Letter pubs.acs.org/NanoLett
Far-Field Emission Patterns of Nanowire Light-Emitting Diodes Junichi Motohisa,* Yoshinori Kohashi, and Satoshi Maeda Graduate School of Information Science and Technology, Hokkaido University, North 14 West 9, Sapporo 060-0814, Japan S Supporting Information *
ABSTRACT: We investigated far-field (FF) emission patterns of nanowire lightemitting diodes (NW-LEDs). NW-LEDs were fabricated using vertical InP-NW arrays with axial pn-junctions grown on InP (111)A substrates, and the emission intensity of NW-LEDs was measured as a function of view angle θ, where θ = 0° indicates the direction normal to the substrate or that along the NWs. For NW arrays with pitch a of around 1 μm, we found a clear dip in the emission intensity at θ = 0°, which was explained by an analogy with dipole antenna, or a smaller contribution of the lowest order guided modes for emission as compared with higher order guided and free-space radiation modes. Results of the simulation of radiation patterns by the finite-difference time-domain method and near-field to far-field transformation are also described. They also confirm that the dip at θ = 0° is specific to light emission from NWs. We also investigated the dependence of the FF pattern on the pitch of the NW array, and the observation was qualitatively explained by the relative contribution of the guided and free-space radiation modes. KEYWORDS: nanowire, metal-organic vapor phase epitaxy, light-emitting diode, far-field pattern
S
electronic structures further contribute to the polarization dependence, together with the structural anisotropy.12,13 Another interesting feature in NWs originating from shape anisotropy is the antenna effect. This includes the enhancement of light emission and absorption under particular conditions and the appearance of specific emission patterns as well as the polarization-sensitive optical response described above. Chen et al.14 studied the antenna effect in a light-scattering experiment in detail and showed that the intensity ratio of the backscattered light for incident electric field either parallel or perpendicular to the NW shows clear enhancement determined by d/λ, where d is the NW diameter and λ is the wavelength of light. They further showed the dependence of the intensity on the light polarization with respect to the NW axis, and that the observed polarization-angle-dependent intensity of the scattered light is reasonably explained by taking into account the radiation pattern of the NW antenna. Recently, Grzela et al.15 have also reported the antenna effect in the light emission of NWs. Notably, they showed that the emission from NW was maximum at a particular direction with respect to the principal axis of the NWs, and a single NW exhibited a specific emission pattern. Such peculiar far-field (FF) emission patterns have actually been predicted by Maslov and co-workers.16,17 The above experimental lines of evidence, together with the results of theoretical analysis, imply that the FF emission pattern is very important in device applications, because the emission properties of NW light-emitting diodes (LEDs) are different from those of conventional planar structures and they should
emiconductor nanowires (NWs) have recently attracted attention as a new class of materials having nanometerscale cross sections and one-dimensional shape. Because of their unique electrical and optical properties, the application of NWs to various nanoscale devices has been reported and it has been shown that they can improve the performance or offer novel functions compared with conventional devices based on thin films.1−4 The uniqueness of the electronic properties of NWs is indeed in their anisotropic one-dimensional shape, and they exhibit completely different properties from bulk materials of macroscopic sizes in three dimensions, or quantum dots and nanoparticles of nanometer scale in all three dimensions. For instance, electron transport has no restriction along the NWs but the electrons are strongly confined in the direction perpendicular to NWs, and if the diameter of NWs is sufficiently narrow electrons and holes are subject to quantum confinement, leading to enormous advantages for the fieldeffect-transistor applications of the NWs. The same is true for the optical properties of NWs. That is, the propagation of light is allowed along the NWs, while the light is confined in their orthogonal directions. This means that NWs possess properties of both a cavity5 and a waveguide.6 By utilizing the end facets of NWs, a Fabry-Pérot-type cavity has also been demonstrated,7 and hybridization of the optical modes along and normal to the NWs offers cavities with large Q-factors.8 Shape anisotropy also results in polarizationsensitive light absorption and emission, depending on whether the electric field of the light is either parallel or perpendicular to the NWs.9−11 Furthermore, if heterostructures or crystal structures with noncubic symmetry (for instance, hexagonal structures) are introduced into the NWs, anisotropy in the © 2014 American Chemical Society
Received: April 17, 2014 Published: May 12, 2014 3653
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depend on the diameter and length of NWs, pitch of the NW array, and wavelength. For example, when one tries to couple the emission from a single NW into optical fibers, it is necessary to optimize the extraction and coupling efficiency by designing a suitable size of the NWs. It is also important in NW-LED arrays, because the divergence of the beam or viewing angle becomes important when they are used for illumination or display applications. Furthermore, maximizing light extraction is important for achieving a high external quantum efficiency. Emission properties are also important for NW-based lasers, for instance, for reducing the lasing threshold. In this Letter, we report our experimental and theoretical investigation on the FF emission patterns in NW-LED arrays. NW-LEDs were fabricated using InP NW arrays formed vertically on the substrates. We found characteristics emission patterns of the NW-LED arrays in which the emission in the direction normal to the substrate (or the direction parallel to the NWs) is minimal. This result can be intuitively explained by the emission of electromagnetic field from a dipole antenna. Explanations based on the guided modes and free-space radiation modes, which respectively contribute to the emission from the NW edge and the sidewalls, and a simulation study are also described to gain further insight into the light emission properties of NWs. We also investigated the dependence of FF emission on emission wavelength λ and the pitch a of the NW array, and the latter is qualitatively explained by the relative contribution of emission from the NW edge and sidewalls. Experimental Procedure. The structure of InP NW-LEDs used in the present study is schematically shown in Figure 1a. The growth of InP NWs is reported in refs 18 and 19 and the fabrication of NW-LEDs is reported in ref 20. In short, we grew InP nanowire arrays by selective-area metal−organic vaporphase epitaxy (SA-MOVPE). The nanowires were grown on ptype (111)A-oriented InP substrates, and had an axial pnjunction (see Supporting Information for details). The top
contact for the n-type layer was formed with transparent indium tin oxide (ITO) after a polarization process using transparent polymer resin (benzocyclobutene, BCB). The contact for the back was formed with AuZn. A cross-sectional SEM image of the NW-LED is shown in Figure 1b. We prepared NW arrays with different NW diameter d, NW length l, and pitch a between the NWs. The average d ranged from 190 to 460 nm. The length l was not measured accurately for each NW-LED and was found to be dependent on a and d. It was in the range of 2 to 4 μm (see Supporting Information for the assessment of NW size). Emission intensity in the far field was measured with the experimental setup shown in Figure 1c. Here, electroluminescence (EL) from a LED was collected using an optical fiber located about 4−6 cm from the sample surface and was analyzed with a spectrometer equipped with a cooled charge-coupled device. EL spectra and intensity were measured as a function of on view angle θ. Figure 2 shows typical characteristics of a NW-LED measured at room temperature. Clear rectifying characteristics
Figure 2. Typical characteristics of NW-LED investigated in this study. (a) I−V (blue dashed line) and I−L (red solid line) characteristics and (b) electroluminescence spectra under forward bias for voltage from 0 to 2 V in 0.2 V steps. Photoluminescence spectrum is also shown in (b) by the blue dotted line.
with negligible forward bias leakage were confirmed in current− voltage (I−V) characteristics, as shown by the dashed blue line in Figure 2a. Linearity in current−light-output (I−L) characteristics (red solid line) was also confirmed except for the lowcurrent-injection regime. The turn-on voltage was about 1.2 V, which is slightly lower than the band gap energy of InP. Spectral shape is independent of the applied bias voltage V above the turn-on voltage. Two peaks were observed in the EL spectra in Figure 2b, and they are thought to be due to the formation of InP with different crystal structures. The position of the peak at the shorter wavelength (838 nm) is the same as that in the photoluminescence (PL) spectrum (blue dotted line), and originates from the wurtzite (WZ) portion of InP NW, while the EL peak at 900 nm is from the zincblende (ZB) portion. The difference between the PL and EL spectra are already reported in our previous paper,20 and it was attributed to the regions of emission in NWs being different between PL and EL experiments. Some nonuniformity was noted in the emission image, as shown in the inset of Figure 2a. The overall characteristics are not as good as those reported in our previous study,20 presumably because of the nonuniformity of the size of the NWs in the array, and the poor ohmic contact between nInP and ITO. This nonuniformity of the emission intensity indicates that the measured FF emission pattern is an average one of all NWs within the arrays. Similar I−V and I−L
Figure 1. (a) Schematic illustration of nanowire-LED. (b) Measurement setup of far-field pattern. 3654
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characteristics were obtained in other NW-LEDs with different a. The spectral shapes differ slightly between NW-LEDs with different a, and the difference was mainly in the relative EL intensity from ZB and WZ sections. Measurement Results. We first show a result of FF measurement for a sample with a = 1 μm and d = 230 nm in Figure 3. In Figure 3a, the EL intensity is shown in a color plot
Figure 3. (a) Color plot of EL intensity at constant bias voltage (2 V), as a function of wavelength λ and view angle θ. Brighter color represents stronger intensity. (b) Polar plot of spectrally integrated EL intensity. Integrated spectra in the two wavelength ranges indicated by arrows in (a) are shown. Figure 4. Polar plots of spectrally integrated EL intensity for NWLEDs with different pitch a of NW arrays; a is (a) 0.6, (b) 0.8, and (c) 1.5 μm. Panels on the right show the EL spectra at bias voltage of 2 V and at θ = 0°. The two arrows indicate the wavelength regions for spectral integration.
as a function of wavelength λ and view angle θ. The spectral shape was nearly independent of the view angle θ and only the intensity changes with θ. Note that the intensity change is not monotonic. That is, emission intensity at the far field for θ ∼ 0° was weaker than that for θ ∼ 0°. To see this dependence more clearly, we show a polar plot of the integrated EL intensity in two wavelength ranges in Figure 3b. The wavelength ranges of integration are shown by arrows in Figure 3a and correspond to emission from WZ (red) or ZB (blue) sections. Clear dips in the FF emission intensity at θ = 0° exist for both peaks. Thus, emission from NW arrays in the vertical direction is not as strong as in the slightly inclined direction. To verify the dip at θ = 0°, we studied various LEDs consisting of NW arrays with different NW pitches. Figure 4 shows the results of FF measurement for a = 0.6, 0.8, and 1.5 μm in panels a, b, and c, respectively. The dip in intensity at θ = 0° is observed in all the samples. It is noted that the dip becomes more clear with the increase of a, and angle of maximum emission intensity also increases with a. We actually have some difference in the NW size d in these samples because of the nature of the NW growth in SA-MOVPE,19 but the tendency on the magnitude of the dip on a was observed in our series of the NW-LEDs investigated in the present study and systematically changes with a (see Supporting Information). So, gathering all the data, we think the difference on the dip depth
is mainly attributable to the pitch of NW arrays. Furthermore, as shown in Figure 4c (a = 1.5 μm), the inversion of integrated intensity and the difference in the angle of maximum intensity around the two emission peaks were observed. Similar dependence on wavelength and θ were also observed in a series of samples with a = 1.5 μm. This clearly indicates the dependence of emission pattern on the emission wavelength for a sample with a = 1.5 μm, whereas it is not very obvious in samples with a ≤ 1 μm. Furthermore, we show the results of FF measurement for samples with a = 3 μm and d = 400 nm in Figure 5. We can follow the shift of the angular peak with the emission wavelength λ. The direction of the shift is both toward smaller and larger angles with the increase of λ. The shift of the maximal emission angle is also observed in a sample with a = 3 μm and d = 460 nm, but the direction is toward the lower angle with the increase of λ (see Supporting Information). We also investigated the dependence on the polarization of the detection. For polarization-dependent measurement, a polarizer is placed between the sample and the optical fiber, and 3655
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becomes different from each other. Nevertheless, because of the anisotropic shape and large dielectric mismatch of the surroundings, the emission of the z-polarized component is enhanced, resulting in the dip at θ = 0 for FF patterns, implying the antenna effect in emission from NWs. NW as a Waveguide. The aforementioned selective excitation for the dipole emission in analogy with infinitesimal dipole antenna model explains the absence of vertical emission. In reality, it is necessary to take into account of dipoles of other directions, particularly for thick NWs. When photons are created by the recombination of electrons and holes, some of them couple to the guided modes of the waveguide and propagate toward the two ends of the NWs; then the light is emitted at the edges of the NWs. This mainly contributes to the emission in the direction parallel to the NWs (in reality, the electromagnetic (EM) field distribution of the guided mode affects the radiation pattern, as we will describe later). The rest, which are not coupled to the guided mode, can leak out from the sidewalls of the NW. This is referred to as the free-space radiation mode and is thought to emit light predominantly in the direction normal to NWs, or θ = π/2. Therefore, the total FF emission pattern consists of the contribution of the guided and free-space radiation modes, and our experimental results suggest that the contribution of the guided mode to the FF intensity is smaller than that of the free-space radiation mode. To be more specific, the NW waveguide emits light from its end following the electromagnetic field distribution of each mode. To determine what types of modes are supported in NWs, we calculated the dispersion relation of a NW waveguide and show the results in Figure 6. Here, for simplicity we assume
Figure 5. Color plot of the EL intensity of samples with a = 3 μm and d = 400 nm at constant bias voltage (2 V), as a function of wavelength λ and view angle θ.
two types of polarization, namely, s- and p-polarizations with respect to the substrates, were measured. However, the FF patterns are very similar even with different d, and no clear polarization dependence was observed, although both polarizations are expected to have a large impact on the emission patterns. This is presumably due to the size inhomogeneity (in both the length and the diameter) of NW arrays; we will not discuss this further in the present investigations. Analogy with Dipole Antenna. To understand the FF emission patterns of NW LEDs, we start with an intuitive explanation for the dip at θ = 0° and show that it is an intrinsic property in the emission of NWs. That is, NWs are anisotropic dielectric medium and the optical transition is enhanced when the electric field, or the electric dipoles for the optical transition, are parallel to NWs. The emission pattern from a single point electric dipole oriented in the z-direction is expressed as21,22 F0(θ ) ∝ sin 2 θ
(1)
indicating that the radiation intensity is zero for θ = 0 and maximum for θ = π/2. Thus, if the dipoles are predominantly lay along NWs, we have a suppression of emission for θ = 0. Although the maximum at θ = π/2 cannot be observed owing to the presence of the substrate and the practical limit in the experimental geometry, this selective excitation of dipoles parallels to NWs should a main reason for the absence of vertical emission in NWs. This reminds us a similarly between NWs and dipole antenna. The far-field pattern of the emission from an infinitesimal dipole is also given by eq 1, because the direction of current and the dipoles associated with it are restricted to the direction of the antenna. Thus, both NWs and wire antenna are considered to be anisotropic medium for electromagnetic waves, and for this reason this effect is referred to as antenna effect in NWs, as discussed in refs 14 and 15. In this discussion, we assume that the thickness d and the length l of the NWs is much smaller than the wavelength λ and the dipole is oriented only in the z-direction. Because the present NWs have diameter comparable to λ, we must consider dipoles in arbitrary directions, and the emission pattern is given by the average of the contribution of dipoles in the three directions. This makes emission intensity more or less omnidirectional and isotropic. It should also be noted that the surface current and its distribution along the wire becomes important for a finite-length dipole antenna, so the situation
Figure 6. Dispersion of a cylindrical waveguide with core index n1 = 3.6 and with clad index 1. Here, r0 is cylinder radius and is given by d/ 2. The bottom colored region indicates normalized frequency ωr0/c for d = 200−290 nm and λ = 800−950 nm, and upper colored regions indicates ωr0/c for d = 400 nm and λ = 800−950 nm.
that the NWs form a lossless cylindrical core (with radius r0) surrounded by an air cladding. The refractive index n1 of the NW core is 3.6. The actual cross-sectional shape of the NW core is hexagonal, but we expect the difference in the shape to be compensated by taking the same geometric cross-sectional size for circular and hexagonal waveguides, as discussed in the case of hexagonal dielectric resonators;23 this is sufficiently 3656
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within the range of experimental error for the determination of the NW size. Because the difference in the refractive index between the core and cladding is large, the weakly guiding approximation is not valid. For this reason, we use HEmn, TE0n, and TM0n for the notation of the modes, where m and n respectively represent azimuthal and radial mode numbers. From the results in Figure 6, NWs with diameter d = 200−290 support a few guided modes, for instance, HE11, TM01, and TE01 modes, in our wavelength range of interest (800−950 nm). The lowest order mode, HE11, which has no cutoff frequency, has a maximum field intensity at its center and is known to exhibit maximum intensity for FF at θ = 0.24 On the other hand, higher order modes, such as TM0n, TE0n, and HEmn (with m ≥ 2), have a node at the center in their EM field distributions. Thus, these higher modes show minimal intensity at θ = 0 in the FF. These mode-dependent FF patterns for the edge emission have already been reported in cylindrical waveguides.16,24 The FF pattern is a superposition of each mode, and the emission from the sidewall gives an additional contribution to the pattern. These are thought to result in the position of the maximum FF intensity being slightly apart from the direction θ = 0. Numerical Analysis. The measurement of FF patterns in NW-based LEDs was first reported by Svensson et al.25 They reported the dip at θ = 0 as well as a broad radiation pattern. They attributed the dip to metal seed particles that resided on the NW, because the dip disappeared after the removal of the seed particles. As described above, our observed dip at θ = 0 is a characteristic feature of NWs. Theoretically determined specific FF patterns for a single NW have been shown by Maslov and co-workers16,17 and experimental ones by Grzela et al.15 Our results are consistent with these reports for single NWs. In addition to this specific dip at θ = 0, we confirmed that the emission pattern also depends on the wavelength, as typically shown in Figures 4c and 5. Furthermore, we found that the emission pattern is dependent on the pitch of NWs, as shown in Figure 4. The dependence on the wavelength of the FF pattern is expected, as discussed in ref 17 but has not yet been addressed experimentally. Below, we discuss the dependence of the FF pattern on the wavelength and the pitch of the NW array on the basis of numerical analysis, in which finitedifference time-domain (FDTD)26 simulation and near-field to far-field (NF2FF) transformation in the framework of surface equivalence theorem27 are combined. This method is a very general one for obtaining FF patterns of antennas and was first adapted to the emission from NW by Maslov and Ning.16 The geometry for the simulation in the case of full threedimensional (3D) calculation in Cartesian coordinates is shown in Figure 7a (see also Figure 1a for the corresponding geometry for NW-LEDs). For simplicity, we assumed a freestanding NW (refractive index n1) in air oriented in the zdirection of cylindrical shape with diameter d and length l. The center of the NW is set at the origin. The observation point for the FF is expressed in spherical coordinates and is given by the polar angle θ, which is defined with respect to the z-axis, and the azimuthal angle ϕ. The calculation procedure is as follows. First, FDTD simulations were performed using a freely available software package.28 An excitation point source p for FDTD was set at the center of the NW, unless otherwise specified. The polarization of the electromagnetic field is determined by the direction of p and can be expressed using polar angle Θ and azimuthal angle Φ. Since the polarization in the arbitrary
Figure 7. (a) Geometry of nanowires and surfaces for NF2FF calculation in three-dimensional representation. The dashed box represents the surface S of integration for NF2FF. Surface S is also divided into three sections, namely, sidewall S0, bottom S1, and top S2 surfaces. (b) Polar plot of calculated far-field intensities in various directions of polarization of excitation sources. Red, z-polarized excitation; blue, x-polarized excitation in the ϕ = 0° direction; and green, y-polarized excitation in the ϕ = 0° direction. (c) Full threedimensional representation of far-field emission pattern of x-polarized point source. The calculation was carried out for d = 230 nm, λ = 838 nm, and l = 2 μm. (d) Comparison of emission intensity from polarized and polarization-averaged (or unpolarized) sources.
direction is given by an appropriate superposition of three orthogonal polarizations, we consider x (Φ = 0, Θ = π/2), y (Φ = π/2, Θ = π/2), and z (Θ = 0) polarizations. We will also describe the results of a circular polarization (c-polarization) within the xy plane (Θ = π/2), which is given by a superposition of x- and y-polarized sources having a phase difference of π/2. The calculated EM field was then used to perform surface integration close to the NW in the framework of NF2FF. A rectangular cubic surface was taken as the integration surface so that it encompasses the cylindrical NW. Finally, the r-component of the Poynting vector of the far-field EM is calculated as the far-field emission intensity F(θ, ϕ). FDTD simulation and NF2FF transformation were also performed in cylindrical coordinates with azimuthal dependence exp(imϕ) for EM fields to save computation time, where m is an azimuthal mode number.16,29 The accuracy of FDTD and NF2FF was confirmed with an FF pattern of a point 3657
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electric dipole in vacuum. We note that our calculation model of isolated free-standing NWs is different from the realistic situations of our NW-LEDs. That is, the NWs are formed on the substrate, embedded with polymer resin, and capped with an ITO electrode. In addition, the measurement was also conducted for NW-LED arrays. Nevertheless, we mainly present the numerical results for an isolated single NW, because many of the essential features of the FF emission are found to be derived in this situation (see Supporting Information for details of the calculation procedure, the relevance between simulations in 3D with Cartesian coordinates and in two dimensions (2D) with cylindrical symmetry, the effect of averaging over the orientations of the dipoles, and the comparison of the results taking into account more realistic situations). Typical results of the simulation are shown in Figure 7b. The parameters for the simulation were d = 230 nm, l = 2 μm, wavelength λ = 830 nm, and refractive index of NW n1 = 3.6. In Figure 7b, polar plots of the FF intensity in the z, x, and ypolarizations are shown for ϕ = 0. For the excitation with the zpolarized point source, the FF intensity Fz is independent of azimuthal angle ϕ. For x- and y-polarized sources, the results are dependent on ϕ, but the x-polarization (Φ = 0) Fx with ϕ = 0 and y-polarization (Φ = π/2) Fy with ϕ = π/2 are identical. A full emission pattern Fx(θ,ϕ) for x-polarization is reproduced in Figure 7c. As we can see in Figure 7b, emission at θ = 0 is zero for z-polarization. This is in agreement with intuitive explanations given at the beginning of this section. For ypolarization, the dip at θ = 0 can also be observed. In the experiment, the FF pattern is given by the average of the contribution of dipoles in the NWs. If the orientation (Φ,Θ) of the dipoles is isotropic and random, as in the case of NWs with the ZB crystal structure, the result of averaging, or the result for an unpolarized source, is expressed as Fxyz(θ ) =
1 1 [Fx(θ , 0) + Fy(θ , 0)] + Fz(θ , ϕ) 4 2
Figure 8. Contribution of FF pattern and intensity from the sidewall and top surface of NWs for z-polarized and c-polarized sources. (a) Contribution of sidewall surface (S2) with z-polarization (or mode m = 0), (b) contribution of top surface (S2) with c-polarization (or mode m = 1), (c) contribution of S0 with z-polarization, and (d) contribution of S2 surface with c-polarization. The far-field intensity is shown as color plots as a function of view angle θ and wavelength λ. The calculation parameters are d = 230 nm, l = 2 μm, and n1 = 3.6.
the TE0n/TM0n and HE1n modes. Among these modes, HE11, TE01, and TM01 modes are guided for d = 230 nm, as shown in Figure 6. It is also noted that because the total FF intensity (integrated emission intensity in the whole surface S) depends strongly on λ, the visualized FF intensity is normalized with the total emission intensity in each wavelength. It is clear that the emission in the direction θ = 0 is mainly from the top edges of NWs (surface close to S2) with c-polarization. The separation of contributions to the FF pattern from each surface also explains the dependence on the pitch a of NW arrays, as shown in Figure 9. When the pitch is sufficiently large, the FF contribution is from all surfaces of NWs. In the actual experimental geometry, the NW is formed on a substrate, which has the same refractive index as the NW, and the effect of the substrate should be taken into account. The substrate does indeed affect the FF total emission. However, we confirmed that the effect of the substrate does not change our main conclusion that the emission toward the direction θ = 0 is due to the emission of HE11 mode at the NW edges and that the sidewall emission contributes to other directions. Hence, the experimental results are reasonably explained by the antenna and waveguide models, and the numerical analysis is discussed above even with the presence of the substrate. Now, if one looks at Figure 8 one can note that the dominant contribution for intermediate θ is that from the sidewall with z-polarization. It is thus likely that the dip at θ = 0 becomes less pronounced when the contribution of the emission from the sidewall becomes less important. Therefore, if the distance between NWs becomes short, the emission from the sidewalls is hindered or reabsorbed by adjacent NWs and only the surface S2 contributes to the FF intensity. This leads to the decrease in the dip depth at θ = 0, as observed in the experiment shown in Figure 4. Next, we will discuss the dependence on the emission wavelength λ. In the experimental results shown in Figure 5, we see some peaks that change the emission angle when the
(2)
where Fi(θ, ϕ) represents the FF intensity for a dipole polarized in the i-direction (i = x,y,z). The result is also shown in Figure 7d. It is noted that the FF intensity becomes independent of ϕ. In Figure 7d, results for z-polarization Fz(θ), c-polarization Fc(θ), and averaged polarization Fxy(θ) within the xy-plane, which is over Φ and is given by Fxy(θ) = [Fx(θ,0) + Fy(θ,0)]/2, are also plotted. As expected, within the range of numerical accuracy, Fc(θ) and Fxy(θ) give the same result. Note that in all the cases presented here we can see the dip at θ = 0. As we have discussed the FF pattern in the context of the emission from waveguides, the dipoles inside the NW excite guided and free-space radiation modes, and they are related to the emission from the edges and the sidewalls of the NWs, respectively. To discriminate the origin of FF emission the surface S for NF2FF was divided into three surfaces, namely, sidewall S0, bottom S1, and top S2 surfaces, as shown in Figure 7a, and their contributions to the FF intensity were calculated independently. The surfaces S0, S1, and S2 are thought to contribute to the FF emission pattern mainly from the sidewall, bottom, and top surfaces of NWs, respectively. The results of the calculation are summarized in Figure 8. Here, normalized FF intensity calculated in the 2D geometry with cylindrical symmetry is shown in color plots, as a function of view angle θ and wavelength λ for z- or c-polarized excitation sources. Note that excitation with z- and c-polarized sources is realized with m = 0 and 1, respectively. Thus, they are respectively related to 3658
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It is possible that the periodic arrangement of the NWs results in the formation of photonic bandstructure (photonic crystal effect), which modifies the emission pattern and the wavelength shift with respect to the view angle. Indeed, Wierer et al.30 experimentally reported spectral features associated with the photonic bandstructure in the angular-dependent emission patterns from the LED having periodic air-hole patterns. The directional emission from NW arrays was also discussed in terms of the coupling with the Bloch modes in periodic structure.31 In our present experiment, however, the electroluminescence spectrum showed rather broad two peaks, thus it seems hard to identify such effect in the spectrum domain. This is thought to be because of the nonuniformity in the size and length of the nanowires. For example, the size variation Δd of the NW is in the range of Δd/d = 10−15%, which is large enough to destroy such effects associated with photonic crystals. It should also be noted that our wavelength range (λ ∼ 850 nm), and thus the normalized frequency (a/λ > 1), is far above the first band of the photonic crystals that sits below a/λ ∼ 0.5; therefore it is hard to correlate them with the photonic bandstructure. So we think the effect of photonic crystal is not very important in our experiment. To get more insight for the effect of photonic crystal, we also carried out calculation of emission from NW arrays based on the scattering-matrix treatment of periodic layered structures.32 This calculation also showed that the emission toward the direction θ = 0 can be minimal, but in general the emission pattern showed very complicated dependence on the wavelength, pitch a of the NW array (some of the calculation results are described in the Supporting Information). Some of the peaks shift systemically with respect to θ and λ, but because of the higher bands are involved in the emission spectrum in our wavelength range of interest it is very hard to correlate the emission peak with the photonic bandstructure. Further study is necessary to elucidate the origin of the shift of emission angle with respect to wavelength λ in the negative direction. Finally, we would like remark on the total emission intensity. In reality, the size and the wavelength have a large effect on the total emission intensity, Furthermore, our calculation indicated that the position of the excitation source also influences the total emission intensity and FF pattern. These are due to the variation in the local density of states (LDOS) in the NWs. It is known that the total emitted power from a point dipole in a medium is proportional to the LDOS33 in it, and in general the LDOS at point r becomes large where the electric field at r of a particular mode is large. Furthermore, at the resonance frequency of the cavity, enhancement of the LDOS at the antinode of the electric field results in the large enhancement of the spontaneous emission rate, which is known as the Purcell effect. Enhancement of the spontaneous emission in NWs has already been reported by Bleuse et al.34 and by Bulgarini et al.,35 but in our experiment the enhancement of emission originating from LDOS was not observed. This is partly due to the size fluctuations of the NWs and, more importantly, to the weak cavity effect in NWs on the substrates, whereby the reflectivity on the substrate side of the NW is much smaller than at its other edge. In fact, in the aforementioned references the NWs were placed vertically on a gold mirror to obtain higher reflectivity. Confirmation of the enhancement of spontaneous emission is thus challenging in the present geometry and still requires further investigation. In summary, we carried out experimental and simulation studies of the far-field emission in InP-NW LEDs. We found a
Figure 9. Model used to explain emission from NW array and its dependence on pitch a. (a) Isolated free-standing single NW. The emission from NW can be in all directions. The surface for NF2FF transformation can be taken as whole surface S encompassing the NW. (b) NW array in which part of the emission is hindered by adjacent NWs. For this reason, emission from the sidewall is less important. To take into account this situation, far-field intensity based on NF2FF transformation was determined considering a contribution only of top surface S2.
wavelength is changed, and there are different types of peak shift, namely, peak shift toward smaller or larger angles, when λ is increased. It is noted that peak shift toward larger angles with increasing λ is observed in this sample with d = 400 nm at around λ ∼ 800 nm, but is not clear in samples with much larger d (see Supporting Information). This positive peak shift in the emission angle with increasing λ was first noted by Maslov et al.17 Their explanation for the shift toward larger angles with the increase of λ (decrease of frequency ω) is as follows. Above a cutoff frequency (or below the cutoff wavelength) of the waveguide, the EM field created by the dipoles couples to the corresponding guided mode and propagates toward the edges of the NWs. Thus, the emission from the edges is dominant. As the frequency becomes smaller (or the wavelength becomes longer), the EM field becomes less confined inside the NW and is concentrated more along the circumferences of the NWs. At the cutoff frequency, it becomes a free-space radiation mode and starts to leak out from the NW sidewalls. However, if the frequency is close enough to the cutoff, it is likely that the EM wave propagates along the NW with a small decay rate. In other words, it is considered that the modes close to the cutoff behave as the leaky mode. The further the frequency is from the cutoff, the more EM waves can leak out from the waveguide, and they leak out more in the outer directions, resulting in the increase in the angle of emission. We explain the shift of the angular position of the peak at λ ∼ 800 nm in Figure 5 on the basis of cutoff-related emission to the cutoff frequency, because the cutoff wavelength for the HE31 mode is 871 nm (ωr0/c = 1.429 and r0 = 200 nm), which, given the accuracy of the measured NW diameter, is reasonably close to 800 nm. On the other hand, the shift in the opposite direction is not well understood. We carried out a number of calculations with a different set of parameters for NW size and wavelength but were unable to conclude whether the decrease in the angular position with the increase in the wavelength is a general tendency for the NWs with this diameter. 3659
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Nano Letters
Letter
clear dip in the emission intensity at θ = 0, that is, the direction parallel to the NW (or sample surface). The simulation in which the finite-difference time-domain method and near-field to far-field transformation are combined reproduces the dip at θ = 0. Furthermore, we confirmed that the FF emission pattern also depends on the emission wavelength. These specific features in the emission from NWs were explained in terms of on the antenna emission, waveguide emission, and the results of a numerical analysis of the electromagnetic field in the far field.
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(17) Maslov, A.; Bakunov, M.; Ning, C. J. Appl. Phys. 2006, 99, 024314−024314−10. (18) Mohan, P.; Motohisa, J.; Fukui, T. Nanotechnology 2005, 16, 2903. (19) Kitauchi, Y.; Kobayashi, Y.; Tomioka, K.; Hara, S.; Hiruma, K.; Fukui, T.; Motohisa, J. Nano Lett. 2010, 10, 1699−1703. (20) Maeda, S.; Tomioka, K.; Hara, S.; Motohisa, J. Jpn. J. Appl. Phys. 2012, 51, 02BN03. (21) Jackson, J. D. Classical Electrodynamics, 3rd Ed.; John Wiley & Sons: New York, 2007. (22) Balanis, C. A. Antenna Theory: Analysis and Design; John Wiley & Sons: New York, 2005. (23) Nobis, T.; Grundmann, M. Phys. Rev. A 2005, 72, 63806. (24) Boncek, R. K.; Rode, D. L. J. Lightwave Technol. 1991, 9, 18−21. (25) Svensson, C. P. T.; Mårtensson, T.; Trägårdh, J.; Larsson, C.; Rask, M.; Hessman, D.; Samuelson, L.; Ohlsson, J. Nanotechnology 2008, 19, 305201. (26) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method; Artech House: Boston, 2000. (27) Balanis, C. A. Advanced Engineering Electromagnetics; John Wiley & Sons: New York, 1989. (28) Oskooi, A. F.; Roundy, D.; Ibanescu, M.; Bermel, P.; Joannopoulos, J. D.; Johnson, S. G. Comput. Phys. Commun. 2010, 181, 687−702. (29) Li, B.; Liu, P. IEEE J. Quantum Electron. 1997, 33, 1489−1491. (30) Wierer, J. J.; David, A.; Megens, M. M. Nat. Photonics 2009, 3, 163−169. (31) Fontana, Y.; Grzela, G.; Bakkers, E. P. A. M.; Rivas, J. G. Phys. Rev. B 2012, 86, 245303. (32) Whittaker, D.; Culshaw, I. Phys. Rev. B 1999, 60, 2610. (33) Oskooi, A.; Johnson, S. G. arXiv:1301.5366 [physics.comp-ph] (accessed 2013). (34) Bleuse, J.; Claudon, J.; Creasey, M.; Malik, N.; Gerard, J. M.; Maksymov, I.; Hugonin, J.; Lalanne, P. Phys. Rev. Lett. 2011, 106, 103601. (35) Bulgarini, G.; Reimer, M.; Zehender, T.; Hocevar, M.; Bakkers, E.; Kouwenhoven, L.; Zwiller, V. Appl. Phys. Lett. 2012, 100, 121106− 121106−4.
ASSOCIATED CONTENT
S Supporting Information *
Growth procedures for nanowires, summary of the measurement of far-field emission patterns, and numerical method including extended numerical analysis are described. 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]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Professor Takashi Fukui, Professor Kenji Hiruma, Professor Shinjiro Hara, and Dr. Katsuhiro Tomioka for support in the experiments and Professor Hajime Igarashi, Professor Masayuki Ikebe, and Professor Kunimasa Saitoh for fruitful discussion. The work is partly financially supported by a Grant-in-Aid for Scientific Research, supported by the Ministry of Education, Culture, Sports, Science, and Technology, Japan.
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