You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. Alright, so something a little bit different. So we're going to talk about measuring these transport parameters. So if you grow semiconductor materials, one of the things you commonly want to do is to characterize the properties of your materials. All of the discussions that we've had about theory help you interpret the measurements and find out something about the material you've grown and how good it is, what its properties are, what determines its mobility, what limits its mobility, what you might do to make it better. So I'm going to just try to give you an introduction to how you do some of these measurements. So as I mentioned, this is commonly used to characterize materials.  Now an important point is that the results can be clouded. There are lots of things that can get in the way of determining what the properties of the material are. One of them is contacts. Another is these thermoelectric effects might be in there in the measurement and if you're not careful to account for them, you might be misinterpreting what you're actually measuring. It's very common also to do measurements in the absence of an electric field and then turn on an electric field and do measurements of Hall Effect and combine the two. So we'll talk about that also. So actually I have quite a few slides that I'm going to try to get through, but it is a brief introduction and I'll point you to some standard references if you want more details. So here's the plan for the lecture and let's dive right in. So we have these current equations. Remember the fundamental equation for diffusive transport and that's all I'm going to talk about in this lecture. Is current, is conductivity times gradient of the electrochemical potential. I've divided by a Q so that gradient is in units of volts per centimeter. If the carrier density is uniform, the gradient of the electrochemical potential is the electric field and I've sort of been going back and forth between those two without mentioning it sometimes. If you have a material that has diffusion currents you just replace the electric field by the gradient of the electrochemical potential and nothing else changes. And we're interested in measuring the conductivity because it should be a material dependent property. It shouldn't depend on the size of the sample or whatever as long as we stay away from the ballistic regime. So these are our fundamental descriptions. We know the theoretical description of how we compute the conductance of a sample. We have this familiar expression now for the conductance. It's 2Q squared over H, the quantum of conductance and there's an integral involving the number of channels at energy E. The probability that an electron will transmit across. And then this minus the FDE which we can think of as a window function. It's only peaked near certain energies and it selects out certain energies that participate in current flow. And then we do an integral over all of the energies. Now as I said I'm not really going to talk about ballistic transport. I'm going to stay in the diffusive regime but if I wanted to include ballistic transport I would just replace the actual mean free path by an apparent mean free path and it would be the shorter of the sample length or the physical mean free path. In that case the conductivity would be sample dependent. It would depend on the length of the resistor. We're normally when we're characterizing the property of a material we want samples that are long compared to the scattering length so we can characterize the properties of the material. If we're in that regime we get material with independent properties. And it's frequently convenient to express these results in terms of mobility and that happens because the conductivity depends on where the Fermi energy is located. And where the Fermi energy is located is controlled usually by the carrier density either by doping the semiconductor and getting a certain carrier density that puts the Fermi level in the conduction band or below the conduction band or by having an electrostatic gate like a field effect transistor where you can move the Fermi energy up and down or the band up and down. So it's very common then to measure both the conductivity and the carrier density because from the carrier density we can deduce where the Fermi energy is and then that'll help us interpret the conductivity. But it also means that people will frequently quote the results in terms of a mobility because they'll frequently instead of using our expression they'll say sigma is nq mu and they've measured n and they've measured sigma so frequently they'll just tell you what mu is and they won't report to sigma. So we'll frequently deal with measured data that is mobility. But we're going to need measurement techniques, we're going to need to measure two things. It's not enough just to measure the conductivity because if we want to understand it we have to relate it back to this theory and then we have to know where the Fermi energy is. So we have to measure the carrier density both. So we want measurement techniques that will measure the resistivity or the conductivity and the carrier density separately and then we put the two together and we try to sort out what's going on. Now let me just briefly mention here one thing. Let's say in a 3D resistor, in my 3D resistor, the conductance is proportional to the conductivity or the conductance is proportional to the conductivity times area divided by length. Now what if I have a thin film? So in this thin film, if the film is not too thin such that the electrons are not quantum confined, there are three dimensional electrons free to move in all three directions but I have a thin film. Then I'll use my 3D expression. I'll say the conductance is conductivity times cross-sectional area divided by length. The cross-sectional area now of this thinner film, it's got a different aspect ratio. The cross-sectional area is the width of that times the thickness T. So I'll write the conductivity as sigma times cross-sectional area which is W times T divided by L. Now if I bring the T outside, the conductivity is proportional to width divided by length which is what we expect in a 2D resistor. The product of those quantities is conductivity times thickness, that's the heat conductance. The units are just ohms. So that's one way I could think about that. But I have to be careful because if my film is a thin film but the electrons are three dimensional electrons and not quantum confined, then I would use this expression. I would use my three dimensional expression for conductance and I would account for the thickness of that sample in the vertical area. But if the layer is very thin like an inversion layer of a MOSFET or a quantum well, then the electrons are two dimensional electrons. They're only free to move in two dimensions. Then I would just go ahead and when I compute the conductivity I would use the two dimensional number of modes. That would be different than the three dimensional. So when you're interpreting data you would have to decide which theory do you use to use a theory for three dimensional electrons or two dimensional electrons and that would depend on how thin T is compared to the de Broglie wavelength of the electrons. So in measuring mobility the way that we would do it was we would first measure the conductivity, then we would measure the sheet carrier density, then we would deduce the mobility from this familiar expression and then we would try to understand the mobility by relating the nq mu to our expression for the conductivity and then we could understand things like how are the number of channels affecting the mobility and how is this scattering rate affecting the mobility and things like that. So we've discussed near equilibrium transport. There are basically three transport parameters. There's a conductivity. There's a C-beck and Peltier coefficient but they're really the same thing. They're related by this relation and there's a thermal conductivity and we'd like to measure all three. As Professor Fischer mentioned measuring the thermal conductivity is very difficult. People do it but it's not easy. So we frequently invoke the Weidmann-Fran's law to try to deduce what it is from the measured electrical conductivity. Measuring the C-beck coefficient also requires some special capability but people do that as well but the conductivity and sheet carrier density are frequently done.  So first one I'll discuss is how we would measure the resistivity or the conductivity. So you think it's very simple. We just take this material that we're interested in determining its resistivity. We put two contacts on it. We force a current and we measure the voltage across it. Trouble is that the contacts probably have some resistance, R sub C. So the voltage that I measure between those two contacts would be the sum of the resistance of each of the two contacts plus the resistance of the material in between which is what I'm trying to characterize. So this creates a big problem. If I make a very long sample that's very highly resistive and I have reasonably close contacts then maybe it's negligible but it's something that we always have to worry about. This we would call a two probe measurement. There are only two contacts there. Now there are various ways that people have for dealing with this and one is called a transmission line measurement or a transfer length measurement and the idea here is this. This is our resistor. You can see the width of the contacts along the length but we space them by different distances and then we just go ahead and we measure the resistance between those contacts and we plot the measured resistance versus the spacing between the contacts. And if you do that you'll get a straight line and the intercept of that straight line will give you the contact resistance and the slope of that line will be related to the resistance and the sheet conductance. Actually the x-intercept has a meeting too. It's called the transfer length and I'll talk about what that means as well. So this is a very common technique you can see from the date of the paper that describes it that it's been around and in use for a long time. People will use it all the time. Here's a side view. Those are the contacts and let me just blow up two of those contacts and look at what happens. So the current comes in one contact, goes into the semiconductor, spreads out, flows through the semiconductor and comes back out the other contact. So actually this is not a completely trivial problem to analyze. It's a two-dimensional flow. I'm going to get back to trying to analyze how the 1772 paper was all about the reference on the previous slide, how you analyze those 2D flow patterns and how they affect the measured resistance. But let's do an easier problem. If I just put a metal contact on a semiconductor and the flow is vertical in its uniform, I could think about what's the contact resistance. So if I think about this as a cross-sectional view, this is a big semiconductor with some cross-sectional area coming out of the page. When I make a contact between a metal and a semiconductor, there's bound to be some interfacial layer there. So it might just be the depletion region of the Schottky barrier. Or there might be a little bit of native oxide or something that's there. But there's some kind of interfacial layer there that has some kind of thickness and some kind of resistivity to the surface. So if I want to know what the resistance of that contact is, the resistance is always resistivity times the length of the contact in the direction of current flow divided by the cross-sectional area of the resistor. So it's rho of the interfacial layer times the thickness of the interfacial layer divided by the cross-sectional area. So the thickness and resistivity of that interfacial layer are a little bit difficult to pin down independently. But resistivity is ohm centimeters and thickness is centimeters. So people lump those together into a number that they frequently measure in quote. And it's called rho sub C, the specific contact resistivity. Its units are ohm centimeters squared. And a good number is 10 to the minus eighth. A reasonable number is 10 to the minus sixth. But you'll frequently see contacts that are 10 to the minus fifth, 10 to the minus third are much higher. So if I wanted to compute what the resistance of that contact is, it's just going to be the specific contact resistivity in ohm centimeters squared divided by the area of the contact in 10 centimeters squared. So let's say I have a contact on a small MOSFET. It might be a tenth of a micron in one direction and maybe it's a wide MOSFET so it's one micron in the other direction. And let's say I have a good contact, a specific contact resistivity of 10 to the minus seventh. So you just plug numbers in and that's going to give you 100 ohms, which is not insignificant at all. So this is one of the big challenges if you look in the semiconductor technology roadmap, as you scale transistors shorter and shorter the channel resistance is getting smaller and smaller and if the contact resistance is big compared to that smaller and smaller resistance it's really going to degrade the performance of the device. So people would really like to find ways to get the specific contact resistivity even lower than 10 to the minus eighth. But that's very challenging. So that's an easy problem to do because the current flow is all vertical, easy, uniform. Now the problem that we're dealing with has a two dimensional current flow problem. And what happens is that the current is coming along, let's say it's been injected from another contact out here on the right that I'm not really showing carefully, it's flowing uniformly through this layer that we're trying to characterize and then it comes underneath the contact and it has to flow up and out the contact. So you can kind of see what's going to happen if the resistivity of that material is very high then the current would like to flow up and get in the metal where the resistivity is low and it will very quickly go up. But if the resistivity of the material is very low then there's no particular reason for the current to get out into the metal where the resistivity is also low. It'll depend then on the ratio of those two resistivities. The length that it takes underneath that contact for all of the current to get up out of the semiconductor is called the transfer length. That was that intercept on the plot that I showed you before. And the transfer length if you go to that reference that I gave you from the burger paper where you solve this two-dimensional flow problem, that transfer length is the square root of the specific contact resistivity over the sheet resistance of the material. So be careful about dimensions here. Specific contact resistivity, that had the units of ohm centimeters squared. Sheet resistance has the units of ohms per square. Square doesn't have any dimensions. So it's a square root of centimeters squared so it has the dimensions of length. And you can see if the contact resistivity is very small then the transfer length is very small. The current can quickly get up into the metal and it wants to do that. If the contact resistivity is very big or if the sheet resistance is very low then the current prefers to stay in the semiconductor and the contact resistance is very long. So you can see that what's happened here is that the area of the contact, really the area that matters is much smaller than the area of the contact. Because the current is only flowing through some fraction of the area of the contact. So it's less than the physical area. Alright, so when you solve this two-dimensional current flow problem you find that the contact resistance is given by this expression involving the hyper-polycotangent. But if you look at that in two limits, in the limit that the length of the contact direction is much less than that transfer length then it's like all of the current is flowing uniformly through the contact. In that length, in that limit where the contact is much smaller than the transfer length the current just flows uniformly up into the contact and the contact resistance is the specific resistivity divided by the area of the contact. But in the other limit where the length of the one that I sketched here where the length of the contact is much longer than the transfer length then when you compute the contact resistance the area that you use is the width of the contact coming out of the page times the length of that transfer length. So it's much smaller. Okay, so this transfer length method is very nice because we plot this, the slope gives us the sheet resistance of the material that we're trying to characterize, intercept gives us the contact resistance independently and the intercept on the x-axis gives us that transfer length so we really characterize things very well. So people will do this both to characterize the material and to characterize the contact. So that's what transfer length measurements are all about. Now if you're primarily interested just in finding out what are the properties of the material, that's what I'm interested in then there are simpler ways to do that and people will commonly do something called the four probe measurement. So the idea here is that we'll force a current through a semiconductor and there will be some contact resistance when we put those probes down. If I were to measure the voltage between contacts one and four it would include the resistance of the contacts. But if I put a high impedance voltmeter down and those are the probes two and three and if I measure the voltage between some section there and if I make sure that no current flows through those probes because it's a high impedance voltmeter there will be no volt drop across the contact resistance and I'll only get the potential drop along the channel. So a four probe measurement is a way to measure the properties of the semiconductor and get the contacts out of the picture. So people will commonly do this with something called a whole bar geometry. So we're looking down on a thin film of semiconductor it might be an n-type layer sitting on a p-type semiconductor in that way that the current will be confined by the p-n junction to just flow in this layer that we're interested in it might be a layer sitting on top of a SiO2 insulating film or something got some thickness T coming out of the page and we'll define a pattern like this by lithography. So contacts zero and five we're going to force a current through those contacts and we've got some redundant contacts that we don't need here but if I go in the direction of current flow and measure the voltage between contacts two and contact one if I do that with a high impedance voltmeter that will give me my four probe measure bit. Now later on we'll talk about doing Hall effect measurements if I measure the voltage in the orthogonal direction while I'm applying a B field normal to the page I'll be getting a Hall voltage that we discussed before the break and that gives me some very useful information about this film. So in terms of just measuring the resistivity the voltage between contacts one and two give me that without any contact resistance. So remember we said we have to measure resistivity and we have to measure carrier density both in order to characterize the material and figure out what's going on. So let's talk about Hall effect measurements because it's a convenient way to measure the carrier density. Now if we're doing gated structures like MOSFETs where there are channels then we can frequently control the carrier density with a gate voltage and that's very nice then we know the carrier density but if we have samples in which we can't control the carrier density that way then we can use Hall effect measurements and I'll show you how this works. So here's the basic idea we have a B field pointing in the Z direction out of the page and we force a current in the X direction so we're going to have some voltage drop in the X direction that's related to the resistivity but if I put a voltmeter across the Y direction you know it's open circuited so I'm measuring an open circuit voltage there I'm going to measure a voltage and that voltage is going to be positive for an N type sample and I discuss this before the break so there will be a pile up of negative charge on the bottom and positive charge on the top that's the Hall effect and by measuring that Hall voltage I can learn some things about the properties of the semiconductor so it's been known for a long time people use it not just to characterize semiconductors but for magnetic field sensors and other things so the physics I guess we talked about in the previous lecture so I won't go through this again you know you can just think about the electrons moving at some average drift velocity think about the Lorentz force minus QV cross B they get deflected charge piles up you build up an electric field in the Y direction therefore you have a voltage across the Y direction now if we want to do a quantitative analysis of this we can take the current equation that we derived and remember this current equation it was the first term is what we had in the absence of a magnetic field, sigma times the electric field so I'm not writing gradient of electrochemical potential because I'm assuming that I have a uniform density of carriers here and the sigma is NQ mu now the term that involves the cross product of the electric field and the magnetic field has a sigma in it but it also has a mu in it but it also has this annoying numerical factor this whole factor, this average of tau squared divided by the average tau quantity squared which is some numerical factor that depends on details of the energy dependence scattering in the band structure which people commonly hope is between one and two but there's no guarantee and people actually sometimes try to pin it down but it takes a lot of work so we can do an analysis of it like this we can take our current equation and we can look at the X component of the current and we'll just evaluate the X component that's the first line there the magnetic field has a small effect so the X directed current is just sigma times electric field in the X direction I can also evaluate the Y directed current but the experimental condition is that I'm open circuited in the Y direction so no current can flow in the Y direction so I set JY equals zero and I just take the electric field in the Y direction and I take the cross product in the Y direction and I get the second line so now I can solve for that Y directed field and I can see it's proportional to the X directed field and to the B field and the X directed field is proportional to the injected current from the first line so when I get all done I'm going to see that the ratio of the Y directed field to the current density in the X direction and the magnetic field that I've applied in the Z direction is something that I'm going to measure now you have to get terminology straight that's the Hall coefficient the little r is the Hall factor the Hall coefficient is what you would measure so you would go in the lab and you would force an electric current in the X direction so you know J sub X you would apply a magnetic field in the Z direction so you know B sub Z is what the electric field in the Y direction is and the ratio of those quantities is capital R sub H the measured quantity and the theory says that that measured quantity is this Hall factor little R sub H this numerical factor of scattering times divided by 1 over Q times the carrier density and the Hall factor the Hall coefficient has different signs for N and P type because the Lorentz force is in different directions for N and P type so we can tell the sign of the semiconductor so the idea in doing the analysis then is if I take this Hall factor if I multiply the top and the bottom by the width of that sample then I get the Hall voltage on top which is current per unit width so I get the actual current on the bottom so what I actually measure is the Hall voltage and I divide it by the injected current in the applied magnetic field the theory says that that quantity is Hall factor over QN sub S remember that the Hall factor is this ratio of scattering times now if I define this quantity now I call the Hall concentration because what I can deduce now you can see from that middle expression capital R sub H I know what its numerical value is if I knew what little R sub H was I could solve that equation for the carrier density which is what I'm trying to do I'm trying to determine the carrier density R H precisely and it's very hard to know it precisely so we'll define this quantity which is the carrier density divided by that factor and we call that the Hall concentration so when people do Hall effect measurements they will frequently say I've measured the carrier concentration what they mean is I've measured the Hall concentration I know the carrier concentration within a factor of maybe 1 or 2 if I have some unusual scattering mechanisms or band structure it might be even worse than that but what you actually measure is the Hall concentration so we go back I asked my colleague over here in the Burke Center to give me some data they make Hall bars like this with dimensions where the length is typically 100 micrometers it's nice and easy to do without any fancy lithography they'll typically inject a current of about a microamp a common magnet can easily give you 2 kilo Gauss that's a small magnetic field Gauss is not an MKS unit remember that 10,000 Gauss is one Tesla that's an MKS unit so this is 0.2 Tesla that's the value I would put in the expressions to get the right units and if you go in the lab, first thing you would do is turn the magnetic field off actually to first order the magnetic field doesn't affect this voltage and you just measure the voltage between contacts 1 and 2 let's say it's 4 tenths of a millivolt and then you turn the magnetic field on and apply it normal to the page and you'd measure a voltage between contacts 2 and 4 let's say that's 13 microvolts so then we should be able to deduce three things the resistivity the sheet carrier density and the mobility at least those are the three things we would like to deduce so we go back in let's do the resistivity the resistivity is kind of easy because people will say that this voltage that I measured in the direction of current flow they'll call RXX it's the voltage dropping the X direction when I've injected a current in the X direction and that's just that V sub 2 1 divided by the injected current that's 400 ohms in this case and we know that that's resistivity times length divided by width and we know the dimensions of the hall bar so we have a material that is 200 ohms per square that's what we were trying to measure okay let's see if we can get the sheet carrier density so for the sheet carrier density we look at the hall we look at the measured hall voltage and the theory says that the actual sheet carrier density divided by this unknown hall factor this thing we're calling the hall concentration is just the known current that I injected the known B field that I applied divided by Q times the voltage that I measured in the hall voltage plug numbers in there and we get a hall concentration of 9.6 times 10 to the 12th per square centimeter and we hope that that's close to the real carrier density but we're always a little bit uneasy because we're not quite sure now mobility, how do I get mobility so you think that well I have resistivity and I have carrier density resistivity is 1 over NQ mu so I should be able to get mobility but we don't quite have the carrier density we only have the hall carrier density so if I look at conductivity which is 1 over 200 ohms per square and I set that equal to NQ mu well I can divide by the hall factor because the quantity that I know is the actual supply by the hall factor again and then I can plug in the hall concentration that we just deduced I can take the sheet conductance 1 over 200 ohms per square that we just measured and I can solve but all I'm going to be able to solve for is R sub H times mu sub N so the only thing I can solve for is the product of the actual mobility and the hall factor so we call that the hall mobility so that's what comes out and we kind of hope that it's close to the drift mobility that we've been dealing with before but we're never really sure but it's the best that we can do so these are widely used every lab that does semiconductor work will have a little magnet that can give you a few kilo Gauss and they'll be set up to do these kind of measurements the terminology to keep straight is what you measure is what's called the hall coefficient R sub H there is this statistical factor which depends on details of band structure and scattering which is little R sub H and the two things that you get out of these measurements are the hall concentration and the hall mobility okay, yes for the little R sub H do you have an idea of how much it would vary in a given material you know across different conditions like doping or things like that can you take it as being pretty awesome so the question is do we have any typical numbers for R sub H and actually I should know these I think if you put in ionized impurity scattering in a simple semiconductor with parabolic bands I think that number is 1.93 and if you put in acoustic phonon scattering in a parabolic band I think it's close to 1 I think so this is why people say well it's generally between 1 and 2 so if your material is heavily doped and you think you're dominated by ionized impurity scattering you might guess that it's closer to 2 but if it's slightly doped and you're dominated by phonon scattering you might guess that it's close to 1 but you know I've seen papers where people have tried to figure out what is this for the valence band that has a very complex structure I think of 4 or more sometimes you can estimate it and feel confident that you've got a good estimate but there are cases where you could be quite far off the other thing I was saying was how much you think it changes around some center point because usually you're looking at things like mobility versus doping or something so as long as you don't think it changes well so that's why you know the common answer is it varies between 1 and 2 so when you're doped lightly it's closer to 1 and when you're doped heavily it's closer to 2 alright something like that alright now I want to talk about another way of doing these measurements just because you'll hear about it frequently and it's commonly used and it's really a neat idea and very old idea so it's called the van der paugh technique it is just another way to do resistivity and hall effect measurements in a different geometry if you don't want to take the time to do lithography and make one of these hall bars if you just want to break off a little chunk of your sample and maybe put for little fodder dots around the edges you can do hall effect and resistivity measurements on that and the amazing thing is even though you don't know precisely what the shape of the sample is you can still get the right answer if the contacts if you satisfy a few conditions like so this is a 2D film it doesn't have any precise shape but it has to be homogeneous so its properties aren't varying spatially it can't have any holes inside it and the contacts have to be small and they have to be on the perimeter if you can do that then you can do this measurement and the way the measurement works is something like this just pick two of those contacts and you scored a current in one and out the other so here I'm forcing a current in contact M and its coming out contact N so there will be some 2D flow of current through this sample now if I measure the voltage between the other two contacts there will be some potential difference between those two contacts due to that 2D flow of current that 2D flow of current will set up a potential variation in there I'll call that voltage V sub PO P is where I measure my positive side and O is the negative side and then I'll divide that measured voltage by the current that I injected so that gives me something with the units of ohms and I'll call that R sub M N comma OP it's the resistance when I've injected the current in contact M and take it out of contact N and when I measure the voltage between contacts O and P somehow that ought to be related to the resistivity of the sample and you would think that it's probably related in some kind of complicated way but it's not so complicated now you can also do Hall effect measurements on these samples and they work this way now you inject the current in one contact and you take it out of another but you don't do it in adjacent contacts you skip one and then you measure the voltage across the other two contacts so in this case I would label the resistance I take the voltage I measure divided by the current we've injected we call that a resistance and I would call that resistance R sub MO the current comes in contact M goes out contact O comma N P measuring the voltage between contacts N and P now that it turns out is related to the Hall voltage so we can get both of the things that we want from this sample so let me just quickly go through this I'm not going to go through all of the details but it's not too difficult to do so let me just give you a flavor for how it works so if I'm doing the Hall effect we're just going to take our current equation that we developed yesterday and the current is going to flow in the XY plane so my top two equations then are just expressions for the current in the X direction and in the Y direction I'm interested in what is that measured voltage drop between contacts N and P so that's going to depend on the electric field inside the sample so I can solve those two equations for the electric field in the X and the Y directions and then I can take an integral from contact N to contact P of the electric field along a path that goes between those two contacts and it will just be minus the integral of EX DX plus EY DY and I could evaluate what that voltage is that would be if I've applied an electric field pointing out of the page I would call that B sub P N B sub Z that's what I would measure now if I also measure if I reverse the direction of the magnetic field and measure that voltage and if I subtract the two quantities and take one half of them that's going to turn out to be the all voltage that I'm after okay so so if you do that Algebra what you'll find is that I'm simply going to take those expressions for the electric field insert them into that integral subtract the two quantities with opposite sign of a magnetic field the expression that we'll get for that voltage is that now if you look at this what is the integral of J X DY minus integral of J Y DX well it turns out that that is the integral of J dot NDL so if I think of a dotted line going between N and P that's an integral of the normal component of the current along that line so all of the current that goes in has to go across that imaginary line that I'm integrating from N to P to to get the voltage and come out the other contact so that integral is just the total current that I injected in so when you get all done you know you would measure in that is resistivity times Hall mobility B field that you've applied times the current that's what we got from the Hall Bar geometry and you would get that from this same geometry and you don't even have to be careful about where you place these contacts or what the shape of it is so that's very nice so we can do Hall Effect measurements on these samples without doing lithography and etching and making Hall Bars okay now how do you do resistivity so let me give you an idea of how this works too so doing resistivity on that sample is actually kind of a complicated problem it's not as easy as the Hall Effect problem was I'm going to get some 2D flow of current through that pattern I'm going to set up some 2D potential drop and I want to know what the potential drop is between contacts P and O well here's a sample that's much easier to solve an infinite half plane let's do the infinite half plane with 4 contacts along the boundary because it turns out the answer is the same okay so I'll have my 4 contacts again they'll be spaced by distances A, B and C and I'll label them M, N, O and P and I'll do I'll inject the current in contact M, I'll take it out in contact N and I'll measure a voltage between the other two alright and I'll call that R sub M, N, O, P just like before so if you look at this if I inject a current in contact M this is a small point contact it's just going to spread out radially in this half plane so the current density would be the current per unit length in 2D it would just be the total current I put in divided by the circumference of that half circle and the current density is conductivity times the radial electric field so that would give me the radial electric field that's coming out and now if I want to compute the potential drop anywhere in this sample I would just integrate minus E dot DR get the potential drop between any two points so if I integrate 1 over R I get a logarithm R I have to pick some location in the sample as a reference I always need a reference point for the voltage so I'll pick some arbitrary location R0 and that will be my reference location and I'll just integrate minus integral of E dot DR and we'll get that expression for the potential drop ok so what do we do with that if I go over and I look at contact P contact P is located a distance from contact M of A plus B plus C so I put A plus B plus C in the numerator how about contact O that's located at a distance A plus B away and I have this arbitrary reference location somewhere if I subtract this to arbitrary location drops away because I'm just asking for the voltage difference between those two points so my voltage difference is related to the current that is injected, the resistivity and then the logarithm of the spacing of these contacts so the spacing of the contacts is important now I also have to remember that I have a current that I'm pulling out from contact N what does that do in this infinite half plane I can just superimpose the results I have a contact going in one contact that gives me a potential drop that we just computed we're pulling a contact current out from contact N that gives me an electric field in the opposite direction so I just get a different sign and now the distance between contact N and contact P is just B plus C and the distance between contact N and contact O is just B an expression for that ok so this resistance that I measure is just I think there's a minus sign there it's the voltage difference between those two contacts divided by the current and the result is going to be that expression A plus B times B plus C divided by B times A plus B plus C ok now I could also do a similar one I could also inject the current in contact N take the current out contact O measure the voltage between contact P and contact M and go through the same analysis and I would get the expression here ok now it's actually quite simple if you multiply each of those by pi over R sub S and then exponentiate each of those and then add them up you'll see that they add up to 1 ok so that means that if I can solve that equation the two measured quantities are R sub M N O P and R sub N O comma P M those I measure then if I put it into that equation and if I guess the sheet resistance it ends up solving that equation I've got the right sheet resistance so I could just solve that equation I think you have to solve that equation iteratively I don't think there's an analytical solution to it so that's very nice now I don't know how Vanderpoth thought this up thought to do all of this but it's beautiful now he took it one step further he said ok if we could do that on an infinite half plane he knew some complex analysis and he knew how to do conformal transforms and he knew that he could map that infinite half plane onto any arbitrary shape as long as there weren't any holes inside or anything so the same equation holds for an arbitrarily shaped sample so again all I do is inject the current in two contacts measure the voltage at the other two contacts divide by the injected current that's one of those measured resistances then I inject the current in the other two contacts do the same measurement they have to satisfy that equation and if I solve that equation for rosa best I get the result ok so this was done in like the 1950s as I said I don't because it was very hard to do lithography and make hall bars if you could just break off a little chunk of sample that worked very nicely but it also works very nice people frequently now will do this with lithography in a hall bar they'll just make a little square sample and put four small contacts around the edges the same equation applies except in this case because of symmetry those two resistances are equal and now you can easily solve the equation and the sheet resistance is just pi over log 2 times v over i I think it's 4.631 about 40 years ago I used to do these measurements I always would inject 4.631 milliamps when I did this because then the answer would come out in ohms anyway something like that ok so that's the van der pohe technique so just to summarize so you know about this because there are two common ways to do these techniques they both in the end give you the same answers either you make a hall bar geometry and you do it or you make a van der pohe geometry you get the same information ok when we're doing the hall effect measurement we'll inject current in alternate contacts apply an electric field out of the page measure the voltage in the other two alternate contacts we'll flip the sign of the magnetic field measure the same voltage again subtract the two and multiply by one half that gives us the hall voltage when we're doing the resistivity we inject current in two adjacent contacts the voltage in the other two adjacent divide by the injected current that gives us the resistance do it for the other two contacts and then solve this equation and we get the resistivity ok so that's the van der pohe method so we have two different ways different labs like to use different approaches but both of them are commonly used so we can get the sheet resistance and the density and the mobility if we want so let me talk a little bit about temperature dependent measurements so a lot of times the equipment that people have set up to do these measurements will frequently have a temperature controlled stage so that you can do all of these measurements as a function of temperature because if you do it as a function of temperature you might be able to do something about what's going on in your material so if you extract the mobility and plot it versus temperature people will produce plots like this now they won't always be careful what this probably is is the Hall mobility because it was probably measured by the Hall effect so you have to read the paper and find out exactly how did they deduce the mobility not always in a MOSFET you don't have to use Hall effect in order to do this but you'll frequently get a characteristic that looks like this at low temperatures as you start heating things up the mobility will improve but if you go to high temperatures it will turn around and start to drop and there's a simple interpretation of that the decreasing mobility at high temperatures suggest that there's more and more phonons or lattice vibrations that are scattering and the increased mobility at low temperatures suggests the presence of charged impurity scattering so experimentalists will do these measurements they'll see a characteristic like this they'll say I'm dominated by charged impurity scattering at low temperatures and I'm dominated by phonons at high temperatures and why does charged impurity scattering have that remember we talked about how the randomly located charges put a roughness on the conduction band edge which scatters electrons yet the electrons have higher energy they don't see that roughness as much and they scatter less and their kinetic energy is then related to KT so as you heat it up they have more kinetic energy their higher energy they don't see the roughness of that conduction band edge as much and they scatter less frequently so the mobility increases and it's also easy to understand we expect the phonon scattering rate to be proportional to the number of phonons that are there the number of phonons is given by this Bose-Einstein factor and there's a temperature in the denominator so you can see that as the temperature increases the number of phonons is going to increase and they're going to scatter more electrons so let's say you have a series of samples that you've grown by different techniques and you might get some characteristics that look like this and generally what will happen is that the purer samples are the ones that are plotted over to the right so what happens here at high temperatures is they all look alike because they're all dominated by phonon scattering but the samples that are doped less heavily or that have fewer unintentional will display different characteristics so as you cool the sample and you get less and less phonon scattering you'll reach a point where now the ionized impurity scattering is dominant and that has a different temperature characteristic so you'll turn around and go down if you're more pure and there are fewer dopants then you have to cool it more until you get to the point where the charged impurity scattering becomes important so the maximum mobility that you can achieve is related to how pure your sample is so a lot of times when people are growing crystals and they're trying to understand how pure it is you know how many unintentional dopants or charged defects do I have in it they'll do a measurement like this and from the peak mobility they can tell something about what the concentration of either intentional or unintentional dopants is there ok so a couple of other things I'm on borrowed time here now so I'm going to do a couple of other things here quickly just because I think you should be acquainted with them one of the things I wanted to talk about are possible errors in hall effect measurements you know good experimentalists are just enormously careful about things when they do these measurements or anything that can happen you know so if you look at this we're injecting a current in contact 0 and taking it out contact 5 ok you know and I was thinking I'm doing this measurement at some temperature 300 k or maybe I cooled it down to 77 k or something to look at how things change with temperature ok but we understand thermoelectric effects and we know that if we inject the current in contact 0 and take it out in contact 5 we should have peltier cooling on one end and peltier heating on the other end now maybe we put this thing on a good thermal conductor and we're trying to hold its temperature constant but we we might not be able to hold it perfectly constant there may be a temperature gradient in the x direction you know how would that temperature gradient in the x direction affect what we've measured ok and we know who are careful experimentalists worry about that ok so I just briefly you know let me this is going to go by a little bit quickly but let me just give you a flavor for what happens we have this magneto conductivity tensor so I can write that expression on the top as a sum you know my rule for matrix multiplication is sigma sub ij times e sub j and I sum over j ok that's what's going to give me the ith component ok now what people will frequently do is they won't show you the sum and they'll make this rule that when they say sigma sub ij times e sub j whenever an index is repeated you're supposed to do the sum that's called the summation convention so that's just another way of writing the equation on the top we'll call that indicial notation because it shows the indices and the other one is in vector matrix notation now I can write our current equation in indicial notation also but then I have to learn how to write a cross product and the trick for writing a cross product is introducing this thing called an alternating unit tensor epsilon sub ijk and it has the property that if the indices are in the proper order x, y, z or it doesn't matter it could be y, z, x as long as it's increasing in that order it'll be plus 1 if they're in the opposite order like z, y, x it'll be negative 1 and if any index is repeated it'll be 0 and if you follow those rules and figure out which components come out of that expression on the lower right you'll see that that's e cross b what we had in our right so what that means is that we can take our expressions for our transport equations we saw that all of those parameters resistivity, c-beck coefficient peltier coefficient, thermal conductivity become 2 by 2 tensors we can write them in this indicial notation and then I can if I apply a b field off diagonal components that came from the cross product from the Lorentz force they're all going to look similar to the conductivity and they'll all have expressions like that they'll all look like that ok so now let's get back to the problem that I'm trying to solve let's assume that we're doing this Hall effect measurement but we've got an inadvertent temperature gradient in the x direction and the question is how is it affecting our measurement ok so in the Hall effect measurement so we'll start with our current equation in the Hall effect measurement we're trying to deduce what the electric field is in the y direction in the Hall bar so we'll just write this expression for the electric field in the y direction we'll just expand that out ok now if I look at that expression first of all my experimental condition is that I'm open circuited in the y direction where j sub y is 0 if I look at the second part I'm applying an electric a magnetic field in the z direction I'm only allowing current to flow in the x direction so the only possibility let's see j sub j is j sub x ok so epsilon sub y jz becomes epsilon sub yxz those indices are in the wrong order so that's a minus 1 because they're going in anti-cyclic order I have a C back coefficient that also depends on that cross product and that depends on the temperature gradient in the x direction and the magnetic field in the z direction and that's a yxz that's also in the wrong direction ok so this is what I get from the expression there will be a contribution due to the Hall voltage that's what we measured before when we did the Hall analysis but there's an extra part of the voltage now due to that temperature gradient in the x direction that temperature gradient in the x direction has ended up giving me a contribution to the voltage in the y direction that's called the Nernst voltage we want to deduce the Hall voltage in the Hall concentration but there might be a little bit of Nernst voltage you can see that the Hall voltage is proportional to the magnetic field times the current the Nernst voltage is proportional to the magnetic field times the temperature gradient what if I quickly switch the magnetic field and the current at the same time well the Hall voltage won't change sign but the Nernst voltage will change sign and the temperature gradient is going to take some time to change in reverse so frequently you'll see in Hall effect measurements they're all set up to do a measurement and then quickly switch the current in the voltage and do another measurement again and then average the two because you can cancel out various extraneous effects that way ok so there's a bunch of these effects there's another one called the Rage Ladouk effect they all have to do with these various thermoelectric effects in the presence of a magnetic field the annoying part is there's one called the Eddingshausen effect that has the same dependence as the Hall voltage and it can't be subtracted out with this switching but at least it should know that it's there ok so I want to end by just saying a little bit about small magnetic fields hard large magnetic fields just because I think you ought to be acquainted with even though we can't do justice to it you ought to be acquainted with it a little bit early on in the lecture before the break I made an assumption that we were dealing with small magnetic fields and remember there was a cyclotron radius cyclotron frequency which was the frequency at which the electron orbits the applied magnetic field it's just qb divided by m and I said I could throw away all of those extra terms if I said that frequency times the scattering time is much less than 1 now you can also see that I could write omega c tau as q times tau over m times b so I could also write it as mobility times b so another way of expressing the low field criteria is q times b is much less than 1 so first question is what is the physical meaning of a small magnetic field so we have this b field pointing out of the page we have electrons that are orbiting the b field at the frequency omega sub c but every now and then they scatter and it deflects them and what the low field criterion means is that it scatters many times it scatters frequently before it can complete an orbit in the high frequency regime the frequency of orbit would be much shorter than the scattering time so you could actually complete an orbit that's the difference between low and high magnetic fields ok so let's look at some numbers if you do silicon and if we take a typical laboratory magnet it's a few thousand kilo gauss and if we compute numbers then omega c tau or mu times b is much less than 1 we're in the low field regime so for the common magnets that are inexpensive and easy to find and that people include in their labs for a material like silicon it would be very hard to get to the high field regime ok ok you know just to calibrate you the kind of magnets we have over here at the high technology center are somewhere between 1 and 8 tesla 1 tesla is 10,000 gauss alright so we're frequently in the low field regime if you want to get really high magnetic fields you go to somewhere like the national magnet lab where you have a superconducting magnet then you might be able to get up to 45 tesla then you could get to the high magnetic fields now but there are materials that have very high or low scattering rates where you can get to high magnetic fields with common laboratory magnets and these tend to be three fives so if I look at a material like a modulation doped indium aluminum arsenide ingest structure at room temperature its mobility might be roughly 10,000 now this product is approaching 1 if I had a magnetic field that was 1 tesla or a little more I might be able to get there at room temperature if I cool it down and go to liquid nitrogen temperature I can get greater than one so with these high mobility materials and common laboratory magnets you can get into the high field regime that we've avoided so far what happens in the high field magnetic regime so some really interesting things happen here and so we have this electron orbiting and now it can complete an orbit because we're in the high magnetic field so it's like a harmonic oscillator but we know from quantum mechanics that harmonic oscillators have quantized energy levels and this is a classic problem you solve in quantum mechanics and you find that the energy levels are quantized n plus one half times h bar omega the one half is the zero point energy and then you can have quanta of energy above that so this motion, these energy levels have to be quantized these are called Landau levels so what does that do so we have this 2D film the density of states was constant it was the same at any energy but now when we put the high magnetic field on we can only have quantized energies that can only be discrete energies where the electrons are that these this continuous density of states has been broken up into a number of Landau levels so they're all separated by omega c the density of states is now a series of delta functions just located at these different energies now you might ask what's the strength of the delta function how many states are there in each delta function so we haven't created any new states we've just rearranged them all of the states now have to fit into these Landau levels and if I look now the Landau levels are separated by h bar omega sub c my 2D density of states was m over pi h bar squared that's the number of states per EV so in that white region there there h bar omega c times the 2D density of states states in that region all of those states have collapsed into that one Landau level so the number of states in that Landau level is 2 q over b divided by h so they're just all divided up there now you know that let's ask what happens if you do scattering you know what happens if there is a little bit of scattering it makes an orbit or two and then it scatters well those levels are going to broaden and that should be expected because we have this uncertainty relation delta e delta t is greater than or equal to h bar so if there's some uncertainty in the amount of time that it's in a Landau level the Landau level is going to broaden in the energy so the uncertainty is the scattering time so the width of those levels is going to be on the order of h bar over tau so if we want to observe these Landau levels then the separation of the Landau levels has to be bigger than their broadening they'll all merge together and we won't see any individual ones well actually the criteria that the separation of Landau levels is bigger than the broadening of an individual Landau level is omega c tau is much much bigger than one it's just the criteria that we've been using for high magnetic fields so if we have a high enough magnetic fields these Landau levels are separate and you know you can see some numbers here let's say we have one Tesla we can compute the density of states in each Landau level we can do say a modulation doped film let's say it has 5 times 10 to the 11th per square centimeter of electrons then you just divide that number by the density of states of each Landau level and you can see that 10 of these are completely filled and the last one is 40% filled now how high at a mobility would you need to observe these well let's see you need mu b greater than 1 so basically for a one Tesla field you need mobility greater than about 10,000 but we can do that in these high mobility materials you know typically they are done by techniques called modulation doping so these are the kind of things that you might measure and you know just I'm not going to go in detail through this but we'll discuss some of it there's a very nice reference I have at the end okay so we sweep the magnetic field along the bottom so you can see it's going from 0 Tesla up to 8 Tesla that's a reasonable regime those are the magnets we have in our lab you don't have to go to a superconducting magnet in order to get those kind of fields on the vertical axis to the left we're measuring the Hall voltage remember that the Hall voltage was proportional to the magnetic field and you can see down near the origin that line going up at about 45 degrees that's just the measured Hall voltage proportional to the magnetic field that's the way it's supposed to work under low magnetic fields that's what we derived now the axis on the right is the measured voltage in the x direction in the direction of transport that gives us the resistivity now under low magnetic fields those were the diagonal elements of the magnetoconductivity tensor the magnetic field didn't affect them we just measured the resistivity if we had a low magnetic field nothing changed there you can see that that line is constant and independent of magnetic field to begin with for small magnetic fields then it starts to oscillate in these beautiful oscillations that's happening when these levels are developing and as the Landau level as you increase the magnetic field you increase the separation between these Landau levels sometimes you have a finite number filled sometimes your Fermi level goes inside a Landau level and then it goes between a Landau level so you just get this oscillation from the frequency of that oscillation you can determine the carrier density and this is a common technique that people use in these samples to get the carrier density instead of the Hall effect now you see some even more interesting things going on if you continue to crank the magnetic field up you can see that the voltage drop in the direction of current transport starts to go down to zero there is no resistance there's no voltage drop there you can see that the Hall voltage starts to acquire steps every time the resistance in the longitudinal direction goes to zero the Hall voltage reaches a plateau and you can see the numbers there 12, 10, 8, 6, 5 those are the number of Landau levels that are occupied as you get the stronger and stronger magnetic fields they're separated more and more and fewer of them are occupied what is this right so this is the Quantum Hall effect you know to explain this is a little more involved it involves things like edge states you have electrons tend to move in one direction along one edge of the bar and the ones moving in the opposite direction move along the opposite edge of the bar which makes it very difficult to back scatter because the only way it can back scatter is to go to the other edge so it wasn't too long ago where a Nobel Prize was awarded for the discovery of this effect but it's something that is very commonly observed now people will routinely do it in the lab to measure the Hall effect when people started working on graphene one of the first things they did was to look at the Quantum Hall effect and it displays unusual features that come from the band structure of graphene okay these oscillations that you see before the onset of the Quantum Hall effect when the resistance drops to zero these periodic oscillations in magnetic field are called Shubnikov-De Haas oscillations it's a commonly used characterization technique in the lab and from it you can deduce the number of electrons that are there okay so just to summarize you can use either Hall Bar or Van der Pohe geometries to measure resistivity and Hall effect the temperature dependent measurements are typically done too to give you more information about scattering mechanisms careful experimentalists really worry a lot about extraneous effects that can be getting in the way of what you're trying to measure so those thermoelectric effects are an example of them and for the most part I've discussed low magnetic fields are accessible when you're dealing with high mobility materials and some really interesting things happen there I haven't talked at all about how you measure CBAT coefficients or electronic heat conductivity those are a little more difficult to measure okay so let me just point you reference one here is a very comprehensive if you need to work on characterization and you want a very comprehensive book that covers most of the commonly used techniques that's a very good reference you can look at this the standard reference for the Van der Pohe technique is still this 1958 paper by Van der Pohe and I mean he solved it correctly and people still use it okay so that's it we're almost at lunchtime but if you're if you have any questions we can take a few questions before lunch yes sir all measurements we have set up with a maximum magnetic field some electromagnetic now if you want to make measurements on a sample which is very low amount of coping say 10-15 and 10-14 now one issue would be to make homing contacts but in all these discussions we are assuming that we have a homing contact yeah well yeah I don't so your question is are we assuming that we have omic contacts here now if I'm doing a hall bar geometry I may have some bad contacts for the current injecting ones but I'm not measuring a voltage across those contacts so it doesn't really matter whether they're bad right and I might have some bad contacts when I put my two probes down to measure the and the four probe measurement but no current is flowing through those contacts so yeah I mean if it's a nonlinear voltage characteristic yeah but in the hall bar geometry even if that's nonlinear you're forcing a current through that bar and then you're just measuring a voltage at two points inside the bar so I don't think the contacts are really hurting you in that kind of measurement yeah if suppose we have substantial coping and we have made in homing so what we usually homing is that the carriers are now going through because the depletion region is very thin there so then you can see the thermal electricity because it's not now going all over the conduction so your question is if you've doped the semiconductor heavily in order to try to make a good contact would you see these strong thermoelectric effects so I think the answer would be a heavily doped semiconductor has a low C-beck coefficient means it also has a low Peltier coefficient so no it would be less important for heavily doped semiconductors I think there's a question near the back you're asking about this transfer length method so it's this method here and your question is does it well so here we probably are assuming that we have omic contacts here and it doesn't really tell me anything about the details of the contact it just tells me what the contact resistance is but the contact does have to be an omic one it can't be a rectifying shot key barrier so the philosophy is a lot of times the contact resistance is something really important that you want to know so then you use a technique like this that will tell you not just the sheet resistance of the material but also the contact resistance if you're mostly interested in the material and you're not so interested in the contacts then you use these four probe measurements or hall bars that just allows you to measure the material you're interested in ok