EE 5340Semiconductor Device TheoryLecture 06 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

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Review the Following• R. L. Carter’s web page:

– www.uta.edu/ronc/• EE 5340 web page and syllabus. (Refresh

all EE 5340 pages when downloading to assure the latest version.) All links at:– www.uta.edu/ronc/5340/syllabus.htm

• University and College Ethics Policies– www.uta.edu/studentaffairs/conduct/

• Makeup lecture at noon Friday (1/28) in 108 Nedderman Hall. This will be available on the web.

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First Assignment

• Send e-mail to ronc@uta.edu– On the subject line, put “5340 e-mail”– In the body of message include

• email address: ______________________• Your Name*: _______________________• Last four digits of your Student ID: _____

* Your name as it appears in the UTA Record - no more, no less

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Second Assignment

• Submit a signed copy of the document posted at

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

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Schedule Changes Due to University Weather

Closings• Make-up class will be held Friday, February 11 at 12 noon in 108 Nedderman Hall.

• Additional changes will be announced as necessary.

• Syllabus and lecture dates postings have been updated.

• Project Assignment has been posted in the initial version.

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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, then

Rtotal = S Ri (series Rs)• The individual scattering

mechanisms are: Lattice, ionized impurity, etc.

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Net intrinsicmobility• Considering only lattice scattering

only, , 11

is mobility total the

latticetotal

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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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Net extrinsicmobility• Considering only lattice and

impurity scattering

impuritylatticetotal

111

is mobility total the

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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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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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Figure 1.17 (p. 32 in M&K1) Low-field mobility in silicon as a function of temperature for electrons (a), and for holes (b). The solid lines represent the theoretical predictions for pure lattice scattering [5].

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Exp. m(T=300K) modelfor P, As and B in Si1

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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Exp. mobility modelfunction for Si1

Parameter As P Bmmin 52.2 68.5

44.9mmax 1417 1414

470.5Nref 9.68e16 9.20e16

2.23e17a 0.680 0.711

0.719

ref

a,d

minpn,

maxpn,min

pn,pn,

N

N1

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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 mp = 189 cm2/V-s

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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)

Figure 1.15 (p. 29) M&K Dopant density versus resistivity at 23°C (296 K) for silicon doped with phosphorus and with boron. The curves can be used with little error to represent conditions at 300 K. [W. R. Thurber, R. L. Mattis, and Y. M. Liu, National Bureau of Standards Special Publication 400–64, 42 (May 1981).]

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Net silicon extrresistivity (cont.)• Since

r = (nqmn + pqmp)-1, and

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

rp > rn, for the same NI

• 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

Figure 1.16 (p. 31 M&K) Electron and hole mobilities in silicon at 300 K as functions of the total dopant concentration. The values plotted are the results of curve fitting measurements from several sources. The mobility curves can be generated using Equation 1.2.10 with the following values of the parameters [3] (see table on next slide).

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

extrinsic and compensated materials

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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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Carrier velocity2

carriervelocity vs Efor Si,Ge, andGaAs(afterSze2)

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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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References

M&K and 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986.– See Semiconductor Device

Fundamen-tals, by Pierret, Addison-Wesley, 1996, for another treatment of the m model.

2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.

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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.