Question #43465

An LED and a photodiode are fabricated from the same intrinsic semiconductor material. The energy bandgap $E_g$ (in $\text{J}$) of the material depends on the absolute temperature $T$ (in $\text{K}$) as $E_g(T) = E_0 - \alpha T$, where $E_0$ (in $\text{J}$) and $\alpha$ (in $\text{J/K}$) are positive constants. The threshold voltage for the LED is given by $V_{th} = E_g/e$ (where $e$ is the elementary charge in $\text{C}$). The dark current $I_d$ (in $\text{A}$) of the reverse-biased photodiode is proportional to $\exp\left(-\frac{E_g}{2k_BT}\right)$, where $k_B$ is the Boltzmann constant (in $\text{J/K}$). Assuming the proportionality constant for $I_d$ is independent of temperature, which of the following correctly represents the pair $\left( \Delta V_{th}, \; \frac{\Delta I_d}{I_d} \right)$ when the temperature is increased by a small amount $\Delta T$?