1. Introduction

Alignment of the impurity centers and complexes by the linearly polarized light is well-known phenomenon [1,2]. It has been observed in centrosymmetric crystals such as alkali halides or KTaO₃ for centers with lower symmetry than the crystal matrix [2–5]. Alignment of such centers results particularly in occurrence of induced anisotropy of light absorption. Axially symmetric centers or complexes often demonstrate noticeable permanent dipole moment (below we call such centers as dipolar). Alignment of these dipoles in a polar manner along axis determined by the plane of the light polarization (i.e., dipole orientation) is symmetrically forbidden in centrosymmetric crystals. However, in crystals lacking inversion symmetry this prohibition is lifted. Consequently, appearance of macroscopic polarization of the crystal due to optical orientation of dipolar centers (OODC) can be observed. This effect somehow resembles optical orientation of electron spins [6]. To the best of our knowledge, possibility of such effect occurrence and its microscopic origin has not been discussed so far.

In this paper we consider the macroscopic electric polarization P induced by optical orientation of dipolar centers in non-polar piezoelectrics and a microscopic model of the effect. The experimental manifestation of the OODC, first of all transient electric current arising under non-stationary illumination, is also discussed. Estimated value of the current and its frequency-response function is shown to give an opportunity for ultrafast optoelectronic applications of the OODC effect.

2. Phenomenological description

Static electric polarization, P, of a noncentrosymmetric crystal exposed by an electromagnetic radiation may emerge as a consequence of the nonlinear optical effect, namely, optical rectification. As known, this effect is described by the second-order nonlinear susceptibility tensor χ _ijk [7]. Since OODC also relates the dc polarization to the oscillating field of the optical wave, the lowest tensor phenomenologically describing this effect is of the third rank, as well:

where d_ijk is the material tensor isomorphic with χ _ijk , I is the light intensity, and e_j and e_k are components of the unit polarization vector of incident light. Despite the similarity of phenomenological description of both effects, they have different physical origin from the microscopic point of view. Optical rectification has generally no relations to resonant optical transitions and it exists in the region of optical transparency of the crystal. In contrast, both OODC and optical alignment stem from real optical transitions accompanied by the relaxation process towards the equilibrium distribution of the centers. Therefore in opposite to optical rectification, OODC is fundamentally related to dissipative processes. Note that the tensor d_ijk is invariant to time reversal because its components are linked with the product of two dissipative constants: relaxation time and optical absorption of the dipolar centers.

3. Microscopic model

There are two main microscopic mechanisms that can be realized for OODC occurrence like it takes place for the effect of optical alignment. The first mechanism is orientation-sensitive optical excitation of centers (including intra-center transitions) and their subsequent real reorientation with the reorientation time in the excited state being different from that in the ground state [2]. The second is repopulation of differently oriented centers resulting from orientation-sensitive release of electrons (holes) from the centers due to charge-transfer transitions and its following orientation-insensitive trapping (not necessarily by the same centers) [3]. Although the latter mechanism does not involve real reorientation process it, nevertheless, results in preferred orientation of the dipole moment of the centers. In general, both mechanisms can be simultaneously realized with higher or smaller efficiency in a crystal. Therefore, below we examine a model of the OODC in which both mechanisms take part to a certain extent.

Typical dipolar centers in crystals are off-center impurity ions and impurity complexes arising as a consequence of the principle of the local charge compensation of excess impurity charge. The microscopic origin of OODC is illustrated here by the example of a crystal with zinc-blende structure (the symmetry class of 4̄3m) that contains dipolar complexes participating in charge-transfer optical transitions of electrons into the conduction band. Similar model of the acceptor-donor-pair complex was previously considered to describe the microscopic origin of the linear photogalvanic current [8]. Note that this current is also described by the third-rank material β_ijk [9]. It was demonstrated that J_LPG is generated only when the acceptor center (A) and donor center (D) occupy the adjacent positions in different sub-lattices as shown in Fig. 1 [8]. Therefore, the dipole moment μ of the donor-acceptor pair is parallel to any of the third-order symmetry axis (C₃ ). Starting from this model, we additionally use the following simplifying assumptions:

 

Fig. 1. Dipolar complexes within the unit cell of a crystal with zink-blende structure. Four possible orientations of A and D centers occupied near-neighbor positions are indicated by a through d. Blue and red atoms refer to two sublattices of the crystal structure. Inset on the left shows propagation direction of the light and its polarization vector, e.

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In the dark, the dipolar complexes on average are homogeneously distributed among four possible directions of μ (Fig. 1). This is provided, on the one hand, by orientation-independent thermal excitation of electrons from the complexes, and on the other hand by the orientation-independent relaxation process, which involves complex reorientation and capture of free electrons by neutral A-centers via nonradiative and/or radiative recombination. Therefore, the total dipole moment of the crystal as a whole is zero in the dark. Let now the linearly polarized light beam with the polarization vector e parallel to the [110] axis propagates along the [001] axis of the crystal (Fig. 1). It is well-known that the absorption cross-section of a dipolar center, σ, is strongly anisotropic [1]. For the separate center, σ depends on the angle θ between the dipole moment μ and the vector e as following:

where σ _⊥ and σ _∥ are the absorption cross-sections for e perpendicular and parallel to μ, respectively. For pairs with D-centers occupying positions a or b in Fig. 1 (we call these complexes as of the type I) e is perpendicular μ. If D-center is either in position c or d (complexes of the type II), the angle between e and μ satisfies the relation of cos² θ = 2/3 . It is known that usually σ _⊥ is much smaller than σ _∥ [1]. Therefore in the geometry shown in Fig. 1, the light will be preferably absorbed by the complexes of the type II. Since the relaxation of the complexes does not depend on their orientation, the equilibrium between pairs of type I and II is broken towards predominance of the former complexes. This evidently results in appearance of the macroscopic dipole moment P directed along the [001] axis (see Fig. 1).

4. Effect estimation

The steady-state value of P and its kinetics can be derived from the balance equations for photo-excited electrons and for pairs of the type I and II. Assuming that the concentration of the photo-excited electrons, n, is much smaller than the concentration of neutral A-centers, these equations, supplemented by the relation for the total number of dipolar complexes under the illumination, are

Here N ₁ and N ₂ are the number of pairs with the ionized A-center of the type I and II, respectively, γ is the recombination coefficient, hv is the photon energy, and Γ = (3τ/2)^-1 is the hopping rate accompanied by the reorientation of one type of the complexes to another. Absorption cross-sections σ ₁ = σ _⊥ and σ ₂ = 2/3 σ _∥ +1/3 σ _⊥ are assumed to satisfy the relation of Δσ = (σ ₂-σ ₁)≪σ ₁,σ ₂ that allows replacement of (σ ₁ N ₁ + σ ₂ N ₂) by σ ₀ N with σ ₀ = (σ ₁ + σ ₂)/2 in Eq. (5). The absorption cross-section of unpaired A-centers is assumed to be equal σ ₅, as well. In equations (3–5) thermal excitation of electrons and their trapping by donor centers are neglected.

Considering that n ≪ $N_A^0$ ≪ N and variations of N ₁ + N ₂ are negligible, Eq. (5) yields n = 2σ ₀ NI / ${\gamma N}_A^0$ hv and the system of Eqs. (3)–(6) is reduced to

which results in the following expression for the steady-state value of ΔN:

with the relaxation time of

When the light intensity I is much smaller than I_TH = 2$N_A^0$Γhv/σ ₀ N , the relaxation time is mainly determined by real reorientation of the complexes and the steady-state difference of complexes concentrations is ΔN_ST = ΔσNI /2Γhv. When the light intensity exceeds I_TH , the concentration difference ΔN_ST is saturated at the level of ΔN_SAT = Δ${\sigma N}_A^0$/σ ₀. In this region kinetics ΔN is governed by the redistribution of electrons between A-centers constituting the complexes of different types which leads to predomination of complexes of the type I though the process of real reorientation of complexes is insignificant. One should take into account that at very high light intensity supposed relationship of n ≪ $N_A^0$ may be violated. To determine the validity limits of the proposed model and expected value of the light-induced polarization, numerical estimation is given below.

It is reasonable to accept the following values for the crystal parameters: N ≈ 10¹⁹ cm^-3, N/$N_A^0$ ≈ 10 ,σ ₀ =10^-17cm², and Δσ/σ ≈ 0.1. In frameworks of the conventional barrier-limited model, the hopping time is τ = τ ₀ exp(-E_A /k_BT), where pre-exponential factor τ₀ is in the range of 10^-13÷10^-11s; E_A , the activation barrier; k_B , the Boltzmann constant; and T, temperature. Taking τ ₀ = 5×10^-13 s and E_A = 0.25 eV, we get Γ ≈ 10⁸s^-1 at the room temperature. Assuming simultaneous processes of radiative and nonradiative recombination, the value of γ is taken as about 10^-9 cm^-3 s^-1, which corresponds to the electron lifetime of τ_n = (${\gamma N}_A^0$)^-1 =10^-9s. At last, the value of dipole moment, μ, is chosen to be about 1.6×10^-29Cm.

With chosen parameters the intensity I_TH is about 4×10⁵ Wcm^-2 at hv = 1.25eV. Consequently, at small intensity (I≪I_TH ) the steady-state polarization is P_ST = μΔN_ST /√3= d ₁₄ I, where d ₁₄ = 1.5×10_-16 V^-1 s is the only non-vanishing component of the tensor d_ijk in a crystal of the symmetry class 4̄3m . For comparison in typical nonlinear crystals, the components of the optical rectification tensor are much smaller: of the order of 10^-19 V^-1s [7]. Saturation of the concentration difference, ΔN_SAT = 10¹⁷ cm^-3, which generates polarization of 10^-6 Cm^-2, is reached at intensity of I≥ I_TH . It is worth noting that inequality n ≪ $N_A^0$ is well satisfied even at the intensity I equal to I_TH .

Non-steady-state illumination of the crystal naturally leads to generation of a transient current density J _d = dP/dt. To separate this current from conventional photovoltaic current, one can use the polarization modulation technique. The OODC current is generated because switching the linear polarization of the light beam to the orthogonal one leads to the change of the sign of P and of J _d . However, the linear photogalvanic current J_LPG possessing the same polarization feature [9] can compete with J_d in such experiment. The problem of separating J_d and J_LPG , which was ignored so far, can be solved studying the difference of the frequency response function (FRF) of these currents. Dispersion of tensor β_ijk (Ω) responsible for generation of J_LPG is β_ijk (Ω)∝ (1 +iΩτ_P )^-1 , where τ_P is the relaxation time of the electron’s momentum, τ_P ~10_-14÷10^-12 s [9]. Thus the current J_LPG is independent on the frequency up to Ω ≈ ${\tau }_P^{-1}$ and then drops off as Ω^-1. Dispersion of d_ijk (Ω) is evidently similar to β_ijk (Ω) with substitution of τ_P by τ_P and, correspondingly, J_d (Ω)∝Ωτ_rel /(1 + iΩτ_rel ). Therefore in contrast to J_LPG (Ω), the current of OODC, J_d (Ω), increases linearly with the frequency up to Ω ≈ ${\tau }_{\mathit{\text{rel}}}^{-1}$and reaching further a plateau.

To compare the order of values of J_d (Ω) and J_LPG (Ω) we use the following expression for the latter current [9]:

with e, the electron charge; l ₀, electron free length; and ζ^ex , the parameter of photo-excitation asymmetry of electrons from the dipolar centers [9]. Taking other parameters as above, for medium values of l₀ ~; 3×10^-6 cm, ξ^ex ~ 10^-3, we obtain (at I ≪ I_TH ) the ratio J_d (Ω)/J_LPG (Ω)≈ Ωτ_rel / [1 + (Ω τ_rel )²]^1/2. This implies that in the plateau region J_d can reach the same order of value as J_LPG . Note that in the frameworks of the proposed model both currents have the same sign.

For experimental observation of the OODC-effect one should take special care of separation J_d from other photo-voltaic effects. Abovementioned polarization-modulation technique is a good tool for separation from convenient photo-voltaic currents. To achieve further separation from .J_LPG -current one can analyze RFR-function of the current measured by the polarization-modulation technique. Existence of a linearly growing part of the function would confirm presence of sought OODC-effect. These currents can be also separated in a time-resolved experiment, for example by measuring current relaxation after fast switching-off the illumination light. The relaxation time of J_LPG is expected to be much shorter than that of J_d since the former is defined by the momentum relaxation of an excited free-electron, while the latter is characterized by either recombination time of an electron at the center or the center reorientation time.

It should be noted that the effect of optical generation of macroscopic electric polarization in polar crystals due to impurity light absorption was studied in the seventies [10,11]. The mechanism of this effect involves optical excitation of dipolar impurities with different dipole moments in the ground and excited states without any orientation of impurities. In contrast with the proposed OODC effect, the exciting light in this case may be non-polarized since the most of dipole moments of the centers are oriented along the polar axis of the crystal.

The FRF of the current caused by macroscopic polarization change in polar crystals (excluding relatively slow pyroelectric component) is similar to J_d (Ω) [10,11]. The generation of picosecond current pulses up to 4A was experimentally observed in LiTaO₃:Cu crystal irradiated by laser pulses with duration of few picoseconds [11]. Considering similarity of FRF, we expect that optical generation of ultrashort electric pulses might be also efficient in non-polar piezoelectrics due to proposed OODC effect. When the laser pulse is much shorter than τ_rel , the relaxation terms in Eq. (7) can be neglected that results in following evolution of ΔN(t) and, consequently, P(t) on the time scale of the pulse length, Δt_p :

As a result, the polarization current of J_d (t) ∝ dP/dt ∝ I(t) practically repeats the laser pulse shape. Note, that for generation of J_d in the time scale of t ≪ τr_el the difference of dipole moments in ground and excited states is the necessary condition. Considering that exhausting of electron population of A-centers occurs at the intensity of 10¹⁰ W/cm² for Δt_P ≈ 10^-12 s (same intensity was used in Ref 11), we estimate at this intensity the peak value of generated current density of about 4×10² A/cm².

5. Conclusion

In conclusion, we have predicted a new nonlinear optical phenomenon of static electric polarization of non-polar piezoelectrics irradiated by linearly polarized light, which arises due to population change of differently oriented dipolar centers. Estimation based on the proposed macroscopic model of the effect shows that expected value of the induced polarization is much higher than that caused by optical rectification. The frequency response function of the transient current generated under non-steady-state illumination and its amplitude allow us to consider the predicted effect as the base for efficient generation of utrashort electric pulses.