Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 341-342

Section Noncentrosymmetric crystals

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: Noncentrosymmetric crystals

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Special care is required in treating the scattering by those optical phonons in the 21 noncentrosymmetric (polar) crystal classes (1, 2, m, 222, mm2, 4, [{\bar 4}], 422, 4mm, [{\bar 42m}], 3, 32, 3m, 6, [{\bar 6}], 622, 6mm, [{\bar 6m2}], 23, 432, [\bar 43m]) that are simultaneously infrared-active. Since these polar modes carry a nonzero macroscopic effective charge ([link], they contribute to the total polarization in the crystal, hence also to the macroscopic electric field, which in turn leads to a coupling between these modes. The polarization being a polar vector, the modes that contribute have the same symmetry character, i.e. they must also transform like the components of polar vectors.

An important consequence of the macroscopic field associated with polar modes in the crystal is the partial lifting of the degeneracies of the long-wavelength ([{\bf q}\approx 0]) mode frequencies (so-called TO–LO splitting). Since the macroscopic field in the crystal is longitudinal, it must be proportional to the longitudinal component of the polarization. Hence, the equations of motion for all polar modes carrying a nonzero longitudinal polarization (i.e. [{\bf P}_j\cdot{\bf q}\neq 0]) become coupled by the field and, consequently, their frequencies depend on the direction of q. This phenomenon is called directional dispersion and is connected with the fact that in the electrostatic approximation the dynamical matrix with long-range Coulomb forces shows non-analytic behaviour for [{\bf q}\to 0]. In lattice dynamics, the limit can be treated correctly by taking into account the retardation effects in the range where cq becomes comparable to [\omega{_j}({\bf q})], i.e. in the crossing region of free photon and optical phonon dispersion curves. As a result, one finds that for small q the true eigenmodes of the system – polaritons – have a mixed phonon–photon character and their frequencies show strong dispersion in the very close vicinity of [{\bf q}=0]. Experimentally, this polariton region is partially accessible only in near-forward Raman scattering [see ([link]]. For larger scattering wavevectors in the usual right-angle or back-scattering geometries, the electrostatic approximation, [cq\gg\omega], is well applicable and the excitations behave like phonons. Owing to the coupling via the longitudinal macroscopic electric field, however, the directional dispersion of these phonon branches remains.

Detailed analysis is complicated in the general case of a low-symmetry crystal with more polar modes (see e.g. Claus et al., 1975[link]). In crystals with at least orthorhombic symmetry, the principal axes of the susceptibility tensor are fixed by symmetry and for the wavevectors oriented along these principal axes the polar optic modes have purely transverse (TO) or longitudinal (LO) character with respect to the associated polarization. The character of a mode is usually mixed for a general direction of the wavevector.

Strictly speaking, conventional symmetry analysis in terms of irreducible representations of the factor group (point group) of the crystal, though giving a true description of polaritons at [{\bf q }= 0], cannot account for the lifting of degeneracies and for the directional dispersion of polar modes. A correct picture of the symmetries and degeneracies is, however, obtained by taking into account the finiteness of the wavevector q and classifying the vibrations according to the irreducible (multiplier) corepresentations of the point group of the wavevector [G{_0}({\bf q})], which is a subgroup of the factor group. Compatibility relations of the representations at [{\bf q}\to 0] can then be used to establish a correspondence between the two approaches.

The oscillating macroscopic field associated with long-wavelength LO polar modes acts as another source of modulation of the susceptibility. In addition to the standard atomic displacement contribution connected with the mechanical displacements of atoms, one also has to consider that the transition susceptibility also contains the electro-optic term arising from the distortion of electron shells of atoms in the accompanying macroscopic field E. Separating both contributions, we may write [\eqalignno{\delta \chi _{\alpha \beta }^{(j)}({\bf q}\approx 0,\omega _I) &= {{{\rm d}\chi _{\alpha \beta } }\over {{\rm d}Q_j }}Q_j &\cr&= {{\partial \chi _{\alpha \beta }}\over {\partial Q_j }}Q_j + {{\partial \chi _{\alpha \beta }}\over {\partial E_\gamma }}E_\gamma ^j &\cr&= \left({{{\partial \chi _{\alpha \beta } }\over {\partial Q_j }}+ {{\partial \chi _{\alpha \beta } }\over {\partial E_\gamma }}{{{\rm d}E_\gamma }\over {{\rm d}Q_j }}}\right)Q_j, &\cr&&(}]or, in terms of the Raman tensor, [\delta \chi = \textstyle\sum\limits_j {{\bf R}^j Q_j} = \textstyle\sum\limits_j {({\bf a}^j Q_j + {\bf bE}^j)} =\textstyle\sum\limits_j \left[{\bf a}^j + {\bf b}({\rm d}{\bf E}/{{\rm d}Q_j })\right]Q_j ,]where we introduce the notation [{\bf a}{^j}] and [{\bf b}({\rm d}{\bf E}/{\rm d}Q{_j})] for the atomic displacement and electro-optic contributions to the Raman tensor [{\bf R}{^j}]. As usual, [Q{_j}] stands for the normal coordinate of the jth mode and E for the total macroscopic electric field resulting from the longitudinal polarization of all optic modes. The modes that contribute to E are only LO polar modes; they transform as Cartesian components of polar vectors (x, y, z). Hence the electro-optic term contributes to the Raman cross section only if [{\bf E}{^j} = ({\rm d}{\bf E}/{\rm d}Q{_j})\neq 0], i.e. if the mode has at least partially longitudinal character. Hence, not only the frequencies but also the scattering cross sections of the TO and LO components of polar modes belonging to the same symmetry species are, in general, different.

Nevertheless, in view of the fact that the macroscopic electric field associated with LO polar phonons transforms in the same way as its polarization vector, the symmetry properties of both the atomic displacement and the electro-optic contributions to the Raman tensors of polar modes are identical. They correspond to third-rank polar tensors, which have nonzero components only in piezoelectric crystals. The symmetry-restricted form of these tensors can also be derived from Table[link] by combining the matrices corresponding to the x, y and z components. Note that these may belong to different irreducible representations in lower-symmetry classes (e.g. z cannot mix with x, y), and that in some uniaxial classes the z component is missing completely. Finally, in the noncentrosymmetric class 32 of the cubic system, the Raman tensors of the triply degenerate polar modes ([{\rm F}{_1}]) are purely antisymmetric; therefore all components of the piezoelectric tensor also vanish.

Example: To illustrate the salient features of polar-mode scattering let us consider a crystal of the [4mm] class, where of the Raman-active symmetry species the modes [{\rm A}{_1}(z)] and [{\rm E}(x,y)] are polar. According to Table[link], their ([{\bf q}=0]) Raman tensors are identical to those of the [{\rm A}_{1g}] and [{\rm E}{_g}] modes in the preceding example of a [4/mmm]-class crystal. Owing to the macroscopic electric field, however, here one has to expect directional dispersion of the frequencies of the long wavelength ([{\bf q}\approx 0]) [{\rm A}{_1}] and E optic phonon modes according to their longitudinal or transverse character. Consequently, in determining the polarization selection rules, account has to be taken of the direction of the phonon wavevector (i.e. the scattering wavevector) q with respect to the crystallographic axes. Since for a general direction of q the modes are coupled by the field, a suitable experimental arrangement permitting the efficient separation of their respective contributions should have the scattering wavevector q oriented along principal directions. At [{\bf q}\parallel{\bf z}], the [{\rm A}{_1}] phonons are longitudinal ([{\rm LO}{_\parallel}]) and both E modes ([2{\rm TO}_{\perp}]) are transverse, remaining degenerate, whereas at [{\bf q}\parallel{\bf x}] or [{\bf q}\parallel{\bf y}], the [{\rm A}{_1}] phonons become transverse ([{\rm TO}_{\perp}]) and the E phonons split into a pair of ([{\rm TO}{_\perp}], [{\rm LO}{_\perp}]) modes of different frequencies. The subscripts [\parallel] or [\perp] explicitly indicate the orientation of the electric dipole moment carried by the mode with respect to the fourfold axis ([4\parallel{\bf c}\equiv{\bf z}]).

Schematically, the situation (i.e. frequency shifts and splittings) at [{\bf q}\approx 0] can be represented by[\matrix{ & {\bf q}\parallel{\bf z}& {\bf q}\parallel{\bf x} &\cr &&-&{\rm A}_1({\rm TO}_\parallel)\cr {\rm A}_1({\rm LO}_\parallel) &- &&\cr &&-& {\rm E}_x({\rm LO}_\perp)\cr {\rm E}(2{\rm TO}_\perp)&-&-&{\rm E}_y({\rm TO}_\perp)\cr}]

For a general direction of q, the modes are of a mixed character and their frequencies show directional (angular) dispersion. The overall picture depends on the number of [{\rm A}{_1}] and E phonons present in the given crystal, as well as on their effective charges and on the ordering of their eigenfrequencies. In fact, only the [{\rm E}({\rm TO}_\perp)] modes remain unaffected by the directional dispersion.

Table[link] gives the corresponding contributions of these modes to the cross section for several representative scattering geometries, where subscripts TO and LO indicate that the components of the total Raman tensor may take on different values for TO and LO modes due to electro-optic contributions in the latter case.

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Raman selection rules in crystals of the 4mm class

Scattering configurationCross section for symmetry species
[{\bf q}\parallel{\bf z}] [z(xx)z, z(yy)z] [\sim|a_{\rm LO}|^2]
[{\bf q}\perp{\bf z}] [x(zz)x, x(zz)y] [\sim|b_{\rm TO}|^2]
[\bar y(xz)y, \bar x(yz)x] [\sim|f_{\rm TO}|^2]
[x'(zx')y', x'(y'z)y'] [{\textstyle{1 \over 2}}|f_{\rm TO}|^2 + {\textstyle{1 \over 2}}|f_{\rm LO}|^2]


Claus, R., Merten, L. & Brandmüller, J. (1975). Light scattering by phonon-polaritons. Berlin, Heidelberg, New York: Springer.

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