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Surface magnetic polaritons on uniaxial antiferromagnets
Solid State Communications, Vol. 42, No. 3, pp. 233-238, 1982. Printed in Great Britain.
0038-1098/82/150233-06503.00/0 Pergamon Press Ltd.
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS C. Shu* and A. Caill6 D6partement de physique, Universit6 de Sherbrooke, Sherbrooke, Qu6bec, Canada, J 1K 2R1 (Received 30 October 1981 by M.F. Collins)
The dispersion relation of surface polaritons for a uniaxial antiferromagnet are calculated taking account of retardation. In the presence of an external magnetic field, one finds that inequivalent propagation occurs in the Voigt configuration(+ t~ x Ho). For a high frequency permeability smaller than a critical value, the surface polaritons in the --t~ × Ho direction, where ~ is the normal to the interface, merges in the same frequency domain as the bulk polaritons.
THE EFFECT of dipolar magnetic fields, resulting from the spin fluctuations themselves, on the dispersion relation of long wavelength magnons in magnetically ordered ferromagnetic and antiferromagnetic crystals has been discussed by various authors [ 1]. The resulting magnetic polaritons is the analogue of the wellknown phenomenon of coupling between photons and optical phonons (dielectric polariton) in an ionic crystal [2]. There exists now clear experimental evidence for the existence of a bulk antiferromagnetic polariton as obtained recently by Sanders et al. [3] in a high resolution far-infrared transmission study of the antiferromagnetic resonance of FeF2. Moreover, as a tool to study surfaces, the complementary surface polaritons, both electric [4] and ferromagnetic [5, 6], have been discussed for semi-infinite and lamellar systems. Hartstein et al. [5] have studied the surface magnetic transverse electric (TE) polariton in a uniaxial ordered ferromagnet, with the external magnetic field and the magnetization parallel to the surface. They found, for the Voigt configuration where the wave vector k is perpendicular to the extemal magnetic field Ho, that the surface polaritons exist between the two branches of the bulk polariton dispersion relation and made evident the inequivalent propagation in the -+t~ x tto directions where ~ is the outside normal to the surface. Karsono et al. [6] studied the surface magnetic polaritons in a ferromagnetic slab in the same geometry as Hartstein et al. [5] and showed that the reciprocal nature of the propagation with respect to direction is restored. More recently, Marchand et al. [7] studied the dispersion relations and reflection symmetry properties of the guided magnetic polaritons for a uniaxial ordered ferromagnetic slab in the Voigt configuration. * Permanent address: Nanjing Institute of Technology, China. 233
In this communication, we study the dispersion relations for surface antiferromagnetic polaritons in a uniaxial ordered antiferromagnetic material in the Voigt configuration. Figure 1 represents schematically the physical system with the choice of axes. The basic limitations of the model are the following: (a) we consider surface excitations with penetration depth sufficiently large that pinning effects of the surface may be neglected and that the bulk magnetic permeability may be used, and (b) excitations with wave number Ikl small enough that the dispersion of the magnons arising from exchange may be neglected so that the energy of the spin excitation receives its dominant contribution from the external Zeeman field and the demagnetizing fields set up by the spin motion. In the study of the magnetic polaritons, the fluctuations of magnetization (in the x - y plane) are related to the fluctuations of dipolar electromagnetic fields using the transverse susceptibility response. This response is obtained using the linearized torque equations for the two sublattices where exchange dispersion terms are neglected. The gyromagnetic permeability tensor for the uniaxial antiferromagnet [8] in an external static magnetic field Ho = 11o2 is:
la=
,--/.t.y
#..
0
\o
0
/g~
,
(i)
where tt~x = Uo[1 + 2 ~ m w a ( ~ ] -- co~ -- 6o2)/Aa ],
(2a)
tlx~ = 4il~060ra ~OOCOa6olA A ,
(2b)
lazz = /a0.
(2c)
The intrinsic frequencies con, ~a and Wo are given by 4~rTMs/~t0, "/Ha and 7//o respectively where Ha is the uniaxial anisotropy field in the z-direction, Ms[Iz o is the
234
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS Vol. 42, No. 3
magnetic dipolar field outside the crystal which is linked to the field inside using the usual boundary conditions. For the TE surface modes, in the Voigt configuration, we look for solutions of equation (5) of the form:
Y
HI = (HlxY¢ + H i , p ) exp (-- t~l y) exp i(kx -- 600,
y > 0, H~ = (H2x~ + H2~p) exp (a2y)ex p i(kx -- 600,
VOCUUm
y < O:
X
/~oo ~ 41z
(6a)
~
(6b)
Substitution of equations (6) into equations (5) leads to: a l = k 2 -- 602/c2,
(7a)
a~ = k 2 -- ~e602/ c 2,
(7b)
antiferromaqnet
where/av is the magnetic permeability in the Voigt configuration
Z
Fig. 1. Schematic representation of the physical system. The semi-infinite space y > 0 is empty. The semi-infinite space y < 0 is occupied by a two sublattice antiferromagnetic material o f sublattice magnetizations M l z and M2~ and scalar dielectric constant. screened saturation magnetization in the z-direction and ~, is the absolute value of the gyromagnetic ratio. #o is the high frequency permeability constant (/ao > I) caused by magnetic dipolar excitations other than the sublattice magnetic spin wave excitations (e.g. optical magnons). 6ol is the antiferromagnetic resonance frequency [9], 601 = 60a(60. + 260e),
(3)
where we is the exchange frequency (tOe 7He) for the z-component of the exchange field equal to He. The denominator AA is:
/4, = /axx +/a~r t-txx
[0.)2 -- (60 -- O.)0)2110)22 -- (60 + 600)21 [601
(4)
Since no attenuation mechanism is invoked, the longitudinal response/axx is purely real while the transverse response/axy is purely imaginary. Note that equation (1) gives the frequency dependence of the magnetic response probing only the long wavelength spin wave excitations. The dipolar induced magnetic fields of the antiferromagnetic polaritons have to satisfy Maxweil's equations which for the Voigt configuration of Fig. 1 become:
0)2] + 2CO 60o('O2 __602 __60
and
(8)
6o~ = 602 + 260m60a.
(9)
Unlike the ferromagnet in the Voigt configuration [5], the generalized Voigt permeability/av(60 ) has two poles and two zeros which are denoted by (coO), 60(3)) and (60(2), 60(4)) respectively, 6020,3) = (601 + 603 + 60m60a) + [46026002
=
£~.4 = [602 _(60 + c00)2][co2 _(60 _60o)21.
O) ][602 __(60 +
+ 460m60am2 + C°mWal" 2 .2,1/2,
60(2,4) = + 600 + [603 + 260,nWa]1/2.
V:H-- V(V- H)
c2 at 2 = 0, c2 at 2
- 0,
y>0, y ~<0,
(5a) (5b)
where we have retained retardation. For surface excitations, the spin motions inside the crystal set up a
(10b)
Frequency gaps (gv < 0) for the bulk antiferromagnetic polaritons exist between the frequencies 60o) to 60(2) and 60(3) to co(4) in agreement with the calculations of Bose et al. [8]. For most uniaxial antiferromagnets 60ra60a/601 "~ 601, such that the frequency gaps are very narrow on the scale of the characteristic frequency 601. The frequency gaps arise because of the photonmagnon coupling and the induced magnetic displacement currents in analogy with the Lyddane-SachsTeller effects in ionic crystals [2]. The dispersion relation of the surface modes, obtained from the divergence equations
1 02H
V2H--V(V'H)
(10a)
V'H1 = 0
y>0,
(lla)
V'(pH2) = 0,
y~<0,
(lib)
and the boundary conditions (at y = 0), H l x = H2x, HIt
= -- ~txrH2x + btxxH2y ,
(12a) (1 2b)
Vol. 42, No. 3
SURFACE MAGNETIC POLARITON ON UNIAXIAL ANTIFERROMAGNETS
235
S2
i{
'..~ ".
"0
1.5
U"
# •
~., " •
x
.....
"
'
D~
k". "~
,/'/" . . . . . . .
,/
!~,,,K,~(,t a I
....
,',//
l
,~o
X
/
c~2)1-J.o
..
"\
/
.3,
... ,,
/
,p ................
..
."
/,,
• N~.
."
'. '\
/"
". '.
~ '.; \/. i - I REDUCED
...............................
/
/ //~@ /~\
-0.5 N
-2
/
i /
\.
-3
,2
- ( 4
...................
\
///"
." / ..'~'
';
I
0 WAVENUMBER
I K
I 2
-"
K 3
Fig. 2. Schematic representation of the dispersion relation of the surface antiferromagnetic polariton. The frequencies and wave-numbers are expressed in reduced units 12 = co/co~ and K = c k / c o l . The full drawn curves are for the surface modes where the dashed lines represent the top two bulk modes. The dotted lines (I2 = + K ) are the lines of light for the external medium and the dashed-dotted curves (I2 = + K [ x / e ) are the lines of light for the medium of dielectric constant e. The physical parameters used are H a = 8 5 x 103 Gauss,H~ = 5.2 x 10 $ Gauss, Ms = 60 x 10 2 (De,Ho = 1.0 x 104 Gauss, e = 4.0 and/~o = 1.25. For the sake of clarity, the forbidden gaps for the bulk polaritons are enlarged. is given by or2 = -- (xl #v -- ik#x~, /laxx .
Non-radiative surface wave solutions exist when (13)
o q ( k c o ) and ot2(kco) are larger or equal to zero. In the
frequency-wave number plane (co, k), there exist two such regions; both are bounded on the low frequency side by the poles of/h,(co) (60(1) and coO)), on the high frequency side by the dispersion curves of the bulk k2c 2 polaritons [a2(kco) = 0] and on the left-hand side (for co2 - {[#2oA(c°)-- B ( c o ) l [ l a ~ A ( c o ) - - etaoC(co)] the positive k mode) by the line of light oq(kco) = 0. In 2 2 2 2 2 contrast to the isotropic dielectric where the surface - - 32/.tocom~ocoaco polariton is located inside the gap of the bulk polariton, g 8UOcomCOaCOo~[(1 + ez)/.tgC2(co) in a gyromagnetic material, it is not limited to this frequency domain. Symmetrical regions exist for the - e u o C ( ~ X a ( ~ ) + u~,a(~))p'2}/ positive and negative wavenumber k. Comparison of -- 16#ocomcoacooco }, (14) equations (8) and (15c) shows that the poles of ~(co) are the roots of C(co). It is then easy to see that the where starting coordinates of the surface antiferromagnetic A(co) = [~22 --(co -- coo)a][co~ --(co + coo)a], (15a) polariton in the frequency gaps which are (kv(1), coO)) a(co) = [co~ - ( ~ -coo)2][co~ -(co + coo)2], (15b) and (kvO), 6o(3)) respectively for the lower and upper gaps, are given by: Using equations (7) and (13), the dispersion relation of the surface antiferromagnetic polariton is:
c(co) = a(co) + 2 co,,, co. ( co~ - ~o~ - coal.
(15c)
236
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS
Vol. 42, No. 3
S2 1.10951 %
m
N~\~
I.10949'
C= m
"11 m 0 I"11 z ..<
.10947-
I K
-30
I
\LI\0945-
Reduced wavenurnber-20 K
-tO
Fig. 3. Bulk and surface polaritons near the upper gap in the -- k direction. The full drawn curve is the surface polariton and the dashed curve is a bulk polariton. The parameters used are the same as in Fig. 2 except that now #o = 1.0 1.
k$o = 60O)G(60"))/c,
(16a)
k(,,~) = _ 60(~)G(60(~))/c,
(16b)
G2(60) =
~u]A(60)[ts~A(eo) -- B(60)] -- 32t~o260:m60~60~w2 -- 64~o60ra60a60o60
(17)
Figure 2 is a sketch of the dispersion relation for surface polaritons i n M , F2 where/.t o = 1.25. The non-reciprocal nature of the propagation in the fz x Ho and -- ~ x Ho directions is quite evident. In and near the lower frequency gap, along the + k direction, the dispersion curve starts at (60(1), k(vO) and goes asymptotically to 60 = 60.(~) when k -* ~o. Along the -- k direction, for the same frequency region, there is only a small segment starting at ~wv('.'(2),-vz'(2)'~son the line o f light 60 = -- c k and ending on the dispersion relation for the bulk polariton. This mode has a spin precession in the non-resonant circular polarization and is driven by its large photon content (near the lines o f light). In (and near) the upper gap, along + k direction, there is only a small segment
t .(4), '~v v(4)~) on the line of light 60 = c k and starting at ~'~v ending on the dispersion curve o f bulk polariton. For the same frequency region, along the - k direction, the curve starts at (60(3), kv(3))being asymptotic to co(s~,) when k ~ -- o~. Harstein et al. [5] have discussed the nonreciprocal property of the surface polaritons of the semiinfinite ordered ferromagnet. Unlike the ferromagnet, the uniaxial antiferromagnet, has two sublattices with opposite magnetization. If we consider each of these sublattices as the "lattice o f a ferromagnet", each o f them would have its own non-reciprocal dispersion curves, so we should have two sets of these curves in (and near) the lower and upper gaps for an antiferromagnet .Because o f the opposite orientations of the two sublattice magnetizations in the external magnetic field H0, the non-reciprocal properties of these two sets o f curves are opposite to each other. The asymptotic frequencies 60(~) (k ~ +- oo) for the surface polaritons are given by the poles of the right hand side o f equation (14). In the limit/ao = 1, the obtained frequencies are:
Vol. 42, No. 3 I.O
237
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS /z{o-)
I.O5
130
r
~
~
~o
A simple calculation, to first order in the small parameter 60m 60a/60~, shows that go(-) ~- 1 + 4 n 60a 3' M8
(21)
600 (600 + 601)
ne lad nn Z u.I <
--IOO
:::) tad n.-150
which is larger than unity. For tto > ta~-), the surface polariton is entirely contained inside the upper frequency gap of the bulk polariton. For go < go(-), the surface polariton is located in the same frequency range as the bulk polariton. Figure 3 shows that for go < / ~ - ) , the surface polariton starts on the bulk polariton dispersion relation and converges asymptotically to w S, p( - ) (~.wait7 ,(-) < 60(3)). This result agrees with the magnetistatic limit calculated by Camley [10] for #o = 1 where the --fJ x Ho direction surface mode in the Voigt configuration is below the bulk mode. When the surface polariton exists in the same frequency domain as the bulk polariton, the former radiates energy into the bulk mode. The resulting leaky surface polariton acquires a finite lifetime. Figure 4 shows the wavenumber kvO) for the frequency ~2O ) as a function of go. It is also simple to show that /a~*) = 1
4 lr 60a 7Ms 600 (601 - 600)
(22)
which as indicated previously is smaller than unity. Surface ferr omagnetic polaritons have been observed using Brillouin back scattering experiments [ 1 1] from an opaque material. That experiment, as well as micro60 (2) lue = z- 600 + [60~ + 60.60o] ½. (18) wave resonance [1 2], is limited to wave numbers far from the line of light. In the above magnetostatic limit, the This result agrees with the frequencies obtained by proposal of Camley [ 10] for the surface antiferromagnetic Camley [ 10] for the long-wavelength surface spin waves polariton is still waiting for experimental observation. Our of a uniaxial antiferromagnet treated in the magnetostatic treatment of this problem has added the changes to be limit where retardation is neglected. It is easily verified expected in the dispersion relation of the surface polariton that as a result of retardation. We suggest a study of the sur÷ face antiferromagnetic polaritons for a material like 60~ < 60(~), (19a) FeF2 [3] where the characteric frequency 60t is in the far 60~-) < 60(4). (19b) infrared and where bulk antiferromagnetic polaritons have been observed in a high resolution far-infrared laser For tto ~ 1, it may be shown using simple algebra that transmission experiment [3]. In order to probe the sur60~) > 60(1). (20) face mode directly, we suggest using attenuated total reflection [13] or a grating method [13] on FeF2 in It is concluded that for the + k direction, the lower branch surface polariton is entirely located inside the gap the far-infrared frequency domain. These methods allow of the bulk polariton for physically accessible parameters for changing the wave number parallel to the suface and measure the effect of retardation. (/.to 1> 1). The situation for the upper branch in the - - k direction is more complex and is now studied in A c k n o w l e d g e m e n t s - This work was partially supported detail. by the National Sciences and Engineering Research Let got-) be the pole of G(60(a)). If go goes to/~o (-) Council of Canada and le Fonds FCAC du Qu6bec. from above, the corresponding starting wave number One of the authors (C.S) wishes to express his gratitude kvO) approaches -- oo. Remembering that the waveto the Department of Physics of the Universit6 de Sherbrooke for their hospitality while he was on leave number of the bulk polariton goes to -- oofor frequencies from the Nanjing Institute of Technology, China. close to 600); for go = go(a), the surface polariton for 6o = 60(3) joins up with the bulk polariton at k -+ -- o o . Fig. 4. The reduced wavenumber kvO) as a function of go. The parameters are the same as in Fig. 2.
238 1. 2.
3. 4. 5. 6.
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS Vol. 42, No. 3 REFERENCES D.L. Mills & E. Burstein,Rep. Prog. Phy~ 37, 817 (1974) and references contained therein. E. Burstein, A. Harstein, T. Schoenwald, A.A. Maradudin, D.L. Mills & R.F. Wallis, Polaritons (Edited by E. Burstein and F. de Martini). Pergamon, New York (1974). R.W. Sanders, R.M. Belanger, M. Motokawa, V. Jaccarino & S.M. Rezende, Phy~ Rev. B23, 1190 (1981). K.L. Kliewer & R. Fuchs, Phy~ Rev. 1444, 495 (1966). A. Hartstein, E. Burstein, A.A. Maradudin, R. Brewer & R.F. Wallis,J. Phy~ C6, 1266 (1973). A.D. Karsono & D.R. TiUey,J. Phy~ C11,3487 (1978).
7. 8. 9. 10. 11. 12.
13.
M. Marchand & A. Caill6, Solid State Commun. 34,827 (1980). S.M. Bose, E.N. Foo & M.A. Zuniga, Phyg Rev. B12, 3885 (1975). C. Kittel, Phy~ Rev. 82, 565 (1951). R.E. Camley, Phy~ R ev. Lett. 45,283 (1980). J.R. Sandercock & W. Wettling, J. Appl Phy~ 50, 7784 (1979). J.A. Duncan, B.E. Storey, A.D. Tooke & A.P. Crack_nell, ./. Phy~ C (Solid State Phys) B, 2079 (1980); S.C. Kondal & M.S. Seehra, Phys. Rev. B22, 5482 (1980). A. Otto, FestkOrperprobleme (Edited by H.J. Quiesser), Vol. XIV, pp. 1-37. Pergamon Vieweg (1974).
0038-1098/82/150233-06503.00/0 Pergamon Press Ltd.
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS C. Shu* and A. Caill6 D6partement de physique, Universit6 de Sherbrooke, Sherbrooke, Qu6bec, Canada, J 1K 2R1 (Received 30 October 1981 by M.F. Collins)
The dispersion relation of surface polaritons for a uniaxial antiferromagnet are calculated taking account of retardation. In the presence of an external magnetic field, one finds that inequivalent propagation occurs in the Voigt configuration(+ t~ x Ho). For a high frequency permeability smaller than a critical value, the surface polaritons in the --t~ × Ho direction, where ~ is the normal to the interface, merges in the same frequency domain as the bulk polaritons.
THE EFFECT of dipolar magnetic fields, resulting from the spin fluctuations themselves, on the dispersion relation of long wavelength magnons in magnetically ordered ferromagnetic and antiferromagnetic crystals has been discussed by various authors [ 1]. The resulting magnetic polaritons is the analogue of the wellknown phenomenon of coupling between photons and optical phonons (dielectric polariton) in an ionic crystal [2]. There exists now clear experimental evidence for the existence of a bulk antiferromagnetic polariton as obtained recently by Sanders et al. [3] in a high resolution far-infrared transmission study of the antiferromagnetic resonance of FeF2. Moreover, as a tool to study surfaces, the complementary surface polaritons, both electric [4] and ferromagnetic [5, 6], have been discussed for semi-infinite and lamellar systems. Hartstein et al. [5] have studied the surface magnetic transverse electric (TE) polariton in a uniaxial ordered ferromagnet, with the external magnetic field and the magnetization parallel to the surface. They found, for the Voigt configuration where the wave vector k is perpendicular to the extemal magnetic field Ho, that the surface polaritons exist between the two branches of the bulk polariton dispersion relation and made evident the inequivalent propagation in the -+t~ x tto directions where ~ is the outside normal to the surface. Karsono et al. [6] studied the surface magnetic polaritons in a ferromagnetic slab in the same geometry as Hartstein et al. [5] and showed that the reciprocal nature of the propagation with respect to direction is restored. More recently, Marchand et al. [7] studied the dispersion relations and reflection symmetry properties of the guided magnetic polaritons for a uniaxial ordered ferromagnetic slab in the Voigt configuration. * Permanent address: Nanjing Institute of Technology, China. 233
In this communication, we study the dispersion relations for surface antiferromagnetic polaritons in a uniaxial ordered antiferromagnetic material in the Voigt configuration. Figure 1 represents schematically the physical system with the choice of axes. The basic limitations of the model are the following: (a) we consider surface excitations with penetration depth sufficiently large that pinning effects of the surface may be neglected and that the bulk magnetic permeability may be used, and (b) excitations with wave number Ikl small enough that the dispersion of the magnons arising from exchange may be neglected so that the energy of the spin excitation receives its dominant contribution from the external Zeeman field and the demagnetizing fields set up by the spin motion. In the study of the magnetic polaritons, the fluctuations of magnetization (in the x - y plane) are related to the fluctuations of dipolar electromagnetic fields using the transverse susceptibility response. This response is obtained using the linearized torque equations for the two sublattices where exchange dispersion terms are neglected. The gyromagnetic permeability tensor for the uniaxial antiferromagnet [8] in an external static magnetic field Ho = 11o2 is:
la=
,--/.t.y
#..
0
\o
0
/g~
,
(i)
where tt~x = Uo[1 + 2 ~ m w a ( ~ ] -- co~ -- 6o2)/Aa ],
(2a)
tlx~ = 4il~060ra ~OOCOa6olA A ,
(2b)
lazz = /a0.
(2c)
The intrinsic frequencies con, ~a and Wo are given by 4~rTMs/~t0, "/Ha and 7//o respectively where Ha is the uniaxial anisotropy field in the z-direction, Ms[Iz o is the
234
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS Vol. 42, No. 3
magnetic dipolar field outside the crystal which is linked to the field inside using the usual boundary conditions. For the TE surface modes, in the Voigt configuration, we look for solutions of equation (5) of the form:
Y
HI = (HlxY¢ + H i , p ) exp (-- t~l y) exp i(kx -- 600,
y > 0, H~ = (H2x~ + H2~p) exp (a2y)ex p i(kx -- 600,
VOCUUm
y < O:
X
/~oo ~ 41z
(6a)
~
(6b)
Substitution of equations (6) into equations (5) leads to: a l = k 2 -- 602/c2,
(7a)
a~ = k 2 -- ~e602/ c 2,
(7b)
antiferromaqnet
where/av is the magnetic permeability in the Voigt configuration
Z
Fig. 1. Schematic representation of the physical system. The semi-infinite space y > 0 is empty. The semi-infinite space y < 0 is occupied by a two sublattice antiferromagnetic material o f sublattice magnetizations M l z and M2~ and scalar dielectric constant. screened saturation magnetization in the z-direction and ~, is the absolute value of the gyromagnetic ratio. #o is the high frequency permeability constant (/ao > I) caused by magnetic dipolar excitations other than the sublattice magnetic spin wave excitations (e.g. optical magnons). 6ol is the antiferromagnetic resonance frequency [9], 601 = 60a(60. + 260e),
(3)
where we is the exchange frequency (tOe 7He) for the z-component of the exchange field equal to He. The denominator AA is:
/4, = /axx +/a~r t-txx
[0.)2 -- (60 -- O.)0)2110)22 -- (60 + 600)21 [601
(4)
Since no attenuation mechanism is invoked, the longitudinal response/axx is purely real while the transverse response/axy is purely imaginary. Note that equation (1) gives the frequency dependence of the magnetic response probing only the long wavelength spin wave excitations. The dipolar induced magnetic fields of the antiferromagnetic polaritons have to satisfy Maxweil's equations which for the Voigt configuration of Fig. 1 become:
0)2] + 2CO 60o('O2 __602 __60
and
(8)
6o~ = 602 + 260m60a.
(9)
Unlike the ferromagnet in the Voigt configuration [5], the generalized Voigt permeability/av(60 ) has two poles and two zeros which are denoted by (coO), 60(3)) and (60(2), 60(4)) respectively, 6020,3) = (601 + 603 + 60m60a) + [46026002
=
£~.4 = [602 _(60 + c00)2][co2 _(60 _60o)21.
O) ][602 __(60 +
+ 460m60am2 + C°mWal" 2 .2,1/2,
60(2,4) = + 600 + [603 + 260,nWa]1/2.
V:H-- V(V- H)
c2 at 2 = 0, c2 at 2
- 0,
y>0, y ~<0,
(5a) (5b)
where we have retained retardation. For surface excitations, the spin motions inside the crystal set up a
(10b)
Frequency gaps (gv < 0) for the bulk antiferromagnetic polaritons exist between the frequencies 60o) to 60(2) and 60(3) to co(4) in agreement with the calculations of Bose et al. [8]. For most uniaxial antiferromagnets 60ra60a/601 "~ 601, such that the frequency gaps are very narrow on the scale of the characteristic frequency 601. The frequency gaps arise because of the photonmagnon coupling and the induced magnetic displacement currents in analogy with the Lyddane-SachsTeller effects in ionic crystals [2]. The dispersion relation of the surface modes, obtained from the divergence equations
1 02H
V2H--V(V'H)
(10a)
V'H1 = 0
y>0,
(lla)
V'(pH2) = 0,
y~<0,
(lib)
and the boundary conditions (at y = 0), H l x = H2x, HIt
= -- ~txrH2x + btxxH2y ,
(12a) (1 2b)
Vol. 42, No. 3
SURFACE MAGNETIC POLARITON ON UNIAXIAL ANTIFERROMAGNETS
235
S2
i{
'..~ ".
"0
1.5
U"
# •
~., " •
x
.....
"
'
D~
k". "~
,/'/" . . . . . . .
,/
!~,,,K,~(,t a I
....
,',//
l
,~o
X
/
c~2)1-J.o
..
"\
/
.3,
... ,,
/
,p ................
..
."
/,,
• N~.
."
'. '\
/"
". '.
~ '.; \/. i - I REDUCED
...............................
/
/ //~@ /~\
-0.5 N
-2
/
i /
\.
-3
,2
- ( 4
...................
\
///"
." / ..'~'
';
I
0 WAVENUMBER
I K
I 2
-"
K 3
Fig. 2. Schematic representation of the dispersion relation of the surface antiferromagnetic polariton. The frequencies and wave-numbers are expressed in reduced units 12 = co/co~ and K = c k / c o l . The full drawn curves are for the surface modes where the dashed lines represent the top two bulk modes. The dotted lines (I2 = + K ) are the lines of light for the external medium and the dashed-dotted curves (I2 = + K [ x / e ) are the lines of light for the medium of dielectric constant e. The physical parameters used are H a = 8 5 x 103 Gauss,H~ = 5.2 x 10 $ Gauss, Ms = 60 x 10 2 (De,Ho = 1.0 x 104 Gauss, e = 4.0 and/~o = 1.25. For the sake of clarity, the forbidden gaps for the bulk polaritons are enlarged. is given by or2 = -- (xl #v -- ik#x~, /laxx .
Non-radiative surface wave solutions exist when (13)
o q ( k c o ) and ot2(kco) are larger or equal to zero. In the
frequency-wave number plane (co, k), there exist two such regions; both are bounded on the low frequency side by the poles of/h,(co) (60(1) and coO)), on the high frequency side by the dispersion curves of the bulk k2c 2 polaritons [a2(kco) = 0] and on the left-hand side (for co2 - {[#2oA(c°)-- B ( c o ) l [ l a ~ A ( c o ) - - etaoC(co)] the positive k mode) by the line of light oq(kco) = 0. In 2 2 2 2 2 contrast to the isotropic dielectric where the surface - - 32/.tocom~ocoaco polariton is located inside the gap of the bulk polariton, g 8UOcomCOaCOo~[(1 + ez)/.tgC2(co) in a gyromagnetic material, it is not limited to this frequency domain. Symmetrical regions exist for the - e u o C ( ~ X a ( ~ ) + u~,a(~))p'2}/ positive and negative wavenumber k. Comparison of -- 16#ocomcoacooco }, (14) equations (8) and (15c) shows that the poles of ~(co) are the roots of C(co). It is then easy to see that the where starting coordinates of the surface antiferromagnetic A(co) = [~22 --(co -- coo)a][co~ --(co + coo)a], (15a) polariton in the frequency gaps which are (kv(1), coO)) a(co) = [co~ - ( ~ -coo)2][co~ -(co + coo)2], (15b) and (kvO), 6o(3)) respectively for the lower and upper gaps, are given by: Using equations (7) and (13), the dispersion relation of the surface antiferromagnetic polariton is:
c(co) = a(co) + 2 co,,, co. ( co~ - ~o~ - coal.
(15c)
236
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS
Vol. 42, No. 3
S2 1.10951 %
m
N~\~
I.10949'
C= m
"11 m 0 I"11 z ..<
.10947-
I K
-30
I
\LI\0945-
Reduced wavenurnber-20 K
-tO
Fig. 3. Bulk and surface polaritons near the upper gap in the -- k direction. The full drawn curve is the surface polariton and the dashed curve is a bulk polariton. The parameters used are the same as in Fig. 2 except that now #o = 1.0 1.
k$o = 60O)G(60"))/c,
(16a)
k(,,~) = _ 60(~)G(60(~))/c,
(16b)
G2(60) =
~u]A(60)[ts~A(eo) -- B(60)] -- 32t~o260:m60~60~w2 -- 64~o60ra60a60o60
(17)
Figure 2 is a sketch of the dispersion relation for surface polaritons i n M , F2 where/.t o = 1.25. The non-reciprocal nature of the propagation in the fz x Ho and -- ~ x Ho directions is quite evident. In and near the lower frequency gap, along the + k direction, the dispersion curve starts at (60(1), k(vO) and goes asymptotically to 60 = 60.(~) when k -* ~o. Along the -- k direction, for the same frequency region, there is only a small segment starting at ~wv('.'(2),-vz'(2)'~son the line o f light 60 = -- c k and ending on the dispersion relation for the bulk polariton. This mode has a spin precession in the non-resonant circular polarization and is driven by its large photon content (near the lines o f light). In (and near) the upper gap, along + k direction, there is only a small segment
t .(4), '~v v(4)~) on the line of light 60 = c k and starting at ~'~v ending on the dispersion curve o f bulk polariton. For the same frequency region, along the - k direction, the curve starts at (60(3), kv(3))being asymptotic to co(s~,) when k ~ -- o~. Harstein et al. [5] have discussed the nonreciprocal property of the surface polaritons of the semiinfinite ordered ferromagnet. Unlike the ferromagnet, the uniaxial antiferromagnet, has two sublattices with opposite magnetization. If we consider each of these sublattices as the "lattice o f a ferromagnet", each o f them would have its own non-reciprocal dispersion curves, so we should have two sets of these curves in (and near) the lower and upper gaps for an antiferromagnet .Because o f the opposite orientations of the two sublattice magnetizations in the external magnetic field H0, the non-reciprocal properties of these two sets o f curves are opposite to each other. The asymptotic frequencies 60(~) (k ~ +- oo) for the surface polaritons are given by the poles of the right hand side o f equation (14). In the limit/ao = 1, the obtained frequencies are:
Vol. 42, No. 3 I.O
237
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS /z{o-)
I.O5
130
r
~
~
~o
A simple calculation, to first order in the small parameter 60m 60a/60~, shows that go(-) ~- 1 + 4 n 60a 3' M8
(21)
600 (600 + 601)
ne lad nn Z u.I <
--IOO
:::) tad n.-150
which is larger than unity. For tto > ta~-), the surface polariton is entirely contained inside the upper frequency gap of the bulk polariton. For go < go(-), the surface polariton is located in the same frequency range as the bulk polariton. Figure 3 shows that for go < / ~ - ) , the surface polariton starts on the bulk polariton dispersion relation and converges asymptotically to w S, p( - ) (~.wait7 ,(-) < 60(3)). This result agrees with the magnetistatic limit calculated by Camley [10] for #o = 1 where the --fJ x Ho direction surface mode in the Voigt configuration is below the bulk mode. When the surface polariton exists in the same frequency domain as the bulk polariton, the former radiates energy into the bulk mode. The resulting leaky surface polariton acquires a finite lifetime. Figure 4 shows the wavenumber kvO) for the frequency ~2O ) as a function of go. It is also simple to show that /a~*) = 1
4 lr 60a 7Ms 600 (601 - 600)
(22)
which as indicated previously is smaller than unity. Surface ferr omagnetic polaritons have been observed using Brillouin back scattering experiments [ 1 1] from an opaque material. That experiment, as well as micro60 (2) lue = z- 600 + [60~ + 60.60o] ½. (18) wave resonance [1 2], is limited to wave numbers far from the line of light. In the above magnetostatic limit, the This result agrees with the frequencies obtained by proposal of Camley [ 10] for the surface antiferromagnetic Camley [ 10] for the long-wavelength surface spin waves polariton is still waiting for experimental observation. Our of a uniaxial antiferromagnet treated in the magnetostatic treatment of this problem has added the changes to be limit where retardation is neglected. It is easily verified expected in the dispersion relation of the surface polariton that as a result of retardation. We suggest a study of the sur÷ face antiferromagnetic polaritons for a material like 60~ < 60(~), (19a) FeF2 [3] where the characteric frequency 60t is in the far 60~-) < 60(4). (19b) infrared and where bulk antiferromagnetic polaritons have been observed in a high resolution far-infrared laser For tto ~ 1, it may be shown using simple algebra that transmission experiment [3]. In order to probe the sur60~) > 60(1). (20) face mode directly, we suggest using attenuated total reflection [13] or a grating method [13] on FeF2 in It is concluded that for the + k direction, the lower branch surface polariton is entirely located inside the gap the far-infrared frequency domain. These methods allow of the bulk polariton for physically accessible parameters for changing the wave number parallel to the suface and measure the effect of retardation. (/.to 1> 1). The situation for the upper branch in the - - k direction is more complex and is now studied in A c k n o w l e d g e m e n t s - This work was partially supported detail. by the National Sciences and Engineering Research Let got-) be the pole of G(60(a)). If go goes to/~o (-) Council of Canada and le Fonds FCAC du Qu6bec. from above, the corresponding starting wave number One of the authors (C.S) wishes to express his gratitude kvO) approaches -- oo. Remembering that the waveto the Department of Physics of the Universit6 de Sherbrooke for their hospitality while he was on leave number of the bulk polariton goes to -- oofor frequencies from the Nanjing Institute of Technology, China. close to 600); for go = go(a), the surface polariton for 6o = 60(3) joins up with the bulk polariton at k -+ -- o o . Fig. 4. The reduced wavenumber kvO) as a function of go. The parameters are the same as in Fig. 2.
238 1. 2.
3. 4. 5. 6.
SURFACE MAGNETIC POLARITONS ON UNIAXIAL ANTIFERROMAGNETS Vol. 42, No. 3 REFERENCES D.L. Mills & E. Burstein,Rep. Prog. Phy~ 37, 817 (1974) and references contained therein. E. Burstein, A. Harstein, T. Schoenwald, A.A. Maradudin, D.L. Mills & R.F. Wallis, Polaritons (Edited by E. Burstein and F. de Martini). Pergamon, New York (1974). R.W. Sanders, R.M. Belanger, M. Motokawa, V. Jaccarino & S.M. Rezende, Phy~ Rev. B23, 1190 (1981). K.L. Kliewer & R. Fuchs, Phy~ Rev. 1444, 495 (1966). A. Hartstein, E. Burstein, A.A. Maradudin, R. Brewer & R.F. Wallis,J. Phy~ C6, 1266 (1973). A.D. Karsono & D.R. TiUey,J. Phy~ C11,3487 (1978).
7. 8. 9. 10. 11. 12.
13.
M. Marchand & A. Caill6, Solid State Commun. 34,827 (1980). S.M. Bose, E.N. Foo & M.A. Zuniga, Phyg Rev. B12, 3885 (1975). C. Kittel, Phy~ Rev. 82, 565 (1951). R.E. Camley, Phy~ R ev. Lett. 45,283 (1980). J.R. Sandercock & W. Wettling, J. Appl Phy~ 50, 7784 (1979). J.A. Duncan, B.E. Storey, A.D. Tooke & A.P. Crack_nell, ./. Phy~ C (Solid State Phys) B, 2079 (1980); S.C. Kondal & M.S. Seehra, Phys. Rev. B22, 5482 (1980). A. Otto, FestkOrperprobleme (Edited by H.J. Quiesser), Vol. XIV, pp. 1-37. Pergamon Vieweg (1974).