{"id":145,"date":"2018-12-06T09:58:39","date_gmt":"2018-12-06T09:58:39","guid":{"rendered":"http:\/\/msp08.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=145"},"modified":"2018-12-06T10:40:41","modified_gmt":"2018-12-06T10:40:41","slug":"landauer-buttiker-formalism-for-conduction-in-confined-structures","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/chapter\/landauer-buttiker-formalism-for-conduction-in-confined-structures\/","title":{"rendered":"Landauer-B\u00fcttiker Formalism for Conduction in Confined Structures"},"content":{"raw":"
\r\n
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

This module provides a description of Landauer approach, which is an important technique in the physics of small systems. The Landauer-B\u00fcttiker approach considers electrons transport as a transmission problem at Fermi level, where ohmic contacts are considered as reservoirs to inject and collect current, wherein the inelastic scattering is assumed to be taking place exclusively. The measured conductances can then be described as functions of transmission probabilities at the Fermi level between the reservoirs. The zero-field conductance quantization in a perfect 1D conductor, and the magnetic field assisted smooth transition to the quantum Hall effect, directly follow from the fact that a reservoir in equilibrium injects a current which is shared equally by all the propagating modes (which may be 1D subbands or magnetic edge channels). Second significant assumption considers the system to be connected to the reservoirs via perfect quantum wires (or nanowires) that behave as waveguides for the electron waves.<\/p>\r\n \r\n\r\nLandauer-B\u00fcttiker formalism<\/strong>\r\n\r\n \r\n

The Landauer formula named after Rolf Landauer, is a relation between the electrical resistance of a quantum conductor and its scattering properties.<\/p>\r\n \r\n\r\nConduction in a macroscopic conductor is given by the Ohm\u2019s law:\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n

here ? is cross-section area of a conductor of length \u2018?\u2019 having the conductivity of ?. From this relation ? \u2192 0 as ? \u2192 0 implying that the conductance vanishes as the conductor becomes narrower. Also, ? \u2192 \u221e for very small conductors (as ? \u2192 0). Such behavior is based on the simple assumption that conductivity of a conductor does not rely on its dimensions since it is the macroscopically described quantity, and is imagined to be uniform over the length of the conductor. As the dimensions of the conductor approach atomic scales, this uniformity in the conductivity is disturbed, and Ohm\u2019s law ceases to apply. This failure in Ohm\u2019s law has been observed in experiments on quantum point contacts and the atomic-sized wires, wherein a staircase type of behavior has been seen in conductance during the narrowing of point contact and the elongation of wire, respectively.<\/p>\r\n \r\n

One significant challenge is defining the characteristic length (dimensions) of the conductor at which\u00a0<\/span>macroscopic explanation fails. The significant <\/span>length scales deciding this dimension are \u2013 (a) de Broglie\u00a0<\/span>wavelength associated with kinetic energy of electrons; (b) mean free <\/span>path, or average distance traveled\u00a0<\/span>by the electron before altering the momentum, for example, scattering with impurities leads to <\/span>change in\u00a0<\/span>momentum; and (c) phase-relaxation length, or average distance traveled by the electron before\u00a0<\/span>destroying its phase. The phase loss occurs because of inelastic scattering attributable to electron-electron\u00a0<\/span>and electron-phonon interactions (Figure 1).<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 1 Schematic representation of the phase randomization of the electrons. L<\/em><\/strong>m<\/em><\/strong> denotes the ballistic transport limit and L<\/em><\/strong>\u03c6<\/em><\/strong> represents the coherent transport limit.<\/strong><\/p>\r\n \r\n

A conductor demonstrates ohmic behavior when its size\/dimension is sufficiently bigger than these lengths. This characteristic length depends on the type of material, and get affected by magnetic field or temperature. If dimensions of given conductor are in between the microscopic and macroscopic, it is termed mesoscopic conductor with a dimension of few nanometres. Microscopic conductors are of nanometre size, and are often termed as atomic-sized conductors (Figure 2<\/strong>).<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n\r\n \r\n

Figure 2 Schematic representation of an atomic wire. An electron coming in from left lead is\u00a0<\/strong>partially transmitted by the right lead. The electronic transport is described by the <\/strong>quantummechanical\u00a0transmission probability amplitudes <\/strong>tn,m, and can be computed at Fermi level for low<\/strong>\r\napplied voltages (V). The conductance of wire ? = ??\/?? can be described in terms of normalized<\/strong>\r\ntransmission probability amplitudes <\/strong>tn,m by \u2018Landauer-Buttiker\u2019 relation.<\/strong><\/p>\r\n \r\n

Since the phase-relaxation length in atomic-scaled and mesoscopic conductors is greater than their dimensions and therefore the electron transport may be considered as coherent implying that full quantum-mechanical consideration is needed for satisfactory characterization of the electron transport in these conductors. Landauer first suggested the quantum-mechanical treatment of the electron propagation in small conductors and gave a simple formula correlating electron transmission probability and conductance in 1D conductors. Buttiker further extended Landauer formula to multi-probe devices.<\/p>\r\n \r\n

The Landauer- B\u00fcttiker approach is based on the quantum-mechanical scattering technique to calculate electrical conductance in very small devices. For an elementary description of quantum conduction effects, 1D mesoscopic semiconductor structure such as quantum wires are ideal candidates (Figure 2). If the length of the wire is less than the electronic mean free path, there is no scattering and the transport is ballistic. Let us assume (Figure 2) that the 1D quantum wire is connected by ideal leads, which do not cause scattering, to the reservoirs (leads) having Fermi levels ??1 and ??2. Suppose a voltage ? is applied across the leads. Therefore, the potential energy, between two leads is, ??= ??1 \u2212 ??2. Noticeably, at low bias voltages (?), only the electrons near the Fermi energy contribute to the current in linear response region. The bias voltage is assumed to be equal to the difference in the chemical potentials of the left and right electrodes. The current across the wire is determined by the product of the concentration of electrons (calculated from DoS ?1?(?), in energy difference ??), the electron velocity ?(?), and the unit electronic charge as:<\/p>\r\n \r\n\r\n? = ??1?<\/sub>(?)?(?)??\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n \r\n

\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nWhich interestingly, does not depend on the velocity of the carriers. The conductance (? = ?\/?) becomes:\r\n\r\n\"\"\r\n

Note that the conductance in quantum wires does not depend on the wire length. This is contrary to macroscopic wires wherein conductance scales inversely with the length of the conductor.<\/p>\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

R can be determined experimentally. Additionally, the quantity 2?2<\/sup>\/\u210e is generally called the fundamental conductance.<\/p>\r\n \r\n

The second generalization involves the energy subbands in low-dimensional semiconductors. Higher order subbands than those corresponding to the first quantization level can participate in transport if the either the concentration of the electrons or their energy are considerably high.<\/p>\r\n \r\n\r\nLandauer formula for coherent transport with many channels<\/em><\/strong>\r\n\r\n \r\n\r\nTwo leads with many channels or subbands<\/em><\/span>\r\n\r\n \r\n\r\nFigure 3 <\/strong>shows a system with extensions beyond one-dimension. There are two leads with scattering centre positioned between them, with each lead having several subbands arising from transverse states. These subbands are also called modes or channels.\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

Figure 3 Coherent transport through a system with two leads, each with multiple propagating states.<\/strong><\/p>\r\n \r\n

It is essential that the transverse potential of each lead remains constant along its length. This is part of the definition of a perfect lead, and any regions with variable potential must be considered as part of a scattering <\/span>centre. The wave function within the lead can be written as:<\/span><\/p>\r\n\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

A matrix ? can be constructed from the coefficients ???<\/sub> in above equation, giving the transmission magnitude for an electron incident from the left in mode ? to be transmitted on the right in mode ?. We restrict ? to propagating states, giving dimensions of ????\u210e?<\/sub> \u00d7 ?????.<\/sub><\/p>\r\n \r\n

One reason of using ? is that it contains sufficient information to find the conductance. Suppose electrons are injected in the mode ?. The electrons emerging in mode ? contribute (2?2<\/sup>\/\u210e)|???<\/sub>|2<\/sup> to the conductance. The velocity of different modes is taken into account by the normalization and does not clutter this result. The total conductance is obtained by summing over all input and output modes:<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

This relation describes the conductance ? = ??\/?? as a sum over the normalized transmission probability amplitudes ??,? between all right going modes n and m in the left and right leads (probes), respectively. The factor of \u201c2\u201d in this relation is due to the electron spin degeneracy. When the conductor is connected with semi-infinite ideal crystalline leads, the modes can be expressed as Bloch functions (these are the plane waves commonly used to model metallic electrodes). The conductance can be written in <\/span>more compact form by using the Hermitian-conjugate matrix of ?, defined by (?\u2a25)?? = (???)\u2217. Then<\/span><\/p>\r\n\r\n

\r\n\r\n\"\"\r\n

This is usually quoted result, where \u2018??\u2019 is the trace of the matrix (the sum of its diagonal elements). Also, the product ??\u2a25 is square, although neither ? nor ?\u2a25 need be, and that the two expressions for the trace are equal even when ??\u2a25 and ?\u2a25? may not be of the same size.<\/p>\r\n \r\n

The Landauer- B\u00fcttiker formalism takes into consideration a single particle picture of the electronic transport. In other words, it assumes electrons as non-interacting entities. However, the electron-electron interactions can be partially considered in a mean field approach, where the transmission amplitudes can be determined from the single-particle equation as:<\/p>\r\n\"\"\r\n

Here, the effective potential ????<\/sub> (?) accounts for the interactions of an electron with the average field of other electrons and nuclei.<\/p>\r\n \r\n\r\nMany leads with many channels or subbands<\/em><\/span>\r\n\r\n<\/div>\r\n\"\"<\/em>\r\n

\r\n

Figure 4 Geometry of a sample for coherent transport with many leads. The general case is shown in (a) with specific examples of (b) a T-junction, (c) four-terminal measurement of longitudinal resistance, and (d) a microscopic Hall bar.<\/strong><\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

All expressions for current involve similar sums over the modes in both leads. It is similar to the trace (calculated above) with the additional subscripts on transmission coefficients to label the leads. The transmission and reflection coefficients which absorb these traces are given as:<\/p>\r\n\"\"\r\n\r\n \r\n

The net current injected into lead ? is given by the incident current minus the reflected current. The total\r\nincident current, summed over all propagating modes is (2?2<\/sup>\/\u210e)??<\/sub>??<\/sub>, thus the net current is:<\/p>\r\n \r\n

???<\/sub> = (2?2<\/sup>\/\u210e)(??<\/sub> \u2212 ??<\/sub>)??<\/sub><\/p>\r\n \r\n

Conservation of current requires that this be equal to the total current injected from lead ? that leaves the\r\nsample though other leads, i.e., ???<\/sub> = \u03a3?,?\u2260?<\/sub> ???<\/sub>. This leads to a sum rule on the transmission coefficients:<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n\r\n \r\n\r\nWhich is a simple generalization of ? + ? = 1.\r\n\r\n \r\n

We have now calculated the currents due to electrons injected from lead ?. Now the summation of all leads\u2019 contributions is required. Lead ? causes a current \u2013 (2?2<\/sup>\/\u210e)???<\/sub>??<\/sub> in lead ?, so the total current in lead ? is:<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

This is the Landauer- B\u00fcttiker formula for the conductance of a system with many leads. It can also be written in terms of a square conductance matrix whose dimension is given by the number of leads:<\/p>\r\n\"\"\r\n\r\n \r\n\r\nApplications of Landauer-B\u00fcttiker formula<\/em><\/strong>\r\n

    \r\n \t
  1. Quantum point contact for a 2DEG<\/li>\r\n \t
  2. Electronic transport in single-molecule junctions<\/li>\r\n \t
  3. Longitudinal electronic transport in carbon nanotubes, and 2D graphene<\/li>\r\n \t
  4. Electronic transport in nanoparticles and chains of nanoparticles<\/li>\r\n \t
  5. Multiterminal quantum point contacts<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n
    you can view video on Landauer-B\u00fcttiker Formalism for Conduction in Confined Structures<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0 \u00a0<\/strong>References & Suggested Reading:<\/strong>\r\n
      \r\n \t
    1. T. Pradeep, A Textbook of \u201cNanoscience and Nanotechnology\u201d, McGraw Hill Education,\u00a0<\/span>2012.<\/li>\r\n \t
    2. https:\/\/nanoyou.eu\/<\/li>\r\n \t
    3. https:\/\/en.wikipedia.org\/wiki\/Ballistic_conduction<\/li>\r\n \t
    4. Supriyo Datta (1997). Electronic Transport in Mesoscopic Systems. Haroon Ahmad, Alec Broers, Michael Pepper. New York: Cambridge University Press. pp. 57\u2013111.<\/li>\r\n \t
    5. www2.ece.rochester.edu\/projects\/bdt\/files\/Landauer_buttiker_formalism<\/li>\r\n \t
    6. http:\/\/tfp1.physik.uni-freiburg.de\/teaching\/Seminar2010\/talks\/LandauerFormula_Praesentation110510.pdf<\/li>\r\n \t
    7. http:\/\/www.phy.bme.hu\/~ujsaghy\/StudentSeminar\/1.pdf<\/li>\r\n \t
    8. https:\/\/hal.archives-ouvertes.fr\/hal-00943060\/document<\/li>\r\n<\/ol>","rendered":"
      \n
      \"epgp<\/a>
      \n<\/span><\/div>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      This module provides a description of Landauer approach, which is an important technique in the physics of small systems. The Landauer-B\u00fcttiker approach considers electrons transport as a transmission problem at Fermi level, where ohmic contacts are considered as reservoirs to inject and collect current, wherein the inelastic scattering is assumed to be taking place exclusively. The measured conductances can then be described as functions of transmission probabilities at the Fermi level between the reservoirs. The zero-field conductance quantization in a perfect 1D conductor, and the magnetic field assisted smooth transition to the quantum Hall effect, directly follow from the fact that a reservoir in equilibrium injects a current which is shared equally by all the propagating modes (which may be 1D subbands or magnetic edge channels). Second significant assumption considers the system to be connected to the reservoirs via perfect quantum wires (or nanowires) that behave as waveguides for the electron waves.<\/p>\n

       <\/p>\n

      Landauer-B\u00fcttiker formalism<\/strong><\/p>\n

       <\/p>\n

      The Landauer formula named after Rolf Landauer, is a relation between the electrical resistance of a quantum conductor and its scattering properties.<\/p>\n

       <\/p>\n

      Conduction in a macroscopic conductor is given by the Ohm\u2019s law:<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      here ? is cross-section area of a conductor of length \u2018?\u2019 having the conductivity of ?. From this relation ? \u2192 0 as ? \u2192 0 implying that the conductance vanishes as the conductor becomes narrower. Also, ? \u2192 \u221e for very small conductors (as ? \u2192 0). Such behavior is based on the simple assumption that conductivity of a conductor does not rely on its dimensions since it is the macroscopically described quantity, and is imagined to be uniform over the length of the conductor. As the dimensions of the conductor approach atomic scales, this uniformity in the conductivity is disturbed, and Ohm\u2019s law ceases to apply. This failure in Ohm\u2019s law has been observed in experiments on quantum point contacts and the atomic-sized wires, wherein a staircase type of behavior has been seen in conductance during the narrowing of point contact and the elongation of wire, respectively.<\/p>\n

       <\/p>\n

      One significant challenge is defining the characteristic length (dimensions) of the conductor at which\u00a0<\/span>macroscopic explanation fails. The significant <\/span>length scales deciding this dimension are \u2013 (a) de Broglie\u00a0<\/span>wavelength associated with kinetic energy of electrons; (b) mean free <\/span>path, or average distance traveled\u00a0<\/span>by the electron before altering the momentum, for example, scattering with impurities leads to <\/span>change in\u00a0<\/span>momentum; and (c) phase-relaxation length, or average distance traveled by the electron before\u00a0<\/span>destroying its phase. The phase loss occurs because of inelastic scattering attributable to electron-electron\u00a0<\/span>and electron-phonon interactions (Figure 1).<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Figure 1 Schematic representation of the phase randomization of the electrons. L<\/em><\/strong>m<\/em><\/strong> denotes the ballistic transport limit and L<\/em><\/strong>\u03c6<\/em><\/strong> represents the coherent transport limit.<\/strong><\/p>\n

       <\/p>\n

      A conductor demonstrates ohmic behavior when its size\/dimension is sufficiently bigger than these lengths. This characteristic length depends on the type of material, and get affected by magnetic field or temperature. If dimensions of given conductor are in between the microscopic and macroscopic, it is termed mesoscopic conductor with a dimension of few nanometres. Microscopic conductors are of nanometre size, and are often termed as atomic-sized conductors (Figure 2<\/strong>).<\/p>\n<\/div>\n

      \"\"<\/p>\n

      \n

       <\/p>\n

      Figure 2 Schematic representation of an atomic wire. An electron coming in from left lead is\u00a0<\/strong>partially transmitted by the right lead. The electronic transport is described by the <\/strong>quantummechanical\u00a0transmission probability amplitudes <\/strong>tn,m, and can be computed at Fermi level for low<\/strong>
      \napplied voltages (V). The conductance of wire ? = ??\/?? can be described in terms of normalized<\/strong>
      \ntransmission probability amplitudes <\/strong>tn,m by \u2018Landauer-Buttiker\u2019 relation.<\/strong><\/p>\n

       <\/p>\n

      Since the phase-relaxation length in atomic-scaled and mesoscopic conductors is greater than their dimensions and therefore the electron transport may be considered as coherent implying that full quantum-mechanical consideration is needed for satisfactory characterization of the electron transport in these conductors. Landauer first suggested the quantum-mechanical treatment of the electron propagation in small conductors and gave a simple formula correlating electron transmission probability and conductance in 1D conductors. Buttiker further extended Landauer formula to multi-probe devices.<\/p>\n

       <\/p>\n

      The Landauer- B\u00fcttiker approach is based on the quantum-mechanical scattering technique to calculate electrical conductance in very small devices. For an elementary description of quantum conduction effects, 1D mesoscopic semiconductor structure such as quantum wires are ideal candidates (Figure 2). If the length of the wire is less than the electronic mean free path, there is no scattering and the transport is ballistic. Let us assume (Figure 2) that the 1D quantum wire is connected by ideal leads, which do not cause scattering, to the reservoirs (leads) having Fermi levels ??1 and ??2. Suppose a voltage ? is applied across the leads. Therefore, the potential energy, between two leads is, ??= ??1 \u2212 ??2. Noticeably, at low bias voltages (?), only the electrons near the Fermi energy contribute to the current in linear response region. The bias voltage is assumed to be equal to the difference in the chemical potentials of the left and right electrodes. The current across the wire is determined by the product of the concentration of electrons (calculated from DoS ?1?(?), in energy difference ??), the electron velocity ?(?), and the unit electronic charge as:<\/p>\n

       <\/p>\n

      ? = ??1?<\/sub>(?)?(?)??<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

       <\/p>\n

      \n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Which interestingly, does not depend on the velocity of the carriers. The conductance (? = ?\/?) becomes:<\/p>\n

      \"\"<\/p>\n

      Note that the conductance in quantum wires does not depend on the wire length. This is contrary to macroscopic wires wherein conductance scales inversely with the length of the conductor.<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      R can be determined experimentally. Additionally, the quantity 2?2<\/sup>\/\u210e is generally called the fundamental conductance.<\/p>\n

       <\/p>\n

      The second generalization involves the energy subbands in low-dimensional semiconductors. Higher order subbands than those corresponding to the first quantization level can participate in transport if the either the concentration of the electrons or their energy are considerably high.<\/p>\n

       <\/p>\n

      Landauer formula for coherent transport with many channels<\/em><\/strong><\/p>\n

       <\/p>\n

      Two leads with many channels or subbands<\/em><\/span><\/p>\n

       <\/p>\n

      Figure 3 <\/strong>shows a system with extensions beyond one-dimension. There are two leads with scattering centre positioned between them, with each lead having several subbands arising from transverse states. These subbands are also called modes or channels.<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      Figure 3 Coherent transport through a system with two leads, each with multiple propagating states.<\/strong><\/p>\n

       <\/p>\n

      It is essential that the transverse potential of each lead remains constant along its length. This is part of the definition of a perfect lead, and any regions with variable potential must be considered as part of a scattering <\/span>centre. The wave function within the lead can be written as:<\/span><\/p>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      A matrix ? can be constructed from the coefficients ???<\/sub> in above equation, giving the transmission magnitude for an electron incident from the left in mode ? to be transmitted on the right in mode ?. We restrict ? to propagating states, giving dimensions of ????\u210e?<\/sub> \u00d7 ?????.<\/sub><\/p>\n

       <\/p>\n

      One reason of using ? is that it contains sufficient information to find the conductance. Suppose electrons are injected in the mode ?. The electrons emerging in mode ? contribute (2?2<\/sup>\/\u210e)|???<\/sub>|2<\/sup> to the conductance. The velocity of different modes is taken into account by the normalization and does not clutter this result. The total conductance is obtained by summing over all input and output modes:<\/p>\n<\/div>\n

      \"\"<\/p>\n

      This relation describes the conductance ? = ??\/?? as a sum over the normalized transmission probability amplitudes ??,? between all right going modes n and m in the left and right leads (probes), respectively. The factor of \u201c2\u201d in this relation is due to the electron spin degeneracy. When the conductor is connected with semi-infinite ideal crystalline leads, the modes can be expressed as Bloch functions (these are the plane waves commonly used to model metallic electrodes). The conductance can be written in <\/span>more compact form by using the Hermitian-conjugate matrix of ?, defined by (?\u2a25)?? = (???)\u2217. Then<\/span><\/p>\n

      \n

      \"\"<\/p>\n

      This is usually quoted result, where \u2018??\u2019 is the trace of the matrix (the sum of its diagonal elements). Also, the product ??\u2a25 is square, although neither ? nor ?\u2a25 need be, and that the two expressions for the trace are equal even when ??\u2a25 and ?\u2a25? may not be of the same size.<\/p>\n

       <\/p>\n

      The Landauer- B\u00fcttiker formalism takes into consideration a single particle picture of the electronic transport. In other words, it assumes electrons as non-interacting entities. However, the electron-electron interactions can be partially considered in a mean field approach, where the transmission amplitudes can be determined from the single-particle equation as:<\/p>\n

      \"\"<\/p>\n

      Here, the effective potential ????<\/sub> (?) accounts for the interactions of an electron with the average field of other electrons and nuclei.<\/p>\n

       <\/p>\n

      Many leads with many channels or subbands<\/em><\/span><\/p>\n<\/div>\n

      \"\"<\/em><\/p>\n

      \n

      Figure 4 Geometry of a sample for coherent transport with many leads. The general case is shown in (a) with specific examples of (b) a T-junction, (c) four-terminal measurement of longitudinal resistance, and (d) a microscopic Hall bar.<\/strong><\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      All expressions for current involve similar sums over the modes in both leads. It is similar to the trace (calculated above) with the additional subscripts on transmission coefficients to label the leads. The transmission and reflection coefficients which absorb these traces are given as:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      The net current injected into lead ? is given by the incident current minus the reflected current. The total
      \nincident current, summed over all propagating modes is (2?2<\/sup>\/\u210e)??<\/sub>??<\/sub>, thus the net current is:<\/p>\n

       <\/p>\n

      ???<\/sub> = (2?2<\/sup>\/\u210e)(??<\/sub> \u2212 ??<\/sub>)??<\/sub><\/p>\n

       <\/p>\n

      Conservation of current requires that this be equal to the total current injected from lead ? that leaves the
      \nsample though other leads, i.e., ???<\/sub> = \u03a3?,?\u2260?<\/sub> ???<\/sub>. This leads to a sum rule on the transmission coefficients:<\/p>\n<\/div>\n

      \"\"<\/p>\n

       <\/p>\n

      Which is a simple generalization of ? + ? = 1.<\/p>\n

       <\/p>\n

      We have now calculated the currents due to electrons injected from lead ?. Now the summation of all leads\u2019 contributions is required. Lead ? causes a current \u2013 (2?2<\/sup>\/\u210e)???<\/sub>??<\/sub> in lead ?, so the total current in lead ? is:<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      This is the Landauer- B\u00fcttiker formula for the conductance of a system with many leads. It can also be written in terms of a square conductance matrix whose dimension is given by the number of leads:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      Applications of Landauer-B\u00fcttiker formula<\/em><\/strong><\/p>\n

        \n
      1. Quantum point contact for a 2DEG<\/li>\n
      2. Electronic transport in single-molecule junctions<\/li>\n
      3. Longitudinal electronic transport in carbon nanotubes, and 2D graphene<\/li>\n
      4. Electronic transport in nanoparticles and chains of nanoparticles<\/li>\n
      5. Multiterminal quantum point contacts<\/li>\n<\/ol>\n\n\n\n
        you can view video on Landauer-B\u00fcttiker Formalism for Conduction in Confined Structures<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        \u00a0 \u00a0<\/strong>References & Suggested Reading:<\/strong><\/p>\n

          \n
        1. T. Pradeep, A Textbook of \u201cNanoscience and Nanotechnology\u201d, McGraw Hill Education,\u00a0<\/span>2012.<\/li>\n
        2. https:\/\/nanoyou.eu\/<\/li>\n
        3. https:\/\/en.wikipedia.org\/wiki\/Ballistic_conduction<\/li>\n
        4. Supriyo Datta (1997). Electronic Transport in Mesoscopic Systems. Haroon Ahmad, Alec Broers, Michael Pepper. New York: Cambridge University Press. pp. 57\u2013111.<\/li>\n
        5. www2.ece.rochester.edu\/projects\/bdt\/files\/Landauer_buttiker_formalism<\/li>\n
        6. http:\/\/tfp1.physik.uni-freiburg.de\/teaching\/Seminar2010\/talks\/LandauerFormula_Praesentation110510.pdf<\/li>\n
        7. http:\/\/www.phy.bme.hu\/~ujsaghy\/StudentSeminar\/1.pdf<\/li>\n
        8. https:\/\/hal.archives-ouvertes.fr\/hal-00943060\/document<\/li>\n<\/ol>\n","protected":false},"author":3,"menu_order":8,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-s-s-islam"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-145","chapter","type-chapter","status-publish","hentry","contributor-dr-s-s-islam"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":4,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/145\/revisions"}],"predecessor-version":[{"id":166,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/145\/revisions\/166"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/145\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/media?parent=145"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapter-type?post=145"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/contributor?post=145"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/license?post=145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}