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Impurity resistance in ferromagnetic metals
Journal of Magnetism and Magnetic Materials 23 (1981) 35-40 © North-Holland Publishing Company
IMPURITY RESISTANCE IN FERROMAGNETIC METALS I_M. JIANG and T.M. WU Department of Physics, Applied Physics and Astronomy, State University of New York at Binghamton, Binghamton, NY 13901, USA Received 25 August 1980; in revised form 31 October 1980 We studied the electrical resistivity of the ferromagnetic metal due to nonmagnetic impurities by considering the different densities of states at the Fermi surface and the different screened potentials for -+spin electrons. Contrary to Nieminen's result which shows that the electrical resistivity decreases when the magnetization increases, we find that the electric resistivity increases monotonically with increasing magnetization. We also do not find the resistivity minimum predicted by Kim and Schwartz. The electrical resistance in a ferromagnetic metal has received attention both experimentally and theoretically [1-11]. Kim and Schwartz [8] (KS) studied the residual resistivity caused by nonmagnetic impurities in ferromagnetic metals using a single parabolic band model for the metallic electrons. They concluded that the conductivities of the spin up and the spin down electrons are sensitive to changes in parameters of the dielectric function and that the total resistance depends on magnetization. They found that the electric conductance of the majority spin electrons can be smaller than that of the minority spin electrons. Below the Curie temperature, the total resistance changes with the magnetization and in some cases exhibits a relative minimum. Their explanation is that the spin-dependent screening makes the scattering cross section of majority spin electrons much larger than that of minority spin electrons and counteracts the effect of the large density of states at the Fermi surface of the majority spin electrons. Recently, Nieminen [9] used the phase shifts obtained from a self-consistent spin-densityfunctional solution for screening potentials and calculated the residual resistivity due to point charges embedded in a spin polarized electron gas. Contrary to KS, Nieminen concluded that the conductance of the majority spin electrons is always greater than that of the minority spin electrons and that the resistivity decreases monotonically with increasing magnetization, without ever exhibiting a relative minimum. In this paper, we analyze the electrical resistance in ferromagnetic metals with a scheme similar to KS but instead of applying ladder approximation, we take all orders of electron-electron interactions into consideration to compute the polarization and vertex corrections. Following KS [12-15] we write the Hamiltonian for a ferromagnetic metal with nonmagnetic impurities as ~'
'
H = ~
ekC~aCko +½ ~:~k'q V(q)C~oC~'o'Ck'-qa'Ck+qo +
kG
U(q)C~+qoCko =He +H'.
(1)
kqo oo t
He, which stands for the first two terms, is an unperturbed electron gas Hamiltonian. H', the third term, is the perturbation due to nonmagnetic impurities. C~o and Cka are the creation and annihilation operators of an itinerant electron with momentum k and spin o. V(q) = 41re2/q 2 is the long-range Coulomb interaction between electrons. U(q) = 47rZeZ/q 2 is the potential energy due to a point charge impurity Ze. The prime on the summation indicates that q = 0 is excluded from the summation. The conductivity for itinerant -+ spin electrons is given by e 2 ['r±(ek±)v2. .af(ek±). , , o2 = - -~ j I,eg±) -z------~v ±Lek±) dek±,
(2)
Oek±
35
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals
36
where r+, u, f and N+_ are, respectively, the life time, the velocity, the Fermi distribution function and the density of states of + spin electrons with m o m e n t u m k and energy ek+-. The life time r_+ can be expressed as follows 1
7/"
r+(ek+) -
'tr
h C~+-(ek+) ~f [Ueff+(2k sin ½0)]2(1 - cos 0) sin 0 dO,
-
(3)
0
where Ci is the number of impurities per unit volume and Ueff+, is the effective potential due to a single impurity. At low temperature we approximate the Fermi distribution function by a step function, which is permissible since kBTlekF+_ < < 1 (egv+ is the Fermi energy of the -+ spin electrons). Assuming a parabolic energy band for the itinerant electrons, the conductivity becomes a+_ = e 2 n +r+ (ekF+)/m*,
(4)
where n+- and m* are the densities and effective mass of the electrons. Using the above equations, the dependence of conductivity on relative magnetization can be expressed as /
o+ :oo(1
/ #
J
,in'01
s i n 2 ( ½ 0 ) } - 2 ( ' - cos 0)sin 0 dO,
(5)
0
where oo = 4ne~F/lrZ 2 e2 h 2 kFCi, ~ = (n+ - n _ )l (n+ + n _ ) = (n+ -- n _ )/n is the relative magnetization and 6+_(q) = U(q)/Ueff , (q)
(6)
is the dielectric constant for spin electrons. The effective potential Ueff+_ (q) due to impurities, which are shown in fig. 1a, can be written as
Ueff+-(q) -v- U(q)r+_(q)/ (l
-
-
(7)
V(q)[~+(q) + ~ _ ( q ) ] } ,
(a)
X--
+
(b)
(:~
....
=
(~
+
. . .
-I- ( ~
(c)
(d)
~'"
+--'-7-._~_
i,,' + ',~,' + '~i + ~
+ !i
+ °''
Fig. 1. Diagrams for t h e impurity scattering in a ferromagnetic electron gas. The electron propagator including the exchange selfenergy is represented by a line. The interaction of electrons with an impurity is represented by a broken line attached to a cross. T h e horizontal and vertical dotted lines represent interactions between electrons. The double lines stand for effective interactions.
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals
37
in which the polarization term is given by f dak dw i G ( q ) = - a ( 2 - - ~ ia±(k +q, co)iG± (k, co)
3k dco d3k ' dco' (2n)--~ia±(k +q,w)iG±(k, co)(-i)Veff(k',co'-w)iG.z_(k
fd~ r r 7
+q+k,w')iG±(k
'
,
(8)
and the vertex correction is given by ('dak ' d c o ' r ± ( q ) = 1 + J -~r~ iG._(k +q + k', co') iG±(k + k', co')(-i) Veff(k',co'-co)
(9)
in which
G±(k, co) = 1/(co - ek± + i5)
(10)
is the Fourier transform of Green's function for spin electrons. Instead of using ladder approximation, the effective electron-electron interaction Veff is included in the polarization and vertex correction. In order to simplify equations for ~+ and F±, we follow KS [ 1 2 - 1 4 ] and define ffaf±(q) as
f
d3k dco' ~ r r y Veff(k', co' - 60) iG± (k + q + k', co') iG± (k + k', co') -- vole±
(q)f dak(~),dco iO± (k + q, co) ia± (k, co).
(11)
Hence, the eqs. (8) and (9) can be written as if± (q) = rr°-+(q) - rr°-+(q) ~eff± (q) 71.O(q),
(12)
F+_(q) = 1 - rr° (q) [~eff+ (q),
(13)
where the bare polarization n o (q) is defined as ilr° (q) = _ f d3k dw d (270 4 iG±(k +q, co)iG±(k, co).
(14)
Therefore, the dielectric constant for -+ spin electrons can be written as
1 e+_(q) = 1 - Veff±(q)rr°(q) {1 - V(q)[(rr°+(q) - rr°(q)~'eff+(q)rr°+(q)) + (rr°(q) - r~°(q)Veff_(q)r~°_(q))]}.
(15)
In order to compute dielectric constants and conductivities, one has to know the explicit form of the effective interaction IYeff±(q), which can be related to the uniform exchange field Veff(O). Following KS, we assume the effective interaction to be of Lorentzian form
Veff+(q)- V(0)/(1
+ Cq2/k~±)
(16)
where C stands for the range parameter. C = 0 corresponds to a 6-function type exchange interaction and C = 1 corresponds to an exchange interaction with the range approximately equal to an interelectronic distance between parallel spins. The uniform exchange field c a n b e found from Green's function for the + spin electrons iG±(k, co) =
' dco' iG°+(*,co) +f dak(2n) 4
iG°.(k, co) iG °_(k ÷ k', co')(-i)Veff(k', w' - co) iG+_(k, co).
(17)
LM. Jiang, T.M. lgu /Impurityresistance in ferromagnetic metals
38
Using eq. (11), eq. (17) can be expressed as
G+(k, co)= G°(k, co) - G°(k, co) rzeff(O)n°+_G+_(k,col
(18)
where n ° is the total number of -+ spin electrons in the unperturbed state. G ° (k, co) is the electron Green's function of the unperturbed system and is given as
G° (k, co) = - 1 / ( c o - ek + i8).
(19)
Substituting eq. (19) into eq. (18), one gets
G+_(k, w) = 1/[w - (ek - 17err(0) n°+_)+ i8],
(20)
i.e. the energy of the + spin electron with wave number k is given as (21)
ek+ = e k -- V e f f ( 0 ) n O .
At the Fermi surface, ekE+ = ekE_, by eq. (21), the uniform exchange field can be expressed as 17eff(0) = (ekF/n)[(l + ~)2/3 _ (1 --'~')2/B ]/~-,
(22)
where ekF is the Fermi energy for ~"= 0. To make it easier to compare these various results, we use an electron density corresponding to rs = 3.6, and the Fermi wave number, kF = 10 a/cm, as KS did. The dielectric constant for -+ spin electrons, e+ (q) and e_ (q) are plotted in fig. 2 for various values of relative magnetization ~', assuming + spin is the majority electrons. The conductivities for +- spin electrons and the resistivity, p = (o+ + o _ ) -t , with respect to the relative magnetization, are shown respectively in figs. 3 and 4. The result of numerical calculation for the dielectric constant e_+ is shown in fig. 2. With the increase of magnetization, the dielectric constant of the majority spin decreases, hence the effective scattering potential Ueff+ for majority spin electrons becomes larger. On the other hand, the dielectric constant of the minority spin increases, hence the effective scattering potential Ueff_ for minority spin electrons becomes smaller. As was pointed out.by KS [8], this can be understood rather easily. The itinerant electrons of metal will be attracted to the positive charge impurity to screen the point charge. Since the density of majority spin electrons is higher I0 \
b
~=0.9
....
\x\
~ { - 0 . 1
C
\\
C_
\
',
\\,,
,,
,,
"'-.. 0
I
.2
I
i
.4
.6 q/2k
Fig. 2. Plot o f the spin dependent dielectric constant
,
.8
_
iI
.
F
e+(q) and
e _ ( q ) f o r various relative magnetization ~.
LM. Jiang, T.M. Wu /lmpurity resistance in ferromagnetic metals
39
than that of minority spin electrons, more of the majority spin electrons will be attracted to the impurity than the minority spin electrons. Therefore, an incoming electron will see electron spin polarization as well as electron charge polarization around the point charge impurity. As is noted in eq. (21), the effective interaction between electrons can be viewed as an effective attraction between electrons of the same spin polarization. In the ferromagnetic case where more of the majority spin electrons are around the positive charge impurity, an incoming majority spin electron will be scattered more readily than an incoming minority spin electron. Therefore, as magnetization increases, the dielectric constant of the majority spin electron decreases and that of the minority spin electrons increases. However, although the dielectric constants for majority and minority spin electrons vary differently with respect to magnetization, they always stay as positive quantities in contrast to KS' result [13] which shows that the dielectric constant for the minority spin electrons has a singular behavior and becomes negative for small q. The conductivities for majority and minority spin electrons are shown in fig. 3. With the increase of magnetization, the conductivity of majority spin electrons increases monotonically while that of minority spin electrons decreases to zero as the system is fully magnetized. This result is in good agreement with that of Nieminen's work [9], yet differs greatly with that of KS [8]. We show that the conductivity of majority spin electrons is always larger than that of minority spin electrons, in contrast to KS' result which indicates that in some case the conductivity of minority spin electrons can be larger than that of majority spin electrons. We fred that the increase (decrease) in scattering potential caused by a spin-dependent screening is not large enough to compensate the effect of the higher (lower) density of states at the Fermi surface. In fig. 4, the resistivity drawn from our calculation is juxtaposed with that of KS' result [8]. We see that the resistivity increases monotonically with the increase of magnetization, contrary to Nieminen's result in which the resistivity is a monotonically decreasing function of magnetization. We do not get the resistivity minimum either. One also notices that the resistivity obtained from our calculation is always smaller than that of KS' result. This
o.a
C=I B C=O- - -
.-_+ (¢)
j
*" (o)
#+
~" I~
~"
"~
/
:,-0.5 I.-
N B Q Z 0
0
0
I
0.2
I
I
I
0.4 0.6 0.8 MAGNETIZATION
I.O
Fig. 3. Impurity-limited conductivities for -+spin electrons in a ferromagnetic electron gas as functions of the relative magnetization. z(0) stands for the overall conductivity in the paramagnetic case.
40
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals / r
=3.6
•~
2
-.
.
.
.
.
/
RPA
.
/
/
------KaS - OURS .
I
/
/
/
/
/ "-,- C =O.O
"\
/
/
/
//
\\
/ C = I.O.,.J
c=oto
Y
P (¢) Po
...........................................
oI
I
.2
I
.4
~
.
l
.8
MAGNETIZATION
Fig. 4. Residual resistivity as a function of the relative magnetization for two values of the range of the effective interaction.
discrepancy may be attributed to an over-correction in the vertex part F and electron polarization ff made by KS. The effective interaction Veff(q) has already included all order interactions. Therefore, any attempt to modify it will produce an over-counting in the F e y n m a n diagram. We conclude that the anomalous resistance behavior in some impure ferromagnets [ 2 - 6 ] cannot be explained by nonmagnetic impurity scattering only, as Nieminen cor~cluded in his research. Other processes, such as magnetic impurity and s p i n - o r b i t scattering might be responsible for the anomalous behavior of resistance in ferromagnetic metals. One should keep in mind the simplification used in present calculation. In real metals the detailed band structure and the local spin splitting should also be considered.
References [1] J. Bass, Advan. Phys. 21 (1972) 431. [2] S.C.H. Lin, J. Appl. Phys. 40 (1969) 2173. [3] D.J. Gillespie, C.A. Macklist and A.I. Schindler, in: Amorphous Magnetism, eds. H.O. Hooper and A.M. de Graaf (Plenum, New York, 1973) p. 343. [4] R. Hasegawa and J.A. Dermon, Phys. Lett. A42 (1973) 407. [5] S. Ogawa, S. Waki and T. Teranishi, Intern. J. Magnetism 5 (1974) 349. [6] T. Yoshiic, K. Yamakawa and E. Fujita, J. Phys. Soc. Japan 37 (1974) 572. [7] Y. Shapira and R.L. Kautz, Phys. Rev. B10 (1974) 4781. [8] D.J. Kim and B.B. Schwartz, Phys. Rev. B15 (1977) 377. [9] R.M. Nieminen, Phys. Rev. B17 (1978).5036. [10] S. Inagaki, J. Phys. Soc. Japan 45 (1978) 1253. [11] K.H. Fischer, Phys. Rept. 47 (1978) 225. [12] D.J. Kim and B.B. Schwartz, Phys. Rev. Lett. 28 (1972) 310. [13] D.J. Kim, B.B. Schwartz and H.C. Praddaude, Phys. Rev. B1 (1973) 205. [14] D.J. Kim, Phys. Rev. B9 (1974) 3307. [15] D.J. Kim, H.C. Praddaude and B.B. Schwartz, Phys. Rev. Lett. 23 (1969) 419.
IMPURITY RESISTANCE IN FERROMAGNETIC METALS I_M. JIANG and T.M. WU Department of Physics, Applied Physics and Astronomy, State University of New York at Binghamton, Binghamton, NY 13901, USA Received 25 August 1980; in revised form 31 October 1980 We studied the electrical resistivity of the ferromagnetic metal due to nonmagnetic impurities by considering the different densities of states at the Fermi surface and the different screened potentials for -+spin electrons. Contrary to Nieminen's result which shows that the electrical resistivity decreases when the magnetization increases, we find that the electric resistivity increases monotonically with increasing magnetization. We also do not find the resistivity minimum predicted by Kim and Schwartz. The electrical resistance in a ferromagnetic metal has received attention both experimentally and theoretically [1-11]. Kim and Schwartz [8] (KS) studied the residual resistivity caused by nonmagnetic impurities in ferromagnetic metals using a single parabolic band model for the metallic electrons. They concluded that the conductivities of the spin up and the spin down electrons are sensitive to changes in parameters of the dielectric function and that the total resistance depends on magnetization. They found that the electric conductance of the majority spin electrons can be smaller than that of the minority spin electrons. Below the Curie temperature, the total resistance changes with the magnetization and in some cases exhibits a relative minimum. Their explanation is that the spin-dependent screening makes the scattering cross section of majority spin electrons much larger than that of minority spin electrons and counteracts the effect of the large density of states at the Fermi surface of the majority spin electrons. Recently, Nieminen [9] used the phase shifts obtained from a self-consistent spin-densityfunctional solution for screening potentials and calculated the residual resistivity due to point charges embedded in a spin polarized electron gas. Contrary to KS, Nieminen concluded that the conductance of the majority spin electrons is always greater than that of the minority spin electrons and that the resistivity decreases monotonically with increasing magnetization, without ever exhibiting a relative minimum. In this paper, we analyze the electrical resistance in ferromagnetic metals with a scheme similar to KS but instead of applying ladder approximation, we take all orders of electron-electron interactions into consideration to compute the polarization and vertex corrections. Following KS [12-15] we write the Hamiltonian for a ferromagnetic metal with nonmagnetic impurities as ~'
'
H = ~
ekC~aCko +½ ~:~k'q V(q)C~oC~'o'Ck'-qa'Ck+qo +
kG
U(q)C~+qoCko =He +H'.
(1)
kqo oo t
He, which stands for the first two terms, is an unperturbed electron gas Hamiltonian. H', the third term, is the perturbation due to nonmagnetic impurities. C~o and Cka are the creation and annihilation operators of an itinerant electron with momentum k and spin o. V(q) = 41re2/q 2 is the long-range Coulomb interaction between electrons. U(q) = 47rZeZ/q 2 is the potential energy due to a point charge impurity Ze. The prime on the summation indicates that q = 0 is excluded from the summation. The conductivity for itinerant -+ spin electrons is given by e 2 ['r±(ek±)v2. .af(ek±). , , o2 = - -~ j I,eg±) -z------~v ±Lek±) dek±,
(2)
Oek±
35
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals
36
where r+, u, f and N+_ are, respectively, the life time, the velocity, the Fermi distribution function and the density of states of + spin electrons with m o m e n t u m k and energy ek+-. The life time r_+ can be expressed as follows 1
7/"
r+(ek+) -
'tr
h C~+-(ek+) ~f [Ueff+(2k sin ½0)]2(1 - cos 0) sin 0 dO,
-
(3)
0
where Ci is the number of impurities per unit volume and Ueff+, is the effective potential due to a single impurity. At low temperature we approximate the Fermi distribution function by a step function, which is permissible since kBTlekF+_ < < 1 (egv+ is the Fermi energy of the -+ spin electrons). Assuming a parabolic energy band for the itinerant electrons, the conductivity becomes a+_ = e 2 n +r+ (ekF+)/m*,
(4)
where n+- and m* are the densities and effective mass of the electrons. Using the above equations, the dependence of conductivity on relative magnetization can be expressed as /
o+ :oo(1
/ #
J
,in'01
s i n 2 ( ½ 0 ) } - 2 ( ' - cos 0)sin 0 dO,
(5)
0
where oo = 4ne~F/lrZ 2 e2 h 2 kFCi, ~ = (n+ - n _ )l (n+ + n _ ) = (n+ -- n _ )/n is the relative magnetization and 6+_(q) = U(q)/Ueff , (q)
(6)
is the dielectric constant for spin electrons. The effective potential Ueff+_ (q) due to impurities, which are shown in fig. 1a, can be written as
Ueff+-(q) -v- U(q)r+_(q)/ (l
-
-
(7)
V(q)[~+(q) + ~ _ ( q ) ] } ,
(a)
X--
+
(b)
(:~
....
=
(~
+
. . .
-I- ( ~
(c)
(d)
~'"
+--'-7-._~_
i,,' + ',~,' + '~i + ~
+ !i
+ °''
Fig. 1. Diagrams for t h e impurity scattering in a ferromagnetic electron gas. The electron propagator including the exchange selfenergy is represented by a line. The interaction of electrons with an impurity is represented by a broken line attached to a cross. T h e horizontal and vertical dotted lines represent interactions between electrons. The double lines stand for effective interactions.
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals
37
in which the polarization term is given by f dak dw i G ( q ) = - a ( 2 - - ~ ia±(k +q, co)iG± (k, co)
3k dco d3k ' dco' (2n)--~ia±(k +q,w)iG±(k, co)(-i)Veff(k',co'-w)iG.z_(k
fd~ r r 7
+q+k,w')iG±(k
'
,
(8)
and the vertex correction is given by ('dak ' d c o ' r ± ( q ) = 1 + J -~r~ iG._(k +q + k', co') iG±(k + k', co')(-i) Veff(k',co'-co)
(9)
in which
G±(k, co) = 1/(co - ek± + i5)
(10)
is the Fourier transform of Green's function for spin electrons. Instead of using ladder approximation, the effective electron-electron interaction Veff is included in the polarization and vertex correction. In order to simplify equations for ~+ and F±, we follow KS [ 1 2 - 1 4 ] and define ffaf±(q) as
f
d3k dco' ~ r r y Veff(k', co' - 60) iG± (k + q + k', co') iG± (k + k', co') -- vole±
(q)f dak(~),dco iO± (k + q, co) ia± (k, co).
(11)
Hence, the eqs. (8) and (9) can be written as if± (q) = rr°-+(q) - rr°-+(q) ~eff± (q) 71.O(q),
(12)
F+_(q) = 1 - rr° (q) [~eff+ (q),
(13)
where the bare polarization n o (q) is defined as ilr° (q) = _ f d3k dw d (270 4 iG±(k +q, co)iG±(k, co).
(14)
Therefore, the dielectric constant for -+ spin electrons can be written as
1 e+_(q) = 1 - Veff±(q)rr°(q) {1 - V(q)[(rr°+(q) - rr°(q)~'eff+(q)rr°+(q)) + (rr°(q) - r~°(q)Veff_(q)r~°_(q))]}.
(15)
In order to compute dielectric constants and conductivities, one has to know the explicit form of the effective interaction IYeff±(q), which can be related to the uniform exchange field Veff(O). Following KS, we assume the effective interaction to be of Lorentzian form
Veff+(q)- V(0)/(1
+ Cq2/k~±)
(16)
where C stands for the range parameter. C = 0 corresponds to a 6-function type exchange interaction and C = 1 corresponds to an exchange interaction with the range approximately equal to an interelectronic distance between parallel spins. The uniform exchange field c a n b e found from Green's function for the + spin electrons iG±(k, co) =
' dco' iG°+(*,co) +f dak(2n) 4
iG°.(k, co) iG °_(k ÷ k', co')(-i)Veff(k', w' - co) iG+_(k, co).
(17)
LM. Jiang, T.M. lgu /Impurityresistance in ferromagnetic metals
38
Using eq. (11), eq. (17) can be expressed as
G+(k, co)= G°(k, co) - G°(k, co) rzeff(O)n°+_G+_(k,col
(18)
where n ° is the total number of -+ spin electrons in the unperturbed state. G ° (k, co) is the electron Green's function of the unperturbed system and is given as
G° (k, co) = - 1 / ( c o - ek + i8).
(19)
Substituting eq. (19) into eq. (18), one gets
G+_(k, w) = 1/[w - (ek - 17err(0) n°+_)+ i8],
(20)
i.e. the energy of the + spin electron with wave number k is given as (21)
ek+ = e k -- V e f f ( 0 ) n O .
At the Fermi surface, ekE+ = ekE_, by eq. (21), the uniform exchange field can be expressed as 17eff(0) = (ekF/n)[(l + ~)2/3 _ (1 --'~')2/B ]/~-,
(22)
where ekF is the Fermi energy for ~"= 0. To make it easier to compare these various results, we use an electron density corresponding to rs = 3.6, and the Fermi wave number, kF = 10 a/cm, as KS did. The dielectric constant for -+ spin electrons, e+ (q) and e_ (q) are plotted in fig. 2 for various values of relative magnetization ~', assuming + spin is the majority electrons. The conductivities for +- spin electrons and the resistivity, p = (o+ + o _ ) -t , with respect to the relative magnetization, are shown respectively in figs. 3 and 4. The result of numerical calculation for the dielectric constant e_+ is shown in fig. 2. With the increase of magnetization, the dielectric constant of the majority spin decreases, hence the effective scattering potential Ueff+ for majority spin electrons becomes larger. On the other hand, the dielectric constant of the minority spin increases, hence the effective scattering potential Ueff_ for minority spin electrons becomes smaller. As was pointed out.by KS [8], this can be understood rather easily. The itinerant electrons of metal will be attracted to the positive charge impurity to screen the point charge. Since the density of majority spin electrons is higher I0 \
b
~=0.9
....
\x\
~ { - 0 . 1
C
\\
C_
\
',
\\,,
,,
,,
"'-.. 0
I
.2
I
i
.4
.6 q/2k
Fig. 2. Plot o f the spin dependent dielectric constant
,
.8
_
iI
.
F
e+(q) and
e _ ( q ) f o r various relative magnetization ~.
LM. Jiang, T.M. Wu /lmpurity resistance in ferromagnetic metals
39
than that of minority spin electrons, more of the majority spin electrons will be attracted to the impurity than the minority spin electrons. Therefore, an incoming electron will see electron spin polarization as well as electron charge polarization around the point charge impurity. As is noted in eq. (21), the effective interaction between electrons can be viewed as an effective attraction between electrons of the same spin polarization. In the ferromagnetic case where more of the majority spin electrons are around the positive charge impurity, an incoming majority spin electron will be scattered more readily than an incoming minority spin electron. Therefore, as magnetization increases, the dielectric constant of the majority spin electron decreases and that of the minority spin electrons increases. However, although the dielectric constants for majority and minority spin electrons vary differently with respect to magnetization, they always stay as positive quantities in contrast to KS' result [13] which shows that the dielectric constant for the minority spin electrons has a singular behavior and becomes negative for small q. The conductivities for majority and minority spin electrons are shown in fig. 3. With the increase of magnetization, the conductivity of majority spin electrons increases monotonically while that of minority spin electrons decreases to zero as the system is fully magnetized. This result is in good agreement with that of Nieminen's work [9], yet differs greatly with that of KS [8]. We show that the conductivity of majority spin electrons is always larger than that of minority spin electrons, in contrast to KS' result which indicates that in some case the conductivity of minority spin electrons can be larger than that of majority spin electrons. We fred that the increase (decrease) in scattering potential caused by a spin-dependent screening is not large enough to compensate the effect of the higher (lower) density of states at the Fermi surface. In fig. 4, the resistivity drawn from our calculation is juxtaposed with that of KS' result [8]. We see that the resistivity increases monotonically with the increase of magnetization, contrary to Nieminen's result in which the resistivity is a monotonically decreasing function of magnetization. We do not get the resistivity minimum either. One also notices that the resistivity obtained from our calculation is always smaller than that of KS' result. This
o.a
C=I B C=O- - -
.-_+ (¢)
j
*" (o)
#+
~" I~
~"
"~
/
:,-0.5 I.-
N B Q Z 0
0
0
I
0.2
I
I
I
0.4 0.6 0.8 MAGNETIZATION
I.O
Fig. 3. Impurity-limited conductivities for -+spin electrons in a ferromagnetic electron gas as functions of the relative magnetization. z(0) stands for the overall conductivity in the paramagnetic case.
40
LM. Jiang, T.M. Wu / Impurity resistance in ferromagnetic metals / r
=3.6
•~
2
-.
.
.
.
.
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MAGNETIZATION
Fig. 4. Residual resistivity as a function of the relative magnetization for two values of the range of the effective interaction.
discrepancy may be attributed to an over-correction in the vertex part F and electron polarization ff made by KS. The effective interaction Veff(q) has already included all order interactions. Therefore, any attempt to modify it will produce an over-counting in the F e y n m a n diagram. We conclude that the anomalous resistance behavior in some impure ferromagnets [ 2 - 6 ] cannot be explained by nonmagnetic impurity scattering only, as Nieminen cor~cluded in his research. Other processes, such as magnetic impurity and s p i n - o r b i t scattering might be responsible for the anomalous behavior of resistance in ferromagnetic metals. One should keep in mind the simplification used in present calculation. In real metals the detailed band structure and the local spin splitting should also be considered.
References [1] J. Bass, Advan. Phys. 21 (1972) 431. [2] S.C.H. Lin, J. Appl. Phys. 40 (1969) 2173. [3] D.J. Gillespie, C.A. Macklist and A.I. Schindler, in: Amorphous Magnetism, eds. H.O. Hooper and A.M. de Graaf (Plenum, New York, 1973) p. 343. [4] R. Hasegawa and J.A. Dermon, Phys. Lett. A42 (1973) 407. [5] S. Ogawa, S. Waki and T. Teranishi, Intern. J. Magnetism 5 (1974) 349. [6] T. Yoshiic, K. Yamakawa and E. Fujita, J. Phys. Soc. Japan 37 (1974) 572. [7] Y. Shapira and R.L. Kautz, Phys. Rev. B10 (1974) 4781. [8] D.J. Kim and B.B. Schwartz, Phys. Rev. B15 (1977) 377. [9] R.M. Nieminen, Phys. Rev. B17 (1978).5036. [10] S. Inagaki, J. Phys. Soc. Japan 45 (1978) 1253. [11] K.H. Fischer, Phys. Rept. 47 (1978) 225. [12] D.J. Kim and B.B. Schwartz, Phys. Rev. Lett. 28 (1972) 310. [13] D.J. Kim, B.B. Schwartz and H.C. Praddaude, Phys. Rev. B1 (1973) 205. [14] D.J. Kim, Phys. Rev. B9 (1974) 3307. [15] D.J. Kim, H.C. Praddaude and B.B. Schwartz, Phys. Rev. Lett. 23 (1969) 419.