P2

where YA(YB) are step functions which are unity if r corresponds to an A layer (to a B layer) and VA(VB) is the potential in the A(B) layer.

The envelope function in the 2 direction, xn(z), of an A/B layer is the solution of the Schrodinger equation h2

2m* dz2

where Vn(z) is the heterostructure potential determined by the band discontinuity: m*nz = m*n(z) = m*n A(m*n B) and Vn(z) = 0 in the A(B) layer.

Let us see the general case of a superlattice. In a given A/B superlattice there is a new periodicity along the growth direction z. The SL period d (sum of the layer thickness of the A and B layers, dA + dB) is longer than the lattice constants of the constituents, so that a new mini-Brillouin zone is formed in the wavevector range of -w/d < kz < w/d. The mini-Brillouin zone formation results in generation of miniband structures in the conduction and valence bands.

We consider the miniband structures of the r6 conduction band and the r8 valence band (heavy hole, \J, MJ) = |3/2, ±3/2) and light hole, \3/2, ±1/2)). The total wave-function in each layer can be written in the form nA'B(r) = E fAf(r)Un,k=o(r)

where we can use the following form of the envelope function:

Then fA'B(z) is the envelope function determined by the SL potential and the effective masses, and q is the SL wavevec-tor parallel to z. The periodicity of the SL results in the Bloch theorem for the envelope function:

where m is integer.

The envelope function of an A/B SL is the solution of the Schrodinger equation h2

where Vn(z) = 0(Aen) in the A(B) layer. For convenience we will omit the band index n in the following treatment. The form of the envelope function can be written as fs(z) =

f A(z) = aA exp(ikqz) + ßAq x exp(- ikAz) fBB(z) = aB exp(ikBz) + 0

x exp(- ikBz) in the B layer kA = -y/2m*Ae/h kB = y 2m*B(e — Ae)/h (54)

Considering the continuity of the probability density, \fr(z)\2, for a stationary state, we obtain the following boundary condition of the envelope function at the A/B interface for (zA — zB) ^ 0:

mA dz mB dz

Finally, the miniband dispersion relation, which is similar to the well known Kronig-Penney relation, is given by cos( qd) = cos(kAdA)cos(kBdB)

This equation has been widely used to calculate the miniband energies.

Figure 6 shows (a) the potential structure and the envelope function profiles of the electron and heavy hole in a

Figure 6. Potential structure and the envelope function profiles in superlattice. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

GaAs(3.2 nm)/AlAs(0.9 nm) SL and (b) the dispersion relations of the electron and heavy hole minibands, calculated using Eq. (56). In Figure 6a, the solid and dashed lines indicate the envelope functions at the mini-Brillouin-zone center (q = 0,F point) and the mini-Brillouin-zone edge (q = ir/d, 7T point), respectively. In this calculation, the following effective-mass parameters are used [63]: m* = 0.0665m0, m*hh = 0.34m0, and m*lh = 0.117m0 for the electron, heavy hole, and light hole GaAs, and ml = 0.15m0, m\h = 0.4m0, and m\h = 0.18m0 for AlAs, where m0 is the free-electron mass. The miniband dispersions are tailored by changing the layer thickness, potential height, and effective masses, which are structural parameters in crystal growth. Figure 7 shows the n = 1 and n = 2 electron miniband of the energies at F and 7T points in a GaAs(3.2 nm)/AlAs(d5 nm) SL as a function of the AlAs layer thickness. This figure exhibits a typical example of the miniband tailoring. The miniband width increases with the decrease of the AlAs (barrier) thickness, which results from the increase of the coupling strength of wavefunctions because of the increase of the tunneling probability. In the case of the disappearance of the miniband width, the electronic state corresponds to a quasi-two-dimensional state in an isolated quantum well (QW).

The z dependence of the effective mass means, in principle, that it is different in different layers but in general may vary even in a given medium if no abrupt interfaces are considered. Then, the kinetic energy term in Eq. (52) should be substituted by the more general Ben Daniel and Duke [64] hermitian form

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