Analysis of kink in plot of photon energy flux vs. temperature for semiconductor

kink.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kink\n",
    "====\n",
    "The purpose of this notebook is to understand the kink in the the value of `tec.electrode.SC.photon_energy_flux()` method vs. the `temp` attribute."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Initialize environment\n",
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from astropy import units, constants\n",
    "import tec\n",
    "import datac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def photon_energy_flux(temp, emissivity, barrier, bandgap):\n",
    "    \"\"\"\n",
    "    Wrapper around `photon_energy_flux` method\n",
    "    \"\"\"\n",
    "    eem = 1. * constants.m_e\n",
    "    hem = 1. * constants.m_e\n",
    "\n",
    "    s = tec.electrode.SC(temp=temp, richardson=120, emissivity=emissivity, barrier=barrier, bandgap=bandgap, electron_effective_mass=eem, hole_effective_mass=hem)\n",
    "    return s.photon_energy_flux().value\n",
    "\n",
    "def thermoelectron_energy_flux(temp, emissivity, barrier, bandgap):\n",
    "    \"\"\"\n",
    "    Wrapper around `thermoelectron_energy_flux` method\n",
    "    \"\"\"\n",
    "    eem = 1. * constants.m_e\n",
    "    hem = 1. * constants.m_e\n",
    "\n",
    "    s = tec.electrode.SC(temp=temp, richardson=120, emissivity=emissivity, barrier=barrier, bandgap=bandgap, electron_effective_mass=eem, hole_effective_mass=hem)\n",
    "    return s.thermoelectron_energy_flux().value\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "temps = np.linspace(700, 900, 100)\n",
    "\n",
    "ds1 = datac.Datac(photon_energy_flux, \"temp\", temps, emissivity=1., barrier=1., bandgap=3.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x108da3390>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108ab4650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.semilogy(ds1.abscissae, ds1.ordinates, \"kx-\")\n",
    "plt.title(\"`tec.electrode.SC.photon_energy_flux` method.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ds2 = datac.Datac(thermoelectron_energy_flux, \"temp\", temps, emissivity=1., barrier=1., bandgap=3.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1091b2a50>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "IBTwRUQKQgFfRKQgFPBFRAri/wFHwuHpCDh+yAAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108d74fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.semilogy(ds2.abscissae, ds2.ordinates, \"kx-\")\n",
    "plt.title(\"`tec.electrode.SC.thermoelectron_energy_flux` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Analysis\n",
    "========\n",
    "It looks like the kink is happening in the `photon_energy_flux` method and not the `thermoelectron_energy_flux` method. Thus, the problem likely is occuring in the `ibei` package."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Investigating `ibei`\n",
    "==================="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import ibei"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def target_uibei(order, energy_lo, temp, chem_potential):\n",
    "    \"\"\"\n",
    "    Annoyingly wrap `ibei.uibei` due to limitations in astropy\n",
    "    \"\"\"\n",
    "    target = ibei.uibei(order, energy_lo, temp, chem_potential)\n",
    "    return target.value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1094f2150>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "DMzMDPj/KwVfBhWvoAkAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093fd2d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds3 = datac.Datac(target_uibei, \"temp\", temps, order=3, energy_lo=units.Quantity(3., \"eV\"), chem_potential=0)\n",
    "plt.semilogy(ds3.abscissae, ds3.ordinates, \"kx-\")\n",
    "plt.title(\"`ibei.uibei` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lower order (=2)\n",
    "================"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1097c5b10>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "ZmZ9zEHAzKyPOQiYmfWx/weG5wA1aWgSmAAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097324d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds4 = datac.Datac(target_uibei, \"temp\", temps, order=2, energy_lo=units.Quantity(3., \"eV\"), chem_potential=0)\n",
    "plt.semilogy(ds4.abscissae, ds4.ordinates, \"kx-\")\n",
    "plt.title(\"`ibei.uibei` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Result\n",
    "======\n",
    "Similar shape, different values.\n",
    "\n",
    "Within the `uibei` method is a call to `sympy.mpmath.polylog`; I suspect the problem is occuring there. All of the other calculations are pretty simple basic mathematical functions. I will try to implement just the relevant calculation and plot the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy.mpmath import polylog\n",
    "\n",
    "def target_polylog(order, energy_lo, temp):\n",
    "    \"\"\"\n",
    "    Just the polylogarithm part of the `uibei` method\n",
    "    \"\"\"\n",
    "    temp = units.Quantity(temp, \"K\")\n",
    "    energy_lo = units.Quantity(energy_lo, \"eV\")\n",
    "    \n",
    "    kT = temp * constants.k_B\n",
    "    reduced_energy_lo = energy_lo / kT\n",
    "    expt = (-reduced_energy_lo).decompose()\n",
    "    real_arg = np.exp(expt.value)\n",
    "    \n",
    "    return float(polylog(order, real_arg))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "order = 1\n",
    "========="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10aa1c110>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a932fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds5 = datac.Datac(target_polylog, \"temp\", temps, order=1, energy_lo=units.Quantity(3., \"eV\"))\n",
    "plt.semilogy(ds5.abscissae, ds5.ordinates, \"kx-\")\n",
    "plt.title(\"`target_polylog` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "order = 2\n",
    "========="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10ac11790>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ab2d5d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds6 = datac.Datac(target_polylog, \"temp\", temps, order=2, energy_lo=units.Quantity(3., \"eV\"))\n",
    "plt.semilogy(ds6.abscissae, ds6.ordinates, \"kx-\")\n",
    "plt.title(\"`target_polylog` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "order = 3\n",
    "========="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10acc3790>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a95bfd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds7 = datac.Datac(target_polylog, \"temp\", temps, order=3, energy_lo=units.Quantity(3., \"eV\"))\n",
    "plt.semilogy(ds7.abscissae, ds7.ordinates, \"kx-\")\n",
    "plt.title(\"`target_polylog` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Analysis\n",
    "--------\n",
    "I could keep going by increasing the order, but it looks like the problem **is** occuring in the polylog function of order 1. What does the value of `real_arg` look like over the temperature range?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def target_real_arg(order, energy_lo, temp):\n",
    "    \"\"\"\n",
    "    Just the polylogarithm part of the `uibei` method\n",
    "    \"\"\"\n",
    "    temp = units.Quantity(temp, \"K\")\n",
    "    energy_lo = units.Quantity(energy_lo, \"eV\")\n",
    "    \n",
    "    kT = temp * constants.k_B\n",
    "    reduced_energy_lo = energy_lo / kT\n",
    "    expt = (-reduced_energy_lo).decompose()\n",
    "    real_arg = np.exp(expt.value)\n",
    "    \n",
    "    return real_arg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10b0a6350>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "jYGZmbkxMDMzNwZmZoYbAzMzw42BmZnhxsDMzHBjYGZmwP8DFC3MDbuJnu8AAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10aabd550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ds8 = datac.Datac(target_real_arg, \"temp\", temps, order=3, energy_lo=units.Quantity(3., \"eV\"))\n",
    "plt.semilogy(ds8.abscissae, ds8.ordinates, \"kx-\")\n",
    "plt.title(\"`target_real_arg` method.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the values passed to the `sympy.mpmath.polylog` method are very small, ranging from 1e-22 to 1e-16."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}

requirements.conda

# This file may be used to create an environment using:
# $ conda create --name <env> --file <this file>
# platform: osx-64
_license=1.1=py27_0
abstract-rendering=0.5.1=np19py27_0
anaconda=2.2.0=np19py27_0
appscript=1.0.1=py27_0
argcomplete=0.8.4=py27_0
astropy=1.0.1=np19py27_0
bcolz=0.8.1=np19py27_0
beautiful-soup=4.3.2=py27_0
binstar=0.10.1=py27_3
bitarray=0.8.1=py27_0
blaze-core=0.7.3=np19py27_0
blz=0.6.2=np19py27_0
bokeh=0.8.1=np19py27_1
boto=2.36.0=py27_0
cdecimal=2.3=py27_0
certifi=14.05.14=py27_0
cffi=0.9.2=py27_0
clyent=0.3.4=py27_0
colorama=0.3.3=py27_0
configobj=5.0.6=py27_0
cryptography=0.8=py27_0
curl=7.38.0=0
cython=0.22=py27_0
cytoolz=0.7.2=py27_0
datac=0.4.0.dev1=1_g0be8f78
datashape=0.4.4=np19py27_1
decorator=3.4.0=py27_0
docutils=0.12=py27_0
dynd-python=0.6.5=np19py27_0
enum34=1.0.4=py27_0
fastcache=1.0.2=py27_0
flask=0.10.1=py27_1
freetype=2.5.2=0
funcsigs=0.4=py27_0
futures=2.2.0=py27_0
gevent=1.0.1=py27_0
gevent-websocket=0.9.3=py27_0
greenlet=0.4.5=py27_0
grin=1.2.1=py27_1
h5py=2.4.0=np19py27_0
hdf5=1.8.14=0
ibei=1.0.5=19_g3b30dc7
ipython=3.0.0=py27_0
ipython-notebook=3.0.0=py27_1
ipython-qtconsole=3.0.0=py27_0
itsdangerous=0.24=py27_0
jdcal=1.0=py27_0
jedi=0.8.1=py27_0
jinja2=2.7.3=py27_1
jpeg=8d=1
jsonschema=2.4.0=py27_0
launcher=1.0.0=2
libdynd=0.6.5=0
libpng=1.5.13=1
libsodium=0.4.5=0
libtiff=4.0.2=1
libxml2=2.9.0=1
libxslt=1.1.28=2
llvmlite=0.2.2=py27_1
lxml=3.4.2=py27_0
markupsafe=0.23=py27_0
matplotlib=1.4.3=np19py27_1
mistune=0.5.1=py27_0
mock=1.0.1=py27_0
multipledispatch=0.4.7=py27_0
networkx=1.9.1=py27_0
nltk=3.0.2=np19py27_0
node-webkit=0.10.1=0
nose=1.3.4=py27_1
numba=0.17.0=np19py27_0
numexpr=2.3.1=np19py27_0
numpy=1.9.2=py27_0
odo=0.3.1=np19py27_0
openpyxl=1.8.5=py27_0
openssl=1.0.1k=1
pandas=0.15.2=np19py27_1
pathlib=1.0.1=py27_0
patsy=0.3.0=np19py27_0
pep8=1.6.2=py27_0
physicalproperty=1.0.1=np19py27_0
pillow=2.7.0=py27_1
pip=6.1.1=py27_0
ply=3.4=py27_0
psutil=2.2.1=py27_0
ptyprocess=0.4=py27_0
py=1.4.26=py27_0
pyasn1=0.1.7=py27_0
pyaudio=0.2.7=py27_0
pycosat=0.6.1=py27_0
pycparser=2.10=py27_0
pycrypto=2.6.1=py27_0
pycurl=7.19.5.1=py27_0
pyflakes=0.8.1=py27_0
pygments=2.0.2=py27_0
pyopenssl=0.14=py27_0
pyparsing=2.0.3=py27_0
pyqt=4.11.3=py27_0
pytables=3.1.1=np19py27_2
pytest=2.6.4=py27_0
python=2.7.9=1
python-dateutil=2.4.2=py27_0
python.app=1.2=py27_3
pytz=2015.2=py27_0
pyyaml=3.11=py27_0
pyzmq=14.5.0=py27_0
qt=4.8.6=0
readline=6.2=2
redis=2.6.9=0
redis-py=2.10.3=py27_0
requests=2.6.0=py27_0
rope=0.9.4=py27_1
runipy=0.1.3=py27_0
scikit-image=0.11.2=np19py27_0
scikit-learn=0.15.2=np19py27_0
scipy=0.15.1=np19py27_0
setuptools=15.0=py27_0
sip=4.16.5=py27_0
six=1.9.0=py27_0
sockjs-tornado=1.0.1=py27_0
sphinx=1.2.3=py27_0
spyder=2.3.4=py27_1
spyder-app=2.3.4=py27_0
sqlalchemy=0.9.9=py27_0
sqlite=3.8.4.1=1
ssl_match_hostname=3.4.0.2=py27_0
statsmodels=0.6.1=np19py27_0
sympy=0.7.6=py27_0
tec=1.0.0.dev1=25_gfa3f894
terminado=0.5=py27_0
tk=8.5.18=0
toolz=0.7.1=py27_0
tornado=4.1=py27_0
ujson=1.33=py27_0
unicodecsv=0.9.4=py27_0
werkzeug=0.10.1=py27_0
xlrd=0.9.3=py27_0
xlsxwriter=0.6.7=py27_0
xlwings=0.3.4=py27_0
xlwt=0.7.5=py27_0
yaml=0.1.4=1
zeromq=4.0.4=0
zlib=1.2.8=0