EE 5340Semiconductor Device TheoryLecture 5 - Fall 2009

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

L 05 Sept 08

Second Assignment

• Please print and bring to class a signed copy of the document appearing at

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

2

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Classes ofsemiconductors• Intrinsic: no = po = ni, since Na&Nd << ni,

ni2 = NcNve-Eg/kT, ~1E-13 dopant level !

• n-type: no > po, since Nd > Na

• p-type: no < po, since Nd < Na

• Compensated: no=po=ni, w/ Na- = Nd

+ > 0

• Note: n-type and p-type are usually partially compensated since there are usually some opposite- type dopants

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n-type equilibriumconcentrations• N ≡ Nd - Na , n type N > 0

• For all N,no = N/2 + {[N/2]2+ni

2}1/2

• In most cases, N >> ni, so

no = N, and

po = ni2/no = ni

2/N, (Law of Mass Action is al-

ways true in equilibrium)

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p-type equilibriumconcentrations• N ≡ Nd - Na , p type N < 0

• For all N,po = |N|/2 + {[|N|/2]2+ni

2}1/2

• In most cases, |N| >> ni, so

po = |N|, and

no = ni2/po = ni

2/|N|, (Law of Mass Action is al-

ways true in equilibrium)

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Intrinsic carrierconc. (MB limit)

• ni2 = no po = Nc Nv e-Eg/kT

• Nc = 2{2m*nkT/h2}3/2

• Nv = 2{2m*pkT/h2}3/2

• Eg = 1.17 eV - T2/(T+)

= 4.73E-4 eV/K = 636K

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Drift Current

• The drift current density (amp/cm2) is given by the point form of Ohm LawJ = (nqn+pqp)(Exi+ Eyj+ Ezk), so

J = (n + p)E = E, where

= nqn+pqp 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,

= nqn + pqp

• So the resistivity, = 1/ = 1/(nqn + pqp)

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Drift currentresistance (cont.)• Consequently, since

R = l/AR = (nqn + pqp)-1(l/A)

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

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

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

and multiple scattering mechanisms, since

R = l/(nqnA) or l/(pqpA), and

(n or p total)-1 = i-1, then

Rtotal = 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 lattice is the lattice scattering mobility due to thermal vibrations

• Simple theory gives lattice ~ T-3/2

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

• Consequently, the model equation is lattice(T) = lattice(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 = (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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Ionized impuritymobility function• The impur is the scattering mobility

due to ionized impurities

• Simple theory gives impur ~ T3/2/Nimpur

• Consequently, the model equation is impur(T) = impur(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. (T=300K) modelfor P, As and B in Si1

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Exp. mobility modelfunction for Si1

Parameter As P Bmin 52.2 68.5 44.9

max 1417 1414 470.5

Nref 9.68e169.20e162.23e17

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 max models 1/lattice

the thermal collision rate

• The parameters min, Nref and model 1/impur the impurity collision rate

• The function is approximately of the ideal theoretical form:

1/total = 1/thermal + 1/impurity

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Carrier mobilityfunctions (ex.)• Let Nd

= 1.78E17/cm3 of phosphorous, so min = 68.5, max = 1414, Nref = 9.20e16 and = 0.711. – Thus n = 586 cm2/V-s

• Let Na = 5.62E17/cm3 of boron, so min

= 44.9, max = 470.5, Nref = 9.68e16 and = 0.680. – Thus p = 189 cm2/V-s

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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)

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 = (nqn + pqp)-1, and

n > p, ( = q/m*) we have

p > n, for the same NI

• Note that since1.6(high conc.) < p/n < 3(low conc.), so

1.6(high conc.) < n/p < 3(low conc.)

L 05 Sept 08 24

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

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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References

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 model.

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