Debye-Waller like broadening of the susceptibility peak in CeRu2Si2

(1)

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Debye-Waller like broadening of the susceptibility peak in CeRu2Si2

L. Puech

To cite this version:

L. Puech. Debye-Waller like broadening of the susceptibility peak in CeRu2Si2. Journal de Physique

I, EDP Sciences, 1991, 1 (7), pp.979-984. �10.1051/jp1:1991176�. �jpa-00246389�

(2)

Classification

PhysicsAbsmacis

75.30M-63.20K-63.20L

Show Communication

Debye.Wailer like broadening of the susceptibility peak in

CeRu2S12

L. Puech

Centre de Recherches sur les ltBs Basses

lbm~dratures,

BP 166J~ 38042 Grenoble Cedex, France

(Received 5Apfil

199«

accepted

13

Mqy 1991)

R6sum6. Los

phonons thermiques

sont

peut

^dtre

responsables

de

l'dlargissement

de la transition

pseudo-mdtamagndtique,

observ6

lorsque

la temperature augmente, ^dans ^le

compass

fermions lourds

CeRu2S12.

Nous montrons que la

largeur

^de ^la ^transition ^observde ^rdcemment ^{k trds} basse

temp6rature

_sur la

magn6tostriction,

est en fait _en

T~;

_avec _un

pr6facteur compatible

avec les fluctuations de densit6 dues aux

phonons thermiques.

^Nous montrons que ^le

couplage magnon-phonon

est trds fort la

transition,

si bien que les

phonons

de

point

z6ro peuvent y

jouer

un role

important.

Abstract. Thermal

phonons

may be

responsible

for the observed

broadening,

with

increasing T,

of the

pseudolmetama gnetic

transition in the

heavy

fermion

compound CeRu2S12.

The width of the

peak

in the

magnetcstriction coefficient, recently

measured at low temperature, is shown _to be

proportional

to

T~;

the

prefactor being compatible

with the

density

fluctuations due to thermal

phonons.

The

phonon-magnon coupling

is shown to be

extremely large

at the transition, sc that _zero

point phonons

may

play

there _a central role.

1. Introduction.

Since the

pionneering

work of references

[1-3],

many

experiments

have been devoted to the

study

of

thermodynamic, microscopic

and

transport properties

of the

heavy

^fermion

compound CeRu2S12.

At low

temperature

^and zero

applied magnetic field,

the

system

enters the so called

heavy

fermion

regime, according

to the

large

linear term 7 of the

specific

heat

(7

=0.35

J/mol K~) [1,4].

^No

long

_range order has been detected in the pure

compound,

but short range anti-

ferromagnetic (A~F)

correlations exist between Ce ions

[5].

^When a

magnetic

field is

applied,

the

susceptibility

_x exhibit _a

large

increase around

Bc,

this is called the

pseudo metamagnetic

transition

[2]. (Bc

=7.8 T at the lowest

temperatures). Correspondingly

^the ^A~F fluctuations

rapidly disappear

when

crossing Bc [5].

Much work has been done to

clarify

what

happens

at

Bc.

It is well established that 7 increases

by

at least

609b,

and that _x

apparently gets extremely large

at

Bc, provided

T is low

enough [6],

suggesting

the

possibility

of _a transition at zero

temperature,

^for ^which _x ^would

diverge

at T ₌ 0.

(3)

980 JOURNAL DE PHYSIQUE I N°7

Theoretical

approaches generally require

the

vicinity

of some

unstability

due to volume

changes

_[7j

or

towards'magnetically

ordered

phases [8,9],

^for

predicting

the

pseudo-metamagnetism.

In these

cases there is _no transition and 7 and _X should

extrapolate

to finite

values,

but in _a

temperature

range ^that ^has ^still ^to ^be

predicted.

Among

the

experiments performed

_on the

compound,

^those

involving

ma

gneto-volumic

effects have been very

spectacular

and

useful;

we summarize them here.

Apparently,

the main effect of the

pressure

^P on the

magnetic properties

of thin

compound

h _a

strong

renormalisation of the energy scales

[3,10,11], especially

of the

magnetic

^field B necessary ^to obtain _a

given magnetization

M

usually expressed

in Bohr _ma

gneton (pB)

_per Ce ion:

m =

I

= m

l~olp~) ⁽ⁱ⁾

In thb

expression,

the field scale B"

(P), changes

with P

by large

amounts, caracterized

by

Q ₌ fij~O

~ ⁼ ¹⁷⁰

MBar~~

_B"

may ^be ^chosen ^as

Bc(P)

_or I

/xo(P),

the inverse of the

susceptibility

B" P

at low field.

Equivalently,

this

unusually _strong

_pressure _or volume

dependence

^of ^B"

(P)

_may

be

expressed using

a

(dimensionless)

Griineisen

parameter

^r =

j~()

=

-~,

_where

_Ito

_is

~

the

compressibility

at zero

magnetic

field. At the

pseudo-metamagnetic

transition around

Bc,

the

susceptibility

_[2] _x, the

compressibility [13] It,

and the

magnetostriction

coefficient

ill] I[

$,

=

exhibit

strong increase,

in

quantitative agreement

with the

Scaling

Ansatz

equation (I) [11,12]. Up

to now there is _no evidence for saturation of the

height

of these maxima at

Bc

when

lowering T; especially

for the recent measurements [6] ^of

I[

down to 0.12 K These maxima are known to broaden

rapidly

when T

increases,

while their

height

decreases.

Performing

thermal

expansion experiments,

the authors of references

[6, 12]

^could ^determine as a function of

B,

a

caracteristic

temperature TM (B)

at which _av

(T, B)

has _an _extremum

(maximum

for B _<

Bc

and minimum for B _>

Bc).

Well below

TM(B)

the

system

enters a

degenerate

Fermi

liquid

behaviour where _av is linear in

T,

while above

TM

no well defined Fermi

liquid

is observed. This _cross-over

temperature

^is ^shown to have _a

sharp

minimum at

Bc.

It decreases from 9K _at B

=

0,

to less than

0.5K around

B~

and then increases

again.

The

extrapolated

^value of _av

IT

increases

sharply

when B increases to

Bc, suddenly changes sign

for _a field

step

of less than 19b of

Bc,

and then

slowly

increases back to zero. In the narrow range ^where av

IT changes sign,

it is

yet

^not ^known ^wether

av

IT

is _a discontinuous function of

B,

and _a determination of

TM(B)

has to be done.

The aim of this letter is to

give

an

explanation

of the facts

concerning

the T

dependence

^of

the maxima of _x and

I[,

and for the low values of

TM (B

-~

Bc).

It relies _on the fact

that,

since

FB

^h

large, relatively

small

density

fluctuations _can induce

strong

fluctuations in B*

(P).

Since It increases around

Bc,

fluctuations due to thermal

phonons

are

strongly

enhanced at that

field,

and the

temperature

^{for which} ^these

phonons

^become

important

^decreases. The main effect of these fluctuations will be to

spread Bc

in _a range ^{of fields} ^of ^width

proportional

to

FT2ItSH,

_that _can _be

predicted

from first

principles.

A similar

approach

_[6] was used to show that the

broadening

of the

anomally

_upon

doping

^the

compound

with La _on the Ce

sites,

_may be due _to statbtical fluctuation of the La concentration within the

sample.

2. A model for the effect of

photons.

16 investi

gate

^the ^effect ^of ^thermal

phonons

on the electronic

properties,

_we assume an adiabatic- like

response

^of ^the

heavy

electrons to

density changes,

and a

Debye approdmation

for the lon-

gitudinal phonons:

(4)

In _a

region

of the

sample

where the

density change

is

given by

z =

~~,

there will be _a local P

renormafisation of B"

by

_a factor

exp(-Fz),

and the

magnetization

will be

given by m(B exp(+Fz))

where

m(B)

b the

magnetization

for _z ₌ 0 and T ₌ 0.

The

probability density P(z)

of _z b assumed _to be

Gaussian,

_as

given by

^the

Debye theory

for T much less than the

Debye temperature (-~100 K), neglecting

the zero

point

motion of the atoms in the

crystaL

The

magnetization

b then

given by:

m(B, T)

₌

/ dzP(z)m (Be~~,

T ₌

0,

_z ₌

0) (2)

There

P(z)

b _a Gaussian of width

(z2)

₌

^~~

^~

~

~~~(~,

c

being

the

longitudinal

sound

velocity

10 Mc

(hc)

and V ₌ 50

cm~/mol,

M

= 0.4

kg/mol

the molar volume and mass, and

kB

and h the Boltzmann and Planck constants. We conclude that _an

anomaly

at B

=

Bc

will be

spread

on the field axis in _a width: S

=

~j

^~ =

F2 (z2) ar(T~c~5

lvith the values

given

above and _c ₌ I

km/s

this amounts to S

= 1.6 _x

10~~T~.

The above

assumptions

indeed

imply

that thermal

phonons

^have a

wavelength

and _a

period

much

larger

than the characteristic

length

and time involved in the fluctuations of the

spin

_sys-

tem

(wavelength,

correlation

length, frequency

and relaxation

time). Actually

these

assumptions

cannot be

justified

on

experimental grounds;

if

they

are wrong ma gnons ^and

phonons

have to be

treated _as

coupled exdtations,

an

approach

which _was not

attempted

in this kind of

compounds.

We

finally

note that _a rather

big problem

arises when _one looks _at _zero

point phonons.

^A _quan-

titative estimate of their influence is

given by

the

spreading _Szp

due to zero

point

motion. If the entire

spectrum up

^to ^the

Debye

wavector

kD

is taken into acccun~

Szp

^is

given by: _Szp

₌

F(

^~

8

~~

^This ^number ^is very

large: _Szp

₌ 20 for

FB

₌ 200 and _c ₌ I

km/s.

This

large spreading

is

strongly exagerated by

c the fact that the adiabatic

assumption

made above is

certainly

not

tjue

^at

kD.

If _a cut-off wavevector

kc

is

introduced,

the above result has to be

multiplied by ~

D

3. Technical details.

Equation (2) implicitely

relates measurements of the

magnetization

at different

temperatures.

In the framework of the S.A~ _an identical

equation

can be written

replacing

_m

by I[ (which

is

proportional

to B

~j _).

We _assume this

formula,

and _use it for

determining

in _a

systematic

way a

broadening

^of the

peak

in

I[

of reference

[fl.

This h done in the

following

_way: we define the _new variables _y

= Fz and b_: B

=

Boexp(b)

where

Bo

is _a reference

magnetic

field.

Equation

(2)

then becomes _a

genuine

convolution

product:

(5)

where A normalizes the Gaussian _on _y. For

relating

measurements at a

temperature T2

to the results at a reference

temperature Ti,

we use the fourier transform _1)~

(q, Ti)

of

I[ (b, Ti)

We have:

I[ _(b, _T2)

=

dq exp(iqb)

_exp

(-Sq~/2) ii, (q, 7i) (3)

where S

= S

(T2)

^S

(Ti)

Note that neither S

(Ti)

_nor

I[(T

=

0)

have to be known for per-

forming

this

analysis,

but that

they

cannot be found from it. The

quality

of the fit

given by (3)

is shown below to be

quite satisfactory provided

the

proper

^value ^of ^{S is} ^chosen. ^We

performed

this

analysis using

the

experimental

results at

Ti

_" ^0.34

K,

the lowest

temperature

for which _a wide range ^of ^fields was

investigated

in reference

[fl.

We use a 512

point FFT

and choose _a range ^of field

larger

than the

peak

in

I[

at 4.2

Xj

but

sufficiently

small for

having

a reasonable accuracy

on B

Bo

₌ 5 T _< B _< 11

T, yielding

to a relative accuracy ^of 1.5x

10~~

on B. We evaluated

l~,~,

the

right

hand of

equation (3)

and compare

l~~~

^to ^the ^measured

magnetostriction 1[~~~

for

T2 =0.12, 0.2, 0.6, 0.8,

1.3 and 4.2 K and for _a very ^broad range ^of S

=

52 Si

We define

(eas aic)~)

a reduced error A

= ~, ^~ that we

plot

_versus S for each

T2 (Fig. I).

We _can then

meas)

find the value of

Sbest,

that

yields

to the smallest

possible

_error

Abest,

and the _error bar _on

Sbest,

defined as the

region

where A is less than I-I

Abest.

The

quality

of the fits for

T2

_>

Ti

_may be

appreciated

in the inset of

figure

I. In any case

(Abest)~~

is about

0.I, comparable

to the error bar on the measured

I[.

For

T2

<

Ti, equation (3) magnifies

^the

noise,

so that _no well defined minimum

appears

ⁱⁿ

A; thus,

_we choose Sbest as the values for which _a linear

regression of1[~~~

verms

1[,~ yields

a

slope equal

to 1+0.I. In this latter case,

Sbest

is

certainly

dominated

by

S

(Ti)

~ T=1.3

T=~.2K

0.8

I

z T=0.8

~ r4~

fl

~

~j

~

~ ~

IT1 io

io-6

~o-5

io-'

_1o-3

io-2 io-i

s

Fig.

1. The relative error A _on

I[ (see text)

^is

plotted

ver~us the

spread

S for the temperatures as indicated. From this

plot

we find the best fit Sbe~i of the

magnetostriction

and the _error

bar,

_as indicated

by

^the

horizontal dashes. Inset: the fit for T

= 0.8 K.

I[

is

plotted

as fttnction of B. The

experimental

curve can be

distinguished

from the fit,

only

because the noise is

slightly

visible.

(6)

4. Results and discussion.

The results of this

analysis

_are

reported

ⁱⁿ

figure

2 as

Sbesi

₌ S

(T2)

^S

(Ti)

_versus

Tf T).

It is

readily

seen that the

T~

_law _for

S(T)

is

nicely obeyed

^for ^S

ranging

from -1.7x

10~~

_at 0.12 K to 2~5_x

10~~

_at _{4.2 K~}

yielding

to

S(T)

₌

^{10~~ T~} (1+0.5).

^This value

imply

at

Bc

a sound

velocity

_c = I

km/s.

This b smaller than the _one

reported [13],

at 1.3 K but _we

expect

^from ^the

S.A~ in view of the

magnetostriction

results

[fl,

that _a much more

significant softening

exist at low

temperature,

^for

good samples.

These facts

imply

that thermal

phonons play

a central role in the

broadening

of the

metamagnetic

transition in

CeRu2S12,

a conclusion in

strong

contrast to the

generally accepted

_one, that the

spreading

is due to

exeptionnally

low

energy

scale in the

heavy

electron

problem.

We

emphasize

that this mechanism of

spreading

is

significant

because of its

strong dependence

on F: it is

proportional

to

r~/c5,

and if the S.A~ is true, ^the

compressibility

at

Bc

is

essentially proportional

to

F2

_so _that _S

«

F7.

_With _F ₌₂₀₀ _we

_get

_an enhancement of the thermal

phonons

contrlution of about 10~~

compared

to nonnal

problems! Finally,

since the

spreading

due to zero

point phonons

is

potentially

_very

large,

this

approach

rises the

question

of the role

played by

the _zero

point density fluctuations,

in the

magnetic properties

of the

compound.

Namely,

_we ask whether _zero

point

motion of the Ce ions b

responsible

for the lack of order at low

field,

^for the width of the transition at

Bc,

and for the

suppression

of _an eventual first order

transition at

Bc

driven

by

volume

changes [7,12].

10"~T~

041 0.I 10 100

T~ K'l

Fig.

2. The

spread (Asbest(

is

plotted

venus

(T~ Tf(

ⁱⁿ

log-log

scales. The

T~

_law _is _well _verified. _The

straight

line

corresponds

to

10~~T'.

_See _text _and

_figure

_for _the

error bars.

Acknowledgements.

J.

Flouquet,

P Haen and A~ Lacerda _are

gratefully acknowledged

for

having

involved the author in this field of research.

(7)

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