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Department of Electronics Nanoelectronics 13 Atsufumi Hirohata 12:00 Wednesday, 25/February/2015 (P/L 006)

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Quick Review over the Last Lecture Input Output Gate Voltage FM / SC hybrid Structures Magnetic tunnel junctions (MTJ) All metal and spin valve structures Interface Ohmic / Schottky barriers Tunnel barriersOhmic (no barrier) Spin carriersSCTunnel barriersNon-magnetic metals Device applications FM / 2DEG Schottky diodes Spin FET Spin LED Spin RTD MOS junctions Coulomb blockade structures SP-STM Supercond. point contacts Spin RTD Johnson transistors Spin valve transistors * After M. Johnson, IEEE Spectrum 37, 33 (2000). ( Spin-transfer torque )

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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 V. Nanodevices (08, 09, 12, 15 ~ 18) 08 Tunnelling nanodevices 09 Nanomeasurements 12 Spintronic nanodevices

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13 Quantum Statistics 1 Classical distribution Fermi-Dirac distribution Bose-Einstein distribution

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Classical Distribution Maxwell-Boltzmann distibution : In the classical theory, the same types of particles can be distinguished. Every particle independently fills each quantum state. In order to distribute N particles to the states 1, 2,... with numbers of N 1, N 2,... To distribute N i particles to the states G i is G i Ni, and hence the whole states are Microscopic states in the whole system W is By taking logarithm for both sides and assuming N i and G i are large, we apply the Stirling formula.

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Classical Distribution (Cont'd) In the equilibrium state, Again, by applying the Stirling formula, Here, with using By using and Now, the partition function is

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Classical Distribution (Cont'd) Therefore, By substituting this relationship into Here, = 1/k B T G i : degree of degeneracy of eigen enrgy E i * http://www.wikipedia.org/

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Fermi-Dirac / Maxwell-Boltzmann Distribution Electron number density : Maxwell-Boltzmann distribution (small electron number density) Fermi-Dirac distribution (large electron number density) Fermi sphere : sphere with the radius k F Fermi surface : surface of the Fermi sphere * M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989). Increase number density classical quantum mechanical

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Fermi Energy Fermi-Dirac distribution : E T = 0 T ≠ 0 Pauli exclusion principle EFEF At temperature T, probability that one energy state E is occupied by an electron : : chemical potential (= Fermi energy E F at T = 0 ) k B : Boltzmann constant 1 1/2 f(E) E 0 T = 0 T 1 ≠ 0 T 2 > T 1

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Fermi velocity and Mean Free Path Under an electrical field : Electrons, which can travel, has an energy of ~ E F with velocity of v F For collision time , average length of electrons path without collision is Mean free path Fermi wave number k F represents E F : Fermi velocity : Density of states : Number of quantum states at a certain energy in a unit volume g(E) E 0

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Density of States (DOS) and Fermi Distribution Carrier number density n is defined as : E 0 g(E) f(E) EFEF T = 0 E 0 T ≠ 0 g(E) EFEF f(E) n(E)

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Fermi-Dirac Distribution In the Fermi-Dirac distribution, 1 quantum state can be filled by 1 particle. Fermi particle (spin 1/2) For the entire wavefunction : Particle exchange induces a sign change of the wavefunction. For a 2-Fermi-particle system, due to the exchange characteristics, the wavefunction is expressed as If the 2 particles satisfy the same wavefunction, Pauli exclusion principle

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Bose-Einstein Distribution In the Bose-Einstein distribution, 1 quantum state can be filled by many particles. Bose particle (spin 1,2,...) For the entire wavefunction : Particle exchange does not induce a sign change of the wavefunction. For a 2-Bose-particle system, due to the exchange characteristics, the wavefunction is expressed as If the 2 particles satisfy the same wavefunction, f(E) E 0

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Bose-Einstein Distribution (Cont'd) Difference between Bose-Einstein and Maxwell-Boltzmann distributions : * http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disbex.html

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Quantum Statistics In quantum mechanics, the same types of particles cannot be distinguished : By exchanging coordinate systems, Hamiltonian (operator) does not change : P : coordinate exchange operator Diagonalization between H and P By assuming the eigenvalue for P to be, Since twice exchange of the coordinate systems recover the original state, Therefore, (symmetric exchange) Bose-Einstein statistics Bosons integer spins (asymmetric exchange) Fermi-Dirac statistics Fermions half-integer spins

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Chapter 19: Fermi-Dirac gases. 19.1 The Fermi energy Fermi-Dirac statistics governs the behavior of indistinguishable particles (fermions). Fermions have.

Chapter 19: Fermi-Dirac gases. 19.1 The Fermi energy Fermi-Dirac statistics governs the behavior of indistinguishable particles (fermions). Fermions have.

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