L14_5340_Sp11

advertisement
EE 5340
Semiconductor Device Theory
Lecture 14 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc
S-R-H net recombination rate, U
• In the special case where tno = tpo = to
= (Ntvthso)-1 the net rec. rate, U is
dp
dn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  n, and p  po  p, (n  p)
©rlc L14-08Mar2011
2
S-R-H “U” function
characteristics
• The numerator, (np-ni2) simplifies in
the case of extrinsic material at low
level injection (for equil., nopo = ni2)
• For n-type (no > n = p > po = ni2/no):
(np-ni2) = (no+n)(po+p)-ni2
= nopo - ni2 + nop + npo + np
~ nop (largest term)
• Similarly, for p-type, (np-ni2) ~ pon
©rlc L14-08Mar2011
3
S-R-H rec for
excess min carr
• For n-type low-level injection and net
excess minority carriers, (i.e., no > n
= p > po = ni2/no),
U = p/tp, (prop to exc min carr)
• For p-type low-level injection and net
excess minority carriers, (i.e., po > n
= p > no = ni2/po),
U = n/tn, (prop to exc min carr)
©rlc L14-08Mar2011
4
Minority hole lifetimes
Mark E. Law, E. Solley,
M. Liang, and Dorothea
E. Burk, “SelfConsistent Model of
Minority-Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991
The parameters used in
the fit are
τo = 10 μs,
Nref = 1×1017/cm2, and
CA = 1.8×10-31cm6/s.
τp 
©rlc L14-08Mar2011
τo
1  ND Nref  τ oC AND2
5
Minority electron lifetimes
Mark E. Law, E. Solley,
M. Liang, and Dorothea
E. Burk, “SelfConsistent Model of
Minority-Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991
The parameters used in
the fit are
τo = 30 μs,
Nref = 1×1017/cm2, and
CA = 8.3×10-32 cm6/s.
τn 
©rlc L14-08Mar2011
τo
1  ND Nref  τ oC AND2
6
Minority Carrier Lifetime, Diffusion Length and
Mobility Models in Silicon
A. [40%] Write a review of the model equations for minority
carrier (both electrons in p-type and holes in n-type
material) lifetime, mobility and diffusion length in silicon.
Any references may be used. At a minimum the material
given in the following references should be used.
Based on the information in these resources, decide which
model formulae and parameters are the most accurate for
Dn and Ln for electrons in p-type material, and Dp and Lp
holes in n-type material.
B. [60%] This part of the assignment will be given by 10/12/09.
Current-voltage data will be given for a diode, and the
project will be to determine the material parameters (Nd,
Na, charge-neutral region width, etc.) of the diode.
©rlc L14-08Mar2011
7
References for Part A
Device Electronics for Integrated Circuits, 3rd ed., by Richard S.
Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons,
New York, 2003.
Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “SelfConsistent Model of Minority-Carrier Lifetime, Diffusion Length,
and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8,
AUGUST 1991.
D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE
SIMULATION”, Electron Devices Meeting, 1990. Technical Digest.,
International 9-12 Dec. 1990 Page(s):357 – 360.
David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling
and Measurement of Minority-Carrier Lifetime versus Doping in
Diffused Layers of n+-p Silicon Diodes”, IEEE TRANSACTIONS ON
ELECTRON DEVICES, VOL. ED-29, NO. 2, FEBRUARY 1982, pages
284-291.
M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination
in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp. 577-597,
1983. Download a copy at Tyagi.pdf.
©rlc L14-08Mar2011
8
S-R-H rec for
deficient min carr
• If n < ni and p < pi, then the S-R-H net
recomb rate becomes (p < po, n < no):
U = R - G = - ni/(2t0cosh[(ET-Efi)/kT])
• And with the substitution that the
gen lifetime, tg = 2t0cosh[(ET-Efi)/kT],
and net gen rate U = R - G = - ni/tg
• The intrinsic concentration drives the
return to equilibrium
©rlc L14-08Mar2011
9
The Continuity
Equation
• The chain rule for the total time
derivative dn/dt (the net generation
rate of electrons) gives
dn n n dx n dy n dz




.
dt t x dt y dt z dt
The definition of the gradient is
      
n  
i
j
k n,
x
y
z 

©rlc L14-08Mar2011
10
The Continuity
Equation (cont.)
The definition of the vector velocity is
dx  dy  dz 
v 
i
j
k.
dt
dt
dt

 
Since A B  AxBx  AyBy  AzBz ,

dn n
then

 n  v
dt t
©rlc L14-08Mar2011
11
The Continuity
Equation (cont.)
The gradient operator can be distributed



as   n v  n  v  n  v .
Considering the second term on the RHS,
 dx  dy  dz
 v 


 0, since
x dt y dt z dt

 dx d x

 0, etc.
x dt dt x
©rlc L14-08Mar2011
12
The Continuity
Equation (cont.)
Consequently, since

Jn

 qn v , we have


n 1
dn n
   J n . So
  n v 

t q
dt t


dp p 1
dn n 1
   Jp

   J n , and

dt t q
dt t q
are the " Continuity Equations".
©rlc L14-08Mar2011
13
The Continuity
Equation (cont.)
dn dp
 The LHS,
or
 -U, of the Continuity Eq.
dt
dt
represents the Net Generation Rate of n
or p at a particular point in space (x, y, z).
n p
 The first term on the RHS,
or , is
t
t
the " explicit" Local Rate of Change of n or
p at (x, y, z).
©rlc L14-08Mar2011
14
The Continuity
Equation (cont.)

1
The second term on the RHS,    J n
q

1
or    J p is the local rate of n or p
q
concentrations flowing " out of" the
point (x, y, z). Note the difference in
signs for electrons (-q) and holes (  q).
©rlc L14-08Mar2011
15
The Continuity
Equation (cont.)
So, we can re - write the continuity equations as :

p dp 1
dp
δp

   J p , where
 U   and
t dt q
dt
τp

n dn 1
dn
δn

   J n , where
 U  
t dt q
dt
τn
Which can be interprete d as :
Local rate of change 
©rlc L14-08Mar2011
net generation rate  rate of inflow
16
Review of depletion
approximation
EFp
qVbi
Ec
EFn
EFi
Ev
-xpc -xp 0 xn
©rlc L14-08Mar2011
•
•
•
•
xnc x •
Depletion Approx.
pp << ppo, -xp < x < 0
nn << nno, 0 < x < xn
0 > Ex > -2Vbi/W,
in DR (-xp < x < xn)
pp=ppo=Na & np=npo=
ni2/Na, -xpc< x < -xp
nn=nno=Nd & pn=pno=
ni2/Nd, xn < x < xnc
17
Review of
D. A. (cont.)
-xpc-xp
Ex
xn
xnc
2Vbi  Va 
W
, W  xp  xn ,
qNeff
x
Neff
NaNd

, Na xp  Ndxn ,
Na  Nd
Ex  0, x  xp
q
Ex  - Na x  xp , xp  x  0,

q
Ex  Na x  xn , 0  x  xn ,

Ex  0, x  xn

-Emax
©rlc L14-08Mar2011

18
Forward Bias Energy Bands

nnon equil  ni expEFn  EFi  / kT   n p  n p 0 eVa Vt  1
q(Vbi-Va)
Imref, EFn
Ec
EFN
EFi
EFP qVa
Imref, EFp




pnon equil  ni exp EFi  EFp / kT  pn  pn 0 eVa
-xpc
©rlc L14-08Mar2011
-xp
0
xn

Ev
Vt
1
x

xnc
19
References
1 and M&KDevice
Electronics for Integrated
Circuits, 2 ed., by Muller and Kamins, Wiley,
New York, 1986. See Semiconductor Device
Fundamentals, by Pierret, Addison-Wesley,
1996, for another treatment of the m model.
2Physics of Semiconductor Devices, by S. M. Sze,
Wiley, New York, 1981.
3 and **Semiconductor Physics & Devices, 2nd ed.,
by Neamen, Irwin, Chicago, 1997.
Fundamentals of Semiconductor Theory and
Device Physics, by Shyh Wang, Prentice Hall,
1989.
©rlc L14-08Mar2011
20
Download