Journal of Magnetism and Magnetic Materials 329 (2013) 142–145

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Journal of Magnetism and Magnetic Materials

0304-88

http://d

n Corr

E-m

kuzmin

journal homepage: www.elsevier.com/locate/jmmm

Hybridization of electromagnetic, spin and acoustic waves in magnetichaving conical spiral ferromagnetic order

Igor V. Bychkov a, Dmitry A. Kuzmin a,n, Vladimir G. Shavrov b

a Chelyabinsk State University, 454001 Chelyabinsk, Br. Kashirinyh Street, 129, Russiab The Institute of Radioengineering and Electronics of RAS, 125009 Moscow, Mokhovaya Street, 11-7, Russia

a r t i c l e i n f o

Article history:

Received 30 June 2012

Received in revised form

6 October 2012Available online 23 October 2012

Keywords:

Electromagnetic wave

Spin wave

Acoustic wave

Conical spiral ferromagnetic order

Reflection coefficient

Acoustic Faraday effect

53/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.jmmm.2012.10.021

esponding author.

ail addresses: bychkov@csu.ru (I.V. Bychkov),

da89@gmail.com (D.A. Kuzmin).

a b s t r a c t

The spectrum of hybrid electromagnetic–spin–acoustic waves for magnetic having conical spiral

ferromagnetic structure defined by heterogeneous exchange and relativistic interactions has been

received. The possibility of resonant interaction of spin, electromagnetic and acoustic waves has been

shown. The electromagnetic waves reflectance from the half-infinity layer of magnetic having conical

spiral ferromagnetic order has been calculated for different values of external magnetic field (angle of

spiral). The acoustic Faraday effect has been considered.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Recently, helicoidal (spiral) magnetic materials have attractedresearchers’ attention for their unusual physical properties [1,2].The spiral magnetic structures contribute a number of features inthe spectrum and dynamics of spin excitation in magneticmaterials: band structure is observed, the nonreciprocity effectis manifested, i.e. difference between the velocity of wavetransmission along and against the spiral axis. Previously, thespin–wave spectrum was calculated without taking into accountof the effects of the electromagnetic retardation, and the electro-magnetic wave spectrum was calculated without taking into accountof the effects of the dynamic interaction of the electromagneticfield with the oscillations of the spins in the ferromagnetic spiralstructure [3,4]. Earlier had been investigated the hybrid electro-magnetic–spin, electromagnetic–spin–acoustic waves in the mag-netic having simple spiral structure [5,6], and the hybridelectromagnetic–spin waves in the magnetic having conical spiralferromagnetic structure, also termed ‘‘ferromagnetic spiral’’ [7].However the spectrum and dynamic properties of magnets in aphase ferromagnetic spiral are not studied enough. In the presentwork the spectrum of the hybrid electromagnetic–spin–acousticwaves in spiral magnetic structure of type ferromagnetic spiral isinvestigated. Also the reflection of electromagnetic waves from a

ll rights reserved.

surface of half-infinity magnetic material with a ferromagneticspiral depending on the angle of spiral determined by an externalmagnetic field and Faraday effect are considered. Researches ofspectrum of the coupled fluctuations in the modulated magneticstructures are spent in approach Lba, where L¼ 2p=q the spiralperiod, q the wave number of spiral, a the lattice constant.

2. The spectrum of hybrid spin, acoustic and electromagneticwaves

The ground state of a crystal is described by a vector ofmagnetization with components:

M0x ¼M0 sin y cos qz, M0y ¼M0 sin y sin qz, M0z ¼M0 cos y,

ð1Þ

where M0 is the magnetization of saturation, q the wave numberof a spiral, y an angle between a direction of magnetization and aspiral axis z. y is defined by the value of an external magneticfield. When y¼ p=2 the magnetic transforms from the phase ofthe ferromagnetic spiral into simple spiral, when y¼ 0 in theferromagnetic phase.

The free energy of the crystal phase of ferromagnetic spiral hasthe form:

F ¼a2

dM!

dxi

!2

þFinþb1

2M2

z þb2

2M4

z�HMzþbijlmMiMjulmþcijlmuijulm,

ð2Þ

I.V. Bychkov et al. / Journal of Magnetism and Magnetic Materials 329 (2013) 142–145 143

where M!

the magnetization of the crystal, uij ¼ ð@ui=@xjþ@uj=@

xiÞ=2 the tensor of deformations; u!

the displacement vector,a,b,b,c the constants of inhomogeneous exchange, anisotropy,magnetostriction and elastic constant.

The term Fin, which causes inhomogeneous magnetization inthe ground state for crystals with exchange spiral structure is

Fin ¼g2

d2M!

dx2i

!2

ð3Þ

and for magnetics with a relativistic helicaloidal structure

Fin ¼ a1M!

rot M!

, ð4Þ

where g and a1 are constants of inhomogeneous exchangeinteraction and inhomogeneous relativistic interaction. In (2) itis taken into account that the external magnetic field is directedalong the axis of symmetry.

From the minimum of free energy with (1) we obtain expres-sions for determining the angle y through an external magneticfield H.

H¼M0 cos y½ ~b1þhmeþð~b2þhme=M2

0ÞM20 cos y2

þaq2þ ~D�, ð5Þ

where ~b1 and ~b2-the constants of anisotropy renormed bymagnetostriction:

~b1 ¼ b1�c33c11�c2

13

Dðc11�c12Þðb11�b12Þ

2M20�

c13

Dðb33�b31Þðb11�b12ÞM

20

þc33

Dðb13�b12Þðb11�b12ÞM

20þ

b244M2

0

2c44,

~b2 ¼ b2�c33c11�c2

13

Dðc11�c12Þðb11�b12Þ

2þ

c11�c12

2Dðb33�b31Þ

2

þc33

Dðb33�b31Þðb11�b12Þ�

c11

Dðb13�b12Þðb11�b12Þ

�b2

44

2c44�

2c13

Dðb33�b31Þðb13�b12Þ,

D¼ c33ðc11þc12Þ�2c213:

For a spiral with the exchange interaction we have, g40,ao0, hme ¼ ðb11�b12Þ

2M20=ðc11�c12Þ, q¼ ð�a=2gÞ1=2, ~D ¼ gq4.

In the case of relativistic spiral, a1a0, a40, hme ¼ b2M20=2m,

q¼ a1=a, ~D ¼�2a1q.Note that the magnetoelastic coupling is not affected by the

value of the wave number of the spiral q.The tensor of the equilibrium deformations is

u0xx ¼M2

0 �c33

2Dðb11�b12Þsin2 y

1

D½c33ðb13�b12Þ�c13ðb33�b31Þ� cos2 y

� �,

u0yy ¼ u0

xx, u0zz ¼�

2c13

c33u0

xx�1

c33ðb33�b31ÞM

20 cos2 y,

u0xz ¼�

b44

4c44M2

0 sin 2y cos qz, u0yz ¼�

b44

4c44M2

0 sin 2y sin qz, u0xy ¼ 0:

ð6Þ

For solving a problem of getting the spectrum of hybrid wavesone have to take into account the system of Maxwell’s, Landau–Lifshitz and motion of an elastic medium equations:

@M!=@t¼ g½M

!H!eff

�, H!eff

¼�dF=dM!

,

r €ui ¼ @sik=@xk, sik ¼ @F=@uik,

rot E!¼�

1

c

@

@tðH!þ4pM!Þ, rot H

!¼ec

@ E!

@t,

divðe E!Þ¼ 0, divðH

!þ4pM!Þ¼ 0: ð7Þ

The linearized system of Eq. (7) for Fourier components is

7om7 ðkÞ ¼ cos y½o72k þ

12ome4 sin2 y�m7 ðkÞ

þ12ome4 sin2 y cos ym7 ðk82qÞ�o1k7q sin ymzðk8qÞ

þ igb44M20k½12 �

32 sin2 y�u7 ðkÞ� i

2gM20b44 sin2 yðk72qÞu7 ðk82qÞ

�igðb33�b31ÞM20 sin 2yðk7qÞuzðk8qÞþgM0 sin yhzðk8qÞ

�gM0 cos yh7 ðkÞ,

omzðkÞ ¼ 12 sin y½o�2k�qm�ðk�qÞ�oþ2kþqmþ ðkþqÞ�

þ12gM0 sin y½hþ ðkþqÞ�h�ðk�qÞ�

� i4gb44M2

0 sin 2y½ðk�qÞu�ðk�qÞ�ðkþqÞuþ ðkþqÞ�,

½o2�s2t k2�u7 ðkÞ ¼ i

rkb44M0½sin ymzðk8qÞþcos ym7 ðkÞ�,

½o2�s2l k2�uzðkÞ ¼�2iðb33�b31ÞkM0 cos ymzðkÞ=r,

½o2�k2v2�h7 ðkÞ ¼ �o24pm7 ðkÞ, hzðkÞ ¼�4pmzðkÞ: ð8Þ

Here, we introduce the following notation: v¼ c=ffiffiffiep

velocityof propagation of electromagnetic waves in a magnetic, e dielec-tric constant, st ¼

ffiffiffiffiffiffiffiffiffiffiffiffic44=r

p, sl ¼

ffiffiffiffiffiffiffiffiffiffiffiffic33=r

pvelocity of propagation of

transversal and longitudinal acoustic waves, respectively,

o1k ¼o10þgM0 sin2 yLJðkÞ, o72k ¼o20þgM0L?ðkÞ,

L7? ðkÞ ¼ �aðq

2�k2Þ�gðq4�k4

Þþ2a1ðq8kÞ,

LJðkÞ ¼�aðq2�k2Þ�gðq4�k4

Þþ2a1q,

ome4 ¼ gM0hme4 ¼ gb244M3

0=c44, o20 ¼ome4 cos2 y,

o10 ¼ gM0½hme4�sin2 yð ~b1þM20 cos2 yð ~b2þ2b2Þþhme sin2 yÞ�: ð9Þ

In system of Eq. (8) we have to add the condition of constancyof the modulus of the magnetization 9M

!9¼ const, what for the

Fourier components of the magnetization is

siny½m�ðk�qÞþmþ ðkþqÞ�þ2mzðkÞ cos y¼ 0: ð10Þ

Using the ordinary values of the constants for magnetic withexchange spiral (TbMn2O5) g ¼ 2� 107 s�1 erg�1, M0 � 103 Oe,a1 � 10�28 cm4, a��10�14 cm2, q� 107 cm�1, and in the case ofrelativistic spiral (CsCuCl3) a� 10�12 cm2, b� 1, a� 10�8 cm,q� 104 cm�1, from Eq. (8), we obtain the spectrum of coupledelectromagnetic–spin–acoustic waves.

Changing y the range 0ryrp=2, we can calculate thespectrum for the ferromagnetic spiral. Fig. 1 shows the depen-dence oðkÞ for y¼ p=4 in the case of relativistic spiral.

To get more details and study the dynamic of changing bandgaps width with changing external magnetic field value let usconsider the dependence oðkÞ for different values of y near k¼0(Fig. 2).

It is seen that all spectra have a band structure. At certainfrequencies the gap (window opacity) is observed as for electro-magnetic such for acoustic waves. These band gaps appear due tothe resonant interaction of spin, acoustic and electromagneticwaves in a magnet. From Fig. 2 we can see that with decreasingangle the electromagnetic band shifts toward lower frequenciesand its width decreases. Calculations show that in the case ofexchange spiral, a band of opacity is much narrower than inthe case of relativistic one. Note also that the magnitude ofthe interaction of spin, acoustic and electromagnetic wavesdepends on the angle y.

Fig. 1. Spectrum of coupled oscillations for y¼ p=4.

Fig. 2. Spectrum of coupled oscillations for different y near k¼0.

Fig. 3. Frequency dependances of reflectance for different y.

I.V. Bychkov et al. / Journal of Magnetism and Magnetic Materials 329 (2013) 142–145144

3. The reflection of electromagnetic waves from a magnetichaving conical spiral ferromagnetic order

Let us now investigate the reflection of electromagnetic wavesfrom a half-infinity layer of magnetic material in phase offerromagnetic spiral. Consider the normal incidence of electro-magnetic wave kJqJz. Restrict ourselves to small wave numbersk5q. The system of boundary conditions include the continuityof normal components of the magnetic and electric fields, thetangential component of the electric and magnetic fields, thevanishing of the derivative of the magnetization and the absenceof stress at the boundaries of the magnet:

HðeÞt ¼HðiÞt , EðeÞt ¼ EðiÞt , BðeÞn ¼ BðiÞn , DðeÞn ¼DðiÞn ,

@m!

@xknk ¼ 0, sðiÞjk nk ¼ 0:

Indexes (i) and (e) denotes quantities inside and outside themagnet, respectively, n

!normal to the surface. Taking into

account the number of roots of dispersion equation in thisapproximation, the system of boundary conditions in the cyclicalcomponents of the magnetic field becomes

h07 þhR7 ¼X3

j ¼ 1

hj7 , h07�hR7 ¼X3

j ¼ 1

kj7

k0ehj7 ,

X3

j ¼ 1

hjz ¼ 0,

X3

j ¼ 1

kj7 ðo2�v2k2j7 Þhj7 ¼ 0,

X3

j ¼ 1

kj7 hjz ¼ 0,

cos yX3

j ¼ 1

o2�v2k2j7

o2�s2t k2

j7

hj7 þsin yX3

j ¼ 1

o2

o2�s2t k2

j7

hjz ¼ 0,

X3

j ¼ 1

s2l k2

j7

o2�s2l k2

j7

hjz ¼ 0: ð11Þ

Fields define wave reflected from the surface of the magnetic, ki7

solutions of the dispersion equation. Solving (11) with (8), we find thereflection coefficient of electromagnetic waves R7 ¼ 9hR7 =h07 92

.Fig. 3 shows the frequency dependence of the reflection

coefficient of electromagnetic waves from an interface of half-infinity magnetic material with a ferromagnetic spiral at differentangles of spiral. From the frequency dependence for differentangles y can be seen that with increasing angle y of ferromagneticspiral (and thus decrease the external magnetic field) increasesthe band gap (window opacity) and shift it to higher frequencies.Also there is a peak at frequencies of resonant spin–acousticinteraction, which shifts to higher frequencies with increasing y.

4. Acoustic Faraday effect

Let us now consider the acoustic Faraday effect, i.e. dependence ofthe angle of rotation of the polarization plane of acoustic wave fromthe external magnetic field. Let the magnet in a phase of ‘‘ferromag-netic spiral’’ falls linearly polarized acoustic wave. It can be repre-sented as a superposition of two acoustic waves of different circularpolarization. From (8) for waves of different polarizations obtaindifferent wave vectors. Since the wave vectors of the waves aredifferent, so the refractive indices also will be different, therefore, willbe observed the rotation of the polarization plane by an amountDj¼Dkl, where Dk¼ ðkþ� k�Þ=2. Based on the solutions (8), itdepends on the frequency of the incident acoustic wave and reachesmaximum values near the acoustic band gap. The dependence of theangle of rotation of the polarization plane from the external magneticfield for magnetic layer thickness of 1 cm is shown in Fig. 4.

5. Conclusions

Studies on the hybrid electromagnetic–spin–acoustic waves inmagnetic materials with a helical magnetic structure, defined by

Fig. 4. The dependence of the angle of rotation of the polarization plane from the

external magnetic field H.

I.V. Bychkov et al. / Journal of Magnetism and Magnetic Materials 329 (2013) 142–145 145

the inhomogeneous exchange and relativistic interactions in thephase of ferromagnetic spiral, showed that the spectrum of thecoupled waves has band structure. The band gap depends on theangle of the ferromagnetic spiral, and hence on the external

magnetic field. Increase in the angle (decrease in the magneticfield) leads to an increase in band gap, the maximum gap(window opacity) is observed at y¼ p=2, i.e. at the phase transi-tion ferromagnetic spiral simple spiral. The possibility of resonantinteraction of spin, acoustic and electromagnetic waves has beenshown. The value of the interaction of the waves depends on y.The frequency dependence of the reflection coefficient of electro-magnetic waves from the plate of the magnetic with a ferromag-netic spiral at different angles of the helix has been calculated. Asthe angle increases the opacity region broadens and shifts tohigher frequencies. The angle of rotation of the polarizationincreases near the band gap. There is a peak of rotation of thepolarization plane, which shifts to lower fields if the frequencyincreases.

References

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Magnetic Materials 118 (1993) 169.[7] I.V. Bychkov, V.D. Buchelnikov, D.A. Kuzmin, V.V. Shadrin, Solid State Phenomena

190 (2012) 257–260.