2.4 Self-assembled SiGe nanostructures
K. Tillmann, H. Trinkaus and W. Jager
May 1999
A INTRODUCTION
Self-organization and spontaneous formation of nanostructures are synonyms which are
used to describe methods for preparing nanoscale structures with tailored electronic properties by direct heteroepitaxial growth. Especially self-assembled island structures are of
interest as they are expected to exhibit the carrier connement properties of quantum
dots [1, 2, 3]. These islands may have a well dened shape but their random nucleation
as well as ripening and coalescence during growth often result in a broad distribution of
island sizes dependent on the growth conditions. Since densely packed arrays of islands of
the same size, shape and composition with a regular spatial arrangement are desired for
practical applications, e.g. single electron transistors and resonant tunneling diodes [4],
the identication of common growth mechanisms leading to a narrowing of their structural properties is of great importance. Thus, the nucleation process and the nal shape
of the islands are the two crucial elements that inuence such regularity and uniformity.
This datareview summarizes the present knowledge about self-assembled nanostructures
in the SiGe/Si system characterized by lattice mismatches 0 f 0.04012 dependent on
the chosen germanium content. Emphasis is put on the growth parameters determing the
structural properties of coherent three-dimensional islands. A brief overview is given on
theoretical concepts describing ordering phenomena in single and multilayer structures.
Information on heterostructures prepared by molecular beam epitaxy (MBE), low-pressure
chemical vapour deposition (LPCVD) and liquid phase epitaxy (LPE) results mainly from
transmission electron microscopy (TEM), scanning tunneling microscopy (STM), atomic
force microscopy (AFM), X-ray diraction (XRD) and reection high-energy electron
diraction (RHEED) analyses.
B STRANSKI-KRASTANOW GROWTH OF SIGE/SI(001)
In the present context, a crucial point is that initially planar epitaxial layers under stress
can lower their elastic energy by morphological changes [5, 6, 7, 8]. The spatial scales
of this morphological instability are controlled by the interplay between the stabilizing
inuence of the surface energy and the destabilizing eect of the mist induced strain
1
energy. In case of low-mismatched SiGe layers (f 0.015) on Si(001) substrates, it has
been observed that roughening takes place by the formation of undulated surfaces with
discrete facet orientations [9, 10, 11, 12]. An analysis of such roughening shows that
it is a thermally activated process with an energy barrier being an extremely sensitive
function of the layer strain and the surface conditions [13, 14, 15]. Thus, mainly dependent on mist and growth temperature, roughening may occur spontaneously, or it may
be totally suppressed on the laboratory time scale. However, in the high mist regime
(f 0.015) the nucleation barrier for roughening is suciently small and layers follow
the Stranski-Krastanow [16] growth mode under typical growth conditions, i.e. at growth
temperatures 500o C TS 700o C and at growth rates 0:2
As,1 3:0
As,1. Several theoretical models, based either on continuum or atomistic approaches, have been
developed to predict the evolution of the layer morphology and defect structure dependent on growth parameters [17, 18, 19, 20, 21]. In agreement with experimental results
these models demonstrate three stages of growth with an increasing amount of deposited
material:
Growth starts with the formation of a two-dimensional wetting layer until a critical
thickness hc is reached. At that thickness a transition occurs and three-dimensional
islands are formed. STM, TEM and RHEED experiments demonstrate that for
the deposition of Ge layers on Si(001) substrates hc is in the order of two to ve
monolayers [22, 23, 24, 25, 26, 27, 28]. The observed hc values are increasing with
decreasing Ge content [29, 30, 31] and models indicate that hc scales as hc / f ,4
[32, 33]. The formation of islands, however, requires a suciently high growth
temperature since otherwise metastable at epilayers with an eectively increased
critical thickness for island formation are formed [34].
Above hc, three-dimensional islands nucleate on top of the wetting layer which
remain free of dislocations in their earliest formation stages, only partially relieving
the strain energy by an expansion of the germanium lattice near the top of the islands
and by the distortion of the silicon substrate beneath the islands [36]. Calculations
yield a lowering of elastic strains hf i in the order of 50 % compared to the nominal
mismatch f of the heterostructures [37, 38, 39, 35] depending on the island shape
as shown in gure 1.
Finally the initially coherent islands evolve into larger islands by ripening and/or
coalescence, in which strain relaxation takes place by the introduction of mist
dislocations [40, 41, 42, 43].
Despite these commonly observed stages of growth the islands undergo a shape transition
during the second stage of growth and the nal topography of an ensemble of islands is
mainly inuenced by the nominal layer thickness hnom and the growth temperature. Table
1 gives an overview on resulting island geometries for a variety of epitaxial parameters.
If growth temperature and nominal layer thickness are chosen suciently small, epitaxial
2
FIGURE 1: Eective lattice mismatch hf i of a pyramidal square-based germanium island on silicon in dependence on the height-to-extension-ratio p and the facet orientation
[hkl]. The calculated values () gained by nite element simulations may be adjusted
to an exponential function hf i = f exp[,hkl p] yielding 113 = 4.45, 111 = 5.25 and
100 = 6.49 [35].
growth results in a quite narrow size distribution of islands as shown in gure 2a. However, in general two basic types of islands, which may simultaneously exist during growth,
each characterized by aspect ratios in the range 0.05 p 0.15, are observed in dierent
experiments. Small square based pyramids, frequently referred to as hut-clusters [49, 50],
with lateral base extension 30 nm hli 40 nm along h100i directions and exhibiting
four f105g facets are transient structures which nally disappear when greater volume
domes with 140 nm hli 200 nm are formed. Recent observations reveal a bimodal
population of both island types, with a signicant discontinuity in the size distribution
separating the pyramids and the domes [51, 52]. During a stage of coexistence both types
are either characterized by quite narrow size distributions which are nearly independent
of the amount of deposited material, i.e. with increasing nominal epilayer thickness, the
island density but not the island size is increased. Up to now the origin of the bimodal
size distribution in not well understood but two mechanisms have been suggested.
3
FIGURE 2: (a) Monomodal island size distribution of small pyramids resulting from an
AFM analysis of a sample grown by LPCVD (TS = 500o C, hnom = 0.7 nm, = 3.0 As,1).
(b) Trimodal distribution of island diameters d of a sample grown by LPCVD (TS = 700o C,
hnom = 5.9 nm, = 3.0 As,1) according to [44].
Medeiros-Ribeiro et al. [53] argue that both island shapes are metastable1 over some
regime of surface coverage due to at least local minima in their energies. The nal survival of the domes is explained by assuming the simultaneous existence of monolayer
height islands on top of the wetting layer acting as a reservoir for the ensemble of threedimensional islands. In this open system energy and atoms are exchanged between the
monolayer height and the three-dimensional islands. The pyramids nucleate and grow
until their volume reaches a maximum value that is smaller than the minimum volume
for which the domes are more stable than the pyramids plus an additional germanium
monolayer on top of the wetting layer. After a particular pyramid has reached its maximum size, additional adatoms deposited in its environment form monolayer height islands
until the pyramid plus this reservoir of atoms can form a dome, and then the transition
from pyramid to dome occurs abruptly.
Alternativley, Ross et al. [54] performing in situ TEM experiments observed a signicant
ripening of coherent three-dimensional islands during epitaxial growth with a spontaneous
shrinkage and the nal disapperence of pyramids which occurs simultaneously with the
formation and enlargement of domes. This observation is explained by the detachment
of adatoms from the metastable pyramids and their diusion on top of the wetting layer
towards the stable domes at a well dened pyramid volume at which the dome energy
In this datareview classications as stable, metastable or unstable refer to the interplay between
surface energy and strain energy contributions with respect to the total energy of the heterostructures but
not to the thermal stability with regard to a possible compositional intermixing of silicon and germanium
during growth.
1
4
Growth Technique / hnom TS hhi hli Ref.
o
,
1
8
,
2
Analysis Method
[mls] [ C] [
As ] [nm] [nm] [10 cm ]
MBE/TEM
12 700 0.2
10 200 6.3
[45]
MBE/TEM
6
700 0.2
7
200 1.6
[45]
MBE/AFM
5
745 0.2
25 200 0.6
[30]
MBE/AFM
6
745 0.2
8
100 9.3
[30]
MBE/SEM
3.5 550 0.3
36.3 6
[46]
MBE/SEM
3.5 550 0.9
34.7 8
[46]
MBE/SEM
3.5 450 0.3
19.4 9
[46]
MBE/SEM
3.5 450 0.9
28.9 8
[46]
MBE/AFM
6
500 0.2
4.5
60
[47]
MBE/AFM
5
500 0.2
4.0
60
[47]
MBE/TEM
3
500 0.06
40
[36]
LPCVD/TEM
6
700 3.0
14 140 5.5
[35]
LPCVD/TEM
5
700 3.0
10 150 2.2
[48]
LPCVD/AFM
5
700 3.0
11 150 2
[44]
LPCVD/AFM
5
500 3.0
3
33 90
[44]
TABLE 1: Experimentally measured average height hhi, lateral base extension hli and
areal density of three-dimensional coherent Ge islands on Si(001) dependent on the
nominal layer thickness hnom given in monolayer units (mls), the growth temperature TS,
the growth rate and the growth technique.
becomes lower than the pyramid energy.
Moreover, Goryl et al. [44] showed that the island size distribution may also become trimodal if incoherent islands are formed, cf. gure 2b. In this case the smallest island size
belongs to the pyramids, the medium one to coherently strained domes and the largest
one to plastically relaxed islands. For a xed nominal layer thickness a variation of the
growth temperature in the range between 550o C and 700o C does, however, only have a
minor inuence on the sizes of the coherent pyramids and domes, indicating that they are
energetically stable. In contrast, an Arrhenius-like temperature dependence of the lateral base extension of the incoherent larger islands is found demonstrating that ripening
plays an important role in the growth of plastically relaxed islands. This observation is in
agreement with calculations predicting a distinctly larger growth rate for incoherent than
for coherent islands caused by a decrease of the lattice strain at the island edges due to
the formation of mist dislocations [55, 56, 57].
Beyond direct island formation during growth, postdeposition annealing of an unstable
two-dimensional layer represents a kinetic route to grow coherent islands with a uni5
form size distribution as shown by Chen et al. [58]. In their experiments, an overcritical (hnom > hc) Si0 5Ge0 5 layer was grown on Si(001) at a rather low temperature of
TS = 400o C thus remaining in a planar state at rst. Annealing below 560o C has nearly
no eect on the layer morphology independent of the annealing time and a sharp transition to three-dimensional island formation is observed at this temperature, once more
indicating the existence of an energy barrier to island formation as stated before. At
585o C and an annealing time of 6 min a narrow distribution of islands sizes (hli 40 nm,
hhi 5 nm) is observed, while annealing at 680o C results in a pronounced ripening and
the formation of incoherent islands.
:
:
C ORDERING OF EPITAXIAL ISLANDS
Island formation in single Ge epilayers on Si(001) substrates is a spontaneous process
usually resulting in a random spatial arrangement of the islands. Potential technological
applications of islands as quantum dot structures with controled electronic properties require the fabrication of ordered island arrangements at a high volume density. Concepts
that have been applied to achieve regularity in the spatial arrangement of an ensemble
of islands employ self-organized lateral ordering in single layers and vertical ordering in
mutlilayers. The common idea of all approaches is to create a regular array of regions at
the growth surface where the nucleation barrier for island formation is locally reduced.
This may be attained by decreasing the activation enthalpy for island nucleation, e.g. by
employing the elastic anisotropy of the crystal or by locally reducing the lattice mismatch.
C1 LATERAL ORDERING OF ISLANDS
Calculations by Shchukin et al. [59] show that the elastic strain elds between adjacent
islands may interact suciently strongly through the underlying substrate to achieve a
two-dimensional ordering of an ensemble of islands in single layer structures. The main
part of this interaction is an elastic dipole-dipole repulsion and the interacting islands
represent a system of elastic domains for which the minimum strain energy corresponds
to a periodic domain structure. For square based islands with lateral base extensions
along the h100i directions, as for the pyramids, the elastic anisotropy of cubic crystals
results in a minimum of the total energy for a periodic square lattice of islands on (001)
substrates with primitive lattice vectors along the elastically soft [100] and [010] directions. However, with increasing island spacing this elastic interaction may become too
weak for ordering. In fact, the observation that lateral ordering of islands does not occur
in single Ge layers on Si(001) but only in periodic SiGe/Si multilayers [60, 61] indicates
that this mechanism must be assisted by other eects and up to now, a pronounced lateral
ordering of islands in single layer structures has only been observed experimentally for
6
FIGURE 3: (a) AFM image of Ge islands grown by MBE (TS = 500o C, hnom = 5 mls,
= 0.2 As,1) on a Si(001) substrate tilted 1:5o towards the [110] direction and the
corresponding two-dimensional Fourier transformation of the image intensity distribution
showing a clear six-fold symmetry. (b) Under same conditions growth on a substate tilted
2:0o towards the [100] direction results in a rectangular arrangement of the Ge islands.
Island heights in the AFM images may be depicted from the grey-scale legends [47].
InAs islands on GaAs(001) substrates [62].
A strategy for the fabrication of regular arrays of islands is the use of vicinal Si(001)
substrates as templates for the growth of Ge islands which represents, in a sense, a hybrid between deliberate nanopatterning and self-organization. In case of substrates tilted
towards the [110] direction, monolayer height triangular islands evolve during growth
of the Ge wetting layer above substrate-tilt induced plateaus. Under step-ow growth
conditions, one side of these islands is parallel to the substrate steps and their opposite
apex terminates close to the neighbouring substrate step as shown by Chen et al. [63].
Three-dimensional islands excessively nucleate at the apex regions which is explained by
7
a reduced nucleation barrier for island formation compared to regions on top of the monolayer height islands and thus results in a chain-like arrangement of islands along the [110]
direction [63]. If step distances are chosen suciently small by increasing the substrate tilt
angle (2o 5o) and step heights are increased by step-bunching prior to the growth of
the germanium epilayer, a pronounced regular hexagonal island arrangement is observed
[47] as shown in gure 3a. In this case, pyramids with lateral base extensions parallel to
the h100i direction nucleate at the [110] steps and the elastic interaction between islands
belonging to dierent (110) planes results in the observed island arrangement. Analogous, growth on substrates tilted towards the [100] direction produces a checkerboard-like
topography (cf. gure 3b) since islands nucleate along the [100] steps and the anisotropic
elastic interaction (cf. above) results in a second set of island chains perpendicular to the
rst one [47].
Another way to achieve regularity is using the strain eld of a regular mist dislocation
network to control the nucleation of islands [64]. The key idea is to grow a plastically
relaxed low-mist SiGe buer layer as well as a silicon capping layer prior to the growth
of the Ge layer forming the islands. The elastic strain eld of a mist dislocation located
at the substrate buer layer interface may then locally enlarge the silicon lattice parameter at the growth front thus reducing the energy barrier for island nucleation directly
above the dislocation. This results in a spatial island arrangement being a replica of the
dislocation network. The approach, however, is limited to low island densities since a
regular array of interfacial mist dislocations without the formation of a high density of
threading dislocations penetrating the buer layer is only attainable for quite low-mist
SiGe buer layers. The mean mutual island distance in such cases is typically in the
order of 500 nm [64, 65],
q while spontaneous formation of islands on unpatternend substrates may yield = 1= 100 nm, cf. table 1.
C2 VERTICAL ORDERING OF ISLANDS IN MULTILAYERS
Compared to an increase of the areal island density by the further optimization of epitaxial parameters during single layer growth, island formation in periodic Ge/Si multilayers
represents an additional pathway to increase the overall volume density of islands. Moreover, under properly chosen conditions, the Si layer, capping a Ge layer with islands, tends
to planarize and a pronounced vertical ordering of islands along the growth direction is
observed [60, 66, 67] as can be seen from the micrograph in gure 4. This ordering results from an eective mismatch reduction caused by the elastic strain elds of the buried
islands [35, 61, 68, 69, 70], i.e. the mechanism of vertical ordering is comparable to the
nucleation of islands above a dislocation network as described in the previous section.
In order to characterize the vertical ordering of islands dependent on the layer geometry
8
FIGURE 4: Cross-sectional TEM micrograph of a Ge/Si multilayer grown on Si(001) by
LPCVD (TS = 700o C, hnom = 0.8 nm, dSi = 40 nm, = 3.0 As,1). Vertical ordering of
islands along the [001] growth direction results from a local mismatch reduction above a
buried island of the rst Ge layer [35].
quantitatively it is suitable to dene an island correlation function:
C=
r ,q
1,q
(1)
where q denotes the areal fraction of circular areas dened in a new growth surface by
imaginary cones of inuence with opening angles 2 above buried Ge islands and r denotes the fraction of islands forming within these circular areas. A theoretical analysis
demonstrates that the degree of ordering in the multilayers at a given growth temperature
TS is primarily inuenced by the volume Vbur 2 hnom of buried Ge islands (or more
precisely the volume of the Ge atoms in the islands), the volume Vnuc of a newly formed
island nucleus at the growth surface and the Si interlayer thickness dSi separating two Ge
9
layers. An analytical approximation for the island correlation function is [69]:
"
#
6
f 2 VburVnuc tan2 ()
C = 1 , exp ,
with:
kTS d3Si
(2)
4G (1 + )2
(3)
1,
where G = 68.14 GPa is the isotropically averaged shear modulus and = 0.2174 is the
Poisson's ratio of silicon. Figure 5 shows this correlation function for = 20o as a function
of the reduced interlayer thickness:
=
= dSi s
3
kTS
2
6f Vbur tan2()
(4)
together with experimental values C ( ) extracted from various experiments. The best t
of C (; Vnuc) to the experiments is gained for an island nucleus volume of Vnuc 4 nm3
which is in good agreement with energetic considerations on the equilibrium volume of
an island nucleus [32, 35]. Similar results are found when careful attention is paid to the
anisotropic material properties of Si and Ge by application of nite element simulations
[35].
The growth of multilayers may not only result in a vertical ordering of islands but can also
be used to enhance the uniformity of island sizes and shapes as well as the regularity in
the spatial arrangement of an ensemble of islands by one order of magnitude compared to
single layer structures. Experimental [67] and theoretical [73] studies demonstrate a signicant increase in the uniformity of the sizes of the islands and their mutal distance with
an increasing number of multilayer periods, i.e. regardless of the initial random in-plane
island arrangement in the rst Ge layer, the lateral island distances and sizes narrow after
growth of many Ge/Si periods. This behaviour is attributed to the dierent inuence of
elastic strain elds of buried islands on the nucleus formation at the growth surface for
dierent initial island arrangements during multilayer growth. For initially large mutual
distances between buried islands a nucleus will not only form above buried islands as
shown in gure 4 but also at positions between the projected island positions thus locally increasing the average island density compared to those of the buried Ge layer. For
initially small mutual distances between buried islands a growth stage may be reached
for which the superposition of strain elds of two or several buried islands leads to an
enhanced nucleation probability at one position between the projected island positions
resulting in the formation of a single island thus locally decreasing the average island
density. By this means a regular array of nucleation centres develops with an increasing
number of periods. When ripening is suppressed a regular three-dimensional array of
islands of high geometrical regularity evolves [72].
Frequently, island and wetting layer compositions are found to deviate considerably from
the nominal stoichiometry [35]. For specimens grown by MBE at TS = 700o C quantitative
10
FIGURE 5: Island correlation function C ( ) according to equation 2 in dependence
on parameter dened by equation 4. Experimental values have been extracted from
various experiments (( [69] with Vbur = 104 103 nm3, TS = 700o C ), ( [71] with
Vbur = 12 103 nm3, TS = 720o C ), (/ [66] with Vbur = 0:2 103 nm3, TS = 460o C ), (
[67] with Vbur = 6 103 nm3, TS = 550o C ), (. [72] with Vbur = 76 103nm3, TS = 670o C ))
yielding a best t for an island nucleus volume Vnuc 4 nm3.
TEM analyses show a signicant reduction of the average germanium content hxi of the
islands in the range between 40 % and 70 % for the deposition of Ge and Ge0 85Si0 15
layers on Si(001). Table 2 gives an overview on experimentally measured hxi values for
buried islands obtained by various techniques. Furthermore, the measured average germanium content of individual layers formed by the deposition of Ge is observed to increase
monotonically with the number of periods [66]. The lower hxi values of layers grown at
rst during the growth process of a multilayer are attributed to an eectively enhanced
interdiusion with silicon.
:
:
D CONCLUSIONS
In summary, self-assembled island formation in the Stranski-Krastanow growth mode represents a promising approach to the fabrication of nanosize quantum dot structures by
direct epitaxial growth. Depending on the nominal layer thickness hnom in the range be11
hxHRTEM i hxLACBED i hxEELSi hxPL i
6
1.00 0.27 0.06 0.42 0.07 0.34 0.05
12
1.00 0.42 0.07 0.38 0.08 0.35 0.07 0.45
12
0.85 0.49 0.07
TABLE 2: Measured average Ge content hxi of three-dimensional buried islands depenhnom [mls]
x
dent on the nominal layer thickness hnom and the nominal Ge content x for Si/Gex Si1,x /Si
heterostructures grown by MBE at TS = 700o C at a rate of 0.072 micron per hour. Results obtained from a quantitative analysis of high-resolution TEM images [74] are compared with data from large angle convergent beam (LACBED) [45], electron energy loss
specrometry (EELS) [45] and photoluminescence (PL) [26] measurements.
tween 3 and 6 germanium monolayers and the growth temperature (500o C TS 700o C),
coherent pyramid-shaped (hli 40 nm, hhi 5 nm) and dome-shaped (hli 150 nm,
hhi 15 nm) islands may be realized in case of single Ge layers on Si(001) substrates.
These structures are metastable over a certain range of surface coverage and show a quite
narrow island size distribution with the amount of deposited material determining the
island density but not the absolute island size.
Spatial ordering of an ensemble of islands is primarily inuenced by the existence of
a regular array of nucleation centres, for which the probability for island formation is
locally increased. This ordering may be achieved by the use of vicinal substrates, by
growth above a regular dislocation network, by considering the elastic anisotropy of the
crystals or by the elastic strain elds of buried islands in periodic multilayers. Especially
in the latter case the uniformity in island shapes, sizes and mutual distances may be
tuned and a complete three-dimensional ordering may be realized with a narrowing of
island properties with an increasing number of multilayer periods. However, interdiusion
of silicon and germanium during growth usually results in a signicant modication of
island compositions with respect to the nominal layer stoichiometry.
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