MOSFETs in the Sub-threshold Region (i.e. a bit below VT) Clifton Fonstad, 10/28/09 In the depletion approximation for n-channel MOS structures we have neglected the electrons beneath the gate electrode when the gate voltage is less than the threshold voltage, VT. We said that it is only when the gate voltage is above threshold that they are significant, and that they are then the dominant negative charge under the gate. Furthermore, we say that above threshold all of the gate voltage in excess of VT induces electrons in the channel; thus our model is that the sheet charge density under the gate, qN*, is ! qN"=0 for vGC# VT$oxtoxvGC%VT( ) for VT# vGC& ' ( ) ( (1)As MOS integrated circuit technology has evolved to exploit smaller and smaller device structures, it has become increasingly important in recent years to look more closely at the minority carriers present under the gate when the gate voltage is less than threshold, i.e. in what is called the “sub-threshold” region. These carriers cannot be totally neglected, and play an important role in device and circuit performance. At first they were viewed primarily as a problem, causing undesirable “leakage” currents and limiting circuit performance. Now it is recognized that they also enable a very useful mode of MOSFET operation, and that the sub-threshold region of operation is as important as the traditional cut-off, linear, and saturations regions of operation. To begin our study of the sub-threshold region, we will first quickly review the electrostatics of the MOS capacitor, and the electrostatic potential profile predicted by the depletion approximation model. Then we will use this result to derive a more accurate expression than that in Equation 1 for qN* below threshold, and use the resulting expression to, among other things, assess the assumption that the contribution of the mobile electrons underneath the gate to the net charge density in the depletion region is negligible compared to the contribution from the ionized acceptors. Finally we will look at the current-voltage characteristic of a MOSFET operating in the sub-threshold region, and merge it with our earlier model so that we then have a model in which the mobile electron charge is taken into account and the drain current is no longer identically zero when vGS is less than VT.6.012 Supplementary Notes: MOSFETs in the Sub-threshold Region (i.e. a bit below VT) The Electrostatics of the MOS Capacitor with vBC = 0 Consider the MOS capacitor with vBC = 0 illustrated in Figure 1, the same structure we used when we first looked at the MOS capacitor using the depletion approximation. In the depletion p-Sin+BCGSiO2+–vGC(= vGB)FIGURE 1 A MOS capacitor connected as a two-terminal capacitor with vGC = vGB = 0. approximation, we assume that Equation 1 holds and that the net charge density profile, ρ(x), under the gate for VFB < vGC < VT can be approximated as: ! "(x) =qNA for 0 # x # xD0 for xD# x$ % & (2)With this assumption, we found that the electrostatic potential profile is: ! "(x) ="p+qNAx # xD( )22$Si for 0 % x % xD"p for xD% x& ' ( ) ( (3)This expression is plotted in Figure 2, which also continues the plot through the oxide to the gate, from which we can also get the expression relating the depletion region width, xD, to vGB and VFB: ! vGB" VFB=qNAxDtox#ox+ qNAxD22#Si (4)26.012 Supplementary Notes: MOSFETs in the Sub-threshold Region (i.e. a bit below VT) FIGURE 2 A sketch of φ(x) from the metal on the left, through the oxide, and into the p-type semiconductor in an n-channel MOS capacitor for an applied gate bias, vGB, in the weak-inversion, sub-threshold region. Equation 4 is useful because it can be solved explicitly for xD, and the result can be used to obtain an expression for φ(x) as a function of vGB. However, it will turn out that what is most important to us is φ(0), the value of the potential at the interface, and φ(0) is much easier to relate to vGB than is φ(x) at an arbitrary x. To do so we first find xD in terms of φ(0): ! xD=2"Si#(0) $#p[ ]qNA (5)Using this in Equation 4 gives us an equation relating φ(0) and vGB that will be useful to us shortly: ! vGB" VFB=tox#ox2#SiqNA$(0) -$p[ ]+ $(0) -$p[ ] (6)Sub-threshold Electron Sheet Charge Density, vGC = 0 Returning to our original goal, which was to find the electron population density, n(x), under the gate, and then the electron sheet charge density, qN*, we note that the Boltzman relationship between the electrostatic potential and carrier population holds under the gate of the MOS capacitor in Figure 1 because the current in the x-direction is zero. Thus we have: 36.012 Supplementary Notes: MOSFETs in the Sub-threshold Region (i.e. a bit below VT) ! n(x,vGB) = nie"x,vGB( )"t (7a) =ni2NAe"x,vGB( )#"p[ ]"t (7b)p(x,vGB) = nie#"x,vGB( )"t (8a) = NAe#"x,vGB( )#"p[ ]"t (8b)In these equations we use φt for the thermal voltage, kT/q, and we have indicated the dependency on vGB to emphasize that these populations depend on the gate voltage as well as on position, x. To obtain the Equations 7b and 8b, we have used po = NA = ni exp(-φp/φt) to get expressions explicitly including the quantity [φ(x,vGB) - φp], which also appears in Eqs. 5 and 6. Note: In many texts, [φ(0,vGB) - φp] is identified as VB(vGB), the voltage drop between the silicon bulk and the oxide-silicon interface, i.e. VB(vGB) ≡ [φ(0,vGB) - φp]. We can calculate the electron sheet charge density, qN*, by multiplying n(x) by –q and integrating with respect to x from the interface, x = 0, into the silicon until x = xi, where xi is defined as the depth at which φ(x) = 0, and thus where we have n(xi) = p(xi) = ni: ! qN"vGB( )= #qn(x,vGB) dx0xivGB( )$
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