Part II Theory
8.4 Results
obtained from statistical thermodynamic relationships. For example, the thermal equation of stateP (T , V )can be deduced via
P (T , V )= −∂F (T , V )
∂V T
, (8.2)
from which the V(T,P) relationship was obtained by interpolation, and the thermal expansion coefficient was deduced from
α= 1 V
∂V
∂T
P. (8.3)
To study anharmonic effects with VASP, we performed ab initio Born-Oppenheimer molecular dynamics (MD) calculations at 7, 100, 200, 300, 450, 600, and 750 K, and we also performed frozen phonon calculations. The MD simulations used a 108-atom supercell, and temperature was controlled by Nosé thermostats. For each temperature, the system was first equilibrated and then simulated for 5 ps with a time step of 5 fs. The QHA and MD simulations were used to identify modes corresponding to experimental spectral features having anomalous temperature dependencies. The vibrational potentials of these modes were obtained through frozen phonon calculations on the minimum supercell determined by symmetry.
The first Brillouin zone and its high symmetry points are shown in Fig. 8.5. Lower energy modes have more complex dispersions and result in broader peaks in phonon DOS. Judging from the mass difference between fluorine and scandium atoms, the fluorine-dominated phonon modes should be relatively separated from the scandium modes and this is mostly true. Naturally, one expects the lighter atoms like fluorine to dominate the high-energy modes while heavier atoms like scandium dominate the lower-energy modes. However, the motions of F atoms dominate both the higher- and lower-energy parts of the DOS, and the majority of Sc-dominated modes are between 40 and 60 meV. This can be explained if the low-energy “rigid unit modes”, where ScF6 octahedra pivot about corner-shared F atoms, have much larger effective mass than individual atoms.
As thermal expansion is closely related to Grüneisen parameters, we show in Fig. 8.4c the mode Grüneisen parameters. Some phonon modes have negative Grüneisen constants, such as the low-energy modes at R and M with anomalous Grüneisen constants of –371 and –84. It is likely these are the softest modes that might contribute to the negative thermal expansion. In what follows we show that these modes have quartic potentials, so these Grüneisen constants are not meaningful and the QHA is not reliable. Figure8.14shows the thermal expansion calculated with the QHA equation of state from Eq. 8.1, compared with recent measurements. [143] Some difference at the highest temperatures could be caused by the creation of defects. For low temperatures, the QHA underestimates the NTE.
The ScF6 octahedra are more flexible than their oxide counterparts—our MD simulations showed that the F atoms in an octahedron executed largely indepen- dent and uncorrelated motions, as shown in the attached animation, and by the pair, radial, and angular distribution functions. The pair distribution functions (PDF) (Fig. 8.6) and radial distribution functions (RDF) (Fig. 8.7) calculated by first-principles MD show a large broadening of the second peak (nearest F-F dis- tance) compared to the first peak (nearest Sc-F distance) at higher temperatures.
The calculated RDFs agree surprisingly well with the results from X-ray diffrac-
RXMRXM (1/meV) Figure8.4:(a)Calculatedphonondispersionsalonghigh-symmetrydirectionsofScF3at0K.X=(1,0,0)π/a;M= (1,1,0)π/a;R=(1,1,1)π/a.(b)TotalandpartialphononDOScurvesat0Kfromfirst-principlescalculation,neutron- weightedphononDOSwithinstrumentbroadeningat120meVadded,andexperimentalneutron-weightedphononDOSat 7K.(c)Grüneisenparameters(γ)calculatedwiththeQHAformodesalonghigh-symmetrydirections.Colorscorrespond tothephonondispersionsina.
R X M Z Γ
x y
z
Figure 8.5: The first Brillouin zone of ScF3 and its high-symmetry points tion experiments using synchrotron sources. Similarly, the angular distribution function for the F-Sc-F bond angle at 300 K (Fig. 8.8) shows a broad distribution with a FHWM of 10 degrees. Both of these results are consistent with uncorre- lated motions of fluorine atoms. This is also evident in the animation, in which relative motions of two nearby F atoms undergo frequent changes in phase and frequency. The structural geometry of the DO9 structure offers little constraint on the transverse modes of F atom motion, also suggested by the first-principle MD calculations.
Figure 8.9shows the plane-projected atomic trajectories of Sc and F atoms at 300 K. Fluorine atoms execute large excursions in the two directions transverse to the Sc-F bond. The distributions of atom centers, projected onto one axis and binned into a histogram, were satisfactorily fit to Gaussian functions. The full- width-half-maxima (FWHM) were 0.124 Å for Sc (isotropic) and 0.124 Å for F along the z-axis (longitudinal), and 0.270 Å for F along the x- and y-axes (transverse).
This anisotropy of F-atom motions decreases with temperature, but the average transverse amplitude of the F-atom motion is more than 10% of the Sc-F bond length at 300 K.
Although the large-amplitude F atom displacements occur largely indepen- dently, the rocking modes of ScF6octahedra are useful for analyzing the dynamics of a periodic structure. We performed frozen phonon calculations for the five
1 2 3 4 5 6 7 0
5 10 15 20 25 30
Pairdistributionfunction(a.u.)
R (angstrom)
7K
100K
200K
300K
450K
600K
750K
F-Sc F-F
Figure 8.6: Pair distribution functions from first-principle MD calculations at various temperatures
0 1 2 3 4 5 6 7 8 9
-6 -4 -2 0 2 4 6 8 10 12 14 16
G(r)(angstrom
-2 )
r (angstrom)
MD
XRD
Figure 8.7: Radial distribution functions from first-principle MD calculations compared to the results from X-ray diffraction by Wilkinson et al. Lower: 300 K;
Upper: 750 K
70 75 80 85 90 95 100 105 110 0
2000 4000 6000 8000 10000
300K
Fitted
ADF(a.u.)
Figure 8.8: The angular distribution function for F-Sc-F angle at 300 K from first-principle MD and its Gaussian fit
modes R1, R3, R4-, R5, and R4+ at the R point with energy decreasing in that order, as shown in Fig. 8.11. Modes R1 and R3 can be easily classified as “breathing"
modes, which involve the change of bond lengths inside octahedra without a change of bond angles. For mode R4-, the scandium atoms are displaced while fluorine atoms are fixed. Mode R5 only involves the change of F-Sc-F angles inside the octahedra. Most of these modes were fit well to a quadratic potential within the range of atomic displacements predicted by the MD calculations, but the R4+
mode (the mode of lowest energy calculated with the harmonic approximation), depicted in detail in Fig. 8.12, was found to have a nearly pure quartic potential.
The R4+ mode, with its quartic potential, is not expected to be modeled well with a harmonic potential, especially when the amplitudes of displacement are large.
If the NTE depends on the dynamics of this mode, it is not surprising that the quasiharmonic approximation would fail to account quantitatively for the thermal expansion. Many other NTE crystals are associated with anharmonicity, but the relationship has been unclear. Nevertheless, considering the large excursion of F atoms, anharmonicity seems to be important.
Figure 8.9: First principles MD trajectories and their projections onto x-y, y-z, and x-z planes for Sc (top) and F (bottom) at 300 K. Distances are in Å. The direction of Sc-F bond is ˆx.
Figure 8.10
Figure 8.11: Phonon modes in ScF3at R-point (in the order of R1, R3, R4-, R4+, and R5) and their frozen phonon potentials and quadratic (harmonic) fits to the frozen phonon potentials. The ranges of the quadratic fits vary, but are comparable.
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Figure 8.12: Phonon mode R4+, its frozen phonon potential, quadratic (harmonic) and quartic fit to the frozen phonon potential. The range of the quadratic fit is from -0.1 to 0.1 Å for transverse displacements of F atoms.
The Grüneisen parameter for theith mode,γi, is
γi = − V ωi
∂ωi
∂V = − V ωi
∂ωi
∂T
∂T
∂V = − V ωi
∂ωi
∂T 1
α, (8.4)
in which αis the volume expansion coefficient, V is the volume, andωi is the frequency of theith mode. Sinceα < 0 in this system, Grüneisen parameters should also be negative in their frequency dependence on temperature,∂ωi/∂T, but experimental results suggest differently. The calculations also predict a lattice parameter 1% larger than the room-temperature experimental value, at least part of which could be explained by NTE in the ground state.
The failure of the QHA is a good indicator that the dynamics and anharmonicity, more than of the static lattice structure, play an important role. Two effects could account for the “potential-induced" anharmonicity: one is an implicit effect due to the change in volume with temperature or pressure, and the other one is an explicit effect from the increase in vibration amplitude with temperature [152].
QHA calculations were performed at zero temperature, and they usually only