How to Calculate the Threshold voltage of a MOSFET

 

Base formula:

 

Vt=Vt-mos +Vfb

 

Vt-mos is the ideal threshold voltage (no work function difference between the gate and substrate materials) and Vfb is the flatband voltage.

 

Vt-mos=2Φb+Qb/Cox   where Cox= eox/tox

 

Φb=kT/q ln(NA/ni)        where   kT/q=.26mV at room temperature (300οK)

                                                k=1.38x10-23 J/οK  Boltzmann’s constant

                                                T is the temperature in kelvins

                                                q=1.602x10-19 Coulombs, the electron charge

                                                NA=density of doped carriers (given)

                                                ni=1.45x1010cm3 carriers in intrinsic silicon

                                                eox=3.9x8.85x10-14 F/cm2, tox is given

                                                Vsb is substrate bias (given)

 

Qb=√(2*esi *q* NA*(2*Φb+|Vsb|))       where esi=1.06x10-12 F/cm  permittivity of silicon

 

The complete formula so far where Vsb = 0:

 

Vt-mos=2 kT/q ln(NA/ni) + √(2esiqNA2 Φb)*1/(eox/tox)  (this is Vt0 in formula 2.30 3rd ed.

                                                                                    Also, his γ is Qb/Cox and Φs=2Φb)

 

Note: Vt-mos is positive for nMOS and negative for pMOS.

 

The flatband formulas:

 

            Vfb=Φms-Qfc/Cox               where   Φms is work function difference ‘twn gate and wafer

                                                            Qfc is a fixed charge for surface states (given)

 

            Φms=-(Eg/2±Φb)           where   sign is determine by the following rule:

                                                            + if device is an nMOS

- if device is a pMOS

                                                            Eg is band gap energy of silicon… 1.1eV or…

 

Eg=(1.16-.704x10-3(T2/(T+1108))   (for silicon band gap energy at other than

room temp)

 

Complete flatband formula:

 

Vfb= -((1.16-.704x10-3(T2/(T+1108)))/2± kT/q ln(NA/ni))-Qfc/(eox/tox)

 

The entire Vt formula:

 

Vt=2 kT/q ln(NA/ni) + √(2esiqNA2 Φb)/(eox/tox)-((1.16-.704x10-3(T2/(T+1108)))/2± kT/q ln(NA/ni))-Qfc/(eox/tox)