Deriving the Landauer Formula: Source & Drain Connected by Wire

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In summary: This is an important step in deriving the Landauer formula for this type of system.In summary, when a voltage is applied to a source and drain connected by a wire, the Fermi functions for each region will be different due to the difference in their chemical potentials. This causes electrons to flow from the source to the drain, resulting in a current. The current is calculated as an integral of the Fermi functions, taking into account the flow of electrons between the two regions. This is a crucial step in deriving the Landauer formula for this system.
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Consider a source and a drain connected by a wire. When a voltage is applied the chemical potential of source and drain are shifted relative to each such that:
μ_source = μ_drain + eV
now, one crucial step in deriving the Landauer formula for a system like this is to, as indicated in the link http://www.gianlucafiori.org/qpc/node7.html, to realize that the fermi functions for source and drain are different and that the current going to the right and left respectively are given as integrals of the fermi function. Now, why is this exactly? It is not like the source has more electrons than the drain. Is it because they are moving faster?
 
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The reason why the Fermi functions for source and drain are different is because of the difference in their chemical potentials. When a voltage is applied, electrons will flow from the source to the drain in an attempt to equalize the chemical potentials of the two regions. The Fermi function describes the probability that an electron will be present at a certain energy level in a given region, so the difference in chemical potentials between the source and the drain will cause the Fermi functions of the two regions to be different. This difference in Fermi functions means that there will be more electrons present in the source than in the drain, which is the source of the current. The current is then given as an integral of the Fermi functions because this integral takes into account all of the electrons flowing from the source to the drain.
 

1. What is the Landauer formula?

The Landauer formula is a mathematical equation used to calculate the conductance of a system connected to two electrodes, such as a source and drain in a wire. It is often used to understand the flow of electrons in electronic devices.

2. How is the Landauer formula derived?

The Landauer formula is derived from the Landauer-Büttiker theory, which states that the conductance of a system is determined by the number of available electronic states at the Fermi level. The formula takes into account the energy of the electrons, the temperature, and the transmission probability of the system.

3. What is the significance of the source and drain being connected by a wire in the Landauer formula?

The source and drain are connected by a wire in the Landauer formula because the formula is specifically used to calculate the conductance of a system with two electrodes. The wire represents the channel through which electrons flow from the source to the drain.

4. How does the Landauer formula relate to quantum mechanics?

The Landauer formula is based on the principles of quantum mechanics, specifically the concept of electron transmission through a system. It takes into account the wave-like behavior of electrons and the probability of them passing through a system at a given energy level.

5. What are some practical applications of the Landauer formula?

The Landauer formula is commonly used in the field of nanoelectronics to understand and predict the conductance of electronic devices, such as transistors and diodes. It is also used in the study of quantum computing and in the development of new technologies for energy-efficient electronics.

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