The MOS Transistor in Weak Inversion

Transcription

1 MOFE Operation in eak and Moderate nversion he MO ransistor in eak nversion n this section we will lore the behavior of the MO transistor in the subthreshold regime where the channel is weakly inverted. his will allow us to model transistors operating with small gate voltages, where the strong inversion model erroneously predicts zero current. Remember, the strong inversion MOFE model makes the assumption that the inversion charge Q goes to zero when the gate voltage drops below the threshold voltage. e saw that this is not quite true. Below threshold, the channel charge drops onentially with decreasing gate voltage. Q slope C ox 0 B epletion eak inversion Moderate inversion trong inversion EE 570/niversity of tah 1

2 MOFE Operation in eak and Moderate nversion his onential relationship becomes clearer if we redraw the above figure with a logarithmic y axis: log Q Q (actual) Q -( B ) Q -C ox ( B 0 ) eak inversion 0 Moderate inversion trong inversion B e can think of weak inversion as the region where Q is an onential function of gate voltage, strong inversion as the region where Q is a linear function of gate voltage, and moderate inversion as a transition region between the two. n weak inversion, the inversion layer charge is much less than the depletion region charge: Q << Q B in weak inversion ince the substrate is weakly doped, Q B is small, and there is not enough charge in the channel to generate a significant electric field to pull electrons from the source to the drain. Current flows by diffusion, not drift. e shall once again consider our fluid model of a transistor. et s take another look at a transistor operating in the subthreshold regime: EE 570/niversity of tah

3 MOFE Operation in eak and Moderate nversion reference voltage electron energy voltage Clearly, this model is insufficient to account for weak inversion since the channel charge is zero. e have to add another degree of realism to our model to account for subthreshold current flow: water vapor. Electrons, like water molecules, can be excited by thermal energy into higher energy levels. Although most water molecules have a potential energy at or below the level of the liquid in the source and drain tanks, some water molecules have gained enough energy (thermally) to rise above this level as vapor. imilarly, a small fraction of electrons in the source and drain acquire significantly more energy than the majority of charge carriers in the conduction band. he density of water vapor above a liquid follows a decaying onential with height (i.e., potential energy). imilarly, air pressure drops onentially with altitude. Electrons in a solid obey Fermi-irac statistics which leads to this onential distribution according to energy. his is similar to the Maxwell- Boltzmann statistics obeyed by atoms in a gas. et s add water vapor to our fluid model: EE 570/niversity of tah 3

4 MOFE Operation in eak and Moderate nversion reference voltage energy electron energy Q voltage Now it s clear that the inversion charge in the channel, while small, is an onential function of the barrier height. he barrier height represents the surface potential ψ s. As we mentioned before, in weak inversion the surface potential is flat it does not change over the length of the channel. he surface potential can be modeled fairly accurately by considering the capacitive divider between the oxide capacitance C ox and the depletion capacitance C dep : poly-i io C ox n + n + ψ s p - silicon ψ s C dep B B sing the equation for a capacitive divider and assuming that B 0, we find that: ψ s where kappa the gate coupling coefficient represents the coupling of the gate to the surface potential: C ox C ox + C dep EE 570/niversity of tah 4

5 MOFE Operation in eak and Moderate nversion he depletion capacitance stays fairly constant over the subthreshold region, and kappa is usually considered to be constant, although it increases slightly with gate voltage. n modern CMO processes, kappa ranges between 0.6 and 0.8. t can have slightly different values for pmo and nmo devices. A good, all-around approximation for kappa (unless another value is given) is: 0.7 ome texts use n or ζ (zeta) instead of, where n ζ (1/) 1.4. Now let s return to the fluid model for > 0: reference voltage electron energy diffusion voltage he important parameter is the concentration of carriers at channel level (and above). ince the source tank is higher than the drain tank, the carrier concentration is higher where the source meets the channel than where the drain meets the channel. Electrons diffuse from the source to the drain. he charge concentration (at channel level) in the source (x 0) and the drain (x ) is given by: Q 0 Q here is the thermal voltage: k q 6m at room temperature (e use instead of for voltage to avoid confusion with the threshold voltage.) EE 570/niversity of tah 5

6 MOFE Operation in eak and Moderate nversion From our previous discussion of diffusion, we know that particle motion is proportional to the concentration gradient. he concentration of electrons decreases linearly from the source to the drain (i.e., concentration gradient is constant), so we can write an ression for the drain current: ( ) ( ) n n Q Q Q Q 0 0 µ his leads us (details omitted) to the ression for drain current in a subthreshold MOFE: e e e 0 or, writing it larger: 0 where 0 is a process-dependant constant. For nfes, n ox n n C 0 0 µ ypical values of 0n range from A to 10-1 A. e can rearrange terms and rewrite the ression for drain current as: 1 0 Notice that when (- / ) << 1, the last term is approximately equal to one, and can be ignored. his occurs (to within %) for > 4, since e he ression for drain current then simplifies to: 0 for > 4 (saturation) At room temperature, 4 100m, an easy value to remember. t is quite easy to keep a subthreshold MOFE in saturation, and the required to do so does not depend on as is the case above threshold. his is very advantageous for low-voltage designs. EE 570/niversity of tah 6

7 MOFE Operation in eak and Moderate nversion 4 100m 0.4 aturation region riode region his saturation behavior is easy to understand if we look at our fluid model. f the drain tank is significantly lower than the source tank, the concentration of carriers in the drain at channel level will be much, much small than the concentration of carriers in the source at channel level, so the exact level doesn t matter. Another difference between subthreshold and above threshold operation is the way changes as we increase. n a weakly-inverted FE, the current increases onentially. n a strongly-inverted FE, the current increases quadratically (square law). his can be understood by looking at a plot of vs. in two ways: with a linear axis and with a logarithmic axis: EE 570/niversity of tah 7

8 MOFE Operation in eak and Moderate nversion grows as ( 0 ) log 0 factor of e grows as ( / ) slope e-fold/( /) ( /) 40m 0 For a pfe, we have to consider the gate, drain, and source potentials with respect with the well potential. nlike the substrate, the well will not be at ground, so we need to write it in licitly: 0 p ( ) ( ) ( ) µ pcox 0 p ypical values of 0p range from A to A. 0 p EE 570/niversity of tah 8

9 MOFE Operation in eak and Moderate nversion n saturation: ( ) ( ) 0 for > 4 his can be rewritten as: 0 [ + ( 1 ) ] here (1 - ) is the back-gate coefficient. Notice that the body effect is licit in the weak-inversion model without having to add a fudge factor like the variable threshold voltage in the strong inversion model. he transconductance of a subthreshold MOFE is easily derived: n weak inversion: g m ubthreshold MOFEs behave similarly to bipolar junction transistors (BJs). he collector current of an npn bipolar transistor exhibits an onential dependence on base-to-emitter voltage: C A bipolar transistor has a transconductance of g m C /, which is equivalent to the ression for a subthreshold MOFE if we set 1. Of course, a MOFE doesn t pull any current through its gate like a bipolar transistor pulls through its base. his can make circuit design much easier. BE EE 570/niversity of tah 9

10 MOFE Operation in eak and Moderate nversion Moderate nversion A transistor does not switch immediately from an onential, weak-inversion behavior to a quadratic, strong-inversion behavior. here is a smooth transition between the two extremes where drift and diffusion generate the current with neither effect dominating. Modeling this area is extremely difficult, but the behavior is easily understood as a hybrid between weak- and strong-inversion behavior. reference voltage electron energy voltage he boundaries between weak, moderate, and strong inversion can be approximated in terms of voltages or currents. Of course, the boundaries between the areas are fuzzy, with no clean breaks. Here are two approximations sometimes used: > + 100m + 100m > > 100m < 100m strong inversion moderate inversion weak inversion (his assumes that the threshold voltage has been adjusted for the body effect, if necessary.) EE 570/niversity of tah 10

11 MOFE Operation in eak and Moderate nversion t is more common to design circuits with bias currents in mind. Following is a good estimate of the inversion boundaries in terms of drain current: > > > 0.1 < 0.1 strong inversion moderate inversion weak inversion where is called the moderate inversion characteristic current: µ C ox ypical values of range from 100nA to 500nA for nfes with / 1, and 40nA to 10nA for pfes with / 1. Of course, for large / ratios, the weak inversion region can extend well into the microamp range. Example: n AM s 0.5µm CMO process, µ n C ox 116µA/ and µ p C ox 38µA/. Estimate the boundaries between weak, moderate, and strong inversion (in terms of drain currents) for nmo and pmo transistors with / 1 and / 100. Assume 0.7. nfes: n µ C n ox 0nA for / 1 o for / 1, the boundary between weak and moderate inversion is around na, and the boundary between moderate and strong inversion is around.µa. For / 100, the boundary between weak and moderate inversion is around.µa, and the boundary between moderate and strong inversion is around 0µA. pfes: p µ C p ox 73nA for / 1 o for / 1, the boundary between weak and moderate inversion is around 7.3nA, and the boundary between moderate and strong inversion is around 0.73µA. For / 100, the boundary between weak and moderate inversion is around 0.73µA, and the boundary between moderate and strong inversion is around 73µA. EE 570/niversity of tah 11

12 MOFE Operation in eak and Moderate nversion A More Accurate Model of Kappa he subthreshold gate coupling coefficient is rarely reported in process data. ometimes, people will report the subthreshold slope /. f it is not given, can be calculated by: where 1+ γ ( 1+ α ) Φ F 1 γ Φ F qεn C ox k ln q C ox he α parameter should be set between zero for extreme weak inversion (near depletion mode) and one for the boundary between weak and moderate inversion to account for the slight change in depletion capacitance. sually, α 0.5 is a good value to use for general-purpose use. ε t o all we really need to know is the oxide thickness (to determine C ox ) and the channel doping (N sub ) to estimate kappa. n PCE models, oxide thickness is called OX (units m) and the channel doping is called NB or NCH (units cm -3 ). ox ox sub N n sub i Example: An nfe has a substrate doping level of 1.7 x cm -3 and an oxide thickness of 139Å. Compute, using α 0.5. t ox cm C ox 0.48µF/cm γ / Φ F EE 570/niversity of tah 1

13 MOFE Operation in eak and Moderate nversion he EK Model n 1995, Enz, Krummenacher and ittoz proposed a relatively simple MOFE model valid in all regions of operation: weak, moderate, and strong inversion. his has come to be known as the EK model. heir basic equation for drain current (in saturation) is given by ln 1+ ( ) 0 where µ C ox An ression for transconductance g m valid in all regions of operation is given by g m ( ) where ( ) e 1. Note that / is the inversion coefficient and that ( ) approaches unity in weak inversion. EE 570/niversity of tah 13

14 MOFE Operation in eak and Moderate nversion Alternate ression for g m he ression used in the previous section for computing transconductance has the significant disadvantage that it cannot be solved to find the inversion coefficient / as a function of the required transconductance and drain current. A simpler (and, it is claimed, more accurate) interpolation function for transconductance is given by g m ( ) where ( ) 1 1 ( ) or ( ) ( ) Either function seems to work well. he latter ression is a bit easier to calculate. EE 570/niversity of tah 14

15 MOFE Operation in eak and Moderate nversion Models vs. ata Circles represent measured data from a real transistor. Note that in the moderate inversion region, both strong and weak inversion models overestimate the true transconductance EE 570/niversity of tah 15