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Guangcun's Blog分享 http://blog.sciencenet.cn/u/gcshan 哥伦比亚大学访问学者,香港城市大学校董 (2011-2012)

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How to calculate Green's function

已有 2728 次阅读 2014-2-27 01:41 |个人分类:科学札记|系统分类:科研笔记

Green'sfunction calculation


For a given k-point $k$,spin component $.sigma$,complex energy $E_c=(E_r,E_i)$,and denoting PLs by $i$or $j$,let us define the following matrices:

$O(i,j,k,.sigma)$ $.Rightarrow$k-space Overlap matrix linking PLs $i$and $j$($i+1.geq j .geq i-1$must hold).$H(i,j,k,.sigma)$ $.Rightarrow$k-space Hamiltonian matrix linking PLs $i$and $j$($i+1.geq j .geq i-1$must hold).$F(i,j,E,k,.sigma)=E .cdot O(i,j,k,.sigma) - H(i,j,k,.sigma)$ $.Rightarrow$k-space secular matrix linking PLs $i$and $j$($i+1.geq j .geq i-1$must hold).$G(i,j,E,k,.sigma)=[F]^{-1}(i,j,E,k,.sigma)$ $.Rightarrow$Retarded Green's function matrix linking PL $j$to $i$.$T(i,j,E,k,.sigma)=G(i,j,E,k,.sigma) .; [G(j,j,E,k,.sigma)]^{-1}$ $.Rightarrow$Retarded propagator matrix linking PL $j$to $i$.$S(i,j,E,k,.sigma)=[G(i,i,E,k,.sigma)]^{-1} .; G(i,j,E,k,.sigma)$ $.Rightarrow$Advanced propagator matrix linking PL $i$to $j$.$.Sigma^.pm(i,E,k,.sigma)= F(i,i.pm1,E,k,.sigma) T(i.pm1,i,E,k,.sigma)$ $.Rightarrow$Self-energy matrix for PL $i$with respect to its right/left PLs.In the AO representation, the element ($.alpha$,$.beta$)of any of the above matrices, $A$,will be denoted by superscripts: $A^{.alpha,.beta}$.

The construction of the Green's functionfor a given BL will depend on its type. Below, we omit the $E_c$, $k$and $.sigma$dependance for all matrices.

  • UC


    • No Green's function is evaluated in thiscase. The electronic structure is directly obtained after diagonalizingthe Hamiltonian at UC $i$:.begin{displaymath}H(i) .; .vec u^m(i) = E^m .; O(i) .; .vec u^m(i).end{displaymath}
    where $m$stands for the band index, $E^m$gives the band structure and $.vec u^m(i)$holds the eigenvectors.

  • PL


    • There are two methods for calculatingthe Green's function of any PL (see option [method]in card "%block")::
 


[method=spectral] [method=inversion]
First solve the diagonalizationproblem to obtain the eigevalues, $E^m$and eigenvectors, $.vec u^m(i)$:
.begin{displaymath}H(i) .; .vec u^m(i) = E^m .; O(i) .; .vec u^m(i).end{displaymath}


Next, for each complex energy $E_c$,the Green's function is constructed from its spectral descomposition:

.begin{displaymath}G(i,i)=.sum_{m} .vec u^m .frac{1}{E-E^m} [.vec u^m]^.dagger.end{displaymath}
Directly invert the secular equation:

.begin{displaymath}G(i,i)= [F(i,i)]^{-1}.end{displaymath}

Both methods yield identical results, theirmain difference being the compulational resources that each requires. For[method=spectral],the main cost is the diagonallzation and the RAM required to store allthe eigenvectors but, in turn, $G(i,i)$is evaluated very fast at any energy. For [method=inversion],a matrix inversion is always required at each energy. Hence, for calculations
involving several energy points, [method=spectral]should be preferable as long as there are no RAM memory resctrictions.
 

  • BULK


    • A bulk BL is built by an infinitely repeatedstack of the same PL, $i$.Due to translational symmetry, we have the following relations, valid forany PL in the infinite bulk BL:
      $T(i,i.pm 1)$aresolved using the iterative scheme of L. Sancho et al (appendix  inSTM1 paper).
      $.Sigma^.pm(i) = F(i,i.pm 1) .; T(i.pm 1,i)$
      $G(i,i)=[ G^0(i,i)+.Sigma^+(i)+ .Sigma^-(i) ]^{-1}$
      $G(i,i.pm 1)=T(i,i.pm 1) .; G(i,i)$
      $S(i,i.pm 1)=G(i,i.pm 1).; [G(i,i)]^{-1}$
    By far, the most time consuming part is the $T(i,i.pm 1)$,where many matrix inversions are involved.
 

  • Matching 2 BLs


    • Assume that the sub_BL on the left contains $N_l$PLs, and the sub_BL on the right$N_r$PLs. The type of each sub_BL may be any: PL, Slab,Surface or Bulk.Since we only consider nearest PL interactions, the interface PLs are $N_l$and $N_{l+1}$(we will use capital letters ,$I$or $J$,to denote them in the table below).

    We use the so-called Surface Green's FunctionMatching (SGFM) technique. If one denotes by $G^0$, $S^0$and $T^0$the Green's function matrices for the isolated sub_BLs, the Dyson equation:

    .begin{displaymath}G = G^0 - G^0.;F.;G.end{displaymath}
    allows to solve all $G$, $S$and $T$matrices for the coupled BL.
    When the interactions between PLs $N_l$and $N_{l+1}$are weak (this is usually the case in STM for the tip-sample interactions),then one may expand the Dyson equation up to first order:
    .begin{displaymath}G= G^0 - G^0.;F.;G^0.end{displaymath}
    leading to matrix inversion-free expressions.

    By projecting the Dyson equation at allPLs, and after a bit of manipulation of the formulas, the final expressionsfor all $G(i,j)$, $S(i,j)$, $T(i,j)$and $.Sigma^.pm(i)$matrices are the following:
     
     




    PLs  ($i$,$j$)[method=sgfm][method=sgfm_1]
    Interface PLs:
    $I=N_l+1$, $J=N_l$
    or
    $I=N_l$, $J=N_l+1$
    $T(I,J)=G^0(I,I) .; F(I,J)$
    $.Sigma^I(J) = F(J,I).; T(I,J)$
    $G(J,J)= .left[ [G^0(J,J)]^{-1} + .Sigma^I(J).right]^{-1}$
    $G(I,J)=T(I,J) .; G(J,J)$
    $S(J,I)=[G(J,J)]^{-1} .; G(J,I)$
    $T(I,J)=G^0(I,I) .; F(I,J)$
    $.Sigma^I(J) = 0$
    $G(J,J)= [G^0(J,J)]^{-1}$
    $G(I,J)=T(I,J) .; G(J,J)$
    $S(J,I)=[G(J,J)]^{-1} .; G(J,I)$
    Diagonal G's: $i=j$with
    $i<I=N_l$or $i>I=N_l+1$
    $G(i,i)= G^0(i,i) + T^0(i,I) .left( G(I,I)-G^0(I,I).right) S^0(I,i)$
    $G(i,i)= G^0(i,i)$
    Propagationaway from the
    interface PLs:
    $i<j.leq N_l$or $i>j.geq N_l+1$
    $G(i,j) = T^0(i,j) .; G(j,j)$
    $T(i,j) = T^0(i,j)$
    $S(i,j) = [G(i,i)]^{-1} .; G(i,j)$
    $G(i,j) = G^0(i,j)$
    $T(i,j) = T^0(i,j)$
    $S(i,j) = S^0(i,j)$
    Propagationtowards the interface:
    $j < i .leq N_l$or $j > i .geq N_l+1$
    $G(i,j) = G(i,i) .; S^0(i,j)$
    $T(i,j) = G(i,j) .; [G(j,j)]^{-1}$
    $S(i,j) = S^0(i,j)$
    $G(i,j) = G^0(i,j)$
    $T(i,j) = T^0(i,j)$
    $S(i,j) = S^0(i,j)$
    Propagationfrom interface PL
    to the oppositesub_BL:
    $i<I=N_l$, $j=N_l+1$
    or
    $i>I=N_l+1$, $J=N_l$
    $G(i,J) = T^0(i,I) .; G(I,J)$
    $T(i,J) = G(i,J) .; [G(J,J)]^{-1}$
    $S(i,J) = [G(i,i)]^{-1}.; G(i,J)$
    $G(i,J) = T^0(i,I) .; G(I,J)$
    $T(i,J) = G(i,J) .; [G(J,J)]^{-1}$
    $S(i,J) = [G(i,i)]^{-1}.; G(i,J)$
    Propagationfrom one sub_BL
    to the other:
    $i.leq N_l$, $j>J=N_l+1$
    or
    $i.geq N_l+1$, $j <J= N_l$
    $G(i,j) = G(i,J).; S^0(J,j)$
    $T(i,j) = G(i,j) .; [G(j,j)]^{-1}$
    $S(i,j) = [G(i,i)]^{-1} .; G(i,j)$
    $G(i,j) = G(i,J).; S^0(J,j)$
    $T(i,j) = G(i,j) .; [G(j,j)]^{-1}$
    $S(i,j) = [G(i,i)]^{-1} .; G(i,j)$

 




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