solid state lectures
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TemperatureDependenceof
CarrierConcentrations(IntrinsicLecture08
.
MdJawaidAlam JIIT,Sect62,Noida,India
Intrinsiccarrierconcwithtempraturefor
Ge,SiandGaAs.RefB
G
Streetman
Fig
318
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TemperatureDependenceofCarrierConcentrations
(ExtrinsicSemiconductors)
Lecture08
Thetemperaturedependenceofelectronconcentrationin
adopedsemiconductorcanbevisualizedasshowninthis
Fig
concentrationNDof1015
cm3.
Atver lowtem eratures
(large1/T),negligible
intrinsicEHPsexist,and the
donorelectronsareboundo e onora oms.
RefBGStreetmanFig319
Between100400Kwheneveryavailable
extrinsicelectronhasbeen transferredtothe
conductionband,n isvirtuallyconstantwith
At higher temperatures ni is
much greater than Nd, and the
MdJawaidAlam JIIT,Sect62,Noida,India
temperature untiltheconcentrationofintrinsic
carriersnibecomescomparabletotheextrinsicconcentrationNd.
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CompensationandSpaceChargeNeutrality Lecture08
D i A i,only approximation no=Ndand po= NA is valid.
The exact relationship among the electron, hole, donor, and acceptor
concentrations can be obtained by considering the requirements for space charge
neutrality
If the material is to remain electrostaticall neutral the sum of the ositive
charges (holes and ionized donor atoms) must balance the sum of the negative
charges (electrons and ionized acceptor atoms).
MdJawaidAlam JIIT,Sect62,Noida,India
Ifthematerialisdopedntype(n0p0)then
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* 10 3
Lecture08
temperature then
a) findtheequilibriumelectronandholeconcentrationinthedoped
semiconductor.
b) Shiftin
Fermi
level
with
respect
to
intrinsic
Fermi
level.c) If wedopeBinsteadofAsthenrepeatparta)andb).
Prob 2:FindthedopingconcentrationofAsorB(whicheverbesuitable)
sothatFermilevelinthedopedSiliesexactlymiddleofthebandgap.
MdJawaidAlam JIIT,Sect62,Noida,India
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Lastlecture.Lecture09
MdJawaidAlam JIIT,Sect62,Noida,India
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Lastlecture.Lecture09
MdJawaidAlam JIIT,Sect62,Noida,India
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TemperatureDependenceofCarrierConcentrations
(IntrinsicSemiconductors)Lecture09
MdJawaidAlam JIIT,Sect62,Noida,India
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Drift and DiffusionDrift and Diffusion Lecture09
We now have some idea of the number density of
semiconductor material from the work we covered inthe last lectures.
ncecurren s e ra e o ow o c arge , we s abe able calculate currents flowing in real devices
since we know the number of charge carriers. There are two current mechanisms which cause
charges to move in semiconductors. The twomechanisms we shall stud in due course are dri tdri tandand diffusiondiffusion.
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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CarrierDriftCarrierDrift Lecture09
fieldsince thefield exert a forceon charge carriers (electrons and holes).
These movements result a current of ; F E=d
d d
dV
Anq
d
:dV
chargeoftheelectrondriftvelocityofchargecarrier :q
areaofthesemiconductor:A
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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CarrierMobility,CarrierMobility, Lecture09
dV E=
:E appliedfieldmobilit ofchar ecarrier
2
[ ]
=SecV
cm isaproportionalityfactor
=E
d
So is ameasurehoweasilychargecarriersmoveundertheinfluenceof .
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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ThemobilityhastwocomponentsThemobilityhastwocomponents Lecture09
Themobilityhastwocomponent
Impurityinteraction
component
component
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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ThermalvelocityThermalvelocity Lecture09
ssume t at s c crysta s at t ermo ynam c equ r um .e. t ere s noapplied field). What will be the energy of the electron at a finitetemperature?
The electron will have a thermal energy ofkkTT//22 per degree of freedom. So, in 3D, electron will have a thermal energy of
* 23 1 3 3th th
kT kT kT
E m V V = = =
: electronovelocitthermalV
1
2
1
TV th
2mV th
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RandommotionRandommotionresultresultnocurrentnocurrent.. Lecture09
Since there isno applied field,the movement of thechar e carriers will be com letel random. This
randomness resultno net current flow. As a result of
thermal energy there are almostan equal number ofcarriers moving right as left, in as out or up as down.
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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CalculationCalculation Lecture09
Calculate the velocity of an electron in a piece of ntype silicon
due to its thermal energy at RT and due to the application of an
*= = =
e ec r c e o m across e p ece o s con.
2? 1000 / 0.15 /( )
t e
dV E V m m V s= = =
53 1.08 10 / secth th
kTV V x m
= =
150 / secV E V m= =
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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MicroscopicunderstandingofmobilityMicroscopicunderstandingofmobility?? Lecture09
Howlon doesacarriermoveintimebeforecollision?
Theaveragetimetakenbetweencollisionsiscalledasrelaxation
time, (ormeanfreetime)
v
collision?
Theaveragedistancetakenbetweencollisionsiscalledasmean
freepath, . l
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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CalculationCalculation Lecture09
Driftvelocity=AccelerationxMeanfreetime
*dFV
m=
Forceisduetotheappliedfield, F=qE
* *d
qV
m m = =
d
qV E
= =
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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CalculationCalculation Lecture09
a cu a e e mean ree me an mean ree pa or e ec ronsin a piece of ntype silicon and for holes in a piece of ptypesilicon.
*
2 2
? ? 1 .1 8 0 .5 90 .1 5 / 0 .0 4 5 8 /
e o h ol m m m m
m V s m V s
= = = == =
1 2 1 31 0 sec 1 .5 4 1 0 sece e h he h
m mx
q q
= = = =
5 51 .0 8 1 0 / 1 .0 5 2 1 0 /elec h o leth th
v x m s v x m s= =
5 1 2 7(1 .0 8 1 0 / )(1 0 ) 1 0elece th e
l v x m s s m = = =
5 1 3 8 .h o l eh th h
. .
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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Saturated DriftSaturated DriftVelocities(EffectatHighElectricField)Velocities(EffectatHighElectricField)Lecture09
dV E=
Soonecanmakeacarriergoasfastaswelikejustbyincreasing
theelectricfield!!!
dV forefore
or o esor o es
E E
a mp ca ono a oveeqn. a ura on r ve oc y
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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Saturated DriftVelocitiesSaturated DriftVelocities Lecture09
The equation of does not imply thatVVdd increasesEVd .=linearly with applied fieldEE..
dd
at some value ofVVdd which is closeVVthth at higher values ofEE.
Any further increase in EE after saturation point does not
increaseVVdd instead warms up the crystal.
Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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LastLectureLecture10
I n V A=
dV E=
So is ameasurehoweasilychargecarriersmoveundertheinfluenceof
anappliedfieldor determineshowmobile the chargecarriersare.
Themobilityhastwocomponent
Impurityinteraction
component
Latticeinteraction
component
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MobilityvariationwithtemperatureMobilityvariationwithtemperature Lecture10
L I
TTTT
Hightemperature Lowtemperature
)ln(1 1 1
Peakdependsonthedensityof
impuritiesI
LT L I
=
ln(T)ln(T)Thisequationiscalled as
Mattheisensrule.Md Jawaid Alam JIIT,Sec62,Noida,INDIA
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For lightly doped samples (e.g., the
MobilityvariationwithtemperatureMobilityvariationwithtemperature Lecture10
sample with doping of 1014 cm3), the
lattice scattering dominates, and the
mobility decreases as the temperature
For heavily doped samples, the effect
of impurity scattering is most
pronounced at low temperatures. The
mobility increases as the temperature
increases, as can be seen for the19
For a given temperature, the mobility
decreases with increasing impurity
Electronmobilityinsiliconversustemperaturefor
impurity scattering
var ous onorconcen ra ons. nser s ows e
theoreticaltemperaturedependenceofelectron
Mobility
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ConceptofMeanfreetimeandMeanFreePathlengthLLecture10
Driftvelocity=AccelerationxMeanfreetime
*dFV
m=
Forceisduetotheappliedfield, F=qE
* *d
qV
m m = =
d
qV E
= =
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Conductivity Lecture10
Anq
d
I
d d dI nqV A V E= =
dJ
A m
= =
d d2
x d x x
nqJ nqV nq E J E
m
= = =
2 1n x xm = = =
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OhmsLaw Lecture10
1x
x xx
I IV VJ E
A L A L = = =
VA V
L R= =
Prob: Assumin that mobilit of electron is three times mobilit of holes insilicon at 300K prove that a silicon sample has maximum resistivity when it isslightly ptype. Also find the doping concentration for maximum resistivity.
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CalculationCalculationLecture10
a cu a e e mean ree me an mean ree pa or e ec ronsin a piece of ntype silicon and for holes in a piece of ptypesilicon.
*
2 2? ? 1 .1 8 0 .5 9
0 .1 5 / 0 .0 4 5 8 /
e o h ol m m m mm V s m V s
= = = == =
1 2 1 31 0 sec 1 .5 4 1 0 sece e h he h
m mx
q q
= = = =
5 51 .0 8 1 0 / 1 .0 5 2 1 0 /elec h o leth th
v x m s v x m s= =
5 1 2 7(1 .0 8 1 0 / )(1 0 ) 1 0elece th e
l v x m s s m = = =
5 1 3 8 .h o l eh th h
. .
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HallEffect Lecture11
The carrier concentration in a semiconductor may be different
from the impurity concentration, because the ionized impurity
density depends on the temperature and the impurity energy
level.
To measure the carrier concentration directly, the most
.
Hall measurement is also one of the most convincing methodsto show the existence of holes as charge carriers, because the
measurement can ive directl the carrier t e.
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HallEffectCont Lecture11
Anelectricfieldappliedalongthexaxis
AmagneticfieldappliedalongthezaxisHallCoefficient
Consideraptypesemiconductorsample
where
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HallEffectCont Lecture11
AmeasurementoftheHallvoltageforaknowncurrentandmagneticfieldyields
whereallthequantitiesinthelighthandsideoftheequationcanbemeasured.
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n(x)DiffusionCurrent Lecture11
F1ofelectronscrossingplane
x=0fromtheleftisthen
Similarly,theaveragerateofelectronflow
perun area 2o e ec ronsa x= cross ng
planex=0fromtherightis
xWhere
* 23 1 3 3th th
kT kT kT E m V V
= = =
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Lecture11
Thenetrateofcarrierflowfromlefttorightis
firsttwotermsofaTaylorseriesexpansion,
weobtain
Where
is calledthediffusioncoefficientalsocalled
F
.
Becauseeachelectroncarriersacharge q
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LastLecture Lecture12
Prob:Assumethat,inanntypesemiconductoratT
=300
K,
the
electron
concentration
variesLinearlyfrom1x1018 to7X1017cm3 overadistanceof0.1cm.Calculatethe
diffusioncurrentdensityiftheelectrondiffusioncoefficientisDn=22.5cm2/s.
Current L t 12
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Current
TotalCurrentdensity Lecture12
n(x)Electrondiffusion
x
Current
p(x)
x
40 L t 12
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EinsteinRelation40 Lecture12
40
L t 13
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Module4: Excesscarriersgeneration&recombination
processesandsemiconductordeviceequations.
Lecture13
Intrinsic Doped
OnlyinthermalequilibriumtheTheprocessofintroducingexcesscarriersis
i
.
calledcarrier
injection.
Wecanintroduceexcesscarriersby
Optica excitation
forwardbiasingapnjunction
Electronbombardment
Lecture 13
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DirectRecombination Lecture13
Numberofelectronholepairsgeneratedpercm3 per
second: GthNumberofelectronholepairsRecombinedper
cm3 persecond:Rth
andatthermalequilibriumRth=Gth
nn0nn0 nn =nn0+ nn0
Thermalequilibrium
pn0pn =pn0+ pn0
n0
Lecture 13
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DirectRecombination Lecture13
therateofchangeofholes withtimeinthisdynamicsystemwillbe
Insteadystate,
whereUisthenetrecombinationrate.
Lecture 13
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Lecture13
And
u
For lowlevel in ection
Where
Lecture 13
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Thephysicalmeaningoflifetimecanbestbe
Lecture13
adeviceafterthesuddenremovalofthelight
source.
Consideranntypesample,asshown
inFigthatisilluminatedwithlightand
inwhichtheelectronholepairsare
generateduniformlythroughoutthe
samplewithagenerationrateGL
Insteadystate GLmustbeequaltoNet
recombinationU
Lecture13
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Lecture13
Ifatanarbitrarytime,say
t=0,thelightissuddenly
Willreduceto
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andthesolutionis Lecture13
Prob:ASisamplewithnn0=1014 cm3 isilluminatedwithlightand1013 electronhole
pairs/cm3arecreatedeverymicrosecond.Iftn=tp=1uSec,findthechangeintheminority
carrierconcentration.
Ifthelightisturnedoffatt=0sec,i.e GL=0att=0, thenfind theholeconcentrationatt=1u
sec,
t=2u
sec,
t=4u
sec
and
t=10u
sec
Indirect Recombination Lecture14
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IndirectRecombination
recombinationcenters
Lecture 14
Traplevel
NetRecombination
rate
(U)
depends
on ee ec ronor o ecap urecross
section.
Auger recombination Lecture14
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Augerrecombination Lecture 14
Usually,Augerrecombinationisimportantwhenthecarrierconcentrationisveryhigh
asaresultofeitherhighdopingorhighinjectionlevel.BecausetheAugerprocess
vo ve eep e , o u o o x
TheproportionalityconstantBhasastrongtemperaturedependence
CONTINUITY EQUATION Lecture14
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CONTINUITYEQUATION Lecture 14
In the revious lectures we considered individual effects such as drift due to an
electricfield,diffusionduetoaconcentrationgradient,andrecombinationofcarriersthroughintermediatelevelrecombinationcenters
, ,
simultaneouslyinasemiconductormaterial.Thegoverningequationiscalledthe
continuityequation.
CONTINUITY EQUATION Lecture14
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CONTINUITYEQUATION
Toderivetheonedimensionalcontinuityequationforelectrons,consideraninfinitesimal
Thenumberofelectronsintheslicemay
thesliceandthenetcarriergeneration
intheslice
Theoverallrateofelectron
increaseisthealgebraicsumof
fourcomponents:thenumberof
,
minusthenumberofelectrons
flowingoutatx+dx,plustherate
atwhichelectronsaregenerated,
minustherateatwhichtheyare
recombined withholesintheslice.
CONTINUITY EQUATION Lecture14
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CONTINUITYEQUATION
atx+dx inTaylorseriesyields
Wethusobtainthebasiccontinuityequation
forelectrons:
Asimilarcontinuityequationcanbederived
forholes,
CONTINUITYEQUATION Lecture14
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Q
Intermsofmobilityanddiffusiontheonedimensionalcaseunderlowinjectioncondition,continuityequationsforminoritycarriersbecomes
YouMustkeepinmindthebasic
requirementofPoissonsEquation
Lecture15SteadyStateInjectionfromOneSide
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IntheFigureanntypesemiconductor
whereexcesscarriersareinjectedfromonesideasaresultofillumination.Itis
assumedthatlight penetrationisnegligibly
small i.e. the assum tions of zero field
andzerogenerationforx>0).
thereisaconcentrationgradientnearthe
sur acean tw sat s y ont nu ty
equationgivenby.
Atsteadystatesolutionofthedifferentialequationfor
t em nor tycarr ers ns et esem con uctor s
Lecture15
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o ut ono t e or er erent a equat on s
Thelength isequalto andiscalledthediffusionlength.ppDL p
Minority Carriers at the Surface Lecture15
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MinorityCarriersattheSurface
Lecture15
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Assignment1Dateofsubmission:onorbefore23rdofFebruary
Answer
Lecture15
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W
Uniqueness ofFermilevelinbulksemiconductor
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