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Relationship between electronic conduction states and indirect magnetic exchange
~
Solld State Communications, Vol. 69, No. 2, pp.167-169, 1989. Printed in Great Britain.
0038-I098/89 $3.00 + .00 Pergamon Press pie
RELATIONSHIP BETWEEN ELECTRONIC CONDUCTION STATES AND INDIRECT MAGNETIC EXCHANGE
P.Monachesi* Institut de Physique ThEorique, Universit6 de Lausanne, CH 1015 Lausanne, Switzerland
P.Giannozzi IRRMA-EPFL, CH 1015 Lausanne, Switzerland
M.Acquarone Dipartimento di Fisica, G.N.S.M., Universita' di Parma, 1 43100 Parma, Italy (Received 6 October 1988 by B. Miihlschlegel)
An attempt to find a quantitative relationship between magnetic transition temperature and conduction states properties in metallic materials is presented. The exchange interaction among magnetic moments is assumed to be of the Anderson's s-f type and the conduction states have s-d orbital character. The results are analyzed vs.different amount of s-d hybridization within LCAO scheme.
eu1
Introduction J(rij) = J0
We present new results aiming to ascertain the importance of electronic conduction bands in determining the magnetic properties of metallic materials where the Anderson's s-f mixing mechanism is effective 1. This mechanism assumes a weak hybridization between a localized state and the conduction bands. The magnetic coupling between the localized states is then provided by the perturbative theory of Gon~alves da Silva and Falicovz. We shall henceforth assume the states bearing the magnetic moments to be f states. Previous results3 obtained with very simple conduction band models pointed out the crucial role played by the position of the Fermi energy EF relative to the main features of the density of states (DOS). Presently we perform calculations using a set of 6 bands obtained within a tight-binding scheme including s and d orbitals centered on the two atoms of the unit cell in the CsC1 structure. The energies involved in the Anderson's mechanism are the virtual excitation energies of a hole (Eh) or an electron (E e) from the Fermi surface to the localized magnetic f orbital and the width A acquired by the f state due to hybridization with the band states. The former energies must be small enough to let the Anderson's mechanism dominate over the s-f direct one, but larger than A in order to prevent valence fluctuations. The experimental determination of Eh and Ee for Rare Earths and Actinides has been done by XPS and BIS techniques, respectively4. They do not exceed typically 1 eV in some compounds of light Rare Earths (e.g. Ce) and light Actinides (e.g. U) ions. Within these boundaries it is possible to express the interaction between the magnetic moments via the conduction states as an effective Heisenberg Hamiltonian whose coupling parameters are given by2
eu2
E f deml J" dem2 F(eml'rq)F(em2'rll) ml,m2 eF eL2 (eF-eh-eml)2 O(eF-em2) x [ eml---em2
O(em2--eF) ~ eF-eh--em2j
(1)
with O step function, L, U lower and upper band limits. In (1) all energies are scaled by the bandwidth W. The assumptions are made that Ee>>E h and that the hybridization amplitude Vm k, between the f-state and the m th band, is independent oia'm and k. V is related to the f enhancement since V2= 1/re A/p(Ef) where p(Ef) is the DOS of the conduction electrons at the energy Ef = EF of the f level. The constant J0 = 2/S ~j V4/W3 fixes the energy scale, p.j depends on the ground state J-multiplet of the f state. The function F(em,rij), with K = ka, rlj= Rij/a (a lattice constan0 is: F(em,rxj) --_
1 ;8(em-e m) cos(g r0) d31¢. (2703 BZ
where em is the dispersion relation of the m~ band. K
Results Since the magnetic properties depend essentially on the Fermi surface we have tried to set up a simplified conduction band scheme where one can easily vary the amount of different orbital admixing at EF. Our conduction states, although not truly realistic, are built up having in mind the CsC1 compounds of Ce with Ag. Zn, Mg, for which the Anderson s mechanism applies 5. In all these materials the outmost electronic configuration contains s and d electrons on either atom in the unit cell. Band calculations of isoelectronic compounds with heavier Rare
* Permanent address: Dxpartimento dl Fisica, Unlversit~t. dell'Aquila, 1-67100 L'Aquila (Italy). 167
ELECTRONIC CONDUCTION STATES AND INDIRECT MAGNETIC EXCHANGE
168
Earths are available and show that the Fermi level intersects bands of mainly s-d character 6,7. To simplify the problem, however, we have built up Bloch sums of s states centered on the magnetic ion and d states on the metallic one, including only the shortest lattice translation. Hence the 6x6 tight-binding secular determinant factorizes into a 4x4 s-T2~ and a 2x2 Eg block due the symmetry properties of th~ Slater-Koster z~ energy and overlap integrals. These integrals, 14 altogether, are the input parameters of our calculation. A first set of parameters has been estimated from the LaAg band calculation of Hasegawa 7. Fig. 1 displays two different band structures differing in the parameters of the s-T2g block. In the upper panel the s-d hybridization is weak and the lower band of about 0.4 Ry width is of mainly s character corresponding to a rather flat DOS (Fig.2a). In the lower panel the stronger s-d interaction flattens the lower band with consequent rise of the DOS in the interval 0-0.2 Ry and a narrowing of the higher peak (Fig.2b). We have evaluated the parameters in (1) up to fourth nearest neighbors and with them the maximum of their Fourier transform Jq=max J(~), with J(K)=~, J(rij ) exp(iK r~j)
60
Vol. 69, No. 2
a)
40 o) 20L~
J.
u)
0 80-
b)
60
for different eF. The q vector in the Brillouin Zone which gives the maximum Ja yields the stable spin configuration, whilst Jq is related to the transition temperature
0.0
0.2
0.4
0.6
L (Ry)
0.8
TN=2S(S+ 1)Jq/3KB
Fig.2
with S total spin of the f state and KB Boltzman's constant. eF was fixed through the number of conduction electrons n and the DOS's of Fig.2 with a step of n=0.2. This corresponds to calculate the maximum Jq(n) for 60 values
o.e".L
g - ~x
~
..,%=
x
a
o°
"~xx%
x
x
.,, .,,,
x
~
% : , , . . o..... ,.... ° " x %= ~ , ~ ,
,.B.,,,~: /
0.4
>..,
x. 0.2
x
0.0
.
~
N
xx •
~ -x x
~
x
x
xx
x
x
x 1¢
'%- ."
=¢
01 ~)
'=e° • o= :Xllx =elm
t,~
-
ax
o~
•
=
Xxaag
o~e = .0= man I0~
xa
,,oo
,o
.o=
II BR x
0.4 x
=
x x
x
x
"> :::x
0.2
=x
~
x
,<
x
x
xXXXx
r
M
x
R
Density of states relative to the upper (a) and lower (b) panels of Fi~.l. The DOS were obtained using a cubic mesh of 120J k-point in 1/8 of the Brillouin Zone (BZ) and an energy step of 1/60 Ry. Dotted lines indicate Fermi levels corresponding to n = 2,4,6,8,10 electrons in the conduction bands.
X
M
r
R
Fig.1 Band structures for the 6 conduction s,d states in a CsCl structure. Upper and lower panel differ by (sscr),(sd(r) Slater-Koster integrals. Crosses and squares mchcate s-T2g and Eg components, respectively. The f state is not shown.
of EF. The resulting patterns, shown in Fig.3, give the behavior of Ja(n) as a function of n in umts of J0. Similar oscillating b~havior has been observed previously~ suggesting that the structures in the DOS enhance or suppress a regularly modulated background. A comparison with the panels of Fig.2 indicates an overall correspondence between DOS and transition temperature vs n or EF. In fact in Fig.3a,b larger temperatures correspond to higher DOS' values in Fig.2a,b, respectively. The effect of the stronger s-d admixing (Fig.2b) reflects itself in the occurrence of the peak below n=2 as well as in the enhancement of the scale in Fig.3b with respect to Fig.3a. The behavior of Jo seems to be well accounted for by the corresponding DO$'s also in the interval 4-10 electrons in both cases, the differences between Fig.3a and b being possibly explained by the rough doubling of the scale and the steeper DOS' behavior in case b with respect to a. The stable spin configurations associated to the maxima Ja(n) are given in Fig.3 for some n values, q vectors representing symmetry points in the BZ (F, R, M, X) describe commensurate structures, whereas points along the symmetry lines (T,A) are incommensurate. Finally, to have an order-of-magnitude estimate of Ieal temperatures we fix V=0.1 eV and S=5/2 (as for Ce~+), obtaining the highest values of TN = 9 and 24 K for case a and b of Fig.3, respectively, occurring for 8
Vol. 69, No. 2
ELECI'RONIC CONDUCTION STATES AND INDIRECT ~ I C
EXCHANGE
% C
o A
CT
x.
r
A
M
I-
R
I
b)
0
4 0 A
c CT
"n 2
O
0
2
~
4
6
electrons
8
10
12
r'l
Fig. 3 Plot of the maximum Fourier transform Ja as a function of n. The calculations have been done for e h = 0.1 W in Eq.(1). Jq is proportional to the magnetic transition temperature: TN = 0.008 Jq/J9 with the parameters fixed in the text. a) and b) panels refer, respectively, to upper and lower panels of Figs.1 and 2. The BZ q vectors of the stable spin configurations are reported for n = 2, 4,6,8,10
a) T : t c = (1 1 0 . 7 6 ) ~ ; b) A : r = (1 1 1) ~/4.
A : w= (1 1 1) n'x/-3/3
REFERENCES
1. P.W. Anderson, Phys. Rev. 11, 97 (1961). 2. C.E.T. Gon~alves da Silva and L.M. Falicov, J. Phys. C 5, 63 (1972). 3. M. Aequarone and P. Monaehesi, Phys. Rev. B 38, 2555 (1988). 4. LK. Lang, Y. Baer, P.A. Cox, J. Phys. F 11,121, (1981); Y. Baet in Ham~ook on the Physics and Chemistry of the Actinides, vol. 1 (North Holland, Amsterdam, 1984).
5. R.M. Galera, J. Pierre, J. Voiron, G. Dampne, Solid State Commun. 46, 45 (1983); A. Eiling and J.S. Schilling, Phys. Rev. Lett. 46, 364 (1981). 6. D.K. Ray, M. Belakhovsky, Solid State Commun. 16, 1015 (1975); M. Belakhovsky, J. Pierre and K. Ray, J. Phys. F 5, 2274 (1975). 7. A. Hasegawa, Z. Physik B22, 231 (1975). 8. J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 (1954).
169
Solld State Communications, Vol. 69, No. 2, pp.167-169, 1989. Printed in Great Britain.
0038-I098/89 $3.00 + .00 Pergamon Press pie
RELATIONSHIP BETWEEN ELECTRONIC CONDUCTION STATES AND INDIRECT MAGNETIC EXCHANGE
P.Monachesi* Institut de Physique ThEorique, Universit6 de Lausanne, CH 1015 Lausanne, Switzerland
P.Giannozzi IRRMA-EPFL, CH 1015 Lausanne, Switzerland
M.Acquarone Dipartimento di Fisica, G.N.S.M., Universita' di Parma, 1 43100 Parma, Italy (Received 6 October 1988 by B. Miihlschlegel)
An attempt to find a quantitative relationship between magnetic transition temperature and conduction states properties in metallic materials is presented. The exchange interaction among magnetic moments is assumed to be of the Anderson's s-f type and the conduction states have s-d orbital character. The results are analyzed vs.different amount of s-d hybridization within LCAO scheme.
eu1
Introduction J(rij) = J0
We present new results aiming to ascertain the importance of electronic conduction bands in determining the magnetic properties of metallic materials where the Anderson's s-f mixing mechanism is effective 1. This mechanism assumes a weak hybridization between a localized state and the conduction bands. The magnetic coupling between the localized states is then provided by the perturbative theory of Gon~alves da Silva and Falicovz. We shall henceforth assume the states bearing the magnetic moments to be f states. Previous results3 obtained with very simple conduction band models pointed out the crucial role played by the position of the Fermi energy EF relative to the main features of the density of states (DOS). Presently we perform calculations using a set of 6 bands obtained within a tight-binding scheme including s and d orbitals centered on the two atoms of the unit cell in the CsC1 structure. The energies involved in the Anderson's mechanism are the virtual excitation energies of a hole (Eh) or an electron (E e) from the Fermi surface to the localized magnetic f orbital and the width A acquired by the f state due to hybridization with the band states. The former energies must be small enough to let the Anderson's mechanism dominate over the s-f direct one, but larger than A in order to prevent valence fluctuations. The experimental determination of Eh and Ee for Rare Earths and Actinides has been done by XPS and BIS techniques, respectively4. They do not exceed typically 1 eV in some compounds of light Rare Earths (e.g. Ce) and light Actinides (e.g. U) ions. Within these boundaries it is possible to express the interaction between the magnetic moments via the conduction states as an effective Heisenberg Hamiltonian whose coupling parameters are given by2
eu2
E f deml J" dem2 F(eml'rq)F(em2'rll) ml,m2 eF eL2 (eF-eh-eml)2 O(eF-em2) x [ eml---em2
O(em2--eF) ~ eF-eh--em2j
(1)
with O step function, L, U lower and upper band limits. In (1) all energies are scaled by the bandwidth W. The assumptions are made that Ee>>E h and that the hybridization amplitude Vm k, between the f-state and the m th band, is independent oia'm and k. V is related to the f enhancement since V2= 1/re A/p(Ef) where p(Ef) is the DOS of the conduction electrons at the energy Ef = EF of the f level. The constant J0 = 2/S ~j V4/W3 fixes the energy scale, p.j depends on the ground state J-multiplet of the f state. The function F(em,rij), with K = ka, rlj= Rij/a (a lattice constan0 is: F(em,rxj) --_
1 ;8(em-e m) cos(g r0) d31¢. (2703 BZ
where em is the dispersion relation of the m~ band. K
Results Since the magnetic properties depend essentially on the Fermi surface we have tried to set up a simplified conduction band scheme where one can easily vary the amount of different orbital admixing at EF. Our conduction states, although not truly realistic, are built up having in mind the CsC1 compounds of Ce with Ag. Zn, Mg, for which the Anderson s mechanism applies 5. In all these materials the outmost electronic configuration contains s and d electrons on either atom in the unit cell. Band calculations of isoelectronic compounds with heavier Rare
* Permanent address: Dxpartimento dl Fisica, Unlversit~t. dell'Aquila, 1-67100 L'Aquila (Italy). 167
ELECTRONIC CONDUCTION STATES AND INDIRECT MAGNETIC EXCHANGE
168
Earths are available and show that the Fermi level intersects bands of mainly s-d character 6,7. To simplify the problem, however, we have built up Bloch sums of s states centered on the magnetic ion and d states on the metallic one, including only the shortest lattice translation. Hence the 6x6 tight-binding secular determinant factorizes into a 4x4 s-T2~ and a 2x2 Eg block due the symmetry properties of th~ Slater-Koster z~ energy and overlap integrals. These integrals, 14 altogether, are the input parameters of our calculation. A first set of parameters has been estimated from the LaAg band calculation of Hasegawa 7. Fig. 1 displays two different band structures differing in the parameters of the s-T2g block. In the upper panel the s-d hybridization is weak and the lower band of about 0.4 Ry width is of mainly s character corresponding to a rather flat DOS (Fig.2a). In the lower panel the stronger s-d interaction flattens the lower band with consequent rise of the DOS in the interval 0-0.2 Ry and a narrowing of the higher peak (Fig.2b). We have evaluated the parameters in (1) up to fourth nearest neighbors and with them the maximum of their Fourier transform Jq=max J(~), with J(K)=~, J(rij ) exp(iK r~j)
60
Vol. 69, No. 2
a)
40 o) 20L~
J.
u)
0 80-
b)
60
for different eF. The q vector in the Brillouin Zone which gives the maximum Ja yields the stable spin configuration, whilst Jq is related to the transition temperature
0.0
0.2
0.4
0.6
L (Ry)
0.8
TN=2S(S+ 1)Jq/3KB
Fig.2
with S total spin of the f state and KB Boltzman's constant. eF was fixed through the number of conduction electrons n and the DOS's of Fig.2 with a step of n=0.2. This corresponds to calculate the maximum Jq(n) for 60 values
o.e".L
g - ~x
~
..,%=
x
a
o°
"~xx%
x
x
.,, .,,,
x
~
% : , , . . o..... ,.... ° " x %= ~ , ~ ,
,.B.,,,~: /
0.4
>..,
x. 0.2
x
0.0
.
~
N
xx •
~ -x x
~
x
x
xx
x
x
x 1¢
'%- ."
=¢
01 ~)
'=e° • o= :Xllx =elm
t,~
-
ax
o~
•
=
Xxaag
o~e = .0= man I0~
xa
,,oo
,o
.o=
II BR x
0.4 x
=
x x
x
x
"> :::x
0.2
=x
~
x
,<
x
x
xXXXx
r
M
x
R
Density of states relative to the upper (a) and lower (b) panels of Fi~.l. The DOS were obtained using a cubic mesh of 120J k-point in 1/8 of the Brillouin Zone (BZ) and an energy step of 1/60 Ry. Dotted lines indicate Fermi levels corresponding to n = 2,4,6,8,10 electrons in the conduction bands.
X
M
r
R
Fig.1 Band structures for the 6 conduction s,d states in a CsCl structure. Upper and lower panel differ by (sscr),(sd(r) Slater-Koster integrals. Crosses and squares mchcate s-T2g and Eg components, respectively. The f state is not shown.
of EF. The resulting patterns, shown in Fig.3, give the behavior of Ja(n) as a function of n in umts of J0. Similar oscillating b~havior has been observed previously~ suggesting that the structures in the DOS enhance or suppress a regularly modulated background. A comparison with the panels of Fig.2 indicates an overall correspondence between DOS and transition temperature vs n or EF. In fact in Fig.3a,b larger temperatures correspond to higher DOS' values in Fig.2a,b, respectively. The effect of the stronger s-d admixing (Fig.2b) reflects itself in the occurrence of the peak below n=2 as well as in the enhancement of the scale in Fig.3b with respect to Fig.3a. The behavior of Jo seems to be well accounted for by the corresponding DO$'s also in the interval 4-10 electrons in both cases, the differences between Fig.3a and b being possibly explained by the rough doubling of the scale and the steeper DOS' behavior in case b with respect to a. The stable spin configurations associated to the maxima Ja(n) are given in Fig.3 for some n values, q vectors representing symmetry points in the BZ (F, R, M, X) describe commensurate structures, whereas points along the symmetry lines (T,A) are incommensurate. Finally, to have an order-of-magnitude estimate of Ieal temperatures we fix V=0.1 eV and S=5/2 (as for Ce~+), obtaining the highest values of TN = 9 and 24 K for case a and b of Fig.3, respectively, occurring for 8
Vol. 69, No. 2
ELECI'RONIC CONDUCTION STATES AND INDIRECT ~ I C
EXCHANGE
% C
o A
CT
x.
r
A
M
I-
R
I
b)
0
4 0 A
c CT
"n 2
O
0
2
~
4
6
electrons
8
10
12
r'l
Fig. 3 Plot of the maximum Fourier transform Ja as a function of n. The calculations have been done for e h = 0.1 W in Eq.(1). Jq is proportional to the magnetic transition temperature: TN = 0.008 Jq/J9 with the parameters fixed in the text. a) and b) panels refer, respectively, to upper and lower panels of Figs.1 and 2. The BZ q vectors of the stable spin configurations are reported for n = 2, 4,6,8,10
a) T : t c = (1 1 0 . 7 6 ) ~ ; b) A : r = (1 1 1) ~/4.
A : w= (1 1 1) n'x/-3/3
REFERENCES
1. P.W. Anderson, Phys. Rev. 11, 97 (1961). 2. C.E.T. Gon~alves da Silva and L.M. Falicov, J. Phys. C 5, 63 (1972). 3. M. Aequarone and P. Monaehesi, Phys. Rev. B 38, 2555 (1988). 4. LK. Lang, Y. Baer, P.A. Cox, J. Phys. F 11,121, (1981); Y. Baet in Ham~ook on the Physics and Chemistry of the Actinides, vol. 1 (North Holland, Amsterdam, 1984).
5. R.M. Galera, J. Pierre, J. Voiron, G. Dampne, Solid State Commun. 46, 45 (1983); A. Eiling and J.S. Schilling, Phys. Rev. Lett. 46, 364 (1981). 6. D.K. Ray, M. Belakhovsky, Solid State Commun. 16, 1015 (1975); M. Belakhovsky, J. Pierre and K. Ray, J. Phys. F 5, 2274 (1975). 7. A. Hasegawa, Z. Physik B22, 231 (1975). 8. J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 (1954).
169