 Let us continue our discussion of the surface potential and threshold based solutions of the drain source current in terms of the various bias voltages for a large bulk mass field. A quick recap of the expressions for surface potential based model. The drain to source current is given by a diffusion component and a drift component. The diffusion current expression is as shown here. This is the drift current expression. This is the expression for the surface potential in terms of the gate to bulk voltage and the channel voltage V. When you substitute source to bulk voltage in the channel voltage you get the surface potential at the source and similarly when you substitute the drain to bulk voltage in the channel voltage V you get the surface potential at the drain. For the diffusion current the average surface mobility is given by this particular formula in terms of the surface potential at the source. On the other hand for the drift current you have a somewhat longer expression for the average surface mobility and this average surface mobility depends on the drain to source voltage because you have to take into account both the transfers as well as longitudinal field effects on the surface mobility. If you do calculations using the expressions of the previous slide you will get an behaviour as shown here. Ideas on a log scale versus the gate to bulk voltage VGD. In this behaviour the diffusion component plot looks like the black curve here. So this diffusion current actually matches the total current in the device for lower gate voltages. For higher gate voltages you have the drift current shown by the red curve. So at lower voltages a diffusion current dominates over drift at higher voltages drift current dominates over diffusion. Now in this context the threshold based model is obtained as follows. You partition these behaviour into weak inversion, moderate inversion and strong inversion where the partitioning voltages are V dash Tb that is the threshold voltage expression given by, so V dash Tb is flat band voltage plus phi s phi f plus Vsb plus gamma square root phi s phi f plus Vsb. Now as against this the VTb that is if you remove the dash that expression will be given by this formula with the change that twice phi f will be replaced by twice phi f plus 6 Vt right. Here also twice phi f will be replaced by twice phi f plus 6 Vt. Then you get the expression VTb. So we will refer to the threshold voltage expression V dash Tb as the expression based on the twice phi f criterion whereas the VTb is the threshold voltage based on the twice phi f plus 6 Vt criterion okay. So V dash Tb partitions the behaviour into weak and moderate inversion. On the other hand VHB partitions the behaviour into moderate and strong inversion. So this partitioning is the first step in a threshold based model. As we have remarked earlier it is a regional model for a different regions you will use different expressions. So the part of the behaviour in the weak inversion okay here on a semi-lock plot is a straight line and therefore it can be shown to be an exponential okay an approximate exponential for diffusion current we will show that you can get a simple exponential approximation. Similarly for the drift current we will show that we can get a square law approximation and that is shown by this particular dash red line and in fact this point here is very close to the VTb okay the threshold voltage based on the twice phi f plus 6 Vt criterion. So let us begin with the derivation. So let us start with the weak inversion regime first here Vgb less than V dash Tb. This is our starting expression based on the surface potential based model for the drain to source current. Now how do we approximate this expression so that we can express this part here in terms of gate to bulk voltage, drain to bulk voltage and source to bulk voltage. So let us write this expression here and create some space so that we can do the derivation. Now first step is that we will introduce a variable u it is a normalized potential u so this is a surface potential expression in which we have used a variable u and what is u? So u is psi s minus channel voltage V plus twice phi f by Vt u suffix l will denote the normalized potential for psi s equal to psi sl and channel voltage V equal to Vdb. On the other hand u suffix 0 will represent the condition for psi s equal to psi s naught and V equal to Vsb so this is the condition at the source. Now let us see what approximation we can make so using psi s equal to Vgb minus Vfb minus gamma into square root psi s into 1 plus Vt e power u by 2 psi s. How do we make this approximation for psi s? So using this particular expression we want to approximate this particular term. So we want to represent this expression for psi s by this approximation this is actually very straight forward let me quickly take you through the steps so psi s plus Vt e power u. Now in weak inversion u is less than 0 when Vgb is less than Vdb u is less than 0 u is this term psi s is less than V plus twice phi f in weak inversion. Please recall this therefore this is negative okay now once this term is negative exponential of a negative quantity will be less than 1 and therefore this term Vt into e power u will be much less than psi s so this quantity is much less than psi s. Now this is our square root term so if this quantity is much less than psi s I can write this as follows. I can take this root psi s out and I can write this as 1 plus Vt into e power u by psi s and this quantity is much less than 1 because this is much less than psi s and then I can approximate square root 1 plus x is nothing but approximately 1 plus x by 2 so called Taylor series approximation so I am going to use this and this can be written as square root psi s into 1 plus half of Vt into e power u by psi s okay so that is what is actually written here so what kind of an approximation will result so now what we could do is you could further expand this so that this will become actually gamma square root sorry square root psi s so the same thing this is equal to square root psi s plus half of Vt e power u by root psi s so this root psi s cancels root psi s here this is what you get okay and then I will shift this gamma square root psi s term to the left hand side so this is what we are writing psi s plus gamma square root psi s is given by this right hand side term now why are we writing it like this because you can see here I have psi s l plus gamma square root psi s l psi s naught plus gamma square root psi s naught so this part I can substitute this here with u replaced by u l and this part I can simply substitute the same thing with u replaced by u naught so Vgb minus Vap term will then get cancelled when you take the difference now that is your result after this this part of the equation comes here as it is and this part has been approximated in this form now let us move it up create some space here so that we can further manipulate this expression we will remove the negative sign here and interchange these two terms now next what we do is we take this e power u naught by square root psi s naught term out of the brackets when you do that inside the bracket you will have 1 minus so here when you take this square root psi s naught out square root psi s naught will go up and you will get square root psi s naught by psi s l and e power u naught will come here in the denominator and therefore u l minus u naught will come in the exponent further we will slightly do a rearrangement we will keep this e power u naught term out and club this square root psi s naught along with gamma Vt by 2 here we will shortly see why because we will be able to physically interpret these terms let us move that expression up here so that we can do further manipulation with this now what we need to do is we need to express psi s naught this psi s naught here psi s l u l and u naught in terms of the applied biases so here also you have a u naught okay so u l u naught psi s l psi s naught in terms of applied biases that is what we need to do let us look at the behavior of psi s l and psi s naught as a function of Vgb now when we are plotting this please recall what is the expression we are using to plot okay so psi s is equal to Vgb minus Vfb minus gamma into square root psi s plus Vt into e power u where this u is psi s minus V plus twice phi f upon Vt this is the expression we are plotting now we are plotting this with Vgb so when I want to plot psi s l when I want to plot psi s l I should put V equal to Vdb here okay and then this psi s will become psi s l so this will be psi s l this will be psi s l notice look at this expression okay so when your Vgb is Vfb Vgb is Vfb this quantity is 0 okay this quantity is 0 you can easily show that your psi s l will have to be 0 for that condition okay when Vgb is Vfb psi s l is 0 now when Vgb starts increasing your psi s l starts increasing but ultimately the psi s l saturates why does it happen you can again see this expression you are going on increasing the Vgb psi s l goes on increasing but if psi s l exceeds this value Vdb plus twice phi f okay then this quantity becomes positive in other words exponential of a positive quantity and if this quantity u starts rising then exponential of this quantity will start rising very fast however this entire term comes here with a negative sign on the left hand side you have the same psi s l so right hand side here will start increasing very rapidly if psi s l exceeds Vdb plus twice phi f too much and that will try to suppress a variation in psi s l so that is why psi s l saturates beyond what point so you can see here beyond Vgb equal to V dash Tb which is a function of Vdb okay similarly one can argue about psi s not so all that you need to do here is you replace this Vdb by Vsb then this psi s l becomes psi s not here this becomes psi s not and here this becomes psi s not okay and you solve this so you will get a lower saturation value evidently why because Vsb is less than Vdb okay so this quantity is less than that your quantity at the drain and therefore the saturation value also will be less you can see here it saturates for some small voltage beyond Vsb plus twice phi beyond this value similar arguments apply here now to clarify some of these points further we have expanded this variation here and shown the behavior more clearly okay when you increase your Vgb first your device is in depletion so until the surface potential becomes equal to channel voltage plus twice phi f okay your V dash Tb term here is written here okay this is a threshold voltage value at any point in the channel this is your MOSFET when Vsb and Vdb are applied your conditions vary from source to drain so this is your inversion layer here you have the gate this is Vgb this is the V type substrate now this is some point in the channel between drain and source here your channel voltage is V so therefore at this point what is your threshold voltage okay as we have remarked in the qualitative model the threshold voltage and MOSFET goes on changing from source to drain along the channel so this is the threshold voltage V dash Tb based on the twice phi criterion at any point here okay at any point here okay so let us look at this graph further so you go on increasing your Vgb your surface potential exceeds V plus twice phi f then you enter the weak inversion because at the surface your electron concentration becomes more than ni once your Vgb exceeds V dash Tb this threshold voltage based on the twice phi f criterion then you enter the moderate inversion now at this point your surface potential is V plus twice phi f increase your Vgb further in moderate inversion this surface potential starts saturating now and this distance difference is 6 Vt and this point here is actually Vtb so this point here is Vtb which is same as this V dash Tb but with twice phi f plus 6 Vt criterion so here you put twice phi f here you put twice phi f plus 6 Vt here you put twice phi f plus 6 Vt then this dash goes off okay that is your threshold voltage then once you exceed Vgb equal to Vhb you are entering the strong inversion and there you have almost total saturation of the surface potential okay so this is your psi s versus Vgb behaviour in detail let us proceed further now now we are interested in the so called depletion and weak inversion okay because we are talking about the diffusion current for Vgb less than V dash Tb okay so our Vgb lies in this region and here we can see that this psi s versus Vgb behaviour whether at drain or at source okay is almost the same for both these cases in this regime and further this is almost straight line approximately straight line okay and therefore we are going to approximate this behaviour by a linear equation now this is what is said here in weak inversion and depletion regions psi s l is approximately equal to psi s not you can see that here psi s l and psi s not they are both almost same and at source okay you can write the following expression psi s not minus Vsb plus twice phi f by Vgb minus V dash Tb which is a function of Vsb is approximately 1 by n so this is the slope okay 1 by n is the slope of this straight line let us see how we write this expression so take let us sketch that curve here this is your straight line portion okay this is Vsb plus twice phi f this is Vgb this is V dash Tb let us take any point here to write this straight line expression okay now this is psi s 0 now the numerator here psi s 0 minus Vsb plus twice phi f that is actually this part this is the difference so in fact this is you can recognize this as Vsb plus twice phi f minus psi s not at any point here right so this quantity psi s not minus Vsb plus twice phi f is negative so you probably will like to put this in the reverse form that is you write it as Vsb plus twice phi f minus psi s not you can write like this this is this difference divided by this difference now this is your Vgb general Vgb right so what is this difference so that difference is V dash Tb minus Vgb now what we are saying is no matter wherever you take your point this ratio will be constant because this is a straight line so that constant we are writing as 1 by n now why cannot we write it as n well you will shortly see why we will get the final expression in a more convenient form if you write the slope as 1 by n so now numerator you take the negative of the numerator as well as the denominator then your psi s not will come here minus this so you bring Vgb here to move it from there okay so this is what is the expression written here this is the expression for the straight line so let us put our results on the slide in this region psi s not psi s not are almost the same therefore you write psi s not by psi s not approximately 1 so this quantity here would be approximately 1 then what is ul minus u not so ul minus u not can be shown to be minus of Vds by Vt well this is straight forward so ul is psi s not minus Vdb plus twice phi f by Vt and u not so minus this minus psi s not minus Vsb plus twice phi f and the Vt is common now but psi sl and psi s not are almost the same therefore I can cancel this when subtracting twice phi f also gets cancelled so minus of minus of Vsb is plus Vsb so this is minus Vdb plus Vsb or minus of Vdb minus Vsb so Vdb minus Vsb is nothing but Vds okay so that is what you are getting here now what about this u not term okay so this ul minus u not we have written in this form minus Vds by Vt what about this u not term here so u not is psi s not minus Vsb plus twice phi f by Vt which we can write using this linear approximation for psi s not so you can see here psi s not minus Vsb plus twice phi f is this numerator and this numerator can be written as I can shift Vgb minus Vdb to the numerator on the right hand side okay so this quantity becomes simply Vgb minus Vdb by n that is what is done here Vgb minus Vdb by n so this is what you have got right it is a very important result and but very simple that u not varies linearly as Vgb in other words in other words psi s not varies linearly with Vgb okay that is what you get now let us substitute these here so this is one ul minus u not is minus Vds by Vt that is what is put here and exponential of u not is exponential of Vgs minus Vdb by n Vt now it was Vgb minus Vdb by n Vt all that we have done is since both these voltages are with respect to bulk we could as well take them with respect to source another common point so this becomes Vgs and we have told earlier that whenever we are taking threshold voltage with respect to source we do not put the suffix s because that is the threshold voltage that is normally used so that is how you get this term here and remaining part of the expression remains the same some further manipulations rearrange this expression as shown here where this Vt is multiplied by this Vt and you get Vt square and then this C ox is clubbed with gamma so you get C ox gamma by 2 square root psi s not you can easily show and that is left to you as an assignment that this entire quantity is nothing but the depletion capacitance Cd of the MOSFET ok so what is the depletion capacitance Cd since we are talking about weak inversion the depletion region is constant until the drain right so this is your n plus and this is your depletion region now so you have a certain depletion with Xd epsilon s so epsilon s by Xd is the Cd here that is the Cd ok so that is the depletion capacitance you can show that this term is nothing but the depletion capacitance this psi s not is a potential drop here over this depletion with Xd so you are getting a physical interpretation right and your expression is also coming out in a very simple form so your diffusion current is W by L into the average surface mobility into Vt square into the depletion capacitance into exponential of Vgs minus V dash t by n Vt into 1 minus exponential of minus Vds by Vt in this form the expression is easy to remember ok so let me put the statement of the assignment show that C ox gamma by 2 square root psi s not is equal to Cd where Cd is the capacitance of the depletion with under the gate near the source now before we move on further let me give you some dimensional insight into this expression right how do you remember this now all of you recall that in the first level course you have come across the square law model of the drain to source current and therefore you know that the right hand side has a square of voltage term in the current expression ok however that model was for drift but we are looking at it dimensionally ok so you know that dimensionally your drain to source current expression is of this form a W by L term then a mobility term then a capacitance term and a square of voltage term I am saying dimensionally ok this is not to be taken exact expression this is capacitance so I will put C dimensionally this capacitance and then you have mobility so dimensionally if you know these are the terms which are to be there now in light of this it is easy to remember the expression for diffusion current you have the W by L term here you have the mobility term here and this capacitance is the depletion capacitance when you are talking about diffusion and this square of V is the thermal voltage square ok that will come here remaining part of the expression has no dimensions right now this remaining part of the expression you can see that there is an exponential of the gate to source bias and the drain to source bias appears with a negative sign in the exponent ok where have you seen an expression like this so let me remind you when you take the diode IV characteristics ideal diode model the reverse saturation current ok this has this particular form if you take the IV expression ideal IV expression I equal to I naught exponential V by VT minus 1 where this V is negative for reverse bias ok so that is minus of some modulus of V here and the current itself is negative so if I want to write the magnitude of I I can write this as magnitude of the right hand side can be written as I naught into 1 minus exponential of model minus or modulus of V by VT now you can see that this expression is of the same form as the expression here except that V is replaced by VDS ok mod V is replaced by VDS so 1 minus exponential minus VDS by VT so as a function of VDS this diffusion current will saturate just as this diode current saturates so it is like a reverse saturation current of the diode the form of the expression however the current will increase rapidly with VGS ok because you are getting this exponential VGS minus VT term so this diffusion current is likened to the current in a bipolar transistor ok because let me remind you about the bipolar transistor current you have emitter, base, collector and your excess carry concentration in the bipolar transistor is of electrons it goes like this where it becomes very small near the collector and this is forward biased so this is the excess carry concentration in the N plus side this is the forward bias so this part of the device is like the source bulk drain part of the MOSFET ok so analogy between bipolar transistor and MOSFET for the diffusion current or current in weak inversion and inversion charge varies from source to drain so this variation of the inversion charge is analogous to this variation of the electron concentration here and the VGB is providing the inversion layer of electrons and therefore this is analogous to the forward bias which injects these electrons in a bipolar transistor ok then VDS that is the voltage between drain and source so this is difference between VDB and VSB that is VDS that is like collector to emitter voltage of the bipolar transistor ok now you know that as your collector to emitter voltage changes bipolar transistor current saturates so this is the one to one analogy between the diffusion current in a MOSFET in weak inversion and the current in a bipolar transistor now as you increase your forward bias in a bipolar transistor the current increases exponentially because these concentrations increase exponentially similarly when you increase the VGB in a MOSFET the diffusion current increases exponentially ok so instead of VGB you have VGS because you have VGB minus V dash TB and you have taken both these terms with respect to source ok so that is how it is VGB minus VSB so like you have VDB minus VDB minus VSB you have VGB minus VSB ok deciding the concentration of inversion charge here a word about the average surface mobility this is calculated using this formula now this result is simply repetition from our earlier slides the only addition here is that the psi s naught can be taken as twice phi f right because the device is in weak inversion if it is in strong inversion then you know you take the psi s naught to be twice phi f plus 6 VT now let us move on to the threshold voltage based model for VGB greater than VSB that is in strong inversion now before we go for the approximations in this regime I would like you to do the assignment so that you really build up the motivation for the approximations which we are going to discuss for the strong inversion so this assignment is the following write the complete IDS expression as a function of VGB VDB and VSB and derive the equation for the saturation drain to source voltage or drain to bulk voltage VDB sat in the 3 by 2 power law model from the condition that DID by DVDB is equal to 0 at VDB equal to VDB sat this is the expression for the 3 by 2 power law model surface potential base 3 by 2 power law model ok here are the 3 by 2 power law terms now in this expression you are going to substitute psi s naught as VSB plus phi T where phi T is twice phi f plus 6 VT and psi SL that is this term here these psi SL terms will be replaced by VDB plus phi T where phi T is again twice phi plus 6 VT when VDB is less than VDB sat but when VDB exceeds VDB sat you will pin the surface potential at the drain at VDB sat plus phi T now if you would like to know why are we replacing psi s naught in terms of VSB using this formula and psi SL in terms of VDB or VDB sat just want to remind you ok that the surface potential versus channel voltage it increases and then saturates ok and the saturation value of the surface potential here is close to V plus phi T the saturation value being close to V plus phi T so if this is VDB it will be VDB plus phi T if this is VSB that is at the source then this will also be VSB ok now this saturation value can also be seen from our previous slide recall the psi SL and psi s naught versus VGB so you can see that the saturation value here is close to VDB plus twice phi f for psi SL and VSB plus twice phi f for psi s naught ok and even detailed behaviour was shown here ok so here you can see that in fact you have to go little bit beyond this VDB plus twice phi f to get the saturation value here that is shown you have to go beyond about 6 VT ok so there is this difference between this value and this line so in that difference you have 6 VT and a little bit more so you can take V plus twice phi f plus 6 VT as the value close to saturation so that is a rationale for using this expression for psi s naught and this expression for psi SL now when you do this derivation also note that the mobility depends on VDS and EY effective 0 this is the expression for mobility so you can see that it is going to be a fairly challenging exercise ok and where we using DID by DVDB equal to 0 as the saturation criterion because you know that if you use this 3 by 2 power law model and sketch ok you are going to get something like this IDS versus VDB and then this is VSB now your saturation current is this value ok and this is your VDB SAT so beyond this point you have to assume the current to saturate right because in practice the current does not decrease we know that so this is unphysical this part of the mathematical segment is unphysical so that is why we are using the criterion DIDS by DVDB is 0 so here this is written as DID you should put an S there ok let us summarize our lecture what have we achieved in this particular lecture so in this lecture we have approximated the surface potential based model by a simple exponential expression in the weak inversion regime or for the diffusion current ok so putting it in a graphical form this IDS versus VGB for any VDB it looks something like this in a semi-lock plot so this is on a lock scale and we have taken this part this is the so-called weak inversion and depletion part ok so we have taken this part and we have said this straight line portion on a semi-lock plot can be approximated as follows so IDS is some terms into exponential of VGS minus V dash T by VT into 1 minus exponential of minus VDS by VT ok and there is an term N here which is related to the slope of the straight line approximation of the psi SVGB plot so this is a straight line approximation for depletion and weak inversion this is VFB so slope of this is 1 by N so that is this 1 by N now what are the terms coming here as coefficient so there we said you remember the dimensional analysis of the drain to source current so we have the W by L then we have a surface mobility expression then we have a capacitance and here it is going to be the depletion capacitance in the MOSFET and you have a square of the voltage term and that is a VT square term now what we did not do is that we did not derive an expression for N so in the summary let me give you some pointers to how we can derive an expression for N so we have said the slope of psi S versus VGB in this linear portion is 1 by N that is so actually this is psi S0 ok psi S0 and psi SL both are same in this region you know this if you plot this this is how it is this is psi SL and this is psi S0 so I can really say that 1 by N is equal to D psi S0 by D VGB ok and we will evaluate it let us say at this point at this point what is your psi S0 so this is VSB plus twice phi F and this is V dash DB so what is the expression for surface potential we can use that expression that is psi S0 is equal to VGB minus VFB minus gamma into square root psi S0 plus VT into exponential of psi S0 minus VSB plus twice phi F in brackets by VT right now when we are in weak inversion here our psi S0 is less than VSB plus twice phi F and so the exponential of negative quantity small so we can remove this so we will be ending up with this simple expression ok so in other words this part is not there like this now if you differentiate this D psi S0 by D VGB differential of VGB here will give you 1 VFB is a constant so you would not have anything here differential of this with respect to psi S0 will give you gamma by 2 square root psi S0 ok into D psi S0 by D VGB so now I can move this whole term to the left hand side ok as 1 plus gamma by 2 square root psi S0 into D psi S0 by D VGB is equal to 1 or D psi S0 by D VGB is simply reciprocal of this so but this is 1 by N so N will be I remove this minus 1 here 1 plus gamma by 2 into square root psi S0 and what is the value of psi S0 so this psi S0 is twice phi F plus VSB so this is your formula for N we will take up the further developments of the model in this regime ok in the strong inversion in the next class so this is the partitioning voltage VHB whereas this is the partitioning voltage V dash TB towards the end of the lecture we have given you an assignment where you have to show that it is really difficult to obtain a simple expression in the strong inversion regime using the 3 by 2 power law model right that should build you that should build a motivation within you to actually look for some good approximations and simplify the expression so we will show how the 3 by 2 power law model can be simplified to a square law model ok now we want to record the next lecture we want to record the next lecture how much time is over.