Defining the Magnetic Field Parameter ($\gamma$)

1. Competing Interactions: Coulomb vs. Magnetic

The physics of an exciton in a magnetic field is fundamentally a story of two competing interactions:

1. The Coulomb Interaction: This is the attractive force between the electron and the hole. It tries to bind them into a hydrogen-like atom. The characteristic energy scale of this interaction is the exciton binding energy, which for the ground state is the effective Rydberg energy, $R_y^{\ast }$. The corresponding length scale is the effective Bohr radius, $a_B^{\ast }$.

2. The Magnetic Interaction: This is the Lorentz force exerted by the external magnetic field, $\mathbf{B}$. It tries to force the electron and hole into circular cyclotron orbits in the plane perpendicular to $\mathbf{B}$. The characteristic energy scale here is the cyclotron energy, $\hslash {\omega }_c$, where ${\omega }_c=eB/\mu$ is the cyclotron frequency for a particle with the exciton's reduced mass $\mu$. The associated length scale is the magnetic length, $l_B=\sqrt{\hslash /eB}$, which represents the radius of the ground-state cyclotron orbit.

The exciton's behavior—its energy spectrum, wavefunction, and optical properties—is determined by which of these two interactions wins the "tug-of-war."

2. Defining the Dimensionless Field Strength, $\gamma$

To quantify the competition between these two effects, we define a dimensionless parameter, $\gamma$, as the ratio of their characteristic energy scales.

The following resource provides a clear introduction to this concept.

As the text explains, the dimensionless parameter $\gamma$ is formally defined as:

$\gamma =\frac{\hslash {\omega }_c}{2R_y^{\ast }}$

The factor of 2 in the denominator is a convention that conveniently sets the crossover point at $\gamma \approx 1$. Based on the value of $\gamma$, we can classify the exciton's behavior into three distinct regimes:

• Low-Field Regime ($\gamma \ll 1$): The cyclotron energy is much smaller than the exciton binding energy ($\hslash {\omega }_c\ll R_y^{\ast }$). The Coulomb interaction dominates. The exciton is essentially a hydrogen atom that is slightly perturbed by the magnetic field. This is the regime we studied in Module 7. From the perspective of length scales, this corresponds to the case where the Bohr radius is much smaller than the magnetic length ($a_B^{\ast }\ll l_B$).

• High-Field Regime ($\gamma \gg 1$): The cyclotron energy is much larger than the exciton binding energy ($\hslash {\omega }_c\gg R_y^{\ast }$). The magnetic interaction dominates. The electron and hole are forced into tight cyclotron orbits, forming quasi-1D structures along the magnetic field lines. The Coulomb force now acts as a perturbation on these magnetically-defined states (Landau levels). We will begin developing the theory for this regime in our next lesson. Here, the magnetic length is the smallest length scale in the problem ($l_B\ll a_B^{\ast }$).

• Intermediate-Field Regime ($\gamma \approx 1$): The cyclotron and Coulomb energies are comparable. Neither interaction dominates, and neither can be treated as a simple perturbation. The Hamiltonian is no longer separable, and the system's dynamics become complex, often exhibiting features of quantum chaos. This fascinating regime, which is the subject of Module 9, is where you see phenomena like level repulsion and avoided crossings in the energy spectrum.

3. The Critical Magnetic Field, $B_{cr}$

The transition between these regimes is centered around $\gamma =1$. The magnetic field strength at which this crossover occurs is called the critical magnetic field, $B_{cr}$. We can find an expression for it by setting $\gamma =1$:

$1=\frac{\hslash {\omega }_c}{2R_y^{\ast }}=\frac{\hslash (e{B}_{cr}/\mu )}{2R_y^{\ast }}⟹B_{cr}=\frac{2R_y^{\ast }\mu }{\hslash e}$

A more intuitive form relates $B_{cr}$ directly to the exciton's size. Given your experience with theoretical physics, you will appreciate how fundamental constants and material parameters combine to define this critical value.

As the paper states, the critical field can be expressed very elegantly as:

$B_{cr}=\frac{\hslash }{e(a_B^{\ast }{)}^2}$

This equation reveals why the high-field regime is readily accessible in semiconductors but not for a free hydrogen atom. The key is the squared effective Bohr radius in the denominator.

• For a hydrogen atom: $\mu =m_e$, ${\epsilon }_r=1$, and $a_B\approx 0.53$ Å. This gives a colossal critical field of $B_{cr}\approx 2.35\times 10^5$ T, far beyond laboratory capabilities.
• For a semiconductor (e.g., GaAs): The reduced mass is smaller ($\mu \approx 0.05m_e$) and the dielectric constant is larger (${\epsilon }_r\approx 12.9$). Both effects significantly increase the Bohr radius ($a_B^{\ast }=a_B\frac{{\epsilon }_r}{m^{\ast }/m_e}\approx 13$ nm). This leads to a much smaller critical field of $B_{cr}\approx 6.7$ T, which is easily achievable.

This is precisely why semiconductors are such powerful systems for studying the fundamental physics of atoms in extreme magnetic fields—they are, as the Cu₂O paper notes, convenient "analogs to a hydrogen atom" where high-field physics occurs at moderate field strengths.

4. $\gamma$ for Rydberg Excitons: A State-Dependent Parameter

A crucial subtlety, particularly relevant to your research in Cu₂O, is that $\gamma$ is not a single value for a given material. It depends on the specific excitonic state you are probing. For a Rydberg exciton series, the binding energy and Bohr radius scale with the principal quantum number $n$:

• $R_y^{\ast }(n)=R_y^{\ast }/n^2$
• $a_B^{\ast }(n)=a_B^{\ast }{n}^2$

Therefore, the dimensionless field strength for the $n$-th state, ${\gamma }_n$, is:

${\gamma }_n=\frac{\hslash {\omega }_c}{2R_y^{\ast }(n)}=\frac{\hslash {\omega }_c}{2R_y^{\ast }/n^2}={\gamma }_1{n}^2$

This has a profound consequence: a single magnetic field can be simultaneously "weak" for the ground state (${\gamma }_1\ll 1$) and "strong" for a high-lying Rydberg state (${\gamma }_n\gg 1$). This is exactly why Rydberg excitons in materials like Cu₂O are such a fascinating platform. Even with $B_{cr}$ for the ground state being very high, you can access the intermediate and high-field regimes by looking at states with large $n$, all within the same experiment and at the same magnetic field.

Conclusion

In this lesson, we have established the crucial parameter that will guide our exploration of excitons in magnetic fields.

Key Takeaways:

• The behavior of an exciton in a magnetic field is determined by the competition between the Coulomb binding energy ($R_y^{\ast }$) and the magnetic cyclotron energy ($\hslash {\omega }_c$).
• The dimensionless parameter $\gamma =\hslash {\omega }_c/(2R_y^{\ast })$ quantifies this competition.
• The value of $\gamma$ defines three physical regimes: the low-field ($\gamma \ll 1$, Coulomb-dominated), high-field ($\gamma \gg 1$, magnetically-dominated), and intermediate-field ($\gamma \approx 1$, non-perturbative) regimes.
• The critical magnetic field $B_{cr}$, where $\gamma =1$, is much lower for semiconductors than for the hydrogen atom due to their smaller effective masses and larger dielectric constants, making the high-field regime experimentally accessible.
• For Rydberg excitons, $\gamma$ scales as $n^2$, allowing a single magnetic field to probe multiple physical regimes simultaneously across the excitonic series.

Preview of the Next Lesson

Now that we have a clear criterion for the high-field limit, our next step is to develop the theoretical tools to describe it. In the next lesson, we will derive the Landau-like quantization of exciton states in the limit $\gamma \gg 1$. We will introduce the adiabatic approximation, which separates the fast in-plane cyclotron motion from the slow 1D Coulomb motion along the magnetic field, providing the foundation for understanding the high-field energy spectrum.