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Transverse surface and slab modes in semiconductor superlattices
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Solid State Communications, Vol. 55, No. 4, pp. 373-376, 1985. Printed in Great Britain.
TRANSVERSE SURFACE AND SLAB MODES IN SEMICONDUCTOR SUPERLATTICES N. Raj and D.R. Tilley Department of Physics, University of Essex, Colchester CO4 3SQ, UK (Received 8 April 1985 by R.A. Cowley)
It is shown from the exact equations for transverse electromagnetic waves propagating in a superlattice that in the long-wavelength limit the superlattice has the optical properties of a conventional uniaxial medium. This result is used to derive the dispersion equations for polaritons at single and double interfaces between superlattices and ordinary media. THE RECENT PUBLICATION by Zucker et al. [1] of Raman scattering data from modes in the reststrahl frequency region of GaAs-Gat_xAlxAs supedattices has aroused interest in the so-called slab modes of superlattices. Zucker et al. refer to the well-known work of Fuchs and Kliewer [2]. They, however, deal only with slabs whose dielectric function is isotropic. The macroscopic symmetry of a superlattice is uniaxial, with the normal to the layers as the uniaxis. The superlattice is therefore birefringent, and the slab modes differ significantly from those of an isotropic material. We give here some general results for the transverse slab and surface modes in the long-wavelength limit ~ >> A, where ), is the optical wavelength and A the superlattice period. We start from the exact formal results of Yeh, Yariv and Hung [3-5] for optical propagation in an infinite superlattice. We take the z-axis as the normal to the layers, which have thicknesses dr, d2 and isotropic dielectric functions ex, e2, both frequency-dependent. The spatial period is A = d ~ +d2. The dispersion equation for a mode of wavenumber (qx, q,) is [3, 4]
implications of this for the surfaces of level frequency in the (qx, qz) plane are discussed in [5]. In general both s- and p-waves are extraordinary in the sense that the effective refractive index depends upon the direction of propagation. Equation (1) can be simplified usefully for q~A< 1, q x A < 1, k t A < l and k 2 A < l , in which case the surfaces of level frequency do not approach the zone edges. Expansion of (1) then gives for s-waves
cos q z A = cos k l d t cos k 2 d
with
2
q2 + q2 = (w2/c2)(eldl + e2d2)/(dx + d2),
and for p-waves
q~(dl + d2) 2 + q~(eld~ + e2d2)(e2dx + eld~)/ete2 = (w2/e2)(etdt + e2dD(dt + d2).
q, = O,
--gs,p sin kxdt sin k2d2, (1)
etdt+e2d2
= 0,
(7)
propagating strictly parallel to the layers, and another solution -t 2 -1 2 exxqz + ezzqx = co2/c 2,
exx = (etdl + e2d2)l(dt + d2),
(8)
(9)
(2)
ezlz = (e~td, + e~ld2)/(dt + d2).
gs = ½(k2/kl + kt/k2),
(3)
gp = ½(e2ka/eak2 + elk2/e2kl).
(4)
It is seen from (5) and (8) that in the limit under discussion the superlattice has the properties of a conventional uniaxial medium, with principal values of the dielectric tensor e~z along the uniaxis and exx in the transverse plane. The s-waves are ordinary, and the p-waves extraordinary. Equations (9) and (10) were derived long ago by Rytov [6] for the particular case of propagation in the x- and z-directions; they give good agreement with infrared reflectivity measurements in the reststrahl frequency region. In the rest of this paper we consider only the long wavelength limit, so that the dielectric properties of the superlattice are modelled by (9) and (10).
kt~
=
(eata)2[C 2 --
for t~ = 1,2,
(6)
Equation (6) has one solution
where qz2)1/2
(5)
The factor gs applies for s-polarisation (E in y-direction) and gp applies for p-polarisation (E in x - z plane). It should be noted that the modes described by (1) are transverse, since the electric field amplitude within each layer is given by a linear combination of the transverse solutions of Maxwell's equations. In (1), q~ is a Bloch wave number, and (1) has the property that Brillouin zone edges with associated frequency stop-bands appear at values qz = mr/A. The 373
(10)
374
TRANSVERSE SURFACE AND SLAB MODES
The frequency dependence of ezz and exx is central to a discussion of the slab modes. It is seen that exx has a pole where either el or e2 has a pole, while ezz has a zero where either el or e2 has a zero. For the particular case of GaAs and Gal_xAlxAs the frequency dependence of e~ and e2 in the reststrahl region was established from infrared reflectivity measurements by Kim and Spitzer [7]. In both cases the frequency dependence is well fitted by simple resonance expressions with small damping; GaAs has a single resonance at 269.2 cm -1 , while Ga~-xAlxAs exhibits two-mode behaviour with slightly x-dependent resonant frequencies around 265 and 360 cm -~. Graphs to illustrate the frequency dependence of exx and ezz are shown in Fig. 1. In this figure and subsequently we neglect the small damping [7] of e~ and e2. This proximity of the poles of the dielectric functions of GaAs and Gal_xAl~As around 2 7 0 c m -1 leads to a very rapid variation with frequency of exx, and in consequence the dielectric properties of the superlattice are highly anisotropic between 260 and 300 cm -~ . There is also substantial anisotropy in the vicinity of the second Gal _ xAl~As pole at 359 cm -~ . We now consider a surface polariton on a semiinfinite superlattice in which the surface is one of the superlattice planes. The dispersion relation for this case is [8-101
(02 ezz(exx -- eM) q~ = - ~ - e M , e x x e z z - - e2M
(11)
where eM is the dielectric constant of the bounding medium and qx is the wave number in the surface plane. The form of the dispersion curve is illustrated in Fig. 2, drawn like Fig. 1 for d~ = d2 and x = 0.14. It is seen that because of the complicated frequency dependences of exx and e~z there are a number of distinct branches on the dispersion curve. It should be noted that two of the branches (the fourth and the sixth counting upwards in frequency) have the property that co is a decreasing function of qx. Figure 2 was obtained essentially by careful numerical exploration, and we found that all the branches extend from the light line q~ = eM(0~/C 2 to infinity. Thus with the parameters chosen we have not found the virtual-excitation surface polaritons that may occur on anisotropic media [8]. Superlattice specimens are often deposited on substrates of Gal_ xAlxAs. It is therefore of interest to derive the special case of (11) when eM is equal to the dielectric constant of one of the superlattice media, eM = e2 say. Equations (9), (10) and (11) then give (0 2
ele2
q~x -- c 2 el + e 2 "
(12)
It may be noted that this is independent ofd~ and d2.
Vol. 55, No. 4
Some interest attaches to the limit q2x ~ ~ of (11) or (12), since this gives the surface modes in the nonretarded limit. It is seen that for (12) these frequencies are independent of the values of dl and d2, being given by the zeros of el + e 2 . For a GaAs-Gal_xAlxAs superlattice with x = 0.14, for example, the data of [7] gives these frequencies as 267.9, 290.7 and 362.5 cm -1 . We may now turn to slab modes, still retaining the long-wavelength limit of (9) and (10). We consider a uniaxial medium b of thickness L bounded by uniaxial media a and c; this is slightly more general than in the experiments of [1], where the bounding media were cubic. For our purposes, it is sufficient to consider the special case where the dielectric principal axes of all media are parallel, and in addition the normal to the interface as well as the direction of propagation of the wave coincide with principal directions. The dispersion equation of the (TM polarized) slab polaritons is then [101 a
b
c
b
(exxkb -- exxka)(exxkb -- exxkc) + e~:,ke) ( e ~ x k b + ex,~ka)(e:,xkb b c
exp (--2ikbL ),
(13)
where
k] = ejzz
__
j
e2 e z z - - q x
2
for
] = a,b,c.
(14)
In order for (13) to describe a mode localised in the slab, the frequency and wavenumber qx must be such that k a and k e are both imaginary:
q2x > (02~z/c~
for
] = a,c.
(15)
k b can be real or imaginary, the former corresponding to guided wave polaritons [ 1 1 - 1 3 ] and the latter to double-interface polaritons. For imaginary kb, (13) factorises as L ~ ~ into
elxxkb--ebxxkj = 0
for
] = a,c.
(16)
This equation, from which (11) can be derived, describes a single-interface polariton at one or other of the bounding surfaces of medium b. For imaginary k b and large L, IkbLI >> 1, (13) describes a tight-binding type of interaction between the surface polaritons at the upper and lower interfaces. These polaritons therefore form bonding and anti-bonding combinations, one of which has a lower and one a higher frequency than the singleinterface polariton. This is well-known for isotropic media [12, 13]. In the unretarded limit q~ ~ , (14) shows that Ikbl ~ , so it is clear from (13) that the bonding and anti-bonding frequencies approach the limit found for the single-interface polariton. However, the approach to the limit is rather slow, as is seen from [121 and [13].
TRANSVERSE SURFACE AND SLAB MODES
Vol. 55, No. 4 20O
375
200
Ex:,c, ~zz
Ezz
100
100
i
0
f!3~o
30
(,'
I
j]
j
2so
~oo
~/2rtc
Iso
2~o
o/2rrc (tin-))
(cm-))
_100I
-200
Fig. 1. (a) Frequency dependence of ex~ (solid line) and (dashed line). (b) exx on expanded frequency scale. (c) ezz on expanded frequency scale. All curves drawn for d l = d2 a n d x = 0.14 and for zero damping.
Fig. l(a).
ezz
2°01
380
t
I
I
100
/
/
i1
360 /
350 310
I I
I I
30(
280 .)50
iI
to
2rr--~ (cm-')
I /
2dO
I
tol2rrc (cm-'}
/
I
/ 280
i
-100
275
Fig. l(b). For a symmetrically bounded slab, medium a = medium c, (13) factorises into equations representing an even and an odd mode. With /ca = iota, k b = it~b, the results are eax Otb
i
325
350
375
i
~-00
I
i
/*25 ~50 ~ x/2rr (cm-)}
Fig. 2. Dispersion curves, 6o vs. qx, for the interface between a semi-infinite superlattice and vacuum. The curves are drawn from equations (9), (10) and (11) with dl = d 2 , x = 0.14 and eM = 1. The dashed line shows the vacuum light line q~ = 6o'~/cz. Note the break in the vertical scale between 310 and 3 5 0 c m -1.
.200
e~xo~
i
300
{ tanh abL)"
(17)
coth (½
In this note we have discussed the slab modes, that
is to say, the polaritons, of a superlattice within the long-wavelength approximation described above (5) and (6). Even in this limit, because of the complicated frequency dependence of the effective dielectric function, illustrated in Fig. 1, the dispersion graphs for the polaritons are quite complicated, as illustrated in Fig. 2. At this stage comparison with experiment is not really
376
TRANSVERSE SURFACE AND SLAB MODES
possible. First we have dealt only with transverse modes, whereas the experiments may also detect longitudinal modes. It seems to us that a proper theoretical account of longitudinal modes of a superlattice is quite difficult, and is not yet available. One might speculate that in the long-wavelength limit the longitudinal modes would be those of an anisotropic medium with the dielectric tensor of (9) and (10), and that (7) represents a longitudinal mode propagating parallel to the layers. However, this is not really adequate, since as noted (9) and (10) were derived from a treatment which from the outset was based only on transverse modes. A second difficulty concerns the nature of the specimens, which typically comprise a substrate, usually Gal_xAlxAs, on which first the superlattice then a capping layer of Gal _ x,Alx,As are deposited. The capping layer is not thick on the scale of the wavelengths involved, so even with the approximate characterization of the dielectric properties of the superlattice that we have used, it would be necessary to treat the four-layer system substratesuperlattice-capping layer-vacuum. It would in fact be very helpful if experiments could be performed on a three-layer system such as substrate-superlattice-vacuum or substrate-superlattice-thick capping layer, since (13) would then apply. Finally, it will be recalled that the light-scattering results show strong polarization selection rules; since we have dealt only with dispersion curves we have no useful comments on these selection rules. A number of theoretical questions remain to be answered. Both the single- and double-interface polaritons should be investigated for all frequencies and wavenumbers, not just those for which a long-wavelength approximation holds. The dispersion equation would be found from a semi-infinite or finite set of difference equations for the field amplitudes in each component of the superlattice; the techniques for dealing with such sets of equations are well known from work on surface phonons [15] and magnons [16]. Yeh et al.
Vol. 55, No.4
[3] identify band-gap related surface modes which arise at frequencies in the stop bands even when both dielectric constants are positive; the relationship between these and the surface-polariton type modes that have been discussed here need clarification.
Acknowledgements - We have benefitted from discussions with many colleagues, including J. Sapriel, M. Babiker, S.R.P. Smith, H. Abu Hassan and B.K. Ridley. NR is grateful for support from a SERC studentship. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.E. Zucker, A. Pinczuk, D.S. Chelma, A. Gossard & W. Wiegmann,Phys. Rev. Lett. 53, 1280 (1984). R. Fuchs & K.L. Kliewer, Phys. Rev. 140, A2076 (1965). P. Yeh, A. Yariv & C.S. Hong, J. Opt. Soc. Am. 67,423 (1977). A. Yariv & P. Yeh, Optical Waves in Crystals, Wiley, New York (1984). A. Yariv & P. Yeh, J. Opt. Soc. Am. 67, 438 (1977). S.M. Rytov, Soy. Phys. JETP 2, 466 (1956). O.K. Kim & W.G. Spitzer, J. Appl. Phys. 50, 4362 (1979). A. Hartstein, E. Burstein, J.J. Brion & R.F. Wallis, Solid State Commun. 12, 1083 (1973). G. Borstel & H.J. Falge, Phys. Status Solidi (b) 83, l l (1977). G. Borstel & H.J. Falge, Appl. Phys. 16, 211 (1978). K.L. Kliewer & R. Fuchs, Phys. Rev. 144, 495 (1966). S. Ushioda, in Progress in Optics XIX, (Edited by E. Wolf), North-Holland, Amsterdam (1981). S. Ushioda & R. Loudon in Surface Polaritons, (Edited by V.M. Agranovich & D.L. Mills), NorthHolland, Amsterdam (1982). S.R.P. Smith & H. Abu Hassan, private communication. R.F. Wallis, Phys. Rev. 105,540(1957). M.G. Cottam, J. Phys. C: Solid State Phys. 9, 2121 (1976).
Solid State Communications, Vol. 55, No. 4, pp. 373-376, 1985. Printed in Great Britain.
TRANSVERSE SURFACE AND SLAB MODES IN SEMICONDUCTOR SUPERLATTICES N. Raj and D.R. Tilley Department of Physics, University of Essex, Colchester CO4 3SQ, UK (Received 8 April 1985 by R.A. Cowley)
It is shown from the exact equations for transverse electromagnetic waves propagating in a superlattice that in the long-wavelength limit the superlattice has the optical properties of a conventional uniaxial medium. This result is used to derive the dispersion equations for polaritons at single and double interfaces between superlattices and ordinary media. THE RECENT PUBLICATION by Zucker et al. [1] of Raman scattering data from modes in the reststrahl frequency region of GaAs-Gat_xAlxAs supedattices has aroused interest in the so-called slab modes of superlattices. Zucker et al. refer to the well-known work of Fuchs and Kliewer [2]. They, however, deal only with slabs whose dielectric function is isotropic. The macroscopic symmetry of a superlattice is uniaxial, with the normal to the layers as the uniaxis. The superlattice is therefore birefringent, and the slab modes differ significantly from those of an isotropic material. We give here some general results for the transverse slab and surface modes in the long-wavelength limit ~ >> A, where ), is the optical wavelength and A the superlattice period. We start from the exact formal results of Yeh, Yariv and Hung [3-5] for optical propagation in an infinite superlattice. We take the z-axis as the normal to the layers, which have thicknesses dr, d2 and isotropic dielectric functions ex, e2, both frequency-dependent. The spatial period is A = d ~ +d2. The dispersion equation for a mode of wavenumber (qx, q,) is [3, 4]
implications of this for the surfaces of level frequency in the (qx, qz) plane are discussed in [5]. In general both s- and p-waves are extraordinary in the sense that the effective refractive index depends upon the direction of propagation. Equation (1) can be simplified usefully for q~A< 1, q x A < 1, k t A < l and k 2 A < l , in which case the surfaces of level frequency do not approach the zone edges. Expansion of (1) then gives for s-waves
cos q z A = cos k l d t cos k 2 d
with
2
q2 + q2 = (w2/c2)(eldl + e2d2)/(dx + d2),
and for p-waves
q~(dl + d2) 2 + q~(eld~ + e2d2)(e2dx + eld~)/ete2 = (w2/e2)(etdt + e2dD(dt + d2).
q, = O,
--gs,p sin kxdt sin k2d2, (1)
etdt+e2d2
= 0,
(7)
propagating strictly parallel to the layers, and another solution -t 2 -1 2 exxqz + ezzqx = co2/c 2,
exx = (etdl + e2d2)l(dt + d2),
(8)
(9)
(2)
ezlz = (e~td, + e~ld2)/(dt + d2).
gs = ½(k2/kl + kt/k2),
(3)
gp = ½(e2ka/eak2 + elk2/e2kl).
(4)
It is seen from (5) and (8) that in the limit under discussion the superlattice has the properties of a conventional uniaxial medium, with principal values of the dielectric tensor e~z along the uniaxis and exx in the transverse plane. The s-waves are ordinary, and the p-waves extraordinary. Equations (9) and (10) were derived long ago by Rytov [6] for the particular case of propagation in the x- and z-directions; they give good agreement with infrared reflectivity measurements in the reststrahl frequency region. In the rest of this paper we consider only the long wavelength limit, so that the dielectric properties of the superlattice are modelled by (9) and (10).
kt~
=
(eata)2[C 2 --
for t~ = 1,2,
(6)
Equation (6) has one solution
where qz2)1/2
(5)
The factor gs applies for s-polarisation (E in y-direction) and gp applies for p-polarisation (E in x - z plane). It should be noted that the modes described by (1) are transverse, since the electric field amplitude within each layer is given by a linear combination of the transverse solutions of Maxwell's equations. In (1), q~ is a Bloch wave number, and (1) has the property that Brillouin zone edges with associated frequency stop-bands appear at values qz = mr/A. The 373
(10)
374
TRANSVERSE SURFACE AND SLAB MODES
The frequency dependence of ezz and exx is central to a discussion of the slab modes. It is seen that exx has a pole where either el or e2 has a pole, while ezz has a zero where either el or e2 has a zero. For the particular case of GaAs and Gal_xAlxAs the frequency dependence of e~ and e2 in the reststrahl region was established from infrared reflectivity measurements by Kim and Spitzer [7]. In both cases the frequency dependence is well fitted by simple resonance expressions with small damping; GaAs has a single resonance at 269.2 cm -1 , while Ga~-xAlxAs exhibits two-mode behaviour with slightly x-dependent resonant frequencies around 265 and 360 cm -~. Graphs to illustrate the frequency dependence of exx and ezz are shown in Fig. 1. In this figure and subsequently we neglect the small damping [7] of e~ and e2. This proximity of the poles of the dielectric functions of GaAs and Gal_xAl~As around 2 7 0 c m -1 leads to a very rapid variation with frequency of exx, and in consequence the dielectric properties of the superlattice are highly anisotropic between 260 and 300 cm -~ . There is also substantial anisotropy in the vicinity of the second Gal _ xAl~As pole at 359 cm -~ . We now consider a surface polariton on a semiinfinite superlattice in which the surface is one of the superlattice planes. The dispersion relation for this case is [8-101
(02 ezz(exx -- eM) q~ = - ~ - e M , e x x e z z - - e2M
(11)
where eM is the dielectric constant of the bounding medium and qx is the wave number in the surface plane. The form of the dispersion curve is illustrated in Fig. 2, drawn like Fig. 1 for d~ = d2 and x = 0.14. It is seen that because of the complicated frequency dependences of exx and e~z there are a number of distinct branches on the dispersion curve. It should be noted that two of the branches (the fourth and the sixth counting upwards in frequency) have the property that co is a decreasing function of qx. Figure 2 was obtained essentially by careful numerical exploration, and we found that all the branches extend from the light line q~ = eM(0~/C 2 to infinity. Thus with the parameters chosen we have not found the virtual-excitation surface polaritons that may occur on anisotropic media [8]. Superlattice specimens are often deposited on substrates of Gal_ xAlxAs. It is therefore of interest to derive the special case of (11) when eM is equal to the dielectric constant of one of the superlattice media, eM = e2 say. Equations (9), (10) and (11) then give (0 2
ele2
q~x -- c 2 el + e 2 "
(12)
It may be noted that this is independent ofd~ and d2.
Vol. 55, No. 4
Some interest attaches to the limit q2x ~ ~ of (11) or (12), since this gives the surface modes in the nonretarded limit. It is seen that for (12) these frequencies are independent of the values of dl and d2, being given by the zeros of el + e 2 . For a GaAs-Gal_xAlxAs superlattice with x = 0.14, for example, the data of [7] gives these frequencies as 267.9, 290.7 and 362.5 cm -1 . We may now turn to slab modes, still retaining the long-wavelength limit of (9) and (10). We consider a uniaxial medium b of thickness L bounded by uniaxial media a and c; this is slightly more general than in the experiments of [1], where the bounding media were cubic. For our purposes, it is sufficient to consider the special case where the dielectric principal axes of all media are parallel, and in addition the normal to the interface as well as the direction of propagation of the wave coincide with principal directions. The dispersion equation of the (TM polarized) slab polaritons is then [101 a
b
c
b
(exxkb -- exxka)(exxkb -- exxkc) + e~:,ke) ( e ~ x k b + ex,~ka)(e:,xkb b c
exp (--2ikbL ),
(13)
where
k] = ejzz
__
j
e2 e z z - - q x
2
for
] = a,b,c.
(14)
In order for (13) to describe a mode localised in the slab, the frequency and wavenumber qx must be such that k a and k e are both imaginary:
q2x > (02~z/c~
for
] = a,c.
(15)
k b can be real or imaginary, the former corresponding to guided wave polaritons [ 1 1 - 1 3 ] and the latter to double-interface polaritons. For imaginary kb, (13) factorises as L ~ ~ into
elxxkb--ebxxkj = 0
for
] = a,c.
(16)
This equation, from which (11) can be derived, describes a single-interface polariton at one or other of the bounding surfaces of medium b. For imaginary k b and large L, IkbLI >> 1, (13) describes a tight-binding type of interaction between the surface polaritons at the upper and lower interfaces. These polaritons therefore form bonding and anti-bonding combinations, one of which has a lower and one a higher frequency than the singleinterface polariton. This is well-known for isotropic media [12, 13]. In the unretarded limit q~ ~ , (14) shows that Ikbl ~ , so it is clear from (13) that the bonding and anti-bonding frequencies approach the limit found for the single-interface polariton. However, the approach to the limit is rather slow, as is seen from [121 and [13].
TRANSVERSE SURFACE AND SLAB MODES
Vol. 55, No. 4 20O
375
200
Ex:,c, ~zz
Ezz
100
100
i
0
f!3~o
30
(,'
I
j]
j
2so
~oo
~/2rtc
Iso
2~o
o/2rrc (tin-))
(cm-))
_100I
-200
Fig. 1. (a) Frequency dependence of ex~ (solid line) and (dashed line). (b) exx on expanded frequency scale. (c) ezz on expanded frequency scale. All curves drawn for d l = d2 a n d x = 0.14 and for zero damping.
Fig. l(a).
ezz
2°01
380
t
I
I
100
/
/
i1
360 /
350 310
I I
I I
30(
280 .)50
iI
to
2rr--~ (cm-')
I /
2dO
I
tol2rrc (cm-'}
/
I
/ 280
i
-100
275
Fig. l(b). For a symmetrically bounded slab, medium a = medium c, (13) factorises into equations representing an even and an odd mode. With /ca = iota, k b = it~b, the results are eax Otb
i
325
350
375
i
~-00
I
i
/*25 ~50 ~ x/2rr (cm-)}
Fig. 2. Dispersion curves, 6o vs. qx, for the interface between a semi-infinite superlattice and vacuum. The curves are drawn from equations (9), (10) and (11) with dl = d 2 , x = 0.14 and eM = 1. The dashed line shows the vacuum light line q~ = 6o'~/cz. Note the break in the vertical scale between 310 and 3 5 0 c m -1.
.200
e~xo~
i
300
{ tanh abL)"
(17)
coth (½
In this note we have discussed the slab modes, that
is to say, the polaritons, of a superlattice within the long-wavelength approximation described above (5) and (6). Even in this limit, because of the complicated frequency dependence of the effective dielectric function, illustrated in Fig. 1, the dispersion graphs for the polaritons are quite complicated, as illustrated in Fig. 2. At this stage comparison with experiment is not really
376
TRANSVERSE SURFACE AND SLAB MODES
possible. First we have dealt only with transverse modes, whereas the experiments may also detect longitudinal modes. It seems to us that a proper theoretical account of longitudinal modes of a superlattice is quite difficult, and is not yet available. One might speculate that in the long-wavelength limit the longitudinal modes would be those of an anisotropic medium with the dielectric tensor of (9) and (10), and that (7) represents a longitudinal mode propagating parallel to the layers. However, this is not really adequate, since as noted (9) and (10) were derived from a treatment which from the outset was based only on transverse modes. A second difficulty concerns the nature of the specimens, which typically comprise a substrate, usually Gal_xAlxAs, on which first the superlattice then a capping layer of Gal _ x,Alx,As are deposited. The capping layer is not thick on the scale of the wavelengths involved, so even with the approximate characterization of the dielectric properties of the superlattice that we have used, it would be necessary to treat the four-layer system substratesuperlattice-capping layer-vacuum. It would in fact be very helpful if experiments could be performed on a three-layer system such as substrate-superlattice-vacuum or substrate-superlattice-thick capping layer, since (13) would then apply. Finally, it will be recalled that the light-scattering results show strong polarization selection rules; since we have dealt only with dispersion curves we have no useful comments on these selection rules. A number of theoretical questions remain to be answered. Both the single- and double-interface polaritons should be investigated for all frequencies and wavenumbers, not just those for which a long-wavelength approximation holds. The dispersion equation would be found from a semi-infinite or finite set of difference equations for the field amplitudes in each component of the superlattice; the techniques for dealing with such sets of equations are well known from work on surface phonons [15] and magnons [16]. Yeh et al.
Vol. 55, No.4
[3] identify band-gap related surface modes which arise at frequencies in the stop bands even when both dielectric constants are positive; the relationship between these and the surface-polariton type modes that have been discussed here need clarification.
Acknowledgements - We have benefitted from discussions with many colleagues, including J. Sapriel, M. Babiker, S.R.P. Smith, H. Abu Hassan and B.K. Ridley. NR is grateful for support from a SERC studentship. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.E. Zucker, A. Pinczuk, D.S. Chelma, A. Gossard & W. Wiegmann,Phys. Rev. Lett. 53, 1280 (1984). R. Fuchs & K.L. Kliewer, Phys. Rev. 140, A2076 (1965). P. Yeh, A. Yariv & C.S. Hong, J. Opt. Soc. Am. 67,423 (1977). A. Yariv & P. Yeh, Optical Waves in Crystals, Wiley, New York (1984). A. Yariv & P. Yeh, J. Opt. Soc. Am. 67, 438 (1977). S.M. Rytov, Soy. Phys. JETP 2, 466 (1956). O.K. Kim & W.G. Spitzer, J. Appl. Phys. 50, 4362 (1979). A. Hartstein, E. Burstein, J.J. Brion & R.F. Wallis, Solid State Commun. 12, 1083 (1973). G. Borstel & H.J. Falge, Phys. Status Solidi (b) 83, l l (1977). G. Borstel & H.J. Falge, Appl. Phys. 16, 211 (1978). K.L. Kliewer & R. Fuchs, Phys. Rev. 144, 495 (1966). S. Ushioda, in Progress in Optics XIX, (Edited by E. Wolf), North-Holland, Amsterdam (1981). S. Ushioda & R. Loudon in Surface Polaritons, (Edited by V.M. Agranovich & D.L. Mills), NorthHolland, Amsterdam (1982). S.R.P. Smith & H. Abu Hassan, private communication. R.F. Wallis, Phys. Rev. 105,540(1957). M.G. Cottam, J. Phys. C: Solid State Phys. 9, 2121 (1976).