Append16

Transcript

Appendix 16: Equation Sheet Chapter 1: Review of Modern Physics 1.2. Quantum mechanics   h p E ph  h    hc   2 d 2  ( x)  V ( x ) ( x )  E  ( x ) 2m dx 2 h2 n 2 ( ) , with n  1, 2, ... En  * 2m 2 L x En   m0 q 4 8 02 h 2 n 2 1.3. Electromagnetic theory dE ( x )  ( x )  dx  d 2 ( x ) dx 2  d ( x )  E ( x ) dx  ( x)  A16-1 , with n  1, 2, ... Chapter 2: Semiconductor Fundamentals 2.3. Energy bands E g (T )  E g (0)  T 2 T  2.4. Density of states g c (E)  1 dN 8 2 * 3 / 2  m E  E c , for E  E c L3 dE h3 g c ( E )  0, for E  E c 2.5. Carrier distribution functions f (E)  1 1 e ( E  E F ) / kT f donor ( E d )  1 1  1 e ( E d  E F ) / kT f acceptor ( E a )  2 f BE ( E )  1 f MB ( E )  e ( E  EF ) / kT  1 1 1  4e ( Ea  E F ) / kT 1 e ( E  EF ) / kT 2.6. Carrier densities  no   g c ( E ) f ( E )dE po  no  N c e E F  Ec kT ni  N c N v e Ei   g v ( E )[1  f ( E )]dE  Ec 2 2  m*    no  3  2   2  Ev 3/ 2 ( E F  E c ) 3 / 2 , for E F  E c and T = 0 K with N c  2 [  E g / 2kT 2 me* kT 3 / 2 ] h2 po  N v e Ev  E F kT n o  p o  n i2 Ec  Ev 1 N E  Ev 3 m*  kT ln( v )  c  kT ln( h* ) Nc 2 2 2 4 me A16-2 with N v  2 [ 2 mh* kT 3 / 2 ] h2 no  ni e ( EF  Ei ) / kT p o  ni e ( Ei  EF ) / kT n E F  E i  kT ln o ni p E F  E i  kT ln o ni E c  E d  13.6 no  N d  N a 2 * m cond m 0  r2 eV N d  N a 2  ( )  ni2 2 n  no   n  ni exp( Fn  Ei ) kT N a  N d N a  N d 2 po   ( )  ni2 2 2 p  p o   p  ni exp( Ei  F p kT ) 2.7. Carrier Transport  | v | q    |E | m    min    J  qn n E J  q nv e  q p v h  q ( n  n  p  p ) E  J  E  q(n n  p  p ) v (E )  1   N 1 ( ) Nr  1 q(n n  p  p ) E E 1 J n  q Dn Dn   n   max   min v sat dn dx J p  q D p kT   nVt q J n  qn n E  q Dn dn dx Dp   p dp dx kT   pVt q J p  q p p E  q D p I total  A( J n  J p ) A16-3 dp dx 2.8. Carrier recombination and generation U n  Rn  Gn  n p  n p0 p  pn0 U p  Rp  Gp  n n p pn  ni2 N t vth Ei  Et p  n  2ni cosh( ) kT U b  b  b( np  ni2 ) U SHR  U Auger  n n(np  ni2 )   p p(np  ni2 ) G p ,light  G n,light   d Popt ( x) dx Popt ( x) E ph A  Popt ( x) 2.9. Continuity equation  n ( x, t ) 1  J n ( x, t )  Gn ( x, t )  Rn ( x, t )  q t x  J p ( x, t )  p ( x, t ) 1  G p ( x, t )  R p ( x, t ) q x t  2 n p ( x , t ) n p ( x, t )  n p 0  n ( x, t )  Dn  t n  x2  2 p n ( x, t ) p n ( x, t )  p n 0  p ( x, t )  Dp  t p  x2 0  Dn d 2 n p ( x) dx 2  n p ( x)  n p 0 d 2 p n ( x) 0  Dp n dx 2 p ( x)  p n0  n p 2.10. The drift-diffusion model   q( p  n  N d  N a ) dE   dx  d  E dx dE i  qE dx n  ni e ( Fn  E i ) / kT p  ni e J n  qn n E  q Dn  Jn 0 1  q x dn dx np  ni2 ( E i  F p ) / kT J p  q p p E  q D p 1 E  Ei  ) n  p  2ni cosh( t kT 01 q A16-4 Jp x  dp dx np  ni2 1 E  Ei  ) n  p  2ni cosh( t kT Chapter 3: Metal-Semiconductor Junctions 3.2. Structure and principle of operation  B   M   , (n-type semiconductor) , i   M    Ec  E F , n , q n - type B  Eg q     M , (p-type semiconductor) i    Ec  E F , p q - M , p - type 3.3. Electrostatic analysis d 2 dx 2 d 2 dx 2    q   ( p  n  N d  N a ) s s 2q ni s (sinh  ( x )  qN d E ( x)     F Vt 0  x  xd , qN d s (x d - x) E ( x)  0 E ( x  0)   xd    sinh F ) Vt 0  x  xd xd  x qN d x d s Q  d s 2 s ( i  V a ) qN d N a  N d  sinh F  Vt 2ni and  ( x )  0  ( x)   ( x)  xd  x qN d 2 [ x d  (x d -x) 2 ] 2 s qN d x d2 xd  x 2 s  i  V a   ( x  0)  Cj  dQ d  dV a 0  x  xd qN d x d2 2 s  q s N d  s 2( i  V a ) x d 3.4. Schottky diode current V  J n  q n E max N c exp( B )[exp( a )  1] Vt Vt B V J n  qv R N c exp( )[exp( a )  1] Vt Vt E max  vR  J n  q vR n  A16-5 2q (i  Va ) N d s  * 3/ 2  kT   4 2qm  B and   exp  E  2 m  3    3.5. Metal-Semiconductor Contacts 3.5.1. Contact resistance to a thin semiconductor layer d 2 I ( x) I ( x)  2 with   dx 2  I ( x)  I 0 sinh  c Rs  V ( x)  I 0 d  Rs W  coth W Rs dx sinh Rc  c d R2   c Rs W cosh dx sinh  Rs  d  L , for d >>  W 3.6. Metal-Semiconductor Field Effect Transistor (MESFETs) VT   i  qN d d 2  i  VP 2 s I D  q n N d d I D , sat 2  (  VG  V D ) 3 / 2 ( i  VG ) 3 / 2 W V D   i  3  L V VP P   (i  VG ) 3 / 2 W 2  q n N d d VG  VT  3 VP   L VP     for VG > VT     with V D , sat  VG  VT   Chapter 4: p-n Junctions 4.2. Structure and principle of operation  i  Vt ln Nd Na ni2    i  Va 4.3. Electrostatic analysis of a p-n diode xd  xn  x p   q ( p  n  N d  N a )  q ( N d  N a ), for  x p  x  x n A16-6 , for x   x p  ( x)  0  ( x)  qN a  ( x)  qN d  ( x)  0 , for  x p  x  0 , for 0  x  xn , for xn  x Q n  qN d x n Q p   qN a x p dE(x)  ( x ) q   ( N  ( x )  N a ( x )), for  x p  x  x n dx s s d E ( x)  0 E ( x)   E ( x)  , for x   x p qN a ( x  x p ) , for  x p  x  0 s qN d ( x  xn ) , for 0  x  x n s , for x n  x E ( x)  0 E ( x  0)   qNa x p s  q N d xn s E ( x  0)   2( i  Va ) xd N d xn  N a x p xn  xd Na Na  Nd  i  Va  q N d xn2 2 s Nd Na  Nd x p  xd  q N a x 2p 2 s xd  2 s 1 1 (  )( i  Va ) q Na Nd 2 s N d 1 ( i  Va ) q Na Na  Nd xn  2 s N a 1 ( i  Va ) q Nd Na  Nd xp  Cj  q s Na Nd 2( i  Va ) N a  N d Cj  s xd 4.4. The p-n diode current p n ( x  xn )  p n0 eVa / Vt n p ( x   x p )  n p 0 eVa / Vt A16-7 pn ( x  xn )  pn 0  Ae  ( x  xn ) / L p n p ( x   x p )  n p0  Ce  Be  ( x  x p ) / Ln ( x  xn ) / L p  De ( x  x p ) / Ln p n ( x  x n )  p n 0  p n 0 (e Va / Vt  1)[cosh n p ( x   x p )  n p 0  n p 0 (e Va / Vt L p  D p p Ln  Dn n x  xn w' x  xn ]  coth n sinh Lp Lp Lp  1)[cosh x xp Ln  coth w 'p Ln sinh x xp Ln ] w'p  w p  x p wn'  wn  xn I  A[ J n ( x   x p )  J p ( x  xn )  J r ]  I s (eVa / Vt  1) I s  q A[ I s  q A[ Dn n p 0 Ln Dn n p 0 Ln coth(  w'p Ln D p pn0 Lp ) D p p n0 Lp wn' coth( )] (general case) Lp ] (“long” diode) J bb  qni2 bw(eVa / Vt  1) I s  q A[ J SHR  Dn n p 0 w 'p  D p pn0 wn' ] (“short” diode) qni x ' Va / 2Vt (e  1) 2 V a *  V a  IR s J  J s e Va / Vt 1 log(e) slope   Vt slope 59.6 mV/decade Q p  I s , p (eVa / Vt  1) p I s, p  q Cd , p  Cd , p  I s , p eVa / Vt  p Vt I s , p eVa / Vt t r , p Vt A pn 0 D p Lp (“long” diode) (“short” diode) with t r , p  A16-8 w 'p 2 2D p 4.5. Reverse bias breakdown E br  4 x10 5 V/cm 1  13 log( N / 1016 cm - 3 ) 2 E  Vbr   i  br s 2qN 1 M V 1 a Vbr n x d ,br  E br  s qN , where 2  n  6  4 2 m* E 3 / 2  g    exp   3 q E    J n  q vR n  4.6. Optoelectronic devices I  I s (eVa / Vt  1)  I ph I ph,max  q Pin h I ph  (1  R)(1  e  d )  i 2  2 q I f I V Fill Factor  m m I scVoc Roundtrip amplification  e 2 gL R1 R2  1 g Pout   qPin h 1 1 ln 2 L R1 R 2 h ( I  I th ) q Chapter 5: Bipolar Junction Transistors 5.2. Structure and principle of operation w B'  w B  x p, BE  x p, BC w E'  w E  x n, BE wC'  wC  x n, BC x n , BE  2 s ( i , BE  V BE ) N B q NE  1   NB  NE    x p , BE  A16-9 2 s ( i , BE  VBE ) N E q NB  1   NB  NE    2 s ( i , BC  VBC ) N C q NB x p , BC  i , BE  Vt ln   1    N B  NC  NB NE 2 s (i , BC  VBC ) N B   1   q N C  N B  N C  xn , BC  i , BC  Vt ln ni2 N B NC ni2 I E  IC  I B I E  I E ,n  I E , p  I r ,d I C  I E ,n  I r , B I B  I E , p  I r , B  I r ,d  IC IE   IB I E ,n  I r , B r  IE,n   1 I E.n I E ,n  I E , p E    T  E  r T  IC I E  I r ,d IE 5.3. Ideal transistor model  D n, B I E ,n  qni2 AE   N w'  B B ni2 Qn, B  q AE NB I E ,n   w'  V  exp( BE )  1 B Vt  2  I r,B  tr n 2 w B'  1 T  1 n 2 D n, B  n Dn, B N E w E' D p, E N B w B' Q n, B tr D p, E N B w B' D n, B N E w E'    exp(V BE )  1    Vt   w' tr  B 2 D n, B 1 1  D p, E I E , p  qni2 AE   N w'  E E 2 Qn, B E      exp(V BE )  1    Vt   , if    E A16-10 VA  QB  C j , BC qN B wB' s x p , BC  x n , BC n V 1  1  t C j , BE d ln I C QB Vt dVBE A16-11 Chapter 6: MOS Capacitors 6.3 MOS analysis VFB   M   S M  S  M    Eg  Vt ln( 2q Poly-silicon:  poly   S  Vt ln( V FB   MS  Eg Na N  Vt ln( d ) (pMOS) ) (nMOS)  M   S   M    2q ni ni N a, poly Na ) (p-type)  poly   S  Vt ln( Qi 1 tox    ox ( x) x dx C ox  ox 0 Qinv  C ox (VG  VT ) for VG  VT  F  Vt ln Na (nMOS) ni VG  V FB   s  VG  V FB   s  xd  C ox C ox 4 s qN a F Cox x d ,T  (nMOS) 1 x 1  d ,T Cox  s Nd (pMOS) ni (nMOS) (pMOS) 4 s F qN a VT  V FB  2 F  4 s qN d  F C ox (pMOS) 1 C LF  C HF  , for VFB  VG  VT xd 1  Cox  s C LF  C HF  C ox , for VG  VFB C LF  Cox and C HF  t ox for VFB > VG > VT and 0   s  2 F 2 s  s , for 0   s  2 F qNa ) (n-type)  ox for VFB < VG < VT and 0   s  2 F 2 s qN d  s N d , poly N a Qinv  0 for VG  VT  F  Vt ln 2 s qN a s VT  V FB  2 F  with C ox  ni2 , for VG  VT C FB  A16-12 1 L 1  D Cox  s with LD   s Vt qN a Chapter 7: MOS Field Effect Transistors 7.3. MOSFET analysis Linear model I D   C ox W (VGS  VT )V DS , for | V DS | (VGS  VT ) L Quadratic model ID V DS2 W   C ox [(VGS  VT )V DS  ] , for V DS  VGS  VT L 2 I D , sat   C ox 2 W (VGS  VT ) , for V DS  VGS  VT 2 L g m, quad   C ox W V DS L g m, sat   C ox g d , quad   C ox W (VGS  VT  V DS ) L g d , sat  0 W (VGS  VT ) L Channel length modulation W (VGS  VT ) 2 I D, sat   Cox ( 1  V DS ) , for VDS  VGS  VT L 2 Variable depletion layer model ID   n C oxW L W 2  n 3 L (VGS  V FB  2 F  2 s qN a ((2 F  V DB ) 3 2  (2 F  V SB ) 3 2 ) V DS , sat  VGS  V FB  2 F  g m, sat   n* C ox V DS )V DS 2 qN a  s W (VGS  VT ) L 2 C ox { 1 2 2 C ox (VGB  V FB )  1} qN a  s    1  n*   n 1  2  2(2 F  V SB )C ox 1  qN a  s  A16-13        7.4. Threshold voltage VT   ( (2 F  VSB )  2 F )  7.7. Advanced MOSFET issues V  VT ) I D  exp( G Vt  surface  E 1 / 3 A16-14 2 s qN a C ox