Drift velocity and temporal phase fluctuations of sliding charge density waves in Rb0.3MoO3

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Drift velocity and temporal phase fluctuations of sliding

charge density waves in Rb0.3MoO3

P. Butaud, P. Ségransan, A. Jánossy, C. Berthier

To cite this version:

Drift

_velocity

and

_{temporal phase}

fluctuations of

_{sliding charge}

density

waves

in

Rb0.3MoO3

P.

_Butaud,

P.

_Ségransan,

A.

_Jánossy

_(*)

and C. Berthier

Laboratoire de

_{Spectrométrie Physique}

_(associé

au

_C.N.R.S.),

Université

Joseph

Fourier, Grenoble 1, B.P. 87, 38402 St Martin d’Hères, France

(Reçu

le 19

_septembre

1989,

accepté

le 20

_septembre

₁₉₈₉₎

Résumé. 2014 Nous observons le mouvement de l’onde de densité de

_{charge dépiégée}

sous l’effet

d’un

_{champ électrique}

_{par des} mesures de _transport, de tension de bruit et _par une étude RMN, effectuées sur le même monocristal de

_Rb0,3MoO3.

L’observation de bandes latérales dans le

spectre RMN en

_présence

du mouvement de l’onde constitue une mesure directe de

03BDd, la

fréquence

locale d’oscillation de la modulation de la

charge

de l’onde. L’étude de

l’amplitude

de l’écho de

spin

permet une mesure résonante de la même

_quantité

03BDd. La

comparaison

avec les mesures de tension de bruit

_{qui présentent}

un

_pic

d’intensité à

03BDn, confirme nos résultats antérieurs montrant _{que 03BDd et 03BDn} sont

_{égaux à}

10 %

près.

Les _spectres RMN relevés à

_{E/ET ~}

20

_(ET

est le

_{champ électrique}

seuil de

_{dépiégeage)}

et dans la

_plage

de

température

40-60 K sont très bien

_reproduits

_{par les simulations}

_numériques

assimilant la densité

spectrale

de bruit à la distribution

_spatiale

de la vitesse de l’onde. Nos résultats

_{s’interprètent}

bien

en considérant le cristal comme une collection de domaines. A l’intérieur de

_chaque

domaine la modulation du

_champ

_hyperfin

reste cohérente

_pendant

environ 25 à 50

_périodes.

Enfin nous

discutons des fluctuations

temporelles

de la

_phase

de l’onde de densité de

charge

qui

deviennent

importantes

à haute

_température

et

_{plus spécialement près}

du

_{champ électrique}

seuil.

Abstract. 2014 The

_dynamics

of CDW-s

_{depinned by}

an electric field is

_{investigated by}

_transport,

voltage

noise and NMR measurements on a

_{Rb0.3MoO3 single crystal.}

A direct measurement of the bulk

_winding

rate _03BDd of the

sliding

CDW

_phase

is

_{provided by}

the observation of sidebands in the

_motionally

narrowed NMR _spectra and

_by

the oscillations of the

_spin

echo

_amplitude

as a

function of the

_delay

time.

_Comparison

with the noise _spectra confirms our earlier statement that

03BDd is

equal

to the

_voltage

noise

_frequency

_03BDn with an _{accuracy of 10 %.}

_Experimental

NMR

spectra at a field of

_{E/ET ~}

20

_(ET

is the

_{depinning threshold)}

in the _{temperature range of 40} to 60 K have been fitted with

_high

_accuracy to

_lineshapes

calculated under the

_assumption

that the noise spectrum is related to the

spatial

current distribution in a

_simple

_{way. Most of the}

experiments

can be

_{interpreted by considering}

the

_crystal

as a collection of domains. In each domain the current is coherent for

_{long times,}

we found in some cases a

_temporal

coherence of 25-50 oscillation

_periods

of the

_hyperfine

field. We discuss stochastic variations of the CDW

_phase

which become

important

at

_high

_{temperatures,}

_especially

at fields near threshold.

Classification

Physics

Abstracts 71.45L - 72.15N

76.60C

(*)

Permanent address : Central Research Institute for

_Physics,

H 1525,

Budapest

114,

Hungary.

1. Introduction.

Sliding Charge Density

Waves

_(CDW),

sometimes called the Frôhlich

_mode,

is a collective

excitation of the

_{electron-phonon}

_system

of some materials with a

_quasi

one dimensional

electronic structure

_[1].

Its most obvious

_signature

_{is the appearance of} a non-linear current

above a threshold

_applied

field.

_By

now there is no doubt that the extra current observed first in

_NbSe3

_[2],

and since in a few other

systems,

is due to the

_sliding

of

_{charge density}

waves as a

_whole,

the best evidence is the motional

narrowing

[3]

and the appearance of sidebands

[4]

of the Nuclear

_Magnetic

Resonance

_(NMR)

spectrum

under current. A number of

_questions

regarding

the Frôhlich mode remain however unsettled. In

_particular

it was realized soon

_[5]

that a constant CDW current is

_{accompanied by}

a

_{periodic voltage}

_oscillation,

but the

_origin

of these oscillations and in

_particular

the role of CDW

_{phase slip}

lines has not

_yet

been

clarified.

The

present

work is an extension of earlier studies

[3b, 4], it

deals with electrical

_transport

and

87Rb

NMR of a

_Rbo.3Mo03

_crystal

in the

_sliding

CDW state. The

_emphasis

is on the

temporal

variation of the CDW

_{phase. Rbo.3Mo03}

is

_{representative}

in most

_respect

of all

sliding

CDW

_{systems ;}

its

_crystal

and CDW structure

_together

with the

_transport

_properties

have been

_{thoroughly investigated}

[1].

After a brief survey of the basic

_quantities

measured

(Sect. 2)

and an account of the

_{experiment techniques}

_{(Sect. 3)}

we deal with the

_transport

properties

(Sect. 4).

The

_presentation

of NMR data is

_{preceded by}

a _{survey of the}

_theory

of

the NMR

_lineshapes

_{(Sect. 5),}

the discussion is limited to the conditions of the

_experiment.

NMR in CDW

systems

has been reviewed

_recently

in reference

_[6].

In section 6 free

precession

and

_spin

echo Fourier Transform

_(FT)

_spectra

_together

with

_spin-spin

relaxation time data obtained at various

_temperatures

and

_applied

electric fields are

_presented.

The

emphasis

is on the measurement of the drift

_velocity

and the random

_phase

fluctuations of the CDW. A more

_{complete description}

of NMR in

_sliding

CDW

_systems

will be

published

elsewhere

_[7].

The drift

_velocity

is measured in a direct resonant _way

_{by spin-echo.}

In the discussion

_{(Sect. 7)}

we correlate the

_transport

and NMR data obtained on the same

_crystal.

We confirm our earlier assertion that the CDW

winding

rate

_equals

the

_frequency

of the

voltage

noise. We

_separate

effects due to

_temporal

fluctuations of the CDW

_phase

from those due to

_{spatial inhomogeneity}

of the current. At low

temperatures

temporal

random fluctuations of the local field are measurable but not _very

_important.

At

_temperatures

above 70 K and for moderate electric fields we arrive at the

_unexpected

conclusion that the

_phase

fluctuates with

_{large amplitude}

even in

_regions

of the

_sample

where,

due to

_{imperfections,}

the average current is zero.

2. The

_sliding

CDW state.

A one dimensional uniform metallic chain is unstable

against

a

_periodic

distortion with wavevector 2

_ir/À

=

2 kF

where

kF

is the Fermi momentum. Weak interactions between

chains lead to a three

_{dimensionally}

ordered distorted state below the Peierls transition

temperature

Tp

where the

_{charge density}

wave has the form :

here p o is the uniform

density,

p

the

CDW

amplitude,

in the mean field

theory proportional

to the

_magnitude

of the order

parameter.

The

_component

of the wavevector

_along

the chain

qy

= 2

kF

is determined

by

the band

filling ;

_{qx, qz}

depend

on weak interactions between the

with the lattice fixes cp if the

system

is commensurate i.e. if

_qy

= _qo

=n/m

2n/ b

where b is the

lattice constant

_along

the uniform

chain, n

and m are small

_integers.

For band

_{fillings leading}

to

_nearly

commensurate wavevectors the CDW is

_expected

to consist of commensurate

regions interrupted by

discommensurations where cp is

varying.

There are two extreme cases

for this

_{« nearly}

commensurate »

phase

where discommensurations are :

_i)

an _{array of 2D}

phase slip

narrow walls

_separated

by 1

2 7T

or

_ii)

broad

_walls,

a weak modulation of the

m

_{Î q - qo}

phase

cp with

period

2 7r

.

lq - qol

l

If the lattice

_potential

is not

_strong

and the

_system

is incommensurate the CDW is

represented

to a

_{good approximation by}

a

_plane

wave with a slow variation of the

_phase

determined

_{by impurities}

or other

_{imperfections.}

In the

_theory

of

_Fukuyama

and Lee

_[8]

and Lee and Rice

_[9]

the

_{impurity pinning}

is called

_strong

if ç has the same value at all

_impurity

sites or,

_{alternatively,}

weak if the

_phase

at _any

_point

in the

_{crystal depends}

on the collective

action of a

_large

number of

_impurities.

In

_{high purity Ko.3Mo03}

the

_phase

_{may be coherent}

over more than 1 )JLm

along

the chain and

nearly

1 )JLm transverse to the chain

_[10].

In cases

where neither the lattice nor

_impurities

are

_important

the main source of

_pinning

_{may be}

macroscopic

defects

_{(e.g. inclusions)}

or the boundaries of the

_crystal.

A small electric field

_polarizes

the CDW. Above a threshold value

_ET

the electric energy

gained by displacing

the CDW

_by

a

_wavelength

exceeds the energy

_{gained by adjusting}

the

phase

to the

_{pinning potential}

and the CDW becomes

_depinned

and slides with a drift

velocity :

Vd may be measured

by

the current

_jcDw

associated to the Frôhlich mode

where _Pc is the condensed electron

_density

which at

_tempeatures

much below

_{Tp equals}

the uniform electron

_density

Po = 2 À e n, n

is the

_density

of

_conducting

_{chains per unit} area

perpendicular

to their direction times the band

_degeneracy.

Alternatively,

the

_{phase winding}

rate

may be measured

directly by

NMR

_{by observing}

the

_temporal

variation of the

_hyperfine

field

at sites on or

_nearby

the chain. In the static case the NMR

spectrum

is

_{inhomogeneously}

broadened

_by

the

_spatial

variation of the

_hyperfine

field which follows the variation of the CDW. The

_drifting

CDW modulates

_periodically

the

_hyperfine

field and the

_{corresponding}

change

in the NMR

_spectrum

allows a direct measurement _{of v d. We} note that one could measure _{v d}

_by

_{the energy}

_change

of neutron diffraction

_superlattice

reflexions

_{(Doppler shift)}

[11].

This method would be

_equivalent

to the NMR

_method,

it also measures _vd.

If the CDW is an ideal

_plane

wave driven

_by

a constant force vd is constant and

_equal

to the

periodicity

of the local

_hyperfine

field variation.

If, however,

the CDW is

_{inhomogeneous}

or

the forces

_acting

on _{it vary with}

_{time v d}

will also fluctuate. For

example

a

_sliding

CDW

interacting

with a

_{single impurity}

encounters a

_{potential varying periodically}

in time. Periodic

In all

_sliding

CDW

_systems,

even for a constant external

_driving

field,

a fluctuation is

superimposed

on the CDW drift. The average

_frequency

of this current oscillation is

proportional

to _{the average} current. Several mechanisms have been

_proposed

to

_explain

this

phenomenon.

In the

_impurity

model

_[8]

it is assumed that the boundaries of the

_crystal

are

unimportant

but rather the

_system

_{may be}

_thought

of

_being

divided into domains with

dimensions

_equal

to the

_{corresponding}

coherence

_lengths.

In the

_sliding

state in each domain the overall

_pinning

force of the

_impurities

varies

_periodically

with the difference between the

equilibrium

and instantaneous

_phases.

In the vortex model

_{[9, 12]}

the current oscillations are due to a

_{periodic generation}

and annihilation of CDW

_phase

dislocation lines at the interface of

_pinned

and

_{unpinned regions.}

Both models

_predict

that the fundamental current oscillation

_frequency

_{v n}

_equals

the

_phase

winding

rate.

Bardeen

_[13]

_suggested

a

_theory

where P,, = 2 Vd and until

recently

[4]

experiments

measuring

vd

by

the nonlinear current could not

_provide

an

_unambiguous

answer

_[14].

As we

show,

a measurement of the local field _{variation proves}

_(2.5)

with

_good

_accuracy.

In _{this paper} we show that in addition to the

_periodic

fluctuation of

_ICDW

there is a

temperature

dependent

random fluctuation. In

_principle,

random fluctuations broaden the

frequency

spectrum

measured

_by

a noise

_spectrum

_analyser.

However the

_broadening

of the noise

_spectrum

_peak

of a

_single

domain with a

_homogeneous

current is difficult to resolve from an

_{inhomogeneous broadening}

due to a

_spatial

distribution of the current of an

imperfect crystal. By

NMR the

_{spatial inhomogeneity}

_{of Vd}

is

_{distinguished}

from local random

temporal

fluctuations of the

_phase.

3.

_{Experimental.}

All new measurements

_reported

_{in this paper} were

_performed

on the same

_{single crystal}

Rbo.3Mo03

(sample # 2)

grown

by electrolysis

from a melt of

_Mo03

and

_Rb2Mo04

_by

Marcus

_[15].

It was cut

_by

a wire saw into an

_{approximately rectangular shape ;}

the

_length

along

the

_{crystallographic}

axis b was 3.60 mm, the cross-section of the ends

perpendicular

to b

was 0.45 mm2. The NMR

_coil,

had a cross-section

_only

_{slightly larger}

than the

_crystal

and was

about as

_long.

After

_{being placed}

in the NMR coil the ends of the

_crystal

were

_polished,

copper

plated

and contacted with silver

_paint

to thin

_gold

wires.

_Judging

from the current

voltage

(I-V)

characteristics and the

_voltage

noise

_spectra,

the

_crystal

_quality

was somewhat

better than those used in

_previous

studies

_[4].

1-V characteristics and

_voltage

noise

_spectra

were recorded in situ in the NMR coil.

_Usually

the current to the

_sample

was continuous

_except

for measurements at 75 and 87 K where the NMR

_spectra

were taken

_by

a

_{pulsed technique}

to avoid

_heating.

1-V characteristics were

measured

_by

a

_quasi

four

_point

method.

_Although

this method does not eliminate contact

resistance we believe this has no influence on the data. At room

_temperature

the resistance of

the

_{sample together}

with contacts was less that 0.1 S2.

The noise

_spectra

were taken with an HP 8568 A

_spectrum

_analyser

connected to an

HP 9836

_computer.

A baseline taken with field below threshold was subtracted and then the

noise

_intensity

_{V2(V )/BW}

calculated

_(here

V is the

_amplitude

of the

_voltage

fluctuations and BW the instrumental

_bandwidth).

The

_spectra

were distorted

_by

50

cycles

interference at

frequencies

below 1 kHz. A small variation of the

_gain

of the

_preamplifier

and transmission line was

_noted,

this affects somewhat the

_intensity

data for

_{large frequency}

_{spans but is}

The free induction

_decay

_(and

in some cases the

_spin

_echo)

of the

₍

-

1/2,

+

1/2)

transition of site

₍₂₎

87 Rb

_(in

the notation of Ref.

_[16])

was recorded at _vo = 80.132 MHz

by

a

BRUKER CXP-100

_{spectrometer.}

The extemal

_magnetic

field

_Ho

= 5.175 T _was

applied

along

the c* axis

_{(perpendicular}

to the a-6

_plane).

The

_amplitude

of the

_rotating

field

Hl

was 70

G,

and the dead time of the receiver was less than 6 ws.

Signals

were

averaged

and

Fourier transformed

_by

a NICOLET LAS 12/70

_{signal analyser.}

To

_improve

the

S/N

ratio a

_{systematic phase}

altemation of

_{the 2}

pulses 2 - -f

1

was

_applied

combined

2 2 je 2

with

_sign

inversion of the

_acquired

data. Also a reasonable offset was chosen between

vo and the center of the

_spectrum

to avoid the formation of a

_spurious

FFT

_image

in the

frequency

range of interest. Final

phase adjustments

were made

_by

an HP 9836

_computer.

4. Measurement of

_{transport properties.}

4.1 CURRENT-VOLTAGE CHARACTERISTICS. - The threshold

voltage VT

for the onset of non-linear conduction was measured between 43 and 78 K

_{(Fig. 1).}

At all

_temperatures

the threshold is well defined

_although

at lower

_temperatures

a small

_uncertainty

arises from the difference between

_increasing

and

_decreasing

field measurements. Between 43 and 60 K

ET

increases

_only

_by

a few

_percent,

at

_higher

T the increase is much faster. These results are in

agreement

with those

_reported

in the literature

_[1].

A

_{typical conductivity}

versus electric field characteristics is shown in

_figure

2. The

non-linear CDW current is determined

_{by subtracting}

the normal current

_extrapolated

from the low field value :

Fig. 1.

Fig. 2.

Fig.

1. -

Variation of the threshold

_voltage

_{(sample # 2)}

versus _temperature T. It is measured in situ in the NMR coil.

’

Fig.

2. -

This

_{extrapolation}

may not be

_entirely

valid for two reasons :

_i)

_Rn,

the normal resistance

depends

on electric

history

and may be somewhat different in the

depinned

_{state ;}

ii)

the onset of the CDW current _{may be}

_{accompanied by}

a normal current backflow

_[17].

We

assume in the

analysis

of data that these effects are not

_important.

The normal

_conductivity

has an activated behaviour as a function of

_temperature

with

activation _energy

_E’

= 490 K

(Fig. 3).

This value is the same as found in other

_crystals

indicating

that the

_crystal

_{is pure}

_enough

to be an intrinsic semiconductor above 40 K. The

CDW

_conductivity

_ICDW/V

is also

_activated,

_{the energy}

_dépends

somewhat on the

field,

at

V = 200 mV

E aCDW =

743 K while at V = 600 mV

EaCDW =

837 K

(Fig. 3).

Although

these

energies

were determined in a

_relatively

narrow

_temperature

_range

₍₄₃

to 55

_K)

_they

still

convincingly

show that

_{E a CDW}

is

_{substantially larger}

than

_Ea.

Fig.

3. -

CDW

_(circles)

and normal

_(triangles)

current densities as a function of the inverse temperature. The data in the inset

_{(sample # 1)}

were taken at a field of 1.5

V/cm,

and those of the main

figure

(sample # 2)

at 1.67 V/cm. The normal current is

_extrapolated

from low field data.

The CDW is not

_depinned

in the entire

_sample

at

_VT,

and due to the

_{inhomogeneity}

of the

current

_{density jcDw,}

the I V characteristics do not

_give

the correct

_dependence

of

jcDw

on E.

_{Nevertheless,}

the

_similarity

of the characteristics at different

_temperatures

suggests

that the distribution

_{of jcDw/IcDw}

within the

_crystal

does not

_change

_{significantly}

4.2 VOLTAGE NOISE SPECTRA. -

Applying

a constant current

_through

the

_sample

above threshold induces a

_voltage

V

(t)

_fluctuating

in time. In

_NbSe3,

in some cases where the CDW current _{may be}

_homogeneous

within the

_sample

_{V (t )}

is

_{nearly periodic}

and the

_frequency

spectrum

V2(v)

of the fluctuations of

_V2(t)

consists of a narrow line with harmonics. In the

Rbo.3Mo03 sample investigated

the

_voltage

fluctuation

_spectra

are

_{complex mainly}

due to an

inhomogeneity

of the CDW current over the cross-section.

Voltage

spectra

were recorded at

_temperatures

between 40 and 87 K at various bias fields. The form of the

_spectra

_{depends strongly}

on the

applied voltage

but for a

given voltage,

spectra

taken at different

_temperatures

_{may be scaled} on to each other.

Figure

4 shows noise

_intensity

_spectra

at a fixed

temperature

(53.5 K)

at various

_applied

fields V

_(35.4,

70,

250,

600

_mV).

For V

_just

above threshold a few narrow

_peaks

_appear, at

twice threshold a

_large

number of

peaks

are

_resolved,

at even

_higher

fields narrow lines are no more resolved. At V = 600 mV the

_spectrum

consists of a broad

_peak

with some structure

and a tail

increasing sharply

at low

_frequencies.

Fig.

4. -

Noise power spectra taken at T = 53.5 K in

sample #

2. The

shape

of these _spectra

radically

changes

with

_applied

field.

Spectra

at a fixed field

_(V

= 600

mV)

_at

42,

50 et 55 K demonstrate the

scaling

with

temperature

(Fig. 5).

Although

the average

frequencies

differ

_by

a factor of 70 the noise

intensity

distributions are

_quite

similar. At low

_temperatures

the low

_frequency

tail could not

be resolved at V = 600

mV,

increasing

the field _to

higher

values _{it appears,}

_however,

with

large

intensity.

For

_{understanding}

the

_origin

of noise the variation of its time

_{averaged amplitude}

Fig.

5. -

Noise power spectra taken at E = 1.67 V/cm in

sample #

2. The first harmonic _component has been subtracted from the raw data. Note the

_similarity

of the _spectra

_despite

a

_change

of a factor 60 in the

frequency

scale. The

voltage

_power

_V

_(vn)12

changes

inversely

to the mean

_frequency

so that the total

_intensity

remains constant.

are difficult to

_perform

since the transmission line

_connecting

the

_sample

to the

_spectrum

analyzer

is somewhat

_frequency

_dependent.

_Also,

the

_change

of the current distribution with field renders the

_{interpretation}

uncertain.

_{Nevertheless,}

_{it may be concluded from}

_figure

6 that the noise

_amplitude

does not _vary

_strongly

with field and

_temperature.

Fig.

6. - Noise

_power

_{1 V (t ) 12}

taken at different values of E and T.

_Despite

the variation of the

5.

Survey

of NMR in

_Rbo.3Mo03.

Before

_presenting

the

_experimental

results a brief

_description

of NMR and

_spin

echo

spectra

in

_quasi

one dimensional CDW

_systems

is

_given.

The discussion is limited to

_aspects

relevant

to our

_experiments

on

.B7Rb

in

_Rbo.3Mo03.

5.1 NMR SPECTRUM, STATIC CDW. - The interaction of the nucleus at site

_Ri

with its

surroundings

is

_{given by}

Under our

_experimental

conditions the

_quadrupolar,

_magnetic

_hyperfine

and

dipolar

interactions

_(second

to fourth terms

_{respectively)}

are a

_perturbation

of the Zeeman term

Hz.

The Rb nucleus is not included in the

_conducting

chain and the

_{magnetic hyperfine}

term is

negligible compared

to the

_{quadrupolar coupling}

even above the Peierls transition. The

dipolar

term determines the

_homogeneous

_part

of the linewidth in both the metallic

_phase

and below the Peierls transition without an

_applied

electric field. The

_quadrupolar

term shifts the Zeeman levels and

_splits

the threefold

_degeneracy

of the transition

_frequencies

of the i

= 3/2 87Rb

nuclei. We consider in the

_following

the - 1/2 -> + 1/2 central transition

_only.

The

_quadrupolar

interaction is

_proportional

to the electric field

_gradient

tensor

The

_{parameters vQ}

=

e2 qQ/2 h

with eq

=

1 Vzz

1

and q =

(VXX-

V yy )/V zz

together

with the directions of the

_principle

axis

_{X, Y,}

Z were determined for the Rb sites in the

metallic

_{phase by Douglass et}

al.

_[18].

The resonance

_frequency

is shifted in second order of

_PQIPO

from the Larmor

_frequency

vo =

yHo

(with

Ho

the

applied

static

magnetic

field and _y the

gyromagnetic

ratio)

[19] :

where 0

_{and 0}

are the

_polar

and azimuthal

_angles

of the external field with

_respect

to the

principal

axis and

_{f (,q, 0, 0 )}

can be found in reference

[19].

Above the Peierls transition in the

_geometry

of the

_{experiment 0}

=

0

= Ir ,

from

reference

_[8]

ry = 0.5 and thus

2

We

_{decompose à v}

into two

_{parts : à v}

= à v 0 +

Sv

where à v 0

arises from the undistorted lattice while Sv

_represents

the

_frequency

shift due to the CDW.

The EFG

_originates

from distant ions and delocalized electrons and from electrons on the site of interest in the case where wave functions are

_hybridized

with the conduction band. The effect of distant

_charges

is

_{amplified by}

the Sternheimer

_{antishielding}

factor

_[20].

For on site electrons the

_{antishielding}

factor is close to 1 but the contribution to the EFG

The contribution to the EFG due to the

_overlap

of delocalized electrons of the Mo-O clusters with the Rb sites is not

_{necessarily negligible compared}

to the ionic

contribution ;

in the case of

NaxW03

for

example

the

overlap

of d-band conduction electrons with the Na

wavefunction

_gives

rise to well observable

_Knight

shift and

_Korringa

relaxation of the

23Na

nuclei

_[21].

In the metallic state there are two Rb sites

_Rb(1)

and

_Rb(2),

with

_symmetries

2/m

and m

_{respectively,}

which have different resonance

_frequencies.

In this work

_Rb(2)

was

investigated.

Below the Peierls transition a _{CDW appears}

_coupled

to a

_periodic

lattice

distortion

_(PLD)

and the

_position

of

_{site j}

is

_displaced

_{from the average}

_position

Ri

by :

in the

_plane

wave

_{approximation.}

The modulation vectors

_41,2

were determined for

Ko.3MO03

(a

compound

very similar in all

respect

to

_Rbo.3Mo03)

_by

Shuttle and de Boer

_[22].

The

_{displacements}

of various sites have similar

_temperature

_dependences

and are

proportional

to the order

_parameter

of the CDW

_{given by equation}

_(2.1).

At T = 100 K the order of

_magnitude

_of

_{1 Àll, 21 [}

is

10- 2

A

_[22].

The wavevector q is found to be

(0,

qb,

1/2)

in

reciprocal

lattice units

with qb

close to 3/4 below T = 100 K

[23].

The CDW and PLD modulates the EFG _{tensor, v Q, 11} and the directions of the

_principal

axes _vary

_periodically

between well defined values for sites

_along

the

crystallographic

direction b. A first

_principles

calculation of the modulation of 8 v as a function of

_position

is unfeasible. A

_{point charge}

model

_taking

into account the

_displacement

of distant ions

_given

by

(5.1.5)

would

_neglect

the covalent nature of the Mo-O bonds and also the variation of the

on site contribution

_(5.1.4).

The resonant

_frequency

varies

_periodically

with

_wavelength

À

= 2 q 7T

along

the b axis and

to second order in the order

parameter

the modulation of à v is :

where v 1 and V2 are

_proportional

to the order

_parameter

and to _{its square}

_{respectively.}

v 1, v 2, cp

1 and

(P 2 are determined

_by

the CDW induced

change

of all

_parameters

of the EFG.

A variation in the

_largest

component

of the EFG tensor results in a

_change

of

with

dvO

is

_{given by equation}

_(5.1.3).

The term

_quadratic

in 5

_{v Q}

is

_negligible

and the shift Sv =

(2

d v 0/ v g)

ô

vQ (R; )

is linear in

the

_{displacement despite}

that the EFG

_perturbes

the

_energies

in second order

_only.

For a

_long

wavelength 2q

» b a local

approximation

_{may be valid and} cpl

(= cp2)

may be

equal

to the

_phase

of the local

_{displacement [24].}

In

_special

cases where the site has a

particular

symmetry

VI is zero. This is not the case in our

_study.

We have checked

experimentally

that for the

_Rb(2)

site of interest the dominant term is linear in the order

parameter

( v

>

V2),

since the

_splitting

of the

_lineshape

below

_TP

evolves

_{symmetrically}

with

respect

to the

_position

of the line in the undistorted

_phase

_[24].

From

_(5.1.6)

the resonance

_{lineshape resulting}

from the distribution of the EFG is

readily

If _V2 is

_negligible

and as

_long

_{as q is incommensurate :}

where v is measured from the

_unperturbed

resonance

_frequency

_Po

+ Av 0

For

nearly

commensurate wavevectors

_large

deviations from the

_plane

wave PLD

given by

(5.1.6)

are

_expected

due to the interaction of the CDW with the

_periodic

lattice

_potential.

In the so called «

nearly

commensurate »

_phase,

observed in some two dimensional

systems,

the CDW is

_composed

of commensurate

_{regions interrupted by}

discommensurations in which the

phase

of the CDW varies

_rapidly

_[25].

In

_Rbo.3Mo03

below 100 K the deviation from the

commensurate wavevector _qo =

(0,

3/4,

1/2)

is very small

[26]

and it _was

suggested

[27]

that an

incommensurate commensurate

_(IC-C)

transition takes

_place.

Such a commensurate

_phase

has not been found

_by

NMR

_[6,

_28,

_29]

_although

indications of a C

_phase

in

_part

of the

_crystals

below 50 K has been found

_by

ESR

_[30].

In a commensurate

_system

there are

_only

a finite

number of

_inequivalent

sites and the NMR

_spectrum

consists of a small number of lines instead of the distribution

_(5.1.8).

There are

_{32 Rb(2)}

and

_{16 Rb(1)}

sites in the

_expected

commensurate unit but due to the

_symmetry

of the PLD for both sites

_only

8 have different

resonance

_frequencies

for an

_arbitrarily

oriented

_magnetic

field. A careful

_study

of the

_Rb(l)

lineshape

at T = 77 K has shown that if there _are

any discommensurations in the

system

their width is

_comparable

to their

_separation

_[28].

Figure

7a shows a fit of

_(5.1.8)

to a

_experimental

_spectrum

of

_Rb(2)

at 77 K. The values of

VI and v2 are 4.9 kHz and 0.7 kHz

_{respectively. Figure}

7b shows the best fit obtained

_by

varying

the

width a -

of

the walls of a discommensuration lattice

where,

f,

the distance between two

_walls,

was chosen

_{arbitrarily equal}

to 1 000 interatomic distances

[28].

5.2 NMR SPECTRA FOR A SLIDING CDW. - In the

_following

we use an adiabatic

approximation

and assume that for a

_depinned

CDW the resonance

_frequency

at site R is

determined

_by

the instantaneous

_phase

_{o (R,t)}

of the CDW.

Thus in

_(5.1.6)

_8v(R)

is

_{replaced by}

In the

_simplest

case the

_{phase velocity}

is constant _{in time and uniform in space and}

with

The

_{simple uniformly sliding plane}

wave

_{approximation}

_may not be

_entirely

correct in

describing

the CDW motion. As

_{suggested by}

Lee and Rice

_[9]

the current _{may be}

transported by

a series of

loop

dislocations. Far from the dislocation center the

_change

of the

amplitude

of the order

parameter

may be

neglected

and the variation of cp is non uniform but

still

_periodic.

The situation is similar to the motion of a discommensuration lattice.

Although

the dislocation

_loops

or discommensurations _{may be}

_{widely spaced,}

the

periodicity

of the

oscillations of the EFG at a

_given

site is the same as for a

_plane

wave CDW

_provided

the same

(average)

current is

_flowing.

The

_phase

_winding

rate in

_general

for an _{array of}

discommensu-rations with finite width

_moving

with

_{velocity Vdc is}

_(supposing

the

system

to be fourth order

commensurate)

where a n

depends

on the form and width of the discommensuration and

8 q

= q -qo is the deviation

of q

from the commensurate value.

The exact solution of the sine Gordon

_equation,

in the limit of broad discommensurations

can be shown to be cp

(R)

=

a 8 qR

₊

/3

cos 4

_{8 qR}

_[28].

For discommensurations of

_large

wall widths or far from

_{phase slip}

dislocation lines the

a "-s are small and the main term is the time

_averaged

_{phase winding}

rate :

For

_equal

current, the

_{phase winding}

rate of a

_sliding

_plane

wave and that of a

discommensuration lattice are

_equal.

Near dislocation

_lines,

if dislocations are narrow, the

harmonics in

_(5.2.3)

_{may be}

_important

in the calculation of the NMR

_spectra.

In this case the

assumption

of a constant

_amplitude

of the order

_parameter

_{is, however,}

_unjustified.

The

periodic generation

and anihilation of

_phase

dislocations at the boundaries or the uneven

motion due to the

_impurity

_{pinning potential}

also

_give

rise to harmonics like those in

_(5.2.3)

but,

except

for very small currents, the

_amplitude

of the current oscillations is small

_compared

to _{the average CDW} current and do not influence much the NMR

_spectrum.

In the

_following

we discuss the NMR

_spectra

for a

_plane

wave CDW. We first consider a

We first retum to

_(5.2.2)

where

_{dcp Idt}

= 2

1TVd is uniform in space and time. The

correlation function of the transverse

_{magnetization}

_{(Ix(t) 1x(0)>}

is :

Neglecting

the

_quadratic

term and

_using

the formula

and

_assuming

an incommensurate CDW for which

we find

The NMR line

_shape

is the Fourier transform of

_{G (t )}

These

_expressions

were first

_derived,

in a

_slighty

different way

_by

_{Kogoj et}

al.

[31].

It is natural that in

_(5.2.8)

the correlation function

_{G (t )}

is

_periodic

in time since we have

considered _up to now a deterministic

_periodic

local

_hyperfine

field modulation. The NMR

spectrum

(5.2.9),

is

_composed

of a central line

(n

=

0)

and sidebands at v = _±

nvd. The central line is

_analogous

to the usual motional

_narrowing

_(except

that it has zero

_width).

The sidebands are

_analogous

to those observed in

_High

Resolution NMR in solids where the

sample

is

_spining

at a

_{magic angle}

to _average out the

_dipolar

fields. Just as the

_position

of the sidebands is a direct measure of the

_{angular velocity}

of the rotor in

_{Magic Angle Spinning,}

the

position

of the sidebands in

_G(v)

for a

_{uniformly sliding}

CDW measures the CDW

_phase

velocity

Vd

= v d 2 ’TT .

The

_amplitude

of the sidebands decreases

_rapidly

with

_increasing

q

vd since

J n (x)

oc x’ for small x

(Fig. 8a),

for v d > _v

_nearly

all the

intensity

is in the central

line.

The effect of the

_quadratic

term in

_(5.2.6)

on the

_lineshape

is found

_by

a

_{straightforward}

calculation :

The exact

_expression

for

_An

is

_given

elsewhere

_{[7, 28].}

For

_Vd/V1>

2,

90 % of the

_intensity

is included in the central line and the

_only

effect of the

_quadratic

term is a shift

_by

v2/2

to

_higher

_frequencies.

In the actual

_crystals

the current is not

_homogeneous

_{in space and instead of} a

_single

drift

velocity

we have to consider a distribution

_P(vd).

We have in mind a

_crystal

with a

_large

Fig.

8. 2013

(a)

Theoretical NMR

_lineshape

in présence of a CDW

sliding

with a uniform

velocity

V =

Vd À.

The local

hyperfine

shift is

v(R)

= _v, cos

_(qR

+ 2 _{7rVd t} +

_{01)’ (b)}

Variation of the NMR

lineshape

as a function of the width of a Gaussian

velocity

distribution P

_(Pd)

=

exxpp --

f

("d -

_{Vd )2}}

at

lineshape

as a function ofthe width ofa Gaussian

velocity

distribution P

(vd)

=

exp -

( _ , )2

at

a fixed

_{value vd}

= _V1.

vd. The form of the central line n = 0 is _not affected

by

_a

velocity

distribution since it is

independent

of Vd in contrast to sidebands which are broadened

_{(Fig. 8b)}

and we find :

The

_lineshape

of both the central and sideband

components

are broadened

_by

the

_dipolar

interaction between

_neighboring

nuclei,

characterized

_by

the

_spin-spin

relaxation rate

Tî 1.

Perturbation of delocalized

electrons,

strains around defects and

_mosaicity

are also of

importance,

the linewidth above the Peierls transition is

_{inhomogeneous}

and much

_larger

than

_expected

from the

_dipolar

interaction alone. The

_dipolar

contribution to the linewidth is somewhat decreased

_by

a static

CDW,

neighbouring

Rb nuclei at sites i

_{and j}

become

the CDW leads to an effective

_hyperfine

field which is the same for all nuclei and causes the

flip-flop

term to _{reappear. The value of}

_T2

is then

_expected

to be similar to that measured

_just

above the Peierls

transition,

a situation where

_{inhomogeneous broadening}

due to the

screening

of

_{charged impurities}

have

_{already developed.}

5.3 SPIN-ECHO IN PRESENCE OF A SLIDING CDW. -

_Spin-echo

_{is the traditional way} to measure the

_{homogeneous spin-spin}

correlation time

_T2.

It enables the distinction between

homogeneous

and

_{inhomogeneous}

contributions to the NMR linewidth. The

_sliding

of the CDW

_{dramatically changes}

the echo and this offers a method to measure in a direct way the

drift

_velocity

_Vd.

In the static case the form of the echo centered at 2 T

(where

T is the

_delay

between the

zr /2

and ir r.f.

pulses)

is

_given

in the

_plane

wave

_{approximation}

(neglecting v2)

_{by :}

and the echo

_amplitude

has the usual

_exponential

form. The onset of the CDW motion modifies the

_amplitude

_{[7, 28] :}

The echo

_amplitude

oscillates as a function of T with a

_period

2

_{Ir/ Y d.}

These oscillations are more

_pronounced

for low drift

_velocities,

if _Vd 1.6 _v

_{1 the amplitude}

becomes even

_negative

in some intervals of T. The effect is somewhat smeared

_by

a distribution

_{of Vd :}

but as we show in section 6.2 the effect remains well observable.

For

_large

values of t

_(i.e.

7TVd T

large),

(5.3.3)

leads to a constant value where

_aO(vd)

is the

_{zero-frequency}

Fourier

_component

of the

_periodic

function This

constant

is of course

_{multiplied by}

e - t/T2

where

T2

is the

_spin-spin

relaxation time which includes the

_{spin-spin dipolar}

interaction and the effect of the stochastic fluctuations of the CDW

_velocity.

Further information on the CDW motion is

_gained

from an

_analysis

of the

_shape

of the echo for t > 2 T

_{[7, 28] :}

where

As we see the Fourier transform of

_{Ge (2}

T +

_{t )}

is similar to

_(5.2.9)

in the sense that it is

components

differ from those obtained from the Fourier transform of the free induction

_decay

and

_depend

on T. Under

_special

conditions the

_intensity

of the central line may even be zero,

e.g. if

v1/Vd

= 0.8 and vd T =

1/2 ;

the

intensity

of the sidebands is increased

by

40 %.

5.4 STOCHASTIC FLUCTUATIONS OF THE _{VELOCITY ;} SPIN-SPIN RELAXATION TIME.

_{- Up}

to

now we have considered a constant or a

_periodically

modulated CDW

_velocity.

We have seen

that this leads to a

_periodic

correlation function for the transverse

_{magnetization}

G (t )

and in this case the

_damping

is due to the nuclear

_magnetic

_{dipole-dipole}

interaction

characterized

_by

the relaxation time

_Tx.

_{In the presence of} a static CDW we found

T2d

= 4.6 _ms in

Rbo.3Mo03 ;

it is

slightly

shorter if the CDW is

sliding

with a fast constant

velocity

as mentioned in section 5.2.

Random fluctuations of the CDW

_phase

_may

_drastically

increase the

_damping

of

G(t)

and have a much more

_pronounced

effect than the

slight change

of the

_dipolar

interaction due to the CDW motion. In

_general

we would like to describe the effect of a time

and

_{position dependent phase}

cp

(R, t )

with a

_periodic

and a non

_periodic

_component.

We first

neglect

the

_periodic

variation of cp and assume that it is varied in time

_by

a _{Markovian process.}

For

_example,

we consider a nucleus in a

_Fukuyama

Lee Rice domain

which,

due to a random

fluctuation _{of the average}

_{phase, experiences}

a

_hyperfine

shift

v (t1)’ v (t2 ),

...

v (ti) at

times

t1, t2,

...,

t; ... ,

with

v

(ti) [

: V1

and

a

jumping

rate of

7-C 1

to

jump

from one

frequency

to

another. These conditions

_correspond

to the usual stochastic process of motional

narrowing

of

a

_hyperfine

structure

_[32].

For

_{T; 1 2 ’TTV1}

1 the

hyperfine broadening

of the NMR line

remains intact but the

_spin-spin

relaxation rate

_T2

increases

_{proportionally}

to _Te. For

7»c

2 ’TT V 1 the

hyperfine

structure of the

spectrum

is

_destroyed

and

_T2

reaches its maximum

of

₍₂

_{’TT v 1)-}

which would be a _{very short value of the order of 30} us in our

_particular

case.

For

_{7- c1}

»

2 ’TT V 1 the

spectrum

consists of a

single

line centered at the first moment of the

hyperfine

distribution and

_T2

becomes

_proportional

to

₍₂

’TT v 1)Z Tc.

Clearly

the above

_description

is not

_adequate

for

_describing

the

_experimental

NMR

lineshape

under current since the motional

_narrowing

due to the

_{phase change}

with constant rate is

_neglected.

Indeed the observed

_T2

does not follow the above

_description.

In the

_following

we outline the behaviour when both the random and constant

_phase

variations are taken into account. Let us define the random function :

and the

_periodic

function :

The correlation function of the transverse

_{magnetization}

is

_{given by}

where the bracket denotes a statistical _average over various

_physical

realizations of the

random function

_{cp (R, t ).}

A

_general

calculation of

_{G (t )}

is

_complex,

we confine the discussion to some

_limiting

cases.

fluctuations _{is small and that its variation may be}

_decoupled

from the

_spatial

_dependence,

i.e.

Supposing

for

all t,

the function

may be

approximated by

2 which overestimates the effect of fluctuations.

Assuming

a Gaussian

_stationary

_process, we find for times t > _Te the correlation time of the

phase,

the usual result

_{[32] :}

while for

We

_emphasize

that the above results are obtained

_by

_employing

severe

approximations

on the motion and in

_particular

we assumed that the drift

_velocity

is fast. The calculation is limited to

electric fields E >

_ET

where fluctuations due to the

_pinning

are small

_compared

to the

constant

_winding

of the

_phase.

An

_{approximation}

which is more

_adequate

to fields close to

threshold is the

_{following :}

We do not consider the

_amplitude

of the fluctuations to be small. Thus we assume the

motion to be coherent for short times

_{(cp (t) = 2 ’TT V d t)}

but with a

_probability

_{T per}

unit time

that an event

_changes

the motion

_drastically.

We assume

8W(R, t )

_changes

so much that

nuclei

at the

_position

where the event

_happens

contribute no more to the echo. Thus the

number of nuclei refocused at time 2 T

_(for

a

_pulse

sequence 2 -

T -

ir)

is

N (2 T ) = N (0)

e- 2 TT

and the

_amplitude

of the echo is

More

_generally,

in

_expression

_{(5.2.11) Gs(v)}

has to be

_{replaced by}

_{GD(v), a}

function

_taking

into account

_broadening

from the

_phase

fluctuations as well as the static

_{imperfections}

and the

dipolar

relaxation rate.

Finally

let us recall the case treated

_{by Kogoj et}

al.

_[31]

where the CDW is

_depinned

thermally

and Vd

= 0. Here

also,

_one

distinguishes

small and

large amplitude

fluctuations.

For

1) cp

(R, t

₂

the static

_lineshape

is

_observed,

while for

_{1) cp}

_{(R, t )}

_{> 2}

and for short t the motional

_narrowing

is

_{complete. Approximating}

_{ôp(/P, t )}

_{by :}

where

_fo

_{is the average distance between}

_pinning

_{centers, wo} is of the order of the lowest

phason

frequency

and B is the root mean _{square thermal fluctuation}

_amplitude,

one finds a

reduction of the

_splitting

of the

_{edge divergences}

of the static

_spectrum

as B increases from

motion where sidebands appear, the

spectrum

in this

_picture

is

_strictly

confined to the

frequency

interval

[ ô v

[

v 1.

6. NMR results.

In a

_previous

set of

_experiments

_{[4] (on}

_{sample #}

₁₎

we studied the variation of the NMR

lineshape

in the

_temperature

_{range of} 40-60 K at a fixed

_applied

field E = 15

ET.

The _new results

_(on

_{sample #-}

₂₎

are in

_agreement

with the

_previous

ones.

_Figure

9

_displays

the

variation of the NMR

_lineshape

with

_temperature

of the

_{previous crystal.}

At 40

_K,

_although

the current

_voltage

characteristics and the noise

_spectrum

show that the CDW is

_depinned,

the CDW

_velocity

is too small to

_change

the

_lineshape.

As

_ICDW

increases with

_increasing

the

temperature

the

_edges

of the

spectrum

broaden and diminish and a

_peak

_emerges at the

center

_frequency

of the static

spectrum. On further

increase of

_jCDW

_(by

_increasing

_7J

the central

_peak

becomes the most

_prominent

_part

of the

_spectrum.

At the intermediate current

of

_ICDW

= 0.405

A/cm2 additional

_steps

arise at the sides of the

_spectrum

outside the range of

the static line.

In

_presenting

the new NMR results on the

_sample

characterised in sections 3 and 4 we focus

on three

_{aspects :}

i)

the variation with

_temperature

and field of the fraction

_f,

for which the CDW remains

static ;

ii)

the variation with

_temperature

and field of the

_spin-spin

relaxation time due to the stochastic fluctuation of the CDW

_{phase ;}

iii)

a more detailed account is

_given

of the sidebands in the free

_precession

_spectrum.

Their

origin

is confirmed

_by

a

_{spin-echo experiment}

under the same

applied

current.

6.1 STATIC FRACTION OF THE CDW AND THE SPIN-SPIN RELAXATION.

_{- Figure}

10 shows

fS,

the fraction of the

_crystal

in which the CDW is

_pinned,

at various electric fields and

temperatures.

f

is

obtained

_by

a fit of the free

_{precession lineshapes}

G(v)

to the

_expression

where

_GS

is deduced from a fit of the

_lineshape

with no

_applied

current to the theoretical

expression

and

_{GD(V )}

is a Gaussian as shown in the insert.

fs

_depends

on both field

(normalized

to

_ET)

and

_temperature.

At 64 K a clear increase of the static fraction is observed as the field is decreased towards

_ET.

On the other hand for

_E/ET

between 2.5 to 3 the

tendency

of a decrease of the static fraction with an increase of

_temperature

above 64 K is well

observed.

The

_spin-spin

relaxation

_time,

_T2,

data measured at various

temperatures

and fields are

plotted

in

_figure

11. At

_high

electric fields

_{(E 1 ET}

>

3 )

the

decay

is

exponential

while at lower

fields

_(E/ET

₃₎

it is

_composed

of two

_{exponentials.}

An

_example

is shown in

_figure

12. The

fast

_decay

_{TZF corresponds}

to nuclei in domains where the CDW is

_sliding

_(or

at least

fluctuating)

while the slow

_{decay TZL}

_corresponds

to domains with a static CDW. In

_figure

11

T 2F

is shown.

_{T 2L}

is

_equal

to 4.2 ms at all

_temperatures

i.e. it

_equals

the relaxation rate in

absence _{of any} current

_applied

to the

_sample.

Fig.

9. - Variation of the NMR

_lineshape

and

_{computer-fitted}

curves _as functions of the CDW current

7cDw

for

sample #

1. The field is fixed at E = 1.5 V/cm

(-

15

ET )

and the _temperature is varied _to

change jcDw

for

_{sample #}

1. For the computer simulated spectra the relative CDW

velocity

distribution is determined from the

_experimental

noise _spectrum at 59.2 K. For each simulated _spectrum

Fig. 10.

Fig.

11.

.

_Fig.

10. Variation of the- static fraction

_f,

S

(in

which the CDW is

pinned)

as a function of

E/ET

at various _{temperatures.}

Fig.

11. -Variation of the

_{dynamic spin spin}

lattice relaxation time

_T2F

versus _temperature. The

number on the

_right

of each

_point

is the ratio

_E/ET.

Fig.

12. - Variation of the

_amplitude

of the

_spin-echo

as a function of the

_delay

time T at T = 73 K and V = 110 mV

(.- 3 VT).

The

longest

time constant for the

decay corresponds

to nuclei in

regions

of the

_crystal

were the CDW is

_pinned,

whereas the short one

_corresponds

to nuclei in

_regions