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ARTICLE IN PRESS Physica B 368 (2005) 76–81 www.elsevier.com/locate/physb The effect of hydrostatic pressure on the photoionization cross-section and binding energy of impurities in quantum-well wire under the electric field E. Kasapoglua,, U. Yesilgüla, H. Saria, I. Sökmena,b a Physics Department, Cumhuriyet University, 58140 Sivas-Turkey b Physics Department, Dokuz Eylül University,İzmir-Turkey Received 26 May 2005; received in revised form 20 June 2005; accepted 29 June 2005 Abstract Using a variational approach, we have calculated the hydrostatic pressure and electric field effects on the donorimpurity related photoionization cross-section and impurity binding energy in GaAs/GaAlAs quantum well-wires. Both the results of impurity binding energy as a function of the impurity position and photoionization cross-section for a hydrogenic donor impurity placed at the center of the quantum well-wire as a function of the normalized photon energy in the quantum well-wire under the hydrostatic pressure and electric field which are applied to the z-direction for two different wire dimensions are presented. r 2005 Elsevier B.V. All rights reserved. PACS: 71.55.Eq; 71.55.i Keywords: Photoionization; Quantum-well wires; Impurity; Hydrostatic pressure 1. Introduction The problem of an electron bound to an impurity atom in the presence of the external electric and magnetic field plays a fundamental role in understanding the optical properties of impurities in semiconductors, and the impurity-related photoCorresponding author. Tel.: +903462191010; fax: +903462191186. E-mail address: [email protected] (E. Kasapoglu). ionization cross-section is mainly used in the characterization of impurities in semiconductors. The photoionization cross-section of hydrogenic impurities in bulk semiconductors is firstly investigated by Lax [1]. In recent years, work has been done on the photoionization cross-section of hydrogenic impurities in structures of reduced dimensionally such as quantum wells (QWs), wires (QWWs) and quantum dots (QDs) [2–10]. The hydrostatic pressure effects on the electronic and impurity states in low-dimensional 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.06.039 ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 368 (2005) 76–81 heterostructures such as QWs, QWWs and QDs have been studied in several theoretical works [11–17]. Elabsy [11] has calculated the effects of the hydrostatic pressure on the binding energy of donor impurities in QWs, finding that the binding energy increases with increasing hydrostatic pressure for a certain well thickness and temperature. Oyoko et al. [16] have studied the effects of an unaxial stress on the binding energy of shallow impurities in parallelpipe-shaped GaAs/GaAlAs QDs. They have found that the binding energy increases almost linearly with applied stress and diminishes with the size of the structure. Correa et al. [17] have calculated the effects of hydrostatic pressure on the binding energy and photoionization cross-section in spherical QDs for different dimensions of the structure and radial impurity position. They have found that the binding energy and photoionization cross-section are affected by the pressure. In this paper, we report a calculation of the photoionization cross-section and the binding energy of a shallow donor impurity in GaAs/ GaAlAs QWW under the electric field and hydrostatic pressure as a function of the wire dimension and impurity position, for incident light polarized along the axis of the wire. 2. Theory In low-dimensional electronic systems, the photoionization process is described as an optical transition that takes place from the impurity ground state as the initial state to the conduction subbands, which requires sufficient energy in order for the transition to occur. The excitation energy dependence of the photoionization cross-section associated with an impurity, starting from Fermi’s golden rule in the well-known dipole approximation, as in the bulk case is [1,8] " # xeff 2 nr 4p2 sð_oÞ ¼ aFS _o x0 ðPÞ 3 X 2 c j~ dðE f E i _oÞ, ð1Þ i rjcf f 77 where nr is the refractive index of the semiconductor, aFS ¼ e2 =_c the fine structure constant, and _o the photon energy. xeff/x0 is the ratio of the effective electric field xeff of the incoming photon and the average field x0 in the medium [18]. ci j~ rjcf is the matrix element between the initial and final states of the dipole moment of the impurity. In the effective mass approximation, the Hamiltonian describing the interaction of an electron with a hydrogenic impurity placed at the position ~ r ¼ ð0; 0; zi Þ in a QWW in the presence of an electric field F applied in the z-direction, may be written as H¼ p2y P2x P2z þ V ðz; PÞ þ V ðxÞ þ þ 2me 2me 2me ðPÞ þ eFz e2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðPÞ x2 þ y2 þ ðz zi Þ2 ð2Þ where, the first terms in Eq. (2) are independent from pressure since the pressure is applied only to the z direction. The quantities e(P) is the static dielectric constant as a function of pressure, me is the effective mass, e is the elementary charge, F is the electric field and V(x) and V(z, P) are the confinement potentials for the electron in the x and z-directions, respectively. The pressure dependence of the effective mass of the electron in GaAs is determined from expression [19–21] me ðPÞ ¼ 1þ E Gp h m0 ð2=E Gg ðPÞÞ þ ð1=E Gg ðPÞ þ D0 Þ i, (3) where m0 is the mass of the bare electron, E Gp ¼ 7:51 eV, D0 ¼ 0:341 eVand E Gg is the variation of the energy gap (in eV) for a GaAs QW at the G-point with the hydrostatic pressure in units of kbar, which in turn is expressed [19–22] as E Gg ðPÞ ¼ a þ bP þ cP2 , (4) where a ¼ 1:425 eV, b ¼ 1:26 102 eV=kbar and c ¼ 3:77 105 eV=kbar2 . ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 368 (2005) 76–81 78 The pressure dependence of static dielectric constant is expressed as [23] ðPÞ ¼ ð0ÞedP (5) 3 where d ¼ 1:73 10 k=bar and ð0Þ ¼ 13:18 is the value of the dielectric constant at the atmospheric hydrostatic pressure(static dielectric constant is assumed to be same GaAs and GaAlAs, charge image effects have not been considered). The barrier potential which confine the electron in the QW in the z-direction, given by [24] 8 V ðPÞ; zo LðPÞ=2; > < 0 0; LðPÞ=2ozoLðPÞ=2; V ðz; PÞ ¼ (6) > : V ðPÞ; z4LðPÞ=2; 0 where V 0 ðPÞ ¼ Qc DE Gg ðx; PÞ (7) is the barrier height and Qc( ¼ 0.6) is the conduction band offset parameter [24], DE Gg ðx; PÞ is the band gap difference between QW and the barrier matrix at the G-point as a function of P, which for an aluminum fraction x ( ¼ 0,3) is given by DE Gg ðx; PÞ ¼ DE Gg ðxÞ þ PDðxÞ (8) where DE Gg ðxÞ 2 ¼ ð1:155x þ 0:37x Þ eV (9) is the variation of the energy gap difference and D(x) is the pressure coefficient of the band gap [25], given by DðxÞ ¼ ½ð1:3 103 Þx eV=kbar (10) In Eq. (6) LðPÞ ¼ Lð0Þ½1 ðS 11 þ 2S12 ÞP , (11) where S11 ¼ 1:16 103 k=bar and S12 ¼ 3:7 104 k=bar are the elastic constants of the GaAs [19–21] and L(0) is the original width of the confinement potentials in the z-direction. The investigation of the photoionization cross-section first needs to know the envelope functions of the initial ground state and the final state. In order to get the binding energy, we follow a variational method and we take the following trial wave function that satisfies the Hamiltonian in Eq. (2) ci ðrÞ ¼ wðxÞwðzÞfðy; lÞ, (12) where the wave function in the y-direction f(y,l) is chosen to be Gaussian-type orbital function [26–28] 1 2 1=4 y2 =l2 fðy; lÞ ¼ pffiffiffi e (13) l p in which l is variational parameter. With the choice of f(y, l), the degrees of freedom are limited to one dimension along the axis of the wire. Our experience with variational calculations of the hydrogenic binding energy in QWs and quantum wires suggests that very simple Gaussian-type function gives quite accurate results in the case of the moderate and strong fields [28 and references there in]. w(x) and w(z) are the first subbands wave functions of the electron, which are exactly obtained from the one-dimensional Schrödinger equation in the x and z-directions, respectively. The ground state impurity binding energy is given by E B ¼ E x þ E z min ci jH jci (14) l where Ex and Ez are the lowest donor electron subband energies related to the w(x) and w(z) wave functions, respectively. Because the electron motion along the y-axis is free, without the impurity potential the eigenstates associated with the Hamiltonian (2) of an electron emitted to the subbands nx, nz relative to the x and z-directions of the quantum-well wire, are given by 1 cf ðrÞ ¼ pffiffiffiffiffiffi wnx ðxÞwnz ðzÞeiky y , Ly (15) where Ly is the length of the wire and ky is the onedimensional wave vector of the electron along the axis of the wire. wnx ðxÞ and wnz ðzÞ are chosen to be solutions of one-dimensional Hamiltonian for electron in the x and z-directions, respectively. The final state energy corresponding to the wave function Eq. (15) is Ef ¼ _2 2 k þ E nx þ E nz . 2me y (16) ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 368 (2005) 76–81 where x ¼ _o=E S and E S ¼ E nx þ E nz E i ¼ E B , ðxeff =x0 Þ ¼ 1. In producing the sð_oÞ expression, we accounted for all ky values in the final state by converting the summation to a one-dimensional integral along the axis of the QWW of length Ly. 40 P = 30 kbar P = 10 kba r P=0 20 10 -1.0 -0.5 0.0 zi / L (a) 0.5 1.0 24 P = 10 kbar 3. Results and discussion 20 Eb ( meV) The variation of the binding energy as a function of the normalized impurity position for different hydrostatic pressure values is given in ( and in Fig. 1(b) Fig. 1(a) for L ¼ Lx ¼ Lz ¼ 50 A ( for L ¼ Lx ¼ Lz ¼ 200 A. The maximum binding energy is obtained for impurity located at the center of QWW and P ¼ 30 kbar pressure value. As the pressure increases, the well width and dielectric constant decrease, the effective mass of electron increases, leading to more confinement in the well in the z-direction of the impurity electron and so the impurity binding energy increases for all impurity positions. The binding energy for the impurity positions closed to the barriers is lower than for on-center impurity since, Coulomb interaction between the electron and impurity decreases. When the electric field is applied, since the electron shifts to the left side of the well in the z-direction with the effect of electric field the probability of finding the electron in the left side of the well increases, while it decreases in the right side of the well so, we observe that for impurity position close to the left side of the well the binding energy is higher than in right side. As seen in Fig. 1(b), for quantum-well wires with large L x = L z = L = 50 F= 0 ----F = 50 kV/cm 30 Eb ( meV) For the incident light polarized along the axis of the wire, y-direction, the transition to the first subbands nx ¼ 1, nz ¼ 1 is dipole allowed. Thus in that case, wnx ðxÞ and wnz ðzÞare taken to be the same as in the initial ground state, i.e., the first subbands wave functions relative to the x and z-directions of the wire, respectively. The photoionization crosssection for the incident light polarized along the axis of the wire is given by " # xeff 2 nr aFS l5 pm E S 3=2 sð_oÞ ¼ x0 ðPÞ 3 _2 pffiffiffiffiffiffiffiffiffiffiffi x x 1 exp½m ðl=_Þ2 E S ðx 1Þ , ð17Þ 79 P = 30 kbar L x = L z = L = 200 F=0 ------ F = 50 kV/cm 16 12 P=0 8 4 -0.5 (b) -0.3 0.0 zi / L 0.3 0.5 Fig. 1. The variation of the binding energy as a function of the normalized impurity position for different hydrostatic pressure ( and (b) values (a) wire dimension L ¼ Lx ¼ Lz ¼ 50 A ( F ¼ 0 (solid line) and F ¼ 50 kV=cm L ¼ Lx ¼ Lz ¼ 200 A. (dashed line). dimensions the impurity binding energy decreases, since the probability of finding the electron and impurity in the same plane decreases, while the localization of the electron in the well-wire increases. The field dependence of the binding ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 368 (2005) 76–81 2.5 L x = L z = 50 F=0 ------- F = 50 kV/cm zi= 0 P=0 2.0 σx10-15 (cm2) energy in narrow well-wire dimensions is very weak, since the geometric confinement is predominant. But in the wider QWWs, the binding energy is more sensitive to the external electric field. Since, when the wire dimension increases, the energy of electron becomes weaker and the electron that approaches the well bottom begins to become sensitive to the electric field strength. Finally, in Fig. 2, we present the photoionization cross-section as a function of the normalized photon energy of donor impurity placed at the center of QWW for several hydrostatic pressure ( and (b) values for (a) L ¼ Lx ¼ Lz ¼ 50 A ( L ¼ Lx ¼ Lz ¼ 200 A, respectively. Solid line shows F ¼0 and dashed line shows F ¼ 50 kV=cm. As the pressure increases, the magnitude of the cross-section becomes much smaller, the value of _o associated with the peak of the cross-section moves the higher photon energies and the photoionization cross-section decreases for higher photon energies. Hydrostatic pressure leads to increment of the binding energy and hence to the optical photoionization threshold energy. As a consequence the excitation of an electron bound to a donor impurity to the first conduction subband by absorption of a photon requires higher photon energies in order for the transition to occur. The electric field on the photoionization cross-section is not effective in norrow wire dimensions. Photoionization crosssection increases since the binding energy decreases with the electric field effect for Pa0 pressure values but this increment is not very important. As known, the binding energy is very sensitive to the electric field in large wire dimensions. In large wire dimensions, the binding energy becomes much smaller and thus photoionization cross-section increases evidently for P ¼ 0 pressure value in Fig. 2(b). For Pa0 pressure values, photoionization cross-section shows the same behavior with in Fig. 2(a). Photoionization crosssection is not very affected from pressure since the geometric confinement is predominant. In summary, we have calculated the hydrostatic pressure and electric field effects on the donorimpurity related photoionization cross-section and impurity binding energy in GaAs/GaAlAs QWWs. The electric field on the photoionization cross- 1.5 P = 10 kba r 1.0 0.5 P = 30 kba r 0.0 1.0 1.5 (a) 2.0 hω / Eb 14 2.5 3.0 L x = L z = 200 F=0 ------- F = 50 kV/cm zi= 0 12 10 σx10-15 (cm2) 80 8 P=0 6 4 P = 10 kbar 2 P = 30 kbar 0 1.0 (b) 1.5 2.0 hω / Eb 2.5 3.0 Fig. 2. The photoionization cross-section as a function of the normalized photon energy of donor impurity placed at the center of QWW for several hydrostatic pressure values for (a) ( and (b) L ¼ L ¼ L ¼ 200 A. ( Solid line L ¼ Lx ¼ Lz ¼ 50 A x z corresponds to F ¼ 0, dashed line corresponds to F ¼ 50 kV=cm. section is not effective in norrow wire dimensions. In large wire dimensions, the binding energy becomes much smaller and thus photoionization cross-section increases evidently for P ¼ 0 pressure value. Photoionization cross-section is not very ARTICLE IN PRESS E. Kasapoglu et al. / Physica B 368 (2005) 76–81 affected from pressure where the geometric confinement is predominant. Both for the results of impurity binding energy as a function of the impurity position and photoionization cross-section for a hydrogenic donor impurity placed at the center of the QWW as a function of the normalized photon energy in the QWW under the hydrostatic pressure and electric field for two different wire dimensions are presented. We have found that the photoionization cross-section decreases with the applied hydrostatic pressure. Additionally, photoionization cross-section increases or decreases depending on size of the wire and the impurity positions. The measurement of photoionization in low-dimensional systems would be of great interest for understanding the optical properties of carriers in QWWs for which the photoionization can give information about the impurity position and the distribution inside the heterostructure. To the best of our knowledge, there are no experimental reports relating to the photoionization cross-section in QW, QWWs, and QDs. References [1] M. 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