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Semiconductor Device Modeling and CharacterizationEE5342, Lecture 6-Spring 2002

Professor Ronald L. Carterronc@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,

– 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).

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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, ED-29, 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 n(N,T)

and p(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 n

AHR(N,T) and pAHR(N,T).

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Question 2:

• Determine the values of the model [n

AHR(N,T) and pAHR(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 n

AHR(N,T) for all nine conditions described in the 3x3 matrix of N,t values, and a similar table will be developed for the p

AHR(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. 1677-83, 1993.) This model will be referred to as n

MBCR(T,E,N) and pMBCR(T,E,N)

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Question 3:

• Determine the same tables defined in Question 2 for the models n

MBCR(T,E,N), p

MBCR(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 |nMBCR(T,E,N) - n

AHR(N,T)|

nAHR(N,T) and

rdp |pMBCR(T,E,N) - p

AHR(N,T)|/

pAHR(N,T)

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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 n(T,N) and a p(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 forp- and n-type s/c

p-type

Ec

Ev

EFi

EFp

qp= kT ln(ni/Na)

Ev

Ec

EFi

EFnqn= kT ln(Nd/ni)

n-type

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Making contactin 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.: q = 4.05 eV (Si),

and q = qEc - EF

Eo

EcEf Efi

Ev

q (electron affinity)

qF

q(work function)

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Band diagram forp+-n jctn* at Va = 0

Ec

EfNEfi

Ev

Ec

EfP

Efi

Ev

0 xn

x-xp

-xpc xnc

qp < 0

qn > 0

qVbi = q(n - p)

*Na > Nd -> |p| > n

p-type for x<0 n-type for x>0

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• A total band bending of qVbi = q(n-p) = kT ln(NdNa/ni

2) is necessary to set EfP = EfN

• For -xp < x < 0, Efi - EfP < -qp, = |qp| so p < Na = po, (depleted of maj. carr.)

• For 0 < x < xn, EfN - Efi < qn, so n < Nd = no, (depleted of maj. carr.)

-xp < x < xn is the Depletion Region

Band diagram forp+-n at Va=0 (cont.)

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DepletionApproximation• Assume p << po = Na for -xp < x < 0, so

= q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Na for -xpc < x < -xp, so = q(Nd-Na+p-n) = 0, -xpc < x < -xp

• Assume n << no = Nd for 0 < x < xn, so = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Nd for xn < x < xnc, so = q(Nd-Na+p-n) = 0, xn < x < xnc

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Depletion approx.charge distribution

xn

x-xp

-xpc xnc

+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp = Ndxn

[Coul/cm2]

[Coul/cm2]

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Induced E-fieldin 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

QQAdxEAdVdSE 'p

'n

xx

xxx

VS

n

p

0

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Induced E-fieldin the D.R.

xn

x-xp-xpc xnc

O-O-O-

O+O+

O+

Depletion region (DR)

p-type CNR

Ex

Exposed Donor

ions

Exposed Acceptor Ions

n-type chg neutral reg

p-contact N-contact

W

0

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Review of depletion approximation

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 < xncxxn xnc-xpc -xp 0

Ev

Ec

qVbi

EFi

EFnEFp

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Review of D. A. (cont.)

nx

nnax

ppax

px

ndpada

daeff

npeff

bi

xx ,0E

,xx0 ,xxNq E

,0xx ,xxNq

- E

xx ,0E

,xNxN ,NN

NNN

,xxW ,qN

VaV2W

xxn xn

c

-xpc-xp

Ex

-Emax

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Depletion Approxi-mation (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/ni

2}, 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 --> x-

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Debye length• 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 = / --> d Ex/dx = q(Nd - n),

and since Ex = -d/dx, we have-d2/dx2 = q(Nd - n)/ to be solved

n

xxn

Nd

0

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Debye length (cont)• Since the level EFi is a reference for

equil, we set = Vt ln(n/ni)

• In the region of xn, n = ni exp(/Vt), so d2/dx2 = -q(Nd - ni e

/Vt), let = o + ’, where o = Vt ln(Nd/ni) so Nd - ni e/Vt = Nd[1 - e/Vt-o/Vt], for - o = ’ << o, the DE becomes d2’/dx2 = (q2Nd/kT)’, ’ << o

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Debye length (cont)• So ’ = ’(xn) exp[+(x-xn)/LD]+con.

and n = Nd e’/Vt, x ~ xn, where LD is the “Debye length”

material. intrinsic for 2n and type-p

for N type,-n for N pn :Note

length. transition a ,q

kTV ,

pnqV

L

i

ad

tt

D

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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,

bi

efft

da0V

dDaDV2

NV

N1

N1

W

NLNL

a

• For Va=0, & 1E13 < Na,Nd < 1E19 cm-3

13% < < 28% => DA is OK