3.5 deformation potential .pdf

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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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