In this section we will look upon an electron as a propagating wave, as first suggested by de Broglie. A wave propagating in a crystal can be disturbed by Bragg reflection. The probability to find an electron at a certain location can be calculated using the Schrodinger equation. At Bragg reflections wavelike solutions to the Schrodinger equation do not exist.
The Bragg condition is:
In one dimension the condition becomes:
where n=1,2,3..... and a is the lattice constant.
The first reflection and the first energy gap occurs at n=1.
Figure 1 Left: Plot of energy versus wavevector k for a free
Figure 2 One dimensional periodic potential
Figure 3 Distribution of probability density in the periodic potential
for standing wave 1 and 2.
The standing wave 2 piles up electrons around the positive ion cores, which means that the average potential energy will be lower than for a free traveling wave (constant probability density). The potential energy corresponding to standing wave 1 will have higher potential energy than a free traveling wave, since it piles up electrons between the ion cores (not compensated by positive ions). The energy difference between the standing waves is the origin to the energy gap Eg.
The relation between the electron energy and the electron wave vector is called the band structure. The band structure is directly related to the crystal structure of the material. In the following section we will calculate the band structure for a square-well periodic potential. Band structure calculation of a real lattice is much more complicated and this example is only a simple demonstration.
Figure 4 One dimensional periodic potential model.
We will use the limit where U0 is approaching positive infinity and b approaches zero (a series of delta functions). Solving the Schrodinger equation for this potential structure gives the following band structure.
Figure 5 Normalized energy versus normalized wave vector for the
potential structure shown in figure 4.
Figure 6 Normalized energy versus normalized wave vector in the first Brillouin zone for the potential structure shown in figure 4.
Figure 7a The first conduction band in Silicon.
Figure 7b The second conduction band in Silicon.
Figure 8 Constant energy surface for the first conduction band in
Figure 9 Constant energy surface for the first valence band in
Figure 10 Constant energy surface for the second valence band in
Click on this link in order to examine band structures of other materials:
Newtons equations for an electron in a semiconductor crystal can be written as:
Notice that the electric field will change the electron position in the reciprocal space and the corresponding movement in real space depends on the gradient of the band structure.
Band structure models
The relationship between the momentum k and the kinetic energy can be accounted for according to different levels of approximation:
(a) Parabolic Band
where is the effective mass at the conduction band minimum (or valence band maximum). The particle velocity is evaluated as
which verifies the simple relation for the mechanical momentum
(b) Non-parabolic Band
where is the coefficient of non-parabolicity and has the dimensions of an inverse energy. The solution of the second order equation (4) is
The velocity is
and using (5)
The coefficient of non-parabolicity is given by
where is the electron mass in vacuum, and is the energy gap between valence and conduction band.
So far, a spherically symmetric band has been implicitly assumed, with isotropic effective mass. This is normally a good approximation for the valley of GaAs, centered about the (0,0,0) point of k-space. In the case of Ge and Si, the bandgap is indirect, therefore in the conduction band there are a number of equivalent valleys, and is described by an ellipsoid rather than a sphere. Ge has eight valleys with the minimum centered approximately (in units of the reciprocal lattice constant) at the (1,1,1) point of the Brillouin zone, called the L-point, while Si has six valleys with minimum at the (1,0,0) or X-point. In GaAs, the first satellite valley is Ge-like and will be called the L-valley in the following. The second satellite valley is Si-like, and will be called the X-valley. In an ellipsoidal valley the mass is not isotropic, therefore it is described by a tensor with components
where i and j correspond to combinations of the coordinates x, y, and z. Cross-terms arise because the axes of the ellipsoids are not going to be parallel to the cartesian axes, in general. For simplicity, one may consider a spherical valley with an average conductivity effective mass, to average the effect of different valley orientation. This is more appropriate for 1-D simulations, but may be acceptable for 2-D simulations, as long as the neglect of anisotropy of the effective mass is not critical.
(c) Full Band Structure The function is determined numerically with detailed band structure calculations or from experiments. The calculations for the particle motion become fully numerical. The inclusion of band structure in Monte Carlo simulations is discussed in Shichijo and Hess.
The non-parabolic band approximation is probably still the best trade-off for many device applications, since a full band structure approach is very time consuming. In the case of very high field transport, which requires the better physical model of the full band structure, it is also possible to use a hybrid approach which treats the less energetic particles with the non-parabolic band formulation. This is effective for device simulation, since only a small fraction of electrons are likely to be in the high field region (most of the simulated particles are usually in the highly doped contact regions).
Kubinio, sfalerito Td - simetrijos ZnSe kristalo juostinės elektroninės struktūros išklotinė jo Briliujeno zonoje, išilgai atskirų aukštos simetrijos krypčių.
Kubinio sfalerito tipo (Td ) kristalo Briliujeno zona.