Adiabatic Quantization of Excitons in High Magnetic Fields

1. The Adiabatic Approximation: Separating Fast and Slow Motion

Let's start with the Hamiltonian for the relative motion of the electron-hole pair in a magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$, using the symmetric gauge $\mathbf{A}=\frac{1}{2}(\mathbf{B}\times \mathbf{r})$:

${\hat{H}}_{\text{rel}}=\frac{1}{2\mu }{\left(\mathbf{p}+e\mathbf{A}(\mathbf{r})\right)}^2-\frac{e^2}{4\pi ϵr}$

In the high-field limit, the kinetic energy term, which includes the strong magnetic confinement, is much larger than the Coulomb potential term. This suggests a separation of timescales:

• Fast Motion: A rapid cyclotron-like motion in the $xy$-plane (perpendicular to $\mathbf{B}$).
• Slow Motion: A much slower drift along the $z$-axis (parallel to $\mathbf{B}$), influenced by the Coulomb attraction.

This situation is analogous to the Born-Oppenheimer approximation in molecular physics, where one solves the electronic motion for fixed nuclei. Here, we'll solve the fast planar motion for a fixed $z$ and then use that result to find an effective potential for the slow motion along $z$. This is the essence of the adiabatic approximation.

We formalize this by proposing a separable wavefunction:
$\Psi (\mathbf{r})=\Psi (\rho ,\varphi ,z)\approx {\Phi }_{NM}(\rho ,\varphi )f(z)$
where $\rho =\sqrt{x^2+y^2}$ and $\varphi$ are the polar coordinates in the $xy$-plane.

• ${\Phi }_{NM}(\rho ,\varphi )$ is the wavefunction for the fast in-plane motion.
• $f(z)$ is the wavefunction for the slow motion along the field.

The resources for today's lesson provide an excellent description of this approach.

With this physical picture in mind, let's proceed step-by-step.

2. Step 1: Solving the Fast In-Plane Motion and Landau Levels

Our first step is to solve for the fast motion in the $xy$-plane, momentarily neglecting the Coulomb potential. The Hamiltonian for this 2D problem is:
${\hat{H}}_{2D}=\frac{1}{2\mu }{\left(p_x-\frac{eBy}{2}\right)}^2+\frac{1}{2\mu }{\left(p_y+\frac{eBx}{2}\right)}^2$
This is the standard problem of a charged particle (with reduced mass $\mu$) in a uniform magnetic field. As you know from your quantum mechanics courses, the solution gives a set of discrete, highly degenerate energy levels known as Landau levels.

To refresh your memory on the derivation, please watch the following video from MIT OpenCourseWare. While it uses the Landau gauge ($\mathbf{A}=(-By,0,0)$), the resulting energy spectrum is, of course, gauge-independent and thus identical to what we get in the symmetric gauge.

The key result is that the energy of the in-plane motion is quantized:
$E_{N,M}^{\text{2D}}=\left(N+\frac{1}{2}\right)\hslash {\omega }_c$
where:

• ${\omega }_c=eB/\mu$ is the cyclotron frequency associated with the exciton's reduced mass.
• $N=0,1,2,\dots$ is the Landau level index. It corresponds to the principal quantum number of the 2D harmonic oscillator.
• In the symmetric gauge, one also finds another quantum number, $M$, for the $z$-component of angular momentum, but the energy is degenerate in $M$ for a simple non-interacting particle.

These Landau levels form the backbone of our high-field energy spectrum.

3. Step 2: The Effective 1D Coulomb Problem

Now we reintroduce the Coulomb interaction, $-e^2/(4\pi ϵr)$, as a perturbation acting on the slow motion along $z$. Following the adiabatic principle, we average the 3D Coulomb potential over the fast in-plane motion described by the Landau wavefunction ${\Phi }_{NM}(\rho ,\varphi )$. This yields an effective 1D potential that governs the motion along the $z$-axis:

$V_{NM}^{\text{eff}}(z)={\int }_0^{2\pi }d\varphi {\int }_0^{\infty }\rho d\rho |{\Phi }_{NM}(\rho ,\varphi ){|}^2\left(\frac{-e^2}{4\pi ϵ\sqrt{{\rho }^2+z^2}}\right)$

This effective potential confines the electron-hole pair along the $z$-axis. The Schrödinger equation for the slow part of the wavefunction, $f(z)$, becomes a purely 1D problem:

$\left(-\frac{{\hslash }^2}{2\mu }\frac{d^2}{d{z}^2}+V_{NM}^{\text{eff}}(z)\right)f_{\nu }(z)=E_{NM\nu }^{1D}{f}_{\nu }(z)$

The paper on Cu₂O (LINK) presents this equation clearly. Their Equation (36) is precisely this 1D Schrödinger equation, where their effective potential $V_{n_{\rho }m}(z)$ (Eq. 37) is our $V_{NM}^{\text{eff}}(z)$.

Solving this equation yields a new set of quantized energy levels, $E_{NM\nu }^{1D}$, indexed by a third quantum number $\nu =0,1,2,\dots$, which labels the bound states within the 1D potential. These energies are negative, representing bound states that form a hydrogen-like series below each Landau level.

4. The Complete High-Field Energy Spectrum

Finally, we combine the energies from the fast and slow motions to get the total exciton energy in the high-field limit.

As stated in Equation (19) of the resource, the total energy is the sum of the Landau level energy and the energy of the 1D Coulomb problem:

$E_{N,M,\nu }\approx E_g+\left(N+\frac{1}{2}\right)\hslash {\omega }_c+E_{NM\nu }^{1D}$

Here, $E_g$ is the band gap energy, and we have established the three quantum numbers that describe the magneto-exciton:

• $N=0,1,2,\dots$: The Landau level index, describing the quantized cyclotron motion. This determines the dominant energy, which scales linearly with $B$.
• $M$: The magnetic quantum number for orbital angular momentum, which can cause smaller splittings.
• $\nu =0,1,2,\dots$: The principal quantum number for the 1D Coulomb problem along $z$. This determines the binding energy of the series of states attached to each Landau level.

This result predicts that the absorption spectrum in a high magnetic field should consist of a series of "fan lines" corresponding to transitions to different Landau levels ($N=0,1,2,\dots$). Each of these main lines is the head of a sub-series of weaker absorption lines corresponding to the different 1D bound states ($\nu =0,1,2,\dots$).

Conclusion

In this lesson, we have successfully derived the structure of exciton energy levels in the high magnetic field limit.

Key Takeaways:

• The adiabatic approximation is the essential tool for solving the exciton problem when $\gamma \gg 1$. It allows us to separate the Hamiltonian into parts describing fast in-plane motion and slow motion along the field.
• The fast in-plane motion is quantized into Landau levels, with energies $E_N=(N+1/2)\hslash {\omega }_c$. This provides the dominant, linearly B-dependent structure of the energy spectrum.
• The slow motion along the field is governed by an effective 1D Coulomb potential, $V_{\text{eff}}(z)$, which is the 3D potential averaged over the 2D Landau wavefunctions.
• The complete energy spectrum is a superposition of these two solutions: a fan of Landau levels, each acting as a convergence limit for its own series of 1D hydrogen-like bound states.

Preview of the Next Lesson

We have successfully derived the form of the energy spectrum. However, to get the actual numerical energy values, we still need to solve the 1D Schrödinger equation for the effective potential $V_{NM}^{\text{eff}}(z)$. In the next lesson, we will do just that: solve for the 1D Coulomb problem along the magnetic field direction to find the full energy spectrum, discussing the approximate analytical and numerical methods used to find the energies $E_{NM\nu }^{1D}$.