ee 5340 semiconductor device theory lecture 13 – spring 2011 professor ronald l. carter...
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EE 5340Semiconductor Device TheoryLecture 13 – Spring 2011
Professor Ronald L. [email protected]
http://www.uta.edu/ronc
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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 dopingprofile (cont.)
p
n
j
3j
j
j
n
j
nd
ndj
p
n2j
n
p2
n
j
xNxN
1
dV
'dCq
'C
'CdVd
q
'C
xd
'Cd N with
, dV
'CddC'xd
qNdVxd
qNdVdQ'
'C further
,xN
xN1
'C
dx
dx1
Wdx
'dC
3
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Arbitrary dopingprofile (cont.)
)V(C
x and ,
dVC
1dqA
2xN
and NxNxNN
when area),( A and V, , 'CAC ,quantities measuredof terms in So,
jn
2j2
nd
0rapnd
jj
ε
ε
εεε
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Arbitrary dopingprofile (cont.)
,VV2
qN'C where , junctionstep
sided-one to apply Now .
dV'dC
q
'C xN
profile doping the ,xN xN orF
abij
3j
n
pn
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Arbitrary dopingprofile (cont.)
bi0j
bi
23
bi
a0j
23
bi
a30j
V2qN
'C when ,N
V1
VV
121
'qC
VV
1'C
N so
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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 mm7
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Reverse biasjunction 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 8
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Reverse biasjunction 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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effbimax
eff
bi
xa
abinx
pxx
NVaV2qE
and ,qN
VaV2W
are Solutions .E reduce to tends V to
due field the since ,VVdxE
that is now change only The
Effect of V 0
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Reverse biasjunction breakdown
8/3
4/3g
Si0crit
4/3B
2/3g]2[
i
2critSi0
i
16E1/N
1.1/EqNV 120E so
,16E1/N
1.1/EV 60BV gives ,Casey
BV usually , qN2
EBV
D.A. the and diode sided-one a Assuming
εε
φεε
φ
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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 curvatureeffect 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 carriergen/recomb
gen rec
-
+ +
-
Ev
Ec
Ef
Efi
E
k
Ec
Ev
(Excitation can be by light)
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Direct gen/recof 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 forlow-level injection• Define low-level injection as
dn = dp < no, for n-type, and dn = dp < po, for p-type
• The recombination rates then areR’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 minority-carrier lifetimes
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Shockley-Read-Hall Recomb
Ev
Ec
Ef
Efi
E
k
Ec
Ev
ET
Indirect, like Si, so intermediate state
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S-R-H trapcharacteristics*• 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 trapchar. (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 recom-bination rate, U• In the special case where tno = tpo
= to = (Ntvthso)-1 the net rec. rate, U is
)pn( ,ppp and ,nnn where
kTEfiE
coshn2np
npnU
dtpd
dtnd
GRU
oo
oT
i
2i
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S-R-H “U” functioncharacteristics• The numerator, (np-ni
2) simplifies in the case of extrinsic material at low level injection (for equil., nopo
= ni2)
• For n-type (no > dn = dp > po = ni
2/no):
(np-ni2) = (no+dn)(po+dp)-ni
2 = nopo - ni
2 + nodp + dnpo + dndp ~ nodp (largest term)
• Similarly, for p-type, (np-ni2) ~
podn
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References1 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.