In this video we will discuss Hall effects. Hall effects refers to a phenomenon where you observe a building up of an additional electric field when a semi conductor, or any metal, any solid actually, any conducting solid, is subject to an electric field and magnetic field simultaneously. So, here is an example geometry. So the applied electric field is given by this voltage, Vx, and that produces an electric field inside the semiconductor along the positive x direction. So applied electric field here is along the positive x direction, and then you have a magnetic field applying along the z direction. And in this situation, you will see an electric field building up along the y direction. So depending on the type of your semiconductor it could be pointing negative y direction or the positive y direction. So this is Hall effect, and it is a very useful phenomenon for semiconductor physics and semiconductor devices, because you can use this phenomenon to determine the type of the semiconductor, carrier concentration, and their mobilities. So the microscoping mechanism that drives Hall effect is Lorentz force. So Lorentz force is a force exerted on a particular, charged particle, moving inside in the presence of a magnetic field. So Lorentz force is given by this equation. It's a product of the charge that particular charger carrier carries times the cross product of its velocity and the magnetic fields. So once again, going back to this geometry here, your E field is along x direction, produced by this voltage here. And flow for electrons, if we consider n-type semiconductor here, electrons are the majority carriers. So electrons, these guys will then move along the negative x direction. So this is your electron velocity. Then the cross product of the magnetic field and this velocity is going to give you a vector pointing to positive y direction. But there is a negative charge for the electron that flips the force direction, and the electron is therefore pushed along the negative y direction. And that leads to a buildup of negative charge along the bottom surface. And at the same time, positive charge will build up along the top surface, and this charge will then produce additional electric field, which we call the whole field, E sub H. So in the case of anti semiconductor, with electrons being the majority carriers, Hall field will build up along the negative y direction, as shown here. And when this Hall field, this additionally electric field, is then going to influence the flow of the electrons. So the Hall field is going to push the electron along the positive y direction, which is in the opposite direction to the Lorentz force. So at some point the Hall field will balance the Lorentz force and reach the steady state. And under the steady state condition your Hall field is equal to the magnitude of the Lorentz force. So E sub H is equal to Vx times the magnetic field along the z direction. Now, the drift current due to the applied electric field, this guy here, V sub x once again, is given by the standard drift current equation that we derived previously. And in terms of the current, instead of the current density, we simply multiply the cross sectional area. So W is the width of your semiconductor, and d here is the thickness of the semiconductor. So that's the cross-sectional area. And so, this is the x direction of the electric current. And the voltage due to Hall, if a Hall field is simply the Hall field E sub H times the distance along the direction of the field, which is W again. So combining these two, we can write the carrier concentration was already carrier concentration, and in terms of the current, which you control. This is the current driven by your voltage here. And then the magnetic field, something you also control, you apply. And the Hall field, V sub H, or Hall voltage, I'm sorry, V sub H is something that you measure. And q is the constant, and d is the thickness of the film, so it's something that you also know or control. So by measuring the Hall voltage in this set up, you can determine the carrier concentration. Similarly, for holes, you can derive this equation for hole concentration p. Now, by measuring Hall voltage you can determine the carrier concentration. Now we can take a step further by writing the drift velocity in terms of mobility times the applied electric field. And by plugging this into the drift current equation, then you can derive an expression for the mobility. Again,this time in terms of the voltage that you apply. And the geometry parameter, L is the length along the x direction, and W and D is the cross-sectional area. So W is once again the width, D is the thickness, and n carrier concentration is something that we determined in the previous slide. V sub x is a voltage that you apply. So from this you can determine your mobility, and again, similar expression for holes can be described. So using these equations, and by measuring Hall voltage for a given applied voltage, V sub x, and applied magnetic field, B sub z, you can calculate. You can determine carrier concentration and mobility. And this is a standard technique that any semiconductor companies on research institution use to determine carrier concentrations and mobilities. And the standard technique that people use is something called the van der Pauw technique. And the van der Pauw technique looks like this. You prepare a square shaped sample, and you make ohmic contacts, resisted contacts on four corners. So 1, 2, 3, 4 here. And then, you apply a constant current here, and then measure the voltage along the same direction. You do this along the direction of 1 and 2 here, and measure the voltage along the same direction between 3 and 4. And you repeat the measurements for the perpendicular direction. You apply the current, constant current between one and four, and you measure voltage between 2 and 3. So these give you the resistance, two resistance values along the two different directions, R sub A and R sub B, and from those you can determine the sheet resistance of this film. And once you know your sheet resistance, the bulk resistivity is simply the sheet resistance times the thickness. And the resistivity, of course, is a reciprocal, or inverse, of the conductivity. And then you measure the whole voltage now. So in this case, you apply current, constant current, along a diagonal direction, and you measure voltage along the upper to diagonal direction, which is perpendicular to your direction of your current. So this is a measurement of your Hall voltage. And once you know your Hall voltage using the equations that we derived earlier, you can determine the sheet carrier concentration, the two dimensional carrier concentration, here, N sub S. And once you know your N sub S, then you can determine the actual carrier concentration by simply dividing the sheet carrier concentration by the thickness. And from the carrier concentration, then you can again, using the equation that we derived earlier, you can determine the mobility of the carrier.
