Plasmon-polaritons spectra in quasi-periodic superlattices

PHYSICS LETTERS A

PhysicsLettersA 181 (1993) 409—412 North-Holland

Plasmon-polaritons spectra in quasi-periodic superlattices E.L. Albuquerque Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil

Received 4 May 1993; revised manuscript received 26 August 1993; accepted for publication 30 August 1993 Communicated by L.J. Sham

A comparative theoretical analysis is carried out for the spectra of plasmon-polaritons of multiple semiconductorlayers in the periodic arrangement and in a quasi-periodic arrangement obeying the Thue—Morse sequence. We consider building blocks composed of two-dimensional electron gas (2DEG) separated by dielectric media. The spectra of both bulk and surface plasmonpolaritons are conveniently derived by using a transfer matrix treatment, with a model frequency-dependent dielectric function, including the effect ofretardation. Numerical results are presented and their features discussed.

As a result of recent advances in experimental techniques, there is a continuing interest to investigate the properties of collective excitations, such as plasmon-polaritons modes, in multilayered systems (for a review see ref. [1]). In particular, the physical properties of a new class of artificial layered media, the so-called quasi-periodic superlattice, have also attracted a lot of attention recently. The fabrication ofsuch structures was pioneered by Merlin et al. [2], who grew a quasi-periodic GaAs—AlAs superlattice from two distinct building blocks, each having one or more layers of different materials with different thickness, arranged according to a Fibonacci Sequence. Theoretical plasmon-polaritons spectra in these structures were then successfully reported by some authors [3—6]. In this communication we investigate the spectra of both bulk and surface plasmon-polaritons in another type of quasi-periodic structurewhich obey the Thue—Morse sequence. This structure has been reported by Humlicek et al. [7], which presented measurements of the reflectance spectra for the Thue— Morse GaAs—AlAs quasi-periodic layered films deposited on the GaAs substrate. They were concerned with the electron—hole excitations in this structure. Later, Tao and Singh [81 presented a comparative theoretical analysis of the reflectance and transmission spectra of multiple layers arranged in the pe~ riodic and the Thue—Morse quasi-periodic pattern,

with a good agreement between their

theory and the

experimental spectra obtained by Humlicek et al. We consider the two building blocks a and fi, as shown in fig. 1, to set up a quasi-periodic superlattice of Thue—Morse type. Each block consists of a two-dimensional electron gas (2DEG) charge sheet and a layer of medium A or B, which may have different thickness a and b, and dielectric functions aA(w) and C~(W).They may also have a volume density of charge. The Thue—Morse sequence is here described in terms of a series of generations that obey the following recursion relations, C’ C’ c’T ‘-“s = ‘‘n— I ‘~ n—I,

C’T C’T C’ 3n —1, ‘3 n = ‘~n—I ‘

____________

A 1%

____________

a

b

a

I

I

I

b

/3 .

Fig. 1. The two building blocks a and fi of the Thue—Morse quasiperiodic superlattice. Here A and B are dielectric layers, with thickness a and b, and the 2DEG layers have carrier concentra-

tions ~a and ~b per unit area.

0375-9601/93/S 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

409

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for n~lwith S0=a and S~’=/J. In order to find the bulk and surface plasmonpolaritons spectra, we follow the general formalism

2]~2, a1= [k~—e~(co/c) = [Cj(0)/C) 2_ k~]1/2,

of ref. [91, where a convenient description of the spectra is carried out by using a transfer matrix treatment to simplify the algebra, which can be otherwise

J=

quite involved. We consider a semi-infinite superlattice structure where the cartesian axes are chosen in such a way that the z-axis is normal to the plane of the layers. The structureis terminated at the plane z=0, with the half-space z ~ 0 filled with a material that has a frequency-independent dielectric function Let us assume that the electromagnetic mode is ppolarized in the absence of any external magnetic field. The 2DEG charge sheet at each interface is considered to be due to the presence of a current density

ifk~>e,w/c, if k~
exp ( a3j)

.

(9) (10)

In general, we assume for media A and B that e~may be frequency dependent, having the form e~=e~01{l [w~,/w(w+if,)]} —

,

(11)

where w~is the plasma frequency, 1 is a damping factor, and a~,is the background dielectric function. The implicit dispersion relation for the surface modes can now be obtained through the application of the usual electromagneticboundary conditions at z = 0 and at each of the interfaces between layers to yield T11

—

T22 + T1 2A T21 ,~ ‘ —

= 0,

(12)

with

J~~=iw8oa~E3~,

(2) (13)

where

2 n3e m7w2e

=

Here 0’

1= a or b.

(3)

Here, n~is the carrier concentration per unit area, e is the electronic charge, and m3 is the effective mass of the charge carrier; e0 is the vacuum permittivity. For the periodic system aflafi..., the bulk dispersion relation is simply given by [8] cos(QL)=Tr(Tap)

Here, the matrices N~and M 1

)

N~=( 1 1 \c—c~. —e;—~~ M~ =

(~ 6~

~

(1= a, b) are defined (6)

(7)

where =

410

~ /a~

are elements of the lap matrix.

We consider now the quasi-periodic Thue—Morse superlattice. The general results (4) and (12), for the bulk and surfaces plasmon-polaritons dispersion relations, respectively, still hold, provided we consider the size of the unit cell for the S,, sequence L equal to n (a+ b), and a suitable transfer matrix T5~is determined. For example, for the S2 = a/?fla Thue—Morse sequence,

(4)

,

where Q is the Bloch wavenumber, L=a+b is the size of the periodic superlattice unit cell, and Tap is a unimodular transfer matrix defined by Tap=N;’MbNc’Ma. (5)

by

Tmn

(8)

IS2 =N~TaTpTpMA,

(14)

and,=N~ in general ‘TaTPTP...TaTPTPMB, TSa

fornodd

~

= N;

(n)~3),

(15)

‘TaTpTp...TpTaTa MA,

forneven Here,

(n~~4).

(16)

T~=Nj~M1, (17) wherey=aforj=a,andy=$forj=b. A similar analysis can be carried out for the spolarized mode, where the electrical field is paralell to the y-axis [5]. Now, we present some numerical examples to il-

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lustrate the bulk and surface plasmon-polariton spectra. In what follows we assume physical parameters typical of electron concentrations in GaAs— A1GaAs superlattices, and we assume the medium outside the superlattice to be vacuum. We take thicknesses corresponding to a=40 nm and b/a=2. Also it is considered 6a= 12.9, C~b 12.3, flaflb 6x l0’~m2, and m 1=6.4x lO_32 kg in (J=a, b). We 0)pa to be different of zero only medium A, allow corresponding to a volume charge density in the medium. The damping factor is taken to be zero in both mcdi a. Figure 2 shows the plasmon-polariton spectra for the periodic a/Jan... case. The bulk bands are the shadow areas bounded by QL = 0 and x, L being the size of the superlattice unit cell, while the surface modes are represented by dashed lines. Here we plot a reduced frequency w/Q against a dimensionless wavevector k~a,where nae

25 October 1993

1.2

0.8

_______

0.4

00.2

0.5

0.8

1.1 kxa

1.4

1.7

2

Fig. 3. Plasmon-polaritons spectra, p-polarization, for the Thue— Morse sequence S 2. and the parameters given in the text.

1.2 ~

0.8

_________

________

0.4

(18)

)I/2

0

0.2

The plasmon-polariton spectra for the quasiperiodic case are now shown in figs. 3 and 4, representing n even (here equal to 2) and odd (here equal to 3), respectively. We avoid to consider a large number for n to illustrate our theoretical computation, since there is no important qualitative difference among the spectra as a direct implication of the increase of n (the main difference being the increase of the number of bulk bands and surface modes in the spectra, which of course means a more rich spectra, already discussed in previous papers on this sub-

1.2 0.8

0.8

1.1

1.4

1.7

2

kx a Fig. 4. Same as in fig. 3, but for the Thue—Morse sequence S3.

ject (for an update on this subject see ref. [10])). However, as we can see comparing figs. 3 and 4 with fig. 2, there is an important qualitative difference, not yet pointed out so far. This is concerned with the lowest plasmon-polariton surface mode in the periodic case and in the quasi-periodic sequence for n even. In the former, and in the quasi-periodic Thue— Morse sequence for n odd, this mode is quite apart from the lowest bulk band, while for n even they are very close, with an almost not noticeable separation. This fact is not presented in the Fibonacci case [5],

__________________________________

and it is due to the fact that when n is odd, the first and last layers of S, are the building blocks a and fi, respectively, although, for n even, the first and last

~‘‘~

0 0.2

0.5

0.5

0.8

1.1

k

1.4

1.7

2

a

Fig. 2. Plasmon-polaritons spectra, p-polarization, for the periodic superlatticeaflafi.... The physical parameters used here are described in the main text.

layers of 5,, are of kind a. A direct comparison between figs. 3 and 4 shows that the overall aspect of the plasmon-polariton dispersion relation is also affected by the two different Thue—Morse configurations (n odd and even, respectively). This fact is a consequence of the different analytical expressions for the transfer matrix given by eqs. (15) (for n odd) and (16) (for n 411

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even), which plays a fundamental role in the dispersion relation, as one can infer from eqs. (4) (bulk case) and (12) (surface case), respectively. Furthermore, as a consequence of the difference between the number of interfaces for the two configurations, as can be noted from eq. (1), there are more surface modes (and bulk bands) in fig. 4 than in fig. 3, although in the latter it is hard to count these modes because they lie very close to the bulk band. In summary, we have presented a concise theoretical calculation for the plasmon-polariton spectra in quasi-periodic Thue—Morse sequences. This work can be considered as an extension of the previous ones done for quasi-periodic Fibonacci-type superlattices [5,6]. We thank Professor N.S. Almeida and Professor A.S. Carrico for helpful discussions. This research was in part financed by the Brazilian Research Council CNPq.

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References [1] M.G. Cottam and D.R. Tilley, Introduction to surface and superlattice excitations (Cambridge Univ. Press, Cambridge, 1989). [2] R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang and P.K. Bhattacharya, Phys. Rev. Lett. 55 (1985) 1768; J. Todd, R. Merlin, R. Clarke, K.M. Mohanty and J.D. Axe, Phys. Rev. Lett. 57(1986) 1157. [3] B.L. Johnson and R.E. Camley, Phys. Rev. B 44 (1991) 1225. [4] D.-H. Huang, J.-P. Peng and S.-X. Zhou, Phys. Rev. B 40 (1989) 7754. [5] E.L. Albuquerque and M.G. Cottam, Solid State Commun. 81(1992) 383. [6] E.L. Albuquerque and M.G. Cottam, Solid State Commun. 83 (1992) 545. [7] J. Humlicek, F. Lukes and K. Ploog, in: Proc. tnt. Conf. on Semiconductor physics, Berlin, 1990; J. Humlicek, F. Lukes, K. Navratil, M. Garrigaand K. Ploog, Appl. Phys. A 49 (1989) 407. [8] Z.C. Tao and M. Singh, Solid State Commun. 81(1992) 383. [9] GA. Farias, M.M. Auto and E.L. Albuquerque, Phys. Rev. B43 (1988) 12540. [10] E.L. Albuquerque and M.G. Cottam, Phys. Rep. 233 (1993) 67.