Introduction

The photovoltaic effect, the spontaneous generation of electric current in materials upon optical excitation, is widely used in energy harvesting, sensing, and imaging devices. It also serves as a unique probe of versatile quantum phases of matter1. In particular, recent studies have uncovered the close relation between photocurrents arising from nonlinear processes and the quantum geometry and topology of electron wavefunctions2,3. A representative example of such nonlinear photocurrent phenomena is the shift current, a steady-state photocurrent arising from the second-order nonlinear optical process in materials lacking inversion symmetry. As opposed to the conventional photocurrent carried by drift and diffusion of photoexcited free charges, the shift current is driven by shift of intracell coordinates during the optical excitation, which is given by the difference in the Berry connections between the two Bloch wavefunctions before and after the transition4,5,6,7,8,9. Therefore, shift current is closely related to the quantum geometry of Bloch electrons in solid, although not directly related to the topology. Shift current intrinsically possesses high robustness against carrier localization due to scattering and trapping as well as ultrafast responsivity to pulsed light due to the geometric origin10,11,12, which have attracted significant interest both from fundamental and technological perspectives.

Establishing deeper insights into the microscopic mechanism will lead to the pathways to improve the photovoltaic response and application development. It has been elucidated that the materials having topologically non-trivial band structures can host large shift current responses, as exemplified in Weyl semimetals TaAs and TaIrTe413,14,15,16. Recent theoretical studies have also revealed that many-body interactions play an important role in the enhancement of shift current17,18,19,20,21,22,23. One typical example is exciton, a bound state of electrons and holes via Coulomb interaction. Excitons are formed by absorbing photons with energies that are smaller than the bandgap by their binding energy. In conventional photovoltaic processes, however, charge-neutral excitons do not yield photocurrent unless dissociated into free charge carriers. The dissociation process generally requires heterointerfaces with a built-in potential difference to overcome the exciton binding energy organic bulk heterojunction solar cells24. By contrast, theory predicts that the creation of excitons directly accompanies photocurrent generation without their dissociation in the shift current process, and its magnitude is strongly enhanced by the quantum geometry taking into account exciton effects.

To date, the experimental demonstration of such exciton shift current remains elusive. Although a few papers have reported resonant behaviors of shift current at exciton transition energies25,26, they are detected by terahertz (THz) emission spectroscopy, in which a sample is excited with ultrashort pulsed light and electromagnetic wave is detected at the THz frequency emitted from the transient photocurrent. More direct demonstration would be direct current (DC) measurements under continuous-wave (CW) light irradiation. However, such direct measurements require thorough optimization of contact electrodes because the Schottky barrier formed at the sample-electrode interface drastically reduces the shift current extracted to the external circuit27. Moreover, in the previous studies, the exciton state largely overlaps with the electron-hole continuum state due to the small exciton binding energy25, leaving ambiguity as to whether exciton creation alone can induce the shift current apart from the free electron–hole excitation.

A potential target compound for more distinct observations of exciton shift current is cuprous iodide (CuI), a prototypical p-type wide bandgap semiconductor with a zinc-blende (ZB) structure28. The ZB structure has no polarity but broken inversion symmetry, allowing the shift current response by irradiating linearly polarized light. Moreover, the electronic band structures of ZB semiconductors are simple and well-established, as shown in Fig. 1a. Similar to many other ZB semiconductors, the valence bands have p-orbital-like nature, and they are split into j = 3/2 and j = 1/2 bands (j is the total angular momentum) due to the spin-orbit interaction by ΔSO = 640 meV. The j = 3/2 bands are four-fold degenerated at the Γ-point, composed of the light hole (lh) and heavy hole (hh) bands, both of which are doubly degenerated. These degeneracies are lifted other than the Γ-point due to the noncentrosymmetry and spin-orbit interaction. As for the optical characteristics, the compound has a direct bandgap of 3.1 eV and a large exciton binding energy of 62 meV29,30. The exciton in CuI shows a Bohr radius as small as 15 Å31, only a few times larger than the lattice constant (6.05 Å), as depicted in Fig. 1b. Due to the strongly electron-hole bound nature, the exciton transitions are clearly observable even at room temperature as sharp peak structure in absorption spectra, and the peak intensities strongly enhances with lowering temperatures. The two sharp transitions at lower (higher) energies arise from the formation of excitons composed of j = 3/2 (j = 1/2) holes and s = 1/2 electrons, which are referred to as the Z1,2 (Z3) excitons, as depicted in Fig. 1a.

Fig. 1: Exciton states and photocurrent detection in CuI thin films.
figure 1

a Schematic band structure of cuprous iodide (CuI). The degenerated j = 3/2 bands consist of the light hole (lh) and heavy hole (hh) bands. The transition from the valence band with j = 3/2 (j = 1/2) to the s = 1/2 conduction band forms the Z1,2 (Z3) exciton. b A schematic comparing the exciton Bohr radius (15 Å) and the lattice structure in the (111) plane of CuI. The dashed line represents the unit cell. c A cross-sectional schematic of the thin film structure to realize single-domain CuI thin films. d A schematic of the experimental set-up for shift current measurements. The samples were irradiated by linearly polarized light from CW light sources and the polarization was rotated with a half-wave plate. The photocurrent was measured along the two orthogonal crystal axes in the (111) plane, \([11\bar{2}]\) and \([\bar{1}10]\), which are labeled as x and y axes, respectively. e Excitation spectra of zero-bias photocurrent in the x direction at 50 K. The spectra are obtained at the incident light polarization angle θ = 0 and 90 degrees, and normalized by the light intensity. f Absorption spectra at 50 K. The peaks at 3.05 eV and 3.7 eV correspond to the resonances of Z1,2 and Z3 exciton transitions.

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For the simple band structure, high exciton stability, and large exciton oscillator strength, CuI provides an ideal platform for the fundamental studies of exciton shift current. However, the quite strong exciton absorption results in a small light penetration depth (only several tens of nanometers), demanding high-quality epitaxial thin films. In this paper, we report on the experimental demonstration of exciton shift current in epitaxial CuI thin films. By direct photocurrent measurement, we observe a significant enhancement of shift current response at exciton resonances. First-principles calculations elucidate that a tiny strain on the thin films critically affects the sign and magnitude of the shift current from the excitons. These results indicate that excitons dramatically increase the magnitude and strain sensitivity of shift current, which will promote establishing deeper insight into light–matter interaction in the nonlinear regime.

Results

The structure of the thin-film samples is shown schematically in Fig. 1c. We grew (111)-oriented CuI thin films by molecular beam epitaxy (MBE) on CaF2 (111) substrates (Methods). X-ray diffraction (XRD) reveals that the thin films grown directly on CaF2 substrates have mixed domains rotated 180 degrees relative to each other in the (111) plane (Supplementary Fig. 1). The presence of such misoriented domains has to be avoided because they cause the cancellation of shift current as experimentally verified (Supplementary Fig. 4). We presume that the sizable lattice mismatch ( ~ 10 %) and the difference in crystal structure between CuI (ZB) and CaF2 (Fluorite) causes this mixed domain formation. To mitigate these two factors, we inserted a 2 nm-thick PbF2 layer having the same fluorite structure with CaF2 and a much smaller lattice mismatch with CuI ( ~ 2 %). By introducing the PbF2 buffer layer, the misoriented domains in the CuI layer are mostly suppressed (Supplementary Fig. 1). Finally, the surface of the CuI films was capped with PbF2 to prevent surface degradation due to moisture and oxygen in the air. It is revealed that the fabricated thin films have significantly high lattice coherence and atomically sharp interfaces as shown in high-resolution transmission electron microscope (TEM) images (Supplementary Fig. 2) as well as cross-sectional elemental mapping (Supplementary Fig. 3). Since CaF2 and PbF2 have the same centrosymmetric fluorite structure, shift current emerges only in the CuI layer. Hereafter, the thin films with (without) the PbF2 buffer layer are referred to as single-domain (mixed-domain) samples.

Using the fabricated thin films, we measured zero-bias DC photocurrent under the irradiation of linearly polarized monochromatic CW light as shown in Fig. 1d. The (111) plane of CuI has \([11\bar{2}]\) and \([\bar{1}10]\) axes in the plane, which are crystallographically inequivalent and orthogonal to each other, as shown in Fig. 1b. We attached stripe-shaped Au electrodes along the two axes as depicted in Fig. 1d, where \([11\bar{2}]\) and \([\bar{1}10]\) axes are defined as x and y axes. Fig. 1e shows the excitation spectra of zero-bias photocurrent in the single-domain samples along the x axis at 50 K measured using monochromatized light from a broadband CW light source (Methods). The measurements were performed at the incident polarization angle θ = 0 and 90 degrees, which correspond to the polarization directions parallel to the \([11\bar{2}]\) and \([\bar{1}10]\) axes as illustrated in Fig. 1d. The spectra show an almost symmetric sign flipping between the two polarization angles. Furthermore, they exhibit peak structures at around 3.05 and 3.7 eV. At the same photon energies, the absorption spectrum shown in Fig. 1f also has peak structures corresponding to the Z1,2 and Z3 exciton resonances. The photocurrent peaks at the exciton resonances imply that the exciton creation leads to the photocurrent generation. The fact that rotating the incident polarization by 90 degrees causes the photocurrent to flow in the opposite direction suggests that the internal electric field is not the origin of the photocurrent. This is concrete evidence that the observed photocurrent is exciton-mediated, not carried by photoexcited free carriers.

To examine the origin of the observed photocurrent in more detail, we measured the incident polarization dependence of zero-bias DC photocurrent for the single-domain sample in the x and y directions employing CW lasers operating at 3.3 eV and 3.05 eV that corresponds to the photon energy above the bandgap and that of Z1,2 exciton resonance, respectively, as shown in Fig. 2a and b (Methods). At both photon energies, the observed photocurrent exhibits two-fold sinusoidal θ dependence with a phase shift by 45 degrees between the x and y directions. Furthermore, the sign of the photocurrent is opposite for the respective photon energies, consistent with the excitation spectra shown in Fig. 1e. Here, we consider the photocurrent arising from the second-order nonlinear optical process. From the symmetry of the ZB structure, the nonlinear photocurrent density flowing in the (111) plane of CuI thin films is expected to have the following incident polarization dependence (Methods):

$${j}_{x}=\frac{\sqrt{6}}{3}\sigma {E}_{0}^{2}\cos 2\theta,\,\,{j}_{y}=-\frac{\sqrt{6}}{3}\sigma {E}_{0}^{2}\sin 2\theta$$
(1)

where \(\sigma\) is the second-order nonlinear conductivity and E0 is the amplitude of the electric field of the incident light. The experimental results are in perfect agreement with Eq. (1), suggesting that the observed polarization-dependent photocurrent arises from second-order nonlinear processes. In particular, only the shift current mechanism is plausible, given the fact that the nonlinear photocurrent is generated solely by exciton creation (with no free electron-hole pairs). Note that other major mechanisms for bulk photovoltaic effects such as injection current and ballistic current rely on the creation of free photocarriers.

Fig. 2: Incident polarization dependence of shift current.
figure 2

a, b Incident polarization dependence of zero-bias DC photocurrent in the x (red) and y (blue) directions at 50 K excited by CW lasers operating at 3.3 eV (a) and 3.05 eV (b). The solid lines are the results of fitting with \({J}_{x}\propto \cos 2\theta\) and \({J}_{y}\propto -\sin 2\theta\). c Incident polarization dependence of the intensity of second harmonic generation (SHG) polarized in the x and y directions at 300 K. The solid lines are the results of fitting with \({I}_{x}^{\text{SHG}}\propto {\cos }^{2}2\theta\) and \({I}_{y}^{\text{SHG}}\propto {\sin }^{2}2\theta\).

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We also performed the same measurements for the mixed-domain sample. The oscillation amplitude is much smaller in the multi-domain sample, implying that cancellation of photocurrent occurs when the rotational domains exist (Supplementary Fig. 4). This result is consistent with the symmetry consideration that the sign of the shift current is opposite for domains rotated by 180 degrees. Symmetry of the sample is also verified by the second harmonic generation (SHG), another second-order nonlinear optical effect, with employing a Ti: Sapphire femtosecond laser system (Methods). Since shift current and SHG have the same tensor form of the nonlinear response function, the intensity of SHG polarized in the x and y directions is expected to vary with the incident polarization as \({I}_{x}^{{\rm{SHG}}}\propto {\cos }^{2}2\theta\) and \({I}_{y}^{{\rm{SHG}}}\propto {\sin }^{2}2\theta\). The observed SHG shown in Fig. 2c exhibits four-fold oscillations and a phase shift by 45 degrees between the x and y directions consistent with the above formula.

Since the polarization-dependent photocurrent following Eq. (1) is the shift current component, we derive the excitation spectrum of shift current as shown in Fig. 3a by antisymmetrizing the spectra at θ = 0 and 90 degrees given in Fig. 1e to eliminate minor contributions independent of the polarization (Supplementary Fig. 5). The magnified view of the shift current spectrum as well as the absorption spectrum near the Z1,2 transition are shown in Fig. 3b, c, respectively. The exact agreement of the peak position between shift current and absorption is verified. The shift current peak at the Z1,2 transition is well isolated below the bandgap, clearly manifesting that the current is produced solely by exciton formation not by continuum excitation. Some more intriguing features of exciton shift current are seen in Fig. 3b. First, the shift current components due to the exciton and interband excitation have opposite signs near the absorption edge. We will later discuss the theoretical interpretation of this sign reversal. Second, another small negative peak (E2 in Fig. 3b) is observed near the absorption edge. The energy difference between this peak and the fundamental Z1,2 transition (E1) is 46 meV which corresponds to 3/4 of the exciton binding energy (62 meV). Therefore, the small peak can be assigned to the n = 2 exciton Rydberg state, whose energy is given by \({E}_{n}={E}_{\text{g}}-{E}_{\text{b}}/{n}^{2}\) with n being the principal quantum number and Eg and Eb the bandgap and exciton binding energies, respectively, as denoted in Fig. 3b. The appearance of such a Rydberg state is a hallmark of the superior crystallinity of the thin films and more definite evidence of the exciton shift current.

Fig. 3: Enhanced shift current response at exciton resonances.
figure 3

a Excitation spectra of shift current in the x direction at 50 K derived from antisymmetrizing the spectra shown in Fig. 1e. b Enlarged view of the shift current spectra around the Z1,2 exciton transition. E1 and E2 stand for the energies of the n = 1 and 2 Rydberg states of the Z1,2 exciton, and Eg and Eb denote the band gap and exciton binding energies, respectively. c Enlarged view of the absorption spectrum at 50 K around the Z1,2 exciton transition. d Temperature dependence of second-order nonlinear conductivity \(\sigma\) at the Z1,2 and Z3 exciton resonances as well as that at 3.5 eV. The filled and open circles represent positive and negative values. e Experimentally measured Glass coefficients for various materials12,13,32,33.

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Temperature dependence of the shift current response is shown in Supplementary Fig. 6, where the measured shift current is converted to the second-order nonlinear conductivity (σ) using the transmittance and reflectance spectra to compare quantitatively the spectra at different temperatures (Methods and Supplementary Fig. 7). The shift current response is observed even at room temperature, and the observed spectra show similar profiles for all temperatures, including the negative peaks at the two exciton resonances and positive current in the continuum regime between the two. However, the exciton and interband shift current have different trends in temperature dependence. To visualize it, we show in Fig. 3d the temperature dependence of σ at the Z1,2 and Z3 exciton resonances as well as that at 3.5 eV. It reveals that the responsivity of the exciton shift current varies significantly with temperature and is strongly enhanced at low temperatures. The temperature dependence for interband transition at 3.5 eV is much smaller because the shift current in the continuum regime reflects the band structure itself that least changes with the temperature in the present compound. Note that the slight decrease in shift current conductivity below 50 K may be due to uncertainties in the absorption and reflectance used for the conversion from the observed photocurrent to conductivity.

Now, we compare the magnitude of shift current in CuI with other compounds. Equation (6) in Methods indicates that the magnitude of shift current is proportional to \(\sigma /\alpha\), which is called the Glass coefficient. The σ value for the Z3 exciton increases up to ~20 μA V−2 at low temperatures, as seen in Fig. 3d, and the corresponding Glass coefficient is calculated to be 1.0 × 10−7 cm V−1. As shown in Fig. 3e, this value is much larger than those in many representative compounds such as ferroelectrics and piezoelectrics with electronic polarization and even comparable to narrow-bandgap topological materials such as TaAs12,13,32,33. Considering that the responsivity of shift current is theoretically formulated as inversely proportional to the bandgap15,34, the Glass coefficient obtained for such a wide bandgap semiconductor CuI is exceptionally large.

It is considered that exciton shift current originates from the transfer of the spectral weight partly from the continuum state in the noninteracting system to the exciton resonance below the bandgap17. Therefore, the exciton and interband shift currents are expected to have the same sign in general, but this is not the case in the present CuI thin films. We examined the mechanism for the opposite signs between exciton and interband shift currents from first-principles calculations. The generalized gradient approximation (GGA) was employed for band-structure calculations, and exciton effects in linear and nonlinear optical responses were computed by solving the Bethe−Salpeter equation (BSE)18,19 (see Methods and Supplementary Fig. 8). The calculations were performed for a strain-free state and −0.1 % strain (very small compressive strain) states. As discussed later, it is revealed that our thin films are subjected to tiny strain, and symmetry lowering associated with the strain is crucial for the strong enhancement and its sign of the exciton shift current.

Figure 4a, b shows the spectra of shift current conductivity (σ) and linear dielectric constant (\({\varepsilon }_{2}\)) calculated without including the exciton effect for strain-free and −0.1 % strain states (more detailed strain dependences of σ and \({\varepsilon }_{2}\) are shown in Supplementary Fig. 9). The shift current conductivity exhibits a negative peak near the band edge, but the peak is broad and very small, not consistent with the experimental results. Then, we examined exciton effects by performing GGA + BSE calculations, as shown in Fig. 4c, d. The linear dielectric constant exhibits two clear peaks corresponding to the Z1,2 and Z3 exciton resonances. By setting the broadening factor to 5 meV, the peak height and width of the Z1,2 exciton resonance are well reproduced (Supplementary Fig. 10). The shift current conductivity calculated using the same broadening factor shows a small positive peak in the strain-free state at the Z1,2 exciton resonance, and the peak is strongly enhanced in the negative direction at −0.1 % strain. The spectrum under strain also exhibits an additional negative peak near the main peak corresponding to the n = 2 Rydberg exciton and positive sign in the continuum state. All these features of the calculated shift current spectra under −0.1 % strain are very consistent with the experimental result. The consistency is not only in the signs but also in their magnitudes (Supplementary Fig. 10). Note that the n = 2 Rydberg exciton state is clearly observed only in the shift current spectra in both experiments and calculations. This is perhaps because the small signal from the Rydberg exciton is pronounced in shift current spectra due to the opposite signs of shift currents from exciton and continuum states.

Fig. 4: First-principles calculation of shift current in CuI films.
figure 4

a–d Calculated spectra of shift current conductivity (σ) (a, c) and imaginary part of the dielectric constant (\({\varepsilon }_{2}\)) (b, d) for strain-free and −0.1 % strained CuI. a and b are calculated using the generalized gradient approximation (GGA) without exciton effects, whereas c and d are calculated incorporating exciton effects by solving the Bethe−Salpeter equation (BSE). The red and blue lines are spectra for strain-free and −0.1 % strained states. The inset of a is an enlarged view near the band edge. The spectra in c were calculated only near the lowest band edge (E−Egap < 0.1 eV) due to the quite large computational complexity, and the higher energy side is shown as dashed lines. The spectral broadening η is set to 5 meV in the calculations. e–h, Distribution of shift vector Ryy (k) around the Γ-point on the kx-ky plane at kz = 0 (e, g) and on the ka-kz plane (f, h) for strain-free and −0.1 % strained states. Here, kx (or ky) lies within the film plane, kz is along the out-of-plane direction, and ka is within the kx-ky plane as denoted in e and g. The shift vector is calculated at the energy of the negative peak top (Epeak) as denoted in a. Note that only k-points having the interband transition energies within ±10 meV of the Epeak would appear in these plots.

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The above results suggest that the strain is crucial for the exciton shift current. The presence of strain in the present CuI thin films on CaF2 substrates is verified by the reflectance spectrum, which shows two peaks at the Z1,2 exciton resonance that indicate the split of the lh and hh bands due to symmetry lowering from cubic to rhombohedral structure by strain (Supplementary Figs. 8 and 11). The strain value estimated from the band splitting (3 meV) is about −0.1% (Supplementary Table 1). This strain probably results from the mismatch of the thermal expansion coefficient between CuI and CaF2 (CuI: 1.6 × 10−5 K−1, CaF2: 1.8 × 10−5 K−1), which results in about −0.1% strain at low temperatures.

Discussion

Hereafter, we discuss the reason for the negative sign and the high strain sensitivity of the exciton shift current based on a momentum-space distribution of the shift vector. Although shift current is not determined solely by shift vector35,36, it is a good quantity to roughly estimate the sign of shift current. Figure 4e–h display the distribution of shift vector around the Γ-point on the kx-ky plane at kz = 0 (Fig. 4e, g) and ka-kz plane (Fig. 4f and h), where kx (or ky) lies within the film plane, kz is along the out-of-plane direction, and ka is within the kx-ky plane as denoted in Fig. 4e, g (see more details in Supplementary Fig. 12). The shift vector is calculated at the energy of the top of the negative peak as denoted in the inset of Fig. 4a (E = Epeak). In these calculations, exciton effects are not included to simplify the calculations and extract the essence of the underlying physics. In the strain-free state, shift vector in the kx-ky plane is positive while negative in the kz direction (Fig. 4e, f). The anisotropic shift vector distribution is related to the degeneracy of the valence bands at the Γ-point, where the degenerate two bands have opposite contributions to the shift vector (see Supplementary Note 13 and Fig. 13). Thus, the integral of the shift vector around the Γ-point cancels out to nearly zero. When the strain is applied, the degeneracy is lifted by the symmetry reduction, and the shift vector becomes finite near the Γ-point (it is negative in compressive strain), as shown in Fig. 4g, h.

Excitons in CuI are strongly localized in the momentum space near the Γ-point, as shown in Supplementary Fig. 14. Therefore, shift current from the Z1,2 exciton is expected to reflect the integrated shift vector around the Γ-point, nearly zero at strain-free and large negative value under compressive strain. This agrees well with the result of extensive first-principles calculations considering all electron-hole interactions shown in Fig. 4c. The present theoretical calculations indicate that the symmetry reduction due to strain and strong localization of exciton in the momentum space play a crucial role in the large negative enhancement of shift current at the Z1,2 exciton resonance observed in the experiment.

Recently, the manipulation of nonlinear optical effects by strain has been extensively studied37,38,39,40,41,42. The present study reveals that the combination of the quantum geometry of degenerated bands and the many-body interactions strongly enhances the magnitude and strain sensitivity of shift current. This finding might open a way for more efficient manipulation of nonlinear optical effects required for next-generation optoelectronic devices.

Methods

Thin film growth and characterization

The thin film samples were fabricated on CaF2(111) substrates by molecular beam epitaxy (MBE). Prior to the thin film growth, the substrate was annealed at 300 °C in the MBE chamber to remove the surface contamination. Then, 2 nm-thick PbF2 was grown at 100 °C as a buffer layer. Subsequently, CuI thin films were grown first at 100 °C with a thickness of 10 nm and then at 210 °C with a thickness of 110 nm. This two-step growth is crucial to obtaining high-quality CuI thin films, which has been established in our previous works43. Finally, 5 nm-thick PbF2 was deposited at 30 °C as a surface capping layer. During the film growth, the flux of CuI and PbF2 was provided by the sublimation of high-purity CuI and PbF2 powders (nominal purity of 5 N) from Knudsen cells. The cell temperatures were 300 °C for CuI and 450 °C for PbF2.

The crystalline structure of the fabricated thin films was characterized by X-ray diffraction (XRD) and transmission electron microscopy (TEM). XRD measurements were carried out with a high-resolution four-circle x-ray diffractometer (SmartLab, Rigaku), as shown in Supplementary Fig. 1. High-resolution TEM analyses were conducted using a standard electron microscope (JEM2800, JEOL) at an acceleration voltage of 200 kV, as shown in Supplementary Fig. 2. The EDS data was captured using a scanning TEM (STEM) at operating 200 kV (Talos F200X G2, FEI) equipped with Super-X detectors, as shown in Supplementary Fig. 3.

Polarization dependence of shift current and SHG in (111)-oriented CuI thin films

Shift current density along the i-axis direction \({j}_{i}^{{\rm{shift}}}\) is given by

$${j}_{i}^{\text{shift}}=\sum _{j,k}{\sigma }_{{ijk}}{E}_{j}(\omega ){E}_{k}(-\omega )$$
(2)

where \({\sigma }_{{ijk}}\) is the second-order nonlinear conductivity tensor and \({E}_{j}(\omega )\) is the electric field of incident light polarized in the j-axis direction. The zinc-blende (ZB) structure belongs to the point group \(\bar{4}\) 3m (Td) that has the following six non-zero tensor components:

$$\sigma={\sigma }_{{xyz}}={\sigma }_{{yzx}}={\sigma }_{{zxy}}={\sigma }_{{xzy}}={\sigma }_{{yxz}}={\sigma }_{{zyx}}$$
(3)

where x, y, and z are set to [100], [010], and [001] axes of the ZB structure, respectively. For a (111)-oriented sample, the tensor is expressed as the following form

$${\sigma }_{{ijk}}=\frac{1}{3}\sigma \left(\begin{array}{cccccc}\sqrt{6} & -\sqrt{6} & 0 & 0 & -\sqrt{3} & 0\\ 0 & 0 & 0 & -\sqrt{3} & 0 & -\sqrt{6}\\ -\sqrt{3} & -\sqrt{3} & 2\sqrt{3} & 0 & 0 & 0\end{array}\right)$$
(4)

where x, y, and z are set to \([11\bar{2}]\), \([\bar{1}10]\), and [111] axes of the ZB structure, respectively. From Eq. (4), the incident polarization dependence of the shift current in the (111)-oriented CuI thin films is derived as Eq. (1). Similarly, the electric field of SHG is written as

$${E}_{i}^{\text{SHG}}\propto \sum _{j,k}{\chi }_{{ijk}}{E}_{j}(\omega ){E}_{k}(\omega )$$
(5)

The nonlinear optical tensor \({\chi }_{{ijk}}\) has the same symmetry as \({\sigma }_{{ijk}}\). Because SHG is detected as the intensity of light, namely, \({I}_{i}^{\text{SHG}}={({E}_{i}^{\text{SHG}})}^{2}\), the oscillation period of the polarization dependence is half compared to that for shift current.

Measurements of light-polarization dependent shift current

The shift current response at a fixed photon energy in the electron-hole continuum regime was measured employing a CW laser diode operating at 375 nm (3.3 eV) and 406.6 nm (3.05 eV), which correspond to the energies above the bandgap and Z1,2 exciton resonance, respectively. Since the exciton resonance is quite sharp, the temperature of the laser diode was controlled so that the emission energy was centered on the exciton resonance peak. The light was first polarized with a linear polarizer, and then the polarization was rotated with a motorized half-wave plate. The beam spot size was shaped to approximately 1 mm in diameter through an iris. The spot size was set larger than the electrode distance (0.5 mm) in order to avoid the generation of convection current in the unirradiated area which reduces the shift current amplitude to be extracted outside27. The light power was typically 1 mW. The light intensity was modulated by a chopper, and the generated photocurrent was measured by a lock-in amplifier.

For the measurement of photocurrent excitation spectra, we employed a monochromatized light from a Xenon lamp. As above, the polarization of the light was controlled with a linear polarizer and a half-wavelength plate. The light was focused onto the samples by a lens to a size of about 1 mm square. The light intensity varied smoothly from 15 μW to 5 μW between 3 eV and 4 eV, and the observed photocurrent was normalized with the light intensity. The spectral resolution is about 1 nm that corresponds to 8 meV near the band edge of CuI.

In exciton resonances, the reflectance and transmittance exhibit significant changes with photon energy and temperature, as shown in Supplementary Fig. 7a, b. Therefore, in order to quantitatively compare the shift current responses from different origins (exciton or interband transitions) and at different temperatures, the observed shift current (\({J}_{x}^{\text{shift}}\)) is converted to the second-order nonlinear conductivity (σ) using the following equation that accounts for the influences of the transmittance and reflectance,

$${J}_{x}^{\text{shift}}=\frac{\sqrt{6}}{3}\frac{\sigma }{\alpha }(1-{e}^{-\alpha d})(1-R)w{E}_{0}^{2}\cos 2\theta$$
(6)

where α is the absorption coefficient, R the reflectance, d the film thickness, and w the diameter of the optical spot. The excitation spectra of σ shown in Supplementary Fig. 6 are calculated using Eq. (6). In the calculation of σ, the fluctuation in the measured shift current and uncertainty in the transmittance and reflectance can cause errors, which are shown in Fig. 3d in the main text.

SHG measurement

For the SHG measurements, the sample was excited by a Ti: Sapphire femtosecond laser system (120 fs, operating at 1 kHz) centered at 800 nm. It was performed with a reflection geometry with the incident laser normal to the sample at room temperature. The reflection SHG signal was collected by a photomultiplier after passing through a Glan-Laser prism, color filters, and a monochromator. To study the SHG polarimetry, the polarization of the incident laser was rotated by a motorized half-wave plate, and SHG signals polarized in the x or y directions were detected.

Transmittance and reflectance spectra

Transmittance and reflectance spectra of the thin films were measured with a double-beam spectrophotometer (JASCO MSV-5700). The energy resolution of the spectra is 0.2 nm that corresponds to 1.6 meV near the band edge of CuI. An Al mirror was used as a reference for the reflectance of the film. The measured transmittance and reflectance spectra are shown in Supplementary Fig. 7a, b. The absorption coefficient (\(\alpha\)) is calculated from the following relation,

$$\alpha=-\frac{1}{d}\mathrm{ln}\frac{(1-{R}_{{\rm{s}}}){T}_{{\rm{f}}}}{(1-{R}_{{\rm{f}}})}$$
(7)

where Tf, Rf, and Rs are the transmittance, reflectance of the film, and reflectance of the substrate, respectively, and d is the thickness of the film. The obtained spectra of \(\alpha\) are shown in Supplementary Fig. 7c.

First-principles calculation

The ground state band structures are calculated based on the density functional theory with the GGA (see Supplementary Note 8 for the computational details) and are displayed in Supplementary Fig. 8. The shift current conductivity and dielectric constant are then calculated within independent particle approximation15, using the calculated GGA band structures. Here we adopt the efficient Wannier function interpolation scheme44 to calculate these linear and nonlinear response functions. The Wannier orbitals used are Cu spd and I p orbitals and these 24 orbitals are optimized by fitting them to the GGA band structures with the Wannier90 program45. A very dense k-point grid of 300 × 300 × 300 is used in the Brillouin zone integration.

The Bethe−Salpeter equation (BSE) calculations are carried out with the BerkeleyGW package46,47,48. A k-point grid of 16 × 16 × 16, and a dielectric energy cutoff of 15 Ry and 300 bands are used in the dielectric function calculation. The frequency dependence of the dielectric screening is computed using the Hybertsen-Louie generalized plasmon pole model47. Six valence bands, two conduction bands, and a coarse k-grid of 16 × 16 × 16 in the BSE Hamiltonian are used, which is interpolated to a finer grid in a patch of radius 0.1 Å−1 centered at the Γ point. The final grid density is equivalent to that of a uniformly sampled 150 × 150 × 150 k-grid. We obtain a band gap value of 1.032 eV from the GGA calculation. To better compare with the experimental absorption spectrum, we made a scissor correction by applying an overall shift of 2.08 eV on the conduction band energy to match the experimental band gap (3.112 eV) and then solved the Bethe−Salpeter equation for excitons. The scissor shift leads to an overall shift of the kinetic energy of the electron-hole pair and is described by a constant shift in the diagonal part of the BSE Hamiltonian matrix elements in numeric. Such shift leads to a constant shift of all exciton eigenenergies and the absorption spectrum. In the calculations, we set the broadening factor at 5 meV so that the calculated linear optical spectrum reproduces the experimental result at the Z1,2 exciton resonance (Supplementary Note 10).

First-principles calculations of the shift currents at the BSE level are technical and computationally challenging19,20. We recently developed a computationally efficient approach20 that combines the BSE and sum-over-state formalism derived from the perturbative density-matrix approach with the mean-field approximation49, which has been applied to study the excitonic effects on the shift currents in BN nanotubes and single BN sheet20. Therefore, this computationally efficient approach is adopted here (see Ref. 20 for more details).