MATERIALS SCIENCE AND CHEMISTRY

Introduction

Reverse piezoelectricity describes the deformation of materials in the presence of an external electric field. To act piezoelectric, the material has to belong to the ferroelectrics, which have non-centrosymmetric crystal phases. The missing inversion symmetry allows that a stable net dipole moment or remanent polarization can form. Well known is PZT (Lead Zirconate Titanate), which is used in a broad range of actuator and sensor applica- tions. But for most ferroelectrics the remanent polarization becomes lost in small structures and thin films. Therefore, the discovery of ferroelectric crystal phases in mixed HfO₂ and ZrO₂ thin film capacitors a few years ago was a breakthrough in micro- and nanoelectronics.

While the material as such was already present in form of a high-k gate and capacitor dielectric in transistor and DRAM, the question of silicon compatibility was proven. The additional nonvolatile functionality of the stabilized ferroelectric phase promises the next big step in chip technology as non-volatile gate or capacitor would combine the fast-operating speeds of transistor and DRAM with the non-volatility of solid-state drives. Whereas the optimum remanent polarization is obtained for mixture of Hf and Zr in equal proportions, with increasing Zr-content the material properties shift from ferroelectric to antiferroelectric, allowing for the exploitation of the giant piezoelectric effects during phase-transitions.

HfO₂, ZrO₂ and the mixtures Hf₁-xZrxO₂ stabilize in different crystal phases depending on various effects such as temperature, crystal size and doping. The main crystal phases are:

• monoclinic, at low temperatures in bulk
• cubic, at high temperature in bulk
• orthorhombic ferroelectric, in thin films
• tetragonal dielectric, in thin films

Exploiting the relation between the tetragonal and the ferroelectric phases is a current research topic, showing that the transition can be achieved with an electric field.

The main issue from simulation side is the misfit between the energy landscape derived from 0 Kelvin ab initio calculations and the experimental observations. In these simulations the ferroelectric phase is, due to its lower symmetry, more stable than the tetragonal phase, contradicting the experimental observation of antiferro- electricity (AFE) at room temperature, which is based on the tetragonal phase being more stable. Because experiments show a loss of AFE at thicker film sizes, the suspected reason for this deviation are size and entropy effects. Therefore, the simulation methods needed to be adjusted.

Results and Methods

Since ab initio simulations are based on the solution of the Kohn-Sham-Equation for each structural calculation or time step in molecular dynamics, the system shows a N³ scaling for calculation time versus the number of atoms. This limits the simulation time to a few picoseconds and system size to few 100 atoms.

An emerging approach to overcome this problem is to utilize machine learning. Using the computing resource of the SuperMUC-NG we were able to calculate the energies of more than 200,000 distorted crystal structures, which were subsequently used to train a neural net. This provides a mapping of the atomic coordinates to the energy and forces, which is the energy landscape. After the considerable investment in the creation of the training data for the neural network, the calculation of the energy of a structure is now 10⁴ times faster and almost as accurate as with an ab initio calculation. Furthermore, because modern high-performance GPUs are tailored to the evaluation of neural networks, the calculation of crystal structures of several nanometers size is possible with up to 100,000 atoms, matching realistic grain sizes in thin films.

Experimental measurements discovered an interlayer region between the electrodes and the ZrO₂, which interpolates between the dielectric and the ferroelectric phase. The properties of this interlayer and its stoichiometry somewhat depend on the used electrode which is typically tungsten or titanium nitride. But in a first approach we modeled in molecular dynamics calculations the crystal boundary condition as a one-unit cell wide tetragonal phase layer with oxygen atoms fixed in the polarization direction. The effect on the oxygen atoms is a kind of oxygen stress extending into the ferroelectric, which leads to a shift in the Curie-temperature from 1,000 K for the periodic boundary condition infinite ferroelectric phase to 600 K for 10 nm thick films and below room temperature for thin films of 5nm thickness. To obtain tetragonal phase for very thin films is a nice match, but there remains a discrepancy for slightly thicker film, because polycrystalline ZrO₂ film retaining their antiferroelectricity up to more than 10nm thickness. By applying not only a BE-like boundary condition in the thickness direction but also in the transversal directions, modeling polycrystalline grain boundary instead of epitaxial grain boundary, the Curie-temperature for films thinner than 14 nm is shifted below room temperature. The detailed insight how to achieve tetragonal thin ZrO₂- films at room temperature is a progress and brings experiments and simulation closer together.

Ongoing Research/Outlook

We aim to fully reconstruct the energy-landscape of ZrO₂ and to include the detailed effects of different BE boundary conditions representative for electrodes. We will extend the analysis to Hf₁-xZrxO₂ mixtures and to slightly oxygen deficit mixtures which are relevant in microelectronics. Further miniaturization of Hf₁-xZrxO₂- based devices depends on the final length scales that still allow the ferroelectric functionality to be switched on and off. We find that at a few nm the ferroelectric phase becomes more difficult to stabilize. We will investigate in further research how this can be counteracted.

References and Links

[1] M. Falkowski et al., Appl. Phys. Lett. 2021, doi.org/10.1063/5.0029610
[2] R. Ganser, Phys. Rev. Applied 2022, doi.org/10.1103/PhysRevApplied.18.054066
[3] P. Lomenzo et al., Adv. Funct. Mat. 2023, doi.org/10.1002/adfm.202303636