Electron-phonon interactions theory: Difference between revisions

Revision as of 14:44, 9 April 2019

Electron-phonon interactions from statistical sampling

The probability distribution of finding an atom within the coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \kappa +d\kappa (where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \kappa denotes the Cartesian coordinates as well as the atom number) at temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): T in the harmonic approximation is given by the following expression^[1]^[2]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): dW_{{\nu }}(\kappa ,T)={\frac {1}{2\pi \langle u_{{\nu \kappa }}^{{2}}\rangle }}e^{{-\kappa ^{{2}}/(2\langle u_{{\nu \kappa }}^{{2}}\rangle )}}d\kappa ,

where the mean-square displacement of the harmonic oscillator is given as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle u_{{\nu \kappa }}^{{2}}\rangle ={\frac {\hbar }{2M_{{\kappa }}\omega _{{\nu }}}}\coth {{\frac {\hbar \omega _{{\nu }}}{2k_{{B}}T}}}.

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): M_{{\kappa }} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \nu and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \omega _{{\nu }} denote the mass, phonon eigenmode and phonon eigenfrequency, respectively. The equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): dW is valid at any temperature and the high (Maxwell--Boltzmann distribution) and low temperature limits are easily regained. In order to obtain an observable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): O(T) at a given temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): T</math,theaverageoftheobservablesampledatdifferentcoordinatesets<math>x_{{T}}^{{{\textrm {MC}},i}} with sample size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): n is taken

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle O(T)\rangle ={\frac {1}{n}}\sum \limits _{{i=1}}^{{n}}O(x_{{T}}^{{{\textrm {MC,i}}}}).

Full Monte-Carlo sampling

Each set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): i is obtained from the equilibrium atomic positions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): x_{{{\textrm {eq}}}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): x_{{T}}^{{{\textrm {MC,i}}}}=x_{{{\textrm {eq}}}}+\Delta \tau ^{{{\textrm {MC,i}}}}

with the displacement

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Delta \tau ^{{{\textrm {MC,i}}}}={\sqrt {{\frac {1}{M_{{\kappa }}}}}}\sum \limits _{{\nu }}^{{3(N-1)}}\varepsilon _{{\kappa ,\nu }}{\mathcal {N}}.

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \varepsilon _{{\kappa ,\nu }} denotes the unit vector of eigenmode Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \nu on atom Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \kappa . The magnitude of the displacement in each Cartesian direction is obtained from the normal-distributed random variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {N}} with a probability distribution according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): dW .

ZG configuration (one-shot method)

Motivated by the empirical observation that for increasing super-cell sizes the number of required structures in the MC method can be decreased, M. Zacharias and F. Giustino^[3] proposed a one-shot method where only a single set of displacements is used

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Delta \tau ^{{{\textrm {OS}}}}={\sqrt {{\frac {1}{M_{{\kappa }}}}}}\sum \limits _{{\nu }}^{{3(N-1)}}(-1)^{{\nu -1}}\varepsilon _{{\kappa ,\nu }}\sigma _{{\nu ,T}},

where the summation over the eigenmodes runs in an ascending order with respect to the values of the eigenfrequencies, and the magnitude of each displacement is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \sigma _{{\nu ,T}}={\sqrt {(2n_{{\nu ,T}}+1){\frac {\hbar }{2\omega _{{\nu }}}}}}.

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): n_{{\nu ,T}}=[{\mathrm {exp}}(\hbar \omega _{{\nu }}/k_{{B}}T)-1]^{{-1}} denotes the Bose-Einstein occupation number. In this way the sum for the observable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \langle O(T)\rangle is reduced to a single calculation. In Ref. ^[3] it was shown that for super-cell sizes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): N\rightarrow \infty the structural configuration obtained using the ZG configuration should lead to equivalent results as fully converged MC calculations. In practice, it was shown that already relatively small super-cell sizes are sufficient to achieve good accuracy, but the convergence with respect to the cell size can vary between different materials. In Ref. ^[4] have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \pm 1 . This was only necessary, when calculating volume dependent ZPR, since the modes sometimes change the order as the volume changes. Using alternating signs for the displacement then causes small discontinuities in the ZPR volume curve of the order of 5 meV for carbon diamond. By averaging over many random phases this problem can be eliminated. Nevertheless 5 meV changes in most calculations are considered as accurate enough.

References

[bloch:zfp:1932-1] F. Bloch, Zeitschrift fuer Physik. 74, 295 (1932).

[landau:ctp:1959-2] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics: Statistical Physics vol 5, (London: Pergamon), (1959).

[zacharias:prb:2016-3] M. Zacharias and F. Giustino, Phys. Rev. B 94, 075125 (2016).

[karsai:njp:2018-4] F. Karsai, M. Engel, E. Flage-Larssen, and G. Kresse, New J. of Phys. 20, 123008 (2018).

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@@ Line 54: / Line 54: @@
 Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
-In this way the sum for the observable <math>\langle O(T)\rangle</math>  is reduced to a single calculation. In Ref. {{TAG|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> the structural configuration obtained using the ZG configuration should lead to equivalent results as fully converged MC calculations. In practice, it was shown that already relatively small super-cell sizes are sufficient to achieve good accuracy, but the convergence with respect to the cell size can vary between different materials. In Ref. {{cite|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. This was only necessary, when calculating volume dependent ZPR, since the modes sometimes change the order as the volume changes. Using alternating signs for the displacement then causes small discontinuities in the ZPR volume curve of the order of 5 meV for carbon diamond. By averaging over many random phases this problem can be eliminated. Nevertheless 5 meV changes in most calculations are considered as accurate enough.
+In this way the sum for the observable <math>\langle O(T)\rangle</math>  is reduced to a single calculation. In Ref. {{cite|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> the structural configuration obtained using the ZG configuration should lead to equivalent results as fully converged MC calculations. In practice, it was shown that already relatively small super-cell sizes are sufficient to achieve good accuracy, but the convergence with respect to the cell size can vary between different materials. In Ref. {{cite|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. This was only necessary, when calculating volume dependent ZPR, since the modes sometimes change the order as the volume changes. Using alternating signs for the displacement then causes small discontinuities in the ZPR volume curve of the order of 5 meV for carbon diamond. By averaging over many random phases this problem can be eliminated. Nevertheless 5 meV changes in most calculations are considered as accurate enough.
 == References ==