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Anisotropy and magnetic order effects on the Europium resistivity
Solid State Communications, Vol. 55, No. 3, pp. 219-221, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON THE EUROPIUM RESISTIVITY L. D'Onofrio, F.P. Marin and J.D. Mujica Dpto. de Fisica, Fac. de Ciencias, Universidad Central de Venezuela, A.P. 21201, Caracas 1020 A, Venezuela
(Received 1 June 1984; in revised form 18 January 1985, by E.F. Bertaut) We introduce and magnetic order effects, into the Kim's approach to the spin disorder resistivity, in order to discuss the Eu spin-only resistivity. INTRODUCTION THE ANOMALOUS BEHAVIOUR of the electrical resistivity of Europium metal has been the subject of several studies [ 1, 2, 3]. In this article, some attempts to fit the experimental data have been made considering the critical fluctuation of the spins and its peculiar magnetic structure. Europium rare earth is different from its lanthanides neighbours. It is divalent is the metallic state and has a b c c structure with a lattice parameter of 4.83,~. Neutron diffraction measurements [4] show that Eu goes from the paramagnetic state to a planar helimagnetic one at around 90.4K, with the magnetic moments parallel to a cube face and the rotation axis along the (1 0 0) direction. The interlayer turn angle is of 54 ° and the observed magnetic moment is 5.9/~n per atom, somewhat less than the theoretical value of 7/a B per atom. Magnetization measurements of McEwen et al. [5] show that (1 1 0) is the easy direction and (1 0 0) is the hard direction, and, as expected, the anisotropy is quite small. The experimental resistivity data has been reported by Curry et al. [1] in 1960. It exhibits a sharp cusp at 90.4 K which corresponds to the magnetic ordering. The spin-only resistivity above Neel temperature shows a slow decrease towards a constant value at around 200 K, which is attributed to the effects of short range order. A short review has been given by Legvold et al. [6] in 1980. In the next section, we discuss a theoretical approach based on Kim's ferromagnetic considerations [7] of the magnetic resistivity, modified by the planar helimagnetic order and anisotropy effects. In the last section, a discussion of the agreement with the experimental data is presented. CALCULATIONS The model Hamiltonian which described particular order, present in Eu, is given by:
the
H = - - ~ ~ JijSi" S./+D~S~z,
(1)
i
j~i
where Si is the Eu spin at the site i • Jij is the coupling constant between ions at sites i and j and it is usual RKKY conduction electron induced interaction which is essential to the magnetic order. The second term of the Hamiltonian represents the anisotropy of the system and D is positive in order to confine the spins in a plane perpendicular to the z axis, e.g. the (1 0 0) direction. We follow Kim's approach to the calculation of the ferromagnetic contribution to the electrical resistivity modifying it to take into account the helical order and the anisotropy of Au. This is expressed normalized to the resistivity at infinite temperature as [7] :
p(T) 4 2;Fdqq(2h~.f p(oo) - N S ( S + I ) ~ d~'2q
•f ~
(S(q) • S(--q)).
(2)
N is the total number of Eu sites, S is the Eu spin which already includes the spin-orbit coupling (S = 7/2), KF is the Fermi wave number and S(q) is defined by:
S(q) - Z S. e-iq'R""
(3)
n
Rn indicates the position of site n. ( ) means the statistical mechanical average as taken with the Hamiltonian written above and p(oo) is the high temperature resistivity which does not contain the effect of short range order as given by Kim [7]. The Kim's point of view relates the spin correlation function to the static susceptibility through the fluctuation-dissipation theorem [8] and gets the latter by making a RPA and a mean field approximation [8] above and below to the transition temperature, respectively. We recalculate the static susceptibility suitable to Eu and get the spin correlation function:
(S(q) "S(--q))T<.TN = ½NS(S+ 1).
i
219
220
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON EUROPIUM
aS(S+ I)D = ~S(S+ 1) T ~ % 12To
(I/2)(T/To) F ~ NS(S +1) • h(q, Q) +S(S + 1)D 3To (l/2)(T/To) ~_( Q ~ _ T N -- T h(q, Q) + To
Q),
3' = - -
0 = Q/2KF (4)
h(q, Q) +
3To
(1/2)(T/To)
+
(13)
,+l
+ 1-2 [02 --3c¢1 In
~ NS(S + 1)
T-- TN
I 1+0 2+1
1 2 0 In
I
(S(q) • S(--q))T>~TN = ~NS(S + 1)
s(s + 1)D
1 [
(1-0)
2 +a
[(1 + 0 ) ~ +oq [ 1 - C ) ) ~ +,~1 [(~2 + c~l 2
a 3/2\
To
4- (Q -~ -- Q).
(5) (14)
T--T N h (q, Q) + - To In (4) there is a factor two in front of (TN -- To)]To in Kim's article [7] (formula 7) which results from a Kim's wrong expansion around TN. To is a characteristical temperature, above it, short range order is negligible and h(q, Q) is given by:
s ( s + l) h(q, Q) = N [ J ( Q ) - - J ( q ) ]
3To
'
(6)
where J(q) is the Fourier transform of Jij and Q is the wave number associated to the magnetic order. We assume that critical spins fluctuation of wave length close to 27r/Q are the important contributions to the Eu spin-only resistivity, an assumption which becomes consistent with the fitted data. That amounts to approximate h(q, Q) as: [q_Q[2 . h(q, Q) =
aK~. '
a > o.
(7)
DISCUSSION Using the well known values of Eu; namely, the turn angle = 54 ° , the lattice constant = 4.83A [6] and KF = 1.065) k-1 (estimated within a spherical Fermi surface model); we get: O = 0.092.
p(rN) 0(oo)
= 43' [~ 1(0, X) + ~I(0, 0)]
p(Tu)/p(oo) = 1.36
p ( T < Tu) = 43'(1 + e) [~I(O, X) + ~ I((~,--3'e)]
7 ~ 4•(0.092,0)=
o(o°)
(9) where
(10)
ru 3"
=
arN 4To
3TN 3"S(S + 1)
(18)
and
(11)
0.670.
(19)
That yields: D ~ 25.7K
TN
(17)
1.36 (8)
P(T>~ TN) = 43'(1 + e) [~ I ( 0 , X + 3'e) + ~I(0,3'e)1,
T--
(16)
in, Eu [1]. Nagamiya et al. [9] have pointed out that anisotropy effects are quite small; so, we set X = 0 in (16) in order to get an approximated value of 3', which also means that:
D "~
_
(15)
Also,
Then, after a lengthy but straightforward integration of (2), we finally get the temperature dependent spin-only resistivity:
0(oo)
(12)
and
I(Q'a) = 6
(1/2)(T/To)
Vol. 55, No. 3
(20)
(20) is in accord with the previous analysis of Nagamiya et al. [9]. Now, we introduce the value of 3, (19) in (8) and (9) and choose the optimum value of the anisotropy parameter D which results to be a small value after the computer numerical analysis. It turns out that the theoretical curve is insensitive to values of X smaller than 0.01 e.g. D smaller than --0.26K. Then, we set 3' = 0.01
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON EUROPIUM
Vol. 55, No. 3 1.~
I
I
1.2
~
/
221
spin wave excitations; a fact which is well-known even though the fitting does not look so bad. It would be interesting to see whether other materials can have a spin-only resistivity described by formulas such as (8) and (9). So far, we do not know any effort in that direction.
I
~
7
Acknowledgement - The authors thank Dr. R. Iraldi for a critical reading of this article.
,0'o
,;o
260
REFERENCES
25o
T/K
1.
M.A. Curry, S. Legvold & F.H. Spedding, Phys.
Rev. 117,953 (1960). Fig. 1. Eu spin-only resistivity vs temperature. The dashed line is the experimental curve [1] and the full line is the theoretical one.
2. 3.
in formulas (8) and (9). As a matter of curiosity, we point out that values of Q smaller than 0.092 yield a better fitting of the experimental data, which is also related to the uncertainty of the Fermi wave vector, where Q = 0 gives the best one. Nevertheless, it does not make any physical sense because this last value correspons to a ferromagnetic order. As we can see in Fig. 1, the disagreement is tess than 10% above T N . The fitting is quite well below Tw in the range T > 80K. We do not expect a fine agreement at low temperatures because a mean field theory, as it happens to be in the Kim's approach, does not describe
N.V. Volkenshtein, G.V. Fedorov & V.E. Startsev,
Bulletin of the Academy of Sciences of the USSR 28,447 (1964). G.T. Meaden,
N.H.
Sze
&
J.R.
Johnston,
Dynamical Aspects of Critical Phenomena, (Edited 4. 5. 6. 7. 8. 9.
by J.I. Budnick & M.P. Kawatra), Gordon & Breach (1972). N.G. Nereson, C.E. Olsen & G.P. Arnold, Phys. Rev. 135, A176 (1964). K.A. McEwen & P. Touborg, J. Phys. (Paris) 32, C1-484 (1971). S. Legvold, Ferromagnetic Materials, (Edited by E.P. Wohlfarth) North Holland (1980). D.J. Kim,Prog. Theor. Phys. 31,921 (1964). R.M. White, Quantum Theory of Magnetism. Springer Verlag (1983). T. Nagamiya, K. Nagata & Y. Kitano, Prog. Theor. Phys. (Kyoto) 27, 1253 (1962).
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON THE EUROPIUM RESISTIVITY L. D'Onofrio, F.P. Marin and J.D. Mujica Dpto. de Fisica, Fac. de Ciencias, Universidad Central de Venezuela, A.P. 21201, Caracas 1020 A, Venezuela
(Received 1 June 1984; in revised form 18 January 1985, by E.F. Bertaut) We introduce and magnetic order effects, into the Kim's approach to the spin disorder resistivity, in order to discuss the Eu spin-only resistivity. INTRODUCTION THE ANOMALOUS BEHAVIOUR of the electrical resistivity of Europium metal has been the subject of several studies [ 1, 2, 3]. In this article, some attempts to fit the experimental data have been made considering the critical fluctuation of the spins and its peculiar magnetic structure. Europium rare earth is different from its lanthanides neighbours. It is divalent is the metallic state and has a b c c structure with a lattice parameter of 4.83,~. Neutron diffraction measurements [4] show that Eu goes from the paramagnetic state to a planar helimagnetic one at around 90.4K, with the magnetic moments parallel to a cube face and the rotation axis along the (1 0 0) direction. The interlayer turn angle is of 54 ° and the observed magnetic moment is 5.9/~n per atom, somewhat less than the theoretical value of 7/a B per atom. Magnetization measurements of McEwen et al. [5] show that (1 1 0) is the easy direction and (1 0 0) is the hard direction, and, as expected, the anisotropy is quite small. The experimental resistivity data has been reported by Curry et al. [1] in 1960. It exhibits a sharp cusp at 90.4 K which corresponds to the magnetic ordering. The spin-only resistivity above Neel temperature shows a slow decrease towards a constant value at around 200 K, which is attributed to the effects of short range order. A short review has been given by Legvold et al. [6] in 1980. In the next section, we discuss a theoretical approach based on Kim's ferromagnetic considerations [7] of the magnetic resistivity, modified by the planar helimagnetic order and anisotropy effects. In the last section, a discussion of the agreement with the experimental data is presented. CALCULATIONS The model Hamiltonian which described particular order, present in Eu, is given by:
the
H = - - ~ ~ JijSi" S./+D~S~z,
(1)
i
j~i
where Si is the Eu spin at the site i • Jij is the coupling constant between ions at sites i and j and it is usual RKKY conduction electron induced interaction which is essential to the magnetic order. The second term of the Hamiltonian represents the anisotropy of the system and D is positive in order to confine the spins in a plane perpendicular to the z axis, e.g. the (1 0 0) direction. We follow Kim's approach to the calculation of the ferromagnetic contribution to the electrical resistivity modifying it to take into account the helical order and the anisotropy of Au. This is expressed normalized to the resistivity at infinite temperature as [7] :
p(T) 4 2;Fdqq(2h~.f p(oo) - N S ( S + I ) ~ d~'2q
•f ~
(S(q) • S(--q)).
(2)
N is the total number of Eu sites, S is the Eu spin which already includes the spin-orbit coupling (S = 7/2), KF is the Fermi wave number and S(q) is defined by:
S(q) - Z S. e-iq'R""
(3)
n
Rn indicates the position of site n. ( ) means the statistical mechanical average as taken with the Hamiltonian written above and p(oo) is the high temperature resistivity which does not contain the effect of short range order as given by Kim [7]. The Kim's point of view relates the spin correlation function to the static susceptibility through the fluctuation-dissipation theorem [8] and gets the latter by making a RPA and a mean field approximation [8] above and below to the transition temperature, respectively. We recalculate the static susceptibility suitable to Eu and get the spin correlation function:
(S(q) "S(--q))T<.TN = ½NS(S+ 1).
i
219
220
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON EUROPIUM
aS(S+ I)D = ~S(S+ 1) T ~ % 12To
(I/2)(T/To) F ~ NS(S +1) • h(q, Q) +S(S + 1)D 3To (l/2)(T/To) ~_( Q ~ _ T N -- T h(q, Q) + To
Q),
3' = - -
0 = Q/2KF (4)
h(q, Q) +
3To
(1/2)(T/To)
+
(13)
,+l
+ 1-2 [02 --3c¢1 In
~ NS(S + 1)
T-- TN
I 1+0 2+1
1 2 0 In
I
(S(q) • S(--q))T>~TN = ~NS(S + 1)
s(s + 1)D
1 [
(1-0)
2 +a
[(1 + 0 ) ~ +oq [ 1 - C ) ) ~ +,~1 [(~2 + c~l 2
a 3/2\
To
4- (Q -~ -- Q).
(5) (14)
T--T N h (q, Q) + - To In (4) there is a factor two in front of (TN -- To)]To in Kim's article [7] (formula 7) which results from a Kim's wrong expansion around TN. To is a characteristical temperature, above it, short range order is negligible and h(q, Q) is given by:
s ( s + l) h(q, Q) = N [ J ( Q ) - - J ( q ) ]
3To
'
(6)
where J(q) is the Fourier transform of Jij and Q is the wave number associated to the magnetic order. We assume that critical spins fluctuation of wave length close to 27r/Q are the important contributions to the Eu spin-only resistivity, an assumption which becomes consistent with the fitted data. That amounts to approximate h(q, Q) as: [q_Q[2 . h(q, Q) =
aK~. '
a > o.
(7)
DISCUSSION Using the well known values of Eu; namely, the turn angle = 54 ° , the lattice constant = 4.83A [6] and KF = 1.065) k-1 (estimated within a spherical Fermi surface model); we get: O = 0.092.
p(rN) 0(oo)
= 43' [~ 1(0, X) + ~I(0, 0)]
p(Tu)/p(oo) = 1.36
p ( T < Tu) = 43'(1 + e) [~I(O, X) + ~ I((~,--3'e)]
7 ~ 4•(0.092,0)=
o(o°)
(9) where
(10)
ru 3"
=
arN 4To
3TN 3"S(S + 1)
(18)
and
(11)
0.670.
(19)
That yields: D ~ 25.7K
TN
(17)
1.36 (8)
P(T>~ TN) = 43'(1 + e) [~ I ( 0 , X + 3'e) + ~I(0,3'e)1,
T--
(16)
in, Eu [1]. Nagamiya et al. [9] have pointed out that anisotropy effects are quite small; so, we set X = 0 in (16) in order to get an approximated value of 3', which also means that:
D "~
_
(15)
Also,
Then, after a lengthy but straightforward integration of (2), we finally get the temperature dependent spin-only resistivity:
0(oo)
(12)
and
I(Q'a) = 6
(1/2)(T/To)
Vol. 55, No. 3
(20)
(20) is in accord with the previous analysis of Nagamiya et al. [9]. Now, we introduce the value of 3, (19) in (8) and (9) and choose the optimum value of the anisotropy parameter D which results to be a small value after the computer numerical analysis. It turns out that the theoretical curve is insensitive to values of X smaller than 0.01 e.g. D smaller than --0.26K. Then, we set 3' = 0.01
ANISOTROPY AND MAGNETIC ORDER EFFECTS ON EUROPIUM
Vol. 55, No. 3 1.~
I
I
1.2
~
/
221
spin wave excitations; a fact which is well-known even though the fitting does not look so bad. It would be interesting to see whether other materials can have a spin-only resistivity described by formulas such as (8) and (9). So far, we do not know any effort in that direction.
I
~
7
Acknowledgement - The authors thank Dr. R. Iraldi for a critical reading of this article.
,0'o
,;o
260
REFERENCES
25o
T/K
1.
M.A. Curry, S. Legvold & F.H. Spedding, Phys.
Rev. 117,953 (1960). Fig. 1. Eu spin-only resistivity vs temperature. The dashed line is the experimental curve [1] and the full line is the theoretical one.
2. 3.
in formulas (8) and (9). As a matter of curiosity, we point out that values of Q smaller than 0.092 yield a better fitting of the experimental data, which is also related to the uncertainty of the Fermi wave vector, where Q = 0 gives the best one. Nevertheless, it does not make any physical sense because this last value correspons to a ferromagnetic order. As we can see in Fig. 1, the disagreement is tess than 10% above T N . The fitting is quite well below Tw in the range T > 80K. We do not expect a fine agreement at low temperatures because a mean field theory, as it happens to be in the Kim's approach, does not describe
N.V. Volkenshtein, G.V. Fedorov & V.E. Startsev,
Bulletin of the Academy of Sciences of the USSR 28,447 (1964). G.T. Meaden,
N.H.
Sze
&
J.R.
Johnston,
Dynamical Aspects of Critical Phenomena, (Edited 4. 5. 6. 7. 8. 9.
by J.I. Budnick & M.P. Kawatra), Gordon & Breach (1972). N.G. Nereson, C.E. Olsen & G.P. Arnold, Phys. Rev. 135, A176 (1964). K.A. McEwen & P. Touborg, J. Phys. (Paris) 32, C1-484 (1971). S. Legvold, Ferromagnetic Materials, (Edited by E.P. Wohlfarth) North Holland (1980). D.J. Kim,Prog. Theor. Phys. 31,921 (1964). R.M. White, Quantum Theory of Magnetism. Springer Verlag (1983). T. Nagamiya, K. Nagata & Y. Kitano, Prog. Theor. Phys. (Kyoto) 27, 1253 (1962).