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Negative differential resistance in aperiodic semiconductor superlattices
Physica A 303 (2002) 493–506
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Negative di!erential resistance in aperiodic semiconductor superlattices M.H. Tyc, W. Salejda ∗ Institute of Physics, Wroclaw University of Technology, Wybreze ˙ Wyspianskiego 27, 50-370 Wroclaw, Poland Received 30 May 2001
Abstract In/uence of external DC electric 4eld on the electron tunnelling through aperiodic semiconductor superlattices is studied numerically in the framework of generalized Kronig–Penney model and Landauer formalism. Spatial dependence of electron e!ective mass and dielectric constant as well as band non-parabolicity e!ects and non-abrupt interfaces are taken into account. In/uence of model parameters on the Landauer resistance is investigated. Areas of negative di!erential c 2002 Elsevier Science resistance are found and presented as two-dimensional grayscale maps. B.V. All rights reserved. Keywords: Fibonacci superlattices; Landauer resistance; Di!erential resistance; Kronig–Penney model
1. Introduction Recently, we have studied electronic transport properties of aperiodic semiconductor multilayers [1–3]. We have derived, using the Kronig–Penney model and dynamical maps approach, an analytical expression for Landauer conductance of Fibonacci-type semiconductor superlattices [1]. Extensive numerical studies performed [2,3] have shown that Landauer conductance exhibits resonant nature. We have worked out very eBcient numerical tools allowing us to analyse the quasi-one-dimensional e!ective mass equation which is the basis of the formalism applied here [4]. The main task of the present paper is to study the in/uence of external DC electric 4eld on the electron tunnelling through aperiodic semiconductor superlattices using more accurate numerical models than ones used so far in the literature [5,6]. ∗
Corresponding author. Fax: +48-71-3283696. E-mail addresses: [email protected] (M.H. Tyc), [email protected] (W. Salejda).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 4 9 5 - 2
494
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506 Table 1 A few initial GFSs for di!erent values of n and m l
2 Sl+1 = Sl Sl−1
1 2 3 4 5
B A ABB ABBAA ABBAAABBABB
l
Sl+1 = Sl2 Sl−1
1 2 3 4 5
B A AAB AABAABA AABAABAAABAABAAAB
In particular, we investigate the phenomenon of negative di5erential Landauer resistance in the framework of generalized Kronig–Penney model [7,8], including spatial dependence of electron e!ective mass and non-rectangular shape of potential barriers. The objects of our studies are so-called generalized Fibonacci superlattices (GFSs) [9], which are generated by an iterative process according to the following rule: Sl+1 = (Sl )n · (Sl−1 )m ;
(1)
where n and m are natural numbers; the index l de4nes so-called generation number of generalized Fibonacci superlattices (GFS), (Sl )n and (Sl−1 )m mean n and m repetitions of Sl and Sl−1 , respectively, and “·” denotes concatenation of strings. We start the construction of GFS with S1 = B, S2 = A, where A and B are di!erent types of superlattice segments, made of di!erent semiconducting materials. Table 1 presents a few initial generations of GFSs of two di!erent kinds. Let us comment brie/y on the internal structure of GFSs. The chains consist of B component subsequences of length lB = m and two kinds (excluding the chain ends) of A component subsequences of lengths lA = n and lA = n + m. Increase of m for 4xed n makes “short” and “long” A subsequences more distinct from each other (so the GFS becomes less ordered) whereas with increasing n for 4xed m the ratio lA =lA tends to 1; these chains correspond to more ordered structures. In the absence of external electric 4eld, the superlattice made from semiconducting layers may be considered, in the 4rst approximation, as the quasi-one-dimensional system of rectangular quantum wells separated by potential barriers in the conduction band (Figs. 1 and 2). External voltage bias V induces electric 4eld ˜ = [E(z); 0; 0]; E
E(z) = −
d (z) ; dz
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
495
Fig. 1. The one-dimensional model of the superlattice.
Fig. 2. Scheme of the superlattice potential in uniform electric 4eld.
which is approximately (when the spatial dependence of dielectric constant is neglected) uniform: E(z) = − V=L;
z (z) = V : L
The generalized Kronig–Penney model used here di!ers from the standard one in the following aspects: • In heterostructures studied here, the electron e!ective mass m(z) is position-dependent and, instead of the ordinary SchrNodinger equation, we use e!ective mass equation [10,11]: 2 ˝ d 1 d − + U (z) − e (z) (z) = E(z) : (2) 2 d z m(z) d z • The dielectric constant = (z) is also position-dependent and the Poisson equation −
d d (z) (z) = − en(z) dz dz
(3)
has to be solved in order to obtain electric potential (z) appearing in (2). We assume that free-carrier density n is suBciently small (n(z) 0) and does not change signi4cantly the external electric potential. • Band non-parabolicity e!ects are included in a simple way, assuming linear dependence of electron e!ective mass on its energy over the bottom of conduction
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M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
Fig. 3. Superlattice potential in external DC 4eld with spatial-dependent dielectric constant and non-rectangular barriers; the di!erence between dielectric constants in barriers and wells is exaggerated in comparison with GaAs=Al0:3 Ga0:7As semiconductor structures.
band [11]:
E − U (z) m(z; E) = mO (z) 1 + = m(0) (z) + m(1) (z)E : Eg (z) (0)
The parameters mO (0) , Eg , m(0) , m(1) depend (linearly in approximation) on the z coordinate through material composition, described by, e.g., mole fraction c(z) of Al in Alc Ga1−c As alloy. • The potential barriers are not necessarily ideally rectangular, as the potential U (z) depends on the material composition c(z), which can be modelled by any continuous function of z (Fig. 3).
2. Landauer resistance Let us consider the electronic transport in the superlattice placed between two metallic electrodes, as shown in Fig. 4. The Landauer resistance L [1,2] of considered system is equal to RL =
j0 h R = ; jt e 2 T
(4)
where T and R are transmission and re/ection probability, respectively. Eq. (4) was derived 4rst by Landauer [12–15] and is valid in zero temperature and under assumption that electron energy is preserved. In order to calculate Landauer resistance (4) let us consider the incident, re/ected and transmitted electron wavefunctions (0 , r and t , respectively) and corresponding current densities [5,16] given by 0 = eikz ;
r = eikz ;
t = eik z ;
e˝k e˝k e˝k ; jr = ||2 ; jt = ||2 ; (5) m m m where k = 2mE=˝2 , k = 2m(E + V )=˝2 , E is energy of incident electron and m is electron e!ective mass. j0 =
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
497
Fig. 4. The quantum model of electronic transport in superlattice; j0 , jr and jt denote the incident, re/ected and transmitted current densities, respectively.
We point out that re/ection and transmission coeBcients are equal to: k : (6) k We will calculate the Landauer resistance L solving Eq. (2) describing the system in the e!ective mass approximation, where the electrostatic potential is obtained from the solution of the Poisson equation (3). T = ||2
R = ||2 ;
3. Numerical algorithm We solve (2) numerically on the discrete grid of points with coordinates zn = ns;
n = 0; 1; : : : ; N ;
where s = L=N is called integration step and L denotes the total length of superlattice. The e!ective mass equation (2) can be discretized as follows [4,11]: 1 1 1 1 − + (7) n−1 + n− n+1 + Un n = E n ; mn−1=2 mn−1=2 mn+1=2 mn+1=2 where n = (zn ), mn±1=2 = m(zn±1=2 ), Un = U (zn ) and = 2m0 l20 W0 s2 =˝2 is dimensionless scale parameter [17,18]; m0 , l0 and W0 are the units of mass, length and energy, respectively. A similar discretization scheme is used in the case of Poisson equation (where 1=m is replaced with ). Eq. (7) can be written in transfer matrix form: (n) (n) P11 P12 n+1 n n (n) =P = ; (8) (n) (n) P21 P22 n n−1 n−1 where (n) P11 =1 +
mn+1=2 + mn+1=2 (Vn − E); mn−1=2
(n) P21 = 1;
(n) P22 =0 :
(n) P12 =−
mn+1=2 ; mn−1=2
Relations (7) and (8) form a new algorithm, which is an extension (to the case of position-dependent e!ective mass) of one proposed in Ref. [16].
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M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
For the whole given aperiodic superlattice, we have N +1 N
= P(N ) P(N −1) · · · P(1) P(0) =
P11
P12
P21
P22
0 −1
0 −1
=P
0 −1
:
Multiplying matrices P(n) and using the fact, that its product P is a real unimodular matrix when m−1=2 = mN +1=2 , we obtain from (5) and (6) the re/ection and transmission coeBcients: P e−iks + P − P eik s − P ei(k −k)s 2 12 11 21 22 R= ; P11 e−iks + P12 − P22 eik s − P21 ei(k −k)s T =
2 2 sin qs ; s −k)s ik i(k −iks + P12 − P22 e − P21 e P11 e which allows us to compute the Landauer resistance (4). The di!erential resistance rL = dV=dI was then calculated by numerical di!erentiation of RL = V=I using the formula RL ; rL 1 − (V=RL )(PRL =PV ) where PRL = RL (V + PV ) − RL (V ).
(9) (10)
4. Numerical results We have calculated the Landauer resistance (4) and di!erential resistance (9) of chosen GFSs as a function of incident electrons energy and applied external electric 4eld in a wide range of model parameters. We have performed our calculations for models of GaAs=Al0:3 Ga0:7 As generalized Fibonacci superlattices, assuming electron e!ective masses 0:067me and 0:092me , respectively, and conduction band discontinuity of 262 meV (according to Ref. [19]). As the “A” component of the structures (well) we took 10 monolayers thick GaAs layer, and as the “B” (“Barrier”) component—3 monolayers thick GaAs=Al0:3 Ga0:7 As layer (the thickness of the single monolayer is 0:28 nm). We investigated the in/uence of the following model parameters: (i) aperiodicity type (de4ned in Eq. (1) by the parameters l; m; n); (ii) energy E of incident electrons, (iii) external voltage bias V on the Landauer resistance and di!erential resistance of studied systems. We tested also the e!ect of including in our model band non-parabolicity and spatial dependence of dielectric constant.
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
499
Fig. 5. Landauer resistance (left) and di!erential resistance (right) maps. Dark areas correspond to small resistance and large negative resistance, respectively. The scheme of the studied system (GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 1) is shown below the legend.
Fig. 6. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 2.
500
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Fig. 7. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 3.
Fig. 8. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 2, n = 1.
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
501
Fig. 9. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 3, n = 1.
Fig. 10. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 6, m = 1, n = 2.
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M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
Fig. 11. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 2, n = 2.
Fig. 12. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 6, m = 2, n = 2.
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503
Fig. 13. Landauer resistance (solid line) and di!erential resistance (dashed line) graphs for the GFS from Fig. 6 for external voltage V = 75 mV. Resistance is shown in dimensionless units [DU]. Let us note that the changes of rL sign are associated with rL jumps from +∞ to −∞ and vice versa.
Fig. 14. Landauer resistance (solid line) and di!erential resistance (dashed line) graphs for the GFS from Fig. 6 for incident electron energy E = 30 meV. Resistance is plotted in dimensionless units [DU].
Selected results of our numerical calculations are presented in Figs. 5 –12 as twodimensional resistance and di!erential resistance maps in grayscale and as conventional graphs in Figs. 13–17.
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M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
Fig. 15. I –V characteristics of the GFS from Fig. 6 for incident electron energy E = 30 meV. Dimensionless units [DU] are used for current I . Each extremum corresponds to a change in di!erential resistance sign in Fig. 14.
Fig. 16. Landauer resistance of a GFS for external voltage V = 30 meV within parabolic (dashed line) and non-parabolic (solid line) band model. The above plot corresponds to a section of the rL grayscale map in Fig. 10.
5. Conclusions In the present paper, the quantum-mechanical model of electron ballistic transport in aperiodic superlattices has been formulated. The e!ects of position dependence of electron e!ective mass and dielectric constant as well as band non-parabolicity have been taken into account.
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505
Fig. 17. Landauer resistance the GFS from Fig. 6 as a three-dimensional plot.
From the results of the calculations performed, we can draw the following conclusions: (i) The Landauer resistance RL of aperiodic GFSs with external voltage imposed shows resonant nature; external electric 4eld shifts the resistance peaks towards lower electron energies (Figs. 5 –12). (ii) Regions of large negative di!erential Landauer resistance rL are observed near the resistance peaks (Figs. 14 and 15). Absolute values of RL and rL grow with increasing l (Figs. 5 –12). The highest |rL | values appear at high-voltage sides of the RL minima. Negative di!erential resistance is not observed at very low external voltages. (iii) The structure of the RL and rL maps complicates remarkably with increasing the l; m; n parameters, which manifests by growth of the number of resistance peaks and negative di!erential resistance areas. In particular, increase of n (Figs. 5 –7) causes rapid growth of the number of wells and the resistance peaks, whereas increase of m (Figs. 5, 8 and 9) causes mainly growth of barrier widths and less resistance peaks appear. (iv) For 4xed aperiodicity type, i.e., m; n = const in Eq. (1), and increasing l, the number of resistance peaks increases because the number of electron quasi-stationary states in the structure grows. The widths of the resistance peaks decrease due to fractal character of electron energy levels in GFSs [2] (Figs. 6, 10 and 11, 12). (v) Band non-parabolicity e!ects cause shifts of resonance peaks for higher electron energies (Fig. 16). In studied structures, the order of changes in RL due to spatial dependence of dielectric constant and the deviation from rectangular barrier shape is lower than 1%. (vi) The resistance and di!erential resistance maps are a convenient form of presentation of the computational results in comparison with three-dimensional plots (Fig. 17) and its two-dimensional sections (Figs. 13, 14 and 16). Using this kind of plots, one can easily localize the areas of negative di!erential resistance in (E; V ) variables.
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References [1] W. Salejda, Physica A 232 (1996) 769. [2] W. Salejda, P. Szyszuk, Physica A 252 (1998) 547. [3] W. Salejda, M. Kubisa, J. Misiewicz, K. Ryczko, M.H. Tyc, Proceedings of XXVIIth International School on Physics of Semiconducting Compounds “Jaszowiec ’98”, UstroWn-Jaszowiec, 7–12.06.1998, Acta Phys. Pol. A 94 (1998) 514. [4] W. Salejda, M.H. Tyc, J. Andrzejewski, M. Kubisa, J. Misiewicz, M. Just, K. Ryczko, Acta Phys. Pol. A 95 (1999) 881. [5] M. Castro, F. DomWXnguez-Adame, Physics Lett. A 225 (1997) 321. [6] R. Farchioni, G. Grosso, Phys. Rev. B 56 (1997) 1981. [7] C.L. Roy, A. Khan, Phys. Rev. B 49 (1994) 14 979. [8] C.L. Roy, C. Basu, A. Khan, Phys. Lett. A 196 (1995) 346. [9] G. Gumbs, M.K. Ali, Phys. Rev. Lett. 60 (1988) 1081. [10] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de Physique, Les Ulis Cedex, 1988. [11] P. Harrison, Quantum Wells, Wires and Dots, Theoretical and Computational Physics, Wiley, Chichester, 2000. [12] R. Landauer, Philos. Mag. 21 (1970) 863. [13] M. BNuttiker, Y. Imry, R. Landauer, S. Pinhas, Phys. Rev. B 31 (1985) 6207. [14] R. Landauer, Z. Phys. B, Cond. Matt. 68 (1987) 217. [15] P. ErdNos, R.C. Hendron, Adv. Phys. 31 (1982) 65. [16] B. MWendez, F. DomWXnguez-Adame, Am. J. Phys. 62 (1994) 143. [17] W. Salejda, M. Just, M.H. Tyc, Comput. Meth. Sci. Technol. 6 (2000) 73. [18] W. Salejda, M.H. Tyc, M. Just, Metody algebraiczne rozwi\azywania rWownania SchrNodingera (Algebraic Methods of Solving SchrNodinger Equation), an academic textbook available in Polish in electronic form at http:==www.if.pwr.wroc.pl=˜ssalejda=MARRS=. [19] S. Adachi, J. Appl. Phys. 58 (1985) R1.
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Negative di!erential resistance in aperiodic semiconductor superlattices M.H. Tyc, W. Salejda ∗ Institute of Physics, Wroclaw University of Technology, Wybreze ˙ Wyspianskiego 27, 50-370 Wroclaw, Poland Received 30 May 2001
Abstract In/uence of external DC electric 4eld on the electron tunnelling through aperiodic semiconductor superlattices is studied numerically in the framework of generalized Kronig–Penney model and Landauer formalism. Spatial dependence of electron e!ective mass and dielectric constant as well as band non-parabolicity e!ects and non-abrupt interfaces are taken into account. In/uence of model parameters on the Landauer resistance is investigated. Areas of negative di!erential c 2002 Elsevier Science resistance are found and presented as two-dimensional grayscale maps. B.V. All rights reserved. Keywords: Fibonacci superlattices; Landauer resistance; Di!erential resistance; Kronig–Penney model
1. Introduction Recently, we have studied electronic transport properties of aperiodic semiconductor multilayers [1–3]. We have derived, using the Kronig–Penney model and dynamical maps approach, an analytical expression for Landauer conductance of Fibonacci-type semiconductor superlattices [1]. Extensive numerical studies performed [2,3] have shown that Landauer conductance exhibits resonant nature. We have worked out very eBcient numerical tools allowing us to analyse the quasi-one-dimensional e!ective mass equation which is the basis of the formalism applied here [4]. The main task of the present paper is to study the in/uence of external DC electric 4eld on the electron tunnelling through aperiodic semiconductor superlattices using more accurate numerical models than ones used so far in the literature [5,6]. ∗
Corresponding author. Fax: +48-71-3283696. E-mail addresses: [email protected] (M.H. Tyc), [email protected] (W. Salejda).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 4 9 5 - 2
494
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506 Table 1 A few initial GFSs for di!erent values of n and m l
2 Sl+1 = Sl Sl−1
1 2 3 4 5
B A ABB ABBAA ABBAAABBABB
l
Sl+1 = Sl2 Sl−1
1 2 3 4 5
B A AAB AABAABA AABAABAAABAABAAAB
In particular, we investigate the phenomenon of negative di5erential Landauer resistance in the framework of generalized Kronig–Penney model [7,8], including spatial dependence of electron e!ective mass and non-rectangular shape of potential barriers. The objects of our studies are so-called generalized Fibonacci superlattices (GFSs) [9], which are generated by an iterative process according to the following rule: Sl+1 = (Sl )n · (Sl−1 )m ;
(1)
where n and m are natural numbers; the index l de4nes so-called generation number of generalized Fibonacci superlattices (GFS), (Sl )n and (Sl−1 )m mean n and m repetitions of Sl and Sl−1 , respectively, and “·” denotes concatenation of strings. We start the construction of GFS with S1 = B, S2 = A, where A and B are di!erent types of superlattice segments, made of di!erent semiconducting materials. Table 1 presents a few initial generations of GFSs of two di!erent kinds. Let us comment brie/y on the internal structure of GFSs. The chains consist of B component subsequences of length lB = m and two kinds (excluding the chain ends) of A component subsequences of lengths lA = n and lA = n + m. Increase of m for 4xed n makes “short” and “long” A subsequences more distinct from each other (so the GFS becomes less ordered) whereas with increasing n for 4xed m the ratio lA =lA tends to 1; these chains correspond to more ordered structures. In the absence of external electric 4eld, the superlattice made from semiconducting layers may be considered, in the 4rst approximation, as the quasi-one-dimensional system of rectangular quantum wells separated by potential barriers in the conduction band (Figs. 1 and 2). External voltage bias V induces electric 4eld ˜ = [E(z); 0; 0]; E
E(z) = −
d (z) ; dz
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
495
Fig. 1. The one-dimensional model of the superlattice.
Fig. 2. Scheme of the superlattice potential in uniform electric 4eld.
which is approximately (when the spatial dependence of dielectric constant is neglected) uniform: E(z) = − V=L;
z (z) = V : L
The generalized Kronig–Penney model used here di!ers from the standard one in the following aspects: • In heterostructures studied here, the electron e!ective mass m(z) is position-dependent and, instead of the ordinary SchrNodinger equation, we use e!ective mass equation [10,11]: 2 ˝ d 1 d − + U (z) − e (z) (z) = E(z) : (2) 2 d z m(z) d z • The dielectric constant = (z) is also position-dependent and the Poisson equation −
d d (z) (z) = − en(z) dz dz
(3)
has to be solved in order to obtain electric potential (z) appearing in (2). We assume that free-carrier density n is suBciently small (n(z) 0) and does not change signi4cantly the external electric potential. • Band non-parabolicity e!ects are included in a simple way, assuming linear dependence of electron e!ective mass on its energy over the bottom of conduction
496
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
Fig. 3. Superlattice potential in external DC 4eld with spatial-dependent dielectric constant and non-rectangular barriers; the di!erence between dielectric constants in barriers and wells is exaggerated in comparison with GaAs=Al0:3 Ga0:7As semiconductor structures.
band [11]:
E − U (z) m(z; E) = mO (z) 1 + = m(0) (z) + m(1) (z)E : Eg (z) (0)
The parameters mO (0) , Eg , m(0) , m(1) depend (linearly in approximation) on the z coordinate through material composition, described by, e.g., mole fraction c(z) of Al in Alc Ga1−c As alloy. • The potential barriers are not necessarily ideally rectangular, as the potential U (z) depends on the material composition c(z), which can be modelled by any continuous function of z (Fig. 3).
2. Landauer resistance Let us consider the electronic transport in the superlattice placed between two metallic electrodes, as shown in Fig. 4. The Landauer resistance L [1,2] of considered system is equal to RL =
j0 h R = ; jt e 2 T
(4)
where T and R are transmission and re/ection probability, respectively. Eq. (4) was derived 4rst by Landauer [12–15] and is valid in zero temperature and under assumption that electron energy is preserved. In order to calculate Landauer resistance (4) let us consider the incident, re/ected and transmitted electron wavefunctions (0 , r and t , respectively) and corresponding current densities [5,16] given by 0 = eikz ;
r = eikz ;
t = eik z ;
e˝k e˝k e˝k ; jr = ||2 ; jt = ||2 ; (5) m m m where k = 2mE=˝2 , k = 2m(E + V )=˝2 , E is energy of incident electron and m is electron e!ective mass. j0 =
M.H. Tyc, W. Salejda / Physica A 303 (2002) 493–506
497
Fig. 4. The quantum model of electronic transport in superlattice; j0 , jr and jt denote the incident, re/ected and transmitted current densities, respectively.
We point out that re/ection and transmission coeBcients are equal to: k : (6) k We will calculate the Landauer resistance L solving Eq. (2) describing the system in the e!ective mass approximation, where the electrostatic potential is obtained from the solution of the Poisson equation (3). T = ||2
R = ||2 ;
3. Numerical algorithm We solve (2) numerically on the discrete grid of points with coordinates zn = ns;
n = 0; 1; : : : ; N ;
where s = L=N is called integration step and L denotes the total length of superlattice. The e!ective mass equation (2) can be discretized as follows [4,11]: 1 1 1 1 − + (7) n−1 + n− n+1 + Un n = E n ; mn−1=2 mn−1=2 mn+1=2 mn+1=2 where n = (zn ), mn±1=2 = m(zn±1=2 ), Un = U (zn ) and = 2m0 l20 W0 s2 =˝2 is dimensionless scale parameter [17,18]; m0 , l0 and W0 are the units of mass, length and energy, respectively. A similar discretization scheme is used in the case of Poisson equation (where 1=m is replaced with ). Eq. (7) can be written in transfer matrix form: (n) (n) P11 P12 n+1 n n (n) =P = ; (8) (n) (n) P21 P22 n n−1 n−1 where (n) P11 =1 +
mn+1=2 + mn+1=2 (Vn − E); mn−1=2
(n) P21 = 1;
(n) P22 =0 :
(n) P12 =−
mn+1=2 ; mn−1=2
Relations (7) and (8) form a new algorithm, which is an extension (to the case of position-dependent e!ective mass) of one proposed in Ref. [16].
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For the whole given aperiodic superlattice, we have N +1 N
= P(N ) P(N −1) · · · P(1) P(0) =
P11
P12
P21
P22
0 −1
0 −1
=P
0 −1
:
Multiplying matrices P(n) and using the fact, that its product P is a real unimodular matrix when m−1=2 = mN +1=2 , we obtain from (5) and (6) the re/ection and transmission coeBcients: P e−iks + P − P eik s − P ei(k −k)s 2 12 11 21 22 R= ; P11 e−iks + P12 − P22 eik s − P21 ei(k −k)s T =
2 2 sin qs ; s −k)s ik i(k −iks + P12 − P22 e − P21 e P11 e which allows us to compute the Landauer resistance (4). The di!erential resistance rL = dV=dI was then calculated by numerical di!erentiation of RL = V=I using the formula RL ; rL 1 − (V=RL )(PRL =PV ) where PRL = RL (V + PV ) − RL (V ).
(9) (10)
4. Numerical results We have calculated the Landauer resistance (4) and di!erential resistance (9) of chosen GFSs as a function of incident electrons energy and applied external electric 4eld in a wide range of model parameters. We have performed our calculations for models of GaAs=Al0:3 Ga0:7 As generalized Fibonacci superlattices, assuming electron e!ective masses 0:067me and 0:092me , respectively, and conduction band discontinuity of 262 meV (according to Ref. [19]). As the “A” component of the structures (well) we took 10 monolayers thick GaAs layer, and as the “B” (“Barrier”) component—3 monolayers thick GaAs=Al0:3 Ga0:7 As layer (the thickness of the single monolayer is 0:28 nm). We investigated the in/uence of the following model parameters: (i) aperiodicity type (de4ned in Eq. (1) by the parameters l; m; n); (ii) energy E of incident electrons, (iii) external voltage bias V on the Landauer resistance and di!erential resistance of studied systems. We tested also the e!ect of including in our model band non-parabolicity and spatial dependence of dielectric constant.
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Fig. 5. Landauer resistance (left) and di!erential resistance (right) maps. Dark areas correspond to small resistance and large negative resistance, respectively. The scheme of the studied system (GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 1) is shown below the legend.
Fig. 6. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 2.
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Fig. 7. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 1, n = 3.
Fig. 8. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 2, n = 1.
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Fig. 9. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 3, n = 1.
Fig. 10. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 6, m = 1, n = 2.
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Fig. 11. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 5, m = 2, n = 2.
Fig. 12. Landauer resistance (left) and di!erential resistance (right) for a GaAa=Al0:3 Ga0:7 As GFS with l = 6, m = 2, n = 2.
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Fig. 13. Landauer resistance (solid line) and di!erential resistance (dashed line) graphs for the GFS from Fig. 6 for external voltage V = 75 mV. Resistance is shown in dimensionless units [DU]. Let us note that the changes of rL sign are associated with rL jumps from +∞ to −∞ and vice versa.
Fig. 14. Landauer resistance (solid line) and di!erential resistance (dashed line) graphs for the GFS from Fig. 6 for incident electron energy E = 30 meV. Resistance is plotted in dimensionless units [DU].
Selected results of our numerical calculations are presented in Figs. 5 –12 as twodimensional resistance and di!erential resistance maps in grayscale and as conventional graphs in Figs. 13–17.
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Fig. 15. I –V characteristics of the GFS from Fig. 6 for incident electron energy E = 30 meV. Dimensionless units [DU] are used for current I . Each extremum corresponds to a change in di!erential resistance sign in Fig. 14.
Fig. 16. Landauer resistance of a GFS for external voltage V = 30 meV within parabolic (dashed line) and non-parabolic (solid line) band model. The above plot corresponds to a section of the rL grayscale map in Fig. 10.
5. Conclusions In the present paper, the quantum-mechanical model of electron ballistic transport in aperiodic superlattices has been formulated. The e!ects of position dependence of electron e!ective mass and dielectric constant as well as band non-parabolicity have been taken into account.
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Fig. 17. Landauer resistance the GFS from Fig. 6 as a three-dimensional plot.
From the results of the calculations performed, we can draw the following conclusions: (i) The Landauer resistance RL of aperiodic GFSs with external voltage imposed shows resonant nature; external electric 4eld shifts the resistance peaks towards lower electron energies (Figs. 5 –12). (ii) Regions of large negative di!erential Landauer resistance rL are observed near the resistance peaks (Figs. 14 and 15). Absolute values of RL and rL grow with increasing l (Figs. 5 –12). The highest |rL | values appear at high-voltage sides of the RL minima. Negative di!erential resistance is not observed at very low external voltages. (iii) The structure of the RL and rL maps complicates remarkably with increasing the l; m; n parameters, which manifests by growth of the number of resistance peaks and negative di!erential resistance areas. In particular, increase of n (Figs. 5 –7) causes rapid growth of the number of wells and the resistance peaks, whereas increase of m (Figs. 5, 8 and 9) causes mainly growth of barrier widths and less resistance peaks appear. (iv) For 4xed aperiodicity type, i.e., m; n = const in Eq. (1), and increasing l, the number of resistance peaks increases because the number of electron quasi-stationary states in the structure grows. The widths of the resistance peaks decrease due to fractal character of electron energy levels in GFSs [2] (Figs. 6, 10 and 11, 12). (v) Band non-parabolicity e!ects cause shifts of resonance peaks for higher electron energies (Fig. 16). In studied structures, the order of changes in RL due to spatial dependence of dielectric constant and the deviation from rectangular barrier shape is lower than 1%. (vi) The resistance and di!erential resistance maps are a convenient form of presentation of the computational results in comparison with three-dimensional plots (Fig. 17) and its two-dimensional sections (Figs. 13, 14 and 16). Using this kind of plots, one can easily localize the areas of negative di!erential resistance in (E; V ) variables.
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