The influence of chemical impurities and X-ray induced defects on the single-particle and spin-density wave conductivity in the Bechgaard salts

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defects on the single-particle and spin-density wave conductivity in the Bechgaard salts

S. Tomić, J. Cooper, W. Kang, D. Jérome, K. Maki

To cite this version:

S. Tomić, J. Cooper, W. Kang, D. Jérome, K. Maki. The influence of chemical impurities and

X-ray induced defects on the single-particle and spin-density wave conductivity in the Bechgaard

salts. Journal de Physique I, EDP Sciences, 1991, 1 (11), pp.1603-1625. �10.1051/jp1:1991229�. �jpa-

00246439�

Classification

Physics

Abstracts

72,15N 75.30F

The influence of chemical impurities and X-ray induced defects

on

the single-particle and spin-density

_wave

conductivity in the

Bechgaard salts

S. Tomib

(I),

J. R.

Cooper (I),

W.

Kang (2),

D. Jkrome

f)

and K. Maki

(3)

(1) Institute of

Physics

of the

University,

POB 304, 41001

Zagreb,

Croatia,

Yugoslavia

~3) Laboratoire de

Physique

^des

Solides,

Universitd de Paris,Sud, 91405

Orsay,

France (3)

Department

of

Physics, University

of Southem Califomia, Los Angeles, ^CA 90089-0484,

U-S-A-

(Received16 May 1991, accepted in

final form 30July 1991)

Abstract. We present some recent

experimental

and theoretical results obtained _on the

single particle

and

spin-density

wave

(SDIAj conductivity

in the

Bechgaard

salts

(TMTSFJ2NO~

and

(TMTSF~~PF~.

Particular

emphasis

is

paid

to the clearest

example (TMTSF~~PF~

for which there is

experimental

evidence,

namely

the absence od 2

k~

diffuse scattering at low temperatures, for _a

pure SDW state below 12 K. A theoretical

analysis starting

from the

anisotropic

Hubbard model and

taking

into _account the influence of

long

_range Coulomb interactions _can account for _many features of the

experimental

^data. The

magnitude

and temperature

dependence

^of the threshold field and to some extent the

magnitude

of the SDW

conductivity

_are well accounted for by this

theory. Comparison

with

experimental

data shows that for most batches of

samples

the SDW is

weakly

pinned to

randomly

distributed

impurities

or defects. Defects _were then introduced in _a controlled way

by X-ray

irradiation. This caused a substantial increase in threshold field and a

changeover

_to the behaviour

expected

for _a

strongly pinned

SDW. For _a

particular

batch of

samples

the temperature

dependence

of the threshold field _was unusual. A detailed theoretical

analysis

of

commensurability pinning

leads _us to conclude that in this _case the SDW _was

commensurate with the lattice. Some recent data dor the SDW

conductivity

in the field induced SDW

phases

of

(TMTSF~~PF~

under pressure are also

reported

and discussed within the _same

theoretical framework.

1. Inwoducfion.

Various

highly anisotropic conductors,

both

inorganic

and

organic,

_are ideal

systems

^for

studying

collective transport

phenomena [I]. Depending

_on the material and

applied

pressure, there is

usually

_a

phase

transition _{to a}

superconducting (SC),

_a

charge-density

_wave

(CDW),

_{or a}

spin-density

_wave

(SDIV~ ground

state at low temperatures. ^These

ground

states are established below _a well defined transition

temperature (T~).

Within _mean field

theory

(*)

Present adiress 966-5

Daechi,Dong,

Kangnam-Ku, Youngdong P.O. Box 1922, Seoul 135-280, South Korea.

J~

and the

single-particle

_gap

(2 A)

_are related

through

the BCS

expression

2

A/kB

T~ ₌ ^3.5.

In

practice

the value of 2

A/kB

T~ ^is much

larger

than 3.5 for _a CDW-Peierls transition.

According

to the standard

explanations

this _can be due to the

suppression

of the _mean field transition temperature

by

fluctuations which _{are more}

pronounced

in low dimensional

systems,

or to strong

coupling

interactions

[2]. However,

_we believe that the

large

2

A/k~

T observed in CDW systems are due to the

imperfect nesting [3].

In

general,

the effect of fluctuations is small and the mean-field

theory

^is reliable _even for CDW and SDW

systems.

The characteristic feature of the CDW _or the SDW state is the modulation of the

charge

_or

spin density, respectively.

It is _now well established that the translational mode of the CDW

ground

state

couples

to _an

applied

electric field and

gives

_a novel contribution to the electrical

conductivity.

In _a

perfect material,

such _a motion of the collective mode could in

principle

lead _to

superconductivity,

_as

originally proposed by

Fr6hlich

[4]. However,

in real materials this motion is restricted

by

various

pinning

mechanisms such _as lattice

defects, impurities

and

commensurability potentials

and

only

sets in above _a certain threshold field

(E~) [I]. Hence,

nonlinear

current-voltage

characteristics

accompanied by

broad and _narrow band noise, with

sharp

threshold fields of the order of 10-1000

mV/crn,

_are a

signature

of CDW motion.

Theoretically,

^similar behaviour

might

be

expected

for _a SDW state, because collective transport does not

depend

_on the nature of the

underlying

interaction mechanism

(electron-

electron rather than

electron-phonon) [5].

^In

particular,

it is the total electric

charge

in the condensate which

couples

to the

applied

electric field and _as there is _a second order local

charge

modulation associated with the

SDW,

it _can also be

pinned by impurities.

Model SDW systems are some of the

Bechgaard

salts

(TMTSF~~X

in which the SDW

nature of the

ground

state, with _a critical

temperature

^of about

10K,

has been

firmly

established

by

various

magnetic

measurements

[6-9]. However,

efforts to detect SDW order

by

neutron diffraction measurements have failed _up to _now

presumably

due to the smallness of the SDW

amplitude [10, 11].

The first

attempts

to measure the

electric-field-dependent

response in the SDW state failed to

identify

intrinsic nonlinear

behaviour,

because of

experimental problems

related to contacts and

heating

effects.

Although

the

conductivity

of

(TMTSF~~PF~ crystals

_was found to be

nonlinear,

there _{was no} threshold field and the

magnitude

of the

non-linearity

seemed _to correlate with resistance

jumps developed during cooling [12,13]. However,

_some

preliminary

evidence for

non-linearity arising

^from SDW conduction _was

reported by

another _group

[14].

Three years ago, non-ohmic

transport

was

reported

in the

magnetic-field-induced

^SDW state of

(TMTSF~~CI04

and _was

interpreted

_as

evidence for SDW

sliding

due to the

applied voltage exceeding

_a threshold value

[15].

However,

because the measured resistance

actually

increased with

voltage,

it _was

suggested

that the

nonlinearity

occurred

only

in the _transverse component ^of the

conductivity.

Theoretically,

_no

nonlinearity

^is

expected

in the transverse

conductivity.

An additional

problem

_was that the measured threshold field _{was very} small and undetectable in _some

samples.

In _{a more} recent

investigation

of the

magnetic-field

induced

spin-density

_wave states of

(TMTSF~~PF~

under pressure, we showed

unambiguous non-linearity

in the

longitudinal

conductivity

and _no effect in the transverse

conductivity [16].

On the other

hand,

the first

frequency-dependent conductivity

measurements in the SDW state of

(TMTSF~~PF~

_were

interpreted

in tennis of _a collective SDW mode with

pinning

as for

a CDW

[14, 17].

Recent

investigations

conducted _{over a} broad

spectral

_range

identify

both the

single particle

excitations _across the _gap and the collective mode contribution well below the gap

[18,19].

In

addition,

_narrow band noise has been also found in the SDW state of

several

Bechgaard

salts

[20-22]. Namely,

coherent effects like intrinsic

voltage

oscillations _or interference between them and _an

externally applied

_ac current _are well established distinctive features of

sliding

CDW

[2]. Therefore,

their presence in SDW systems

gives

another

important piece

of evidence of _a collective SDW response.

However,

_no

quantitative

determinations

of,

_par

example,

number of condensed electrons _or

periodic length

of the

pinning potential,

have been

possible yet

due to the

inhomogenity

of the SDW current flow.

The _purpose of this _paper is _to

present

^{and discuss}

experiments

_we have

performed

in the last three years in order to look for _one of the

properties

of _a

possible

SDW

current-carrying

state :

namely

_an increase in the dc electrical

conductivity

above a finite threshold field

[23- 26].

The paper is

organized

_as follows. We describe the measurement

techniques

used with

special emphasis

_on

sample mounting

and

self-heating problems (Sect. 2).

In section 3 _we review _our

experimental

results. Section 3,I deals with the low-field

resistivity

behaviour and the SDW

phase

transition itself. In section 3.2 _we review

investigations

of the influence of

pinning

centers on the threshold field. This includes the field and

temperature dependence

of

the _excess

conductivity

in

pristine samples

from different chemical batches and in

samples

with _a controlled amount of

X-ray

irradiation induced defects. In

addition,

_we

_present

the main results

concerning

non-ohmic

transport

in the

magnetic-field

induced SDW

phases

of

pristine (TMTSF~~PF~ samples.

Section 4 is dedicated to

theory.

^In

particular,

_we will address the

following points

_:

(a)

The

dependence

of the threshold field _on

temperature

and

sample purity

and the relation of this behaviour with the

underlying pinning potential, ~b)

the

importance

of the

commensurability potential

for SDW

pinning, (c)

the

temperature dependence

of the _excess

conductivity

and the effect of Coulomb interactions _on the _excess

conductivity.

In section 5 _we discuss _our results in the framework of the

theory

described in section 4 and outline _some future prospects.

2.

Experimental techniques.

2.I SAMPLE PREPARATION. Good

quality single crystals

of

(TMTSF~~PF~

and

(TMTSF~~NO~

used in this

study

have been _grown

electrochemically using

the usual method

[27].

2.2 SAMPLES WITH X-RAY IRRADIATION INDUCED DEFECTS.

Single crystals

of

(TMTSF~2PF~

were selected from the _same batch

(referred

later in the text _as the standard

batch).

We have used unfiltered radiation from _a Cu

X-ray

tube

(35 kV,

24 and 20

mA)

to

produce

radiation

damage

in _a controlled _manner. The concentrations of irradiation-induced

defects _were determined

using'a TMTSF-DMTCNQ crystal

_as a reference

sample.

The

correspondence

between

damage

rates and

resulting

resistance

changes

of TMTSF-

DMTCNQ

_was established in

previous

studies

[28, 29].

2.3 SAMPLE MOUNTING. All measurements _were

performed

on

single crystals

of

(TMTSF~2N03

and

(TMTSF~~PF~

from dilTerent batches with various distances between

voltage

contacts

(0.5-2.5 mrn)

and cross-sections in the _range of 0.004-0.015

mrn2.

_Gold

pads

were

evaporated

on to the

samples

to mininfize contact resistances which _were

typically

1- 5 Ohms. Two dilTerent

mounting techniques

_were used. Electrical _contacts

arranged

in the

four-probe configuration

_were made either

by

silver

paint

_or

by

mechanical

clamping

of fine annealed

gold

wires

[30]. Using

the latter

technique samples

_can be cooled down without

appearance of resistance

jumps

which _are

usually

encountered when the standard

mounting technique

is used.

Initially

_we used the standard

mounting technique

and very slow

cooling

rates

(2-6K

_per

hour).

In

particular, (TMTSF~~NO~ crystals

are less

fragile

than

PF~

and _on

cooling

_we observed either _no cracks _at all _{or some} very small _{ones as} reflected

by only

_{one or} two small resistance

jumps

_near 100 K. The total increase in resistance caused

by

the cracks _never exceeded 0.5fli of the

sample

resistance at 100K. As far _as the

PF~ samples

are

concerned,

all ambient pressure data

presented

in this paper have been

JOURNAL DE PHYSIQUE I T I,M II, NOVEMBRE lwl 63

obtained for

crystals

mounted in the strain-free fashion

(mechanical clamping

of the

gold wires)

and which therefore did not suffer from

externally

induced defects _on

cooling.

2.4 MEASUREMENT TECHNIQUES. Low-field

resistivity

measurements were

performed

using

_a standard low

frequency

_a-c-

technique

and lock-in detection. The electric-field-

dependent conductivity

_was measured

by

_a

short-dc-pulse technique together

with _a

bridge

circuit to subtract the ohmic

component

^{of the}

conductivity [31].

The standard

pulse length

was 40 ~cs with a dead time of about 5 ~cs and a

repetition

time of 20 _ms.

Regular

checks _were

performed

for

sample heating

to rule out

spurious

elTects. Possible

heating during

_a

single pulse

could be detected

by monitoring

the out-of-balance

signal

from the

bridge

_versus time

on the

oscilloscope,

from 5 to 40 _~cs after the start of the

pulse.

If this

signal

increased

linearly

with

time,

corrections for

heating

could be made

by extrapolating

back to the onset of the

pulse. However,

above certain fields

(of

the order of

100mV/cm

at 4.2K and about 40

mV/cm

above 6

K)

the

heating

after 5-20 _~Ls became too

large

for this

procedure

to be

accurate.

Consequently,

_we could not obtain data outside this field and

temperature

_range.

Any

overall increase in the

sample temperature

was ruled out

by varying

the

repetition

rate of the

pulses. Furthermore,

under favorable circumstances checked

by

the tests described above for

sample heating,

_we used _a standard

dynamic

resistance measurement

technique

to

determine the value of threshold field _more

accurately.

The _ac current could be varied between I and 10 ~LA.

Superposed

_on this _ac current _a dc current _was swept

linearly

with time. The dilTerential resistance of the

sample

_was measured with _a lock-in

amplifier working

at a low

frequency (typically

70

Hz).

The output ^{of the lock-in} was recorded _{as a} function of the dc

voltage

_across the

sample

_{on an x y} recorder. In

addition,

at very low temperatures and for

high

resistances _a standard dc

technique

_was also used.

In summary,

solving problems

related to

sample self-heating

and

externally

induced

defects,

_{we are} confident that

artificially

induced nonlinear effects _can be ruled _out.

3.

Experimental

results.

3.I LOW-FIELD RESISTIVITY BEHAVIOUR, SDW _PHASE TRANSITION.

3.I.I Pristine

samples.

The

organic

conductors

(TMTSF~~X

_are

single

chain

systems

ⁱⁿ which _a

nominally quarter-filled

conduction band is created

by charge

delocalization

along

the

organic

chains. The

spatial anisotropy

of the

overlapping

molecular orbitals leads to _an open

Fermi surface and to strong

anisotropy

in the electronic

properties.

There is _a rich

variety

of

ground

states

ranging

from metallic

(and

often

superconducting)

to

insulating.

The latter state can be due to anion

ordering (AO)

_or to the formation of _a

spin-density

_wave

(SDW~ phase.

At ambient pressure both the

PF~

and

NO~ compounds

exhibit _a metal-to-semiconductor transition to a SDW

ground

state.

X-ray

studies for the

PF~

salt show that it,is _{a pure} SDW state in that the one-dimensional 2

kp scattering,

which is the precursor of _a CDW

instability, disappears

below 50K

[32]. Magnetic

measurements such _as

antiferromagnetic

_resonance

(AFR) [7], proton

nuclear

magnetic

resonance

(NMR) [33],

static

susceptibility [6], magnetic

anisotropy

and determinations of the

spin-flop

field

[8]

were

performed only

for the

PF~

^salt and confirm the AF nature of the

ground

state. The

magnetic-distortion

_wave vector

(Q

was estimated to be close _to the commensurate value

Q

=

(0.5 a*,

0.24 _± 0.03 b*

)

and

(0.5 a*,

0.20 _± 0.05 b*

), by

two dilTerent groups

[10, 34], ignoring

the

third, weakly coupled

c* direction.

We have used the electron

spin

resonance

(ESR) technique

to characterize the

magnetic

transition of both

compounds [9,35].

In

particular,

the critical behaviour of the ESR linewidth

(AH)

_was

investigated

close to the SDW transition

(TN).

The

vanishing

ESR

(TMTSF)~ NO~

2~

~H19) aHlg) log(bH)

~

i I

h~

I

( a

-l 0

o

~~ ~'

-j

o

~°f~

~

° ~~~~~~2

~§

~

°

(

~' °

°~

° log(bH) ^° °

°

~ ^o

° °

30 20 30 40

T(K) _T(K)

a) b)

Fig.

I. ESR linewidth

(AH)

_versus temperature

(7~

for

(TMTSF~2NO~

(a) and

(TMTSF~2PF6 (b)

with the

corresponding log-log plots.

susceptibility

and concomitant critical

divergence ^if

the ESR linewidth confirm the AF nature of the transition

(Fig. I).

Note that the critical

exponent

^for

NO~ (p

= 0.5 _± 0.I

)

dilTers

considerably

from the value of _p =1.5 ±0.I observed for the

PF~

salt and _some other

organic

conductors. The dilTerence

might

be

assigned

to the dilTerent

anisotropy

in the

spin degrees

of

freedom,

_as well _as to dilTerent relative

magnitudes

of the

dipole-dipole

interaction and

spin-orbit coupling. Alternatively, they

could be connected with the presence of anion

ordering

in the

NO~ compound leading

to a broader ESR line in the semimetallic

region

between

TN

^and TAO.

The

NO~ compound

is the best conductor of the

family

in the whole temperature range

[27].

Furthermore,

_an anion

ordering

transition with _{a wave} vector

(0.5 a*, 0,

0 is observed at 45 K

[32].

This _wave vector does not

give nesting

and

probably

leaves

large pockets

of electrons and holes which have _a metallic

(or semimetallic) conductivity. Indeed,

the electrical

resistivity

falls at the anion

ordering

transition

[23].

In contrast, the SDW _wave vector is

presumably

_{very near} to the

optimal nesting

vector and therefore leads to _a

large

increase in

resistivity.

As discussed

below,

the gap may still not be uniform _over the whole Fermi

surface,

but may be very small _{or even zero} at certain

points.

Namely,

careful

analysis

of the

log

R _versus inverse temperature

plots

below the SDW transition

(TN)

reveals _a very low activation _energy

(A

= 8

K)

and hence _a smaller ratio

2A/TN

than the mean-field BCS value.

Moreover,

the curvature of the

logR

vs.

I

IT plot

indicates that the

ground

state may be

semimetallic,

rather than

semiconducting (Fig. 2).

In contrast, for the

PF~ compound

the activation _energy is rather well defined

(A

= 20

K)

and the ratio 2

Al

_TN

equals

the BCS value of 3.5

(Fig. 3).

A _common feature is _an

extremely sharp

transition. The width of the

transition,

^defined as the full width _at half maximum _{on a}

plot

of d

log R/d(I In

_vs.

T,

is

only

0.2 K. In table I _we summarise _some

parameters

^{of the SDW transition obtained for}

NO~

^and

PF~ crystals.

rr denotes the

resistivity

ratio

pRT/p~;~,

where _pRT and p~;~ are the resistivities measured at room temperature

(RT)

and at the temperature ^{where the}

resistivity

reaches its minimum value before the transition.

We _{can use rr as a} relative _measure of the

crystal purity,

however it is difficult to say to which kind of defects _rr is sensitive. As far _as the

NO~ compound

is concerned _we have studied

samples

from two different batches which

typically

showed _rr of about

150, apart

^from one

crystal

which had _an

extremely high value,

rr ₌ 750. More elTort has been devoted _to the

/

- ,'

(TMTSF)~PF~

,/

(TMTSF)~ N0~ ^RI~J

,11"

ooo o o o II

aaoo°° II

°° '

/ ll~

o II

o° _/

o° _/

a° _/

~

/ ~

/" ^,/~~

a_b

o

-I _c

o o

lYU/T

200 ,o~

Fig.

2.

Fig.

3.

Fig.

2.

Logarithm

of the low-field resistance _versus inverse temperature for _a

(TMTSF)2N03

crystal.

Fig. 3.

Logarithm

of the low-field resistance _versus inverse temperature for

(TMTSF)~PF~

crystals from three

representative

batches standard

(a), particular

(b) and P273C

(c).

PF~

material for which _we have found _some marked differences

depending

_on the

preparation

batch. In table I _we

give

the characteristics of three different

nominally

_pure batches.

Table I.

(a)

Some

resistivity

and

(b)

ESR parameters

for pristine (TMTSF~~NO~

and

PF~ samples.

_«~~, rr, TN,

bTN

^and A _are the _room temperature

conductivity, resistivity

ratio,

SDW transition temperature, width

of

the SDW transition and the activation energy,

respectively. T~;~

is the temperature where ESR linewidth AH attains its minimum value

AH~;~, AHN

is the value

of

AH _at TN ^and _p is the critical exponent

defined

_as

AH

= cst

((T TN) /TN)~

^~

(a)

«RT

(Q cm)~

_rr

TN (K) bTN (K)

2 A

(K)

2

A/TN

d

log R/d(1/7~

NO~

000-1 800 150-750 9.5 0.2 16 1.7

PF~

batch S 500 250 11.1 0.2 39 3.5

batch P 500 90-150 11.9 0.2 39 3.3

P273C 100 35 11.7 0.2 34 2.9 0.5

(b)

T~;~ (K) TN (K) AH~~~ (gauss) AHN (gauss)

_p

NO~

9.8 8.35 5.1 21.5 0.5 _± 0.I

PF~

12.5 11.3 3.5 136 1.5 _± 0,1

Common features _are the

following.

There is _a metal-to-semiconductor transition at TN ^with a concomitant

jump

in the

resistivity,

followed

by

_an anomalous

resistivity

behaviour

down to about 4.2

K,

below which _an activation _energy is well defined. In contrast to two other

batches,

the standard batch

~batch S,

_we call it standard because _we find the _same results for

crystals

from several different batches apart ^{from this}

one)

shows the lowest TN ^{with the}

largest jump

in _p and with the _{same or}

slightly larger

activation energy. As far _as

the anomalous

region

is

concerned,

it is

clearly

most

pronounced

for the standard

batch,

while it is smeared out for _one

particular

batch

~batch P).

On the other hand the temperature at which the inflection

point

is situated

(1j)

does not _seem to be sensitive _on the

preparation

batch.

In

figures

4a and 4b _we show the

resistivity

_versus temperature

squared

for

PF~ samples

from three distinct batches and for three

samples

from batch

P, respectively.

The law p

=po+BT~

_{is valid below about 35K down to the SDW} trantition. Note that po is close to _zero for the standard batch

sample,

while it becomes finite for batch P and is

exceptionally large

for batch P273C. Note also the

spread

of _po values _among

samples

from the _same batch.

Furthernlore,

the

parameter

B is not the _same for all

samples

measured and is

always larger for larger

values of _po. In

addition,

the _rr is

larger

for the standard batch than for P and P273C. The latter also showed rather small RT

conductivity.

From these data _we

can infer that the batch S is _more pure than the batch P and that batch P273C is

exceptionally impure.

It is not clear for _us

why

it is _so because

preparation

conditions

during electrosynthesis

_were not

particularly

dilTerent. In

figure

5 _we show

T~ plots

for two

NO~ samples

_up to RT with the low

temperature region

_{as an} inset.

Again,

the

sample

with

exceptionally large

_rr

(m 750)

has po close to _zero and very small B, while the

sample

with _a

more

typical

_rr

m 130 still has very small po, but the value of B is similar to that found for the standard

PF~ samples

in the _same low

temperature region.

la1 lbl

j 1'

~

~__ ^g

°

O

~

> ,

~ j

>

I

$ L0

$

~

-

W W

~ ~ -~ _

l~~ ~ ^"~

300 600 300 600 900 1200

T~/K~ T~/K~

Fig.

4.

Resistivity

versas temperature squared for

(a) (TMTSF~~PF~ samples

from three batches standard (A),

partcular

(B) and P273C (C) and (b) three

(TMTSF~~PF~ samples

from

particular

batch.

3.1.2

X-ray

^irradiated

samples.

As noted in section 2.2 _we have chosen the standard batch of the

PF~ samples

to

study

the effects of

X-ray

irradiation-induced defects. The defect concentration

(c)

_was varied from 0.002 fli mole _to 0.04 fli mole _~. Results _are summarized in

figure6

and tableII. All

samples

had the _same RT

conductivity «~~=500±

100

(Q cm)~

~. However, rr values decrease

strongly

^{with the} defect

concentration,

_a linear

relationship

is satisfied up ^to 0.008 fb mole

Also,

the transition remains

sharp

for small

;.«

(TMTSF)~ NO~ :."

,.":

.°

;.

;.°

:°

..' ».

," _a

,«o

;. ,o.°

.. "

,:'

;. _:

,."

defect concentrations

(c

<0.008fb mole

~~)

and

TN

shifts toward

higher temperature by

about _one

degree.

At

higher

doses

(c

_m 0.008 fb mole

~)

rr tends to saturate, the transition broadens

substantially

and

TN

^decreases

again.

The

broadening

is reflected

by

_an increase of the transition width

(& TN)

and _a decrease of the

peak height

of d

log R/d(I In

_vs. T. The

activation energy

already

becomes smaller _at low defect

concentrations,

while for

large

concentrations A cannot be determined

accurately

because of the curvature of the

log

R _vs. I

IT plot.

As far _as the anomalous

region

^is concerned it is

barely

visible for low defect concentrations

(c

=

0.002 and 0.004 fb mole _~) and it is

completely

smeared out for

higher

concentrations. It is worth

noting

the resemblance of

log

R _vs, I

/T

_curves for batch P and the _c

= 0.002 and 0.004 fb mole

samples.

In

figure

7 _we

display resistivity

_versus T~

plots

for the pure and irradiated

samples

from the standard batch. The

parameter

^{B increases}

strongly

with the defect

concentration,

while po does not _seem to

change

_very much.

200

I

~~°

~

$

_l~

$

~

ioo o o ax

:"~

_o

$

li ~~"

_,-,

~

_...."

~~

^~

~

^{50 ,:.'" o oo2x}

'~' pure

~0

_{300 600 900 1200}

T~/K~

Fig.

7.

Resistivity

_versas temperature

squared

for _pure and irradiated

samples

from standard batch of

(TMTSF)~PF~.

3.2 ELECTRIC-FIELD-DEPENDENT CONDUCTIVITY.

3.2.I Pristine

samples.

The

electric-field-dependent conductivity

observed in the

NO~

and

PF~ compound

is shown in

figure

8. In the metallic state the

conductivity _stays

constant in the whole field _range measured

(up

to about 0.7

V/cm). However,

in the SDW state, ^the

conductivity

is constant until _a threshold field is

reached,

above which the

conductivity

increases. Values of the threshold field measured at 4.2 K _are 40

mV/cm

for the

NO~ samples,

and 8 and 5

mV/cm

for batches S and

P, respectively.

The

sharpness

of the threshold field _was checked

by dynamic

resistance measurements

(Fig. 9). E~

is rather

sharp

for the

NO~ sample

and for the

PF~ samples

from batch

S,

while the _onset is rounded for other

two

PF~

batches. For both

NO~

and

PF~

the value of the threshold field is

temperature independent

below about

TN/2.

For the

latter,

we also established the overall temperature

dependence

of

E~

_as shown in

figure10.

For the standard batch _we found the

following

behaviour.

E~ displays

_a

steady

and rather

large

increase above 5K towards

TN.

E~(0.9 TN) /E~(T~;~)

=

2.6,

where

T~;~

^is the lowest

temperature

^reached in the

experiment.

However,

for the batch P

E~

_was found to decrease _on

approaching TN

^{and then it increased}

o 100K (a) (TMTSF)2°F~ ^tbl

a SK

Q,)

a ~2

' o 42

. ~8 _~o

a 25K

a 7 °.°

a &3 o~ j ,a x

x %5K ~#~& ^{,' i}

Oo J ~ ~~

o, a a x

0 O PO ^{' ° ~ °}

"'°°"°°°~"'~~

Elmv/cmJ E(mv/cmJ

b 4.6K

(C

O 6.0K

A 7/K

~

~fa

^°

. 8.7K _u .a ^°

Q 12,7K D

.A~

o

D ., _o

au

~.,

~a

U .,a ^O _~

D

~.,

~ ~n

o

g~

~ ~

. Da Aoowi la &

E(mV/cml

Fig.

8. Non-ohInic

conductivity ((«(E)

_«

(0)/«(0)))

_versus logarithm ^of electric field (E) for

(TMTSF~~NO~ (a)

and

(TMTSF~~PF~,

batch S

(b)

and batch P

(c).

sharply

at TN. ^In

addition,

_we note that the _excess

conductivity

and associated current are

smaller in

samples

with _a lower _rr

(Fig,

I

I).

Plots of the

conductivity

versus inverse temperature ^{show that the} excess

conductivity

has _a

thernlally

activated behaviour similar to that of the nornlal component

(Fig. 12),

and that both

conductivity

^{channels have the} same activation _energy. Its value is

slightly

smaller in the

temperature region

between 5 and 10 K.

Further,

there is _a clear break in the activation

plots

for batch S _at the temperature ^which

corresponds

to

(

as defined in section 3.I.I.

Actually,

both channels become less conductive below 5 K. That is not the _case for batch P. In

addition,

the activated behaviour

implies

that the _excess current decreases with

decreasing

temperature.

3.2.2

X-ray

irradiated

samples.

The influence of

X-ray

irradiation induced defects for the

PF~ samples

from the standard batch is shown in

figures

10 and 13.

Figure

13 shows the field-

dependent conductivity

for

pristine

and irradiated

samples

at 4.2 K _vs. the

logarithm

of the electric field. The value of

E~

increases with the defect concentration and the

magnitude

of the extra

conductivity

becomes smaller. The temperature

dependence

of

E~

for dilTerent

defect concentrations is

given

in

figure10.

In the

low-temperature

_range 1.2-4.2K

E~

is

temperature independent

for both _pure and irradiated

samples. Already,

_{a very} small defect concentration of 0.002 fb mole ^~~ is sufficient to diminish the rise of

E~

towards TN

significantly: E~ stays

constant until =0.8

TN

and then increases

only slightly:

E~(TN)/Er(T~,~)

=1.35±0.15. It is also

important

to note that

E~

does not

diverge.

However,

_as Joule

heating prevented

us from

checking

this feature for other

samples

and irradiation

doses,

_we cannot rule out this

possibility completely.

The inset of

figure10 displays

the low

temperature

^{threshold field} as a function of defect concentration

showing

_a

linear

relationship

_up to about 0.02 fb mole ~. A defect concentration of 0.04 fb mole is

already high enough

to smear the SDW transition almost

completely

and the

development

of

the SDW order

parameter

at low temperatures ^{becomes much} more

gradual (see Fig. 6).

(TMTSF)~N0~

a)

T=1.5K

loo

(mv/cm)

b)

C)

is

is

.". _io

(TMTSF)2PF6

5 (TMTSF)IPFG

T=4.2K T=4.2K

-io -s o s io is -lo -5 0 s lo ls

E<mV/cm) E <mV/cm)

Fig.

9.

Dynamic

resistance

(dV/dI)

_versus electric field

(E)

for

(TMTSF)~NO~ (a)

and

(TMTSF~~PF~

crystals from batch S

(b)

and batch P

(c).

s

-' ,D

,' D

$

~ ~." ,' ^,'S

~ ."' "'

~

,j'

~

~,'~'

>~

' ,

r'w3 ,,1WO

,

'(i' '

~,

[

~,,

.-~"'4"~'~z"f'~'l'~

_.$---- ~ ~ ,

f..f --f-f-

T(K)

Fig.

10. Threshold field

(E~)

_versus temperature

(7~

for

(TMTSF)~PF~. Open

and full

points

for

noIninally

_pure

samples

from standard and

particular batch, respectively. Open

and full

triangles

and open squares for 0.002 fb, 0.008 fb and 0.02 fb of molar concentration of defects. Dashed lines _are fits based _on Maki's

theory (see text),

^where W

corresponds

to weak

pinning

and S to strong

pinning.

The inset shows the

low-temperature

value of the threshold field _versus defect concentration

(c).

)

E ^~~~

< ^~

_@ a

o

C~ (Q)

~ _a

o o ^o

~

C~ a ° .

b ~ _o

' a . .

$

a .

o .

b .

~ . ^.

o

a .

o .

o . o

o ^: o

o Oo

o .

~

Fig.

ll. (a) Non-ohmic conductivity ((«(E) _« (0)/«(0))) and

(b)

excess current Q~~) versas

electric field

(E)

for two

samples

of

(TMTSFJ~NO~

at 1.5 K with different

resistivity.

ratio

(rr). Open

and close circles for _rr m170 and 60,

respectively.

/Tii/K) o

o~

Fig,

12. Ohmic

(«o,

_open

points)

and non-ohmic

((« «o),

closed

points) conductivity

_versas

inverse temperature ^for

(TMTSF~2PF6 samples

^from (a) standard, and

(b)

particular batch. The electric field is 2

ET

and 1.5

E~, respectively.

Further,

_we have irradiated _one

(TMTSF~~PF~ sample

from

particular

batch.

Again,

0.002 fb mole of defects _was

enough

to

give

_a

higher

value of the threshold field at low temperatures :

E~(4.2 K)

_m 9.5

mV/crn. Moreover,

the

temperature dependence

of

E~

_was

qualitatively

the _{same as} the _one found for irradiated

samples

from standard batch.

However,

the total increase _was

larger

_:

E~(TN)/fir(T~~)

_m 2.2.

Activation

plots

of the

conductivity

for 0.002 fb mole ~' are shown in

figure

14.

First,

note that the _same activation behaviour for both

conductivity

channels is

preserved. However, already

for 0.002 fb mole of defects the break at

f

is rather smeared out

(as

for _pure batch

P),

while for 0.008 fb mole it is

completely

washed _out.

3.2.3

Magnetic-field-induced spin-density

_wave

phases

in

(TMTSF~~PF~.

We have searched for the non-linear

conductivity

in

high

electric fields in the

magnetic-field-induced spin-

TMTSF)~_PF~

~% _"o

c= 0.002 mdf' °fi

',

°"

%

~o

". ""O

O pwre

O ". ".

b 0.002$~ O°

-

"...

'". _~

°

0.00~$~ '".._ "".,__

~'~

~~~

^~~

fi~~~~~~ "",~'

^'"

O ~

odor.wl~#~°°-

~

. . .*..l 1 "~~ ~~

"._

Fig.

13. Fig, 14.

Fig.

13. Non-ohmic

conductivity ((« «o)/«o)

_versus

logarithm

of electric field

(E)

for pure and irradiated

(TMTSFJ~PF~ samples

from standard batch at 4.2 K.

Fig. 14. Ohmic

(«o,

_open

points)

and non-ohmic

((« «o),

closed

points) conductivity

_versas

inverse temperature ^for irradiated

(TMTSF~2PF~ samples

with 0.002

fb/mole

of defects.

density

_wave

phases

in

(TMTSF~~PF~ pristine samples

under _an

applied

_pressure of 10 kbars.

We have found

unambigous

non-ohmic contribution to the

conductivity along

the _most

conducting

_a axis. Some

previously unpublished

results _are summarized in

figures

15 and 16.

j,5_)

⁴ _,) ^{3 2}

~~

^H=ll6kG

T=035K

j~

^a

# ^~

~

Q

& #

E aa

(

p

b~

E

j

DO (

o o

2 3 4

MAGNETIC FIELD (kG) T lKl

Fig.

15. Fig. 16.

Fig.

15.

Magnetic

field

(H) dependence

of threshold electric field

(ET)

at 0.35 K. Different

symbols

are from different field _sweeps. Error bars _are smaller than

symbol

size unless

given explicitly.

Numbers above the _{curve are}

corresponding

_quantum numbers of each

phase.

The

positions

of the transitions _are

marked

by

vertical bars.

Fig.

16.

Temperature (7~ dependence

of threshold field

(ET)

at ll6kG. Measurement _was

perforrned

while

increasing

temperature after

applying

the

magnetic

field at the lowest temperature.

4. Theoretical model.

In order to describe the threshold electric field

ET,

^{the elTect of}

commensurability

and the temperature

dependence

of the _excess

conductivity

_we shall consider _an

anisotropic

Hubbard model with

impurities.

The

anisotropic

Hubbard model _as introduced

by Yamaji [36].

^This

model not

only

describes the

pressure-temperature phase diagram

of

(TMTSF~~X [37]

but also

predicts [38-40]

the _appearance of _a

magnetic

field induced SDW _as observed

experimentally

in _a number of

Bechgaard

salts

[41-44].

4, I THRESHOLD _{ELECTRIC FIELD.} Within mean-field

theory

the

phase

Hamiltonian which

describes the

dynamics

of the

phase

of the order parameter ^is

given by [45]

H(

4l

)

=

d~x No fj (@4l

j@t)~ +

i~(@ 4lj@x)~

₊

uj(@4l

/@y)~ +

+

uj(@4lj@z)~

+ 4

eu4lEj

₊

_V~;~(4l) (1)

where

No

=

(arubc)~~

is the electronic

density

of states at the Fermi surface per

spin,

§

=

(I

₊

tf)~/~u,

_u~

=

/ _bt~,

u~ =

/

ct~ ^and

tf

=

UNO

^{with U the Hubbard}

potential,

E is _an

applied

electric field in the most

conducting (a) direction,

_{u, u~} and v~ are the

anisotropic

Fermi velocities in the _a, b and _c

directions, respectively,

and t~, t~ and t~ the

corresponding tight binding

transfer

integrals (reduced

units _are used with h

=

I).

Here

fj

is the static condensate

density

which has the _same temperature

dependence

as

p~( nip

in the BCS

theory

of

superconductivity. Finally, V~,~(4l)

^{is the}

^{pinning potential}

when the

pinning

is due to

impurities [46-48]

V~i~(4l)

=

((w/2) No V)~ A(T)

tanh

((A(T~/2 T~)

_x

x

£

cos

(2(Q~

₊ 4~

(, )) (x

_x,

) (2)

where V is the

impurity potential

and the _sum is _over the

impurity

sites ~. In

deriving equation (2)

_we

neglect

the elTect of

imperfect nesting

for

simplicity. Fortunately

this effect is

negligible

for

(TMTSF~~PF~

at ambient _pressure.

Equation (I) generalizes

the

early phase

Hamiltonian considered

by Fukuyama-Lee [49]

to SDW and to all

temperatures.

Now

following Fukuyama,

Lee and Rice

[49, 50],

_we derive the threshold electric field. In the

strong pinning

limit _we obtain

Ei(°)

"

(Qle)(ni/n)l'rNo V)~ do (3)

and

E](T)/El(0)

=

(A(T~jAo)

tanh

(A(T)j2

_T~

fj1 (4)

while in the weak

pinning

^limit

(3D) Ei(o)

=

(36/21°) (Qle) (

_~ ⁱ

n;/n)2 ( «No v)8 N] at ₍₅₎

and

Ei(n/w(o)

=

(Ei(n/Ei(o))4 ₍₆₎

where n; and _{n are} the

impurity density

and the electron

density, Q

= 2 p~

~pp

^{is the Ferrni}

momentum), do

=

A(0)

and

1~ ⁼ u~ u~/u~ is the

anisotropy parameter.

The temperature

dependences

of

equations (4)

and

(6)

_are evaluated

numerically,

and _are shown in

figure17.

We note that for T~

TN, E~(T~

is almost constant. As T increases

2

further

E~(T~

increases

monotonically reaching E~(TN) /ET(0)

=

1.33 and 3.13 for the strong and weak

pinning limits, respectively.

For reasonable

parameters

we obtain

I)

the weak

pinning

limit is most

likely

when

No

V

=

0,1, 2)

it

gives

_a threshold electric field of the order of1-10

mV/cm.

The detailed

comparison

with

experiments

is

given

in section 5.

w

~

~

~ s

~~

For t~ = 3 000

K, do

=

20 K and

tf

=

UNO

₌

I,

_we find that

E)(0)

in

equation (8)

is of the

order of

lmv/cm.

Further

E((T~

decreases _as T increases and vanishes

linearly

in

(TN T)

at T

=

TN,

as shown in

figure17.

We

point

out here that

commensurability

is relevant

only

for N

=

3 and 4. In

TTF-TCNQ

under pressure

[31]

there is _a drastic increase in

E~

at third order

commensurability (N

=

3).

For N

=

2 _as in

trans-polyacetylene

the

commeilsurability

is too strong. ^In this _case CDW _or SDW is

completely

locked to the

crystalline

lattice. On the other

hand,

for N _m

5,

the

conunensurability

is _too weak. For TN =

10 K and _t~

= 0.I eV the threshold field is of the order of10

~LV/cm, though

_no such

example

is known.

4.3 ExcEss CONDUCTIVITY.- From the

phase

Hamiltonian

given

in

equation (I),

the

equation

of motion for 4l is found to be

d~4l/@t~

+

r~(@4l/@t)

=

F~(@~4l/@~)

+

v((@~4l/@y~)

+

u((@~4l/@z~)

² eu

(E E~) (10)

where _we have inserted _a

phason damping

constant

r~ (due

to

impurities). Further,

_we have

replaced

the 4l

-dependent pinning

term

by

_a constant

E~.

Then for E

~

E~, equation (10)

has

a

simple

solution

am

pat

=

(2 ev)jr~

_x

(E E~) (i i)

J~ (E)

=

eQ

_n

fj

_(@4l

/@t)

= «o A

(E ET) (12)

«(E)

=

J/E

=

«o(I

+ A

(I E~/E) 8(E E~)) (13)

where

«~ =

e~n/m(I fj) rj ('~~)

A

=

~fi/(' -fi)) rn/rp ('4b)

&(E E~)

₌ I for Em

E~

^and 0 for E _<

E~

^and

r~

is the

quasi-particle damping

constant.

According

to

equation(13)

the non-ohmic term increases

exponentially (I.e.

«o A = exp

(flA)

and

fl

=

(k~ T)~

_~) at low

temperatures

contrary to the observation.

As

already

discussed for CDW

[51, 52],

the

long

_range Coulomb interaction

couples

the

phason

with the

quasi-particles.

Then the

quasi-particle damping

dominates the

phason damping

for all temperatures ^T _< TN. ^This

implies

that

r~

in

equations (10), (11)

and

(14)

has to be

replaced by [53]

rp

+

rn fl/(1 fl) (15)

which

yields

A

=

(i

₊

(i fj)/fj

_x

r~/r~)-1 (16)

The

temperature dependence

of A is evaluated

numerically

for _a few

rj

and

r~

and is shown in

figure18. rj

and

r~

_are the forward and the backward

scattering

rate due _to

impurities [54].

In

figure

17. TNO is the

hypothetical

Neel temperature ^{in the absence of the}

impurity scattering (I.e. rj

=

r~

=

0).

For small

impurity scattering

TN ^is

given by [54, 55]

TN= TNO-I (rj+ _~r~). (17)

Below T

= TNO, A starts from

0,

increases

rapidly,

and reaches

unity

for Tw 0.8 TNO.