Journal of Metals, Materials and Minerals, Vol. 30, No. 2, pp. 31-41, 2020

In search for near-room-temperature superconducting critical temperature of metal superhydrides under high pressure: A review

Udomsilp PINSOOK*

Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, THAILAND

*Corresponding author e-mail: [email protected]

Received date: Abstract 6 May 2020 An overview and the latest status of the of metal superhydrides Revised date: 15 May 2020 under high pressure are discussed in this review article. The searching for near-room- Accepted date: temperature superconductors have been one of the most enthusiastic fields in physics. 10 June 2020 This is because of several key factors in both theoretical and experimental sides. By advanced experiment innovation, pressure exceeded that of the earth’s core can be generated in laboratories. This allows scientists to explore new physics of materials Keywords: Superconductors under high pressure. In the synergy form, the theoretical calculation gives accountable Metal superhydrides predictions on the structural and electronic properties which can be served as a Near-room-temperature practical map for experimentalists. In this review, I also give a brief overview on the Critical temperature existing theory of superconductivity which leads to the calculation of the High pressure superconducting critical temperature, 푇 . The key element for calculating 푇 is stemmed from the so-called spectral function, which can now be evaluated from the

density functional theory. In order to obtain insight information and gain deeper understanding, I model the spectral function with a simple constant function. This analysis gives a powerful suggestion on the way to search for a higher value of 푇.

1. Introduction and strontium, which are not a superconductor at ambient pressure. The search for a high value of 푇 is A superconductor is a phase of materials which to look for large 휃 and large 푁 . Sometimes, large exhibit zero electrical resistivity and perfect 푁 can be found if van Hove singularity [6] occurs diamagnetism at temperature lower than a critical around the Fermi level. However, 푉 is hard to be value, called the superconducting critical temperature, identified in practice. The BCS theory valids for the 푇 . The external applied magnetic field is expulsed weak electron-phonon coupling regime only. Before from a superconductor. This is the so-called Meissner- the new millennium, the highest 푇 in the conventional Ochsenfeld effect [1]. The first known superconductor superconductors was found just below 39 K [3]. At the is Hg with 푇 = 4.15 K, discovered in 1911 by Onnes turning of the new millennium, MgB2 was discovered [2]. He received the Nobel prize in 1913. Since then, to be a convention superconductor with 푇 = 39 K [7]. large number of materials, such as a number of metals This gave a boost in the field of superconductors as [3] and intermetallic compounds [4], had been MgB2 can be easily synthesized and non-toxic. It discovered to be a superconductor. In 1957, Bardeen, opens large number of possibilities in terms of Cooper and Schriffer announced the theory that gives superconducting technology and applications. Despite the convincing explanation to the superconducting of a large number of modifications, MgB2 had been a phenomenon [5]. This is the so-called BCS theory. 푇 champion in the class of the conventional They received the Nobel prize in 1972. The BCS superconductors for more than a decade. theory gives many powerful predictions, such as the There is another class of superconductors, called non- superconducting gap and the critical temperature, 푇, conventional or high- 푇 superconductors. which can be expressed as Surprisingly. they are mostly a ceramic, of which normal state is an insulator. The first known superconductor of this class is a BaLaCuO compound 푇 = exp − . (1) . with 푇 = 35.1 K, discovered in 1986 by Bednorz and Müller [8]. They were awarded the Nobel Prize in The BCS prediction of 푇 is based on three key Physics in 1987. Since then, large number of ceramic physical quantities, such as the Debye temperature, compounds have been found to be a high- 푇 휃, the density of electrons at the Fermi energy, 푁, superconductors. The famous examples are a and the effective electron interaction, 푉 . The YBaCuO compound with 푇 = 93 K [9] and a condition for the BCS theory to be valid is that the HgBaCaCuO compound with 푇 = 130 K [10]. In normal state of materials needs to be a metal. This type Thailand, the theory for high-T_c superconductors has of materials is called a conventional superconductor. been pioneered by Prof.Suthat Yoksan [11]. In 2010, However, there are also some metals, such as calcium Udomsamuthirun et. al. [12] have discoveried new

Metallurgy and Materials Science Research Institute (MMRI) Chulalongkorn University DOI: 10.14456/jmmm.2020.18 32 PINSOOK, U. composition of YBaCuO compounds with T_c = 94 K calculations is the search for metallic . [12]. Up till now, the highest Tc of this class of Wigner and Huntington predicted the existence of superconductors is around 150 K. Unfortunately, the in 1935 [32]. At that time, its BCS cannot give a complete description of this class correct structure at high pressure was unknown. Since of materials. The applicable theory for this class of then, there have been a large number of experimental materials is really complicated and far from complete. and theoretical developments in order to identify the This is beyond the scope of this review. correct structure and the precise transition pressure of Pressure is an important thermodynamic variable. the metallic hydrogen [33]. Only recently, the A device that controls high pressure in experiments is experiments have just observed a possible candidate of the so-called anvil cell, pioneered by Bridgman [13]. the metallic hydrogen at pressure between 425-495 His device can produce pressure as high as 10 GPa, GPa [34,35]. From Eq. (1), the BCS theory suggested 100,000 times higher than the ambient pressure. He that high 휃 could also give high 푇 . As H is the was awarded the Nobel Prize in Physics in 1946. In lighest atom, it has a potential to give the highest 휃. 1959, the first diamond anvil cell (DAC) was invented In 1968, Ashcroft predicted that the metallic hydrogen [14]. Since then, high pressure has become a powerful could become a high-temperature superconductor tool for studying the new physics of materials. For [36]. Up till now, the closest realization of Ashcroft’s example, pressure can induce structural phase prediction is the hydrogen-rich compounds. In 2014, a transitions, and some physical properties are H3S compound was predicted to be a conventional dependent on the crystal structure of materials. The superconductor at 200 GPa with 푇 ≈ 200 퐾 [37,38]. more explicit example is calcium, which is not a In 2015, the H3S compound was observed in a high- superconductor at ambient pressure, but was found to pressure experiment [39]. It was confirmed that it is a be a superconductor under high pressure [15-17]. Its conventional superconductor with 푇 = 203 퐾 at 155 highest 푇 of 29 K was found at 216 GPa [16]. This is GPa [39]. These researches have opened a new so far the highest 푇 among the elemental metals. direction of searching for new hydrogen-rich Likewise, strontium is not a superconductor at ambient compounds which exhibit a high value of 푇 under pressure, but becomes a superconductor under high high pressure. pressure [18,19]. Strontium has been shown to exhibit I conclude this section with Table 1 which series of structural phase transitions [20] and some summarizes 푇 of several metals and compounds from specific structures under high pressure are responsible experiments. In the next section, I will give a brief for its superconductivity [21]. At the present time, the overview of a new class of compounds, called metal periodic table included 푇 of all elements both at superhydrides. Some of these compounds exhibit 푇 ambient and under pressure is almost fully constructed that is exceeding those of the non-conventional [22]. superconductors, and the classification word “high- Apart from experimental studies, the theoretical Tc” is about to be disrupted. methods have also been developed. These theories are based on the quantum behavior of materials. One of Table 1. 푇 of some selected metals and compounds the most powerful tools for calculating the quantum from experiments. effects in many-body systems is the so-called density functional theory (DFT), pioneered by Hohenberg, 푬푿푷 Superconductors P (GPa) 푻풄 (K) Reference(s) Kohn and Sham [23,24]. Kohn received his Nobel Hg 0 4.16 [1] prize in 1998. However, the practical DFT calculations were realized by several superb state-of- V3Si 0 17.1 [4] the-art softwares, such as the Vienna Ab initio MgB2 0 39 [7]

Simulation Package (VASP) [25], the Cambridge BaxLa5−x 0 30 [8] Serial Total Energy Package (CASTEP) [26], and the Cu5O5(3−y) Quantum Espresso (QE) [27]. Together with a YBaCuO7-x 0 93 [9] powerful structure searching algorithm, such as HgBaCaCuO 0 130 [10] Universal Structure Predictor: Evolutionary Xtallography (USPEX) [28] and Ab initio random Sr 58 8.0 [19] structure searching (AIRSS) [29], the search for the Ca 216 29 [16] most energetically favorable structure at each pressure ThH10 175 161 [45] and its associating physical properties, such as H3S 155 203 [39] superconductivity becomes computationally feasible. YH 166 224 [50] As of now, we enter into the era of machine learning, 6 some useful outputs that involving structural YH6 237 227 [51] properties and their superconductivity have also YH9 201 243 [51] emerged from the combination of DFT and machine LaH10 180- 260, 280 [48], [49] learning [30-31]. 200 The most explicit example of the long-term interplay between experiments and theoretical

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In search for near-room-temperature superconducting critical temperature of metal 33 superhydrides under high pressure: A review

2. Metal Superhydrides structure with the P63/mmc space group at 80 GPa. Its Tc has not been measured but it was estimated to be There is a large class of metal polyhydrides, MHx, 117 K at 200 GPa [52]. At the same time, the existence where M is a metal element and x > 2. If x is large, of CeH9 has been confirmed by another independent sometimes the compounds are called metal superhydrides. experiment by Li, et al. [53]. However, its There were several excellent theoretical and superconductivity is yet to be confirmed by experimental reviews led by Peng et al. [40], Zurek experiments. and Bi [41], Semenok et al. [42], and Flores-Livas et There are a large number of metal polyhydrides al. [43]. They gave a comprehensive overview of the that were predicted to be a superconductor but current achievement on the structural prediction of experimental confirmation is still awaiting. I give here these metal superhydrides and their superconducting only a limited list of examples. The first examples are properties under high pressure. Some of these the MgHx compounds. The MgH2 compound has compounds appear to be a metal in their normal states attracted much of attention as its application on the at room temperature and a conventional superconductor hydrogen storage [54,55]. Its phase transitions under high pressure have been studied [56,57] and a phase at temperature below 푇. Also, there have been many metal superhydrides that can be synthesized under with superconductivity was found at 180 GPa with 푇 high pressure by using DAC. Furthermore, there are = 23 K [57]. Its superhydride cousin, MgH6, was several members of this metal superhydride class that predicted to have a sodalite-like structure and exhibit exhibit superconductivity with Tc > 150 K, see Table superconductivity with 푇 as high as 263-384 K [58- 2 for theoretical prediction, and some have been 60], depending on the models of 푇 . Another close confirmed by experiments, see Table 1. Thus, we have cousin of MgH6 is CaH6. It was reported to be a arrived at the age of synchronizing interplay between superconductor with a similar lattice structure as of experiments and theory. The most surprising finding MgH6. At 150 GPa, CaH6 was predicted with 푇 = 235 is that 푇 is now getting closer to the room K [61]. More of the CaHx compounds have been temperature. studied [62] and the existence of CaH6 is still A large number of metal superhydrides share debatable. The last of my list is the real example of the common host-guest clathrate structures. In these word “superhydride”. It is the family of actinium structures, hydrogen atoms form a hydrogen cage superhydrides, AcHx, where x = 10-16 [63]. The which hosts a metal atom inside. The first example is AcH10 compound exhibits 푇 = 204-251 K at 200 GPa, ThH10 with an Fm-3m structure. It was predicted that and the AcH16 compound exhibits 푇 = 199-241 K at Tc = 176-241 K at 100 GPa and Tc = 166-228 K at 200 150 GPa [63]. GPa [44]. The Fm-3m-ThH10 was confirmed by an The binary compounds have been experimental observation, and it was reported that Tc diligently investigated for a few decades. A nearly =159-161 K at 170-175 GPa [45] in excellent complete periodic table of the stable binary agreement with the previous prediction. The most under high pressure has been established. The next famous examples are the prediction of step is the making of ternary hydride compounds, superconducting LaH10 and YH10, as the predicted Tc AxByHz. It seems like a straightforward modification is approaching the room temperature. In LaH10, it was but some tedious works must also be resolved. A large predicted that 푇 = 274-286 K at 210 GPa [40,46,47] number of possibilities of how to combine xA and yB and it was reported by experiments that LaH10 is and zH is a handful computational task. Nevertheless, indeed a conventional superconductor with 푇 of 260 K some examples have already been theoretically at 180-200 GPa [48], and the latest report has shown investigated. The first few examples are MgGeH6 the evidence that its 푇 could be as high as 280 K [49]. which is a superconductor with 푇 = 66.6 K at 200 GPa In YH10, it was predicted that 푇 could be as high as [64], CaYH12 with 푇 = 226-258 K at 200 GPa [65], 305-326 K at 250 GPa [46]. This theoretical prediction and MgCaH12 with 푇 = 205-288 K at 200 GPa [66]. In has been very fascinating as, at around 300 K, it is the 2019, there was a theoretical suggestion on the so-called room temperature. The YHx compounds increment of the electron density of states at the Fermi have been synthesized under high pressure [49-51]. level by using a metal substitution method. The most The existence of superconducting YH6 and YH9 famous example is the substitution of Li into the compounds have been confirmed with 푇 = 227 K at MgH16 clathrate structure [67]. Its 푇 was reported to 237 GPa [51] and 푇 = 243K at 201GPa [51], be as high as 473 K at 250 GPa. This finding has respectively. However, YH10 and its superconductivity coined the term “hot superconductors”, which are yet are yet to be confirmed. to be confirmed by experiments. Another example is the CeHx compounds which I conclude this section with Table 2 which have been synthesized by Salke et al. [52] using a summarizes 푇 of several metal hydride compounds laser-heated DAC. They obtained the CeH2, CeH3 and from the theoretical predictions. CeH9 compounds at different pressures. The structure of CeH9 was identified as a hexagonal clathrate

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34 PINSOOK, U.

Table 2. The parameters 휔 and 휆, and 푇 of some selected binary compounds and ternary compounds from theoretical predictions. In literature, 푇 have been estimated using different models, i.e. McM is the McMillan equation (modified by Allen and Dynes, Eq. (13)), AD is the strong coupling Allen-Dynes equation (Eq. (15)) and E is from the numerically solved Eliashberg equation [71].

Compounds P (GPa) λ 흎풍풏 (K) 푻풄 (K) (Model) Reference

MgH2 180 0.58 1111 23 (McM) [57]

CaH6 150 2.69 n/a 235 (E) [61]

YH6 165 2.24 929 184 (AD) [50] 240 (E) [50]

MgH6 300 3.29 1450 263 (McM) [58] 384 (AD) [59]

CeH9 200 2.30 740 117 (McM) [52]

LaH10 210 3.41 848 238 (McM) [46] 286 (E) [46]

YH10 250 2.58 1282 265 (McM) [46] 326 (E) [46]

MgGeH6 200 1.16 773 66.6 (McM) [64]

CaYH12 200 2.2 1230 226 (AD) [65] 258 (E) [65]

MgCaH12 200 2.53 1400 205 (McM) [66] 288 (AD) [66]

Li2MgH16 250 ~4.0 n/a 473 (E) [67]

3. Electron-Phonon Interaction Im Σ(풌, 휀) = −휋 푑휔 (훼 퐹풌 (휔)[푏(휔) Superconductivity is stemmed from the electron- + 푓(휔 + 휀)] (5) phonon interaction. There are several excellent + 훼 퐹풌 (휔)[푏(휔) reviews on the subject [68,69]. I will only give a brief + 푓(휔 − 휀)]), description. Migdal [70] and Eliashberg [71] used quantum where 푏(휔) and 푓(휔 + 휀) are boson’s and fermion’s field theory to solve the system with FrÖlich number distribution, respectively. The spectral Hamiltonian, which describes the interaction between function can be calculated from electrons and phonons. The FrÖlich Hamiltonian can be written as ± 훼 퐹 (휔) ∝ ∑ 훿휔 − 휔 ∑ 푔 훿(휀 − 휀 ± 풌 풒 풒 풌 풒,풌 ,풌 풌 휔), (6) 퐻 = 퐻 + 퐻 + 퐻 , (2)

where 풒 are the phonon modes, 풌, 풌 are the electron where 퐻 describes electrons, 퐻 describes phonons states, and the electron-phonon coupling matrix is and 퐻 describes the electron-phonon interaction given by via the electron-nucleus potential, 푉(풓 − 푹 ), with the electron’s coordinates, 풓 , and nuclear’s 푔 = 풌 + 풒퐻 풌. (7) 풒,풌 ,풌 coordinate, 푹 , and nucleus displacements, 풖. Thus, 퐻 can be explicitly written as The available electron states are confined in the vicinity of the Fermi surface only. This imposes a 퐻 = 풖 ∙∇푉(풓 − 푹 ) (3) selection rule on the transfering states due to the interaction. From this theory, the largest The lattice vibrations and hence all phonon modes are superconducting gap is proportional to the quantity embeded in 푢. By treating 퐻 as perturbation, the solution can be derived in terms of Green’s function as 훼퐹(휔) 휆 = 2 푑휔 , (8) 휔 퐺(풌, 휀) = , (4) 풌(풌,) which is sometimes called the isotropic coupling where the self-energy part, Σ(풌, 휀), can be written in constant, and 훼퐹(휔) is closely related to Eq. (6). terms of the so-called spectral function, 훼퐹(휔), as Most of all, the spectral function can be calculated by follows;

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In search for near-room-temperature superconducting critical temperature of metal 35 superhydrides under high pressure: A review using DFT method [27]. Thus 휆 can be derived from For the strong-coupling, in which 휆 > 1.5 , the the DFT results. Eq. (8) can be interpreted that 휆 equation for 푇 can be modified further to measures the average strength of the electron-phonon coupling. Hopfield [72] analysed this equation and .() 푇 = 푓푓 exp − ∗ , (15) concluded the relation between 휆 and 푁 , as . (. )

where the correction factor 휆 = , (9) 휆 where 푔 is the effective coupling related to 푓푓 = 1 + 2.46(1 +3.8휇∗) 푔,, . This form of Eq. (9) already suggests that 휆 (16) 휔 and 휔 are inversely proportional. This parameter 휆 ( − 1)휆 휔 can also be measured from the electronic specific heat, 1 + , ∗ 휔 퐶(푇) , as suggested by Allen [3]. The relation 휆 + 3.31(1 + 6.3휇 ) 휔 between 퐶(푇) and 휆 can be written as and, 퐶(푇) = 푁(1+ 휆) 푇. (10) / 휔 = ∫ 푑휔 휔 훼퐹(휔) . (17) 4. Empirical Formula of Tc () If we think of as a distribution function, As discussed earlier, 푇 from the BCS theory, Eq. (1), has some limitations. This is because Eq. (1) can 휔 is just the square root of the average 휔 over the be used in the weak electron-phonon coupling regime distribution function. Eq. (15) is known as the strong coupling Allen-Dynes equation, denoted as AD in only. Furthermore, 푉 is very difficult to be Table 2. evaluated. Thus, a more powerful model of 푇 is The analytic equation of 푇 is now explicitly needed. From section 3, it has been shown that 휆 can related to two important quantities, 휆 and 휔 . In order be evaluated in a more practical way, such as from to find a high value of 푇, it seems that we need both DFT. Also, the effective interaction in the BCS theory large 휆 and large 휔 . However, from Eq. (14), they can be related to 휆 as [3] are not independent. Thus if we need to improve 푇, we might need to look into one step deeper. The next ∗ 푁푉 → 휆 − 휇 , (11) section, I will discuss the spectral function from which

both 휆 and 휔 are derived. where 휇∗ is effective Coulomb parameter, of which the value is limited in the range of 0.1-0.2 only. If we 5. Simple Model of Spectral Function insert this relation into Eq. (1), the equation resembles

For any system, if we look for a high value of 푇, 푇 = exp − , (12) it is natural to search for conditions that give large 휔 . ∗ and 휆. However, from Eq. (8) and (14), 휆 and 휔 are This led McMillan [73] and subsequent development not entirely independent. There is no guarantee that 휆 of Allen and Dynes [74] to make a modification of Eq. and 휔 will always be large at the same time. This led us to make a deeper detail analysis. As both 휆 and 휔 (12) into an empirical formula of the form are stemmed from 훼 퐹(휔), we need to elucidate its .() impacts in more detial. In order to give a simplest view 푇 = exp − , (13) . ∗(.) of 훼 퐹(휔) and to gain insight into the improvement of 푇, Allen and Dynes proposed the Einstein Model Eq. (13) is the so-called McMillan equation if the for analyzing the spectral function [74], written as factor is replaced by . Subsequently, Allen and . . 훼퐹 (휔) = 훿(휔 − 휔 ). (18) Dynes proposed that 휃 should be replaced by

It can be shown easily that Eq. (18) gives 휔 → 휃 , 휔 = exp ∫ ln(휔) 훼 퐹(휔), (14) where 휃 is the celebrated Einstein temperature. It turns out that the model in Eq. (18) is too simple. which measures the average frequency by using In this review, I assume that 훼퐹(휔) can be estimated distribution function based on the spectral function. by a constant function Eq. (13) has been widely accepted and also known as Allen-Dynes modified McMillan equation, denoted as 푐 ; 휔 ≤ 휔 ≤ 휔 훼퐹(휔) = . (19) McM in Table 2. 0; 푒푙푠푒푤ℎ푒푟푒

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36 PINSOOK, U.

This is a straightforward modification of Eq. (18). It From the bandwidth function, Eq. (19), we can can be thought of as a sum of a series of many different integrate Eq. (8) and (14) analytically for 휆 and 휔 Einstein oscillators, instead of just one type of the respectively. The results are Einstein oscillator as in Eq. (18). This is because, by inspection, I find that the spectral function in most 휆 =2푐 ∫ =2푐 ln , (20) metal hydrides under high pressure is dominated by the oscillation part of the hydrogen atoms. A typical and 2푐 푑휔 spectral function from DFT [66] is shown in Figure 1, 휔 = exp ln(휔) 휆 휔 where the yellow line shows the contribution from the metal atoms and the green line shows the contribution ( ( ) ( )) from the hydrogen atoms. Let us call the function in 푐 ln 휔 − ln 휔 (21) = exp 휔 Eq. (19) as the bandwidth function. The constant 푐 2푐 ln 휔 can be related to the average magnitude of the electron-phonon interaction, and subscript H = 휔휔. emphasises the importance of the hydrogen atoms. The analytic solutions give us some insight Typically, the bandwidth function starts at 휔 >0 as the low-frequency vibrations of metal atoms information as follows; contribute much less to 훼퐹(휔), comparing with the 1. According to Eq. (20), 휆 will be large if 푐 is large and the bandwidth function is wide, i.e. 휔 ≫ high-frequency vibration of hydrogen atoms, and 휔 has a cutoff roughly around the order of 70 THz 휔. (equivalent to approximately 3,500K). For a pure 2. According to Eq. (21), 휔 will be large if both 휔 and 휔 are large. Hence, 휔 will be largest for a hydrogen lattice, 휔 could be as high as 85 THz (equivalent to approximately 4,100K) [43]. Table 3 given set of parameters if 휔 = 휔 . In the other words, 휔 will be large if the bandwidth function is gives the values of 휔 , 휔 and 푐 of several compounds, roughly extracted from the DFT narrow with a large value of 휔 and 휔. In addition, calculations [43,66]. 휔, derived from Eq. (21), is independent of 푐. 3. Under some influential factors, i.e. under high pressure, sometimes 푐 is only slowly varying in a given compound. If the influential factors tend to make a higher 휔 by narrowing the bandwidth function, 휆 will definitely tend to reduce. Figure 2 shows typical plots between 휔 and 휆 . For comparison, 휔 and 휆 of some selected compounds, calculated from DFT [60], are shown in Table 4. It clearly shows the inverse proportion between 휔 and 휆. By further analyzing Eq. (17) and (21), the relation between 휔 and 휔 can be established and they are positively correlated. Figure 2 also shows the positive correlation between 휔 and 휔 . This implies that increasing of 휔 will also tends to diminish 휆. This finding is also in fair agreement with Hopfield analysis, Eq. (9) in that 휆 and 휔 are inversely proportional. Figure 1. Shows a typical spectral function of MgCaH12 at 250 GPa, calculated from DFT [66]. The yellow line 4 The limitation of the above analysis is that 푐 is shows the contribution from metal atoms and the green assume to be slowly varying. If it is largely varied line shows the contribution from hydrogen atoms. The between different compounds, or different structures red dashed lines show the definition of model of the same compound, the above analysis might be parameters of Eq. (19). These parameters are chosen in invalid. order to conserve the area under the original spectral function curve, denoted by 퐴 .

Table 3. Parameters for the model of the spectral function, Eq. (19), of several compounds, roughly extracted from the DFT calculations of Ref. [43] and [66].

Compounds 흎풂 (K) 흎풃 (K) 풄푯 흀 Eq. (20) 흎풍풏 (K) Eq. (21) 푨푺푭 (K) Tc (K) H3S [43] 812.32 2901.13 0.64 1.63 1535.13 1336.84 214 LaH10 [43] 580.23 2669.04 0.72 2.20 1244.45 1503.95 234 MgCaH12 [66] 1100 2800 1 1.87 1754.99 1700 277 H [43] 928.36 4061.58 0.56 1.65 1941.81 1754.60 282

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In search for near-room-temperature superconducting critical temperature of metal 37 superhydrides under high pressure: A review

the same structure but slightly different thermodynamic conditions, we would arrive at searching for the largest area under the 훼 퐹(휔) curve, 퐴. In the other words, 퐴 must be positively correlated with 푇 . Tables 3 and 4 show the values of 휔 and 휆 , and 푇 . The relation among these three quantities is rather ambiguous. However, the relation between 퐴 and 푇 is clearly established, as shown in Figure 4. According to the Leavens-Carbotte analysis, they suggested that 푇 =푇 +푆×퐴 [60,75]. In Figure 4, the red dotted line is the Leavens-Carbotte prediction with 푆= 0.148 [60,75]. The open triangles are the results taken Figure 2. shows typical plots between 휔 (light red line) and 휆 (light grey line), and also shows the from ref [60]. The open circles are the results from Table 3. The blue dashed line is the best linear fitting positive correlation between 휔 and 휔 (light purple dots). with 푆 = 0.172. The discrepancy is possibly from the crudeness of the model, Eq. (19), and also from the different methods of finding 푇 . Surprisingly, the Table 4. The inverse proportion between 휔 and 휆 in correlation between 푇 and 퐴 also establishes across several compounds. All data are taken from Ref. [60]. different compounds.

Compounds P (GPa) 흎풍풏 (K) 흀 Tc (K) H3S 220 1415.75 2.08 229

H3S 250 1531.80 1.82 218

H3S 280 1589.82 1.61 199 CaH6 150 1044.41 2.53 200 CaH6 200 1241.68 1.95 190 CaH6 250 1357.73 1.69 180 CaH6 300 1415.75 1.59 175 LaH10 250 1067.62 2.46 206 LaH10 300 1334.52 1.80 189

At this stage, I can now discuss the impact of the bandwidth function, Eq. (19), on the value of 푇. For ∗ 푐 =1 and 휇 = 0.1 , The contour plot of 푇 , Eq. Figure 3. shows contour plot of Tc, Eq. (15), as a (15), as a function of 휔 and 휔 is shown in Figure 3. function of 휔 and 휔, all in units of Kelvin. Some special cases can be discussed as follows; 1. If the path A (green arrow A) in Figure 3 is taken, where 휔 is fixed and 휔 increases, 푇 will increase accordingly. 2. If the path B (green arrow B) in Figure 3 is taken, where 휔 is fixed and 휔 increases, 푇 will appear to decrease. 3. If the path C (green arrow C) in Figure 3 is taken, where both 휔 and 휔 increases, there exists a special path that 푇 appears to be a constant. 4. Surprisingly, in order to achieve a highest 푇, the present model favors a widest bandwidth function, as shown at the end point of the path A in Figure 3. This finding also implies that a large 휆 is more favorable than a large 휔. This is also in good agreement with the data in Table 4. If we look at the same compound, Figure 4. Shows Leavens-Carbotte analysis of 푇 , i.e. the highest comes from the largest . However, for 푇 휆 푇 =푇 +푆×퐴 [60,75]. The red dotted line is the different compounds, a more careful analysis must be Leavens-Carbotte prediction with 푆 = 0.148 [60,75]. taken into account. The open triangles are the results taken from ref [60]. As searching for the largest 푇 is related to The open circles are the results from Table 3. The blue searching for the widest bandwidth function. If is 푐 dashed line is the best linear fitting with 푆 = 0.172. not varied much, such as in the same compound with

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38 PINSOOK, U.

6. Conclusions [10] A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, “Superconductivity above 130 K in the I gave a brief review on the superconductivity in Hg-Ba-Ca-Cu-O system,” Nature, vol. 363, pp. general in both experimental observation and 56-58, 1993. theoretical prediction. Then, I focused on a limited list [11] See for example, S. Yoksan, “Isotope effect in of metal superhydrides under high pressure and their high-Tc superconductors,” Solid State Communications, vol. 78, pp. 233-236, 1991. near-room-temperature 푇 . I also emphasised on the important of the spectral function in order to give a [12] P. Udomsamuthirun, T. Kruaehong, T. Nilkamjon, S. Ratreng, “The new superconductors of YBaCuO prediction on 푇 . I proposed a simple model that materials,” Journal of Superconductivity and provides insight information in the behavior of 휔 and 휆. Finally I pointed out where to look for a higher Novel Magnetism, vol. 23, pp. 1377-1380, 2010. 푇. [13] Prof. Percy W. Bridgman - Nobel Lecture. 7. Acknowledgement https://www.nobelprize.org/prizes/physics/194 6/bridgman/lecture/

[14] C. E. Weir, E. R. Lippincott, A. van I dedicate this work to my great mentor, Prof. Valkenburg, and E. N. Bunting, “Infrared Suthat Yoksan. I thank my best colleagues, Prof. Thiti studies in the 1- to 15-micron region to 30,000 Bovornratanaraks, Dr. Prutthipong Tsuppayakornaek, atmospheres,” Journal of Research of the and Dr. Wiwittawin Sukmas. National Bureau of Standards Section A, vol.

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