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7/29/2019 EC3220_DM - Characteristc Times and Lengths
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At the end of this module, you should be able to
State the characteristic times and lengths associated with
- the bulk carrier population under equilibrium
Module 3
Characteristic Times and Lengths
- the relaxation of disturbance in
* carrier momentum and energy
* EHP generation / recombination
* space-charge
- the transit of an average carrier across the device length
7/29/2019 EC3220_DM - Characteristc Times and Lengths
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At the end of this module, you should be able to
State the situation, defining differential equation, and
boundary conditions associated with each characteristic
Module 3
Characteristic Times and Lengths
length and time
Derive the defining differential equations associated
with dielectric relaxation time, Debye length and diffusion
length
7/29/2019 EC3220_DM - Characteristc Times and Lengths
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At the end of this module, you should be able to
Decide, for analyzing a given situation,
- which equation to start with
Module 3
Characteristic Times and Lengths
3
- what approximations are possible for decoupling,
simplifying or eliminating any of the equations
Express the qualitative analysis of a modeling problem
using graphs of n, p, Jn, Jp, E, versus xand t
7/29/2019 EC3220_DM - Characteristc Times and Lengths
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At the end of this module, you should be able to
State the order of magnitude of, and factors governing,
the characteristic times and lengths
Module 3
Characteristic Times and Lengths
4
State how each characteristic time and length is useful
in establishing the validity range of some concept or an
approximation of a physical situation / transport equation
7/29/2019 EC3220_DM - Characteristc Times and Lengths
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RMS velocity or thermal velocity, vth
De-Broglie wavelength of thermal average carrier th= h/mn,pvth
th= h/kTC
Mean free path between collisions (length AB), lc (>th)
Bulk Carrier Population in a Large
Semiconductor under Equilibrium
5
Mean free time between collisions (time AB), c=lc / vth (>th) Minority carrier lifetime (time GR), minority (>c)
n-type semiconductor
A B
R
G
h+e-
A
B
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Relaxation of Small Disturbances in
Carrier Momentum and Energy
t= 0 t 3 t 3
p0
Momentumrelaxation time
Energyrelaxation time
momentum
Many collisions elastic, i.e. do not affect carrier energy (E) energy relaxes later than momentum, i.e. M
7/29/2019 EC3220_DM - Characteristc Times and Lengths
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p0
Relaxation of Small Disturbances in
Carrier Momentum and Energy
t= 0 t 3 t 3
M, Edepend on scattering options which are functions
of p0or KE and derivable from quantum mechanics
Momentumrelaxation time
Energyrelaxation time
momentum
7
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Flow Creation Continuity
Jn
Current density equations Continuity equations
n n nJ = qD n + qn E ( ) ( )1 = +t nn q J G - n i
DD Transport Model for Our Course
Jp
E
Electrostatic equations
E =
Gauss law
E = /i
+n pJ = J J= I J dS 00
n = n - n
p = p - p
minority
= - E d l
p p pJ = -qD p + qp E ( ) ( )1 = +t pp - q J G - p i
8
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Volume generation of excessEHPs in n-type semicon. at t= 0
Relaxation of Disturbance in EHP G/R
n-type
Steady state surfacegeneration of excess EHPs
9
t= 0 t 3p
Minority carrierdiffusion length
0 3Lp x
p- Minority carrier lifetime
0 3p t
p
p+p0
p0
p p p
L = D
p+p0
p0
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Assignment-3.1
Refer to the previous slide. Sketch n, p, Jn, Jp, Eand in an n-type semiconductor as a function ofx for two different t when excess EHP concentration
Relaxation of Disturbance in EHP G/R
10
generated in the semiconductor volume at (a) t = 0,and (b) x = 0, relaxes to equilibrium.
In each case, show(i) each of the pairs n, p and Jn, Jp on the same plot;(ii) p, n on both semi-log and linear plots.
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Assignment-3.2
The quasi-neutrality approximation is generallyvalid in a uniformly doped semiconductor regionwith excess EHPs, even if the EHP concentration
Relaxation of Disturbance in EHP G/R
n
11
varies with distance or time. Establish the validityof this approximation in the n-region (see figure
below) having surface generation of EHPs due toillumination at one end, where an electric fielddevelops out of the need to maintain |Jn| = |Jp|.
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Assignment-3.3
Consider the following two cases (see figure)
(a)n-region of a forward biased p+n junction;(b)n-region with surface generation of EHPs
Relaxation of Disturbance in EHP G/R
12
ue to um nat on.Point out similarities anddifferences in the steady
state distributions of n, p,Jn, Jp, E, for the two
cases.
n
np++ _
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Relaxation of Disturbance in Space-charge
Injection of carriers of one polarity into a
semiconductor volume at t= 0
0 3 t
n(0)
n0
Example n-type semiconductor
13
0 3d t0
(0)
Majority carrier injection at t= 0
t= 0 t d
Dielectric relaxation time d =
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Injection of carriers of one polarity into a
semiconductor volume at t= 0
+++
++
+
+
+
+
++
+
+
+
++
+n-type
Relaxation of Disturbance in Space-charge
14
Minority carrier injection at t= 0
++ + + + +
t= 0 t3d t3(d+p)
Space-chargeneutralization bymajority carriers
Decay of excesscarriers by EHPrecombination
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Assignment-3.4
Refer to the previous slide. Sketch the semi-log-
Minority carrier injection at t= 0
Relaxation of Disturbance in Space-charge
15
,semiconductor as a function of time, whenminority carriers, i.e. holes, are injected into
the semiconductor volume at t = 0. Show nand p on the same plot.
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Surface Electric Field, E Abrupt change in doping
+++ nn+
N
n-typeE
n(0)
Relaxation of Disturbance in Space-charge
16
( )+D t DL = V qNDebye length
n
0 ~3LD x
0
s
(0)
0 3LD xn0
3LD x0
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(a)Sketch the distributions of n, p, E, and as
a function of x within the semiconductor forthe surface field condition shown in the figure.
Assignment-3.5
Relaxation of Disturbance in Space-charge
17
n-type+++
++E
(b) Sketch the distributions of and as afunction of x within the semiconductor whenthe doping changes abruptly as shown.
+++ nn+
(a) (b)
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Express the depletion width of a p+-n junctionin terms of the LD of the lightly doped region.Estimate how man times L is the de letion
Assignment-3.6
Relaxation of Disturbance in Space-charge
width of a typical p+-n junction.
18
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Transit Time
It is the duration in which
- an average carrier moves across the device length, L
or
- a charge, Q, equal to that in the device volume within L
19
under the assumptions of
1) steady state 2) unipolar flow 3) no G/R within L
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Transit Time
Independent of the transport mechanism
( )
= =
2
2
1tr1 2
1
A q p x d xdx Q =
v(x) I I
20
=2
tr L V for drift across length L dropping a voltage V
for diffusion across length L= 2tr L 2D
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Consider the holes diffusing
across the n-region of aforward biased long p-nunction. The exponential
Assignment-3.7n
p(0)
p
p+
Transit Time
+_
21
0 3L
px
hole distribution implies ahole current I = qADp(p(0)-p0)/Lp injected fromthe p+ region into the n-region and an excess holecharge of Q = qA(p(0)-p0)Lp. Application of theformula tr= Q/I yields tr= Lp2/Dp. Comment on
the validity of this transit time derivation.
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Derive the transit time, tr, of holes drifting fromsource to drain via the inversion layer of an p-channel MOSFET having channel length, L, biasedon the verge of saturation by VGS > VTand VDS=
Assignment-3.8
Transit Time
VDSat. Estimate tr assuming L = 0.5m, p = 100cm2V-1s-1, VGS-VT= 1 V, and using a suitable
approximation for the variation of the inversion
charge with distance along the channel. Compareyour answer with the tr of holes across the 0.5mbasewidth of a p-n-p BJT, using asuitable value of Dp.
22
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Relationships among Characteristic
Times and Lengths
In a variety (not all) of semiconductor devices
th< c
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Considering the geometry and doping of amodern n-p-n bipolar transistor, estimate
Assignment-3.9
Relationships among Characteristic
Times and Lengths
and compare the dand tr of the base region.State what implications this might have onmodeling of the high frequency operation of
this device.
24
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( ) ( ) + + = + n p t d t B = J = J J E E E
Validity Range of the Quasi-static Approximations
( ) ( )1 = +t x n 0 minorityn q J G - n - n
Approximation Valid for
( )-1
minorityf
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104
103
102
105
104
103l
ength(
nm)
Drift-diffusion
Silicon GaAs
Validity of Transport Equations
26
0.01 0.1 1 10 100 1000
10
1
0.1
102
10
1
Characteristic time (ps)
Ch
aracterist
i
Boltzmann Transport
Quantum Transport
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A carrier can be regarded as a particle rather than wave if
1) lc >> th = h / mn,pvth, i.e. the carrier momentum is large
enough to allow sharp localization within lc; n (Si) = 120
Ao GaAs = 240 Ao at 300 K
Validity Range of the Particle Approximation for
Carriers Between Collisions
27
2) c >> th= h / kTC, i.e. the carrier energy is large enough
to allow sharp localization within c, or, the carrier
remains in a state long enough to have a well defined
energy (you get this relation from =-1 and = E / h
where E =Energy of average thermal carrier = kTC).
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A carrier can be regarded as a particle rather than wave if
3) Potential experienced by the carrier varies little over
- length = th= h / mn,pvth
Validity Range of the Particle Approximation for
Carriers Between Collisions
28
- me =th = C, .e.
fapplied voltage
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In modeling of the wheel movement, the wavy nature of
the road can be ignored if R >> , hof the wavyness
Analogy Explaining the Conditions for
Validity of the Particle Approximation
29
R
R
h
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v = + t r p t collf f F f f + s(r, p, t)i i
Validity Range of the BTE
1) Conditions allowing the particle approximation hold
2) Device dimensions >> lc and signal varies over time
30
interval >> c
so as to include many scattering
events.
Validity Range of the Band Structure
Device dimensions >> a