On the Possibility of Metastable Metallic Hydrogen

Jeﬀrey M. McMahon

Department of Physics & Astronomy

March 14, 2017

Jeﬀrey M. McMahon (WSU) March 14, 2017 1 / 19 The Problem of Metallic Hydrogen

In 1935, Wigner and Huntington predicted1 that suﬃcient pressure would dissociate hydrogen molecules; and any Bravais lattice of such atoms would be metallic.

Initial interest was primarily related to astrophysical problems ...... later to fundamental physics; also with practical applications: • It is expected to have remarkable properties: I High-temperature superconductivity2,3 I Novel types of quantum ﬂuids3 I Powerful rocket fuel • Theoretical capabilities have progressed4; properties can be accurately calculated from ﬁrst principles • ... As have experimental ones; it is now possible to cre- ate in the laboratory • It may be metastable6

1E. Wigner and H. B. Huntington, J. Chem. Phys. 3, 764–770 (1935) 2N. W. Ashcroft, Phys. Rev. Lett. 21, 1748–1749 (1968) solid metallic hydrogen5 3E. Babaev, A. Sudbo, and N. W. Ashcroft, Nature 431, 666-668 (2004) 4J. M. McMahon, M. A. Morales, C. Pierleoni, and D. M. Ceperley, Rev. Mod. Phys. 84, 1607–1653 (2012) 5R. P. Dias and I. F. Silvera, Science (2017) 6E. G. Brovman, Yu. Kagan, and A. Kholas, JETP 34, 1300–1315 (1972)

Jeﬀrey M. McMahon (WSU) Introduction March 14, 2017 2 / 19 Metastable Metallic Hydrogen

There is little doubt that hydrogen becomes metallic at high pressures.

For practical (terrestrial) applications, ...... the signiﬁcant, outstanding1,2 question is whether it is metastable.

Note: The answer to this presupposes solutions to the following: (1) Determination of the minimum-energy crystal structure(s), ...... and proof that it lies at a stationary point (2) Proof of (dynamic) stability (3) Analysis of the relation between the ground- and metastable-state structures of metallic hydrogen, and the molecular phase (4) Determination of the lifetime of the metastable state

It is the purpose of this talk to answer this question.

1E. G. Brovman, Yu. Kagan, and A. Kholas, JETP 34, 1300–1315 (1972) 2H. Y. Geng, H. X. Song, J. F. Li, and Q. Wu, J. Appl. Phys. 111, 063510 (2012)

Jeﬀrey M. McMahon (WSU) Introduction March 14, 2017 3 / 19 Metastable Metallic Hydrogen (Continued)

Brief answer:

Metallic hydrogen is metastable ...

... but, only to (approximately) 250 GPa.

Jeﬀrey M. McMahon (WSU) Introduction March 14, 2017 4 / 19 Outline

This talk is outlined as follows:

• Phase diagram of hydrogen • Crystal structure(s) of metallic hydrogen I Ground-state structures of atomic metallic hydrogen I Searching for (new) metastable metallic structures

• Dynamic stabilities

• Concluding remarks

Jeﬀrey M. McMahon (WSU) Introduction March 14, 2017 5 / 19 Phase Diagram of Hydrogen

1500

liquid H2 1000 liquid H

500 I I'? Temperature (K) IV V VI solid H III H2-PRE II 0 10 100 1000 10000 Pressure (GPa)

Figure: Combined experimental (lines) and theoretical (lines with points) phase diagram of hydrogen.

Data compiled from several sources

Jeﬀrey M. McMahon (WSU) Phase Diagram March 14, 2017 6 / 19 Ground-state Structures of Atomic Metallic Hydrogen

We earlier found1 several candidate structures of atomic metallic hydrogen:

(Not shown are: • High-pressure molecular phases, since found2 • Ultrahigh-pressure phases, since m found2 m • Nearly-degenerate, lower-symmetry distortions3, not originally reported in Ref. [1]) Figure: Ground-state enthalpies of the crystal structures of atomic metallic hydrogen relative to fcc ∆H, not including proton zero-point en-

1 ergy. The inset shows an expanded view of the J. M. McMahon and D. M. Ceperley, Phys. Rev. Lett. 106, 165302 (2011) ultrahigh-pressure region. 2H. Liu, H. Wang, and Y. Ma, J. Phys. Chem. C 116, 9221–9226 (2012) 3H. Y. Geng, H. X. Song, J. F. Li, and Q. Wu, J. Appl. Phys. 111, 063510 (2012)

Jeﬀrey M. McMahon (WSU) Atomic Metallic Hydrogen March 14, 2017 7 / 19 Tetragonal Family of Structures

The most promising candidates structures for higher-pressure, metastable metallic hydrogen belong to the tetragonal family.

Examples: x c/a y

Diamond β-Sn Cs-IV

Fd-3m I41/amd I41/amd (c/a = 1.414) (c/a ∼ 0.55 < 1) (c/a ∼ 3.73 > 1) (These structures diﬀer only in their c/a ratio.)

Jeﬀrey M. McMahon (WSU) Tetragonal Structures March 14, 2017 8 / 19 0 K Phase Diagram of the Tetragonal Structures

The 0 K phase diagram of the tetragonal structures reveals general, qualitative features of metastable metallic hydrogen:

200 Regions of distinct qualitative behavior are

100 indicated with dotted lines: • > 300 GPa: The (relative) enthalpies re- 0 main relatively ﬂat (with the exception of -100 diamond). • 200–300 GPa: The enthalpies of the β-

(meV/proton) -200

H c/a << 1 Sn, diamond, and Cs-IV structures become ∆ β-Sn -300 diamond nearly degenerate; those of c/a 1 and Cs-IV c/a >> 1 1 begin to decrease and increase, re- -400 spectively. 0 100 200 300 400 500 P (GPa) • < 200 GPa: The enthalpies of the β-Sn, diamond, and Cs-IV structures becomes Figure: 0 K phase diagram of the tetragonal degenerate; those of c/a 1 and 1 structures of metallic hydrogen. decrease and increase sharply.

Jeﬀrey M. McMahon (WSU) Tetragonal Structures March 14, 2017 9 / 19 Energies of the Tetragonal Structures

This behavior is understood by considering the energies of the tetragonal structures over all c/a: Notice that:

-13 • At 500 GPa: The diamond structure forms the energy barrier (a maximum) between the β-Sn and Cs-IV structures. -14 • ... Between 300–200 GPa: The energy barrier collapses (appreciably).

(eV/proton) • ... By 100 GPa: The energy barrier has E

-15 3 1.15017 Å (500 GPa) collapsed, leaving a (single) minimum. 3 1.40361 Å (300 GPa) 3 1.62966 Å (200 GPa) 3 2.08067 Å (100 GPa) General feature: The collapse of en- 3 4.72262 Å (0 GPa) ergy barriers, between 200–300 GPa. -16 0 2 4 6 8 10 c a / • The increase in energy as c/a → 0 and Figure: Energies of the tetragonal structures of → ∞ are due to the coming together of metallic hydrogen, as a function of c/a. Fixed Brillouin planes with like charges. volumes are shown, calculated as an average of • ... This explains the sharp decrease in the each structure at the pressure indicated in paren- c/a 1 structure (nearest-neighbor dis- thesis. The c/a 1, β-Sn, diamond, Cs-IV, tance: 0.9900Å): and 1 structures are indicated with arrows (in General feature: The tendency to form order of increasing c/a). molecules, below 200 GPa.

Jeﬀrey M. McMahon (WSU) Tetragonal Structures March 14, 2017 10 / 19 Energy of Metallic Hydrogen

There is no guarantee that the high-pressure structures are representative of those (potentially) metastable at lower pressures.

Consider the energy E (in the adiabatic approximation):

E = Estatic + EZPE

• Estatic (static energy) favors anisotropic structures (in metallic hydrogen). • EZPE (zero-point energy) favors symmetric structures.

• At low pressures: The energy is determined largely by Estatic. • With increasing pressure: The importance of EZPE increases.

We performed searches1 for metallic (atomic and/or mixed atomic/molecular) structures with the lowest Estatic at 0 GPa.

1C. J. Pickard and R. J. Needs, J. Phys. Condens. Matter 23, 053201 (2011)

Jeﬀrey M. McMahon (WSU) Energy of Metallic Hydrogen March 14, 2017 11 / 19 Relative Enthalpies of Candidate Structures

Our searches revealed several candidate structures of (potentially) metastable metallic hydrogen. The results are (qualitatively) understood, and further support the prior results, by considering only the most stable structures. Example:

100 0 Regions of distinct behavior (dotted lines): -100 • < 200 GPa: Molecular structures become -200 favored. Indicated by sharply decreasing -300 enthalpies, with a magnitude proportional Cs-IV to the number of atoms in the search.

(meV/proton) -400 1 H H 3 H ∆ • 200–300 GPa: Transition region. Energy -500 5 H 7 H 9 H barriers between (atomic) metallic phases -600 form. Indicated by crossovers. -700 0 100 200 300 400 500 • > 300 GPa: Energy barriers between metal- P (GPa) lic phases exist. Structures found at 0 GPa are also no longer representative. In- Figure: Relative enthalpies ∆H (to Cs-IV) of the dicated by lack of trend. most stable metallic structures, from searches over diﬀerent numbers of atoms in the unit cell.

Jeﬀrey M. McMahon (WSU) Candidate Structures March 14, 2017 12 / 19 The Tendency Towards Molecules

Below 200 GPa: All structures found show a tendency towards the formation of molecules.

Example: As the number of atoms increases, there is a greater proportion of molecules (to atoms) formed:

(a) C2 (5) (b) C2/m (7) (c) Pm (9)

Figure: Most stable candidate structures of metastable metallic hydrogen at ambient conditions, for searches with diﬀerent numbers of atoms (shown in parenthesis).

Jeﬀrey M. McMahon (WSU) Candidate Structures March 14, 2017 13 / 19 The Tendency Towards Molecules (Continued)

Example: This tendency is seen even in the (completely) atomic structures (found in searches over a lower number of atoms):

(a) Pmmm (b) Cmmm (c) Immm (d) P6/mmm

Figure: Most stable (nearly degenerate) candidate structures of metastable atomic metallic hydrogen at ambient conditions.

(Notice the proximity of atoms; distances are on the order of 0.9914 Å.) These results suggest that the atomic hydrogen at lower pressures has no region(s) of stability.

Jeﬀrey M. McMahon (WSU) Candidate Structures March 14, 2017 14 / 19 Dynamic Stability

A crystal is dynamically (mechanically) stable, if it executes a stable os- cillatory motion about its equilibrium conﬁguration.

By solving the equation of motions for the normal modes of vibration, ...... this is equivalent to the condition that their frequencies ω satisfy:

2 ωj(q) ≥ 0

for all phonon branches j and wavevectors q. (For an imaginary frequency means that the system, subject to a small displacement, will disrupt exponentially with time.)

Dynamic stabilities were determined by calculating the phonon density of states F(ω) for each structure.

M. Born and K. Huang, “Dynamical Theory of Crystal Lattices” (1954)

Jeﬀrey M. McMahon (WSU) Dynamic Stabilities March 14, 2017 15 / 19 Dynamic Stabilities of the Tetragonal Structures

The Cs-IV structure is dynamically stable below molecular dissociation:

0.004

500 GPa 0.008 200 GPa 350 GPa 100 GPa 250 GPa 50 GPa 0.003 0 GPa 0.006

0.002

) / proton ) / proton 0.004 ω ω ( ( F F 0.001 0.002

0 0 0 500 1000 1500 2000 2500 3000 -2000 -1000 0 1000 2000 3000 -1 -1 ω (cm ) ω (cm ) (a) stable region (b) unstable region

Figure: Phonon density of states F(ω) of Cs-IV, at several pressures. Pressure regions of (a) stability and (b) instability are shown. The dotted line in (b) is used to separate the stable from unstable frequencies. The arrow also in this plot indicates the growing instability, with decreasing pressure. ... but not to ambient conditions.

(All other tetragonal structures are dynamically unstable.)

Jeﬀrey M. McMahon (WSU) Dynamic Stabilities March 14, 2017 16 / 19 Dynamic Stabilities of the Candidate Structures

The candidate structures are dynamically unstable, at all pressures:

0.020

Pmmm 0.0005 Cmmm 0.0020 Cmmm Immm Immm P6/mmm 0.015 P6/mmm 0.0004 0.0015 0.0003 0.010

0.0002 ) / proton

) / proton 0.0010 ω ( ω ( F 0.0001 F 0.005 0.0005 0

0.000 0.0000 -6000 -4000 -2000 0 2000 4000 -2000 -1000 0 1000 2000 3000 4000 -1 -1 ω (cm ) ω (cm ) (a) 0 GPa (b) 200 GPa

Figure: Phonon density of states F(ω) of candidate structures of metastable metallic hydrogen, at two pressures. An inset of the imaginary-frequency region is shown in (a). The dotted line in (b) is used to separate the stable from unstable phonon frequencies. (The imaginary frequencies in (a) are not anomalous1.)

6E. G. Brovman, Yu. Kagan, and A. Kholas, JETP 34, 1300–1315 (1972)

Jeﬀrey M. McMahon (WSU) Dynamic Stabilities March 14, 2017 17 / 19 Summary and Open Questions

Summary • The Cs-IV structure of atomic metallic hydrogen is metastable, to approximately 250 GPa • Candidate structures for metastable metallic hydrogen were predicted • ... None were dynamically stable (below 250 GPa) • The processes involved with these phases, as the pressure is changed, was analyzed

Open Questions • Properties (e.g., superconductivity) • Quantitative corrections (calculation uncertainties, anharmonic eﬀects, etc.) • Thermal eﬀects and lifetimes (considering zero-point energy)

C. M. Tenney, K. L. Sharkey, and J. M. McMahon, In Preparation (2017)

Jeﬀrey M. McMahon (WSU) Concluding Remarks March 14, 2017 18 / 19 Acknowledgments

Members of the McMahon Research Group on the hydrogen project:

This project: Other projects: • Jeevake Attapattu • Zachary Croft

Craig M. Tenney Keeper L. Sharkey

Start-up support:

Department of Physics & Astronomy

Jeﬀrey M. McMahon (WSU) Concluding Remarks March 14, 2017 19 / 19