Lattice dynamics of a quasi-2D layered TlCo₂Se₂ with a helical magnetic structure

Abstract

The vibrational properties of a quasi-two-dimensional layered TlCo₂Se₂ compound exhibiting a helical magnetic order were investigated using density functional theory and an approximation of the harmonic phonons. The helical magnetic structure of this ternary dichalcogenide was shown to be stabilized by phonons. The dynamical properties of the TlCo₂Se₂ lattice conform with the quasi-2D layered structure of this compound. The intensities of the Raman and infrared active phonon modes of TlCo₂Se₂ remained sensitive to magnetic interactions, suggesting a rather strong spin–phonon coupling within the Co–Se layers. A detailed analysis of the calculated phonon, Raman and infrared absorption spectra of TlCo₂Se₂ which was carried out with respect to the spectra of its ferromagnetic (KCo₂Se₂) and paramagnetic (KNi₂Se₂) counterparts enabled the essential differences between the dynamical properties of the quasi-2D layered transition-metal dichalcogenides to be elucidated.

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1 Introduction

The ternary layered compounds from the AT₂X₂ family (A = K, Rb, Cs, Tl; T = Fe, Co, Ni; X = S, Se, Te) have unique properties such as the superconductivity of Fe-based compounds^1,2 and the charge-density waves and heavy-fermion behavior of Ni-based ones.^3–5 These materials often exhibit interesting magnetic properties, which are attributed to their quasi-two-dimensional (quasi-2D) layered structure, which is shown in Fig. 1(a). The layers are built of edge-sharing TX₄ tetrahedra extending two-dimensionally in the tetragonal plane, while the A ions which are coordinated by eight X ions are located between the layers.

Fig. 1 (a) Tetragonal structure of the AT₂X₂ compounds (space group I4/mmm, no. 139). Wyckoff positions: A (2a), T (4d), and X (4e). (b) Schematic of the TlCo₂Se₂ magnetic helix with a turn angle Θ ∼ 121° running along the crystal c-axis.

In general, localized magnetism is absent in ANi₂X₂ compounds, whereas in AFe₂X₂ materials an interplay between the superconducting (SC) metallic state and the non-superconducting, insulating antiferromagnetic (AF) state is observed.² On the other hand, Co-based ternary dichalcogenides are metallic conductors showing mostly a ferromagnetic (FM) order.^6–10 There are, however, two compounds within the ACo₂X₂ system which order antiferromagnetically, namely CsCo₂Se₂ (ref. 11–13) and TlCo₂Se₂.^14–18 The latter remains unique among the Co-based ternary dichalcogenides as it exhibits an incommensurate helical magnetic structure^14,15 below T_N = 80–90 K, in which the Co spins order ferromagnetically within each layer containing Co ions (basal tetragonal plane) and the magnetic moments of the adjacent Co layers are rotated by about 121° with respect to each other. This leads to an incommensurate helix structure extending along the crystallographic c-axis, as is schematically shown in Fig. 1(b).

So far, TlCo₂Se₂ has been the subject of few theoretical studies which have focussed mainly on its electronic and magnetic structures.^15,16,18 This work, based on state-of-the-art density functional theory (DFT), has provided insight at the atomic level into the phonon dynamics of the ternary dichalcogenide TlCo₂Se₂, and supplements the existing experimental and theoretical information on the fundamental properties of this quasi-2D layered material which exhibits an unusual magnetic order. The results of this research may serve as a guide for future neutron inelastic scattering, Raman and infrared experiments.

2 Methodology

Calculations were carried out within the spin-polarized density functional theory (DFT) implemented in the VASP code.^19,20 Electron–ion interaction was represented by the projector augmented wave (PAW) method. The generalized gradient approximation with parametrization of Pedrew, Burke and Ernzerhof (GGA-PBE)^21,22 was applied for the exchange and correlation potential. Plane waves up to an energy cutoff of 351 eV were used as basis functions. Reference configurations for valence electrons were (6s² 6p¹) for Tl, (3d⁸ 4s¹) for Co, and (4s² 4p⁴) for Se. The incommensurate helical magnetic structure was approximated within the 1 × 1 × 3 supercell which consisted of 30 atoms and Θ = 120°, as indicated by previous experimental studies,¹⁴ and suggested an approximate repeat distance of 40 Å (3c). Lattice constants and internal atomic positions of this supercell were fully optimized with convergence criteria for the residual Hellmann–Feynman (HF) forces and the system’s total energy of 10⁻⁵ eV Å⁻¹ and 10⁻⁷ eV, respectively. The k-point mesh of 12 × 12 × 12, generated according to the scheme of Monkhorst–Pack, was used for the Brillouin zone integration. Phonon calculations were performed within the direct method approach^23,24 and harmonic approximation. The HF forces were obtained by displacing the symmetry non-equivalent Tl, Co and Se atoms from their equilibrium positions by ±0.02 Å in the 2 × 2 × 3 supercell containing 120 atoms. The HF forces were calculated with a reduced k-point mesh of 6 × 6 × 2. The total number of calculated displacements amounted to 12. The peak intensities of the non-resonant Raman spectrum (in Stokes process) were calculated from the well-known expression:²⁵ I ∝  |e_sRe_i|²ω⁻¹(n + 1), where (n + 1) is the population factor for Stokes scattering with n = [exp(ℏω/k_BT) − 1]⁻¹ denoting the Bose–Einstein thermal factor, e_i (e_s) is the polarization of the incident (scattered) radiation, R is the Raman susceptibility tensor. The components of R tensor (α_ij) were determined from derivatives of the electric polarizability tensor over the atomic displacements.^24,26,27 The electric polarizabilities were calculated²⁸ for each symmetry non-equivalent atom displaced from its equilibrium position by ±0.01 Å. The intensities of the peaks in the infrared (IR) absorption spectra were computed according to the formalism described in ref. 24, 29 and 30. The intensities of the infrared absorption spectrum of the transverse optical phonons having frequencies ω_T at the Brillouin zone center take on the form: where κ is the refractive index, Z* denotes the Born dynamical charge tensor, and j runs over the vibrational modes. Details of the calculations can also be found elsewhere.^31,32

3 Results and discussion

3.1 Structural and magnetic properties

The parameters of the TlCo₂Se₂ structure calculated at the ground state together with the results of the neutron diffraction measurements of powder¹⁴ and single-crystalline samples¹⁵ are summarized in Table 1. The optimized lattice constants (a, c) and the z_Se coordinate of the selenium atom closely correspond to those determined in the most recent experiments on TlCo₂Se₂ (ref. 14 and 15) and TlCo₂Se_2−xS_x (ref. 17) with x = 0 as well as to those determined in earlier experiments on Tl_1−xK_xCo₂Se₂ (ref. 6) and TlCo_2−xNi_xSe₂ (ref. 9) at x = 0. The largest deviation between the results of our DFT calculations and the experimental data (∼2.5%) was observed for the c lattice constant. This affects the interlayer distance d^inter_Co–Co which is overestimated in comparison with the results of measurements by the same amount. On the other hand, the difference between the calculated and experimental lattice constant a, which is below 1%, has a negligible effect on the intralayer Co–Co distance d^intra_Co–Co. The remaining interatomic distances i.e. d_Co–Se, d_Se–Se, d_Tl–Co, and d_Tl–Se, which also depend on z_Se, correlate with the experimental results within ∼1%.

Table 1 Structural parameters, interatomic distances, magnetic moment (M) on Co ion, spin spiral wave vector q = (0, 0, q_z), and the angle Θ between Co moment in adjacent layers determined for TlCo₂Se₂ at the ground state, from neutron diffraction of powder samples¹⁴ at 10 K and single crystals¹⁵ at 45 K

Quantity	Present	10 K	45 K
a (Å)	3.8265	3.8316	3.8315
c (Å)	13.7612	13.4254	13.4400
zSe	0.3471	0.3537	0.3537
d^intraCo–Co (Å)	2.7057	2.7094	2.7093
d^interCo–Co (Å)	6.8806	6.7127	6.7200
dCo–Se (Å)	2.3337	2.3682	2.3691
dSe–Se (Å)	3.8030	3.8851	3.8872
dTl–Co (Å)	3.9365	3.8646	3.8678
dTl–Se (Å)	3.4276	3.3464	3.3476
M (μB)	0.78	0.46	0.70
qz	0.337	0.324	0.309
Θ	119.4	121.7	124.4

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The calculated helical arrangement of spin is characterized by the propagating vector q = (0, 0, q_z) with q_z = 0.337 and the corresponding helical turn angle between the two adjacent planes containing Co ions Θ = 119.4°. Both theoretical values remain in quite good agreement with the respective experimental data^14,15 (see Table 1) and the results¹⁶ for the linear muffin-tin orbital method implementing atomic spheres approximation (LMTO-ASA) which gives q_z = 0.287 and Θ = 128.3°. There is, however, significant discrepancy between the magnetic moment on the Co ion gained from the present work (M = 0.78 μ_B) and that deduced¹⁴ from the neutron powder diffraction (NPD) experiments (0.46 μ_B). We note that significantly higher M = 0.72 μ_B has also been obtained in the LMTO-ASA calculations.¹⁶ On the contrary, both theoretical approaches produce magnetic moments conforming to the results of the neutron diffraction measurements on single-crystalline^15,16 TlCo₂Se₂ which yield M = 0.7 μ_B. We should mention that, a too low value of M found in the NPD experiments can result from weak magnetic contributions (low intensities of the magnetic peaks) to the powder diffraction pattern.¹⁴ Thus, the neutron diffraction of TlCo₂Se₂ single-crystals seems to provide more reliable values for the magnetic moments of Co ions as compared to the NPD experiments, this is also because the latter technique is unable to give sufficient spatial information.¹⁵

A non-commensurate helical magnetic structure with q_z ∼ 0.3 was shown to be more stable than both the ferromagnetic and antiferromagnetic solutions within the LMTO-ASA first principles calculations.¹⁶ Unfortunately, this result has not been confirmed by the full-potential augmented plane wave method (FP-APW + lo), in which the antiferromagnetic state minimizes the energy of the TlCo₂Se₂ system.¹⁵ Therefore, we have performed additional calculations for the system with the interlayer AF order and intralayer FM alignment of the Co spins, as suggested by some early experiments.^6,9 The calculated difference in energies between the TlCo₂Se₂ structure with the AF ordering of Co spins and the structure with Co spins conforming to the helical magnetic order ΔE₀ = E^AF₀ − E^HLX₀ amounts to 0.29 meV per formula unit (f.u.), indicating that the former is less energetically favourable than the latter one. Furthermore, the helical magnetic structure is additionally stabilized by phonons as the difference in the energies ΔE = ΔE₀ + ΔE_vib, which takes into account the zero-point energy motions of the respective lattices ΔE_vib = E^AF_vib − E^HLX_vib, becomes equal to 5.9 meV per f.u.

3.2 Phonon spectra

Fig. 2 displays the total and atomic-projected phonon densities of the states calculated for the helical magnetic structure of TlCo₂Se₂. The phonon spectrum which covers the frequency range up to about 33 meV can be divided into three bands. The lowest band (I) extends to about 10 meV and remains dominated by the vibrations of the Tl-sublattice, while the phonons associated with vibrations of the Co- and Se-sublattices are only a small admixture in this band. The bottom of band I corresponds mostly to the acoustic phonon modes, whereas the top of this band is constituted by the low-lying transverse optical (TO) modes.

Fig. 2 (a) Total phonon density of states and (b) partial densities of phonon states for the Tl, Co, and Se sublattices in TlCo₂Se₂ with helical magnetic structure.

Both the middle and the highest-frequency bands, which range respectively from 10 meV to 22 meV (band II) and from 22 meV to 33 meV (band III) are solely due to mixed vibrations of the Co and Se atoms within the CoSe₄ tetrahedral units. Transverse optical phonon modes are the main constituents of the band II. This band remains dominated by two intense peaks, both originating from vibrations of the Se and Co atoms along the c-axis i.e. in the direction perpendicular to the Co–Se layers, with the lower (higher) frequency peak arising from movements of Se (Co) atoms. Here, the phonon peaks resulting from atomic displacements within the Co–Se layers are significantly smaller in comparison with those perpendicular to the layers. Vibrations of the Co and Se atoms within the layers become enhanced in band III and they give rise to the high-frequency transverse and longitudinal optic (LO) modes. The LO phonons set up the top of band III.

The location of phonon bands in TlCo₂Se₂ is very similar to that found for its ferromagnetic counterpart³¹ KCo₂Se₂, nevertheless the phonons of the Tl-sublattice belong to band I, while phonons of the K-sublattices are confined to band II. In both of the Co-based ternary dichalcogenides no gap separating the middle-frequency band II from the high-frequency band III exists, unlike that in the Pauli paramagnetic KNi₂Se₂ (with negligible effective magnetic moment on the Ni-sublattice) where a small gap (∼2.5 meV) appears.³² Such a difference between the FM (or AF) compounds and the paramagnetic one is due to the spin–phonon interaction affecting the atomic bonding, and hence the force constants which determine the phonon dynamics of particular systems. We should mention that the spin–phonon interaction has a meaningful impact on the phonon frequencies, as was shown³³ for the antiferromagnetic Fe-based ternary dichalcogenides KFe₂Se₂ or K_xFe_2ySe₂.

The phonon DOS can be used to determine the Debye temperature (Θ_D) of the TlCo₂Se₂ compound by applying the relationship:³⁴ where k_B is the Boltzmann constant, E = ℏω, and with g(E) denoting the total phonon density of states. The resulted Θ_D = 135 K.

The phonon spectrum of TlCo₂Se₂ can be measured by inelastic neutron scattering (INS) experiments. However, a direct comparison of the calculated and the INS spectra requires evaluation of the so-called generalized phonon density of states (GDOS) i.e. the quantity which is usually extracted from the measured scattering function.³³ The GDOS differs from the bare phonon DOS because of different scattering efficiencies (σ/M) of the atomic species constituting particular sublattices of a system.³⁵ Here, σ and M are the total scattering cross section and atomic mass, respectively. For TlCo₂Se₂ the weighting factors amount to³⁶ 0.048 barn per a.m.u. for Tl, 0.095 barn per a.m.u. for Co, and 0.105 barn per a.m.u. for Se. Thus, the GDOS obtained according to relation: with c_i and g_i(ω) denoting respectively the concentration and partial phonon DOS of the i-th atom, represents predominantly the dynamics of the Co–Se layers. This results from particular contributions to the GDOS spectrum which amount to 42% for Co, 47% for Se and 11% for Tl sublattices. Therefore, a somewhat different INS experimental spectrum of band I should be expected because the high-intensity peaks yielded by vibrations of the Tl-sublattice would be considerably suppressed in comparison to those shown in Fig. 2.

3.3 Raman and infrared spectra

The primitive unit cell of the AT₂X₂ (TlCo₂Se₂) compound with the space group I4/mmm contains 5 atoms, and hence 15 phonon branches are expected for an arbitrary wave vector q. A factor group analysis also provides 15 modes at the Brillouin zone center. The zone-center phonon modes can be decomposed into the irreducible representations of the point group D¹⁷_4h as follows: A_1g ⊕ B_1g ⊕ 2E^(II)_g ⊕ 3A_2u ⊕ 3E^(II)_u. Among them 3 modes (A_2u ⊕ E^(II)_u) are lattice translational modes. Because of the presence of the inversion symmetry the modes are either Raman-active (gerade) or infrared (IR)-active (ungerade). The modes with symmetries A_1g, B_1g, and E_g are Raman-active, whereas modes A_2u and E_u are IR-active. The E_g and E_u remain doubly degenerate. There are no optically inactive (silent) modes at the Γ-point. Contributions of particular atomic sites to the zone-center phonons of the AT₂X₂ system are given in Table 2.

Table 2 Contributions of each atomic site to the Raman and IR-active phonons in the AT₂X₂ system

Atomic site	Irrep.
A (2a)	A2u ⊕ Eu
T (4d)	A2u ⊕ B1g ⊕ Eg ⊕ Eu
X (4e)	A1g ⊕ A2u ⊕ Eg ⊕ Eu

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In the Raman-active modes the A (Tl) ions are at rest as they are located at inversion centers. The A_1g and B_1g phonons involve vibrations of the T (Co)- and X (Se)-sublattices along the crystallographic c-axis, while vibrations of these sublattices within the tetragonal ab-plane give rise to the E_g modes, as schematically illustrated in Fig. 3. We may regard the phonons with symmetries of A_1g and B_1g as the breathing modes, whereas the E_g phonons as interlayer shear modes. The frequencies of particular Raman phonon modes predicted by our calculations for TlCo₂Se₂ are as follows: ω(E⁽¹⁾_g) = 13.77, ω(B_1g) = 19.88, ω(A_1g) = 25.66, ω(E⁽²⁾_g) = 31.42 meV.

Fig. 3 Atomic displacements associated with the Raman-active modes in the AT₂X₂ system (space group I4/mmm). Frequencies of the calculated Raman-active modes in TlCo₂Se₂ (in cm⁻¹): ω(E⁽¹⁾_g) = 111, ω(B_1g) = 160, ω(A_1g) = 207, ω(E⁽²⁾_g) = 253 (1 meV = 8.0585 cm⁻¹).

The Raman tensors for the particular phonon modes in the AT₂X₂ system have only the following non-zero components:

The polarization selection rules³⁷ for the point group D¹⁷_4h allow the polarized Raman scattering experiments to distinguish the single gerade from twofold degenerate ungerade modes. Thus, in the backscattering geometry, where the wave vector of incident (k_i) and scattered (k_s) radiations remain antiparallel (k_i = [001] and k_s = [001̄], k_i = [010] and k_s = [01̄0] or k_i = [100] and k_s = [1̄00]), the mode of A_1g symmetry always appears in the polarized backscattering Raman spectra when the incident light polarization vector (e_i) and the scattered light polarization vector (e_s) are parallel (e_i ∥ e_s), while it disappears for the crossed polarization configuration (e_i ⊥ e_s). The intensity of this mode depends on the angle ϕ between e_i and e_s i.e. I(A_1g) ∝ |a|² cos² θ. However, in the parallel polarization configuration, such as e_i ∥ x, where x = [100] is the crystal axis (for example z(xx)z̄ or y(xx)z̄ in Porto’s notation), the A_1g mode is assisted by the B_1g mode, the latter having an intensity dependent not only on the ϕ angle but also on the sample orientation defined by the angle ψ = ∠(e_i,x). The intensity of the B_1g phonon is given as I(B_1g) ∝ |c|² cos(ϕ + 2ψ). Therefore, in the parallel polarization configuration (ϕ = 0) and sample orientation ψ = 0 (e_i ∥ x), I(B_1g) is maximum, as shown in Fig. 4. At this scattering geometry, the I(A_1g) is constant and treated here as the scaling factor, while the I(B_1g) gradually decreases with increasing ψ and reaches null value at ψ = 45°. The calculated intensity ratio I(B_1g)/I(A_1g) = 1.376 for ψ = 0 in the parallel polarization configuration. An opposite behavior of the I(B_1g) is observed in the crossed polarization configuration (ϕ = 90°), where I(B_1g) gradually increases with increasing ψ and becomes maximal at ψ = 45°.

Fig. 4 Variation of the relative intensities I(B_1g)/I(A_1g) and I(B_1g) versus sample orientation ψ = ∠(e_i,x) in the parallel (ϕ = 0) and crossed (ϕ = 90°) polarization configurations. The parallel (crossed) polarization configuration, defined by ϕ = ∠(e_i,e_s), is indicated by circles and solid curve (squares and dashed curve). The maximal value of I(B_1g) at ϕ = 90° and ψ = 45° is taken as the reference.

The A_1g mode can be the only mode detected in the Raman spectrum measured at the y(zz)ȳ scattering geometry. One may also observe a single peak associated with the A_1g phonon by performing measurements from the crystal ab-plane, but such a configuration requires setting the incident light polarization along the crystal axis, and then one may work with the z(x′x′)z̄ scattering geometry. The Raman spectrum can also reveal a single phonon peak due to the B_1g mode in the crossed polarization configuration, such as z(x′y′)z̄, where Determination of the E_g symmetry phonon modes is possible when the Raman scattering measurements are carried out for the crystal ac-plane and a crossed polarization configuration, e.g. y(zx)ȳ scattering geometry.

Typical polarized Raman spectra of the TlCo₂Se₂ crystal simulated at the scattering configurations z(xx)z̄ and y(zx)ȳ are displayed in Fig. 5. The spectrum calculated at scattering geometry z(xx)z̄ shows two peaks which are attributed to the phonons with the B_1g symmetry (19.88 meV) and A_1g symmetry (25.66 meV). In TlCo₂Se₂ the B_1g phonon mode is more intense than the A_1g, whereas opposite effects are encountered in the ferromagnetic³¹ KCo₂Se₂ and paramagnetic^32,38 KNi₂Se₂, where the phonons of B_1g symmetries exhibit significantly lower intensities as compared to the phonons of A_1g symmetries. The E⁽¹⁾_g (13.77 meV) and E⁽²⁾_g (31.42 meV) which can be observed in the Raman spectrum of TlCo₂Se₂ at the y(zx)ȳ scattering geometry are shown to have lower and higher intensities, respectively, and the intensity ratio I(E⁽¹⁾_g)/I(E⁽²⁾_g) ∼ 0.24. However, for the ferromagnetic KCo₂Se₂ compound the intensity ratio of the E⁽¹⁾_g and E⁽²⁾_g modes is almost null, and hence its Raman scattering spectrum is characterized³¹ by a single peak arising from the higher frequency E⁽²⁾_g phonon mode. On the other hand, the E⁽²⁾_g mode is hardly observed in the Raman spectrum of the paramagnetic KNi₂Se₂ due to its low intensity and the spectrum shows a single peak resulting from the quite intense E⁽¹⁾_g mode having a lower frequency.^32,38

Fig. 5 Backscattering Raman spectra of TlCo₂Se₂ crystal calculated at scattering geometries (a) z(xx)z̄ and (b) y(zx)ȳ. Spectra were recorded at 50 K and with a laser excitation wavelength of 514.5 nm. Peaks are represented by Lorentzians with artificial FHWMs of 0.25 meV each.

The unpolarized Raman scattering spectrum calculated from the crystal ab-plane as well as the backscattering spectrum of the polycrystalline TlCo₂Se₂ (not shown) reveal two-peak structures originating from the B_1g and A_1g phonons. The I(B_1g)/I(A_1g) for monocrystalline and polycrystalline systems equal about 1.8 and 1.2, respectively. Although both E_g modes are also present in these spectra, they have incomparably weaker intensities compared to the strong B_1g and A_1g modes, and thus they are not expected to be detected experimentally.

In the AT₂X₂ system, the infrared active phonons of A_2u and E_u symmetries correspond to the oscillations of the dipole moment along and perpendicular to the T–X layers, respectively. These modes are schematically presented in Fig. 6. The respective frequencies of the IR-active modes for the TlCo₂Se₂ compound are as follows: ω(E⁽¹⁾_u) = 5.20, ω(A⁽¹⁾_2u) = 5.92, ω(E⁽²⁾_u) = 30.29, ω(A⁽²⁾_2u) = 31.98 meV. The A_2u and E_u modes can be observed in the IR absorption spectra when the incident radiation E is parallel and perpendicular to the crystallographic c-axis.

Fig. 6 Polarization vectors of IR-active modes in the AT₂X₂ system (space group I4/mmm). Frequencies of the calculated IR-active modes in TlCo₂Se₂ (in cm⁻¹): ω(E⁽¹⁾_u) = 42, ω(A⁽¹⁾_2u) = 48, ω(E⁽²⁾_u) = 244, ω(A⁽²⁾_2u) = 258 (1 meV = 8.0585 cm⁻¹).

In order to calculate the intensities of the absorption peaks associated with the transverse optical phonons of the A_2u and E_u symmetries, one needs to determine components of the Born effective charge (Z*) and dielectric (ε) tensors. In TlCo₂Se₂, both tensors consist of two independent components which are perpendicular and parallel to the crystal c-axis i.e. and (ε_⊥, ε_∥). Our computations produce ε (401.41, 93.95) and the Born dynamical charges of particular atomic species as follows: and The anisotropy between ε_⊥ and ε_∥ is about two times larger in TlCo₂Se₂ than in KCo₂Se₂, which is mainly due to the fact that the value of ε_∥ for TlCo₂Se₂ is more than two times smaller than that for KCo₂Se₂.³¹ Nevertheless, the structural features of these two ternary transition-metal dichalcogenides are similar, they exhibit a reversed relation between the perpendicular and parallel components of the Z* tensors. In TlCo₂Se₂ we found for Co and Se ions and for Tl ions, while in KCo₂Se₂ the respective relations are completely opposite. This effect may result from the magnetic interactions which are antiferromagnetic in TlCo₂Se₂ and ferromagnetic in KCo₂Se₂, but it has a negligible influence on the simulated IR absorption spectra which are found to be quite similar for both compounds.

The absorption spectrum of TlCo₂Se₂ (see Fig. 7), likewise that of KCo₂Se₂ displays small intensity A⁽¹⁾_2u and E⁽¹⁾_u peaks and high intensity A⁽²⁾_2u and E⁽²⁾_u peaks, independently of the scattering configuration. The strong A⁽²⁾_2u and E⁽²⁾_u peaks are almost ten times higher than the weak A⁽¹⁾_2u and E⁽¹⁾_u peaks. On the other hand, the unpolarized IR absorption spectra of either monocrystalline or polycrystalline TlCo₂Se₂ systems contain all of the IR-active modes, but only the E_u ones have intensities large enough for detection in experiments. Again, the absorption peak of the lower frequency E⁽¹⁾_u mode is about eight times smaller than the absorption peak of the higher frequency E⁽²⁾_u mode.

Fig. 7 (a) Polarized infrared absorption spectra of TlCo₂Se₂ crystal at E ∥ c (dashed curve) and E ⊥ c (solid curve) configurations. (b) Unpolarized infrared absorption spectra of TlCo₂Se₂ polycrystal. Peaks are represented by Lorentzians with artificial FHWMs of 0.25 meV each.

4 Summary and conclusions

The present work reports on the structural, magnetic and dynamical properties of the quasi-2D layered ternary transition metal dichalcogenide TlCo₂Se₂ with helical magnetic ordering. Results of the present theoretical research based on the DFT method indicate that the structure, with a spin spiral characterized by a propagation wave vector (0, 0, 0.337) and a turn angle of 119°, is more energetically stable than the structure with a conventional AF order of type-I. Moreover, the helical magnetic order of TlCo₂Se₂ is additionally stabilized by phonons which increases the energy difference between structures with helical and AF magnetic order. The phonon spectra determined from inelastic neutron scattering experiments are expected to provide information on atomic vibrations within and perpendicular to the Co–Se layers. The predicted Raman and IR absorption spectra which were analyzed in terms of the I4/mmm space group symmetries are supposed to reflect magnetic interactions within the Co–Se layers, where the spin–phonon coupling seems to be quite strong. Results of our theoretical studies could be useful for the interpretation of TlCo₂Se₂ spectra measured in future INS, Raman and IR experiments.

Acknowledgements

Interdisciplinary Center for Mathematical and Computational Modeling (ICM), Warsaw University, Poland and the IT4Innovations National Supercomputing Center, VSB-Technical University, Ostrava, Czech Republic are acknowledged for providing the computer facilities under Grants No. G28-12 and LM2015070.

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