Novel high-κ dielectrics for next-generation electronic devices screened by automated ab initio calculations

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, high-κ materials that exhibit a larger permittivity and band gap are introduced as gate dielectrics to enhance both the capacitance and block leakage simultaneously. Currently, HfO₂ is widely used as a high-κ dielectric; however, a higher-κ material remains desired for further enhancement. To find new high-κ materials, we conduct a high-throughput 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 high-κ 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 random-access 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 long-term reliability (in flash memory). In past decades, silicon dioxide (SiO₂) has been used as an archetypical dielectric material because it allows for defect-free, high-quality thin-film growth. As the integration level of microelectronic devices is currently exponentially increasing, the thickness of SiO₂ has decreased to maintain the device performance. However, if the SiO₂ 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 SiO₂ with insulators that possess high dielectric constants (high-κ).^{1, 2} With high-κ dielectrics, the dielectric thickness can be increased at the same capacitance, thereby suppressing the leakage current.³ Currently, the favored high-κ dielectrics are HfO₂ (as the gate dielectric in CPU),² ZrO₂ (as the capacitor dielectric in DRAM)⁴ and Al₂O₃ (as the blocking oxide in charge-trap flash memory).³

In addition to a large dielectric constant, the high-κ 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 high-κ dielectrics should possess both large E_g and κ. Notably, when E_g and κ of well-known oxides are plotted (see Figure 1), the trade-off 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.

Figure 1

The experimental band gap and dielectric constant for well-known oxides. The property region ideal for dielectrics is also shown.

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 high-throughput computational screening.^{5, 6, 7, 8} The machine-learning approach presented by G. Pilania⁹ 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 high-κ 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 high-temperature or high-pressure conditions, provided that they are theoretically stable at zero temperature and pressure conditions, because the metastable structures can exist at ambient conditions through do**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 Package¹³ is adopted as the core engine for the ab initio calculations. Regarding the exchange-correlation 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 hybrid-functional calculation for the band-edge 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.

Figure 2

Schematic automation strategy to collect computational data on the band gap and static dielectric constant. The initial structures collected from the ICSD were filtered for the ensuing ab initio computations.

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 k-points 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 plane-wave 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 high-temperature or high-pressure conditions often have higher symmetry compared with low-temperature 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 high-κ candidates because of their stability issue.

Computation of band gap

We first computed energy levels along the lines connecting the high-symmetry k-points within GGA.¹⁶ On the basis of the results, the k-points 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 hybrid-functional (HSE06) calculation¹⁷ (without further structural relaxation) and calculate the energy levels at the band-edge k-points 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 hybrid-functional calculations. For oxides that include heavy elements such as Tl, Pb and Bi, the spin-orbit 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 large-gap 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 high-throughput 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.

Figure 3

Comparison of theoretical and experimental data for the (a) band gap (E_g) and (b) static dielectric constant (κ); some reference data from other ab initio studies (LDA(ref) and GGA(ref)) are also shown as open symbols.^{20, 33, 34, 35, 28} Therefore, we reasonably expect that c-BeO can be stabilized by do** or strain and will exhibit physical properties similar to the present calculations.

Figure 5

(a) The unit-cell structure of c-BeO and the lowest phonon mode indicated by arrows. The frequency of this mode (ω) is also noted. (b) The phonon dispersion curve of c-BeO. The lowest phonon mode (threefold degenerate) is marked by a circle. The unit-cell structure of (c) NbOCl₃ and (d) Na₂SO₄. The lowest phonon modes that are responsible for most of κ are shown in each structure with the vibrational directions indicated by arrows.

Except for c-BeO, we could not find any outstanding high-κ dielectrics with either E_g or κ larger than those of the HfO₂ thin films currently used in CPU or DRAM (E_g~6.0 eV and κ~20–25; see t-HfO₂ 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 higher-κ dielectrics with κ>30.²⁹ We note the candidate high-κ materials that satisfy this condition and have larger f_FOM than that of t-HfO₂ (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 Al₂O₃, the currently favored high-κ blocking oxide in the charge-trap 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 high-κ/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_vacs for the candidate materials are listed in Table 1. Because the E_vac of SiO₂ 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 ZrO₂ (E_vac=5.1 eV) exhibits a stability issue on Si substrates,³⁰ whereas HfO₂ (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.

Table 1 New candidate materials suitable for high-κ dielectrics selected from Figure 3

Table 2 All high-κ candidate materials for DRAM and CPU from the present property database, which satisfy f_FOM>210, and E_g>3 eV (for DRAM) or E_g>4 eV (for CPU)

Table 3 All high-κ candidate materials for flash memory application from the present property database, which satisfy f_FOM>80 and E_g>6 eV

Discussion

In Figures 5c and d, the structures of two candidate oxides, AlO(OH) and Na₂SO₄, are displayed together with the lowest phonon mode, indicated by arrows. These structures exhibit the two representative features of high-κ ternary oxides. In the lowest phonon modes of AlO(OH) (Figure 5c), cations vibrate within the octahedral cage formed by anions. In contrast to face-sharing octahedra in Al₂O₃, H⁺ ions in AlO(OH) lead to corner sharing, which softens the vibrational mode of Al³⁺. NbOCl₃ in Table 1 also has a similar feature of cationic vibration within the corner-sharing octahedra. However, all of the other ternary oxides in Table 1 have another common feature, which can be represented by Na₂SO₄ in Figure 5d: they usually contain non-metal oxygen units such as (SO₄)²⁻, (PO₄)³⁻ and (IO₃)⁻. 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 Na₂SO₄ 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 instability³¹. However, considering the lower ionic conductivity in the Cmcm structure compared with that for other phases³², we cautiously believe that the high-κ phase of Na₂SO₄ could be employed in microelectronic devices.

On the basis of these observations, high-κ ternary oxides that simultaneously exhibit large E_g and κ might be identified through two models: (1) cationic vibration within the corner-sharing octahedral cage of anions and (2) channeled structures formed by a combination of metal ions and various non-metal 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 high-throughput ab initio calculations of the band gap and static dielectric constant with the aim to find new candidate high-κ 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 high-κ materials and develo** 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 high-throughput study on oxygen vacancy defects in oxides is also in progress.