Zeeman Splitting and Effective g-Factors of Excitons

Hello! Welcome back to our exploration of excitons in magnetic fields.

In our last lesson, we focused on the term in the exciton Hamiltonian that is quadratic in the magnetic field, $\propto B^{2}$ . Using perturbation theory, we derived the diamagnetic shift, an effect that increases the energy of an exciton state and is related to the "squeezing" of its wavefunction. We saw that for s-states, this is the dominant correction in the weak-field limit.

Today, we will turn our attention to the term in the Hamiltonian that is linear in the magnetic field, $\propto B$ . This will lead us directly to the linear Zeeman effect, which lifts the degeneracy of states with different angular momentum projections. Our goal is to analyze this linear splitting and understand how to determine the effective g-factors that govern its magnitude. This is a critical step in interpreting magneto-spectroscopy data, including the kind you are collecting for Cu₂O, as it's the Zeeman splitting that often provides the most direct insight into the spin and orbital character of the electronic states.

1. The Complete Zeeman Interaction

Let's start with the full Hamiltonian for the interaction of our exciton with a magnetic field $B = B z^$ . In the previous lesson, we isolated the orbital part related to the relative motion. The full interaction, however, must also include the intrinsic spins of the electron and the hole.

The total magnetic moment of the exciton has three contributions:

The electron spin magnetic moment: $μ_{e} = - g_{e}^{0} μ_{B} \frac{s _{e}}{ℏ}$
The hole spin magnetic moment: $μ_{h} = + g_{h}^{0} μ_{B} \frac{s _{h}}{ℏ}$
The orbital magnetic moment of the relative motion: $μ_{or b} = - \frac{e}{2 μ} l$

Here, $g_{e}^{0} \approx 2$ is the free-electron g-factor, and $g_{h}^{0}$ would be the g-factor for the hole's spin. $μ_{B} = e ℏ/2 m_{0}$ is the Bohr magneton. Note the positive sign for the hole's moment, as it behaves like a positive charge.

The Zeeman Hamiltonian is $H_{Z} = - μ_{t o t a l} \cdot B$ . Summing these contributions and taking the dot product with $B = B z^$ , we get:

H^_{Z} = μ_{B} B (g_{e}^{0} \frac{s _{e, z}}{ℏ} - g_{h}^{0} \frac{s _{h, z}}{ℏ}) + \frac{e B}{2 μ} l_{z}

It's often more convenient to express the orbital term using $μ_{B}$ , by multiplying and dividing by $m_{0}$ :

H^_{Z} = μ_{B} B (g_{e}^{0} \frac{s _{e, z}}{ℏ} - g_{h}^{0} \frac{s _{h, z}}{ℏ} + \frac{m _{0}}{μ} \frac{l _{z}}{ℏ})

From Free-Particle to Effective g-factors in a Crystal

The expression above is for particles in a vacuum. Inside a crystal, the situation is more complex. The electron and hole are not free particles; they are quasiparticles whose wavefunctions are Bloch states. Spin-orbit coupling and interactions with other bands modify their response to a magnetic field.

As you've seen in your advanced quantum and condensed matter courses, this complexity is elegantly packaged into effective g-factors. Instead of using the free-electron g-factor $g_{e}^{0}$ , we use an effective electron g-factor $g_{c}$ (for the conduction band) and an effective hole g-factor $g_{v}$ (for the valence band). These effective g-factors incorporate not just the intrinsic spin but also the orbital angular momentum contributions from the periodic part of the Bloch functions, $u_{n, k} (r)$ .

The theoretical tool for deriving these g-factors from first principles is k·p theory, which you've encountered before. The magnetic field interaction is introduced into the k·p Hamiltonian, and the resulting linear-in-B terms are grouped to define the g-factors.

To give you a concrete sense of where these g-factors come from, let's look at the formal magnetic Hamiltonian derived using the method of invariants for a system with the symmetries of Cu₂O.

Rydberg excitons in external fields - RosDok

This excerpt from the PhD thesis 'Rydberg excitons in external fields' by E. Krüger shows the magnetic part of the Suzuki-Hensel Hamiltonian, which is a powerful k·p model for materials like germanium and Cu₂O.

Please read the short paragraph starting 'In the presence of a magnetic field B...' on page 22, and focus on Eq. (2.47). You don't need to digest every term, but observe that the magnetic Hamiltonian H(B) is constructed from terms like A_4 I_x, B_4 \sigma_x, etc. These parameters A_4, B_4, etc., are the so-called Luttinger parameters, and they are what determine the numerical values of the effective g-factors we use.

With this in mind, we can write a much more practical Zeeman Hamiltonian for the exciton, using the effective g-factors for the electron ( $g_{c}$ ) and hole ( $g_{v}$ ):

H^_{Z} = μ_{B} B (g_{c} j_{c, z} + g_{v} j_{v, z}) + envelope orbital term

Here, $j_{c, z}$ and $j_{v, z}$ represent the total angular momentum (spin + Bloch orbital) for the conduction and valence band edges. In many practical cases, the problem is simplified by considering the separate contributions from the electron spin, hole spin, and the envelope orbital angular momentum, each with its own effective g-factor. For an exciton state with envelope function $∣ n, l, m_{l} ⟩$ , the Zeeman Hamiltonian is often expressed as:

H^_{Z} = μ_{B} B (g_{c} s_{z, e} + g_{v} s_{z, h} + g_{or b} l_{z})

where $s_{z, e}$ and $s_{z, h}$ are now spin-1/2 operators (eigenvalues ±1/2), and $g_{or b}$ is an effective orbital g-factor for the envelope function, which can be related back to the effective masses.

2. The Exciton Effective g-factor

In the weak-field regime, the various angular momenta within the exciton couple together. Typically, the electron and hole spins ( $s_{e}, s_{h}$ ) form a total exciton spin $S = s_{e} + s_{h}$ . This total spin then couples with the envelope orbital angular momentum $L$ to form a total angular momentum $J = L + S$ . The energy levels are then labeled by the quantum numbers $J$ and $m_{J}$ .

This is directly analogous to L-S or Russell-Saunders coupling in atomic physics. The magnetic field is a small perturbation that acts on this coupled system. To find the energy splitting, we need to find the expectation value of $H^_{Z}$ in the coupled basis $∣ J, m_{J}, L, S ⟩$ .

The projection theorem (a special case of the Wigner-Eckart theorem) tells us that for states within a given $J$ manifold, the expectation value of any vector operator is proportional to the expectation value of $J$ . This allows us to define a single effective exciton g-factor, $g_{e x}$ , such that the energy shift is simply:

Δ E_{Z} = g_{e x} μ_{B} B m_{J}

The value of $g_{e x}$ is given by the Landé g-factor formula, adapted for our exciton system. The derivation involves projecting the individual magnetic moment contributions onto the total angular momentum vector $J$ .

The following video provides an excellent derivation of the Landé g-factor for an atom. The logic is identical for an exciton.

2 Ways to Derive the Landé g-Factor | Easy → HARD

The video '2 Ways to Derive the Landé g-Factor' from the channel Pretty Much Physics will walk you through the derivation. It presents both a quick, intuitive vector-based approach and a more rigorous one using perturbation theory.

Please watch from the beginning to 4:01. The first section (until 2:22) sets up the problem, defining the magnetic moments and the goal. The second section (2:22 - 4:01) presents the elegant vector projection argument. Pay close attention to how the individual angular momenta \mathbf{L} and \mathbf{S} are projected onto the conserved total angular momentum \mathbf{J} to find the effective g-factor.

Following the logic in the video, we can write the Landé formula for an exciton as:

g_{e x} = g_{L} \frac{J ( J + 1 ) + L ( L + 1 ) - S ( S + 1 )}{2 J ( J + 1 )} + g_{S} \frac{J ( J + 1 ) - L ( L + 1 ) + S ( S + 1 )}{2 J ( J + 1 )}

Here, $g_{L}$ is the effective g-factor for the envelope orbital motion, and $g_{S}$ is the effective g-factor for the total exciton spin $S$ . This $g_{S}$ itself must be calculated by combining the electron and hole g-factors ( $g_{c}, g_{v}$ ).

For example, for a $1 s$ exciton, $L = 0$ , so $J = S$ . The formula correctly simplifies to $g_{e x} = g_{S}$ . The splitting is then determined by the electron and hole g-factors. The total spin $S$ can be 0 (para-exciton) or 1 (ortho-exciton).

Para-exciton (S=0, $m_{S} = 0$ ): This is a spin singlet. It is non-magnetic ( $g_{S} = 0$ ) and does not split in a magnetic field. It is an optically "dark" state.
Ortho-exciton (S=1, $m_{S} = - 1, 0, + 1$ ): This is a spin triplet. It is magnetic and will split into three levels. The effective g-factor $g_{S}$ for the triplet state can be shown to be related to the difference, $g_{S} \approx g_{c} - g_{v}$ .

Test your understanding!

Consider a p-like exciton state ( $L = 1$ ) in a material where the electron and hole spins combine to form a triplet state ( $S = 1$ ). Assume $g_{L} = 1$ and $g_{S} = 1.5$ . What are the possible values of the total angular momentum $J$ , and what is the effective g-factor $g_{e x}$ for the $J = 2$ state?

Show answer

For $L = 1$ and $S = 1$ , the possible values of $J$ are given by the triangle rule: $∣ L - S ∣, ..., L + S$ . So, $J$ can be 0, 1, or 2.

For the $J = 2$ state, we use the Landé formula with $J = 2, L = 1, S = 1$ , $g_{L} = 1$ , and $g_{S} = 1.5$ :

g_{e x} (J = 2) = 1 \cdot \frac{2 ( 3 ) + 1 ( 2 ) - 1 ( 2 )}{2 \cdot 2 ( 3 )} + 1.5 \cdot \frac{2 ( 3 ) - 1 ( 2 ) + 1 ( 2 )}{2 \cdot 2 ( 3 )}

g_{e x} (J = 2) = 1 \cdot \frac{6}{12} + 1.5 \cdot \frac{6}{12} = 0.5 \cdot (1 + 1.5) = 1.25

So, the $J = 2$ state would split with an effective g-factor of 1.25.

3. Determining g-factors: The Case of Cu₂O

This theoretical framework is precisely what is used to analyze experimental magneto-optical data. By measuring the energy splitting of absorption or photoluminescence peaks as a function of the magnetic field, one can extract the effective g-factors.

Let's ground this in the material you are studying, Cu₂O. The following paper provides a masterclass in this process.

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

The paper 'Magneto-absorption spectra of hydrogen-like yellow exciton series in cuprous oxide' provides a detailed analysis of magneto-absorption in Cu₂O, resolving long-standing questions about its electronic parameters. This is directly relevant to your research.

Please read two key sections: 'Fit to experimental data': Focus on Eq. (16), which presents the spin Hamiltonian used in their model: ({\hat{H}}{{ m{spin}}}={\mu }{{ m{B}}}H({g}{c}{j}{e}^{z}+{g}{v}{j}{v}^{z})). Notice how they discuss the mixing of ortho- and para-exciton states and use this model to fit the experimental data (Fig. 4) to extract 'bare' exciton energies. 'Discussion': This section pulls everything together. Read the paragraphs explaining how the two sets of absorption lines are fit. Pay close attention to how they determine the values g_c = 2.1 and g_v = -0.1 by fitting the observed Zeeman splitting. This is the direct answer to our learning outcome: how g-factors are determined.

The key points from this paper are:

The Zeeman splitting is modeled using a simple spin Hamiltonian with two parameters: the electron g-factor $g_{c}$ and the hole g-factor $g_{v}$ .
The orthoexciton states split linearly with the field. The states with $S_{z} = \pm 1$ shift by $\pm \frac{1}{2} (g_{c} + g_{v}) μ_{B} B$ , while the $S_{z} = 0$ state mixes with the para-exciton state, and their energies depend on $∣ g_{c} - g_{v} ∣$ .
By carefully measuring the energy shifts for different light polarizations, which selectively probe different transitions, the authors are able to uniquely determine both $g_{c}$ and $g_{v}$ .
They find $g_{c} \approx 2.1$ , which is close to the free electron value as expected for an s-like conduction band, and $g_{v} \approx - 0.1$ , which is close to zero due to a cancellation of spin and orbital contributions in the d-like valence band of Cu₂O. This agreement provides strong confidence in the model.

This process of postulating a Hamiltonian, calculating the expected spectral features, and fitting them to experimental data to extract fundamental parameters like g-factors is the cornerstone of magneto-spectroscopy.

Conclusion

Today we have built a complete picture of the linear Zeeman effect for excitons in the weak-field limit.

Key Takeaways:

The linear Zeeman splitting arises from the interaction of the magnetic field with the electron spin, hole spin, and the envelope orbital angular momentum.
In a crystal, the complex band structure effects are captured by effective g-factors for the electron ( $g_{c}$ ) and hole ( $g_{v}$ ), which can be formally derived from k·p theory.
In the weak-field limit, exciton angular momenta couple similarly to atoms (L-S coupling), and the splitting of a state with total angular momentum $J$ is described by an effective exciton g-factor, $g_{e x}$ , given by the Landé formula.
By measuring the linear energy splitting of spectral lines versus magnetic field strength ( $Δ E = g_{e x} μ_{B} B m_{J}$ ), these effective g-factors can be precisely determined, as demonstrated for Cu₂O.

Preview of the Next Lesson

We have now analyzed both the quadratic diamagnetic shift and the linear Zeeman splitting. We've seen that a magnetic field splits the exciton energy levels, but which of these transitions are actually observable? The answer lies in magneto-optical selection rules. In our next lesson, we will derive these rules for different experimental geometries (Faraday and Voigt) and light polarizations. This will complete the toolkit you need to fully interpret magneto-photoluminescence spectra.

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