Magnetic ordering and hybridisation in YbAuCu 4

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Magnetic ordering and hybridisation in YbAuCu 4

P. Bonville, B. Canaud, J. Hammann, J. Hodges, P. Imbert, G. Jéhanno, A.

Severing, Z. Fisk

To cite this version:

P. Bonville, B. Canaud, J. Hammann, J. Hodges, P. Imbert, et al.. Magnetic ordering and hybridisation in YbAuCu 4. Journal de Physique I, EDP Sciences, 1992, 2 (4), pp.459-486. �10.1051/jp1:1992157�.

�jpa-00246500�

(2)

J. Phys. ^I ^France 2

(1992)

459-486 APRIL 1992, PAGE 459

Classification

Physics Abstracts

71.28 75.30M 76.80

Magnetic ordering and hybridisation in YbAuCu4

P.

Bonville(~),

B.

Canaud(~),

J.

HamnJann(~),

J.A.

Hodges(~),

P.

Imbert(~),

G.

Jdhanno(~),

A.

Severing(~)

and Z.

Fisk(~)

(1)

^Service ^de ^Physique ^de ^l'Etat

Condens6,

D6partement de Recherche _sur l'Etat Condens6, les Atomes et les Moldcules, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette, ^France (~) ^Institut

Laue-Langevin,

38042 Grenoble, France

(~) Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U-S-A-

(RecHved

25 October 1991, accepted in final form 18

December1991)

RAsumA Nous pr6sentons _une 4tude

par spectroscopie M6ssbauer _sur

l~°Yb

_des _propr16t6s _h

basse temp6rature du compos6 cubique k 61ectrons lourds YbAuCu4. Dans la phase antiferroma- gn6tique (TN

# 1

K),

_nous _mesurons _une r4duction de 20 $l du moment spontan6 h saturation

de

Yb~+

par rapport _au moment de l'4tat fondamental T7 de champ cristallin, et _nous mettons en 4vidence _une anisotropie notable de l'interaction magn4tique entre les ions. Nous interpr4tons

ce comportement en termes d'hybridation _entre les 41ectrons 4f et les 41ectrons de conduction, qui rend compte des propri4t4s de type 41ectron lourd. L'dchelle d'4ner#e kBTo de l'hybridation

est estim4e de l'ordre de 0,3 K. Les spectres M6ssbauer dans les phases ordonn4e et parama-

gn4tique

pr4sentent des 41argissements de rare inhomog+nes qui sont interpr6t4s _en termes de distribution de contraintes locales suivant les _axes _< ii1> du cristal, d'4nergie magn4to-41astique

moyenne 0,2 o,4 K.

Abstract. We report on a

"°Yb

_M6ssbauer spectroscopy investigation of the low tem-

perature magnetic properties of the cubic heavy electron material YbAuCu4. In the antiferro-

magnetically ordered phase (TN

# 1

K),

_we

measure a 20 $l reduction of the Yb~+ _saturated

spontaneous moment with respect to that of the crystal field T7 ground state and _we evidence _a sizeable anisotropy of the interionic magnetic interaction. We interpret these features in terms of 4f electron conduction electron

hybridisation,

which accounts for the heavy electron

properties.

The energy scale kBTo of hybridisation is estimated to be o.3 K. The M6ssbauer spectra ⁱⁿ the ordered and paramagnetic phases show inhomogeneous line broadenings which

are interpreted

as

arising from _a distribution of local strains along the

ill ii

crystal _axes, of _mean magneto-elastic

energy o.2 oA K.

JOURNAL DE PHYSIQUE I T 2, N'4, APR~ lW2

(3)

1 Introduction.

The intermetallic

compounds

with

Ce,

Yb _or U _can present non-standard transport ^and _mag- netic

properties

at low temperature ^which are

thought

to be related to the presence of

hybridi-

sation between the localised f electrons and the itinerant band electrons

[1,2].

The _energy scale for

hybridisation

is called the Kondo temperature

Toi

^for T <

To,

the

properties

of the material

strongly

deviate from those of _an _array oflocalised trivalent Ce _or Yb ions

(with

total

angular

momentum

J), weakly coupled

to band electrons

(with spin s) through

the conventional k f

exchange

interaction:

7ikf _"

-2Jkfs.J, (1)

where Jkf is the "atomic"

exchange integral.

For temperatures below

To, hybridisation

leads _to the formation of _a _narrow band located

near the Fermi level and of width of order

kBTo,

where the

quasi-particles

have

predominantly

4f character and _are

strongly

correlated.

Depending

on the structure and symnJetry

properties

of the _energy

bands,

the

charge

carriers at the Fermi level _may show

high

effective _masses,

varying roughly

_as

I/Toi

in the so-called

"heavy

electron"

compounds

where the Kondo temperature is

low,

the _mass enhancement _can reach values of _a few 100.

In the limit of weak

hybridisation, charge

fluctuations _are

negligible

and the

compound

is called _a Kondo

lattice,

the _rare earth ions

being

trivalent to _a

good approximation.

The

ground

state of _a Kondo lattice

depends

_on the relative

magnitudes

of To and of the _energy scale of the interionic

magnetic coupling,

as was first shown

by

Doniach [3]. ^The ^interionic

coupling

induced

by hybridisation

has _a RKKY-like

dependence

_upon distance [4] ^and can be

considered _m _an effective interionic

exchange

TRKKY it favours _a

ground

state with

long

_range

ordering

of the 4f

magnetic

moments, whereas

hybridisation,

^which ^induces local quantum fluctuations of the 4f moment with time scale

h/kBTo,

has _an on-site

demagnetising

effect.

So,

when To > TRKKY, a

non-magnetic ground

state

results,

which behaves like _a Fermi

liquid

for T _< To- A

magnetically

ordered

ground

state _occurs when the quantum fluctuations do not

entirely

_smear out intersite

magnetic correlations,

I-e- when To £

TRKKy.

In this _case, the

magnetically

ordered

phase

_can show unconventional features: the actual

ordering

temperature TN _can be

drastically

reduced with respect to TRKKYi ^the 4f saturated

magnetic

moment _can

be smaller than that

expected

for the

crystal

electric field

(C.E.F.) ground

state of the

ion;

the

anisotropy

of the interionic interaction _can be

strong

even in

compounds

with cubic symmetry;

the

magnetic ordering

_can show unusual structures. In the present state of

knowledge, however,

the _occurence of _one _or _more of these

properties

in _a

given

material cannot be

predicted

with

certainty.

The determination of the saturated

magnetic

moment, of the

magnetic

_energy kBTRKKY and of the

magnetic anisotropy

in

hybridised compounds

is thus of great ^interest ^for a better

understanding

of the

magnetic ground

state formation. In Yb based

compounds,

_a local

mea-

surement

technique

such _as

~~°Yb

M6ssbauer spectroscopy can

provide

such

information,

_as

demonstrated

by

the M6ssbauer

investigation

of the cubic

monopnictides

YbP and YbAs [5,

6],

^which are

hybridised

materials with _a

magnetic ground

state (TN Cf 0.7

K).

In these _com-

pounds, hybridisation

leads to _a reduction of the Yb saturated moment and of the transition temperature

TN,

but the

magnetic anisotropy

is found to be much smaller than the

exchange

interaction.

In another series of cubic Yb based

intermetallics, YbXCu4,

where X

=

Ag,

Au and

Pd,

_spe-

cific

heat, resistivity

and

magnetic

measurements have evidenced

"heavy

electron" behaviour [7].

Among

these

compounds, YbAgCu4

_appears to have the

highest

Kondo temperature

(To " 110

K)

[8] and it has a

non-magnetic ground

state. In contrast,

YbAuCu4

and YbPdCu4

(4)

N°4 MAGNETIC ORDERING AND HYBRIDATION IN YbAuCu4 461

show _a

specific

heat

peak

below I

K,

attributed to the _occurence of _a

magnetic phase

transition.

Hence _we expect ^that

hybridisation

is much less

important

in these two

compounds

than in

YbAgcu~

and that the C.E.F. _energy livels of the

Yb~+

_ion

are well defined in

YbAuCu4

and

YbPdCu4. Crystal

field transitions have indeed been observed and

analysed

in these materials

by

_an inelastic neutron

scattering study

_{[9, 10].} The cubic C.E.F. interaction

partially

lifts the

degeneracy

of the J ₌

7/2 ground spin-orbit multiplet

of the Kramers

Yb~+

_ion _into _two _dou- blets r6 and

T7,

and _a quartet T8. The C.E.F, level scheme

proposed

in reference [9] ^for

Yb~+

in YbAuCu4 consists of _a r7

ground

state, the excited states

being

r8 and r6

respectively

at

energies

45 K and 80 K above the

ground

state.

We present in this work ~"Yb M6ssbauer spectroscopy measurements

together

with _mag- netic

susceptibility

and

magnetisation

data at low temperature in

YbAuCu4i

_we confirm the

presence of _an

antiferromagnetic

transition at TN _t I K, and _we

analyse

the

properties

of

the

magnetically

ordered

phase

in terms of

hybridisation

of the

ground

state. We could esti- mate the Kondo temperature ^of

YbAuCu4,

the interionic

exchange

_energy and _we evidenced the _presence of _a sizeable

magnetic anisotropy.

We

analyse hybridisation

in terms of _a model

derived from the

Noncrossing Approximation (N.C.A.)

solution of the Anderson Hamiltonian [11].

Another feature of M6ssbauer spectroscopy is that it is _very sensitive to small deformations of the ionic

environment;

the M6ssbauer spectra in

YbAuCu4

show noticeable line

broadenings

which _we attribute to _a distribution of small distortions of the Yb site in _our

sample.

This _paper is

organized

_as follows: section 2 deals with the

sample

characterisation

by

means

of _x-ray diffraction spectra. ^Section 3 describes the

experimental

setup, and section 4 reports

on the _zero

applied

field M6ssbauer spectra in the

magnetically

ordered

phase.

In section 5, the T

= 0.05 K M6ssbauer spectrum ^is

thoroughly analysed

in terms of

crystal

field distortions and weak

hybridisation

of the Yb ion 4f electrons with band electrons. Section 6 contains the M6ssbauer data obtained in

applied magnetic

fields in the

antiferromagnetic phase

and

their

interpretation.

In section 7 _we describe the results obtained in the

paramagnetic phase:

M6ssbauer spectra ⁱⁿ zero

applied

field _as well _as in

high magnetic fields,

and

magnetisation

and

susceptibility

measurements.

Finally,

section 8 contains _a discussion of the results.

2

Sample

characterisation.

The

YbAuCu4 sample

^used for the present

experiments

is part ^of ^the

sample prepared

for the neutron inelastic

scattering experiments

_[9].

Following

the _x-ray and _neutron diffraction

patterns

[10],

^all ^three

compounds YbAgcu~, YbAuCu4

and

YbPdCu4 crystallize

in the cubic Clsb _structure

(space

_group

F13m).

The _x-ray

powder

pattern of YbAuCu4

using

filtered Cu

radiation shows _a weak

impurity

line _at _an

angle

29 _ci

50°,

which does _not

pertain

to the diffraction pattern ^of ^the

oxyde Yb203.

^The

impurity

content of the

YbAuCu4 sample

is thus of the order of _a few percent.

The Clsb _structure is

closely

related _to the C15

MgCu2-type

structure

(space

_group

Fd3m);

the Yb and metal

(Ag,

Au _or

Pd)

atonw,

occupying respectively

the

positions (0,0,0)

and

(1/4,1/4,1/4)

in the cubic

cell,

form two fcc lattices translated from each other

by

the vector

(1/4,1/4,1/4)

_a, where _a is the fcc cell basis vector. The Cu _atoms _occupy the

positions (xxx), (Tee), (exe)

and

(ttz),

where _z is _a free parameter.

According

to x-ray

intensity calculations,

the value of _x in the three

compounds

should not be _very different from its value

5/8

^{in the} C15 structure. The

(200)

diffraction

line,

which is not allowed in the C15 structure, but

possible

in the Clsb structure, _appears

generally

in the x-ray diffraction patterns of

YbAgcu~

and

YbPdCu4,

but is absent in that of YbAuCu4 (9]. ^This êxtinction îs âccidental ând ^due to the

(5)

fact that the _x-ray diffusion factors of

Yb~+

_and _Au

are not very different:

f(Yb~+)

₌ 63 and

f(Au)

₌ 70 at small reflection

angles.

In the Clsb structure

ofYbAuCu4,

the Yb site has cubic symmetry; ^with z = 5

/8,

its nearest

neighbours

_are 12 Cu atoms at a distance

a

@/8

_ci _0Ala. The next-nearest

neighbours

_are

4 Au atoms located _on the vertices of _a tetraedron

along

the <ill>

directions,

at _a distance

a

v%/4

_m 0A3a from the Yb site. The Yb ion is thus surrounded

by

16 almost

equidistant

metal atoms; this

high

coordination environment has the < it I> directions _m symmetry axes.

The cubic lattice parameter

reported

in reference [9] for

YbAuCu4

is _a

=

7.05191.

_We

performed

_a _more detailed structural

analysis

of _our

sample.

An x-ray diffractometer spectrum

of100 _mg of

finely powdered

material shows that the lines at

high

reflection

angles

_are

relatively broad,

which is indicative of _a distribution of lattice parameters. Ân êstimation ôf the width of this distribution leads to: 7.033

1

_<

a < 7.055

1.

_However

a

Debye-Scherrer

_pattern

performed

on a small

fragment

of the

sample

shows that the material is _very well

crystallised

with _a

unique

lattice parameter, ^this

parameter depending

_on the chosen

fragment.

We

tentatively

attribute this parameter distribution to

slight stoichiometry

variations within the

sample,

which could be due to the

large

number of

meltings

_necessary to prepare the

quantity

of material

(50 g) required

for the neutron measurements [9, 10].

Therefore,

there exists _a small parameter

inhomogeneity

from

grain

to

grain

in _our

YbAuCu4 sample,

^each

grain being

well

crystallized

with _a well-defined cubic lattice parameter.

3.

Experiu~ental

setup.

The

magnetic

measurements _were

performed using

_a standard extraction

technique.

The magnetisation _curve _was obtained at T ₌ 4.2 K for fields _up _to 4.6 T. The

magnetic susceptibility

measurements _were

performed

between 0.5 K and 30 K at low fields: H

= 13.6 Oe for T < 3 K and H ₌ 136 Oe at

higher

temperatures.

The ~"Yb _M6ssbauer _transition has _an _energy of 84.3 kev and links the

ground

nuclear state with

spin Ig

₌ ⁰ ^and ^the ^excited ^state ^with

spin

1

= 2. The M6ssbauer spectra were

recorded

using

_a neutron-irradiated

Tm°B12

_7-ray _source mounted _on _a

triangular velocity

drive. The linewidth of the

Tm°B12

_source

versus a reference

YbA13

absorber is 2.7 _mm

Is (for

~~°Yb,

I _turn

Is

= 68

MHz).

The M6ssbauer

experiments

in _zero

applied magnetic

field _were

performed

in the tempera-

ture range 0.05 60 K.

Experiments

with _an

applied magnetic

field _up to 7 T _were

performed

at T ₌ 0.05 K and T ₌ 4.2 K. The field direction _was

parallel

to the _7-ray

propagation

direction.

Spectra

below I K _were obtained

using

_a

~He-~He

dilution

refrigerator.

4. M6ssbauer spectra ⁱⁿ ^the

magnetically

ordered

phase

ⁱⁿ _zero

applied

field.

4.I SPECTRUM _AT T ₌ 0.05 K. The ~7°Yb

absorption

spectrum at T ₌ 0.05 K

(Fig.I)

shows five resolved lines

indicating

the _presence of _a

hyperfine

field at the

nucleus,

and thus of

magnetic ordering

of the

Yb~+ magnetic

moments at this temperature. Such _a

spectrum,

^where the _energy differences between

adjacent

lines _are almost the _same, _can be fitted _to _a

hyperfine

Hamiltonian

consisting

of _a dominant

magnetic

term and _a small electric

quadrupolar

term:

lihf _"

~gn~nlZHhf

+

a[ (II ^~/ l(2)

(6)

-w~~(

J ~

j

° °o$°

~B ol

c~

_4*2f~~

fl~

* ~~°~

- °'

,OiO/

fi- $

§JO

o , °S% it

j

w jo

~l ) ~i i'z

m~

~ h hi ) _I

Op

I

~j ^~ ^~ ^~3

, ~O oO

L

O w

" (

'

-Z -I O 2

v(cm/s)

Fig, 1. ~~°Yb _M6ssba~ler absorption spectrum in YbA~1Cu4 at T

= 0.05 K; the solid line is _a fit with hyperfine Hamiltonian

(2)

using linearly correlated Gaussian distributions of Hhf and

a(

with A = 3.1 X

io~~ mm/s/T

and B

= -4A

mm/s (see text).

In this

expression,

the

Z-quantisation

axis is taken to be

along

the

hyperfine

field Hhf and

gn~nI

is the

magnetic

moment of the excited nulear state

(1= 2).

The

quadrupolar

parameter

a(

is defined _as:

o(

=

~~~~,

where

Q

is the nuclear electric

quadrupole

moment and

~

8

Vzz

"

l~z.~~°~

^~ is the

projection

onto the

hyperfine

field direction OZ of the

principal

2

component

l~z

of the electric field

gradient (E.F.G.)

tensor at the Yb

site,

8

being

the

angle

between OZ and the

principal

axis _z of the E-F-G- tensor. Such _an

expression

for the

hyperfine quadrupolar

interaction holds

only

when it _can be considered _as _a

perturbation

_on the

magnetic hyperfine

interaction. The small value found for the

quadrupolar

_parameter in

YbAuCu4

shows

that the Yb ion lies at a site with cubic _or

quasi-cubic

symmetry.

The fitted parameters at 0.05 K _are:

Hhf

_"

(148.5

+

3)T

and

a(

=

(0.25

+

0.I)mm/s.

The

hyperfine

field in _a metallic host _comes

mostly

from the 4f Yb

electrons,

and is

proportional

to the 4f

magnetic

moment ~ :

H((

= C ~, where C

=

(102

+

3)T/~B

for

I~°Yb~+[12].

There is _an _extra contribution

coming

from the conduction electron

polarisation

which is _an order of

magnitude

smaller than

H((,

and which will be shown _to be

negligible

in

YbAuCu4 (see

Sect.

7.3). Therefore,

the saturated spontaneous

magnetic

moment of the

Yb~+

_ion _is:

~(0.05 K)

=

(1.46

+

0.07)~B.

The cubic C.E.F.

eigenstates

r6 and r7 have

magnetic

moments

~(r6)

_"

1.33pB

and

~(r7)

=

1.715~B,

and _zero associated E-F-G- tensor. In the _presence of _an

exchange (or Zeeman) interaction, mixing

of the r8 state into the

ground

doublet enhances the saturated _moment and creates _a small

positive

^induced E-F.G, at the Yb site. At first

approximation,

the

experimental

values of _~ and

a(

_are consistent either with _a

r6 ground

state or with _a

r7 ground

state with

a reduced

magnetic

moment. The latter

possibility

is in

agreement

with the inelastic _neutr6n

scattering

determination [9, 10] ^and ^the ^reduction ^{of the} ^saturated ^moment ^could ^then ^be ^due

to 4f electron-band electron

hybridisation.

The

possibility

of _a r8 quartet as the

ground

state will be examined and ruled out in section 8 and the nature of the

Yb~+ ground

state will be

further discussed in section 6.

Another feature of the _T

= 0.05 K spectrum ^is ^the presence of

inhomogeneous broadening

of the lines _as _can be _seen in

figure

I. This is the halimark of _a distribution of

hyperfine

parameters. In the _case of

only

_a distribution of the

hyperfine fields,

the

broadenings

would be

symmetrical

with respect to the central

line;

the observed asymmetry of the linewidths is

caused

by

_a correlated distribution of

hyperfine

fields and electric field

gradients.

This property

(7)

is

easily

understood

using

the

expressions

of the

hyperfine energies

of Hamiltonian

(2):

E(m)

₌

-gnPnHhrm

+

al(m~ 2), (3)

where _m is the z-component of the nuclear

spin (-2

< _m <

2). Then,

for _a small variation of

Hhf

and tY$

I~E(m)

#

-gn~nRll~Hhf

+ (Rl~

2)l~tY$, (4)

with gnpn ^> 0.

Inspection

of this formula shows

that,

if

AHhf

and

ho(

_are

correlated,

for instance

positively,

then for the less

energetic

line

(m

₌

2),

the two contributions to

AE(m)

are of

opposite sign,

whilst for the most

energetic

line

(m

=

-2) they

have the _same

sign.

This leads to

broadenings

of the type

experimentally observed,

for instance the linewidth

G(m

₌

2)

₌ 3.2

mm/s

is much smaller than

G(m

=

-2)

= 6.2

mm/s.

We have fitted the T

= 0.05 K spectrum

using

correlated distributions of Hhf and

a(

with Gaussian

shape.

The correlation is taken to be linear:

a(

= A Hhr +

B, (5)

which,

_as will be shown in section

5,

is the form to first order

perturbation

in _presence of distortions from cubic symmetry ^of ^the ^Yb ^site. ^The ^solid curve in

figure

I, which represents such _a fit where the intrinsic M6ssbauer linewidth is fixed at the minimal

experimental

value 2.7

mm/s,

shows that the

inhomogeneous broadenings

_are

correctly reproduced.

The fitted

gaussian

distribution of

hyperfine

fields has the

following

characteristics: _mean value

H(~

₌

148.5T and _mean _square deviation

AHhr

_"

(19

+

2.5)T.

The A and B parameters ^of ^the correlation _can be allowed to vary

substantially

without much

affecting

the

quality

of the fit.

One finds that the

acceptable

values for A

(in mm/s/T)

and B

(in mm/s)

_are

linearly

10~A, (6)

the coefficient A

varying

^{in the} _range: I-S _x

10~~

_< A < 3.2 _x

10~~, yielding

-4.5 < B < -2

(see Fig. 3).

The center of the distribution of

quadrupolar

parameters is

quasi-independant

of the A and B values and is worth:

a(

= 0.25

mm/s;

the _mean _square deviation

Aa~

varies from 0.34

mm/s (for

A

= I-S _x

10~~)

to 0.52

mm/s (for

A ₌ 3.2 _x

10~~).

We

performed

detailed

perturbation

calculations of the A and B coefficients _as _a function of the type ^of C-E-F- distortions for the two cases of _a r6 or r7 C-E-F-

ground

state: these calculations and

comparison

with

experiment

will be

explained

in section 5.

4.2 THERMAL _VARIATION _OF _THE _HYPERFINE _FIELD _AND _THE _MAGNETIC TRANSITION.

The

hyperfine

^field

smoothly

decreases from 148.5 T at 0.05 K to l10 T at 0.8

K,

its thermal variation

matching

with the _mean field _curve for S

=

1/2

with TN ₌ 1+ 0.I K

(Fig.2).

Above 0.8 K the spectra ^show no resolved structure and the overall linewidth decreases upto 1.2 K. The

magnetic

^transition ^is _very

probably

of second

order,

^the transition temperature

lying

between 0.8 K and 1.2 K. The low temperature

magnetic susceptibility

shown in the

insert of

figure

2 confirms this conclusion: it presents _a kink characteristic of _a transition to

an

antiferromagnetic phase

with TN

= I K. This result is at variance with that

reported

in reference [7] ^where ^the

specific

heat

peak

found in the YbAuCu4

sample

is located _near 0.6 K.

(8)

iso

125

")

~~ ~'~~0 ₂ ₃

T '

~

0 0.25 0.50 0.?5 1.00

T (K)

Fig. 2. Thermal variation of the hyperfine field _on ~~°Yb _in _YbAuCv4i the dashed line is the S ₌

1/2

_mean field _curve with

H(~

= 148.5T and TN ₌ i K. Insert: low temperature magnetic

susceptibility in YbAuCu4

(H

=

130e).

5

hybr.Y~, ^str.sdb <11l>

n Y~, str.sts

~

m str.exh

Y~, str.sxls <100>

D

~0.5 1.0 1.5 2.0 2.5 3.0 3.5

A _x 10~~ (mm/s/T)

Fig. 3. Coefficients of the linear correlation between H and

a(

_in _YbAuCu4 _at _T

= o.05 K

:

a~

= AHhf + B; the solid line represents the experimentally acceptable values of A and B

(expression (5i);

the squares represent ^the ^calculated ^values ^for _a r6 _or _a r7 ground state and for _a strain axis

along <loo> _or <iii>

(see

Tab.

II).

5.

Analysis

of the spectrum at T=o,05

K; hybridisation

and C.E.F. distortions.

In this

section,

_we shall examine the combined effects of the

exchange

interaction and of small distortions from cubic symmetry

(strains)

_on the

~~°Yb hyperfine

spectrum in the

magnetically

ordered

phase.

We shall show that the

inhomogeneous broadenings

of the T

= 0.05 K spectrum

can be

interpreted by assuming

the presence of _a distribution of strains at the Yb site

along

_a

well defined symmetry axis.

We treat the

exchange

interaction 7iex and the C-E-F- distortion interaction 7idis in perturbation with respect to the cubic C-E-F- interaction. This is

justified

since the lowest C-E-F-

excitation has _an _energy of 45 K [9], 7iex _~w

kBTN

" I K and 7idis, as is shown

below,

is of the

order of _a few 0.I K. To first order

perturbation,

7iex and 7idis _can be treated

independantly.

(9)

As will be shown in section

7.I,

the _mean value of 7id;s ^is 2ero:

(7ld;s)

" 0, ^I-e- ^there ^is no

average non-zero strain in the

sample.

We _can then

analyse

the _mean values of Hhf ^and

o(

in the saturated state at T ₌ 0.05 K in terms of the

exchange

interaction alone and then examine the effects of the strains described

by

7idis.

5. I MIXING DUE TO THE EXCHANGE FIELD. At T ₌ 0.05

K,

the

ground

doublet is

split

by

the

exchange

interaction:

7iex _"

gJ~B3.Hex> (7)

where Hex is the

exchange

field which _we _assume is directed

along

the

antiferromagnetic

sublattice moment direction OZ.

Mixing

of the r8 state into the

ground

doublet

(there

is _no matrix element of J between

r6

and

r7)

to first order

perturbation

enhances the

magnitude

of the

magnetic

moment p = -g jpB

(Jz)

in _an

isotropic

_way:

p =

p(r;)(i

₊

Fe)

I ₌ 6,7.

(8)

In this

expression,

_e is the

perturbation

parameter: e =

gjpBHex/A8,

As

being

the r8 excitation _energy, and P is _a number

independent

of the direction of Hex but

depending

_on the

nature

(r6

_or

r7)

of the

unperturbed ground

state.

Mixing

also induces _an

axially symmetric

E-F-G- at the Yb

site,

with its

principal

_component

along

the direction of Hex the associated

quadrupolar

parameter

is,

to first order

perturbation:

a(

=

AQ (3J( J(J

+

I))

=

R(u)AQe, (9)

where

AQ

₌ ^0.276

mm/s

for ~~°Yb and

R(u) depends

_upon the orientation of the

exchange

field

represented by

the unit vector _u and _on the nature of the

ground

state.

We shall _compare the

experimental

values _~ ₌

(l.46

+

0.07)~B

and

a(

=

(0.25

+

0.I)mm/s

with

(8)

and

(9)

in the _two

hypotheses

of _a r6 _or r7

ground

doublet. Details

concerning

the calculation of P and

R(u)

_are

given

in

Appendix

I.

doublet, p(r7)

_"

1.715pB

and P

= 4. The

quantity R(u)

has its maximum value 48 when Hex ^is

along

<ill> and its minimum value 18 when

Hex

is

along

<100>. The

experimental

_~ value is IS it smaller than the bare

p(r7)

and is _even smaller than the

exchange

enhanced value

~(r7)(1+4e). Hybridisation

of 4f electrons with band

electrons,

which is known to be present in

YbAuCu4

_[7], _can account for such _a reduction. We shall _use here _a

simplified

model

[iii

in order _to compute ^the moment reduction. In this

model,

the effect of _a weak

hybridisation

is

merely

to

assign

_an

occupation

^number _n; to each

hybridised

electronic

level;

this

occupation

number is

given,

at T

= 0

K, by

the

expression:

~'

/To ~~~l'

^~~~~

where To ^is ^the ^Kondo

temperature,

r is the

"hybridisation width",

_nr is the 4f-shell hole

occupancy, which is _very close to I in _a

weakly hybridised

material like

YbAuCu4,

and A; is

the _energy of the electronic level from the

ground

state. We shall consider here

only

^the two

exchange split

states of the

ground

Kramers doublet and

neglect

all excited states.

Labelling

(10)

N°4 MAGNETIC ORDERING AND HYBRIDATION IN YbA~1Cu4 467

n and fi the

occupation

^numbers ^of these two states,

separated by

_an _energy A

=

2~(r?)Hex,

it _can

easily

be shown that:

Iii i ill(()(

⁺

_t~~

⁺ ^*~i

_(i1)

Introducing

_an

antiferromagnetic

molecular field constant 1 _<

0,

the

exchange

field is Hex

=(

1 _~ and the self consistent

equation (11)

^for _~

yields:

~ =

~(r7)(n fi)/ 1 _4~~~~ (/

^~^~~~~

_(n

₊

^i)j ₍₁₂₎

As _n fi < I and _n ₊ fi _ci

I,

these

equations

show that

hybridisation

introduces _a reduction factor for both the

magnetic

moment and the E-F-G-- To calculate the moment

value,

it is _necessary to also take into account the effect of

exchange

enhancement. Estimations indicate that values for _n fi in the _range 0.75 < _n fi <

0.8,

1 values in the _range 0.5

T/~B

< I < I

T/~B, yielding

0.73 T < Hex < 1.46

T, give

a

good

agreement with the

experimental

_p and

a(

values when the

exchange

field direction is _< II I> _or <l10>. The

Kondo temperature To can be estimated

using

the two relations derived &om

(10):

1_

^~ ^~

^r(i

^K

^nr) ^To(To

^A⁺

^A)

_~i~~

r(i nr)

2To + A

~ ~ ~

x

To(To

+

A)

Introducing

the moment reduction ratio _r ₌

~/p (r7)

and

defining TRKKy through:

kBTRKKY I p

(r7)~,

_we _get,

using

n + fi ₌ nr ci I and

taking

into account the

exchange

enhancement

(12)

^{of the}

magnetic

moment:

To _nr

~~~~~

~~~~)

'~~~~~

^~~ ^~

~~~~

As TRKKY _"

(1.60

+

0.35)

K and _r

=

0.85,

_we obtain for the Kondo temperature:

To _"

(0.30

+

0.10)

K.

To summarize this

section,

it _can be concluded that the _mean values of the

hyperfine

field and of the induced E-F-G- at T ₌ 0.05 K in

YbAuCu4

_are

compatible

either with _a bare r6

state.

5.2 EFFECT OF STRAINS ON THE MAGNETIC HYPERFINE SPECTRUM. Small deviations

from cubic symmetry ^{of the}

crystal

electric field _at

a Yb site _can be

expanded

_up to second order in J

using

_a basis of

angular

momentum

operator-equivalents

which transform

according

to the

representations

r3 and r5 under the

operations

of the cubic group. The two r3-like operators _are

[13]:

O(r3g,o)

_"

3J) _J(J

₊ ₁₎

Olr3g,~)

₌

#(J(

₊

_{J~ ),}

^~~~~

(11)

and the three

r5-like

_operators _are:

O(r~~,;)

=

j(J~JI

⁺

^JIJ~),

^I,

^k,

ⁱ = z, y, z and I

#

k

#

1.

(16)

A natural choice for the z-axis is _one of the

high symmetry

cubic _axes; in the

following

we

present calculations where the z-axis is either the

tetragonal

<100> axis _or the

trigonal

< II I>

axis.

The distortion

(or strain)

Hamiltonian writes therefore:

7idiS _~

~(

~~~'~

~~~~~'~

~

i~>Z ~'~~~'~

_~~~~

The B coefficients _are assumed to be small with respect to the first C-E-F- excitation energy

(45 K)

which allows 7idis to be treated in

perturbation

with respect to the cubic C-E-F- interaction.

To first order

perturbation,

it turns out that the

only

term in 7idis

yielding

corrections to the

magnetic

moment is the

axially symmetric

term

7i$(j°

₌ _B3g~o

[3J) J(J

+

I))

This type of distortion is

equivalent

to an

elongation (or compression)

of the Yb environment

along

the

local

z-axis,

I-e- the <100> _or <ill> axis.

The axial distortion

7i$[°

has matrix elements between all three _states

r6, r7

and

rs.

The first order

perturbation

_parameters will be

expressed

via

fl

₌

3B3g,g/As

and 6

=

As/A67

=

0.57,

where

A67

is the _energy

separation

between

r6

and r?.

Mixing by 7i$(°

creates _an

axially symmetric

E-F-G, whose

principal

axis lies

along

the

z-elongation

axis.

Assumin£

in _a first step that the

magnetic

moment, ^I-e- ^the

exchange field,

lies

along

the

elongation axis,

_we obtain the

following perturbed

values _{for ~} ₌ _pz and

a[

:

~ =

p(r;)(i

+

Qfl)

«j

₌

AQS f(6)fl, (18)

where

Q

and S _are numbers

depending

_on the direction of the

z-elongation

axis and _on the type of

ground

doublet and

f(b)

is _a function of the ratio

A8/A67.

Details

concerning

the

calculation of

Q,

S and

f(b)

are

given

in

Appendix

I. Their values _are

given

in table 1.

Table I.

Coejficients for

the

first

order _correction _to _p and

a[

^due to the axial distortion

7i$(j° (see (18) ),

when the

exchange field

is

parallel

to the strain axis Oz.

r6

ground

state r7

ground

qp(i

+ 1.5

sin2 o)]

p~ =

p(r;)

sin _«

(i qp~

^{~ ~} ^~°~~

°j

,

~~~~

2

(12)

which

gives

the

magnitude

of the moment to first order:

p =

p(r;) [i

⁺

^QP~ ~"(° ^j (20)

The

angle

R' between _~ and Oz is

given by:

tan R'

=

(1-3Qfl)tan

_R; to first order

perturbation,

it is

equal

to R. The

component

of the E-F-G- tensor

along

the

hyperfine

field

direction,

I-e the

magnetic

moment

direction,

which

corresponds

to the

experimentally

measured

quadrupolar

parameter, ^writes then:

~ ~~2 _f

~Z

_tyz ^~ ^~~~ ^~

(21)

"

~Q~

_~~~~~

2

Including

the

exchange

field correction obtained in the

previous paragraph,

_we get ^the

complete perturbation expression

for the

hyperfine

field Hhf _"

Cp

and for the

Z-component

^ofthe E-F-G- for a

given

strain value

(I.e. ^fl value):

~r(p)

=

H~r(r;) i

+ Fe +

Qfl~ ~"(

^~

a((fl)

=

AQ R(u)e

+ S

f(6)fl~ ^~"~

^~

j

^~~~~

2

Inspection ofexpressions (22)

leads _to the

following

conclusions:

I) ^the ^axial ^distortion

7i$[°

induces first order corrections to Hhf and

a(

ifthe

angle

between the

exchange

^field and the

elongation

axis is difTerent from the

"magic angle"

Rm ₌

Arcos(I/vi).

ii)

whatever the

angle

8

(except em)

between the

exchange

field and the

elongation

axis _on

a

given site,

there exists _a linear

relationship

^between Hhf and

a(;

_for

_instance, eliminating fl°

^~ ^between

equations (22):

2

~Y[lfl)

=

AQ

~

~~~ (())() _ij

⁺

AQ6 Rlu)

@fib)j ¹²³⁾

The

quantities Q,S

and

f(b) depend only

_on the direction of the distortion axis whereas the

quantity R(u) depends only

_on the direction of the

exchange

field.

iii) therefore,

in the _presence of _a distribution of axial

strains,

I-e- of

fl values,

the

resulting

distributions of

Hhf

and

a(

values _are

linearly

correlated if the strain axis is the _same for all

sites,

and if the

exchange

field lies at _a fixed

angle,

different from the

"magic angle",

with respect to the strain axis. Even for _a random orientation of the

exchange field,

the correlation would still hold in first

approximation,

^but ^would be

slightly

smeared

by

the random values taken

by

the

quantity R(u) which,

as is shown

below, gives only

a small contribution to

expression (23).

The

experimentally

observed

significant

distribution of

hyperfine

parameters ⁱⁿ

YbAuCu4

at T

= 0.05 K and the linear correlation between them _can therefore be

interpreted

_as

arising

from a distribution of axial C-E-F- distortions of the Yb site

along

a common symmetry

axis;

furthermore,

the _easy

magnetisation

direction of the

antiferromagnetic

structure cannot lie at

an

angle

em ₌

Arcos(I/vi)

_from _the _strain _axis. _This

means that if the distortion is

along

11<

ill _>

C-E-F-

hybr.

A _x 10~ 2.13 0.71 2.48 3.10

(mm/s/T)

B(mm/s)

-2.90 -1.24 -4.33 -4.46

In order _to _compare the coefficients A and B of the linear

correlaiion: a((fl)

₌ A

Hhf(fl)+

B calculated in

(23)

with the

experimental values,

_we

computed

them for the two _cases of _an

elongation

axis

along

<100> _or

<ill>,

and in the two

hypotheses

of _a r6 _or _a r7

ground

doublet

(see

Tab.

II).

As _can be _seen _on

expression (23),

the

slope

A does not

depend

on the

exchange

field

quantities

_u and _e; in contrast, ^the ^B coefficient has _a contribution:

eAQ R(u)

~~

(b)j

which

depends

_on Hex for which _we know neither its exact

magnitude Q

nor its direction.

However,

^this

e-dependant

term is

small,

of the order of 0.1- 0.2

mm/s.

The numbers

given

in table II _are

computed

with the

assumption

that Hex is lT and lies

along

the

elongation

^axis.

We _can

proceed

_now with the

comparison

of the

experimental

correlation coefficients obtained in section 4.I

(see expression (6))

with the calculated values of table

II;

all these values

are

represented

in

figure

3. For the _case of _a

r6 ground

state, the calculated A and B values with the

elongation

axis

along

<100> match

precisely

the

experimental

values. For the _case of _a r7

ground

state, the calculated correlation with the

elongation

axis

along

<100> does

not match the

experimental data,

whereas the A and B values obtained when the

elongation

axis is

along

< II I> are not far from the

experimental

correlation.

However,

we have shown in section S-I

that,

if the

ground

state is

r7, hybridisation

must be taken into account. A

simple

calculation

using expressions (II)

combined with

(18) yields

the

hybridisation

corrected A and B values:

Bhyb

₌

eAQ Rlu)In

fi)

^~~

())~~(

^~~~

^AQ

^~

_(~~ ^In

⁺

^fi)

~ ~ S

fib)

_n + fi ~~~~

hyb

Q~

_~

_j~) 'fi

hi I

Using

n + fi ₌ I and the

representative

values _n fi ₌ 0.8 and Hex=

lT,

_we

get: Bhyb

=

-4.6

mm/s

and

Ahyb

_" ^3.I x

10~~ mm/s/T,

which _are _very close to

expqrimentally acceptable

values

(see Fig. 3).

From the measured _mean _square deviation of the

hyperfine

field distribution

AHhf,

_we can

deduce the _mean _square deviation of the

a(

distribution

through

relation:

ha(

= A

AHhf,

and then obtain the _mean _square deviation

AB3g,o

^of ^the ^axial ^distortion distribution

through

relation

(22). Actually

_we _can

only

^derive the

quantity:

~~~~'°

2

Hhf(r;) 3QAn

^~~~~