## Bandgap Physics

J. C. Daly
December 1, 2005

The voltage drop across a pn junction carring the current, I  is,

 (1)
where VT  is the thermal voltage.
 (2)
Is depends on temperature, physical constants and fabrication parameters.
 (3)
where A is the emitter area, q is the electronic charge, Dn is the diffusion constant for electrons, NAB is the acceptor doping in the base, and WB is the width of the base. mn is the electron mobility in the base.
 (4)

The temperature dependence of mobility depends on doping.

 (5)
where n is 1.5 in lightly doped silicon and n is -1.5 in heavily doped silicon. In the lightly doped transistor base n may be about 1.

ni  , the intrinsic carrier concentration, is strongly temperature dependent.

 (6)
where Ego is the silicon bandgap voltage extrapolated to 0 oK.

Plugging Equations 4, 5, and 6 into Equation 3 results in the following temperature dependence of Is.

 (7)
where B is a temperature independent constant.

Plugging Equation 7 into Equation 1,and assuming ,

 (8)
where B2 is a temperature independent constant.
 (9)
The output voltage is the sum of Vbe that decreases with temperature and a voltage proportional to VT that increases with temperature, since VT is proportional to the absolute temperature, T.
 (10)
where B3 is a temperature independent constant.

Plugging Equation 9 into Equation 10,

 (11)
The desired bandgap output is to a first order independent of temperature. Taking the derivative of Vout,
 (12)
Setting the derivative of Vout with respect to temperature equal to zero.
 (13)
Using Equation 13 to eliminate B2 and B3 from Equation 11,
 (14)
For example

If

• the current is proportional to the absolute temperature (PTAT), a = 1
• n = 1
• T = To
• VT = 0.0259V
• then.
Vout = Ego + 2 VT
Since Ego = 1.205V,
Vout = 1.25 V

Figure 1   The bandgap output voltage (Rainbow curve) is shown,
where n = 1, a = 1, VT = KT/q, To = 300, and Ego = 1.205V.