Temperature Models

Temperature Models

The effect of temperature on the thermodynamic properties of a solid mostly manifests through vibrations. An exact treatment of the vibrational contributions to the free energy is not possible. Instead, gibbs2 uses the quasiharmonic approximation (QHA), which applies the harmonic approximation at every volume, thus including some degree of anharmonicity. Another further approximation is that the system is constrained to move on the static minimum-energy path, i.e., the effect of temperature on thermodynamic properties and internal parameters such as atomic positions is modeled only indirectly via the change in volume.

A complete statically constrained QHA calculation requires the energy-volume curve and the phonon density of states calculated at each of the volumes. In many cases, this is unfeasible due to computational limitations. Simpler temperature models can be used in this case. It is important to note that these simplified thermal models have their limitations, so it is fundamental to check whether they can be applied to your system by comparing their predictions with experimental results.

Gibbs2 implements several models for the inclusion of vibrational effects. These models cover a range of complexity (and accuracy), with the Debye model in Slater’s implementation being the simplest and the full quasiharmonic approximation being the most complex. More complex temperature models require more data. For instance, the Debye model with Slater’s formula for the Debye temperature requires only the static E(V) curve in the input (and optionally the experimental Poisson ratio). In contrast, the full QHA model requires the phonon spectrum calculated at each grid volume, which is much more computationally expensive.

The different temperature models are accessed using the TMODEL option to the PHASE keyword:

PHASE ... [TMODEL {STATIC|DEBYE_INPUT|DEBYE_POISSON_INPUT|DEBYE|
                   DEBYE_EINSTEIN [FREQG0 file.s]|
                   DEBYE_GRUNEISEN {SLATER|DM|VZ|MFV|a.r b.r}|
                   {QHAFULL|QHA}}]
          [PHFIELD ifield.i] [DOSFIELD i1.i i2.i]|}] [PREFIX prefix.s]
          [POISSON sigma.r] [FSTEP step.r]

The temperature model is selected with one of the following keywords:

  • STATIC: do not calculate thermal properties of this phase.

  • DEBYE: Debye model. The Debye temperature at each volume is calculated from the bulk modulus and the Poisson ratio using Slater’s formula. By default, the Poisson ratio is assumed to be constant and equal to 1/4. Its value can be changed using the POISSON keyword. Besides the Poisson ratio, this model requires only the static energy-volume curve.

  • DEBYE_INPUT: Debye model where the Debye temperatures are read from the input. Each line in the energy-volume data must contain an additional field with the Debye temperature at the corresponding volume.

  • DEBYE_POISSON_INPUT: Debye model where the Poisson ratio at each volume is read from the input. Each line in the energy-volume data must contain an additional field with the Poisson ratio at the corresponding volume.

  • DEBYE_GRUNEISEN: Debye model where the Debye temperature is calculated for the equilibrium volume as in DEBYE but the evolution with volume of the Debye temperature is given by an approximate formula chosen by the user. The Grüneisen gamma is given by:

    \[\gamma_D = -\frac{\partial\ln\Theta_D}{\partial\ln V} = a - b\frac{d\ln B_{\text{sta}}}{d\ln V}\]

    where \(a\) and \(b\) are parameters chosen by the user with a keyword following DEBYE_GRUNEISEN. The available keywords are:

    Keyword Name a b
    SLATER Slater -1/6 1/2
    DM Dugdale-McDonald -1/2 1/2
    VZ Vaschenko-Zubarev -5/6 1/2
    MFV Mean free volume -0.95 1/2

    The DEBYE_GRUNEISEN SLATER temperature model is the same as DEBYE. Alternatively, the user can also follow the DEBYE_GRUNEISEN keyword with two numbers (a.r and b.r). In that case, the first number is interpreted as \(a\) and the second as \(b\) in the formula above.

  • DEBYE_EINSTEIN: Debye model for the acoustic branches and 3n-3 Dirac deltas representing the optical part of the phonon spectrum. The Debye model is applied as in DEBYE, including Slater’s formula, but only for the acoustic branches.

    This model requires the frequencies at the Brillouin zone center (the Gamma point). There are two options two give them to gibbs2. If the FREQG0 option is used followed by a file name, the 3n-3 frequencies at Gamma at the equlibrium volume are read from the file. Otherwise, an additional column from the external data file is read. The column must contain the path of the file containing the frequencies at Gamma at the corresponding volumes.

  • QHAFULL or QHA: full quasiharmonic approximation. This model implements the statically constrained quasiharmonic approximation. It is the most accurate temperature model in gibbs2. It requires the phonon density of states (phDOS) at each volume, which makes it also the most expensive temperature model. Each line in the energy-volume data must contain an additional field with the location of file containing the phDOS. The phDOS files must contain two columns. The first column is the frequency (default units: Hartree) and the second column is the DOS (default units: 1/Hartree). The default column from which the phDOS file is read as well as the various units can be changed with additional optional keywords.

    It is often the case that some points in the energy-volume grid have a substantial amount of their phonon density of states in the negative-frequency region. This could be due to numerical error in the calculation (if the phDOS decays exponentially below zero) but more often it indicates that those points are dynamically unstable (i.e. a symmetry-breaking relaxation starting at that point would decrease the enthalpy). By default, gibbs2 eliminates (“deactivates”) the points with substantial negative-frequency phDOS from the energy-volume grid so they do not interfere with the calculation of thermodynamic properties.

In absence of a TMODEL keyword, the default is DEBYE.

Optional PHASE options

The following optional keywords can be used in PHASE to control how gibbs2 carries out the equation of state fitting for a particular phase.

Debye model

[POISSON sigma.r]

The Poisson ratio of the phase. This value is used in the calculation of the Debye temperature when using the Debye model. The default value is 0.25, the Poisson ratio of a Cauchy solid.

Debye-Einstein model

PRINTFREQ|PRINTFREQS

Print the calculated frequencies in the Debye-Einstein model for all static volumes in input. Writes a file with extension .gammafreq. Only valid when FREQG0 is used.

Full QHA model

PHASE ... [PREFIX prefix.s]

Indicates the prefix used to find files containing the vibrational density of states or frequencies. For instance, this input:

PHASE MgO FILE mgo.dat PREFIX ../mgo-b1 TMODEL QHA [...]

reads the energy-volume data and the location of the phonon density of states files from the external file mgo.dat. This file may contain:

81.8883583665837   -73.5171659350000 001/001.phdos
86.0358791612784   -73.5360133400000 002/002.phdos
[...]

The path to the phDOS files is build by concatenating the prefix (../mgo-b1) with the names of the files (001/001.phdos). In this case, gibbs2 expects the phDOS files to be at ../mgo-b1/001/001.phdos, ../mgo-b1/002/002.phdos, etc. An absolute path may also be used in PREFIX. By default, PREFIX is the current directory (.).

PHASE ... [PHFIELD ifield.i]

Indicates which column in the external data file contains the location of the phDOS files. By default, they assumed to be in the third column (as in the example above).

PHASE ... [DOSFIELD i1.i i2.i]

Indicates which columns gibbs2 reads from each individual phDOS file. By default, the first column (i1.i) is interpreted as the frequencies and the second column (i2.i) as the density of states.

PHASE ... [FSTEP step.r]

In the QHA thermal model, gibbs2 needs to interpolate the phonon DOS at arbitrary volumes. To do this, all phDOS read from input need to be expressed on the same frequency grid. By default, the step of this grid is the difference in frequency between the two highest frequencies at the first volume in the grid. By using FSTEP, the user can set an explicit value for the phDOS step, equal to step.r.

Optional global keywords

The following optional keywords can be used in the gibbs2 input to control the EOS fitting for all phases. All the following keywords apply to the QHA model.

ACTIVATE {ALL|v1.i v2.i v3.i...}

In the QHA temperature model, gibbs2 automatically deactivates a volume when the phonon density of states in input contains negative frequencies. However, because of numerical errors in the DFPT calculation and posterior Fourier interpolation, it is possible to have a small region of negative frequencies. In such cases, it is possible to activate manually the use of those volumes in the dynamic calculation with ACTIVATE. With the ALL keyword, all the volumes become active. Alternatively, the user can input the volume integer identifier (the position of the volume in the input grid). This identifier is written to the output when a volume is deactivated because of negative frequencies.

SET PHONFIT {LINEAR|SPLINE}

Type of interpolation of the phDOS with volume. It can be either linear (LINEAR) or a cubic not-a-knot spline (SPLINE). Linear interpolation is almost equivalent to the spline for a reasonably fine volume grid, and much faster, so LINEAR is the default.

SET IGNORE_NEG_CUTOFF inegcut.r

When negative phonon frequencies are passed to gibbs2, the program assumes that the point does not correspond to a stable structure and deactivates its use in the calculation of thermodynamic properties. A deactivated volume is dropped from the energy-volume grid, as if it had not been given in the input.

Sometimes, small negative frequencies can be the result of numerical errors in the program used to calculate the phonon frequencies. This keyword sets a cutoff for the negative frequencies. If a frequency is less than -abs(inegcut.r), the point is deactivated. If a frequency is between -abs(inegcut.r) and 0, the point remains active, but that frequency is discarded (and the phDOS renormalized to the correct value). The units of inegcut.r are cm-1. The default value is 20 cm-1.

SET NORENORMALIZE

In the QHA thermal model, eliminate the negative frequencies from the phonon density of states but do not renormalize. This is used for testing purposes only.