Designs for high-efficiency electrically pumped photonic nanowire single-photon sources

: We propose and analyze three electrically-pumped nanowire single-photon source structures, which achieve output efficiencies of more than 80%. These structures are based on a quantum dot embedded in a photonic nanowire with carefully tailored ends and optimized contact electrodes. Contrary to conventional cavity-based sources, this non-resonant approach provides broadband spontaneous emission control and features an improved fabrication tolerance towards surface roughness and imperfections. Using an element-splitting approach, we analyze the various building blocks of the designs with respect to realistic variations of the experimental fabrication parameters.


Introduction
The development of solid-state single-photon sources (SPSs) is a major challenge in the context of quantum communication, optical quantum information processing, and metrology [1]. Ideally, such a source must implement a stable single-photon emitter with a high radiative yield. For practical reasons, it is also highly desirable to trigger the single-photon emission by an electrical pulse. In this context, a semiconductor quantum dot (QD) is an attractive singlephoton emitter, which has a nearly perfect radiative yield and a stable emission (no blinking or bleaching). Moreover, it benefits from the maturity of semiconductor technology and can be incorporated into the intrinsic region of a p-i-n structure, to realize a single-photon LED [2][3][4][5][6][7]. For all potential applications, the efficiency ε of the source, defined here as the probability to collect a photon into the first lens of the optical setup, is a key figure of merit. Unfortunately, for a QD embedded in a semiconductor material, the large refractive index of the semiconductor prevents an efficient photon extraction and limits ε to below a few percent. The collection of photons can be strongly improved by inserting the QD in a resonant optical microcavity and, up to very recently, this strategy has provided the most efficient SPS. Thanks to their spectrally-narrow emission lines at cryogenic temperatures, the spontaneous emission (SE) of QDs can be controlled through cavity quantum electrodynamical effects. The Purcell effect that arises when a QD is inserted in a resonant optical microcavity with a high-quality factor (Q) and low-volume dynamically funnels a fraction β of its SE into the cavity mode [8][9][10][11][12]. The photon then escapes through a specifically designed loss channel to be collected by external optics with a probability γ. In this simple picture, the photon outcoupling efficiency ε 0 is given by ε 0 = ηβγ, where η the radiative quantum efficiency of the QD exciton. The total source efficiency reads ε = ε 0 ξ, where ξ is the pumping efficiency (defined here as the probability to inject at least one electron-hole pair in the QD per command pulse). In the past few years, fabrication progress has led to an impressive improvement of the microcavity figures of merit [13] which in turn brought β close to 100% [14], while η is usually close to 100% for self-assembled QDs at low temperature. However, ε remains so far limited to about 44% [10,11,15] for optically pumped SPSs and 34% for electrically pumped devices [7].
In fact, the far-field emission pattern of high-Q cavities is very sensitive to fabrication imperfections [16]. Thus, defects scatter photons in random directions, which limits γ. The issue is even more serious for electrically pumped structures, which require doping and contact electrodes, potential sources of additional optical losses. Moreover, combining a high efficiency with pure single-photon emission is desirable for most SPS applications. Under non-resonant optical pumping [10,11] or electrical pumping [7], most of the cavity-based SPSs exhibit an autocorrelation function at zero time delay g (2) (τ = 0) well above 0, when the QD is saturated. The origin of the detrimental multiphoton pulses under strong pumping is not yet fully elucidated. It could be related to non-resonant cavity feeding [17][18][19][20] or recapture of carriers if the injection of electrons and holes in the dot is not fast enough compared to the QD radiative decay. Finally, the exploitation of the Purcell effect is restricted to quasimonochromatic emitters and is effective only over the narrow bandwidth of the cavity resonance.
In this context, the non-resonant SE control provided by a one-dimensional photonic system [21-25] constitutes an appealing alternative to cavity designs. Recently, an optically pumped source design, relying on an InAs QD embedded in a GaAs photonic nanowire was proposed [26]. This geometry ensures a broadband and nearly perfect SE control for an emitter located on the wire axis; β exceeds 90% in a 200 nm wavelength range and features a reasonable tolerance towards off-axis positioning [27]. Moreover, an additional optimization of the far-field emission, using a SiO 2 -gold mirror and a smooth tapering of the upper extremity of the wire, led to a record-high measured efficiency of 72% [28]. The broadband SE control in photonic wires has also been recently used to demonstrate a 40% SPS efficiency at room temperature for a color center embedded in a diamond photonic nanowire [29].
In this work, we investigate the feasibility of high efficiency, electrically pumped, SPSs based on a quantum dot in the photonic nanowire geometry. Our focus is the photon extraction efficiency of the SPS, and we assume in the following that ξ = η = 1 and thus ε = ε 0 = βγ. Possible deviations from this ideal situation are discussed in section 6 of the paper. The InAs/GaAs material system, which is the most technologically mature and studied system, is chosen. The paper addresses in particular the associated issue of charge injection electrodes compatible with efficient photon extraction. We investigate three different structures, one inspired by Ref [26]. and two new designs that we believe more adapted to the realization of a practical, electrically pumped source.
The paper is organized as follows. In section 2 we describe the three electrically pumped structures that are analyzed in the paper. In section 3 we present the simplified model based on an element-splitting approach employed to determine the efficiency. We analyze the performance of the various elements of the structures in section 4. Initial experimental results, which confirm the technological feasibility of these designs, are presented in section 5. A discussion of the predicted efficiencies follows in section 6, and finally a conclusion and an outlook are given in section 7.

Nanowire SPS structures
In the following we study the GaAs nanowire with an embedded InAs QD emitting light at a wavelength of λ = 950 nm, and all subsequent computations are performed at this wavelength. We assume that the QD is positioned in the center of the nanowire and has an in-plane dipole orientation. The QD is approximated with a point dipole and we neglect the small perturbation induced by the refractive index profile of the finite size QD on the mode properties of the waveguide structures. We investigate the three electrically pumped SPS designs illustrated in Fig. 1.
All the designs are rotationally symmetric. The materials used are GaAs, Si 3 N 4 , polymer, gold and indium-tin-oxide (ITO). The polymer electrically isolates the contact sections and the GaAs nanowire is doped to create a p-i-n diode. The refractive indices of the materials employed in our simulations are given in Table 1. An absorption loss of κ = 10 cm −1 in the inverted tapers of designs B and C due to free-carrier scattering from n-doping with a carrier concentration of ~10 18 cm −3 [30] is included. The doping concentration may be increased locally near the contacts to ensure good electrical conductivity, however we do not expect local doping to have a major influence on the calculated losses in the tapers.
An optically pumped nanowire single-photon source design featuring a nanowire, a conical taper and a bottom metal mirror was described in Ref [26], and the design A represents the most simplistic extension of this design to accommodate electrical pumping. It features a nanowire with a quantum dot in section I ensuring a high β [36], a bottom mirror in section II reflecting light emitted towards the substrate [37], a regular conical tapering in section III-A for controlling the far field radiation pattern [38] and a top contact region in section IV-A.  In Ref [26]. the metal in the bottom mirror in section II is silver and a thin SiO 2 layer is sandwiched between the nanowire and the metal to improve the reflectivity [37]. For electrical pumping, we replace the electrically isolating SiO 2 layer with a conducting ITO layer, and, to avoid the rapid oxidation of silver in air, we replace silver by a gold bottom layer which is used for the bottom electrical contact. For the top electrical contact in section IV-A we choose ITO instead of gold, as the fundamental mode of the nanowire extends into the cladding material and propagation through a gold cladding results in a large loss reducing efficiency.
While the design A is conceptually simple, the fabrication of the side top contact without damaging the conical taper poses an experimental challenge. For this reason we propose the designs B and C, where well-established technology [7,39] from micropillar SPSs is employed for the top contact.
The design B features the same nanowire and bottom metal mirror sections I and II, however the strategy for controlling the far field radiation pattern is based on the inverse taper of section III-B rather than a regular conical one. In this design, the fundamental mode is adiabatically expanded inside the taper, which also guides electrical current to the quantum dot. The contact section IV-B consists of an easily deposited uniform layer of ITO serving the dual purpose as a top contact and an anti-reflection coating.
Finally, we propose the design C, where a gold ring slightly overlapping the GaAs cylinder is employed for the top contact [7,39]. The gold contact also scatters the fundamental mode, but for small d and large R top , this scattering can be negligible. The design C also features an air pocket surrounding the QD nanowire section. This air pocket leads to an increased index contrast and thus a higher β. These effects are further analyzed and discussed below.
Combinations of the elements from the designs B and C, e.g. a top ITO planar contact with an air pocket, can also be imagined.

SPS efficiency in the element-splitting approach
To determine the SPS efficiency ε we should place the point dipole inside the full structure and compute the corresponding far field emission. Though this approach is rigorous, it does not give a clear physical insight into the physical mechanisms governing the efficiency. For this reason we instead employ a single-mode model based on an element-splitting approach. This simplified model, the validity of which was demonstrated in Ref [26], allows us to analyze and optimize the various elements separately.
The model includes two approximations. First we assume that light scattered out of the fundamental mode into higher-order modes is not coupled back to the fundamental mode and does not reach the collection optics. This is a good approximation for a high-efficiency SPS, where the coupling of the HE 11 mode to higher-order modes is weak. Second, we assume that the SE rate into radiation modes, g Rad , for the full structure is identical to that for the isolated nanowire element I. This assumption will be justified by the agreement of the results from the simplified model with those obtained using an exact computation, as shown later.
We consider the propagation of the fundamental HE 11 mode in the full SPS structure, illustrated in Fig. 1 above. The coefficient r 11 , describes the amplitude reflection cofficient of the bottom mirror, and γ t and γ c are intensity transmission coefficients of the HE 11  (1) where γ = γ t γ c and g Total is the total emission rate. For the QD in the infinitely long nanowire the amplitude coefficients are given by c u = c d = c 0 ° sqrt(g HE11 / 2), where g HE11 is the SE rate into the fundamental mode. In the presence of the metal mirror we still have c d = c 0 , however the coefficient c u is modified to include the reflection of the downward propagating mode such that Here, φ is the total phase change for the downward propagating reflected mode, however we will assume that the QD is placed in an antinode of the optical field such that φ = 0. The total SE rate g Total is given by ( ) Insertion of Eqs. (2) and (3) in Eq. (1) and using β = g HE11 / (g HE11 + g Rad ) results in Equation (4) is a transparent expression for the total efficiency ε. The element-splitting approach thus allows us to analyze the four building blocks separately and directly study the influence of the characteristic parameters β, r 11 , γ t and γ c on the SPS efficiency ε.

Element analysis
We discuss the performance of the various elements in the following subsections. These studies subsequently allow us to determine the total efficiency coefficient ε using our simplified model based on the element-splitting approach.
The numerical simulations have been performed using the eigenmode expansion technique (EET) [40] with improved perfectly matched layers [41], and the tapering sections of elements IV and V are modelled using a staircase approximation [42].
The analysis of the various elements mainly consists of computing scattering coefficients for the eigenmodes, and the EET directly produces scattering matrices relating incoming and outgoing modes. Even though the staircase approximation is less elegant than the modeling of a tapered section performed in Ref [43], the extraction of scattering coefficients from the finite elements method or the finite-difference time-domain technique is not immediate, and the EET thus remains a natural choice for analyzing the separate building blocks.

Element I: QD section
The first element is the GaAs nanowire with an embedded QD surrounded by a cladding material illustrated in Fig. 2(a). We here study the β factor, that is the power coupled to the fundamental mode relative to total power radiated by the QD. To this purpose we approximate the QD with an in-plane oriented dipole placed in the center. For a radially oriented emitter closer to the center than the sidewalls, values of β larger than 75% are expected [27], however a detailed study of the influence of the position of the QD in the nanowire is beyond this work. We now compute the SE power emission rate g HE11 into the fundamental mode as well as the rate g Rad into radiation modes. The total SE rate is then given by g Total = g HE11 + g Rad , and in Fig. 2(b) we plot β = g HE11 / g Total as function of R nw for various cladding indices. We observe in Fig. 2(b) that the maximum value of β depends on the index contrast. To understand this dependence we note that the high β value is the result of a screening effect, which dampens the field amplitude of radiation modes inside the nanowire. The magnitude of the effect is proportional to the index contrast between core and cladding. For the nanowire surrounded by air, we obtain a β value of 96.3% for a R nw ~120 nm, while in the geometry with polymer cladding (n clad = 1.57) the index contrast is smaller and the maximum β value is reduced to 83.3% for R nw ~110 nm. In the following we consider these two parameter sets for designs A and B and design C, respectively.
Regarding the height of element I, the only requirement to maintain SE control is φ = 0 in (2). With this condition met, the distance between the emitter and the adjacent elements should be minimized to reduce electrical and optical losses.

Element II: Metal mirror
Choosing the correct nanowire radius R nw ensures a high β value, however half of the generated photons still propagate towards the substrate and to achieve high efficiency a bottom mirror with a high reflection coefficient of the fundamental HE 11 mode is thus required. The reflectivities of a DBR and a metal mirror were analyzed in Ref [37]. for a lowdiameter GaAs nanowire. It was shown that the small diameter results in strong diffraction effects in the DBR mirror leading to poor reflectivity. On the other hand the metal mirror provides good reflection in the entire diameter regime. A weakness of the metal mirror is the coupling to surface plasmon polaritons propagating in the transverse plane. This coupling results in a lowering of the reflectivity, but can be avoided by adding a thin intermediate dielectric layer between the nanowire and the metal. The effect of this layer is to perturb the condition for plasmon coupling while maintaining the high reflectivity coefficient of the GaAs-metal interface. The metal mirror is illustrated in Fig. 3(a).
The optically pumped photonic nanowire demonstrated in [28] employed silica for the dielectric layer. However, for electrical pumping a conducting dielectric material is required, and a transparent conducting oxide such as ITO is a natural choice. The reflection coefficient R 11 = |r 11 | 2 of the fundamental HE 11 mode is shown in Fig. 3(b) as function of the thickness t ITO of the bottom ITO layer. In the absence of the intermediate ITO layer, the reflectivity is only ~56 -60% due to coupling to surface plasmon modes [37]. For large values of t ITO , the value of R 11 drops due to diffraction and destructive Fabry-Perot effects. We observe maximum modal reflectivity coefficients of ~89.5% and ~91.6% for air and polymer claddings, respectively. These values are rather close to the plane-wave reflection coefficient of 94.7% for a GaAs-gold interface under normal incidence conditions.
As long as the thickness of the gold layer is sufficiently large to prevent tunnelling of light into the GaAs substrate, the exact thickness does not influence the reflectivity and should instead be chosen to ensure good electrical conductivity.

Element III-A: Top conical tapering
Even though most of the photons are propagating towards the top of the nanowire, a low divergence of the output beam is necessary to ensure a high collection efficiency of the detection optics. The far field emission profile is related to the Fourier transform of the mode profile, and for small nanowire radii the far field profile is wide leading to a low γ. In our earlier work [38] we proposed to improve γ by implementing the conical tapering sketched in Fig. 4(a).
The tapering has two beneficial effects. First, it serves to adiabatically expand the fundamental mode such that the divergence of the output beam profile is reduced. Second, the mode is expanded into the air cladding and the reflection at the semiconductor-air interface is strongly reduced. The total transmission into free space as well as the relative collection efficiency γ t for 0.5 and 0.8 numerical aperture (NA) lenses are illustrated as function of taper opening angle α in Fig. 4(b) for a perfectly sharp tip, characterized by a truncation radius R tr = 0, see Fig. 4(a). Here, the total transmission and γ t are both normalized to the power of the forward propagating guided mode at the interface of elements III and IV. For large opening angles, the transition through the tapering is not adiabatic and the relative collection efficiency γ t is low. For an adiabatic transition a small opening angle is required, and we observe that γ t → 1 as the opening angle α is decreased.
Obviously, a small α leads to a long tapering section and for this reason we also study the collection efficiency of a truncated taper. We have computed the collection efficiency γ t as function of opening angle α for four truncation radii R tr , and the results are shown in Fig. 4(c). We observe that for R tr = 90 nm, γ t is poor, however for a truncation radius R tr of 75 nm the collection effiency is much improved and for 60 nm, γ t is almost identical to that for the perfectly sharp tip. A truncation parameter of R tr = 60 nm thus allows a reduction of the tapering height by a factor of two while maintaining the good collection efficiency.

Element III-B/C: Inverted tapering section
In all three designs a low divergence of the output beam is required to maintain a high collection efficiency. Whereas a conical taper is employed to adiabatically expand the mode into air in design A, designs B and C feature an inverted tapering which instead expands the mode inside the GaAs nanowire. The tapering sections are sketched in Figs. 5(a) and 5(d). The bottom radius R nw is fixed at 110 and 120 nm for designs B and C respectively.
We first study the transmission coefficient γ t of the fundamental HE 11 mode as function of sidewall angle θ for various values of R top for polymer and air claddings. The results are given in Figs. 5(b) and 5(e). Inspecting the figures, we observe two distinct domains. For a sidewall angle θ below ~5°, the transmission is near-adiabatic with values of γ t above 95%, whereas an oscillatory behavior of the transmission for θ above ~5° is observed. For a fixed angle, the taper height h increases with R top and due to the free-carrier scattering loss the transmission generally deteriorates for increasing R top . We are mainly interested in the near-adiabatic regime for θ < 5°, where the value of γ t is above ~95%. Generally, a decreasing sidewall angle results in an improved adiabatic transition and a increasing transmission, however for θ → 0 the taper height h becomes substantial and the free-carrier absorption loss then reduces the transmission. We observe that the optimum transmission is obtained for sidewall angles between ~2° and 5°.

Element IV-A: Side ITO contact
The fourth element in design A is the side ITO contact placed on top of the polymer cladding shown in Fig. 6(a) below. The contact is used to provide electrical pumping of the QD, however its influence on the optical mode should be minimized. To examine this influence we have computed the transmission γ c of the HE 11 fundamental mode through the contact section as function of ITO layer thickness t ITO and of nanowire radius R nw . The results are presented in Fig. 6(b).
For small R nw the mode is predominantly situated outside the nanowire. In the absence of the ITO layer the transmission coefficient γ c drops towards the plane-wave value for the polymer-air interface as R nw → 0, and the addition of the ITO contact further deteriorates this transmission. For large R nw the confinement to the nanowire is strong. When R nw increases, the influence of the cladding variation on the mode propagation decreases and the transmission approaches 100%. For R nw = 110 nm we observe that γ c is above 95% for all three ITO contact thicknesses considered. At this small radius the HE 11 field profile has some overlap with the cladding but not enough to severely compromise the transmission. Also, the figure reveals that even a fairly thick ITO layer only reduces γ c by ~2%. The exact thickness of the ITO contact is thus not a critical issue in fabrication. We note that for R nw = 110 nm and t ITO = 50 nm, the transmission γ c is ~97.7%. To justify our material choice of ITO for the side contact in element IV-A, we consider the transmission through the corresponding element with a side contact made of gold instead of ITO. The transmission for this gold contact is shown in Fig. 6(c). We observe that for R nw = 110 nm, the transmission is below 50% for both contact thicknesses considered. This poor transmission is due to the HE 11 mode extending into the gold cladding and experiencing strong absorption and scattering by the metal. Thus, the implementation of an ITO contact instead of a gold contact improves the total efficiency ε of the SPS by a factor of 2.

Element IV-B/C: Top ITO/gold contacts
In design B the top ITO contact serves two purposes, it allows for electrical pumping and it also serves as an anti-reflection (AR) coating to reduce the reflection from the semiconductorair interface. The top contact element is sketched in Fig. 7(a). The thickness of the ITO contact layer is fixed at the AR optimum of one-quarter optical wavelength in bulk ITO, and we have computed the total transmission and relative collection efficiency γ c for 0.5 and 0.8 NA lenses as a function of nanowire top radius R top . The results are illustrated in Figs. 7(b) and 7(c) for polymer and air claddings. The total transmission and the collection efficiency γ c are, as before, normalized to the power of the forward propagating mode at the interface between elements III and IV. We observe that for R top ~110 nm, the transmission is less than 30% for both lenses. This low transmission is caused by the narrow HE 11 mode waist resulting in a wide far field emission profile.
To achieve a high collection efficiency, the HE 11 mode waist should be extended and for this reason we introduce the inverted tapering section described above. The real part 1.75 of n ITO is close to the ideal AR coating index of sqrt(3.54) ≅ 1.88, however as R top increases the total transmission approaches an upper bound of ~95% rather than 100% due to the weak material absorption of the ITO contact. As R top increases, the collection efficiency is improved, however we keep in mind that a large top nanowire radius requires a long inverted tapering section.
The top gold contact element IV-C for design C is illustrated in Fig. 7(d). A gold contact ring similar to that used to electrically pump micropillars in Refs [7,39]. provides electrical current and Si 3 N 4 is used in the central part as an AR coating. Again, the thickness of the AR coating is chosen to be one-quarter optical wavelength in bulk Si 3 N 4 material. The collection efficiency γ c for this element is given in Figs. 7(e) and 7(f) as function of R top and the contact ring overlap parameter d for polymer and air claddings. We observe that for R top = 750 nm the efficiency γ c is above 85% and approaches ~99% for R top approaching 1500 nm for all values of d considered. For the value R top = 1500 nm, the overlap of the HE 11 mode with the contact ring is so small that the influence on the collection efficiency is negligible.

Fabrication issues
In the following, we briefly discuss the feasibility of fabricating the tapering sections III-A and III-B/C. These sections represent the most challenging part in the experimental realization of the proposed SPS designs, and we present initial fabrication results to demonstrate that the successful fabrication of the structures is indeed realistic. The fabrication of an optically-pumped SPS featuring a top taper of the III-A type has been demonstrated in Ref [28]. The associated process begins with an e-beam lithography step followed by a reactive ion etching step (SF 6 plasma) that define a conical Si 3 N 4 hard-mask. This conical shape is then transferred into the GaAs layer, using another reactive ion etching step (SiCl 4 -Ar plasma). Finally, the sidewall angle transferred in GaAs is the initial angle of the Si 3 N 4 mask divided by the GaAs-Si 3 N 4 selectivity of the last etching step. This process has allowed reaching an opening angle α ~5°. A typical SEM picture is shown in Fig. 8(a): note that due to the presence of a lateral passivation layer deposited during the etching, the exact shape of the high-refractive index GaAs core is not accessible on such a picture. In the future, further improvements of the process may lead to a decrease of the tapering angle, resulting in a higher γ.
Following a similar process with different etching parameters, it is also possible to fabricate type III-B/C tapers. The sidewall angle is then directly controlled by the etching parameters. Figure 8(b) shows that structures featuring a 10 µm height and a sidewall angle θ ~3° are already achievable. Such parameters are close to the ones discussed in this paper. Experimental work is on-going to further optimize the process and obtain a wider top facet and further details of the associated fabrication process is beyond the scope of this paper.
Regarding design C, the air pocket could be defined with the following process. After planarization of the sample with a polymer and deposition of the top ring contact and contact pad, the polymer is etched with a highly selective dry etching step (O 2 plasma). The contact pad and the top facet of the wire will act as an etching mask, and a slight chemical component in the etching can be used to under-etch the polymer, to define an air pocket around the photonic wire.

Discussion of efficiency
With the various element coefficients β, r 11 , γ t and γ c computed, we can now estimate the total efficiency of the proposed electrically-pumped SPS designs. The optimum parameters for the bottom part of the designs are fixed and summarized in Table 2. However we have the liberty of choosing α, R tr and t ITO in design A and R top , h (or θ) and d in designs B and C. In the following we have chosen R tr = 60 nm, t ITO = 50 nm and d = 100 nm, and we study the total efficiency as function of the remaining free parameters. The QD is positioned 261 nm (269 nm) above the metal mirror in design B (design C) in an antinode of the field such that φ = 0. The total SPS efficiency ε as function of opening angle α for design A for a 0.5 and a 0.8 NA lens and as function of R top for designs B and C for the taper heights h = 10 µm and h = 20 µm is given in Fig. 9. The results of calculations using the element-splitting model as well as results from exact computations are shown.
Inspecting Fig. 9(a) we observe that for the experimentally realistic opening angle of α = 3°, the total efficiency ε for design A is ~82% for the 0.8 NA lens. Good agreement between the results obtaining using the simplified model and those from a full computation is observed, validating the element-splitting approach. A small oscillatory deviation for the 0.8 NA curves is observed, which we attribute to an increasing reflection of the HE 11 mode reaching the element III-A as the opening angle α is increased. This reflection is not taken into account in the simplified model.
For design B, Fig. 9(b) shows that the efficiency ε increases with R top towards a maximum value of ~81% obtained for a top nanowire radius R top of ~800 nm for the 0.8 NA lens. At this radius, the collection efficiency γ t for the element IV-B is no longer improved by increasing R top and the total efficiency ε clamps.
A similar behavior is observed in Fig. 9(c) where the efficiency ε for design C increases towards 89% as obtained for R top ~900 nm. Figure 7(f) reveals that the collection efficiency γ c for the element IV-C is slightly improved for R top approaching 1500 nm, however for R top > 750 nm the taper transmission coefficient γ t starts to drop. There is thus a trade-off, between a large R top to provide a high collection efficiency and a small R top to maintain a high transmission through the inverted taper. For our choice of parameters the optimum value is ~900 nm for h = 20 µm. In both Figs. 9(b) and 9(c) we observe a slight deviation for R top > 750 nm for the h = 10 µm curves computed using the simplified and the exact models. For design C, the efficiency obtained by the exact model actually increases beyond 89% as R top approaches 1500 nm, whereas ε for the simplified model clearly decreases, reflecting the drop in transmission observed in Fig. 5(f) for R top approaching 1500 nm. This discrepancy can be understood by considering the breakdown of the adiabatic transition of the HE 11 mode as it propagates through the inverted taper. For a fixed taper height h, the side wall θ increases with R top and for R top > 1000 nm there is significant coupling to higher order modes HE 1m (m > 1). In the simplified model this coupling is treated as a loss, however even though these higher order modes generally have a more divergent far field emission profile, a significant part of the light from these modes can be picked up by the 0.8 NA lens. In this multi-mode picture, the breakdown of adiabaticity means that the power initially carried by the HE 11 mode is distributed over various HE 1m modes at the top part of the taper, and for sufficiently large values of R top all this power is still coupled to the lens. Depending on the application, this distribution of power among several HE 1m modes may, or may not, be acceptable. E.g. if subsequent coupling to a single-mode fiber is required, the power in the higher order modes will be lost and the efficiency ε will be correctly predicted by the simplified single-mode model presented here.
We have chosen parameter values based on initial fabrication tests and it is not yet completely clear which values for the parameters α, R trunc , t ITO , h, R top and θ can be achieved in the fabrication of the complete structures. However, we can estimate the upper bounds of the efficiency under the assumption that arbitrary geometric parameters are possible. All three designs are limited by their β factor and the reflection coefficient R 11 of the bottom mirror. Furthermore, the designs are limited by the transmission through their respective contact elements. This results in an upper bound of 86.7% for the design A. Neglecting absorption loss in the inverted taper due to free carrier scattering, the corresponding bounds are 84.3% and 95.5% for designs B and C respectively.
The design C benefits from the high β value of 96.3% due to the large GaAs-air index contrast and a transmission of 99.7% in the plane-wave limit through the Si 3 N 4 AR coating. The main bottlenecks for the total efficiency for design C are thus absorption losses in the inverted taper and the reflectivity coefficient of 89.5% of the metal mirror. The absorption losses may be reduced by carefully optimizing the doping profile of the inverted taper. The reflectivity of the metal mirror is limited by the plane-wave reflection coefficient of 94.7% for the GaAs-gold interface. Even though this coefficient can be slightly increased by replacing gold with silver, a DBR mirror is most likely necessary to obtain 100% reflectivity. A regular DBR mirror performs poorly at small nanowire diameters [26], however the mode-matched DBR structure presented in [44] could be considered. This element represents a substantial design complication and a thorough analysis is outside the scope of this paper.
Finally, we discuss the possibility of non-ideal carrier-capture efficiency, that could in turn contribute to the upper limit on ε. In Ref [7], values of ξ between 20% and 72% were reported for micro-cavity devices. These low values were attributed to non-radiative recombination at the sidewalls and associated carrier losses. This effect can in principle be compensated by the injection of a stronger current. However, in the micro-cavity case, this solution is not ideal due to the marked degradation of the g (2) as the pumping current increases. The photonic nanowire SPS has demonstrated a nearly perfect g (2) at saturation of the X transition, under non-resonant optical pumping [28]. Thus, it is likely that a stronger current up to X saturation is possible in an electrically pumped device, without the burden of detrimental multi-photon pulses. Another solution proposed in [7], which is not specific to photonic nanowires is the passivation of the sidewalls. This helps to decrease non-radiative recombination at weak currents, as demonstrated successfully in the context of electrically pumped lasers based on nanocavities [45].

Conclusion and outlook
We have proposed and analyzed three electrically-pumped SPS structures. The designs are based on a GaAs nanowire with an embedded QD and on ITO and gold contacts. The operating principle ensuring a high β relies on a geometrical screening of radiation modes rather than the Purcell effect. This choice avoids the high sensitivity to geometry imperfections, which limits the efficiency of traditional high-Q cavity-based SPS designs.
The various elements of the designs have been analyzed and the influence of geometrical parameters on the efficiency has been quantified. For realistic geometrical parameters we predict the total efficiencies of the designs A, B and C to be 82%, 81% and 89% respectively. These values are well above reported values for state-of-the-art electrically-pumped SPS's based on cavity effects. For all the designs, the tapering sections are critical elements, and we have demonstrated the feasibility of experimental fabrication of both the top conical and the inverted tapering. We have discussed possible strategies for further increasing the total efficiency towards 100%.
Finally, some advanced applications of SPS (such as quantum computing using single photon Qbits) require single photon indistinguishability, which is possible only if the emitter's dephasing time is limited by the SE time. Although SE enhancement using the Purcell effect is, generally speaking, favorable to reach this regime, we note that the emission for indistinguishable photons has already been demonstrated for electrically driven QD devices in the absence of measurable Purcell effect [46,47]. Also, "fast" emitters such as QDs in the giant-oscillator strength regime [48] or donor impurities [49] are good candidates for the generation of indistinguishable photons without the help of cavity effects. Embedding such emitters in photonic wires appears therefore as a promising route toward efficient and practical sources of indistinguishable single photons.