Let's analyze the properties of a pure semiconductor, also known as an intrinsic semiconductor, focusing on the movement of charge carriers within the material. In a pure semiconductor, the number of electrons in the conduction band is equal to the number of holes in the valence band. These electrons and holes are the primary charge carriers responsible for electrical conductivity.

Understanding Charge Carrier Mobility in Pure Semiconductors

Mobility ($\mu$) is a measure of how easily charge carriers move through a material when an electric field is applied. It is defined as the magnitude of the drift velocity ($v_d$) per unit electric field ($E$):

$$ \mu = \frac{|v_d|}{E} $$

The drift velocity itself depends on the charge ($q$), the electric field ($E$), the effective mass ($m^*$) of the carrier, and the scattering time ($\tau$), roughly given by $v_d = \frac{qE\tau}{m^*}$. Thus, mobility can also be related to the effective mass and scattering time:

$$ \mu = \frac{q\tau}{m^*} $$

This formula shows that mobility is inversely proportional to the effective mass of the charge carrier. A lower effective mass generally leads to higher mobility.

Comparing Electron Mobility and Hole Mobility in Pure Semiconductors

In a pure semiconductor like silicon or germanium, both electrons and holes contribute to conductivity. However, their mobilities are not equal. Electrons in the conduction band move relatively freely, while holes represent the absence of an electron in the valence band. The movement of a hole is essentially the sequential movement of electrons filling these vacancies.

Crucially, the effective mass of an electron ($m_e^*$) in a semiconductor is generally smaller than the effective mass of a hole ($m_h^*$). This difference arises from the specific band structure of the semiconductor material. Electrons in the conduction band and holes in the valence band interact differently with the crystal lattice, leading to different effective masses.

Since mobility is inversely proportional to effective mass ($\mu \propto 1/m^*$), and the effective mass of electrons is less than that of holes ($m_e^* < m_h^*$), the electron mobility ($\mu_e$) is typically higher than the hole mobility ($\mu_h$) in a pure semiconductor.

Also, electrons in the conduction band experience fewer collisions with the crystal lattice compared to holes in the valence band, which also contributes to higher electron mobility.

Analyzing the Options

Let's evaluate the given options based on our understanding of charge carrier mobility in a pure semiconductor:

1. Holes and electrons have equal mobility: This is incorrect. As discussed, electron mobility is generally higher than hole mobility.
2. holes have higher mobility: This is incorrect. Electron mobility is higher than hole mobility.
3. electrons have higher mobility: This aligns with our understanding that the effective mass of electrons is less than that of holes, leading to higher electron mobility.
4. only electrons are mobile: This is incorrect. Both electrons and holes are charge carriers and contribute to conductivity in a pure semiconductor.

Therefore, in a pure semiconductor, electrons have higher mobility compared to holes. This difference in electron mobility and hole mobility is a fundamental property of most semiconductor materials.