## Tunneling through square barrier

An electron accelerated through a potential difference of $E=1.5 eV$ approaches a potential barrier having a height of $U = 3 eV$ and width of $d =2 nm$. Write down the continuity equations for the wave function in the regions 1, 2 and 3 and find the probability of tunneling (transmission probability).

The continuity conditions at interface 1 are
$\psi_1(0)=\psi_2(0)$   and $\frac{d\psi_1(0)}{d x}=\frac {d\psi_2(0)} {d x}$
At interface 2 the continuity conditions are

$\psi_2(d)=\psi_3(d)$   and $\frac{d\psi_2(d)}{d x}=\frac {d\psi_3(d)} {d x}$

The tunneling probability is just (from continuity conditions):
$P=|\psi_3(d)/\psi_1(0)|^2=|(\psi_2(d))/\psi_2(0))|^2=exp{(-2*k_2*d)}$
Since the wave function is decreasing exponential in the barrier. K2 is the wave number in the barrier region 2.
$k_2 = 2\pi/\lambda_2 =(2\pi p_2)/h=\sqrt{(2m(U-E))}/\hbar$
U is the barrier height, E is the total energy.
$P = exp{{-2[\sqrt{2m(U-E)}/\hbar]*d}}$
For U =3 eV, E =1.5 eV and d =2 nm for an electron we have $P=1.31*10^{-11}$

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