Large Enhancements in Optical and Piezoelectric Properties in Ferroelectric Zn1-xMgxO Thin Films through Engineering Electronic and Ionic Anharmonicities

Multifunctionality as a paradigm requires materials exhibiting multiple superior properties. Integrating second-order optical nonlinearity and large bandgap with piezoelectricity could, for example, enable broadband, strain-tunable photonics. Though very different phenomena at distinct frequencies, both second-order optical nonlinearity and piezoelectricity are third-rank polar tensors present only in acentric crystal structures. However, simultaneously enhancing both phenomena is highly challenging since it involves competing effects with tradeoffs. Recently, a large switchable ferroelectric polarization of ~ 80 uC cm-2 was reported in Zn1-xMgxO films. Here, ferroelectric Zn1-xMgxO is demonstrated to be a platform that hosts simultaneously a 30% increase in the electronic bandgap, a 50% enhancement in the second harmonic generation coefficients, and a near 200% improvement in the piezoelectric coefficients over pure ZnO. These enhancements are shown to be due to a 400% increase in the electronic anharmonicity and a ~200% decrease in the ionic anharmonicity with Mg substitution. Precisely controllable periodic ferroelectric domain gratings are demonstrated down to 800 nm domain width, enabling ultraviolet quasi-phase-matched optical harmonic generation as well as domain-engineered piezoelectric devices.


Introduction
[3][4][5][6][7][8] A single multifunctional material that could achieve both effective strain tuning of materials properties and exceptional optical response would be ideal from an engineering design perspective, but relatively few candidates are known, and of these, many are limited to bulk single crystals. [9,10]th piezoelectricity and optical second harmonic generation (SHG), the subjects of this study, are third-rank tensor properties with the same tensor form; both require acentric structures with broken spatial inversion symmetry.Since the most direct way to break inversion symmetry is in a polar structure with a spontaneous polarization, Ps, many of the well-known piezoelectric and SHG crystals are also polar and often ferroelectric.Ferroelectricity induces an anharmonic ionic potential well and, in turn, an anharmonic electronic potential well.Therefore, conventional wisdom suggests that enhancing Ps would favor both piezoelectric and SHG tensors, though the magnitudes of the coefficients depend not just on the polarization, but also the polarizability.This picture however is oversimplified.It is often possible to enhance the piezoelectricity by increasing the dielectric constant, and in many cases, this is coupled with a concomitant decrease in the spontaneous polarization, for example, in the wurtzite structures [9] as well as in numerous normal and relaxor ferroelectric perovskites. [11,12]Further, although molecules with a larger electric dipole, [13,14] and crystals with larger Ps, [15][16][17][18][19] often exhibit larger SHG responses, many commercial nonlinear optical crystals belong to the nonpolar point group 4 � 2. [20]It is shown in this work that instead of spontaneous polarization, the anharmonicities of the electronic and ionic potential wells determine the SHG and piezoelectric properties.For optical frequency conversion such as SHG, a broader frequency range of optical transparency is desired, which requires a larger electronic bandgap, Eg.However, a larger bandgap is known to reduce the SHG coefficients dramatically, limiting high conversion efficiency in the ultraviolet range. [20,21]Thus, increasing the piezoelectric tensor,   Piezo , the SHG tensor,   SHG , and the electronic bandgap, Eg, simultaneously in one material is a challenging balancing act.
Nonetheless, each of these phenomena, namely the   Piezo , [22][23][24] the   SHG , [25,26] the bandgap, [27][28][29] and in addition, the electro-optic tensor, [30] have been separately reported to be enhanced with Mg substitution in ZnO.This work revisits and confirms the first three of these experimental measurements in a single set of films, though the magnitudes of enhancement reported here significantly differ from the literature for   SHG .Importantly, this work presents a unified theoretical framework for understanding these counterintuitive simultaneous enhancements.Finally, it reports for the first time, precise ferroelectric domain engineering on the sub-micrometer scale.
ZnO is a well-known wide-bandgap semiconductor (Eg ~ 3.2 eV).Recently, Zn1-xMgxO was shown to have a giant switchable spontaneous polarization of ~ 80 μC/cm 2 . [31]In this work, it is demonstrated that the electronic bandgap, the SHG coefficient, and the piezoelectric coefficient are simultaneously enhanced by Mg substitution in Zn1-xMgxO.Mg substitution induces a 30% increase in the bandgap and a simultaneous 50% enhancement in the SHG coefficients, contrary to the well-known inverse relationship between the two properties in the literature. [20,21]This is shown to be due to a 400% increase in the electronic anharmonicity that offsets the increasing bandgap and results in a net increase in the  SHG .The ferroelectric polarization Ps decreases with higher Mg concentration (i.e., it anticorrelates to the increasing  SHG ), in contrast to the reported proportionality between Ps and SHG response. [15,16]Near 200% enhancement in the piezoelectric coefficient is observed in Zn1-xMgxO with increasing Mg addition, which is attributed to an increase in the low-frequency dielectric constant arising in part due to the softening of the wurtzite structure (reduced c/a ratio) [32,33] and a decrease in the anharmonicity of the ionic well.Firstprinciples calculation and Landau theory provide a basis for understanding these trends.The piezoelectricity is enhanced by a small anharmonicity of the ionic potential well, while SHG is enhanced by a large anharmonicity of the electronic potential well.Here, harmonicity refers to the potential energy of the electron being proportional to the square of the displacement of the electron from its equilibrium position within the well, whereas in this work, anharmonicity will refer specifically to the potential energy term proportional to the cubic power of such electron displacements.(Higher order terms in such a Taylor expansion of the electron potential energy versus its displacement are also considered anharmonic but they will not be considered here).This work highlights the design paradigm of achieving both enhanced nonlinear optical properties and piezoelectric response by engineering anharmonicity differently in different frequency regimes.
Because the polarization in Zn1-xMgxO is switchable, precise domain control can be achieved in these thin films down to 800 nm domain size, which enables optical quasi-phase-matched (QPM) SHG and domain-engineered electro-optic and piezoelectric devices with a materials growth process that is CMOS-compatible. [34,35]

Results and Discussion
Zn1-xMgxO films (x = 0, 16, 23, 28, 37 mol%) with a thickness of 150 nm were epitaxially grown at room temperature along the c-axis on (111)-Pt//(0001)-Al2O3 via RF magnetron sputtering from metal targets. [31]Over this range of x, Zn1-xMgxO adopts the wurtzite structure (Figure 1a) in PVDgrown thin films. [22,27,28]Mg substitutes on the Zn site due to their similar ionic radii and electronegativity, providing a local distortion of the bond lengths and angles. [31,36]Figure 1b shows the X-ray diffraction (XRD) and the film stack of (0001)-Zn1-xMgxO//(111)-Pt//(0001)-Al2O3.Zn1-xMgxO maintains the wurtzite structure and high crystallinity without forming a secondary rocksalt phase. [22,31,36]The (0002)-Zn1-xMgxO peak shifts gradually toward a higher 2 in XRD with increasing Mg concentration.This indicates a systematic contraction in the c-lattice constant by Mg substitution, which agrees well with the previous studies. [22,27,31]The films are epitaxial to the underlying Pt electrode and have an out-of-plane full-width-half-max value in the omega x-ray circle between 1.5° and 2.0° at the lowest and highest Mg concentrations, respectively.An expansion along the a axis and an overall reduction in the c/a ratio have also been reported for comparable films. [22,27,31]Scanning electron microscopy (SEM) was performed to evaluate the surface microstructure, while surface roughness was measured by atomic force microscopy (AFM).Results reveal uniform faceted surface grains with an average diameter of ~ 100 nm and smooth surface (RMS roughness ~1.3 nm) (Figure S1, Supporting Information).
These observations illustrate high-quality films for further optical and electromechanical studies.
Figure 1(c) exhibits the hysteresis loop for Zn0.63Mg0.37O,demonstrating that the ferroelectricity emerges beyond the critical concentration (~34%). [31]Although the polarization in ZnO is not switchable, alloying with Mg has been proven to be an effective way to access the ferroelectricity without further external stimuli.The coercive field is found to be around 3 MV/cm, similar to that reported in previous studies. [31] bandgap measurements are discussed first, then the SHG measurements, followed by the piezoelectric measurements.The complex linear optical refractive indices ( � =  + ) (Figure 1d) were studied using spectroscopic ellipsometry from 1800 nm to 200 nm (equivalent to 0.69 eV to 6.2 eV).Mg substitution reduces the refractive indices and pushes the band edge towards higher energy.Since the experimental bandgaps of ZnO and MgO are reported to be 3.2 eV and 7.8 eV, [37,38] respectively, Mg addition is expected to increase the bandgap.The refractive indices below the band edges were fitted using the birefringent Cauchy dispersion relation due to the uniaxial structure of Zn1-xMgxO, and the Cauchy parameters are summarized in Table S1 (Supporting Information).Using the Tauc method [39,40] and the measured n and k of Zn1-xMgxO, a direct band transition was confirmed throughout the Zn1-xMgxO series from x = 0 to 0.37.As expected, adding Mg reduces the refractive indices and pushes the band edge towards higher energy.The Tauc fitting for x=0.37 is shown in Figure 1e, which results in the largest bandgap in the series.The extracted bandgaps as a function of Mg concentration from the experiment (Figure 1f) show a linear dependence on the Mg concentration, which agrees well with previous studies and is in agreement with the DFT predictions. [28,29]cond harmonic generation (SHG) is a nonlinear optical process that converts two photons at ω frequency to one photon at 2ω frequency. [41]SHG has been widely applied in coherent lasing sources, structural characterization, and biological imaging.Due to the strong reflection and absorption of  and 2 frequencies in the Pt layer, multireflection and absorption of both the fundamental  and second harmonic 2 waves (Figure 2b) need to be considered to correctly model and extract second-order nonlinear susceptibilities.
[46] Excluding multiple reflections in the optical analysis would lead to a failure of available SHG models. [47,48]Kleinman's symmetry forces  31 SHG =  15 SHG in Voigt notation in the Zn1-xMgxO system which could be problematic and therefore is not assumed a priori in the analysis presented in this current study. [41,49]Failure to take all necessary effects into account can result in one to two orders of magnitude error in the estimation of the SHG coefficients as compared with bulk single crystals.An advanced modeling tool named ♯SHAARP was employed, which fully accounts for multiple reflections, interference, and the complete anisotropic SHG tensor. [49]e polarized second harmonic intensities (dots) and fitting (solid lines) are shown in

Figure 2c
, where blue and red represent p-and s-polarized intensity, respectively.The powerdependent SHG response shows a quadratic dependence between pump power and SHG intensity,  S2, and Equation S7-25 (Supporting Information).A 0.5mm thick ZnO single crystal was also studied to verify and benchmark the analysis.The absolute SHG susceptibility  33 SHG of the ZnO single crystal is found to be 7.30 pm V -1 , which agrees well with reported values in the literature. [45,46,50]This indicates a robust and reliable methodology using #SHAARP [49] for the characterization of nonlinear optical response (Figure S3, Supporting Information).Figure 2d summarizes the absolute SHG susceptibilities of Zn1-xMgxO as a function of Mg concentration, where   increases monotonically with the Mg concentration.Interestingly, a nearly 50% enhancement of  33 SHG is found in x = 0.16 and 0.23 as compared to pure ZnO films and single crystals.This enhancement is significantly less than the enhancement of 420% reported in literature; [25] Such large previously reported SHG coefficients (approaching ~50pm/V) for wide bandgap semiconductors such as ZnO are unusual.A slightly different wavelength or the details of the film growth are likely minor contributors to this discrepancy since the pure ZnO film in our current study exhibits SHG coefficients similar to the bulk crystal.The most likely source of the discrepancy we surmise is the modeling of the SHG response in the thin film.For example, if we assume Kleiman's symmetry in our case, it can erroneously result in ~50 pm/V of  33 SHG for both x=0.28 and 0.36, suggesting a possible reason for such discrepancy with the previous literature (Figure S6, Supporting Information).Given our confidence in taking care of all the relevant details in modeling using the proven #SHAARP code and benchmark analysis using both films and the single crystal, the prior reported enhancement needs to be revisited. [25]Further Mg substitution beyond 23% Mg tends to suppress  33 SHG before reaching the maximum solubility.On the other hand, both the raw  31 SHG and  15 SHG exhibit a monotonic reduction with increasing Mg concentration; nonetheless as shown below, the electronic anharmonicity increases with the Mg concentration for these coefficients as well.
In the classical nonlinear spring model, the energy dispersion of the SHG nonlinear susceptibility is given by, , where  SHG ,  e, ,   and  photon are the SHG susceptibility, anharmonicity of the nonlinear spring, bandgap, and probing photon energy. [21,41]In particular,  e, is the strength of the lowest power anharmonic term in the energy profile for a bound electron oscillating with respect to its nuclei.The anharmonicity,  e, , describes the intrinsic origin of SHG response and captures the physics underlying design methods for highly efficient NLO crystals, including triangle-planar anion groups, second-order Jahn-Teller effects, and lone electron pair cations. [15,51]Based on this dependence,  SHG tends to be suppressed by the larger bandgap at higher Mg concentrations. [52]wever, nearly 50% enhancement of  33 SHG up to 23% Mg indicates a significant enhancement in the SHG coefficients, which is unexpected.By correcting the raw   SHG data in Figure 2d for the , one can obtain a quantity proportional to the anharmonicity of the electronic well,  e, as shown in Figure 2e.The increase of  e, for all three susceptibilities suggests that Mg substitution promotes the anharmonicity of the electronic potential well along both the ordinary and the extraordinary polarization directions.In particular, a roughly 400% enhancement of the  e,33 is observed, indicating a more significant influence on the electronic potential well along the polar direction.Simultaneously, the spontaneous polarization, Ps decreases as a function of Mg concentration experimentally (Figure 2f); [31] DFT calculations attribute this trend mainly to the reduced Born effective charge of Zn and Mg, from 2.17 in ZnO to an average of 2.03 in Zn0.61Mg0.39Oalong the c direction.Thus, the substantial enhancement of  33 SHG and  e,33 in this material system are both found to be inversely correlated to the change in the spontaneous polarization, contrary to conventional expectations, [53][54][55] suggesting that it is the electronic anharmonicity rather that the spontaneous polarization that determines the SHG coefficients.materials. [20]With increasing bandgaps, the magnitudes of nonlinear optical susceptibilities reduce dramatically.Remarkably, Zn1-xMgxO with x from 0% to 23% clearly demonstrates a substantial enhancement of both  SHG and   , contrary to the overall general trend (highlighted in grey) observed across a broad range of material families.
To develop some intuitive understanding of the mechanism that supports this contrary trend, let us consider a classical anharmonic electronic potential well given by   = where   ,   ,   , , ℏ, and  are, respectively, the electronic potential energy, electronic bandgap, anharmonicity of the oscillator, effective mass, reduced Planck constant, and the relative position between the electron and nuclei. [21,41]Using the classical theory of anharmonicity, one can derive expressions for the dielectric susceptibility (  ) and  SHG as a function of   and   for the non-resonant SHG process as shown in Equation ( 1) and (2): [21,41]   = where  0 is the vacuum permittivity,  e is the number of dipoles per unit volume,  is the electron charge, and  photon is the experimental pumping energy at 0.8 eV.Experimental observation has confirmed a ~16% increase in the   and 400% enhancement in the  ,33 from x = 0% to 23%.length, as confirmed by the lattice parameters. [31]Since the decrease in the bond length will promote the Coulomb repulsion between Zn(Mg) and O, it is postulated that the distorted O-2pz and the Zn 3d and Mg 2p orbitals are likely the driving force for the enhanced nonlinearity in the Zn1-xMgxO system.These orbitals dominate the density of states near the band edge, and thus are the major source for the non-resonant SHG response near it. [56,57][24] The discrepancy in the magnitude of  33 Piezo between our work and other studies is likely due to the clamping effect from the substrate in our thinner epitaxial films.
[60] Starting from the free energy of a system, one can derive the piezoelectric coefficients, given as , where   is the linear ionic susceptibility tensor, Q is the electrostrictive tensor and   is the anharmonicity tensor of the ionic potential well of the Zn or Mg ions in the oxygen tetrahedral cage (Equation S1-6 and Figure S5, Supporting Information).According to the Gibbs free energy of ferroelectrics,   represents the magnitude of cubic term in the Taylor expansion of the ionic potential energy versus ionic displacement away from its equilibrium ionic position at  =   .
Based on the changes in  Piezo ,   and   , the ionic anharmonicity and electrostrictive coefficient can be quantitatively analyzed. [31]Figure 4b highlights the changes in ,   and the corresponding  Piezo among Zn1-xMgxO concentrations.Between x=0 and x= 37%,  varies only slightly (less than 5%), indicating that the electrostrictive response is not the major factor enhancing the piezoelectric response.On the other hand, Ps from DFT exhibits a ~2% reduction (Figure 2f) and the reported ionic dielectric susceptibility,   , exhibits a ~75% enhancement, [31] resulting in ~40% reduction in the anharmonicity,   of the ionic potential well between x = 0 and 0.37.(  = 1 2 0     as derived in the supplementary section 7, Supporting Information) The reduced ionic anharmonicity combined with a slight increase in the electrostriction produces a near 200% enhancement in the piezoelectric coefficient.64][65] Applying LGD theory in Zn1-xMgxO also predicts the lowering of the energy barrier of ferroelectric switching with increasing Mg concentration (Figure S5c, Supporting Information).This further provides insight into the polarization switching mechanism from non-ferroelectric ZnO towards ferroelectric Zn1-xMgxO.That is, the softening of the wurtzite structure (smaller c/a ratio) and flattened double well potential lowers the energy barrier of the polarization reversal, facilitating cation motion through the oxygen at the base of the tetrahedron to reverse the spontaneous polarization.This theoretical framework agrees well with experimental observations of enhanced dielectric constant, [31] increased piezoelectric response, reduction in spontaneous polarization, and switchable ferroelectric polarization with increasing Mg concentration in Zn1-xMgxO, as demonstrated in this study.
To highlight the potential technological interest in this material, piezoelectric force microscopy was utilized to demonstrate fine control over domain patterning in ferroelectric Zn1-xMgxO.Domain reversal with electric fields is far simpler and preferred as compared with domain reversal during synthesis using engineered surface termination in AlN, [66,67] and polarity control in ZnO. [68]Coherence length ( c ) is the size of the domain in a domain grating of period 2 c required for quasi-phase-matched SHG, defined as  c =  ω /4( 2ω −  ω ), where  and  are wavelength and refractive index, and superscript ω represents the corresponding frequency. [69]The calculated  c as a function of fundamental wavelength  ω in Zn0.63Mg0.37O is illustrated in Figure 4c.The  c is found to be 800 nm when a wavelength of  ω = 650 nm is halved to  2ω = 325 nm.The theoretical limit of  c is found to be ~650 nm when  2ω approaches the band edge, the limit of the useful range for nonlinear optics.As proof of the feasibility of QPM in Zn1-xMgxO, a periodic poled pattern with a domain width of 800 nm is demonstrated in Zn0.63Mg0.37O.The BE-PFM amplitude (Figure 4d) and phase (Figure 4e) show a clear domain contrast with opposite polarization, demonstrating the realization of precise and controllable QPM for any targeted wavelength below the bandgap.(e) are 0.8 μm.

Conclusions
In summary, Zn1-xMgxO brings exceptional optical, electrical, and electromechanical responses to life in a single multifunctional platform.The simultaneous increase in both the second-order nonlinear optical susceptibility and the optical bandgap in Zn1-xMgxO suggests that increasing the anharmonicity of the electron potential well can offset a bandgap increase for designing NLO materials in the ultraviolet spectral region.On the low-frequency end, adding Mg reduces the anharmonicity of the ionic potential well, leading to an enhancement of the piezoelectric response.
This study thus demonstrates that the anharmonicity can be independently engineered in the optical and low-frequency regimes through chemical pressure.This may be key to optimizing exceptional piezoelectric and nonlinear optical response in one material.Moreover, precisely controlled periodic ferroelectric domain patterns for optical quasi-phase-matching (QPM) were demonstrated in Zn1-xMgxO, which opens new possibilities for NLO optical waveguides for efficient nonlinear optical conversion and all-optical switching schemes deeper into the ultraviolet range.The presence of ferroelectricity with large polarization, high ionic dielectric susceptibility, improved piezoelectric response, large tunable bandgap, and significantly enhanced SHG response make Zn1-xMgxO a promising candidate for applications such as microelectromechanical systems, integrated optics, nonlinear photonics, and strain-tunable photonics.

Experimental Section
Growth of Zn1-xMgxO films, XRD, SEM, EDS, AFM: Zn1-xMgxO thin films were grown via a radio frequency (RF) magnetron co-sputtering technique using metallic Zn and Mg targets.where  , ℎ,  ,  and   are the absorption coefficient, Planck's constant, photon frequency, proportionality constant, and bandgap. [40] is the measure of direct or indirect transition and  is set to 0.5 for the best fitting condition in this study.
SHG Measurement: SHG has been widely used to confirm noncentrosymmetric structures, ferroelectric response, and wavelength conversion efficiency.The polarization-resolved SHG measurements were carried out at room temperature in 45-degree reflection mode on samples.The SHG measurement is an all-optical technique where two photons of frequency ω with fields Ej and Ek and polarization directions j and k, respectively, interact with a material with a non-zero dijk tensor and generate a polarization   2ω of frequency 2ω in the i direction.The SHG intensity, I 2ω , was detected with a Hamamatsu photomultiplier tube.A Ti-sapphire laser (Spectra-Physics) with an output of 800 nm, 80 fs pulses at 2 kHz frequency was used.The fundamental light with a central wavelength at 1550 nm is generated through an optical parametric amplifier after the Coherent Libra Amplified Ti: Sapphire femtosecond laser system (85 fs, 2 kHz).Here, (, , ) is the lab coordinate system where experiments are performed.The plane of incidence is defined in the x-z plane, and z corresponds to the surface normal.The surface normal of Zn1-xMgxO is the (0001) plane.
[72][73][74][75][76][77][78] Supercell structures were generated using the Atomic Simulation Environment (ASE) module. [79]To consider atomic fraction of x = 0-40%, a supercell size of 3 × 3 × 2 was considered with minimal Mg clustering.The k-point spacing in the first Brillouin zone was set to 0.05 Å −1 , and the kinetic energy cutoff for the electronic wavefunctions was set to 80 Ry with a charge density cutoff of 320 Ry.The lattice parameters and c/a ratios of the unit cell of ZnO and the supercell structures of Zn1−xMgxO were calculated using geometry optimization.The total energy and force thresholds were set to 10 −5 Ry and 10 −4 Ry/bohr, respectively.Band gaps were calculated within the DFT+U approximation at fixed (PBE) geometry. [80][83] A detailed description of this nonempirical DFT+U framework is provided in the supplementary information.
Using this approach, the spontaneous polarization was calculated via the Berry phase method. [84,85]S-PFM, BEPFM measurement, periodic poling: Interferometric displacement sensing PFM measurements were taken using an Oxford Instruments Cypher atomic force microscope equipped with a Polytec OFV-5000 Modular Vibrometer routed to the tip for measuring tip displacements, i.e., the piezoelectric coefficient ( 33 Piezo ).AC voltages ranging from 2 to 10 V at 40 kHz were used for a linear extraction of  33 Piezo , i.e. the slope of IDS-PFM amplitude versus applied AC voltage.
The band excitation PFM measurements were acquired using an Oxford Instruments Cypher atomic force microscope with an imaging AC voltage of 2V, and periodic poling voltage of ±50V.
Details describing band excitation functionality can be found elsewhere. [86]For all atomic force microscopy measurements, Budget Sensor Electri Muli75-G Cr/Pt coated probes were used.
Remanent Polarization Measurement: Remanent polarization values for Pt/ZMO/Pt capacitors were extracted from bipolar Polarization-Electric field (P-E) hysteresis loops driven at ≥ 4 MV/cm with a 100 Hz triangle wave at room temperature.A precision Multiferroic II tester (Radiant Technologies) was used to measure the P-E loops and to extract the remanent polarization values.
Each reported value is averaged over 10 measurements on 5 capacitors.

Landau-Ginzburg-Devonshire (LGD) Theory for Ionic Response
[3] The Gibbs free energy can be written in the form indicated below, ,  ,  ,  ,  , and ,  are the free energy potential, polarization, compliance, electrostriction, stress, and Landau expansion coefficients.The subscript  represents the contributions from ions.Since higher-order terms tend to play less effect on the energy, for the scope of the study, we keep the form up to fourth-order term  4 .The equilibrium state refers to the polarization state with minimum energy (), and the corresponding polarization refers to the spontaneous polarization ( , ) of the system.Based on the expression of the energy potential well, five key parameters, namely, polarization ( , ), energy barrier (  ), dielectric susceptibility (  ), anharmonicity (  ), and piezoelectric coefficient ( Piezo ) can be obtained following Equation S1.
Under zero stress, the magnitude of the spontaneous polarization ( , ) can be obtained following .Taking the factor of  3 using Taylor expansion at  , can reveal the anharmonicity (   ) of the potential well.The origin of piezoelectricity using double well potential has been laid out by Haun and Damjanovic [4,5,1,6] .
Therefore, five properties as a function of Landau expansion coefficients can be summarized below [4] , where  0 represents the vacuum permittivity.and (c) show that both  , and   will decrease consistently with higher Mg concentration.Both DFT and experiment [7] confirms that  , reduces as increasing Mg concentration.Moreover, experimental observation [7] reveals the emerging ferroelectricity as substituting more Zn with Mg, indicating the reduction of the energy barrier.Figure S5(d) predicts the enhancement of the dielectric constant, which is also well supported by the experimental results [7] .This agreement highlights that   is decreasing in the Zn1-xMgxO system.On the other hand,   is expected to increase in the Zn1-xMgxO, since it is the measure of the slope of outer energy wall and describes the repulsive energy as Zn ions move toward the oxygen atom along the c axis.Since the c axis shrinks with more Mg, the closer distance between Zn(Mg) and O along the c axis indicates higher repulsive energy, leading to a higher   .Figure S5(e) and (f) further illustrate changes of   and  Piezo in Zn1-xMgxO.In particular, the inverse proportionality between   and  Piezo emphasizes that a less unsymmetric potential well at  , is essential for a large  Piezo , in contrast to the  SHG where an increase in the electronic nonlinearity is required.However, utilizing anharmonicity at distinct frequencies lays out a possible route for achieving enhancements of both SHG and piezoelectricity simultaneously.

SHG Fitting Analysis
In the SHG polarimetry analysis, three incident polarization angles are representative for uniquely determining nonlinear optical coefficients.10] The measurement was performed at 45-degree reflection geometry.The measured SHG intensity was then compared with a reference crystal, a wedged X-cut LiNbO3 in this study, to calibrate the absolute nonlinear optical coefficients.Both films and LiNbO3 are aligned such that the optical axes are in the plane of incidence.The  33 SHG of LiNbO3 was 19.5 pm/V, calibrated using Miller's rule from Shoji et al.'s work, [9,11] and the SHG expression of wedged X-cut LiNbO3 can be found from our previous work. [12]The thicknesses of films are set to 150 nm, as confirmed by ellipsometry.

Figure 3a summarizes
Figure3asummarizes the  SHG and   among state-of-art nonlinear optical (NLO)

Figure 3b and c illustratesEquation 1 and 2 .
Figure 3b and c illustrates the relative changes of   and  SHG as compared with pure ZnO following Equation 1 and 2. With increasing   , the   decreases due to the reduction of the

Figure 4 .
Figure 4. PFM results and QPM of Zn1-xMgxO.(a)  33 Piezo acquired from interferometric displacement sensing PFM on Zn1-xMgxO series.(b) Contour plot of  Piezo as a function of  and A 2-inch (0001)-Al2O3 wafer was rinsed successively in isopropanol, methanol and acetone for 1 min, followed by an ultraviolet ozone treatment for 10 min.The wafer was placed in a sputtering chamber until the base pressure was reduced to 1 × 10 -7 Torr at 300 o C. Deposition of a 200 nm thick (111)-Pt layer on (0001)-Al2O3 was performed at 10 mTorr in Ar atmosphere.Zn1-xMgxO thin films were then sputtered on the (111)Pt//(0001)-Al2O3 substrate at room temperature.During depositions, a gas flow of Ar and O was fixed to 16 sccm and 4 sccm respectively, and a total pressure was kept to 4 mTorr.The Mg content was controlled by changing the Mg target power between 0 to 48 W, while the Zn target power was fixed at 23 W. The chemical composition and surface morphology were confirmed by energy dispersive spectroscopy (EDS) and scanning electron microscope (SEM) in Zeiss Sigma.Crystal structure was investigated via x-ray diffraction (XRD) using Cu-Kα1 (1.5406 Å) in a Panalytical Empyrean, and the surface roughness was measured using atomic force microscope (AFM) in Asylum MFP3D.Optical Spectroscopic Ellipsometry: Optical spectroscopic ellipsometry was conducted using Woollam M-2000 and Woollam M-2000F spectroscopic ellipsometers.Woollam M-2000F has a fixed incident angle of around 60 o , and three incident angles (55 o , 65 o , and 75 o ) were collected using Woollam M-2000.The two results are cross confirmed and combined to reveal the complex refractive indices from 200 nm -1800 nm using the same model.Tauc method for optical bandgap: The Tauc equation is expressed as (ℎ) 1/ = (ℎν −   ),

Figure S5 outlines the
FigureS5outlines the ionic potential wells and five properties as a function of Landau coefficients. )

Figure S5 .
Figure S5.Ionic tdouble-well potential and polarization-related properties as a function of Landau

Table S1 .
Cauchy parameters of Zn1-xMgxO films.A, B, and C are Cauchy parameters.Subscripts o and e represent ordinary and extraordinary parts.