How do you calculate Hall conductivity?
2 Answers
Hall conductivity (\(\sigma_{xy}\)) is a measure of how an electric current flows perpendicular to an applied electric field in the presence of a magnetic field. It is calculated from the Hall effect, a phenomenon observed when a magnetic field is applied perpendicular to a current-carrying conductor. Here's a detailed explanation of the calculation process:
---
### **1. Hall Effect Basics**
When a conductor or semiconductor carries an electric current (\(I\)) in the presence of a perpendicular magnetic field (\(B\)), a transverse voltage (\(V_H\), Hall voltage) develops across the sample. This voltage arises due to the Lorentz force deflecting charge carriers to one side of the conductor.
- **Lorentz Force**: \(F = q(v \times B)\), where \(q\) is the charge of the carrier and \(v\) is its velocity.
- The accumulation of charge on one side creates an electric field that opposes the Lorentz force, leading to a steady Hall voltage.
---
### **2. Relationship Between Hall Voltage and Hall Conductivity**
Hall conductivity (\(\sigma_{xy}\)) is related to the Hall resistance (\(R_H\)):
\[
\sigma_{xy} = \frac{1}{R_H}
\]
Hall resistance (\(R_H\)) is defined as:
\[
R_H = \frac{V_H}{I}
\]
where:
- \(V_H\) is the Hall voltage (measured between the two transverse sides of the conductor).
- \(I\) is the current through the conductor.
---
### **3. Hall Conductivity in Terms of Carrier Density**
In a material with charge carriers (electrons or holes) of density \(n\), charge \(q\), and thickness \(d\), the Hall voltage is:
\[
V_H = \frac{B I}{n q d}
\]
From this, the Hall resistance becomes:
\[
R_H = \frac{V_H}{I} = \frac{B}{n q d}
\]
The Hall conductivity, which is the inverse of \(R_H\), is:
\[
\sigma_{xy} = \frac{1}{R_H} = \frac{n q d}{B}
\]
---
### **4. Quantum Hall Effect (if applicable)**
In two-dimensional systems (like 2D electron gases in semiconductors), the Hall conductivity quantizes under low temperatures and strong magnetic fields. The quantized Hall conductivity is given by:
\[
\sigma_{xy} = \frac{e^2}{h} \cdot \nu
\]
where:
- \(e\) is the electron charge.
- \(h\) is Planck's constant.
- \(\nu\) is the filling factor, an integer or fractional value determined by the Landau levels.
---
### **5. Summary of Steps to Calculate Hall Conductivity**
1. **Measure Hall Voltage (\(V_H\))**: Measure the transverse voltage across the conductor.
2. **Measure Current (\(I\))**: Determine the current flowing through the conductor.
3. **Measure Magnetic Field (\(B\))**: Ensure it is perpendicular to the plane of the conductor.
4. **Determine Material Properties**: Obtain the charge carrier density (\(n\)), charge (\(q\)), and thickness (\(d\)) of the conductor.
5. **Apply the Formula**:
- Classical Hall Effect: \(\sigma_{xy} = \frac{n q d}{B}\)
- Quantum Hall Effect: \(\sigma_{xy} = \frac{e^2}{h} \cdot \nu\), if quantization conditions apply.
This calculation provides insight into the electronic properties of materials, especially in condensed matter physics and semiconductor technology.
---
### **1. Hall Effect Basics**
When a conductor or semiconductor carries an electric current (\(I\)) in the presence of a perpendicular magnetic field (\(B\)), a transverse voltage (\(V_H\), Hall voltage) develops across the sample. This voltage arises due to the Lorentz force deflecting charge carriers to one side of the conductor.
- **Lorentz Force**: \(F = q(v \times B)\), where \(q\) is the charge of the carrier and \(v\) is its velocity.
- The accumulation of charge on one side creates an electric field that opposes the Lorentz force, leading to a steady Hall voltage.
---
### **2. Relationship Between Hall Voltage and Hall Conductivity**
Hall conductivity (\(\sigma_{xy}\)) is related to the Hall resistance (\(R_H\)):
\[
\sigma_{xy} = \frac{1}{R_H}
\]
Hall resistance (\(R_H\)) is defined as:
\[
R_H = \frac{V_H}{I}
\]
where:
- \(V_H\) is the Hall voltage (measured between the two transverse sides of the conductor).
- \(I\) is the current through the conductor.
---
### **3. Hall Conductivity in Terms of Carrier Density**
In a material with charge carriers (electrons or holes) of density \(n\), charge \(q\), and thickness \(d\), the Hall voltage is:
\[
V_H = \frac{B I}{n q d}
\]
From this, the Hall resistance becomes:
\[
R_H = \frac{V_H}{I} = \frac{B}{n q d}
\]
The Hall conductivity, which is the inverse of \(R_H\), is:
\[
\sigma_{xy} = \frac{1}{R_H} = \frac{n q d}{B}
\]
---
### **4. Quantum Hall Effect (if applicable)**
In two-dimensional systems (like 2D electron gases in semiconductors), the Hall conductivity quantizes under low temperatures and strong magnetic fields. The quantized Hall conductivity is given by:
\[
\sigma_{xy} = \frac{e^2}{h} \cdot \nu
\]
where:
- \(e\) is the electron charge.
- \(h\) is Planck's constant.
- \(\nu\) is the filling factor, an integer or fractional value determined by the Landau levels.
---
### **5. Summary of Steps to Calculate Hall Conductivity**
1. **Measure Hall Voltage (\(V_H\))**: Measure the transverse voltage across the conductor.
2. **Measure Current (\(I\))**: Determine the current flowing through the conductor.
3. **Measure Magnetic Field (\(B\))**: Ensure it is perpendicular to the plane of the conductor.
4. **Determine Material Properties**: Obtain the charge carrier density (\(n\)), charge (\(q\)), and thickness (\(d\)) of the conductor.
5. **Apply the Formula**:
- Classical Hall Effect: \(\sigma_{xy} = \frac{n q d}{B}\)
- Quantum Hall Effect: \(\sigma_{xy} = \frac{e^2}{h} \cdot \nu\), if quantization conditions apply.
This calculation provides insight into the electronic properties of materials, especially in condensed matter physics and semiconductor technology.
The **Hall conductivity** (\(\sigma_{xy}\)) is a fundamental property in condensed matter physics, describing how a material conducts current in a direction perpendicular to an applied electric field and magnetic field. It is a key concept in understanding the **Hall effect**. Below is a detailed explanation of how to calculate Hall conductivity.
---
### **1. Key Equations and Concepts**
Hall conductivity relates to the Hall resistivity (\(\rho_{xy}\)) and the components of the conductivity tensor (\(\sigma_{ij}\)). It can be derived from experimental or theoretical principles, depending on the situation.
#### **Relation to Hall Resistivity:**
The Hall resistivity \(\rho_{xy}\) is the inverse of the Hall conductivity in a simple case:
\[
\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2},
\]
where:
- \(\rho_{xx}\) is the longitudinal resistivity (resistance in the direction of the current),
- \(\rho_{xy}\) is the Hall resistivity (resistance in the transverse direction due to the magnetic field).
#### **Conductivity Tensor:**
In tensor notation, the conductivity in the presence of a magnetic field has components:
\[
\sigma_{ij} = \sigma_{xx}\delta_{ij} + \sigma_{xy}\epsilon_{ij},
\]
where:
- \(\delta_{ij}\) is the Kronecker delta (identity component),
- \(\epsilon_{ij}\) is the antisymmetric tensor related to the Hall effect.
---
### **2. Classical Approach to Hall Conductivity**
The classical Hall conductivity can be determined using the motion of charge carriers under the influence of a magnetic field.
#### **Setup:**
- A conductor or semiconductor is placed in a magnetic field \(\mathbf{B}\) (along the \(z\)-axis, say).
- An electric field \(\mathbf{E}\) is applied, causing charge carriers to move.
- The magnetic field causes the carriers to deflect (via the Lorentz force), inducing a transverse voltage (the Hall voltage).
#### **Key Expression:**
The Hall conductivity is:
\[
\sigma_{xy} = \frac{q n}{B},
\]
where:
- \(q\) is the charge of the carrier (e.g., \(q = -e\) for electrons),
- \(n\) is the carrier density,
- \(B\) is the magnetic field strength.
#### **Derivation:**
1. The Lorentz force on a charge carrier is:
\[
\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),
\]
where \(\mathbf{v}\) is the carrier velocity.
2. In steady-state conditions, the forces balance, giving rise to a transverse voltage (Hall voltage \(V_H\)) and a current density \(J\).
3. Relating the Hall voltage to the carrier velocity, you find the Hall conductivity as above.
---
### **3. Quantum Hall Effect (QHE)**
For 2D systems at low temperatures and in strong magnetic fields, the **quantized Hall effect** occurs. The Hall conductivity is given by:
\[
\sigma_{xy} = \nu \frac{e^2}{h},
\]
where:
- \(\nu\) is an integer (integer quantum Hall effect) or fraction (fractional quantum Hall effect),
- \(e\) is the elementary charge,
- \(h\) is Planck’s constant.
#### **Quantum Explanation:**
- In 2D electron gases, electrons form discrete energy levels called Landau levels.
- The filling of these levels determines the value of \(\nu\), leading to quantized steps in Hall conductivity.
---
### **4. Experimental Measurement of Hall Conductivity**
#### **Procedure:**
1. Place the sample in a magnetic field (\(\mathbf{B}\)).
2. Measure the Hall voltage (\(V_H\)) across the sample while applying a current (\(I\)).
3. Compute the Hall resistivity using:
\[
\rho_{xy} = \frac{V_H}{I/B}.
\]
4. Use the relationship between resistivity and conductivity to calculate \(\sigma_{xy}\):
\[
\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2}.
\]
---
### **5. Summary of Important Cases**
| **Scenario** | **Formula for \(\sigma_{xy}\)** | **Key Features** |
|-------------------------------|---------------------------------------------------------------------|----------------------------------------------------------|
| Classical Hall Effect | \(\sigma_{xy} = \frac{qn}{B}\) | Carrier density-based; depends on \(B\). |
| Quantum Hall Effect (QHE) | \(\sigma_{xy} = \nu \frac{e^2}{h}\) | Quantized, observed in 2D electron gases at low \(T\). |
| Relation to Resistivity | \(\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2}\) | Uses measured resistivity tensor components. |
---
### **6. Applications**
- Understanding charge transport in semiconductors and metals.
- Identifying material properties like carrier density and mobility.
- Precision measurements of fundamental constants (e.g., \(e\) and \(h\)) using the QHE.
- Designing Hall sensors for magnetic field detection.
---
### **1. Key Equations and Concepts**
Hall conductivity relates to the Hall resistivity (\(\rho_{xy}\)) and the components of the conductivity tensor (\(\sigma_{ij}\)). It can be derived from experimental or theoretical principles, depending on the situation.
#### **Relation to Hall Resistivity:**
The Hall resistivity \(\rho_{xy}\) is the inverse of the Hall conductivity in a simple case:
\[
\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2},
\]
where:
- \(\rho_{xx}\) is the longitudinal resistivity (resistance in the direction of the current),
- \(\rho_{xy}\) is the Hall resistivity (resistance in the transverse direction due to the magnetic field).
#### **Conductivity Tensor:**
In tensor notation, the conductivity in the presence of a magnetic field has components:
\[
\sigma_{ij} = \sigma_{xx}\delta_{ij} + \sigma_{xy}\epsilon_{ij},
\]
where:
- \(\delta_{ij}\) is the Kronecker delta (identity component),
- \(\epsilon_{ij}\) is the antisymmetric tensor related to the Hall effect.
---
### **2. Classical Approach to Hall Conductivity**
The classical Hall conductivity can be determined using the motion of charge carriers under the influence of a magnetic field.
#### **Setup:**
- A conductor or semiconductor is placed in a magnetic field \(\mathbf{B}\) (along the \(z\)-axis, say).
- An electric field \(\mathbf{E}\) is applied, causing charge carriers to move.
- The magnetic field causes the carriers to deflect (via the Lorentz force), inducing a transverse voltage (the Hall voltage).
#### **Key Expression:**
The Hall conductivity is:
\[
\sigma_{xy} = \frac{q n}{B},
\]
where:
- \(q\) is the charge of the carrier (e.g., \(q = -e\) for electrons),
- \(n\) is the carrier density,
- \(B\) is the magnetic field strength.
#### **Derivation:**
1. The Lorentz force on a charge carrier is:
\[
\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),
\]
where \(\mathbf{v}\) is the carrier velocity.
2. In steady-state conditions, the forces balance, giving rise to a transverse voltage (Hall voltage \(V_H\)) and a current density \(J\).
3. Relating the Hall voltage to the carrier velocity, you find the Hall conductivity as above.
---
### **3. Quantum Hall Effect (QHE)**
For 2D systems at low temperatures and in strong magnetic fields, the **quantized Hall effect** occurs. The Hall conductivity is given by:
\[
\sigma_{xy} = \nu \frac{e^2}{h},
\]
where:
- \(\nu\) is an integer (integer quantum Hall effect) or fraction (fractional quantum Hall effect),
- \(e\) is the elementary charge,
- \(h\) is Planck’s constant.
#### **Quantum Explanation:**
- In 2D electron gases, electrons form discrete energy levels called Landau levels.
- The filling of these levels determines the value of \(\nu\), leading to quantized steps in Hall conductivity.
---
### **4. Experimental Measurement of Hall Conductivity**
#### **Procedure:**
1. Place the sample in a magnetic field (\(\mathbf{B}\)).
2. Measure the Hall voltage (\(V_H\)) across the sample while applying a current (\(I\)).
3. Compute the Hall resistivity using:
\[
\rho_{xy} = \frac{V_H}{I/B}.
\]
4. Use the relationship between resistivity and conductivity to calculate \(\sigma_{xy}\):
\[
\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2}.
\]
---
### **5. Summary of Important Cases**
| **Scenario** | **Formula for \(\sigma_{xy}\)** | **Key Features** |
|-------------------------------|---------------------------------------------------------------------|----------------------------------------------------------|
| Classical Hall Effect | \(\sigma_{xy} = \frac{qn}{B}\) | Carrier density-based; depends on \(B\). |
| Quantum Hall Effect (QHE) | \(\sigma_{xy} = \nu \frac{e^2}{h}\) | Quantized, observed in 2D electron gases at low \(T\). |
| Relation to Resistivity | \(\sigma_{xy} = \frac{\rho_{xy}}{\rho_{xx}^2 + \rho_{xy}^2}\) | Uses measured resistivity tensor components. |
---
### **6. Applications**
- Understanding charge transport in semiconductors and metals.
- Identifying material properties like carrier density and mobility.
- Precision measurements of fundamental constants (e.g., \(e\) and \(h\)) using the QHE.
- Designing Hall sensors for magnetic field detection.