This can be solved in the usual manner; that is,
F(r) = 1/√ arbitrary normalization area. Note that the spatially dependent elec-tron mass introduces ak||-dependent potential term, so that the in-plane and perpendicular motions are not entirely decoupled. This introduces nonparabolicity to the conduction subbands that broadens the intersub-band absorption spectrum. If this effect is ignored, eq. 2.16 becomes the Schrödinger equation for a particle in a finite square well with varying mass. Such an equation can be solved by the usual method of matching solutions in each region with the appropriate current conserving bound-ary conditions. Some examples of this technique can be found in any textbook on quantum mechanics. Of course, this technique works only when the solutions are known in each region, restricting its application to square quantum wells, perhaps with applied electric fields.
When a substantial number of free carriers are present in the well, they will induce band bending. In this case, the effective mass equation will not take the simple form 2.16, but will contain a space-charge po-tential. If the analytic solutions in each region are not known, one needs to employ numerical calculation techniques.
2.1.3 Conduction band states in quantum well su-perlattices
Let us consider a sequence of the same quantum wells (materialA, thick-ness lA, lattice constant aA) spaced by the equal quantum barriers (ma-terial B, thickness lB, lattice constant aB=aA). Such a heterostructure reveals an additional to the elemental crystalline cell of a bulk mate-rial periodicity along the growth direction z. Since that, the size of an elemental cell in such a heterostructure along z is not an aB anymore, but d ≡ lA+lB. In another words, such a heterostructure becomes to
-d 0 d 2d -π/d 0 π/d V
(a)
z
Energy (arb. units)
∆1
∆2 (b)
kz
m=2
m=1
Figure 2.3: Miniband structure of a superlattice in energy–real space coordinates (a) and energy–k-space (b).
be an artificial anisotropic crystal. The size of elemental cell of such a superlattice can vary from the infinity down to aB scale. In the upper limit case, we will basically deal with the macroscopic pieces of individual materials, attached to each other, where each of them reveals individual three-dimensional properties. In the lower limit case we obtain a ho-mogeneous alloy of materials A and B with isotropic three-dimensional properties. Highly anisotropic properties of a superlattice one obtains when the periodd is on the scale of the de Broglie wavelength in individ-ual materials. Particular properties of energy bands of such short period superlattices we are going to consider in current subsection.
To describe qualitatively the new specific features of superlattices, let us ignore the spatial dependence of effective mass me(z)in the effec-tive mass equation 2.14, and consider the Schrödinger equation for the envelope function F(r)
[− ~2 2me( d2
dx2 + d2
dy2 + d2
dz2) +VSL(z)]F(r) =EF(r), (2.17) where VSL(z) is the periodic superlattice confinement potential, as de-picted on fig. 2.3a. We consider a space independent effective mass separately for well and for barrier material. The lateral and perpendicu-lar motions of electron in such a case are decoupled. And one looks for solution in simplified form of eq. 2.15:
F(r) = 1
√Aei(kxx+kyy)fm(z). (2.18) Let us consider a single electronic band and, hence, ignore the index m. Applying the periodical Bastard’s boundary conditions (19) on the
heterointerfaces one obtains the transcendental equation
cos(kzd) = cos(kAlA) cos(kBlB)−1
The resulting band structureEn(kz), which is a solution of eq. 2.21, is called minibands. In contrast, purely two-dimensional quantum confined states are usually called subbands. The crucial quantity distinguishing two-dimensional subband from three-dimensional minibands behavior is the ratio of minibands width to the strength of some broadening mecha-nisms such as scattering rate or potential fluctuations. If this broadening is larger than the minibands width, the electrons do not feel the miniband dispersion and are effectively confined in z direction. Figure 2.3 shows schematically a superlattice band structure in real space and in k-space.
Figure 2.4 shows the density of states, DOS, in minibands. The DOS does not have the real gap, but takes the constant value m/π~2 between the minibands. It is useful to have an explicit expression for the superlat-tice energy dispersion. This is available through the tight binding model, where one starts with the energy levels of separated quantum wells and takes into account the interaction with the nearest neighbouring wells, leading to
En(k) = En± ∆n
2 cos(kzd) + ~2(kx2+ky2)
2m . (2.25)
Here En and ∆n are the middle and the width respectively of the n-th miniband (n ≥1). The minus sign in eq. 2.25 holds for odd minibands, the plus sign is for even ones. The effective mass for the in-plane motion is assumed to be identical to the mass in the well, since the electrons have a much higher probability amplitude there than in the barriers. Equation 2.25 can also be obtained by expanding eq. 2.22, 2.23 and 2.24 in the vicinity of the quantum well energy levels (24).
0 50 100 150 200 250 300 350 0
1 2
m=2
DOS ( 2π(2me)½ /h )
Energy (meV) m=1
Figure 2.4: DOS for SL minibands (solid line) comparing the DOS of the same thickness single QW.
Within the tight-binding model the effective mass along the z direc-tion (evaluated at kz = 0, i.e. the band bottom edge) is given by
1
mSL = 1
~2
∂2E
∂kz2 = ∆d2
2~2 cos(kzd)|kz=0= ∆d2
2~2 , (2.26) and the velocity by
υ(kz) = 1
~
∂E
∂kz = ∆d
2~ sin(kzd). (2.27) Here we obtain the novel feature of the miniband, which is a pos-sibility of a negative values of effective mass along the z-direction at kz > π/2d. It means that in case of applied external force,F, an electron will (according the 2nd Newton’s law) perform the oscillatory motion.
First, it accelerates along the vector F till it reaches the negative mass kz-space. There electron slows down till the kz = π/d, where it stops (υ(kz = π/d) = 0) and finally goes backwards. Such an oscillatory be-havior of electron within a miniband is considered in subsection 2.3.1.