Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 759 764 c International Academic Publishers Vol. 43, No. 4, April 15, 2005 Transient Intersubband Optical Absorption in Double Quantum Well Structure WU Bin-He State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, the Chinese Academy of Sciences, Shanghai 200050, China (Received July 28, 2004) Abstract The microscopic equations of motion including many-body effects are derived to study the intersubband polarization in the double quantum well structure induced by an ultrafast pumping infrared light. Based on the selfconsistent field theory, the transient probe absorption coefficient is calculated. These calculations are beyond the previous steady-state assumption. Transient probe absorption spectra are calculated under different pumping intensity and various pump probe delay. PACS numbers: 78.67.De, 42.65.-k Key words: optical absorption, double quantum well, intersubband 1 Introduction Coherent effects in semiconductor structures on a nanometer scale have many novel electronic and optoelectronic properties, such as the Bloch oscillation, [1 4] electromagnetic induced transparency, [5 7] infrared light emission due to coherent charge oscillations, [8] coherent population trapping and coherent control and enhancement of the refractive index, [9] and other nonlinear transport phenomena in the existence of far-infrared radiations. [10,11] The unique properties of these low dimensional semiconductor materials are, generally speaking, originated from the quantum confinement of the motion of electrons. In the case of quantum well structures, the conduction band will be quantized to several subbands along the growth direction due to the quantum confinement. Intersubband transitions in the quantum confined structures have attracted a great deal of attention in recent years, both from the theoretical and the experimental point of view. Novel optoelectronic devices like the quantum cascade lasers [12] and the quantum well infrared photodetectors [13] for the infrared range are based on the optical intersubband transition in the same band. Far infrared or terahertz emissions via electrically [14 17] or optically [18,19] pumping semiconductor intersubband lasers have been achieved. Coherent transitions in the semiconductor suffer from the short scattering times, usually less than one picosecond. Ultrafast laser pulse should be employed in exploring the coherent dynamics in the semiconductor material. Optical absorption has proven to be one of the simplest and most applicable methods to characterize the materials. Description of the coherent pump probe optical absorption between subbands in the semiconductor quantum structures can be based on the Green s function or the density matrix method. [20 22] However, taking into account the many-body effects in the semiconductor quantum well is nontrivial. The effect of many-body interactions on the collective response of confined electrons in the quantum well structure to intense far infrared radiation was theoretically investigated recently. [23] The ultrafast pump probe coherent transition induced optical absorption between two energy levels have been described successfully by the semiconductor Bloch equations. [22] When multisubband was included in the pump probe scheme, electron dynamics in the quantum well subbands were investigated under the steady state model for simplicity. [5,7,9] Quantum interference and nonlinear optical properties in the quantum well driven by infrared fields have been extensively explored under the steady state assumption. Experimental results exploring the transient intersubband absorption including two conduction subbands by femtosecond pump probe experiments have been reported, [24] where the nonlinear optical response reflects both the intersubband scattering and intrasubband redistribution. Due to the ultrafast spectroscopy, the steady state assumption used before is not valid in this case. The transient electron dynamics including multi-subbands has been investigated using the self-consistent-field theory [25] to analyze the pump probe absorption properties, where the time-averaged optical spectrum was calculated to show the coherent transition induced quantum interference in the system. In this paper, we will derive the equations of motion for the unipolar carrier in three subbands within the conduction band. First, the electron Hamiltonian coupling to the radiation field is introduced. Then, using the Heisenberg equation of motion, we derive a set of equations that describes the dynamics of subband population and intersubband polarization under the random phase approximation. These equations will be numerically solved and The project supported by the National Fund for Distinguished Young Scholars of China (60425415), the Special Funds for Major State Basic Research of China (G20000683), and the Shanghai Municipal Commission of Science and Technology (03JC14082)
760 WU Bin-He Vol. 43 the transient absorption spectra will be obtained from the Fourier transform. 2 General Theory The N-type doped double quantum well structure considered here consists of two GaAs quantum wells separated by a 1.13 nm width of Al 0.35 Ga 0.65 As barrier. The coupled quantum wells are 7.0 nm and 3.5 nm wide respectively. Each coupled double quantum well is separated by a 30 nm wide Al 0.35 Ga 0.65 As barrier. The central of the wide barrier is n-type doped with the doping density to be 1 10 11 cm 2. The asymmetric design is intended to break the parity, so that the transition from the ground subband to the second excited subband is permitted. The wave function for confined electrons in the conduction band can be expressed as the product of the rapidly varying Bloch functions of the band extrema, u c (r), and the slowly varying envelopes, φ µ (r) ψ µ = u c (r)φ µ (r) = u c (r)ϕ µ (z) e ik r A, (1) where µ is the subband index, k is the in-plane wave vector, z is the material growth direction, and A is the quantization area of the double quantum well. In the absence of the external fields, the eigen energies and the wave functions can be obtained from self-consistently solving Schrödinger s equation and Poisson s equation under effective mass approximation. [26,27] Fig. 1 Calculated effective potential for the double quantum well structure. The wave functions of the three subband are schematically plotted. The barrier and well materials are Al 0.35Ga 0.65As and GaAs, respectively. The solid thick arrow represents the pumping pulse coupled to two subbands, E 1 and E 3. The dashed thick arrow stands for the probe light. The calculated wave functions and the structure scheme are displayed in Fig. 1. The lowest three conduction subbands are taken into consideration in our model. A strong ultrafast infrared light will couple the ground subband and the third subband. A weak probe light with the frequency around the energy difference of the upper two levels will be used to detect the absorption spectrum. Thus forms a so-called Λ configuration. [28] 2.1 Hamiltonian The Hamiltonian of an electron in the static semiconductor is the sum of the kinetic and Coulomb interaction energies. When a light or electric potential is imposed, the interaction of the electron and the external field will contribute to the total Hamiltonian. Thus the electron Hamiltonian from which the equations of motion will be derived is the sum of the above terms: H = H 0 + H Coul + H I, (2) where H 0 represents the single particle Hamiltonia, H I is the light-semiconductor interaction, and the Coulomb interaction term is denoted as H Coul. Here, we have neglected scattering mechanics such as the electron-phonon interactions, impurity scatterings and exciton effects, for simplicity. Under the second quantization formulation, these Hamiltonian can be expressed in the form H 0 = µ,k E µ,k a µ,k a µ,k, (3) where E µ,k = E µ,0 + h 2 k 2 /2m e is the kinetic energy of an electron in the µ-th subband with the in-plane wave vector k, E µ,0 is the eigen energy of the µ-th subband, and m e is the electron effective mass in the conduction band. The effective masses of subbands are taken to be constant in our model. This is sound for GaAs/AlGaAs quantum wells. In other cases such as the InAs/AlSb quantum wells, nonparabolic effect is not negligible. a µ,k and a µ,k are the electron creation and annihilation operators, respectively. a µ,k creates an electron in the µ-th subband with the inplane wave vector k, while a µ,k is the Hermitian conjugate of a µ,k, which denotes the annihilation of an electron. Under the dipole approximation, the light-semiconductor interaction is given by H I = (µ µν,k E(t)a µ,k a ν,k {µ,ν} k + µ µν,ke (t)a ν,k a µ,k), (4) where µ µν,k is the optical dipole matrix elements, and the external electric field of the imposed infrared laser light has the form E(t) = 1 2 E(t)( e iωt + e iωt ). (5) The light frequency is ω. E(t) is the time-dependent electric amplitude of the external field. This envelope is assumed to be Gaussian with a half width of the pulse t centered at t = 0: E(t) = E 0 e ( t2 /2 2 t ). (6)
No. 4 Transient Intersubband Optical Absorption in Double Quantum Well Structure 761 The Coulomb interaction has the form [22,29] H Coul = 1 V µνν µ q a µ,k+q 2 a ν,k q a ν,k a µ,k, (7) µνν µ kk q where the Coulomb matrix element = V q dzdz ϕ µ (z)ϕ µ (z) V µνν µ q e q z z ϕ ν (z )ϕ ν (z ). (8) V q is the Fourier transform of the Coulomb potential in two dimensions e 2 V q = 2Aɛ 0 ɛ b q, (9) where ɛ b is the background dielectric constant. The manybody interactions in an electron plasma will lead to plasma screening. Thus the Coulomb potential should be replaced by the dynamically screened Coulomb potential. [22] In the static Lindhard formula, [22] the screened Coulomb potential is Vq s = V q ɛ(q), (10) where the dynamic dielectric function for the static limit has the form ɛ(q) = 1 + κ q, (11) κ = m ee 2 2πɛ 0 ɛ b h 2 f k=0, (12) where κ is the inverse screening length in quasi-two dimensions and f k=0 is the Fermi Dirac distribution of the electrons. In the above discussion, µ, µ, ν, and ν in the footnotes are the subband indices. The subband indices with the primed quantities denote the initial subbands and those without the prime quantities denote the final subbands. According to the initial and final positions of the interaction electrons, the nonzero Coulomb interaction terms for two subbands can be categorized to several cases: [30] (i) µ = µ = ν = ν. The two electrons are in the same subband. This intraband interaction devotes to the subband gap renormalization. (ii) µ = µ ν = ν. The electrons are in the different subbands. After interaction, they fall to the same subband. (iii) µ = ν µ = ν. This is the depolarization term. The depolarization contribution grows with the electron density and the width of the well. [30] In our case the depolarization term becomes negligible. The direct and exchange terms are dominant in the Coulomb interactions. 2.2 Equations of Motion The subband electron distribution and intersubband polarization are defined as the expectation of the two operator products given by for the electron distribution and n µ,k = a µ,k a µ,k (13) P µν,k = a ν,k a µ,k (14) for the intersubband polarization. In our configuration, the pumping photon energy hω c is around the energy separation of the ground subband and the third subband hω 31. The probe frequency is near resonant with the upper two levels, that is hω p = hω 32. The polarization between the subbands can be expressed as a fast varying part resonant with the external field times a slowly varying envelope: P 31,k P 31,k e iωct, P 32,k P 32,k e iωpt. (15) When the pumping pulse imposed on the double quantum well structure is taken to be much stronger than the probe light, the electrons in the system are mainly driven by the pumping laser light coherently. The weak probe field effects can be neglected. Under the rotating wave approximation, using the Heisenberg equation of motion and the Hamiltonian given before, the motion equations for the expectation values are written as i h t n 1,k = 2i Im (µ 31 E c (t)p 31,k) i h t n 3,k = 2i Im (µ 31 E c (t)p31,k) + i h ( t P 31,k = h(ω c ω 31 )P 31,k µ 31 E c (t)(n 1,k n 3,k ) + V 3113 k k (P 31,kP 31,k P 31,k P 31,k ) i hn 1,k n T 1,k T 1, (16) V 3113 k k (P 31,kP 31,k P 31,k P 31,k ) i hn 3,k n T 3,k T 1, (17) V k k 1111 n 1,k V k k 3333 n 3,k )P 31,k V k k 3113 n 1,kP 31,k + V k k 3333 n 3,kP 31,k i h P 31,k, (18) T 2 where T 1 and T 2 are the phenomenological dephasing time constants. This damping effect is due to the incoherent scattering in the system, such as collisions between electrons and electron-phonon scattering. [22,30] These scatterings tend to relax the nonequilibrium electrons and polarization to the thermal equilibrium states. Here n T 3,k denotes
762 WU Bin-He Vol. 43 the thermal equilibrium electron distribution. In deriving the equations above, random phase approximation has been employed to factorize the fourth-order expectations to the products of second-order expectations. Equations (16) (18) have been solved by the fourth-order Runge Kutta method with equilibrium state initial conditions. 2.3 Absorption Coefficient When the probe field is included, P 32,k will be coupled to the probe light and it is no longer zero. The presence of the probe field is considered as an additional perturbation to the system. By using the self-consistent-field theory and neglecting the secondary Coulomb interaction terms, [25] P 32,k satisfies the following equation: i h ( t P 32,k = h(ω p ω 32 )P 32,k µ 32 E p (t)(n 2,k n 3,k ) + V k k 2222 n 2,k V k k 3333 n 3,k )P 32,k V k k 3223 n 2,kP 32,k + V k k 3333 n 3,kP 32,k i h P 32,k. (19) T 2 The time-dependent polarization P 32,k can be solved by the Runge Kutta method with Eqs. (16) (18). The polarization function is the sum of all momenta: P 32 (t) = µ 32 P 32,k (t). (20) Method to get the time averaged absorption coefficient is presented in Ref. [25], where the absorption coefficient is the average value of the probe pulse duration. In our calculations, the absorption coefficient can be obtained from the simple relation given by α(ω) = k 2ω ( P (ω) ɛ 0 ɛ b Lc Im E(ω) ), (21) where L is the length of the quantum well, ω is the frequency of the light field and P (ω) and E(ω) are the Fourier transform of the polarization function P (t) and the electric field E(t). When the absorption coefficient is negative, light gain is achieved. 3 Numerical Results The structural parameters of the double quantum well structure are presented in the preceding section. In Fig. 1, the wavefunction of each subband and the electron potential are plotted by self-consistently solving the Schrödinger s equation and the Poisson s equation. [26,27] The energy separation between the ground subband and the third subband is 237.48 mev. The energy separation between the upper two subbands is 131.78 mev. The solid thick double headed arrow indicates the coupling of the pumping pulse and the quantum double well structure. The dashed thick double headed arrow stands for the weak probe light. We numerically solve Eqs. (16) (18) for the strong pumping field. The half width of the pumping light is set to be 100 fs. In our calculation, the dephasing times T 1 and T 2 are set to be 1 mev for simplicity. Absorption coefficient is calculated from Eqs. (19) and (21). In Fig. 2, the probe pulse profile and the polarization between the upper two subbands are shown. The real part, Re (P 32 ) and the imaginary part Im (P 32 ) are coupled to each other and numerically solved simultaneously. The decay of the polarization reflects the presence of the phenomenological damping coefficient T 2, which is the result of incoherent scatterings. Fig. 2 The polarization between the upper two subbands as a function of time and the probe pulse. The probe pulse has a delay of 1 ps compared with the pump light. Fig. 3 Absorption properties with various pumping intensity. The negative absorption represents the optical gain of the probe field.
No. 4 Transient Intersubband Optical Absorption in Double Quantum Well Structure 763 The absorption coefficient is strongly related to the pumping intensity. In Fig. 3, absorption spectra at different pumping intensities are presented. The coupling frequency is set to be 5 mev higher than ω 31. The pumping pulse imposed on the double quantum well structure will pump the electrons in the ground level to the higher lasing level. Population inversion can be expected between the upper two subbands. In our result, the absorption coefficient is negative, thus optical gain centered at the frequency ω 32 is obtained. With the growing of the pumping electric field, the optical gain increases. These results are obtained under the condition that the probe light has a time delay of 1 ps. In ultrafast pump-probe experiments, a pump pulse excites carriers in the sample. These excited carriers modify the absorption and the refractive index, which is probed by a subsequent probe pulse. The measurement of the differential probe transmission versus time delay between pump and probe pulse monitors the dynamics of the carrier population. Our model is based on the microscopic dynamics equations without the continuous coupling laser and probe field were assumed. The transient absorption coefficients as a function of the delay time can be calculated. In Fig. 4, we show the calculated absorption spectra at different delay times. The absorption at E 32 is plotted as a function of the delay time in the same figure. The pumping and various scattering mechanism induced population redistribution among the three subbands play an important role in the absorption coefficient spectra. The exponential decay of absorption coefficient with the delay time reflects the dynamics of electrons in the subbands. A detailed analysis of the time evolution of the nonequilibrium electrons can reveal the complex information about the carrier relaxation mechanics. The ultrafast optical pulse can act as coherent carrier control. The results presented above have the advantage of exploring the coherent dynamics of the carriers in the double quantum well structures under external field. Fig. 4 Relation between the absorption spectra and delay times. The lines are the absorption spectra as a function of detuning energy at different delay times (bottom x and left y axis). The line with symbols represents the absorption coefficient as a function of the delay time (top x and right y axis). The pumping electric field is set to be 15 kv/cm in these calculations. 4 Conclusion We have numerically investigated the absorption spectra induced by an ultrafast infrared pulse on the double quantum well structure. The microscopic equations of motion are established from the Heisenberg equation of motion including many-body effects. These equations are numerically solved. The intersubband polarization is calculated from the self-consistent field theory. The transient intersubband absorption spectra are obtained. References [1] J. Feldmann, et al., Phys. Rev. B46 (1992) 7252. [2] M.M. Dignam, Phys. Rev. B59 (1999) 5770. [3] J.C. Cao and X.L. Lei, Phys. Rev. B59 (1999) 2199. [4] J.C. Cao and X.L. Lei, Phys. Rev. B60 (1999) 1871. [5] S.M. Sadeghi, S.R. Leffler, and J. Meyer, Phys. Rev. B59 (1999) 15388. [6] G.B. Serapiglia, et al., Phys. Rev. Lett. 84 (2000) 1019. [7] L. Silvestri, et al., Euro. Phys. J. B27 (2002) 89.
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