Electric Current, Drift Velocity, and Mobility

Electric Current

Electric current is the rate of flow of electric charge. It is a fundamental concept in understanding the behavior of electricity. When we talk about electric current, we are essentially describing how much charge passes through a particular point or cross-sectional area in a given amount of time.

The SI unit of electric current is the Ampere (A), named after the French physicist André-Marie Ampère. One Ampere is defined as the flow of one Coulomb of charge per second. Mathematically, current (I) is expressed as:

$I = \frac{dQ}{dt}$

Where $I$ is the current, $dQ$ is the amount of charge, and $dt$ is the time interval over which the charge flows. If the charge flow is uniform, the formula simplifies to:

$I = \frac{Q}{t}$

In most metallic conductors, the charge carriers are free electrons. These electrons are in constant random motion within the material. However, when an electric field is applied across the conductor, these electrons experience a force and acquire a net directional motion, resulting in an electric current. The direction of the conventional current is defined as the direction of flow of positive charge, which is opposite to the direction of flow of electrons.

The magnitude of the current depends on the number of charge carriers, the charge of each carrier, and their average drift velocity. A higher density of charge carriers or a faster drift velocity leads to a larger current.

Factors Affecting Current

• Charge Carrier Density: More charge carriers per unit volume mean more charge can flow.
• Charge of Carrier: The magnitude of the charge on each carrier. For electrons, this is the elementary charge, $e$.
• Drift Velocity: The average velocity attained by the charge carriers due to the applied electric field.

Drift Velocity

When a voltage is applied across a conductor, an electric field ($E$) is established within it. This electric field exerts a force on the free charge carriers (usually electrons in metals). The force on a charge carrier with charge $q$ is given by $F = qE$. In metals, $q = -e$ for electrons.

This force causes the electrons to accelerate. However, as they move, they collide with the atoms and ions in the conductor's lattice. These collisions impede their motion, causing them to lose the energy gained from the electric field. The net effect of acceleration and frequent collisions is that the electrons acquire a small, constant average velocity in the direction opposite to the electric field. This average velocity is called the drift velocity ($v_d$).

The motion of electrons can be visualized as a 'random walk' with a superimposed slow drift. Even though individual electrons move randomly at high speeds (thermal velocity), their net displacement in the direction of the electric field is very small.

The drift velocity is related to the electric field ($E$) and the average time between collisions ($\tau$), also known as the relaxation time. The acceleration of an electron due to the electric field is $a = \frac{F}{m} = \frac{qE}{m}$, where $m$ is the mass of the electron. If an electron starts with velocity $u$ and undergoes collisions, its velocity changes. Considering the average time between collisions, the average velocity gained due to the field is $v_d = a\tau$.

For electrons in a metal, where $q = -e$, the drift velocity is:

$v_d = -\frac{eE\tau}{m_e}$

The negative sign indicates that the drift velocity of electrons is opposite to the direction of the electric field. The magnitude of the drift velocity is:

$|v_d| = \frac{eE\tau}{m_e}$

Here, $e$ is the elementary charge, $E$ is the magnitude of the electric field, $\tau$ is the average relaxation time, and $m_e$ is the mass of the electron.

The drift velocity is typically very small, on the order of millimeters per second, even for currents that can be readily measured. This is because the density of free electrons in metals is very high.

Relationship between Drift Velocity and Current

We can establish a direct relationship between the drift velocity of charge carriers and the electric current flowing through a conductor. Consider a cylindrical conductor of length $L$ and cross-sectional area $A$. Let $n$ be the number of free charge carriers per unit volume, and let $q$ be the charge of each carrier.

When an electric field $E$ is applied, the charge carriers drift with an average velocity $v_d$. In time $dt$, each charge carrier drifts a distance $dx = v_d dt$.

Consider a small segment of the conductor of length $v_d dt$. The volume of this segment is $A \cdot v_d dt$. The number of charge carriers in this volume is $n \cdot (A \cdot v_d dt)$.

The total charge ($dQ$) passing through the cross-sectional area $A$ in time $dt$ is the product of the number of charge carriers in the segment and the charge per carrier:

$dQ = (n A v_d dt) q$

The electric current $I$ is the rate of flow of charge, $I = \frac{dQ}{dt}$. Substituting the expression for $dQ$:

$I = \frac{n A v_d dt q}{dt}$

$I = n A v_d q$

For a conductor where charge carriers are electrons, $q = -e$. The current is conventionally taken as positive in the direction of positive charge flow. Since electrons drift in the opposite direction of the electric field, the current direction is the same as the electric field direction. The magnitude of the current is:

$I = n A v_d e$ (where $v_d$ is the magnitude of drift velocity)

This equation shows that the current is directly proportional to the charge carrier density ($n$), the cross-sectional area ($A$), the magnitude of the charge of each carrier ($e$), and the drift velocity ($v_d$).

Mobility

Mobility is a measure of how easily a charge carrier can move through a material under the influence of an electric field. It is defined as the magnitude of the drift velocity acquired per unit electric field.

The symbol for mobility is $\mu$ (mu). Mathematically, it is expressed as:

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

From the drift velocity equation, $|v_d| = \frac{eE\tau}{m_e}$, we can substitute this into the mobility formula:

$\mu = \frac{\frac{eE\tau}{m_e}}{E}$

$\mu = \frac{e\tau}{m_e}$

This formula shows that the mobility of charge carriers depends on the nature of the material (through the relaxation time $\tau$ and the mass of the carrier $m_e$) and the charge of the carrier $e$. It is independent of the electric field strength, provided the field is not excessively strong (which could lead to effects like velocity saturation).

The SI unit of mobility is $\text{m}^2/\text{V}\cdot\text{s}$ (square meters per volt-second).

Mobility is an important parameter for semiconductor devices. Higher mobility means charge carriers can move faster, leading to faster device operation and higher conductivity. Materials like silicon and germanium have different mobilities for electrons and holes, which influences their electronic properties.

We can also express conductivity ($\sigma$) in terms of mobility. Recall that $I = n A v_d e$ and $v_d = \mu E$. Substituting $v_d$:

$I = n A (\mu E) e$

$I = (n e \mu) A E$

We also know Ohm's law in its microscopic form: $J = \sigma E$, where $J$ is the current density ($J = I/A$). So, $I = J A = \sigma E A$.

Comparing the two expressions for $I$:

$\sigma E A = (n e \mu) A E$

This gives us the relationship between conductivity and mobility:

$\sigma = n e \mu$

For materials with multiple types of charge carriers (like semiconductors with both electrons and holes), the total conductivity is the sum of the conductivities due to each type of carrier:

$\sigma = n_e e \mu_e + n_h e \mu_h$

Where $n_e$ and $n_h$ are the densities of electrons and holes, and $\mu_e$ and $\mu_h$ are their respective mobilities.

Properties of Mobility

• Temperature Dependence: For metals, mobility generally decreases with increasing temperature because increased thermal vibrations lead to more frequent collisions ($\tau$ decreases). For semiconductors, mobility often increases with temperature up to a certain point, as lattice scattering becomes less dominant than impurity scattering at lower temperatures, but eventually decreases at higher temperatures due to increased lattice scattering.
• Material Dependence: Different materials have different crystal structures and atomic arrangements, affecting the relaxation time and hence mobility.
• Carrier Type: Electrons and holes have different masses and interactions within a crystal lattice, leading to different mobilities. Electrons typically have higher mobility than holes in most semiconductors.

Example Calculation

Consider a copper wire with a cross-sectional area of $1 \text{ mm}^2$ carrying a current of $1 \text{ A}$. The number density of free electrons in copper is approximately $n = 8.5 \times 10^{28} \text{ m}^{-3}$. Calculate the drift velocity of the electrons.

Given: $A = 1 \text{ mm}^2 = 1 \times 10^{-6} \text{ m}^2$ $I = 1 \text{ A}$ $n = 8.5 \times 10^{28} \text{ m}^{-3}$ $e = 1.602 \times 10^{-19} \text{ C}$

Using the formula $I = n A v_d e$:

$v_d = \frac{I}{n A e}$

$v_d = \frac{1 \text{ A}}{(8.5 \times 10^{28} \text{ m}^{-3}) \times (1 \times 10^{-6} \text{ m}^2) \times (1.602 \times 10^{-19} \text{ C})}$

$v_d = \frac{1}{1.3617 \times 10^{13}} \text{ m/s}$

$v_d \approx 7.34 \times 10^{-14} \text{ m/s}$

This result, though very small, demonstrates the concept. The drift velocity is extremely slow.

Example: Mobility and Conductivity

In silicon at room temperature, the electron mobility is $\mu_e \approx 1400 \text{ cm}^2/\text{V}\cdot\text{s}$ and the hole mobility is $\mu_h \approx 450 \text{ cm}^2/\text{V}\cdot\text{s}$. If the concentration of electrons and holes are both $n_e = n_h = 10^{10} \text{ cm}^{-3}$ (intrinsic silicon), calculate the conductivity.

First, convert mobilities and concentrations to SI units: $\mu_e = 1400 \times 10^{-4} \text{ m}^2/\text{V}\cdot\text{s} = 0.14 \text{ m}^2/\text{V}\cdot\text{s}$ $\mu_h = 450 \times 10^{-4} \text{ m}^2/\text{V}\cdot\text{s} = 0.045 \text{ m}^2/\text{V}\cdot\text{s}$ $n_e = n_h = 10^{10} \times (10^2)^3 \text{ m}^{-3} = 10^{16} \text{ m}^{-3}$ $e = 1.602 \times 10^{-19} \text{ C}$

Total conductivity $\sigma = n_e e \mu_e + n_h e \mu_h$

$\sigma = (10^{16} \text{ m}^{-3}) \times (1.602 \times 10^{-19} \text{ C}) \times (0.14 \text{ m}^2/\text{V}\cdot\text{s}) + (10^{16} \text{ m}^{-3}) \times (1.602 \times 10^{-19} \text{ C}) \times (0.045 \text{ m}^2/\text{V}\cdot\text{s})$

$\sigma = (1.602 \times 10^{-19}) \times (0.14 + 0.045) \times 10^{16} \text{ S/m}$

$\sigma = (1.602 \times 10^{-3}) \times (0.185) \text{ S/m}$

$\sigma \approx 2.96 \times 10^{-4} \text{ S/m}$

This calculation shows how mobility directly contributes to the conductivity of a material.