Transmission probability for a rectangular barrier:
\[T = \left(1 + \frac{V_0^2 \sinh^2(kL)}{4E(V_0-E)}\right)^{-1}\]
Where the wave number \(k\) in the barrier is:
\[k = \frac{\sqrt{2m(V_0-E)}}{\hbar}\]
For \(E < V_0\), quantum tunneling occurs, allowing particles to pass through the barrier.
Quantum tunneling is a quantum mechanical phenomenon where particles penetrate through potential energy barriers that would be impossible to overcome according to classical physics.
The time-independent Schrödinger equation for a particle of mass \(m\) in a potential \(V(x)\):
\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\]
For a rectangular barrier of height \(V_0\) and width \(L\), the solutions are:
Region I (before barrier):
\[\psi_I(x) = A e^{ik_1x} + B e^{-ik_1x}\] \[k_1 = \frac{\sqrt{2mE}}{\hbar}\]
Region II (inside barrier, for \(E < V_0\)):
\[\psi_{II}(x) = C e^{-k_2x} + D e^{k_2x}\] \[k_2 = \frac{\sqrt{2m(V_0-E)}}{\hbar}\]
Region III (after barrier):
\[\psi_{III}(x) = F e^{ik_1x}\]
The transmission coefficient is then calculated by matching the wave functions and their derivatives at the boundaries.
These devices utilize quantum tunneling to create negative differential resistance regions in their I-V characteristics, enabling high-frequency oscillators and fast switching.
STM uses the tunneling current between a sharp tip and a sample surface to image individual atoms with sub-angstrom precision.
Modern flash memory devices use Fowler-Nordheim tunneling or direct tunneling to program and erase floating-gate transistors.
TFETs utilize band-to-band tunneling to achieve subthreshold swing below the thermal limit of 60 mV/decade, enabling ultra-low power operation.
Tunneling is exploited in various quantum computing architectures, including Josephson junction-based superconducting qubits.
Coulomb blockade and single-electron tunneling effects enable precise control of individual electrons, forming the basis for single-electron transistors and pumps.