Direct and indirect exciton mixing in a slightly asymmetric double quantum well
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2 /2
|l|+1
1/2
r eiቤተ መጻሕፍቲ ባይዱφ ( )|l| i
(6)
Ln−l/2−|l|/2 (gB r 2 ),
|l |
where gB = qB , and only l = 0 functions are considered. The electron wave functions 2¯ h defined in the five regions of z , shown in Fig. 1, are given in terms of sin(ze ) , cos(ze ), and ±exp(ze ). The hole wave functions are given by similar expressions. The Coulomb interaction produces off-diagonal terms by mixing our basis states. In order to obtain the system of equations for the coefficients in expansion (5), we need to evaluate the Coulomb integrals
where p, µ and mh,x−y are the momentum operator, reduced mass, and hole mass respectively, defined on the x − y plane. Vcoul (r, |ze − zh |) is the Coulomb potential between electrons and holes, including an effective dielectric constant for the system. We expanded the solutions of the Hamiltonian (1), as a linear combination of products of eigenfunctions of the magnetic Hamiltonian in the x − y plane (4), and eigenfunctions of the electron and hole Hamiltonians in the z direction [(2) and (3)], Ψexc n =
PACS numbers: 73.20.Dx, 73.23.-b, 78.20.Ls,78.66.-w
Submitted to: J. Phys.: Condens. Matter
1. Introduction In this work, we studied the excitonic energy levels in an asymmetric vertical double quantum well [1]-[9] as a function of the magnetic and electric field strengths. Within the effective-mass approach we expanded the excitonic wave function in an orthogonal basis formed by products of electron and hole wave functions along the crystal growth direction z , and one-particle solutions of the magnetic Hamiltonian in the x-y plane. The Coulomb potential between electrons and holes produces off-diagonal terms thus mixing our basis states. We obtained the energy spectra and wave functions by diagonalizing the excitonic Hamiltonian in a truncated basis. We applied our method to study the excitonic states in a GaAs Al.33 Ga.67As DQW, for the specific case of a DQW composed of a left well of 10.18 nm, a barrier of 3.82 nm, and a right well of 9.61 nm. The effects of external electric and magnetic fields on the luminescence (PL) intensity for this heterostructure has been recently studied experimentally by Krivolapchuk et al.[10], [11].
arXiv:cond-mat/0007346v1 [cond-mat.mes-hall] 21 Jul 2000
Direct and indirect exciton mixing in a slightly asymmetric double quantum well
Francisco Vera
Universidad Cat´ olica de Valpara´ ıso, Av. Brasil 2950, Valpara´ ıso, Chile Abstract. We studied, theoretically, the optical absorption spectra for a slightly asymmetric double quantum well (DQW), in the presence of electric and magnetic fields. Recent experimental results for a 10.18/3.82/9.61 nm GaAs Al.33 Ga.67 As DQW show clearly the different behavior in the luminescence peaks for the indirect exciton IX and left direct exciton DX as a function of the external electric field. We show that the presence of a peak near the DX peak, attributed to an impurity bound left DX in the experimental results, could be a consequence of the non-trivial mixing between excitonic states.
and Ve (ze ) (Vh (zh )) is the potential that defines the double quantum well for electrons (holes) in the five regions of z . We included the electric field in Ve (ze ) and Vh (zh ) by a shift in the potential in stair steps similar to Fig. 1. Hmag (r ) is the magnetic Hamiltonian in the symmetric gauge, which depends on the relative coordinates of electrons and holes in the x − y plane, Hmag = qB (p − q A)2 + lz , 2µ mh,x−y (4)
Direct and indirect exciton mixing in a slightly asymmetric DQW 2. Formalism
2
The effective-mass Hamiltonian for excitons in a double quantum well in the diagonal approximation [9] and in the presence of a magnetic field pointing towards z , can be written as H = H0 (ze ) + H0 (zh ) + Hmag (r ) + Vcoul (r, |ze − zh |), where H0 (ze ) is the one-dimensional Hamiltonian for electrons, H0 (ze ) = p2 ze /2mze + Ve (ze ), H0 (zh ) is the one-dimensional Hamiltonian for holes, H0 (zh ) = p2 zh /2mzh + Vh (zh ), (3) (2) (1)
νr ,νe ,νh n Cν ψ (r, φ)ψνe (ze )ψνh (zh ), r ,νe ,νh νr
(5)
in which, in the symmetric gauge, ψνr ,l 1 2(n − l/2 − |l|/2)gB = 2π (|l|/2 + n − l/2)! × e−gB r
∗ dφdrdze dzh ψν ′ ψν ′ ψν ′ Vcoul (r, |ze − zh |)ψνr ψνe ψνh . e r h
(7)
Direct and indirect exciton mixing in a slightly asymmetric DQW
|l|+1
1/2
r eiቤተ መጻሕፍቲ ባይዱφ ( )|l| i
(6)
Ln−l/2−|l|/2 (gB r 2 ),
|l |
where gB = qB , and only l = 0 functions are considered. The electron wave functions 2¯ h defined in the five regions of z , shown in Fig. 1, are given in terms of sin(ze ) , cos(ze ), and ±exp(ze ). The hole wave functions are given by similar expressions. The Coulomb interaction produces off-diagonal terms by mixing our basis states. In order to obtain the system of equations for the coefficients in expansion (5), we need to evaluate the Coulomb integrals
where p, µ and mh,x−y are the momentum operator, reduced mass, and hole mass respectively, defined on the x − y plane. Vcoul (r, |ze − zh |) is the Coulomb potential between electrons and holes, including an effective dielectric constant for the system. We expanded the solutions of the Hamiltonian (1), as a linear combination of products of eigenfunctions of the magnetic Hamiltonian in the x − y plane (4), and eigenfunctions of the electron and hole Hamiltonians in the z direction [(2) and (3)], Ψexc n =
PACS numbers: 73.20.Dx, 73.23.-b, 78.20.Ls,78.66.-w
Submitted to: J. Phys.: Condens. Matter
1. Introduction In this work, we studied the excitonic energy levels in an asymmetric vertical double quantum well [1]-[9] as a function of the magnetic and electric field strengths. Within the effective-mass approach we expanded the excitonic wave function in an orthogonal basis formed by products of electron and hole wave functions along the crystal growth direction z , and one-particle solutions of the magnetic Hamiltonian in the x-y plane. The Coulomb potential between electrons and holes produces off-diagonal terms thus mixing our basis states. We obtained the energy spectra and wave functions by diagonalizing the excitonic Hamiltonian in a truncated basis. We applied our method to study the excitonic states in a GaAs Al.33 Ga.67As DQW, for the specific case of a DQW composed of a left well of 10.18 nm, a barrier of 3.82 nm, and a right well of 9.61 nm. The effects of external electric and magnetic fields on the luminescence (PL) intensity for this heterostructure has been recently studied experimentally by Krivolapchuk et al.[10], [11].
arXiv:cond-mat/0007346v1 [cond-mat.mes-hall] 21 Jul 2000
Direct and indirect exciton mixing in a slightly asymmetric double quantum well
Francisco Vera
Universidad Cat´ olica de Valpara´ ıso, Av. Brasil 2950, Valpara´ ıso, Chile Abstract. We studied, theoretically, the optical absorption spectra for a slightly asymmetric double quantum well (DQW), in the presence of electric and magnetic fields. Recent experimental results for a 10.18/3.82/9.61 nm GaAs Al.33 Ga.67 As DQW show clearly the different behavior in the luminescence peaks for the indirect exciton IX and left direct exciton DX as a function of the external electric field. We show that the presence of a peak near the DX peak, attributed to an impurity bound left DX in the experimental results, could be a consequence of the non-trivial mixing between excitonic states.
and Ve (ze ) (Vh (zh )) is the potential that defines the double quantum well for electrons (holes) in the five regions of z . We included the electric field in Ve (ze ) and Vh (zh ) by a shift in the potential in stair steps similar to Fig. 1. Hmag (r ) is the magnetic Hamiltonian in the symmetric gauge, which depends on the relative coordinates of electrons and holes in the x − y plane, Hmag = qB (p − q A)2 + lz , 2µ mh,x−y (4)
Direct and indirect exciton mixing in a slightly asymmetric DQW 2. Formalism
2
The effective-mass Hamiltonian for excitons in a double quantum well in the diagonal approximation [9] and in the presence of a magnetic field pointing towards z , can be written as H = H0 (ze ) + H0 (zh ) + Hmag (r ) + Vcoul (r, |ze − zh |), where H0 (ze ) is the one-dimensional Hamiltonian for electrons, H0 (ze ) = p2 ze /2mze + Ve (ze ), H0 (zh ) is the one-dimensional Hamiltonian for holes, H0 (zh ) = p2 zh /2mzh + Vh (zh ), (3) (2) (1)
νr ,νe ,νh n Cν ψ (r, φ)ψνe (ze )ψνh (zh ), r ,νe ,νh νr
(5)
in which, in the symmetric gauge, ψνr ,l 1 2(n − l/2 − |l|/2)gB = 2π (|l|/2 + n − l/2)! × e−gB r
∗ dφdrdze dzh ψν ′ ψν ′ ψν ′ Vcoul (r, |ze − zh |)ψνr ψνe ψνh . e r h
(7)
Direct and indirect exciton mixing in a slightly asymmetric DQW
