The Bohr model of the Hydrogen atom provides the energy levels of electrons as w

Question

The Bohr model of the Hydrogen atom provides the energy levels of electrons as well
as the radius of the electron orbit at each energy level.

a. Apply this model to shallow lying donor levels in a semiconductor and find the activation energy, ED, given that the relative dielectric constant is 12 and the effective mass m* = 0.3 m0, where m0 is the free electron mass.

b. Find the radius of the electron orbit in the ground state of the donor and compare this value to the spacing of a unit cell in the crystal.

Explanation / Answer

m* = 0.3 m0, er = 12
a)    By Bohr Model
   En = -(k*e^2/er)^2 * m* * 4 * pi^2/(hn)^2 = -13.6*(m*/m0)/(er*n)^2
   For shallow lying donor levels n is small, so let n=1
   E1 = -13.6*0.3/144 at n=1
   E1 = -0.028 eV
b)   r1 = (h/2*pi)^2*er/(k * e^2 * (m*/m0))
   = 5.29 * 10^-11 * 12/0.3 m
   = 2.12 * 10^-9 m      
   Unit cell of Silicon is 5.43 * 10^-10 m
   Radius found by this method is significantly larger than the spacing of unit cell of silicon

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