short information on EEM

ENERGY BANDS IN SOLIDS:

§  CONDUCTION BAND: The energy band associated with the free conducting electrons is called conduction band. It is either partially filled or vacant.

§  FORBIDDEN BAND:  The energy gap between valence electrons and conducting electrons is called forbidden band. When energy equal to this gap is supplied electrons from valence band move to conduction band.

§  VALENCE BAND:   The energy band associated with the valence shell of atom is called valence band. It is generally filled (partially or completely).

SEMICONDUCTOR:

Elements of group (iv) in periodic table.

Eg:si,ge

1.       INTRINSIC SEMICONDUCTOR: It is semiconductor (si or ge) in pure crystalline form (without impurity).

Ge(32):K(2),L(8),M(18),N(4)

Each germanium atom makes four covalent bonds with other four neighboring atoms. That is, in pure crystalline form, there are no free electrons. But the conductivity in intrinsic semiconductor is due to thermal agitation which breaks down the covalent band and makes the electron free.

2.       EXTRINSIC SEMICONDUCTOR: The conductivity of the intrinsic semiconductor is very small. So, the conductivity is increased by mixing the crystalline form with impurities such a process of increasing conductivity is called doping and the corresponding semiconductor is extrinsic.Doping is made by mixing one impure atom (example: boron, proton, etc) in ten millions of Ge or Si atoms. Since, the impure atom (dopant) regularly replace the atoms from the semiconductor, the original crystalline structure doesn’t change.

                                I.            N-type semiconductor:

In n-type, the doping elements are of group (v) in periodic table.

Example:  Antimony (sb), arsenic (as),   phosphorus (p), etc.

The doping elements have five electrons neighboring Ge or Si atoms, and one electron is unable to make the covalent bond. So, this extra electron acts as free and helps in conduction. In n-type semiconductors, the majority charge carries are free electrons and minority charge carries are thermally formed holes (positive charge).

Donor level in n-type semiconductor:  donor level in n type semiconductor.
the energy corresponding to the free (extra) electron in n-type semi conductor is called donor level. The donor level is about 0.01 ev below the conduction band.

2  P-type semiconductor:
    The doping element for the p-type semiconductors are of third group of periodic table.
example: Gallium, boron, indium, etc.
boron is trivalent impurity.it has three electrons in its outer most shell. There exists three covalent bonds but there is insufficient of one electron to make the covalent bond.this create a hole and seeks another electron that is, hole shows a positive charge. the majority charge carriers are thermally generated electrons. Since the holes have the tendency to accept electrons, the trivalent impurities semiconductors are called acceptors.

Acceptor level in p-type semiconductor:- The energy corresponding to the holes of p-type semiconductor is called acceptor level. It lies about 0.01ev above the valence band.that is, the acceptor level is near to valence band then conduction band. If we supply energy of about 0.01ev is easily obtained at a temperature of about 50k.

Fermi energy (Fermi level): The corresponding to the topmost valence shell of any atom at 0k (zero kelvin) is called Fermi energy. Suppose we intend to accommodate seven electrons in different energy levels. The pauli exclusion principal avoids two electrons in a single quantum state. So, n=1 state has two electrons with opposite spins and this is the same case for n=2 and n=3. But the n=4 state has only one electrons. Thus, for a system of seven electrons the Fermi-energy corresponds to n=3 which is the top most filled energy level at ok (absolute temperature).

Schrodinger wave equation (⎊)s: The three dimensional time independent wave equations is,

⎊^2Ψ+2m/h^2(e-v) Ψ=0

Where, ⎊^2= {(Ƌ^2/Ƌx^2) + (Ƌ^2/Ƌy^2) + (Ƌ^2/Ƌz^2)}, laplacian equation.

 For one dimensional wave equation, {(Ƌ^2/Ƌx^2) 2m/h^2 e-v) Ψ=0……….. (1)

Where, ђ=h/2Л, h=6.67*10^-34 js (is planks constant).

M= mass of particles (i.e electrons)

V=potential of the well (potential energy provided by nucleus).

Ψ= wave function

It has the property of different physical quantities like energy, momentum, velocity, angular momentum etc. The quantity Ψ doesn’t have a fixed meaning unless we introduce the probability density. If  Ψ is a given wave function.

Ψ*=complex conjugate of Ψ.

Probability density (ϸ) = ΨΨ*

∫_a^b▒〖(ϸ)dx=∫_a^b▒(ΨΨ*) 〗 dx=1

The equation (A) is the condition for normalization and the corresponding wave function. Ψ is said to be normallised.

                                            …………*********…………..

Electron in one dimensional  infinite well: Consider an electron of mass, m is confined to a potential well. So, that it is certainly found between the walls enclosed by x=0 and x=1.

V(x)=0, 0<x<1

V(x)=∞, 0>=x; x>=1

The one dimensional equation,

{(Ƌ^2/Ƌx^2) 2m/ђ^2 e-v) Ψ=0 ………. (1)

Since, the potential inside the well is zero, v=0

{(Ƌ^2/Ƌx^2) 2m/ђ^2 e-v) Ψ=0

2mE/ ђ^2 = 2m/ђ^2(1/2)mv^2

mv^2/ђ^2 = {(p) /ђ}^2 = k^2

where, k=p/h is propagation constant.

{(Ƌ^2/Ƌx^2)+k^2 Ψ=0…………………….(3)

The solution of second order linear differential equation can be written as,

Ψ(x) = asinkx+bcoskx …………………….. (4)

At, x=0, Ψ(x) = Ψ (0) =0

Ψ (0) = asin0 + bcos0

0 = b ………………….. (5)

Equation (4) reduces to, Ψ(x) = asinkx …………….. (6)

Again, at x=1, Ψ(x) = Ψ (l) = 0,

Thus from equation (6),

Ψ (L) = asinkl or, 0 = asinkl  or, sinkl = 0 = sinnЛ, n = 01,2,3,……… or, kl = nЛ

Therefore, k = nЛ/l, n = 0,1,2,……

Since, k = p/ ђ, momentum,p = ђk  = ђ nЛ/l

At n = 0, p becomes zero. but, moment of the electron can’t be zero, so, the value n = 0 is not acceptable.

Therefore, k = nЛ/l, n = 1,2,3,……

Thus, the wave function is written as, Ψ(x) = asinkx = asin nЛ/lx, n = 1, 2,….

This wave function must be normallised,

i.e, ∫_0^l〖(ΨΨ*)dx = 1

a^2∫_0^lsin^ 2(nЛ/l x)dx = 1

a^2∫_0^l2sin^ 2(nЛ/l x)dx = 1

a^2∫_0^l(1-cos2(nЛ/l x))dx = 1

a^2*l= 1

a= (2/l)^1/2