EE 5340Semiconductor Device TheoryLecture 13 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

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Doping Profile• If the net donor conc, N = N(x),

then at x, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(x)dx

• The increase in field, dEx =-(qN/e)dx, by Gauss’ Law (at x, but also all DR).

• So dVa=-xddEx= (W/e) dQ’• Further, since qN(x)dx, for both xn

and xn, we have the dC/dx as ...

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Arbitrary dopingprofile (cont.)

e

e

e

pn

j

3j

j

j

n

j

nd

ndj

pn

2j

np

2n

j

xNxN1

dV'dCq

'C'Cd

Vdq'C

xd'Cd N with

, dV'Cd

dC'xdqNdV

xdqNdVdQ''C further

,xNxN1'C

dxdx1

Wdx'dC

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Arbitrary dopingprofile (cont.)

)V(C

x and ,

dVC

1dqA

2xN

and NxNxNN

when area),( A and V, , 'CAC ,quantities measuredof terms in So,

jn

2j2

nd

0rapnd

jj

ε

ε

εεε

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Arbitrary dopingprofile (cont.)

,VV2qN'C where , junctionstep

sided-one to apply Now .dV

'dCq

'C xN

profile doping the ,xN xN orF

abij

3j

n

pn

e

e

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Arbitrary dopingprofile (cont.)

bi0j

bi23

bia0j

23

bia30j

V2qN'C when ,N

V1

VV12

1'qC

VV1'C

N so

e

e

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Example• An assymetrical p+ n junction has

a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)?

Vbi=0.816 V, Neff=9.9E15, W=0.33mm

• What is C’j0? = 31.9 nFd/cm2

• What is LD? = 0.04 mm7

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Reverse biasjunction breakdown• Avalanche breakdown

– Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons

– field dependence shown on next slide• Heavily doped narrow junction will

allow tunneling - see Neamen*, p. 274– Zener breakdown 8

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Reverse biasjunction breakdown• Assume -Va = VR >> Vbi, so Vbi-Va--

>VR

• Since Emax~ 2VR/W = (2qN-VR/(e))1/2, and VR = BV when Emax = Ecrit (N- is doping of lightly doped side ~ Neff)

BV = e (Ecrit )2/(2qN-)

• Remember, this is a 1-dim calculation

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e

e

effbimax

effbi

xa

abinx

pxx

NVaV2qE

and ,qNVaV2W

are Solutions .E reduce to tends V to

due field the since ,VVdxE

that is now change only TheEffect of V 0

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Reverse biasjunction breakdown

8/3

4/3g

Si0crit

4/3B

2/3g]2[

i

2critSi0

i

16E1/N

1.1/EqNV 120E so

,16E1/N

1.1/EV 60BV gives ,Casey

BV usually , qN2

EBV

D.A. the and diode sided-one a Assuming

εε

φεεφ

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Ecrit for reverse breakdown [M&K]

Taken from p. 198, M&K**

Casey 2model for Ecrit

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Table 4.1 (M&K* p. 186) Nomograph for silicon uniformly doped, one-sided, step junctions (300 K). (See Figure 4.15 to correct for junction curvature.) (Courtesy Bell Laboratories).

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Junction curvatureeffect on breakdown• The field due to a sphere, R, with

charge, Q is Er = Q/(4per2) for (r > R)

• V(R) = Q/(4peR), (V at the surface)• So, for constant potential, V, the

field, Er(R) = V/R (E field at surface increases for smaller spheres)

Note: corners of a jctn of depth xj are like 1/8 spheres of radius ~ xj

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Direct carriergen/recomb

gen rec

-

+ +

-

Ev

EcEfEfi

E

k

Ec

Ev

(Excitation can be by light)

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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

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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 areR’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

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Shockley-Read-Hall Recomb

Ev

EcEfEfi

E

k

Ec

Ev

ET

Indirect, like Si, so intermediate state

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S-R-H trapcharacteristics*• 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”

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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

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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 = (Ntvthsp)-1, where

sn,p~p(rBohr,n.p)2

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S-R-H net recom-bination rate, U• In the special case where tno = tpo

= to = (Ntvthso)-1 the net rec. rate, U is

)pn( ,ppp and ,nnn wherekT

EfiEcoshn2npnpnU

dtpd

dtndGRU

oo

oTi

2i

dddd

t

dd

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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 > dn = dp > po =

ni2/no):

(np-ni2) = (no+dn)(po+dp)-ni

2 = nopo - ni

2 + nodp + dnpo + dndp ~ nodp (largest term)

• Similarly, for p-type, (np-ni2) ~ podn

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References1 and M&KDevice Electronics for Integrated

Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the m model.

2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.

3 and **Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997.

Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.