Why does the base-emitter voltage of a BJT decrease with temperature?

Question

According to Sedra/Smith Microelectronic Circuits, \$v_{BE}\$ changes by \$-2\text{mV}/\text{°C}\$. I cannot understand how this could possibly be the case given the equations I am familiar with.

With all currents kept constant, we have:

\$\large{i_E = \frac{I_s}{\alpha}e^{v_{BE}/V_T}}\$

To keep \$i_E\$ constant, any change in \$V_T\$ would have to be accompanied by a change of the same factor in \$v_{BE}\$, otherwise \$\alpha\$ or \$I_s\$ would have to change, which as far as I understand is not possible.

So, how can \$v_{BE}\$ be inversely proportional to \$V_T\$?

Answer 1

\$I_S \$ is highly temperature dependent. As the temperature of the material increases, more electron-hole pairs are thermally generated, increasing \$I_S \$. Here's a link that gives the formula SPICE uses for \$I_S\$, albeit with a typo.

Temperature appears explicitly in the exponential terms of the BJT and diode model equations. In addition, saturation currents have a built-in temperature dependence. The temperature dependence of the saturation current in the BJT models is determined by:

The corrected formula is:

$$ I_S(T_1) = I_S(T_0) \left[\dfrac{T_1}{T_0}\right]^{XTI} \exp\left[ \dfrac{E_g q (1{\rm\,V})}{k} \left(\dfrac{1}{T_0}-\dfrac{1}{T_1} \right) \right], $$

where \$E_g\$ is in electron-Volts, remaining quantities are in SI units, and \$XTI=3\$ unless the transistor model specifies otherwise.

I believe that \$ I_S \$ is roughly cubic in \$ T \$.

Answer 2

"So, how can vBE be inversely proportional to VT?"

I think, this leads to a false understanding of the effect to be observed. With other words: Base-emitter voltage does NOT decrease (automatically) with rising temperature.

The effect is as follows: For rising temperature the collector current Ic increases (because of Is temperature dependence). That means: To keep this current Ic on the same level the base-emitter voltage must be (externally !) decreased. Hence, the data sheet says that for constant Ic the well-known value −2mV/K applies.

EDIT: I like to add that in the following link the temperature dependence of Is is derived as well as a formula for the „magic“ value of -2mV/K .

web.mit.edu/klund/www/Dphysics.pdf

It is interesting to note that this derivation is based on transistor physics only - and without using the base current and the current gain at all.

In this context, I remember some - often controversal debated - questions whether the bipolar transistor is controlled by the current Ib or the voltage Vbe. For me, the derivation contained in the said document is a further clear evidence that the transistor - physically speaking - is controlled by the applied voltage Vbe.

Answer 3

As temperature increases , Ibo or reverse saturated current through emitter junction increases and hence overall current through Je reducess, and hence Vbe also get reduces proportionally.