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Department of Electronics Nanoelectronics 14 Atsufumi Hirohata 10:00 Tuesday, 3/March/2015 (B/B 103)

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Quick Review over the Last Lecture Fermi-Dirac distributionBose-Einstein distribution Function Energy dependence

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Contents of Nanoelectonics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scholar and vector potentials III. Basics of quantum mechanics (04 ~ 06) 04 History of quantum mechanics 1 05 History of quantum mechanics 2 06 Schrödinger equation IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) 07 Quantum well 10 Harmonic oscillator 11 Magnetic spin 13 Quantum statistics 1 14 Quantum statistics 2 V. Nanodevices (08, 09, 12, 15 ~ 18) 08 Tunnelling nanodevices 09 Nanomeasurements 12 Spintronic nanodevices

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14 Quantum Statistics 2 Boson Fermion Bose-Einstein condensation Fermi degeneracy Fermi liquid Ideal quantum gas

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Fermions / Bosons In particle physics, Fermions : 1 fermion occupies 1 quantum state (Fermi-Dirac statistics) Particles, which consist of materials; e.g., protons, neutrons and electrons If electrons were bosons...If protons / neutrons were bosons... Bosons : Several bosons occupy 1 quantum state (Bose-Einstein statistics) Particles, which transfer particle-interactions; e.g., photons and gluons * If photons were fermions...

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Fermions / Bosons (Cont'd) * Particles consisting of even numbers of fermions : bosons e.g., He 4 Particles consisting of odd numbers of fermions : fermions e.g., He 3

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Low-Temperature Generation By using 3-He and 4-He, a dilution refrigerator is realised (< 100 mK) : *

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Dilution Refrigerator Oxford Instruments SM with Heliox TL : Temperature : < 350 mK Magnetic field : ± 8 T

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Bose-Einstein Condensation - He 3 superfluid In boson systems, all bosons fall into the lowest energy state at very low temperature. Bose-Einstein condensation * e.g., superfluid of He 4 (1938 Fritz W. London) e.g., superfluid of He 3 (1972 Douglas D. Osheroff, Robert C. Richardson and David M. Lee) 2 He-3 fermions form a Cooper pair quasi-particle superconducting state

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Bose-Einstein Condensation - Rb atom In 1995, B-E condensation observed in Rb atom by Eric A. Cornell and Carl E. Wieman : *** R. Baierlein, Thermal Physics (Cambridge University Press, Cambridge, 1999). Reduced speed by photon-momentum absorption ** *

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Bose-Einstein Condensation - Na atom In 1995, Wolfgang Ketterle also observed B-E condensation in Na atom : ** *

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Bose-Einstein Condensation - Superconductor Cooper pair in a superconductor : ** *

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Superconductor and Heavy-Electron System Cooper pair in a superconductor : ** * Heavy-electron system : Lattice deformation Quadra-pole rigid lattice Dipole Cooper pair Mott insulator

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Magnetic Quantum Critical Point Phase diagram : * Fermi liquid : Frustrated spin system : *** ** Mott insulator Conventional metals

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Fermi Liquid In 2D and 3D, Landau proposed electron behaviour as liquid : Electron states can be defined by the competition among ； Kinetic energy Liquid Coulomb interaction Solid crystal Potential randomness Amorphous glass At very low temperature, Coulomb interaction becomes negligible. Fermi liquid

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Fermi Degeneracy Fermi-Dirac distribution : E T = 0 T ≠ 0 Pauli exclusion principle EFEF Fermi degeneracy Free electrons in a metal at room temperature » Higher energy states are occupied. » Heat capacity becomes smaller than expected from the classical model. » Magnetic susceptibility becomes smaller (Pauli paramagnetism). Star cores even at 10 9 K » Very high density degeneracy pressure » After nuclear fusion, › white dwarf stars by electron degeneracy pressure › neutron stars by electron degeneracy pressure black holes

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Ideal Quantum Gas Equation of states for ideal gas : where P : gas pressure, V : volume, N : number of particles, k B : Boltzman constant and T : temperature Energy E satisfies : T E Fermions Bosons Fermions : Compared with the classical model E increases / P increases (classical model) = Repulsive force between particles Pauli exclusion law Bosons : Compared with the classical model E decreases / P decreases = Attractive force between particles

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