Uniform excitation; Non-uniform conditions · PDF fileLecture 2 - Uniform Excitation; Non-uniform conditions ... Diffusion current; total current (drift plus diffusion) Fick's 2nd

Clif Fonstad, 9/15/09 Lecture 2 - Slide 1
6.012 - Electronic Devices and CircuitsLecture 2 - Uniform Excitation; Non-uniform conditions
Announcements Review
Carrier concentrations in TE given the doping levelWhat happens above and below room temperature?Drift and mobility - The full story.
Uniform excitation: optical generationGeneration/recombination in TEUniform optical generation - external excitationPopulation excesses, p' and n', and their transientsLow level injection; minority carrier lifetime
Uniform excitation: applied field and opticalgeneration
Photoconductivity, photoconductors Non-uniform doping/excitation: diffusion, continuity
Fick's 1st law; diffusionDiffusion current; total current (drift plus diffusion)Fick's 2nd law; carrier continuity

Extrinsic Silicon, cont.: solutions in Cases I and II
Case I - n-type: Nd > Na:, (Nd - Na) >> ni "n-type Si"
Define the net donor concentration, ND:
!
ND" (N
d# N
a)
!
no" N
D, p
o = n
i
2(T) /n
o" n
i
2(T) /N
D
We find:
In Case I the concentration of electrons is much greater thanthat of holes. Silicon with net donors is called "n-type".
!
no
>> ni>> p
o
Case II - p-type: Na > Nd:, (Na - Nd) >> ni "p-type Si"
Define the net acceptor concentration, NA:
!
NA" (N
a# N
d)
!
po" N
A, n
o = n
i
2(T) / p
o" n
i
2(T) /N
A
We find:
In Case II the concentration of holes is much greater than thatof electrons. Silicon with net acceptors is called "p-type".
!
po
>> ni>> n
o

Variation of carrier concentration with temperature (Note: for convenience we assume an n-type sample)
Around R.T.Full ionizationExtrinsic doping
!
Nd
+" N
d, N
a
#" N
a
Nd
+# N
a
#( ) >> nin
o" N
d# N
a( ), po = ni2
no
At very high TFull ionizationIntrinsic behavior
!
Nd
+" N
d, N
a
#" N
a
ni>> N
d
+# N
a
#
no" p
o" n
i
At very low TIncomplete ionizationExtrinsic doping, but withcarrier freeze-out
!
Nd
+

Uniform material with uniform excitations (pushing semiconductors out of thermal equilibrium)
A. Uniform Electric Field, Ex , cont.Drift motion:
Holes and electrons acquire a constant net velocity, sx,proportional to the electric field:
At low and moderate |E|, the mobility, , is constant.At high |E| the velocity saturates and deceases.
Drift currents:Net velocities imply net charge flows, which imply currents:
Note: Even though the semiconductor is no longer in thermalequilibrium the hole and electron populations still have theirthermal equilibrium values.
!
sex
= "eE
x, s
hx=
hE
x
!
Jex
dr= "q n
os
ex= q
en
oE
xJ
hx
dr= q p
os
hx= q
hp
oE
x

Conductivity, o:Ohm's law on a microscale states that the drift current density islinearly proportional to the electric field:
The total drift current is the sum of the hole and electron driftcurrents. Using our early expressions we find:
From this we see obtain our expression for the conductivity:
Majority vs. minority carriers:Drift and conductivity are dominated by the most numerous, or"majority," carriers:
n-type
p-type
!
Jx
dr= J
ex
dr+ J
hx
dr= q
en
oE
x+ q
hp
oE
x= q
en
o+
hp
o( )Ex!
Jx
dr="
oE
x
!
"o
= q en
o+
hp
o( ) [S/cm]
!
no
>> po"#
o$ q
en
o
po
>> no"#
o$ q
hp
o

Resistance, R, and resistivity, o:Ohm's law on a macroscopic scale says that the current andvoltage are linearly related:
The question is, "What is R?"
We have:
Combining these we find:
which yields:
where
Note: Resistivity, o, is defined as the inverse of the conductivity:
!
Jx
dr="
oE
x
with Ex
=v
AB
land J
x
dr=
iD
w # t
!
vab
= R iD
!
iD
w " t=#
o
vAB
l
l
t
w
iD+
vAB
!
vAB
=l
w " t
1
#o
iD
= R iD
R $l
w " t
1
#o
=l
w " t%
o=
l
A%
o
!
"o# 1
$o
[Ohm - cm]

Clif Fonstad, 9/15/09Lecture 2 - Slide 7
Velocity saturationThe breakdown of Ohm'slaw at large electric fields.
Above: Velocity vs. fieldplot at R.T. for holesand electrons in Si (log-log plot). (Fonstad, Fig. 3.2)
Left: Velocity-field curvesfor Si, Ge, and GaAs atR.T. (log-log plot).(Neaman, Fig. 5.7)
Silicon
102105
106
107
108
103 104 105 106
Car
rier d
rift v
eloc
ity (c
m/s
)
Electric field (V/cm)
Ge
Si
GaAs (electrons)
T = 300KElectronsHoles
Figure by MIT OpenCourseWare.

Variation of mobility withtemperature and doping
e vs T in Si at several doping levels vs doping for Si, Ge, and GaAs at R.T.(Neaman, Fig. 5.3)
1019
1018
1017
1016
ND = 1014 cm-3
(T)-3/2(T)3/2
n (c
m3
/V -
S)
Log
n
50
102
103
104
100 200 500 1000T(K)
Log T
Impurityscattering
Latticescattering
Si
104
103
102
104
103
102
104
103
1021014 1015 1016 1017 1018 1019
Mob
ility
(cm
2/V-
s)
p
p
p
Ge
Si
GaAs
Impurity concentration (cm-3)
n
n
n
T = 300 K

Having said all of this,it is good to be aware that the mobilities vary with dopingand temperature, but in 6.012 we will
1. use only one value for the hole mobility in Si, and one for theelectron mobility in Si, and will not consider the variation withdoping. Typically for bulk silicon we use
e = 1600 cm2/V-s and h = 600 cm2/V-s
2. assume uniform temperature (isothermal) conditions androom temperature operation, and
3. only consider velocity saturation when we talk about MOSFETscaling near the end of the term.

Uniform material with uniform excitations (pushing semiconductors out of thermal equilibrium)
B. Uniform Optical Generation, gL (t) The carrier populations, n and p:
The light supplies energy to "break" bonds creating excess holes, p',and electrons, n'. These excess carriers are generated in pairs.Thus:
Generation, G, and recombination, R:In general:
In thermal equilibrium: G = R
!
G = go
R = nop
or
" # $
G = R % go
= nop
or = n
i
2r
!
dn
dt=
dp
dt= G " R
G > R #dn
dt=
dp
dt> 0
G < R #dn
dt=
dp
dt< 0
$
% &
' &
!
Electron concentration : n
o" n
o+ n'(t)
Hole concentration : po" p
o+ p'(t)
# $ %
with n'(t) = p'(t)

B. Uniform Optical Generation, gL (t), cont.
With uniform optical generation, gL(t):
thus
!
dn
dt=
dp
dt= G " R = g
o+g
L(t) " (n
o+ n')(p
o+ p')r
!
G = go
+ gL(t)
R = n p r = no
+ n'( ) po + p'( ) r
The question: Given Nd, Na, and gL(t), what are n(t) and p(t)?
To answer: Using
gives one equation in one unknown*:
* Remember: no and po are known given Nd, Na
!
(1) dn
dt=
dp
dt=
dn'
dt=
dp'
dt
(2) go
= no
por
(3) n' = p'
!
dn'
dt= g
L(t) " (p
o+ n
o+ n')n'r

Special Case - Low Level Injection: assume p-type, po >> noLLI: n'

Important facts about min and recombination:
The minority carrier lifetime is a gauge of how quickly excesscarriers recombine in the bulk of a semiconductor sample.
Recombination also occurs at surfaces and contacts.(Problem x in P.S. #2 deals with estimating the relative importanceof recombination in the bulk relative to that at surfaces and contacts.)
In silicon: the minority carrier lifetime is relatively very long, the surface recombination can be made negligible, and the only significant recombination occurs at ohmic contacts
(Furthermore, the lifetime is zero at a

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Uniform excitation; Non-uniform conditions · PDF fileLecture 2 - Uniform Excitation; Non-uniform conditions ... Diffusion current; total current (drift plus diffusion) Fick's 2nd