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Electronic structure of spin density wave states in transition metals
Journal of Magnetism and Magnetic Materials 104-107 (1992) 749-750 North-Holland
Electronic structure of spin density wave states in transition metals K. Hirai Department of Physics, Nara Medical University, Kashihara, Nara 634, Japan The electronic structure of spin density wave states that have finite local magnetic moments is calculated for fcc Fe and Cr. By comparing energies of the states with different wave vectors, a state of lowest energy is investigated• The result accounts well for the observed states of fcc Fe and Cr. We discuss spin density wave (SDW) states of fcc Fe and Cr on the basis of electronic structure calculations of S D W states. Our aim is to investigate the magnitude of local magnetic moments and the wave vector of the stable SDW. The S D W of fcc Fe has recently been observed in some -yFe precipitates in a Cu matrix such as those with a small amount of Co [1]. In these precipitates, lattice distortion, which most 7 F e precipitates suffer, is suppressed and the magnetic structure is not one that is associated with the lattice distortion but helical S D W (HSDW). As for the S D W of Cr, which is a sinusoidal S D W (SSDW), systematic and exhaustive experiments have been carried out up to the present time (see, e.g., ref. [2]). As is well known, the mechanism of S D W formation in Cr has so far been discussed from the point of view of electronic structure of a non-magnetic state [2]. However, electronic structure calculation of the S D W states with finite local magnetic moments itself has hardly been carried out for Cr or for fcc Fe. This is because the dimensions of the S D W unit cell become rather large for the calculation when the period of the S D W is long. In addition, there are several inequivalent atoms that have different magnitudes of the magnetic m o m e n t for the case of SSDW, which causes a considerable difficulty in the calculation. On account of such difficulties, we abandon the calculations by first-principle methods. Instead, we use a Hamiltonian modeled on te L M T O method in the tight-binding representation [3]. The Hamiltonian is written as HSDW = nnon + E
E
Y'~ (Di)crcr,atimaaim~r,,
i m~do',~'
(1)
where H,o n is the tight-binding Hamiltonian of a nonmagnetic state obtained by the L M T O method, and
l[cos(Q'Ri) Di= --URMd sin(Q " Ri)
sin(Q'Ri) ] - c o s ( Q . Ri)
--URMadCOS(Q'Ri)I[~ _~]
fcc Fe
.... it
20
(2)
(3)
0.0 A. . ". ".
,, ..D'""
z
-I .5
o . . . . o , . .'o
".
1.0
0--.. 0
•
""%
A..
z
0.0
W.
~W ""~V""".V"
-3.0
'.&o "•'A
0.5
.O"
""O",. O
UR
©
Oo-"
A_ . .A"" .O .a'"
..ql
v
z o
"A.. "'A"
O.
© 1.5 .v'-..v...q'"'v'"'v 1.j
".
a 84 v 80 O76 A 72
" '.......A
f-O Z r.o - 4 . 5
3 ".
.O...,a
-6.0 I
for H S D W and Di =
for SSDW. H e r e Q denotes the wave vector and R i is the position vector of the ith site. With a parameter UR, which represents electron correlation among the d electrons, we self-consistently determine the d-component of the magnitude of the magnetic moments, M d for H S D W and the d-component of the amplitude of the first-harmonic, Mid for SSDW; in SSDW, the magnitude of the magnetic moment at the ith site is expressed as M 1 cos(Q • R i) + M 3 cos(3Q • R i) + ... in terms of the harmonics M~. We have assumed that Mid is of primary importance and have neglected the self-consistency with respect to the other harmonics in the first approximation. We show in fig. 1 the magnitude of the magnetic moments M and the energy E of H S D W states with Q = a* (1, q, 0) for fcc Fe, and in fig. 2 the amplitude of the first-harmonic M 1 and the energy E of S S D W
l
o
I 1/4-
I
I 1/2
I
!
I
I/4
I
I
1/2
q of WAVE VECTOR a*(1,q,O) Fig. 1. Magnitude of the magnetic moments M and energy E of the HSDW states with Q = a* (1, q, 0) (q = 0, 1/8, 1/4, 3/8, 1/2) for fcc Fe (n e =8). The value indicated in the figure represents the value of UR (mRy).
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
K. Hirai / Electronic structure of spin density waues
750 Cr
n~ = 5.92
n. =5.96 u
I
~1.5
u
I
I
ne = 5.92 I
l
l 0.0
1.2
Z
. . .V :'. :
I
~
0.9
...-1.-
v
73
n 7~
0 72
v73
&7t
-0. 1
•"
-0.2
-A.-.. "..i
0.3
x" a 78
v
.
:
z . - . . . . --"..::
":.b-
.
1
//
: /
A.....A-
. 7/8
.
i
.
-I
1
7/8
I,
,
1
,
-0.5
. .... n
q of WAVE VECTOR a*(q,O,O)
I
.....
#v
o'" :~" .
..-
.:
o ,"
"
: )"
"
1
7/8
I " ,:.'.'"
.:
.:
: "-
I
o,:
-
-0.4
&75
,~o.o
:g
:
~. ". i : .
y77 o 76
..:::--
V
-0.3
o.. ':""'b
I A .....
."i"i'." '. ,~i.'.'i :-:
"-..-
":
"'a :"
....- ... -~,
©
l
o
". "~'"'"$ '. .'Y •
"-
....., -'.,
I
-
~
'O
-
°-
-=o.8 ~
-A
...... ':
I
.~
.o .
n 74
76
A 75
9""
..-
l
n, = 6
"'.
O.
v"
ne = 5.96
J'
UR 0
~
n.=6
I
7/8
I
1
I
I
7/8
I
1
I
I
7/8
q of WAVE VECTOR a* (q, 0, 0)
Fig. 2. Amplitude of the first harmonic M t and energy E of the SSDW states with Q = a*(q, 0, 0) (q = 1, 15/16, 7/8) for Cr. The cases of n~ = 5.92, 5.96 and 6 are shown. In the case of an antiferromagnetic state (q = 1), M is shown as M~. The value indicated in the figures represents the value of UR (mRy).
states with Q = a * (q, 0, 0) for Cr. H e r e a* = 2 v / a with a being a lattice constant and E is m e a s u r e d from that of the c o r r e s p o n d i n g non-magnetic state. As for Cr, we present cases of a valence n e = 5.92, 5.96 and 6 to see a tendency with respect to no; there is almost no essential difference with respect to n e for fcc Fe. We choose these wave vectors in consideration of an optimum wave vector for the a p p e a r a n c e of infinitesimal magnetic moments, which is o b t a i n e d from the maximum of the u n e n h a n c e d spin susceptibility of a nonmagnetic state [4]. It is to be n o t e d that this o p t i m u m wave vector c o r r e s p o n d s well with the observed one for both fcc Fe and Cr; the observed wave vectors are a * (1, 0.123, 0) for fcc F%~Co 3 precipitates [1] and a * (0.95, 0, 0) for Cr [2]. Now let us discuss a state of lowest energy. For fcc Fe the H S D W state with q = 1 / 8 has the lowest energy, almost i n d e p e n d e n t of U R and accordingly of M. For Cr the S S D W state with q = 1 5 / 1 6 has the lowest energy when M~ is small for every case of ne. We, however, expect that q of the lowest energy state is b e t w e e n 7 / 8 and 15/16 for n c = 5.92, around 15/16 for n¢ = 5.96, and b e t w e e n 15/16 and 1 for n e = 6, judging from the energy difference b e t w e e n these t h r e e states. We note that when M or M 1 is small the wave vector of the lowest energy state agrees with the optimum wave vector m e n t i o n e d above and hence the observed one for both fcc Fe and Cr. W h e n U R and accordingly M or M1 b e c o m e s large, there appears a striking contrast b e t w e e n fcc Fe and
Cr. The wave vector of the lowest energy state remains in almost the same position for fcc Fe, while it approaches rapidly the wave vector of the antiferromagnetic state for Cr. This rapid change and also its n~ d e p e n d e n c e , which is seen in Cr, may be due to the fact that the S D W of Cr is ascribed to the nesting of the Fermi surface; we concluded from the u n e n h a n c e d spin susceptibility calculation that the nesting plays a decisive role in d e t e r m i n i n g the o p t i m u m wave vector for Cr although not for fcc Fe [4]. We suppose that the features of the Fermi surface of a non-magnetic state fade away, and the nesting b e c o m e s obscure when M~ b e c o m e s large. It is to be a d d e d that this difference b e t w e e n fcc Fe and Cr does not originate from the difference b e t w e e n H S D W and SSDW. In conclusion, we have shown that the p r e s e n t calculation of S D W states explains the observed S D W states of fcc Fe and Cr. We have also illustrated a difference in the characteristics of the S D W states b e t w e e n fcc Fe and Cr. References
[1] Y. Tsunoda, J. Phys. F 18 (1988) L251; J. Phys.: Condens. Matter. 1 (1989) 10427. [2] E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [3] O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [4] K. Hirai, J. Phys. Soc. Jpn. 58 (1989) 4288; Prog. Theor. Phys. Suppl. 101 (1990) 119.
Electronic structure of spin density wave states in transition metals K. Hirai Department of Physics, Nara Medical University, Kashihara, Nara 634, Japan The electronic structure of spin density wave states that have finite local magnetic moments is calculated for fcc Fe and Cr. By comparing energies of the states with different wave vectors, a state of lowest energy is investigated• The result accounts well for the observed states of fcc Fe and Cr. We discuss spin density wave (SDW) states of fcc Fe and Cr on the basis of electronic structure calculations of S D W states. Our aim is to investigate the magnitude of local magnetic moments and the wave vector of the stable SDW. The S D W of fcc Fe has recently been observed in some -yFe precipitates in a Cu matrix such as those with a small amount of Co [1]. In these precipitates, lattice distortion, which most 7 F e precipitates suffer, is suppressed and the magnetic structure is not one that is associated with the lattice distortion but helical S D W (HSDW). As for the S D W of Cr, which is a sinusoidal S D W (SSDW), systematic and exhaustive experiments have been carried out up to the present time (see, e.g., ref. [2]). As is well known, the mechanism of S D W formation in Cr has so far been discussed from the point of view of electronic structure of a non-magnetic state [2]. However, electronic structure calculation of the S D W states with finite local magnetic moments itself has hardly been carried out for Cr or for fcc Fe. This is because the dimensions of the S D W unit cell become rather large for the calculation when the period of the S D W is long. In addition, there are several inequivalent atoms that have different magnitudes of the magnetic m o m e n t for the case of SSDW, which causes a considerable difficulty in the calculation. On account of such difficulties, we abandon the calculations by first-principle methods. Instead, we use a Hamiltonian modeled on te L M T O method in the tight-binding representation [3]. The Hamiltonian is written as HSDW = nnon + E
E
Y'~ (Di)crcr,atimaaim~r,,
i m~do',~'
(1)
where H,o n is the tight-binding Hamiltonian of a nonmagnetic state obtained by the L M T O method, and
l[cos(Q'Ri) Di= --URMd sin(Q " Ri)
sin(Q'Ri) ] - c o s ( Q . Ri)
--URMadCOS(Q'Ri)I[~ _~]
fcc Fe
.... it
20
(2)
(3)
0.0 A. . ". ".
,, ..D'""
z
-I .5
o . . . . o , . .'o
".
1.0
0--.. 0
•
""%
A..
z
0.0
W.
~W ""~V""".V"
-3.0
'.&o "•'A
0.5
.O"
""O",. O
UR
©
Oo-"
A_ . .A"" .O .a'"
..ql
v
z o
"A.. "'A"
O.
© 1.5 .v'-..v...q'"'v'"'v 1.j
".
a 84 v 80 O76 A 72
" '.......A
f-O Z r.o - 4 . 5
3 ".
.O...,a
-6.0 I
for H S D W and Di =
for SSDW. H e r e Q denotes the wave vector and R i is the position vector of the ith site. With a parameter UR, which represents electron correlation among the d electrons, we self-consistently determine the d-component of the magnitude of the magnetic moments, M d for H S D W and the d-component of the amplitude of the first-harmonic, Mid for SSDW; in SSDW, the magnitude of the magnetic moment at the ith site is expressed as M 1 cos(Q • R i) + M 3 cos(3Q • R i) + ... in terms of the harmonics M~. We have assumed that Mid is of primary importance and have neglected the self-consistency with respect to the other harmonics in the first approximation. We show in fig. 1 the magnitude of the magnetic moments M and the energy E of H S D W states with Q = a* (1, q, 0) for fcc Fe, and in fig. 2 the amplitude of the first-harmonic M 1 and the energy E of S S D W
l
o
I 1/4-
I
I 1/2
I
!
I
I/4
I
I
1/2
q of WAVE VECTOR a*(1,q,O) Fig. 1. Magnitude of the magnetic moments M and energy E of the HSDW states with Q = a* (1, q, 0) (q = 0, 1/8, 1/4, 3/8, 1/2) for fcc Fe (n e =8). The value indicated in the figure represents the value of UR (mRy).
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
K. Hirai / Electronic structure of spin density waues
750 Cr
n~ = 5.92
n. =5.96 u
I
~1.5
u
I
I
ne = 5.92 I
l
l 0.0
1.2
Z
. . .V :'. :
I
~
0.9
...-1.-
v
73
n 7~
0 72
v73
&7t
-0. 1
•"
-0.2
-A.-.. "..i
0.3
x" a 78
v
.
:
z . - . . . . --"..::
":.b-
.
1
//
: /
A.....A-
. 7/8
.
i
.
-I
1
7/8
I,
,
1
,
-0.5
. .... n
q of WAVE VECTOR a*(q,O,O)
I
.....
#v
o'" :~" .
..-
.:
o ,"
"
: )"
"
1
7/8
I " ,:.'.'"
.:
.:
: "-
I
o,:
-
-0.4
&75
,~o.o
:g
:
~. ". i : .
y77 o 76
..:::--
V
-0.3
o.. ':""'b
I A .....
."i"i'." '. ,~i.'.'i :-:
"-..-
":
"'a :"
....- ... -~,
©
l
o
". "~'"'"$ '. .'Y •
"-
....., -'.,
I
-
~
'O
-
°-
-=o.8 ~
-A
...... ':
I
.~
.o .
n 74
76
A 75
9""
..-
l
n, = 6
"'.
O.
v"
ne = 5.96
J'
UR 0
~
n.=6
I
7/8
I
1
I
I
7/8
I
1
I
I
7/8
q of WAVE VECTOR a* (q, 0, 0)
Fig. 2. Amplitude of the first harmonic M t and energy E of the SSDW states with Q = a*(q, 0, 0) (q = 1, 15/16, 7/8) for Cr. The cases of n~ = 5.92, 5.96 and 6 are shown. In the case of an antiferromagnetic state (q = 1), M is shown as M~. The value indicated in the figures represents the value of UR (mRy).
states with Q = a * (q, 0, 0) for Cr. H e r e a* = 2 v / a with a being a lattice constant and E is m e a s u r e d from that of the c o r r e s p o n d i n g non-magnetic state. As for Cr, we present cases of a valence n e = 5.92, 5.96 and 6 to see a tendency with respect to no; there is almost no essential difference with respect to n e for fcc Fe. We choose these wave vectors in consideration of an optimum wave vector for the a p p e a r a n c e of infinitesimal magnetic moments, which is o b t a i n e d from the maximum of the u n e n h a n c e d spin susceptibility of a nonmagnetic state [4]. It is to be n o t e d that this o p t i m u m wave vector c o r r e s p o n d s well with the observed one for both fcc Fe and Cr; the observed wave vectors are a * (1, 0.123, 0) for fcc F%~Co 3 precipitates [1] and a * (0.95, 0, 0) for Cr [2]. Now let us discuss a state of lowest energy. For fcc Fe the H S D W state with q = 1 / 8 has the lowest energy, almost i n d e p e n d e n t of U R and accordingly of M. For Cr the S S D W state with q = 1 5 / 1 6 has the lowest energy when M~ is small for every case of ne. We, however, expect that q of the lowest energy state is b e t w e e n 7 / 8 and 15/16 for n c = 5.92, around 15/16 for n¢ = 5.96, and b e t w e e n 15/16 and 1 for n e = 6, judging from the energy difference b e t w e e n these t h r e e states. We note that when M or M 1 is small the wave vector of the lowest energy state agrees with the optimum wave vector m e n t i o n e d above and hence the observed one for both fcc Fe and Cr. W h e n U R and accordingly M or M1 b e c o m e s large, there appears a striking contrast b e t w e e n fcc Fe and
Cr. The wave vector of the lowest energy state remains in almost the same position for fcc Fe, while it approaches rapidly the wave vector of the antiferromagnetic state for Cr. This rapid change and also its n~ d e p e n d e n c e , which is seen in Cr, may be due to the fact that the S D W of Cr is ascribed to the nesting of the Fermi surface; we concluded from the u n e n h a n c e d spin susceptibility calculation that the nesting plays a decisive role in d e t e r m i n i n g the o p t i m u m wave vector for Cr although not for fcc Fe [4]. We suppose that the features of the Fermi surface of a non-magnetic state fade away, and the nesting b e c o m e s obscure when M~ b e c o m e s large. It is to be a d d e d that this difference b e t w e e n fcc Fe and Cr does not originate from the difference b e t w e e n H S D W and SSDW. In conclusion, we have shown that the p r e s e n t calculation of S D W states explains the observed S D W states of fcc Fe and Cr. We have also illustrated a difference in the characteristics of the S D W states b e t w e e n fcc Fe and Cr. References
[1] Y. Tsunoda, J. Phys. F 18 (1988) L251; J. Phys.: Condens. Matter. 1 (1989) 10427. [2] E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [3] O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [4] K. Hirai, J. Phys. Soc. Jpn. 58 (1989) 4288; Prog. Theor. Phys. Suppl. 101 (1990) 119.