The motion of a single electron moving in one dimension is represented by time-independent Schrödinger s equation (1)

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1 International Journal Journal of Electronics of Electronics and Communication and Communication Engineering & Technology (IJECET), ISSN 0976 Engineering & Technology (IJECET) ISSN (Print), ISSN (Online) Volume, Number 1, Jan April (011), pp IAEE, IJECET I A E E NUERICAL COPUTATION OF EIGENENERGY AND TRANSISSION COEFFICIENT OF SYETRIC QUANTU DOUBLE BARRIER STRUCTURE WITH VARIABLE EFFECTIVE ASS IN PRESENCE OF ELECTRIC FIELD Arpan Deyasi Department of Electronics & Communication Engineering RCCIIT, Kolkata, India & Swapan Bhattacharyya Department of Computer Science & Engineering Asansol Engg College, Asansol, India ABSTRACT Theoretical computation of eigenenergy and transmission coefficient for symmetric quantum double barrier structure considering GaAs/Al x Ga 1-x As material composition has been carried out using transfer matrix method to study the resonant tunneling phenomenon under 1-D confinement which is a quantum-coherent mechanism, and also to study about the existence of quasi-bound states when the device is subjected to electric field. Device is made dimensionally asymmetric to observe the variation of tunneling probability in presence and absence of electric field to compute probability of resonant tunneling at specific energy values less than barrier potential.. Effective mass mismatch at junctions are considered throughout the analysis by varying the mole fraction of Al to estimate near accurate values of eigenenrgies and also of the transmission probabilities. Application of negative bias makes the possibility of quasibound states near zero energy. Keywords: Double Barrier, Transfer atrix ethod (T), Transmission Coefficient, Quasi-bound states INTRODUCTION With the emergence of nanotechnology, it is already found out by intense research activities that the existing problems of present VLSI-based electronics can be solved by quantum-confined semiconductor structures where miniaturization is possible beyond the existing saturation point. Confinement of electrons along reduced dimensions which are comparable to the electron wavelength such as quantum wells, wires and dots have led to numerous significant improvements in semiconductor physics which have been used already in several micro- and optoelectronics applications [1]. The region of confinement, usually nomenclature as quantum system, is coupled to the external world through tunneling barriers which vividly reflects the dominance of the quantum effects to understand the physical properties of heterostructure devices, and the discrete electronic states become resonant one. The likelihood that the electron will pass through the barrier 4

2 is given by the transmission coefficient, and this type of finite potential barrier problem demonstrates the phenomenon of quantum tunneling; which can be predicted by better mathematical modeling and numerical computation. From the viewpoint of low-bias application, resonant tunneling in quantum well-like structures produced considerable interest amongst theoretical researchers []-[4]. Easki and Tsu first have proposed a semiconductor symmetric double barrier structure [5] where electronic transport proceeds via resonant tunneling mechanism, which is constructed by growing two or more different semiconducting materials leading to a potential distribution along growth directions [6]; producing a series of energy levels and associated subbnads due to the quantization of carriers in the direction of confinement. Computation of the transmission coefficient for this type of structure is possible by solving Schrödinger s equation through potential barriers, and coupling between electronic states localized in semiconductor nanostructures is evaluated for understanding the performance of heterostructure devices. Chanda & Eastman [7], and later Christodoulides [8] calculated transmission coefficient for triangular barrier, whereas Scandella calculated the probability for trapezoidal barrier. In all the analysis, care was not taken for material parameters, which was first reported by Chang etc [9], and later Read etc [10], who computed resonant tunneling probability in semiconductor double barrier structure for different material parameters. Computation of transmission coefficient and corresponding eigenenergy for resonance transmission can be analyzed by solving time-independent Schrödinger s equation with proper boundary conditions so that a complex transcendental equation is formed having roots as the complex eigenenergies of the heterostrucutre [11]. DBRT structure subjected to electric field can be analyzed by several numerical techniques such as Variational ethod [1], Airy s function approach [13]-[15], Finite Element ethod [16], Transfer atrix Technique [1], [3 ], [13]-[15], [17]-[19], Weighted Potential ethod [0]. Comparing all these methods, TT is considered as one of the effective and accurate method by eminent researchers. Resonant tunneling phenomenon for these devices under applied field [1], [], [1]-[] provides a theoretical estimation about transmission coefficient. Existence of bound and quasibound states [11] speaks in favor of that statement. The present paper provides a comprehensive theoretical analysis of transmission coefficient based on tunneling probability of electrons between two barriers through the well in presence and absence of electric field using transfer matrix method. Introduction of the concept of variable effective mass in the barrier and the well layers makes the analysis more realistic compared to some earlier theoretical researches [], [5]-[6], [9]. As effective mass and barrier potential, both are function of the mole fraction for GaAs/Al x Ga 1-x As heterostructure, so computation of transmission coefficient can be considered as closer to the physical solution, which speaks in favor of existence of quasi-bound states in addition with bound states. Here potential profile of the device is assumed to take the form of alternate rectangular barriers at CB & VB edges along the same direction, and differences in barrier potentials are considered in calculations by changing the material compositions in barrier regions. A gradual change of applied bias from low to moderate value shows the consequent alteration in transmission coefficient for different potential conditions, and corresponding variation in eigenenergy. Logarithmic scale of transmission coefficient is chosen to study the generated profiles. 5

3 THEORETICAL ANALYSIS The motion of a single electron moving in one dimension is represented by time-independent Schrödinger s equation h m d + V ( z) = Eψ ( z) dz (1) For the double barrier structure under consideration as shown in Figure, we consider the solutions to Schrödinger s equation within each region for E<V = Aexp[ iκ1z] + B exp[ iκ z] for z<i 1 (.1) = C exp[ κ1z] + Dexp[ κ z] for I 1 <z<i (.) = F exp[ iκ1z] + Gexp[ iκ z] for I <z<i 3 (.3) = H exp[ κ1z] + J exp[ κ z] for I 3 <z<i 4 (.4) ψ z) = K exp[ iκ z] + Lexp[ iκ ] for I 4 <z (.5) ( 1 z where κ 1 & κ are defined as: m w E κ 1 = (3.1) h m ( ) & b V E κ = (3.) h The positions of interfaces have been labeled I 1, 1, I 3 and I 4 respectively. Using standard BenDaniel-Duke boundary conditions at each interface, and introducing transfer matrix technique, A = B K 8 L (4) We assume that there are no further heterojunctions to the right of the structure, so that no further reflections can occur and wave function beyond the structure can only have a traveling wave component moving to the right, i.e. the coefficient L must be zero. Thus equation (4) can be modified as: A K = (5) B 0 The probability interpretation of the wave function implies that A 7 = B 1 = 0, which gives 6

4 ( E) = 0 (6) This condition gives the eigenenergy for resonance transmission. Transmission coefficient can be given by- KK 1 T ( E) = = (6) AA Schrödinger s equation subjected to applied electric field of strength F along the quantized direction is given by: h m d + V ( z) qf( z) = Eψ ( z) dz Considering suitable substitutions, solution can be approximated as: ψ ( z ') = A' ] + B' ] for z<i 1 (8.1) ψ ( z ') = C' ] + D' ] for I 1 <z<i (8.) ψ ( z ') = F' ] + G' ] for I <z<i 3 (8.3) ψ ( z ') = H ' ] + J ' ] for I 3 <z<i 4 (8.4) ψ ( z ') = K' ] + L' ] for I 4 <z (8.5) (7) where 1/ 3 m V ( z) E 3 = ( qf) 1/ z / 3 h ( qf) (9) Similarly, using standard BenDaniel-Duke boundary conditions at each interface, and introducing transfer matrix technique, A' K ' = 1 ' ' 3 ' 4 ' 5 ' 6 ' 7 ' 8 ' (10) B' L' Similarly transmission coefficient can be obtained as: K' K' 1 T ( E) = = (11) A' A' ' ' and eigenenergy is obtained as- ' = ( E) 0 (1) NUERICAL RESULTS Theoretical investigation of symmetric quantum double barrier heterostructure starts with GaAs/Al x Ga 1-x As system having assumption that both the barriers have equal material composition; where it is also considered that barrier potential is solely a function of the material parameters, and effective masses of barrier region and in well region have a mismatch as it 7

5 depends on mole fraction of Al in barrier regions. These variations of effective mass at junctions are regarded throughout the analysis to compute eigenenergies and transmission probabilities. Figure 1 gives the comparative study of ground state and the next higher state eigenenergy values in presence of constant electric field and also the energies without field for varying thickness of barrier regions. It is observed that eigenenergy remains almost constant for unbiased condition, but varies when device is subjected to electric field. Figure 1: Comparative analysis of eigenenergy for first two states of symmetric double barrier structure in presence and absence of electric field with varying barrier width When the well width is varied even if at unbiased condition, energy values start decreasing for the ground state, and for the next state, it reduces after well dimension attains a particular value. Nature of the profile remains unchanged at biasing condition, as shown in Figure. Figure : Comparative analysis of eigenenergy for first two states of symmetric double barrier structure in presence and absence of electric field with varying well width 8

6 A change of barrier potential by varying mole fraction of Al also affects the effective mass mismatch at heterojunction which shows that nature of the eigenenrgy profiles for first two states remain unaltered even if bias is applied. Ground state energy remains constant whereas first higher state shows a linear incremental slope at first, and then attains constant value. It is evident from Figure 3. Figure 3: Comparative analysis of eigenenergy for first two states of symmetric double barrier structure in presence and absence of electric field with varying Al mole fraction in barrier region With increasing electric field, eigenenergy starts deceasing, as the device is bended more compared to the unbiased condition. For ground state, the rate of decrement is linear, whereas nonlinearity is encountered for higher states after field becomes moderate, as shown in Figure 4. Figure 4: Comparative analysis of eigenenergy for first two states of symmetric double barrier structure for different electric field 9

7 Transmission coefficient analysis starts in absence of electric field where it can be observed that away from resonance, an increasing barrier height leads to a decrease in the transmission coefficient, which is expected also from classical physics also. The resonance energies increase with increasing barrier height due to confinement effects, and the appearance of the second resonance at higher energy values speaks in favor of the existence of a second quasi bound state, where localized wavefunctions of these states are non- stationary. Figure 5: Comparative analysis of transmission coefficient of symmetric double barrier structure without electric field with varying barrier thickness Figure 6: Comparative analysis of transmission coefficient of symmetric double barrier structure without electric field with varying well thickness 30

8 A simultaneous study of Figure 5 & Figure 6 reveals the fact that with increasing well width, tunneling probabilities can be estimated at lower energy values. But by increasing barrier width, tunneling probability decreases. Increase of mole fraction of Al in barrier region increases the potential height, and simultaneously effective mass mismatch at heterojunctions increases, which causes a shift of resonance energies at higher energy ranges, as evident from Figure 7. Figure 7: Comparative analysis of transmission coefficient of symmetric double barrier structure without electric field with varying Al mole fraction In presence of electric field, nature of variation of transmission probability remains same where we have neglect the transmission probability foe E<0. Similarly, well width variation provides a similar change compared to the case when field is absent. Figure 8: Comparative analysis of transmission coefficient of symmetric double barrier structure in presence of electric field with varying barrier thickness 31

9 Figure 9: Comparative analysis of transmission coefficient of symmetric double barrier structure in presence of electric field with varying well thickness Variation of electric field affects the origin of resonance transmission. Higher field speaks about possibility of transmission at lower energy values, i.e, quasi-bound states exist. With decrease of negative bias, transmission is delayed w.r.t energy, and more bias gives origin of less peaks. Figure 10: Comparative analysis of transmission coefficient of symmetric double barrier structure for different electric field 3

10 Similarly, in presence of electric field, higher barrier potential shifts the resonance condition towards higher energy values, as shown in Figure 11, neglecting transmission probabilities for negative energy values. Figure 11: Comparative analysis of transmission coefficient of symmetric double barrier structure in presence of electric field with varying Al mole fraction CONCLUSION The composition of material in a heterostructure device is very important in determination of the electrical and optical properties. In the present paper, concentration of al is varied to observe the effect of changing barrier potential on eigenenergy and on transmission coefficient, and simultaneously effective mass mismatch at junctions are also taken into account to make the investigation more realistic, which is ignored in several earlier works. Peaks of transmission coefficient graphs show the resonance values, where a measure of the lifetime of such a localized resonance is given by the increase of the width of resonance. With increase of Al concentration, transmission is initiated at higher energies. Applied bias pulls down the second barrier to a large extent, and hence transmission probability is also modified. Existence of quasibound states can be verified by origin of resonance peaks, and varying well width and barrier thickness gives pictorial information of these states subjected to constant electric field. This analysis can be extended further to study the performance of complex quantum structures. REFERENCES 1. A.R.Sugg & J.P.C.Leburton, odeling of odulation-doped ultiple-quantum-well Structures in Applied Electric Fields using The Transfer-atrix Technique, IEEE J.Quantum Electron, 7,, Y.Guoa, B.L.Gu, J.Z.Yu, Z.Zeng & Y.Kawazoe, Resonant Tunneling in Step-Barrier Structures Under an Applied Electric Field, J. App. Phys, 84,, S.Vatannia & G.Gildenblat, Airy s Function Implementation of the Transfer-atrix ethod for Resonant Tunneling in Variably Spaced Finite superlattices, IEEE J.Quantum Electron, 3, 6,

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