The temperature dependence of resistance of Cu and undoped Si in the temperature range 300 - 400 K is best described by:

We begin by recalling how the resistance of a metal varies with temperature. For any ordinary metal such as copper, the empirical relation is

$$R = R_0\left(1 + \alpha (T-T_0)\right),$$

where $$R_0$$ is the resistance at a reference temperature $$T_0$$ and $$\alpha$$ is the temperature coefficient of resistance. This expression is of the form $$mT + c$$, which is a straight-line graph. Because $$\alpha$$ for metals is positive, the slope is positive. Hence the resistance of copper rises linearly as the temperature increases from 300 K to 400 K.

Now we analyse undoped (intrinsic) silicon. The intrinsic carrier concentration $$n_i$$ in a semiconductor is given by the well-known relation from band-gap theory,

$$n_i = A\,T^{3/2}\exp\!\left(\frac{-E_g}{2kT}\right),$$

where $$A$$ is a material constant, $$E_g$$ is the band-gap energy, and $$k$$ is Boltzmann’s constant. The resistivity $$\rho$$ of an intrinsic semiconductor satisfies

$$\rho = \frac{1}{q\mu_e n_i}$$

with $$q$$ the electronic charge and $$\mu_e$$ the (approximately weakly temperature-dependent) electron mobility. Because $$n_i$$ contains the factor $$\exp\!\left(-\dfrac{E_g}{2kT}\right)$$ in the denominator of $$\rho$$, we obtain

$$\rho \propto \exp\!\left(\frac{E_g}{2kT}\right).$$

Resistance $$R$$ is proportional to resistivity $$\rho$$, so

$$R \propto \exp\!\left(\frac{E_g}{2kT}\right).$$

As $$T$$ increases from 300 K to 400 K, the exponent $$\dfrac{E_g}{2kT}$$ decreases, making the whole exponential expression shrink rapidly. Therefore the resistance of intrinsic silicon shows a pronounced exponential decrease with rising temperature.

Putting the two results together, we see that

• copper: linear increase in $$R$$ with $$T$$, • intrinsic silicon: exponential decrease in $$R$$ with $$T$$.

This behaviour matches Option A.

Hence, the correct answer is Option A.