Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice
A review of peculiarities of magnetic properties and spin kinetics of a heavy-fermion Kondo lattice revealed by electron spin resonance (ESR) experiments and their theoretical analysis is given. Among the issues discussed in some detail are the renormalization of spin kinetics coefficients due to th...
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| Cite this: | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice / B.I. Kochelaev // Физика низких температур. — 2016. — Т. 43, № 1. — С. 93-103. — Бібліогр.: 44 назв. — англ. |
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| citation_txt | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice / B.I. Kochelaev // Физика низких температур. — 2016. — Т. 43, № 1. — С. 93-103. — Бібліогр.: 44 назв. — англ. |
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| description | A review of peculiarities of magnetic properties and spin kinetics of a heavy-fermion Kondo lattice revealed by electron spin resonance (ESR) experiments and their theoretical analysis is given. Among the issues discussed in some detail are the renormalization of spin kinetics coefficients due to the Kondo effect, formation of the collective spin modes of the Kondo ions and wide-band conduction electrons, unexpected behavior of ESR parameters as functions of temperature and magnetic fields. Special attention is focused on the possible role of the Kondo effect for the ESR signal existence at low temperatures.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1, pp. 93–103
Magnetic properties and spin kinetics of a heavy-fermion
Kondo lattice
B.I. Kochelaev
Institute of Physics, Kazan Federal University, Kremlevskaya 18, Kazan 42008, Russian Federation
E-mail: bkochelaev@gmail.com
Received July 8, 2016, published online November 25, 2016
A review of peculiarities of magnetic properties and spin kinetics of a heavy-fermion Kondo lattice revealed
by electron spin resonance (ESR) experiments and their theoretical analysis is given. Among the issues discussed
in some detail are the renormalization of spin kinetics coefficients due to the Kondo effect, formation of the col-
lective spin modes of the Kondo ions and wide-band conduction electrons, unexpected behavior of ESR parame-
ters as functions of temperature and magnetic fields. Special attention is focused on the possible role of the Kon-
do effect for the ESR signal existence at low temperatures.
PACS: 76.30.Kg Rare-earth ions and impurities;
71.27.+a Strongly correlated electron systems; heavy fermions;
75.40.Gb Dynamic properties (dynamic susceptibility, dynamic scaling).
Keywords: magnetic resonance, heavy fermions, Kondo effect.
1. Introduction
Heavy fermion compounds like 2 2YbRh Si and
2 2YbIr Si have very peculiar magnetic, thermal, transport
and spin kinetics properties. They are determined by the
interplay of the strong repulsion of 4f electrons on the rare-
earth ion sites, their hybridization with wide-band conduc-
tion electrons and the influence of the crystalline electrical
field. Experimental observations are consistent with a me-
tallic behavior with very heavy charge carriers having the
properties of a Landau Fermi liquid (LFL). With decreas-
ing of temperature these materials experience the Kondo
effect. At very low temperatures a long-range antiferro-
magnetic (AF) order appears with 70 mKNT = in
2 2YbRh Si , which is suppressed by an external magnetic
field at the quantum critical point (QCP) with
0.06 T,cH = 0.T → Near the QCP at higher magnetic
fields and up to surprisingly high temperatures, a new
phase appears exhibiting non-Fermi-liquid (NFL) behav-
ior. In this temperature region also some features of ferro-
magnetic fluctuations were observed. The electrical resis-
tivity in 2 2YbRh Si linearly increases with temperature and
the Sommerfeld coefficient of the electronic specific heat
diverges logarithmically upon cooling down to 0.3 K. The
phase diagram based on these experimental findings is
presented in Fig. 1 [1].
The discovery of electron spin resonance (ESR) in the
2 2YbRh Si compound at temperatures below the thermo-
dynamically determined Kondo temperature ( 25 K)KT =
became a great surprise for the condensed matter physics
community [2]. According to the common belief, based on
the single ion Kondo effect, the ESR signal should not be
Fig. 1. (Color online) Experimental phase diagram of 2 2YbRh Si
for c⊥H from [1]. KT and H∗ mark the crossover from local
moment to heavy fermion behavior, derived from the zero-field
specific heat and low-temperature magnetization. LFL and NFL
denote the Landau Fermi-liquid and non-Fermi-liquid regions,
respectively. The broad pink line specifies the position of crossover
in the isothermal Hall resistivity, isothermal magnetostriction,
magnetization and longitudinal resistivity. AF means the antiferro-
magnetic phase.
© B.I. Kochelaev, 2017
B.I. Kochelaev
observable by at least two reasons. Firstly, the magnetic
moments of the Kondo ions should be screened by the
conduction electrons at temperatures below ;KT secondly,
the ESR linewidth was expected to be of the order of ,KT
i.e. 500 GHz. The experimental results were completely
opposite: at X-band (9.4 GHz) and 1.6 KT = a linewidth of
0.3 GHz was observed (see Fig. 2 from [3], see also [4])
and the ESR intensity corresponds to the participation of
all Kondo ions with a temperature dependence following a
Curie–Weiss law [5]. In addition, the angular dependence
of the resonant magnetic field reflects the tetragonal sym-
metry of the crystal electric field at the position of the Yb
ion with extremely anisotropic g factors 3.56,g⊥ =
0.17g =
at 5 KT = [2]. Similar results were obtained
later for 2 2YbIr Si [6,7], and the main features of the ESR
phenomenon were confirmed for 2 2YbRh Si at very high
frequencies up to 360 GHz [8]. This discovery stimulated
considerable efforts, both theoretical and experimental, in
order to understand the nature of the ESR existence and
Kondo lattice systems in general.
The first attempt was made by Zvyagin et al. using a
model of metallic systems with multichannel Kondo mag-
netic impurities [9] and the Anderson lattice model [10]
correspondingly, relating them to the actual compound on
a qualitative level. Abrahams and Wölfle [11,12] studied
the ESR in heavy-fermion systems using a Fermi-liquid
description in the framework of the Anderson model,
where the local magnetic moment is that of a quasi-
localized f electron. Schlottmann [13] suggested an expla-
nation of the ESR signal existence based on the Kondo
lattice model with an isotropic interaction between the
conduction electrons and the Kondo ions. However, both
of these approaches do not take into account the strong
spin orbital interaction and the crystal electric field (CEF)
effects which result in the anisotropy of the Kondo interac-
tion similar to that of the Zeeman energy. On the contrary
a semi-phenomenological theory presented by Huber [14]
takes into account the anisotropy of the static and dynam-
ical susceptibilities. The author was able to describe the
ESR data in Yb heavy-fermion compounds, especially
their angular dependence, but the analysis did not touch the
problem of the ESR signal existence, assuming it a priori.
This paper presents a short review of magnetic properties
and spin kinetics of a heavy-fermion Kondo lattice including
a problem of the ESR existence. This analysis is based on a
model of localized 4f electrons coupled to the wide-band
conduction electrons as a starting point. It was found that the
key ingredient of the ESR signal existence in a heavy-fer-
mion Kondo lattice in the NFL state is a formation of a col-
lective spin mode of quasi-localized f electrons and wide-
band conduction electrons even in a strongly anisotropic
system. In opposite to a common belief, it was found that the
Kondo effect can support the ESR existence.
2. Crystal electric field and magnetic susceptibility
The angle resolved photoemission spectroscopy
(ARPES) measurements on 2 2YbRh Si revealed a very
narrow 4f band near the Fermi energy [15]. It is an addi-
tional argument confirming the quasi-localized nature of
the f-electron motion (in the case of absolutely localized
electrons their width of energy band as a function of the
wave vector reduces to zero). So, it was reasonable to in-
vestigate the energy spectra, g factors of the ground state
and the static magnetic susceptibility of the 3Yb + ions in
the 2 2YbRh Si and 2 2YbIr Si compounds with localized
f electrons. A full solution of this problem was given in the
papers [16–18]. Here we describe a scheme of calculations
and the main results.
A free 3Yb + ion has the electronic 134f configuration
with one term 2 .F The spin-orbit interaction splits the 2F
term into two multiplets: 2
7/2F with 7/2J = and 2
5/2F
with 5/2,J = where J denotes the total momentum
= +J L S with L and S as the orbital and spin momentum
of the ion. The excited multiplet 2
5/2F is separated from
the ground one 2
7/2F by about 1 eV. Since this value is
much larger than the energy of the crystal electric field, we
consider the ground multiplet only. The potential of the te-
tragonal crystal field for an ion can be written as
( ) ( )0 0 0 0 4 4 0 0 4 4
2 2 4 4 4 4 6 6 6 6 .V B O B O B O B O B O= α +β + + γ + (1)
The crystal field parameters are .q
kB The operators q
kO and
coefficients ,α ,β γ are standard and given in the book by
Abragam and Bleaney [19]. To define energy levels and
wave functions of the 3Yb + ion one has to diagonalize the
matrix of the operator (1) on the states of the ground
Fig. 2. Representative ESR spectrum in 2 2YbRh Si at 1.6 KT =
(from [3]).
94 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
Magnetic properties and spin kinetics of a heavy fermion Kondo lattice
multiplet 2
7/2.F As a result, the crystal field splits the low-
er multiplet into four Kramers doublets. Inelastic neutron
scattering (INS) experiments give the following values for
the excited energy levels in 2 2YbRh Si : 1 17 meV,∆ =
2 25 meV,∆ = 3 43 meV∆ = relative to the ground doublet
[20]. These results together with mentioned ARPES and
ESR data gave an opportunity to find the single set of te-
tragonal CEF parameters for 3Yb + ion in 2 2YbRh Si , pre-
sented in the Table 1 [18].
The Zeeman energy of the Yb ion ZJH for the ground
multiplet 2
7/2F can be expressed via the total momentum
of the ion. Using the Lande g factor 8/7,Jg = we have
Z ( 2 ) ,J B i i J B i
i i
H g= µ = µ∑ ∑L + S H J H (2)
where H is the external magnetic field and Bµ is the Bohr
magneton. The calculations of the static magnetic suscep-
tibility can be performed in a straightforward way. In the
case of low enough temperatures B kk T ∆ we have the
standard expressions for two following contributions to the
magnetic susceptibility C VV:χ = χ + χ
02
2
,
( )
( )
(C) ;
2
(VV) ( ) .
J B
B
J B
n l
n l
CN g
l J l l J l
k T T
l J n n J l
N g
E E
αβ
αβ α β
′νν
α β
αβ
′νν
≠
µ
′ ′χ = ν ν ν ν ≡
′ ′ν ν ν ν
χ = µ
−
∑
∑
(3)
Here N is the density of the Yb ions in the crystal. Indexes
,n ,ν ′ν denote the Kramers doublets and their correspond-
ing components; lν is the ground Kramers doublet. The
first term, (C),αβχ corresponds to the Curie susceptibility
proportional to the inverse temperature, while the second
term, (VV),αβχ corresponds to the Van Vleck susceptibil-
ity, which does not depend on temperature. According to
the tetragonal symmetry from all the Curie constants sur-
vive two ones only: 0
||C and 0 .C⊥
Before a comparison of these results with an experiment,
one should take into account their renormalization due to
interaction of the Yb ions with conduction electrons and
their indirect exchange interaction via these electrons, so-
called RKKY interaction. One could expect, that the tem-
perature dependence should become a Curie–Weiss type:
||,
, ||,
||,
(VV).
C
T
⊥
⊥ ⊥
⊥
χ = + χ
+ θ
(4)
Here ||,⊥θ are the Weiss constants.
It is appropriate to mention at this point, that there is a
different approach to calculations of the static magnetic
susceptibility in these materials, so-called a locally quan-
tum critical scenario [21,22]. One of the hallmarks of this
local criticality is a generalized Curie–Weiss law, which
for a wave-vector-dependent magnetic susceptibility can
be written in the form
( ), CT
Tα α
χ =
+ θq
q (5)
with an exponent 1.α < In particular, experimental results
for the 2 2YbRh Si magnetic susceptibility in the tempera-
ture region 0.3 K < T < 10 K was related to this scenario
for 0,=q with 0.6,α = 0θ = [22].
Figure 3 shows the experimental results for the static
magnetic susceptibility in 2 2YbRh Si in the temperature
region 0.02 K 3.6 KT< < from [22] together with a fitting
according formulas (4) and (5). The fitted constants in (4)
were 6 3 12.31 10 m mol KC − −
⊥ = ⋅ ⋅ ⋅ and 0.22 K⊥θ = (solid
line). One can see that the temperature dependence of
magnetic susceptibility is better described in the frame of a
simple and transparent local model of the 3Yb + ions in the
crystal electric field in comparison with a sophisticated
scenario of the local quantum criticality. However, in spite
of the success of this entirely local approach for the static
magnetic susceptibility of 2 2YbRh Si , such a model is in-
sufficient for a proper theoretical understanding of the dy-
namical susceptibilities as observed by ESR. In the follow-
ing sections this problem will be considered by taking into
account an interaction of the 3Yb + ions with wide-band
conduction electrons and their collective response to the
resonant magnetic alternating field (the bottleneck regime).
This interaction will be considered in details, including a
role of the Kondo effect in a formation of the collective
spin modes.
Table 1. Crystal electric field parameters q
kB (meV) for 3Yb +
ion in 2 2YbRh Si [18]
0
2B 0
4B 4
4B 0
6B 4
6B
22.4 −6.6 ±29.6 −2.3 ±63.9
Fig. 3. (Color online) Fitting of the magnetic ac-susceptibility
data [22] by expression (4) (solid line) and by (5) (dashed line)
with parameters given in the text (from [16]).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 95
B.I. Kochelaev
3. Dynamical magnetic susceptibility
3.1. Theoretical model
To consider the spin dynamics in the Kondo lattice un-
der discussion we use the model including the kinetic en-
ergy of conduction electrons, the Zeeman energy, the Kon-
do interaction between Yb ions and conduction electrons,
and the coupling between the Yb ions via conduction elec-
trons (RKKY interaction) [23–30]. The physics of low-
energy spin excitations at temperatures 200 KT can be
described by the lowest Kramers doublet. It means that the
Zeeman energy (2) has to be projected onto this state.
Within the ground Kramers doublet, described by the wave
functions of the type , ,l ↑ ↓ the Zeeman energy can be
represented by the effective spin Hamiltonian with an ef-
fective spin 1/2:S =
( ) ;
, 2 .
yx z
ZS x i y z ii
i
J x y J z
H g B S B S g B S
g g l J iJ l g g l J l
⊥
⊥
= + +
= ↑ + ↓ = ↑ ↓
∑
(6)
Here the z axis is along the crystal symmetry and we have
introduced .BB H= µ We can see that starting from the
Zeeman energy with the isotropic Lande g factor, we came
to the anisotropic expression.
The Kondo exchange coupling of the rare-earth ion
with the conduction electrons occurs due to the hybridiza-
tion of their wave functions at the ion site. An exchange
integral can be written in the form
2
* 4 1( , ) ( , ) ( ,..., ,..., )f i n
ii
eA ′ ′= ψ Ψ ×
′−∑∫k k r k r r r
r r
4 1 1( , ) ( ,..., ,..., ) ... .i f n nd d d′ ′ ′×ψ Ψr k r r r r r r (7)
Here ( , )ψ r k is the Bloch wave function of the conduction
electrons. The wave function of the 4f electrons 4 fΨ is
represented by the determinant constructed from the one-
electron wave functions of the type 4 3( ) ( ).m
fR r Y r Expand-
ing the Bloch functions and 1
i
−′−r r in spherical harmon-
ics, one can obtain ( , )A ′k k as an expansion in multipoles.
The small parameter of this expansion is the value
4 1,F fk r〈 〉 the product of the wave vector of the conduc-
tion electron at the Fermi surface and an average radius of
the 4f electron. The Kondo interaction corresponding to the
zero order of this expansion is isotropic and can be written
in the form
( )0 0 1 ,KJ i i J i i
i i
H A A g= = −∑ ∑S σ J σ (8)
where iσ is the spin density of the conduction electrons at
the ion site. The next terms of the expansion in multipoles
are much smaller, and we neglect them. The projection of
the isotropic Kondo interaction (8) involves the same ma-
trix elements of the total momentum J as in the case of the
Zeeman energy. Hence, after projection onto the ground
Kramers state the total Kondo interaction can be expressed
via the g factors, given above
( ) ||
0 || 0 ||
;
1 1
, .
y yx x z z
K i i i ii i
i
J J
J J
H J S S J S
g g
J A g J A g
g g
⊥
⊥ ⊥
= σ + σ + σ
− −
= =
∑
(9)
The same arguments can be used to reveal the anisotro-
py of the RKKY interaction between the Kondo ions. Alt-
hough this interaction appears in the second order of the
Kondo interaction (8), it is convenient to consider it inde-
pendently. Starting with the isotropic exchange Hamiltoni-
an for two Kondo ions ex ex ,ij ij i jH I= J J we arrive after
projection to the effective anisotropic exchange interaction
( )
( ) ( )
||
ex
22 ||ex ex
||
1 ;
2
, .
y yx x z z
ij i j i ji j ij
ij
ij J ij J ijij
H I S S S S I S S
I g g I I g g I
⊥
⊥
⊥
= + +
= =
∑
(10)
This expression was given earlier in [31]. As a matter of
fact all these results are simply consequences of the well-
known Wigner–Eckart theorem.
The kinetic energy of the conduction electrons and their
Zeeman energy can be written as
( )
, ;
e .j
c Z j
j
i
j
H c c H g
c c
+
ν ν σ σ
ν
′− +
′ ′ ′νν ν ν
′ ′νν
= ε =
=
∑ ∑
∑
k k k
k
k k r
k k
kk
Bσ
σ s
(11)
The energy band of conduction electrons is assumed sym-
metric about the Fermi surface, and their energy relative to
the Fermi level εk is running within limits ( , ).W W− Here
,c+νk c νk are the creation and annihilation operators of the
conduction electrons with the wave vector and spin projec-
tion ( , ),νk gσ denotes their g factor, ′ννs are the matrix
elements of the electron spin operator for 1/2.s = The
Kondo interaction in this representation takes the form
( )1
2K j j
j
H J S c c S c c+ + − +
⊥ ′ ′↑ ↓↓ ↑
′
= + +∑ k kk k
kk
( ) ( )
|| e .jiz
jJ S c c c c ′−+ +
′ ′↑ ↓↑ ↓
+ −
k k r
k kk k (12)
Here .yx
j j jS S iS± = ±
3.2. Spin dynamics
The most detailed ESR experiments with the 2 2YbRh Si
samples were performed for the static and alternating mag-
netic fields oriented perpendicular to the crystal symmetry
96 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
Magnetic properties and spin kinetics of a heavy fermion Kondo lattice
axis, since the g factor along this axis is very small
( 0.2).g <
So, as a first step we consider in detail the spin
dynamics of the system for this case with the z axis along
the static magnetic field. After a unitary transformation to
a new coordinate system, the Hamiltonian takes the form
0 ex:KH H H H= + +
( )
( )
( )
0
ex
,
,
1 .
2
z z
i i
i
y yx x z z
K i i i i i i
i
y yx x z z
ij i j i j ij i j
ij
H c c B g S g
H J S S J S
H I S S S S I S S
+
ν ν ⊥ σ
ν
⊥
⊥
= ε + + σ
= σ + σ + σ
= + +
∑ ∑
∑
∑
k k k
k
(13)
The ESR response is given by the transverse (relative to z
axis) dynamical susceptibility for a total magnetization of
the Yb ions and conduction electrons
( ) ( ); , , .sαβ
αβ
χ ω = χ ω α β = σ∑ (14)
Partial susceptibilities ( )αβχ ω can be represented by the
Green functions
2
2
, ,
, .
ss s
s
g S S g g S
g g S g
− + − +
⊥ σ ⊥ σ
− + − +
σ σ ⊥ σσ σ
χ = − χ = − σ
χ = − σ χ = − σ σ
(15)
Here A B is the Fourier transform of the retarded
commutation Green function
[ ]
0
exp ( ) ( ), (0) .A B i dt i t A t B
∞
= − ω∫ (16)
,S σ are the total spin operators of Yb ions and conduction
electrons; ( ),x yS S iS± = ± ( )x yi±σ = σ ± σ , respectively,
and ... means the statistical average at temperature T.
The Green functions can be calculated by the perturba-
tion method. As a result a set of coupled equations appears
which is convenient to write in matrix form
2
2
0
0
z
ss s ss s
zs s
g Sa a
a a g
⊥σ σ
σ σσ σ σσ σ
χ χ = χ χ σ
(17)
with
2
2
, ,
, .
z
ss s ss s s
z
s s ss
ga a g S
g
g
a g a
g
⊥
σ ⊥ σ
σ
σ
σ σ σ σ σσ
⊥
= ω−ω +Σ = λ − Σ
= σ λ − Σ = ω−ω +Σ
(18)
Here ( )/ ,J J g g⊥ ⊥ σλ = +
/4( ),zS g B T⊥= − + θ
/2,z g Bσσ = − ρ ρ is the conduction electron density of
states at the Fermi surface. The resonance frequencies of
Yb ions and conduction electrons, containing the first order
Knight shifts due to the Kondo and RKKY interactions are
given by
( ) 4 ,
.
z z
s
z
g B J S
g B J S
⊥ ⊥
σ σ ⊥
ω = + σ + θ
ω = +
(19)
Here θ is the Weiss temperature due to the RKKY interac-
tion in a molecular field approximation
1 .
4 ij
j
Iθ = ∑ (20)
The imaginary part of the kinetic coefficient Im( )ss ssΓ = Σ
describes the spin relaxation of Yb ions toward conduction
electrons, which remain in a thermodynamically equilibri-
um state (the Korringa relaxation rate). The Overhauser
relaxation rate (conduction electrons relax toward Yb spin
system, being in the equilibrium with the thermal bath) is
described by the kinetic coefficient Im( ).σσ σσΓ = Σ Calcu-
lations of these coefficients up to the third order in the
Kondo interaction give [24]
( )
( ) ( )
2 2 2 2
|| ||
2 2 2
|| ||
3 8 ln / ,
4
3 8 ln / ,
8
ss T J J J J W T
g T
J J J J W T
g T
⊥ ⊥
σ
σσ ⊥ ⊥
⊥
π Γ = ρ + + ρ
ρπ Γ = + + ρ + θ
(21)
where W is a half of the band width of the conduction
electrons. Besides, for a correct analysis of a stationary
solution one has to take into account the spin relaxation of
Kondo ions and conduction electrons toward the thermal
bath (“lattice”). Correspondingly, the relaxation rates ,ssΓ
σσΓ in equations (18) should be replaced by ss sLΓ +Γ and
Lσσ σΓ + Γ , respectively.
4. Collective spin modes of localized moments and
conduction electrons
An equilibrium state approximation for the conduction
electron spin system is not valid to study the ESR response
of the samples with a high concentration of Kondo ions. It
is especially the case for a Kondo lattice. Instead, one has
to treat the coupled kinetic equations (17) for both magnet-
ic moments of Kondo ions and conduction electrons. Two
additional kinetic coefficients sσΣ and sσΣ couple the ki-
netic equations of motion for the transverse magnetizations
of localized moments and conduction electrons. Their im-
aginary parts represent additional relaxation rates:
( ) ( ) ( ) ( )
( ) ( ) ( )2
1 ln / ,
4
1 ln / .
2
s
s
g T J J J J J W T
g T
TJ J J J J W T
⊥
σ ⊥ ⊥ ⊥
σ
σ ⊥ ⊥ ⊥
ρπ Γ = + +ρ + + θ
π Γ = ρ + +ρ +
(22)
To study the ESR response of the total system we have to
find solutions of the system (17). The poles of the total
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 97
B.I. Kochelaev
dynamical susceptibility are determined by the condition
0.ss s sa a a aσσ σ σ− = As the result we obtain two complex
roots: their real parts represent resonance frequencies, their
imaginary parts represent the corresponding relaxation
rates. We are interested in a solution close to the Kondo
ion resonance frequency .sω
The coupling between the two systems is especially im-
portant if the relaxation of conduction electrons toward the
Kondo ions is much faster than to the lattice and the reso-
nant frequencies are close to one another (a relaxation
dominated “bottleneck” regime):
( ) ,ss sL L sσ σΓ Γ + Γ ω −ω . (23)
In the case of an isotropic system and equal Larmor
frequencies g g g⊥ σ= =
the ESR linewidth in the bottle-
neck regime is greatly narrowed due to conservation of the
total magnetic moment. In this case the operator of the
total spin commutes with the isotropic Kondo interaction
and the latter disappears from the effective relaxation rate
(see the review by Barnes [32]). Similar conclusions can
be made, if the real parts of the kinetic coefficient sσΣ and
sσΣ satisfy the conditions (23) (a field dominated bottle-
neck regime). In the opposite case of a strongly anisotropic
Kondo interaction the results in the second order in the
Kondo interaction do not show any sufficient narrowing of
the ESR linewidth in the bottleneck regime [33].
The third order of the perturbation theory for the relaxa-
tion rates contain terms proportional to ln( / ),W T what
gives a divergent contribution at low temperatures 0.T →
It is especially important for the antiferromagnetic type of
the Kondo interaction ( , 0).J J⊥ >
The same terms appear
in the scattering amplitudes of conduction electrons by mag-
netic impurities, what results in the Kondo effect in a resis-
tivity. This phenomenon is also of a crucial importance for
the ESR experiments. To reveal it one has to improve the
perturbation theory for the spin relaxation rates.
5. Renormalization of the spin relaxation rates
To develop a satisfactory theory in the low-temperature
regime one has to remove the logarithmic divergences by
summing somehow the higher order terms in the perturba-
tion expansion. Abrikosov [34] carried out a summation of
the leading logarithmic terms for the resistivity applying
the Feynman diagrams technique to the s–d model. Later
Anderson [35] proposed another method known as “Poor
Man’s Scaling” that allows one to extend the lowest order
perturbation results to sum the leading order logarithmic
terms on the basis of the Dyson equation. The main idea of
the “Poor Man’s Scaling” approach is to take into account
the effect of the high-energy excitations on the low energy
physics by a renormalization of coupling constants. Here
we use this idea starting from the second order of the per-
turbation theory in the operator form suggested by
Bogolubov and Tyablikov which was developed for the
case of a degenerate ground energy level [36]. We apply it
for the Kondo interaction in the form (12) in the absence of
the external magnetic field
( )( ) ( )1
0 0 .K K K K KH P H PH P H E H PH P P−δ = − − − −
(24)
Here P is a projecting operator of exited states onto the
space of wave functions of the ground energy level. Substi-
tuting the explicit expression of the Kondo interaction, we
obtain products of the Yb spin operators of the type .i jS Sβα
In the case i j≠ we obtain the RKKY interaction which
was already taken into account by exH (10). In the case
( , 1/2)i j S= = these products can be reduced to linear
forms of spin operators:
( )
( ) ( )
2
2 2
1 1 1, , ,
2 2 4
10, ,
2
1 .
2
z z z
j j j j j j j
z z
j j j j j j j
z z
j j j j j
S S S S S S S
S S S S S S S
S S S S S
+ − − +
+ − − − −
+ + −
= + = − =
= = = − =
= − = −
(25)
The terms, which do not contain the Yb spin operators, can be
related to renormalized conduction electrons energy. The rest
operator of the second order contribution for the j-site is
the following:
_____________________________________________________
{ }
( )
( )
2 1 4 3
1 41 4 1 4
3 41 2 3 4
2 4 2 41 4 1 3 1 3
1 4 2 2 4 2 41 1 3 1 3 1
2
.... ..
..
..
1 e ;
2
,
2
ji
z z
Kj j j j
z
H J S F J J S F S F
F c c c c c c c c
F c c c c c c c c c c c c
− + −
+ − − +
⊥ ⊥
+ + + +
↑ ↓ ↓ ↑↓ ↑ ↑ ↓
− + + + + + +
↑ ↓ ↑ ↑ ↑ ↓↑ ↓ ↓ ↓ ↑ ↓
δ = + + ε − ε
= −
= − − −
∑ ∑
k k k k r
k kk k k k
k kk k k k
k k k kk k k k k k
k k k k k k kk k k k k k
( )
( ) ( )
2
2 4 2 2 2 41 4 1 3 1 1 1 3..
,
2 .F c c c c c c c c c c c c+ + + + + + +
↓ ↑ ↓ ↑ ↓ ↓↑ ↑ ↓ ↑ ↓ ↑
= − − −
k
k k k k k kk k k k k k k k
(26)
________________________________________________
We reduce the fourfold products of the creation and an-
nihilation operators of conduction electrons to quadratic
forms using the following type of approximations for terms
diagonal in spin projections:
98 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
Magnetic properties and spin kinetics of a heavy fermion Kondo lattice
( )
1 42 4 4 21 3 1 3
2 3 2 43 1
1 4 1 2 3 32 43 1
1 .
c c c c c c c c
c c c c
f c c f c c
+ + + +
↑ ↓ ↓ ↑↓ ↑ ↓ ↑
+ +
↑ ↓↑ ↓
+ +
↑ ↓↑ ↓
≈ δ +
+δ =
= −δ + δ −
k kk k k kk k k k
k k k kk k
k k k k k kk kk k
(27)
Here fk is the Fermi function. For nondiagonal terms we
make similar approximations
( )
( )
2 2 41 1 3
1 4 1 21 3
2 3 2 43 1
1 4 1 2 3 32 43 1
1 .
c c c c c c
c c c c
c c c c
f c c f c c
+ + +
↑ ↓ ↑↑ ↓ ↓
+ +
↑ ↑↑ ↓
+ +
↓ ↑↓ ↓
+ +
↑ ↑↓ ↓
− ≈
≈ δ −
−δ =
= −δ −δ −
k k kk k k
k k k kk k
k k k kk k
k k k k k kk kk k
(28)
The result of these approximations is the following:
_____________________________________________________
( ) ( ) 2 1
2 2 2 21 1 1 1
1 2
3 3 3
3 2 3 1 3 23
( )
||
||
|| ||
1 e ;
2
1 21 .
2
jiz
Kj j j jH J S c c S c c J S c c c c
f f f J
J J J J
J
−+ + − + + +
⊥ ↑ ↓ ↑ ↓↓ ↑ ↑ ↓
⊥ ⊥
⊥
= δ + + δ −
−
δ = − + = δ
ε − ε ε − ε ε − ε
∑
∑
k k r
k k k kk k k k
k k
k k k
k k k k k kk
(29)
________________________________________________
The operator structure of this result is exactly the same as
the starting Kondo interaction (12). The divergent contri-
bution to Jδ at 0T → comes from the first term in the se-
cond line in (29). Taking into account the relation
1 2 ( ) tanh( /2 ),f T− ε = ε we make the following approxima-
tion for this function: tanh x x≈ ( 1),x < and tanh 1x ≈
( 1).x > An approximate result for the sum in the second
line in (29) is the following:
3
3 23 2
2 2
20
2 2
2 2 2 2
2 20 2
1 21 tanh ( /2 )( )
2 tanh ( /2 )( )
2 2 ln .
2
W
W
W
T W
T
f Td
N
Td
Wd d
T T
−
− ε
= ερ ε =
ε − ε ε − ε
ε ε
= ερ ε ≈
ε − ε
ρ ε ε ≈ ε + ρ ε ≈ ρ
ε − ε ε − ε
∑ ∫
∫
∫ ∫
k
k kk
(30)
Here ρ is the density of states at the Fermi surface; the
populated level is expected to be 2 2 .Tε < The main con-
tribution comes from the second integral in the range
2 .T W< ε < The other terms can be neglected in a compar-
ison with the divergent one. As the result we have the fol-
lowing equations for the contribution of the second order:
||2
|| ||ln , .
2
JWJ J J J
T J⊥ ⊥
⊥
δ = ρ δ = δ
(31)
As the next step we divide the conduction electrons band
into the low and high energy states
0 , ,W W W< ε < < ε <k k (32)
where ,W W are the initial and the running half of the
bandwidth, respectively. Using the Anderson’s idea, we
want to incorporate the contribution of the high-energy
levels into renormalized values of ,J⊥ ||.J According to
(31) it gives the following relations:
2 2
||
|| ||
ln ln ln ,
2 2
.
W W WJ J J
T T W
J J J J
⊥ ⊥
⊥ ⊥
δ = − = ρ
δ = δ
(33)
These two equations represent the scaling law. From calcu-
lations (30) it follows that equations (33) are valid only for
the range (2 ).T W W< < The projection of the original
Kondo interaction (12) on the low-energy states yields a new
Hamiltonian of the same operator structure, but with renor-
malized coefficients. For a solution of the equations (33) we
start from the second one. A simple integration gives
2 2
|| const.J J⊥ − = (34)
This relation was obtained first by Anderson for 0T = [35].
Since this equation does not depend on the value of the
bandwidth, we choose the original values: 2 2
||const .J J⊥= −
As the next step we take a small difference .W W W− = δ It
is convenient to introduce dimensionless values U J⊥ ⊥= ρ
and || ||.U J= ρ Then according to (33) and (34) we have
( )
2 2
||
2 2 2 2 2
|| 0 ||
ln ,
.
W W WU U U
W W
U U U J J
⊥ ⊥
⊥ ⊥
+ δ δ
δ = ≈
− = = ρ −
(35)
After substitution of the second equation into the first
one we obtain a closed differential equation, which can be
easily integrated:
||
||
||
2 2
|| 0 2
|| ||
0 0 0
;
1 arccot arccot ln .
2
U W
U T
U W
WU U
U U W
U U U T
δ δ
=
+
− =
∫ ∫
(36)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 99
B.I. Kochelaev
Here U
is the final result of the renormalization process.
Let us introduce the values
|| ||
GK
0 0 0
1arccot , exp arccot .
2
U UWT
U U U
ϕ = = −
(37)
Then the final result of the renormalization process can be
presented in the following form:
|| 0 0
1cot , .
sin
U U U U⊥= ϕ =
ϕ
(38)
Since the Zeeman Hamiltonian of Yb ions (6) has the same
operator structure as the Kondo interaction (9) with param-
eters, satisfying the relation || ||/ / ,g g J J⊥ ⊥= we accept the
same relation between the renormalized g factors
|| || ||/ / / .g g J J U U⊥ ⊥ ⊥= =
(39)
The characteristic temperature GKT is obviously independ-
ent of the initial and the running values of parameters, repre-
senting a universal energy scale to govern all the low tem-
perature physics involving the ground Kramers doublet. All
physical quantities can be expressed in terms of the ratios
GK( / ).T T For the isotropic case with g g⊥ =
the character-
istic temperature GKT reduces to the well-known formula
for the Kondo temperature iso
GK ( /2)exp ( 1/ ).T W J= − ρ
This formula follows from the logarithmic approximation;
a more accurate result for the isotropic case is
1/2( ) exp ( 1/ )KT JW J− ρ [37]. It is appropriate to mention
that the scaling equations (33) can be represented in an
alternative form, if to introduce a variable ln ( / ).t W W=
Then from (33) we have
2 2
|| ||
|| 2
||
, ;
, .
WU U U t U U U t
W
dU dUU U U
dt dt
⊥ ⊥ ⊥ ⊥
⊥
⊥ ⊥
δ
δ = = δ δ = δ
= =
(40)
In this form the scaling equations were given by Belov
et al. [25].
The temperature-dependent parameters are logarithmi-
cally divergent at GK:T T→ || GK, 1/ ln ( / ),U U T T⊥
the
perturbative scaling approach begins to break down. Con-
sequently, all results derived by this method are only valid
for temperatures above GK.T
6. The bottleneck regime in presence of the Kondo effect
Using the expressions (38) for the temperature-
dependent Kondo couplings we find the renormalized ki-
netic coefficients
( )
( )
2 2
0
2
0 2
3cot , ,
4 2
1, .
2 4 sin ( /2)
ss ss
s s s
gU T
g T
g U T
g T
⊥
σσ
σ
⊥
σ σ σ
σ
Γ = π ϕ+ Γ = Γ ρ + θ
π
Γ = Γ Γ =
ρ + θ ϕ
(41)
One can see that all kinetic coefficients logarithmically
diverge at GK:T T→ to the leading order in logarithmic
terms they are of the form
( )
( )
2
GK
,
ln /
.
2
ss s
s ss
T
T T
g
g T
σ
⊥
σσ σ
σ
Γ = Γ = π
Γ = Γ = Γ
ρ + θ
(42)
One could think that such a behavior confirms the common
belief that the ESR linewidth of Kondo ions is expected to
be too large for its detection. However, the coupling be-
tween two spin systems makes the situation quite different.
It is interesting to analyze a solution of the coupled equa-
tions (17) under the condition of the bottleneck regime
(23), taking into account the renormalization of kinetic
coefficients. The relaxation rate of the collective spin mode
with a frequency close to the Kondo-ion resonance now is
the following:
eff eff
coll
eff eff
2
;
, .
sL ss L
s s s s
ss ss L L
σ
σ σ σ σ
σ σ
σσ σσ
Γ = Γ + Γ + Γ
Γ Γ Γ Γ
Γ = Γ − Γ = Γ
Γ Γ
(43)
It is important to consider the asymptotic behavior of an
effective Korringa relaxation rate eff
ssΓ and an effective
spin relaxation rate of conduction electrons to the lattice
eff
LσΓ at temperatures approaching to GK.T In this case their
expressions are essentially simplified to be written explicit-
ly as functions of temperature:
( )
( )
eff 4 2
0 GK
eff
ln / ,
8
2
.
ss
L L
U T T T
g
T
g
σ
σ σ
⊥
π
Γ =
Γ = ρ + θ Γ
(44)
This result is remarkable: instead of being divergent ac-
cording to (42), the effective Korringa relaxation rate is
greatly reduced and goes to zero at GK.T T→ Although the
Kondo interaction is strongly anisotropic, the Kondo effect
leads to the common energy scale GKT regulating the tem-
perature dependence of different physical quantities. As a
result of this common energy scale, all divergent terms in
the kinetic coefficients (41) experience their complete mu-
tual cancelation in the collective spin mode even in the
case of a strongly anisotropic system. The effective relaxa-
tion rate of the conduction electrons to the lattice is also
greatly reduced, becoming proportional to temperature
similar to the usual Korringa relaxation rate. This reduction
supports the conditions for the bottleneck regime (23).
It is reasonable to mention that the divergent physical
values appear due to approximate calculations. Neverthe-
less these divergences indicate strongly increasing values
and stimulate more accurate calculations. At the same time
we can conclude that although the used scaling procedure
did not remove divergent terms from the initial relaxation
100 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
Magnetic properties and spin kinetics of a heavy fermion Kondo lattice
rates (22), it was able to remove them from the relaxation
rates of the final solution for the collective modes having
real physical sense.
Concerning the broadening of the ESR linewidth which
is represented by the kinetic coefficient ,sLΓ one could
expect an obvious contribution from the distribution of
effective local magnetic fields due to spin-spin interactions
of the Yb ions. In particular, the usual magnetic dipole-
dipole interactions yield approximately loc 700 Oe,H∆ ≈
while the observed ESR linewidth in 2 2YbRh Si at the X
and Q bands is ESR 200 OeH∆ ≈ at 5 K.T = Therefore, it
is evident that some narrowing mechanism for these type
of contributions should exist. At the same time, it is well
known that the broadening of the ESR line by the distribu-
tion of local fields can be reduced in the bottleneck regime
due to fast reorientation of the Kondo ion spins caused by
the Korringa relaxation [32], the more as being greatly
increased due to the Kondo effect. Another reason for the
relaxation rate sLΓ can be related to the spin-phonon inter-
actions. At temperatures above a few K the main contribu-
tion comes from the two-phonon processes. The Orbach
process via the excited energy level at a given crystal field
splitting ∆ yields the temperature dependence
[ ] 1Orb exp ( / ) 1 .sL T −Γ ∆ − (45)
This type of contribution does not need any scaling procedure.
In the case of the ESR resonance frequency of the col-
lective mode, the situation is somewhat different. For a
single Kondo ion it is well known that besides the usual
Knight shift of the ESR resonance frequency, the Kondo
effect results in a divergent logarithmic term. The same
happens with the resonance frequency of the conduction
electrons. We have found that all divergent parts of the
ESR resonance frequency cancel each other in the collec-
tive mode similar to the relaxation rate described above.
However, the RKKY interaction provides an additional
local field at the Yb ion and the Weiss constant θ in the
spin susceptibility. In the molecular field approximation
involving both the Kondo and RKKY interactions, θ be-
comes also a subject of the Kondo renormalization. As a
result, the ESR resonance frequency contains a divergent
logarithmic term even for the collective spin mode. For the
corresponding effective g factor effg⊥ the following relation
was obtained for the magnetic field perpendicular to the
symmetry axis [23]:
0eff
0
11 1 arccot cot .
2
Ug U U
T Ug
⊥
⊥
θ = + + + − ϕ
(46)
Next we consider the variation of the ESR parameters
with the orientation of the microwave magnetic field hav-
ing the angle η relative to the plane perpendicular to the c
axis [25]. The static magnetic field is still perpendicular to
the c axis and microwave field. It was found that the g fac-
tor g⊥ and the relaxation rates (41) do not depend on the
orientation of the microwave magnetic field. It is obvious
that the relaxation rate of the collective spin mode given
by (43) and the resonant frequency are also independent of
the angle .η
Concerning the ESR intensity, the situation is quite dif-
ferent. It is known that the intensity is determined by the g
factor component along the direction of the microwave
magnetic field: 2 2 2 2
|| sin cos .I g g⊥η+ η Taking into ac-
count the renormalization of the g-factor components in
accordance with (39), it was obtained for renormalized
ESR intensity
( ) ( ) 2 2, 0, 1 sin sin ,I T I T η = − ϕ η
(47)
where ϕ is defined by (37). The ratio (0, ) ( /2, )I T I T= π is
reduced to 21/ cos ϕ instead of the values 2 2
||/ 14g g⊥ ≈ and
2 2
||/ 400g g⊥ ≈ in the cases of 2 2YbIr Si and 2 2YbRh Si , re-
spectively.
It is reasonable to expect, that since the all divergent
terms at GKT T→ were mutually canceled in kinetic coef-
ficients of the collective modes, the obtained result can be
valid even at temperatures close to the GK.T
7. Experimental ESR results
As a matter of fact, the ESR measurements with very
interesting results were performed first at every step and
then have stimulated the theoretical investigations de-
scribed in the previous sections. At first, we use the diver-
gent logarithmic term in the g factor (46) to reveal the
characteristic temperature GK.T As the starting value for
the temperature dependence of the g factor was taken tem-
perature-independent experimental result at high tempera-
tures 3.65,g⊥ = the density of states can be related to the
band width of the conduction electrons as 1/ .Wρ = The
result of the fitting is given in Fig. 4(a) with 0 0.18,U =
0.2 Kθ ≈ and GK 0.36 K;T = the latter is by two orders of
magnitude smaller than the Kondo temperature KT derived
thermodynamically [38] and by transport measurements [39].
The revealed value GKT was used to fit the temperature
and frequency dependences of the ESR linewidth with help
of equations (43) in order to see whether the theory is self-
consistent:
Orb
theor coll const.sLΓ = Γ + Γ + (48)
Here const represents the local field distribution which is
greatly reduced as discussed above. The results are given
in Fig. 4(b). The fitting of the temperature dependence of
the ESR linewidth gave 0U about the same and 198 K.∆ =
The latter coincides with the first excited energy level of
the Yb ion [20], confirming that the Orbach processes
dominate in the spin-phonon relaxation. Having the value
0 0.18,U = we can now estimate the Korringa relaxation
rate without the bottleneck regime. According to equa-
tion (41) this value yields 50 GHzssΓ ≈ at 5 K. Such
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 101
B.I. Kochelaev
a linewidth would leave no chance to observe the ESR
signal, neither at X or Q bands nor at higher frequencies
without formation of the collective spin mode.
Next we compare the developed theory with the exper-
imental data for the case of an arbitrary orientation of the
microwave field with respect to basal plane [25]. The static
field was always perpendicular to the c axis (η is varying).
The resonant frequency and the relaxation rate were pre-
dicted to be independent of the angle what is in full agree-
ment with experimental data of 2 2YbRh Si and 2 2YbIr Si
(see Fig. 5). The angular dependence of the ESR intensity
is fitted with the theoretical expression (47). The results of
the fitting are given in Fig. 5: the intensity of both compounds
can be described with the same value 2sin 0.5.ϕ ≈ The pa-
rameters 0U and GKT could not be determined unambiguous-
ly because the experimental data on the angular dependence
of the ESR intensity were obtained only for 5 K.T =
8. Concluding remarks
The discovery of the electron spin resonance in the
Kondo lattice with heavy fermions gave not only new op-
portunities to study these interesting materials but also
stimulated new approaches in understanding of their com-
plicated properties. It is evident that the conception based
on the single Kondo ion effect does not work in the case of
Kondo lattice. It is remarkable that the Kondo effect, being
responsible for a suppression of the ESR signal on para-
magnetic impurities in a metal at low temperatures, crucially
supports it in the Kondo lattice due to the formation of the
collective spin mode with a dramatic narrowing of the ESR
linewidth. It happens even in the case of a strongly aniso-
tropic Kondo interaction in contradiction with the case of a
small concentration of the Kondo ions as impurities.
It is interesting to note that there is a deep similarity of
this phenomenon with another one related to the ESR on
paramagnetic impurities in superconductors. The first ob-
servations of the ESR have shown a sharp increase of the
spin relaxation rate after transition into the superconduct-
ing state [40,41]. It was in a full agreement with results of
the nuclear magnetic resonance (NMR) and the Bardeen–
Cooper–Schrieffer (BCS) theory. The well-known Hebel–
Slichter peak appears due to increasing of the spin relaxa-
tion rate to conduction electrons in consequence of sharp
increasing of their density of states at the energy gap near
the Fermi level and a coherence factor of the Cooper par-
ing. However, very soon an unexpected phenomenon was
observed by ESR on erbium impurities in lanthanum: the
ESR linewidth was sharply narrowed instead of an increase
after the transition to the superconducting state [42]. In
order to understand this phenomenon it was suggested that
an increase of the coupling between the spin systems of
impurities and superconducting electrons resulted in a tran-
sition into the bottleneck regime [43]. This idea was
proved later by the direct analysis using the Feynman dia-
gram technic [44]. In the case of the NMR the bottleneck
regime cannot appear because of a great difference of elec-
tron and nuclear magnetic moments. As a matter of fact the
phenomenon of the ESR narrowing in superconductors and
in the Kondo lattice have a common nature: a formation of
a collective modes of two spin subsystems due to enhanced
coupling between them caused on the one hand by a transi-
tion to the superconducting state and by the Kondo effect
on the other hand.
It is appropriate to mention some questions still waiting
their elucidation. In particular, it is not clear how the char-
acteristic temperature for the ground Kramers doublet GKT
Fig. 4. (Color online) Temperature dependence of Q band (a) g
factor and (b) ESR relaxation rate Γ of 2 2YbRh Si (open cir-
cles) from [23]. Solid lines denote data fitting: effg⊥ (Eq. (46))
and theorΓ (Eq. (48)) with two contributions collΓ and Orbach
sLΓ
as indicated. Inset: frequency dependence of theorΓ (solid line)
fitted to Γ (T = 10 K).
Fig. 5. Variation of the ESR intensity of 2 2YbRh Si and
2 2YbIr Si by changing the angle η only and keeping the static
field always perpendicular to the c axis (from [25]). The solid
lines denote data fitting according to Eq. (47) with the ratio
( )/ (0).I Iη The upper and lower graphs show the corresponding
linewidth H∆ and the g factor ,g⊥ respectively. Details on the
intensity determination are given in [7].
102 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
Magnetic properties and spin kinetics of a heavy fermion Kondo lattice
relates to the thermodynamical Kondo temperature .KT An
analysis of the phase transition from the NFL state to the
AFM state needs additional ESR experiments at very low
temperatures and magnetic fields and theoretical efforts.
There is no common understanding of the phase transition
from the AFM state to the LFL state near the QCP.
Acknowledgments
I am very grateful to Ilya M. Lifshitz for his support at
the beginning of my scientific carrier. Many thanks to
J. Sichelschmidt, S.I. Belov, and A.S. Kutuzov for a help
in preparation of the paper. This work is supported by the
subsidy allocated to Kazan Federal University for the state
assignment in the sphere of scientific activities.
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1. Introduction
2. Crystal electric field and magnetic susceptibility
3. Dynamical magnetic susceptibility
3.1. Theoretical model
3.2. Spin dynamics
4. Collective spin modes of localized moments and conduction electrons
5. Renormalization of the spin relaxation rates
6. The bottleneck regime in presence of the Kondo effect
7. Experimental ESR results
8. Concluding remarks
Acknowledgments
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| id | nasplib_isofts_kiev_ua-123456789-129356 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:06:26Z |
| publishDate | 2017 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Kochelaev, B.I. 2018-01-19T13:58:50Z 2018-01-19T13:58:50Z 2017 Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice / B.I. Kochelaev // Физика низких температур. — 2016. — Т. 43, № 1. — С. 93-103. — Бібліогр.: 44 назв. — англ. 0132-6414 PACS: 76.30.Kg, 71.27.+a, 75.40.Gb https://nasplib.isofts.kiev.ua/handle/123456789/129356 A review of peculiarities of magnetic properties and spin kinetics of a heavy-fermion Kondo lattice revealed by electron spin resonance (ESR) experiments and their theoretical analysis is given. Among the issues discussed in some detail are the renormalization of spin kinetics coefficients due to the Kondo effect, formation of the collective spin modes of the Kondo ions and wide-band conduction electrons, unexpected behavior of ESR parameters as functions of temperature and magnetic fields. Special attention is focused on the possible role of the Kondo effect for the ESR signal existence at low temperatures. I am very grateful to Ilya M. Lifshitz for his support at the beginning of my scientific carrier. Many thanks to J. Sichelschmidt, S.I. Belov, and A.S. Kutuzov for a help in preparation of the paper. This work is supported by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 100-летию со дня рождения И.М. Лифшица Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice Article published earlier |
| spellingShingle | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice Kochelaev, B.I. К 100-летию со дня рождения И.М. Лифшица |
| title | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice |
| title_full | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice |
| title_fullStr | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice |
| title_full_unstemmed | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice |
| title_short | Magnetic properties and spin kinetics of a heavy-fermion Kondo lattice |
| title_sort | magnetic properties and spin kinetics of a heavy-fermion kondo lattice |
| topic | К 100-летию со дня рождения И.М. Лифшица |
| topic_facet | К 100-летию со дня рождения И.М. Лифшица |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129356 |
| work_keys_str_mv | AT kochelaevbi magneticpropertiesandspinkineticsofaheavyfermionkondolattice |