6.3 Flat band voltage calculation

In this Section

1. Workfunction difference
2. Flat band voltage calculation
Reading: Neamen 10.1.3, 10.1.4 pp 428-434

Required background: 6.2 Energy band diagram of an MOS capacitor

Next: 6.4 The MOS inversion layer charge

6.3.1 Workfunction difference

If there is no charge present in the oxide or at the oxide-semiconductor interface, the flat band voltage simply equals the workfunction difference between the gate metal and the semiconductor.

The workfunction is the voltage required to extract an electron from the fermi energy to the vacuum level. This voltage is between 4 and 5 Volt for most metals. It should be noted that the actual value of the workfunction of a metal deposited onto silicon dioxide is not exactly the same as that of the metal in vacuum. The figure below provides experimental values for the workfunction of different metals as obtained from a measurement of a MOS capacitor as a function of the measured workfunction in vacuum.

oxphif.gif
Fig. Workfunction of metals as obtained from I-V and C-V measurements on MOS structures as a function of the workfunction of those metals measured in vacuum. (Mg = Magnesium, Al = Aluminum, Cu = Copper, Ag = Silver, Ni = Nickel and Au = Gold)

The workfunction of a semiconductor requires some more thought since the fermi energy varies with the doping type as well as with the doping concentration. This workfunction equals the sum of the electron affinity, the difference between the conduction band energy and the intrinsic energy divided by the electronic charge and the bulk potential as expressed by the following equation:

(mf26)
where the bulk potential is given by:

(mf27)
As can be seen from the above equations, the bulk potential is positive for p-type substrates and negative for n-type substrates.

For MOS structures with a highly doped poly-silicon gate one must also calculate the workfunction of the gate based on the bulk potential of the poly-silicon.

6.3.2 Flat band voltage calculation

The flat band voltage of real MOS structures is further affected by the presence of charge in the oxide or at the oxide-semiconductor interface. The flat band voltage still corresponds to the voltage which when applied to the gate electrode yields a flat energy band in the semiconductor. The charge in the oxide or at the interface changes this flatband voltage. For a charge, Qi, located at the interface between the oxide and the semiconductor, and a charge density, rox, distributed within the oxide, the flat band voltage is given by:
(mf28a)
where the second term is the voltage across the oxide due to the charge at the oxide-semiconductor interface and the third term is due to the charge density in the oxide.

The actual calculation of the flat band voltage is further complicated by the fact that charge can move within the oxide, while the charge at the oxide-semiconductor interface due to surface states also depends on the position of the fermi energy.

Since any additional charge affects the flat band voltage and thereby also the threshold voltage, great care has to be taken during fabrication to avoid the incorporation of charged ions as well as creation of surface states.

6.2 ¬ ­ ® 6.4

© Bart J. Van Zeghbroeck, 1996, 1997