semiconductor device modeling and characterization ee5342, lecture 5-spring 2005
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L5 February 02 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2005
Professor Ronald L. Carterronc@uta.edu
http://www.uta.edu/ronc/
L5 February 02 2
Equipartitiontheorem• The thermodynamic energy per
degree of freedom is kT/2Consequently,
sec/cm10*m
kT3v
and ,kT23
vm21
7rms
thermal2
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
L5 February 02 4
vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a)
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
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
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
L5 February 02 8
Diffusion ofCarriers (cont.)
L5 February 02 9
Current densitycomponents
nqDJ
pqDJ
VnqEnqEJ
VpqEpqEJ
VE since Note,
ndiffusion,n
pdiffusion,p
nnndrift,n
pppdrift,p
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
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
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
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
L5 February 02 14
Direct carriergen/recomb
gen rec
-
+ +
-
Ev
Ec
Ef
Efi
E
k
Ec
Ev
(Excitation can be by light)
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
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
L5 February 02 17
Shockley-Read-Hall Recomb
Ev
Ec
Ef
Efi
E
k
Ec
Ev
ET
Indirect, like Si, so intermediate state
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”
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
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
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
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
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
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
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)
L5 February 02 26
Minority hole lifetimes. Taken from Shur3, (p.101).
L5 February 02 27
Minority electron lifetimes. Taken from Shur3, (p.101).
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 ?
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
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
L5 February 02 31
The ContinuityEquation (cont.)
vntn
dtdn then
,BABABABA Since
.kdtdz
jdtdy
idtdx
v
is velocity vector the of definition The
zzyyxx
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
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
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
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
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
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.
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