Transcript EE 5340 Semiconductor Device Theory Lecture 06 – Spring 2011 Professor Ronald L.

EE 5340
Semiconductor Device Theory
Lecture 06 – Spring 2011
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc
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 [email protected]
– 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


I   J  dS
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Drift current
resistance
• 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 current
resistance (cont.)
• Consequently, since
R = rl/A
R = (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 current
resistance (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 intrinsic
mobility
• Considering only lattice scattering
the total mobility is
1
m total
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
1
mlattice
, only,
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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 extrinsic
mobility
• Considering only lattice and impurity
scattering
the total mobility is
1
m total
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
1
mlattice

1
mimpurity
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Net silicon extr
resistivity (cont.)
• Since
r = (nqmn + pqmp)-1, and
mn > mp, (m = qt/m*) we have
rp > rn
• Note that since
1.6(high conc.) < rp/rn < 3(low conc.), so
1.6(high conc.) < mn/mp < 3(low conc.)
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Ionized impurity
mobility 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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Mobility (cm^2/V-sec)
Exp. m(T=300K) model
for P, As and B in Si1
1500
1000
P
As
500
B
0
1.E+13
1.E+15
1.E+17
1.E+19
Doping Concentration (cm^-3)
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Exp. mobility model
function for Si1
max
min
min mn, p  mn, p
mn, p  mn, p 
a
 Nd, a 


Parameter
mmin
mmax
Nref
a
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1

N
 ref 
As
P
B
52.2
68.5
44.9
1417
1414
470.5
9.68e16 9.20e16 2.23e17
0.680
0.711
0.719
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Carrier mobility
functions (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 mobility
functions (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 (extrinsic) 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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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 extr
resistivity (cont.)
• Since
r = (nqmn + pqmp)-1, and
mn > mp, (m = qt/m*) we have
rp > rn, for the same NI
• Note that since
1.6(high conc.) < rp/rn < 3(low conc.), so
1.6(high conc.) < mn/mp < 3(low conc.)
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Net silicon (compensated) 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 is
r = (nqmn)-1 = [Nqmn(NI)]-1
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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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Equipartition
theorem
• The thermodynamic energy per
degree of freedom is kT/2
Consequently,
1
2
mv
2
vrms
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thermal
3
 kT, and
2
3kT
7

 10 cm / sec
m*
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Carrier velocity
saturation1
• 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
carrier
velocity
vs E
for Si,
Ge, and
GaAs
(after
Sze2)
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Carrier velocity
saturation (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 Fundamentals, 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.
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