single crystal growth and magnetic properties of rrhin5 compounds

Doctoral Thesis

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


Abstract
A series of ternary compounds RRhIn5 (R: rare earths) was grown in the single crystalline form by means of the flux method. Magnetic properties of these compounds were
investigated by measuring the lattice parameter, electrical resistivity, specific heat, magnetic susceptibility and magnetization. All the compounds crystallize in the tetragonal
HoCoGa5 -type structure. Most of these compounds order antiferronmagnetically at low
tempertures, except for R= Y, La, Pr and Yb.
The observed temperature dependence of the specific heat and the anisotropic features in the magnetic susceptibility and magnetization were analyzed on the basis of the
crystalline electric field (CEF) model. It is suggested that the overall splitting energy of
RRhIn5 in the CEF scheme decreases as a function of 4f -electron number from 330K in
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 B20 in the CEF parameters.
For the former compounds, we observed characteristic metamagnetic transitions. Below a N´eel temperature TN = 11.6K, the magnetization of NdRhIn5 reveals two metamagnetic 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 metamagnetic
transitions correspond to the change of the magnetic structure. TbRhIn5 , DyRhIn5 and

HoRhIn5 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 magnetic fields are proposed by considering the exchange interactions based on the crystal
structure.
Furthermore, we observed the de Hass-van-Alphen (dHvA) oscillation of PrCoIn5 ,
PrRhIn5 and PrIrIn5 to clarify the Fermi surface properties. The detected topology of
the 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-Fermi
surface of PrIrIn5 , which was not observed previously in LaRhIn5 . This is mainly due to
the high-quality single crystal sample of PrIrIn5 .


Contents
1 Introduction

4

2 Magnetic Properties of Rare Earth Compounds
2.1 Magnetic properties of rare earth ions and metals . .
2.2 Crystalline electric field (CEF) effect . . . . . . . . .
2.3 Kondo effect and heavy fermions . . . . . . . . . . .
2.4 Magnetic properties of RIn3 and RRhIn5 compounds
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Motivation of the Present Study

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4 Single Crystal Growth and Measurement Methods

4.1 Single crystal growth . . . . . . . . . . . . . . . . . .
4.2 Measurement methods . . . . . . . . . . . . . . . . .
4.2.1 Electrical resistivity . . . . . . . . . . . . . . .
4.2.2 Specific heat . . . . . . . . . . . . . . . . . . .
4.2.3 Magnetic susceptibility and magnetization . .
4.2.4 High field magnetization . . . . . . . . . . . .
4.2.5 de Haas-van Alphen effect . . . . . . . . . . .
4.2.6 Neutron scattering . . . . . . . . . . . . . . .

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5 Experimental Results and Discussion
5.1 Magnetic properties and CEF scheme in RRhIn5 . . . . . . . . . . .
5.2 Fermi surface and magnetic properties of PrTIn5 (T: Co, Rh and Ir)
5.3 Unique magnetic properties of RRhIn5 (R: Nd, Tb, Dy, Ho) . . . .
5.4 Neutron scattering study in RRhIn5 (R: Nd, Dy , Ho) . . . . . . . .

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6 Conclusion

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Acknowledgments

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References

158

Publication List

165

3


1

Introduction

The rare earth compounds indicate a variety of electronic states such as magnetic
ordering, quadrupole (multipole) ordering, charge ordering, heavy fermions, Kondo insulators 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 tetragonal
crystal 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 with
the quasi-two dimentional electronic state. CeCoIn5 and CeIrIn5 are superconductors
at ambient pressure, with the superconducting transition temperature Tsc =2.3 K and
0.4K, respectively. On the other hand, CeRhIn5 indicates an antiferromagnetic ordering
with the N´eel temperature TN = 3.8K but becomes superconductive above 1.6GPa. The
uniaxially distorted AuCu3 -type layers of RIn3 and RhIn2 layers in RRhIn5 (R: rare earths)
are stacked sequentially along the [001] direction (c-axis). The Fermi surface properties of
LaRhIn5 and CeRhIn5 5, 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 of
LaRhIn5 .
These findings motivated us to undertake more investigations of the magnetic properties of RRhIn5 series in the single crystal form. Namely, the present study is to clarify
the fundermental magnetic properties of localized 4f -electrons, such as 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 were
carried out with a commercial SQUID magnetometer. The specific heat was measured
by the quasi-adiabatic heat-pulse method and commercial PPMS. The high-field magnetization was also measured by the standard pick-up coil method, using a long-pulse
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 magnetic properties of RRhIn5 series.
The present thesis consists of the following contents. In Chap. 2, the fundamental
4


2
2.1

Magnetic Properties of Rare Earth Compounds
Magnetic properties of rare earth ions and metals

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 5s2 5p6 6s2 . 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 (4f 1 5d1 6s2 ) 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 (RudermanKittel-Kasuya-Yosida interaction)9–11 and Kondo effect .12 Under the Hund rule, the

Ionic Radius ( Å )

1.3

1.2

1.1

1.0

0.9
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

R3+
Fig. 2.1 Ionic radius of R3+

6


2.1. MAGNETIC PROPERTIES OF RARE EARTH IONS AND METALS

7

1.0
4f

Ce (4f 15d 1 6s 2 )

|rR(r)|

2

0.8

5p

0.6
0.4

5d

6s

0.2
0
0

1

2

3

4

5

r [ a. u. ]

Fig. 2.2 Radial wave function of Ce (4f 1 5d1 6s2 ) with and without the relativistic effect.8
J = 72
8 fold

Ce3+
14 fold
~3000 K

doublet
J = 52

6 fold
spin-orbit
interaction

doublet
doublet

∆2
∆1

CEF

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


8

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

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 and
Fig. 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 and

magnetic structures are very complex in rare earth metals, as shown in Table 2.II and
Fig. 2.5. The double hexagonal close packed (dhcp) crystal structure is typical, possessing
both 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 J(J + 1) and de Gennes
factor (g-1)2 J(J + 1).
R
La3+
Ce3+
Pr3+
Nd3+
Pm3+
Sm3+
Sm2+
Eu3+
Eu2+
Gd3+
Tb3+
Dy3+
Ho3+
Er3+
Tm3+
Tm2+
Yb3+
Yb2+
Lu3+

Z
57
58
59
60
61
62
62
63
63
64
65
66
67
68
69
69
70
70
71

4fn
4f0
4f1
4f2
4f3
4f4
4f5
4f6
4f6

4f7
4f7
4f8
4f9
4f10
4f11
4f12
4f13
4f13
4f14
4f14

S
0
1/2
1
3/2
2
5/2
3
3
7/2
7/2
3
5/2
2
3/2
1
1/2
1/2

0
0

L
0
3
5
6
6
5
3
3
0
0
3
5
6
6
5
3
3
0
0

J = L ±S
0
5/2
4
9/2
4

5/2
0
0
7/2
7/2
6
15/2
8
15/2
6
7/2
7/2
0
0

spetroscopy
1 0
S
2 5/2
F
3 4
H
4 9/2
I
5 4
I
6 5/2
H
7

F0

8 7/2

S
F6
6 15/2
H
5 8
I
4 15/2
I
3 6
H
7

2

F7/2
1 0

S

g
0
6/7
4/5
8/11
3/5
2/7

0
0
2
2
3/2
4/3
5/4
6/5
7/6
8/7
8/7
0
0

gJ
0
2.14
3.2
3.27
2.40
0.71
0
0
7
7.0
9.0
10
10
9.0
7.0

4.0
4.0
0
0

µeff
0
2.54
3.58
3.62
2.68
0.85
0
0
7.96
7.94
9.72
10.65
10.61
9.58
7.56
4.54
4.54
0
0

(g-1)2 J(J + 1)
0
0.178
0.80

5.11
3.20
4.46
0
0
15.75
15.75
10.50
7.08
4.50
2.55
1.17
0.32
0.32
0
0


2.1. MAGNETIC PROPERTIES OF RARE EARTH IONS AND METALS

9

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(4f 7 5s2 4p6 6s2 ). Yb is also not trivalent, but divalent, indicating the nonmagnetic property. The ordering temperture is shown in Fig. 2.6.

Table 2.II Crystal structure, lattice constant and easy axis at 4.2K in rare earth metals.
Z
57
58

59
60
61
62
63
64
65
66
67
68
69
70
71
21
39

crys.struct.
dhcp
fcc
dhcp
dhcp
dhcp
rhomb
bcc
hcp
hcp
hcp
hcp
hcp
hcp

fcc
hcp
hcp
hcp

a(˚
A)
3.7740
5.1610
3.6721
3.6582
3.65
3.629
4.5827
3.6336
3.6055
3.5915
3.5778
3.5592
3.5375
3.4848
3.5052
3.4088
3.6482

c(˚
A)
12.171
11.8326
11.7996

11.65
26.207
5.781
5.6966
5.6501
5.6178
5.5851
5.554
5.5494
5.5268
5.7318

easy axis (4.2K)
[100]
[010]
[100]
[110]
[010]
[100]
[010]
[001]
-

L, S and J

R
La
Ce
Pr
Nd

Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Sc
Y

3+

R

Fig. 2.4 L ,S and J of R3+ based on the Hund rule.


10

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

R

(a)

(b)

(c)

(d)

Temperature ( K )

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: conelike helical structure, CM: modulated structure along the c-axis, F: ferromagnetism, FR:
ferrimagnetism, P: paramagnetism and HE: helical structure.


2.1. MAGNETIC PROPERTIES OF RARE EARTH IONS AND METALS

11

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 Hex between the total spin S of the 4f electrons and the spin s of
conduction electrons:
Hex = −2Jcf s · S
(2.1)
where Jcf is the magnitude of the exchange interaction. In the indirect exchange model,
the 4f electron spin Si at Ri interacts locally with the spin of the conduction electrons,
which then interact in turn with the 4f electron spin Sj at Rj . This approach is needed
because the 4f wave functions have insufficient overlap to give a direct Heisenberg exchange. The exchange interaction between Si and Sj is thus expressed as follows:
-J(Rij )Si · Si ,

where J(Rij ) contains a so-called Friedel oscillation of F (x)= (x cos x - sin x)/x4 , as
shown in Fig. 2.7. Here, Rij =Rj -Ri = 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 element changes from Ce to Gd or reversely from Yb to Gd in the compound, the magnetic
moment 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 : Si = (g − 1)Ji . The following equation can be obtained:
-J(Rij )(g − 1)2 Ji Jj
The magnetic ordering temperature Tord is thus propotional the√de Gennes factor
(g − 1)2 J (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


CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

Conduction electron
spin polarization

12

Friedel Oscillation
F(x) = (x cos x - sin x)/x

+
-

-

4

+

Distance

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 ( =Rij ) from
the central ion.

Theory

B

µeff (µ )

Experiment

3+

R

Fig. 2.8 Effective magnetic moment in the rare earth metals and the theoretical one in
R3+ .


Ordered moment (µ B)

2.1. MAGNETIC PROPERTIES OF RARE EARTH IONS AND METALS

13

Theory
Experiment

Rare earth metals

Fig. 2.9 Ordered moment in the rare earth metals and the corresponding theoretical
one in R3+ .


14

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

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:
∑
qi
V (r) =
,

(2.2)
|r
−
R
|
i
i
where r is the positional vector of the f electron, qi is the charge of the six-coordinated
negative ions and Ri 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
V (x, y, z) = Vx + Vy + Vz
where Vx = q[ √

1
(r2 +a2 −2ax)

+√

1
]
(r 2 +a2 +2ax)

and Vy = q[ √

(2.3)
1
(r 2 +a2 −2ay)

+√

1
]
(r 2 +a2 +2ay)

z

'
C
࡮
2
Z[\

࡮

T

࡮
x

#
C

࡮
$
C

&
C

Q

࡮
%
C

࡮(
C

y

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

1
(r2 +a2 −2az)

+√

1
]
(r 2 +a2 +2az)

with r2 = x2 + y 2 + z 2 .


2.2. CRYSTALLINE ELECTRIC FIELD (CEF) EFFECT

15

Insert Vx , Vy and Vz into eq. (2.3) and expand by the Taylor expansion, and we get the
following equation:
{
}
( 4
) 3 4
6q
4
4
+ D4 x + y + z − r
V (x, y, z) ≃
a
5
{
(
) 15 ( 2 4
+ D6 x 6 + y 6 + z 6 +
x y + x2 z 4 + y 2 x4 + y 2 z 4 + z 2 x4
4
}
)
15
+ z 2 y 4 − r6 .
(2.4)
14
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:
∫
ρ(r)V (r)d3 r.
(2.5)

V (r) can be expressed by the angular momentum operator based on the WignerEckart’s theorem in quantum mechanics. For example:
∫
⟨ ⟩{
}
(3z 2 − r2 )ρ(r)d3 r = αJ r2 3Jz2 − J (J + 1)
⟨ ⟩
= αJ r2 O20 .
(2.6)

We can represent the CEF Hamiltonian HCEF based on eq. (2.5) by the WignerEckart’s theorem as follows:
(
)
(
)
HCEF = B40 O40 + 5O44 + B60 O60 − 21O64 ,
(2.7)
where Stevens equivation operators such as Onm : O40 , O44 , O60 , O64 and O04 are expressed in
the form of matrix by Hutchings.18
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 = 25 , 32 , 12 ,− 12 ,− 32 ,− 25 ,
with the Lande factor g = 1 + J(J+1)+S(S+1)−L(L+1)
= 67 . Therefore, the multiplet J = 5/2
2J(J+1)
is degenerated by sixfold of 2J + 1 = 6, and this sixfolded degenerate state splits into the
4f levels by the CEF effect. For Jz = 5/2, O60 = O64 are zero and O40 , O44 can be expressed
as follows:
O40 = 35Jz4 − 30J(J + 1)Jz2 + 25Jz2 − 6J(J + 1) + 3J 2 (J + 1)2

1 4
O44 =
(J + J−4 ),
2 +

(2.8)
(2.9)


16

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

where J± = Jx ± iJy . The operator Onm can be expressed by (6 × 6)-matrix. Therefore,
the CEF Hamiltonian of the cubic Ce3+ is expressed as follows:
⟩
⟩
⟩
⟩
⟩
⟩
5
3
1
− 12
− 32
− 25
2
2
2


√ 0
⟨5 
0
60B
0
0
0
60
5B
0
4
4
√ 0
⟨ 23 
0
0
−180B
0
0
0
60
5B4 

4
⟨ 21 

0
0
0

120B
0
0
0


4
2
⟨
.
HCEF =
(2.10)
1 
0
− 
0
0
0
120B4
0
0

⟨ 23  √ 0


0
0
0
0
−180B40

−  60 5B4
√ 0
⟨ 25
0
−2
0
60 5B4
0
0
0
60B4
The wave function |i⟩ and its enery Ei at the CEF-4f level are expressed:
HCEF |i⟩ = Ei |i⟩.
By using eq. (2.10), we obtain the 4f state |i⟩ and its energy Ei as follows:
⟩ √
⟩ 
|Γα7 ⟩ = √16 52 − 56 − 32 
0
⟩ √ 5 3 ⟩  EΓ7 = −240B4
β
1
5
|Γ7 ⟩ = √6 − 2 − 6 2
√
⟩
⟩ 
5 5
√1 − 3 
+
|Γν8 ⟩ =



√6 2 ⟩ 6 2⟩ 

5
5
1 3
κ
√
|Γ8 ⟩ =
−
+
EΓ8 = 120B40
6
2
6 2
⟩


|Γλ8 ⟩ = 21 ⟩



µ
1
|Γ8 ⟩ = − 2

(2.11)

(2.12)

(2.13)

The energy level of −240B40 is named Γ7 and the energy state 120B40 is named Γ8 . The
splitting energy between Γ7 and Γ8 is 360B40 .
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 expands
along 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, indicating
that the Γ8 state becomes an excited state. In general, the CEF Hamiltonian for the
lanthanide ions can be expressed as follows:
∑
HCEF =
Bnm Onm ,
(2.14)
n,m

If the number of the f electron is odd, namely, J has the semi-integer: Ce3+ , Nd3+ ,
Sm , Dy3+ , Er3+ and Yb3+ , the 4f energy level always have the doublet (called Kramers
3+


2.2. CRYSTALLINE ELECTRIC FIELD (CEF) EFFECT

17

(a)

(c)

(b)

Fig. 2.11 Characteristic shapes of electron clouds in cubic Ce3+ : (a) Γα7 and Γβ7 , (b) Γλ8
and Γµ8 , (c) Γν8 and Γκ8 .

doublet). Kramers degeneration is based on the time reversal symmetry and the doublet ground state always holds even if the crystal structure is changed into the lower
symmetrical structure. The magnetic properties are different whether the number of the
electrons is odd or even. When the magnetic field is applied to the system, all the degenrate 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 magnetic
field is as follows:
H = HCEF − gJ µB H,

(2.15)

where the second term corresponds to the Zeeman energy. Consider 4f -energy state,
where |i⟩ is the 4f state of the level state i, Ei is the eigenvalue and µi is the magnetic
moment of the energy level. When applying the magnetic field, the energy state of the
every
level scheme is influence by the other energy scheme. We represent this state as
⟩
˜i and Ei (H) as its eigenvalue. Namely, we calculate the every states under magnetic
⟩
field ˜i and Ei (H), diagonalizing the matrix of the Hamiltonian shown in eq (2.15).
⟩
We calculate the magnetization and the magnetic susceptibility by using ˜i and Ei (H).


18

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

Here, the Helmholtz free energy F can be expressed by the partion function Z as follows:
F = −kB T ln Z,
∑ − Ei (H)
Z =
e kB T ,

(2.16)
(2.17)

i

The magnetization M is expressed as the difference of F by magnetic field:
∂F
M = −
∂H
∑
µzi e−Ei (H)/kB T
=

∑

i

e−Ei (H)/kB T

i

≡ ⟨µzi ⟩ ,

(2.18)

⟩
where µzi is the magnetic moment of the state ˜i .
∂Ei (H)
∂H
= gJ µB ⟨˜i|Jz |˜i⟩.

µzi = −

(2.19)

Namely, the magnetization M correspond to the average ⟨µzi ⟩ of the magnetic moment
µzi . The magnetic susceptibility χ is the difference of magnetization ∂M/∂H(H → 0) :
(⟨(
)2 ⟩ ⟨
⟩2 ) ⟨ 2
⟩
∂Ei (H)
∂Ei (H)
∂ Ei (H)
1
−
−
,
(2.20)
χ=
kB T

∂H
∂H
∂H 2

In the case of the calculation of the magnetic susceptibility, we can treat the Zeeman
energy −gJ µB HJz as the perturbation. The energy Ei (H), with the second perturbation,
can be expressed as follows:
∑ ⟨j|Jz |i⟩ 2
,
Ei (H) = Ei − gJ µB H⟨i|Jz |i⟩ + (gJ µB ) H
Ej − Ei
2

2

(2.21)

j(̸=i)

By using eq. (2.20), eq. (2.21) is obtained as follows:


2
∑
∑
|⟨j|Jz |i⟩| 
2
(gJ µB )2
e−Ei /kB T  ⟨i|Jz |i⟩ + 2kB T
Ej − Ei

i
j(̸=i)
∑
χ=
kB T
e−Ei /kB T ,
i

(2.22a)


2.2. CRYSTALLINE ELECTRIC FIELD (CEF) EFFECT

19

The expression eq. (2.22a) is the general expression of the magnetic susceptibility
under consideration for the CEF, and another expression is often used:

∑
2 −Ei /kB T
⟨i|J
z |i⟩ e
−Ei /kB T
∑∑
(gJ µB )2 
− e−Ej /kB T 
2e
.
 i
χ= ∑

+
⟨j|J
|i⟩
z


kB T
E
−
E
j
i
e−Ei /kB T
i j(̸=i)
i

(2.22b)

The first term is the Curie term which can be determined by the diagonal element of
the matrix Jz and the second term is related to the non-diagonal element. Namely, it is
the Van-Vleck term, which is related to the transition between the states. It is known
from 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 Jz for the cubic Ce3+ . The Jz matrix element can be expressed as follows:
 |Γα7 ⟩
⟨Γα7 |

⟨Γβ7 | 

ν 

Jz = ⟨Γ8 | 

⟨Γκ8 | 

⟨Γλ8 | 
⟨Γµ8 |

− 65
0
√

2 5
3

0
0
0

⟩

|Γν8 ⟩

|Γκ8 ⟩

Γλ8

0

5
6

0

√
−235

0

11
6

0
0

0
0
0

0
0
0
0

Γβ7
0

√
−235

√

2 5
3

0
− 11
6
0
0

1
2

0

⟩

|Γµ8 ⟩ 
0
0
0
0
0
− 12






,






(2.23)

⟩
We evaluate the magnetic moment as −5/7 µB for |Γα7 ⟩ and +5/7 µB for Γβ7 from
gJ = 6/7. The summation over
⟩ the two degenerated states of the Γ7 is zero. The magnetic
moments for |Γν8 ⟩, |Γκ8 ⟩, Γλ8 and |Γµ8 ⟩ 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
and EΓ8 − EΓ7 = ∆):


25 65 −∆/kB T
(
)

2

−∆/kB T 
+
e
40 1 − e
(gJ µB )
36 18
+
,

(2.24)
χz =

1 + 2e−∆/kB T 
kB T
9∆



Figure 2.12 (a) and (b) show the temperature dependence of the inverse magnetic susceptibility and magnetization, respectively, for three cases: no CEF, Γ7 ground state and


20

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS
400

χ ( mol-Ce/emu )

Ce3+ cubic

without CEF

2

300

200
1

without CEF
100
(a)
0

0

200
100
Temperature ( K )

(b)
300

0

0

500

1000

Magnetic Field ( kOe )

Fig. 2.12 (a) inverse magnetic susceptibility and (b) magnetization for ∆ = 200 K in
cubic Ce3+ .
Γ8 ground state with ∆ = 200 K. No CEF corresponds to ∆ → 0: χz = 35
(gJ µB )2 /3kB T .
4
Furthermore, the case, which is ∆ → 0, is equivlent to the expression kB T ≫ ∆ and approaches 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 magnetic
field 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 calculate
the splitting energy, magnetic susceptibility and magnetization curves based on the CEF
effect. Using again eq. (2.14) for the tetragonal symmetry, we can write the CEF Hamiltonian as follows:
HCEF = B20 O20 + B40 O40 + B44 O44 + B60 O60 + B66 O66 .

(2.25)

The total orbital angular momentum, total spin momentum and total angular momentum are L=3, S=1/2 and J=5/2, respectively, and the magnetic angular momentum:
m = 25 , 32 , 12 , − 12 ,− 32 , − 52 . 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


2.2. CRYSTALLINE ELECTRIC FIELD (CEF) EFFECT

21

effect. In the case of J = 5/2, O60 = O66 are zero. O20 , O40 and O44 are expressed as follows:
O20 = 3Jz2 − J(J + 1)
O40 = 35Jz4 − 30J(J + 1)Jz2 + 25Jz2 − 6J(J + 1) + 3J 2 (J + 1)2
1 4
O44 =
(J + J−4 )
2 +

(2.26)
(2.27)

(2.28)

Other parameters are shown in the case of the Ce3+ cubic symmetry case. The
Hamiltonian for the tetragonal Ce3+ :
⟩
⟩
⟩
⟩
⟩
⟩
5
3
1
− 21
− 23
− 25
2
2
2

√ 0
⟨5 
A
0
0
0
12
5B
0
4

√ 0
⟨ 23 
0
B
0
0
0
12
5B4 

⟨ 21 

0
0
C
0
0
0


HCEF = ⟨ 2 1 
. (2.29)
−2 
0
0
0
C
0
0


⟨ 3  √ 0


− 2  12 5B4
0
0
0
B
0
√ 0
⟨ 5
−2
0
12 5B4
0
0
0
A
with A = 10B20 + 60B40 , B = −2B20 − 180B40 and C = −8B20 − 120B40 . As for the magnetic
susceptibility data in CeRhIn5 with the tetragonal structure, the CEF parametters were
estimated by Takeuchi et al19 as follows: B20 = −16, B40 = 0.55 and B44 = 0.61. Therefore,
the above matrix can be expressed:
⟩
⟩
⟩
⟩
⟩
⟩
3
1

5
− 21
− 32
− 52
2
2
2

⟨5 
−127
0
0
0
16.368
0
2
⟨3

−67
0
0
0
16.368 

⟨ 21  0

 0
0
194
0

0
0
,
(2.30)
HCEF = ⟨ 2 1 

0
0
194
0
0 

⟨− 23  0

0
0
0
−67
0 
⟨− 25 16.368
−2
0
16.368 0
0
0
−127

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:
⟩

⟩ }
0.247142 + 23 ⟩
|Γα7 ⟩ = 0.968978 − 52 −
⟩
EΓ7 = 0
(2.31)
|Γβ7 ⟩ = −0.968978 + 52 + 0.247142 − 23
⟩
⟩ }
|Γα8 ⟩ = −0.247142 − 52 ⟩ 0.968978 + 32 ⟩
EΓ8 = 68.3494
(2.32)
|Γβ8 ⟩ = −0.247142 + 52 + 0.968978 − 23
⟩ }
|Γα9 ⟩ = −⟩12
EΓ9 = 325.175,
(2.33)
|Γβ9 ⟩ = 12


22

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

The Jz matrix element can be expressed as follows:
⟩
β
α
Γ7
|Γα8 ⟩

 |Γ7 ⟩

Γβ8

⟩

⟨Γα7 | 2.25568
0
−0.9579
0
⟨Γβ7 | 
0
−2.25568
0
0.9579

ν 
⟨Γ
|
0
−1.25568
0
8  −0.9579
Jz =
κ 
⟨Γ8 | 
0
0.9579
0
1.2568

⟨Γλ8 | 
0
0
0
0
µ
⟨Γ8 |
0
0
0
0

⟩

|Γα9 ⟩

Γβ8

0
0
0
0
0.5
0

0
0
0
0
0

0.5






,




(2.34)

Therefore, the ground state is the doublet Γ7 , and the first and second excited states
are also doulets Γ8 , Γ9 , respectively, which are shown in Fig. 2.13.
doublet

Ce3+

excited state 2

J = 52
∆2 = 325 K

doublet
doublet

excited state 1
∆1 = 68 K

ground state

Fig. 2.13 Splitting energies with three doublets in Ce+3 teragonal structure under CEF
effect: the doublet Γ7 ground state, the first excited state double Γ8 and the second
excited state double Γ9 .
The magnetic susceptibility and magnetization under magnetic field also can be calculated
by eq. (2.22). The Hamiltonian is:
⟩
⟩
⟩
⟩
⟩
⟩
5
3
1
− 21
− 23
− 25
2
2
2

√ 0
⟨5 
′
A
0
0
0

12
0
5B
4
√ 0
⟨ 23 
′
0
B
0
0
0
12
5B4 

⟨ 21 

′
0
0
C
0
0
0


HCEF = ⟨ 2 1 
, (2.35)
′
− 

0
0
0
C
0
0

⟨ 32  √ 0

′

0
0
0
B
0
− 2  12 5B4
√
⟨ 5
′
0
−2
0
12 5B4
0
0
0
A

2.2. CRYSTALLINE ELECTRIC FIELD (CEF) EFFECT

23

with A′ = 10B20 + 60B40 + 52 gµB H, B ′ = −2B20 − 180B40 + 23 gµB H and C ′ = −8B20 −
120B40 + 21 gµB H. We can calculate energy eigenvalues, wavevalue function and Jz matrix
element in the same way as below. Finally, magnetic susceptibility and magnetization
can be calculated by using eq. (2.22) and eq. (2.18), which are shown in Fig. 2.14.
3+

0
-100

2

2

T = 1.3K
4.2K
10K

0
0

1

500
1000
Magnetic Field ( kOe )

1500
8

300

6
4

100

2

0

H = 10 kOe
0

(a)

100
200
Temperature ( K)

0
300

4
3

g·J = 2.14

H//[001]

[100]
T = 1.3K
500
1000
Magnetic Field ( kOe )
H = 10 kOe
[100]

1500
3
2

H//[001]

2

1

1
0
0

100
200
Temperature ( K)

0
300

(b)

Fig. 2.14 Six-fold 4f -level in the magnetic field, magnetization curves in H//[001]
and [100] and the temperature dependent of χ and 1/χ in CeRhIn5 with the tetragonal
structure: (a) without CEF and (b) CEF effect.

-2

200

5

CeRhIn5 with CEF

χ (x10 mol/emu )

400

0
0

2

1

χ ( mol/emu )

1/χ ( emu/mol )

Magnetization ( µΒ/Ce )

-200

300
200
100
0
-100
-200
-300

Magnetization ( µΒ/atom )

Energy ( K )

Ce tetragonal
100 without CEF

1/χ ( x 10 emu/mol )

Energy ( K )

200


24

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

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 the
origin of the long standing problem of the minimum resistivity. This is the start of the
Kondo 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 kB TK . Here the Kondo temperature TK is the single
energy scale. In other words, disappearance of the localized moment is thought to be
due to the formation of a spin-compensating cloud of the electrons around the impurity
moment.
Kondo-like behavior was observed in lanthanide compounds, typically in Ce and Yb
compounds.21–23 For example, the electrical resistivity in Cex La1−x Cu6 increases logarithmically with decreasing temperature for all the x-values,24 as shown in Fig. 2.15. The
Kondo 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 Cex La1−x Cu6 .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 = 25 ) and the 4f energy
level splits into the three doublets by the crystalline electric field (CEF) effect, namely
possessing the splitting energy of ∆1 and ∆2 .25


2.3. KONDO EFFECT AND HEAVY FERMIONS

The Kondo temperature is given as follows:26
(
)
1
h
TK = D exp −
3|Jcf |D(EF )

25

when T > ∆1 , ∆2 ,

(2.36)

(
)
D2
1
TK =
D exp −
when T < ∆1 , ∆2 .
∆1 ∆2
|Jcf |D(EF )

(2.37)

and

Here D, |Jcf | 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 obtained, which is compared to the S = 21 - Kondo temperature of 10−3 K defined as
TK0 = D exp(−1/|Jcf |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 TK0 but TK shown in eq. (2.37).
On the other hand, the magnetic ordering temperature is about 5 K in the Ce compounds, which can be simply estimated from the de Gennes relation under the consideration of the Curie temperature of about 300 K in Gd. Therefore, it depends on the
compound whether magnetic ordering occurs at low temperatures. The ground state
properties of the dense Kondo system are interesting in magnetism, which are highly
different from the dilute Kondo system. In the cerium intermetallic compounds such as
CeCu6 , cerium ions are periodically aligned whose ground state cannot be a scattering
state but becomes a coherent Kondo-lattice state. The electrical resistivity ρ decreases
steeply with decreasing the temperature, following a Fermi liquid behavior as ρ ∼ AT 2
27
with a large
√ value of the coefficient A.
The A value is proportional to the effective mass of the carrier m∗ and thus inversely
proportional 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:
∫ TK
C
R log 2 =
dT,
(2.38)
T
0
C = γT,
(2.39)
thus
γ=

5.8 × 103
R log 2
=
(mJ/K2 ·mol).
TK
TK

(2.40)

It reaches 1600 mJ/K2 ·mol for CeCu6 28 because of a small Kondo temperature of
4-5 K. The conduction electrons possess large effective mass and thus move slowly in the
crystal. Actually in CeRu2 Si2 , an extremely heavy electron of 120 m0 was detected from
the de Haas-van Alphen (dHvA) effect measurements.29, 30


26

CHAPTER 2. MAGNETIC PROPERTIES OF RARE EARTH COMPOUNDS

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 moments order. The latter effect may be approximated by a band-folding procedure where
the paramagnetic Fermi surface is folded into smaller Brillouin zone based on the magnetic
unit cell, which is larger than the chemical one. If the magnetic energy gaps associated

with the magnetic structure are small enough, conduction electrons undergoing cyclotron
motion in the presence of magnetic field can tunnel through these gaps and circulate the
orbit 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 ordering
such as CeB6 , the Kondo effect is expected to have minor influence on the topology of
the Fermi surface, representing that the Fermi surfaces of the Ce compounds are roughly
similar to those of the corresponding La compounds, but are altered by the magnetic
Brillouin zone boundaries mentioned above. Nevertheless the effective masses of the conduction 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 , CeRu2 Si2 , CeNi and CeSn3 , the magnetic
susceptibility follows the Curie-Weiss law with a moment of Ce3+ , 2.54µB /Ce, has a
maximum at a characteristic temperature Tχmax , and becomes constant at lower temperatures. This characteristic temperature Tχmax 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 ordering is clarified to be itinerant at low temperature from the dHvA experiments and energy
band calculations.