Magnon-polaritons and magnetic reststrahlen-bands of uniaxial antiferromagnetic crystals

32

Journal of Magnetism and Magnetic Materials 50 (1985) 32-38 North-Holland, Amsterdam

M A G N O N - P O L A R I T O N S AND MAGNETIC R E S T S T R A H L E N - B A N D S O F UNIAXIAL A N T I F E R R O M A G N E T I C CRYSTALS A. L E H M E Y E R and L. M E R T E N Physikalisches lnstitut der Westfiffischen Wilhelms- Universiti~t, DomagkstraBe 75, D- 4400 MUnster, Fed. Rep. Germany Received 27 June 1984

The magnitudinal and directional dispersion of the two kinds of magnon-polaritons (magnitons) in uniaxial antiferromagnetic crystals is studied. In particular a proof is given for the existence of a purely longitudinal magnon propagating perpendicular to the optic axis. Furthermore, the corresponding magnetic reststrahlenbands for crystal cuts are discussed.

1. Introduction

Magnon-polaritons are quasi-particles in magnetic crystals consisting of a photon coupled with a magnon. In uniaxial antiferromagnetic crystals two kinds of magnon-polaritons are possible, denoted as magnetically extraordinary and magnetically ordinary magnon-polaritons (magnitons). The dispersion function t0(k) of the extraordinary magnon-polaritons is determined by (see e.g. ref. [1], formula (9a)) to,'

) =

In eqs. (1) and (2) ~ denotes the angle between the wave vector k and the optical axis. c ±, ~ll, /~ ± a n d /.tll are the elements of the dielectric tensor

,.(to) ,(to) =

, ± (to)

(3)

,,(to) and the permeability tensor

l(to) (to) =

c2k 2

l(to)

(4) l(to)

to2

=,±(to)

~ ± (to)/~ll(to) /~ ~_(to) sin 2 ~ +/~11(~0) cos 2

;

(1)

and the dispersion to(k) of the ordinary magnonpolaritons by c2k 2

= to2

= l(to)

, . (to),,(to) , ± (to) sin 2 ~ + ,,,(to) cos 2~

(2) (see, e.g. ref. [1], formula (9b)).

Combined with the formulae of the magnetic reststrahlenbands for crystal cuts derived in section 4 eqs. (1) and (2) determine also the shape and the directional dependence of the reststrahlenbands. From this follows a strong connection between the dispersion curves of the magnonpolaritons and the corresponding reststrahlenbands, as pointed out in detail in the following. Especially, in section 3 we prove that eq. (1) contains a longitudinal magnon for ~ - - 9 0 °. The existence of a longitudinal magnon in antiferromagnetic crystals first was discussed by Manohar and Venkataraman [2] and subsequently by Kaganov and T'yau [3].

0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A. Lehmeyer, L Merten / Magnitons in uniaxial antiferromagnetic crystals

2. Dispersion of magnon-polaritons

frequency 60TO for the transverse waves propagating along the optical axis:

To demonstrate the essential features of the magnitudinal and directional dependence of the dispersion curves according to eqs. (1) and (2) we assume the validity of the model of linear spin wave theory for simplicity. Then /~ ± is represented by a function of the Kurosawa type (see, e.g. ref. [4]) 602 _ 0~2

/~ ± (~o) - -~ 2 - _ ~2

(5a)

and

~oR is the resonance frequency and o)A the so-called antiresonance frequency. (o R is identical with the 100,

T

E u

80.

100- / 80- ~,~, /

Hlk/O=O°

G0

3

t.0. 20"

z6o

~ 3

40. 20"

60-

40.

20

0

-~ 3

260 ~6o k/2~ (cm-1)

'::l

/o.oo

£11=2.

/o.,oo

-:,oF_

~ go- f,(UA /

40- /

3

20

2C10 k/2~ (cm-1 }

460

400

o

260

460

80,

;80

O =90 °

{LO-Magnon

2,,k 260

26o

4;0

o

k / 2 K (cm -I )

Fig. 1. Dispersion of magnetically extraordinary magnonpolaritons.

zGo

k/2~ [cm-1 )

,60

1000=90 o

O =8S ° 80"

~80'

60'

TuE60•

3 40"

-~ L0 ,

f~Y/_t=R

3

2o.

20.

,60

°"

20.

100"

0

0 = 80 °

3 40.

.

k/2K (cm -I )

//~-t ight fine

~b0

100'

20' S 0

2Go k/2n. (cm -I)

0 = 60 °

_~_ _ j _ = ~ "3 40. . . . . .

680"

4o,

zoo k/2n, (cm -I]

k/2~ (cm -I ) 100

20-

o

-- 60

/

G0- ~t~A

0

£±=5,

These values are arbitrary, but comparable with the values of a real reststrahlenband of CoF z [4]. Fig. 1 shows the dispersion of magnetically extraordinary magnon-polaritons according to eq. (1). Obviously, the frequency gap decreases with increasing 0. For 0 = 0 ° there is a gap between o~R and o~A, whereas the lower branch is enhanced

80,

/~Aj t

o

/ 0=850

0

0:A=50cm -t,

100 0=80

80-

60,

80-

~R=40cm -],

2()0 ' ~6o k/2~ (cm-1 )

0

1000=60°

80'

100"

Furthermore, we suppose £± and (, to be constant in the frequency region of the magnon dispersion. For evaluation of the dispersion curves the following input values are used:

-:,o......

/-0-

0

,6o

100.

3

(5c)

20k/2~ (cm-I )

.x

,0To = ,0R.

-':00: I

O=30 °

60" 3

33

200 k/2Tt (cm-1 )

4~

o

200 4b0 k/2w. (era-1)

Fig. 2. Dispersion of magnetically ordinary magnon-polaritons.

A. Lehmeyer, L. Merten / Magnitons in uniaxial antiferromagnetic crystals

34 I

I

l

I

'

For # = 9 0 ° ( s ± = s i n ~ = l , (6) simplifies to

I

~8

/~

to 1.~1 44

c2k2

- -

sll = cos # = 0). Eq.

/zll ) = 0 ,

± E±692

with the two solutions:

692 _ 692

~2

1) /~± ( 6 9 ) - - -

692 _ 0)2

=0

~0 [

0*

,

i

20*

,

I

J

~0"

I

60*

,

or

I

80" 90*

69 = 69A

Fig. 3. Directional dispersion for k --, ~ (magnons).

and

in frequency above 69R for k --, oo with increasing v~. The physical meaning of the two branches for wave propagation perpendicular to the optical axis (v~ = 90 °) is treated in the next chapter. The directional dispersion of the lower branch for k ~ oo (magnon region) is sketched in fig. 3. The dispersion of magnetically ordinary magnon-polaritons is represented in fig. 2. For this case (cf. eq. (2)) we have a gap of the width A = 6 9 A - 69R independent of ~. Moreover, the polariton branches show directional dispersion due to the dielectric anisotropy of n TH (see eq. (2)). From an algebraic point of view the band gaps are characterized by negative values of n2TE and n2H, i.e. riTE and nTn are imaginary quantities. These properties of the refractive indices are used in section 4 for the interpretation of the corresponding reststrahlenbands.

2)

3. L o n g i t u d i n a l m a g n o n s

In the following it is proved that in the magnetically extraordinary magnon-polaritons a longitudinal magnon is contained as a limiting case. We start with an equivalent form of eq. (1):

/~ ±

c2k2 E±

692

/

c2k 2

(69)=0

or 69 =

ck

(7b)

For the interpretation of the horizontal line we should remember that horizontal lines are characteristic for longitudinal quasi-particles, because longitudinal quasi-particles do not couple with photons (within a linear theory) and therefore show no dispersion. Hence the horizontal line must be assigned to a longitudinal magnon propagating perpendicular to the optic axis, i.e. in the basal plane. Now, by regarding the polarization vectors (eigenvectors) we will give an exact proof that the horizontal line 69 =69A for ~ = 90 ° indeed represents a longitudinal magnon defined by H l t k . If we fix the coordinate system as in ref. [1] the wave vector k for 0 = 90 ° in the basal plane is represented by

k = (kx, 0, 0).

(8)

The magnetic vector H on the other hand has for general angles ~ the form (see eqs. (16) and (17) in ref. [1])

/~11) sin2~

{ c2k2 --~t±]cos21~= 0,

(7a)

(6)

n = ( H I , 0, H3),

(9)

A. Lehmeyer, L. Merten / Magnitons in uniaxial antiferromagnetic crystals eq. (9) we therefore have the result

where -/3 ~

35

].t/ -- _ _

tan 0 H1

(10a)

#11

=-(#±

tan 2 o ) c o t o H 1.

(10b)

#ll

We now can show that * lim (#J" tan20)

H[Ik,

(16)

and also Milk because of H + 4,~M = 0 for this direction. In words: The quasi-particle is a pure longitudinal magnon. The electric vector vanishes for all k, which follows directly from the Maxwell equation (see e.g. eqs. (4a) and (15) in ref. [1])

(11)

o-,90 o \ #jp

is finite for all k (except for k = ~cx (60A)#,(60a) × (60A/C)). AS COt O = 0 for 0 = 90 ° then follows from (10) that lim H 3 = 0.

(12)

0--, 90 o

60Lo = 60A,

(18)

Eq. (5a) can be rewritten as Therefore, in the limit 0 = 90 ° we have H = ( H 1, 0, 0),

If we introduce the notation 60LO for the frequency of the longitudinal magnon, i.e.

(13)

. (60)

-

60 o - 602

(19)

(,~2TO __ 602 "

i.e. H and k are parallel. To show that the expression (11) is finite in the limit 0 = 90 ° we write eq. (6) in the form # . tan20 = /~11

_

c2k2/¢ ± 602 _ # ± c2k2/e ± 602 -/~11

(14)

In the limit 0 = 90 ° we get lim

/~±tan2# = -

0--*90° /'~ll

lim

c2k2/e±602-P'±

To summarize: For 0 = 90 ° beside the light wave there exists a longitudinal magnon with HlInllk and E = 0, H is a magnetostatic field. This longitudinal magnon does not interact with the light wave. For completeness we should mention that on the contrary the polaritons represented by eq. (9b) in ref. [1] cannot contain a longitudinal magnon, because for these polaritons

v~--'90° ¢ 2 k 2 / ¢ ± 6 0 2 - - ~ 1 1

H_l_ k c2k2/~- _L(60A)

602

is valid for all 0 (see p. 328 in ref. [1]).

( ,,,,, ) _ ¢2 k 2/ , ± (60,,)602

1 ± (60A),,(60A)

60

/c2k 2 - 1

(15) which obviously is finite (except for k = ~'±(%)#11(60A)(60A/c) which corresponds to the value of k of the crossing point of the light line and the horizontal line ~0 = 60A)- Combined with

* The simple factor ( # ± / # l l ) t a n 0 of eq. (10a) is not suitable for the proof, because in the limit 0 = 90 ° it leads to an undefined expression 0 × oo.

4. M a g n e t i c reststrahlenbands for crystal cuts

The directional dependence of the dispersion curves can be measured directly by infrared magnetic restrahlenbands for normal incidence, if one uses crystal cuts. There is a strong correspondence between the directional dependence of the dispersion curves and the reststrahlenbands of the magnon-polaritons. The geometry of the crystal cuts is illustrated in fig. 4. The optical axis is directed parallel to the z-axis of the coordinate system. The angle between the optical axis and the normal of the surface is

A. Lehmeyer, L. Merten / Magnitons in uniaxial antiferromagnetic crystals

36

tions read:

.zlle

E' +E" =E

(24)

and

H'+H"=(e.H)e,

(25)

or

E' + E" = E

(26)

and H ' + H " = e- H = cos 0 H x - sin 0 / / . . H x and //~ can be substituted Maxwell's field equation

Fig. 4. Illustration of the employed geometry.

by E,

(27) using

U = nvE~,-'(s × E). denoted by O. This geometrical arrangement enables a clear separation of magnitudinal and directional dispersion effects: for a given v~ only the magnitudinal dispersion m a y be studied. If we introduce s ' as wave normal vector of the incident wave, s " as normal vector of the reflected wave, and s as wave normal vector of the wave inside the crystal, the following relation is valid s' = s = - s " = ( - sin O, 0, - cos 0 ) .

(20)

(28)

D e c o m p o s i n g H in its components, we get: H x = nTEs:E v =

#±

riTE COS 0 E, /~±

(29)

nTE sin 0 E ,

H . = nTE s x E ~

(30)

E= fv" Introducing H X and H. in (27) and regarding /-I' = E ' ,

(31)

Besides the wave normal vectors we introduce a unit vector e lying in the surface and in the xz-plane

H" = -E".

(32)

e = (cos 0, 0, - s i n 0 ) .

E ' - E " = F/TE

(21)

Eq. (27) finally reads

We first treat the case that the electric vector E ' of the incident wave is directed perpendicular to the plane of incidence (xz-plane): E'--- (0, E ' , 0).

(22)

Then the magnetic vector H ' is directed parallel to

/~llCOS2v~ +/x ± sinZv~ F/TE

H'e.

(23)

T h e b o u n d a r y conditions require the continuity of the tangential c o m p o n e n t s of E and H . Inside the crystal also a wave with E perpendicular to the plane of incidence (xz-plane) is possible (TEpolarized wave). Regarding that five of the six vectors lie in the surface the two b o u n d a r y condi-

hi' ± ~11

E.

(33)

As nZE=

e:

U'=

( c°s20 + sin2~) E ~± ~11

#-#11 2 e± /~ ± sin2v~ + #iNos 0

(34)

(see e.g., eq. (9a) in ref. [1]).Eq. (33) reduces to E'-E"=

E

" E.

/'/TE

F r o m (26) and (35) finally one gets 2E'=E

( ") 1+

riTE

(35)

A. Lehmeyer, L. Merten / Magnitons in uniaxial antiferromagnetic crystals

and

where

2E"=E(l-CZ) "nT E

/'/2H = /~ j.

E " 2= nTE--~. 2. RTE =

~7-

(36)

riTE _k £ ±

If inside the crystal the wave with H perpendicular to the incident plane (TH-polarized wave) shall be excited the magnetic vector H ' instead of E' of the incident wave must be chosen perpendicular to the xz-plane. The result then is: 2 RTH

=

nTn

(37)

- - /J' ±

/'/TH -~- ~ ±

UT~

( ± (II

sinZt~+

(.

The reflectivity RTE therefore is given by

'

.mLO

"

The proof of eq. (37) is completely analogous to the proof given above for the TE-polarized wave. For evaluation of RTE and RTH the same input values are used as for the dispersion curves (see section 2). The results are illustrated in figs. 5 and 6. Obviously, the width of the magnetic reststrahlenbands for the TE-polarisation (Fig. 5) and the TH-polarisation (fig. 6) agrees with the width of the gap of the dispersion curves given in figs. 1 and 2. The TE-band width decreases with increasing angle 0. For 0 = 90 ° the band vanishes completely because in this case magnons and pho-

/ULO

/mLO

inTO

R (el 0.8"

=

UTO 1.0

R (m) 0.8'

0.8'

0.6.

0.6,

0.6

0.4-

0.4 -

0.4.

Off-.

0.2-

0,2-

0.2

o.o 0

0.0

o

4o

do

/-0

~LO

1.0 R (u) 0.8-

0 = 60 °

/

0.20.0

o

~

do

0.6. 0.4.

o

~

4'o

~ ( c m -1 )

0.2-

do

0.0

- -

~ = 850

1.0R (~) 0.8-

0.6

0.6-

0.4,

0,4.

0.2

0.2

o.o

1o

8'0

(~[cm -1)

o.o o

.0' = 90 °

io

~o m (cm "I )

Fig. 5. Directional dependence of the magnetic reststrahlenband (TE-polarization).

,

,

t2 ,

,

1,0Rim) 0.8-

0./.0.2,

0.0

,

i

i

i

i

40 80 m(cm -1 )

wTO /mLO

inTO / ~ L O 1.0-

R (ul 0.8-

0.6-

0.4-

0.4_

0.2-

0.2-

0

/~LO

0.6.

0.6-

O.0

dO

Rim] 0.8'

~0 80 m{cm -1 )

m (cm -1 )

¢oLO 1.0R (m) 0.8

UTO

1.0,

0.8"

0.40.2 -

~.o

¢OTO /mLO

@= 80 °

i

40 to (era -1 )

1,0-

R (m)

0.8-

,

m(cm -1 )

0.6 -

0.0

0.0 ,

8'0

,~o

/~LO

1.0. RIm)

0.6 -

0.4 -

8'0

0.2 ¸

--~,,

~ ( c m -1 )

micro -11

/mLO

¸

R(m}

0.6-

0.0 ~ l

(38)

(IICOS2~

1.0' R {el 0.8-

37

4o

u)lcm -1 )

~io

0.0 0

'

40 ' 80 u ( c m -1 )

Fig. 6. Directional dependence of the magnetic reststrahlenband (TH-polarization).

38

A. Lehmeyer, L. Merten / Magnitons in uniaxial antlferromagnetic crystals

tons are decoupled (see fig. 1). As to be seen in fig. 6 the TH-band shows directional dependence due to the dielectric anisotropy of the crystals. Furthermore, the TH-band width is independent of direction: We have total reflection (R = 1) between to ro and to LOIn general algebraically total reflection (RTE(to)= 1, R-re(to)= 1) follows from the fact that for frequencies within the band gap the refractive indices n-rE and nTH, respectively, are imaginary quantities (see section 2).

Acknowledgements We thank Prof. J. Brandmi~ller and Dr. K.M. H~iussler for stimulating discussions. We are also

indebted to the Deutsche Forschungsgemeinschaft for financial support.

References [1] J. Brandmi~ller, A. Lehmeyer, K.M. Haussler and L. Merten, Phys. Stat. Sol. (b) 117 (1983) 323. [2] C. Manohar and G. Venkataraman, Phys. Rev. B5 (1972) 1993. [3] M.I. Kaganov and N.K. T'yau, Sov. Phys. Solid State 17 (1976) 1909. [4] K.M. H~ussler, A. Lehrneyer and L. Merten, Phys. Stat. Sol. (b) 111 (1982) 513.