Calculation of metamorphic two-dimensional quantum energy system:Application to wetting layer states in InAs/InGaAs metamorphic quantumdot nanostructuresL. Seravalli, G. Trevisi, and P. Frigeri Citation: J. Appl. Phys. 114, 184309 (2013); doi: 10.1063/1.4830021 View online: http://dx.doi.org/10.1063/1.4830021 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i18 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

Calculation of metamorphic two-dimensional quantum energysystem: Application to wetting layer states in InAs/InGaAsmetamorphic quantum dot nanostructures

L. Seravalli, G. Trevisi, and P. FrigeriIMEM-CNR Institute, Parco Area delle Scienze, 37/A 43100 Parma, Italy

(Received 18 June 2013; accepted 28 October 2013; published online 13 November 2013)

In this work, we calculate the two-dimensional quantum energy system of the In(Ga)As wetting

layer that arises in InAs/InGaAs/GaAs metamorphic quantum dot structures. Model calculations

were carried on the basis of realistic material parameters taking in consideration their dependence

on the strain relaxation of the metamorphic buffer; results of the calculations were validated

against available literature data. Model results confirmed previous hypothesis on the extrinsic

nature of the disappearance of wetting layer emission in metamorphic structures with high In

composition. We also show how, by adjusting InGaAs metamorphic buffer parameters, it could be

possible: (i) to spatially separate carriers confined in quantum dots from wetting layer carriers, (ii)

to create an hybrid 0D-2D system, by tuning quantum dot and wetting layer levels. These results

are interesting not only for the engineering of quantum dot structures but also for other applications

of metamorphic structures, as the two design parameters of the metamorphic InGaAs buffer

(thickness and composition) provide additional degrees of freedom to control properties of interest.VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4830021]

I. INTRODUCTION

In lattice-mismatched structures with layers thicker

than the critical thickness for plastic relaxation, the lattice

parameter changes from the value of the underlying sub-

strate to the free-standing value of topmost material. From

this perspective, these layers, known as metamorphic buf-

fers (MBs), behave as virtual substrates where the lattice

parameter can be controlled without acting on the composi-

tion of the alloy. The density of dislocations that allow

the relaxation of the strain in the layer can be reduced by

orders of magnitude by an adroit design of the composition

profile of the MB, keeping them separated from the active

region of the device.

Metamorphic semiconductor structures grown on GaAs

substrates have been actively researched for a variety of

applications, ranging from triple-junction1 and quantum dot

(QD)2 solar cells to light-emitting devices,3,4 from High

Electron Mobility Transitors (HEMTs)5 to Semiconductor

Optical Amplifiers (SOAs).6 However, despite the relevant

success of metamorphic structure, little attention has been

paid to consider in depth their basic physical properties and

their potentialities in terms of advanced design possibilities.

From a theoretical point of view, although transport proper-

ties of metamorphic structures have been investigated,7,8 to

the best of authors knowledge, no work has been conducted

on the effects of strain and composition on the electronic

properties of low-dimensional metamorphic structures of in-

terest for applications in photonics.

In this theoretical work, we study InAs/InGaAs meta-

morphic systems, by calculating the wetting layer (WL)

states. The results of this work, beyond a relevant interest for

QD-based structures, can be considered of general validity

for 2D metamorphic systems.

Metamorphic InAs/InGaAs QDs have been proven to

be interesting structures and rather different from pseudo-

morphic InAs/GaAs,9 in particular for redshifting the emitted

light in the 1.3–1.55 lm optical window, of great interest for

fiber-optic telecommunication devices.10–12 In addition to

this technological interest, the metamorphic system has the

unique advantage of providing two independent design pa-

rameters (composition and thickness) that allow to engineer

the quantum system on a wide range: this has been studied in

depth for the QD system,13,14 but not considered so far for

2D structures.

It is widely known that the thin InAs layer that is present

below InAs QDs, termed WL affects substantially QD

properties.15–17 This is true both for confined carrier dynam-

ics, as WL states are channels for carrier relaxation,18,19 and

for Coulomb interaction effects between QD and WL car-

riers, in particular for structure where optical emission from

single QD is studied.20,21 Hence, an in-depth theoretical

study of the metamorphic WL system is justified, to explore

the possibilities of this system to control the WL energy lev-

els, both in terms of energy values and of carrier localization.

We emphasize that in this work, we do not calculate QD lev-

els that have been already modelled in previous works, as

here our interest is devoted to the WL quantum system.

We used Tibercad software to calculate InAs WL states

grown on InxGa1–xAs partially relaxed meta-buffers. We

focused in particular on the effects of composition and

thickness of the MBs on 2D energy states and on carrier

wavefunctions. As WL parameters such as thickness, compo-

sition, and strain are not easily available for metamorphic

structures—X-Ray Diffraction (XRD) and Transmission

Electron Microscopy (TEM) characterization are difficult due

to the presence of defects—model calculation were based on

reasonable assumptions that, nevertheless, can give important

0021-8979/2013/114(18)/184309/8/$30.00 VC 2013 AIP Publishing LLC114, 184309-1

JOURNAL OF APPLIED PHYSICS 114, 184309 (2013)

semi-quantitative indications, as already done in literature.22,23

We considered two growth methods used to obtain metamor-

phic QDs: the mainstream Stranski-Krastanow (SK) one,

which results in structures with an high density of QDs, for

which an InAs WL has been considered and the one recently

introduced based on the post-growth annealing of sub-critical

InAs coverages, for which an InGaAs WL has been consid-

ered, as discussed in depth in section “Theoretical Model.”

The latter approach allows to obtain very low density of QDs

of great interest for the development of single photon

sources.24,25

Model results were validated by the comparison with

available literature data for WL energy states, also consider-

ing structures where InAs has been overgrown with InAlAs

and InyGa1�yAs layers, to explore how this more complex

design may open the way to new engineering possibilities.

The results of our work aim at answering some questions

that could give new precious hints to the design of complex

quantum structures: (i) is it possible to create hybrid 0D-2D

systems using MBs, where QD and WL levels might reso-

nate in energy?, (ii) is it possible to push carriers out of the

WL, far from the QDs, to reduce the effects of Coloumb

interactions?, and (iii) what is the reason of the recently

reported WL "disappearance" in In-rich metamorphic struc-

tures: is it design or material related?

II. THEORETICAL MODEL

For the calculation of the WL quantum confined levels for

the system composed by the GaAs layers, the InxGa1�xAs MB,

the In(Ga)As WL, the InyGa1�yAs cap layer, and the InAlAs

layers (when considered), we used the software Tibercad.26

The geometry of the structure (schematically shown in Fig. 1)

was divided in different regions; and subsequently, a computa-

tional mesh was used to discretize the partial differential equa-

tions representing the physical models to be solved.

The one dimensional Schroedinger equation was solved

with a single-band effective mass model for the conduction

band (CB) and with a 6 band-kp model for the valence band

on the discretized mesh geometry. Band profiles are shown

in Fig. 1. The eigenvalue problem was solved by calculating

first the strain condition of the structure relatively to a

“virtual” InGaAs material; calculation of material parame-

ters is described below. Then, a quantum region is selected

where the Schroedinger equation was solved for electrons

and heavy holes by an iterative process with open boundary

conditions: eigenvalues and eigenfunctions were derived by

values of convergence of solutions. In Table I, we indicate

parameters used to carry on the calculation on the Tibercad

software.

The QD effect on WL levels has been disregarded in

this calculation, experimental work recently showed that this

is a reasonable hypothesis.27 The lattice parameter and the

energy gap of the InGaAs MB depend on its thickness t and

composition x, thus a “virtual” material InGaAs has been

considered for every t/x couple. The in-plane lattice parame-

ter ax of the InxGa1� xAs MB is related to the in-plane strain

e by e¼ (ax - a0)/a0, where a0 is the InGaAs free standing lat-

tice parameter. The dependence of e on the t and x parame-

ters was derived by following the model of Mare�e et al.,28

whose predictions were experimentally confirmed in Refs.

29 and 30.

FIG. 1. (Top) Scheme of the structure used for the model calculation, grown

on GaAs (001) substrate and composed by a GaAs buffer layer, an

InxGa1�xAs MB of thickness t, the In(Ga)As WL, the InyGa1�yAs capping

layer of thickness 20 nm, and the InAlAs additional layers (when considered).

The dashed line indicates the extension of the quantum region considered for

solving the eigenvalues problem. (Bottom) Profiles of CB and VB along the

vertical direction of the structures considered, dotted lines indicate the band

profiles due to additional InAlAs layers, while dashed lines refer to confined

levels for electrons and holes in the WL.

TABLE I. Solver parameters used to carry on calculation with Tibercad software.26

Parameter Description and unit Value

Eigensolver tolerance Numerical eigensolver tolerance used as a convergence criteria (meV) 0.1

Number of eigenstates Number of eigenstates to be computed 3

Model–electrons Number of bands for electrons 1 (single band)

Model–holes Number of bands for kp model for holes 6

Mesh–units Maximum distance between geometrical points in mesh grid (nm) 0.1

Quantum region Extension of quantum region (nm) 40

Temperature Temperature of the system (K) 10

184309-2 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

The energy gaps for different values of x and t were

derived from the deformation potential theory31 in which the

gap E0HH between the minimum of the CB and the maximum

of the heavy hole valence band (HH-VB) can be expressed

in terms of e as

E0HH ¼ E0 þ dEH � dES=2; (1)

where dEH¼ 2ae (C11�C12)/C11 and dES¼ 2be (C11þ 2C12)/

C11; a and b are the hydrostatic and the shear deformation

potentials and Cij the elastic stiffness constant.

The value of the exciton binding energy was set at

10 meV,32 while values of effective masses, deformation

potentials, elastic constants, and other relevant parameters

for the “virtual” material fed in the Tibercad model were

derived from the available literature data, including the

bowing effects for different values of x.33 On the basis of

the derived lattice parameter of the MB, prior to the calcula-

tion of quantum confined levels, a strain calculation of

the InxGa1�xAs/In(Ga)As/InyGa1�yAs (InAlAs) system is

executed by the software to consider the correct value of

the mismatch f between the MB and the WL, defined as

f¼ (aInAs - ax)/ax, where aInAs is the lattice parameter of

InAs. See Table I of supplemental material for values of mis-

match, lattice parameter, strain, and energy gap for different

InGaAs MBs.34

With respect to the structural parameters of the thin

In(Ga)As layer that constitutes the WL, namely thickness dand In composition, it should be noticed that substantial dif-

ferences were reported when growing it in the SK regime or

with the subcritical coverage approach, thus different hy-

pothesis have been put forward in the two cases.

A. Modeling of InAs WL in the Stranski-Krastanowregime

In SK structures, the WL was considered as an InAs

square well (SQW), a reasonable assumption as discussed

in Ref. 23; in Ref. 27, it was shown that WL energy calcu-

lated under this approximation deviates form the experi-

mental value less than 10 meV. d was taken as equivalent

to the critical thickness h for the transition from two-

dimensional to three-dimensional growth; in Ref. 35, it was

discussed how h depends on the mismatch f. Experimental

values of h were considered when available in the litera-

ture,36,37 otherwise values extracted from the curve

reported in Ref. 35 were used. Values of h range from

1.2 ML (for MBs with high x and/or small t) to 1.4 ML (for

MBs with low x and/or large t) and are reported in Table II

of supplemental material.34

A second assumption of this calculation is that the In

composition of the WL does not vary when x and t are

changed, as experimentally demonstrated for QDs grown on

InGaAs MBs, due to the agreement between experimental

PL emissions of QDs and model calculations where the QD

In composition was fixed in the whole range x/t.13

As the lattice parameter of the metamorphic layer on

which InAs is deposited depends on both x and t, it was nec-

essary to calculate the tetragonal distortion due to different

mismatch between InGaAs MBs and InAs WLs, using the

relation

aWL perpð Þ ¼ aWL freeð Þð1� 2 � eWL � c12=c11Þ; (2)

where aWL(perp) is the out-of-plane lattice parameter of the

InAs layer, aWL (free) is the free-standing lattice parameter

of InAs, and eWL is the in-plane strain of the InAs layer,

given by (ax - aWL(free))/aWL(free). See Table I of supple-

mental material for values of quantities for representative xand t.34

B. Modeling of InGaAs WL in the sub-critical regime

In the case of the deposition of subcritical coverages of

InAs on MBs, the structural parameters of the WL have been

considered differently, due to the experimental evidences of

substantial changes from the SK case.38 As a true critical

thickness does not exist in this growth regime, d has been

derived by considering the deposited InAs sub-critical cover-

ages: experimental data based on XRD characterization of

structures with InAs sub-critical coverage of 1.5 ML showed

that the WL was 1.0 nm thick with an In composition of

0.30.38

As we showed that the sub-critical deposition needed to

obtain QDs on MBs is larger than on GaAs,25 here we assume

a linear proportionality between the subcritical coverage and

the WL thickness. By considering the tetragonal distortion

discussed above, the data of Ref. 25 may be converted in

the following WL thicknesses: 1.5 ML ¼> 1 nm for x¼ 0,

1.55 ML ¼> 1.1 nm for x¼ 0.15, 1.65 ML ¼> 1.3 nm for

x¼ 0.22, and 1.8 ML¼> 1.4 nm for x¼ 0.30.

With respect to the WL composition, as it was proved

experimentally that in this growth regime the WL is

much poorer in In that in the SK case, we considered an

InzGa1�zAs WL where the change in z is linear with x, i.e.,

z¼ 0.30þ x.

This assumption is reasonable if one considers the

effects of both QD/substrate intermixing and In segregation

in the MBs. As often discussed for InAs WLs grown on

GaAs,23,39,40 it is known that InAs/GaAs intermixing takes

place when growing QDs structures. This leads to the incor-

poration of Ga from the substrate into the WL. In the case of

MBs, the fraction of Ga in the growth front is lower, then a

lower Ga content (i.e., higher In content) in the WL should

be expected with respect to the growth on GaAs. Second, In

segregation is known to take place during the epitaxial

growth of InGaAs alloys,41,42 with the consequence of an In

enrichment of the growth front that should increase with the

In content in the MB.

Although these might be strong assumptions, they should

be indeed reasonable and allow to give semi-quantitative indi-

cations: more refined calculations could be possible with

structural characterization of metamorphic samples to give

precise measurements of composition and dimensions of WL.

Anyhow, it should be noted that such information is not easily

obtained on these kind of structures, where the effect of sur-

face roughness, cross-hatch morphology, and defect density

has to be taken into account.

184309-3 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

III. RESULTS

A. SK structures

In order to validate the model, we calculated WL energy

levels for metamorphic QD structures grown in the SK re-

gime and compared them with available experimental data

from the literature. First of all, we considered structures

where the QDs are grown on InxGa1�xAs MB and capped by

an InxGa1�xAs layer (symmetric structures); and as a second

step, we also considered the comparison with experimental

data from asymmetric structures where the QDs are capped

by an InyGa1�yAs or InyAl1�yAs layer.

In Figure 2, we show the WL emission energies as meas-

ured by PR43 and by Photoluminescence Excitation

Spectroscopy (PLE)44 as functions of the mismatch between

InAs WL and InxGa1�xAs MBs with x ranging from 0.09 to

0.35 (see Table II of supplemental material for additional

structural parameters);34 model calculations of WL energy

levels for same structures are represented by the solid lines

that go from a pseudomorphic layer (with same lattice param-

eter of GaAs and, hence, f¼ 7.14%) to a completely relaxed

layer (with lattice parameter corresponding to the

free-standing one of InGaAs and with the minimum possible

value of f). For PR data at temperature different from 10 K, a

Varshni shift from RT to 10 K of 63 meV was used. The errors

for the model results (reported as vertical bars) stem from

uncertainties in transition thickness (that can be estimated in

60.1 ML) and in the In composition of layers (63%).14

The WL emissions are reduced with decreasing mis-

match due to two effects: (i) the reduction of the strain in the

WL layers that causes a lowering of the InAs energy gap and

(ii) the slight increase of the critical thickness (hence, of the

WL thickness) that has been discussed in Ref. 35.

Experimental works did not show WL emissions in struc-

tures with x¼ 0.35, a feature explained by increase surface

roughness that might inhibit WL formation.

It can be seen how for almost all data, the agreement

between model calculation and experimental data is less than

20 meV, a noteworthy result if one considers the strong start-

ing hypothesis on the absence of change in WL composition

for different x/t values.

However, it can be noticed that in general the WL PL

emissions are at energy values consistently lower than the

model calculations, with a larger discrepancy for structures

with higher x: this could be an indication that there is a

change in structural parameters of WL grown on high-xsurfaces, a conclusion reached also in Ref. 35.

For structures with x� 0.28 and large MB thickness

(i.e., low f), it has been reported that no WL levels are detect-

able by optical techniques.43,44 Present calculations allow to

conclude if this effect is due to intrinsic properties of the

quantum system, such as band discontinuities between MB

and WL too low to confine carriers in the 2D states. In

Fig. 3, we report the model calculation for the electron and

heavy holes states of the MB and of the WL for x¼ 0.28,

0.31, and 0.35, showing their dependence on the mismatch f.In the same graph, we indicate with filled squares structures

where WL levels have been detected and with open squares

structures where no WL levels have been detected.43,44

It can be seen how by increasing x WL states become very

close to MB states, in particular for electrons: however, WL

levels were not detectable in structures with same x but with

larger d (i.e., smaller mismatch). It can be clearly seen that,

if x is unchanged, a reduction of the mismatch results in an

increase of the energy difference between WL and MB

states, due to effect of increase of the WL thickness dis-

cussed above. Hence, from a theoretical point of view, there

FIG. 2. Energy of WL transitions as deduced by PR (Ref. 43—circles) and

by PLE (Ref. 44—squares) for InAs/InxGa1�xAs metamorphic structures as

functions of the WL-MB mismatch against model calculation (solid lines).

Dashed vertical bars indicate errors in model results due to uncertainties in

input parameters.

FIG. 3. Model calculation of WL (dashed line) and MB (solid line) electron

(leftside vertical scale) and heavy holes (rightside vertical scale) energy lev-

els for InAs/InxGa1�xAs metamorphic structures as functions of the WL-MB

mismatch for x¼ 0.28, 0.31, and 0.35. Open squares indicate structures

where no WL states have been detected, while filled squares refer to struc-

tures where WL states have been detected.43,44

184309-4 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

is no intrinsic reason that explains why WL levels are not

detected for large thickness: therefore, previous hypothesis

on extrinsic factors related to the InGaAs materials, such as

rough surfaces, composition inhomogeneity, and higher

defect density are reinforced.9,45

In order to further validate the predictability of the

model on metamorphic structures, we considered more com-

plex structures where the capping layers is composed by a

different material from the MB (asymmetric structures), tak-

ing available data from different works appeared in the liter-

ature: (i) with 10 nm-thick InyGa1�yAs capping layers

(y> x)36,37 and (ii) with 6 nm-thick InyAl1�yAs layers below

and above InAs WL (see Fig. 1 for structure schematics and

band profiles).43

In Table III of supplemental materials,34 we report the

structure parameters, where it can be noticed how the critical

thicknesses for 2D-3D transition reported in the published

works in some cases are different from those reported in

Ref. 35, conceivably due to different growth conditions.

Nevertheless, in both cases, the agreement between model

calculation and experimental data is quite good. For struc-

tures of case (i), the reported experimental WL energy level

for structures with x¼ 0.16, y¼ 0.30 was 1.088 eV (Ref. 36)

and present calculation give a WL level at 1.090 eV;

for x¼ 0.20, y¼ 0.40, the experimental data was 1.123 eV

(Ref. 37) against a 1.110 eV calculated value; and for

x¼ 0.25, y¼ 0.45, the experimental data was 1.077 eV

(Ref. 37) while the model calculated value was 1.070 eV. In

case (ii) for x¼ 0.15 and y¼ 0.20 and 0.10, the experimental

WL energy was 1.340 eV and a 1.355 eV, respectively,43

comparing with model values of 1.345 eV and 1.350 eV.

These results confirm that the model can help predict meta-

morphic 2D states also for more complex structure design.

Incidentally, we note that structures with InAlAs layers

are a good example of the usefulness of this model to control

the quantum energy system by an adroit design engineering

of the metamorphic structure: from these results, it is proven

that it is possible to raise WL levels above InxGa1�xAs MB

energy states by using InAlAs embedding layers below and

above QDs, with interesting consequences for QD carrier

dynamics.

FIG. 4. Energy of WL transitions as

deduced by PL (Ref. 25)—full circles)

against model calculation (open

squares) as functions of x. Values of y

of upper layers (Ref. 25) are indicated.

Inset: 10 K PL spectra for structures

with x¼ 0.22 and x¼ 0.22 (red),

y¼ 0.30 (blue). The WL value calcu-

lated by the model is indicated by

dashed lines.

FIG. 5. Model calculation of WL energy (continuous line) and experimental

QD peak emission (open squares) as deduced by PL (Ref. 25) in sub-critical

structures as functions of x. Vertical bars indicate FWHM values of QD

emission. Dashed line is a guide for the eye.

184309-5 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

B. Sub-critical coverages structures

The interest in these structures relies in the very low den-

sity of QDs that is attainable, making them attractive for the

development of single photon sources in the 1.3–1.55 lm

range. From this perspective, the properties of WL levels and

their dependence on MB parameters are of great interest due

to the fact that, beside effect on the thermal properties of QD

emission, WL-confined carriers levels are known to influence

single photon characteristics via Coulomb interaction.21

In Figure 4, we present comparison of model calcula-

tion of WL states and experimental PL emission of WL lev-

els as functions of the MB composition x for structures

where the MB thickness was fixed to 500 nm (hence with

fixed MB-QD mismatch for same x), while InyGa1�yAs

capping layers with y> x were used to further red-shift the

QD emission.25 Due to the low density of QDs, standard PL

characterization allows to detect light emission also from

WL states, in contrast with SK structures. The agreement

between model calculation and PL peaks is always less than

15 meV (except for the x¼ 0.15 y¼ 0.45 case). Data are

taken from Ref. 25 and from unpublished data for x¼ 0.15,

y> x (see supplemental material for experimental details

and for structural parameters).34 For x¼ 0.30, no emission

has been reported from WL, similarly to SK structures,

although the model predicts to have WL levels at the

1.030 eV for x¼ y¼ 0.30 at about 70 meV from QD peak

emission. Previous indications of effect of high-x, high-tsurfaces on WL formation that have been discussed in

depth in the previous sections are confirmed also for struc-

tures deposited by sub-critical coverages.

From the comparison of WL levels calculation and PL

experimental data, it can be concluded that the model is vali-

dated also for structures with a low density of QDs (with dis-

crepancies in the range of 15 meV) and it could be used for

advanced metamorphic structure design, for example, to

engineer 2D carrier energy and localization.

As a very interesting case of study, let us consider struc-

tures with x¼ 0.22 (whose PL spectra are shown in the inset

of Fig. 4), where the expected WL emission falls in the tail

of the large QD emission (in low density structures, QD pres-

ent very wide emission due to higher dishomogeneities of

QD size and/or compositions). This suggests that it could be

possible to have small energy difference between energy lev-

els for carriers confined in some isolated QDs (emitting in

the high-energy tail) and 2D WL levels: this could result in

an increase of the electronic coupling between single elec-

trons in QDs and the 2D electron gas in the WL, up to a pos-

sible establishment of an hybrid 0D-2D system, a topic of

great recent interest.46–48

In order to make some more general assessment on this

possibility, let us consider, in Fig. 5, the PL emission of WL

and of sub-critical QDs25 presented as functions of x. It can

be noticed how the energy difference between QD and WL

emission reduces with the increase of the MB In composi-

tion. This dependence allows to predict that in the high-xranges (x� 0.25), the WL emission energy difference with

QDs could become less than the full width half maxima

(FWHM) of the QD PL spectrum (indicated in Figure 5 as a

vertical bar of about 100 meV): this means that a good frac-

tion of isolated QDs could be in resonance with WL levels.

However, it should be kept in mind that there is the limita-

tion on x related to the absence of WL states discussed

above: for this reason, an hybrid 0D-2D systems composed

by an isolated QD and the WL could be realized for values

of x ranging from 0.20 to 0.30. As a second example of the

usefulness of this model to predict properties of metamor-

phic 2-dimensional quantum system, let us consider the spa-

tial localization of carriers confined in the WL.

In Fig. 6, we show the band profiles and the squared val-

ues of the wavefunctions of carriers for the metamorphic sys-

tem with x¼ 0.15, using capping layers going from y¼ 0.15

to 0.35, for structures grown by the deposition of subcritical

FIG. 6. Plots along growth direction of: VB profile (thicker blue line), CB profile (thicker red line), ground electron (red line) and heavy hole (blue line) energy

levels, first electron (dashed red line), and heavy hole (dashed blue line) excited levels (leftside vertical scale) calculated by the model. Plots along growth

direction of model calculations of squared absolute values of electron (blue line) and heavy hole (red line) wavefunctions (right handside).

184309-6 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

InAs coverages. In the case of x¼ 0.25 and 0.35, a mixed

WL-QW system emerges, where the spatial localization of

carriers may greatly differ. This is a system that has been

explored recently, as a possible mean to control the Coulomb

interaction between QD and WL carriers via their spatial

separation.27

When y¼ 0.15, carriers in the ground levels are confined

within the WL; while in the y¼ 0.25 case, the electron

spreads into the cap layer and the heavy hole has still a

greater probability of being in the WL than in the InGaAs

capping layer. When y¼ 0.35, the situation is totally differ-

ent as both carriers are delocalized in the upper layer and the

WL does not seem to have any effect in confining them. This

is a very interesting result, as it has been very recently exper-

imentally proved that heavy holes cannot be completely

delocalized from the WL in pseudomorphic InAs/InGaAs

QD structures.38

In Fig. 7, we present the overlap integral of the heavy

hole-electron wavefunctions for the ground states in the WL

and the carrier probability of being found in the WL as func-

tions of y in the whole range 0� y� 0.35 (x value remains

constant at 0.15).

It is interesting to notice that while electrons are weakly

localized within the WL even for y¼ 0.15 and spread in the

capping layer (for y> x) or in the MB (for y< x), heavy

holes become completely delocalized from the WL only

when y¼ 0.35, effectively spatially separating the 2D system

from the 0D QD system. Also the overlap integral (propor-

tional to the radiative recombination probability) changes

consistently, reducing by almost the half when y¼ 0 (due to

the delocalization of electrons) and by much less when

y¼ 0.35 (due to the concomitant delocalization of electron

and heavy holes from the WL to the InGaAs capping layer).

The different dependence of 2D carrier localization on the

structure design suggests the possibility to engineer the inter-

action between QD and WL carriers, also by selecting nega-

tive or positive 2D charge carriers to electrically interact

with QD carriers. This is a very interesting example of how

the metamorphic system is a flexible system, with a wider

degree of engineering possibilities with respect to the

pseudomorphic system, for advanced design of structures for

single photon sources. As showed above, these possibilities

include: (i) engineering of WL carrier spatial localization in

structures for single photon emission, to reduce the Coulomb

interactions with QD confined carriers, (ii) separation of

electrons and heavy holes in the 2D WL system, thus reduc-

ing the radiative recombination processes and influencing

the QD carrier dynamics, and (iii) find conditions for strong

electronic coupling between QD-WL states of great interest

for single photon devices. In structures where carriers are

electrically driven into QDs, the 0D–2D system could be

switched on/off by simple application of external bias.

Hence, from these results, it can be concluded that the meta-

morphic QD system is not only a viable approach to obtain

single photon emission in the 1.3–1.55 lm range, but can

provide useful tools to control electronic interactions

between 0D and 2D carriers.

IV. CONCLUSIONS

In conclusion, we developed a simple model for the

calculation of quantum 2D states in metamorphic buffers

and applied it to the case of InAs WL in metamorphic

InAs/InGaAs QD nanostructures. The model was validated

against experimental data from different research groups,

for structures grown in the Stranski-Krastanow regime and

with sub-critical coverages. The analysis of model results

highlighted the unique properties of metamorphic system of

having two independent design parameters, namely the MB

thickness and composition that determine the MB-WL mis-

match. The effect of the change of mismatch and of MB

composition on WL properties was considered, allowing to

reach interesting conclusions.

It was confirmed that the previously reported WL disap-

pearance in InAs/InGaAs system for high x/high t is an

extrinsic effect, probably related to the surface roughness on

which InAs is deposited.

For structures with a low density of QDs, the effect of the

change in composition of the capping layer on the spatial deloc-

alization of 2D carriers (that influence the Coulomb interaction

between QD and WL carriers) and on the electron-heavy hole

spatial separation in the QW-WL system (that determines the

probability of radiative recombination) was studied and allowed

to draw interesting conclusions. In metamorphic InAs/InGaAs

system, it could be possible to expel both carriers from the WL,

a feature not obtainable in pseudomorphic GaAs-based struc-

tures, where only the electron can be delocalized from the WL.

Moreover, model calculations suggest that in these systems, it

could be possible to have QD and WL levels resonate in energy

(at least for a fraction of the QD ensemble). On the basis of

these model results, it seems viable to propose the use of meta-

morphic InAs/InGaAs QDs as: (i) a possible system to have

hybrid 2D–0D quantum levels and (ii) a scheme for QD-based

single photon sources where the interaction between WL and

QD carrier can be engineered. More in general, one could think

at the metamorphic approach as a system that allows to engi-

neer two properties of the QW system: this could be a feature

not limited to QD structures, but it could also have applications

in other metamorphic 2-dimensional structures (QW lasers,

FIG. 7. (Left handside) Overlap of ground electron and heavy hole wave-

functions in WL-QW levels as function of y from model calculations

(crosses). (Right handside) Probability of finding a carrier within 1 nm of the

WL as functions of y from model calculations for heavy holes (circles) and

electrons (squares). Lines are guides for the eye.

184309-7 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)

high mobility devices, such as MHEMT, semiconductor optical

amplifiers, and QW solar cells).

1W. Guter, J. Schoene, S. P. Philipps, M. Steiner, G. Siefer, A. Wekkeli, E.

Welser, E. Oliva, A. W. Bett, and F. Dimroth, Appl. Phys. Lett. 94,

223504 (2009).2J. S. Gandhi, C. U. Kim, and W. P. Kirk, in Conference Record of theIEEE Photovoltaic Specialists Conference (2012), p. 789.

3I. Tangring, H. Ni, B. P. Wu, D. H. Wu, Y. Xiong, S. Huang, Z. Niu, S. M.

Wang, Z. H. Lai, and A. Larsson, Appl. Phys. Lett. 91, 221101 (2007).4Y. Xin, L. Vaughn, L. R. Dawson, A. Stintz, Y. Lin, L. F. Lester, and

D. L. Huffaker, J. Appl. Phys. 94, 2133 (2003).5S.-G. Ihn, S. J. Jo, and J.-I. Song, Appl. Phys. Lett. 88, 132108 (2006).6J. A. Czaban and D. A. Thompson, Nanotechnology 21, 134005 (2010).7C. P. Jiang, Z. M. Huang, S. L. Guo, J. H. Chu, L. J. Cui, Y. P. Zeng, Z. P.

Zhu, and B. Q. Wang, Appl. Phys. Lett. 79, 1909 (2001).8M. Boudrissa, E. Delos, C. Gaquiere, M. Rousseau, Y. Cordiere, D. Theron,

and J. C. De Jaeger, IEEE Trans. Electron Devices 48, 1037 (2001).9L. Seravalli, P. Frigeri, L. Nasi, G. Trevisi, and C. Bocchi, J. Appl. Phys.

108, 064324 (2010).10G. Balakrishnan, S. Huang, T. J. Rotter, A. Stintz, L. R. Dawson, K. J.

Malloy, H. Xu, and D. L. Huffaker, Appl. Phys. Lett. 84, 2058 (2004).11Z. Mi and P. Bhattacharya, J. Appl. Phys. 98, 023510 (2005).12L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, Appl. Phys. Lett. 92,

213104 (2008).13L. Seravalli, M. Minelli, P. Frigeri, S. Franchi, G. Guizzetti, M. Patrini, T.

Ciabattoni, and M. Geddo, J. Appl. Phys. 101, 024313 (2007).14L. Seravalli, P. Frigeri, M. Minelli, P. Allegri, V. Avanzini, and S.

Franchi, Appl. Phys. Lett. 87, 063101 (2005).15M. Hugues, M. Teisseire, J.-M. Chauveau, B. Vinter, B. Damilano, J.-Y.

Duboz, and J. Massies, Phys. Rev. B 76, 075335 (2007).16C. Cornet, C. Platz, P. Caroff, J. Even, C. Labb�e, H. Folliot, A. Le Corre,

and S. Loualiche, Phys. Rev. B 72, 35342 (2005).17S. Lee, O. L. Lazarenkova, P. von Allmen, F. Oyafuso, and G. Klimeck,

Phys. Rev. B 70, 125307 (2004).18S. Sanguinetti, D. Colombo, M. Guzzi, E. Grilli, M. Gurioli, L. Seravalli,

P. Frigeri, and S. Franchi, Phys. Rev. B 74, 205302 (2006).19G. Gelinas, A. Lanacer, R. Leonelli, R. A. Masut, and P. J. Poole, Phys.

Rev. B 81, 235426 (2010).20M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E.

L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoglu, Phys. Rev.

Lett. 103, 207403 (2009).21N. Chauvin, C. Zinoni, M. Francardi, A. Gerardino, L. Balet, B. Alloing,

L. H. Li, and A. Fiore, Phys. Rev. B 80, 241306 (2009).22Y. Chen and J. Washburn, Phys. Rev. Lett. 77, 4046 (1996).23G. SeRk, K. Ryczko, M. Motyka, J. Andrzejewski, K. Wysocka, J. Misiewicz,

L. Li, A. Fiore, and G. Patriarche, J. Appl. Phys. 101, 063539 (2007).24H. Song, T. Usuki, Y. Nakata, N. Yokoyama, H. Sasakura, and S. Muto,

Phys. Rev. B 73, 115327 (2006).25L. Seravalli, G. Trevisi, and P. Frigeri, Cryst. Eng. Commun. 14, 6833 (2012).26See www.tibercad.org for TiberLab.

27L. Seravalli, C. Bocchi, G. Trevisi, and P. Frigeri, J. Appl. Phys. 108,

114313 (2010).28P. Maree, J. Barbour, J. Vanderveen, K. Kavanagh, C. Bullelieuwma, and

M. Viegers, J. Appl. Phys. 62, 4413 (1987).29M. Geddo, G. Guizzetti, M. Patrini, T. Ciabattoni, L. Seravalli, P. Frigeri,

and S. Franchi, Appl. Phys. Lett. 87, 263120 (2005).30V. Bellani, C. Bocchi, T. Ciabattoni, S. Franchi, P. Frigeri, P. Galinetto,

M. Geddo, F. Germini, G. Guizzetti, L. Nasi, M. Patrini, L. Seravalli, and

G. Trevisi, Eur. Phys. J. B 56, 217 (2007).31F. Pollak and M. Cardona, Phys. Rev. 172, 816 (1968).32P. Wang, N. Ledentsov, C. Sotomayor Torres, I. Yassievich, A.

Pakhomov, A. Egovov, P. Kop’ev, and V. M. Ustinov, Phys. Rev. B 50,

1604 (1994).33I. Vurgaftman, J. Meyer, and L. Ram-Mohan, J. Appl. Phys. 89, 5815

(2001).34See supplemental material at http://dx.doi.org/10.1063/1.4830021 for ma-

terial parameters and experimental details.35L. Seravalli, G. Trevisi, and P. Frigeri, Cryst. Eng. Commun. 14, 1155

(2012).36E. Y. Lin, C. Y. Chen, T. E. Tzeng, S. L. Chen, D. J. Y. Feng, and T. S.

Lay, Physica E 42, 2544 (2010).37N. V. Kryzhanovskaya, A. Y. Gladyschev, S. Blokhin, Y. G. Musikhin,

A. E. Zhukov, M. V. Maksimov, N. Zakharov, A. Tsatsul’nikov, N. N.

Ledentsov, P. Werner, F. Guffarth, and D. Bimberg, Semiconductors 38,

833 (2004).38L. Seravalli, G. Trevisi, P. Frigeri, F. Rossi, E. Buffagni, and C. Ferrari,

J. Phys. D: Appl. Phys. 46, 315101 (2013).39J. Ib�a~nez, R. Cusc�o, S. Hern�andez, L. Art�us, M. Henini, A. Patane, L.

Eaves, M. Roy, and P. A. Maksym, J. Appl. Phys. 99, 043501 (2006).40C. Heyn, A. Bolz, T. Maltezopoulos, R. Johnson, and W. Hansen, J. Cryst.

Growth 278, 46 (2005).41D. Litvinov, D. Gerthsen, A. Rosenauer, M. Schowalter, T. Passow, P.

Fein€augle, and M. Hetterich, Phys. Rev. B 74, 165306 (2006).42R. R. Pel�a, L. K. Teles, M. Marques, and S. Martini, J. Appl. Phys. 113,

033515 (2013).43L. Seravalli, G. Trevisi, P. Frigeri, S. Franchi, M. Geddo, and G. Guizzetti,

Nanotechnology 20, 275703 (2009).44L. Seravalli, G. Trevisi, P. Frigeri, R. J. Royce, and D. J. Mowbray,

J. Appl. Phys. 112, 034309 (2012).45E. S. Semenova, A. E. Zhukov, S. Mikhrin, A. Y. Egorov, V. A.

Odnoblyudov, A. P. Vasil Ev, E. V. Nikitina, A. R. Kovsh, N. V.

Kryzhanovskaya, A. Y. Gladyschev, S. A. Blokhin, Y. G. Musikhin, M. V.

Maksimov, Y. M. Shernyakov, V. M. Ustinov, and N. N. Ledentsov,

Nanotechnology 15, S283 (2004).46Y. I. Mazur, V. G. Dorogan, M. E. Ware, E. Marega, Jr., M. Benamara,

Z. Y. Zhuchenko, G. G. Tarasov, C. Lienau, and G. J. Salamo, Nanoscale

4, 7509 (2012).47V. G. Dorogan, Y. I. Mazur, E. Marega, G. G. Tarasov, M. E. Ware, and

G. J. Salamo, J. Appl. Phys. 105, 124304 (2009).48M. Syperek, J. Andrzejewski, W. Rudno-Rudzi�nski, G. SeRk, J. Misiewicz,

E.-M. Pavelescu, C. Gilfert, and J. P. Reithmaier, Phys. Rev. B 85, 125311

(2012).

184309-8 Seravalli, Trevisi, and Frigeri J. Appl. Phys. 114, 184309 (2013)