The Quantum Tunneling Detector · AI Generator #GEN-C87C7F02

Physics · quantum-mechanics

Problem

A novel quantum device uses electron tunneling through a carefully engineered potential barrier to detect minute changes in electric fields. The device consists of two metallic electrodes separated by a thin insulating barrier of width a=2.0  nma = 2.0\;\text{nm} and height V0=8.0  eVV_0 = 8.0\;\text{eV}. When an external electric field is applied, it tilts the potential barrier, creating a triangular potential well on one side. When a uniform electric field E=5.0×108  V/mE = 5.0 \times 10^8\;\text{V/m} is applied across the device, electrons with kinetic energy K=6.0  eVK = 6.0\;\text{eV} approach the barrier from the left. Due to the applied field, the potential energy varies linearly across the barrier region according to V(x)=V0−eExV(x) = V_0 - eEx for 0≤x≤a0 \leq x \leq a, where x=0x = 0 is at the left edge of the barrier. (a) Find the position xcx_c within the barrier where the electron's total energy equals the local potential energy. (b) Using the WKB approximation, calculate the tunneling probability for an electron to pass through this tilted barrier. Express your answer as a numerical value. Given: ℏ=1.055×10−34  J⋅s\hbar = 1.055 \times 10^{-34}\;\text{J·s}, me=9.109×10−31  kgm_e = 9.109 \times 10^{-31}\;\text{kg}, e=1.602×10−19  Ce = 1.602 \times 10^{-19}\;\text{C}
Intuitive explanation

Quantum tunneling through a tilted barrier is fundamentally about finding where the electron's energy intersects the sloping potential wall—this classical turning point divides the classically forbidden region. The WKB approximation then lets you calculate the exponential suppression of tunneling by integrating the "penetration depth" across that forbidden zone, turning a hard quantum problem into manageable calculus. The key insight is that the tilted barrier creates an asymmetry: electrons can only tunnel through the narrower forbidden region, making the probability much higher than through a uniform barrier.

Intuitive explanation

Quantum tunneling through a tilted barrier is fundamentally about finding where the electron's energy intersects the sloping potential wall—this classical turning point divides the classically forbidden region. The WKB approximation then lets you calculate the exponential suppression of tunneling by integrating the "penetration depth" across that forbidden zone, turning a hard quantum problem into manageable calculus. The key insight is that the tilted barrier creates an asymmetry: electrons can only tunnel through the narrower forbidden region, making the probability much higher than through a uniform barrier.

Related theory

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