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Author's personal copy Superlattices and Microstructures 47 (2010) 661–675
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Electronic spectra of one-dimensional nano-quasi-periodic systems under bias M.T. Pérez-Maldonado a,∗ , G. Monsivais b , V. Velasco c , R. Rodríguez-Ramos d , C. Stern e a
Facultad de Física, Universidad de La Habana, San Lázaro y L, Vedado, La Habana, CP 10400, Cuba
b
Instituto de Física, Universidad Nacional Autónoma de México, P.O. box 20-364, 01000 México, D. F., Mexico Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain
c d
Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, La Habana, CP 10400, Cuba
e
Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México, D. F., Mexico
article
info
Article history: Received 17 August 2009 Received in revised form 30 December 2009 Accepted 15 April 2010 Available online 11 May 2010 Keywords: Electronic spectra Electrified quasi-periodic systems Stark ladders
∗
abstract We investigate the properties of the energy spectra of quantum one-dimensional nano-quasi-crystals in the presence of external electric fields. These systems are modelled by means of finite sequences, ordered according to a Fibonacci rule which constituted of two blocks A (constant potentials of different heights defined on finite intervals) and B (delta potentials of different intensities). We use the electric field ability of producing Stark ladders in periodic systems to obtain well separated energy levels and to study the evolution of these levels when disorder is introduced. We show that this effect also allows us to predict the approximate position of the levels in the disordered system, in spite of its chaotic appearance at first view. We show, against the usual belief, that the nth Stark ladder in general is not formed exclusively from the levels of the nth band. The disorder is introduced in two different ways: by changing the distribution of the blocks or by changing the values of the delta potential intensities. In both cases we start from electrified periodic structures which are gradually perturbed to obtain electrified quasi-periodic structures. We show that the use of Fibonacci sequences as a particular case is not crucial and one can use the electric field to analyze any other type of quasiperiodic systems. © 2010 Elsevier Ltd. All rights reserved.
Corresponding author. Tel.: +53 7 8788956. E-mail addresses:
[email protected] (M.T. Pérez-Maldonado),
[email protected] (G. Monsivais).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.04.005
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1. Introduction The study of the electronic properties of quasi-crystals has been a very active field in the last years. Among the interesting properties related to these structures we can quote the localization phenomena, the fractal properties of the electronic spectra and the lack of translational symmetry, which could have practical applications. The Fibonacci system, a linear lattice constructed recursively, is the one-dimensional (1D) version of the quasi-crystals, and it has been the subject of intensive theoretical study as a model of these structures. These systems whose structural order is described by means of deterministic sequences, can be seen as an intermediate case between periodic and disordered one-dimensional systems. The electronic structure of the Fibonacci system has been investigated mainly in the single-band tightbinding limit. In these studies it was found that the energy spectrum is self-similar, and the energy band divides into three sub-bands, each of which further subdivides into three and so on [1–4], thus producing a singular continuous spectrum [5] which in the infinite limit reduces to a Cantor-like spectrum with dense energy gaps everywhere [6–9]. A much more realistic study was presented in [10] by using an empirical tight-binding (ETB) sp3 s∗ Hamiltonian [11], with no spin–orbit coupling, and arranging one monolayer of GaAs and one monolayer of AlAs in a Fibonacci sequence. The authors of [10] obtained the electronic structure of a superlattice having as period a 12th Fibonacci generation, containing 144 GaAs and 89 AlAs monolayers. Fibonacci generations ranging from the second to the fifth, but including more monolayers in each constituent block, were studied in [12,13] by using a similar ETB Hamiltonian, now including the spin–orbit coupling [14]. It was found in these more realistic studies [10,12,13,15] that the Fibonacci spectrum could only be observed for some energy ranges and for wavevectors in the vicinity of the superlattice 0 point. It was found also that the lower conduction and higher valence bands exhibited a selective spatial localization in the thickest GaAs slabs forming the superlattices and quantum wells. We follow here a different path to study the properties of the electronic spectra of one-dimensional quasi-crystals. We shall employ the capability of the electric fields to produce Stark ladders in periodic systems, thus having well spaced energy levels. As is well known the Stark ladders are formed by a series of energy levels equally spaced appearing in the energy spectrum of a particle under the simultaneous influence of a periodic potential and an external uniform electric field. They were predicted theoretically by Wannier [16] in 1962 and they were found experimentally years later [17]. Effects of the electric field on the transport properties of Fibonacci systems have been considered [18,19]. The localization induced by the electric field has been studied by means of the tight-binding method and a diagonal model for Fibonacci systems [20] and a disordered Anderson model [21]. The effect of the electric field on the states of a quasi-periodic system described by the Harper model has also been considered [22]. Due to computational requirements we shall study here systems up to the eighth Fibonacci generation only. Moreover, this allows us to keep the analysis as simple as possible. As we shall see, the main features discussed in the paper are independent of the precise size of the system. Because of the finite size of the systems considered here, similar to real structures with dimensions of the order of a few nanometers, we cannot hope to obtain perfect Stark ladders with perfect equally spaced levels. This will be even worse when the systems are partially disordered. Nevertheless, we shall see that if the disorder is not too big, as in the case of the one-dimensional quasi-crystals, the Stark ladders exhibit a reasonable behavior allowing its use. At the same time and due to the system’s finite size the fractal properties will be absent of their energy spectra. Our study will be focused on the individual evolution of the energy levels as a function of the disorder by using electric fields. It has been seen for acoustic waves that the intermediate structures when going from the periodic system to the Fibonacci one present variations in the frequency spectrum that can have potential applications in filtering [23]. The theoretical model employed here is based on sequences formed by different blocks A and B, the disorder being introduced in two different ways as discussed in detail in Section 3. In both cases we start from periodic structures under the influence of an electric field and they are gradually perturbed until we arrive to quasi-periodic structures. The model and the theoretical method to study the problem is presented in Section 2. The results are presented and discussed in Section 3. Conclusions are presented in Section 4.
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V(x) β5 β4
straight line of slope f β2
β1
β3
h3
x0=0
x1
x2
x3
x4
x5
xN
x
Fig. 1. Potential V (x), given by Eq. (1).
2. Theoretical model We will consider the problem of an electron of charge e = 1 in the one-dimensional piecewiseconstant potential represented in Fig. 1. The mathematical expression for this potential is
V ( x) =
∞ N −1 X
x≤0 N −1
βi δ(x − xi ) +
i=1 ∞
X
hi θ(x − xi )
0 < x < xN
i =1
x ≥ xN
xi ∈ [0, xN ], i = 0, . . . , N ,
(1)
where δ(x − xi ) and θ(x − xi ) are the Dirac delta function and the Heaviside step function, respectively, both having a discontinuity at x = xi . We will call interval number i the interval [xi , xi+1 ] and its length xi+1 − xi will be denoted by li . We note that when the lengths li and the intensities βi of the delta potentials are set equal to constants l0 and β1 ∀i, respectively, the potential (1) can be seen as the superposition of a periodic potential plus a linear term fx describing a potential due to a constant and uniform applied electric field of intensity f . However, there are other combinations of the values βi and the lengths li from which we can obtain periodic potentials (see for example the configuration discussed later in connection with Fig. 5(a)). The parameters hi are related to the electric field intensity f by means of the expression hi = li f = (xi+1 − xi )f .
(2)
This mathematical function is a very simplified model for potentials arising in real nanostructures, even in the absence of an external electric field ( f = 0). Following this idea of simplicity, in the case f 6= 0 we have not taken into account slopes within the intervals but an average slope f . The inclusion of slopes instead of flat intervals only introduces more complicated solutions in each interval (in terms of Airy functions) and this fact should not affect the general results of this work. For example, using typical values of [17], in a structure composed of quantum wells of depth 250 meV and width 30 Å, an electric field of intensity 104 V/cm will produce an increase of only 3 meV in the height of the bottom of the well at its right end. It is interesting to note that not always the flat intervals constitute an approximation. For example, the superposition of a ‘‘saw-tooth’’ potential with teeth of slope −f in each interval (xi , xi+1 ) and the straight line fx results in a potential of the form depicted in Fig. 1. In the following we will consider the potential V (x) as a function built from the superposition of Pi blocks of type Ai and blocks of type Bi . Blocks Ai are flat intervals of height Vi ≡ m=1 hm and length lA , independent of i. Blocks Bi are delta potentials of intensity βi (see Fig. 2). When f = 0 we have Vi = 0∀i and all the blocks Ai are equal to a given block A. In this case we can obtain periodic structures if we take, for example, all the blocks Bi equal to a given block B and we take the configuration AB
AB
AB
AB
AB . . . .
(3)
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Fig. 2. Schematic representation of blocks Ai , Bi .
We will study the evolution of the electronic spectrum (mainly when the system is in the presence of an external field) when the potential (1) becomes disordered by changing the distribution of the blocks or the values βi of the delta potential intensities. In both cases we will start from perfectly periodic structures which are gradually perturbed to obtain quasi-periodic structures. Here we will analyze only the more typical examples of quasi-periodic structures: the well-known Fibonacci sequences {Sj }. For the case f = 0 we built these sequences from the two building blocks A and B defined above and using the rule Sj+1 = {Sj , Sj−1 } with the initial conditions S1 = A and S2 = AB, being j the generation number. For example S3 = ABA, S4 = ABAAB, S5 = ABAABABA, etc. The sequence S8 appears explicitly in expression (27). We notice, for example, that both the periodic sequence AB AB AB and the Fibonacci sequence S4 can be written as AB
AB0
AB
(4)
where B0 is a block consisting in a delta potential of intensity β0 that, for the periodic case, is equal to the intensity β of the delta function of block B (and therefore B0 = B) and for S4 is equal to zero. It is clear that when β0 is equal to zero we can drop the block B0 from expression (4) and then it becomes the expression for S4 given above. In Section 4.3 we will write the periodic sequences as in expression (4) when varying continuously the value of β0 from β to zero, in order to transform continuously a periodic sequence into a Fibonacci sequence. We will also use this procedure in the more general case f 6= 0 which is the main objective of this paper. This case produces, in special situations, the Stark ladders. 3. Method of calculation. Transfer matrix approach We will solve the Schrödinger equation
−
1 d2 ψ 2 dx2
+ V (x)ψ = E ψ,
(5)
for the potential V (x) given in Eq. (1). We have used units such that h¯ = m = e = 1. Pj In the jth interval (xj < x < xj+1 ), where V (x) = i=1 hi ≡ Vj , Eq. (5) has solutions of the form
Aj exp[κj (x − xj )] + Bj exp[−κj (x − xj )] ψj (x) = Aj sin[kj (x − xj )] + Bj cos[kj (x − xj )] Aj (x − xj ) + Bj
if E < Vj if E > Vj
(6)
if E = Vj ,
where
κj =
p
kj =
p
2(Vj − E )
(7)
2(E − Vj ).
(8)
A relation between the coefficients Aj , Bj and Aj−1 , Bj−1 exists as a consequence of the boundary conditions satisfied by the solutions (6) and their derivatives. This relation can be expressed in matrix form:
Aj Bj
= Mj,j−1
Aj−1 Bj−1
,
(9)
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where Mj,j−1 is a matrix which relates the coefficients associated to the interval j − 1 with the coefficients associated to the interval j. This matrix has different forms depending on the values of E − Vj and E − Vj−1 . The explicit forms of the matrix elements for the different cases are given in the Appendix. Expression (9) allows us to find a relation between AN −1 , BN −1 and A0 , B0 :
A N −1 BN −1
=M
A0 B0
,
(10)
where M = MN −1,N −2 MN −2,N −3 . . . M3,2 M2,1 M1,0
≡
M11 M21
M12 M22
.
(11)
As a consequence of the boundary condition ψ0 (0) = 0 we obtain B0 = −A0 , and if we use as a normalization condition A0 = 1, Eq. (10) implies AN −1 = M11 − M12
(12)
BN −1 = M21 − M22 . Finally from the boundary condition ψN (0) = 0 we obtain the equation
(M11 − M12 ) exp[ikN −1 (xN − xN −1 )] + (M21 − M22 ) exp[−ikN −1 (xN − xN −1 )] = 0,
(13)
whose roots are the eigenenergies we are looking for. 4. Results and discussion All the potentials analyzed in this paper have a finite number of cells enclosed in a box with infinite potential outside, that is, V (x) = ∞ for x ≤ 0 and x ≥ xN (see Fig. 1). For brevity, in the following we use the phrase periodic system or finite periodic system to denote a section of a periodic system inside the big potential well. Due to the finite size of these systems we are not involved with the fractal properties of the spectrum of quasi-periodic potentials. Here we are interested in the evolution of individual levels as a function of the disorder and, as mentioned before, we use electric fields in order to have well separated levels. 4.1. Periodic systems For future reference we start by discussing the well-known effect of an electric field on the spectrum of finite periodic systems of the form given by expression (3). The period of these systems is p = lA . Fig. 3(a) and (b) show the evolution of their energy levels as a function of the electric field intensity f . Each line corresponds to an energy level. The characteristics of this spectrum were discussed many years ago in [24]. The only difference is that now we have used other form of the cells and we have depicted continuous lines showing the evolution of each level. For f = 0 we see that all the levels are grouped in bands as they must be, since in this case the systems are periodic. Each band in Fig. 3(a) has 21 levels because the potential considered in this figure has exactly 21 cells (the system has 21 blocks A and 21 blocks B). The system considered in Fig. 3(b) has only 8 cells and therefore the number of levels in each band is 8. It is easy to prove that the upper limit of the nth band of the finite sequence (3) is equal to the upper limit En of the nth band of the infinite Kronig–Penney Model. Furthermore, it is also equal to the nth level of an infinite well of length p which is given by [25] En =
n2 π 2 2p2
.
(14)
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Fig. 3. First and second bands of the electronic spectra of finite periodic structures as functions of the electric field intensity. (a) Structure with 21 cells, (b) Structure with 8 cells.
For the lower level of the nth band of sequence (3) there is not a closed expression but it is near to the lower limit en [26] of the nth band of the infinite Kronig–Penney Model which satisfies the expression [27] cos
p
2en p + β
√ √
sin( 2en p) 2en
= ±1.
(15)
This limit can be as close to the upper limit as wished, as long as β is made big enough. We have verified that the values of E1 , E2 , e1 , and e2 obtained from these formulas are in agreement with the plotted values on the left sides of Fig. 3(a) and (b), where we have used p = 1 and p = 2 respectively. At f = 0.55 we observe that the levels have been separated due to the presence of the electric field. In Fig. 3(a) we clearly see that all the levels coming from the first band, that is, the band of lower energy, appear equally spaced (except the two levels of the extremes that are a little bit more separated). This structure of equally spaced electronic levels is the well-known Stark ladder, first predicted by Wannier [16], whose existence was controversial for about 20 years [16,28–31]. According to Wannier, the nearest-neighbor energy level spacing 1E for an electron moving in an infinite periodic potential of period p and in the presence of an external electric field of intensity f must be equal to the product pf , i.e., 1E = pf . However, because our systems are finite, we cannot expect that the Stark ladders be perfect. So, the levels of the ladders are not exactly equally spaced. This is frequently more evident, with the extreme levels of the ladders, as occurs with the first ladder of Fig. 3(a) at f = 0.55. If one excludes these two levels the rest of the levels have a quite reasonable behavior. In fact, according with Wannier, since p = 1, the value of 1E must be equal to 0.55, which is in agreement with the average separation equal to 0.549 that has been calculated for these levels (excluding the two end levels). We have observed that the lower Stark ladder is formed for smaller values of f than the other Stark ladders. For example, the second Stark ladder of Fig. 3(a) is not yet well formed for f = 0.55. Higher values of f were needed in order to get a well defined ladder. Because the system considered in Fig. 3(b) has fewer cells (8 of them) the ladders are even more imperfect. However, we can see that they are reasonably well formed. In this case we have that the average separation between levels at f = 0.55 is equal to 1.096 while the theoretical value 1E, with p = 2, is equal to 1.1. In this figure one can also see the first level of the third band, which we have marked with an arrow.
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Fig. 4. First band of the electronic spectra of finite periodic structures as functions of the electric field intensity. (a) Structure with 34 cells, (b) Structure with 55 cells.
It is usually believed that the nth Stark ladder comes from the nth band [24]. For example, a superficial analysis of Fig. 3(b) could yield to the conclusion that the second Stark ladder comes from the second band. However, this is not true. Indeed, a detailed analysis of the lines shows that the fifth level of the first band is bending at f = 0.44 (due to level repulsion) to give rise to the first level of the second ladder. As a matter of fact, things can get more complicated. For example, the sixth level of the first band is bending at f = 0.34 to become the first level of the second ladder, but at f = 0.44 it is bending again to become the fifth level of the first ladder, etc. In order to simplify our discussion of the figures, we will speak as if the nth ladder is generated from the nth band valid. In Fig. 4(a) and (b) we show again results for the first band but now for the cases of 34 and 55 cells respectively. So, the number of levels of the band is equal to 34 and 55 respectively. The upper levels rise more rapidly as compared to those of Fig. 3(a) since at f = 0.55 all the levels must be separated by the quantity 1E ∼ 0.55. Due to this effect, the upper levels of Fig. 4(a) ‘‘cross’’ the first levels of the second band more rapidly than in Fig. 3(a). This effect is clearer in Fig. 4(b) than in Fig. 4(a) since in Fig. 4(b) we have more levels and they must rise even more rapidly in order to be separated by the quantity 1E = 0.55 at f = 0.55. In these figures we can see again the repulsion level phenomenon which is now even more involved. 4.2. Effects of disorder (I) In the following we analyze the effects of disorder. We have mentioned that because our systems are finite, we cannot expect the Stark ladders to be perfect. Things become more complicated when disorder is introduced [32–37]. However we will find that for the type of disorder shown by the Fibonacci sequences, it is possible to deduce a systematic behavior. In the six insets of Fig. 5 we show respectively six configurations of potentials for the case f = 0. The inset of Fig. 5(a) shows a finite periodic potential with 8 wide cells and 8 thin cells distributed as Cw Ct
Cw Ct
Cw Ct
Cw Ct
Cw Ct
Cw Ct
Cw Ct
Cw Ct
(16)
where Cw represents a wide cell of length lw and Ct a thin cell of length lt . We are taking lw = 2, lt = 1. Therefore the period p is equal to 3 and the number of periods is equal to 8. The above sequence of cells can be constructed by assigning the sequence of blocks AAB with the cells Cw and the sequence
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Fig. 5. Effect of disorder on the electronic spectra of finite structures as functions of the electric field intensity. The structures at f = 0 are schematically shown in the insets.
AB with the cells Ct . So, the sequence of blocks AABABAABABAABABAABABAABABAABABAABABAABAB
(17)
is equivalent to the sequence of cells (16). Now we will transform step by step the above periodic system into a Fibonacci sequence. In the inset of Fig. 5(b) we have moved the eight thin cells to the right in order to have the configuration Cw Cw Cw Cw Cw Cw Cw Cw
Ct Ct Ct Ct Ct Ct Ct Ct
(18)
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which is equivalent to: AABAABAABAABAABAABAABAABABABABABABABABAB.
(19)
Next (Fig. 5(c)) we have dropped three thin cells. So, this inset corresponds with the configuration Cw Cw Cw Cw Cw Cw Cw Cw
Ct Ct Ct Ct Ct
(20)
or, equivalently, AABAABAABAABAABAABAABAABABABABABAB.
(21)
Now we have two possibilities. In one of them, one transforms the periodic sequence of cells into a Fibonacci sequence of cells. In the other, one transforms the periodic sequence of blocks into a Fibonacci sequence of blocks. In this paper we only present the results related with the second possibility because it permits a more direct comparison with the results of Figs. 3, 4 and 6. However, the main conclusions are similar in both cases. Thus, in the inset of Fig. 5(d) we have moved one of the couples AB to the place that it would have in a Fibonacci sequence of blocks. This movement gives the configuration of cells Ct Cw Cw Cw Cw Cw Cw Cw Cw
Ct Ct Ct Ct
(22)
and the configuration of blocks ABAABAABAABAABAABAABAABAABABABABAB.
(23)
In the inset of Fig. 5(e) we have moved a second couple AB to the place that it would have in a Fibonacci sequence. This gives the configurations Ct Cw Ct Cw Cw Cw Cw Cw Cw Cw
Ct Ct Ct
(24)
equivalent to ABAABABAABAABAABAABAABAABAABABABAB.
(25)
At the end of this process we obtain the inset of Fig. 5(f) which corresponds to the configuration of cells Ct Cw Ct Cw Cw Ct Cw Ct Cw Cw Ct Cw Cw
(26)
whose configuration of blocks is the Fibonacci sequence S8 with 8 wide cells and 5 thin cells ABAABABAABAABABAABABAABAABABAABAAB.
(27)
In Fig. 5 we show the energy spectra associated with the respective insets. The set of points corresponding to the value f = 0 form the electronic spectra associated with the potentials of the insets. The evolution of the energy levels as a function of f is shown. At f = 0.55 we see a very interesting structure that will be discussed later. On the left of Fig. 5(a), corresponding to the periodic sequence of expression (16) and depicted in the inset, we see at least two groups of levels which evolve to a complex structure when the intensity f is increased. Since the system has a total amount of 16 cells, each band must have 16 levels. However, because each period has an internal structure consisting of the two cells Cw and Ct , each band must have an internal structure consisting of two sub-bands, each of them having 8 levels. This is so because there are 8 cells of each type. The 8 levels of lower energy shown in Fig. 5(a) are just the levels of the first sub-band of the first band B1 . According to the values of the parameters associated with this figure, the other 8 levels forming the second sub-band of B1 are in the same zone as the first sub-band of the second band B2 (see next paragraph). Therefore the second group of levels shown on the left of this figure has 16 levels. The second sub-band of B2 is not shown in the figure. The sub-band with the lower energy is in the zone where a periodic sequence formed uniquely of thin cells (whose length is lt = 1) has no levels as can be deduced from Eq. (14). Therefore, this first sub-band must be associated with the wide cells (whose length is lw = 2) and the second sub-band with the thin cells. We recall
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Fig. 6. Evolution of the electronic spectra of finite structures as functions of the electric field intensity from the periodic case to the Fibonacci sequence.
that the wider a well is, the lower its energy levels are. The first sub-band of B1 generates the Stark ladder shown by means of thick points on the right of the figure. Since the period of this potential is equal to 3, the theoretical value of 1E at f = 0.55 must be equal to 1.65, which is very close to the measured average value 1.647. As mentioned before, at f = 0 the second sub-band of B1 (which is associated with the thin cells) overlaps with the first sub-band of B2 (which is associated with the wide cells). This is so, because the relation between the width of the cells is lw = 2lt . In particular, from Eq. (14) one obtains that the upper level E2w of the second band of a periodic system formed uniquely of wide cells is equal to the upper level E1t of the first band of a periodic system formed uniquely of thin cells. Their value is E1t = E2w = 4.934, as can be seen on the left of Fig. 5(a). However, the 16 levels of this group evolve to form two different Stark ladders. One of them is due to the thin cells and the other to the wide cells.
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We have attached crosses at the right ends of the levels forming one of these Stark ladders and arrows to the levels forming the other Stark ladder. We observe that each level associated with a cross is always beside a level associated with an arrow. So, we have a composed ladder in which each of its rungs is indeed a couple of nearby rungs. This effect comes from a broken degeneracy due to the perturbation introduced inside an original simple cell of length equal to 3 that now has internal structure consisting of two subcells Cw and Ct . Therefore, an original Stark ladder associated with a periodic potential of period p = 3 becomes a composed Stark ladder. We also observe that this distribution of levels is similar to the distribution of the cells, that is, a wide cell is always besides a thin cell. This last effect is more clearly seen in Fig. 5(b) where we have put together the 8 wide cells and the 8 thin cells as shown in expression (18) and depicted in the inset. On the right of this figure we see, in the zone of lower energies, a first Stark ladder generated by the first band of the wide cells and with a nearest-neighbor level spacing of the order of 1.1 at f = 0.55. However, for higher energies we observe a more complicated structure. It consists of two Stark ladders separated by a distance 1S and having 8 levels and 7 spaces each of them. The nearest-neighbor level spacing for the first ladder is of the order of 1.1 at f = 0.55 and for the other ladder of the order of 0.55 at f = 0.55. One can understand this characteristic if one observes that now the local period associated with the wide cells is equal to 2 and therefore the levels associated with these cells must be separated by a distance of the order of 1.1 at f = 0.55. Similarly, the local period associated with the thin cells is equal to 1 and therefore their levels must be separated a distance of the order of 0.55 at f = 0.55. These two Stark ladders are separated one from each other because the cells are also distributed in that way. In the figure we can see that the distance 1S between these two Stark ladders is of the order of 0.55 at f = 0.55. This property can be also understood by means of the following argument. Let us suppose that cells Cw and Ct have infinite walls and lw = 2lt . We have proven before that the second level of Cw is equal to the first level of Ct , that is, E2w = E1t . If now the bottom of Ct is risen a quantity equal to flt the level E1t will be above the level E2w by the same quantity (that is, the nearest-neighbor spacing of the levels of the Stark ladder associated with the cells Ct ). If instead of rising the cell Ct we rise the cell Cw a quantity equal to flw then the level E2w will be above the level E1t by a quantity equal to flw (that is, the nearest-neighbor spacing of the levels of the Stark ladder associated with the cells Cw ). The cells have not actually infinite walls but we can expect that the mentioned property remains approximately valid. Then, we can expect that the separation 1S between the second ladder of the cells Cw and the first ladder of the cells Ct arranged as in expression (18) is equal to flt . On the other hand, if the configuration of the cells were Ct Ct Ct Ct Ct Ct Ct Ct
Cw Cw Cw Cw Cw Cw Cw Cw ,
(28)
the value of 1S would be equal to flw . Fig. 5(c) shows the results for the configuration of expression (20) depicted in the inset. It has 3 thin cells less than expression (18). Therefore, the number of levels associated with these cells is only 5 and they form a ladder with 5 rungs. However, we observe in this figure essentially the same characteristics as in Fig. 5(b). Fig. 5(d) shows the results for the configuration of expression (22) depicted in the inset. Here we have moved one of the thin cells to the place that it has in the final sequence (26). In this case there are only 4 thin cells together. This implies in turn that now there are only 4 levels in their associated ladder instead of 5 as was the case in Fig. 5(c). The level associated with the moved cell is now the first level of this group of 13 levels. It is separated from the second ladder associated with wide cells a quantity of the order of flw due to the argument just discussed. So, we can see a ladder with 9 levels with a nearest-neighbor spacing equal to flw . Fig. 5(e) shows the results for the configuration of expression (24) represented in the inset. Here we have moved a second thin cell to the place that it has in the final sequence (26). In this case there are only 3 thin cells together and therefore there are only 3 levels in their associated ladder. The level associated with the cell just moved is the third level of this group of 13 levels. Its separation from the second level is of the order of flt and its separation from the fourth level is of the order of flw as expected from the argument just discussed.
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When one continues moving the other thin cells to the place that they have in the final sequence (26) shown in the inset, one obtains, at the end, the behavior of the levels shown in Fig. 5(f). On the right side of this figure we have the corresponding spectrum of the electrified Fibonacci sequence S8 . This figure shows a rather complex structure which appears as chaotic. However, by using the reasoning discussed above it is easy to explain such structure. The number of levels shown in this figure is equal to 21 and they are just the 21 levels of the first band discussed in Fig. 3(a). In order to see that it is so, we proceed now to study the system from another point of view. In the following we start from a periodic structure and we decrease step by step the intensity of some delta potential in order to arrive at expression (27) shown in the inset of Fig. 5(f) as explained in connection with expression (4).
4.3. Effects of disorder (II) Let us consider again the periodic system considered in Fig. 3(a) which is of the form given by expression (3) with 21 cells or periods. As mentioned above, we can alternatively write expression (3) as AB
AB0
AB
AB
AB0
AB
AB0
AB
AB
AB0 . . .
(29)
where β0 is equal to β . If we now decrease the value of β0 from β to zero, the periodic system (29) is transformed in the Fibonacci sequence S8 of expression (27) whose behavior is shown in Fig. 5(f). In Fig. 6 we show the spectra of the system (29) for six different values of β0 . Only the first band levels are shown. In Fig. 6(a) we show the results for β0 = β . In this case the system is the same as the one considered in Fig. 3(a). Therefore, both figures are equal except that in Fig. 6(a) we have excluded the second band. In Fig. 6(f) we show the results for β0 = 0. In this case the system is the same as the one considered in Fig. 5(f), therefore, both figures are equal. In Fig. 6(b)–(e) we have taken intermediate values of β0 in order to see the evolution from the periodic case to the Fibonacci structure. Since the system of Fig. 6(a) has 21 cells its first band (shown on the left of the figure) has also 21 levels. Instead of that, the system of Fig. 6(f) has only 13 cells: 8 wide cells and 5 thin cells. Therefore one must expect that for f = 0 the levels of Fig. 6(f) appear in groups of 13 elements. Furthermore, Fig. 6(b)–(e) must show in some way this evolution from 21 levels to 13 levels. This in fact occurs. At f = 0 we see how the original band of 21 levels evolves until it is divided in 3 groups. In the zone of lower energies we have the first group which has 8 levels. These levels are associated with the 8 wide cells since they are in the zone of the first band of a periodic chain of wide cells. The second group has 5 levels and it is in the zone of higher energies. These levels are associated with the 5 thin cells since they are in the zone of the first band of a periodic chain of thin cells. The third group has 8 levels and it overlaps with the previous group of 5 levels. These 8 levels are associated with the 8 wide cells since they are in the zone of the second band of a periodic chain of wide cells. The last two groups overlap due to the effect discussed in connection with Fig. 5(a). The number of levels associated with the first two groups is equal to 13 as expected.
5. Conclusions In this paper we have used an external electric field to analyze theoretically some effects of disorder on the energy spectrum of one-dimensional quantum systems. The advantage of using an electric field for this analysis is that the field produces an arrangement of well separated energy levels that make easier to follow the modifications introduced by the disorder. When the system is periodic this arrangement of levels forms the so called Stark ladder which has the property that the levels are equally spaced by a quantity proportional to the electric field intensity. This allows to have a separation between levels as large as wished, as long as f is made large enough. This is a very convenient property because when the levels are sufficiently well separated it is easier to follow carefully their evolution as a function of disorder. However, we have been able not only to follow
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this evolution, but also to explain why the levels acquire the observed arrangement, in spite of the arrangement appearing as chaotic at first view. Although we have used rather small systems in order to simplify the analysis, the main idea is equally applicable for larger systems. This is so, because the separation between the levels is independent of the size of the system. In this paper we have used as an example of quasi-periodic structures the Fibonacci sequences. However, the use of Fibonacci sequences as a particular case was not crucial and one can use the electric field to analyze any other type of quasi-periodic systems such as the Thue–Morse, Godreche–Luck, etc. However, the arguments discussed in this paper permit to have an approximate idea of how the levels will be modified before the calculations are made. This comment is also applicable for the different possibilities to construct Fibonacci sequences, i.e., by means of cells or by means of blocks as was mentioned in the text in connection with the insets of Fig. 5(c) and (d). Although the effect of separating the energy levels forming the Stark ladder by means of an electric field was controversial for many years, it is currently well established and the phenomenon has been observed in three-dimensional systems [17]. Therefore the possibility to use an electric field to analyze carefully the effect of disorder, as proposed in this paper, could be also used in real three-dimensional multilayers.
Acknowledgements This work was supported in part by DGAPA-UNAM under project IN119509, and CONACYT project 82474. MTPM thanks for the warm hospitality received at Instituto de Física, Universidad Nacional Autónoma de México. Appendix. Explicit form for transfer matrices Mj ,j −1 A.1. Case E − Vj−1 < 0 and E − Vj < 0 In this case, the solutions of the Schrödinger equation (5) have the form
ψi (x) = Aj exp[κi (x − xi )] + Bi exp[−κi (x − xi )].
(30)
Boundary conditions at x = xj imply that 1 Aj + Bj = Aj−1 ∆j−1 + Bj−1 ∆− j −1
(31)
1 (κj − 2βj )Aj − (κj + 2βj )Bj = κj−1 ∆j−1 Aj−1 − κj−1 ∆− j−1 Bj−1 ,
where
∆j = exp(κj lj ).
(32)
These conditions can be written in matrix form
Aj Bj
= Mj,j−1
Aj−1 Bj−1
,
(33)
where
κj−1 βj ∆ 1+ +2 j −1 κj κj 1 = 2 κj−1 βj ∆j−1 1 − −2 κj κj
Mj,j−1
1 ∆− j −1
1 ∆− j −1
κj−1 βj 1− +2 κj κj κj−1 1+ κj
. βj −2 κj
(34)
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A.2. Case E − Vj−1 > 0 and E − Vj < 0 In this case we have the solutions
ψj−1 (x) = Aj−1 sin[kj−1 (x − xj−1 )] + Bj−1 cos[kj−1 (x − xj−1 )]
(35)
in the interval (xj−1 , xj ), and
ψj (x) = Aj exp[κj (x − xj )] + Bj exp[−κj (x − xj )]
(36)
in the interval (xj , xj+1 ). Boundary conditions imply that
Aj Bj
Aj−1 Bj−1
= Mj,j−1
,
(37)
where
kj−1
P sin Ωj−1 + cos Ωj−1 1 j κj Mj,j−1 = kj−1 2 Q sin Ω − cos Ω j−1
j
j −1
κj
Pj cos Ωj−1 − Qj cos Ωj−1 +
kj−1
κj
kj−1
κj
sin Ωj−1 sin Ωj−1
,
(38)
where Pj = 1 + 2
βj , κj
Qj = 1 − 2
βj , κj
Ωj = kj lj .
(39)
A.3. Case E − Vj−1 > 0 and E − Vj > 0 In both intervals (xj−1 , xj ) and (xj , xj+1 ) we have solutions of the form
ψi (x) = Ai sin[ki (x − xi )] + Bi cos[ki (x − xi )],
(40)
where i = j − 1, j. Boundary conditions lead to
Aj Bj
Aj−1 Bj−1
= Mj,j−1
,
(41)
where
βj
kj−1
2 sin Ωj−1 + cos Ωj−1 Mj,j−1 = kj kj sin Ωj−1
2
βj kj
cos Ωj−1 −
kj−1 kj
cos Ωj−1
sin Ωj−1
.
(42)
A.4. Case E − Vj = 0. Matching at x = xj In this case the solution in the interval (xj , xj+1 ) is
ψj = Aj (x − xj ) + Bj .
(43)
Analogously we obtain
Aj Bj
= Mj,j−1
Aj−1 Bj−1
,
(44)
where 2βj sin Ωj−1 + kj−1 cos Ωj−1 Mj,j−1 = sin Ωj−1
2βj cos Ωj−1 − kj−1 sin Ωj−1 . cos Ωj−1
(45)
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A.5. Case E − Vj = 0. Matching at x = xj+1 In this case the relation
A j +1 Bj+1
= Mj+1,j
Aj Bj
(46)
takes place, with
βj+1 lj + +2 lj 1 κj+1 κj+1 = βj+1 1 2 lj − −2 lj κj+1 κj+1
Mj+1,j
1
βj+1 1+2 κ j +1 . βj+1 1−2 κ j +1
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