Transcript EE 5340 Semiconductor Device Theory Lecture 13 – Spring 2011 Professor Ronald L.

EE 5340
Semiconductor Device Theory
Lecture 13 – Spring 2011
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc
Doping Profile
• If the net donor conc, N = N(x), then
at x, the extra charge put into the
DR when Va->Va+dVa is dQ’=-qN(x)dx
• The increase in field, dEx =-(qN/e)dx,
by Gauss’ Law (at x, but also all DR).
• So dVa=-xddEx= (W/e) dQ’
• Further, since qN(x)dx, for both xn
and xn, we have the dC/dx as ...
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Arbitrary doping
profile (cont.)
2
C
'
dxp 
Nxn  
e 
j 

  2  1 
1 
,

dxn
dxn 
e  N  xp 
W 
dxn
dxn dC'
dQ'
further Cj' 
 qNd
 qNd
,
dV
dV
dC' dV
dCj'


dCj' Cj' dV
C'j3 
Nxn 
with N 

1 
dCj'  N  xp
dxn q dCj'

eq
dV

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



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Arbitrary doping
profile (cont.)
So, in termsof measuredquantities,
C j  A  C' j  , V, andA ( area),when
Nd  Nx n   Nx p   Na andε  ε r ε 0
2
d 1
Nd x n   
C
εA q
dV
2
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j
ε
, andx n 
C j (V)
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Arbitrary doping
profile (cont.)


For Nxn   N  xp , the doping profile
Nxn   
C'j3
dC'
eq
dV
. Now apply to one - sided
step junction , where C'j 
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eqN
,
2Vbi  Va 
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Arbitrary doping
profile (cont.)
so N   
3

 Va  2
3
C'j0 1 

 Vbi 
3

 1   Va  2 
eqC'j0    1  
V
 2

 N  , when C'j0 
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bi 
1 
 

 Vbi 
eqN
2Vbi
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Example
• An assymetrical p+ n junction has a
lightly doped concentration of 1E16
and with p+ = 1E18. What is W(V=0)?
Vbi=0.816 V, Neff=9.9E15, W=0.33mm
• What is C’j0? = 31.9 nFd/cm2
• What is LD? = 0.04 mm
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Reverse bias
junction breakdown
• Avalanche breakdown
– Electric field accelerates electrons to
sufficient energy to initiate
multiplication of impact ionization of
valence bonding electrons
– field dependence shown on next slide
• Heavily doped narrow junction will
allow tunneling - see Neamen*, p. 274
– Zener breakdown
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Reverse bias
junction breakdown
• Assume -Va = VR >> Vbi, so Vbi-Va-->VR
• Since Emax~ 2VR/W = (2qN-VR/(e))1/2,
and VR = BV when Emax = Ecrit (N- is
doping of lightly doped side ~ Neff)
BV = e (Ecrit )2/(2qN-)
• Remember, this is a 1-dim calculation
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Effect of V  0
The only change now is that
xn
  Exdx  Vbi  Va , since the field due
 xp
to Va tends to reduce Ex . Solutions are
2eVbi  Va 
W 
, and
 qNeff 
2qVbi  Va Neff 
Emax  

e


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Reverse bias
junction breakdown
Assuminga one - sideddiodeandtheD.A.
εε E
BV  φi 
2qN
2
0 Si crit

, usuallyBV  φi
60 V Eg / 1.1
, gives BV 
3/ 4
NB / 1E16
3/4

120V qN Eg / 1.1
3/2
[2]
Casey
soEcrit 
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ε 0 ε Si
N

/ 1E16
,
3/8
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Ecrit for reverse
breakdown [M&K]
Taken from p. 198, M&K**
Casey 2model for Ecrit
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Table 4.1 (M&K* p. 186) Nomograph for silicon
uniformly doped, one-sided, step junctions (300 K).
(See Figure 4.15 to correct for junction curvature.) (Courtesy Bell
Laboratories).
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Junction curvature
effect on breakdown
• The field due to a sphere, R, with
charge, Q is Er = Q/(4per2) for (r > R)
• V(R) = Q/(4peR), (V at the surface)
• So, for constant potential, V, the
field, Er(R) = V/R (E field at surface
increases for smaller spheres)
Note: corners of a jctn of depth xj are
like 1/8 spheres of radius ~ xj
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Direct carrier
gen/recomb
(Excitation can be by light)
-
gen
+
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rec
+
E
Ec
Ef
Efi
Ec
Ev
Ev
k
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Direct gen/rec
of excess carriers
• Generation rates, Gn0 = Gp0
• Recombination rates, Rn0 = Rp0
• In equilibrium: Gn0 = Gp0 = Rn0 = Rp0
• In non-equilibrium condition:
n = no + dn and p = po + dp, where nopo=ni2
and for dn and dp > 0, the recombination
rates increase to R’n and R’p
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Direct rec for
low-level injection
• Define low-level injection as
dn = dp < no, for n-type, and
dn = dp < po, for p-type
• The recombination rates then are
R’n = R’p = dn(t)/tn0, for p-type, and
R’n = R’p = dp(t)/tp0, for n-type
• Where tn0 and tp0 are the minoritycarrier lifetimes
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Shockley-ReadHall Recomb
Indirect, like Si, so
intermediate state
ET
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E
Ec
Ef
Efi
Ec
Ev
Ev
k
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S-R-H trap
characteristics*
• The Shockley-Read-Hall Theory
requires an intermediate “trap” site in
order to conserve both E and p
• If trap neutral when orbited (filled)
by an excess electron - “donor-like”
• Gives up electron with energy Ec - ET
• “Donor-like” trap which has given up
the extra electron is +q and “empty”
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S-R-H trap
char. (cont.)
• If trap neutral when orbited (filled)
by an excess hole - “acceptor-like”
• Gives up hole with energy ET - Ev
• “Acceptor-like” trap which has given
up the extra hole is -q and “empty”
• Balance of 4 processes of electron
capture/emission and hole capture/
emission gives the recomb rates
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S-R-H
recombination
• Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
sn (capture cross sect for electrons),
sp (capture cross sect for holes), with
tno = (Ntvthsn)-1, and
tpo = (Ntvthsp)-1, where sn,p~p(rBohr,n.p)2
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S-R-H net recombination rate, U
• In the special case where tno = tpo = to
= (Ntvthso)-1 the net rec. rate, U is
ddp
ddn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  dn, and p  po  dp, (dn  dp)
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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 > dn = dp > po = ni2/no):
(np-ni2) = (no+dn)(po+dp)-ni2
= nopo - ni2 + nodp + dnpo + dndp
~ nodp (largest term)
• Similarly, for p-type, (np-ni2) ~ podn
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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.
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