Symmetry-broken crystal structure of elemental boron at low temperature

1
Symmetry-broken crystal structure of elemental
boron at low temperature
With Marek Mihalkovic (Slovakian Academy of
Sciences)

• Outline
• Cohesive energy puzzle (E? lt E? ?)
• Optimization of partial occupancy in ?
• Symmetry-restoring ???? phase transition

2
a
Bond lengths
b
Occupancy
100
75
9
27
7
7
4
3
The structure of elemental Boron
?-B.hR12 McCarty (1958, powder,
red) ?-B.tP50 Hoard (1958, 56 reflections,
R0.114) ?-B.hR105 Geist (1970, 350 reflections,
R0.074) ?-B.hR111 Callmer (1977, 920
reflections, partial occ. R0.053) ?-B.hR141 Slack
(1988, 1775 reflections, partial occ. R0.041)
The energies of elemental Boron (relaxed DFT-GGA)
?-B.hR12 ?E 0.00 (meV/atom) ?-B.tP50 ?E
91.91 ?-B.hR105 ?E 25.87
105 atoms/105 sites ?-B.hR111 ?E 0.15
106 atoms/111 sites ?-B.hR141 ?E
?0.86 107 atoms/141
sites ??-B.aP214 ?E ?1.75
214 atoms/214 sites
3rd law of thermodynamics!
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Stability of ?-Boron

• Possibility of Finite T phase transition (Runow,
  1972 Werheit and Franz, 1986)
• Vibrational entropy can drive ??? transition
  (Masago, Shirai and Katayama-Yoshida, 2006)
• Quantum zero point energy can stabilize ? (van
  Setten, Uijttewaal, de Wijs and de Groot, 2007)
• Symmetry-broken ground state ??, symmetric ?
  phase restored by configurational entropy (Widom
  and Mihalkovic, 2008)

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? cell center, partial occupancy
Clock model
Optimal sites
All sites
Occupancy
100
75
27
100
7
4
9
7
6
Structure and fluctuations
Optimized structure
Molecular dynamics
T2000K, duration 12ps
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2x1x1 Supercell
Clock Model Time shows occupancies Optimal
times 0220 and 1000 Other times are low-lying
excited states
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Symmetry-restoring phase transition of clock model
C
TS
? all distinct clock configurations in 2x1x1
supercell
U
?? degeneracy of configuration ?
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Conclusions

• E? gt E? conflicts with observation of ? as
  stable
• Optimizing partial occupancy brings E?? lt E?
• Symmetry broken at low temperature (3rd law)
• Symmetry restored through ???? phase transition
• ? stabilized by entropy of partial occupation