Fermi-Dirac Integrals (FDINT)

FDINT is a free, open-source python package that provides fast, double precision (64-bit floating point) approximations to the Fermi-Dirac integrals of integer and half integer order, based on the work by Prof. Fukushima [1-3]. FDINT is written predominantly in Cython, which is compiled to native code through an intermediate C source, resulting in C-like performance.

[1]	T. Fukushima, "Precise and fast computation of Fermi-Dirac integral of integer and half integer order by piecewise minimax rational approximation," Applied Mathematics and Computation, vol. 259, pp. 708-729, May 2015. DOI: 10.1016/j.amc.2015.03.009

[2]	T. Fukushima, "Precise and fast computation of inverse Fermi-Dirac integral of order 1/2 by minimax rational function approximation," Applied Mathematics and Computation, vol. 259, pp. 698-707, May 2015. DOI: 10.1016/j.amc.2015.03.015

[3]	T. Fukushima, "Precise and fast computation of generalized Fermi-Dirac integral by parameter polynomial approximation," 2014. DOI: 10.13140/2.1.1094.6566

The source code and documentation (coming soon) are graciously hosted by GitHub.

Installation

In order to use FDINT, you must have a working Python distribution installed. Python 3 support has not yet been tested, so Python 2.7 is suggested. You will also need to install Numpy before proceeding. If you're not familiar with Python, you might consider installing a Python distribution that comes prepackaged with Numpy.

From PyPi

This is the recommended method for installing FDINT. PyPi is the python package index, which contains many python packages that can be easily installed with a single command. To install FDINT from PyPi, open up a command prompt and run the following command:

pip install fdint

From Github

To install the latest release of FDINT from Github, go to the FDINT releases page, download the latest .zip or .tar.gz source package, extract its contents, and run python setup.py install from within the extracted directory.

Testing

Once installed, you can test the package by running the following command:

python -m fdint.tests

If you have Matplotlib installed, you can also plot a sample of the available functions by running the following command:

python -m fdint.examples.plot

Tutorial

First, start up an interactive python shell from the command line:

$ python

Next, import everything from the fdint package:

>>> from fdint import *

Now you can access the Fermi-Dirac integral and derivative convenience functions, fdk and dfdk:

>>> fdk(k=0.5,phi=-10)
4.0233994366893939e-05
>>> fdk(0.5,-10)
4.0233994366893939e-05
>>> fdk(k=0.5,phi=5)
7.837976057293096
>>> fdk(k=0.5,phi=50)
235.81861512588432
>>> dfdk(k=0.5,phi=-10) # first derivative
4.0233348580568672e-05

You can also pass in numpy arrays as phi:

>>> import numpy
>>> fdk(k=0.5,phi=numpy.linspace(-100,10,3))
array([  3.29683149e-44,   2.53684104e-20,   2.13444715e+01])

If you request an order or derivative that is not implemented, a NotImplementedError is raised:

>>> fdk(1,0)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "fdint/__init__.py", line 50, in fdk
    raise NotImplementedError()
NotImplementedError

For semiconductor calculations, parabolic, dparabolic, iparabolic, nonparabolic, and dnonparabolic are provided:

>>> parabolic(0)
0.7651470246254078
>>> dparabolic(0)
0.6048986434216304
>>> iparabolic(.7)
-0.11156326391089397
>>> nonparabolic(0,0)
0.7651470705342294
>>> nonparabolic(0,0.07) # InAs
1.006986898726782
>>> dnonparabolic(0,0.07) # InAs
0.8190058991462952

Benchmarks

Below are a few benchmarking runs. First, numpy.exp:

$ python -m timeit -s "import numpy; from numpy import exp; x=numpy.linspace(-100,10,10000)" "exp(x)"
10000 loops, best of 3: 72.6 usec per loop

The same arguments to the Fermi-Dirac integral of order k=1/2, fdint.fd1h, takes only ~2.2x the runtime:

$ python -m timeit -s "from fdint import fd1h; import numpy; x=numpy.linspace(-100,10,10000)" "fd1h(x)"
10000 loops, best of 3: 158 usec per loop

Similarly, the inverse Fermi-Dirac integral of order k=1/2, fdint.ifd1h, takes only ~2.4x the runtime of numpy.log:

$ python -m timeit -s "import numpy; from numpy import exp,log; x=numpy.linspace(-100,10,10000);y=exp(x)" "log(y)"
10000 loops, best of 3: 69.9 usec per loop
$ python -m timeit -s "from fdint import fd1h,ifd1h; import numpy; x=numpy.linspace(-100,10,10000);y=fd1h(x)" "ifd1h(y)"
10000 loops, best of 3: 178 usec per loop

The generalized Fermi-Dirac integrals are also quite fast. For order k=1/2 with zero nonparabolicity, fdint.gfd1h takes only ~3.7x the runtime of numpy.exp for zero nonparabolicity:

$ python -m timeit -s "from fdint import gfd1h; import numpy; x=numpy.linspace(-100,10,10000);b=numpy.zeros(10000);b.fill(0.)" "gfd1h(x,b)"
1000 loops, best of 3: 266 usec per loop

However, if there is significant nonparabolicity, fdint.gfd1h can take a up to ~10x longer than numpy.exp:

$ python -m timeit -s "from fdint import gfd1h; import numpy; x=numpy.linspace(-100,10,10000);b=numpy.zeros(10000);b.fill(0.1)" "gfd1h(x,b)"
1000 loops, best of 3: 467 usec per loop

$ python -m timeit -s "from fdint import gfd1h; import numpy; x=numpy.linspace(-100,10,10000);b=numpy.zeros(10000);b.fill(0.3)" "gfd1h(x,b)"
/usr/local/Cellar/python/2.7.8_2/Frameworks/Python.framework/Versions/2.7/lib/python2.7/timeit.py:6: RuntimeWarning: gfd1h: less than 24 bits of accuracy
1000 loops, best of 3: 696 usec per loop

The full calculation for a nonparabolic band takes ~5-17x longer than numpy.exp, depending on the level of nonparabolicity (Note: for some reason the timing for this command is unreasonably high when timed from the command line. When timed inside of ipython, it works fine):

$ ipython
In [1]: from fdint import *

In [2]: import numpy

In [3]: phi = numpy.linspace(-100,10,10000)

In [4]: %timeit numpy.exp(phi)
10000 loops, best of 3: 72.9 µs per loop

In [5]: %timeit parabolic(phi)
10000 loops, best of 3: 165 µs per loop

In [6]: alpha = numpy.empty(10000); alpha.fill(0.0) # parabolic

In [7]: %timeit nonparabolic(phi, alpha)
1000 loops, best of 3: 346 µs per loop

In [8]: alpha = numpy.empty(10000); alpha.fill(0.07) # InAs

In [9]: %timeit nonparabolic(phi, alpha)
1000 loops, best of 3: 695 µs per loop

In [10]: alpha = numpy.empty(10000); alpha.fill(0.15) # InSb

In [11]: %timeit nonparabolic(phi, alpha)
/usr/local/bin/ipython:257: RuntimeWarning: nonparabolic: less than 24 bits of accuracy
1000 loops, best of 3: 1.26 ms per loop

Documentation

The documentation (coming soon) is graciously hosted by GitHub.