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Ferromagnetic dense Kondo behavior of Ce-Si system
0038-lO98/82/27Ol93-05$03.00/O
Solid State Communicaitons, V01.43,No.3, pp.l93-197, 1982. Printed in Great Britain.
Pergamon Press Ltd.
FERROMAGNETIC DENSE KONDO BEHAVIOR OF Ce-Si SYSTEM H. Yashima, H. Mori and T. Satoh Department of Physics, Faculty of Science Tohoku University, Sendai, 980 Japan K. Kohn School of Science and Engineering, Waseda University Tokyo, 160, Japan (Received 3 March 1982 by J. Kanamori) In order to clarify the nonmagnetic-magnetictransition found in Ce-Si system, magnetization measurements have been made to reveal a large reduction of the magnetic moment of Ce atom. Taking into account the results and also the reduction of the magnetic entropy, we propose a dense Kondo model to interpret our Ce-Si system. Kondo temperatures are deduced from various physical quantities and a comparison is made between them. Our Ce-Si system is the first example of a ferromagnetic dense Kondo system.
A valence fluctuation state is believed to occur due to the proximity of the 4f level to the Fermi energy level. There have been many efforts to find and study systems in which one can control the relative position of these two energy levels. We have recently reported1) a systematic investigationon CeSi, in the composition range 1.55 Ix 52.00, where the system remains.in the same crystal structure of the a-ThSi2 type. Measurements of the magnetic susceptibility and the specific heat revealed that in the composition range 1.85 -
the entropy of the magnetic moment system by forming a singlet ground state whereas the latter exhausts it by a magnetic ordering. The interpretation of our case is that the Kondo effect reduces the magnetic moment and, at lower temperatures the RKKY interaction dominates to make a magnetic ordering. If this is the true picture of CeSil.80 and CeSil.70, our system is the first example of a ferromagnetic dense Kondo system. TO substantiate our picture we make an estimation of the Kondo temperatures, TK, of our system including the samples which exhibit no magnetic order. Expressions for the single impurity problem are used throughout because there is no established theoretical treatment for a dense Kondo system. From the high temperature magnetic susceptibilityTK is obtained with the relation2) C’
X(T) = T+2TK
(1)
where C' is the Curie constant. The TK thus obtained is a half of 0 tabulated in the previous reportl). The specific heat data below T, can be used to estimate TK as follows. The magnetic entropy associated with an ordering AS is reduced from RLn2, corresponding to a ground doublet, by an amount already exhausted above Tc by the Kondo effect. A Kondo effect, when only a ground doublet is involved at low temperatures, also exhausts an entropy of RP.n2. Therefore,
The reduction in the magnetic moment and the magnetic entropy suggests a Kondo effect. In the present case the Ceions form a periodic array and one has to consider a dense Kondo system, in which the Kondo effect and the RKKY interaction coexist. The former effect exhausts
As an analytical expression for CKondo is not available, we replace CKondo with a two-level Shottky specific heat with an energy splitting kBTK. The ratio of TK and T, is then expressed 193
194
FERROYAGSXIC Tab.1
y
: coefficient
x(O) a
of
of
: magnetic
where
TC
s
I(C CeSix-
0
is 1.70
’ ‘LaGe
is
2 a part
is
not
the
m.I
term of
in the
magnetic
the
specific
heat.
susceptibility.
low temperature
susceptibility
temperature. below
Tc obtained
with
the
formula
)/T]dT
CLaSi specific
subtracted. of
entropy
is
of
LaSi2.
almost It
is
The specific
same as that likely
from upper
a x
x (0) 2)
heat
energy
that
of
LaSi2.
AS thus
levels,
heat
the
of For
obtained amount of
certain.
10
3
(k-*)
(2)
mole-K
CeSi2.00
104
4.2~10-~
-0.42
C%.90
151
7.oX1o-3
-0.55
234
4.0x10
-2
CeSil.
85
CeSil.
80
9.0
3.41
CeSil.
70
10.9
4.53
7.0
5.74
CeGe2
with
state
5 x < 2.00
Y
(
T term
SYSTE?l
2
for
contains which
2
entropy
C LaSi2
LaSix6) CeCe2
T
transition
AS : magnetic AS =
linear
ground
7 = i((O) [l+aT-1.
x(T) Tc
the
: extrapolated
: coefficient
OF Ce-Si
DENSE KOKDO BEHAVIOR
AS as
-TK/T, AS = R{.Zn(l+e ?iext we extend our picture of the dense Kondo state to the non-magnetic CeSix(1.85’x< 2.00). Here we consider the Kondo effect to dominate the RKKY interaction, thereby favoring a non-magnetic ground state with strongly exTK can change-enhanced Fermi liquid behavior.
-2.7
be obtained with eq.(l) The estimation of TK is specific heat coefficient expression3) for S=1/2.
for x=1.90 and x=1.85. made from the linear Y using Yoshimori’s
,Attempts to interprete the large Y value of as the result of a Kondo Ce compounds effect have been proceeded by several authors4).
FEXRO!UGNETIC DE?ISEKONDO BEHAVIOB OF Ce-Si SYSTEM
Vol. 43, No. 3
-0.6 aJ
P’O
-
-Cl----
-J
CeGe,
/
Y .3 IO.4
.a
/” .
k
.*-
_
,x-A--x--
X-x
SC----* I
_.%-&-A-+
x-x--x-
-A--
CeS
i, .7~
CeS i, .80
T=4.2
K I
10
5
I
l5 H(kOe)
Fig.1 Magnetization of CeSil 80, CeSil 7. and CeCe2 at 4.2K as a function of the applied field measured using the pendulum type magnetometer.
CeGe,
T =4.2
1 0
20
40
K
6o H (kCe)e’
Fig.2 Magnetization of CeGe at 4.2K measured with the fluxmetric 2 method up to 80 Koe.
FERROMAGNETICDENSE KONDOBEHAVIOR OF Ce-Si
196 The low temperature also used to deduce state susceptibility x(0)
= NA.
magnetic susceptibility is TK. The theoretical ground x(O) is expressed as (Peff
u’B)%(s+l)
; Avogadro’s
where S=1/2 tive g-value
(S)
number,
and we tentatively adopt the effecof the I-‘7 doublet in the cubic
;;:::;;
;;e:;eg~~~i;
Another sions)
way of x(T)
;;;zel;l
;~c;;;;lt~,~;;;en_
obtaining
TK is _ _ T’ T2 = x(D) (l-? --$ TK
to use
an expres-
43,.No.
In conclusion, we have presented the first example of a dense Kondo system which orders ferromagnetically. We have also shown that the Kondo temperature changes smoothly with the Si concentration in the CeSi, system. The Ce ion in the present system can be regarded as an
.
The values of TK obtained with these formulae are listed in Table II.
Vol.
Kondo effects and also that spin-orbit coupling is not fully taken into account, the agreement between the TK ‘s deduced from different physical quantities should be regarded rather good. The monotonic change of TK with x , especially the smooth variation of TK across the magnetic ordering region and the non-magnetic region seems to support our picture of the Ce-Si system One notes the general as a dense Kondo system. trend that, in the non-magnetic region, the TK’s ontained from x(O) are smaller than those from reasonable because Y . This is qualitatively x(0) is actually enhanced over a single-impurity value by an interatomic interaction.
3kBTK
NA
SYSTEM
various
Tab. II ::ondo
temperature
TK1 TK3 TK5
deduced
from various
: with eq. (1))
T
: with eq.(6),
T
K2
:
K4
with .
physical
quantities.
eq.(3)
: with eq. (4)
: with eq.(S)
CeSi2.G0
TK3
TK4
TK5
(K)
(R)
(K)
89
132
I
69
77
35
38
I
CeSil.
80
cesil.
70
q---T
-’
In Fig.3 we plot the Kondo temperatures estimated by various methods as a function of that we the Si concentration x . Considering have applied the expressions for single-impurity
almost stable trivalent ion. The position of the 4f level is below the Fermi level and the separation becomes larger with the reduction of the Si concentration. An approach from a dense
3
VOL! 13,
x0.
FERROMGNETIC DENSE KONDO BEtiVIOR OF Ce-Si
3
CeS
SYSTEM
197
ix
o TKI A TKZ x
G3
q
TKL
l
TKS
0
0 *
x 0
0 . 0
.
x
z
Q
0
1.9
1.8
1.7
Fig.3
Kondo temperatures are
plotted
: with eq.(l), TK1 : with eq.(6), TK3 TK5
deduced
as a function
from of
various
the
2.0 X
Concentration
physical
Si concentration
quantities x.
: with eq.(3), TKZ : with eq.(4), TK4
: with es.(S).
Kondo model is promising although a quantitative analysis must await more theoretical work.
We wish to thank Prof. T. Kasuya for able discussions and Prof. T. Ohtsuka for nuous encouragement during the study.
valuconti-
References
1. H. Yashima and T. Satoh; to appear in Solid State Communication 2. H. R. Krishna-murthy, K. G. Nilson and J. W. Wilkins; Phys. Rev. Letters -35 (1975) 1101. 3. A. Yoshimori; Progr. Theor. Phys. -55 (1976) 67.
4
5 6
C. D. Bredl, F. Steglich and K. D. Schotte; Z. Physik B29 (1978) 327. A. Benoit,T Flouquet and M. Ribault; J. de Physique CS (1979) 328. K. D. Schotte ax V. Schotte; Phys. Lett. 55A (1975) 38. H. Yashima and T. Satoh (unpublished).
Solid State Communicaitons, V01.43,No.3, pp.l93-197, 1982. Printed in Great Britain.
Pergamon Press Ltd.
FERROMAGNETIC DENSE KONDO BEHAVIOR OF Ce-Si SYSTEM H. Yashima, H. Mori and T. Satoh Department of Physics, Faculty of Science Tohoku University, Sendai, 980 Japan K. Kohn School of Science and Engineering, Waseda University Tokyo, 160, Japan (Received 3 March 1982 by J. Kanamori) In order to clarify the nonmagnetic-magnetictransition found in Ce-Si system, magnetization measurements have been made to reveal a large reduction of the magnetic moment of Ce atom. Taking into account the results and also the reduction of the magnetic entropy, we propose a dense Kondo model to interpret our Ce-Si system. Kondo temperatures are deduced from various physical quantities and a comparison is made between them. Our Ce-Si system is the first example of a ferromagnetic dense Kondo system.
A valence fluctuation state is believed to occur due to the proximity of the 4f level to the Fermi energy level. There have been many efforts to find and study systems in which one can control the relative position of these two energy levels. We have recently reported1) a systematic investigationon CeSi, in the composition range 1.55 Ix 52.00, where the system remains.in the same crystal structure of the a-ThSi2 type. Measurements of the magnetic susceptibility and the specific heat revealed that in the composition range 1.85 -
the entropy of the magnetic moment system by forming a singlet ground state whereas the latter exhausts it by a magnetic ordering. The interpretation of our case is that the Kondo effect reduces the magnetic moment and, at lower temperatures the RKKY interaction dominates to make a magnetic ordering. If this is the true picture of CeSil.80 and CeSil.70, our system is the first example of a ferromagnetic dense Kondo system. TO substantiate our picture we make an estimation of the Kondo temperatures, TK, of our system including the samples which exhibit no magnetic order. Expressions for the single impurity problem are used throughout because there is no established theoretical treatment for a dense Kondo system. From the high temperature magnetic susceptibilityTK is obtained with the relation2) C’
X(T) = T+2TK
(1)
where C' is the Curie constant. The TK thus obtained is a half of 0 tabulated in the previous reportl). The specific heat data below T, can be used to estimate TK as follows. The magnetic entropy associated with an ordering AS is reduced from RLn2, corresponding to a ground doublet, by an amount already exhausted above Tc by the Kondo effect. A Kondo effect, when only a ground doublet is involved at low temperatures, also exhausts an entropy of RP.n2. Therefore,
The reduction in the magnetic moment and the magnetic entropy suggests a Kondo effect. In the present case the Ceions form a periodic array and one has to consider a dense Kondo system, in which the Kondo effect and the RKKY interaction coexist. The former effect exhausts
As an analytical expression for CKondo is not available, we replace CKondo with a two-level Shottky specific heat with an energy splitting kBTK. The ratio of TK and T, is then expressed 193
194
FERROYAGSXIC Tab.1
y
: coefficient
x(O) a
of
of
: magnetic
where
TC
s
I(C CeSix-
0
is 1.70
’ ‘LaGe
is
2 a part
is
not
the
m.I
term of
in the
magnetic
the
specific
heat.
susceptibility.
low temperature
susceptibility
temperature. below
Tc obtained
with
the
formula
)/T]dT
CLaSi specific
subtracted. of
entropy
is
of
LaSi2.
almost It
is
The specific
same as that likely
from upper
a x
x (0) 2)
heat
energy
that
of
LaSi2.
AS thus
levels,
heat
the
of For
obtained amount of
certain.
10
3
(k-*)
(2)
mole-K
CeSi2.00
104
4.2~10-~
-0.42
C%.90
151
7.oX1o-3
-0.55
234
4.0x10
-2
CeSil.
85
CeSil.
80
9.0
3.41
CeSil.
70
10.9
4.53
7.0
5.74
CeGe2
with
state
5 x < 2.00
Y
(
T term
SYSTE?l
2
for
contains which
2
entropy
C LaSi2
LaSix6) CeCe2
T
transition
AS : magnetic AS =
linear
ground
7 = i((O) [l+aT-1.
x(T) Tc
the
: extrapolated
: coefficient
OF Ce-Si
DENSE KOKDO BEHAVIOR
AS as
-TK/T, AS = R{.Zn(l+e ?iext we extend our picture of the dense Kondo state to the non-magnetic CeSix(1.85’x< 2.00). Here we consider the Kondo effect to dominate the RKKY interaction, thereby favoring a non-magnetic ground state with strongly exTK can change-enhanced Fermi liquid behavior.
-2.7
be obtained with eq.(l) The estimation of TK is specific heat coefficient expression3) for S=1/2.
for x=1.90 and x=1.85. made from the linear Y using Yoshimori’s
,Attempts to interprete the large Y value of as the result of a Kondo Ce compounds effect have been proceeded by several authors4).
FEXRO!UGNETIC DE?ISEKONDO BEHAVIOB OF Ce-Si SYSTEM
Vol. 43, No. 3
-0.6 aJ
P’O
-
-Cl----
-J
CeGe,
/
Y .3 IO.4
.a
/” .
k
.*-
_
,x-A--x--
X-x
SC----* I
_.%-&-A-+
x-x--x-
-A--
CeS
i, .7~
CeS i, .80
T=4.2
K I
10
5
I
l5 H(kOe)
Fig.1 Magnetization of CeSil 80, CeSil 7. and CeCe2 at 4.2K as a function of the applied field measured using the pendulum type magnetometer.
CeGe,
T =4.2
1 0
20
40
K
6o H (kCe)e’
Fig.2 Magnetization of CeGe at 4.2K measured with the fluxmetric 2 method up to 80 Koe.
FERROMAGNETICDENSE KONDOBEHAVIOR OF Ce-Si
196 The low temperature also used to deduce state susceptibility x(0)
= NA.
magnetic susceptibility is TK. The theoretical ground x(O) is expressed as (Peff
u’B)%(s+l)
; Avogadro’s
where S=1/2 tive g-value
(S)
number,
and we tentatively adopt the effecof the I-‘7 doublet in the cubic
;;:::;;
;;e:;eg~~~i;
Another sions)
way of x(T)
;;;zel;l
;~c;;;;lt~,~;;;en_
obtaining
TK is _ _ T’ T2 = x(D) (l-? --$ TK
to use
an expres-
43,.No.
In conclusion, we have presented the first example of a dense Kondo system which orders ferromagnetically. We have also shown that the Kondo temperature changes smoothly with the Si concentration in the CeSi, system. The Ce ion in the present system can be regarded as an
.
The values of TK obtained with these formulae are listed in Table II.
Vol.
Kondo effects and also that spin-orbit coupling is not fully taken into account, the agreement between the TK ‘s deduced from different physical quantities should be regarded rather good. The monotonic change of TK with x , especially the smooth variation of TK across the magnetic ordering region and the non-magnetic region seems to support our picture of the Ce-Si system One notes the general as a dense Kondo system. trend that, in the non-magnetic region, the TK’s ontained from x(O) are smaller than those from reasonable because Y . This is qualitatively x(0) is actually enhanced over a single-impurity value by an interatomic interaction.
3kBTK
NA
SYSTEM
various
Tab. II ::ondo
temperature
TK1 TK3 TK5
deduced
from various
: with eq. (1))
T
: with eq.(6),
T
K2
:
K4
with .
physical
quantities.
eq.(3)
: with eq. (4)
: with eq.(S)
CeSi2.G0
TK3
TK4
TK5
(K)
(R)
(K)
89
132
I
69
77
35
38
I
CeSil.
80
cesil.
70
q---T
-’
In Fig.3 we plot the Kondo temperatures estimated by various methods as a function of that we the Si concentration x . Considering have applied the expressions for single-impurity
almost stable trivalent ion. The position of the 4f level is below the Fermi level and the separation becomes larger with the reduction of the Si concentration. An approach from a dense
3
VOL! 13,
x0.
FERROMGNETIC DENSE KONDO BEtiVIOR OF Ce-Si
3
CeS
SYSTEM
197
ix
o TKI A TKZ x
G3
q
TKL
l
TKS
0
0 *
x 0
0 . 0
.
x
z
Q
0
1.9
1.8
1.7
Fig.3
Kondo temperatures are
plotted
: with eq.(l), TK1 : with eq.(6), TK3 TK5
deduced
as a function
from of
various
the
2.0 X
Concentration
physical
Si concentration
quantities x.
: with eq.(3), TKZ : with eq.(4), TK4
: with es.(S).
Kondo model is promising although a quantitative analysis must await more theoretical work.
We wish to thank Prof. T. Kasuya for able discussions and Prof. T. Ohtsuka for nuous encouragement during the study.
valuconti-
References
1. H. Yashima and T. Satoh; to appear in Solid State Communication 2. H. R. Krishna-murthy, K. G. Nilson and J. W. Wilkins; Phys. Rev. Letters -35 (1975) 1101. 3. A. Yoshimori; Progr. Theor. Phys. -55 (1976) 67.
4
5 6
C. D. Bredl, F. Steglich and K. D. Schotte; Z. Physik B29 (1978) 327. A. Benoit,T Flouquet and M. Ribault; J. de Physique CS (1979) 328. K. D. Schotte ax V. Schotte; Phys. Lett. 55A (1975) 38. H. Yashima and T. Satoh (unpublished).