An Emphasis of Electron Energy Calculation in Quantum Wells

Transcription

1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 3, September 15, 2004 An Emphasis of Electron Energy Calculation in Quantum Wells GAO Shao-Wen, CAO Jun-Cheng,, and FENG Song-Lin State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, the Chinese Academy of Sciences, Shanghai , China (Received November 4, 2003) Abstract We investigate various methods for the calculation of the electron energy in semiconductor quantum wells and focus on a matrix algorithm method. The results show better fitness of the factor h2 1 than that of 2 m (z) h2 1 2 in the first part of the Schrödinger equation. The effect of nonparabolicity in the conduction band is 2 m (z) 2 also discussed. PACS numbers: Px, Fg, Ea Key words: quantum wells, nonparabolicity, matrix algorithm 1 Introduction Quantum energy-band engineering of electronic energy states and wavefunctions using ultrathin layers of semiconductor compounds, with different compositions, allows designing novel semiconductor devices. With the development of epitaxial techniques, such as molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD), lattice mismatched heteroepitaxy could be operated precisely. Thus, determining exactly the electronic band structure is of the essence for understanding and modifying the optical and transport properties of the semiconductor devices. G. Bastard, [1 3] D.F. Nelson, [4,5] and S.L. Chuang, [6] et al., have excellent discussion in this area. Novel devices have been developed such as quantum cascade (QC) laser [7,8] and quantum well infrared photodetector (QWIP) [9] based on these theories. In our recent research work of quantum cascade laser emitting at the terahertz range, we find it is very important to calculate the electronic band structure precisely since one terahertz energy is approximate to 4 mev (micro electron volt). We investigated various methods for calculation of the electron energy levels in the conduction band of semiconductor quantum wells with the parabolic or nonparabolic approximation. To GaAs/Al x Ga 1 x As quantum wells based on the GaAs substrate, lattice mismatch is negligible. Solving one-dimensional effective mass Schrödinger equation along the material growth direction is the direct way to obtain electron energy levels and wavefunctions based on slowly varying envelope function approximation. In this paper, we discuss a matrix algorithm to calculate the electronic states and wavefunctions in the quantum wells. We compare the two cases where the first part of the Schrödinger equation is expressed as h m (z) 2 and h2 2 m (z) respectively. The different cases are derived from different selections of kinetic energy operator or the neglect of the abruptness of the electron mass at the boundaries. The results show that the later case (the matrix algorithm II) gives results that compare more favorably with the empirical Bastard s model and the TMM than the first one (the matrix algorithm I) does. The effect of nonparabolicity in the conduction band is also discussed. The results of the matrix algorithm II agree well with the previous observation that band nonparabolicity raises the lowest conduction band energy level of the quantum well by a small amount while lowering the higher energy levels. 2 Matrix Algorithm Model In a quantum well structure assuming the material growth direction is the z axis, the complete envelope function of electron is the product of a plane wave describing translational motion in the x y plane multiplied by a slowly varying function ϕ(z), which describes the confined motion of the electron along z axis. The envelope function ϕ(z) satisfy the one-dimensional effective mass Schrödinger equation, h2 2 ( 1 m (z) ) ϕ(z) + V b (z)ϕ(z) = Eϕ(z), (1) where m (z) is the effective mass along z direction, and V b (z) is the finite quantum well potential. The Schrödinger equation is solved using the finite difference method on an evenly spaced mesh. Specially, we expand the first part of the equation and consider it as a whole body in the bracket. However, in most of The project supported by the Foundation of the State Key Research Program under Grant Nos CCA02800 and G , and the Special Funds of the Research and Development Foundation of Shanghai Municipal Commission of Science and Technology under Grant No

2 436 GAO Shao-Wen, CAO Jun-Cheng, and FENG Song-Lin Vol. 42 papers, [10,11] the authors neglected the abruptness of the electron mass at the boundaries and obtained the result as h m ϕ(z). From Eq. (1), we get (z) 2 [ h2 1 ( 1 2δz 2 m ϕ i+1 i+0.5 m + 1 ) i+0.5 m ϕ i i ] m ϕ i 1 + V b,i ϕ i = Eϕ i, (2) i 0.5 where δz is the space between mesh points, i = 0, 1, 2,..., N labels the mesh points along the z axis. On the boundary of a quantum well structure, the wave function vanishes, ϕ 0 = ϕ N = 0. We assume m i+0.5 = (m i+1 + m i )/2 and m i 0.5 = (m i 1 + m i )/2. Finally, we get a matrix equation as M (N 1) (N 1) Ψ (N 1) 1 = EΨ (N 1) 1. (3) Here, M is an (N 1) (N 1) matrix of electron effective mass and V potential derived from Eq. (2), Ψ is an (N 1) 1 matrix of envelope function ϕ, E is the electron energy. Equation (3) is a matrix equation and can be solved easily. The eigenvalue is the electron energy level E, and the corresponding eigenvector is the envelope function. When there exits an external electric field along z axis, V b is substituted by V i = V b,i + e F z i, e is the electronic charge, and F is the external electric field. In the case of nonparabolic band, m (z) is replaced by m (E, z) in Eq. (1). We use the same expression of electron effective mass as the empirical Bastard s model: [4] m i (E) = m (0)[1 + E/(E gi + i /3)] (i = w, b), where E gi is the energy gap between the conduction band and light-hole valence band that is nonparabolicity parameter related. i is the energy splitting between the light-hole band and the split-off valence band. It seems impossible to use the form of Eq. (3) to calculate the electron energy levels and envelope functions in this case, then the recursion relation is used. 3 Numerical Results and Discussions We calculate the electron energy levels for a quantum well structure composed of GaAs/Al 0.37 Ga 0.63 As materials in the case of parabolic and nonparabolic energy band approximation, respectively. The electron mass in the GaAs well material is taken as m 0, and m 0 in Al 0.37 Ga 0.63 As barrier material. The conduction band offset is taken as V b = ev. The case of parabolic energy band is shown in Table 1 for the well width d w = 0.5, 5, and 10 nm, respectively. The energy levels in the parabolic one-band model (OBM) is calculated by the energy dispersion relation in the materials and the boundary conditions. [4] The transfer matrix method (TMM) [6] solved the Schrödinger equation using a propagation approach, which is similar to that used in electromagnetic wave reflection in a multilayered medium. From Table 1, we notice that the matrix algorithm II gives results in excellent agreement with the one-band parabolic mode and the TMM, however, the matrix algorithm I gives higher energy levels than all the other three models. The difference is thought to be due to the treatment of [1/m (z)]/ as a differential function in the matrix algorithm II while as a δ function in the matrix algorithm I. Table 1 Conduction band energy levels (mev) for single GaAs/Al 0.37Ga 0.63As quantum well (nm) in the parabolic one-band model (OBM), the TMM, the matrix algorithm methods I and II in the case of parabolic energy band approximation. OBM TMM Matrix algorithm I Matrix algorithm II d w = d w = d w = Table 2 Conduction band energy levels (mev) for single GaAs/Al 0.37Ga 0.63As quantum well (nm) in the empirical Bastard model (EBM), the TMM, the matrix algorithm methods I and II in the case of nonparabolic energy band approximation. EBM TMM Matrix algorithm I Matrix algorithm II d w = d w = d w =

3 No. 3 An Emphasis of Electron Energy Calculation in Quantum Wells 437 Table 2 shows the energy levels of single GaAs/ Al 0.37 Ga 0.63 As quantum well with the nonparabolic effect, calculated by the empirical Bastard model, the TMM, the matrix algorithm methods I and II. The empirical Bastard model [4] was derived from the Bastard three-band model by recasting it into energy-dependent effective mass formulation, while the two-band model was obtained crudely by neglecting the interaction of distant bands and the three-band model corrects it by including the split-off component. Here, we obtain the same results as D.F. Nelson has done using the empirical Bastard model as we take the same energy gap as E g = 0.6(1.425x 0.9x x 3 ), the effective mass as m (0) = E g /1.625, and the nonparabolicity factor as γ = m 2. The transfer matrix results are copied from Ref. [4] for comparison. The matrix algorithms I and II take the same energy-dependent effective mass m (E, z) as the empirical Bastard model. From Table 2, we notice that, comparing to the matrix algorithm I, the matrix algorithm II gives better agreement with the empirical Bastard model and the transfer matrix method. From both Tables 1 and 2, we find that the matrix algorithm II gives good description of the electron energy levels in the case of nonparabolic energy band approximation, raising the lowest conduction band energy level of the quantum well by a small amount while lowering the higher energy levels. However, the matrix algorithm I lowers all energy levels compared to the parabolic energy band case. The results demonstrate the validity of matrix algorithm II which expend the first part of the Eq. (1), considering it as a whole body in the bracket, and solved one-dimensional Schrödinger equation as the form of Eq. (3). Fig. 1 Energy levels as functions of well width in GaAs/Al 0.37Ga 0.63As quantum well. The dashed line, solid line and the triangles are the results of the matrix algorithm I, the matrix algorithm II, and the one-band model, respectively, in the case of parabolic band approximation. The dotted line is the result of matrix algorithm II with nonparabolic band approximation. Cn (n = 1, 2, 3.) denote the electron energy levels in the conduction band. The inset is an amplified part of the figure. Figure 1 illustrates the energy variation with the well width in GaAs/As 0.37 Ga 0.63 As quantum well. The superposition of solid line and the triangles shows that the matrix algorithm II gives almost the same results with the one-band model on the case of parabolic band approximation. The dashed line is the result of the matrix algorithm I, which is higher than all the other models for the same energy level. The dotted line is the energy levels in nonparabolic band approximation calculated by the matrix algorithm II, which gives lower energy of Cn (n 2) than the parabolic band approximation explicitly and raises the C1 energy by a small amount illustrated in the inset. 4 Applications Using the matrix algorithm II, we calculate the energy levels in a strained GaInAs/GaInAsP quantum well and a multiple GaAs/AlGaAs quantum well structures, respectively. The conventional theoretical approach for strained semiconductors is based on the Luttinger Kohn Hamiltonian, taking into account the strains, or called the Pikusbir Hamiltonian. We should solve a 4 4 Hamiltonian assuming the heavy and light hole mixing for eigenvalues with corresponding eigenvectors. An improvement on the solutions of the 4 4 Hamiltonian can be obtained by a unitary transformation such that the 4 4 Hamiltonian is block diagonalized. [12] In the special case when k x = k y = 0, the heavy hole and light hole bands are decoupled, H = h 2 2m 0 (γ 1 2γ 2 )k 2 z + ζ 0 0 h 2 2m 0 (γ 1 + 2γ 2 )k 2 z ζ + V h (z). (4) Here, we define the Hamiltonian by assuming all energies measured upward. V h (z) is the valence band potential profile including the strain effect. ζ = b(1 + 2C 12 /C 11 )ε is the strain parameter, where C 11, C 12 are the elastic constants. h, m 0, b, ε, and k z are the Planck constant, the free electron mass, the potential constant, the elastic strain due to the lattice difference and the wave vector along the z axis, respectively. γ 1, γ 2 are the Luttinger parameters. Then, we can find the energy levels and corresponding wave envelopes of the heavy and light holes, respectively. For a strained Ga 0.8 In 0.2 As/Ga 0.8 In 0.2 As 0.62 P 0.38 quantum well structure, we obtain the results using the matrix algorithm II. Figure 2 shows the energy band diagram of the quantum well structure. All energies measured upward assuming the conduction and the valence band potential profiles, in the well, are zero before the strain effect consideration.

4 438 GAO Shao-Wen, CAO Jun-Cheng, and FENG Song-Lin Vol. 42 One electron level (C1), four heavy hole levels (HH1, HH2, HH3, HH4) and two light hole levels (LH1, LH2), with corresponding wave envelopes modulus, are displayed. The strain effect is illustrated from the different potential profiles in the conduction band, the heavy hole and light hole valence bands compared to the case regardless of the strain effect. The calculated results give the electron transmission emitting wavelength at µm from the first electron subband (C1) to the first heavy hole subband (HH1) of this strained Ga 0.8 In 0.2 As/Ga 0.8 In 0.2 As 0.62 P 0.38 quantum well structure, which is designed as the active region of a 980 nm pump laser. [13] Figure 3 shows the conduction band structure of a GaAs/Al x Ga 1 x As quantum well laser calculated by the matrix algorithm II with the parabolic band approximation. This is a three-quantum-well structure, which is usually calculated by the transfer matrix method (TMM). Here, our model shows the results in well agreement with the TMM, E 43 = mev, E 32 = mev, and E 41 = mev. The structure is used by R.W. Kelsall et al. [14] as a terahertz solid source emitting at 11.7 THz (25.6 µm). We also discuss here with a few remarks about the application of the matrix algorithm method. (i) The method is an extended model to transfer matrix method and can be used to calculate the electron and hole energy levels in a multilayer quantum well or supperlattice structure, even to quantum well structure with irregular potential profiles. (ii) The method is applicable to self-consistent solution for doped quantum wells. For one-dimensional Poisson s equation [ ε(z) φ ] = ρ(z), (5) where ε(z) is the dielectric constant, and φ is the electrostatic potential, and ρ is the charge distribution, the left part is also expended as a whole body in the bracket using matrix algorithm method with the corresponding boundary conditions (with and without external electric field). Fig. 2 Potential profile, energy levels and wavefuction modulus of a single strained Ga 0.8In 0.2As/ Ga 0.8In 0.2As 0.62P 0.38 quantum well structure with the well width 7.5 nm based on the GaAs substrate. Dashed lines denote the light hole. E(C1) = mev, E(HH1) = 7.8 mev, λ = µm. Fig. 3 Calculated conduction band structure of a tearhertz quantum well laser. The layers thicknesses (nm), from left to right, are 20, 7.8, 3.5, 3.9, 2.1, 4.0, 20. Al 0.24Ga 0.76As barriers thicknesses are typed with bold face, GaAs quantum wells normal. References [1] G. Bastard, Phys. Rev. B25 (1982) [2] G. Bastard, Phys. Rev. B24 (1981) Conclusions In conclusion, we discussed a matrix algorithm to calculate the energy levels in semiconductor quantum wells. The comparison emphasizes that the expression of the first part of Schrödinger equation as h2 1 2 m (z) gives results in good agreement with the data obtained by the one-band parabolic model, the TMM, and the empirical model. On the other hand, the expression as h m (z) 2 gives higher energy levels than the other models and different description of the energy states in the case of nonparabolic approximation. The discussion results are efficient for us to select correct method to calculate the electron energies in the quantum wells. [3] G. Bastard, in Molecular Beam Epitaxy and Heterostructures, Proceedings of the NATO Advanced Study Institute on Molecular Beam Epitaxy and Heterostructures, eds.

5 No. 3 An Emphasis of Electron Energy Calculation in Quantum Wells 439 L.L. Chang and K. Ploog, Nijhoff, Amsterdam (1983) p [4] D. F. Nelson, R.C. Miller, and D.A. Kleinman, Phys. Rev. B35 (1987) [5] K.H. Yoo, L.R. Ram-Mohan, and D.F. Nelson, Phys. Rev. B39 (1989) [6] S.L. Chuang, Physics of Optoelectronic Devices, John Wiley and Sons, New York (1995). [7] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, and A.Y. Cho, Science 264 (1994) 533. [8] Rüdeger Köhler, Alessandro Tredicucci, Fabio Beltram, Harvey E. Beere, Edmund H. Linfield, A. Giles Davies, David A. Ritchie, Rita C. Iotti, and Fausto Rossi, Nature 417 (2002) 156. [9] H.C. Liu and Federico Capasso, Intersubband Transitions in Quantum Wells: Physics and Devices Applications II, Academic Press, San Diego (2000). [10] Yang Quan-Kui and Li Ai-Zhen, Chin. Phys. Lett. 16 (1999) 443. [11] A.V. Kuznetsov, G.D. Sanders, and C.J. Stanton, Phys. Rev. B52 (1995) [12] S.L. Chuang, Phys. Rev. B43 (1991) [13] Gao Shao-Wen, Cao Jun-Cheng, and Feng Song-Lin, Commun. Theor. Phys. (Beijing, China) 39 (2003) 327. [14] R.W. Kelsall, P. Kinsler, and P. Harrison, Physica E7 (2000) 48.