[MUSIC] In previous lecture, we learned the concept of a MOSFET transistor. Especially, we learned that y saturation region is forming IV characteristic, IV VD IV curve. Now, let's solve the IV equation more mathematically. Now let's assume that you apply drain voltage to the drain VD on drain region. And the source region, potential is the 0 and drain region is VD. Those VD is apply from the 0 to drain of region L like this. At certain point of the channel region, let's say they're here, let's say the DL region in certain of the channel region. Then all data as to region, then there will be potential also changing in a dVx. So those dVx is increasing like this. And then potential here is, Vx. This Vx is between the 0 to VD. Then Q inversion. Inversion charging here is equal to the Q is to CV. So Ci, insulator capacitance, times, Gate voltage minus bridge threshold voltage because up to the threshold voltage, there's no inversion. And there is -VX of this region because you applying VD then drain region potential is increased. They are a negative effect, right? So here, as I saying previously, VG minus VTH is 3 volt, and you applying 3 volt, then there's the no inversion, right? So those should be subtracted. Why we have a negative? Because electron charge in a MOSFET is negative. So this is the positive charge, that's why we put the negative. Then current in this small fraction with ID, with dx is mobility time channel width and Q inversion time dVx. Z is the width. This is the equation that we learned in the previously. IDS is the channel width times Q inversion Z carrier velocity and CI region minus VTH. And carrier mobility is mobility times electric field. This is the extremely small drain voltage that drain voltage is not influencing the inversion charging channel. If the drain voltage is enough voltage that influencing the charging channel, then this should be considered as Vx and should be subtracted in addition to the threshold voltage. Then total current can be calculated from the integration from 0 to L, 0 to channel length L IVdx, and mobility channel width Ci in the integration of the 0 to VD, and VG minus VTH and Vs per dVx. The final equation is ID equal mobility ZCi per l VG minus VTH and VD minus the 0.5 VD square. So in small VD, a string in small VD, let's say that VD equals 0.1. Then VD square is 0.01, then neglect it. So small VD, you can calculate ID width like this. But not small VD, in any VD other than small VD, then this factor is not neglectable. So let's say that, what about the VD saturation which is not small VD? Then VD saturation, as I said, is occurred at pinch-off, where the VG minus VD(sat) equal to the VG minus VD(sat) as threshold voltage. Then you should insert VG minus VTH here. And VG minus VTH is VD(sat). This becomes the VD(sat) square minus 0.5 VD(sat). Then conclusion is ID(sat) is 1/2 mobility Ci Z per L (VG minus VTH), which is the VD(sat). In first slide of MOSFET transistor, I said ID(sat) is proportional to the square of the VD(sat), which is the quadratic relationship. So this is the transfer curve, ID-VG curve. And then, ID is, previously, we calculated this number. And then slope of the ID-VG curve, which is the transfer curve important to calculate the mobility, this slope in transfer curve called GM, transconductance. GM is the slope of the transfer curve, which is the derivative of ID per VG. Then if you derivative of this ID with VG, then this is the final value. From this value, mobility comes with this equation. Think about this equation. So you made a MOSFET transistor, you know the channel lengths, since you made it, and then you know the channel width. And then, so you made a MOSFET, then you know the channel width. And then Ci is depending on your thickness of gate dielectric. And Vd is the constant here, let's say the Vd is 0.1. Then you know the Vd, then if you measure the IV curve, you can calculate the gm. So if you know the device structure and transconductance, which is the slope of a transfer curve, then you can calculate the mobility. So actually, if you get a transfer curve like this with the Excel data, then you can get the slope by the delta x per delta y, then the slope becomes like this. You can put the gm to here, or you just can get the maximum slope of the transfer curve, you can insert them in here. Let's solved some equation problem for n channel MOSFET device gate with oxide thicknesses of 10 nanometers, Vth is 0.6, W is 25, L is 1 micron. Calculate the drain current at Vg 5 volt and Vd is 0.1, mobility is 200. Then current equation, Is this. So we know the Z, channel width, and L, mobility, and Ci is the, depending on the thickness, oxide insulator, (3.9)(8.85) x 10 to the -4. Thickness is 10 nanometer, which is 10 to the -6. Then Ci, we got it. Then Ci, Vg, 5 volt, minus Vth is the 0.6, Vd, 0.1, 0.1 squared, then current. So this Vg 3 volt, Vd equals 3. 5 volt is the saturation region, because the Vd, Vd is above the Vg minus Vth, which is the here, Vd is the 5 volt, Vg is the 3 volt, threshold voltage, 0.6. So let's calculate the saturation current first. Because this is a saturation region, once we know the saturation current, this is the same thing. Saturation current at VG is 3 voltage when drain voltage 5 volt, and saturation drain voltage is VG minus VTH when drain voltage is 2.4. So using this VD stat of 2.4, we can calculate ID equal this value, and then here's the 2.4 square minus 0.5(2.4 square), and this is the saturation current. So Vg, 3 volt, Vd, 5 volt, where the saturation region, ID, is not increasing in the same value as this. What you learned so far is the long channel MOSFET where channel length is very long. And so in long channel device, Id is saturation after pinch-off, and current is quadratically dependence on Ig on Vg. So ID stat is quadratically, Quadratic relationship. However, in most of MOSFET is the short channel device where our channel length is very short. In short channel device, however, it's not. Those characteristics no longer apply. So then first, before we go, what is the definition of the short channel and long channel? People may have a different view, but many people saying that channel lengths below the 1 micrometer. Now we have 20 nanometers and extremely one. And channel lengths reached below the 1 micrometers around 1990. However, channel lengths below the 1 micro, we can call the short channel MOSFET, where the very weird phenomenon occurs which is not equal to the long channel device. In short channel device, ID is no longer saturating, they are slightly increasing after the saturation region. Also, instead of quadratic relationship, short channel device, current are linearly increasing. ID(sat) is linearly increasing to the VD(sat). The phenomenon of the short channel effect is a little complicate above the barriers parameter. However, in the most important cause is like this. In short channel device, carrier velocity is reached the saturation velocity before pinch-off. So when the electron comes out from the source and before reaching up the pinch-off where saturation currents occur, the velocity of carrier is reached the saturation velocity and does not increase. Then what happen is like this. So if at the critical electric field, Ec, then velocity of carrier starting to maximize at the 10 to the 7 centimeter per sec. And velocity of carrier is expressed by the mobility electric field over the 1 plus mobility electric field per Vs. Vs is the 10 to the 7. Some of you already probably seen that velocity saturation is like this. That's actually log scale, in linear scale velocity expressed like this equation and this graph. So in high electric field, velocity's one over, Boost Vs, therefore becomes the saturation velocity. So, in this saturation region, final equation becomes ID equal 10 over Q inversion and per carrier velocity. And then Q inversion is the CI BG minus Vth, and the velocity is no longer electric field or drain voltage related, but they reached the saturation velocity. Therefore, ID is proportional to the VD(sat), that quadratic relationship.
