Raman study of the vibrational modes in ZnGeN₂ (0001)

ZnGeN₂ is a heterovalent ternary analogue of wurtzite GaN, obtained conceptually by substitution of each pair of Ga atoms, belonging to group III, by a Zn (group II) and Ge (group IV) atom. There has been increasing interest lately in heterovalent ternary nitrides. This interest is motivated in part by the search for alternatives to the binary nitrides that would be composed entirely of abundant elements. In addition, the increased complexity of the compounds offers opportunities for developing doping strategies, engineering defects, and tailoring properties that are not available in the simpler, binary nitrides. The prospect of combining the group-III nitrides with heterovalent II–IV nitrides in heterostructures offers additional opportunities. For example, while the band gap of ZnGeN₂ is within 0.1 eV or so of that of GaN, there is a sizable type-II band offset.¹ This situation offers the prospect of designing novel nitride heterostructure devices with desirable properties, taking advantage also of the close lattice match and similar optimal growth temperatures of the two materials.² 

The increased complexity of the lattice, compared to the binary nitrides, affects the materials' fundamental properties such as electronic band structure,³ phonons,^4,5 and defects.⁶ Depending on the growth conditions, ZnGeN₂ may exhibit either a disordered (wurtzite-like) or ordered structure.⁷ In a recent paper, it was shown that even the disordered phase may obey locally the octet rule of having exactly two Zn atoms and two Ge atoms as nearest neighbors of each N atom.⁸ This phase involves a disordered mixture at the atomic scale of two stacking arrangements of rows of atoms in the basal plane, each row containing alternating Zn and Ge atoms, corresponding to two simple ordering schemes with space groups Pna2₁ and Pmc2₁. To date, only the Pna2₁ ordering has been observed. The Pna2₁ structure of ZnGeN₂ was observed first by neutron diffraction.⁹ 

Because the Pna2₁ structure of ZnGeN₂ has 16 atoms per unit cell, its vibrational spectrum is much more complex than that of the wurtzite structure, and has not been fully determined experimentally. The crystal structures of orthorhombic Pna2₁ and wurtzite are related as follows: $ao=2a1w, bo=2a2w+a1w, co=cw$⁠, and the atoms all occur in 4a Wyckoff positions. Figures of the ordering of the atoms, the relation between the crystal structures, and details about the structure can be found in Refs. 3, 4, and 8. The vibrational modes at Γ can be labeled according to the irreducible representation of the point group $C2v$⁠. There are 11 A₁, 11 B₁, 11 B₂, and 12 A₂ modes, not counting the zero frequency acoustic modes. All modes are Raman active. The Raman selection rules were discussed in Peshek et al.¹⁰ In summary, the A₂ modes correspond to the xy element in the Raman tensor, while A₁ corresponds to xx, yy, zz, B₁ to xz and B₂ to yz. The A₁ modes are also infrared active for light polarized along z = c, the B₁ for light polarized along x = a, and the B₂ for y = b. All modes except the A₂ modes exhibit LO-TO splitting.

The calculated vibrational spectra of the Zn-IV-N₂ materials with IV = Si, Ge, and Sn were presented in a series of papers.^4,5,11 The first report of a measured Raman spectrum of ZnGeN₂ was for polycrystalline material. The unpolarized spectrum showed no well-resolved Raman peaks.¹² Subsequently, Raman spectra were measured for hexagonally faceted single crystals of diameters of a few microns and lengths of tens of microns along the c axis by Peshek et al.¹⁰ Using this geometry and exploiting polarization-dependent selection rules, it was possible to measure only the A₁ and B₂ TO modes. Additional features were found in the measured spectra and assigned to peaks in the phonon density of states, indicating some effects of disorder-induced Raman scattering.

In a recent paper,⁷ we reported a new growth procedure for ZnGeN₂, which leads to platelet-shaped crystallites with the plane of the platelet being the c-plane. This geometry offers the opportunity to measure the A₂ modes, which were not reported before, and, for the A₁ symmetry, should give LO rather than TO modes. Our primary goal is to establish measured values for these modes and to compare them with calculations. Calculations with an improved pseudopotential are presented in this paper. At the same time, a Raman study provides information on the quality of the crystals obtained by the new growth method. The c-direction is also the preferred growth direction for heterostructures and films grown on substrates. Since the same orientation will occur in films grown on the basal plane, it is important to establish the corresponding Raman spectrum experimentally, as this information could become useful in monitoring the quality of the material in thin film growth.

By using cross-polarized as well as parallel-polarized spectra for different orientations relative to the hexagonally shaped plates, we were able to identify the A₂ modes and the A₁ modes. Good correspondence between theory and experiment was obtained for the A₂ modes and those A₁ modes that show weak LO-TO splitting. However, for the few modes that show a larger LO-TO splitting, the LO peaks were not visible in the experimental spectra. This result is explained in terms of overdamped LO-plasmon coupling.

ZnGeN₂ was synthesized in a quartz tube furnace by exposing a Ge-Zn-Sn liquid to gaseous ammonia at 758 °C. To form the liquid, a small amount of Sn (50 mg) was placed on a [111] oriented Ge wafer. A small amount of Ge melted to form a liquid alloy that was in equilibrium with the underlying solid Ge. The Zn was supplied by a heated Zn crucible upstream of the Ge wafer, which maintained a Zn pressure in the growth chamber of approximately 0.03 atm. Based on equilibrium data, the predicted liquid composition was 31 at. % Sn, 15 at. % Zn, and 54 at. % Ge. The NH₃ pressure was maintained at 0.31 atm and the H₂ carrier gas pressure was 0.63 atm. The growth time was 4.0 hours. Platelet-shaped crystals approximately 20 microns in diameter formed on the surface of the melt. The lattice parameters of the ZnGeN₂ platelets were measured using x-ray diffraction (a = 6.44 ± 0.01 Å, b = 5.467 ± 0.006 Å, and c = 5.191 ± 0.004 Å). The material has the orthorhombic lattice distortion associated with Pna2₁ cation ordering.⁷ 

Micro-Raman measurements were performed using a 633 nm HeNe laser focused through a 50× objective to a spot size of approximately 1 micron. The incident laser power was 8mW. The scattered light was collected in reflection using the same objective, filtered, and detected by a spectrometer equipped with a liquid-nitrogen-cooled CCD. A polarizer was located in the laser line before entering the objective. The sample was rotated in order to change the direction of incident polarization with respect to the crystallographic axes. The second polarizer was located after the scattered light left the microscope.

The vibrational modes were calculated using density functional perturbation theory,^13,14 also called linear response theory, within a plane wave basis set pseudopotential method using the ABINIT code.¹⁵ The calculations were performed in the local density approximation (LDA),¹⁶ and used the relativistic Hartwigsen, Goedecker, Hutter (HGH) norm-conserving pseudopotentials.¹⁷ As will be shown, these give a slight improvement in the agreement with the experiment compared to the previously used pseudopotentials.^4,5,10 A large plane wave cut-off of 80 Hartree was used and the Brillouin zone integration used a $4×4×4$ mesh. For Zn, the 3d electrons were treated as valence electrons while for Ge they were included in the pseudized core.

The Raman spectra were recorded for a hexagonal platelet for several polarization directions as indicated in Fig. 1. The scattering geometry has the wavevector of the laser light at nearly normal incidence to the plane of the platelets, that is, along the c-direction. When the incoming and scattered light has parallel polarizations, the A₁ symmetry modes are excited. The Raman tensor for A₁ symmetry has different components for the xx, yy, and zz polarizations of the incoming and scattered light. Since the wavevector is along z and a vector's z component belongs to the A₁ irreducible representation of the $C2v$ point group, the spectrum measured in this way should correspond to the longitudinal optical modes. Although the A₁ modes contain displacements of the atoms along x and y as well as z, only their z-components contribute to the Raman tensor. For example, the first-order Raman tensor element

involves the displacements eigenvector $uiαm$ of the m-th mode of A₁ symmetry, with α the Cartesian component and i the index of the atom in the unit cell. Although there is a sum over i and α, because of the third rank tensorial nature of the derivative of the susceptibility versus atomic displacements only the xxz elements of the tensor are non-zero and thus only the z displacements contribute.

On the other hand, the A₂ modes correspond to the R_xy Raman tensor and should be measurable with crossed polarizers. However, we do not have an independent measure of which is the x and which is the y direction on the platelet. We assume that x must point either to the corner of the hexagonal plate or to the middle of the side. Although the platelets look like regular hexagons, the underlying crystal structure is orthorhombic and thus not all of the corners or flat sides are equivalent. Therefore, four different directions were measured. We utilized the selection rules to identify the x or y directions. For a general direction in the plane, and parallel polarizers, the intensity should be proportional to

with $ϕ$ the azumuthal angle measured from the x axis. For parallel polarizations, if $ϕ$ is in a direction between x and y, both A₁ and A₂ modes will be present in the spectrum. Only when $ϕ=0$ or $ϕ=90°$ does the spectrum contain no A₂ modes.

To determine which are the x and y axes, we first show the spectra under parallel polarizers in Fig. 2 for the four directions shown in Fig. 1. The peaks at 572, 658, and 753 cm⁻¹ are the three strongest measured A₂ modes. According to Equation 2, when $ϕ$ is in the x or y direction, no A₂ modes should be present. Clearly for directions 1 and 4, the modes at 572, 658, and 753 cm⁻¹ are almost completely suppressed, and thus we identify one of the directions 1 or 4 as the x direction and the other as the y direction, is which will be determined below on the basis of the A₁ modes.

The full spectrum under crossed polarizers with incoming polarization along one of these directions is shown in Fig. 3. We can now clearly identify all twelve A₂ modes. The measured values are compared with the present calculation and a previous one⁴ in Table I, and the differences between these and the experimental values is noted. The new calculation using a more accurate pseudopotential gives a slightly smaller maximum and average difference. Most frequencies are obtained to within 1% and the maximum difference between experimental and calculated values is 2%. The small peak at 617 cm⁻¹ may result from incomplete suppression of the strong A₁ peak by the crossed polarizers, but the small peak at 726 cm⁻¹ cannot be accounted for in this manner and is unexplained.

In Fig. 4, we show the spectra under parallel polarizers for directions 1 and 4 compared with calculated Raman spectra for the yy and xx Raman tensor components, respectively, for the longitudinal and transverse A₁ modes. First, we note that the spectrum for direction 1 best corresponds to the calculated spectrum for the $A1yy$ Raman tensor and for direction 4 to the $A1xx$ Raman tensor. The collection times for the measured spectra in directions 1 and 4 were identical and the spectra were scaled by the same factor in the figure. The four calculated spectra were scaled by a common factor in order to make the measured and calculated intensities roughly comparable. We can see that the absolute intensity as well as the ratio of the strongest peak (at 614.5 cm⁻¹) to the other peaks is higher for the experimental direction 1 and for the calculated yy component. Second, the details of the spectrum in the low frequency range also match best when identifying direction 4 with x and direction 1 with y. The calculated peak positions for A₁ and their tentative association with experimental peaks are listed in Table II. For completeness' sake, we also report the calculated B₁ and B₂ modes in Table III, although it was not possible to measure these with the present samples and scattering geometry.

Although X-ray diffraction (XRD) shows that the material has clearly Zn-Ge cation ordering according to the Pna2₁ structure,⁷ this ordering may not be perfect. In addition, point defects or any other degradation of the momentum selection rule may also lead to disorder-induced Raman scattering and produce features in the experimental spectrum corresponding to the phonon density of states (DOS). First, we note that the presence of DOS features in the Raman spectra of the present material is much less pronounced than in the spectra of the material grown by Peshek et al.¹⁰ Here, only a few peaks appear to correlate better with the calculated DOS than with specific modes at Γ: namely, the peaks at 157, 219, 283, and 536 cm⁻¹. These peaks are labeled with a D in the measured spectra in Fig. 4. The reason for this assignment is that these peaks appear insensitive to the polarization direction and their intensities do not correlate as well with the calculations. The D peak at 219 cm⁻¹ is actually fairly close to the calculated 4th A₁ mode at 225.2 cm⁻¹ and is also predicted to have similar intensities in xx and yy tensors, so it could have a contribution from that mode as well as from DOS. The 5th A₁ mode at 306 cm⁻¹ is predicted to be weak and is not clearly seen. The nearby D peak at 283 cm⁻¹ is rather strong and therefore does not match mode 5 neither in position nor in intensity and is therefore labeled D. The 7th A₁ LO mode calculated at 528 cm⁻¹ is close to the 536 cm⁻¹ peak but the latter also corresponds to a peak in the DOS. This peak might be the only clearly distinct LO mode seen in the spectrum. However, in view of the fact that we do not see the other distinct LO modes (see below), we still consider this assignment tentative and assign this peak to be DOS-like instead. The predicted DOS in the range 500–700 cm⁻¹ could contribute to the background in this whole region and the highest peaks in DOS could contribute to the step-like feature in the experimental spectrum ending at about 850 cm⁻¹.

In Table II, we have assigned each LO mode to a corresponding TO mode. Strictly speaking, one cannot make such a one-to-one correspondence between TO and LO modes of the same symmetry. The LO modes result from diagonalizing a force constant matrix, which includes long-range forces resulting from the coupling of the Born effective charges to the electric field produced by the dipoles for longitudinal modes in the limit of the wavevector $q→0$⁠. These forces are absent for transverse modes. Since the frequencies with (LO) and without (TO) long-range forces result from diagonalizing different 12 × 12 matrices (including one zero eigenvalue) in the two cases, there is no a priori reason why the eigenvalues and eigenvectors should match up on a one-to-one basis. Nonetheless, there is a significant similarity in eigenvectors between most mode pairs and hence, in an approximate way, one could identify the modes according to the overlap between the LO and TO eigenvectors rather than simply ordering them according to increasing frequency. This similarity in their eigenvectors is manifested in the Raman tensor values, and the LO and TO mode correspondences were assigned accordingly. The Born effective charges obtained here agree with the ones reported in Paudel et al.⁵ to within an uncertainty of 0.1 due to the use of different pseudopotentials in the two cases.

The chosen LO-TO mode correspondences also become apparent when considering the calculated oscillator strengths of the modes, listed in Table II. The oscillator strength is proportional to the long-range electric field set up in the crystal as a result of the normal mode oscillation and so should be related to the size of the LO-TO splitting. Three of the eleven modes have significantly larger calculated oscillator strengths than the other modes and accordingly, the LO-TO shifts associated with these assigned mode pairs are the largest of the eleven mode pairs. We note that in crystals with inversion symmetry, the modes are either Raman active or IR active but not both. Although in the present case we do not have inversion symmetry, it is interesting to note that there still appears to be clear distinction between modes that are strong in IR and weak in Raman or vice versa. Calculated IR spectra were reported in Ref. 5, Fig. 1, and show indeed only three strong peaks in the $Im(ε)$ spectrum, corresponding to TO modes, and three corresponding peaks in the $−Imε−1$ spectra corresponding to LO modes. To further analyze the correspondence between TO and LO modes, we also examined the normal mode vibration patterns and these confirm the identification of the corresponding TO-LO mode pairs. Specifically, we note that the LO mode at 676 cm^–1 corresponds to the TO mode at 586 cm⁻¹ rather than to the 626 cm⁻¹ mode. These modes have thus crossed in the sense that if one were to vary the wave vector $k$ from normal to the plane to in-plane one would expect to see the LO-TO-mixed mode gradually change from longitudinal at 676 cm⁻¹ to transverse at 585 cm⁻¹ and cross the 626 cm⁻¹ mode.

Two of the three LO modes that have a significant LO-TO frequency splitting are not present in the measured spectra: the 668 cm⁻¹ and 839 cm⁻¹ calculated LO modes. The third one at 528 cm⁻¹ was already discussed to be close to a DOS peak and can therefore not be assigned unambiguously. Generally, the LO modes lie close to DOS peaks and could be hidden by them, but the DOS features in the 650–700 cm⁻¹ do not stand out above the background and the predicted LO peak at 839 cm⁻¹ lies definitely above the highest DOS peaks. We thus need to find another explanation for why these modes are not observed.

We hypothesize that these LO modes are suppressed due to an overdamped plasmon coupling. To gain insight, we compare our situation to a recent study of this effect in GaN,¹⁸ where the fairly low carrier mobility dampens the coupled LO-plasmon mode. In GaN doped with Si to a concentration of 2.5 × 10¹⁸cm⁻³, the LO A₁ mode is already broadened to the point where it cannot be separated from the background noise.¹⁸ In Kozawa et al.¹⁸ a plasmon broadening factor of 930 cm⁻¹ was extracted for the highest carrier concentration sample where the peak is still visible. This result corresponds to a carrier mobility of order 50 cm²/Vs. We may expect reasonably that a similar or even larger broadening and carrier concentration occur in our ZnGeN₂ samples. The carrier concentrations of these small ZnGeN₂ platelets could not be measured, but ZnGeN₂ rods grown using a similar method and at a similar temperature were measured and a carrier concentration of order 10¹⁹cm⁻³ was estimated.¹⁹ We have measured the spectra up to 1230 cm⁻¹ and did not find any indication of an upper LO-plasmon coupled peak in this range. This result indicates that the plasmon frequency could be even higher and outside the range measured which would imply either that the carrier concentration is larger than $2×1019$ cm⁻³ or that an even larger damping occurs. With a broadening factor γ of order 1000–2000 cm⁻¹ as obtained by Mohajerani et al.²⁰ for GaN nanorods, our simulations of the Raman spectra following the approach described by Kozawa et al.¹⁸ indicate that the upper branch is very weak and broad compared to the lower branch and likely not detectable. For a carrier concentration of order $5×1019$ cm⁻³, the plasmon mode would already be above 2000 cm⁻¹. The lower branches in this case are so close to the TO mode that they become indistinguishable from it.

With the scattering geometry used, in principle we expect to only see longitudinal A₁ modes. However, because we used a high numerical aperture objective, there was some portion of the incident light with wavevector in directions other than the z direction. Because of this situation we might expect to see some intensity from TO modes. For example, the small peak in the direction 4 spectrum at 760 cm⁻¹ could well be a trace of the strong TO $A1xx$ peak predicted to be at 776 cm⁻¹. Similarly, the small peak in the direction 1 spectrum at 580 cm⁻¹ could stem from the strong TO $A1yy$ peak predicted at 585 cm⁻¹. These two peaks are labeled with a T in the measured spectra of Fig. 4. The third TO peak which is predicted to have a large LO-TO splitting is located at 491.4 cm⁻¹, just to the higher energy side of the large peak at 478.6 cm⁻¹. In the measured spectrum in direction 4, this TO peak might be contributing to the slight asymmetry of this large feature at 473.0 cm⁻¹. These small traces of the TO Raman peaks observed further support our assignment of the crystallographic orientation of the measured platelet. As mentioned in the previous paragraph, the observation of these traces of the TO modes could also be explained by the fact that they coincide with the $L−$ lower branches of the plasmon coupled modes. Since the latter are so close to the pure TO modes, they cannot be distinguished in the present spectra. In the work on GaN rods by Mohajerani et al.,²⁰ the $L−$ modes could be distinguished from the pure TO mode as broad peaks (almost hidden as subtle changes in the background) thanks to careful fitting over a range of samples with systematically changing carrier concentration. Unfortunately, we are not yet in a position to carry out such a precise analysis in the present samples and thus our conjecture of the presence of LO-plasmon coupling is still indirectly based on the non-observation of LO peaks.

ZnGeN₂ was grown with a preferential c-oriented platelet type of crystal habit by exposing a Ge-Sn-Zn liquid to ammonia at 758 °C. Micro-Raman measurements were carried out on these platelets for different orientations of the polarization directions with respect to the crystallite orientation in the plane and with parallel or crossed incident versus scattered polarization. Combined with first-principles calculations these measurements allowed us to determine which were the x and y directions on the platelet and to measure the A₂ and A₁ Raman modes. Most of the predicted peak locations were within 5 cm⁻¹ of the measured values, and the predicted relative intensities agreed semi-quantitatively with the measured peaks. The LO peaks of the three modes that were predicted to have large LO-TO frequency splittings were not detected, likely because of over-damped coupling to plasmons. This result is consistent with the expected high level of unintentional n-type doping.

This work was supported by the National Science Foundation under Grant Nos. DMR-1006132, DMR-1409346 (E. B and K. K.), and DMR-1533957 (M. H. and W. R. L.). Calculations made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Case Western Reserve University.