HAL Id: jpa-00228980
https://hal.archives-ouvertes.fr/jpa-00228980
Submitted on 1 Jan 1988
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
THE PHONON SELF-ENERGY IN AN
ANISOTROPIC HEISENBERG FERROMAGNET
WITH SINGLE-ION ANISOTROPY
J. Tucker
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988
THE PHONON SELF-ENERGY IN AN ANISOTROPIC HEISENBERG
FERROMAGNET WITH SINGLE-ION ANISOTROPY
J. W. Tucker
Department of Physics, University of Shefield, Shefield, S37RH, G.B.
Abstract.
-
The phonon self-energy in an anisotropic Heisenberg ferromagnet with single-ion anisotropy is derived using diagrammatic theory. The result is an improvement over those hitherto reported in that the optical branch of the magnetic excitations arising from the single-ion anisotropy is fully accounted for.1. Introduction
In this paper we report the result of a theoret- ical study of the spin-phonon contribution t o the self-energy of the phonon Green function in a spin-
1 Heisenberg ferromagnet having both anisotropic ex- change and single-ion anisotropy. The dominant spin- phonon interaction was taken as the contribution aris- ing from the modulation of the exchange interactions that is linear in the atomic displacements. A diagram-
matic technique based on a generalization of the VLP
approach to spin-phonon systems was used and the calculations were carried out using a semi-invariant expansion of the spin Green function in which the crystal field anisotropy was incorporated exactly [I]. The semi-invariants were evaluated using standard ba- sis operators. The diagrams were classified according to the high density expansion (1
/
z),
where z is the number of spins with which a given spin interacts. Un- til now, the single spin anisotropy has only been ac- counted for by use of an effective anisotropy field, or by the adoption of a phenomenological interaction Hamil- tonian, or by working within the Holstein-Primakoff representation of the spin operators. Such devices ne- glect the optical spin wave excitations in the energy spectrum and over simplify the spin dynamics, partic- ularly at elevated temperatures. By using as the basic building blocks in the construction of the perturba- tion expansion, transverse and longitudinal spin Green functions and effective exchange interactions that have already been renormalised to lowest order, full account of the optical spin wave spectrum was included.2. The Hamiltonian
The system we studied was the coupled spin-phonon system described by the Hamiltonian H = H,
+
H L+
Hs-,
with,i, j,
H, is the Hamiltonian for a spin-1 ferromagnet hav-
ing both uniaxial anisotropy and single-ion anisotropy.
J0 ( R j ) and J+ ( R j ) are the longitudinal and trans- verse exchange interactions between a pair of spins at sites R i and R j , and D is the single-ion anisotropy.
H L is the Hamiltonian for the lattice in the harmonic
approximation where and a q j are creation and an- nihilation operators for phonons of wave vector q in the jth branch with energy hwW. H,-, is the lowest-order spin-phonon interaction arising from the modulation of the exchange integral among the spins by the lattice vibrations, with
I a ( R j ) = [ a ~ ( R i
-
R j ) /a
( E
-
R ; ) ] ~ . (2) The atomic displacement operators are expressed In terms of phonon operators byu. (Ri) =
x
(h/
2 p ~ w q , ) I" xq , j
xe. (qj) (aqj
+
a? q,) exp ( i q R ) (3)where e (qj) is a unit vector and p and V are the den- sity and volume of the system respectively.
3. The Green function
The quantity of interest is the phonon Green func- tion
A (qj, i a ; q'j', ia') =
-
1'
1'
d71d7zx
xexp {i (071
+
6 ' 7 2 ) ) X (T,& (TI) J q / j t (72)) (4)1/2
where
4%
= (wqj/
2 ~ ) (a*+
akq).
The operators are in the imaginary-time Heisenberg representation and the angular brackets denote a ther- mal average. T, is the Wick's time-ordering oper- ator, and iff is an imaginary boson frequency that takes plus and minus even integer multiples of i?r/
P.
,t? r ti
/
~ B T . In diagrammatic theory a Dyson like equation may be readily established for this Green function, namely,A (qj, iff ; j') = Ao (qj, iu) 6j, j r +
+PA0 (qj, iu) A (qj, i a ; j") A (qj", i a ; j') (5)
C8 - 1614 JOURNAL DE PHYSIQUE
in which A (gj, i a ; j") is an irreducible self-energy consisting of all those diagrams that cannot be broken into two parts by the cutting of a zeroth-order internal phonon line. A0 is the zeroth-order Green function
A0 ( a , i a ; j') = w$6ii,
/
[V (w&+
02)].
(6)The importance of the self-energy A (qj, i o ; j") is that it determines the excitation frequency and the damping of the phonon excitations. Equation (5) is of course a maxtrix equation with respect to the po- larization indices, so the self-energy is not necessarily diagonal with respect to these indices, and the pure phonon modes are coupled. Usually, in experimental situations, such as in an ultrasonic experiment, a ge- ometry is selected where the self-energy is diagonal, or may be approximated so. In this case one is only inter- ested in the diagonal component A (qj, io) whose real and imaginary parts determine the energy shift and damping of the phonon excitations. It is this quantity that we have calculated.
4. Results
The diagrammatic theory adopted was that used by Chakraborty and Tucker [I] in a recent study of the magnetic excitations in an anisotropic ferromag- net with spin-phonon interactions. A similar approach has been used by Kamenskii [2], and Jones and Cot- tam [3], for the isotropic magnet, and by Kamensky [4], for the ferromagnet with anisotropic exchange. How- ever, in those cases the situation was somewhat sim- pler because of the absence of any single-ion terms in the Hamiltonian. In this paper we report the low tem- perature result for the self-energy. Space limiations prevent us from giving any details of the calculation here. We simply remark that the result follows from diagrams that are the analogue of those in figure 2 of [I]? used to obtain the expression for the polarization part of the spin Green function at low temperatures. To order (1
/
z ) we obtain for the self-energy,x
{pi
(q) k, ~ i - ~ , EL) B; (q, k,~ i ,
EL-^)
+
C.c.1 D (io, EL,+
[.i (q, k ) ~ t - ~ , EL) 8; (q, k,EL,
E c - ~ )+
c.C.1 D (io, ~ t , Ei-q)+
[.i (q, k1 ck-ql EL) B; (9, k,EL,
+
c.c.] D (io, c i ,EL-^)
+
[B, (q, k, EL) B; (q, kl&ti
E G - ~ ) +c.c.] D (iu, EL, Ekeq)} (7)with
a
+
[I+" (k)-
I+" (k-
q)] [b (y-
y)-
DP])and
D (io, x, y)
=
coth (By/
2)-
coth (px/
2) i a + y - xThe E$ are the excitation frequencies of the non-
interacting spin-wave spectrum first established by Ginzburg [5], namely 5 1 Ek = y
-
- b J + ( k ) f 2++
[b2 J+ (k12+
4D2+
~DPJ' ( k ) ] ''l (8) with y = wo+
J0 (0) (SZ), (9) and b = (SZ), ; P = 2-
3 ((s")~), (I0)The importance of our result, equation (7), is that it includes the single-ion anisotropy in a rigorous way. In the isotropic limit, when J+ = J0 and D vanishes, it is
seen that the optical branch drops out and
~i
reduces to the familiar spin-wave energy, ~k = y - bJ (k) of the isotropic magnet. The result of [3] for the phonon self-energy is thenrecovered. To extend the calculation to higher temperatures, other diagrams involving two momentum summations, but still of order 1/
z , have to be included. Diagrams of this type which had to be accounted for in our discussion of the magnetic excita- tions near the Curie temperature are very cumbersome to evaluate (see Eqs. (38) to (80) of Ref. [I]. Work is in progress on the evaluation of these diagrams and a full account of our work including numerical estimates of the phonon lifetime over the whole temperature range will be published in due course.[I] Chakraborty, K. G. and Tucker, J. W., Physica A 146 (1987) 582.
[2] Kamenskii, V. G . , Zh. Eksp. Teor. Fiz. 59
(1970) 2244 [Sow. Phys.-JETP 32 (1971) 12141. [3] Jones, M. J. and Cottam, M. G., Phys. Status
Solidi B 67 (1975) 75.
[4] Kamensky, W. G., Physica 70 (1973) 493. [5] Ginzburg, S. L., Fiz. Tverd. Tela 1 2 (1970) 1805.
