Semiconductor Device Modeling
and Characterization – EE5342
Lecture 7 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
First Assignment
• e-mail to listserv@listserv.uta.edu
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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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Schedule Changes Due to the
University Closures last week
• Plan to meet until noon some days in
the next few weeks. This way we will
make up the lost time. The first
extended class will be Wednesday,
February 9.
• The MT will be postponed until
Wednesday, February 16. All other due
dates and tests will remain the same.
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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*
5
Carrier velocity
1
saturation
• 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
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[cm/s]
vs. E
[V/cm]
(Sze2, fig. 29a)
7
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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Diffusion of
carriers
• In a gradient of electrons or holes,
p and n are not zero
• Diffusion current,`J =`Jp +`Jn (note
Dp and Dn are diffusion coefficients)

 p p
p 
Jp  qDpp  qDp  i 
j  k 
z 
 x y

 n
n
n 
Jn   qDnn   qDn  i 
j  k 
x y
z 

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Diffusion of
carriers (cont.)
• Note (p)x has the magnitude of
dp/dx and points in the direction of
increasing p (uphill)
• The diffusion current points in the
direction of decreasing p or n
(downhill) and hence the - sign in the
definition of`Jp and the + sign in the
definition of`Jn
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Diffusion of
Carriers (cont.)
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Current density
components

Note, since E  V



Jp,drift   pE  pqm pE  pqm pV



Jn,drift  nE  nqmnE  nqmn V

Jp,diffusion   qDpp

Jn,diffusion   qDn n
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Total current
density
The total current density is driven by
the carrier gradients and the potential
gradient





Jtotal  Jp,drift  Jn,drift  Jp,diff.  Jn,diff.

Jtotal    p  n V  qDpp  qDnn

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
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Doping gradient
induced E-field
•
•
•
•
•
If N = Nd-Na = N(x), then so is Ef-Efi
Define f = (Ef-Efi)/q = (kT/q)ln(no/ni)
For equilibrium, Efi = constant, but
for dN/dx not equal to zero,
Ex = -df/dx =- [d(Ef-Efi)/dx](kT/q)
= -(kT/q) d[ln(no/ni)]/dx
= -(kT/q) (1/no)[dno/dx]
= -(kT/q) (1/N)[dN/dx], N > 0
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Induced E-field
(continued)
• Let Vt = kT/q, then since
• nopo = ni2 gives no/ni = ni/po
• Ex = - Vt d[ln(no/ni)]/dx
= - Vt d[ln(ni/po)]/dx
= - Vt d[ln(ni/|N|)]/dx, N = -Na < 0
• Ex = - Vt (-1/po)dpo/dx
= Vt(1/po)dpo/dx
= Vt(1/Na)dNa/dx
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The Einstein
relationship
• For Ex = - Vt (1/no)dno/dx, and
• Jn,x = nqmnEx + qDn(dn/dx) = 0
• This requires that
nqmn[Vt (1/n)dn/dx] = qDn(dn/dx)
• Which is satisfied if
Dp
Dn kT

 Vt , likewise
 Vt
mn
q
mp
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Direct carrier
gen/recomb
(Excitation can be by light)
-
gen
+
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rec
+
E
Ec
Ef
Efi
Ec
Ev
Ev
k
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Direct gen/rec
of excess carriers
• Generation rates, Gn0 = Gp0
• Recombination rates, Rn0 = Rp0
• In equilibrium: Gn0 = Gp0 = Rn0 = Rp0
• In non-equilibrium condition:
n = no + dn and p = po + dp, where nopo=ni2
and for dn and dp > 0, the recombination
rates increase to R’n and R’p
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Direct rec for
low-level injection
• Define low-level injection as
dn = dp < no, for n-type, and
dn = dp < po, for p-type
• The recombination rates then are
R’n = R’p = dn(t)/tn0, for p-type, and
R’n = R’p = dp(t)/tp0, for n-type
• Where tn0 and tp0 are the minoritycarrier lifetimes
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Shockley-ReadHall Recomb
Indirect, like Si, so
intermediate state
ET
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E
Ec
Ef
Efi
Ec
Ev
Ev
k
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S-R-H trap
1
characteristics
• The Shockley-Read-Hall Theory
requires an intermediate “trap” site in
order to conserve both E and p
• If trap neutral when orbited (filled)
by an excess electron - “donor-like”
• Gives up electron with energy Ec - ET
• “Donor-like” trap which has given up
the extra electron is +q and “empty”
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S-R-H trap
char. (cont.)
• If trap neutral when orbited (filled)
by an excess hole - “acceptor-like”
• Gives up hole with energy ET - Ev
• “Acceptor-like” trap which has given
up the extra hole is -q and “empty”
• Balance of 4 processes of electron
capture/emission and hole capture/
emission gives the recomb rates
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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.
• 3 Physics of Semiconductor Devices, Shur, PrenticeHall, 1990.
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