EFERMI NEDOS

EFERMI_NEDOS = [integer]
Default: EFERMI_NEDOS = 21 

Description: Number of Gauss–Legendre integration points used to evaluate the Fermi–Dirac distribution and determine the Fermi level at finite temperature using the tetrahedron method only with ISMEAR = −14 or -15 .

Mind: Available as of VASP 6.5.0

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EFERMI_NEDOS sets the number of points in the Gauss–Legendre grid used to integrate the Fermi–Dirac distribution for determining the Fermi level within the tetrahedron method when ISMEAR = −14 or -15 . Larger values improve accuracy, especially at low temperatures or with sharp features in the electronic DOS, but also increase computational cost. A brief convergence test is recommended in case very accurate occupancies are required, e.g., in the context of transport calculations.

Implementation details

At [math]\displaystyle{ T=0 }[/math], the integrated and differential densities of states are $$ n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}). $$

At finite temperature, $$ N_e(\epsilon_F,T)= \sum_{n\mathbf{k}}f(\epsilon_{n\mathbf{k}}-\epsilon_F,T) =\int g(\epsilon)f(\epsilon-\epsilon_F,T)\,d\epsilon. \tag{1} $$

With the substitution [math]\displaystyle{ x = 1 - 2f(\epsilon-\epsilon_F,T) }[/math], $$ \epsilon = k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F, \qquad d\epsilon = -k_BT\,\frac{2}{x^2-1}\,dx, $$ Eq. (1) becomes $$ N_e(\epsilon_F,T)= \frac{1}{2}\int_{-1}^{1} n\!\left(k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F\right)\,dx. $$

In practice, this integral is discretized as $$ N_e(\epsilon_F,T)\simeq \frac{1}{2}\sum_{i=1}^{N}w_i\, n\!\left(k_BT\ln\!\frac{1+x_i}{1-x_i}+\epsilon_F\right), $$ where [math]\displaystyle{ w_i }[/math] and [math]\displaystyle{ x_i }[/math] are Gauss–Legendre weights and abscissas. The step functions \(\theta(\epsilon-\epsilon_{n\mathbf{k}})\) entering \(n(\epsilon)\) are evaluated using the tetrahedron method, with the number of energy points [math]\displaystyle{ N }[/math] given by EFERMI_NEDOS.

Related tags and articles

ISMEAR, SIGMA, Smearing technique, K-point integration