time_derivative_poisson.ipynb

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      "## Time derivatives of Poisson equation\n",
      "\n",
      "Starting with total current equation (Maxwell-Ampere law),\n",
      "\n",
      "$$\n",
      "\\nabla \\times H = j_n + j_p + \\epsilon \\frac{\\partial E}{\\partial t} = j_t\n",
      "$$\n",
      "\n",
      "where $\\nabla \\times H$ can be interpreted as the total current ($j_t$): the summation of the particle currents ($j_n$ and $j_p$) and the displace current ($j_d = \\epsilon \\frac{\\partial E}{\\partial t}$) caused by dielectric relaxation. I don't know whether this is correct but I think about the displacement current as being the media's delayed response in reacting to the movement of change carriers.\n",
      "\n",
      "If we now take the divergence of the above equation we get,\n",
      "\n",
      "$$\n",
      "\\nabla \\cdot (\\nabla \\times H) = \\nabla \\cdot \\left( j_n + j_p + \\epsilon \\frac{\\partial E}{\\partial t} \\right) = 0\n",
      "$$\n",
      "\n",
      "where $\\nabla \\cdot (\\nabla \\times H)=0$ (for any vector field) so we end up with,\n",
      "\n",
      "$$\n",
      "\\nabla \\cdot \\left(-\\epsilon\\frac{\\partial E}{\\partial t} \\right) = \\nabla \\cdot \\left( j_n + j_p \\right)\n",
      "$$\n",
      "\n",
      "If we now *exchange* the order of the partial derivatives on the l.h.s.,\n",
      "\n",
      "$$\n",
      "-\\frac{\\partial}{\\partial t} \\left( \\nabla (\\epsilon E) \\right) = \\nabla \\cdot \\left( j_n + j_p \\right)\n",
      "$$\n",
      "\n",
      "That Poission equation is,\n",
      "\n",
      "$$\n",
      "\\nabla E = -\\nabla^2 \\phi\n",
      "$$\n",
      "\n",
      "Therefore the term on the l.h.s. is just the Poisson equation multiplied by the dielectric permittivity (assuming constant),\n",
      "\n",
      "$$\n",
      "\\nabla (\\epsilon E) = -\\epsilon\\nabla^2 V\n",
      "$$\n",
      "\n",
      "So in one-dimension we recover,\n",
      "\n",
      "$$\n",
      "\\frac{\\partial}{\\partial t} \\left(\\epsilon \\frac{\\partial^2 V}{\\partial x^2 } \\right) = \\frac{\\partial}{\\partial x} \\left( j_n + j_p \\right)\n",
      "$$\n",
      "\n",
      "### Writing in the same form as J. Rogel-Salazar et al.\n",
      "\n",
      "A similar expression appears in J. Rogel-Salazar et al. *\"An efficient method-of-lines simulation procedure for organic\n",
      "semiconductor devices\"* (2009),\n",
      "\n",
      "$$\n",
      "\\boldsymbol{D}_2 \\frac{\\partial V}{\\partial t} = C \\left( \\frac{\\partial n}{\\partial t} - \\frac{\\partial p}{\\partial t}\\right)\n",
      "$$\n",
      "\n",
      "where $C$ is a constant containing $\\epsilon$ and other parameters, and $\\boldsymbol{D}_2$ is an finite-difference matrix operating on a vector to give the second partial derivative.\n",
      "\n",
      "We can try to find equivalence between the two time dependence equations. If we again exchange the order of the partial derivatives on the l.h.s. get,\n",
      "\n",
      "$$\n",
      "\\frac{\\partial^2}{\\partial x^2} \\left( \\epsilon\\frac{\\partial V}{\\partial t } \\right) = \\frac{\\partial}{\\partial x} \\left( j_n + j_p \\right)\n",
      "$$\n",
      "\n",
      "The l.h.s. now has the same form as reported by J. Rogel-Salazar et al.. We now turn our attention to the r.h.s.. \n",
      "\n",
      "The continuity equations are,\n",
      "\n",
      "$$\n",
      "\\frac{\\partial n}{\\partial t } = -\\frac{1}{z_n|q|}\\frac{\\partial}{\\partial x }\\left(j_n\\right) + R\\\\\n",
      "\\frac{\\partial p}{\\partial t } = -\\frac{1}{z_p|q|}\\frac{\\partial}{\\partial x }\\left(j_p\\right) + R\\\\\n",
      "$$\n",
      "\n",
      "where $z_i$ is the valence of species $i$, $|q|$ is the modulus of the electronic charge."
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      "\n",
      "Rearranging for to isolate the divergence of the current,\n",
      "\n",
      "$$\n",
      "-z_n|q|\\left(\\frac{\\partial n}{\\partial t } - R\\right) = \\frac{\\partial}{\\partial x }\\left(j_n\\right)\\\\\n",
      "-z_p|q|\\left(\\frac{\\partial p}{\\partial t } - R\\right) = \\frac{\\partial}{\\partial x }\\left(j_p\\right)\\\\\n",
      "$$\n",
      "\n",
      "If we now let the valancies become, $z_n=-1$, $z_p=1$,\n",
      "\n",
      "$$\n",
      "|q|\\left(\\frac{\\partial n}{\\partial t } - R\\right) = \\frac{\\partial}{\\partial x }\\left(j_n\\right)\\\\\n",
      "-|q|\\left(\\frac{\\partial p}{\\partial t } - R\\right) = \\frac{\\partial}{\\partial x }\\left(j_p\\right)\\\\\n",
      "$$\n",
      "\n",
      "Substituting into the expression gives,\n",
      "\n",
      "$$\n",
      "\\frac{\\partial^2}{\\partial x^2} \\left( \\frac{\\partial V}{\\partial t } \\right) = \\frac{|q|}{\\epsilon} \\left( \\frac{\\partial n}{\\partial t } - \\frac{\\partial p}{\\partial t } \\right)\n",
      "$$\n",
      "\n",
      "This is the same form as J. Rogel-Salazar et al. To be consistent with we simply use the constants, $\\boldsymbol{D}_{2} = \\frac{\\partial^2}{\\partial x^2}$, $C=\\frac{|q|}{\\epsilon}$."
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