Optical Transitions in Quantum Wells: Gain and Spontaneous Emission

From Electronic States to Light

In the previous tutorial on parabolic quantum-well bands, we started with the confined solutions of Schrödinger's equation and showed how each solution becomes an electronic subband when motion parallel to the semiconductor layers is included. We also calculated the photon energy needed to connect an electron state to a hole state.

Knowing which transitions are possible is only the beginning. To calculate the light absorbed or emitted by a quantum well, we must also ask how likely each transition is and how many electrons and holes are available to take part. This is where Fermi's golden rule and optical matrix elements enter the calculation.

We will build the calculation one step at a time: first consider a single transition, then calculate its probability, sum over all the electron and hole subbands, and finally introduce Lorentzian broadening to obtain a smooth optical spectrum.

Three Ways Light Interacts with a Semiconductor

Consider an electron in a conduction-band state and an available state in the valence band. There are three closely related optical processes:

• Absorption: an incoming photon excites an electron from an occupied valence-band state into an empty conduction-band state.
• Stimulated emission: an incoming photon stimulates a conduction-band electron to fill an empty valence-band state (a hole), producing an additional photon.
• Spontaneous emission: an electron recombines with a hole and emits a photon even when no incoming photon is present.

All three processes depend on the energy difference between the initial and final electronic states and on how strongly light couples those states. Their crucial difference is the occupation of the states, and whether a photon must already be present.

Start with One Electron–Hole Transition

Let us select electron subband \(n\) and hole subband \(m\). For a simple parabolic quantum well, a transition between states having the same in-plane wavevector has an energy difference

\[ E_{nm}(k_{\parallel})= E_g+\varepsilon_{e,n}+\varepsilon_{h,m} +\frac{\hbar^2 k_{\parallel}^{2}}{2\mu}, \]

where \(E_g\) is the well material's band gap, \(\varepsilon_{e,n}\) and \(\varepsilon_{h,m}\) are the electron and hole confinement energies, and \(\mu\) is the in-plane reduced effective mass:

\[ \frac{1}{\mu}=\frac{1}{m_e^{*}}+\frac{1}{m_h^{*}}. \]

For a direct optical transition the in-plane crystal wavevector is approximately conserved, because a photon carries relatively little momentum. An electron and hole therefore contribute to a vertical transition between the two subband curves in ??.

If the photon energy matches the separation of the states, the transition is energetically possible:

\[ \hbar\omega=E_{nm}(k_{\parallel}). \]

However, matching the energies does not tell us how frequently that transition takes place. For this, we need to calculate the interaction between the electron and the electromagnetic field.

Fermi's Golden Rule: How Likely Is a Transition?

Fermi's golden rule gives the transition rate between an initial quantum state \(i\) and a final state \(f\) when a relatively weak interaction connects them:

\[ \boxed{ W_{i\rightarrow f}= \frac{2\pi}{\hbar} \left|\langle f|\hat H_{\mathrm{int}}|i\rangle\right|^2 \delta(E_f-E_i) } \]

This equation contains two particularly important pieces:

• The matrix element, \(\langle f|\hat H_{\mathrm{int}}|i\rangle\), measures how strongly the initial and final states are coupled by the optical field. Squaring its magnitude gives the factor controlling the transition probability.
• The delta function, \(\delta(E_f-E_i)\), enforces conservation of energy. When the initial and final states include photons, their photon energies are included in \(E_i\) and \(E_f\).

A useful way to think about this is that two conditions must be met: the transition must have the correct energy, and the light must be able to couple the two electronic states. The first is determined by the subband separation; the second is determined by the matrix element.

What Is an Optical Matrix Element?

When light interacts with an electron, its electric field can drive transitions between quantum states. In a common description, the relevant interaction is proportional to the electron momentum operator dotted with the optical field's polarisation. The corresponding momentum matrix element has the form

\[ p_{nm}= \langle\Psi_{c,n}|\,\mathbf e\!\cdot\!\hat{\mathbf p}\,|\Psi_{v,m}\rangle, \]

where \(\mathbf e\) is the light polarisation, and \(\Psi_{c,n}\) and \(\Psi_{v,m}\) represent the electron states associated with the conduction and valence bands. A large magnitude means that the optical field couples the states strongly; a zero matrix element means that this transition is optically forbidden within the model being used.

In the simple parabolic-band approximation, we can separate the rapidly varying atomic-scale Bloch functions from the more slowly varying quantum-well envelope wavefunctions. The optical coupling then contains two factors:

\[ \boxed{ |p_{nm}|^2\simeq |\mathbf e\!\cdot\!\mathbf p_{cv}|^2 \left|\int \psi_{e,n}^{*}(z)\psi_{h,m}(z)\,dz\right|^2 } \]

The first factor, \(\mathbf p_{cv}\), describes the optical coupling between the underlying semiconductor bands. The second is the envelope overlap, which tells us how well the electron and hole wavefunctions overlap across the quantum well.

For example, two states concentrated in the same region of the well can have a large overlap. If an electric field pulls the electron towards one interface and the hole towards the other, the overlap can become much smaller. Importantly, the relevant quantity is the integral of the wavefunctions, not simply the overlap of their separate probability-density plots: wavefunction signs and phases can cause contributions to cancel.

This also explains why the polarisation of the light can matter. For example, TE- and TM-polarised light probe different components of the momentum matrix element, and more advanced multiband models are needed when band mixing and polarisation selection rules are important. These effects can be explored in the GaAs/AlGaAs eight-band zincblende tutorial and the GaN/AlGaN eight-band wurtzite tutorial.

An Allowed Transition Does Not Mean It Will Happen

Even a transition with the correct energy and a large matrix element cannot occur unless its initial state is occupied and its destination is available. Let \(f_c\) be the probability that the relevant conduction-band state contains an electron, and \(f_v\) the probability that the relevant valence-band state contains an electron. The probability that the valence state contains a hole is therefore \(1-f_v\).

The available electron populations give different factors for absorption and emission:

\[ \begin{aligned} \text{Absorption:}\qquad & f_v(1-f_c),\\[4pt] \text{Emission:}\qquad & f_c(1-f_v). \end{aligned} \]

Absorption needs a filled valence-band state and an empty conduction-band state. Emission needs exactly the reverse: an electron in the conduction band and a hole in the valence band. These occupation factors will determine whether a given transition contributes net absorption or net optical gain.

From One Transition to an Optical Spectrum

A real quantum well has more than one electron and hole subband, and electrons can occupy different in-plane wavevectors along each parabola. To calculate a spectrum, we must add the contribution from every relevant combination of electron subband \(n\), hole subband \(m\) and in-plane wavevector \(k_{\parallel}\).

Schematically, a spectral absorption calculation has the form:

\[ \alpha(\hbar\omega)\ \propto \sum_{n,m}\int\!\frac{d^2k_{\parallel}}{(2\pi)^2} |p_{nm}(k_{\parallel})|^2 f_v(1-f_c) \delta\!\left[E_{nm}(k_{\parallel})-\hbar\omega\right]. \]

The sum runs over subband pairs; the integral collects all in-plane states. The matrix element weights each transition by its optical strength, the occupation factors tell us whether the required carriers are present, and the delta function selects transitions with the correct photon energy. The proportionality leaves out the common electromagnetic normalisation factors.

For a parabolic two-dimensional subband, the number of available states per unit energy has a particularly simple form: its joint density of states is constant above the transition threshold in the ideal model. Consequently, a subband pair begins to contribute when the photon energy reaches its minimum allowed transition energy. Further subband pairs introduce further thresholds in the spectrum.

When Does Absorption Become Optical Gain?

An incoming beam can both be absorbed and stimulate emission. Optical gain is the balance between these two competing processes, not simply the total rate of electron-hole recombination. Subtracting absorption from stimulated emission gives a particularly useful simplification:

\[ \begin{aligned} f_c(1-f_v)-f_v(1-f_c) &=f_c-f_v. \end{aligned} \]

The spectral material gain therefore has the same sum over transitions as the absorption calculation, but uses the population difference:

\[ \boxed{ g(\hbar\omega)\ \propto \sum_{n,m}\int\!\frac{d^2k_{\parallel}}{(2\pi)^2} |p_{nm}(k_{\parallel})|^2 (f_c-f_v) \delta\!\left[E_{nm}(k_{\parallel})-\hbar\omega\right] } \]

If \(f_v\) is greater than \(f_c\), absorption dominates and the gain contribution is negative. If \(f_c\) is greater than \(f_v\), stimulated emission dominates: this is population inversion, and the transition contributes positive gain.

In a complete calculation, the proportionality factor includes quantities such as the optical frequency, refractive index, electromagnetic constants and the thickness used to normalise the quantum-well states. The gain within the active material should also be distinguished from the modal gain of a laser, which depends on how strongly the optical mode overlaps the quantum-well region.

What About Spontaneous Emission?

Spontaneous emission uses the same electronic transition energies and optical matrix elements, but it is not the same quantity as net gain. To see the difference, imagine a particular photon mode already contains \(N\) photons. The quantum-mechanical emission rate into that mode contains a factor of \(N+1\):

\[ \underbrace{(N+1)}_{\text{emission}} =\underbrace{N}_{\text{stimulated}} +\underbrace{1}_{\text{spontaneous}}. \]

The stimulated part grows with the number of photons already present. The additional \(1\) means emission can still occur when the initial photon number is zero. Absorption, in contrast, requires an incoming photon and is proportional to \(N\).

To calculate spontaneous emission, we sum the transition probabilities over the available electronic states and the photon modes into which light can be emitted. Schematically, the spontaneous-emission spectrum is

\[ \boxed{ R_{\mathrm{sp}}(\hbar\omega)\ \propto D_{\mathrm{ph}}(\omega) \sum_{n,m}\int\!\frac{d^2k_{\parallel}}{(2\pi)^2} |M_{nm}(\omega,k_{\parallel})|^2 f_c(1-f_v) \delta\!\left[E_{nm}(k_{\parallel})-\hbar\omega\right] } \]

Here, \(D_{\mathrm{ph}}\) represents the available photon modes and \(M_{nm}\) is the appropriately normalised electron–photon interaction matrix element for spontaneous emission. This notation makes explicit that the photonic mode density and optical normalisation must be included when calculating an absolute emitted-photon rate.

Notice the occupation factor: spontaneous emission uses \(f_c(1-f_v)\), not the population difference \(f_c-f_v\) used for net gain. A quantum well can therefore exhibit spontaneous emission even when its net optical response is absorptive. We should not obtain spontaneous emission by simply setting negative gain values to zero.

Why Do We Need Lorentzian Broadening?

Up to this point, Fermi's golden rule has used a delta function to represent exact energy conservation. In a real semiconductor, however, optical transitions do not appear as infinitely sharp lines. Scattering and the finite coherence time of an electronic transition give it a finite spectral width.

One common way to represent this is to replace the delta function by a normalised Lorentzian lineshape:

\[ \boxed{ L_{\gamma}(\Delta E)= \frac{1}{\pi} \frac{\gamma}{(\Delta E)^2+\gamma^2} } \]

Here, \(\Delta E\) is the difference between the electronic transition energy and photon energy, and \(\gamma\) is the Lorentzian half-width at half-maximum (HWHM). Its full width at half-maximum is \(2\gamma\). Because the Lorentzian is normalised over energy, broadening redistributes a transition's spectral weight rather than arbitrarily changing its integrated contribution.

For each transition, we therefore replace \(\delta(E_{nm}-\hbar\omega)\) with \(L_{\gamma}(E_{nm}-\hbar\omega)\). For example, the broadened gain spectrum becomes

\[ \boxed{ g(\hbar\omega)\ \propto \sum_{n,m}\int\!\frac{d^2k_{\parallel}}{(2\pi)^2} |p_{nm}(k_{\parallel})|^2 (f_c-f_v) L_{\gamma}\!\left[E_{nm}(k_{\parallel})-\hbar\omega\right] } \]

The same replacement is made in the spontaneous-emission calculation, but its occupation factor remains \(f_c(1-f_v)\). A larger broadening parameter produces smoother, wider spectral features; a smaller value preserves sharper transition thresholds. Lorentzian broadening is a useful model of homogeneous broadening, rather than a universal description of every physical broadening mechanism.

Putting the Calculation Together

We can now summarise how a quantum-well optical spectrum is built from the electronic states:

1. Find the confined states. Solve Schrödinger's equation for the electron and hole energies and wavefunctions across the quantum well.
2. Construct the subbands. Include motion parallel to the layers to obtain the electronic transition energies at different in-plane wavevectors.
3. Calculate the optical coupling. Evaluate the appropriate matrix elements, including the electron–hole envelope overlap and, where necessary, light polarisation.
4. Include the carrier populations. Use the occupations of the electron and valence-band states to distinguish absorption, stimulated emission and spontaneous emission.
5. Sum over transitions. Add the contributions from the electron and hole subbands and the available in-plane states.
6. Broaden the spectrum. Replace ideal delta-function transitions with a suitable lineshape, such as a Lorentzian.

These are the principles connecting OghmaNano's quantum-well electronic solvers to calculations of optical gain and spontaneous emission. The simple parabolic model makes the physics easier to follow; multiband k·p theory extends the electronic states and optical matrix elements when band coupling and polarisation dependence become important.