ECE 4430 Analog Integrated Circuits and Systems

ECE 4430 Analog Integrated Circuits and Systes Prof. B. A. Minch s lecture notes in Cornell University on Septeber 21, 2001 1 MOS Transistor Models In this section, we shall develop large-signal odels for an nmos transistor in all of its various operating regies. Because the MOS transistor is a four-terinal device, the current depends on three potential differences, which are typically taken to be V GS, V DS, and V BS. Here, the gate, drain, and bulk i.e., substrate voltages are all referenced to that of the source. Although this convention is ubiquitous, it is not at all necessary. For exaple, it is possible to reference the gate, source, and drain potentials instead to the bulk potential. In this case, the odel would be expressed in ters of V GB, V SB, and V DB [1]. With this convention, the channel current of an MOS transistor can be expressed as I = W L I s fv GB,V SB - fv GB,V DB, 1 where I s is related to the channel current of a square transistor at threshold and f is a functional that assues an exponential for below threshold and a quadratic for above threshold. Such MOS odels are called bulk-referenced for obvious reasons or source/drain-syetric because the source and drain voltages coe into the odel in a syetric fashion. With the proper choice of f, this single equation describes the channel current in all regions of operation, transitioning soothly fro subthreshold to above threshold and fro the ohic region to the saturation region. For an nmos transistor fabricated in an n-well technology, as shown in Fig. 1.1, the p-type bulk is connected to ground, so the three potentials in the odel would siply be V G, V D, and V S. Our intent in developing these odels will not be physical rigor. Instead, our concern is to develop siple large-signal odels that are useful fro a circuit-design point of view rather than a device-physics one. We will define soe notions associated with MOS transistor operation, such as threshold and the onset of saturation in slightly unconventional ways so that we can develop the odels with a iniu of physical detail and so that the concepts can be applied consistently in all operating regions. In our discussion, we will ake several siplifying assuptions. First, we shall be assuing a long-channel device. Second, we shall assue that the device behavior is entirely unifor along the direction of its width and that the obile charge is located at the surface of the device, so that we can consider current flow in one spatial diension, along the length of the channel. Third, we shall neglect assue that there is no charge trapped in surface states. Fourth, we shall neglect contact potentials that exist between different aterials. 1.1 Channel Capacitance We shall take the conduction band edge deep in the silicon substrate to be the zero of potential. When the gate voltage is equal to the flatband potential, V fb, then the conduction and valence bands are flat fro deep within the silicon substrate all the way to the surface and there is no depletion region beneath the gate. Likewise, the bands are flat within the oxide. If we apply a voltage to the gate that is ore positive than V fb, then the difference between the applied gate voltage and V fb is 1

I V G I V S V D W z yx p + n + n + p - L Figure 1: Siplified structure of an nmos transistor. partially dropped across the oxide and partially across a depletion region that fors beneath the gate, such that V G V fb = ψ s + ψ ox, 2 where ψ s is the potential drop across the depletion region beneath the gate and ψ ox is the potential drop in the oxide. If there is no charge within the oxide, then Gauss law iplies that the electric field within the oxide, E ox, is constant and is given by E ox = ψ ox, 3 t ox where t ox is the thickness of the oxide layer. Integrating the charge density in the depletion layer and the inversion layer fro deep within the silicon to the surface, we find the electric field at the surface of the silicon, E Si,tobe E Si = Q dep + Q, 4 ɛ Si where Q dep is the charge per unit area in the depletion layer beneath the gate, Q is the obile charge per unit area in the channel, and ɛ Si is the dielectric constant of silicon. The displaceent vector i.e., the product of the dielectric constant and electric field is conserved across the silicon/oxide interface, which iplies that ɛ Si E Si = ɛ ox E ox. Substituting Eqs. 3 and 4 into this equation and solving for ψ ox,wefindthat ψ ox = t ox Q dep + Q Qdep + Q =, 5 ɛ ox C ox where C ox ɛ ox /t ox is the capacitance per unit area of the oxide. Substituting this equation into Eq.2,wefindthat V G V fb = ψ s Q dep + Q. 6 C ox 2

Differentiating this equation with respect to distance along the channel, we find that 0 = ψ s 1 Q dep ψ s C ox ψ s x 1 Q C ox x = ψ s + C dep C ox ψ s x 1 C ox Q x, where C dep Q dep / ψ s is the increental capacitance per unit area of the depletion layer beneath the gate. By rearranging this equation, we find that Q x = C ox + C dep ψ s x = C ψ s x, 7 where C C ox + C dep = Q / ψ s is the increental capacitance per unit area of the channel. 1.2 Channel Current The channel current in an MOS transistor flows both by drift and by diffusion. In soe operating regies, the current flows priarily by drift, in others it flows priarily by diffusion. However, in any cases of interest, both current coponents are of roughly equal agnitudes. The predoinant carrier transport echanis can also change as a function of position along the channel in soe regions of operation. Thus, to odel the channel current in all regies, we need to include both a drift and a diffusion ter in the current-flow equation. Thus, we write the channel current, I,asa function of position along the channel, as The drift coponent of I is given by Ix = I drift x + I diff x. 8 ψ s I drift x = WQ µe = WµQ x, 9 where W is the width of the MOS transistor note that WQ gives the obile charge density per unit length along the channel, µ is the effective low-field obility of electrons in the channel, and E is the coponent of the electric field parallel to the channel. The diffusion coponent of I is given by I diff x = D x WQ Q =WµU T x, 10 where D is the diffusion constant of electrons in the channel, and U T is the theral voltage, kt/q. Thus, we can rewrite Eq. 8 using Eqs. 9 and 10 as ψ s Ix = WµQ x + WµU Q T x ψ s = Wµ Q x +U Q T. 11 x 3

To obtain an expression for the channel current that is independent of position along the channel, we integrate both sides of Eq. 11 fro source to drain. Doing so, we write L L ψ s Ix dx = Wµ Q 0 0 x +U Q T dx. x Now, conservation of charge iplies that the total channel current ust be constant along the channel, so the left-hand side of this equation becoes L 0 Ix dx = I L 0 dx = IL. Thus, we can write the channel current as I = W L L µ ψ s Q 0 x +U Q T dx x = W L L µ ψ s Q Q 0 Q x +U Q T dx x = W L L µ Q 0 C +U Q T x dx = W QD L µ Q Q S C +U T dq = W QD L µ Q Q S C +U T dq + WL 0 µ Q Q D C W 0 L µ Q Q D C +U T dq = W 0 L µ Q C +U T dq W 0 L µ Q C Q S } {{ } I F Q D +U T dq } {{ } I R + U T dq, 12 where Q S is the obile charge per unit area at the source end of the channel and Q D is the obile charge per unit area at the drain end of the channel. Note that we have ade use of Eq. 7 in going fro the second step to the third step in the above derivation. We have expressed the channel current as the difference between a forward current coponent, I F, and a reverse current coponent, I R, which have identical functional fors, except that I F depends on Q S and I R depends on Q D. The obile charge density at the source end of the channel, in turn, will depend on the gate-to-bulk voltage and the source-to-bulk voltage and not on the drainto-bulk voltage. In the sae way, the obile charge density at the drain end of the channel should depend on the gate-to-bulk potential and on the drain-to-bulk potential and not on the source-tobulk potential. Moreover, we should expect that Q S depends on V SB in precisely the sae way that Q D depends on V DB. Note that this MOS transistor odel has the source/drain syetric for expressed in Eq. 1. In such odels, the priary channel current dependence on the drain voltage is contained wholly within I R. If the forward current coponent is uch larger than the reverse current coponent, then the channel current no longer depends significantly on the drain voltage, 4

and the transistor is saturated. The saturation current is siply given by I F. On the other hand, if the agnitudes of I F and I R are coparable, then the channel current depends strongly both on the source voltage and on the drain voltage. In this case, the transistor is in the ohic region. To ake progress on the MOS transistor odel beyond Eq. 12, we introduce an approxiation, which was ade initially by Maher and Mead [2,3], that will allow us to evaluate the integrals Eq. 12 in closed for. The channel capacitance per unit area, C, defined in Eq. 7, is a weak function of position along the length of the channel in oderate and strong inversion, the depletion layer gets thicker closer to the drain end of the channel, aking C dep and, hence, C saller nearer to the drain. We shall take C to be constant as a function of position, with C dep replaced by an average value. With this approxiation, we have that I = W L µ Q2 0 0 2C + U T Q WL µ Q2 0 0 Q S Q S 2C + U T Q Q D Q D = W L µ Q 2 S 2C U TQ S W L µ Q 2 D 2C U T Q D = W L µ Q 2 2C S 2CU T Q S W }{{} L µ Q 2 2C D 2CU T Q D. 13 }{{} I F I R Note that the quadratic ters in I F and I R in Eq. 13 originated with the drift coponent of the channel current, whereas the linear ters steed fro the diffusion coponent. When Q S 2CU T, the diffusion ter in I F is doinant over the drift ter. Conversely, when Q S 2CU T,then the drift ter is uch larger than the diffusion ter. When Q S is equal to 2CU T, then these two current coponents are equal. This condition on the obile charge density corresponds to the usual notion of threshold. Below threshold, the obile charge density in the channel is sall and the channel current is carried priarily by diffusion. Above threshold, the obile charge density is large, the channel current is carried ainly by drift. At threshold, the channel current flows both by drift and by diffusion. All of the sae stateents can also be ade for Q D and I R. Thus, we shall take the agnitude of the obile charge density at threshold to be given by 2CU T. Equation 13 represents a siple, closed-for expression for the channel current flowing in an nmos transistor in ters of the obile charge densities at the source and drain ends of the channel. This odel equation is valid in all regions of noral MOS transistor operation, transistioning continuously fro subthreshold to above threshold and fro the ohic region to the saturation region. Unfortunately, we would like to have the channel current explicitly in ters of the terinal voltages, V G, V S,andV D. The dependence of the channel current on the terinal voltages coe in through the dependence of Q S on V G and V S and the dependence of Q D on V G and V D. Unfortunately, no physically exact, closed-for expressions derived fro first principles exist for these dependencies. In the next two sections, we shall explore two extree liits of this odel where siple approxiate relationships between the obile charge densities and the terinal voltages exist. 5

1.3 Subthreshold Operation In the subthreshold region of operation, the aount of obile charge in the channel is negligible copared to the aount of charge exposed in the depletion layer beneath the gate. Thus, we should expect that the presence of the obile charge will have a inuscule effect on the electrostatics in the channel region. If the electrostatics are only deterined by the substrate potential, the gate potential, the gate charge, and the depletion charge, then, because the gate and the substrate are both isopotential, we would expect that the surface potential, ψ s, should also be constant along the channel. Because the electric field is given by the gradient of the potential, a constant surface potential iplies that there is no electric field in the direction of the channel. The absence of an electric field along the channel, in turn, iplies that any current flow ust be by diffusion rather than by drift, which is consistent with our conclusions at the end of Section 1.2 steing fro Eq. 13. Therefore, to obtain a subthreshold odel for MOS transistor operation, we should be able to neglect the quadratic ters in this equation, but we ust still relate the obile charge densities at the source and drain ends of the channel to the applied terinal voltages. We know that, in subthreshold, there ust be a relatively substantial energy barrier between the source and drain regions and the channel, otherwise there would be a substantial nuber of charges in the channel. The nuber of carriers in the source that have sufficient energy to surount the energy barrier at the source end of the channel will follow the Boltzann distribution, being exponential in the height of the energy barrier, which, in turn, is given by the difference between the source potential, V S, and the surface potential, ψ s. Likewise, the nuber of carriers in the drain that have sufficient energy to surount the energy barrier at the drain end of the channel will be exponential in the difference between the V D and ψ s. Thus, we expect that Q S e ψ s V S /U T and Q D e ψ s V D /U T. Unfortunately, we need to have these charge densities in ters of the applied gate voltage and we will need to know what the constant of proportionally to use. Toward this end, we shall again ake use of Eq. 6 to deterine how increental changes in the gate voltage affect the surface potential. By neglecting the obile charge ter and differentiating with respect to ψ s,wefindthat V G = 1 1 Q dep ψ s C ox ψ s = 1 + C dep C ox = C ox + C dep, C ox which iplies that, in subthreshold, the increental voltage gain fro the gate to the surface is given by κ ψ s V G = C ox C ox + C dep. 14 In this regie, the situation can be thought of intuitively as a capacitive voltage divider between the oxide capacitance above the channel and the effective capacitance of the depletion layer beneath the channel this paraeter is siply the capacitive divider ratio. Because the thickness of the 6

depletion layer beneath the channel increases with increasing gate voltage, C dep will get saller for larger values of V G, which iplies that κ will increase with increasing V G.However,κis only a slowly-varying function of V G, and even for oderate changes in V G, it is reasonable to assue that the value of κ is constant with a value between 0.5and0.9. For relatively sall changes about soe operating point, we can expand ψ s in a Taylor series and truncate after the linear ter. In order to also get the constant of proportionally in the obile charge densities, we shall choose to expand the surface potential around the point V G = V T0,thezero-bias threshold voltage. When the gate voltage is equal to V T0 with the source and drain grounded, Q S and Q D should both be equal to 2CU T. Thus, we have that the obile charge densities at the source and drain end of the channel are given approxiately by Q S 2CU T e κv G V T0 V S /U T and Q D 2CU T e κv G V T0 V D /U T. 15 Retaining only the ters in Eq. 13 that ste fro diffusion and substituting the obile charge densities given in Eq. 15, we find that the channel current in an nmos transistor operating in subthreshold is given by I = W L µu T Q S W L µu T Q D = W L 2µCU2 T eκv G V T0 V S /U T W L 2µCU2 T eκv G V T0 V D /U T = W L 2µCU2 T eκv G V T0 /U T e V S /U T e V D/U T = W L 2µC oxut 2 e κv G V T0 /U T e V S /U T e V D/U T κ = W L I se κv G V T0 /U T e V S /U T e V D/U T, 16 where we have used Eq. 14 to express C as C ox /κ and we have introduced I s 2µC ox UT 2/κ. Equation 16 represents a coplete odel for the operation of an nmos transistor in subthreshold, covering both the ohic region and the saturation region. To see that it does, we can rearrange Eq. 16 to obtain I = W L I se κv G V T0 V S /U T 1 e V DS /U T = I sat 1 e V DS /U T I sat when V DS is larger than about 4U T or 5U T. Thus, the channel current saturates, becoing independent of V DS for V DS 4U T.ForsallV DS, to see that the odel predicts ohic behavior, we can expand the e V DS/U T ter in this equation in a Taylor series around V DS = 0, retaining only the linear ter. Doing so, we find that I = I sat 1 e VDS/U T = I sat 1 1 V DS + 1 2 VDS U T 2 7 U T

I sat 1 1 + V DS U T = I sat U T V DS = g DS V DS, where g DS I sat /U T represents the increental conductance of the channel deep in the ohic region. The subthreshold odel given in Eq. 16 is forally identical to the ones presented by Mead [4], Vittoz [5], and Bult [6]. 1.4 Above-Threshold Operation Above threshold, the obile charge density in the channel exceeds the charge density in the depletion layer, and the channel charge has a large effect on the surface potential along the channel. As discussed at the end of Section 1.2, for the levels of obile charge density that occur above threshold, the current flow is predoinantly by drift. Consequently, in developing an above-threshold odel for the MOS transistor, we shall only retain the quadratic ters in Eq. 13. In this regie, the energy barriers at the source and drain ends of the channel have been reduced to such an extent that nearly every additional charge that we place on the gate is balanced by additional obile charges in the channel rather than by uncovering ore charge in the depletion layer beneath the channel. Moreover, the inversion layer basically serves as the botto plate of a parallel plate capacitor between the gate and the channel whose capacitance per unit area is just C ox. To obtain expressions for the obile charge densities at the source and drain end of the channel, we shall assue that all of the gate charges that go into raising the gate voltage up to the threshold voltage are balanced by fixed charges in the depletion layer beneath the channel and that all of the gate charges that go into raising the gate voltage above the threshold voltage are balanced by additional obile charges the channel. Thus, we have that the obile charge densities above threshold are given by Q S = C ox V G V T V S and Q D = C ox V G V T V D, 17 where V T V S and V T V D are the bulk-referenced threshold voltages at the source and drain ends of the channel, respectively. The threshold voltage at the source end of the channel represents the gate voltage that we need to apply given the source voltage in order for the obile charge density at the source end of the channel to just equal 2CU T. We can obtain a siple approxiate expression for V T V S using the subthreshold expression for Q S given in Eq. 15 by deterining the value of V G that akes Q S equal to 2CU T. Doing so, we find that V T V S = V T0 + V S κ. 18 Note that we are here taking into account the threshold-voltage increase norally associated with the body effect using a siple linear approxiation, siilar to the one introduced by Wallinga and Bult [7]. Thus, the above-threshold odel that we are developing accounts directly for the body effect to first order via the κ paraeter without any auxiliary equations and without adding too uch coplexity to the odel. Siilarly, the threshold voltage at the drain end of the channel is the 8

gate voltage that we need to apply given the drain voltage so that the Q D is just equal to 2CU T. Using the sae approach as we just took for V T V S,wefindthat V T V D =V T0 + V D κ. 19 Retaining only the ters in Eq. 13 that ste fro drift and substituting the obile charge densities and threshold voltages given in Eqs. 17, 18, and 19, we find that the channel current in an nmos transistor operating above threshold is given by I = W L µ 2C Q2 S W L = W L = W L µ 2C µ 2C Q2 D C ox V G V T0 V S κ 2 C ox V G V T0 V 2 D κ µ 2C C2 ox κ VG κ 2 V T0 V S 2 κ V G V T0 V D 2 κ VG V T0 V S 2 κ V G V T0 V D 2. 20 = W L µc ox 2κ Unfortunately, Eq. 20 only captures the behavior of the MOS transistor in the ohic region above threshold. To see that it does, we rearrange Eq. 20 slightly to obtain I = W L µc ox 2κ = W L µc ox 2κ κ VG V T0 V S 2 κ V G V T0 V S + V S V D 2 κ VG V T0 V S 2 κ V G V T0 V S V DS 2 = W L µc ox 2κ κ V G V T0 V S 2 = W L µc ox 2κ = I sat 1 V 2 DSsat 1 where V DSsat κv G V T0 V S and 1 V DS V DSsat 1 2 1 V DS κv G V T0 V S 2 1 V 2 DS, 21 V DSsat I sat W L µc ox 2κ V DSsat 2. Figure 2 shows a typical above-threshold drain characteristic along with a plot of Eq. 21. The two curves agree for V DS V DSsat, but the odel equation turns around while the drain characteristic saturates for V DS > V DSsat. This deviation occurs because at V DS = V DSsat,thevalueofQ D given in Eq. 17 is equal to zero and for V DS > V DSsat, it would becoe positive, which does not happen physically in this region of operation because the obile charges in the channel are electrons. Rather, what actually happens is that the drain end of the channel goes subthreshold and Q D transitions soothly over to the exponential for given in Eq. 15. Understanding that the 9

I I sat above-threshold drain characteristic above-threshold odel V DSsat V DS Figure 2: Typical above-threshold drain characteristic. second ter in Eq. 20 coes fro Q D akes it clear how the transistor transitions fro the ohic region to the saturation region above threshold. In saturation, the first ter in Eq. 20 is uch larger than the second one, so we can neglect the second one and the current is described by the first one by itself. This point is alost copletely obscured by conventional source-referenced odels of the above-threshold MOS transistor. The odel given in Eq. 20 is forally identical to the one given by Wallinga and Bult [7] and by Vittoz [5]. 1.5 The EKV Model The Enz-Kruenacher-Vittoz EKV odel [5,8] of the MOS transistor provides a siple approxiate closed-for expression for the channel current of a MOS transistor in ters of the terinal voltages, each of which is referenced to the transistor s bulk voltage. It is valid in all regions of noral MOS transistor operation i.e., when the drain-bulk and the source-bulk junctions are reversed biased, transitioning between the continuously. In its siplest for, the EKV odel expresses the channel current in an nmos transistor as I = W L I s log 2 1 + e κv G V T0 V S /2U T log 2 1 + e κv G V T0 V D /2U T, 22 where all of the ters have their previously defined eanings. The function log 2 1 + e x/2 interpolates soothly between an exponential i.e., subthreshold behavior when x < 0 and a quadratic i.e., above-threshold behavior when x > 0. To see that it does, first we suppose that x < 0. Then, it follows that e x/2 1 and we have that log 1 + e x/2 e x/2, because log 1 + y y for sall y 1. Finally, because e x/2 2 = e x, we have that log 2 1 + e x/2 e x for x < 0. Conversely, suppose that x > 0. Then, it follows that e x/2 1and1+e x/2 e x/2. Finally, because log e x/2 = x,wehavethatlog 2 1+e x/2 x/2 2 for x > 0. 10

Consequently, if both V G < V T0 + V S /κ and V G < V T0 + V D /κ, then Eq. 22 becoes I = W L I s log 2 1 + e κv G V T0 V S /2U T log 2 1 + e κv G V T0 V D /2U T W L I s e κv G V T0 V S /U T e κv G V T0 V D /U T = W L I se κv G V T0 /U T e V S /U T e V D/U T, and we recover the subthreshold odel given in Eq. 16. Conversely, if both V G > V T0 + V S /κ and V G > V T0 + V D /κ, then Eq. 22 becoes I = W L I s log 2 1 + e κv G V T0 V S /2U T log 2 1 + e κv G V T0 V D /2U T W κvg L I V T0 V 2 S κvg V T0 V 2 D s 2U T 2U T = W L I s κ VG V T0 V S 2 κ V G V T0 V D 2 4U 2 T = W L µc ox κ VG V T0 V S 2 κ V G V T0 V D 2, 2κ and we recover the above-threshold odel given in Eq. 20. Note that, if V G > V T0 + V S /κ but V G < V T0 + V D /κ, then Eq. 22 becoes I = W L I s log 2 1 + e κv G V T0 V S /2U T log 2 1 + e κv G V T0 V D /2U T W κvg L I V T0 V 2 S s e κv G V T0 V D /U T 2U T W L I s 4U 2 T κ V G V T0 V S 2 = W L µc ox 2κ κ V G V T0 V S 2, which is just the above-threshold odel in the saturation region. The EKV odel switches between these fors soothly with no discontinuities. Figure 3 shows a seilog plot of the saturation currents predicted by the subthreshold odel, the above-threshold odel, and the EKV odel. Note that both the subthreshold odel and the abovethreshold odel deviate substantially fro the EKV odel for V G V T0, which indicates that, if we were to assue either of these two odels in the oderate-inversion region, our conclusions based on such an assuption would be highly suspect. The siple EKV odel represents an excellent tool for reasoning about CMOS circuits in all regions of operation, including oderate inversion, which is becoing increasingly iportant for CMOS circuit design. 11

I log scale subthreshold odel above-threshold odel I s log 2 2 EKV odel V T0 V G Figure 3: Saturation currents predicted by various MOS odels. References [1] J. E. Meyer, MOS Models and Circuit Siulation, RCA Review, vol. 32, no.??, pp. 42 63, 1971. [2] M. A. Maher and C. A. Mead, A Physical Charge-Controlled Model for MOS Transistors, in ARVLSI: Proceedings of the 1987 Stanford Conference, P. Losleben, Ed., pp. 211 229. MIT Press, Cabridge, MA, 1987. [3] M.A. Maher, A Charge-Controlled Model for MOSTransistors, Ph.D. thesis, Caltech, Pasadena, CA, 1989. [4] C. Mead, Analog VLSI and Neural Systes, Addison-Wesley, Reading, MA, 1989. [5] E. A. Vittoz, Micropower Techniques, in Design of Analog-Digital Circuits for Telecounications and Signal Processing, J. E. Franca and Y. Tsividis, Eds., pp.?? 96. Prentice-Hall, Englewood Cliffs, NJ, 1994. [6] K. Bult, Basic CMOS Circuit Techniques, in AnalogVLSI Signal and Inforation Processing, M. Isail and T. Feiz, Eds., pp. 12 56. McGraw-Hill, New York, 1994. [7] H. Wallinga and K. Bult, Design and Analysis of CMOS Signal Processing Circuits by Means of a Graphical MOST Model, IEEE Journal of Solid-State Circuits, vol. 24, no. 3, pp. 672 680, 1989. [8] C. C. Enz, F. Kruenacher, and E. A. Vittoz, An Analytical MOS Transistor Model Valid in All Regions of Operation and Dedicated to Low-Voltage and Low-Current Applications, Analog Integrated Circuits and Signal Processing, vol. 8, no. 1, pp. 83 114, 1995. 12