# Effect of Oxide Charges on Energy Band Structure: Flat Band Voltage

In order to determine the total shift in the flat band voltage (Д *V _{fh})* by various oxide charges, let us consider

*Q*as the charge per unit area at any point

_{ax}(x)*x*within the oxide of thickness

*T*Then from Gauss’ law [Equation 2.75], we can show

_{ox}.where:

*С ox* = *K _{ox}e_{t)}IT_{ox}* is the oxide capacitance per unit area as defined in Equation 3.1

*dV*is the voltage drop in oxide due to the oxide charges

The voltage drop *dV* defines the shift in the flat band voltage *V _{fh}* of an MOS capacitor and is given by

where:

*Q _{f}* and

*0„*are located at or near the Si/SiO, interface

*(x*=

*T*On the other hand,

_{m}).*Q,„(x)*and

*Q„,(x)*are distributed throughout the oxide. Therefore, after integration, we get

For the simplicity of circuit CAD, Equation 3.19 is expressed as

where:

*Q _{a}* is the

*equivalent interface charge*located at the Si/SiO, interface and causes the same effect as that of the actual charges of unknowm distribution

*Q _{a}* is always positive for both

*p-*and и-type substrates. Д

*V*is the gate voltage that is needed to cause

_{fh}*Q*to be imaged in the gate electrode so that none is induced in the silicon. However, w'hen gate “floats” or the gate electrode is absent, the oxide charges will seek all their image charges in the silicon.

_{a}In Figures 3.4(a), 3.5, and 3.6, w'e have showrn the band bending of an MOS capacitor due to workfunction difference between the metal and semiconductor. The corresponding flat band voltage is given by Equation 3.6. Now, the shift in workfunction due to additional band bending by oxide charges is given by Equation 3.20. Then, combining Equations 3.6 and 3.20, the overall *V _{fh}* due to <£>

_{ms}and

*Q*is given by

_{a}Typically, Qq/C_{ox} is much smaller than Ф„,_{л} in Equation 3.21. Therefore, for an MOS capacitor on /Муре substrate with the value of MG workfunction close to silicon СВ-edge, the value of *V _{fh}* is a negative number since Ф„

_{в}is negative from Equation 3.10. On the other hand, for an MOS capacitor with и-substrate and MG workfunction close to silicon VB-edge,

*V*is positive since Ф,,,, is positive from Equation 3.12.

_{fh}The band bending at the Si/SiO, interface due to *V _{fh}* induces a potential on the substrate at the Si/SiO, interface referred to as the

*surface potential.*The surface potential is an important parameter to discuss the performance and mathematical analysis of FET devices. In Section 3.2.5, we define the surface potential of an MOS capacitor.

# Surface Potential

Now, let us consider an MG-SiO_{r}(/Mype) silicon MOS capacitor to discuss the effect of band bending at the silicon surface on the surface behavior of MOS capacitors.

We know that the concentration of holes in а /Муре substrate (assuming complete ionization of acceptor atoms) is given by [Equations 2.30 and 2.32]

The band structure of the system can be shown as in Figure 3.8. Since *E,* is continuous within the system, it is evident from Figure 3.8 that as the bands bend downwards, the energy difference (£) - *E _{f})* gradually decreases as we approach the silicon surface at x = 0 from the bulk (x=oc). Then from Equation 3.22, the decrease in (£, -

*E*results in a decrease in the hole concentration

_{f})*p.*This implies that the holes are depleted at the surface giving rise to a space charge region for an MOS system with Ф„„ < Ф

_{(}. On the other hand, in the case of an MOS capacitor with Ф,„ > Ф„ the bands bend upwards and, therefore, the value of (£, -

*E*increases at the surface resulting in an increase in the hole concentration (accumulation) at the surface.

_{f})From Equation 3.22, it is seen that even without an applied external voltage the carrier concentration at the surface of an MOS capacitor differs from that in the bulk due to *Ф _{ш}* and

*Q*This change in the concentration sets up an electric field at the surface and hence a voltage difference between the silicon surface and bulk. This voltage difference is referred to as the

_{n}.*surface potential*ф

_{(}and represents the electrostatic potential at the surface measured from the bulk intrinsic level £,• as shown in Figure 3.8. Thus, ф, is the difference between

*Efx*= 0) at the surface and £,(x = «) at a point deep into the substrate. As shown in Figure 3.8, ф, is a measure of the amount

FIGURE 3.8 MOS capacitor system: band bending near the silicon surface showing surface potential ф, at the surface of a *p*-type silicon; here, *x* is the distance from the insulator/sub- strate interface, *x* = 0 into the substrate.

of total band bending at the silicon surface. And, at a depth *x* into the surface, the potential is given by ф(л).

The band bending described above can be compensated by applying an external gate voltage *V _{fh},* defined in Section 3.2.4 as the

*flat band voltage*and is given by Equation 3.21. The condition to achieve the flat bands at the surface is called the

*flat band condition.*Thus,

*V*The flat band condition is often used as a reference state along with

_{fb}is the applied gate voltage to have zero surface potential with flat energy bands over the entire semiconductor surface.*V*as a reference voltage and thus, can be considered as an important figure-of-merit for an MOS capacitor system.

_{fb}