Question

a sample of germanium shows no Hall effect. If the mobility of electrons in germinium is 3500 cm2/V.sec and that of holes is 1400 cm2/V.sec. What fraction of the current in the sample is carried by electrons ?

Answer 1

Answer on Question#53461 - Solid State Physics

A sample of Germanium shows no Hall Effect. If the mobility of electrons in Germanium is $\mu_{e} = 3500\frac{\mathrm{cm}^{2}}{\mathrm{V}\cdot\mathrm{s}}$ and that of holes is $\mu_{h} = 1400\frac{\mathrm{cm}^{2}}{\mathrm{V}\cdot\mathrm{s}}$. What fraction of the current in the sample is carried by electrons?

Solution:

The speed of charge carriers is related to their mobility by means of the following expression

$$v = \mu E,$$

where $E$ – is the electric field applied across the sample of Germanium. Thus the net current due to electrons is given by

$$I_{e} = v_{e} \cdot e \cdot n \cdot A = \mu_{e} \cdot E \cdot e \cdot n \cdot A,$$

where $A$ – is the cross-sectional area of the sample, $n$ – is the concentration of the electrons in the sample, $e$ – elementary electric charge. If the sample is electrically neutral, the concentration of holes is also $n$ and the current due to holes is given by

$$I_{h} = v_{h} \cdot e \cdot n \cdot A = \mu_{h} \cdot E \cdot e \cdot n \cdot A$$

Therefore the fraction of the current in the sample carried by electrons is

$$\frac{I_{e}}{I} = \frac{I_{e}}{I_{e} + I_{h}} = \frac{\mu_{e} \cdot E \cdot e \cdot n \cdot A}{(\mu_{e} + \mu_{h}) \cdot E \cdot e \cdot n \cdot A} = \frac{\mu_{e}}{\mu_{e} + \mu_{h}} = \frac{3500 \frac{\mathrm{cm}^{2}}{\mathrm{V} \cdot \mathrm{s}}}{3500 \frac{\mathrm{cm}^{2}}{\mathrm{V} \cdot \mathrm{s}} + 1400 \frac{\mathrm{cm}^{2}}{\mathrm{V} \cdot \mathrm{s}}} = \frac{5}{7}$$

Answer: $\frac{5}{7}$.