SPIN-WAVE LIFETIME IN ANTIFERROMAGNETS

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SPIN-WAVE LIFETIME IN ANTIFERROMAGNETS

E. Balcar

To cite this version:

E. Balcar. SPIN-WAVE LIFETIME IN ANTIFERROMAGNETS. Journal de Physique Colloques,

1971, 32 (C1), pp.C1-701-C1-702. �10.1051/jphyscol:19711247�. �jpa-00214076�

JOURNAL DE PHYSIQUE

Colloque C I, supplkinent au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 701

SPIN-WAVE LIFETIME IN ANTIFERROMAGNETS

E. BALCAR

Physics Institute, Reactor Center Sejbersdorf, A-2444 Austria

R&um6.

On deduit le temps de vie d'un magnon pour un antiferromagnetique decrit par un Hamiltonien de Dyson-Maleev. A cette fin on commence par les equations de mouvement des fonctions de Green thermiques, exprimkes par les operateurs des ondes de spin. Un simple procede de decouplement menant a la renormalisation de Hartree-Fock dans l'equation de premier ordre est Btendu aux equations de second ordre, et donne la valeur du temps de vie d'un magnon.

Une analyse 9. basses temperatures ( ~ B T < wr < ^{2 rJS)} fournit une largeur d'etat proportionnelle a kT4.

Abstract.

The magnon lifetime for an antiferromagnet described by a Dyson-Maleev Hamiltonian is derived in the spin-wave formalism from the equations of motion of thermal Green functions. A simple decoupling procedure lead- ing to the Hartree-Eock renormalization in first order is extended to second-order equations and yields an expression for the magnon lifetime. A low temperature expansion for ksT < wl; < ² rJS produces an energy-width proportional to kT4.

1. Introduction. - Recent work on the energy width of spin-waves in antiferromagnets (Harris et al. [I], Cottam and Stinchcombe [2]) has led to results contradicting earlier theories (Tani [3], Solyom [4]).

Here we apply the equation of motion method and introduce a decoupling procedure suitable for low temperatures. The higher-order Green functions are split into a diagonal and an offdiagonal part and neglect of the offdiagonal parts in third-order func- tions breaks the chain of equations in second order.

The resulting one-magnon thermal Green function (Zubarev [5]) has the form

Here w, denotes the free spin-wave energy for wave- vector k. The self-energy term Z(k, w) in (1.1) yields an energy width

r ( k , w ) = - 2 I m { Z ( k , w + i ~ ' ) ) . (1.2) The algebra of including the second-order equations of motion can be simplified, if only the offdiagonal parts are considered (Balcar [6]). A further simpli- fication stems from the fact, that we retain only those terms in the complicated interaction part of the spin- wave Hamiltonian, which contribute to in the Hartree- Fock approximation.

2. Theory.

Taking the antiferromagnetic Hei- senberg Hamiltonian with exchange interaction J between nearest neighbours on opposite sublattices and a reduced anisotropy field h, as starting point, we change to Bose operators applying a Dyson-Maleev transformation (Maleev [7]). We assume a s. c.

antiferromagnet consisting of two f. c. c. sublattices, each with different magnetic orientation, containing N atoms of spin S. Each atom has r nearest neighbours.

Consequently two kinds of Bose creation- and annihi- lation-operators arise ; a + , a refer t o one and b f , b t o the other sublattice. The Dyson-Maleev Hamil- tonian has the form

X = 2 ~ J S Z [(l+hA) (ak+ ak+bkfbk)+

k

The functions y, depend on the geometry of the crystal and are defined as usual (see f. i. Kittel [8]). 8a.+8-

is a Kronecker &function.

Neglecting the interaction terms in (2.1) for the moment, we see that the remaining part of the Hamil- tonian, X,, does not describe free antiferromagnetic spin-waves. A further transformation is necessary to obtain magnon creation- and annihilation-operators a', a and B', P where

The real coefficients u and w are chosen to make the transformation (2.2) canonical and to diagonalize X, (see f.i. Keffer [9]). The interaction term in (2.1) becomes even more complicated because each of the three terms contributes sixteen terms in the new operators. With the reduced energy

given by

we write X in the form

x (wa a,C +

Pa) (w, a: +

Pk) + + t yp(ua a,f +

Pa) ( ~ k + wk Pk) x

The large number of terms in the interaction part leads in the equation of motion for Gk

< ak ; a:

' 3 P

y,k

to several higher-order Green functions arising from

f

4 yP(ap b: ba bk+a,f a,f ay bl)]

(2.1) the commutator in (2.5). Green functions like

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711247

C

- 702 E. BALCAR 4 a: ap/3: ; a: % o r < ~ a f $"P, ;a: % etc. give no contribution in the Hartree-Fock approximation.

Functions which contribute are decoupled into a diagonal and an offdiagonal part :

= < Ps > ^{4 my ; a: 4} + ^Baayk(~).

Neglect of the offdiagonal parts AaSyk(~) and BaSYk(w) yields the Hartree-Fock approximation for the spin- wave energy-shift (Keffer [9]).

In our considerations we have retained only those terms in the Hamiltonian (2.4) which lead to the Green functions (2.61, if we take the commutator in eq. (2.5).

Evidently, we keep only terms like a + a + aa, fl' /3+ pfl, a+ alp+ or permutations in the interaction part of (2.4). We will use the equations of motion for the Green functions (2.6) to derive an approximate expression for AaSYk(m) and Bapyk(w). We decouple the second-order equations and keep only the diagonal parts of the third-order Green functions. To obtain ihe equations for Aapyk(w) and Bapyk(w) we need to consider only those terms, which in the end are propor- tional to 8a-,-y+k (Balcar [6]). Both equations can be solved easily and we insert the expressions for AaSyk(~) and BaSyk(~) into (2.6) and into the first- order equation of motion (2.5).

To simplify the notation for the resulting self- energy term Z1(k ,w) we define

U~lfi ⁼

Y p

W~/J =

(2.7) and obtain, finally, besides the Hartree-Fock energy- shift Z0(k), an additional contribution to the self- energy, Z1(k, a), from Aapyk(w) and BaByk(w) :

+ npYa(~) C(ua, Ukp) (wy, Wpk) + (wa, qVky) (wp,

)

(2.8) In (2.8) we use the abbreviations (< a,f a, >

n,) napy(w)

[na(l + ^np + ^n,)

np n,] (w + ^{wa- wp}

^coy)-'

(2.9) and

(XU, YBy) =

Ypy +

YBy-a (2.10) Z1(k, w) depends explicitly on the energy w and with formula (1.2) gives rise to the lowest-order contribu- tion to the energy-width r ( k , w) of antiferromagnetic magnons.

We expand the imaginary part of Z1(k, w) for small temperatures T and small spin-wave momenta in the isotropic case (i. e. for h,

0). Assuming

k, T -4 2 r ~ ~ k ~ / & -4 2 r J S

we obtain with w

o, @ is the nearest neighbour distance) :

1

2 r J n: k,T

r ^{(k, m,) =} - s

³²

(rJs) . ^(2.11)

This expression does not agree with the results given by Tani [3] and Solyom [4]. It is however in agreement with the results derived by Harris et al. [1] and Cottam and Stinchcombe [2].

Acknowledgement. - The author is grateful to Dr. A. B. Harris for making his results available prior to publication. Thanks are due to Drs. W. Marshall and S. W. Lovesey of AERE Harwell for many helpful discussions.

References

[I] HARRIS (A. B.), KUMAR (D.), HALPERIN (B. I.) and [5] ZUBAREV (D. N.), SOV. Phys. Uspekhi, 1960, 3, 320 HOHENBERG (P. C.), to be published. [6] BALCAR (E.), Acta Physica Austriaca, 1970, 31, 300.

[2] COTTAM (M. G.) and STINCHCOMBE (R. B.), Proc. Int. [7] MALEEV (S. V.), SOV. Phys. J. E. T. P . , 1958, 6 , 776.

Magnetism Conf., Grenoble, 1970. [8] KITTEL (C.), Quantum Theory of Solids, John Wiley, [3] TANI (K.), Progr. Theor. Phys., 1964, 31, 335. 1963, 59.

[4] SOLYOM (J.), SOV. Phys. J. E. T. P., 1969, 28, 1251. [9] KEFFER (F.), Handbuch der Physik, 18, Springer, 1966.