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Semiconductor Device Modeling and CharacterizationEE5342, Lecture 4-Spring 2002

Professor Ronald L. Carterronc@uta.edu

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

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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

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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)

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Net silicon extrresistivity (cont.)

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.E+13 1.E+15 1.E+17 1.E+19

Doping Concentration (cm̂ -3)

Res

isti

vity

(oh

m-c

m)

P

As

B

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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.)

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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

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Equipartitiontheorem• The thermodynamic energy per

degree of freedom is kT/2Consequently,

sec/cm10*m

kT3v

and ,kT23

vm21

7rms

thermal2

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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

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Carrier velocity2

carriervelocity vs Efor Si,Ge, andGaAs(afterSze2)

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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

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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

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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

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Diffusion ofCarriers (cont.)

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Current densitycomponents

nqDJ

pqDJ

VnqEnqEJ

VpqEpqEJ

VE since Note,

ndiffusion,n

pdiffusion,p

nnndrift,n

pppdrift,p

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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

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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

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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

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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 ift

pt

n

n Vp

D likewise ,V

qkTD

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Direct carriergen/recomb

gen rec

-

+ +

-

Ev

Ec

Ef

Efi

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 + n and p = po + p, where

nopo=ni2

and for n and p > 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

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

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Shockley-Read-Hall Recomb

Ev

Ec

Ef

Efi

E

k

Ec

Ev

ET

Indirect, like Si, so intermediate state

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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”

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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),

n (capture cross sect for electrons),

p (capture cross sect for holes), with

no = (Ntvthn)-1, and

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

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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

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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 > 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

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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

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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

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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)

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

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

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Parameter example

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

36cm6Ni2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

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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.