Lecture3: Fermi Level and Fermi Energy.pdf

The Fermi level, often referred to as the Fermi energy or Fermi energy level, is a concept in condensed matter physics and quantum mechanics that plays a crucial role in understanding the behavior of electrons in a solid-state material, such as a metal, semiconductor, or insulator. It is named after the Italian physicist Enrico Fermi.

1. Solid State Devices-I
Dr. Vaishali V. Deshmukh
Dept. of Physics
Shri Shivaji Science College,
Amravati

2. Fermi Level and Fermi Energy
Lecture -3

3. Fermi Level in Intrinsic and Extrinsic Semiconductors
Fermi Level
01
Expression for electrical conductivity
Mobility and Conductivity
03
Theory of Hall Effect
Hall Effect
04
Outline
Drift Velocity
Drift current in Semiconductors
02

4. Fermi Level and Fermi Energy
Fermilevelisthehighestenergystateoccupiedbyelectronsinamaterialatabsolutezero
temperature.Fermilevelisalsodefinedastheworkdonetoaddanelectrontothesystem.
ThevalueoftheFermilevelatabsolutezerotemperature(−273.15°C)isknownastheFermienergy.
ItisalsothemaximumkineticenergyanelectroncanattainatT=0K.Fermienergyisconstantforeach
solid.TheFermienergyisaconceptinquantummechanicsusuallyreferringtotheenergydifference
betweenthehighestandlowestoccupiedsingle-particlestatesinaquantumsystemofnon-interacting
fermionsatabsolutezerotemperature.
EnricoFermi,thephysicistwhofirstproposedFermilevel.Itisimportantindeterminingthe
electricalandthermalpropertiesofsolids.
Theterm“Fermilevel”comesfromFermi-Diracstatistics,whichdescribesadistributionofparticles
overenergystatesinsystemsconsistingoffermions(electrons)thatobeythePauliexclusionprinciple.
In semiconductors the position of the Fermi level is within the band gap, approximately in the middle of
the band gap.

5. Fermi level of intrinsic semiconductor
Foranintrinsicsemiconductor,everytimeanelectronmovesfromthevalencebandtotheconductionband,itleavesa
holebehindinthevalenceband.Thedensityofelectronsintheconductionbandequalsthedensityofholesinthe
valenceband.AttemperatureT(K),theelectrondensity‘n’isequaltoholedensity‘p’inanintrinsicsemiconductor.
HereNcistheeffectivedensityofstatesinthe
conductionband,Nvistheeffectivedensityof
statesinthevalenceband,E
FistheFermi
energy,Ecisthelowerenergylevelofconduction
band,Evisthelowerenergylevelofvalence
band,K
BisBoltzmann'sconstant,andTisthe
temperatureinK.
Conduction Band
Valence Band
E
F
Fermi level
Band Energy
Band Gap
TheFermienergyisinthemiddleofthebandgap
(Ec+Ev)/2plusasmallcorrectionthatdepends
linearlyonthetemperature.Thecorrectiontermis
smallatroomtemperaturesinceEg~1eV
whilekBT~0.025eV.ForSiandGe,Nc>Nvand
thecorrectiontermisnegativewhilefor
GaAsNc<Nvandthecorrectiontermispositive.
E
V
E
C
E
g

6. Fermi level of extrinsic semiconductor
Inextrinsicsemiconductor,thenumberofelectronsintheconductionbandandthenumberofholesinthevalence
bandarenotequal.Hence,theprobabilityofoccupationofenergylevelsinconductionbandandvalencebandarenot
equal.Therefore,theFermilevelfortheextrinsicsemiconductorliesclosetotheconductionorvalenceband.
Fermilevelinn-typesemiconductor
Inn-typesemiconductorpentavalentimpurityisadded.Eachpentavalentimpuritydonates
afreeelectron.Theadditionofpentavalentimpuritycreateslargenumberoffreeelectrons
intheconductionband.Atroomtemperature,thenumberofelectronsintheconduction
bandisgreaterthanthenumberofholesinthevalenceband.Hence,theprobabilityof
occupationofenergylevelsbytheelectronsintheconductionbandisgreaterthanthe
probabilityofoccupationofenergylevelsbytheholesinthevalenceband.This
probabilityofoccupationofenergylevelsisrepresentedintermsofFermilevel.
Therefore,theFermilevelinthen-typesemiconductorliesclosetotheconductionband.
TheFermilevelforn-typesemiconductorisgivenas,
Conduction Band
Valence Band
E
F
Fermi level
Band Energy
Band Gap
E
V
E
C
Fig: Fermi level in n-type semiconductor
Where,E
Fisthefermilevel,E
Cistheconductionband,K
BistheBoltzmannconstant,Tistheabsolute
temperature,N
Cistheeffectivedensityofstatesintheconductionband,N
Dtheconcentrationofdonaratoms.

7. Fermi level of extrinsic semiconductor
In p-type semiconductor trivalent impurity is added. Each trivalent impurity creates a hole in the valence band and
ready to accept an electron. The addition of trivalent impurity creates large number of holes in the valence band.
Fermilevelinp-typesemiconductor
Atroomtemperature,thenumberofholesinthevalencebandisgreaterthanthenumber
ofelectronsintheconductionband.Hence,theprobabilityofoccupationofenergylevels
bytheholesinthevalencebandisgreaterthantheprobabilityofoccupationofenergy
levelsbytheelectronsintheconductionband.Thisprobabilityofoccupationofenergy
levelsisrepresentedintermsofFermilevel.Therefore,theFermilevelinthep-type
semiconductorliesclosetothevalenceband.TheFermilevelforp-typesemiconductoris
givenas
Conduction Band
Valence Band
E
F
Fermi level
Band Energy
Band Gap
E
V
E
C
Fig: Fermi level in p-type semiconductor
Where,N
Vis the effective density of states in the valence band, N
Ais the concentration of acceptor atoms.

8. Current density in Semiconductors

9. Drift current in Semiconductors
Theflowofcharge(ie)currentthroughasemiconductormaterialareoftwotypesnamelydrift&diffusion.
Thenetcurrentthatflowsthrougha(PNjunctiondiode)semiconductormaterialhastwocomponents
(i)Driftcurrent
(ii)Diffusioncurrent
Whenanelectricfieldisappliedacrossthesemiconductormaterial,thechargecarriersattainacertaindriftvelocity
Vd,whichisequaltotheproductofthemobilityofthechargecarriersandtheappliedElectricFieldintensityE;
DriftvelocityVd=mobilityofthechargecarriersXAppliedElectricfieldintensity.
Holesmovetowardsthenegativeterminalofthebatteryandelectronsmovetowardsthepositiveterminalofthe
battery.Thiscombinedeffectofmovementofthechargecarriersconstitutesacurrentknownasthedriftcurrent.
Thusthedriftcurrentisdefinedastheflowofelectriccurrentduetothemotionofthechargecarriersunderthe
influenceofanexternalelectricfield.
Driftcurrentduetothechargecarrierssuchasfreeelectronsandholesarethecurrentpassingthroughasquare
centimeterperpendiculartothedirectionofflow.
Itispossibleforanelectriccurrenttoflowinasemiconductorevenintheabsenceoftheappliedvoltageprovideda
concentrationgradientexistsinthematerial.
Inasemiconductormaterialthechargecarriershavethetendencytomovefromtheregionofhigherconcentration
tothatoflowerconcentrationofthesametypeofchargecarriers.Thusthemovementofchargecarrierstakesplace
resultinginacurrentcalleddiffusioncurrent.

10. Drift current in Semiconductors
Whenanelectricfieldisappliedtoaconductoratroomtemp,electronmovetowardsthe+Veterminaloftheappliedvoltbutthey
continuouslycollidewithatomsalongtheway.Eachtimetheelectroncollideswithanatom,itreboundsinarandomfashion.Ateach
collision,theelectronlosessomekineticenergy,thenacceleratesagain,gainscertaincomponentofvelocityinthedirectionof–Eand
losesitsenergyatthenextcollision.Obiviously,presenceoftheelectricfielddoesnotstopcollisionsandrandommotionbutitdoes
causetheelectronstodrifttowardsthe+VeterminaloftheappliedvoltageV.Consequently,theelectronsgainanaveragedirecteddrift
velocityvwhichisdirectlypraportionaltoE.
V = µ
eEWhere, µ
eis called electron mobility.
V (meter/second)
E(volt/meter)
µ
e =
The resulting flow of electrons carrying negative charge at drift velocity V constitutes electric
current called drift current.
Let n = number of electrons per unit volume of the conductor
A = conductor cross-section (m
2
)
I = conductor length (m)
E= V/l = applied electric field (V/m)
I = n ×(v ×A) Toalno. of electrons which cross any plane P of cross-section A in one second
Charge carried by them per second is = e n v A

11. Drift current in Semiconductors
It represents the drift current I = v e n A
Substituting the value of v, we get,I= n e A µ
eE = n e A µ
e
V
l
Now,R= 1
e
Vl
lAneµ



Þ = l
A
Resistivity þ1
neue

Conductivity = neµ
esiemens/m

12. Mobility and Conductivity
Theabilityofanelectrontomovethroughasemiconductor,inthepresenceof
appliedelectricfieldisknownaselectronmobility.
V
n=μ
nE,Therefore,Mobilityofelectronμ
n=V
n/E(SIunitofmobilitym
2
/(V.s)
Theabilityofanholetomovethroughasemiconductor,duetoappliedelectricfield
isknownasholemobility.
V
p=μ
pE,Therefore,Mobilityofelectronμ
p=V
p/E(SIunitofmobilitym
2
/(V.s)
In intrinsic semiconductors, the number of electrons is equal to the number of holes.
n
e= n
p
σ= n
ee μ
n + n
pe μ
p= n
ie (μ
n + n
p)

13. HALL EFFECT
Whenamagneticfieldisappliedtoacurrentcarryingconductor
inadirectionperpendiculartothatoftheflowofcurrent,a
potentialdifferenceortransverseelectricfieldiscreatedacrossa
conductor.ThisphenomenonisknownasHallEffect.
HallEffectwasdiscoveredbyEdwinHallin1879.
Thevoltageorelectricfieldproducedduetotheapplicationof
magneticfieldisalsoreferredtoasHallvoltageorHallfield.
Whenavoltageisappliedtoaconductororsemiconductor,
electriccurrentstartsflowingthroughit.Inconductors,the
electriccurrentisconductedbyfreeelectronswhereasin
semiconductors,electriccurrentisconductedbybothfree
electronsandholes.
Thefreeelectronsinasemiconductororconductoralwaystrytoflowinastraightpath.However,becauseofthe
continuouscollisionswiththeatoms,freeelectronsslightlychangetheirdirection.Butiftheappliedvoltageisstrong
enough,thefreeelectronsforcefullyfollowthestraightpath.Thishappensonlyifnootherforcesareappliedtoitinother
direction.Ifweapplytheforceinotherdirectionbyusingthemagneticfield,thefreeelectronsintheconductoror
semiconductorchangetheirdirection.

14. HALL EFFECT
Consideramaterial,eitherasemiconductororconductorasshowninthebelow
figure.Whenavoltageisapplied,electriccurrentstartsflowinginthepositive
x-direction(fromlefttoright).Ifamagneticfieldisappliedtothiscurrent
carryingconductororsemiconductorinadirectionperpendiculartothatofthe
flowofcurrent(thatisz-direction),anelectricfieldisproducedinitthatexerts
forceinthenegativeydirection(downwards).
MathematicalexpressionfortheHallvoltageisgivenby
V
H=
IB
qnd
Where,
V
H= Hall voltage
I = current flowing through the material
B = magnetic field strength
q = charge
n = number of mobile charge carriers per unit volume
d = thickness of the material
HallVoltageisdirectlyproportionaltothe
electriccurrentandappliedmagneticfield.

15. Derivation of Hall Coefficient
e E
H = e Bv
E
H = Bv
J
X = nev V =
Thus,E
H=
E
H=R
HJ
XB
R
H= =
R
H=
J
X
n
e
BJ
X
n
e
E
H
J
X B
1
n
p
Ifvisthevelocityofelectronsatrightanglestothemagneticfield,thereisa
downwardforceactsoneachelectronofmagnitudeBev.Thisresultsintheelectron
currenttobedeflectedinadownwarddirectionandcausesanegativechargeto
gatheronthebottomfaceofthespecimen.Apotentialdifferenceistherefore
generatedfromtoptobottomofthespecimen.Thispotentialdifferencecausesa
fieldE
Hinthenegativey-direction,andthusaforceofeE
Hactingintheupward
directionontheelectron.Henceatequilibriumcondition,theforcedownwardsdue
tomagneticfieldwillbeequaltotheupwardelectricforce.
TheHalleffectisdescribedbymeansoftheHallcoefficientR
Hdefinedinterms
ofcurrentdensity
Negativesignindicatesthattheelectricfielddevelopedisinthenegativey-
direction.
Incaseofp-typespecimen,whencurrentiscompletelyduetoholes,
-1
n
e

16. Applications of hall effect
HallEffectisusedtofindwhetherasemiconductorisN-typeorP-type.
HallEffectisusedtofindcarrierconcentration.
HallEffectisusedtocalculatethemobilityofchargecarriers(free
electronsandholes).
HallEffectisusedtomeasureconductivity.
HallEffectisusedtomeasurea.c.powerandthestrengthofmagnetic
field.
HallEffectisusedinaninstrumentcalledHallEffectmultiplierwhich
givestheoutputproportionaltotheproductoftwoinputsignals.

17. Chargecarriersinsemiconductorsarenegativelychargedelectronsandpositively
chargedholes.Holesareessentiallyvacanciesinanotherwisefilledbandsothatwhenan
electronmovestosuchavacancy,theeffectisequivalenttomovementofaholeinreverse
direction.
Intheabsenceofanelectricfield,chargecarriersmoverandomlysothattheiraverage
velocityiszero.Whenanelectricfieldisapplied,thepositvechargecarriersmoveinthe
directionofthefieldandthenegativechargecarriersmoveagainstthefielddirection.
Thisdirectedmotionissuperimposedovertherandomdirectionandiscalleddrift.
Driftvelocityisproportionaltothedirectionofthefield,theconstantofproportionality
isthemobilityofthecarrier.
Ifamagneticfieldisappliedonaflatstripofsemiconductorinadirectionperpendicular
tothedirectionofcurrentflow,avoltagedevelopsinadirectionwhichisperpendicularto
boththedirectionofthefieldandthemagneticfield.ThisisknownasHallvoltage.
HallvoltagearisesduetoLorentzforcethatactsonchargecarriersandprovidesadirect
meansofverifyingexistenceofholes.

18. Without semiconductors, the
world would be a very
different place; we would
have no electronic hand
calculators, microwave
ovens, digital alarm clocks,
cellphones, personal
computers, electronically
controlled transmissions, or
washing machines.

19. Anyone who
has never
made a
mistake has
never tried
anything
new.
Albert Einstein

20. THANK YOU