Magneto-Optical Selection Rules: Faraday & Voigt Geometries

Hello! Welcome to our next lesson on excitons in magnetic fields.

In our last two lessons, we dissected the exciton Hamiltonian in a weak magnetic field. We first analyzed the term quadratic in $B$ , deriving the diamagnetic shift, and then explored the term linear in $B$ , which gives rise to the linear Zeeman splitting. We saw that these effects lift the degeneracy of the exciton states and that their magnitude is governed by effective g-factors, which can be determined experimentally.

This leaves a crucial question unanswered: now that the magnetic field has created a new ladder of energy levels, which transitions to these levels are actually observable? The answer lies in magneto-optical selection rules. Today, we will derive these rules, which dictate which light polarizations can create which exciton states in different experimental geometries. This is the final piece of the puzzle you need to begin interpreting real-world magneto-photoluminescence spectra.

Our learning outcome for this session is to derive the magneto-optical selection rules for transitions in the Faraday and Voigt geometries.

1. The Principle of Optical Selection Rules

As you know from your quantum mechanics courses, the probability of an optical transition between an initial state $∣ i ⟩$ (the crystal ground state) and a final state $∣ f ⟩$ (an exciton state) is proportional to the square of the matrix element:

W_{i \to f} \propto ∣ ⟨ f ∣ H^_{in t} ∣ i ⟩ ∣^{2}

In the electric dipole approximation, the interaction Hamiltonian is $H^_{in t} = - e E \cdot r$ , where $E$ is the electric field of the light wave and $r$ is the position operator. The selection rules emerge because this matrix element is non-zero only if the symmetry of the final state $∣ f ⟩$ "matches" the symmetry of the operator acting on the initial state, $H^_{in t} ∣ i ⟩$ .

The polarization of the light is encoded in the vector $E$ . Let's assume the light propagates along the z-axis ( $k ∥ z^$ ).

Linearly polarized light (e.g., along x-axis): $E = E_{0} x^$ . The operator is proportional to $x$ .
Circularly polarized light: $E \propto (x^\pm i y^)$ . The operators are proportional to $x \pm i y$ .

Recalling the spherical harmonics, $x \pm i y \propto r sin θ e^{\pm i ϕ} \propto Y_{1}^{\pm 1}$ . This tells us that circularly polarized light carries one unit of angular momentum ( $Δ m = \pm 1$ ). Therefore, $σ^{+}$ polarized light creates states with $m_{J} = + 1$ , and $σ^{-}$ polarized light creates states with $m_{J} = - 1$ .

Our task is to identify the exciton eigenstates in a magnetic field and determine their angular momentum character to see which ones couple to which light polarization.

2. Exciton Spin States and Experimental Geometries

For simplicity, let's consider excitons where the envelope function is s-like ( $L = 0$ ). The total angular momentum comes from the combination of the electron spin ( $s_{e} = 1/2$ ) and hole spin ( $s_{h} = 1/2$ ). This gives rise to:

A spin-singlet para-exciton with total spin $J = 0$ .
A spin-triplet ortho-exciton with total spin $J = 1$ and projections $m_{J} = - 1, 0, + 1$ .

In many materials with cubic symmetry, at zero magnetic field, the $m_{J} = \pm 1$ states are degenerate and optically active ("bright"), while the $J = 0$ and $J = 1, m_{J} = 0$ states are optically inactive ("dark").

We will analyze the selection rules in two standard magneto-optical configurations:

Faraday Geometry: The magnetic field $B$ is parallel to the light propagation vector $k$ ( $B ∥ k$ ).
Voigt Geometry: The magnetic field $B$ is perpendicular to the light propagation vector $k$ ( $B ⊥ k$ ).

A diagram illustrating the Faraday geometry (left), where the magnetic field is parallel to the light's wavevector, and the Voigt geometry (right), where the magnetic field is perpendicular to the wavevector. The light polarization is always transverse to the direction of propagation.

The following resource provides a clear and rigorous framework for deriving the selection rules in both cases. Although the paper focuses on perovskites, the underlying spin Hamiltonian formalism is general for cubic systems and directly applicable here.

Exciton spin structure in lead halide perovskite ...

The paper 'Exciton spin structure in lead halide perovskite...' by Yakovlev et al. provides an excellent theoretical treatment of the exciton spin Hamiltonian in a magnetic field. We will use its results to derive the selection rules.

Please study the following three sections from this paper: Section III (Exciton spin structure in magnetic field for cubic symmetry): Read the entire section (up to the figures). Focus on the total Hamiltonian (Eq. 7), the Zeeman term (Eq. 8), and the explicit matrix forms for the Faraday (Eq. 9) and Voigt (Eq. 11) geometries. Understand how the 'bright' (g_X) and 'dark' (g_{DX}) exciton g-factors are defined. Section IV.1 (Exciton spin dynamics in longitudinal magnetic field): This section covers the Faraday geometry. Pay close attention to how circularly polarized light selectively excites specific eigenstates (Eq. 12) and how the dark states remain dark (Eq. 16). Section IV.2 (Exciton spin dynamics in transverse magnetic field): This section covers the Voigt geometry. Notice how the transverse field mixes all states (Eqs. 17 & 19), leading to different selection rules. Don't worry about the spin dynamics (time-dependent oscillations); focus on which states are excited by which light polarization.

Let's now use the insights from the paper to derive the selection rules for each geometry. We will work in the basis of exciton states that are active for linear polarization at zero field: $∣1, X ⟩$ , $∣1, Y ⟩$ , and the dark states $∣1, Z ⟩$ (longitudinal) and $∣0, 0 ⟩$ (singlet).

3. Selection Rules in Faraday Geometry ( $B ∥ k$ )

Let's set $k ∥ z^$ and $B ∥ z^$ . Light polarization is in the x-y plane.

The Hamiltonian matrix from the paper (Eq. 9) shows a crucial feature: it is block-diagonal. There is no mixing between the bright ${∣1, X ⟩, ∣1, Y ⟩}$ subspace and the dark ${∣0, 0 ⟩, ∣1, Z ⟩}$ subspace.

Bright States:
The 2x2 block for the bright states diagonalizes to give the eigenstates $ψ_{I I I}$ and $ψ_{I V}$ (Eq. 12). As we traced in the previous lesson (and you can verify), these correspond to the circularly polarized states $∣1, - 1 ⟩$ and $∣1, + 1 ⟩$ , respectively. Their energies split linearly with the field:

E_{I I I, I V} = Δ_{X} \mp \frac{1}{2} g_{X} μ_{B} B

where $g_{X} = g_{e} + g_{h}$ is the bright exciton g-factor.

Dark States:
The dark states $∣0, 0 ⟩$ and $∣1, Z ⟩$ mix with each other, but they do not gain any contribution from the bright states. They remain optically forbidden.

The Selection Rules:
The consequence is straightforward. Since the eigenstates in the magnetic field are the pure circularly polarized states, only circularly polarized light can excite them.

$σ^{+}$ polarized light (angular momentum $+ 1$ ) excites a transition to the $∣1, + 1 ⟩$ state ( $ψ_{I V}$ ).
$σ^{-}$ polarized light (angular momentum $- 1$ ) excites a transition to the $∣1, - 1 ⟩$ state ( $ψ_{I I I}$ ).
Transitions to the dark states $ψ_{I}$ and $ψ_{I I}$ are forbidden for any light polarization.

In an absorption experiment in the Faraday geometry, you would therefore see two peaks, one for each circular polarization, whose separation increases linearly with the magnetic field $B$ .

4. Selection Rules in Voigt Geometry ( $B ⊥ k$ )

Now, let's set $k ∥ z^$ but apply the field transversely, $B ∥ x^$ . Light polarization remains in the x-y plane, meaning we can probe with light polarized parallel ( $π$ -pol, $E ∥ B ∥ x^$ ) or perpendicular ( $σ$ -pol, $E ⊥ B$ , so $E ∥ y^$ ) to the magnetic field.

The Hamiltonian matrix (Eq. 11) looks very different. The magnetic field now introduces off-diagonal terms that mix states differently:

The dark state $∣0, 0 ⟩$ mixes with the bright state $∣1, X ⟩$ .
The dark state $∣1, Z ⟩$ mixes with the bright state $∣1, Y ⟩$ .

This mixing is the key. The formerly "dark" states now acquire some "bright" character, and all four resulting eigenstates can become optically active.

The Eigenstates and Selection Rules:
Let's analyze the new mixed states described in the paper (Eqs. 17 & 19):

States $ψ_{I}$ and $ψ_{I I}$ : These are mixtures of $∣0, 0 ⟩$ and $∣1, X ⟩$ .
- The $∣1, X ⟩$ component has an optical dipole moment along the x-axis.
- Therefore, these states are excitable by light polarized parallel to the magnetic field ( $π$ -polarization, $E ∥ x^$ ).
States $ψ_{I I I}$ and $ψ_{I V}$ : These are mixtures of $∣1, Z ⟩$ and $∣1, Y ⟩$ .
- The $∣1, Y ⟩$ component has an optical dipole moment along the y-axis. The longitudinal state $∣1, Z ⟩$ is dark for light propagating along z.
- Therefore, these states are excitable by light polarized perpendicular to the magnetic field ( $σ$ -polarization, $E ∥ y^$ ).

In an absorption experiment in the Voigt geometry, you would see:

With $π$ -polarized light, two absorption peaks corresponding to transitions to states $ψ_{I}$ and $ψ_{I I}$ .
With $σ$ -polarized light, two different absorption peaks corresponding to transitions to states $ψ_{I I I}$ and $ψ_{I V}$ .

The ability to "brighten" dark states by applying a transverse magnetic field is a powerful tool in spectroscopy, as it provides access to states that are otherwise invisible.

Test your understanding!

You are performing a magneto-PL experiment on a sample with cubic symmetry at low temperature. You observe a bright emission peak at zero field.

In Faraday geometry, you apply a field and see the peak split into two components, but only when you detect circularly polarized light. Why does linearly polarized detection show no splitting?
You switch to Voigt geometry. With the detector polarized parallel to the B-field, you see a new peak appear that was not there at B=0. What is the origin of this peak?

Show answer

In Faraday geometry, the eigenstates are the circularly polarized states $∣1, \pm 1 ⟩$ . A linear polarizer detects a superposition of $σ^{+}$ and $σ^{-}$ light. Since both transitions are allowed and the states are populated, the detector sees emission from both split levels simultaneously. Unless you perform time-resolved measurements to see quantum beats (as hinted at in the paper), a simple intensity measurement will show a single, possibly broadened, peak. You need a circular polarizer to select only the $σ^{+}$ or $σ^{-}$ emission and resolve the splitting.
In Voigt geometry, the transverse field mixes the dark singlet state $∣0, 0 ⟩$ with the bright linearly polarized state $∣1, X ⟩$ . This gives the originally dark singlet state some oscillator strength, making it "bright." The new peak corresponds to a transition from this mixed state, which is now weakly allowed and couples to light polarized parallel to the field.

5. A Group Theory Perspective for Cu₂O

Since your work involves Cu₂O, it's valuable to see how these rules can be derived using the formal language of group theory. This approach is more abstract but also more powerful, especially in complex crystal structures.

Nonlinear optical spectroscopy of exciton-polaritons in ...

The PhD thesis 'Nonlinear optical spectroscopy of exciton-polaritons' by M. Mund provides a detailed group-theoretical derivation of selection rules, specifically for Cu₂O. This offers a complementary and highly rigorous perspective on our topic.

Please read Appendix B, 'Group theory' (pages 105-106). This section is dense but walks through the exact procedure for Cu₂O. Focus on the subsections: 'Symmetries of electron bands': See how the conduction (\Gamma_6^+) and valence (\Gamma_7^+) bands get their symmetry labels. 'Symmetries of excitons': Understand how the symmetries of the bands combine to give the exciton symmetries. For the 1S exciton, you get \Gamma_6^+ \otimes \Gamma_7^+ = \Gamma_2^+ \oplus \Gamma_5^+. The \Gamma_2^+ is the para-exciton (triplet, dark) and \Gamma_5^+ is the ortho-exciton (singlet-triplet mixture, dark). 'Symmetries of photons': Note that the electric dipole operator has \Gamma_4^- symmetry. (Note: In Cu₂O, the para-exciton is the J=1 triplet state \Gamma_2^+, and the ortho-exciton is the J=0,1 mixed state \Gamma_5^+. Both are parity-forbidden at k=0, but the ortho-exciton can be weakly quadrupole-allowed. The magnetic field breaks these symmetries, mixing states and altering the rules as we've discussed).

The core idea of group theory is that the transition matrix element $⟨ exciton ∣ H^_{in t} ∣ ground ⟩$ is non-zero only if the product of the symmetries of the three components contains the totally symmetric representation ( $Γ_{1}^{+}$ ).
A magnetic field is a vector and has a specific symmetry ( $Γ_{4}^{+}$ in the $O_{h}$ group). Its presence changes the total symmetry of the Hamiltonian, which in turn mixes states of different symmetries, leading to a modification of the selection rules. For example, a field can mix a dipole-forbidden state (like $Γ_{5}^{+}$ ) with a dipole-allowed state ( $Γ_{4}^{-}$ ), making the forbidden transition weakly allowed. This is the group-theoretical underpinning of the state mixing we saw in the Voigt geometry.

Conclusion

We have now established the fundamental rules governing which exciton transitions are observable in a magnetic field.

Key Takeaways:

Selection rules are determined by which exciton eigenstates have a non-zero dipole matrix element for a given light polarization.
Faraday Geometry ( $B ∥ k$ ): The bright eigenstates are the circularly polarized states $∣1, \pm 1 ⟩$ . The selection rules are $Δ m_{J} = \pm 1$ , corresponding to $σ^{\pm}$ light. Dark states remain dark.
Voigt Geometry ( $B ⊥ k$ ): The transverse field mixes bright and dark states. All four exciton levels can become optically active, coupling to specific linear polarizations ( $π$ or $σ$ ) relative to the magnetic field direction.
The ability to selectively probe transitions with polarized light in different geometries is a cornerstone of magneto-spectroscopy, allowing for the complete characterization of the exciton spin structure.

Preview of the Next Lesson

We have built a complete theoretical model for excitons in a weak magnetic field: we can predict their energy shifts (diamagnetic and Zeeman) and the selection rules for observing them. In our next lesson, we will put it all together and correlate these Zeeman splittings and diamagnetic shifts observed in low-field magneto-PL spectra with our theoretical predictions, bridging the gap between theory and your experimental work.

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