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3/12/2015 3.5.1 Deformation Potential Theory http://www.iue.tuwien.ac.at/phd/windbacher/node31.html 1/6 Next: 3.5.2 The k.p Method Up: 3.5 Strain and Bulk Previous: 3.5 Strain and Bulk Subsections 3.5.1.1 Strain Induced Conduction Band Splitting 3.5.1.2 Strain Induced Degeneracy Lifting at the Point 3.5.1.3 Strain Induced Valence Band Splitting 3.5.1 Deformation Potential Theory Bardeen and Shockley [165 ] originally developed the deformation potential theory. Herring and Vogt [166 ] generalized this theory. Bir and Pikus [161 ] studied various semiconductors via group theory and showed how to calculate strain effects on the band structure with deformation potentials. A short introduction into the deformation potential theory is given subsequently. The deformation potential theory introduces an additional Hamiltonian , that is attributed to strain and its effects on the band structure. This Hamiltonian is based on first order perturbation theory and its matrix elements are defined by (3.15) denotes the deformation potential operator which transforms under symmetry operations as second rank tensor [167 ] and describes the strain tensor component. The subscripts in denote the matrix element of the operator . Due to the symmetry of the strain tensor with respect to and , also the deformation potential operator has to obey this symmetry and thus limits the number of independent deformation potential operators to six. In the case of cubic semiconductors the edges of the conduction band and the valence band are located on symmetry lines. These symmetries are reproduced in the energy band structure and in the basis states. Furthermore, the symmetry of the basis states allows to describe the deformation potential operator of a particular band via two or three deformation potential constants [166 ]. Although, theoretically the deformation potential constants can be calculated via the empirical pseudo potential method or by ab initio methods, it is more convenient to fit the deformation potentials to experimental results obtained by electrical, optical, microwave techniques, or by analyzing stress induced absorption edges. Even though, theoretical predictions and measurements match quite well, deformation potentials in literature and found by different methods deviate from each other [168 ]. 3.5.1.1 Strain Induced Conduction Band Splitting Cubic crystalls exhibit a strain induced energy shift for the nondegenerate energy levels of the conduction band. Along the symmetry line it is sufficient to describe the deformation potential operators as scalars by one or two independent constants. The energy shifts of the conduction band edge of valleys along the and directions is determined by two independent

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  • 3/12/2015 3.5.1 Deformation Potential Theory

    http://www.iue.tuwien.ac.at/phd/windbacher/node31.html 1/6

    Next:3.5.2Thek.pMethodUp:3.5StrainandBulkPrevious:3.5StrainandBulk

    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

    http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Herring1956http://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Bardeen1950http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Bir1974http://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hinckley1990http://www.iue.tuwien.ac.at/phd/windbacher/node32.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node32.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Fischetti1996ahttp://www.iue.tuwien.ac.at/phd/windbacher/node4.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Herring1956

  • 3/12/2015 3.5.1 Deformation Potential Theory

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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,

    http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hensel1965http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Bir1974http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Balslev1966http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hensel1965http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hensel1965http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Laude1971http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Bir1974

  • 3/12/2015 3.5.1 Deformation Potential Theory

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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).

    http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hensel1965http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Laude1971

  • 3/12/2015 3.5.1 Deformation Potential Theory

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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 .

  • 3/12/2015 3.5.1 Deformation Potential Theory

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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:

    http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Cardona1966http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Hensel1965http://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Bir1974

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    (3.24)

    denotesthe matrix

    (3.25)

    Inthecaseofthevalencebandthedescriptionofthestraininducedshiftsoftheheavyhole,lighthole,andthesplitoffbandaremorecomplex[169].

    Footnotes

    ...constants3.1neglectingstraininducedsplittingofthedegenerateconductionbands and atthe

    point

    Next:3.5.2Thek.pMethodUp:3.5StrainandBulkPrevious:3.5StrainandBulk

    T.Windbacher:EngineeringGateStacksforFieldEffectTransistors

    http://www.iue.tuwien.ac.at/phd/windbacher/node32.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node76.html#Balslev1966http://www.iue.tuwien.ac.at/phd/windbacher/node4.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node32.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.htmlhttp://www.iue.tuwien.ac.at/phd/windbacher/node30.html


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