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Subtitles generated by robot

00:07

hi welcome
we're on to one of the final sections of
the course BCS theory this is non
aníbal before we get going I would like
to impart some wisdom as to how to
survive physics during the lockdown and
I have two words of advice chocolate
good chocolate preferably and coffee
excellent
okay so BCS theory of superconductivity

00:42

recall from the beginning of the course
superconductivity was discovered in in
in 1911 by chemically notice and the
full microscopic theory of
superconductivity wasn't developed until
the late 1950s 1957 by bardeen cooper
and schrieffer and maybe I'll even write
that out because those are important
names bcs bardeen cooper and schrieffer
so Bardeen was the senior professor
Cooper was the post talkin and and

01:14

schrieffer was the graduate student is
undoubtedly one of the great
intellectual achievements of the last
century Bardeen had already become quite
famous because of the invention of the
transistor Bell Laboratories in nineteen
nineteen forty-seven he was actually
pushed out of Bell Laboratories by his
manager and also sometimes collaborator
William Shockley
were more or less prevented him from
ever working on transistor physics at
Bell Labs ever again so he moved from

01:46

Bell Labs to Urbana and in 1951 began
his his work trying to understand it a
superconductivity
now we sort of we we already sort of
know what the answer is gonna kind of
look like we know that a super
conductivity is gonna have something to
do with pairing of electrons into into
bosons take two fermions and you stick
them together to form a a pair that pair
will be a boson and this idea had been

02:17

floated before the bcs theory of
superconductivity but it wasn't very
popular and matter-of-fact people didn't
like it at all some people were
extremely opposed to
the idea of electron pairing including
the great live landau and there was good
reason why people opposed this idea of
super conductivity due to both ionic
pairs of electrons
first of all electrons repel quoting Lev
Landau you can't repeal Coulomb's law
and secondly even if you could get them

02:51

to two pair
so pairing two electrons equal one boson
that this is actually to make it really
a boson to be really a boson really a
boson we would need to have the binding
radius of the boson so the radius of the
boson work which we call it binding
radius a maken s community a binding

03:23

radius radius we would need that to be
less than the distance between bosons
less than distance between otherwise if
there if the binding of the of the of
the two particles into a boson is is
greater than the distance between the
bosons then it's not like you have
bosons it's like you have individual
electrons so the problem is to get the
binding radius this small you would need

03:56

a binding energy binding energy to be on
the order of a rid berg and on the order
of affirming Fermi energy binding energy
would have to be on the order on the
order at least on the Fermi energy which
is a huge number ten thousand Kelvin or
light or something like that and
certainly even if you can generate some
sort of attraction mechanism between
between electrons something to try to
glue the electrons together it's going
to be very hard to imagine gluing coming
up with an attraction mechanism that

04:28

will come up with glue that's strong so
for this reason no one really liked the
idea of binding electrons together in
into bosons so in in our study he
the way in our study we're going to
first discuss what the attraction
mechanism comes from and how it forms
pairs and then we'll go to the full
full-blown bcs theory in the in the next
lecture so the attraction mechanism

04:59

mechanism well actually there are
different kinds of superconductors this
is what we call the conventional
superconductors sometimes you know as
the bcs superconductor and in
conventional superconductors
conventional the attraction mechanism
conventional mechanism is an attraction
due to phonons or lattice vibrations
phonons and so this is some sort of
interaction between the electrons and

05:31

the and the positive ions left behind
that vibrate to create the the phonons
and this was even in the night early
1950s a number of people had started
thinking about whether phonons could
somehow be important for
superconductivity including Fineman and
and frolic and another number of other
great scientists in fact in the 1950s a
lot of the greatest minds in in the
world a lot of great physics minds in
the world were thinking about the
problem of superconductivity landau

06:03

Fineman was sort of a period in history
when people had sort of lost faith in
high-energy physics and they thought
maybe there wasn't anything interesting
left to be done
in high-energy physics so let's see if
we can attack all some of these problems
that have been sticking around for a
long time like super conductivity okay
so in the conventional so-called bcs
superconductors
the mechanism that is going to pair bind
the electrons together is going to be
phonons but there are unconventional

06:35

mechanisms which can come from other
sources so there are many problems not
as many but there are many modern
superconductors which with
unconventional pairing mechanism
this includes the high TC peroxide
superconductors the more recently
discovered nicked ID superconductors
helium-3 as a superfluid little when you
that the two helium-3 atoms can actually
pair together to to form a boson as well

07:06

and these are and in the case of healing
with me there are no phonons it's just
the helium-3 atoms there's no background
of lattice whatsoever so there obviously
there are other mechanisms that can can
do the job as well and many of these
unconventional mechanisms are more
controversial but more and still have a
lot of people researching them even in
the current current era so in the case
of high TC superconductors the peroxide
superconductors there's probably on the

07:38

order of 1 million research papers
that's not an exaggeration trying to
figure out something about the mechanism
of high TC superconductors and still
there's no agreement as to what the
mechanism is but it is believed it is
not phonons so as I was saying even in
the early 1950s people kind of feeling
that phonons were going to be important
and there was a real important clue to
this to see that the lattice vibrations

08:10

or the phonons what is going to be so
let's see in the case of conventional
superconductors the important clue is
known as the isotope effect dope effect
which so remember isotopes of atoms are
different versions of the same atom
where you change the number of neutrons
you keep the same number of protons you
keep the same number of electrons you
just change the number of neutrons the
chemistry of the atom stays completely
same the electronic properties of the
atom stays completely the same the only
thing that's different is the mass of

08:44

the nucleus and what was found is that
the critical temperature for super
conductivity depends on the mass of the
nucleus where alpha is approximately
equal to 0.5 it varies a little bit
depending on the
on details but there is a distinct
dependence of the critical temperature
on the mass of the nucleus and the only
thing that we could do that is if the
actual vibration of the nucleus

09:16

you know the motion of the nucleus
moving back and forth is somehow
involved in in the mechanism if the
nucleus was completely stationary then
there would be no place in top effect
whatsoever so by adding some neutrons to
the nucleus you see a change in the
critical temperature which tells you
that the vibrational frequency of the
which depends on the mass the rational
frequency of the of the nucleus is
somehow getting into the game and that

09:47

tells us that the phonons are somehow
important so so what we're going to find
that that this phonon mechanism is going
to be strong enough to actually induce
this BCS pairing and give us super
conductivity and there's a couple of
important lessons to be learned about
this the first important lesson was that
land that was actually wrong that you I
mean there were a number of cases we

10:17

already found this case of you know of
vortices he didn't like the idea of
vortices so he was wrong about that and
he was also wrong about not being able
to repeal Coulomb's law it's it's true
that you can't repeal Coulomb's law but
you can screen it by Thomas from
rescreening which we discussed in in
previous lectures to make it a lot less
strong over long distances and once
you've screamed at the Coulomb
interaction there are other mechanisms

10:49

which although they're weak like the
coupling from of electrons to phonons is
sufficient to bind the electrons
together now there's still this question
is the binding strength going to be
large enough we're not going to find
enormous binding strength on the order
when I've written up here we're not
going to find binding strengths on the
order of
the Fermi energy tip of getting on these
four typical metals we won't but we will
find a binding potential nonetheless
okay

11:19

now the actual mechanism that gives an
attraction between between electrons via
phonons is important enough that I'm
going to derive it in two different ways
the first way so this is phonon mediated
attraction this is what I'm going to
drive next phonon mediated attraction
electron electron attraction so I'm
going to drive this in two ways the

11:51

first way is to use second quantization
which we're all familiar with and we're
going to try to derive an effective
low-energy Hamiltonian which shows an
effective interaction between electrons
and then we're going to read arrive it
in a sort of more semi classical way
which will maybe give a little bit more
intuition so okay so let's do it we what
we usually do here we'll write our
Hamiltonian is two pieces H 0 and H 1
and H 0 is going to be the bare electron

12:23

Hamiltonian and their phonon Hamiltonian
so all right is there electrons and
their phonons not interacting with each
other just the Hamiltonian for them as
if they were alone and H 1 is going to
be the interactions between interactions
between electrons and phonons okay so
it's let's actually write out what we
mean by these so H naught I'm going to
take a really simplified form of of H

12:55

naught here we'll write it as sum over K
epsilon K C dagger KCK supressing spin
indices here which should be there and
then for phonons we'll take a really
simple form of phonon Hamiltonian it
will be sum over Q a dagger q AQ just
counts the number of photons are bosons
at wave vector Q and gives each of them
an energy
H bar Omega so these are n stein phone

13:26

hunts meaning all phones have the same
frequency it's not crucial that we do
this we could have phones with different
frequencies just makes the calculation a
little bit more complicated but not too
much more complicated but we might as
well keep it simple as possible
I've also simplified this a bit by not
keeping track of the polarization of
phonons phones have multiple different
polarizations and I've dropped those
indices which really should be there as
well now one thing you'll notice about

13:57

this this Hamiltonian we've written is
that there's no Coulomb interaction
between the electrons I'm gonna ignore
that so you can write that down no
Coulomb electric no electron-electron
interaction and the reason for this is
because we're assuming that the Coulomb
interaction between electrons is going
to be screened with Thomas from
rescreening over a fairly short length
scale so we can we can ignore it anyway
what we're interested in is actually the

14:28

effect of the electron phonon coupling
okay so I said you write out the
electron foreign coupling that were
interested in studying and there's
various different ways we could write
this let me write it in a fairly simple
form sum over Q M sub Q and this is a
matrix element that's why we use M Rho
hat minus Q a dagger Q plus relation
conjugate and Rho hat here is the
electron density operator we've seen him

14:58

before and M is is this coupling as a
function of Q and just again for
simplicity we'll just take the coupling
to be independent of Q so all Q's have
the same coupling between the electron
density and the creation of a of a
phonon so this we can rewrite since we
know what the electron density operator
is we can rewrite this as sum of Q and K
we're going to take m to be a constant

15:29

and then we'll have C dagger K minus Q C
kay a dagger q+ formation conjugate and
the sort of Fineman diagram way to to
draw this is that you have a electron
coming in with wave vector K you destroy
that via this is C CK and you create a
electron with K minus Q and in the
process you also create a phonon with

16:00

wave vector Q so that overall momentum
is is conserved okay now what we'd like
to do is we'd like to figure out what
the effect is of this coupling between
the electrons and the phonons with the
assumption that the coupling m is is
weak and the technique for doing this is
or at least one technique for doing this
is kannada is known as canonical
transformation transform and the idea

16:32

here is to transform your Hamiltonian in
the following way we'll make a script H
the transform Hamiltonian either that -
s H e to the S and the idea is that once
we make this transformation this
transform Hamiltonian we will have
decoupled the electrons from the phonons
but as at at the price of introducing a
direct electron-electron interaction
which is what we're actually interested

17:03

in so we are going to find the effect of
the of the phonons by seeing once we try
to decouple the electrons from the
phonons how did how do the what is the
effective electron-electron interactions
left over if you're a field theorist and
you like to use path integrals what
we're doing here is integrating out the
phonon so you have some path integral
that includes electrons and phonons and
you do an integral that gets rid of all
the phonons
and as a price you introduce a

17:35

interaction a direct interaction between
the
electrons and that's the interaction
we're going to be interested in which we
believe is going to be attractive okay
so um how do we deal with this well
order by order and s we can write this
out using the famous Campbell Baker how
I think this is Campbell Baker house
door isn't this plus one-half H as
successive commutator is I'm not drawing
this very well successive commutator

18:06

Zeng comma s comma s plus dot dot dot
okay
and the idea is then to solve this order
by order in this coupling constant M
which we know is small by choosing an S
that removes the interaction of the
electrons from the photons okay so how
do we do that well let's assume that
both that s is going to be small why is

18:39

s going to be small well our idea is to
choose an s such that it be decoupled
electrons from the photons if M is small
then you can choose s to be 0 because if
M if M is actually 0 then you can choose
s to be 0 because there are 80 couples
there's no interaction between them if
you make em nonzero then s has to be
only a little non zero to decouple them
so when M is small s is also small so
let's rewrite this series keeping track
of of the terms in in order of their

19:10

smallness so something is small if it
contains an M or if it contains an S
okay so the only term that has no MS or
s is just H 0 which comes from the
regular H here ok then it will have two
terms which is order 1 in small and that
will be H 1 that has an M H 1 s and M
and then it will also have h not
commuted with s ok so H 1 comes from

19:41

this term and H nought commuted with S
comes from from this term it's not
renewing this and then it next order so
then so these terms here are
order one in small or one or order small
okay and then the next order term of our
order - and small and we can have order
- and small by looking at the h1
commuted with s term that's when H 1 is
sitting here remember H is made up of

20:11

buh a CR on H 1 H is out H equals H 0
plus h 1 um so when H 1 is in here we
have an h1 community that's that's order
2 and small and then we also have plus
1/2 H 0 comma s comma s and that's also
ordered to and small ok so these terms
here are or you're a small squared
squared ok now our scheme here is to try
to get rid of things order by order and

20:43

small so we want to choose an S choose
an S to get rid of the order small terms
so that these things are 0 these things
here are 0 ok and that will remove the
interaction at leading order and s then
we remove H 1 so to do that we are
choosing H 1 plus h 0 comma s equals 0
and then these terms here these leading

21:14

order order 1 in small or too small to
the first terms all vanish ok and then
we plug the same result in here same
result into the order small squared and
you'll notice that both these terms are
of the same form and if I wrote it
correctly this is then going to be yes
so this term will exactly have cancel

21:46

this term and we'll be left with H
equals just H not the decoupled term
plus 1/2 H 1 comma s and this is H 1
common s within each one page 1 comma s
okay so each one come s so this term
this H 1 comma s term is going to be the
effective interaction that's left over

22:17

once we did once we so this is H int its
left over once we decouple the electrons
from the phonons at first order and
what's left over here is order small
cubed which we throw away entirely okay
okay good so how do we figure out what s
is supposed to be that will solve this
equation here well the way to do this is
to take the matrix element of of this

22:49

equation between two arbitrary states
and in home so we can write n h1 m
equals I guess we'll move the other one
to the other side and H naught s
like this and I can well we know H
naught applied to things just give me
the energy so this will be minus y n
minus e M times n s m and that means I

23:19

can write the matrix elements of s and s
M equals and H naught divided by VM
minus en so now I've defined what the
operator s needs to be okay and then
once we know what this what this s thing
is we can figure out we want to ask what
does this H interaction look like
between two states in our system maybe

23:53

even states that don't have any phones
at all so let's consider States States
and B which have no phones no phone on
this and I want to figure out what is
the effective interaction between them
so a H int
be so remind me what when h int is h int
is is 1/2 H 1 commutative s so this

24:23

thing is going to be 1/2 a h1 commuters
s B and I can well I know what I see
here is I know s here is we wrote it
right up here so I can then write out
the interaction between a and B all
right it is 1/2
I can I can insert a complete set so
that I have a h1 si si s B minus ay s c

25:04

c h1 being is writing out the commutator
and then I'm going to use my expression
here to plug in for this matrix element
of s and this gives me 1/2 sum over see
a h1c c h1 b and then the energy
denominator is EB minus EC and the other
way around it's a minus EC okay so this

25:38

is the with a and B have no phonons in
them this is the effective interaction
between a and B so so what's actually
going on here so remember that h1 is the
term that that scatters an electron and
creates a phonon so so if we start in a
state B over here let's starting to
state B over here with no photons this
term it scatters to a state C with a

26:11

phonon so this has 1 phonon and then
scatters back to a term a with no phone
on this and in the process there's an
energy denominator associated with the
energy with the energy see so in fact we
can be a little bit more specific about
this we can even draw a diagram so let's
let B equal so we want to have say two

26:45

electrons added to the Fermi sea want it
to have momentum K and when a momentum P
and then C will using going all the way
back up to our expression for H one so
it scatters an electron from some K to
some K minus Q and creates a phonon in
the process so C can be of the form C

27:14

dagger K minus Q and then it has the
same C dagger P and an a dagger q on the
Fermi sea and then it's going to scatter
back get rid of the the phonon back to
state a which will be C dagger K minus Q
C dagger P plus Q on the Fermi sea and
let me draw a diagram of this so it has
K coming in and it has P coming in and
they exchange a full path say this is P

27:48

coming in to two electrons one with wave
vector K one will a vector P they
exchange a phonon with wave vector Q
this guy comes out with K minus Q this
one comes out with P plus Q P plus Q
okay now the key thing to realize here
is that if the phonon energy that's EC
here well EC is actually the energy of

28:20

this whole thing here but if the energy
of the phonon is larger than the energy
of the electronic excitations then then
this whole expression here is going to
be negative sorry I have a
that should be a plus so both of these
terms will be will be will be negative
and that corresponds to an attractive
interaction so let me write it out in a
little bit more detail trying to maybe

28:51

make it a little smaller so you can
still see what's going on here so H
effective H int maybe it's effective
will look like 1/2 sum over P KQ c
dagger of P plus Q CP C dagger K minus q
ck this is a matrix element squared and
then we'll have energy denominators 1

29:23

over e k minus e K minus Q minus H bar
Omega o Geer run out of space well put
here plus 1 over e P plus Q minus e P
minus H bar Omega okay and again if if
the energy of excitation of the
electrons as these energies here these
energy differences is smaller than h-bar

29:55

Omega then this whole thing is negative
so H int I call it h int + int is less
than 0 meaning we have an attractive
interactions in particular if ek minus
ek minus Q is less than h-bar Omega then
we have an attractive interaction in a
matter of fact we can we can simplify it
even more let's assume that it's much
much greater ek where as the ek minus Q

30:27

is much much less than h-bar Omega if
that's true then we can throw away these
energies epsilon in the denominator and
only keep the h-bar Omega in which case
what we have is H int equals minus M
Squared
over h-bar Omega some over some over Q
of Rho Q Rho minus Q and this

30:58

corresponds to a delta function
interaction Delta function interaction
between between electrons and is
attractive okay so the fact that we're
getting a delta function interaction is
is is a feature of the fact that we we
took em to be independent of of Q and we
took Omega to be independent of Q as
well that's those don't have to be have
to be true but the fact that we're

31:28

getting attractive interaction is coming
from the fact that the energy of the of
the phonon is greater than the energy of
the excitation of the electrons so so
this this calculation from second
quantization it seems print potentially
a little mysterious because it you know
involves creating these virtual phonons
so you create a phonon then you absorb
the phonon again and you have an
effective interaction between between

31:59

electrons that goes via this virtual
phonon but it's it's something that's
hard to develop intuition for so I'm
going to do sort of a more classical
derivation of the same effect so this
classical derivation is going to rely on
the kind of screening calculation we did
when we discuss plasmas and RP a in
previous lecture this particular
calculation is going to someone
overestimate the strength of the
attraction but it will give you some
idea of how this works

32:29

so let's recall the response of a
physical system it has the charge
density we'll call it Chi naught times
gamma phi let's write it Delta Rho here
it's charge density I guess when we
wrote down Chi note before we were
looking at density rather than number
density rather charge density so let me
an extra factor of
of a floating around here but kind not
here for if we're looking at responses
of electrons kind not at the high

33:02

frequency was of the form Q squared
average density in this e squared Omega
squared M which we can rewrite as Q
squared epsilon naught Omega Omega
plasma for electrons squared divided by
Omega squared we're call we derived the
plasma energies the plasma energy plasma

33:35

frequency is square root of n bar e
squared over gamma epsilon nought and
that's for electrons it's a very high
energy scales on the order of an
electron volt it's very very very high
frequency for this type of response now
what you can do is we can make a really
crude model of let so this this model

34:07

here was really just the electrolyte we
derive this actually in three different
ways but one way the simplest way was
just to think about the motion of a
charged fluid so you're making the
charge fluid oscillate back and forth it
builds up a charge and then you know
swings back and forth due to the
build-up of the charge and that's what
plasma is now you can do the same thing
with the back with the charge of the
background ions background ions ions

34:37

also move has a fluid so this is a
really really crude approximation where
instead of thinking about our background
ions as being you know stuck on a
lattice we're going to think of them as
being a free fluid of of ions so you get
the same result but the mass of the
electron becomes the mass of nucleus
which of course is huge and
the the charge changes sign it's not not
so so important

35:09

so we'll get KY not for the ions is
going to be Q squared epsilon naught
Omega ion a plasma and for the ions
squared over Omega squared where Omega
ion plasma equals square root of n bar Z
squared over the mass of the nucleus
times e naught well the mass of the

35:41

nucleus is is huge compared to the mass
of the electron first of all remember
the proton has a mass of 1,800 times
that of an electron and furthermore the
most nuclei have a lot of protons in
them they have you know even for a small
atom like like aluminum you have you
know what's that topic number of
aluminum 20 something whatever it so
it's atomic weight is 30 40 whatever
nucleons and so it's 30 or 40 times

36:11

1,800 factor in the denominator and this
this ion frequency of the ion
oscillation frequency just treating it
as a fluid comes out to be on the order
of the phonon Debye frequency both of
these things are describing the
characteristic frequency scale on which
these nuclei respond and this is much
much lower so the Debye frequency is on

36:42

the order of say 100 Kelvin for typical
materials it could be 200 Kelvin or
something like that so it's a much much
lower frequency as compared to the
plasma frequency for electrons which is
up on the ten thousand or 40,000 Kelvin
scale okay now at these very low
frequencies at these frequencies
the electrons I mean this looks like
it's completely static for an electron
electrons move really really fast so the
response of electrons is just the static

37:20

response that we calculate the static
compressibility okay so this is the
limit we're gonna be working in where
the where the ions are you know
oscillating at the plasma frequency but
the electrons are screaming as if the
the system is is is static so it's it's
a low frequency ET scale for the
electrons but a high frequency scale for
the for the ions okay so now we're going

37:51

to do the same kind of thing that we did
in in the previous lecture we are going
to see what the the interaction between
the two the two subsystems are by
writing some total potential is going to
be an externally applied potential plus
the induced potential and okay so this
will be the external potential plus the
Coulomb interaction times the density

38:21

you build up of the electrons plus the
density you build up of the of the ions
okay
now exactly as we did before will then
write the density of electrons as the
electrons response okay and now I had
put the alright well let me put this
upstairs zero downstairs zero means
we're treating non-interacting electrons
we're not worrying about their

38:50

interaction with each other this is
exactly what we did in our PA they well
the electrons respond as if they were
not interacting but they are responding
to the total potential and part of the
total and potential is whatever
potential the other electrons create and
then the ion density is going to be the
ion response
time's the total potential and I guess
yeah okay right I brought it correctly

39:24

here KY ion superscript zero okay so
then we just want to take this equation
and solve it for Phi total in terms of
Phi external it's easy enough so Phi
total equals Phi external divided by one
minus MV twiddle of Q kind not electrons
plus Chi not ions in-kind out of ions is
actually a function of Q and Omega kind
of electron we just took it to be a

39:55

number just this compressibility because
it's static and similarly the effective
interaction between any two charged
objects is going to be screen the same
way so this will be V of Q divided by
well okay so for that the
electron-electron interaction is V
twiddle of Q between two electrons and
it gets screened exactly the same way
beat will of Q kind of E Plus Chi not

40:28

ion motion of Q Omega and we will now
plug in the expressions we just derived
for both of these quantities and we'll
get if this is V twiddle of Q is
screened by 1 plus K times Fermi squared
over Q squared that that comes from this
term only and then the ion term is minus
Omega plasma ion squared divided by

41:01

Omega squared okay so there are a couple
things to notice about this expression
for the effective interaction between
electrons first of all if you throw away
the plasmas altogether just drop this
last term you just get Thomas Fermi
screaming of interaction between
electrons so at wave vectors that are
longer than this humming Thomas from
your weight screening length the the
interaction gets
it's damped ok this will give you the

41:32

Yukawa form that we derived in a
previous lecture of interaction between
charges in your in your system if you
drop this term with the plasmons a
second thing to notice is that if you go
to frequencies which are much much
higher than the Debye frequency of the
phonons then this term with the ions
will vanish altogether some you know
maybe I'll write that down maybe I'll
write all these points down one with no
phones no phones we get Thomas for

42:05

phones phones phones we get Thomas for
me for me and two at Omega much greater
than it'll make it to by then we have
the phonons don't matter phonons don't
matter because this whole term will get
will drop to zero as well three but if

42:40

you take omega equals zero then you get
perfect screening why is this okay let's
look at this for a second if you take
omega equals zero then this term down
here in the denominator blows up and the
effective interaction drops completely
to zero so this small Omega going to
zero the denominator blows up the entire
interaction goes to zero and the reason
for this is because if you go to zero
frequency then my my eye on the fluid
can come in and perfectly screen any

43:13

charges I put in the system so I put an
electron at some position in the system
the ion fluid can come in and completely
neutralize it so that they so that the
interaction is completely destroyed but
the last part last point for is that for
small Omega you'll notice that the
attractive the interaction is attractive
attractive here
right notice that you that this term
will dominate the denominator and it

43:44

comes with a minus sign okay so the
picture that we should have in our head
is that the an electron a charge
polarizes the lattice polarizes lattice
and then moves away from that region the
lattice responds but very slowly once
the lattice responds the this becomes a
region which is now net positive I mean
that it started to move because it was
trying to scream the presence of an

44:14

electron but by the time the the lattice
has moved there to build up a positive
charge the electron is gone but another
electron can be attracted to that region
so electron polarized as the lattice
leaves the region leaves region with
plus charge behind another electron
attracted to that region
okay now this screening picture that we

44:53

have of the attractive interaction
between electrons by a perfect phonon
fluid is actually it was shown somewhat
later that it predicts an attraction
which is actually much too strong which
is a little little surprising but it
gives you an idea of how the how the
mechanism works a cartoon picture that
you often hear talked about is I mean
I'll give the words because you hear
people say it all the time is the idea

45:26

of a matress model and I don't think
this is actually a good model but you
imagine two objects sitting on a
mattress so like this okay here imagine
putting two bowling balls or two people
on a mattress like this that so this is
two objects which polarize the
background they distort the background
and then these two objects will actually
be attracted towards each other because
of the distortion of the background I

45:58

don't think this is quite a good model
because this is not dynamical and it's
very essential that the that the model
be dynamical that people still talk
about this is Nationals model is its
distorting a background and inducing an
interaction because of it okay maybe
it's it's good to stop this lecture here
and we'll start the next lecture with
the the issue of balance States and the
Cooper problem
all right welcome back I stopped my

46:32

coffee so this lecture we're going to
discuss bound States and what's known as
the Cooper problem which described
binding of electrons the in the previous
lecture we explained how phonons can in
principle create a an attraction between
electrons in the first calculation we
did we saw that the attraction was weak
its order of the coupling between
electrons and phonon squared but at
least there's an attraction that can be

47:04

generated and the first question we want
to ask is an important question is if we
have an attractive interaction if we
have we have an attractive potential
attractive interaction maybe with
instead of thinking about two electrons
let's think about just one electron in a
well so you know you've got going to
from if I'm talking about two particles

47:34

by going to relative coordinates I can
think about it as a single particle and
well instead of two particles attracting
each other so we have an attractive
potential in D dimensions and potential
is weak and potential is weak is weak
is there a bound state
okay simple question about quantum

48:10

mechanics if you haven't seen this
before the answer is to this we'll
derive it is in D less than two the
answer is always yes no matter how weak
the potential is you can find a bound
state for any attractive potential in D
less than two in D greater than two it's
no or these not for very weak
interactions for strong enough
interaction of course you can have a
bound state like a hydrogen atom but for

48:42

an arbitrarily weak interaction that you
don't generally get a bound state D
equals two is tricky tricky but the
answer is is usually yes so we're not
gonna treat the D equals 2 case because
that one's complicated but we'll do all
other dimensions so it's um proving this
in general so take a case of general
dimensions although I'll draw everything
in one dimension because I can't draw
higher dimensions so we're going to

49:09

imagine a general potential potential
which with the constraint that it goes
to 0 which is 0 outside of some radius
outside of radius R
okay and the magnitude of the potential
magnitude of the potential is is less
than V zero inside inside our inside the

49:45

radius R with visa or small okay so we
can use any this proof more or less
works for any shape potential but you'll
get the general idea of how it works if
we just take a box potential particle in
a box potential it's good enough not
infinitely hard at all so we find out
hard wall so let's let's draw the draw
the picture here in one dimension so
here's that so this is energy on this

50:22

axis this is X on this axis here's could
be radius I guess here's a position zero
and then I'll give a box potential of
draw like this so zero is exactly zero
energy outside of some radius exactly
zero energy outside of some radius and
then within that radius it has a depth
so this is radius R here and it has a

50:57

depth V naught okay so that's the the
example potential we're going to we're
going to use now if we want to ask if
there is a balance state so a bound
state my own state means means that
energy is less than zero and if you
imagine a state with energy down here
less than zero that just by using wkb

51:31

approximation the wave function must
decay exponentially as when you're
outside of this this radius so you go
you're so tunneling once you're below
the energy of the potential here once
the energy is below that the energy of
the potential the wave function must
decay exponentially as you as you
penetrate into this into this what what
is acting like a a wall a confinement

52:02

potential here and so the wave function
must must have can't like this so that
maybe that's even draw with the what the
wave function kind of looks like it
looked like something like this it's
gonna decay exponentially okay good so
outside of the potential it will look it
will look like this inside of the
potential it will have some you know
peak maybe like this like edge on here
and we must also must have I mean I

52:36

guess if the we should say one more
thing that if energy is very small so
it's very weakly bound small weakly
bound then this length scale is long
very long so this length scale of a
binding can get very very long if the
energy is bound just just below zero so
I'm thinking about very weakly bound
Balan States so we also have to have

53:06

must have normalization normalization
the wave function so integral size
squared D D of R has to equal one and
that implies that the magnitude of the
wave function if if the length scale the
wave function is some number oops
yeah we call it L health being the
length scale of the wave function
I'll write that here at the links scale

53:40

the wave function is L here longer than
R so if the length scale the wave
function is is L then the magnitude the
wave function has to be 1 over L to the
D over 2 such that when you do or to
such that when you square it and you
integrate it in D dimensions you can get
one ok alright so now let's just do an
order of magnitude estimate of the terms
in the shorter equation so that the

54:13

terms in the shorter equation if Ella is
is long there's a potential term in this
kinetic term so the potential term is
well okay so it has an energy minus V
naught and it's minus V naught over a
region DDR of size squared over a region
of size R only goes up to a ball size R
to the D so this will be roughly V

54:46

naught R over L to the D power and the
are over 1 over L comes from the north
from the size of the wave function so
we're just plugging in this expression
for sy square and we're integrating out
to a radius R maybe just call it okay
that gives us an estimate of the
potential energy on the other hand the
kinetic energy is going to be H bar

55:19

squared over 2m l squared because L if L
is the decay length that's the more or
less one over L is the ISM is the it's a
wave vector so that the kinetic energy
plus the potential energy is going to be
H bar squared over 2m l squared minus V
naught R over L to the D and it's easy
to see that for D less than 2 for large
enough large enough L this is negative

55:52

and that means a bound state okay
because this term will will dominate
it's not large enough l4d greater than 2
then for very large el large el it's
positive so you get no bound State so no
bound state now for strong enough
potential you can um you can still make
a balanced state but for very weak

56:24

potential what is that
[Music]
hold on okay someone didn't know how to
turn off their their cell phone ringer
so as I was saying the point here is
that for D greater than two you cannot
get a balanced state just by going to
large enough L and generically that
means that for a week
V for weak attractive potential you do
not get a bound state in dimensions

57:00

greater than two of course for a strong
enough potential like for example
hydrogen atom you can get a you can get
a bound state in any dimension but not
for arbitrary weak potential so let me
actually write down the conclusion in
and we're interested in in three
dimensions in particular in D equals
three no bound state for very weak
potentials for weak potential and that

57:30

actually sounds pretty bad for our
attempt to try to form a bound state
from the weak attractive interaction
which is induced by by phonons from the
phone on the weak attraction attraction
between electrons which is induced by
their interaction with phonons it seems
like it's gonna be a pretty tough
problem to try to figure out how are we
going to bind electrons together so the

58:01

whole story of superconductivity was
broken open by leon cooper and what is
now called the cooper problem leon cupra
1956 where he showed that the issue of
whether you form a bound state or not is
completely different if you're thinking
about two single electrons just hanging
out by themselves or if you're thinking
about two electrons and a Fermi sea what

58:33

he concluded was the two electrons on
top of a Fermi sea on top of a Fermi sea
a Fermi sea
do form a bound state a bound state for
arbitrarily weak arbitrarily weak
attractive interactions
and the we'll go through the calculation
because it's it's important but the key
piece of information is that if you're

59:18

making a wave function for fermions
above a Fermi sea all momentum that you
use must be greater than KF so your wave
function can only include momento
greater than KF because all the melenz
eccleston KF are already filled with
other electrons okay so how do we show
this to be true the way to do this is
you write a trial wave function for two
electrons or two electrons so we'll

59:48

write it like this sign of our 1 Sigma 1
R 2 Sigma 2 and we'll let write it as
spin 1 up spin down to minus down one up
two so this is a spin singlet singlet
and then a spatial part times 1 over the
volume sum over K greater than KF e to
the I K dot R 1 minus R 2

01:00:22

thank you sub ok ok so this part here is
some sort of G of r1 r2 and I've just
written it in a Fourier notation so a
couple things about about this wave
function first of all it has to be
antisymmetric because we're talking
about electrons then this singlet is
already antisymmetric
so that means the spatial part should be
symmetric ok
in order to get the spatial part to be
symmetric we should have G of K equal G

01:00:55

of minus K and if we wanted to be
rotationally invariant as well we might
even go further and say that G should be
a function of absolute K only second
thing I should comment is that this
argument all of BCS Theory does not rely
on pairing electrons together in
singlets it turns out that the vast
majority of electro
of superconductors known do have
electrons that pair together in in

01:01:24

singlets rather than in in triplets so
it makes sense to do the do the
calculation for singlets but you can
actually you can also do the calculation
for triplets maybe a tiny bit more
complicated but not not too much more
complicated another thing to realize
here is that on the right hand side this
wave function is a function of r1 minus
r2 but is not a function not a function
of the center of mass of our center of

01:01:55

mass is r1 plus r2 over to that doesn't
enter at all and what that's telling us
is that we're taking this pair that
we've wrote written and put it in a a
plane wave at K equals zero so we put
the center of mass momentum to be zero
there's a maybe a slicker way to write
this this trial wave function is to do
it in in second quantized notation will

01:02:26

write sy is 1 over the volume some of
our loops sum over K greater than KF g
sub k c VAR k up c dagger minus k down
on the Fermi sea and the fermionic anti
commutation z' guarantee that this
combination is creating a singlet and it
also guarantees that it has the right
poly anti symmetry okay so then with
this trial wave function we are going to

01:02:59

assume some attractive interaction which
might be weak attraction U of R 1 minus
R 2 which may be weak
what and we would like to solve the
Schrodinger equation solve Schrodinger
which
we'll write it in the following way the
kinetic energy plus the potential energy
- the eigenenergy on sy equals zero okay

01:03:31

so let's try to plug in our wave
function into this expression and and to
do that let me just actually copy our
wave function here amazing I can do this
I'd say this every time it's so amazing
modern technology never get tired okay
paste okay there it is so that's the way
functionally we'd like to plug in so so

01:04:01

I'll write the plugging this in I'll
write the what we get as follows so it
will be a sum over K G of K we have the
e to the I K dot R 1 minus R 2 like this
the kinetic energy term well each time
we create an electron with with a plane
wave momentum K we get some energy EF K
and in fact there's two electrons with
the same energy we've created two of

01:04:32

them the same energy okay - campus have
the same energy so the kinetic energy
term is - EF k here the potential energy
term is going to have U of R 1 minus R 2
and then minus e equals 0 ok it's just
from plugging in the wave function into
the shorter equation okay now to solve
this we for any transform integrate with
it ok with D and D dimensions of our

01:05:03

one-way sir all right 1 minus R 2 a
relative coordinates that they were
Fourier transforming so even at Q R 1
minus R 2 and at the same time we're
going to define the Fourier transform of
U of R U of R the potential one more
volume sum over Q u twiddle Q either
that I Q dot R ok
so making that Fourier transform we get

01:05:33

to energy of Q minus e times G of Q plus
the Fourier transform of this term here
is going to pick up well we're gonna get
some one over volume sum over K greater
than KF you twiddle of Q minus K GK all
equals zero so how did that happen well
okay
there was a quedar here but there's also
Q dot R here so we end up getting Q

01:06:09

minus K GK over here if I didn't think
did that carefully no - there miss -
okay good
and this equation is what we're gonna
work with but it's still rather
difficult analytically so what we should
do is think about simplifying this even
if it's simplify even if you you know
the simplification looks like it might

01:06:41

be an oversimplification so remember
that when we when we consider the the
physical phonon attraction for phonons
the actual thing we're interested in we
found attraction for frequency for
energies which are less than the phonon
frequency so attraction for energy less
than the phonon frequency so h2 by so

01:07:14

let's just take a model is Cooper's
model of interaction potential U which
is extremely crude and it actually looks
it even a little bit unphysical
will write it this way instead of
writing it as a function of Q minus K
I'll write it as a function of Q and K
and I'll write it as minus u naught if
both energy of K and
q are both within both within H bar

01:07:49

Omega 2 by H bar Omega 2 by the Fermi
surface and 0 otherwise
so if both P and Q are close to the
Fermi surface closer than this maybe so
this is really I really I should have
written here - yeah less than H bar

01:08:27

Omega by so if if both of these things
are close to the Fermi surface you get
minus u and if not you get you get zero
now it's a little bit of a weird
interaction because it's not you know
really it should be a function of K
minus Q naught K and Q separately so it
looks it looks kind of kind of strange
it sort of has some of the properties
that we want of of an interaction but
the reason we use this is because it's
analytically tractable and that's a huge

01:08:57

advantage so let's plug in this form
into this equation we do that we get to
epsilon Q minus e times G sub Q P sub Q
- I guess it's u naught over volume sum
over K greater than K F G sub K equals
zero subject to the statement that Y sub

01:09:28

K must be less than EF plus h bar will
make it to buy okay so good and this
also must have he sub Q closed has to be
close to the Fermi surface also there's
no Q close to for me
it's also okay for this equation to have
been derived alright so what do I do

01:10:00

with this so here I'm going to switch
from a sum over from a similar K to an
integral over energy
so I'll write it as u not density of
states at the Fermi surface integral the
energy from the Fermi energy so the
Fermi energy plus h-bar Omega Dubai and
the thing I'm going to be integrating is
is G G sub K but let's K is going to be
a function of E and we're gonna assume

01:10:32

that G is a function of absolute K all
right as G of K and K is a function of E
and G of K of e like this G vo maybe we
can just write it as G of e if we like
so so all I did here was to replace this
sum over K s with an integral of
reckoner G times the density of states
so this just to be be careful this D
here is the density of states per unit
volume and we're gonna assume that

01:11:05

that's close to a constant you're near
the Fermi energy so it just ends who
stays in EF okay so good so we can then
solve this equation for G of Q by moving
this factor under this fact moving move
this whole thing to the other side and
then divide through by this factor so I
get G of K of energy here is then you

01:11:35

not div F integral EF EF plus h-bar
Omega 2 by de G of K of e maybe I'll
write that is e prime because we have
I'm sitting over over here yeah
so I want to write this at this one
probably should have sort of been Q you
have Q of e QV that's from this cute

01:12:08

here you have QV and then the thing that
comes over here I move this all over to
this side
e Q minus e I can then drop this Q so
this is sorry I'm going to make an
eraser and I shouldn't make a razor
transform let me draw exactly the same
equation here
copy paste it again paste so G is now
gonna be a function of e only G is a
function of E and then G over here is

01:12:41

also a function of e this good let us
define this numerator to be a number
called C so C is you not then CBF
integral of e f to he f + H bar Omega 2
by the e prime G of e Prime and so that
means that this equation can be written
as G of e is like a C over 2 e this is

01:13:16

another form strike it's just e yep so
this is now 2 e ha ha 2 epsilon - I can
evaluate and then the way to solve this
is to realize that I can calculate C
again by plugging this G in here so we
get G C equals u naught d AF integral e
f TF + H bar Omega 2 by epsilon prime C

01:13:51

over 2 e - e we can cancel see on both
sides and then maybe move the the U
naught over to the other side so we get
1 over u naught
India VF is integral EF d e te f + H bar
Omega by 1 over 2 e should be a prime up
there - Yi this we can do that integral

01:14:25

to get 1/2 log of 2 EF minus H bar Omega
by minus e / - EF - e which basically
gets me to the oops come back gets me to
the solution exponentiating both sides
we then have e to the let's move the 2
over from this side to this side so we

01:14:58

have e to the 2 over you not density
stays to the Fermi surface equals the
exponential in this log is 1 + 2 H bar
will make it to by over to EF minus e
good so what is this this is our final
equation it's important for small you
small attractive interaction for small

01:15:30

unit - what we have small unit you know
at the right the left hand side over
here is huge the only way the left hand
side is it can be huge
is if the right hand side here has - EF
- e it is small but positive so this
happens if - EF - E is small but

01:16:00

positive and that means E is less than 2
EF which means we have a bound State why
well
the energy of the electrons that we put
in sorry the momentum of the electrons
that you put in are above EF so we put
in electrons above EF but when they
interacted with each other they ended up
having an energy below EF which means we
have binding that the the net energy of

01:16:32

interaction is below is zero and so we
have have a bound state now we can go a
little bit further for small u naught u
naught we can we know that both sides of
this equation are large and so we can
drop the one and rewrite this equation
as e to the two over you know density of
states in Kiev equals to H bar or make
it to by over to EF minus minus E or

01:17:05

another way of saying is the binding
energy to a EF minus e is 2h bar I'll
make it two by e to the minus two
divided by u naught density of states
that at the Fermi surface so this is the
strength of the binding and it is
exponentially small in one over the
attraction strength u naught and it's
it's quite important to realize that

01:17:37

this is non perturbative what do I mean
by that it is if you think about the
function e to the minus 1 over X minus 1
over X and you try Taylor expanding that
around around x equals zero it it has no
radius of convergence has no radius no
radius of convergence of convergence
at x equals zero so no order in no order

01:18:14

Taylor series will ever represent e to
the minus one over X if you're expanding
it around XE x equals zero that's
exactly what we have here we have a
binding energy maybe this is not
supposed to be divided by binding they
must come right this way this is the
binding so that we have a binding energy
here which goes is e to the minus
something over u knot and unit is small
which means that no order in

01:18:45

perturbation theory no order in
perturbation Theory will predict this
binding
which is one of the reasons why it took
so long to to figure out BCS theory it's
also important to realize that the
pharmacy is crucial Fermi sea is crucial
particular the density of states at EF

01:19:23

must be not equal to zero here otherwise
I mean if the density of states here was
zero you would get no no binding so it's
really coming from the finite density of
states at the Fermi sea which assures
that you can have binding energy even if
it's exponentially small and this is you
know arbitrarily weak you not will still
give you binding even though it's
exponentially exponentially weak binding

01:19:54

but is binding nonetheless so this
actually was the the hint that launched
BCS theory and the reason that you can
sort of understand what's going to
happen here so you start with a Fermi
sea let's draw from EC there we go
you start with a friend EC and it's
filled and then some of the electrons
and the Fermi sea realized that it's
ones right at the Fermi surface realize
that they can come out of the Fermi
surface
I guess it's they form a pair one with

01:20:27

plus cameras - can they form some some
pair here but they pair together oops
these guys pair together with each other
this guy pairs with this guy and they
manage to lower the energy if they were
right at the at the Fermi surface that
energy was - EF but by coming above the
Fermi sea pairing together via this
phonon interaction they lowered their
energy so now they they now they have a
lower energy and then another pair can
come out of the Fermi surface and you

01:21:00

know form a pair like this and then
these to lower their energy and you
realize that that what's gonna happen is
you're going to have a complete
instability of the Fermi surface
stability a Fermi surface
and this is what is going in signals the
the presence of this this pairing this
ability towards a super conductivity so
in the next lecture we'll try to
understand the what happens when you
have all of the electrons trying to pair

01:21:31

with each other okay until next time

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