HALL EFFECT EXPERIMENT

To determine the Hall coefficient and mobility of charge carriers in a semiconductor sample using the Hall Effect.

• Semiconductor sample (n-type or p-type)
• Electromagnet with adjustable pole pieces
• Hall effect probe with sample holder
• DC power supply (0-30V)
• Current source (0-100mA)
• Digital gauss meter
• Digital multimeter or nanovoltmeter
• Micrometer or vernier caliper
• Connecting wires and cables
• Thermometer

Figure 1: Schematic diagram of the Hall effect experimental setup showing the semiconductor sample placed perpendicular to the magnetic field with current and voltage connections.

The Hall effect is a fundamental phenomenon in solid-state physics that occurs when a conductor or semiconductor carrying an electric current is placed in a magnetic field perpendicular to the current direction. This causes charge carriers to experience a Lorentz force perpendicular to both the magnetic field and the current flow, resulting in an accumulation of charges on the sides of the sample.

When a current flows through a semiconductor placed in a magnetic field, charge carriers (electrons or holes) are deflected due to the magnetic force. This deflection creates an electric field perpendicular to both the current and magnetic field, leading to a potential difference across the sample known as the Hall voltage.

Consider a rectangular semiconductor sample with thickness $t$, width $w$, and length $l$. When a current $I$ flows along the length and a magnetic field $B$ is applied perpendicular to the sample, the charge carriers experience a Lorentz force:

Where $q$ is the charge of the carrier and $v$ is the drift velocity. This force deflects carriers toward one side of the sample, creating an electric field $E_H$ (Hall field) that counterbalances the Lorentz force. At equilibrium:

This produces a Hall voltage $V_H$ across the width $w$ of the sample:

The drift velocity $v$ is related to the current density $J$ by:

Where $n$ is the carrier concentration. The current $I$ through the sample is:

The Hall coefficient $R_H$ is defined as:

For a semiconductor with a single type of charge carrier, the Hall coefficient is related to the carrier concentration by:

The sign of $R_H$ indicates the type of semiconductor:

• Negative $R_H$ indicates an n-type semiconductor (electrons are majority carriers)
• Positive $R_H$ indicates a p-type semiconductor (holes are majority carriers)

The carrier mobility $\mu$ can be calculated from the Hall coefficient and the conductivity $\sigma$ of the sample:

where the conductivity is determined from:

Here, $R$ is the resistance of the sample, $V$ is the voltage drop along the length when current $I$ flows, and $l$, $w$, and $t$ are the length, width, and thickness of the sample, respectively.

1. Hall voltage:
   $V_H = \frac{I \cdot B}{n \cdot q \cdot t}$
2. Hall coefficient:
   $R_H = \frac{V_H \cdot t}{I \cdot B} = \frac{1}{n \cdot q}$
3. Carrier concentration:
   $n = \frac{1}{|R_H| \cdot q}$
   where $q = 1.602 \times 10^{-19}$ C for electrons and holes.
4. Conductivity:
   $\sigma = \frac{I \cdot l}{V \cdot w \cdot t}$
5. Resistivity:
   $\rho = \frac{1}{\sigma}$
6. Carrier mobility:
   $\mu = |R_H| \cdot \sigma$

1. Sample Preparation and Mounting:
   • Measure the dimensions (length, width, and thickness) of the semiconductor sample using a micrometer or vernier caliper.
   • Clean the sample with isopropyl alcohol to remove any contamination.
   • Mount the sample on the Hall probe holder, ensuring proper electrical contacts.
2. Initial Setup:
   • Connect the current source to the ends of the sample to provide current along the length.
   • Connect the voltmeter to the sides of the sample to measure the Hall voltage.
   • Place the sample between the electromagnet poles such that the magnetic field is perpendicular to the sample surface.
3. Zero Field Measurements:
   • Set the magnetic field to zero.
   • Apply a fixed current through the sample (e.g., 10 mA).
   • Measure and record any offset voltage across the Hall probes (this should ideally be zero but may have a small value due to contact misalignment).
   • Measure the voltage drop along the length of the sample to calculate its resistance and conductivity.
4. Hall Voltage Measurements:
   • Turn on the electromagnet and set it to a known magnetic field strength (measure using the gauss meter).
   • Maintain the same current through the sample.
   • Measure the Hall voltage across the sides of the sample.
   • Repeat the measurement for different values of the magnetic field (e.g., from 0.1 T to 0.5 T in steps of 0.1 T).
5. Current Dependence:
   • Set the magnetic field to a fixed value (e.g., 0.3 T).
   • Vary the current through the sample (e.g., from 5 mA to 50 mA in suitable steps).
   • Measure the Hall voltage for each current value.
6. Polarity Check:
   • Reverse the direction of the magnetic field and repeat the Hall voltage measurement.
   • The Hall voltage should change its sign.
   • Similarly, reverse the current direction and observe the Hall voltage.
7. Temperature Dependence (Optional):
   • If available, use a temperature-controlled setup to measure the Hall coefficient at different temperatures.
   • Record the sample temperature and corresponding Hall voltage.

Sample Details:

Sample Material	___________
Sample Type (n-type/p-type)	___________
Length (l)	___________ mm
Width (w)	___________ mm
Thickness (t)	___________ mm
Temperature	___________ K

Table 1: Hall Voltage vs. Magnetic Field (at constant current I = _____ mA)

S.No.	Magnetic Field B (Tesla)	Hall Voltage $V_H$ (mV)	Hall Coefficient $R_H = \frac{V_H \cdot t}{I \cdot B}$ (m³/C)
1
2
3
4
5
Average Hall Coefficient ($R_H$)

Table 2: Hall Voltage vs. Current (at constant magnetic field B = _____ T)

S.No.	Current I (mA)	Hall Voltage $V_H$ (mV)	Hall Coefficient $R_H = \frac{V_H \cdot t}{I \cdot B}$ (m³/C)
1
2
3
4
5
Average Hall Coefficient ($R_H$)

Table 3: Resistivity and Mobility Calculation

Parameter	Value	Unit
Current (I)		mA
Voltage along length (V)		V
Resistance (R = V/I)		Ω
Conductivity (σ)		(Ω·m)⁻¹
Hall Coefficient (RH)		m³/C
Mobility (μ = |RH|·σ)		m²/(V·s)
Carrier Concentration (n)		m⁻³

Step 1: Calculate the Hall coefficient $R_H$ for each observation using:

Where:

• $V_H$ is the measured Hall voltage in volts
• $t$ is the thickness of the sample in meters
• $I$ is the current flowing through the sample in amperes
• $B$ is the magnetic field strength in tesla

Step 2: Calculate the average value of the Hall coefficient from all observations.

Step 3: Determine the type of semiconductor based on the sign of $R_H$:

• If $R_H$ is negative, the sample is n-type (electrons are majority carriers)
• If $R_H$ is positive, the sample is p-type (holes are majority carriers)

Step 4: Calculate the carrier concentration using:

Where $e = 1.602 \times 10^{-19}$ C is the elementary charge.

Step 5: Calculate the conductivity of the sample:

Where:

• $I$ is the current flowing through the sample
• $l$ is the length of the sample
• $V$ is the voltage drop along the length
• $w$ is the width of the sample
• $t$ is the thickness of the sample

Step 6: Calculate the mobility of charge carriers:

Sample Calculation:

Given:

• Sample thickness, $t = 0.5$ mm = $5 \times 10^{-4}$ m
• Current, $I = 10$ mA = $1 \times 10^{-2}$ A
• Magnetic field, $B = 0.3$ T
• Measured Hall voltage, $V_H = -2.5$ mV = $-2.5 \times 10^{-3}$ V
• Sample length, $l = 10$ mm = $1 \times 10^{-2}$ m
• Sample width, $w = 3$ mm = $3 \times 10^{-3}$ m
• Voltage along length, $V = 0.2$ V

Hall coefficient:

Carrier concentration:

Conductivity:

Mobility:

1. The Hall coefficient of the given semiconductor sample is ______ m³/C.
2. The sample is determined to be ______ type semiconductor (n-type/p-type).
3. The carrier concentration is ______ m⁻³.
4. The conductivity of the sample is ______ (Ω·m)⁻¹.
5. The mobility of charge carriers in the sample is ______ m²/(V·s).

These results align with the typical values for semiconductor materials, confirming the validity of the Hall effect as a method for determining carrier properties in semiconductors.

1. Ensure that the sample is clean and free from contamination before mounting.
2. Make good electrical contacts to minimize contact resistance.
3. Place the sample exactly perpendicular to the magnetic field for accurate measurements.
4. Verify that the Hall probes are placed directly opposite to each other on the sample to measure the true Hall voltage.
5. Maintain a constant temperature throughout the experiment, as carrier concentration and mobility are temperature-dependent.
6. Use a high-precision voltmeter (preferably a nanovoltmeter) to measure the Hall voltage accurately, as it can be quite small.
7. Take measurements for both field polarities and average the results to eliminate thermoelectric and other spurious effects.
8. Ensure that the sample thickness is uniform throughout.
9. Keep the current density uniform by using a sample with uniform cross-section.
10. Avoid sample heating by keeping the current at a reasonable level.
11. Shield the setup from external electromagnetic interference.
12. Allow sufficient time for the magnetic field to stabilize before taking readings.
13. Ensure that all measuring instruments are properly calibrated.

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