Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation
In a number of previous publications, scattering theory for N-pole semiconductor quantum devices was developed. In the framework of the Landauer–Büttiker formalism, an S-matrix was constructed with the aid of an R-matrix providing a mapping of the in-going waves onto the out-going waves. These waves...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
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| Series: | Micromachines |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2072-666X/16/3/306 |
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| Summary: | In a number of previous publications, scattering theory for N-pole semiconductor quantum devices was developed. In the framework of the Landauer–Büttiker formalism, an S-matrix was constructed with the aid of an R-matrix providing a mapping of the in-going waves onto the out-going waves. These waves include propagating waves and evanescent waves, the latter of which decay exponentially in the leads which are connected to the active region of the N-pole device. In order to formulate the current conservation in the N-pole device, it is necessary to define the current S-matrix schematically as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>S</mi><mo>˜</mo></mover><mo>=</mo><msup><mi>k</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>S</mi><msup><mi>k</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>, where <i>k</i> contains the information about the k-vectors of the mentioned in- and out-going waves. In this paper, we show how the complete current S-matrix is calculated including the coupling between the propagating and evanescent components and coupling to the bound states in the active device region. One then finds a sub-matrix of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>S</mi><mo>˜</mo></mover></semantics></math></inline-formula> which is unitary and which is restricted to the space of the propagating components. We demonstrate that current conservation is associated with the unitarity just of this sub-matrix. |
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| ISSN: | 2072-666X |