2.2 Carrier Statistics

2.2 Carrier Statistics - ECE3080, Chapter 2.2 1 May 24,...

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Unformatted text preview: ECE3080, Chapter 2.2 1 May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 1 of 27 2.2 Carrier Statistic 2.2.1 Density of States 2.2.2 Fermi Function & Fermi Energy Physical Interpretation Characteristics 2.2.3 Carrier Densities Intrinsic/Extrinsic Semiconductor Intrinsic Fermi Energy Mass Action Law Temperature Dependence 2.2.4 Charge Neutrality Relationship 2.2.5 Non-Complete Ionization Anderson, Chapter 2.7-2.13, page 71-103 Anderson, Appendix D4, page 768-770 Pierret, Chapter 2.4-2.6, page 40-68 May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 2 of 27 S C (E)dE = 1 2 ! 2 2m dse * ! 2 " # $ % & 3/2 E ( E C dE ) E ( E C ( ) 1/2 S V (E)dE = 1 2 ! 2 2m dsh * ! 2 " # $ % & 3/2 E V ( E dE ) E V ( E ( ) 1/2 2.2.1 Density of States From quantum mechanics, we not only obtain the band structure, i.e., the E(k) relations, but also the density of states S(E)dE, i.e., how many allowed states are in the range EE+dE: ECE3080, Chapter 2.2 2 May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 3 of 27 Density of States Band Gap E C E V E S C (E) S V (E) S(E) Units of S(E)dE are [cm-3 ] May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 4 of 27 Derivation of S(E) Imagine electron in 3-D potential well (our crystal) with dimensions L x , L y and L z This problem yields the wave function and the following quantized wave numbers This yields the following number of spaces per unit volume in the K space (factor 2 attributes for spin) and the number of states in K per unit volume ! (x,y,z) = A e j(K x x + K y y + K z z) K x = 2 ! n x L x K y = 2 ! n y L y K z = 2 ! n z L z L x 2 ! " L y 2 ! " L z 2 ! " 2 S(K x ,K y ,K z ) = 1 2 ! " # $ % & 3 ( 2 ECE3080, Chapter 2.2 3 May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 5 of 27 Derivation of S(E) How many states do we find between K and K+dK? Evaluate spherical shell with thickness dK and radius K (see Anderson, Figure D.3) In order to find S(E)dE we need to form with S(K)dK = S(K x ,K y ,K z ) 4 ! K 2 dK S(E)dE = S(K(E)) dK dE dE E ! E C = ! 2 K 2 2m and dE dK = ! 2 K m S(E)dE = 2 1 2 ! " # $ % & 3 4 ! 2m(E ( E C ) ! 2 m ! 2 ! 2 2m(E ( E C ) dE = 1 2 ! 2 2m ! 2 " # $ % & 3/2 E ( E C dE q.e.d. May 24, 2007 ECE 3080: Chapter 2.2 Device Fundamentals O. Brand, 6 of 27 2.2.2 Fermi Function & Fermi Energy What determines whether an allowed state is occupied by an electron or not?...
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2.2 Carrier Statistics - ECE3080, Chapter 2.2 1 May 24,...

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