Transcript EE 5340 Semiconductor Device Theory Lecture 05 – Spring 2011 Professor Ronald L.

EE 5340
Semiconductor Device Theory
Lecture 05 – 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.
©rlc L05-08Feb2011
2
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
©rlc L05-08Feb2011
3
Second Assignment
• Submit a signed copy of the document
posted at
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
©rlc L05-08Feb2011
4
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 will
be updated in the next 24 hours.
• Project Assignment will be posted in
the next 36 hours.
©rlc L05-08Feb2011
5
Intrinsic carrier
conc. (MB limit)
•
•
•
•
ni2 = no po = Nc Nv e-Eg/kT
Nc = 2{2pm*nkT/h2}3/2
Nv = 2{2pm*pkT/h2}3/2
Eg = 1.17 eV - aT2/(T+b)
a = 4.73E-4 eV/K
b = 636K
©rlc L05-08Feb2011
6
Classes of
semiconductors
• 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
©rlc L05-08Feb2011
7
Equilibrium
concentrations
• Charge neutrality requires
q(po + Nd+) + (-q)(no + Na-) = 0
• Assuming complete ionization, so
Nd+ = Nd and Na- = Na
• Gives two equations to be solved
simultaneously
1. Mass action,
no po = ni2, and
2. Neutrality
po + N d = n o + N a
©rlc L05-08Feb2011
8
Equilibrium
conc (cont.)
• For Nd > Na (taking the + root)
no = (Nd-Na)/2 + {[(NdNa)/2]2+ni2}1/2
• For Nd >> Na and Nd >> ni, can use the
binomial expansion, giving
no = Nd/2 + Nd/2[1 + 2ni2/Nd2 + … ]
• So no = Nd, and po = ni2/Nd in the limit
of Nd >> Na and Nd >> ni
©rlc L05-08Feb2011
9
n-type equilibrium
concentrations
• N ≡ Nd - Na , n type  N > 0
• For all N,
no = N/2 + {[N/2]2+ni2}1/2
• In most cases, N >> ni, so
no = N, and
po = ni2/no = ni2/N,
©rlc L05-08Feb2011
(Law of Mass Action is always true in equilibrium)
10
Position of the
Fermi Level
• Efi is the Fermi level
when no = po
• Ef shown is a Fermi
level for no > po
• Ef < Efi when no < po
• Efi < (Ec + Ev)/2,
which is the midband
©rlc L05-08Feb2011
11
p-type equilibrium
concentrations
• N ≡ Nd - Na , p type  N < 0
• For all N,
po = |N|/2 + {[|N|/2]2+ni2}1/2
• In most cases, |N| >> ni, so
po = |N|, and
no = ni2/po = ni2/|N|,
(Law of Mass Action is always true in equilibrium)
©rlc L05-08Feb2011
12
Position of the
Fermi Level
• Efi is the Fermi level
when no = po
• Ef shown is a Fermi
level for no > po
• Ef < Efi when no < po
• Efi < (Ec + Ev)/2,
which is the midband
©rlc L05-08Feb2011
13
EF relative
to Ec and Ev
• Inverting no = Nc exp[-(Ec-EF)/kT]
gives Ec - EF = kT ln(Nc/no)
For n-type material:
Ec - EF =kTln(Nc/Nd)=kTln[(Ncpo)/ni2]
• Inverting po = Nv exp[-(EF-Ev)/kT]
gives
EF - Ev = kT ln(Nv/po)
For p-type material:
EF - Ev = kT ln(Nv/Na)
©rlc L05-08Feb2011
14
EF relative
to Efi
• Letting ni = no gives  Ef = Efi
ni = Nc exp[-(Ec-Efi)/kT], so
Ec - Efi = kT ln(Nc/ni). Thus
EF - Efi = kT ln(no/ni) and for n-type
EF - Efi = kT ln(Nd/ni)
• Likewise
Efi - EF = kT ln(po/ni) and for ptype
Efi - EF = kT ln(Na/ni)
©rlc L05-08Feb2011
15
Locating Efi in
the bandgap
• Since
Ec - Efi = kT ln(Nc/ni), and
Efi - Ev = kT ln(Nv/ni)
• The 1st equation minus the 2nd gives
Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv)
• Since Nc = 2.8E19cm-3 > 1.04E19cm-3 =
Nv, the intrinsic Fermi level lies below
the middle of the band gap
©rlc L05-08Feb2011
16
Example
calculations
• For Nd = 3.2E16/cm3, ni = 1.4E10/cm3
no = Nd = 3.2E16/cm3
po = ni2/Nd , (po is always ni2/no)
= (1.4E10/cm3)2/3.2E16/cm3
= 6.125E3/cm3 (comp to ~1E23 Si)
• For po = Na = 4E17/cm3,
no = ni2/Na = (1.4E10/cm3)2/4E17/cm3
= 490/cm3
©rlc L05-08Feb2011
17
Sample
calculations
• Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv), so
at 300K, kT = 25.86 meV and Nc/Nv =
2.8/1.04, Efi is 12.8 meV or 1.1% below
mid-band
• For Nd = 3E17cm-3, given that
Ec - EF = kT ln(Nc/Nd), we have
Ec - EF = 25.86 meV ln(280/3),
Ec - EF = 0.117 eV =117meV
~3x(Ec - ED) what Nd gives Ec-EF =Ec/3
©rlc L05-08Feb2011
18
Equilibrium electron
conc. and energies
 no 
 Ef  Ec 
no
 exp
, or Ef  Ec  kT ln ;
Nc
 kT 
 Nc 
 no 
 Ef  Efi 
no
 exp
, or Ef  Efi  kT ln ;
ni
 kT 
 ni 
 noNv 
 Nv 
and Ef  Ev  kT ln 2   kT ln 
 po 
 ni 
©rlc L05-08Feb2011
19
Equilibrium hole
conc. and energies
po
 po 
 Ev  Ef 
 exp
, or Ev  Ef  kT ln ;
Nv
 kT 
 Nv 
 po 
po
 Efi  Ef 
 exp
, or Efi  Ef  kT ln ;
ni
 kT 
 ni 
 poNc 
 Nc 
and Ec  Ef  kT ln 2   kT ln 
 no 
 ni 
©rlc L05-08Feb2011
20
Carrier Mobility
• In an electric field, Ex, the velocity
(since ax = Fx/m* = qEx/m*) is
vx = axt = (qEx/m*)t, and the displ
x = (qEx/m*)t2/2
• If every tcoll, a collision occurs which
“resets” the velocity to <vx(tcoll)> = 0,
then <vx> = qExtcoll/m* = mEx
©rlc L05-08Feb2011
21
Carrier mobility
(cont.)
• The response function m is the
mobility.
• The mean time between collisions,
tcoll, may has several important causal
events: Thermal vibrations, donor- or
acceptor-like traps and lattice
imperfections to name a few.
• Hence mthermal = qtthermal/m*, etc.
©rlc L05-08Feb2011
22
Carrier mobility
(cont.)
• If the rate of a single contribution to
the scattering is 1/ti, then the total
scattering rate, 1/tcoll is
1
all
1

, and the



tcoll collisions  ti 
total mobility m is given by
1
all
1


m total collisions  mi 
©rlc L05-08Feb2011
23
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).
©rlc L05-08Feb2011
24
m  m min 
m max  m min

1  N i N ref

a
Parameter
Arsenic
Phosphorus
Boron
μmin
μmax
52.2
1417
68.5
1414
44.9
470.5
Nref
α
9.68 X 1016
0.680
9.20 X 1016
0.711
2.23 X 1017
0.719
Figure 1.16 (cont. M&K)
©rlc L05-08Feb2011
25
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
©rlc L05-08Feb2011
26
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)
©rlc L05-08Feb2011
27
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)
©rlc L05-08Feb2011
28
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.
©rlc L05-08Feb2011
29
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.
©rlc L05-08Feb2011
30