Zeeman Splitting and Diamagnetic Shifts in Magneto-PL

Hello! Let's dive into today's lesson.

In our last few sessions, we've systematically built a theoretical toolkit for understanding how excitons behave in a weak magnetic field. We have explored:

The diamagnetic shift, a quadratic effect ( $\propto B^{2}$ ) that increases the exciton's energy.
The linear Zeeman splitting ( $\propto B$ ), which lifts the degeneracy of spin states.
The magneto-optical selection rules, which dictate which of these new states can be accessed with different light polarizations and experimental geometries (Faraday vs. Voigt).

Today, we will put all these pieces together. Our goal is to use this theoretical framework to interpret real-world experimental data. You will see how magneto-photoluminescence (or magneto-absorption) spectra are not just qualitative pictures but are rich quantitative sources of information from which we can extract fundamental material parameters like effective g-factors and masses.

This lesson directly addresses the learning outcome: to correlate Zeeman splitting and diamagnetic shifts observed in low-field magneto-PL spectra with theoretical predictions.

1. From Theory to Spectrum: A Case Study in Cu₂O

Given your research interests, there is no better material for our case study than Cuprous Oxide ( $Cu_{2} O$ ). We will analyze a high-quality magneto-absorption dataset and see exactly how the theory we've developed is applied.

The following paper provides a detailed study of the yellow exciton series in $Cu_{2} O$ in magnetic fields up to 32 T. It's a fantastic example of correlating experimental spectra with numerical calculations based on the exciton Hamiltonian.

Magneto-absorption spectra of hydrogen-like yellow ...

Let's begin by studying 'Magneto-absorption spectra of hydrogen-like yellow excitons...' by Bazak et al. This paper is a perfect illustration of today's topic and is directly relevant to your work on Cu₂O.

Please start by reading the 'Experimental results' section and examining Figures 1 and 2. Figure 2, in particular, provides a stunning visualization of the data. As you look at it, try to visually identify the two key phenomena we've studied: the overall blue-shift of the exciton lines with increasing field, and their splitting into multiple components.

As you can see from Figure 2 in the paper, applying a magnetic field has a dramatic effect. The dark traces, representing high absorption, clearly shift to higher energies and split apart as the field strength $B$ increases.

The overall upward trend in energy for each branch is the diamagnetic shift.
The splitting of a single zero-field line (like the $n = 3$ or $n = 4$ state) into a fan of new lines is the Zeeman effect.

Now, how can we quantitatively explain this "fan diagram"?

2. Modeling the Spectrum

The process involves fitting the experimental data points with the energy levels predicted by the exciton Hamiltonian in a magnetic field. Let's revisit the full Hamiltonian for the relative motion of the electron-hole pair:

H = \frac{p ^{2}}{2 μ} - \frac{e ^{2}}{ε r} + \frac{e}{2 μ c} (γ L \cdot B) + \frac{e ^{2}}{8 μ c ^{2}} [B \times r]^{2} + μ_{B} (g_{c} j_{e} + g_{v} j_{v}) \cdot B

This includes the kinetic energy, Coulomb potential, orbital Zeeman effect (related to mass asymmetry $γ$ ), the diamagnetic term, and the spin Zeeman effect.

The authors of the paper do exactly this. They construct this Hamiltonian and then numerically solve for its eigenvalues as a function of the magnetic field $B$ .

Magneto-absorption spectra of hydrogen-like yellow ...

To understand the theoretical model used to fit the data, let's look at how the authors construct their Hamiltonian.

Please read the section 'Motion of electron-hole pair in magnetic field', focusing on the derivation leading to Equation (14). You'll see it matches the orbital terms we've discussed. Then, skim the section 'Fit to experimental data' to see how they incorporate the spin Hamiltonian (Equation 16) and a practical lineshape fitting model (Equation 18) to extract the 'bare' exciton energies from the raw spectra. This step is crucial for accurate comparison with theory.

The key steps in the modeling process are:

Define the Hamiltonian: The Hamiltonian includes terms for kinetic energy, Coulomb interaction, diamagnetic confinement, and Zeeman splitting (both orbital and spin).
Choose a Basis: The problem is solved by diagonalizing the Hamiltonian matrix. A suitable basis must be chosen. The authors use two different bases for weak and strong field limits (hydrogen-atom wavefunctions and Landau-level-like functions, respectively) and ensure they match in the intermediate regime. Your experience with computational physics should give you an appreciation for the numerical heavy lifting involved here.
Extract Experimental Peak Positions: As noted in the paper, exciton-phonon coupling can create asymmetric absorption lineshapes. They first fit the experimental peaks to an asymmetric Lorentzian function to find the true, "bare" energy $E$ of the excitonic state, removing the influence of the lineshape.
Fit Theory to Experiment: The calculated eigenvalues (energies) depend on several material parameters: the reduced mass $μ$ , the mass asymmetry parameter $γ$ , and the g-factors $g_{c}$ and $g_{v}$ . These are treated as fitting parameters. The goal is to vary them until the calculated energy levels (the lines) match the experimentally determined peak positions (the points).

3. Correlating Theory and Experiment: The Result

The final result of this procedure is a plot that overlays the calculated energy levels on the experimental data. This is the moment where theory and experiment confront each other.

Magneto-absorption spectra of hydrogen-like yellow ...

Now for the main event. Let's examine the result of the fitting procedure.

Please read the 'Discussion' section and carefully study Figure 5. This figure is the centerpiece of the paper. Pay close attention to how the authors use the selection rules for circularly polarized light to separate the transitions into different 'sets' (panels a and b). This is how they can uniquely determine the parameters.

Figure 5 is the crucial result. The dashed lines are the theoretical predictions from the diagonalized Hamiltonian, and they show remarkable agreement with the experimental absorption data.

This correlation is not just a simple overlay. The authors leverage the selection rules we derived in the last lesson to make sense of the complex fan of states:

They know that for a given circular polarization (e.g., right-circularly polarized light), only transitions with specific changes in angular momentum are allowed.
In the paper, they state that for $σ^{+}$ light, the absorption spectrum is formed by two sets of states: one with spin projection $S_{z} = - 1$ and orbital projection $m = 0$ , and another with $S_{z} = 0$ and $m = + 1$ .
By fitting the splittings and shifts for these distinct families of transitions separately, they can unambiguously extract the underlying parameters.

From the slopes of the splittings (Zeeman effect) and the overall curvature (diamagnetic shift), they extract:

Electron and hole g-factors: $g_{c} = 2.1$ and $g_{v} = - 0.1$ .
Electron and hole effective masses: $m_{e} = 1.0 m_{0}$ and $m_{h} = 0.7 m_{0}$ .

This is a powerful demonstration of magneto-spectroscopy in action. By applying a magnetic field and analyzing the resulting spectra through the lens of our theoretical model, we can probe and measure some of the most fundamental properties of charge carriers in a semiconductor.

4. A Broader View: The General Method

The method is general, even if the specifics of the selection rules in Cu₂O are complex. Let's look back at the perovskite paper from our last lesson, "Exciton spin structure in lead halide perovskite..." (ID: LINK), which shows a more direct case.

In Figure 11(b) of that paper, the authors plot the measured Larmor frequency (which is the Zeeman splitting energy $Δ E_{Z}$ divided by $ℏ$ ) versus the magnetic field $B$ . The relationship is beautifully linear, exactly as predicted by our low-field theory:
$ℏ ω = Δ E_{Z} = ∣ g_{X} ∣ μ_{B} B$
The slope of this line directly gives the bright exciton g-factor, $∣ g_{X} ∣$ . They perform this analysis in both Faraday (Fig. 10) and Voigt (Fig. 11) geometries to fully characterize the g-factor tensor. This is a more direct application of the same principle: measure energy splitting vs. field, and the slope gives you the g-factor.

Test your understanding!

Imagine you are analyzing magneto-PL data from a direct-gap semiconductor. You apply a field in the Faraday geometry and record the energies of the two Zeeman-split peaks, $E_{+}$ and $E_{-}$ , for $σ^{+}$ and $σ^{-}$ emission.

The theoretical energy for these peaks is given by:
$E_{\pm} (B) = E_{0} \pm \frac{1}{2} g_{e f f} μ_{B} B + σ_{d} B^{2}$

You decide to create two plots from your data:

The average energy, $(E_{+} + E_{-}) /2$ , as a function of $B$ .
The energy splitting, $E_{+} - E_{-}$ , as a function of $B$ .

What should be the functional form of each plot (e.g., linear, quadratic, constant)? How would you extract the effective g-factor $g_{e f f}$ and the diamagnetic coefficient $σ_{d}$ from these plots?

Show answer

Average Energy:
$\frac{E _{+} + E _{-}}{2} = \frac{( E _{0} + \frac{1}{2} g _{e f f} μ _{B} B + σ _{d} B ^{2} ) + ( E _{0} - \frac{1}{2} g _{e f f} μ _{B} B + σ _{d} B ^{2} )}{2} = E_{0} + σ_{d} B^{2}$
This plot should be a parabola. The coefficient of the $B^{2}$ term in a polynomial fit directly gives you the diamagnetic coefficient $σ_{d}$ .
Energy Splitting:
$E_{+} - E_{-} = (E_{0} + \frac{1}{2} g_{e f f} μ_{B} B + σ_{d} B^{2}) - (E_{0} - \frac{1}{2} g_{e f f} μ_{B} B + σ_{d} B^{2}) = g_{e f f} μ_{B} B$
This plot should be a straight line passing through the origin. The slope of this line is $g_{e f f} μ_{B}$ . Since the Bohr magneton $μ_{B}$ is a known constant, you can directly calculate the effective g-factor $g_{e f f}$ from the slope.

This technique is commonly used to separate and quantify the linear Zeeman effect and the quadratic diamagnetic effect from experimental data.

Conclusion

Today we closed the loop, connecting the abstract Hamiltonian we've been working with to tangible, measurable features in experimental spectra.

Key Takeaways:

Magneto-optical spectroscopy is a precise quantitative tool for determining fundamental semiconductor parameters.
The Zeeman splitting of excitonic lines provides a direct measure of the effective g-factors of the electron and hole.
The diamagnetic shift provides information about the spatial extent of the exciton wavefunction and is related to the reduced effective mass.
A successful correlation between theory and experiment relies on:
1. An accurate theoretical model (the Hamiltonian).
2. High-quality experimental data.
3. Correct application of the magneto-optical selection rules to assign observed transitions to specific theoretical states.

Preview of the Next Lesson

We have now completed our study of excitons in the low-field regime, where the magnetic field acts as a small perturbation. But what happens when the field becomes very strong? In this situation, the magnetic forces can become dominant over the Coulomb interaction, and our perturbative approach breaks down entirely.

In our next lesson, we will begin Module 8, Excitons in a High Magnetic Field. We will introduce a dimensionless parameter, $γ$ , which quantifies the ratio of Coulomb to magnetic energy, and explore the fascinating physics of the high-field limit ( $γ ≫ 1$ ), where the exciton's behavior becomes dominated by Landau quantization. This is a critical step towards understanding the high-field phenomena you are investigating in your own research.

Recommended resources

1. From Theory to Spectrum: A Case Study in Cu₂O

2. Modeling the Spectrum

3. Correlating Theory and Experiment: The Result

4. A Broader View: The General Method

Conclusion

Preview of the Next Lesson