Introduction to Solid State Electronics part-1

8/13/2019 Introduction to Solid State Electronics part-1

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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


7/26

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.


14/26

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.


16/26

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.


17/26

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.


18/26

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


20/26

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


21/26

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.


22/26

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.


24/26

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.


25/26

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


26/26

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

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Introduction to Solid State Electronics part-1

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