ec3220_dm - characteristc times and lengths

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


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State the situation, defining differential equation, and

boundary conditions associated with each characteristic

Module 3


length and time

Derive the defining differential equations associated

with dielectric relaxation time, Debye length and diffusion

length


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Decide, for analyzing a given situation,

- which equation to start with

Module 3


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


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State the order of magnitude of, and factors governing,

the characteristic times and lengths

Module 3


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


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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


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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


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


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


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


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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


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,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)


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( )+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


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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


width of a typical p+-n junction.

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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

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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

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=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.

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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

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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

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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

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interval >> c

so as to include many scattering

events.

Validity Range of the Band Structure

Device dimensions >> a

ec3220_dm - characteristc times and lengths

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