Semiconductor Device
Modeling and Characterization
EE5342, Lecture 6-Spring 2002
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
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General Instructions:
• All projects should be submitted on 8.5"
x 11" paper with a cover sheet attached,
or electronically as a single document
file which will print as such. If submitted
as a paper project report, it should be
stapled only in the upper left-hand
corner and no other cover or binder or
folder should be used.
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Format
• The cover sheet
– your name,
– the project title,
– the course name and
number, and
– your e-mail address.
• The report includes
– purpose of the project
and the theoretical
background,
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– a narrative explaining
how you did the project,
– answers to all questions
asked in the project
assignment, and a
– list of references used in
the order cited in the
report (the reference
number should appear
in the report each time
the reference is used).
3
All figures and tables should be clearly marked with a
figure or table number and caption. The caption
and labels on the figures should make the
information in the figure comprehensible without
reading further in the text of the report. Circuits
used should be shown in the text. Auxiliary
information (such as SPICE data outputs, etc.)
should be included in appropriate Appendices at
the end of the report. Be sure to describe exactly
how all results were obtained, giving enough
information for anyone who understands EE 5342
to repeat your work. All work submitted must be
original. If derived from another source, a full
bibliographical citation must be given. (See all of
Notes 5 and 6 in the syllabus.)
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• The temperature dependence of the mobility of
carriers in silicon (the Arora model - see Arora,
Hauser and Roulston, Electron and Hole Mobilities
in Silicon as a Function of Concentration and
Temperature, IEEE Trans. Electron Devices, ED29, p. 292, ff., 1982) is quoted by Casey (Devices
for Integrated Circuits : Silicon and III-V Compound
Semiconductors, by H. Craig Casey, John Wiley,
New York, 1999, p. 75) and also quoted by Muller
and Kamins (Device Electronics for Integrated
Circuits, 2nd ed., by Richard S. Muller and
Theodore I. Kamins, John Wiley and Sons, New
York, 1986, p. 35).
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Question 1:
• Careful examination of the form of
mn(N,T) and mp(N,T) (N = doping
concentration, T = temperature) will
reveal that Casey and Muller and
Kamins do not agree. Resolve the
differences and determine the correct
equation for the model. This model will
be referred to as mnAHR(N,T) and
mpAHR(N,T).
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Question 2:
• Determine the values of the model
[mnAHR(N,T) and mpAHR(N,T)] for the 3x3
matrix of values of T= 0, 30, and 60C and
N=1E15, 3E16, and 1E18 cm-3. Show your
results in table format, i.e., one table will be
values of mnAHR(N,T) for all nine conditions
described in the 3x3 matrix of N,t values,
and a similar table will be developed for the
mpAHR(N,T) values.
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• Another model is discussed by
Mohammad, Bemis, Carter and
Renbeck (Temperature, Electric field and
Doping Dependent Mobilities of
Electrons and Holes in
Semiconductors”, Solid-State
Electronics, Vol. 36, No. 12, PP. 167783, 1993.) This model will be referred to
as mnMBCR(T,E,N) and mpMBCR(T,E,N)
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Question 3:
• Determine the same tables defined in
Question 2 for the models mnMBCR(T,E,N),
mpMBCR(T,E,N)] for the case where E = 0.
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Question 4:
• Determine the tables of values for the
conditions defined in Question 2 for the
relative differences between the models
when E = 0. Use the following
definitions for the relative differences:
rdn  |mnMBCR(T,E,N) - mnAHR(N,T)|
 mnAHR(N,T) and
rdp  |mpMBCR(T,E,N) - mpAHR(N,T)|/
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 mpAHR(N,T)
Questions 5 and 6
• 5: Comment on the results of Question
4. What possible reasons can you give
for the differences between the two
models?
• 6: Comment on the application of a
mn(T,N) and a mp(T,N) model to
determine a R(T,N) model for an
integrated circuit resistor. For one thing,
what additional modeling issues would
need to be considered?
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Energy bands for
p- and n-type s/c
p-type
n-type
Ec
EFi
EFp
Ev
qfn= kT ln(Nd/ni)
qfp= kT ln(ni/Na)
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Ec
EFn
EFi
Ev
12
Making contact
in a p-n junction
• Equate the EF in
the p- and n-type
materials far from
the junction
• Eo(the free level),
Ec, Efi and Ev must
be continuous
N.B.: qc = 4.05 eV (Si),
and qf = qc + Ec - EF
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qc (electron
Eo
affinity)
qf
(work function)
Ef
qfF
Ec
Efi
Ev
13
Band diagram for
p+-n jctn* at Va = 0
Ec
qf p < 0
Efi
EfP
Ev
*Na > Nd -> |fp| > fn
p-type for x<0
-xpc
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-xp 0
qVbi = q(fn
-
fp)
EfN
Efi
qfn > 0
n-type for x>0
xn
Ec
Ev
xnc
x
14
Band diagram for
p+-n at Va=0 (cont.)
• A total band bending of
qVbi = q(fn-fp) = kT ln(NdNa/ni2)
is necessary to set EfP = EfN
• For -xp < x < 0, Efi - EfP < -qfp, = |qfp|
so p < Na = po, (depleted of maj. carr.)
• For 0 < x < xn, EfN - Efi < qfn,
so n < Nd = no, (depleted of maj.
carr.)
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31Jan02< x < x
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p
n is the Depletion Region
Depletion
Approximation
• Assume p << po = Na for -xp < x < 0, so
r = q(Nd-Na+p-n) = -qNa, -xp < x < 0,
and p = po = Na for -xpc < x < -xp, so
r = q(Nd-Na+p-n) = 0, -xpc < x < -xp
• Assume n << no = Nd for 0 < x < xn, so r
= q(Nd-Na+p-n) = qNd, 0 < x < xn, and n
= no = Nd for xn < x < xnc, so
r=
q(Nd-Na+p-n) = 0, xn < x < xnc
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Depletion approx.
charge distribution
r
+Qn’=qNdxn
[Coul/cm2]
+qNd
-xp
-xpc
xn
-qNa
Qp’=-qNaxp
[Coul/cm2]
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xnc
x
Charge
neutrality =>
Qp’ + Qn’ = 0, =>
Naxp = Ndxn
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Induced E-field
in the D.R.
• The sheet dipole of charge, due to Qp’
and Qn’ induces an electric field which
must satisfy the conditions
• Charge neutrality and Gauss’ Law*
require that Ex = 0 for -xpc < x < -xp
and Ex = 0 for -xn < x < xnc
 x  xn




  E  dS   rdV  A  Exdx   A Qn' + Qp'
 x   xp 
S
V


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 


Induced E-field
in the D.R.
Ex
p-contact
p-type
CNR
- O
+
O
n-type chg
+
O
O neutral reg
+
- O
O
Depletion
Exposed
Acceptor Ions region (DR)
W
-xpc
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N-contact
-xp 0
xn
Exposed
Donor ions
xnc
x
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Review of depletion
approximation
qVbi
EFp
Ec
EFn
EFi
Ev
-xpc -xp 0 xn
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xnc x
•
•
•
•
•
Depletion Approx.
pp << ppo, -xp < x < 0
nn << nno, 0 < x < xn
0 > Ex > -2Vbi/W,
in DR (-xp < x < xn)
pp=ppo=Na & np=npo=
ni2/Na, -xpc< x < -xp
nn=nno=Nd & pn=pno=
ni2/Nd, xn < x < xnc
20
Review of
D. A. (cont.)
-xpc-xp
Ex
xn
xnc
2Vbi  Va 
W
, W  xp + xn ,
qNeff
x
Neff
NaNd

, Na xp  Ndxn ,
Na + Nd
Ex  0, x  xp
q
Ex  - Na x + xp , xp  x  0,

q
Ex  Na x  xn , 0  x  xn ,

Ex  0, x  xn

-Emax
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
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Depletion Approximation (Summary)
• For the step junction defined by
doping Na (p-type) for x < 0 and Nd,
(n-type) for x > 0, the depletion width
W = {2(Vbi-Va)/qNeff}1/2, where
Vbi = Vt ln{NaNd/ni2}, and
Neff=NaNd/(Na+Nd). Since Naxp=Ndxn,
xn = W/(1 + Nd/Na), and
xp = W/(1 + Na/Nd).
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One-sided p+n
or n+p jctns
• If p+n, then Na >> Nd, and
NaNd/(Na + Nd) = Neff --> Nd, and
W --> xn, DR is all on lightly d. side
• If n+p, then Nd >> Na, and
NaNd/(Na + Nd) = Neff --> Na, and
W --> xp, DR is all on lightly d. side
• The net effect is that Neff --> N-,
(- = lightly doped side) and W --> xL06 31Jan02
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Debye
length
Nd
0
n
xn
x
• The DA assumes n changes from Nd to
0 discontinuously at xn, likewise, p
changes from Na to 0 discontinuously
at -xp.
• In the region of xn, Poisson’s eq is
=*E = r/ --> d Ex/dx = q(Nd n), and since Ex = -df/dx, we have
-d2f/dx2 = q(Nd - n)/ to be
solved
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Debye length
(cont)
• Since the level EFi is a reference for
equil, we set f = Vt ln(n/ni)
• In the region of xn, n = ni exp(f/Vt), so
d2f/dx2 = -q(Nd - ni ef/Vt), let
f = fo + f’, where fo = Vt ln(Nd/ni)
so Nd - ni ef/Vt = Nd[1 - ef/Vt-fo/Vt],
for f - fo = f’ << fo, the DE becomes
d2f’/dx2 = (q2Nd/kT)f’, f’ << fo
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Debye length
(cont)
• So f’ = f’(xn) exp[+(x-xn)/LD]+con.
and n = Nd ef’/Vt, x ~ xn, where
LD is the “Debye length”
Vt
kT
LD 
, Vt 
, a transition length.
qn + p
q
Note : n + p  Nd for n - type, Na for
p - type and 2ni for intrinsic material.
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Debye length
(cont)
• LD estimates the transition length of
a step-junction DR (concentrations Na and
Nd with Neff = NaNd/(Na +Nd)). Thus,
LD Na  + LD Nd   1
1  VtNeff


+

WV  0 
N
N
2Vbi
 a
d
a
• For Va=0, & 1E13 < Na,Nd < 1E19 cm-3
13% <  < 28% => DA is OK
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