Electronic Properties in a Hierarchical Multilayer Structure

Transcription

1 Commun. Theor. Phys. (Beijing, China) 35 (2001) pp c International Academic Publishers Vol. 35, No. 3, March 15, 2001 Electronic Properties in a Hierarchical Multilayer Structure ZHU Chen-Ping 1,2 and XIONG Shi-Jie 1 1 National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing , China 2 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing , China (Received December 6, 1999; Revised March 15, 2000) Abstract We investigate electronic properties of a hierarchical multilayer structure consisting of stacking of barriers and wells. The structure is formed in a sequence of generations, each of which is constructed with the same pattern but with the previous generation as the basic building blocks. We calculate the transmission spectrum which shows the multifractal behavior for systems with large generation index. From the analysis of the average resistivity and the multifractal structure of the wavefunctions, we show that there exist different types of states exhibiting extended, localized and intermediate characteristics. The degree of localization is sensitive to the variation of the structural parameters. Suggestion of the possible experimental realization is discussed. PACS numbers: Ph, w, h, y Key words: hierarchical structure, multilayer, multifractal 1 Introduction One-dimensional (1D) deterministic but aperiodic systems are a sort of objects with physical characteristics between that of periodic and disordered systems. Examples are the quasiperiodic structures which can be defined by the generating matrices or obtained from the projection of the periodic lattices in higher dimensions. [1] Fibonacci lattice and the new versions of the model with the similar distributive rules have been extensively investigated. [2] In recent years, the investigations on the quasiperiodic systems are no longer restricted to the structures of only two components, three or even arbitrary K-component structures are adopted to search the possible new properties induced by the additional components. [3] In general, the quasiperiodic or aperiodic structures fill in a portion of the intervenient blank zone between the periodic and disordered systems, and exhibit rich features in the electronic, optical and acoustic properties. Investigations on such systems are helpful in exploring the crossover processes from the typical Bloch behavior of the states to the Anderson localization. In this paper we present a new deterministic aperiodic model, a system with hierarchically distributed multiquantum wells, which exhibits distinguishing characteristics in electronic properties. The structure of the model is intrigued from a quasi-1d model of stretched polymers [4] by replacing every network pattern with a component structure of multi-quantum wells. The structure is formed in a sequence of generations, each of which is constructed with the same pattern but with the previous generation as the basic building blocks. The physical characteristics are strongly related to the motion of electrons in the layered direction. In this direction, we calculate the transmission spectrum of electrons which shows the multifractal behavior for systems with large generation index. From the analysis of the average resistivity and the multifractal structure of the wavefunctions, we show that there exist different types of states exhibiting extended, localized and intermediate characteristics. The degree of localization is sensitive to the variation of the structural parameters. The paper is arranged as follows. In the next section, we describe the structure of the hierarchical multiquantum wells. In the third section, we employ the transfer matrix method to calculate the transmission spectrum and discuss the general features from the multifractal analysis. In the fourth section, we calculate the average resistivity from the Landauer formula. In the fifth section, we investigate the wavefunctions of different types from the multifractal analysis of them. The last section is devoted to the summary of results and the discussion of possible experimental realization. 2 Structure of the Hierarchical Aperiodic Multi-quantum Wells Consider a step multi-quantum well structure with a 1D aperiodic potential, which consists of an alternative stacking of quantum wells of width d w and barriers of width d b. The width of a unit containing one well and one barrier is a constant (d = d w + d b ) but the heights of the barriers are hierarchically distributed. The profile of the potential in the growth direction (x-axis) can be described by the following formula, The project supported by National Natural Science Foundation of China under Grant No , and the China State Key Projects of Basic Research (G )

2 340 ZHU Chen-Ping and XIONG Shi-Jie Vol. 35 v 0, for d i + d w x d i+1 and mod (i,n) 0, v(x) = p l v 0, for d i + d w x d i+1 and mod (i,n) = 0, l = int(lni/ln n), 0, for d i < x < d i + d w, (1) where integer n and real number p are parameters describing the hierarchical structure, and v 0 is the basis barrier height. For the N th generation in the hierarchical series, there are n N units in the system and the unit index 1 i n N. The (N + 1)th generation is constructed by n-fold stacking of the N th generation and changing the height of both terminate barriers by multiplying the value with p. The first generation is formed by n units with the same height v 0 of all the barriers except for the last one which has height pv 0. In order to restore the reflection symmetry of the system, for every generation we add a barrier with the same height as the last one to the left end of the structure. Thus, in the Nth generation there are n N + 1 barriers separated by n N wells. Figure 1 illustrates an example of such a structure with n = 3, N = 3, and p = 1.5. By denoting a unit with barrier height p l v 0 as A l and a trivial unit with barrier height v 0 as B, the generations of the hierarchical series with n = 3 can be rewritten as S 1 = A 1 BA 0 BA 0 BA 1, S 2 = A 2 BA 0 BA 0 BA 1 BA 0 BA 0 BA 1 BA 0 BA 0 BA 2,, S N+1 = A N+1 [S N ] 3 [A N ] 1 A N+1,. Fig. 1 A hierarchical structure of alternative stacking of barriers and wells. The parameters are n = 3, p = 2.0, d b = d w, N = 3. In general, we have the following iteration rule for constructing the generations, S N+1 = A N+1 [S N ] n [A N ] 1 A N+1, S 1 = A 1 [BA 0 ] n [A 0 ] 1 A 1. (2) This is another type of Kronig Penney model with hierarchical aperiodic structure, and one may expect that there are some new features different from those of the periodic or quasiperiodic Kronig Penney model. [5] 3 Transmission Spectrum In the aperiodic multi-quantum wells the parallel wave vectors are still good quantum numbers. For the given parallel wave vectors the motion of electrons in the x direction is described by the 1D Schrödinger equation. It is well known that the central problem of 1D system is the degree of localization of electrons which is usually energy-dependent and can be affected by other factors. This behavior is directly reflected from the transmission spectrum. We assume that the system is connected to two semi-infinite leads of the well material at the ends. Imagine an electron is injected from the left lead, its wavefunction within a well (labeled as j) can be defined with a j and b j, corresponding to the amplitudes of forward and backward plane waves, respectively. The continuous conditions of the wavefunction and its derivatives at interface between the jth and the (j +1)th layers give two relations between (a j,b j ) and (a j+1,b j+1 ), which can be written in the transfer-matrix form [6] ) ( aj+1 b j+1 ( ) aj = T j, (3) b j where T j is a 2 2 transfer matrix. For the whole system, we have a product of matrices to connect the amplitudes at the left and right leads, ( ) ( ) ar al = T 2nN +1T 2n N T 1 T 0 b R ( T11 T 12 T 21 T 22 )( al b L b L ), (4) where T ij (i,j = 1,2) is an element of the gross transfer matrix. Note that in the injection lead there exists a reflective wave, while in the transmission lead there is only the transmitted one. The transmission probability T of the tunneling electron is therefore expressed as T = 1/ T 22 2, (5)

3 No. 3 Electronic Properties in a Hierarchical Multilayer Structure 341 which is energy-dependent. [7] For simplicity we merely consider the range 0 < E < v 0 in the present paper. generation. For all the generations it is the necessary condition for the transmission that the electron can transmit through the system of the first generation. This confines the transmitting peaks of all the generations within the peaks of the first one. Since every transmission peak in any generation undergoes n-fold splitting, with the generation index increased the spectrum becomes more complicated but the hierarchical feature remains. It approaches to a Cantor-set-like distribution in the limit of N. Meanwhile, the position and width of peaks are sensitive to parameters N, v 0, d b, d w, n and p. For instance, the basic units of the structure are changed by increasing n, one can investigate the effect of n in details. We calculate the transmission coefficient as a function of energy by keeping all other parameters unchanged and only changing n from 2 to 10. The transmission spectra form a pattern of attractors, where several forbidden areas for the transmission appear (See Fig. 3). Fig. 2 Transmission coefficient versus electronic energy. The parameters are n = 3, p = 1.5, d b = 0.5, d w = The unit of energy is v 0. (a) N = 1 (dashed line), N = 2 (dotted line); (b) N = 3; (c) N = 4. A group of typical numerical results of transmission spectra are shown in Fig. 2. One can see that for the structure of the first generation (N = 1), transmission peaks are quite broad. As the generation index N increases, the peaks become narrower and denser. Meanwhile, the distribution of the peaks is within the energy range where T(E) of the previous generation is nonzero. In general, one peak in the transmission spectrum of a given generation is split into n peaks in the next generation. Such an n-fold splitting of the peaks by increasing the generation index originates from the hierarchical structure which distinguishes itself from other structures. If we denote the number of peaks in the Nth generation as m(n), we have a relation m(n) = nm(n 1) = n N 1 m(1). From Eq. (3) it can be seen that the system of a given generation is constructed from the previous generation as the basic building block. Thus, it is a necessary condition for an electron to transmit through the system of a given generation that it should be able to transmit through the system of the previous Fig. 3 Transmission coefficient as a function of energy E for n = 3 9. The other parameters are d b = 0.5, d w = 10.0, p = 1.5 and N = 1. It can be seen from Fig. 2 that transmission spectra exhibit fractal feature when the generation index N is large. In order to study its general characteristics, we use the multifractal analysis [8] to investigate the nature of the obtained spectrum. We denote the weight of the transmission coefficient within the ith energy window for a given generation as p i, i.e., / N E p i = T (i) 2 T (i) 2, (6) i=1 where N E is the total number of energy windows in the

4 342 ZHU Chen-Ping and XIONG Shi-Jie Vol. 35 range between 0 and v 0, T (i) is the average transmission amplitude within energy range (E i,e i +δe), and δe is the step length in the calculation. The generalized partition function is defined as N E Z t (q) = p q i. (7) i=1 The multifractal exponent for the transmission spectrum can be calculated as The singularity function is 1 d α t = Z t (q)ln N E dq Z t(q). (8) f(α t ) = [ln Z t (q)/lnn E ] + qα. (9) Following Ref. [8], we calculate f(α t ) for various generations. Figure 4 demonstrates typical results of the f(α t ) curves. We can see that when generation index N grows, the distribution range of α t where the singularity function f(α t ) has significant values is broadened, implying the increasing degree of multifractal of the spectrum. Moreover, the distance from the summit of curve f(α t ) to the unity characterizes how singular the transmission spectrum is. We can see from Fig. 4 that the summit of curves is lowered by increasing N, reflecting the increase of number of gaps and consequently the increase of degree of singularity in T(E) spectrum. As N is large enough (N = 9 in Fig. 4), the transmission spectrum shows the complex Cantor-setlike behavior, and the singularity function f(α t ) displays a typical multifractal feature. Fig. 4 The curves of the singularity function f(α t) versus the multifractal exponent α t of the transmission spectrum for different generations of the hierarchical aperiodic structure. Hereafter, the parameters n, p, d w and d b are the same as those in Fig Average Resistivity The transfer matrices for transmission spectrum can be used in calculating average resistivity which is energydependent and is a plausible physical quantity for estimating the localization of the electronic states. This technique combined with the Landauer formula is a powerful approach used in 1D quasicrystal and disordered systems. [8] We assume that an N j -layer segment of the Nth generation of the hierarchical structure is embedded between two semi-infinite and perfect leads, and the transfer matrix for the embedded system is denoted as T N (N j ). Then, the Landauer formula for the energy-dependent dimensionless resistance R(E) of this finite segment is R N (E,N j ) = T N12 (N j ) 2. (10) As an effective criterion of the localization, we examine the average resistivity ρ N for this segment which is defined as ρ N (N j ) = N j i=1 ρ N (i) N j = 1 N j [ RN (E,1) 1 + R N(E,2) + + R N(E,N j ) ] (11) 2 N j for 1 N j n N + 1. The scaling behavior of the system can be investigated by examining the variation of ρ N (N j ) in changing N j. As pointed out in Ref. [9], for an extended state, ρ N (N j ) 0 as N j. Therefore, when we consecutively calculate the average resistivity by increasing N j, the average resistivity should monotonically decrease for the extended states and ρ N (N j ) for the localized ones. We indeed find these two types of states in the present hierarchical structure (See Figs 5a and 5b, respectively). Generally speaking, the function ρ N (E,N j ) N j for localized states and has an energydependent slope, and for some values of energy the slope remains the same for different generations which is shown in the inset of Fig. 5b. This means that the average resistivity shows a power-law dependence on the system size with unity exponent, suggesting a less degree of localization than that in other systems of weak localization. [9] It is notable that this behavior emerges at different generations N for different energy values. For example, ρ N (0.962,N j ) has power-law behavior when N 8 (See Fig. 5c) which is higher than the corresponding value of N in Fig. 5b. The intermediate behavior of states whose average resistivity displays long range oscillations can be seen from curves in Fig. 5a for the state N = 10,

5 No. 3 Electronic Properties in a Hierarchical Multilayer Structure 343 E = , and in Fig. 5b for N = 6, E = wavefunctions of the present hierarchical multi-well structure, which makes further specification of electronic states. In Fig. 6 we plot a set of amplitudes of wavefunctions as functions of layer index. For a given energy, by increasing the generation index the band structure of the system is changed, so we cannot determine the localization nature in the usual sense. However, from the figures one can intuitively see the increasing tendency of localization from lower generations to higher ones and different degrees of energy-dependent localization of the states in a given generation. A lot of states are intermediate, which are neither strictly localized nor extended. Similarly to the case of the generalized Fibonacci model, there are two types of the intermediate states: [10] one has the tendency to be extended (Fig. 6a, for example), the other has the tendency to be localized (Fig. 6b, for example). Many of the states show evident self-similar structure. A specific example is shown in Figs 6c 6e. We employ multifractal analysis to investigate different types of wavefunctions, and obtain more evidences for different localized degrees of electronic states. We define the scaling index α p of a wavefunction by where 1 d α p = NZ p (q)ln n dq Z p(q), (12) Z p (q) = n N j=1 P q j (13) with P j = ψ j 2 being the probability amplitude of the wavefunction at the jth geometric unit. Here n N is the total number of units of the Nth generation. And the singularity function is defined as f(α p ) = lnz p (q)/n lnn + qα. (14) Fig. 5 (a) Average resistivity as a function of the system size for state with energy E = in generations N = 2 to 10. (b) Average resistivity as a function of the system size for state with energy E = in generations N = 4, 5 and 6. While in the inset N = 7, 8 and 9. (c) Average resistivity as a function of the system size for state with energy E = in generations N = 9 and 10. In the whole range of energy, the results of the calculations of the average resistivity suggest the existence of extended, localized and intermediate states. For a large enough N, they may coexist in a small energy range, in consistence with the complex Cantor-set-like transmission spectrum. 5 The Multifractal Analysis of Wavefunctions Transfer-matrix method is also used in calculating It is well known that for an extended wavefunction the probability amplitudes P j 1/n N, so there is no multifractal structure and f(α p ) is defined only at a single point, f(α p = 1) = 1. On the other hand, for an extremely localized wavefunction the probability amplitudes are nonzero only on a finite number of sites, so one has f(α p = 0) = 0, and f( ) = 1. A multifractal structure appears only in the intermediate wavefunctions with a smooth distribution of α p and finite values of f(α p ) in a finite interval of α p. We can determine relative degree of localization by comparing the widths of the intervals [α p min,α p max ] of the singularity function. Curves in Fig. 7a display f(α p ) functions for states with different energies in a structure with N = 9, which indicates that the wider f(α p ) peak corresponds to the larger multifrac-

6 344 ZHU Chen-Ping and XIONG Shi-Jie Vol. 35 tal strength. with the generation index. Fig. 7 (a) Curves of f(α p) versus α p of wavefunctions with energy E = for generations N = 9, 10, 11 and 12. (b) Curves of f(α p) versus α p of wavefunctions with energy E = , , and in generation N = 9. Fig. 6 Wavefunctions for different energies and generations. (a) E = , N = 8; (b) E = , N = 10; (c) E = 0.75, N = 6; (d) E = 0.75, N = 7; (e) A segment of wavefunction in (d), with the same spatial variation as that in (c), displaying translational self-similarity of the wavefunction. The amplitudes of wavefunctions are in arbitrary units. In the same way we determine the nature of multifractal of states in different generations. Generally the interval becomes wider for increasing N. An example of the curves of f(α p ) (E = ) is shown in Fig. 7b, implying that the localization degree of certain state (or the localization tendency of the intermediate state) increases In the calculation, a careful extrapolation by varying the generation index is necessary to distinguish different types of states, because numerical results for a finite system always show smooth curves of f(α p ). To avoid uncertainty, we follow the approach of multifractal analysis of Hiramoto and Komoto [11] for finite systems. In Fig. 8 we plot f(α p min,n) and α p min (N) versus 1/N for different values of energy. It was pointed out that, if the wavefunction has a tendency to be extended or localized, the representative points of α p min (N) and f min (N) could turn out to be linear with respect to 1/N, but generally not in a sole slope. In Fig. 8a the representative points can be aligned in various lines converging into point α p min = 1 and f(α p min ) = 1 at N, corresponding to the decreasing of average resistivity in Fig. 5a. This implies that the state is extended or tending to be extended. Figure 8b shows another type of behavior where the representative points for small N can be aligned in lines converging into

7 No. 3 Electronic Properties in a Hierarchical Multilayer Structure 345 point α p min = 1 and f(α p min ) = 1, but those points for large enough N can be aligned in lines converging into point α p min = 0 and f(α p min ) = 0, corresponding to the increasing of ρ(e,n) in Fig. 5c. Without doubt this state can be viewed as a localized one or an intermediate one tending to be localized. However, there are some states for which the calculated representative points do not show any trace of such an asymptotic convergence, corresponding to some kind of intermediate behavior. Fig. 8 Representative points of α p min (squares) and f(α p min) (circles or stars) versus 1/N for wavefunctions. (a) E = ; (b) E = Finally, we would point out that the wavefunctions of many states in such a hierarchical aperiodic structure exhibit the feature of self-similarity. In these states there appear repeating segments with similar shapes in different length scales. The amplitudes of wavefunction in a higher generation are spatially modulated in quite the same way of that in the corresponding lower generation, and are reminiscent of the substructures of the hierarchical barrier-well system itself. This characteristic actually reflects the rudimental translational symmetry of the system. 6 Discussions and Conclusions In summary, we have presented a new structure with hierarchically distributed barriers and wells. By comparison with the Fibonacci sequence, there are two competitive effects in the present system: the rudimental translational symmetry reflected by the stacking of super-cells made of the previous generation, which tends to make states extended; and the hierarchical arrangement of the super-cells resulting in band splitting and increasing of localization trend. When one increases parameters n, p, or d b /d w, the former effect is strengthened, the gap ranges increase and the states are more localized. We have calculated numerically the transmission spectrum, average resistivity, and wavefunctions. It is found that in this system there exist electronic states with different extents of localization. Many of the states exhibit self-similar feature in wavefunctions. For large enough N, the transmission spectrum shows multifractal behavior. Moreover, the multifractal analysis for wavefunctions can be used to classify the different types of states and the results are compatible with the calculations of the average resistivity. All the investigated characteristics show that the present model could flexibly cover a large fraction of the gap between the 1D periodic systems and the disordered ones, making itself as another sample among the quasiperiodic and aperiodic models. The present model of hierarchical aperiodic structure can be applied to multiple quantum wells, superlattices, and quantum-dot arrays. [12] In this paper we confine our discussion on semiconductor multilayers. In recent years, GaN/AlGaN multiple quantum wells has been fabricated. The gap of GaN is larger than any other semiconductor materials, and the height of electronic barrier can be adjusted in a quite large range of energy. [13] We hope that this enables the experimental realization of the hierarchical structure discussed here. References [1] V. Elser, Acta Cryst. A42 (1986) 36. [2] R. Merlin, K. Bajema, R. Clarke, F.Y. Yuang and P.K. Bhattacharya, Phys. Rev. Lett. 55 (1985) 1768; L.X. HE, X.Z. LI and K.H. KUO, Phys. Rev. Lett. 61 (1988) 1116; M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett. 50 (1983) 1870; G. Gumbs and M.K. Ali, Phys. Rev. Lett. 60 (1988) [3] LIU You-Yan, LUAN Chang-Fu, DENG Wen-Ji and HAN

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