ee5342 semiconductor device modeling and characterization lecture 05 spring 2010 l.
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EE5342 – Semiconductor Device Modeling and Characterization Lecture 05-Spring 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ First Assignment Send e-mail to ronc@uta.edu On the subject line, put “5342 e-mail” In the body of message include Your email address

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ee5342 semiconductor device modeling and characterization lecture 05 spring 2010

EE5342 – Semiconductor Device Modeling and CharacterizationLecture 05-Spring 2010

Professor Ronald L. Carter

ronc@uta.edu

http://www.uta.edu/ronc/

first assignment
First Assignment
  • Send e-mail to ronc@uta.edu
    • On the subject line, put “5342 e-mail”
    • In the body of message include
      • Your email address
      • Your Name as it appears in the UTA Record - no more, no less
      • Last four digits of your Student ID: _____
      • The name you would like me to use when speaking to you.
second assignment
Second Assignment
  • e-mail to listserv@listserv.uta.edu
    • In the body of the message include subscribe EE5342
  • This will subscribe you to the EE5342 list. Will receive all EE5342 messages
  • If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.
direct carrier gen recomb

E

-

-

Ec

Ec

Ef

Efi

gen

rec

Ev

Ev

+

+

k

Direct carriergen/recomb

(Excitation can be by light)

direct gen rec of excess carriers
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 + dn and p = po + dp, where nopo=ni2

and for dn and dp > 0, the recombination rates increase to R’n and R’p

direct rec for low level injection
Direct rec forlow-level injection
  • Define low-level injection as dn = dp < no, for n-type, and dn = dp < po, for p-type
  • The recombination rates then are R’n = R’p = dn(t)/tn0, for p-type, and R’n = R’p = dp(t)/tp0, for n-type
  • Where tn0 and tp0 are the minority-carrier lifetimes
slide7

Shockley-Read-Hall Recomb

E

Indirect, like Si, so intermediate state

Ec

Ec

ET

Ef

Efi

Ev

Ev

k

s r h trap characteristics 1
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”
s r h trap char cont
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
s r h recombination
S-R-H recombination
  • Recombination rate determined by:

Nt (trap conc.),

vth (thermal vel of the carriers),

sn (capture cross sect for electrons),

sp (capture cross sect for holes), with

tno = (Ntvthsn)-1, and

tpo = (Ntvthsn)-1, where sn~p(rBohr)2

s r h recomb cont
S-R-Hrecomb. (cont.)
  • In the special case where tno = tpo = to the net recombination rate, U is
s r h u function characteristics
S-R-H “U” functioncharacteristics
  • The numerator, (np-ni2) simplifies in the case of extrinsic material at low level injection (for equil., nopo = ni2)
  • For n-type (no > dn = dp > po = ni2/no):

(np-ni2) = (no+dn)(po+dp)-ni2 = nopo - ni2 + nodp + dnpo + dndp ~ nodp (largest term)

  • Similarly, for p-type, (np-ni2) ~ podn
s r h u function characteristics cont
S-R-H “U” functioncharacteristics (cont)
  • For n-type, as above, the denominator = to{no+dn+po+dp+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is tono, giving U = dp/to as the largest (fastest)
  • For p-type, the same argument gives U = dn/to
  • Rec rate, U, fixed by minority carrier
s r h net recom bination rate u
S-R-H net recom-bination rate, U
  • In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is
s r h rec for excess min carr
S-R-H rec forexcess min carr
  • For n-type low-level injection and net excess minority carriers, (i.e., no > dn = dp > po = ni2/no),

U = dp/to, (prop to exc min carr)

  • For p-type low-level injection and net excess minority carriers, (i.e., po > dn = dp > no = ni2/po),

U = dn/to, (prop to exc min carr)

parameter example
Parameter example
  • tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2
  • For Nd = 1E17cm3, tp = 25 msec
    • Why Nd and tp ?
s r h rec for deficient min carr
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/(2t0cosh[(ET-Efi)/kT])

  • And with the substitution that the gen lifetime, tg = 2t0cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/tg
  • The intrinsic concentration drives the return to equilibrium
the continuity equation
The ContinuityEquation
  • The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives
energy bands for p and n type s c
p-type

Ec

Ec

Ev

EFn

qfn= kT ln(Nd/ni)

EFi

Ev

Energy bands forp- and n-type s/c

n-type

EFi

qfp= kT ln(ni/Na)

EFp

making contact in a p n junction

Eo

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.: qc = 4.05 eV (Si),

and qf = qc + Ec - EF

qc(electron affinity)

qf

(work function)

Ec

Ef

Efi

qfF

Ev

band diagram for p n jctn at v a 0

EfN

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

Ec

qVbi = q(fn -fp)

qfp < 0

Ec

Efi

EfP

Ev

Efi

qfn > 0

*Na > Nd -> |fp|> fn

Ev

p-type for x<0

n-type for x>0

x

-xpc

xn

0

-xp

xnc

band diagram for p n at v a 0 cont
Band diagram forp+-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.)

-xp < x < xn is the Depletion Region

depletion approximation
DepletionApproximation
  • Assume p << po = Nafor -xp < x < 0, so r = q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Nafor -xpc < x < -xp, so r = q(Nd-Na+p-n) = 0, -xpc < x < -xp
  • Assume n << no = Ndfor 0 < x < xn, so r = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Ndfor xn < x < xnc, so r = q(Nd-Na+p-n) = 0, xn < x < xnc
poisson s equation
Poisson’sEquation
  • The electric field at (x,y,z) is related to the charge density r=q(Nd-Na-p-n) by the Poisson Equation:
poisson s equation33
Poisson’sEquation
  • For n-type material, N = (Nd - Na) > 0, no = N, and (Nd-Na+p-n)=-dn +dp +ni2/N
  • For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = dp-dn-ni2/N
  • So neglecting ni2/N, [r=(Nd-Na+p-n)]
depletion approx charge distribution
Depletion approx.charge distribution

r

+Qn’=qNdxn

+qNd

[Coul/cm2]

-xp

x

-xpc

xn

xnc

Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn

-qNa

Qp’=-qNaxp

[Coul/cm2]

induced e field in the d r
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 thatEx = 0 for -xpc < x < -xp and Ex = 0 for -xn < x < xnc

h0

induced e field in the d r39

O

O

O

O

O

O

+

+

+

-

-

-

Induced E-fieldin the D.R.

Ex

N-contact

p-contact

p-type CNR

n-type chg neutral reg

Depletion region (DR)

Exposed Donor ions

Exposed Acceptor Ions

W

x

-xpc

-xp

xn

xnc

0

1 dim soln of gauss law
1-dim soln. ofGauss’ law

Ex

-xp

xn

xnc

-xpc

x

-Emax

depletion approxi mation summary
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 = {2e(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).
references
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.