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Electron–TO-phonon interaction in polar crystals
Physica B 406 (2011) 1586–1591
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Electron–TO-phonon interaction in polar crystals Aleksandr Pishtshev Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2010 Received in revised form 31 January 2011 Accepted 1 February 2011 Available online 22 February 2011
In the present work, the interaction between electrons and long-wavelength transverse optical (TO) vibrations in a polar insulator is considered. The crystal model employed for the theoretical analysis includes classical potentials and electronic polarizabilities. A significant enhancement of the strength of the electron–TO-phonon interaction in ferroelectrics has been found. A microscopic justification of this effect is given. A bridge that relates the interaction of electrons with the polar long-wavelength TO modes of lattice vibrations to the long-range dipole–dipole interaction is established. As an application of our analysis, a novel method for quantitative predictions of the electron–TO–phonon interaction constants in polar crystals via the data of experimental studies is suggested. & 2011 Elsevier B.V. All rights reserved.
Keywords: Electron–phonon interaction Optical lattice vibrations Polar crystals
1. Introduction In polar insulating compounds, there exist a number of the different lattice-dynamical aspects that are of great theoretical interest both because of their connection with fundamental dielectric properties and because they may directly be related to a unique combination of lattice and electronic degrees of freedom [1]. Considering displacements of the ions participating in longitudinal and transverse vibrations, it is quite natural to assume that just long-wavelength optical vibrations should be considered as a key factor for relating electronic and lattice (structural) properties. One main example among those, where longitudinal components of optical vibrational modes play a principal role, is polaron physics within the framework of which many impressive developments have been made (e.g., Refs. [2,3]). As regards transverse optical (TO) phonons, the vibronic theory (e.g., Refs. [4–7] and references therein) in which the interaction between band electrons and the zone-center polar TO lattice vibrations is the leading driving force of a ferroelectric instability represents the other important example. The studies of the interaction of electrons with longitudinal optical (LO) phonons are characterized by quite a long history— due to LO phonons a number of essential physical properties of insulators and semiconductors have been rigorously recognized and widely studied (e.g., Refs. [8–11]). At the same time, strictly speaking, many central aspects of the electron–TO–phonon (el–TO–ph) interaction have remained poorly understood. In particular, with only a few exceptions [12,13], no attempts have been made to consider the problem of determining the el–TO–ph
E-mail address: ap@eeter.fi.tartu.ee 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.02.003
interactions in polar crystals with a due account for their peculiar (polar) properties. Moreover, new developments in the understanding of different properties of oxide ferroelectrics and multiferroics [5,7,14–16] have emphasized the considerable practical importance of the theoretical studies of el–TO–ph interactions. At present, there remains a number of important issues, which should be investigated in detail, such as the microscopic explanation of a large strength of the el–TO–ph interaction in ferroelectrics or, which have not been addressed at all, such as an importance of the screening effects or a general question concerning the specific properties of the el–TO–ph interaction. For instance, unlike LO phonons, due to little theoretical knowledge about the el–TO–ph interaction, until the appearance of Ref. [17], it was typically difficult to realize quantitative predictions for the el–TO–ph interaction constants in polar compounds. In the present paper, using the characteristic parameters of a polar crystal in an explicit way, we investigate the main features of the el–TO–ph interaction. In particular, we give a microscopic justification of a significant enhancement of the el–TO–ph interaction strength in ferroelectric compounds. Within a first-principles methodology, we show how to link the interaction of electrons with polar long-wavelength TO phonons to the long-range dipole–dipole interaction. This, in turn, provides a nontrivial relationship between the el–TO–ph interaction constants and the macroscopic material parameters, such as dipole oscillator strengths and the forbidden gap. Being independent on the specific details of how the coupling was established, the relationship (which does not involve any adjustable parameters and requires only some appropriate amount of information usually available experimentally) gives us a practical framework for estimating the values of the el–TO–ph interaction constants in a polar material for which no data have been available so far. In a more general context, this leads to a deeper
A. Pishtshev / Physica B 406 (2011) 1586–1591
understanding of displacive structural instabilities because, by analyzing what factors related to material properties may be responsible for the strength of the el–TO–ph interaction, our results can be used to probe into the ferroelectric properties of polar insulators.
2. A model for electron–TO–phonon interaction As our focus is the el–TO–ph interaction, we are interested in finding the change in the electronic charge density induced by a polar TO vibration of a long wavelength. Following the work [12], we represent a polar crystal by a model based on classical potentials and contributions of the electronic polarization effects via the relative displacements of the electronic shells. Within the standard procedure of the dipole approximation (expanding to first order in the ionic and electronic displacements, entering into the q-representation, and going to the normal coordinates uqj), the perturbation of the host crystal due to the interactions with the polar lattice TO vibrations (qj) is given by [12] X tr dUext ðrÞ ¼ N 1=2 Vq j ðrÞ uq j , ð1Þ q,j
where Vq j ðrÞ ¼ v Pðq jÞ Fq ðrÞ, and Fq ðrÞ ¼ ie
4p X G ei v G a 0 jq þ Gj2
ðq þ GÞ r
:
ð2Þ
Pðq jÞ is an amplitude of the dipole polarization (dipole moment per unit volume) associated with the polar TO mode [1], G denotes the reciprocal lattice vector, v is the volume of the unit cell, and N is their number. The physical meaning of the quantity Fq ðrÞ is that in a polar crystal it defines the electric field at a point r inside the bulk as a part of the internal field associated with the TO phonon displacement [12]. This suggests a macroscopic characterization of the quantity Vq j ðrÞ of Eq. (1) as an electrostatic interaction of this electric field and the dipole polarization. Note that the scalar production Pðq jÞ G reflects the selection of the relevant transverse component of the optical vibrational modes splitting. The derivation of Eq. (1) accounts also for the assumption that in the long-wavelength limit one can take the corresponding Fourier transformation of the Coulomb potential to be independent of the ion’s position in the unit cell. The accuracy of this approximation was investigated previously in Refs. [12,13]. The important advantage of Eq. (1) is that it considers the changes in the ion properties that are caused by the relevant changes in the ionic environment, and, hence, in accordance with Ref. [18] contains, in principle, not only pairwise interactions via the electron–ion potential, but also many-body effects caused by electron correlations. This follows from the definition of the dipole polarization as [1,12] Pðq jÞ ¼ PðiÞ ðq jÞ þ PðeÞ ðq jÞ, where PðeÞ ðq jÞ is the electronic contribution which is connected with the distortion of the electronic charge distribution (e.g., Refs. [12,19]). The corresponding ionic contribution, PðiÞ ðq jÞ, is related to the displacements of the ions participating in pffiffiffiffiffiffi P the TO vibrations [1]: PðiÞ ðq jÞ ¼ v1 s ðZs = Ms Þwðs,q jÞ, where Zs denotes the effective charge [20] of an ion with the mass Ms placed in the cite s, and wðs,q jÞ is the polarization vector of the normal TO vibration. Following Ref. [21], we can say that the effect of the dipole polarization is to change the center of force between a ‘‘shell’’ of an ion’s electrons and the other force centers in the crystal. Keeping in mind correctness in the description of an insulated state, let us assume that the wave functions csk and energies Es ðkÞ are the set of one-particle Bloch eigenfunctions and eigenvalues corresponding to the band structure, which are determined within the framework of DFT-LDA methods with the correct accounting for the quasiparticle effects [22]. Eq. (1), providing a basis for studying the el–TO–ph interaction,
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corresponds in the second-quantization formalism to the Hamiltonian [4] XX Helph ¼ N 1=2 gssu ðk,q jÞ asþk asukq uq j , ð3Þ s, suj k,q
which characterizes the dynamic mixing of an electron state csk with another state csukq caused by the TO phonon. Here a + (a) are the creation (annihilation) operators for electronic states in the valence and conduction bands (s, su), gssu ðk,q jÞ denotes the matrix elements of the el–TO–ph interaction defined at the equilibrium high-symmetry configuration of the lattice by Z csk Vq j ðrÞ csukq dt: ð4Þ gssu ðk,q jÞ ¼ Expression (4) involves both interband ðs a suÞ and intraband ðs ¼ suÞ matrix elements. In the long-wavelength limit, the intraband matrix elements vanish due to inversion symmetry and, correspondingly, the coupling of electrons and TO lattice vibrations is completely characterized by interband matrix elements. In order to achieve the objectives of the present study, the further work is divided into two main steps: The first one is a comparative first-principles analysis of the contribution of Eq. (3) into the lattice dynamics of TO lattice vibrations; the second one handles the interband matrix elements, and develops the formulas that allow us to estimate el–TO–ph interaction strengths in terms of macroscopic material parameters for a wide range of polar insulators.
3. Electron–phonon interaction and TO lattice vibrations 3.1. On the dynamics of TO lattice vibrations Let us employ the microscopic formulation of the el–TO–ph interaction to express its contribution to the dynamics of the polar long-wavelength TO vibrations. Since the real part of the self-energy of the TO vibrational mode specifies the deviation of the unperturbed phonon states due to external perturbation, we represent the square of the TO phonon frequency O2q j as a sum of the unperturbed ~ 2q j and the additional term Do2q j which accounts for el–TO– term o ~ 2q j þ Do2q j . We ph coupling governed by Hamiltonian (3): O2q j ¼ o 2 ~ expect the ‘‘normal’’ part o q j , which may be regarded as the bare TO phonon frequency with respect to the el–TO–ph coupling, to be ~ 2q j free of softening anomalies. As usual [4,23], it is assumed that o also involves the contributions from the phonon–phonon interactions, which are needed to describe the dependence of the TO phonon frequency on temperature. From the results of Ref. [4] it follows that the electronic contribution Do2q j can be represented to the lowest-order in g2 as the corresponding harmonic term. The other important point is to account for the Coulomb interaction among electrons as well. Following the standard procedure (e.g., Refs. [4,24,25]) and using expressions for the matrix elements gssu ðk,q jÞ given by Eq. (4), we can write the equation for Do2q j as X Do2q j ¼ 4pv Bab ðqÞPa ðq jÞ Pb ðq jÞ, ð5Þ a, b
where Bab ðqÞ ¼
w^ el ¼
X v Ga vc ðq þGÞwel ðq þ G,q þ GuÞ vc ðq þGuÞ Gub , 4pe2 G,Gu a 0
^0 P 4pe2 , vc ðq þGÞ ¼ , ^0 ^ v^ c P vjq þGj2 I
1 XX fs ðkÞfsu ðkþ qÞ N s, su k Esu ðk þ qÞEs ðkÞ Z Z csk eiðq þ GÞr csukq dt csukq eiðq þ GuÞru csk dtu:
ð6Þ
ð7Þ
P0 ðq þ G,q þ GuÞ ¼
ð8Þ
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A. Pishtshev / Physica B 406 (2011) 1586–1591
Here the matrix elements of the microscopical dielectric susceptibility of the electron subsystem wel ðq þ G,q þ GuÞ are expressed in the random phase approximation in terms of the polarization ^ 0 [25,26], I^ is the identity matrix, v^ c is the bare Coulomb operator P interaction and f are the occupation numbers. With the help of expression (6), which is written in terms of the microscopic electronic dielectric susceptibilities and the Coulomb potentials, it is not difficult to confirm that Eq. (5) describes the corresponding electronic contribution to the square of the TO phonon frequency O2q j (i.e., according to the microscopic theory of lattice vibrations [27–29], the contribution which uniquely involves the wel ðq þ G,q þGuÞ matrix elements with both G and Gu different from zero). 3.2. The role of el–TO–ph interaction A formal decomposition being applied to the structure of Eq. (5) allows us to distinguish two conceptual factors contributed to Do2q j . The first one is the ‘‘kernel’’ Bab ðqÞ. The other factor, which is represented by the product of two polarizations Pa ðq jÞPb ðq jÞ, plays a role as a term of the model interaction similar to a dipole–dipole one. If now we let q ¼0, then according to Refs. [30,31] this similarity certainly implies that in the longwavelength limit we may put Eq. (5) on the same footing as an effective contribution of the long-range dipole forces. Let us consider the particular case of q¼0 in further detail. It is convenient to employ the following relations for the polarization components [1,12]: PðiÞ ð0jÞ ¼
3
e1 þ 2
Pð0jÞ,
PðeÞ ð0jÞ ¼
e1 1 Pð0jÞ, e1 þ 2
ð9Þ
where e1 is the electronic (static-high-frequency) dielectric permittivity. Recall that in a polar insulator, in the limit of zero wavevector, the displacement of charges from their equilibrium positions creates dipoles, which interact with long-range forces [1]. Moreover, as it was argued in Ref. [32], the dipoles in a crystalline medium are not represented simply by the ionic polarizations P(i)(0j), but due to the dipole–dipole interaction they acquire an external moment which transforms P(i)(0j) into P(0j) as Pð0jÞ ¼ ½ðe1 þ2Þ=3PðiÞ ð0jÞ with the corresponding proportionality coefficient given by the local-field factor (see Eqs. (9)). Thus, expressing the product Pa ð0jÞPb ð0jÞ in terms of the relevant ionic components and taking into account Ref. [32], we can classify Eq. (5) at q ¼ 0 as a contribution of a long-range dipole–dipole interaction. In order to clarify this correspondence, let us determine whether the result given by Eqs. (5) and (6) in the longwavelength limit matches its counterpart, which is caused by the long-range dipole–dipole interaction [30,31]. For this purpose, we compare Eqs. (5) and (6) at q ¼0 with the corresponding formulas of the lattice dynamics theories of polar insulators [1,20,27,30,31]. As we shall see below, such a comparison procedure also gives us unique ability to establish a bridge that relates the interaction of electrons with the polar long-wavelength TO modes to the long-range dipole–dipole interaction. As a measure of the dipolar activity of the given zone-centre TO vibration, let us introduce the dipole oscillator strength Sab ðjÞ defined by Sab ðjÞ ¼ 4pvPa ð0jÞPb ð0jÞ. The classical description of Sab ðjÞ is well known from the previous theoretical studies of lattice dynamical models (e.g., Refs. [1,20,31]). In particular, the quantities Sab ðjÞ correspond to the contributions of the TO vibrational modes in the infrared (IR) part of the Lorentz model of the dielectric matrix (e.g., Refs. [12,27,33]). In view of the definitions of Sab ðjÞ and PðiÞ ð0jÞ, it is convenient to express Sab ðjÞ with the help of Eqs. (9) as follows: ! ! X wb ðt,0jÞ 4pe2 X wa ðs,0jÞ Zs pffiffiffiffiffiffi Zt pffiffiffiffiffiffi Sab ðjÞ ¼ : ð10Þ v Mt Ms s t
Here the result of each such summation in the round brackets defines the TO mode effective charge [34], Zns are the Born transverse dynamical effective charges determined by [1,30,35] Zs ¼ ½ðe1 þ 2Þ=3Zs . At q ¼0, Eq. (5) can be rewritten in terms of Sab ðjÞ as X Do20j ¼ Bab ð0Þ Sab ðjÞ, ð11Þ a, b
where the tensor Bab ð0Þ is defined by X v Ga vc ðGÞwel ðG,GuÞvc ðGuÞ Gub Bab ð0Þ ¼ 4pe2 G,Gu a 0
ð12Þ
and wel ðG,GuÞ ¼ limq-0 wel ðq þ G,q þ GuÞ. Introducing according to Ref. [36] the representation of the susceptibility function wel ðG,GuÞ in terms of matrix elements of the microscopic polarizP ability tensor agZ ðG,GuÞ as wel ðG,GuÞ ¼ gZ Gg agZ ðG,GuÞ GuZ , the evaluation of Eq. (12) becomes Bab ð0Þ ¼
GuZ Gub 4pe2 X X Ga Gg agZ ðG,GuÞ : v G,Gu a 0 g, Z jGj2 jGuj2
ð13Þ
What is important to note here is that Eq. (13) provides useful information about the basic role of Bab ð0Þ: it emphasizes not only the importance of the polarizability effects but also it well illustrates how the quantity Bab ð0Þ is responsible for the softening of the zone-center TO vibration mode. In Eq. (11), the dipole oscillator strengths Sab ðjÞ serve as a direct mediator of the long-range dipole forces. The only puzzle that remains is how to efficiently reproduce a result of summations in Eq. (13) in order to calculate Bab ð0Þ. One simple and quicker way to find the proper evaluation of Bab ð0Þ is by comparison of Eq. (11) to a similar expression which is obtained from the same ‘‘source’’, Eq. (1), by means of the macroscopic formalism [1,12]. As a comparison parameter, it is reasonable to choose the product Pa ð0jÞPb ð0jÞ due to its ability to measure the long-range dipole–dipole interaction [32]. By using the corresponding result of Ref. [12], ‘‘macroscopic’’ counterpart of Eq. (11) for the case of the cubic (or tetrahedral) lattice symmetry can be represented in terms of the product Pa ð0jÞPb ð0jÞ (or Sab ðjÞ) as follows: X dab X dab Pa ð0jÞPb ð0jÞ ¼ S ðjÞ: e þ 2 e þ 2 ab a, b 1 a, b 1
Do20j ¼ 4pv
ð14Þ
A one-to-one correspondence between Eqs. (11) and (14) provides the resulting expression for the quantity Bab ð0Þ in the following form: Bab ð0Þ ¼ dab =ðe1 þ 2Þ:
ð15Þ
Note that Eq. (15) reproduces exactly the same result that was previously found in Ref. [31]. Now it is obvious that Eq. (11) is properly consistent with the previous studies. A comparative analysis of the obtained results allows us to draw several general conclusions as follows. (i) Following Ref. [37], with the help of Eqs. (10) and (13) one can identify Eq. (11) as the long-range Coulomb contribution resulting from the induced dipole interactions through the electronic polarizability. (ii) Eqs. (11) and (15) reproduce the similar results derived within the exact accounting of the dipole–dipole interaction in TO vibration mode dynamics [31]. (iii) In the macroscopic sense, the physical meaning of the expression represented by the product of three factors Bab ð0ÞPa ð0jÞPb ð0jÞ is determined by the regular (at q¼ 0) contribution of the long-range dipole forces. Accordingly, this implies the explicit correspondence between the model specified by Eqs. (1)–(4) and the first-principles approach proposed and developed for polar crystals in Refs. [30,31]. The last two conclusions represent a novel result of principal importance: we showed and confirmed the full correspondence
A. Pishtshev / Physica B 406 (2011) 1586–1591
between the results of two descriptions of the dynamics of the zone-centre TO vibrational mode—one in terms of the el–TO–ph interaction and the other one in terms of the regular at the q ¼0 part of the long-range dipole–dipole interaction [30,31]. In particular, Eq. (11) not only demonstrates the equivalence of the interpreting power of both descriptions, but also can serve as a link between the two approaches. Thus, our findings give a clear understanding of the role of the el–TO–ph interaction in the conversion of the zone-centre TO vibrational mode O0j to the ferroelectric soft-mode. First, the interaction of electrons with polar zone-centre TO phonons is directly associated with the long-range Coulomb interaction. In particular, Eq. (11) shows how the changes in the electron density (or the changes in the electronic states) induced by the local displacements are sufficient to yield a considerable softening of the TO phonons around the G point (note that Bab ð0Þ o 0). Second, the magnitude of the destabilizing contribution determined by Eq. (11) is highly dependent on dipole oscillator strengths. The latter reflects the fact that the zone-centre TO vibrational modes may exceptionally strongly be coupled to valence band electrons.
4. On macroscopic features of electron–TO–phonon interaction in ferroelectrics 4.1. Parametrization of electron–TO phonon interaction at q ¼ 0 The microscopic approach considered above is quite general; its distinguishing feature is that it allows us to treat the electronic correlations on the same footing as the electron–phonon interaction. This is very important (and, as we shall see below, necessary) for determining the electron–TO–phonon interaction constants accurately. However, even in the long-wavelength limit, Eqs. (5)–(8) are rather complicated for a further theoretical treatment. In the present section, we shall derive a reduced model of el–TO–ph interaction, which may provide analytical solutions and, on the other hand, may help to improve the applicability of the vibronic theory as well. Let us pass to the limit q-0 in Eqs. (5)–(8). The main complication in evaluating the matrix elements of wel ðq þ G,q þ GuÞ at q ¼ 0 is that one needs to exclude the irrelevant contributions associated with the electronic displacements (since these contributions are by 2 definition already incorporated into gss u ). Let us take into account that the dielectric susceptibility w^ relates a variation of the electronic density dr^ to the relevant perturbation potential dv^ ext as [25,38]
dr^ ¼ w^ dv^ ext , e^
1
¼ I^ þ v^ c w^ ,
ð16Þ
1
where e^ denotes the inverse dielectric matrix. Then the problem is solved in two steps: (i) by finding a change of the electronic charge density associated with interaction (1) and (ii) by applying Eqs. (16) in a similar way as was described in Ref. [30] regarding the analysis on the internal electric field effects. Acting in this manner, and taking into account Eqs. (9), we obtain that in the long-wavelength 0 limit, the reduced susceptibility w^ el turns out to be related to w^ el by the following equation: q-0 :
3 w^ el ¼ w^ 0 : e1 þ 2 el
ð17Þ
Assuming that in insulators ðfs ðkÞfsu ðkÞÞ ¼ 1, we can now write the long-wavelength limit of Eqs. (5)–(8) in terms of the constants of the j el–TO–ph interaction gssu ðk,0jÞ gss u ðkÞ as follows:
Do20j ¼
j 3 1 X X jgssu ðkÞj2 , e1 þ 2 N s a su k jEsu ðkÞEs ðkÞj
ð18Þ
1589
X v2 X Pa ð0jÞ Pb ð0jÞ Ga vc ðGÞvc ðGuÞGub 2 e a, b G,Gu a 0 Z Z csk eiGr csuk dt csuk eiGuru csk dtu:
j 2 jgss u ðkÞj ¼
ð19Þ
A remarkable property of Eq. (18) is that it can be represented j as a product of two independent factors ðl Þ2 P 0 ð0Þ, one of j 2 1 j 2 which, ðl Þ ¼ jg j ½ðe1 þ2Þ=3 , characterizes an effective interaction of electrons with polar zone-centre TO vibrations, the other one, P 0 ð0Þ, is the so-called non-interacting susceptibility. Following the vibronic theory prescription (e.g., Ref. [4]), this allows us to constitute a reduced one-parameter version of the el–TO–ph Hamiltonian (3) in the following model form: X X j ðredÞ Helph ¼ N 1=2 lssu ðkÞ asþk asuk u0j : ð20Þ s a su
k
Thus, in the calculations outlined above, we have performed a mapping of the original microscopic model (3) onto the effective model given by Eq. (20), which permitted an accurate incorporation of Coulomb renormalization effects. As we can see, the el– TO–ph interaction presented in Eq. (18) is already renormalized due to the interelectronic response (screening): the square of the screened el–TO–ph interaction constant is inversely related to the local field factor ðe1 þ2Þ=3. Since local fields and their effects become more prominent in compounds with a mixed ionic– covalent character of chemical bonds (e.g., Refs. [25,39,40]), this implies an important improvement in accuracy of determining the constants of the el–TO–ph interaction for polar materials. We refer to Eq. (20) as a vibronic-type model, since it represents the long-range electron-lattice dynamic hybridization of the electronic bands of opposite parities, which takes into account the relevant s-, p- and d-channels (e.g., Refs. [4–7] and references therein). For the last several decades, the model of type (20) was a useful prototype for intensive studies of various properties of both typical insulating perovskites [5,14,41–44] and the AIVBVI narrow-gap semiconductors and their alloys [45–49]. For instance, as applied to a family of the ferroelectric ABO3 perovskite oxides, hybridization (20) involves a significant mixing between the filled O 2p and the empty d0 (Ti4 + , Nb5 + , Ta5 + , W6 + , etc.) electronic states caused by the IR-active TO F1u soft vibrations [5,15,41]. 4.2. Evaluation of electron–TO–phonon interaction at q¼0 Let us explain why one can expect significant enhancement of the el–TO–ph interaction strength in ferroelectric compounds. By analogy with the scattering of an electron due to LO phonons [12], we first examine the expression for the amplitude of the interband el–TO–ph interaction defined by jgssu ðkÞj ¼ ðlimq-0 jgssu ðk,q jÞj2 Þ1=2 . Fixing j, using Eq. (4) and the macroscopic expression for the dipole oscillator strength of the chosen TO mode [12], we can obtain the following equation: rffiffiffiffiffiffiffiffiffi ve1 jgssu ðkÞj ffi DOLT ð21Þ j/skjF0 ðrÞjsukSj, 4p where, according to Ref. [50], we introduced (for the given polarization) the corresponding difference (the LO–TO splitting) DOLT ¼ ðO2LO O2TO Þ1=2 between the longitudinal ðOLO Þ and the transverse ðOTO Þ frequencies (i.e., the difference in long-range fields given by longitudinal and transverse modes as q-0Þ. The question is now under which conditions does a material have a strong el–TO–ph interaction. First of all, let us show that the dynamic hybridization described by Eq. (3) leads to distortions of the charge density distribution, which are caused by the internal electric field F0(r). From Eq. (2) it follows that for the central point q¼0 the quantity F0(r) satisfies the following
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equation (l denotes lattice vectors): ! X e F0 ðrÞ ¼ = ¼ = fe ðrÞ: jrlj l
ð22Þ
In accordance with Refs. [51,52], Eq. (22) defines a field generated by the Coulomb electrostatic potential fe ðrÞ which is connected (at given boundary conditions) through the Poisson equation with P a charge distribution dre ðrÞ ¼ e l dðrlÞ. Consequently, changes in the electron charge densities are represented by the relevant distortions owing to the electric field gradients at certain lattice sites of a polar crystal. On the other hand, the field F0(r) provides, due to Eq. (22), the long-range character of the el–TO–ph coupling at the G point. Secondly, looking at Eq. (21), we can see that the magnitude of the splitting DOLT serves as an enhancement factor of the matrix elements (integrals) associated with F0(r). Therefore, in the longwavelength limit, the el–TO–ph interaction in polar crystals is distinguished from that of most other insulators by the following features: (i) it is long-range, (ii) it is controlled by the internal electric field and (iii) it is essentially sensitive to the values of the LO–TO difference (i.e., the polar strength of long-wavelength optical vibrational modes). Hence, it is clear that Eq. (21) can be viewed as indicative of the strength of the el–TO–ph coupling in polar materials. In particular, one can see that an essential prerequisite for manifesting a large magnitude of jgssu ðkÞj is directly determined by the great values of the LO–TO splitting DOLT . The latter is a general feature that is observed for ferroelectric materials [34]. For example, the experimental difference between the corresponding LO and TO modes in the ferroelectric BaTiO3 is about 530 cm 1 [53] to be compared with the values of 100, 69, and 48 cm 1 obtained for the differences in the compounds NaCl, KCl, and RbCl, respectively [54]. In a polar lattice, the LO–TO splitting depends generally on Born’s transverse dynamical effective charges Zn of the lattice ions and the screening of the Coulomb interaction as [34]: DO2LT pjZ j2 =e1 . Recall that the nature of these charges is associated with the existence of large dipole–dipole interactions [55]. Thus, one can conclude that the presence of anomalously large Born’s effective charges is the key signature that the interband el–TO–ph coupling is particularly strong in a ferroelectric material. This result corresponds to the main assumption of the vibronic theory [4–7] suggesting the existence of sufficiently strong interband el–TO–ph coupling in displacive ferroelectrics. Hence, the strength of the interband el– TO–ph interaction can serve as a direct indicator which measures how close a crystal lattice is to a possible ferroelectric instability. Further details concerning with the quantitative characteristics of the electron–TO–phonon coupling are given in our separate paper [17]. 4.3. Electron–TO–phonon interaction via macroscopic parameters As shown in the present work, one can associate the interaction between electrons and polar long-wavelength TO phonons with the long-range dipole-dipole interaction. This link, which was not anticipated by previous theoretical considerations, helps us to express the strength of the el-TO-ph interaction via macroscopic parameters of a polar crystal, which in turn can be deduced from the analysis of the relevant experimental data. Let us introduce the bare constant g0j which represents the effective k-independent interband el–TO–ph interaction defined by a proper average of the squared el–TO–ph matrix elements over the given electronic states: 2 g0j ¼
X Mj X j Eg , jgssu ðkÞj2 jEsu ðkÞEs ðkÞj su 4 s 2N k
ð23Þ
where Mj is the reduced mass of the TO vibration and Eg is a bondgap energy. According to Eq. (18), accounting for the local-fieldseffects-induced partial screening of the bare el–TO–ph interaction is described by the renormalization: 2 2 g0j !g 2j ¼ 3g0j =ðe1 þ2Þ:
ð24Þ
On the other hand, according to Eqs. (11) and (15), the contribution of the linear el–TO–ph interaction to the square of the zonecentre TO vibrational mode frequency O20j can be decomposed in two factors, the first of which is Bab ð0Þ, which accounts for the corresponding electronic contribution, and the other one, Sab ðjÞ, represents the dipole oscillator strength of the given TO vibration. The bare constant of the el–TO–ph interaction g0j can thus be determined from matching Eqs. (11) and (15) together. As a P result, we obtain the following relationship ðSðjÞ ¼ a Saa ðjÞÞ: 2 1 ¼ 12 Eg Mj SðjÞ: g0j
ð25Þ
Eq. (25) is an important result of the present work. For each zone-center TO vibration of the branch j and of the reduced mass Mj, it relates the bare constant of the corresponding el–TO–ph coupling at the G point, g0j, to the macroscopic material constants, the values of which can be obtained from the experiment. Taking from the IR spectra analysis the relevant information about the IR-active TO phonons and the dielectric function behavior, and using Eqs. (25) and (24) in combination with the experimental data for e1 and Eg, we can directly evaluate the el–TO–ph coupling constants for polar materials of interest. A more detailed demonstration of the practical usefulness of Eq. (25), together with the numerical results for a wide number of selected polar insulators and semiconductors, can be found in our separate publication [17]. In particular, the calculations of the interband el–TO–ph interaction constants and the further comparative analysis showed that the large interband el–TO–ph interaction is a special microscopic feature of ferroelectric materials. In contrast, in non-ferroelectrics, as it has been demonstrated, the strength of the el–TO–ph interaction is not necessarily high enough due to their lower polar nature.
5. Conclusions In this paper, focusing mostly on the fundamental contribution of the electron subsystem to the dynamics of polar long-wavelength TO vibrations and using a first-principles methodology, we provided a systematic description of the el–TO–ph interaction in a polar insulator. The study is based on the model of a polar crystal with classical potentials, which takes into account electronic polarizability effects. By analyzing the electronic contribution to the TO vibrational mode in terms of the el–TO–ph coupling, we established a bridge which allowed us (a) to link the model under consideration to the microscopic lattice dynamics and (b) in the long wavelength limit, to relate the interaction of electrons with polar TO vibrations to the long-range dipole–dipole interaction. Our results highlight the importance of the el–TO–ph interaction for the genesis of the longwavelength TO vibrations in ferroelectrics, thereby giving a fundamental support at the microscopic level for the applicability of the vibronic theory. Within a first-principles methodology, we found and explained the significant increase of the constants of the el–TO–ph coupling in ferroelectric materials, we showed how the el–TO–ph interaction constants might be dependent on macroscopic material parameters, and we obtained the analytical equations allowing us to estimate the el–TO–ph interaction strengths in a wide range of polar insulators. In particular, the effective charge of the zone-center TO vibrational mode can be considered to be the key macroscopic parameter of the el–TO–ph coupling strength. In the materials where the el–TO–ph coupling is operative, this can be
A. Pishtshev / Physica B 406 (2011) 1586–1591
verified by the analysis of the corresponding IR spectroscopic measurements.
Acknowledgments The author would like to thank N. Kristoffel for helpful discussions, A. Sherman for attention to this work. The work was supported by the Estonian Science Foundation Grants No. 6918 and No. 7296.
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Physica B journal homepage: www.elsevier.com/locate/physb
Electron–TO-phonon interaction in polar crystals Aleksandr Pishtshev Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2010 Received in revised form 31 January 2011 Accepted 1 February 2011 Available online 22 February 2011
In the present work, the interaction between electrons and long-wavelength transverse optical (TO) vibrations in a polar insulator is considered. The crystal model employed for the theoretical analysis includes classical potentials and electronic polarizabilities. A significant enhancement of the strength of the electron–TO-phonon interaction in ferroelectrics has been found. A microscopic justification of this effect is given. A bridge that relates the interaction of electrons with the polar long-wavelength TO modes of lattice vibrations to the long-range dipole–dipole interaction is established. As an application of our analysis, a novel method for quantitative predictions of the electron–TO–phonon interaction constants in polar crystals via the data of experimental studies is suggested. & 2011 Elsevier B.V. All rights reserved.
Keywords: Electron–phonon interaction Optical lattice vibrations Polar crystals
1. Introduction In polar insulating compounds, there exist a number of the different lattice-dynamical aspects that are of great theoretical interest both because of their connection with fundamental dielectric properties and because they may directly be related to a unique combination of lattice and electronic degrees of freedom [1]. Considering displacements of the ions participating in longitudinal and transverse vibrations, it is quite natural to assume that just long-wavelength optical vibrations should be considered as a key factor for relating electronic and lattice (structural) properties. One main example among those, where longitudinal components of optical vibrational modes play a principal role, is polaron physics within the framework of which many impressive developments have been made (e.g., Refs. [2,3]). As regards transverse optical (TO) phonons, the vibronic theory (e.g., Refs. [4–7] and references therein) in which the interaction between band electrons and the zone-center polar TO lattice vibrations is the leading driving force of a ferroelectric instability represents the other important example. The studies of the interaction of electrons with longitudinal optical (LO) phonons are characterized by quite a long history— due to LO phonons a number of essential physical properties of insulators and semiconductors have been rigorously recognized and widely studied (e.g., Refs. [8–11]). At the same time, strictly speaking, many central aspects of the electron–TO–phonon (el–TO–ph) interaction have remained poorly understood. In particular, with only a few exceptions [12,13], no attempts have been made to consider the problem of determining the el–TO–ph
E-mail address: ap@eeter.fi.tartu.ee 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.02.003
interactions in polar crystals with a due account for their peculiar (polar) properties. Moreover, new developments in the understanding of different properties of oxide ferroelectrics and multiferroics [5,7,14–16] have emphasized the considerable practical importance of the theoretical studies of el–TO–ph interactions. At present, there remains a number of important issues, which should be investigated in detail, such as the microscopic explanation of a large strength of the el–TO–ph interaction in ferroelectrics or, which have not been addressed at all, such as an importance of the screening effects or a general question concerning the specific properties of the el–TO–ph interaction. For instance, unlike LO phonons, due to little theoretical knowledge about the el–TO–ph interaction, until the appearance of Ref. [17], it was typically difficult to realize quantitative predictions for the el–TO–ph interaction constants in polar compounds. In the present paper, using the characteristic parameters of a polar crystal in an explicit way, we investigate the main features of the el–TO–ph interaction. In particular, we give a microscopic justification of a significant enhancement of the el–TO–ph interaction strength in ferroelectric compounds. Within a first-principles methodology, we show how to link the interaction of electrons with polar long-wavelength TO phonons to the long-range dipole–dipole interaction. This, in turn, provides a nontrivial relationship between the el–TO–ph interaction constants and the macroscopic material parameters, such as dipole oscillator strengths and the forbidden gap. Being independent on the specific details of how the coupling was established, the relationship (which does not involve any adjustable parameters and requires only some appropriate amount of information usually available experimentally) gives us a practical framework for estimating the values of the el–TO–ph interaction constants in a polar material for which no data have been available so far. In a more general context, this leads to a deeper
A. Pishtshev / Physica B 406 (2011) 1586–1591
understanding of displacive structural instabilities because, by analyzing what factors related to material properties may be responsible for the strength of the el–TO–ph interaction, our results can be used to probe into the ferroelectric properties of polar insulators.
2. A model for electron–TO–phonon interaction As our focus is the el–TO–ph interaction, we are interested in finding the change in the electronic charge density induced by a polar TO vibration of a long wavelength. Following the work [12], we represent a polar crystal by a model based on classical potentials and contributions of the electronic polarization effects via the relative displacements of the electronic shells. Within the standard procedure of the dipole approximation (expanding to first order in the ionic and electronic displacements, entering into the q-representation, and going to the normal coordinates uqj), the perturbation of the host crystal due to the interactions with the polar lattice TO vibrations (qj) is given by [12] X tr dUext ðrÞ ¼ N 1=2 Vq j ðrÞ uq j , ð1Þ q,j
where Vq j ðrÞ ¼ v Pðq jÞ Fq ðrÞ, and Fq ðrÞ ¼ ie
4p X G ei v G a 0 jq þ Gj2
ðq þ GÞ r
:
ð2Þ
Pðq jÞ is an amplitude of the dipole polarization (dipole moment per unit volume) associated with the polar TO mode [1], G denotes the reciprocal lattice vector, v is the volume of the unit cell, and N is their number. The physical meaning of the quantity Fq ðrÞ is that in a polar crystal it defines the electric field at a point r inside the bulk as a part of the internal field associated with the TO phonon displacement [12]. This suggests a macroscopic characterization of the quantity Vq j ðrÞ of Eq. (1) as an electrostatic interaction of this electric field and the dipole polarization. Note that the scalar production Pðq jÞ G reflects the selection of the relevant transverse component of the optical vibrational modes splitting. The derivation of Eq. (1) accounts also for the assumption that in the long-wavelength limit one can take the corresponding Fourier transformation of the Coulomb potential to be independent of the ion’s position in the unit cell. The accuracy of this approximation was investigated previously in Refs. [12,13]. The important advantage of Eq. (1) is that it considers the changes in the ion properties that are caused by the relevant changes in the ionic environment, and, hence, in accordance with Ref. [18] contains, in principle, not only pairwise interactions via the electron–ion potential, but also many-body effects caused by electron correlations. This follows from the definition of the dipole polarization as [1,12] Pðq jÞ ¼ PðiÞ ðq jÞ þ PðeÞ ðq jÞ, where PðeÞ ðq jÞ is the electronic contribution which is connected with the distortion of the electronic charge distribution (e.g., Refs. [12,19]). The corresponding ionic contribution, PðiÞ ðq jÞ, is related to the displacements of the ions participating in pffiffiffiffiffiffi P the TO vibrations [1]: PðiÞ ðq jÞ ¼ v1 s ðZs = Ms Þwðs,q jÞ, where Zs denotes the effective charge [20] of an ion with the mass Ms placed in the cite s, and wðs,q jÞ is the polarization vector of the normal TO vibration. Following Ref. [21], we can say that the effect of the dipole polarization is to change the center of force between a ‘‘shell’’ of an ion’s electrons and the other force centers in the crystal. Keeping in mind correctness in the description of an insulated state, let us assume that the wave functions csk and energies Es ðkÞ are the set of one-particle Bloch eigenfunctions and eigenvalues corresponding to the band structure, which are determined within the framework of DFT-LDA methods with the correct accounting for the quasiparticle effects [22]. Eq. (1), providing a basis for studying the el–TO–ph interaction,
1587
corresponds in the second-quantization formalism to the Hamiltonian [4] XX Helph ¼ N 1=2 gssu ðk,q jÞ asþk asukq uq j , ð3Þ s, suj k,q
which characterizes the dynamic mixing of an electron state csk with another state csukq caused by the TO phonon. Here a + (a) are the creation (annihilation) operators for electronic states in the valence and conduction bands (s, su), gssu ðk,q jÞ denotes the matrix elements of the el–TO–ph interaction defined at the equilibrium high-symmetry configuration of the lattice by Z csk Vq j ðrÞ csukq dt: ð4Þ gssu ðk,q jÞ ¼ Expression (4) involves both interband ðs a suÞ and intraband ðs ¼ suÞ matrix elements. In the long-wavelength limit, the intraband matrix elements vanish due to inversion symmetry and, correspondingly, the coupling of electrons and TO lattice vibrations is completely characterized by interband matrix elements. In order to achieve the objectives of the present study, the further work is divided into two main steps: The first one is a comparative first-principles analysis of the contribution of Eq. (3) into the lattice dynamics of TO lattice vibrations; the second one handles the interband matrix elements, and develops the formulas that allow us to estimate el–TO–ph interaction strengths in terms of macroscopic material parameters for a wide range of polar insulators.
3. Electron–phonon interaction and TO lattice vibrations 3.1. On the dynamics of TO lattice vibrations Let us employ the microscopic formulation of the el–TO–ph interaction to express its contribution to the dynamics of the polar long-wavelength TO vibrations. Since the real part of the self-energy of the TO vibrational mode specifies the deviation of the unperturbed phonon states due to external perturbation, we represent the square of the TO phonon frequency O2q j as a sum of the unperturbed ~ 2q j and the additional term Do2q j which accounts for el–TO– term o ~ 2q j þ Do2q j . We ph coupling governed by Hamiltonian (3): O2q j ¼ o 2 ~ expect the ‘‘normal’’ part o q j , which may be regarded as the bare TO phonon frequency with respect to the el–TO–ph coupling, to be ~ 2q j free of softening anomalies. As usual [4,23], it is assumed that o also involves the contributions from the phonon–phonon interactions, which are needed to describe the dependence of the TO phonon frequency on temperature. From the results of Ref. [4] it follows that the electronic contribution Do2q j can be represented to the lowest-order in g2 as the corresponding harmonic term. The other important point is to account for the Coulomb interaction among electrons as well. Following the standard procedure (e.g., Refs. [4,24,25]) and using expressions for the matrix elements gssu ðk,q jÞ given by Eq. (4), we can write the equation for Do2q j as X Do2q j ¼ 4pv Bab ðqÞPa ðq jÞ Pb ðq jÞ, ð5Þ a, b
where Bab ðqÞ ¼
w^ el ¼
X v Ga vc ðq þGÞwel ðq þ G,q þ GuÞ vc ðq þGuÞ Gub , 4pe2 G,Gu a 0
^0 P 4pe2 , vc ðq þGÞ ¼ , ^0 ^ v^ c P vjq þGj2 I
1 XX fs ðkÞfsu ðkþ qÞ N s, su k Esu ðk þ qÞEs ðkÞ Z Z csk eiðq þ GÞr csukq dt csukq eiðq þ GuÞru csk dtu:
ð6Þ
ð7Þ
P0 ðq þ G,q þ GuÞ ¼
ð8Þ
1588
A. Pishtshev / Physica B 406 (2011) 1586–1591
Here the matrix elements of the microscopical dielectric susceptibility of the electron subsystem wel ðq þ G,q þ GuÞ are expressed in the random phase approximation in terms of the polarization ^ 0 [25,26], I^ is the identity matrix, v^ c is the bare Coulomb operator P interaction and f are the occupation numbers. With the help of expression (6), which is written in terms of the microscopic electronic dielectric susceptibilities and the Coulomb potentials, it is not difficult to confirm that Eq. (5) describes the corresponding electronic contribution to the square of the TO phonon frequency O2q j (i.e., according to the microscopic theory of lattice vibrations [27–29], the contribution which uniquely involves the wel ðq þ G,q þGuÞ matrix elements with both G and Gu different from zero). 3.2. The role of el–TO–ph interaction A formal decomposition being applied to the structure of Eq. (5) allows us to distinguish two conceptual factors contributed to Do2q j . The first one is the ‘‘kernel’’ Bab ðqÞ. The other factor, which is represented by the product of two polarizations Pa ðq jÞPb ðq jÞ, plays a role as a term of the model interaction similar to a dipole–dipole one. If now we let q ¼0, then according to Refs. [30,31] this similarity certainly implies that in the longwavelength limit we may put Eq. (5) on the same footing as an effective contribution of the long-range dipole forces. Let us consider the particular case of q¼0 in further detail. It is convenient to employ the following relations for the polarization components [1,12]: PðiÞ ð0jÞ ¼
3
e1 þ 2
Pð0jÞ,
PðeÞ ð0jÞ ¼
e1 1 Pð0jÞ, e1 þ 2
ð9Þ
where e1 is the electronic (static-high-frequency) dielectric permittivity. Recall that in a polar insulator, in the limit of zero wavevector, the displacement of charges from their equilibrium positions creates dipoles, which interact with long-range forces [1]. Moreover, as it was argued in Ref. [32], the dipoles in a crystalline medium are not represented simply by the ionic polarizations P(i)(0j), but due to the dipole–dipole interaction they acquire an external moment which transforms P(i)(0j) into P(0j) as Pð0jÞ ¼ ½ðe1 þ2Þ=3PðiÞ ð0jÞ with the corresponding proportionality coefficient given by the local-field factor (see Eqs. (9)). Thus, expressing the product Pa ð0jÞPb ð0jÞ in terms of the relevant ionic components and taking into account Ref. [32], we can classify Eq. (5) at q ¼ 0 as a contribution of a long-range dipole–dipole interaction. In order to clarify this correspondence, let us determine whether the result given by Eqs. (5) and (6) in the longwavelength limit matches its counterpart, which is caused by the long-range dipole–dipole interaction [30,31]. For this purpose, we compare Eqs. (5) and (6) at q ¼0 with the corresponding formulas of the lattice dynamics theories of polar insulators [1,20,27,30,31]. As we shall see below, such a comparison procedure also gives us unique ability to establish a bridge that relates the interaction of electrons with the polar long-wavelength TO modes to the long-range dipole–dipole interaction. As a measure of the dipolar activity of the given zone-centre TO vibration, let us introduce the dipole oscillator strength Sab ðjÞ defined by Sab ðjÞ ¼ 4pvPa ð0jÞPb ð0jÞ. The classical description of Sab ðjÞ is well known from the previous theoretical studies of lattice dynamical models (e.g., Refs. [1,20,31]). In particular, the quantities Sab ðjÞ correspond to the contributions of the TO vibrational modes in the infrared (IR) part of the Lorentz model of the dielectric matrix (e.g., Refs. [12,27,33]). In view of the definitions of Sab ðjÞ and PðiÞ ð0jÞ, it is convenient to express Sab ðjÞ with the help of Eqs. (9) as follows: ! ! X wb ðt,0jÞ 4pe2 X wa ðs,0jÞ Zs pffiffiffiffiffiffi Zt pffiffiffiffiffiffi Sab ðjÞ ¼ : ð10Þ v Mt Ms s t
Here the result of each such summation in the round brackets defines the TO mode effective charge [34], Zns are the Born transverse dynamical effective charges determined by [1,30,35] Zs ¼ ½ðe1 þ 2Þ=3Zs . At q ¼0, Eq. (5) can be rewritten in terms of Sab ðjÞ as X Do20j ¼ Bab ð0Þ Sab ðjÞ, ð11Þ a, b
where the tensor Bab ð0Þ is defined by X v Ga vc ðGÞwel ðG,GuÞvc ðGuÞ Gub Bab ð0Þ ¼ 4pe2 G,Gu a 0
ð12Þ
and wel ðG,GuÞ ¼ limq-0 wel ðq þ G,q þ GuÞ. Introducing according to Ref. [36] the representation of the susceptibility function wel ðG,GuÞ in terms of matrix elements of the microscopic polarizP ability tensor agZ ðG,GuÞ as wel ðG,GuÞ ¼ gZ Gg agZ ðG,GuÞ GuZ , the evaluation of Eq. (12) becomes Bab ð0Þ ¼
GuZ Gub 4pe2 X X Ga Gg agZ ðG,GuÞ : v G,Gu a 0 g, Z jGj2 jGuj2
ð13Þ
What is important to note here is that Eq. (13) provides useful information about the basic role of Bab ð0Þ: it emphasizes not only the importance of the polarizability effects but also it well illustrates how the quantity Bab ð0Þ is responsible for the softening of the zone-center TO vibration mode. In Eq. (11), the dipole oscillator strengths Sab ðjÞ serve as a direct mediator of the long-range dipole forces. The only puzzle that remains is how to efficiently reproduce a result of summations in Eq. (13) in order to calculate Bab ð0Þ. One simple and quicker way to find the proper evaluation of Bab ð0Þ is by comparison of Eq. (11) to a similar expression which is obtained from the same ‘‘source’’, Eq. (1), by means of the macroscopic formalism [1,12]. As a comparison parameter, it is reasonable to choose the product Pa ð0jÞPb ð0jÞ due to its ability to measure the long-range dipole–dipole interaction [32]. By using the corresponding result of Ref. [12], ‘‘macroscopic’’ counterpart of Eq. (11) for the case of the cubic (or tetrahedral) lattice symmetry can be represented in terms of the product Pa ð0jÞPb ð0jÞ (or Sab ðjÞ) as follows: X dab X dab Pa ð0jÞPb ð0jÞ ¼ S ðjÞ: e þ 2 e þ 2 ab a, b 1 a, b 1
Do20j ¼ 4pv
ð14Þ
A one-to-one correspondence between Eqs. (11) and (14) provides the resulting expression for the quantity Bab ð0Þ in the following form: Bab ð0Þ ¼ dab =ðe1 þ 2Þ:
ð15Þ
Note that Eq. (15) reproduces exactly the same result that was previously found in Ref. [31]. Now it is obvious that Eq. (11) is properly consistent with the previous studies. A comparative analysis of the obtained results allows us to draw several general conclusions as follows. (i) Following Ref. [37], with the help of Eqs. (10) and (13) one can identify Eq. (11) as the long-range Coulomb contribution resulting from the induced dipole interactions through the electronic polarizability. (ii) Eqs. (11) and (15) reproduce the similar results derived within the exact accounting of the dipole–dipole interaction in TO vibration mode dynamics [31]. (iii) In the macroscopic sense, the physical meaning of the expression represented by the product of three factors Bab ð0ÞPa ð0jÞPb ð0jÞ is determined by the regular (at q¼ 0) contribution of the long-range dipole forces. Accordingly, this implies the explicit correspondence between the model specified by Eqs. (1)–(4) and the first-principles approach proposed and developed for polar crystals in Refs. [30,31]. The last two conclusions represent a novel result of principal importance: we showed and confirmed the full correspondence
A. Pishtshev / Physica B 406 (2011) 1586–1591
between the results of two descriptions of the dynamics of the zone-centre TO vibrational mode—one in terms of the el–TO–ph interaction and the other one in terms of the regular at the q ¼0 part of the long-range dipole–dipole interaction [30,31]. In particular, Eq. (11) not only demonstrates the equivalence of the interpreting power of both descriptions, but also can serve as a link between the two approaches. Thus, our findings give a clear understanding of the role of the el–TO–ph interaction in the conversion of the zone-centre TO vibrational mode O0j to the ferroelectric soft-mode. First, the interaction of electrons with polar zone-centre TO phonons is directly associated with the long-range Coulomb interaction. In particular, Eq. (11) shows how the changes in the electron density (or the changes in the electronic states) induced by the local displacements are sufficient to yield a considerable softening of the TO phonons around the G point (note that Bab ð0Þ o 0). Second, the magnitude of the destabilizing contribution determined by Eq. (11) is highly dependent on dipole oscillator strengths. The latter reflects the fact that the zone-centre TO vibrational modes may exceptionally strongly be coupled to valence band electrons.
4. On macroscopic features of electron–TO–phonon interaction in ferroelectrics 4.1. Parametrization of electron–TO phonon interaction at q ¼ 0 The microscopic approach considered above is quite general; its distinguishing feature is that it allows us to treat the electronic correlations on the same footing as the electron–phonon interaction. This is very important (and, as we shall see below, necessary) for determining the electron–TO–phonon interaction constants accurately. However, even in the long-wavelength limit, Eqs. (5)–(8) are rather complicated for a further theoretical treatment. In the present section, we shall derive a reduced model of el–TO–ph interaction, which may provide analytical solutions and, on the other hand, may help to improve the applicability of the vibronic theory as well. Let us pass to the limit q-0 in Eqs. (5)–(8). The main complication in evaluating the matrix elements of wel ðq þ G,q þ GuÞ at q ¼ 0 is that one needs to exclude the irrelevant contributions associated with the electronic displacements (since these contributions are by 2 definition already incorporated into gss u ). Let us take into account that the dielectric susceptibility w^ relates a variation of the electronic density dr^ to the relevant perturbation potential dv^ ext as [25,38]
dr^ ¼ w^ dv^ ext , e^
1
¼ I^ þ v^ c w^ ,
ð16Þ
1
where e^ denotes the inverse dielectric matrix. Then the problem is solved in two steps: (i) by finding a change of the electronic charge density associated with interaction (1) and (ii) by applying Eqs. (16) in a similar way as was described in Ref. [30] regarding the analysis on the internal electric field effects. Acting in this manner, and taking into account Eqs. (9), we obtain that in the long-wavelength 0 limit, the reduced susceptibility w^ el turns out to be related to w^ el by the following equation: q-0 :
3 w^ el ¼ w^ 0 : e1 þ 2 el
ð17Þ
Assuming that in insulators ðfs ðkÞfsu ðkÞÞ ¼ 1, we can now write the long-wavelength limit of Eqs. (5)–(8) in terms of the constants of the j el–TO–ph interaction gssu ðk,0jÞ gss u ðkÞ as follows:
Do20j ¼
j 3 1 X X jgssu ðkÞj2 , e1 þ 2 N s a su k jEsu ðkÞEs ðkÞj
ð18Þ
1589
X v2 X Pa ð0jÞ Pb ð0jÞ Ga vc ðGÞvc ðGuÞGub 2 e a, b G,Gu a 0 Z Z csk eiGr csuk dt csuk eiGuru csk dtu:
j 2 jgss u ðkÞj ¼
ð19Þ
A remarkable property of Eq. (18) is that it can be represented j as a product of two independent factors ðl Þ2 P 0 ð0Þ, one of j 2 1 j 2 which, ðl Þ ¼ jg j ½ðe1 þ2Þ=3 , characterizes an effective interaction of electrons with polar zone-centre TO vibrations, the other one, P 0 ð0Þ, is the so-called non-interacting susceptibility. Following the vibronic theory prescription (e.g., Ref. [4]), this allows us to constitute a reduced one-parameter version of the el–TO–ph Hamiltonian (3) in the following model form: X X j ðredÞ Helph ¼ N 1=2 lssu ðkÞ asþk asuk u0j : ð20Þ s a su
k
Thus, in the calculations outlined above, we have performed a mapping of the original microscopic model (3) onto the effective model given by Eq. (20), which permitted an accurate incorporation of Coulomb renormalization effects. As we can see, the el– TO–ph interaction presented in Eq. (18) is already renormalized due to the interelectronic response (screening): the square of the screened el–TO–ph interaction constant is inversely related to the local field factor ðe1 þ2Þ=3. Since local fields and their effects become more prominent in compounds with a mixed ionic– covalent character of chemical bonds (e.g., Refs. [25,39,40]), this implies an important improvement in accuracy of determining the constants of the el–TO–ph interaction for polar materials. We refer to Eq. (20) as a vibronic-type model, since it represents the long-range electron-lattice dynamic hybridization of the electronic bands of opposite parities, which takes into account the relevant s-, p- and d-channels (e.g., Refs. [4–7] and references therein). For the last several decades, the model of type (20) was a useful prototype for intensive studies of various properties of both typical insulating perovskites [5,14,41–44] and the AIVBVI narrow-gap semiconductors and their alloys [45–49]. For instance, as applied to a family of the ferroelectric ABO3 perovskite oxides, hybridization (20) involves a significant mixing between the filled O 2p and the empty d0 (Ti4 + , Nb5 + , Ta5 + , W6 + , etc.) electronic states caused by the IR-active TO F1u soft vibrations [5,15,41]. 4.2. Evaluation of electron–TO–phonon interaction at q¼0 Let us explain why one can expect significant enhancement of the el–TO–ph interaction strength in ferroelectric compounds. By analogy with the scattering of an electron due to LO phonons [12], we first examine the expression for the amplitude of the interband el–TO–ph interaction defined by jgssu ðkÞj ¼ ðlimq-0 jgssu ðk,q jÞj2 Þ1=2 . Fixing j, using Eq. (4) and the macroscopic expression for the dipole oscillator strength of the chosen TO mode [12], we can obtain the following equation: rffiffiffiffiffiffiffiffiffi ve1 jgssu ðkÞj ffi DOLT ð21Þ j/skjF0 ðrÞjsukSj, 4p where, according to Ref. [50], we introduced (for the given polarization) the corresponding difference (the LO–TO splitting) DOLT ¼ ðO2LO O2TO Þ1=2 between the longitudinal ðOLO Þ and the transverse ðOTO Þ frequencies (i.e., the difference in long-range fields given by longitudinal and transverse modes as q-0Þ. The question is now under which conditions does a material have a strong el–TO–ph interaction. First of all, let us show that the dynamic hybridization described by Eq. (3) leads to distortions of the charge density distribution, which are caused by the internal electric field F0(r). From Eq. (2) it follows that for the central point q¼0 the quantity F0(r) satisfies the following
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equation (l denotes lattice vectors): ! X e F0 ðrÞ ¼ = ¼ = fe ðrÞ: jrlj l
ð22Þ
In accordance with Refs. [51,52], Eq. (22) defines a field generated by the Coulomb electrostatic potential fe ðrÞ which is connected (at given boundary conditions) through the Poisson equation with P a charge distribution dre ðrÞ ¼ e l dðrlÞ. Consequently, changes in the electron charge densities are represented by the relevant distortions owing to the electric field gradients at certain lattice sites of a polar crystal. On the other hand, the field F0(r) provides, due to Eq. (22), the long-range character of the el–TO–ph coupling at the G point. Secondly, looking at Eq. (21), we can see that the magnitude of the splitting DOLT serves as an enhancement factor of the matrix elements (integrals) associated with F0(r). Therefore, in the longwavelength limit, the el–TO–ph interaction in polar crystals is distinguished from that of most other insulators by the following features: (i) it is long-range, (ii) it is controlled by the internal electric field and (iii) it is essentially sensitive to the values of the LO–TO difference (i.e., the polar strength of long-wavelength optical vibrational modes). Hence, it is clear that Eq. (21) can be viewed as indicative of the strength of the el–TO–ph coupling in polar materials. In particular, one can see that an essential prerequisite for manifesting a large magnitude of jgssu ðkÞj is directly determined by the great values of the LO–TO splitting DOLT . The latter is a general feature that is observed for ferroelectric materials [34]. For example, the experimental difference between the corresponding LO and TO modes in the ferroelectric BaTiO3 is about 530 cm 1 [53] to be compared with the values of 100, 69, and 48 cm 1 obtained for the differences in the compounds NaCl, KCl, and RbCl, respectively [54]. In a polar lattice, the LO–TO splitting depends generally on Born’s transverse dynamical effective charges Zn of the lattice ions and the screening of the Coulomb interaction as [34]: DO2LT pjZ j2 =e1 . Recall that the nature of these charges is associated with the existence of large dipole–dipole interactions [55]. Thus, one can conclude that the presence of anomalously large Born’s effective charges is the key signature that the interband el–TO–ph coupling is particularly strong in a ferroelectric material. This result corresponds to the main assumption of the vibronic theory [4–7] suggesting the existence of sufficiently strong interband el–TO–ph coupling in displacive ferroelectrics. Hence, the strength of the interband el– TO–ph interaction can serve as a direct indicator which measures how close a crystal lattice is to a possible ferroelectric instability. Further details concerning with the quantitative characteristics of the electron–TO–phonon coupling are given in our separate paper [17]. 4.3. Electron–TO–phonon interaction via macroscopic parameters As shown in the present work, one can associate the interaction between electrons and polar long-wavelength TO phonons with the long-range dipole-dipole interaction. This link, which was not anticipated by previous theoretical considerations, helps us to express the strength of the el-TO-ph interaction via macroscopic parameters of a polar crystal, which in turn can be deduced from the analysis of the relevant experimental data. Let us introduce the bare constant g0j which represents the effective k-independent interband el–TO–ph interaction defined by a proper average of the squared el–TO–ph matrix elements over the given electronic states: 2 g0j ¼
X Mj X j Eg , jgssu ðkÞj2 jEsu ðkÞEs ðkÞj su 4 s 2N k
ð23Þ
where Mj is the reduced mass of the TO vibration and Eg is a bondgap energy. According to Eq. (18), accounting for the local-fieldseffects-induced partial screening of the bare el–TO–ph interaction is described by the renormalization: 2 2 g0j !g 2j ¼ 3g0j =ðe1 þ2Þ:
ð24Þ
On the other hand, according to Eqs. (11) and (15), the contribution of the linear el–TO–ph interaction to the square of the zonecentre TO vibrational mode frequency O20j can be decomposed in two factors, the first of which is Bab ð0Þ, which accounts for the corresponding electronic contribution, and the other one, Sab ðjÞ, represents the dipole oscillator strength of the given TO vibration. The bare constant of the el–TO–ph interaction g0j can thus be determined from matching Eqs. (11) and (15) together. As a P result, we obtain the following relationship ðSðjÞ ¼ a Saa ðjÞÞ: 2 1 ¼ 12 Eg Mj SðjÞ: g0j
ð25Þ
Eq. (25) is an important result of the present work. For each zone-center TO vibration of the branch j and of the reduced mass Mj, it relates the bare constant of the corresponding el–TO–ph coupling at the G point, g0j, to the macroscopic material constants, the values of which can be obtained from the experiment. Taking from the IR spectra analysis the relevant information about the IR-active TO phonons and the dielectric function behavior, and using Eqs. (25) and (24) in combination with the experimental data for e1 and Eg, we can directly evaluate the el–TO–ph coupling constants for polar materials of interest. A more detailed demonstration of the practical usefulness of Eq. (25), together with the numerical results for a wide number of selected polar insulators and semiconductors, can be found in our separate publication [17]. In particular, the calculations of the interband el–TO–ph interaction constants and the further comparative analysis showed that the large interband el–TO–ph interaction is a special microscopic feature of ferroelectric materials. In contrast, in non-ferroelectrics, as it has been demonstrated, the strength of the el–TO–ph interaction is not necessarily high enough due to their lower polar nature.
5. Conclusions In this paper, focusing mostly on the fundamental contribution of the electron subsystem to the dynamics of polar long-wavelength TO vibrations and using a first-principles methodology, we provided a systematic description of the el–TO–ph interaction in a polar insulator. The study is based on the model of a polar crystal with classical potentials, which takes into account electronic polarizability effects. By analyzing the electronic contribution to the TO vibrational mode in terms of the el–TO–ph coupling, we established a bridge which allowed us (a) to link the model under consideration to the microscopic lattice dynamics and (b) in the long wavelength limit, to relate the interaction of electrons with polar TO vibrations to the long-range dipole–dipole interaction. Our results highlight the importance of the el–TO–ph interaction for the genesis of the longwavelength TO vibrations in ferroelectrics, thereby giving a fundamental support at the microscopic level for the applicability of the vibronic theory. Within a first-principles methodology, we found and explained the significant increase of the constants of the el–TO–ph coupling in ferroelectric materials, we showed how the el–TO–ph interaction constants might be dependent on macroscopic material parameters, and we obtained the analytical equations allowing us to estimate the el–TO–ph interaction strengths in a wide range of polar insulators. In particular, the effective charge of the zone-center TO vibrational mode can be considered to be the key macroscopic parameter of the el–TO–ph coupling strength. In the materials where the el–TO–ph coupling is operative, this can be
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verified by the analysis of the corresponding IR spectroscopic measurements.
Acknowledgments The author would like to thank N. Kristoffel for helpful discussions, A. Sherman for attention to this work. The work was supported by the Estonian Science Foundation Grants No. 6918 and No. 7296.
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