Abstract
As the scale of transistors and capacitors in electronics is reduced to less than a few nanometers, leakage currents pose a serious problem to the device’s reliability. To overcome this dilemma, highκ materials that exhibit a larger permittivity and band gap are introduced as gate dielectrics to enhance both the capacitance and block leakage simultaneously. Currently, HfO_{2} is widely used as a highκ dielectric; however, a higherκ material remains desired for further enhancement. To find new highκ materials, we conduct a highthroughput ab initio calculation for band gap and permittivity. The accurate and efficient calculation is enabled by newly developed automation codes that fully automate a series of delicate methods in a highly optimized manner. We can, thus, calculate >1800 structures of binary and ternary oxides from the Inorganic Crystal Structure Database and obtain a total property map. We confirm that the inverse correlation relationship between the band gap and permittivity is roughly valid for most oxides. However, new candidate materials exhibit interesting properties, such as large permittivity, despite their large band gaps. Analyzing these materials, we discuss the origin of large κ values and suggest design rules to find new highκ materials that have not yet been discovered.
Introduction
The dielectric insulator is a key component in microelectronic devices such as the central processing unit (CPU), dynamic randomaccess memory (DRAM) and flash memory. The basic function of the dielectric material is to enhance the capacitive coupling between adjacent metals and semiconductors, although it should also suppress the leakage current between electrodes, which undermines the energy consumption (in CPU and DRAM) or longterm reliability (in flash memory). In past decades, silicon dioxide (SiO_{2}) has been used as an archetypical dielectric material because it allows for defectfree, highquality thinfilm growth. As the integration level of microelectronic devices is currently exponentially increasing, the thickness of SiO_{2} has decreased to maintain the device performance. However, if the SiO_{2} layer becomes thinner than ~1 nm, the leakage current due to the quantum tunneling effect begins to dominate,^{1} which causes serious problems in power consumption and device performance. This technical obstacle has been overcome by replacing SiO_{2} with insulators that possess high dielectric constants (highκ).^{1, 2} With highκ dielectrics, the dielectric thickness can be increased at the same capacitance, thereby suppressing the leakage current.^{3} Currently, the favored highκ dielectrics are HfO_{2} (as the gate dielectric in CPU),^{2} ZrO_{2} (as the capacitor dielectric in DRAM)^{4} and Al_{2}O_{3} (as the blocking oxide in chargetrap flash memory).^{3}
In addition to a large dielectric constant, the highκ dielectric is required to have a large band gap (E_{g}) to suppress the charge injection from electrodes into dielectrics that cause the leakage current. Therefore, the ideal highκ dielectrics should possess both large E_{g} and κ. Notably, when E_{g} and κ of wellknown oxides are plotted (see Figure 1), the tradeoff relation is clearly noticeable. That is, materials are abundant with large E_{g} (>~8 eV) or high κ (>~20); however, no material has been discovered that satisfies both conditions simultaneously. Because this observation is based on a limited set of materials, one may question whether a material possessing both large E_{g} and κ may indeed exist if the search space is expanded.
Obtaining material information on E_{g} and κ, particularly the static dielectric constant, would be prohibitive if only experimental measurements were utilized. However, with the recent advances in computational methods and facilities, it is now feasible to compute ab initio various physical properties of the bulk phase in a relatively short period. Recently, several attempts have been made to find an optimal functional material using highthroughput computational screening.^{5, 6, 7, 8} The machinelearning approach presented by G. Pilania^{9} is a more accelerated approach to predict properties of large numbers of polymers; however, the prediction accuracy for E_{g} and κ is low for extending the method to other systems. In this study, we perform ab initio calculations on ~1800 oxides (except for 3d transition metal oxides) that cover most binary and ternary oxides identified to date and suggest novel candidate highκ dielectrics suitable for each device type. To this end, we set up computational machinery that automatically fetches structures from the database, prepares input files and reliably performs the ab initio calculations.
Materials and methods
Figure 2 presents a brief scheme of our automation strategy for calculating E_{g} and κ. First, all of the structures that contain specific cations are collected from the Inorganic Crystal Structure Database.^{10} We exclude 3d transition metal atoms because the electronic correlation effects are strong and E_{g} is moderate (<3 eV). The structures are further screened to avoid duplicative calculations on the same structure. The structures with partial atomic occupation are excluded because of the difficulty in computational modeling. In addition to the structures that are stable at ambient conditions, we consider structures that are characterized under hightemperature or highpressure conditions, provided that they are theoretically stable at zero temperature and pressure conditions, because the metastable structures can exist at ambient conditions through doping^{11} or in nanocrystalline states.^{12} The atomic positions and lattice parameters are then relaxed, and the theoretical equilibrium structures are obtained. For the computational code, the Vienna Ab Initio Simulation Package^{13} is adopted as the core engine for the ab initio calculations. Regarding the exchangecorrelation functional between electrons, we employ the generalized gradient approximation (GGA)^{14} for E_{g} and local density approximation (LDA)^{15} for κ. Because GGA and LDA underestimate the band gap, we also perform a hybridfunctional calculation for the bandedge points identified by GGA. The detailed procedure for each step is provided in the Methods section. The reliability of the present automation procedure is confirmed by comparing the extant experimental data with previous calculations.
Structural relaxation
All experimental structures should first be relaxed theoretically. This step is mandatory because when computing a dielectric constant, the structures are assumed to be at the local equilibrium. The kpoints in the first Brillouin zone are carefully and automatically sampled such that the total energy and stress tensor components are converged within 5 meV per atom and 10 kbar, respectively. The energy cutoff for the planewave basis set is selected based on the atomic species and pseudopotential types. The structural relaxation is performed until the atomic force and stress tensor are reduced to below 0.02 eV Å^{−1} and 5 kbar, respectively. Many structures reported at hightemperature or highpressure conditions often have higher symmetry compared with lowtemperature phases. When these structures are relaxed with the symmetry maintained, the final structures often possess unstable phonon modes. We exclude these structures from consideration as new highκ candidates because of their stability issue.
Computation of band gap
We first computed energy levels along the lines connecting the highsymmetry kpoints within GGA.^{16} On the basis of the results, the kpoints corresponding to the valence top and conduction minimum are determined. Because the band gap underestimation is severe in the semilocal functional, we additionally perform a hybridfunctional (HSE06) calculation^{17} (without further structural relaxation) and calculate the energy levels at the bandedge kpoints identified by GGA (this scheme is called HSE@GGA hereafter). This process assumes that the band structure rigidly shifts upon application of the hybrid functional. Supplementary Figure S1 in the Supplementary Information shows that the band structure from the hybrid functional is approximately a rigid shift of the GGA result. By adopting the HSE@GGA scheme, we could significantly reduce the computational cost compared with the full hybridfunctional calculations. For oxides that include heavy elements such as Tl, Pb and Bi, the spinorbit coupling is included in computing E_{g}.
To ensure the reliability of the present scheme, test calculations on several oxides are performed and compared with the experiment and previous GGA calculations (see Figure 3a). The estimated energy gaps are in good agreement with the experiment, although sizeable errors of up to 1 eV are noticeable for oxides with E_{g} larger than ~4 eV. These errors are due to the fixed fraction of the exact exchange term in the HSE06 functional, which should be increased in the largegap materials.^{18} This finding implies that more sophisticated methods such as GW calculations can be utilized to obtain more precise values of E_{g}. We note that the GW calculations are too expensive to be incorporated into the highthroughput screening. In addition, it is not yet known which level of GW approximations can be universally applied to every class of oxides.^{19} Nevertheless, the accuracy of HSE@GGA is sufficient to screen promising dielectrics in the present study.
Computation of dielectric constant
To compute the electronic permittivity, the linearresponse method based on the density functional perturbation theory is used to obtain Born effective charges and phonon modes at the zone center.^{20} Because the linearresponse computation is sensitive to the kpoint sampling, we double the kpoint density along each direction. The static dielectric constant () is then calculated using the following formula:
where , Ω, ω_{m} and denote the dielectric tensor contributed by electrons, the unitcell volume, the frequency of the infraredactive phonon with the mode number of m and the modeeffective Born effective charges, respectively. The subscripts α and β in equation (1) indicate the directions. The dielectric constant (κ) is obtained by averaging the diagonal components of . The theoretical κ of some oxides determined by GGA and LDA are compared with the experiment (see Figure 3b). The mean absolute errors for materials with κ<30 are 2.06 and 1.37 for the GGA and LDA results, respectively. The GGA results exhibit larger errors than those of LDA because the ionic part of κ is sensitive to the lowfrequency phonon modes that are significantly softened when the lattice parameters are expanded in GGA. Therefore, we employ LDA for computing κ.
Computational cost
The average computational cost was ~70 CPU hours per structure on the 24core cluster. The most timeconsuming part was the hybridfunctional calculation in the HSE@GGA scheme (50%) followed by the density functional perturbation theory calculations for the dielectric constant (27%). The structure relaxation and bandedge searching consumed ~15% and 8% of the total computational time, respectively.
Results
Total property map
We calculated 1762 oxides in total and obtained a large property database. Figure 4 presents the property map of E_{g} versus κ for 1158 binary and ternary oxides. We excluded the structures that were metallic or unstable under the ambient condition according to the results. First, the inverse relation between E_{g} and κ was roughly valid, and the oxides satisfying the ideal condition were scarce. To select candidate highκ oxides from the database, we defined a figure of merit for reducing the leakage current. The leakage current density by direct tunneling (J_{DT}) can be expressed using the following semiempirical formula:^{21}
where q and m_{eff} are the charge and tunneling effective mass, respectively, of the electron or hole, Ф_{b} is the injection barrier, and t_{ox,eq} represents the equivalent oxide thickness. We define (m_{eff} Ф_{b} )^{1/2}κ in the exponent as the figure of merit (f_{FOM}), which indicated that the tunneling current is exponentially suppressed with f_{FOM}. One can compute m_{eff} and Ф_{b} theoretically,^{22} but the process requires demanding calculations that are not compatible with the highthroughput approach. Here we make a crude but reasonable assumption that the two parameters are roughly proportional to E_{g},^{23} which approximates f_{FOM} as simply E_{g}·κ. Each point in Figure 4 is color coded according to f_{FOM}.
New candidate highκ materials
Among the oxides in Figure 4, cBeO, the highpressure phase of BeO in the rocksalt structure (see Figure 5a), is particularly prominent as it has the unusual combination of 10.1 eV and 275 for E_{g} and κ, respectively. Consequently, its f_{FOM} is considerably beyond those of other materials. The wurtzite BeO (wBeO), which is stable at ambient conditions, has already been used in dielectric applications as it has the largest band gap among all the oxides. The atomiclayerdeposited wBeO was fabricated on Si substrates and employed as a diffusion barrier of oxygen between Si and highκ dielectrics such as HfO_{2}.^{24} However, wBeO has a very small κ of ~7. However, cBeO has not been applied in microelectronic devices to date as far as we are aware. The origin of the large κ of cBeO is its soft optical phonon mode (~3.5 THz), wherein the Be and O atoms vibrate in opposite directions (see Figures 5a and b). In cBeO, the Be–O bond length is longer than that in wBeO by 0.11 Å, which softens the optical phonon mode. The computed total energy indicates that cBeO is less stable than wBeO by 0.483 eV per atom. The large energy difference implies that cBeO would be difficult to stabilize under ambient conditions. However, we pay attention to the experiments in Adelmann et al.^{25}, Kita et al.^{26} and Tsipas et al.^{27}, which indicate that the hightemperature phases of HfO_{2} and ZrO_{2} can be synthesized as thin films by external doping or strain. More importantly, the doped phases possessed increased dielectric constants, as predicted by theory.^{28} Therefore, we reasonably expect that cBeO can be stabilized by doping or strain and will exhibit physical properties similar to the present calculations.
Except for cBeO, we could not find any outstanding highκ dielectrics with either E_{g} or κ larger than those of the HfO_{2} thin films currently used in CPU or DRAM (E_{g}~6.0 eV and κ~20–25; see tHfO_{2} in Figure 4). Nevertheless, we identified several candidates that are noteworthy and list them in Table 1. These materials could be important in the future as the main material shifts from Si to Ge and GaAs and a more diverse selection of gate dielectrics is highly demanded to meet chemical conditions that are different from Si. Here we exclude oxides that have been studied previously. In the last column of Table 1, we mark the appropriate device type according to the material properties; for CPU and DRAM devices, the international technology roadmap of semiconductors states that further device scaling requires higherκ dielectrics with κ>30.^{29} We note the candidate highκ materials that satisfy this condition and have larger f_{FOM} than that of tHfO_{2} (f_{FOM}~210). We also limited E_{g}>4 eV for CPU and E_{g}>3 eV for DRAM, considering that the ideal Φ_{b} should be at least >1.5 eV. For the blocking oxides used in flash memory, large values of E_{g} are more crucial than large κ to satisfy the more stringent leakage current specification (<~10^{−9} Acm^{−2}) compared with DRAM (<~10^{−7} Acm^{−2}) and CPU (<~10^{−1} Acm^{−2}). We, therefore, impose the condition of E_{g}>6 eV and f_{FOM}>80, which is larger than the values for Al_{2}O_{3}, the currently favored highκ blocking oxide in the chargetrap flash memory. The total candidate lists including those for DRAM and Flash are provided in Tables 2 and 3, respectively. For the gate dielectric for CPU, we further consider that the stability of highκ/Si interfaces is important to prevent the unintentional oxidation of the Si substrate. One metric of oxide stability is the formation energy of oxygen vacancies (E_{vac}) because it reflects the strength of metal–oxygen bonding. The computed E_{vac}s for the candidate materials are listed in Table 1. Because the E_{vac} of SiO_{2} is 5.6 eV, oxides with E_{vac} values larger than this value may form a stable interface with Si. For example, it is known that ZrO_{2} (E_{vac}=5.1 eV) exhibits a stability issue on Si substrates,^{30} whereas HfO_{2} (E_{vac}=6.9 eV) is stable. This behavior is reflected in selecting candidate oxides for CPU in Table 1. ΔE per atom in Table 1 denotes the total energy per atom relative to the most stable phase identified from the present computation and indicates the relative thermal stability of the given phase.
Discussion
In Figures 5c and d, the structures of two candidate oxides, AlO(OH) and Na_{2}SO_{4}, are displayed together with the lowest phonon mode, indicated by arrows. These structures exhibit the two representative features of highκ ternary oxides. In the lowest phonon modes of AlO(OH) (Figure 5c), cations vibrate within the octahedral cage formed by anions. In contrast to facesharing octahedra in Al_{2}O_{3}, H^{+} ions in AlO(OH) lead to corner sharing, which softens the vibrational mode of Al^{3+}. NbOCl_{3} in Table 1 also has a similar feature of cationic vibration within the cornersharing octahedra. However, all of the other ternary oxides in Table 1 have another common feature, which can be represented by Na_{2}SO_{4} in Figure 5d: they usually contain nonmetal oxygen units such as (SO_{4})^{2−}, (PO_{4})^{3−} and (IO_{3})^{−}. These units form a rigid bond in the compounds, whereas the other type of cation is loosely bound to oxygen and yields soft phonon modes (see Na atoms in Figure 5d). Furthermore, we find that the unblocked cation channel is critical for a large dielectric constant. For example, two polymorphs of Na_{2}SO_{4} have common orthorhombic phases but slightly different atomic arrangements with a crystal symmetry of Cmcm and Pbnm, respectively (see Supplementary Figure S2 in Supplementary Information). The coordination number and oxidation state of each atom are similar, and E_{g} is thus almost the same. However, the two κ values significantly differ (Cmcm: 20.7 vs Pbnm: 5.9).
For the Cmcm structure, as shown in Figure 5d, all of the cations in the soft mode vibrate coherently along certain passages without being blocked by other species of atoms. However, such an unblocked channel does not exist in the Pbnm phase. Consequently, the Na atoms oscillate in rather random directions, and the contributions to dielectric polarization cancel each other out, resulting in smaller κ. Note that the ionic conduction of Na^{+} can cause device instability^{31}. However, considering the lower ionic conductivity in the Cmcm structure compared with that for other phases^{32}, we cautiously believe that the highκ phase of Na_{2}SO_{4} could be employed in microelectronic devices.
On the basis of these observations, highκ ternary oxides that simultaneously exhibit large E_{g} and κ might be identified through two models: (1) cationic vibration within the cornersharing octahedral cage of anions and (2) channeled structures formed by a combination of metal ions and various nonmetal oxide units. Considering the numerous possible combinations to form ternary oxides, we expect that several ideal dielectric materials that have these features could be identified in the future.
In summary, we screened ~1800 binary and ternary oxides using highthroughput ab initio calculations of the band gap and static dielectric constant with the aim to find new candidate highκ dielectrics that can be used in various microelectronic devices such as CPU, DRAM and flash memory. From the obtained property database, we generated a materials map of the band gap versus static dielectric constant and identified new candidate materials that have not been considered in previous studies. From the detailed analysis on the atomic structure and phonon mode, we identified key factors that correlate with the large dielectric constant. By suggesting new candidate highκ materials and developing a large material property database covering most binary and ternary oxides, the present work will contribute greatly to selecting functional oxides that are optimal for specific applications. An automated highthroughput study on oxygen vacancy defects in oxides is also in progress.
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Acknowledgements
This research was supported by the EDISON program (NRF2012M3C1A6035307). The computations were performed at the KISTI supercomputing center (KSC2014C3012).
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Yim, K., Yong, Y., Lee, J. et al. Novel highκ dielectrics for nextgeneration electronic devices screened by automated ab initio calculations. NPG Asia Mater 7, e190 (2015). https://doi.org/10.1038/am.2015.57
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DOI: https://doi.org/10.1038/am.2015.57
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