EMS Measurement of the Valence Spectral Function of Silicon - a test of Many-body Theory

0.53 MB·English

by  C. Bowles

1 EMSMeasurementoftheValenceSpectralFunctionofSil-
icon - a test of Many-body Theory
C.Bowles,A.S.Kheifets,V.A.Sashin,M.Vos,E.Weigold1
AtomicandMolecularPhysicsLaboratories,ResearchSchoolofPhysicalSciences
andEngineering,AustralianNationalUniversity,Canberra,0200Australia
4
0
F.Aryasetiawan
0
2
Research Institute for Computational Sciences, AIST, Tsukuba Central 2, Umezono 1-1-1,
n
TsukubaIbaraki305-8568Japan
a
J

8

1
v
3 1.1 Introduction
3
1
Theelectronicpropertiesofthegroundstatesofsemiconductorshavebeenstudiedbothexper-
1
0 imentallyandtheoreticallyformanyyears. Thusangleresolvedphotoelectronspectroscopy
4 (ARPES), especially in combination with tunable synchrotron light sources, has been ex-
0 tensively used to map the dispersion of bands in single crystals. However, in the past the
/
t experimentalworkhasconcentratedalmostexclusivelyonthemeasurementofenergies,and
a
thetheoreticalvalence-bandstructurecalculationshavebeentestedessentiallyonlyinterms
m
of their predictions of eigenvalues. In contrast, relatively little attention has been given to
-
the wave functionof the electrons, despite the fact that wave function informationprovides
d
n a much more sensitive way of testing the theoretical model under investigation. Although
o thewavefunctioncannotbemeasureddirectlyitiscloselyrelatedtothespectralmomentum
c density. Intheindependentparticlemodelitissimplyproportionaltothemodulussquareof
:
v theone-electronwavefunction,
i
X
A(q,ω)= n |Φ (q)|2δ(q−k−G)δ(ω−ε ), (1.1)
r j,k j,q jk
a XGk
where Φ(q) is the momentum-spaceone-electronwave function, j is the band index, k the
crystalwavevector,andn andε aretheoccupationnumberandenergyofthecorrespond-
jk jk
ingone-electronstate. ThereciprocallatticevectorGtranslatesthemomentumq tothefirst
Brillouinzone. Foraninteractingmany-electronsystemthefullspectralelectronmomentum
density(SEMD)isgivenby
1 1 1
A(q,ω)= G−(q,ω)= (1.2)
π π[ω−h−Σ(q,ω)]
1e-mail:[email protected]
2 1 EMSMeasurementoftheValenceSpectralFunctionofSilicon-atestofMany-bodyTheory
Here G−(q,ω) is the interacting single-hole (retarded) Green’s function of the many-
electron system, Σ is the self energyand h is the one-electronoperator, which includesthe
kinetic energy and the Coulomb potential from the nuclei and the average of the electron
chargeclouddensity(theHartreepotential).PresumingthattheGreen’sfunctioncanbediag-
onalizedonanappropriatebasisofmomentum-spacequasiparticlestatesφ (q)(e.g. orbitals
j
inatoms,Blochwavesincrystals)thenforacrystalittakestheform[1]
1
A(q,ω)= |φ (q)|2δ ImG−(k,ω). (1.3)
j q,(k+G)π j
jX,k,G
Intheabsenceofelectron-electroninteractionsthenon-interactingGreen’sfunctionissimply
adeltafunctionandeq.(1.3)reducestoeq.(1.2).TheinteractingSEMDcontainsmuchmore
informationthan simply the band dispersion. The main feature describes the probabilityof
quasiparticle in band j having momentum k and energy ω. The center of the quasiparticle
peakisshiftedwithrespecttotheone-electronenergyε anditacquiresawidthduetothe
jk
finitequasiparticlelifetime. Inadditionelectroncorrelationeffectscangiverisetosignificant
satellitestructures.
The full SEMD can be measured by electron momentum spectroscopy(EMS) [1, 2], in
which the energies E , E and E and momenta k , k and k of the incident (subscript
0 1 2 0 1 2
0) and two outgoing electrons (subscripts 1 and 2) in high-energyhigh-momentum-transfer
(e,2e) ionizing collisions are fully determined. From energy and momentum conservation
one can determine for each (e,2e) event the binding (or separation) energy of the ejected
electron
ω =E −E −E , (1.4)
0 1 2
andtherecoilmomentumoftheionizedspecimen
q =k +k −k . (1.5)
1 2 0
Thedifferentialcrosssectionisgivenby[2].
σ(k ,k ,k ,ω)=(2π)4k−1k k f A(q,ω). (1.6)
0 1 2 0 1 2 ee
Heref istheelectron-electronscatteringfactor,whichisconstantinthenon-coplanarsym-
ee
metric high-energy (e,2e) kinematics used in the spectrometer at the Australian National
University[3,4]. Thusthe(e,2e)crosssectionisdirectlyproportionaltothefullinteracting
SEMD. Since the EMS measurements involve real momenta, the crystal momentum k not
appearingin the expressionfor the crosssection, EMS canmeasure SEMDsfor amorphous
andpolycrystallinematerialsaswellasforsinglecrystals.
Theprototypesemiconductorsiliconhasbeenused asa test-bedtoinvestigatethe influ-
enceofelectroncorrelationsontheSEMD,A(q,ω). Manyfirst-principlescalculationshave
beencarriedoutonbulksilicon,seee.g. refs[5,6,7,8,9,10,11,12]. Themajorityofthese
calculationsarebasedontheGW approximationtotheinteractingGreen’sfunction[13,14].
ThedispersionofthebulkbandsinsiliconhasbeenstudiedwithARPESalonghighsymme-
trydirections(seee.g.refs[15,16,17]). However,therehasbeenessentiallynoexperimental
1.2 ExperimentalDetails 3
dataavailableontheshapesofthequasiparticlepeaks(i.e. quasiparticlelifetimes)asafunc-
tion of momentum, and the satellite density as a functionof energyand momentum. These
propertiesoftheSEMDarisedirectlyfromelectroncorrelationeffectsandprovidestringent
testsforapproximationstothemany-electronproblem. First-principlescalculationsofmany
physicalquantitiesofinterestrequiretheinteractingone-particleGreen’sfunctionasinput.It
isthereforeimportanttohavereliablemethodsforaccuratelycalculatingandtestingthereal
andimaginarypartsoftheselfenergy,andhenceofA(q,ω).
ThereareseveredifficultiesinextractingthefullA(q,ω)fromexperimentaldataobtained
byothertechniques. InARPESthesedifficultiesincludeknowingthespecificsofthetransi-
tionsinvolved,suchastheuntanglingoffinal-stateeffectsfromtheinitial-stateones,andthe
strongenergyandmomentumdependenceofthematrixelements[13,18].ARPESisalsovery
surfacesensitive,whichcanobscuredetailsofthebulkelectronicstructure(seee.g. [17]). In
additionthereisusuallyasignificantbackgroundunderlyingthephoto-electronspectrumand
thiscanhideanycontinuoussatellitecontributions. CertainComptonscatteringexperiments
in which the struck electron is detected in coincidence with the scattered photon, so-called
(γ,eγ) experiments, can in principle map out the spectral function [19, 20]. However, the
energyresolutionissuchthatitisextremelydifficulttoresolveeventhevalencecontributions
fromthatofthecoreelectrons[21].
In this paper we presentEMS measurementsof the valence spectralmomentumdensity
for the prototypical and most studied semiconductor, silicon, and compare the results with
calculationsbasedontheindependentparticleapproximationaswellascalculationsbasedon
many-bodyapproximationstotheinteractingone-particleGreen’sfunction.Insection1.2we
discussthe experimentaltechnique. The theoreticalmodelsareoutlinedin section 1.3. The
results are discussed and compared with the FP-LMTO calculations and the first-principles
many-bodycalculationsinsection1.4and,whereappropriate,withpreviousARPESdata. In
section1.4theroleofdiffractionisalsodiscussedandinthelastsectionabriefsummaryand
conclusionisgiven.
1.2 Experimental Details
Anoutlineoftheexperimentalapparatus(describedfullyin[3,4])andthecoordinatesystem
isshowninFig. 1.1. Anelectrongunemitsahighlycollimated25keVelectronbeam,which
entersthesampleregioninsideahemisphereheldat+25kV.Thus50keVelectronsimpinge
on the target sample along the z−direction, the diameter of the beam being 0.1 mm. The
emerging pairs of electrons with energies near 25 keV are decelerated and focussed at the
entranceoftwosymmetricallymountedhemisphericalelectrostaticanalyzers. Theanalyzers
detectelectronsemergingalongsectionsofa conedefinedbyΘ = 44.3◦, whichischosen
s
so thatif all threeelectronsare in the same planethenthere isno momentumtransferredto
thetarget(i.e. q = 0). Intheindependentparticlepicturethiscorrespondstoscatteringfrom
astationaryelectron. Iftheelectronsarenotinthesameplane(i.e. φ 6= φ )thenthereisa
1 2
y−componentofmomentumwithq =q = 0,thatisonlytargetelectronswithmomentum
x z
qdirectedalongthey−axiscancauseacoincidenceevent.Theelectronsaredetectedbytwo-
dimensionalposition-sensitiveelectrondetectors,mountedattheexitplanesoftheanalyzers,
whichmeasuresimultaneouslyoverarangeofenergiesandy−momenta(i.e. rangeofangles
4 1 EMSMeasurementoftheValenceSpectralFunctionofSilicon-atestofMany-bodyTheory
a
b c d
Figure 1.1: Schematics of the experimental arrangements. Incident electrons of momentum
k0 alongthez−axisejectanelectronfromathinself-supportingSicrystal. Thescatteredand
ejectedelectronsemergingalong theshaded portionsof theconedefined byΘ = 44.3◦, are
detectedincoincidencebytwoenergyandanglesensitiveanalysers. Inthebottompanelsthe
sampleisindicatedasablockwithsidesparalleltotheh010iandh001isymmetrydirections.
Thusin(b) thespectral momentum densityismeasured along theh010i direction. In(c)the
densityismeasuredalongtheh110idirection,asthecrystalhasbeenrotatedaboutthesurface
normalby45◦. In(d)thecrystalhasbeentiltedby35.3◦ relativetothepositionin(c)sothat
thedensityismeasuredalongtheh111idirection.
φ).
Twopairsofdeflectionplatesmountedinsidethehigh-voltagehemispherealongthesec-
tions of the cone can be used to change the effective scattering angle by up to 1◦. In this
way [3] one can select nonzero values for the x− and/or the z−componentof momentum,
the y−componentalways lying in the range 0-5 a.u. (atomic units are used here, 1 a.u. =
1.89A˚−1). This allowsone to probethe fullthree-dimensionalmomentumspace. The two
double deflectors are also used to check that the measured momenta correspond to the ex-
pectedones[3,22],ensuringthattherearenooffsetsinq orq asaresultofanysmallpossi-
x z
1.2 ExperimentalDetails 5
blegeometricalmisalignments. Theorientationofthetargetspecimencanbedeterminedby
observingthediffractionpatternofthetransmittedelectronbeamonaphosphorusscreen. A
specificdirectionofthethincrystaltargetisthenalignedwiththey−axisofthespectrometer
sothatwemeasuretheenergy-resolvedmomentumdensityalongthatdirection. Thesample
orientationcan be changedwithoutremovingthe sample from the vacuumby means of the
manipulatormountingarrangement[3].
Thesinglecrystaltargethastobeanextremelythinself-supportingfilm. Theinitialpart
ofthetargetpreparationfollowedtheprocedureofUtteridgeetal[23]. Firstaburiedsilicon
oxidelayerwasproducedbyionimplantationinacrystalwithh100isurfacenormal.Acrater
wasthenformedonthebackofthecrystalbywetchemicaletching.Theoxidelayerservesto
stoptheetching. ItisthenremovedbyaHFdipandthesampleistransferredtothevacuum.
At this stage the thicknessof the thinnedpartof the crystal is around200nm. Low energy
(600eV)argonsputteringisthenusedtofurtherthinthesample.Thethinpartofthecrystalis
completelytransparent.Thethicknessismonitoredbyobservingthecolourofthetransmitted
lightfromanincandescentlampplacedbehindthesample.Thecolourchangeswiththickness
duetotheinterferenceofthedirectlytransmittedlightwiththatreflectedfromthefrontand
back silicon-vacuuminterfaces. The thinning is stopped when the thickness reaches 20 nm
(correspondingto a grey-greenishlight). Thesample is thentransferredto the spectrometer
underUHV. A thin amorphouslayercouldbepresentonthe backsideduetothe sputtering,
whereas the frontside (facing the analyzers) is probablyhydrogenterminatedas a result of
theHFdip. Thebasepressureinthesputteringchamberwasoftheorderof10−9Torrandin
thespectrometertheoperatingpressurewas2×10−10Torr.
The experimentalenergy and momentum resolutions were discussed in detail by Vos et
al [1]. The full width at half-maximum (FWHM) energy resolution is 1.0 eV, whereas the
FWHM momentumresolutions are estimated to be (0.12, 0.10, 0.10 a.u.) for the q ,q ,q
x y z
momentumcomponentsrespectively.
Evenforaverythintargetmultiplescatteringbytheincomingoroutgoingelectronshas
tobetakenintoaccount.Thesefastelectronscanlooseenergybyinelasticcollisions,suchas
plasmonexcitation, or transfer momentumbyelastic collisions(deflectionfromthe nuclei).
Thesescatteringeventsmoverealcoincident(e,2e)eventsto”wrong”partsofthespectrum,
aseithertheenergyormomentumconservationequations(eqs. 1.4and1.5)areusedincor-
rectly. Forpolycrystallineoramorphoussolidsthesemultiple-scatteringeventscanbemod-
eledbyMonteCarlosimulations[24]. Inthecaseofinelasticscatteringthebindingenergyas
inferredfromeq.1.4willbetoohigh.Inelasticmultiplescatteringeventscanbedeconvoluted
fromthedatabymeasuringanenergylossspectrumfor25keVelectronspassingthroughthe
sample. Thisdeconvolutionisdonewithoutusinganyfreeparameters[25]. Thisapproachis
usedinthepresentcase. Figure1.2showstheenergylossspectrumobtainedwiththepresent
siliconcrystaltarget. Forsinglecrystalselasticscatteringfromthenucleiaddsupcoherently
(diffraction). The change of the incoming or outgoing momenta by diffraction changes the
outcomeofthemeasurementbyareciprocallatticevector.Wedemonstratelaterinthispaper
howdifferentmeasurementscanbeusedtodisentanglethediffractedcontributionsfromthe
primary(non-diffracted)contribution.
6 1 EMSMeasurementoftheValenceSpectralFunctionofSilicon-atestofMany-bodyTheory
s)
nit
u
b.
ar
y (
sit
n
e
nt
I
x5
0 5 10 15 20 25 30 35 40
Electron Energy Loss (eV)
Figure 1.2: The energy loss spectrum observed in the spectrometer for the 20 nm thick Si
crystalsamplewith25keVincidentelectrons. Thefittedcurveisusedtodeconvolutethedata
forinelasticscattering.
1.3 Theory
1.3.1 Independent ParticleApproximation
The local density approximation (LDA) of density functional theory (DFT) has long been
established as a very useful tool for investigating ground-state properties of bulk semicon-
ductors from first principles [26, 27]. The advantage of DFT for approximate calculations
inmany-bodysystemsisthatoneextractstheneededinformationfromaone-bodyquantity,
theelectrondensityn(r). Althoughtheone-particleeigenvaluesinthetheoryhavenoformal
justificationasquasiparticleenergies,inpracticetheyturnouttobesurprisinglyaccurate[28].
We employed here the linear-muffin-tin-orbital(LMTO) method [29] within the frame-
workofDFT.TheLMTOmethodisjustoneofmanycomputationalschemesderivedwithin
theframeworkoftheDFT.ThegreatpracticaladvantageoftheLMTOmethodisthatonlya
minimalbasissetofenergy-independentorbitals(typically9-16peratom)isneededtoobtain
accurate eigenvalues (band energies). In the present study we implemented a full-potential
version(FP-LMTO)ofthemethod[30]. Wewritetheone-electronwavefunctioninacrystal
inthetight-bindingrepresentationastheBlochsumofthelocalisedMTorbitals:
Ψ (r)= eik·t ajkφ (r−R−t). (1.7)
jk Λ Λ
Xt XΛ
Herekisthecrystalmomentum,jbandindex,tlattice(translation)vectorandRbasisvector.
The label Λ defines a MT orbital centered at a given site R and it comprises the site index
R and a set of atomic-likequantum numbers. The expansioncoefficientsajk are foundby
Λ
solvingtheeigenvalueproblemusingthestandardvariationaltechnique.
MomentumspacerepresentationofthewavefunctionΨ isgivenbytheFouriertrans-
jk
1.3 Theory 7
formoftheBlochfunction:
Φ (q)= e−iq·rΨ (r)dr. (1.8)
jk jk
Z
Due to the periodic nature of the charge density the only non-zero contributionsto Φ (q)
jk
occurwhenq =k+GwithGareciprocallatticevector.
Φ (q)= cj δ(q−k−G). (1.9)
jk G,k
XG
The contributionscj , the Bloch wave amplitudes, are expressedthroughthe (Fourier)
G,k
integrals:
cj = Ω−1 e−i(k+G)·rΨ (r)dr (1.10)
G,k Z jk
= Ω−1 [ajke−i(k+G)·R e−i(k+G)·rφ (r)dr]
Λ Z Λ
XΛ
HereitisassumedthatthewavefunctionΨ isnormalizedintheunitcellofthevolumeΩ.
jk
The limits of the three-dimensionalintegration indicates symbolically the whole coordinate
space. The EMD in the occupiedpart of the bandj is proportionalto the modulussquared
Blochamplitudes:
Ω2
ρ (q)= n |Φ (k+G)|2δ (1.11)
j (2π)3 jk j q,k+G
XG
where n is the occupationnumber. The EMD (1.11) is normalizedto the totalnumberof
jk
valenceelectronsperunitcell:
2 dqρ (q)=n . (1.12)
j e
Z
Xj
1.3.2 Electron CorrelationModels
TheholeGreen’sfunctionenteringequation1.3canbecalculatedbythemany-bodyperturba-
tiontheory(MBPT)expansionontheBlochwavebasis(1.7). Takingthefirstnon-vanishing
term in the MBPT leads to the so-called GW approximation [31, 32]. In this acronym G
stands for the Green’s function and W denotes the screened Coulomb interaction. The GW
approximationisknowntogiveaccuratequasiparticleenergies[14]. However,itsdescription
ofsatellitestructuresisnotsatisfactory. Inalkalimetals,forexample,photoemissionspectra
showthepresenceofmultipleplasmonsatelliteswhereastheGW approximationyieldsonly
oneattoolargeanenergy. ThisshortcomingoftheGW approximationhasbeenresolvedby
introducingvertexcorrectionsintheformofthecumulantexpansiontotheGreen’sfunction
[33,34,35]. Thisallowedtheinclusionofmultipleplasmoncreation. Asaresultthecalcu-
latedpeakpositionsoftheplasmonsatelliteswerefoundtobeinmuchbetteragreementwith
theexperimentthanthosepredictedbytheGW schemeitself[36,37,38].
8 1 EMSMeasurementoftheValenceSpectralFunctionofSilicon-atestofMany-bodyTheory
Formally, the cumulant expansion for the one-hole Green’s function can be derived as
follows. We choose the time representationforthe Green’sfunction, dropthe bandindex j
forbrevity,andwriteitas
G(k,t<0)=iθ(−t)e−iωkt+Ch(k,t), (1.13)
whereω istheone-electronenergyandCh(k,t)isdefinedto bethecumulant. Expanding
k
theexponentialinpowersofthecumulantweget
G(k,t)=G (k,t) 1+Ch(k,t)+ 1 Ch(k,t) 2+... , (1.14)
0
(cid:20) 2 (cid:21)
(cid:2) (cid:3)
whereG (k,t)=iθ(−t)exp(−iω t). Intermsoftheself-energy ,theGreenfunctionfor
0 k
theholecanbeexpandedas P
G=G +G G +G G G +... . (1.15)
0 0 0 0 0 0
X X X
TolowestorderinscreenedinteractionW,thecumulantisobtainedbyequating
G Ch =G G , (1.16)
0 0 0
X
where = =iG W. Thefirst-ordercumulantistherefore
GW 0
P P
∞ ∞
Ch(k,t)=i dt′ dτeiωkτ (k,t). (1.17)
Zt Zt′ X
ThisisthenputbackintoEq.(1.13)yieldingmultipleplasmonsatellites.Theenergy-momentum
representationoftheGreen’sfunctioncanberestoredbythetimeFouriertransform.
1.4 Results and Discussions
1.4.1 Band Structure
Wefirstdiscussthedispersionε (q),i.e.thedependenceoftheenergyoftheBlochfunction
jk
Φ (q) of bandj onits crystalmomentumk = q +G. The bandwith the largestbinding
jk
energyis labeled1, thenextone2 etc. Inmomentumspace theBloch functionwith crystal
momentumkisnon-zeroonlyatmomentumvaluesk+Gwithamplitudecj (seeeq.1.9).
G,k
The band structure and momentum densities obtained by Kheifets et al. [30] in a FP-
LMTOcalculation(basedon theDFT-LDA) areshownin Fig. 1.3, the bandsbeinglabeled
asdiscussedabove. Thebandsareperiodicinqspace,withband1havingamaximuminthe
bindingenergyattheΓpoints.However,theonlyΓpointwithsignificantmomentumdensity
inband1istheonecorrespondingtozeromomentum(seelowerpanelinFig. 1.3). Thusthe
functionwiththelowestenergyisaBlochfunctionwithk = 0,c1 ≃ 1,andtheother
(0,0,0),k
c1 ≃0.
G,k
ThecutofthefirstfourBrillouinzonesofsiliconalongtheq =0planeisshowninFig.
z
1.4. Forafreeelectronsolidthewavefunctionsareplanewavesandinthegroundstatethe
1.4 ResultsandDiscussions 9
Direction [100] Direction [110] Direction [111]
0
V) 2 3,4 4 3,4
Binding energy (e 18640 12 21 3 12
12
Γ X Γ X Γ X K X Γ Γ L Γ L

1
Momentum density (a.u.) 0000....2468 1 2 1 3 1 2
0
0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5
Momentum (a.u.) Momentum (a.u.) Momentum (a.u.)
Figure1.3: FP-LMTOcalculations[30]ofthedispersion(toppanel)andmomentumdensity
(bottompanel)ofSifordifferenthighsymmetrycrystallographicdirections. Thetotalmomen-
tumdensityissplitupintothedensitiesoftheindividualbandsasindicated.
occupiedstatesarewithintheFermispherewithradiusk i.e. |k| < k . Theintersectionof
f f
this sphere with the q = 0 plane is indicated by the dashed circle in Fig. 1.4. The lattice
z
potential of silicon can be viewed as a perturbationon the free electron picture, so that the
wave functions are Bloch functions with more than one |cj |2 > 0. The semiconductor
Gk
silicon has 8 valence electrons per unit cell, and hence the first 4 bands are fully occupied
withonespinupandonespindownelectronperband. Wewillnowdiscusstheresultsofthe
measurementofsiliconandemphasizethattheelectrondensityforbandj isatitsmaximum
inBrillouinzonej.
Thesamplewasathin(≃20nm)singlesiliconcrystalwithh001isurfacenormal,which
is first alignedwith the z direction(i.e. alignedwith k , see Fig. 1.1(a)). Rotatingaround
0
thesurfacenormal,measurementsweretakenwiththesampleh100idirectionalignedalong
the y−axis, then the h110i direction and 4 intermediate directions were aligned along the
y−axisas shownin Fig. 1.4. In all these cases the potentialson the sets of deflectorplates
weresettoensurethatthemeasurementspassedthroughzeromomentum(correspondingto
Γ ). Theexperimentallyobserveddensitydistributions,togetherwiththeresultsofthe
(0,0,0)
FP-LMTOcalculation,areshowninFig. 1.5forthe6directionsmeasuredthroughΓ .
(0,0,0)
The calculationswere broadenedwith the experimental1 eV energyresolutionand split up
intothe4occupiedbands.
Forthemeasurementofmomentadirectedalongthe h100idirection(0◦ inFig. 1.4)the
theorypredictsbands1and2occupied.InthefirstBrillouinzoneband1isoccupied,changing
abruptly to band 2 at 0.61 a.u. (see also Fig. 1.4 and left panel of Fig. 1.3). There is no
band gap in the dispersion on crossing the first Brillouin zone. This is due to additional
symmetryofthediamondlattice(seee.g. [39]). AfterleavingthesecondBrillouinzonethe
10 1 EMSMeasurementoftheValenceSpectralFunctionofSilicon-atestofMany-bodyTheory
Figure1.4: The cut through the reciprocal lattice of silicon along the q = 0 plane with the
z
first 4 Brillouin zones labeled. The Brillouin zone boundaries are labeled by the indices of
thereciprocallatticeitbisects. ThedashedcircleistheFermisphereforafreeelectronsolid
withthesameelectrondensityassilicon. Differentmeasurementsthroughtheq = 0pointare
indicatedbythedashedlines.
calculateddensitydropsonlygraduallytozero. Themeasureddensity(leftpanelofFig. 1.5)
has the same behaviour , but also shows an additional branch at smaller binding energies,
whichmergeswiththemainfeatureat1.2a.u. Thisadditionalbranchcanalsobeseeninthe
calculatedbandstructureforthecasewherethecrystalhasbeenrotatedby8.5◦, andcomes
from band 4. From the shape of the Brillouin zone shown in Fig. 1.4 it is clear that the
measurementalongtheh100idirectionjustmissesBrillouinzone4. Duetofinitemomentum
resolutionitisobviousthatthemeasurementwillpickupcontributionsfromthiszone,giving
risetotheextrabranchintheobservedintensity.
Intheh110isymmetrydirection(reachedbyarotationby45◦ alongthesurfacenormal)
thefirstBrillouinzonecrossingistwoplanes,(the(111)and(111)planes),makinganangle
of±54.35◦withtheq =0plane.Thusthebandswitchesfrom1to3atthedoublecrossing.
z
Thisgivesrisetotheclassicbandgapbehaviour,band1havingaminimuminbindingenergy
(maximum in energy) at the Brillouin zone crossing, with its density petering out after the
crossing. Band3slowlyincreasesinintensityfromzeromomentumuptothefirstBrillouin
zone boundary, where it has a maximum in binding energy, with increased density as one
passesthroughBrillouinzone3. Thenextextremuminenergy,whichcorrespondstoamini-
muminbindingenergy,iswhenband3crossesthenextsetofBrillouinzoneboundaries,i.e.
onleavingBrillouinzone3,anditsintensitydecreasesthereafterasthemomentumincreases.
The calculations and measurements are in quite good agreement with each other for these
generalfeatures. Fortheintermediateanglesband1remainsdominant,band4isprominent,
band 2 makes a significant contribution for directions not far from the h100i direction, and
band3makesmallcontributionsclosetotheh110idirection.
AlsoshowninFig. 1.5(bottompanels)isthespectralmomentumdensityobtainedalong
theh111idirection.Thisdirectionwasreachedbytiltingtheh110ialignedsampleover35.3◦
(see Fig. 1.1(d)). Here the density is due to bands 1 and 2 with a large band gap at the