Aharonov-Bohm effect

Image [CC-BY-SA-3.0](https://creativecommons.org/licenses/by-sa/3.0/deed.en), original by [Kismalac](https://commons.wikimedia.org/wiki/File:AharonovBohmEffect.svg)

In this tutorial, we are going to discuss one of the cornerstone of quantum nanoelectronics, the Aharonov-Bohm effect. We are going to calculate the conductance of a ring through which one applies a magnetic field. One interesting aspect of the Aharonov-Bohm effect is that the magnetic field can actually vanish in the sample itself. All that is required is that the potential vector does not. This is a very nice proof that in quantum mechanics the electron motion indeed couples to the potential vector, not the magnetic field. This can be seen from the following Gauge transformation: let us consider the Schrodinger equation $H\Psi = E\Psi$ with following Hamiltonian,

$$H = \frac{1}{2m} [ P - eA(R)]^2 + V(R)$$

where $B(r) = \nabla \times A(r) = 0$ inside the sample itself (but not in the hole of the ring). Then the function

$$F(r) = \int_{r_-}^r A(r).dr $$

is well defined independently of the path from $r_-$ to $r$. Let us define,

$$\tilde \Psi(r) = e^{iF(r)} \Psi(r)$$

We find that $\tilde \Psi$ obeys the free equation with $H = P^2/2m + V(R)$ to which we add the boundary condition,

$$\tilde \Psi(r_+) = e^{iF(r_+)} \tilde \Psi(r_-)$$

with $$F(r_+) = \frac{B S}{\hbar/e}$$



In [ ]:

    
from math import pi

%run matplotlib_setup.ipy
from matplotlib import pyplot
import numpy as np
import kwant



In [ ]:

    
lat=kwant.lattice.square()
L,W=30,16
def myshape(R): return ( 
        (R[0]**2 + R[1]**2) > (L-W/2)**2 and                 
        (R[0]**2 + R[1]**2) < (L+W/2)**2)
H=kwant.Builder()
H[lat.shape(myshape,(L,0) )]=4


H[lat.neighbors()]=1

def Aharonov_Bohm(site1,site2,phi): return np.exp(-2j*pi*phi)
    
for hop in H.hoppings():
    if hop[0].tag[0]==1 and hop[0].tag[1]>0 and hop[1].tag[0]==0: 
        H[hop]=Aharonov_Bohm

sym=kwant.TranslationalSymmetry(lat.vec((1,0)))
def lead_shape(R): return abs(R[1]) < W/2 and abs(R[0]) <3
Hlead =kwant.Builder(sym)
Hlead[lat.shape(lead_shape,(0,0) )]=4
Hlead[lat.neighbors()]=1
H.attach_lead(Hlead)
H.attach_lead(Hlead.reversed())
kwant.plot(H);



In [ ]:

    
Hf=H.finalized()
data = []
phis = np.linspace(0,1.,50)
for phi in phis:
    smatrix = kwant.smatrix(Hf, 3.3,args=[phi])
    data.append(smatrix.transmission(1, 0))
pyplot.plot(phis, data,'o');
pyplot.xlabel('$\phi = BS/(h/e)$')
pyplot.ylabel('g in unit of $(2e^2/h)$');
pyplot.title('Aharonov-Effect')

We see that the Aharonov-Bohm effect contains several harmonics $$ g = g_0 + g_1 cos(\phi) + g_2 cos(2\phi) + ...$$

Your turn:

How can we get just one harmonics (as in most experiments)?
Try L = 100 and W= 12, what do you see?
The results should not depend on the position of the gauge transform, can you check that?

Your turn:

Can you make a different sample with an assymetric shape, say an ellipse shape?

Real magnetic field

In real life it is difficult to put the field only in the hole so experimentalists tend to apply a uniform field everywhere. Let us modify the script to just do that. You need now to pick up a phase $\phi$ on each small square. This can be done with the following piece of code - try to modify your code to incorporate this new pieces.

Your turn:



In [ ]:

    
L,W=100,12

def Field(site1,site2,phi):
    x1,y1=site1.pos
    x2,y2=site2.pos
    return -np.exp(-0.5j * phi * (x1 - x2) * (y1 + y2))

H[lat.neighbors()] = Field

Now run it, don't forget to change the x-scale of the plot.

Do you understand why the x - scale is so much smaller?
Do you happen to know what will happen at higher field?



In [ ]:

    
phis = np.linspace(0.,0.0005,50)