•

Chapter 1

Introduction

1.1 Preliminary Remarks

The field of Condensed Matter Physics has proved to be extremely fertile in giving rise to

new and unexpected phenomena which are quite often associated with new ground states of

the electronic system. The flurry of experimental and theoretical activity over the last two

decades has shown that systems containing rare earths (mainly Ce) and actinides (mainly

U) ions lie at the intersection of large number of problems. Some of them are cited below.

1 Virtual bound state problem, where an atom with partially filled d or f shell preserves

its magnetic moment even when dissolved in a simple metal. This happens essentially

due to slight delocalization of d or f shell and formation of admixture with conduction

electron states of the host.

2. Kondo effect resulting out of the coupling of such a local moment and conduction

electron.

3. Kondo lattice problem, when there is at least one such local moment per unit cell

4. Spin fluctuations or paramagnons may scatter the single particle states, either in virtual

bound state or narrow bands, to alter the resistivity and specific heat behaviour.

5. Intermediate valence problem where the f electron hops to and fro from f level to

conduction band.

6. The interaction of magnetic moments of solute via conduction electrons: RKKY inter-

action which forms the central concept of magnetic ordering in rare earths and dilute

alloys.

7. The onset of coherence at very low temperatures leading to large density of states at

the Fermi level presents us with the problem 0. Heavy Fermions. IP

1 •

-or

8. The problem of superconductivity where phonons no longer mediate the electron-electron interaction and should the electrons have opposite spins?

Many of these problems are yet to be understood fully. In this thesis we have made an attempt to understand one such problem of Kondo lattice. First let us review some of the important concepts and the present status of this problem. We have mainly concentrated on cerium based systems although some mention is made about some uranium compounds which also show similar characteristics.

1.2 Normal State Behaviour of the Rare Earth Sys-tems

Before we review the exotic phenomena exhibited by the rare earth (RE) systems let us briefly discuss their normal behaviour. Table 1.1 gives the various parameters for a free RE ion. In the RE compounds 4f electrons couple together according to Russel-Saunders coupling and Hund's rule. Thus the total angular momentum I is built up of the vectors 1: and S. L is the total angular momentum resulting from the magnetic coupling of angular momenta of the individual 4f electron (1 = 3). g is the total spin vector due to the intra atomic exchange coupling of the individual spins. The magnetic moment is given by gjJJ(J + 1) in the units of Bohr magneton.

Table 1.1: Various parameters of free rare earth ions Rare

earth ion 4f confi- guration

Basic Level

gj gjO(J + 1) de Gennes factor G

La3+Ce4+ 4f° 1 S0 - 0 0 Ce3+ 4f 1 2 F5/ 2 6/7 2.54 0.011 Pr3+ 4f2 3H4 4/5 3.58 0.051 Nd3+ 4f3 4 19/ 2 8/11 3.62 0.116 Pm3+ 4f4 'Li 3/5 2.68 0.217 Sm3+ 4f5 6 115/2 2/7 0.84 0.283

Eu3+Sm2+ 4f6 7 F0 0 0 0 Gd3+Eu2 + 4f7 8 S7/2 2 7.94 1

Tb3+ 4f8 7F6 3/2 9.72 0.667 Dy3+ 4f9 6 H 15/ 2 4/3 10.63 0.450 Ho3+ 4r° 5 18 5/4 10.60 0.386 Er3+ 4f11 4 I 15 / 2 6/5 9.59 0.162 Tm3+ 4f12 3 H6 7/6 7.57 0.074

Yb3+Tm2+ 4f13 2 F7/2 8/7 4.54 0.020

Lu3+Yb2+ 4f14 1st) - 0 0

•

2 •

In a crystalline environment, a feature that determines the magnetic properties of the rare

earth compounds is the interaction of the 4f level with the electric field due to surrounding

ions [1] called the crystalline electric field (CEF). The crystal field (CF) interaction leads to • partial removal of the (2J+1)-fold degeneracy. This is due to the fact that the 4f electrons

charge distributions of the (2J+1) wave functions are dissimilar. In general, the removal of

degeneracy is more complete if the symmetry of the CEF is lower.

The 4f level in the RE lies deep below the Fermi level. Due to its very small spatial extent

the direct f-f interaction is almost ruled out. Only in few cases, the magnetic dipole-dipole

interaction brings about magnetic order. The magnetic order in the rare earth intermetallics

is believed to be mediated through the conduction electrons, known as the Rudermann-

Kittel-Kasuya-Yoshida (RKKY) indirect exchange interaction. In this approach the following

assumptions are made.

• The linear response in the exchange interaction between local moments and the con-

duction electrons (that is only terms linear in exchange integral J(k', k) involving

conduction electron orbitals) is assumed;

• The conduction electron wave functions can be represented by free electron Bloch

orbitals;

• The exchange matrix elements are functions only of q = k' —k. Here k is the conduction

electron wave vector and J(q) is assumed to be constant.

The spin polarization at a distance R from the scattering center is given by

P(R) = —97n 2 N/2EFVJ < Sz >a „ F(2kFR) (1.1)

where J is the exchange integral between 4f and conduction electrons. N is the total number

of atoms present in the volume V, n is the average conduction electron per atom, EF is the

Fermi energy, kF is the Fermi wave vector and < Sz > is the expectation value of the spin.

Due to the function F(X) = (sinX — XcosX)I X 4 , where X = 2kFR, the spin polarization

is oscillatory in nature and falls of at large distances as shown in Figure 1.1. At a given

lattice site, the actual spin polarization consists of the contributions of all the moments

present in the crystal. If one chooses this site as the origin, the local conduction electron

polarization, P, at this site equals E, P(R2 ), where the summation is over all magnetic

moment sites. This sum contains positive contributions from some neighbouring atoms and

negative contributions from others. A given localized moment will orient its spin according

to the sign of conduction electron polarization P, at this site and determine the ultimate

type of magnetic ordering. Extending this idea with molecular field model, de Gennes [2]

has shown that Curie temperature Tc (or Neel temperature, TN ), Op is given by

Tc = Op = —( 31rn 2 1J1 2 IkB EF)(yv — 1) 2 JP + F(2kFRi ) (1.*

3 •

EF = h 2 k2F /2m* where m* is the effective mass. It has been generally assumed that J is

constant for a given rare earth series. The factor (gj — 1) 2 J(J + 1), is the well known as de

Gennes factor, G. Within this RKKY formalism Tc , TN and Or are ideally proportional to

the concentration of the magnetic ion.

F(X)

2kR

Figure 1.1: Schematic representation of the oscillatory nature of RKKY interaction

In the presence of thermal disorder of the rare earth magnetic moment direction, the

above coupling leads to 4f contribution (p 4 f ) to the resistivity p, which increases from zero

at T = OK to a maximum value at the ordering temperature.Within this validity of the

first order perturbation calculations, the maximum value of (psf ) at temperatures T > Tc remains constant (assuming full degeneracy) and is given by the expression

P31 = (371-m* 12he2 EF)NG (1.3)

The magnetic ordering temperature (Tc or TN), Op and psi, are expected to scale with the

de Gennes factor as shown in Figure 1.2.

In the above discussion we have assumed that J is constant. However, its dependence on

impurity quantum number M or M' [3, 4, 5] causes an anisotropy in the RKKY interaction.

The dominant part of the RKKY interaction comes from states with M = +1/2 i.e. with

the z-component of the orbital angular momentum, 1, = 0. These are the states that pile

up charge along the quantization axis, which is taken to be the bonding axis. Siemann and

Cooper [5] have shown that this anisotropy favours formation of ferromagnetic planes. The

anisotropy becomes more pronounced when the polarization of the conduction band due to

hybridization with the 4f state is taken into account. The density of states changes due

to the hybridization with the f level and depends on M. The strength and the symmetfy

•

0.8

Tc, 0.6 TN, or , Psd 0.4

0.2

0

•

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

Figure 1.2: de Gennes scaled value of Tc , TN, O, and psd for different rare earth metals normalized at Gd.

of the RKKY interaction again depend on the density of states at the Fermi level through

m*. This will lead to a "bootstrapping" process determining the strength and anisotropy of

the two ion interaction [6]. This kind of magnetocrystalline anisotropy has been confirmed

experimentally for cubic monopnictides (eg . CeSb [7]) and this could be extended even to a

tetragonal CeT 2 X2 compounds even though the calculations were performed on a cubic case

[8]. The main difference, when the theory was extended to systems with more than one f

electrons is that there the electrons do not pile up their charges along the interionic axis.

Hence there is a different angular dependence of the RKKY interaction, which will vary

with the total number of f electrons. All the above results were obtained in the so called

Kondo limit of almost completely localized moments. However as pointed out by Zhang and

Levy [9], hybridization effects will greatly influence these results for short f-f distance. These

authors have shown that in the case of non-negligible hybridization, RKKY interaction does

not decay by 1/R3 but at a much slower rate between 1/R 3 and 1/R2 .

1.3 Mix Valence Phenomena

The mixed valent state can be thought of as a mixture of 4P and 4f' ions the energies

of which are nearly degenerate. At any lattice site the 4f electron fluctuates back and forth

from the conduction band to 4f level. These fluctuations occur on a time scale Tvf - valence

fluctuation time or Inter Configuration Fluctuation (ICF) time. In the case of Ce, the

ambivalent tendency is more due to the fact that the 4f orbital is more spatially extended

than for the other rare earths. These materials are also dubbed as homogeneous mixed

5 •

valent as there is only one charge state present [10, 11]. Figurel.3 shows the schematic

representation of density of states at Fermi level, EF in cerium metal.

•

•

•

5d band Ef EF

-y-Ce

Ef

a-Ce

Figure 1.3: Schematic density of states diagram for cerium

•

7 •

Quantum mechanically the situation can be understood in terms of hybridization of the

two configurations as

a n ifn > +an-ilfn-1 > (1.4)

The more appropriate description in terms of band structure is given as follows: Under the application of pressure, the conduction band comes closer to the 4.1' level. When the conduction band comes in contact with the .4f level the later empties its electrons into the band. In the mixed valent state the f level is thus pinned to the Fermi level. It takes a width

O due to hybridization (inverse of this width is the valence fluctuating time). The hybrid

level is only half filled in non integral state. In case of Ce the f level is already somewhat

hybridized with the sd band but is far below EF to ensure complete occupancy.

The key feature of the Intermediate Valent (IV) compounds is that they exhibit a non

magnetic ground state. In Ce metal, the -y-state susceptibility is of Curie-Weiss form which

aprops trivalence while that in a-state is Pauli paramagnetic; much too large to represent

simple tetravalence but also non diverging as T 0 indicating that cerium moments are

quenched by valence fluctuations [12, 13].

The mixed valent systems with a localized level degenerate and hybridizing with conduc-

tion band and in the close proximity to the Fermi level bear a strong resemblance to the

single impurity Kondo problem with expected differences. But many properties of the ICF

materials can be understood through this analogy. As in the case of one impurity system,

there is also a cross-over from Curie-Weiss behaviour at high temperature to Pauli param-

agnetism at low temperature in valence fluctuating systems. The cross-over temperature is

called the spin fluctuation temperature, 71,f . Further support to this behaviour comes from

the resistivity measurements which implies that the quenching of moment is due to spin

fluctuations resulting from strong sf exchange. The later presumably occurs from virtual 4f

charge fluctuations where the 4f electron hops to the Fermi level and returns back with a

opposite spin. Whereas the Mossbaur spectroscopy shows only a single line corresponding

to a single charge state [14], the high energy spectroscopic measurements like XPS and XAS

clearly show two peaks corresponding to 4fn and 4f" --.1 ground states. This is due to the fact that the fluctuation time (10 -12 ) is slower than the x-ray spectroscopic time scale. XAS

and XPS techniques have been extensively used by many researchers to calculate the all

important valence of Ce in these compounds [15, 16, 17, 18, 19].

Neutron inelastic scattering experiments have been performed on many mixed valent

compounds. These experiments essentially reveal the underlying dynamics of the fluctuating

4f moments as well as yield crystal field splittings where relevant [20, 21, 22, 23]. The observed

line width of the quasielastic peak is a direct measure of the spin fluctuation temperature

Tsf . The fractional occupancy of the f level can be evaluated from neutron scattering by solving [20]

(1.5) da(Q,w) N jp + 1) I dw dw

•

•

This gives x(0) ti Ni (gtiB) 2 J(J + 1)/r ti nf cir where nf is the valence fraction. The line

width of the quasielastic peaks is finite at T 0 and is essentially temperature independent

[21].

Theoretically all properties of the mixed valent system can be understood by using the

Anderson Lattice Hamiltonian [24].

1.4 Kondo Effect

In some of the dilute magnetic alloys a departure from the normal magnetic behaviour was

seen. Especially in the resistivity behaviour a low temperature resistance minimum was

found and below this minimum the resistivity showed a In T dependence. Also the magnetic

susceptibility showed a departure from the normal Curie-Weiss behaviour and the coefficient

of the electronic specific heat had a much larger value than that in normal metals.[25]. Figure

1.4 shows typical properties of such materials. This behaviour is similar to that observed in

the dilute transition metal alloys e.g. Fe in Au. The first theoretical model explaining this

unconventional behaviour was put forward in 1964 by J. Kondo [26]. According to this model

the logarithmic rise in resistivity below T < Tmin is the result of the conduction electron

spin flip scattering by the localized magnetic moments of the impurity (3d or 4f) atoms. The

temperature Tmin is called the Kondo Temperature TK and the resistivity was given by

p4f (Kondo) = p3 f [1 — 2N(E F)Jln(T ITF)] (1.6)

where TF is the Fermi temperature.

At temperatures T TK ti TFexp(-1 /(N(EF)J) the perturbation theory used by

Kondo is divergent and could not be applied to find the resistivity dependence at T TK. This divergence is due to the formation of a narrow (r- kB TK) many-body resonance near

Fermi level at low temperatures T < TK [27, 28, 29], when a cross-over from local magnetic

moment (LMM) (T > TK) to strong interaction and the formation of the quasi bound state

(T < TK ) take place. At T << TR- a Kondo system can be considered within the framework

of a Fermi liquid model [30]. The magnetic and transport properties have been extensively

computed. The electrical resistivity [31], magnetoresistivity [32], thermoelectric power [33],

Hall effect [34], thermal conductivity [35] have been calculated using the Hamiltonian which

describes both the Kondo effect for cerium impurities [3] and crystal field effects [31]. Thus

the decrease of magnetic resistivity as 1nT at sufficiently high temperatures, after passing

through a maximum corresponding approximately to CF splitting A is now considered as

a clear signature of the Kondo effect in the magnetic cerium impurities. But perturbation

theory completely fails to account for the low temperature properties. The single impurity

Kondo problem was solved exactly at T = OK by Wilson [36] using the renormalization group

technique for the S = 1/2 classical exchange Hamiltonian

H = —Js.S (withJ < 0)

9 •

p

T

r Enhanced Pauli paramag

— Curie-Weiss

T

Figure 1.4: Some typical properties of single impurity Kondo system

10 •

Wilson showed that at T = OK nonmagnetic singlet state is formed by antiparallel cou-

pling between impurity spin and the spins of the conduction electrons. This also established

that the low temperature properties are those of a Fermi liquid [30]. An analytical solution

to the Kondo problem using Bethe Ansatz has been worked out by many workers [37, 38, 39]

using both classical exchange Hamiltonian and the Coqblin-Schrieffer Hamiltonian. A new

method using a auxiliary boson field called the 'Slave Boson' has also been introduced to

describe the low temperature properties of Cerium Kondo compounds [40, 41].

The low temperature properties of the single Kondo impurity can be summarized as

• The magnetic susceptibility x and electronic specific heat constant y are greatly en-

hanced and the ratio R = xl-y is a constant with its value ranging between 1 and

2.

• The Kondo temperature TK define a smooth transition from Fermi liquid to Free ion

behaviour at higher temperatures.

• The 4f density of states shows two peaks : one broad peak centered at the position of

4f level and one extremely narrow peak at the Fermi level with a width of the order of

TK/N where N is the degeneracy of the 4f level.

1.5 The Kondo Lattice

In systems having higher concentration of magnetic impurities, to have a single impurity type

behaviour, the direct interaction between the magnetic impurities should be absent. A similar

observation was made in concentrated rare earth alloys [25] wherein the high temperature

properties of these alloys resemble the single impurity behaviour. This is because of the

extremely small localization radius of the 4f shell (r 4f ,-- 0.2 — 0.3 A) and hence the strong

intersite interaction will begin only at concentrations greater than Avogadro number (e lf ti

1025 cm-3). Such systems are called the concentrated Rondo systems or Kondo lattice systems

where there is at least one magnetic impurity atom per unit cell. Figure 1.5 shows a schematic

representation of a Concentrated Rondo System. These systems have features of the single

impurity in an enhanced form, roughly multiplied by the ion concentration to the amplitude

in single ion case. However, there is one essential difference, the original Kondo problem

treats the impurity spin as an external quantity embedded in an conduction electron system.

The formation of singlet state takes place by a compensation of the impurity spin through the

spins of conduction electrons. In case of concentrated Kondo systems this picture becomes

inadequate. One cannot any longer consider the spin of the impurity as an external quantity

and hence the f electrons to be perfectly localized. Instead it is more appropriate to consider

the number of f electrons nf as a non-integer so that f electrons can participate in the

formation of the Fermi surface. As one goes down in temperature, below a temperature T*, which is of the order of tens of Kelvin, the effective mass of conduction electrons becomes 'a

11 •

few hundred times larger than the free electron mass. Due to presence of such quasiparticles

at T <T*, these systems are also called as Heavy Fermion systems. Several review articles

and conference proceedings exist in literature highlighting the novelties of Heavy Fermion

compounds [42, 43, 44, 45, 46, 47, 48, 49, 50]. The Kondo Hamiltonian is therefore no

longer valid and one has to work with a Hamiltonian which allows the variation of f electron

number. Such a Hamiltonian is the Anderson Hamiltonian [51]

,yt J_ t T T Vki t H -=EckL4k .k i,11 71 fn +E a f ak c.c. f,f' k,f \57

(1.8)

where ak and af are the destruction operators of conduction and f electrons respectively,

Vkf is the matrix element of hybridization between k and f states (degeneracy N1), and

Uff describes the Coulomb interaction between f states. Here as the 4f level is not well

localized, there will always be a finite hybridization of the 4f electron on the rare site with

the conduction electrons on the neighbouring site. J is related to the hybridization (Vk f ),

to the position of the 4f level with respect to the Fermi level EF and to the Hubbard

intrasite Coulomb repulsion U of the two electrons with opposite spins in the same magnetic

4f ion. According to Schrieffer Wolff transformation [52] J in the asymmetric case (U >> Vkf, EF E4 is given by

Vkf

J = 77, (1.9) — Elf!

When Vkf /REF — E4 f I > 1 i.e. the 4f level is completely hybridized with the conduction

band then one is dealing with the valence fluctuating systems. In the other case when

Vk2f /1EF — E4 f I < 1 or when 4f level lies on the verge of delocalization limit, apart from the

Kondo interactions which depend exponentially on J;

T1 a exp( - 1/ISJI (1.10)

where S is the density of conduction states at the Fermi energy, leading to non magnetic

ground state, there is also the indirect RKKY interaction characterized by temperature

TRKKY whose dependence on J is given by

TRKKy CC 8,1 2

(1.11)

which leads to magnetically ordered ground state. The competition between the quenching

of the local magnetic moments and the intersite magnetic interactions in Ce compounds has

attracted the attention of several experimental and theoretical researchers [53, 54, 55]. The

two mechanisms compete in the formation of the ground state with their respective energy

scales kBTK and kBTRKKY • We will discuss the implications of this competition in detail.

•

12 •

g(E)

E4 f

EF

E41 -I-U

E

Figure 1.5: Schematic representation density of states in a concentrated Kondo system

13 •

•

Irplc Irpl

Figure 1.6: Qualitative description of competition between RKKY and Kondo interactions

•

14 •

•

1.5.1 Kondo versus RKKY interaction

As mentioned above, in periodic Kondo lattices, the addition to the onsite Kondo interaction,

which leads to drive the compound to a non-magnetic ground state, the exchange interaction

(intersite interaction) which governs magnetic ordering also exists. These two interactions

compete and the ultimate ground state is decided by the relative strength of these two interac-

tions. Figure 1.6 projects the well known qualitative description of the competition between

these two interactions in Ce system. In this figure TK 0 represents the Kondo temperature for

a single impurity and TN 0 the Neel (or Curie) temperature as if there were no Kondo effect.

It is clear from the figure that for smaller values of J, TNowill dominate TK O and the system

will tend to order magnetically. This implies that for a well localized Ce or U impurities

where J is small RKKY interactions will dominate Kondo interactions. However, for a large

J values TK 0 is larger than TN 0 and the system will tend to become non magnetic. Hence the

real ordering temperature TN will increase firstly with increasing J, then passes through a

maximum and tends to zero at a critical value, J. This was first theoretically supported by

Doniach [53]. In a mean field approximation of j = 1/2 moments, he proved that a critical

value IJN(EF)Ic separates a spin compensated state IJN(EF)I > IJAT(EF)i c from that of

magnetic order. This derivation has been checked by several theoretical approaches such as

one dimension real space renormalization group method [56, 57] or the slave boson approach

[58]. The problem of two magnetic Kondo impurities has also been studied by numerical

renormalization group method used by Wilson for single impurity problem [59]. Moreover,

the curve of the ordering temperature presenting a maximum versus IJN(EF)1 as shown in

Figure 1.6 has been experimentally observed in several cases: the Curie temperature T c of

the CeAg compound passes through a maximum with increasing pressure [60]. In the two

systems CeSi. [61] and CeNi xPt i _x [62] the Tc increases first slowly with increasing x and

then passes through a maximum; after its maximum, Tc decreases and Kondo temperature

TKincreases, both very rapidly. These experimental results are in good agreement with the

theoretical curves of Figure 1.6.

The above description appears much too simplified. Firstly the cerium compounds cor-

responding to smaller I JN(EF)1 values have clearly much larger gamma values and would

consequently have to exhibit smaller TK values corresponding to smaller IJN(EF)1 values.

This contradicts the picture in Figure 1.6. For example in CePt 2 Sn2 [63] for which TK es-

timated from the low temperature C/T value was 1K and the value of TN was observed to

be 0.9K. This suggests that this compound lies in the low coupling limit of Kondo Necklace

model. The values of TK and TN suggest that magnetic order should quench the development

of heavy mass state contrary to the experimental findings. Secondly, the understanding of

magnetic ordering phenomena is not so simple as can be visualized from Figure 1.6. Some

kind of magnetic ordering and magnetic correlations have been detected in the compounds

like CeA1 3 , CeCu6 , CeRu2 Si 2 or CeInCu 2 by different experimental techniques. Muon spin

rotation experiments have provided evidence for static magnetic correlations in CeA1 3 , ar‘d

-r 15 •

these results are interpreted on the basis of weak long range antiferromagnetic ordering with

a small net magnetic moment of the order of 0.1pB below 2K but there is no evidence for a

macroscopic phase transition [64, 65]. The NMR experiments on Al n in CeA13 [66, 67] Cu.' in CeInCu 2 [68] or of Cu' in CeCu 2 Si 2 [69] under a magnetic field have been also analyzed in

terms of weak antiferromagnetic like ordering at very low temperature in these compounds.

Extensive neutron scattering work has been recently done on the single crystals of

CeCu6[70] CeInCu 2 [71] CeRu2 Si2 [70, 72] and Cei,La,Ru2Si2 [73, 74] at very low tem-

peratures. Two points to be noticed are : one, the neutron scattering spectra of CeCu 6 and CeRu2 Si2 are considered as sums of a q independent, single site quasielastic contribution,

associated with the single site Kondo effect and of an intersite q independent contribu-

tion associated with short-range magnetic correlations between different Ce sites. Thus at

low temperatures short range magnetic correlations develop between two Ce neighbours in

CeCu6 , CeRu2 Si2 , CeInCu2 and other compounds. These correlations which are charac-

terized by commensurate or incommensurate wavevectors, are destroyed by application of

magnetic field (25 kOe in CeCu 6 and 83 kOe in CeRu2Si2) [70]. Second, the elastic neutron

diffraction experiments show simultaneous onset of sharp antiferromagnetic peaks and dif-

fuse magnetic correlations corresponding to short range magnetic correlations below 2.3K.

Another implicit consequence of the competition between RKKY and Kondo effect is the

proximity of a magnetic instability [75]. Even by small changes in chemical composition,

long range magnetic order with fairly large magnetic moments (Ipl 0.5pB/fatom) can be induced in the most heavy Fermion compounds. For example in Ce 1 _„La„Ru 2 Si 2 [76] long range antiferromagnetic order is found at x > 0.07 and for CeCu6_„Aux long range occurs for x > 0.1 [77]. The emergence of a magnetic instability is explained by an increase in volume

induced alloying. In general increase in volume will lead to weaker hybridization i.e. J will decrease. Thus at smaller values of J, RKKY interactions will dominate leading to magnetic

order. However, as most of the heavy Fermions have strongly anisotropic hybridization, J will not always be governed by volume effect. For instance in case of UPt 3,Pdx series, long range antiferromagnetic order with a maximum in TN , of 5.8K, has been found in concentra-tion range 0.02 < x < 0.07 [78]. In this case since Pd is smaller ion its substitution should

have decreased the volume giving an opposite effect, however the localization of f atoms

occur presumably due to subtle change in c/a ratio in the hexagonal lattice. An important

observation that can be made for heavy Fermions is that only a part of the fluctuating mo-

ment orders, that is heavy Fermion properties survive even in the ordered state and therefore

the entropy involved in ordering is very small. In the case of CePd 2 Si2 [79] which lies just at the maximum of TN (J) curve of Doniach's diagram a reverse effect is observed. Substitution of Ce by Y or La in CePd 2 Si 2 leads to decrease in ordering temperature. A similar thing is observed in URu 2 Si 2 , doping by both Rh [80] and Tc [81] leads to suppression of TN. Obvi-ously therefore volume is not the only dominant parameter, dilution effects and anisotropic

hybridization must also be considered in any attempt to explain these properties.

In a recent study of volume and electronic concentration effects in cerium Kondo systenf's

16 •

the authors, Sereni et al, [82] have concluded that a maximum in ordering temperature is

observed when there is a range of substitutions in which the eventual (weak) hybridization

strength does not affect the intensity of the ground state magnetic moment. They furthrr

say that the changes caused in volume and electronic concentration by substitution are not

equivalent to a applied chemical pressure because the first one implies inhomogeneity in

the Ce-ligand spacings while the second one induces a homogeneous change in the chemical

potential. Also the substitution of Ce ligand by a hole like atom (e.g. Pd substitution by Ni

or Rh) can be differentiated from an electron like atom substitution (e.g. Ag for Pd) from

the changes in magnetic structure of the system, the latter happens because the local change

in sign of the electronic charge of the partner breaks the spatial charge periodicity. However,

these conclusions, the authors say, need to be confirmed by similar studies on other such

systems.

This suggests that in the Ce based compounds the experimental situation is still not

clear. On one side we have weak magnetic ordering in large y compounds like CeA1 3 and

CeInCu2 and on the other side magnetic interactions always exist in the compounds with

nonmagnetic ground state which lie close to non-magnetic - magnetic transition and the

effect of correlations becomes important when one approaches this transition. A correct

theoretical interpretation with a non zero correlation function of < s 1 .S2 > type was obtained

by Jones and Varma [59] for two magnetic impurity problem. But a full description would

also be necessary to improve the understanding of magnetic correlations in cerium Kondo

compounds.

1.5.2 Heavy Fermion Superconductivity

The discovery of superconductivity in CeCu 2 Si2 about 16 years ago [48] gave birth to a new

field of Heavy Fermion superconductivity. Since then five to six new superconductors in

Uranium series and one at high pressure in Ce compounds have been discovered but the

original question posed then, namely the coexistence of magnetism and superconductivity,

still remains a central challenge for the understanding of these materials even today [49]. As

this phenomenon does not form a part of this thesis we shall not discuss it any further.

1.5.3 Kondo Insulators

The recent discovery of CeNiSn, a valence fluctuating cerium compound by Takabatake et al

[83] has renewed the interest in the several decades old hybridization gap problem. Generally

the valence fluctuating compounds have metallic character but a few such as SmB 6 , gold

SmS, TmSe, and YbB 12 showed semiconductor like behaviour [84]. This behaviour is due

to formation of a small gap at low temperatures near Fermi level. Since the discovery of

CeNiSn few others like CeRhSb, Ce 3 Bi4Pt3 , Ce3 Sb4 Pt4 , CeFe4 P 12 have been classified as

Kondo insulators. Theoretically the problem is still not understood. Aeppli and Fisk [85] •

17 •

argue that a simply formulated lattice Anderson Hamiltonian describes the entire physics of

all hybridization gap materials. The highly correlated nature of the problem disappears in

the ground state and a simple effective Hamiltonian that for a band insulator is obtained..

1.5.4 Multichannel Kondo Effect and Non Fermi Liquid Behaviour

The multichannel Kondo model is being studied for over a decade [86], even with essentially

exact methods using numerical renormalization group [87], the Bethe Ansatz [88], confor-

mal field theory [89] and more recently the non crossing approximation [90]. The recent

discovery of non Fermi liquid behaviour in 1.1 1 ,Y„Pd3 [91] and U i ,ThxRu2 Si2 [92] as well

as Ce i ,Lax Cu2 Si2 [93] has renewed interest in M = 2, S = 1/2 model (Two channel). The

physics has been argued to be derived from two channel quadrupolar (U) and magnetic (Ce)

Kondo model [94]. The breakdown of the Fermi liquid behaviour has also been seen in a

Heavy Fermion at a magnetic instability [95]. Experimentally it has now been stated that

non-Fermi liquid behaviour is rather a general feature which occurs when the non-magnetic

(Kondo compensated) and the magnetic ground state are nearly degenerate i.e. at the prox-

imity to a long range magnetic order.

The occurrence of such exotic phenomena coupled with the long standing problem like

Heavy Fermion superconductivity, and its coexistence with magnetic order has maintained

the interest of condensed matter physicists in this field. we believe that a detailed experi-

mental work with theoretical understanding of the problem may provide a key to the solution

of this problem.

1.6 Scope and Organization of the Thesis

It is clear from the above survey that a detail study of the Kondo lattice problem to un-

derstand the various anomalous ground states that these materials exhibit is very essential.

The scope of this problem is rather vast with almost every new system presenting a unusual

characteristic of its own. From the voluminous work that is done in this field, one thing that

has emerged out is the competition between the Rondo and the RKKY interactions which

compete with each other in their respective energy scales. The two energy scales depend on

the density of states at Fermi level and the exchange interaction constant. Apart from these

s- two interactions crystalline electric field set up by the surrounding ions also plays an impor-

tant role in deciding the ground state of these systems. It was therefore decided to study

one cerium based concentrated Kondo system through its magnetic and transport proper-

ties. The crystal field splitting of the ground state is studied by inelastic neutron scattering.

We have also attempted to phenomenologically calculate the transport properties from the

neutron quasielastic line width. The thesis is organized as follows. Chpater 2 deals with

the basic concepts of neutron scattering. Starting from the derivation of the "master for-

mula" we have derived the double differential cross-section for the magnetic scattering arPd

18 •

the magnetic inelastic neutron scattering. We have then applied this expression to under-

stand various phenomena like crystalline electric field, spin fluctuations, excitation of the 4f

electron to conduction band etc. In the last section we have described two phenomenological

models that employ neutron linewidths to calculate various transport properties of cerium

based Kondo systems.

In Chapter 3 a brief description of the apparatus and the underlying principle of this

apparatus, used for measuring the various magnetic and transport properties and the Triple

Axis Spectrometer used to record the inelastic neutron spectra are given.

The method of sample preparation, CeSi 2 _,,Ga„, and the characterization of these samples

by X-ray diffraction (XRD) are presented in detail in this Chapter. The XRD results show

that the samples upto x = 1.3 are tetragonal. The lattice constants show very little or no

variation in the region 0.7 < x < 1.3. From x = 1.4 upwards the compounds have hexagonal

structure and the lattice constants show a systematic increase with x.

The results of magnetization (2 K - 300 K) and heat capacity (2 K - 300 K) measurements

are presented in Chapter 4. All the three tetragonal compounds, that is x = 0.7, 1.0

and 1.3, order ferromagnetically at around 10 K or below. The analysis of susceptibility

curve shows that cerium in all these compounds is in trivalent state. The ferromagnetic

ordering temperature Tc, first increases as the concentration is varied from x = 0.7 to x

= 0.9 and then decreases rapidly as x is further increased to 1.3. The variation of Tc and

the Kondo temperature TK indicates that there is a strong competition between magnetic

ordering and Kondo effect. The heat capacity plots also show a Schottky anomaly near

the ordering temperature. The magnetic entropy suggests that the ground state is split by

crystalline electric field and it is a doublet. The relationship between various parameters

like magnetic moment (t), coefficient of electronic specific heat (y) etc. derived from both

the measurements contribute to our understanding of hybridization effects involved in this

system.

Volume and electronic concentration effects have been shown to affect the behaviour

of different parameters like, magnetic ordering temperature, valence of Ce ion, etc. with

concentration in different ways. In the case of CeSi 2 _„Gax , Ga being a larger ion than Si

as well as it has a different outer electronic configuration than that of Si both the effects

should be present simultaneously. In the tetragonal compounds, as mentioned above, the

cell volume shows very little or no variation with Ga concentration but the variation in Tc

with x, which shows a peak, is indicative of strong volume effects. While in the hexagonal

compounds the cell volume increases with x indicating the presence of volume effects, but

the behaviour of magnetic ordering temperature, To , with x, which is a smooth line hints

at electronic concentration effect. Possible reason for such a behaviour is discussed in this

Chapter.

Neutron inelastic measurements were performed on a few compounds belonging to the

series CeSi 2 _„Gax . In order to get familiarized with the technique itself and the data analysis

procedures we first studied a well characterized compound, CeSn 2In. Our results of this

19 0

measurement and the details of data analysis procedure especially the phonon subtraction

method are described in the first section of Chapter 5. The magnetic neutron spectra of

CeSn2 In shows a single inelastic peak at about 8 meV with a width of about 7meV. This has

been analyzed using the Kuramoto Milller-Hartmann analytical expression. In this system

all the 4f electrons with energy greater than a characteristic energy, EF get an additional

channel to decay into the conduction band coupled with magnetic excitation.

In Section II of this Chapter we present results of our investigation on the compounds

belonging to the series CeSi2,Ga.. The spectra were recorded for three compounds, viz, x =

0.7, 1.0 and 1.3 at different temperatures from 10 K to 100 K using a close cycle refrigerator.

All these spectra show both quasielastic and inelastic peaks. The variation of quasielastic

linewidth with temperature in all the three compounds has been studied. These results

again indicate a strong competition between Kondo and RKKY interactions. The cross-

over from RKKY dominating region to Kondo dominating region is due to the increase of

the k-f coupling constant with increasing x over its critical value above which the intra-site

Kondo interactions dominate the intersite RKKY interactions. The inelastic spectra have

been analyzed using the crystal field Hamiltonian for Ce ion in tetragonal point symmetry

and the CEF parameters are calculated.

Chapter 6 deals with the measurement of transport properties. We have measured

resistivity (4 K - 300 K). The resistivity curves show the anomalies around Tc in the case of tetragonal compounds. The resistivity behaviour changes as the concentration is varied

from 0.7 to 1.0. In the case of hexagonal compounds the resistivity curves are similar to that of CeGa2 probably due to strong CEF effects.

We have also calculated resistivity using our data of neutron scattering line widths in

a phenomenological manner. The results shed some light on the nature of hybridization in this system.

In Chapter 7 a short summary of this study and suggestions for further work to be

carried out on such systems at very low temperatures and with better experimental technique are presented.

•

20 •

•

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