Author: Frederick Dooley
Institution: School of Physics and Astronomy, University of Nottingham
Course: PHYS3003 — Third Year Experimental Physics Project (Project No. 19)
Supervisor: Dr Chris Mellor
Date: Autumn Semester 2025 — submitted January 2026
This repository contains the complete experimental record, data, and analysis code for a laboratory-scale demonstration of the integer quantum Hall effect (IQHE) in a GaAs/AlGaAs two-dimensional electron gas (2DEG). Magnetotransport measurements were performed at temperatures between 3 K and 5 K in perpendicular magnetic fields up to 5 T, revealing the defining signatures of the IQHE: a quantised Hall resistance plateau at filling factor ν = 2 accompanied by strong suppression of the longitudinal resistance.
The Hall resistance on the ν = 2 plateau was measured to deviate from the ideal quantised value h/2e² by tens of parts per million. Carrier densities extracted from four independent classical and quantum methods agree to within 0.3%, demonstrating strong internal consistency of the analysis.
| Quantity | Value |
|---|---|
| Carrier density n_s | (1.57 ± 0.01) × 10¹⁵ m⁻² |
| Carrier density agreement | ~0.3% across four independent methods |
| Filling factor observed | ν = 2 |
| Plateau centre field B_{ν=2} | (3.23 ± 0.01) T |
| Hall resistance on plateau | (12905.9 ± 0.3) Ω |
| Deviation from h/2e² | Tens of parts per million |
| Carrier mobility μ | (9.5 ± 1.4) × 10⁵ cm² V⁻¹ s⁻¹ |
| Transport lifetime τ_tr | ~3.5 × 10⁻¹¹ s |
| Measurement temperature range | 3 K – 5 K |
This repository contains the complete software workflow used to acquire, process, analyse, and visualise magnetotransport data from a GaAs/AlGaAs two-dimensional electron gas quantum Hall experiment.
The project also includes the machine-facing acquisition code and the analysis pipeline used to generate the reported results.
The codebase is split into two main parts:
src/acquisition/— real-time machine interaction, instrument communication, and data loggingsrc/analysis/— data processing, parameter extraction, uncertainty-aware analysis, and figure generation
The acquisition software communicates with laboratory instruments using PyVISA, records synchronised voltage and magnetic-field monitor data, and writes timestamped CSV files for later analysis. The analysis code then converts raw voltages into Hall and longitudinal transport quantities, extracts carrier density using multiple independent methods, calculates mobility and transport lifetime, constructs resistivity and conductivity tensors, and regenerates the final report figures.
The acquisition script controls and records from multiple laboratory instruments during a magnetic-field sweep:
python src/acquisition/qhe_data_acquisition.pyThe software interfaces with:
| Instrument | Measurement role | Interface |
|---|---|---|
| SR830 lock-in amplifier | Longitudinal voltage, V_xx |
GPIB |
| SR830 lock-in amplifier | Hall voltage, V_xy |
GPIB |
| Keithley 2100 digital multimeter | Magnetic-field monitor voltage, V_B |
USB/VISA |
During acquisition, the program:
- queries all instruments once per second
- records lock-in
X,Y,R, and phase channels - records the field-proportional monitor voltage
- writes timestamped CSV files
- displays live plots of voltage versus time and magnetic-field monitor voltage
- produces raw datasets suitable for direct downstream analysis
This means the repository contains the original software used to interact with the experimental hardware, not just post-processed data and final plots.
The analysis is implemented as a reproducible Python workflow rather than manual spreadsheet processing.
Example usage:
python src/analysis/qhe_analysis_pipeline.py
python src/analysis/qhe_final_analysis.pyThe pipeline performs:
- loading and cleaning of raw CSV files
- magnetic-field calibration
- Hall and longitudinal resistance calculation
- quantum Hall plateau identification
- Shubnikov–de Haas oscillation analysis
- Landau fan fitting
- carrier density extraction using multiple methods
- mobility and transport lifetime estimation
- resistivity and conductivity tensor construction
- automatic export of publication-style figures
The final analysis script stores the main constants, calibration choices, and frozen analysis parameters in one place, so the reported figures can be regenerated from the committed dataset.
- Python
- NumPy
- SciPy
- Matplotlib
- PyVISA
- GPIB / USB instrument communication
- SR830 lock-in amplifiers
- Keithley 2100 digital multimeter
- CSV-based experimental data pipeline
The quantum Hall effect arises in a two-dimensional electron gas subjected to a strong perpendicular magnetic field at low temperature. Landau quantisation breaks the continuous energy spectrum into discrete levels separated by ΔE = ℏω_c = ℏeB/m*. When the Fermi level lies between Landau levels, in a region of disorder-localised states, the Hall resistance becomes precisely quantised at R_xy = h/(νe²) while the longitudinal resistance vanishes.
This quantisation is topologically protected and extraordinarily robust — insensitive to sample geometry and microscopic disorder — which is why it underpins the international resistance standard for the ohm.
In this experiment, the transition from classical Hall transport (linear R_xy vs B) to quantised Hall transport is traced through the emergence of Shubnikov–de Haas oscillations in the longitudinal resistance and the development of the ν = 2 Hall plateau at higher fields.
Figure 5: Hall resistance R_xy (upper) and longitudinal resistance R_xx (lower) as functions of perpendicular magnetic field at T = 3 K. The staircase structure in R_xy and the oscillatory behaviour in R_xx are the defining signatures of the integer quantum Hall regime. The shaded region marks the ν = 2 plateau window.
Figure 6: Magnified view of the Hall resistance R_xy(B) in the vicinity of the ν = 2 plateau. The shaded region indicates the common plateau window over which R_xy remains approximately constant and close to the ideal quantised value h/2e² (dashed line).
Figure 8: Deviation of the measured Hall resistance from the ideal quantised value h/2e² across the ν = 2 plateau, shown for both increasing and decreasing field sweeps. The close agreement between traces demonstrates reproducibility and negligible hysteresis. The deviation remains within tens of parts per million across the plateau window.
Figure 9: Resistivity tensor components ρ_xy(B) (upper) and ρ_xx(B) (lower) from Runs 3 and 4. At low magnetic field, ρ_xy follows the classical Hall relation (dashed line), while at higher fields it develops a plateau accompanied by strong suppression of ρ_xx, indicating the transition to quantised Hall transport. The shaded region marks the ν = 2 plateau window.
Figure 10: Conductivity tensor components σ_xy(B) (upper) and σ_xx(B) (lower) obtained by inversion of the resistivity tensor. The low-field region near B = 0 is masked due to numerical instability of the inversion as ρ_xy → 0. Outside this region, minima in σ_xx coincide with plateaux in σ_xy at integer multiples of e²/h, consistent with the localisation picture of the integer QHE.
Figure 19: Landau fan diagram constructed from Shubnikov–de Haas oscillation extrema in the longitudinal resistance. The linear relationship between extremum index n and inverse field 1/B_peak confirms a single well-defined carrier population. The slope of the linear fit yields n_s independently of the absolute field calibration.
Figure 12: Longitudinal voltage V_xx(B) measured at T = 3 K, 4 K, and 5 K in the common positive-field window. The monotonic decrease in Shubnikov–de Haas oscillation amplitude with increasing temperature is consistent with thermal broadening of Landau levels (Lifshitz–Kosevich damping). The oscillation positions remain unchanged, confirming that the carrier density is constant across this temperature range.
Figure 4: Hall-bar geometry and measurement wiring configuration. The diagram shows the current injection path (black), longitudinal voltage probes V_xx (blue, GPIB:8), and Hall voltage probes V_xy (green, GPIB:9), and the lock-in detection scheme used to extract R_xx = V_xx/I and R_xy = V_xy/I at f = 67 Hz.
Figure 21: Annotated schematic of the Hall-bar showing the channel width W and longitudinal probe separation L used to determine the geometry factor W/L = 0.0610 ± 0.0094 for conversion of resistance to resistivity. The 15.4% relative uncertainty in W/L is the dominant systematic contribution to the mobility.
Cryogenics: Laboratory helium insert cryostat (DEWAR 01) cooled with liquid helium. A vacuum leak prevented stable operation below ~3 K; active thermal stabilisation was used throughout, limiting the base temperature to approximately 3 K.
Magnet: Superconducting solenoid, 0–7 T. Field calibration: B = 1.3445 × V_B (T/V), where V_B is the power supply monitor voltage.
Measurement electronics: Two Stanford Research Systems SR830 lock-in amplifiers operating at f = 67 Hz with a 1 MΩ series resistor, providing an excitation current of approximately 0.84 µA. A Keithley 2100 digital multimeter read the field-proportional monitor voltage. All instruments communicated via GPIB/USB using PyVISA.
Contact configuration: Current injection through pads 2 and 9; Hall voltage V_xy through pads 5 and 10; longitudinal voltage V_xx through pads 6 and 4. Pads 8 and 11 had bonding failures and were not used.
Key instrument correction: A 5% gain offset was identified on the upper SR830 (V_xy channel) and corrected uniformly across all datasets before analysis. See Appendix A.2 of the report.
The device studied is a modulation-doped GaAs/AlGaAs heterostructure two-dimensional electron gas, identified on the MBE growth sheet as sample NU1783 (provided by Dr C. Mellor, University of Nottingham).
The layer structure from top to bottom is:
| Layer | Material | Thickness |
|---|---|---|
| Cap | GaAs | 17 nm |
| Donor layer | n-Al₀.₃₃Ga₀.₆₇As (n = 1.3×10¹⁸ cm⁻³) | 40 nm |
| Spacer | Undoped Al₀.₃₃Ga₀.₆₇As | 40 nm |
| 2DEG | GaAs/AlGaAs interface | — |
| Buffer | GaAs | 500 nm |
| Superlattice | GaAs/Al₀.₃₃Ga₀.₆₇As | 250 nm |
| Buffer | GaAs | 1000 nm |
| Substrate | Semi-insulating GaAs (100) | — |
The 2DEG forms at the GaAs/AlGaAs interface through band bending, which creates a triangular potential well confining electrons in the growth direction while allowing free motion in the plane. Spatial separation of carriers from the ionised donors in the AlGaAs layer (modulation doping) suppresses impurity scattering, enabling the high mobilities required for observation of quantum Hall transport.
Four independent methods were used to determine n_s, providing a stringent internal consistency check spanning both classical and quantum transport regimes:
| Method | n_s (×10¹⁵ m⁻²) |
|---|---|
| ν = 2 Hall plateau position | 1.563 ± 0.005 |
| SdH oscillation periodicity Δ(1/B) | 1.574 ± 0.004 |
| Landau fan diagram slope | 1.572 ± 0.003 |
| Low-field Hall slope | 1.562 ± 0.006 |
| Mean (adopted value) | 1.57 ± 0.01 |
The maximum fractional spread across all four methods is approximately 0.3%, confirming that a single well-defined carrier population governs transport across the full magnetic field range and validating the magnetic-field calibration and filling-factor assignment.
quantum-hall-effect/
│
├── README.md — this file
├── .gitignore
├── requirements.txt
│
├── report/
│ └── QHE_Report_Submission.pdf — full laboratory report (30 pp.)
│
├── diary/
│ └── lab_diary.pdf — complete project diary
│
├── data/
│ ├── session1_legacy/ — Session 1 PNG screenshots and raw CSV data
│ │ ├── README.md
│ │ └── *.png (8 files)
│ └── session2/ — Session 2 primary dataset (all CSVs)
│ ├── README_runs.md — full run log with file descriptions
│ └── QHE_mergedDATA_*.csv
│
├── src/
│ ├── acquisition/
│ │ └── qhe_data_acquisition.py — real-time instrument control and logging
│ └── analysis/
│ ├── qhe_analysis_pipeline.py — step-by-step modular analysis (4 sections)
│ └── qhe_final_analysis.py — definitive one-pass figure export
│
└── figures/
├── report_figures/ — main report figures (PNG)
└── appendix_figures/ — appendix diagnostic figures (PNG)
Real-time data acquisition script. Communicates simultaneously with both SR830 lock-in amplifiers (via GPIB) and the Keithley 2100 multimeter (via USB) using PyVISA. Logs V_xx, V_xy (X, Y, R, θ for each), and V_B to a timestamped CSV at 1 s intervals. Displays live matplotlib plots of both signals against time and against field voltage during acquisition.
Requires instruments connected and a VISA backend installed (NI-VISA was used). Not required for analysis only.
Step-by-step modular analysis script organised into four sequential sections. Figures are displayed interactively as each section completes.
| Section | Content |
|---|---|
| 1 | Carrier density and excitation current — Hall plateau (Run 4), SdH Δ(1/B), Landau fan (Run 3) |
| 2 | Carrier mobility and transport lifetime — zero-field R_xx (Run 1), sheet resistance, μ, τ_tr |
| 3 | Conductivity tensor and plateau quality — ρ and σ tensors; ppm deviation of R_xy from h/2e² |
| 4 | Temperature dependence — V_xx overlays and SdH amplitude at 3 K, 4 K, 5 K |
Definitive one-pass pipeline that reproduces every figure in the submitted report.
All input parameters are frozen in a single frozen{} dictionary at the top of
the script. Exports all figures as both PNG (300 dpi) and PDF to
figures/report_figures/ and figures/appendix_figures/.
1. Install dependencies
pip install -r requirements.txt2. Clone and navigate to the repo
git clone https://github.com/freddiedooley/quantum-hall-effect.git
cd quantum-hall-effect3. Confirm the data directory — both analysis scripts default to:
DATA_DIR = "data/session2" # correct if running from repo root4. Run the step-by-step pipeline (interactive, plots shown as each section completes):
python src/analysis/qhe_analysis_pipeline.py5. Export all report figures (one-pass, saves PNG + PDF):
python src/analysis/qhe_final_analysis.pyFigures will be written to figures/report_figures/ and figures/appendix_figures/.
| Issue | Detail | Effect on results |
|---|---|---|
| 5% V_xy gain offset | Upper SR830 configuration setting; corrected uniformly by ×(1/0.95) before all analysis | Corrected — no effect on reported results |
| Cryostat vacuum leak | Prevented stable operation below ~3 K | Restricts temperature range to 3–5 K; reduces visibility of higher ν plateaux |
| Excitation frequency discrepancy | Acquisition script issued FREQ70 (70 Hz) but front panel was set to 67 Hz | No effect on analysis |
| Run 6 filename bug | Some intermediate scripts used 163959.csv (aborted run) instead of 164654.csv |
Fixed in both final scripts |
| Session 1 CSVs lost | No CSV files survive from 6 November 2025 session (later recovered) | Eight PNG screenshots preserved in data/session1_legacy/ |
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[7] R. E. Prange and S. M. Girvin (eds.). The Quantum Hall Effect. Springer, New York, 1987.
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[9] M. Büttiker. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Physical Review B, 38(14):9375–9389, 1988.
[10] L. Landau. Diamagnetismus der Metalle. Zeitschrift für Physik, 64:629–637, 1930.
Project partner: Christina Mooney
Supervisor: Dr Chris Mellor, School of Physics and Astronomy, University of Nottingham, who provided the sample, cryogenic apparatus, and guidance throughout the project.