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On the residual resistivity of CeAl2 and CeCu6. implications for the Kondo lattice model
Volume 138, number 9
PHYSICS LETTERS A
17 July 1989
ON THE RESIDUAL RESISTIVITY OF CeAI2 AND CeCu6. IMPLICATIONS FOR THE KONDO LATTICE MODEL B. BARBARA, J. BEILLE, B. CHEIATO, A. NAJIB and S. ZEMIRLI Laboratoire Louis Née!, CNRS, 166X, 38042 Grenoble Cedex, France Received 20 September 1988; revised manuscript received 18 March 1989; accepted for publication 25 April 1989 Communicated by D. Bloch
High pressure resistivity experiments performed on CeAI2 and CeCu6 show that the residual resistivity Pa intervening in P POll + a ( T/ 7’s) 2_••• is a function of the characteristic temperature T~.A scaling form is proposed to fit the low temperature resistivity of heavy fermions.
It is well established that on non-magnetic Kondo 2 term their low temperacompounds show a T ture resistivity [1]. In most experiments [2,3] and theories (see e.g. ref. [4]) resistivity PM fits the law PM=PO[1~(T/TC)—..i,
(1)
tively the effect resonant level to an in-a crease broaden of T~. This must giveleading in particular Kondo contribution to the residual resistivity, function of T~: pg’=p
(2)
0_-p8=f(T~).
where Pu is a residual resistivity and T~the characteristic Kondo-lattice temperature. In principle, it should be possible to extract this temperature from expression (1); however, it is not clear whether the parameter Poa intervening in A =Po a! T~is dependent on T~or not. It might be argued, following ref. [2], where it is shown that A/y~= const, that Poa does not depend on T~.However, this argument assumes that Yo 1 / T~,i.e. a result of the single-ion Kondo theory [6,71. Although such a theory gives sometimes the right order of magnitude for thermodynamical variables, it cannot be used without caution in a Kondo lattice. In particular, the singleion Kondo theory for electrical transport cannot explain the low temperature resistivity of a Kondo lattice: the residual resistivity ofan assembly ofKondo impurities with concentration c should be equal to cpa, where p~is the unitary limit [81, whereas in a perfect Kondo lattice it should decrease at low ternperature. In real Kondo compounds the weak disorder due to impurities must break the penodicity of the many-body resonance near the Fermi energy, i.e. reduce the coherence length ~ to approximately the mean distance 1,. between impurities and correla-
Here Pu and p8 represent respectively the total (PM ( T= 0)) and non-Kondo (pressure independent) residual resistivity. In order to test the sensitivity of any parameter upon variations of the ternperature Tv., it is convenient to apply high hydrostatic pressures. We have performed resistivity experiments on single crystals of CeAl2 and CeCu6 down to 1.5 K, up to 150 kbar [9,101. In this Letter, we show that A depends on 7’,. according to the proportion A T~2.5; a generalized scaling form to describe the low temperature resistivity of a Kondo lattice in the presence of weak disorder is proposed. Using the quadratic behaviour of the resistivity of CeA12 above 30 kbar (see refs. [9,101) shown in fig. I we have determined two rows of values for T,. (F), using the following forms for the magnetic resistivity: -~
2
Pm(1~)=Po(1~)E1~ao(T/T) —...]
(3)
or 2
pm(P)=Po(P)+ao(T/Tc)
—....
(4)
a0 and a are temperature and pressure independent. These numbers have been evaluated by adjusting the measured values of T,. in such a way that,
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
‘~
509
Volume 138, number 9
PHYSICS LETTERS A 200
/
/
17 July 1989
___________
T,IK)
~(mr~ 14~
CeAI
/
2
iso.
12L I 101
~1
t
:
~
~
/ -
100
(b)8=R,
~: ~
0
ii
20
30
40
50
60
70
50
Fig. 2. Comparative evolution of T~(P)determined from eq. (3) (•) or eq. (4) (-1-) with 7~)(P)(0) (see ref. (91 for the definition of this line). Insert: schematic variation of the resistivity ofheavy fermions. Since TK and 7~.are related to each other, they 0
1000
2000
3000
4000
i2 (is) 5000 6000
define the same energy scale.
2above30kbar. Fig. 1, Magnetic contribution of the resistivity of CeA12 versus T
very close to unity,coefficient a~(CeAlof (3) is found to be The dimensionless 2)~ 1.1 ±0.2.It is important to mention that, proceeding exactly in the
in the limit of P—p 0, T,. equals the Kondo-lattice ternperature of the same system at normal pressure; for the Kondo lattice CeAI2, we have taken T,.(P-~0)~ ~ K. (This temperature is usually called TK; we prefer to take another denomination to distinguish it from the single-ion Kondo temperature, TK.) Then comes our first interesting result: in the pressure dependent case (eq. (3), A=ct0p0(P)/T~) the line T,.(P) lies well below the cross-over line T0(P) separating the low temperature Kondo-lattice regime from the high temperature impurity-like regime ~‘ (fig. 2). On the other 2, hand in T,.(P)~T the second hypothesis (4), 4= we find a~/T,. 0(P),which is not physically realistic since the Kondo-lattice (Ta), the TM) and the cross-over Kondo-impurity like (TK (T 0) temperatures must satisfy the inequalities
same way with CeCu4, we have reached the same conclusions; in particular ct0 (CeCu6) 1.3 ±0.5 (see also ref. [10]). Let us now come to our second result: we have observed that the total residual resistivity of CeA12 and CeCu6 vary very strongly with pressure. More precisely, since we know the variation of TJP), we can plotp~versus T,. for both systems (fig. 3). The result is rather simple with CeAI2 where the “magnetic” residual resistivity can be expressed as a power law in T,.:
T,. < T0 ~ TM (5) (see the inset in fig. 2), TM is here defined as the temperature for which the resistivity is maximum. ~ The cross-over temperatures T0 defined from the inflection point of isotherms or of isobars are about the same (see ref. [9]).
510
p°p°(T/D)—~
(6)
withbe c~ 0.5, p~ i j.d~cm residual and D0 resistivity 2300 K (pg, can also called “magnetic” by reference to the case of disordered magnetic systems (see e.g. ref. [11]).) The behaviour of the residual resistivity of CeCu4 under pressure is more complex, but understandable: at the lowest pressures the decreasing of p~( Tj is very similar to that of CeAl2 with a slope —0.5. However, due to the proximity, in the P—i’ plane, of a monoclinic distortion [12,13] the residual resistivity starts to increase near 40 kbar. The
Volume 138, number 9
PHYSICS LETTERS A
2
I I
pressure-dependent contribution which might vary proportionally to the square root of the Kondo-lat-
a
Ln(~,_P)
17 July 1989
CeAI
the impurities (say a few ppm) as well as the atomic tice remark naturesysof sitestemperature. they occupy We are must unknown in that thesethe “pure” tems. The effect of impurities on heavy fermions will be soon the object of another report. To conclude, let us put on the same scaling form
2
•
1.
.\ •
0~.
our CeCu6: Pu0 results~[l+(T/T)21 concerning the resistivity p of CeAl2 and PPo (7)
LnTK 1,5
2
3
4
5
2, ~ ____________________________________ Ln ~ b CeCu6 2,L.
-
2,2
-
0,5
-
2,0
tortion) we find that the value co~i0.5is consistent with p8 ~ 1 J.L~cm. A better determination of p~ is needed to evaluate D0. We believe that the scaling form (7) might be general for “simple” Kondo lattices, i.e. for those systems which do not present extra phenomena such as
-
__
\\
-
phenomena such weak softening of elastic the occurrence ofas a crystallographic distortion (as in CeCu6, at high pressures), or even less drasticmodes; elastic
-
one must mention that it is not proven that the distortion appearing in CeCu6 is not related to the fluctuation mechanism responsible for the coherence of this system [111. Besides, the log—log plots of eqs. (3) and (4) do not exclude other types of plots; they simply show that if p~( T,.) is a power law, then the
-
LnTK
18
I
1
2
3
4
5
6
In CeAl2 above 30 kbar (to prevent magnetic order), p8 the pressure-independent resistivity is of the order of 1 ~ cm, D0 2300 K should be related to the conduction electron band width cut-off and the coefficient ci0, not very accurate, is of the order of unity. For CeCu6 below 30 kbar (to avoid the lattice dis-
7
Fig. 3. Log—log plot of the magnetic residual resistivity pg’ versus T~in CeAl2 (a) and CeCu6 (b).
maximum observed near highest pressures suggests that, at even higher pressures, where the distortion should5 stabilize, but one with should againtimes the behaviour a p8observe about four T~° This scenario for CeCu larger. 6 together with the rather unambiguous result for CeAI2 strongly suggests that the residual resistivity of Kondo cornpounds (real Kondo lattices) contain an important
exponent is ~. In such a case this exponent should characterize the low temperature behaviour of heavy ferrnions weakly coupled to the lattice. Interestingly, a recent theoretical paper [5] came to the same conclusion p~ T~ but with ~= 1 instead of ~0.5. It would be of interest to verify this effect and especially to evaluate the exponent c in other systems. This might the current these systems. Asadd an to example let usunderstanding go back to theofbeginning of this paper where we mentioned the well admitted 2 result (see e.g. ref. [2]): A~y 0 (8) .
511
Volume 138, number 9
PHYSICS LETTERS A
Our conclusions, not necessarily in contradiction with this result, give (from (7)): 2~. (9) A~ T~ in
Then the relation A/’,’~=const (2) indicates that a Kondo lattice one should have
y
(10)
12, 0xT,~
which differs markedly from the single-ion relation
17 July 1989
References [1~G.R. Steward, Rev. Mod. Phys. 56 (1984) 455. [2]K. Kadowaki and SB. Woods. Solid State Commun. 58 (1986) 507. [3IL. Puech, J.M. Mignot, P. Lejay, P. Haen and J. Flouquet. J.LowTemp.Phys.70(1988)3. 141 P. Morin, C. Vettier, J. Flouquet, M. Konczykowski. Y. Lassaly and J.M. Mignot, J. Low Temp. Phys. 70 (1988)
y 0xT~’. Finally we should mention that this exponent a = 1.2 for the linear specific heat term, ~‘ x ~ can be also determined [9,10,14] by two other independent ways: the plot of Yo ( T,.) versus A( T,.) in CeAI2 under pressure and in the plot of the high temperature specific heat (data from ref. [15 1) y( T) T~ of CeCu6 and CeAI3. To conclude, our results suggest that, when the concentration ofimpurities increases in a Kondo lattice, the rise in the residual resistivity Po and lowering in T,. are non-independent, but connected to each other (likely through the relation (6)). As a consequence the relationship between the low ternperature specific heat and the reciprocal characteristic temperature (Ta’ ) should not be linear, contrary to what is generally admitted, ~
512
377. [5]E. P. Fetisov and DI. Khomskii. Soy. Phys. JETP 43 (1986) 933. [61K.D. Schotte and U Schotte. Phys. Lett. A 55 (1975) 38. 171 N. Andrei and J.H. Lowenstein, Phys. Rev. Len. 46 (1981) 356. [81 P. Nozibres, J. Low Temp. Phys. 17 (1974) 31. [918. Barbara, J. Beille, B. Cheiato, J.M. Laurant, M.F. Rossignol, A. Waintal and S. Zemirli, Phys. Lett. A II 3 (1986) 381;J. Phys. (Paris) 48(1987)635. 110] S. Zemirli, Thesis, U niversity of Grenoble (1986) [11] A.J. Decker, J. App!. Phys. 36 (1965) 906. [121 E. Gratz. E. Bauer, H. Novotny. H. Mueller, S. Zemirli and B. Barbara, J. Magn. Magn. Mater. 63/64 (1987) 312. and references therein. [13] A. Shibata, G. Oonu, Y. OnukiandT. Komatsubara,J. Phys. Soc. Japan 55 (1986) 2086. [14] B. Barbara and S. Zemirli, Solid State Commun., to be published. [l5]G.E. Brodale, R.A. Fisher, N.E. Phillips and J. Flouquet. Phys. Rev. Lett. 56 (1986)390.
PHYSICS LETTERS A
17 July 1989
ON THE RESIDUAL RESISTIVITY OF CeAI2 AND CeCu6. IMPLICATIONS FOR THE KONDO LATTICE MODEL B. BARBARA, J. BEILLE, B. CHEIATO, A. NAJIB and S. ZEMIRLI Laboratoire Louis Née!, CNRS, 166X, 38042 Grenoble Cedex, France Received 20 September 1988; revised manuscript received 18 March 1989; accepted for publication 25 April 1989 Communicated by D. Bloch
High pressure resistivity experiments performed on CeAI2 and CeCu6 show that the residual resistivity Pa intervening in P POll + a ( T/ 7’s) 2_••• is a function of the characteristic temperature T~.A scaling form is proposed to fit the low temperature resistivity of heavy fermions.
It is well established that on non-magnetic Kondo 2 term their low temperacompounds show a T ture resistivity [1]. In most experiments [2,3] and theories (see e.g. ref. [4]) resistivity PM fits the law PM=PO[1~(T/TC)—..i,
(1)
tively the effect resonant level to an in-a crease broaden of T~. This must giveleading in particular Kondo contribution to the residual resistivity, function of T~: pg’=p
(2)
0_-p8=f(T~).
where Pu is a residual resistivity and T~the characteristic Kondo-lattice temperature. In principle, it should be possible to extract this temperature from expression (1); however, it is not clear whether the parameter Poa intervening in A =Po a! T~is dependent on T~or not. It might be argued, following ref. [2], where it is shown that A/y~= const, that Poa does not depend on T~.However, this argument assumes that Yo 1 / T~,i.e. a result of the single-ion Kondo theory [6,71. Although such a theory gives sometimes the right order of magnitude for thermodynamical variables, it cannot be used without caution in a Kondo lattice. In particular, the singleion Kondo theory for electrical transport cannot explain the low temperature resistivity of a Kondo lattice: the residual resistivity ofan assembly ofKondo impurities with concentration c should be equal to cpa, where p~is the unitary limit [81, whereas in a perfect Kondo lattice it should decrease at low ternperature. In real Kondo compounds the weak disorder due to impurities must break the penodicity of the many-body resonance near the Fermi energy, i.e. reduce the coherence length ~ to approximately the mean distance 1,. between impurities and correla-
Here Pu and p8 represent respectively the total (PM ( T= 0)) and non-Kondo (pressure independent) residual resistivity. In order to test the sensitivity of any parameter upon variations of the ternperature Tv., it is convenient to apply high hydrostatic pressures. We have performed resistivity experiments on single crystals of CeAl2 and CeCu6 down to 1.5 K, up to 150 kbar [9,101. In this Letter, we show that A depends on 7’,. according to the proportion A T~2.5; a generalized scaling form to describe the low temperature resistivity of a Kondo lattice in the presence of weak disorder is proposed. Using the quadratic behaviour of the resistivity of CeA12 above 30 kbar (see refs. [9,101) shown in fig. I we have determined two rows of values for T,. (F), using the following forms for the magnetic resistivity: -~
2
Pm(1~)=Po(1~)E1~ao(T/T) —...]
(3)
or 2
pm(P)=Po(P)+ao(T/Tc)
—....
(4)
a0 and a are temperature and pressure independent. These numbers have been evaluated by adjusting the measured values of T,. in such a way that,
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
‘~
509
Volume 138, number 9
PHYSICS LETTERS A 200
/
/
17 July 1989
___________
T,IK)
~(mr~ 14~
CeAI
/
2
iso.
12L I 101
~1
t
:
~
~
/ -
100
(b)8=R,
~: ~
0
ii
20
30
40
50
60
70
50
Fig. 2. Comparative evolution of T~(P)determined from eq. (3) (•) or eq. (4) (-1-) with 7~)(P)(0) (see ref. (91 for the definition of this line). Insert: schematic variation of the resistivity ofheavy fermions. Since TK and 7~.are related to each other, they 0
1000
2000
3000
4000
i2 (is) 5000 6000
define the same energy scale.
2above30kbar. Fig. 1, Magnetic contribution of the resistivity of CeA12 versus T
very close to unity,coefficient a~(CeAlof (3) is found to be The dimensionless 2)~ 1.1 ±0.2.It is important to mention that, proceeding exactly in the
in the limit of P—p 0, T,. equals the Kondo-lattice ternperature of the same system at normal pressure; for the Kondo lattice CeAI2, we have taken T,.(P-~0)~ ~ K. (This temperature is usually called TK; we prefer to take another denomination to distinguish it from the single-ion Kondo temperature, TK.) Then comes our first interesting result: in the pressure dependent case (eq. (3), A=ct0p0(P)/T~) the line T,.(P) lies well below the cross-over line T0(P) separating the low temperature Kondo-lattice regime from the high temperature impurity-like regime ~‘ (fig. 2). On the other 2, hand in T,.(P)~T the second hypothesis (4), 4= we find a~/T,. 0(P),which is not physically realistic since the Kondo-lattice (Ta), the TM) and the cross-over Kondo-impurity like (TK (T 0) temperatures must satisfy the inequalities
same way with CeCu4, we have reached the same conclusions; in particular ct0 (CeCu6) 1.3 ±0.5 (see also ref. [10]). Let us now come to our second result: we have observed that the total residual resistivity of CeA12 and CeCu6 vary very strongly with pressure. More precisely, since we know the variation of TJP), we can plotp~versus T,. for both systems (fig. 3). The result is rather simple with CeAI2 where the “magnetic” residual resistivity can be expressed as a power law in T,.:
T,. < T0 ~ TM (5) (see the inset in fig. 2), TM is here defined as the temperature for which the resistivity is maximum. ~ The cross-over temperatures T0 defined from the inflection point of isotherms or of isobars are about the same (see ref. [9]).
510
p°p°(T/D)—~
(6)
withbe c~ 0.5, p~ i j.d~cm residual and D0 resistivity 2300 K (pg, can also called “magnetic” by reference to the case of disordered magnetic systems (see e.g. ref. [11]).) The behaviour of the residual resistivity of CeCu4 under pressure is more complex, but understandable: at the lowest pressures the decreasing of p~( Tj is very similar to that of CeAl2 with a slope —0.5. However, due to the proximity, in the P—i’ plane, of a monoclinic distortion [12,13] the residual resistivity starts to increase near 40 kbar. The
Volume 138, number 9
PHYSICS LETTERS A
2
I I
pressure-dependent contribution which might vary proportionally to the square root of the Kondo-lat-
a
Ln(~,_P)
17 July 1989
CeAI
the impurities (say a few ppm) as well as the atomic tice remark naturesysof sitestemperature. they occupy We are must unknown in that thesethe “pure” tems. The effect of impurities on heavy fermions will be soon the object of another report. To conclude, let us put on the same scaling form
2
•
1.
.\ •
0~.
our CeCu6: Pu0 results~[l+(T/T)21 concerning the resistivity p of CeAl2 and PPo (7)
LnTK 1,5
2
3
4
5
2, ~ ____________________________________ Ln ~ b CeCu6 2,L.
-
2,2
-
0,5
-
2,0
tortion) we find that the value co~i0.5is consistent with p8 ~ 1 J.L~cm. A better determination of p~ is needed to evaluate D0. We believe that the scaling form (7) might be general for “simple” Kondo lattices, i.e. for those systems which do not present extra phenomena such as
-
__
\\
-
phenomena such weak softening of elastic the occurrence ofas a crystallographic distortion (as in CeCu6, at high pressures), or even less drasticmodes; elastic
-
one must mention that it is not proven that the distortion appearing in CeCu6 is not related to the fluctuation mechanism responsible for the coherence of this system [111. Besides, the log—log plots of eqs. (3) and (4) do not exclude other types of plots; they simply show that if p~( T,.) is a power law, then the
-
LnTK
18
I
1
2
3
4
5
6
In CeAl2 above 30 kbar (to prevent magnetic order), p8 the pressure-independent resistivity is of the order of 1 ~ cm, D0 2300 K should be related to the conduction electron band width cut-off and the coefficient ci0, not very accurate, is of the order of unity. For CeCu6 below 30 kbar (to avoid the lattice dis-
7
Fig. 3. Log—log plot of the magnetic residual resistivity pg’ versus T~in CeAl2 (a) and CeCu6 (b).
maximum observed near highest pressures suggests that, at even higher pressures, where the distortion should5 stabilize, but one with should againtimes the behaviour a p8observe about four T~° This scenario for CeCu larger. 6 together with the rather unambiguous result for CeAI2 strongly suggests that the residual resistivity of Kondo cornpounds (real Kondo lattices) contain an important
exponent is ~. In such a case this exponent should characterize the low temperature behaviour of heavy ferrnions weakly coupled to the lattice. Interestingly, a recent theoretical paper [5] came to the same conclusion p~ T~ but with ~= 1 instead of ~0.5. It would be of interest to verify this effect and especially to evaluate the exponent c in other systems. This might the current these systems. Asadd an to example let usunderstanding go back to theofbeginning of this paper where we mentioned the well admitted 2 result (see e.g. ref. [2]): A~y 0 (8) .
511
Volume 138, number 9
PHYSICS LETTERS A
Our conclusions, not necessarily in contradiction with this result, give (from (7)): 2~. (9) A~ T~ in
Then the relation A/’,’~=const (2) indicates that a Kondo lattice one should have
y
(10)
12, 0xT,~
which differs markedly from the single-ion relation
17 July 1989
References [1~G.R. Steward, Rev. Mod. Phys. 56 (1984) 455. [2]K. Kadowaki and SB. Woods. Solid State Commun. 58 (1986) 507. [3IL. Puech, J.M. Mignot, P. Lejay, P. Haen and J. Flouquet. J.LowTemp.Phys.70(1988)3. 141 P. Morin, C. Vettier, J. Flouquet, M. Konczykowski. Y. Lassaly and J.M. Mignot, J. Low Temp. Phys. 70 (1988)
y 0xT~’. Finally we should mention that this exponent a = 1.2 for the linear specific heat term, ~‘ x ~ can be also determined [9,10,14] by two other independent ways: the plot of Yo ( T,.) versus A( T,.) in CeAI2 under pressure and in the plot of the high temperature specific heat (data from ref. [15 1) y( T) T~ of CeCu6 and CeAI3. To conclude, our results suggest that, when the concentration ofimpurities increases in a Kondo lattice, the rise in the residual resistivity Po and lowering in T,. are non-independent, but connected to each other (likely through the relation (6)). As a consequence the relationship between the low ternperature specific heat and the reciprocal characteristic temperature (Ta’ ) should not be linear, contrary to what is generally admitted, ~
512
377. [5]E. P. Fetisov and DI. Khomskii. Soy. Phys. JETP 43 (1986) 933. [61K.D. Schotte and U Schotte. Phys. Lett. A 55 (1975) 38. 171 N. Andrei and J.H. Lowenstein, Phys. Rev. Len. 46 (1981) 356. [81 P. Nozibres, J. Low Temp. Phys. 17 (1974) 31. [918. Barbara, J. Beille, B. Cheiato, J.M. Laurant, M.F. Rossignol, A. Waintal and S. Zemirli, Phys. Lett. A II 3 (1986) 381;J. Phys. (Paris) 48(1987)635. 110] S. Zemirli, Thesis, U niversity of Grenoble (1986) [11] A.J. Decker, J. App!. Phys. 36 (1965) 906. [121 E. Gratz. E. Bauer, H. Novotny. H. Mueller, S. Zemirli and B. Barbara, J. Magn. Magn. Mater. 63/64 (1987) 312. and references therein. [13] A. Shibata, G. Oonu, Y. OnukiandT. Komatsubara,J. Phys. Soc. Japan 55 (1986) 2086. [14] B. Barbara and S. Zemirli, Solid State Commun., to be published. [l5]G.E. Brodale, R.A. Fisher, N.E. Phillips and J. Flouquet. Phys. Rev. Lett. 56 (1986)390.