Erbium arsenide nanoparticles (ErAs NPs) grown using a two-step modified diffusion length growth process are studied. The process consists of a low diffusion length (LDL) deposition step for nucleation followed by a high diffusion length (HDL) deposition step for the growth of existing nuclei without forming new nuclei. With the LDL and HDL growth conditions properly coordinated, independent control over the NP size is demonstrated over a wide range of NP densities (14–309 NP/μm2). This control is attributed to the change of Péclet number between the LDL and HDL steps, which enables complete capture of the adatoms by the existing nuclei. The appropriate LDL and HDL growth conditions are identified for practical applications of the process. Given the uniqueness of the process, the ErAs nucleus critical size (∼7 Er atoms) is derived in the nucleation regime in the LDL step exhibiting consistency with the previous observation for ErAs NP growth.

Control of the nanoparticle (NP) growth is of paramount importance for many applications such as in emitters and detectors,1 metasurfaces using plasmonics,2 or hybrid structures3 in which strong coupling between two states is desired. The NP density and size distribution capture the essence of a growth process. In a normal continuous deposition, adatoms deposited on the surface either form new nuclei or are captured by existing nuclei causing them to grow. It is quite evident that two factors cause variability in the size. One is the nonuniformity of the nuclei on the surface, which causes nuclei to capture different numbers of adatoms. The second source is the variability in time the nuclei are formed. Even if the atoms somehow had a uniform capture rate, nuclei that formed sooner would be larger than ones that formed later. In the case of quantum dot (QD) formation, a change in the geometry causes the growth rate of QDs to slow down so that the nuclei formed later can catch up in size.4 However, this approach to increase QD uniformity has only shown utility at high QD densities and does not prevent the additional growth once a random QD changes shape.

In typical growth, nuclei act as sinks for existing adatoms causing a region around them to be devoid of nuclei.5 The adatom concentration increases with distance from a nucleus and becomes maximum at the midpoint between nuclei. As the growth rate is increased, the concentration at all points increases. However, no nucleation occurs until the concentration exceeds a critical value preventing nucleation in a region around existing nuclei. Farther from the existing nuclei where the concentration is larger than the critical concentration, nucleation occurs. Nucleation continues to occur until the total area is comprised of only regions with less concentration than the critical value. This competition between nucleation and growth drives the NP density and size distribution and intimately links them together for a particular growth condition. This linkage makes it difficult to achieve a specific NP density with a particular size distribution.

Previous studies looking at the balance between deposition rate and diffusion have shown that a useful quantity to consider is the Péclet number, L2F/D, which specifies the type of growth mode depending on its magnitude.6 Here, L is the distance between nuclei (sinks), F is the flux or growth rate, and D is the diffusion coefficient. In a typical growth where F and D are fixed, the Péclet number decreases as additional nucleation occurs, which reduces the spacing between nuclei. Nucleation continues until the Péclet number is much less than 1, at which point it ceases.

Recently, we reported on a two-step modified diffusion length growth process of ErAs NP on GaAs(001) that allows separate control of the ErAs NP density and size.7 First a short duration high flux was used to produce a low diffusion length (LDL) deposition step. This was followed by a long duration low Er flux to produce a high diffusion length (HDL) deposition step. Nucleation occurs in the LDL step. In the HDL step, a key requirement is to choose conditions such that the Er-atom diffusion length prevents additional nucleation and only allows growth of preexisting NP nuclei. In this article, we extend this study to cover a much broader range of conditions and demonstrate this independent control. We further discuss our process in terms of Péclet number changing between the LDL and HDL steps. The uniqueness of this process is demonstrated by enabling the study of nucleation in the low coverage nucleation regime in the LDL step that allows us to derive the critical size of an ErAs nucleus. Furthermore, we previously reported a roughening effect, which was seen to occur during HDL growth of ErAs on GaAs.7 In this study, we discuss how this roughening can be prevented using the modified diffusion process.

The ErAs NP samples were grown on a GaAs (001) substrate in a V80H MkII MBE system with the same growth procedure as described in a previous study.7 All samples were grown using an As2 overpressure of 3.3 × 10−6 Torr. The Er diffusion cell temperature was set to 1150 and 1011.4 °C in the LDL and HDL deposition steps, respectively, producing an LDL deposition rate of 0.4 ML/min and an LDL flux (FLDL) to HDL flux (FHDL) ratio of 16.4. Note that this ratio was actually obtained from beam equivalent pressure (BEP) measurements, however, since the same species is involved in the ratio the BEP ratio and flux ratio are equivalent. We varied the LDL deposition time from 1 to 16 s to adjust the nucleation density, while keeping the HDL deposition time fixed at 1200 s. As a comparison, we further lowered the Er cell temperature to 973 °C (or 935 °C) to obtain a flux ratio of 40 (or 100) to produce an ultrahigh diffusion length for the HDL step. We used these conditions in cases where an ultralow nucleation density was present after a 1.5 s or 2 s LDL step. ErAs was deposited at a substrate temperature of 640 °C for both LDL and HDL deposition steps. As always, when the growth was changed from the LDL step to HDL step, the substrate temperature was ramped down at 50 °C/min to a lower temperature of 580 °C and was maintained under a relatively low As overpressure until the Er cell stabilized at the new temperature setting. Then, the substrate was reramped to 640 °C for the HDL Er deposition step. A 2 × 4 surface reconstruction was kept during growth indicating an As-rich growth condition. Morphological analyses were performed using cross-sectional transmission electron microscopy (TEM) and tapping mode atomic force microscopy (AFM).

The initial study was comprised of samples grown with 1–16 s LDL deposition followed by a 1200 s HDL deposition. Here, the LDL growth rate is 0.4 ML/min and the LDL to HDL flux ratio is 16.4. This yields an ErAs surface coverage of 0.007–0.107 ML corresponding to the time of deposition in the LDL step. The AFM images of a 1 × 1 μm2 areas are shown in Fig. 1 and allows a comparison of the surface morphology. We see significant changes in the morphological features as the LDL time was increased. We see roughened areas appear on the low nucleus density [1–2 s LDL corresponding to Figs. 1(a)1(c)] surface. We have previously seen that growth using the HDL conditions causes the surface to roughen.7 Our basic understanding is that the high density of nuclei [4–16 s LDL corresponding to Figs. 1(d)1(f)] capture the diffusing adatoms to keep the adatom surface concentration lower than the threshold for surface roughening, whereas for a low density of nuclei (1–2 s LDL), the separation is larger and therefore not capable of preventing the increase of the surface adatom concentration. The NP density increase seen in Figs. 1(a)1(f) is due to a higher density of nuclei with a longer LDL deposition time. The nucleus density increase has been separately confirmed by a TEM investigation performed on a sample where the 1–16 s LDL depositions were embedded, as shown in Fig. 2.

The low density of nuclei makes the adatoms unable to diffuse to the nuclei and thus drives the adatom surface concentration higher at larger distances from the nuclei, ultimately increasing it above the critical concentration. A diagram of this situation is given in Fig. 3 for a different Péclet number that changes with flux (F), diffusion coefficient (D), or nuclei separations (L). We consider situations in which growth is above and below two critical concentration, C*nuclei and C*rough, for nucleation and roughening, respectively, under conditions where D is constant. Under normal growth, the deposition rate is so large that the concentration becomes larger than the C*nuclei in the regions between nuclei. In the diagram, this region occurs between points (a) and (b). Therefore, nucleation can occur in this region. As F or L is decreased, the concentration falls below the critical value for C*nuclei and nucleation of additional nuclei is not possible. This is the HDL condition we have used in all the growths (see Fig. 1). However, in some growths, the spacing or flux is large enough that the concentration is larger than C*rough so roughening can occur in the region between (a′) and (b′). This is the case for the 1–2 s LDL depositions in Figs. 1(a)1(c). It seems this roughening process is extremely slow since it takes 20 min to deposit 0.5 ML in the HDL step in which roughening occurs. We have not seen this roughening in the case of normal nucleation and growth, indicating that the rate of additional nucleation or other factors, such as critical area, prevent this from being present.

Recall from the introduction that Péclet number, L2F/D, is a useful quantity to compare different growth condition. From the above discussion, the modified diffusion length process in a broader sense involves the change of the Péclet number from a high value in the LDL step to a value significantly less than 1 in the HDL step. Since the growth temperature and the As overpressure are fixed, the diffusion coefficient for these growths should be constant. A decrease in the growth rate by a factor of 4 would increase L by a factor of 2 at a constant Péclet number driving the growth to a more diffusive condition. This reasoning suggests lowering the growth rate or flux in the HDL step to eliminate the surface roughening in the case of low nucleus density. Here, we cut the FHDL to study its effect on surface roughening in the HDL step. Figure 4 shows the AFM images for 2 s [Fig. 4(a)] and 1.5 s [Fig. 4(b)] LDL deposition after the FHDL was cut from 1/16 to 1/40 and 1/100 of the FLDL, respectively. Comparing Figs. 1(c) and 4(a), where both have the same LDL deposition times but different FLDL:FHDL (16.4 vs 40), we see that both depositions had nearly the same NP density (39 vs 36 NP/μm2); however, the roughened area disappeared when the FHDL was cut (FLDL:FHDL ∼ 40). This indicates that all the Er adatoms are now able to diffuse to the nuclei while keeping the local concentration less than C*rough. For similar examples, one can compare Fig. 1(b) with Figs. 4(b) and 4(c), where all have 1.5 s LDL deposition but the flux ratios were both cut to 1/100. It is important to point out that by increasing HDL deposition time from 2000 s in Fig. 4(b) to 6000 s in Fig. 4(c), one can significantly change the NP sizes without disturbing the NP density. This demonstrates again the process’s capability for separate control over the NP density and size. In Fig. 5, we listed different growths using this process that show consistency in keeping NP density unchanged while increasing the deposition amount in the HDL step. Note that we have varied the deposition time as well as the condition of LDL step to achieve different NP densities, and our process allows us to decouple the NP density and size growth in a very wide range from 14 to 309 NPs per μm2.

In order to apply the modified diffusion length, one must be able to adjust the conditions such that no additional nucleation occurs beyond the LDL step. The quantity L2F/D mentioned in the introduction shows that several different factors can be changed to influence growth. Explicitly, L can be tuned by the ratio of D/FHDL. Lowering FHDL is a simple way to enhance the adatom diffusion length. However, the diffusion coefficient, D, is a more comprehensive parameter that links many different factors involved in growth. D can be influenced by the substrate temperature, surfactants, and anion/cation flux ratios, indicating that the modified diffusion process for the growth of NPs should be quite adaptable to many situations.

A critical step for the practical implementation of the two-step modified diffusion length process is the determination of the HDL condition, which prevents additional nucleation for given LDL step. It is clear that any HDL condition that causes surface roughening as opposed to nucleation satisfies the criteria for no additional nucleation. However, in general, when surface roughening does not occur but nucleation is still present, determining an appropriate HDL condition is more complicated. We present two procedures to accomplish this. The first procedure involves the comparison of the steady state density (also the maximum density) achieved using an HDL only growth with the NP density produced in the LDL step. In this case, an HDL condition is appropriate for any LDL step provided the HDL NP density is lower than the corresponding LDL NP density. As a second procedure, one could perform an LDL deposition and then do multiple growths using a particular proposed HDL condition and see if the NP density changes. If it remains the same, the HDL condition is appropriate. If NP density changes, then the proposed HDL condition needs to be modified to lower the density. This could be accomplished by lowering FHDL, increasing the growth temperature or applying an appropriate surfactant.

In practice, the LDL condition should have a steady state nucleation density significantly higher than the HDL value to keep the LDL nuclei small in the LDL step, thereby having the size of the NPs being predominantly determined by the HDL step. Moreover, the overall inhomogeneity of the NPs should be improved compared to single step growth. Nuclei size variation in our case should be limited to the initial size variation in the small nuclei in the LDL step and the growth rate variation in the HDL step due to different distances between the nuclei. While in a single flux growth, the nuclei form at different times and then begin to grow, small NPs grow slower than big NPs due to difference in capture cross sections, which aggravate the inhomogeneity. We note that when a high flux is used in the LDL step, a high nucleation rate may compromise the spatial homogeneity of the NPs, which may in turn impact the size homogeneity. However, it seems that by comparing the changes in the size uniformity with different deposition conditions, these two effects could be studied.

It is important to point out the significance of the two-step modified diffusion length growth process for studying NP nucleation. Since the HDL deposition step only enlarges the preexisting nuclei without forming new nuclei, the NP density reflects the preexisting nucleus density formed by the LDL deposition step. Figure 6 is the plot of NP density versus the LDL deposition time. The NP density appears very low for 1 s LDL deposition, 0.4 NP/μm2, following roughly an increment of 33 NP for every additional second up to 8 s and reaches 388 NP/μm2 for 16 s. According to the point island growth model,8,9 in the low coverage nucleation regime, the island density Nisl is low and its dependence on the coverage θ follows Nisl ∼ θi+2. Fitting the first two data points (1 and 1.5 s) with a power law dependent equation gives i ≈ 7, suggesting that the critical size ErAs nucleus has nearly two unit cells (four atoms in a unit cell).10,11 It is very interesting to note that the two-unit-cell ErAs nucleus quantitatively agrees with the observation by Schultz and Palmstrøm12 that an ErAs nucleus needs 3–4 ML in height to attain a rock salt structure. Two vertical aligned ErAs unit cells equal exactly 4 ML. Without the HDL growth, finding these nuclei comprised of two unit cells, presumably half-buried in the case of ErAs would be virtually impossible. In our case with 1 or 1.5 s deposition, there is 1 or 14 nuclei in a 1 μm2 area. With the additional HDL growth, they are easily visible, enabling the study of early nucleation in the LDL step.

We have systematically investigated ErAs NP growth using the two-step modified diffusion length process involving a LDL step that controls nucleation density and a HDL step that controls NP size while preserving the NP density. By properly adjusting the HDL flux, we were able to mitigate the surface roughening observed when the NP density is low. An independent control over the NP size over a wide range of NP density was established. A critical island size of seven ErAs atoms using the LDL conditions was estimated, which demonstrates that this process opens up the possibility for studying nucleation in the early stage. We discuss the possibility of using other conditions, such as temperature and flux, to modify the HDL step and point out two methods to determine the FHDL to prevent additional nucleation after the LDL step. The process outperforms conventional single step growth process by isolating the nucleation and growth process and has high adaptability owing to a wide parameter space during the growth.

This material is based on the work supported by the Air Force Office of Scientific Research under Award No. FA9550-17RXCOR427 managed by Gernot Pomrenke.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
Wei
Zhang
,
Michael
Saliba
,
Samuel D.
Stranks
,
Yao
Sun
,
Xian
Shi
,
Ulrich
Wiesner
, and
Henry J.
Snaith
,
Nano Lett.
13
,
4505
(
2013
).
2.
N.
Meinzer
,
W.
Barnes
, and
I.
Hooper
,
Nat. Photonics
8
,
889
(
2014
).
3.
Frank
Szmulowicz
and
Kurt G.
Eyink
,
J. Nanophotonics
9
,
093078
(
2015
).
4.
I.
Mukhametzhanov
,
Z.
Wei
,
R.
Heitz
, and
A.
Madhukar
,
Appl. Phys. Lett.
75
,
85
(
1999
).
5.
J. W.
Evans
,
P. A.
Thiel
, and
M. C.
Bartelt
,
Surf. Sci. Rep.
61
,
1
(
2006
).
6.
Jeffrey Y.
Tsao
,
Materials Fundamentals of Molecular Beam Epitaxy
(
Academic
,
London
,
1993
), Chap. 6.
7.
Yuanchang
Zhang
,
Kurt G.
Eyink
,
Madelyn
Hill
,
Brittany
Urwin
, and
Krishnamurthy
Mahalingam
,
Thin Solid Films
692
,
137586
(
2019
).
8.
Yong
Han
,
Émilie
Gaudry
,
Tiago J.
Oliveira
, and
James W.
Evans
,
J. Chem. Phys.
145
,
211904
(
2016
).
9.
Jacques G.
Amar
,
Fereydoon
Family
, and
Pui-Man
Lam
,
Phys. Rev. B
50
,
8781
(
1994
).
10.
Harald
Brune
,
Holger
Röder
,
Corrado
Boragno
, and
Klaus
Kern
,
Phys. Rev. Lett.
73
,
1955
(
1994
).
11.
Ivo
Doudevski
,
William A.
Hayes
, and
Daniel K.
Schwartz
,
Phys. Rev. Lett.
81
,
4927
(
1998
).
12.
B. D.
Schultz
and
C. J.
Palmstrøm
,
Phys. Rev. B
73
,
241407
(
2006
).