Table Of ContentQuantum Confinement in Si and Ge Nanostructures
E.G. Barbagiovanni,1,∗ D.J. Lockwood,2 P.J. Simpson,1 and L.V. Goncharova1
1Department of Physics and Astronomy, The University of Western Ontario, London, Ontario, Canada, N6A 3K7
2
National Research Council Ottawa, Ontario, Canada, K1A 0R6
(Dated: January 4, 2012)
Weapplyperturbativeeffectivemasstheoryasabroadlyapplicabletheoreticalmodelforquantum
confinement(QC)in allSiandGenanostructuresincludingquantumwells (QWs),wires(Q-wires)
and dots (QDs). Within the limits of strong, medium, and weak QC, valence and conduction
2 band edge energy levels (VBM and CBM) were calculated as a function of QD diameters, QW
1
thicknesses and Q-wire diameters. Crystalline and amorphous quantum systems were considered
0
separately. Calculated band edge levels with strong, medium and weak QC models were compared
2
with experimental VBM and CBM reported from X-ray photoemission spectroscopy (XPS), X-ray
n absorption spectroscopy (XAS) or photoluminescence (PL). Experimentally, the dimensions of the
a nanostructuresweredetermineddirectly,bytransmissionelectronmicroscopy(TEM),orindirectly,
J by x-ray diffraction (XRD) or by XPS. We found that crystalline materials are best described by
2 a medium confinement model, while amorphous materials exhibit strong confinement regardless
of the dimensionality of the system. Our results indicate that spatial delocalization of the hole in
] amorphousversuscrystallinenanostructuresistheimportantparameterdeterminingthemagnitude
l
l of the band gap expansion, or the strength of the quantum confinement. In addition, the effective
a
masses of theelectron and holeare discussed as a function of crystallinity and spatial confinement.
h
-
s PACSnumbers: 73.21.-b,73.22.-f,78.67.-n,61.46.-w,81.07.-b
e
m
I. INTRODUCTION mechanism.[7] The lifetime associated with the recombi-
.
t nation event can be altered by the excitation power.[8]
a
m Semiconductornanostructures(NSs)exhibitincreased (For a review of general properties of low-dimensional
- oscillator strength due to electron hole wave function structures, see Refs. 2, 4, and 9. For a discussion of
d overlap, and band gap engineering due to the effect of other higher order effects in NSs, see Ref. 9.) This ar-
n quantumconfinement(QC).Thus,materialslikeSiarea ticle is concerned with the electron/hole recombination
o viable option for opto-electronics, photonics, and quan- process in amorphous (a) versus crystalline (c) NSs with
c
[ tum computing.[1–3] QC is defined as the modification different dimensions.
in the free particle dispersion relation as a function of Several theoretical models (e.g. see Refs. 10–12) have
2
a system’s spatial dimension.[4] If a free electron is con- been applied to NS; all models are empirical and no one
v
4 fined within a potential barrier, a shift in the band gap model can model all semiconductor NSs. Since the pa-
1 energyisobserved,whichisinverselyproportionaltothe rameters of a NS system are dependent upon the prepa-
0 systemsizesquared,intheeffectivemassapproximation. rationmethodforaparticularmaterial,acomprehensive
2 Asaresult,theemittedphotonenergyisdirectlypropor- theoreticalunderstandingmusttestalongthisdimension
1. tional to the gap energy (EG). QC often manifests itself as well. In this article, we consider a relatively simple
1 in optical experiments when the dimension of the sys- model of direct e-h recombination using a ‘particle in a
1 tem is systematically reduced and an increase in the ab- box’ type model as a perturbation to the effective mass
1 sorbed/emitted photon energy is measured correspond- theory. We use no adjustable parameters [? ] and in-
v: ing to electron transitional states, i.e. in semiconductor cludecorrectionstothemodeldependentontheprepara-
i NSs. tion method as known experimentally and/or computa-
X
For practical applications, utilizing QC effects in NSs tionallywhenneeded,thusachievingtransparencyinthe
ar requiresanunderstandingofthebandstructureofalow- physicsinvolved. Theonlyparametertestedinthiswork
dimensionalmaterial,howthe method ofpreparationef- is the crystallinity, which is shown to effect the strength
fects the final properties of the NS, and the kinetics/ ofconfinement (defined in Sec. II), because ofthe differ-
dynamics of the absorption/emission process. The con- ent symmetry properties of the electron and hole.
finement potentialis determined by the alignmentofthe
The model is applied to experimental results on crys-
respective Fermi levels when a material of a E is sur-
G1
talline and amorphous Si and Ge NSs, including quan-
roundedbyamaterialofaE ,withE <E .[5]The
G2 G1 G2
tum wells (QWs), wires (Q-wires) and dots (QDs). Sys-
preparationtechniquecanintroducestressinthesystem,
tems of regular shape are chosen to ensure crystallinity
which changes the band gap energy.[6] For indirect gap
is the primary parameter. For example, data obtained
materials phonon processes can effect the recombination
by van Buuren et al. [13] for high quality ‘star-shaped’
samplesaredifficulttoanalysetheoretically. Parameters
relevant to a particular system are discussed and energy
∗ [email protected] correctionsaregivenwhenneeded. Briefly,wecomparea
few theoretical models with experiment, thus, illustrat- This length scale corresponds to the systems considered
ing the need to categorically understand experimental here; therefore, theoretically it is valid to assume direct
parameters. Results are discussed and a mechanism for e-h recombination without phonon-assistance.
the differences between the strength of confinement in In the ‘particle in a box’ model the bulk E is taken
G
the amorphous and crystalline system is proposed. as the ground state energy. The effect of reduced di-
mension is considered as a perturbation to the bulk E .
G
Therefore, we consider the general field Hamiltonian for
II. THEORY asystemofCoulombicinteractingparticlesgivenby(de-
tails are given in Ref. 17):
In this work, we use the effective mass approximation
−~2
(EMA)basedontheBlochperiodicfunction. Theessen- H= d3rψ†(r) ▽2 ψ(r)+
tial features of the model are discussed below. Z (cid:18) 2m (cid:19)
The Bohrradius of anelectron(e), hole (h) or exciton 1 e2
d3rd3r´ψ†(r)ψ†(r´) ψ(r´)ψ(r), (1)
(X) is given by, in SI units: 2Z 4πǫ|r−r´|
4πǫ~2 where ψ(r) is the field operator, m is the mass of the
a = ,
e(h)(X) m∗ e2 electron or hole, ǫ is the dielectric constant of the sur-
e(h)(X) roundingmediumandeistheelectriccharge. Wedonot
m∗ is the effective mass of the e, h or X, respec- consider the spin-orbit interaction here, because the fine
e(h)(X) structure is negligible at the energies considered here.
tively, e is the electric charge and ǫ is the dielectric con-
The field operatorsareexpandedin a two-bandmodel
stant. Dependingontheeorheffectivemass,theX-Bohr
for the conduction band C and the valence band V as:
radiusis4.5nmforSiand24nmforGe. TheBohrradius
defines the spatial dimension of the particles, which de-
ψ(r)= a ϕ (r) (i∈C,V), (2)
k,i k,i
terminestherangeofsizesforwhichQCcanbeobserved.
Xk
We define three regimes of confinement here:[4]
where k represents a summation over momentum states.
• Weakconfinement: Whenthedimensionofthesys-
Theϕ (r)basissetinEq. (2)isexpandedtoreflectthe
k,i
tem is much larger than a and a . In this situa-
e h useofaninfiniteconfinementpotentialwithaBlochbasis
tion, the appropriate mass in the kinetic term is
u . Infinite confinement is a reasonable assumption for
M = m∗+m∗. The energy term is dominated by k,i
e h thesystemsweareconsidering,becausethematrixmate-
the Coulomb energy.
rialhas a E severaleV higher than the nano-structure;
G
however,wecannotdiscusshoppingorothersuchhigher
• Medium confinement: When the dimension of the
order effects. Bloch states reflect the periodic nature of
system is much smaller than a , but larger than
e
the crystal (Luttinger-Kohn representation), while the
a ,thenonlyelectronswillexperienceconfinement.
h
Therelevantmassissimplym∗forthekineticterm. boundaryconditions ofa NS do notreflectthis same pe-
e riodicity. However, in many NSs the transitions we are
Most materials belong to this class.
interested in happen near the Brillouin zone centre, e.g.
• Strong confinement: When the dimension of the the Γ-point. This statement may not be strictly true in
system is much smaller than a and a . Here both the case of weak confinement, because k-selection rules
e h
electronsandholesexperienceconfinementandthe arenotasstronglybrokenasinthecaseofstrongconfine-
relevant mass is the reduced mass, µ, with 1 = ment. Nonetheless,k·pperturbationtheoryconsidersex-
µ
1 + 1 . Inthisregime,theCoulombtermissmall pansions about the Brillouin zone minimum, ko. There-
m∗e m∗h fore, we may justify the use of Bloch states through the
and can generally be treated as a perturbation.
use of the slowly varying wave approximation whereby
Belowwewillusetheterms‘weak,’‘medium’and‘strong’ only the ko=0 states are retained.
to refer to the different regimes ofconfinement discussed For indirect gap materials the exciton is Wannier-like,
above. in the limit k ≪ aπc (ac is the lattice spacing) and we
Si and Ge are both indirect gap materials, meaning can drop the exchange term, which goes to zero quickly.
that, in principle, phonon scattering events are essential Equation(1)issolvedintheexcitonbasisusingthestate
to maintain momentum and energy conservation during Φ defined as an e-h pair above the ground state, Φ0, as:
a radiative event. This situation is true in the case of Φ = k1k2Ck1k2a†k1b†k2ΦV, & ΦV = bk3bk4···bkNΦ0,
a bulk material; however, as the dimension of the sys- wherePak (bk)referstoelectrons(holes)intheconduction
tem is reduced,the uncertainty inthe momentum kvec- (valence)band. Expandinginlowlyingk-statesnearthe
tor is increased. Therefore, it is possible to break the band edge, we solve E (D)=hΦ|H|Φi, which gives the
G
k selection rules making the E ‘pseudo-direct,’ allow- variation of gap energy with nano-structure size.
G
ing for direct e-h recombination.[14] The length scale at For the mass terms in Eq. (1), we use the effective
which this ‘pseudo-direct’ phenomenon becomes impor- masses calculatedusing the density of states.[18] The ef-
tant is typically less than a few nanometres.[12, 15, 16] fective mass is related to the parabolicity of the band
2.5
TABLE I. Parameter A given in Eq. (3) for 3D, 2D, 1D
confinement and for ∆ECBM, ∆EVBM. 2
Si Ge
3D Strong 3.57 7.88 V) 1.5
e
Medium (∆ECBM) 1.39 2.69 p ( Expt. VBM shift
Weak 0.91 1.77 y Ga 1 Expt. CBExMp ts. hPiLft
∆EVBM -2.64 -5.19 nerg 1D: Medium - 1CDB: MSt rsohnifgt
2D Strong 2.09 4.62 E 0.5 1D: VBM shift
Medium (∆ECBM) 0.81 1.58
0
Weak 0.53 1.04
∆EVBM -1.55 -3.04
-0.5
1D Strong 0.89 1.97 1 1.5 2 2.5 3
Si Layer Thickness (nm)
Medium (∆ECBM) 0.35 0.67
Weak 0.23 0.44
FIG. 1. Disordered Si-QW data and theoretical fit. Experi-
∆EVBM -0.66 -1.30 mental data from Ref. 27. Theoretical fit using A=0.89 and
EGap(∞) = 1.6 eV in Eq. (3). NB: The CBM shift is offset
bythe EGap(∞).
structure, which is not expected to change in a nano-
structure compared to a bulk material at the Γ-point.
[20]orx-rayphoto-emissionspectroscopy(XPS).[21]Fur-
Therefore, we assume the effective mass from the bulk
thermore, QC in Ge has been a greater challenge for
system. For Si the effective masses at room temperature
are: m → m∗ = 1.08m and m → m∗ = 0.57m . researchers to observe than in Si, because of the ten-
For Gecthe effcective massoes are: mV → mV∗ = 0.56mo dency to form defects, interfacial mixing and sub-oxide
and m → m∗ = 0.29m . These dcefinitiocns yield theo states.[22–25] Therefore, only limited results on Ge are
V V o discussed here. However, there is recent progress in this
equation:
area, showing very promising results.[26]
A
E (D)=E (∞)+ eV·nm2. (3)
Gap Gap D2
III. EXPERIMENT
E (∞) is the band gap of the bulk material and D
Gap
represents the QD diameter, the QW thickness or the
We cite the results of several experimental works in-
Q-Wire diameter in what follows. The calculation was
cluding our own from the University of Western Ontario
carried out for confinement in 1D, 2D with cylindrical
and from the NationalResearchCouncil Ottawa, in Sec.
coordinates and 3D with spherical coordinates. The pa-
IV. The essential features of each experiment are given
rameter A is given for Si and Ge in the strong, medium
here. The details of the experiments canbe found in the
andweakconfinementregimesin Table I. The changein
references provided.
energy of the CBM (∆E ) due to QC is labelled as
CBM
‘medium confinement’ in Table I, because a ∆E is
CBM
equivalent to QC of the electron only as defined by our IV. RESULTS
model, where onlythe electronmassis consideredin Eq.
(1). The changeinenergyofthe VBM(∆E )due to
VBM A. Silicon
QC is also listed in Table I, which is calculated by con-
sideringconfinementoftheholeonly,whereonlythehole
massisconsideredinEq. (1). Theotherfixedparameter 1. Quantum Well
is the appropriate E (∞) of the bulk system and one
G
could argue for the use of a renormalized effective mass Si/SiO superlattice Si-QWs have been grown using
2
with dimension of the system, which is discussed in Sec. molecular beam epitaxy, determined to be disordered
V. viaRamanscatteringmeasurements,andtheir thickness
Finally, it is important to note that theoretical mod- foundusingTEMandXRD.[27,28]Thechangeintheva-
elling can be further complicated by the accuracy of NS lence band maximum (VBM) and conduction band min-
size determination. Transmission electron microscopy imum (CBM) position was measured using XPS and Si
(TEM) is the direct method to determine NS size; how- L edgeabsorptionspectroscopy,respectively,androom
2,3
ever, if the contrast between the matrix and the nano- temperaturePLspectroscopywasmeasured. Fig. 1plots
structure is poor, then the size uncertainty can be on the model predictions with the experimental data.
the order of 1 nm.[19] Indirect size determinations can InRef. 27theauthorsusedafittingprocedureaccord-
be used as well, such as with x-ray diffraction (XRD) ing to the effective mass theory for the ∆E ,
VBM(CBM)
2.5 3
3D: Strong E (∞)=1.12 eV
3D: Medium EGap(∞)=1.12 eV
3D: Strong EGap(∞)=1.56 eV
2 3D: Medium EGap(∞)=1.56 eV
Gap
Expt. Ion-Implantation SiO PL
2.5 Expt. MPD SiO2 PL
2
1.5 Expt. PVCD SiN PL
V) V) Expt. PVCD SiN Absorption
e e
p ( y (
Energy Ga 0 .15 ExEEpxxtp.p tCt.. 1BVCDMBBE:x MM+SpVt trss.Bo hhPnMiiLffgtt Gap Energ 2
1D: Medium - CBM shift
1.5
1D: VBM shift
0
-0.5 1
0.5 1 1.5 2 2.5 3 1.5 2 2.5 3 3.5 4 4.5 5
Si Layer Thickness (nm) QD Diameter (nm)
FIG. 2. Crystalline Si-QW data and theoretical fit. Experi- FIG. 3. Crystalline and amorphous Si-QD data and theo-
mental data from Ref. 21. Experimental PL data from Ref. retical fit. ‘Expt. Ion-Implantion SiO2’ refers to crystalline
29. Theoretical fit using A=0.35 and EGap(∞)=1.12 eV in Si QDs embedded in SiO2 from Ref. 30. ‘Expt. microwave
Eq. (3). NB: The CBM shift is offset by theEGap(∞). plasma decomposition (MPD) SiO2’ refers to crystalline Si
QDs embedded in SiO2 from Ref. 20. ‘Expt. plasma en-
hanced chemical vapour deposition (PCVD) SiN’ refers to
amorphous Si QDs embedded in SiN from Ref. 34. Theoret-
ical fit using A=3.57 and 1.39 and EGap(∞) = 1.12 or 1.56
resultingin∆E =−0.5/D2 and∆E =0.7/D2, eV(aslabeled) inEq. (3). NB:Theabsorption datais offset
where D is the tVhBicMkness ofthe QW.OurCmBoMdel predicts bythe EGap(∞).
∆E = −0.66/D2 and ∆E = 0.35/D2. The
VBM CBM
trend for ∆E is more accurately given in Ref. 27.
CBM
In Ref. 28, the change in E was fitted with A = 0.7 2. QDs
G
and E (∞)=1.6 eV, as in Eq. (3). The fit also de-
Gap
termined the effective mass to be m∗ ≈1. The model First we consider Si QDs formed by ion implantation
h(e)
uses EGap(∞)=1.6 eV to fit the experimental PL data in SiO2 films, followed by high-temperature annealing
well when employing the curve for strong confinement in N2 and forming gas.[30] Ref. 30 reports the QD di-
with A=0.89. ameter and crystalline structure observed by TEM, and
roomtemperature PL measurements. TEM data show a
Nextwelookatc-Si/SiO QWsfabricatedbychemical Gaussiandistributioninthe Si-QDdiameterwithdepth,
2
andthermalprocessingofsilicon-on-insulatorwafers.[21] resulting in a stretched exponential PL dynamic.[30, 31]
The same methods described above were used to deter- We compare ion-implanted Si-QDs with Si QDs in a
mine experimentally the ∆EVBM(CBM) and the change SiO2 matrix prepared by microwave plasma decompo-
in the gap energy including the total electron yield for sition (MPD) creating ultrafine and densely packed Si
a better signal to noise ratio. The thickness of the Si QDs[20] (implying that tunnelling effects are important
layer was determined by XPS using a mean free path here [32]). The crystallinity and size was determined by
in Si of ∼1.6 nm. Note that a thickness of 0.5 nm corre- TEMimagingandXRD,respectively. InRef. 20,theau-
spondstoasingleunitcellofSi. Therefore,experimental thors note that PL was not observed unless the Si QDs
data below ≈ 1 nm should be treated with caution. In were oxidized, implying that surface bonds were passi-
a parallel study, these c-Si/SiO QWs were investigated vatedwithsuboxidestateseventuallyformingasurround
2
optically.[29] SiO2 matrix.
Fig. 3 shows the experimental PL data for ion-
Fig. 2 compares experimental measurements and the implantated and MPD Si QDs together with our cal-
model results for c-Si-QWs. The E (∞) in the model is culated curves for strong and medium confinement.
G
1.12eVandthe ∆E isnotsignificantbelow1.5nm. Above 3 nm both sets of experimental data follow
VBM
The ∆E , ∆E , and the experimental PL closelythemodelofstrongconfinementwithA=3.57and
CBM CBM+VBM
are all well fitted by the curve for medium confinement, E (∞)=1.12 eV. This indicates that for sample diame-
G
withA=0.35. InRef. 29itwasfoundthatthereisasec- terslargerthanthissizetunnellingeffectsaresignificant,
ond PL peak fixed with respect to the Si layer thickness implying a de-localization of carrier states. Iacona et al.
at 1.8 eV. This second peak was associated with inter- measured a similar trend for experimental PL data. [33]
facestates. Therefore,wecanassigntheexperimentalPL Below 3 nm, when QC effects are particularly strong,
datainFig. 2withdirecte-hrecombinationmodelledby the ion-implantation data follows the curve for medium
medium confinement. confinement, with A=1.39.
Next we consider a-Si QDs embedded in a SiN
Expt. por-Si Wire PL
matrix.[34]TheSiQDswerefabricatedusingplasmaen- 3.5 Expt. Si Wires STS
Expt. por-Si QD PL
hanced chemical vapour deposition. The size and amor- Expt. por-Si QD Absorption
phous structure were measured using TEM and the PL 2D: Strong
3 2D: Medium
was taken at room temperature. Absorption data was 3D: Strong
V) 3D: Medium
takenbyultraviolet-visibleabsorptionspectroscopy. The e
value for the bulk band gap given by the authors is 1.56 gy ( 2.5
er
eV, whichis obtainedvia a fitting procedure. This value En
p
is known to vary between 1.5→1.6 eV, for Si samples Ga 2
prepared similarly.[34]
We can see in Fig. 3 that the experimental data
1.5
for absorption and PL of a-Si QDs embedded in SiN
lies between the curve for medium (A=1.39) and strong
(A=3.57)confinement,withE (∞)=1.56. Usingafit- 1
Gap 1 2 3 4 5 6 7
ting procedure, the authors of Ref. 34 found A=2.40. Q-Wire Diameter/ QD Diameter (nm)
The authors further conclude that by observing the fact
that the experimental absorption data lies close to the FIG. 4. Crystalline Si Q-wire and QD data and theoreti-
PL data, one can conclude that the PL data for these cal fit. Experimental por-Si wire data from Ref. 38. Ex-
perimental Si Wire data from Ref. 39, using scanning tun-
samples is a good measure of the actual change in the
nelling spectroscopy (STS). Experimental por-Si QD data
E (D).[34] Notice that this situation is similar to that
G from Ref. 37 and 41. Theoretical fit using A=1.39 and 0.81
observed for Si-QWs (see Fig. 1 and 2).
andEGap(∞)=1.12eVinEq. (3). NB:Theabsorptiondata
isoffsetbytheEGap(∞). NB:‘QD’herereferstospheroids.
3. Quantum Wires
E (∞)=1.12 eV. Notice that the experimental data
Gap
Due to inherent complications in the fabrication pro- also lie close to the curve for 3D medium confinement
cess of Si or Ge wires with a diameter below the Bohr withA=1.39. Thisobservationmaybeareflectionofthe
radius,few studies onQC innano-wiresexistandweare idea that these structures are between dots and wires.
only able to report on c-Si-Q-wires. On the other hand, On the other hand, the data from Ma et al. lie close
por-Si studies are widely cited in the literature. With to the curve for 3D strong confinement, using the same
suitable control of the etchant, por-Si QDs can become EGap(∞) and A=3.57. We also note that recently Si-Q-
elongated,[35]thusbreakingconfinementinonedirection wireshavebeenproduced[40]withresultsnearlyidentical
implying they aremore wire-like;a detaileddiscussionis to those of Ma et al.
providedinRef. 36. Inthis case,they arecalledpseudo- Experimentaldata onpseudo-por-Si-QDsfor both ab-
por-Si-QDsorinthecasetheybehavelikeinterconnected sorption and PL are taken from Ref. 37 and 41. Raman
dots, spherites.[37] andTEMmeasurementswereusedto determine the size
Anodically grown por-Si samples were prepared by andthe‘spherite’natureofthesamples,respectively. PL
Schuppler et al.[38] X-ray absorption measurements de- measurements were performed at roomtemperature and
termined the structures to be closer to c-Si than to a-Si. at 4.2K, with very little difference in the two measure-
TEM was used to determine the size and PL measure- ments. Optical absorption was performed at room tem-
ments were performed at room temperature. The por-Si perature. It is also noted in Ref. 41 that, for por-Si,
structures are said to be H-passivated and O-free; how- interfacestatesandphononeventsaresignificant. Fig. 4
ever, samples were exposed to air. shows the PL and absorptionexperimentaldata for por-
SiQ-wireswereproducedbyMa etal. usinganoxide- Si-QDs. Here the experimentaldataaremodelledby the
assisted growth method with SiO powders.[39] Subse- curve for 3D strong confinement, with A=3.57 and the
quently, the wires were cleaned with HF to remove the same gap energy as above. Compared to absorptionand
oxide, thus forming a H-terminated surface. Scanning PL data for a-Si-QDs in Fig. 3 and the Si-QWs in Fig.
tunnelling microscopy was used to determine the diame- 1 and 2, there is a significant shift between the absorp-
terofthewires. TheformationofSiH andSiH wasob- tion data and the PL data, indicating a Stokes shift in
2 3
servedonthefacetsoftheQ-wires,whichwasattributed the emission.[41] Furthermore, as noted in Ref. 37, the
to bending stressesin the wires. The energy gapwas de- experimental PL data are nearly identical to Takagi et
termined using scanning tunnelling spectroscopy, which al., shown in Fig. 3.
also indicated doping levels in the wires as seen by an
asymmetrical shift of the E around 0 V.
G
B. Germanium
The experimental data from Ma et al. and Schuppler
etal. canbeseeninFig. 4. Below3nmtheexperimental
data from Schuppler et al. (‘por-Si Wire PL’) lie close The first observation of QC in Ge was by Takeok et
to the curvefor2DstrongconfinementwithA=2.09and al.[42] In this study, they produced Ge QDs using an
1.8 siveargumentsappearintheliteratureconcerningtheva-
Expt. PL
Expt. Absorption lidityoftheEMAanditsk·pgeneralization. Ontheone
3D: Strong
1.6 3D: Medium hand,itisarguedanddemonstratedthatthe EMAover-
3D: Weak estimates the E [4, 10]; however, Sec. IV demonstrated
G
that in some cases the EMA can underestimate the E .
V) 1.4 G
e Inpart,thisisbecauseduetoQCtheparabolicnatureof
gy ( thebandsispossiblyremoved. Anothercomplicationcan
Ener 1.2 arise from the fact that the envelope functions may not
p
Ga be slowly-varying over the unit cell, which is essentially
1 complicatedbytheboundaryconditions. Acentralprob-
lemforEMAisinitsapplicabilitytoa-materials,because
0.8 it is based on the assumption of translationalsymmetry.
Street has argued that while it is strictly not justified in
the a-system, due to nonspecifically-defined k vectors, it
1 2 3 4 5 6
QD Diameter (nm) is still widely used albeit with differing assumptions.[45]
We will discuss further the application of the EMA to
FIG. 5. Crystalline Ge-QD data and theoretical fit. Exper- both amorphous and nanostructured-systems below.
imental Ge QDs data from Ref. 42. Experimental Ge ab-
On the other hand, it has been argued by S´ee et al.
sorptiondatafromRef. 43. TheoreticalfitusingA=2.69and
that the EMA is well justified and produces agreement
EGap(∞)=0.66 eV in Eq. (3). with the tight-binding method.[46] Such arguments re-
side in the fact that it is not clear what all the relevant
parametersareinanano-structuredsystemofaparticu-
rf co-sputtering method followed by thermal annealing.
lar material. In general, the boundary conditions of the
ThesizeoftheGeQDswascontrolledbyvaryingtheini-
system become very important, which is a problem for
tial Ge concentrationand was later determined by TEM
all theories.[47] If the Fourier components of the enve-
imaging,whichalsoshowedthattheGeQDswerehighly
lope function are centred around the the Brillouin zone
crystalline. PL was performed at room temperature.
centre,then envelopefunctions can be justified. In addi-
Inamorerecentstudy,GeQDswereproducedbycon-
tion,thisjustificationhasbeenextendedtoconsiderthat
densationoutofthegasphaseontoaSisubstratecleaned
iftheinterfaceisdefectfreethentheEMAisjustified.[47]
byHF.[43]TheGeQDsweredeterminedtobeinthebulk
OtherconsideredcorrectionstotheEMAuseafourthor-
diamondcrystallinephase. X-rayabsorption(XAS)data
der term in k.[4] The advantage of the EMA is that it
weretakenandcanbeseeninFig. 5. XASexcitestheGe
is straightforward in its application, thus allowing one
2p electron into the conduction band; therefore, the re-
to highlight key features of individual systems. Pertur-
searchersobtaineddataforthe changeinthe conduction
bations in the NS system are naturally treated in the
band.
k·pmethodanddefectstateseasilycalculated[18]. Com-
The experimental data from Refs. 42 and 43 are pre-
pared to empirical methods,[10, 48] which produce a di-
sented in Fig. 5. Note that the absorption data are mensionaldependence ofD−1.39, the EMA hasthe units
obtained by shifting the Bostedt et. al. data by the D−2 (see (3)). In addition, it has been shown that the
EG(∞) of Ge at 0.66 eV. Further note that above 3 nm k·p Hamiltonian can be made to reproduce multiband
there is a nearly identical departure from the medium
couplingeffectsandthe correctsymmetryoftheQD.[49]
confinement curve into strong confinement as was seen
Toemphasizetheimportanceofaccuratelyparametriz-
with Si-QDs in Fig. 3. In general, both sets of experi-
ingthepreparationmethod,wecompareourresultswith
mental data are well modelled by the curve for medium
a few theoretical models with respect to experiment. In
confinement with A=2.69 and E (∞) = 0.66 eV. For
Gap the work by Bulutay[50], the variation in the E (D) is
the smaller sizes (below 2.5 nm) the behaviour appears G
calculated using an atomistic pseudopotential method.
to deviate from medium confinement. This result may
The result is given in Fig. 4 of Ref. 50 and is repro-
be because for the smaller sizes the authors only esti-
duced here in Fig. 6. References for the works listed in
mated the sizes.[42] In Ref. 43 the Ge-QD diameter was
figure caption are given in Ref. 50. The top curve in
determinedusingatomicforcemicroscopy,whichcanpo-
Fig. 6 is for Si, which we compare with our Fig. 3 here
tentiallygivealargeruncertaintyindeterminingthesize
and the bottom curve for Ge, which we compare with
of the dot.[44] Therefore, if the QDs are not symmetric
our Fig. 5 here. It is clear that the band gaps shown in
then the diameter measurements could be inaccurate.
Fig. 6 are consistently larger than what we present in
this manuscript. This result is easy to explain.
In the case of Si, the experimental data in Fig. 6 is
V. DISCUSSION
from Furukawa et al. In this work, they produce Si QDs
using magnetron rf sputtering. It is demonstrated that
We start our analysis by giving a justification of the the QDs are surrounding by H and composed of Si:H.
EMA,while highlighting someof the limitations. Exten- The incorporation of H in the QDs causes an increase
lutay,thereis no explicitinclusionofa stresscomponent
in the Hamiltonian, instead it is implicitly fitted in the
pseudo-potential, while the other three methods ignore
stress altogether. Therefore, these four methods do not
explicitly consider the experimentaldetails, insteadthey
are fitted to experiment. The empirical nature of the
theoreticalmethods canbe furtherseenwhencomparing
with similar pseudopotential calculations that produce
different results from those shown here.[10]
Considering the results for Ge in Fig. 6, the situation
is essentiallythe sameasfor Siabove. The experimental
data from Kanemitsu et al. is associated with defect PL
only,makingitbeyondthescopeofthispaper. Niquetet
al. uses ansp3 tightbinding method, while Tsolakidiset
al. uses time-dependent density functional theory in the
adiabaticlocaldensityapproximation. Theexperimental
datafromTakeokaetal. isfittedinthismanuscript(Fig.
5)andnotinthereferencesofFig. 6. Furthermore,these
calculations for Ge are similar to Si, which implies the
materialproperties are not being properly accounted for
theoretically. Thus, it is necessaryto quantify eachterm
intheHamiltonianaccordingtothepreparationmethod.
Finally, we comment on the relevant energy scales for
the experiments considered above. The electron and
hole can form a hydrogenic or positronium-like exciton,
a bound state of the constituent particles, thus modify-
ingthephotonenergyduringtherecombinationeventby
FIG. 6. Comparison of the EG as a function of QD diameter the Coulomb interaction between the electron and hole.
forSiandGe. ReprintedfigurewithpermissionfromRef. 50. The Coulombenergyis onthe orderofhundreds of meV
Copyright (2007) by the American Physical Society. Refer- (→ 1/R, R=NS dimension), the exchange energy is on
ences for the works listed in figure caption are given in Ref. the order of 0.1 meV (→ 1/R3), while the gap energy
50. is on the order of several eV. Due to the large number
of competing parameters in any real system, the exact
value of the above parameters is not known, and these
in the E (see Fig. 1 of Ref. [51]). Furthermore, the are important for precision control of a device.
G
Raman peak of the QDs is measured to be 514 cm−1, as To summarize the comparisons made in Sec. IV, we
opposed to the bulk value at 520 cm−1, which indicates first consider the relationship between experimental ab-
that the system is under stress. To verify this claim, the sorption and PL data. In the case of disordered-Si-QWs
authors measure a 2% extension in the bond length us- (Fig. 1),c-Si-QWs(Fig. 2)anda-Si-QDsinSiN(Fig. 3)
ing x-ray diffraction. Thus, the QDs are under tensile theabsorptioncurvefollowscloselywiththePL.Asmen-
stress, which increases the EG[52], see below for more tioned in Sec. IVA2, this result indicates that the PL
detail. Finally,Furukawaetal. explainthattheoriginof measurementisanaccuratemeasureofE (D). Further-
G
thestressisthroughtheincorporationoftheabovemen- more, in the case of Si-QWs the VBM does not change
tioned H in the Si QD lattice through a plasma-assisted significantly. Therefore, we conclude that the model de-
crystallizationprocess. Whereas,if H was acting only to pendencebetweenthese threesystemsdoesnotlieinthe
passivate the dangling bonds there would be no change change in the VBM.
in the EG.[53] Therefore, the experimental results of Fu- Consideringtheabsorptiondatafrompor-Si-QDs(Fig.
rukawaetal. arehigherthanwhatispresentedhere(Fig. 4),thereisasignificantshiftbetweentheabsorptiondata
3) because of an additional stress component, which in- and PL data, which was noted in Sec. IVA3. In addi-
crease that EG beyond that of QC alone and which is tion,thepor-SiQDdataarenearlyidenticaltotheMPD
introduced because of the preparation method. Si-QDs (Fig. 3), which indicates that these systems are
The remaining results in Fig. 6 for Si are theoretical structurally similar with similar decay dynamics. In the
results. The work by O¨˘gu¨t et al. uses a real-space pseu- caseofpor-Siithasbeenfoundthatthissystemisunder
dopotential method. Vasiliev et al. uses linear-response tensilestress.[54]Tensilestress,whichisafunctionofthe
within the time-dependent local density approximation. thicknessofoxide,isknowntoincreasethebandgap.[52]
Garoufalis uses time-dependent density functional the- Itisknownthatthesurroundingoxidehasastrongeffect
ory. IncludingtheworkofBulutay,allfourmethodsgive on the resulting PL in por-Si.[55] The resulting Si-O-Si
approximately the same result. However,looking at Bu- bonds due to the oxidation process place large stresses
onthe por-Sicrystallites. In addition,it has beenshown disordered-Si-QWs,c-Si-QWsanda-Si-QDsinSiNalldo
that the dominant PL comes from surface states.[41] At not show a large variation in the VBM.
the surface or interface states, it has been shown that Amechanismforpinningoftheholestatesinc-Si-QDs
band bending on the order of 0.2→0.3 eV can occur.[56] was discussed in the work of Sa’ar et al. as a function
Such a shift in energy corresponds with the discrepancy of the hole coupling with vibrons.[58] However,this phe-
shown in Fig. 3 and 4. nomenon does not account for the fact that the hole be-
For the c-Si-Q-wires measured by STS (Fig. 4), the comes more delocalized in the a-system,it is well known
data are modelled by strong confinement. This is be- that band-tail states play a very important role in the
causeofthe stressesobservedinthe systemandpossibly band structure of a-materials, even though the popula-
becauseofthedoping;botharefactorsthatdochangethe tion density is relatively low.[45] Kanemitsu et al. (and
nature of electronic structure. In Sec. IVA3, we men- Refs. within), report the experimental observation that
tioned that these structures experience bending stress, the band-tail states become strongly delocalized in the
which has a tensile component. Furthermore, Fig. 4 il- a-system, while the hole remains relatively localized in
lustratesthatc-Si-Q-wiresareidenticalinenergytopor- the c-system.[59] This observation accounts for what is
Si; therefore, the analysis of these systems is similar. By observed in this work.
contrast, the por-Si wire PL data (Fig. 4) behaves more Another critical factor to discuss is the effective mass
wire-like, which may be a result of the fact the authors concept, particularly in the a-system. Recall from Sec.
tookcaretominimizeoxygenexposure(seeSec. IVA3). II, the bulk effective mass is used in the calculations.
It is possible that this parameter is not well-justified in
FromFig. 3and5,bothionimplantedSi-QDsandGe-
the a-system[45] and is simply not valid in the nano-
QDshavethesamebehaviourabove3nm. Theylieclose
structured system, in the worst possible case, or it is
to the curve for strong confinement, similar to the case
size-dependent.[60–62]
of por-Si, indicating that possible stresses or interface
The electron (e) and hole (h) interact differently with
states are important in this regime. Ge is known to ex-
the atomic structure. The s-like electron has Γ6 symme-
periencestressinaSiO2 matrix.[57]Tensilestresscanbe c
try; the p-like hole is contained in Γ8. Therefore, holes
relieved depending on the nature of the interface bonds v
interact more strongly with the acoustical lattice vibra-
and the surface to volume ratio of Si:SiO .[52] In the
2
tions. The electron has approximately twice the mass of
work of Ref. 17 it was found from Raman spectroscopy
the hole, which is dependent on the gap energy. Hence
that ion-implanted QDs are not under stress for diame-
the crystallinity will effect the properties of the particles
ters smaller than 3 nm. Therefore, c-Si-QDs produced
differently and recombination events are dependent on
by ion implantation and c-Ge-QDs are well modelled by
such properties.
medium confinement below 3 nm.
The a-system has typically 80%[63] of the density of
Finally, a-Si-QDs in SiN (Fig. 3) lie between medium
the c-system, while disordered Si generally refers to a
andstrongconfinement(seeSec. IVA2). SiNhasaband
density ≈ 98%[64] of that of c-Si, and these values can
gap of 5.3 eV versus SiO at 9.2 eV, which allows for
2 varywidelybasedonthepreparationmethod.[65]There-
tunnelling of carrier states.[34] More importantly, if we
fore, short and medium range structural order does re-
consider the nucleation process during thermal anneal-
main in both of these systems. Although the long-range
ingandconsiderthebondenthalpiesfordiatomicspecies
orderisnotwell-definedinthea-systemalongwiththek-
(SiN at 470 kJ/mol and SiO at 799 kJ/mol), it is easier
vectors,alternativeapproachestothisconcepthavebeen
to break SiN bonds, thus allowing for a greater degree
extensively presented. In an earlier work, Kivelsonet al.
of intermixing at the QD-matrix interface. Therefore, a
definedanalternativeapproachtothisconcept.[66]They
SiNmatrixactsmorelikeafinitepotentialbarrier,which
formulatedthe assumptions (i) the structure of the solid
lowers the gap energy from the infinite case. A numeri-
can be approximated by a rigid continuous random net-
calcomputationindicatesthatthedifferencebetweenthe
workthatishomogeneousonthescaleoftheslowlyvary-
case of finite versus infinite confinement potential is be-
ing envelope,and(ii) the band canbe measuredby aset
tween10%and15%dependingonthesizeofthe system.
of linearly independent orbits, which are not necessarily
This difference exactly corresponds with the difference
orthogonal. Furthermore, Kivelson et al. used a tight-
we see in Fig. 3. Therefore, we conclude that a-Si-QDs
bindingapproachwithapproximateeigenvaluestoobtain
in SiN are well modelled by strong confinement.
theeffectivemassHamiltonian. Inanotherapproach[67],
From the results above and considering modifications
Singh lookedat the effective masses in the extended and
thatmustbemadetoourmodeltoaccountfornon-direct
tail-statesaroundthemobilityedgedirectlyusingareal-
e-hrecombinationphenomena,itisclearthatstrongcon-
space formulation. The electron energy eigenvalues are
finement describes a-materials and medium confinement
given in terms of the probability amplitude, which can-
describes c-materials. Therefore, since QC of a particle
not be defined as in the case of a c-material in terms of
is a function of the delocalization of that particle with
k-vectors. Instead, the probability amplitude is defined
respect to the dimension of the system, we need to ac-
as[67]:
countforthefactthattheholebecomesmoredelocalized
in the a-system than in the c-system. This fact may or 2m∗(E−E )
may not be seen as a shift in the VBM. As noted above, C1l =N−1/2exp(ise·l), withse(E)=r e ~2 C
where E defines the mobility edge; therefore, the ef- the electron according to the ∆E measure-
C CBM(VBM)
fective mass is defined above the mobility edge in the ments, described above. Although, there may still be
extended states and is imaginary in the tail states. In a slight hole contribution in this ideal approximation,
either approach described here, the result is that the whichneedsfurtherstudy. Inaddition,inourtheoretical
effective mass calculated is lower than in the bulk sys- modelling, we have consistent results for strong confine-
tem. This observation implies that the Bohr radius of ment in the amorphous samples, because both the elec-
the holeinthe a-materialislargerthaninthe c-material tron and hole effective mass decrease implying confine-
and hence the hole is more delocalizedin the a-material, ment of both, due to spatial de-localization. Although,
thus the observed strong confinement. It is clear that therelativecontributionfromtheelectronversusthehole
this is the dominant mechanism for strong confinement is not clear and needs further study. Furthermore, it is
in the amorphous system, since the pinning discussion clearthattheeffectivemasspredictionfor∆E
CBM(VBM)
above does not describe all the systems considered here. isnotcorrect,unlessarenormalizedeffectivemassisused
Therelativemagnitudeofthetwomechanismsneedsfur- accordingto systemdimension. These results area clear
ther analysis. indicationthattheuseofthebulkeffectivemassisonlya
The size dependence of the effective mass inc-systems first order approximation. Nevertheless, very good over-
is reported in Refs. 60–62. Experimentally, the effective allagreementisobtainedbetweenexperimentandathe-
mass is reported to decrease with size in Ref. 61 and orywithessentiallynoadjustableparametersforbothSi
62. In one theoretical report, the hole effective mass in- and Ge nanostructures.
creases, while the electron effective mass decreases.[60]
Themagnitudeofchangeinthe effectivemassisroughly
the same for the electron and the hole, and consider-
ing the effective mass of the electron in the bulk sys-
VI. CONCLUSION
tem is roughly twice that of the hole, it is not likely
that the change will be within experimental resolution.
Overall, the effective mass in the a-system and in the We have studied the effect of confinement dimensions
nano-structured system is understood to decrease, but and crystallinity on the magnitude of the band gap ex-
the magnitude of the decrease is unclear. Therefore, in pansion (as a function of decreasing size) in group IV
terms of the calculations presented here, if the effective semiconductor NSs (quantum wells (QWs), wires (Q-
massis loweredthanwe shouldexpectto see anincrease wires) and dots (QDs). Medium and strong confinement
in the calculated E and hence our curves will shift up- modelsprovidethebestfittoexperimentalresults;more-
G
wards. However,we wouldalsoexpect to see anincrease over crystalline materials exhibit medium confinement,
in E from the experimental results. Since the exact while amorphous materials exhibit strong confinement
G
magnitude of these changes is not known it is difficult regardless of the confinement dimensions of the system.
to evaluate the errorincurredby using the bulk effective This difference in confinement strength was explained
mass. by considering the extent of spatial delocalization of the
Thisissueofthecorrecteffectivemassismorepoignant hole. A possible explanation is hole pinning due to cou-
when considering the ∆E . In this work, pling with the vibronic states.[58] It has previously been
CBM(VBM)
∆E > ∆E , which is understood, because the reported[59] that band tail states become strongly delo-
VBM CBM
effective mass of the hole is smaller than the electron. calized in the amorphous system compared to the crys-
However,experiment consistently shows the opposite ef- talline system. This hole delocalization would partially
fect, see Fig. 1,2 and see Ref. 68 and 69. This ob- account for the trends observed in our work. The con-
servation implies that experiment is measuring a larger cept of the effective mass was reviewed for the amor-
decrease in the electron effective mass than the hole, or phous system. We argue that the effective mass can still
possiblyarelativeincreaseinthe holemasscomparedto be defined in the amorphous material around the mobil-
the electron. This observation is nearly consistent with ity edge.[66, 67] A lower value of the effective mass is
Ref. 60, where they predict a nearly symmetric change. reported for the amorphous system, which accounts for
Furthermore, recall that experiment reports a decrease the trends observed in our work, while the hole mass in-
in the electron effective mass.[61, 62] creases and the electron mass decreases as a function of
Therefore, the decrease in the electron effective mass spatialconfinement.[60–62]Withthediminishedeffective
and increase in the hole effective mass is consistent for mass(the absolute valueofthis changeis notpossibleto
the crystalline system with our observation of medium estimate, and more work is needed in this area), we ex-
confinement,becausetheholeismorespatiallylocalized. pect an increase in E , and our calculated curves of
Gap
In being consistent with experiment, we drop the hole energyversusdiameterwillbeshiftedupwards. However
contribution for the crystalline system in the ideal ap- the general trends observed in this work will remain the
proximation, because this term is not as significant as same.
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