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

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 343-344

Section Electric-field-induced scattering

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: Electric-field-induced scattering

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Expanding the linear dielectric susceptibility into a Taylor series in the field, we write [\chi _{\alpha \beta }({\bf E}) = \chi _{\alpha \beta }(0) + {{\partial \chi _{\alpha \beta }}\over {\partial E_\gamma }}E_\gamma + {{\partial ^2 \chi _{\alpha \beta }}\over {\partial E_\gamma \partial E_\delta }}E_\gamma E_\delta +\ldots. \eqno (]The coefficients of the field-dependent terms in this expansion are, respectively, third-, fourth- and higher-rank polar tensors; they describe linear, quadratic and higher-order electro-optic effects. The corresponding expansion of the Raman tensor of the jth optic mode is written as [{\bf R}{^j}({\bf E}) =] [{\bf R}^{j0}] [+] [{\bf R}^{jE}{\bf E}] [+] [{\textstyle{1 \over 2}}{\bf R}^{jEE}{\bf EE}] [+] [\ldots].

Since the representation [\Gamma({\bf E}) = \Gamma_{\rm PV}], the coefficients of the linear term in the expansion for [\bf \chi], i.e. the third-rank tensor [b_{\alpha}{_\beta}{_\gamma} = (\partial {\chi}{_\alpha}{_\beta}/{\partial}E_{\gamma})], transform according to the reducible representation given by the direct product: [[{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}]{_S} \otimes {\Gamma}_{\rm PV}.]First-order field-induced Raman activity (conventional symmetric scattering) is thus obtained by reducing this representation into irreducible components [{\Gamma}(j)]. Higher-order contributions are treated analogously.

It is clear that in centrosymmetric crystals the reduction of a third-rank polar tensor cannot contain even-parity representations; consequently, electric-field-induced scattering by even-parity modes is forbidden in the first order (and in all odd orders) in the field. The lowest non-vanishing contributions to the field-induced Raman tensors of even-parity modes in these crystals are thus quadratic in E; their form is obtained by reducing the representation of a fourth-rank symmetric polar tensor [[{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}]{_S} \otimes [{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}]{_S}] into irreducible components [{\Gamma}(j)]. On the other hand, since the electric field removes the centre of inversion, scattering by odd-parity modes becomes allowed in first order in the field but remains forbidden in all even orders. In noncentrosymmetric crystals, parity considerations do not apply.

For completeness, we note that, besides the direct electro-optic contribution to the Raman tensor due to field-induced distortion of the electronic states of the atoms in the unit cell, there are two additional mechanisms contributing to the total first-order change of the dielectric susceptibility in an external electric field E. They come, respectively, from field-induced relative displacements of atoms due to field-induced excitation of polar optical phonons [Q{_p}({\bf E}) \sim {\bf E}] and from field-induced elastic deformation [{\bf S}({\bf E})={\bf dE}] (piezoelectric effect, d being the piezoelectric tensor). In order to separate these contributions, we write formally [\chi({\bf E}) = \chi({\bf E}, Q{_p}({\bf E}),{\bf S}({\bf E}))] and get, to first order in the field,[\eqalignno{\delta \chi ({\bf E}) &= (\partial \chi /\partial {\bf E}){\bf E}+ \textstyle\sum\limits_p (\partial \chi /\partial Q_p)Q_p ({\bf E}) + (\partial \chi /\partial {\bf S}){\bf S}({\bf E}) &\cr&= \textstyle\sum\limits_j {\bf R}^{jE}{\bf E}Q_j \hbox{, where we define}&\cr {\bf R}^{jE} &= (\partial {\bf R}^j / \partial {\bf E}) +\textstyle\sum\limits_p (\partial {\bf R}^j /\partial Q_p)({\rm d}Q_p/{\rm d}{\bf E})+ (\partial{\bf R}^j/\partial{\bf S}){\bf d}. &\cr&&(}]

The first term in these equations involves the susceptibility derivative [{\bf b}=(\partial\chi/\partial{\bf E})] at constant [Q{_p}] and S. The second term involves the second-order susceptibility derivatives with respect to the normal coordinates: [{\chi}^{(j,p)}=(\partial^2\chi/\partial Q_j\partial Q_p)=(\partial R^j_{\alpha\beta}/\partial Q_p)].  Since [Q_p({\bf E})\sim Z_{p\nu}E_\nu], where the quantity [{\bf Z}{_p}=(Z_{p \nu})] is the effective charge tensor ([link] of the normal mode p, its nonzero contributions are possible only if there are infrared-active optical phonons (for which, in principle, [{\bf Z}{_p}\neq 0]) in the crystal. The third term is proportional to the field-induced elastic strain [{\bf S}({\bf E})={\bf dE}] via the elasto-optic tensor [{\bf p}=(\partial\chi/\partial{\bf S})] and can occur only in piezoelectric crystals.

Example: As an illustration, we derive the matrix form of linear electric-field-induced Raman tensors (including possible antisymmetric part) in a tetragonal crystal of the 4mm class. The corresponding representation [[{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}] \otimes {\Gamma}_{\rm PV}] in this class reduces as follows: [\displaylines{[{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}] {_S} \otimes {\Gamma}_{\rm PV}=3{\rm A}{_1} \oplus {\rm A}{_2} \oplus 2{\rm B}{_1} \oplus 2{\rm B}{_2} \oplus 5{\rm E},\cr [{\Gamma}_{\rm PV} \otimes {\Gamma}_{\rm PV}] {_A} \otimes {\Gamma}_{\rm PV}={\rm A}{_1} \oplus 2{\rm A}{_2} \oplus {\rm B}{_1} \oplus {\rm B}{_2} \oplus 2{\rm E}.}]Suitable sets of symmetrized (s) and antisymmetrized (a) basis functions (third-order polynomials) for the representations of the 4mm point group can be easily derived by inspection or using projection operators. The results are given in Table[link]. Using these basis functions, one can readily construct the Cartesian form of the linear contributions to the electric-field-induced Raman tensors [{\bf R}{^j}({\bf E})={\bf R}^{jE}{\bf E}] for all symmetry species of the [4mm]-class crystals. The tensors are split into symmetric (conventional allowed scattering) and antisymmetric part.[\matrix{&\hbox{Symmetric}\hfill&\hbox{Antisymmetric}\hfill\cr {\rm A}{_1}:\hfill &{\pmatrix{ {a_1^{}E_z }&. & {a_2^{}E_x } \cr. & {a_1^{}E_z }& {a_2^{}E_y } \cr {a_2^{}E_x }& {a_2^{}E_y }& {b_1^{}E_z } \cr }}\hfill & +{\pmatrix{. &. & {a_3^{}E_x } \cr. &. & {a_3^{}E_y } \cr {- a_3^{}E_x }& {- a_3^{}E_y }&. }}\hfill\cr \cr{\rm A}{_2}:\hfill &{\pmatrix{. &. & {c_2^{}E_y } \cr. &. & {- c_2^{}E_x } \cr {c_2^{}E_y }& {- c_2^{}E_x }&. \cr }}\hfill & +{\pmatrix{. & {c_1^{}E_z }& {c_3^{}E_y } \cr {- c_1^{}E_z }&. & {- c_3^{}E_x } \cr {- c_3^{}E_y }& {c_3^{}E_x }&. \cr }}\hfill\cr {\rm B}{_1}: \hfill&{\pmatrix{ {d_1^{}E_z }&. & {d_2^{}E_x } \cr. & {- d_1^{}E_z }& {- d_2^{}E_y } \cr {d_2^{}E_x }& {- d_2^{}E_y }&. \cr }}\hfill& + {\pmatrix{. &. & {d_3^{}E_x } \cr. &. & {- d_3^{}E_y } \cr {- d_3^{}E_x }& {d_3^{}E_y }&. \cr }}\hfill\cr {\rm B}{_2}:\hfill&{\pmatrix{. & {e_1^{}E_z }& {e_2^{}E_y } \cr {e_1^{}E_z }&. & {e_2^{}E_x } \cr {e_2^{}E_y }& {e_2^{}E_x }&. \cr }}\hfill &+ {\pmatrix{. &. & {e_3^{}E_y } \cr. &. & {e_3^{}E_x } \cr {- e_3^{}E_y }& {- e_3^{}E_x }&. \cr }}\hfill\cr {\rm E}:\hfill & {\pmatrix{ {(f_1 + f_2)E_x }& {f_4 E_y }& {f_5 E_z } \cr {f_4 E_y }& {(f_1 - f_2)E_x }&. \cr {f_5 E_z }&. & {f_3 E_x } \cr }}\hfill &+ {\pmatrix{. & {g_4^{}E_y }& {g_5^{}E_z } \cr {- g_4^{}E_y }&. &. \cr {- g_5^{}E_z }&. &. \cr }}\hfill\cr &{\pmatrix{ {(f_1 - f_2)E_y }& {f_4 E_x }&. \cr {f_4 E_x }& {(f_1 + f_2)E_y }& {f_5 E_z } \cr. & {f_5 E_z }& {f_3 E_y } \cr }}\hfill& + {\pmatrix{. & {- g_4^{}E_x }&. \cr {g_4^{}E_x }&. & {g_5^{}E_z } \cr. & {- g_5^{}E_z }&. \cr }}\hfill}]

Table| top | pdf |
Symmetrized (s) and antisymmetrized (a) sets of trilinear basis functions corresponding to symmetry species of the 4mm class

SpeciesBasis functionsSymmetry
A1 [(x{_1}x{_2}+y{_1}y{_2})z{_3}]; [z{_1}z{_2}z{_3}]; [(x{_1}z{_2}+z{_1}x{_2})x{_3}] [+] [(y{_1}z{_2} +z{_1}y{_2})y{_3}] (s)
[(x{_1}z{_2}-z{_1}x{_2})x{_3}] [+] [(y{_1}z{_2}-z{_1}y{_2})y{_3}] (a)
A2 [(x{_1}z{_2}+z{_1}x{_2})y{_3}][(y{_1}z{_2}+z{_1}y{_2})x{_3}] (s)
[(x{_1}y{_2}-y{_1}x{_2})z{_3}]; [(x{_1}z{_2}-z{_1}x{_2})y{_3}][(y{_1}z{_2}-z{_1}y{_2})x{_3}] (a)
B1 [(x{_1}x{_2}-y{_1}y{_2})z{_3}]; [(x{_1}z{_2}+z{_1}x{_2})x{_3}][(y{_1}z{_2}+z{_1}y{_2})y{_3}] (s)
[(x{_1}z{_2}-z{_1}x{_2})x{_3}][(y{_1}z{_2}-z{_1}y{_2})y{_3}] (a)
B2 [(x{_1}y{_2}+y{_1}x{_2})z{_3}]; [(x{_1}z{_2}+z{_1}x{_2})y{_3}] [+] [(y{_1}z{_2}+z{_1}y{_2})x{_3}] (s)
[(x{_1}z{_2}-z{_1}x{_2})y{_3}] [+] [(y{_1}z{_2}-z{_1}y{_2})x{_3}] (a)
E [[(x{_1}x{_2}+y{_1}y{_2})x{_3},(x {_1}x{_2}+y{_1}y{_2})y{_3}]]; [[z{_1}z{_2}x{_3}, z{_1}z{_2}y{_3}]]; [[(x{_1}z{_2}+z{_1}x{_2})z{_3}, (y{_1}z{_2}+z{_1}y{_2})z{_3}]]; [[(x{_1}x{_2}-y{_1}y{_2})x{_3},] [-(x{_1}x{_2}-y{_1}y{_2})y{_3}]]; [[(x{_1}y{_2}+y{_1}x{_2})y{_3}], [(x{_1}y{_2}+y{_1}x{_2})x{_3}]] (s)
[[(x{_1}z{_2}-z{_1}x{_2})z{_3}, (y{_1}z{_1}-z{_1}y{_2})z{_3}]]; [[(x{_1}y{_2}-y{_1}x{_2})y{_3}], [-(x{_1}y{_2}-y{_1}x{_2})x{_3}]] (a)

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