ECE Purdue Semiconductor Fundamentals L2.3: Quantum Mechanics - Tunneling and Reflection

Summary

This lecture delves into quantum mechanical tunneling and reflection in semiconductors, highlighting their significance in electron behavior when encountering potential barriers. Unlike classical particles, electrons can tunnel through barriers they cannot overcome classically, with probabilities that depend on barrier properties and electron mass. The lecture explains these phenomena using the wave equation and boundary conditions, offering insights into their impact on semiconductor devices. Tunneling, while often detrimental to device performance, can sometimes be harnessed for technological benefits, as seen in resonant tunneling and MOSFET designs. The session concludes by emphasizing the relevance of quantum effects in modern semiconductor technologies.

Takeaways

• 😀 Quantum tunneling and quantum reflection are important quantum mechanical phenomena that occur in semiconductors.
• 😀 Classically, an electron with energy less than the barrier height cannot pass through, but quantum mechanics allows tunneling even for electrons with lower energy.
• 😀 The probability of quantum tunneling depends on the thickness, height of the barrier, and the mass of the electron.
• 😀 Electrons in a semiconductor can behave as waves, meaning their wave function may reflect or transmit depending on the energy and the barrier characteristics.
• 😀 When an electron's energy is lower than the barrier's potential, the wave function exhibits exponential decay, indicating a possibility for tunneling.
• 😀 Transmission probability decreases exponentially with increasing barrier thickness, height, and the particle's effective mass.
• 😀 Quantum mechanical reflection occurs when electrons reflect off a barrier even if they have enough energy to pass over it.
• 😀 In semiconductor devices, tunneling can lead to performance degradation, such as increased gate leakage in MOSFETs due to very thin insulator layers.
• 😀 A very thin insulating layer (about 1.1-1.2 nm) in MOSFETs can cause tunneling, which affects the functionality and performance of modern transistors.
• 😀 Resonant tunneling can occur when two barriers are present, where interference effects lead to a perfect transmission at certain energies.
• 😀 To avoid quantum mechanical reflections, the potential must vary smoothly on the scale of the electron’s de Broglie wavelength (around 10 nm in silicon).

Q & A

What is quantum tunneling, and how does it differ from classical behavior?

Quantum tunneling occurs when an electron passes through a potential barrier even though its energy is lower than the barrier height. Classically, an electron with insufficient energy would be unable to overcome the barrier. However, quantum mechanically, the electron has a nonzero probability of tunneling through the barrier due to its wave-like nature.

What role does the thickness of the barrier play in quantum tunneling?

The probability of quantum tunneling decreases exponentially as the thickness of the barrier increases. A thicker barrier makes tunneling less likely, as the wavefunction of the electron decays more rapidly within the barrier.

How does the energy of an electron affect its transmission through a barrier?

If the energy of an electron is above the potential barrier, it can pass through with a probability of 1. If its energy is less than the barrier height, the electron may still tunnel through, but the probability of doing so decreases exponentially with the barrier's thickness and height.

What is the significance of the de Broglie wavelength in quantum mechanical reflections?

The de Broglie wavelength determines the scale at which quantum mechanical effects, such as reflection, become significant. To avoid reflections at barriers, the potential must change slowly over a distance comparable to the electron's de Broglie wavelength, typically around 10 nm in silicon.

What is the mathematical form of the wave equation when the energy is less than the barrier's potential?

When the energy is less than the barrier's potential, the wave equation leads to solutions in the form of exponentially growing or decaying functions. This results in a decaying wavefunction inside the barrier, indicating a reduced probability of finding the electron there.

What is the boundary condition that must be applied to the wave function at the interfaces of the barrier?

At the interfaces of the barrier, the wave function and its derivative must be continuous. This ensures that the probability density and current remain consistent across the boundary, preventing infinite values in the wavefunction or its derivative.

What is resonant tunneling, and how can it be used to advantage in semiconductor devices?

Resonant tunneling occurs when electrons pass through two potential barriers, with the probability of transmission increasing significantly at specific energy levels due to constructive interference between reflected waves. This phenomenon can be used in the design of quantum devices where control over tunneling behavior is needed.

How does the mass of the particle affect the tunneling probability?

The tunneling probability decreases exponentially as the mass of the particle increases. A larger mass results in a smaller de Broglie wavelength, which reduces the likelihood of tunneling through a given barrier.

Why is quantum tunneling a concern in modern MOSFET devices?

In modern MOSFETs, the insulating layer between the gate and the semiconductor has been made extremely thin to improve performance. However, as the layer thickness approaches the nanoscale, quantum tunneling causes leakage currents between the gate and the semiconductor, degrading device performance.

What are the key features of tunneling that engineers must consider when designing semiconductor devices?

Engineers must consider that tunneling probabilities decrease exponentially with the barrier's thickness, height, and the mass of the particle. They must also ensure that the potential changes smoothly to avoid quantum mechanical reflections, particularly as device sizes shrink and quantum effects become more pronounced.