Question

1. The potential energy difference between the tip of an electron scanning microscope and the surface is 2.09x10^-18 J. The energy of an electron is 1.02x10^-18 J. What length must the tip be from the surface for the probability of an electron tunneling from the tip to the surface is 0.75?

Answer 1

The probability of tunneling T is given by:

T = exp ( - 2GL)

which on rearranging gives:

L = - ln(T) / 2G

where ;

L = tunneling length

T = tunneling probability = 0.75

G = ( 2m (U - E) / h^2 )^0.5

where;

m = mass of electron = 9.1 X 10^-31 kg

U = potential energy barrier = 2.09 X 10^-18 J

E = energy of electron = 1.02 X 10^-18 J

h = reduced Planck's constant = 1.05 X 10^-34 J / s

Therefore;

G =  ( 2 X 9.1 X 10^-31 X (2.09 X 10^-18 - 1.02 X 10^-18) / (1.05 X 10^-34)^2 )^0.5

which gives;

G = 1.33 X 10^8

Substituting this in the equation;

L = - ln(T) / 2G

L = - ln(0.75) / ( 2 X 1.33 X 10^8)

L = 1.082 X 10^-9 m

L = 1.082 nm

which is the required length the tip must be from the surface.