Quantum Matter Lecture 19

Quantum Matter Lecture 19

SUBTITLE'S INFO:

Language: English

Type: Robot

Number of phrases: 1467

Number of words: 10713

Number of symbols: 45250

DOWNLOAD SUBTITLES:

DOWNLOAD AUDIO AND VIDEO:

SUBTITLES:

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

DOWNLOAD SUBTITLES: