6.012 - Electronic Devices and Circuits
Lecture 2 - Uniform Excitation; Non-uniform conditions
• Announcements
• Review
Carrier concentrations in TE given the doping level
What happens above and below room temperature?
Drift and mobility - The full story.
• Uniform excitation: optical generation
Generation/recombination in TE
Uniform optical generation - external excitation
Population excesses, p' and n', and their transients
Low level injection; minority carrier lifetime
• Uniform excitation: applied field and optical
generation
Photoconductivity, photoconductors
• Non-uniform doping/excitation: diffusion, continuity
Fick's 1st law; diffusion
Diffusion current; total current (drift plus diffusion)
Fick's 2nd law; carrier continuity
Clif Fonstad, 9/15/09
Lecture 2 - Slide 1
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:
We find:
N D " (N d # N a )
n o " N D , po = n i2 (T) /n o " n i2 (T) /N D
n o >> n i >> po !
In Case I the concentration of electrons is much greater than
that of!holes. Silicon with net donors is called "n-type".
!
Case II - p-type: Na > Nd:, (Na - Nd) >> ni
"p-type Si"
Define the net acceptor concentration, NA:
N A " (N a # N d )
We find:
po " N A , n o = n i2 (T) / po " n i2 (T) /N A
po >> n i >> n o !
In Case II the concentration of holes is much greater than that
of electrons.
Silicon with net acceptors is called "p-type".
!
Clif Fonstad, 9/15/09
!
Lecture 2 - Slide 2
Variation of carrier concentration with temperature
(Note: for convenience we assume an n-type sample)
•
Around R.T.
Full ionization
Extrinsic doping
N d+ " N d , N a# " N a
(N d+ # N a# ) >> ni
n o " ( N d # N a ), po = n i2 n o
•
At very high T
Full ionization
Intrinsic behavior
!
N d+ " N d , N a# " N a
n i >> N d+ # N a#
n o " po " n i
•
!
At very low T
Incomplete ionization
Extrinsic doping, but with
carrier freeze-out
Clif Fonstad, 9/15/09
N d+ << N d ( assuming n " type)
N d+ " N a" >> n i
n o # ( N d+ " N a" ) << ( N d " N a ), po = n i2 n o
Lecture 2 - Slide 3
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:
sex = " µ e E x , shx = µ h E x
At low and moderate |E|, the mobility, µ, is constant.
At high |E| the velocity saturates and µ deceases.
!
Drift currents:
Net velocities imply net charge flows, which imply currents:
J exdr = "q n o sex = qµ e n o E x
J hxdr = q po shx = qµ h po E x
Note: Even though the semiconductor is no longer in thermal
!
equilibrium the hole and electron populations still have their
thermal equilibrium values.
Clif Fonstad, 9/15/09
Lecture 2 - Slide 4
Conductivity, σo:
Ohm's law on a microscale states that the drift current density is
linearly proportional to the electric field:
J xdr = " o E x
The total drift current is the sum of the hole and electron drift
currents. Using our early expressions we find:
J xdr = J exdr + J hxdr = qµ e n o E x + qµ h po E x = q (µ e n o +µ h po ) E x
!
From this we see obtain our expression for the conductivity:
" o = q (µ e n o +µ h po )
!
[S/cm]
Majority vs. minority carriers:
Drift and conductivity are dominated by the most numerous, or
"majority,"
carriers:
!
n-type
p-type
Clif Fonstad, 9/15/09
n o >> po " # o $ qµ e n o
po >> n o " # o $ qµ h po
Lecture 2 - Slide 5
!
Resistance, R, and resistivity, ρo:
Ohm's law on a macroscopic scale says that the current and
voltage are linearly related: v ab = R iD
−
w
The question is, "What is R?"
vAB
J xdr = "!o E x
v
i
with E x = AB and J xdr = D
l
w#t
σο
We have:
Combining these we find:
iD
v AB
= #o
w"t
l
!
which yields:
v AB =
l
+
iD
l
1
iD = R iD
w " t #o
where
!
t
R$
l
1
l
l
=
%o =
%o
w " t #o
w"t
A
Note: Resistivity, ρo, is defined as the inverse of the conductivity:
Clif Fonstad, 9/15/09
!
" o # 1$
o
[Ohm - cm]
Lecture 2 - Slide 6
Velocity saturation
The breakdown of Ohm's
law at large electric fields.
108
Silicon
Carrier drift velocity (cm/s)
GaAs (electrons)
Ge
107
Above: Velocity vs. field
plot at R.T. for holes
and electrons in Si (loglog plot). (Fonstad, Fig. 3.2)
106
Si
105
102
103
T = 300K
Electrons
Holes
104
Electric field (V/cm)
105
Left: Velocity-field curves
for Si, Ge, and GaAs at
R.T. (log-log plot).
106
Figure by MIT OpenCourseWare.
Clif Fonstad, 9/15/09
(Neaman, Fig. 5.7)
Lecture 2 - Slide 7
Variation of mobility with
temperature and doping
104
104
Log µn
Si
103
-3/2
(T)
T = 300 K
µp
Ge
102
Lattice
scattering
Log T
104
Mobility (cm2/V-s)
103
(T)3/2
Impurity
scattering
1016
µn (cm3 /V - S)
µn
ND = 1014 cm-3
1017
µn
103
Si
µp
102
1018
104
µn
102
50
100
200
GaAs
103
1019
T(K)
500
1000
102
1014
µp
1015
1016
1018
1019
Impurity concentration (cm-3)
Figure by MIT OpenCourseWare.
Figure by MIT OpenCourseWare.
µe vs T in Si at several doping levels
Clif Fonstad, 9/15/09
1017
µ vs doping for Si, Ge, and GaAs at R.T.
(Neaman, Fig. 5.3)
Lecture 2 - Slide 8
Having said all of this,…
…it is good to be aware that the mobilities vary with doping
and temperature, but in 6.012 we will
1. use only one value for the hole mobility in Si, and one for the
electron mobility in Si, and will not consider the variation with
doping. Typically for bulk silicon we use
µe = 1600 cm2/V-s
and
µh = 600 cm2/V-s
2. assume uniform temperature (isothermal) conditions and
room temperature operation, and
3. only consider velocity saturation when we talk about MOSFET
scaling near the end of the term.
Clif Fonstad, 9/15/09
Lecture 2 - Slide 9
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:
#
Electron concentration : n o " n o + n'(t)
$ with
Hole concentration : po " po + p'(t) %
Generation, G, and recombination, R:
In general:
$
!
&G > R #
dn
dp
=
= G " R %
dt
dt
&G < R #
'
n'(t) = p'(t)
dn
dp
=
> 0
dt
dt
dn
dp
=
< 0
dt
dt
In thermal equilibrium: G = R
!
Clif Fonstad, 9/15/09
!
G = go
R = n o po r
"
# G = R % go = n o po r = n i2 r
$
Lecture 2 - Slide 10
B. Uniform Optical Generation, gL (t), cont.
With uniform optical generation, gL(t):
G = go + gL (t)
R = n p r = ( n o + n')( po + p') r
thus
dn
dp
=
= G " R = g o +gL (t) " (n o + n')( po + p')r
dt
dt
!
The question: Given Nd, Na, and gL(t), what are n(t) and p(t)?
! answer: Using
To
(1)
(2)
(3)
dn
dp
dn'
dp'
=
=
=
dt
dt
dt
dt
g o = n o po r
n' = p'
gives one equation in one unknown*:
!
dn'
= gL (t) " ( po + n o + n')n' r
dt
* Remember: no and po are known given Nd, Na
Clif Fonstad, 9/15/09
!
Lecture 2 - Slide 11
B. Uniform Optical Generation, gL (t), cont.
This equation is non-linear: It is in general hard to solve
dn'
= gL (t) " ( po + n o + n')n' r
dt
Special Case - Low Level Injection:
!
assume p-type, po >> no
LLI: n' << po
When LLI holds our equation becomes linear, and solvable:
dn'
" gL (t) # po n' r
dt
n'
= gL (t) #
with $ min % 1 po r
$ min
This first order differential equation is very familiar to us. The
homogeneous solution is:
!
Clif Fonstad, 9/15/09
n'(t) = Ae"t # min
Lecture 2 - Slide 12
Important facts about τmin and recombination:
The minority carrier lifetime is a gauge of how quickly excess
carriers 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 importance
of 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 well built ohmic contact, and
any excess carrier reaching a contact immediately recombines, so
the excess population at an ohmic contact is identically zero.)
In most other semiconductors:
– both bulk and surface recombination are likely to be important,
but it is hard to make any further generalizations
Clif Fonstad, 9/15/09
Lecture 2 - Slide 13
Uniform material with uniform excitations
(pushing semiconductors out of thermal equilibrium)
C. Photoconductivity - drift and optical generation
When the carrier populations change because of optical
generation… gL (t) " n(t) = n o + n'(t), p(t) = po + n'(t)
Used : p'(t) = n'(t)
...the conductivity changes:
!
" (t) = q[µe n(t) + µh p(t)]
= q[µe n o + µh po ]!+ q[µe + µh ]n'(t)
= " o + " '(t)
This change is used in photoconductive detectors to sense light:
gL(t)
w # d!
iD (t) = [" o + " '(t)]
VAB = ID + id (t)
l
w#d
with id (t) = " '(t)
VAB
l
The current varies in response to
the light
gL (t) " id (t)
!
Clif Fonstad, 9/15/09
w
−
VAB
d
σ(t)
+
iD = ID + id(t)
l
Lecture 2 - Slide 14
An antique photoconductor at MIT:
A Stanley Magic
Door with a lensed
photoconductorbased sensor unit.
Do you know
where it is on
campus?
Clif Fonstad, 9/15/09
Lecture 2 - Slide 15
Modern photoconductors - mid-infrared sensors, imagers*
mid-infrared: λ = 5 to 12 µm, hν = 0.1 to 0.25 eV
Electron
energy
Electron
energy
Conducting
states
Ed
−
+
Eg ≈ 0.1
- 0.25 eV
Ed
Ed ≈ 25 meV
hν @ 5 - 12 µm
Conducting
states
−
Ed ≈ 0.1 - 0.25 eV
+
Eg ≈
1.1 eV
hν @ 5 - 12 µm
Bonding
states
Bonding
states
Density of states
Intrinsic, band to band option:
• small energy gap; ni very large
• n' (= p') small relative to no
• signal very weak
Clif Fonstad, 9/15/09
Density of states
Extrinsic, donor to band option:
• large energy gap; ni very small
• n' (= ND+) large relative to no
• signal much stronger
* Night vision, deep space imaging, thermal analysis
Lecture 2 - Slide 16
Photoconductors - quantum well infrared photodetectors
QWIPs
Photoexcited
electron
Pho
tocu
Emitter
Energy
+++
Donor
atoms
rren
t
+++
+++
Distance
© Lockheed Martin and BAE Systems. All rights reserved.
This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/fairuse.
Collector
Figure by MIT OpenCourseWare.
Upper right: A QWIP imager
photograph
Lower right: Spectral response
of a typical IR QWIPs
-3V
80
60
Blue
QWIP
40
-3V
Red
QWIP
-1V
20
0
Clif Fonstad, 9/15/09
T = 77K, TBB = 800K, f/4
100
Responsivity (mA/W)
Above: Schematic illustration of
QWIP structure and function.
-1V
3
4
5
6
7
8
9
10
Wavelength (microns)
Ref: Lockheed-Martin (now BAE
Systems), Nashua, N.H.
Figure by MIT OpenCourseWare.
Note: 5 µm ≈ 0.25 eV
Lecture 2 - Slide 17
8 µm ≈ 0.15 eV
Non-uniform doping/excitation: Diffusion
When the hole and electron populations are not uniform we have
to add diffusion currents to the drift currents we discussed before.
Diffusion flux (Fick's First Law):
Consider particles m with a concentration distribution, Cm(x).
Their random thermal motion leads to a diffusion flux density:
#Cm (x,t)
Fm (x,t) = " Dm
particles/cm2 - s]
[
#x
where Dm is the diffusion constant of the particles.
Note that the diffusion flux is down the gradient.
!
Diffusion current:
Diffusion depends only on the random thermal motion of the
particles and has nothing to do with fact that they may be charged.
However, if the particles carry a charge qm, the particle flux is also
an electric current density:
#C (x,t)
J m (x,t) = " qm Dm m
A/cm2 ]
[
#x
Clif Fonstad, 9/15/09
!
Lecture 2 - Slide 19
Non-uniform doping/excitation, cont.: Diffusion
Hole Diffusion Fluxes and Currents:
The hole concentration is p(x,t), each hole carries a charge +q,
and the hole diffusion constant is Dh. The hole diffusion flux and
current densities are :
Fh (x,t) = " Dh
#p(x,t)
#x
J h (x,t) = " qDh
Electron Diffusion Fluxes and Currents:
Similarly for electrons using n(x,t), -q, and De :
!
Fe (x,t) = " De
#n(x,t)
#x
J e (x,t) = qDe
#p(x,t)
#x
#n(x,t)
#x
Total Current Fluxes:
Adding the diffusion and drift current densities yield the total
currents:
!
Holes:
Electrons:
Clif Fonstad, 9/15/09
!
#p(x,t)
J h (x,t) = qµ h p(x,t)E(x,t) " qDh
#x
"n(x,t)
J e (x,t) = qµ e n(x,t)E(x,t) + qDe
"x
Lecture 2 - Slide 20
Non-uniform doping/excitation, cont.: Diffusion...
Continuity
Total Current Fluxes, cont.:
An important different between the drift and diffusion currents:
Holes:
Electrons:
#p(x,t)
#x
"n(x,t)
J e (x,t) = qµ e n(x,t)E(x,t) + qDe
"x
Drift depends on
J h (x,t) = qµ h p(x,t)E(x,t) " qDh
total carrier
concentration
!
Diffusion depends on
the concentration
gradient
! Relationships (Fick's Second Law):
Continuity
Another consequence of non-uniform doping/excitations is that
fluxes can vary in space, leading to concentration increases or
decreases with time:
"Fm (x,t)
"Cm (x,t)
=#
"x
"t
This effect must be added to generation and recombination when
counting carriers.
Clif Fonstad, 9/15/09
!
Lecture 2 - Slide 21
Non-uniform doping/excitation, cont.: Continuitity
Continuity, cont.:
For holes and electrons, Fick's Second Law translates to:
Holes:
Electrons:
"Fh (x,t)
1 "J h (x,t)
"p(x,t)
=
= #
"x
q "x
"t
"Fe (x,t)
1 "J e (x,t)
"n(x,t)
=
= #
"x
#q "x
"t
! factors the total expressions for the dp/dt and dn/dt are:
With these
Holes:
!
Electrons:
"p(x,t)
1 "J h (x,t)
= G#R#
"t
q "x
"n(x,t)
1 "J e (x,t)
= G#R+
"t
q "x
These can also be written as:
"p(x,t) 1 "J h (x,t) "n(x,t) 1 "J e (x,t)
#
=G#R
! "t + q "x =
"t
q "x
Clif Fonstad, 9/15/09
We'll come back to this in Lectures 3 and 6.
Lecture 2 - Slide 22
Non-uniform doping/excitation, cont.: Summary
What we have so far:
Five things we care about (i.e. want to know):
Hole and electron concentrations:
Hole and electron currents:
Electric field:
p(x,t) and n(x,t)
J hx (x,t) and J ex (x,t)
E x (x,t)
And, amazingly, we already have five equations relating them:
Hole continuity:
Electron continuity:
Hole current density:
Electron current density:
Charge conservation:
Clif Fonstad, 9/15/09
"p(x,t) 1 "J h (x,t)
+
= G # R $ Gext (x,t) # [ n(x,t) p(x,t) # n i2 ] r(t)
"t ! q "x
"n(x,t) 1 "J e (x,t)
#
= G # R $ Gext (x,t) # [ n(x,t) p(x,t) # n i2 ] r(t)
"t
q "x
"p(x,t)
J h (x,t) = qµ h p(x,t)E(x,t) # qDh
"x
"n(x,t)
J e (x,t) = qµ e n(x,t)E(x,t) + qDe
"x
" [&(x)E x (x,t)]
%(x,t) =
$ q[ p(x,t) # n(x,t) + N d (x) # N a (x)]
"x
So...we're all set, right? No, and yes.....
We'll see next time that it isn't easy to get a general solution, but we can prevail.
!
Lecture 2 - Slide 23
6.012 - Electronic Devices and Circuits
Lect 2 - Excitation; Non-Uniform Profiles - Summary
• Uniform excitation: optical generation
In TE, go(T) = nopo r(T)
Uniform illumination adds uniform generation term, gL(t)
Populations increase: no → no + n', po → po + p', and n' = p'
dn'/dt = dp'/dt = go(T) + gL(t) – np r(T) = gL(t) – [np – nopo]r(T)
focus is on minority ≈ gL(t) – n'/τmin with τmin ≡ [po r(T)]-1 if LLI holds
• Uniform excitation: both optical and electrical bb
Photoconductivity: σo → σo + σ' = q [µe (no + n') + µh (po + p')]
= σo + q (µe + µh) p'
Photoconductors: an important class of light detectors
• Non-uniform doping/excitation: diffusion added bb
Fick's first law: Fmx = - DmdCm/dx
[Jmx = -qmDmdCm/dx]
Diffusion currents: Jex,df = qDedn/dx, Jhx,df = -qDhdp/dx,
Total currents: Jex = Jex,dr + Jex,df = qnµeEx + qDedn/dx
Jhx = Jhx,dr + Jhx,df = qpµhEx − qDhdp/dx
Fick's second law: dCm/dt = -dFmx/dx [dCm/dt = -(1/qm)dJmx/dx]
Continuity: dn/dt - (1/q)dJex/dx = dp/dt + (1/q)dJhx/dx = G - R
Clif Fonstad, 9/15/09
Lecture 2 - Slide 24
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6.012 Microelectronic Devices and Circuits
Fall 2009
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