Electron Tunneling Probability

Classical Mechanics Level 2

Suppose an electron with energy $1 \text{ eV}$ encounters a one-dimensional barrier potential of height $3 \text{ eV}$ and width $1 \text{ nm}$ .

What is the probability that the electron tunnels through this barrier?

0

1.81 \times 10^{-6}

7.24 \times 10^{-4}

3.62 \times 10^{-3}

1 solution

Matt DeCross
May 10, 2016

Relevant wiki: Quantum Tunneling

The tunneling probability in terms of the energies is given by:

$T = \left(1+ \frac{V_0^2}{4E(V_0 -E)} \sinh^2 \left(\frac{L}{\hbar} \sqrt{2m(V_0 - E)}\right)\right)^{-1} .$

Plugging in numbers with $L = 1 \text{ nm}$ , $E = 1 \text{ eV}$ , and $V_0 = 3 \text{ eV}$ , using $m$ equal to the electron mass, obtains the result of $T = 1.81 \times 10^{-6}$ . The tunneling probability is very small, as expected given these numbers.

i must be doing something wrong, because i can't get this value from this equation... are the energies meant to be kept in electron volts including h bar?

Haroon Shami - 2 years, 8 months ago

You do need to be careful that the units of everything have been chosen to make everything inside the sinh^2 dimensionless.

Matt DeCross - 2 years, 7 months ago

how to compute sinh^2 in excel?

zul hamdi - 2 years ago

@Zul Hamdi – No idea if that's possible. I recommend using the WolframAlpha website if you don't have computing software.

Matt DeCross - 1 year, 10 months ago

Can anyone give complete solution?

Uma S - 2 years, 8 months ago

You should see the linked wiki for the derivation of the formula used, which is fairly extensive. After obtaining the formula, the problem is just to plug in the numbers to see the size of the effect for electrons.

Matt DeCross - 1 year, 10 months ago

Electron Tunneling Probability

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