Transcript Lecture 13

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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Minority Carrier Lifetime
Fig 5.24
Low-level photoinjection into an
n-type semiconductor in which
nn > n0
• Consider an n-type semiconductor doped
5E16 cm-3 that is uniformly illuminated with
an appropriate l of light to photogenerate
EHPs
• In n-type semiconductors, electrons are the
majority carriers and holes are the minority
carriers
• When we illuminate the semiconductor, EHPs
are created
• Then, at an instant during illumination, the
semiconductor will be comprised of EHP
generated e- and h+, plus what was already
there at thermal equilibrium
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.25
• nno: majority carrier concentration (econcentration in an n-type
semiconductor) in thermal equilibrium
in the dark. These e- are thermally
ionized from the donors
• pno: minority carrier concentration (h+
concentration in an n-type
semiconductor) in thermal equilibrium
in the dark. These h+ are thermally
generated across the bandgap
• In both cases, the subscript no refers to
an n-type semiconductor and thermal
equilibrium conditions, respectively.
• Thermal equilibrium means that mass
action law is obeyed:
Low-level injection in an n-type
semiconductor does not significantly affect nn
but drastically affects the minority carrier
concentration pn.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• The electron and hole concentrations at
any instant during illumination are
denoted by nn and pn. These are defined
as the intstaneous majority (e-) and
minority (h+) concentrations,
respectively.
• At any instant and at any location in the
semiconductor, we define the departure
from equilibrium by excess
concentrations as follows:
Dnn is excess e- (majority) concentration:
Fig 5.25
Low-level injection in an n-type
semiconductor does not significantly affect nn
but drastically affects the minority carrier
concentration pn.
Dpn is excess h+ (majority) concentration:
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• So, under illumination, at any instant:
• Photoexcitation creates EHPs or an
equal number of e- and h+ such that
and mass action law is obeyed
• Recall that:
Fig 5.25
Low-level injection in an n-type
semiconductor does not significantly affect nn
but drastically affects the minority carrier
concentration pn.
since nno and pno depend only on temp
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• So, under illumination, at any instant:
• Photoexcitation creates EHPs or an
equal number of e- and h+ such that
and mass action law is obeyed
• Recall that:
Fig 5.25
Low-level injection in an n-type
semiconductor does not significantly affect nn
but drastically affects the minority carrier
concentration pn.
Since nno and pno depend only on temp
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Light is switched on (B), then
off again (C)

Fig 5.26
Illumination of an n-type semiconductor results in excess electron and hole concentrations.
After the illumination, the recombination process restores equilibrium; the excess electrons
and holes simply recombine.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Minority Carrier Lifetime
Recombination time depends
on the semiconductor
material, impurities, crystal
defects, temp, etc.
• Right after the light is turned off the condition pn = Dpn must eventually go back to
the dark case, pn = pno, thus the excess minority carriers, Dpn, and excess majority
carriers, Dnn, must be removed
• This removal occurs by recombination
• Excess h+ recombine with available e• However, this process takes time because the electrons and holes need to find each
other
• Thus, minority carrier lifetime, th, describes the rate of recombination
• This is defined as the mean time the h+ is free in the VB before recombining with
an e• Alternatively, an equivalent definition is that 1/th is the average probability per unit
time that a hole will recombine with an electron.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Excess Minority Carrier Concentration
• Ex: Let’s say minority carrier lifetime is 10 s, and there are ~1000 holes. Then, these
excess holes will be disappearing at a rate of 100 per second
• The rate of recombination of excess minority carriers is simply: Dpn/tn
• Thus, at any instant:
Rate of increase
in excess h+
=
Rate of
photogeneration
–
Rate of recombination
of excess h+
dDpn
Dpn
 Gph 
dt
th
This assumes weak
injection: Dpn < nno
Dpn = excess hole (minority carrier) concentration in n-type
t = time
Gph = rate of photogeneration
th = minority carrier lifetime (mean recombination time)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.27
Illumination is switched on at time t = 0 and then off at t= toff.
The excess minority carrier concentration Dpn(t) rises exponentially to its steady-state value
with a time constant th. From toff, the excess minority carrier concentration decays
exponentially to its equilibrium value.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Definition of Particle Flux
Example typically used: diffusion
If concentration gradient, then diffusion occurs from high to low concentration
DN

ADt
 = particle flux, DN = number of particles crossing A in a time interval Dt, A =
area, Dt = time interval
Definition of Current Density
J  Q
J = electric current density, Q = charge of the particle,  = particle flux
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.29
(a) Arbitrary electron concentration n (x, t) profile
in a semiconductor. There is a net diffusion
(flux) of electrons from higher to lower
concentrations.
(b) Expanded view of two adjacent sections at x0.
There are more electrons crossing x0 coming
from the left (x0-) than coming from the right
(x0+)
• Suppose e- concentration in a
semiconductor, at some point t,
decreases in the x direction and
has a profile n(x,t)
• Electron motion is random when
there is no E-field, caused by
lattice vibration and impurity
scattering
• Recall l is the mean free path
and t is mean free time between
collisions/scattering events
• So, an e- moves a mean distance
l in the +x or –x direction and
then scatters
• Mean speed along x is vx = l/t
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.29
(a) Arbitrary electron concentration n (x, t)
profile in a semiconductor. There is a net
diffusion (flux) of electrons from higher to
lower concentrations.
(b) Expanded view of two adjacent sections at
x0. There are more electrons crossing x0
coming from the left (x0-) than coming
from the right (x0+)
• Let’s evaluate the flow of electrons in
the –x and +x directions through
plane at xo to thereby find the net
flow in the +x direction
• Divide the x axis into segments of
length l corresponding to a mean free
path
• During one mean free time, half of
the e- in xo -l would move toward xo,
and the other half away from xo
• In time t, half will then reach xo and
cross
• If n1 is the concentration of e- at
xo-½l, then the # of e- moving toward
the right to cross xo is ½ n1Al
• Similarly, half of the e- in xo+½l
would be moving toward the left and
in time t would reach xo
• Their number is ½ n2Al where is the
concentration at xo+½l
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• The net # of e- crossing xo per unit
time per unit area in the +x direction
is the electron flux e,
Fig 5.29
(a) Arbitrary electron concentration n (x, t)
profile in a semiconductor. There is a net
diffusion (flux) of electrons from higher to
lower concentrations.
(b) Expanded view of two adjacent sections at
x0. There are more electrons crossing x0
coming from the left (x0-) than coming
from the right (x0+)
• Because the mean free path l is small,
calculus allows us to calculate n2-n1
from the concentration gradient
using:
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fick’s First Law
dn
e   De
dx
Relationship between the
net particle flux and driving
force (i.e., the gradient)
e = electron flux, De = diffusion coefficient of electrons, dn/dx = electron
concentration gradient
Electron Diffusion Current Density
J D,e
dn
 ee  eDe
dx
JD, e = electric current density due to electron diffusion, e = electron flux, e =
electronic charge, De = diffusion coefficient of electrons, dn/dx = electron
concentration gradient
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
• Similar situation if it were holes
involved:
J D ,h
Fig 5.30
Arbitrary hole concentration p (x, t) profile in a
semiconductor.
There is a net diffusion (flux) of holes from
higher to lower concentrations. There are more
holes crossing x0 coming from the left (x0-)
than coming from the right (x0+).
dp
 eh  eDh
dx
JD, h = electric current density due to
hole diffusion, e = electronic charge,
h = hole flux, Dh = diffusion
coefficient of holes, dp/dx = hole
concentration gradient
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 5.31
When there is an electric field and also a concentration gradient, charge carriers move both by
drift and diffusion.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Total Electron Current Due to Drift and Diffusion
dn
J e  en eE x  eDe
dx
Je = electron current due to drift and diffusion, n = electron concentration, e =
electron drift mobility, Ex = electric field in the x direction, De = diffusion
coefficient of electrons, dn/dx = electron concentration gradient
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Total Hole Current Due to Drift and Diffusion
dp
J h  ep hE x  eDh
dx
Jh = hole current due to drift and diffusion, p = hole concentration, h = hole drift
mobility, Ex = electric field in the x direction, Dh = diffusion coefficient of holes,
dp/dx = hole concentration gradient
Einstein Relation
De
kT

e
e
Dh
kT

h
e
• Diffusion coefficients are a measure of
the ease with which the diffusing
charge carriers move in the medium.
• So is drift mobility
• Both quantities are related through this
expression
De = diffusion coefficient of electrons, e = electron drift, Dh = diffusion coefficient
of the holes, h = hole drift mobility
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Time-Dependent Continuity Equation
• Many semiconductor devices operate on the principle of excess carriers being injected
• This injection of carriers by external means upsets the equilibrium concentration
• To determine the carrier concentration at any instant we need to solve the continuity
equation, which accounts for the total charge at that location in the semiconductor
pn(x,t)
n-type
Fig 5.33
Consider an elemental volume Adx in
which the hole concentration is pn(x, t)
• Consider infinitesimally thin elemental
volume Adx in which h+ is pn(x,t)
• The current density at x due to h+
flowing into volume is Jh and that due
to h+ flowing out at x+d x is Jh +d Jh
• There is a change in hole current
density Jh where Jh(x,t) is not uniform
along x. (assume it does not change in
y and z directions)
• If d Jh is negative, then the current
leaving the volume is less thanthat
entering the volume, which leads to an
increase in the h+ concentration in Adx
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Time-Dependent Continuity Equation
• Therefore,
Rate of increase in h+ concentration due
to the change in Jh
pn(x,t)
n-type
Fig 5.33
Consider an elemental volume Adx in
which the hole concentration is pn(x, t)
• The negative sign ensures that d Jh
leads to an increase in pn
• Recombination taken place in Adx
removes h+ from this volume
• In addition, photogeneration at x at
time t could be happening
• Therefore, the net rate increase in the
h+ concentration pn in Adx = Rate of
increase due to decrease in Jh – Rate
of recombination + Rate of
photogeneration
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Continuity Equation for Holes
Rate of increase due
to decrease in Jh
Net rate of
increase
Rate of
Recomb.
Rate of
Photogen.
pn
1  Jh  pn  pno
 

 Gph
t
e  x 
th
pn = hole concentration in an n-type semiconductor, pno = equilibrium minority
carrier (hole concentration in an n-type semiconductor) concentration, Jh = hole
current due to drift and diffusion, th = hole recombination time (lifetime), Gph =
photogeneration rate at x at time t, x = position, t = time
Continuity Equation with Uniform Photogeneration
Dpn
Dpn

 Gph
t
th
Dpn = pn  pno is the excess hole
concentration
• Photogeneration and current density do not
vary with distance along the sample length, so
first term above = 0
• If Dpn is the excess concentration, then the
time derviative of pn is the same as Dpn
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Steady-State Continuity Equation for Holes
1  J h 
pn  pno


e  x 
th
Jh = hole current due to drift and diffusion, pn = hole concentration in an n-type
semiconductor, pno = equilibrium minority carrier (hole concentration in an n-type
semiconductor) concentration, th = hole recombination time (lifetime)
Steady-State Continuity Equation with E = 0
2
d Dpn Dpn
 2
2
dx
Lh
Dpn = pn  pno is the excess hole concentration, Lh = diffusion length of the holes
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
(a) Steady state excess carrier concentration profiles in
an n-type semiconductor that is continuously illuminated
at one end.
(b) Majority and minority carrier current components in
open circuit.
Total current is zero.
Fig 5.34
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Minority Carrier Concentration, Long Bar
 x
Dpn (x)  Dpn (0)exp   
 Lh 
Dpn = pn  pno is the excess hole concentration, Lh = diffusion length of the holes
Steady State Hole Diffusion Current
Ih  ID,h
 x
dpn (x) AeDh
  AeDh

Dpn (0)exp  
dx
Lh
 Lh 
Ih = hole current, ID, h = hole diffusion current, A = cross-sectional area, Dh =
diffusion coefficient of holes, pn(x) = hole concentration in an n-type
semiconductor as a function of position x, Lh = diffusion length of holes, Dpn = pn 
pno is the excess hole concentration
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Majority Carrier Concentration, Long Bar
 x
Dnn (x)  Dnn (0)exp   
 Le 
Dnn(x) = the excess electron concentration, x = position, Le = diffusion length of the
electrons
Electron Diffusion Current
ID, e
 x
dnn (x)
AeDe
 AeDe

Dnn (0)exp  
dx
Le
 Le 
ID, e = electron diffusion current, De = diffusion coefficient of electrons, nn(x) =
electron concentration in an n-type semiconductor as a function of position x, Le =
diffusion length of the electrons, Dnn = the excess electron concentration
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Electron Drift Current: Use the Open Circuit Condition
I drift ,e  I D ,e  I D ,h  0
Idrift, e = electron drift current, ID, e = electron diffusion current, ID, h = hole diffusion
current,
Electric Field
E
I drift ,e
Aenno e
E = electric field, Idrift, e = electron drift current, nno = equilibrium majority carrier
(electron concentration in an n-type semiconductor) concentration, e = electron
drift mobility
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Optical absorption generates electron-hole pairs.
Energetic electrons must lose their excess energy to lattice vibrations until their average
energy is (3/2)kT in the CB.
Fig 5.35
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Absorption of photons within a small elemental volume of width
Fig 5.36
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
x
Definition of Optical Absorption Coefficient
dI
 
Idx
 = absorption coefficient, I = light intensity, dI = change in the light intensity in a
small elemental volume of thickness dx at x
Beer-Lambert Law
I ( x)  I o exp( x)
I(x) = light intensity at x, Io = initial light intensity,  = absorption coefficient, x =
distance from the surface (location) where I = Io. Note: Light propagates along x.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The absorption coefficient depends on the photon energy h and hence on the wavelength.
Density of states increases from band edges and usually exhibits peaks and troughs. Generally
increases with the photon energy greater than Eg because more energetic photons can excite
electrons from populated regions of the VB to numerous available states deep in the CB.
Fig 5.37
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Piezoresistivity and its applications. (a) Stress m along the current (longitudinal) direction
changes the resistivity by
. (b) Stresses L and T cause a resistivity change. (c) A force
applied to a cantilever bends it. A piezoresistor at the support end (where the stress is large)
measures the stress, which is proportional to the force.
Fig 5.38
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
(d) A pressure sensor has four piezoresistor
R1, R2, R3, R4 embedded in a diaphragm.
The pressures bends the diaphragm, which
generates stresses that are sensed by the
four piezoresistors.
Fig 5.38
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)