BAND THEORY OF SOLIDS
According to quantum free electron theory of metals, a electron in a metal experiences constant(or zero) potential and free to move inside the crystal but will not come out of the metal because an infinite potential exists at the surface. Bloch Theorem: According to this theorem, the periodic potentials due to the positive ions in metal have been considered. (i.e. the electrons moves in a periodic potential provided by lattice). If the electron moves through these ions, it experiences varying potentials. The potential of an electron at the positive ion site zero and is maximum in between two ions. i.e. the potential experienced by an electron varies periodically with the same period as the lattice. The potential is negative because of attractive force between electrons and positive ions.

Along X- direction in the crystal, the potential function V(x) has the periodicity of lattice given by V(x) = V(x + a ) where, ‘x’ is distance of the electron from the core and ‘ a’ is the periodicity of the lattice. The energies of electron can be known by solving Schreodinger’s time independent wave equation for motion of an electron along x-direction is given by

Bloch showed a type of solution for the above equation, given by ψ(x) = uk (x) eikx where uk(x) represents periodic function given by uk (x) = uk(x+a) Here, K = 2π/λ = propagation vector and λ = wavelength of debroglie’s wave associated with the moving electron and eikx represents a plane wave.
However, it is extremely difficult to solve the Schrödinger’s equation with periodic potential described above. Hence the Kronig – Penney Model is adopted for simplification.
Kronig – Penney Model In 1930, Kronig- penny proposed a one dimensional model for the shape of rectangular potential wells and barriers having lattice periodicity. According to this theory, the electrons move in a periodic potential produced by the positive ion cores. The potential of electron varies periodically with periodicity of ion core ‘a’ which is nothing but inter-atomic spacing. It is assumed that the potential energy of the electrons is zero near nucleus of the positive ion core and maximum when it is lying between the adjacent nuclei as shown fig.

Fig. One dimensional periodic potential The width of the potential well and barrier are ‘a’ and ‘b’ respectively. The potential energy of an electron in the well is zero and in the barrier is V0. The periodicity of the potential is a+b. This model is an highly artificial, but it illustrates many of the characteristics features of the behavior of electrons in periodic lattice. The energies and wave functions of electrons associated with model can be calculated by solving the Schrödinger’s wave equation for two regions I and II.

The Schrödinger’s equations are for region-I 0 < x < a and for region-II -b < x < a Since E less than Vo, define two +ve quantities, α2 = 2mE/ ħ2 and β2 = 2m(V0– E )/ħ2 for region 0 < x < a for region -b < x < a
According to Bloch Theorem, the solutions of above equations can be written as Ψ(x) = uK (x) eiKx
The consists of a plane wave eiKx modulated by the periodic function uK(x), where this UK(x) is periodic with the periodicity of the lattice. uK(x) = uK(x+a) where K is propagating constant along x-direction and is given by K =2π/λ is a Propagation wave vector. In order to simplify the computations, an assumption made regarding the potential barrier. As Vo increases the width of the barrier ‘b’ decreases so that the product Vob remains constant. After tedious calculations, the possible solutions for energies are obtain from the relation Psin∝aαa+cos∝a=cosKa ------------ (1) where P = mVo ab /ħ2 is scattering power of the potential barrier. It is a measure with which electrons in a crystal are attracted to the ions on the crystal sites.
The left hand side of the equation (1) is plotted as a function of αa for the value of P=3π/2. The right side of the equation imposes a limitations on the values of left side function, i.e., between -1 to +1 as indicated by the horizontal lines in fig. Hence only certain range of α are allowed.

Fig. a) For large value of P b) P⟶∞ c) P⟶0
The permitted values of energy are shown as solid lines between white portions. This gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a.
Conclusions:
1. The energy spectrum of electron consists of alternate regions of allowed energy and un allowed regions i.e. The motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions. 2. As the value of a increases, the width of allowed energy bands also increases and the width of the forbidden bands decreases. 3. If the potential barrier strength P is large, the function described by the left hand side of the equation crosses +1 and -1 region at a steeper angle. Thus the allowed bands are narrower and the forbidden bands are wider. 4. IfP⟶∞, the the allowed band reduces to one single energy level corresponding to the discrete energy level of an isolated atom.
P is follows from equation (1) that Sina=0, a = n ; 2 = .
∴

Here E is independent of k. The energy levels in this case are discrete and the result is similar to the energy levels of a particle in a constant potential box of atomic dimensions. This is expected because for large P tunneling through the barrier becomes almost improbable. 5. when P0, leads to
Cos a = cos Ka
Thus = K
i.e. 2 = K2
K2 = 2 =
E = E =

E =
This energy corresponds to a completely free particles and no energy levels exist i.e. all energies are allowed to the electrons.
Thus by varying P from zero to infinity, we find that the completely free electron becomes completely bound electron.
Brillouin Zones: (Energy(E) versus Wave number (K)
The motion of electrons in a periodic lattice is characterized by the bands of allowed energy regions separated by forbidden regions. The right side of the eqn becomes ±1 for values of K= ±nπ/a. Hence the discontinuities in the E versus K graph occur at K= ±nπ/a, where n= ±1,±2, ±3, …..etc… It is clear that the relation between E and K is parabolic. From the graph, the electron has allowed energy values in the regions or zones extending from K= -π/a to K= +π/a. the zone is called the First Brillouin zone. After a break in the energy values called the forbidden region or band or zone, we get another allowed zone of energy values in the region from K= -π/a to -2π/a and +π/a to +2π/a. This zone is called the Second Brillouin zone. Similarly the other higher order Brillouin zones can be defined.

The discontinuities occur at the boundaries of the Brillouin zones. Each portion of the curve gives a no. of allowed band of energies. The curves are horizontal at the top and bottom and they are parabolic near the top and bottom with curvatures in opposite directions. As P increases, the discontinuous E-K graph will reduce to a continuous parabolic graph and forbidden bands disappear. Then, the energy values are practically continuous.
Effective mass of electron (m*)
An electron in crystal may behave as if it had a mass different from the free electron mass m0. There are crystals in which the effective mass of the carriers is much larger or much smaller than m0. The effective mass may be anisotropic, and it may even be negative.
Def: When an electron in a periodic potential of lattice is accelerated by an electric field or magnetic field, then the mass of the electron is called effective mass (m*).
Let us consider an electron of charge ‘e’ and mass ‘m’ moving inside a crystal lattice of electric field E. The acceleration a =eE/m is not a constant in the periodic lattice of the crystal. It can be considered that its variation is caused by the variation of electron’s mass when it moves in the crystal lattice ∴accleration=aEm*
The electrical force on the electron, F= m*a ---------------(1)
Considering the free electron as a wave packet, the group velocity vg corresponding to the particle’s velocity can be written as vg=dωdk=2πdνdk=2πhdEdk=1ℏdEdk------(2) where the energy E=hν and ℏ=h/2π fig(a) E
Since ℏk=p, ℏdkdt=dpdt=F, dkdt= Fℏ v
∴a= or ---------(3) fig(b)
Comparing (1) and (3), we get m* Fig© -π/a 0 k0 +π/a
a) Variation of E with k : The variation of E with k for the first allowed band (in Fig(a)) and from equation (2), velocity can be calculated.
b)Variation of v with k: The variation of velocity with k is illustrated in the Fig.(b). We find that for k=0 the velocity is zero and as the value of k increases (i.e.,the energy E increases), the velocity increases reaching its maximum value at k=k0. k0 corresponds to the point of inflexion on the E-k curve. Beyond this inflexion point the velocity begins to decrease and finally assumes the zero value at k=π/a.
c) Variation of m* with k: It is shown in Fig.(c). Near k=o, the effective mass approaches m. As the value of k increases, m* increases, reaching its maximum value at the point of inflection on the E-k curve. Above the point of inflection, m* is negative and as k tends to π/a, it decreases to small negative value.
Experimentally, Effective mass can be obtained from cyclotron resonance experiment.
From Fig.(b), it is clear that beyond the inflection point k0 velocity decreases i.e., acceleration negative. This means that in this region of k the lattice exerts a large retarding force on the electron. This means that in this region of k the electron behaves as a positively charged particle referred to as hole.
Concept of hole: The effective mass m* is negative near the zone edges of almost filled valence bands. Physically speaking the electrons in these regions are accelerated in a direction opposite to the direction of the applied force. This is called the negative mass behavior of the electrons.
The electrons with negative mass can be considered as a new entity having the same positive mass of that electron and the same positive charge as the numerical value of the electron’s charge. The new entity is given the name ‘hole’.
The holes are not real particles like electrons but its is only a way of looking at the negative mass electrons behavior near the zone edge. The hole concept is used to explain the motion of the effective negative mass electrons in a nearly filled band.
Origin of Energy Bands in Solids:
♣ A solid contain a large number of atoms. In an isolated atom the electrons are tightly bound and have discrete energy levels. The atoms are very close to each other.
♣ Due to close packing of atoms, the orbital’s of outer shell electrons overlap to produce strong atomic forces. Due to this the no. of permissible energy levels increases.
♣ As the inter atomic distance decreases the energy levels of one atom overlaps with those of neighboring atoms. ♣ Hence in a solid the energy level corresponding to one quantum number splits up into many closely spaced levels. ♣ A set of such closely spaced energy levels, is called an Energy Band.
♣ The electrons in the higher energy bands of solids are important in determining many physical properties of solids. ♣ Overlapping of these bands occur for smaller equilibrium spacing (ro) .
Example: Formation of energy bands in Silicon (Si):

a) If interatomic spacing of atoms is very large i.e., r = d>>a, there is no interatomic separation. Each atom in the crystal behaves as free atom. Take for example silicon whose electronic configuration is 1s2 2s2 2p6 3s2 3p2. If N atoms were to be considered in silicon crystal, then there will be 2N electrons filling 2N possible energy levels in 3s, 6N possible levels in 3p of which only 2N is completely filled.
b) When the spacing is progressively decreased i.e., c<r<d, there is no visible splitting of energy levels.
c) When r = c, the 3s and 3p electrons of neighbouring silicon atoms becomes appreciable. The energy of electrons of each atom starts changing, whereas the energies of electrons in the inner shell do not change.
d) When r lies between b and c, the energy levels get slightly changed and instead of a single 3s or 3p levels, we get a large number of closely packed levels. This collection of closely spaced energy levels is called an energy band.
e) When r = b>a, the gap between 3s and 3p completely disappear and the 8N energy levels are (2N of 3s and 6N of 3p sub shells) continuously distributed. In this stage 4N levels are filled and 4N levels are empty.
f) When r = a i.e., actual spacing in the crystal the 4N filled energy levels are separated from 4N unfilled energy levels. This gap or separation is called the forbidden gap. E.g., the lower completely filled band is called valence band and upper unfilled band is called conduction band.

♣ The gap between these two allowed bands is called forbidden energy gap or energy gap ♣ The conduction band is vacant at 0k. ♣ According to the width of the gap between the bands, all solids can be classified into three groups namely, metals, semi conductors and insulators.