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Department of Electronics

Nanoelectronics 13. Atsufumi Hirohata. Department of Electronics. 10:00 24/February/2014 Monday (G 001). Quick Review over the Last Lecture. Gate Voltage. Input. Output. ( Spin-transfer torque ). * After M. Johnson, IEEE Spectrum 37 , 33 (2000). Contents of Nanoelectonics.

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Department of Electronics

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  1. Nanoelectronics 13 Atsufumi Hirohata Department of Electronics 10:00 24/February/2014 Monday (G 001)

  2. Quick Review over the Last Lecture Gate Voltage Input Output ( Spin-transfer torque ) * After M. Johnson, IEEE Spectrum37, 33 (2000).

  3. 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

  4. 13 Quantum Statistics 1 • Classical distribution • Fermi-Dirac distribution • Bose-Einstein distribution

  5. 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 N1, N2, ... To distribute Ni particles to the states Gi is GiNi, and hence the whole states are Microscopic states in the whole system W is By taking logarithm for both sides and assuming Ni and Gi are large, we apply the Stirling formula.

  6. 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

  7. Classical Distribution (Cont'd) Therefore, By substituting this relationship into Here, =1/kBT Gi : degree of degeneracy of eigen enrgy Ei * http://www.wikipedia.org/

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

  9. Fermi Energy T ≠ 0 f(E) 0 E Fermi-Dirac distribution : E T = 0 EF Pauli exclusion principle At temperature T, probability that one energy state E is occupied by an electron : T = 0 1  : chemical potential (= Fermi energy EF at T = 0) kB : Boltzmann constant T1 ≠ 0 1/2 T2 > T1 

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

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

  12. 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

  13. Bose-Einstein Distribution f(E) 0 E 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,

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

  15. 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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