Exciton Mixing and Optical Selection Rules

A Quick Review: Ortho- and Para-excitons in Cu₂O

As you know from your background, an exciton is a bound state of an electron (spin $s_e=1/2$) and a hole (which has an effective spin $s_h=1/2$). Their spins can combine to form two types of excitons:

• Para-excitons: Total spin $S=0$ (antiparallel spins, singlet state).
• Ortho-excitons: Total spin $S=1$ (parallel spins, triplet state).

At zero magnetic field, the electron-hole exchange interaction splits these states in energy. In many materials, dipole selection rules mean that only singlet excitons can be created by a single photon, as the crystal ground state is also a singlet.

However, cuprous oxide (Cu₂O) is a fascinating and more complex case. Due to the specific symmetries of its crystal lattice and band structure, the optical selection rules are quite different. As we'll review in the following reading, the lowest energy transitions are parity-forbidden for electric dipole coupling. For the higher-lying "yellow" exciton series ($n\ge 2$), which are formed from $p$-like envelope functions, transitions are weakly allowed via electric quadrupole coupling. Crucially, due to the symmetries of the bands involved, these rules lead to a situation where the para-exciton is optically forbidden, while the ortho-exciton is the one that is weakly "bright".

So, at zero magnetic field, we expect to see absorption peaks corresponding to the ortho-exciton states, while the para-exciton state should be invisible. This clean separation, however, is about to be disrupted by the magnetic field.

The Mixing Mechanism: How a Magnetic Field Couples Spin States

Let's construct the mechanism for this disruption. The interaction of the exciton's spin with an external magnetic field $\mathbf{B}=B\hat{z}$ is described by the Zeeman Hamiltonian:
$H_Z={\mu }_{B}B(g_c{S}_{ez}+g_v{S}_{hz})$
where $g_c$ and $g_v$ are the g-factors for the conduction band electron and valence band hole, respectively, and ${\mu }_B$ is the Bohr magneton.

The crucial insight is that this Hamiltonian does not necessarily commute with the total spin squared operator, ${\mathbf{S}}^2$, where $\mathbf{S}={\mathbf{S}}_e+{\mathbf{S}}_h$. The operator $H_Z$ conserves total spin only in the special case where $g_c=g_v$. If the g-factors are different, the magnetic field can induce transitions between states of different total spin—specifically, it can mix singlet and triplet states.

Let's prove this by examining the effect of $H_Z$ on the $S_z=0$ states:

• Para-exciton ($S=0$): |0,0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle_e_h - |\downarrow\uparrow\rangle_e_h)
• Ortho-exciton ($S=1$): |1,0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle_e_h + |\downarrow\uparrow\rangle_e_h)

The Zeeman Hamiltonian has a non-zero off-diagonal matrix element between these two states:
$⟨0,0|H_Z|1,0⟩=\frac{{\mu }_{B}B}{2}⟨\uparrow \downarrow -\downarrow \uparrow |(g_c{S}_{ez}+g_v{S}_{hz})|\uparrow \downarrow +\downarrow \uparrow ⟩$
After applying the spin operators and simplifying, we find:
$⟨0,0|H_Z|1,0⟩=\frac{1}{2}(g_c-g_v){\mu }_{B}B$
This non-zero off-diagonal element is the key. It demonstrates that the magnetic field acts as a coupling term that directly mixes the $S=0$ and $S=1$ manifolds, as long as $g_c\ne g_v$.

The Hamiltonian restricted to this two-level subspace $\{|1,0⟩,|0,0⟩\}$ can be written as a 2x2 matrix:

where $E_{ortho}$ and $E_{para}$ are the zero-field energies. Diagonalizing this matrix gives the new energy eigenvalues and eigenstates in the presence of the field. These new states are no longer pure ortho- or para-excitons, but superpositions of them.

The paper you are reading confirms this analysis perfectly.

The Optical Signature: Relaxation of Selection Rules

Now we can connect the state mixing to its observable consequence. At $B=0$, the para-exciton state $|0,0⟩$ is optically dark. The oscillator strength for a transition is proportional to the square of the transition matrix element, $⟨{\psi }_f|\hat{O}|{\psi }_i⟩$, where $\hat{O}$ is the light-matter interaction operator. For the para-exciton, this is zero.

In the magnetic field, the new eigenstates are:
$|{\psi }_{+}⟩=\cos \alpha |1,0⟩+\sin \alpha |0,0⟩$
$|{\psi }_{-}⟩=-\sin \alpha |1,0⟩+\cos \alpha |0,0⟩$
The mixing angle $\alpha$ is determined by the field strength and the zero-field energy splitting.

Consider the transition to the $|{\psi }_{-}⟩$ state, which is predominantly para-like at small fields. Its transition matrix element is:
$⟨{\psi }_{-}|\hat{O}|\text{ground}⟩=-\sin \alpha ⟨1,0|\hat{O}|\text{ground}⟩+\cos \alpha ⟨0,0|\hat{O}|\text{ground}⟩$
Since $⟨0,0|\hat{O}|\text{ground}⟩=0$, this simplifies to:
$⟨{\psi }_{-}|\hat{O}|\text{ground}⟩=-\sin \alpha ⟨1,0|\hat{O}|\text{ground}⟩$
The oscillator strength for this transition is now $f_{-}\propto {\sin }^2\alpha \cdot f_{ortho}$, where $f_{ortho}$ is the oscillator strength of the pure ortho-exciton.

The dark state has "borrowed" intensity from the bright state. As the magnetic field $B$ increases, the mixing angle $\alpha$ grows, and the initially forbidden transition becomes progressively brighter. This phenomenon is what we call the relaxation of optical selection rules. It's a direct, spectroscopic signature of the state mixing induced by the field.

This principle of field-induced mixing is quite general. The image below shows an elegant experiment where a microwave field, instead of a static magnetic field, is used to mix Rydberg exciton states of different orbital parity ($p$-states and $s/d$-states) in Cu₂O. The mechanism is analogous: the external field provides the coupling that mixes states of different symmetries, making a dark state visible.

Conclusion

Today we have seen a direct physical manifestation of the Hamiltonian's non-integrability in a magnetic field. The same underlying interactions that lead to chaotic level statistics also cause a mixing of states with different symmetries. This has profound and measurable consequences for the optical properties of the system.

Key Takeaways:

• In materials like Cu₂O, zero-field selection rules can render certain exciton states, like the para-exciton, optically dark.
• An external magnetic field introduces a Zeeman term to the Hamiltonian. If the electron and hole g-factors differ ($g_c\ne g_v$), this term does not conserve total spin and acts as a coupling between ortho- and para-exciton states.
• This field-induced mixing creates new eigenstates that are superpositions of the original "pure" spin states.
• This mixing leads to a relaxation of selection rules, where optically dark states "borrow" oscillator strength from bright states, causing formerly forbidden transitions to appear in absorption and photoluminescence spectra.

Preview of the Next Module

We have now completed our deep dive into the behavior of a single exciton, from its basic hydrogenic model to its complex dynamics in a magnetic field. We are now prepared to consider what happens when excitons are no longer isolated. In the next module, "Excitonic Complexes: Trions and Biexcitons," we will begin exploring the fascinating world of many-body exciton physics. Our first lesson will focus on describing the formation and binding of trions (charged excitons, $X^{-}$ or $X^{+}$) and biexcitons (excitonic molecules, $XX$).