Emergence of superconductivity in doped H2O ice at high pressure

We investigate the possibility of achieving high-temperature superconductivity in hydrides under pressure by inducing metallization of otherwise insulating phases through doping, a path previously used to render standard semiconductors superconducting at ambient pressure. Following this idea, we study H2O, one of the most abundant and well-studied substances, we identify nitrogen as the most likely and promising substitution/dopant. We show that for realistic levels of doping of a few percent, the phase X of ice becomes superconducting with a critical temperature of about 60 K at 150 GPa. In view of the vast number of hydrides that are strongly covalent bonded, but that remain insulating up to rather large pressures, our results open a series of new possibilities in the quest for novel high-temperature superconductors.

Unfortunately, many (if not most) chemical compounds containing hydrogen only metallize at extremely high pressures.The paradigmatic case is pure hydrogen, whose metallic state is the ground-state structure only above 500 GPa [44][45][46][47][48][49] .There are certainly other phases that are metallic at lower pressure, but these are often thermodynamically unstable, and therefore difficult, if not impossible, to access experimentally.
A possible, but until now overlooked, solution is doping.It is well known that by introducing enough electronor hole-donating impurities one can render a semiconducting system metallic and even superconducting.This strategy was already successful in inducing superconductivity in diamond (doped with boron) in 2004 50 , silicon (doped with boron 51 ), germanium (doped with gallium 52 ), and silicon carbide (doped with boron 53 or alu-minum 54 ).Transition temperatures are unfortunately rather low, remaining below 4 K.
In this work we follow this strategy, and investigate if the combination of doping and high pressure can be used to obtain high-temperature superconductivity in hydrides.We select as an example one of the most abundant, and also one of the best studied, hydrides, namely H 2 O.Note that undoped H 2 O remains insulating up to the terapascal range of pressures.In fact, its metallization was predicted to occur beyond 5 TPa [55][56][57][58] .The paper is organized as follows: in Sec.II, we present the discussion of the structural transition from molecular ice to the covalent phase under pressure, followed, in Sec.III, by the results for the electronic structure of doped ice.In Sec.IV, we discuss the emergence of superconductivity in ice under pressure.Finally, we present our conclusions in Sec.V.
At ambient pressure and low temperatures ice assumes 64 its phase I, where oxygen has four hydrogen neighbors: two covalently bonded (forming the H 2 O molecule) and two connected by hydrogen bonds to neighboring H 2 O molecules.Below 200 K, phase I transforms to phase XI, and under compression to phase IX, stable in the range from 0.1 to 1 GPa.Under further compression, and at very low temperatures, the phase VIII dominates up to 60-80 GPa.This molecular crystal can be seen as an ordered and symmetrized version of phase VII that occurs at high temperatures.At 80-90 GPa was The molecular crystal transforms to a fully covalent phase-X at 120 GPa.Bottom panel: Calculated enthalpy for the VIII and X phases of H2O ice.Values are given with respect to the elemental decomposition H2 + O.The reference structures for hydrogen are P 63m (0-120 GPa) and C2/c (120-300 GPa) from Ref. 45 .The reference phase for oxygen is C2/m, ζ structure [59][60][61] .The experimental transition region between the two structures is marked with lines according to Goncharov et al. 62 .From our calculations, the transition from a molecular ice-crystal to the fully symmetric covalent phase X occurs at 120 GPa.
reported the emergence of the cuprite-type ice-X, characterized by static, symmetric O-H bonds 62,81,82 .
Figure 1 shows our theoretical phase diagram under pressure.The energies, atomic forces and stresses necessary to construct this figure were evaluated within density functional theory with the Perdew-Burke-Erzernhof (PBE) 83 approximation to the exchange-correlation functional.A plane wave basis-set with a high cutoff energy of 1000 eV was used to expand the wave-functions together with the projector augmented wave (PAW) method as implemented in the Vienna Ab Initio Simulation Package vasp 84 .Geometry relaxations were performed with tight convergence criteria such that the forces on the atoms were less than 2 meV/Å and the stresses were less than 0.1 meV/Å 3 .
At low pressure between 20 to 110 GPa we have phase VIII (see Fig. 1, in agreement with the experimental phase diagram).Above 110 GPa it undergoes a transition to the proton-symmetric and experimentally confirmed phase X.It has been shown that due to proton symmetrization, quantum effects and anharmonicity no longer play a major role [85][86][87] at higher pressure.Despite the pressure shift, our calculations (Fig. 1) are in good agreement with experiments.Phase X is the dominant structure of ice up to at least 300 GPa.This phase is extremely interesting from our point of view, as it is no longer a molecular crystal and exhibits a complete covalent character, as indicated by the behavior of the electron localization function 88 (see top panel of Fig. 1).This is absolutely essential for the appearance of doping induced superconductivity, otherwise impurities would just introduce localized states that can not participate in the formation of Cooper pairs.

III. ELECTRONIC STRUCTURE OF DOPED ICE UNDER PRESSURE
In order to study a realistic doping in ice, we have created supercell structures of ice-X under pressure for a wide range of doping values H 2 O 1−x Dopant x , with x = 25%, 12.5%, 6.25%, and 4.16% (x = 100% indicates the removal of one electron per formula unit of H 2 O).Full structural relaxation were then carried out for the supercells (12 atoms cell for x = 25%, 24 atoms cell for 12.5%, 48 atoms cell for 6.25% and 72 atoms cell for 4.16% ).For low doping (4-6%), we only find fairly small modifications of the crystal structure of ice-X.On the contrary, larger doping levels lead to a considerable deformation of the local environment.
Figure 2 depicts the electronic band structure obtained for boron, carbon, nitrogen and phosphorous used as dopant in the ice-X at 150 GPa.In these plots the color scale represents the overlap of the Kohn-Sham states on the atomic orbitals of the dopant: red means large overlap (dopant states projected), gray intermediate, while blue means small (hydrogen and oxygen states projected).For large doping with boron and phosphorous (12% and 25%), the dispersive band coming from the dopant completely closes the gap in ice-X, while for 4% and 6% the dopant states form impurity molecularlike bands.These atoms are therefore not suitable to hole-dope ice-X.The case of carbon is intermediate: at low doping we see again the formation of impurity bands, while at higher levels we do see some hybridization between the carbon and the ice bands at the top of the valence.I n general we find a higher density of states at the Fermi level with B, C and P acting as a dopant, however the band structure shows mostly localized molecular states which are detrimental to superconductivity.It is important to mention that, since the ice-X is highly symmetric (cubic structure, P − 43m, space group 215) there is only one site symmetry to substitute for high doping levels (i.e. 25 and 12 percent) and for a reasonably small supercell.For lower doping levels, however, many other doping sites become available.Nevertheless, a study of all possible site/defect substitutions of oxygen by dopants is clearly beyond the scope of this work (and would probably require techniques such as cluster expansion).However, our results provide a clear general trend of the physics of doped ice under pressure.
Conversely, to what the other elements have shown, nitrogen clearly induces hole doping in ice-X (seeing in Figure 2) and, as shown in Ref. 89 (supplemental) it is the statistically most likely non iso-valent element able to substitute oxygen.The bottom panel in Fig. 3 shows in detail the electronic structure of ice-X.Phase-X is an insulator with a PBE electronic band gap about 10 eV and does not undergo major modifications at least up to 300 GPa.According to our calculations, the insulator-metal transition is achievable starting from values of 4% nitrogen doping (as seen in the projected-states band plot Figure 2), we can conclude that holes are indeed populating the Fermi level.For 25% doping, the top valence is mostly dominated by nitrogen states forming a fairly dispersive band.For values between 4% and 12%, oxygen and nitrogen strongly hybridize in the top valence forming the metallic states (gray colors in the plot).
Electronically, it is clear that nitrogen doping introduces holes in the ice-X crystal.These hole states are, as intended, hybrid O-N states, as can be seen from the projected density of states in Fig. 3.At high doping (12.5% and 25%) the states at the Fermi level are homogeneous O-N hybrids, meaning that the density of N states is simply proportional to the fraction of N atoms.On the other hand, at lower doping (4.16% and 6.25%) the N projected DOS, although overall smaller, is larger than the N/O fraction and shows a sharp peak close to the Fermi level, indicating that the induced holes are more localized on the N sites.We note that the two last values are realistic doping values and similar to the doping values used to render diamond or silicon superconducting at ambient pressure 90 .There are many possible ways to study theoretically the effect of doping in the superconducting properties.The simplest way is by shifting rigidly the Fermi level, leaving both Kohn-Sham eigenvalues and eigenfunctions unchanged.The resulting phonon spectrum and electronphonon scattering amplitude can then be used within an Eliashberg 91,92 scheme to compute the superconducting critical temperature as a function of the position of the Fermi level.For the ice-X of H 2 O at 150 GPa (2 formula unit cell) we compute, within this procedure, an astonishingly high phononic superconducting coupling, leading to room-temperature superconductivity already at a doping of a few percent!Although widely used in the literature, we can not expect that such a rigid shift of the Fermi level yields more than an estimate for the order of magnitude of the critical temperature upon doping.In fact, the extreme electron-phonon coupling obtained by the rigid shift would induce a strong electronic response, leading to a complete breakdown of the rigid shift approximation.Moreover, this method does not account for important physical effects stemming from the metallic part of the electronic screening, such as the mechanism responsible for Kohn anomalies 93 that can significantly modify the spectrum of phonons.A more realistic way to study theoretically the effect of doping in the superconducting properties is to calculate the phonon and electronic-phonon matrix elements 94 in the supercell of nitrogen doped ice-X.As we have seen, doping turns out to have a dramatic effect in the electronic structure of ice, and in the phonon spectrum (not shown) and in the Eliashberg spectral functions, shown in Fig. 4 as a function of doping at 150 GPa.

IV. EMERGENCE OF SUPERCONDUCTIVITY IN N-DOPED ICE
Comparing the supercell calculations with results obtained with a rigidly shifted Fermi level (see Fig. 4), we observe a complete restructuring of the phonon energies and coupling strength.The metallization provides a significant screening both causing a softening of the phonon frequencies and a reduction of the deformation potential.Nevertheless we still observe a significant electronphonon coupling, as can be seen from the Eliashberg functions as well as from the logarithmic average of the phonon frequency ω log 92,95 (see Fig. 5).We furthermore calculated the phonon band-structure for B, C and P doping.We find that all systems are highly unstable with large imaginary frequencies, with the only exception nitrogen, being dynamically stable in the doped range studied.
For low doping (< 12%) two major contributions to λ can be distinguished: i) the low frequency optical phonons (oxygen vibrations) that couple with 2p nitro-gen states; and ii) the mid-frequency range 150-200 meV (1100-1800 cm −1 ) that couples with the covalent oxygennitrogen hybridized state.For 25% doping all phonon branches contribute significantly to the e − p coupling, since at this doping limit the structure adopts a completely metallic character.
From these parameters we can easily estimate the critical temperature by means of the McMillan-Allen-Dynes 96 formula (assuming µ * =0.1).This gives T C in the range from 20 to 60 K, with the maximum value reached for a doping level of 6.25% (see Fig. 5).Although lower than the astonishing value found in sulphur hydride (200 K) and considerably lower than the rigid shift prediction (300 K) this is still a sizable value, much larger than the T C 4 K found in doped semiconductors at ambient pressure.
A possible path to reach the synthesis of the nitrogen doped ice-X and the superconducting state is to start from a high pressure synthesis similar to what is used to obtain H 2 +H 2 O clathrates.In 1993, Vos et al. 75 reported the formation of filled-ice clathrates.They succeed to fill, at room temperature, H 2 molecules inside H 2 O crystalline C 1 (clathrate) phase at 0.7 GPa, this structure is also viewed as ice-II phase.Experimentally, the unit cell of the C 1 phase contains 36 water molecules, in a channel-like arrangement, which can host up to six hydrogen molecules 97,98 .In short we propose to start the synthesis from the analogous H 2 +H 2 O clatrhate (C 1 ) but filled with N 2 molecules where the percentage of filled N 2 molecules will determine the doping level at high pressure, and the percentage of filling of molecules can be controlled at ambient conditions 99 .

V. CONCLUSIONS
In conclusion, we have investigated the possibility of inducing high-temperature superconductivity by doping insulating hydrides under pressure.By taking the phase X of ice as an example, we studied how the phonon spectra and the electron-phonon coupling evolve as a function of nitrogen doping.Despite the number of elements and the range studied to dope H 2 O it is clear that only nitrogen, favors thermodynamically the synthesis of the insulator-metal transition of ice under pressure.It turns out that for rather reasonable values of doping, one can reach superconducting transition temperatures as high as 60 K at 150 GPa.Considering the vast number of hydrides that remain insulating under pressure and that can be doped, this result opens a number of possibilities for the exploration of high-temperature superconductivity in these unique systems.

Figure 1 .
Figure 1.(Color online) Top panel: Calculated electron localization function (ELF) at n =0.6 for ice as a function of pressure.The molecular crystal transforms to a fully covalent phase-X at 120 GPa.Bottom panel: Calculated enthalpy for the VIII and X phases of H2O ice.Values are given with respect to the elemental decomposition H2 + O.The reference structures for hydrogen are P 63m (0-120 GPa) and C2/c(120-300 GPa) from Ref.45 .The reference phase for oxygen is C2/m, ζ structure[59][60][61] .The experimental transition region between the two structures is marked with lines according to Goncharov et al.62 .From our calculations, the transition from a molecular ice-crystal to the fully symmetric covalent phase X occurs at 120 GPa.

Figure 2 .
Figure 2. (Color online) Electronic band structure for different dopants (B, C, N and P) in the phase-X of ice at 150 GPa.The Fermi level is set to 0 eV and the color scale represents the overlap of the Kohn-Sham states on the atomic orbitals of the dopant: red means large overlap (dopant states projected), gray intermediate, while blue means small (hydrogen and oxygen states projected).

Figure 3 .
Figure 3. (Color online) Density of states (DOS) and Fermi surfaces of undoped and N-doped phase X of H2O at 150 GPa.Thick black lines are the total DOS (a scaling factor is applied for plotting convenience).Solid red curves are the projection on nitrogen atomic states.All doped systems are metallic featuring one or two small electron pockets around the gamma point (blue/green) an one or two large open surfaces (red/green).

Figure 4 .
Figure 4. (Color online) Eliashberg spectral function (black lines) and integration curve of the electron-phonon coupling constant λ(ω) (red lines) for hole-doped H2O in its phase X at 150 GPa.Doping level is indicated in each panel.

Figure 5 .
Figure 5. (Color online) Top panel: calculated critical temperatures with the McMillan-Allen-Dynes formula as a function of doping of H2O in its phase X at 150 GPa.Lower panel: electron phonon coupling constant λ (red left triangles and left axis) and average phonon frequency ω log (green right triangles and right axis).Solid lines are a guide to the eye.

Figure 7 .
Figure 7. (Color online) Left panel shows the formation enthalpy for the doped structures at 150 GPa.Right panel shows the free energy including the zero point energy correction.Bottom panels represent the line of stability of the ternary phase diagram between H2O and an hypothetical H2N composition.

Table I .
Crystal structure parameters for the supercell created to dope the phase-X of ice (relaxations at indicated pressure).