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Spin-phonon interaction in Heisenberg antiferromagnet
PHYSICS LETTERS
Volume 44A, number 7
SPINPHONON
INTERACTION
13 August 1973
IN HEISENBERG
ANTIFERROMAGNET
P. CLUCK* Physics Department,
KingS College, London,
U.K.
Received 21 May 1973 Using a combination of reduced temperature and interaction range as small parameters, the magnon-phonon scattering part of sound attenuation in antiferromagnets is calculated in the intermediate temperature range, resulting in qualitative agreement with experimental frequency and temperature dependences.
The present discussion results from an attempt to improve on an earlier treatment of spin-phonon coupling in’a Heisenberg magnet based on RPA, in which the sound attenuation coefficient vanished identically [ 11. By utilizing
the recent work of Mattuck [2] on the relationship between decoupling and diagrammatic methods we were able to systematize and improve the equation of motion method in El], by employing the diagram technique developed by Vaks et al. [3]. The consequences for the magnetic properties of a ferromagnet will be described elsewhere [lo]. Here a similar treatment is used to finding the sound attenuation oq for a compressible antiferromagnet, for frequencies w4 exceeding inverse spin relaxation times and satisfying wq > Dq2 (D = diffusion constant [4]). The treatment is based entirely on that of ref. [3] for ferromagnets. The hamiltonian for the two sublattice model is
+
Hphonon + g C (ai, olh
where
the gradient
aoh - 1)2
form arising from expanding
(1) J(cy-a’)
in powers
of atomic
displacements
u, describes
volume
Hphononis the lattice hamiltonian
in harmonic approximation. The last term [3] is included to facilitate the diagram expansion and effectively excludes states (a, h) for which the number of particles per site is not unity (g + m at end of calculation). The problem has been considered in a number of works [4-71, using decoupling and other methods, as well as experimentally [S]. The idea here is to order the perturbation series for the phonon correlation function D,(q,W,) in powers of the parameter a = u.-’ e-112, where E = 1 - T/TN and u. = ri is the interaction = - i ({u(o$ u(o$]+), volume, basic to the ideas in [3]. a is small except when the interaction is nearest neighbour only and very near TN where e-1/2 dominates. The validity of results will thus be in the range between those of Pytte [9] (low 7’) and the scaling results of [7], since as shown in [3] weakly damped magnons exist for all T satisfying E > u02, beyond the Dyson range. The constant o+ = l’,/c(q) (c = sound velocity) is given by the imaginary part r4 of the pole of the matrix phonon propagator (i,j = 1,2 denote sublattices) Dii(q,W") = Dt + D$ C(q,wv) Dji. We sum the subset of all irreducible diagrams for the self energy Z(q, w) which are of first order in a, basing the expansion on the free phonon propagator for vYhor,,c,n and on the zero order, magnon propagator G = /3(Jo (S2) - iw,)-l in which (.S2) IS replaced by B,,the Brrl oum function. The modification to the rules of [3] for transcribing diagrams arise from the presence of two sublattices (i,j = 1,2), the presence of phonon-magnon scattering and the shifting of the ordering temperature calculated in a self-consisted way [ 1, lo]. For the singly-connected diagrams with two vertices magnetostriction,
* On leave from Bar-Ban University, Israel.
533
Volume
44A, number
PHYSICS
7
13 August
LETTERS
1973
for the spin correlation functions K:y, K;--, the vertices can be from either sublattice (denote by ,~:and x). ‘The ) and F vertices (- - - -) correspond to effective interactions interaction lines connectingS+,S.vertices (accounting for particle correlations and spin waves, .G ~-(k, w, and Jff (k, wv). Thus for example
and similarly for JF (k, uv). There are additional modified interaction lines (-t, -w- -) due to the gradient term ‘JJ, corresponding to effective interactions for a cl-phonon of qjm (k, wv) q *k/q and Jy (k. w, ) q *k/q in thus approximation. Typical terms for C(q, w) are --
‘-+--’
The resulting summation
for C to first order in m gives
J; B,2d3 C(q,w
=
w4)=
__--
q/3
s
d3k
u2(k2cos20m%._ .~~~
L/‘)
_
_
f$$+rl
&?k(‘X ~~~
+E _
[l + (ck+
+k)(y+k~~Sf)) !mmmm. ~_
+ fk)-*/$I
where fk is the spin-wave energy 131. Evaluating (3) for a simple cubic lattice in the long-wave w4 limit in the Debey approximation, with Jk = 2J C:= 1 (I-cos kid) (kro < I), one finds via 04= Im C/C (C = longitudinal sound velocity) (y x “2,-312
0
DebeylC2
(4)
(Jo T&)1’*
The w2 dependence is the experimentally verified behaviour [8], in agreement with [4j and at variance with some previous work [S]. Corrections to account for spin-correlations, possibly by utilizing the Migdal-Polyakov techniques [ 11j, are necessary to lower the temperature index towards experimental values. Recent calculations [4j tended to be also uncertain on this aspect of cr, owing to lack of definite calculation of the diffusion coefficient’s temperature dependence. It is of interest to calculate the effect of spin-phonon interactions on line broadening in antiferromagnets by the same techniques and we hope to report on this later.
References [l] P. Gluck and 0. Wohlman-Entin, Phys. Stat. Sol. (b) 52 (1972) 323. 121 R.D. Mattuck and A. Theumann, Adv. in Phys. 20 (1971) 722. 131 V.G. Vaks, A.I. Larkin and S.A. Pikin, JETP 26 (1968) 188; V. Kamenski, JETP 32 (1971) 1214. 141 H.S. Bennett and E. Pytte, Phys. Rev. 164 (1967) 712. IS] M. Papoular, Compt. Rend. 258 (1964) 5598. [6] K. Tani and H. Mori, Phys. Letters 19 (1966) 627. [7] K. Kawasaki, Phys. Letters 26A (1968) 543; Prog. Theoret. Phys. 39 (1968) [Sj B. Liithi and R. Pollina, Phys. Rev. 177 (1969) 841. [9] E. Pytte, An. Phys. (N.Y.) 32 (1915) 377. IlO] P. Gluck, to be published. [ll] A.M. Poyakov, JETP 28 (1969) 533; A.A. Migdal, JETP 28 (1969) 1036.
534
285.
Volume 44A, number 7
SPINPHONON
INTERACTION
13 August 1973
IN HEISENBERG
ANTIFERROMAGNET
P. CLUCK* Physics Department,
KingS College, London,
U.K.
Received 21 May 1973 Using a combination of reduced temperature and interaction range as small parameters, the magnon-phonon scattering part of sound attenuation in antiferromagnets is calculated in the intermediate temperature range, resulting in qualitative agreement with experimental frequency and temperature dependences.
The present discussion results from an attempt to improve on an earlier treatment of spin-phonon coupling in’a Heisenberg magnet based on RPA, in which the sound attenuation coefficient vanished identically [ 11. By utilizing
the recent work of Mattuck [2] on the relationship between decoupling and diagrammatic methods we were able to systematize and improve the equation of motion method in El], by employing the diagram technique developed by Vaks et al. [3]. The consequences for the magnetic properties of a ferromagnet will be described elsewhere [lo]. Here a similar treatment is used to finding the sound attenuation oq for a compressible antiferromagnet, for frequencies w4 exceeding inverse spin relaxation times and satisfying wq > Dq2 (D = diffusion constant [4]). The treatment is based entirely on that of ref. [3] for ferromagnets. The hamiltonian for the two sublattice model is
+
Hphonon + g C (ai, olh
where
the gradient
aoh - 1)2
form arising from expanding
(1) J(cy-a’)
in powers
of atomic
displacements
u, describes
volume
Hphononis the lattice hamiltonian
in harmonic approximation. The last term [3] is included to facilitate the diagram expansion and effectively excludes states (a, h) for which the number of particles per site is not unity (g + m at end of calculation). The problem has been considered in a number of works [4-71, using decoupling and other methods, as well as experimentally [S]. The idea here is to order the perturbation series for the phonon correlation function D,(q,W,) in powers of the parameter a = u.-’ e-112, where E = 1 - T/TN and u. = ri is the interaction = - i ({u(o$ u(o$]+), volume, basic to the ideas in [3]. a is small except when the interaction is nearest neighbour only and very near TN where e-1/2 dominates. The validity of results will thus be in the range between those of Pytte [9] (low 7’) and the scaling results of [7], since as shown in [3] weakly damped magnons exist for all T satisfying E > u02, beyond the Dyson range. The constant o+ = l’,/c(q) (c = sound velocity) is given by the imaginary part r4 of the pole of the matrix phonon propagator (i,j = 1,2 denote sublattices) Dii(q,W") = Dt + D$ C(q,wv) Dji. We sum the subset of all irreducible diagrams for the self energy Z(q, w) which are of first order in a, basing the expansion on the free phonon propagator for vYhor,,c,n and on the zero order, magnon propagator G = /3(Jo (S2) - iw,)-l in which (.S2) IS replaced by B,,the Brrl oum function. The modification to the rules of [3] for transcribing diagrams arise from the presence of two sublattices (i,j = 1,2), the presence of phonon-magnon scattering and the shifting of the ordering temperature calculated in a self-consisted way [ 1, lo]. For the singly-connected diagrams with two vertices magnetostriction,
* On leave from Bar-Ban University, Israel.
533
Volume
44A, number
PHYSICS
7
13 August
LETTERS
1973
for the spin correlation functions K:y, K;--, the vertices can be from either sublattice (denote by ,~:and x). ‘The ) and F vertices (- - - -) correspond to effective interactions interaction lines connectingS+,S.vertices (accounting for particle correlations and spin waves, .G ~-(k, w, and Jff (k, wv). Thus for example
and similarly for JF (k, uv). There are additional modified interaction lines (-t, -w- -) due to the gradient term ‘JJ, corresponding to effective interactions for a cl-phonon of qjm (k, wv) q *k/q and Jy (k. w, ) q *k/q in thus approximation. Typical terms for C(q, w) are --
‘-+--’
The resulting summation
for C to first order in m gives
J; B,2d3 C(q,w
=
w4)=
__--
q/3
s
d3k
u2(k2cos20m%._ .~~~
L/‘)
_
_
f$$+rl
&?k(‘X ~~~
+E _
[l + (ck+
+k)(y+k~~Sf)) !mmmm. ~_
+ fk)-*/$I
where fk is the spin-wave energy 131. Evaluating (3) for a simple cubic lattice in the long-wave w4 limit in the Debey approximation, with Jk = 2J C:= 1 (I-cos kid) (kro < I), one finds via 04= Im C/C (C = longitudinal sound velocity) (y x “2,-312
0
DebeylC2
(4)
(Jo T&)1’*
The w2 dependence is the experimentally verified behaviour [8], in agreement with [4j and at variance with some previous work [S]. Corrections to account for spin-correlations, possibly by utilizing the Migdal-Polyakov techniques [ 11j, are necessary to lower the temperature index towards experimental values. Recent calculations [4j tended to be also uncertain on this aspect of cr, owing to lack of definite calculation of the diffusion coefficient’s temperature dependence. It is of interest to calculate the effect of spin-phonon interactions on line broadening in antiferromagnets by the same techniques and we hope to report on this later.
References [l] P. Gluck and 0. Wohlman-Entin, Phys. Stat. Sol. (b) 52 (1972) 323. 121 R.D. Mattuck and A. Theumann, Adv. in Phys. 20 (1971) 722. 131 V.G. Vaks, A.I. Larkin and S.A. Pikin, JETP 26 (1968) 188; V. Kamenski, JETP 32 (1971) 1214. 141 H.S. Bennett and E. Pytte, Phys. Rev. 164 (1967) 712. IS] M. Papoular, Compt. Rend. 258 (1964) 5598. [6] K. Tani and H. Mori, Phys. Letters 19 (1966) 627. [7] K. Kawasaki, Phys. Letters 26A (1968) 543; Prog. Theoret. Phys. 39 (1968) [Sj B. Liithi and R. Pollina, Phys. Rev. 177 (1969) 841. [9] E. Pytte, An. Phys. (N.Y.) 32 (1915) 377. IlO] P. Gluck, to be published. [ll] A.M. Poyakov, JETP 28 (1969) 533; A.A. Migdal, JETP 28 (1969) 1036.
534
285.