Lec 10

[Music] [Music] in the previous lectures we had discussed inversion charge and depletion charge linearized model of inversion charge and a linearized model of depletion charge with respect to the surface potential and we saw inversion charge in Weak inversion and moderate inversion and strong inversion its relation with the surface potential so in this lecture we would be discussing on the relationship between inversion charge and the surface potential when strong inversion approx imation is valid to begin with let us recall the function f that we had derived and so it was exponential of CH s - 25f minus V the local channel potential divided UT minus exponential of minus CH by UT plus CH by UT square root so here we had said that that this represents the inversion charge this accumulation and this depletion so in this what one would see is that whenever the value of CH is larger than or equal to 25f + V what what would happen you would have value one for the first term and after that it would exponentially increase first term in F has a value 1 so it is still much smaller than CH by UT so for example typically for CH being 25f or nearly 25f you will have for our example device that we have been seeing you would have this to be quite large it would be of much larger than one so therefore you can though it is said conventionally that in general that when ch is equal to 25f a strong inversion sets in but it doesn't really because actually the inversion charge density is comparable to the bulk charge density at just at the interface but the inversion charge is much smaller than the depletion charge present when you look at the per area charge from the top of the device so if you look from the top of the device and look at the per area charge per unit area charge the inversion charge even when CH is exactly equal to 25 would not be comparable to the depletion charge depletion charge would be much larger and that would much orders of magnitude larger or an order of magnitude larger so you would uh find that uh though the inversion charge density per uh per unit volume just at the interface is almost equal to the doping density or depletion charge density per unit volume just at the interface for CH is equal to 25f the actual inversion charge density which contributes to current that is per unit area inversion charge density would be much smaller than the depletion charge density so if you want to really Uh define threshold you would say that the surface potential at which strong inversion occurs is slightly larger than 25f how much larger than 25 that we'll discuss in this lecture so we would say that surface potential is some CH not plus v v is the local channel potential Channel local channel potential so where do you where do you see that so suppose this is V so here you would have ch not so this would be CH not plus v and this CH not is the potential surface potential at which we consider the inversion strong inversion to have setting strong inversion set now we'll uh determine the value of chin not as we go ahead so we could have written an expression for inversion charge as this is just the charge across the oxide capacitance VG gate voltage minus flatband voltage so a plus v now what is this so what have we done this is the surface potential CH is at this point this is the value of surface potential CH not plus v and here the surface potentials value is CH not itself okay so so what is this if we observe this entire thing is what we have defined as vtb vtb of value of surface potential okay so now if we look at this vtb we would now we would write it as vfb plus the surface potential plus gamma b square root of now so what is vtb vtb is that value of gate voltage at which the inversion charge can be considered to be zero so for a given local surface potential at that point the inversion charge can be considered to be zero now there is a caveat to it so the surface potential would be CH not plus v and the value of V should be such that the inversion charge can be considered to be zero because when V is zero for the same gate voltage you would have inversion charge so now when V is zero what is the value of vtb when V is zero so that is just vfb plus chot plus gamma B CH so what is this value so when gate voltage is equal to VTO when gate voltage is equal to VTO what is the value of inversion charge so inversion charge is not really zero because there is inversion charge sufficient even in moderate inversion there there is always an inversion charge present but using strong inversion approximation we know that when we use strong inversion approximation that we had discussed in the previous lecture Qi we had Qi b c Ox we had approximated it to be VP minus V by 2 so in this case for V equal to VP inversion charge comes out to be zero so this is strong inversion approximation this is an approximation so we will now like to interpret this vtb in terms of strong inversion approximation so at VP what happens 2 Qi + Qi logarithm of Qi is 0 Qi is not zero but in the but we would consider using strong inversion approximation what we get is that at VP so when you use strong inversion approximation what you get at is that at VP at VP Qi is zero so we will since so looking at uh so using strong inversion approxim what is the value of gate voltage so that at for a local channel potential V the the value of inversion charge is zero so inversion charge is zero now when we use strong inversion approximation and what is the value of the local channel potential at which this happens or what is the gate voltage for which at for a given local channel potential V the inversion charge density is zero according to strong inversion approximation so that VG we can write as pfb plus CH + V + gamma V Square < TK of CH + V isn't so you can also write this as VTO + V + gamma B to square Ro TK of CH + V minus just we have R written it using vtu now what is the or now so what we are saying is that for a given value of voltage CH local channel potential V what is the gate voltage at which inversion charge is zero now let us write it in another form let us say that for a given gate voltage what is the local channel potential at which this happens so for a given gate voltage VG the local channel potential at which inversion charge density according to strong inversion approximation is zero that is at VP so instead of V you just replace V with VP because at pinch of potential we had seen that uh the according to strong inversion approximation the inversion charge density is zero because because Qi is VP minus V by2 in strong inversion approximation so so what do we observe that the relationship of VG versus VG versus VP is similar to the relationship of vtb versus CH or vtb versus CHP all these relationships have a similar form so we will use this further so now let us invert let us from this obtain an expression for from this expression of VG versus VP let us also obtain an expression for VP so what do we see so we rewrite the same expression as as a quadratic equation so that we can solve for the roots of chin not plus VP so written the same expression so now solve this as a quadratic equation and obtain the value roots for this and then Square it so once you do that what do you get you get that sin not + VP minus GMA b b 2 these are roots of uh so if you call this one roots of one so in this Roots we have neglected one of the roots we have dropped one of the roots because that becomes larger than VG so we cannot be larger than so this gate voltage let us comment more on this gate voltage this gate voltage is the this is the value of gate voltage for which according to strong inversion according to the strong inversion approximation this is the value of VP so for a given value of the gate voltage what is the value of pinch of potential in strong inversion approximation so near the source edge of the the channel the potential is CH not as you move towards so there is strong inversion there and as you move towards the drain the local surface potential keeps on changing because of the drain to Source voltage and when it reaches this VP according to strong inversion approximation the channel is inversion charge is zero in reality the inversion charges the value of Qi where 2 Qi plus natural logarithm of QA is z okay now let us look at the slope function n so how did we Define n in the first place so it is the rate of change of vtp with respect to the surface potential say yes but we have seen in the form of in the expression for vtb for a given value of V local channel potential the expression looks similar the only difference is that instead of CH you have ch not plus v so you could also say that you get the same expression since CH not we are saying is a constant we will justify that it's a constant as we go ahead it's a constant where inversion strong inversion approximation can be used okay so where you can see that our expression for strong inversion approximation is valid from for a value of CH is which is equal to CH not so you will obtain you will therefore have that n is also equal to D vtb by DV and since VG versus VP function which we had just now discussed so what was that function we were saying that for a given gate voltage at VG if strong inversion has set in then at the source edge of the channel at the for the edge of the channel which is is close to Source the surface potential would be chin not but as you move towards the drain for a positive and for a nonzero VDS As you move towards the drain the inversion charge density would keep on reducing so according to strong inversion approximation when we when uh the value of surface potential is chin not plus VP pinch of potential the inversion charge Den would be zero so so we are trying to find out the value of VP so we had obtained the value of VP for that given value of gate voltage so now we are and the and if we write the value of VG in terms of VP what would you have you would say that that value of VG is vfb flatband voltage plus VP + CH + gamma b s otk of VP VP plus CH the at the source edge of the channel the surface potential is CH not as you move towards drain it is now V plus CH not and we are saying that for this gate voltage what is the value of pinch of potential VP where according to strong inversion approximation you can neglect the you can assume the inversion charge density to be zero so this expression is also similar to that of vtb so therefore if we plot this gate voltage and its corresponding VP you will also get an the same slope Factor n so here we have plotted VG versus VP or vtb versus VP okay so what is this inversion charge density for a given VG now this is for even CH local channel potential small smaller than even VP so it's in strong inversion approximation so what is the value of this Qi so if we can observe that if we really plot Qi by Cox versus VP you will have a slope minus n between 0 and VP when it is zero it is actually the value of VTO which is the value of vtb value of vtb is VTO when when VP is zero okay so from here what is the value of VP so you can say that VP is VG minus VT by n so this and Qi by coox is n * VP minus V so this you can since here you have V so therefore n * VP minus V would give me the inversion charge density at VP according to strong inversion approximation the value of inversion charge density is zero now you would say how can we assume it to be zero in moderate inverse at VP it is actually 2 Q i+ Plus L logorithm of Qi but when we should observe here that as inversion strong inversion sets in for a small increase in the gate voltage or a small increase in the surface potential there would be a large increase in the inversion charge density and therefore when you compare for a value of V smaller than VP then inversion charge density with respect to that at VP the inversion charge density would surely be I mean Qi of V we can write it as Qi of V is much larger than Qi of VP even when you don't consider strong inversion approximation so when strong inversion is approximation is not considered even in that case Qi of V is much larger than Qi of VP even when strong inversion approximation is not considered therefore it it is Justified that you can see that you can use strong inversion approximation when strong inversion actually sets in because there the relationship between inversion charge and surface potential is now exponential so so Qi of V would be much larger than Qi of VP so it is safe to assume that Qi of VP is zero so therefore VP can be given by this expression now let us look at obtaining the value of CH not let us write the expression for VP by expanding the expression for VTO so VP we had written as VG minus VT minus gamma b square < TK of V G minus V gamma B by 2 < TK of CH minus gamma B by 2 + < TK of CH so expand on vtu in this so we get VG minus vfb + CH + gamma b square root of ch ch minus gamma B the second term vg- vfb - CH - gamma b square Ro TK of CH + gamma b² by 4 so I have expanded this Square here I just expanded that square so here I have just this is just I'm writing rewriting this so several of the terms cancel out as we can see like that this we can see is actually you can this you can see is we have Rewritten VP is equal to CHP minus CH not so if if you write if you look at the expression for VP in this we have re written it as VP is equal to CHP minus CH once you cancel out terms and you find and work out the algebra you find this so if we now expand by writing VP and expanding on VTO we get this and here within the square root when we expand with you we write this we just expand this Square term we get this and then we find that there are lot of some of the Expressions get cancelled and we land up with VP is equal to CHP minus CH not so you can write that CH is equal to vp+ CHP minus VP plus CH so or in other words in normalized terms TR not normalized value would be equal to normalized CHP minus VP so we have and and this we know would be 25f + vsh so it is 25f plus v vsh which is the value of CH not so that is where we can assume strong inversion to have set in if we use strong inversion approximation so in this lecture we have discussed strong inversion approximation and we related the inversion charge density with respect to surface potential we also find a value of surface potential CH not at which surface potential value value of the surface potential CH not for which strong inversion approximation is valid or in other in other words uh we say that strong inversion actually sets in and all the Expressions which we simplified expressions of strong inversion approximation would stand for the surface potential being equal to CH not and that is larger than 2 slightly larger than 25 okay in the next lecture we would be discussing the slope Factor n how to obtain its value and we will also try to understand the linearization of inversion charge or depletion charge in moderate inversion and in strong inversion by solving a few numerical problems thank you [Music]