Nanowire growth and sublimation: CdTe quantum dots in ZnTe nanowires
The role of the sublimation of the compound and of the evaporation of the constituents from the gold nanoparticle during the growth of semiconductor nanowires is exemplified with CdTe-ZnTe heterostructures. Operating close to the upper temperature limit strongly reduces the amount of Cd present in the gold nanoparticle and the density of adatoms on the nanowire sidewalls. As a result, the growth rate is small and strongly temperature dependent, but a good control of the growth conditions allows the incorporation of quantum dots in nanowires with sharp interfaces and adjustable shape, and it minimizes the radial growth and the subsequent formation of additional CdTe clusters on the nanowire sidewalls, as confirmed by photoluminescence. Uncapped CdTe segments dissolve into the gold nanoparticle when interrupting the flux, giving rise to a bulb-like (pendant-droplet) shape attributed to the Kirkendall effect.
Fabricating semiconductor quantum dots (QD) embedded in a nanowire (NW) constitutes a more flexible process than the usual Stranski-Krastanow growth mode: it permits to combine various materials with various strain configurations, and adjustable shape and size. One interest of adjusting the aspect ratio and the strain configuration Niquet (); Zielinski () is that it allows one to shape the hole state (”heavy hole” or ”light hole”), without involving complex structuresHuo (). This has a strong impact on the optical selection rules: emission diagram for classical photonic devices, photovoltaics, and quantum devices for optical manipulation of the electron or hole state in a doped quantum dot - these aspects are particularly attractive for III-V and II-VI material combinations. It also governs the spin properties.
The growth of NWs by molecular beam epitaxy (MBE) is generally achieved thanks to a nano-sized catalyst, most often a gold droplet (the so-called vapor-liquid-solid growth mode, or VLS): the elements constituting the NW are trapped in the droplet where the nucleation to form the semiconductor is more efficient than in the neighboring vapor phase. The QD is inserted by switching the impinging flux for a certain time, which controls the QD height, while the diameter of the contact area between the catalyst and the NW controls the diameter of the QD. For a good control of the QD shape, sharp interfaces are needed, meaning that the problem of ”reservoir effect” (the progressive change of composition of the droplet when switching the flux) has to be circumvented. Various solutions were proposed. Silicon-germanium axial heterostructures were fabricated using a solid catalyst instead of a liquid one (vapor-solid-solid growth, or VSS, instead of VLS): the solubility is expected to be lower Wen () and a model was proposed which assumes that surface and interface diffusion takes place instead of diffusion through the catalyst nanoparticle Cui (). In III-V heterostructures, the reservoir effect is small, a few %, when dealing with column-V atoms Priante2015 (); Johansson2017 () but the residual pressure in the MBE chamber has to be mastered. The reservoir effect is more severe when dealing with column-III atoms, and elaborate sequences were designed to make it as small as possible Dick2012 (); Priante2016 ().
We address the growth of QDs embedded in NWs made of II-VI semiconductors. Several aspects make such structures particularly attractive. Light-hole emission was reported in such CdTe QDs in ZnTe NWs Jeannin (). Single photon emission was observed up to room temperature in CdSe QDs in ZnSe NWs Bounouar (). II-VI QDs in NWs can incorporate a dilute magnetic semiconductorSzymura (): in that case one can expect the formation of a magnetic polaron around a hole trapped in the QD, which constitutes a model system for ultimate magnetic objects where the superparamagnetic state is induced by one or a small number of carriers, so that it can be controlled by a low applied bias. Again, the anisotropy is governed by the hole state - a feature reminiscent of carrier induced ferromagnetism in a quantum well. Finally, CdTe NWs have been used recently Kessel () as a template for the growth of HgTe NWs in the quest for 1D topological insulators.
It has been recognized for some time that the growth window for CdTe NWs, using MBE with a gold catalyst, is extremely narrow, with a high temperature limit attributed to the sublimation of CdTe Wojnar2017 (). Indeed a dramatic drop of the growth rate is observed when the temperature is increased by a few degrees above C. Our goal here is to make this observation more quantitative and to unravel the role played in the growth of CdTe by the sublimation of CdTe and the evaporation of Cd from the gold catalyst, and the consequences for the properties of CdTe QDs.
Ii Experimental methods and samples
ii.1 Growth by molecular beam epitaxy
The details of the conditions used for the sample preparation and the growth of NWs have been reported previouslyRueda growth (). The growth was achieved by MBE in a Riber 32 chamber, using solid catalyst particles formed by dewetting a fraction of a monolayer of gold deposited on a ZnTe buffer on a GaAs substrate. All fluxes were calibrated prior to growth by measuring the RHEED oscillations on a ZnTe or CdTe (001) test substrate: for instance the Zn flux (and the Te flux) from the ZnTe cell is measured by the growth rate of a ZnTe layer, in ML s, with an additional Zn flux from a Zn cell Arnoult (). The sample holder was aligned on the horizontal line containing the CdTe cell (angle of incidence in the horizontal plane, ) while our ZnTe cell is on the other cell line ().
ii.2 EDX and TEM
We used energy dispersive x-ray (EDX) spectrometry coupled to a FEI Tecnai Osiris scanning transmission electron microscope (TEM) equipped with four Silicon Drift Detectors and operated at 200 kV. The NWs were removed mechanically from the as-grown samples and deposited on a holey carbon-coated copper grid. The EDX signal is an hypermap where each pixel corresponds to the x-ray emission of atoms along the electron beam path. We used the QUANTAX-800 software from Bruker for background correction and deconvolution to extract the contributions of the L lines of Te, Cd, Au, and K lines of Zn and O. The absorption correction - for the typical size of the NWs - was estimated to be negligible for Te, Zn, Cd and less than 10% for O. The cross-sections for each element are deduced from the so-called -factors directly measured on our equipment at the same operating conditions using reference samples of known composition and thickness Lopez-Haro ().
High angle annular dark field (HAADF) high resolution scanning TEM images were realized on a probe corrected FEI Titan Themis operated at 200 kV.
In this paper we discuss three samples grown by MBE on a ZnTe buffer layer, in the VSS mode. They incorporate a series of CdTe-based insertions, see Fig. 1a, in a micrometer-long ZnTe NW: sample I to test the CdTe-ZnTe interfaces, sample T to test the effect of growth temperature, and sample R to measure the growth rate by testing the effect of the duration of the CdTe pulse.
The three samples are:
Sample T (temperature) involves a series of CdTe pulses of identical duration, but the temperature was decreased by steps of C at the middle of each ZnTe sequence, down to C for the last insertion.
Sample R (rate) was designed to measure the growth rate by testing the effect of the duration of the CdTe pulse: the duration was increased by a factor 2 for each insertion; the nominal growth temperature was C.
The last sequence was ZnTe for sample I and CdTe for sample T and R. The position of the insertions along the NW is such that the growth of the CdTe insertion and of the surrounding shells is essentially due to the flux impinging onto the catalyst and onto the sidewalls of the NW, with no contribution of adatoms diffusing from the substrate Rueda growth (). In all cases we observe both zinc-blende NWs, with a cone shape, and wurtzite NWs, with a cylinder shape Rueda2014 (). The local composition has been determined by a quantitative modeling Rueda2016 () of EDX spectrometry, and by the geometrical phase analysis (GPA) hytchGPA (); rouviereGPA () of TEM images.
Section III describes the experimental results obtained on these three samples. Based on these results, section IV proposes a modified version of the model previously used to describe the growth of ZnTe NWs Rueda growth (): we incorporate the sublimation of CdTe and the evaporation of Cd from the gold nanoparticle. Finally section V discusses the consequences for the axial and radial growth rates and the influence on the luminescence properties of a CdTe dot in ZnTe NW; it also describes the impact on the nanoparticle, which acquires a bulb-like shape which we attribute to the Kirkendall effect induced by a redissolution of CdTe.
Iii Experimental results
Figure 1 displays the result of the EDX study of one NW from sample I. The profile (Fig. 1e) is not symmetrical and features a tail towards the NW tip; the Cd content of CdZnTe at position can be fitted with two interfaces separated by a distance , , each interface being broadened by a Gaussian of width and an exponential of length extending along the growth direction towards the tip. While a more sophisticated model of the interface could be elaborated Priante2015 (); Priante2016 (), as suggested by the different growth rates of CdTe and ZnTe and the different values of their formation energy, it would require an accurate knowledge of the thermodynamics of Cd, Zn and Te in the gold nanoparticle and this is beyond the scope of the present study. Going on with the simple hypothethis of an exponential tail, the convoluted profile is obtained by a straightforward calculation as
Figure 1d shows two examples of such a profile, the one which provides a good fit to the measured EDX profile (Fig. 1e), and the other one with a ten-times thicker insertion, which allows a better identification of the role of the exponential tails.
Finally, Fig. 1c and 1e show that the same individual profile, with the same parameters, well describes the overall axial profile of the NW, provided we take into account the presence of the radial ZnTe-CdTe multishell structure, as schematized in the bottom part of Fig. 1a. This multishell structure is due to lateral growth: for instance, during the growth of a CdTe insertion, Cd adatoms which are present on the NW sidewalls and do not reach the gold nanoparticle may induce some lateral growth of CdTe and form a CdTe shell over the previously grown sections of the NW; of course this shell will be absent from the sections grown subsequently. In a symmetrical way, a ZnTe shell is formed during the growth of each ZnTe section. The resulting structure, with an increasing number of individual shells from the NW tip to its base, is schematically displayed at the bottom of Fig. 1a, where the lateral growth has been exaggerated. The EDX analysis provides the average Cd content (with respect to Cd+Zn) averaged over the electron beam path. The presence of Zn shells creates the tapering and this additional Zn content along the electron beam path decreases the signal measured at each insertion in Fig. 1c; the fit assumes a factor of 1.6 between the diameter along the electron beam path at the basis and at the tip of the NW. The presence of Cd shells slightly contributes to the tapering, and creates the rising background between the insertions in Fig. 1c: the fit assumes that a CdTe shell of thickness 0.2 nm (half a monolayer) is added by the growth of each insertion.
This NW from sample I had been transferred onto a grid for the EDX study and it may have been broken during the harvesting process. Other NWs were observed by TEM on a cleaved edge Rueda growth () so that we are sure that they grew perpendicular to the substrate, and that they are complete although their basis is hidden by the 2D layer grown at the same time as the NWs. Profiles were obtained by GPA: GPA measures the crystal plane distance, also averaged over the electron beam path, with respect to its value measured in the ZnTe sequence. An example is displayed in Fig. 1f. A good fit of the experimental profile is obtained with slightly lower values of the parameters, and nm and nm. The larger values of the parameters measured by EDX are probably partly of instrumental origin (GPA requires a perfect alignment of the NW in the TEM, EDX does not); they may be due also to a slight angle of the NW axis with respect to the substrate normal during the growth: notice the rather strong tapering, and the asymmetric shell surrounding the CdTe insertions. We conclude that this kind of profile is validated by EDX and GPA, and it will be used in the following.
Figure 2 displays the results of the EDX study of a NW from sample T, with CdTe insertions grown at different temperatures: TEM image, EDX map, axial EDX profiles for Cd and Zn (total=1), and its fit with Eq. 1, nm , nm and different values of . The CdTe pulse was the same for all insertions, about 4 times larger than in sample I. It is clear that the CdTe dot grown at C is much longer than the dot at , while the dot at almost disappeared. This trend was confirmed on 6 other NWs from sample T, with either wurtzite or zinc-blende structure, with the results gathered in Fig. 2c. Two additional NWs in the series are long enough to confirm that dots grown at C are beyond the detection limit by EDX.
Figure 3 displays the results obtained on sample R, where the length of the CdTe pulse was progressively increased. The EDX map (Fig. 3a) indeed suggests that the length of the CdTe insertion correspondingly increases, but for the last, uncapped insertion. The axial profile (Fig. 3b) and the plot in Fig. 3c confirm the length-duration dependance.
The radial profiles (Fig. 3d) reveal interesting features. Summing all contributions, we can plot the local thickness of the NW, which is best fitted assuming an hexagonal cross section. In the analysisRueda2016 (), the fitting parameters are the overall orientation of the facets with respect to the electron beam (the vertical axis in the schemes of Fig. 3d, which was not oriented with respect to the crystal structure in the NW) and the distance between opposite facets. Profile d(2) reveals a ZnTe core with a shell enriched in CdTe; the thickness of the shell is difficult to ascertain, but the total amount of CdTe corresponds to less than a monolayer of pure CdTe. Profile d(1) points to an almost pure CdTe core with a thin shell enriched in ZnTe. Profile d(3) demonstrates an increase of the thickness by lateral overgrowth (tapering), with a ZnTe-rich shell; the apparent Cd content of the core is reduced, part of it is due to the presence of the thicker ZnTe shell, but we cannot exclude some radial diffusion of Cd from the core to the shell. Note that all inter-facet distances increase from d(1) to d(2) and d(2) to d(3), but not identically, so that the shape of the hexagons is changed ( 10% with respect to the regular hexagon). Although we have no definitive interpretation, we have already noted Rueda growth () that the shape of the nanoparticle is affected by random changes during the growth. This affects both radial and axial growth (and it probably accounts for a part of the length fluctuations observed in Fig. 2c and Fig. 3c). It does not invalidate the general conclusion of the present study.
The temperature dependance of the CdTe growth rate, as demonstrated in Fig. 2c, is completely different from the weak temperature dependance that we found for ZnTe Rueda growth (). In the following we show that this difference is due for a part to the cell-NW geometry, and for the greatest part to the evaporation of Cd from the nanoparticle and the sublimation of CdTe from the sidewalls, which are both significant in this temperature range when compared to the impinging flux.
Elaborating on the model previously used to describe the growth of ZnTe NWs Rueda growth (), we consider a NW with a quasi-sphere nanoparticle of diameter . The NW diameter at the tip, noted , may be different: in the case of ZnTe, our previous analysis of the growth rate Rueda growth () lead us to assume . In the case of CdTe, we will keep the same values as a reasonable starting point. The normal component of the incident flux, , has been measured from RHEED (reflexion high energy electron diffraction) oscillations during growth on a test substrate; it is expressed as the growth rate of a 2D layer, in nm s. The NW is normal to the substrate and the angle of incidence of the flux is : in our MBE chamber, =0.21 for the CdTe cell and 0.48 for the ZnTe cell. The NW length is significantly larger than the diffusion length along the sidewalls ( nm at C for ZnTe Rueda growth ()), so that the diffusion of adatoms from the substrate is negligible. Hence there are two contributions to the NW growth: (1) the direct flux to the nanoparticle, (the quasi-full sphere intercepts the total flux and not only its normal component ), and (2) the flux to the NW sidewalls, , integrated over the NW diameter and the diffusion length , . Their contribution to the growth rate of the NW of cross section area is thus
respectively. We neglect the contribution of the flux reflected or re-emitted from the substrate Ramdani ().
We now estimate the effect of the two negative contributions, evaporation from the nanoparticle and sublimation of CdTe adatoms from the sidewalls, to finally propose a global balance.
Cadmium is a volatile species: the vapor pressure above Cd at C is 40 Pa (compared to 2 Pa for Zn and 0.4 Pa for Te ). The flux evaporating from a unit surface area is Glas2013 () with the mass of a Cd atom (112 g for atoms, with the Avogadro constant), the Boltzmann constant, the temperature, and a factor representing barrier effects (in principle smaller than 1 but often omitted). In the case of a quasi-full sphere, this flux is integrated over the surface area, , and it reduces the growth rate of the NW of cross section area by a volume per Cd atom, where =0.648 nm is the CdTe lattice constant. The final reduction of the growth rate is thus . With =40 Pa at C, mm s (millimeter per second). This illustrates the benefit of a gold catalyst, which will reduce the Cd vapor pressure to with an activity much smaller than the Cd concentration , and small enough to limit the evaporation of Cd to reasonable values: low values indeed were reported Filby (); Bartha () for the activity of Cd in gold at low concentration (8 to 10%), as low as 10 for 8% Cd in gold wires Bartha (). With such values, the loss of material due to the evaporation is reasonable, but not negligible. The temperature and concentration dependance is discussed in the supplementary material Sup (): a reasonable extrapolation in the frame of the regular solution model is
That means that at C, the equilibrium between our impinging flux and the evaporation of Cd in the nanoparticle is realized at 2% Cd; the equilibrium temperature rises above C at 1% Cd. If the Cd concentration stays around these values, we can neglect its influence on the value of the activation energy in Eq. 3. Actually the order of magnitude of can be obtained if we assume that all the Cd contained in the exponential tail of length describing the interface (Eq. 1) is due to the reservoir effect in the gold nanoparticle. Distributing the Cd content of a cylinder of CdTe of height nm and diameter 7.5 nm, into a gold sphere of diameter 13 nm (diameters measured on the TEM image) results in a concentration equal to 1.5% (a similar value, 2%, is calculated for the NW studied by EDX in Fig. 1). This is well below the concentration allowing the formation of Cd-Au compounds. Note that this is an average concentration; it gives no information about the distribution: uniform, or with an axial gradient, or a radial one, or even limited to the surface.
The sublimation of CdTe in the same temperature range is significant. In addition to several reports on the Cd and Te vapor pressure above CdTe, the sublimation rate from the (001) surface was measured by RHEED oscillations Arias (); Waag (); Tatarenko () together with the growth rate. The sublimation rate is Arnoult () (in (001) ML s: , with an activation energy equal to 1.86 eV. This is close to the activation energy of vapor pressure measured above CdTe Yang (), , hence under congruent sublimation. Although the sublimation rate is expected to depend on the orientation of the surface (its anisotropy gives rise to the Wulff construction of the shape of a sublimating crystal), we will take the (001) rate as a first estimate of the sublimation from the NW sidewalls. It is worth at this point to remind the interpretation of the growth rate, which is smaller than the incident flux (incorporation less than unity): the difference involves two contributions, the sublimation, and the desorption of the adatoms which do not reach a nucleation center. In the case of 2D growth, nucleation centers are steps or stable nuclei (larger than the critical size). In the classical Burton-Cabrera-Frank analysis, which was applied to CdTe Peyla (); Pimpinelli (), the flux of adatoms towards the steps is proportional to , where is the incident flux and the desorption rate, and the effective diffusion length takes into account the true diffusion length (desorption) and the average distance between nucleation centers (incorporation): . In the case of a NW sidewall, the gold nanoparticle forms another trap, and if its efficiency is similar to that of the steps, it receives the same adatom flux . As a result, the sublimation makes the flux smaller, through the factor instead of . itself if quite small if the flux-NW angle is small, and the lateral growth makes smaller either by the propagation of steps formed at the NW-substrate interface Filimonov () or by the formation of critical nuclei on the sidewalls as in 2D growth Peyla (); Pimpinelli ().
The contribution to the NW growth - positive if the diffusion takes place from the sidewall to the nanoparticle, or negative if from the nanoparticle to the sidewalls - is thus
With the present growth conditions (=0.5 ML s and =0.21 ), the impinging flux is fully compensated by the sublimation around C.
Finally, we should calculate the concentration in the nanoparticle by writing the equilibrium between:
the sum of the four previous contributions, , where the first two terms result from the direct flux to the nanoparticle and the evaporation from the nanoparticle, and the last two terms are the contribution of the diffusion between the sidewalls and the nanoparticle.
the nucleation at the NW tip. In the case of the self-catalyzed growth of GaAs NWs, the knowledge about the NW-nanodroplet interface is good enough, that classical nucleation theory can be fully developed and applied to calculate the nucleation rate as a function of the droplet composition Glas2013 (). It features a fast increase as a function of the difference of chemical potential - i.e, as a function of the logarithm of the activity of As in the Ga droplet. We will use a crude approximation, by assuming that the nucleation rate is zero below a threshold (which reflects the equilibrium concentration and the effect of the nucleation barrier) and then rises linearly with . We ignore the stochastic nature of the nucleation Glas2017 () since we calculate an average growth rate.
The result is shown in Fig. 4b, where we plot the growth rate of ZnTe and CdTe segments in sample T and a similar sample grown at lower temperatures, as a function of temperature. For ZnTe we show only the segments at high growth rate, which are associated with quasi-full-sphere nanoparticles Rueda growth (), and the fitting parameter is the diffusion length along the NW sidewalls. A good fit is obtained by assuming an activation energy 0.95 eV. A slow increase of the diffusion length with temperature was already noted Rueda growth () in association with a smaller tapering. For CdTe we keep the same geometrical parameters and diffusion length value, correct for the different angle of incidence, and we add the sublimation of CdTe and the evaporation of Cd from the gold nanoparticle, taken from the literature as discussed previously.
v.1 Growth rates
The four contributions to the growth rate are detailed in Fig. 4c. At low temperature, the only significant contribution is the direct flux to the nanoparticle. As the temperature increases, the diffusion length also increases and so does the contribution from the flux to the sidewall. However, this positive flux is rapidly compensated by the evaporation from the nanoparticle. In this temperature range, the flux is positive from the NW sidewalls to the nanoparticle. At higher temperature, the sublimation from the NW sidewalls becomes significant, so that the adatom density decreases (blocking the formation of lateral dots) and finally the flux is reversed, the Cd concentration in the nanoparticle goes down, and the axial growth is stopped. It appears that the evaporation of Cd from the gold nanoparticle, the small angle of incidence on the sidewalls and the sublimation of CdTe from the sidewalls conspire to achieve a low growth rate, a low concentration of Cd in the nanoparticle (resulting in sharp interfaces, much sharper than reported in earlier studies Dluzewski () realized in the VLS mode with Au-Ga liquid droplets at C), and a low adatom density on the sidewalls (resulting in a weak lateral growth, as confirmed in the next subsection).
Other reports on the growth of CdTe NWs by MBE Wojnar2017 (); Kessel () mention higher values of flux (larger values of the beam equivalent pressure on each of a Cd and a Te cells, than that from a single CdTe cell in our case). Additionally, the angle of incidence to the NW may be Wojnar2017 (), or is on purposeKessel (), larger than ours. The temperature dependance of each of the four contributions, calculated using the present model and for such conditions, is shown in Fig. 4d. The result is a higher growth rate at C and a similar growth rate at C. It is worth noting that the impact of the sublimation from the sidewalls is significantly decreased (it is now competing with a lateral flux with a higher intensity at relative incidence closer to the normal) so that the adatom density is expected to be larger than under our present growth conditions: in Fig. 1 each CdTe insertion contributes by less than one monolayer. This makes a sharp contrast with the thick shell (equivalent to 2 nm of pure CdTe) measured by EDX Rueda2016 () on a sample we have grown previously using a high flux from a Cd cell with a large angle of incidence. Such a strong lateral growth is likely to give rise to the formation of additional quantum dots which complicate the photoluminescence spectra and indeed have been identified by cathodoluminescence Wojnar2016 ().
As an example, Fig. 4e shows the photoluminescence spectrum of a CdTe-QD in ZnTe NW. The initial NW was grown at C to reduce tapering, then the CdTe insertion was grown at C as a compromise between a reasonably fast axial growth and a reasonably slow radial growth. The whole structure was capped with a (Zn,Mg)Te shell. The photoluminescence spectrum features three well separated bands which we associate (for increasing energies) to the long CdTe dot resulting from axial growth, to a parasitic CdTe inclusion resulting from lateral growth, and to ZnTe. Note that the polarization diagram shown in the inset is consistent with a light-hole exciton Jeannin (), while the polarization of the intermediate line (not shown) is oblique, suggesting that the principal axis of the insertion is not along the NW axis, and the polarization of ZnTe (not shown) is consistent with a heavy-hole exciton. Cathodoluminescence on the same NW (not shown) confirms that the three lines originate from well-separated areas along the NW.
v.3 Negative growth and nanoparticle shape
The unstable character of the CdTe segments at such temperatures (above C) is also confirmed by the smaller length we systematically observed on the final, uncapped CdTe insertions in samples R and T. The CdTe segment is shorter while keeping the same diameter, an effect similar to what has been reported as ”negative growth” upon annealing of GaAs NWs Dubrovskii2009 (). In the present case, it indicates that CdTe is re-dissolved into the gold nanoparticle as soon as the impinging flux has been stopped, creating a gradient of concentration, , directed on average from the interface towards the apex of the nanoparticle. This gradient drives the diffusion of Cd through the nanoparticle, with a flux where is the diffusion coefficient of Cd in Au, and the atomic volume in gold (). The same gradient induces self-diffusion of gold in the opposite direction, with a flux . Diffusion in a substitutional solid solution is accompanied by the so-called vacancy drift, which is not balanced if the diffusion coefficients of the two atomic species are different. In the present case, they are indeed very different ( at 1000K) Decroupet () so that a number of vacancies approximately equal to the number of atoms dissolved is driven towards the interface, where they have to diffuse to the surface. As a result the nanoparticle tends to assume a bulb-like shape: this is indeed what we observe in Fig. 5 for a NW with the wurtzite structure. This is a manifestation of the Kirkendall effect, which has been known for years to create voids in alloys, and has been used recently to fabricate hollow nanostructures Gusak (); Gonzalez (). Elongated shapes have been observed during redissolution of GaAs, InAs or SnO NWs into gold nanoparticles upon heating Persson (); Pennington (); Hudak (). We occasionally observe elongated nanoparticles, but the shape in Fig. 5 is more complex and well approximated by the shape of a pendant droplet Gassin (); Rotenberg (); Hansen (), with a value of the characteristic parameter =0.037. This is the signature of a quasi-uniform force field, directed along the axis, induced within the nanoparticle by the surface energy: in the pendant droplet Gassin (); Rotenberg (); Hansen (), the force per unit volume is where is the radius of curvature at the apex, is the energy per unit surface area; this force compensates the difference of weight per unit volume between the liquid in the droplet and the gas outside. The force corresponds to the gradient of the Laplace pressure which is uniform in a spherical droplet. In the context of nanowire growth, the equivalent of the Laplace pressure is the shift of the chemical potential within the catalyst droplet or nanoparticle (Gibbs-Thomson effect) Glas2013 (): in a spherical droplet, the shift is uniform, equal to . The surface energy along the pendant-droplet shape results in a gradient of this chemical potential, equal to , which decreases the Cd flux and increases the Au flux. The equilibrium shape is achieved when the vacancy drift vanishes, i.e., when the Cd (or Cd and Te) and Au currents compensate each other. Within the regular solution model, the gradients of chemical potentials become and , respectively. Balancing the diffusion flux of Cd and Au implies , hence . With a few , , nm, =1.5 J m for gold Vitos (), and =50 meV, we obtain : the concentration drop between the interface and the apex is a (significant) fraction of the concentration estimated previously. This estimate is certainly oversimplified (we totally ignore the role of Te, and the actual flux distributions are more complex) but it shows that the orders of magnitude are realistic.
CdTe QDs have been grown in ZnTe NWs, with abrupt interfaces and a controlled aspect ratio, and reduced parasitic growth on the sidewalls. This is achieved in VSS growth by MBE using a gold catalyst and a low flux of Cd and Te, quasi-parallel to the NW axis, at temperatures where a simple model shows that the sublimation of CdTe and the evaporation of Cd from the gold nanoparticle are significant. Similar concepts should apply to the growth of other systems, for instance NWs incorporating CdSe or HgTe. Very peculiar shapes of the gold nanoparticle are ascribed to the redissolution of CdTe under these specific conditions which - through the Kirkendall effect - favors the build up of a gradient of the chemical potential along the nanoparticle.
Acknowledgements.This work was performed in the joint CNRS-CEA group ”Nanophysique & semiconducteurs”, the team ”Laboratory of Material Study by Advanced Microscopy”, and the team ”Materials, Radiations, Structure”. We benefitted from the access to the Nano-characterization facility (PFNC) at CEA Minatec Grenoble. We acknowledge funding by the French National Research Agency (projects Magwires, ANR-11-BS10-013, COSMOS, ANR-12-JS10-0002, Espadon, ANR-15-CE24-0029 and labex LANEF ANR-10-LABX-51-01).
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