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Theory of plasmon-polaritons in fibonacci-type superlattices with two-dimensionl electron gas layers
~
Solid State Communications, Vol. 81, No. 5, pp. 383-386, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press plc
THEORY OF PLASMON-POLARITONS IN FIBONACCI-TYPE SUPERLATTICES WITH TWO-DIMENSIONAL ELECTRON GAS LAYERS E.L. Albuquerque Centre for Chemical Physics, (Received
and
M.G. Cottam
University of Western Ontario, Nov. 26, 1991
London, Ontario,
Canada N6A 3K7
by C.E.T. Conqalves da Silva)
The spectra of plasmon-polaritons are evaluated in semi-infinite superlattices composed of two-dimensional electron gas (2DEG) layers separated by dielectric media. The 2DEG charge densities and/or the properties of the dielectric spacers are taken to vary in accordance with a Fibonacci sequence. Using a transfer-matrix approach, we include the effects of retardation and an applied magnetic field. Our calculations generalize previous work on this subject.
As a result of recent advances in experimental techniques, the physical properties of quasiperiodic structures have attracted a lot of attention. The fabrication of such structures 1 was pioneered by Merlin et al, who grew a quasiperiodic GaAs-AIAs superlattice with an incommensurate ratio, equal to the golden mean number • = (I+~5}/2, for some of the length parameters. This superlattice structure involved two distinct building blocks, each having one or more layers of different materials with different thicknesses, arranged according to a Fibonacci sequence. Structures with length ratios other than the golden mean T can also be investigated. 2'3 The above developments have motivated theoretical and experimental studies of excitations in Fibonacci superlattices, including, for 4 5 example, the electronic magnetic and acous• 6 . ' . ' tlc properties. In partlcular some calculat'7 ions were very recently reported for bulk and surface plasmons in a layered system derived from a specific Fibonacci sequence number. Other plasmon-related work for quasiperiodic structures has been concerned with more complicated materials, such as HgTe-CdTe (see Ref. 8). In this communication we investigate the plasmon-polariton spectra in Fibonacci-type superlattices with two-dimensional electron gas (2DEG) layers. The corresponding properties for simple periodic 2DEG superlattices are well known (e.g., see Refs. 9-13). The aim of this work is two-fold: first, we extend the work of Johnson and Camley 7 by considering the effects of retardation and surface 2DEG sheets and also by evaluating the excitation spectra for higherorder Fibonacci sequence numbers. Second, we generalize previous superlattice work in this field as a particular case of our general result for the plasmon-polariton dispersion relations. We consider the two building blocks ~ and ~, as depicted in Fig. I, of a Fibonacci superlattice. Each block consists of a 2DEG charge sheet and a layer of medium A or B, which may t
A na ~.
a
)
B nb b
<
)
Fig. I. The two building blocks of the Fibonacci superlattice, where A and B are dielectric layers (with thicknesses a and b) and the 2DEG layers have carrier concentrations n a and n b p e r unit area.
have different thickness and dielectric function and may also (for generality) have a volume density of charge• The Fibonacci sequence is here described in terms of a series of generations that obey the following rule (e.g., see Ref. I):
s0=
[ ~ ] ,
sj=
[e~
....
st:
l e ] ,
[ e l , s4=
Sz=
[e~e
S~ = [ S~_ I i s~_ z ]
. . . .
[ ~ I ~ ] , [ e~ (~
I , z 2).
The spectra of bulk and surface plasmonpolaritons can be conveniently derived by using a transfer matrix treatment to simplify the algebra. For this, we follow the general formalism of Refs. 14 and 15, and so only the final expressions are quoted• First, the bulk dispersion relation in an infinitely-extended superlattice is given by :
Permanent address: Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072 Natal RN, Brazil.
383
384
THEORY
cos(QL)
=
~-- T r ( T ) 2
.
OF PLASMON-POLARITONS (1)
Here T is a unimodular matrix which relates the coefficients of the electric and magnetic fields in one cell to those in the preceding one, Q is the Bloch wave number, and L is the size of the superlattice unit cell. Next, in order to find the dispersion relation of the surface m o d e s , we c o n s i d e r a semiinfinite superlattice structure which is termina t e d a t t h e p l a n e z = O, w i t h t h e h a l f - s p a c e z < 0 filled with a material that has a frequencyindependent dielectric constant c s. The periodicity in the z direction is now absent a n d we can no longer assume the Bloch ansatz. As w e l l as the above bulk superlattice modes, there can be surface superlattice modes, in which the electric and magnetic fields in each superlattice cell have an exponential decay into the material. The application of the usual electromagnetic boundary conditions at z = 0 and at each of the interfaces between layers yields: T
- T
II
22
+ T
t
12
- T
21
t-1
= 0
{2)
with
J
,
{3)
(j = a or s )
= c /a ] J
[k:-
cjCo/c)2]
1/z
,
if k x
(4)
(5) i[cj (m/c)2 - k2] I/2x
- %
%
f
if k x < cjo/c
Here Tmn are eiements of the T matrix, and £j is the dielectric function of the medium under consideration. Also k x is the common wave-vector component parallel to the layers, w is the angular frequency, and c is the velocity of the light in vacuum. Equation (2) represents the dispersion relation for the surface modes along with another equation analogous to (I) in which we replace cos(QL) by cosh(#L), with Re(~) > 0 for attenuation. To determine the T matrix of each Fibonacci sequence, we follow the lines of Ref. 14. Let us initially assume p-polarizatlon for the electromagnetic mode and the absence of any external magnetic field. We consider that the 2D charge sheet at each interface is due to the presence of a current density, which can be written as (JJ)x = ioc0~j(Ej)x where
, ]
(8)
- %
the additional
notation
= exp (a J) . J
J
(9)
In general, we assume for media A and H that c] may be frequency dependent, having the form e] = e j[l
,
- {o~jlwCw+iFl)}]
ClO)
where Opj is the plasma f r e q u e n c y , r j is a damping factor (which may be different from ~j), and e is the background dielectric constant. ~J From the above results, we get the following expressions for the T matrix in three cases of interest: (a)
for
S = ~ or
S
= ~
2
TS 2 = Na - I (C)
for
= ~
I
:
= Nj - 1 M J
TsI{TSo) for
S
0
any
j
= a
(b)
;
(11)
:
(12)
Mb Nb - I Ma ; higher
sequence
TSk+2 = TSk TSk+I
> c Jo / c
% :
[,
where we introduce
(b)
7~ = { c 'a + c ~ ) / ( C ' a - C s ) c'
= %
Vol. 81, No. 5
,
:
(k a 1 ) .
(13)
A similar analysis can be carried out for the s-polarlzed modes (i.e., where the electric field E in each layer is in the y direction, rather than in the xz plane as above). It can be shown that the above formal results for the dispersion relations still hold provided we m a k e the following replacements in Eqs. (7) and (g):
e'] --~ ~ J / g O
~'j --> (i~go/~j)~ ~. .
and
(14)
The results may straightforwardly generalized to include a static external magnetic field (I.e., to the magnetoplasmon-polariton regime). We assume the magnetic field to be along the y-axis, perpendicular to the in-plane wave vector k w The main effect of the magnetic field in this geometry Is that the dielectric functions of layers A and B become matrix quantities {if the plasma frequency is nonzero). Taking, for simplicity, the case of op being nonzero only for material A (and e b of material B equal to ¢ ) we h a v e mb
c
0
0
ca
0
0
c
]
-ic 2
m ¢T~ = n j e 2 /
mlo{w+i~j)c 0 ,
(j
= a or
b).
{6)
Here nj is the c a r r i e r concentration eper unit area, e is the electronic charge, mj is the effective mass of the charge carrier, Tj is a damping factor, and c o is the vacuum permittivity. We define in each layer the matrices Mj and N] given by (j = a or b):
fj =
Mj
f-1 J
~JfJ
_c,f- I
JJ
teXaCO) =
ic
(iS)
I
where
~I = c c
2
= c
[I + ~ p ~ / (
Oc2 _ o 2 ) ]
o o 2 / o C w 2 - o 2)
~a C pa
c 3 = c a[1 -
]
2
,
c
(Opa/O)2]
.
,
,
(16) (17)
(18)
(7)
with o c denoting the cyclotron frequency corresponding to material A. It is found that the
Vol. 81, No. 5
THEORY OF PLASMON-POLARITONS
1.4!
,
' --
I
0.8 3
0.6
0.4
/,/"
t
0.5
0.0
1.0
1.5
ka X
Fig. 2. Plot of reduced p l a s m o n - p o l a r i t o n frequencies, ~/~, aEainst kxa for the case of Fibonacci sequence $3, p-polarization, and zero a p p l i e d m a g n e t i c field. The bulk-mode regions are shown shaded (and are b o u n d e d by QL = 0 and Q L = ~) and the surface modes are the d a s h e d lines. See the text for parameter values. d i s p e r s i o n r e l a t i o n equations are f o r m a l l y the same as in the absence of a m a E n e t i c field, p r o v i d e d we make the f o l l o w i n E replacements:
c
~
c
a
1
a
at
I n Eq. (20) t h e + s i g n r e f e r s t o t h e Hi2 and Ni2 m a t r i x elements, w h i l e the - siln refers to M22 and N22. Next we present some numerical e x a m p l e s to illustrate the bulk and surface plasmonp o l a r i t o n d i s p e r s i o n relations and their d e p e n d ence on the Pibonacci sequence number. In what follows, we assume p a r a m e t e r values that w o u l d be typical of e l e c t r o n c o n c e n t r a t i o n s in G a A s AIGaAs superlattices. We take thicknesses corre s p o n d i n g to a = 40 nm and b = 80 nm and assume the m e d i u m o u t s i d e the s u p e r l a t t i c e to be vacuum. Also c = 12.9, c = 12.3, n a = n b = 6 x Wa
0.2 0.0
- c 21c
+_ = k c ~xl
2
2L
2
(19)
1
x
1
= (n e2/mic c a
,,1j
'
a
0
a) I12 ,
(22)
~a
is p l o t t e d aEainst kxa. In this case, fl/2~ = 23.0 THz. The bulk b a n d s are the shaded areas, b o u n d e d b y QL = 0 and Q L = ~ w i t h L = 2a+b, w h i l e the surface modes are r e p r e s e n t e d b y solid lines. For comparison, we show in Fi E . 3 the c o r r e s p o n d i n E case of S 4 = [cq8~I(*48] w i t h L = 3a+2b. In b o t h cases we have taken ~pa = O, so the p l a s m o n - p o l a r i t o n modes are e n t i r e l y due to
1.5
,<2
~b
o
10 *5 m -2, and mj = 6.4 x 10 -32 k E ( j = a,b). We a l l o w opa to be n o n z e r o in some cases, corre s p o n d i n E to a volume charEe d e n s i t y in medium A. For simplicity, we neElect the e f f e c t s of d a m p i n E. F i E u r e 2 shows the p l a s m o n - p o l a r i t o n spectrum, in p - p o l a r i z a t i o n and in the absence of an a p p l i e d m a E n e t i c field, for the Fibonacci sequence S 3 = [¢~I¢]. Here the reduced f r e q u e n c y ~/~, w i t h ~ d e f i n e d by
2
l
1j-Lx
385
I
~.
i
I
1.0
1.5
(20)
1.4
1.0
' "''''''t
1.2
3 3
1.0
0.5
:3
~'"
0.6
0.4-
0.0 0.0
0.2 0.0 0.0
0.5
2.0
k aX 0.5
1.0 k
8L
Fi E . 3. As in Fi 8. 2 but for the Fibonacci sequence S4.
Fi E . 4. Plot of reduced p l a s m o n - p o l a r i t o n frequencies, ~/opa, aEainst kxa for the case of nonzero a p p l i e d m a E n e t i c field and n o n z e r o ~paThe results are for Fihonacci sequence S 3 and p-polarization. See the text for p a r a m e t e r values.
386
T H E O R Y OF P L A S M O N - P O L A R I T O N S
the 2DEG sheets at the interfaces. As expected, the numbers of bulk bands and surface branches increase with the number of building blocks in the Fibonaccl sequence, and Fig. 3 shows a much richer spectrum. The effect of an applied magnetic field on the plasmon-polariton spectrum is illustrated in Fig. 4 for the Fibonacci sequence S 3. We have assumed a magnetic field such that wc/2n = 20.0 THz and taken wpa/2~ = 40 THz and Wpb = 0 for the bulk plasma frequencies. A comparison of this spectrum with Fig. 3 shows the removal of the degeneracy of the surface modes, as a consequence of the magnetic field, g ~ i n g rise to the magnetoplasmon-polariton modes. In summary, we have presented a concise
Vol. 81, No. 5
theoretical derivation of the plasmon-polariton spectrum in a quasiperiodic layered system obeying a Fibonacci sequence, 7generalizing earlier calculations for plasmons. We note also that our work encompasses previous results for the plasmon-polariton spectrum in superlattlces, in the sense that any periodic structure can be cast in the S 2 Fibonacci sequence with appropriate choice of the ~ and B building blocks.
A c k n o w l e d g m e n t s - We a r e g r a t e f u l t o Coordenaqfio de Aperfelqoamento de Pessoal de Nivel Superior (CAPES) of Brazil, the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Centre for Chemical Physics (University of Western Ontario) for partial support.
References
I.
2. 3. 4.
5. 6. 7.
R. M e r l i n , K. Bajema, R. C l a r k e , F.-Y. J u a n g , and P.K. B h a t t a c h a r y a , Phys. Rev. L e f t . 55, 1768 (1985); J. Todd, R. M e r l i n , R. C l a r k e , K.M. Mohanty, and J.D. Axe, Phys. Rev. L e t t . 57, 1157 (1986). G. Gumbs and M.K. A l l , Phys. Rev. L e f t . 60,1081 (1988). M.K. A l l and G. Gumbs, Phys. Rev. B 38, 7091 (1988). M, Inoue, T. Takemori, and H. Miyazaki, Solid S t a t e Comm. 77, 751 (1991). M. K o l a r and M.K. A l l , Phys. Rev. B 39, 426
(1989). A.P. Mayer, J. Phys. Condens. Mat. (1989). B . L . J o h n s o n and R.E. Camley, Phys. 44, 1225 (1991).
8. 9. 10. II.
12. 13.
14. I,
3301 15.
Rev.
B,
D.-H. Huang, J.-P. Peng, and S.-X. Zhou, Phys. Rev. B 40, 7754 (1989). A.L. Fetter, Ann. Phys. 88, 1 (1974). S. Das Sarma and J.J. Quinn, Phys. Rev. B 25, 7603 (1982). G.F. Giuliani and J.J. Quinn, Phys. Rev. L e t t . 51, 919 (1983). N.C. C o n s t a n t l n o u and M.G. Cottam, J. Phys. C 19, 739 (1986). M.G. Cottam and D.R. T i l l e y , I n t r o d u c t i o n t o S u r f a c e and S u p e r l a t t i c e Excitations (Camb r i d g e U n i v e r s i t y P r e s s , Cambridge, 1989). G.A. F a r i a s , M.M. Auto, and E.L. A l b u q u e r que, Phys. Rev. B 38, 12540 (1988). E.L. A l b u q u e r q u e , P. F u l c o , G.A. F a r l a s , M.M. Auto, and D.R. T i l l e y , Phys. Rev. B 43, 2032 (1991).
Solid State Communications, Vol. 81, No. 5, pp. 383-386, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press plc
THEORY OF PLASMON-POLARITONS IN FIBONACCI-TYPE SUPERLATTICES WITH TWO-DIMENSIONAL ELECTRON GAS LAYERS E.L. Albuquerque Centre for Chemical Physics, (Received
and
M.G. Cottam
University of Western Ontario, Nov. 26, 1991
London, Ontario,
Canada N6A 3K7
by C.E.T. Conqalves da Silva)
The spectra of plasmon-polaritons are evaluated in semi-infinite superlattices composed of two-dimensional electron gas (2DEG) layers separated by dielectric media. The 2DEG charge densities and/or the properties of the dielectric spacers are taken to vary in accordance with a Fibonacci sequence. Using a transfer-matrix approach, we include the effects of retardation and an applied magnetic field. Our calculations generalize previous work on this subject.
As a result of recent advances in experimental techniques, the physical properties of quasiperiodic structures have attracted a lot of attention. The fabrication of such structures 1 was pioneered by Merlin et al, who grew a quasiperiodic GaAs-AIAs superlattice with an incommensurate ratio, equal to the golden mean number • = (I+~5}/2, for some of the length parameters. This superlattice structure involved two distinct building blocks, each having one or more layers of different materials with different thicknesses, arranged according to a Fibonacci sequence. Structures with length ratios other than the golden mean T can also be investigated. 2'3 The above developments have motivated theoretical and experimental studies of excitations in Fibonacci superlattices, including, for 4 5 example, the electronic magnetic and acous• 6 . ' . ' tlc properties. In partlcular some calculat'7 ions were very recently reported for bulk and surface plasmons in a layered system derived from a specific Fibonacci sequence number. Other plasmon-related work for quasiperiodic structures has been concerned with more complicated materials, such as HgTe-CdTe (see Ref. 8). In this communication we investigate the plasmon-polariton spectra in Fibonacci-type superlattices with two-dimensional electron gas (2DEG) layers. The corresponding properties for simple periodic 2DEG superlattices are well known (e.g., see Refs. 9-13). The aim of this work is two-fold: first, we extend the work of Johnson and Camley 7 by considering the effects of retardation and surface 2DEG sheets and also by evaluating the excitation spectra for higherorder Fibonacci sequence numbers. Second, we generalize previous superlattice work in this field as a particular case of our general result for the plasmon-polariton dispersion relations. We consider the two building blocks ~ and ~, as depicted in Fig. I, of a Fibonacci superlattice. Each block consists of a 2DEG charge sheet and a layer of medium A or B, which may t
A na ~.
a
)
B nb b
<
)
Fig. I. The two building blocks of the Fibonacci superlattice, where A and B are dielectric layers (with thicknesses a and b) and the 2DEG layers have carrier concentrations n a and n b p e r unit area.
have different thickness and dielectric function and may also (for generality) have a volume density of charge• The Fibonacci sequence is here described in terms of a series of generations that obey the following rule (e.g., see Ref. I):
s0=
[ ~ ] ,
sj=
[e~
....
st:
l e ] ,
[ e l , s4=
Sz=
[e~e
S~ = [ S~_ I i s~_ z ]
. . . .
[ ~ I ~ ] , [ e~ (~
I , z 2).
The spectra of bulk and surface plasmonpolaritons can be conveniently derived by using a transfer matrix treatment to simplify the algebra. For this, we follow the general formalism of Refs. 14 and 15, and so only the final expressions are quoted• First, the bulk dispersion relation in an infinitely-extended superlattice is given by :
Permanent address: Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072 Natal RN, Brazil.
383
384
THEORY
cos(QL)
=
~-- T r ( T ) 2
.
OF PLASMON-POLARITONS (1)
Here T is a unimodular matrix which relates the coefficients of the electric and magnetic fields in one cell to those in the preceding one, Q is the Bloch wave number, and L is the size of the superlattice unit cell. Next, in order to find the dispersion relation of the surface m o d e s , we c o n s i d e r a semiinfinite superlattice structure which is termina t e d a t t h e p l a n e z = O, w i t h t h e h a l f - s p a c e z < 0 filled with a material that has a frequencyindependent dielectric constant c s. The periodicity in the z direction is now absent a n d we can no longer assume the Bloch ansatz. As w e l l as the above bulk superlattice modes, there can be surface superlattice modes, in which the electric and magnetic fields in each superlattice cell have an exponential decay into the material. The application of the usual electromagnetic boundary conditions at z = 0 and at each of the interfaces between layers yields: T
- T
II
22
+ T
t
12
- T
21
t-1
= 0
{2)
with
J
,
{3)
(j = a or s )
= c /a ] J
[k:-
cjCo/c)2]
1/z
,
if k x
(4)
(5) i[cj (m/c)2 - k2] I/2x
- %
%
f
if k x < cjo/c
Here Tmn are eiements of the T matrix, and £j is the dielectric function of the medium under consideration. Also k x is the common wave-vector component parallel to the layers, w is the angular frequency, and c is the velocity of the light in vacuum. Equation (2) represents the dispersion relation for the surface modes along with another equation analogous to (I) in which we replace cos(QL) by cosh(#L), with Re(~) > 0 for attenuation. To determine the T matrix of each Fibonacci sequence, we follow the lines of Ref. 14. Let us initially assume p-polarizatlon for the electromagnetic mode and the absence of any external magnetic field. We consider that the 2D charge sheet at each interface is due to the presence of a current density, which can be written as (JJ)x = ioc0~j(Ej)x where
, ]
(8)
- %
the additional
notation
= exp (a J) . J
J
(9)
In general, we assume for media A and H that c] may be frequency dependent, having the form e] = e j[l
,
- {o~jlwCw+iFl)}]
ClO)
where Opj is the plasma f r e q u e n c y , r j is a damping factor (which may be different from ~j), and e is the background dielectric constant. ~J From the above results, we get the following expressions for the T matrix in three cases of interest: (a)
for
S = ~ or
S
= ~
2
TS 2 = Na - I (C)
for
= ~
I
:
= Nj - 1 M J
TsI{TSo) for
S
0
any
j
= a
(b)
;
(11)
:
(12)
Mb Nb - I Ma ; higher
sequence
TSk+2 = TSk TSk+I
> c Jo / c
% :
[,
where we introduce
(b)
7~ = { c 'a + c ~ ) / ( C ' a - C s ) c'
= %
Vol. 81, No. 5
,
:
(k a 1 ) .
(13)
A similar analysis can be carried out for the s-polarlzed modes (i.e., where the electric field E in each layer is in the y direction, rather than in the xz plane as above). It can be shown that the above formal results for the dispersion relations still hold provided we m a k e the following replacements in Eqs. (7) and (g):
e'] --~ ~ J / g O
~'j --> (i~go/~j)~ ~. .
and
(14)
The results may straightforwardly generalized to include a static external magnetic field (I.e., to the magnetoplasmon-polariton regime). We assume the magnetic field to be along the y-axis, perpendicular to the in-plane wave vector k w The main effect of the magnetic field in this geometry Is that the dielectric functions of layers A and B become matrix quantities {if the plasma frequency is nonzero). Taking, for simplicity, the case of op being nonzero only for material A (and e b of material B equal to ¢ ) we h a v e mb
c
0
0
ca
0
0
c
]
-ic 2
m ¢T~ = n j e 2 /
mlo{w+i~j)c 0 ,
(j
= a or
b).
{6)
Here nj is the c a r r i e r concentration eper unit area, e is the electronic charge, mj is the effective mass of the charge carrier, Tj is a damping factor, and c o is the vacuum permittivity. We define in each layer the matrices Mj and N] given by (j = a or b):
fj =
Mj
f-1 J
~JfJ
_c,f- I
JJ
teXaCO) =
ic
(iS)
I
where
~I = c c
2
= c
[I + ~ p ~ / (
Oc2 _ o 2 ) ]
o o 2 / o C w 2 - o 2)
~a C pa
c 3 = c a[1 -
]
2
,
c
(Opa/O)2]
.
,
,
(16) (17)
(18)
(7)
with o c denoting the cyclotron frequency corresponding to material A. It is found that the
Vol. 81, No. 5
THEORY OF PLASMON-POLARITONS
1.4!
,
' --
I
0.8 3
0.6
0.4
/,/"
t
0.5
0.0
1.0
1.5
ka X
Fig. 2. Plot of reduced p l a s m o n - p o l a r i t o n frequencies, ~/~, aEainst kxa for the case of Fibonacci sequence $3, p-polarization, and zero a p p l i e d m a g n e t i c field. The bulk-mode regions are shown shaded (and are b o u n d e d by QL = 0 and Q L = ~) and the surface modes are the d a s h e d lines. See the text for parameter values. d i s p e r s i o n r e l a t i o n equations are f o r m a l l y the same as in the absence of a m a E n e t i c field, p r o v i d e d we make the f o l l o w i n E replacements:
c
~
c
a
1
a
at
I n Eq. (20) t h e + s i g n r e f e r s t o t h e Hi2 and Ni2 m a t r i x elements, w h i l e the - siln refers to M22 and N22. Next we present some numerical e x a m p l e s to illustrate the bulk and surface plasmonp o l a r i t o n d i s p e r s i o n relations and their d e p e n d ence on the Pibonacci sequence number. In what follows, we assume p a r a m e t e r values that w o u l d be typical of e l e c t r o n c o n c e n t r a t i o n s in G a A s AIGaAs superlattices. We take thicknesses corre s p o n d i n g to a = 40 nm and b = 80 nm and assume the m e d i u m o u t s i d e the s u p e r l a t t i c e to be vacuum. Also c = 12.9, c = 12.3, n a = n b = 6 x Wa
0.2 0.0
- c 21c
+_ = k c ~xl
2
2L
2
(19)
1
x
1
= (n e2/mic c a
,,1j
'
a
0
a) I12 ,
(22)
~a
is p l o t t e d aEainst kxa. In this case, fl/2~ = 23.0 THz. The bulk b a n d s are the shaded areas, b o u n d e d b y QL = 0 and Q L = ~ w i t h L = 2a+b, w h i l e the surface modes are r e p r e s e n t e d b y solid lines. For comparison, we show in Fi E . 3 the c o r r e s p o n d i n E case of S 4 = [cq8~I(*48] w i t h L = 3a+2b. In b o t h cases we have taken ~pa = O, so the p l a s m o n - p o l a r i t o n modes are e n t i r e l y due to
1.5
,<2
~b
o
10 *5 m -2, and mj = 6.4 x 10 -32 k E ( j = a,b). We a l l o w opa to be n o n z e r o in some cases, corre s p o n d i n E to a volume charEe d e n s i t y in medium A. For simplicity, we neElect the e f f e c t s of d a m p i n E. F i E u r e 2 shows the p l a s m o n - p o l a r i t o n spectrum, in p - p o l a r i z a t i o n and in the absence of an a p p l i e d m a E n e t i c field, for the Fibonacci sequence S 3 = [¢~I¢]. Here the reduced f r e q u e n c y ~/~, w i t h ~ d e f i n e d by
2
1j-Lx
385
I
~.
i
I
1.0
1.5
(20)
1.4
1.0
' "''''''t
1.2
3 3
1.0
0.5
:3
~'"
0.6
0.4-
0.0 0.0
0.2 0.0 0.0
0.5
2.0
k aX 0.5
1.0 k
8L
Fi E . 3. As in Fi 8. 2 but for the Fibonacci sequence S4.
Fi E . 4. Plot of reduced p l a s m o n - p o l a r i t o n frequencies, ~/opa, aEainst kxa for the case of nonzero a p p l i e d m a E n e t i c field and n o n z e r o ~paThe results are for Fihonacci sequence S 3 and p-polarization. See the text for p a r a m e t e r values.
386
T H E O R Y OF P L A S M O N - P O L A R I T O N S
the 2DEG sheets at the interfaces. As expected, the numbers of bulk bands and surface branches increase with the number of building blocks in the Fibonaccl sequence, and Fig. 3 shows a much richer spectrum. The effect of an applied magnetic field on the plasmon-polariton spectrum is illustrated in Fig. 4 for the Fibonacci sequence S 3. We have assumed a magnetic field such that wc/2n = 20.0 THz and taken wpa/2~ = 40 THz and Wpb = 0 for the bulk plasma frequencies. A comparison of this spectrum with Fig. 3 shows the removal of the degeneracy of the surface modes, as a consequence of the magnetic field, g ~ i n g rise to the magnetoplasmon-polariton modes. In summary, we have presented a concise
Vol. 81, No. 5
theoretical derivation of the plasmon-polariton spectrum in a quasiperiodic layered system obeying a Fibonacci sequence, 7generalizing earlier calculations for plasmons. We note also that our work encompasses previous results for the plasmon-polariton spectrum in superlattlces, in the sense that any periodic structure can be cast in the S 2 Fibonacci sequence with appropriate choice of the ~ and B building blocks.
A c k n o w l e d g m e n t s - We a r e g r a t e f u l t o Coordenaqfio de Aperfelqoamento de Pessoal de Nivel Superior (CAPES) of Brazil, the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Centre for Chemical Physics (University of Western Ontario) for partial support.
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