The organic permeable base transistor is a vertical transistor architecture that enables high performance while maintaining a simple low-resolution fabrication. It has been argued that the charge transport through the nano-sized openings of the central base electrode limits the performance. Here, we demonstrate by using 3D drift-diffusion simulations that this is not the case in the relevant operation range. At low current densities, the applied base potential controls the number of charges that can pass through an opening and the opening is the current limiting factor. However, at higher current densities, charges accumulate within the openings and in front of the base insulation, allowing for an efficient lateral transport of charges towards the next opening. The on-state in the current-voltage characteristics reaches the maximum possible current given by space charge limited current transport through the intrinsic semiconductor layers. Thus, even a small effective area of the openings can drive huge current densities, and further device optimization has to focus on reducing the intrinsic layer thickness to a minimum.

The Organic Permeable Base Transistor (OPBT) is a transistor concept using a vertical current flow perpendicular to the substrate, therefore eliminating the difficulties in downscaling of conventional Organic Field-Effect Transistors (OFETs).1 The OPBT consists of three parallel electrodes, as shown in Figure 1. The current between the outer electrodes acting as emitter and collector can be controlled by the potential of the inner base electrode. To work properly, charge carriers need to be able to pass the base electrode. This transmission of charge carriers is explained by a perforated grid-like base layer.2,3 Research on OPBTs covers experimental techniques and methods for improving the performance of n-type as well as p-type OPBTs.2–14 Klinger et al. recently showed easy-to-produce OPBTs based on the n-type semiconductor C60, reaching high current densities above 10 A/cm2 and transit frequencies in the MHz-regime at low voltage.8 The devices are fabricated by a layer by layer vacuum deposition, and after processing of the base electrode, the device is exposed to air for several minutes, leading to a thin native aluminum oxide film around the base electrode.

The basic operation principle of the OPBT can be understood as follows: For n-type devices, a low base potential represents a high energy barrier for electrons and almost no electrons can pass through the base, leading to the off-state (cf. Figure 2(b)). For high base potentials, the device is in the on-state (cf. Figure 2(c)). The opposite is the case for p-type devices.

In order to improve those devices even further, a key factor is to develop a comprehensive understanding of the operation mechanism and charge transport in OPBTs. Inorganic permeable base transistors based on GaAs have been simulated already by Bozler and Alley in 1980.15 Instead of an insulator between base and semiconductor, they modelled a Schottky barrier. Therefore, these devices can only operate at low base-emitter voltages as base currents would rise too fast otherwise, thus the operation mechanism at high base potentials could not be studied. Simulation on OPBTs studying the effect of certain parameters, e.g., on device speed can also be found in the literature.16 However, they do not discuss the different transport regimes that are important for the OPBT operation. Additionally, they are using rather large openings with a size of 20 nm. Such large openings are not visible in SEM images of OPBTs fabricated with nano-sized self-structuring of the base electrode, like the high performance devices presented recently.8,13 The openings in those devices are much smaller, only a few nanometers in diameter, and the distance between two openings is large compared to the opening itself.17 Nevertheless, very high current densities above 10 A/cm2 are reported.8 At this point, the question arises, how such high current densities are possible if the total area of all openings is only a small fraction of the active device area.

In this work, we consider structures with very small openings and a large distance between the openings, and we use simulation results to derive a detailed understanding of the complete OPBT operation and the processes taking place inside the device. Therefore, we study the behaviour of the OPBT characteristics and the current limitation mechanisms. We show that the aluminum oxide around the base does not only act as an insulator that reduces the base leakage currents but also ensures that many charge carriers can be transported towards the nearest opening in a charge accumulation channel in front of the insulator. This is a crucial process that enables high current densities and leads to an advantage over similar structures.7 Devices can only benefit from this effect, if they have a semiconductor in front of the base insulation. Additionally, we address the question if the openings in the base layer are a bottleneck for the total driving current as well as for stability of the OPBT with respect to power dissipation.

The operation principle of the OPBT is studied by three-dimensional simulations of the carrier flow based on drift-diffusion modeling. We consider n-type devices and neglect holes and all recombination processes reducing the van Roosbroeck system18 to its unipolar version consisting of a continuity equation for the electron density n coupled to the Poisson equation for the electrostatic potential φ

· ( ε 0 ε r φ ) = q ( N D n ) , n t 1 q · j n = 0.
(1)

Here ε r , ε 0 , and q are the relative dielectric constant of the semiconductor, the vacuum permittivity, and the elementary charge, respectively. N D denotes the doping profile. The current density jn is described by a drift-diffusion form

j n = q μ n n φ + q D n n ,

where μn is the electron mobility and the diffusion coefficient Dn is obtained by the Einstein relation, D n = μ n k B T / q , with Boltzmann's constant kB and temperature T.

To understand the electron transport through the grid-like permeable base with many openings, we consider the carrier flow through a single opening (cf. Figure 1). The simulation domain uses a small representative unit cell of the device, reaching from top to bottom. Due to the assumed radial symmetry around one opening, the simulation domain is restricted to a pie slice geometry (cf. Figure 1). This ansatz means that there are no currents through the edges or sides of the unit cell except for the top/bottom interfaces to the contacts.

The vertical layer structure of the OPBT between emitter and collector contact consists of a 20 nm thick n-doped C60 layer, a 100 nm thick intrinsic C60 layer, a 15 nm thick oxidized Al base layer, another 100 nm thick intrinsic C60 layer, and again 20 nm thick n-doped C60 layer. The oxidized Al base layer is assumed to be formed by a 2 nm thick AlOx insulator oxide wrapped around the base. The radius of the opening in the base layer is assumed to be 2 nm, and the opening is filled with intrinsic C60. The n-doped layers have a donator concentration of N D + = 10 18 cm 3 .

The simulation domain consists of the region filled with the organic semiconductor and of the oxidized part around the base. The metal contacts are implemented by boundary conditions described below. The Poisson equation is solved on the whole simulation domain, whereas the drift-diffusion equation is considered in C60 only, because the insulator is assumed perfect and does not contribute to charge transport. The boundary conditions at the emitter and collector contacts correspond to Ohmic contacts defined by both the applied voltage and local charge neutrality. The base contact is realized by an insulating gate-like contact defined by the applied voltage at the metal-insulator interface (cf. Figure 1(c), red line). Along the vertical pie slice surfaces, we assume zero-flux boundary conditions for the drift-diffusion equation. At the non-contact boundary (cf. Figure 1(c), green line), the condition for the electrostatic potential enforces the normal component of the electric field strength to vanish.

Due to the n-doping at the top and bottom contacts, a built-in voltage of 0.6 V is introduced, defined by the voltage drop between emitter and base in equilibrium. As the device is symmetric with respect to emitter and collector, this will only lead to a shift of the base potential. For the simulations, the emitter potential is fixed at 0 V, while the collector remains at 1 V. The base potential is varied between −0.5 V and 1.5 V, therefore, the resulting characteristic is called base sweep.

The drift-diffusion equation and the Poisson equation on the simulation domain are solved self consistently using the WIAS software Oskar3 (Ref. 19) which is based on a finite-volume approach in combination with the Scharfetter-Gummel scheme.20 The spatial discretization of the simulation domain by tensor product meshes exhibits fine resolution near to the base contact. The motivation for our simulations is experimental studies on C60-OPBTs.8,13 Following Ref. 21, we assume the mobility to be 0.1 cm2/(Vs). While higher mobilities for C60 can be found in the literature, those mainly refer to an enhanced field effect mobility, whereas here, the bulk mobility at lower charge density is the appropriate value. Furthermore, the mobility is assumed to be constant. All physical and geometric parameters used in the simulations are collected in Table I.

The 3D electronic simulations provide besides IV-characteristics spatially resolved data on current density, electrostatic potential, and carrier density and allow for an in-depth study of the device operation.

Due to the n-doped layers at the top and bottom contacts, an Ohmic injection of electrons is given at those contacts. At the same time, this prevents holes from being injected into the device and makes the OPBT a unipolar transistor. As the base is surrounded by an insulator, no charge injection or extraction can occur at that contact. During operation of the n-type OPBT, the collector will always have a higher electric potential than the emitter. It means that electrons get injected at the emitter, are transmitted through an opening in the base layer, and finally gathered by the collector.

The simulation of a base sweep can be seen in Figure 2(a). This characteristic reveals three different regions: Region I with an exponential increase in collector current with increasing base potential, up to a base-emitter voltage VBE = 0.6 V. In region II, the collector current rises with a lower slope. This part can be seen as a transition between region I and region III. The latter is the region where the collector current stays constant and does not rise further, independently of the potential applied to the base. In Subsections III B–III G, we will cover all regions and explain the IV-characteristics by studying the processes which define the current transport.

A low base potential, as shown in the energy diagram in Figure 2(b), does not allow electrons to flow to the base, representing the off-state. Instead, the built-in potential leads to an electric field pushing electrons away from the base region and thus preventing a significant current flow.

Up to a base-emitter voltage VBE of 0.6 V, the potential inside the opening is almost spatially constant and changes like the base potential itself, as the opening has a radius of only 2 nm. From emitter to base, the electrons flow by diffusion only (see Figures 3(a) and 3(d)), behind the base, the electrons are transported to the collector by the electric base-collector field. A higher electron density in the opening will lead to a higher current through the base opening. However, the base potential controls the concentration of charge carriers in the opening, similar to the subthreshold region in a field effect transistor, where the gate controls the charge carrier density in the channel. As the electrons can be approximated by Boltzmann statistics in this region, the number of charge carriers that can overcome the energy barrier of the base increases exponentially. Inside the opening, the electric potential is constant and therefore also the charge carrier density is constant in region I, which is shown in Figure 4(a)). As the conductivity scales with the charge carrier density, also the current increases following an exponential law, showing a subthreshold slope of 60 mV/decade. Therefore, we can relate the exponential behaviour in region I to the limitation by the opening and the applied base potential.

The exponential increase in region I in Figure 2(a) does not continue above 0.6 V. Two mechanisms could lead to such a saturation, which must not be confused with the saturation known from the output characteristics of field effect transistors. First, the opening is no longer the limiting factor. Instead, the transport in the intrinsic semiconductor between the emitter and the base is limiting for a given voltage, so no more charge carriers can arrive at the base. Second, the number of charge carriers in the opening could be at a limit, preventing them from rising any longer and therefore no further conductivity increase can occur. The latter can be excluded by having a look at the profile of the charge carriers in the opening (Figures 4(b) and 4(c)). Even at highest current densities, i.e., in the on-state, the electron density does not reach its limit: In contrast to the charge carrier distribution in region I, the electron density is much higher and electrons have their highest concentration at the edge of the opening, whereas in the middle, fewer electrons are present. Obviously, the number of charge carriers could be larger, but the flow into the base region is not sufficient. Due to the Ohmic injection at the emitter, injection limitations can be excluded. The overall current in this case must thus be limited by the charge transport in the intrinsic semiconductor regions. The transport limited by a semiconductor bulk with constant mobility is either Ohmic or follows a power-law as discussed below, but it is not expected to show an exponential current-voltage law. Consistently, a lower slope is observed in region II.

At this point, it should be mentioned that not only the intrinsic semiconductor which is directly above the opening contributes to the current. The current between the emitter and the base flows over the whole available area. Accordingly, many electrons arrive in front of the base insulator. Together with the high electric potential at the base, this leads to an accumulation of charge carriers in front of the insulator. That way, a highly conductive charge channel is formed in front of the insulator, which can be seen in Figure 3(b). This channel allows those charge carriers, which do not arrive directly at the opening, to be transported in that channel, in a lateral direction parallel to the base. Due to the high charge carrier concentration and conductivity, this is easily possible without substantial voltage drops.

In order to understand the current saturation in region III, we consider the collector part of the device as well. As no current can flow into or out of the base, emitter and collector currents must be the same. The collector part of the OPBT also contains a 100 nm intrinsic semiconductor and as such needs to be able to transport the same amount of current as the intrinsic semiconductor at the emitter. With the highly conductive accumulation of charge carriers around the base, the emitter and collector part of the device can be understood as a series connection of two semiconductor resistors. Thus, in order for the OPBT to transport the maximum current, both need half of the operation voltage.

This means that the beginning of region III, where the current saturates, is exactly at the point, where the potential around the base is in the middle between the potentials at emitter and collector. When reading the externally applied voltages, the built-in potential has to be added. In this example, half of the operation voltage is 1 / 2 · V CE = 0.5 V and the built in voltage is Vbi = 0.6 V. Therefore, we expect the transition to region III to be at 1 / 2 · V CE + V bi = 1.1 V . This value is well confirmed by Figure 2(a), and the relation can also be applied for different operation voltages (see supplementary material, Figure S2). The potentials of emitter, base, and collector in region III will be discussed below, see Figure 8(c). The base is close to the expected value of 1.1 V. This corresponds to the point where region III starts (cf. Figure 2(a)), which is at 1.2 V. The small deviation can be attributed to the voltage drop across the insulator. This calculation is valid only if both intrinsic layers at emitter and collector are equal. And this condition is met, since material as well as thickness is the same by design. Also, the collector intrinsic semiconductor contributes with the whole available area in region III, like the emitter.

This can also be seen by visualizing the current flow in the device (see supplementary material, Figure S1): While at the emitter, the current flows to the base vertically and from there to the opening, the behaviour is different at the collector. In regions I and II, electrons flow directly or in a diagonal way towards the collector, and in the shadow behind the base, the smallest current density can be observed. In region III, in contrast, electrons spread laterally behind the base across the whole simulated area. That way also the complete collector area takes part in current transport.

Even when the base potential is increased above 1.2 V, never does more than half of the operation voltage VCE = 1 V drop over the emitter part of the device for the following reason: Due to the series connection, the current and consequently the voltage over both equally thick intrinsic layers must be equal. In a theoretical situation where the voltage over the emitter part would be larger, the current flowing from the emitter to the base would also be larger. The remaining voltage for the collector part of the OPBT would be smaller than half of the operation voltage, and as the collector obeys the same IV-law as the emitter part, the collector would only transport a lower current, which would be a violation of current continuity in the steady state. More and more electrons would accumulate in front of the base, shielding the higher base potential and therefore pushing the channel potential back to half the operation voltage. This effect can be seen in Figure 5, which shows the average of the potential in the channel around the base in dependence of the externally applied base voltage: While the channel potential follows the base voltage up to 1.2 V, where region II turns into region III, the channel potential saturates towards 1.2 V for higher base potentials. Accordingly, the voltage drop across the insulator dramatically increases in region III, which has already been observed qualitatively in Figure 3(f). In a real device, where the insulator is not perfect, this leads to an increase of the base leakage current.

While this proves that the whole volume of the OPBT contributes to the current, an interesting question is which areas contribute most current towards the current flowing through one opening. Figure 6 shows a cross section through the OPBT where each colored area shows the regions of the OPBT where 10% of total current are flowing in the on-state. While the current is using all available space until the edge of simulation unit cell at 25 nm, one can see that most current even comes from the outer parts, that are farther away from the opening. This is however not very surprising and simply results from the circular symmetry. The outer rings just have a larger area, i.e., there are more areas that are farther away from one opening, than there are areas close to one opening. In terms of current density, however, there is an equal distribution across the whole OPBT.

While the simulation presented here covers a radius of 25 nm, even charge carriers from a larger distance could be collected towards one opening. If we consider that the electron density in the channel in front of the insulator is higher by a factor of 1000 compared to intrinsic charge carrier density (cf. Figures 4(c) and 8(f)), also the conductivity scales with that factor. Consequently, the width would need to be 1000 times as long as the intrinsic layer in order to become a dominant resistance. Therefore, we can estimate for the case of openings with a radius of 2 nm, that even very few openings with distances between neighbouring openings in the range of 100 μm would be sufficient to avoid current limitation by the base. This estimation assumes a perfect insulator, i.e., charge carriers do not leak into the base. In a real device, the requirements for the density of openings will be accordingly higher. In the devices previously published in Ref. 8, we have seen that real devices operate well within the described range. Furthermore, a change in the ratio between opening area and device area will lead to a shift of the exponential law in region I.

In region III, the complete base insulator is surrounded by a highly conductive charge channel, as shown in Figure 3(c). This allows the use of the whole area, and the base layer does not influence the current transport any more in region III. Therefore, in the on-state (region III), the OPBT can be seen as a series connection of two electron-only semiconductor resistors (nin-stack), where the base with its extremely high charge carrier density behaves like a highly doped or metallic layer, which is able to screen the electric field. In principle, those two quasi-nin-elements, one describing the current transport from emitter to base and the other from base to collector, share the same behaviour, as they can be considered to be identical. Also the distribution of electron density and electric potential show the same profile between emitter and base as well as between base and collector (see Figures 8(c) and 8(f)).

We conclude that, in region III, the current is only given by the thickness of the intrinsic layers and the applied operation voltage VCE. This dependence is shown in Figure 7. For low operation voltages VCE < 0.3 V, a linear IV law is observed. It results from the organic semiconductor having a uniform conductivity due to the charge carriers that are in the semiconductor, either intrinsically or brought in by doping. At higher operation voltages, the current increases faster than linear, going over towards a square-law, as shown by the double logarithmic plot in Figure 7. At this point, the OPBT reaches the SCLC regime, which shows a jV2 dependence according to the Mott-Gurney-law

j = 9 8 ϵ 0 ϵ r μ V 2 L 3 .
(2)

As the base acts like a metallic contact, the Mott-Gurney-law must be applied separately to emitter and collector part of the device, which forms a series connection. Because emitter and collector both have a thickness of the intrinsic semiconductor layer of 100 nm and in region III both have a voltage drop of half the operation voltage, we can calculate the current density in region III as

j III = 9 8 ϵ 0 ϵ r μ ( 0.5 V CE ) 2 L i 3 ,
(3)

where VCE is the operation voltage and Li is the thickness of the intrinsic semiconductor layer at the emitter or collector side of the device. The material parameters ϵr and μ can be found in Table I. While for an operation voltage of 10 V, the calculated j III = 1.0 × 10 6 mA cm 2 matches the simulated on-state current density very well (cf. Figure 7 and supplementary material Figure S2), the equation leads to j III = 1.0 × 10 4 mA cm 2 for V CE = 1 V . This is lower than the simulated value of 2.25 × 10 4 mA cm 2 (cf. Figure 2(a)) and only matches the square law contribution to the current (cf. Figure 7, ∼V2 line). The reason for the difference is the additional linear contribution arising from the conductivity due to the charge carriers that are present in the semiconductor even under unbiased conditions. By fitting the current density in Figure 7 with a linear plus a square law, we obtain j III = 15.8 A cm 2 V · V CE + 8.6 A cm 2 V 2 · V CE 2 . We can now compare the square law contribution to Equation (3) and cross-check by recalculating the corresponding thickness Li using the material parameters from Table I. The result of Li = 105 nm is very close to the simulated thickness of 100 nm. The minor deviation can be explained by the diffusion of charge carriers at the interface of the doped and intrinsic semiconductor layers, resulting in a small electric field due to the remaining ionized donor molecules which is not considered in the Mott-Gurney-Law.

Due to the large number of electrons injected from the metal into the semiconductor and the Coulomb interaction, the charge carriers repel each other. This manifests in the electric potential and electron density profiles typical for SCLC as shown in Figure 8. In region I (Figures 8(a) and 8(d)), SCLC does not play a role, as the transport from emitter to base is limited by the low base potential and the current is diffusion dominated. Between the base and the collector, the low current density can be transported easily due to the high electric field between base and collector. In region II (Figures 8(b) and 8(e)), the applied base potential limits the available base-emitter voltage, and the overall current is defined by an SCLC law in the emitter region, while the collector part is at higher voltage and transports all charge carriers that arrive by Ohmic transport. In region III (Figures 8(c) and 8(f)), the SCLC defines the current transport in emitter and collector part of the device and the externally applied base potential is no longer related to the potential in the semiconductor.

At the injecting emitter contact, charge carriers diffuse into the semiconductor. In Figures 8(b) and 8(c), it can be seen that, however, the electric field at the injecting contact is not zero, as usual for SCLC. Instead, there is a small region where the potential has a concave shape, which can be explained with the positive charges of the ionized donors in the doped layer. Closer to the extracting contact, the current transport is mainly a drift current.

Reaching SCLC means that the device transports the maximum current that the semiconductor can provide, i.e., the OPBT is at the upper limit of current transport. Even more, the high charge carrier density at the base which separates the OPBT into two SCLC zones leads to an even higher current density than what would be expected from a single nin-device with an intrinsic layer thickness of total 200 nm, as SCLC shows an j L 3 dependence of the current density on the layer thickness.

Considering that the opening area is smaller by a factor of (25 nm/2 nm)2 = 156.25, the current density in those openings is higher by this factor and current densities in the openings exceed several kA/cm2. Such high current densities can be achieved in organic semiconductors, as shown by Matsushima and Adachi22 However, the question arises, whether that leads to the formation of hot spots which threaten the stability of the device. As shown in Figures 4(c) and 8(f), the charge carrier density in the opening is higher by almost three orders of magnitude, therefore also the conductivity scales with that factor. Hence, the power dissipated per unit length in the opening is even smaller than in the intrinsic semiconductor. Additionally, the mobility in organic semiconductors rises with charge carrier density and is significantly enhanced when charge carrier densities like in the opening of the OPBT are reached.23,24 This has not been considered in the previous discussion and will increase the mobility in the opening, reducing the power dissipation in the opening even further.

Using a constant mobility instead of a charge carrier density dependent model is a good approximation. In region I, a limitation by the opening is observed, where charge carrier densities are rather low. In regions II and III, extremely high charge carrier densities are present in the doped layers and around the base, while the intrinsic layers define the current that can flow. In those intrinsic regions, however, the charge carrier density does not rise substantially. Therefore, the exact mobility for very high charge carrier densities is not required and also the Boltzmann statistics is a valid approximation. The channel around the base is already highly conductive, so an increased mobility or charge carrier statistics would indeed make the channel even more conductive, but that would not influence the overall characteristics that are shown in Figure 2(a).

Figure 9 shows an experimental base sweep measurement of an OPBT. The device uses C60 as organic semiconductor and 1 wt. % W2(hpp)4 as dopant. The thickness of each layer is equal to the thickness used in the simulation (cf. Table I). Further experimental details can be found elsewhere.8 

The discussed regions can be recognized: For low base potentials, an exponential increase of the current density can be observed (region I). This is followed by region II, and finally at high base-emitter voltages, the current density stays almost constant.

In the on-state, the experiment reaches a current-density of almost 104 mA/cm2. This is slightly lower than the simulation as the mobility in the experiment does not reach 0.1 cm2/(Vs). Also, the transition between the regions (Figures 2(a) and 9, dashed lines) shows small deviations. This could be related to several effects in the experiment that are not considered in the simulation: The presence of interface states near the insulator interface could not only lead to a lower subthreshold-slope but also to a shift of the base potential. Also, at higher currents, the electrode resistance will lead to a voltage drop, so that the actual base-emitter voltage is smaller and the transition between region II and III seems higher than it is. Finally, the experimental device does not have a perfect circular symmetry. Instead both, the opening size and the distance towards the next opening are angular dependent, leading to characteristics that correspond to a superposition of several devices with slightly varying geometry. As a consequence, the transition between the regions is rather a broader range than a single voltage point, making the definition of the transition point imprecise.

The comparison shows that the simulation and the experiment have the same features. The experiment therefore confirms the qualitative characteristics obtained from the simulations, which were used to derive a detailed understanding of the operation mechanism.

In conclusion, we have shown by 3D drift-diffusion simulations that the operation of OPBTs can be divided into three regions based on the base potential. For low base potentials (region I), the opening with a potential defined by the base limits the device current, yielding an exponential law. Region II can be understood as a transition region, where the charge channel around the base is forming. For high base potentials (region III), the externally applied base potential has no influence any more, it is shielded by a charge accumulation around the base insulator, and the current is defined by the charge transport through the intrinsic semiconductor. We conclude that neither the size of the openings nor the distance between neighbouring openings needs to be precisely controllable. The openings are not a bottleneck neither in terms of current nor in terms of power dissipation. Therefore, excellent performance can be achieved with OPBTs,8 as the whole device volume contributes to the current. Also the intrinsic semiconductor layers provide the highest possible current densities allowed by SCLC, reaching the space charge limitation. Therefore, future device optimization should address the intrinsic semiconductor layers rather than improving the perforation of the base electrode.

See supplementary material for a visualization of the current flow in the OPBT and simulated characteristics for different operation voltages.

The work leading to these results has received funding from the European Community's Seventh Framework Programme under Grant Agreement No. FP7-267995 (NUDEV). This work was partly supported by the German Research Foundation (DFG) within the Cluster of Excellence “Center for Advancing Electronics Dresden” and within the collaborative research center 787 Semiconductor Nanophotonics. A.G. received funding from the Einstein Foundation Berlin within the Research Center MATHEON under the ECMath Project SE2 “Electrothermal modeling of large-area OLEDs.” K.L. thanks the Canadian Institute for Advanced Research (CIFAR) for support. The authors would like to thank Klaus Gärtner for many helpful comments, inspiring discussions and assistance with the device simulation software WIAS-Oskar3.

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Supplementary Material