FLOW-K

Please note that this project is still under development and will be updated on a regular basis.

Here we provide a package which calculates the transport properties of a N dimensional device subject to an external electromagnetic field. Using a model developed by Martinez et. al. (see https://journals.aps.org/prb/abstract/10.1103/PhysRevB.77.075339), a recursive scheme was implemented to calculate the current flowing through an electronic device subject to an electric field when attached to electron resoirvoirs.

This project was developed as part of a Theoretical Physics Final Year Project at Trinity College Dublin.

Features

• Measure the current flowing throughout an electronic device when attached to two electron resoirvoirs
• Investigate transport properties of an electronic device when an external potential is applied
• Determine how current flow through various systems, such as DNA wires is effected by an external potential

Future Updates

• Tutorial for Quantum Ratchets
• Tutorial for DNA Wire
• Package code such that its easy to download and install

Tutorial: Double Quantum Dot

First we import in the functions and packages required.

from FLOW_K import E_field as E_field
from FLOW_K import Electrons as Electrics

Our first step is to define the dimensionality and Hamiltonian of the system we wish to investigate. In this example we will use a double Quantum Dot. For this example we normalise all parameters to the hopping parameter in our Hamiltonian and set the onsite energies of the quantum dots to 5.

Using these parameters, the Hamiltonian of the double quantum dot under investigation is given as

n = 2
Hamiltonian = [[-5, 1], [1, 5]]

We normalise all other parameters with regards to the hopping parameter $t$ set all other parameters as follows:

Name	Symbol	Value
Boltzmann Temperature	kT	0.01
Applied Bias	V	0
Self Energy (Leads)	Gamma	0.5
Frequency Range of Efield	E_to_scan	-40 - 0.1
Number of Floquet Modes	m_max	10
Magnitude of E-field	C	6
Number of points	m	200
Dimensionality of Hamiltonian	n	2

T = 0.01 
V = 0.0 
Gamma = 0.5
C =[6]
m_max = 10
m = 20 
E_to_scan = np.linspace(-40, 0.1, m)

After setting all our parameters, we now create the class which will contain both our external electric field and central electronic system

H = Electrons(n, Hamiltonian, V, T);  
E = E_field( -E_to_scan[0], -E_to_scan[-1], E_to_scan, Gamma, C/2, C/2)

Our next step is to set up the simulation and determine the current for a range of different frequencies for the external electric field. To do so, we use the function .coupling2

current = coupling2(E, Hamiltonian, m, m_max) 

We can plot the results of the current to visualise the current for a variety of different frequcies

plt.plot(E_to_scan, current[0], linestyle="-", linewidth=1, marker = "o", markersize = 0)
plt.ylabel("Normalised Current")
plt.xlabel(" Applied Bias accross Leads [t]")
plt.grid("on")
plt.legend(loc='upper left', bbox_to_anchor=(1, 1))
plt.style.use('ggplot')
plt.show()