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CH274 - Electrons in Solids and Materials

Exercise 4.1 – n-doping

Model answer

(a)

In Bohr’s model, the electronic ground state of a hydrogen atom is given by

m e4  

H            8h20(2)

The donor binding energy can be obtained from EH, provided the electron mass is substituted by the electron effective mass in Si and the dielectric constant in vacuum by the Si dielectric constant. This results in

E   = 一   me(*)e4        = E   me(*)  1   = -42.5 meV,

d               8h220(2)            H  me  2

where the values for Si me(*)  = 0.43me and  = 11.7 have been used.

(b)

The radius of the electron’s ground state in Bohr’s model is given by

rH  =  = 5.3  10- 11 m = 0.53 Å .

The radius of the donor electron orbit can be obtained from this expression, provided the electron mass is substituted by the electron effective mass in Si and the dielectric constant in vacuum by the Si dielectric constant. This results in

rd  =  = rH   = 1.44 10-9 m = 14.4 Å .

(c)

The number density of Si can be obtained from its crystal structure as

  = 5.0 10-2 Å-3 .

As a consequence, in a sphere of radius 14.4 Å there will be

 (14 .4)3  5.0 10-2 ~ 625 Si atoms.

We note that this corresponds to a significantly delocalised electron.