Another Origin of Bimodal Size Distribution in InAs Self-Assembled Quantum Dots
Bhavtosh Bansal$^*$
M. R. Gokhale
A. Bhattacharya and B. M. Arora
The evolution of InAs quantum dots grown on InP substrates by metal-organic vapour phase %%@ epitaxy is studied as a function of InAs coverage. Under specific growth conditions, the %%@ onset of the two- to three-dimensional transition is seen to proceed via two distinct %%@ pathways: through (i) an abrupt appearance of quantum dots as expected in the usual %%@ Stranski-Krastanov growth picture and (ii) a continuous evolution of small surface %%@ features into well developed quantum dots. The average size of the features in both these %%@ families increases with coverage, leading to a bimodal distribution in dot sizes at an %%@ intermediate stage of growth evolution that eventually becomes a unimodal distribution as %%@ more material is deposited. Complementary information obtained from independent %%@ measurements of photoluminescence spectra and surface morphology is correlated and is %%@ found to be independently consistent with the picture of growth proposed.
qdot_coverage_final.TEX
\documentclass[twocolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
\usepackage{graphicx}
\usepackage{dcolumn}
\renewcommand{\thefootnote}{\alph{footnote})}
\usepackage{bm}% bold math
\begin{document}
\title{Another Origin of Bimodal Size Distribution in InAs Self-Assembled Quantum Dots}
\author{Bhavtosh Bansal$^*$ , M. R. Gokhale, A. Bhattacharya and B. M. Arora}
\affiliation{Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai-400005,
India}
{
\date{\today}
\begin{abstract}
The evolution of InAs quantum dots grown on InP substrates by metal-organic vapour phase %%@
epitaxy is studied as a function of InAs coverage. Under specific growth conditions, the %%@
onset of the two- to three-dimensional transition is seen to proceed via two distinct %%@
pathways: through (i) an abrupt appearance of quantum dots as expected in the usual %%@
Stranski-Krastanov growth picture and (ii) a continuous evolution of small surface %%@
features into well developed quantum dots.  The average size of the features in both these %%@
families increases with coverage, leading to a bimodal distribution in dot sizes at an %%@
intermediate stage of growth evolution that eventually becomes a unimodal distribution as %%@
more material is deposited. Complementary information obtained from independent %%@
measurements of photoluminescence spectra and surface morphology is correlated and is %%@
found to be independently consistent with the picture of growth proposed.  
\end{abstract}
\pacs{68.65.Hb, 78.67.Hc, 81.07.Ta, 68.37.Ps, 81.15.Gh}
\maketitle
The semiconductor heterostructures grown via the strain-mediated Stranski-Krastanov (S-K) %%@
route\cite{stangl, joyce review} have been of special interest. The resulting self %%@
assembled clusters are extremely small ($\sim$10nm, hence called quantum dots), coherently %%@
strained (hence optically active) and have a size dispersion that may be acceptable for %%@
their use in real optoelectronic devices, e.g., \cite{laser refs, detectors}. The S-K %%@
transition has been systematically studied in most III-V semiconductors of interest (e.g. %%@
InAs/GaAs \cite{petroff_prb, {Ramachandran}, {patella}, da silva, duarte}, %%@
In$_{1-x}$Ga$_{x}$As/GaAs\cite{leon-fafard, cullis}, InAs/InP\cite{ponchet}, %%@
GaSb/GaAs\cite{muller-kirsch}). While the growth route may be rationalized by energetic %%@
considerations that predict an initial two-dimensional growth followed by an abrupt %%@
appearance of three-dimensional clusters upon the `wetting layer'\cite{stangl}, the actual %%@
process of self-assembly is complicated by kinetic effects, substrate conditions and %%@
orientation, a non-quiescent wetting layer during growth evolution, alloying and %%@
long-ranged substrate-mediated elastic interactions. For example, it has also been %%@
observed, though not consistently by different groups, that there may be redistribution of %%@
matter leading to a decrease in the quantum dot (QD) size with coverage. Often a bimodal %%@
size distribution \cite{petroff_prb, ponchet} has also been observed at an intermediate %%@
growth stage. Although many of these phenomena have been described in terms of elastic %%@
energy barriers corresponding to change in shape and a maximum permissible size for %%@
quantum dots beyond which they are dislocated, the complex growth evolution of %%@
self-assembled dots, especially under far from equilibrium (usual) growth conditions,  is %%@
still very much an open problem despite more than a decade of extensive work. 

In this paper, we focus on InAs/InP self-assembled QDs grown by metal-organic vapour phase %%@
epitaxy (MOVPE)\cite{movpe refs1, movpe refs2}. The samples are studied at different %%@
stages of evolution by independently studying and quantitatively correlating the %%@
morphology and the optical emission spectra. Contrary to many of previous reports, we %%@
have, on samples grown under more non-equilibrium conditions (lower temperature, %%@
450$^\circ$C and higher growth rate $\sim$1-2 monolayers(ML) per second) where surface %%@
kinetics may be an important limiting step, observed that the 2D-3D transition is not %%@
completely abrupt and proceeds via two different pathways. This study also yields a %%@
different picture for bimodality in the QD's size distribution.

Growth was carried out using low pressure (100 torr) MOVPE on n+ doped (001) ``epi-ready" %%@
InP substrates in a horizontal reactor with hydrogen as the carrier gas. Group III and V %%@
sources were trimethyl-indium and arsine and phosphine respectively. Prior to InAs growth, %%@
a InP buffer layer was grown first at 625$^\circ$C and then with the temperature %%@
continuously ramped down and stabilized to 450$^\circ$C. InAs layers were grown at a %%@
relatively low temperature of 450$^\circ$C at a growth rate\cite{footnote growth rate} of %%@
approximately 1.8ML/s. A pair of samples was grown with identically deposited InAs layer %%@
in two growth runs. In the first case, the sample was taken out of the reactor after InAs %%@
deposition itself to enable a study of surface morphology and in the second case a InP cap %%@
layer was grown for samples used for photoluminescence study. For these samples, about 50 %%@
\AA$\,$  InP was deposited at the InAs deposition temperature to avoid any further %%@
ripening during the higher temperature overgrowth. Surface features were studied ex-situ %%@
using an atomic force microscope (AFM) in contact mode, typically within a few hours of %%@
sample growth.  The PL spectra were measured at $\sim$25 K at low enough excitation power %%@
($\sim 0.5 W/cm^2$) to preclude any subband filling and were corrected for the system %%@
response using a standard black body source. 
\begin{figure}[!h]
\begin{center}
\resizebox{!}{10cm} {\includegraphics{afm_and_hists.eps}}
\caption{\label{fig:fig2}{\it Histogram of island heights inferred from $1\mu m \times %%@
1\mu m$ AFM images on samples with increasing InAs coverage. The coverage is %%@
approximately: (a) 4.5ML, (b) 6.5ML (c)9ML (d)14.5ML. The coverage is calibrated against %%@
the WL PL peak in Fig.\ref{fig:fig3}(a). See text. The dotted line depicts the separation %%@
of the islands into two families, evidently evolved from two different routes (see Fig.3). %%@
This leads to a bimodal size distribution (a)-(c) which merges into a single broad %%@
distribution (d).}}     
\end{center}
\end{figure}

Fig. \ref{fig:fig2} (inset) shows $1\mu m \times 1\mu m$ AFM scans for samples with %%@
progressively increasing InAs coverage in Fig. \ref{fig:fig2}(a) to (d). The histograms of %%@
the island heights corresponding to these AFM images are shown in the main %%@
Fig.\ref{fig:fig2}. All well-separated ($\sim$20nm) convex features with heights above two %%@
ML were counted here. Fig.\ref{fig:fig2} (a) depicts an early stage of growth. Here we %%@
observe that along with very few ($\sim 5\times 10^8$ cm$^{-2}$) well developed (height %%@
$>$ 8 nm) QDs, there are many small surface features of minimum height of $\sim$ 14\AA %%@
$\,$and a mean height of 26\AA $\,$contributing to a large number of counts at low %%@
heights. Furthermore, in Fig. \ref{fig:fig2} (b)-(c), we observe that  these small surface %%@
features are  stable and continuously grow in size as more material is deposited. %%@
Therefore, 
\begin{figure}[!h]
\begin{center}
\resizebox{!}{10cm} {\includegraphics{coverage_pl_log_energy.eps}}
\caption{\label{fig:fig3}{\it (Solid line) 25K PL Spectra. The growth conditions for %%@
samples whose response is depicted in (a)-(d) are the same as their counterparts in Fig. %%@
\ref{fig:fig2}, except for the InP overgrowth,.  (Squares) Calculated transition energies %%@
\cite{holm-pryor} scaled with the heights histogram (Fig. \ref{fig:fig2}) in an attempt to %%@
reconstruct the PL spectra. The expected response from only the features higher than the %%@
cut-off depicted by dotted lines in Fig. \ref{fig:fig2} is shown. Note that there is no %%@
fitting parameter. The higher energy feature is fitted to two Gaussians. (Dotted line) %%@
Wetting layer contribution to PL. PL from the wetting layer appears at 1.02$-$1.03 eV in %%@
(a)-(c) and shifts to 1.10 eV for (d). See text.  The other fit corresponding to the %%@
evolution of small surface features is not shown here for brevity. }}
\end{center}
\end{figure}
with incremental coverage, Fig.\ref{fig:fig2}(b)-(d), the morphology evolves as (i) matter %%@
is accreted by all pre-existing features making them grow in size (ii) new well developed %%@
QDs of height $\sim$ 8nm are spontaneously generated (usual S-K transition). Growth via %%@
two independent pathways naturally leads to a bimodal size distribution at an intermediate %%@
stage of growth, Fig.\ref{fig:fig2}(c). The role of existing (two- and %%@
quasi-three-dimensional) surface features during the S-K transition has been much %%@
discussed\cite{Ramachandran, patella, priester}. Here these surface corrugations are only %%@
being connected to dots which continuously evolve in size and we have not examined their %%@
role, if any, as precursors to abruptly generated well developed dots (S-K transition).  

The above mentioned picture of growth evolution is independently observed in optical %%@
properties. Fig.\ref{fig:fig3} shows the low temperature PL spectra measured on the four %%@
similarly grown but capped samples. Along with the characteristic PL from well-developed %%@
dots, the PL spectrum also shows a wetting layer peak around 1.02$-$1.03 eV that is %%@
broadened towards lower energy by the emission from these optically active %%@
quasi-three-dimensional clusters. The spectrum in Fig.\ref{fig:fig3}(c) shows almost three %%@
distinct peaks corresponding to the wetting layer and the two kinds of quantum dots. %%@
Finally, in the saturation regime (when the number of well resolved QDs does not increase %%@
with coverage), the distribution of the dot sizes is again roughly unimodal, but with a %%@
large dispersion. 

The optical signature from the wetting layer was inferred by fitting two Gaussians, %%@
(corresponding to signals from WL and quasi-three dimensional clusters) to the high energy %%@
PL peak. We observe that the wetting layer peak is constant to within $10meV$ in %%@
Fig\ref{fig:fig3}(a)-(c). This low temperature PL peak at around 1.02$-$1.03eV corresponds %%@
to an approximately 4.5 ML strained InAs/InP quantum well\cite{leonelli}. This value may %%@
be used to calibrate the thickness with the deposition time, which is otherwise difficult %%@
to estimate with sub-ML precision in a typical MOVPE set-up without in-situ diagnostic %%@
tools. It is worth pointing out that there has been a wide variation in the reported value %%@
of the wetting layer thickness for InAs/InP, with the lower limit being $\sim$1ML %%@
\cite{girard, movpe refs3}. With a lower growth temperature, the wetting layer thickness %%@
is expected to be (exponentially) larger than its equilibrium value %%@
\cite{Johansson_and_seifert_kinetics1}, since the former is inversely dependent on the %%@
diffusion length \cite{snyder-mansfield-orr}. Because 450$^\circ$C is among the lower %%@
reported growth temperatures for InAs dots, a thicker wetting layer is naturally expected. %%@
Remarkably, a recent reflection high energy electron diffraction study\cite{gendry} on %%@
2D-3D transition in MBE grown InAs/InP films, reports for growth at 450$^\circ$C, a value %%@
of the wetting layer thickness that is very similar to what we have inferred from the PL %%@
spectra.

Finally, in Fig. \ref{fig:fig3}(d), the wetting layer PL shifts to a higher energy, %%@
$\sim$1.10 eV indicating the well known narrowing of wetting layer in the saturation %%@
regime \cite{wetting-layer-thinning}. 

The observed PL spectra from QDs should in principle be derivable from the quantitative %%@
analysis of the AFM images \cite{footnote_afm_pl_scaledifference}. Using published ground %%@
state energy calculations\cite{holm-pryor} for InAs/InP QDs, the expected emission spectra %%@
from {\em large} dots are shown in Fig.\ref{fig:fig3}(open squares).  While there is a %%@
good quantitative agreement without any fitting parameter for the expected and observed %%@
spectral features from large dots (those above the size cut-off depicted by dotted lines %%@
in Fig. \ref{fig:fig2}), the PL from the smaller features in each histogram did not agree %%@
as well and is therefore not shown in Fig.\ref{fig:fig3}. It has been previously %%@
established that the capping process can considerably change the morphology of the %%@
individual quantum dots and it is likely that the smaller dots are more affected by %%@
capping\cite{morphology change on overgrowth1, morphology change on overgrowth2} and %%@
compositional changes due to the As/P exchange during the early stage of InP overgrowth. 

\noindent
{\it Implications:} 
We have observed that the 2D-3D transition is not abrupt and the bimodality in the QD's %%@
size distribution at an intermediate stage of growth is a consequence of the growth %%@
evolving via two distinct pathways. These two observations are contrary to previous %%@
studies on InAs/GaAs\cite{petroff_prb, Ramachandran} and In$_x$Ga$_{1-x}$As/GaAs %%@
\cite{leon-fafard} 
where the 2D-3D transition had been found to be abrupt and thermodynamically first-order %%@
with the surface coverage playing the role of a critical parameter\cite{petroff_prb}. An %%@
abrupt appearance of quantum dots beyond a critical coverage has also been theoretically %%@
reproduced within a rate equation based model \cite{dobbs}. 

Bimodality in dot sizes also implies that the QDs ensemble cannot be characterized by a %%@
single length scale (e.g.,  mean island size)\cite{joyce review} during most of the %%@
evolution and rules out the very attractive possibility of data collapse onto a universal %%@
scaling\cite{ebiko} function during intermediate stages of growth. Absence of scaling at %%@
an intermediate growth stage has actually been observed by Krzyzewski, et al. (e.g., see %%@
ref. \cite{stangl}). The issue of bimodality in dot sizes at intermediate coverage, %%@
although not fully resolved, is typically explained by the presence of energy %%@
barriers\cite{bimodal equilibrium} corresponding to another (shape-change, from pyramid to %%@
dome) first order transition that has been experimentally studied in SiGe/Si \cite{sige %%@
science paper},  GaN/AlN \cite{GaN/AlN} and InAs/GaAs \cite{costantini-kern}. 
\begin{figure}[!h]
\begin{center}
\resizebox{!}{6cm} {\includegraphics{coverage_graph.eps}}
\caption{\label{fig:fig4}{\it Variation of the island density with coverage for the two %%@
families of islands corresponding to the two modes in the heights distribution in Fig. 1. %%@
Solid square represents the late stage when the distribution is umimodal (Fig. %%@
\ref{fig:fig2}(d)).}}
\end{center}
\end{figure}

It is instructive to (visually) separate \cite{da silva} the heights histograms in Fig. %%@
\ref{fig:fig2} at the minima between the two modes (dotted lines in Fig.\ref{fig:fig2}). %%@
Fig.\ref{fig:fig4} is the corresponding plot of cluster densities in these two families %%@
(S-K dots and continuously evolving clusters) as a function of coverage. The increase in %%@
the density of larger dots with coverage is abrupt, whereas the number of smaller features %%@
decreases with coverage due to coalescence. At large enough coverage the distribution %%@
becomes unimodal. Without invoking the existence of energy barriers, this is most simply %%@
understood as being due to the difference in growth rates of larger and smaller dots (due %%@
to their different volumes). 

Among the various studies of growth evolution, the results in references \cite{da silva, %%@
duarte}, although on MBE grown InAs/GaAs, are qualitatively most similar to ours. %%@
Nevertheless there are some important differences both in data and interpretation of %%@
results. Firstly, the quasi-three dimensional clusters were thought to be precursors to %%@
all the quantum dots giving a common origin to the dots in both the families.
Possibly due to a smaller strain in the InAs/InP system as compared to InAs/GaAs, we %%@
observe an uninhibited increase in size and a corresponding redshift in PL with coverage.  %%@
This is in contrast with InAs/GaAs QDs, where a pronounced barrier seems to restrict the %%@
maximum dot size to $\sim$ 8nm. This supports the general observation of larger dispersion %%@
in dot sizes and low temperature PL linewidths ($\sim 100meV$) in InAs/InP. 

The differences observed in growth evolution by different groups may be a result of %%@
differing substrate/buffer layer conditions. It is possible for the local roughness in the %%@
wetting layer surface to stabilize pre-existing quasi-three dimensional surface structures %%@
and determine the extent of bimodality at an intermediate growth stage. Many of the %%@
thermodynamic arguments used to describe the QD self-assembly are relevant only under %%@
quasi-equilibrium conditions of growth which many such previous experiments tried to %%@
maintain. Our results, on the other hand, are obtained under much higher growth rates, %%@
more typical growth conditions of quantum dots. 

\noindent
{\em Conclusions}: Studying the growth evolution of MOVPE grown InAs/InP self-assembled %%@
quantum dots, we have observed that an alternate pathway for the 2D-3D transition exists %%@
that naturally explains the often observed bimodality in the quantum dots size %%@
distribution at an intermediate growth stage. Independent evaluations of the morphology %%@
and the photoluminesence spectra are consistent with the picture of growth presented.

We gratefully acknowledge J. John and Sandip Ghosh for their help with the AFM and PL %%@
measurements and Sandeep Krishna for his help with the development of the image processing %%@
software.\\  
}
\noindent
\small{$^*$Electronic Address: bhavtosh@tifr.res.in}

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cond-mat0505305.extracted_bib
\bibitem{stangl}J. Stangl, V. Holy and G. Bauer, Rev. Mod. Phys. {\bf 76}, 725 (2004)

\bibitem{joyce review}B. A. Joyce, D. D. Vvedensky, Mat. Sci. Eng. {\bf R 46} ,127 (2004).

\bibitem{detectors}H. C. Liu, M. Gao, J. McCaffrey, Z. R. Wasilewski, and S. Fafard, 
Appl. Phys. Lett. {\bf 78}, 79 (2001).

\bibitem{laser refs}Y. Qiu, D. Uhl, R. Chacon, and R. Q. Yang, 
Appl. Phys. Lett. {\bf 83}, 1704 (2003).

\bibitem{petroff_prb}D. Leonard, K. Pond, and P. M. Petroff, Phys. Rev. B {\bf 50}, 11687 %%@
(1994). 

\bibitem{Ramachandran}A. Madhukar, p26 {\it Nano-Optoelectronics} Ed. M. Grundmann %%@
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}
