Auxetic piezoelectric effect in heterostructures

Inherent symmetry breaking at the interface has been fundamental to a myriad of physical effects and functionalities, such as efficient spin–charge interconversion, exotic magnetic structures and an emergent bulk photovoltaic effect. It has recently been demonstrated that interface asymmetry can induce sizable piezoelectric effects in heterostructures, even those consisting of centrosymmetric semiconductors, which provides flexibility to develop and optimize electromechanical coupling phenomena. Here, by targeted engineering of the interface symmetry, we achieve piezoelectric phenomena behaving as the electrical analogue of the negative Poisson’s ratio. This effect, termed the auxetic piezoelectric effect, exhibits the same sign for the longitudinal (d33) and transverse (d31, d32) piezoelectric coefficients, enabling a simultaneous contraction or expansion in all directions under an external electrical stimulus. The signs of the transverse coefficients can be further tuned via in-plane symmetry anisotropy. The effects exist in a wide range of material systems and exhibit substantial coefficients, indicating potential implications for all-semiconductor actuator, sensor and filter applications.


Note S2: Crystallographic orientation dependent piezo-coefficients in bulk materials
Manipulating crystallographic orientation has also been utilized to optimize piezoelectric coefficients and electromechanical coupling factor in conventional piezoelectric materials with bulk inversion asymmetry, such as quartz (S37) and BaTiO3 (S38).Here we take tetragonal BaTiO3 crystal as an example and show that its longitudinal piezoelectric coefficient  "" and transverse coefficients  "# &  "$ are always of opposite signs measured at any cutting orientation directions.The piezoelectric tensor of tetragonal BaTiO3 in its conventional coordinate axis is of the form: The definition of  and  is shown in Fig. S2a.The coordinate set is first rotated along -axis by  and then rotated along the x-axis by .
Therefore, the cutting orientation dependent piezoelectric coefficient  %&( ) (, ) can be calculated by following equation: We found that

Note S4: Frequency dependent piezoelectric response
As shown in Fig. S4a, the current induced by the interface piezoelectric effect in both (001)-and ( 111)-oriented Au/Nb:SrTiO3 junctions increases linearly while increasing stress frequency.This is consistent with the equation: (S40) where  " is the amplitude of current density output by Schottky junctions,  is the frequency of the stress and  % is the amplitude of the stress.At all frequencies the phase of piezo-current generated by  "" of (001)-oriented Au/Nb:SrTiO3 and both  "" and  "# of (111)-oriented Au/Nb:SrTiO3 stays at ~90°, while that of  "# in (001)-oriented remains constant of −90°.
This is consistent with that shown in Fig. 2 of the main text.The physical properties of the Schottky junctions can be characterized by classical semiconductor measurements.The dielectric permittivity  " and built-in potential  .%can be determined from the current-voltage ( vs ) and capacitance-voltage ( vs ) curves based on following equations: (S41) where  is the current density,  * is the Richardson constant,  is temperature,  0 is the Boltzmann constant, Φ 0 is the potential barrier height,  is the ideality factor,  is the capacitance per unit area and  2 is the relative dielectric permittivity.The ideality factor  can be acquired by fitting the ln    as shown in the insets of Fig. S5a, B. Given the values of  1 and , the effective permittivity  + can be obtained by  -$   fitting as shown in the insets of Fig. S5c, d.
The dopant density  1 can be approximated as the effective carrier density in Nb:SrTiO3 measured by the Hall effect.From standard Hall measurements at room temperature with magnetic field up to 2 T, the carrier density of (001)-and ( 111)-oriented Nb:SrTiO3 crystals are respectively 2.16 × 10 $'  -" and 2.43 × 10 $'  -" .Based on these parameters, the values of  "# and  "" of (001)-and ( 111)-oriented junction can be predicted based on Eq. ( 1) of the main text.All these values are given in Table S2.
Table S2.Values of physical parameters of (001)-and ( 111)-oriented Au/Nb:SrTiO3 junctions The values of the longitudinal and transverse piezoelectric coefficients, i.e.,  "# ,  "$ and  "" , of (110)-oriented Au/Nb:SrTiO3 junction are quantitatively characterized by using sinusoidal stress wave (see Fig. S6).The stress-induced current increases linearly with the amplitude of the applied stress, confirming their piezoelectric nature.The phase of the piezoelectric current of both  "# and  "" is 90° with respect to the dynamic stress, indicating its positive sign of their values, while the current phase of  "$ is about −90°, indicating its negative sign.Based on these measurements, the values of  "# ,  "$ and  "" are 6.2   ⁄ , −6   ⁄ and 9.3   ⁄ , respectively.
The electrical properties of (110)-oriented Au/Nb:SrTiO3 have also been characterized by current-voltage curve and capacitance-voltage curve (see Fig. S7).The fitted values of its parameters are shown in Table S3.
Table S3.Values of physical parameters of (110)-oriented Au/Nb:SrTiO3 junctions Based on these measurements, the electrical parameters of these (112)oriented Au/Nb:STO junctions derived using Eq.S8 and S9 in the Supplementary Information are summarized in Table S4.With these parameters, its piezoelectric coefficients can be predicted using Equation 1 of the main text, which are given as: Table S4.Values of physical parameters of (112)-oriented Au/Nb:SrTiO3 junctions 1.46 1.26 328 1.14 -1.4 1.5 6.3According to the modern theory of polarization (S46, S47), only the change of polarization is well defined and can be associated with the experimentally measurable quantity, i.e., switching current.Hence, we calculate the polarization as the interface charge transfer per area given by following equation: where ∆ is the interface charge transfer which can be obtained by employing the Bader charge analysis (S12), and  is the interface area.Interface piezoelectricity can be described by the piezoelectric strain coefficient, which is defined as the ratio of interface polarization  " with respect to an applied uniaxial stress  % , i.e., or by the piezoelectric stress coefficient, which is defined as the ratio of interface polarization  " with respect to an applied uniaxial strain  % , i.e., "% =  "  %
The atomic structures of the SrTiO 4+ -terminated Au/SrTiO3 (110) interface are shown in Fig. S9a.Fig. S9b shows the dependence of its interface polarization on uniaxial stress (upper panel,  "% ), on uniaxial strain with ions relaxed (middle panel,  "% ) and with ions clamped (lower panel,  "% (!) ).Meanwhile, the calculation results of the (111)-oriented Au/SrTiO3 interface with Ti 4+ -termination are given in Fig. S10.The interface polarization  " exhibits good linear relationship with the uniaxial stress and uniaxial strain applied along all the directions.The piezoelectric strain coefficient  "% and piezoelectric stress coefficient  "% can respectively be estimated by performing linear fitting between the polarization  " and the uniaxial stress  % and uniaxial strain  % (see Table S5).It is remarkable that the calculated piezoelectric strain coefficients are of the same signs and same order of magnitudes with both experimental and phenomenological results.
Interestingly, these values have the same signs with the piezoelectric strain coefficients and more importantly, their relative magnitudes are more consistent with experimental results as  "# ≈ − "$ for (110) interface and  "# ≈  "$ for (111) interface.We propose that in our heterostructure models, due to the existence of metal layer which has better malleability than semiconductor layer, the application of uniaxial stress unavoidably introduces a relatively large strain in the other two lattice directions.It implies that in comparison to uniaxial stress condition, the uniaxial strain condition (with the lattice fixed) is closer to the experimental reality.Hence, the calculated piezoelectric stress coefficients have a better consistency with experimental results than the piezoelectric strain coefficients.Furthermore, the piezoelectric stress coefficients can be decomposed into the clamped-ion and internal-strain contributions.
we find that all the positive interface piezoelectricity, i.e.,  "# and  "" of (110) interface,  "# ,  "$ and  "" of (111) interface, arises from the domination of positive internal-strain term.In contrast, the clamped-ion term only dominates the negative  "$ of (110) interface.Table S5.Calculated piezoelectric strain coefficients   (in units of pC/N) and piezoelectric stress coefficients   (in units of μC/cm 2 ).The clamped-ion" (  "% (!) ) and "internal-strain" ( "% (%) ) contributions of SrTiO 4+ -terminated Au/ STO(110) and Ti 4+ -terminated Au/STO (111) interfaces have also been resolved.To conform the general nature of the auxetic piezoelectric effect and demonstrate its thickness-limitation free nature, we systematically studied both (001)-and (111)-oriented [SrTiO3/Ba0.6Sr0.4TiO3/BaTiO3]tricolour superlattice with a total thickness of about 100 nm (see Fig. S13).Each layer is set to have a nominal thickness of about 3 nm and their structures have been characterized by XRD measurements shown in Fig. S14a, b.The electrical properties of the superlattices have been detected by the capacitance vs voltage characteristics (see Fig. S14c, d).It indicates that there exists built-in potential of about 2.2 V and 0.86 V in (001)-and ( 111)-oriented superlattices, respectively.Such strong built-in fields induce substance electric polarization in the superlattices pointing from bottom electrode (i.e., Nb:SrTiO3 substrate) to top electrode (i.e., Pt layer) and thus, give rise to the piezoelectric effect.
The piezoelectric property of the tricolour superlattices has been characterized using both the direct effect (i.e., applying stress and measuring short-circuit current) and the converse effect (i.e., applying voltage and measuring deformation/strain).As shown in Fig. S15,  "" and  "# of the (001)-oriented superlattice respectively show phase contrast of about −90° and 90° with respect to the applied AC stress, indicating positive and negative signs of these two piezo-coefficients.Based on the linear fitting of the current density as function of the applied stress amplitude, we obtain the values of the piezoelectric coefficients as  "" = 18.4 / and  "# = −6.6 /.In contrast, both  "" and  "# of the (111)-oriented superlattice exhibit phase values of about −90° .The piezo-coefficient of the (111)-superlattice is measured as  "" = 6.6 / and  "# = 3.3 /, which corresponding to the auxetic piezoelectric effect.

Note S11. In-plane orientation dependent piezoelectric coefficients
Rotating the in-plane axis  ) and  ) along the  ) -axis in the newly oriented crystal will also modulate the form and value of the electrostriction tenor and piezoelectric coefficients.
Among the three crystallographic orientations of interest, both (001)-and ( 111)-oriented SrTiO3 crystals exhibit in-plane isotropic physical properties due to their  7 and  " in-plane rotational symmetry.However, the (110)-oriented SrTiO3 shows in-plane anisotropic properties due to its  $ in-plane rotation symmetry.The electrostriction coefficients  """" ) ,  ##"" ) and  $$"" ) of ( 110)-oriented SrTiO3 crystal by rotating the  ) &  ) along the  ) -axis are given as where  is the angle between the [110] crystallographic direction and the  ) -axis that is rotated

Note S12: Preparation and characterization of Mo/4H-SiC junctions
Mo/4H-SiC planar Schottky diodes were fabricated on highly n-type (nitrogen-doped), 4° off-axis 4H-SiC substrates that had a 35 μm epitaxial layer (1x10 15 cm -3 nitrogen-doped) grown on top of it.All chips underwent a standard RCA1 / HF (10%) / RCA2 / HF (10%) routine, after which the individual active device areas were defined and patterned using a conventional photolithography and mesa-isolation (dry etch) process.To assist the isolation between devices on each chip, a 1 μm thick silicon dioxide (SiO2) layer was deposited by means of low-pressure chemical vapour deposition (LPCVD) using tetraethyl orthosilicate (TEOS) as Si precursor.This was then followed by the backside deposition of Ti (30nm) / Ni (100nm), leading to the ohmic contact formation after a rapid thermal anneal at 1000°C for 2 minutes in Ar (5 slm).The Schottky contacts were subsequently formed by opening up a window in the thick field oxide layer and evaporating 100 nm of Mo before annealing them at 500 °C in Ar 5 slm ambient.Processing was finished after a 1 μm thick Al metal overlay was evaporated on top of the device, serving as a field plate.

Fig. S2 .
Fig. S2.Crystallographic orientation dependent piezoelectric coefficients of BaTiO3 crystal.a), Schematic showing the geometrical correlation between new coordinate set {  } and default set { !,  !,  !}.In this rotation configuration,  -axis retains in the ( ! ! ) plane. refers to the angle between  -axis and  !-axis. refers to the angle between -axis and  !-axis.3D-dimensional spherical polar plot of b)  "" , c)  "# , and d) "$ .Here angle  is set in the range of (0, /2).The light-yellow indicates positive value while cyan refers to negative value of piezoelectric coefficients.

Fig. S4 .
Fig. S4.Frequency dependent piezoelectric response of (001)-and (111)-oriented Au/Nb:SrTiO3 junctions.The stress for all measurements is set as 1 MPa.The solid curves in A are linear fits.

Fig. S7 .
Fig. S7.Electrical characterization of (110)-oriented Au/Nb:SrTiO3 junction.a), Current-voltage curves.Inset is the ln(J) vs voltage curve and its linear fit.b), The capacitance-voltage curve.Inset is the  -$ vs voltage curve and its linear fit.
The experimental characterization of the piezoelectric effect of the (112)-orientations are shown in Fig.S8 c-f.The piezoelectric coefficients are measured as  "" = 4.2   ⁄ ,  "# = −1.0  ⁄ ,  "$ = 0.93   ⁄ , which are fairly close to the predicted values.Thus, the experimental results are consistent with our phenomenological theory that further consolidates our work.

Fig. S8 .
Fig.S8.Electrical and piezoelectric characterization of (112)-oriented Au/Nb:SrTiO3 junction.a, Current vs voltage characterization and b, capacitance vs voltage curve of the junction.Stress dependent amplitude of current density induced by c)  "" , e)  "# and g)  "$ .The phase dependence on stress is shown respectively in d), f) and h).

Fig. S9 .
Fig. S9.DFT calculation results of (110)-oriented Au/SrTiO3 interface.a), Schematic illustrates the atomic structure of the heterostructure.b), Dependence of interface polarization on uniaxial stress (upper panel), on uniaxial strain with ions relaxed (middle panel) and with ions clamped (lower panel).The Sr, Ti, O, and Au atoms are depicted as green, light blue, red, and golden spheres, respectively.The polarization of unstrained interface is set to zero.

Fig. S10 .
Fig. S10.DFT calculation results of (111)-oriented Au/SrTiO3 interface.a), Schematic illustrates the atomic structure of the heterostructure.b), Dependence of interface polarization on uniaxial stress (upper panel), on uniaxial strain with ions relaxed (middle panel) and with ions clamped (lower panel).The Sr, Ti, O, and Au atoms are depicted as green, light blue, red, and golden spheres, respectively.The polarization of unstrained interface is set to zero.
Fig. S16.Effect of rotating the  ) &  ) along the  ) -axis on electromechanical properties of (110)-Au/Nb:SrTiO3 junctions.a), Angle  dependent electrostriction coefficients and b) variation of piezoelectric coefficients as functions of .

Fig. S21 .
Fig. S21.Demonstration of the converse auxetic piezoelectric effect in Schottky junctions.a), Schematic shows the conventional piezoelectric effect induced cantilever bending.b), Schematic shows the auxetic piezoelectric effect induced cantilever bending.The layer with white arrows refers to the piezoelectric active layer of the cantilever, which is the depletion region in the case of Schottky junctions.The dependence of bending amplitude and vibration phase of (001)-oriented Nb:SrTiO3 junction cantilever on c) the amplitude of the 1.33kHz AC voltage and d) the frequency of AC voltage with an amplitude of 0.25 V. e) and f) respectively show the vibration dependence of the (111)-oriented Nb:SrTiO3 junction cantilever on AC voltage amplitude and frequency.The AC frequency used in e) is 1.33kHz and the amplitude used in f) is 0.25 V.