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Microscopic theory of dynamical matrix in itinerant model of antiferromagnetism
59
Journal of Magnetism and Magnetic Materials 31-34 (1983) 59-60 MICROSCOPIC THEORY OF DYNAMICAL MATRIX IN ITINERANT MODEL OF ANTIFERROMAGNETISM Seiju A M I , N . A . C A D E a n d W. Y O U N G
Department of Physics, Queen Mary College, London E1 4NS, England
The dynamical matrix and the elastic constants are derived for an itinerant antiferromagnet. An orbital representation is used which bypasses the problem of large matrix inversion in reciprocal space. We show that exchange enhancement and antiferromagnetic ordering leads to softening of some of the elastic constants.
Recently Lowde et al. [1] observed acoustic phonon softening in MnNi alloys in the antiferromagnetic (AF) state. Subsequent neutron scattering revealed a martensitic transformation from cubic to tetragonal symmetry at around the same temperature. The experiments strongly suggest that the mechanism for the transformation is associated with the long range ordered nature of the A F state. Experimentally, there have been many indications of the connection between magnetic and elastic properties in transition metals. This connection has also been given theoretical support by Kim [2] who pointed out that it may be explained by an exchange interaction mechanism. Including the exchange contributions, he has shown that the Bohm-Staver formula for the sound velocity squared is modified to become inversely proportional to the bulk longitudinal spin suscesptibility. This leads to phonon softening because the susceptibility tends to diverge as we approach the elastic transition temperature from above. Exchange enhancement is not the only consequence of the A F state: the density of states is also modified when the A F gapping appears. We expect therefore that the elastic properties are even more profoundly modified. The calculation of the elastic constants involves that of the dynamical matrix which, according to Sham [3] depends on the dielectric susceptibility X
x ( q + ¢;, q + ~ ' ) = ~ ( q +
+ U*(q + G , ) x ( q + G I, q + G2)U(q + G2) ]
(2) from which the q = 0 values should be subtracted and where, as in the rest of the paper, numerically subscripted variables are summed or integrated over their values domains. The susceptibility, X, is calculated using the eigenvalues and eigenfunctions of the one-electron Hamiltonian, H(r), describing the A F state. To highlight the magnetic aspects of the system we take
H ( r ) = riP(r) - ½i( ~)[ po(~) - 4 ( , ) ] , where HP(r) is the paramagnetic Hamiltonian which is modified through the exchange-correlation vertex l(r) by the selfconsistent magnetisation [ p o ( r ) - ~ ( r ) ] . In order to simplify both the one electron problem and the implicit matrix inversion in obtaining ;~ from X, we used a basis of Andersen's [4] linear muffin tin orbitals for H r', giving the eigenfunctions:
4~.k(r ) = B.t,.(k)[ q~t,,,(r) - (¢.k - c.)t~tm(r)]
(3)
for r within the atomic cell and having energies o.k. In the Andersen scheme * is the energy derivative of the wavefunction q~ at the reference energy c~. In terms of such a basis we obtain a 2 × 2 (super) matrix for the A F Hamiltonian
~ , q + c;')
+ ~ ( q + G, q ) ~ f g ( q ,
(1) where G, G ' are reciprocal lattice vectors and 2 is the dielectric matrix obtained by exchange, correlation and short range Coulomb enhancement of the usual Lindhard susceptibility, X. The singularity of the long range Coulomb potential, V(q) is cancelled by that of the dielectric function c ( q ) = 1 - V(q)~(q, q). In terms of the ion-ion, ion-electron interaction W(q), U(q) and the dielectric susceptibility, the dynamical matrix is
C~'qq) = (q + Gi)~'(q + G2)" [ W(q + G,),$a,,~ 2 0304-8853/83/0000-0000/$03.00
gO
q + G')
HP(k+Q)
where the reduced symmetry of the magnetic part of the exchange leads to a coupling of paramagnetic k states differing by the spin-density-wave vector Q. In terms of the matrix, U, diagonalising this A F Hamiltonian matrix and B, ¢, and q, of eq. (3) we obtain X in the form
X°(q + G, q + G')
= fy'~; (q + G ) l~i~i.,.,( q, o )f~'2 (q + G'),
(4)
where
f.~.(p) = ~.(~,)ei...,, +co~.,(,. ),
© 1983 N o r t h - H o l l a n d
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S. Ami et aL / Dynamical matrix in antiferromagnetism
60
# is a composite index denoting lm and energy derivative index and a is the sublattice lable. Then, suppressing all but those indices characteristic of the AF state, we have
i""'(o)=la~"(o)+l""'(o)(y,,~+'v $°°',l'q";(ol), IC J~(iO~l -
\
x x \ x \\
(6)
where Yx~, Y~ are the exchange correlation, short range Coulomb components of the effective electron-hole interation obtained as the p(r) derivatives of H. Explicitly,
\ 1
\\
,,,(
\ ,.¢
( Yx~)o of = _ 8oo,8,,,,,~ t f , ( rl ) I ( r I ) f ( r, ), where vat is the atomic cell volume. The integral equation (6) is most easily inverted by using the subband representation given by the (Fourier) transform 0
S~'~(q) = 2-1/2 exp[i(q + ~O).ed ] with a and ~ taking values 0, 1. The effect of the "t~c may be first included by noting the simple spin dependences of 1(o) via the eigenvectors, U, of the AF Hamiltonian matrix. The spin independent Coulomb enhancement may then be included to give l ~ = [1 + P ( I - 2 " y c ) ] - IP for ~ = 0 with
p(q)
= (~(q) - ~ ( q ) I [ l
+ (~(q + Q)I]-16,,~ (q + Q),
(7) where the matrices C and ~ are given by
~.~...(q) = l- p~p ' v,p~ , ~ ,,-z
+ I~,~,~Lt+ x~[
~, - - - p l ~ ' v l ,
Q)
and 1~ ~ (~2(,x
l t ~ , 1 2 ~ ( - + Q),
(8)
with the spin indices of 1 and ] summed over. Apart from minor corrections, the corresponding quantities were derived by Yamada [5] for the jellium model. Because we would like to deal with actual band structures we have cast our quantities in terms of those obtained from an L M T O band structure calculation. Work is in progress to compute these quantities for y-Mn. In the absence of the results of computation, we can make some general remarks about the likely results and verify them for a simplified one-band model. From eqs. (1) and (7) we note that the dielectric function is related to the density of states P enhanced by the interaction l-2yc. Because of the temperature dependence of the wavefunctions U~x and of the gap function implicit in U, and hence A and ~ , P(q) is very temperature dependent. We note also that the AF gap equation is effectively 1 + t ~ ( Q ) I = 0 near T~. Therefore, in the bulk
1 T/T N
Fig. I. The screening length >, ( L T F = Thomas-Fermi value). (i) Dashed curve, perfect nesting; (ii) full curve, imperfect nesting.
limit of q--, 0 and near to the N6el temperature TN, P(0) can assume a whole range of values provided ~ , giving the degree of mixing of particle-hole momenta q and q + Q is non-zero. If the Fermi level lies in the gap, = 0 because ofparticle hole symmetry, P(q)= (~(q) from eq. (7) and system behaves undramatically. However, in all other cases ~ * 0. For the case of simplified Fedders and Martin band structure our calculations show that this strong temperature dependence leads to elastic softening in that the screening length ), ()k-2a100) may tend to zero. In contrast to the perfect nesting case ( ~ = 0) where ~, is continuous at TN and increases as the temperature is lowered, imperfect nesting (~5 = 0) leads to a discontinuity at TN and decreases as shown in fig. l to be zero at Tm given by 1 + ( I - 2~,c)P(0) = 0. This softening leads, via eqs. (1), (2) and (4) to a softening of all elastic constants and to a vanishing of the sound velocity at TM if the lattice (G = 0) is ignored in eq. (2). These short range (G * 0) elasticities will modify the various elasticities but, despite its simplicity, this calculation shows a very general softening mechanism. References
[1] R.D. Lowde, R.T. Harley, G.A. Saunders, M. Sato, R. Schern and C. Underhill, Proc. R. Soc. London A374 (1981) 87. [2] D.J. Kim, J. Phys. Soc. Japan 40 (1976) 1244. [3] L.J. Sham, Phys. Rev. B6 (1972) 3581. [4] O.K. Andersen, Phys. Rev. BI2 (1975) 3060. [5] H. Yamada, Solid State Commun. 37 (1981) 841.
Journal of Magnetism and Magnetic Materials 31-34 (1983) 59-60 MICROSCOPIC THEORY OF DYNAMICAL MATRIX IN ITINERANT MODEL OF ANTIFERROMAGNETISM Seiju A M I , N . A . C A D E a n d W. Y O U N G
Department of Physics, Queen Mary College, London E1 4NS, England
The dynamical matrix and the elastic constants are derived for an itinerant antiferromagnet. An orbital representation is used which bypasses the problem of large matrix inversion in reciprocal space. We show that exchange enhancement and antiferromagnetic ordering leads to softening of some of the elastic constants.
Recently Lowde et al. [1] observed acoustic phonon softening in MnNi alloys in the antiferromagnetic (AF) state. Subsequent neutron scattering revealed a martensitic transformation from cubic to tetragonal symmetry at around the same temperature. The experiments strongly suggest that the mechanism for the transformation is associated with the long range ordered nature of the A F state. Experimentally, there have been many indications of the connection between magnetic and elastic properties in transition metals. This connection has also been given theoretical support by Kim [2] who pointed out that it may be explained by an exchange interaction mechanism. Including the exchange contributions, he has shown that the Bohm-Staver formula for the sound velocity squared is modified to become inversely proportional to the bulk longitudinal spin suscesptibility. This leads to phonon softening because the susceptibility tends to diverge as we approach the elastic transition temperature from above. Exchange enhancement is not the only consequence of the A F state: the density of states is also modified when the A F gapping appears. We expect therefore that the elastic properties are even more profoundly modified. The calculation of the elastic constants involves that of the dynamical matrix which, according to Sham [3] depends on the dielectric susceptibility X
x ( q + ¢;, q + ~ ' ) = ~ ( q +
+ U*(q + G , ) x ( q + G I, q + G2)U(q + G2) ]
(2) from which the q = 0 values should be subtracted and where, as in the rest of the paper, numerically subscripted variables are summed or integrated over their values domains. The susceptibility, X, is calculated using the eigenvalues and eigenfunctions of the one-electron Hamiltonian, H(r), describing the A F state. To highlight the magnetic aspects of the system we take
H ( r ) = riP(r) - ½i( ~)[ po(~) - 4 ( , ) ] , where HP(r) is the paramagnetic Hamiltonian which is modified through the exchange-correlation vertex l(r) by the selfconsistent magnetisation [ p o ( r ) - ~ ( r ) ] . In order to simplify both the one electron problem and the implicit matrix inversion in obtaining ;~ from X, we used a basis of Andersen's [4] linear muffin tin orbitals for H r', giving the eigenfunctions:
4~.k(r ) = B.t,.(k)[ q~t,,,(r) - (¢.k - c.)t~tm(r)]
(3)
for r within the atomic cell and having energies o.k. In the Andersen scheme * is the energy derivative of the wavefunction q~ at the reference energy c~. In terms of such a basis we obtain a 2 × 2 (super) matrix for the A F Hamiltonian
~ , q + c;')
+ ~ ( q + G, q ) ~ f g ( q ,
(1) where G, G ' are reciprocal lattice vectors and 2 is the dielectric matrix obtained by exchange, correlation and short range Coulomb enhancement of the usual Lindhard susceptibility, X. The singularity of the long range Coulomb potential, V(q) is cancelled by that of the dielectric function c ( q ) = 1 - V(q)~(q, q). In terms of the ion-ion, ion-electron interaction W(q), U(q) and the dielectric susceptibility, the dynamical matrix is
C~'qq) = (q + Gi)~'(q + G2)" [ W(q + G,),$a,,~ 2 0304-8853/83/0000-0000/$03.00
gO
q + G')
HP(k+Q)
where the reduced symmetry of the magnetic part of the exchange leads to a coupling of paramagnetic k states differing by the spin-density-wave vector Q. In terms of the matrix, U, diagonalising this A F Hamiltonian matrix and B, ¢, and q, of eq. (3) we obtain X in the form
X°(q + G, q + G')
= fy'~; (q + G ) l~i~i.,.,( q, o )f~'2 (q + G'),
(4)
where
f.~.(p) = ~.(~,)ei...,, +co~.,(,. ),
© 1983 N o r t h - H o l l a n d
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S. Ami et aL / Dynamical matrix in antiferromagnetism
60
# is a composite index denoting lm and energy derivative index and a is the sublattice lable. Then, suppressing all but those indices characteristic of the AF state, we have
i""'(o)=la~"(o)+l""'(o)(y,,~+'v $°°',l'q";(ol), IC J~(iO~l -
\
x x \ x \\
(6)
where Yx~, Y~ are the exchange correlation, short range Coulomb components of the effective electron-hole interation obtained as the p(r) derivatives of H. Explicitly,
\ 1
\\
,,,(
\ ,.¢
( Yx~)o of = _ 8oo,8,,,,,~ t f , ( rl ) I ( r I ) f ( r, ), where vat is the atomic cell volume. The integral equation (6) is most easily inverted by using the subband representation given by the (Fourier) transform 0
S~'~(q) = 2-1/2 exp[i(q + ~O).ed ] with a and ~ taking values 0, 1. The effect of the "t~c may be first included by noting the simple spin dependences of 1(o) via the eigenvectors, U, of the AF Hamiltonian matrix. The spin independent Coulomb enhancement may then be included to give l ~ = [1 + P ( I - 2 " y c ) ] - IP for ~ = 0 with
p(q)
= (~(q) - ~ ( q ) I [ l
+ (~(q + Q)I]-16,,~ (q + Q),
(7) where the matrices C and ~ are given by
~.~...(q) = l- p~p ' v,p~ , ~ ,,-z
+ I~,~,~Lt+ x~[
~, - - - p l ~ ' v l ,
Q)
and 1~ ~ (~2(,x
l t ~ , 1 2 ~ ( - + Q),
(8)
with the spin indices of 1 and ] summed over. Apart from minor corrections, the corresponding quantities were derived by Yamada [5] for the jellium model. Because we would like to deal with actual band structures we have cast our quantities in terms of those obtained from an L M T O band structure calculation. Work is in progress to compute these quantities for y-Mn. In the absence of the results of computation, we can make some general remarks about the likely results and verify them for a simplified one-band model. From eqs. (1) and (7) we note that the dielectric function is related to the density of states P enhanced by the interaction l-2yc. Because of the temperature dependence of the wavefunctions U~x and of the gap function implicit in U, and hence A and ~ , P(q) is very temperature dependent. We note also that the AF gap equation is effectively 1 + t ~ ( Q ) I = 0 near T~. Therefore, in the bulk
1 T/T N
Fig. I. The screening length >, ( L T F = Thomas-Fermi value). (i) Dashed curve, perfect nesting; (ii) full curve, imperfect nesting.
limit of q--, 0 and near to the N6el temperature TN, P(0) can assume a whole range of values provided ~ , giving the degree of mixing of particle-hole momenta q and q + Q is non-zero. If the Fermi level lies in the gap, = 0 because ofparticle hole symmetry, P(q)= (~(q) from eq. (7) and system behaves undramatically. However, in all other cases ~ * 0. For the case of simplified Fedders and Martin band structure our calculations show that this strong temperature dependence leads to elastic softening in that the screening length ), ()k-2a100) may tend to zero. In contrast to the perfect nesting case ( ~ = 0) where ~, is continuous at TN and increases as the temperature is lowered, imperfect nesting (~5 = 0) leads to a discontinuity at TN and decreases as shown in fig. l to be zero at Tm given by 1 + ( I - 2~,c)P(0) = 0. This softening leads, via eqs. (1), (2) and (4) to a softening of all elastic constants and to a vanishing of the sound velocity at TM if the lattice (G = 0) is ignored in eq. (2). These short range (G * 0) elasticities will modify the various elasticities but, despite its simplicity, this calculation shows a very general softening mechanism. References
[1] R.D. Lowde, R.T. Harley, G.A. Saunders, M. Sato, R. Schern and C. Underhill, Proc. R. Soc. London A374 (1981) 87. [2] D.J. Kim, J. Phys. Soc. Japan 40 (1976) 1244. [3] L.J. Sham, Phys. Rev. B6 (1972) 3581. [4] O.K. Andersen, Phys. Rev. BI2 (1975) 3060. [5] H. Yamada, Solid State Commun. 37 (1981) 841.