The Schr¨odinger equation (Eq. 2.1) is rewritten in a more convenient form using the effective mass and the envelope function approaches. The calculation of confinement energies and wavefunctions in heterosystems is further supported by two following con-siderations:
Chapter 2. Quantum confinement in two-dimensional systems 19
• The effective mass is written as a function of growth direction (m∗=m∗(z)) [80].
• Boundary conditions across a heterojunction are used [81, 82]. The most popu-lar boundary conditions on envelope function solutions are the Ben Daniel-Duke conditions, which state that
both ξ(z) and 1 m∗(z)
∂
∂zξ(z) are continuous (2.15) across a heterojunction [83]. This is incorporated into our analysis of the Schr¨odinger equation through the kinetic energy operator, which is rewritten for this purpose as
With these considerations the stationary Schr¨odinger equation for conduction band electron states in a heterosystem reads
where EC is the conduction band potential including strain-induced energy shifts and externally applied potentials, and Ei (ξi) is the eigenenergy (envelope function) of i-th order.
Solutions to the Schr¨odinger equation are found numerically using the shooting method, which discretizes Eq. 2.17 and approximates iteratively the correspondent so-lutions (envelope functions and eigenenergies). For this purpose, we parametrize the z-coordinate through a small1step lengthδz, resulting in
1The step length is usuallyδz=1 ˚A. For more details on the discretization of the Schr¨odinger equation see for example the work of P. Harrison [68].
19
Chapter 2. Quantum confinement in two-dimensional systems 20
ξi(z+δz)
m∗(Ei,z+δz) =ξi(z) 2δ2z
¯
h2 (EC(z)−Ei) + 1
m∗(Ei,z+δz/2)+ 1 m∗(Ei,z−δz/2)
− ξi(z−δz) m∗(Ei,z−δz/2) .
(2.18) For each quantum numberi, the shooting equation 2.18 allows the reconstruction of the entire envelope function ξi(z) in small steps of length δz. That means, by known ξi atz−δz andz, we calculate the value of ξi atz+δz. As next, this value is used to calculateξiatz+2δz, and so, iteratively, the entire envelope function. Two initial values are however needed to start off this procedure. If the definition range ofziszmin−zmax withzmaxδz, then the starting conditions
ξi(zmin) =0, ξi(zmin+δz) =1 (2.19) can be used.
The first expression in Eq. 2.19 states that the wavefunction extinguishes at the bound-aries of the confinement potential. The second expression states that the envelope func-tion has a finite value atzmin+δz. The exactly magnitude ofξiatzmin+δzis not important due to the linear character of the Schr¨odinger equation. Furthermore, after the calculation ofξi over the definition range ofz,zmin−zmax, a normalization procedure is executed in order to get normalized functions.
The iterative method described above assumes a known eigenenergyEifor each quan-tum numberi. This energy is however unknown and needs also to be determined. For this purpose, the shooting method usually starts using an arbitrary (some meV) value for Ei. Then, the optimal value (up to some level of accuracy) for Ei is found evaluating if ξ(Ei,zmax) =0. Again, this condition states that the wavefunction should extinguish at the boundary of the confinement potential. If this condition is not fulfilled, a new value ofEi is generated and a new functionξi(z)is calculated. Finding the root ofξi(Ei,zmax) with respect toEiis further accelerated generatingEivalues using the Newton-Raphson
Chapter 2. Quantum confinement in two-dimensional systems 21
Direction of confinement, z [nm]
Figure 2.6: Calculated solutions ξ1(z) for the electron ground state of a 7 nm-thick In0.73Ga0.27As quantum well surrounded by 14 nm-thick AlAs barriers. The solid line represents the optimal solution for the eigenenergy of E1=94.74 meV. Deviations from this value lead to divergent values of ξ1 at the boundary zmax. Energies are given relative to the conduction band edge of In0.73Ga0.27As.
Figure 2.6 illustrates the process of finding optimal eigenenergies. We represent here a single quantum well consisting on a thin In0.73Ga0.27As layer sandwiched between two wide layers of AlAs. We assume pseudomorphic growth on InP. Application of the shoot-ing method to find the ground state solution for a conduction band electron leads to the characteristic shape represented by the solid line in Fig. 2.6. The correspondent eigenen-ergy is calculated toE1=94.74 meV. Small deviations from this value lead to divergent values ofξ1at the boundaryzmax=21 nm. The correct eigenenergy lies between values forE1, at whichξ1(zmax)changes its sign.
Figure 2.7 shows calculated envelope functions squared for the single quantum well (QW) of Fig. 2.5. For graphical representation purposes, we shift the|ξi(z)|2 values in the energy axis up to the correspondent eigenenergy Ei. We see that the shape of the envelope functions squared is similar to analytic solutions for electron states confined in
21
Chapter 2. Quantum confinement in two-dimensional systems 22 an infinitely deep QW, which have the form sin2(πiz/lw)withlwthe well width.
-14 -7 0 7 14 21
Direction of confinement, z [nm]
2
Figure 2.7: Calculated conduction band solutions for the quantum well system of Fig. 2.5.
Figure 2.8a illustrates the effect of varying the quantum well width. The confinement energies are pushed down (up) as the QW becomes wider (narrower). These calculations take into consideration strain-induced energy shifts as well as band non-parabolicity. Ne-glection of band non-parabolicity would led to an energy shift of the order of few tens of meV (Fig. 2.8b).
The accurate calculation of envelope functions and confinement energies for electrons in semiconductor heterostructures in very important for an effective device design pro-cess. Consider for example the active region of QCLs, which includes superlattices and multiple quantum wells. The correct calculation of energy levels is here, for exam-ple, crucial for the accurate prediction of the emission wavelength. Finally, envelope functions and confinement energies play an important role in several (radiative and non-radiative) scattering mechanisms of charge carriers in semiconductor heterostructures (chapter 5).
Chapter 2. Quantum confinement in two-dimensional systems 23
0.0 0.4 0.8 1.2
2 4 6 8 10 12 14
0 10 20
E