single crystal growth and magnetic properties of rrhin5 compounds

Most of these compounds order antiferronmagnetically at lowtempertures, except for R= Y, La, Pr and Yb.crys-The observed temperature dependence of the specific heat and the anisotropic tu

Single Crystal Growth and

Magnetic Properties of RRhIn5 Compounds

( R: Rare Earths )

Nguyen Van Hieu

Department of Physics, Graduate School of Science

Osaka University, Japan

January, 2007

A series of ternary compounds RRhIn5 (R: rare earths) was grown in the single talline form by means of the flux method Magnetic properties of these compounds wereinvestigated by measuring the lattice parameter, electrical resistivity, specific heat, mag-netic susceptibility and magnetization All the compounds crystallize in the tetragonalHoCoGa5-type structure Most of these compounds order antiferronmagnetically at lowtempertures, except for R= Y, La, Pr and Yb.

crys-The observed temperature dependence of the specific heat and the anisotropic tures in the magnetic susceptibility and magnetization were analyzed on the basis of thecrystalline electric field (CEF) model It is suggested that the overall splitting energy ofRRhIn5 in the CEF scheme decreases as a function of 4f -electron number from 330K in

fea-CeRhIn5 to 44K in ErRhIn5, which might be correlated with the c/a value in the lattice

constant

The antiferromagnetic easy-axis corresponds to the [001] direction in RRhIn5 (R=

Nd, Tb, Dy and Ho), while it is in the (001) plane for R= Ce, Sm, Er and Tm It is

noticed that this might be related to the sign of B0

2 in the CEF parameters

For the former compounds, we observed characteristic metamagnetic transitions low a N´eel temperature TN= 11.6K, the magnetization of NdRhIn5 reveals two metamag-

Be-netic transitions at Hm1= 70 kOe and Hm2= 93 kOe for the magnetic field along the [001]

direction The saturation moment of 2.5 µB/Nd is in good agreement with the staggered

Nd moment determined by the neutron diffraction experiment These metamagnetictransitions correspond to the change of the magnetic structure TbRhIn5, DyRhIn5 andHoRhIn5 are found to be the similar antiferromagnets with TN = 47.3, 28.1 and 15.8 K,respectively The magnetization curves of these compounds are also quite similar to those

of NdRhIn5, revealing two metamagnetic transitions The magnetic structures in netic fields are proposed by considering the exchange interactions based on the crystalstructure

mag-Furthermore, we observed the de Hass-van-Alphen (dHvA) oscillation of PrCoIn5,PrRhIn5 and PrIrIn5 to clarify the Fermi surface properties The detected topology ofthe Fermi surface is found to be the same as that of LaRhIn5, consiting of two kinds

of corrugated cylindrial (bands 14 and band 15) Fermi surfaces and a lattice-like Fermi

surface We detected an inner orbit named ϵ1 in the band 13-lattice-like hole-Fermisurface of PrIrIn5, which was not observed previously in LaRhIn5 This is mainly due tothe high-quality single crystal sample of PrIrIn5

1 Introduction 4

2.1 Magnetic properties of rare earth ions and metals 6

2.2 Crystalline electric field (CEF) effect 14

2.3 Kondo effect and heavy fermions 24

2.4 Magnetic properties of RIn3 and RRhIn5 compounds 27

3 Motivation of the Present Study 43 4 Single Crystal Growth and Measurement Methods 44 4.1 Single crystal growth 44

4.2 Measurement methods 51

4.2.1 Electrical resistivity 51

4.2.2 Specific heat 53

4.2.3 Magnetic susceptibility and magnetization 55

4.2.4 High field magnetization 57

4.2.5 de Haas-van Alphen effect 59

4.2.6 Neutron scattering 66

5 Experimental Results and Discussion 73 5.1 Magnetic properties and CEF scheme in RRhIn5 73

5.2 Fermi surface and magnetic properties of PrTIn5 (T: Co, Rh and Ir) 98

5.3 Unique magnetic properties of RRhIn5 (R: Nd, Tb, Dy, Ho) 120

5.4 Neutron scattering study in RRhIn5 (R: Nd, Dy , Ho) 137

3

The rare earth compounds indicate a variety of electronic states such as magneticordering, quadrupole (multipole) ordering, charge ordering, heavy fermions, Kondo in-sulators and anisotropic susperconductivity These phenomena are closely related to

hybridization of almost localized 4f electrons with the conduction electrons The 4f electrons in the rare earth atom are pushed deeply into the interior of closed 5s and 5p shells This is a reason why the 4f electrons possess an atomic-like character even

in the compounds On the other hand, the tail of their wave function spreads to the

outside of the closed 5s and 5p shells, which is highly influenced by the potential energy,

the relativistic effect and the distance between the lathanide atoms This results in the

hybridization of the 4f electrons with the conduction electrons These cause the various

phenomena mentioned above

Recently, the family of rare earth 115 compounds with the HoCoGa5-type tetragonalcrystal structure1 attracts strongly interests in the field of condensed master physics,after the dicovery of heavy fermion superconductivity in CeTIn5 (T: Co, Rh, Ir)2–4 withthe quasi-two dimentional electronic state CeCoIn5 and CeIrIn5 are superconductors

at ambient pressure, with the superconducting transition temperature Tsc=2.3 K and0.4K, respectively On the other hand, CeRhIn5 indicates an antiferromagnetic orderingwith the N´eel temperature TN= 3.8K but becomes superconductive above 1.6GPa Theuniaxially distorted AuCu3-type layers of RIn3and RhIn2layers in RRhIn5(R: rare earths)

are stacked sequentially along the [001] direction (c-axis) The Fermi surface properties of

LaRhIn5 and CeRhIn55, 6 were studied via the de Haas-van Alphen(dHvA) experiments

The Fermi surface of a non-4f reference compound LaRhIn5 is quasi-two dimensional,reflecting the unique tetragonal structure The topology of the Fermi surface in CeRhIn5

is similar to that of LaRhIn5, but the cyclotron mass in CeRhIn5 is larger than that ofLaRhIn5

These findings motivated us to undertake more investigations of the magnetic erties of RRhIn5 series in the single crystal form Namely, the present study is to clarify

prop-the fundermental magnetic properties of localized 4f -electrons, such as prop-the crystalline

electric field (CEF) scheme, together with the magnetic exchange interactions between

the 4f electrons in rare earth atoms (ions) via the conduction electrons.

The single crystals of RRhIn5 were grown by the self-flux method using In as flux.Structural parameters of RRhIn5 were determined by the single-crystal x-ray diffraction

experiments with the Mo-Kα radiation The electrical resistivity was measured by the

4-probe DC method The magnetic susceptibility and magnetization measurements werecarried out with a commercial SQUID magnetometer The specific heat was measured

by the quasi-adiabatic heat-pulse method and commercial PPMS The high-field netization was also measured by the standard pick-up coil method, using a long-pulse

mag-magnet We also measured the dHvA oscillation using a so-called 2ω detection of the

field modulation method From the results of these measurements, we clarified the netic properties of RRhIn5 series

mag-The present thesis consists of the following contents In Chap 2, the fundamental

4

2.1 Magnetic properties of rare earth ions and

met-als

First we will explain the magnetic properties of rare earth atoms (ions) Rare earth(R) atoms include 15 elements of lanthanide series, scandium (Sc) and yttrium (Y) La,

Ce, Pr, Nd,(Pm), Sm and Eu are called the light rare earths We also put the name

of heavy rare earths for Gd, Tb, Dy, Ho, Er Tm and Yb The magnetic properties

change systematically and regularly because of the 4f -electronic configuration : Xe shell 4f n 5s25p66s2 The atomic radius of R3+shrinks monotonically from cerium to ytterbium,

as shown in Fig 2.1 This is well known as ”lanthanide contraction” The 4f electron

in the Ce atom is, for example, pushed deeply into the interior of the closed 5s and 5p shells because of the strong centrifugal potential l (l + 1)/r2, where l = 3 holds for the

f electron This is a reason why the 4f electrons possess an atomic-like character in

the crystal (rare earth metal and rare earth compound).7 Figure 2.2 shows the radial

wave function of Ce (4f15d16s2) with and without the relativistic effect On the other

hand, the tail of their wave function spreads to the outside of the closed 5s and 5p shells,

which is highly influenced by the potential energy, the relativistic effect and the distance

between the lanthanide atoms This results in the hybridization of the 4f electrons with

the conduction electrons This causes the various phenomena such as RKKY Kittel-Kasuya-Yosida interaction)9–11 and Kondo effect 12 Under the Hund rule, the

(Ruderman-1.3 1.2 1.1 1.0 0.9

∆ 1

∆ 2

spin-orbit interaction

CEF

Fig 2.3 Level scheme of the 4f electron in Ce3+

strong spin-orbit coupling of the 4f electrons in rare earth ion leads to a low magnetic moment for the light rare earths with J = |L − S|, while J = L + S for the heavy rare

earths Here, J is the total angular momentum, L is the total orbital angular momentum and S is the total spin momentum Namely, the 4f multiplets, which obey the Hund rule in the LS-multiplets, split into the J -multiplets (J = 52 and J = 72 in Ce3+) by the

spin-orbit interaction, as shown in Fig 2.3 Moreover, the J -multipltes split into the 4f

levels based on the crystalline electric field (CEF) effect We also show in Table 2.I andFig 2.4 the electronic configuration and the fundermental magnetic parameters in the

rare earth ions The 4f electrons possess an atomic-like character even in the rare earth

metals and the rare earth compounds

Next we describe the magnetic properties of the rare earth metals The crystal andmagnetic structures are very complex in rare earth metals, as shown in Table 2.II andFig 2.5 The double hexagonal close packed (dhcp) crystal structure is typical, possessingboth the cubic symmetry sites and hexagonal symmetry sites

Table 2.I Electronic configuration of 4f shell and general magnetic parameters of rare

earth ion: spin moment (S), orbital moment (L), total moment J , spectroscopy state, Lande factor g, gJ , effective magnetic moment of free ion µeff=g√

The helical and cone-like helical structures are also typical in the magnetic structure.

As show in Fig 2.5, Eu is magnetic, meaning that the valence is not trivalent, but

divalent: Eu(4f75s24p66s2) Yb is also not trivalent, but divalent, indicating the magnetic property The ordering temperture is shown in Fig 2.6

non-Table 2.II Crystal structure, lattice constant and easy axis at 4.2K in rare earth metals.

R Z crys.struct a(˚ A) c(˚ A) easy axis (4.2K)

Fig 2.5 Magnetic structures in rare earth metals: (a) helical, (b) cone-like helical, (c)

modulated along the c-axis and (d) helical structure in Er.

Rare earth metals

Fig 2.6 Magnetic properties of rare earth metals AF: antiferromagnetism, CH:

cone-like helical structure, CM: modulated structure along the c-axis, F: ferromagnetism, FR:

ferrimagnetism, P: paramagnetism and HE: helical structure

Here, the magnetic ordering in rare earth metals including the rare earth compounds

is mainly based on the RKKY interaction We pay attention on the Hamiltonian of

exchange interaction H ex between the total spin S of the 4f electrons and the spin s of

conduction electrons:

where J cf is the magnitude of the exchange interaction In the indirect exchange model,

the 4f electron spin S i at Ri interacts locally with the spin of the conduction electrons,

which then interact in turn with the 4f electron spin S j at Rj This approach is needed

because the 4f wave functions have insufficient overlap to give a direct Heisenberg

ex-change The exchange interaction between S i and S j is thus expressed as follows:

-J (R ij )S i · S i,

where J (R ij ) contains a so-called Friedel oscillation of F (x)= (x cos x - sin x)/x4, as

shown in Fig 2.7 Here, R ij =R j -R i = x In other words, the mutual magnetic interaction between the 4f electrons occupying different atomic sites cannot be of a direct type such as 3d metal magnetism, but should be indirect one, which occurs only through the

conduction electrons

When the number of 4f electrons increases in such a way that the lanthanide

ele-ment changes from Ce to Gd or reversely from Yb to Gd in the compound, the magneticmoment becomes larger and the RKKY interaction stronger, leading to the magnetic

ordering At this point, it may be called that the total angular momentum J is a good quantum number and is better to use the projection of spin on J , as suggested by de Gennes We replace S with J : S i = (g − 1)J i The following equation can be obtained:

-J (Rij)(g − 1)2 J i J j

The magnetic ordering temperature Tord is thus propotional the de Gennes factor

(g − 1)2J (J + 1) The effective magnetic moment is closed to µeff=g√

J (J + 1) [µB], as

shown in Fig 2.8 and the ordered moment is also close to gJ [µB], as shown in Fig 2.9

Fig 2.7 Spatial variation of the RKKY interaction F (x)=(x cos x-sin x)/x4 The

arrows indicate the position of several localized moments at a distance x ( =R ij ) fromthe central ion

Theory Experiment

Rare earth metals

Fig 2.9 Ordered moment in the rare earth metals and the corresponding theoretical

one in R3+

2.2 Crystalline electric field (CEF) effect

We memtioned in Sec 2.1 that the LS multiplet based on the Hund rule splits into the J multiplets due to the spin-orbit interaction Moreover, the J -multiplets split into 4f -levels due to the crystalline electric field (CEF) On basic of a simple point-charge ionic model, we will consider the electrostatic potential V (r) due to the surrounding

negative ions and construct the CEF scheme

V (r) can be expressed as follows:

i

q i

where r is the positional vector of the f electron, q i is the charge of the six-coordinated

negative ions and R i is its positional vector We consider the following configuration

where the negative ions of the charge q is located on the corners of an octahedron: (a, 0, 0) , ( −a, 0, 0) , (0, a, 0),(0, −a, 0) and (0, 0, a) , (0, 0, −a) in Fig 2.10 Then

Fig 2.10 Six-coordinated negative ions and the 4f electron at the point P(x,y,z).

Insert V x , V y and V z into eq (2.3) and expand by the Taylor expansion, and we get thefollowing equation:

where D4 and D6 are D4 = 35q/4a5 and D6 = −21q/2a7 Next, we consider the charge

distribition of the f electron ρ(r) The static potential energy can be expressed as follows:

∫

V (r) can be expressed by the angular momentum operator based on the

Wigner-Eckart’s theorem in quantum mechanics For example:

Next, for an easy example, we will discuss the CEF scheme of Ce3+ with the cubic

symmetry There is only one electron in the 4f -shell Therefore, the orbital angular momentum, spin momentum and total angular momentum are L=3, S=1/2 and J =5/2, respectively, and the magnetic angular momentum : m = 52,32,12,−1

2,−3

2,−5

2,

with the Lande factor g = 1 + J (J +1)+S(S+1) 2J (J +1) −L(L+1) = 67 Therefore, the multiplet J = 5/2

is degenerated by sixfold of 2J + 1 = 6, and this sixfolded degenerate state splits into the 4f levels by the CEF effect For J z = 5/2, O0

where J ± = J x ± iJ y The operator O m

n can be expressed by (6× 6)-matrix Therefore,

the CEF Hamiltonian of the cubic Ce3+ is expressed as follows:

The wave function |i⟩ and its enery E i at the CEF-4f level are expressed:

By using eq (2.10), we obtain the 4f state |i⟩ and its energy E i as follows:

|Γ α

7⟩ = √1

6¯¯5 2

⟩

6¯¯−3 2

⟩

|Γ β

7⟩ = √1

6¯¯−5 2

⟩

6¯¯3 2

⟩+√1

6¯¯−3 2

⟩

|Γ κ

8⟩ = √5

6¯¯−5 2

⟩+√1

6¯¯3 2

The energy level of −240B0

4 is named Γ7 and the energy state 120B0

4 is named Γ8 Thesplitting energy between Γ7 and Γ8 is 360B0

4

We show in Fig 2.11 the charge distribution of every states The quartet Γ8 wave

function expands along the x, y, z directions On the other hand, the doublet Γ7 expandsalong the ⟨111⟩ direction so as to avoid these principal directions If the negative ions

approach to the cerium ion along the principal directions, the Coulomb energy of the 4f

electron is perferable to the Γ7 ground state, compared to the Γ8 ground state, indicatingthat the Γ8 state becomes an excited state In general, the CEF Hamiltonian for thelanthanide ions can be expressed as follows:

n,m

If the number of the f electron is odd, namely, J has the semi-integer: Ce3+, Nd3+,

Sm3+, Dy3+, Er3+ and Yb3+, the 4f energy level always have the doublet (called Kramers

dou-genrate 4f -states, including the Kramers doublet, split into singlets.

We can obtain the mangetic moment of the f electron by measuring the magnetic

susceptibility or magnetization under magnetic field The Hamiltonian under magneticfield is as follows:

where the second term corresponds to the Zeeman energy Consider 4f -energy state,

where |i⟩ is the 4f state of the level state i, E i is the eigenvalue and µ i is the magneticmoment of the energy level When applying the magnetic field, the energy state of theevery level scheme is influence by the other energy scheme We represent this state as

¯¯˜i⟩

and E i (H) as its eigenvalue Namely, we calculate the every states under magnetic

field ¯¯˜i⟩

and E i (H), diagonalizing the matrix of the Hamiltonian shown in eq (2.15).

We calculate the magnetization and the magnetic susceptibility by using ¯¯˜i⟩

and E i (H).

Here, the Helmholtz free energy F can be expressed by the partion function Z as follows:

Namely, the magnetization M correspond to the average ⟨µ zi ⟩ of the magnetic moment

µ zi The magnetic susceptibility χ is the difference of magnetization ∂M/∂H(H → 0) :

The expression eq (2.22a) is the general expression of the magnetic susceptibilityunder consideration for the CEF, and another expression is often used:

The first term is the Curie term which can be determined by the diagonal element of

the matrix J z and the second term is related to the non-diagonal element Namely, it isthe Van-Vleck term, which is related to the transition between the states It is knownfrom eq (2.22) that the magnetic susceptibility can be determined from the states of the

f electron without magnetic field and its energy eigenvalue For example, we calculate

the J z for the cubic Ce3+ The J z matrix element can be expressed as follows:

gJ = 6/7 The summation over the two degenerated states of the Γ7is zero The magneticmoments for |Γ ν

8⟩, |Γ κ

8⟩, ¯¯Γλ

8

⟩and |Γ µ

8⟩ are 11/7 µB, −11/7 µB , 3/7 µB and −3/7 µB,respectively Eq (2.22) can be expressed (Γ7 is the ground state, Γ8 is the excited state

sus-Ce 3+ cubic without CEF

Magnetic Field ( kOe )

(b)

0 1 2

Γ8 ground state with ∆ = 200 K No CEF corresponds to ∆ → 0: χ z = 354 (gJµB)2/3kBT

Furthermore, the case, which is ∆ → 0, is equivlent to the expression kBT ≫ ∆ and

ap-proaches the Curie law, which ignores the CEF effect at high temperatures When Γ7

is the ground state, the magnetic moment is 0.7 − 0.8 µB On the other hand, it is

1.7 − 1.8 µB when Γ8 becomes the ground state If the Zeeman energy of the magneticfield is larger than the CEF splitting energy, the magnetization becomes the saturated

magnetic moment gJ

Next we show the CEF scheme of Ce3+ in the tetragonal symmetry We calculatethe splitting energy, magnetic susceptibility and magnetization curves based on the CEFeffect Using again eq (2.14) for the tetragonal symmetry, we can write the CEF Hamil-tonian as follows:

HCEF = B20O20+ B40O40+ B44O44+ B60O60+ B66O66. (2.25)

The total orbital angular momentum, total spin momentum and total angular

mo-mentum are L=3, S=1/2 and J =5/2, respectively, and the magnetic angular momo-mentum:

m = 52, 32, 12, −1

2,−3

2,−5

2 Therefore, the multiplet which has J = 5/2 degenerate sixfold

of 2J + 1 = 6 and this sixfold degenerate state splits into three doublets by the CEF

effect In the case of J = 5/2, O0

We calculated the energy values with three-doublets: E1 = E2 = 0, E2 = E3 =

68.3494 and E5 = E6 = 325.175 We also get the wave functions as follows:

The J z matrix element can be expressed as follows:

doublet

excited state 1

∆ 2 = 325 K

∆ 1 = 68 K excited state 2

ground state

Fig 2.13 Splitting energies with three doublets in Ce+3 teragonal structure under CEFeffect: the doublet Γ7 ground state, the first excited state double Γ8 and the secondexcited state double Γ9

The magnetic susceptibility and magnetization under magnetic field also can be calculated

by eq (2.22) The Hamiltonian is:

Magnetic Field ( kOe )

T = 1.3K

4.2K 10K

86420

1000

5000

Magnetic Field ( kOe )

1000

Temperature ( K)

543210

2.3 Kondo effect and heavy fermions

The Kondo effect was studied for the first time in a dilute alloy where a ppm range

of the 3d transition metal was dissolved in a pure metal of copper Kondo showed the

transtion impurity diverges logarithmically with decreasing temperture, and clarified theorigin of the long standing problem of the minimum resistivity This is the start of theKondo problem, and it took ten years for theorists to solve this divergence problem.The many-body Kondo bound state is now understood as follows: For the simplest

case of no orbital degeneracy, the localized spin S( ↑) is coupled antiferromagnetically with

the conduction electrons s( ↓) Consequently, the singlet state {S(↑) · s(↓) ± S(↓) · s(↑)}

is formed with the binding energy kBTK Here the Kondo temperature TK is the singleenergy scale In other words, disappearance of the localized moment is thought to bedue to the formation of a spin-compensating cloud of the electrons around the impuritymoment

Kondo-like behavior was observed in lanthanide compounds, typically in Ce and Ybcompounds.21–23 For example, the electrical resistivity in CexLa1−xCu6 increases loga-

rithmically with decreasing temperature for all the x-values,24as shown in Fig 2.15 TheKondo effect occurs independently at each Ce cite even in a dense system Therefore,this phenomenon was called the dense Kondo effect The Kondo temperature in the Ce

Fig 2.15 Temperature dependence of the electrical resistivity in CexLa1−xCu6.24

(or Yb) compounds is largely compared to the magnetic ordering temperature based on

the RKKY interaction For example, the Ce ion is trivalent (J = 52) and the 4f energy

level splits into the three doublets by the crystalline electric field (CEF) effect, namelypossessing the splitting energy of ∆1 and ∆2.25

The Kondo temperature is given as follows:26

Here D, |J cf | and D(EF) are the band width, the magnetic exchange interaction and

the density of states at the Fermi energy EF, respectively If we assume TK ≃ 5 K,

for D = 104 K, ∆1 = 100 K and ∆2 = 200 K, the value of TKh ≃ 50 K is

ob-tained, which is compared to the S = 12 - Kondo temperature of 10−3 K defined as

TK0 = D exp( −1/|J cf |D(EF)) These large values of the Kondo temperature shown in

eqs (2.36) and (2.37) are due to the orbital degeneracy of the 4f levels Therefore, even

at low temperatures the Kondo temperature is not T0

K but TK shown in eq (2.37)

On the other hand, the magnetic ordering temperature is about 5 K in the Ce pounds, which can be simply estimated from the de Gennes relation under the consid-eration of the Curie temperature of about 300 K in Gd Therefore, it depends on thecompound whether magnetic ordering occurs at low temperatures The ground stateproperties of the dense Kondo system are interesting in magnetism, which are highlydifferent from the dilute Kondo system In the cerium intermetallic compounds such asCeCu6, cerium ions are periodically aligned whose ground state cannot be a scattering

com-state but becomes a coherent Kondo-lattice com-state The electrical resistivity ρ decreases steeply with decreasing the temperature, following a Fermi liquid behavior as ρ ∼ AT2

with a large value of the coefficient A.27

The√

A value is proportional to the effective mass of the carrier m ∗and thus inverselyproportional to the Kondo temperature Correspondingly, the electronic specific heat

coefficient γ roughly follows the simple relation γ ∼ 104/TK (mJ/K2·mol) because the

Kramers doublet of the 4f levels is changed into the γ value in the Ce compound:

Therefore the Kondo-lattice system is called a heavy fermion or heavy electron system.

The Ce Kondo-lattice compound with magnetic ordering also possesses the large γ value

even if the RKKY interaction overcomes the Kondo effect at low temperatures For

example, the γ value of CeB6 is 250 mJ/K2·mol,31 which is roughly one hundred times

as large as that of LaB6, 2.6 mJ/K2·mol.32

In the 4f -localized system, the Fermi surface is similar to that of corresponding La compound, but the presence of the 4f electrons alters the Fermi surface through the 4f -electron contribution to the crystal potential and through the introduction of new Brillouin zone boundaries and magnetic energy gaps which occur when 4f -electron mo-

ments order The latter effect may be approximated by a band-folding procedure wherethe paramagnetic Fermi surface is folded into smaller Brillouin zone based on the magneticunit cell, which is larger than the chemical one If the magnetic energy gaps associatedwith the magnetic structure are small enough, conduction electrons undergoing cyclotronmotion in the presence of magnetic field can tunnel through these gaps and circulate theorbit on the paramagnetic Fermi surface If this magnetic breakthrough (breakdown)effect occurs, the paramagnetic Fermi surface may be observed in the dHvA effect even

in the presence of magnetic order For Kondo-lattice compounds with magnetic orderingsuch as CeB6, the Kondo effect is expected to have minor influence on the topology ofthe Fermi surface, representing that the Fermi surfaces of the Ce compounds are roughlysimilar to those of the corresponding La compounds, but are altered by the magneticBrillouin zone boundaries mentioned above Nevertheless the effective masses of the con-duction carriers are extremely large compared to those of La compounds In this system

a small amount of the 4f electron most likely contributes to make a sharp density of

states at the Fermi energy Thus, the energy band becomes flat around the Fermi energy,which brings about the large mass

In some Ce compounds such as CeCu6, CeRu2Si2, CeNi and CeSn3, the magneticsusceptibility follows the Curie-Weiss law with a moment of Ce3+, 2.54µB/Ce, has a

maximum at a characteristic temperature T χmax, and becomes constant at lower

temper-atures This characteristic temperature T χmax corresponds to the Kondo temperature TK

A characteristic peak in the susceptibility is a crossover from the localized-4f electron to

the itinerant one The Fermi surface is highly different from that of the corresponding

La compound The cyclotron mass is also extremely large, reflecting a large γ-value of

γ ≃ 104/TK (mJ/K2·mol) The 4f electron in these compounds without magnetic

order-ing is clarified to be itinerant at low temperature from the dHvA experiments and energyband calculations

2.4 Magnetic properties of RIn3 and RRhIn5

com-pounds

Rare earth intermetallic compounds RIn3 (R: rare earths) crystallizes in the AuCu3type cubic structure Figure 2.16 shows the crystal structure of RX3where the corner-sitesare occupied by the R atoms and the face-centered sites are occupied by the X atoms

-First we describe the localized 4f -electron system with antiferromagnetic ordering This

ordering is caused by the so-called RKKY interaction Namely, a localized spin S i of

the 4f -electrons at the site i interacts with a conduction electron with spin s, which

leads to a spin polarization of the conduction electron This polarization interacts with

another spin S j at the site j and therefore creates an indirect interaction between the

spins S i and S j This indirect interaction extends to the far distance and damps with

a sinusoidal 2kF oscillation, where kF is half of the caliper dimension of the Fermi surface.The metamagnetic transition, which is based on the spin-flip mechanism, is oftenobserved in the antiferromagnets The antiferromagnetic state at zero field is changedinto the field-induced ferromagnetic or paramagnetic state at high magnetic fields It

is possible to investigate a changes of magnetic properties by increasing the number of

4f electrons in the series of RIn3 The magnetic susceptibility of RIn3 compounds were

measured between 4.2K and 500K with magnetic field up to 30kOe by Buschow et al.34

AuCu -type cubic structure3

In

R

Fig 2.16 AuCu3-type cubic structure of RIn3 compounds Larger spheres withoutpattern and small spheres with pattern show the rare earth atoms and indium atoms,respectively

Almost of the RIn3 compounds, exception of R= La, Sm and Yb, Curie-Weiss havior is observed in the high temperature region At low temperatures, the magneticbehavior for compounds in RIn3 (R=Ce, Nd, Sm, Gd, Tb, Dy, Ho , Er and Tm) is

be-ascribed to antiferromagnetic ordering Gorlich et al 35 reported that EuIn3 orders

anti-ferromagnetically below TN= 10K, but this compound does not exist in a binary phasediagram

The N´eel temperature,TN is observed, indicating onset of antiferromagnetic ordering.PrIn3 possesses no magnetic ordering, with the singlet CEF ground state in the CEFscheme In SmIn3, the magnetic susceptibility does not indicate the simple Curie-Weiss

law, where the J multiplets of J =52 and J =72 are considered because the splitting energy

between J =5

2 and J =7

2 is not large The magnetic susceptibility contains a independent Van Vleck-type paramagnetic susceptibility In the case of YbIn3, Yb isdivalent, possessing no magnetic ordering Lattice contants, effective moments, N´eeltemperature and other parameters are shown in Table 2.III

temperature-50403020100

( b )

4.804.754.704.654.604.554.50

Table 2.III Lattice contants, effective moments, N´eel temperature TN and

paramag-netic Curie temperature θ p of RIn3 compounds34

Specific heat study on RIn3 were carried out by Satoh et al.,36 as shown in Fig 2.18.

The obtained γ values of magnetic RIn3 are found to be larger than that of LaIn3 Thisenhancemant is due to the electron-magnon correlation and especially due to Kondo effect

in CeIn3 Both the nuclear specific heat(CN) from In nuclei and rare earth nuclei possesseffective magnetic fields and electronic field gradients In LaIn3 and PrIn3, CN is due tothe quadrupole splitting of In nuclear levels.38 Moreover, the value of other RIn3 (R= Nd,

Sm, Tb and Ho) show the large CN that originates from Beff at rare earth sites GdIn3

does not have the orbital angular momentum Hence, it does not show a large CN

Fig 2.18 Specific heat of RIn3 between 0.2 and 2.2K by Satoh et al36 : (a) a conducting transition at 0.77K in LaIn3, with the singlet ground state of CEF scheme inPrIn3 and antiferromagnetic dense Kondo compound in CeIn3,(b) the least-squares fitswith C = A/T2+γ T+ DT3 ln (−∆ /T) +βT3 at low temperatures for NdIn3 and SmIn3,

super-(c) shows plot of a C/T versus T 1/2in GdIn3 and (d) indicate the large CN in HoIn3,TbIn3

The Fermi surfaces of RIn3 are found to be almost that same as those of LaIn3 by deHass-van Alphen measurements.39–44

In Fig 2.19(a), dHvA branches a, d and j are shown in LaIn3 The same dHvAdata are obtained in PrIn3 with a singlet ground state Fig 2.19(b) The branch d-Fermi

surface in CeIn3 is roughly spherical, possessing no arms (Fig 2.19(c)) On the other

hand, branches a, d and j are also found in the antiferromagnetic state of NdIn3 but the

dHvA amplitude of branch is strongly damped (Fig 2.19(d)) The similar branches a and j are obtained in the antiferromagnetic state of SmIn3 and GdIn3 but not appear

in TbIn3 in Fig 2.19(g,h) Figure 2.19(e) shows hole and electron Fermi surfaces inLaIn3.45–47 The stippled regions indicate the cross-sections of the hole Fermi surface inthe (100) and (110) planes

(a)

(h)

Fig 2.19 Angular dependence of the dHvA frequency in RIn3 (R= La, Ce, Pr, Nd, Gd,

Sm and Tb) indicate the change of the Fermi surface.37 Hole and electron Fermi surface

in LaIn3 are shown in (e)

We present here a change of the electronic state, namely, the Fermi surface in NdIn3viathe dHvA experiment.48 NdIn3with the cubic crystal structure is an antiferromagnet with

a N´eel temperature TN = 5.9 K The magnetic structure in the antiferromagnetic state,determined by neutron scattering experiments,49 is shown in Fig 2.20 The magneticstructure is not the same as the original cubic structure, but is tetragonal

Figure 2.21 shows the dHvA oscillation for the field along the ⟨100⟩ direction and the

corresponding magnetization curve obtained by Czopnik et al.50 Three metamagnetictransitions at 7.82, 8.85 and 11.14 T, which are indicated by the vertical broken lines,are clearly reflected in both data With increasing field, the compound goes from anantiferromagnetic to a paramagnetic state, passing through an intermediate-magneticstate We show in Fig 2.22 the corresponding fast Fourier transform (FFT) spectra

in the antiferromagnetic (AF), intermediate-magnetic and paramagnetic states Here

we note ”branch a” with the dHvA frequency of 7.9 × 103 T The dHvA frequency F

is proportional to the extremal (maximum or minimum) cross-sectional area SF of theFermi surface

The dHvA oscillation due to ”branch a” possesses a substantial amplitude in eachstate The dHvA frequency is degenerated corresponding to up- and down-spin states

in the antiferromagnetic state This is almost the same as in the intermediate-magneticstate, but it is split into the up- and down-spin states in the paramagnetic state, asshown in Fig 2.23 The spin splitting of the dHvA frequency is clearly seen in the secondharmonic in Fig 2.22

8 6

Magnetic Field (T)

NdIn3 H // <100>

1 2

Fig 2.21 Magnetization curve and dHvA oscillation of NdIn3

1510

50

dHvA Frequency (103 T)

AF (6 - 7.5 T)

Intermediate(9 - 10.5 T)

Para (12.5 - 14.5 T)

a (2nd)

a (2nd)

aNdIn3

H // <100>

0.45K

Fig 2.22 FFT spectra of NdIn3

Fig 2.23 Magnetic field dependence of the dHvA frequency in NdIn3.

Similar experiments were done for the field along the ⟨110⟩ direction Two

transi-tions are found at 7.31 and 9.31 T ”Branch a” is, however, extremely small in amplitude

in the antiferromagnetic and intermediate-magnetic states When crossing the secondmetamagnetic transition at 9.31 T, the oscillation due to ”branch a” becomes dominant.Figure 2.24 shows the angular dependence of the dHvA amplitude for ”branch a” in theantiferromagnetic state The amplitude is strongly reduced when the field is tilted byabout 10 ◦ from⟨100⟩ to ⟨110⟩ in the {100} and {110} planes.

We discuss ”branch a” on the basis of the topology of the Fermi surface of LaIn3 TheFermi surface of LaIn3 consists of a complicated hole Fermi surface and a nearly sphericalelectron surface ”Branch a” corresponds to the latter surface The Fermi surfaces inthe paramagnetic state of NdIn3 are expected to be similar to those of LaIn3 because the

4f -electrons in NdIn3 are localized as determined from the neutron measurements.49 Infact, the angular dependence of the dHvA frequency in the paramagnetic state of NdIn3

is almost the same as that of LaIn3.51 Antiferromagnetic order, however, changes thetopology of the Fermi surface because the size of the magnetic Brillouin zone is reduced.The magnetic unit cell is tetragonal, as mentioned above Therefore, the volume of themagnetic Brillouin zone becomes half of the chemical Brillouin zone, as shown in Fig 2.25

From the angular dependence of the dHvA frequency and amplitude for ”branch a”,

a nearly spherical electron Fermi surface in the paramagnetic state of NdIn3 or LaIn3, asshown in Fig 2.25(a), is changed into a multiply-connected Fermi surface with necks in theantiferromagnetic state, as shown in Fig 2.25(b) Note that no necks exist along the [100]

6.3 -7.3 T0.45K

<110> <100>

Fig 2.24 Angular dependence of the dHvA amplitude for ”branch a” in NdIn3

and [010] directions Therefore, the cubic symmetry is broken in the antiferromagneticstate

If we follow the Fermi surface shown in Fig 2.25(b), the ”branch a” oscillation isnot expected to be detectable for the field along ⟨110⟩, but is experimentally detected

even in the antiferromagnetic and intermediate-magnetic states, although its amplitude

is extremely small This is because an electron can circulate along a closed orbit bybreaking through the necks of the Fermi surface

The antiferromagnetic Fermi surface is thus reconstructed on the basis of the magneticBrillouin zone A nearly spherical Fermi surface in the paramagnetic state, corresponding

to a band 7-electron Fermi surface of LaIn3, is changed into a multiply-connected Fermisurface with necks in the antiferromagnetic state In other words, this antiferromagneticFermi surface is changed into the paramagnetic surface after the metamagnetic transition,

surface, denoted a, is modified in the antiferromagnetic state The cyclotron masses of

RIn3 are enhanced by twice or three times larger than those of LaIn3 by the magnon interaction

electron-2) RRhIn5 compounds

CeTIn5 (T: Co, Rh, Ir) and AcTGa5 (Ac: U, Np, Pu) are interesting compounds.CeCoIn5 and CeIrIn5 are well known as heavy fermion superconductors with the super-

conducting transition temperature Tsc = 2.3 K and 0.4 K, respectively.3 The electronic

specific heat coefficient γ in CeCoIn5 is 1070 mJ/K2·mol52 and an extremely large

cy-clotron mass of 83 m0 was detected in the de Haas-van Alphen (dHvA) experiment.76

On the other hand, CeRhIn5 orders antiferromagnetically below a N´eel temperature TN=

3.8 K but becomes superconductive under pressure, with Tsc = 2.2 K.2 ity in CeRhIn5 under pressure is closely related to the heavy fermion state, which wasrecently clarified by the dHvA experiment under pressure.54 UTG5 (T: Co, Rh, Ir) andNpTGa5 (T: Co, Rh) in the 5f -electron systems are, however, not superconductors but

Superconductiv-Pauli paramagnets and antiferromagnets, respectively On the other hand, PuCoGa5 andPuRhGa5 are relatively high-Tsc superconductors with Tsc = 18.5 K55 and 8.7 K,56 re-

spectively The d-wave superconductivity was recently clarified for both compounds.57, 58

A characteristic feature in these CeTIn5 and AcTGa5 compounds is quasi-two sionality in the electronic state, which is closely related to the unique crystal structure.The uniaxially distorted AuCu3-type layers of RIn3 and RhIn2 layers are stacked sequen-

dimen-tially along the [001] direction (c-axis), as shown in Fig 2.26.

diffrac-The topology of the Fermi surface of a non-4f reference compound LaRhIn5 is

approx-imately cylindrical, as shown in Fig 2.27 The 4d electrons of Rh in LaRhIn5 hybridize

with the 5p electrons of In, which results in a small density of states around the Fermi

energy.5 This means that there are very few conduction electrons in the RhIn2 layerand hence the Fermi surface mainly consists of two kinds of corrugated cylindrical Fermi

surfaces A relatively large Tsc value in CeTIn5 and PuTGa5 is closely related to this

quasi-two dimensionality, which is compared to a low Tsc value of Tsc = 0.2 K in a threedimensional compound CeIn3 under pressure.59

Here we note the magnetic structure of CeRhIn5 with TN= 3.8K.2 It was clarifiedfrom the neutron scattering experiment.61 The magnetic moment of about 0.8µ B/Ce62resides on the Ce ion The (001) plane is an easy plane, where magnetic moments form

a antiferromagnetic structure on a square lattice and spiral transversely along the [001]direction, indicating a helical spin structure, as shown in inset of Fig 2.28

Field Angle (Degrees) [110] [001] [100] [110]

0 dHvA Frequency (x108 Oe)

Magnetic Field ( kOe ) 200

400600

to a change from the helical spin structure to the fan structure The fan structure in

changed to the field-induced ferromagnetic (paramagnetic) state above H0(=500kOe).5

From the results of the magnetic susceptibility and magnetization, the CEF scheme isproposed, as shown Fig 2.30

The topology of the main Fermi surfaces in an antiferromagnet CeRhIn5 is in goodagreement with that of LaRhIn5, as shown in Fig 2.29, indicating that the 4f electrons

do not contribute to the volume of the Fermi surface The cyclotron mass in CeRhIn5

is, however, relatively large, ranging from 0.93 to 6.1 m0.5 The cyclotron mass of mainbranches in CeRhIn5is seven to nine times large than the corresponding cyclotron mass inLaRhIn5 This is approximately consistent with γ value: γ ≃ 50(mJ/K2·mol) in CeRhIn5

and 57(mJ/K2·mol) in LaRhIn5

300306090

Field Angle (Degrees)

Fig 2.29 (a) dHvA oscilation, (b) its FFT spectrum and (c) angular dependence of the

dHvA frequency in CeRhIn5.5

Fig 2.30 Temperature dependence of the magnetic susceptibility χ in CeRhIn5 The

inset shows T vs M/H under the field Solid lines express the CEF calculation.19

Moveover, the magnetic properties of RRhIn5 were studied, specially for NdRhIn5

with TN=11K,60SmRhIn5 with TN=15K,64GdRhIn5 with TN= 40K64 and TbRhIn5 with

TN= 45.5K.63The magnetic structure of NdRhIn5 and TbRhIn5 was determined from theneutron scattering and resonant x-ray photoemission experiments, respectively, and wasfound to be a commensurate antiferromagnetic structure with a magnetic wave vector

q = (12, 0,12) below TN, as shown in Fig 2.26 The taggered Nd moment at 1.6K is 2.5

µB/Nd aligned along the [001] direction The magnetic properties of YbRhIn5 was alsoclarified, indicating a non-magnetic divalent compound We also indicate the correspond-ing magnetic properties of RCoIn5 (R: Tb, Dy, Ho, Er, Yb).66 These compounds orderantiferromagnetically, except a divalent YbCoIn5, with the N´eel temperatures 30.2K,20K, 10.5K and below 2K in NdCoIn5, DyCoIn5, HoCoIn5 and ErCoRh5, respectively, asshown in Fig 2.31 Interestingly, the magnetic easy-axis is found to be the [001] direc-

tion (c-axis) in R= Tb, Dy and Ho, as shown in Fig 2.31 and indicates a characteristic

metamagnetic transition curve, as shown in Fig 2.32

In the present study, we studied mainly the magnetic properties of RRhIn5(R: Y,La-Yb) The similar metamagnetic transition was observed in high-field magnetizationcurves A change of the magnetization process will be discussed precisely