Piezoelectricity in hafnia

Because of its compatibility with semiconductor-based technologies, hafnia (HfO2) is today’s most promising ferroelectric material for applications in electronics. Yet, knowledge on the ferroic and electromechanical response properties of this all-important compound is still lacking. Interestingly, HfO2 has recently been predicted to display a negative longitudinal piezoelectric effect, which sets it apart from classic ferroelectrics (e.g., perovskite oxides like PbTiO3) and is reminiscent of the behavior of some organic compounds. The present work corroborates this behavior, by first-principles calculations and an experimental investigation of HfO2 thin films using piezoresponse force microscopy. Further, the simulations show how the chemical coordination of the active oxygen atoms is responsible for the negative longitudinal piezoelectric effect. Building on these insights, it is predicted that, by controlling the environment of such active oxygens (e.g., by means of an epitaxial strain), it is possible to change the sign of the piezoelectric response of the material.

where V 0 is the driving ac voltage amplitude and Q is the quality factor associated with the cantilever dynamics at resonance [1]. The calibration procedure used to evaluate the quality factor and to quantify d 33,eff is described in section C.
The PFM phase signal, on the other hand, is related to the sign of the piezoelectric coefficient. PFM measures the sample contraction or expansion via the converse piezoelectric effect, where the strain developed in the out-of-plane direction, η 3 , due to the applied electric field in this direction, E 3 , is given by Eq. 2: where 0 and r are vacuum permitivity and the relative permitivity of the sample, Q 33 is the electrostriction coefficient and P s is the spontaneous polarization [2]. As illustrated In Supplementary Figure 1, in a material with a positive d 33,eff , such as PZT [3], the sample will expand (contract) when the applied field and the polarization are in the same (opposite) direction. In PFM, a small ac driving voltage is used to probe the sample expansion and contraction using the lock-in amplifier technique. For a material with positive d 33 , when the polarization is oriented downward, the sample will expand during the positive half-cycle and contract during the negative half-cycle, thus the sample deformation and the driving ac signal are in phase (top panel in Supplementary Figure 1(a)), while for the upward polarization, there is an 180 • phase difference between the driving ac and the sample deformation (bottom panel in Supplementary Figure 1(a)). This scenario was observed in the PZT capacitors ( Figure 2(a) in the main text). Following a similar argument, the phase signal will be opposite in a material with a negative d 33,eff -the sample oscillations and the driving field will be in-phase when the polarization is oriented upward and 180 • out of phase for the downward polarization (Supplementary Figure 1 (b))-. This scenario is relevant to the PFM phase signal measured in the PVDF film, a well-known negative piezoelectric material [4,5] (Figure 2(b) in the main text). Under the same measurement conditions, a similar PFM phase signal behavior was observed in the 20-nm-thick La:HfO 2 capacitors (Figure 2(c) in the main text), from which it can be inferred that the sign of the d 33,eff is negative. This behavior was reproducibly observed in multiple capacitors while using different types of cantilevers.
B. Identification of the phase offset and determining the correct PFM phase During the PFM hysteresis measurement, there could be an uncontrollable instrument-related parasitic phase offset contributing to the measured PFM phase signal, which is typically constant for the same AFM tip and measurement parameters. To identify this phase offset and find out the actual PFM phase related to the piezoelectric deformation of the sample, we have adopted two different approaches: the first one uses a reference sample with a known d 33 value, while the second approach uses the electrostatic signal from the differential signal between the bias-on and bias-off PFM loops.
In the first approach, a reference sample, a PZT capacitor, is used to obtain the PFM hysteresis loop in the bias-off mode. In general, a measured raw piezoresponse (PR) signal can be represented as PR = cos(ϕ − ∆ϕ) (4) where A and ϕ are the amplitude and phase signals due to the genuine piezoelectric response, respectively, and ∆ϕ is a parasitic phase offset. For PZT, which has a positive d 33 coefficient, the PR signal is in phase with the ac modulation signal for the downward polarization (or out of phase for the upward polarization), so that ϕ should be equal to zero at the far positive dc bias of the PFM hysteresis loop corresponding to the downward polarization. The parasitic phase offset ∆ϕ can be found from the actually measured PFM phase signal ϕ − ∆ϕ and then deduced from all subsequent measurements carried out with the same AFM tip and the same measurement parameters.
In the second approach, PFM hysteresis loops in the bias-on (measured when dc bias is present) and bias-off (measured when dc bias is not present) regimes are measured simultaneously for the sample under investigation. Typically, the AFM cantilever arm is held outside of the top electrode during the measurement, a strong electrostatic interaction between the cantilever arm and the sample will be contributing to the bias-on PFM loop, where the raw piezoresponse bias-on signal (PR on ) is a sum of the bias-off PR signal (PR off ) and the electrostatic signal, i.e., where k is the AFM cantilever spring constant, dC/dz the cantilever-sample capacitance gradient, V dc the dc voltage, and V 0 the ac amplitude. The term that has a linear dependence on V dc is due to the electrostatic interaction when dc voltage is on, which gives rise to a negative slope versus V dc due to the negative sign of dC/dz. The ∆ϕ is chosen to satisfy either ϕ on − ∆ϕ = 0, or ϕ on − ∆ϕ = 180 • on the far negative V dc side of the hysteresis loop, whichever gives a negative slope for the differential signal (PR on − PR off ), i.e., take the phase offset to be the bias-on phase at the far negative V dc side, and use Eqs. 4 and 5 to calculate the P R on and PR off signal. If the differential signal (PR on − PR off ) results in a negative slope, this would be the correct phase offset. Otherwise, take ϕ on − ∆ϕ = 180 • (use the bias-on phase at the far negative V dc side), which would then introduce a negative sign for the (PR on − PR off ) signal. This phase offset ∆ϕ is then subtracted from the raw PFM phase to obtain the actual PFM phase signal. Supplementary Both of the two approaches to figure out the correct PFM phase worked well and gave consistent results during our measurements.

C. Calibration procedure for evaluation of the piezoelectric coefficient
The PFM amplitude signal is proportional to the d 33,eff (Eq. 1). An order of magnitude estimate of the d 33,eff for an unknown material can be obtained by comparing the PFM amplitude of the unknown material with that of a reference material [6]. Here, we use the PZT capacitor as a reference sample to calibrate the piezoelectric coefficient of La:HfO 2 in the quasi-static regime, i.e. at a frequency much lower than that used in the dynamic PFM measurements. First, the quasi-static strain loops were measured in the IrO 2 /PZT/Pt capacitors by monitoring the deflection signal of the AFM cantilever during application of a triangular voltage sweep at 1 Hz ( Supplementary Figure 3(a)). The deflection signal was then converted to the actual displacement of the cantilever through calibration of the optical lever sensitivity using the force-distance curves. From the slope of the quasi-static strain loops, the d 33,eff for the IrO 2 /PZT/Pt capacitor was calculated to be 48 pm V −1 , which agrees well with values reported in literature [7]. This value is then used to obtain the quality factor of the cantilever using Eq. 1. This allowed us to obtain the calibrated piezoresponse PR loops or the d 33,eff -voltage loops for the IrO 2 /PZT/Pt capacitors (Supplementary Figure 3(b)). The same cantilever was then used to measure the PFM hysteresis loops in the TiN/La:HfO 2 /TiN capacitors. The d 33,eff is calculated from Eq. 1 using the obtained quality factor for the specific AFM tip. The obtained d 33,eff value of 2 − 5 pm V −1 in the TiN/La:HfO 2 /TiN capacitors (Supplementary Figure 3(c)) matches the reported d 33,eff values obtained by means of the interferometric techniques [8][9][10] and PFM [6].
In addition, the piezoresponse loops will have anti-clockwise rotation in the material with positive d 33,eff (similar to the polarization-voltage loops), while it will exhibit a clockwise rotation in a negative d 33,eff . Supplementary  Figure 3(c) shows that the PR loop in the TiN/La:HfO 2 /TiN capacitors corresponds to a negative d 33,eff coefficient.

SUPPLEMENTARY NOTE 2: LINEAR-RESPONSE FORMALISM
Here we mainly follow the formalism for the response properties of a material introduced in Ref. [11]. Let us consider a reference structure of an insulating crystal with cell volume Ω 0 that is in equilibrium at vanishing macroscopic electric field. We can expand the energy per unit cell volume E(u, η, E) as a function of atomic displacements u m , homogeneous strain η j , and applied electric field E α around the reference state as follows: where E 0 is the energy density of the reference structure. In this equation the first-order coefficients A u m , A E m and A η m correspond to the forces (F m = −Ω 0 A u m ); polarizations (P m = −A E m ) and stresses (σ m = A η m ), respectively. We assume the atomic coordinates and strains are fully relaxed in the reference system. Hence, the coefficients A u and A E are zero. The second order coefficients are defined as follows. The force constant matrix: The purely-electronic elastic tensor:C the Born effective charges: the force-response internal strain tensor: and the purely-electronic piezoelectric tensor:ē The barred quantitiesC mn andē mn represent the purely-electronic response, and are computed with ionic coordinates to their values at the reference state.

Relaxed-ion tensors
To obtain the physical static response properties one must consider the contribution due to ionic relaxations. We compute such relaxed-ion quantities by introducing the functional obtained from Eq. 6 by setting ∂E/∂u n = 0. For details on the full derivation see Ref. [11]. Defining and we get the physical elastic and piezoelectric tensors by using Eqs. (7)-(11) In Eq. (16), e αj is also known as the piezoelectric stress coefficient which is defined under the conditions of controlled E and η. The piezoelectric strain coefficient (d αj ) is defined under the conditions of controlled E and σ, and is typically what is directly accessible in experiments. d αj is defined as and is related to e αj as follows: where S jk (= C −1 jk ) is the compliance tensor.   Table 6. Λ tensors for the symmetry-inequivalent atoms of the ferroelectric phase of PbTiO3 (in eVÅ −1 ). The 3 rows correspond, respectively, to the 3 spatial directions; the 6 columns correspond, respectively, to the 6 strain indices in Voigt notation.
Supplementary Figure 1. Relationship between the applied ac field and the sample oscillation due to the piezoelectric deformation for materials with positive d33 (a) and negative d33 (b). (a) In a material with positive d33, the sample oscillations and the applied field are in-phase (180º out of phase) when the polarization is pointing downward (upward). (b) In a material with negative d33, the sample oscillations and the applied ac field are in-phase (180º out of phase) when the polarization is pointing upward (downward). The black cuboids indicate the un-deformed state of the domains when no external field is applied, while the red dashed cuboids indicate the deformed state of the domains under an external field due to the converse piezoelectric effect. The blue arrows denote the direction of the applied field, E, and the black arrows denote the direction of the spontaneous polarization, Ps.

E p i t a x i a l s t r a i n ( % )
Supplementary Figure 6. Computed polarization as a function of epitaxial strain in HfO2. The obtained increase of P3 upon epitaxial compression may seem at odds with the negative value of e33: an epitaxial compression (η1 = η2 = ηepi < 0) leads to an out-of-plane lattice expansion (η3 > 0), which yields a positive change of P3 because e33 < 0. Note, however, that the total change in P3 also depends on η1 and η2, on account of the (also negative) e13 and e12 coefficients. Indeed, these transversal piezoelectric effects are responsible for the observed increase of P3 upon epitaxial compression.