

Cl04

sait is _more studied than trie

PF6

for both FISDW'S and for

superconductivity

even

though

both

phenomena

_were discovered first in the

PF6. Cl04

is an anibient pressure

supercondiictor

and FISDW

system

while

PF6 requires

_a _pressure of

greater

^thon ⁶ ^kbar.

This makes it much barder _to _meurt,

align,

and rotate trie

PF6 samples

and makes most

thermodynamic

and elastic

properties

almost

impossible

to _measure. Ail of these measurements bave been done _on

Cl04 (4T48j.

Cl04

has _an _anion

ordenng

transition at 24K which leads to _a

doubling

of trie unit oeil in

trie b direction

[49j.

^It was well known that trie low

temperature properties

of this

sait,

its resistance, its

superconducting

transition

temperature

etc.

depended strongly

on how

rapidly

trie

sample

was cooled

through

this anion

ordenng

transition.

However~

thé _extreme

sensitivity

of trie

quantum

^Hall steps,

especially

trie

change

_in _sign _or "Ribault" anomali

[50j,

was un-

expected.

For _many _years trie

only

Hall data _was from

Cl04

and masked trie

acceptance

of

(18)

~.~

b'

(TVTSF)~PF,

~'~~

fi

a 5

~'~~

2 3 ' Sompie #1

0.50 P ^* il kb°~

i o 0.25

~~'~°Ù

~'~~ ^~~~

ç~

0 5

p = 8.3 kbor -

O'~

_(~)

So~np,e il o o

~~~

~ ~~ ~~~

o.z5 - -o 5

o oo ^~~~

~~~~j~ ~~

_~ ~~ ^p ^« ^BS ^kbar

~'-~

Tm 380 mK

.? ~'/~ oiz

p~= h/2

e'

p~~ /Po

_0.08

0.6 (d) -

0.04

$

ù_~ ^somp,e ;2 o.oo

li

_t

~ ~ ^-

~

je) O-O

5 lO 15 20

MagneLic

field

(Lesla)

Fig.

15.

(TMTSF)2

PF6 exhibits

"uegative"

Hall steps ai pressures below about 9 kbar [51], ^similar to what, bas been observed eallier in

(TMTSF)2CI04s by

Ribault et ai. [50].

They

have been attributed to _a

specific

_warpmg of the Fermi surface. po =

H/2e~.

trie

QHE

in trie

Bechgaard

salts until several _groups

(10]

^looked ^at

PF6

and

got

^trie ^data ^hke that shown in

Figure

8. More recent data _on bath

PF6 (3, 51j, Figure

15 and

Cl04

_(4]

land

_on

Re04) _Î52j

at different _pressures show that trie

sign changes

_con be found in both materials and _are most

probably

associated with

changes

in the

nesting

vector

resulting

from subtleties

in the bandstructure

[29].

The lack of _a

simple

staircase for the Hall resistance is

Orly

_Due of _many anomalous behaviors of trie FISDW in

Cl04. Although expenments

have been carried out to very

high

fields

(50

T at

~w

il

trie N

= 0

insulating

state bas

yet

to be found. At low

temperature

^there ^is a

very wide field

region

from 7.5 to 27 Tesla where trie Hall resistance is flot and trie

longitudinal

resistance is

dearly

semi-metallic.

Beyond

^{27 T} there is _a first order transition to _a state with very

large

oscillations _in resistance, Hall effect and

thermodynamic properties [53].

For many years this _was considered _a reentrant metallic

phase,

but trie most recent results

suggest

^that the

phase diagram (Fig. 16)

at ambient _pressure _is

yet

more

complicated

^with _some ^FISDW

phases

endosed in other