Alloying effects on the optical properties of Ge$_{1-x}$Si$_x$ nanocrystals from TDDFT and comparison with effective-medium theory

0.25 MB·English

by  Silvana Botti

Alloying effects on the optical properties of Ge Si nanocrystals from TDDFT and
1−x x
comparison with effective-medium theory
Silvana Botti,1,2,3 Hans-Christian Weissker,1,3 and Miguel A. L. Marques2,3
1Laboratoire des Solides Irradi´es, E´cole Polytechnique, CNRS, and CEA-DSM, F-91128, Palaiseau, France
2Laboratoire de Physique de la Mati`ere Condens´e et Nanostructures,
Universit´e Lyon I and CNRS, F-69622 Villeurbanne Cedex, France
3European Theoretical Spectroscopy Facility (ETSF)
(Dated: January 30, 2009)
9
We present the optical spectra of Ge1−xSix alloy nanocrystals calculated with time-dependent
0
density-functional theory in the adiabatic local-density approximation (TDLDA). The spectra
0
change smoothly as a function of the composition x. On theGe side of thecomposition range, the
2
lowestexcitationsattheabsorptionedgearealmostpureKohn-Shamindependent-particleHOMO-

n LUMOtransitions,whileforhigherSicontentsstrongmixingoftransitionsisfound. WithinTDLDA
a thefirstpeakisslightlyhigherinenergythaninearlierindependent-particlecalculations. However,
J
theabsorptiononsetandinparticularitscompositiondependenceissimilartoindependent-particle

9 results. Moreover, classical depolarization effects are responsible for a very strong suppression of
2 the absorption intensity. We show that they can be taken into account in a simpler way using
Maxwell-Garnett classical effective-medium theory. Emission spectra are investigated by calculat-
] ingtheabsorptionofexcitednanocrystalsattheirrelaxedgeometry. Thestructuralcontributionto
i
c theStokesshiftisabout0.5eV.Thedecompositionoftheemissionspectraintermsofindependent-
s particletransitionsissimilartowhatisfoundforabsorption. Fortheemission,veryweaktransitions
-
l are found in Ge-rich clusters well below the strong absorption onset.
r
t
m PACSnumbers: 71.15.Mb,78.67.-n,78.67.Bf
.
t
a
I. INTRODUCTION the HOMO-LUMO transition,thus resulting inshortra-
m
diative lifetimes of Ge clusters.8,13
-
d The semiconductors silicon and germanium can form Therefore, when mixing both materials two questions
on aersinugbstthiteutwiohnoallesoralindgseolouftioconmopfothsietifoonrsmxG.e1P−uxSreixSciovis- atrroisnei:cpwrhoapteritsietshoefetffheecntaonfotchryesitnatlse,rmanixdihnogwonarethteheeldeicf--
c
[ the mostwidely usedmaterialfor electronicapplications ferent characters of the two materials combined. While
since many yearsandits fabricationtechnologyis highly theanswertothesequestionsisessentiallywellknownfor
1 developed. However, the indirect band gap of bulk Si the bulk alloy, there are still few investigations concern-
v presents a problem for light-emitting applications. A so- ingthemixednanostructures. Mostexperimentalstudies
6
lution that has been proposed to circumvent this prob- useStranski-Krastanovgrowth,(see,e.g.,Ref.14),which
0
7 lem is nanostructurizationof Si in structures comprising results in relatively large structures in which it is safe to
4 porousSi,1 nanowires,2 aswellasSinanocrystals. More- assume that the effects of confinement and alloying act
. over, ample use has been made of the fact that Ge can independently. However, for smaller structures this can-
1
0 easily be combined with Si in heterostructures. In addi- not be taken for granted. For small nanocrystals, pho-
9 tion, Ge nanocrystalsina matrixofSiO2, SiC,sapphire, toluminescence experiments15 have been compared with
0 orSihavebeeninvestigatedbymanygroups.3,4,5Inmany theoreticalresults.16 Furthermore, Ge-Si nanowireshave
: ofthosecases,intermixingbetweenGeandSiisfoundin been investigated experimentally17 and theoretically.18
v
i the nanostructures. Previous theoretical studies have focused on the
X The two materials show different properties upon interplay of confinement and alloying.16,18 However,
ar nanostructurization. While Si retains its character of an these ab initio calculations were performed within the
indirect material,6,7,8 the work of several groups showed independent-particle approximation based on the Kohn-
that, depending on the structure, strain, etc., Ge nanos- Sham scheme of static density-functional theory (DFT).
tructures canbecome quasi-direct,i.e., they exhibit very Therefore, important many-body effects have been ne-
strong transitions at or very close to the HOMO-LUMO glected, viz., the self-energy corrections describing the
transition.6,7,8,9,10,11,12Thedifferentbehaviorreflectsthe effect of the excitation of the electrons or holes individ-
different character of the band gaps in the bulk materi- ually, as well as the electron-hole interaction. (Nonethe-
als. While both are indirect, the minimum gap in Si lies less, these two effects are found to cancel each other to
between Γ and a point near the X point, and the direct a large extent for many systems.19) Furthermore, these
gap is much larger. In Ge, the minimum gap between calculations miss the very important depolarization or
Γ and L is energetically very close to the direct gap at crystal local-field effects.
Γ. The effects of confinement and structural relaxation The way we choose to improve upon the independent-
result in a strong contribution of the Γ–Γ transition to particle results is provided by time-dependent
2
DFT20,21,22 (TDDFT) in which the many-body ef-
fects neglected in static DFT are introduced by the
so-called exchange-correlation kernel f . With respect
xc
to the independent-particle results, the excitation
energies are corrected, and the transitions between
the independent-particle states are mixed. The de-
gree of the mixing indicates the degree to which the
independent-particle approximation fails to provide a
good description of the system.
Within TDDFT, we use the adiabatic local-density
approximation (ALDA, also known as TDLDA) of the
exchange-correlation kernel. The choice is motivated by
thefactthatTDLDAyieldsgoodresultsforopticalspec- FIG. 1: (Color online) Example of the studied nanocrystal
tra of isolated systems23 as well as for non-zero momen- structures: Ge48Si35H108. The colors are: Si (yellow), Ge
tum transfers.24 Note, however, that the ALDA is well (blue),and H (white).
knowntofailinsomecases,themostimportantofwhich
are perhaps extended systems.22,25 Within TDDFT, the
dination. The outer bonds were saturated by hydrogen
random-phase approximation (RPA) is obtained by ne-
atoms. Alloying between Ge and Si was introduced by
glecting the exchange-correlation kernel, i.e., by setting
randomly exchanging Ge atoms by Si. The surface was
f =0. Fromthis, the independent-particle approxima-
xc then passivated with H atoms to saturate the remaining
tion results from the neglect of the microscopic terms of
dangling bonds.
the variation of the Hartree potential. In other words,
the difference between the independent-particle approxi- InRef.16,thestudyofGe1−xSixnanocrystalswiththe
same number of atoms for ten different atomic configu-
mationandthe RPAarethedepolarizationeffectswhich
rations demonstrated that those with nearly uniformly
are due to the inhomogeneity of the system.
distributed Ge and Si atoms possess the lowest total
The simplest way to account for depolarizationeffects
energies and nearly equal excitation energies. On the
within an approximated classical picture is to apply the
other hand, nanocrystals with deliberately clustered Si
Maxwell-Garnetteffective-mediumtheorytothecomplex
atoms and, hence, rather different excitation energies
dielectric function of the bulk alloy crystals. We applied
give rise to total energies substantially higher than the
the effective medium theory to our alloy clusters in or-
average. As their probability of occurrence is, conse-
der to check if the depolarization effects are dominant
quently, small, the configurational average can be re-
with respect to confinement effects and further many-
placed by the study of only one nanocrystal with nearly
body corrections. In fact, if it is the case, the spectra
uniformlydistributedSiandGeatomsforeachcomposi-
obtained using the effective-medium theory are in good
tion x. We selected nanocrystals with a total number of
agreement with full TDLDA results. These model cal-
83atomsofGeandSi,havingadiameterofabout1.5nm.
culations thus give a good overall description of the op-
This radius corresponds to a sphere of the volume occu-
tical spectra at a much lower computational cost than
pied by 83 atoms in the bulk. These nanocrystals are
TDLDA.
large enough to exhibit the characteristics of nanocrys-
In the present paper we first present TDLDA results
tals asopposedto muchsmaller structureswhichshow a
for Ge1−xSix nanocrsytals,focusing on alloying effects. molecular-like behavior.6,13,26 Moreover, they are small
The results are then compared with the previous
enough to present significant confinement effects on the
independent-particle calculationsofRef.16,highlighting
electronicstates. Anexampleoftheatomicarrangement
the effects of the depolarization, as well as of the mix-
is shown in Fig. 1 for Ge Si H .
48 35 108
ing of transitions. We will also consider depolarization
To obtain the relaxed geometries of the Ge1−xSix
effects alone, decoupled from confinement effects, by ap- nanocrystals, we used the plane-wave code VASP27
plying Maxwell-Garnett classical effective-medium the-
within the local-density approximation (LDA) in the
ory. Finally, the emission properties of the nanocrystals parametrization of Perdew and Zunger28 and the
are investigated by considering the geometries obtained projector-augmented wave (PAW) method.29 This com-
after excitation of an electron-hole pair.
putational set-up is the same as the one employed in
Ref.16.
Starting from the relaxed geometries we obtained
II. CALCULATION DETAILS the optical spectra at zero temperature using TDDFT
as implemented in the computer code octopus.30,31
We used the same modeling scheme as used in Ref. 16 The electron-ion interaction is described through norm-
consideringGe1−xSixnanocrystalswithafixedsizemade conserving pseudopotentials32 and the LDA28 is em-
of83GeandSiatoms. Quasi-sphericalnanocrystalswere ployed in the adiabatic approximation for the xc poten-
built starting from one atom and adding nearest neigh- tial (i.e., we applied the TDLDA). The time-dependent
bors shell by shell, assuming bulk-like tetrahedral coor- Kohn-Sham equations, in this code, are represented in
3
0.5
12 0.5
2.4
0.42
0.4 10
2) 0.6 0.60 2.02)
Å Å
( (
n n
ent 0.3 0.30 8 ptio ent 0.7 1.6 ptio
nt or nt 0.72 or
Si co 0.2 0.19 6 A Abs Si co 1.2 A Abs
0.8
D D
0.15 4 DL 0.84 0.8 DL
0.12 T T
0.1 0.9
0.07 2 0.93 0.4
0.98
0.00 1.00
0 0 1 0
2 2.5 3 3.5 2 2.5 3 3.5
w (eV) w (eV)
FIG.2: (Color online)AbsorptionspectraasafunctionoftheSicontentoftheclusters: Independent-particlespectra(dashed
lines) compared with TDLDA results (solid lines). The green arrows (arrows at lowest energy) mark the HOMO-LUMO gap,
whiletheredarrows (middlearrows) markthe∆SCFexcitation energies andthebluearrows (arrowsat highestenergy)mark
the first transitions of theCasida’s analysis. The independent-particlecurves are divided by a factor of 15. Note the different
scale in thetwo panels.
a real-space regular grid, using a spacing of 0.275˚A at spectra obtained by time propagation by means of the
whichthecalculationsareconverged. Thesimulationbox octopuscode. No substantialdifferences were found be-
isconstructedbyjoiningspheresofradius4.5˚A,centered tween the results.
around each atom. Due to the interest in the luminescence of this kind
To calculate the optical response we excite the system of systems, the lowest excitation energies are of partic-
from its ground state by applying a delta electric field ular importance. As a consequence, we decided to com-
E δ(t). The real-time response to this perturbation is pare three different quantities. Besides the lowest ex-
0
Fourier transformed to get the dynamical polarizability citations of the TDLDA absorption spectra, we present
α(ω) in the frequency range of interest. The absorption the Kohn-Sham HOMO-LUMO gap of the ground state
cross section σ(ω), is then obtained from the relation: and the ∆SCF excitation energies calculated as the dif-
ference E =E(N,e–h)−E(N), where E(N) is the
∆SCF
σ(ω)= 4πω Imα(ω), (1) totalenergyofthe ground-stateandE(N,e–h) isthe to-
c tal energy of the excited-state configuration where one
electron has been promoted from the HOMO into the
where c is the velocity of light in vacuum. For a tech-
LUMO. In this way, an excited configuration is mod-
nical description of this method we refer to Refs. 30
eled and the electron-hole interaction is partially taken
and 33. To obtain the RPA spectra in this approach we
into account. This approach enables also a subsequent
just kept the exchange-correlationpotential fixed during
ionicrelaxationwiththeelectron-holepairpresent,which
the propagation (which amounts to making f = 0).
xc yieldsthedescriptionoftheexcited-stategeometries. Us-
The independent-particlespectrawerecalculatedby fur-
ingtheexcited-stategeometrieswewerealsoabletoeval-
ther fixing the Hartree potential to its initial value. A
uate emission spectra.
time stepof0.0055~/eVanda totalpropagationtime of
37.5~/eV were sufficient to ensure a stable propagation.
We estimate ournumericalprecisionin the spectrato be
III. RESULTS
better than 0.05eV. The mixing of independent-particle
transitions in the spectra has subsequently been investi-
gated using the Casida’s formulation of linear-response A. Ground-state geometries: absorption
TDDFT.34,35
In order to test the coherence of our calculations, we Results for the photoabsorption of the Ge1−xSix
compared independent-particle spectra computed using nanocrystallites are presented in Fig. 2. In the figure we
matrix elements calculated with the VASP code6,36,37 to comparetheindependent-particleresponse(dashedlines)
4
with full TDLDA calculations (solid lines) for the whole
range of compositions. The dependence of the curves on
thecompositionturnsouttobequitesmooth. Whengo-
12
ingfromGe-richclusterstoSi-richclusters,weobservea
shift to higher energies of the onset and a suppressionof 1.00
the absorptionstrengthofthe firstpeak (note the differ- 1 10
ent scales of both panels). Moreover, the dependence
)
2
othfethsaempeeaikn tphoesitiniodnepoenndtehnet-cpoamrtpicolseitaionnd xtheisTroDuLgDhlAy ent 0.8 8 n (Å
sthcheeminedse.peFnodretnhte-pcalrutsictelerssspteucdtriead,isthsterotontgalyl isnutepnpsrietsyseodf Si cont 0.6 0.60 6 sorptio
in TDLDA (the independent particle curves are divided b
A
byafactorof15inFig.2). Indeed,thisquenchingofthe 0.4 4
absorptionisawellknowneffectthatisduetotheinclu- 0.30
sion of classical depolarization effects, and not from the
0.2 2
exchange-correlationeffectsaccountedforbytheTDLDA
kernel. Thiscanbeverifiedbycalculatingtheabsorption
0.00
spectrumwithintheRPA,whichincludesallclassicalef- 0 0
fects due to the variations of the Hartree potential but 2 2.5 3 3.5 4 4.5
neglects quasi-particle and excitonic effects. Indeed, the w (eV)
spectra we calculated within RPA are so close to the
TDLDA spectra shown in Fig. 2 that we chose not to
FIG.3: (Coloronline)Absorptionspectraasafunctionofthe
show them. Si content of the clusters: Effective-medium theory (dashed
lines) using the dielectric function of bulk Ge1−xSix , com-
pared with TDLDA results (solid lines).
B. Transition analysis and excitation energies
the absorption edge. In the intermediate energy range
For Ge-rich nanoparticles, both the position and the
and for all compositions, excitations can be decomposed
composition dependence of the peaks are already well
as a sum of many contributions.
describedattheleveloftheindependent-particleapprox-
In Fig. 2 we also show the HOMO-LUMO gap (green
imation. This perhaps surprising fact can be explained
arrows), the lowest excitation calculated within the
intermsofcompensationofquasiparticlecorrectionsand
∆SCF approximation (red arrows), and the first exci-
binding energies of the excitons. For small x at the ab-
tationwithinTDLDA(bluearrows,theseexcitationsare
sorptionedge,the firstpeakis strongandappearsessen-
dark on the Si side of the composition). For all cases
tially at the HOMO-LUMO energy gap. The absorption
we find that the first ∆SCF excitation is, as expected,
edgehasacompletely differentnatureinSi-richclusters.
blue-shiftedwith respectto the HOMO-LUMOgap,and
In fact, in this case the peaks of lowest energy have a
that the first TDLDA transition is at slightly higher en-
vanishing oscillator strength in TDLDA, thereby blue-
ergies. The differences are, however, quite small, and of
shifting the absorptionedge with respect to the HOMO-
the order of one tenth of an eV. This is known for this
LUMO gap.
class of systems,38 part of it being due to the cancella-
In order to analyze the origin of the different peaks
tion of self-energy and excitonic effects.19 However, the
in the spectra as a function of the composition of the
differences appear to be slightly larger in the region of
Ge1−xSix alloy, we decomposed the excitations in sums intermediate composition x, i.e., of a greater degree of
of Kohn-Sham particle-hole transitions through the so-
structural disorder. Similarly, Degoli et al. found in Si
lution of Casida’s equation.34,35 We found that on the
nanocrystals that the differences become larger in cases
Ge side of the composition range, the lowest peak of
of stronger localization.38 It is also clear from the plot
the spectra which defines the absorption edge is pro-
how the first transition becomes forbidden while going
duced essentially by a strong, pure transition between
from Ge-rich to Si-rich clusters — this behavior is remi-
theKohn-ShamHOMOandLUMO.Thisiscertainlyone
niscent of the different character of the band gap in the
more reason for the similarity between the independent-
parent bulk Ge and Si.
particle and the TDLDA spectra. The large peak with
HOMO-LUMO character decreases in intensity with in-
creasingpercentage ofSi in the Ge1−xSix alloy,until the
composition of about x = 0.2 when it disappears. For C. Absorption from classical effective-medium
even smaller x the absorptionat the onsetis determined theory
by a strong mixture of transitions between states close
to HOMO and LUMO. However, the very lowest transi- A much simpler approach to model the absorption
tions are forbidden, producing a significant blue shift of cross section of a Ge1−xSix nanocrystal is to start from
5
thecomplexdielectricfunctionofthecorrespondingbulk can then be used to calculate the spectra of the crys-
alloy crystal and to apply the effective-medium the- tallites in a different environment.26 In this sense, com-
ory.39,40 This classical approach is based on Maxwell’s parisonofthe TDLDA resultswith the presenteffective-
equations and neglects completely the microscopic de- medium approachgives an idea of the importance of the
tails, such as atoms and bonds. Of course, this assump- confinement effects on the overall spectra.
tionisbetterjustifiedwhenthesizeofthesystemislarge. We note that these model calculations can be per-
However,italwayshandlescorrectlytheboundarycondi- formed at negligible computational cost, and therefore
tionsfortheMaxwell’sequationsattheinterfaces,which provide a simple and fast method to obtain reasonable
give the very important contributions to the dielectric spectra for medium and large nanocrystallites.
responsethroughthe classicaldepolarizationeffects. Of- Giventhestronginfluenceofthedepolarizationeffects,
ten, these classical contributions are enough to describe the question arises as to why the independent-particle
the physics of the dielectric response of a composite sys- spectrahavebeensuccessfully comparedpreviouslywith
tem made of objects embedded in some matrix.41,42,43 experiment.6 This agreement is, in fact, not a fortuitous
Our clusters can be considered as a family of spheres coincidence,butduetotheexperimentalconditions. The
of volume Vobj cut from a Ge1−xSix bulk alloy. The experiment has been done on Ge NCs inside a matrix
Maxwell-Garnett expression39 in the specific case of an of sapphire.46 This reduces strongly the depolarization
isolated spherical object in vacuum yields: effects because it reduces the inhomogeneity of the sys-
tem. Thecalculations,ontheotherhand,treatedNCsin
ω Imǫ(ω)
σ(ω)=9 V , (2) vacuum but neglected the depolarization effects. There-
c obj[Reǫ(ω)+2]2+[Imǫ(ω)]2 fore, together with the cancellation between self-energy
effects and electron-hole interaction mentioned above,
where ǫ is the experimental complex dielectric function
the independent-particle approximation provides in fact
of the bulk alloy. To represent well the extension of the
agooddescriptionofthespectraofthisparticularexper-
polarizable nanocrystals, we took an average distance of
iment.
thefurthermostsaturatinghydrogenatomstoobtainthe
radiusofthe cluster,andconsequentlyV . This results
obj
in a radius slightly larger than the one mentioned above
D. Excited-state geometries: emission
which takes into account only the Ge and Si atoms. We
used the value of R =9.1˚A, but the results are fairly
obj In order to calculate the emission properties, we con-
insensitivetoa(reasonable)choiceofthisvalue. Theex-
sider the geometry of the relaxed nanocrystals where
perimental dielectric function of the bulk alloy with pre-
the ionic relaxation has been carried out after transfer-
ciselytheneededcompositionxhasbeenobtainedfroma
ring an electron from the HOMO to the LUMO Kohn-
discretesetofmeasurements44usingarecentlydeveloped
interpolationscheme.45ThisschemeinterpolatesImǫ(ω) Shamorbital. Astheradiativelifetimesareusuallymuch
longer than the times that the electrons (holes) take to
making use of the screening sum rule and the positive
definitenessofthespectra. Reǫ(ω)wassubsequentlyob- relax to the LUMO (HOMO), we can assume thermal-
ized electron-hole pairs. As their lifetimes are then de-
tained by means of the Kramers-Kronig relations after
termined by the exponential factor describing their dis-
fitting appropriatetails to the imaginary parts. The lat-
tribution,47,48 it is the onset of the emission which re-
ter procedure was tested on the input curves to insure
flects the emission properties, while the higher parts of
the quality of the real part.
the spectrum are suppressed.
InFig.3weshowspectracalculatedusingtheeffective-
Stimulated emission spectra can be easily obtained
medium theory for selected compositions. Comparing to
withinourformalismbycalculatingtheabsorptioncross-
the first-principles TDLDA curves, we can see that, al-
section σ˜ (ω) at the excited-state geometry. Lumi-
readyforthisrelativelysmallsizeofclusters,theclassical abs
nescence spectra can then be calculated from the van
theory gives a quite good overall description of the ab-
Roosbroeck-Shockleymodel49
sorption spectrum. As expected, effective-medium the-
ory is not capable of describing the peaks of the individ- 1
ual transitions, but it describes correctly the intensity σlum(ω)∼ω2e~ω/kbT −1σ˜abs(ω), (3)
and the trends of the spectrum. Once again these re-
sults confirm that the dependence on the compositionof where k is the Boltzmann constant and T is the tem-
b
the optical spectra is smooth and that the confinement perature.
andalloyingeffects actindependenttoalargedegree. In In Fig. 4 we compare the absorption cross sections
fact, within this classical scheme, only the alloying ef- for the ground-state (solid lines) and the excited-state
fects determine the variation of the absorption response (dashed lines) geometries for Ge1−xSix clusters in all
asafunctionofx. Inparticular,theconfinement-induced the composition range. The energy difference between
opening of the HOMO-LUMO gaps and the resulting the onsetsofabsorptionandemissioncorrespondsto the
blue-shift of the absorption onset are not accounted for. structural contribution to the Stokes shift.
The confinement effects could be described by introduc- In Fig. 5 we compare the independent-particle re-
ing a size-dependent crystallite dielectric function which sult with the TDLDA for the absorption σ˜ (ω) at the
abs
6
0.5
12 0.5
2.4
0.42 2) 2)
Å Å
0.4 10n ( 0.60 n (
o 0.6 2.0 o
si si
s s
nt 0.3 0.30 8 Emi nt 1.6 Emi
nte on/ nte 0.7 0.72 on/
Si co 0.2 0.19 6 sorpti Si co 0.8 1.2 sorpti
b b
0.15 4 A A 0.84 0.8 A A
D D
0.1 0.07 2 TDL 0.9 0.93 0.4 TDL
0.00 1.00
0 0 1 0
2 2.5 3 3.5 2 2.5 3 3.5
w (eV) w (eV)
FIG.4: (Coloronline)Absorption(solidlines)andemission(dashedlines)spectraasafunctionoftheSicontentoftheclusters.
Note thedifferent scales.
excited-state geometry. The conclusions with respect that even though severaltheoretical predictions coincide
to their similarity drawn for the ground state remain inthatGe nanostructuresshouldhavestrongtransitions
valid for the excited-state geometries. The same ap- at the absorptiononset, few experiments have been able
plies to the transition analysis using Casida’s equation. todetectluminescencefromexcitonsinGenanocrystals.
The character of the emission onset on the Ge side of
the composition range is already well described within
the independent-particle approximation. The decompo- IV. CONCLUSIONS
sition of the excitations as a sum of Kohn-Sham tran-
sitions provides a picture strictly analogous to the one
for the absorption spectra: the lowest transitions of the The absorption spectra of free Ge1−xSix have been
calculatedwithintime-dependentdensity-functionalthe-
Ge-rich nanocrystals correspond to almost pure Kohn-
ory in the adiabatic local-density approximation. The
Sham transitions, while for large x and for the higher
changes of the spectra upon changing composition x are
transitions, independently of x, strong mixing is found.
smooth. In particular at the absorption onset, the po-
However,itisimportanttonotethataveryweakpeak sition and the composition dependence of the spectra
appears at about 1.95eV. This is red-shifted by about is found to be already well represented by independent-
0.5 eVwith respectto the ground-statecalculation. The particle results. The analysisofthe solutionsofCasida’s
peak,whichcanbeeasilyseeninFig.5,isclearlypresent equation show that this is due to the fact that the first
in both the independent-particle result and the TDLDA transition of Ge-rich nanocrystals corresponds to an al-
result. ItoccursforallcompositionsontheGesideofthe most pure independent-particle transition. The TDLDA
x range and is almost composition independent. It ap- onsets are slightly blue-shifted with respect to their
pearstobeconnectedwiththeloweringofthesymmetry independent-particle counterpart, their composition de-
ascomparedtothegroundstatewherethepureSiorGe pendence at the Ge side of the compositional range is
nanocrystalswithoutalloyinghaveT symmetryandthe practically the same. For higher Si contents, a mixing of
d
HOMO-LUMO transition is threefold degenerate. The many independent-particle transitions is found.
symmetry breaking due to the geometry relaxation un- Depolarization effects are strong and their inclusion
der excitation has therefore a much stronger effect than alone, even within a simplified classical model, on top of
the introduction of the alloying, which at the Ge-rich anindependent-particlecalculationallowstogetthecor-
side splits the degeneracy only slightly and which does rect physical picture of the optical response. They can
not change the character of the strong absorption onset beapproximatelytakenintoaccountatanegligiblecom-
at the HOMO-LUMO transition. Due to the argument putational cost using Maxwell-Garnett effective-medium
above, cf. Eq. 3, the appearance of this weak peak will theory.
strongly increase the radiative lifetimes of the systems. Further many-body terms do not modify significantly
We conjecturethatthis mightbe responsibleforthe fact the spectra due to the cancellation of opposite contri-
7
20 2
TDLDA abs
TDLDA em
IP abs
15 1.5
IP em
)
2
Å
2Å) 10 1 A (
P ( LD
I D
T
5 0.5
0 0
1.5 2 2.5 3
w (eV)
FIG. 5: (Color online) Absorption (red) and emission (black) spectra of the pure Ge nanocrystal in the independent-particle
approximation (dashed) and theTDLDA (solid). Note thedifferent scales.
butions given by quasiparticle corrections and excitonic were performed at the Laborat´orio de Computac¸a˜o
effects. As a consequence, the effects of the excitation Avanc¸ada of the University of Coimbra. The authors
of an electron-hole pair do not alter the comparison be- were partially supported by the EC Network of Excel-
tween the independent-particle spectra and the TDLDA lenceNANOQUANTA(NMP4-CT-2004-500198)andthe
results. ETSF e-I3 (INFRA-2007-211956). S. Botti acknowl-
Emission spectra have been investigated using the ge- edges financial support from French ANR (JC05 46741
ometry of excited nanocrystals. A Stokes shift of about and NT05-3 43900). H.-Ch. W. acknowledges sup-
0.5eV is found. Very weak peaks appear at the absorp- port from the European Union through the individ-
tion onset for all systems on the Ge side which suppress ual Marie Curie Intra-European Grant No. MEIF-CT-
the radiative transition probability and lead to long ra- 2005-025067. M. A. L. Marques acknowledges par-
diative lifetimes. tial support from the Portuguese FCT through the
project PTDC/FIS/73578/2006 and from the French
ANR (ANR-08-CEXC8-008-01).
V. ACKNOWLEDGMENTS
The authors would like to thank Lucia Reining for
many fruitful discussions. All TDDFT calculations
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