Scanning Tunneling Microscope

(a) What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a Scanning Tunneling Microscope probe if the work function is 4.0 eV?
(b) The probe passes over an atom that is 0.050 nm ”tall.” By what factor does the tunneling current increase?
(c) If a 10% current change is reliably detectable, what is the smallest height change the STM can detect?

Answers
Basically the electron is described by a wave function psi. Since it needs to go through a barrier of potential the wave function will decrease exponentially in the barrier with the barrier thickness x.
$Psi(x) = Psi(0)*exp(-k x)$

$k$  is the wave number so that ($k x)$ is non-dimensional.
$k =1/\lambda = P/\hbar= \sqrt(2*m*E)/ \hbar$
E is the work function, m is the electron mass
Thus
$k = \sqrt(2*9.1*10^{-31}*4*1.6*10^{-19})/10^{-34} =10^{10} m^{-1}$

The probability of tunneling is just
$P = |\psi(x)/\psi(0) |^2 = exp(-2*k*x) = exp(-2*10^{10}*5*10^{-10}) =exp(-10) =4.54*10^{-5}$

b) The probability of tunneling in this case is again
$P2 = exp(-2*k*(x+d x)) = exp(-2*(5-0.5)) =exp(-9)$
$P2/P =exp(-9)/exp(-10) = exp(1) = e^1 =2.718$
The tunneling current increases with the same factor as the probability does (2.718)

c)
$I2/I1 = 1.1$
$exp(-2*k*x2)/exp(-2*k*x) = 1.1$
$exp(-2*k*(x-x2)) =1.1$
$\Delta(x) = x2-x =ln(1.1)/(2k) =ln(1.1)/(2*10^10) =4.765*10^{-12} m =0.00477 nm$