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Abstract
Energy harvesting from ambient light can be used to power wireless sensors and other standalone electronic devices. The intensity of light used for illumination is 300-3000x lower than sunlight and the spectrum of artificial light is normally narrowly concentrated in the visible range. As a result, the optimal design of photovoltaic devices for energy harvesting from ambient light differs from conventional solar cells. We calculate the maximum efficiency for Si photovoltaic devices operating under conditions expected indoors as a function of the cell thickness, taking into account the relevant properties of Si. The optimum thickness for devices operating under 250 lux illumination produced by white LED’s is 1.8 µm and the efficiency is 29.0%, whereas for direct sunlight, the optimum thickness is much larger at 109 µm, while the maximum efficiency is almost the same (29.7%). The relative efficiency increases logarithmically with light intensity at 8.5% per decade.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Energy harvesting from ambient lighting with photovoltaic devices is a convenient way to power sensors and other standalone electronic devices with capability for low power wireless communication [1,2]. Applications of standalone devices include the internet of things, personal health monitoring, and sensors for smart buildings [1–4]. Although photovoltaic devices can be made from a number of different semiconductor materials [5–7], Si has special importance for light harvesting because intelligent sensors need to be made from Si anyway in order to accommodate the necessary circuitry [3].
The light intensity used for indoor illumination is 300-3000 times weaker than direct sunlight [2] and in some outdoor settings, such as sidewalk lighting at night, the light intensity is 3x104 times weaker than direct sunlight [8]. The spectrum of light used in illumination is also different from sunlight. For maximum efficiency the radiation used in lighting is typically more narrowly concentrated in the visible part of the spectrum, depending on the type of light source used, whereas sunlight approximates a blackbody with a broad emission spectrum. A variety of light sources are used for lighting including discharge lamps, hot tungsten filaments, fluorescent lamps and light emitting diodes [5]. In this paper we explore how the design of a Si photovoltaic device must be changed for optimal performance in light harvesting applications and how the maximum efficiency compares with conventional Si solar cells operating in sunlight. We calculate the maximum possible efficiency allowed by thermodynamics and the physical properties of Si, independent of device design, following a methodology developed earlier for conventional solar cells [9,10]. An interesting result of this study is that the optimum thickness for a Si light harvesting device is 1.8 μm, whereas the optimal thickness of a conventional Si solar cell is 109 μm. This reduction in thickness increases the maximum efficiency of the energy conversion by 18% under indoor light harvesting conditions.
Efficiency is important for light harvesters just as it is for conventional solar cells because it determines the light level required to operate the sensor device, the size of the light collecting element and hence impacts on the cost. It is instructive to compare the light intensity available in an office environment with the power required by a wireless sensor chip. As one example, a motion sensor for biomedical purposes consisting of a system on chip with integrated radio has been demonstrated with 6.45 μW power consumption [3]. If the office is illuminated at a level of 250 lux with LED lights, the light intensity is about 80 μW/cm2, as discussed below.
2. Photovoltaic device model
As an idealized version of a Si photovoltaic device, we consider a high purity intrinsic Si slab covered with an SiO2 surface passivation layer with electrical contacts provided by small p- and n-type point contacts in openings in the surface passivation layer. The SiO2 surface layer and point contacts reduce surface recombination and minimize leakage currents. The front surface is randomly textured and the back surface is coated with a high reflectivity layer in order to scatter the incoming light and trap it inside the Si. Light is efficiently trapped because of the small critical angle for total internal reflection in Si due to the high index of refraction. Taking into account multiple internal reflections, the absorbance A of the semiconductor slab can be described approximately as follows [9]:
as a function of the semiconductor thickness L, absorption coefficient α and refractive index n. This simple expression for the absorbance can be expected to be a good representation for the absorbance when the film thickness is greater than an optical wavelength in the material which is about 0.3 µm for bandgap radiation in Si, taking into account the refractive index. This expression has been tested experimentally for Si films in the 0.75-1.4 µm thickness range [11].The short circuit current (ISC) is the maximum current that can be drawn out of an illuminated solar cell slab at zero output voltage. Mathematically it is equal to the integral of the product of the absorbance and the spectrum of the incident radiation. At zero output voltage all of the electron hole pairs generated by the incident radiation can in principle be extracted as current into the external circuit. At non-zero output voltage the electron-hole pair concentration inside the semiconductor material exceeds the equilibrium concentration and some of the energy stored in the photo-generated electrons and holes and internal luminescence is lost through intrinsic non-radiative recombination processes or by emission of luminescent photons from the front surface of the semiconductor slab. Non-radiative recombination mediated by defects and impurities (Shockley-Read-Hall recombination) can be neglected in the ideal limit in which defects and chemical impurities do not exist. The remaining losses which cannot be avoided by improving material quality are Auger recombination, external luminescence and free carrier absorption.
The maximum output voltage VOC occurs in the open circuit condition when the current extracted by the external circuit is zero. The maximum output power can be found by solving for the maximum value of the product µf where µ is the separation between the electron and hole chemical potentials (also known as quasi-Fermi levels) and f is the fraction of the photogenerated electron-hole pairs that is extracted as current in the external circuit. In the case of a device with ideal n and p-type contacts the output voltage V is equal to the separation between the electron and hole quasi-Fermi levels µ/e. We assume that the chemical potential is constant throughout the cell or in other words that the solar cell is thin compared to a carrier diffusion length. The chemical potential µ of the electron-hole pairs is also equal to the chemical potential of the luminescent photons in the Si, which is non-zero when the Si is under illumination. µ and f are related to each other by the following nonlinear integral equation derived from detailed balance [9]:
The electron density n in this equation is related to the chemical potential as follows:where ni is the intrinsic carrier concentration. In the case of undoped Si and for a solar cell that is operating in a high injection condition the Auger recombination rate C(n) can be approximated by [12],The non-integer exponent in Eq. (4) takes into account electron-electron correlation effects in the Auger process. In the equations in this paper the refractive index n is distinguished from the electron density which has the same symbol, by their places in the equations. The Si material parameters needed to compute the various terms in Eq. (2) are provided in Table 1.
Table 1. Material parameters for crystalline silicon, GaAs and CdTe used in this paper
There is a term on the left hand side of Eq. (2) for each of the loss processes: α1 is the free carrier absorption which is proportional to the carrier concentration, bn(E,T) is the spectrum of room temperature black body radiation and C(n) is the Auger recombination rate. Auger recombination is the most important loss mechanism at high illumination levels when the carrier density is high. In addition to Auger recombination the photo-generated electron-hole pairs can recombine radiatively and produce luminescence inside the semiconductor. This radiation can be re-absorbed or depending on the thickness of the semiconductor it can be lost by escaping out the front surface of the semiconductor. In thick solar cells most of the recombination radiation is re-absorbed and for thin solar cells most of the recombination radiation escapes out the front surface. Emission of radiation from the solar cell is the second important loss process in conventional Si solar cells and dominates at low illumination levels when Auger recombination is not important. Electrons and holes can also absorb photons inside the semiconductor material through free carrier absorption and dissipate the photon energy into heat. This loss process is weak in Si, especially at low light levels.
Figure 1 shows the light spectra of two common indoor light sources, a white fluorescent lamp (standard F3 illuminant) and a white LED (2700 K color temperature), plotted together with the air mass 1.5 global solar spectrum. The white LED and fluorescent light mainly emit in the visible wavelength range in contrast to the solar spectrum which spreads over a large range from the UV to the IR. The solar spectrum in Fig. 1 is obtained from the National Renewable Energy Lab for an Air Mass 1.5 solar spectrum measured at a 37° tilt including both direct and scattered light from the sky [17]. The spectra for the indoor light sources, were taken from the OSRAM Company’s ColorCalculator [18]. The OSRAM ColorCalculator also calculates the photometric parameters of the spectra in accordance with International Commission on Illumination color space of 2017. According to the ColorCalculator, the color rendering index of the white fluorescent lamp and the white LED shown in Fig. 1 are 57 and 82 respectively.
Fig. 1 The spectra of three different light sources: sunlight (AM1.5G), white LED (2700 K) and F3, white fluorescent lamp, standard illumination equivalent to 3450 K. The spectra are normalized to their peak values. The solar spectrum is from the National Renewable Energy Lab website [17]. The white LED and F3 spectra are taken from the Osram ColorCalculator software [18].
A typical illumination level in a school classroom, building lobby or an office in which ‘easy work’ is being carried out is 250 lux [19]. If the illumination level of the light sources in Fig. 1, are normalized to 250 lux then the efficiency of a solar cell can be compared for different light spectra at the same illumination level. The mathematical formulation for the solar cell output from reference [9] was implemented in Matlab codes and the solar cell efficiency calculations were carried out with crystalline Si material parameters. At high carrier concentrations the optical absorption and hence the radiative recombination coefficient depends on carrier density. In this paper we use the low carrier concentration value for the absorption coefficient of Si with no loss of accuracy since we are interested in indoor photovoltaic devices which operate at low light intensity [20]. The solar cell efficiencies as a function of Si layer thickness for the light sources in Fig. 1 and an illumination level of 250 lux are plotted in Fig. 2. In the case of the solar spectrum the best solar cell performance occurs when the solar cell is thick (109 µm) whereas for illumination by the white LED and fluorescent lamps at lower intensity the optimum thickness are 1.8 µm and 1.2 µm, respectively. A solar cell operating in sunlight needs to be thick because the solar spectrum contains significant radiation in the infrared region for which the Si absorption coefficient is weak. In the case of the two indoor light sources selected, the IR content of the radiation is small, all of the radiation is strongly absorbed and a thin layer of Si is sufficient. The efficiency with the F3 light source in Fig. 2 is lower than for LED light because the F3 spectrum has more blue content. Figure 2 gives an idea of the range of maximum energy conversion efficiencies and optimum thicknesses to be expected with other efficient artificial light sources.
Fig. 2 The efficiency of Si solar cells as a function of thickness with randomly textured antireflection coated front surface and perfectly reflecting back surface with three different light sources: sunlight (AM1.5G), white LED (2700 K), and F3 white fluorescent lamp. The light source intensities were all normalized to 250 lux, an appropriate illumination level for an easy office environment.
The limiting efficiencies and optimum device thicknesses are summarized in Table 2 below. It is noteworthy that the limiting efficiency of 29.7% and optimum thickness of 109 μm for Si solar cells operating in sunlight are close to the corresponding values obtained by Richter et al. [21] of 29.43% and 110 μm with a more comprehensive model.
Table 2. Calculated maximum solar cell efficiencies for sunlight (AM1.5G) and white LED (2700 K) illumination. The Si solar cell thicknesses are optimized for the light source and illumination level while the GaAs and CdTe solar cells have a fixed thickness of 0.5 µm. Because of strong absorption at the bandgap and abrupt absorption edges the GaAs and CdTe efficiencies have a weak thickness dependence above a threshold thickness. Sun is the AM1.5G spectrum, F3 is a white fluorescent lamp, and LED is a white LED with a colour temperature of 2700K.
As discussed above the output voltage is equal to the separation between the electron and hole quasi-Fermi levels, in which case the voltage increases as the carrier density increases. Thin solar cells have higher output voltages because the carrier concentration is higher in a thin layer but at the same time the output current is lower, because thin layers are not as effective at absorbing the incident radiation. The tradeoff between these two effects leads to an optimum thickness at which the output power has a maximum. The thickness dependence of the efficiency in Fig. 2 shows the importance of matching the solar cell thickness with the light source of interest. Figure 2 also shows that the narrower spectrum of indoor light concentrated in the visible part of the spectrum, increases the maximum energy conversion efficiency. Near optimal performance for indoor solar cells is obtained with solar cells that are 0.8-2.0 µm thick and for outdoor solar cells the optimum thickness is in the 20-300 µm range. The efficiency improvement with indoor light sources has been demonstrated experimentally earlier [22].
Figure 3 shows the magnitude of the various loss processes obtained from Eq. (2) for a Si solar cell operating under AM 1.5 solar illumination and white LED illumination at 250 lux, both at their respective optimum solar cell thicknesses. The values for the various loss processes for the solar cell with AM 1.5 sun illumination (Fig. 3(a)) are in good agreement with the earlier results in reference [9] but not identical because the material parameters for Si in Table 1 have been updated. Auger recombination increases approximately as the cube of the carrier concentration and is the most important loss process at the high illumination level in Fig. 3(a) characteristic of direct sunlight. Auger recombination is still significant at the low illumination level characteristic of indoor illumination but is not the largest loss mechanism. The thin Si layer in Fig. 3(b) is less effective at re-absorbing the internal luminescence than the thick Si layer in Fig. 3(a), with most of the luminescent photons escaping in Fig. 3(b). The external luminescence emission is the most important loss process for the indoor photovoltaic device in Fig. 3(b). Free carrier absorption is almost negligible in both cases.
Fig. 3 Summary of the relative magnitudes of the various loss processes in Si solar cells under (a) AM 1.5 global solar illumination and b) white LED at an illumination of 250 lux at the maximum power point. The losses are represented as currents.
3. Intensity dependence of limiting efficiency
The solar cell efficiency increases at high light intensities because the carrier density is higher which increases the output voltage, while the extracted current increases almost linearly with the illumination level. Figure 4 shows the effect of the illumination level on the efficiency-thickness curves. Three and six orders of magnitude reduction in the solar intensity, drops the solar cell efficiency to 23.8% and 17% respectively. The optimum thickness of the Si layer also decreases somewhat at lower illumination levels, as shown in Fig. 4. At high intensity Auger recombination becomes more important, which favors thicker layers for which the carrier density is lower.
Fig. 4 The calculated efficiency of silicon solar cells illuminated by the AM1.5G solar spectrum at three different intensities, as indicated. The maximum efficiency and optimum solar cell thickness increase with light intensity. The low intensity curve (10−6x one sun) is approximately the intensity of moonlight.
The maximum output power for Si solar cells was calculated for the six order of magnitude range in illumination levels from direct sunlight to full moon for two different illumination spectra, namely sunlight and white LED light. The intensity dependence of the output power shown in Fig. 5 follows a power law to a high degree of accuracy with exponent 1.037 over six orders of magnitude in intensity. In other words the limiting efficiency of Si solar cells depends on intensity I as:. Equivalently, the relative efficiency increases approximately logarithmically with light intensity at 8.5% per decade. The power output is consistently higher at the same illumination level for sunlight in Fig. 5 because the IR and UV components of sunlight which do not contribute to the illumination, do contribute to the output power. The power output as a function of illumination level has been measured for several different solar cell materials and illumination spectra before [22,23] but there has been no consideration given to the optimum thickness for energy harvesting devices in the literature. Figure 6 shows the optimum thickness of a Si solar cell as a function of illumination level for sunlight and white LED light sources. The optimum Si thickness for white LED illumination is almost two orders of magnitude smaller than for sunlight, and therefore solar cells optimized for outdoor use are not optimal for indoor energy harvesting. Table 2 summarizes the limiting efficiencies and optimum thickness of Si solar cell for several different light sources and illumination intensities.
Fig. 5 Calculated power output as a function of illumination intensity for Si photovoltaic cells illuminated by a light spectrum matching AM1.5G, triangles, and by a white LED (2700K), circles. The electrical output is a power law function of the light intensity with exponent 1.037.
Fig. 6 Optimum thickness as a function of illumination intensity for photovoltaic cells illuminated by a light spectrum matching AM1.5G and by a white LED (2700K).
In energy harvesting applications from intermittent light sources it is convenient to use capacitors for energy storage. In this case the output voltage is an important factor in the system design. The maximum achievable output voltage at the maximum power point is shown in Fig. 7 as a function of illumination intensity for both the solar spectrum and the LED spectrum. The output voltage increases approximately logarithmically with light intensity with a slope of 19 meV per factor of e in the case of the LED data in Fig. 7. Naively one might have expected this slope to be kT/q, however the increasing optimum thickness with light intensity means that the carrier density does not increase linearly with light intensity. Also the output voltage for the optimized device under LED illumination is higher than in sunlight with the same brightness because the indoor device is much thinner which increases the carrier density and the voltage.
Fig. 7 Calculated output voltage at maximum power as a function of illumination intensity for silicon photovoltaic cells illuminated by a light spectrum matching AM1.5G (triangles) and by a white LED (2700K) (circles).
4. Comparison with CdTe and GaAs solar cells
In Table 2 we show for comparison the limiting efficiency of GaAs and CdTe thin film solar cells that are 0.5 µm thick calculated in the same way as for Si. The relevant electronic properties for CdTe and GaAs are provided in Table 1 with references. Because CdTe and GaAs are direct bandgap semiconductors with strong optical absorption just above the bandgap, the thickness dependence of the cell performance is less important than in Si. The calculated efficiency of 32.8% for a 0.5 μm GaAs cell in sunlight in Table 2 is the same as the result in ref [10]. As noted previously in ref [23]. the higher bandgap of CdTe and GaAs means that solar cells made from these semiconductors can convert the visible light characteristic of indoor illumination into electricity significantly more efficiently (43.7% and 40.3% respectively) than in the case of Si (29.0%) even though their performance in sunlight is only marginally better than Si.
5. Conclusions
We have calculated the maximum efficiency of Si, GaAs and CdTe solar cells allowed by the known electronic and optical properties of the materials under conditions characteristic of indoor energy harvesting applications where the radiation is mainly in the visible part of the spectrum and the intensity is lower than in the case of sunlight. We show how the optimum design and performance of Si solar cells depends on light intensity at low light levels. The optimum thickness of a Si solar cell for indoor applications is found to be 1.8 µm, significantly thinner than the optimum design for outdoor applications in sunlight. In addition, the optimal thickness increases with light intensity. Because the optimal thickness for indoor energy harvesting is in the micron range it should be possible to design efficient thin film Si solar cells using integrated circuit type manufacturing technologies, rather than the bulk wafer production methods used for conventional solar cells. In addition a thin photovoltaic device can be fully depleted and the response time fast enough to detect rf modulated optical signals for wireless communication. The maximum efficiency of Si, GaAs and CdTe solar cells illuminated by indoor light, in which the light is concentrated in the visible part of the spectrum, is higher than at similar illumination levels in the case of the wider blackbody spectrum characteristic of sunlight. The maximum efficiency of Si solar cells is found to have a power law dependence on light intensity with exponent 0.037, over six orders of magnitude in light intensity.
Funding
NSERC.
Acknowledgments
We thank Tom Pinnington of Cooledge Lighting for helpful discussions on illumination and for introducing us to the Osram ColorCalculator. We thank NSERC for financial support.
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