NNSE508_EM-L11-semiconductors

NNSE508_EM-L11-semiconductors - 1 Lecture contents...

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NNSE 508 EM Lecture #11 1 Lecture contents Semiconductor statistics Transport
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NNSE 508 EM Lecture #11 2 Statistics of carriers: General Electron concentration in the energy range E to E+dE close to the conduction band minimum: Total electron concentration in the conduction band   1 exp 2 ) ( 2 1 3 2 2 3 * T k E E dE E E m dE E n B F C e   c E B F C e T k E E dE E E m n 1 exp 2 2 1 3 2 2 3 * T k E E x B C T k E E B C F c     0 2 1 2 3 3 2 2 3 * 1 2 c B e x dx x T k m   0 2 1 2 3 2 * 1 2 2 2 c B e x dx x T k m General equation for carrier concentration (effective density of states) x (Fermi integral of ½ order): ) ( 2 / 1 c C N n ) ( ) ( ) ( E f E N E n c Electron concentration at the energy E (Density of states) x (distribution function):
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NNSE 508 EM Lecture #11 3 Statistics of carriers: General The same is true for holes in the valence band: Effective density of states of electrons (or holes)   0 2 1 2 3 2 * 1 exp 2 2 2 V B h x dx x T k m p ) ( 2 / 1 V V N p ) ( 2 / 1 Concentration of mobile (band) carriers Fermi level position One-to-one correspondence
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NNSE 508 EM Lecture #11 4 Statistics of carriers: Non-degenerate system If all the C.B. energies are far from Fermi level: E C – E F >> k B T (> 3 k B T ) : ) ( 2 / 1 V V N p   2 1 exp 0 2 1 0 2 1 ) ( 0 2 1 c c c e dx x e e dx x e x dx x x x c   1  c x   0 2 1 1 2 c C x dx x N n General equation: T k E E B C F c T k E E N n B C F C T k E E N p B F V V T k E E x B C Concentration of band carriers Non-generate system: General case: ) ( 2 / 1 c C N n 2 i B g V C n T k E N N np Definition of intrinsic carrier concentration
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NNSE 508 EM Lecture #11 5 Carriers in intrinsic semiconductors Fermi level position in intrinsic semiconductor
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NNSE 508 EM Lecture #11 6 Effective mass approximation   ) ( ) ( ) ( 2 0 2 r E r r V r V m p i i One-electron Schrödinger equation with weak and slow varying perturbation V i (Effective mass approximation): Small perturbation of periodicity: shallow impurities, most of “hand-made” structures, external forces And as usual build a solution as a wave packet of Bloch wavefunctions : ) ( ) ( ) ( , r u e k c r nk ikr k n n Bloch wave packet: With dimensions in real and k-space 0 1 a k r  ) ( ) ( ) ( 0 r u r F r i V Large perturbation of periodicity other bands need to be considered: deep impurities Depending on sign of the perturbation, the top- most or bottom-most state splits from the band :
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NNSE 508 EM Lecture #11 7 Example of EMA: Hydrogen-like impurity (donor)
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NNSE508_EM-L11-semiconductors - 1 Lecture contents...

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