Optimization of the short-circuit current in an InP nanowire array solar cell through opto-electronic modeling

We solve for the diffraction of light in the nanowire array using the Maxwell equations [23, 24]. In this modeling, we include the three-dimensional geometry of the nanowire array. The optical response of the constituent materials is described by their complex valued refractive indexes, and we use tabulated values for InP and GaP [25], respectively. Note that for simplicity we neglect possible variations in the refractive index due to doping in the n and p segments. Such effects are expected to cause only a minor impact on the resulting short-circuit current for the doping concentrations considered in this study.

We solve for normally incident light, that is, for incident light parallel to the axis of the nanowires, and we use periodic boundary conditions with one nanowire per unit cell. As a result, we obtain the electric field E(r) throughout the system. From this electric field, we can calculate the local absorption of incident photons in the nanowires and the substrate, which is proportional to Re(n(r))Im(n(r))|E(r)|² [2]. For a given incident intensity spectrum I(λ) we can then calculate G(r), the number of electron–hole pairs photogenerated per unit volume per unit time. This G(r) enters the drift-diffusion modeling for the electron–hole transport (equation (1)). For the calculation of G(r), we use circular polarized light and an incident intensity given by the AM1.5D solar spectrum (ASTM G173-03) with a total input power of 900 W m⁻².

We solve the electron–hole transport with the drift-diffusion formalism (see, e.g., ref. [26]):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\nabla }}\cdot (-\varepsilon {\rm{\nabla }}\varphi ) & = & q(p-n+{N}_{{\rm{d}}}-{N}_{{\rm{a}}})\\ {\rm{\nabla }}\cdot {J}_{{\rm{n}}} & = & {\rm{\nabla }}\cdot (-q{\mu }_{{\rm{n}}}n{\rm{\nabla }}{\varphi }_{{\rm{n}}})=q(R-G)\\ {\rm{\nabla }}\cdot {J}_{{\rm{p}}} & = & {\rm{\nabla }}\cdot (-q{\mu }_{{\rm{p}}}p{\rm{\nabla }}{\varphi }_{{\rm{p}}})=-q(R-G).\end{array}\end{eqnarray} \tag{ 1 }$$

Here, ε is the (static) dielectric constant, q is the elementary charge, μ_n/p is the electron/hole mobility, n(r)/p(r) is the electron/hole concentration, φ(r)/φ_n(r)/φ_p(r) is the electrostatic potential/electron quasi-Fermi potential/hole quasi-Fermi potential, N_d/a(r) is the concentration of ionized donors/acceptors, R(r) is the net recombination rate and G(r) is the optical generation rate defined in section 2.1. For R in the bulk of the nanowire, we use:

$$\begin{eqnarray}&&\begin{array}{c}R={R}_{{\rm{S}}{\rm{R}}{\rm{H}}}+{R}_{{\rm{r}}{\rm{a}}{\rm{d}}}+{R}_{{\rm{A}}{\rm{u}}{\rm{g}}}\\ \,=\,\left(\displaystyle \frac{{\rm{A}}}{n+p+2{n}_{i}}\right.+B+C(n+p))(np-{n}_{i}^{2}\phantom{\rule{}{3.3ex}})\end{array}\end{eqnarray} \tag{ 2 }$$

where A/B/C is the recombination coefficient of the SRH/radiative/Auger recombination, and n_i is the intrinsic carrier concentration [26, 27]. Here, we assume that the electron and hole concentrations are equal to n_i in the case where the Fermi level coincides with the trap level responsible for SRH recombination, and we assume an equal SRH lifetime for electrons and holes [28]. Regarding the Auger recombination rate, the same coefficient C is assumed for electrons and holes. Since it is very challenging to include wave-optical effects in light emission, and as we focus on the short-circuit current, we neglect radiative recombination in this study. That is, we use B = 0 in equation (2). See online supplementary data section S1, however, for a discussion of the possible effects of radiative recombination. The surface recombination, at the surface of the nanowire, is included through the term:

$$\begin{eqnarray}&&{R}_{{\rm{s}}{\rm{u}}{\rm{r}}{\rm{f}}{\rm{a}}{\rm{c}}{\rm{e}}}=\displaystyle \frac{{v}_{{\rm{s}}{\rm{r}}}}{n+p+2{n}_{i}}(np-{n}_{i}^{2})\end{eqnarray} \tag{ 3 }$$

where v_sr is the surface recombination velocity [29].

In the drift-diffusion equations, the carrier concentrations n and p are calculated by:

$$\begin{eqnarray}\begin{array}{rcl}n & = & {N}_{{\rm{c}}}{F}_{\displaystyle \frac{1}{2}}\left(\displaystyle \frac{{E}_{{{\rm{F}}}_{0}}-{\varphi }_{{\rm{n}}}-{E}_{{\rm{c}}}}{{k}_{{\rm{b}}}T}\right)\\ p & = & {N}_{{\rm{v}}}{F}_{\displaystyle \frac{1}{2}}\left(\displaystyle \frac{{E}_{{\rm{v}}}-{E}_{{{\rm{F}}}_{0}}+{\varphi }_{{\rm{p}}}}{{k}_{{\rm{b}}}T}\right)\\ {N}_{{\rm{c}}/{\rm{v}}} & = & 2{\left(\displaystyle \frac{{m}_{{\rm{e}}/{\rm{h}}}^{* }{k}_{{\rm{b}}}T}{2\pi {{\hslash }}^{2}}\right)}^{\displaystyle \frac{3}{2}}\end{array}\end{eqnarray} \tag{ 4 }$$

where k_b is the Boltzmann constant, m_e/h* is the effective mass of electrons/holes, $\hslash$ is the reduced Planck constant, and E_F0 is the Fermi level at zero bias voltage and zero optical generation:

$$\begin{eqnarray}&&{E}_{{{\rm{F}}}_{0}}={E}_{{\rm{c}}{\rm{n}}}+\left(\displaystyle \frac{{k}_{{\rm{b}}}T}{q}\right){F}_{\displaystyle \frac{1}{2}}^{-1}\left(\displaystyle \frac{{N}_{{\rm{d}}{\rm{n}}}}{{N}_{{\rm{c}}}}\right)\end{eqnarray} \tag{ 5 }$$

where N_dn is the ionized donor concentration which equals the n doping level and E_cn is the conduction band edge, both calculated at the n-type contact.

In equations (4) and (5), the functions F_1/2 and ${{{\rm{F}}}_{\mathrm{1/2}}}^{-1}$ are, respectively, the ½ order Fermi integral and its inverse function:

$$\begin{eqnarray}&&{F}_{\displaystyle \frac{1}{2}}(\eta )=\displaystyle \frac{2}{\sqrt{\pi }}{\int }_{\infty }^{0}\displaystyle \frac{{x}^{\mathrm{1/2}}}{1+\exp (x-\eta )}{\rm{d}}x.\end{eqnarray} \tag{ 6 }$$

In addition, the band edges are calculated as

$$\begin{eqnarray}&&{E}_{{\rm{v}}}={\rm{\Delta }}VBO(r)-\varphi \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{E}_{{\rm{c}}}={E}_{{\rm{g}}}(r)+{\rm{\Delta }}VBO(r)-\varphi \end{eqnarray} \tag{ 8 }$$

where ΔVBO(r) = VBO(r)-VBO_InP with VBO_InP the value for InP, and E_g (r) is the band gap.

The specific values for the parameters used in the drift-diffusion modeling are listed in table 1. Note that for simplicity in the modeling, we do not include doping-induced band gap narrowing, or strain-induced band offset in the lattice mismatched heterostructures like the GaP/InP system in section 3.5. Strain effects are only important close to the interface between InP and GaP, and the strain relaxes away from the interface. For the length of the GaP segment we use in section 3.5, the strain is negligible at the top.

Regarding the geometry used in this drift-diffusion modeling, we consider a single nanowire on top of a substrate. Note, however, that the G(r) is calculated for the nanowire array. We place an ohmic contact at the top of the nanowire, and in the drift-diffusion modeling, the substrate has a lateral size equal to the unit cell size of the array and a depth of 300 nm. We also placed an ohmic contact at the bottom of the substrate and ascertained that its thickness did not affect the resulting short-circuit current. Unless otherwise stated, we used a 100 nm long n segment and a 300 nm long p segment in the modeling, and both segments have an ionized doping concentration of 10¹⁸ cm⁻³.

The ohmic boundary condition at the short-circuit current is defined as:

$$\begin{eqnarray}&&[\varphi ,{\varphi }_{{\rm{n}}},{\varphi }_{{\rm{p}}}]=\begin{array}{c}[0,0,0]\,({\rm{n}}\,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}{\rm{a}}{\rm{c}}{\rm{t}})\\ [-{V}_{0},\mathrm{0,0}]\,({\rm{p}}\,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{t}}{\rm{a}}{\rm{c}}{\rm{t}})\end{array}\end{eqnarray} \tag{ 9 }$$

where V₀ is

$$\begin{eqnarray}&&\begin{array}{c}{V}_{0}={E}_{{\rm{g}},InP}+VB{O}_{{\rm{p}}}-VB{O}_{{\rm{n}}}+\left(\displaystyle \frac{{k}_{{\rm{b}}}T}{q}\right){F}_{\displaystyle \frac{1}{2}}^{-1}\left(\displaystyle \frac{{N}_{{\rm{d}}{\rm{n}}}}{{{\rm{N}}}_{{\rm{c}}}}\right)\\ \,\,+\left(\displaystyle \frac{{k}_{{\rm{b}}}T}{q}\right){F}_{\displaystyle \frac{1}{2}}^{-1}\left(\displaystyle \frac{{N}_{{\rm{ap}}}}{{N}_{{\rm{v}}}}\right)\end{array}\end{eqnarray} \tag{ 10 }$$

where N_ap/dn is the p/n doping level at the p/n contact and VBO_p/VBO_n is the valence band offset at the p/n contact.

For a more detailed analysis of the charge separation mechanism within the nanowires, we used the spatially resolved internal quantum efficiency (SIQE). The SIQE is given by the spatially resolved probability S(r) of an electron–hole pair, injected at r, to contribute an elementary charge to the short-circuit current at a zero bias voltage. Practically, we calculate S(r) by using a generation rate density of G_SIQE = 10²⁸ m⁻³ s⁻¹ in a cubic box of volume V_SIQE with a 10 nm side length, centered at r (we ascertained that the resulting SIQE did not vary noticeably with varying G_SIQE around this value); that is, V_SIQE = (10 nm)³. Next, using the drift-diffusion equations we calculate the short-circuit current I_sc,SIQE that flows through the nanowire due to this injection. Finally, we obtain the SIQE as: S(r) = (1/q) I_sc,SIQE/(G_SIQEV_SIQE). Note that a similar SIQE is used in ref. [32] for the analysis of silicon nanowire photovoltaic devices.

In wave-optics, the absorption performance of the nanowire array depends on the nanowire diameter, the nanowire length, and the array pitch [13]. When considering the absorption of sunlight, we typically look at the short-circuit current J_sc. This current results from the absorption of photons and the consecutive separation of the photogenerated electrons and holes over the p-i-n junction. For the 900 W m⁻² direct and circumsolar AM1.5D solar spectrum, we obtain an upper limit of J_sc,max = 31 mA cm⁻² for InP with a band gap E_g = 1.34 eV by assuming that each incident photon with an energy of E_ph > E_g is absorbed, and that each photogenerated electron–hole pair contributes to the short-circuit current. Below, we consider the two effects that decrease J_sc from J_sc,max: (1) less than 100% absorption and (2) less than 100% probability of separating the photogenerated charges over the p-i-n junction.

Both modeling [33] and measurements [34] have shown that nanowire arrays can absorb incident light that would travel between the nanowires in a ray-optics description [33]. This absorption beyond the geometrical cross-section of each nanowire occurs due to the delocalized wave-nature of light [35]. By optimizing the geometry of the array, the nanowires can absorb almost all of the incident light, incident on the whole nanowire array with energy above the band gap energy of the semiconductor [13]. Therefore, the short-circuit current has the potential to approach J_sc,max.

For nanowire arrays, the short-circuit current shows diameter-dependent peaks [13] due to diameter-dependent absorption resonances in the nanowires. The diameter that optimizes the absorption depends on the band gap E_g of the nanowire material. For InP with E_g = 1.34 eV, the smallest diameter that optimizes the absorption performance is D = 180 nm. We fix D = 180 nm throughout this study, and we choose a nanowire length of L = 1400 nm to match the experiments [2].

In figure 2(a), J_sc is plotted as a function of the remaining free parameter, the pitch of the square array. In the drift-diffusion modeling (solid line in figure 2(a)), we find a maximum value of 25.8 mA cm⁻² when the pitch is 330 nm. We also find J_sc = 24.2 mA cm⁻² for the pitch P = 470 nm, which has been used in previous experiments for InP nanowire arrays [2]. Both these short-circuit values are lower than the upper limit J_sc,opt from absorption modeling (dashed line in figure 2(a)), which predicts an upper limit of J_sc,opt = 28 mA cm⁻² for J_sc assuming a 100% contribution from all the photogenerated carriers to the current. In contrast, drift-diffusion modeling includes loss mechanisms that lead to a less than 100% probability of splitting the photogenerated electron–hole pairs over the p-i-n junction.

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To better understand the absorption properties of the nanowires, we turn to the spatial distribution of the optical generation rate in the nanowire (figure 2(b)). First, we find a radial dependence where the optical generation rate peaks between the center and the surface of the nanowire. We assign this radial profile to the radial pattern of the optical eigenmodes of the nanowire array [23, 24]. Next, we find a very strong dependence of the generation as a function of the position along the axis of the nanowire. For example, 85% of the generation occurs in the top half of it. Also, the maximum generation rate in the nanowire is two orders of magnitude higher than in the substrate. This strong generation in the top part of the nanowires indicates that it is very important to maximize the probability of extracting photogenerated carriers from this segment. Thus, in the next subsection we investigate how the design of the p, i, and n segments affects this extraction probability.

To investigate the possible effect of the design of the p, i, and n segment on the short-circuit current, we varied the segment lengths for the above fixed nanowire length L = 1400 nm. In this way, the photogeneration inside the nanowire does not change. Thus, any variation in J_sc originates from variations in the efficiency of splitting electron–hole pairs over the p-i-n junction.

Figure 3(a) shows the short-circuit current when the n and p segment lengths are varied. We varied the p segment length for a fixed n segment length of 300 nm (blue line in figure 3(a)). Similarly, we varied the n segment length for a fixed p segment length of 300 nm (black line in figure 3(a)). We find that J_sc is much more sensitive to a variation in the length of the top n segment than to a variation in the length of the bottom p segment. An increase of the bottom segment length from 25 nm to 900 nm decreases the J_sc by just 2.7 mA cm⁻², from 21.0 mA cm⁻² to 18.3 mA cm⁻². In contrast, J_sc decreases by 16.5 mA cm⁻², that is, by more than 60%, when the n segment length is increased by a similar amount. Note that such a strong dependence on the length of the top n segment is to be expected from the much stronger optical generation rate in the top part of the nanowire rather than in the bottom part (figure 2(b)).

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To obtain a better, quantitative, understanding of this strong dependence on the top n segment length, we studied the SIQE (solid lines in figure 3(b)) to vary the length of the n segment. That is, we studied the spatially resolved probability S(r) of splitting electron–hole pairs over the p-i-n junction (see section 2.3 for technical details). To aid the analysis, we also show the optical generation rate in the nanowire (dashed red line in figure 3(b)). First and foremost, we find a rapid decrease in the SIQE when moving the n segment from the edge of the i segment toward the top contact. We find a similar drop in the p segment when moving down, away from the edge of the i segment. In contrast, the SIQE is close to 100% in the i segment. This drastically different behavior originates from the existence of an electric field in the i segment, which is lacking in the n and p segment. In the i segment, the electric field splits the electron and the hole rapidly and efficiently in opposite directions, leading to a high SIQE. In the n and p segment, the electron and the hole diffuse instead. For example, if a hole generated in the n segment, which is the minority carrier, diffuses into the i segment, it will be swept toward the p side by the electric field and contributes to the current over the p-i-n junction. However, if the hole diffuses into the ohmic top contact, it contributes to current leakage and recombines in the contact.

The probability of the hole diffusing from the n segment into the i segment depends on the distance to the i segment and the distance to the ohmic top contact. Therefore, the SIQE decreases from close to 100% to close to 0% when the photogeneration position shifts from the edge of the i segment to the edge of the ohmic contact. Note that a similar behavior/drop has recently been observed in the electron-beam-induced current measurements of individual, as-grown nanowires [1]. Thus, we believe that our modeling accurately represents the loss mechanism in short-circuit currents in such experiments.

The number of holes that are photogenerated in the top n segment increases on increasing the length of the n segment. Hence, on increasing the segment length, more holes are available for diffusion to the top contact, leading to an increasing drop in J_sc in figure 2(a). Furthermore, the optical generation rate is high in the topmost 300 nm of the nanowire (dashed red line in figure 3(b)). Therefore, an increase of the n segment length from 25 nm to 300 nm causes a rapid drop in J_sc (marked by region 1 and region 2 in figure 3(a)). When the n segment length is further increased from 300 to 1100 nm, the decrease in J_sc is slower (marked by region 3 and region 4 in figure 3(a)), since the optical generation rate is lower here than in the topmost 300 nm.

Thus, a shorter n segment length leads to a higher J_sc by decreasing the number of holes that can diffuse into the ohmic top contact. The effect of increasing J_sc on decreasing the top n segment length has already been reported from the experiments in [2], although it has not been assigned to the diffusion of minority carriers into the top contact. Nevertheless, in the experiments, a drastic decrease in the n segment length can become impractical due to problems in contacting nanowires with a short top segment [2]. Below, in order to maximize J_sc, we present two alternative designs for limiting the diffusion of holes into the top contact.

Here, in order to maximize J_sc, we discuss the potential of using a gradient in the doping profile in the n segment to prevent photogenerated holes from reaching the top contact. Note that such a doping scheme is often used at the bottom side of planar solar cells in order to reduce the diffusion and consecutive loss of minority carriers into the back contact [36].

We consider a linear increase in the doping concentration from 1 × 10¹⁸ cm⁻³ at the edge of the i segment to n_contact at the edge of the ohmic contact. Here, we assume that the dopants are fully ionized. This gradient introduces an electric field to the n segment, which prevents the holes from reaching the top contact. In terms of the band diagram, this doping profile bends the valence band edge down toward the ohmic contact.

The J_sc is shown in figure 4(a) as a function of the doping concentration increase from the i segment to the top contact. We find that J_sc increases initially with an increasing gradient in the doping concentration. When n_contact increases from 1 × 10¹⁸ to 1.5 × 10¹⁹ cm⁻³, J_sc increases from 25.8 mA cm⁻² to 27.1 mA cm⁻², that is, by 5%.

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To explain the shape of the curve shown in figure 4(a), we show in figure 4(b) the different loss current mechanisms, again, as a function of the doping concentration increase. As expected, the minority carrier current, that is, the hole current, to the top contact decreases monotonously with an increasing doping gradient, that is, with an increasing n_contact (dashed line in figure 4(b)). However, when the n_contact > 1.5 × 10¹⁹ cm⁻³, the further increase actually leads to a decrease in J_sc (figure 4(a)). This decrease originates from the increasing recombination losses in the n segment due to the increasing carrier concentration with increasing doping.

Initially, when 1 × 10¹⁸ < n_contact < 1 × 10¹⁹ cm⁻³, surface recombination is the dominant recombination mechanism, even in the case of the low surface recombination velocity of 200 cm s⁻¹ used here (see online supplementary data section S2 for a discussion of the effect of increasing surface recombination velocity and the online supporting information section S1 for a discussion of the possible effects of radiative recombination). In this case, the total recombination rate is several orders of magnitude lower than the corresponding minority carrier leakage into the top contact, as seen in figure 4(b). Thus, for a low doping level, the major loss mechanism in J_sc is the diffusion of holes into the top contact.

However, with an increasing n_contact, Auger recombination, which scales as the carrier concentration to the third power, in contrast with the surface recombination that scales as the carrier concentration to the first power, sets in (see solid line in figure 4(b)). Auger recombination becomes significant when n_contact = 10¹⁹ cm⁻³ and increases sharply for a higher n_contact. For example, for n_contact = 2 × 10¹⁹ cm⁻³, the Auger recombination amounts to 2% of the optical generation rate in the whole nanowire and starts to limit J_sc noticeably. Note that the high doping level does not cause a net Auger recombination in itself, but induces a high Auger recombination velocity for the optically generated holes, which strongly reduces the benefit of gradient doping at very high doping levels.

Thus, by using a gradient doping profile in the top n segment, we can prevent the minority carriers—that is, the holes—from reaching the top ohmic contact. With an increasing doping gradient, we deflect the minority carriers away from the contact more and more efficiently. However, at the same time, we open ourselves up to Auger recombination of the photogenerated carriers, which decreases the overall current. Note that when such a gradient doping profile is used at the back side of a planar solar cell [36], we do not expect similar problems with increased recombination, since the photogeneration rate is expected to be low on the bottom side of the solar cell.

To avoid the increased recombination losses introduced by the gradient doping profile above, we consider here a design with a semiconductor heterostructure. Our idea is to use a semiconductor with a higher band gap for the top n segment, in which only a small amount of holes is generated. In this way, only a small amount of holes can diffuse into the top contact. After consideration of both the band structure [37] and growth possibilities [38], we selected GaP, which has an indirect band gap, to be used for the n segment. We keep the total length of the nanowire at the previously chosen L = 1400 nm. We have calculated the photogeneration rate for the case of a GaP top segment (dashed line in figure 5(a)), which should be compared to the case of an InP top segment in figure 3(b). The optical generation rate in this GaP top n segment is 2.3 mA cm⁻², which is only 9% of that in the InP i segment below. At the same time, we find a charge separation of close to 100% in the i segment—that is, SIQE ≈ 100% (solid line in figure 5(a)). The concentration of photogeneration into the i segment with an SIQE ≈ 100% leads to a high J_sc, with values close to the absorption limited J_sc,opt.

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Finally, when we compare the J_sc for varying n-segment length (figure 5(b)), we find much higher current levels with the GaP top segment than with the InP top segment. For a 100 nm long GaP segment, we reach a short-circuit current of 27.6 mA cm⁻². This current level is very close to the above-stated absorption limited J_sc,opt = 27.9 mA cm⁻² in a 1400 nm long InP nanowire, when all the photogenerated carriers are extracted as current. Note that the overall absorption in the nanowire tends to decrease with an increasing length of GaP segment. For example, for a GaP segment of 500 nm in length, the drop in the overall absorption is equivalent to a photocurrent of 1.7 mA cm⁻² (see supplementary data section S3).

When the GaP segment length is increased from 100 nm to 500 nm, the current drops from 27.6 to 24.0 mA cm⁻². In contrast, a similar increase in the n segment length when using InP for the top segment causes a much larger drop—from 25.7 to 15.9 mA cm⁻². Thus, for the 500 nm long top segment, the use of GaP gives a 50% higher short-circuit current. In this way, with the GaP top segment, we can allow for a relatively long top segment without losing current detrimentally.