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Anomalous resistivity near curie temperature due to the critical scattering
Solid State Communications,
Vol. 14, pp. 253—256, 1974.
Pergamon Press.
Printed in Great Britain
ANOMALOUS RESISTWITY NEAR CURIE TEMPERATURE DUE TO THE CRITICAL SCATFERING T. Kasuya and A. Kondo Department of Physics, Tohoku University, Sendai, Japan (Received 3 September 1973 by Y. Toyozawa)
Critical behavior of the temperature derivative of the electrical resistivity at the Curie temperature is treated in general. It is shown by treating the sum rule in the correlation function rigorously that, for a large Fermi wave vector, the divergence is similar to that of specific heat both below and above the Curie temperature, but for a small Fermi wave vector, the de Gennes—Friedel region appears at higher temperatures. Comparison with other divergent mechanisms are also done. IT WAS shown first by de Gennes and Friedel1 that, due to the critical fluctuation of the localized spins (hereafter called fspins for simplicity), the s—f exchange interaction causes a divergence of the electncal resistivity p at the Curie temperature T~for the bottom of the conduction band in the first Born approximation. For a metal with a fairly large Fermi wave vector k~,the temperature derivative of the resistivity p’ shows a logarithmic divergence in the same approximation as shown in Fig. 2. The Ornstein— Zernike form was used for the correlation function ~ which is, however, not applicable in the critical region. Based on the more sophisticated correlation function, on the other hand, Fisher and Langer2 showed that the divergence of ~ neat at q = 0 is masked because of the finite life time of the conduction electron, and then the divergence of p’ above the Curie temperature is of the same character as the magnetic specific heat Cm of the f-spin system when the nearest neighbour exchange interaction is dominated, Below the Curie temperature they expected the divergence of p’ in the form t2~” which comes from the uniform magnetization part of 7q and ~ is the critical index of the magnetization. Mannari also obtained the similar result at paramagnetic region. For the antiferromagnets, considering the finite life time, Suezaki and Mon, and Takada expected another critical index for the resistivity,
It is shown in this letter that the sum rule in 7q1 where i is the direction index, ~7qi ~(~SnSmiC~~’nm = NS(S + 0~(1) in
which have not treated rigorously in the above mentioned theories, is essentially important for both p’ and Cm because the divergent character of ~ near at q = 0 is reflected through the sum rule to the region of large q value which is connected directly to C~and p’. It is also shown that, by using the correct Wy,, the divergence of ~ at q = 0 has no effect on the divergence of p’ irrespectively of the finite life time, and the divergences of C~and p’ have the same index quite in general. Other divergent mechanisms are also considered and compared with the present mechanism. We assume here a typical s—fmodel, that is well localized f-spins and conduction electrons with the s—f exchange interaction. The effective f—f interaction is assumed to be of the isotropic Heisenberg type. Then the magnetic specific heat of the f spin system may be given by C~n = ~jfr~7qi, (2) in which J~is the Fourier component of the f—fcxchange constant I and I is put zero namely = 0, ‘y~jis the temperature derivative of 7~j andfq assumed to be temperature independent. Owing to the sum rule, the summation of y~on q and i is zero. —
253
~54
ANOMALOUS RESiSTIVITY NEAR CURIE TEMPERATURE
Vol. 14, No.3
The expected behavior of 7~(t)for a small posi. tive value oft = T TCITC is shown in Fig. I together with that of the normalized Ornstein—Zernike form,
constant and the sum of J~on q is zero.
Z(t)i~
the electrical resistivity p is veryone complicated. approximation of the effective relaxation When time the model is applicable, however, the resistivity due to the s—f scattering Pm may be approximated by the following form
—
0~,in which Z(t)6 is thedetailed normalization constant The numerical calcu. to ensure the nearest sum rule. lation on the neighbour Heisenberg model ~ shows that the real space correlation function i(Rnm) diverges uniformly with the same index as the specific heat, a, which is usually very small, for Rnm <~,in which ~ is the correlation length and its index v is usually about 2/3. Therefore, we expect from the scaling law that 7~ diverges in general with the index a for q larger than q~,which is substantially larger than r1.Thus, the qspace is divided into the regions! and II by qc~In region I, the summation of ‘yq on the negative region is estimated using the scaling law to diverge with the index 2j3 I while the summation over the region I should diverge with the index a. Therefore, because of the sum rule, ~ 0, the positive peak cancels the main divergence of the negative peak. As was shown by the detailed calculation by Fisher group,7 the position of q to divide the negative and the positive peaks decreases proportionally to ~“ as temperature decreases to T~,and thus at T,~,the both peaks should con—
dense at q = 0 cancelling the main7qi divergence that becomessoanisothe region I disappears. Below Ti,, tropic and, when the magnetization is in the z-direction, q~for y~increases again roughly proportional to l~’~ as temperature decreases beyond l~in which, however, the negative part is kept to condense at q = 0 with the same index 2j3 I making the magnetization and thus the positive peak cancels the main divergence, making the situation fairly symmetrical for both sides of 7,. On the other hand, for x andy directions, the region I is kept to disappear because the correlation length is kept to be infinity. Note that the molecular field approximation gives inconsistent results below T~.As the summary of the above consideration, the following formula are applicable for both sides of ~ —
=
=
—
r’~c0(T),
y~,~0(7)t~ (in the region III)
(3) (4)
in which C~(7)and y>(7) are slowly varying functions. The divergent part of the specific heat per magnetic atom is now given by =
(J0c0 (t)
—
~Jq4jo(t))C’~’IN,
(5)
in which the second term in the bracket is usually much smaller than the first term because ‘Yijo is nearly
For an arbitrary Fermi surface, the expression of
Pm = ~f(q,t)T~~ 11(q) j2, (6) in which 1(q) is the s—f exchange constant and f(q, t) is a slowly varying function of q and proportional to q for a small value of q. At first, the effect of ~ on p~is considered. The q-space in equation (6) is again divided into the regions land!!. Then, the divergent character which comes from the region I is much weaker than t~ due to the q-dependence off(q, t). For example, if the scaling law is applicable and q is scaled by the correlation length, the temperature dependence in the region I is t~’~’ or at most t2~ and thus the divergence disappears in usual case. The divergent part of p’ comes from the region II and is written as p(~(t)= ~ ~“f(q,r) y~ 10(t)I 1(q) j2, (7) The magnetic resistivity well above the critical ternperature is given approximately by S(S + 1) p,,, 0 3 ~f(q,t) 11(q) 12. (8) -
Therefore, by using Pmo and C~,
~
C~ (9) p,,,., S(S + 1 )J0 The same expression was also obtained by Mannari ~ in a different method for the paramagnetic region. In the present case, equation (9) is applicable for both sides of T~,which is the essential difference from the result of Fisher and Langer. Their result is caused by .
overlooking the effect of the positive peak in the region 1. The above consideration is applicable only for the case that the Fermi wave vector is much larger than the wave vectors in the region I, that is, kf ~> q~,.When k1 is small, the situation becomes complicated. Then we may expect a case that at higher temperature kf is well smaller than ~ so that all the scattering belongs to the region I. in this case, p’ is apparently negative. At sufficiently high temperature, the normalized
ANOMALOUS RESISTIVITY NEAR CURIE TEMPERATURE
~o1. 14, No.3
255
PP’
•
r~-. T,
_____
-t-2
T
7 FIG. 2. Schematic plot for the divergent behaviors of resistivity p and its temperature derivative p’ as functions of temperature Tin Gd doped Eu-chalcogenides. The dotted lines are expected from the de Gennes— Friedel theory, (after reference 6).
-~
t~’2
.
volume
4
q
2
q~
___________ ti’
volume
T
~2/3_I
_t_v_ I FIG. 1. Schematic picture of 7~(t)with a small positive value oft as a function of q for the actual case, the solid line, and the normalized Ornstein—Zemike form, the dotted line. The order of magnitudes of y~and q at the peak points, the volume f7’qq2dq of the negative region and the divergent character at the large q region are indicated. Ornstein—Zemike form shown in Fig. I by the dotted curve may be rather applicable. Typical examples are seen in the Eu-chalcogenides doped with trivalent rare earth atoms and the agreement with the experiments and the calculation is fairly good.6 When temperature decreases to T~,the boundary of the negative region decreases in the same way as and becomes smaller than k~at a critical temperature T~ For T< 7’1, p’ becomes positive and the treatment for a large Fermi wave vector is applicable and p’ diverges positively at T~.A typical example is given in Fig. 2. In Eu-chalco-
r’
.
to the changes in the average velocity and the area of the Fermi surface. In the pararnagnetic region, the self energy due to the second order s—f exchange is calculated. Then, the region I has no contribution on the divergence of p’ because of the additive q-dependence. The contribution from the region II gives t~divergence on p but it is smaller about the order of (ISlE 2, in which EQ is the order of the Fermi energy, than the 0) divergence considered before. For the case of the small Fermi energy such as an electron in magnetic semiconductors, the perturbation collapses and the formation of magnetic polaron occurs.8 Below T~,due to the character of broken symmetry, effects of the magnetic moment become dominant. Then the change off(q.r) is proportional to the square of the magnetic moment and p’ by this effect may be written as = a(lS~~2i~ (10) Pd TC E
genides doped with Gd, this behavioris seen very near to 7, but, as T1 is very close to T~,which is estimated to be .about (T1 T,,)/7”~ 0.03 using the calculations in reference 7, detailed quantitative comparison with the theory is impossible.
01 in which p4, is the resistivity near at 7~and a is a constant of the order unity with any sign. The ratio with pj in equation (9) is P~d (S(S + l)aN) (Is) 2~ ~ (11) ~ E~ Pm0
So far, we have considered the effect of the scattering matrix element. In the next, the change of the electronic states due to the critical fluctuation and the magnetization, which is included inf(q.t) in equation (6), is considered. The main effects may be attributed
in which 2(1+ a—1 is, in usual case, about —1/3. Therefore, for very small value of t, the divergence of p’ is determined by the present mechanism. However, in usual case, the first factor in equation (11) is near unity for T~of 10 to 100 and the second factor the
—
~-
Pa
ANOMALOUS RESISTIVITY NEAR CURIE TEMPERATURE
256
order of 102. Therefore, the divergence due to p~ overcomes that of p~only when r is smaller than I0~, which is beyond the limit of present experiments. For small T~and small E0 such as the cases in slightly doped
ledge on
Vol. 14, No.3
ap~/ae~, this is not yet definite.
The same treatment is possible even for the anisotropic case given by
Eu-chalcogenides, p~is expected to be observed. Note that, for the simple free electron model, a is positive.
Em =
—
q~JQi7Qi
(13)
Another origin for the divergence of p’ is due to the critical magneto-elastic effect. Then p’ is given by
in which J~is anisotropic. The only difference is that the divergencewhen occurs only for of thethe easy direction. In particular, thenow differences Curie tem-
~
peratures in the different directions are sufficiently large so that at the Curie temperature the correlations
PneI
=
(~)
e~
(12)
in which em,, is the v-th mode of the strain due to the magneto-elastic effect. The temperature derivative of em,,, that is e~,,,is proportional to the magnetic speciftc heat C,, and thus make divergence of t~ in the both sides of the Curie temperature. Therefore, this divergence is exactly the same as that due to the critical s—f scattering for a system with a large Fermi wave vector. However, in usual case, P,,ei is much smaller than p’~because 8p,/ae~is small. So far the c-axis resistivity of Gd was claimed to be due explained the 9 However, to poorbyknowmagneto-elastic effect.
for the non-easy directions are still not in the critical region, we can treat them by the molecular field approximation. The limiting case of the present situation is the so called Ising model. In this case, instead of equation (9), we have 2J Pi’iIPmo C~/S 0. (14) Acknowledgements Valuable discussions with A. Yanase and S. von Molnar are greatly acknowledged. —
REFERENCES 1.
DE GENNES P.G. and FRIEDEL J.,J. Phys. Chem. Solids 4,71(1958).
2.
FISHERM.E. and LANGER J.S.,Phys. Rev. Lett. 20,665 (1968).
3.
MANNARI I.,Phys. Lett. 26A, 134 (1968). SUEZAKI Y. and MORI H.,Prog. Theor Phys. 41, 1177(1969). TAKADAS.,Prog. Theor.Phys. 46,15(1971). VON MOLNAR S. and KASUYA T.,Phys. Rev. Lett. 21, 1757 (1968). VON MOLNAR S. and SHAFER MW.,
4. 5. 6. 7.
I. Appi. Phys. 41, 1093 (1970). FISHER M.E. and BURFORD R.J., Phys. Rev. 156, 583 (1967). RITCHIE D.S. and FISHER ME., Phys. Rev. B5, 2668 (1972).
8.
For example, KASUYA 1., Proc.
9.
ZUMSTEG F.C., CADIEU F..L, MARCELJA S. and PARKS R.D.,Phys. Rev. Lett. 25, 1204 (1970).
mt. Con!. Semiconductor in
Warsaw (1972) 141, and references therein.
Die kritische Verhaltung der Temperaturableitung des elektrisches Wider-
stand bei der Curie-Temperatur wird in Aligemeinen untersucht. Die genaue Behandlung der Summenregel in der Korrelationsfunktion ist besonders wichtig. Für das System mit dem gro1~enFermi-Wellenvektor, ist die Divergenz ahnlich der von der spezifische Wärme beide unter und über der CurieTemperatur. Für das System mit dem kleinen Fermi-Wellenvektor, gibt es das Gebiet von de Gennes—Friedel bei hohen Temperaturen.
Vol. 14, pp. 253—256, 1974.
Pergamon Press.
Printed in Great Britain
ANOMALOUS RESISTWITY NEAR CURIE TEMPERATURE DUE TO THE CRITICAL SCATFERING T. Kasuya and A. Kondo Department of Physics, Tohoku University, Sendai, Japan (Received 3 September 1973 by Y. Toyozawa)
Critical behavior of the temperature derivative of the electrical resistivity at the Curie temperature is treated in general. It is shown by treating the sum rule in the correlation function rigorously that, for a large Fermi wave vector, the divergence is similar to that of specific heat both below and above the Curie temperature, but for a small Fermi wave vector, the de Gennes—Friedel region appears at higher temperatures. Comparison with other divergent mechanisms are also done. IT WAS shown first by de Gennes and Friedel1 that, due to the critical fluctuation of the localized spins (hereafter called fspins for simplicity), the s—f exchange interaction causes a divergence of the electncal resistivity p at the Curie temperature T~for the bottom of the conduction band in the first Born approximation. For a metal with a fairly large Fermi wave vector k~,the temperature derivative of the resistivity p’ shows a logarithmic divergence in the same approximation as shown in Fig. 2. The Ornstein— Zernike form was used for the correlation function ~ which is, however, not applicable in the critical region. Based on the more sophisticated correlation function, on the other hand, Fisher and Langer2 showed that the divergence of ~ neat at q = 0 is masked because of the finite life time of the conduction electron, and then the divergence of p’ above the Curie temperature is of the same character as the magnetic specific heat Cm of the f-spin system when the nearest neighbour exchange interaction is dominated, Below the Curie temperature they expected the divergence of p’ in the form t2~” which comes from the uniform magnetization part of 7q and ~ is the critical index of the magnetization. Mannari also obtained the similar result at paramagnetic region. For the antiferromagnets, considering the finite life time, Suezaki and Mon, and Takada expected another critical index for the resistivity,
It is shown in this letter that the sum rule in 7q1 where i is the direction index, ~7qi ~(~SnSmiC~~’nm = NS(S + 0~(1) in
which have not treated rigorously in the above mentioned theories, is essentially important for both p’ and Cm because the divergent character of ~ near at q = 0 is reflected through the sum rule to the region of large q value which is connected directly to C~and p’. It is also shown that, by using the correct Wy,, the divergence of ~ at q = 0 has no effect on the divergence of p’ irrespectively of the finite life time, and the divergences of C~and p’ have the same index quite in general. Other divergent mechanisms are also considered and compared with the present mechanism. We assume here a typical s—fmodel, that is well localized f-spins and conduction electrons with the s—f exchange interaction. The effective f—f interaction is assumed to be of the isotropic Heisenberg type. Then the magnetic specific heat of the f spin system may be given by C~n = ~jfr~7qi, (2) in which J~is the Fourier component of the f—fcxchange constant I and I is put zero namely = 0, ‘y~jis the temperature derivative of 7~j andfq assumed to be temperature independent. Owing to the sum rule, the summation of y~on q and i is zero. —
253
~54
ANOMALOUS RESiSTIVITY NEAR CURIE TEMPERATURE
Vol. 14, No.3
The expected behavior of 7~(t)for a small posi. tive value oft = T TCITC is shown in Fig. I together with that of the normalized Ornstein—Zernike form,
constant and the sum of J~on q is zero.
Z(t)i~
the electrical resistivity p is veryone complicated. approximation of the effective relaxation When time the model is applicable, however, the resistivity due to the s—f scattering Pm may be approximated by the following form
—
0~,in which Z(t)6 is thedetailed normalization constant The numerical calcu. to ensure the nearest sum rule. lation on the neighbour Heisenberg model ~ shows that the real space correlation function i(Rnm) diverges uniformly with the same index as the specific heat, a, which is usually very small, for Rnm <~,in which ~ is the correlation length and its index v is usually about 2/3. Therefore, we expect from the scaling law that 7~ diverges in general with the index a for q larger than q~,which is substantially larger than r1.Thus, the qspace is divided into the regions! and II by qc~In region I, the summation of ‘yq on the negative region is estimated using the scaling law to diverge with the index 2j3 I while the summation over the region I should diverge with the index a. Therefore, because of the sum rule, ~ 0, the positive peak cancels the main divergence of the negative peak. As was shown by the detailed calculation by Fisher group,7 the position of q to divide the negative and the positive peaks decreases proportionally to ~“ as temperature decreases to T~,and thus at T,~,the both peaks should con—
dense at q = 0 cancelling the main7qi divergence that becomessoanisothe region I disappears. Below Ti,, tropic and, when the magnetization is in the z-direction, q~for y~increases again roughly proportional to l~’~ as temperature decreases beyond l~in which, however, the negative part is kept to condense at q = 0 with the same index 2j3 I making the magnetization and thus the positive peak cancels the main divergence, making the situation fairly symmetrical for both sides of 7,. On the other hand, for x andy directions, the region I is kept to disappear because the correlation length is kept to be infinity. Note that the molecular field approximation gives inconsistent results below T~.As the summary of the above consideration, the following formula are applicable for both sides of ~ —
=
=
—
r’~c0(T),
y~,~0(7)t~ (in the region III)
(3) (4)
in which C~(7)and y>(7) are slowly varying functions. The divergent part of the specific heat per magnetic atom is now given by =
(J0c0 (t)
—
~Jq4jo(t))C’~’IN,
(5)
in which the second term in the bracket is usually much smaller than the first term because ‘Yijo is nearly
For an arbitrary Fermi surface, the expression of
Pm = ~f(q,t)T~~ 11(q) j2, (6) in which 1(q) is the s—f exchange constant and f(q, t) is a slowly varying function of q and proportional to q for a small value of q. At first, the effect of ~ on p~is considered. The q-space in equation (6) is again divided into the regions land!!. Then, the divergent character which comes from the region I is much weaker than t~ due to the q-dependence off(q, t). For example, if the scaling law is applicable and q is scaled by the correlation length, the temperature dependence in the region I is t~’~’ or at most t2~ and thus the divergence disappears in usual case. The divergent part of p’ comes from the region II and is written as p(~(t)= ~ ~“f(q,r) y~ 10(t)I 1(q) j2, (7) The magnetic resistivity well above the critical ternperature is given approximately by S(S + 1) p,,, 0 3 ~f(q,t) 11(q) 12. (8) -
Therefore, by using Pmo and C~,
~
C~ (9) p,,,., S(S + 1 )J0 The same expression was also obtained by Mannari ~ in a different method for the paramagnetic region. In the present case, equation (9) is applicable for both sides of T~,which is the essential difference from the result of Fisher and Langer. Their result is caused by .
overlooking the effect of the positive peak in the region 1. The above consideration is applicable only for the case that the Fermi wave vector is much larger than the wave vectors in the region I, that is, kf ~> q~,.When k1 is small, the situation becomes complicated. Then we may expect a case that at higher temperature kf is well smaller than ~ so that all the scattering belongs to the region I. in this case, p’ is apparently negative. At sufficiently high temperature, the normalized
ANOMALOUS RESISTIVITY NEAR CURIE TEMPERATURE
~o1. 14, No.3
255
PP’
•
r~-. T,
_____
-t-2
T
7 FIG. 2. Schematic plot for the divergent behaviors of resistivity p and its temperature derivative p’ as functions of temperature Tin Gd doped Eu-chalcogenides. The dotted lines are expected from the de Gennes— Friedel theory, (after reference 6).
-~
t~’2
.
volume
4
q
2
q~
___________ ti’
volume
T
~2/3_I
_t_v_ I FIG. 1. Schematic picture of 7~(t)with a small positive value oft as a function of q for the actual case, the solid line, and the normalized Ornstein—Zemike form, the dotted line. The order of magnitudes of y~and q at the peak points, the volume f7’qq2dq of the negative region and the divergent character at the large q region are indicated. Ornstein—Zemike form shown in Fig. I by the dotted curve may be rather applicable. Typical examples are seen in the Eu-chalcogenides doped with trivalent rare earth atoms and the agreement with the experiments and the calculation is fairly good.6 When temperature decreases to T~,the boundary of the negative region decreases in the same way as and becomes smaller than k~at a critical temperature T~ For T< 7’1, p’ becomes positive and the treatment for a large Fermi wave vector is applicable and p’ diverges positively at T~.A typical example is given in Fig. 2. In Eu-chalco-
r’
.
to the changes in the average velocity and the area of the Fermi surface. In the pararnagnetic region, the self energy due to the second order s—f exchange is calculated. Then, the region I has no contribution on the divergence of p’ because of the additive q-dependence. The contribution from the region II gives t~divergence on p but it is smaller about the order of (ISlE 2, in which EQ is the order of the Fermi energy, than the 0) divergence considered before. For the case of the small Fermi energy such as an electron in magnetic semiconductors, the perturbation collapses and the formation of magnetic polaron occurs.8 Below T~,due to the character of broken symmetry, effects of the magnetic moment become dominant. Then the change off(q.r) is proportional to the square of the magnetic moment and p’ by this effect may be written as = a(lS~~2i~ (10) Pd TC E
genides doped with Gd, this behavioris seen very near to 7, but, as T1 is very close to T~,which is estimated to be .about (T1 T,,)/7”~ 0.03 using the calculations in reference 7, detailed quantitative comparison with the theory is impossible.
01 in which p4, is the resistivity near at 7~and a is a constant of the order unity with any sign. The ratio with pj in equation (9) is P~d (S(S + l)aN) (Is) 2~ ~ (11) ~ E~ Pm0
So far, we have considered the effect of the scattering matrix element. In the next, the change of the electronic states due to the critical fluctuation and the magnetization, which is included inf(q.t) in equation (6), is considered. The main effects may be attributed
in which 2(1+ a—1 is, in usual case, about —1/3. Therefore, for very small value of t, the divergence of p’ is determined by the present mechanism. However, in usual case, the first factor in equation (11) is near unity for T~of 10 to 100 and the second factor the
—
~-
Pa
ANOMALOUS RESISTIVITY NEAR CURIE TEMPERATURE
256
order of 102. Therefore, the divergence due to p~ overcomes that of p~only when r is smaller than I0~, which is beyond the limit of present experiments. For small T~and small E0 such as the cases in slightly doped
ledge on
Vol. 14, No.3
ap~/ae~, this is not yet definite.
The same treatment is possible even for the anisotropic case given by
Eu-chalcogenides, p~is expected to be observed. Note that, for the simple free electron model, a is positive.
Em =
—
q~JQi7Qi
(13)
Another origin for the divergence of p’ is due to the critical magneto-elastic effect. Then p’ is given by
in which J~is anisotropic. The only difference is that the divergencewhen occurs only for of thethe easy direction. In particular, thenow differences Curie tem-
~
peratures in the different directions are sufficiently large so that at the Curie temperature the correlations
PneI
=
(~)
e~
(12)
in which em,, is the v-th mode of the strain due to the magneto-elastic effect. The temperature derivative of em,,, that is e~,,,is proportional to the magnetic speciftc heat C,, and thus make divergence of t~ in the both sides of the Curie temperature. Therefore, this divergence is exactly the same as that due to the critical s—f scattering for a system with a large Fermi wave vector. However, in usual case, P,,ei is much smaller than p’~because 8p,/ae~is small. So far the c-axis resistivity of Gd was claimed to be due explained the 9 However, to poorbyknowmagneto-elastic effect.
for the non-easy directions are still not in the critical region, we can treat them by the molecular field approximation. The limiting case of the present situation is the so called Ising model. In this case, instead of equation (9), we have 2J Pi’iIPmo C~/S 0. (14) Acknowledgements Valuable discussions with A. Yanase and S. von Molnar are greatly acknowledged. —
REFERENCES 1.
DE GENNES P.G. and FRIEDEL J.,J. Phys. Chem. Solids 4,71(1958).
2.
FISHERM.E. and LANGER J.S.,Phys. Rev. Lett. 20,665 (1968).
3.
MANNARI I.,Phys. Lett. 26A, 134 (1968). SUEZAKI Y. and MORI H.,Prog. Theor Phys. 41, 1177(1969). TAKADAS.,Prog. Theor.Phys. 46,15(1971). VON MOLNAR S. and KASUYA T.,Phys. Rev. Lett. 21, 1757 (1968). VON MOLNAR S. and SHAFER MW.,
4. 5. 6. 7.
I. Appi. Phys. 41, 1093 (1970). FISHER M.E. and BURFORD R.J., Phys. Rev. 156, 583 (1967). RITCHIE D.S. and FISHER ME., Phys. Rev. B5, 2668 (1972).
8.
For example, KASUYA 1., Proc.
9.
ZUMSTEG F.C., CADIEU F..L, MARCELJA S. and PARKS R.D.,Phys. Rev. Lett. 25, 1204 (1970).
mt. Con!. Semiconductor in
Warsaw (1972) 141, and references therein.
Die kritische Verhaltung der Temperaturableitung des elektrisches Wider-
stand bei der Curie-Temperatur wird in Aligemeinen untersucht. Die genaue Behandlung der Summenregel in der Korrelationsfunktion ist besonders wichtig. Für das System mit dem gro1~enFermi-Wellenvektor, ist die Divergenz ahnlich der von der spezifische Wärme beide unter und über der CurieTemperatur. Für das System mit dem kleinen Fermi-Wellenvektor, gibt es das Gebiet von de Gennes—Friedel bei hohen Temperaturen.