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Phenomenological consideration of spin-wave spectrum in noncollinear antiferromagnet Mn3NiN
Journal of Magnetism and Magnetic Materials 86 (1990) 301-306 North-Holland
301
P H E N O M E N O L O G I C A L C O N S I D E R A T I O N O F S P I N - W A V E S P E C T R U M IN N O N C O L L I N E A R A N T I F E R R O M A G N E T Mn3NiN E.V. G O M O N A J and V.A. LVOV Metal Physics Institute of the Ukrainian SSR, Academy of Sciences, 36 Vernadsky Boulevard, 252142 Kiev, USSR
Received 9 February 1989; in revised form 4 October 1989
A spin-wave spectrum in noncollinear cubic antiferromagnet Mn3NiN of perovskite structure type is investigated theoretically using the phenomenologicalLagrangian method. The activation of one branch of!a spin-wave spectrum is shown to vanish at the points of spin-rotational phase transitions examined experimentally in this crystal. The activation of two other branches does not vanish at the points of phase transitions. The splitting of these branches is proportional,to the magnitude of the weak ferromagnet moment detected experimentally in the Mn3NiN crystal.
1. Introduction The spatial symmetry of metallic perovskite Mn3NiN is described by a group O~. Manganese magnetic atoms are centred in cubic unit cell faces. The neutron diffraction data [1] point to the existence of a noncollinear antiferromagnetic structure with k = 0 in this crystal. At a temperature T < 180 K the atomic magnetic moments are oriented along crystallographic directions [110], [101] and [011]. Within the temperature range 180 K < T < 266 K the magnetic moments rotate in the (111) plane and at T = 266 K they appeared to be directed along [112], [121], [211] as shown in fig. 1. It has been revealed by precise magnetic measurements [1] that the low-temperature phase (phase 1) is pure antiferromagnefi¢ while the intermediate and high-temperature phases (phase 2 and 3, respectively) are weak ferromagnetic with a magnetic moment up to 2 × 10-3ttB/moL A theoretical analysis of the cubic perovskite magnetic structure is carried out in ref. [2] and symmetry classification of spin-wave spectrum branches is given in ref. [3]. It seems to be interesting to study the spin-wave spectrum in Mn3NiN in more detail. The first cause of our interest to this problem is the absence of the magnetoelastic gap in the soft mode of spin-wave spectrum (symmetry argumentation of this fact is presented in ref. [4]). It means that the activation of the soft magnon branch is zero at the points of above mentioned phase transitions, i.e. at T - - T1 = 180 K and at T = T2 -- 266 K. Therefore, in the vicinity of the transition points one can to carry out the antiferromagnetic resonance experiment~ at extremely low frequencies (the only theoretical restriction is that the resonance frequency to be greater
Fig. 1. Spin-oriented phase transitions in Mn3NiN. (a) phase 1, T<180 K; (b) phase 2, 180 K < T< 266 K; (c) phase 3, T= 266 K. 0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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302
than the damping coefficient). The second point of no little interest is that the splitting of two other magnon branches in the long-wave limit is caused exclusively by weak ferromagnetism. This fact allows, in principle, to obtain additional experimental information concerning the magnitude and the nature of ferromagnetic moment in Mn3NiN. The present paper intends to construct a phenomenological spin-wave Lagrangian (see refs. [5,6]) of cubic perovskite and, based on it, to calculate the low-frequency magnon spectra branches referred to, in some works, as "hydrodynamic" branches. (The idea of a "hydrodynamical" approach to a spin-wave description elaborated first in refs. [7,8] is related idealogically with the phenomenological Lagrangian method.)
2. Phenomenological Lagrangian and spectrum Mn3NiN crystal magnetic ordering is characterized by two antiferromagnet vectors 11 and 12 which form the basis of the two-dimensional irreducible representation of the permutation group of manganese atoms and a ferromagnet vector m which is the basic vector of a unitary representation of this group (for more details see ref. [9]). A phenomenological expression for perovskite exchange energy can be developed as a series in powers of ferro- and antiferromagnet vector components. The expansion invariant with respect to the exchange group is: w(e') = ½Jt( I 2 + 122) + ½Jmmz + ~ l m
4 + ½~zm2( i 2 + ! 2) + ~ 3 ( t 2 + 122)2 + ~4[/,, t2] 2
+½~5 { (/,m) 2 + (12m) 2 } + ¼-@6{( t , m ) ( 1 2 - 1 2 ) -- 2 ( t 2 m ) ( t , 1 2 ) } .
(1)
This can be substantially simplified via the common convention =
=
= s= = coast
<2)
Basic vectors of irreducible representations of a permutation group are related to atomic spins ~1, ~2, ~3 in such a way: m =
m0 ~--~-(s I + s 2 + s 3 ) ,
m0 I1 = ~"-~-(2s 3 - s 2 - s l ) ,
m0 12 = ~-~-(s 1 - s = ) ,
(3)
where sl,2.3 = (SLz.3), m0 is the average magnetic moment value per manganese atom. Using the formula (2) and (3) it is easy to obtain the normalization constraints for basic vectors
m2 +i? +12--- 3m2, (t,, t2)=C~(t2m),
12-12= 2VC2(llm).
(4)
Then, by means of (4), one can write expression (1) in a simple form: W ''x, = 1jmZ + ~-~'m" - ½ . ~ { ( l l m ) 2 + (/zm)2},
(5)
where J=J,,-J,+3m~(~2-~3-~,),
~=4~,-~+¢~,
~'=~1-2~2+~3+~4.
The spatial orientation of ferro- and antiferromagnet vectors is fixed by relativistic interactions. The expression for the energy of such interactions is w(r)
~
~1 a l p
2
+ ~1 b l p 4 + lb2(/n/)2
+
½b3p2q 2 -I,-d(q/~ll).
(6)
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303
Expression (6) is distinct from the similar one used in ref. [2] by the presence of the term d(qm) responsible for weak ferromagnetism. In (6) the following symbols are employed:
px=-½(12x+lay+v~12z),
qx=½(12y-ilx-V~llz),
Py=½(t2y-l,x+V/-21,~), qy=½(12x+ll,-~/-212~), 1
(7)
1
pz='-~(lly--12x),
qz=---~(llx+12y),
axis z of rectangular coordinates is directed along the third order crystallographic axis and axis y is directed along the second order axis. When calculating spin wave spectra it is also necessary to take into account the inhomogeneous exchange energy:
+2v~--~
0y
Ox
+2¢2--~--[-~+-~1j.
(8)
Following ref. [6], let us go over from antiferromagnet vector components to variables ¢p~= n~tg(O/2) and ~p~= dcpJdt, that describe the rotation of atomic magnetic moments through an angle O(xi) about some direction n (n 2 = 1). For this purpose we use the well known formula [6] which relates antiferromagnet vectors with variables %: !1.2( ~ )
it0) +
2
-- •
S1.2
1,2 .[ } ,
(9)
where/~o), /2(0) are antiferromagnet vectors in equilibrium. As it was previously shown [2] the equilibrium values P0, q0 of variables p , q defined by (7), are the order parameters of phase transitions from phase 1 to phase 2 and then to phase 3. It is easy to see that _qo/~/-~, t¢o)_ ~ for all phases. To find equilibrium relations lt0)_tto) z ~'2z ~ 0, l(, J x°) ~ t(o)= S2y " l y - - _ltO)_po/V "2x -between order parameters and phenomenological coefficients of expansion ( 6 ) w e substitute into (6) the expressions (7), (9) and m = - d q / J . Then the requirement for the absence of terms linear with respect to ~ in energy expansion results in q2 = 0,
p2 __ 3m 2
in phase 1,
q2 = a / b ,
p2 = 3m 2 _ a / b
in phase 2,
q02 = 3m 2,
p02 = 0
in phase 3.
(10)
Here a -- a I + 3m2(b1 - b2 - b 3 + d 2 / j ) , b = b 1 - 2b 2 - 2b 3. Parameter a is nggative in phase 1 and is positive in phases 2 and 3. To the points of phase transitions from phase 1 to phase 2 and from phase 2 to phase 3 there correspond the values a -- 0 and a -- 3rn2b, respectively. If the energy expression is specified, one can construct a phenomenologl(~l Lagrangian, using the procedure explained in ref. [6]. In the case of small amplitude spin oscillation pBrameters ~i are small, so to find the spectra it suffices to leave in the Lagrangian only the term quadratic in ¢pi. The Lagrangian
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E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn sNiN
corresponding to the energy (5), (6), (8) is the following 1 2 2 2 L = (2/g2)[(~p~/J) + (~+qb_/J')] - i(2/g)Fmo(~p+ep_- ~p_~p+) + ~Amo~+ep_+ 2Bmo~P~
- eqmg[(V+~p+)(V_~p_) + (V_cP+)(V+cp_) + (V~P+I(vtP-) + 4(V+cp~l(v_q%) + 2(Vzq%)z] - a z m ~ { 7z+ [(V+q0_) 2 + 4(V_q~_)(V~cp_)] + 7~[(V_q0+) 2 + 4(V+cP+)(VtP+)I },
(11)
where g is the gyromagnetic ratio, V± = ~/2%/~(x -T-iy), V~ = O/i~z, ~ ±= eO~+ i~y, J ' = J - 3..~mg/2. Lagrangian parameters A, B, F and 7 ± are connected with phenomenological constants a, b, d. Such a connection can be written for each of phases 1-3 as follows: Phase 1
A = a + 3 m ~ b z + 3 ~ ( m o d ) E / 2 J J ',
B=a,
F=O,
7+ = 7_ = 1 .
(12)
Phase 2
A = 3mgb 2 + (d2~/2JJ')(3m2o + 8a/b), F= d(a/3m~b//2(J'
+ 3J')/aJJ',
B= -2a(1 -a/3m~b), (13)
7 ± = (1 - a/3m~b) 1/2 + i(a/3m~b) 1/2.
Phase
A = 3m2o(b + b2) - a + 3 ~ ( 3 m o d ) 2 / 2 j J ", B=3m2b-a, F=d(J'+3J")/4JJ', 7+=+i.
(14)
(In expressions (13), (14) we denote J " = J + 3~m~/2.) It can be easily seen that A < 0 in all three phases, coefficient B ~<0 vanishes in the phase-transition points and coefficient F - d / J is nonzero because of the presence of the Dzyaloshinsky-like term in the energy (6) which, as it was pointed out in ref. [1], is of exchange-relativistic nature. Variational equations for Lagrangian (11) are the following
( 1 / g 2 j ) i~z - Bm~cpz - alm~( V~ + 2V+ V_ )q% = 0, (2/g 2J , )¢p_- 12AmoeP__ 2 alm~( V~ + 2V+ V_)qo_ . . l(4/g)Fmoq~ . . - 2ot2m~7~ ( V 2_+ 4VzV+) ~+ = O,
(15)
(2/g2j,)ik+ + i ( 4 / g ) F m o ~ + _ !2AmocP + z _ alm~( V: + 2 V + V _ ) ~ + - 2a2mo272+(V2+ + V,V_)cp_ =0. The compatibility requirement of eqs. (15) with ~ , cp± - exp{i(kr - tot)} has given rise to three branches ws(k), w+(k) and w_(k) of spin-wave spectra. In a long wave limit (a2 k2 ,~: j 2 F 2 ) these are (oa,/O~o)z = - J ( B - o t , k Z ) ,
2 x] 1 / 2
(o~±/6Oo) = + J ' V + [ ( J ' F ) z - ( J ' / 4 ) ( A - 2 a , k )]
,
(16)
where ~o =gmo. From (16) and (13) the frequency o:,(0)- B 1/2- al/Z(3m~b- a) 1/z vanishes at the phase transition points, i.e. at T-- T1 -- 180 K (a -- 0) and at T = T2 = 266 K(a = 3m~b) and attains its maximal value hy 2 °asm = ~,.,2~.,2 2""o'"o"" when a = 3~mob. The frequencies to ±(0) have finite values at the phase transition points. Their splitting A~ = oa+(0)- w_(0) is caused only. by weak ferromagnetism, i.e. AoJ---0 in phase 1 and (Aa~/o0) - dqo/m o in phases 2 and 3. At a temperature appreciably more than 180 K Aoa ~ o~+(0) ~ ~_(0) if d 2 ~ 3m2ob2J ' and A~0 :~ a~±(0), in the inverse limiting case.
E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn 3NiN
305
If k is nonzero the inhomogeneous exchange interaction also contributes to the frequency splitting. To find the structure of this contribution we take into account in (15) only exchange terms, i.e. we assume A -- B -- F - - 0. Then the requirement of compatibility of two last equations in (15) gives
2602
(
8
~+- - =a2J" k2+-~k~k_) 60~
,1/2[
8
[k2+-~k~k+
)1/2
(17)
It results from (17) that the exchange contribution to frequency splitting is zero if the wave propogates along the third order crystalline axis, i.e. when k+= k _ = O. If, on the contrary, the wave propogates perpendicular to the third order axis, then 2 602= ot2Jt(g,00k)2. 60+-
(18)
3. Another way of calculations
To substantiate the chosen form of Lagrangian (11) it is useful to calculate the spectra (16) in a different way, based, in contrast to the previous one, on the relation (3) of ferro- and antiferromagnetic vectors with atomic spins. It is convenient to operate with the linear combinations of spin components which form irreducible representations of the unitary subgroup of crystal magnetic group [10]. In this case these linear combination are expressed through the values (7) in a very simple way: r ~ ) = p ±+ iq ± , r t2)--±p ±q: iq±,
r~ 1) = pz, rz(2)-- qz,
m ±,
(19)
= m z.
(Here p += Px + ipy, q ±= qx + iqy, m ±= m x + imy.) The unitary subgroup for phases 1-3 is the group S~. The values Fzt~) form unitary representation and F t=) ± form two-dimensional physically irreducible representation E of this group. When calculating spin-wave spectra it is advantageous to follow the method suggested in ref. [10], namely, to substitute operators for average spin values in r t') and to use quantum equations i h r t,) = pc,),~_ ~ , t ~ )
(20)
with "Hamiltonian" ,,~ which is the sum of (5), (6) and (8) with average spin values replaced by operators. For the sake of simplicity we assume the atom magnetic moments to be of pure spin nature (i.e. = (2Pa/C~-)(sl + s2 + s3), Pa = Bohr magneton), then hating linearized eq. (20) in the random phases assumption [10], one can obtain formula (16) for spin-wave spectra. The Lagrangian (11) results in correct expressions for spin-wave spectra. There can be given a sufficiently simple description of the nonlinear spin dynamics in Mn3NiN crystal by including in the Lagrangian the additional terms with q0~, n > 2. As a result there may appear interesting details [11] caused by the terms linear in qbi. But this problem requires separate consideration.
Acknowledgements The authors are greatly indebted to academician V.G. Bar'yakhtar for the statement of the problem and I.M. Vitebsldi and B.A. Ivanov for fruitful discussions.
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E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn 3NiN
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
D. Fruchart, E.F. Bertaut, R. Madar, G. Lorthioir and R. Fruchart, Solid State Commun. 9 (1971) 1793. Yu.A. Izyumov, V.E. Naish, Yu.N. Skrjabin and V.N. Syromjatnikov, Fiz. Tverd. Tela 23 (1981) 1101. Y.A. Izyumov and S.V. Petrov, Fiz. Mat. Met. 55 (1983) 24. V.G. Bar'yakhtar, I.M. Vitebskii, Yu.G. Pashkevich, V.L. Sobolev and V.V. Tarasenko, preprint ITF-84-39R. D.V. Volkov, A.A. Zheltukhin and Yu.P. Blioch, Fiz. Tverd. Tela 13 (1971) 1668. A.F. Andreev and V.I. Marchenko, Uspekhi Fiz. Nauk 130 (1980) 39. B.I. Halperin and P.C. Hohenberg, Phys. Rev. 188 (1969) 898. B.I. Halperin and W.M. Saslov, Phys. Rev. B 16 (1977) 2155. E.V. Gomonaj and V.A. Lvov, preprint ITF-85-111R. V.G. Bar'yakhtar, I.M. Vitebskii and D.A. Yablonsldi, Zh. Eksp. Teor. Fiz. 76 (1979) 1381. E.V. Gomonaj, B.A. Ivanov, V.A. Lvov and G.K. Oksjuk, Zh. Eksp. Teor. Fiz. 97 (1990) 307.
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P H E N O M E N O L O G I C A L C O N S I D E R A T I O N O F S P I N - W A V E S P E C T R U M IN N O N C O L L I N E A R A N T I F E R R O M A G N E T Mn3NiN E.V. G O M O N A J and V.A. LVOV Metal Physics Institute of the Ukrainian SSR, Academy of Sciences, 36 Vernadsky Boulevard, 252142 Kiev, USSR
Received 9 February 1989; in revised form 4 October 1989
A spin-wave spectrum in noncollinear cubic antiferromagnet Mn3NiN of perovskite structure type is investigated theoretically using the phenomenologicalLagrangian method. The activation of one branch of!a spin-wave spectrum is shown to vanish at the points of spin-rotational phase transitions examined experimentally in this crystal. The activation of two other branches does not vanish at the points of phase transitions. The splitting of these branches is proportional,to the magnitude of the weak ferromagnet moment detected experimentally in the Mn3NiN crystal.
1. Introduction The spatial symmetry of metallic perovskite Mn3NiN is described by a group O~. Manganese magnetic atoms are centred in cubic unit cell faces. The neutron diffraction data [1] point to the existence of a noncollinear antiferromagnetic structure with k = 0 in this crystal. At a temperature T < 180 K the atomic magnetic moments are oriented along crystallographic directions [110], [101] and [011]. Within the temperature range 180 K < T < 266 K the magnetic moments rotate in the (111) plane and at T = 266 K they appeared to be directed along [112], [121], [211] as shown in fig. 1. It has been revealed by precise magnetic measurements [1] that the low-temperature phase (phase 1) is pure antiferromagnefi¢ while the intermediate and high-temperature phases (phase 2 and 3, respectively) are weak ferromagnetic with a magnetic moment up to 2 × 10-3ttB/moL A theoretical analysis of the cubic perovskite magnetic structure is carried out in ref. [2] and symmetry classification of spin-wave spectrum branches is given in ref. [3]. It seems to be interesting to study the spin-wave spectrum in Mn3NiN in more detail. The first cause of our interest to this problem is the absence of the magnetoelastic gap in the soft mode of spin-wave spectrum (symmetry argumentation of this fact is presented in ref. [4]). It means that the activation of the soft magnon branch is zero at the points of above mentioned phase transitions, i.e. at T - - T1 = 180 K and at T = T2 -- 266 K. Therefore, in the vicinity of the transition points one can to carry out the antiferromagnetic resonance experiment~ at extremely low frequencies (the only theoretical restriction is that the resonance frequency to be greater
Fig. 1. Spin-oriented phase transitions in Mn3NiN. (a) phase 1, T<180 K; (b) phase 2, 180 K < T< 266 K; (c) phase 3, T= 266 K. 0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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302
than the damping coefficient). The second point of no little interest is that the splitting of two other magnon branches in the long-wave limit is caused exclusively by weak ferromagnetism. This fact allows, in principle, to obtain additional experimental information concerning the magnitude and the nature of ferromagnetic moment in Mn3NiN. The present paper intends to construct a phenomenological spin-wave Lagrangian (see refs. [5,6]) of cubic perovskite and, based on it, to calculate the low-frequency magnon spectra branches referred to, in some works, as "hydrodynamic" branches. (The idea of a "hydrodynamical" approach to a spin-wave description elaborated first in refs. [7,8] is related idealogically with the phenomenological Lagrangian method.)
2. Phenomenological Lagrangian and spectrum Mn3NiN crystal magnetic ordering is characterized by two antiferromagnet vectors 11 and 12 which form the basis of the two-dimensional irreducible representation of the permutation group of manganese atoms and a ferromagnet vector m which is the basic vector of a unitary representation of this group (for more details see ref. [9]). A phenomenological expression for perovskite exchange energy can be developed as a series in powers of ferro- and antiferromagnet vector components. The expansion invariant with respect to the exchange group is: w(e') = ½Jt( I 2 + 122) + ½Jmmz + ~ l m
4 + ½~zm2( i 2 + ! 2) + ~ 3 ( t 2 + 122)2 + ~4[/,, t2] 2
+½~5 { (/,m) 2 + (12m) 2 } + ¼-@6{( t , m ) ( 1 2 - 1 2 ) -- 2 ( t 2 m ) ( t , 1 2 ) } .
(1)
This can be substantially simplified via the common convention =
=
= s= = coast
<2)
Basic vectors of irreducible representations of a permutation group are related to atomic spins ~1, ~2, ~3 in such a way: m =
m0 ~--~-(s I + s 2 + s 3 ) ,
m0 I1 = ~"-~-(2s 3 - s 2 - s l ) ,
m0 12 = ~-~-(s 1 - s = ) ,
(3)
where sl,2.3 = (SLz.3), m0 is the average magnetic moment value per manganese atom. Using the formula (2) and (3) it is easy to obtain the normalization constraints for basic vectors
m2 +i? +12--- 3m2, (t,, t2)=C~(t2m),
12-12= 2VC2(llm).
(4)
Then, by means of (4), one can write expression (1) in a simple form: W ''x, = 1jmZ + ~-~'m" - ½ . ~ { ( l l m ) 2 + (/zm)2},
(5)
where J=J,,-J,+3m~(~2-~3-~,),
~=4~,-~+¢~,
~'=~1-2~2+~3+~4.
The spatial orientation of ferro- and antiferromagnet vectors is fixed by relativistic interactions. The expression for the energy of such interactions is w(r)
~
~1 a l p
2
+ ~1 b l p 4 + lb2(/n/)2
+
½b3p2q 2 -I,-d(q/~ll).
(6)
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303
Expression (6) is distinct from the similar one used in ref. [2] by the presence of the term d(qm) responsible for weak ferromagnetism. In (6) the following symbols are employed:
px=-½(12x+lay+v~12z),
qx=½(12y-ilx-V~llz),
Py=½(t2y-l,x+V/-21,~), qy=½(12x+ll,-~/-212~), 1
(7)
1
pz='-~(lly--12x),
qz=---~(llx+12y),
axis z of rectangular coordinates is directed along the third order crystallographic axis and axis y is directed along the second order axis. When calculating spin wave spectra it is also necessary to take into account the inhomogeneous exchange energy:
+2v~--~
0y
Ox
+2¢2--~--[-~+-~1j.
(8)
Following ref. [6], let us go over from antiferromagnet vector components to variables ¢p~= n~tg(O/2) and ~p~= dcpJdt, that describe the rotation of atomic magnetic moments through an angle O(xi) about some direction n (n 2 = 1). For this purpose we use the well known formula [6] which relates antiferromagnet vectors with variables %: !1.2( ~ )
it0) +
2
-- •
S1.2
1,2 .[ } ,
(9)
where/~o), /2(0) are antiferromagnet vectors in equilibrium. As it was previously shown [2] the equilibrium values P0, q0 of variables p , q defined by (7), are the order parameters of phase transitions from phase 1 to phase 2 and then to phase 3. It is easy to see that _qo/~/-~, t¢o)_ ~ for all phases. To find equilibrium relations lt0)_tto) z ~'2z ~ 0, l(, J x°) ~ t(o)= S2y " l y - - _ltO)_po/V "2x -between order parameters and phenomenological coefficients of expansion ( 6 ) w e substitute into (6) the expressions (7), (9) and m = - d q / J . Then the requirement for the absence of terms linear with respect to ~ in energy expansion results in q2 = 0,
p2 __ 3m 2
in phase 1,
q2 = a / b ,
p2 = 3m 2 _ a / b
in phase 2,
q02 = 3m 2,
p02 = 0
in phase 3.
(10)
Here a -- a I + 3m2(b1 - b2 - b 3 + d 2 / j ) , b = b 1 - 2b 2 - 2b 3. Parameter a is nggative in phase 1 and is positive in phases 2 and 3. To the points of phase transitions from phase 1 to phase 2 and from phase 2 to phase 3 there correspond the values a -- 0 and a -- 3rn2b, respectively. If the energy expression is specified, one can construct a phenomenologl(~l Lagrangian, using the procedure explained in ref. [6]. In the case of small amplitude spin oscillation pBrameters ~i are small, so to find the spectra it suffices to leave in the Lagrangian only the term quadratic in ¢pi. The Lagrangian
304
E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn sNiN
corresponding to the energy (5), (6), (8) is the following 1 2 2 2 L = (2/g2)[(~p~/J) + (~+qb_/J')] - i(2/g)Fmo(~p+ep_- ~p_~p+) + ~Amo~+ep_+ 2Bmo~P~
- eqmg[(V+~p+)(V_~p_) + (V_cP+)(V+cp_) + (V~P+I(vtP-) + 4(V+cp~l(v_q%) + 2(Vzq%)z] - a z m ~ { 7z+ [(V+q0_) 2 + 4(V_q~_)(V~cp_)] + 7~[(V_q0+) 2 + 4(V+cP+)(VtP+)I },
(11)
where g is the gyromagnetic ratio, V± = ~/2%/~(x -T-iy), V~ = O/i~z, ~ ±= eO~+ i~y, J ' = J - 3..~mg/2. Lagrangian parameters A, B, F and 7 ± are connected with phenomenological constants a, b, d. Such a connection can be written for each of phases 1-3 as follows: Phase 1
A = a + 3 m ~ b z + 3 ~ ( m o d ) E / 2 J J ',
B=a,
F=O,
7+ = 7_ = 1 .
(12)
Phase 2
A = 3mgb 2 + (d2~/2JJ')(3m2o + 8a/b), F= d(a/3m~b//2(J'
+ 3J')/aJJ',
B= -2a(1 -a/3m~b), (13)
7 ± = (1 - a/3m~b) 1/2 + i(a/3m~b) 1/2.
Phase
A = 3m2o(b + b2) - a + 3 ~ ( 3 m o d ) 2 / 2 j J ", B=3m2b-a, F=d(J'+3J")/4JJ', 7+=+i.
(14)
(In expressions (13), (14) we denote J " = J + 3~m~/2.) It can be easily seen that A < 0 in all three phases, coefficient B ~<0 vanishes in the phase-transition points and coefficient F - d / J is nonzero because of the presence of the Dzyaloshinsky-like term in the energy (6) which, as it was pointed out in ref. [1], is of exchange-relativistic nature. Variational equations for Lagrangian (11) are the following
( 1 / g 2 j ) i~z - Bm~cpz - alm~( V~ + 2V+ V_ )q% = 0, (2/g 2J , )¢p_- 12AmoeP__ 2 alm~( V~ + 2V+ V_)qo_ . . l(4/g)Fmoq~ . . - 2ot2m~7~ ( V 2_+ 4VzV+) ~+ = O,
(15)
(2/g2j,)ik+ + i ( 4 / g ) F m o ~ + _ !2AmocP + z _ alm~( V: + 2 V + V _ ) ~ + - 2a2mo272+(V2+ + V,V_)cp_ =0. The compatibility requirement of eqs. (15) with ~ , cp± - exp{i(kr - tot)} has given rise to three branches ws(k), w+(k) and w_(k) of spin-wave spectra. In a long wave limit (a2 k2 ,~: j 2 F 2 ) these are (oa,/O~o)z = - J ( B - o t , k Z ) ,
2 x] 1 / 2
(o~±/6Oo) = + J ' V + [ ( J ' F ) z - ( J ' / 4 ) ( A - 2 a , k )]
,
(16)
where ~o =gmo. From (16) and (13) the frequency o:,(0)- B 1/2- al/Z(3m~b- a) 1/z vanishes at the phase transition points, i.e. at T-- T1 -- 180 K (a -- 0) and at T = T2 = 266 K(a = 3m~b) and attains its maximal value hy 2 °asm = ~,.,2~.,2 2""o'"o"" when a = 3~mob. The frequencies to ±(0) have finite values at the phase transition points. Their splitting A~ = oa+(0)- w_(0) is caused only. by weak ferromagnetism, i.e. AoJ---0 in phase 1 and (Aa~/o0) - dqo/m o in phases 2 and 3. At a temperature appreciably more than 180 K Aoa ~ o~+(0) ~ ~_(0) if d 2 ~ 3m2ob2J ' and A~0 :~ a~±(0), in the inverse limiting case.
E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn 3NiN
305
If k is nonzero the inhomogeneous exchange interaction also contributes to the frequency splitting. To find the structure of this contribution we take into account in (15) only exchange terms, i.e. we assume A -- B -- F - - 0. Then the requirement of compatibility of two last equations in (15) gives
2602
(
8
~+- - =a2J" k2+-~k~k_) 60~
,1/2[
8
[k2+-~k~k+
)1/2
(17)
It results from (17) that the exchange contribution to frequency splitting is zero if the wave propogates along the third order crystalline axis, i.e. when k+= k _ = O. If, on the contrary, the wave propogates perpendicular to the third order axis, then 2 602= ot2Jt(g,00k)2. 60+-
(18)
3. Another way of calculations
To substantiate the chosen form of Lagrangian (11) it is useful to calculate the spectra (16) in a different way, based, in contrast to the previous one, on the relation (3) of ferro- and antiferromagnetic vectors with atomic spins. It is convenient to operate with the linear combinations of spin components which form irreducible representations of the unitary subgroup of crystal magnetic group [10]. In this case these linear combination are expressed through the values (7) in a very simple way: r ~ ) = p ±+ iq ± , r t2)--±p ±q: iq±,
r~ 1) = pz, rz(2)-- qz,
m ±,
(19)
= m z.
(Here p += Px + ipy, q ±= qx + iqy, m ±= m x + imy.) The unitary subgroup for phases 1-3 is the group S~. The values Fzt~) form unitary representation and F t=) ± form two-dimensional physically irreducible representation E of this group. When calculating spin-wave spectra it is advantageous to follow the method suggested in ref. [10], namely, to substitute operators for average spin values in r t') and to use quantum equations i h r t,) = pc,),~_ ~ , t ~ )
(20)
with "Hamiltonian" ,,~ which is the sum of (5), (6) and (8) with average spin values replaced by operators. For the sake of simplicity we assume the atom magnetic moments to be of pure spin nature (i.e. = (2Pa/C~-)(sl + s2 + s3), Pa = Bohr magneton), then hating linearized eq. (20) in the random phases assumption [10], one can obtain formula (16) for spin-wave spectra. The Lagrangian (11) results in correct expressions for spin-wave spectra. There can be given a sufficiently simple description of the nonlinear spin dynamics in Mn3NiN crystal by including in the Lagrangian the additional terms with q0~, n > 2. As a result there may appear interesting details [11] caused by the terms linear in qbi. But this problem requires separate consideration.
Acknowledgements The authors are greatly indebted to academician V.G. Bar'yakhtar for the statement of the problem and I.M. Vitebsldi and B.A. Ivanov for fruitful discussions.
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E.V. Gomonaj, V.A. Lvov / Spin-wave spectrum in antiferromagnet Mn 3NiN
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