semiconductor device modeling and characterization ee5342, lecture 5-spring 2005
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Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2005. Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc/. Equipartition theorem. The thermodynamic energy per degree of freedom is kT/2 Consequently,. Carrier velocity saturation 1. - PowerPoint PPT PresentationTRANSCRIPT
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L5 February 02 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2005
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
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L5 February 02 2
Equipartitiontheorem• The thermodynamic energy per
degree of freedom is kT/2Consequently,
sec/cm10*m
kT3v
and ,kT23
vm21
7rms
thermal2
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L5 February 02 3
Carrier velocitysaturation1
• The mobility relationship v = E is limited to “low” fields
• v < vth = (3kT/m*)1/2 defines “low”
• v = oE[1+(E/Ec)]-1/, o = v1/Ec for Si
parameter electrons holes v1 (cm/s) 1.53E9 T-0.87 1.62E8 T-0.52
Ec (V/cm) 1.01 T1.55 1.24 T1.68
2.57E-2 T0.66 0.46 T0.17
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L5 February 02 4
vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a)
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L5 February 02 5
Carrier velocitysaturation (cont.)• At 300K, for electrons, o = v1/Ec
= 1.53E9(300)-0.87/1.01(300)1.55 = 1504 cm2/V-s, the low-field mobility
• The maximum velocity (300K) is vsat = oEc = v1 = 1.53E9 (300)-0.87 = 1.07E7 cm/s
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L5 February 02 6
Diffusion ofcarriers• In a gradient of electrons or holes,
p and n are not zero
• Diffusion current,J =Jp +Jn (note Dp and Dn are diffusion coefficients)
kji
kji
zn
yn
xn
qDnqDJ
zp
yp
xp
qDpqDJ
nnn
ppp
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L5 February 02 7
Diffusion ofcarriers (cont.)• Note (p)x has the magnitude of
dp/dx and points in the direction of increasing p (uphill)
• The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition ofJp and the + sign in the definition ofJn
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L5 February 02 8
Diffusion ofCarriers (cont.)
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L5 February 02 9
Current densitycomponents
nqDJ
pqDJ
VnqEnqEJ
VpqEpqEJ
VE since Note,
ndiffusion,n
pdiffusion,p
nnndrift,n
pppdrift,p
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L5 February 02 10
Total currentdensity
nqDpqDVJ
JJJJJ
gradient
potential the and gradients carrier the
by driven is density current total The
npnptotal
.diff,n.diff,pdrift,ndrift,ptotal
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L5 February 02 11
Doping gradient induced E-field• If N = Nd-Na = N(x), then so is Ef-Efi
• Define = (Ef-Efi)/q = (kT/q)ln(no/ni)
• For equilibrium, Efi = constant, but
• for dN/dx not equal to zero,
• Ex = -d/dx =- [d(Ef-Efi)/dx](kT/q)= -(kT/q) d[ln(no/ni)]/dx= -(kT/q) (1/no)[dno/dx]= -(kT/q) (1/N)[dN/dx], N > 0
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L5 February 02 12
Induced E-field(continued)• Let Vt = kT/q, then since
• nopo = ni2 gives no/ni = ni/po
• Ex = - Vt d[ln(no/ni)]/dx = - Vt d[ln(ni/po)]/dx = - Vt d[ln(ni/|N|)]/dx, N = -Na < 0
• Ex = - Vt (-1/po)dpo/dx = Vt(1/po)dpo/dx = Vt(1/Na)dNa/dx
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L5 February 02 13
The Einsteinrelationship• For Ex = - Vt (1/no)dno/dx, and
• Jn,x = nqnEx + qDn(dn/dx) = 0
• This requires that nqn[Vt (1/n)dn/dx] =
qDn(dn/dx)
• Which is satisfied if tp
tn
n Vp
D likewise ,V
qkTD
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L5 February 02 14
Direct carriergen/recomb
gen rec
-
+ +
-
Ev
Ec
Ef
Efi
E
k
Ec
Ev
(Excitation can be by light)
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L5 February 02 15
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 + n and p = po + p, where
nopo=ni2
and for n and p > 0, the recombination rates increase to R’n and R’p
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L5 February 02 16
Direct rec forlow-level injection• Define low-level injection as
n = p < no, for n-type, andn = p < po, for p-type
• The recombination rates then areR’n = R’p = n(t)/n0, for p-type,
and R’n = R’p = p(t)/p0, for n-type
• Where n0 and p0 are the minority-carrier lifetimes
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L5 February 02 17
Shockley-Read-Hall Recomb
Ev
Ec
Ef
Efi
E
k
Ec
Ev
ET
Indirect, like Si, so intermediate state
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L5 February 02 18
S-R-H trapcharacteristics1
• 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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L5 February 02 19
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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L5 February 02 20
S-R-H recombination• Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
n (capture cross sect for electrons),
p (capture cross sect for holes), with
no = (Ntvthn)-1, and
po = (Ntvthn)-1, where n~(rBohr)2
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L5 February 02 21
S-R-Hrecomb. (cont.)• In the special case where no = po
= o the net recombination rate, U is
)pn( ,ppp and ,nnn where
kTEfiE
coshn2np
npnU
dtpd
dtnd
GRU
oo
oT
i
2i
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L5 February 02 22
S-R-H “U” functioncharacteristics• The numerator, (np-ni
2) simplifies in the case of extrinsic material at low level injection (for equil., nopo = ni
2)
• For n-type (no > n = p > po = ni2/no):
(np-ni2) = (no+n)(po+p)-ni
2 = nopo - ni
2 + nop + npo + np ~ nop (largest term)
• Similarly, for p-type, (np-ni2) ~ pon
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L5 February 02 23
S-R-H “U” functioncharacteristics (cont)• For n-type, as above, the denominator
= o{no+n+po+p+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is ono, giving U = p/o as the largest (fastest)
• For p-type, the same argument gives U = n/o
• Rec rate, U, fixed by minority carrier
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L5 February 02 24
S-R-H net recom-bination rate, U• In the special case where no = po
= o = (Ntvtho)-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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L5 February 02 25
S-R-H rec forexcess min carr• For n-type low-level injection and net
excess minority carriers, (i.e., no > n = p > po = ni
2/no),
U = p/o, (prop to exc min carr)
• For p-type low-level injection and net excess minority carriers, (i.e., po > n = p > no = ni
2/po),
U = n/o, (prop to exc min carr)
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L5 February 02 26
Minority hole lifetimes. Taken from Shur3, (p.101).
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L5 February 02 27
Minority electron lifetimes. Taken from Shur3, (p.101).
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L5 February 02 28
Parameter example
• min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni
2
• For Nd = 1E17cm3, p = 25 sec
– Why Nd and p ?
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L5 February 02 29
S-R-H rec fordeficient 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/(20cosh[(ET-Efi)/kT])
• And with the substitution that the gen lifetime, g = 20cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/g
• The intrinsic concentration drives the return to equilibrium
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L5 February 02 30
The ContinuityEquation• The chain rule for the total time
derivative dn/dt (the net generation rate of electrons) gives
n,kz
jy
ix
n
is gradient the of definition The
.dtdz
zn
dtdy
yn
dtdx
xn
tn
dtdn
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L5 February 02 31
The ContinuityEquation (cont.)
vntn
dtdn then
,BABABABA Since
.kdtdz
jdtdy
idtdx
v
is velocity vector the of definition The
zzyyxx
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L5 February 02 32
The ContinuityEquation (cont.)
etc. ,0xx
dtd
dtdx
x
since ,0dtdz
zdtdy
ydtdx
xv
RHS, the on term second the gConsiderin
.vnvnvn as
ddistribute be can operator gradient The
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L5 February 02 33
The ContinuityEquation (cont.)
.Equations" Continuity" the are
Jq1
tp
dtdp and ,J
q1
tn
dtdn
So .Jq1
tn
vntn
dtdn
have we ,vqnJ since ly,Consequent
pn
n
n
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L5 February 02 34
The ContinuityEquation (cont.)
z).y,(x, at p
or n of Change of Rate Local explicit"" the
is ,tp
or tn
RHS, the on term first The
z).y,(x, space in point particular a at p or
n of Rate Generation Net the represents
Eq. Continuity the of -V,dtdp or
dtdn LHS, The
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L5 February 02 35
The ContinuityEquation (cont.)
q).( holes and (-q) electrons for signs
in difference the Note z).y,(x, point
the of" out" flowing ionsconcentrat
p or n of rate local the is Jq1
or
Jq1
RHS, the on term second The
p
n
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L5 February 02 36
The ContinuityEquation (cont.)
inflow of rate rate generation net
change of rate Local
:as dinterprete be can Which
Jq1
dtdp
tp
:as holes the for equation
continuity the write-re can we So,
p
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L5 February 02 37
References
• 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.
• 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.