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Magneto-electric resonance in antiferromagnets
Solid State Communications,
771—773
Vol. 9, pp.
1 1971.
Pergamon Press.
Printed in Great Britain
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS R.S. Tripathi, R.P.S. Kushwaha and K.G. Srivastava Department of Physics, U.P. Agricultural University, Pantnagar (Nainital) India
(Received 30 October 1970; in revised form 12 December 1970 by hR. Verma)
An expression for antiferromagnetic resonance frequencies under the influence of simultaneous electric and magnetic fields is derived by using the Green function method in random-phase approximation. The fission of antiferromagnetic resonance is calculated in correlation with the applied fields. It is suggested that this might be utilized in estimating the spin-orbit coupling coefficients.
5 single change in ‘g’ interaction factor alongterm withinRado’s ion spin-orbit the Hamiltonian and calculated the susceptibilities in molecular field approximation again. %*iith some minor changes. Rado’s results were confirmed. 8 included interRecently Yatom and Englrnan action terms, arising from all possible mechanisms in the Hamiltonian and calculated the thermodynamic properties of the magnetoelectric chromium oxide by using the two-time Green’s function method in random phase and Callen
IN THEinrecent years, of considerable interest has grown the problem magnetoelectric effect. The effect was first predicted by Landau and Lifts hitz.’ Dzyaloshinski 2 demonstrated theoretically, on symmetry grounds, its possibility in Cr 203. Astrov ~ and Rado independently susmeasured the magnetic and~ magnetoelectric ceptibilities (XM, XME) by applying the magnetic and electric fields along and perpendicular to the spin orientation in the single crystal of chromium-oxide, and confirmed the effect in Cr 203. At present we know several other antiferromagnets like LiMnPO4 etc. which are known to show the magnetoelectric effect,
decoupling approximations, and compared the resuits with those obtained by the molecular field approximation.
The first microscopic explanation5 for was magnetoelectric effect given on by a Rado based on spin-orbit coupling single ion. He included this interaction term in the Hamiltonian, and calculated the magnetic and magnetoelectric susceptibilities in the molecular field approximation. A good agreement with the experimental observations was obtained. Date et al.6 attributed the effect to the exchange interaction between two spins in the same sublattice via an applied electric field. Later Hornreich and Shtrikman7 included the term showing the
In the present paper, we have considered another aspect of the magnetoelectric effect, It is well known that the spin wave resonance frequency splits into two branches when static magnetic field is applied along the spin orientation of an antiferromagnet. The splitting of the two modes depends on the applied magnetic field, while the resonance frequency depends on the exchange and anisotropic fields. Our interest here is to see the modification in the characteristic resonance in the magnetoelectric antiferromagnets. Through the modification, we 771
772
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS
expect to get some insight into the various coupling coefficient parameters. In the present calculation, we assume that the electric and magnetic fields are both applied along the spin direction (the so called parallel case). The antiferromagnetic resonance (AMFR) frequencies are calculated in the temperature dependent magnon formalism. The method has been used sucessfully by Nagai and Tanaka ~ for the calculation of AFMR frequencies in FeF 2 and MnF2 which agree very well with the experimental data close to the Née! temperature. We write down the Hamiltonian for the twosublattice antiferromagnet, magnetic and electric field being parallel to the direction of the spin orientation.
2
—
H 1 ~ ~ S~‘S3 ‘a (S) /.LH 2~ Sj2+~_a~,HzEz~e(j)S 2 ÷~a~E2 ~2 (j)(S) +~ ~E 2~~[(i)+ E(j)] a~(ij)S~’S~ (1) =
—
~
Notations are the same as given in reference (8). In order to study the spin wave frequencies, we now use Hoistein—Primakoff transformation and write Hamiltonian in magnon-creation and annthilation operator formalism. In the transformed Hamiltoniari, we retain terms quadratic and quartic in magnon operators. As such the diagonalisation of the transformed Hamiltonian is difficult. To facilitate the evaluation of the magnon energy, following Nagai and Tanaka we use random phase approximation and decouple the quartic terms into terms quadratic in magnon operators. As an example,
+
<
Cjc~Ck2 Ck3 dk4
<
C~tC~3 > ~ -~-
>
=
+
c~,Ck2 C~,ç2 < <
> Ck3dk4
Ck3dk4>
C,~C~3< C~2dk4 >
we have neglected non-diagonal terms like < G~dk4 > and < Ck2CkS >. The notation K> stands for the statistical average over grand canonical ensemble. Finally we obtain the following Hamiltonian.
H
const.
=
+ ~
Vol.9, No.11
t}l/~C~Ck + Akd,dk
±
Ic
14 (Ckck
+
C,,~d~)]
(2)
where C~CIc and d~,dIc are magnon creation and annihilation for the two sublattices, A~ andoperators ‘4 are given below
A~ A~C
=
Ak
=
‘4
S/.ik,
—
=
—
u
J~S± j~tH~ S(J8 ±~a”E2)[1 —
+
IaS(1
~ 2~LSa~~E1
~s)
—
—
-
w’)
+
y~(k)1
aHzEa
~_.
—
8E =
2(1,~~ 2~a~
2) U
(3)
~f8and ±1iE2a~’)(v u)[1 y8(k)J + Jd~U + w’) W’ are the statistical averages defined
+ v, U
—
by =
2
NS
v
w’
=
Ic
~y2(k)
~
<
CICI,,
<
C~C~>
~ y,~(k)(
=
>~
> +
Other symbols carry the same meaning as given in reference 8. Here we have assumed the averages < ~ > and < d~dk> to be field independent, and equal. A rigorous calculation shows that they are field dependent and not equal, but in the first approximation without any loss of generality they can be treated to be equal and field independent. The Hamiltonian (2) can now be diagonalised easily using the standard methods. %~/eget the following expression for the energy associated with the excitation spectra of the spin waves. W~ k
— =
2
A~~
I [(4~
A~)2
‘42]
(4)
As in the case with magnetic field, the spin wave energy consists of two branches. Antiferromagnetic resonance (AFMR) frequencies are obtamed by setting k = 0 in equation (5). In analogy with usual antiferromagnetic resonance, the second term of equation (5) gives the
Vol. 9, No. 11
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS
intrinsic magnetoelectric, and the first term causes splitting into the spin wave modes. The difference of the two resonance frequencies in terms of the sublattice magnetisation < ni > (per spin) is given below. 1 = j~H~ — 2~aYsE~ 1 —
+
2~iE2aY’s(l
—
K
m
>
S)
(5)
We observe here that \w is temperature aependent. The contribution to L\w 4’5 occurs andfrom the single term two ionion term duepropounded to Date et by al.6Rado, does not give any explicit contribution. Through equation (3) it
773
is very niuch obviot~sthat only a~’, appears inside the square root of equation (4). This demonstrates the importance of the single ion term due 4’5 We hope that the measurement of to Rado. A’~ and the resonance frequencies in the simu1taneous presence of electric and magnetic fields applied along the spin orientation of magnetoelectric antiferromagnet like Cr 203, LiMnPO4 etf should be able to give an estimate of the spin-orbit coupling coefficient a~, and the g-shift factor a~,. It was remarked by Alexander and Shtrikmaii° the magnetoelectric ation may be morethat accurate than those by evaluESR methods.
REFERENCES 1.
LANDAU L.D. and LIFSHITZ E.M., Electrodynamics of Continuous Media pp. 119, Addison—Wesley, Reading, Mass. (1960).
2.
DZYALOSHINSKI I.E., J.E.T.P. 10, 628 (1960).
3.
ASTROV P.N., J.E.T.P., 11, 708 (1960).
4.
RADO G.T., Phys. Rev. Leit., 6, 609 (1961).
5.
RADO G.T., Phys. Rev., 128, 2546 (1962).
6.
DATE M., KANAMORI
7.
HORNREICH R.H. and SHTRIKMAN S., Phys. Rev., 161, 506 (1967).
8.
YATOM H. and ENGALMAN R., Phys. Rev. 188, 793 (1969).
9.
NAGA! 0. and TANAKA T., Phys. Rev. 188, 821 (1969).
10.
J.
and TACHIKI M., J. Phys. Soc. Japan, 16, 2589 (1961).
ALEXANDER S. and SHTRIKMAN S., Solid State Commun., 4, 115 (1966).
Em Ausdruck für antiferromagnetische Resona’izfrequenzen unter dem Einfluss gleichzeitiger elektrische und magnetische Felder wird abgeleitet unter Benutzung der Greenschen-f unktions Methode in Random-Phasen Naherung. Die Spaltung antiferromagnetische Resonanz als Funktion der angewandte Felder wird gerechnet. Es wird vergeschlagen man moge diese zur Schatzubg des Spin-Orbit Kopplings-Koeffizienten benutzen.
771—773
Vol. 9, pp.
1 1971.
Pergamon Press.
Printed in Great Britain
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS R.S. Tripathi, R.P.S. Kushwaha and K.G. Srivastava Department of Physics, U.P. Agricultural University, Pantnagar (Nainital) India
(Received 30 October 1970; in revised form 12 December 1970 by hR. Verma)
An expression for antiferromagnetic resonance frequencies under the influence of simultaneous electric and magnetic fields is derived by using the Green function method in random-phase approximation. The fission of antiferromagnetic resonance is calculated in correlation with the applied fields. It is suggested that this might be utilized in estimating the spin-orbit coupling coefficients.
5 single change in ‘g’ interaction factor alongterm withinRado’s ion spin-orbit the Hamiltonian and calculated the susceptibilities in molecular field approximation again. %*iith some minor changes. Rado’s results were confirmed. 8 included interRecently Yatom and Englrnan action terms, arising from all possible mechanisms in the Hamiltonian and calculated the thermodynamic properties of the magnetoelectric chromium oxide by using the two-time Green’s function method in random phase and Callen
IN THEinrecent years, of considerable interest has grown the problem magnetoelectric effect. The effect was first predicted by Landau and Lifts hitz.’ Dzyaloshinski 2 demonstrated theoretically, on symmetry grounds, its possibility in Cr 203. Astrov ~ and Rado independently susmeasured the magnetic and~ magnetoelectric ceptibilities (XM, XME) by applying the magnetic and electric fields along and perpendicular to the spin orientation in the single crystal of chromium-oxide, and confirmed the effect in Cr 203. At present we know several other antiferromagnets like LiMnPO4 etc. which are known to show the magnetoelectric effect,
decoupling approximations, and compared the resuits with those obtained by the molecular field approximation.
The first microscopic explanation5 for was magnetoelectric effect given on by a Rado based on spin-orbit coupling single ion. He included this interaction term in the Hamiltonian, and calculated the magnetic and magnetoelectric susceptibilities in the molecular field approximation. A good agreement with the experimental observations was obtained. Date et al.6 attributed the effect to the exchange interaction between two spins in the same sublattice via an applied electric field. Later Hornreich and Shtrikman7 included the term showing the
In the present paper, we have considered another aspect of the magnetoelectric effect, It is well known that the spin wave resonance frequency splits into two branches when static magnetic field is applied along the spin orientation of an antiferromagnet. The splitting of the two modes depends on the applied magnetic field, while the resonance frequency depends on the exchange and anisotropic fields. Our interest here is to see the modification in the characteristic resonance in the magnetoelectric antiferromagnets. Through the modification, we 771
772
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS
expect to get some insight into the various coupling coefficient parameters. In the present calculation, we assume that the electric and magnetic fields are both applied along the spin direction (the so called parallel case). The antiferromagnetic resonance (AMFR) frequencies are calculated in the temperature dependent magnon formalism. The method has been used sucessfully by Nagai and Tanaka ~ for the calculation of AFMR frequencies in FeF 2 and MnF2 which agree very well with the experimental data close to the Née! temperature. We write down the Hamiltonian for the twosublattice antiferromagnet, magnetic and electric field being parallel to the direction of the spin orientation.
2
—
H 1 ~ ~ S~‘S3 ‘a (S) /.LH 2~ Sj2+~_a~,HzEz~e(j)S 2 ÷~a~E2 ~2 (j)(S) +~ ~E 2~~[(i)+ E(j)] a~(ij)S~’S~ (1) =
—
~
Notations are the same as given in reference (8). In order to study the spin wave frequencies, we now use Hoistein—Primakoff transformation and write Hamiltonian in magnon-creation and annthilation operator formalism. In the transformed Hamiltoniari, we retain terms quadratic and quartic in magnon operators. As such the diagonalisation of the transformed Hamiltonian is difficult. To facilitate the evaluation of the magnon energy, following Nagai and Tanaka we use random phase approximation and decouple the quartic terms into terms quadratic in magnon operators. As an example,
+
<
Cjc~Ck2 Ck3 dk4
<
C~tC~3 > ~ -~-
>
=
+
c~,Ck2 C~,ç2 < <
> Ck3dk4
Ck3dk4>
C,~C~3< C~2dk4 >
we have neglected non-diagonal terms like < G~dk4 > and < Ck2CkS >. The notation K> stands for the statistical average over grand canonical ensemble. Finally we obtain the following Hamiltonian.
H
const.
=
+ ~
Vol.9, No.11
t}l/~C~Ck + Akd,dk
±
Ic
14 (Ckck
+
C,,~d~)]
(2)
where C~CIc and d~,dIc are magnon creation and annihilation for the two sublattices, A~ andoperators ‘4 are given below
A~ A~C
=
Ak
=
‘4
S/.ik,
—
=
—
u
J~S± j~tH~ S(J8 ±~a”E2)[1 —
+
IaS(1
~ 2~LSa~~E1
~s)
—
—
-
w’)
+
y~(k)1
aHzEa
~_.
—
8E =
2(1,~~ 2~a~
2) U
(3)
~f8and ±1iE2a~’)(v u)[1 y8(k)J + Jd~U + w’) W’ are the statistical averages defined
+ v, U
—
by =
2
NS
v
w’
=
Ic
~y2(k)
~
<
CICI,,
<
C~C~>
~ y,~(k)(
=
>~
> +
Other symbols carry the same meaning as given in reference 8. Here we have assumed the averages < ~ > and < d~dk> to be field independent, and equal. A rigorous calculation shows that they are field dependent and not equal, but in the first approximation without any loss of generality they can be treated to be equal and field independent. The Hamiltonian (2) can now be diagonalised easily using the standard methods. %~/eget the following expression for the energy associated with the excitation spectra of the spin waves. W~ k
— =
2
A~~
I [(4~
A~)2
‘42]
(4)
As in the case with magnetic field, the spin wave energy consists of two branches. Antiferromagnetic resonance (AFMR) frequencies are obtamed by setting k = 0 in equation (5). In analogy with usual antiferromagnetic resonance, the second term of equation (5) gives the
Vol. 9, No. 11
MAGNETO-ELECTRIC RESONANCE IN ANTIFERROMAGNETS
intrinsic magnetoelectric, and the first term causes splitting into the spin wave modes. The difference of the two resonance frequencies in terms of the sublattice magnetisation < ni > (per spin) is given below. 1 = j~H~ — 2~aYsE~ 1 —
+
2~iE2aY’s(l
—
K
m
>
S)
(5)
We observe here that \w is temperature aependent. The contribution to L\w 4’5 occurs andfrom the single term two ionion term duepropounded to Date et by al.6Rado, does not give any explicit contribution. Through equation (3) it
773
is very niuch obviot~sthat only a~’, appears inside the square root of equation (4). This demonstrates the importance of the single ion term due 4’5 We hope that the measurement of to Rado. A’~ and the resonance frequencies in the simu1taneous presence of electric and magnetic fields applied along the spin orientation of magnetoelectric antiferromagnet like Cr 203, LiMnPO4 etf should be able to give an estimate of the spin-orbit coupling coefficient a~, and the g-shift factor a~,. It was remarked by Alexander and Shtrikmaii° the magnetoelectric ation may be morethat accurate than those by evaluESR methods.
REFERENCES 1.
LANDAU L.D. and LIFSHITZ E.M., Electrodynamics of Continuous Media pp. 119, Addison—Wesley, Reading, Mass. (1960).
2.
DZYALOSHINSKI I.E., J.E.T.P. 10, 628 (1960).
3.
ASTROV P.N., J.E.T.P., 11, 708 (1960).
4.
RADO G.T., Phys. Rev. Leit., 6, 609 (1961).
5.
RADO G.T., Phys. Rev., 128, 2546 (1962).
6.
DATE M., KANAMORI
7.
HORNREICH R.H. and SHTRIKMAN S., Phys. Rev., 161, 506 (1967).
8.
YATOM H. and ENGALMAN R., Phys. Rev. 188, 793 (1969).
9.
NAGA! 0. and TANAKA T., Phys. Rev. 188, 821 (1969).
10.
J.
and TACHIKI M., J. Phys. Soc. Japan, 16, 2589 (1961).
ALEXANDER S. and SHTRIKMAN S., Solid State Commun., 4, 115 (1966).
Em Ausdruck für antiferromagnetische Resona’izfrequenzen unter dem Einfluss gleichzeitiger elektrische und magnetische Felder wird abgeleitet unter Benutzung der Greenschen-f unktions Methode in Random-Phasen Naherung. Die Spaltung antiferromagnetische Resonanz als Funktion der angewandte Felder wird gerechnet. Es wird vergeschlagen man moge diese zur Schatzubg des Spin-Orbit Kopplings-Koeffizienten benutzen.