L4 January 29 1

Semiconductor Device Modeling and CharacterizationEE5342, Lecture 4-Spring 2004

Professor Ronald L. Carterronc@uta.edu

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

L4 January 29 2

Web Pages* You should be aware of information

at• R. L. Carter’s web page

– www.uta.edu/ronc/

• EE 5342 web page and syllabus– www.uta.edu/ronc/5342/syllabus.htm

• University and College Ethics Policies– www2.uta.edu/discipline/– www.uta.edu/ronc/5342/Ethics.htm

• Submit a signed copy to Dr. Carter

L4 January 29 3

First 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.

L4 January 29 4

Equilibriumconcentrations• Charge neutrality requires

q(po + Nd+) + (-q)(no + Na

-) = 0

• Assuming complete ionization, so Nd

+ = Nd and Na- = Na

• Gives two equations to be solved simultaneously

1. Mass action, no po = ni2, and

2. Neutrality po + Nd = no + Na

L4 January 29 5

Equilibrium electronconc. and energies

o

v2i

vof

i

ofif

fif

i

o

c

ocf

cf

c

o

pN

lnkTn

NnlnkTEvE and

;nn

lnkTEE or ,kT

EEexp

nn

;Nn

lnkTEE or ,kT

EEexp

Nn

L4 January 29 6

Equilibrium hole conc. and energies

o

c2i

cofc

i

offi

ffi

i

o

v

ofv

fv

v

o

nN

lnkTn

NplnkTEE and

;np

lnkTEE or ,kT

EEexp

np

;Np

lnkTEE or ,kT

EEexp

Np

L4 January 29 7

Mobility Summary

• The concept of mobility introduced as a response function to the electric field in establishing a drift current

• Resistivity and conductivity defined

• Model equation def for (Nd,Na,T)

• Resistivity models developed for extrinsic and compensated materials

L4 January 29 8

Drift currentresistance• Given: a semiconductor resistor with

length, l, and cross-section, A. What is the resistance?

• As stated previously, the conductivity, = nqn + pqp

• So the resistivity, = 1/ = 1/(nqn + pqp)

L4 January 29 9

Exp. mobility modelfunction for Si1

Parameter As P Bmin 52.2 68.5 44.9

max 1417 1414470.5

Nref 9.68e16 9.20e16 2.23e17

0.680 0.711 0.719

ref

a,d

minpn,

maxpn,min

pn,pn,

N

N1

L4 January 29 10

Exp. mobility modelfor P, As and B in Si

L4 January 29 11

Carrier mobilityfunctions (cont.)• The parameter max models 1/lattice

the thermal collision rate

• The parameters min, Nref and model 1/impur the impurity collision rate

• The function is approximately of the ideal theoretical form:

1/total = 1/thermal + 1/impurity

L4 January 29 12

Carrier mobilityfunctions (ex.)• Let Nd

= 1.78E17/cm3 of phosphorous, so min = 68.5, max = 1414, Nref = 9.20e16 and = 0.711. Thus n = 586 cm2/V-s

• Let Na = 5.62E17/cm3 of boron, so

min = 44.9, max = 470.5, Nref = 9.68e16 and = 0.680. Thus

p = 189 cm2/V-s

L4 January 29 13

Lattice mobility

• The lattice is the lattice scattering mobility due to thermal vibrations

• Simple theory gives lattice ~ T-3/2

• Experimentally n,lattice ~ T-n where n = 2.42 for electrons and 2.2 for holes

• Consequently, the model equation is lattice(T) = lattice(300)(T/300)-

n

L4 January 29 14

Ionized impuritymobility function

• The impur is the scattering mobility due to ionized impurities

• Simple theory gives impur ~ T3/2/Nimpur

• Consequently, the model equation is impur(T) = impur(300)(T/300)3/2

L4 January 29 15

Net silicon (ex-trinsic) resistivity• Since = -1 = (nqn +

pqp)-1

• The net conductivity can be obtained by using the model equation for the mobilities as functions of doping concentrations.

• The model function gives agreement with the measured (Nimpur)

L4 January 29 16

Net silicon extrresistivity (cont.)

L4 January 29 17

Net silicon extrresistivity (cont.)• Since = (nqn + pqp)-1, and

n > p, ( = q/m*) we have

p > n

• Note that since1.6(high conc.) < p/n < 3(low conc.), so

1.6(high conc.) < n/p < 3(low conc.)

L4 January 29 18

Net silicon (com-pensated) res.• For an n-type (n >> p) compensated

semiconductor, = (nqn)-1

• But now n = N = Nd - Na, and the mobility must be considered to be determined by the total ionized impurity scattering Nd + Na = NI

• Consequently, a good estimate is = (nqn)-1 = [Nqn(NI)]-1

L4 January 29 19

Equipartitiontheorem• The thermodynamic energy per

degree of freedom is kT/2Consequently,

sec/cm10*m

kT3v

and ,kT23

vm21

7rms

thermal2

L4 January 29 20

Carrier velocitysaturation1

• The mobility relationship v = E is limited to “low” fields

• v < vth = (3kT/m*)1/2 defines “low”

• v = oE[1+(E/Ec)]-1/, o = v1/Ec for Si

parameter electrons holes v1 (cm/s) 1.53E9 T-0.87 1.62E8 T-0.52

Ec (V/cm) 1.01 T1.55 1.24 T1.68

2.57E-2 T0.66 0.46 T0.17

L4 January 29 21

vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a)

L4 January 29 22

Carrier velocitysaturation (cont.)• At 300K, for electrons, o = v1/Ec

= 1.53E9(300)-0.87/1.01(300)1.55 = 1504 cm2/V-s, the low-field mobility

• The maximum velocity (300K) is vsat = oEc = v1 = 1.53E9 (300)-0.87 = 1.07E7 cm/s

L4 January 29 23

Diffusion ofcarriers• In a gradient of electrons or holes,

p and n are not zero

• Diffusion current,J =Jp +Jn (note Dp and Dn are diffusion coefficients)

kji

kji

zn

yn

xn

qDnqDJ

zp

yp

xp

qDpqDJ

nnn

ppp

L4 January 29 24

Diffusion ofcarriers (cont.)• Note (p)x has the magnitude of

dp/dx and points in the direction of increasing p (uphill)

• The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition ofJp and the + sign in the definition ofJn

L4 January 29 25

Diffusion ofCarriers (cont.)

L4 January 29 26

Current densitycomponents

nqDJ

pqDJ

VnqEnqEJ

VpqEpqEJ

VE since Note,

ndiffusion,n

pdiffusion,p

nnndrift,n

pppdrift,p

L4 January 29 27

Total currentdensity

nqDpqDVJ

JJJJJ

gradient

potential the and gradients carrier the

by driven is density current total The

npnptotal

.diff,n.diff,pdrift,ndrift,ptotal

L4 January 29 28

Doping gradient induced E-field• If N = Nd-Na = N(x), then so is Ef-Efi

• Define = (Ef-Efi)/q = (kT/q)ln(no/ni)

• For equilibrium, Efi = constant, but

• for dN/dx not equal to zero,

• Ex = -d/dx =- [d(Ef-Efi)/dx](kT/q)= -(kT/q) d[ln(no/ni)]/dx= -(kT/q) (1/no)[dno/dx]= -(kT/q) (1/N)[dN/dx], N > 0

L4 January 29 29

Induced E-field(continued)• Let Vt = kT/q, then since

• nopo = ni2 gives no/ni = ni/po

• Ex = - Vt d[ln(no/ni)]/dx = - Vt d[ln(ni/po)]/dx = - Vt d[ln(ni/|N|)]/dx, N = -Na < 0

• Ex = - Vt (-1/po)dpo/dx = Vt(1/po)dpo/dx = Vt(1/Na)dNa/dx

L4 January 29 30

The Einsteinrelationship• For Ex = - Vt (1/no)dno/dx, and

• Jn,x = nqnEx + qDn(dn/dx) = 0

• This requires that nqn[Vt (1/n)dn/dx] =

qDn(dn/dx)

• Which is satisfied if tp

tn

n Vp

D likewise ,V

qkTD

L4 January 29 31

Direct carriergen/recomb

gen rec

-

+ +

-

Ev

Ec

Ef

Efi

E

k

Ec

Ev

(Excitation can be by light)

L4 January 29 32

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 + n and p = po + p, where

nopo=ni2

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

L4 January 29 33

Direct rec forlow-level injection• Define low-level injection as

n = p < no, for n-type, andn = p < po, for p-type

• The recombination rates then areR’n = R’p = n(t)/n0, for p-type,

and R’n = R’p = p(t)/p0, for n-type

• Where n0 and p0 are the minority-carrier lifetimes

L4 January 29 34

Shockley-Read-Hall Recomb

Ev

Ec

Ef

Efi

E

k

Ec

Ev

ET

Indirect, like Si, so intermediate state

L4 January 29 35

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”

L4 January 29 36

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

L4 January 29 37

S-R-H recombination• Recombination rate determined by:

Nt (trap conc.),

vth (thermal vel of the carriers),

n (capture cross sect for electrons),

p (capture cross sect for holes), with

no = (Ntvthn)-1, and

po = (Ntvthn)-1, where n~(rBohr)2

L4 January 29 38

S-R-Hrecomb. (cont.)• In the special case where no = po

= o the net recombination rate, U is

)pn( ,ppp and ,nnn where

kTEfiE

coshn2np

npnU

dtpd

dtnd

GRU

oo

oT

i

2i

L4 January 29 39

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 > n = p > po = ni2/no):

(np-ni2) = (no+n)(po+p)-ni

2 = nopo - ni

2 + nop + npo + np ~ nop (largest term)

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

L4 January 29 40

S-R-H “U” functioncharacteristics (cont)• For n-type, as above, the denominator

= o{no+n+po+p+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is ono, giving U = p/o as the largest (fastest)

• For p-type, the same argument gives U = n/o

• Rec rate, U, fixed by minority carrier

L4 January 29 41

S-R-H net recom-bination rate, U• In the special case where no = po

= o = (Ntvtho)-1 the net rec. rate, U is

)pn( ,ppp and ,nnn where

kTEfiE

coshn2np

npnU

dtpd

dtnd

GRU

oo

oT

i

2i

L4 January 29 42

S-R-H rec forexcess min carr• For n-type low-level injection and net

excess minority carriers, (i.e., no > n = p > po = ni

2/no),

U = p/o, (prop to exc min carr)

• For p-type low-level injection and net excess minority carriers, (i.e., po > n = p > no = ni

2/po),

U = n/o, (prop to exc min carr)

L4 January 29 43

Minority hole lifetimes. Taken from Shur3, (p.101).

L4 January 29 44

Minority electron lifetimes. Taken from Shur3, (p.101).

L4 January 29 45

Parameter example

• min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni

2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

L4 January 29 46

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.