3/12/2015 3.5.1 Deformation Potential Theory
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Subsections
3.5.1.1StrainInducedConductionBandSplitting3.5.1.2StrainInducedDegeneracyLiftingatthe Point3.5.1.3StrainInducedValenceBandSplitting
3.5.1DeformationPotentialTheory
BardeenandShockley[165]originallydevelopedthedeformationpotentialtheory.HerringandVogt[166]generalizedthistheory.BirandPikus[161]studiedvarioussemiconductorsviagrouptheoryandshowedhowtocalculatestraineffectsonthebandstructurewithdeformationpotentials.Ashortintroductionintothedeformationpotentialtheoryisgivensubsequently.
ThedeformationpotentialtheoryintroducesanadditionalHamiltonian ,thatisattributedtostrain
anditseffectsonthebandstructure.ThisHamiltonianisbasedonfirstorderperturbationtheoryanditsmatrixelementsaredefinedby
(3.15)
denotesthedeformationpotentialoperatorwhichtransformsundersymmetryoperationsassecondranktensor[167]and describesthe straintensorcomponent.Thesubscripts in
denotethematrixelementoftheoperator .Duetothesymmetryofthestraintensorwith
respectto and ,alsothedeformationpotentialoperatorhastoobeythissymmetry
andthuslimitsthenumberofindependentdeformationpotentialoperatorstosix.
Inthecaseofcubicsemiconductorstheedgesoftheconductionbandandthevalencebandarelocatedonsymmetrylines.Thesesymmetriesarereproducedintheenergybandstructureandinthebasisstates.Furthermore,thesymmetryofthebasisstatesallowstodescribethedeformationpotentialoperatorofaparticularbandviatwoorthreedeformationpotentialconstants[166].
Although,theoreticallythedeformationpotentialconstantscanbecalculatedviatheempiricalpseudopotentialmethodorbyabinitiomethods,itismoreconvenienttofitthedeformationpotentialstoexperimentalresultsobtainedbyelectrical,optical,microwavetechniques,orbyanalyzingstressinducedabsorptionedges.Eventhough,theoreticalpredictionsandmeasurementsmatchquitewell,deformationpotentialsinliteratureandfoundbydifferentmethodsdeviatefromeachother[168].
3.5.1.1StrainInducedConductionBandSplitting
Cubiccrystallsexhibitastraininducedenergyshiftforthenondegenerateenergylevelsoftheconductionband.Alongthe symmetrylineitissufficienttodescribethedeformationpotentialoperators asscalarsbyoneortwoindependentconstants.Theenergyshiftsoftheconductionbandedgeofvalleysalongthe and directionsisdeterminedbytwoindependent
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deformationpotentialconstants3.1[169]:
(3.16)
describestheuniaxialand thedilatationdeformationpotentialconstantsforvalleysofthetype
. denotestheunitvectorparalleltothe vectorofvalley .The conductionband
minimumvalleyshiftcanbedeterminedfromasingledeformationpotentialconstant
(3.17)
Viathetworelationsfromabovethevalleysplittingfromuniaxialstressalongarbitrarydirectionscanbecalculated.
3.5.1.2StrainInducedDegeneracyLiftingatthe Point
Additionallytostraininducedenergyshiftsofenergylevelsoftheconductionbandedges,therecanalsobeapartiallyorcompleteliftingofdegeneracyfordegeneratebands,causedbythereductionofsymmetry.Duetothespecialsymmetryofthediamondstructure(threeglidereflectionplanesat
, and ),thelowesttwoconductionbands and touchatthe
zoneboundary .Shearstrain duetostressalong reducesthesymmetryofthediamond
crystalstructureandproducesanorthorhombiccrystal.Theglidereflectionplane isremoved
bytheshearstraincomponentandthusthedegeneracyofthetwolowestconductionbands and
atthesymmetrypoints islifted[161,170].Itshouldbementionedthatin
biaxiallystrained layersgrownon substratesandforuniaxiallystrained/stressed
alongafourfoldrotationaxis theglidereflectionsymmetryispreserved.
BirandPikusfoundfromk.ptheory,thatwhenthedegeneracyatthezoneboundary islifted,arelativelylargechangeintheenergydispersionoftheconductionbandminimumlocatedclosetothispointarises[161].Thiseffectwasexperimentallyprovedfor byHenselandHasegawa[170],whomeasuredthechangeineffectivemassforstressalong ,andbyLaude[171],whoshowedthe
effectviatheindirectexcitonspectrum.
Therefore,inordertotaketheliftingofthedegeneracyofthetwolowestconductionbands and
atthe points intoaccount,(3.16)hastobeadapted[170]
(3.18)
where denotesanewdeformationpotential,
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(3.19)
Thesolutionsoftheeigenvalueproblemlooklike:
(3.20)
whichshowsthatatthe points thebandshiftsbyanamountof (likebeforein
(3.16))plusanadditionalsplittingof ,whichliftsthedegeneracy.(3.19)showstheproportional
dependenceonshearstrain forthesplitting
(3.21)
Avalueof eVhasbeenpredictedbyHenselforthesheardeformationpotential [170].
Laude[171]confirmedthisvaluebyhismeasurementof eVviatheindirectexcitonspectrumof
.
Thesplittingisalreadystronglypronouncedforshearstrain .Duetotheliftingofthedegeneracy
the conductionbandisdeformedclosetothesymmetrypoints (Fig.3.2).
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Figure3.2:Energydispersionoftheconductionbands and nearthe
zoneboundary pointalong .For theconductionbandsare
degenerateatthezoneboundary.Introductionofshearstrain liftsthis
degeneracyandopensupagap.Theenergyseparation betweenthebands
becomeslargerwithincreasingstrain .Atthesametimethetwominimaofthe
lowerconductionband moveclosertothezoneboundarywithrisingstrain
,untiltheymergeatthezoneboundaryandstaythereforfurtherincreasing
strain.i
Anonvanishingshearstraincomponent hasthefollowingeffectsontheenergydispersionofthe
lowestconductionband:
Thebandedgeenergyofthevalleypairalong directionshiftsdownwithrespecttothe
otherfourvalleysalong and .
Theeffectivemassofthevalleypairalong changeswithincreasing .
Theconductionbandminimaalong movetothezoneboundary pointsat
withincreasing .
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Figure:3.3Energydispersionofthetwolowestconductionbandsatthezoneboundaries and .Thebandseparationof
unstrained attheconductionbandedge isdenotedby
.Contrarytotheconductionbandsalong theconductionbandsalong
and arenotaffectedbyshearstrain .
Fordifferingstrains( ),theconductionbandminimaalongthe axesare
differentintheirenergies,causingarepopulationbetweenthesixconductionbandvalleys.Thiskindofeffectisnotcoveredwith(3.16),duetothenegligenceofpossibledegeneracyliftingsbyshearstrainandbyignoringapossiblerepopulationofenergystates.
Themodelpresentedshowsnochangeintheconductionbandsnearthezoneboundariesand forashearcomponent (Fig.3.3).However,shear
componentslike or liftthedegeneracyat or .
Applyingadegeneratek.ptheoryatthezoneboundary point[161,170]enablesananalyticaldescriptionforthevalleyshiftalongthe direction.Shearstrain causesanenergyshiftbetween
theconductionbandvalleysalong / andthevalleysalong .Thisshiftisdescribedby
(3.22)
isadimensionlessparameterand denotesthebandseparationbetweenthelowest
twoconductionbandsattheconductionbandedge
(3.23)
denotesthepositionofthebandedgeintheunstrainedlattice.
3.5.1.3StrainInducedValenceBandSplitting
Causedbythedegeneracyatthemaximumofthevalencebandsthedeformationpotentialisdifferentthanthatoftheconductionbands.Thedeformationpotentialoperators arenolongerscalarsandhavetobeexpressedas matrices.Usingsymmetriesthesixindependentoperatorscanbe
describedviathreeindependententries,commonlynamed or ,relatedtotheappliedset
ofeigenfunctions[172].Forthebasis , , ,with denotingthespinstate,the
perturbationHamiltoniantakesthefollowingform:
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(3.24)
denotesthe matrix
(3.25)
Inthecaseofthevalencebandthedescriptionofthestraininducedshiftsoftheheavyhole,lighthole,andthesplitoffbandaremorecomplex[169].
Footnotes
...constants3.1neglectingstraininducedsplittingofthedegenerateconductionbands and atthe
point
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