Gyrotropic module

doc/tutorial/tutorial.tex

@@ -2578,6 +2578,121 @@ \subsection*{Further ideas}
 %\cleardoublepage
 
 
+\sectiontitle{24: Tellurium -- gyrotropic effects}
+
+
+\begin{itemize}
+\item{Outline: {\it Calculate the gyrotropic effects in trigonal right-handed Te} Similar to the calculations of \cite{tsirkin-arxiv17}}
+\item{Directory: {\tt examples/example24/}}
+\item{Input files}
+\begin{itemize}
+\item{ {\tt Te.scf} {\it The \pwscf\ input file for ground state
+    calculation}}
+\item{ {\tt Te.nscf}  {\it The \pwscf\ input file to obtain Bloch
+    states on a uniform grid}} 
+\item{ {\tt Te.pw2wan}  {\it The input file for {\tt pw2wannier90}}}
+\item{ {\tt Te.win} {\it The {\tt wannier90} input
+      file}}
+\end{itemize}
+\end{itemize}
+
+To make things easy, the example treats Te without spin-orbit
+
+
+One may skip steps 1-4, by copying the files {\tt Te.eig, Te.mmn, Te.amn, Te.uHu} 
+from the directory {\tt examples/example24/wan\_files}. However, this steps are needed 
+to perform a calculation on a denser grid, or include spin-orbit coupling.
+
+\begin{enumerate}
+\item Run \pwscf\ to obtain the ground state of tellurium\\
+{\tt pw.x < Te.scf > scf.out}
+
+\item Run \pwscf\ to obtain the Bloch states on a uniform {\tt 3x3x4} k-point
+  grid\\ 
+{\tt pw.x < Te.nscf > nscf.out}
+
+\item Run \wannier\ to generate a list of the required overlaps (written
+  into the {\tt Te.nnkp} file).\\
+{\tt wannier90.x -pp Te}
+
+\item Run {\tt pw2wannier90} to compute:
+  \begin{itemize}
+
+  \item[{\bf --}] The overlaps $\langle u_{n{\bf k}}\vert u_{m{\bf k}+{\bf
+          b}}\rangle$ (written in the {\tt Te.mmn} file)
+
+  \item[{\bf --}] The projections for the starting guess (written in the {\tt
+        Te.amn} file)
+
+  \item[{\bf --}] The matrix elements $\langle u_{n{\bf k}+{\bf b}_1}\vert
+      H_{\bf k}\vert u_{m{\bf k}+{\bf b}_2}\rangle$ (written in the
+      {\tt Te.uHu} file)
+
+  \item[{\bf --}] The spin matrix elements $\langle \psi_{n{\bf
+        k}}\vert \sigma_i\vert \psi_{m{\bf k}}\rangle$ (would be written in the
+    {\tt Te.spn} file, but only if spin-orbit is included, which is not the case for the present example)
+
+  \end{itemize}
+{\tt pw2wannier90.x < Te.pw2wan > pw2wan.out}
+
+\item Run \wannier\ to compute the MLWFs.\\
+{\tt wannier90.x Te}
+
+\item  Add the following lines to the {\tt wannier90.win} file:\\
+{\tt gyrotropic=true  \\
+gyrotropic\_task=-C-dos-D0-Dw-K \\
+fermi\_energy\_step=0.0025\\
+fermi\_energy\_min=5.8\\
+fermi\_energy\_max=6.2\\
+gyrotropic\_freq\_step=0.0025\\
+gyrotropic\_freq\_min=0.0\\
+gyrotropic\_freq\_max=0.1\\
+gyrotropic\_smr\_fixed\_en\_width=0.01\\
+gyrotropic\_smr\_max\_arg=5\\
+gyrotropic\_degen\_thresh=0.001\\
+gyrotropic\_box\_b1=0.2 0.0 0.0\\
+gyrotropic\_box\_b2=0.0 0.2 0.0\\
+gyrotropic\_box\_b3=0.0 0.0 0.2\\
+gyrotropic\_box\_center=0.33333 0.33333 0.5\\
+gyrotropic\_kmesh=50 50 50
+}
+
+
+
+\item Run \postw\ \\to compute the gyrotropic properties: tensors $D$, $\widetilde{D}$, $K$, $C$ (See the User Guide):.\\
+  {\tt postw90.x Te} (serial execution)\\
+  {\tt mpirun -np 8 postw90.x Te} (example of parallel execution with
+  8 MPI processes) \\
+
+
+The integraton in the $k$-space is limited to a small area around the H point. Thus it is valid only for Fermi levels near the band gap. 
+And one needs to multiply the results by 2, to account forthe H' point. To integrate over the entire Brillouin zone, one needs to remove the 
+{\tt gyrotropic\_box\_$\ldots$} parameters
+
+\item Now change the above lines to \\ {\tt
+gyrotropic=true\\
+gyrotropic\_task=-NOA\\
+fermi\_energy=5.95\\
+gyrotropic\_freq\_step=0.0025\\
+gyrotropic\_freq\_min=0.0\\
+gyrotropic\_freq\_max=0.3\\
+gyrotropic\_smr\_fixed\_en\_width=0.01\\
+gyrotropic\_smr\_max\_arg=5\\
+gyrotropic\_band\_list=4-9\\
+gyrotropic\_kmesh=50 50 50\\
+}
+
+and compute the interband natural optical activity\\
+
+  {\tt postw90.x Te} (serial execution)\\
+  {\tt mpirun -np 8 postw90.x Te} (example of parallel execution with
+  8 MPI processes) \\
+
+
+\end{enumerate}
+
+
+
 \bibliographystyle{apsrev4-1}
 \bibliography{../wannier90}
 

doc/user_guide/berry.tex

@@ -35,6 +35,7 @@ \section{Background: Berry connection and curvature}
 \begin{equation}
 {\bf A}_{nm}({\bf k})=\langle u_{n{\bf k}}\vert i\bm{\nabla}_{\bf k}\vert
 u_{m{\bf k}}\rangle={\bf A}_{mn}^*({\bf k}),
+\label{eq:berry-connection-matrix}
 \end{equation}
 %
 and write the curvature as an antisymmetric tensor,

doc/user_guide/gyrotropic.tex

@@ -0,0 +1,159 @@
+\chapter{Overview of the {\tt gyrotropic} module \label{ch:gyrotropic}}
+
+
+The {\tt gyrotropic} module of {\tt postw90} is called by setting {\tt
+  gyrotropic = true} and choosing one or more of the available options for
+{\tt gyrotropic\_task}. The module computes the quantities, studied in 
+ \cite{tsirkin-arxiv17}, where more details may be found.
+
+\section{{\tt berry\_task=-d0}: the Berry curvature dipole  }
+
+The traceless dimensionless tensor
+\begin{equation}
+\label{eq:D_ab}
+D_{ab}=\int\dk\sum_n
+\frac{\partial E_n}{\partial{k_a}}
+\Omega_n^b
+\left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n},
+\end{equation}
+
+
+\section{{\tt berry\_task=-dw}: the finite-frequency generalization of the Berry curvature dipole  }
+
+\begin{equation}
+\label{eq:D-tilde}
+\widetilde{D}_{ab}(\ww)=\int\dk\sum_n
+\frac{\partial E_n}{\partial{k_a}}\widetilde\Omega^b_n(\ww)
+\left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n},
+\end{equation}
+
+where $\widetilde{\bm\Omega}_{\kk n}(\ww)$ is a finite-frequency
+generalization of the Berry curvature:
+%
+%
+\begin{equation}
+\label{eq:curv-w}
+\widetilde{\bm\Omega}_{\kk n}(\ww)=-
+\sum_m\,\frac{\ww^2_{\kk mn}}{\ww^2_{\kk mn}-\ww^2}
+\im\left({\bm A}_{\kk nm}\times{\bm A}_{\kk mn}\right)
+\end{equation}
+Contrary to the Berry
+  curvature, the divergence of $\tilde{\bm\Omega}_{\kk n}(\ww)$ is
+  generally nonzero. As a result, $\wt D(\ww)$ 
+can have a nonzero trace at finite frequencies, $\tilde{D}_\|\neq-2\tilde{D}_\perp$ in Te.
+
+\section{{\tt berry\_task=-C}: the ohmic  conductivity }
+
+In the constant relaxation-time
+approximation  the ohmic conductivity is expressed as
+ $\sigma_{ab}=(2\pi e\tau/\hbar)C_{ab}$,  with
+%
+\beq
+\label{eq:C_ab}
+C_{ab}=\frac{e}{h}\int\dk\sum_n\,
+\frac{\partial E_n}{\partial{k_a}} \frac{\partial E_n}{\partial{k_b}}
+\left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n}
+\eeq
+a positive quantity with
+units of surface current density (A/cm).
+
+
+\section{{\tt berry\_task=-K}: the kinetic magnetoelectric effect (kME) }
+
+A microscopic theory of the intrinsic kME effect in bulk crystals was
+recently developed~\cite{yoda-sr15,zhong-prl16}.  
+
+The response is described by
+\beq
+\label{eq:K_ab}
+K_{ab}=\int\dk\sum_n\frac{\partial E_n}{\partial{k_a}} m_n^b 
+\left(-\frac{\partial f_0}{\partial E}\right)_{E=E_n},
+\eeq
+%
+which has the same form as \eq{D_ab}, but with the Berry
+curvature replaced by the intrinsic magnetic moment ${\bm m}_{\kk n}$
+of the Bloch electrons, which has the  spin and orbital components
+ given by~\cite{xiao-rmp10} 
+%
+\bea
+\label{eq:m-spin}
+m^{\rm spin}_{\kk n}&=&-\frac{1}{2}g_s\mu_{\rm B} \me{\psi_{\kk
+      n}}{\bf \sigma}{\psi_{\kk n}}\\
+\label{eq:m-orb}
+{\bm m}^{\rm orb}_{\kk n}&=&\frac{e}{2\hbar}\im
+\bra{{\bm\partial}_\kk u_{\kk n}}\times
+(H_\kk-E_{\kk n})\ket{{\bm\partial}_\kk u_{\kk n}},
+\eea
+%
+where $g_s\approx 2$ and we chose $e>0$. 
+
+\section{{\tt berry\_task=-dos}: the density of states }
+
+The density of states is calculated with the same width and type of smearing, as the other properties of the {\tt gyrotropic} module
+
+\section{{\tt berry\_task=-noa}: the interband contributionto the natural optical activity }
+
+Natural optical rotatory power is given by \cite{ivchenko-spss75}
+%
+\beq
+\label{eq:rho-c}
+\rho_0(\ww)=\frac{\ww^2}{2c^2}\re\,\gamma_{xyz}(\ww).
+\eeq
+%
+for light propagating ling the main symmetry axis of a crystal $z$. Here $\gamma_{xyz}(\ww)$
+is an anti-symmetric (in $xy$) tensor with units of length, which has both inter- and intraband contributions.
+
+Following Ref.~\cite{malashevich-prb10} for the interband contribution we writewe write,
+with $\partial_c\equiv\partial/\partial k_c$,
+%
+\begin{multline}
+\re\,\gamma_{abc}^{\mathrm{inter}}(\ww)=\frac{e^2}{\varepsilon_0\hbar^2}
+\int[d\kk]
+\sum_{n,l}^{o,e}\,
+\Bigl[ \frac{1}{\ww_{ln}^2-\ww^2} 
+\re\left(A_{ln}^bB_{nl}^{ac}-A_{ln}^aB_{nl}^{bc}\right) \\
+-\frac{3\ww_{ln}^2-\ww^2}{(\ww_{ln}^2-\ww^2)^2} 
+\partial_c(E_l+E_n)\im\left(A_{nl}^aA_{ln}^b\right)   
+\Bigr].
+\label{eq:gamma-inter}
+\end{multline}
+%
+The summations over $n$ and $l$ span the occupied ($o$) and empty
+($e$) bands respectively, $\ww_{ln}=(E_l-E_n)/\hbar$,
+and  ${\bm A}_{ln}(\kk)$ is given by (\ref{eq:berry-connection-matrix}) Finally, the matrix
+$B_{nl}^{ac}$ has both orbital and spin contributions given by
+%
+\beq
+\label{eq:B-ac-orb}
+B_{nl}^{ac\,({\rm orb})}=
+  \bra{u_n}(\partial_aH)\ket{\partial_c u_l}
+ -\bra{\partial_c u_n}(\partial_aH)\ket{u_l}
+\eeq
+%
+and
+%
+\beq
+\label{eq:B-ac-spin}
+B_{nl}^{ac\,({\rm spin})}=-\frac{i\hbar^2}{m_e}\epsilon_{abc}
+\bra{u_n}\sigma_b\ket{u_l}.
+\eeq
+%
+The spin matrix elements contribute less than 0.5\% of the total
+$\rho_0^{\rm inter}$ of Te.  Expanding
+$H=\sum_m \ket{u_m} E_m \bra{u_m}$ we obtain for the orbital matrix
+elements
+\beq
+B_{nl}^{ac\,({\rm orb})}=-i\partial_a(E_n+E_l)A_{nl}^c \sum_m \Bigl\{ (E_n-E_m) A_{nm}^aA_{ml}^c -(E_l-E_m) A_{nm}^cA_{ml}^a \Bigr\}.
+\label{eq:Bnl-sum}
+\eeq
+%
+This reduces the calculation of $B^{\text{(orb)}}$ to the evaluation
+of band gradients and off-diagonal elements of the Berry connection
+matrix. Both operations can be carried out efficiently in a
+Wannier-function basis following Ref.~\cite{yates-prb07}.
+
+
+\section{{\tt berry\_task=-spin}: compute also the spin component of NOA and KME }
+
+Unless this task is specified, only the orbital contributions are calcuated in NOA and KME, thus contributions from \eqs{m-spin}{B-ac-spin} are omitted.
+