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UNCORRECTED
PROOF
PHYSE5182
p p : 1íí7 (col.fig.: Nil)
PROD. TYPE: COM
ED: JSS
PAGN: Murthy.N íí SCAN: Profi
Physica E
000( 2001) 000--000
www.elsevier.com/locate/physe
1
Electron spectrum and infrared transitions in semiconductor
superlattices with a unit cell allowing for quasiílocalised
3
carrier states
A.V. Dmitriev a ; # , R. Keiper b , V.V. Makeev a
5
a Department
of Low Temperature Physics, Faculty
of Physics, Moscow State University, Moscow, 119899, Russia
b
Institutf #
ur Physik, Humboldt--Universit# at zu Berlin, Invalidenstr. 110, 10115 Berlin, Germany
7
Abstract
We studied theoretically, the electron spectrum and infrared transitions in a superlattice with a unit cell allowing for
9
quasiílocalised carrier states. The dispersion relation and the band structure of such a system have been found. We calculated
the dipole matrix element for interísubband carrier infrared transitions. The wave functions and the electron spectrum in this
11
superlattice show a peculiarity when the energy of a band state approaches the energy of the quasiílocalised state in the
single cell. The absorption strength peaks up at the respective frequencies. c
# 2001 Published by Elsevier Science B.V.
13
PACS: 78.66.-w; 73.20.Dx; 71.15.Ap
Keywords: Superlattice; Resonant states; IRítransitions
15
1.I ntroduction
Usually, one assumes that in semiconductors, along
17
with other crystals, an electron state belongs to one
of the two possible kinds. Namely, it can be either
19
a Bloch band state, or a localised state residing in
the forbidden gap. However, a third kind of carrier
21
states, resonant or quasiílocalised [1], has been shown
to play a signi#cant role in a number of occasions.
23
These states, long before known both in optics [2] and
# Corresponding author. Fax: +7í095í932í88í76.
Eímail address:
dmitriev@1t.phys.msu.su( A.V. Dmitriev).
in quantum mechanics [3], appear in semiconductors 25
e.g., when an impurity level, split o# from one band,
overlaps with another allowed band, or when a deep 27
impurity level overlaps with one of the allowed ení
ergy bands. In certain conditions, resonant states may 29
signi#cantly a#ect the kinetic properties of a semiconí
ductor [4,5]. 31
Quasiílocalised states may be present in arti#cially
prepared heterostructures, e.g. in twoíbarrier quaní 33
tum well systems. We have shown earlier [6] that in
their presence the absorption coe#cient signi#cantly 35
increases in the frequency range of the intraband traní
sitions into the resonant state. Since the latter state 37
1386í9477/01/$ í see front matter c
# 2001 Published by Elsevier Science B.V.
PII: S 1386 í
9477( 01 ) 00220 í X

UNCORRECTED PROOF
2 A.V. Dmitriev et al. / Physica E 000 (2001) 000--000
PHYSE5182
formally belongs to the continuum, one can expect
1
it to decay easily into a delocalised wave, so that no
strong electric #eld would be required to add the exí
3
cited electron to an observable photocurrent. Hence,
heterostructures with quasiílocalised states sound
5
quite appealing as candidates for selective quantum
well infrared
photodetectors( QWIPs). These detecí
7
tors would combine high spectral selectivity with
low dark current because of low bias applied. Such
9
combination is hardly attainable with conventional
QWIPs, where the working transition goes to a bound
11
state, or in the opposite case, to a plain continuum
state( see the review article [7]).
13
In the cited paper [6], we discussed infrared optical
properties of a single quantum well system. However,
15
arrays of quantum wells or superlattices are normally
used for experimental purposes and practical applicaí
17
tions. Thus, we thought it relevant to consider a perií
odic structure composed of quantum wells with resoí
19
nant states. How the adjacent resonant states interfere
with each other and with the continuum states may
21
be a matter of independent theoretical interest. To the
best of our knowledge, neither infrared optical properí
23
ties nor subband spectrum of this kind of superlattices
have been considered before. This is the goal of the
25
present paper.
The theoretical approach to the electronic spectrum
27
of superlattices is well
developed( see e.g. Refs. [8,9]).
In Ref. [10] a detailed spectral analysis of a coní
29
ventional AB
superlattice( two alternating layers) has
been demonstrated. We use a similar approach to analí
31
yse the electronic spectrum of a superlattice with a
more complex unit cell.
33
2. Model
We consider one nonídegenerate band, let it be
35
the conduction band, of a semiconductor superlattice,
where each cell is described within the e#ectiveímass
37
approximation by a oneídimensional model potential
as
follows( see also Fig. 1):
39
U( x) =
#
-V; 0 á x á a
0; a á x á b
#
+#
[#( x) +
#( x -
a)];( 1)
where x is the growth direction of the superlattice,
a and b are the well width and the structure period,
41
Fig. 1. The considered model potential. Please note the additií
nal #íbarriers surrounding the well. Several lower subbands are
marked. One subband is supposed to remain below the top of the
main barriers, the rest being above.
respectively, V is the well depth. #ílike barriers on
the well's edges represent a simpli#ed approximation 43
of additional real barriers of #nite width and height
that would surround each well. The parameter # thus 45
represents the reverse tunnel transparency of the real
barrier. The main barriers of
width( b - a) separate 47
the wells. The potential in
Eq.( 1) is assumed to be 0
at the top of the main barrier. 49
It is clear that the in#nitely high and in#nitely thin
#íbarriers, surrounding the wells, cannot be grown up 51
in a real heterostucture. In real structures, all barriers
have #nite height and width. However, the #ífunction 53
approximation of real barriers adopted in
Eq.( 1) is
a wellíknown simpli#ed method used in a number 55
of quantumímechanical
problems( see, for example
[11]). It corresponds to a very high and thin barrier 57
with #nite penetrability. From the point of view of
the problem, we consider here, the main di#erence beí 59
tween the #íbarrier and the real one is an in#nite height
of the former. As a result, our model system has an in#í 61
nite set of quasiílocalised
states( resonances) whereas
a real structure hardly can produce more than one or 63
two of them. But as far as we are interested in the
properties of a single resonance, the #íapproximation 65
leads to qualitatively correct and physically meaningí
ful results. 67
If all the structure consisted of only one quaní
tum well with
potential( 1), we might speak of 69
quasiílocalised electronic states in its spectrum.
These states appear on an energy scale close to 71
truly localised states that would exist in the well,

UNCORRECTED PROOF
PHYSE5182
A.V. Dmitriev et al. / Physica E 000 (2001) 000--000 3
if the additional
barriers( walls) were absolutely
1
impenetrable. Finite penetrability of the walls transí
forms the truly localised sizeíquantised states into
3
quasiílocalised states. Of course, this matters only for
the excited states lying above the top of the main barí
5
riers, like the two higher levels in Fig. 1. The lower
ground state in the single quantum well is always
7
localised.
Turning back to the periodic heterostructure, let us
9
see how the resonant states a#ect the properties of the
whole system. The envelope wave functions may be
11
represented as
# ( x) =
#
A 1 e iqx + A 2 e -iqx ; 0 á x á a;
B 1 e i#x + B 2 e -i#x ; a á x á b;
# ( x + b) = e
ikb# ( x);
( 2)
where q
=( 1=Ô)
#
2m( E + V ), #
=( 1=Ô) # 2mE, E
13
is the particle energy counted from the top of the
main barrier, m is the e#ective mass, kb is the
15
phase shift of the envelope function, resulting from
a oneílatticeíperiod displacement along the growth
17
direction. Ignoring the changes in the e#ective mass
across the superlattice layers, we obtain a convení
19
tional boundary condition on the leftíhand border of
the
well( x = 0):
21
#
#
## ( x)| 0+
0- = 0;
d
dx
ln# ( x)| 0+
0- =#:
Having written similar boundary conditions for the
rightíhand
border( x = a), we come to a homogeí
23
neous system of equations de#ning the coe#cients in
Eq.( 2):
25
#
# #
##
e ikb e ikb
-e i#b
-e -i#b
( q + i#)e
ikb( -q + i#)e ikb
-#e i#b #e -i#b
e iqa e -iqa -e i#a
-e -i#a
( -q + i#)e
iqa( q + i#)e -iqa #e i#a
-#e -i#a
#
# # # #
½
#
#
##
A 1
A 2
B 1
B 2
#
# #
# = 0:
A nonízero solution of this system exists only if the
system determinant is zero, hence we obtain the disí 27
persion relation
cos kb =
# 2
- q 2
- # 2
2#q sin
#( b - a) sin qa
+ #
q cos
#( b - a) sin qa
+ #
#
sin
#( b-a) cos qa+cos
#( b-a) cos
qa:( 3)
The energy intervals, where the absolute value of the 29
rightíhand side of
Eq.( 3) does not exceed unity, corí
respond to allowed subbands of our superlattice. Uní 31
fortunately, no analytic solution for the wave function
coe#cients in
Eq.( 2) can be obtained at arbitrary k , 33
so further spectrum calculations were performed nuí
merically. 35
3. Spectrum, wave functions and momentum
matrix element 37
Fig. 2 depicts the envelope wave functions of seví
eral adjacent subbands. The lower plot represents a 39
wave function belonging to the lowest subband; this
band originates from the well's ground state. Natuí 41
rally, electronic density concentrates within the well's
limits. The shape of the wave functions within the well 43
practically does not depend on k; only the phase shift
between adjacent cells changes with k. 45
The rest of the wave functions in Fig. 2 corresponds
to positive energy values. Most of these have elecí 47
tronic density concentrated just outside the wells. We
can roughly infer that these functions originate from 49
electronic states residing over the barriers. The addií
tional #íbarriers, surrounding the wells, prevent the 51
particles from entering the latter.
Note that the functions on the edges of each subband 53
have de#nite parity when viewed from both well cení
tre or barrier centre, in agreement with general rules 55
established in Ref. [12] for wave functions in periodic
structures with symmetric potential. The wave funcí 57
tions on the edges of the ground subband are both even
about the well centre, but when viewed from the barí 59
rier centre, the k = 0 function is even and the k = #=b
one is odd. This can be easily understood in full 61

UNCORRECTED PROOF
4 A.V. Dmitriev et al. / Physica E 000 (2001) 000--000
PHYSE5182
Fig. 2. The subband structure of the superlattice for b=a = 4,
V = 1:47 and #= 8. The energy unit is Ô 2 =ma 2 . The subband
edges are marked by horizontal dotted lines. Solid curves represent
the envelope functions at the band edges, of which the states
with k = 0 are marked with rhombuses. Two thick solid curves
represent the functions of the `resonant' subband. In this subband,
electron density resides mainly within the wells, and both subband
edge states are odd about the well centre. The unit cell potential
is shown below.
analogy to the tightíbinding model [13] with the lowí
1
est electron states in the wells taken as a basis. On the
contrary, in most higher subbands both wave funcí
3
tions on the band edges have the same parity about
the barrier centre and di#erent parities about the well
5
centre. This is because they are made up mainly of
the electron states that reside over the barriers as exí
7
plained above. For de#niteness, further on we speak
of parity about the well centre.
9
However, there is an excited subband, with energy
close to the resonant value in the wells, with propí
11
erties that resemble the ground subband. Let us call
this subband resonant. Here electron density is large
13
within the well limits, and the envelope functions have
the same parity on the band edges. Their structure reí
15
sembles the structure of the functions in the lowest
subband, which originated from the localised states in
17
the wells. Similarly, the resonant band is built from
the quasiílocalised electron states between the addií
19
tional barriers. The quasiílocalised states are mainly
concentrated within the wells, so the structure of the
21
corresponding resonant subband is much like that of
the ground subband. 23
Henceforth, we can expect the dipole matrix eleí
ment of the optical transition between these two bands, 25
ground and resonant, to be anomalously large, because
of high overlap between the wave functions in the 27
two bands. Then the absorption coe#cient would also
increase. The energy of corresponding transitions in 29
common superlattices lies in the infrared range.
In our calculations, we used |p
n( k)| 2 , the momení 31
tum matrix element squared, as a convenient straightí
forward parameter, characterising the absorption per 33
one electron in the ground
subband( see Section 4).
|p
n( k)| 2 is the matrix element between wave functions 35
in the lowest and nth subbands taken at one Bloch vecí
tor value
k( because of negligible photon's momení 37
tum, we can consider the electron transitions vertical).
The derivative parameters, such as absorption probaí 39
bility or absorption coe#cient #, are proportional to
|p
n( k)| 2 . 41
Figs. 3 and 4 depict the dependence of |p n | 2 on
the energy of the #nal electron state at two di#erent 43
values of superlattice period. The variation of other
superlattice
parameters( a, #, V ) does not change the 45
qualitative picture. We can see #rst that the transition
matrix element goes up in a number of subbands in the 47
area of resonance. Secondly, absorption is maximum
at one edge and drops almost to zero at another edge 49
of the subband. This is true for all subbands except
the resonant. While before the resonance absorption 51
Fig. 3. The momentum matrix element squared for the same
system as in Fig. 2. Absorption in subbands preceding the resonant
subband drops from the lower edge to the higher. When the
resonant band is passed, the picture reverses. The energy unit is
Ô 2
=( ma 2 ), and the matrix element squared is measured in Ô 2 =a 2 .

UNCORRECTED PROOF
PHYSE5182
A.V. Dmitriev et al. / Physica E 000 (2001) 000--000 5
Fig. 4. The same dependence as in Fig. 3, but for thicker main
barriers: b=a = 10. The well depth V = 1:47. Full squares show the
ratio of #|p n | 2
#=( E n -E n-1 ), that is, the averaged over an energy
interval matrix element squared, for the superlattice. Smooth curve
represents the product of square of the momentum operator matrix
element by the density of #nal states, |p if | 2 #, in a single quantum
well according to [6].
monotonously goes down from the lower edge to the
1
upper, after the resonance the picture become reversed.
Matching the picture with Fig. 2, we see that when
3
absorption is maximum, the #nal wave function has
`proper' parity, i.e. the opposite to the parity of the
5
ground state. In the resonant band parity is `proper' on
both band edges, and band absorption spectrum has
7
di#erent shape.
The picture re#ects the hybrid structure of the elecí
9
tronic spectrum of the considered superlattice. In the
given con#guration, where main barriers are thicker
11
than wells, the excited subband spectrum is formed
mainly by barrier levels. An `intrusion' of the resonant
13
level from the well confuses the monotonous pattern
and upturns parity switching order.
15
As the interwell distance increases, the transition
matrix elements drop down but simultaneously the
17
density of subbands per energy interval increases so
that if one considers the absorption averaged over an
19
energy interval containing many bands
#( !)#Ô! #
#
n##Ô!
#|p n | 2
#;
then this quantity varies only weakly. Here #|p n | 2
# is
21
the transition matrix element averaged over all states
in the nth subband, and summation goes over all subí
23
bands that enter the #Ô!íwide interval of #nal ení
ergies. Remember that the number of singleíparticle
25
states in a subband is determined only by the number
of the superlattice periods, and not by the band width.
27
Fig. 5. The change of |p n | 2 for the system of Fig. 4 under variation
of V , the well
depth:( a) V =
1:33,( b)
1.433,( c)
1.47,( d) 1.52.
It is interesting to observe the superlattice energy
spectrum and absorption variations over the parameí 29
ter region where the resonant state leaves one subband
and enters another. Fig. 5 illustrates this process at 31
the variation of the well depth. One can see how the
absorption peak moves from one subband to another 33
and follow the corresponding changes of the absorpí
tion band shapes: the property to be `resonant' goes 35
from the subband to its neighbour.
4. Absorption strength 37
Using the momentum matrix element data shown
above one can easily calculate such physical quantity 39
of interest as the absorption probability due to an elecí
tron transition from the ground subband to a higher 41
subband n. The standard perturbative approach for a
transition probability at one photon
absorption( see, 43

UNCORRECTED PROOF
6 A.V. Dmitriev et al. / Physica E 000 (2001) 000--000
PHYSE5182
for example, [14,15]) gives the absorption probabií
1
lity as
W # = 2#
Ô | Ó
H int
# | 2
#( # f - # i - Ô!)
= 2#e 2
m 2 c 2 Ô A 2
0 |p # | 2
#( # f - # i - Ô!)
=
# 2#e
m
# 2
N( !)
!c |p # | 2
#( # f - # i - Ô!)
=
# 2#e
m
# 2
I( !)
Ô! 2 c |p # | 2
#( # f - # i -
Ô!);( 4)
where subscripts i, f stand for initial and #nal state,
3
respectively;
Ó
H int
=- e
mc A Ó
p
is the interaction Hamiltonian with electromagnetic
5
#eld, A being the vector potential of the latter with the
amplitude A
0( !), and Ó
p being the electron momentum
7
operator; the photon #ux density
N( !) satis#es the
relation A 2
0( !)=( 2#Ôc=!)N( !);
I( !)=
Ô!N( !) is
9
the radiation spectral intensity. It was assumed in Eq.
( 4) that the superlattice length is small as compared
11
with the radiation wavelength, which seems reasoní
able for the infrared intraband transitions we considí
13
ered here.
On the other hand, as the superlattice we consider
15
is not too long, we will assume that each subband
Bloch level can be spectrally resolved separately from
17
the others. Assuming also that the light beam specí
tral width covers only one possible transition from a
19
Bloch state in the ground subband into another Bloch
state in an excited subband, after the integration over
21
the incident radiation frequency one obtains, taking
into account also the initial and #nal electron state deí
23
generacy due to the
perpendicular( iníplane) electron
motion 1 .
25
W( ! # ) = 2 #
k y k z
P
# 2#e
m
# 2
I( ! # )
Ô 2 ! 2
# c |p # | 2
;( 5)
where ! #
=( # f -# i )=Ô is the transition frequency, and
P is the statistical factor describing the electron Fermi
27
1 The degeneracy is connected with the neglect of the electron
e#ective mass di#erence in the layers of the superlattice. As a
result, in plane energy dispersion laws are similar in all subbands.
This is true for doped superlattices, and for compositional ones
this is an approximation.
Fig. 6. Light absorption probability vs photon energy for the
system of Fig. 3. Each point corresponds to a transition between
two Bloch electron states, one in the ground subband, the other in
an excited one. The line is a guide for the eye. The energy unit
is Ô 2
=( ma 2 ), the probability is in arbitrary units.
distribution in the ground subband. The electron mo
mentum conservation at an optical transition has been 29
taken into account in this equation. Factor 2 re#ects
the spin degeneracy. 31
Performing the elementary summation in
Eq.( 5),
one comes #nally to the expression for the light ab 33
sorption probability at the electron transition between
two Bloch subband states, one in the ground subband 35
and the other in an excited one.
W( ! # ) = 2#
c
# e
Ô 2 ! # m
# 2
I( ! # ) p 2
F# |p x | 2
=
# 2#e
Ô! # m
# 2
I( ! # )
c 2d |p x | 2
;( 6)
where p F# is the 2D in plane Fermi momentum of 37
electrons in the ground subband; 2D is the corre
sponding sheet electron density in a layer of the super 39
lattice; the light beam is directed along the superlat
tice layers with its polarisation parallel to the growth 41
axis x to ensure maximum absorption.
Formula( 6) gives the connection between the ab 43
sorption strength and |p z | 2 and clearly shows that the
absorption strength re#ects all the peculiarities of the 45
matrix element discussed
above( see Fig. 6). Other
physical quantities such as the cross section of the 47
photon absorption, absorption coe#cient, etc., can be
calculated similarly [6].

UNCORRECTED
PROOF
PHYSE5182
A.V. Dmitriev et al. / Physica E 000 (2001) 000--000 7
5. Comparison with properties of a single well
An analytic calculations have been performed in
Ref. [6] for a single well heterostructure with the same
3
model potential as in
Eq.( 1). One could expect that
the current results should #t to the conclusions of Ref.
5
[6] in the limit of remote wells, i.e. for a long period
superlattice with b=a#1.
7
We can employ |p n | 2 # as a variable characterising
optical absorption per one electron in a single quaní
9
tum well, where |p n | 2 is the momentum matrix eleí
ment squared, and # is the density of #nal states. An
11
analogous parameter for a superlattice is #|p n | 2
#=( E n -
E n-1 ), where #: : :# stands again for the averaging over
13
the states in nth subband, and E n is the energy of the
middle state in the
subband( when kb = #=2). Thus
15( E n - E n-1 ) is approximately the distance between
adjacent bands. This parameter characterises the abí
17
sorption in the area of nth subband, averaged over an
energy interval. As it is evident from Fig. 4, the two
19
variables coincide reasonably well already at b=a = 10.
6. Conclusion
We considered a superlattice with a unit cell alí
lowing for resonant states. In this system, the dipole
23
matrix element of the transitions between the lowest
subband and one of the excited subbands signi#cantly
25
increases when the #nal subband approaches the ení
ergy of the resonant state, peaking up in the resonant
27
subband. However, transitions to all subbands except
the resonant one have a zero matrix element at one of
29
the subband edges. The intraband absorption strength
will demonstrate similar behaviour. The shape of
31
the absorption peak corresponding to the resoí
nant subband is strongly a#ected by the `intrusion'
33
of the quasiílocalised states into the superlattice 35
spectrum.
Acknowledgements 37
A.V.D. and R.K. would like to thank Prof. W. Noltí
ing for his interest in this work and valuable discusí
sions. A.V.D. is grateful to DAAD for the support of
his stay at Humboldt University Berlin. A.V.D. and
V.V.M. acknowledge partial support of the Russian
Foundation for Basic Research and of the Foundation
`Universities of Russia: Basic Research'.
References
[1] V.P. Kaidanov, Yu. I., Ravich, Uspekhi Fiz, Nauk
145( 1985) 39
51.
[2] R.W. Pohl, Einf# uhrung in die Optik, Springer, Berlin, 1948, 41
Section 122.
[3] W. Heitler, The Quantum Theory of Radiation, Oxford 43
University Press, Oxford, 1944, Section
15( Chapter III).
[4] S.D. Beneslavskii, A.V. Dmitriev, N.S. Salimov, JETP 92 45
( 1987) 305.
[5] A.V. Dmitriev, Solid State Commun.
74( 1990) 237. 47
[6] A.V. Dmitriev, R. Keiper, V.V. Makeev, Semiconductor Sci.
Technol.
11( 1996) 1791. 49
[7] B.F. Levine, J. Appl. Phys.
74( 1993) R1.
[8] G. Bastard, Phys. Rev. B
24( 1981) 5693. 51
[9] G. Bastard, Phys. Rev. B
25( 1982) 7584.
[10] HungíSik Cho, P.L. Prucnal, Phys. Rev. B
36( 1987) 3237. 53
[11] S. Fl# ugge, Practical Quantum Mechanics, I and II, Springer,
Berlin, 1971. 55
[12] W. Kohn, Phys. Rev.
115( 1959) 809.
[13] N.W. Ashcroft, N.D. Mermin, Solid State Phys, New York 57
Holt, Rinehart and Winston, New York,
1976( Chapter 10).
[14] W. Heitler, The Quantum Theory of Radiation, Oxford 59
University Press, Oxford, 1944, Section
10( Chapter III).
[15] H.A. Bethe, Intermediate Quantum Mechanics, Benjamin, 61
New York, Amsterdam
1964( Chapter 12).