A phenomenological phase transition model for heavy particles

~

Solid State Conmunlcatlons, Printed in Great Britain.

Vol. 71, No. 8, pp. 637-641,

1989.

0038-I098/8953.00+.00 Pergamon Press plc

A P H E N O M E N O L O G I C A L P H A S E TRANSITION M O D E L FOR HEAVY PARTICI~S

B. BARBARA and S. ZEMIRLI Laboratoire

Louis N4el, CNRS,

166X, 38042 Grenoble Cedex France Received

: September 19 th 1989

In revised form : April 3 rd 1989 by P. Burlet.

We provide thermodynamical heavy

fermions

supposed

to

a phenomenological and

transport

model

for

properties

of

in w~ich the coherence

be

the

analog

of

the

length is

correlation

length in a phase transition occuring at zero Kelvin. Many consequences are derived which can be tested experimenzally. Although has

been

active

the problem

for

several

investigations,

of heavy

years

the

it

still

is

being understood. In particular, difficults to find a link between experiments experiments.

and

also

heavy a

which

model,

based

different

with

on

the

at T - 0 to link as

for

the

and

strong analogy

effects

coupling. with

of

some

kind

These each

interferences

of

involve

wave

coupled,

last are therefore other

through

indirectly

their

in a

coupled

interaction

In the simplest

a

specific

i).

atomic

phase

electrons

in as

heat,

with

to the

case of two

TK is

2n

a

electrons

singlet

; the

it

total

angular

or

holes

(see

problem ; below function. this

TK,

~(T)

model

we

assume

phase

transition

occurring

at

zero

sites

Impurity")

is

related

to

temperature particle :

T

function

localized

sites

i).

Such

a

that

the

evolution of the coherence length ~(T) is similar to that of the correlation length in a

atomic

(T S TK), when the

is

must be a decreasing

~/a

between

be n~/a

here consider the case n - i, where N - ~/a is taken as the "coherence length" of our

involving

temperature

-

momentum

figure

diffused coherently) is fK ~ I/TK" Therefore, the spheres of radius ~K of neighbouring impurities overlap incoherently with each interactions

N

"propagating Kondo singlet" might be considered as a boson for two electrons and as n coupled bosons for 2n electrons (with n > i). We shall

such a phase transition critical will be only associated with contributions.

At lower

will

involving

only for an even number of correlated

In

At high temperatures (T Z TK, where Kondo temperature) a Kondo lattice

For to

sites

conserved

Kelvin, allows us, many experiments

susceptibility,

in

(figure

equivalent

generally described as an assembly of non-correlated impurities. The spatial extension of the singlet ground state of each impurity (where conduction electrons are

other.

now

more atomic sites per In other words N atomic

spin-density

spin density wave.

many other approaches.

magnetic

are

time dependant way, with conduction electrons.

This

impurities. This model is not a Fermi liquid model and in that sense it differs markely from

the

electrons

spins fluctuate coherently through the diffusion of conduction electrons leading to

a very simple

resistivity, de Haas-Van Alphen .... for heavy fermions at different temperatures, magnetic fields, pressure or with the addition of

is

constructive

includes more and conduction electron.

of

different

the conduction

in a coherent way between such spheres

several local magnetic moments. Therefore, when the temperature decreases each "Kondo impurity"

from

properties

in terms of coherence

lattice

transition principle,

far

it is very theories and

between

we provide

acco~mts

fermions

Kondo

subject

relevant,

diffused

electrons, this picture reduces to that of a Kondo singlet involving (~/a) atomuic sites 1

In this paper, model I

fermions

become

The binding

637

c

energy

Kelvin.

of a Kondo

of

impurity

("Giant a the

In

quantities non-linear

Kondo

characteristic size

of

the

A P H E N O M E N O L O G I C A L PHASE TRANSITION M O D E L FOR HEAVY P A R T I C L E S

638 T

= Tc(f)

c

have

O n the other hand, proportional

to

dimensionality where d =

the

i)

substituting

(i) the n u m b e r of such GKI b e i n g

f-d

(where

of a GKI

coherence

the

(4)

:

d is the E u c l i d i a n

F = f-D = ruD ~(TcT-I )

takes

place

a l o n g a line,

a

(5)

w h e r e T c = l ( ~ ) "dz - IA -v/~ and

I =

D----d ' ~ d z

impurity 2 of

We

size

is the

f.

identical

shall

same

as

r a p i d l y above, written

now

in

The to

scaling form function

that

of

admit

(6)

the phase

everything

scaling

function

of

the

(Tc/T).

need

F(TcT'I , hT °~°)

T vD

=

:

(7)

is

i.e. the free energy c a n be also we

a

In the p r e s e n c e of a m a g n e t i c f i e l d h

2

transition described

which

is

r e d u c e d K o n d o lattice t e m p e r a t u r e

F that

dimensionality,

introduce

i

f is

ref.

F = f-D, w h e r e f = T "u ~(AT "¢) (D is a n

effective

-

(2)

= Tf(Tc(f)T'1)

necessarily

a

above

~ must as

g i v e n for a "true" impurity.

is

that

c o n s e q u e n c e the free e n e r g y c a n b e w r i t t e n

:

Fi(f)

an

the

find

and

F(x )

not

we given

; in the s i m p l e s t case

F - ~-d Fi(f )

of

(2)

form

the free e n e r g y of the a s s e m b l y o f GKI

w i l l be

where

in

scaling

Vol. 71, No. 8

and therefore all the t h e r m o d y n a m i c a l v a r i a b l e s such as

the susceptibility,

etc.., can be d e d u c e d

to

the specific heat,

:

X = T -7 X(TcT-*),

C = T x'= C(TcT'X)

(8)

to pay the coast of our assumption,

scaling

function

and

A

a

characteristic

etc...

energy). Different As

for

critical flipping

the

slowing time ~

of the impurity c

= To

Here the

(~)dz

well-known down,

the

phenomenon

should increase w i t h

c

of

characteristic the size

:

limits

asymptotic f(x) = when

I +

T

~

x

TO

is

X

a

reciprocal

flipping

time

exchange

of

energy

(r 0 - h/A) and z is a dynamical exponent.

= A(~) -dz ; z > 0 a c

(4)

limits

easily,

even

field.

Since

heavy

fermions

x

<<

C/T

i.

-

In p a r t i c u l a r

T "~ f c '

T "~ c '

can

also

be

obtained very

in the p r e s e n c e of a m a g n e t i c we

are :

2

Dv

- ~

dealing with non magnetic

the o r d e r p a r a m e t e r ~ s h o u l d be

n e g a t i v e or null C o r r e l a t i v e l y the K o n d o e n e r g y is : T

for

=TcT,

(3)

characteristic the

o b t a i n e d f r o m the u s u a l -a - x for x > > I a n d

R = C/xT = T2(Du'l)etc... c

; z > 0

of

f(x)

+...

0,

Other

order

are

forms

=

- dv + i = ~ + fl - 7 + 2fl

with

(9) ~ 0, vdz = i It

is

worthwhile

noticing

that

these

exponents refer to T and T c (h - 0). N o w we can define the

same

way

as

for

an e f f e c t i v e mass,

the

flipping

time,

in by

i n t r o d u c i n g the e x p o n e n t p :

= mo (~)pD , p > 0

mc

m o is the effective by

an

isolated

extension

fK"

mass

Kondo

In

an

(i0)

of e l e c t r o n s d i f f u s e d impurity

ideally

of

rigid

spatial

picture

p

should be equal to u n i t y and in any case p > 0. Clearly

enough

our

quantity

;

propagates

it

electrons, heavy

"coherent FIGURE

I

and

particle

is

near

is

a

static

conduction

equivalent

a null

dominated the

not

with

therefore

(having

regime"

fluctuations

GKI

by

spin).

to

a

In

this

critical

spin

absolute

zero

of

Vol.

71, No. 8

temperature (i0) is :

A PHENOMENOLOGICAL

the

form

PHASE TRANSITION

of ~ to be substituted

= T -~ f(Tc T . 1) However this

regime

essentially nothing

can

also

use

when

spin

fluctuations

but

the

scale.

Kondo

(I0)

the

act

where

f

is

of assemblies

or "giant" resistivity

(a)

(~ > can

; P > 0

~K ) be

(13)

For

single

non- interacting

impurities,

the

number of particles is constant and equal to n o the number of Kondo sites ; furthermore the (12)

Note transitions

that our simple analogy with phase at zerc Kelvin indicate that T and c T K are connected to each other : T c = T K - A I. The expressions (ll) and (12) simply are the limits

near the phase

(ii)

and

well

characteristic

transition

above

time

for elastic

(12)

of

reflects

by

balistic collisions,

the

the

;

the

problem

"width

The scaling form of the for these two limits are then :

low

temperature

mK

=

regime

of

is the

affeetive

mass

(TcT-I , hT - ~ ° ) for the where

flipping

time z.

temperatures

f > fK'

and

(T ~ O) will not be time

T but by the c different site corresponds to

between This case

T < T K.

m A

(~a)-b .

f > f K ' p = Pc = ~ Since

Pc

must

decrease

GKI increases,

when

b=D(Z-p)-d

the

size

(14)

( of the

b > 0. In this case T = T . c

regime".

m c = mo T -pDv Mc

diffusions

the

at zero K e l v i n

it

temperature

simply

critical

classes

iP

m

< ~K' P = PK = ~

dominated

Tc

main

639

is

: = T-* ~ (TKT-I)

TK ;

two

of incoherent (f _< ~K ) Kondo impurities, the written in this model :

beyond

This regime

regime

For

in

(ii)

at the atomic

else

given by

one i.e.

MODEL FOR HEAVY PARTICLES

Using the forms of ~K(T) and ~(T) given above we deduce the scaling (and the low temperature limits) forms of the resistivity : Far of the critical

regime

:

(ii) PK ~ T'P PK (TKT-I)

m0 T'P

temperature mass

~(TKT-I, regime

increases

between

hT -4°)

where

when

for

~ S ~K"

the

the

PK ~ TKP

decreases

In the critical

:

For T = Te, m e = mo Tc-PDv

(15)

The effective

temperature

T K and T c. In particular

high

regime

:

Pc o~ T ub Pc (TcT-1) Pc a TUbc

and

(16)

(12) T = T K , m K = mo T K p The

other

limits

and field dependant

Well

which

are

temperature

can also be evaluated.

critical unitary

The

resistivity

impurity

is

given,

at

p - p0(l (T/TK)~) limit s . Our physical allows

an extension

in which the

the

size

of

v e r y different

low

single

Kondo

temperature,

of this Fermi liquid

o~

Kondo

a

by

limit GKI.

lattice

theory

is renormalized

to

Therefore,

the

is

approach

in

our

from a simple Fermi liquid.

number

of

heavy-particles

= lifetime

where

critical

where

the

depends

on

"~-renormalized the

temperature

the

~(x) - x -a - x -(a+2) length function

same +

asymptotic

form

... for the correlation

at Tc < T << T K and T << Tc :

p(T~T K) = p0(T K)

(I-P(~K)2+...)

(17)

= p0(T c) ( l + b ( ~ ) 2 - . . . ) c

(18)

and

and

- Zc(~) : flipping time of a GKI = time of cohereHt conduction of Kondo electrons within the heavy particle

regime limit"

Using

p(T~Tc)

where b and pd are both positive. The change of sign of the T 2 coefficient between a Fermi liquid

=

region

low

Starting from the simplest Boltzmann expression for the resistivity associated with a flow of particles p = m * /ne 2 r, we just put * m = mc(f) mass of an heavy-particle, n = (~)-d~

the

P0 ~ TVb~ (TcT-*)-

where P0 is the unitary picture for heavy fermions

unitary

~

temperature

of

above

effects are sizable, the system is in a Fermi liquid limit. This is no longer the case in the

of this particle.

near

TK

and

the

Kondo

lattice

near

T c = TK-A , is in this model simply due to the fact that in the latest the time of coherent (non dissipative) conduction within the "heavy particles" r = ~0 (~/a) Dz increases c

640

A PHENOMENOLOGICAL

dramatically well

when

Lelow

nothing spin

T K.

else

the

We

but

model

relate

quantities. to

critical

phase

write

are

the

quantum

transition

set it

this

check

of

b > 0,

(14),

a

the

moment

The

the

the T -I one

T

relative

the

effective mass w i t h temperature

Furthermore

our

which,

the correlation

in

=

0,

new

of

rate), (see

eq.

14).

always

is not

Many

our

experiments

;

the

3.

All

the

if we

From

our

This

(or holes) of

~,

origin

size the

or

electrons bounding might

of

f

b = 31

always

the number

volume =

values

of

exponent

~.

not

the

or

holes

mechanism

experiments the moment

phase

satisfied >

0.

(see

During

below

them

containning

the

scheme the

a

figure

2b)

renormalization

critical

of non correlated

volume b y the clusters)

divided

the inequality fd > (D

temperature,

moments

can be w r i t t e n

Vexcl = 1 - ~-[ D - 4 = 1 - t~ where

~ > 0, Vexcl

is an increasing

the

(a)

evaluated,

~

bounded

an

even

and

value

the following values

of

into a Kondo picture

cluster of

being

of

quantum

of

bounded

it

the

is

between

given

can give

d.

general

whatever is),

for

a pair

number

of our exponents

will be we

are

be

in which

fluctuations

any

staggered

the sample is schematically of

on the

usal

in

include

for

(in a more

to

take

at

initial

in r e l a t i o n

In

spatial

antiferromagnetic

due

determination

the

:

=included

negative

have b e e n done

might

a value picture,

into a fluctuating size

of

should therefore

does

exponents

assumed

starting

electrons

to length

dimensionality

of other contributions 4 .

now,

singlet

related

(excluded :

P0 a cb/s

effective

be

to

of

that Po a c (at least if c

of our clusters

equal

the

to I/T.

shall conclude by a d i s c u s s i o n

concept

magnetization

is

scattering

As a consequence

dimensionality existence

that

show a low

(21)

(see e.g. ref (i) and therein

large).

model

low value

A(fo/a) "2d

=

just

showing

too

the

process

in h e a v y fermions ref)

correlation

length should be limited to the

gives the residual resistivity

shows also

only

transitions

picture

diffusion

to C/T

term must

is

~

strong

a

limiting

the

because

(with

the

ratio

defects,

~D ~ c'Z/s where c is the c o n c e n t r a t i o n

defects

indicates to

into f-d boxes, each cluster of volume fD ; of

. c

A T s contribution

coupling, and

R

or pressure

presence

or

or the

value

changes

physical

the

example

(T T "2)

but proportional

v i r t u a l excitations

(d Ln mc/dT = d Ln mc/dT c = - pDu).

according

heat

for

X (TcT-=),

the

(19)

little

up

upturn,

Wilson

exchange

with

size

specific

We

only

(Tc/A)2e.

=

X0 =

constraint

i - ud + z P < I + wd therefore

R

susceptibility

Tr3/2,-

can

C/T - T "1 C

upturn

such as :

and

on

X = T-3

heat

temperature

The

we

concerning

mass-exponent

ratio

temperature

therefore

the

values

Vol. 71, No. 8

the

verifying

At

exponents.

implies

be

these

susceptibility

specific

linear

of the n o n proven

model 4.

these

now

with

and

only give some indications

values

will

exponents

relations

an a-posteri

measurable

comparison

the

Wilson

at

of exponents

many

that

by

different

of

a

other

believe

scaling

hypotheses shall

GKI

to

each

find,

experiments, proposed providing

a

provides

to We

possible

near

of

From

decreases

the

(for TK=~).

This which

that

clusters

fluctuations

zero Kelvin

temperature

recall

PHASE TRANSITION M O D E L FOR HEAVY PARTICLES

physical

clear i

and

that 3.

(b)

d

The

in relation with

in another

paper.

from the relations

l

At

(9),

:

= 0, Dv = i, 40 = i, 7 = i, dv = ~ << i = -I, Dv = I, 4o = 2, 7 = 3, d~ = ¢ << i

FIGURE 2

a

function of

A PHENOMENOLOGICAL PHASE TRANSITION MODEL FOR HEAVY PARTICLES

Vol. 71, No. 8

In the present model where ~ < O, the cluster volume fD is larger than the volume allowed by each bc,x ~d, leading to an overlap between different clusters (figure 2b), and consequently to a "cluster included volume" Vinci = I - El D-4

subjected to the destructive interference of overlapping clusters (it is clear that there is no phase coherence between different clusters).

which

dynamical

constant

;

volume of cluster cores in

correlations

the

core

f(fd-D-l)I/D.-

This

decrease

down

to

f'fcore'

which

are

size

correlation

zero

over

re]presents

essentially

is

~core

length

the distance

the

thickness

=

will

represents therefore

the ~ n increase

be

considered

can be very

as

particles

small

of

size

if ~ is large

(contrarily to the excluded volume of usual phase transitions the included volume is a decreasing function of f). In particular, we always have the inequality fcore << fK' the usual Kondo impurity spatial extension in a diluted system (without overlap). These bags of spin fluctuating coherently can be considered as "Kondo droplets" or more generally "spin fluctuating quantum droplets "I. They are a direct consequence of the negative value of the staggered magnetization exponent ~ (This point will be discussed in more details in ref 4).

6 = of a

continous transit:ion region of dynamical moments separing different clusters of coherent quantum fluctuations (dynamical domain wall characterized by a gradient of coherence of spin fluctuations). The total proportion of moments belonging to walls is ~ fD-d .

can

~core which

This volume measures the proportion of magnetic moments belonging to a given cluster and fluctuating coherently without being

Vinci represents ~ e

temperature decreases or the exchange interaction A increases (physically due to increasingly destructive interferences volumes associated with larger exchange couplings between localized sites). Clearly enough our CKI D

(22)

641

these quantum dynamical I ; this volume also fluctuating sites which with ~ i.e. when the

A C K N O ~ S

We are grateful to C.M. Varma who encouraged us to develop this model at the ICAREA conference and to C. Bastide, M. Cyrot and P. Stamp for their interest and critical remarks. More recently M. Papoular read in details our manuscript. He is gratefully acknowledged for his suggestions. We are also pleased to thank Drs J. Vannimenus and N. Ashcroft for interesting discussions.