Dielectric-properties

Density functional perturbation theory is not limited to atomic displacement perturbations, and may also be used to calculate other response properties with respect to an electric field perturbation of an insulating system¹. These are the dielectric permittivity

$\epsilon_{\alpha,\beta}^{\infty} = \delta_{\alpha,\beta} - \frac{4 \pi}{\Omega}\frac{\partial^2 E}{\partial \varepsilon_{\alpha} \partial \varepsilon_{\beta}} \;,$

the “molecular” polarizability

$\alpha_{\alpha,\beta}^{\infty} = - \frac{1}{\Omega}\frac{\partial^2 E}{\partial \varepsilon_{\alpha} \partial \varepsilon_{\beta}} \;,$

and the Born effective charges

$Z^{*}_{\alpha,\beta} = \Omega \frac{\partial P_{\beta}}{\partial u_{\kappa,\alpha}} = \frac{\partial F_{\kappa,\alpha}}{\partial \varepsilon_{\beta}} = \frac{\partial^2 E}{\partial u_{\kappa,\alpha}\partial \varepsilon_{\beta}}$

which play a strong role in lattice dynamics of crystals, and in particular governs the frequency-dependent dielectric response in the infra-red region (Gonze and Lee 1997).

An electric field response calculation is selected using the task keyword in the .param file.

task : EFIELD

which computes the optical frequency dielectric permittivity tensor and the low frequency (ionic lattice) response to a time-varying field in the regime of the phonon modes and the Born charges. Alternatively

task : PHONON+EFIELD

performs both dielectric and phonon tasks.

The low and near-infrared frequency contributions to the permittivity and polarizability and the Born effective charges are printed to the .castep output file. An extract for the same Wurtzite BN calculation as earlier is shown in figure [efield-out]. An additional output file seedname.efield is also written and contains the frequency-dependent permittivity over the entire range, with a spacing determined by parameter efield_freq_spacing, and a Lorentzian broadening governed by a fixed $Q$ , efield_oscillator_q.

The low-frequency contribution of the phonons to the dielectric polarizability and permittivity is only well-defined when all mode frequencies are real and positive. In the presence of any imaginary mode or one of zero frequency the low-frequency dielectric and polarizability tensors are not calculated and are reported as “infinity”. By default the three lowest frequency modes are assumed to be acoustic modes and not included in the calculation. To support molecule-in-supercell calculations, the parameter efield_ignore_molec_modes may be set to molecule, which excludes the 6 lowest frequency modes from the dielectric calculation. Allowed values are CRYSTAL, MOLECULE and LINEAR_MOLECULE which ignore 3, 6 and 5 modes respectively.

In fact CASTEP usually performs an electric field response calculation even for a task : PHONON calculation because the permittivity tensor and Born charges are required to calculate the LO/TO splitting terms. Conversely a pure task : EFIELD calculation also performs a $\Gamma$ -point phonon calculation which is needed to compute the ionic contribution to the permittivity and polarizability. Consequently the only real difference between any of the tasks PHONON, EFIELD and PHONON+EFIELD lies in what is printed to the output file as the same computations are performed in each case. Only if one or other of those properties is specifically turned off with one of the parameters

phonon_calc_lo_to_splitting : FALSE
efield_calc_ion_permittivity : FALSE

is a “pure” phonon or E-field calculation ever performed.

It is well documented that the LDA tends to overestimate dielectric permittivities - by over 10% in the case of Si or Ge (Levine and Allan 1989). It is possible to include an ad-hoc correction term to model the missing self-energy, by applying the so-called “scissors operator”, which consists of a rigid upshift of all conduction band states. This was incorporated into DFPT electric field response calculations by Gonze and Lee (Gonze and Lee 1997). The parameters keyword

excited_state_scissors 1.0 eV

is used to model the effect of a 1 eV (in this example) upshift of conduction band states and will include the effects on dielectric permittivity and Born charges (but not phonons). The value to use must be determined from high-level calculations or empirically.

 ===============================================================================
        Optical Permittivity (f->infinity)             DC Permittivity (f=0)
        ----------------------------------             ---------------------
         4.50788     0.00000     0.00000         6.64363     0.00000     0.00000
         0.00000     4.50788     0.00000         0.00000     6.64363     0.00000
         0.00000     0.00000     4.63846         0.00000     0.00000     7.15660
 ===============================================================================

=============================================================================== Polarisabilities (A**3) Optical (f->infinity) Static (f=0) --------------------- ------------- 6.52844 0.00000 0.00000 10.50324 0.00000 0.00000 0.00000 6.52844 0.00000 0.00000 10.50324 0.00000 0.00000 0.00000 6.77146 0.00000 0.00000 11.45793 =============================================================================== =================================================== Born Effective Charges ---------------------- B 1 1.84636 0.00000 0.00000 0.00000 1.84636 0.00000 0.00000 0.00000 1.94523 B 2 1.84636 0.00000 0.00000 0.00000 1.84636 0.00000 0.00000 0.00000 1.94523 N 1 -1.85371 0.00000 0.00000 0.00000 -1.85371 0.00000 0.00000 0.00000 -1.94009 N 2 -1.85371 0.00000 0.00000 0.00000 -1.85371 0.00000 0.00000 0.00000 -1.94009 ===================================================: Figure 6 Extract from the .castep output file generated from the hexagonal BN run of figure [example-gamma], with task : EFIELD. The Born effective charges are laid out with the columns representing the X,Y,Z electric field directions and the rows the X,Y,Z displacement directions.

Non-linear optical susceptibility

In addition to the linear response properties calculated with task EFIELD or phonon+efield the non-linear dielectric susceptibility may be computed if the parameter

efield_calculate_nonlinear : CHI2

is set. The calculation uses the “2n+1 theorem” extension of DFPT (Baroni et al. 2001; Miwa 2011) to compute the static response, $\chi^{(2)}$ . The results are reported in the .castep file in reduced tensor form as the “d-matrix” (Boyd 2003). See figure [nlo].

This is not activated by default, because it is substantially more expensive than a baseline E-field linear response calculation. (It requires the calculation of three sets of response functions with a full k-point set in P1 symmetry, even for crystals with higher, even cubic symmetry.)

=========================================================================== Nonlinear Optical Susceptibility (pm/V) --------------------------------------- 1.13621 -1.13625 0.00001 0.00002 -7.73417 0.00002 0.00002 -0.00003 -0.00008 -7.73421 0.00002 -1.13625 -7.73417 -7.73421 -30.12197 -0.00008 0.00001 0.00002 ===========================================================================: Figure 7 Nonlinear optical susceptibility $\chi^{(2)}$ expressed as a d-matrix for LiNbO3.

Because the response of a metallic system to an applied field is the generation of a current flow, the dielectric permittivity diverges and the Born charges become zero. CASTEP will check that the system has a band gap before proceeding with an E-field response calculation and abort if there is none. ↩