Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2002
Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002
description
Transcript of Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002
![Page 1: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/1.jpg)
L04 24Jan02 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 4-Spring 2002
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
![Page 2: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/2.jpg)
L04 24Jan02 2
Summary
• The concept of mobility introduced as a response function to the electric field in establishing a drift current
• Resistivity and conductivity defined
• Model equation def for (Nd,Na,T)
• Resistivity models developed for extrinsic and compensated materials
![Page 3: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/3.jpg)
L04 24Jan02 3
Net silicon (ex-trinsic) resistivity• Since
= -1 = (nqn + pqp)-1
• The net conductivity can be obtained by using the model equation for the mobilities as functions of doping concentrations.
• The model function gives agreement with the measured (Nimpur)
![Page 4: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/4.jpg)
L04 24Jan02 4
Net silicon extrresistivity (cont.)
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.E+13 1.E+15 1.E+17 1.E+19
Doping Concentration (cm̂ -3)
Res
isti
vity
(oh
m-c
m)
P
As
B
![Page 5: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/5.jpg)
L04 24Jan02 5
Net silicon extrresistivity (cont.)• Since = (nqn + pqp)-1, and
n > p, ( = q/m*) we have
p > n
• Note that since1.6(high conc.) < p/n < 3(low conc.), so
1.6(high conc.) < n/p < 3(low conc.)
![Page 6: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/6.jpg)
L04 24Jan02 6
Net silicon (com-pensated) res.• For an n-type (n >> p) compensated
semiconductor, = (nqn)-1
• But now n = N = Nd - Na, and the mobility must be considered to be determined by the total ionized impurity scattering Nd + Na = NI
• Consequently, a good estimate is = (nqn)-1 = [Nqn(NI)]-1
![Page 7: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/7.jpg)
L04 24Jan02 7
Equipartitiontheorem• The thermodynamic energy per
degree of freedom is kT/2Consequently,
sec/cm10*m
kT3v
and ,kT23
vm21
7rms
thermal2
![Page 8: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/8.jpg)
L04 24Jan02 8
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
![Page 9: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/9.jpg)
L04 24Jan02 9
Carrier velocity2
carriervelocity vs Efor Si,Ge, andGaAs(afterSze2)
![Page 10: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/10.jpg)
L04 24Jan02 10
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
![Page 11: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/11.jpg)
L04 24Jan02 11
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
![Page 12: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/12.jpg)
L04 24Jan02 12
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
![Page 13: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/13.jpg)
L04 24Jan02 13
Diffusion ofCarriers (cont.)
![Page 14: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/14.jpg)
L04 24Jan02 14
Current densitycomponents
nqDJ
pqDJ
VnqEnqEJ
VpqEpqEJ
VE since Note,
ndiffusion,n
pdiffusion,p
nnndrift,n
pppdrift,p
![Page 15: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/15.jpg)
L04 24Jan02 15
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
![Page 16: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/16.jpg)
L04 24Jan02 16
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
![Page 17: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/17.jpg)
L04 24Jan02 17
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
![Page 18: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/18.jpg)
L04 24Jan02 18
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 ift
pt
n
n Vp
D likewise ,V
qkTD
![Page 19: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/19.jpg)
L04 24Jan02 19
Direct carriergen/recomb
gen rec
-
+ +
-
Ev
Ec
Ef
Efi
E
k
Ec
Ev
(Excitation can be by light)
![Page 20: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/20.jpg)
L04 24Jan02 20
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
![Page 21: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/21.jpg)
L04 24Jan02 21
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
![Page 22: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/22.jpg)
L04 24Jan02 22
Shockley-Read-Hall Recomb
Ev
Ec
Ef
Efi
E
k
Ec
Ev
ET
Indirect, like Si, so intermediate state
![Page 23: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/23.jpg)
L04 24Jan02 23
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”
![Page 24: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/24.jpg)
L04 24Jan02 24
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
![Page 25: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/25.jpg)
L04 24Jan02 25
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
![Page 26: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/26.jpg)
L04 24Jan02 26
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
![Page 27: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/27.jpg)
L04 24Jan02 27
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
![Page 28: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/28.jpg)
L04 24Jan02 28
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
![Page 29: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/29.jpg)
L04 24Jan02 29
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
![Page 30: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/30.jpg)
L04 24Jan02 30
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)
![Page 31: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/31.jpg)
L04 24Jan02 31
Minority hole lifetimes. Taken from Shur3, (p.101).
![Page 32: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/32.jpg)
L04 24Jan02 32
Minority electron lifetimes. Taken from Shur3, (p.101).
![Page 33: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/33.jpg)
L04 24Jan02 33
Parameter example
• min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-
36cm6Ni2
• For Nd = 1E17cm3, p = 25 sec
– Why Nd and p ?
![Page 34: Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002](https://reader035.fdocuments.us/reader035/viewer/2022062408/56813fd0550346895daab089/html5/thumbnails/34.jpg)
L04 24Jan02 34
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.