Nonlinear Intersubband THz Absortion in Asymmetric Quantum Well Structures

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(1)Nonlinear Intersubband THz Absorption in Asymmetric Quantum Well Structures Trabajo para la obtención del grado de maestría en Física. Mauricio David Bedoya Saavedra Departamento de Física, Universidad de los Andes e-mail: [email protected]. Dirigido por: Dra. Angela Camacho Departamento de Física, Universidad de los Andes. 2004.

(2) Contents 1 Introduction. 11. I. 13. Theory and numerical methods. 2 Basic calculations for Quantum Structures 2.1. 2.2. 2.3. 15. Complex Eigenenergies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.1.1. General formulation. 15. 2.1.2. Quasibound electron states[16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. Transfer Matrix Method with exponential functions . . . . . . . . . . . . . . . . . . . . .. 17. 2.2.1. The transfer matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 2.2.2. Dipole moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. Transfer Matrix Method with Airy functions. . . . . . . . . . . . . . . . . . . . . . . . .. 3 Nonlinear absorption calculation 3.1. 3.2. 3.3. II. Density matrix formalism. 19. 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.1.1. Formalism of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 3.1.2. Density matrix. 24. 3.1.3. Perturbation solution of the density matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Calculations of the susceptibility perturbation terms. . . . . . . . . . . . . . . . . . . . .. 25 27. 3.2.1. Density matrix calculation of the linear susceptibility. . . . . . . . . . . . . . . .. 27. 3.2.2. Density matrix calculation of the second order susceptibility . . . . . . . . . . . .. 29. 3.2.3. Density matrix calculation of the third order susceptibility. Optical absorption coecient. . . . . . . . . . . . .. 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. Results. 35. 4 Linear and nonlinear absorption and geometry. 39. 4.1. Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.2. Transitions. 41. 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 4.3.1. Wavefunctions. First resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 4.3.2. Second resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 4.4. Dipole moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.5. Absorption. 48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 3.

(3) 4. CONTENTS. 5 Linear and nonlinear absorption and external eld. 57. 5.1. Energies and transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 5.2. Wavefunctions. 60. 5.3. Dipole moments. 5.4. Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. 6 Analysis of resonance tuning and escape times 6.1. 6.2. Resonance tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.1.2. Dipole moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71 71 71 75. Escape times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78. 6.2.1. Barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 6.2.2. Barrier. B2 = 20Å . B2 = 60Å .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80. 7 Conclusions. 87.

(4) List of Figures 2.1. Scheme of the biased potential showing the energy ranges for bound and quasibound states [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. 16. Scheme of a semiconductor heterostructure. The potential is subdivided in regions of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 2.3. constant value [16]. Scheme of the biased potential [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 4.1. Potential prole of an ADQW. 4.2. B = 20Å and zero electric eld (F = 0kV /cm) . . . . . . . . Eigenenergies vs with B = 60Å and zero electric eld (F = 0kV /cm) . . . . . . . . Eigenenergies vs with B = 100Å and zero electric eld (F = 0kV /cm) . . . . . . . Eigenenergies vs for several values of B with zero electric eld (F = 0kV /cm) . . . Transition energies vs W2 with B = 20Å and zero electric eld (F = 0kV /cm) . . . . . Transition energies vs W2 with B = 60Å and zero electric eld (F = 0kV /cm) . . . . . Transition energies vs W2 with B = 100Å and zero electric eld (F = 0kV /cm) . . . . Eigenfunctions for F=0 kV/cm, B = 20Å, 60Å and 100Å at resonance . . . . . . . . . Eigenfunctions for F=0 kV/cm, B = 20Å, 60Å and 100Å when W2 is 5Å shorter than. 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10. Eigenenergies vs. W2 W2 W2 W2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. with. 39. .. 40. .. 40. .. 41. .. 41. .. 42. .. 42. .. 43. .. 44. its resonance value (below resonance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 4.11 Eigenfunctions for F=0 kV/cm,. B = 20Å, 60Å. resonance value (above resonance). and. 100Å, W2 W2. is. 5Å. larger than its. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. B = 20Å, 60Å and 100Å in the second resonance Intersubband dipole moments vs W2 , B = 20Å, F = 0kV /cm . . . . . . . . . . . Intersubband dipole moments vs W2 , B = 60Å, F = 0kV /cm . . . . . . . . . . . Intersubband dipole moments vs W2 , B = 100Å, F = 0kV /cm . . . . . . . . . . . Intersubband dipole moment µ23 vs W2 , B variable, F = 0kV /cm . . . . . . . . . Linear absorption for B = 20Å and F = 0kV /cm . . . . . . . . . . . . . . . . . . Absorption for B = 20Å, F = 0kV /cm and E = 0kV /cm above resonance . . . . Absorption for B = 20Å, F = 0kV /cm and E = 1, 2, 5, 10kV /cm . . . . . . . . . Linear absorption for B = 60Å and F = 0kV /cm . . . . . . . . . . . . . . . . . . Absorption for B = 60Å, F = 0kV /cm and E = 1, 2, 5, 10kV /cm . . . . . . . . . Linear absorption for B = 100Å and F = 0kV /cm . . . . . . . . . . . . . . . . . Absorption for B = 100Å, F = 0kV /cm and E = 1, 2, 5, 10kV /cm . . . . . . . . .. 46. 4.12 Eigenfunctions for F=0 kV/cm,. . . . .. 47. 4.13. . . . .. 48. . . . .. 48. . . . .. 49. 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 5.1 5.2 5.3 5.4. . . . .. 50. . . . .. 50. . . . .. 51. . . . .. 51. . . . .. 52. . . . .. 53. . . . .. 54. . . . .. 54. Eigenenergies for. . . . . . . . . .. 58. Eigenenergies for. . . . . . . . . .. 58. . . . . . . . . .. 59. . . . . . . . . .. 59. B = 20Å vs F , using W2 = 38Å . . . . . . . . . . . . . B = 60Å vs F , using W2 = 38Å . . . . . . . . . . . . . Eigenenergies for B = 100Å vs F , using W2 = 38Å . . . . . . . . . . . . Transitions for B = 20Å, B = 60Å and B = 100Å vs F , using W2 = 38Å 5.

(5) 6. LIST OF FIGURES. 5.5. W2 = 38Å, B = 20Å, 60Å and 100Å at resonance, respectively the F = 28kV /cm, F = 18kV /cm and F = 14kV /cm . . . . . . . . Dipole moments vs F for B = 20Å . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dipole moments vs F for B = 60Å . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dipole moments vs F for B = 100Å . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption for B = 20Å, E = 0kV /cm vs F . . . . . . . . . . . . . . . . . . . . . . . . Absorption for B = 20Å vs F and E = 0.1, 0.2, 0.4, 1kV /cm . . . . . . . . . . . . . . . Nonlinear absorption vs. F from two points of view (B = 20Å) . . . . . . . . . . . . . Absorption for B = 60Å, E = 0kV /cm vs F . . . . . . . . . . . . . . . . . . . . . . . . Absorption for B = 60Å vs F and E = 0.1, 0.2, 0.4, 1kV /cm . . . . . . . . . . . . . . . Nonlinear absorption vs. F from two points of view (B = 60Å) . . . . . . . . . . . . . Absorption for B = 100Å, E = 0kV /cm vs F . . . . . . . . . . . . . . . . . . . . . . . Absorption for B = 100Å vs F and E = 1, 2, 5, 10kV /cm . . . . . . . . . . . . . . . . . Nonlinear absorption vs. F from two points of view (B = 100Å) . . . . . . . . . . . . . Nonlinear absorption vs. E (kV /cm) for B = 20Å, W2 = 38Å and F = 14kV /cm . . . Nonlinear absorption vs. E (kV /cm) for B = 60Å, W2 = 38Å and F = 22kV /cm . . . Nonlinear absorption vs. E (kV /cm) for B = 100Å, W2 = 38Å and F = 18kV /cm . . . Eigenfunctions for. intensity of the eld is. 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12. E32 in meV vs (W2 , F ) for B = 20Å (3D) . . . . . . E32 in meV vs (W2 , F ) for B = 20Å (Intensity plot) E32 in meV vs (W2 , F ) for B = 60Å (3D) . . . . . . E32 in meV vs (W2 , F ) for B = 60Å (Intensity plot) E32 in meV vs (W2 , F ) for B = 100Å (3D) . . . . . . E32 in meV vs (W2 , F ) for B = 100Å (Intensity plot) µ32 in meV vs (W2 , F ) for B = 20Å (3D) . . . . . . µ32 in meV vs (W2 , F ) for B = 20Å (Intensity plot) . µ32 in meV vs (W2 , F ) for B = 60Å (3D) . . . . . . µ32 in meV vs (W2 , F ) for B = 60Å (Intensity plot) . µ32 in meV vs (W2 , F ) for B = 100Å (3D) . . . . . . µ32 in meV vs (W2 , F ) for B = 100Å (Intensity plot). 6.13 Potential prole of an ADQW. .. 60. .. 61. .. 61. .. 62. .. 62. .. 63. .. 64. .. 64. .. 65. .. 66. .. 66. .. 67. .. 68. .. 68. .. 69. .. 69. . . . . . . . . . . . . . . . . . . . .. 72. . . . . . . . . . . . . . . . . . . . .. 72. . . . . . . . . . . . . . . . . . . . .. 73. . . . . . . . . . . . . . . . . . . . .. 73. . . . . . . . . . . . . . . . . . . . .. 74. . . . . . . . . . . . . . . . . . . . .. 74. . . . . . . . . . . . . . . . . . . . .. 75. . . . . . . . . . . . . . . . . . . . .. 76. . . . . . . . . . . . . . . . . . . . .. 76. . . . . . . . . . . . . . . . . . . . .. 77. . . . . . . . . . . . . . . . . . . . .. 77. . . . . . . . . . . . . . . . . . . . .. 78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78. 6.14 Potential prole with an electric eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. F = 0kV /cm . . . . . . . . . . . . . . . Escape times for states vs W2 in resonance with B = 20Å and F = 0kV /cm . . . . . . . Escape times for states in resonance (E2 and E3 ) vs W2 with B = 20Å and F = 10kV /cm Energies vs W2 with B = 60Å and F = 0kV /cm . . . . . . . . . . . . . . . . . . . . . . Escape times for states in resonance vs W2 with B = 60Å and F = 0kV /cm . . . . . . . Wavefunctions for B2 = 60Å and F = 0kV /cm for W2 = 28Å (at resonance) . . . . . . . Escape times for states in resonance vs W2 with B = 60Å and F = 2kV /cm . . . . . . . Wavefunctions for states 2 and 3 with B = 60Å and F = 2kV /cm in resonance (W2 = 28Å) Wavefunctions for states 2 and 3 with B = 60Å and F = 2kV /cm below resonance (W2 = 22Å) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavefunctions for states 2 and 3 with B = 60Å and F = 2kV /cm above resonance (W2 = 38Å) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Escape times for states in resonance vs W2 with B = 60Å and F = 10kV /cm . . . . . . Escape times for states in resonance vs W2 with B = 60Å and F = 22kV /cm . . . . . . Escape times for states in resonance vs F with B2 = 60Å and W2 = 34Å . . . . . . . . .. 6.15 Energies vs 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27. W2. in resonance with. B = 20Å. and. 79 79 80 80 81 81 82 82 83 83 83 84 84 85.

(6) 7. LIST OF FIGURES. 6.28 Escape times for states in resonance vs. F. with. B2 = 60Å. and. W2 = 38Å. . . . . . . . . .. 85.

(7) List of Tables 3.1. Parameters for all the cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 4.1. Parameters for ADQW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 5.1. Parameters of the ADQW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 6.1. Parameters of the ADQW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 9.

(8) Chapter 1. Introduction Semiconductor devices are a very important source of electromagnetic (EM) waves in daily life applications. In one extreme, the visible and infrared (IR) lasers are at the core of information technology. In the other, microwave and radio frequency emitters make possible wireless communications [1]. But in the THz region of the EM spectrum, that spans from 100GHz to 10THz, there is an scarcity of compact solid-state sources that operate at room temperature[2]. But in present times there is a need for such devices as a relatively long list of applications has been envisioned for the THz range: chemical detection, astronomy, wireless communications, atmospheric sensing, dental, biological and medical imaging [1, 2] are among the possible applications for this region. Another set of interesting experiments in the THz range are the THz Time Domain Spectroscopy (THz TDS) experiments [3, 4, 5]. These works are important in both fundamental research and realworld applications. Characterization of the optical properties of materials, study of the dynamics of phonon and photocarriers and THz imaging are some of the applications foreseen. Recently THz laser devices have been reported [1, 6], these are Quantum Cascade Lasers (QCL) [7], but no ecient or fast terahertz detectors are yet available [8, 9].. Also THz emitters based on. intersubband transitions in simple and asymmetric quantum wells have been proposed [10, 11, 12]. In particular, compound semiconductors (III-V) and (II-VI) are used to build multiple quantum wells and superlattices, by growing layers of ternary material together with layers of binary material. very popular material is gallium arsenide compound. Alx Ga1−x As,. of the element.. Al. A. a binary compound which is layered with the ternary. x is the mole fraction GaAs band gap is lower than the value of the AlAs gap. When GaAs crystal the value of the gap is between the values of the pure. gallium arsenide with aluminum impurities, where. The value of the. impurities are added to a. compounds.. GaAs,. Therefore, it is possible to control the height of the barriers between quantum wells. varying the concentration. x. of. Al. in a. Al[x] Ga[1−x] As. material.. Asymmetric multiple quantum wells are suited to the design of THz devices because their energies can be tuned so that the intersubband transitions in the conduction band lie in the THz range. Besides the value of these transitions can be designed changing the geometry of the well and also introducing a DC electric eld in the growth direction of the layers. The aim of the work presented here is to study systematically the nonlinear absorption of Asymmetric Double Quantum Wells (ADQWs) designed for the THz range. The dependence of the absorption on dierent parameters of the quantum wells is studied.. In particular, dierent values of the bar-. rier and wells width are considered, which account for the geometrical coupling of the wells and the corresponding eigenfunctions of the system. Also, a DC eld in the direction of growth of the wells 11.

(9) 12. CHAPTER 1.. INTRODUCTION. is introduced in the calculations [13, 28], as a second interaction for the 2D electronic gas in each subband, and the nonlinear absorption is obtained for dierent DC and AC eld intensities.. The. second and third nonlinearities are strong depending on the geometry and the electric eld applied showing well dened ranges of the parameters, for which the nonlinear absorption could be used in device applications. To acquire a better understanding of the absorptions calculated the changes in the energies, wavefunctions and dipole moments and absorption for dierent parameter values are shown in the results. An exact numerical method is used to calculate the energies, wavefunctions and dipole moments. The nonlinear absorption is obtained by using a perturbation calculation of the nonlinear susceptibility up to the third order within a density matrix approach. The absorption spectra are studied near the anticrossing of the two sub-bands in resonance. Special attention is given to the nonlinear peaks due to the intensities of both elds. For each set of geometric parameters an unique DC electric eld is found at which giant second and third order nonlinearities are found. Some of them are of negative absorption or gain. In chapter 2 the basic theory of quasibound states is introduced and the method to calculate the eigenenergies, the wave functions and the dipole moments of Asymmetric Quantum Wells (AQW) is explained. The theory and method to calculate the linear and nonlinear absorption for an Asymmetric Quantum Well (AQW) is introduced in chapter 3. In order to get the absorption a perturbative method based on the density matrix is presented. In chapter 4 the results for energies, wavefunctions, dipole moments and linear and nonlinear absorption are presented using dierent geometries without electric eld. Chapter 5 contains the results for energies, wavefunctions, dipole moments and linear and nonlinear absorption as a function of the intensity of the external DC electric eld. An analysis of the transition between the second and third states (E2 and. E3 ). at resonance is. presented in chapter 6. In particular, the energy of transition between the states (E23. = E3 − E2 ). and the dipole moment (µ23 ) for dierent geometries and dierent values of the applied DC electric eld are analysed. Finally, in the same chapter the escape times for the eigenstates of the ADQW are presented for dierent values of the parameters..

(10) Part I. Theory and numerical methods. 13.

(11) Chapter 2. Basic calculations for Quantum Structures In this chapter the theory and method to calculate the eigenenergies and the wave functions of an Asymmetric Quantum Well (AQW) will be explained.. Also, it is necessary to calculate the dipole. moments between the eigenstates using the eigenfunctions to obtain the absorption.. 2.1 Complex Eigenenergies 2.1.1. General formulation. In fundamental quantum mechanics it is known that the Hamiltonian generates the time evolution of quantum states. If. |ψ (t)i. is the state of the system at time t, then the insertion of this state in the. Schrödinger equation gives us the following equation,. Ĥ |ψ (t)i = if the state at. t = 0. is known to be. |ψ (0)i. i ∂ |ψ (t)i h̄ ∂t. and. Ĥ. (2.1). is independent of time eq.. 2.1 has a known. solution:. |ψ (t)i = e−iĤt/h̄ |ψ (0)i and if. |ψ (0)i. is an eigenstate of the operator. Ĥ. (2.2). the solution is:. |ψ (t)i = e−iEt/h̄ |ψ (0)i where. E. is the eigenvalue of. |ψ (0)i.. (2.3). The time evolution of the wave function can be obtained from. eq. 2.3:. if the eigenenergy value. E. hr|ψ (t)i = hr| e−iEt/h̄ |ψ (0)i. (2.4). ψ (r, t) = e−iEt/h̄ ψ (r, 0). (2.5). is complex valued. E = Er − iEi 15. then,.

(12) 16. CHAPTER 2.. BASIC CALCULATIONS FOR QUANTUM STRUCTURES. ψ (r, t) = e−i(Er −iEi )t/h̄ ψ (r, 0) −iEr t/h̄ −Ei t/h̄. ψ (r, t) = e. e. ψ (r, 0). (2.6) (2.7). From eq. 2.7 it can be observed that the state occupation decreases as a function of time for every. r.. Then the probability of nding the system in that state decreases exponentially as a function of the. imaginary part of the complex eigenenergy:. |ψ (r, t)| = e−Ei t/h̄ |ψ (r, 0)| 2. (2.8). 2. |ψ (r, t)| = e−2Ei t/h̄ |ψ (r, 0)|. (2.9). Thus the following relation between the state lifetime and the imaginary part of the complex eigenenergy can be derived:. τ= 2.1.2. h̄ 2Ei. (2.10). Quasibound electron states[16]. Figure 2.1: Scheme of the biased potential showing the energy ranges for bound and quasibound states [16]. The possible eigenenergies of a biased heterostructure can be classied as bound states or quasibound states as seen in g.. 2.1.. Bound states are the states in which the carriers are conned between. innitely thick left and right barriers, the quasibound states can be classied in two types: States whose eigenenergy is higher than the potential energy in all directions are called quasibound states type 1 and states whose eigenenergy is higher than the potential energy at one boundary are called quasibound states type 2. The eigenenergies of bound states are real valued, so the lifetime of these states are in theory innite.. When an electron is in a bound state external energy has to be given to the electron to. escape. But the eigenenergies of quasibound states are complex, then in theory the lifetime of these states is nite. An electron in a quasibound state leaks out from the heterostructure by itself..

(13) 2.2.. 17. TRANSFER MATRIX METHOD WITH EXPONENTIAL FUNCTIONS. When an electron leaks out of the structure its eigenenergies are no longer discrete, but when it is conned its eigenenergies are quasi-discrete since the energy spectrum of these quasi-conned particles consists of a series of broadened energy levels which can be related to the imaginary part of the eigenenergy, the energy line broadening is given by:. Γ = 2Ei. (2.11). And the lifetime of an electron in a quasibound state is given by eq. 2.10.. 2.2 Transfer Matrix Method with exponential functions 2.2.1. The transfer matrix. The Transfer Matrix Method (TMM) is a very known method for the calculation of the eigenenergies of a Multiple Quantum Well (MQW) [15, 16]. The idea is to solve the one dimensional Schrödinger equation with an arbitrary potential prole. First we will review the TMM for a variable potential prole that have segments of constant value. Lets introduce the one dimensional Schrödinger equation.. − V (z)is the potential.. h̄2 d 1 d ϕ (z) + V (z) ϕ (z) = Eϕ (z) 2 dz m (z) dz. (2.12). This prole is divided in regions of constant value and constant eective mass. as shown in the gure 2.2.. Figure 2.2:. Scheme of a semiconductor heterostructure.. The potential is subdivided in regions of. constant value [16]. The value of the potential is. di = zi − zi−1 .. Vi. for the region. zi−1 ≤ z ≤ zi. and the length of this layer is. The single-band eective mass Schrödinger equation for the. ith. region can be written. as:. −. h̄2 d2 ϕi (z) + Vi (z) ϕi (z) = Eϕi (z) 2mi dz 2. (2.13). we can rewrite the equation 2.13 in a more clear form,. d2 ϕi (z) + ki ϕi (z) = 0 dz 2. (2.14).

(14) 18. CHAPTER 2.. where. ki. is the wave vector in the. BASIC CALCULATIONS FOR QUANTUM STRUCTURES. ith. region,. r ki = ± mi ith. is the eective mass and. ϕi (z). 2mi (E − Vi ) h̄2. (2.15). represents the one dimensional envelope wave function in the. layer. The solution of 2.14 is a superposition of a right and a left traveling wave.. ϕi (z) = Ci+ ejki (z−zi−1 ) + Ci− e−jki (z−zi−1 ) In the extreme regions (i. = 0. and. i = N + 1). (2.16). we have to take only one term in 2.16 to ensure. that the wavefunctions of the bound states decay to zero. In the boundary between regions the wave function. z = zi. ϕ (z). and the function. (1/m (z)) dϕ (z) /dz. must be continuous, applying this condition in. we obtain,. Now we can express. + − Ci+1 + Ci+1 = Ci+ ejki di + Ci− e−jki di. (2.17). ki ki+1 + − Ci+1 − Ci+1 = Ci+ ejki di − Ci− e−jki di mi+1 mi. (2.18). Ci+. and. Ci−. in terms of. . Where. Mi,i+1. Ci+ Ci−. + Ci+1. and. . = Mi,i+1. − Ci+1. + Ci+1 − Ci+1. in a matrix form,. (2.19). is the backward transfer matrix dened as,. Mi,i+1.  1  1 + = 2 1−. . mi ki+1 −jki di mi+1 ki e mi ki+1 +jki di mi+1 ki e. . 1− 1+. . mi ki+1 −jki di mi+1 ki e mi ki+1 +jki di mi+1 ki e.  . We can use eq. 2.19 to obtain the relation between the coecients of the rst region those of the last region. The product. (i = N + 1). + + CN +1 C0 = M0,1 M1,2 ...MN,N +1 − C0− CN +1. M0,1 M1,2 ...MN,N +1 is a 2 × 2 matrix, + + CN +1 C0 m11 (E) m12 (E) = − m21 (E) m22 (E) C0− CN +1. To obtain the coecients of. (2.20). (i = 0). and. (2.21). (2.22). ϕi (z), the proper boundary conditions must be applied at extremes.. In. the case of a bound state the coecients in the extremes are chosen to assure a decaying wavefunction, then. − C0+ = CN +1 = 0. (2.23). This assumption gives what is called the one wave function method (OWM) [16]. The eq. 2.23 is satised when,. m11 (E) = 0. (2.24).

(15) 2.3.. 19. TRANSFER MATRIX METHOD WITH AIRY FUNCTIONS. In the quasibound states case the term in eq. 2.24 is complex valued, so in this case it is better to consider the following condition to obtain the complex eigenenergies:. |m11 (E)| = 0. (2.25). In this case the OWM method is applied, but the wavefunctions of the rst and/or the last region represent traveling waves as quasibound states appear. (see section 3.1.1). 2.2.2. Dipole moments. Using either the eq. 2.24 or eq. 2.25 we can obtain the values of the eigenenergies, and then we can solve eq. 2.22 to obtain. + CN +1. assuming a constant value for. ± 2.19 can be applied to obtain the coecients Ci ,. C0− = C .. (2 ≤ i ≤ N ). constant C , the value. These coecients are proportional to the. Finally the set of equations. of the constant is obtained when. the normalization condition is used:. Z. ∞. 2. |ϕ (z)| dz = 1. (2.26). |ϕ (z)| dz = KC 2 ⇒ KC 2 = 1. (2.27). −∞. Z. ∞. 2. −∞. C = K −1/2. (2.28). The dipole moments are calculated from the normalized wave functions using the following equation:. Z. ∞. µij = e. ϕ∗i (z) zϕj (z) dz. (2.29). −∞. ϕi (z) is the wavefunction if the ith state with ϕi (z) and ϕj (z) are real valued wavefunctions µij = µji . where. energy. Ei .. In general,. µij = µ∗ji. but when. 2.3 Transfer Matrix Method with Airy functions When a DC electric eld is applied in the growth direction of the quantum wells (z direction) the potential. V (z)is. biased:.

(16) 20. CHAPTER 2.. BASIC CALCULATIONS FOR QUANTUM STRUCTURES. Figure 2.3: Scheme of the biased potential [16]. The value of the potential is. Vi − |e| F z. for the region. zi−1 ≤ z ≤ zi where F ith region can be. electric eld. The eective mass Schrödinger equation for the. −. is the intensity of the written as[18]:. h̄2 d2 ϕi (z) + (Vi (z) − |e| F z) ϕi (z) = Eϕi (z) 2mi dz 2. (2.30). This equation can be transformed into the Airy dierential equation making the following variable substitution[21]:. ηi (z) = −. !1/3. 2mi 2. (eh̄F ). (E − Vi + |e| F z). (2.31). then eq. 2.30 transforms in the following equation,. d2 ϕi (ηi ) − ηi ϕi (ηi ) = 0 dηi2. (2.32). this is the Airy equation, then its solution is:. ϕi (z) = Ci+ A (ηi (z)) + Ci− B (ηi (z)) where. A (x)and B (x). (2.33). are the solutions of the Airy equation.. To simplify the calculation of the eigenstates of the biased quantum structure the potential is considered to be constant in the extremes, so. ϕ0 (z) = C0+ ejk0 z + C0− e−jk0 z. (2.34). + jkN +1 z ϕN +1 (z) = CN + C0− e−jkN +1 z +1 e. (2.35). In the interface between regions the wave function. ϕ (z)and. the factor. (1/m (z)) (dϕ (z) /dz). be continuous, applying these boundary conditions we obtain the transfer matrices:. must.

(17) 2.3.. TRANSFER MATRIX METHOD WITH AIRY FUNCTIONS. M0,1 =. 1 2. 1 (A (η1 (z0 )) − K1 A0 (η1 (z0 ))) eik0 z0 1 (A (η1 (z0 )) + K1 A0 (η1 (z0 ))) e−ik0 z0. 1 (B (η1 (z0 )) − K1 B 0 (η1 (z0 ))) eik0 z0 1 (B (η1 (z0 )) + K1 B 0 (η1 (z0 ))) e−ik0 z0. 21. (2.36). 1 Mi,i+1 = (A(ηi (zi ))B 0 (ηi (zi ))−A 0 (η (z ))B(η (z ))) i i i i 0 0 0 0 Ai+1,i Bi,i − Ki+1 Ai+1,i Bi,i Bi+1,i Bi,i − Ki+1 Bi+1,i Bi,i × 0 −Ai+1,i A0i,i + Ki+1 A0i+1,i Ai,i −Bi+1,i A0i,i + Ki+1 Bi+1,i Ai,i. (2.37). 1 MN,N +1 = (A(ηN (zN ))B 0 (ηN (zN ))−A 0 (η (z ))B(η (z ))) N N N N 0 −ikN +1 zN 0 e BN,N − KN +1 BN,N e BN,N + KN +1 BN,N × −eikN +1 zN A0N,N − KN +1 AN,N −e−ikN +1 zN A0N,N + KN +1 AN,N. (2.38). ikN +1 zN. where,. A0 =. dB (z) dA (z) 0 ,B = dz dz. Ai,j = A (ηi (zj )) , Bi,j = B (ηi (zj )). K1 =. . iKm0 2/3. ,K =. k0 m1. Ki+1 =. mi mi+1. 2 |e| F h̄2. (2.39). (2.40). 1/3 (2.41). 2/3 K. (2.42). and,. 2/3. KN +1 =. imN kN +1 KmN +1. (2.43). As in section 2.2.1 we can use equations 2.36, 2.37 and 2.38 to obtain the relation between the. (i = 0) and those of the last region (i = N + 1). + + CN +1 C0 = M M ...M 0,1 1,2 N,N +1 − C0− CN +1 + + CN +1 C0 m11 (E) m12 (E) = − m21 (E) m22 (E) C0− CN +1. coecients of the rst region. (2.44). (2.45). Also we can use the OWM method to obtain the complex eigenenergies, this leads to the following condition:. |m11 (E)| = 0. (2.46).

(18) Chapter 3. Nonlinear absorption calculation In this chapter the theory and method to calculate the linear and nonlinear absorption for an Asymmetric Quantum Well (AQW) will be introduced. In order to get the absorption a perturbative method based on the density matrix is presented. To apply the method we will use the eigenenergies and the dipole moments of the structure.. 3.1 Density matrix formalism 3.1.1. Formalism of quantum mechanics. For the nonlinear absorption calculation a perturbative solution of the density matrix is necessary. First a brief formal introduction is presented [17]. To solve by perturbation theory the Schrödinger equation,. ih̄ the Hamiltonian. Ĥ. ∂ψs (r, t) = Ĥψs (r, t) ∂t. is separated in two terms. Ĥ = Ĥ0 + V̂ (t). (3.1) where. Ĥ0. is independent of time and. its eigensolutions are known,. Ĥ0 un (r) = En un (r). (3.2). also the eigenfunctions are assumed to be orthonormalized,. Z. u∗m (r) un (r) d3 r = δmn. (3.3). Every solution of eq. 3.1 can be expressed as a linear combination of the eigenfunctions of. ψs (r, t) =. X. Cns (t) un (r). Ĥ0 , (3.4). n introducing 3.4 in eq. 3.1 we obtain a set of dierential equations for the coecients. ih̄. X d s Cm (t) = Hmn Cns (t) dt n 23. s Cm , (3.5).

(19) 24. CHAPTER 3.. NONLINEAR ABSORPTION CALCULATION. where,. Z Hmn =. u∗m (r) Ĥun (r) d3 r. (3.6). s Cm (t) terms, together with the functions un (r), contain all the information necessary wavefunctions ψs (r, t).. Note that the to know the. Other useful information contained in these functions is the average value of any observable,. Z hAi =. ψs∗ Âψs d3 r. (3.7). In Dirac notation,. D E D E hAi = ψs Â ψs = s Â s and in terms of the. s Cm (t). (3.8). terms,. hAi =. X. D E s∗ Cm (t) Cns (t) Amn ; Amn = um Â un. (3.9). mn In conclusion, the knowledge of the eigenfunctions of the part of the Hamiltonian that is time. s using eq. 3.5. Then these coecients can Cm Hamiltonian Ĥ and average value of any observable.. independentĤ0 can be used to obtain the coecients used to calculate the wavefunctions of the. 3.1.2. be. Density matrix. We have seen in section 3.1.1 that if we know that a system is in state s with wave function then we can calculate the expectation value of any observable. Â.. ψs (r, t). But in many situations the actual. state of the system is unknown, then it is at least necessary to know the probability distribution of nding a system in a particular state. Assuming that the probability of nding a system in the state s is. p (s),. then the ensemble average. of an observable is,. ¯ = hAi. X. p (s) hAis. (3.10). s where. hAis. is the average value of the observable. hAis =. X. Â. when the system is in the state. s∗ Cm (t) Cns (t) Amn. s, (3.11). mn Then,. ! ¯ = hAi. X s. p (s). X. s∗ Cm. (t) Cns. (t) Amn. E D ; Amn = um Â un. (3.12). mn. At this point it is useful to introduce the density matrix, dened as,. ρmn =. X s. s∗ p (s) Cm (t) Cns (t). (3.13).

(20) 3.1.. 25. DENSITY MATRIX FORMALISM. ρmn : X ¯ = hAi ρmn Amn. then eq. 3.12 can be expressed in terms of. (3.14). mn and in a compact form,. ! ¯ = hAi. X X n. ρmn Amn. =. X. m. ρ̂Â. = T r ρ̂Â. (3.15). nn. n. ¯ = T r ρ̂Â hAi. (3.16). So all the information necessary to calculate the expected values of any observable is in the density matrix. Now we will concentrate on it. The time evolution of the density matrix elements. ρnm. with the restriction. dp (s) /dt = 0. can be. easily obtained using equations 3.5 and 3.13:. i X (ρnν Hνm − Hnν ρνm ) h̄ ν i ρ̂Ĥ − Ĥ ρ̂ ρ̇nm = h̄ nm i −i h ρ̇nm = Ĥ, ρ̂ h̄ nm. ρ̇nm =. In general. dp (s) /dt 6= 0,. (3.17). (3.18). (3.19). to account with this case some phenomenological damping terms are. introduced in the equation 3.19:. ρ̇nm =. i −i h Ĥ, ρ̂ − γnm (ρnm − ρeq nm ) h̄ nm. (3.20). where,. ρeq nm : γnm : 3.1.3. is the equilibrium value of is the decay rate of. ρnm. ρnm. Perturbation solution of the density matrix. From the previous section we can take the following relations:. ρ̇nm =. i −i h Ĥ, ρ̂ − γnm (ρnm − ρeq nm ) h̄ nm Ĥ = Ĥ0 + V̂ (t). (3.21). (3.22). then,. ρ̇nm =. i i −i h i h − − γnm (ρnm − ρeq Ĥ0 , ρ̂ V̂ , ρ̂ nm ) h̄ h̄ nm nm. (3.23). the perturbation term for the dipole moment approximation is,. V̂ (t) = −û.E (t) ; û = −er̂. (3.24).

(21) 26. CHAPTER 3.. An orthonormal base of. Ĥ0. NONLINEAR ABSORPTION CALCULATION. is used to construct the density matrix, the. Ĥ0. matrix representation. is,. H0,nm = Enm δnm. (3.25). i Ĥ0 , ρ̂ = Ĥ0 ρ̂ − ρ̂Ĥ0 nm P nm = ν (H0,nν ρνm − ρnν H0,νm ) = En ρnm − Em ρnm = (En − Em ) ρnm. (3.26). and then,. h. To simplify the equations we can use the following denition:. ωnm =. En − E m h̄. (3.27). Finally,. i i h V̂ , ρ̂ − γnm (ρnm − ρeq nm ) h̄ nm. (3.28). i X (Vnν ρνm − ρnν Vνm ) − γnm (ρnm − ρeq nm ) h̄ ν. (3.29). ρ̇nm = −iωnm ρnm − ρ̇nm = −iωnm ρnm −. The exact solution for this equation is not easy, then we will nd its perturbation solution. First, the perturbation term is multiplied by a factor expansion of. ρnm in. powers of. λ, Vij → λVij .. Then we substitute in eq.. 3.29 an. λ: (1) 2 (2) ρnm = ρ(0) nm + λρnm + λ ρnm + · · ·. (3.30). this expression is substituted in eq. 3.29 and then the terms on each side of the equality with the same power of. λ. are matched.. And the following coupled equations are obtained:. (0) (0) (0) ρ̇nm = −iωnm ρnm − γnm ρnm − ρeq nm h i (1) (1) ˆ ρ̇nm = − (iωnm + γnm ) ρnm − h̄i V̂ , ρ(0) h inm (2) (2) ˆ ρ̇nm = − (iωnm + γnm ) ρnm − h̄i V̂ , ρ(1) nm. . . . The. (0). ρnm. (3.31). term corresponds to the steady-state of the system, without external eld, then we take. a constant value for it:. (eq) ρ(0) nm = ρnm where,. (eq). ρnm = 0. for. n 6= m. [17].. To obtain. (1). ρnm (t). (3.32) we use eq.. 3.31 and make the following. substitution,. (1) −(iωnm −γnm )t ρ(1) nm (t) = Snm (t) e nally,. (3.33).

(22) 3.2.. 27. CALCULATIONS OF THE SUSCEPTIBILITY PERTURBATION TERMS. −(iωnm −γnm )t ρ(1) nm (t) = e. Z. t. dt0. −∞. i 0 −i h V̂ (t0 ) , ρ̂(0) e(iωnm −γnm )t h̄ nm. (3.34). This procedure can be repeated with higher order terms of the density matrix, in general the following recursion equation is obtained:. ρ(q) nm (t) =. Z. t. −∞. i 0 −i h V̂ (t0 ) , ρ̂(q−1) e(iωnm −γnm )(t −t) dt0 h̄ nm. (3.35). 3.2 Calculations of the susceptibility perturbation terms 3.2.1. Density matrix calculation of the linear susceptibility. For the calculation of the linear susceptibility eq. 3.34 is used with the following perturbation term:. V̂ (t0 ) = −û.E (t0 ) , û = er̂. (3.36). with the electric eld represented as a wave superposition,. E (t) =. X. E (ωp ) e−iωp t. (3.37). p. i V̂ (t) , ρ̂(0) , h i i P h (0) (0) V̂ (t) , ρ̂(0) = ν V (t)nν ρνm − ρnν V (t)νm i P h (0) (0) = ν ûnν ρνm − ρnν ûνm .E (t) (0) (0) = − ρmm − ρnn ûnm .E (t). First lets calculate the commutator. h. The simplication of the last equation takes into account the fact that. (3.38). (0). ρnm. is diagonal.. The substitution of the commutator and eq. 3.37 in eq. 3.34 gives,. ρ(1) nm. (t) = h̄. −1. . ρ(0) mm. −. ρ(0) nn. . ûnm .. X. −(iωnm −γnm )t. Z. t. E (ωp ) e. 0. dt0 e(i(ωnm −ωp )+γnm )t. (3.39). −∞. p the integral is easy to calculate,. X û .E (ω ) e−iωp t nm p −1 (0) ρ(1) ρ(0) nm (t) = h̄ mm − ρnn (ω − ω nm p ) − iγnm p. (3.40). Now we can calculate the expectation value for the dipole moment:. X hu (t)i = tr ρ̂(1) û = ρ(1) nm umn. (3.41). nm. hu (t)i =. X u [u .E (ω )] e−iωp t mn nm p (0) h̄−1 ρ(0) mm − ρnn (ω − ω ) − iγnm nm p nm p. X. (3.42).

(23) 28. CHAPTER 3.. hu (t)ican. NONLINEAR ABSORPTION CALCULATION. be expressed as a sum of frequency components:. hu (t)i =. X. hu (ωp )i e−iωp t. (3.43). p and the linear susceptibility is dened by the following equation:. P (ωp ) = N hu (ωp )i = χ(1) (ωp ) .E (ωp ). (3.44). where N denotes the atomic number density. Following the equations 3.42-3.44 the susceptibility tensor is:. χ(1) (ωp ) =. N X (0) umn unm ρmm − ρ(0) nn h̄ nm (ωnm − ωp ) − iγnm. (3.45). This tensor can be written in Cartesian form,. uimn ujnm N X (0) ρmm − ρ(0) nn h̄ nm (ωnm − ωp ) − iγnm. (3.46). N X (0) uimn ujnm N X (0) uimn ujnm ρmm − ρnn h̄ nm (ωnm − ωp ) − iγnm h̄ nm (ωnm − ωp ) − iγnm. (3.47). (1). χij (ωp ) = then,. (1). χij (ωp ) =. N X (0) uimn ujnm N X (0) uinm ujmn − ρmm ρmm h̄ nm (ωnm − ωp ) − iγnm h̄ nm (ωmn − ωp ) − iγmn N X (0) uimn ujnm uinm ujmn (1) − ρ χij (ωp ) = h̄ nm mm (ωnm − ωp ) − iγnm (ωmn − ωp ) − iγmn. (1). χij (ωp ) =. using the fact that. ωnm = −ωmn. (1). χij (ωp ) =. and. (3.48). (3.49). γnm = γmn ,. uimn ujnm uinm ujmn N X (0) ρmm + h̄ nm (ωnm − ωp ) − iγnm (ωnm + ωp ) + iγmn. (3.50). In the one dimensional case we consider that the perturbation and the electric eld are in a xed direction. z,. then only one term of the tensor. χ(1) zz omitting the. z. N X (0) (ωp ) = ρ h̄ nm mm. . (1). χij. is considered:. uzmn uznm uznm uzmn + (ωnm − ωp ) − iγnm (ωnm + ωp ) + iγmn. (3.51). index,. χ(1) (ωp ) =. N X (0) umn unm unm umn ρmm + h̄ nm (ωnm − ωp ) − iγnm (ωnm + ωp ) + iγmn. It is also useful to conserve the tensor. χ(1) (ωp ) =. (1). χij. (3.52). in the form of eq. 3.46:. N X (0) umn unm ρmm − ρ(0) nn h̄ nm (ωnm − ωp ) − iγnm. (3.53).

(24) 3.2.. CALCULATIONS OF THE SUSCEPTIBILITY PERTURBATION TERMS. 3.2.2. 29. Density matrix calculation of the second order susceptibility. The second order term of the density matrix is (see eq. 3.35):. ρ(2) nm. −(iωnm −γnm )t. Z. t. dt0. (t) = e. −∞. i 0 −i h V̂ (t0 ) , ρ̂(1) e(iωnm −γnm )t h̄ nm. (3.54). (1) ûnν ρ(1) νm − ρnν ûνm .E (t). (3.55). in this case the commutator is:. h. V̂ (t) , ρ̂(1). i. =− nm. X ν. the rst order terms of the density matrix are written with dummy indexes,. X u .E (ω ) e−iωp t νm p −1 (0) ρ(1) ρ(0) νm (t) = h̄ mm − ρνν (ω − ω νm p ) − iγνm p. (3.56). X u .E (ω ) e−iωp t nν p −1 (0) (0) ρ(1) (t) = h̄ ρ − ρ nν νν nn (ω − ω ) − iγnν nν p p. (3.57). after substituting these terms in the commutator,. h. V̂ (t) , ρ̂(1). i. P (0) (0) P [uνm .E(ωp )][ûnν .E(ωq )] −i(ωp +ωq )t ρ − ρ e mm νν ν pq (ωνm −ωp )−iγνm (0) (0) P [unν .E(ωp )][ûνm .E(ωq )] −i(ωp +ωq )t −1 P +h̄ e ν ρνν − ρnn pq (ωnν −ωp )−iγnν. = −h̄−1 nm. (3.58). Equation 3.58 is inserted into 3.54 and the integral is calculated to get,. (2). ρnm (t) =. P P n ν(0) pq(0). ρmm −ρνν [uνm .E(ωp )][ûnν .E(ωq )] [(ωnm −ωp −ωq )−iγnm ][(ωνm −ωp )−iγνm ] h̄2 o ρ(0) −ρ(0) [u .E(ωp )][ûνm .E(ωq )] − νν h̄2 nn [(ωnm −ωpnν −ωq )−iγnm ][(ωnν −ωp )−iγnν ] P P e−i(ωp +ωq )t ≡ ν pq Knmν e−i(ωp +ωq )t. (3.59). Now we calculate the expectation value of the dipole moment,. hui =. X. ρnm umn =. nm. X. hû (ωr )i e−iωr t. (3.60). r. hu (ωp + ωq )i =. XX. Knmν umn. (3.61). nmν (pq) Remember that. Knmν. is the expression between brackets in eq. 3.59.. Re-introducing the polarization:. P(2) (ωp + ωq ) = N hu (ωp + ωq )i = N. XX. Knmν umn. (3.62). nmν (pq). P(2) (ωp + ωq ) =. XX jk (pq). (2). χijk (ωp + ωq , ωp , ωq ) Ej (ωq ) Ek (ωp ). (3.63).

(25) 30. CHAPTER 3.. NONLINEAR ABSORPTION CALCULATION. then the susceptibility tensor is:. P (2) N χijk (ωp + ωq , ωp , ωq ) = n nmν h̄2 (0) (0) uimn ujnν uk νm ρmm − ρνν [(ωnm −ωp −ωq )−iγ nm ][(ωνm −ωp )−iγνm ] o i j (0) (0) umn uνm uk nν − ρνν − ρnn [(ωnm −ωp −ωq )−iγ nm ][(ωnν −ωp )−iγnν ] The. (2). χijk. (3.64). term must have intrinsic permutation symmetry, but eq. 3.64 does not have this, so we. have to introduce some permutation terms into 3.64,. P (2) N χijk (ωp + ωq , ωp , ωq ) = n nmν 2h̄2 h (0) (0) uimn ujnν uk νm ρmm − ρνν [(ωnm −ωp −ωq )−iγnm ][(ωνm −ωp )−iγνm ] i j uimn uk nν uνm + [(ωnm −ωp −ωq )−iγ h nm ][(ωνm −ωqi)−iγjνm ] k (0) (0) umn uνm unν − ρνν − ρnn [(ωnm −ωp −ωq )−iγ nm ][(ωnν −ωp )−iγnν ] io i k j. (3.65). umn uνm unν [(ωnm −ωp −ωq )−iγnm ][(ωnν −ωq )−iγnν ]. To write this equation in a more simple form we can substitute the indexes. (m, ν, n). in the terms that are multiplied by. (2). χijk (ωp + ωq , ωp , ωq ) =. . (0) ρνν. N 2h̄2 n. −. (ν, n, m) by the indexes. . (0) ρnn :. P. . nmν. (0). (0). ρmm − ρνν. . uimn ujnν uk νm [(ωnm −ωp −ωq )−iγnm ][(ωνm −ωp )−iγνm ] i k umn unν ujνm + [(ωnm −ωp −ωq )−iγ nm ][(ωνm −ωq )−iγνm ] uinν ujmn uk − [(ωνn −ωp −ωq )−iγνn ][(ωνm νm −ωp )−iγνm ] o j uinν uk mn uνm − [(ωνn −ωp −ωq )−iγνn ][(ωνm −ωq )−iγνm ] To rewrite this tensor in another simple way, we make the index substitutions. (3.66). (m, ν, n) → (l, m, m). (ωlm , ωln , ωmn ) → (−ωml , −ωnl , −ωnm ): P (2) (0) (0) N χijk (ωp + ωq , ωp , ωq ) = 2h̄ ρ − ρ mm 2 lmn ll n i j k. and the frequency substitutions. uln unm uml [(ωnl −ωp −ωq )−iγnl ][(ωml −ωp )−iγml ] j uiln uk nm uml + [(ωnl −ωp −ωq )−iγ ][(ω nl ml −ωq )−iγml ] ujln uinm uk ml + [(ωnm +ωp +ωq )+iγnm ][(ωml −ωp )−iγml ] o j i uk ln unm uml + [(ωnm +ωp +ωq )+iγ nm ][(ωml −ωq )−iγml ]. In the one dimensional case we can consider the case. χ(2) (ωp + ωq , ωp , ωq ) =. N 2h̄2 n. P. lmn. . (2). χzzz. and omit the. z. (3.67). index,. (0) (0) ρll − ρmm uln unm uml. 1 [(ωnl −ωp −ωq )−iγnl ][(ωml −ωp )−iγml ] 1 + [(ωnl −ωp −ωq )−iγnl ][(ωml −ωq )−iγml ] 1 + [(ωnm +ωp +ωq )+iγnm ][(ωml −ωp )−iγml ] o 1 + [(ωnm +ωp +ωq )+iγnm ][(ωml −ωq )−iγml ]. (3.68).

(26) 3.3.. 31. OPTICAL ABSORPTION COEFFICIENT. 3.2.3. Density matrix calculation of the third order susceptibility. The calculation of the third order susceptibility is done with the same procedure of the second order susceptibility calculation (see section 3.2.2). The result for the third order susceptibility (omitting the tensor indexes since we work in 1D) is [17]:. χ(3). (ω n p + ωq + ωr , ω p , ω q , ω r ) =. N P h̄3 I. (0) νnml (ρll. P. (0). − ρmm )uln unν uνm uml. 1 [(ωnl −ωp −ωq −ωr )−iγnl ][(ωνl −ωp −ωq )−iγνl ][(ωml −ωp )−iγml ] 1 + [(ωnν +ωp +ωq +ωr )+iγnν ][(ωmn −ω p −ωq )−iγmn ][(ωml −ωp )−iγml ] 1 + [(ωnν +ωp +ωq +ωr )+iγnν ][(ωνl −ωp −ωq )−iγνl ][(ωml −ωp )−iγml ] o 1 + [(ωmν −ωp −ωq −ωr )−iγmν ][(ωmn −ω −ω )−iγ ][(ω −ω )−iγ ] p q mn p ml ml. where the. PI. (3.69). term is the permutation operator. It means that the sum between brackets has to. be made over all the permutations of the. ωp , ωq. and. ωr. terms.. 3.3 Optical absorption coecient The absorption coecient in a crystal is the fraction of photons absorbed per unit distance[15]:. α=. N o. of photons absorbed per unit volume per second N o. of photons injected per unit area per second. (3.70). this coecient can be expressed as a function of the refractive index:. ω Im (n (ω)) nr c0. (3.71). p 1 + 4πχ (ω) ' 1 + 2πχ (ω). (3.72). α (ω) = where,. n (ω) = so,. α (ω) = 2π. ω Im (χ (ω)) nr c0. (3.73). In nonlinear optics, the optical response can be described by expressing the polarization as a power series of the eld strength. E (t): P (t) = χ(1) E (t) + χ(2) E2 (t) + χ(3) E3 (t) + · · ·. (3.74). To calculate the absorption we are interested in the susceptibility so we write the relation between the polarization and the susceptibility,. P (t) = χE (t). (3.75). Comparing equations 3.75 and 3.74 we can deduce the expression that have to be used in 3.73 for. χ: χ (ω) = χ(1) (ω) + χ(2) (ω) E (t) + χ(3) (ω) E2 (t) where,. (3.76).

(27) 32. CHAPTER 3.. NONLINEAR ABSORPTION CALCULATION. N X (0) umn unm ρmm − ρ(0) nn h̄ nm (ωnm − ωp ) − iγnm P (0) (0) χ(2) (2ω, ω) = h̄N2 lmn ρll − ρmm uln unm uml n. χ(1) (ω) =. (3.77). 1 [(ωnl −2ω)−iγnl ][(ωml −ω)−iγml ]. + [(ωnm +2ω)+iγnm1][(ωml −ω)−iγml ]. (3.78). o. and,. χ(3). (0) (0) νnml (ρll − ρmm )uln unν uνm uml 1 [(ωnl −3ω)−iγnl ][(ωνl −2ω)−iγνl ][(ωml −ω)−iγml ] 1 + [(ωnν +3ω)+iγnν ][(ωmn −2ω)−iγ mn ][(ωml −ω)−iγml ] 1 + [(ωnν +3ω)+iγnν ][(ωνl −2ω)−iγνl ][(ωml −ω)−iγml ] o 1 + [(ωmν −3ω)−iγmν ][(ωmn −2ω)−iγ mn ][(ωml −ω)−iγml ]. (3ω, 2ω, ω) = n. N h̄3. P. As we are considering only the sub-harmonic terms we have made the substitutions. (ω, ω, ω). (3.79). (ωp , ωq , ωr ) →. [17].. (0) ρll are equal to the equilibrium state occupations. So (0) the terms ρll can be represented by the Fermi-Dirac distribution function fl [19]: The diagonal terms of the density matrix. 1. (0). ρll = fl = where. EF. 1+. is the Fermi level of the system, T is the temperature and. For a given electron density doping, the quasi-Fermi levels. N=. N,. Fc. n and fc. Nn (E) is. KB. is the Boltzmann constant.. which is determined by the injection current and the background. can be obtained using the following equation,. X. Nn =. is the electron density in the. XZ. n dEρ2D e (E) fc (E). (3.81). n. n = occupied subbands where. (3.80). e[(El −EF )/KB T ]. nth. conduction subband,. ρ2D e (E). is the density of states. the energy density. From eq. 3.81 we obtain,. Z Nn =. n dEρ2D e (E) fc (E). also we can use the Fermi-Dirac distribution for. fcn =. (3.82). fcn (E),. 1 e[(En −EF )/KB T ] + 1. (3.83). and using the following identities,. ρ2D e (E) = Z dx. me πh̄2 Lz. 1 = −ln 1 + e−x x 1+e. (3.84). (3.85).

(28) 3.3.. 33. OPTICAL ABSORPTION COEFFICIENT. we obtain:. Nn =. me K B T ln 1 + e[(EF −En )/KB T ] 2 πh̄ Lz. (3.86). where,. me : is the eective mass of electrons in the subband KB : is the Boltzmann constant T : is the temperature LZ : is the eective width of the quantum well EF : is the Fermi energy and En : is the energy of the subband The Fermi energy is calculated using the following expression [15],. EF = E1 + kB T where. nd = 1018 m−2 , ns. Exp. nd LZ −1 ns. (3.87). is obtained by this formula,. ns =. me kB T πh̄2. (3.88). As an example we will calculate the linear absorption. Using equations 3.77, 3.86 and 3.73, the linear absorption can be written:. ω Im α (ω) = π nr c0. N X (0) umn unm ρmm − ρ(0) nn h̄ nm (ωnm − ωp ) − iγnm. ! (3.89). ! me KB T X 1 + e[(EF −Em )/KB T ] umn unm ln πh̄3 Lz nm 1 + e[(EF −En )/KB T ] (ωnm − ωp ) − iγnm umn unm ω me KB T X 1 + e[(EF −Em )/KB T ] α (ω) = ln Im nr c0 h̄3 Lz nm (ωnm − ωp ) − iγnm 1 + e[(EF −En )/KB T ]. ω Im α (ω) = π nr c0. (3.90). (3.91). ! ω me KB T X (ω − ω ) + iγ 1 + e[(EF −Em )/KB T ] nm p nm α (ω) = ln |umn | 2 Im 2 2 nr c0 h̄3 Lz nm 1 + e[(EF −En )/KB T ] (ωnm − ωp ) + γnm γnm 1 + e[(EF −Em )/KB T ] ω me KB T X 2 ln α (ω) = |u | mn 2 2 nr c0 h̄3 Lz nm 1 + e[(EF −En )/KB T ] (ωnm − ωp ) + γnm P In general the absorption is expressed for a transition n → m and the sumation term nm. (3.92). (3.93). can be. dropped, so nally [19, 5]:. ω me K B T h̄γnm ln α (ω) = |umn | 2 2 nr c0 h̄ Lz (En − Em − h̄ωp )2 + (h̄γnm )2. . 1 + e[(EF −Em )/KB T ] 1 + e[(EF −En )/KB T ]. (3.94). The non-linear absorption is calculated in a similar way using the equations 3.73, 3.76, 3.77, 3.78, 3.79 and 3.86..

(29) Part II. Results. 35.

(30) 37. We present a systematic study of the coupling between three excited subbands in an asymmetric double quantum well, in which the coupling parameter is the width of the barrier between the wells. We rst study the variation of wave functions and eigen-energies as the barrier decreases from values of the order of the wide well to very narrow barriers of 10A. For three dierent values of the barrier width which correspond to weak, middle and strong coupling we study the interaction with light by calculating the linear absorption and we follow the behavior of our device as function of intensity of the light. Additionally, we also study the eect of an external electric eld, which can also be weak, middle and strong and discuss the changes in linear and nonlinear absorption. Finally we also study the escape times of the electrons from the device as a consequence of the electric eld. The calculations that were made can be described in a structured way, we have made them changing basically three parameters of the asymmetric double quantum well. 1. The width of the narrow quantum well: The range of values for this well is from 1nm to 15nm in some cases, this range is sucient to sweep two resonance regions. Next, we concentrated our attention in the resonance regions. Finally, we selected the rst resonance to make more detailed calculations. 2. The width of the barrier: In this case we used mainly three values 2nm, 6nm and 10nm. We choose these values to have strong, mean and weak interaction between the wells, and we checked that values under 1nm and above 15 nm do not bring signicant changes. So the results obtained in the 2nm-10nm range are very representative. 3. The strength of the DC electric eld: We used values in the range. 0kV /cm − 32kV /cm. − 80kV /cm).. This. range of values is in the typical range reported in the literature [5, 1, 22](2. For each chapter the results obtained can be described schematically. First, we show the results of the bound/quasibound. 1 eigenenergies of the ADQW by using the OWM method (see: 2.2.1, eq. 2.46),. then the energies of transition are calculated. In the case of quasibound states, the escape rates (see: 2.1, eq. 2.10) were obtained with the OWM method too. Then, the calculation of the wavefunctions of the bound/quasibound states is made using the eigenenergies and the dipole moments are obtained from the wavefunctions. Finally, the nonlinear absorption is calculated from the energies and dipole moments. In general the following parameters are common to all the quantum wells:. • GaAs:. for the wells. • Al0.2 Ga0.8 As:. for the barriers. Thus,. Mass of the electron Eective mass for the wells Eective mass for the barriers Height of the barriers (or depth for the wells). m0 mW mB V0. Table 3.1: Parameters for all the cases. 1 See. sec. 2.1.2. 9.1 ∗ 10−31 kg 0.0665m0 0.0832m0 167meV.

(31) Chapter 4. Linear and nonlinear absorption and geometry The ADQW used for the calculations presented in this chapter have the following characteristics (see g. 4.1):. Figure 4.1: Potential prole of an ADQW. W1 B W2. 100Å {20Å, 60Å, 100Å} {20Å − 150Å} in 1Å. steps.. Table 4.1: Parameters for ADQW. 4.1 Energies The eigenenergies versus. W2. width for diverse values of the barriers are in gures 4.2-4.4.. The variation of the eigenenergies when the barrier is one for a. W2. well of approximately. 30Å. 20Å. and the second one for 39. shows clearly two resonances: the rst. 100Å.. In. 30Å. the states 2 and 3 are in.

(32) 40. CHAPTER 4.. resonance and for. 100Å. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. this condition is encountered for the states 1 and 2. This behavior is known. as anticrossing and in it there is strong interaction or coupling between the two subbands.. Figure 4.2: Eigenenergies vs. In the cases of. W2. with. B = 20Å. and zero electric eld (F. = 0kV /cm). B = 60Å and B = 100Å the same resonances are observed, however the anticrossings. have a smaller gap of energy between the states in resonance. The energy of transition is related to the coupling so we conclude that the coupling between the subbands in resonance is weaker when we have a wider barrier between the wells.. Figure 4.3: Eigenenergies vs. W2. with. B = 60Å. and zero electric eld (F. = 0kV /cm).

(33) 4.2.. 41. TRANSITIONS. Figure 4.4: Eigenenergies vs. W2. with. B = 100Å. and zero electric eld (F. = 0kV /cm). To support the observations made above for the eects of the barrier the gure 4.5 is shown.. Figure 4.5: Eigenenergies vs. W2. for several values of. B. with zero electric eld (F. = 0kV /cm). 4.2 Transitions W2 E31 transition the E32 transition. Now we will focus on the energy transitions. In gures 4.6-4.8 the energy dierences are shown vs.. B = 20Å, B = 60Å and B = 100Å. in the range of 15 − 30T Hz which is. for. In all the cases similar eects are observed. The. is. above the THz gap. Near the rst resonance. has the smaller value and at the second resonance the. E21. transition is smaller.

(34) 42. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. Figure 4.6: Transition energies vs. W2. with. B = 20Å. and zero electric eld (F. In g. 4.6 we can observe that in the rst resonance the value of the and its value is approximately 10THz. The second resonance is near. 100Å. E32. transition is minimum. E21. and. E32. E21 transition W2 = 44T Hz : at. and in it the. is minimum with a value of 2THz. Also another interesting condition is observed at this geometry the. = 0kV /cm). transitions are equal and we can expect a strong absorption for 10THz. radiation.. Figure 4.7: Transition energies vs. W2. with. B = 60Å. E32 is 2THz W2 = 47.5T Hz .. In g. 4.7 the minimum value for the transition 1THz. The equality between. E32. and. E21. is at. and zero electric eld (F. = 0kV /cm). and the minimum for. E21. is below.

(35) 4.3.. 43. WAVEFUNCTIONS. Figure 4.8: Transition energies vs. W2. with. B = 100Å. and zero electric eld (F. = 0kV /cm). In g. 4.8 the minimum value for the transition. E32. is below 1THz and the minimum for. E32. and. E21. too little to be estimated. The equality between. As the barrier is changed we note the following.. E32. transition is about 10 THz, the minimum for a. minimum in 0.5 THz for a. E21. 100Å. For a. 60Å. is at. 20Å. E21. is. W2 = 47.5T Hz . barrier the minimum value of the. barrier is 2 THz and this transition has a. barrier, these values are in the THz gap. The same occurs for the. transition when the barrier is changed. These eects are explained as before by an increase of the. coupling between the states in resonance when the barrier is shrinked. We see that for weak coupling the transitions' energies shift to lower frequencies because when the barrier is thicker there is less coupling between the wells.. Another view of the same eect can be. understood by noting that the states' energies tend to the value they would have if the wells where independent of each other. In the resonances the widths of the wells are such that if they were independent one state of one of the wells would have the same energy of an state of the other. In the ADQW when they are coupled this 'degeneracy' disappears and the energy of one state rises some value and the energy of the other decreases by the same value approximately. The value of the change depends on the coupling in the following manner: when the coupling is weak this value is small and viceversa.. 4.3 Wavefunctions An interesting analysis for the wavefunctions can be made around the resonances and for several barrier widths..

(36) 44. 4.3.1. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. First resonance. Figure 4.9: Eigenfunctions for F=0 kV/cm,. B = 20Å, 60Å. and. In g. 4.9 the wavefunctions for the rst three eigenenergies are plot for the functions for. W2 = 27Å, B = 60Å. and. W2 = 27Å, B = 100Å. 100Å. at resonance. W2 = 29Å and B = 20Å.. Also. are shown. As can be seen in all the. cases in resonance the wavefunctions for the two resonant states have similar forms. Of these forms one is called symmetric and the other antisymmetric form. When the barrier is narrow the symmetry of the functions is not perfect, but when it is thicker an almost perfect symmetry appears (see for example g. 4.9 for. B = 60Å).. A consequence of this perfect symmetry is a bigger value for the dipole. moments at the resonances when the barrier is wider. This will be seen in section 4.4..

(37) 4.3.. 45. WAVEFUNCTIONS. Figure 4.10: Eigenfunctions for F=0 kV/cm,. B = 20Å, 60Å. and. 100Å. when. W2. is. 5Å. shorter than its. resonance value (below resonance). In gs 4.10 and 4.11 the wavefunctions are shown for values of. W2. below and above the resonance. W2 is smaller than its resonance moved 5Å from the resonance value in. value. Consequently these conditions are called below resonance, when value, and above resonance, when. W2. is bigger. In our case we. both directions.. In the below resonance condition (g. 4.10) the wavefunctions of the states that were in resonance are not symmetrical. The lost of the symmetry is not equal for the three barriers considered: for the thinner barrier (B. = 20Å). the symmetry between the wavefunctions is almost the same observed at. resonance, for the barrier of. 100Å. 60Å. the eigenfunctions conserve some symmetry and for the barrier of. the symmetry is almost completely lost..

(38) 46. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. Figure 4.11: Eigenfunctions for F=0 kV/cm,. B = 20Å, 60Å. and. 100Å, W2 W2. is. 5Å. larger than its. resonance value (above resonance). Also, in the above resonance condition (g.. 4.11) the wavefunctions of the states that were in. resonance are not symmetrical. The lost of the symmetry is not equal for the three barriers in the same way observed for the below resonance condition.. It can be seen that the similarity between the wavefunctions in resonance is less clear when we move away from the resonance. In consequence the dipole moments in the below and above resonance conditions are smaller.. When the barrier is thinner (B. = 20Å in our case) the wavefunctions o resonance are almost equal = 100Å) the wavefunctions are very localized resonance and in the intermediate case (B = 60Å), the wavefunctions. to the functions in resonance, if the barrier is wide (B in one well when we are out of. out of resonance tend to be localized in one well but with a little participation in the other well.. As a consequence of previous facts when the barrier is thinner the resonance region is wider but its peak value is smaller and viceversa..

(39) 4.3.. 47. WAVEFUNCTIONS. 4.3.2. Second resonance. Figure 4.12: Eigenfunctions for F=0 kV/cm,. B = 20Å, 60Å. and. 100Å. in the second resonance. In g. 4.12 the wavefunctions for the rst three eigenenergies are shown for. W2 = 100Å, B = 60Å. and. W2 = 100Å, B = 100Å.. These values for. W2. W2 = 103Å, B = 20Å;. were chosen because the. second resonance is centered at them for each barrier, it is seen that the second resonance tends to be at. 100Å. when the barrier is wider.. In this resonance the symmetries for the wavefunctions of the two resonant states is also observed. In this picture is clearer that for a wider barrier the symmetry between eigenfunctions is higher. By comparing the wave function of. E1. to those of. E2. and. E3. we see the symmetry of the functions in the. barrier. This condition has eects on the dipole moment between these states, this can be seen in sec. 4.4.

(40) 48. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. 4.4 Dipole moments. Figure 4.13: Intersubband dipole moments vs. W2 , B = 20Å, F = 0kV /cm. As in previous subsections the dipole moments were calculated for dierent barriers and dierent geometries. In g 4.13 the intersubband dipole moments are shown for the transitions. E21. (barrier=20Å). These moments are labeled. Around the rst resonance the of the. E2. and. E3. µ23. µ13 , µ23. and. µ12. E31 , E32. and. respectively.. moment is maximum, this can be explained by the symmetry. wavefunctions. Lets recall the expression for. Z. µij :. ∞. µij = e. ϕ∗i (z) zϕj (z) dz. (4.1). −∞ The symmetry of a wave function at resonance can be observed if we trace an axis of symmetry in the middle of the barrier, when we draw this axis we see that one wavefunction is odd and the other is even around this axis, but as z is an odd function the product of these three terms is even and so the value of. µij. has to be maximum as the two wavefunctions are more symmetrical in this case that. in the above or below resonance cases. On the second resonance the dipole. µ21. is maximum as the wavefunctions of the states. are very symmetrical. Also there is a weaker symmetry between the states maximum of the dipole. µ23. E2. and. E3. E2. and. E1. that explains the. around the resonance, their wavefunctions are very similar in the barrier.. Figure 4.14: Intersubband dipole moments vs. W2 , B = 60Å, F = 0kV /cm.

(41) 4.4.. 49. DIPOLE MOMENTS. In g 4.14 the intersubband dipole moments are shown for the transitions (barrier=60Å).. Now the dipole. µ23. maximums at the second resonance, as in the energy dierences in g.. 4.3.. with the case of a barrier of. B = 20Å. case, this result is also expected from the. The two pronounced peaks are at. 20Å. E31 , E32 and E21 µ12 and µ23 have. has a maximum in the rst resonance and. 30Å. and. 100Å. and the dierence. is that now the peaks are narrower and higher. The dipole moments. maxima are better dened geometrically.. In the o resonance region the dipoles. µ23. and. µ12. are smaller than the. Figure 4.15: Intersubband dipole moments vs. µ13 .. W2 , B = 100Å, F = 0kV /cm. In g 4.15 the intersubband dipole moments are shown for the transitions. B = 100Å.. The dipoles. µ23 , µ12 and µ13 have. E31 , E32. and. E21. for. maximums as in the previous cases, and again the peaks. are narrower and higher.. In the o resonance region the dipoles. µ23. and. µ12. are almost vanishing and. µ13. has its maximum. positive value. Also the values of the dipoles in this region are almost constant.. For weak coupling there are very well dened regions at which the dipole moments are maximum. But for strong coupling these regions are not so well dened. These features are very important for the design of the device.. To show with more detail the eects of the barrier the values of the barrier.. µ23. dipole is shown in gure 4.16 for several.

(42) 50. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. Figure 4.16: Intersubband dipole moment. µ23. vs. W2 , B. variable,. F = 0kV /cm. 4.5 Absorption The linear absorption for. B = 20Å. in the rst resonance without radiation is shown on gure 4.17. In. this gure three peaks are observed: the rst near 10THz correspond to the (near 15THz) is from. E21. and the third (~25THz) is from the. frequency of the second and third peaks (E21 and. E31 ). E31. E32. transition, the second. transition. We can observe that the. decrease as the narrow well (W2 ) is widened,. this is expected from the transitions.. Figure 4.17: Linear absorption for. B = 20Å. and. F = 0kV /cm. Besides that, another interesting eect appears, the maximum value of the second peak decreases and in the third peak an increase is observed in its value. Thus an equality between the intensity of the third and the second peaks at some value of. W2. is expected.. Also a fusion of the rst and second peak is expected near absorption for. W2. at. 45Å.. W2 = 45Å. (see sec. 4.2). So we plot the.

(43) 4.5.. 51. ABSORPTION. Figure 4.18: Absorption for. B = 20Å, F = 0kV /cm. and. E = 0kV /cm. above resonance. W2 = 47Å there are two peaks that have the same height. E32 and E21 and the second with E31 . At W2 = 49Å the rst peak split. This can be explained by the frequencies of the E32 and E21 transitions: they are in the same frequency for W2 = 47Å and they are slightly separated at W2 = 49Å and at W2 = 45Å (see g. 4.6). But before 47Å the peak E32 is much weaker than the E21 and after 47Å both peaks In g. 4.18 it is observed that about. The rst peak is associated with the transitions. have similar values.. Figure 4.19: Absorption for. B = 20Å, F = 0kV /cm. and. E = 1, 2, 5, 10kV /cm.

(44) 52. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. When an AC eld is applied, the non-linear terms of the absorption become visible, in g. 4.19 the absorption is plotted for. E = 1, 2, 5and 10kV /cm,. where. E. is the intensity of the AC eld. The. absorption at 7.5 THz presents a negative peak when a radiation eld is applied, and the peak is bigger when the radiation is more intense. This is the frequency of the second harmonic of the transition. The non-linear absorption for. B = 60Å. in the rst resonance without radiation is shown on gure. 4.20. In this gure a dominant absorption component appears near 18THz, this peak is from the and. E31. transitions as can be deduced from the transition graphic (see g.. maximum at. E21 .. 4.7).. E21. Here the peak is. W2 = 23Å.. Figure 4.20: Linear absorption for. B = 60Å. and. F = 0kV /cm. When an AC eld is radiated, the non-linear terms of the absorption are more visible, in g. 4.21, the absorption is plotted for. E = 1, 2, 5and 10kV /cm..

(45) 4.5.. 53. ABSORPTION. Figure 4.21: Absorption for. B = 60Å, F = 0kV /cm. and. E = 1, 2, 5, 10kV /cm. The non linear eect is less pronounced for weak coupling (thicker barriers). However a new very small negative absorption is obtained near 1.5 THz, this value corresponds to the second nonlinearity of the. E32. transition. Another negative nonlinearity appear at 9 THz, this frequency corresponds to. the second harmonic of the. E21. transition.. The non-linear absorption for. B = 100Å in the rst resonance without radiation is shown on gure E21 and the E31. 4.22. In this plot can be noted that the peak of absorption is from the sum of the peaks..

(46) 54. CHAPTER 4.. LINEAR AND NONLINEAR ABSORPTION AND GEOMETRY. Figure 4.22: Linear absorption for. B = 100Å. and. F = 0kV /cm. When an AC eld is radiated, the non-linear terms of the absorption are more visible, in g. 4.23, the absorption is plotted for. E = 1, 2, 5and 10kV /cm.. Figure 4.23: Absorption for. B = 100Å, F = 0kV /cm. and. E = 1, 2, 5, 10kV /cm. Now two negative nonlinearities appear near 0.5 and 10 THz. These are the second harmonics of the. E32. and the. E21 − E31. transitions respectively.. The non linear eects are more important in strong coupling, the absorption spectra are richer;.

(47) 4.5.. 55. ABSORPTION. we can distinguish clearly three peaks at 10, 15 and 25 THz coming from the transitions studied in g. 4.6. Moreover negative absorption is obtained at half energy of transitions 1-2 and 2-3. This is a nonlinear absorption arising from the second order susceptibility at 7.5 THz, in the region of THz gap.. χ(2). which shows a gain eect exactly.

(48) Chapter 5. Linear and nonlinear absorption and external eld Now we analyze the change of the physical characteristics of an ADQW when a DC electric eld is applied. The ADQW used in these calculations have the following conguration (see g. 4.1):. W1 B W2. 100Å {20Å, 60Å, 100Å} 38Å. Table 5.1: Parameters of the ADQW. W2 = 38Å case because for this value the eigenenergies are near F = 0kV /cm and as will be seen the structure can be carried to resonance with The intensity of the electric eld F sweeps the range F = 0kV /cm-32kV /cm in steps. We chose the geometry with resonance for a eld of the electric eld. of. 2kV /cm.. 5.1 Energies and transitions The eigenenergies of an ADQW when the applied eld is changed is shown on g. 5.1 for. B = 20Å.. The. resonance region is barely seen but it can be established by numerical calculations that the resonance is at 28kV/cm approximately. 57.

(49) 58. CHAPTER 5.. LINEAR AND NONLINEAR ABSORPTION AND EXTERNAL FIELD. Figure 5.1: Eigenenergies for. When the same graphic is obtained for. B = 20Å. B = 60Å. vs. F,. using. W2 = 38Å. (g. 5.2) the resonance region is clearer and as in. section 4.1 the coupling between the subbands in resonance is weaker. The resonance is at 18kV/cm.. Figure 5.2: Eigenenergies for. In the case of. B = 100Å. B = 60Å. vs. F,. using. W2 = 38Å. (g. 5.3) the coupling between the subbands in resonance is the weakest.. The resonance is at 14kV/cm..