Hydrogen Confined in a Single Wall Carbon Nanotube Becomes a

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Hydrogen confined in a single wall carbon nanotube becomes metallic and superconductive nanowire under high pressure Yueyuan Xia, Bo Yang, Fan Jin, Yuchen Ma, Xiangdong Liu, and Mingwen Zhao Nano Lett., Just Accepted Manuscript • Publication Date (Web): 19 Mar 2019 Downloaded from http://pubs.acs.org on March 19, 2019

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Hydrogen confined in a single wall carbon nanotube becomes metallic and superconductive nanowire under high pressure

Yueyuan Xia†, Bo Yang†, Fan Jin ‡, Yuchen Ma‡*, Xiangdong Liu†, and Mingwen Zhao†* † School

of Physics & State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China

‡ School

of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, China

*Corresponding authors, E-mail: [email protected]; [email protected], Tel. 86-53188366533

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ACS Paragon Plus Environment

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ABSTRACT: Metallic hydrogen is a long-desired material. However, the pressure needed to metalize hydrogen is difficult to access experimentally. We demonstrated that the high-density of hydrogen confined in a (8,0) single-wall carbon nanotube (SWNT) can be metalized at a relative low pressure of 163.5GPa, due to the "physical compression" effect of SWNT. Through mimicking experimental measurements of the specific heat of confined hydrogen nanowire, we showed that the electronic specific heat of the hydrogen has a clear jump around 225K, verifying a superconducting transition at this critical temperature. The superconducting hydrogen can be very well explained by the Eliashberg superconductivity theory for electron-phonon strongcoupling system. Our simulation results open an avenue for study of nano-hydrogen materials at high pressure.

KEYWORDS: superconductivity, electronic specific heat, physical compression, hydrogen, carbon nanotubes

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Studies of the properties of hydrogen at high pressure have been a long standing subject since Wigner and Huntington predicted that hydrogen would become metallic under high compression in 1935. 1 Ashcroft predicted that metallic hydrogen would be a superconductor with high critical temperature (TC).2 High temperature superconductivity of metallic hydrogen is so attractive that the goal of metallization of hydrogen and searching for the structures of hydrogen under ultra-high pressure has been a continuous focus of interest.3,4 A theoretical work based on first-principles study indicated that molecular metallic hydrogen could be a superconductor at pressure of 450 GPa with a superconducting critical temperature TC ∼242K.3 Even TC could be above room temperature for hydrogen in atomic phase at pressure of 500GPa.

5,6

However, metallization of hydrogen is a challenging subject because the pressure needed is as high as 400~500GPa, a value that is difficult to access experimentally. Although, Dias and Silvera reported that they observed metallic hydrogen at a pressure of 495GPa,4 there are some unfavorable comments on their results,7-10 followed by the reply of Silvera.11 It indicates that to metalize hydrogen experimentally is a very difficult task due to the technological barrier. To circumvent this difficulty, Ashcroft studied hydrogen-rich compounds and predicted that these hydrides might become metallic and superconducting at much lower pressure than that needed for pure hydrogen because hydrogen in these compounds was “chemically compressed”.12 This prediction has been a driving force for studies of hydrogen-dominant materials under high pressure.13-24 Hydrogen sulfide, among other hydrides, has attracted special attention. Motivated by the theoretical predictions of high-Tc superconductivity in 3 / 20


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compressed solid H2S,13,14 Drozdov et al. obtained very encouraging experimental results. They observed superconducting H3S with TC=203K at a pressure of 155GPa.15 Recently, a reﬁned phase diagram of H-S, at pressures from 0 to 200 GPa has been built, and the exact decomposition pathway of H2S under pressure was known.16 In recent years, hydrogen-rich rare-earth (RE) hydrides, such as LaH10, YH10,and ThH10, have attracted much research interests due to the room temperature superconductivity predicted for LaH10 and YH10 19-21 and the successful synthesis of LaH10 at 170 GPa.22 While superconducting ThH10 has stable phase at lower pressure of 100 GPa.23 The superconducting phase of AcH16 was also predicted at 110 GPa.24 In this work, we devise a new way to metallize hydrogen, rather than hydrides, and to obtain superconducting hydrogen nanowire confined in single-walled carbon nanotubes (SWNTs), taking advantage of the excellent mechanical properties and inert inner walls of SWNTs. Systematic molecular dynamics simulations (MDSs) study indicated that hydrogen can be injected inside the cages of the SWNTs to form very high-density molecular hydrogen confined inside the SWNTs.25 Therefore, a complex system that consists of two subsystems, the SWNT and the hydrogen nanowire, can be obtained. We use a (8, 0) SWNT, which confines high-density molecular hydrogen in its cage, to study the thermodynamic properties of the system under hydrostatic pressure by using first-principles MDSs. In this way, we can study hydrogen under physical compression rather than chemical compression by heavier elements, as in the case of the hydrides. We demonstrated that it is hydrogen alone that is responsible for the high TC superconducting state. 4 / 20


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The supercell used in the simulations consists of a segment of (8,0) SWNT having H2 molecules are encapsulated (called H@SWNT hereafter), as shown in Figure 1(a). The size of the supercell is carefully chosen to ensure that the energy of the system is accurately predicted, while the computational resources are not wasted. The structure stability of the H@SWNT is discussed in the Supplementary Information. Periodic boundary condition is used to simulate the thermodynamic properties of the complex system under a constant pressure of 163.5 GPa at different temperatures (the details of the simulation method are given in the Supplementary Information). Different from most of the theoretical studies of superconductive properties of hydrogen or hydrogenrich materials, in which the researchers usually searched for special structures and predicted the superconductive properties for these structures by use of density functional theory for superconductivity (DFTSC), we simulate the case of a thermal source connecting to our system to transfer thermal energy to the system causing the temperature and the inner energy changes of the system. Based on the first law of thermodynamics, we determine the specific heat of the system as a function of temperature at a constant pressure P0. In this way, we can mimic the experimental measurements of the specific heat as a function of temperature.26,27 The outstanding advantage of this method is that the values of specific heat at constant pressure are determined by the energy difference at different temperatures., thus the influence of the nuclear quantum effects (NQEs) at high pressure 28 on the specific heat values can be partially reduced. The proper DF-functionals

28-30

were selected in the simulation to

ensure the accuracy of the energies and the pressure obtained (see the Supplement 5 / 20


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Information). If there is a normal-state to superconductive-state transition for this system at the critical temperature TC, a jump of the specific heat at TC must be observed. We really observed a specific heat jump of the system under a hydrostatic pressure of 163.5 GPa at temperature T~225K, confirming a normal-state to superconducting-state transition, since structural phase transition is not observed (see the structures of the confined hydrogen shown in the Supplement Information). Then we use the electronphonon strong-coupling Eliashberg theory31,32 to predict the characteristic parameters of the metallic and superconductive hydrogen nanowire. Figure 1(a) gives the structure of the H@SWNT complex system at temperature T=20K, and under pressure P=163.5GPa, which exhibits two sub-systems without chemical bonds between them. Under the hydrostatic pressure of 163.5 GPa, some of the hydrogen molecules are dissociated and closely packed to form H-clusters, as shown in Figure 1(b)-(d). The electron density (n) of the confined hydrogen is as high as 3.86 ×1023/cm3. The average radius, rs, of the electrons in the confined hydrogen estimated by using the formula

4𝜋 3 3 𝑟𝑠

1

= 𝑛 is rs ≈ 0.85Å, satisfying the Goldhammer-Herzfeld

criterion of metallization rs ≤ 0.88 Å. The electronic band structures of the H@SWNT calculated by using Becke-Lee-Yang-Parr (BLYP) functional are plotted in Figure 1(e). Clearly, the hydrogen confined in the SWNT becomes a metal, while the SWNT itself keeps semiconducting nature. To confirm the metallization of the hydrogen nanowire, we also calculated the band structures by using ab initio many-body perturbation theory within GW approximation. The metallization is also obvious, as shown in Figure 1(f). The high pressure exerted on the system makes the electrons deviate from their ground 6 / 20


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state, thus suppresses the Peierls transition. Therefore, the confined 1D hydrogen nanowire can be metallized, as the electronic energy bands shown in Figure 1(e) and Figure (f). The pressure 163.5 GPa, however, is much lower than those predicted in previous literatures.

3,4

The SWNT and the confined hydrogen form two subsystems.

Although the closest distance between a carbon atom on the tube wall and a hydrogen atom of the encapsulated hydrogen reaches about 1.4 Å, there is no any chemical bond formed between carbon atom and hydrogen atom in the whole temperature range of the simulations. We attributed it to the inertness of the inner wall of SWCNTs. The metallization of the hydrogen is due to the additional “physical compression” provided by carbon atoms of the SWNT (ref. to the discussion in the Supplementary Information). We now turn to the thermodynamic properties of the system under hydrostatic pressure. If a thermal source connecting to the H@SWNT transfers an amount of heat energy dQ to the system under a constant pressure of P0, it will cause a volume change of dV and a change dU of the inner energy with a temperature change of dT. According to the energy conservation law, we have: 𝑑𝑈 = 𝐶𝑃𝑑𝑇 ― 𝑃0𝑑𝑉,

(1)

where Cp is the constant-pressure specific heat. CP as a function of temperature can be obtained from dU, dT and dV at different temperature via differential average. Figure 2(b) gives the molar heat-capacity CP of the confined hydrogen as a function of temperature obtained by subtracting the contribution of empty SWNT (shown in Figure 2(a)) from the complex H@SWNT. The CP values of the hydrogen drastically change with temperature for T < 270K, and there is a clear jump around T~225K. The constant-pressure specific heat CP of the hydrogen consists of different 7 / 20


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contributions: 𝐶𝑃 = 𝐶𝐷 + 𝐶𝑉 = 𝐶𝐷 + 𝐶𝐸 + 𝐶𝐿,

(2)

where CD is the dilation contribution, CV is the constant-volume specific heat, CE is the conduction electronic contribution that includes electron-phonon renormalization effects, and CL is the contribution from the lattice vibrations. CL is written as: 𝒆𝜷ħ𝝎𝝆(𝝎)𝒅𝝎

𝝎

𝑪𝑳 = ∫𝟎 𝒎𝒌𝑩(𝜷ħ𝝎)𝟐 (𝒆𝜷ħ𝝎 ― 𝟏)𝟐 ,

(3)

where 𝛽 = 1/𝑘𝐵𝑇, 𝜔𝑚 is the maximum phonon frequency(energy), kB is Boltzmann constant, and ρ(ω) is the phonon density of states, which is shown in Figure S2(a) of the Supplementary Information. The dilation contribution CD can be obtained from the volume change. Through numerical calculations by use of Eq.(3), we obtained CL. All the contributions to the specific heat in Eq. (2) as a function of temperature are shown in Figure S2(b) of the Supplement Information. The specific heat of the electrons of the confined hydrogen, CE, including the electrons in superconductivity-state and in normal-state, is obtained by subtraction of CL+CD from the CP. The specific heat of the electrons in normal state can be written as : 𝐶𝑁 𝐸 = 𝛾𝑇,

(4)

where γ is the Sommerfeld constant, which is related to the single-spin electronic density of state (DOS) N(0) at the Fermi energy. The relation between N(0) and γ can be written as: 35 𝑠𝑡𝑎𝑡𝑒𝑠

0.212

𝑚𝐽

𝑁(0)[𝑒𝑉.𝐻𝑎𝑡𝑜𝑚 .𝑠𝑝𝑖𝑛] = 1 + 𝜆 𝛾[𝑚𝑜𝑙.𝐾2] ,

(5)

where λ is the mass-renormalization parameter that we will give its value later. The figure of the DOS near the Fermi energy for the confined hydrogen at T=290K is shown 𝑠𝑡𝑎𝑡𝑒𝑠

in Figure 3(b), from which a very high N(0) value of 𝑁(0) = 0.6785𝑒𝑉.𝐻𝑎𝑡𝑜𝑚.𝑠𝑝𝑖𝑛 is

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𝑚𝐽

obtained. The Sommerfeld constant γ = 9.8293 𝑚𝑜𝑙.𝐾2 is calculated by using Eq.(5). Figure 3(a) shows the electronic molar heat capacity of the confined hydrogen as a function of temperature. It is evident that the electronic specific heat has a clear jump at T~ 225K, which can be attributed to superconductive phase transition at the critical temperature TC. There is also a small jump point around T~80K. Of course, it is possible that the fluctuation of the H@SWNT system under the high-pressure may cause some errors of the simulations. However, the amplitude of the main jump is much larger than that of the possible error bars of the simulation. Until now, we proved theoretically that the hydrogen confined in the SWNT becomes metallic and superconductive under pressure of 163.5GPa, and the critical temperature TC is determined without relying on any theory of superconductivity, just as the case of experimental measurements of the specific heats to determine the critical temperatures for the superconductors.26,27 In the following, we will use the electron-phonon strongcoupling Eliashberg theory to predict the characteristic superconductivity parameters of the hydrogen nanowire.31-33 The details of our theoretical calculations are given in the Supplementary Information. Figure 4(a) shows the electron-phonon interaction spectral density α2F(ω) for the hydrogen nanowire obtained by use of the scaling theorem for TC developed by Coombes and Carbotte. 34,35 The details of calculations for the Coulomb pseudo-potential μ* and the electron-phonon interaction spectral density α2F(ω) and its associated parameters, such as the mass-renormalization parameter λ, the characteristic phonon energy 𝜔𝑙𝑛, and 𝜆 to be used in the calculation of TC are given in the Supplementary Information.

The results obtained are listed in Table I. 9 / 20


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The semi-quantitative formula for TC developed by Leavens and Carbotte

36

are

employed to calculate the critical temperature:

[

𝑘𝐵𝑇𝐶 = 1.13𝜔0𝑒𝑥𝑝 ―

],

1+𝜆+𝜆 𝜆 + 𝜇∗

(6)

that gives TC =225K, in good consistence with our simulations. Now, we compare the superconductivity parameters of the hydrogen nanowire with those of the conventional superconductors. Mitrović et al

37

developed a relation

between 2Δ0 /kBTC and the strong-coupling index 𝑇𝐶/𝜔𝑙𝑛, i.e., 2∆0/𝑘𝐵𝑇𝐶 = 3.53[1 + 12.5[𝑇𝐶/𝜔𝑙𝑛]2𝑙𝑛[𝜔𝑙𝑛/2𝑇𝐶]]

(7).

The strong-coupling index 𝑇𝐶/𝜔𝑙𝑛=0.266 is obtained for the hydrogen nanowire, and 2Δ0 /kBTC is thus 5.500 from Eq. (7). Figure 4(b) shows 2∆0/𝑘𝐵𝑇𝐶 as a function of the strong-coupling index 𝑇𝐶 /𝜔𝑙𝑛. The dashed curve is from Eq. (7), and the data points are taken from the data summed in the Table I of a previous literature. 37 It seems this curve is universal since many of the superconductors including the hydrogen nanowire of this work follow this curve very well. Another importance quantity is the electron-specific heat jump at Tc. From Figure 3, it is difficult to determine the accurate jump value ΔC(𝑇𝐶) because of the complicate structure of the electronic specific heat near TC. If we use the averaged value ΔC(𝑇𝐶) = 1.453 cal./mol.K , the ratio specific heat jump ΔC(TC)/CN(TC)= ΔC(TC)/γTC =2.74 can be obtained, which is agree with the theoretical prediction of ΔC(TC)/γTC =2.8 for many conventional superconductors. 36 We summarize the superconductive parameters of the hydrogen nanowire studied in this work in Table I. Obviously, these parameters are consistent with the parameters for many known superconductors. The TC =225K is 10 / 20


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among the high values of all-known superconductors. This is reasonable since the maximum phonon energy 𝜔𝑚= 620meV found for the hydrogen nanowire is remarkable larger than those for other superconductors, the DOS at the Fermi energy and the strong-coupling index 𝑇𝐶/𝜔𝑙𝑛 are large for the hydrogen nanowire. For the same reason, the 2∆0/𝑘𝐵𝑇𝐶 value is large for the hydrogen nanowire.

Table I. The superconductive parameters of the hydrogen nanowire

𝜔0 (𝑚𝑒𝑉) 𝜔𝑙𝑛 (𝑚𝑒𝑉) λ 277

72.611

2.071

𝜆 2.339

𝜇∗

𝑇𝐶(K)

𝑇𝐶/𝜔𝑙𝑛

2∆0/𝑘𝐵𝑇𝐶

0.127

225

0.266

5.500

ΔC(𝑇𝐶)/𝛾𝑇𝐶 2.74

___________________________________________________________

In conclusion, by using first-principles MDSs to mimic experimental measurements of specific heat under high compression at different temperature to verify superconductive phase transition, we have theoretically proved that high density of hydrogen confined in a (8,0) SWNT under pressure of 163.5GPa becomes metallic and superconductive. The pressure felt by the confined hydrogen is much higher than the external exerted value since the additional pressure from the tube wall increase substantially with the external pressure (see the Supplement Information). The superconductive properties of the hydrogen nanowire are fully consistent with the electron-phonon strong-coupling superconductive theory. Some of the confined hydrogen molecules dissociate and form small cluster groups such as H, H3 H4 and H6. But the hydrogen does not react with the SWNT. Thus, a pure physical-pressure from the carbon atoms on the tube wall assists the hydrogen becoming metallic and 11 / 20


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superconductive at much lower pressure needed for metalizing bulk hydrogen. Recently, molecular hydrogen dissociation to form metastable atomic phase under similar pressure region was reported.

38

Although the present simulation results are based on

the simulations for hydrogen confined in an ideal straight SWNT, the validity of the results for hydrogen confined in SWNTs with defects and/or SWNT bundles may be also held (see the discussion in the Supplementary Information). The superconductive nanowire confined in semiconducting SWNTs may find potential application in design of nano-electronic or nano-optoelectronic devices. The molecular reactions induced by high compression to form small clusters may provide information on the properties of hydrogen under extreme conditions of high pressure, which are also very important for other areas such as planetary science concerning the giant planets, inertial fusion and high energy density materials.

ASSOCIATE CONTENT Supplementary Information I. Method and computational details of the simulations; II. The structure of the H@SWNT system and the electronic density distributions of the confined hydrogen; III. The phonon frequency spectrum and the contribution of the lattice vibration to the specific heat; IV. The superconductivity property of the hydrogen nanowire confined in the SWNT; V. Details in GW calculations.

AUTHOR INFORMATION Corresponding author *E-mail: [email protected] (Y. Ma), [email protected] (M. Zhao) 12 / 20


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ORCID Mingwen Zhao: 0000-0002-7583-9682

Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (Nos. 21433006, 21833004 and 11774201).

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Figure 1 (a) The structure of the H@SWNT system at temperature T=20K, and under pressure P=163.5GPa. (b)-(d) the electron density distributions around some H-clusters. (e) Electronic band structures of the H@SWNT system obtained by use of BLYP functional, and (f) by use of many-body perturbation theory within GW scheme. The bands of (8,0) SWNT and the confined hydrogen are indicted by the black dotted lines and read solid lines, respectively. The energy at the Fermi level was set to zero.

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Figure 2. The constant-pressure heat capacities versus temperature obtained by using first-principles MDSs. (a) the molar heat capacity as a function of temperature at a constant pressure P0 = 163.5GPa for the SWNT, and (b) the molar heat capacity as a function of temperature at a constant pressure of P0=163.5GPa for the hydrogen confined in the SWNT.

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Figure 3. (a) The electronic molar heat capacity as a function of temperature for the hydrogen confined in a (8,0)SWNT under hydrostatic pressure of 163.5 GPa, the solid squares are for all the electrons including superconducting electrons and the normalstate electrons , and the straight dashed line for the electrons in normal-state; (b) the density of state of the electrons near the Fermi energy for the hydrogen confined in the SWNT under hydrostatic pressure P=163.5GPa and at T=290K .

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Figure 4. (a) the spectral density α2F(ω) obtained by use of the scaling theorem for the hydrogen nanowire; (b)

2∆0 𝑘𝐵𝑇𝐶

as a function of the strong-coupling index 𝑇𝐶 /𝜔𝑙𝑛

showing the universal behavior of superconductors.

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TOC

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Hydrogen Confined in a Single Wall Carbon Nanotube Becomes a

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