Dimension effects on spin resonance in graphite intercalation compounds

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Dimension effects on spin resonance in graphite

intercalation compounds

M. Saint Jean, C. Rigaux, J. Blinowski

To cite this version:

Dimension effects

on

_spin

resonance

in

_graphite

intercalation

compounds

M. Saint

Jean,

C.

_Rigaux

and J. Blinowski

_(*)

Groupe

de

_Physique

des _Solides, Ecole Normale

Supérieure,

24 rue Lhomond, 75231 Paris Cedex

05, France

(*)

Institute of Theoretical

_Physics,

Warsaw

_University,

U1. _{Hoza 69,} 00-681 _Warsaw, Poland

(Reçu

le 12 octobre 1989, révisé le 19 janvier et le 1 S

_février

1990,

accepté

le 19

_{février 1990)}

Résumé. 2014 Pour

interpréter

les spectres de résonance de

_{spin électronique}

dans les

_composés

intercalaires du

_{graphite (CIG),}

nous avons étendu la théorie de

_Dyson-Kaplan

au cas des conducteurs

_anisotropes

de dimensions finies. Pour les

_composés

_accepteurs, ce modèle

_prédit

que ce sont

_{principalement}

les faces

_parallèles

à l’axe c

qui

absorbent l’onde

hyperfréquence

incidente,

et _que les _spectres ESR

_enregistrés

_peuvent être

_{interprétés}

dans le cadre du modèle de

Dyson correspondant.

Sur la base de ce modèle nous _proposons une

_{procédure expérimentale}

permettant de déterminer les

_{caractéristiques dynamiques}

des _porteurs dans les

_composés

accepteurs : la conductivité

_électrique

suivant l’axe

perpendiculaire

aux

_{plans graphitiques,}

03C3c, la constante de diffusion

_{planaire Da}

et le _temps de relaxation

_T2.

Pour tester la validité du modèle

_théorique,

nous avons étudié

_{systématiquement}

les effets des dimensions réduites de

l’échantillon

_(dimension

_{géométrique}

sur

_épaisseur

de

_peau)

sur la forme de raie. Plusieurs échantillons de

_C8AsF5

de différentes tailles ont été étudiées entre 10 et 300 K. Pour modifier

indépendamment l’épaisseur

de peau, nos mesures ont été effectuées à v = 4 _et 9 GHz. Les

variations

_importantes

de la forme de raie en fonction des dimensions de l’échantillon observées aux deux

_fréquences

et leur

_{complète interprétation}

dans le cadre du modèle

_théorique

confirment sa validité et celle de la

_{procédure proposée.}

Abstract. 2014 To

interpret

the

_experimental

_spectra of Conduction Electron

_Spin

Resonance

(CESR)

in

_Graphite

Intercalation

Compounds (GIC),

we have extended the

Dyson-Kaplan

theory

to the case of

_anisotropic

conductors of finite dimensions. For _acceptor

_compounds,

this model

predicts

that the CESR _spectra can be

_interpreted

within the framework of the

_Dyson

theory considering

the microwave

_{absorption through}

the

_sample

faces

_parallel

to the c-axis.

Using

this _model, we propose an

_{experimental procedure}

to determine the

_dynamic

_parameters of carriers from CESR spectra in acceptor

compounds.

To check the

validity

of this theoretical

model, we have

_{systematically}

_investigated

the effects of the reduced

_sample

dimensions

(geometrical

dimension over skin

depth)

on the CESR

_{lineshape. Many samples}

of

_C8AsF5

with different sizes have been studied in the temperature range 10-300 K. To

modify

the skin

depth

independently,

CESR measurements have been

performed

at different

_{frequencies, 03BD}

= 4 and

9 GHz. The pronounced effects of

_sample

dimensions observed at both

frequencies

and their

complete interpretation

within the framework of the theoretical model confirm its

_validity

and the

applicability

of the

_{proposed procedure.}

Classification

Physics

Abstracts 73.60 - 76.30P - 72.15

1. Introduction.

Conduction Electron

Spin

Resonance

_(CESR)

is one of the most

_{powerful techniques}

for

studying

the

_{dynamic properties}

of carriers in conductors. The

_analysis

of CESR

_lineshape

and linewidth may

yield

information on

_physical

characteristics of the résonance : the electrical

_conductivity

u, the

spin

relaxation

T2,

the

gyromagnetic

factor as well as the carrier

diffusion constant.

In

_Graphite

Intercalation

_{Compounds (GIC),}

as in metallic

_systems,

the resonant

absorption

is

_{accompanied by}

another

_absorption

process due to the

_eddy

current induced

_by

the

_{hyperfrequency electromagnetic}

field. Both

_absorptions

are

_coupled

and determine the field and current distributions in the medium. The direct

magnetic

resonance

_absorption

with

the small variations of the

strong

conduction losses leads to the characteristic

_asymmetric

Dysonian shape

of the resonance line in the derivative

absorption

spectra.

The

_lineshape

is

characterized

_by

three

_experimental

_{parameters :}

the line

asymmetry

ratio

_{A /B ;}

the

peak-to-peak

linewidth OH and the resonant field

_Hres.

These

parameters

are indicated in

_figure

1 for a theoretical

Dysonian lineshape plotted against

the reduced variable X =

gJL B(H -

HO) lh.

The

_precise

determination of

_physical

characteristics of the resonance from these

experimental

parameters

requires

a _proper

_{lineshape analysis procedure.}

In

previous

papers

[1, 2],

we have shown that the

Dyson

theory [3]

and the

currently

used

Feher-Kip

procedure

[4],

which have been

_developed

to

_interpret

the CESR

spectra

in

_isotropic

metals with an

infinite flat

_plate

_geometry,

are not

_appropriate

for the case of GIC. To obtain the correct

lineshape analysis

in the case of

_anisotropic

conductors of finite

dimensions,

we have

extended the

_{Dyson-Kaplan theory by taking}

into account the

anisotropy

of

_conductivity

and

Fig.

1. - ESR line

shape

observed in the derivative spectra

(with

non uniform microwave field inside

diffusion,

and the finite

_sample

dimensions.

_Using

this

model,

we _propose an

_experimental

procedure

to determine the c-axis

_conductivity

of

_acceptor

_{compounds [5, 6].}

In _{this paper,} we

_present

an extensive

_study

of CESR

experiments performed

on first

_stage

CgAsFs

to check the

_validity

of the theoretical model. In these

_{highly anisotropic}

conductors,

we have

systematically investigated

the effects of the reduced

sample

dimensions

(geometrical

dimension over skin

_depth)

on the CESR

_lineshape.

Many

samples

of different sizes have

been studied in the

_temperature

_{range 10-300 K} for both

_{configurations}

of the static

_magnetic

field

_(parallel

and

_{perpendicular}

to the

_c-axis).

To

_modify

the skin

_{depth independently,}

CESR measurements have been

_performed

at different

_{frequencies, v}

= 4 and 9 GHz.

The

_pronounced

effects of

_sample

dimensions observed at both

_frequencies

and their

complete interpretation

within the framework of our

specific procedure

confirm the

_validity

of our theoretical model.

_Using

this

_procedure,

the

_dynamic

characteristics of carriers in

C8AsF5

(c-axis conductivity, spin

relaxation

times,

and

_in-plane

diffusion

_constant)

can be

determined. The determination of these

_physical

_parameters

_requires

CESR measurements

on a set of identical

_samples

of dimensions

_comparable

and

_larger

than the skin

_depth.

This paper is

organized

as follows : the main conclusions of the

_lineshape

model in

anisotropic

conductors of finite dimensions are

_presented

in section 2.

_Experimental

details are described in section 3. In section 4 we

_report

the

_experimental

results

_confirming

the

validity

of the theoretical model obtained at v = 4 and 9 GHz. The

procedure

used for the

Fig.

2. -

quantitative analysis

of these CESR

_spectra

and the

_{complementary experiments}

which confirm its

_{applicability}

are

_presented

in section 5.

2. Theoretical model of

_lineshape.

The

_theory

of the resonance

_absorption

P has been

_developed

for

_anisotropic

metallic

_systems

with finite dimensions

_by

Blinowski et al.

_{[2, 3,}

_5].

In this

_model,

the

sample

has a

_rectangular

parallelepipedic

form with

_length

L,

width f

and thickness d with d

_parallel

to the c-axis of

graphite.

The microwave field

_Hr

is

_{perpendicular}

to the c-axis and

_parallel

to the

_length

L of the

_sample.

The

_sample

dimensions

_(L

= 10 mm considered as

infinite, f

= 0.5-2 _mm,

d = 0.1-0.4

mm) being

small

compared

to the

wavelength

of the microwave field

(A

= 3

cm),

the field

_Hr

is

_practically

uniform in the

_vicinity

of the

_sample.

The

_{particular geometries}

of

sample

and field are shown in

figure

2. The derivations

presented

in this section are limited to

Hol/c

because the extension to the case

_Ho

1 c does not introduce _any

_difficulty

and similar final formulas for the absorbed _power are obtained in both cases.

The absorbed _{power per} unit surface is :

where n is a

_unitary

vector

_{perpendicular}

to the surface and S =

1/2 (E

A

H * ),

the

Poynting

vector. In these

_expressions,

E and H are

_respectively

the electric and the

_magnetic

fields

measured on the

_sample

surface. To determine the absorbed power at _resonance, the derivation of S _may be limited to the contribution of the microwave

_magnetic

field

11,..

In the

_particular

_geometry

_presented

in

_figure

2,

the

_magnetic

field

_Hr

is

_parallel

to the

x-axis ;

only the y

and z

_components

of the electric field contribute to _{the energy flux :}

Using

the Maxwell

_equations,

_{it is easy} to show that in an infinite rod

parallel

to the uniform

oscillating magnetic

field,

the power P absorbed per unit of

length

is

simply

related to the total flux of the

magnetic

induction inside the rod

Hx

and

_Mx

are the

_{complex amplitudes}

of the

x-components

of the

_oscillating

_magnetic

field

H e- iwt

and the

_{magnetization}

M e- iwt

in the

_sample.

To determine the field and

_{magnetization}

distributions

inside

the

_{sample, following}

the

procedure proposed by Kaplan [7],

we have solved the Maxwell

_equations

and the

_generalized

Bloch

_{equations simultaneously.}

The Maxwell

_equations,

in which the

_displacement

current and the

_magnetic

field

dependence

of the

_conductivity

tensor have been

_neglected,

are :

where the

_conductivity

tensor 0=’ is

_diagonal,

u xx =

uyy

= _{O»a, U zz} =

U c where U a and Uc are the

_planar

and the c-axis conductivities

_{respectively.}

The

_{magnetization}

M is related to

the

_magnetic

field H

_by

the Bloch

_equations.

In metallic

_systems,

the

_{magnetization}

is no

motion can _{carry the}

_{magnetization}

at distances over which the

magnetic

field

_changes

considerably.

To account for this

_effect,

it is necessary to

_complete

the Bloch

_{equations by}

a

diffusion term. For each

_i-component

of the

_{magnetization Ml ( i = x, y, z),}

_Torrey

has

introduced a

_{corresponding}

current of

magnetization

_JMI

[8].

In

_anisotropic

_systems,

Jm

is

related to

_{Mi by}

the tensor

_{expression :}

_JM.

₌

D

_{grad Ml}

where

D

is the diffusion tensor. In our case, this tensor is

_{diagonal, Dxx}

=

D yy

=

D a, D zz

=

Dc, Da

and

Dc

are the

_planar

and the c-axis diffusion constant

_{respectively.}

The Bloch

_{equation describing}

the evolution of

M

then have to be

_{completed by -}

div

_Jmi»

The modified

_{equations generalized by}

the diffusion terms take the

_following

form :

where

_{M =}

_{Ho (Ho.}

_{M) / HJ}

and

_{M -}

M -

_Mil

are the

_longitudinal

and transverse compo-nents of the

_{magnetization M. Tl}

and

_T2

are the

_longitudinal

and transverse

_{spin-relaxation}

times,

respectively. Mo

= _{X o}

Ho

is the

steady magnetization

induced

by

the static

magnetic

field

_Ho,

X o is the static Pauli

susceptibility.

After

_eliminating

the electric field we obtain the

followmg

set of

_{equations :}

where

Since it is convenient to limit these derivations

_only

to the linear terms in

_Hr,

we may take

Mz

= 0. The Maxwell

equations

then involve _constant

H,,

and

Ey,

in our

_expressions

we have

chosen

_{Hz = Ey = O.}

To solve these

_equations

we use the fact that the value of the static Pauli

susceptibility

in

paramagnetic

materials is small and

_expand

all fields in a power series in X o

neglecting

quadratic

and

_higher

order terms. These

_equations

show that

_{only Hx}

has a

_nonvanishing

zero

term

_HO,;

since

_Hy,

_Mx

and

_My

start with the first order terms

_Hÿ,

Mx

and

_Mÿ

_{respectively.}

The finite dimensions are introduced

_by

the standard

_boundary

conditions

_imposed

to the solutions at the

_sample

surface. The

_tangential

_component

of the

_magnetic

field must be

continuous,

H°

=

Hr,

Hx

= 0 and the normal

component

of the

magnetization gradient

must

vanish

Moreover,

it is convenient to

_expand

the fields in double Fourier series so as to reduce the

Fourier coefficients

(the

choice of sine or cosine series

depends

on the

_boundary

conditions

imposed

to the field

_[9]).

Since we are

_only

interested in the _{absorbed power} at the resonance, it is

justified

to restrict the calculation of P to the linear term in _Xo,

AP,

the zero order term

_being

constant in the

vicinity

of the resonance. In this

formalism,

we obtain the

_{following expression}

for the absorbed _power at the resonance :

where

_H"(0,

₀₎

and

_M,’,’(0,

₀₎

are the zero cosine Fourier coefficients of the first order of

Hx

and

_Mx.

These two coefficients can be found

_by

the

_following

three

step

procédure :

-

First,

the sine Fourier coefficients of

_Hf

is determined

_{by solving}

the

_{corresponding}

Maxwell

_equations

in which we

_put

_Hx

=

Hf,

Hy

=

H0y

= 0 and

M°

=

M§

= 0 since _{we are}

limited to zeroth order.

Fig.

3. -

A/B

for the derivative spectra as a function of the reduced

sample

width for different reduced

sample

thickness

1)

very slow diffusion

along the y

and z axes

_{(T2 ta, T2 «}

tc) ;

2)

very

rapid

diffusion

_{along the y}

and z axes

_{(T2 » Ta, T2 > Tc) ;}

3a)

very

rapid

diffusion

along the y

axis and very slow diffusion

along

the z axis

_{(Ta T2 tc) ;}

- In the second

step,

the cosine Fourier coefficients of

_Ml

are determined from the Bloch

equations

in which we

_replace

_Hx,

_Hy,

_M,,,

_My

_by

Hf

_0,

M1x

and

M1y

respectively.

- In the third

step,

the sine Fourier coefficients of

_Hx

are derived from the Maxwell

equations

in which

_Hx

=

Hx

and

Mx

=

Mx.

This

_{procedure requires}

to _go over to another Fourier

_{representation}

of a

_given

_{field ;}

this

introduces an infinite summation over two _{indices. The final formula for the absorbed power}

then involves infinite summation over six

indices ;

this is unsuitable for

_practical

calculations.

However,

this

_expression

can be

_simplified

in some

_particular

cases

_{corresponding}

to different

diffusion

_regimes.

The extensive

_presentation

of these calculations and the

_simplified

final

expression

of the absorbed power can be found in references

_[2,

3 and

_5].

We recall here the

main conclusions of these derivations.

For each direction

_(y

or

_z),

the diffusion

_regime

can be characterized

_by

three different

times. For

_instance,

the diffusion

_{regime along}

the

_y-axis

is characterized

_by

the diffusion time

across the skin

_{depth ta}

=

6 2/2

Da

where

l) a

=

c

is the skin

depth

for microwave

2 w

-B/Vcrc

penetration through

the

_{planes parallel}

to the

_{c-axis ;}

the diffusion time across the

_sample

Ta

=

f2 /2D,,

and the

spin

relaxation time

T2.

Corresponding

diffusion

times,

with index

c, can be defined to describe the diffusion

_along

the z-axis.

In the

_{approximation}

of slow

_(T2

«

_ta,

_{T2 « tc),}

fast

_{(T2 > Ta,}

_{T2 »}

_rj,

and mixed

(Ta « T2 « t,

and

_{Tc « T2 « ta)}

diffusion

_regimes,

the line

_{asymmetry parameter}

_A/B

for the field _{derivative of the power absorbed dP}

_/dH

was calculated

_analytically

as a function of the

two reduced

_sample

dimensions A =

f /8 a

and u

=

d / 8 c.

These results are

_displayed

in

figure

3.

_They

can be

_compared

to those obtained

_by

Kodera for an infinite

_plate

conductor of

arbitrary

thickness

_d,

_displayed

in

_figure

4. A detailed

_analysis

of the line

_asymmetry

parameter

shows

_that,

for _very

_large

values of

_{IL / À,}

the

_asymmetry

ratio

_{A /B}

obtained for

anisotropic

conductors with finite dimensions coincide with those obtained for an infinite flat

Fig.

4. -

plate of isotropic

conductor : cases 1 and 3b in

_figure

3 with the lowest curve in

_figure

4 ;

cases

2 and 3a with the

_highest

curve. Thus the infinite

_plate

model for

_isotropic

conductors is

applicable

to finite

samples

of

_anisotropic

conductors in all four

_limiting

cases when one of the

conditions »

/A

>

100, À

1 J.L

> 100 is satisfied

[2, 3].

For _IL

/A

> 100 the

Dyson

formula with

Uc and

Da

is

_appropriate.

For À

_{1 J.L}

> 100 one should use _{ua and}

_D,.

It is natural to

_expect

that these conclusions also

_apply

for the intermediate diffusion case.

Owing

to the

_strong

_{conductivity anisotropy}

in

_acceptor

_compounds,

the skin

_depth

8 a

exceeds 8 c

by

several orders of

_magnitude.

From static

_conductivity

measurements, one can

_{estimate 8c ~}

_{0.5 u}

_{and 8a ~}

0. 5 mm at v = 9 GHz at T = 10 K. This

implies

that the

reduces

_{dimension u =}

_{dl 8 c}

is about

103

while À

_{= f / & ,}

is around 1-10 for the

_{typical sample}

dimension

_investigated

in CESR

_experiments.

Under these

conditions,

the theoretical

lineshape

model

_predicts

that the line

asymmetry parameter

is

_{independent of » (for}

» >

100)

but varies

strongly

with A.

3.

_Experiments.

To check the

_validity

of the theoretical

model,

we have

_{systematically}

studied the effects of

the reduced

_sample

dimensions

_(geometrical

dimension over skin

_depth)

on the CESR

lineshape.

For each

_type

of

_graphite

host,

a series of

_{rectangular parallelepipedic samples}

with

different

_geometrical

dimensions : L = 10 _mm, 0.11 mm « d 0.4 mm and

0.5

mm l

2 mm were

_{investigated (Fig.}

_2).

To

_modify

the skin

_depth

_{independently,}

two

different

_frequencies

were

_used,

4 and 9 GHz.

All CESR

_spectra

have been obtained for first

_stage

Arsenic Pentafluoride

_compounds

of stoechiometric formula

_CgAsFs.

The

_compounds

are

_{synthetized following}

the method of

Falardeau et al.

_{[ 11}

_from

different

_graphite

hosts

_(Union

Carbide HOPG

_type

_{ZYA, ZYB,}

ZYC).

After

intercalation,

the

_stage

and the

_homogeneity

of the

_compounds

are controlled

_by

X-Ray

diffraction

_{analysis [12]}

and the

_samples

are

_encapsulated

in

_quartz

tube filled with

helium gas to avoid

_{sample degradation.}

CESR

_experiments

were

_performed

with a standard reflection method

_using

BRUKER

(4 GHz)

and VARIAN E 115

₍₉

_GHz)

_{spectrometers.}

The

_sample

_geometry

and the field

configurations

are shown in

_figure

2. Measurements were made for the static

_magnetic

field

Ho

either

_parallel

or

_{perpendicular}

to the c-axis. In GIC the

_lineshape

is very sensitive to

experimental

measurement conditions

[2].

In order to eliminate the passage and saturation

effects over the whole

temperature

range, measurements were made at a low microwave power of 10

JLW

at both

_frequencies.

The field modulation

_frequencies

were 1.57 kHz and

35 kHz at 4 and 9 GHz

_{respectively.}

The CESR

_experiments

were carried out between 10 and 300 K at 9 GHz and between 90 and 300 K at 4 GHz. The

_sample

_temperature

was controlled with a _{accuracy of}

T = _± 1 K. All the _{measurements were}

performed during

the

heating

process, the initial

cooling

was achieved after one hour.

4.

_Experimental

results.

All CESR

_spectra

obtained in

_CgAsFs

exhibit an

_{asymmetric Dysonian lineshape}

charac-terized

_by

the

_asymmetry

ratio

_{A /B}

and the

_peak-to-peak

linewidth AH.

_Figure

5 shows a

typical experimental

spectrum.

The studies

_performed

on several different series of

_samples

show that the

_{Dysonian lineshape}

is

_extremely

sensitive to the finite

_sample

dimensions and to

the

_{crystallographic}

nature of the

_graphite

host.

_However,

_although

the absolute values of

A/B and

AH can

_change

from one series to

_another,

some

_general

_{features may be} observed

Fig.

5. -

Experimental Dysonian

spectrum observed for a

_sample

of d = 0.4 _{mm ;} ZYA f = 2 _{mm ; at} v = 9 GHz ; T = 200 K,

Ho

₁ c.

A : theoretical _spectrum calculated for À = 6.4 and R = 0.75.

of the

_experimental

results is restricted to these characteristic behaviors. The best

experimental

illustrations of each have been selected from our

_experimental

results and are

given

below. The

_experimental

results chosen do not all

_necessarily

to the same

sample

serie.

Fig.

6. - Line _{asymmetry parameter}

_{A /B}

vs. _temperature measured at v = 9 GHz on

_samples

of

constant width f = 2 _mm and of different thicknesses.

4.1 LINESHAPE INDEPENDENCE OF d. -

To

_study

the influence of the

_sample

thickness d on the line

_shape,

we first recorded the CESR

_spectra

on identical

_samples

of constant width

(f

= 2

mm)

but with different thicknesses d =

0.11,

0.27 and 0.42 _mm. These _measurements

carried out at v = 9 GHz do _not show a

_{significant dependence}

_of A

_/B

or OH

_parameters

on

sample

thickness. These features are illustrated in

_figures

6 and 7

_{respectively.}

A similar invariance of

_{A IB and}

AH was observed at v = 4 GHz.

Fig.

7. -

Temperature dependence

of OH measured on

_samples

of constant width f = 2 mm and of

différents thicknesses.

Open symbols :

Hollc,

black

_{symbols : Ho}

1 c.

4.2 INFLUENCE

OF f

ON THE LINESHAPE. In _contrast, at both

_frequencies,

at _any

_given

temperature

and

_regardless

of the

sample

series,

the

_{lineshape depends drastically}

on the

sample

width f.

a)

Asymmetry

parameter

A/B.

Measurements

_performed

at v = 9 GHz for the ZYB series

(Fig. 8)

show a non monotonic

variation of

_{A IB}

with

l :

for

instance,

at T =

_{100 K,}

_{A /B}

varies from about 3 for

f

= 0.5 mm to 8 for

f

= 1 mm and decreases to about 3.5

for f

= 2 mm. In this

temperature

range, the

asymmetry

ratio

_{corresponding}

to

f=1.5mm

is constant and

_equal

to

7 ± 0.5. Measurements

_performed

at 4 GHz

_{(Fig. 9)}

on the same

_samples

show an

_opposite

dependence

on

_sample

_size,

_{A /B}

_decreasing

from 6 to 3 as the width is reduced from 2 to

1 mm. This

f-dimension

_dependence

was observed for both series

_{investigated.}

Moreover,

the

_temperature

variations of

_{A /B}

_depend

on the

_sample

width

f. At 9 GHz,

A/B

is

_independent

of

_temperature

between 10-200 K. Above 200

_K,

_A/B

varies

_strongly

with

_temperature,

the direction and

_amplitude

of these variations

_depending

on the

_sample

width

f.

For

_instance,

the

asymmetry

ratio

_{A /B}

_{corresponding}

to

_samples

with f

= 0.5 and

Fig.

8. - Influence of the f-dimension effects _on the line

asymmetry ratio

_{A /B}

observed at 9 GHz on

ZYB series.

o £ = 0.5 mm; .. £ = 1 mm; . £ = 2 mm; d = 0.4 mm.

Fig.

9. - Influence of the f-dimension effects

on the line _asymmetry ratio

_{A IB}

observed at 4 GHz on

ZYB series.

whereas it increases for wide

_samples

_(f

= 2

_mm);

above 260

_K,

_{A IB}

decreases for all

samples.

b)

Linewidth : AH.

Whatever the

_sample

dimensions,

for each

_{configuration}

of the external field

_(Hollc

and

Ho

1

_c)

and at each

_frequency,

the linewidth exhibits the same

_temperature

behavior :

âH1

and

_{AH_L}

decrease with

_increasing

_temperature,

_varying

_typically

from a few tens of gauss at low

_temperatures

to 0.5 G at room

_{temperatures.}

An

_example

of this

_temperature

dependence

of the linewidth is

_given

in

_figure

7.

At low

_{temperatures,}

the linewidth is sensitive to the

_sample

width

f.

For

instance,

at

T = 10 K and v = 9

GHz,

the linewidth measured _on ZYB

samples

_presents

a maximum

around f

= 1 _mm

(Fig. 10).

1

Fig.

10. - Influence of

f-dimension

effects on the linewidth OH observed at 9 GHz on ZYB series,

T = _{10 K,}

_HIlc.

Moreover,

the linewidths

_depend

on the static

_magnetic

field orientation :

_{AH_L}

are

systematically larger

than

_{AHII (Fig. 7).}

_Tbis

linewidth

_{anisotropy AH-L}

_{/ J1HI}

is however

independent

of the

_sample

dimensions and have a

_comparable

value for all

_investigated

compounds.

At 4

GHz,

the linewidth exhibits the same three

_qualitative

features.

These observations confirm

_{experimentally}

the theoretical

_{prediction concerning}

the

lineshape

in the

_acceptor

_compounds.

In

_particular,

the invariance of the

_lineshape

parameters

with the

_sample

width

f

demonstrate

_{experimentally}

_{that all the microwave power} is

_practically

absorbed

_by

the

_sample

faces

_{(zx) parallel}

to the c-axis. We can conclude that

the

_commonly

used Feher and

_{Kip analysis}

based on the

_Dyson

formula with _a-a and

Dc

is

_{completely unjustified}

in

_acceptor

_compounds.

5. Procédure for a

_{quantitative interprétation}

of CESR

_spectra.

5.1 DESCRIPTION OF THE PROCEDURE. - The theoretical model

_being

confirmed

_by

the

experimental

studies,

we have

_developed,

within the framework of the

_Dyson

_theory

involving

uc and

Da,

a

_{procedure adapted}

for the

_{quantitative analysis}

of the CESR

_spectra

in

acceptor

compounds.

This

_analysis

leads to the determination of the

_dynamic

_parameters

of

carriers: u c’ T2

and

_{D a.}

The theoretical

_spectrum

is calculated

_numerically

as a function of the reduced field X = _{g¡..t B}

_{T2(H - Ho)/It}

_using

the theoretical

_expression

obtained

_by

Kodera

_[10]

for the

absorption

derivative. The line

_{asymmetry parameter}

_{A /B}

of the

_absorption

derivative

depends

on two

_{parameters À}

and R =

(Tn/T2)1/2

where is the time

required

for

the carriers to diffuse across the skin

_{depth 8 a.}

The theoretical curve

_{A/B( À ,}

_{R )}

is

reported

in

_figure

_11,

for several values of R. For

_sample

widths

_comparable

to the skin

_depth

Fig.

11. - Theoretical

(A 2.5 ),

the

_magnetic

field inside the

_sample

is almost

_uniform,

the

_asymmetry

_parameter

A/B

does not

_depend

on R. For

_samples

with intermediate

_thicknesses,

the distribution of r.f. field and

_consequently

_{A /B}

_depend

in a

_complex

_way on both A and R.

_{A /B (A )}

exhibits a

sharp

maximum which occurs in the

_region

of A = 3-4. For

larger

values of À

(À

_>

6),

the

spins

which leave the skin

_depth

relax before

_reaching

the middle of the

_sample

so that the

microwave field distribution

_depends

on the diffusion. Then

_{A/B( )}

is

_nearly

constant and exhibits a

_regular

increase with the decrease of R.

Moreover,

for a

_{subsequent analysis}

of the

_linewidth,

we have calculated the theoretical

line width Ax for different values of the

_{parameters A,}

R

_{(Fig. 12).}

Fig.

12. - Theoretical

dependence

of Ox vs. A. The values of R are indicated on each curve.

For a

_{quantitative interpretation}

of the

_spectra,

it is necessary to consider

_samples

of different widths

_satisfying

the condition either À 2.5 or À > 2.5.

For narrow

_samples,

À

_2.5,

_{A /B}

is

_independent

of the diffusion. Thus measurements of

A/B

on

_samples

of dimension

_comparable

to the skin

_{depth provide}

a

_good

means of

determining k

and

_consequently

the c-axis

_conductivity

at the

_frequencies

used in the

measurement.

The c-axis

_conductivity

is deduced from the value of À with the use of the formula :

This

_procedure

does not

_permit

the value of R to be determined.

_However,

the

_spin

relaxation time may be evaluated

_{using the}

theoretical linewidth Ax which is

independent

of R for this class of

_samples,

Notice that for thèse narrow

samples,

the diffusion

constant cannot be evaluated.

constant and identical for a

_given

set

_{of samples}

of the same series

_[6].

Thus we can to deduce the values of À for the

_samples

with various widths

f by

using

the

_{proportionality}

between

À and l.

Therefore the

_{parameter R}

can be estimated from the best

_agreement

between the

experimental

and theoretical values

_{of A /B}

measured on

_{large samples.}

The

spin

relaxation

time

_T2

and the diffusion

_Da

can be determined. The

_spin

relaxation time

T2 is

estimated

_by

comparing

the

_experimental

linewidth AH to the theoretical linewidth Ax calculated for the

parameters

À and R obtained from the

_{lineshape analysis,}

_using

the relation :

where _{IL B} is the Bohr

_magneton.

Notice that it is essential to introduce Ax

_(1.2

Ax « 2.1 for

our

_experiments)

in order to discuss the relaxation process

correctly.

Indeed,

some

_apparent

effects can be observed on OH which

_disappear

when the

_study

is based on

_T2.

After determination of R and

_T2,

the in

_plane

diffusion constant

_{Da is}

extracted

by

using

the

relation :

5.2 EXPERIMENTAL VALIDATION OF THE PROCEDURE. -

_Using

the values of A and R obtained from the

_{quantitative analysis}

of the

experimental

spectra,

the

_absorption

Fig.

13. - Values of À determined from theoretical curve.

A : f

= 0.6

mm ; 0 : f = 0.8 _mm.

derivative

_{dP /dH}

was calculated from the theoretical

_expression

derived

_by

Kodera

_[10].

The

comparison

between an

_experimental

and the

corresponding

theoretical

_spectra

is

_presented

in

_figure

5. The

_good

_agreement

between

_experimental

and theoretical

_lineshapes

indicates that our model

_correctly

describes the CESR

lineshapes

in

_acceptor

_compounds.

In order to select in

_experiment

the desired

_region

of A 2.5 and to check the

_validity

of the

_procedure

of

_analysis

we have

performed

measurements of

_{A /B}

on a series of narrow

samples

with different

f-dimensions.

The values of À obtained from the

_experimental

lineshape

asymmetry

ratio

_using

the theoretical

_dependence

_{A/B (l )}

are

_reported

in

figure

13. The reduced dimensions A are, with excellent accuracy,

proportional

to the

_sample

size

f.

This

_provides

a

convincing

proof

of the invariance of _{u c} for a

_given

set of

_samples

of the

same series.

The

_validity

of the model is further

_{supported by}

the self

_consistency

of the results obtained. In

_particular,

this model

_provides

a

_{quantitative interpretation}

of the influence of

the

_sample

width

e

on the

_asymmetry

ratio

_{A IB}

and the variations

of A /B

with

temperature.

The non-monotonic variations of

_{A /B}

with f

reflect the existence of a maximum in the curve

AI B(À).

For

_{AsFs-graphite compounds,}

at v = 9

_GHz,

the

_{corresponding}

values of A for

l =

0.5 and 2 mm fall on different sides of the

maximum ; for f

= 1 _mm,

A occurs in the

_region

of the maximum. At v = 4

GHz,

the k values

corresponding

to

e

= 1 and 2 mm both fall on the

ascending

_part

of the curve

_{A/B (,k ).}

Thus for

_instance,

the

temperature

dependence

of the line

_{shape reported}

in

_figures

7 and 8 for

_samples

with different widths is

_{consistently explained :}

as the

_temperature

is

raised,

A decreases

implying

an increase

_{of A /B}

for f

= 2 _mm and a decrease

of A /B

for

samples

off

= 0.5 and 1 mm at

9 GHz. At 4

_GHz,

_{A IB}

decreases with

T,

in

_good

_agreement

with

_experimental

features.

Fig.

14. - C-axis

conductivity

obtained from

_{A IB}

measurements on narrow

_samples.

5.3 EXAMPLES OF QUANTITATIVE RESULTS. - This

_{general procedure permit}

a

_complete

interpretation

of CESR

_spectra

measured on

_acceptor

_{graphite compounds}

and a

_precise

determination of the different

_physical

characteristics of the

_compounds

studied :

_T2,

Da.

In this

_section,

we

_present

some of the main results obtained for first

_stage

AsFs-graphite

compounds.

In

_particular,

the CESR measurements

_provide

direct evidence of the

_frequency

depen-dence of the c-axis

_conductivity

which was never observed

previously [6]. Figure

14 presents

this

_{frequency dependence}

of _Uc determined on narrow

samples

ZYB

f

= 0.5 mm at 9 GHz

and ZYB

f

= 1 mm at 4

_GHz,

the c-axis d.c.

_conductivity

is measured on a reference

_sample

synthetized

in the same reaction as the

_{sample prepared}

for CESR

_experiments.

This

observed

_dependence

of the c-axis

_conductivity

indicates the non standard character of the conduction mecanism

_along

the c-axis. Most

_probably,

there is a contribution to the a.c.

conductivity

from carriers whose

_{displacements}

in an electric field are limited

_[13]

or which

obey

a

_hopping

_process

_[14,

_15].

The

_spin

relaxation time determined from CESR measurements reveals two

_general

features,

whatever the

_sample

dimensions or the

frequency :

_T2

increases with the

temperature

and is

_{systematically larger}

for

_Hollc

than for

_Ho

1 c.

Fig.

15.

_{Temperature dependence}

of the

_spin

relaxation time obtained from the

_lineshape

and line width

_{analysis ( v}

= 4 GHz,

Hollc).

An

_example

of such behaviors is

_presented

in

_figure

15 which

_gives

the variations of

T2( T)

for ZYB

_samples

at 4 GHz :

_T2

is

_equal

to 3 x

10- 9

s at 90 K and increases

by

two

orders of

_magnitude

as the

_temperature

rises from 90 K to 300 K. The

_{large uncertainty}

observed on

_T2

at room

_temperature

is due to the

_imprecise

determination of the

experimental

linewidth AH and is not to be attributed to the

_{analysis procedure.}

Moreover,

the

_comparison

of the

_spin

relaxation time determined for

_samples

with

different widths

f

indicates

that,

whatever the resonance

_{frequency, T2}

seems to decrease with the

_sample

width

f

_{(Fig. 15). Although}

more

_{complete experiments}

_{will be necessary} to

confirm this

_sample

dimension

_dependence,

we can

_already

note that the non monotonic

variation

_{of J1H ( f)}

shown in

_figure

10 is not observed in

T2.

This confirms that the linewidth is not the

_right

_parameter

to _{discuss the relaxation processes and that} a determination of

T2 taking

into account the corrective term introduced

_by

the theoretical linewidth X is essential to

_interpret

_{the relaxation processes}

_correctly.

These variations of the

_spin

relaxation time with

_temperature

can be

_{explained qualitatively}

by inhomogeneous broadening

mecanisms

_{[2, 5].}

In a

_previous

_paper, we have

_suggested

that

this

narrowing

is due to

_exchange

interactions between delocalized carriers and other

magnetic

moments in GIC. Several recent

_experimental

results confirm the existence of these localized moments, but their nature and their localization are not

_clearly

established. The variation of the linewidth with the atomic number of the intercalant

species

[16, 17]

suggests

that these localized moments are associated with the intercalant

_layers

whereas recent

_spin

echo results

[18]

and the

_anisotropy

of the linewidth with the orientation of

_{Ho [15]}

suggest

that

_they

are localized in the

_{graphite layers.}

A

_complete

discussion of the

spin

relaxation time

is

_presented

in references

_[2,

_5].

The

_planar

diffusion constant

_Da

obtained from À and R at 9 GHz

presents

a

_systematic

decrease as the

_temperature

increases. For

instance,

_{Da corresponding}

to the

_sample

ZYB

f = 2 mm,

varies from

6 x 104 cm2 s-1

1

at 10 K to

2 x 103 cm2 s-1

1 at 300 K. This

_high

temperature

value seems to be identical for all

_samples,

whatever their

f-dimensions.

In

contrast at low

_temperature,

_Da

increases when the

_sample

width decreases. For

instance,

at

10

K,

the diffusion constant obtained for the

_sample

ZYB

f

= 1 mm is

_equal

to

2.5 x

105

cm2

s-1 whereas

that

corresponding

to ZYB

f

= 2 mm is around 0.6 x

105

cm2

s-1.

Although

very

surprising,

this effect has been observed on different

_sample

series and is not

an artefact of the

_{analysis procedure.}

More

_experimental

and theoretical effort will be

necessary to confirm and

_interpret

this

effect, however,

we can

already

conclude that these

compounds

are not usual two dimensional metallic

_systems.

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BLINOWSKI J., KACMAN P., RIGAUX C., SAINT JEAN M.,

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