-?. The Hall effect and the sign of charge carriers....

-?. The Hall effect and the sign of charge carriers. Consider a rectangular conductor of width $w$ and thickness $t$ carrying a current $I$ in a uniform magnetic field $\mathbf{B}$ perpendicular to the plane of the conductor, as shown in Fig. $14.45 .$ The charge carriers, with charge $q$ and drift velocity $\mathbf{v}$, experience a Lorentz force $\mathbf{F}=q \mathbf{v} \times \mathbf{B}$, which tends to push them to one side of the conductor. The resulting steady-state charge buildup causes a voltage difference $V_{\mathrm{H}}$, called the Hall voltage, between the sides of the conductor. (a) Show that the Hall voltage is given by $$ V_{\mathrm{H}}=\frac{B I}{n q t} $$ where $n$ is the concentration of charge carriers. [Hint: In steady state, there is no net force on the charge carriers transverse to the current direction, so the force due to the transverse $E$ field must be canceled by the force from the $B$ field.] (b) Show, with an appropriate diagram, that the sign of the Hall voltage gives the sign of the charge carriers. Thus, the Hall voltage allows one to determine whether the carriers in a doped semiconductor are electrons or holes. (c) Compute the magnitude of the Hall voltage in an $n$ -type semiconductor sample at room temperature with donor concentration $$ \begin{aligned} &N_{\mathrm{d}}=10^{22} \mathrm{~m}^{-3} \text { and the following parameters: } t=1 \mathrm{~mm} \text { , }\\ &I=0.01 \mathrm{~A}, B=1.0 \mathrm{~T} \end{aligned} $$

-?. The Hall effect and the sign of charge carriers. Consider a rectangular conductor of width $w$ and thickness $t$ carrying a current $I$ in a uniform magnetic field $\mathbf{B}$ perpendicular to the plane of the conductor, as shown in Fig. $14.45 .$ The charge carriers, with charge $q$ and drift velocity $\mathbf{v}$, experience a Lorentz force $\mathbf{F}=q \mathbf{v} \times \mathbf{B}$, which tends to push them to one side of the conductor. The resulting steady-state charge buildup causes a voltage difference $V_{\mathrm{H}}$, called the Hall voltage, between the sides of the conductor.
(a) Show that the Hall voltage is given by $$
V_{\mathrm{H}}=\frac{B I}{n q t}
$$
where $n$ is the concentration of charge carriers. [Hint:
In steady state, there is no net force on the charge carriers transverse to the current direction, so the force due to the transverse $E$ field must be canceled by the force from the $B$ field.] (b) Show, with an appropriate diagram, that the sign of the Hall voltage gives the sign of the charge carriers. Thus, the Hall voltage allows one to determine whether the carriers in a doped semiconductor are electrons or holes. (c) Compute the magnitude of the Hall voltage in an $n$ -type semiconductor sample at room temperature with donor concentration $$
\begin{aligned}
&N_{\mathrm{d}}=10^{22} \mathrm{~m}^{-3} \text { and the following parameters: } t=1 \mathrm{~mm} \text { , }\\
&I=0.01 \mathrm{~A}, B=1.0 \mathrm{~T}
\end{aligned}
$$

Key Concepts

Determination of Charge Carrier Sign

The sign of the Hall voltage, determined by the direction of the accumulated charge, can be used to infer whether the dominant charge carriers in a material are positive or negative. This method is particularly important in semiconductor physics, where understanding the type of charge carrier—electrons or holes—is essential for characterizing and designing electronic devices.

Charge Carrier Concentration

Charge carrier concentration refers to the density of charge carriers, such as electrons or holes, within a material. In the context of the Hall effect, this concentration is a critical parameter in determining the magnitude of the Hall voltage, as it directly influences the net current and the resulting force balance in the presence of a magnetic field.

Semiconductor Doping

Semiconductor doping is the process of adding impurities to an intrinsic semiconductor to modify its electrical properties by increasing either the number of electrons (n-type) or holes (p-type). The Hall effect is a useful experimental technique in doped semiconductors to determine the sign and concentration of the charge carriers, which in turn helps in optimizing the semiconductor's performance in electronic applications.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving through electric and magnetic fields. In the context of the Hall effect, the magnetic component of the Lorentz force causes the charge carriers to deviate from their initial path, leading to a buildup of charges and the creation of a transverse electric field that eventually balances this force.

Hall Effect

The Hall effect is the development of a transverse voltage across an electrical conductor when it is placed in a magnetic field perpendicular to the current flow. This phenomenon arises from the deflection of moving charge carriers due to the magnetic field, leading to a measurable voltage difference across the material, which provides insights into the properties of the charge carriers.

Steady-State Equilibrium

In the Hall effect, steady-state equilibrium is reached when the transverse electric force generated by the accumulated charges balances the magnetic force acting on the moving charge carriers. This balance eliminates any further net deflection of carriers and establishes a constant Hall voltage across the conductor.

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