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Electronic structure and thermodynamic properties of Fe-Pt alloys
Journal of Magnetism and Magnetic Materials 11/4-107 (1992) 703-704 North-Holland
,44"
Electronic structure and thermodynamic properties of F e - P t alloys M. Podg6rny, M. T h o n and D. W a g n e r Theoretische Physik III, Ruhr-Universitiit Bochum, 4630 Bochum 1, Germany We discuss the magnetic properties of the Pt-Fe alloys using a combined approach of the fixed-spin-moment, self-consistent electronic structure energy calculations for T = 0. and the improved spin- and density fluctuation theory for T > 0. For T = 0, we show the total energy and equations of state as functions of moment and volume. Special attention is paid to the issue of high- to low-moment transition for PtFe~ Invar. Further, we discuss some improvements to the previously formulated spin- and density fluctuation theory. This theory, using the T = 0 total energy surfaces as a starting point, allows for a description of the thermal behavior of the magnetic and elastic properties of a magnetic material. We discuss the temperature dependence of the volume magnetostriction and bulk modulus, and the pressure dependence of the critical temperature for PtF%. We compare these quantities with the available experimental data. First we discuss the T = 0 results. Self-consistent total energy electronic structure calculations have b e e n carried out for o r d e r e d P t F e 3 using the so-called " F i x e d - S p i n - M o m e n t " (FSM) m e t h o d [1]. In contrast to the conventional spinpolarized b a n d structure calculations w h e r e the magnetic m o m e n t is one of the output parameters, in the FSM scheme it is an input p a r a m e t e r t r e a t e d on the same footing as the atomic volume. As a result, one obtains a binding surface E ( V , M ) i.e. total energy as a function of both atomic volume and magnetic m o m e n t . Taking the m o m e n t derivative of E ( V , M ) one obtains a magnetic equation of state H ( V , M ) and a path in the M - V plane along which H ( V , M ) vanishes is equivalent to the M ( V ) curve. The volume derivative of the total energy gives the mechanical equation of state, e x t e n d e d to the M - V plane as well. In fig. 1 the contour plot of both state equations is shown [2]. The M ( V ) curve is discontinuous and displays t h r e e magnetic phases: N o n m a g n e t i c (NM), Low-Spin (LS) and High-Spin (HS) ones. The onset of magnetism is of the second order. The N M phase is stable for the W i g n e r - S e i t z radius Sw~ < 2.61 a.u. The LS phase a p p e a r s at this radius and the magnetic m o m e n t slowly increases up to M = 0.37#B, w h e r e the LS phase ends abruptly at Sw~--2.69 a.u. The HS phase appears at Sws slightly below 2.65 a.u. with the magnetic m o m e n t of ~ 1/x B. In the range 2.65 < Sws < 2.69 a.u. PtFe 3 is metamagnetic, with two coexisting phases of a n o n - z e r o magnetic m o m e n t . The pressure contours are s u p e r i m p o s e d onto the H contours. The diagram indicates two local stability points H = P = 0: one on the HS branch of the M ( V ) curve at S,~ = 2.723 a.u. and M = 1.92/.% and second one on the LS branch at Sw~ = 2.657 a.u. and M = 0.21/~B, and two saddle points: one c o r r e s p o n d i n g to the N M phase energy minimum at Sw~ = 2.655 a.u. and second at the crossing point of the P = 0 contour with the unphysical part (dashed line) of the H = 0 contour at M = 0.6p. B. P t F e 3 is the first material for which such an excep-
tional situation has b e e n theoretically found. The total energy calculated along the M ( V ) branches is shown in the inset of fig. 1. The global minimum is found along the HS branch. The N M and LS total energies differ very little as one could expect from the low magnetic m o m e n t of the LS phase. The difference does not exceed 0.05 mRy, but it should be n o t e d that the NM and LS phases are (meta)stable in the nonoverlapping V regions, so that the one or the o t h e r can be realized. From the inset of fig. 2 we find the A E = ENM - - E . s equal to 1.32 toRy. W e see t h e r e f o r e that the calculated energy difference b e t w e e n the N M and HS states of PtFe is of the o r d e r of 200 K. The LS minimum is positioned at M = 0.21#B and Sws = 2.656 a.u. and its energy is 0.03 t o r y lower than that of the NM saddle 2.0 PtF% NM~-5
~o1.5
I
HS
1.o 20.!o
!65
2 7C
2 75
£b
E 0
E
[
%0.5 c
i 2~ 2
0.0
2.54
2.58
2.62 2.66 Sws (a.u.)
2.70
2.74
Fig. 1. Contour plot of the magnetic field in the Sw~- M plane for PtFe 3. Contours are plotted every 1 mRy/p. B. Dashed line indicates the unphysical region of the M(V) curve. Superimposed are the pressure contours for -50, 0, and 50 kbar. Inset: total energy calculated along the HS and LS branches of the M(V)curve and the for the NM phase.
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M. Podg6rny et al. / Electronic structure and thermodynamics of Fe-Pt
704 84.0 25 7
~%.
~
65
~ C:50
C-50
~ 2,0
side improves the previously formulated theory [3]. It describes the correlation ( m : ( r ) u ( r ) ) o of the local m o m e n t and density. The e q u a t i o n of state are
,
' \" \ \ \ \
~
__
c"="s'°21 :g!i
_
aF
H °~_
'
m
,~l~,r
= - P=
0.5 0
~83.0 --~
200
460
~ l
aM
Iv,<,,~> ..... :b.,< ..... '>,,
i!(i)vE) O M,<,,,~),.<:,:'e.,<,,,=, .,, .
(4)
600
Temperature (K)
-
1
C=30 -
8 2 . 5
a(E).
=
~
0
,
,
100
I
~
/
-3",',
"
'
200 300 400 T e m p e r a t u r e (K)
7
500
600
Fig. 2. Temperature dependence of the volume per atom for C=31)...50 mRy ao/Ix]~. D and k,. are calculated to fix T~ = 450 K and V(T~)=82.6a~, see text• The dashed line shows the experimental data for Pt2_sFev5 [4]. Inset: bulk modulus for the same choice of parameters.
point. It is r e m a r k a b l e that a system like PtFe 3 can be b r o u g h t from the HS-state to the LS-state by application of pressure. T h e critical pressure for such a transition is according to o u r calculations 100 kbar. Such a transition has b e e n found recently at the critical pressure of approximately 60 kbar [5]. In the case of T > 0 the FSM-calculations are comb i n e d with a classical fluctuation theory [3], where we use the following H a m i l t o n i a n
~'= v ' f@r{E[V+ + ~I c E
,'(r), M+m(r)]
[ V , m , ( r ) ] 2 + 2' D [ V u ( r ) ] - / .
(1)
/
i,)
Here, M and V are the t h e r m o d y n a m i c m e a n values of m a g n e t i z a t i o n and volume p e r atom; re(r) and u ( r ) are local fluctuations, which are t r e a t e d as r a n d o m variables. FAV, M ) is the energy polynomial in M and V, fitted to the FSM-calculations. T h e a p p r o x i m a t e free energy F is the m i n i m u m of the right h a n d side of the Peierls inequality F ~ c , < m i n ( F 0 + <)Y - #~0)u) =: F ,
(2)
where the Fourier t r a n s f o r m e d quadratic Y 0 is
+(k(rn=kt' k + m : .
kUk)} •
(3)
T h e variational p a r a m e t e r s (-Oik, wk, ~k) are determ i n e d from eq. (2). T h e last term on the right h a n d
In addition, we get four equations of selfconsistency for the fluctuations ( m ~ ) 0 , ( r 2 ) 0 and ( m : u ) o . T h e p a r a m e t e r s of the theory are C, D of eq. (1) and a necessary cutoff k c. In fig. 2 we p r e s e n t the calculations for the volume magnetostriction and the bulk modulus• We take C7 as a free p a r a m e t e r and calculate D and k c to r e p r o d u c e the experimental values for T~ and V(T¢). As the best fit to the m e a s u r e d V(T)-curvc [4] we find C = 35, D = 0.39 and k c = 1.68, where all quantities are expressed in atomic units. For this choice of p a r a m e t e r s we find that the linear expansion coefficient is more or less c o n s t a n t for T < Tc, but unfortunately too large by a factor of 3 for T > T c. This may be traced back to the fact that, including the volume fluctuations • (V ( v-" )( Tc) > _ 5a~), we have to extrapolate the FSM-calculations into a region, where the energy surface has not b e e n calculated. A l t h o u g h the experimental results differ from the theoretical curves in detail, the overall characteristics of the transition with respect to bulk modulus and with respect to the magnetovolume effect is r e p r o d u c e d by theory for the first time. In particular we point to the softening of the bulk modulus below T~ and to the steep increase at the transition which is of the correct o r d e r of m a g n i t u d e B H ( T >~ T ~ ) - B H ( T < _ T c ) = 1 M b a r [6]. For the pressure d e p e n d e n c e of the Curie t e m p e r a t u r e wc get d T J d P ] v,,~ = 2.8 K / k b a r which c o m p a r e s very well with the experimental value of - 3 . 8 K / k b a r [5]. W e also note that inclusion of the extra terms in eq. (3) converts the first-order phase transition at T~ into a s e c o n d - o r d e r transition, at least for the above r e p o r t e d choice of p a r a m e t e r s . References
[1] A.R. Williams, V.L. Moruzzi, J. Kiibler and K. Schwarz, Bull. Am. Phys. Soc. 29 (1984) 278. K. Schwarz and P. Mohn, J. Phys. F 14 (1984) L129. [2] M. Podg6rny, Phys. Rev. B 43 (1991) in press. [3] D. Wagner, J. Phys.: Condens. Mater. 1 (1989) 4635. P. Mohn, K. Schwarz and D. Wagner, Phys. Rev. B 43 (1991) 3318. [4] K. Sumiyama, Y. Emoto, M. Shiga and Y. Nakamura, J. Phys. Soc. Jpn. 5(I (1981) 3296. [5] M.M. Abd-Elmeguid and H. Micklitz, Phys. Rev. B 411 (1989) 7395. [6] G. Hausch, J. Phys. Jpn. 37 (1974) 819.
,44"
Electronic structure and thermodynamic properties of F e - P t alloys M. Podg6rny, M. T h o n and D. W a g n e r Theoretische Physik III, Ruhr-Universitiit Bochum, 4630 Bochum 1, Germany We discuss the magnetic properties of the Pt-Fe alloys using a combined approach of the fixed-spin-moment, self-consistent electronic structure energy calculations for T = 0. and the improved spin- and density fluctuation theory for T > 0. For T = 0, we show the total energy and equations of state as functions of moment and volume. Special attention is paid to the issue of high- to low-moment transition for PtFe~ Invar. Further, we discuss some improvements to the previously formulated spin- and density fluctuation theory. This theory, using the T = 0 total energy surfaces as a starting point, allows for a description of the thermal behavior of the magnetic and elastic properties of a magnetic material. We discuss the temperature dependence of the volume magnetostriction and bulk modulus, and the pressure dependence of the critical temperature for PtF%. We compare these quantities with the available experimental data. First we discuss the T = 0 results. Self-consistent total energy electronic structure calculations have b e e n carried out for o r d e r e d P t F e 3 using the so-called " F i x e d - S p i n - M o m e n t " (FSM) m e t h o d [1]. In contrast to the conventional spinpolarized b a n d structure calculations w h e r e the magnetic m o m e n t is one of the output parameters, in the FSM scheme it is an input p a r a m e t e r t r e a t e d on the same footing as the atomic volume. As a result, one obtains a binding surface E ( V , M ) i.e. total energy as a function of both atomic volume and magnetic m o m e n t . Taking the m o m e n t derivative of E ( V , M ) one obtains a magnetic equation of state H ( V , M ) and a path in the M - V plane along which H ( V , M ) vanishes is equivalent to the M ( V ) curve. The volume derivative of the total energy gives the mechanical equation of state, e x t e n d e d to the M - V plane as well. In fig. 1 the contour plot of both state equations is shown [2]. The M ( V ) curve is discontinuous and displays t h r e e magnetic phases: N o n m a g n e t i c (NM), Low-Spin (LS) and High-Spin (HS) ones. The onset of magnetism is of the second order. The N M phase is stable for the W i g n e r - S e i t z radius Sw~ < 2.61 a.u. The LS phase a p p e a r s at this radius and the magnetic m o m e n t slowly increases up to M = 0.37#B, w h e r e the LS phase ends abruptly at Sw~--2.69 a.u. The HS phase appears at Sws slightly below 2.65 a.u. with the magnetic m o m e n t of ~ 1/x B. In the range 2.65 < Sws < 2.69 a.u. PtFe 3 is metamagnetic, with two coexisting phases of a n o n - z e r o magnetic m o m e n t . The pressure contours are s u p e r i m p o s e d onto the H contours. The diagram indicates two local stability points H = P = 0: one on the HS branch of the M ( V ) curve at S,~ = 2.723 a.u. and M = 1.92/.% and second one on the LS branch at Sw~ = 2.657 a.u. and M = 0.21/~B, and two saddle points: one c o r r e s p o n d i n g to the N M phase energy minimum at Sw~ = 2.655 a.u. and second at the crossing point of the P = 0 contour with the unphysical part (dashed line) of the H = 0 contour at M = 0.6p. B. P t F e 3 is the first material for which such an excep-
tional situation has b e e n theoretically found. The total energy calculated along the M ( V ) branches is shown in the inset of fig. 1. The global minimum is found along the HS branch. The N M and LS total energies differ very little as one could expect from the low magnetic m o m e n t of the LS phase. The difference does not exceed 0.05 mRy, but it should be n o t e d that the NM and LS phases are (meta)stable in the nonoverlapping V regions, so that the one or the o t h e r can be realized. From the inset of fig. 2 we find the A E = ENM - - E . s equal to 1.32 toRy. W e see t h e r e f o r e that the calculated energy difference b e t w e e n the N M and HS states of PtFe is of the o r d e r of 200 K. The LS minimum is positioned at M = 0.21#B and Sws = 2.656 a.u. and its energy is 0.03 t o r y lower than that of the NM saddle 2.0 PtF% NM~-5
~o1.5
I
HS
1.o 20.!o
!65
2 7C
2 75
£b
E 0
E
[
%0.5 c
i 2~ 2
0.0
2.54
2.58
2.62 2.66 Sws (a.u.)
2.70
2.74
Fig. 1. Contour plot of the magnetic field in the Sw~- M plane for PtFe 3. Contours are plotted every 1 mRy/p. B. Dashed line indicates the unphysical region of the M(V) curve. Superimposed are the pressure contours for -50, 0, and 50 kbar. Inset: total energy calculated along the HS and LS branches of the M(V)curve and the for the NM phase.
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M. Podg6rny et al. / Electronic structure and thermodynamics of Fe-Pt
704 84.0 25 7
~%.
~
65
~ C:50
C-50
~ 2,0
side improves the previously formulated theory [3]. It describes the correlation ( m : ( r ) u ( r ) ) o of the local m o m e n t and density. The e q u a t i o n of state are
,
' \" \ \ \ \
~
__
c"="s'°21 :g!i
_
aF
H °~_
'
m
,~l~,r
= - P=
0.5 0
~83.0 --~
200
460
~ l
aM
Iv,<,,~> ..... :b.,< ..... '>,,
i!(i)vE) O M,<,,,~),.<:,:'e.,<,,,=, .,, .
(4)
600
Temperature (K)
-
1
C=30 -
8 2 . 5
a(E).
=
~
0
,
,
100
I
~
/
-3",',
"
'
200 300 400 T e m p e r a t u r e (K)
7
500
600
Fig. 2. Temperature dependence of the volume per atom for C=31)...50 mRy ao/Ix]~. D and k,. are calculated to fix T~ = 450 K and V(T~)=82.6a~, see text• The dashed line shows the experimental data for Pt2_sFev5 [4]. Inset: bulk modulus for the same choice of parameters.
point. It is r e m a r k a b l e that a system like PtFe 3 can be b r o u g h t from the HS-state to the LS-state by application of pressure. T h e critical pressure for such a transition is according to o u r calculations 100 kbar. Such a transition has b e e n found recently at the critical pressure of approximately 60 kbar [5]. In the case of T > 0 the FSM-calculations are comb i n e d with a classical fluctuation theory [3], where we use the following H a m i l t o n i a n
~'= v ' f@r{E[V+ + ~I c E
,'(r), M+m(r)]
[ V , m , ( r ) ] 2 + 2' D [ V u ( r ) ] - / .
(1)
/
i,)
Here, M and V are the t h e r m o d y n a m i c m e a n values of m a g n e t i z a t i o n and volume p e r atom; re(r) and u ( r ) are local fluctuations, which are t r e a t e d as r a n d o m variables. FAV, M ) is the energy polynomial in M and V, fitted to the FSM-calculations. T h e a p p r o x i m a t e free energy F is the m i n i m u m of the right h a n d side of the Peierls inequality F ~ c , < m i n ( F 0 + <)Y - #~0)u) =: F ,
(2)
where the Fourier t r a n s f o r m e d quadratic Y 0 is
+(k(rn=kt' k + m : .
kUk)} •
(3)
T h e variational p a r a m e t e r s (-Oik, wk, ~k) are determ i n e d from eq. (2). T h e last term on the right h a n d
In addition, we get four equations of selfconsistency for the fluctuations ( m ~ ) 0 , ( r 2 ) 0 and ( m : u ) o . T h e p a r a m e t e r s of the theory are C, D of eq. (1) and a necessary cutoff k c. In fig. 2 we p r e s e n t the calculations for the volume magnetostriction and the bulk modulus• We take C7 as a free p a r a m e t e r and calculate D and k c to r e p r o d u c e the experimental values for T~ and V(T¢). As the best fit to the m e a s u r e d V(T)-curvc [4] we find C = 35, D = 0.39 and k c = 1.68, where all quantities are expressed in atomic units. For this choice of p a r a m e t e r s we find that the linear expansion coefficient is more or less c o n s t a n t for T < Tc, but unfortunately too large by a factor of 3 for T > T c. This may be traced back to the fact that, including the volume fluctuations • (V ( v-" )( Tc) > _ 5a~), we have to extrapolate the FSM-calculations into a region, where the energy surface has not b e e n calculated. A l t h o u g h the experimental results differ from the theoretical curves in detail, the overall characteristics of the transition with respect to bulk modulus and with respect to the magnetovolume effect is r e p r o d u c e d by theory for the first time. In particular we point to the softening of the bulk modulus below T~ and to the steep increase at the transition which is of the correct o r d e r of m a g n i t u d e B H ( T >~ T ~ ) - B H ( T < _ T c ) = 1 M b a r [6]. For the pressure d e p e n d e n c e of the Curie t e m p e r a t u r e wc get d T J d P ] v,,~ = 2.8 K / k b a r which c o m p a r e s very well with the experimental value of - 3 . 8 K / k b a r [5]. W e also note that inclusion of the extra terms in eq. (3) converts the first-order phase transition at T~ into a s e c o n d - o r d e r transition, at least for the above r e p o r t e d choice of p a r a m e t e r s . References
[1] A.R. Williams, V.L. Moruzzi, J. Kiibler and K. Schwarz, Bull. Am. Phys. Soc. 29 (1984) 278. K. Schwarz and P. Mohn, J. Phys. F 14 (1984) L129. [2] M. Podg6rny, Phys. Rev. B 43 (1991) in press. [3] D. Wagner, J. Phys.: Condens. Mater. 1 (1989) 4635. P. Mohn, K. Schwarz and D. Wagner, Phys. Rev. B 43 (1991) 3318. [4] K. Sumiyama, Y. Emoto, M. Shiga and Y. Nakamura, J. Phys. Soc. Jpn. 5(I (1981) 3296. [5] M.M. Abd-Elmeguid and H. Micklitz, Phys. Rev. B 411 (1989) 7395. [6] G. Hausch, J. Phys. Jpn. 37 (1974) 819.