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Resistivity anomalies due to charge fluctuations of Ce and Pr impurities
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 249-251 North-Holland, Amsterdam
249
R E S I S T I V I T Y A N O M A L I E S D U E T O C H A R G E F L U C T U A T I O N S O F Ce A N D Pr I M P U R I T I E S A. S L E B A R S K I a, E. Z I P P E R ~l and M. D R Z A Z G A h " lnstvtut FizvkL Uniwer.~vtet Slqski, 40-007 Katowice, Poland i, lnstitut fiir Theoretische Pt~vsik der Universitiit zu KOln, 5000 Cologne 41, Ziilpicher Str. 77, led Rep. Germany
This paper presents a qualitative analysis of the resistivity anomalies of the alloys with unstable Ce and Pr impurities. We show that the anomalies correlate with the temperature dependent fluctuation temperature Tf. Near T = 0 Ap seems to reflect both the charge density contrast between the impurity and the matrix and the life-time smearing of the f level described by Tf.
1. Introduction The resistivity anomaly of dilute alloys with unstable Ce and Pr impurities usually interpreted in terms of the K o n d o effect [1], namely the effective impurity scattering cross sections increases over the normal cross-section due to potential scattering by an instability of 4f shell (positive resistivity anomaly). However, it was recently [2-4] found that the resistivity anomalies of dilute Ce in LaA12, ZrIr 2 in YNi 2 and of Pr in Pd and in ZrIr 2 are negative i.e. that the resistivity increment of the unstable impurities becomes much smaller than that of the stable ones. The 4f instability rather than adding a contribution to the resistivity gives a contribution to the conductivity. It was also suggested in ref. [2] that at T = 0 the resistivity comes from the potential scattering due to the charge density contrast of the impurity against the matrix. In this paper we analyse the resistivity a n o m a lies using the dynamic alloy model [5,6] i.e. by calculating the scattering of conduction electrons (ce) on valence fluctuations (charge fluctuations). Since the model is simple and does not take into account e.g. the scattering on the strain field produced by the impurity we expect mainly the qualitative understanding of the data.
2. Dynamic alloy model for the system with impurities The main idea of this model proposed by Wohlleben [5] is that the periodic intermetallic comp o u n d with unstable ions can be regarded as the dynamic alloy: the minority configuration produces a scattering potential which, however, is not static but fluctuates with some fluctuation time
T t. is the fluctuation temperature which can be measured by inelastic neutron scattering or extracted from other e.g. susceptibility measurements. The second time scale in the model is an inelastic life time % of ce, % = h / k B T . At high T when % << ~'f, ce see the full scattering potential coming from the minority configuration (fully incoherent scattering), but at very low T, when % >> ~-f, ce average over the fluctuating potential and it is " s e e n " as a mean-field potential (fully coherent scattering). At the intermediate temperature range, the scattering is partly coherent and partly incoherent and only the last one is seen in the resistivity. The formula for the resistivity arising from scattering on valence fluctuations in periodic systems has been derived in ref. [6]: •f=h/k~Tf.
m*
j W 1 [2
ne2kB h 1 + 4v2~2(Tr/T)
2 Tr"
1, is the deviation of the valence from integral number, W~ is the one-site scattering potential. In this paper we want to derive the formula for the resistivity a n o m a l y in real alloys e.g. Pr~Zr 1 ,.Ir 2 or Ce, La~ ,.AI 2, Pr and Ce being unstable impurities. One has to notice that when decreasing T ce can average over the P r 3 + - P r 4+ potential fluctuations but cannot average over the P r - Z r potential fluctuations because Pr are real impurities in the Zr matrix. We can visualize it by saying that at high T the system looks like a three-component alloy of Pr 3 +, Pr 4+ and Zr ions whereas when the temperature is decreased ce start to average over Pr 3+ Pr 4+ valence fluctuations and at low T the system looks like a two c o m p o n e n t alloy of Zr and Pr 3+" ions.
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.
A. S:lebarski et al. / ResistiPitv anomalieA
25{)
The formula Ap per i m p u r i t y a t o m should thus consist of two parts:
(2)
At) = Ap, + Ap2.
(LaCe)A[ 2
~2
gB
c~
where Ap 1 is given by eq. (1) = I W ~ , ' - W,~/ I-" for P r ~ Z r 1 , I r 2 Ap2
m*
IW~ I~
ne2kn h
Tr
with
/ /
B -4 2
P
0
(3)
300
2
W M is the scattering p o t e n t i a l of the host ions. A & is the resistivity a n o m a l y of d y n a m i c alloy of Pr ions which " l o o k s " different at high T and at low T (incoherent scattering at high T, increasing coherent scattering with decreasing T at the expense of the incoherent one). Ap 2 is the resistivity a n o m a l y of the real alloy of Pr ~+'' a n d Z r ions - scattering is entirely incoherent at the whole t e m p e r a t u r e region.
g o~
In fig. I we present the A p ( T ) b e h a v i o u r obtained using eqs. (1) (3) for three different systems. The values of Tr's has been extracted from susceptibility m e a s u r e m e n t s [2,3] for ( Z r P r ) I r 2 and P d P r and from the quasielastic n e u t r o n linewidth [8] for (LaCe)A12. The values for v's at T - - 300 K has been taken from LUl measurements. Since the m e a s u r e m e n t s of the valence at low T are not available we a s s u m e d after ref. [7] a r e a s o n a b l e relation v --- Tr for lower T. The resistivity a n o m a l y saturates at high T where Tr(T ) saturates and the full scattering potential is visible in the resistivity m e a s u r e m e n t s because the scattering is entirely incoherent in this t e m p e r a t u r e range. On the other h a n d at low T where Apl dies out we rest with the elastic scattering Ap2. The m i n i m a of A t) correlate roughly with the m a x i m a of Tr's. Near T = 0 (4)
a n d A#(0) reflects both the charge density contrast between the i m p u r i t y with valence (3 + v) and the matrix as well as the life time s m e a r i n g of the f level described by Tr.
%.
( Z_r Pr)]r2
b
-~
-
•.. ................... ~
-
2
0
o.~ /
,
3. Comparison with experiment
I W2 I 2 / T , ( 0 )
0 temperQture{K)
I W2 I2 =]wl;,~+, - m . 12'
=
/
I W1 12
-
Ap(0)
a
h.
~
100 260 t e m p e r a t u r e (K)
......
30[
.........................................
C
Z
P dPr
g 09
F-
3 2 -1
~6o
2bo
30o
temperature(K)
Fig. 1. Resistivity increment AR/~ for: (alCc in LaAL< (b) Pr in Zrlr2: (c) Pr in Pd vs. temperature. The solid lines represent the Ao(T) as calculated by the use of eqs. (1) (3), the dashed lines reflect the temperature dependence of 71. The experimental data are presented as black points.
W e see from (4) that the i m p u r i t y with broad, s m e a r e d (due to valence fluctuations) energy level scatters less than the stable impurity. To give m o r e s u p p o r t to this s t a t e m e n t we present in fig. 2 [2] d a t a on the resistivity i n c r e m e n t of Z r I • Re, ir z with both u n s t a b l e a n d stable impurities. W e see that the smallest a n o m a l y exhibits Ce i m p u r i t y which has the greatest fluctuation temp e r a t u r e ( T r --- 1400 K at 300 K).
A. Slebarski et al. / Resistivity anomalies
8 -~
Zro98REo 02Ir 2 .
-~,
.~-~,'..,:'" ,...
"
,...'..
" .....
Lu
251
We would like to thank D. Wohlleben for helpful discussions. This work was supported by the Polish Academy of Sciences from the problem CPBP 01.04 and through the Deutsche Forschungsgemeinschaft through SFB 125.
~d
References
<2
2
"~
2%
o
u~ e~ EF
100 200 300 T e m p e r o t u r e (K} Fig. 2. Resistivity increment Ap = p ( a l l o y ) - 0 ( Z r l r 2 ) vs. temperature for 2% of La, Ce, Pr, Gd and Lu and For 1.14% of Pr in ZrIr 2. The dashed lines are obtained when the upturn of A p ( T ) of Lu below 100 K is subtracted from A o ( T ) of all other alloy Kondo type anomalies (detailes in ref. [2]).
Summarizing, it seems that the qualitative behaviour of the observed anomalies can be understood inside the dynamic alloy model.
[1] J.R. Schrieffer and P.A. Wolf, Phys. Rev. 149 (1966) 491. J. Kondo, Solid State Phys. 23 (1969) 183. [2] A. Slebarski and D. Wohlleben, Z. Phys. B 60 (1985) 449. [3] A. Slebarski, D. Wohlleben, P. Weichner, J. R/Shler and A. Freimuth, J. Magn. Magn. Mat. 47 & 48 (1985) 595. [4] A. Slebarski, J. Jelonek and D. Wohlleben, Z. Phys. B 66 (1987) 47. [5] D. Wohlleben and B. Wittershagem Adv. in Phys. 24 (1985) 403. [6] E. Zipper and M. Drzazga, J. Magn. Magn. Mat. 71 (1987) 119. [7] D. Wohllebem Phys. and Chemistry of Electrons and Ions in Condensed Matter (D. Reidel, Dordrecht, 1984) p. 85. [8] S. Horn, F. Steglich, M. Loewenhaupt and E. HollandMoritz, Physica B 107 (1981) 103.
249
R E S I S T I V I T Y A N O M A L I E S D U E T O C H A R G E F L U C T U A T I O N S O F Ce A N D Pr I M P U R I T I E S A. S L E B A R S K I a, E. Z I P P E R ~l and M. D R Z A Z G A h " lnstvtut FizvkL Uniwer.~vtet Slqski, 40-007 Katowice, Poland i, lnstitut fiir Theoretische Pt~vsik der Universitiit zu KOln, 5000 Cologne 41, Ziilpicher Str. 77, led Rep. Germany
This paper presents a qualitative analysis of the resistivity anomalies of the alloys with unstable Ce and Pr impurities. We show that the anomalies correlate with the temperature dependent fluctuation temperature Tf. Near T = 0 Ap seems to reflect both the charge density contrast between the impurity and the matrix and the life-time smearing of the f level described by Tf.
1. Introduction The resistivity anomaly of dilute alloys with unstable Ce and Pr impurities usually interpreted in terms of the K o n d o effect [1], namely the effective impurity scattering cross sections increases over the normal cross-section due to potential scattering by an instability of 4f shell (positive resistivity anomaly). However, it was recently [2-4] found that the resistivity anomalies of dilute Ce in LaA12, ZrIr 2 in YNi 2 and of Pr in Pd and in ZrIr 2 are negative i.e. that the resistivity increment of the unstable impurities becomes much smaller than that of the stable ones. The 4f instability rather than adding a contribution to the resistivity gives a contribution to the conductivity. It was also suggested in ref. [2] that at T = 0 the resistivity comes from the potential scattering due to the charge density contrast of the impurity against the matrix. In this paper we analyse the resistivity a n o m a lies using the dynamic alloy model [5,6] i.e. by calculating the scattering of conduction electrons (ce) on valence fluctuations (charge fluctuations). Since the model is simple and does not take into account e.g. the scattering on the strain field produced by the impurity we expect mainly the qualitative understanding of the data.
2. Dynamic alloy model for the system with impurities The main idea of this model proposed by Wohlleben [5] is that the periodic intermetallic comp o u n d with unstable ions can be regarded as the dynamic alloy: the minority configuration produces a scattering potential which, however, is not static but fluctuates with some fluctuation time
T t. is the fluctuation temperature which can be measured by inelastic neutron scattering or extracted from other e.g. susceptibility measurements. The second time scale in the model is an inelastic life time % of ce, % = h / k B T . At high T when % << ~'f, ce see the full scattering potential coming from the minority configuration (fully incoherent scattering), but at very low T, when % >> ~-f, ce average over the fluctuating potential and it is " s e e n " as a mean-field potential (fully coherent scattering). At the intermediate temperature range, the scattering is partly coherent and partly incoherent and only the last one is seen in the resistivity. The formula for the resistivity arising from scattering on valence fluctuations in periodic systems has been derived in ref. [6]: •f=h/k~Tf.
m*
j W 1 [2
ne2kB h 1 + 4v2~2(Tr/T)
2 Tr"
1, is the deviation of the valence from integral number, W~ is the one-site scattering potential. In this paper we want to derive the formula for the resistivity a n o m a l y in real alloys e.g. Pr~Zr 1 ,.Ir 2 or Ce, La~ ,.AI 2, Pr and Ce being unstable impurities. One has to notice that when decreasing T ce can average over the P r 3 + - P r 4+ potential fluctuations but cannot average over the P r - Z r potential fluctuations because Pr are real impurities in the Zr matrix. We can visualize it by saying that at high T the system looks like a three-component alloy of Pr 3 +, Pr 4+ and Zr ions whereas when the temperature is decreased ce start to average over Pr 3+ Pr 4+ valence fluctuations and at low T the system looks like a two c o m p o n e n t alloy of Zr and Pr 3+" ions.
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.
A. S:lebarski et al. / ResistiPitv anomalieA
25{)
The formula Ap per i m p u r i t y a t o m should thus consist of two parts:
(2)
At) = Ap, + Ap2.
(LaCe)A[ 2
~2
gB
c~
where Ap 1 is given by eq. (1) = I W ~ , ' - W,~/ I-" for P r ~ Z r 1 , I r 2 Ap2
m*
IW~ I~
ne2kn h
Tr
with
/ /
B -4 2
P
0
(3)
300
2
W M is the scattering p o t e n t i a l of the host ions. A & is the resistivity a n o m a l y of d y n a m i c alloy of Pr ions which " l o o k s " different at high T and at low T (incoherent scattering at high T, increasing coherent scattering with decreasing T at the expense of the incoherent one). Ap 2 is the resistivity a n o m a l y of the real alloy of Pr ~+'' a n d Z r ions - scattering is entirely incoherent at the whole t e m p e r a t u r e region.
g o~
In fig. I we present the A p ( T ) b e h a v i o u r obtained using eqs. (1) (3) for three different systems. The values of Tr's has been extracted from susceptibility m e a s u r e m e n t s [2,3] for ( Z r P r ) I r 2 and P d P r and from the quasielastic n e u t r o n linewidth [8] for (LaCe)A12. The values for v's at T - - 300 K has been taken from LUl measurements. Since the m e a s u r e m e n t s of the valence at low T are not available we a s s u m e d after ref. [7] a r e a s o n a b l e relation v --- Tr for lower T. The resistivity a n o m a l y saturates at high T where Tr(T ) saturates and the full scattering potential is visible in the resistivity m e a s u r e m e n t s because the scattering is entirely incoherent in this t e m p e r a t u r e range. On the other h a n d at low T where Apl dies out we rest with the elastic scattering Ap2. The m i n i m a of A t) correlate roughly with the m a x i m a of Tr's. Near T = 0 (4)
a n d A#(0) reflects both the charge density contrast between the i m p u r i t y with valence (3 + v) and the matrix as well as the life time s m e a r i n g of the f level described by Tr.
%.
( Z_r Pr)]r2
b
-~
-
•.. ................... ~
-
2
0
o.~ /
,
3. Comparison with experiment
I W2 I 2 / T , ( 0 )
0 temperQture{K)
I W2 I2 =]wl;,~+, - m . 12'
=
/
I W1 12
-
Ap(0)
a
h.
~
100 260 t e m p e r a t u r e (K)
......
30[
.........................................
C
Z
P dPr
g 09
F-
3 2 -1
~6o
2bo
30o
temperature(K)
Fig. 1. Resistivity increment AR/~ for: (alCc in LaAL< (b) Pr in Zrlr2: (c) Pr in Pd vs. temperature. The solid lines represent the Ao(T) as calculated by the use of eqs. (1) (3), the dashed lines reflect the temperature dependence of 71. The experimental data are presented as black points.
W e see from (4) that the i m p u r i t y with broad, s m e a r e d (due to valence fluctuations) energy level scatters less than the stable impurity. To give m o r e s u p p o r t to this s t a t e m e n t we present in fig. 2 [2] d a t a on the resistivity i n c r e m e n t of Z r I • Re, ir z with both u n s t a b l e a n d stable impurities. W e see that the smallest a n o m a l y exhibits Ce i m p u r i t y which has the greatest fluctuation temp e r a t u r e ( T r --- 1400 K at 300 K).
A. Slebarski et al. / Resistivity anomalies
8 -~
Zro98REo 02Ir 2 .
-~,
.~-~,'..,:'" ,...
"
,...'..
" .....
Lu
251
We would like to thank D. Wohlleben for helpful discussions. This work was supported by the Polish Academy of Sciences from the problem CPBP 01.04 and through the Deutsche Forschungsgemeinschaft through SFB 125.
~d
References
<2
2
"~
2%
o
u~ e~ EF
100 200 300 T e m p e r o t u r e (K} Fig. 2. Resistivity increment Ap = p ( a l l o y ) - 0 ( Z r l r 2 ) vs. temperature for 2% of La, Ce, Pr, Gd and Lu and For 1.14% of Pr in ZrIr 2. The dashed lines are obtained when the upturn of A p ( T ) of Lu below 100 K is subtracted from A o ( T ) of all other alloy Kondo type anomalies (detailes in ref. [2]).
Summarizing, it seems that the qualitative behaviour of the observed anomalies can be understood inside the dynamic alloy model.
[1] J.R. Schrieffer and P.A. Wolf, Phys. Rev. 149 (1966) 491. J. Kondo, Solid State Phys. 23 (1969) 183. [2] A. Slebarski and D. Wohlleben, Z. Phys. B 60 (1985) 449. [3] A. Slebarski, D. Wohlleben, P. Weichner, J. R/Shler and A. Freimuth, J. Magn. Magn. Mat. 47 & 48 (1985) 595. [4] A. Slebarski, J. Jelonek and D. Wohlleben, Z. Phys. B 66 (1987) 47. [5] D. Wohlleben and B. Wittershagem Adv. in Phys. 24 (1985) 403. [6] E. Zipper and M. Drzazga, J. Magn. Magn. Mat. 71 (1987) 119. [7] D. Wohllebem Phys. and Chemistry of Electrons and Ions in Condensed Matter (D. Reidel, Dordrecht, 1984) p. 85. [8] S. Horn, F. Steglich, M. Loewenhaupt and E. HollandMoritz, Physica B 107 (1981) 103.