From Excitons to Landau Levels: Tracing Magneto-PL Spectra

From Zero to High Field: A Tale of Two Symmetries

To understand the evolution of the energy levels, we must recall the two distinct physical regimes.

• Low-Field Limit ($\gamma \ll 1$): The Coulomb interaction dominates. The spherical symmetry of the Coulomb potential means the states are best described by the hydrogenic quantum numbers $(n,l,m)$. The magnetic field is a weak perturbation, causing the Zeeman effect (a splitting linear in $B$) and the diamagnetic shift (an upward shift quadratic in $B$). The energy of a state is approximately:
  $E_{nlm}(B)\approx E_g-\frac{R{y}^{\ast }}{n^2}+g_{\text{eff}}{\mu }_{B}B{m}_J+{\sigma }_{nl}{B}^2$

• High-Field Limit ($\gamma \gg 1$): The magnetic interaction dominates. The cylindrical symmetry imposed by the magnetic field means the states are best described by the magneto-exciton quantum numbers $(N,M,\nu )$. The energy is dominated by the cyclotron term, which is linear in $B$:
  $E_{NM\nu }(B)\approx E_g+\left(N+\frac{1}{2}\right)\hslash {\omega }_c+E_{NM\nu }^{1D}(B)$
  The term $E^{1D}$ represents the binding energy of the 1D Coulomb states, which themselves have a weaker, logarithmic dependence on $B$.

The transition from the low-field (n, l, m) states to the high-field (N, M, \nu) states is not simple. Because the fundamental symmetry of the Hamiltonian changes, there is no straightforward one-to-one correspondence between the two sets of quantum numbers. The states mix and rearrange in the intermediate-field regime ($\gamma \approx 1$), a complex process we will explore in the next module. For now, we will focus on visualizing this evolution.

To begin, let's get a concise overview of these concepts from a standard text.

Visualizing the Evolution: The Landau Fan Diagram

The most effective way to visualize the evolution from the Coulomb regime to the magnetic regime is with a fan diagram—a plot of the exciton energy levels as a function of the magnetic field strength, $B$.

1. At $B=0$, the diagram starts with a series of discrete horizontal lines at energies $E_g-R{y}^{\ast }/n^2$ for $n=2,3,4,\dots$.
2. As $B$ increases slightly, these lines split and curve upwards due to the Zeeman and diamagnetic effects. The curvature reflects the $B^2$ dependence of the diamagnetic shift.
3. As $B$ becomes large, the levels regroup into "fans" of lines that rise approximately linearly with $B$. These are the emerging Landau levels, with the slope determined by the cyclotron energy $\hslash {\omega }_c$. The individual lines within each fan correspond to the different 1D Coulomb states (\nu=0, 1, 2, ...) attached to that specific Landau level N.

The resulting plot looks like a fan opening up as the magnetic field increases, hence the name.

From Theory to Reality: Magneto-Spectroscopy of Cu₂O

This theoretical fan diagram is not just a concept; it is precisely what is measured in magneto-absorption or magneto-PL experiments. By measuring the optical spectrum at many different magnetic field values and stacking them side-by-side, one can create a 2D intensity map of absorption/emission versus energy and magnetic field. This map is the experimental realization of a fan diagram.

Given your work with Cu₂O, the following paper is particularly illuminating. It presents state-of-the-art magneto-absorption data that beautifully illustrates these concepts.

Analyzing the Cu₂O Fan Diagram

Let's take a closer look at Figure 5 from the paper you just read. It is a masterful demonstration of theory meeting experiment.

Figure 5 from Assmann et al., Sci. Rep. 8, 7785 (2018), showing the measured magnetoabsorption spectra (color map) with theoretically calculated exciton energies (dashed lines) superimposed. Panels (a) and (b) correspond to different optical selection rules.

Here's how to interpret this plot in the context of our learning:

• Zero-Field Levels: At $B=0$, you can clearly see the absorption peaks corresponding to the hydrogenic p-states (since these are dipole-allowed in Cu₂O) for n=2, n=3, and n=4.
• Low-Field Evolution: Follow one of these lines, for example, the n=2 state. As B increases from zero, the line curves upward, demonstrating the diamagnetic shift. It also splits into different components due to the Zeeman effect and the complex band structure of Cu₂O.
• High-Field Emergence: Look at the top right of the diagram (high energy, high field). You see a series of lines that are almost perfectly straight and fanning out. These are transitions to high-N Landau levels. They exhibit a nearly pure linear dependence on $B$ because the cyclotron energy $\hslash {\omega }_c$ dwarfs the 1D Coulomb binding energy for these states.
• Connecting Theory and Experiment: The dashed lines represent the calculated eigenvalues of the exciton Hamiltonian in a magnetic field. Their excellent agreement with the bright lines of the experimental absorption map validates the entire theoretical framework we have been building. You are literally seeing the solutions E_{NMν}(B) plotted against the data.

Interpreting Fan Diagrams to Extract Material Parameters

Fan diagrams are not merely for visualization; they are powerful quantitative tools. By analyzing the features of the diagram, we can extract fundamental properties of the semiconductor.

As the reading highlights, we can deduce several key parameters:

• Reduced Mass ($\mu$): In the high-field limit, the energy of a transition to the N-th Landau level is approximately $E_N\approx E_g+(N+1/2)\hslash {\omega }_c$. Since ${\omega }_c=eB/\mu$, the energy spacing between adjacent Landau levels is $\Delta E=\hslash eB/\mu$. Therefore, the slope of the high-N lines in the fan diagram is directly proportional to $1/\mu$. This is one of the most reliable methods for measuring effective masses. The authors of the Cu₂O paper used this technique to resolve a long-standing discrepancy in the accepted mass values for that material.
• Band Gap ($E_g$) and Binding Energy ($E_b$): If you extrapolate the linear, high-field portions of the fan lines (for N > 0) back to $B=0$, they converge to a single point. This point represents the energy where the exciton would be ionized, which is the band gap energy, $E_g$. The difference between this extrapolated $E_g$ and the measured energy of the 1s (or lowest) exciton state at $B=0$ gives the exciton binding energy, $E_b$.
• g-factors: The splitting of lines for a given excitonic state, particularly visible in circularly polarized measurements (${\sigma }^{+}$ vs. ${\sigma }^{-}$), is due to the Zeeman effect. The magnitude of this splitting is proportional to the effective g-factors of the electron and hole, allowing for their determination.

Test your understanding!

The image below shows a fan diagram for Rydberg excitons in a 2D material.

Based on this plot, which of the following states shows behavior that is most dominated by the magnetic confinement (i.e., most "Landau-level-like") even at moderate fields? And which state is most dominated by the Coulomb interaction (i.e., most "hydrogenic")? Justify your answer.

Show answer

• Most Landau-level-like: The highest excited states, such as the 5s state (purple triangles). Its energy dependence on the magnetic field is almost perfectly linear across the entire range shown. This indicates that for this large, weakly bound exciton, the magnetic length scale quickly becomes smaller than the exciton radius, and its behavior is dominated by the cyclotron motion even at lower fields.

• Most Hydrogenic: The 1s ground state (black squares). Its energy shift is clearly non-linear at low fields, showing the characteristic quadratic diamagnetic shift. It only begins to approach a linear dependence at the highest fields. This is because the 1s exciton is the most tightly bound and has the smallest radius, so a much stronger magnetic field is required before the magnetic effects dominate the strong internal Coulomb attraction.

Conclusion

In this lesson, we have successfully bridged the gap between the abstract theory of magneto-excitons and concrete experimental data.

Key Takeaways:

• As the magnetic field increases, exciton energy levels evolve from a hydrogenic structure (dominated by Coulomb forces) to a Landau-level structure (dominated by magnetic confinement).
• This evolution is visualized in a Landau-level fan diagram, which plots energy versus magnetic field.
• Magneto-absorption and magneto-PL experiments provide a direct measurement of these fan diagrams.
• By analyzing the slopes, intercepts, and splittings within a fan diagram, we can extract fundamental material parameters like the reduced effective mass, band gap, binding energy, and g-factors.

Preview of the Next Lesson

We have now established a clear understanding of exciton behavior in the high-field limit. The final lesson in this module will put this into practice, where we will calculate the critical field strength ($B_c$) that marks the onset of this regime for specific materials, including GaAs and your material of interest, Cu₂O. This will give us a concrete value for the condition $\gamma \gg 1$. Following that, we will embark on the next module, which explores the fascinating and complex physics of the intermediate-field regime ($\gamma \approx 1$), where the competition between Coulomb and magnetic symmetries leads to the breakdown of simple models and the emergence of quantum chaos.