3.39 Sspd_chapter 1_part 16_fermions_part 17_density of states in  (Page 2/2)

SSPD_Chapter 1_Part 17_Density of States

1.17. Calculation of Density of States in the Conduction Band of metals based on Quantum Mechanics.

Figure(1.88) describes the momentum space in terms of three reference momentum vectors p _x , p _y and p _z . Figure (1.89) describes the potential well created by one atom in (a), describes the overlapping potential wells resulting in periodic variation of potential along a linear array of atoms in (b) and a 3-D potential box in a single crystal solid with periodic potential variations in (c).

Figure(1.88) Momentum Space .

We saw in Chapter 1_Part 8continued_Electron in an infinite potential well that in a 1-D potential well(Eq.1.51):

ka=nπ

where k is the wave vector or wave number, a is the lattice constant and n is the principal quantum number.

And E = ћ ² k ² /(2m*)= (h ² /4π ² )(1/(2m*))(nπ/a) ² = n ² h ² /(8m*a ² )

where m* is effective mass of electron.

Therefore E = n ² h ² /(8m*a ² ) ...................................1.164

In 3-D potential box where length , breadth and height are a, b and c

E = (h ² /8m*)[n _x ² /a ² + n _y ² /b ² + n _z ² /c ² ]

If a=b=c=L then E= (h ² /(8m*L ² ))[n _x ² + n _y ² + n _z ² ]

Or E = (h ² n ² )/(8m*L ² ))

Where n ² = n _x ² + n _y ² + n _z ² .....................................................1.165

At n _x = 1, n _y = 1, n _z = 1, E ₀ = (h ² /(8m*L ² ))(3)

This is the lowest energy state and this is non-degenerate. Therefore the lowest energy state electron is non-degenerate.

But the energy state E ₁ = (h ² /(8m*L ² ))(6) has three fold degeneracy.

The following three states have the same energy namely:

n _x = 2, n _y = 1, n _z = 1

n _x = 1, n _y = 2, n _z = 1

n _x = 1, n _y = 1, n _z = 2

Therefore E ₁ has three fold degeneracy.

Next energy state E ₂ = (h ² /(8m*L ² ))(1+4+9)= (h ² /(8m*L ² ))(14) has 6-fold degeneracy.

As the energy increases the degeneracy increases. From the allowed energy states and the number of electrons present we can determine if the given solid is conductor, semi-conductor or insulator.

1.17.1. Theoretical Formulation of density of permissible energy states.

Now we will do the theoretical formulation of density of permissible energy states.

Suppose

N(E)= permissible energy states per unit electron-volt per unit volume;

Therefore N(E)dE= permissible energy states per unit volume between E and E+dE.

We are studying a cubic volume with equal sides.

From Eq (1.165) we know that:

E= (h ² /(8m*L ² ))[n _x ² + n _y ² + n _z ² ]

Here with positive integer values of wave numbers n _x , n _y , n _z we have quantum states associated. In a 3-D Cartesian Momentum Space with n _x , n _y , n _z chosen as frame of reference, in the positive octet as shown in Figure(1.90), every point with integer coordinates are permissible and possible quantum states.

Figure (1.90) Positive Octet of 3-D momentum space with wave numbers n x , n y , n z as the frame of reference.

In reality n _x , n _y , n _z are discrete integer values but if we regard them to be continuous we will not incur much error. So we will regard n _x , n _y , n _z to be continuous.

Assuming that n _x , n _y , n _z are continuous we have the following relation:

N(E)dE= (1/8)(4πn ² )dn =

permissible energy states in incremental energy dE in positive octet ........................................................ 1.166

Where 4πn ² is the surface area of a sphere of radius n and 4πn ² dn is the volume of the shell of radius n and thickness dn in n _x , n _y , n _z Cartesian Momentum Space. Since we are considering only the positive octet, we are considering only the positive values of n _x , n _y , n _z . Hence (1/8) factor has been introduced.

From Eq(1.165) n ² = (8m*L ² /h ² )E

Square root of this expression is:

n = (8m*L ² /h ² ) ^1/2 E ^1/2 ...................................................1.167

Differentiating n with respect to E we get:

dn/dE= (8m*L ² /h ² ) ^1/2 (1/2)(1/E ^1/2 ) ...............................1.168

Substituting the values of dn/dE and n ² in Eq(1.166)

N(E)dE= (1/8)(4π)( (8m*L ² /h ² )E) (8m*L ² /h ² ) ^1/2 (1/2)(1/E ^1/2 )dE

Simplifying the expression:

N(E)dE= (π/4)( (8m*) ^3/2 / h ³ )(L ³ )E ^1/2 dE

Therefore permissible states per unit volume:

N(E)dE=(4√2)π( (m*) ^3/2 / h ³ )E ^1/2 dE

Since every permissible state has two electrons of opposite spins therefore with every point having integer values of n _x , n _y , n _z has two quantum states associated. Therefore permissible quantum states per unit electron volt and per unit volume is:

N(E)= (8π(√(2m*))/ h ³ )m E ^1/2 ...................................1.169

In 3-D case N(E) = CE ^1/2 --------- as in case of bulk devices.

In 2-D case N(E) = m*/(πћ ² )= uniform---------- as in case of quantum sheet devices

In 1-D case N(E)= C/E ^1/2 -------------------------- as in case of quantum dot devices.

Special Quantum Devices can be fabricated using the fact that Density of States variation law depends on the dimensions.

Figure 1.91. Density of States with respect to E in bulk devices, sheet devices and quantum dot devices.

1.17.2. DETERMINATION OF THE CONDUCTING ELECTRONS IN A METAL.

Let the number of electrons in conduction band be = n per cm ³ ;

Therefore n =∫{dn,0,E _F } = ∫{N(E)dE, 0, E _F };

Substituting Eq.(1.169):

n = ∫{(8π(√(2m*))/ h ³ )m* E ^1/2 dE, 0, E _F }

n = (8π(√(2m*))/ h ³ )m* E _F ^3/2 ×(2/3) ....................................................... 1.170

Rearranging the terms we get:

E _F = (3n/8π) ^2/3 (h ² /2m*) ............................................................1.171

Table 1.33. The Fermi Energy (E F ) and Work Function (W F ) of selected Metals.

[From Principles of Electronic Materials and Devices, Second Edition, S. O. Kasap(© McGraw- Hill, 2002)]