Abstract
Ultrafast optical control of ferroelectricity using intense terahertz fields has attracted significant interest. Here we show that the nonlinear interactions between two optical phonons in SnTe, a twodimensional inplane ferroelectric material, enables a dynamical amplification of the electric polarization within subpicoseconds time domain. Our firstprinciples timedependent simulations show that the infraredactive outofplane phonon mode, pumped to nonlinear regimes, spontaneously generates inplane motions, leading to rectified oscillations in the inplane electric polarization. We suggest that this dynamical control of ferroelectric material, by nonlinear phonon excitation, can be utilized to achieve ultrafast control of the photovoltaic or other nonlinear optical responses.
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Introduction
Light–matter interaction has become the main gateway through which the microscopic quantum material properties are interacted, and hence engineered, with tools of laboratories in the macroscopic world^{1,2,3,4}. Optical responses of noncentrosymmetric insulators are paradigmatic examples in this context^{5}. They can host various photoelectric dynamical phenomena, such as shift current, second harmonic generation, and circular photogalvanic effect^{6,7,8,9}. Moreover, various driven states induced by intense lowfrequency external fields attracted substantial interest within various different context in recent years^{10,11}. For example, the nonlinear Hall current in inversionasymmetric materials shows secondorder responses to lowfrequency driving fields, controlled by asymmetric distribution of Berry curvature (more specifically, the Berry curvature dipole)^{10}.
Defining characteristics of ferroelectricity is the existence of switchable bistable configurations, which share the same space group but with opposite electric polarization^{12,13}. However, recent studies using optical pump, in terahertz or low infrared (IR) band, have expanded the scope of ferroelectricity to the nonequilibrium regime of driven states^{3,4,14,15}. In this field, lowdimensional materials or nanostructured systems are more attractive in view of fewatomscale local switching, which can lead to even enhanced nonvolatility of state^{3,4,11,12,13,14,15,16,17}. An exemplary system in this area is the monolayer SnTe: the inplane ferroelectricity of the atomthick layer has Curie temperature of 270 K, three times higher than that of the corresponding threedimensional ferroelectric bulk structure (100 K)^{13}.
In this study, we investigate the interacting phonon dynamics in this ferroelectric SnTe monolayer. Through realtime ab initio molecular dynamics simulations, we find that the coherently pumped outofplane A_{u} phonon mode, up to nonlinear regimes, spontaneously entails the inplane E_{u,x} vibrations. Remarkably, the entangled phonon dynamics induces the inplane polarization to increase over time in a rectified manner. This nonlinear phonon interaction is explained in terms of the modification of the potential energy surface by the phonon modes and the microscopic contributions to the polarization changes are analyzed on the standpoint of the electronic part of the Born effective charge. We discuss that this dynamically amplified ferroelectricity makes the variations of the Berry curvature field more steep, which results in enhanced Berry curvature dipole and also the increased stability of the ferroelectric phase.
Results
Atomic geometry and electronic structure of monolayer SnTe
Monolayer SnTe consists of alternating Sn and Te atoms in a puckered rectangular lattice, as shown in Fig. 1a. Previous experimental studies revealed that the inplane ferroelectricity of monolayer SnTe originates from the staggered atomic displacement between Sn and Te, which otherwise comprises the cubic rocksalt structure^{13}. In the ferroelectric configuration, the inversion symmetry is lifted along with the removal of the mirror plane perpendicular to the xaxis, whereas the mirror plane along the yaxis (M_{y}) is well preserved, as depicted by the dashed line in Fig. 1a. This equilibrium structure (one of the energy minima shown in Fig. 1b) is to be characterized by the space group of Pmn2_{1}^{13,16,18}. To visualize the doublewell potential energy with respect to the inplane polarization, as depicted in Fig. 1b, the atomic structures are linearly interpolated (or extrapolated) from the two equilibrium geometries at each equilibrium valley. The optimized geometry of monolayer SnTe, obtained by minimizing the total energies of density functional theory (DFT), is distorted by 70 mÅ along the xaxis as shown in Fig. 1a, producing the inplane polarization of P_{x} = ±13.9 × 10^{−12} cm^{−1 }^{16}. The monolayer SnTe is a seminconductor with 0.6 eV indirect band gap, which is an underestimated value by DFT calculation in comparison with the experimental gap (1.6 eV)^{13,19}, between the valence band maximum at a point on the ΓX line and conduction band minimum at a point on ΓY, as indicated by upward and downward arrows in Fig. 1c, respectively. This geometric nature of the monolayer SnTe is well reflected in the distribution of the Berry curvature, which is asymmetric with respect to the same mirror plane (M_{y}), as indicated by the dashed line in Fig. 1d. Previous studies have suggested that this asymmetric Berry curvature can be attributed to the orbital Rashba effect^{16}. Upon hole doping, this asymmetry in the Berry curvature grants sizable Berry curvature dipole (Λ_{y}) in the in the ydirection which can be manifested helicitydependent secondorder electrooptical responses^{10,11}.
Nonlinear lattice dynamics and electric polarization
We now show how the inplane polarization is delicately intertwined with selected phonon dynamics. We first briefly describe the phonon structures of the monolayer SnTe, as depicted in Fig. 2a^{20}. The optical phonons can be classified into three groups: the inplane oscillations at around 1.2 to 1.6 THz, the outofplane modes at higher frequency range from 4.0 to 4.8 THz, and the intermediate bands around 3.0 THz that comes form the mixed motions. In the present work, we focus on the phonons that can be directly coupled with the electronic dipole oscillation. In this monolayer SnTe, the inplane motions (E_{u,x} and E_{u,y}) and the outofplane A_{u} mode are active for IR field (IRactive). The A_{u} phonon mode mainly consists of the alternating atomic motion between the Sn and Te in the outofplane direction with the period of 216 fs period as shown in Fig. 2a (upper). The E_{u,x} phonon is characterized by the alternating motion between Sn and Te along the xdirection, has a period of 742 fs, as shown in Fig. 2a (below). The details of phonon dispersion and LOTO splitting of E_{u} is discussed in Supplementary Note 1 and Supplementary Fig. 1.
To quantitatively gauge the effect of the coherently pumped phonons on the electronic structure, we performed the DFT Born–Oppenheimer (BO) molecular dynamics starting from the groundstate atomic positions displaced along the eigenvector of the Au phonon mode (\({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}=80\) mÅ). Figure 2b shows that the obtained zdirection motions in the BO dynamics indeed prove the Au mode vibration with the frequency of 216 fs. On the different instantaneous atomic configuration along this lattice dynamics, we calculated the electric polarization, of which the inplane component is shown in Fig. 2c. Unexpectedly, instead of vibrating back and forth around the equilibrium point, the induced inplane polarization oscillates in a rectified manner leading to increased polarizations at all time. Furthermore, this polarization oscillation complies with the frequency of the inplane E_{u,x} phonon mode (742 fs) rather than that of the driving of Au phonon mode (216 fs). This dynamical polarization is mainly contributed by the electronic part, whereas the change in the ionic contribution is negligible. Similarly, the atomic displacements in the xdirection, recorded in Fig. 2d, mainly follows the pattern of E_{u,x} phonon: the major oscillation reveal the period of 742 fs, whereas the minor secondary oscillations with higher frequencies are deemed to reflect the beating between the two phonon modes. However, it is worth noting that this induced inplane vibration deviates from the static equilibrium point. The oscillation centers for two Sn atoms shifted positively (indicated by the upward arrow in Fig. 2d), whereas that of two Te atoms are relocated negatively (indicated by the negative arrow in Fig. 2d), constituting the increased dipole moment. In short, our results indicate that a pumping of the outofplane IRactive Au mode (4.6 THz) over a nonlinear regime derives the lowerfrequency inplane vibration of E_{u,x} mode, which in turn gives rise to the inplane oscillation of the polarization near terahertz (1.3 THz).
To analyze in detail the effect of each phonon mode on the electric polarization, we distorted the equilibrium lattice along each of the phonon eigenvectors, and then calculated the electric polarization. As summarized in the Fig. 3a, the variation in the polarization scales almost linearly with the atomic movement in the E_{u,x} mode. In contrast, the atomic displacement along the outofplane A_{u} mode has no apparent effect in the shown window in the upper panel of Fig. 3a. However, when the displacement is further increased, as shown in the bottom panel of Fig. 3a, a trace of a parabolic dependence emerges, which should be attributed to the interaction of the E_{u,x} mode into the A_{u} mode in the nonlinear range. It is worth noting that, as shown in the bottom panel of Fig. 3a, both the positive and negative displacement in the A_{u} mode increases the inplane polarization, irrespective of the phases of A_{u} phonon oscillations, leading to the rectified oscillations.
To access the nonlinear interaction between the E_{u,x} and A_{u} phonon modes, we calculated the total energy of the lattice with the additional displacement along the E_{u,x} mode on top of a given distortion along the A_{u} mode: the potential energy surface along the additional E_{u,x} coordinate hereafter is denoted as \(\Delta {E}_{{Q}_{{A}_{\mathrm{{u}}}}}[{Q}_{{E}_{\mathrm{{u,x}}}}]\), which is plotted in Fig. 3b. It is noteworthy that the potential energy shown in the E_{u,x} coordinate is plotted with respect to the minimum energy of configurations with given \({Q}_{{A}_{\mathrm{{u}}}}\). Irrespective of whether the movement in the A_{u} mode is positive or negative (\({Q}_{{A}_{\mathrm{{u}}}}=+80\) mÅ and \({Q}_{{A}_{\mathrm{{u}}}}=80\) mÅ in Fig. 3b), the energy minimum shifts positively in the E_{u,x} coordinate (as indicated by the positive arrow in Fig. 3b). This can be explained by considering the Coulomb interaction between the asymmetric ionic chain in the primitive cell. At the equilibrium, the Sn and Te atoms alternates in the xaxis with small inplane displacement, as indicated by x in the crosssectional view, shown in Fig. 3c. If the atomic positions are displaced toward the A_{u} phonon mode, depicted by d/2 in the middle panel of Fig. 3c, the bond distances between the pair of Sn and Te become
(b_{−} for left pair (Sn1Te1) and b_{+} for right pair (Sn2Te2) in Fig. 3c), which provide repulsive and attractive forces between Sn and Te atoms, respectively. The repulsive restoring force on the compressed bond is depicted in Fig. 3c and the attractive restoring force is expected from the elongated part. The stronger repulsive force produces the inplane translations of Sn and Te atom from their equilibrium position, resulting in an increased inplane displacement (\({Q}_{{E}_{\mathrm{{u,x}}}}\)). As depicted in the middle and bottom parts of Fig. 3c, the cases of positive and negative \({Q}_{{A}_{\mathrm{{u}}}}\) correspond to mere interchange between the bond pair of the elongated and the compressed: notice the bonds denoted by Sn1Te1 and Sn2Te2. As a result, the atomic displacement along the A_{u} phonon mode always increases the inplane distortions irrespective of whether the \({Q}_{{A}_{u}}\) is positive or negative.
On the other hand, the presence of the mirror plane along ydirection (M_{y}), prohibits the inplane atomic distortion in ydirection. In other words, E_{u,y} phonons would never be excited by the A_{u} vibration, once the lattice already has ferroelectric distortion in the xdirection. This is consistent with the trend shown in Fig. 3a, which reveals that both the positive and negative displacement along the A_{u} mode derives the positivedirection distortion in the E_{u,x} mode eigenvector. Eventually, the vibration in the A_{u} mode intensifies the inplane polarization overall time, regardless of the phases in the phononic oscillation. We explicitly present the doublewell potential energy surface with respect to the inplane electric polarization, as shown in Fig. 3d. Besides the increase in the inplane polarization, the barrier of the double well is substantially increased (ΔE = 0.8 meV) stabilizing the ferroelectric phase. The full range plot of the doublewell potential energy is discussed in Supplementary Note 2 and Supplementary Fig. 2. The dynamical amplification induced by nonlinear phonon interaction can be compared to the polarization rotation discussed numerously in previous literatures. However, as equilibrium geometry has negligible polarization along y and zdirections, this dynamic increase in the polarization should be considered as a distinct mechanism^{14,15}.
We now quantitatively estimate the effect of this dynamical polarization enhancement on the stability of the ferroelectric phase. The Curie temperature for ferroelectric transitions in two and threedimensional structures, such as monolayer groupIV monochalcogenides and perovskite transition metal oxides, have been very accurately predicted from the doublewell potential energy surface by using forthorder Landau theory^{12,21,22}. The free energy (F) can be written in terms of spontaneous electric polarization (P) with positive constants α and β :
At the equilibrium polarization, the free energy minimization condition (dF/dP = 0) requires the Curie temperature to be \({{{T}}}_{\mathrm{{c}}}=\frac{2\beta }{\alpha }{P}^{2}\). By fitting the free energy form to the doublewell potential given in Fig. 1b, we obtained the constants \(\frac{2\beta }{\alpha }=0.967\ {\mathrm{{eV}}}/{(1{0}^{10}\,{\mathrm{{Cm}}}^{1})}^{2}\), and thus we have T_{c} = 287 K, which is well matched with experimental observation (~270 K)^{13}. It is noteworthy that although thermal average phonons make the lattice vibrates with respect to the equilibrium, the coherently pumped A_{u} phonons shifts the center of vibration, granting an enhanced inplane polarization onto lattice overall time. On average, this rapid oscillation can be interpreted as a static increase in the inplane polarization. With \({Q}_{{A}_{\mathrm{{u}}}}=80\) mÅ the average increase in the polarization amounts to 15.9 × 10^{−12} cm^{−1}, which renders the Curie temperature T_{c} = 358 K with \(\frac{2\beta }{\alpha }=1.213\, {\mathrm{{eV}}}/{(1{0}^{10}\,{\mathrm{{Cm}}}^{1})}^{2}\) according to the fourthorder Landau theory.
Charge oscillation and photovoltaic response functions
The time variation of the inplane polarization, as presented in Fig. 2c, indicates that response of the material produces a dipole oscillation with a substantial xcomponent. To quantify these electronic responses within the full ab initio way, we performed realtime time propagation of the Kohn–Sham electronic states with the atomic positions being translated concurrently through the Ehrenfest dynamics, as follows:
Assuming an impulsive kick, the lattice was initially distorted into an A_{u} mode eigenvector (\({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}=40\) mÅ) and the electronic states were prepared in its ground state at t = 0. Detailed techniques related to the construction of the timepropagation unitary operators are well described elsewhere, including our previous works^{23,24}. The obtained Ehrenfest realtime atomic trajectories show almost identical nonlinear phonon interaction to those of BO molecular dynamics as shown in Supplementary Fig. 3. This indicates that the electronic states do not much deviate from the instantaneous ground states and nonadiabatic effects are negligible even in the regime of nonlinear phononic oscillations. The movement of charges were evaluated from the electric current, as computed with the timeevolving Kohn–Sham states^{24}:
The realtime profile of the outofplane and the inplane current (J_{z} and J_{x}) are presented in Fig. 4a, b, respectively. The corresponding Fourier components are presented in Fig. 4c, d. On this initial impulsive kick into the A_{u} phonon eigenvector, the outofplane oscillation (J_{z}(t)) is dominated by the \({\omega }_{{A}_{\mathrm{{u}}}}\) component with a sizable band width (Fig. 4c). On the other hand, the J_{x}(ω) exhibits an apparent \({\omega }_{{E}_{\mathrm{{u,x}}}}\) peak, together with \(2{\omega }_{{A}_{\mathrm{{u}}}}\) signal, as shown in Fig. 4d. The apparent E_{u,x} excitation directly evidences the nonlinear coupling of the E_{u,x} mode to the A_{u} mode, as explained above. The mechanism that underlies the frequency doubling (\(2{\omega }_{{A}_{\mathrm{{u}}}}\)) can be ascribed to the parabolic dependence of the polarization on the A_{u} mode displacement, as shown in the bottom panel of Fig. 3a: on the harmonic vibration in the A_{u} mode (\({Q}_{{A}_{\mathrm{{u}}}}(t) \sim sin({\omega }_{{A}_{\mathrm{{u}}}}t)\)), the parabolic polarization should convey the second harmonic responses in the current from
This nonlinear phononic interaction can be detected from the radiation field by tracing the inplane component of the polarity inplane polarity, as schematically sketched in left panel of Fig. 4e. In realistic experiment, the perfect impulsive kick is hard to realize, but a pulse with a substantial band width centered at \({\omega }_{{A}_{\mathrm{{u}}}}\) can be applied. If the pulse is sufficiently short, which band width covers the range between \({\omega }_{{E}_{\mathrm{{u,x}}}}\) and \({\omega }_{{A}_{\mathrm{{u}}}}\) as depicted in the Fig. 4e, the proposed phenomena can be experimentally observed.
The dynamical amplification of the polarization also has a substantial effect on the photovoltaic optical responses. Among various characteristics of noncentrosymmetric materials, such as second harmonic generation and circular photogalvanic current, which sharply depends on the polarization, here we examine the effect of the nonlinear phononics on the shift current^{5,6,7,8,9}. The response function for shift current can be evaluated as follows:
where f_{nm}, \({R}_{\mathrm{{nm}}}^{a}\), \({r}_{\mathrm{{nm}}}^{b}\), and ω_{nm} are occupation difference, shift vector, Berry connection, and energy difference between nth and mth states, respectively^{6}. The shift vector, which describes the charge center difference between occupied and unoccupied bands, is directly proportional to the internal polarization^{6,9,25} and thus will be most directly affected by this nonlinear phononics of the A_{u} phonon. As an example, the shift current response function σ^{xxx}(ω), the xdirection photoconductivity in response to the incoming light with xdirection polarization, is summarized in Fig. 4f with respect to various distortions into the A_{u} mode. The responses to a particular frequency of incoming light (ω = 0.6 eV) and the maximum of σ^{xxx} over frequencies are presented in Fig. 4f. This enhanced shift current is attributed to the increase in the inplane polarization, which is induced by the nonlinear phononics and thus irrespective of whether the A_{u} phonon is positively or negatively distorted (see Fig. 2c). Whether the bulk photovoltaic effect of solid can be altered by phonons has been questioned previously^{26} and our finding of the rectified oscillation of the polarization and the amplified photovoltaic responses can be considered as a new example in this light of search.
Interaction strength and the effect of hole doping
The larger initial pumping along the A_{u} mode produces the stronger entailing inplane motion in the E_{u,x} mode. Here we quantify the nonlinear coupling strength between the two optical phonons and analyze the electronic origin of the coupling in terms of mode effective charge. The interaction between these two optical phonons can be described through a nonlinear harmonic oscillator coupling model, as treated in previous literature^{12,27}:
Here, M and g are effective mass and the coupling constant, respectively. The equations of motion for \({Q}_{{E}_{\mathrm{{u,x}}}}(t)\) and \({Q}_{{A}_{\mathrm{{u}}}}(t)\) can be derived from the Hamiltonian dynamics, and their time series can be integrated through the Verlet algorithm. As we have assumed an impulsive kick in the ab initio dynamics study above, the time trajectories were evaluated from various initial displacements (\({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}\)) with zero initial velocity. By fitting the obtained \({Q}_{{E}_{\mathrm{{u,x}}}}(t)\) to the BO molecular dynamics results, shown in Fig. 2d, we determined the coupling coefficients : g_{0} = 1.7 Å^{−1} and g_{1} = 220 Å^{−3}, when M = 1.12 × 10^{6} m_{e}. For example, the realtime profile obtained from the displacement of \({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}=40\) mÅ is presented in the inset of Fig. 5a. The amplitude for the induced vibration, denoted as \({Q}_{{E}_{\mathrm{{u,x}}}}^{\mathrm{{ind}}}\), is deduced from the time series \({Q}_{{E}_{\mathrm{{u,x}}}}(t)\) and shown with respect to the various initial displacement of \({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}\), as shown in Fig. 5a. The lattice dynamics and the ensuing polarization variation can be described with various choices of coordinates, such as bond angles as used in the previous literature^{12}. In the present work, to efficiently manifest the interaction between phonon modes, the displacements along the phonon modes are selected. The efficiency of this selection of coordinates can be observed from the molecular dynamics calculation results (Fig. 5a); the dynamical path obtained from the molecular dynamics is quite nicely described with the model Hamiltonian written in terms of these phonon coordinates. A comparison with Fig. 5a and the Fig. 3a explains that the nonlinear coupling is negligible for small \({Q}_{{A}_{\mathrm{{u}}}}\), but increases almost parabolically as \({Q}_{{A}_{\mathrm{{u}}}}\ge 20\) mÅ.
We now show that the coupling strength between the two phonons can be adjusted by an electrostatic gating: the hole doping. We evaluate the same \({Q}_{{E}_{\mathrm{{u,x}}}}^{\mathrm{{ind}}}\), as introduced above, by \({Q}_{{A}_{\mathrm{{u}}}}^{\mathrm{{init}}}=80\) mÅ with the change of Fermi level, as shown in Fig. 5b. Overall, the increase in the hole concentration enhances the nonlinear phononic coupling strength. The electronic origin for this variation with hole doping can be analyzed with the electric part of Born effective charge^{28}:
Assuming the adiabatic evolution of the electronic states along a specific phonon mode Q, one can obtain the kresolved Qmode effective charge (\({{\mathbf{\Omega }}}^{Q}({\bf{k}})=\frac{1}{e}\frac{d{{\bf{P}}}^{\mathrm{{elec}}}({\bf{k}})}{dQ}\))^{29} for the nth band using a Kubo formula^{30,31,32}:
where \(\hat{{\bf{v}}}\), \({\nabla }_{Q}{\hat{H}}_{\mathrm{{KS}}}\), and ϵ are the velocity operator, the gradient of Kohn–Sham Hamiltonian with respect to atomic displacement of the phonon mode (Q) and the eigenvalue of Kohn–Sham state, respectively. The electronic part of the mode effective charge can be obtained by integrating all occupied contributions over the Brillouin zone. For example, the xcomponent of the Q mode effective charge is given by
Figure 5c shows the xcomponent of the A_{u} mode effective charge, which is sharply asymmetric against the sign change in \({Q}_{{A}_{\mathrm{{u}}}}\). As a result, we can deduce the fact that, for a light hole doping (in the range −0.17 ≤ ϵ_{F} ≤ 0 in Fig. 5c), the inplane polarization always increase as \(\Delta {P}_{x}^{\mathrm{{elec}}}={Z}_{x}^{{Q}_{{A}_{\mathrm{{u}}}}}{Q}_{{A}_{\mathrm{{u}}}}{\,> \,0}\) for both positive and negative \({Q}_{{A}_{\mathrm{{u}}}}\), which is in line with the explicit computation shown in the inset of Fig. 3a. On the other hand, the increasing nonlinear phonon coupling strength of the two modes with the hole doping, shown in Fig. 5b, can also be explained by the trend of the local charge distribution and mode effective charge on the hole doping. For example, for a given positive \({Q}_{{A}_{\mathrm{{u}}}}=80\) mÅ as shown in Fig. 5c, the A_{u} mode effective charge decreases with the hole doping, which implies the reduced amount of the electric polarization from \(\Delta {P}_{x}^{\mathrm{{elec}}}={Z}_{x}^{{Q}_{{A}_{\mathrm{{u}}}}}{Q}_{{A}_{\mathrm{{u}}}}\) with a given \({Q}_{{A}_{\mathrm{u}}}\). This change of the inplane polarization should be reflected as a variation in the electronic dipole moment. This modified mode effective charge can be understood by Coulomb interaction of modified local charge distribution by hole doping. The reduction of ionicity decreases the Coulomb attraction between Sn and Te from the local charge distribution at equilibrium geometry (depicted by δ in the inset of Fig. 5c) and thus leads to the increase of the repulsive restoring forces in the bondcompression regime, as depicted in the middle and bottom parts of Fig. 3c. This intensified repulsion between Sn and Te results in the increase inplane displacement, which eventually results in an enhanced responses of the E_{u,x} amplitude with a given \({Q}_{{A}_{u}}\). The effect of doping on polar distortions has also discussed previously for bulk SnTe^{33}. We expect that the modified Coulomb interaction by doping not only alters the ground state configuration but modifies the lattice dynamics and the interaction strength between phonons for both bulk and monolayer SnTe. Variations of mode effective charge and Born effective charge are discussed in Supplementary Note 3 and Supplementary Tables 1 and 2.
Besides optical responses of this inversionasymmetric insulators, as mentioned above, the carrier dynamics in the holedoped material have been intriguing in terms of nonlinear Hall effect and secondorder responses^{10,11,16}. The Berry curvature dipole have attracted intense interests for last couple of years^{10,11}. The coupling between an external electric field and the Berry curvature dipole of the system can result in a secondorder response current of
is Berry curvature dipole (Λ) along adirection, \(\hat{z}\) is the normal unit vector perpendicular to the plane made up of D and E, and f_{occ}, τ are occupation of Bloch state and constant^{10}. As discussed previously, the Berry curvature dipole directly scales with the polarization that measures the amount of inversionsymmetry breaking of the lattice^{10,16}. Here we focus on the fact that the lattice distortion along the A_{u} mode enhances the net inplane polarization and leads to an increase in the Berry curvature dipole. The static DFT calculation of the Berry curvature dipoles are presented in Fig. 5d, which confirms an almost monotonic increase of the Berry curvature dipole with the lattice displacement in A_{u} mode eigenvector. With the enhanced coupling strength, as shown in Fig. 5b, the nonlinear phononic interaction is to be more apparently revealed by the nonlinear Hall current and second harmonic generations of the Fermi level carriers of the holedoped system, which is mediated by the Berry curvature dipole. These results indicate that secondorder optical responses, such as nonlinear Hall effect and shift current, can be enhanced by nonlinear phonon interactions.
Discussion
In summary, we demonstrated that nonlinear couplings between optical phonons in monolayer SnTe, a twodimensional inplane ferroelectric system, produce a polarization oscillations of the inplane polarization in response to the pumped outofplane phonon. We expect that this nonlinear phonon interaction can be generally observed from groupIV monochalcogenide monolayers which shares similar electronic and atomic structures. (Potential energy surfaces with respect lattice distortions of monolayer SnS are discussed in Supplementary Note 4 and Supplementary Fig. 4.) Through firstprinciples dynamical simulations, we revealed that a pumped A_{u} phonon mode over nonlinear regime excites the E_{u,x} modes, which in turn increases the inplane electric polarization. This charge oscillation originated from the nonlinear phonon coupling can be utilized for a dynamical control of ferroelectricities and optical response functions. The coupling strength between the phonons is further enhanced by hole doping and thus the secondorder optical responses of the holedoped carriers can also be pursued to evidence the nonlinear phononic coupling.
Methods
Computational details for firstprinciple calculation
To investigate the electronic structure and effect of phonons of monolayer SnTe, we performed DFT calculation using the Quantum Espresso package^{34}. To describe the electron–electron exchange correlation potential, Perdew–Burke–Ehrenof type generalized gradient approximation is employed^{35}. The fullrelativistic normconserving pseudopotential is considered to describe spin–orbit interaction. The plane wave basis set with 50 Ry energy cutoff is used to describe wavefunction under threedimensional periodic boundary condition. The lattice parameters are employed from the experimental observation with vacuum slab up to 20 Å^{13,16}. The Bloch states are considered with kpoint sampling up to 50 × 50 × 1 and convergence is checked up to 80 × 80 × 1 kpoint mesh. The electric polarization of the twodimensional periodic system is evaluated following equation:
is the electric polarization of the electrons, and Z_{i} and R_{i} are the ionic charge and the position of the ith atom, respectively^{36}. For the BO molecular dynamics simulation, Verlet algorithm is employed with time step dt = 0.48 fs. We also checked on that the total system energy is conserved within 0.1 meV during molecular dynamics time of 2.1 ps. Realtime propagation of Bloch state were performed by using our inhouse computation packages^{23}. Detailed techniques related to the timeintegration of the timedependent DFT and Kohn–Sham equations are well described in our previous works^{23,24}.
Data availability
All relevant data are included in the main manuscript. All the data generated and analyzed during this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We are grateful for helpful discussions with Hosub Jin and Jungwoo Kim. We further acknowledge financial support from the European Research Council (ERC2015AdG694097), the Clusters of Excellence Advanced Imaging of Matter (AIM, EXC 2056, ID 390715994), Grupos Consolidados (IT124919), and SFB925. D.S. acknowledges the support from National Research Foundation of Korea (NRF2019R1A6A3A03031296). N.P. was supported by National Research Foundation of Korea (NRF2019R1A2C2089332). The Flatiron Institute is a division of the Simons Foundation.
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D.S. performed the calculations. D.S., S.S., H.H., and U.D.G. analyzed the data. D.S., N.P., and A.R. wrote the paper. All authors discussed and analyzed the results, and contributed and commented on the manuscript.
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Shin, D., Sato, S.A., Hübener, H. et al. Dynamical amplification of electric polarization through nonlinear phononics in 2D SnTe. npj Comput Mater 6, 182 (2020). https://doi.org/10.1038/s41524020004496
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DOI: https://doi.org/10.1038/s41524020004496
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