ece 606 key equations4
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photovoltaic basic equationsTRANSCRIPT
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ECE-606 Spring 2013 1
ECE-606: Key Equations (for exam 4) Spring 2013 Mark Lundstrom, Purdue University (Revised 4/5/2013) Physical constants: Silicon (T = 300K)
! = 1.055 !10"34 J-s[ ] NC = 3.23!1019 cm-3
m0 = 9.109 !10"31 kg[ ] NV = 1.83!1019 cm-3 kB = 1.380 !10"23 J/K[ ] ni = 1!1010 cm-3 q = 1.602 !10"19 C[ ] Miller Indices: (hkl) {hkl} [hkl] Angle between two planes:
cos! =
h1h2 + k1k2 + l1l2h1
2 + k12 + l1
2 h22 + k2
2 + l22
Spacing between two planes: d = 1
!N = a h2 + k 2 + l2 Time independent wave eq.:
!!2
2m0
"2# x( )"x2
+U x( )# x( ) = E# x( ) ! x,t( ) =" x( )e# i$t
d 2! x( )dx2
+ k 2! x( ) = 0 k 2 =
2m0!2
E !U0"# $% ! x( ) = Ae ikx Infinite quantum well of width, W: !n =
!2kn22m* =
!2n2" 22m*W 2 n = 1, 2, 3, Momentum operator:
p ! !
i""x
Energy operator: E = ! !
i""t
Plane wave: ! x,t( ) = ei kx"#t( ) Momentum of plane wave: p = !k Density of states in k-space: 1D: Nkdk= 2! L 2"( )dk = L "( )dk 2D:
Nkd
2k = 2! A 2"( )2#$% &'(d2k = A 2" 2( )d 2k 3D: Nkd 3k=2! " 8# 3( )d 3k = " 4# 3( )d 3k
Density of states in energy (above the bottom of the conduction band): 1D: D1D (E) =
1!!
2mD*
E " EC 2D: D2 D (E) = mD*!!2 3D: D3D E( ) = mD*( )3/2
2 E ! EC( )" 2!3
Fermi function: f E( ) = 11+ e E!EF( ) kBT
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ECE-606 Spring 2013 2
Electron densities: ( !F = EF " EC( ) kBT ): 1D: nL = NCF !1/2 "F( ) m-1 NC = 1! 2mD* kBT! m-1 2D: nS = NCF 0 !F( ) m-2 NC =
mD* kBT!!2
"
#$%
&' m-2 3D: n = NCF 1/2 !F( ) m-3 NC =
14
2mD* kBT
!!2"
#$%
&'
3/2 m-3 FD Integral:
F j !F( ) = 1"( j +1)
! jd!1+ e!#!F0
$
% For n an integer: !(n) = n"1( )! For non-integer n: ! 1/ 2( ) = " and ! p +1( ) = p! p( ) Space charge neutrality: p ! n + ND+ ! NA! = 0 ND+ND
= 11+ gDe EF!ED( ) kBT NA!NA = 11+ gAe EA!EF( ) kBT
Recombination: Radiative: RRad = B np ! ni2( ) Auger: RAug = Cnn np ! ni2( ) +Cp p np ! ni2( ) SRH: ! "n"t SRH
= ! "p"t SRH
= RSRH =np ! ni2( )
# p n + n1( ) +# n p + p1( )
Current equations: Jn = nn
dFndx
Jn = nqnE x + qDn
dndx
Dn n = kBT q J p = p p
dFpdx
J p = pq pE x ! qDp
dpdx
Dp p = kBT q Semiconductor Equations:
!n!t = "# i
!Jn"q
$%&
'()+Gn " Rn Minority Carrier Diffusion Eqaution
! p!t = "# i
!Jpq
$
%&'
()+Gp " Rp !"p!t = Dp !2"pdx2 # "p$ p +GL
0 = !" i #
!E( ) + $ Lp = Dp! p
Diode Equation: ID = qA
ni2Dn
N ALn+
ni2Dp
N D Lp
!
"#
$
%& e
qVA /kBT '1( ) (long)
ID = qA
ni2Dn
N AWP+
ni2Dp
N DWN
!
"#
$
%& e
qVA /kBT '1( ) (short)
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ECE-606 Spring 2013 3
Bipolar transistors: ! = ICIB
= NDENABDnDp
WEWB
gm = !IC!VBE = ICkBT q( ) ! = IEnIEp + IEn
= 11+WBWE
niE2niB2
NABNDE
ICn =!T IEn " IC !T "1# 12 WBLn$%& '() 2 IC =! dcIE ! dc =!T" !dc = " dc1#" dc tt =
WB22Dn
fT max = 12! tt