Landauer-Büttiker formalism – Electricity

Explore the Landauer-Büttiker formalism, its applications in quantum transport, challenges, and a calculation example.

Introduction

The Landauer-Büttiker formalism offers a powerful theoretical framework for studying electronic transport in mesoscopic systems – structures that are intermediate in size between microscopic (atomic scale) and macroscopic (bulk) structures.

Landauer’s Principle

Proposed by Rolf Landauer in the 1950s, Landauer’s Principle fundamentally asserts that the conductance of a quantum conductor is related to the quantum of conductance and the transmission probabilities of the conductor’s electronic modes. The equation is symbolically represented as: G = G₀ ∑ T_n.

Büttiker’s Extension

Later, in the 1980s, Markus Büttiker extended Landauer’s formalism to multi-terminal conductors, thus making the formalism capable of describing not only conductance but also other transport properties, including shot noise and local density of states.

Impact and Importance

The Landauer-Büttiker formalism is a seminal contribution to the field of condensed matter physics, particularly in the realm of quantum transport in mesoscopic systems. It fundamentally changed the way we understand electrical conduction on a small scale.
It has found extensive use in modeling nano-electronic devices, including quantum dots and single-electron transistors, where quantum mechanical effects are prominent.
This formalism has also been instrumental in the study and development of quantum computing and information theory, specifically in the context of understanding information transfer and its associated dissipation in quantum systems.

Challenges and Criticism

Despite its broad applicability and success, the Landauer-Büttiker formalism is not without challenges. It assumes that the system under consideration is coherent and that the contacts to the leads do not alter the properties of the system. These assumptions may not hold true for all physical situations, particularly for larger systems or systems at higher temperatures.

Conclusion

In conclusion, the Landauer-Büttiker formalism provides a powerful tool for studying electronic transport in mesoscopic systems. Despite certain limitations, it has significantly deepened our understanding of quantum mechanics and propelled advancements in quantum technology.

Example of Calculation Using Landauer-Büttiker Formalism

Let’s consider a simple system of a two-terminal quantum point contact, where the transmission probability is unity (T_n = 1) for one conducting channel and zero for all others. We also consider the system to be at zero temperature.

Step 1: Apply Landauer’s Principle

Given that the quantum of conductance G₀ = 2e²/h, where e is the electron charge and h is Planck’s constant, and using Landauer’s formula G = G₀ ∑ T_n, we find the conductance of the quantum point contact to be G = 2e²/h.

Step 2: Apply Büttiker’s Extension

Büttiker’s extension relates the current I flowing from terminal ‘m’ to ‘n’ through the relationship: I_mn = (2e²/h) T_mn (V_m – V_n), where V_m and V_n are the voltages at terminals ‘m’ and ‘n’ respectively. Given that T_mn is unity and V_m = V while V_n = 0 (since we take ‘n’ as the ground terminal), we find the current I = 2e²/h * V.

Step 3: Understanding the Results

The conductance obtained in step 1 is the maximum conductance possible for a quantum point contact, signifying the perfect transmission of an electron.
The current obtained in step 2 exhibits a linear relationship with voltage V, as expected for Ohm’s law, showing that the quantum point contact behaves like a conventional resistor at this scale.

In conclusion, this example illustrates how the Landauer-Büttiker formalism can be used to calculate basic transport properties of a mesoscopic system, further emphasizing its role in understanding quantum transport phenomena.