EE 5340Semiconductor Device TheoryLecture 12 - Fall 2009

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

L 12 Oct 01

Soln to Poisson’sEq in the D.R.

xnx

-xp

-xpc xnc

Ex

-Emax(V)

dx qN

dxdE

ax qN

dxdE

-Emax(V-V)

W(Va)W(Va-V)

L 12 Oct 01

effbimax

eff

bi

xa

abinx

pxx

NVaV2qE

and ,qN

VaV2W

are Solutions .E reduce to tends V to

due field the since ,VVdxE

that is now change only The

Effect of V 0

L 12 Oct 01

JunctionC (cont.)

xn

x-xp

-xpc xnc

+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp = Ndxn

Qn’=qNdxn

Qp’=-qNaxp

L 12 Oct 01

JunctionCapacitance• The junction has +Q’n=qNdxn (exposed

donors), and (exposed acceptors) Q’p=-qNaxp = -Q’n, forming a parallel sheet charge capacitor.

2da

daabi

da

daabi

a

d

dndn

cm

Coul ,

NN

NNVVq2

,NqN

NNVV2

N

N1

qNxqN'Q

L 12 Oct 01

JunctionC (cont.)• So this definition of the capacitance

gives a parallel plate capacitor with charges Q’n and Q’p(=-Q’n), separated by, L (=W), with an area A and the capacitance is then the ideal parallel plate capacitance.

• Still non-linear and Q is not zero at Va=0.

L 12 Oct 01

JunctionC (cont.)• The C-V relationship simplifies to

][Fd/cm ,NNV2

NqN'C herew

equation model a ,VV

1'C'C

2

dabi

da0j

21

bi

a0jj

L 12 Oct 01

JunctionC (cont.)• If one plots [Cj]

-2 vs. Va

Slope = -[(Cj0)2Vbi]-1

vertical axis intercept = [Cj0]-2 horizontal axis intercept = Vbi

Cj-2

Vbi

Va

Cj0-2

1M31

VVJ C0CJ

VJV

10CJACC

:Equation Model

bi0j

M

jj

,~,~

'

L 12 Oct 01

Junction Capacitance

• Estimate CJO• Define y Cj/CJO• Calculate y/(dy/dV) = {d[ln(y)]/dV}-

1

• A plot of r y/(dy/dV) vs. V has

slope = -1/M, andintercept = VJ/M

L 12 Oct 01

dy/dx - Numerical Differentiation

x y dy/ dx (central diff erence)

x(n-1) y(n-1) [y(n) - y(n-2)]/ [x(n) - x(n-2)]

x(n) y(n) [y(n+1) - y(n-1)]/ [x(n+1) - x(n-1)]

x(n+1) y(n+1) [y(n+2) - y(n)]/ [x(n+2) - x(n)]

x(n+2) y(n+2) [y(n+3) - y(n+1)]/ [x(n+3) - x(n+1)]

L 12 Oct 01

Practical Junctions• Junctions are formed by diffusion or

implantation into a uniform concentration wafer. The profile can be approximated by a step or linear function in the region of the junction.

• If a step, then previous models OK.• If linear, let the local charge density

=qax in the region of the junction.

L 12 Oct 01

Practical Jctns (cont.)

Shallow (steep) implant

N

x (depth)

Box or step junction approx.

N

x (depth)

Na(x)

xj

Linear approx.

NdNd

Na(x)

Uniform wafer con

L 12 Oct 01

Linear gradedjunction• Let the net donor concentration,

N(x) = Nd(x) - Na(x) = ax, so =qax, -xp < x < xn = xp = xo, (chg neu)

xo

-xo

= qa x

Q’n=qaxo2/2

Q’p=-qaxo2/2

x

L 12 Oct 01

Linear gradedjunction (cont.)• Let Ex(-xo) = 0, since this is the edge

of the DR (also true at +xo)

2omax

2

omaxx

x

ox-

x

ox-x

x2qa

E

where ,xx

1E)x(E

so ,axdxq

dE Law, Gauss' By

L 12 Oct 01

Linear gradedjunction (cont.)

x

Ex

-Emax

xo-xo

|area| = Vbi-Va

L 12 Oct 01

Linear gradedjunction (cont.)

31

bi

ajj

31

abio

i

otbi

3oabi

VV

10C'dV

'dQC' Letting

.qa2

VV3x so ,

nax

lnV2V

and ,X3qa2

VVV

L 12 Oct 01

Linear gradedjunction, etc.

2m1

abi

1m

j

mj

31

bi

2

oj

VV2mqB

W0'C

,BxN when 'C for formula general

the suggesting ,V12

qax2

0'C

L 12 Oct 01

Doping Profile

• If the net donor conc, N = N(x), then at x, the extra charge put into the DR when Va->Va+Va is Q’=-qN(x)x

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

• So Va=-xdEx= (W/) Q’

• Further, since qN(x)x, for both xn and xn, we have the dC/dx as ...

L 12 Oct 01

Arbitrary dopingprofile (cont.)

p

n

j

3j

j

j

n

j

nd

ndj

p

n2j

n

p2

n

j

xNxN

1

dV

'dCq

'C

'CdVd

q

'C

xd

'Cd N with

, dV

'CddC'xd

qNdVxd

qNdVdQ'

'C further

,xN

xN1

'C

dx

dx1

Wdx

'dC

L 12 Oct 01

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

L 12 Oct 01

Arbitrary dopingprofile (cont.)

,VV2

qN'C where , junctionstep

sided-one to apply Now .

dV'dC

q

'C xN

profile doping the ,xN xN orF

abij

3j

n

pn

L 12 Oct 01

Arbitrary dopingprofile (cont.)

bi0j

bi

23

bi

a0j

23

bi

a30j

V2qN

'C when ,N

V1

VV

121

'qC

VV

1'C

N so

L 12 Oct 01

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=0.33m

• What is C’j0? = 31.9 nFd/cm2

• What is LD? = 0.04 m

L 12 Oct 01

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

L 12 Oct 01

effbimax

eff

bi

xa

abinx

pxx

NVaV2qE

and ,qN

VaV2W

are Solutions .E reduce to tends V to

due field the since ,VVdxE

that is now change only The

Effect of V 0

L 12 Oct 01

Ecrit for reverse breakdown [M&K]

Taken from p. 198, M&K**

L 12 Oct 01

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

φεε

φ

L 12 Oct 01

Ecrit for reverse breakdown [M&K]

Taken from p. 198, M&K**

Casey Model for Ecrit

L 12 Oct 01

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

>VR

• Since Emax~ 2VR/W =

(2qN-VR/())1/2, and VR = BV when

Emax = Ecrit (N- is doping of lightly

doped side ~ Neff)

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

• Remember, this is a 1-dim calculation

L 12 Oct 01

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

charge, Q is Er = Q/(4r2) for (r > R)

• V(R) = Q/(4R), (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

L 12 Oct 01

BV for reverse breakdown (M&K**)

Taken from Figure 4.13, p. 198, M&K**

Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5

L 12 Oct 01

References

[M&K] Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986.

[2] Devices for Integrated Circuits: Silicon and III-V Compound Semiconductors, by H. Craig Casey, Jr., John Wiley & Sons, New York, 1999.

Bipolar Semiconductor Devices, by David J. Roulston, McGraw-Hill, Inc., New York, 1990.