Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration

Q34P

Expert-verified
Fundamentals Of Physics
Found in: Page 1274

Answers without the blur.

Just sign up for free and you're in.

Illustration

Short Answer

In a simplified model of an undoped semiconductor, the actual distribution of energy states may be replaced by one in which there are NV states in the valence band, all having the same energy EV, and NC states in the conduction band all these states having the same energy Ec. The number of electrons in the conduction band equals the number of holes in the valence band.

  1. Show that this last condition implies that Ncexp(Ec / kT)+1=Nv exp(Ev / kT)+1 in which Ec=Ec-EF and Ev=-(Ev-EF).
  2. If the Fermi level is in the gap between the two bands and its distance from each band is large relative to kT then the exponentials dominate in the denominators. Under these conditions, show that EF=(Ec+Ev)2+kTIn(Nv+Nc)2 and that if NvNc , the Fermi level for the undoped semiconductor is close to the gap’s center.

  1. The number of electrons in the valence band equals the number of holes in the valence band, which implies the condition that Ncexp(Ec / kT)+1=Nv exp(Ev / kT)+1 in which Ec=Ec-EF and Ev=-(Ev-EF) .
  2. Under the given condition, the Fermi energy equation becomes EF=(Ec+Ev)2+kTIn(Nv+Nc)2 for NvNc.
See the step by step solution

Step by Step Solution

Step 1: The given data

  1. In the undoped semiconductor, the number of states in the valence band is with energy Ev, while the number of states in the conduction band is Nc with energy Ec.
  2. The number of electrons in the conduction band equals the number of holes in the valence band.

Step 2: Understanding the concept of density of states

We are given the condition that the number of electrons in the conduction band and the number of holes in the valence band are equal. Thus, using the probability equation in the density of occupied states equation, we can get the individual equations for both cases. This will imply the given condition. Similarly, for the second case, we can consider the condition that the Fermi level in the energy state is relatively large to the value kT.

Formulae:

The density of occupied states, N0E=NEPE (i)

The probability of the condition that a particle will have energy E according to Fermi-Dirac statistics, PE=1eE-EF+1 (ii)

Step 3: a) Calculation of the condition of the equation that the number of holes in the valence band is equal to the number of electrons in the conduction band

The number of electrons occupying the valance band is given by substituting equation (ii) in equation (i) as follows:

Nev=NvPEv =NveEv-EFkT+1

Since there are a total of states in the valence band; the number of holes in the valence band is given by:

Nhv=Nve-Ev-EF/kT+1.................................(a)

Now, the number of electrons in the conduction band is given by substituting equation (ii) in equation (i) as follows:

Nec=NcPECNhv=Nce-EC-EF/kT+1.................................(b)

As we are given that Nhv=Nec

Thus, using equations (a) and (b), we get that

Nve-Ev-EF/kT+1=Nce-EC-EF/kT+1.....................................(c)NveEv-EF/kT+1=NceEC-EF/kT+1

where Ec-Ec-EF and Ev=-Ev-EF.

Step 4: b) Calculation of the further condition for Fermi level being relatively greater than kT

In this case,eEC-EF/kT 1 and e-EV-EF/kT 1

Thus, using this condition in the above equation (c), we get that

NVe-Ev-EF/kT =NVeEC-EF/kT eEv-Ec+2EF/kT=NV/NcEF=EC+EV2=kTInNV/Nc2

Hence, the above condition is proved.

Recommended explanations on Physics Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.