me = ∂2En(0)
∂k2 (2.8)
The effective mass me appears in an equation that is mathematically identical to the Schrödinger equation for a spinless particle in a scalar potential. It is the best to think ofme as a material property. The effec-tive mass simplifies the problem of electron motion in a periodic potential of ion cores. Naturally, such an approximation will break down when the wavelength of the particle (photon or electron) is on the scale of atomic fluctuations (X-ray diffraction, short-wavelength electrons). Fortunately, in direct band gap III-V semiconductors most of the carriers reside in the low-energy band-edge states. And these are represented quite well by the effective mass theory. This is the justification for applying many of the formulas derived for the quantum mechanics of free particles directly to the conduction band electrons by making the simple replacement of the free electron mass m0 by the effective conduction band electron mass me.
2.1.2 Conduction band states in quantum wells
Now we proceed to the quantum well case. Modern epitaxial techniques permit us to fabricate structures with extremely sharp interfaces. Be-cause the composition of each monolayer can be controlled, it is possi-ble to grow materials of widely different bandgaps (classical example is GaAs and AlxGa1−xAs) next to each other, thus creating the so-called heterostructure. This produces a very sharp band gap discontinuity and dramatically affects the carrier behavior. One of the most interesting het-erostructures to study is the quantum well, in which a thin layer (or well) of a narrow band gap material is sandwiched between two thicker layers (or barriers) of a wider band gap material (see fig. 2.1). If the well width is less than the de Broglie wavelength of the carriers in the well material ( 10 nm in most III-V compounds), the carrier is "quantum" confined.
The envelope wave functionF(z), and so the full wave functionf(z)are mostly localized in material A (see fig. 2.2) and do not propagate along the Z-axis. Such structures should properly be regarded as completely new materials, with properties wholly distinct from either well or barrier bulk materials. Because the well width is smaller than the wavelength of the carrier, quantum wells are not truly tree-dimensional structures,
7
0 5 10 15 20 25 0.0
1.5 2.0
∆Ev
∆Ec
H2
H1 E2 E1
Egbarrier Eg
well
Ev
Energy rel. top of valence band (eV)
Depth (nm) Ec
Figure 2.1: Quantum confined states, subbands, of electrons,E1 and E2, and holes H1 and H2 in 4 nm thick quantum well in energy-real space coordinates.
F(z) f(z) Un,k(z)
B B
B B
A A
Figure 2.2: Illustration of the wave function localization in a quantum well. F(z) is the z-direction localized envelope function. f(z) is the full wave function in QW.Un,k(z)represents Bloch function. Lattice constant of materials A and B assumed to be the same.
but are instead quasi-two-dimensional. The change in the dimensional-ity of the system results in dramatic variation in many of the material properties of these structures (such as carrier masses, bandgaps, densities of states) and even suggests device concepts that are not possible with simple bulk materials (QWIPs, QCLs, so on). The band structure of these quantum wells may easily be addressed within the effective mass equation. If we take thez-axis to be the growth direction, assuming that the material composition will vary in thez direction but will be uniform in the x −y plane, we can again use eq. 2.3, but we must allow the band structureEn(k)to be dependent onz as well. Taking the arbitrary reference for En(k) = 0, we can compose two materials A and B with different dispersion relations and form a quantum well of width L by arranging them as
En(k) =EA(k) =EA+~2k2
2mA at |z|< L
2, (2.9)
En(k) =EB(k) =EB+ ~2k2
2mB at |z|> L
2. (2.10) Thus, when EA< EB the electrons see an effective z-dependent "poten-tial" well given by
V0(z) =EA at |z|< L
2, and (2.11)
V0(z) = EB at |z|> L
2. (2.12)
Let us address now the issue of interpretation of the kinetic energy operator E(−i∇) when the material parameters, and hence the band structure En(k), depend on z. The problem arises because dzd and m(z)1 do not commute; hence, the ordering is important. Generally, for a single spherical band, the order is taken as
En(−i∇, z)→ −h2∇ · 1
2me(z)∇
. (2.13)
It must be admitted that this operator cannot be derived, but this is the preferred quadratic form for a number of reasons. First, the operator is Hermitian, which ensures real energy eigenvalues and conservation of probability for the envelope functions. Second, the operator conserves the probability current across the heterointerface, as can be seen by ex-amining the form of the current operator (21). Thomsen et al. (22) and Morrow (23) have shown, that this is the only physically acceptable quadratic form for a kinetic energy operator with spatially varying mass
under very general conditions. Hence, we assume that eq. 2.13 and the effective mass equation for electrons in a quantum well reduces to
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.