 you can follow along with this presentation using printed slides from the nano hub visit www.nano hub.org and download the PDF file containing the slides for this presentation print them out and turn each page when you hear the following sound enjoy the show okay so look at started first of all you all understand that professor alum and professor data these are these set of very high standard right don't expect me to live up to that but I'll do the best I can my my job this afternoon is to try to relate all of the concepts that you heard from professor data yesterday to some real data show you how we apply it as real data how we interpret and understand what's going on I'm going to make use of published data and that's always very dangerous right because you don't know you can't ask the people questions you can't look at them when they're doing the measurements you don't know what they're not showing you things like this but there's been a lot of data that's been shown and it's more or less consistent and there is a general understanding of this is the way things work so the problem that we're talking about is you have a sheet of graphene you apply a small voltage between the two terminals you're making a graphene resistor what is the resistance how do we relate it to band structure and scattering mechanisms and how do we figure out what's going on it's still a field that is evolving so there are still people that are debating about well is this the explanation or is that the explanation there are still some details that haven't been sorted out so I'm gonna I'll try to illustrate where or discuss where some of those are now we're doing some of our own measurements here or Professor Affenzeller and Yang Shui worth Yang doing it in the lab here so I've asked them to sit in and give me a hard time if I'm saying something that's not correct and if we have questions they may be able to to reply from their own perspective of actually being in the lab and doing these measurements and I'd like to thank Dionysus who helped me pull this together and along with several other people Tony Lowe did a lot and as I said it it's going to apply the concepts that Cyprio talked about to this problem of graphene conductance and Professor Affenzeller and Yang Shui we spent a lot of time with there's a set of notes that go along with this lecture so this lecture is pretty descriptive I'm not going to derive anything I'm going to sort of remind you of what you saw with Professor Dada's lectures some of these things integrals can be worked out rather easily but we have a nice set of notes where everything is worked out so if you'd like to see how it's done what you should concentrate here on is getting the main ideas of what's done and you can go back and get the mathematical details with those lecture notes that are in your notebook okay so what are we trying to do I guess we don't need to say anything about graphene we all know what it is there's this nice little tool on the nano hub that runs really quickly and I think later on today we're going to have a session where we will rerun this tool and illustrate some of the band structure concepts that Professor Dada was was discussing yesterday so the point is it has a really unusual electronic structure is different from most semiconductors that has people really excited that this ought to be good for something it's not clear what you know but but I agree that it probably is it is so different there are probably some really some things that are difficult to do by other means that we can that we can put graphene to use for so it's great fun there's all kinds of really interesting science going on and everybody is trying to sort out where are the applications and what do we do with this tomorrow you'll hear a lecture from Professor Oppensauer from you know more of I don't know whether he's an experimentalist or a theorist he's sort of both and he's he's working on device applications and is really pursuing a number of of specific potential device applications and you'll hear about that tomorrow so my objectives are very simple so describe the technique that people commonly use to measure the resistance or conductance of graphene to show some typical results and then to talk about how we understand the characteristics that we see so feel free to interrupt anytime you have some questions so just a little bit of theory as I said not very much we'll just remind you of some things you saw yesterday so we all know now about this band structure of graphene it's two-dimensional so k is a vector in the xy plane we have this we have two equivalent valleys so whenever we're computing conductance and things we have them remember to multiply by two for spin but also two because there are two equivalent valleys and it has this interesting band structure where nominally in pure graphene the Fermi level is right at the intersection of the cone so all the states below it are filled the t equals zero all above it are empty that point is the neutral point where we've got an equal number of positive charges below with their empty states and negative charges above that they're filled and it's also called the Dirac point because of the relation to this Dirac equation that present professor data discussed yesterday so the ek around one of these points can be described by this simple relation so e is plus or minus h bar v f the Fermi velocity which is constant times the magnitude of k and the magnitude of that velocity one of the unusual things is it doesn't matter how far it is above the band minima or where it is in the band the velocity is constant of course if you go far enough away from this point it'll be different but we're going to stick near this point and it's about one times ten to the eighth centimeters per second which is sort of an order of magnitude faster than electrons move when they move very fast in most semiconductors so it has people interested in applications about high-speed electronics and the density of states I think professor data worked this out yesterday it's worked out again in the notes so it goes linearly with energy for this particular e of k you can work out the 2d density of states and it's zero at this neutral point and then it increases with energy when you go above think about that as the conduction band and it increases with energy when you go below linearly think about that like the valence band so we're going to refer to the case when the Fermi level is above the neutral point as n-type graphene and we're going to refer to the case when the Fermi level is below as p-type graphene okay and we're going to be interested in a very simple problem just a resistor so we're looking down at the top we have the sheet of graphene I think professor happens or might show you some real pictures of what these look like and they're not quite as uniform and square but we're going to think about them as a resistor with some length l some width w we have a couple of contacts applied at the end and the contacts are these land our contacts they're big there's a lot of scattering there's a well-defined thermodynamic equilibrium each one has its own Fermi level so there are two different Fermi functions Fermi level is shifted by a little bit we're only going to be considering small biases here low field transport not high field so we begin with this land our formalism where the current is 2q over h we have to do an interval over all those energy states t is the transmission m is a number of conducting channels sort of velocity times density of states professor data talked about that too and then current flows when there's a difference in Fermi levels but we're only going to consider the case when there's a very small difference in Fermi levels because there's a small voltage applied so you can approximate this difference by a derivative that's where that the f the e comes in and f not now is just to remind us that we're near equilibrium so this is the equilibrium Fermi function you can pick contact one or contact two because the bias is so small that everything is basically the same and current is only going to flow at the Fermi level or states very very close to the Fermi level okay so we could plug in things we know what the Fermi function is we saw this expression for the transmission function so this is a number between zero and one so we're thinking about this as in a semi classical picture this is you know there's no phase involved in what we're going to do if the mean free path lambda is very much longer than l then this is one everything transmits across if it's very much smaller than only some small fraction transmits across and we know we have the expression for the number of modes conducting channels I think professor data worked that out yesterday so I put all of these in and do the integral and we've got an expression and if you look at the notes that's done the unpleasant thing is you know you get these Fermi Dirac integrals and things like that if you do it above t equals zero but but at t equals zero things are really simple so let's just talk about t equals zero I think professor data also talked about how this Fermi function goes from one to zero at the Fermi level and if that's reasonably sharp then we can think about that as a delta function at the Fermi level then the integrals become easy because we just pull the integrand out and evaluate everything at the Fermi level so our conductance is 2q squared over h times the transmission at the Fermi level times the number of conducting channels at the Fermi level and what we're going to be interested in now is as we dope graphene differently n type or p type move the Fermi level around how do we change the conductance and how do we understand what we measure so to begin with the Fermi level if it's everything is perfect and there are no potentials applied or stray charges Fermi level is right at the intersection and if we were able to move the Fermi level up in the conduction band then we expect the conductance to increase linearly with the Fermi level because the dense the number of channels increases linearly with energy assuming that this transmission is more or less constant it might have some energy dependence too and if we move the Fermi level down into the valence band all that matters is that we have states around the Fermi level so we're going to have a conductance and everything is symmetrical the density of states increases below the neutral point also so the conductance would increase linearly everything would be symmetrical it'd be ambipolar we'd have conduction if we move the Fermi level up or if we move the Fermi level down okay frequently experimentalists you don't really know directly where the Fermi level is in the semiconductor experimentally you have to have some kind of model and compute it you know things but what you are much more likely to know and we'll talk about in a minute is how many electrons there are or how many holes there are so that's the electron density in sub s per square centimeter so we have this expression for the conductance now if I plot it versus electron density instead of versus Fermi level then I won't get a linear plot like this dash line we showed before it'll look a little bit different and I don't recall now professor data probably did this too you can at t equals zero it's easy to derive a relation between the Fermi level and the and the number of carriers the number of carriers goes as a square of the Fermi level if you forgot that from professor data's lecture that's in the notes also so the important point is that the carrier density goes as Fermi level squared so that means that the conductance and instead of increasing linearly with Fermi energy because the number of modes increases linearly with Fermi energy it's going to increase as the square root of carrier density so if we plot the conductance versus carrier density which will be more convenient to do experimentally we'll get plots that look like this you'll have some curvature again assuming that the transmission is about constant and that'll be affected by the scattering processes and we'll have to talk about that later okay so what about at t equals at t greater than zero then things get a little more complicated I have these integrals that I have to work out with these Fermi functions I get these Fermi-Dirac integrals of different orders that I have to worry about but you can see sort of physically what's going to happen the Fermi function will go from one to zero over some width the energy that's on the order of kt so there'll be some spread there'll be some places where some of the states are empty below the Fermi level some of the states are filled above the Fermi level and I'll just have to do this averaging process and that's what happens when you when you do the integral so if I move it up I'll average over those states that are higher in energy if I move the Fermi level down I'll do some averaging over the states that are down and I expect something if I get very very far above the minimum you know then it begins to look like a metal and I'm back to the case where it looks sort of like a delta function now that spread is very much smaller than the energy that I'm at if I get near the bottom I get some rounding because now I don't go to now I have a few states filled in the conduction band I have a few states that are empty in the valence band and and I have to make some corrections around there so I can think about this as the average transmission times the number of channels the average quantity near the Fermi level and when I do that integral I get the appropriate average now the so the key equations that we're going to work with these t equals zero equations are so simple that these are the only ones I'm going to work with I don't want to work with Fermi direct integrals or anything so we'll just work with the t equals zero relations so the conductance it's a nice easy expression and simple to remember and very physical it's the quantum of conductance times the transmission at the Fermi energy times the number of channels at the Fermi energy okay and it doesn't matter you know there are states everywhere except exactly at E equals zero that direct point and if I have t greater than zero then I have to add the conductance of the conduction band and the conduction of the valence band right there are two contributions we have an expression for the number of modes basically velocity times density of states it's it goes linearly with energy just like the density of states is because when I multiply velocity times density of states for graphene velocity is constant so both of them go linearly with energy in most semiconductors they have different dependencies with energy the transmission is given by this formula that professor data discussed yesterday so we have a simple expression for the conductance at zero kelvin okay now people frequently like to write this you know if you have a large resistor you like to write the conductance as the sheet conductance times width divided by length you make it twice as wide the conductance is twice as big you make it twice as long the conductance is is half okay so that makes sense if you're diffusive in a normal resistor it wouldn't make much it doesn't really make sense if you're ballistic because you make something ballistic twice as long it doesn't matter you know it can go twice as far just as easily but people frequently do this so when you're analyzing experimental data sometimes it's not always this is one of the difficulties in pulling numbers out of the papers because you can't go to the people and ask them okay what's your w and what's your l sometimes they tell you sometimes you have a little piece of graphene that's kind of irregular and it's not exactly clear what's w and what's l so you're kind of trying to get close but people will do that and then they'll extract and frequently what they're plotting is not the raw data which would i over v which is what you would like to see and then make your own decision about what w and l are what they will frequently plot is this sheet conductance or they usually call it the conductivity sigma in 2d that the same thing mose per square so it's that g sub s that we get now if i factor a w over l out of this expression then i can find out what this g sub s is and if you do that it'll will factor this l out so i'll take a l out to multiply by an l on top and i'm going to call that quantity the apparent mean free path it's just some algebra because i insist on interpreting things in this diffusive picture one i shouldn't always do that and that apparent mean free path will be the smaller of two quantities it'll be either the real mean free path in a big piece of graphene the average distance between scattering events or it will be the length between the two contacts whichever one is shorter so it's sort of this picture that professor alum alluded to when you have a ballistic piece of graphene and you're thinking about you know mobility and what's the scattering time or what's the mean free path well it's actually limited by the contacts because you scatter in the first contact you go across the resistor you scatter in the second contact so the mean free path is the length of the channel can't ever be longer than that okay so i'll just mention on site in introductory semiconductor courses we teach students a lot about non-degenerate Fermi a non-degenerate semiconductor has a Fermi level way below the conduction band and you can use Boltzmann statistics and the integrals are easy to work out when it's degenerate usually the Fermi level is around the bottom of the band or a little bit above the bottom it doesn't get too much beyond that and then you have these Fermi drag integrals and things just get messy if you're a metal you can go back and assume that the Fermi function is a delta function and everything gets easy again so in graphene you know almost always your Fermi level is way above the minimum of the conduction band or way below and actually the t equals zero expressions work out really very very well so it's you can go ahead and you can use the t equals zero expressions you can always go ahead and use the full expressions if you want to but the t equals zero expressions work really well and they're much easier to help you understand what's going on there's a little bit of error when you get right near the drag point but there are other reasons that you shouldn't use any of this right near the drag point anyway so we'll stay away from the drag point and we'll talk about that a little bit later so i'm only going to use t equals zero kelvin results and really it works just fine i'm not making much of an approximation okay so the questions that we have are if we're able well how do we do this experiment experimentally how do we do this and some of you i think are probably doing this experiment so you know how it's done but we want to talk briefly about how do you move the Fermi level around while you're measuring the conductance and then we want to look at some of the results and see if we can compare them to what we expect so i just described sort of what you'd expect to see we'll look at some of data and we'll see what we really do see and it won't look exactly the way i sketched and it's because even though graphene has very long mean free paths and the mobilities are very high there is some scattering and you will see that reflected in the shape of the characteristics okay all right so i'm going to say a little bit about the experiments and and we can discuss a little bit more about how that's done so experimentally it's difficult to dope graphene you know in bulk semiconductors you dope them that's how you move the Fermi level in the conduction band or the valence band in graphene you do it with a gate some some people will call this electrostatic doping it's just modulating the potential in the graphene with a gate so that you can change the position of the Fermi level with respect to the neutral point and in a typical experiment we have a layer of graphene that is placed on a layer of silicon dioxide i think professor Appenzauer might talk a little bit about how you make these or come on so the experimentalists have these very clever ways these very sophisticated ways they've talked they've discovered to do this using scotch tape is that right and you know it's interesting you you read these papers and they will even tell you what brand of 3m scotch tape they used you know because you know it's very important to be able to reproduce this that you get the stickiness just right and even that it takes a lot of trial and error before you get the hang of okay okay you know they discovered if you put it on a 300 nanometer thick si-02 then you can see one monolayer of it i think that's 90 nanometers you can also see so you know it's it's really it's actually quite impressive okay so this is what the structures look like this is this is a side view so you take a silicon wafer you know it's only function is to hold everything up right and and serve as a substrate here's an si-02 layer whose thickness is usually picked such that when i put one monolayer of graphene on you can look through a microscope and you can see it you know the interference fringes are just right and then you put a couple of contacts on and you make a resistor the silicon layer is doped on the bottom you know the silicon below it so that's like a it's going to serve as a gate we're going to apply a voltage at the bottom and that's going to change the potential in the graphene on the top if you look at the top view over here on the right you can see the width and and the length and when you see pictures of the real ones you know you take this off by scotch tape and put it down and put some contacts on they don't the W and the L aren't quite as clearly defined but and you can use different metals contacts and different thicknesses of si-02 but it's typically fairly thick si-02 you know those 90 nanometer numbers are very thick you know what are they in a in a CMOS chip these days one point some nanometers yeah so these these are pretty thick layers typically okay now there are also some beautiful experiments that people have done uh at columbia and i think probably one or two other places where you can build this structure and then etch the si-02 out from underneath it so it's suspended it's not sitting on something that might have some stray charges and things in it that could affect the measurements and we'll talk about what happens when you do measurements on this suspended graphene also okay so the measurement is is very simple you can do them two terminal or four terminal so this is a two terminal measurement you apply a voltage across the two contacts measure the current and divide the two and get the conductance and you can plot either the conductance or one over the conductance the resistance and you can do that at a fixed temperature and then you can sweep the voltage on the bottom and when you do that you'll move the potential of that neutral point up or down and you'll move the Fermi level into the conduction band or the valence band and you can plot out these characteristics of conductance versus gate voltage or you can sit at some gate voltage and vary the temperature and see how things vary versus temperature and as i mentioned earlier frequently what's reported is not the direct measurement of i divided by v but it's this sheet conductance so the w and l are factored out now just a word about using the gate voltage to change the direct point so i'm going to draw the pictures this way and you know so the picture will be that we apply a positive gate voltage so physically you can think what's going to happen is the positive gate voltage is going to attract negative electrons in the graphene so it must mean the Fermi level is up into the conduction band so we'll usually draw our pictures like this now the Fermi level is moved up into the conduction band what's really happened is that we've taken the direct point and we've lowered the energy of all the electrons by minus q times a gate voltage or whatever the potential is inside the graphene we we pulled the direct point down and the Fermi level hasn't moved but it's just easier for me to draw the pictures this way all that matters is the relative distance between the Fermi level and the direct point so positive gate voltage the Fermi level goes up negative gate voltage we're going to attract positive holes in the graphene so the Fermi level should be in the valence band and it'll get pulled down now if you want to solve well let's see what we're doing here if you typically do a measurement you'll get something like this ideally if there are no stray charges and there were no work function differences between the metal electrode and the graphene everything would you would go through zero at vg equals zero experimentally it doesn't there's some offset there's some stray charges there are some work function differences just like you have in an m os capacitor you know flat band voltage so people will typically find out what that is and then they will translate that offset to the origin and you'll plot the characteristics like those dash lines okay now c times v is charge so it turns out that for most cases that's a really that's all there is to the story that's a really good approximation so q times the charge in the graphene that's the total amount of charge per square centimeter in the graphene is just the capacitance of that insulating layer epsilon of si o2 divided by that 300 nanometers or 90 nanometers or whatever it is divided so times the gate voltage times vg prime after i've made this shift so it's like vg minus the threshold voltage and the threshold voltage is zero now that doesn't always work um you know if you're solving for this problem it's just like solving an m os capacitor and uh there is a capacitance of the graphene people call the quantum capacitance that's in series with the insulator capacitance and if you solve it you find that this is an approximation that works really well if the oxides are reasonably thick and most of these oxides in these experiments are reasonably thick but if you want to see how it works for like one or two nanometer oxides it'd be a little more involved the relation between ns and vg and that's worked out in the notes too i think that right so if you're actually building a high performance device your gate insulator would be very thin and the relation would be a little more complicated than this but for the these kinds of experiments that we're doing it's very easy so when we plot conductance versus gate voltage we're really plotting conductance versus sheet carrier density in the graphene it's the same thing okay so now we can look at some results and we can see if we can make sense of them so here's a typical measurement if now it's uh it's from a review article that's recent i think the measurements themselves were done what some you probably know three or four years ago which is like ancient history and graphene but it's still it's a nice looking plot so you're seeing things look sort of like what we talked about um in this case they have a nicely structured piece of graphene they must have etched this somehow i don't know how yeah oxygen etched oxygen plasma etched okay i knew there was a good reason i invited you to sit down in front with me so i will rely on you more so you you can see this resistor this is a four probe measurement so here you can pump the current in you can measure the voltage between these two probes the width looks like it's one micrometer the length is five micrometers or five thousand nanometers the oxide sitting underneath this thing is 300 nanometers stick all right typical experiment and we're plotting the sheet conductance here and the temperature is quite low 10 degrees kelvin okay okay so we could try to make sense out of this data we have this basic expression that we showed a little bit earlier which tells me what the conductance should be we plot it versus gate voltage so there is a relation between the gate voltage tells us what the carrier density is the carrier density is related to the Fermi level so i can deduce what the Fermi level is at any point and then i can deduce i can put that in i can take the measured sheet conductance and the Fermi level then and i can deduce what the apparent mean free path is okay so there's a very simple relation i can just solve these two equations for the apparent mean free path and professor data mentioned this yesterday too so i take the measured sheet conductance in units of 2q squared over h i divide by the square root of the carrier density which i determined from the gate voltage and i can find out what the mean free path is very easy and again i'm using these t equals zero expressions but they're really quite good so let's go up here where the gate voltage is high the sheet conductance is about three millisiemens i know the gate voltage it's about a hundred volts with a 300 nanometer thick oxide so we have about seven times 10 to the 12th carriers per square centimeter so are people calibrated to this is that is that a high sheet density yeah so about the highest you can get and say a CMOS transistor if you applied under on current conditions about 10 to the 13th so this is a this is a good hefty charge Fermi level is way above the bottom of the of the way above that direct point that neutral point in fact you know we have all the information it's three tenths of an eb above we can just work that out and we find an apparent mean free path that's 130 nanometers so this thing is 5000 nanometers long and the mean free path is 130 so it's diffusive there's a lot of scattering going on between between the two contacts so the apparent mean free path is the real mean free path because it's the shorter of the length of the resistor and the actual distance between scattering events and graphene now we could go down here we could go at a lower gate voltage and we could repeat the exercise so the sheet conductance there is half because this is a straight line it's 1.5 millisiemens the carrier density is half because it's the same capacitance times half the voltage so it's 3.6 times 10 to the 12th we can deduce but the Fermi level the carrier density goes as a square the Fermi level so the Fermi level isn't half it's instead of being 0.3 it's 0.2 and I can go through the exercise and I can deduce the apparent mean free path and it's 90 nanometers so it's different the mean free path is energy dependent it's different at the higher energy in the band than it is at the lower energy so something some type of scattering is going on now it's it's interesting to note that the ratio of the mean free paths is about two-thirds and the ratio of the Fermi energy when I did the calculation first up here and when they do the calculation now the ratio of those two energies is about two-thirds so what that's telling us is that the mean free path in this sample is proportional to energy the higher the energy if the energy is doubled the mean free path is doubled so that should be telling us something about the scattering processes you know so people start to think what scattering processes are proportional to energy and maybe that's the one that's controlling the performance of this particular resistor okay now mobility so you know people like to talk about mobility and the reason is because we're used to thinking of things like sheet conductance is sheet carrier density times q times mobility now that's always true in the sense that I can always make this definition right and then I can divide the measured conductance by the measured sheet carrier density and q and I'll find out what the mobility is now in this case it makes it makes a lot of sense to talk about a mobility because there is a region quite a long region here where the conductance is proportional to the carrier density so the mobility is just some constant number and you can do the math and it's twelve thousand five hundred centimeters squared per volt second so that's is that a so is that a good mobility pardon me yeah it's it's good but there are other semiconductors that are much higher but this is you know this is a little bit old people can do better than this now but it's it's a pretty high mobility it's like 10 times higher than silicon but it's also at 10 degrees kelvin right so um okay now let's go down here at this minimum point and let's run through the same exercise so you know it's sort of I said that the carrier density is the capacitance times the gate voltage the gate voltage is zero so it says ns is zero so now I really shouldn't use this t equals zero expression anymore because you know now it's important to get this little thermal spread you know but at 10 degrees kelvin there isn't very much thermal spread so if I attempt to do this and even if I attempt to do it rigorously you know using the proper expressions with Fermi Dirac integrals I'll be led to the conclusion that suddenly the mean free path is is big up here at high gate voltage it's dropping linearly and then when it gets in this region it turns around and it goes off to infinity right doesn't make any sense right it really it's what it's an indication of is that something else is happening here and uh the something else that's happening or what people think is happening is that in this region if you have a large piece of graphene and it's sitting on silicon dioxide and it's sitting in the ambient there are stray charges all over the place that are pulling the Dirac point up and down and you have these puddles they call them where some places it's p-type some places it's n-type and the conduction is is different we developed a theory for which everything is uniform the Fermi level is in one place the Dirac point is in one place it's uniform and it's not and so our theory really only works when we get out of this regime and we're well above those fluctuations when we get down in there then there are many papers professor alum alluded to it on you know percolation theory and so something different is going on there and I guess it's a it's an area that we're we're doing some looking at with professor alum and tony lowe and we haven't fully digested all of that yet it's not as professor alum mentioned it's not exactly like classic percolation theory that he talks about where you you have these regions where you have these very big differences between conductivity either conducts or it doesn't conduct we will see in the last lecture that p-unjunctions in graphene don't block current very well so so it's an interesting region lots of papers being written about it for the purpose of this lecture i'm just going to stay away from it it's not the point of this lecture okay so there are lots of people doing doing some really nice experiments so this comes from michael furrier's group in university of maryland and this is doping so and the way it is done is by exposing the graphene to potassium and so my understanding of this is it's not done at a high temperature it's not substitutional doping the way you would dop silicon you know you you put a material on top of graphene you expose it to it there is some charge transfer between that material and the underlying graphene and you effectively dope it so i would think it has something like modulation doping in a semiconductor and what we're seeing here are the pristine curve zero seconds exposure and you can see this nice symmetrical characteristic and you can see it's not linear the way the previous one was different samples showed different characteristics so it's a little bit different and now what you'll observe is that as you increase the exposure to phosphorus you move this neutral point to the left and you know if you had an m os capacitor you would shift the flatband voltage by minus charge over c ox so this suggests that you have some kind of positive charge from the phosphorus the more phosphorus you put down the more positive charge you have so it shifts the neutral point all right that part's easy to understand from m os electrostatics but it also changes the slope and you can see that these curves become more and more linear and the more you expose them to phosphorus the less the slope is pardon me potassium what did i say oh potassium all right all right so that's so that's an observation of what happens experimentally so let's look at this undote sample first of all remember i i showed you a a plot earlier on where we're trying to understand what the shape of these characteristics should be if you plotted them versus carrier density or gate voltage which is the same thing and we said if the transmission was constant it should go as a square root so this sort of looks like it might be going as a square root maybe it's not quite as strong i should probably plot it versus square root of voltage the transmission is constant when you're ballistic it's just one but it might also be constant if you're if you're being free path is constant isn't changing so we could ask ourselves are we anywhere near the ballistic limit in this sample before we expose it to potassium right well you just plug in numbers just like we did before it's 164 nanometers and i think this scale of this i forget it's a micron scale length so it's a diffusive sample so we're not near the ballistic limit um now you can also look at these samples and you can sit at a particular gate voltage and you can measure now here here we're doing one over the conductance so we're plotting the resistivity instead of the conductivity so the vertical axis is so you have to invert it i'm sorry it's only plot I had so you have to invert it your plot and what we're doing here is just two different samples and you're sitting at some particular gate voltage that's away from this minimum where our theory should apply and you're plotting resistivity versus temperature and you can see the resistivity increasing linearly with temperature and then when the temperature gets hot enough around 200 k or so it starts to increase faster than linear and i guess this is being done for probably the different samples that were exposed to to potassium so experimentally we observed that for temperatures below 100 degrees k the resistivity is proportional to temperature and people say that's due to acoustic phonon scattering we'll talk about why in a minute and for higher temperatures the resistivity starts to have an exponential characteristic and people are still arguing about this some people say that it's due to uh polar phonons in the sio2 underneath and those polar phonons have an electric field that then scatters the electrons and the graphene on top of it and other people are saying no it can be just explained with the optical phonons and the graphene itself so you know it just isn't resolved yet but we can understand what's going on here the resistivity that's being plotted is one over the conductance the conductance is proportional to the mean free path so it's one over the mean free path and the mean free path is related to scattering and scattering is going to be related to how many phonons that are there to scatter so we expect the resistivity to be proportional to the phonon occupation number more phonon the hotter it is the more phonons there are the more scattering is going to occur and the phonon occupation number is given by this bozeinstein factor now for acoustic phonons you can go through these energy momentum conservation arguments and find out which acoustic phonons are responsible for scattering and it turns out that unless you go to extremely low temperatures you can think of acoustic phonon scattering as an elastic process the average acoustic phonon that's scattering an electron has an h bar omega that's small compared to kt when people go down to really low temperatures they have to worry about this energy exchange but at reasonable temperatures you can assume acoustic phonon scattering is elastic yeah elastic no no no no change in energy so if h bar omega the phonon energy is much less than kt then i can expand e to the x as one minus x and this typical approximation you make then for acoustic phonons at temperatures that aren't too low is that the number of acoustic phonons is kt divided by h bar omega okay so the number of acoustic phonons is proportional to temperature and the measured resistivity is proportional to temperature so that's why the experimentalists will suspect that in that regime and being dominated by acoustic phonon scattering you calculate the scattering rates i think we even we do that in the notes too don't we we calculate the scattering rate for acoustic phonon scattering it's going to depend on a deformation potential and with some reasonable numbers for the deformation potential you can account for the measured data although there's a lot of spread in what people think the deformation potential is no we haven't talked about impurities yet so one step at a time we'll get all right so optical phonons optical phonons have a lot of energy all right so if you're at low temperatures um the electrons don't have enough energy to excite an optical phonon and there isn't enough thermal energy for them for there to be any population of them but if you increase the temperature enough then this factor can start to become big enough and you can get some population of optical phonons and in that case you would expect to have this exponential characteristic and that explains the rise the thing that's not exactly clear is where those optical phonons are coming as i said there's still some debate this is still a new field some people say they're coming from the sio2 underneath some people say no those are optical we can explain that with the optical phonons that are present in the graph theme and of course people are using different deformation potentials and it's not quite clear what that is so there's still some uncertainty that's being sorted out okay now there are these interesting experiments on suspended remember i talked that there are people who etched the sio2 out from underneath and when you do that you get a curve here that's blue that looks very much like the ones we got before but then if you heat that up and drive the impurities off you have a very clean piece of graphene with no straight charges around it and then you get the red curves so you can see initially they were linear so you kind of suspect when we doped it with potassium the curves became linear when they were on the substrate where there might be a lot of straight charges around the curves were linear before we annealed it when there might be a lot of gunk on the films they're linear but after you anneal that all off they become nonlinear now you can see uh oops before we do that there's a dash line here and there's a solid line and as professor data mentioned people frequently you know don't don't always appreciate that this Landauer formalism can work in the diffusive limit or in the ballistic limit so what the dash line is is a Landauer calculation in the ballistic limit which is what people typically do and the fact that it is very close to the measured results suggests that that annealed sample is operating almost at the ballistic limit but we can just apply our theory this apparent mean free path the apparent mean free path will be the length of the resistor if you're at the ballistic limit and if you deduce a parent mean free path here it's about 1300 nanometers and if you look at this picture here what's up you know here's two nanometers so this distance between these voltage contacts here looks like it's less than two that's about the length of the resistor so it makes sense it's it looks like it could be and one of the differences you'll notice here you know the last data if i were this minimum point here is actually quite high right if i were to extrapolate these linear regions down and go to zero gate voltage there would be a quite a lot of conductance that's residual conductance that's still there and i don't know what that might be is that some leakage around the edges or some shunt conduction path but it's it's quite large in this sample so that's different i i guess we'll talk about that a little more later okay i want to say just a word or two about mobility because a lot of people that do experiments and measure mobility the objective seems to be to report the highest possible mobility but mobility is kind of a fuzzy concept and it's not always means free path is always very clear we know what mean free path is but when conductance isn't proportional to carrier density it's not always obvious exactly what the mobility is you can always make the definition conductance is nq times mobility so you can always do that so um but this expression this is the right expression um and what that says is that the conductance is proportional to the mean free path times the square root of carrier density not times carrier density so i can always make this definition and then i can go ahead and i can if i were to divide my measured conductance by the square root of the carrier density i would get a mobility right so if i were doing that in a sample like this i would try to get the carrier density as small as possible because then i would measure a mobility that's as high as possible but the mean free path itself might be constant the whole time you know it's probably a more meaningful physical quantity so there are kind of two cases here if we have an apparent mean free path which might be energy dependent if it's proportional to energy uh which is one second if the mean free path is proportional to energy then energy is proportional to the square root of carrier density so square root of carrier density times square square root of carrier density that would give me a conductance is proportional to carrier density and then i could pull a mobility off anywhere on that plot and it would be the same and it would make a lot of sense to do it but if the mean free path is constant then the conductance then is proportional to the square root of carrier density and then i get a mobility that depends on carrier density and then you have to be very careful when you're comparing numbers between labs okay what carrier density did you do that at and nothing very complicated is going on it's just a constant mean free path is there a question so this is a sample where first of all the graphene was on the sio2 right and then the sio2 underneath the sample was etched out okay so after it was etched out then this curve was measured but after the etching process you know there may have been straight charges and things that were left on the graphene so then the graphene was heated up and annealed to drive all of that stuff off and the idea was to produce as clean a piece of graphene with as little stuff on it as possible and when after that annealing process then the conductance got much higher and more non-linear oh that's a good question why is this curve not symmetrical so let me ask you to ignore that for now okay um i'm going to mention that on my very last slide as one of the reasons that you should stay for my last lecture of the summer school right let me just defer that but that's so you know what i'm going to do is to say okay this is the good side this is the side i'm always going to analyze because it looks better okay something is going on here and i'll talk about that briefly at the end then i'll talk about it more in the very last lecture yeah they're not always symmetrical okay so just a summary um the low conductance samples we often see a conductance that's proportional to carrier density if we get away from this minimum where we shouldn't be because that requires a different theory the higher conductance samples are frequently non-linear they look like they sort of like the square root of carrier density but not exactly um the conductance decreases with temperature people call that metallic that's sort of what you'd expect more phonons more scattering um but it can become super linear if you go high enough and that means there's phonons with a higher energy let's see the best mobilities if we've got our numbers correct the best ones that we can find these days are on the substrate are around 30,000 but you often do see these asymmetries these experimental effects now after you suspend it you get much more pure graphene and uh before you anneal it it looks like the graphene on sio2 that g is frequently not always but frequently linear proportional to ns after annealing the conductance is much higher and it becomes distinctly non-linear in fact you can then approach the ballistic limit it looks like there's very little scattering at all you can get mean free paths that are on the order of a micron and people quote mobilities now it's not always so meaningful to quote mobilities because it depends on where on that curve you decide you're going to choose a mobility but they get numbers that are quite high and i don't remember now what 70,000 100,000 200,000 you know but it's questionable as to what what that actually means okay so let's see we'll see if we can wrap up here in a little bit oh so we want to i want to relate this a little bit i'm not going to get too deeply into scattering but we'll try to relate it a little bit some of the results that we've seen we just described what's seen you know we haven't described really why that occurs and the different shapes reflect the underlying scattering processes so this lambda is the mean free path so mean free path is the average distance between scattering events now we have to be a little bit careful here because we call this a mean free path for backscattering when people quote mean free paths they do it in somewhat different ways most people do it one way and we do it in a different way right because what comes out naturally when you relate transmission to mean free path is this thing called mean free path for backscattering and this is why professor dada mentioned a factor of two you know the mean free path that he uses in one dimension is not v times tau it's two v times tau because if on average it can scatter anywhere if it scatters forward it doesn't do any damage if it scatters backwards it backscatters and it hurts the conductance so it takes two scattering events on average to do damage so your mean free path for backscattering with an extra factor of two in two dimensions you have to average over angle in three dimensions also and so sometimes when you're comparing our mean free paths to other mean free paths you have to take account of that difference so the idea is an electron comes in with some energy and some wave vector it encounters some scattering potential due to a phonon or due to a charged impurity or due to a defect in the graphene lattice and it scatters out and goes off in another direction at the same or different energy depending on whether it's elastic or or in elastic so one over tau if tau is the average time between scattering events then one over tau is the probability per second that you'll scatter and that's what we call the scattering rate and it depends on energy right high energy ones typically scatter faster than low energy ones and there's a standard process that people use doesn't always work but it's very common in semiconductor work and you just calculate these scattering weights by Fermi's golden rule so you you identify the scattering potential you sandwich that in between the wave function for the initial state and the final state and compute that matrix element and you can compute the scattering rate and once we have the time then we can convert it to a mean free path for backscattering we can stick it in our transmission formula and we can compute the conductance all right so here I mentioned that we're in two dimensions now so when we do this averaging over angle to get the backscattering proper it turns out that the averaging gives you a pi over two so you'll frequently if you compare the beam free paths this way to the beam free paths that people frequently quote artists will be pi over two times bigger all right it's not about the same physics it's just a different definition okay now for many scattering rates scattering processes the scattering rate is proportional to the density of states that makes sense so I have an electron coming in at some energy e it can scatter to some state let's say it's elastic if there are more states to scatter to if there are twice as many states at that energy it'll scatter twice as much assuming that the interaction potential is the same at that energy so it's reasonable to assume that the scattering rate is proportional to the to the density of states in graphene the density of states is proportional to energy so the average time between scattering events for a reasonable scattering mechanism would be proportional to one over energy so high energy ones would scatter more quickly and it would be smaller than low energy ones so the energy the mean free path then would be v times tau with this pi over two so the mean free path would be proportional to one over energy so this happens for acoustic phonon scattering it happens if your scattering potential is a delta function your scattering rate is just proportional to the final density of states okay now it's very interesting in graphene that this this does something very interesting to the conductance that this fellow Andal observed many years ago so so let's look at what would happen this is our expression for the conductance and i'm operating in the diffusive limit now so the apparent mean free path is a real mean free path and the real mean free path for this common type of scattering mechanism is proportional to one over energy so look at i put a one over energy here i had an energy here in my expression for the conductance so the result is that the conductance is independent of the number of electrons in the conduction band right that's not typical you know typically you're used to saying if i have twice as many electrons in the conduction band i have twice as much conductance for this particular type of scattering it would be independent so that works for acoustic deformation potential scattering or for short range scattering like a delta function now what if you have charged impurity scattering so let's say i've got the graphene sitting on s i o 2 and there are charges around maybe they're on top of it maybe they're in the s i o 2 they're all over the place that coulomb potential is going to affect uh the potential in the graphene and what it's going to do is to move the neutral point up and down it's going to fluctuate all over the place depending on whether there's a charge nearby whether it's positive charge whether it's a negative charge so i've got that neutral point bouncing all over the place so an electron wave comes in and it sees all of these fluctuating potentials and it's got a chance to reflect off of them that's the scattering process and you can kind of see that if the electron energy is very very high then these ripples and potential don't have much effect they're they're small compared to the electron energy so there isn't very much reflection or scattering so i would think that the scattering would become less important as the energy becomes higher okay so that means that the mean free path should increase as the energy increases so actually you can go through some simple arguments calculating these scattering rates with Fermi's golden rule and you find out that that is exactly what happens but actually i mean those arguments are a little too simple you really need to include all of the screening and everything but even when you do that more involved calculation you typically find that percharged impurity scattering the mean free path is proportional to energy so what does the a mean free path that's proportional to energy do well we have our expression for conductance if i say the mean free path is proportional to energy then i have an energy here an energy here i have an energy squared but carrier density is proportional to energy so now the conductance goes linearly with carrier density i get these linear plots i can compute a mobility it's nice and constant and this was pointed out some time ago and this is what led people to believe one of the things that led them to believe that if you see these linear plots it's a signature that there is charged impurity scattering going on in this particular sample now as i said there are still things that are being debated people have shown recently that if you have very strong short range scattering instead of a weak scattering potential a very strong scattering potential that's actually reducing some mid-gap states that you can get the same thing that can happen and i think i saw some experiments out of the mariland group just recently where they deliberately iron bombarded graphene to create damage like this and they found these linear characteristics so it's suggested you see these linear characteristics if you haven't intentionally damaged it in that way it probably indicates that it's charged impurity scattering all right okay so uh so that's pretty much the whole story here let me just show you one other way that you can analyze data which seems like a sensible way to do it it's very very common for people to analyze their data and quote you a mobility but what i've suggested is that that's not always a meaningful thing to do but a mean free path is a very physical thing you know we know what that is and there's a very easy way you can take the expression for the conductance we can take the expression for the carrier density and professor data showed you this yesterday if you just divide the conductance by the square root of the carrier density we can deduce experimentally what this apparent mean free path is we can do that in any sample whether the characteristic is linear or whether it's curved in some way we can experimentally deduce what the energy dependent mean free path is and then we can ask ourselves what scattering process would give us that energy dependence and kind of deduce what's going on so if we take this data that we've been talking about again and if we apply that procedure we'll get mean free paths for the red curve the annealed curve that was near ballistic that will look like this and we'll get characteristics for the blue curve that will look like this let's look at that red curve first and first of all remember we should stay away from the region around vg equals zero because our theory doesn't apply there and what you'll see if you go away from that is that the apparent mean free path increases slowly with energy higher in energy they are away from that band minimum then the longer the mean free path is so the apparent mean free path is increasing with energy and eventually it gets to a point where it's almost constant with energy so what you would the way you interpret that is well initially there's some charged impurity scattering when you put the Fermi level high in the conduction band that gets weak and then the apparent mean free path is the shorter of the actual mean free path and the length of the resistor and eventually you just hit the length of the resistor and it doesn't get in longer and you can actually if you know the length of the resistor you can actually extract out from that what the real mean free path through the scattering is and it's quite long in that sample three and a half microns so you can pull that out now if you try to do that on the blue we expect the mean free path is quite short in fact we just argued that in order to get this linear dependence your mean free path has to increase linearly with energy so we'll stay away from this region near zero because the theory doesn't apply but if we go out here the mean free path is not increasing with energy it's decreasing with energy now it turns out this data is complicated by the fact that we're assuming here our expression for conductance assumes that when you go to zero Fermi level you're at the Dirac point and there's no conductance but if you just extrapolate those points down it doesn't go to zero it goes to quite a large value there's a big there's a very big in this particular sample is a very big residual leakage some kind of maybe well maybe along the edges maybe somewhere there's a shunt leakage path if you subtract it out because that's some other effect that we're not quite clear what it is if you subtract that out and then do the analysis of the part that's behaving properly then you'll get a mean free path that increases with energy the way we argued that it should to give us that kind of characteristic and you can see that those mean free paths are a lot shorter than they were in the annealed sample you know instead of being microns they're tenths of microns right just what you would expect all right so our general picture of these curves is something like this if we have a ballistic characteristic then the shape of the curve should go as a square root of the carrier density everything should be symmetrical around the Dirac point it isn't always now if the mean free path is constant you'll get the same kind of shape you'll just get a lower conductance but it'll have the same square root shape now on the other hand if you see a sample where the conductance goes linearly with carrier density most people will tell you that that's a signature of ionized impurity scattering and you have charged impurities in your sample somewhere and that's what's going on if you have acoustic phonon scattering you would get this highly unusual characteristic where the conductance wouldn't depend at all on the carrier density now in practice if you have a little bit of charged impurity scattering and you have a little bit of acoustic phonon scattering you're going to get the sum of these two although it'll be the minimum of whichever those are and you'll get a characteristic that will have some shape and it won't necessarily be square root of encebes but it will be a competition between these two scattering mechanisms and if you think you know deformation potential is well enough you may be able to to pull some of that information out oh and these things you know sometimes the minimum conductance goes close to zero but in that sample that we looked at before it was quite far from zero all right so as i said this lecture is mostly descriptive it's showing you this is what the kind of data that people are reporting and this is how we interpreted in terms of the kinds of concepts that professor data introduced yesterday so the general features you could sort of understand pretty clearly what's going on although people are still debating not everybody agrees and i might put professor Appenzeller on the spot here later by asking you which parts of this he doesn't agree with because they're probably are water too there's a nice way to analyze this data just by extracting the mean free path it seems to make a little more sense than extracting mobilities so i've talked about some very simple theoretical treatments but some of the treatments of these scattering processes get much more sophisticated and people try to account for all of the details of the shapes the actual experiments are frequently you see some non-idealities they're not always symmetrical the way i've assumed that they would be but we have a general framework that understand what's going on most of it makes sense and now to Zhu Fang's question you know what about the contacts you sort of assume you put these contacts on and their ideal ohmic contacts they have low resistance compared to the graphene they don't play any part in the measured conductance versus gate voltage characteristic well this graphene is high quality material right typically its mobilities are reasonably high the contact is some kind of metal to graphene junction as we change the gate voltage we're changing the potential drop across that it may well play a role in fact it probably does play a role and when we talk about junctions on the last day i'll i'll talk a little bit more about how we think about p-n junctions and n-n junctions and why you might argue that you can take the one that's higher and assume that that's the graphene and if the other one is being contaminated because you're seeing some of the junction conductance okay i think that's all i have to say so thank you now are there are there any questions well i mean let me look and see where we stand on our okay yeah so we have do we have any questions or comments you want so for the acoustic bonon yeah so what it is if you look in the notes you can see we use a very classic description of acoustic deformation potential scattering so we're just thinking we're just treating the longitudinal acoustic phonons with a deformation potential approach we work it out you work it out the way you would work out acoustic phonons scattering in any semiconductor but there's one twist and the twist is this two component wave function and the two component wave function i'll talk a little bit more about it on friday but it it suppresses back scattering so what it does is for the same scattering potential it gives you less scattering than you would get in a then you would have gotten if you had not accounted for that that two component wave function but outside of that it's just a standard treatment now the big question is what is this deformation potential and i'm trying to remember there's actually quite a wide variation in what people believe the number to be you know like 18 what is the ev is that the units but there are lots you know that there's quite a wide spread as to as to what people think it might be you know so it's not pinned down very accurately and the scattering rate goes as deformation potential squared so there's a lot of uncertainty there yeah uh which slide are we talking slide 33 okay which is actually it's this one isn't it is this because i honestly don't know why this is this is pretty high do you remember the first one that i showed way earlier that one if you extrapolated that one the two branches it actually went down quite close to zero and but on this slide 33 after you annealed the sample it went even higher that the minimum point went even higher you know there are two different axis two different y axis okay oh are they the y axis are different that's why the curve higher is low that's how i like it i shouldn't chop it no no you should there's a red axis and the oh the red axis actually shows that this is much closer to the then it's comparable yeah go ahead so so one of those of pictures that professor lansom was showing me for was this percolation type picture right and and what it says is if you are drawing a cone for your entire looking structure that's really not capturing what's going on but literally you have to picture you have a lot of little cones in different areas of the trophy now imagine these little cones are offset and you're trying to make the car really small and i create an offset that is rather large between all these cones what are you going to find you're going to find a rather high minimum conductivity if you manage to make all these cones nicely behaving very similarly then i have a much better chance of benefiting from the low density of states close to the point and what am i going to say see i'm going to see a much lower conductivity close to the point so every time you perturb your system a lot a lot meaning i have all these cones digging around a different spot in my routine i have a good chance that my connectivity is going to be high in this minimum point do you mean by this point? yeah yeah so a picture i think that's fair is to say okay make make a typical circle size around these pilots and every circle for me is a little cone and every cone is slightly offset in terms of energy up and down and now try because all these cones are in parallel right these are white structures now you're trying to picture i want to make my current very small but if one cone's always kind of above the or below that matters at the general point you can do whatever you want with your game but you're always going to have flights and connectivity so the more more dirty the more impure you have this is the counter-intuitive state right normally we are we are seeing the more scattering the lower the connectivity but that's that's what we argue all the time here for the minimum point it's upside down right the more things are disturbed the higher the conductivity because you're trying so hard to make the conductivity smaller so you're saying York that the difference between this one and the earlier one is that there's much the the fluctuations and potential are much larger in this one and it's my structure that's at least my take on it and it increases your chances of having always quite some nice conductivity and the thing that's confusing me just a little bit is that i think this mean free path up that i extract up here is about what the other one was and you know and this mean free path is thought to still be affected by those charged impurities but presumably the same charged impurities that are responsible for the puddling and yet the mean free paths end up being the same once you get away from that regime here i guess this is also very similar to the public size that you're having yeah right and if the width of these structures of the order of several micrometers which it often is then i believe there's a good reason to assume that there are always some kind of parallel path through the structure yeah but how can you be so confident about that because even for example in the martinite uh when you put down a dialect your formula was to almost pin that at the uc and i always start the frailty level and that because of that you want to go shut off the due and so one of the biggest differences here i think nobody would debate that there is no band cap available in this structure there is no band cap so so the best thing i can do is really go to the girl point where i have supposedly a very low density of space right and and yes you can argue i i am not moving my bands anymore well i see and i pull a device first but how do i explain that i have two branches and i'm going to a local minimum if i'm pinned at some point right that's also really hard to envision and so i have an easier time to picture that something prevents me from getting very low connectivity rather than assuming that i'm not moving my bands at all then i would assume i have no second breath so i say that because in the martinite case people still see some sort of v-shaped character this is not completely flat and that comes because of brand command coming from the other side too because you have from from from the you talk about the and i pulled up the vice characteristics that are there all the time but keep in mind this is a very small dds region so we are very very close i mean we are pretending that when the gaiting space up and down my drain was much much smaller than that which is also certainly something that we have to take into account closer to the atonement what you are saying is slightly different you have the and i pulled and vice characteristics now as a drain function we can spill off the other part of the character you don't think the greenboard is not high enough for that that is the assumption in this entire talk very very small drain voltage right close to near near equilibrium so this band to band tunneling um so this is the key feature of graphene p and junctions so we'll we'll talk quite a bit about that then on friday typically you see all of these measurements are at really low temperature right zero to fifty k at that temperature so what do you think happens at for measurements at room temperature to uh to the to the hole in electron coordinates which are isolated recombined or do they coexist plus uh you have uh thermal generation of carriers taking place anyways so are these electron or particles going to be contributing to the thermo-generated carousels have you looked at these characteristics at room temperature yes yes we have and so look at the energy scale we are talking about at some point uh professor lancin translated the air concentration energy axis into an energy axis and the largest carry concentration of seven times blah blah times 10 to the 12 corresponds to 300 millilektron volts okay now let's look at kp at room temperature 30 millilektron volts so we're talking about a very small portion very close to the dr point that gets affected at room temperature at all by this generation of carriers and by the pilot effect so because you have such a wide span of energy that you can cover with a gate watch even at room temperature you have a very very easy test to get out of this thermally excited point i would like to uh if you don't mind make a comment about this ballistic versus scattering in context can we lift up one of those oh sure you do you want to use the blackboard yeah so i let's see i apologize i mean i don't want to jump to something but i personally always have this approach as a method i look at our approach and then i ask myself so what do i expect what do i expect out and the professor answer is always much more careful in his statements and i i'm always more outspoken and mean to other people that's why that's why i invited you here yeah i know i know this is why i'm always happily participating in this kind of rum session so imagine you you are an experimentalist right you're measuring an id vgs characteristic and uh it looks like this you have any kind of problem very vgs value now professor answer was to kind of point out to us that there is a square root dependence in the ballistic case right and then he said oh and there is also a combination of a linear dependence and uh current saturation and so if i have both what am i going to end up with well again this right so if i don't look at the current do i have a chance to tell what it is no right i can look at this there's no way okay now we are still safe right because ballistic we know how the current is but what if i'm telling you this is ballistic but there were some contact effects involved now let's say these contact effects for simplicity is is gate faults independent right so i'm saying this wants to be a ballistic curve and now i have some cut off due to contacts not acoustic phonon but those contacts that professor lanstra measured mentioned so what's the superposition of those two things it's again a curve that looks very very similar to what we had before it's still not scattering it's still not acoustic phonon it's still not impurities it still has a linear region it still has a level of region but i call it ballistic and contact effects so please be very very careful with all these seducing arguments about uh oh yeah i know exactly this is this type of sketch mechanism and this is this as i said professor lanstra was much more political than me so right i would say as an experimentalist it's true to say at this point there is no clear cut 100 convincing case and the least thing i would expect you to take home is be very very careful ballistic plus contact maybe as good as all sorts of scattering that you want to include thank you so so this is clearly does not apply thank you for pointing that out so for the for tomorrow measurements i wanted to i know that did you did you see how huge this voltage proofs are in comparison to the spacing between them and that's very typical right normally in a whole large geometry you have that contact and that contact and actually it's pretty much the geometry where we have this distance let's say the same as the context so what am i going to measure with these voltage proofs right i mean they're clearly not just proving of potential here at this one point they are huge they're wide i have scattering going on even on the length scale of the contact size so to be completely honest with you i don't know exactly what i'm measuring in these types of structures as long as i'm not making the geometry somehow smaller in terms of the the the voltage proofs being smaller and then of course we know from from previous work from professor data's work and others that in addition it's very hard to make these voltage proofs actually deduct the class k states and minus k states the way you want that you really measure something that is meaningful in these types of structures so i don't i don't know i mean it's it's very very hard for me to see in a gate voltage dependent measurement where the gate voltage suppose you also impact the contact regions that's at least what we believe underneath the contacts even things get dated and then on top of that now i'm asking the question how what do i measure with the quantum and voltage proofs you're saying there are no quantum defective parts i understand should be suppressed a little bit for the other one and if i see the same fact and then i can argue that the contact may not be as important in if it is if it is very sensitive then i'd say that it must be more effective so so we know that the mean from and again this is open discussion to see this this in this discussion so the contacts themselves the method the reservoirs right also are rather close in these pictures to the channel and and to this region for all of these dimensions are similar we know that the distance over which things verbalize the hot spot in front of the context is of the order of the mean free path the nihilistic mean free path that is so now my question is how unperturbed is this region where i'm testing with the voltage probes by the contacts are they far enough way because i know when i'm building hallbars and i try to do a real good job i make my current probes far away from my voltage probes in particular i want to be a nihilistic mean free path distance from that right there are all these these experiments that show the thermalization occurs somewhat here also in the front so i don't know that these contacts don't matter even in a four-term of the configuration i don't know the geometry is not cleaning up we are going to still show up tomorrow i will yell at you but don't worry that's okay right this is wonderful it's a brand new field there's still lots lots of things to sort out watch it it's like it's the people at the meeting so this is exactly what we want to conduct