ıntroductıon to the quantum theory of solids Energy bands and charge carries in semiconductors

Understand conduction and valance energy bands, and how band gaps are formed. Appreciate the idea of doping in semiconductors Use the density of states and Fermi Dirac statistic to calculate carrier concentrations Calculate drift currents in an electric field in terms of carrier mobility, and how mobility is affected by scattering Discuss the idea of ^effective’’ mass

Allowed and forbidden energy bands The last week, we treated the one-electron or hydrogen atom. That analysis showed that the energy of the bound electron is quantized: only discrete values of electron energy are allowed. The radial probability density for the electron was also determined. This function gives the probability of finding the electron was also determined. This function gives the probability of finding the electron at a particular distance from the nucleus and shows that the electron is not localized at a given radius. We can extrapolate these single –atom results to a crystal and qualitatively derive the concepts of allowed and forbidden energy bands. We can then apply quantum mechanics and Schrodinger wave equation to the problem of an electron in a single crystal. We find that the electronic energy states occur in bands of allowed states that are separated by forbidden energy bands.

The kronig-penney model We know that qualitatively the splitting of allowed electron energies as atoms are brought together to form a crystal. The concept of allowed and forbidden energy bands can be developed more rigorously by considering quantum mechanics and S.C. wave equation. It may be easy for the reader to ‘’get lost’’ in following derivation, but the results forms the basis for energy –band theory of semiconductors.

The potential function of a single, non interacting, one-electron is shown in Figure (a). Figure (b) shows the same type of Potential function for the case when Several atoms are in close proximity Arranged in a one-dimensional array. The potential functions of adjacent Atoms overlap, and the net potential function for this case is shown in figure (c)ıt is this potential function we would need to use in Schrödinger wave equation to model a one-dimensional single-crystal material.

The solution to Schrodinger’s wave equation, for this one-dimensional single crystal lattice, is made more tractable by considering a simpler potential function. Figure 3-6 is the one dimensional Kronig-Penney model of the periodic potential function which is used to represent a one-dimensional single –crystal lattice. We need to solve SC wave equation in each region. As with previous quantum mechanical problems, the more interesting solution occurs for the case when E<V0, which corresponds to a particle being bound within crystal, but the results will illustrate many of the important features of the quantum behavior of electron in a periodic lattice

To obtain the solution to SC wave equation, we make use of a mathematical theorem By Bloch.

The theorem states that all one-electron wave functions, for problems involving periodically varying potential energy functions, must be of the form The parameter k is called a constant of motion and will be considered in more Details as we develop theory. The function u(x) is a periodic function with period (a+b). The total solution to the wave equation is the product of the time-independent Solution and the time-dependent solution , or

This travelling-wave solution represents the motion of an electron in a single –crystal material. The amplitude of the traveling wave is a periodic function and the parameter k is also referred to as wave number.

The k space dıagram To begin to understand the nature of the solution, initially consider the special case for which V0=0. ın this case P’=0, which corresponds to a free particle since there are no potential barriers.

Bonding forces ın solıds The interaction of electrons in neighboring atoms of a solids serves the very important functions of holding the crystal together. For example; alkali halides such as NaCl are typified by ionic bonding. ın the NaCl

Na atom is surrounded by six Nearest neighbor Cl atoms, and vice versa. Four of the nearest neighbors are evident In the two dimensional representation shown in fig a. the electronic structure of Na (Z=11) is 3s1, and Cl (Z=17) has the structure 3s23p5. ın the lattice each atom gives up its outer 3s electron to a Cl atom, so that the crystal is made up of ions.

ın the 1900 s Paul Drüde come up with the ‘’sea of electrons’’ metallic bonding theory by modeling metal as a mixture of atomic cores (atomic cores=positive nuclei+inner shell of electrons) and valence electrons. Metallic bonds occur among metal atoms. Covalent bonding

Energy bands As isolated atoms are brought to form a solid, various interactions occur between neighboring atoms. the force of attraction and repulsion between atoms will find a balance at The proper interatomic spacing for the crystal. ın the process, important changes occur in the electron energy level configurations, and these changes result in the varied electrical properties of solids.

The band gap is the difference in energy between the lowest point of the conduction band and the highest point of the valance band. The lowest point in the conduction band is called conduction band edge, the highest point in the valance band is called band edge.

Figure 2–8 Electronic structure and energy levels in a Si atom: (a) the orbital model of a Si atom showing the 10 core electrons (n = 1 and 2), and the four valence electrons (n = 3); (b) energy levels in the Coulombic potential of the nucleus are also shown schematically.

Fundamental particles in nature have either integer spin and are called bosons (e.g. photons), or half-integer spin and are known as fermions (e.g. electrons). The quantum mechanical wave function of the electron has a spatial (or orbital) part and a spin-dependent part. ıt can be shown that fermion wave functions of a multi-electron system must be anti symmetric. Hence, the two electrons in the bonding orbital have opposite spins, while those in the anti-bonding state have parallel spins. This explains the Pauli exclusion principle. When we try to put two electrons into a quantum state, they must have opposite spins. To determine the energy levels of the bonding and the antibonding states, it is important to recognize that in the region between the two nuclei the Coulombic potential energy V(r) is lowered (solid lines in Fig3-2) compared to isolated atoms (dashed lines)

Dırect and ındırect semıconductor ısolated atoms were brought together to form a solid, is useful In pointing out the existence of energy bands and some of their properties. Other techniques are generally used, however, when quantitative calculations are made of band structures. ın a typical calculation, a single electron is assumed to travel through a perfectly periodic lattice. The wave function of the electron is assumed to be in the form of a plane wave moving, for example, in the x-direction with propagation constant k, also called wave vector.

The space-dependent wave-function for the electron is

The best value of the band gap are obtained by optical absorption ın the direct absorption process a photon is absorbed by the crystal with the creation of an electron and hole ın the indirect absorption process, the minimum energy gap of band structure involves electrons and holes separated by substantial wave vector kc

A direct optical transition is drawn vertically with no significant change of k, because the absorbed photon has very small wavevector. The threshold frequency wg for the absorption by the direct transition determines the energy gap Eg=hwg the indirect transition involves both photon and phonon because The band edges of the conduction and valance bands are widely Separated in k space. The threshold energy for the indirect process in is greater than the true band gap. The absolute threshold for the indirect transition between the Band edges is at hw=Eg+hn, n is the frequency of an emitted phonon of wave vector.

Electrons and holes As the temperature of a semiconductor is raised from 0K, some electrons in valence band receive enough thermal energy to be excited across the band gap to the conduction band. The result is a material with some electrons in otherwise empty conduction band and some unoccupied states An otherwise filled valance band For convenience, an empty state in the valance band is referred to as a hole.

Drift current Current is due to the net flow of charge. ıf we had a collection of positively charged ions with a volume density N (cm-3) and an average drift velocity vd (cm/s), then the drift current density would be; ıf, instead of considering the average drift velocity, we considered the indivii Dual ion velocities, then we could write the drift current density as

Since the electrons are charged particles, a net drift of electrons in the conduction band will give rise to a current. We may write the drift current density due to the motion of electrons as;

example

solutıon

Effectıve mass The movement of an electron in a lattice will, in general, be different from that of an electron in free space. ın addition to an externally applied force, there are internal forces in the crystal due to positively charged ions or protons and negatively charged electrons, which will influence the motion of electrons in the lattice. We can write

Since it is difficult to take into account all of the interanl forces, we will write the equation

To begin, considering the case of a free electron whose E versus k curve was shown in fig. The energy and momentum are related by;

Effective mass of electron; Thus the curvature of the band determines the electron effective mass A particular interesting feature of this graph is that the curvature of d2E/dk2 Is positive at the conduction band minima, but is negative at the valence band Maxima. Thus, the electrons near the top of the valance band have negative Effective mass, according this equa. Valance band electrons with negative charge and negative mass move in an Electric field in the same direction as holes with positive charge and positive mass

For a band centered at k=0, the (E,k) relation near the minimum is usually parabolic: Comparing this relation with the effective mass, the effective mass m* is constant In a parabolic band. On the other hand, many conduction bands have complex (E,k) Relationships that depend on the direction of electron transport with respect to The principal crystal directions. ın this case, the effective mass is a tensor quantity. However, we can use approriate averages over such bands in most calculations.