Letter pubs.acs.org/NanoLett
High Mobility One- and Two-Dimensional Electron Systems in Nanowire-Based Quantum Heterostructures Stefan Funk,†,‡ Miguel Royo,§ Ilaria Zardo,*,†,‡,∥ Daniel Rudolph,† Stefanie Morkötter,† Benedikt Mayer,† Jonathan Becker,† Alexander Bechtold,† Sonja Matich,† Markus Döblinger,⊥ Max Bichler,† Gregor Koblmüller,† Jonathan J. Finley,† Andrea Bertoni,§ Guido Goldoni,§,¶ and Gerhard Abstreiter†,‡ †
Walter Schottky Institut and Physik Department, Technische Universität München, Am Coulombwall 4, D-85748 Garching, Germany ‡ Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2a, D-85748 Garching, Germany § Institute for Nanoscience, CNR NANO S3, I-41125 Modena, Italy ∥ Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands ⊥ Department of Chemistry, Ludwig-Maximilians-Universität München, Munich 81377, Germany ¶ Department of Physics, Information, and Mathematics, University of Modena & Reggio Emilia, Modena, Italy S Supporting Information *
ABSTRACT: Free-standing semiconductor nanowires in combination with advanced gate-architectures hold an exceptional promise as miniaturized building blocks in future integrated circuits. However, semiconductor nanowires are often corrupted by an increased number of close-by surface states, which are detrimental with respect to their optical and electronic properties. This conceptual challenge hampers their potentials in high-speed electronics and therefore new concepts are needed in order to enhance carrier mobilities. We have introduced a novel type of core−shell nanowire heterostructures that incorporate modulation or remote doping and hence may lead to high-mobility electrons. We demonstrate the validity of such concepts using inelastic light scattering to study single modulation-doped GaAs/Al0.16Ga0.84As core-multishell nanowires grown on silicon. We conclude from a detailed experimental study and theoretical analysis of the observed spin and charge density fluctuations that one- and two-dimensional electron channels are formed in a GaAs coaxial quantum well spatially separated from the donor ions. A total carrier density of about 3 × 107 cm−1 and an electron mobility in the order of 50 000 cm2/ (V s) are estimated. Spatial mappings of individual GaAs/Al0.16Ga0.84As core−multishell nanowires show inhomogeneous properties along the wires probably related to structural defects. The first demonstration of such unambiguous 1D- and 2Delectron channels and the respective charge carrier properties in these advanced nanowire-based quantum heterostructures is the basis for various novel nanoelectronic and photonic devices. KEYWORDS: Modulation doped nanowires, GaAs/AlGaAs core−multishell nanowires, inelastic light scattering, high mobility, 1D- and 2D-electron channels
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also NW-based heterostructures can be directly integrated on Si due to their nanosized cross sections, which enable elastic accommodation of strain also in lattice mismatched systems.5−12 This may lead to improved high-speed performances4 and to design novel, more power-efficient electrode-architectures.9,12−16 The fabrication as well as the characterization of high quality MD core−shell (C−S) NWs, however, is a challenging task. Only very few publications exist reporting on global transport experiments with only marginal evidence of charge separation and mobility enhancement, while almost no
he concept of modulation doped (MD) semiconductor heterojunctions1−3 was introduced already 35 years ago and was a breakthrough for developing low-noise and ultrafast high electron mobility transistors that paved the way to various high-frequency applications nowadays.4 In general, impurity scattering is reduced in such devices due to the spatial separation of ionized (donor) impurities from the electron channel. The concept has been realized mainly with heterostructures formed by lattice-matched semiconductor compounds with different band gaps like GaAs/AlxGa1‑xAs1,4 So far, modulation doping was introduced into Si technology only in exploratory studies but did not make it into mass production. Nanowire (NW)-based transistors provide the optimum geometry for best performance and there is hope that © 2013 American Chemical Society
Received: September 24, 2013 Revised: November 9, 2013 Published: November 25, 2013 6189
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NW with an embedded 20 nm thick GaAs coQW. For better visualization (i.e., Z-contrast), we used here a C-MS heterostructure that contains a larger Al fraction [x(Al) = 0.3]. The higher Al content chosen here not only evoke a better contrast between the GaAs and Al0.3Ga0.7As layers but also other features within the Al0.3Ga0.7As shell layers, such as the unique Al-rich corner facets, as reported recently.10,29,30 In addition, we also find that the width of the GaAs coQW varies slightly at different side facets that can be attributed to shadowing effects during NW sidewall growth.10 Figure 1c shows a schematic sketch of the MD C-MS NWs. Figure 2a shows the calculated free electron density of the MD C-MS NW using nominal parameters as indicated in Figure 1c and assuming a dopant density ndop = 1.6 × 1018 cm−3, which results in an estimated one-dimensional (1D) electron density of n = 2.96 × 107 cm−1 within the GaAs coQW. Details of the self-consistent calculations are discussed in the Supporting Information. It is worth emphasizing that the electron density is concentrated at the six corners of the coQW as a consequence of the hexagonal symmetry of the coQW.31−33 In order to expose the formation of these electron channels within the 26 nm thick GaAs coQW, we performed polarization dependent resonant ILS experiments in the center region of individual MD C-MS NWs at 10 K. Here, the excitation energy ℏωi = 1.916 eV is chosen close to resonance with the energy gap between the spin−orbit split-off valence band and the conduction band of GaAs.21 The laser beam impinged the C-MS NWs normal to the top facets. In the depolarized configuration (with excitation polarization parallel to the NW axis and analyzed polarization perpendicular to it), we obtained the spectra of a single MD C-MS NW and an undoped C-MS NW reference displayed as red and black solid lines in Figure 2b, respectively. The sharp peaks at 33.8, 35.9, and 46.7 meV are observed in both spectra with about the same intensities. They are attributed to the GaAs transversal optical (TO), the GaAs-like longitudinal optical (LO), and the AlAslike LO phonon modes, respectively.10 The black trace originating from the undoped sample exhibits a weak broad background peaking at around 40 meV, which corresponds to an absolute energy of about 1.88 eV. This spectral feature is attributed to the GaAs split-off valence band “hot” luminescence.2,21 The background is much stronger for MD C-MS NWs and exhibits in addition three peaks at 20 ± 2, 30 ± 1, and 47 ± 1 meV that are labeled D1, D2, and D3, respectively. We conducted a Gaussian peak fitting of the D3 resonance displayed as a solid red line in the inset of Figure 2b. Here, we assumed a linear background that is displayed as a green dashed line. The width of the resonance is w ≈ 7 meV. Notably, the D1 and D2 resonances have similar widths. Additionally, we performed measurements in a polarized configuration where both the incident and the analyzed radiation fields are polarized parallel to the main NW axis under otherwise identical measurement conditions. The spectra are shown in Figure 2c, where the red and black solid lines originate from the MD and undoped C-MS NW, respectively. Both spectra in Figure 2c show the GaAs TO, the GaAs-like LO, and the AlAs-like LO peaking at the same energetic positions but with a higher absolute intensity with respect to the spectra in the depolarized configuration (Figure 2b). Two additional sharp peaks are observed at 36.7 and 45.4 meV that we identify as the GaAs LO and the AlAs-like TO phonons, respectively.10,21 The increased intensity of the phonon modes in the polarized configuration is in accordance with common
information about spatial inhomogeneities along the NWs is provided.17−19 Here we use resonant inelastic light scattering (ILS) by electronic excitation as a well established, contactless, and nondestructive spectroscopic technique to probe the electronic properties of MD C−S hetero NWs with high spatial resolution. It was demonstrated already 30 years ago that resonant ILS allows separation of, for example, charge density and spin density excitations which in turn lead to information about the electronic structure, the carrier density, the carrier− carrier interactions, the scattering times, and the dimensionality of free charge carriers.20−25 Simultaneously, ILS provides information on structural properties such as crystal phases, defects, and alloy composition ordering of NWs via the dielectric response of crystal excitations with a spatial resolution of few hundred nanometers.10,26−28 In this article, we report on the synthesis as well as the electronic and coupled electronic-structural properties of GaAs/Al0.16Ga0.84As MD core−multishell (C-MS) NWs. We have observed charge and spin density excitations at cryogenic temperatures in such hetero NWs, and in combination with self-consistent calculations we have obtained quantitative information on the density, the mobility, and the spatial distribution of electron gases in hexagon-shaped coaxial quantum wells (coQW) around the core of the nanowires. Figure 1a displays a typical scanning electron microscope (SEM) image of an array of GaAs/Al0.16Ga0.84As C-MS NWs
Figure 1. Metrology of core−multishell nanowires grown on silicon. (a) Scanning electron micrograph of an array of GaAs/Al0.16Ga0.84As C-MS NWs grown on a (111) silicon substrate. (b) False-color STEM-HAADF micrograph of a typical cross-section of a GaAs/ Al0.3Ga0.7As C-MS NW with a 20 nm thick embedded GaAs coQW. (c) Sketch of the cross-section of the investigated GaAs/Al0.16Ga0.84As MD C-MS NW with its estimated dimensions along with the defined axes convention. In panels (b) and (c), GaAs NW core and QW are highlighted in light blue, while the AlxGa1-xAs shells are highlighted in orange.
grown on a (111) Si substrate. The NWs were grown by solid source molecular beam epitaxy (MBE) using a self-catalyzed vapor−liquid−solid growth method (for the NW core) combined with a vapor-solid growth method (for the radial NW shell layers).10 Full details of the fabrication and basic ex situ characterization processes are discussed in the Methods section. Figure 1b shows exemplarily a cross-sectional highangle annular dark field/scanning transmission electron microscope (HAADF-STEM) image of a GaAs/Al0.3Ga0.7As C-MS 6190
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Figure 2. Inelastic light scattering spectra of single core-multishell nanowires. (a) Calculation of the electron distribution at a dopant density ndop = 1.6 × 1018 cm−3. (b) The ILS spectrum of a single MD (undoped) C-MS NW in the depolarized configuration is shown as a red (black) solid line. The broad excitation labeled D3 is displayed in the inset on an expanded scale, where full dots are experimental points, the red solid line a Gaussian fit, and the green dashed line the estimated linear background. The lower panel displays simulations of spin density excitations for various doping densities (see text). (c) ILS spectrum of the identical MD (undoped) C-MS NW measured in the polarized configuration under otherwise identical conditions is shown as a red (black) solid line. The lower panel shows the simulations of charge density excitations for carrier densities as in panel b (see text). Vertical lines in panels b and c are a guide to the eye.
selection rules for arsenide-based NWs.10,21,26,27,34 The polarized spectrum of the MD C-MS NW exhibits in addition broad and partially asymmetric peaks at 25 ± 4, 40 ± 1, and 51 ± 2 meV labeled P1, P2, and P3, respectively. Interestingly, the spectrum has a pronounced dip at 36.7 meV, the spectral position of the GaAs LO mode, close to the P2 spectral feature. Our experimental observations are qualitatively in line with many reports of ILS performed on planar MD GaAs/ AlxGa1−xAs heterostructures.1,2,21,22 In these reports, spectral features recorded in depolarized configurations are associated with spin density excitations (SDEs) and those observed in polarized configurations with charge density excitations (CDEs).21,22 SDEs reflect in good approximation single particle excitations since exchange interaction are usually small in GaAs.21 On the other hand, CDEs (plasmons) carry a macroscopic electric field (the depolarization field) as a consequence of the direct Coulomb interaction that evokes a blueshift of excitations with respect to the bare intersubband spacings as well as coupling with LO phonon modes.21,22 We have performed calculations of the energies and intensities of SDE and CDE modes in a multisubband nonresonant formalism (see the Methods section)35 and compared the results with our experimental observations. We expect a good description of the observed ILS shifts. However, only a qualitative evaluation of ILS cross sections is expected, as resonance effects which are not included in the theory severely affect the relative intensities.21,22 The lower panels of Figure 2b,c show the calculated SDE and CDE spectra, respectively. Results are shown as calculated for different doping concentrations ndop = 1.2, 1.4, 1.6, 1.8, and 2.0 × 1018 cm−3. We find the best agreement between experiments and theory for a dopant density of 1.6 × 1018 cm−3, corresponding to a 1D electron density of 2.96 × 107 cm−1
within the coQW. The widths of the electronic excitations correlate with the electronic damping parameter Γ.36 Γ = 2 meV is found to reproduce best the experimentally observed linewidths (see the P1, D3, and P3 peaks in Figure 2b,c). This electronic damping corresponds to an electron mobility of the order of μ ≈ 50 000 cm2/(V s) considerably higher than previously reported for nanowires.37,38 The increased electron density at the six corners of the hexagon shaped coQW (see Figure 2a), can be regarded as six quasi one-dimensional (q1D) channels in the corners and quasi two-dimensional (q2D) channels in the facet planes. In the following, we will label the resulting electronic subbands as Cn and Fn for 1D-corner and 2D-facet states, respectively, where n denotes the number of nodal surfaces pallel to the coQW interfaces, similarly to a planar QW. Few states are localized in the GaAs NW core, but contribute very little to the calculated ILS, and will not be discussed further. Figure 3a shows the DOS in arbitrary units of a C-MS NW with a dopant density n = 1.6 × 1018 cm−3. Since electrons occupy 1D states, each level i shows the characteristic DOS behavior ∝(E − Ei)−1/2. The lowest energy state stems from the six degenerate states localized in the corners (see inset C0). The next set of states are localized along the facets (see, e.g., inset F0). States between F0 and the next corner state C1 have an increasing number of azimuthal nodes with nodal planes normal to the coQW plane due to quantization of the 2D electron gas wrapped around the NW axis. These states have a small energy separation, and the DOS approximately sums up to the characteristic constant behavior of 2D systems, as highlighted by the red line in Figure 3a. As expected, radial nodes with nodal planes parallel to the coQW plane first show up in corner states (inset C1) and then in facet states (see e.g. inset F1), which build up the second step in the 2D DOS. A 6191
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Figure 3. Calculation of the density of states and respective charge and spin density excitations. (a) Normalized DOS for a MD C-MS NW with a density of dopants ndop = 1.6 × 1018 cm−3. The horizontal gray line indicates the position of the Fermi energy. The insets show 2D maps of the squared envelope functions for selected states. The bold red solid line is a guide to the eye to emphasize the seemingly 2D profile of the DOS. (b) CDE and (c) SDE spectra. Total (red solid lines) and projected (blue, green, black, and orange lines) ILS spectra. Projected excitations are labeled as corner, facet, corner-facet, and core excitations if they refer to Fn → Fm, Cn → Cm, Cn → Fm, or Fn → Cm, core →core single particle excitations, respectively. Red dashed lines represent energy positions of the experimental peaks. The insets show 2D maps of the IDD (see Supporting Information) induced by photon scattering at the indicated energies. The top right inset in panel (c) illustrates the scattering geometry.
CDEs in the top and bottom facets (with one nodal surface parallel to the QW plane) and intrasubband CDEs along the lateral facets (with nodal surfaces perpendicular to the QW plane). Peak P1 is energetically close to the GaAs LO phonon resonance and is strongly affected by the plasmon-phonon coupling.21,22 As a consequence, two plasmon-phonon coupled modes arise on opposite sides of the phonon resonance, while at the GaAs LO phonon CDE modes vanish. This produces the pronounced dip in the experimental spectra (see Figure 2c). In Figure 3b, therefore, peak P2 is interpreted as the mate of P1. This is confirmed by the calculated evolution of the P1 and P2 peaks versus doping, shown in the lower panel of Figure 2c. Here, the P2 peak gains intensity as P1 approaches the phonon resonances with increasing doping. A relevant contribution to this resonance is also given by the C1 → C2 and C1 → F2 transitions, which happen to be at the same energy, resulting in a second radial node in the IDD (see inset). Finally, peak P3 corresponds to a CDE mode associated with the C0 → F2 and F0 → F2 transitions, namely excitations with Δn = 2, located in the top and bottom facets of the coQW, as shown by the corresponding IDD. The IDD also shows that the Δn = 2 transitions occur in association with variations in the number of azimuthal nodes in the side facets. The calculated SDE spectrum, shown in Figure 3c, has two maxima in good agreement with the resonances D1 and D3. Comparing the corresponding IDDs with those in the CDE
further sequence of corner/facet states follows. Clearly, the DOS shows the concurrence of q1D and q2D character of the electronic states. CDEs and SDEs are associated with transitions from occupied to unoccupied states. At the present carrier density C0, F0, and C1 states are below the Fermi energy. In Figure 3b,c, we show the calculated CDE and SDE spectra together with the projected Raman spectra, that is, the contribution to the total spectrum arising from excitations of a specific type Fn→Fm, Cn→Cm, and Cn → Fm plus Fn → Cm screened by all excitations (see Supporting Information). The calculated CDE spectrum of Figure 3b shows three relative maxima matching very well the resonances P1, P2, and P3 observed in the polarized configuration (see Figure 2c). Peak P1 is the strongest in our calculations, and it turns out to be mainly due to the concurrence of two types of CDEs, (i) intersubband transitions with Δn = 1 originating from F0 → F1 and C0 → F1 transitions, the latter contributing to the higher energy part of the peak and (ii) intrasubband transitions, that is, the intrasubband plasmons of the coQWs. Note that in a flat QW, such intrasubband plasmons are forbidden with the laser normal to the QW plane.21,22 In the current NW, intrasubband plasmons are activated through the sidefacets with oblique orientation with respect to the incident photons. The calculated induced density distribution (IDD) (see Supporting Information) at the P1 resonance (see corresponding inset) clearly shows the two types of collective excitations, intersubband 6192
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Figure 4. Electron gas and structural defects along a single modulation doped nanowire. (a) Colormap of ILS spectra normalized to the GaAs TO intensity recorded in the depolarized configuration as a function of spatial position along the MD C-MS NW main axis (step size: 500 nm). (b) Normalized spectra at selected positions recorded in the depolarized configuration. The spectra have been shifted vertically for clarity. (c) Color map of normalized ILS spectra of the same C-MS NW as in panel a but in the polarized configuration (step size: 500 nm). (d) Spectra at selected positions recorded in the polarized configuration. The spectra have been shifted vertically for clarity. A linear background was subtracted from all ILS spectra shown in panels a−d. (e) Spatial PL mapping along the axis of the same MD C-MS NW (step size: 500 nm), as well as a typical spectrum of an undoped NW as a reference. Spectra have been shifted vertically for clarity. Vertical dashed lines and the bold dash-dotted line are a guide to the eye.
following two main observations can be made in the depolarized configuration: (i) at z < 1 μm, the intensity of SDE modes reduces, while the forbidden34 GaAs-like LO and AlAs-like LO phonon excitation gain in relative intensity with respect to the GaAs TO mode; and (ii) at the opposite end (z > 3 μm), the normalized intensity of the D2 resonance gradually increases. The detailed spectrum at z = 4 μm in Figure 4b shows that the D1 resonance vanishes while D2 redshifts by about 2.5 meV. Electronic resonances suppression and enhancement of LO phonon modes are also observed in CDE modes at z < 1 μm (see Figure 4c,d). The reduction of the intensity of all electronic excitations at z < 1 μm can be traced to either a reduced density or a reduced mobility of electrons. These are most probably caused by twinning defects and crystal phase changes26,27,29 as often observed at the ends of the NWs, and as also confirmed by TEM data on NWs grown under similar conditions. Such defects also lead to an enhanced intensity of forbidden Raman modes.34 In order to confirm this interpretation, we performed also spatially resolved PL measurements of the same GaAs/ Al0.16Ga0.84As MD C-MS NWs (see Methods section), shown in Figure 4e. We observe position-dependent PL spectra along the nanowire. At z < 1 μm, we mainly observe PL signals below
spectra shows that peaks D1 and D3 are the SDE counterparts of peaks P1 and P3. The resonance D2 at ∼30 ± 1 meV on the other hand does not seem to have a corresponding peak in the simulation. We recall, however, that valence band resonance effects are not included in the theoretical model. The large intensities of peaks D2 and D3 in the experiment suggest that both peaks involve similar resonantly populated final states. In this regard, we know from the previous analysis of resonances P3 and D3 that such final states are basically of the F2 type. Therefore, transitions of type C1 → F2, which are allowed but are weak in our calculations, would be strongly enhanced by resonance effects. From our calculations, the energy for such an SDE, which is roughly equal to the subband splitting, is ∼30 meV. Our non resonant calculated SDE spectrum indeed shows a weak resonance at this energy, as shown in the inset corresponding to D2. Therefore, we assign peak D2 to a SDE associated with C1 → F2 intersubband transitions. The MD C-MS NWs display spatial variations of the ILS shifts and intensities of SDEs and CDEs along the NWs likely due to structural inhomogeneities along the MD C-MS NWs, as we show next. Spatial profiles recorded in the depolarized and polarized configurations are shown in Figure 4a,c, respectively, in color code. Individual spectra at selected positions z = 0.5, 2.0, and 4.0 μm are shown in Figure 4b,d. The 6193
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the outer interface of the 26 nm GaAs coQW by closing the Ga shutter and opening a Si effusion cell for 10 min. In order to experimentally confirm the nominal values of the shell layer thicknesses, we conducted atomic force microscopy and SEM measurements on single C-MS NWs mechanically transferred on top of a Si substrate. For determining the core thickness, we synthesized GaAs NWs employing identical growth conditions as for the MD and undoped GaAs/Al0.16Ga0.84As C-MS NWs but terminated the growth before initializing the capping sequence described previously. Following this approach, we found an inner core diameter d of 50 ± 3 nm and a total shell thickness of 149 ± 8 nm in excellent agreement with the total nominal shell thickness of 151 nm. For the cross-sectional HRTEM investigations, we additionally fabricated C-MS NWs with higher Al contents (x[Al] = 0.3) in order to improve the material contrast of the AlxGa1−xAs layers with respect to the GaAs layers. The GaAs/Al0.3Ga0.7As C-MS NWs displayed a nominal layer sequence of 30 nm Al0.3Ga0.7As, 20 nm GaAs, 70 nm Al0.3Ga0.7As, and 10 nm GaAs. We mechanically transferred the C-MS NWs on top of a Ptcoated glass substrate and cut lamellas out of carbon-coated CMS NWs ensembles using a Zeiss focused ion beam (FIB) apparatus as reported recently.10 Analyzing the cross sections of nine individual C-MS NWs, we determined a shell layer sequence of 31 ± 2 nm AlxGa1−xAs, 20 ± 1 nm GaAs, 70 ± 1 nm AlxGa1−xAs, and a partially oxidized GaAs layer10 in excellent agreement with the nominal growth conditions. Here, we confirmed the nominal values of the GaAs coQW layer thickness with a statistical error of 1 nm. The homogeneous Al content x along the NWs has been also confirmed by spatially resolved phonon Raman spectroscopy. Inelastic Light Scattering and Photoluminescence. For the ILS and PL investigations, MD and undoped GaAs/ Al x Ga 1−x As C-MS NWs were transferred on top of prepatterned Si substrates in order to facilitate polarizationdependent ILS and PL measurements of individual NWs. The measurements were performed in a He flow cryostat at T = 10 K. For the ILS experiments, the C-MS NWs were subsequently excited with the 1.916 and 1.833 eV lines of a Kr+-laser with power densities of 2 kW/cm2. The incident light with its polarization rotated by a λ/2 plate was focused via an objective lens (NA = 0.75) and the measurements were performed in backscattering geometry. We measured the spot size of the laser beam to be 900 ± 100 nm. The scattered radiation field was analyzed employing a polarization filter in combination with a λ/2 plate. Subsequently, the scattered radiation was collected by a Dilor XY triple Raman spectrometer equipped with a N2 cooled multichannel charge coupled device (CCD). For the PL experiments, we excited the same individual C-MS NWs with the 1.959 eV line of an HeNe laser focused by a NA = 0.5 objective lens. We collected the PL spectra with a Princeton instruments Acton spectrometer equipped with a N2 cooled multichannel CCD. Employing a set of polarization filters and λ/2 plates, incoming and emitted light polarizations were selected to be parallel to the main C-MS NW axes. Calculation of ILS Cross Section. We obtain the ILS cross section of CDEs from the imaginary part of the reducible response function ICDE(Q,ω) ∝ −; [Π̃(Q,ω)], and the cross section of SDEs from the irreducible response function ISDE(Q,ω) ∝ −; [∏(Q,ω)]. Here, ω = ωi − ωs is the ILS shift between the scattered and incoming photon frequency ωs, ωi, respectively. Q = Qi − Qs is the transferred momentum, where Qi and Qs are the incident and scattered photon
the free exciton line in zincblende (ZB) GaAs of 1.51 eV, the dominant one at about 1.476 eV. Spectral features at such lower energies are typical for indirect transitions due to the type II band alignment of wurtzite−zincblende (WZ−ZB) heterostructures.39,40 An additional peak evolves for z > 1 μm above the free exciton line. Moving along the z-direction, this peak shifts in energy up to 1.522 eV at z = 3.5 μm and then moves downward again and disappears at z = 4.5 μm. For comparison, a typical PL spectrum recorded at the center of an undoped C-MS NW is shown in the lower panel of Figure 4e. Here, the FE peak is resolved at 1.517. The FE peak intensity exceeds all PL signals found in the MD NWs by a factor of at least eight. An inhibition of the FE is known to occur due to screening effects caused by a free electron density.41 The additional peak which shifts between 1.519 and 1.522 eV on the other hand indicates a varying free carrier concentration along the NW and is due to a many-body singularity at the Fermi edge that is especially enhanced in 1D systems.41 Spatial variation of the electron density also correlates with the behavior of D1 and D2 excitations along the wire (see Figure 4a,b). The observed redshift of 2.5 meV of resonance D2 in Figure 4b is consistent with calculated spectra at higher energies. Indeed, with increasing EF F1 states become occupied and the F1 → F2 transition, which is about 3 meV lower in energy than the C1 → F2 transition that we previously associated with the D2 resonance, is expected to contribute to the depolarized ILS spectrum. The lower intensity of resonance D1 at z = 4.0 μm also confirms this interpretation. Indeed, when F1 is occupied, C0 → F1 transitions, which contribute to the D1 resonance, are inhibited and the scattering cross section of D1 reduces. On the other hand, the spectral positions of CDE modes depend on the direct Coulomb interaction and blueshift with increasing EF. This is observed indeed comparing the P1 mode at z = 2.0 μm and at z = 4.0 μm, where P1 appears blueshifted by about 8 meV, which is again in fair agreement with the calculated shift of ∼5 meV in calculations at ndop ∼2 × 10−18 cm−3 (see Figure 2c). In summary, we have demonstrated 1D and 2D free electron gases in MD C-MS NWs with a high electron mobility using spatially resolved resonant inelastic light scattering and PL measurements. However, the electronic properties are inhomogeneous along the individual nanowires and high mobility electron gases are only achieved in certain regions along the NW axis.
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METHODS Nanowire Growth and Structural Characterization. The GaAs/Al0.16Ga0.84As C-MS NWs were synthesized in a solid source MBE chamber on a SiO2/Si(111) substrate patterned by nanoimprint lithography.42 First, GaAs core NWs were grown by catalyst-free vapor−liquid−solid (VLS) growth employing a substrate temperature Tsub of 610 °C, a beam equivalent arsenic pressure BEP(As4) of 2 × 10−6 mbar, and a Ga flux of 0.025 nm/s. After decreasing Tsub to 490 °C and increasing BEP(As4) to 3.5 × 10−5 mbar, the GaAs NWs have been overgrown subsequently in situ by Al0.16Ga0.84As, GaAs, Al0.16Ga0.84As, and a final GaAs cap layer with nominal widths of 25, 26, 87, and 13 nm, respectively. Here, the nominal Al content x[Al] as well as the nominal thicknesses di have been obtained from the measured flux ratios and calibrations as described in ref 10. For modulation doping, a Si delta layer has been introduced to the shell at a distance of 25 nm away from 6194
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INDEX and the CINECA award under the ISCRA initiative for the availability of high-performance computing resources and support. M.R. and G.K. acknowledge project Generalitat Valenciana VALi+d Grant (M.R.) and SFB-631, Marie Curie Integration Grant, EU-SOLID, respectively, for financial support.
momenta, respectively. The momentum dependent reducible response function Π̃(Q,ω) is (the irreducible response function is analogously defined) Π̃(Q , ω) =
∑ Π̃ijlm(qz , ω) ∬ dr dr′e−iq(r− r′)φi*(r)φj(r) ijlm
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× φl(r′)φm*(r′)
The φα are the in-plane part of envelope functions that are self-consistently calculated in a density functional theory−local density approximation (DFT-LDA) model described elsewhere31 and q, qz are in-plane and in-wire components of Q = (q, qz), respectively. The matrix elements of Πijlm and Π̃ijlm are obtained from a Dyson-type equation in a standard timedependent LDA formalism.43 The coupling of CDEs with the GaAs LO phonon has been taken into account by means of a phenomenological frequency dependent dielectric constant (see Supporting Information). The self-consistent calculations of the confined states is performed on a triangular grid with 5.26 points/nm2 using material dependent parameters m* = 0.067, ε = 13.8, for GaAs and m* = 0.08, ε = 12.98, for Al0.16Ga0.84As, and a conduction band offset 0.146 eV between the two materials.44 Here, the material modulations in the NW cross-section are taken into account, while translational invariance along the growth direction is assumed. ILS cross sections are calculated including up to 150 levels (all levels contributing to the DOS of Figure 3a) in the response function. Since the experimental excitation energy 1.916 eV is large with respect to the SDEs and CDEs, we approximate |Qi| ≈ |Qs| ≈ 6.9 × 105 cm−1 in the calculation.
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(1) Abstreiter, G.; Ploog, K. Inelastic light scattering from a quasitwo-dimensional system in GaAs-AlxGa1‑xAs heterojunctions. Phys. Rev. Lett. 1979, 42, 1308−1311. (2) Dingle, R.; Störmer, H. L.; Gossard, A. C.; Wiegmann, W. Electron mobilities in modulation-doped semiconductor heterojunction superlattices. Appl. Phys. Lett. 1978, 33, 665−667. (3) Mimura, T.; Hiyamizu, S.; Fujii, T.; Nanbu, K. A new field-effect transistor with selectively doped GaAs/n-AlxGa1‑xAs heterojunctions. Jpn. J. Appl. Phys. 1980, 19, L225−L227. (4) Mimura, T. Development of high electron mobility transistor. Jpn. J. Appl. Phys. 2005, 44, 8263−8268. (5) Glas, F. Critical dimensions for the plastic relaxation of strained axial heterostructures in free-standing nanowires. Phys. Rev. B 2006, 74, 121302(4). (6) Kästner, G.; Gösele, U. Stress and dislocations at cross-sectional heterojunctions in a cylindrical nanowire. Philos. Mag. 2004, 84, 3803− 3824. (7) Martensson, T.; et al. Epitaxial III-V Nanowires on Silicon. Nano Lett. 2004, 4, 1987−1990. (8) Roest, A. L.; et al. Position-controlled epitaxial III-V nanowires on silicon. Nanotechnology 2006, 17, S271−S275. (9) Tomioka, K.; Tanaka, T.; Hara, S.; Hiruma, K.; Fukui, T. III-V nanowires on Si substrate: selective-area growth and device applications. IEEE J. Sel. Top. Quantum Electron. 2011, 17, 1112−1129. (10) Rudolph, D.; et al. Spontaneous Alloy Composition Ordering in GaAs−AlGaAs Core−Shell Nanowires. Nano Lett. 2013, 13, 1522− 1527. (11) Jiang, N.; Gao, Q.; Parkinson, P.; Wong-Leung, J.; Mokkapati, S.; Breuer, S.; Tan, H. H.; Zheng, C. L.; Etheridge, J.; Jagadish, C. Enhanced Minority Carrier Lifetimes in GaAs/AlGaAs Core−Shell Nanowires through Shell Growth Optimization. Nano Lett. 2013, DOI: 10.1021/nl4023385. (12) Tomioka, K.; Motohisa, J.; Hara, S.; Hiruma, K.; Fukui, T. GaAs/AlGaAs Core Multishell Nanowire-Based Light-Emitting Diodes on Si. Nano Lett. 2010, 10, 1639−1644. (13) Huang, Y.; et al. Logic gates and computation from assembled nanowire building blocks. Science 2001, 9, 1313−1317. (14) Ng, H. T.; et al. Single Crystal Nanowire Vertical Surround-Gate Field-Effect Transistor. Nano Lett. 2004, 4, 1247−1252. (15) Schmidt, V.; et al. Realization of a silicon nanowire vertical surround-gate field-effect transistor. Small 2006, 2, 85−88. (16) Yao, J.; Yan, H.; Lieber, M. A nanoscale combining technique for the large-scale assembly of highly aligned nanowires. Nat. Nanotechnol. 2013, 8, 329−335. (17) Lucot, D.; et al. Quasi one-dimensional transport in single GaAs/AlGaAs core-shell nanowires. Appl. Phys. Lett. 2011, 98, 142114. (18) Spirkoska, D.; Fontcuberta i Morral, A.; Dufouleur, J.; Xie, Q.; Abstreiter, G. Free standing modulation doped core-shell GaAs/ AlGaAs hetero-nanowires. Phys. Status Solidi RRL 2011, 5, 353−355. (19) Wirths, S.; et al. Preparation of Ohmic contacts to GaAs/ AlGaAs core/shell nanowires. Appl. Phys. Lett. 2012, 100, 042103. (20) Burstein, E.; Pinczuk, A.; Mills, D. L. Inelastic light scattering by charge carrier excitations in two-dimensional plasmas: theoretical considerations. Surf. Sci. 1980, 98, 451−468. (21) Pinczuk, A.; Abstreiter, G. Spectroscopy of free carrier excitations in semiconductor quantum wells. Top. Appl. Phys. 1989, 66, 153−207 and references therein. (22) Schü ller, C. Inelastic Light Scattering of Semiconductor Nanostructures, Fundamentals, and Recent Advances; Springer: Berlin, 2006.
ASSOCIATED CONTENT
S Supporting Information *
Additional material and information about theoretical model and calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. S.F., M.R., I.Z., A.B., G.G., and G. A. wrote the manuscript. G.A., S.F., and I.Z. conceived the sample design and the experiments. A.B., G.G., and M.R. designed and performed the theoretical calculations. M.B., G.K., and D.R. synthesized the samples. S.F. recorded the ILS spectra and took part in all other measurements. B.M. and D.R. performed the PL and A.B. performed the Magneto-PL experiments. J.B. took the SEM pictures. FIB preparation was performed by S.Ma., HRTEM measurements by M.D. and S.Mo. S.F., M.R., and I.Z. contributed equally. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge financial support by the Deutsche Forschungsgemeinschaft via the Excelence Cluster “Nanosytems Initiative Munich” and the TUM Institute for Advanced Study as well as partial financial support from EU-MC network 6195
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(23) Goni, A. R.; et al. One-dimensional plasmon dispersion and dispersionless intersubband excitations in GaAs quantum wires. Phys. Rev. Lett. 1991, 67, 3298−3301. (24) Strenz, R. Single-particle excitations in quasi-zero- and quasione-dimensional electron systems. Phys. Rev. Lett. 1994, 73, 3022− 3025. (25) Kalliacos, S.; et al. A molecular state of correlated electrons in a quantum dot. Nat. Phys. 2008, 4, 467−471. (26) Zardo, I.; et al. Raman spectroscopy of wurtzite and zinc-blende GaAs nanowires: polarization dependence, selection rules, and strain effects. Phys. Rev. B 2009, 80, 245324. (27) Funk, S.; Li, A.; Ercolani, D.; Gemmi, M.; Sorba, L.; Zardo, I. Crystal Phase Induced Bandgap Modifications in AlAs Nanowires Probed by Resonant Raman Spectroscopy. ACS Nano 2013, 7, 1400− 1407. (28) Zardo, I.; et al. E1(A) Electronic Band Gap in Wurtzite InAs Nanowires Studied by Resonant Raman Scattering. Nano Lett. 2013, 13, 3011−3016. (29) Heiss, M.; et al. Self-assembled quantum dots in a nanowire system for quantum photonics. Nat. Mater. 2013, 12, 439−444. (30) Zheng, C.; Wong-Leung, J.; Gao, Q.; Tan, H. H.; Jagadish, C.; Etheridge, J. Polarity-Driven 3-Fold Symmetry of GaAs/AlGaAs Core Multishell Nanowires. Nano Lett. 2013, 13, 3742−3748. (31) Bertoni, A.; Royo, M.; Mahawish, F.; Goldoni, G. Electron and hole gas in modulation-doped GaAs/Al1‑xGaxAs radial heterojunctions. Phys. Rev. B 2011, 84, 205323. (32) Wong, B. M.; Lonard, F.; Li, Q.; Wang, G. T. Nanoscale effects on heterojunction electron gases in GaN/AlGaN core/shell nanowires. Nano Lett. 2011, 11, 3074−3079. (33) Ferrari, G.; Goldoni, G.; Bertoni, A.; Cuoghi, G.; Molinari, E. Magnetic States in Prismatic Core Multi-Shell Nanowires. Nano Lett. 2009, 9, 1631−1635. (34) Cardona, M. Resonance phenomena. Top. Appl. Phys. 1982, 50, 19−178. (35) See, for example, Kushwaha, M. S. Inelastic electron and light scattering from the elementary electronic excitations in quantum wells: Zero magnetic field. AIP Advances 2012, 2, 032104 and references therein. (36) Pinczuk, A.; Worlock, J. M.; Störmer, H. L.; Gossard, A. C.; Wiegemann, W. J. Vac. Sci. Technol. 1981, 19, 561−563. (37) Ford, C. A.; et al. Diameter Dependent Electron Mobility of InAs Nanowires. Nano Lett. 2009, 9, 360−365. (38) Parkinson, P.; et al. Transient Terahertz Conductivity of GaAs Nanowires. Nano Lett. 2007, 7, 2162−2165. (39) Spirkoska, D.; et al. Structural and optical properties of high quality zinc-blende/wurtzite GaAs nanowire heterostructures. Phys. Rev. B 2009, 80, 245325. (40) Heiss, M.; et al. Direct correlation of crystal structure and optical properties in wurtzite/zinc-blende GaAs nanowire heterostructures. Phys. Rev. B 2011, 83, 045303. (41) Skolnick, M. S.; et al. Observation of a many-body edge singularity in quantum-well luminescence spectra. Phys. Rev. B 1987, 58, 2130−2133. (42) Hertenberger, S.; et al. High composition homogeneity in Inrich InGaAs nanowire arrays on nanoimprinted SiO2/Si(111) grown by catalyst-free molecular beam epitaxy. Appl. Phys. Lett. 2012, 101, 043116. (43) Marques, M. A. et al. Fundamentals of time-dependent density functional theory. Lecture Notes in Physics; Springer: Berlin, 2012; Vol. 837. (44) Levinshtein, M.; Rumyantasev, S.; Shur, M. Handbook Series on Semiconductor Parameters; World Scientific Publishing: River Edge, NJ, 1996; Vols. 1, 2
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High Mobility One- and Two-Dimensional Electron Systems in Nanowire-Based Quantum Heterostructures Stefan Funk,†,‡ Miguel Royo,§ Ilaria Zardo,*,†,‡,∥ Daniel Rudolph,† Stefanie Morkötter,† Benedikt Mayer,† Jonathan Becker,† Alexander Bechtold,† Sonja Matich,† Markus Döblinger,⊥ Max Bichler,† Gregor Koblmüller,† Jonathan J. Finley,† Andrea Bertoni,§ Guido Goldoni,§,¶ and Gerhard Abstreiter†,‡ †
Walter Schottky Institut and Physik Department, Technische Universität München, Am Coulombwall 4, D-85748 Garching, Germany ‡ Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2a, D-85748 Garching, Germany § Institute for Nanoscience, CNR NANO S3, I-41125 Modena, Italy ∥ Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands ⊥ Department of Chemistry, Ludwig-Maximilians-Universität München, Munich 81377, Germany ¶ Department of Physics, Information, and Mathematics, University of Modena & Reggio Emilia, Modena, Italy S Supporting Information *
ABSTRACT: Free-standing semiconductor nanowires in combination with advanced gate-architectures hold an exceptional promise as miniaturized building blocks in future integrated circuits. However, semiconductor nanowires are often corrupted by an increased number of close-by surface states, which are detrimental with respect to their optical and electronic properties. This conceptual challenge hampers their potentials in high-speed electronics and therefore new concepts are needed in order to enhance carrier mobilities. We have introduced a novel type of core−shell nanowire heterostructures that incorporate modulation or remote doping and hence may lead to high-mobility electrons. We demonstrate the validity of such concepts using inelastic light scattering to study single modulation-doped GaAs/Al0.16Ga0.84As core-multishell nanowires grown on silicon. We conclude from a detailed experimental study and theoretical analysis of the observed spin and charge density fluctuations that one- and two-dimensional electron channels are formed in a GaAs coaxial quantum well spatially separated from the donor ions. A total carrier density of about 3 × 107 cm−1 and an electron mobility in the order of 50 000 cm2/ (V s) are estimated. Spatial mappings of individual GaAs/Al0.16Ga0.84As core−multishell nanowires show inhomogeneous properties along the wires probably related to structural defects. The first demonstration of such unambiguous 1D- and 2Delectron channels and the respective charge carrier properties in these advanced nanowire-based quantum heterostructures is the basis for various novel nanoelectronic and photonic devices. KEYWORDS: Modulation doped nanowires, GaAs/AlGaAs core−multishell nanowires, inelastic light scattering, high mobility, 1D- and 2D-electron channels
T
also NW-based heterostructures can be directly integrated on Si due to their nanosized cross sections, which enable elastic accommodation of strain also in lattice mismatched systems.5−12 This may lead to improved high-speed performances4 and to design novel, more power-efficient electrode-architectures.9,12−16 The fabrication as well as the characterization of high quality MD core−shell (C−S) NWs, however, is a challenging task. Only very few publications exist reporting on global transport experiments with only marginal evidence of charge separation and mobility enhancement, while almost no
he concept of modulation doped (MD) semiconductor heterojunctions1−3 was introduced already 35 years ago and was a breakthrough for developing low-noise and ultrafast high electron mobility transistors that paved the way to various high-frequency applications nowadays.4 In general, impurity scattering is reduced in such devices due to the spatial separation of ionized (donor) impurities from the electron channel. The concept has been realized mainly with heterostructures formed by lattice-matched semiconductor compounds with different band gaps like GaAs/AlxGa1‑xAs1,4 So far, modulation doping was introduced into Si technology only in exploratory studies but did not make it into mass production. Nanowire (NW)-based transistors provide the optimum geometry for best performance and there is hope that © 2013 American Chemical Society
Received: September 24, 2013 Revised: November 9, 2013 Published: November 25, 2013 6189
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NW with an embedded 20 nm thick GaAs coQW. For better visualization (i.e., Z-contrast), we used here a C-MS heterostructure that contains a larger Al fraction [x(Al) = 0.3]. The higher Al content chosen here not only evoke a better contrast between the GaAs and Al0.3Ga0.7As layers but also other features within the Al0.3Ga0.7As shell layers, such as the unique Al-rich corner facets, as reported recently.10,29,30 In addition, we also find that the width of the GaAs coQW varies slightly at different side facets that can be attributed to shadowing effects during NW sidewall growth.10 Figure 1c shows a schematic sketch of the MD C-MS NWs. Figure 2a shows the calculated free electron density of the MD C-MS NW using nominal parameters as indicated in Figure 1c and assuming a dopant density ndop = 1.6 × 1018 cm−3, which results in an estimated one-dimensional (1D) electron density of n = 2.96 × 107 cm−1 within the GaAs coQW. Details of the self-consistent calculations are discussed in the Supporting Information. It is worth emphasizing that the electron density is concentrated at the six corners of the coQW as a consequence of the hexagonal symmetry of the coQW.31−33 In order to expose the formation of these electron channels within the 26 nm thick GaAs coQW, we performed polarization dependent resonant ILS experiments in the center region of individual MD C-MS NWs at 10 K. Here, the excitation energy ℏωi = 1.916 eV is chosen close to resonance with the energy gap between the spin−orbit split-off valence band and the conduction band of GaAs.21 The laser beam impinged the C-MS NWs normal to the top facets. In the depolarized configuration (with excitation polarization parallel to the NW axis and analyzed polarization perpendicular to it), we obtained the spectra of a single MD C-MS NW and an undoped C-MS NW reference displayed as red and black solid lines in Figure 2b, respectively. The sharp peaks at 33.8, 35.9, and 46.7 meV are observed in both spectra with about the same intensities. They are attributed to the GaAs transversal optical (TO), the GaAs-like longitudinal optical (LO), and the AlAslike LO phonon modes, respectively.10 The black trace originating from the undoped sample exhibits a weak broad background peaking at around 40 meV, which corresponds to an absolute energy of about 1.88 eV. This spectral feature is attributed to the GaAs split-off valence band “hot” luminescence.2,21 The background is much stronger for MD C-MS NWs and exhibits in addition three peaks at 20 ± 2, 30 ± 1, and 47 ± 1 meV that are labeled D1, D2, and D3, respectively. We conducted a Gaussian peak fitting of the D3 resonance displayed as a solid red line in the inset of Figure 2b. Here, we assumed a linear background that is displayed as a green dashed line. The width of the resonance is w ≈ 7 meV. Notably, the D1 and D2 resonances have similar widths. Additionally, we performed measurements in a polarized configuration where both the incident and the analyzed radiation fields are polarized parallel to the main NW axis under otherwise identical measurement conditions. The spectra are shown in Figure 2c, where the red and black solid lines originate from the MD and undoped C-MS NW, respectively. Both spectra in Figure 2c show the GaAs TO, the GaAs-like LO, and the AlAs-like LO peaking at the same energetic positions but with a higher absolute intensity with respect to the spectra in the depolarized configuration (Figure 2b). Two additional sharp peaks are observed at 36.7 and 45.4 meV that we identify as the GaAs LO and the AlAs-like TO phonons, respectively.10,21 The increased intensity of the phonon modes in the polarized configuration is in accordance with common
information about spatial inhomogeneities along the NWs is provided.17−19 Here we use resonant inelastic light scattering (ILS) by electronic excitation as a well established, contactless, and nondestructive spectroscopic technique to probe the electronic properties of MD C−S hetero NWs with high spatial resolution. It was demonstrated already 30 years ago that resonant ILS allows separation of, for example, charge density and spin density excitations which in turn lead to information about the electronic structure, the carrier density, the carrier− carrier interactions, the scattering times, and the dimensionality of free charge carriers.20−25 Simultaneously, ILS provides information on structural properties such as crystal phases, defects, and alloy composition ordering of NWs via the dielectric response of crystal excitations with a spatial resolution of few hundred nanometers.10,26−28 In this article, we report on the synthesis as well as the electronic and coupled electronic-structural properties of GaAs/Al0.16Ga0.84As MD core−multishell (C-MS) NWs. We have observed charge and spin density excitations at cryogenic temperatures in such hetero NWs, and in combination with self-consistent calculations we have obtained quantitative information on the density, the mobility, and the spatial distribution of electron gases in hexagon-shaped coaxial quantum wells (coQW) around the core of the nanowires. Figure 1a displays a typical scanning electron microscope (SEM) image of an array of GaAs/Al0.16Ga0.84As C-MS NWs
Figure 1. Metrology of core−multishell nanowires grown on silicon. (a) Scanning electron micrograph of an array of GaAs/Al0.16Ga0.84As C-MS NWs grown on a (111) silicon substrate. (b) False-color STEM-HAADF micrograph of a typical cross-section of a GaAs/ Al0.3Ga0.7As C-MS NW with a 20 nm thick embedded GaAs coQW. (c) Sketch of the cross-section of the investigated GaAs/Al0.16Ga0.84As MD C-MS NW with its estimated dimensions along with the defined axes convention. In panels (b) and (c), GaAs NW core and QW are highlighted in light blue, while the AlxGa1-xAs shells are highlighted in orange.
grown on a (111) Si substrate. The NWs were grown by solid source molecular beam epitaxy (MBE) using a self-catalyzed vapor−liquid−solid growth method (for the NW core) combined with a vapor-solid growth method (for the radial NW shell layers).10 Full details of the fabrication and basic ex situ characterization processes are discussed in the Methods section. Figure 1b shows exemplarily a cross-sectional highangle annular dark field/scanning transmission electron microscope (HAADF-STEM) image of a GaAs/Al0.3Ga0.7As C-MS 6190
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Figure 2. Inelastic light scattering spectra of single core-multishell nanowires. (a) Calculation of the electron distribution at a dopant density ndop = 1.6 × 1018 cm−3. (b) The ILS spectrum of a single MD (undoped) C-MS NW in the depolarized configuration is shown as a red (black) solid line. The broad excitation labeled D3 is displayed in the inset on an expanded scale, where full dots are experimental points, the red solid line a Gaussian fit, and the green dashed line the estimated linear background. The lower panel displays simulations of spin density excitations for various doping densities (see text). (c) ILS spectrum of the identical MD (undoped) C-MS NW measured in the polarized configuration under otherwise identical conditions is shown as a red (black) solid line. The lower panel shows the simulations of charge density excitations for carrier densities as in panel b (see text). Vertical lines in panels b and c are a guide to the eye.
selection rules for arsenide-based NWs.10,21,26,27,34 The polarized spectrum of the MD C-MS NW exhibits in addition broad and partially asymmetric peaks at 25 ± 4, 40 ± 1, and 51 ± 2 meV labeled P1, P2, and P3, respectively. Interestingly, the spectrum has a pronounced dip at 36.7 meV, the spectral position of the GaAs LO mode, close to the P2 spectral feature. Our experimental observations are qualitatively in line with many reports of ILS performed on planar MD GaAs/ AlxGa1−xAs heterostructures.1,2,21,22 In these reports, spectral features recorded in depolarized configurations are associated with spin density excitations (SDEs) and those observed in polarized configurations with charge density excitations (CDEs).21,22 SDEs reflect in good approximation single particle excitations since exchange interaction are usually small in GaAs.21 On the other hand, CDEs (plasmons) carry a macroscopic electric field (the depolarization field) as a consequence of the direct Coulomb interaction that evokes a blueshift of excitations with respect to the bare intersubband spacings as well as coupling with LO phonon modes.21,22 We have performed calculations of the energies and intensities of SDE and CDE modes in a multisubband nonresonant formalism (see the Methods section)35 and compared the results with our experimental observations. We expect a good description of the observed ILS shifts. However, only a qualitative evaluation of ILS cross sections is expected, as resonance effects which are not included in the theory severely affect the relative intensities.21,22 The lower panels of Figure 2b,c show the calculated SDE and CDE spectra, respectively. Results are shown as calculated for different doping concentrations ndop = 1.2, 1.4, 1.6, 1.8, and 2.0 × 1018 cm−3. We find the best agreement between experiments and theory for a dopant density of 1.6 × 1018 cm−3, corresponding to a 1D electron density of 2.96 × 107 cm−1
within the coQW. The widths of the electronic excitations correlate with the electronic damping parameter Γ.36 Γ = 2 meV is found to reproduce best the experimentally observed linewidths (see the P1, D3, and P3 peaks in Figure 2b,c). This electronic damping corresponds to an electron mobility of the order of μ ≈ 50 000 cm2/(V s) considerably higher than previously reported for nanowires.37,38 The increased electron density at the six corners of the hexagon shaped coQW (see Figure 2a), can be regarded as six quasi one-dimensional (q1D) channels in the corners and quasi two-dimensional (q2D) channels in the facet planes. In the following, we will label the resulting electronic subbands as Cn and Fn for 1D-corner and 2D-facet states, respectively, where n denotes the number of nodal surfaces pallel to the coQW interfaces, similarly to a planar QW. Few states are localized in the GaAs NW core, but contribute very little to the calculated ILS, and will not be discussed further. Figure 3a shows the DOS in arbitrary units of a C-MS NW with a dopant density n = 1.6 × 1018 cm−3. Since electrons occupy 1D states, each level i shows the characteristic DOS behavior ∝(E − Ei)−1/2. The lowest energy state stems from the six degenerate states localized in the corners (see inset C0). The next set of states are localized along the facets (see, e.g., inset F0). States between F0 and the next corner state C1 have an increasing number of azimuthal nodes with nodal planes normal to the coQW plane due to quantization of the 2D electron gas wrapped around the NW axis. These states have a small energy separation, and the DOS approximately sums up to the characteristic constant behavior of 2D systems, as highlighted by the red line in Figure 3a. As expected, radial nodes with nodal planes parallel to the coQW plane first show up in corner states (inset C1) and then in facet states (see e.g. inset F1), which build up the second step in the 2D DOS. A 6191
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Figure 3. Calculation of the density of states and respective charge and spin density excitations. (a) Normalized DOS for a MD C-MS NW with a density of dopants ndop = 1.6 × 1018 cm−3. The horizontal gray line indicates the position of the Fermi energy. The insets show 2D maps of the squared envelope functions for selected states. The bold red solid line is a guide to the eye to emphasize the seemingly 2D profile of the DOS. (b) CDE and (c) SDE spectra. Total (red solid lines) and projected (blue, green, black, and orange lines) ILS spectra. Projected excitations are labeled as corner, facet, corner-facet, and core excitations if they refer to Fn → Fm, Cn → Cm, Cn → Fm, or Fn → Cm, core →core single particle excitations, respectively. Red dashed lines represent energy positions of the experimental peaks. The insets show 2D maps of the IDD (see Supporting Information) induced by photon scattering at the indicated energies. The top right inset in panel (c) illustrates the scattering geometry.
CDEs in the top and bottom facets (with one nodal surface parallel to the QW plane) and intrasubband CDEs along the lateral facets (with nodal surfaces perpendicular to the QW plane). Peak P1 is energetically close to the GaAs LO phonon resonance and is strongly affected by the plasmon-phonon coupling.21,22 As a consequence, two plasmon-phonon coupled modes arise on opposite sides of the phonon resonance, while at the GaAs LO phonon CDE modes vanish. This produces the pronounced dip in the experimental spectra (see Figure 2c). In Figure 3b, therefore, peak P2 is interpreted as the mate of P1. This is confirmed by the calculated evolution of the P1 and P2 peaks versus doping, shown in the lower panel of Figure 2c. Here, the P2 peak gains intensity as P1 approaches the phonon resonances with increasing doping. A relevant contribution to this resonance is also given by the C1 → C2 and C1 → F2 transitions, which happen to be at the same energy, resulting in a second radial node in the IDD (see inset). Finally, peak P3 corresponds to a CDE mode associated with the C0 → F2 and F0 → F2 transitions, namely excitations with Δn = 2, located in the top and bottom facets of the coQW, as shown by the corresponding IDD. The IDD also shows that the Δn = 2 transitions occur in association with variations in the number of azimuthal nodes in the side facets. The calculated SDE spectrum, shown in Figure 3c, has two maxima in good agreement with the resonances D1 and D3. Comparing the corresponding IDDs with those in the CDE
further sequence of corner/facet states follows. Clearly, the DOS shows the concurrence of q1D and q2D character of the electronic states. CDEs and SDEs are associated with transitions from occupied to unoccupied states. At the present carrier density C0, F0, and C1 states are below the Fermi energy. In Figure 3b,c, we show the calculated CDE and SDE spectra together with the projected Raman spectra, that is, the contribution to the total spectrum arising from excitations of a specific type Fn→Fm, Cn→Cm, and Cn → Fm plus Fn → Cm screened by all excitations (see Supporting Information). The calculated CDE spectrum of Figure 3b shows three relative maxima matching very well the resonances P1, P2, and P3 observed in the polarized configuration (see Figure 2c). Peak P1 is the strongest in our calculations, and it turns out to be mainly due to the concurrence of two types of CDEs, (i) intersubband transitions with Δn = 1 originating from F0 → F1 and C0 → F1 transitions, the latter contributing to the higher energy part of the peak and (ii) intrasubband transitions, that is, the intrasubband plasmons of the coQWs. Note that in a flat QW, such intrasubband plasmons are forbidden with the laser normal to the QW plane.21,22 In the current NW, intrasubband plasmons are activated through the sidefacets with oblique orientation with respect to the incident photons. The calculated induced density distribution (IDD) (see Supporting Information) at the P1 resonance (see corresponding inset) clearly shows the two types of collective excitations, intersubband 6192
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Figure 4. Electron gas and structural defects along a single modulation doped nanowire. (a) Colormap of ILS spectra normalized to the GaAs TO intensity recorded in the depolarized configuration as a function of spatial position along the MD C-MS NW main axis (step size: 500 nm). (b) Normalized spectra at selected positions recorded in the depolarized configuration. The spectra have been shifted vertically for clarity. (c) Color map of normalized ILS spectra of the same C-MS NW as in panel a but in the polarized configuration (step size: 500 nm). (d) Spectra at selected positions recorded in the polarized configuration. The spectra have been shifted vertically for clarity. A linear background was subtracted from all ILS spectra shown in panels a−d. (e) Spatial PL mapping along the axis of the same MD C-MS NW (step size: 500 nm), as well as a typical spectrum of an undoped NW as a reference. Spectra have been shifted vertically for clarity. Vertical dashed lines and the bold dash-dotted line are a guide to the eye.
following two main observations can be made in the depolarized configuration: (i) at z < 1 μm, the intensity of SDE modes reduces, while the forbidden34 GaAs-like LO and AlAs-like LO phonon excitation gain in relative intensity with respect to the GaAs TO mode; and (ii) at the opposite end (z > 3 μm), the normalized intensity of the D2 resonance gradually increases. The detailed spectrum at z = 4 μm in Figure 4b shows that the D1 resonance vanishes while D2 redshifts by about 2.5 meV. Electronic resonances suppression and enhancement of LO phonon modes are also observed in CDE modes at z < 1 μm (see Figure 4c,d). The reduction of the intensity of all electronic excitations at z < 1 μm can be traced to either a reduced density or a reduced mobility of electrons. These are most probably caused by twinning defects and crystal phase changes26,27,29 as often observed at the ends of the NWs, and as also confirmed by TEM data on NWs grown under similar conditions. Such defects also lead to an enhanced intensity of forbidden Raman modes.34 In order to confirm this interpretation, we performed also spatially resolved PL measurements of the same GaAs/ Al0.16Ga0.84As MD C-MS NWs (see Methods section), shown in Figure 4e. We observe position-dependent PL spectra along the nanowire. At z < 1 μm, we mainly observe PL signals below
spectra shows that peaks D1 and D3 are the SDE counterparts of peaks P1 and P3. The resonance D2 at ∼30 ± 1 meV on the other hand does not seem to have a corresponding peak in the simulation. We recall, however, that valence band resonance effects are not included in the theoretical model. The large intensities of peaks D2 and D3 in the experiment suggest that both peaks involve similar resonantly populated final states. In this regard, we know from the previous analysis of resonances P3 and D3 that such final states are basically of the F2 type. Therefore, transitions of type C1 → F2, which are allowed but are weak in our calculations, would be strongly enhanced by resonance effects. From our calculations, the energy for such an SDE, which is roughly equal to the subband splitting, is ∼30 meV. Our non resonant calculated SDE spectrum indeed shows a weak resonance at this energy, as shown in the inset corresponding to D2. Therefore, we assign peak D2 to a SDE associated with C1 → F2 intersubband transitions. The MD C-MS NWs display spatial variations of the ILS shifts and intensities of SDEs and CDEs along the NWs likely due to structural inhomogeneities along the MD C-MS NWs, as we show next. Spatial profiles recorded in the depolarized and polarized configurations are shown in Figure 4a,c, respectively, in color code. Individual spectra at selected positions z = 0.5, 2.0, and 4.0 μm are shown in Figure 4b,d. The 6193
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the outer interface of the 26 nm GaAs coQW by closing the Ga shutter and opening a Si effusion cell for 10 min. In order to experimentally confirm the nominal values of the shell layer thicknesses, we conducted atomic force microscopy and SEM measurements on single C-MS NWs mechanically transferred on top of a Si substrate. For determining the core thickness, we synthesized GaAs NWs employing identical growth conditions as for the MD and undoped GaAs/Al0.16Ga0.84As C-MS NWs but terminated the growth before initializing the capping sequence described previously. Following this approach, we found an inner core diameter d of 50 ± 3 nm and a total shell thickness of 149 ± 8 nm in excellent agreement with the total nominal shell thickness of 151 nm. For the cross-sectional HRTEM investigations, we additionally fabricated C-MS NWs with higher Al contents (x[Al] = 0.3) in order to improve the material contrast of the AlxGa1−xAs layers with respect to the GaAs layers. The GaAs/Al0.3Ga0.7As C-MS NWs displayed a nominal layer sequence of 30 nm Al0.3Ga0.7As, 20 nm GaAs, 70 nm Al0.3Ga0.7As, and 10 nm GaAs. We mechanically transferred the C-MS NWs on top of a Ptcoated glass substrate and cut lamellas out of carbon-coated CMS NWs ensembles using a Zeiss focused ion beam (FIB) apparatus as reported recently.10 Analyzing the cross sections of nine individual C-MS NWs, we determined a shell layer sequence of 31 ± 2 nm AlxGa1−xAs, 20 ± 1 nm GaAs, 70 ± 1 nm AlxGa1−xAs, and a partially oxidized GaAs layer10 in excellent agreement with the nominal growth conditions. Here, we confirmed the nominal values of the GaAs coQW layer thickness with a statistical error of 1 nm. The homogeneous Al content x along the NWs has been also confirmed by spatially resolved phonon Raman spectroscopy. Inelastic Light Scattering and Photoluminescence. For the ILS and PL investigations, MD and undoped GaAs/ Al x Ga 1−x As C-MS NWs were transferred on top of prepatterned Si substrates in order to facilitate polarizationdependent ILS and PL measurements of individual NWs. The measurements were performed in a He flow cryostat at T = 10 K. For the ILS experiments, the C-MS NWs were subsequently excited with the 1.916 and 1.833 eV lines of a Kr+-laser with power densities of 2 kW/cm2. The incident light with its polarization rotated by a λ/2 plate was focused via an objective lens (NA = 0.75) and the measurements were performed in backscattering geometry. We measured the spot size of the laser beam to be 900 ± 100 nm. The scattered radiation field was analyzed employing a polarization filter in combination with a λ/2 plate. Subsequently, the scattered radiation was collected by a Dilor XY triple Raman spectrometer equipped with a N2 cooled multichannel charge coupled device (CCD). For the PL experiments, we excited the same individual C-MS NWs with the 1.959 eV line of an HeNe laser focused by a NA = 0.5 objective lens. We collected the PL spectra with a Princeton instruments Acton spectrometer equipped with a N2 cooled multichannel CCD. Employing a set of polarization filters and λ/2 plates, incoming and emitted light polarizations were selected to be parallel to the main C-MS NW axes. Calculation of ILS Cross Section. We obtain the ILS cross section of CDEs from the imaginary part of the reducible response function ICDE(Q,ω) ∝ −; [Π̃(Q,ω)], and the cross section of SDEs from the irreducible response function ISDE(Q,ω) ∝ −; [∏(Q,ω)]. Here, ω = ωi − ωs is the ILS shift between the scattered and incoming photon frequency ωs, ωi, respectively. Q = Qi − Qs is the transferred momentum, where Qi and Qs are the incident and scattered photon
the free exciton line in zincblende (ZB) GaAs of 1.51 eV, the dominant one at about 1.476 eV. Spectral features at such lower energies are typical for indirect transitions due to the type II band alignment of wurtzite−zincblende (WZ−ZB) heterostructures.39,40 An additional peak evolves for z > 1 μm above the free exciton line. Moving along the z-direction, this peak shifts in energy up to 1.522 eV at z = 3.5 μm and then moves downward again and disappears at z = 4.5 μm. For comparison, a typical PL spectrum recorded at the center of an undoped C-MS NW is shown in the lower panel of Figure 4e. Here, the FE peak is resolved at 1.517. The FE peak intensity exceeds all PL signals found in the MD NWs by a factor of at least eight. An inhibition of the FE is known to occur due to screening effects caused by a free electron density.41 The additional peak which shifts between 1.519 and 1.522 eV on the other hand indicates a varying free carrier concentration along the NW and is due to a many-body singularity at the Fermi edge that is especially enhanced in 1D systems.41 Spatial variation of the electron density also correlates with the behavior of D1 and D2 excitations along the wire (see Figure 4a,b). The observed redshift of 2.5 meV of resonance D2 in Figure 4b is consistent with calculated spectra at higher energies. Indeed, with increasing EF F1 states become occupied and the F1 → F2 transition, which is about 3 meV lower in energy than the C1 → F2 transition that we previously associated with the D2 resonance, is expected to contribute to the depolarized ILS spectrum. The lower intensity of resonance D1 at z = 4.0 μm also confirms this interpretation. Indeed, when F1 is occupied, C0 → F1 transitions, which contribute to the D1 resonance, are inhibited and the scattering cross section of D1 reduces. On the other hand, the spectral positions of CDE modes depend on the direct Coulomb interaction and blueshift with increasing EF. This is observed indeed comparing the P1 mode at z = 2.0 μm and at z = 4.0 μm, where P1 appears blueshifted by about 8 meV, which is again in fair agreement with the calculated shift of ∼5 meV in calculations at ndop ∼2 × 10−18 cm−3 (see Figure 2c). In summary, we have demonstrated 1D and 2D free electron gases in MD C-MS NWs with a high electron mobility using spatially resolved resonant inelastic light scattering and PL measurements. However, the electronic properties are inhomogeneous along the individual nanowires and high mobility electron gases are only achieved in certain regions along the NW axis.
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METHODS Nanowire Growth and Structural Characterization. The GaAs/Al0.16Ga0.84As C-MS NWs were synthesized in a solid source MBE chamber on a SiO2/Si(111) substrate patterned by nanoimprint lithography.42 First, GaAs core NWs were grown by catalyst-free vapor−liquid−solid (VLS) growth employing a substrate temperature Tsub of 610 °C, a beam equivalent arsenic pressure BEP(As4) of 2 × 10−6 mbar, and a Ga flux of 0.025 nm/s. After decreasing Tsub to 490 °C and increasing BEP(As4) to 3.5 × 10−5 mbar, the GaAs NWs have been overgrown subsequently in situ by Al0.16Ga0.84As, GaAs, Al0.16Ga0.84As, and a final GaAs cap layer with nominal widths of 25, 26, 87, and 13 nm, respectively. Here, the nominal Al content x[Al] as well as the nominal thicknesses di have been obtained from the measured flux ratios and calibrations as described in ref 10. For modulation doping, a Si delta layer has been introduced to the shell at a distance of 25 nm away from 6194
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INDEX and the CINECA award under the ISCRA initiative for the availability of high-performance computing resources and support. M.R. and G.K. acknowledge project Generalitat Valenciana VALi+d Grant (M.R.) and SFB-631, Marie Curie Integration Grant, EU-SOLID, respectively, for financial support.
momenta, respectively. The momentum dependent reducible response function Π̃(Q,ω) is (the irreducible response function is analogously defined) Π̃(Q , ω) =
∑ Π̃ijlm(qz , ω) ∬ dr dr′e−iq(r− r′)φi*(r)φj(r) ijlm
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× φl(r′)φm*(r′)
The φα are the in-plane part of envelope functions that are self-consistently calculated in a density functional theory−local density approximation (DFT-LDA) model described elsewhere31 and q, qz are in-plane and in-wire components of Q = (q, qz), respectively. The matrix elements of Πijlm and Π̃ijlm are obtained from a Dyson-type equation in a standard timedependent LDA formalism.43 The coupling of CDEs with the GaAs LO phonon has been taken into account by means of a phenomenological frequency dependent dielectric constant (see Supporting Information). The self-consistent calculations of the confined states is performed on a triangular grid with 5.26 points/nm2 using material dependent parameters m* = 0.067, ε = 13.8, for GaAs and m* = 0.08, ε = 12.98, for Al0.16Ga0.84As, and a conduction band offset 0.146 eV between the two materials.44 Here, the material modulations in the NW cross-section are taken into account, while translational invariance along the growth direction is assumed. ILS cross sections are calculated including up to 150 levels (all levels contributing to the DOS of Figure 3a) in the response function. Since the experimental excitation energy 1.916 eV is large with respect to the SDEs and CDEs, we approximate |Qi| ≈ |Qs| ≈ 6.9 × 105 cm−1 in the calculation.
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ASSOCIATED CONTENT
S Supporting Information *
Additional material and information about theoretical model and calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. S.F., M.R., I.Z., A.B., G.G., and G. A. wrote the manuscript. G.A., S.F., and I.Z. conceived the sample design and the experiments. A.B., G.G., and M.R. designed and performed the theoretical calculations. M.B., G.K., and D.R. synthesized the samples. S.F. recorded the ILS spectra and took part in all other measurements. B.M. and D.R. performed the PL and A.B. performed the Magneto-PL experiments. J.B. took the SEM pictures. FIB preparation was performed by S.Ma., HRTEM measurements by M.D. and S.Mo. S.F., M.R., and I.Z. contributed equally. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge financial support by the Deutsche Forschungsgemeinschaft via the Excelence Cluster “Nanosytems Initiative Munich” and the TUM Institute for Advanced Study as well as partial financial support from EU-MC network 6195
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