Introduction

Nonlinear optical crystals are the key elements in laser frequency conversions, optical parameter oscillators, and optical signal communications. Particularly, there are needs for better infrared nonlinear optical crystals due to the shortcomings of AgGaQ2 (Q = S, Se) and ZnGeP21,2. BaGa4Se7 crystal is a promising nonlinear optical crystal working in the infrared region, and exhibits a second harmonic generation about twice of the benchmark material AgGaS23,4,5. Phonons are the key players in infrared absorptions, especially in middle and far infrared ranges. In addition, the propagation of terahertz phonon-polaritons6 are reported7 and high nonlinear coefficients for terahertz generation are observed in BaGa4Se7 crystals. Both phenomena are the results of resonances between photons and BaGa4Se7 phonons. Therefore, a thorough investigation of the phonon structures of BaGa4Se7 is necessary in order to understand its behaviors ranging from infrared to terahertz.

Here, we study the phonon structures of BaGa4Se7 crystal, with polarized Raman spectroscopy and theoretical calculations. BaGa4Se7 phonon structures show an energy top at about 300 cm−1, which is much smaller than those of most materials. This is consistent with the fact that BaGa4Se7 is softer than most materials. The phonon structures also show a phonon gap. This gap separates the modes of still Ba atoms from the modes of moving Ba atoms. Theoretical calculations give the phonon dispersion curves, density of states (DOS) and vibration modes. We determine nine strongest Raman peaks’ vibration modes and Raman tensors. Our Raman mode assignments and phonon calculations show consistencies in phonon energies, phonon types, and vibration directions. Above knowledge provides a new case example for phonon gaps, offers a complete picture of the phonon structures of BaGa4Se7, and helps us understand its phenomena at infrared and terahertz frequency ranges.

Results

BaGa4Se7 crystal is described by the space group Pc (\(C_3^2\), group No. 7) of the monoclinic system3. In one of this asymmetric unit cell, there is one unique Ba atom, four unique Ga atoms, and seven unique Se atoms. The irreducible representations classify8 the vibration modes of this system into [2A′ + A″] + [34A′ + 35A″], which are 3 acoustic modes (two A′ and one A″) and 69 optical modes (34A′ and 35A″). Due to the low symmetry of this crystal, all those modes are Raman and infrared active. To determine the phonon vibrational configurations and estimate the phonon frequencies, the calculations were carried out within the framework of density functional theory as implemented in the Vienna Ab-initio Simulation Package (VASP)9,10. The exchange or correlation energy was assessed by the generalized gradient approximation in the scheme of Perdew-Burke-Ernzerhof11. The electron–ion interactions were described by means of projector augmented wave12 with 4s24p1, 4s24p4, and 5s5p6s as valence electrons for Ga, Se, and Ba atoms, respectively. We set a 7 × 7 × 7 Monkhorst-Pack11 momentum (k) point mesh and a 500 eV basis set cutoff for the electronic wave functions. Iterative relaxations of cell volume, cell shape, and atomic positions were stopped when the forces generally acting on the atoms were found to be smaller than 0.0001 eV per Å. With this criterion, the change in the total energy between successive steps was less than 0.01 meV per cell. Phonon calculations were performed using the density functional perturbation theory13,14,15 to get the phonon frequencies and vibration modes at the Г point. Real-space force constants were calculated within VASP, and phonon frequencies were calculated from the force constants using the PHONOPY code16,17. The resulting phonon dispersion curves and phonon DOS are plotted in Fig. 1 for all 72 modes; the atomic displacements of 9 modes are shown in Fig. 2 with their type (A′ or A″) and their Г point energies (both from calculations and experiments).

Fig. 1: Calculated phonon dispersions and DOS for BaGa4Se7.
Fig. 1: Calculated phonon dispersions and DOS for BaGa4Se7.
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a Calculated phonon band dispersions along the high-symmetry directions of Brillouin zone for BaGa4Se7. The x axis is the phonon momentum in K space, the y axis shows the phonon energies in units of wavenumbers. The inset gives the definition of high-symmetry points in the momentum space. The blue arrows show the momentum path corresponding to the dispersion displayed. b Phonon density of state (PHDOS) for BaGa4Se7. The red, green, and blue shaded areas are for Ba, Ga, Se atom PHDOS, respectively.

Fig. 2: Main atomic displacements for the optical modes of BaGa4Se7.
Fig. 2: Main atomic displacements for the optical modes of BaGa4Se7.
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The Ga, Se, and Ba atoms are denoted by yellow, blue, and green balls; their displacements are indicated by blue, red, and pink arrows, respectively. The axes in the bottom left corner show the crystallographic axes (abc) and the dielectric frames (xyz). Each block with atom locations and displacement arrows shows one mode and total nine modes are listed here. The A′ or A″ labels below the blocks indicate the mode symmetries in the \(C_3^2\) group notation. The numbers below the blocks are the experimentally measured (in red) and calculated values (in black) phonon energies.

The yellow transparent BaGa4Se7 crystals in this study were grown by the Bridgeman method4,18. Three samples were prepared by cutting along different dielectric frame directions (xyz directions in Fig. 2), gave top surfaces with normal directions in x, y, z directions, and were defined as <100>, <010>, <001> crystals. They have typical size of 13 × 8 × 3 mm3 with the thinnest direction in the normal direction. The polarized Raman experiments were performed on a Horiba HR-800 Raman system with a 532 nm excitation laser. The excitation light at the scale of 1 mW were focused on the top surfaces of the samples with a 100×, NA = 0.9 objective mounted in a backscattering Raman configuration. Polarized Raman measurements were performed with spectra data named accordingly to the configurations; for instance, xyz spectrum means: the sample is <100> crystal and its top surface’s normal direction is in x direction, the incident light’s polarization is in y direction, the analyzer’s polarization is in z direction. Twelve configurations were measured at same excitation powers with xyy, xzz, yxx, yzz, zxx, zyy, xyz under 3 s exposure time and xzy, yxz, yzx, zxy, zyx under 20 s exposure time. All the Raman spectra data were analyzed and fitted with multiple Lorentzian peaks to retrieve Raman intensities for individual modes. Figure 3 shows the xyy spectrum and its fitting peaks; other spectra are shown in Supplementary Figs. 1–11; all 12 spectra were processed with the same method as the xyy. Then, the intensities of individual Raman modes from different polarization configurations were combined together (after normalization for different exposure times) to give the Raman tensors. The relative strengths of the elements give the shapes of the Raman tensors, although their absolute strengths are in arbitrary unit. Nine modes’ resulting Raman tensors are listed in Table 1, together with the modes’ types which are determined from the Raman tensors.

Fig. 3: Raman spectrum and its fitting with multiple Lorentzian peaks of BaGa4Se7 under xyy measurement configuration.
Fig. 3: Raman spectrum and its fitting with multiple Lorentzian peaks of BaGa4Se7 under xyy measurement configuration.
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The center locations of all Lorentzian peaks are listed in the figure. The two dash line rectangles show the locations of the phonon gap, which maintains its positions in the other Raman spectra of different settings.

Table 1 List of nine modes’ energies, Raman matrixes and types.
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BaGa4Se7’s phonon structure

BaGa4Se7’s phonon structure shows a relatively low phonon energy cap and a phonon gap. Figure 1a shows the calculated phonon band dispersions along the high-symmetry directions of Brillouin zone for BaGa4Se7. The inset gives the definitions of high-symmetry points in the momentum space. The blue arrows show the momentum path corresponding to the dispersion displayed. The dispersion curves give the maximum phonon energy about 300 cm−1 and a phonon gap around 150 cm−1; both the top of phonon bands and the phonon gap are visible in Raman spectra (see Fig. 3 for example) at corresponding energy locations. We expect the BaGa4Se7 crystal is soft and fragile as the result of such a low phonon energy cap. The phonon gap around 150 cm−1 is about 45 cm−1 wide. Figure 1b shows the phonon DOS for BaGa4Se7. The red, green, and blue shadowed areas are for Ba, Ga, Se atom phonon DOS, respectively. The Ba atom only has phonon distribution within the low energy part just below the phonon gap. It means that the Ba atom doesn’t move at all in the phonon modes above the gap. For instance, the 180.8 and 230.5 cm−1 modes show no movements of Ba atoms in Fig. 2. All of the upper band phonon modes satisfy the condition of a still Ba atom. The freezing of the Ba atom at upper phonon band probably is due to its place at a high symmetry point of the cell and its heavy weight. Earlier works reported a gap between acoustic and optical phonons in MoS219 and WS220, and phonon gaps in two elements hydrides21. Previous reports show that a phonon gap happens in a two-elements crystal with a very heavy atom and a very light atom and with high symmetric conditions. Here, we showed that a phonon gap can happen in a complex three-elements monoclinic crystal, BaGa4Se7, with very low symmetric conditions. Our finding lowers the requirements for having a phonon gap and suggests that engineering a phonon gap might be achievable in a large amount of different kinds of crystal systems. Also, we feel that the fact, this gap in BaGa4Se7 separates the modes with a still or vibrating Ba atom, is interesting and might be potentially useful for phonon effective mass control and phonon structure engineering. For instance, our calculations show that replacing Ba with a lighter atom, Sr, Ca, or Be will reduce the phonon gap, and replacing Ba with a heavier atom, Ra will enlarge the phonon gap. By engineering two materials with mismatching phonon gaps, we might have a very large interfacial thermal resistance.

Matching of phonon energies

Our phonon calculations and Raman mode assignments show good coincidence in phonon energies. We list nine modes’ atomic displacements in Fig. 2 and their energies, Raman matrixes and types in Table 1. These nine modes are the strongest modes, and can be identified clearly with matching phonon energies, matching phonon types (A′ or A″), and matching main vibration directions between calculations and Raman measurements. In Fig. 2, the Ga, Se, and Ba atoms are denoted by yellow, blue and green balls, whose sizes represent their weights; their displacements are indicated by blue, red and pink arrows, respectively. The axes in the bottom left corner show the crystallographic axes (abc) and the dielectric frames (xyz, used in polarized Raman measurements). Each block with atom locations and displacement arrows shows one mode. The numbers below the blocks are the experimentally measured (in red) and calculated (in black) values of phonon energies at Г point, and they match well between calculations and experiments. We also plotted the experimentally measured (in red) and calculated (in black) values of phonon energies at Г point of these nine phonon modes, for a comparison, in Supplementary Fig. 12. In Table 1, the first column shows those modes’ energies from different polarized Raman setting. Each value is the averages of peak fitting results from two diagonal Raman spectra; only strong spectra and Stokes peaks are used for determining phonon energy for smaller errors. For instance, the first energy value 23.77 cm−1 is the average of Stokes Raman peak fitting results from spectra yxx and zxx; the other two values are the averages from spectra xyy and zyy, yzz, and xzz. The final mode energies are the averages of the values in column one, are listed in column two, and are also shown in Fig. 2 together with calculation values for comparisons. The calculations and experiments show similar values for phonon energies at Г point.

Phonon types

The phonon type is the second indicator of a good Raman mode assignment. There are only two type of phonons, A′ and A″, in the point group Cs(m) (space group Pc). Their Raman tensors must satisfy \({\mathbf{A}}^{\prime} \left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} a & d & 0 \\ d & b & 0 \\ 0 & 0 & c \end{array}} \right)\), \({\mathbf{A}}^{\prime\prime} (z) = \left( {\begin{array}{*{20}{c}} 0 & 0 & e \\ 0 & 0 & f \\ e & f & 0 \end{array}} \right)\). The A′ or A″ labels below the blocks in Fig. 2 indicate the mode symmetries in the \(C_3^2\) group notation. The phonon types (A′ or A″) of those modes are determined independently and are identical, for results both from calculations and experiments. Polarized Raman measurements can give the Raman tensors for modes, thus determine the phonon types. The measured Raman tensors of Stokes modes, for smaller errors, are listed in the third column of Table 1. Each component is the square of Raman matrix elements. For components on the diagonal line, the values are the averages of peak fitting intensity results from two diagonal Raman spectra; for other components, the values are single results from one spectrum. For instance, the first Raman matrix component value, (30 ± 2) × 103, of mode 23.74 cm−1 is the average of Stokes Raman peak fitting results from spectra yxx and zxx, with standard deviation as the error. The component values, (53 ± 9) × 102 and (17 ± 1) × 102, are the averages from spectra xyy and zyy, yzz and xzz, respectively. The other component value, 18 × 10, is from spectrum zyx, and so on. Some component values are too small compared to the strong component values of a same phonon mode, and thus can be neglected as zero with original small values in the brackets. Because phonon type A′ and A″ can only have certain Raman tensor structures, the nine phonon modes’ types are determined from their Raman tensor shapes and are listed in the fourth column of Table 1. Both calculations and Raman measurements give identical phonon types for the nine modes.

Matching of vibration directions

In addition, the main vibration direction of each phonon modes matches well between the calculations and Raman measurements. For A′ phonons, the main vibration directions are where the strongest components are on the diagonal line of Raman tensors. For instance, the strongest component of mode 23.74 cm−1 is (30 ± 2) × 103, and this suggests the main vibration amplitude direction is in x direction. Based on the Raman tensor intensities of those nine modes, the Raman measurements suggest: mode 23.74, 123.5, and 230.4 cm−1 mainly vibrate in x direction; mode 69.9 cm−1 mainly vibrates in y direction; mode 137.0 cm−1 mainly vibrates within xz surface; mode 180.8 cm−1 mainly vibrates within yz surface. For A″ modes, because only xz or yz elements are non-zero, it seems that the main vibration direction has to be in Z direction due to symmetry conditions. In Fig. 2, calculations show atomic displacements of those modes with color arrows; they are indeed mainly in the directions as Raman tensors suggest. For instance, mode 23.74 cm−1 in the second block of Fig. 2 clearly shows mostly vibrating in the x direction. So, the theoretical calculations are in consistence with the experimental findings, not only in phonon energies and phonon types but also in vibration directions.

In terms of Raman mode intensities, the 180.8 cm−1 mode is the outstanding and strongest Raman peak (see Fig. 3). This is also supported by the strongest Raman tensor values, 6 × 104 in yy setting and 5 × 104 in zz setting, which are about twice of the largest values of other modes. The strong intensity of the 180.8 cm−1 mode partly is benefitted from its largest phonon density of states, which shows up as the strongest peak in the Fig. 1b. The rest of Raman peaks in the spectra are much weaker than the 180.8 cm−1 mode.

Discussion

As a promising nonlinear optical crystal in the infrared region, BaGa4Se7 also shows phonon strongly related polariton dynamics with terahertz waves and high nonlinear coefficients for terahertz generation due to phonon resonances. In this work, we studied the phonon structures of BaGa4Se7 crystal, with both polarized Raman spectroscopy and theoretical calculations. Theoretical calculations present the phonon dispersion curves, DOS, and vibration modes. Our Raman mode assignments and phonon calculations show consistencies in phonon energies, phonon types, and vibration directions. We also listed nine strongest Raman peaks’ vibration mode pictures and Raman tensors. Above detailed phonon information will greatly help us to understand BaGa4Se7’s behaviors at terahertz and infrared frequency ranges. In addition, an interesting phonon gap appears in this three-elements monoclinic crystal and separates modes with a still or vibrating Ba atom. This might be potentially useful for phonon effective mass control and phonon structure engineering. For instance, by engineering two materials with mismatching phonon gaps, we might have a very large interfacial thermal resistance. Overall, this study of BaGa4Se7 phonon structures will help us understand phonon gaps, monoclinic crystals, and BaGa4Se7’s interactions with infrared and terahertz frequency light.