L5 February 02 1

Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2005

Professor Ronald L. Carterronc@uta.edu

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

L5 February 02 2

Equipartitiontheorem• The thermodynamic energy per

degree of freedom is kT/2Consequently,

sec/cm10*m

kT3v

and ,kT23

vm21

7rms

thermal2

L5 February 02 3

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

L5 February 02 4

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

L5 February 02 5

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

L5 February 02 6

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

L5 February 02 7

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

L5 February 02 8

Diffusion ofCarriers (cont.)

L5 February 02 9

Current densitycomponents

nqDJ

pqDJ

VnqEnqEJ

VpqEpqEJ

VE since Note,

ndiffusion,n

pdiffusion,p

nnndrift,n

pppdrift,p

L5 February 02 10

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

L5 February 02 11

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

L5 February 02 12

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

L5 February 02 13

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

L5 February 02 14

Direct carriergen/recomb

gen rec

-

+ +

-

Ev

Ec

Ef

Efi

E

k

Ec

Ev

(Excitation can be by light)

L5 February 02 15

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

L5 February 02 16

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

L5 February 02 17

Shockley-Read-Hall Recomb

Ev

Ec

Ef

Efi

E

k

Ec

Ev

ET

Indirect, like Si, so intermediate state

L5 February 02 18

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”

L5 February 02 19

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

L5 February 02 20

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

L5 February 02 21

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

L5 February 02 22

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

L5 February 02 23

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

L5 February 02 24

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

L5 February 02 25

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)

L5 February 02 26

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

L5 February 02 27

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

L5 February 02 28

Parameter example

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

2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

L5 February 02 29

S-R-H rec fordeficient min carr• If n < ni and p < pi, then the S-R-H net

recomb rate becomes (p < po, n < no):

U = R - G = - ni/(20cosh[(ET-Efi)/kT])

• And with the substitution that the gen lifetime, g = 20cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/g

• The intrinsic concentration drives the return to equilibrium

L5 February 02 30

The ContinuityEquation• The chain rule for the total time

derivative dn/dt (the net generation rate of electrons) gives

n,kz

jy

ix

n

is gradient the of definition The

.dtdz

zn

dtdy

yn

dtdx

xn

tn

dtdn

L5 February 02 31

The ContinuityEquation (cont.)

vntn

dtdn then

,BABABABA Since

.kdtdz

jdtdy

idtdx

v

is velocity vector the of definition The

zzyyxx

L5 February 02 32

The ContinuityEquation (cont.)

etc. ,0xx

dtd

dtdx

x

since ,0dtdz

zdtdy

ydtdx

xv

RHS, the on term second the gConsiderin

.vnvnvn as

ddistribute be can operator gradient The

L5 February 02 33

The ContinuityEquation (cont.)

.Equations" Continuity" the are

Jq1

tp

dtdp and ,J

q1

tn

dtdn

So .Jq1

tn

vntn

dtdn

have we ,vqnJ since ly,Consequent

pn

n

n

L5 February 02 34

The ContinuityEquation (cont.)

z).y,(x, at p

or n of Change of Rate Local explicit"" the

is ,tp

or tn

RHS, the on term first The

z).y,(x, space in point particular a at p or

n of Rate Generation Net the represents

Eq. Continuity the of -V,dtdp or

dtdn LHS, The

L5 February 02 35

The ContinuityEquation (cont.)

q).( holes and (-q) electrons for signs

in difference the Note z).y,(x, point

the of" out" flowing ionsconcentrat

p or n of rate local the is Jq1

or

Jq1

RHS, the on term second The

p

n

L5 February 02 36

The ContinuityEquation (cont.)

inflow of rate rate generation net

change of rate Local

:as dinterprete be can Which

Jq1

dtdp

tp

:as holes the for equation

continuity the write-re can we So,

p

L5 February 02 37

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.

• 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.