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00:00

hello everyone welcome back to my
channel in this video I will talk about
how to calculate the band structure of
silicon crystal and in the last video I
have shown you how to calculate the
tensile state of silicon but that is
much simpler than the calculation of the
band structure because the attends of
state is just a one-dimensional function
so x axis is the energy and y axis is
the dance of state and and information
of the of the reciprocal lattice has

00:31

been integrated out but here in in
reality for for every point in the
reciprocal space you get a discrete
array of of energies for example at at
this L point here you can see from the
band structure here at our point you
have an array of discrete energy levels
and that array is basically the energy
of different bands and if you change

01:00

your if you move your K points in the in
the k space those array of those array
of energy will also change with respect
to the position in a tech space so
basically now the problem is not a
one-dimensional function but it is a
three dimensional function and and
several three different dimensional
function so in this case it is it is not
possible to represent all of the

01:33

information on paper so so you need to
come up with a smarter idea how to
represent your calculation so the common
way to do it is to Ursula is to find out
high symmetry points in the in the
Brillion zone and what I show here is is
the first Brillouin zone for FCC lattice
and you know that silicon crystal is two
FCC lattice shifted by one fourth of the

02:05

crystal of the unit cell so so here
the earned the highest symmetry points
are labeled here with different names
for example the L point is the center of
the hexagon here and the gamma point is
the like center of the Brillion zone and
this X point is the center of the square
here and you point is the is the Center
for this edge here so so on right hand
side this is the this is one reference

02:39

of the pen structure for silicon and it
actually goes from our 2 comma 2 X 2 u 2
comma so it goes so it chooses a path
from L to gamma and from gamma to X that
is the center of the square and then
from X to u there's the center of this
edge and then from you directly to come
a straight Lee ok so if you search
search the literature you can find

03:12

different ways to define the K path and
it's totally fine that you that if you
include this K point here or in you
include the W point here indeed you can
find some literature also using other
other K paths so but in today's video I
will never follow the same path as the
reference here from how to command gamma
to X X to you and u2 comma and what we

03:44

notice here is that the size and the
length of the segments of the of this
plot is different so so I also try to
reproduce that by setting the number of
points to be different so we see that in
hacked from X to u that is this in this
segment we set the points to be smaller
so that in the output it is also
proportional to what we see in the
reference so the basic steps are the

04:16

same you first need to do the case of
consistency calculation and then you
need to pants calculation and to do some
post-processing and then to plot the
pant
however things you since when you want
to put a plot depend you need to specify
the Fermi energy and and in the in the
step status that before it is not unlike
the from energy is not automatically
obtained so there are two ways to obtain

04:46

the Fermi energy either you add a
smearing term in self-consistency
calculation part or you you keep you
keep the self-consistency part the same
but you add another known
self-consistency calculation with a
tensor of k k points as we did last time
for the sense of state so if you follow
the second possibility it is and it's
usually more accurate like the fermi
energy and also the north of consistency

05:17

calculation would help a little bit on
an advanced calculation because you you
already refined the calculation then
intenser okay k point grid
however the left possibility with the
smearing has has one step less so that
you don't have to calculate that's K
point so this will be faster so in this
video I will show show you so what does
it mean by adding smearing or not but

05:49

then we will follow the second
possibility to calculate energy pad okay
so as usual I have already written down
the input files and yeah and then we
need to first source the pair of studio
and we first checked the
self-consistency input file and

06:22

everything is is more as the same it's
just that I add and
kind of kinetic energy for the charge to
be eight times thicker WFC here so to be
more accurate and then and then I
specified a number of pens calculated
here so if you don't specify this line
it will it will be a half of the
electrons in half of the total electrons

06:54

in the system and and the total
electrons calculated are specified by by
this pseudo potential file how do you
find out there are how many electrons
are there you can comment out you can so
if you don't know you can comment out
one line in in the input file by this
exclamation mark you know so so
basically though and this line and this
line are are commented out so they will
not be they will not be run okay so so

07:27

we first see if we do it without without
the specification for the number of pens
and then we take a look in the output
file and in the header of the whole
profile you can find the number of
number of electrons here so there are
eight electrons in the in the system and
the band and it only uses half of the

07:59

electrons as the number of bands so for
so in in the case of insulator of
semiconductor it will be exactly half of
the half of the number of electrons but
in the case of of metal or you a dance
at a smearing term it could be a little
bit more than half but here we want to
calculate eight the reason is that if
you just calculate for it is all of them
are the valence band and there is no

08:32

conduction spent if you if you see from
output file still you can find out that
in the end it only specifies the
is occupied arrival because there is no
information about the conduction band so
it doesn't doesn't know what is the what
is the lowest unoccupied level so we can
add this on communities and set bands to
p8 and then we start we run again
and see in our profile here we first go

09:12

and going to the heater to see that the
number number of electrons are still 8
but the band is successfully increased
to 8 bands okay and if we scroll down
there there is more information here
because now we include the conduction
bands
so it knows what is the highest occupied
level what is also what is the lowest
unoccupied level and the the difference
between those two numbers is more or

09:43

less and band gap the reason why I say
more or less instead the K point grid is
still not very dense here so the
estimation could be a little bit off but
well I guess this is more and more or
less there okay but still you see no no
matter whether you you add at a
specification for the band number you
don't get a Fermi energy inside this
file to get the Fermi energy inside this

10:16

file you need a smearing term here the
reason is that if you don't have a
smearing term you are basically
calculate adds at zero temperature at
zero temperature you you know there's a
there's a band gap the Fermi energy
could be anything's at a band gap so to
solve that you need you need some
smearing under Fermi honor Fermi level
and then if you do the calculation again
and it opened on fire and you see that

10:56

it outputs the Fermi energy however here
you see that it doesn't output the
highest occupied level or lowest
unoccupied level the reason is that you
add a smearing so so not everything
below the Fermi energy is occupied or
everything under laughter from an energy
is occupied so this is no longer true
because of the smearing okay so this is
one way that you can get the Fermi

11:27

energy but but in in this video I will I
will just not use that if you if you do
it like that you can skip the SCF
calculation and and then directly go to
the bands and bands P P and plot band
calculation you know
so we commented out and do the
calculation do the self-consistency
calculation again so it's finished and

11:59

we check again yeah so now in it is back
the highest occupied and lowest
unoccupied level and then we want to do
a know Suffolk insist consistency
calculation to calculate the Fermi level
and yeah so this is quite familiar to
you and we just do this unless yeah one
do this calculation
and in the output file you can always

12:44

find the Fermi energy so yeah so the
third step is to run bands calculation
so still the PW tax program and the only
difference is that you change the
calculation type to be bands and
everything else you keep the same except
that you change the K points to the
through capers and points that you
manually that we manually define and
this is the K points the K path that you
want and want to define here okay I need

13:18

to make this so so basically it is the
first high point is l and second is
gamma and xu gamma and it specify the
path from L to gamma from gamma to X X
to u u2 comma and and this number is is
the total number of K points which is
just a number of lines here and then for

13:54

each line the first three numbers are
the K X KY and KZ and the fourth line of
the fourth number is the weight and the
weight here means that missed number of
points starting from that starting from
that path so basically it means that
from L to gamma there are 20 points from
gamma to X there are 30 points from X to
you
there are 10 points from you to gamma
there are 30 points and this trend here

14:25

has no meaning here oh you can set it to
whatever you like okay and then the
question is how do you like you know you
know that you want to specify a path
from our 2 comma 2 from gamma to X and
from X to you from u2 comma but now the
question is that how do you get these
numbers you know so one very useful
software is X Kristin
and you can open the input file or opera
fire whatever you want

14:56

for example the odd profile for the SCF
calculation here for example and it's
just a single point calculation just to
get the structure of of silicon crystal
and then you choose in the tools there's
ok pass a selection yeah and then you
see the first boolean zone so maybe I

15:31

need to move it a little bit so that I
can see yeah I can show you both at the
same time ok so we first so we rotate it
to the to the same way as soon as this
one and then we want to get from from L
to gamma and L is the center of this
hexagons of this one and then to gamma
gamma is the center of the boolean zone

16:00

and from gamma to 2x from gamma to X
exit the center of the square from X to
u use the center of this edge from u2
comma again yeah and then you get
reciprocal coordinates here and an in
principle you could just copy all of the
coordinates from here from here to here

16:34

and of course you see that it's it's
it's different the reason is that there
there is a very high symmetry of the
first Brillouin zone so even if you get
different numbers as here but but they
are more or less symmetric so you will
get the same band structure out you know
ok so I will just close this one
and and this is the k path that we want

17:04

to specify and let's just do the
calculation so to do the calculation you
also have to also use the Peter brutal
x3 no sorry there's not three but you
okay and now it's finished and you know
and now the Barons are calculated but
they are not in a way that is readable

18:03

so we need to post processed the result
and then you need to use another another
or sub program which is called bent dot
X to do post process the result and this
is the input file for bent on X and you
need to specify the name of the of the
output data so this output file will
contain the data of the pants and
afterwards you can you can use gene you
plot you can use Python you can you know

18:36

or you can use as I will show later the
plot bent to a pen etat X in quantum
espresso to plot the depend okay and
then we just run the bandit or X
remember that you need your changes
depend two x plus three
okay I'm in mistake it is not caught
band directs but bends and directs

19:35

okay so now it's finished and you see
this is the fire that contains the data
of the Bands
you know and it is a structured but
still we cannot directly put plotted
what I what I want to use is not this
one is the P is the plot bandit order
and Daleks so for the broadband don't
acts you can do the interact
interactively you can just type in plot

20:08

bandit or X and then it will prompt you
for different inputs what I do here is
that I constructed a input file for the
broadband drugs and and for the input
file the first line is the is the fire
that contains the data of the bands
which is the output of the of the last
step and then the second line specifies
the the window of energy so this is the
minimum energy this is a maximum energy

20:39

plotted so how do you get these numbers
you can go to whatever output file for
example an output file for the SCF or a
tall profile of that unless CF or even
the pants I will just show you the
profile of the pants and then scroll
down and take a look at at the energy of
different K points so basically this
means there are one two three and there
are eight energy bands and at this K

21:10

point and energy is from minus 4a action
bar to 13 electron volt and you may
scroll down a little bit and you will
see that okay it's more as from minus
six to 215 in extreme world so here we
say minus 7 to 16 in action were two to
take into account of everything and this
is the name of the output file and these
are also the name of the output file for
this one you need a special and special
software to be open in whooping to and

21:41

this way you can directly open in open
to
and so basically those are the plots of
the pans and this is the Fermi energy
you have to specify and in our case we
can just copy it from from here from the
output you can either copy it from the
output of the nest and SCF or if you use
smearing in the ass here you can also

22:11

copy it from there okay so this is and
this one is its spacing of the while
APIs you know of the of the labels on
the energy axis and and this number here
we usually set it to the Fermi energy
this is the energy that you you were
ship it to be to be zero energy so the Z
this defines the Fermi energy as a

22:43

street and zero energy in the plot so
then we can do the plot and usually I
don't use MPI run here plot and toe tags
and the name is floor okay yeah

23:25

in this case you shouldn't use - I am P
because it doesn't accept that you need
to use this this lesser sign okay and
then this is the ultra as I adore pen
store PS open this PS file maybe you
want to rotate it and this is actually
the energy band so so this one means the

23:56

spacing of the energy axis so this is a
nine eight seven if you if you change it
to two for example maybe we should cross
this one first
we have 22 you are see it on a spacing
changes to 2 and 0 is the Fermi energy

24:25

okay so so if you see from from here to
here
we successfully reproduced the pen
structure calculation in the reference
and you see this is a this is the Fermi
energy and there is a band gap here and
and the first line is is what we
specified an arrow gamma X U and gamma
and the shape and everything is the same
mass and as the reference okay

25:07

so into this video I have shown you how
to do Ben structure calculation of of
silicon crystal and and I've shown you
how to obtain the Fermi energy in in two
different ways and how to how to obtain
K paths if you know that we want to plot
from our 2 gamma 2 ax 2 u 2 gamma how to
convert the name here to the actual
coordinates by x Kristin okay so thank

25:39

you for watching and I hope to see you
next time

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