The Hall Effect | 霍尔效应

📚 The Hall Effect | 霍尔效应

The Hall effect describes how a magnetic field can generate a measurable transverse voltage across a current-carrying conductor or semiconductor. It is a powerful tool for investigating charge carriers and measuring magnetic fields.

霍尔效应描述了磁场如何在载流导体或半导体中产生可测量的横向电压。它是研究电荷载流子和测量磁场的有力工具。

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1. What Is the Hall Effect? | 什么是霍尔效应？

When a current-carrying conductor is placed in a magnetic field perpendicular to the current, a voltage is produced across the conductor in the direction perpendicular to both the current and the magnetic field. This transverse voltage is called the Hall voltage, named after Edwin Hall who discovered it in 1879.

当载流导体置于与电流垂直的磁场中时，在同时垂直于电流和磁场的方向上会产生一个电压。这个横向电压称为霍尔电压，以 1879 年发现它的埃德温·霍尔命名。

The effect occurs because moving charge carriers experience a Lorentz force F = qvB, which deflects them to one side of the conductor.

该效应发生的原因是运动的电荷载流子受到洛伦兹力 F = qvB，使其向导体的某一侧偏转。

The Hall effect provides evidence that current in metals is carried by negative electrons, and it allows the sign and density of charge carriers to be determined.

霍尔效应为金属中电流由负电子携带提供了证据，并且可以确定电荷载流子的符号和密度。

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2. Experimental Setup and Coordinate System | 实验装置与坐标系

A common arrangement is a thin flat slab of conducting material of width w and thickness t. A current I flows along its length, taken as the x-axis, and a uniform magnetic field B is applied perpendicular to the large face, taken as the z-axis.

常见装置是一片宽度为 w、厚度为 t 的薄平板导电材料。电流 I 沿其长度方向流动，取为 x 轴，均匀磁场 B 垂直施加于大面，取为 z 轴。

The Hall voltage Vₕ is measured across the width of the slab, along the y-axis, between two contacts on opposite edges.

霍尔电压 Vₕ 沿 y 轴跨过薄片宽度测量，在两个相对边缘的触点之间。

It is essential to distinguish thickness t from width w: t is the dimension parallel to B, while w is the dimension across which Vₕ appears.

必须区分厚度 t 与宽度 w：t 是平行于 B 的尺寸，而 w 是 Vₕ 出现的尺寸。

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3. Origin of the Hall Voltage | 霍尔电压的产生机制

Consider positive charge carriers moving with drift velocity v in the +x direction. In a magnetic field B in the +z direction, each carrier experiences a Lorentz force F = qvB directed along the +y direction.

考虑正电荷载流子以漂移速度 v 沿 +x 方向运动。在 +z 方向的磁场 B 中，每个载流子受到沿 +y 方向的洛伦兹力 F = qvB。

This force pushes positive carriers to one edge, leaving a deficit of positive charge on the opposite edge. The charge separation creates a transverse electric field Eₕ across the slab.

这个力将正载流子推向一个边缘，使相对边缘缺少正电荷。电荷分离在薄片内产生横向电场 Eₕ。

For electrons, the direction of deflection is opposite, so the sign of the Hall voltage flips. This difference is used to identify whether conduction is by electrons or holes.

对电子而言，偏转方向相反，因此霍尔电压的符号反转。这种差异可用于判断导电是电子还是空穴。

Equilibrium is reached when the electric force from the accumulated charge balances the magnetic force: qEₕ = qvB, hence Eₕ = vB.

当累积电荷产生的电场力与磁力平衡时达到平衡：qEₕ = qvB，因此 Eₕ = vB。

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4. Deriving the Hall Voltage Equation | 推导霍尔电压公式

From equilibrium, the Hall electric field is Eₕ = vB. Since the field is uniform between the edges, the Hall voltage is Vₕ = Eₕw = vBw, where w is the width across which the voltage is measured.

由平衡条件，霍尔电场为 Eₕ = vB。由于边缘间电场均匀，霍尔电压为 Vₕ = Eₕw = vBw，其中 w 是测量电压的宽度。

The current I is related to the drift velocity by I = nqvA, where n is the number density of charge carriers, q is the charge on each carrier, and A = wt is the cross-sectional area perpendicular to the current.

电流 I 与漂移速度的关系为 I = nqvA，其中 n 是电荷载流子的数密度，q 是每个载流子的电荷，A = wt 是垂直于电流的横截面积。

Substituting v = I/(nqwt) into Vₕ = vBw gives the standard result:

将 v = I/(nqwt) 代入 Vₕ = vBw，得到标准结果：

Vₕ = B I / (n q t)

Notice that the width w cancels, so the Hall voltage depends on the thickness t along the magnetic field, not on the width across which the voltage is measured.

注意宽度 w 消掉了，因此霍尔电压取决于沿磁场方向的厚度 t，而不是测量电压的宽度。

For electrons, q = e, so the magnitude is often written as Vₕ = BI/(net), where e = 1.60 × 10⁻¹⁹ C.

对电子而言，q = e，因此大小常写作 Vₕ = BI/(net)，其中 e = 1.60 × 10⁻¹⁹ C。

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5. The Hall Coefficient and Carrier Density | 霍尔系数与载流子密度

The Hall coefficient Rₕ is defined as Rₕ = 1/(nq). Using this, the Hall voltage can be expressed as Vₕ = RₕBI/t.

霍尔系数 Rₕ 定义为 Rₕ = 1/(nq)。由此，霍尔电压可表示为 Vₕ = RₕBI/t。

For negative electrons, Rₕ is negative; for positive holes, Rₕ is positive. Therefore the sign of the Hall voltage reveals the sign of the charge carriers.

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

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