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Two types of extended states in random dimer barrier superlattices
Superlattices and Microstructures 37 (2005) 292–303 www.elsevier.com/locate/superlattices
Two types of extended states in random dimer barrier superlattices S. Bentata Laboratoire de Physique des Plasmas, des Mat´eriaux conducteurs et leurs Applications, D´epartement de Physique, Institut du Tronc Commun, Universit´e des Sciences et de la Technologie d’Oran (U.S.T.O.), BP1505 El-M’nouar, 31000 Oran, Algeria Received 8 March 2003; received in revised form 27 October 2003; accepted 5 November 2003 Available online 25 March 2004
Abstract We study numerically the effects of short-range correlated disorder on the electronic and transport properties of intentionally disordered GaAs–Alx Ga1−x As superlattices. We consider layers having identical thickness where the Al concentration x takes at random two different values with the constraint that one of them appears only in pairs, i.e. the random dimer barrier. Various physical quantities such as the conductance, the universal fluctuation conductance, the localization length, the resistance and its probability distribution are statistically computed by means of the transfer matrix formalism to discriminate the nature of the electronic states. In spite of the presence of disorder, the system exhibits two kinds of sets of propagating states lying below the barrier due to the characteristic structure of the superlattice. The states close to the resonance can be viewed as consisting of weakly localized states with very large localization length. In the band tails, i.e. for vanishing conductance, the states are strongly localized. The nature of the transition between these two regimes is quantitatively investigated through relevant physical quantities. © 2003 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering paper by Anderson [1], statistics and scaling of physical quantities in disordered electronic systems in the limit of low temperature continue to be a fascinating challenge from both technological and fundamental aspects [2]. Quantum coherence and randomness of microscopic details can induce, in particular, large fluctuations of physical quantities. These mesoscopic systems, containing about 1019 elementary objects like E-mail address: sam [email protected] (S. Bentata). 0749-6036/$ - see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2003.11.001
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electrons or atoms, are expected to be promising in the next generation of electronic nanostructures. The relevant characteristic length associated to the electronic wavefunction is the phase coherence length L φ describing the average distance over which an electron travels before its initial and final phases become incoherent. The quantum conductance fluctuations in mesoscopic systems are universal. Once L φ becomes larger than the system size L, the conductor can no larger be described by material constants. The microscopic details of the conductor play a significant role involving quantum mechanic interference effects. Therefore, even in the presence of disorder, a regime of weak localization can occur [3]. Furthermore, these systems undergo a continuous metal to insulator transition [1] leading to the existence of localized and delocalized phases separated by a critical point determined by a critical conductance according to the overquoted scaling theory [4] for dimension above two. One of the main universal conclusions concerns the one-dimensional case: almost all the eigenstates are exponentially localized in one dimension irrespective of the arbitrary amount of disorder. This is well established and known as a rigorously proven theorem. Obviously, there may be a set of states of zero measure at typical energies that remain extended and also a series of exceptions such as certain pseudo-random lattice Hamiltonians [5]. In this context, surprisingly recent developments on the subject have pointed out the possible suppression of localization in one-dimensional disordered systems induced by nonlinearity [6], correlation in disorder [7, 8] or long-range interactions [9]. One of the tasks of the theory is then to determine the probability distribution of the conductance for ensembles of disorder potentials. In the following we focus on the description of the distribution function of the conductance (or resistance) for onedimensional disordered superlattices (SLs). Recent experimental data have supported the existence of delocalized states in random dimer superlattices [10]. Such a phenomenon has been predicted by Dom´ınguez-Adame and co-workers [11] through a series of papers over the last decade. They have reported the existence of extended states in these systems exhibiting short-range correlation in structural disorder, the so-called random dimer quantum well superlattices (RDQWSL). In such a case, the tunnel effect appears as the physical mechanism breaking down the destructive interferences induced by disorder. To the best of our knowledge very little has been done for the case of cellular disorder [12], namely the case of dimer for which randomness is assumed in the height of the barriers. Up to now, only the one-dimensional Kronig–Penney model, especially for its simplicity, has been treated, i.e. the case of one-dimensional array of regularly spaced δ-function potentials with paired correlated δ-function strengths [13]. Indeed there is no explicit justification for such a limitation although the latter has been shown to be efficient in suppressing localization. Furthermore, the cellular disorder appears to be relevant in creating complex features in the miniband since the periodicity of the lattice is preserved along the growing axis and the potential is more structured. These two ingredients are expected to offer a large range of possibilities in modelling the nature of the minibands. Therefore, this situation has motivated us to examine numerically the effect of random dimer barriers (RDB) on the nature of the eigenstates of one-dimensional disordered superlattices (RDBSL). The paper is organized as follows: in Section 2 the formalism used to calculate the conductance, i.e. the one-parameter scaling of the sample, is developed within the framework of the transfer matrix for the transmission coefficient.
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For definiteness, we have considered samples of three hundred supercells whose length ˚ which is an appropriate size in describing mesoscopic systems. is about 2 × 104 A, Results and discussions are reported in Section 3 through physical quantities such as the conductance, the relative fluctuations of the conductance, the localization length, and the resistance and its probability distribution in the different regimes observed in these systems. Finally, Section 4 concludes the paper with a brief summary of the main results. 2. Formalism In this section, we study the electronic properties of the RDBSL in the stationary case. For definiteness, we consider quantum well-based SL constituted by two semiconductor materials with the same well width dw and barrier thickness db in the whole sample which in turns preserves the periodicity of the lattice along the growing axis; the unit supercell having the period d = dw + db . For an appropriate understanding of the RDB effect on the nature of the electronic and transport properties, the physical picture may be handled through the investigation of states close to the bottom of the conduction miniband with k⊥ = 0. As usual, nonparabolicity effects can be neglected without loss of generality. Under these circumstances, the one-electron one-band effective-mass Hamiltonian provides a satisfactory description 2 d 1 d − + VSL (z) ψ(z) = Eψ(z). (1) 2 dz m ∗ (z) dz Here the SL potential VSL derives directly from the different energies of the conductionband edge of the two semiconductor materials (GaAs and Alx Ga1−x As) at the interfaces. In this model of disordered SL, we consider that the height of the barriers takes at random only two values, namely V and V . These two energies are proportional to the two possible values of the Al fraction in the Alx Ga1−x As barriers, x, for x ≤ 0.45. The sequence of energies is short-range correlated since the V only appears forming pairs, e.g. V V V V V V V V V . . .. In the following treatment, we include the electron effective masses according to the different regions of the potential: m b and m b corresponding to barrier heights V and V , respectively, and m w to the well. 2.1. Analytical study As usual in scattering problems, we assume an electron incident from the left and define the reflection, r N , and the transmission, t N , amplitude by the envelope function at the contacts of the RDBSL: iκz e + r N e−iκz , z < z 1 − db /2, (2) ψ(z) = z > z N + db /2, t N eiκz , where z n denotes the coordinate along the growth direction of the centre of the nth barrier and κ 2 ≡ 2m w E/2 .
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Using Bastard’s continuity conditions [14], the knowledge of the 2×2 transfer matrix of the DBSL, M(N), allows us to relate t N and r N in a closed expression (see, e.g. [15, 16]). Defining A N = 1/t N∗ and B N = r N /t N , we have 1 A N BN Pn ≡ M(N) ≡ . (3) B N∗ A∗N n=N Here the transfer matrix can be recursively computed using the unimodular promotion matrices, αn βn , (4) Pn ≡ βn∗ αn∗ whose elements are given by i m b,n κ m w ηn αn = cosh(ηn db ) + − sinh(ηn db ) eiκdw 2 m w ηn m b,n κ i m b,n κ m w ηn βn = − + sinh(ηn db )e−iκdw . 2 m w ηn m b,n κ
(5)
Here ηn2 ≡ 2m b,n (Vn − E)/2 is the electron momentum in the nth barrier. Two values are allowed, η and η, corresponding to barrier heights V and V respectively. We take into account the variation of the effective mass with the Al mole fraction. Thus, the effective mass in the nth barrier m b,n can take the values m b and m b , corresponding to barrier heights V and V , respectively. It has been shown in the case of RDQWSL [17] that there exists an extended state when Re(αd ) = 0 and |Tr(Pr )| ≤ 2, where αd is the promotion matrix element associated to the defected (dimeric) supercells and Pr is the promotion matrix associated to regular supercells. In a straightforward manner it is easy to demonstrate that a similar condition holds in the case of RDBSL. In the latter, conditions for the existence of an extended state turn out to be formally the same. The following equations can be derived from these conditions cosh(ηdb ) cos(κdw ) − 1 m b κ − m w η sinh(ηdb ) sin(κdw ) ≤ 1, (6a) 2 mw η mbκ mwη 1 mbκ − sinh(ηdb ) sin(κdw ) = 0 (6b) cosh(ηdb ) cos(κdw ) − 2 mw η mbκ for energies below the lowest barrier. Similar equations can be found for energies above the lowest barrier; we consider only the former case for the sake of simplicity. The energy for which the last two conditions hold simultaneously (if it exists) corresponds to an extended state in a regular SL with a single dimer defect. It can be shown numerically that, in fact, this energy corresponds to an extended state for a SL with a randomly distributed ensemble of dimers. Unexpectedly, the DBSL support another type of extended state, its origin being completely different. Let us take a system constructed with two kind of blocks distributed randomly on a lattice. It is evident that the effects of randomness will be removed when, for a given electron energy, the positions of two consecutive different blocks can
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be interchanged [18]. In this case all the blocks of each type can be moved to one of the two sides. We can represent this process as follows A A A A A A A A A A A → A A A A A A A A A A A. This argument can be mapped to the case of a SL within the transfer matrix formalism assuming that each block corresponds to a SL period. Thus the condition for the randomness removing reduces to that of the commutation of the promotion matrices corresponding to the two types of barriers [P, P] = 0. This leads to the following equations ∗
Im(ββ ) = 0,
(7a)
βIm(α) = βIm(α).
(7b)
In the case of the DBSL these equations can be fulfilled simultaneously: Eq. (7a) is reduced to an identity whereas (7b) reduces to the following transcendental equation mbκ mw η sinh(ηdb ) + mwη mbκ mw η mbκ + sinh(ηdb ) mwη mbκ 1 mbκ mwη cosh(ηdb ) sin(κdw ) + − sinh(ηdb ) cos(κdw ) 2 mwη mbκ . (8) = 1 mbκ mwη cosh(ηdb ) sin(κdw ) + − sinh(ηdb ) cos(κdw ) 2 mwη mbκ In order to have an extended state, the energy at which the randomness is removed must lie within the bands corresponding to perfect SL’s formed with blocks of only one type |Tr(P)| ≤ 2,
|Tr(P)| ≤ 2.
(9)
To ascertain whether the states given by expressions (6) and (8) are truly extended states or not, we have performed a numerical study. 2.2. Numerical study For a proper understanding, we have treated the overquoted GaAs/Alx Ga1−x As as semiconductor SL. The fundamental variable in our treatment is the Al concentration x which fixes both the effective masses and the height of the potential barrier VSL within each layer of the structure. In particular, the SL potential may be expressed using the rule 60% for the conduction-band offset, via the relation [19] VSL (x) = 0.6 × 1.247x,
0 ≤ x ≤ 0.45.
(10)
This range of x ensures that Alx Ga1−x As presents a direct gap at the Γ valley. The effective mass in this region is given by m(x) = (0.067 + 0.083x)m 0
(11)
m 0 being the free electron mass. Within the previous description, several parameters can be varied, namely the height of the potential barriers V and V , the thickness of the wells dw and the potential barriers db ,
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the dimer concentration c in the SL, and the length of the system L through the number N of supercells. Each of these parameters has a specific physical bearing: the width dw determines the number of minibands while the thickness db acts on the spread of these minibands by controlling the strength of the interaction between neighbour states belonging to neighbour wells, and the dimer concentration c measures the degree of disorder. We have computed various physical quantities to discriminate the effects of the RDB on the nature of the eigenstates and their transport properties in mesoscopic disordered ˚ (i.e. N = 300). In particular, we have SL having length size up to L ∼ 2 × 104 A, numerically investigated the transmission coefficient τ (E) = |A N |−2 , the dimensionless dc conductance deduced from the Landauer formula [20] σ (E) =
τ (E) , 1 − τ (E)
(12)
the resistance ρ(E) = 1/σ (E), and its probability distribution W (all these functions are taken as dimensionless). All the results reported here correspond to an average over an ensemble of 104 realizations in order to obtain a desired accuracy for fitting the probability distribution of the resistance. 3. Results and discussions This section concerns the statistical description of the electronic transport process in the RDBSL mesoscopic devices, by means of numerical calculations of its transmission coefficient with the corresponding resistance probability distribution. Results will also be confirmed with analytical support resolutions. ˚ db = 26 A, ˚ V = 260 meV and Physical parameter values, such as dw = 26 A, V = 200 meV, are chosen, to obtain allowed minibands lying below the barriers. The corresponding effective masses are taken to be m w = 0.067m 0, m b = 0.096m 0 and m b = 0.089m 0 for respectively the quantum well, host barrier and dimer barrier layers where m 0 is the free electron mass. For convenience, the bottom of GaAs wells has been chosen as the energy reference. 3.1. Transmission coefficient For the above parameters, transmission coefficient versus electron incident energy τ (E) is plotted. Fig. 1 shows the position of the lower and upper band edges of the minibands corresponding to the two ordered superlattices with the two barrier heights V and V . One can observe the existence of one miniband under the well, ranging from 89 up to 199.99 meV for V and from 112 up to 200 for V . In Fig. 2 different dimer concentrations c = 0.3 and c = 0.4 are considered. The RDBSL transmission’s profile differs from the RDQWSL’s one, with the appearance of a second unity peak transmission at the critical energy E c and a mini gap (valley). Notice that the resonant energies lie inside the common region between the allowed minibands of V and V . This provides the existence of different types of eigenstates: those having low resistances near resonant energies and those, with considerable resistances, very far from the quasi perfect energy transparencies (i.e. near
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Fig. 1. Transmission coefficient of incident electron energy E for the two ordered superlattices with the two barrier heights. (a) for V = 200 meV and (b) for V = 260 meV.
band tails and inside the valley mini gap). This result is in perfect accord with that obtained by Bentata et al. [21] and Gomez et al. [16]. The two resonant energies E d Er1 = 131 meV and E c Er2 = 194 meV turn out to be those obtained by solving the analytical Eqs. (8) and (6b). We have to stress on the different origins of such resonances since the first one comes from the transparency of the dimer unit cell inside the host superlattice, while the second one reproduces directly the property of the commutative permutation of the unit cells (i.e. host unit cell and the dimer unit cell). For the last one, an equivalent two adjacent superlattices structure (adjacent sub structure blocks) is consequently given out, which is completely different with the fundamental concept of n-mers. Indeed, the ‘unicity’ of such particular new resonance type (E c ), is justified with the resolution of the analytical resonance equation (8) as shown in Fig. 2.
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Fig. 2. Transmission coefficient of incident electron energy E for different concentrations of dimers.
Moreover, it appears from both c = 0.3 and c = 0.4, that the position of such resonant energies does not depend on the degree of disorder. One can conclude that their positions obey directly to dimer and host unit cell technological characteristics (i.e. aluminium molar fraction x in the Alx Ga1−x As layers). 3.2. Resistance probability distribution The main idea, in this work, is the investigation of the eigenstate nature in RDBSL from the statistical description of the resistance distribution function. It is well know from Melkinov [24] and Berman et al. [22], that in the strong localized regime, when especially elastic scattering processes are taking into account, the function ln ρ has a Gaussian distribution (logarithmically normal in ρ) of a one-dimensional disordered system:
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Fig. 3. Probability distribution W of ln ρ for different energies (E = 120, 200 meV and E v = 164 meV).
W (ln ρ, L s ) = √
(L s −ln ρ)2 1 e− 4L s , 4π L s
L s 1, ln ρ 1
(13)
the variance ω = (2L)1/2 is related to L, which is measured in the mean free path le units. With this in mind, the resistance probability distribution is first calculated for band tails (i.e. E min , E max ), where the strongly localized regime is well known to be established. As shown in Fig. 3, the log normal distribution is obviously reproduced. Such localized investigation, is then applied to the energy valley E v . Indeed, according to the Melkinov [24] and Berman et al. [22] statements, the corresponding probability distribution presents the same characters. E v behaves consequently like a mini gap energy inside the allowed miniband structure [E min and E max ]. Since the first resonant of the transmission coefficient Er1 E d has been extensively examined during the last decade by various theoretical models which prove that this singular extended state is originated from a resonant tunnelling through pairs of barriers representing the dimer [9–23], in this section we have investigated a statistical study near the second resonant Er2 E c which was interpreted differently (the energy at which
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Fig. 4. Probability distribution W of ln ρ for different energies near E c .
the host and dimer blocks remain unchanged under permutation). As shown in Fig. 4, the probability distribution preserves its qualitative character: the Gaussian distribution (logarithmically normal in ρ) still survives, even near the resonance. Nevertheless, we have to stress on the quantitative study of the variance, since it is directly related to the diffusive random media effects, which governs the incident electron eigenstates nature. To this aim, Fig. 5 describes the variance behaviour versus the incident energy inside the allowed miniband neighbour to the second resonant E c . As we can see, Gaussian shape should disappear, providing a quasi Lorentzian one (narrow peak in the probability distribution), since incident energy approaches the resonant conditions. We have to point out that the disappearance of the Gaussian shape could be interpreted as a loss of the purely isolated character. Straightforwardly, the diffusion media response is then described, by giving the mean free path behaviour in the allowed miniband. It is easy to see, that since resonant conditions are satisfied, the incident electron can travel as freely as possible through the whole structure without any elastic diffusion process.
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Fig. 5. Variance ω of the Gaussian distribution versus the incident energy neighbour to the second resonant E c .
4. Conclusion We have analytically and numerically examined the effects of RDBSL on the electronic and transport properties of one-dimensional disordered superlattices and in particular GaAs/Alx Ga1−x As SLs. In the above description, we have introduced two fundamental ingredients to counteract the destructive influence of disorder: the periodicity along the growing axis and the short-range correlation. These two features create the conditions favouring tunnelling of electrons. We have shown the ability of the DBSL in suppressing localization by supporting two kinds of delocalized states lying within the potential structure by properly choosing the parameters of the DBSL. The origins of these delocalized states are completely different: one of them is due to short-range correlations whereas the other one is due to the commuting nature of the transfer matrices describing the system at certain energies. We would like to point out that the kind of commuting delocalized states we are describing are not characteristic of the DBSL in the sense that no dimer correlations are needed at all. It means that a SL with a binary distribution of barrier heights satisfying the
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following correlator Vn Vm = V 2 δnm , should exhibit the same kind of delocalized state. Furthermore, these commuting extended states seem to appear in a number of binary models, for instance the random binary Kronig–Penney model with delta barriers, where Eqs. (7a) and (7b) lead to the same resonant energy observed by Ishii [25] and discussed by Hilke and Flores [26]. The results obtained from the various computed quantities indicate the existence of two types of states; namely extended and strongly localized. The probability distribution of the resistance appears to be a relevant physical quantity. For the strongly localized regime our results confirm the analytical expression of the probability distribution of Melnicov [24]. This conclusion seems to be of universal validity since it has been reported for other types of disorder. The analytical expression of the probability distribution of the resistance deserves further investigation to explain in more detail its physical nature. This is the subject of a forthcoming paper. Acknowledgements The author would like to thank A. Brezini, F. Dominguez-Adame and I. Gomez, for valuable and helpful discussions. References [1] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [2] For a review see M. Janseen, Phys. Rep. 295 (1998) 1. [3] G. Bergmann, Phys. Rev. Lett. 49 (1982) 162; Phys. Rev. Lett. 48 (1982) 1046; Phys. Rev. B 25 (1982) 2937; Phys. Rev. B 28 (1983) 515; Phys. Rev. B 28 (1983) 2914; Phys. Rep. 107 (1984) 2. [4] E.A. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [5] N. Kumar, in: V. Srivastava, A.K. Bhatnagar, D.G. Naugle (Eds.), Ordering Disorder: Prospect and Retrospect in Condensed Matter Physics, AIP Conference Proceeding, vol. 286, 1994, p. 3. [6] Yu.S. Kivshar, S.A. Gredeskul, A. S´anchez, L. V´azquez, Phys. Rev. Lett. 64 (1990) 1693. [7] D. Dunlap, H.L. Wu, P. Phillips, Phys. Rev. Lett. 65 (1990) 88. [8] F.A.B.F. Moura, M.M. Lyra, Phys. Rev. Lett. 81 (1998) 3735. [9] A. Rodr´ıguez, V.A. Malyshev, F. Dom´ınguez-Adame, J. Phys. A: Math. Gen. 33 (2000) L161. [10] V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G.B. Parravicini, F. Dom´ınguez-Adame, R. G´omezAlcal´a, Phys. Rev. Lett. 82 (1999) 2159. [11] E. Diez, A. S´achez, F. Dom´ınguez-Adame, Phys. Rev. B 50 (1994) 14359; E. Diez, A. S´achez, F. Dom´ınguez-Adame, IEEE J. Quantum Electron. 31 (1995) 1919; E. Diez, A. S´achez, F. Dom´ınguez-Adame, G.P. Berman, Phys. Rev. B 54 (1996) 14550. [12] A. S´anchez, E. Maci´a, F. Dom´ınguez-Adame, Phys. Rev. B 49 (1994) 147. [13] L. de Kronig, W.G. Penney, Proc. R. Soc. London Ser. A 130 (1931) 499. [14] G. Bastard, Phys. Rev. B 24 (1981) 5693. [15] E. Maci´a, F. Dom´ınguez-Adame, Electrons, Phonons and Excitons in Low-Dimensional Aperiodic Systems, Complutense University, Madrid, 2000. [16] I. Gomez, F. Dom´ınguez-Adame, E. Diez, Physica B 324 (2002) 235. [17] F. Dom´ınguez-Adame, A. S´anchez, E. Diez, Phys. Rev. B 50 (1994) 17736. [18] E. Maci´a, F. Dom´ınguez-Adame, Phys. Rev. Lett. 76 (1996) 2957. [19] S. Adachi, J. Appl. Phys. 58 (1985) R1. [20] R. Landauer, IBM J. Res. Dev. 1 (1957) 223. [21] S. Bentata, B. Ait Saadi, H. Sidiki, Superlatt. Microstruct. 30 (2001) 297. [22] G.P. Berman, F. Dom´ınguez-Adame, A. S´achez, Physica D 107 (1997) 166. [23] S.B. Haley, P. Erd¨os, Phys. Rev. B 45 (1992) 8572. [24] V.I. Melnicov, ZETP Lett. 32 (1980) 244. Sov. Phys. Solid State 23 (1981) 444. [25] K. Ishii, Prog. Theor. Phys. Suppl. 53 (1973) 77. [26] M. Hilke, J.C. Flores, Phys. Rev. B 55 (1997) 10625.
Two types of extended states in random dimer barrier superlattices S. Bentata Laboratoire de Physique des Plasmas, des Mat´eriaux conducteurs et leurs Applications, D´epartement de Physique, Institut du Tronc Commun, Universit´e des Sciences et de la Technologie d’Oran (U.S.T.O.), BP1505 El-M’nouar, 31000 Oran, Algeria Received 8 March 2003; received in revised form 27 October 2003; accepted 5 November 2003 Available online 25 March 2004
Abstract We study numerically the effects of short-range correlated disorder on the electronic and transport properties of intentionally disordered GaAs–Alx Ga1−x As superlattices. We consider layers having identical thickness where the Al concentration x takes at random two different values with the constraint that one of them appears only in pairs, i.e. the random dimer barrier. Various physical quantities such as the conductance, the universal fluctuation conductance, the localization length, the resistance and its probability distribution are statistically computed by means of the transfer matrix formalism to discriminate the nature of the electronic states. In spite of the presence of disorder, the system exhibits two kinds of sets of propagating states lying below the barrier due to the characteristic structure of the superlattice. The states close to the resonance can be viewed as consisting of weakly localized states with very large localization length. In the band tails, i.e. for vanishing conductance, the states are strongly localized. The nature of the transition between these two regimes is quantitatively investigated through relevant physical quantities. © 2003 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering paper by Anderson [1], statistics and scaling of physical quantities in disordered electronic systems in the limit of low temperature continue to be a fascinating challenge from both technological and fundamental aspects [2]. Quantum coherence and randomness of microscopic details can induce, in particular, large fluctuations of physical quantities. These mesoscopic systems, containing about 1019 elementary objects like E-mail address: sam [email protected] (S. Bentata). 0749-6036/$ - see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2003.11.001
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electrons or atoms, are expected to be promising in the next generation of electronic nanostructures. The relevant characteristic length associated to the electronic wavefunction is the phase coherence length L φ describing the average distance over which an electron travels before its initial and final phases become incoherent. The quantum conductance fluctuations in mesoscopic systems are universal. Once L φ becomes larger than the system size L, the conductor can no larger be described by material constants. The microscopic details of the conductor play a significant role involving quantum mechanic interference effects. Therefore, even in the presence of disorder, a regime of weak localization can occur [3]. Furthermore, these systems undergo a continuous metal to insulator transition [1] leading to the existence of localized and delocalized phases separated by a critical point determined by a critical conductance according to the overquoted scaling theory [4] for dimension above two. One of the main universal conclusions concerns the one-dimensional case: almost all the eigenstates are exponentially localized in one dimension irrespective of the arbitrary amount of disorder. This is well established and known as a rigorously proven theorem. Obviously, there may be a set of states of zero measure at typical energies that remain extended and also a series of exceptions such as certain pseudo-random lattice Hamiltonians [5]. In this context, surprisingly recent developments on the subject have pointed out the possible suppression of localization in one-dimensional disordered systems induced by nonlinearity [6], correlation in disorder [7, 8] or long-range interactions [9]. One of the tasks of the theory is then to determine the probability distribution of the conductance for ensembles of disorder potentials. In the following we focus on the description of the distribution function of the conductance (or resistance) for onedimensional disordered superlattices (SLs). Recent experimental data have supported the existence of delocalized states in random dimer superlattices [10]. Such a phenomenon has been predicted by Dom´ınguez-Adame and co-workers [11] through a series of papers over the last decade. They have reported the existence of extended states in these systems exhibiting short-range correlation in structural disorder, the so-called random dimer quantum well superlattices (RDQWSL). In such a case, the tunnel effect appears as the physical mechanism breaking down the destructive interferences induced by disorder. To the best of our knowledge very little has been done for the case of cellular disorder [12], namely the case of dimer for which randomness is assumed in the height of the barriers. Up to now, only the one-dimensional Kronig–Penney model, especially for its simplicity, has been treated, i.e. the case of one-dimensional array of regularly spaced δ-function potentials with paired correlated δ-function strengths [13]. Indeed there is no explicit justification for such a limitation although the latter has been shown to be efficient in suppressing localization. Furthermore, the cellular disorder appears to be relevant in creating complex features in the miniband since the periodicity of the lattice is preserved along the growing axis and the potential is more structured. These two ingredients are expected to offer a large range of possibilities in modelling the nature of the minibands. Therefore, this situation has motivated us to examine numerically the effect of random dimer barriers (RDB) on the nature of the eigenstates of one-dimensional disordered superlattices (RDBSL). The paper is organized as follows: in Section 2 the formalism used to calculate the conductance, i.e. the one-parameter scaling of the sample, is developed within the framework of the transfer matrix for the transmission coefficient.
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For definiteness, we have considered samples of three hundred supercells whose length ˚ which is an appropriate size in describing mesoscopic systems. is about 2 × 104 A, Results and discussions are reported in Section 3 through physical quantities such as the conductance, the relative fluctuations of the conductance, the localization length, and the resistance and its probability distribution in the different regimes observed in these systems. Finally, Section 4 concludes the paper with a brief summary of the main results. 2. Formalism In this section, we study the electronic properties of the RDBSL in the stationary case. For definiteness, we consider quantum well-based SL constituted by two semiconductor materials with the same well width dw and barrier thickness db in the whole sample which in turns preserves the periodicity of the lattice along the growing axis; the unit supercell having the period d = dw + db . For an appropriate understanding of the RDB effect on the nature of the electronic and transport properties, the physical picture may be handled through the investigation of states close to the bottom of the conduction miniband with k⊥ = 0. As usual, nonparabolicity effects can be neglected without loss of generality. Under these circumstances, the one-electron one-band effective-mass Hamiltonian provides a satisfactory description 2 d 1 d − + VSL (z) ψ(z) = Eψ(z). (1) 2 dz m ∗ (z) dz Here the SL potential VSL derives directly from the different energies of the conductionband edge of the two semiconductor materials (GaAs and Alx Ga1−x As) at the interfaces. In this model of disordered SL, we consider that the height of the barriers takes at random only two values, namely V and V . These two energies are proportional to the two possible values of the Al fraction in the Alx Ga1−x As barriers, x, for x ≤ 0.45. The sequence of energies is short-range correlated since the V only appears forming pairs, e.g. V V V V V V V V V . . .. In the following treatment, we include the electron effective masses according to the different regions of the potential: m b and m b corresponding to barrier heights V and V , respectively, and m w to the well. 2.1. Analytical study As usual in scattering problems, we assume an electron incident from the left and define the reflection, r N , and the transmission, t N , amplitude by the envelope function at the contacts of the RDBSL: iκz e + r N e−iκz , z < z 1 − db /2, (2) ψ(z) = z > z N + db /2, t N eiκz , where z n denotes the coordinate along the growth direction of the centre of the nth barrier and κ 2 ≡ 2m w E/2 .
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Using Bastard’s continuity conditions [14], the knowledge of the 2×2 transfer matrix of the DBSL, M(N), allows us to relate t N and r N in a closed expression (see, e.g. [15, 16]). Defining A N = 1/t N∗ and B N = r N /t N , we have 1 A N BN Pn ≡ M(N) ≡ . (3) B N∗ A∗N n=N Here the transfer matrix can be recursively computed using the unimodular promotion matrices, αn βn , (4) Pn ≡ βn∗ αn∗ whose elements are given by i m b,n κ m w ηn αn = cosh(ηn db ) + − sinh(ηn db ) eiκdw 2 m w ηn m b,n κ i m b,n κ m w ηn βn = − + sinh(ηn db )e−iκdw . 2 m w ηn m b,n κ
(5)
Here ηn2 ≡ 2m b,n (Vn − E)/2 is the electron momentum in the nth barrier. Two values are allowed, η and η, corresponding to barrier heights V and V respectively. We take into account the variation of the effective mass with the Al mole fraction. Thus, the effective mass in the nth barrier m b,n can take the values m b and m b , corresponding to barrier heights V and V , respectively. It has been shown in the case of RDQWSL [17] that there exists an extended state when Re(αd ) = 0 and |Tr(Pr )| ≤ 2, where αd is the promotion matrix element associated to the defected (dimeric) supercells and Pr is the promotion matrix associated to regular supercells. In a straightforward manner it is easy to demonstrate that a similar condition holds in the case of RDBSL. In the latter, conditions for the existence of an extended state turn out to be formally the same. The following equations can be derived from these conditions cosh(ηdb ) cos(κdw ) − 1 m b κ − m w η sinh(ηdb ) sin(κdw ) ≤ 1, (6a) 2 mw η mbκ mwη 1 mbκ − sinh(ηdb ) sin(κdw ) = 0 (6b) cosh(ηdb ) cos(κdw ) − 2 mw η mbκ for energies below the lowest barrier. Similar equations can be found for energies above the lowest barrier; we consider only the former case for the sake of simplicity. The energy for which the last two conditions hold simultaneously (if it exists) corresponds to an extended state in a regular SL with a single dimer defect. It can be shown numerically that, in fact, this energy corresponds to an extended state for a SL with a randomly distributed ensemble of dimers. Unexpectedly, the DBSL support another type of extended state, its origin being completely different. Let us take a system constructed with two kind of blocks distributed randomly on a lattice. It is evident that the effects of randomness will be removed when, for a given electron energy, the positions of two consecutive different blocks can
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be interchanged [18]. In this case all the blocks of each type can be moved to one of the two sides. We can represent this process as follows A A A A A A A A A A A → A A A A A A A A A A A. This argument can be mapped to the case of a SL within the transfer matrix formalism assuming that each block corresponds to a SL period. Thus the condition for the randomness removing reduces to that of the commutation of the promotion matrices corresponding to the two types of barriers [P, P] = 0. This leads to the following equations ∗
Im(ββ ) = 0,
(7a)
βIm(α) = βIm(α).
(7b)
In the case of the DBSL these equations can be fulfilled simultaneously: Eq. (7a) is reduced to an identity whereas (7b) reduces to the following transcendental equation mbκ mw η sinh(ηdb ) + mwη mbκ mw η mbκ + sinh(ηdb ) mwη mbκ 1 mbκ mwη cosh(ηdb ) sin(κdw ) + − sinh(ηdb ) cos(κdw ) 2 mwη mbκ . (8) = 1 mbκ mwη cosh(ηdb ) sin(κdw ) + − sinh(ηdb ) cos(κdw ) 2 mwη mbκ In order to have an extended state, the energy at which the randomness is removed must lie within the bands corresponding to perfect SL’s formed with blocks of only one type |Tr(P)| ≤ 2,
|Tr(P)| ≤ 2.
(9)
To ascertain whether the states given by expressions (6) and (8) are truly extended states or not, we have performed a numerical study. 2.2. Numerical study For a proper understanding, we have treated the overquoted GaAs/Alx Ga1−x As as semiconductor SL. The fundamental variable in our treatment is the Al concentration x which fixes both the effective masses and the height of the potential barrier VSL within each layer of the structure. In particular, the SL potential may be expressed using the rule 60% for the conduction-band offset, via the relation [19] VSL (x) = 0.6 × 1.247x,
0 ≤ x ≤ 0.45.
(10)
This range of x ensures that Alx Ga1−x As presents a direct gap at the Γ valley. The effective mass in this region is given by m(x) = (0.067 + 0.083x)m 0
(11)
m 0 being the free electron mass. Within the previous description, several parameters can be varied, namely the height of the potential barriers V and V , the thickness of the wells dw and the potential barriers db ,
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the dimer concentration c in the SL, and the length of the system L through the number N of supercells. Each of these parameters has a specific physical bearing: the width dw determines the number of minibands while the thickness db acts on the spread of these minibands by controlling the strength of the interaction between neighbour states belonging to neighbour wells, and the dimer concentration c measures the degree of disorder. We have computed various physical quantities to discriminate the effects of the RDB on the nature of the eigenstates and their transport properties in mesoscopic disordered ˚ (i.e. N = 300). In particular, we have SL having length size up to L ∼ 2 × 104 A, numerically investigated the transmission coefficient τ (E) = |A N |−2 , the dimensionless dc conductance deduced from the Landauer formula [20] σ (E) =
τ (E) , 1 − τ (E)
(12)
the resistance ρ(E) = 1/σ (E), and its probability distribution W (all these functions are taken as dimensionless). All the results reported here correspond to an average over an ensemble of 104 realizations in order to obtain a desired accuracy for fitting the probability distribution of the resistance. 3. Results and discussions This section concerns the statistical description of the electronic transport process in the RDBSL mesoscopic devices, by means of numerical calculations of its transmission coefficient with the corresponding resistance probability distribution. Results will also be confirmed with analytical support resolutions. ˚ db = 26 A, ˚ V = 260 meV and Physical parameter values, such as dw = 26 A, V = 200 meV, are chosen, to obtain allowed minibands lying below the barriers. The corresponding effective masses are taken to be m w = 0.067m 0, m b = 0.096m 0 and m b = 0.089m 0 for respectively the quantum well, host barrier and dimer barrier layers where m 0 is the free electron mass. For convenience, the bottom of GaAs wells has been chosen as the energy reference. 3.1. Transmission coefficient For the above parameters, transmission coefficient versus electron incident energy τ (E) is plotted. Fig. 1 shows the position of the lower and upper band edges of the minibands corresponding to the two ordered superlattices with the two barrier heights V and V . One can observe the existence of one miniband under the well, ranging from 89 up to 199.99 meV for V and from 112 up to 200 for V . In Fig. 2 different dimer concentrations c = 0.3 and c = 0.4 are considered. The RDBSL transmission’s profile differs from the RDQWSL’s one, with the appearance of a second unity peak transmission at the critical energy E c and a mini gap (valley). Notice that the resonant energies lie inside the common region between the allowed minibands of V and V . This provides the existence of different types of eigenstates: those having low resistances near resonant energies and those, with considerable resistances, very far from the quasi perfect energy transparencies (i.e. near
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Fig. 1. Transmission coefficient of incident electron energy E for the two ordered superlattices with the two barrier heights. (a) for V = 200 meV and (b) for V = 260 meV.
band tails and inside the valley mini gap). This result is in perfect accord with that obtained by Bentata et al. [21] and Gomez et al. [16]. The two resonant energies E d Er1 = 131 meV and E c Er2 = 194 meV turn out to be those obtained by solving the analytical Eqs. (8) and (6b). We have to stress on the different origins of such resonances since the first one comes from the transparency of the dimer unit cell inside the host superlattice, while the second one reproduces directly the property of the commutative permutation of the unit cells (i.e. host unit cell and the dimer unit cell). For the last one, an equivalent two adjacent superlattices structure (adjacent sub structure blocks) is consequently given out, which is completely different with the fundamental concept of n-mers. Indeed, the ‘unicity’ of such particular new resonance type (E c ), is justified with the resolution of the analytical resonance equation (8) as shown in Fig. 2.
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Fig. 2. Transmission coefficient of incident electron energy E for different concentrations of dimers.
Moreover, it appears from both c = 0.3 and c = 0.4, that the position of such resonant energies does not depend on the degree of disorder. One can conclude that their positions obey directly to dimer and host unit cell technological characteristics (i.e. aluminium molar fraction x in the Alx Ga1−x As layers). 3.2. Resistance probability distribution The main idea, in this work, is the investigation of the eigenstate nature in RDBSL from the statistical description of the resistance distribution function. It is well know from Melkinov [24] and Berman et al. [22], that in the strong localized regime, when especially elastic scattering processes are taking into account, the function ln ρ has a Gaussian distribution (logarithmically normal in ρ) of a one-dimensional disordered system:
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Fig. 3. Probability distribution W of ln ρ for different energies (E = 120, 200 meV and E v = 164 meV).
W (ln ρ, L s ) = √
(L s −ln ρ)2 1 e− 4L s , 4π L s
L s 1, ln ρ 1
(13)
the variance ω = (2L)1/2 is related to L, which is measured in the mean free path le units. With this in mind, the resistance probability distribution is first calculated for band tails (i.e. E min , E max ), where the strongly localized regime is well known to be established. As shown in Fig. 3, the log normal distribution is obviously reproduced. Such localized investigation, is then applied to the energy valley E v . Indeed, according to the Melkinov [24] and Berman et al. [22] statements, the corresponding probability distribution presents the same characters. E v behaves consequently like a mini gap energy inside the allowed miniband structure [E min and E max ]. Since the first resonant of the transmission coefficient Er1 E d has been extensively examined during the last decade by various theoretical models which prove that this singular extended state is originated from a resonant tunnelling through pairs of barriers representing the dimer [9–23], in this section we have investigated a statistical study near the second resonant Er2 E c which was interpreted differently (the energy at which
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Fig. 4. Probability distribution W of ln ρ for different energies near E c .
the host and dimer blocks remain unchanged under permutation). As shown in Fig. 4, the probability distribution preserves its qualitative character: the Gaussian distribution (logarithmically normal in ρ) still survives, even near the resonance. Nevertheless, we have to stress on the quantitative study of the variance, since it is directly related to the diffusive random media effects, which governs the incident electron eigenstates nature. To this aim, Fig. 5 describes the variance behaviour versus the incident energy inside the allowed miniband neighbour to the second resonant E c . As we can see, Gaussian shape should disappear, providing a quasi Lorentzian one (narrow peak in the probability distribution), since incident energy approaches the resonant conditions. We have to point out that the disappearance of the Gaussian shape could be interpreted as a loss of the purely isolated character. Straightforwardly, the diffusion media response is then described, by giving the mean free path behaviour in the allowed miniband. It is easy to see, that since resonant conditions are satisfied, the incident electron can travel as freely as possible through the whole structure without any elastic diffusion process.
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Fig. 5. Variance ω of the Gaussian distribution versus the incident energy neighbour to the second resonant E c .
4. Conclusion We have analytically and numerically examined the effects of RDBSL on the electronic and transport properties of one-dimensional disordered superlattices and in particular GaAs/Alx Ga1−x As SLs. In the above description, we have introduced two fundamental ingredients to counteract the destructive influence of disorder: the periodicity along the growing axis and the short-range correlation. These two features create the conditions favouring tunnelling of electrons. We have shown the ability of the DBSL in suppressing localization by supporting two kinds of delocalized states lying within the potential structure by properly choosing the parameters of the DBSL. The origins of these delocalized states are completely different: one of them is due to short-range correlations whereas the other one is due to the commuting nature of the transfer matrices describing the system at certain energies. We would like to point out that the kind of commuting delocalized states we are describing are not characteristic of the DBSL in the sense that no dimer correlations are needed at all. It means that a SL with a binary distribution of barrier heights satisfying the
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following correlator Vn Vm = V 2 δnm , should exhibit the same kind of delocalized state. Furthermore, these commuting extended states seem to appear in a number of binary models, for instance the random binary Kronig–Penney model with delta barriers, where Eqs. (7a) and (7b) lead to the same resonant energy observed by Ishii [25] and discussed by Hilke and Flores [26]. The results obtained from the various computed quantities indicate the existence of two types of states; namely extended and strongly localized. The probability distribution of the resistance appears to be a relevant physical quantity. For the strongly localized regime our results confirm the analytical expression of the probability distribution of Melnicov [24]. This conclusion seems to be of universal validity since it has been reported for other types of disorder. The analytical expression of the probability distribution of the resistance deserves further investigation to explain in more detail its physical nature. This is the subject of a forthcoming paper. Acknowledgements The author would like to thank A. Brezini, F. Dominguez-Adame and I. Gomez, for valuable and helpful discussions. References [1] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [2] For a review see M. Janseen, Phys. Rep. 295 (1998) 1. [3] G. Bergmann, Phys. Rev. Lett. 49 (1982) 162; Phys. Rev. Lett. 48 (1982) 1046; Phys. Rev. B 25 (1982) 2937; Phys. Rev. B 28 (1983) 515; Phys. Rev. B 28 (1983) 2914; Phys. Rep. 107 (1984) 2. [4] E.A. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [5] N. Kumar, in: V. Srivastava, A.K. Bhatnagar, D.G. Naugle (Eds.), Ordering Disorder: Prospect and Retrospect in Condensed Matter Physics, AIP Conference Proceeding, vol. 286, 1994, p. 3. [6] Yu.S. Kivshar, S.A. Gredeskul, A. S´anchez, L. V´azquez, Phys. Rev. Lett. 64 (1990) 1693. [7] D. Dunlap, H.L. Wu, P. Phillips, Phys. Rev. Lett. 65 (1990) 88. [8] F.A.B.F. Moura, M.M. Lyra, Phys. Rev. Lett. 81 (1998) 3735. [9] A. Rodr´ıguez, V.A. Malyshev, F. Dom´ınguez-Adame, J. Phys. A: Math. Gen. 33 (2000) L161. [10] V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G.B. Parravicini, F. Dom´ınguez-Adame, R. G´omezAlcal´a, Phys. Rev. Lett. 82 (1999) 2159. [11] E. Diez, A. S´achez, F. Dom´ınguez-Adame, Phys. Rev. B 50 (1994) 14359; E. Diez, A. S´achez, F. Dom´ınguez-Adame, IEEE J. Quantum Electron. 31 (1995) 1919; E. Diez, A. S´achez, F. Dom´ınguez-Adame, G.P. Berman, Phys. Rev. B 54 (1996) 14550. [12] A. S´anchez, E. Maci´a, F. Dom´ınguez-Adame, Phys. Rev. B 49 (1994) 147. [13] L. de Kronig, W.G. Penney, Proc. R. Soc. London Ser. A 130 (1931) 499. [14] G. Bastard, Phys. Rev. B 24 (1981) 5693. [15] E. Maci´a, F. Dom´ınguez-Adame, Electrons, Phonons and Excitons in Low-Dimensional Aperiodic Systems, Complutense University, Madrid, 2000. [16] I. Gomez, F. Dom´ınguez-Adame, E. Diez, Physica B 324 (2002) 235. [17] F. Dom´ınguez-Adame, A. S´anchez, E. Diez, Phys. Rev. B 50 (1994) 17736. [18] E. Maci´a, F. Dom´ınguez-Adame, Phys. Rev. Lett. 76 (1996) 2957. [19] S. Adachi, J. Appl. Phys. 58 (1985) R1. [20] R. Landauer, IBM J. Res. Dev. 1 (1957) 223. [21] S. Bentata, B. Ait Saadi, H. Sidiki, Superlatt. Microstruct. 30 (2001) 297. [22] G.P. Berman, F. Dom´ınguez-Adame, A. S´achez, Physica D 107 (1997) 166. [23] S.B. Haley, P. Erd¨os, Phys. Rev. B 45 (1992) 8572. [24] V.I. Melnicov, ZETP Lett. 32 (1980) 244. Sov. Phys. Solid State 23 (1981) 444. [25] K. Ishii, Prog. Theor. Phys. Suppl. 53 (1973) 77. [26] M. Hilke, J.C. Flores, Phys. Rev. B 55 (1997) 10625.