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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2025-03-01
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| author | Jan Kučera Ulrich Wulf George Alexandru Nemnes |
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| description | 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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| institution | Kabale University |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-62d2e34a9d5a43e2b43b5b944b0c8b762025-08-20T03:43:25ZengMDPI AGMicromachines2072-666X2025-03-0116330610.3390/mi16030306Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current ConservationJan Kučera0Ulrich Wulf1George Alexandru Nemnes2Institute of Physics of the Czech Academy of Sciences, Cukrovanická 10/112, 16200 Praha, Czech RepublicFaculty 1, Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, Konrad-Wachsmann-Allee 13, 03046 Cottbus, GermanyFaculty of Physics, University of Bucharest, 077125 Magurele-Ilfov, RomaniaIn 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.https://www.mdpi.com/2072-666X/16/3/306S-matrixLandauer–Büttiker formalismN-pole deviceR-matrixcurrent conservation |
| spellingShingle | Jan Kučera Ulrich Wulf George Alexandru Nemnes Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation Micromachines S-matrix Landauer–Büttiker formalism N-pole device R-matrix current conservation |
| title | Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation |
| title_full | Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation |
| title_fullStr | Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation |
| title_full_unstemmed | Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation |
| title_short | Scattering Theory in an N-Pole Semiconductor Quantum Device: The Unitarity of the Current S-Matrix and Current Conservation |
| title_sort | scattering theory in an n pole semiconductor quantum device the unitarity of the current s matrix and current conservation |
| topic | S-matrix Landauer–Büttiker formalism N-pole device R-matrix current conservation |
| url | https://www.mdpi.com/2072-666X/16/3/306 |
| work_keys_str_mv | AT jankucera scatteringtheoryinannpolesemiconductorquantumdevicetheunitarityofthecurrentsmatrixandcurrentconservation AT ulrichwulf scatteringtheoryinannpolesemiconductorquantumdevicetheunitarityofthecurrentsmatrixandcurrentconservation AT georgealexandrunemnes scatteringtheoryinannpolesemiconductorquantumdevicetheunitarityofthecurrentsmatrixandcurrentconservation |