
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38271339
202307425
10.1073/pnas.2307425121
datasetDatasetresearch-articleResearch Articleapp-physApplied Physical Sciences405
Physical Sciences
Applied Physical Sciences
Emergence of threefold symmetric helical photocurrents in epitaxial low twinned Bi2Se3
Connelly Blair C. blair.c.connelly.civ@army.mil
a 1
Taylor Patrick J. a
de Coster George J. a
aU.S. Army Combat Capabilities Development Command Army Research Laboratory, Adelphi, MD 20783
1To whom correspondence may be addressed. Email: blair.c.connelly.civ@army.mil.
Edited by J. C. Davis, University of Oxford, Oxford, United Kingdom; received May 3, 2023; accepted November 29, 2023

25 1 2024
30 1 2024
25 7 2024
121 5 e230742512103 5 2023
29 11 2023
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

Our work reveals the interplay of topological insulator (TI) surface states and crystal symmetries in light-matter interactions that until now have proven elusive. We explain how observation of many of these symmetry-dependent responses is suppressed and hidden in the TI Bi2Se3 by a common crystallographic twinning defect that introduces additional deleterious crystal symmetries. Our growth method of low-twinning defect thin films overcomes this barrier, leading to the observation of strong circular photon drag effect in Bi2Se3. This effect is characterized by threefold symmetric photocurrents whose direction reverses upon 60-degree sample rotations and/or alternating helicity of light and is an archetypal example of the many nonlinear physical phenomena remaining to be explored in TIs.

We present evidence of a strong circular photon drag effect (PDE) in topological insulators (TIs) through the observation of helicity-dependent topological photocurrents with threefold rotational symmetry using THz spectroscopy in epitaxially-grown Bi2Se3 with reduced crystallographic twinning. We establish how twinned domains introduce competing nonlinear optical (NLO) responses inherent to the crystal structure that obscure geometry-sensitive optical processes through the introduction of a spurious mirror symmetry. Minimizing the twinning defect reveals strong NLO response currents whose magnitude and direction depend on the alignment of the excitation to the crystal axes and follow the threefold rotational symmetry of the crystal. Notably, photocurrents arising from helical light reverse direction for left/right circular polarizations and maintain a strong azimuthal dependence—a result uniquely attributable to the circular PDE, where the photon momentum acts as an applied in-plane field stationary in the laboratory frame. Our results demonstrate new levels of control over the magnitude and direction of photocurrents in TIs and that the study of single-domain films is crucial to reveal hidden phenomena that couple topological order and crystal symmetries.

circular photogalvanic effect
photon drag effect
azimuthal symmetries
Bi2Se3
THz spectroscopy
DOD | Office of the Under Secretary of Defense (OUSD(C)) 100014043 NA Blair C ConnellyPatrick J TaylorGeorge J De Coster
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pmcLight–matter interactions within topological insulators (TIs) and Weyl semimetals (WSMs) yield unique photoresponses due to spin-momentum locked surface and bulk states, respectively (1–12). Specifically, nonlinear optical (NLO) processes such as the photogalvanic effect (PGE) and photon drag effect (PDE) generate strongly polarization-sensitive photocurrents. Several experiments have observed polarization-dependent photocurrents in these materials, with WSMs boasting inherently stronger responsivities due to the participation of bulk states in generating photocurrents (13–15). A large body of research exists on the nature of photocurrents in the (Bi, Sb)2(Se, Te)3 (BSST) family of TIs (16–19). These studies span all-optical contactless probing of photocurrents using ultrafast time-domain THz spectroscopy (TDTS) and direct measurement of photocurrents in fabricated devices (20–27). For BSST TIs, the surface photocurrents generated by PGE and PDE are expected to reflect the underlying threefold rotational symmetries of the crystal, and propagate in opposite directions when generated by left versus right circularly polarized light (8, 9). Observations of threefold symmetric photocurrents are strikingly absent in epitaxially grown thin films. Strong photocurrents that reverse under different helicities of light have been seen in samples derived from single crystals where reversal and large responsivities were engineered through Fermi level control (17) or utilizing the photothermal effect (5). However, only weak rotational dependence of circular polarization sensitive photocurrents ascribed to the circular PDE (CPDE) has been observed, even in samples derived from bulk crystals, which is likely due to large fluences necessary to generate measurable CPDE (1). In this article, we posit the crystallographic twin defects that are prevalent in thin films grown via molecular beam epitaxy (MBE) generally suppress NLO processes that encode the coupling between the threefold rotational symmetry of BSST and its topological surface states (18, 28). Through growth and measurement of low-twinned Bi2Se3, we uncover clear evidence of strong CPDE, a prime example of physical processes that were previously obscured from study and as such assumed altogether absent or weak.

We present TDTS measurement and analysis of high-quality Bi2Se3—grown to preferentially obtain a single twinning domain to significantly reduce twinning defects—which demonstrate a colossal enhancement in helicity-dependent photocurrents over highly twinned samples. Additionally, we report on the emergence of threefold symmetry in polarization-dependent photocurrents, as well as a clear demonstration of the dependence of the photocurrent direction on helicity and plane of incidence. Through reducing this obfuscating mirroring effect, the twinning defect is revealed to be particularly deleterious to the expression of phenomena inherent to the topological nature of the underlying material. Illuminating these fundamental physical responses has ramifications beyond basic research, with implications for novel optoelectronic devices that harness the strong coupling between topology and symmetry. For example, intricate knowledge of these azimuthally and polarization-dependent photoresponses can inform clever contact schemes for a single pixel that enable the complete decoding of the degree of ellipticity and helicity for polarimetric measurement of light.

Bi2Se3 grows in quintuple layers of SeI–Bi–SeII–Bi–SeI that stack in the growth direction z^, with van der Waals bonding between neighboring SeI atoms of the quintuple layers. A perfect Bi2Se3 crystal is symmetric under the operations of the point group D3d, which is generated by threefold rotations about the z-axis, C3, inversion, i, and mirror reflection along the (100) x-axis, σx^. The Bi2Se3 lattice can be “twinned” by mirroring it along the (010) y-axis. A top-down view of the final Se–Bi layers of a particular choice of “twin domain” is presented in Fig. 1A. MBE growth often allows different twinning domains of Bi2Se3 to percolate randomly on a substrate, resulting in a material that effectively has an additional mirror reflection symmetry along the y-axis that induces a sixfold symmetry in addition to the threefold symmetry of perfect Bi2Se3. In this case, the point group describing the effective crystal is D6h and certain threefold symmetric responses will be suppressed in measurements of these thin-film samples (SI Appendix) (21). Judicious substrate choice and growth conditions enable a single domain of Bi2Se3 to dominate in MBE growth (28–32). For this study, we measured a “low-twinned” Bi2Se3 thin film with an approximate twinning ratio of 5:1, grown by MBE as detailed in the Materials and Methods section.

Fig. 1. Experimental setup and bulk crystal symmetries. (A) Time-domain experiment: An 800-nm, ~125-fs laser pulse is prepared in linear horizontal polarization and incident on a Bi2Se3 sample at an angle of incidence of 45°; the ellipticity of the light is controlled using a QWP. The emitted THz is collected with 90° off-axis parabolic mirrors and focused on a ZnTe crystal for detection using electro-optic sampling (not shown here). A WGP is used to filter the emitted polarization for detection in the horizontal and vertical directions (Syz and Sx, respectively). (A, Inset) Top view of outermost Se and Bi layers of the Bi2Se3 crystal structure, oriented with an azimuthal angle, ϕ, of 0°, where the (100) axis is oriented vertically and the (010) axis horizontally. Note: the direction of the shift current occurs along the Se–Bi bond (red arrow). (B) Differential THz signals emitted with Syz polarization for displayed ϕ under excitation with linear horizontal polarization; the direction of increasing ϕ is indicated. (C) Peak-to-peak amplitude of Syz as a function of ϕ (purple triangles); the fitted function ∝ cos(3ϕ-18°) (blue line) demonstrates the threefold crystallographic symmetry. (D) THz waveforms of Sx at ϕ=0∘ (solid lines) and ϕ=180∘ (dashed lines) for QWP angles θ=45∘ (right-hand circular; blue), θ=90∘ (linear horizontal; black), and θ=135∘ (left-hand circular; red). Note, both an inversion of sign between excitations of right and left-circular polarization and for sample rotations of 180∘.

We conducted polarization-sensitive TDTS measurements of photocurrent responses in a low-twinned Bi2Se3 thin film at room temperature in an open-air laboratory environment. Corresponding data for a high-twinned sample are presented in SI Appendix for comparison. The measurement technique is an ideal way to extract photocurrent physics directly and non-destructively without interferences from photothermal currents, whilst maintaining sensitivity to directionality, helicity, and amplitude. As shown in Fig. 1A, the ellipticity of an 800-nm excitation laser pulse (~125-fs pulse width, ~5-µJ/cm2 fluence), is varied between linear horizontal in the yz-plane, and right- and left-circular polarizations (RCP and LCP, respectively) using a quarter wave plate, QWP, and incident upon the sample at a 45° angle of incidence. Here, polarizations of RCP, linear, and LCP are obtained with a QWP angle, θ, of 45°, 90° and 135°, respectively. Excitations within the bulk and between the 1st and 2nd topological surface states (1) generate photocurrents in the sample, where the time-rate-change of the current, ddtji=x,y,z, is proportional to the emitted THz electric field, Si=x,y,z. The time-dependence of the emitted THz is detected using electrooptic (EO) sampling on a 1-mm thick (110) ZnTe crystal (33), and the signal’s horizontal and vertical components are distinguished using a wire grid polarizer (WGP) with 99.9% rejection of the cross-polarized THz signal. The horizontal component, Syz, combines surface-parallel and perpendicular currents in the sample’s yz-plane. The vertical component, Sx, is purely due to a surface-parallel current in the x-direction, which is orthogonal to the laser excitation’s plane of incidence. Data were acquired from the EO signal with a balanced detector using a lock-in amplifier synchronized to the chopped frequency of the pump laser; at each time delay between the pump and EO gating pulse, significant averaging was utilized such that the SD for each data point was less than 2% of the signal. The peak of the Syz signal for each experimental condition was set to a time delay of t=0 ps, for consistent timing across measurements on different days.

Results and Discussion

To interrogate crystal symmetries, we explore the dependence of Syz on the plane of incidence of light by rotating the sample’s azimuthal angle, ϕ, defined by the sample’s cleaved edge. Here, ϕ=0∘ is nominally represented by the top view of the sample in the xy-plane in the Inset of Fig. 1A; increasing ϕ indicates counterclockwise rotations of the sample, as shown. Fig. 1B plots a series of time-dependent THz waveforms from ϕ=0∘ to ϕ=60∘, under linear horizontal polarization excitation (θ=90∘). Fig. 1C plots the peak-to-peak signal amplitude over a full 360° sample rotation. A clear threefold symmetry emerges and the data are well fit by a constant plus cos (3ϕ+3δ) dependence where δ=-6∘.

To interpret the physical processes observed in the THz data, we use symmetry analysis (34, 35) to determine the 2nd-order NLO contributions to the photocurrent density, jP, which can be decomposed into circular (C) and linear (L) PGE and PDE terms, jP=jCPGE + jLPGE + jCPDE + jLPDE. Explicitly presented in SI Appendix, we find the dependence of the PGE and PDE contributions to jP as a function of QWP angle θ and the crystal (100) axis azimuthal orientation φ in our experiment:[1a] jLPGE = ηx4 sin4θ-sin3φ(δ~ + κ~4 cos4θ)+η~4cos3φ sin4θδy + κy4 cos4θ + cos3φ(δ~ + κ~4 cos4θ)-η~4 sin3φ sin4θ δz+ κz4 cos4θ,

[1b] jLPDEq = ξx4 sin4θ + sin3φΔ~ + λ~4 cos4θ+ ξ~4 cos3φ sin4θΔy + λy4 cos4θ + cos3φΔ~ + λ~4 cos4θ- ξ~4 sin3φ sin4θΔz + λz4 cos4θ + cos3φΔ~z + λ~z4 cos 4θ+ ξ~z4 sin3φ sin4θ,

[2a] jCPGE=i  sin2θ η2,0,0T,

[2b] jCPDE=i q sin2θξ2+ξ~2cos3φ,-ξ~2sin3φ,0T.

Here, q is the incoming light’s momentum, and the η, κ, ξ, λ, δ, and Δ coefficients are related to elements of the relevant NLO response tensors and incident electric field, which are defined in SI Appendix. The circular PGE and CPDE (CPGE) and linear PGE and linear PDE (LPGE and LPDE) are distinguished by their 2θ and 4θ-dependence on the QWP angle, respectively. Dependence of jP on frequency and momentum is implicit in the coefficients. We note that our analysis differs from previous works by not taking the DC limit of jP as our photocurrents are strongly transient, which in turn imposes fewer constraints on the elements of the NLO response tensors (1). The analysis reveals that PGE contributions to jP can only arise from the inversion breaking surface of Bi2Se3, as the third rank tensor governing the PGE response is 0 for the inversion symmetric bulk. Conversely, the fourth rank tensor for PDE is invariant under inversion, so both bulk and surface states can contribute to PDE. The additional cos3φ dependencies in jPDE arise as the photon momentum defines a stationary plane of incidence in the laboratory frame that periodically aligns with the crystal axes upon rotation.

When considering linearly polarized light, i.e., θ=0∘ or 90∘, Eq. 1 simplifies to show a cos3φ periodicity in jy and jz. This periodicity is transferred to Syz as Braun et al. (1) show Syz∝Sy + αSz, where α≈0.3 is a weighting term accounting for geometry and index of refraction at THz frequencies. The cos (3ϕ+3δ) dependence of Syz in Fig. 1 B and C is thus captured by the NLO symmetry analysis and shows ϕ-6∘=φ, i.e., our (100) crystal axis is 6∘ from the cleaved sample edge. In highly twinned samples, signatures of threefold periodicity are diminished as competing twin domains and contribute ξ∼2 terms with opposite sign, leading to a near net cancellation (SI Appendix, Fig. S4) (21).

Fig. 1D shows the Sx component of the THz waveforms for RCP (blue), linear-horizontal (black) and LCP (red) polarized light for sample azimuthal angles 0∘ (solid) and 180∘ (dashed). As the CPGE response can only originate from the inversion breaking surface, it is predominantly due to the topological surface state (23). The related photocurrent propagates orthogonal to the plane of incidence and is manifest in the Sx component of the emitted THz. From Fig. 1D, we can see that in low-twinned Bi2Se3, there is a nearly complete reversal of the helical photocurrent when the incident light is switched from RCP to LCP light, creating counter-propagating helical photocurrents. Additionally, we see that as the sample is rotated 180∘, the helical photocurrents reverse direction. This reversal with crystal rotation was seen to a muted effect in earlier experiments (21, 36, 37) and indicates that intrinsic crystal symmetries impact helical photocurrents. Eq. 2 shows jCPGE,x is φ-independent, while jCPDE,x∝ξ2 + ξ~2 cos3φ, and changes sign every Δϕ=Δφ=60∘ provided ξ∼2<ξ2 and minimal jLPDE,x. Additionally, for θ∈{45∘,135∘} and ϕ=0∘ and 180∘, Eq. 1 shows jLPGE,x=jLPDE,x=0. Therefore, CPDE and careful choice of plane of incidence can drive the current reversal in Fig. 1D between the ϕ=0∘ and 180∘ (equivalently φ=- 6∘ and 174∘) data sets.

The presence of threefold symmetry in Syz and current reversal in Sx motivates the investigation of the impact of crystalline symmetries on the Sx data. In Fig. 2, we present a series of Sx data taken at every 15∘ of azimuthal sample orientation for RCP, horizontal linear polarization and LCP, respectively, QWP angles θ=45∘, 90∘, and 135∘. The threefold symmetry of the Sx data is easily observed in Fig. 2, wherein waveforms every 120° of sample rotation are identically colored for emphasis. This is represented in Fig. 3A as a radial plot of the Sx waveforms for θ=90∘, with time on the radial axis and ϕ on the polar axis; analogous radial plots for the RCP and LCP data are provided in Supporting Information. Fig. 2 A and C show the sign of Sx for RCP and LCP light reverses every Δϕ=60∘, which is consistent with the discussion in the previous paragraph and indicates a large ξ∼2 coefficient in jCPDE,x. We observe that the largest helical photocurrents occur for sample orientations ϕ=60∘, 180∘, and 300∘, which correspond to angles where the linear photocurrents are also minimized. At these angles, the incoming light is nearly anti-parallel to the ultrafast shift current that transfers charge along the Se–Bi bond (1, 38) (as shown in Fig. 1A) and correspondingly Syz is minimized in Fig. 1B. This demonstrates that particular sample orientations allow for the observation of helical photocurrent responses. Conversely, a 60° azimuthal rotation leads to a dominant shift current contribution, as the detection is aligned with the Se–Bi bond.

Fig. 2. Azimuthal dependence of surface parallel photocurrents. Vertical/surface THz signal, Sx, as a function of azimuthal angle, ϕ, under excitation with: (A, θ = 45°) Right-handed circular polarization; (B, θ = 90°) Linear horizontal polarization; (C, θ = 135°) Left-handed circular polarization. Threefold symmetric angles with Δϕ = 120° are indicated by the same color. Radial plots can be found in SI Appendix. The reversal of right/left surface currents as a function of azimuthal angle is most evident when comparing 0° to 60°. Notably, at 60° the linear current is minimized, suggesting that this crystal axis would be beneficial for binary circularly polarized light detection.

Fig. 3. Azimuthal angle-dependent Fourier decomposed THz contributions. (A) Data from Fig. 2B as a radial plot of the Sx waveforms for θ=90∘, with time on the radial axis and ϕ=φ+6° on the polar axis to visualize the threefold symmetric response. For each incident polarization in Fig. 2, the THz signal’s φ dependence is separated into time-dependent Fourier series sine and cosine coefficients (sn and cn, respectively), where (B) plots the resultant waveform of the threefold contributions and (C) plots the onefold (Top) and constant (Bottom) contributions.

We analyze the azimuthal rotational symmetries of Sx by computing the waveforms’ transforms with respect to φ. One can express Sx as a Fourier series Sxt,φ,θ=Σn∈ℕ0cnt,θ cos (nφ)+ snt,θ sin(nφ) with the Fourier coefficients formally determined by the discrete Fourier transform ckt,θ + iskt,θ≡ 2/NΣn=1NSxt,φn,θeikφn and c0=1/NΣn=1NSxt,φn,θ, where we have accounted for the offset in the azimuthal angle identified by the Syz data fit in Fig. 1B by incorporating the shift φn=ϕn-6° as explained in SI Appendix. This analysis quantifies the degree of cn≡cosnφ and sn≡sinnφ periodicity present in a waveform at a time t and polarization θ. The threefold c3,s3, onefold (c1,s1) and azimuthally independent c0 coefficients were found to be the dominant contributions to Sx and are plotted in Fig. 3 B and C. The transformed data shows that for RCP/LCP light the threefold coefficients obey c3t,45∘·c3t,135∘≤0 and s3t,45∘·s3t,135∘≥0. This behavior is perfectly captured by the NLO response current in Eq. 2b, which simplifies for RCP/LCP light to jx≡c0±+s3sin3φ±c3cos3φ, for c0±,s3 and c3 time-dependent coefficients. Fig. 3C shows that the coefficients c0 have opposite signs for LCP and RCP light and are smaller than c3 and/or s3. This suggests the CPGE contribution to the photocurrent in Eq. 2a is smaller than the one from CPDE in Eq. 2b and is why we see such a pronounced threefold symmetry in the data.

Fig. 3C further shows that there is a non-trivial onefold periodic contribution from s1 and c1 to Sx, which is not captured by the symmetry analysis where a perfect planar crystal is assumed. The onefold aberration may be induced by the offcut substrate, or as other works found, strain effects in thin films of Bi2Se3 (16) and wandering of the excitation/collection area with sample rotation (39). It is surprising that the onefold contribution outweighs the azimuthally independent one, suggesting that if c3 and s3 were weak, the directional dependence of the photocurrent on RCP and LCP light could be obscured. Twinned Bi2Se3 has an effective sixfold rotational symmetry in which case symmetry analysis predicts s3≈c3≈0 (SI Appendix). The coupled effects of twinning, small CPGE, and onefold aberration provide a plausible explanation for nonuniversal observation of current reversal in MBE-grown Bi2Se3.

To explore the full polarization dependence of the photocurrents, Sx waveforms are captured at discrete sample orientations, and at 15∘ increments of the QWP angle θ from 0∘ to 180∘; Sx waveforms are duplicated for mathematically equivalent QWP angles from 180∘ to 360∘ (SI Appendix, Eq. S3). Contributions from the substrate to the photocurrent have been ruled out through comparison of complementary Sx data, where emission from a bare InP (111) substrate are observed to be an order-of-magnitude smaller (additional details are presented in SI Appendix). Fig. 4 presents the low-twinned Bi2Se3 Sx data with time on the radial axis and θ on the polar axis. Sample angles ϕ=0∘, 90∘, 180∘, 240∘, 270∘, and 300∘ were chosen to visualize the threefold azimuthal symmetry in the mirrored pairs (0∘, 240∘) and (180∘, 300∘), and the suppression of CPGE and CPDE at the orientations 90∘ and 270∘. At ϕ = 90° and 270° where the strongest linear (i.e., θ=90∘) response was observed in Fig. 2B, we see a strong fourfold symmetry in the data, which presents itself as four positive/negative lobes over a 360∘ rotation in θ. Moreover, the photocurrent is seen to reverse direction from ϕ = 90° to 270°, i.e., for a general time t the current changes sign with a Δϕ=180∘ sample rotation. At ϕ = 0°, 180°, 240°, and 300°, a strong twofold modulation of the signal in θ and a sign/photocurrent reversal when switching between RCP and LCP (θ=45∘ and 135∘) is observed, indicating a dominant helical photocurrent response. The ϕ=0∘ and 180∘ panels show reversed photocurrents similar to 90° and 270° For the pair 0∘,180∘, this reversal is most evident at θ=60∘ where the observed photocurrents are largest. Consistent with our analysis of Fig. 2, the strongest twofold modulation with respect to θ is observed at ϕ=180∘ and 300∘.

Fig. 4. Helicity dependence of surface parallel photocurrents. Radial plots of Sx waveforms as a function of the QWP angle, θ, with time on the radial axis and θ on the polar axis; each radial plot is displayed on an independent color scale (as shown). The raw data are taken for QWP angles θ=0∘ through 180∘ in 15∘ steps, and then piecewise interpolated with a third-degree polynomial to achieve a finer plot resolution. The interpolated dataset is then repeated for θ∈(180∘,360∘] to achieve a full radial plot (QWP polarization states are inherently twofold periodic). Pairs of threefold symmetric azimuthal angles (separated by 120°) are shown in the left column for ϕ = 0° and 240° and right column for ϕ=180∘ and 300∘ to demonstrate similarities between signals and twofold dependence on input polarization. The center column plots ϕ=90∘ and 270∘, where detection is aligned and counter-aligned with the Se–Bi bond direction, respectively, and strong fourfold, linear dependence on the input polarization is observed.

To analyze the polarization-dependent response of the photocurrent, we compute the discrete Fourier transforms with respect to θ at each sample orientation ϕ, where Sx(t,ϕ,θ) =Σθ~∈ℕ0c~θ~t,ϕ cosθ~θ+s~θ~ sinθ~θ. Fig. 5 presents the physically significant coefficients c~4t,ϕ, s~4t,ϕ and s~2t,ϕ computed with the discrete Fourier transformation in the same way as for the φ-dependent data in Fig. 3. Note, the θ∈0∘,180∘ measurement range implies we can only calculate even periodic modes over θ∈[0∘,360∘]. In this manner, we have deconvolved the different polarization-dependent responses to isolate the helical and linear photocurrent channels for a given sample orientation. Several observations of the data in Fig. 5 are elegantly matched by the earlier theoretical analysis. For example, Eq. 1 dictates c~4t,ϕ∼sin3φ, which is represented in Fig. 5A wherein c~4t,ϕ=90∘ and c~4t,ϕ=270∘ are the largest signal components and have opposite signs. Additionally, s~4t,ϕ (Fig. 5B) attains its maximum absolute value for ϕ = 180° and 300°, which follows from the functional form s~4t,ϕ∼const. -cos3ϕ+3δ≡ const.-cos3φ imposed by Eq. 1.

Fig. 5. QWP angle-dependent Fourier decomposed THz contributions. The time-dependent Fourier coefficients corresponding to (A) cos(4θ), (B) sin(4θ), and (C) sin(2θ) are displayed for the same azimuthal angles as Fig. 4. Observe that pairs of coefficients separated by Δϕ=120∘ show remarkable similarity, highlighting the threefold rotational symmetry. Deviations from threefold can be accounted for by the presence of substrate-induced onefold periodic corrections to the photocurrent-azimuthal angle relationships. The strength of the helical photocurrent is governed by the sin(2θ) coefficient, which reaches its maximum value at ϕ=180∘.

Fig. 5C reveals one of the most impactful results of this study: the helical photocurrent s~2t,ϕ is dominated by the CPDE contribution. Using Eq. 2 one can show that s~2t,ϕ=iη2+qξ2+qξ~2cos3φsin2θ, where η2 is the CPGE coefficient and ξ2 and ξ^2 are the CPDE coefficients. If CPGE were the dominant photocurrent mechanism, s~2t,ϕ would be azimuthally independent. Instead, we see s~2t,ϕ attains maximum values at ϕ=60∘, 180∘, and 300∘, mostly obeying threefold rotational symmetry. The small onefold ϕ-dependence discussed following Fig. 4C accounts for a directional enhancement along 180∘: s~2t,ϕ=180∘≳s~2t,ϕ=300∘. If one considers the PDE and PGE response of fully twinned Bi2Se3, which acquires an additional mirror symmetry, all azimuthal dependence drops out of the photocurrents. Therefore, it is essential to minimize twinning (as is the case for judicious growth or exfoliation) to see this result.

In this article, we have presented measurements of clear threefold azimuthally symmetric photocurrents in TIs that reverse direction for different helicities of light on low twinned MBE grown samples. Crucially, these all-optical measurements ensure that photothermal currents did not impact helical photocurrent reversal (5, 13, 39). While weak threefold azimuthal dependence has previously been reported in the literature, it was only seen as a small modulation on top of a substrate-generated onefold periodicity and was not accompanied by helical current reversal (1, 21). We establish a key explanation for the lack of this observation: the presence of twinning defects in MBE-grown TIs, which on average endow the crystal structure with an effective sixfold rotational symmetry. The NLO tensor symmetry analysis in this case is presented in SI Appendix and we find that all dependence of the surface current on the azimuthal angle drops out, i.e., jx∝i(η2+ξ2)sin2θ+(ηx4+ξx4)sin4θ. Recalling the discussion of Fig. 3, we found that the intrinsic azimuthally independent contributions to the photocurrent were weaker than extrinsic onefold contributions, which could obscure helical photocurrent reversal in highly twinned materials.

We emphasize that the threefold periodicity originates from CPDE as the jCPGE,x photocurrent is inherently azimuthally independent (23). We suspect multiple effects conspire to provide large CPDE-driven responses in our sample(s). First, given that CPDE comes from a fourth rank tensor, it contains bulk crystal contributions, as even rank tensors do not vanish under inversion symmetry, whereas CPGE can only come from an inversion breaking interface. Second, it has been well reported that the helical photocurrents that arise in Bi2Se3 pumped by 800-nm light come from excitations from initial states in the principal Dirac TSS to a 1.7-eV higher energy Dirac TSS with opposite chirality (40) in the conduction bands (41, 42). Since photoexcitation with RCP or LCP light requires the final and initial states to have opposite quantum numbers, the spin-flip transition rule can be frustrated in a direct transition if the excitation energy is smaller than the Dirac point separation. This is the case for our 1.55-eV excitation pulse, and so an additional momentum transfer to the electron may be necessary to satisfy the selection rules, i.e., PDE (43–46). We anticipate that by changing the excitation wavelength and Fermi level in low twinned BSST TIs—thereby tuning between inter- and intra-cone (and sub bulk bandgap) excitations—one can modulate the strength, directionality, and clarity of the threefold symmetries, achieving complete control over TI photocurrents.

This work has strong implications for future avenues of basic and applied research on TIs. Chiefly, any device that aims to leverage helical photocurrents will benefit from enhanced signals in low-twinned materials, and the contacts of these devices must be aligned to the TI crystallographic axes to ensure the full strength of the helical photocurrents are captured. Similarly, heterostructures of TIs and other materials (such as ferromagnets) used to study the interplay of electronic transport in TIs and adjacent material parameters should see stronger coupling (especially if the transport is optically generated) when the heterostructure is grown by MBE to ensure low-twinning and axis alignment. Finally, the spurious mirror symmetry introduced by twinning has greater consequences than modifying the nonlinear tensors governing PDE and PGE. Nonlinear response tensors of any rank and excitation process will be modified by the presence of mirror symmetry and so low-twinned TIs will be essential to any experiment aiming to study the interplay of topology and threefold rotational symmetry.

Materials and Methods

A 12.4-nm-thick epitaxial film of Bi2Se3 with an approximate twin domain ratio of 5:1 was grown via MBE on a semi-insulating InP(111)B substrate with a vicinal mis-orientation of two degrees (2°) toward the <010> direction at a substrate temperature of 290 °C. Reflection high-energy electron diffraction and X-ray diffraction spectra confirm high-quality epitaxy with an approximate twin domain ratio of 5:1 (SI Appendix, Figs. S1–S3). An in situ capping layer of ~10 nm of cubic EuS was deposited to prevent environmental degradation of the surface states in the atmosphere (27). The large bulk energy gap of 3.1 eV makes the EuS layer transparent to the 1.55-eV excitation laser. Additionally, at room temperature, EuS on Bi2Se3 has been conclusively demonstrated to be a nonmagnetic dielectric and therefore no intrinsic time-reversal symmetry-breaking effects are anticipated to contribute to the experiment (47).

Supplementary Material

Appendix 01 (PDF)

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Dataset S01 (TXT)

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Dataset S02 (TXT)

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Dataset S03 (TXT)

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Dataset S04 (TXT)

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Dataset S05 (TXT)

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Dataset S06 (TXT)

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Dataset S07 (TXT)

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Dataset S08 (TXT)

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Dataset S09 (TXT)

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Dataset S10 (TXT)

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Dataset S11 (TXT)

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Dataset S12 (TXT)

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Dataset S13 (TXT)

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Dataset S14 (TXT)

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Dataset S15 (TXT)

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Dataset S16 (CSV)

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We would like to thank D. Hsieh, D. Rees, M. Khajavikhan, and A. Llopis for helpful conversations. This work was funded by an Applied Research for Advancement of Science and Technology Priorities award from the Office of the Under Secretary for Research and Engineering.

Author contributions

B.C.C., P.J.T., and G.J.d.C. designed research; B.C.C., P.J.T., and G.J.d.C. performed research; B.C.C. and G.J.d.C. analyzed data; P.J.T. developed, grew and characterized material; and B.C.C. and G.J.d.C. wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix and can be downloaded at https://doi.org/10.24435/materialscloud:4p-f2 (48).

Supporting Information

This article is a PNAS Direct Submission.
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