semiconductor device modeling and characterization ee5342, lecture 6-spring 2010

L6 February 03 1

Semiconductor Device Modeling and CharacterizationEE5342, Lecture 6-Spring 2010

Professor Ronald L. Carterronc@uta.edu

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

Project 1A – Diode parameters to use

L6 February 03 2

Param Value UnitsIS 3.608E-16 AN 1IKF 1.716E-08 ARS 10 OhmISR 2.422E-12 ANR 2M 0.5VJ 755 mVCJ0 3.316E-15 FdTMOM 300 KRTH 500

Tasks• Using PSpice or any simulator, plot the i-v curve for

this diode, assuming Rth = 0, for several temperatures in the range 300 K < TEMP = TAMB < 304 K.

• Using this data, determine what the i-v plot would be for Rth = 500 K/W.

• Using this data, determine the maximum operating temperature for which the diode conductance is within 1% of the Rth = 0 value at 300 K.

• Do the same for a 10% tolerance.• Propose a SPICE macro which would give the Rth =

500 K/W i-v relationship.

L6 February 03 3

Example

L6 February 03 4

L6 February 03 5

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

L6 February 03 6

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]

L6 February 03 7

1-dim soln. ofGauss’ law

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

L6 February 03 8

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).

L6 February 03 9

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-

L6 February 03 10

JunctionC (cont.)

xn

x-xp

-xpc xnc

+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp =

Ndxn

Qn’=qNdxn

Qp’=-qNaxp

L6 February 03 11

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

L6 February 03 12

JunctionC (cont.)• If one plots [C’j]

-2 vs. Va

Slope = -[(C’j0)2Vbi]-1

vertical axis intercept = [C’j0]-2 horizontal axis intercept = Vbi

C’j-2

Vbi

Va

C’j0-2

L6 February 03 13

Arbitrary dopingprofile• If the net donor conc, N = N(x), then at xn,

the extra charge put into the DR when Va->Va+Va is Q’=-qN(xn)xn

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

• So Va=-(xn+xp)Ex= (W/) Q’

• Further, since N(xn)xn = N(xp)xp gives, the dC/dxn as ...

L6 February 03 14

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

L6 February 03 15

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

L6 February 03 16

Arbitrary dopingprofile (cont.)

bi0j

bi

23

bi

a0j

23

bi

a30j

V2qN

'C when ,N

V1

VV

121

'qC

VV

1'C

N so

L6 February 03 17

Arbitrary dopingprofile (cont.)

)( and ,

12

and

when area),(A and V, , '

,quantities measured of in terms So,

22

0

VCxN

dV

CdqA

NxNxNN

CAC

jnd

j

rapnd

jj

L6 February 03 18

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, the 1-dim Poisson equation is dEx/dx = q(Nd - n), and since Ex = -d/dx, the potential is the solution to -d2/dx2 = q(Nd - n)/

n

xxn

Nd

0

L6 February 03 19

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

L6 February 03 20

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

L6 February 03 21

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

L6 February 03 22

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’j? = 31.9 nFd/cm2

• What is LD? = 0.04 m

L6 February 03 23

Ideal JunctionTheory

Assumptions

• Ex = 0 in the chg neutral reg. (CNR)

• MB statistics are applicable• Neglect gen/rec in depl reg (DR)• Low level injections apply so that np < ppo for -xpc < x < -xp, and pn < nno for xn < x < xnc

• Steady State conditions

L6 February 03 24

Forward Bias Energy Bands

1eppkT/EEexpnp ta VV0nnFpFiiequilnon

1/exp 0 ta VV

ppFiFniequilnon ennkTEEnn

Ev

Ec

EFi

xn xnc-xpc -xp 0

q(Vbi-Va)

EFPEFNqVa

x

Imref, EFn

Imref, EFp

L6 February 03 25

Law of the junction(follow the min. carr.)

t

bia

n

p

p

na

t

bi

no

po

po

no

po

not

no

pot2

i

datbi

V

V-Vexp

n

n

pp

,0V when and

,V

V-exp

n

n

pp

get to Invert

.nn

lnVp

plnV

n

NNlnVV

L6 February 03 26

Law of the junction (cont.)

t

a

pt

a

n

t

a

t

a

t

bi

t

bia

VV

2ixpp

VV

2ixnn

VV

no

2iV

V

pono

pon

VV

nopoVV-V

pn

ennp also ,ennp

Junction the of Law the

enn

epn

np have We

enn nda epp for So

L6 February 03 27

Law of the junction (cont.)

dnonapop

ppnn

ppopppop

nnonnnon

a

Nnn and Npp

injection level- low Assume

.pn and pn Assume

.ppp ,nnn and

,nnn ,ppp So

. 0V for nnot' eq.-non to Switched

L6 February 03 28

pt

apop

nt

anon

V

V-

pononoV

V-V

pon

t

biaponno

xx at ,1VV

expnn sim.

xx at ,1VV

exppp so

,epp ,pepp

giving V

V-Vexpppp

t

bi

t

bia

InjectionConditions

L6 February 03 29

Ideal JunctionTheory (cont.)

Apply the Continuity Eqn in CNR

ncnn

ppcp

xxx ,Jq1

dtdn

tn

0

and

xxx- ,Jq1

dtdp

tp

0

L6 February 03 30

Ideal JunctionTheory (cont.)

ppc

nn

p2p

2

ncnpp

n2n

2

ppx

nnxx

xxx- for ,0D

n

dx

nd

and ,xxx for ,0D

p

dx

pd

giving dxdp

qDJ and

dxdn

qDJ CNR, the in 0E Since

L6 February 03 31

Ideal JunctionTheory (cont.)

)contacts( ,0xnxp and

,1en

xn

pxp

B.C. with

.xxx- ,DeCexn

xxx ,BeAexp

So .D L and D L Define

pcpncn

VV

po

pp

no

nn

ppcL

xL

x

p

ncnL

xL

x

n

pp2pnn

2n

ta

nn

pp

L6 February 03 32

Excess minoritycarrier distr fctn

1eLWsinh

Lxxsinhnxn

,xxW ,xxx- for and

1eLWsinh

Lxxsinhpxp

,xxW ,xxx For

ta

ta

VV

np

npcpop

ppcpppc

VV

pn

pncnon

nncnncn

L6 February 03 33

CarrierInjection

xn-xpc 0

ln(carrier conc)ln Naln Nd

ln ni

ln ni2/Nd

ln ni2/Na

xnc-xp

x

~Va/Vt~Va/Vt

1enxn t

aV

V

popp

1epxp t

aV

V

nonn

L6 February 03 34

Minority carriercurrents

1eLWsinh

Lxxcosh

LNDqn

xxx- for ,qDxJ

1eLWsinh

Lxxcosh

LN

Dqn

xxx for ,qDxJ

ta

p

ta

n

VV

np

npc

na

n2i

ppcdx

ndnn

VV

pn

pnc

pd

p2i

ncndxpd

pp

L6 February 03 35

Evaluating thediode current

p/nn/pp/nd/a

p/n2isp/sn

spsns

VV

spnnp

LWcothLN

DqnJ

sdefinition with JJJ where

1eJxJxJJ

then DR, in gen/rec no gminAssu

ta

L6 February 03 36

Special cases forthe diode current

nd

p2isp

pa

n2isn

nppn

pd

p2isp

na

n2isn

nppn

WN

DqnJ and ,

WND

qnJ

LW or ,LW :diode Short

LN

DqnJ and ,

LND

qnJ

LW or ,LW :diode Long

L6 February 03 37

Ideal diodeequation• Assumptions:

– low-level injection– Maxwell Boltzman statistics– Depletion approximation– Neglect gen/rec effects in DR– Steady-state solution only

• Current dens, Jx = Js expd(Va/Vt)

– where expd(x) = [exp(x) -1]

L6 February 03 38

Ideal diodeequation (cont.)• Js = Js,p + Js,n = hole curr + ele curr

Js,p = qni2Dp coth(Wn/Lp)/(NdLp) =

qni2Dp/(NdWn), Wn << Lp, “short” =

qni2Dp/(NdLp), Wn >> Lp, “long”

Js,n = qni2Dn coth(Wp/Ln)/(NaLn) =

qni2Dn/(NaWp), Wp << Ln, “short” =

qni2Dn/(NaLn), Wp >> Ln, “long”

Js,n << Js,p when Na >> Nd

L6 February 03 39

Diffnt’l, one-sided diode conductance

Va

IDStatic (steady-state) diode I-V characteristic

VQ

IQ QVa

DD dV

dIg

t

asD V

VdexpII

L6 February 03 40

Diffnt’l, one-sided diode cond. (cont.)

DQ

t

dQd

QDDQt

DQQd

tat

tQs

Va

DQd

tastasD

IV

g1

Vr ,resistance diode The

. VII where ,V

IVg then

, VV If . V

VVexpI

dV

dIVg

VVdexpIVVdexpAJJAI

Q

L6 February 03 41

Charge distr in a (1-sided) short diode

• Assume Nd << Na

• The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/Vt)

• Total chg = Q’p = Q’p = qpn(xn)Wn/2x

n

x

xnc

pn(xn

)

Wn = xnc-

xn

Q’p

pn

L6 February 03 42

Charge distr in a 1-sided short diode

• Assume Quasi-static charge distributions

• Q’p = Q’p =

qpn(xn)Wn/2

• dpn(xn) = (W/2)*

{pn(xn,Va+V) -

pn(xn,Va)}x

n

xxnc

pn(xn,Va)

Q’p

pn pn(xn,Va+V)

Q’p

L6 February 03 43

Cap. of a (1-sided) short diode (cont.)

p

x

x p

ntransitQQ

transitt

DQ

pt

DQQ

taaa

a

Ddx

Jp

qVV

V

I

DV

IV

VVddVdV

dVA

nc

n2W

Cr So,

. 2W

C ,V V When

exp2

WqApd2

)W(xpqAd

dQC Define area. diode A ,Q'Q

2n

dd

2n

dta

nn0nnn

pdpp

L6 February 03 44

General time-constant

np

a

nnnn

a

pppp

pnVa

pn

Va

DQd

CCC ecapacitanc diode total

the and ,dVdQ

Cg and ,dV

dQCg

that so time sticcharacteri a always is There

ggdV

JJdA

dVdI

Vg

econductanc the short, or long diodes, all For

QQ

L6 February 03 45

General time-constant (cont.)

times.-life carr. min. respective the

, and side, diode long

the For times. transit charge physical

the ,D2

W and ,

D2W

side, diode short the For

n0np0p

n

2p

transn,np

2n

transp,p

L6 February 03 46

General time-constant (cont.)

Fdd

transitminF

gC

and 111

by given average

the is time transition effective The

sided-one usually are diodes Practical