Semiconductor Device Modeling and

Characterization – EE5342 Lecture 6 – 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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Drift Current• The drift current density

(amp/cm2) is given by the point form of Ohm Law

J = (nqmn+pqmp)(Exi+ Eyj+ Ezk), so

J = (sn + sp)E = sE, where

s = nqmn+pqmp defines the conductivity

• The net current is SdJI

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Drift currentresistance• Given: a semiconductor resistor

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

• As stated previously, the conductivity,

s = nqmn + pqmp

• So the resistivity, r = 1/s = 1/(nqmn +

pqmp)

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Drift currentresistance (cont.)• Consequently, since

R = rl/AR = (nqmn + pqmp)-1(l/A)

• For n >> p, (an n-type extrinsic s/c)

R = l/(nqmnA)• For p >> n, (a p-type extrinsic s/c)

R = l/(pqmpA)

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Drift currentresistance (cont.)• Note: for an extrinsic

semiconductor and multiple scattering mechanisms, since

R = l/(nqmnA) or l/(pqmpA), and

(mn or p total)-1 = S mi-1,

thenRtotal = S Ri (series Rs)

• The individual scattering mechanisms are: Lattice, ionized impurity, etc.

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Exp. mobility modelfunction for Si1

Parameter As P Bmmin 52.2 68.5 44.9

mmax 1417 1414470.5

Nref 9.68e16 9.20e162.23e17

a 0.680 0.7110.719

ref

a,d

minpn,

maxpn,min

pn,pn,

N

N1

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Exp. mobility modelfor P, As and B in Si

0

500

1000

1500

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

Doping Concentration (cm̂ -3)

Mob

ilit

y (c

m̂2/V

-sec)

P

As

B

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Carrier mobilityfunctions (cont.)• The parameter mmax models

1/tlattice the thermal collision rate

• The parameters mmin, Nref and a model 1/timpur the impurity collision rate

• The function is approximately of the ideal theoretical form:

1/mtotal = 1/mthermal + 1/mimpurity

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Carrier mobilityfunctions (ex.)• Let Nd

= 1.78E17/cm3 of phosphorous, so mmin = 68.5, mmax = 1414, Nref = 9.20e16 and a = 0.711. Thus mn = 586 cm2/V-s

• Let Na = 5.62E17/cm3 of boron, so

mmin = 44.9, mmax = 470.5, Nref = 9.68e16 and a = 0.680. Thus mn = 189 cm2/V-s

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Lattice mobility

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

• Simple theory gives mlattice ~ T-3/2

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

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

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Ionized impuritymobility function• The mimpur is the scattering

mobility due to ionized impurities• Simple theory gives mimpur ~

T3/2/Nimpur

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

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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 m(Nd,Na,T)• Resistivity models developed for

extrinsic and compensated materials

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Net silicon (ex-trinsic) resistivity• Since

r = s-1 = (nqmn + pqmp)-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 s(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

r = (nqmn + pqmp)-1, and

mn > mp, (m = qt/m*) we have

rp > rn

• Note that since1.6(high conc.) < rp/rn < 3(low

conc.), so1.6(high conc.) < mn/mp < 3(low

conc.)

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

compensated semiconductor, r = (nqmn)-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 isr = (nqmn)-1 = [Nqmn(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 = 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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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.