Optical matrix elements — an analytical expression for the dipole matrix element

1. The dipole matrix element of the box in closed form#

The lecture plugs the box states into the dipole integral. With the walls at \(x = \pm L/2\) and the states counted from \(n = 0\),

and the integral can be done by hand. Write \(p = i + 1\) and \(q = f + 1\) for the number of half-wavelengths of the two states (the sine's own quantum number), and shift the origin to the left wall: for \(i \ne f\) the shift adds \((L/2)\langle\psi_i|\psi_f\rangle = 0\), so it does not change the element. The product of sines becomes a difference of cosines, \(2\sin a\sin b = \cos(a - b) - \cos(a + b)\), and with \(\int_0^L x\cos(k\pi x/L)\,dx = (L/k\pi)^2\,[(-1)^k - 1]\) for an integer \(k \ne 0\),

Both brackets vanish when \(p - q\) is even, and both are \(-2\) when it is odd. Up to an overall sign, which depends on the phase convention of the states and drops out of every rate (\(\propto|\mu|^2\)), this is the slide's result:

One point of reading: in this formula \(i\) and \(f\) are the sine quantum numbers \(p, q = 1, 2, \dots\), not the labels \(n = 0, 1, \dots\) of the states; with the labels from zero, \(i = 0\) would make every element vanish. Since \(p - q = i - f\), the selection rule of lecture 22.2 is the same in both countings.

We check the formula against open-nemo. The effective-mass model (Heterostructure1D) of a 10 nm GaAs layer with hard walls is a discretized box: the sines are exact on the grid points, but the energies and the matrix elements carry a discretization error that should fall as the square of the grid spacing \(\Delta\).

The formula and the open-nemo box agree to better than \(10^{-8}\) nm on the 0.02 nm grid, and the relative error falls by 16 each time the spacing halves (the last column is constant). The energies of the discretized box converge only as \(\Delta^2\), but its eigenvectors are the sines sampled exactly at the grid points, so what remains is the error of summing \(\psi_i\,x\,\psi_f\) on the grid instead of integrating it, and its \(\Delta^2\) term vanishes because the integrand and its slope are zero at the walls. Every forbidden element is zero in both.

2. Limiting cases: distant levels couple weakly#

The formula looks awkward, the lecturer admits, but its limits are telling. If the final state is far above the initial one, \(f \gg i\), then \(f^2 - i^2 \approx f^2\) and

large jumps, from a low to a very high quantum number or back down, are allowed by parity but weak. The strongest transitions are those between neighbours, where the first term of the bracket, \(1/(i - f)^2\), is largest. For neighbours \(f = i + 1\) the element tends to a constant,

From the ground state the elements fall as \(1/f^3\): \(|x_{01}|\) is 12.5 times \(|x_{03}|\), and the jump from the ground state to the 40th level is four orders of magnitude weaker than the first. The far-jump asymptote \(8i/(\pi^2 f^3)\) (dotted) is approached from above for every initial state once \(f\) is a few times \(i\). Between neighbours the element grows slowly from \(0.180L\) and saturates at \(2L/\pi^2 = 0.203L\): in a box the strongest transitions are always the steps by one level.

3. Limiting case: small boxes couple weakly#

The second limit is the prefactor: every element is proportional to \(L\). The matrix element is a length, the size of the charge displacement that the light can drive between the two states, and a smaller box has less room for it. We compute \(|x_{01}|\) with open-nemo for boxes from 2 to 20 nm:

\(|x_{01}|/L = 16/(9\pi^2) = 0.180\) for every width: the dipole matrix element is strictly proportional to the box size, as the lecture states. At a given optical field, a smaller box couples more weakly.

The same table shows that this is not the whole story, and this is the exploration of this notebook. The level spacing grows as \(1/L^2\) (from 42 meV at 20 nm to 4.2 eV at 2 nm, where the infinite-wall model has long stopped describing a real well). The oscillator strength combines the two,

and \(L^{-2}\cdot L^2\) makes it independent of the size: \(f_{01} = 256/(27\pi^2)\,m_0/m^* = 0.961\,m_0/m^*\) for every box. The library's optics.oscillator_strengths, computed from the velocity operator, gives exactly that, and the sum over all final states approaches the Thomas–Reiche–Kuhn sum rule \(\sum_f f_{0f} = m_0/m^*\) (the 0 → 1 transition alone carries 96 % of it). Rates that weigh \(|\mu|^2\) with powers of the transition frequency therefore do not shrink with the box. The spontaneous emission rate of lecture 22.1, \(A \propto \omega^3|x|^2 \propto L^{-6}L^2\), even grows as \(L^{-4}\):

So "smaller dots give weaker optical coupling" is a statement about the matrix element itself, the coupling energy \(e\mathcal{E}|x_{if}|\) at a given field, which is what this section is about. The absorption strength per transition, measured by the oscillator strength, does not depend on the size, and the radiative rate grows steeply as the box shrinks, here as \(L^{-4}\) (the fit over 2–20 nm prints the exponent).

4. Summary of section 22#

The lecture closes the section with its summary, which the three notebooks have now computed:

• Light couples states through the dipole matrix element. The oscillating field adds \(H' = e\mathcal{E}_0\cos(\omega t)\,x\), which mixes the confined states; \(\mu_{if} = e\int\psi_i^*\,x\,\psi_f\,dx\) decides whether a transition happens, and the rate goes as \(|\mu_{if}|^2\) (lecture 22.1).
• Parity sets the selection rule. Even ↔ odd: the integrand is even, \(\mu \ne 0\), allowed (0 → 1, 1 → 2). Even ↔ even and odd ↔ odd: the integrand is odd, \(\mu = 0\), forbidden (0 → 2, 0 → 4, 1 → 3). A transition is allowed only when \(f - i\) is odd (lecture 22.2).
• The strength fades with the separation of the states and with the box size. \(\mu_{if} = \frac{8eL}{\pi^2}\frac{if}{(f^2-i^2)^2}\): neighbouring levels couple most strongly, and \(\mu \propto i/f^3\) far apart; \(\mu \propto L\), so smaller boxes give a weaker coupling at a given field (this notebook).

Summary#

• The dipole matrix element of the box is \(\frac{8eL}{\pi^2}\frac{if}{(f^2-i^2)^2}\) for \(f - i\) odd and zero otherwise, with \(i, f\) the sine quantum numbers (state label + 1); its sign is a phase convention.
• open-nemo's discretized box reproduces every element, with an error that falls as \(\Delta^4\) (below \(10^{-8}\) nm on a 0.02 nm grid).
• Far jumps are weak (\(\propto i/f^3\); ground state to level 40: four orders of magnitude below 0 → 1); neighbouring levels are strongest and saturate at \(2L/\pi^2\).
• \(|x_{01}| = 0.180\,L\) at every width, but the oscillator strength \(f_{01} = 0.961\,m_0/m^*\) is size independent (Thomas–Reiche–Kuhn sum \(m_0/m^*\)), and the spontaneous emission rate grows as \(L^{-4}\).

Exercises#

1. Use mu_analytic to show that \(\sum_f |\langle\psi_0|x|\psi_f\rangle|^2 = \langle\psi_0|x^2|\psi_0\rangle\) (closure; the diagonal term vanishes because \(\langle x\rangle = 0\) in the centred box) and evaluate \(\langle x^2\rangle\) numerically from psi.
2. Replace the infinite box by a finite GaAs well in 0.25 eV barriers (layers of an EffectiveMassMaterial with conduction_band_edge=0.25, as in lecture 22.2) and compute \(|x_{01}|\) for widths from 3 to 20 nm. Where does it depart from \(0.180L\), and why does it not keep shrinking?
3. Compute the oscillator strengths \(f_{0f}\) for \(f = 1, 3, 5\) from the formula (\(f_{0f} = \frac{2m^*}{\hbar^2}(E_f - E_0)|x_{0f}|^2\) in units of \(m_0/m^*\)) and compare with optics.oscillator_strengths.
4. For the 1 → 2 transition (labels from 0), find how \(A_{21}\) scales with \(L\) and compare its value with that of 0 → 1 at \(L = 10\) nm.

Further reading#

• Lecture 22.1: the dipole matrix element and Lecture 22.2: allowed and forbidden transitions.
• Optical transitions and absorption of quantum wells: the intersubband oscillator strength of a real GaAs/AlGaAs well against the infinite-well 0.961, and full absorption spectra.
• Excitons in quantum wells and wires: how the Coulomb attraction redistributes the oscillator strength of interband transitions.
• GaAs momentum matrix elements: where the intra-atomic dipole sits.
• L. C. West and S. J. Eglash, "First observation of an extremely large-dipole infrared transition within the conduction band of a GaAs quantum well", Appl. Phys. Lett. 46, 1156 (1985): the intersubband dipole of a real well.
• D. J. Griffiths, Introduction to Quantum Mechanics (Cambridge University Press): the infinite square well and the Thomas–Reiche–Kuhn sum rule.