# Density of carriers in a Conduction band

1. Oct 5, 2012

### cybhunter

1. The problem statement, all variables and given/known data

Find the density of carriers N in the conduction band of intrinsic silicon at room temperature

2. Relevant equations

lower integrand Ec, upper integrand= Ec+1*qe,

Eg=1.12eV,
Ev=0
Ec=Ev+Eg
Ef=Ev+Eg/2

m*n=1.08*(9.11*10^31),
T=300K,
kb=1.38065*10^-23J/K ,
h=(6.625*10^-34)

fF(E)=1/(1+e^((E-Ef)/(kT))
g(E)=4*pi*(2*m*n)^(3/2)/h^3*(E-Ec)^(1/2)

N=integral(Ff(E)*g(E))dE

3. The attempt at a solution

for fF(E): 1/(1+e^(0.56/0.0259)= 4.07239*10^-10

for g(E): 4*pi*(2*m*n)^(3/2)/h^3*(E-Ec)^(1/2)= 1.159*10^56*(E-Ec)^(1/2)

intergating from Ec to Ec+1*qe: 3.23869*10^46*[Ec+1*qe-Ec]

converting the eV values to Joules, and simplifying the interrogation: 3.23869*10^46*[1.6*10^-19]

the value I got is 2.07276*10^18 m^-3

The value I should have received is 1.102*10^16 m^-3

in the process of solving for the density function, I symbolical replaced E-Ec with a delta E and integrated to get 2/3*(E-Ef)^(3/2), and took to the limits between Ec and Ec+1*qe, effectively leaving a difference of 1*qe. Solving just for the density I have no problem. The trouble I'm having lies with the Fermi Dirac probability function.

When solving for the Fermi Dirac, I found online (Microelectronics I: Introduction to the quantum Theory of Solids): e^(Ec-(Ec-Ef))/kT. Since I was given Ef as Eg/2 (0.56eV), I end up getting a value for the probability function of 4.07239*10^-10, which apparently is wrong.

What am I doing wrong?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution