Although carbon, silicon and germanium have same lattice structure and four valence electrons each, their band structure leads to the energy gaps as

Options

A. `E_(g)(Si)ltE_(g)(Ge)ltE_(g)(C )`
B. `E_(g)(Si)gtE_(g)(Ge)ltE_(g)(C )`
C. `E_(g)(Si)ltE_(g)(Ge)gtE_(g)(C )`
D. `E_(g)(Si)gtE_(g)(Ge)gtE_(g)(C )`

Answer

To solve the question regarding the energy gaps of carbon, silicon, and germanium, we need to analyze the energy band structures of these materials and compare their energy gaps. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Concept of Energy Bands
- **Energy Bands**: In solids, electrons occupy energy levels that form bands. The two important bands are the valence band (filled with electrons) and the conduction band (where electrons can move freely and conduct electricity).
- **Energy Gap (EG)**: The energy gap is the difference in energy between the top of the valence band and the bottom of the conduction band. It determines the electrical conductivity of the material.

**Hint**: Remember that the energy gap is crucial for determining whether a material behaves as a conductor, semiconductor, or insulator.

### Step 2: Identify the Energy Gaps for Each Material
- **Silicon (Si)**: The energy gap (EG) is approximately 1.1 eV.
- **Germanium (Ge)**: The energy gap (EG) is approximately 0.7 eV.
- **Carbon (C)**: The energy gap (EG) is approximately 5.4 eV.

**Hint**: Look up the standard values for the energy gaps of these materials to ensure accuracy.

### Step 3: Compare the Energy Gaps
Now that we have the energy gaps:
- **EG for Carbon**: 5.4 eV (largest)
- **EG for Silicon**: 1.1 eV
- **EG for Germanium**: 0.7 eV (smallest)

From this, we can establish the order of energy gaps:
\[ EG_{C} > EG_{Si} > EG_{Ge} \]

**Hint**: When comparing values, it helps to write them down in order to visualize the differences clearly.

### Step 4: Analyze the Options Given
The options provided in the question were:
- A) \( EG_{Si} < EG_{Ge} < EG_{C} \)
- B) \( EG_{Si} > EG_{Ge} < EG_{C} \)
- C) \( EG_{Ge} < EG_{Si} > EG_{C} \)
- D) \( EG_{C} > EG_{Si} > EG_{Ge} \)

From our comparison, the correct order is:
- \( EG_{C} > EG_{Si} > EG_{Ge} \)

Thus, the correct option is **D**.

**Hint**: Always check each option against your established order to find the correct answer.

### Final Answer
The correct relationship of the energy gaps is:
\[ EG_{C} > EG_{Si} > EG_{Ge} \]