Dear Anton,
Thank you for your kind reply, I really appreciate any help given!
From your answer:
The unit of energy everywhere is the same as the one you use when
defining the Hamiltonian.
This does not correspond with what is returned by `kwant.physics.Bands`,
or I am doing some trivial mistake I cannot see. That is because the
numbers do not correspond to what they should be if they were eV versus
nm^{-1}. The following is the simplest code I could possibly imagine:
```python
a = 0.142 #inter-carbon distance in nm
lc = sqrt(3)*a #lattice constant in nm
t = 2.8 #hopping in eV
W = 50 #width in UNSCALED lattice constants, as if sf=1. (multiply by lc
to get it in nm)
sf=1
def bandstructure(sf): #sf = scaling factor
glat = kwant.lattice.honeycomb(sf) #sf*la for lattice constant in nm
A, B = glat.sublattices
# create a lead pointing downwards (easy with pre-defined honeycomb)
def leadshape(pos): #set leads width
x, y = pos
return (0 <= x < W)
sym_lead_vertical = kwant.TranslationalSymmetry(glat.vec((-1*sf,2*sf)))
armchair = kwant.Builder(sym_lead_vertical)
armchair[glat.shape(leadshape, (0, 0))] = 0 #onsite energy is 0
for undoped graphene.
armchair[glat.neighbors()] = -1/sf #hopping. use t/sf for value in eV
bands = kwant.physics.Bands(armchair.finalized())
wavenums = np.linspace(-pi/10 , pi/10, 41)
#To get energy in eV (for any scaling factor) I do:
energies = [bands(k)*(t) for k in wavenums]
# Rescale the wavevectors so that it is measured in nm instead of
lattice constants
wavenums /= sf*lc
# plot dirac dispersion first:
pyplot.plot(wavenums, (3/2)*a*t*wavenums, color="black",
linestyle="dashed")
pyplot.plot(wavenums, -(3/2)*a*t*wavenums, color="black",
linestyle="dashed")
# plot armchair dispersion:
pyplot.plot(wavenums, energies)
pyplot.xlabel(r"Wavevector [nm$^{-1}$]", family = "serif")
pyplot.ylabel(r"Energy [eV]", family = "serif")
pyplot.xlim(-0.4, 0.4)
pyplot.ylim(-0.2, 0.2)
bandstructure(sf)
```
yet the band structure it plots is wrong (see image at the end of the
email). It is not contained within the (correct) Dirac cone, as it
should. If I understand your statement correctly, the line ` energies =
[bands(k)*(t) for k in wavenums]` should give me the energy in eV,
irrespectively of how big the hopping is. (i.e. irrespectively of the
value of `sf`, the scaling factor). In addition, I call the function
with sf=1 so the `sf` by itself should influence nothing.
This code is extremely simple, yet there is obviously something wrong
with it. The problem is that I cannot understand what it is wrong,
precisely because the code is so simple! I simply plot the energy versus
the wavevector, but I do it in units of eV and nm^{-1}...
The dirac cone is correct, as I have compared even with published
papers, but the dispersion for the armchair is incorrect (should be
contained within the dirac cone for such low energies).
I am sorry that my issue is not solved after your answer, but I would
appreciate any input that can make me understand why the armchair
dispersion is not contained within the Dirac cone...
Best,
George
On 10/09/2017 03:51, Anton Akhmerov wrote:
Dear George,
What does not make any assumptions about the units, leaving the
interpretation entirely up to you.
Which is the unit of measurement of length? I always thought that it is simply
`1`! If I write `honeycomb(32.123)` is the unit of measurement of length still
`1` or does it become 32.123, irrespectively of how many nanometers 1 actually
corresponds to? When I get the lattice sites, I know that they are assumed
(i,j) TIMES the lattice constant, so the actual unit of distance is still
(i,j)*32.123*1, and `1` corresponds to whatever units I choose.
The length argument to the honeycomb lattice is the lattice constant.
Creating a lattice does not change the units of length. The site
positions are then calculated according to the lattice definition, so
a site (i, j) belonging square lattice with a lattice constant a would
have coordinates (a*i, a*j).
Now, is the wavenumber represented in units of `1`, or in units of 1/32.123? If
I have a value of `k=3` and my `1` corresponds to 1 nanometer, does this value
of `k` correspond to `3`nm^{-1} or to `3/32.123` nm^{-1} ?
kwant.physics.Bands and kwant.plotter.Bands calculate the spectrum of
a system with a 1D translation invariance as a function of its lattice
wave vector (so, these band structures always have a period of 2*pi).
In other words, they measure momentum in units of inverse period of
the system whose band structure you are calculating. This is different
from kwant.wraparound.plot_2d_bands (introduced in kwant v1.3) that
uses units of inverse length.
What is the input given to the function `bands(k)`, obtained from
`kwant.physics.Bands` ? Does it have to always be within the interval -pi/2 to
pi/2 , meaning that it is always expected in units of 1/lattice const.
irrespectively of which are the units of measurement of the wavenumber?
The input to bands does not have to be within that interval, but it is
measured in the units of the inverse period of the system, indeed.
What is the unit of energy, that is returned by the function `bands()`? Does it
depend on the value that I give to the hopping, or it depends on how I define
`1` ? E.g. when I write `armchair[glat.neighbors()] = -1/sf` is the unit of
energy internally defined to be `1` or `1/sf` ??? It cannot possibly be the
latter, because I could also assign some number to the on-site potential. So it
must be the first, right? But then, how do you answer my question number 4:
Given that I know the length of the lattice constant in nanometers,
and I know the energy of the hopping value in eV, how can I get the
bands in eV versus nm^{-1} ?
The unit of energy everywhere is the same as the one you use when
defining the Hamiltonian.
I hope the above answers help.
Best,
Anton
On Fri, Sep 8, 2017 at 3:12 PM, George Datseris
<george.datse...@ds.mpg.de> wrote:
Hello,
I am having significant trouble understanding which are the units of
measurement of the energy and the wavenumber when using kwant and when using
different lattice constants.
I am trying to reproduce the results of the (probably well-known paper)
"Scalable Tight Binding Model for Graphene", Phys. Rev. Lett. 114, 036601
(2015).
I have read the answers to another question
(https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00069.html)
but I simply cannot understand which is the actual unit of measurement used
internally so I can translate those to nanometers and electronVolts. Even
though I have realized how I can translate internal wavenumbers into
nanometers^{-1}, I cannot understand how and why to use properly
kwants.physics.Bands().
Here is the minimal code that captures my problems:
```python
import kwant
from matplotlib import pyplot
a = 0.142 #inter-carbon distance in nm
lc = sqrt(3)*a #lattice constant in nm
t = 2.8 #hopping in eV
W = 200 #width in UNSCALED lattice constants, as if sf=1. (multiply by lc to
get it in nm)
def bandstructure(sf): #sf = scaling factor
glat = kwant.lattice.honeycomb(sf) #sf*la for lattice constant in nm
A, B = glat.sublattices
# create an armchair lead
def leadshape(pos): #set leads width
x, y = pos
return (0 <= x < W)
sym_lead_vertical = kwant.TranslationalSymmetry(glat.vec((-1*sf,2*sf)))
armchair = kwant.Builder(sym_lead_vertical)
armchair[glat.shape(leadshape, (0, 0))] = 0 #onsite energy is 0 for
Dirac
armchair[glat.neighbors()] = -1/sf #hopping. use t/sf for value in eV
#function that gives the bands (energies) at a given wave vector:
bands = kwant.physics.Bands(armchair.finalized())
#The wavenumbers are measured in units of 1/lattice_const.
#that is why they only need to go from -pi to pi (Bloch):
wavenums = np.linspace(-pi/10 ,pi/10, 41)
#I now want to get the energies, measured in eV. How????
energies = [bands(k/sf)*(t/sf) for k in wavenums] #???? doesn't work no
matter what I change!
pyplot.figure(figsize=(6,8))
# Rescale the wavevectors so that it is measured in nm instead of
lattice constants
wavenums /= sf*lc #this works correctly
# plot dirac dispersion first:
pyplot.plot(wavenums, (3/2)*a*t*wavenums, color="black") #this works
correctly!
# The band structure does not work.
pyplot.plot(wavenums, energies)
pyplot.xlabel(r"Wavevector [nm$^{-1}$]", family = "serif")
pyplot.ylabel(r"Energy [eV]", family = "serif")
pyplot.xlim(-0.4, 0.4)
pyplot.ylim(-0.2, 0.2)
bandstructure(1)
```
From previous answers, I cannot understand what corresponds to the "the
distance units that I have chosen". I have chosen here the distance units
such that `lc -> 1`. When I am creating the lattice with the command
honeycomb(sf), are my distance units still `lc` or are they now `lc*sf` ?
Notice that the above `bandstructure()` should give similar results
independently of scaling factor (see the paper for details on why).
Therefore, the only problem that may remain is the scaling of energy and/or
wavenumber. Let me summarize my questions as clearly as possible, to help
you giving me an easy answer:
Which is the unit of measurement of length? I always thought that it is
simply `1`! If I write `honeycomb(32.123)` is the unit of measurement of
length still `1` or does it become 32.123, irrespectively of how many
nanometers 1 actually corresponds to? When I get the lattice sites, I know
that they are assumed (i,j) TIMES the lattice constant, so the actual unit
of distance is still (i,j)*32.123*1, and `1` corresponds to whatever units
I choose.
Now, is the wavenumber represented in units of `1`, or in units of 1/32.123?
If I have a value of `k=3` and my `1` corresponds to 1 nanometer, does this
value of `k` correspond to `3`nm^{-1} or to `3/32.123` nm^{-1} ?
What is the input given to the function `bands(k)`, obtained from
`kwant.physics.Bands` ? Does it have to always be within the interval -pi/2
to pi/2 , meaning that it is always expected in units of 1/lattice const.
irrespectively of which are the units of measurement of the wavenumber?
What is the unit of energy, that is returned by the function `bands()`? Does
it depend on the value that I give to the hopping, or it depends on how I
define `1` ? E.g. when I write `armchair[glat.neighbors()] = -1/sf` is the
unit of energy internally defined to be `1` or `1/sf` ??? It cannot possibly
be the latter, because I could also assign some number to the on-site
potential. So it must be the first, right? But then, how do you answer my
question number 4:
Given that I know the length of the lattice constant in nanometers, and I
know the energy of the hopping value in eV, how can I get the bands in eV
versus nm^{-1} ?
I would also like to comment that the documentation string of `bands()`
would solve all of these questions by only having 2 more sentences in it.
The sentences that answer questions 2-3-4 would probably be enough. I am
sure that the answers to my questions are almost trivial, but after many
hours of trying to understand what is going on, I am not so confused that I
had to ask. So, sorry in advance for the very easy question!
Best,
George Datseris
MPI For Dynamics & Self-Organization
