Solving Drift-Diffusion Equations

Run the programs as following order:

main_ode_script_mat0.m

• search the easy solutions from weak recombination (controlled by mag_R) and moderate acc (by acc and acc1)

• try-catch is used to prevent the program stopped by failing in convergence. And, this setting is maintained until the statement is patched as

  warnings(s)

• iterative direction is along Vds with Vgs kept fixed one-by-one

• linear projection technique is used along the Vds-iteration, which is packaged as a function interp_sol

  %% solx, (a,b) --> x
  a_vgs=Vgs;
  a_vds=vdslis(i-2);
  a_sol=sollis(i-2);
  
  b_vgs=Vgs;
  b_vds=vdslis(i-1);
  b_sol=sollis(i-1);
  
  x_vgs=Vgs;
  x_vds=vdslis(i);
  x_sol=interp_sol(a_vgs,a_vds,a_sol, ...
                  b_vgs,b_vds,b_sol, ...
                  x_vgs,x_vds);
  

• data structure: matrix: idsmat, timmat, solmat is compressed as vector idsvec, timvec, solvec, respectively, so that vds and vgs are mixed as one dimension.

• usage: decrease mag_R and find the maximum workable value.

main_ode_script_matR.m

• run parfor loop to parallel iterative mag_R as the second dimension.

  parfor i=1:max(14,cub_rows)
  % bias -- def2
  vgsswp=(3:0.3:6);
  vgslen=length(vgsswp);
  vdslis=(0.1:0.1:4.9)';
  vdslen=length(vdslis);
  % assign
  [idx_vds,idx_vgs]=vec2squ(vdslen,vgslen,i);
  vds=vdslis(idx_vds);
  vgs=vgsswp(idx_vgs);
  
  for j=1:mag_Rlen
  mag_Rlis=10.^linspace(-8,0,mag_Rlen);
  mag_R=mag_Rlis(j);
  if j==1
      sol_init=solvec(i);
  else
      sol_init=sol1;
  end
  tic
  sol1=main_ode(acc,acc1,mag_R,gausf(vgs),vgs,vds,sol_init);
  timcub(i,j)=toc;
  x_vgs=vgs;x_vds=vds;
  solw=mean(sol_post(gausf(x_vgs),sol1,x_vgs,x_vds).ids);
  idscub(i,j)=solw;
  end
  
  end
  
  

  collect the data:

  % bias -- def3
  vgsswp=(3:0.3:6);
  vgslen=length(vgsswp);
  vdslis=(0.1:0.1:4.9)';
  vdslen=length(vdslis);
  
  idsedg=idscub(:,end);
  idsmat_edg=idsmat;
  
  for z=1:cub_rows
      [zi,zj]=vec2squ(vdslen,vgslen,z);
      idsmat_edg(zi,zj)=idscub(z,end);
  end
  
  timtri=zeros(vdslen,vgslen,mag_Rlen);
  for z=1:cub_rows
      [zi,zj]=vec2squ(vdslen,vgslen,z);
      for zz=1:mag_Rlen
      timtri(zi,zj,zz)=timcub(z,zz);
      end
  end
  
  timpst=mean(sum(timtri,3),2);

  and then plot

  figure
  plot(vdslis,idsmat_edg)
  
  figure
  semilogy(vdslis,timpst)

  

  

main_ode_script_matR_pst.m

• use the results calculated above to generate the convergence track:

  

main_ode_script_matR_pst_acc.m

• compute the relative error (log)

  

• compute the relative error (linear)