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EE 315/ECE 451 N ANOELECTRONICS I Derived from lecture notes by R. Munden 2010 2 8.1 D ENSITY OF S TATES Imagine the hard walled box, and the energy states available: # states in sphere of radius En: Octant of sphere J. DENENBERG- FAIRFIELD UNIV. - EE315 10/13/2015 D O S 3D S YSTEM 3 Accounting for Spin. Potential and L3 = 1 for density per unit volume: J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 4 8.1.1 D ENSITY OF S TATES IN L OW D IMENSIONS For One Dimensional Quantum Wire With spin and potential: J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 5 8.1.1 D ENSITY OF S TATES IN L OW D IMENSIONS For 2D Quantum Well For 0D Quantum Dots J. DENENBERG- FAIRFIELD UNIV. - EE315 Although real materials are 3D, quantum confinement in small materials can approximate low dimensional structures, like quantum dots. 10/19/2015 6 8.1.2 D ENSITY OF S TATES IN A S EMICONDUCTOR In 3D materials: We can use in semiconductors by substituting effective mass for band structure and Ec for potential In Silicon (with transverse and longitudinal effective mass) and 6 fold symmetry of the conduction band: J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 8.2 C LASSICAL AND Q UANTUM S TATISTICS 7 Classical or Boltzman Distribution (distinguishable particles – e.g.molecules): Fermi-Dirac Distribution (indistinguishable, exclusive particles – e.g. electrons): Bose-Einstein Distribution (indistinguishable, non-exclusive particles – e.g. photons and phonons): J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 F ERMI D ISTRIBUTION 8 “Ripples on the Fermi sea” Occupied J. DENENBERG- FAIRFIELD UNIV. - EE315 Unoccupied 10/19/2015 9 8.2.1 C ARRIER C ONCENTRATION IN M ATERIALS 3D confined box J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 10 8.2.2 T HE I MPORTANCE OF F ERMI E LECTRONS R.MUNDEN - FAIRFIELD UNIV. - EE315 11/1/2010 11 E LECTRON S TATES J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 12 8.2.3 E QUILIBRIUM C ARRIER C ONCENTRATION J. DENENBERG- FAIRFIELD UNIV. - EE315 10/19/2015 8.3 M AIN P OINTS 13 the concept of density of states in various spatial dimensions, and the significance of the density of states; how the density of states can be measured; quantum and classical statistics for collections of large numbers of particles, including the Boltzmann, Fermi-Dirac, and BoseEinstein distributions; the role of density of states and quantum statistics in determining the Fermi level; applications of density of states and quantum statistics to determine carrier concentration in materials, including in doped semiconductors. R.MUNDEN - FAIRFIELD UNIV. - EE315 11/1/2010 8.4 P ROBLEMS 14 R.MUNDEN - FAIRFIELD UNIV. - EE315 11/1/2010