We demonstrate the sensitive measurement of photocarriers in an active layer of a GaAs-based photovoltaic device using time-resolved terahertz reflection spectroscopy. We found that the reflection dip caused by Fabry-Pérot interference is strongly affected by the carrier profile in the active layer of the *p-i-n* structure. The experimental results show that this method is suitable for quantitative evaluation of carrier dynamics in active layers of solar cells under operating conditions.

Solar cells with high energy conversion efficiency have been developed recently.^{1} According to Shockley-Queisser theory, the maximum efficiency is 31% for single-junction solar cells under standard AM1.5 sunlight.^{2} Because it is expected that the maximum efficiency can be achieved for materials with a bandgap energy, *E*_{g}, of around 1.34 eV, solar cells fabricated from materials with ideal bandgap energies have been optimized by evaluating their performance under continuous irradiation. The GaAs-based solar cell is used as a model device because its bandgap of 1.39 eV is close to the ideal value. To reach the maximum conversion efficiency, it is critical to understand the carrier dynamics in the device. In particular, the dynamics involving phenomena such as charge separation in active regions, and information about photocarrier lifetimes including photon recycling, which is the reabsorption of photons generated as a product of radiative recombination,^{3–6} are crucial. Photoluminescence^{7–10} and electroluminescence^{11–17} are convenient tools for characterizing solar cell properties such as minority carrier lifetime, diffusion length, internal conversion efficiency, and charge separation efficiency. Luminescence from a solar cell contains information about electron-hole pairs in an active layer, but the overlapping of luminescence bands from *p*- and *n*-type regions obscures the luminescence from the active layer. Moreover, photon recycling makes analysis complicated. Hence, a combination of different optical methods will provide important insights into the development of solar cells that are more efficient.^{18–21}

Terahertz (THz) spectroscopy is one of the most powerful optical methods for investigating photoexcited free-carrier dynamics in semiconductors. Time-resolved THz spectroscopy (TRTS) has also made it possible to evaluate photocarrier dynamics in various solar cell materials.^{22–29} We can quantitatively determine the photocarrier density, *N*, and the momentum scattering time, *τ*, with the Drude-Lorentz analysis of THz data using a known value of effective mass, $ m \u22c6 $. The technique for evaluating carrier properties in doped bulk materials and epitaxial films from their optical properties is well established.^{30–33} The optical constants can also be evaluated from the reflection spectra through the Fresnel coefficients.^{8} In the THz region, the reflection features are sensitive to the carrier density owing to phonon-plasmon coupling modes, and a number of studies have revealed the effect of doping on the optical constants.^{31–34} However, strong free carrier absorptions and multiple reflections caused by micrometer-scale layers of semiconductors with high doping concentrations on the order of 10^{18} cm^{−3} prevent simple spectroscopic analysis. Additionally, the photocarrier density under AM1.5 sun is smaller than the usual doping concentrations of layered semiconductors. Therefore, it is challenging to monitor photocarriers in an active region of a solar device by THz spectroscopy.

In this letter, we demonstrated that time-resolved THz reflection spectroscopy can be used to monitor the frequency shift caused by Fabry-Pérot interference to sensitively measure the concentration of carriers in the active layer of a GaAs photovoltaic device under open circuit conditions. A theoretical calculation using the Drude-Lorentz model quantitatively reproduced our experimental findings. Taking advantage of the interference-induced phase-sensitive responses, we established a method for precisely evaluating the concentration of photocarriers in an active layer of a solar device. We propose this method as a useful tool for evaluating solar cells under operating conditions.

Figure 1(a) shows a schematic of our GaAs-based photovoltaic device with a single junction. This consists of a 660-nm-thick *p*-type layer, a 500-nm-thick *i*-region, and a 3-*μ*m-thick *n-*layer, all of which were grown by molecular-beam epitaxy on an *n*^{+}-type GaAs substrate with a thickness of 350 *μ*m. The doping level of carbon in the *p*-type layer was 1 × 10^{18} cm^{−3} and that of silicon in the *n*-type layer was 2 × 10^{17} cm^{−3}. In general, the dielectric function, $ \epsilon \u0303 j ( \omega ) $, of a layer labeled with index *j* (= *p*, *i*, *n*, *n*^{+}) in a semiconductor device is expressed by the Drude-Lorentz model as

With the GaAs-based device in this study, the first term, *ε*_{∞} = 10.86, is the high-frequency dielectric constant.^{34,35} The second term is the phonon component, in which the static dielectric constant, *ε*_{s}, is 12.8, the transverse-optical phonon frequency, *ω*_{TO}/2π, is 8.0 THz, and γ is the damping constant.^{34,35} This causes high reflectivity around 8 THz, which is known as the reststrahlen band.^{9} The third term is the free carrier component of the conduction, in which *ε*_{0} is the vacuum permittivity. The optical conductivity, $ \sigma \u0303 j ( \omega ) $, is expressed as

where *N _{j}* is the free carrier density,

*e*is the electron charge,

*τ*is the momentum scattering time, and $ m e , h * $ is the effective mass of carriers. The effective masses of electrons and holes are assumed to be $ m e * $ = 0.067

*m*

_{0}and $ m h * $ = 0.35

*m*

_{0},

^{34,35}respectively, where

*m*

_{0}is the free-electron mass. We have confirmed the availability of these values and the Drude model by measuring chemically doped GaAs substrates and epitaxial films grown by molecular-beam epitaxy method which was also used for fabricating the photovoltaic sample. The density of the photoexcited carriers is smaller than the chemical doping concentrations, and the properties of the photoexcited carriers should be as same as that of chemically doped carriers in the device. Therefore, the availability of the Drude model with the effective mass approximation has been verified from the terahertz spectra of the chemically doped GaAs substrates, epitaxial films, and photovoltaic devices, which is more detailed in the supplementary material. The screened plasma frequency, $ \omega s p , \u2009 j $, of free carriers is expressed as $ \omega s p , \u2009 j = N j e 2 / ( \epsilon b \epsilon 0 m e , h * ) $. The screened plasma frequency of the

*p-*type layer, $ \omega s p , \u2009 p / 2 \pi $, is 4.2 THz, that of the

*n-*type substrate, $ \omega s p , \u2009 n / 2 \pi $, is 4.2 THz using the background dielectric constant of $ \epsilon b = \epsilon s $, and that of the

*n*

^{+}

*-*type substrate, $ \omega s p , \u2009 n + / 2 \pi $, is 18 THz with $ \epsilon b = \epsilon \u221e $. Figure 1(b) shows the simulated reflection coefficients at the interfaces between the air and

*p*-type layer (red dashed curve), between the

*i*- and

*n*-type layers (black dotted curve), and between the

*n*- and

*n*

^{+}-type layers (blue curves) in the GaAs device. The reflectivity is high below the screened plasma frequency, $ \omega s p , \u2009 p $. Because the THz pulse is attenuated less in the

*p*- and

*n*-type layers above the screened plasma frequencies of $ \omega s p , \u2009 p $ and $ \omega s p , \u2009 n $, the THz pulse can propagate in the

*p-i-n*layer, except in the reststrahlen gap. It also causes the strong reflection of the THz pulse at the boundary between the

*n*-layer and

*n*

^{+}-substrate below $ \omega s p , \u2009 n + / 2 \pi $ = 18 THz (Fig. 1(a)). Consequently, the reflection spectrum in the window region between the frequencies of $ \omega s p , \u2009 p $ and $ \omega s p , \u2009 n + $ (light blue area in Fig. 1(b)) contains the information about photoexcited carriers in the

*p-i-n*structure. Furthermore, strong dispersion of the refractive index near the screened plasma frequencies causes Fabry-Pérot interference between both sides of the

*p-i-n*structure, which increases the spectral responsiveness. The spatial distribution of the electric field of the Fabry-Pérot interference is analyzed in detail in Fig. 4(b).

In the experiments, we used a time-resolved terahertz spectroscopy system^{27–29} based on a 1 kHz amplified Ti:sapphire laser with a duration of 35 fs and a center wavelength of 800 nm. We divided the output beam into three using two beamsplitters for optical excitation, THz pulse generation, and detection. The center wavelength of the excitation pulse was 800 nm, and its fluence was varied from 0.88 to 31 *μ*J/cm^{2}. Because the penetration depth was 750 nm, the *p*- and *i*-layers were mainly excited in the photovoltaic sample. THz pulses from two-color (*ω* and 2*ω*) excited air plasma were focused on the sample with *p*-polarization at a 30° incidence angle. The reflected THz pulse was detected with the electro-optic sampling method by using a 300–*μ*m-thick GaP crystal, which had an available frequency range from 0.5 to 7.5 THz, covering the sensitive frequency region of the device. The time profile of the THz pulses was obtained by varying the time delay, *t*, between the THz and sampling pulses. We also measured THz pulses reflected on an Al plane mirror as a reference to evaluate the absolute reflectivity precisely. For the optical excitation, we used unfocused fundamental pulses at normal incidence. The transient reflection response was obtained by varying the pump-probe delay, Δ*t*_{p}. All the measurements were performed under open circuit conditions.

Figures 2(a) and 2(b) show the reflected THz transient from the Al reference mirror *E*_{ref}(*t*), and THz transient, *E*(*t*, Δ*t*_{p}), at the surface of the sample without photoexcitation (blue dotted curve) and 10 ps after photoexcitation (red solid curve). We confirmed that the hot carriers were relaxed thermally by the delay time 10 ps after photoexcitation.^{27} The spectrum of the THz radiation from air plasma extends to over 10 THz; however, the sensitivity of the GaP-based electro-optical (EO) detector is low in the frequency range above 7 THz owing to the phase mismatching of the EO effect. Thus, the oscillation components appear above 8 THz. This oscillation is strongly modulated by the reflection dip of the photovoltaic device. The peak amplitude of reflected THz pulses was increased by the photoexcitation, but the strength of the oscillation around Δ*t*_{p} = 0.3 ps was attenuated. We Fourier transformed these time profiles to extract the complex reflection spectra. Figure 2(c) shows the power reflectance $ R ( \omega ) = | E ( \omega , \Delta t p ) / \u2009 E ref ( \omega ) | 2 $ as a function of frequency. The blue and red circles show the spectra without excitation and 10 ps after photoexcitation, respectively. The reflectance dip appeared at 5 THz without excitation. We numerically simulated the complex total reflection from the sample by a sequential calculation of the Fresnel coefficients, which reproduced the measured results consistently (solid lines). The best parameters of $ N p $, $ N n $, $ N n + $, and *τ* were 1 × 10^{18} cm^{−3}, 1.3 × 10^{17} cm^{−3}, 3 × 10^{18} cm^{−3}, and 80 fs, respectively. These values were consistent with the original carrier density of our sample structure. We evaluated the parameters of the GaAs substrates, thin films and other device structures and confirmed these values are valid. The deviation of $ N n $ from the designed value of 2 × 10^{17} cm^{−3} was within the error possible during the fabrication process. After the photoexcitation, the reflectance dip shifted toward a higher frequency, as shown by the red circles. We also calculated the transient reflection spectrum assuming the ratios of the excess photoexcited carrier density in *p-, i-*, and *n-*regions are distributed exponentially following the Beer-Lambert law distribution, and the simulated spectrum reproduced the measured spectrum well (Fig. 2(c)). This frequency shift could be attributed to the change of the plasma frequency because the reflectance dip appeared near the original plasma frequency of $ \omega s p , \u2009 p $. However, the anomalous phase jump cannot be explained by this simple model. Figure 2(d) shows the phase of the complex reflection coefficient $ \Delta \varphi ( \omega ) = a r g ( E ( \omega , \u2009 \Delta t p ) / E ref ( \omega ) ) $ as a function of frequency. While Δ*ϕ*(*ω*) monotonically increased toward 2π with the frequency without photoexcitation, the direction of the phase jump was changed drastically by the photoexcitation. The change in the detection of phase jump is characteristic of the interference between two waves. Impedance matching and tuning for microwave circuits are performed by monitoring such jumps.^{36,37} In the visible and THz frequency region, high sensitivity-sensing of surface plasmon resonance by observing the large phase change has been reported.^{38,39} In our case, the destructive interference between the returning THz pulse on the surface of the device and the reemitted THz pulse from the resonator, including *p-i-n* structures caused the phase anomaly, which enhanced the response of the photoexcited carrier in the active layers.

The representation of the reflection coefficient in the complex plane supports our observation of the phase jump.^{36–39} Figures 3(a) and 3(b) show the parametric plots of the complex reflective coefficient, $ r \u0303 ( \omega ) $, as a function of the frequency on the complex plane. The green arrow is an example of a vector showing the complex reflective coefficient, where the square of its length and the polar angle correspond to *R*(*ω*) and Δ*ϕ*(*ω*), respectively. The blue and red circles indicate the results without excitation and 10 ps after photoexcitation, respectively. The trajectories start from the first quadrant and turn counterclockwise as the frequency increases. At the center frequency of the reflectance dip, the trajectory passes through the horizontal axis from the positive to the negative region. The intersection of the horizontal axis moves smoothly in the positive direction in the complex plane with increasing photocarrier density, *N*_{ph}, and passes through the origin at *N*_{ph} of 3 × 10^{13} cm^{−2}. When the trajectory goes through the zero reflection point, the 2π phase jump occurs in the phase Δ*ϕ*(*ω*) spectrum. Because the interference-induced response is enhanced near the zero reflection point, as in the balance detection method,^{36} we can monitor the photocarrier density in the *p-i-n* structures with high precision.

Figure 4(a) shows the *N*_{ph} dependence of the frequency shift, Δ*ω*_{dip}/2π (red circles). Although the frequency shift is roughly proportional to *N*_{ph} in the low excitation region below 1 × 10^{13} cm^{−3}, the slope becomes shallow in the higher excitation region. For low excitation densities, the shift is reproduced well by the numerical simulation (solid line, Fig. 4(a)). Hence, we can determine the sheet carrier density of photocarriers precisely with a resolution of the order of 10^{12} cm^{−2}. It corresponds to bulk density of 10^{16} cm^{−3}, assuming reflection loss of the device, the unit quantum efficiency, and the penetration depth of 750 nm. This is lower than the doping concentration in the *p*- and *n*-layers of the device. This resolution is sufficient for measuring photocarrier densities in devices under realistic solar irradiance because the sheet carrier density of 10^{12 }cm^{−2} corresponds to that due to photocarriers in GaAs concentrator solar cell devices. For example, for the solar spectra at 1 sun AM1.5G,^{40} the photon flux, *Φ*, for wavelengths between 280 and 880 nm is estimated as 1.85 × 10^{17 }cm^{−2} s^{−1}. Considering the Fresnel loss, photons with a flux of 1.08 × 10^{17 }cm^{−2} s^{−1} can be absorbed by GaAs. The photocarrier density, *N*, can be written as *N* = *Φ* × *τ*_{life} where, *τ*_{life} is the carrier lifetime. Assuming *τ*_{life} of ∼10 ns, *N* can be estimated as ∼10^{9 }cm^{−2}. For high excitation densities, the measured shifts are smaller than the calculated shifts. We think this is due to the carrier diffusion. Figure 4(b) shows the distribution of the absolute square of the electric field $ | E | 2 $ (red curve) of the lowest interference mode at 5 THz in the *p-i-n* structure calculated by the transfer matrix method. The electric field is the highest near the surface where the *p*- and *i*-layers lie, which allows spatially sensitive monitoring of the photocarrier density. For the high excitation regime, some photocarriers could diffuse into the insensitive region because the ambipolar diffusion behavior of electrons and holes can be enhanced by the many-body effects,^{41} as illustrated in Fig. 4(c). Other possibility is density dependence of the effective mass $ m * $ because the effective mass $ m * $ can become larger at high excitation regime, which can cause underestimation of the evaluated carrier density. Consequently, the evaluated carrier density is smaller for the high excitation regime.

The quantitative characterization of charge carriers in the devices via THz reflectance could be used widely because of the design of direct-gap compound semiconductor solar cells. In general, solar cells consist of a micrometer-scale layered structure owing to their absorption coefficients. The emitter layer is a sub-100-nm-thick thin film with a high doping concentration. The base layer is thick with a low doping concentration. Although the emitter layer is opaque for excitation light due to the strong free carrier absorption, the layer is thin enough for the THz waves to propagate into the *p-i-n* structure. Hence, the time-resolved THz reflection measurement could be a powerful tool for investigating the spatiotemporal dynamics of photocarriers in compound solar cell devices. We have measured the center frequency of the dip as a function of the pump-probe delay and observed the recovery of the frequency shift for the reflection dip with a decay constant of 3 ns, which reflects the carriers' decay in the device. The photocarrier dynamics observed quantitatively in solar cell devices near their operating conditions with TRTS provides a good measure for estimating photocarrier generation efficiency precisely, enabling more accurate device simulations. For indirect gap semiconductor devices, such as single-crystal Si solar cells, the interference effect is less important in the THz region because the multilayer is much thicker owing to the small absorption coefficient.

In conclusion, we have measured transient reflectance in the GaAs photovoltaic device with time-resolved THz reflection spectroscopy. We demonstrated that the photocarrier density in the photovoltaic device can be monitored sensitively by measuring the THz transients of the device, even with the highly doped *p*-type emitter layer. Fabry-Pérot interference dominates the reflection features, and the center frequency of the reflection dip strongly depends on the excitation density. For the low excitation regime, our numerical simulation reproduced the reflection features of the photovoltaic device consistently. We propose that the interference-induced phase-sensitive response can be used for determining the photocarrier density with a high resolution of the order of 10^{16 }cm^{−3}, which is smaller than the doping concentration in typical photovoltaic devices. Such a direct measurement of carrier density in the active region of the device will contribute to the elucidation of carrier dynamics in solar cell devices under operating conditions.

See supplementary material for the derivation for the reflection coefficient of the photovoltaic device and the temporal evolution of the transient reflection coefficient.

This work was supported by Grant-in-Aid for JSPS Fellows (No. 16J05700). Y.K. and H.A. are thankful for the support from JST-CREST.