Soft-mode driven polarity reversal in ferroelectrics mapped by ultrafast x-ray diffraction

Ferroelectrics display a macroscopic electric polarization which originates from their peculiar electronic structure. Crystalline ferroelectric materials show a rich variety of lattice geometries and microscopic distributions of electronic charge within their unit cell.^1,2 Extensive theoretical work has shown that, in contrast to early simplistic concepts, the macroscopic polarization P(r) cannot be derived solely from the time-independent microscopic electron density ρ(r).^3,4 Instead, a variation of charge density as a function of an external parameter or a current is required. In theory, one has to calculate polarization differences between different states of the crystal from the respective electronic wavefunction. The elegant formalism presented in Refs. 4–7 expresses the polarization difference in terms of a quantum (Berry) phase calculated from the cell-periodic part of the electronic wavefunction.⁸ This method has been applied to calculate the stationary macroscopic polarization of a number of prototype ferroelectrics.

In contrast to such sophisticated treatments, experimental work benchmarking the theoretical predictions has remained scarce. Moreover, transient polarizations in ferroelectrics have been discussed in terms of nuclear motions in the crystal lattice, but the interplay of atomic displacements and electronic charge distributions is unknown.^9–11 In this context, the pioneering concept of soft-modes which display a frequency strongly changing with charge relocations has played a key role.^12,13 While soft-mode theory has qualitatively been invoked to explain the macroscopic properties of the ferro- and paraelectric phases of a material, the interplay of lattice motions and transient polarizations is not understood at the microscopic level.

Measurements of time-dependent microscopic charge densities have the potential to address such issues in a direct and quantitative way. X-ray powder diffraction with a femtosecond time resolution has been applied to determine transient charge density maps in crystalline materials.^14,15 In a pump-probe approach, the excitation pulse provides a strong nonresonant electric field,¹⁶ induces lattice motions via resonant pumping,^17,18 or a displacement of the nuclei upon electronic excitation.^19,20 A synchronized hard x-ray probe pulse, diffracted from the excited sample, generates a Bragg diffraction pattern from which the momentary distribution of electronic charge density is derived. In a number of ionic materials, this method has provided detailed insight into charge relocations and their interplay with lattice motions.^16,20,21

Here, ultrafast x-ray powder diffraction is applied for the first time to map the charge relocations connected with nonequilibrium low-frequency lattice excitations in a ferroelectric and to address the relation between transient microscopic electronic structure and macroscopic ferroelectric polarization. The prototype material ammonium sulfate [(NH₄)₂SO₄, AS] is studied in the ferroelectric phase. The transient charge density maps in the electronically excited state of AS reveal periodic large-amplitude charge relocations upon small-amplitude soft-mode vibrations with a frequency of 12 cm⁻¹ (0.36 THz) in the electronically excited state. The macroscopic electric polarization is derived from the transient microscopic charge density and exhibits a reversal of polarity with the 3-ps period of the soft-mode oscillation.

Ferroelectric AS crystallizes in an orthorhombic lattice structure (space group Pna2₁) with four formula units per unit cell [Fig. 1(a)].²² The dimensions of the ferroelectric unit cell are a = 0.78566(3) nm, b = 1.05813(4) nm, and c = 0.59530(2) nm. The experimentally observed ferroelectric polarization has a value of P_fe(T = 222 K) = 6 mC m⁻² slightly below the ferro- to paraelectric phase transition temperature T_C = 223 K.^23,24 The AS samples were prepared fresh every day from finely ground commercially available starting material (Alfa Aesar, purity of 99.999%). Tightly pressed pellets with a thickness of ∼40 μm were placed between a LiF entrance window (∼100 μm thickness) and a polycrystalline diamond exit window (∼20 μm thickness), which are acting as heat sinks, and fixed on a sample holder continuously rotating around an axis vertical to the x-ray beam. The sample temperature was set using a cooled nitrogen gas flow (Oxford Instruments CryojetHT). At the beginning of a measurement series, we slowly cooled down the sample under optical excitation and repeatedly recorded the diffraction pattern at different temperatures until the phase transition of the entire sample (at T_C = 223 K) was indicated by sudden changes of the angular position of selected Bragg reflections.²² The sample was then cooled down by another 20 K to achieve a uniform sample temperature of 200 ± 5 K under the experimental conditions. This is somewhat below the critical temperature T_C = 223 K of the ferro- to paraelectric phase transition of AS and, thus, the sample safely remains in the ferroelectric phase.^23–27 An additional experiment was conducted at a sample temperature of ∼110 K, significantly lower than T_C.

The ultrafast diffraction experiments are based on the optical pump/x-ray probe scheme illustrated in Fig. 1(b) where the sample is optically excited by 70 fs pulses with a center wavelength of 400 nm and probed with hard x-ray pulses similar to earlier experiments with paraelectric AS at room temperature.²⁰ Both pump and x-ray probe pulses are derived from an amplified Ti:sapphire laser system delivering sub-50 fs pulses centered at 800 nm with an energy of 5 mJ and a repetition rate of 1 kHz. The optical pump pulses have an energy of 110 μJ and are focused to a spot size of ∼430 μm, providing a peak intensity I_p ∼ 1 × 10¹² W/cm² at the sample surface. The sample is electronically excited via 3-photon absorption of the pump pulses. The major part (80%) of the 800 nm laser output is focused on a 20 μm thin Cu tape target to generate hard x-ray pulses with a photon energy of 8.04 keV (Cu-K_α) and a duration of roughly 100 fs.¹⁶ The emitted x-ray pulses are collected, monochromatized and focused to a ∼100 μm spot size at the sample position by a Montel multilayer mirror (Incoatec) providing a flux ∼5 × 10⁶ photons/s. Further details of this table-top femtosecond hard x-ray source have been described earlier.²⁸ 

The hard x-ray pulses serve for probing the pump-induced structural dynamics in the photoexcited sample. The Cu-K_α photons diffracted from the sample were recorded in transmission geometry by a large area detector (Pilatus Dectris 1M; pixel size 172 μm × 172 μm) which allows us to determine the intensity of multiple Debye-Scherrer rings at each delay time simultaneously. The pump pulses are chopped mechanically with a frequency of 12.5 Hz and a photodiode (PD) monitors for every x-ray probe pulse if the sample was excited. This experimental concept successfully mitigates the influence of temporal fluctuations of the femtosecond x-ray source on the signal-to-noise ratio, allowing for shot-noise limited measurements and the determination of the absolute change of diffracted intensity for each Bragg reflection.²⁹ For each individual delay time, a total collection time of 140 s was selected, and ∼1000 different delays (i.e., ∼10 fs spacing in-between) within the chosen ∼12 ps long interval were measured in random order over 10 days. All-optical cross-correlations of the pump pulses with 800 nm pulses traveling along the optical path of the x-ray pulses were measured repeatedly to ensure a proper stacking of data from different days. Finally, we sorted all individual data points according to their delay time and then averaged ∼40 neighboring data points (i.e., a time interval of ∼400 fs) with increasing delay time together, to obtain transients with a good signal-to-noise ratio, as shown in Fig. 2(c).

The powder diffraction pattern from an unexcited ferroelectric AS sample at T = 200 K is shown in Fig. 2(a). Integrating over all pixels with identical scattering angle 2θ yields 1D powder diffraction patterns as shown in Fig. 2(b) for an unexcited ferroelectric AS sample at T = 200 K. The result is in good agreement with literature and allows for an assignment of the 15 Bragg peaks to sets of lattice planes.²² 

Upon optical excitation, the angular positions of all observed reflections remain unchanged within the experimental accuracy. Furthermore, no additional Bragg reflections forbidden by the symmetry of the ferroelectric space group Pna2₁ are observed within our experimental sensitivity. Such observations confirm that the ferroelectric lattice geometry is preserved under our experimental conditions. In contrast, the diffracted intensities display pronounced changes ΔI_hkl(t). In Fig. 2(c), the intensity change $ΔIhkl(t)/Ihkl0=(Ihkl(t)−Ihkl0)/Ihkl0$ on selected Bragg peaks is plotted as a function of pump-probe delay [$Ihkl(t)$⁠, $Ihkl0$⁠: intensity diffracted with and without optical excitation]. The data recorded at T = 200 K exhibit pronounced oscillations of diffracted intensity with a period of ∼3 ps and a somewhat delayed onset. As also shown in Fig. 2(c), much faster and rapidly damped oscillations are observed at T = 110 K.

The oscillations of diffracted intensity are due to coherent phonon excitations in the AS sample. There are different mechanisms for exciting coherent phonons as illustrated schematically in Fig. 3. Promotion to the electronically excited state results in a change of the electronic charge distribution and the generation of coherent nonequilibrium phonon displacements. A first mechanism is the direct displacive excitation of coherent optical phonons via the electronic deformation potential of the crystal. The electronic deformation potential in solids corresponds to an origin shift of the electronically excited potential surface vs. that of the electronic ground state as schematically depicted in Fig. 3(a). In this case, photoexcitation by a broadband femtosecond pulse generates a coherent superposition of vibrational quantum states in a quasi-instantaneous way. This nonstationary superposition is equivalent to wavepacket motions along the vibrational coordinate x and results in a cosine-like oscillation with respect to the minimum of the excited-state potential.^19,20

In a different scenario illustrated in Fig. 3(b), low-frequency oscillations can also be induced impulsively by incoherent carrier relaxation on a time scale faster than the oscillation period. In this case, carrier relaxation excites a superposition of low-frequency quantum states either directly or via high-frequency excess phonons which couple anharmonically to low-frequency modes and, thus, induce their displacement. In contrast to the first mechanism, incoherent phonon excitation is possible even with a negligible shift of the excited state vs. ground state potential, and the subsequent electronic relaxation [orange arrow in Fig. 3(b)] leads to a temporally delayed onset of the phonon oscillation. Typical relaxation and, thus, onset times are in the range of up to a few picoseconds. The cosine-like oscillations in the experimental transients at T = 200 K do not set in immediately at t = 0 ps, as would be expected for a direct displacive excitation mechanism, but rather start with a 1–2 ps delay, which is clearly visible in Figs. 2(c) and 5. This suggests a predominance of the incoherent excitation mechanism under the present experimental conditions.

The experimentally determined time-dependent intensity changes $ΔIhkl(t)/Ihkl0$ are related to the transient x-ray structure factors F_hkl(t) according to $ΔIhkl(t)/Ihkl0=(|Fhkl(t)|2−|Fhkl0|2)/|Fhkl0|2$⁠, where $Fhkl0$ are the known structure factors of the unperturbed material. The time-dependent electron density $ρ(r,t)=ηρex(r,t)+(1−η)ρ0(r)$ averaged over all crystallites and its change relative to the unperturbed electron density ρ₀(r) of ferroelectric AS are extracted from the structure factors F_hkl(t) by employing the maximum entropy method as implemented in the BayMEM suite of programs.^30–32 The maximum entropy method maximizes the information entropy S which is defined as $S=−∑(v=1)Nρv(t) log(ρv(t)/ρ0v)$⁠, where the summation runs over a discretized grid of N voxels while fulfilling a set of constraints for the supplied structure factors F_hkl(t).³⁰ The fraction of excited unit cells η = 0.06 ± 0.03 is estimated from the absorbed pump fluence.²⁰ 

In Fig. 4, equilibrium and transient charge density maps are summarized for the (ab)-lattice plane at z/c = 0.5 in the unit cell [highlighted in Fig. 1(a)]. Figure 4(a) shows the equilibrium charge density ρ₀(r) and Figs. 4(b)–4(f) differential charge densities $ηΔρ(r,t)=η(ρex(r,t)−ρ0(r))=ρ(r,t)−ρ0(r)$ for different pump-probe delays. The latter reveal a pronounced modulation of charge density with time, close to the original positions of the lattice atoms, which are indicated by colored circles. It is important to note that all major changes ηΔρ(r, t) are centered on the ground state atomic positions and no charge transfer to previously unoccupied positions in space is observed. This behavior confirms the preservation of the ferroelectric lattice structure and is in striking contrast to paraelectric AS at T = 300 K where a spatial relocation of hydrogen atoms into a channel-like geometry along the c-axis has been observed.²⁰ Within the $SO42−$ anion, charge is shifted in a highly anisotropic way: the sulfur atom and three of the four oxygen atoms jointly exhibit a strong oscillatory increase in ηΔρ(r, t), while the fourth oxygen atom O(4) behaves in an opposite fashion. Additionally, the net charge transfer from the $NH4+$ to the $SO42−$ units enhances the local polarity in the crystal lattice compared with the stationary equilibrium structure.

To derive changes of electronic charge ηΔQ(t) of different structural units, we subdivided the entire unit cell into subvolumes containing one atom each. This partitioning is based on the unperturbed crystal structure and each point in the unit cell is assigned to the atom nearest to it. Transient atomic charges were then obtained by integration over the respective subvolumes. In the case of the two $NH4+$ cations, the charge of the entire unit was added together. The results presented in Fig. 5 as a function of delay time reveal pronounced oscillatory changes of local charge with an oscillation period of ∼3 ps, corresponding to a frequency of 12 ± 2 cm⁻¹ (0.36 ± 0.07 THz). The periodic charge modulations are due to coherent oscillations of a low-frequency lattice mode which is observed here for the first time and displays a frequency much lower than librational motions affecting the orientation of the $SO42−$ and $NH4+$ units.^33–36 These oscillations are severely damped on a time scale of several picoseconds. This is a known phenomenon in disordered powder samples, caused by inhomogeneous broadening of the frequencies of infrared-active phonons (in particular, the low-frequency soft-mode) compared with those measured in single crystals.^37,38

The displacements of the 12 cm⁻¹ mode are connected with a transfer of charge between the sulfur atom and three of the four oxygen atoms of the SO₄ groups [Figs. 5(a) and 5(b)], and the apical oxygen atom O(4) of SO₄ as well as one of the $NH4+$ cations [Figs. 5(c) and 5(d)]. The maximum amplitude of charge increase on the SO₃ subunit [Fig. 5(b)] is close to the sum of the amplitudes of charge decrease on the O(4) atom and the NH₄(1) unit. The second $NH4+$ cation displays much smaller changes of its associated charge [Fig. 5(d)]. These results give evidence of a net charge transfer to the SO₃ unit over a distance of some 100 pm, i.e., a chemical bond length.

Transient atomic positions and interatomic distances are derived from an analysis of the core electron density of the sulfur, nitrogen, and oxygen atoms by fitting a three-dimensional Gaussian distribution to their high core electron density in the transient electron density maps. In contrast to valence electrons, the core electrons spatially follow the respective nucleus without any deformation of their electronic charge distribution. In conjunction with our limited experimental resolution, this gives an upper boundary for the magnitude of the nuclear motions. We find that the 12 cm⁻¹ oscillations are connected with extremely small displacements of practically all heavy atoms in the unit cell, demonstrating a highly delocalized character of this lattice mode with shifts along the a-, b-, and c-axes. As an illustrative example, the bond length change Δd(S-O_ap) is plotted as a function of delay time in Fig. 6(f) and displays a magnitude of up to 100 fm, roughly 1000 times smaller than the S-O(4) distance. The nuclear displacements along the ferroelectric c-axis, shown in Figs. 6(a)–6(e), are in the sub-picometer regime as well.

The electrons of the hydrogen atoms also contribute to the total electron density map. However, due to the limited spatial resolution of the femtosecond x-ray diffraction experiment, there are no discernible electron density maxima at the proton positions and, thus, our experiment does not allow for an accurate determination of the transient proton positions within the ammonium ions. We, thus, assume in the following that the center of gravity of the 4 hydrogens in an $NH4+$ ion has a time-independent distance from the respective nitrogen nucleus and the protons follow the motion of the nitrogen nucleus.³⁹ 

The analysis of charge dynamics shows that electronic charge relocates during the coherent lattice motions over a distance 3 orders of magnitude larger than the individual nuclear displacements [cf. transient ηΔρ(r) maps in Fig. 4]. This observation and the delocalized displacement pattern are a hallmark of soft-mode behavior, i.e., a very strong coupling of the vibrational and electronic degrees of freedom.⁴⁰ This coupling results in strong local electric fields rearranging the electronic charge over large distances. Cochran's soft-mode theory,^12,13 though not allowing for a quantitative description, predicts such a behavior to be particularly pronounced close to the critical temperature of a ferroelectric phase transition, in full agreement with our results. Thus, our results establish the soft-mode character of the newly discovered 12 cm⁻¹ mode, occurring in the electronically excited state of AS.⁴⁰ 

A potential change of the macroscopic polarization ΔP(t) upon excitation of the microscopic soft-mode represents a key issue for the transient ferroelectric behavior. We recall that the electronically excited sample stays in the crystal structure of the ferroelectric phase. Correspondingly, the charge density projected onto the (ab) plane keeps the inversion symmetry during the soft-mode oscillations and therefore the polarization components in the (ab) plane vanish. On the other hand, an inversion center and an (ab) mirror plane perpendicular to the c-axis are absent. The tilt of the NH₄ and SO₄ tetrahedra relative to the c-axis, illustrated in Fig. 7(a), allows for a net electric dipole moment along c. Figure 7(b) displays transient changes ηΔρ(r, t) of the electron density in the (bc) plane at T = 200 K for a pump-probe delay of 2.8 ps. The positions of all $NH4+$ and $SO42−$ ions in the unit cell are projected on the (bc) plane (colored spheres).

In this representation, red/blue contours indicate the spatially resolved areas of charge increase/decrease. This illustrates again that the soft-mode relocates electronic charge within the $SO42−$ units in a highly anisotropic fashion. The out-of-plane oxygen atom O(4) exhibits an oscillatory increase/decrease in charge density, while the sulfur atom and the three remaining oxygen atoms that lie roughly within the same bc-plane behave in an opposite fashion. Figure 7(b) also illustrates an additional and subtle, yet noticeable asymmetry along the polar c-axis that characterizes the redistribution of charge among the three oxygen atoms in the (bc) plane. This induces a time-dependent change of the macroscopic polarization ΔP(t) which is governed by the vibrational period of the 12 cm⁻¹ mode.

It has been shown in the pioneering work of Resta and others that one cannot derive macroscopic electronic polarizations from the cell-periodic charge density maps ρ₀(r) but only from changes of ρ(r, λ) as a function of an external parameter λ.^3–7 In this context, the time-dependent charge density change ηΔρ(r, t) averaged over all unit cells is particularly relevant because it provides the microscopic charge distribution as a function of the parameter time. The time-dependent charge density fulfills the continuity equation $∇·j(r,t)=−∂ρ(r,t)/∂t$ and allows for determining $ΔPcelectron(t)$ if one follows ρ(r, t) in small steps dt as a function of time. In practice, one solves the instantaneous Poisson equation $Δϕ(r,t)=−ρ(r,t)/ε0$ from which the microscopic electric field $E(r,t)=−∇ϕ(r,t)$ can be derived. After averaging the electric field over the unit cell $Ẽ(t)=Vuc−1∫ucdVE(r,t)$ the change of the macroscopic polarization is calculated from the difference $ΔPcelectron(t)=−ε0[Ẽz(t)−Ẽz(t=−∞)]$⁠.

This general procedure can be further adopted in the special case of ferroelectric AS, since it consists of neutral molecular arrangements in layers parallel to the ab-plane of its unit cell, separated by regions of extremely low electron density. Microscopic currents flowing between one molecular layer and the neighboring layers can therefore be ruled out even during the soft-mode oscillation. In the language of Resta, this is called the Clausius-Mossotti-like case.⁴ In such a case, one can choose the (uneven) interface defined by the lowest electron density as a boundary between two adjacent subvolumes in c direction, over which the averaging of the electric field is then performed to calculate the change of the macroscopic polarization $ΔPcelectron(t)$⁠.⁴¹ 

The resulting total transient change of the macroscopic electric polarization parallel to the c-axis is given by $ΔP(t)=ΔPc(t)=ΔPcelectron(t)+ΔPcnuclei(t)$⁠, where $ΔPcelectron(t)$ and $ΔPcnuclei(t)$ represent the electronic and nuclear contributions to the polarization change. While the nuclear motions of the soft mode display a similar time evolution to the electronic polarization [Figs. 6(a)–6(e)], they are associated with only minute displacements of some tens of femtometers. Since these displacements are 3 orders of magnitude smaller than the typical distance of the electronic charge redistribution, the resulting total nuclear contribution to the overall polarization change [Fig. 8(a)] is roughly 30 times smaller than the electronic contribution [Fig. 8(b)], and can safely be neglected.

In Fig. 8(c), we show the resulting total transient change of the macroscopic electric polarization ΔP_c(t) (symbols, solid line) as a function of time in comparison with the amplitude of the spontaneous ferroelectric polarization P_fe(T = 200 K) ≃ 4.5 mC m⁻² indicated by dashed lines.²⁴ The amplitude of $|ΔPc(t)|≈Pfe(T=200 K)$ is close to the full ferroelectric polarization at a lattice temperature of 200 K. In other words, the tiny soft-mode displacements induce an oscillatory reversal of the full amplitude of electric polarization within an ultrashort 3 ps oscillation period. While the sign of P_fe relative to the crystal structure has remained unknown,²⁴ this phonon-driven reversal of electric polarization holds potential for switching applications with ultrahigh processing speed.

In conclusion, our results demonstrate the potential of ultrafast x-ray diffraction for unraveling the microscopic mechanisms behind ferroelectricity and for mapping the intrinsically ultrafast dynamics of electric polarizations upon phonon excitations. The fact that time-resolved charge density maps allow for solving the continuity equation establishes a natural and most direct link between microscopic charge density changes and macroscopic electric polarizations. This insight will allow for benchmarking ab initio quantum theory of ferroelectrics and will, thus, be instrumental for understanding and tailoring ferroelectric materials for a wide range of applications.

We would like to thank Dr. Beatrice Braun-Cula, Humboldt-Universität zu Berlin, for providing beam time for the steady-state single crystal x-ray diffraction experiment. This research was supported by the Deutsche Forschungsgemeinschaft (Grant No. WO 558/13-2).