SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.8. DOPING OF SEMICONDUCTORS 61

Silicon

host atom

Pentavalent

donor impurity

=+

= Silicon-like + Electron-ion

All 4 outer

electrons go into

the valence band

EC

EV

——

— —

——

——

———

— —

— —

—

—

+

—

Figure 2.19: A schematic showing the approach we can take to understand donors in semicon-
ductors. The donor problem is treated as the host atom problem, together with a Coulombic
interaction term.

We have seen that electrons in the crystal can be represented by an effective mass near the
bandedge. We get the effective mass equation for the donor level which has an energy forEdof
[
−^2
2 m∗e

∇^2 −

e^2

4 πr

]

Fc(r)=(Ed−Ec)Fc(r) (2.8.2)

wherem∗eis the conduction bandedge mass andEd−Ecis the impurity energy with respect to
the conduction bandedgeEclevels.
This equation is now essentially the same as that of an electron in the hydrogen atom problem.
The only difference is that the electron mass ism∗and the Coulombic potential is reduced by
 0 /.
The energy solutions for this problem are

Ed=Ec−

e^4 m∗e

2(4π)^2 ^2

1

n^2

,n=1, 2 , ... (2.8.3)

A series of energy levels are produced, with the ground state energy level being at

Ed = Ec−

e^4 m∗e

2(4π)^2 ^2

= Ec− 13. 6

(

m∗

mo

)(

o



) 2

eV (2.8.4)