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Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field
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Numerical computation of eigenenergy and transmission coefficient of symmetric quantum double barrier structure with variable effective mass in presence of electric field

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  • 1. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976International Journal of Electronics and Communication IJECET– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEEngineering & Technology (IJECET)ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online)Volume 2, Number 1, Jan – April (2011), pp. 24-34 ©IAEME© IAEME, http://www.iaeme.com/ijecet.html NUMERICAL COMPUTATION OF EIGENENERGY AND TRANSMISSION COEFFICIENT OF SYMMETRIC QUANTUM DOUBLE BARRIER STRUCTURE WITH VARIABLE EFFECTIVE MASS 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, IndiaABSTRACT Theoretical computation of eigenenergy and transmission coefficient forsymmetric quantum double barrier structure considering GaAs/AlxGa1-xAs materialcomposition has been carried out using transfer matrix method to study the resonanttunneling 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 subjectedto electric field. Device is made dimensionally asymmetric to observe the variation oftunneling probability in presence and absence of electric field to compute probability ofresonant tunneling at specific energy values less than barrier potential.. Effective massmismatch at junctions are considered throughout the analysis by varying the molefraction of Al to estimate near accurate values of eigenenrgies and also of thetransmission probabilities. Application of negative bias makes the possibility of quasi-bound states near zero energy.Keywords: Double Barrier, Transfer Matrix Method (TMM), Transmission Coefficient,Quasi-bound statesINTRODUCTION With the emergence of nanotechnology, it is already found out by intense researchactivities that the existing problems of present VLSI-based electronics can be solved byquantum-confined semiconductor structures where miniaturization is possible beyond theexisting saturation point. Confinement of electrons along reduced dimensions which arecomparable to the electron wavelength such as quantum wells, wires and dots have led tonumerous significant improvements in semiconductor physics which have been usedalready in several micro- and optoelectronics applications [1]. The region of confinement,usually nomenclature as quantum system, is coupled to the external world throughtunneling barriers which vividly reflects the dominance of the quantum effects tounderstand the physical properties of heterostructure devices, and the discrete electronicstates become resonant one. The likelihood that the electron will pass through the barrier 24
  • 2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEis given by the transmission coefficient, and this type of finite potential barrier problemdemonstrates the phenomenon of quantum tunneling; which can be predicted by bettermathematical modeling and numerical computation. From the viewpoint of low-biasapplication, resonant tunneling in quantum well-like structures produced considerableinterest amongst theoretical researchers [2]-[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 bygrowing two or more different semiconducting materials leading to a potential distribution alonggrowth directions [6]; producing a series of energy levels and associated subbnads due to thequantization of carriers in the direction of confinement. Computation of the transmissioncoefficient for this type of structure is possible by solving Schrödinger’s equation throughpotential barriers, and coupling between electronic states localized in semiconductornanostructures is evaluated for understanding the performance of heterostructure devices. Chanda& Eastman [7], and later Christodoulides [8] calculated transmission coefficient for triangularbarrier, whereas Scandella calculated the probability for trapezoidal barrier. In all the analysis, carewas not taken for material parameters, which was first reported by Chang etc [9], and later Readetc [10], who computed resonant tunneling probability in semiconductor double barrier structurefor different material parameters. Computation of transmission coefficient and corresponding eigenenergy for resonancetransmission can be analyzed by solving time-independent Schrödinger’s equation with properboundary conditions so that a complex transcendental equation is formed having roots as thecomplex eigenenergies of the heterostrucutre [11]. DBRT structure subjected to electric field canbe analyzed by several numerical techniques such as Variational Method [12], Airy’s functionapproach [13]-[15], Finite Element Method [16], Transfer Matrix Technique [1], [3 ], [13]-[15],[17]-[19], Weighted Potential Method [20]. Comparing all these methods, TMT is considered asone of the effective and accurate method by eminent researchers. Resonant tunnelingphenomenon for these devices under applied field [1], [2], [21]-[22] provides a theoreticalestimation about transmission coefficient. Existence of bound and quasibound states [11] speaksin favor of that statement. The present paper provides a comprehensive theoretical analysis of transmissioncoefficient based on tunneling probability of electrons between two barriers through the well inpresence and absence of electric field using transfer matrix method. Introduction of the concept ofvariable effective mass in the barrier and the well layers makes the analysis more realisticcompared to some earlier theoretical researches [2], [5]-[6], [9]. As effective mass and barrierpotential, both are function of the mole fraction for GaAs/AlxGa1-xAs heterostructure, socomputation 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. Herepotential 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 incalculations by changing the material compositions in barrier regions. A gradual change ofapplied bias from low to moderate value shows the consequent alteration in transmissioncoefficient for different potential conditions, and corresponding variation in eigenenergy.Logarithmic scale of transmission coefficient is chosen to study the generated profiles. 25
  • 3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMETHEORETICAL ANALYSISThe motion of a single electron moving in one dimension is represented by time-independentSchrödinger’s equation h 2 d 2ψ ( z ) (1) − + V ( z )ψ ( z ) = Eψ ( z ) 2m* dz 2For the double barrier structure under consideration as shown in Figure, we consider the solutionsto Schrödinger’s equation within each region for E<V –ψ ( z ) = A exp[iκ 1 z ] + B exp[−iκ 2 z ] for z<I1 (2.1)ψ ( z ) = C exp[κ 1 z ] + D exp[−κ 2 z ] for I1<z<I2 (2.2)ψ ( z ) = F exp[iκ 1 z ] + G exp[−iκ 2 z ] for I2<z<I3 (2.3)ψ ( z ) = H exp[κ 1 z ] + J exp[−κ 2 z ] for I3<z<I4 (2.4)ψ ( z ) = K exp[iκ 1 z ] + L exp[−iκ 2 z ] for I4<z (2.5)where κ1 & κ2 are defined as: * 2m w Eκ1 = (3.1) h * 2mb (V − E )& κ2 = (3.2) hThe positions of interfaces have been labeled I1, 12, I3 and I4 respectively. Using standardBenDaniel-Duke boundary conditions at each interface, and introducing transfer matrixtechnique, A K (4) B  = M1 M 2M 3 M 4M 5 M 6M 7 M 8 L   −1 −1 −1 −1     We assume that there are no further heterojunctions to the right of the structure, so that no furtherreflections can occur and wave function beyond the structure can only have a traveling wavecomponent moving to the right, i.e. the coefficient L must be zero. Thus equation (4) can bemodified as: A K  (5)  = M  B 0   The probability interpretation of the wave function implies that A7 = B1 = 0, which gives 26
  • 4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEM 22 ( E ) = 0 (6)This condition gives the eigenenergy for resonance transmission. Transmission coefficient can begiven by- KK * 1T (E) = * = * (6) AA M 11 M 11Schrödinger’s equation subjected to applied electric field of strength F along the quantizeddirection is given by: h 2 d 2ψ ( z ) − + V ( z )ψ ( z ) − qF ( z )ψ ( z ) = Eψ ( z ) (7) 2m* dz 2Considering suitable substitutions, solution can be approximated as:ψ ( z ) = A Ai[ z ] + B Ai[ z ] for z<I1 (8.1)ψ ( z ) = C Ai[ z ] + D Ai[ z ] for I1<z<I2 (8.2)ψ ( z ) = F Ai[ z ] + G Ai[ z ] for I2<z<I3 (8.3)ψ ( z ) = H Ai[ z ] + J Ai[ z ] for I3<z<I4 (8.4)ψ ( z ) = K Ai[ z ] + L Ai[ z ] for I4<z (8.5)where 1/ 3  2m *  V ( z ) − E z =  2     2/3 − (qF )1 / 3 z  (9)  h   (qF ) Similarly, using standard BenDaniel-Duke boundary conditions at each interface, and introducingtransfer matrix technique, A   K  (10) B  = M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8  L   −1 −1 −1 −1     Similarly transmission coefficient can be obtained as: K K * 1T (E) = * = * (11) A A M 11 M 11 and eigenenergy is obtained as- M 22 ( E ) = 0 (12)NUMERICAL RESULTS Theoretical investigation of symmetric quantum double barrier heterostructure starts withGaAs/AlxGa1-xAs system having assumption that both the barriers have equal materialcomposition; where it is also considered that barrier potential is solely a function of the materialparameters, and effective masses of barrier region and in well region have a mismatch as it 27
  • 5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEdepends on mole fraction of Al in barrier regions. These variations of effective mass at junctionsare regarded throughout the analysis to compute eigenenergies and transmission probabilities. Figure 1 gives the comparative study of ground state and the next higher stateeigenenergy values in presence of constant electric field and also the energies without field forvarying thickness of barrier regions. It is observed that eigenenergy remains almost constant forunbiased 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 startdecreasing for the ground state, and for the next state, it reduces after well dimension attains aparticular value. Nature of the profile remains unchanged at biasing condition, as shown in Figure2. Figure 2: Comparative analysis of eigenenergy for first two states of symmetric double barrier structure in presence and absence of electric field with varying well width 28
  • 6. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME A change of barrier potential by varying mole fraction of Al also affects the effectivemass mismatch at heterojunction which shows that nature of the eigenenrgy profiles for first twostates remain unaltered even if bias is applied. Ground state energy remains constant whereas firsthigher state shows a linear incremental slope at first, and then attains constant value. It is evidentfrom Figure 3. Figure 3: Comparative analysis of eigenenergy for first two states of symmetric double barrierstructure 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 morecompared to the unbiased condition. For ground state, the rate of decrement is linear, whereasnonlinearity 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 29
  • 7. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Transmission coefficient analysis starts in absence of electric field where it can beobserved that away from resonance, an increasing barrier height leads to a decrease in thetransmission coefficient, which is expected also from classical physics also. The resonanceenergies increase with increasing barrier height due to confinement effects, and the appearance ofthe second resonance at higher energy values speaks in favor of the existence of a second quasibound 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 thicknessFigure 6: Comparative analysis of transmission coefficient of symmetric double barrier structure without electric field with varying well thickness 30
  • 8. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME A simultaneous study of Figure 5 & Figure 6 reveals the fact that with increasing wellwidth, tunneling probabilities can be estimated at lower energy values. But by increasing barrierwidth, tunneling probability decreases. Increase of mole fraction of Al in barrier region increases the potential height, andsimultaneously effective mass mismatch at heterojunctions increases, which causes a shift ofresonance 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 samewhere we have neglect the transmission probability foe E<0. Similarly, well width variationprovides 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. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEFigure 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 fieldspeaks about possibility of transmission at lower energy values, i.e, quasi-bound states exist. Withdecrease of negative bias, transmission is delayed w.r.t energy, and more bias gives origin of lesspeaks.Figure 10: Comparative analysis of transmission coefficient of symmetric double barrier structure for different electric field 32
  • 10. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Similarly, in presence of electric field, higher barrier potential shifts the resonancecondition towards higher energy values, as shown in Figure 11, neglecting transmissionprobabilities 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 fractionCONCLUSION The composition of material in a heterostructure device is very important indetermination of the electrical and optical properties. In the present paper, concentration of al isvaried to observe the effect of changing barrier potential on eigenenergy and on transmissioncoefficient, and simultaneously effective mass mismatch at junctions are also taken into accountto make the investigation more realistic, which is ignored in several earlier works. Peaks oftransmission coefficient graphs show the resonance values, where a measure of the lifetime ofsuch a localized resonance is given by the increase of the width of resonance. With increase of Alconcentration, transmission is initiated at higher energies. Applied bias pulls down the secondbarrier to a large extent, and hence transmission probability is also modified. Existence of quasi-bound states can be verified by origin of resonance peaks, and varying well width and barrierthickness gives pictorial information of these states subjected to constant electric field. Thisanalysis can be extended further to study the performance of complex quantum structures.REFERENCES1. A.R.Sugg & J.P.C.Leburton, “Modeling of Modulation-Doped Multiple-Quantum-WellStructures in Applied Electric Fields using The Transfer-Matrix Technique”, IEEE J.QuantumElectron, 27, 2, 1991.2. Y.Guoa, B.L.Gu, J.Z.Yu, Z.Zeng & Y.Kawazoe, “Resonant Tunneling in Step-BarrierStructures Under an Applied Electric Field”, J. App. Phys, 84, 2, 1988.3. S.Vatannia & G.Gildenblat, “Airy’s Function Implementation of the Transfer-Matrix Methodfor Resonant Tunneling in Variably Spaced Finite superlattices”, IEEE J.Quantum Electron, 32,6, 1996. 33
  • 11. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME4. A.Harwit & J.S.Harris, “Calculated Quasi-Eigenstates and Quasi-Eigenenergies of QuantumWell Superlattices in an Applied Field”, J. Appl. Phys, 60, 3211, 1986.5. L.Esaki and R.Tsu, “Superlattice and Negative differential Conductivity in Semiconductors”,IBM Journal Research Division, 14, 61, 1970.6. L.Esaki & L.L.Chang, “New Transport Phenomenon in Semiconductor Superlattice”, Phys.Rev. Lett, 33, 8, 1974.7. LA.Chanda and L.F.Eastman, “Quantum mechanical reflection at triangular planar-dopedpotential barriers for transistors”, J Appl Phys, 53, 9165, 1982.8. D.N.Christodoulides, A.G.Andreou, R.I.Joseph and C.R.Westgate, “Analytical calculation ofthe quantum-mechanical transmission coefficient for a triangular, planar-doped potential barrier”,Solid State Electron, 28, 821, 1985.9. L.L.Chang, L.Esaki & R.Tsu, “Resonant Tunneling in Semiconductor Double Barriers”, ApplPhys Lett, 24, 12, 1974.10. M.A.Reed, R.J. Koestner & M.W.Goodwin, “Resonant Tunneling Through a HeTe/Hg1-xCdxTe Double Barrier, Single Quantum Well Structure”, Appl Phys Lett, 49, 19, 1986.11. E.Anemogiannis, “Bound and Quasibound State Calculations for Biased/UnbiasedSemiconductor Quantum Heterostrucutres”, IEEE J.Quantum Electron, 29, 2731, 1993.12. G.Bastard, E.E.Mendez, L.L.Chang & L.Esaki, “Variational Calculations on a Quantum Wellin an Electric Field”, Phys. Rev. B, 28, 3241, 1983.13. A.K.Ghatak, K.Thyagarajan & M.R.Shenoy, “A Novel Numerical Technique for Solving theOne-Dimensional Schroedinger Equation using Matrix Approach - Application to Quantum WellStructures”, IEEE J.Quantum Electron, 24, 8, 1988.14. S.Vatannia & G.Gildenblat, “Airy’s Function Implementation of the Transfer-Matrix Methodfor Resonant Tunneling in Variably Spaced Finite Superlattices”, IEEE J.Quantum Electron, 32,6, 1996.15. K. F. Brennan and C.J.Summers, “Theory of Resonant Tunneling in a Variably SpacedMultiquantum Well Structure: An Airy Function Approach”, J. Appl. Phys. 51, 614, 1987.16. K.Hayata, M.Koshiba, K.Nakamura & A.Shimizu, “Eigenstate calculations of Quantum wellStructures using Finite Elements”, Electron. Lett, 24, 614, 1988.17. E.P.Samuel & D.S.Patil, “Analysis of Wavefunction Distribution in Quantum Well BiasedLaser Diode using Transfer Matrix Method”, Progress In Electromagnetics Research Letters, 1,119, 2008.18. K.Talele & D.S.Patil, “Analysis of Wavefunction, Energy and Transmission Coefficients inGaN/AlGaN Superlattice Nanostructures”, Progress In Electromagnetics Research Letters, 81,237, 2008.19. B.Jonsson & S.T.Eng, “Solving the Schrödinger Equation in Arbitary Quantum-Well Profilesusing the Transfer-Matrix Method”, IEEE J.Quantum Electron, 26, 11, 1990.20. Y.Tsuji and M.Koshiba, "Analysis of Complex Eigenenergies of an Electron in Two- andThree-Dimensionally Confined Systems using the Weighted Potential Method", MicroelectronicsJournal, 30, 1001, 1999.21. E.J.Austin & M.Jaros, “Electronic Structure of an Isolated GaAs-GaAlAs Quantum well in aStrong Electric Field”, Phys. Rev. B, 31, 5569, 1985.22. S.S.Allen & S.L.Richardson, “Theoretical investigations of resonant tunneling in asymmetricmultibarrier semiconductor heterostructures in an applied constant electric field”, Phys. Rev. B,50, 11693, 1994.23. Y.J.Hong, J.G.Zhi, Z.Yan, L.W.Wu, S.Y.Chun, W.Z.Guo, X.J.Jun, “Resonant Tunneling inBarrier-In-Well and Well-In-Well Structures”, Chin. Phys. Lett, 25, 4391, 2008. 34

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