Selective observation of enantiomeric chiral phonons in $\alpha$-quartz

We report anomalous circularly polarized Raman spectra of phonons in right- and left-handed quartzes. The phonon branches splitting from the E-mode at a finite wavenumber were found to be chiral with mutually opposite angular momenta. Our analysis reveals the Raman selection rules for chiral phonons. We also find that the conservation of angular momenta should be satisfied between two photons and one chiral phonon in the Raman process. Furthermore, we experimentally show that the helicity of phonons with a certain frequency and wavevector should be the opposite in the other enantiomer.

Chiral materials have been investigated in many studies because they exhibit various properties, such as optical activity [1,2], circular dichroism [1,2], and chiralityinduced spin selectivity [3][4][5]. In particular, chiral phonons in these materials have recently attracted significant attention [6][7][8]. Chiral phonons are phonons with angular momentum, which correspond to lattice vibration modes with the rotation of atomic displacements [9]. They may contribute to the thermal Hall effect, in which heat flow is deflected by a magnetic field [10]. In addition, chiral phonons are expected to relate to interesting phenomena, such as coupling between chiral phonons and circularly polarized light/valley electrons [11][12][13][14], conversion of magnons to chiral phonons [15], phonon thermal Edelstein effects [16], and phonon rotoelectric effects [17]. The angular momentum degrees of freedom can be used for information communication technology [18].
It is well known that the rotation direction of the plane of linear polarization owing to optical activity is opposite in right-and left-handed chiral crystals. In analogy to optical activity, chiral phonons are expected to exhibit opposite properties (e.g., angular momentum and propagation direction) in crystals with different chirality. A typical example in this regard is the recently proposed chiral phonon diode effect [7]. In this effect, chiral phonons along the c-axis of chiral crystals (e.g., α-quartz and Te) belonging to a certain branch can propagate in only either a positive or a negative direction. In addition, it is predicted that the direction along which the chiral phonons can propagate is opposite in right-and lefthanded chiral crystals. Investigation of chiral phonons in crystals with different chirality is significant for controlling angular momentum and propagation direction of chiral phonons. However, to the best of our knowledge, such experimental studies have rarely been reported till date.
In the present study, we achieve the selective observation of enantiomeric chiral phonons in right-and left-handed quartzes by employing circularly polarized Raman spectroscopy. The experimentally deduced Raman selection rules show that phonons having a finite wavenumber can carry angular momenta and that the helicity of phonons with a certain frequency and wavevector should be the opposite in the other enantiomer.
We employed (0001) plates of right-handed quartz (space group P3 1 21; Crystal Base Co., Ltd., Osaka, Japan) and left-handed quartz (space group P3 2 21; CRYSTAL GmbH, Berlin, Germany) with dimensions of 10 × 10 × 0.5 mm 3 synthesized by hydrothermal method. We performed Raman spectroscopy on the crystals at room temperature (297 K) using a 532 nm excitation line from a frequency-doubled Nd:YAG laser (Oxxius LCX-532S-300), a single monochromator (Jobin Yvon HR320 with a 1200 gr/mm grating and frequency resolution of 2.7 cm −1 ), and a double monochromator (Jobin Yvon U1000 with 1800 gr/mm grating and frequency resolution of 0.3 cm −1 ). As shown in Fig. 1, we adopted a backscattering geometry and four polarization configurations (++), (−−), (+−), and (−+), where the first and second signs in the parentheses indicate the sign of the SAM of the incident and scattered photons, respectively. The SAM of the right-/left-circularly polarized light propagating along the direction toward the sample is + / − , respectively, whereas that of the backpropagating right-/left-circularly polarized light is − / + , respectively. In other words, the sign of the SAM of a photon corresponds to the rotation direction of the electric field of circularly polarized light on the sample surface regardless of the propagation direction.
because the Raman shifts of the Stokes-(above 0 cm −1 ) and anti-Stokes scattering (below 0 cm −1 ) coincide for all modes. Our result is consistent with the previous work on α-quartz by Pine and Dressellhaus [20], in which they reported that the lowest-frequency E-mode showed a slight difference depending on the polarization condition; however, they did not mention the handedness of the sample crystal. In addition, the relationship between the positions of the Raman peaks observed in (+−) and (−+) was interchanged in the right-and left-handed quartzes. This trend was observed for all Raman peaks of E-mode phonons shown in Fig. 3.
To investigate the slight difference in the Raman shift, several intense Raman peaks were measured with a higher resolution of 0.3 cm −1 . Figure 5 shows the Raman peaks normalized by the integrated intensity. These peaks were fitted with Voigt functions to obtain the phonon frequencies ν (+−) and ν (−+) . The phonon frequency difference ∆ν ≡ ν (+−) − ν (−+) is shown in the upper left of each spectrum. The sign of ∆ν is seen to be different for each phonon, and it is the opposite for rightand left-handed quartzes in the same frequency region.
Here, we discuss the origin of ∆ν. Under the present experimental conditions, the observed phonons have a wavenumber |k| = 4nπ/532 nm −1 from the Γ point to A point (n is the refractive index of α-quartz). According to the phonon dispersion relation of α-quartz in the previous study [28], the dispersion curves of the E-mode phonons split into two branches at a finite wavenumber. The split widths of the branches at |k| are 1 cm −1 ; this agrees with the observed ∆νs (Fig. 5). Although circular birefringence may also cause the frequency difference, it is negligible because its order is estimated to be 10 −4 cm −1 [29]. Thus, we consider that each pair of Raman peaks is assigned to the two phonon branches having different selection rules. Let H λ be the Hamiltonian of the λ-th phonon mode, and expand it with respect to the wavevector q as S λ is the order-1 tensor of the expansion coefficients, and q ⊗ S λ is from the crystal symmetry of α-quartz [26,27], where B λ is a real constant and q z is the z component of the phonon wavevector. Here, we show only the 2 × 2 matrix related to the E-mode phonon of q ⊗ S λ ; it is essentially a 3 × 3 matrix as it is sufficient to describe the back-scattering experiment along the c-axis, as shown in Fig. 1. By introducing the eigenenergy of the unperturbed Hamiltonian H λ0 of the λ-th E-mode phonon as E λ0 , H λ is expressed as The eigenvectors |u λ and eigenvalues E λ of H λ are respectively. The subscripts of E λ± and |u λ± are in the same order as the double signs. Equation (4) shows the E-mode phonons split into the upper and lower branches at a finite wavenumber. Therefore, we can interpret that the splitting of the E-mode phonons causes ∆νs. In this case, the Raman spectra for (+−) and (−+) show that the Raman selection rules are different for the phonons in the upper and lower branches. In addition, Eq. (5) implies that these phonons are in mutually opposite circular polarization states, that is, they are chiral phonons with opposite angular momenta.
The Raman selection can be described using the Raman tensor α. The scattering intensity I can be written as I = | e s |α|e i | 2 by using the polarization vector of the incident light e i and that of scattered light e s . Loudon [19] reported the real-valued Raman tensors of E(x)-and E(y)-mode phonons (E-mode phonons with normal coordinates of x and y) for the point group D 3 . However, these Raman tensors cannot explain our experimental results because they predict I (+−) = I (−+) for the E(x)-and E(y)-modes, resulting in the same Raman scattering intensities (+−) and (−+); this is not consistent with our results. Therefore, we assume Raman tensors with complex elements and deduce the Raman tensors of chiral phonons from the experimental results. Let the Raman tensor and right-and left-circularly polarized vectors be respectively. Here, a, b, c, and d are complex constants. We derive the Raman tensor of a phonon mode whose Raman scattering intensity is non zero only in (+−) for one example (e.g., the Raman peak (solid line) near 128 cm −1 in left-handed quartz in Fig. 4). The Raman scattering intensity of such a phonon mode takes a finite value in (+−) while it vanishes in other polarization configurations. Therefore, the experimental results can be ex-pressed as follows: where A is a real-valued constant. Solving these equa- Similarly, the Raman tensor of the phonon mode whose Raman scattering in- Here, C λ and D λ are constants. The matrices α λ and α λ * are Hermitian conjugate, reflecting the fact that we observe chiral phonons with mutually opposite angular momenta in (+−) and (−+).
Next, we consider the Raman scattering process of chiral phonons. In a typical Raman scattering process, energy-and momentum-conservation laws are established for two photons and one phonon [19]. In addition, the angular momentum of the photons and phonon are considered because the SAM of the incident and scattered photons differs in (+−) and (−+). We accordingly consider the Stokes Raman processes in (+−) and (−+). The chiral phonons generated in (+−) and (−+) should receive SAMs of +2 and −2 from the incident and scattered photons, respectively. Note that the angular momentum of the phonon is a pseudo-spin angular momentum (PSAM) [8,11,[30][31][32]. Although the PSAM of the phonons appears to be +2 or −2 , it can be − and + , respectively, owing to the Umklapp process of PSAM originating from the symmetry of the threefold rotation of α-quartz [7,31]. Therefore, a chiral phonon with a PSAM of − for (+−) and one with a PSAM of + for (−+) are generated. For example, the PSAMs of the upper and lower branches of the E-mode near 694 cm −1 with the largest ∆ν in left-handed quartz are −2 = + and +2 = − , respectively. Our experimental results are consistent with the PSAM of phonons in left-handed quartz calculated in a recent study [7]. Because the same argument can be made for the anti-Stokes Raman process, the conservation laws of energy, momentum, and pseudo-angular momentum for the Stokes and anti-Stokes Raman processes are expressed as follows: where ν, p, and s are the frequency, momentum, and angular momentum, respectively, and the subscripts i, s, and q indicate the incident photon, scattered photon, and phonon, respectively. + and − on the right-hand sides in Eqs. (6)-(8) represent Stokes and anti-Stokes pro-cesses, respectively. The Stokes and anti-Stokes Raman processes in (+−) are shown in Fig. 6, where h, ν i , ν s , |g , |e , |f + , and |f − are the Planck constant, frequencies of the incident and scattered light, initial state, intermediate state, right-and left-circularly polarized states of the phonon, respectively. Figure 6(a) shows that a chiral phonon with the PSAM − is generated in the (+−) Stokes Raman process. By contrast, Fig. 6(b) shows that a chiral phonon with the PSAM + is annihilated in the (+−) anti-Stokes Raman process. Finally, we discuss the chiral phonons in right-and left-handed quartzes. As shown in Fig. 4, the signs of ∆ν for right-and left-handed quartzes are opposite to each other in the same frequency range. This result suggests that the Raman symmetries of the paired chiral phonons are interchanged according to the chirality of α-quartz. Specifically, the helicities of the paired chiral phonons are switched in the other enantiomer.
To summarize, we selectively observed enantiomeric chiral phonons in right-and left-handed quartzes by circularly polarized Raman spectroscopy. The Raman selection rules of the chiral phonons can be explained by the complex Raman tensors α λ = C λ 1 −i −i −1 and We found that the conservation laws of pseudo-angular momentum, in addition to those of energy and momentum, are established among the photons and chiral phonons in the Raman scattering process. Furthermore, our experimental results imply that the helicity of phonons with a certain frequency and wavevector should be the opposite in the other enantiomer. In the future, the Raman tensor should be derived from a Raman cross section in consideration of phonons with a finite wave number and conservation of angular momentum. Further study is needed to obtain more information on the chirality of elementary excitations. The authors would like to thank Professor Takuya Satoh for helpful discussions on the pseudo-angular momentum.
This work was supported by JST SPRING Grant Number JPMJSP2101 and JSPS KAK-ENHI Grant Numbers JP19K05252, JP19H05618, and JP21H01018.
[29] Circular birefringence is a phenomenon in which the refractive indices n+ and n− for right and left circularly polarized light, respectively, are different (e.g., for αquartz, n+ = 1.54694 and n− = 1.54686.). The refractive indices of light in (+−) and (−+) are n+ and n−, respectively. The relationship between the phonon frequencies and wavenumbers observed in Raman scattering in the backscattering geometry is where νi, c, ni, ns, ν ph , and q are the angular frequency of incident light, speed of light, refractive index of incident light and scattered light, and phonon frequency and wavenumber, respectively. Solving equation (i) and the phonon dispersion relation ν(q) for the sample determines the phonon frequencies observed in Raman scattering. Generally, ν(q) is complicated; however, we can approximate it as ν(q) = aq + ν0, where a and ν0 are the slope of the phonon dispersion relation and the phonon frequency at the Γ point, respectively, because the phonons observed in Raman scattering are near the Γ point. Solving (i) and (ii) for ν ph yields where ν ph = ν(q). From (ii), we calculated the observed phonon frequencies ν (+−) and ν (−+) in the polarization configurations (+−) and (−+), respectively, and estimated the order of ∆ν = ν (+−) − ν (−+) to be approximately 10 −4 cm −1 , where νi is 10 4 cm −1 , ν0 is 10 2 cm −1 , and a is of the order of 10 2 cm −1 /Å from the phonon dispersion relation reported in previous studies [28]. This is much smaller than the observed ∆ν. Therefore, circular birefringence is ruled out as the origin of the observed ∆ν. Dispersion relation for an E-mode phonon in αquartz. At the phonon wavenumber observed in our measurements, the E-mode phonon is split by the order of 1 cm −1 , which is of the same order as the observed ∆ν. Further, phonons in the upper and lower branches are observed in (+−) and (−+), respectively; however, some other E-mode branches obey an inverse relationship. shows that the chiral phonon with a PSAM of − is generated in the Stokes Raman process. In contrast, (b) shows that the chiral phonon with a PSAM of + is annihilated in the anti-Stokes Raman process.