Spatial Variation of Available Electronic Excitations within Individual

(1) Quantum confinement entails a significant modification of the electron and hole ... The onset of inelastic scattering beyond the zero-loss peak re...
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Spatial Variation of Available Electronic Excitations within Individual Quantum Dots Hee Joon Jung,†,⊥ Neil P. Dasgupta,‡,⊥ Philip B. Van Stockum,§ Ai Leen Koh,∥ Robert Sinclair,†,∥ and Fritz B. Prinz*,†,‡ †

Department of Materials Science and Engineering, ‡Department of Mechanical Engineering, §Department of Physics, and ∥Stanford Nanocharacterization Laboratory, Stanford University, Stanford, California 94305, United States S Supporting Information *

ABSTRACT: Quantum dots (QDs) allow for manipulation of the position and energy levels of electrons at sub-10 nm length scales through control of material chemistry, size, and shape. It is known from optical studies that the bandgap of semiconductor QDs increases as their size decreases due to the narrowing of the quantum confinement potential. The mechanism of quantum confinement also indicates that the localized properties within individual QDs should depend on their shape in addition to their size, but direct observations of this effect have proven challenging due to the limited spatial resolution of measurement techniques at this scale and the ability to remove contributions from the surroundings. Here we present experimental evidence of spatial variations in the lowest available electron transition energy within a series of single electrically isolated QDs due to a dome-shaped geometry, measured using electron energy-loss spectroscopy in a (scanning) transmission electron microscope [(S)TEM-EELS]. We observe a consistent increase in the energy onset of electronic excitations from the lateral center of the dot toward the edges, which we attribute purely to shape. This trend is in qualitative agreement with a simple quantum simulation of the local density of states in a domeshaped QD. KEYWORDS: Quantum dots, transmission electron microscopy, electron energy-loss spectroscopy, atomic layer deposition

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epitaxial III−V QDs has revealed states reminiscent of the solutions of the classic particle-in-a-box problem. However, a geometrical interpretation of the effects observed in these studies is obscured by the fact that the QDs are typically in contact with a different semiconductor and tend to be connected to each other by thin wetting layers, both of which can contribute to the signal and lead to nonuniform chemical composition in the dots.7,11,12 In addition, these techniques only measure individual states rather than directly probing interband transitions. Here we report a study of single, electrically isolated, domeshaped QDs using electron energy-loss spectroscopy in a (scanning) transmission electron microscope [(S)TEM-EELS]. This technique measures the energy loss spectrum of inelastically scattered electrons that have traversed the sample, revealing a variety of spectroscopic information related to its material properties, such as plasmon energies, elemental composition, and the joint density of electronic states (JDOS) near the band edges.20 By forming a focused probe in the STEM, localized variations in these properties can be measured with subnm spatial resolution. Using this technique we measure a consistent gradual increase of the lowest available transition energy (LATE)

hen matter is confined to sufficiently small regions of space, its electronic properties can change dramatically through the mechanism of quantum confinement. This principle is realized in quantum dots (QDs), which are semiconductor nanocrystals with dimensions below the Bohr exciton radius of the constituent material.1 Quantum confinement entails a significant modification of the electron and hole wave functions inside a material, leading to the possibility of engineering both the spatial and energetic distribution of charges when designing modern solid-state devices such as transistors or solar cells. Further insight into the nature of quantum confinement at the nanometer scale is needed to exploit the degrees of freedom it may provide for the engineering of devices. By probing interband transitions, optical spectroscopy techniques have demonstrated that the QD bandgap increases with decreasing size.1−4 In addition to these size effects, the bandgap and electronic properties of QDs depend on the nanocrystal shape5,6 and these properties are expected to exhibit spatial variation within a single nanostructure. However, direct observations of local variations within nanostructures remain challenging due to the limited spatial resolution of measurement techniques at this scale. Past studies on localized electronic structure within QDs have primarily been performed using scanning tunneling microscopy (STM)7−12 and magnetotunneling spectroscopy (MTS).13−17 Measurement of the local density of electronic states (LDOS) within single strained © 2012 American Chemical Society

Received: November 29, 2012 Revised: December 23, 2012 Published: December 31, 2012 716

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Figure 1. Measurement of the lowest available transition energy (LATE) and JDOS within a dome-shaped QD. (a) Schematic diagram of the STEM-EELS process for measuring the LATE and JDOS at the center and edge of a dome-shaped QD. (b) Schematic of STEM electron interaction with the quantized QD energy states in the conduction and valence band-edges at the center and edge of the QD. (c) Schematic of inelastic EELS spectra corresponding to the process shown in (a,b) with the onset of the signal given by the LATE. (d) Experimental low-loss inelastic EEL spectrum (blue line) after Gaussian−Lorentzian-fitted zero-loss peak subtraction taken near the center of the single PbS QD shown in (e). (e) Annular dark field-STEM image (high-angle collection >50 mrad) of a dome-shaped 8.5 nm PbS QD. (f) Spatial variation of band-edge EEL spectra from the QD in (e), showing the variation of LATE with lateral position.

from the lateral center of PbS QDs toward the edge, which is in qualitative agreement with the anisotropic confinement effects predicted by a simple quantum simulation of our dome-shaped geometry. The effect is attributed to geometrically induced spatial variations of the supported wave functions, which affect the probability of locally exciting a specific electronic transition between states. The lowest energy transitions (associated with the QD bandgap) become unavailable toward the edges of the dome due to negligible probability density of the low-lying states in those regions. Instead, the lowest available transitions there involve higher energy states that exhibit significant probability density near the edges of the structure. Thus, while the QD has a single set of electronic states that are associated with its geometry, the ability to effectively couple to specific transitions varies as a function of position. This phenomenon has possible implications for devices that depend on electron dynamics near the surfaces of nanostructures, for example, through its effects on the local availability of optical transitions, the likelihood of tunneling events for photogenerated charge carrier extraction, and the availability of surface charge for chemical reactions. The samples reported in this study are dome-shaped PbS QDs fabricated in our laboratory through nucleation and

growth of islands during the initial cycles of atomic layer deposition (ALD)18 (see Supporting Information). This shape is typical of vapor-phase growth of QDs in which the base of the dome is in contact with the substrate surface.19 The advantages of the ALD technique are that isolated QDs can be grown on an insulating SiO2 TEM grid surface without the presence of a wetting layer or surface ligands, and a well-defined dome geometry can be formed by subsequent e-beam irradiation in the TEM.18 Irradiation also leads to the removal of surface oxidation of the dot in vacuum, leaving an uncapped QD of the constituent material (see Supporting Information). Localized variations are therefore expected to result purely from shape as opposed to chemical inhomogeneity. Our process for measuring band-edge JDOS variations in single QDs is illustrated schematically in Figure 1. A monochromated 80 kV electron beam is focused into a 0.5 nm probe diameter, which is scanned across the surface of a QD. An EEL spectrometer is used to measure the kinetic energy of transmitted electrons. A large characteristic “zero-loss peak” (ZLP) is observed due to electrons that interact with the sample without inelastic scattering, followed by information on the localized valence structure of the electrons present in the material. By studying the region of the spectrum with energy 717

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Figure 2. Variations in LATE in dome-shaped QDs of different sizes. (a) ADF-STEM images of five different QDs with base diameters of (from left) 8.5, 10, 11.5, 14, and 18.5 nm. All images are shown at the same scale. (b) Localized EELS line profiles of LATE from the QDs in (a), taken at the positions of the black dots. (c) Summary of the trends from the data in (b) showing the consistent increase in LATE from the center to the edge within the QDs and an increase in the average bandgap with decreasing QD size.

We repeated this EELS line profile experiment on domeshaped PbS QDs of five different base diameters (8.5, 10, 11.5, 14, 18.5 nm), as seen in Figure 2 (see Supporting Information). The color level in the images in Figure 2a indicates ADF contrast, which is proportional to the mass−thickness of the sample when collecting high-angle-scattered electrons (>50 mrad).24 While the absolute value of thickness variations is difficult to determine, the relative contrast variations in the ADF-STEM allowed for a confirmation of the dome-shaped geometry, as well as relative height differences between the different QDs. Figure 2b shows the line profile of lowest available transition energy extracted from the EEL spectra for each QD. From the summary in Figure 2c, it can be seen that the center-point bandgap increases as the dots become smaller (as expected), and in all cases the trend of increasing onset energy of available transitions toward the edge of the structures is confirmed. In order to explore the transition between bulk and quantum-confined behavior, we also extended the size trend to a QD of base diameter ∼50 nm (see Supporting Information). This structure showed the bulk PbS bandgap of 0.41 eV at the center, and an increase in LATE to 0.55−0.59 eV near the edges (where the thickness decreased below the Bohr exciton radius), further confirming the role of quantum confinement in our local electronic excitation measurements. We performed control experiments on a bare a-SiO2 TEM grid and a 20 nm thick PbS film, which should not exhibit quantum confinement. ADF-STEM images of these samples are shown in Figure 3, along with the corresponding EEL spectra. EEL spectra (a) and (b) were taken on the bare TEM grid. The first notable feature of these spectra is the lack of EELS signal at low energies (50 mrad) and each EELS data point was spaced by 1 or 2 nm intervals within a line profile. Further details can be found in the Supporting Information. In order to measure the spatial dependence of allowed electronic excitations within individual QDs, we acquired EEL spectra over several dome-shaped PbS QDs of varying diameters at 1−2 nm intervals. Figure 1d shows the EEL spectrum taken near the center of a QD with a base diameter of 8.5 nm, revealing a contribution from electronic transitions in the PbS starting at 1.1 eV, followed by an increase in intensity of the EEL spectrum at the bandgap of SiO2 at 9.0 eV, which represents the onset of inelastic scattering due to the TEM substrate.22,23 The variation in the JDOS across this QD is shown in the set of EEL spectra in Figure 1f. The variable onset of the spectra at each lateral position on the QD indicates that the LATE over this single nanostructure varies from 1.10 eV at center to 1.21 eV at edges. We ascribe this effect to the narrower confinement potential near the edges of the QD, where the height of the structure is smaller, leading to a decreased occupation probability of the lowest energy states near the band edges. This result provides evidence for the ability to locally modify the available electronic excitations within a single semiconducting nanostructure directly through shape control. 718

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Figure 3. Comparison of representative EEL spectra from an a-SiO2 substrate, PbS bulk films, and a PbS QD. (left) EEL spectra taken within the red line profiles from the samples shown on the right, showing electronic properties such as bandgaps, LDOS, core loss edges, and bulk plasmon resonances. The important energy positions (vertical dotted lines) are numbered. The EEL spectra were taken from a bare 8 nm a-SiO2 TEM grid using a 200 kV beam (a), a bare 8 nm a-SiO2 TEM grid using an 80 kV beam (b), only pure PbS thick bulk film at the edge of a broken SiO2 TEM grid using an 80 kV beam (c), a double-sided PbS thick bulk films with the SiO2 TEM grid using an 80 kV beam (d), and a dome-shape QD at center using a 80 kV beam (e). (right) STEM-ADF images showing the samples and spectrum positions for the EEL spectra on the left. Note that an 80 kV beam in spectra (b−e) was used to diminish the Cerenkov radiation effect observed in spectrum (a) using 200 kV. Note that the bandgap at the center of the QD (1.1 eV in (e)) is higher than the bulk bandgap (0.4 eV in (c,d)), and the PbS bulk plasmon peak from the PbS bulk films ((c,d)) disappears in the PbS QD (e).

plasmon peak at 15.5 eV and Pb core loss edges at ∼19.0 and ∼22.0 eV. The positions of these Pb and PbS features in the spectra are unaffected by the presence of the TEM substrate in the electron beam path as shown by the EEL spectra (c,d) in Figure 3, indicating that the presence of the SiO2 does not affect our ability to accurately measure the properties of the PbS. For direct comparison, the EEL spectrum for a PbS QD with base diameter of 8.5 nm is shown in (e). We observe a bandgap of 1.1 eV, which confirms the ability to measure quantum confinement effects with this technique. The core loss features are still observed in the QD, while the bulk plasmon peak disappeared. This phenomenon was observed in comparisons of QDs to bulk PbS, as well as in locally confined regions in a larger QD (see Supporting Information). In order to gain insight into the quantum mechanical origins of the electronic excitation variations measured in this report, here we discuss the role of the JDOS in the EEL spectrum and describe a basic qualitative model relating its behavior to the effects of quantum confinement. The signal intensity of an EEL spectrum is described by the double differential cross section for inelastic electron scattering, which in the low-loss region is calculated from Fermi’s golden rule.28 The physics involved can be simplified by assuming the traversing electron beam to have infinitesimal width and adopting the result of a similar exercise widely used in STM29 (see Supporting Information). By analogy with role of the local density of states (LDOS) in the STM treatment, we expect the EELS differential cross section

to become proportional to the joint local density of states (JLDOS), which we define as JLDOS(r, ΔE) =

∑ |ψi(r)|2 |ψj(r)|2 δ(Ej − Ei , ΔE) i,j

(1)

Where ψi are the eigenstate wave functions of the sample with energy Ei, evaluated at position r, and δ is the Kroneker delta. The summation is over the states i in the valence band and j in the conduction band. This expression is simply the JDOS weighted by the probability density of finding the transition electrons at a given point in space. It reflects the intuitive notion that a localized traversing electron should only interact with states that exist at its location as it passes through the sample. As discussed earlier, the energy onset of the JLDOS represents the lowest available transition energy of the sample in a particular region of space. To simulate the electronic excitation profile observed in this report, we approximated the JLDOS expected to arise from quantum confinement in dome-shaped PbS QDs. The mechanism of confinement is generally understood to increase the energies of the electron and hole states near the band edges of a semiconductor by an amount similar to the energy of the confined envelope functions of the Bloch states. In our model these envelope functions were calculated under the effective mass approximation30 by numerically solving the Schrödinger Equation for a dome-shaped potential in COMSOL (see Supporting Information). 719

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Figure 4. Simulation of JLDOS variation within a dome-shaped quantum dot. (a) Visualization of three wave functions from particle-in-a-box calculations with a 10 nm wide dome-shaped boundary. Electron probability density is represented by color, increasing from blue to red. Lowerenergy states are concentrated at the center of the structure, while higher-energy states extend closer to the edges. (b) Overhead view of the QD from a, showing partitions representing possible locations of EELS electron beam traversal (normal to the page). (c) JLDOS functions for the partitions from (b) based on electron energy states like those in (a). The variation in the onset of the JLDOS represents the increasing LATE near the edges of the QD. Delta functions from eq 1 are shown as arbitrary-width Lorentzian lineshapes with a logarithmic color scale for ease of visualization.

believe that this approach may open significant pathways toward the miniaturization and enhancement of a wide range of solid-state devices that rely on electron dynamics.

Figure 4a shows the squared envelope wave function for three electron energy eigenstates in the dome-shaped QD. We observe characteristic orbital shapes similar to those of the hydrogen atom with the charge density shifted due to the hemispherical anisotropy. From these wave functions, we can gain immediate insight into the origin of the observed excitation trends by noting that lower energy states have lower amplitude near the edges of the QD, and hence contribute little to the onset of the JLDOS there. Therefore, while the QD has a single set of electronic states throughout its volume, the ability to excite the lowest-energy transitions decreases toward the edges of the dome due to the wave function shape. To represent the chosen locations of the electron beams in the EELS experiment, the hemisphere was partitioned along its width as shown in Figure 4b. The JLDOS contributions from all pairs of eigenstates were then integrated over the volume within each section, yielding the energy-resolved JLDOS function associated with each location. These JLDOS functions are shown in Figure 4c. (Delta functions from eq 1 are instead shown as Lorentzian line shapes of arbitrary width for ease of visualization). It can be seen that the onset of the JLDOS occurs at higher energies near the edges of the QD than near the center, as expected from the shape of the wave functions. This result indicates that the lowest available transition energy in a dome-shaped QD should be expected to increase with cylindrical radius, which is in agreement with our EELS measurements. The measurements presented in this work provide direct evidence for the capability to tune the local electronic properties within individual nanostructures through shape control. When combined with the well-known capability to control overall electronic structure through size control, we

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ASSOCIATED CONTENT

* Supporting Information S

Experimental details of QD fabrication and STEM-EELS measurements. Detailed measurements of QDs of different sizes and control experiments. Comparison of ZLP subtraction methods. Quantum simulation and modeling details. In-situ TEM video of PbS QDs merging on Si NW under electronbeam irradiation. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions ⊥

These authors contributed equally.

Notes

The authors declare no competing financial interests.

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ACKNOWLEDGMENTS H.J.J., P.B.V.S., R.S., and F.B.P. were supported as part of the Center on Nanostructuring for Efficient Energy Conversion, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001060. The authors acknowledge use of equipment of the National Center for Electron Microscopy (NCEM), Lawrence Berkeley National Laboratory (LBNL), which is supported by the U.S. Department of Energy under Contract No. DE-AC02720

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(28) Rafferty, B.; Brown, L. M. Phys. Rev. B 1998, 58 (16), 10326− 10337. (29) Tersoff, J.; Hamann, D. R. Phys. Rev. B 1985, 31 (2), 805−813. (30) Luttinger, J. M.; Kohn, W. Phys. Rev. 1955, 97 (4), 869−883.

05CH11231, and at the Stanford Nanocharacterization Laboratory. The authors would also like to thank members of the Nanoscale Prototyping Laboratory at Stanford University for their assistance, in particular Michael C. Langston. H.J.J. also acknowledges Dr. Velimir Radmilovic and Chengyu Song at NCEM, LBNL, for support and discussion.

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