CPU vs RAM in the Issue of ab initio Simulations of Doped Hafnium Oxide for RRAM and FRAM

c (cid:13) The Authors 2023. This paper is published with open access at SuperFri.org Atomic and electronic structure of dopped HfO 2 is studied using ﬁrst principle simulations. The 96-and 324-atom supercell are used to simulate impurity density in the range of 2–6 . 3 mol.% that is used in real electronic memory devices. The optimal spatial conﬁgurations of impurity atoms with an oxygen vacancy are found. It is shown that there are no defect levels in the band gap dopped HfO 2 with the optimal structures. The electronic structure of additional neutral oxygen vacancy in HfO 2 is equivalent to that of neutral oxygen vacancy in pure HfO 2 . An increase in the size of a supercell predictably leads to an increase in the need for computing resources. At the same time, the need for RAM is growing faster than for CPU power. Doping HfO 2 with Al/La/Y with concentration of up to 6 . 2 mol.% has negligible eﬀect on the electronic structure of neutral oxygen vacancies.


Introduction
Promising candidates for the role of universal memory, which combines the advantages of Random Access Memory (RAM), Hard Drive Disks and Flash memory, are resistive (RRAM) and ferroelectric (FRAM) memories based on hafnium oxide (HfO 2 ) [16,22].In a HfO 2 -based RRAM, the resistive switching between states of different resistance is carried out when exposed to an external electric field due to the formation/breaking of a conductive filament.In the HfO 2 -based FRAM, the information storage is supported by the dielectric polarization in the ferroelectric layer.The switching of polarization is carried out through the application of an external electric field.The performance of such a device is ensured by the ferroelectric phase stabilization in HfO 2 films.It is known that doping HfO 2 with various metals, such as Al, La and Y, leads to the increased performance of RRAM and FRAM cells: reduced forming voltage, increased memory window and increased number of reprogramming cycles [2,9,10,18,21,23].The mechanisms by which the dopant influences the characteristics of RRAM and FRAM have not been established yet.This problem can be solved using first principle simulations within the density functional theory (DFT).However, to do this, first of all, it is necessary to establish the atomic structure of doped HfO 2 .Despite a fairly large number of studies on this issue, they are all limited to the use of model structures, the correctness of which has not been proven yet [3, 6, 13-15, 20, 24-26].
The complexity of the task is determined by two factors.First of all, the use of HfO 2 supercells with the replacement of two Hf atoms by impurity atoms and an oxygen vacancy necessary for the charge compensation of the system, necessitates the search for the most probable (energetically favorable) spatial position of three defects in a supercell.It should be clarified that this particular HfO 2 doped with Al, La, Y model structure is the most popular and justified, since the valency of the considered impurities is one lower than that of hafnium.This requires considering a huge number of nonequivalent defect configurations.Thus, for the 96-atomic supercell of the Supercomputing Frontiers and Innovations monoclinic phase (m-) HfO 2 (with 32 metal atoms and two types of oxygen vacancies), which is mostly used in calculations, obtained by the 2 × 2 × 2 translation of a primitive 12-atomic cell, it is necessary to calculate C 2 32 × 2 = 992 configurations.In the vast majority of existing studies, the authors limit themselves to considering the defect configurations in which a pair of impurity atoms is in close proximity to oxygen vacancy.
The second difficulty is that to simulate HfO 2 with an impurity concentration of about 2 mol.%, corresponding to the best characteristics of real RRAM and FRAM elements based on doped HfO 2 , it is necessary to use supercells of 324 atoms (obtained by a symmetric translation of a 3 × 3 × 3 primitive cell).The use of a 96-atom supercell corresponds to the simulation of an overestimated impurity concentration of 6.25 mol.%.The correctness of the results obtained for 324-atom supercells was not verified due to the need to use a lot of computing resources.It is important to note that despite the absence of serious difficulties in calculating of this scale systems for standard DFT, to correctly reproduce the electronic structure and, in particular, the position of defect levels in the band gap, it is necessary to use a significantly more resourceintensive DFT with hybrid exchange-correlation functionals.This, in turn, requires significantly more computational resources.
Thus, the purpose of this work is to study the atomic and electronic structure of HfO 2 doped with Al, La and Y at low concentrations.The study includes, firstly, determining the optimal atomic structures of HfO 2 doped with Al, La and Y from the point of view of energy efficiency; secondly, studying the influence of the supercell size on the resulting optimal structure and, thirdly, simulations the electronic structure of additional oxygen vacancies in the found structures.Additionally, the problem was formulated as studying the influence of the supercell size on the reproducibility of the calculation results of the atomic and electronic structure of defect complexes.
The article is organized as follows.Section 1 is devoted to structures under study and calculation methods of electronic structure of structure defects in the studied electronic systems.In Section 2 we discuss obtained results.Subsection 2.1 is devoted to the description of the atomic and electronic structures of HfO 2 supercells with mutual arrangement of the impurity atoms and oxygen vacancies.In Subsection 2.2 the required computing resources for the simulations of 96and 324-atom HfO 2 supercells are discussed and compared.Conclusion summarizes the study and points directions for further work.

Methods
The simulations were carried out within the DFT in the approximation of a periodic 3D supercell, with a plane-wave basis and optimized norm-conserving Vanderbilt pseudopotentials [7,8] using the Quantum ESPRESSO (QE) software package [4,5].Two types of exchangecorrelation functionals were used: PBEsol to calculate structural relaxation and B3LYP to calculate the electronic spectrum of optimal HfO 2 :X structures (X in one of Al, Y or La).The simulation was carried out for the m-HfO 2 (P 2 1 /c) phase.This phase is the most stable and closest in physical properties to amorphous HfO 2 , and is also observed in real HfO 2 films used in RRAM and FRAM elements.The optimal structures of HfO 2 :X were found by calculating all possible nonequivalent configurations of the arrangement of oxygen vacancy and a pair of impurity atoms at the Hf substitution position in 96-atom supercells from which configurations with the minimum total energy of the system were selected.For 324-atom supercells, the search for optimal structures was carried out by finding the optimal position of the first impurity atom at a fixed position of the oxygen vacancy, and then of the second one.The oxygen vacancy, in this case, is a structural element of HfO 2 :Al/Y/La providing a charge compensation for the impurity, and, for convenience, is further referred to as a structural vacancy (V O ).The calculations used the plane waves cutoff energy 80 Ry, the Fock exchange operator cutoff energy 100 Ry, the kpoint grid 2×2×2, the Fock operator point grid 1×1×1, the exact exchange fraction for B3LYP 0.175 (which provides the m-HfO 2 bandgap value of 5.8 eV) and the total energy convergence threshold 10 −4 Ry.In the found HfO 2 :X structures, the optimal position of the additional oxygen vacancy with the minimum formation energy (hereinafter denoted V ' O ) was found by calculating and analyzing all possible positions in the supercell.The spatial distributions of atoms and defects in HfO 2 :X supercells were visualized using the XCrySDen program [11,12].The used computing resources and memory distributions were extracted from QE reports.
The energy of V ' O formation (E form ) was calculated using the formula: where E(V O ) is the energy of the HfO 2 :X supercell with neutral V ' O ; E p is the energy of the 'perfect' supercell without V ' O ; µ(O) is the chemical potential on an oxygen atom O.For the convenience of comparing the results with literature data, the µ(O) value was taken equal to half the total energy of the O 2 molecule in the triplet state, which corresponds to the oxygen-enriched limit.

Atomic and Electronic Structures
After calculating all possible spatial configurations of the Al/La/Y atom pair position in supercells of 96-and 324-atoms with oxygen vacancy V O , optimal structures with the lowest total energy were found (Fig. 1).The spread of the total energy of supercells with different defect configurations is about 3 eV, while the structure closest in energy differs from the optimal one by about 0.4 eV.It was established that the features of the atomic structure obtained for 96 and 324atomic supercells coincide.It is obvious that further decrease in the impurity concentration due to an increase in the supercell size will not lead to changes in the optimal mutual arrangement of the impurity atoms and V O .Thus, the use of a 96-atom supercell is sufficient to reproduce the main features of the relative arrangement of impurity atoms in m-HfO 2 .
In all structures, V O is 3-coordinated.In HfO 2 :La and HfO 2 :Y, the impurity atoms are spaced from each other at approximately 6 Å, with one of the impurity atoms located close to V O at a distance of r ≈ 2 Å, and the second -at r ≈ 4.1 Å from V O .For La/Y near V O the coordination number is 6, for the second La/Y it is 7.It is noteworthy that, the total energy of HfO 2 :La and HfO 2 :Y supercells with an optimal structure is lower (by more than 0.4 eV), compared with the total energy of non-optimal structures, that were used in calculations in the works of various authors previously [6, 13-15, 20, 25].The optimal structure of HfO 2 :Al, on the contrary, meets this assumption: V O is located between the Al atoms at r ≈ 2.3 Å from each Al, while the Al atoms are relatively close to each other (r ≈ 4.5 Å).It is probably energetically unfavorable for La and Y atoms to be too close to each other due to their large ionic radius compared to the Hf one: R Hf = 0.85 Å, R La = 1.17 Å, R Y = 1.06 Å [19].In HfO 2 :Al, a pair of Al atoms might be close to each other due to a small Al ionic radius (R Al = 0.68 Å).The total density of states (TDOS) spectra calculated for HfO 2 doped with Al, La and Y with concentrations of 6.25 mol.% and 2 mol.% show that the band gap of all oxides is empty (Fig. 2).This result is consistent with the experimental data, according to which doping HfO 2 with lanthanum does not change the bandgap E g [17].In contrast, the TDOS spectra for nonoptimal structures have an empty level with a depth of about 1 eV [13,15].In doped HfO 2 , as well as in pure HfO 2 , E g = 5.85 eV, which is close to the experimental value E g = 5.7 eV [1].In the case of doping with La, a subband with an energy of about 14 eV below the valence band top E V is formed predominantly by La5p orbitals, which can be seen in the TDOS spectrum.
It was established that the optimal position of an additional oxygen vacancy V ' O in 95and 323-atomic supercells of HfO 2 :Al, HfO 2 :La and HfO 2 :Y, with the minimal E form , is near one of the impurity atoms.As a result, in HfO 2 :La and HfO 2 :Y, both impurity atoms become 6-coordinated.For HfO 2 :La and HfO 2 :Y, the V O -V ' O distance is 5.50 Å, and for HfO 2 :Al it is 3.13 Å.The E form of V '  O weakly depends on the value of the considered defect density and is approximately 0.2 eV less than the E form of the vacancy in undopped HfO 2 .Thus, doping HfO 2 with Al/La/Y facilitates the generation of new oxygen vacancies in the oxide.Recently, the opposite results have been obtained, however, when the non-optimal structures of doped HfO 2 were simulated [20,25].In doped HfO 2 , a neutral V ' O forms a level filled with two electrons just above the middle of the bandgap, as shown in Fig. 3.The position of this level, firstly, weakly depends on the type of dopant (any of Al, La or Y); secondly, it does not depend on the impurity concentration, and, thirdly, it is close to that for a neutral oxygen vacancy in undoped HfO 2 .Thus, one can conclude that doping HfO 2 with Al/La/Y with a concentration of up to 6.2 mol.% has a negligible effect on the electronic structure of neutral oxygen vacancies.

Required Computing Resources
The required computing resources for the simulations of 96-and 324-atom HfO 2 supercells are given in Tab. 1.As the cell size increased, the complexity of the problem increased about 3.4 times.At the same time, CPU time increased about 6-7 times, while the memory require-ments increased more significantly, about 8 times.The memory is mainly used to store data from exact-exchange (EXX) integrals within hybrid functional, wave-functions and β-functions of non-local pseudopotentials.The needs for EXX and β-function data increased 7 times and that is close to the total growth of using RAM resources.Note that the data volume for reducing matrices to the diagonal shape has increased more than 13 times.

Conclusion
This work is devoted to the thorough study on the atomic and electronic structure of HfO 2 doped with aluminum, lanthanum and yttrium.The simulation was carried out for two impurity concentrations, covering the actual range of doping density of films in real electronic devices.Two types of oxygen vacancies are considered, namely, the oxygen vacancy involved in compensating the impurity charge (V O ) and the additional oxygen vacancy (V ' O ).It was established that, in the optimal structure of the doped oxide, La and Y atoms tend to distance themselves from each other at about 6 Å.Just one of the La/Y atom is located near V O , while both Al atoms are located near one common V O and the distance between Al is about 4 Å.It was established that the features of the atomic structure obtained for 96 and 324-atomic supercells coincide.The data obtained for HfO 2 :La and HfO 2 :Y arouse doubts as for all the results previously published on this topic.It was found that there are no defect levels in the bandgap of HfO 2 :Al, HfO 2 :La and HfO 2 :Y with the optimal structures.The formation of additional neutral vacancy V '  O in HfO 2 near Al, Y or La atoms is facilitated, compared to the formation of V O in undoped HfO 2 .The electronic structure of V ' O in HfO 2 :Al, HfO 2 :La and HfO 2 :Y is equivalent to that of neutral V O in pure HfO 2 .
Increasing the size of a supercell leads to an increase in the need for computing resources.At the same time, the need for RAM is growing faster than for the CPU power.Since doping HfO 2 with Al/La/Y with concentration of up to 6.2 mol.% has a negligible effect on the electronic structure of neutral oxygen vacancies, 96-atomic supercells exhibit all features of m-HfO 2 .

Figure 1 .Figure 2 .
Figure 1.Supercells of 96-and 324-atoms of the optimal m-HfO 2 :Al, m-HfO 2 :La and m-HfO 2 :Y structures.Gray color balls are Hf, blue ones are O, green ones are Al, red ones are La and black ones represent O vacancies (removed O atoms)

Figure 3 .
Figure 3. TDOS spectra calculated for HfO 2 , HfO 2 :Al and HfO 2 :La structures with additional V ' O .Zero energy corresponds to the valence band top E V

Table 1 .
Resource consumption per calculation