The probe passes over an atom that is 0.050 nm “tall.” By what factor does the tunneling current increase?

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Step 1: Understand the concept of tunneling current. Tunneling current depends exponentially on the distance between the probe and the surface. The relationship can be expressed as \( I \propto e^{-2 \kappa d} \), where \( I \) is the tunneling current, \( d \) is the distance, and \( \kappa \) is a constant related to the material properties and the energy barrier.

Step 2: Identify the change in distance. The atom is 0.050 nm tall, which means the distance between the probe and the surface decreases by 0.050 nm when the probe passes over the atom.

Step 3: Write the ratio of tunneling currents. The tunneling current increases by a factor \( \frac{I_{new}}{I_{old}} = e^{2 \kappa \Delta d} \), where \( \Delta d \) is the change in distance (0.050 nm).

Step 4: Substitute \( \Delta d = 0.050 \) nm into the formula. The factor of increase in tunneling current is \( e^{2 \kappa \times 0.050} \). Note that \( \kappa \) depends on the material and energy barrier, but its exact value is not provided in the problem.

Step 5: Conclude that the tunneling current increases exponentially by the factor \( e^{2 \kappa \times 0.050} \). To find the exact numerical factor, you would need the value of \( \kappa \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Tunneling

Quantum tunneling is a phenomenon in quantum mechanics where a particle can pass through a potential energy barrier that it classically shouldn't be able to surmount. This occurs due to the wave-like properties of particles, allowing them to exist in a superposition of states. In the context of tunneling current, it refers to the flow of electrons through a barrier, which can be influenced by the height and width of the barrier.

Tunneling Current

Tunneling current is the electric current that results from quantum tunneling, particularly in systems like scanning tunneling microscopes (STM). It is dependent on the distance between the probe and the surface, as well as the energy barrier presented by the atom. As the probe approaches the atom, the tunneling probability increases, leading to a higher tunneling current.

Exponential Dependence

In quantum mechanics, the tunneling current exhibits an exponential dependence on the distance between the tunneling probe and the barrier. This means that even a small change in distance can lead to a significant change in current. The relationship is often described by the equation I ∝ e^(-2κd), where I is the tunneling current, κ is a constant related to the barrier height, and d is the distance.

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What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a STM probe if the work function is 4.0 eV?

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