introduction to solid state electronics part-1
TRANSCRIPT
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Solid State Electronics
Text Book
Ben. G. Streetman and Sanjay Banerjee: Solid State
Electronic Devices, Prentice-Hall of India Private
Limited.
Chapter 4
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Excess Carrier in SemiconductorsThe carriers, which are excess of the thermal equilibrium
carries values, are created by external excitation is called excess
carriers.The excess carriers can be created byoptical excitation or
electron bombardment.
Optical Absorption
Measurement of band gap energy: The band gap energy of a
semiconductor can be measured by the absorption of incident photons
by the material.
In order to measure the band gap energy, the photons of selected
wavelengths are directed at the sample, and relative transmission of thevarious photons is observed.
This type of band gap measurement gives an accurate value of
band gap energy because photons with energies greater than the band
gap energy are absorbed while photons with energies less than band gapare transmitted.
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Excess carriers by optical excitation: It
is apparent from Fig. 4-1 that a photon
with energyhv>Egcan be absorbed in a
semiconductor.
Figure 4-1Optical absorption of a photon with
hv>Eg: (a) an EHP is created during photon
absorption (b) the excited electron gives upenergy to the lattice by scattering events; (c)
the electron recombines with a hole in the
valence band.
Thus the excited electron losses energy to the lattice in scattering events until
its velocity reaches the thermal equilibrium velocity of other conduction bandelectrons.
The electron and hole created by this absorption process areexcess carriers:
since they are out of balance with their environment, they must even eventually
recombine.
While the excess carriers exit in their respective bands, however, they arefree to contribute to the conduction of material.
Since the valence band containsmany electrons and conduction band has
many empty states into which the
electron may be excited, the probability
of photon absorption is high.
Fig. 4-1 indicates, an electron
excited to the conduction band by optical
absorption may initially have more
energy than is common for conduction
band electrons.
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If a beam of photons with hv>Eg falls on asemiconductor, there will be some predictable amount of
absorption, determined by the properties of the material.
The ratio of transmitted to incident light
intensity depends on the photon wavelength and the
thickness of the sample.
let us assume that a photon beam of intensityI0(photons/cm-2-s) is directed
at a sample of thicknesslas shown inFig. 4-2.
)1.4()()( xdxxd II
The beam contains only photons of wavelength selected by
monochromator.As the beam passes through the sample, its intensity at a distance x from
the surface can be calculated by considering the probability of absorption with in
any incrementdx.
The degradation of the intensitydI(x)/dxis proportional to the intensity remaining
atx:
I0 It
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The solution to this equation is)2.4(
0)( xex II
Figure 4-3 Dependence of opticalabsorption coefficient for asemiconductor on the wavelength
of incident light.
and the intensity of light transmitted
through the sample thicknesslis)3.4(
0let
II
The coefficient is called theabsorption coeff icientand has units of
cm-1
.
This coefficient varies with the photon
wavelength and with the material.
Fig. 4-3shows the plot of vs. wavelength.There is negligible absorption at long wavelength (hv small) and
considerable absorptions with energies larger thanEg.
The relation between photon energy and wavelength isE=hc/. IfEis
given in electron volt andis micrometers, this becomesE=1.24/.
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Steady State Carrier GenerationThe thermal generation of EHPs is balanced by the recombination rate that means
[Eq. 3.7] )10.4(00
2)( pnrinrTg
If a steady state light is shone on the sample, an optical generation rate gopwill be
added to the thermal generation, and the carrier concentrationnandpwill increase to
new steady sate values.
)11.4()0
)(0
()( ppnnr
nprop
gTg
For steady state recombination and no traping, n=p; thus Eq. (4.11) becomes
)12.4(]2)00[(00)( nnpnrpnropgTg
Sinceg(T)==rn0p0and neglecting the n2, we can rewrite Eq. (4.12) as
)13.4()/(])00[( nnnpnropg
.theis)00(
1where, timelifecarrier
pnrn
Ifnandpare the carrier concentrations which are departed from equilibrium:
The excess carrier can be written as )14.4(nopgpn
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Quasi-Fermi LevelThe Fermi levelEFused in previous equations is meaningful only when no excess
carriers are present.
The steady state concentrations in the same form as the equilibrium expressions bydefining separatequasi-F er mi levels FnandF
pfor electrons and holes.
The resulting carrier concentration equations
)15.4(/)(
;/)( KTpFiEeinpKTiEnFeinn
can be considered as defining relation for the quasi-Fermi levels.
Example 4-3 and 4-4Let us assume that 103 EHP/cm3 are created optically every
microsecond in a Si sample with n0=1014 cm-3 and n=p =2 msec. Find the
position of the quasi-fermi level for electrons and holesat room temperature.
Solution: Given, optical generation rate, gop= 1013 EHP/cm3; n0=10
14 cm-3;
n=p=2msec,ni=1.51010 cm-3; andkT=0.0259 eVat room temperature.
The steady state excess electron (or hole) concentration is then
n=p= gop n=21013
cm-3
.
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While the percentage change in the majority electron concentration is small, the
minority carrier concentration changes from
p0=ni2/n0= (2.2510
20)/1014=2.25106 cm-3 (equilibrium)
to p= 21013 cm -3 (Steady State)
The steady state electron concentration is
0259.0/)()10105.1(4102.11310214100
iEnFennn
Thus the electron quasi Fermi level positionFn-Eiis found from
eV233.0)3108.0ln(0259.0)(
;3108.0)
10105.1(
14102.10259.0/)(
iEnF
iEnFe
0259.0/)()10105.1(131021310261025.20 pFiEeppp
The steady state hole concentration is
eV186.02.70259.0)3
1033.1ln(0259.0
31033.1)10105.1(
131020259.0/)(
pFiE
pFiEe
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c
v
F=Fn
p
i0.186 eV
0.233 eV
Fig 4-11Quasi-fermi levelsFnandFpfor a Si sample with
n0=1014 cm-3, p=2 ms, and gop=10
3 EHP/cm3-s.
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Diffusion of CarriersAny spatial variation (gradient) in n
and p calls for a net motion of the
carriers from the regions of high carrierconcentration to regions of low carrier
concentration.
This type of motion is calleddiffusion.
The two basic process of current
conduction are diffusion due to a
carrier gradient and drift in an
electric filed.
Carriers in a semiconductor diffuse in a carrier gradient by random thermal motion and
scattering from the lattice and impurities.
Fig. 4-12Spreading of a pulse of electrons
by diffusion.
For example, a pulse of excess electrons injected atx=0 at timet=0 will spread out in
time as shown inFig. 4-12.
Initially, the excess electrons are concentrate atx=0; as time passes, however, electrons
diffuse to regions of low electron concentration until finallyn(x) is constant.
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We can calculate the rate at which the electrons diffusion in one dimensional problem
by considering at arbitrary distributionn(x) such asFig. 4-13(a).
Since the mean free path between collisions is a small
incremental distance, we can divide x into segments
wide, withn(x) evaluated at the center of each segment
(Fig. 4-13b).
l
l
The rate of electron flow in the +xdirection per unit area
(the electron flux densityn) is given by
)18.4()21
(2
)0
( nnt
lxn
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Since the mean free path is a small differential length, the difference in electron
concentration (n1-n2) can be written as)19.4(
)()()21( l
x
xxnxnnn
where,xis taken at the center of segment (1) and x= .l
In the limit of small x, Eq. (4-18) can be written in terms of the
carrier gradientdn(x)/dx:
)20.4()(
2
22)()(
0lim
2)(
dx
xdn
t
ll
x
xxnxn
xt
lx
n
The quantity is called the electron diffusion coefficientwith units cm2/s.tlnD 2/2
The minus sign in Eq. (4-20) arises from the definition of thederivative; it simply indicates thatthe net motion of electrons due to
diffusion in the direction of decreasing electron concentration.
Equ. (4-20) can be written as )21.4()(
)( a
dx
xdn
n
Dx
n
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Similarly, the rate of hole flow can be written as follows:
)21.4()(
)( bdx
xdppDxp
The diffusion current crossing a unit area (the current density) is theparticle flux density multiplied by the charge of the carrier:
)22.4()()(
)()diff.( adx
xdnnqD
dx
xdnnDqnJ
)22.4()()(
)()diff.( bdx
xdppqD
dx
xdppDqpJ
Electrons and holes move together in a carrier gradient [Eqs.
(4.21)], butthe resulting currents are in opposite directions [Eqs.
(4.22)] because of the opposite charge of electrons and holes.
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Diffusion and Drift of carriers
If an electric field is present is
addition to the carrier
gradient, the current densities
will each have a drift
component and a diffusion
component
If an electric field is applied to a semiconductor, the drift currents are obtained as
follows:
)b23.4()(
)()()(
diffusiondrift
)a23.4()(
)()()(
dx
xdppqDxxppqxpJ
dx
xdnnqDxxnnqxnJ
m
m
)()()drift,( xxnn
qxn
J m )()()drift,( xxpp
qxp
J m
(=-dV/dx) is electric field intensity,Vis potential,mis mobility.
It is well known thatthe direction of flow of electron is the opposite direction of
the applied electrical field and the direction of flow of hole is the same direction of
the applied electrical field.
According to above equation it is seen thatthe electron and hole drift currents are
in the same direction of the applied electric field intensity.
and the total current is the sum of the contributions due to electron and holes
)24.4()()()( xpJxnJxJ
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n(x)
p(x)
(x) p(drift)n(drift)
Jp(drift)Jn(drift)
n(diff.)
Jn(diff.)
p(diff.)
Jp(diff)
Fig. 4-14 Drift and
diffusion directions for
electrons and holes in a
carrier gradient and an
electric field. Particle flow
directions are indicated by
dashed arrows, and the
resulting currents are
indicated by solid arrows.
The resulting drift current is in the +xdirection in each case.
The drift and diffusion components of the current are additive for holes when the field is in
the direction of decreasing hole concentration, whereas the two components are subtractive
for electrons under similar condition.
The total current may be due primarily to the flow of electrons or holes depending on the
relative concentrations and the relative magnitudes and directions of electric field and carrier
gradients.
An important result of Eqs. (4-23) is that minority carriers can contribute significantly to the
current through diffusion.
Since the drift terms are proportional to carrier contribution, minority carriers seldomprovide much drift current.
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Assuming an electric field(x) in thexdirection, we can draw the energy bands as
inFig. 4-15, to include the change in potential energy of electrons in the field.
(x)
i
v
c
Fig. 4-15 Energy band diagram of a
semiconductor in an electric field(x).
Since electrons drift in a direction opposite to
the field, we expect the potential energy for
electrons to increase in the direction of thefield.
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Comparison between diffusion anddriftcurrents:
1. Diffusion current produces due to a carrier gradient and drift
current produces due to the applied of an electric filed.
2. Due to the diffusion, the net motion of electrons or holes in the
direction of decreasing electron concentration. Due to appliedelectric field, the net motion of electron is the opposite direction
of the applied electrical field and the direction of flow of hole is
the same direction of the applied electrical field.
3. Due to the diffusion, the electrons and holes currents are inopposite directions.Due to the applied voltage the electron and
holes currents are in same direction.
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Einstein relationThe electrostatic potentialV(x) varies in the opposite direction, since it is defined in
terms of positive charges and is therefore related to the electron potential energy
E(x) displayed in the figure byV(x)=E(x)/(-q).
From the definition of electric field, )25.4()(
)(dx
xdVx
ChoosingEias a reference, the electric
field to this reference can be given by )26.4(
1
)(
)()(
dx
idE
qq
iE
dx
d
dx
xdVx
At equilibrium, no net current flows in a semiconductor.
So, at equilibrium, setting Eq. (4.23b) equal to zero, we obtain the relation for electric
filed as follows:
0)(
)()()(
dx
xdppqDxxppqxpJ m )27.4(
)(
)(
1)(
dx
xdp
xpp
pDx
m
We know that,
dx
xFdE
dx
xidEkTxFExiEein
kTdx
xdp )()([
/)]()([1)(so,
kTxFExiEeinxp /)]()([
)(
D
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)27.4()(
)(
1)(
dx
xdp
xpp
pDx
m
])()(
[/)]()([1
/)]()([
1)(
dx
xFdE
dx
xidEkTxFExiEeinkTkTx
FEx
iE
einp
pDx
m
)28.4(])()(
[1
)(dx
xFdE
dx
xidE
kTp
pDx
m
The equilibrium Fermi level does not vary withx, and the derivative ofEiis given by
Eq. (4.26) reduces to
dx
xidE
kTp
pD
dx
idE
q
)(11
m
q
kT
p
pD
m q
kTD
mgeneral,In
This result is obtained either carrier type. This important equation is called
Einstein relation.
It allows us to calculate eitherDormfrom a measurement of the other.
At room temperature,D/m 0.026 V.
E l 4 5 A i i i Si l i d d i h d f id h h
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Example 4-5An intrinsic Si sample is doped with donors from one side such that
Nd=N0exp(-ax). (a) Find an expression for which Nd>>ni. (b) Evaluate (x) when
a=1(mm)-1.(c) Sketch a band diagram such as in Fig. 4-15 and indicate the direction
of(x).Solusion: (a) From Eq. (4-23a);
0)()()()( dxxdn
nqDxxnnqxnJ m dxxdn
xnnnDx )(
)(1)(
m here )(0
)( axeNxn
)(0
)( axeaNdxxdn ]
)(0
[)(
0
1)( axeaNaxeNn
nDx
m
a
nnDx
m )(
According to Einstein relation qkT
nnD
m a
qkTx )(
V/cm259cm410
1
C19106.1
J19106.10259.0
cm410
1
C19106.1
ev0259.0)()(
aqkTxb
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Diffusion and Recombination:
The Continuity EquationThe effects of recombination must be included in a description of
conduction process, however since the recombination can cause a variation in the
carrier distribution.
Consider a differential length xof a semiconductor sample with areaAin
theyz-plane (Fig. 4-16)
The hole current density leaving the
volume, Jp(x+x), can be larger or
smaller than the current density
entering Jp(x), depending on the
generation and recombination of
carriers taking place within the
volume.The net increase in hole concentration
per unit time, p/t, is the difference
between the hole flux per unit volume
entering and leaving, minus the
recombination rate.
We can convert hole current density to hole
particle flux density byJpdividing byq.
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The current densities are already expressed
per unit area; thus dividing Jp(x)/q by x
gives the number of carriers per unit volume
entering xA per unit time, and (1/q)
Jp(x+x)/x is the number leaving per unitvolume and time:
)30.4()()(1
p
px
xxpJxpJ
qxxxtp
(Rate of hole build up)=(increase of hole concentration in xAper unit time)-(recombination rate)As xapproach zero, we can write the current change in derivative form:
)31.4(1),( ap
pxpJ
qtp
ttxp
The expression (4-31a) is called the
continuity equationfor holes.
For electrons we can write )31.4(1 bnnxnJ
qtn
When the current is carried strictly by diffusion (negligible drift), we can replace
the currents in Eqs. (4-31) by the expressions for diffusion current; for example, for
electron we have
)32.4()diff.( t
n
nqDnJ
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)32.4()diff.(tn
nqDnJ
)31.4(1 bn
nxnJ
qtn
Substitute
into
we obtain the diffusion equationfor electrons,
)33.4(2
2 ann
tnnDt
n
and similarly for holes,
)33.4(
2
2b
p
n
t
p
p
D
t
p
These equations are useful in solving transient problems of diffusion
with recombination.
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Steady State Carrier Injection;
Diffusion LengthIn many problems a steady state distribution of excess carriers is maintained, such
that the time derivatives in Eqs (4-33) are zero.
In the steady state case the diffusion equation become
02
2
02
2
pn
tppD
n
n
t
nnD
ppDn
tp
nnDn
t
n
22
2
2
222
22
2
pLn
ppDn
tp
nL
n
nnDn
t
n
(4.34)
where, is called the electrondiffusion lengthand
is diffusion length of holes.
nnDnL
ppDpL
Let us assume that excess holes are somehow injected into a semi-
infinite semiconductor bar atx=0 and the steady state hole injection
maintains a constant excess hole concentration at the injection point
dp(x=0)=p.
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The injected holes diffuse along the bar, recombining with a
characteristic life timep.
In steady state we expect the distribution of excess hole to decay to
zero for long values ofx, because of the recombination (Fig. 4-17).
Figure 4-17 Injection of holes at
x=0, giving a steady state hole
distribution p(x) and a resulting
diffusion current densityJp(x).
For this problem we use the steady state
diffusion equation for holes, Eq. (4-34b).
The solution to this equation has the form
)35.4(/2
/1
pLxeCpLxeCp
We can evaluateC1andC2from the boundary
conditions.
Since recombination must reduce dp(x) to
zero for large values ofx, dp=0 atx=and
thereforeC1=0.
Similarly, the condition dp=p at x=0 gives
C2=p, and the solution is
)36.4(/ pLxpep
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The injected excess hole concentration dies out exponentially inx
due to recombination, and the diffusion length Lp represents the
distance at which the excess hole distribution is reduced to 1/eof
its value at the point of injection.
Problem: Holes are injected in a very long p-type Si bar with cross-
sectional area = 0.5 cm2 andNa=1017 cm-3 such that the steady state
excess holes concentration is 51016 cm-3 at x = 0. Derive the
analytical expression of hole distribution. Assumemp= 500 cm2/V-s
andp=10-10s.
Hence:Dp= 12.5 cm2/s [Table 4-1]
pLxpep/
ppDpL q
kT
p
pD
m