Scaling in heavy fermions: the case of CeRu2Si2

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Scaling in heavy fermions: the case of CeRu2Si2

M. Continentino

To cite this version:

M. Continentino. Scaling in heavy fermions: the case of CeRu2Si2. Journal de Physique I, EDP

Sciences, 1991, 1 (5), pp.693-701. �10.1051/jp1:1991163�. �jpa-00246364�

Classification

Physics

Abswacti

71.28 75.30M 75.30K

Scaling in heavy fermions: the

_case

of CeRu2S12

M. /L Continentino

Instituto de Fisica, Universidade Federal

Fluminense,

Outeiro de S. J. Batista S/n, Niteroi, 24210, RJ, Brasil

(Received

5 December199( mvbed

l2Mmch1991, accepted l5Mmch1991)

Abstract. A

recently proposed scaling theory

of

heavy

fermions iS used to

analyze experimental

results on

CeRu2S12

under external

applied

_pressure. We examine

Susceptibility

and

resistivity

data up to

approximately

8 kbars, to obtain relations bebveen the exponents

characterizing

the quantum

critical behavior of this system ^due to its

proximity

to a zero temperature

magnetic instability.

The Kondo lattice is _a model Hamiltonian which has been used

successfully

to describe the

magnetic degrees

of freedom of

systems

^{which show} a

competition

between Kondo effect and

long

_range

magnetic

order

[Ii.

This model

gives

a very

good qualitative description

of the

phe-

nomena

occurring

in

heavy

fermions where thin

competition

is the dominant feature. The Kondo

lattice Hamiltonian has two

parameters

^{J and W which stand for the}

coupling

between conduc- tion electrons and localized moments and the bandwidth

respectively.

The

phase diagram

of this

model has been

investigated by

several authors

[1,

_2] and the main result which _comes out from this

analysis

is the existence of _a critical ratio

(J/W)c

at zero

temperature

^which

separates

a

phase

with

long

_range

magnetic

order from _a

non-magnetic

one for

J/W

_>

(J/W)c

where the Kondo effect dominates. Renorrralization group

approaches

to this

problem

_[2] have shown that this

continuous transition is associated with an unstable fixed

point

at zero

temperature (T

₌

0)

and

(J/W)

₌

(J/W)c. Recently

a

scaling theory

of the Kondo lattice has been

proposed

_[3] which makes _use of the

properties

of this zero

temperature

^fixed

point.

An

interesting

feature of this

approach

is to show

unambiguously

the existence of _{a new}

temperature

^scale

T~,

lower than the

single

ion Kondo

temperature TK,

which is characteristic of the lattice. This

temperature

or

line,

in the non-critical

part

^{of the}

T/W

_x

J/W phase diagram,

marks the onset of the Fermi

liquid regime

and has been associated with the _so called coherence transition observed in

heavy

fermions

(Fig. I).

This is _{a crossover} line and its collective _or "lattice" character is evidenced

by

the fact that it h

governed by

the same

exponent

^{of the}

critical,

^Neel

line,

which _occurs for

(J/W)

<

(J/W)c.

The

scaling theory

makes

explicit predictions

for the

scaling

behavior of different

physical

_quan-

tities close to the zero

temperature

transition [3]. ^{We find:}

f

«

lJl~~"fp lT/Tc

,

H/Hcl (la)

694 JOURNAL DE PHYSIQUE I N°5

x «

jjj-~ i~ jT/Tc

,

H/Hcj (i~)

mT « 7 =

C/T

=

lJl~~"~~" fc lT/Tc

,

H/Hcl (lC)

T «

lJl~~~f< lT/Tc

,

H/Hcl (id)

f

_«

lJl~~ fL.lT/Tc

,

H/Hcl (le)

COHERENCE

N££L LINE LINL

i PARAMAGNET

WITH j

T/W M##TS

i i

i i i i i

ANTI j

FERROMAGNET

~ FERMI

~ LIQUID (J/W),

Fig.

I. Phase

diagram

of the Kondo lattice. The critical Neel line is

governed by

the same exponent of the coherence line.

where T h the

temperature

^{and H the external field}

coupled

to the order

parameter.

^Since most of the

heavy

fermions _are close to an

antiferromagnetic instability,

H is _a

staggered magnetic

field.

The _crossover or coherence

temperature

^T~ =

Al j("

and the _crossover field

H~

=

B(j(fl+~.

In the

equations above, f,

_{x, mT} stand for the free energy

density,

^the order

parameter suscepti- bility

and the thermal _mass obtained from the coefficient of the linear term of the

specific

heat

respectively.

_r is the characteristic relaxation time which governs ^the critical

slowing

down and

f

the correlation

length

^which can be measured

by nqutron scattering.

The reduced variable

j

is

given by j

₌

(J/W) (J/W)~. Differently

from

previous approaches

to the Kondo lattice

where a

unique, single ion,

_energy scale controls all the action

[4],

we have now a set of criti- cal

exponents goveming

the critical behavior of the

system

and in

particular

the enhancement of the

susceptibility

and of the thermal _mass. The

exponents

a,

p,

_{7, v, z} are standard critical

exponents

^{associated with the} zero

temperature

^fixed

point

at

(J/W)~

and

obey

usual

scaling

laws as a +

2fl

_{+ 7 =} 1 However due to the

quantum

character of the critical fluctuations at a zero

temperature

^{transition the}

dynamic _exponent

_z

plays

a distinct role [3]. ^{In fact} in the scal-

ing

laws which involve the

dimensionality

d it is

replaced by

d + _z so that d~fl ₌ d + _z acts as an effective

dimensionality _[3].

In

particular

^the

hyperscaling

relation is modified and _we have

v(d

+

z)

₌ 2 _a. This result has

important implications

as we discuss below. We remark that the characteristic field

H~

and the coherence

temperature

T~

represent

crossover values and _are not critical

parameters.

^Also we recall that the critical behavior

along

^{the Neel line is} not

governed by

the zero

temperature

^fixed

point

since

temperature

^is a relevant "field".

Consequently

the

expo-

nents

along

this line _are different from those considered here [5]. ^The

generalized scaling theory [fl predicts

however that the critical Neel line rises in

temperature

^{with the} same

exponent (I /vz)

of the coherence line

(see Fig. I). Finally

notice that the Fermi

liquid regime

attained below the coherence line T~

implies

that for T « T~

,

H =

0,

the

scaling

functions

f~

are

expanded

as

f~ [T/T~]

^Gt

ii

+ a

(T/T~)~

+ b

(T/T~)~

+

..j

^{and assume for T = 0 constant values.}

Since the ratio

(J/W) depends

on the volume of the

system, pressure

^{P is} an

implicit

vari- able in the

equations

above and _can be used to

explore

the

phase diagram

of the Kondo lattice

[7, 8].

^{It also allows} to check

scaling

relations

predicted by

these

equations

as for

example

that the

susceptibility

_x, norrralized

by

^its value at _a maximum

(or

its

limiting

Pauli

value),

will show universal behavior _{as a} function of

pressure I.e., x(T, P) lx [T~(P)]

_{= g}

[T/T~(P)]

This

type

^of

scaling

has in fact been observed [8] in

CeRu2S12. Finally

it _can be

easily

shown from

equations (I)

^that

maxima,

minima _or inflection

points

that _occur in the

quantities given by

these

equations

as a function of

temperature

or field will

always

occur at the characteristic

temperature

or field and this allows to obtain

7~

and

Hc experimentally.

Let _us consider _a

given physical quantity X,

lie the

staggered susceptibility

or the thermal mass, which close to the _zero

temperature

^{transition has it's} enhancement characterized

by

an

exponent

z I.e. X _«

j[-~

₌

((J/W) (J/W)c(~~

For the ratio

J/W

we assume a

dependence

on the volume V which is

given by J/W

₌

(J/W)o

_exp

i-q (V

_Vo)

/Vo]

where

(J/W)o

is the value of this ratio at the

equilibrium

volume

%

_{and q} a

parameter

[9]. ^The Gruneisen

parameter [10] r~

^{associated with the}

physical quantity X,

can be

easily

obtained and h

given by:

~~ ~~~

l

(J/W~j(J/W)o

^~~~

This

equation

^shows that if the

system

^{is close to the}

magnetic instability

such that

(J/W)o

££

(J/W)~,

the Gruneisen

parameters

may ^become very

large.

This is indeed the _case in

heavy

fermions where the

large

Gruneisen

parameters

^lead to an extreme

sensitivity

of these

systems

to

changes

in the volume and

consequently

to

applied

_pressure

[10].

^Another

interesting

as-

pect

^of

equation (2)

is to show that the ratio between Gruneisen

parameters

^{of different}

quan- tities,

for the same

material,

allows to obtain relations between the

exponents characterizing

their critical behavior. ln order to make

explicit

the connection between the Grunehen

parame-

ters and the

applied _pressure P,

we introduce the

compressibility

_~

=

(-l/V)3V/3P

to obtain

696 JOURNAL DE PHYSIQUE I N°5

r~

₌

(-1/~)3LnX/3P.

For _a more direct

comparbon

of the

scaling theory

vith

experiments

un-

der

applied pressure

^{it is convenient to consider the}

following expansion

^{for small}

applied _pres-

sures:

Ln

X(P)

= Ln

X(0)

+

~$

_P

₍₃₎

v=vo

from which _we

get Ln[X (P) IX (0)]

_{= -z}

(MT) P,

where _~o is the

equih~rium compressibility

and r _a

"property independent"

^Gruneisen

parameter

^defined as r

=

(3Ln j/31n Vi

^such that

r~

₌ -zr

(z

_>

0).

The linear behavior of the

logarithm

of the normalized

pressure

^{variation of} a

given quantity

vith

applied _pressure,

for small

pressures [8],

^h illustrated in

figure

2 for different

properties

of

CeRu2S12.

For _a

given

material the

product ~or

is _a constant so that the

slopes

of the lines defined

by

the data in

figure

² are related to the critical

exponent

xi. In

particular

the pressure variation of the

quantities

shown in this

figure

fall on the _same line

indicating

that the

exponents governing

their behavior _assume, within

experimental

_accuracy, the _same numerical values. The

quantities

whose

pressure

^{variation are shown in}

figure

2 are :

I)

^{The normalized}

temperature

^{of the maxima of the}

uniform,

low

field, susceptibility

_xo at different

pressures

[8] ^I.e.

Tc(P)/Tc(0).

These maxima _{occur at} the coherence

temperature Tc

characterized

by

the

exponent

vz.

ii)

The normalized coefficients

[A(0)/A(P)]~/~

of the

T~

_{term of the}

resistivity

defined

by

p pa = A

T~

_where

po is the residual

resistivity [8, 11].

^{For the Ferrri}

liquid

at T «

7~,

we

expect

that A _«

Tp~,

where

Tc

is the coherence

temperature.

iii)

The normalized characteristic uniform field

h~(P)/h~(0)

at which the uniform differential

susceptibility

_{xh =}

dM/dh

has _a maximum

[10]

^for a fixed

temperature

^T «

T~.

iv)

The normalized value of the uniform differential

susceptibility _[10]

at the characteristic uni- form field

h~

for fixed T «

Tc,

I.e. _xh

(h

₌

h~, T,

P

=

0) /xh (h

₌

h~, T, P).

v)

The

normalized, uniform,

^low field

(h

«

hc)

,

susceptibility

_{[8] xo} at the coherence

temper-

ature, I.e. _xo

(h

Gt

0,

T

=

n,

P ₌

0) /xo (h

Gt

0,

T ₌

Tc, P)

The

system CeRu2S12,

as evidenced

by doping _[12] (or negative applied pressure)

is close to an

antiferromagnetic instabili~y

^and

consequently

the relevant field

coupled

to the order

parameter

and which appears ^{in the}

scaling

functions

given

ⁱⁿ

equations (I)

^{is the}

staggered magnetic

field H and not the uniform field h. In order to obtain

scaling

relations for the uniform

susceptibility,

_we

consider _a uniform

magnetic

field and introduce this

quantity

as a variable

scaling

close to the zero

temperature

^fixed

point

as h'

= b«h _or

(h/Jl'

=

b«+~ (h/J)

where is _a

scaling

factor. _a is _{a new}

exponent

^and z the

dynamical

critical

exponent

^which renormalizes the

coupling

J at the T

= 0

fixed

point

_[3]. The normalized uniform

magnetic

^field

h/J

is taken _{as a} relevant field

(a

+ z >

0) implying

_a multicritical character for the _zero

temperature

fixed

point

at

j

=

0,

h

/J

₌

0, T/J

=

0).

The relevance of the

magnetic

field is evidenced

by

the decrease of the uniform

susceptibility

under

pressure

^and

yields

the

following scaling expression

for the total

ground

state energy ^{as a} function of the uniform

magnetic

field

[3]:

f

_«

iJi~-"f~ I(h/JJ/iJi*hl (4J

where

#h

₌

v(a

₊

z). Defining

a uniform characteristic

magnetic

field

h~

«

(J J~)*~

we no- tice that the uniform field will enter the

scaling

functions in the combination

(h/h~)

We _{can now} derive the

scaling

form of the uniform

susceptibility.

^We find that the uniform differential _sus-

ceptlility

_xh «

3~ f/3h~

_«

j(~-"-~*h fh (T/T~, h/h~)

and the uniform low

field, susceptibility

xo ^«

j(~-"-~*h fo (T/T~)

Note that the normalized

pressure

^variations of xh

(h

₌

h~, T)

and xo

(T

₌

_T~)

_are

governed by

the _same

exponents.

^{Then it h} a direct

consequence

^{of the}

scaling approach that, independent

^of the values of the

exponents,

the

susceptibility

ratios defined in

iv)

and

v)

will

present

^the same variation under

applied

_pressure as shown in

figure1

1o

CeRu~si~

+

0

*

~

*

l

+

j

oo o

Pressure (kbars)

Rg.

~ Pressure

dependence

of:

(*) hc(P)/hc(0); (X)

_Xh

(hc,

P

=

0) /Xh (hc, P)

for T _« Tc;

(+) [A(0)/A(P)]~/~; ₍

_~)

Tc(P)/Tc(0); (Q )

Xo

(Tc,

_{p =}

0) /Xo (Tc, P)

for h « hc. For details see text.

The _same behavior under pressure ^{is also}

expected

for

[A(0)/A(P)]~/~

and

Tc(P)/Tc(0)

_as

indeed is found in

figure

^2. This

just

confirms the Fermi

liquid

nature of the state attained

by CeRu2S12

at very ^low

temperatures (T

<

Tc)

I-e- below the coherence line.

A

quantitative

^{discussion of} the

exponents implied by

the results shown in

figure

2

requires

_a

knowledge

of the

universality

class of the Kondo lattice

instability

in

CeRu2S12.

There are two main

points

to consider and which will prove very ^{useful in the} determination of these

exponents.

The first _concerns the nature of the

antiferromagnetic

state obtained

by doping CeRu2S12.

The

Ising

character of these

systems

due to the

large anisotropy

and

strong in-plane

correlations

[12]

suggest

^that these materials behave _as

metamagnets

^whose

phase diagram

^is illustrated in

figure

3.

JOURNAL DE PHYSIQUE i T I, At 5,MAT 1991 ²⁹

698 JOURNAL DE PHYSIQUE I N°5

Notice that the T

= 0 transition in _a finite uniform

magnetic

field is first order and also the presence of _a tricritical

point.

On the other hand the transition line in the h

=

0, T/W

_x

J/W

plane

is second order as obtained

previously ii].

These results

suggest

^the

generalized

Kondo lattice

phase diagram

shown in the _same

figure

and make clear the multicritical character of the fixed

point

at

T/J

₌

0, J/W

=

(J/W)c, h/J

= 0 which _we will _assume behaves _{as a} tricritical

point _[13].

~/v/ ilv/

T

' '

'

,~

"

~

'

',

^M

~/~~/

,"h

'

fl

~j T

h/v/

x

T/v/

Fig.

3.

I)

Generalized

phase diagram (schematic)

of the Kondo lattice for finite uniform

magnetic

field h. N

represents

^{the Neel line shown in}

figure

I and (T~ a line of tricritical

points. II)

^Phase

diagram

of _a metamagnet ^{in the uniform field} vends

temperature plane.

^{The continuous line}

represents

^{second order}

transitions and the traced line is _a line of first order transitions. T is a tricritical

point.

The second main

point

is that for thin T

= 0 tricritical

point

the effective

dimensionality,

d~fl ₌ d _{+ z,} is

certainly larger

than the

upper

^{critical dimension} d~ ₌ 3. This

implies

that the

exponents

^{associated with} the zero

temperature

^fixed

point

take classical values

[3, 13].

^{In fact} if _{we use} the classical tricritical

exponents

_{a =}

1/2,

_{v =}

1/2

and _assume

#h

_{= vz =}

3/2,

with

a tricritical

exponent #i

₌

vz/#h

=

I,

we can account for the pressure

dependence

of all the

quantities

shown in

figure

2 in _a consistent way. We

get xo(T)

_«

j(-~/~ fo (T/T~), xh(T, h)

_«

j (-~/2 fh (T/T~, h/h~)

and

7~

« h~ «

j(~/2

AJso these values for the

exponents

are

compatible

with the

scaling

form for the free energy ^F «

hc(P) f [T/T~(P), h/h~(P)]

with

T~(P)

_«

h~(P)

which _was obtained

previously

on

experimental ground by _[14]

Puech et al. and [8]

Mignot

et al.

Although

this

expression

has been used _even in _a

predictive

_way

[IS,

the

analysis presented

here

put

^{these results in} a more

general

framework

namely,

the

theory

of

quantum

^critical

phenomena.

Notice that the value

#h

₌ vz =

3/2 implies

that the

exponent

a which renorma1i2es the uniform

magnetic

field is _a

= 0 so that h acts as a

parameter

^whose renormalization is controlled

by

that of the

coupling

J at the T ₌ 0 fixed

point.

The _same

applies

for the

temperature[3].

The value _z

= 3 for the

dynamic

critical

exponent

^indicates

ferromagnetic

correlations

[3, 16].

Thin h not

surprising

if _we recall the

phase diagram

shown in

figure

3 and the relevance of these

ferromagnetic

fluctuations _{at a} T

= 0

metamagnetic

transition which for

(J/W)

=

(J/W)~

occurs at

arbitrarily

low fields. Notice that the

exponents

^{above violate} the modified

hyperscaling

relation

as

expected _[13]

since d _{+ z >} d~ ₌ ^{3 for} a tricritical

point.

A direct

consequence

^{of the above values for the critical}

exponents

^{is that the ratio}

A(P)/ni~(P)

=

[T~(P)]~~ /m((P) (the equality being

valid for _a Fermi

liquid)

will tum out to

be _a constant

independent

of pressure. In fact since 2 _a 2vz

=

-3/2

_we find _{mT «}

T71

and the result stated above. The

constancy

of the ratio

Tp~(P) /mT(P)

has been observed

[17j

ⁱⁿ

CeCu6, UBe13

and

UPt3

but has not been verified

yet

in

CeRu2S12

due to the lack of

specific

heat

measurements under

pressure

^{in this}

system.

From the

scaling

form for the uniform

magnetization

M _at low

temperatures (T

<

n),

M _«

(j(2-°-*h f (h/hc)

and the values _a

=

1/2

and

#h

=

3/2,

we deduce M _«

f (h/hc).

Thin

implies

that the

metamagnetic-like

transition in

CeRu2S12

at

hc

will occur, for different pres- sures,

always

at the _same fixed

magnetization.

This has indeed been observed _on

pressure

^studies in thin

system

[8] ^{and led to the above}

scaling expression

for the

magnetization _{[8, 14].}

Thin

scaling

form for

M, provides support

for the values of the

exponents given

above. Another

consequence

of these values is that the "Wilson ratio"

xo/mT

h

independent

of

pressure.

An

unambiguous

confirmation of the above values for the

exponents

^would come from _mea-

surements of the pressure

dependence

of the correlation

length f

^obtained

by

neutron

scattering

[18].

^We

expect

^{the ratio}

[f(P

₌

0,

T

=

0) If (P,

T ₌

0)]~

^with z = 3 _to

yield points

on the line de- fined

by

the data in

figure

2. It should be _more convenient to work with a

doped system (see below),

closer to the

magnetic instability

and

consequently

with a

larger

correlation

length,

in order to

get

better accuracy.

Also,

_we

point

out that

dynamic experiments

can

give

access to the critical relax- ation time _r

given by equation (Id).

Nuclear

magnetic

resonance

probes

the Ce

spin

fluctuations

[19]

^{and these}

experiments

carried under

applied

_pressure can

provide

information _on the expo-

nents. Besides EPR measurements under

pressure

on dissolved

magnetic impurities,

like those

performed by

Schlott and Elschner

[20]

on

CeAl3,

_may

yield equivalent

information. The

scaling

of the nuclear relaxation rate

Tp~

^{has been obtained}

by

Bourbonnais eta!

[21]

^{and the contribu-} tion due to

antiferromagnetic spin

fluctuations is found to be:

Tp~

«

f~/v-d+~

«

T)~~'~~~+~~/~.

In these

equations

_{7, v} and _{z are} the

susceptibility (staggered),

correlation

length

and

dynamic exponents respectively.

d is the

dimensionality

of the

system.

^Since 7 =

1,

v =

1/2

for _a classical

tricritical

point

and d _{= z}

= 3 _as obtained in the

previous analysis,

_we

get Tp~

«

T)~/~

_from

which _{we can} obtain its relative pressure

dependence.

The

scaling theory

allows _to make

predictions concerning

the pressure

dependence

of the Neel

temperature

TN ^and critical field

hN

for

doped systems (see below)

with

J/W

close to but smaller than

(J/W)c

I.e. in the

antiferromagnetic region

of the

phase diagram

shown in

figure

_{I,. In} this

case the ratios

TN(P

₌

0)/TN(P)

and

hN(P

₌

0)/hN(P)

should fall _on the line defined

by

the data in

figure

2. This is a consequence ^of

generalized scaling _[6], independent

of the

particular

exponents,

^which

predicts that,

for

example,

the

temperature exponent

^{of the critical Neel line is}

the _same than the coherence line I-e-

I/vz.

Doping CeRu2S12

with lanthanum

[22]

is

equivalent

to

applying negative

_pressure in this

system.

At a critical concentration _xc of La of

approximately [12]

z~ =

7$l, Cei-zLa~Ru2S12

becomes

7m JOURNAL DE PHYSIQUE I N°5

antiferromagnetic

at T

= 0. The

dependence

of the Neel

temperatures

on z for _{z > z~} has been determined for different concentrations

[12].

^The

system

^{at the critical} concentration _z~,

I.e.,

with

J/W

₌

(J/W)~,

is

particularly interesting.

From the

phase diagram

in

figure

I we notice that this

system

^does not cross the coherence line and

consequently

does not enter the Fermi

liquid regime.

In fact

taking

into account a

regular temperature dependent

term in

(j[, I.e., defining j(T)

₌

J/W (J/W)~

+ aT _we find _{c «}

T~/~

_for

J/W

=

(J/W)~

and T _~ 0. Thin is

clearly

_a

non-Fermi

liquid

behavior

although

we may ^{want to extend thb}

concept

^{and think in} terms of _a

diverging

thermal _{mass mT «}

C/T

_«

^T-2/~

at z~. For the uniform

susceptibility

_{in very} low fields

we

get

_xh « I

IT

and for T _«

7~,

_{xh «}

I/h

at the critical

composition.

We have

presented

a

scaling analysh

of the

non-magnetic heavy

^fermion

system CeRu2S12

under

applied

_pressure. As in _a

previous _[23], although different, scaling approach

we have _at-

tempted

to

explicitly

determine the critical

exponents

associated with the zero

temperature

^fixed

point

and which

give

rise to the

scaling

laws which _were discovered

experimentally.

Based _on considerations _on the nature of the zero

temperature

^{transition and} on the role of the uniform

magnetic

field _we arrived _{at a}

particular

set of

exponents (a

₌

1/2,

_{v =}

1/2, #t

₌

vz/#h

=

I,

z =

3)

which

consistently

describes all the data.

They

allow to make non-trivial

predictions,

which

can be

tested,

like for

example

the relative pressure

dependence

of the correlation

length

and of the nuclear relaxation rate. It would be

interesting

to extend this

analysis

to other

systems

^like

Cei-zsi~,

which is close to a

ferromagnetic instability

and to those

exhibiting superconducting

behavior to look for the existence of different

universality

classes in

heavy

fermion

systems.

Acknowledgements.

1would like to thank Jean-Louis Tholence and Alex Lacerda for useful and

stimulating

dhcussions

on their results _on

CeRu2S12.

I also thank Dr. J.

Mignot

for

calling

_my attention to the careful

investigations

carried _on in this

sytem by

the Grenoble

group. Finally

I thank Dr. J.

Thompson

for

interesting preprints

and

CNPq

of Brasil for

partial

financial

support.

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