Semiconductor Device Modeling and

Characterization – EE5342 Lecture 7 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

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

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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.

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Second Assignment

• Submit a signed copy of the document that is posted at

www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

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Schedule Changes Due to the University Closures last week• Plan to meet until noon some days in the next few weeks. This way we will make up the lost time. The first extended class will be Wednesday, February 9.

• The MT will be postponed until Wednesday, February 16. All other due dates and tests will remain the same.

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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 = mE is limited to “low” fields

• v < vth = (3kT/m*)1/2 defines “low”

• v = moE[1+(E/Ec)b]-1/b, mo = 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

b 2.57E-2 T0.66 0.46 T0.17

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

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Carrier velocitysaturation (cont.)• At 300K, for electrons, mo = 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 = moEc

= 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 of`Jp and the + sign in the definition of`Jn

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

• For equilibrium, Efi = constant, but• for dN/dx not equal to zero, • Ex = -df/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 = nqmnEx + qDn(dn/dx) = 0• This requires that

nqmn[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 + dn and p = po + dp, where

nopo=ni2

and for dn and dp > 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

dn = dp < no, for n-type, and dn = dp < po, for p-type

• The recombination rates then areR’n = R’p = dn(t)/tn0, for p-

type, and R’n = R’p = dp(t)/tp0, for n-type

• Where tn0 and tp0 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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References *Fundamentals of Semiconductor Theory and

Device Physics, by Shyh Wang, Prentice Hall, 1989.

**Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago.

M&K = Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003.

• 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.