8.15 Optical Absorption and Excitons

8.15 Optical Absorption and Excitons#

Elementary Computational Physics
Volume VIII — Electronic Structure and Many-Body Matter Notebook 8.15
What does light see in a crystal? The dielectric function ε₂(ω) of real silicon, computed from the three form factors of §8.11 with proper dipole matrix elements: the famous E₂ peak lands at 4.26 eV with height 39.5 against the measured 4.3 eV and ≈40, and the f-sum rule — checked in absolute units against ℏω_p = 16.6 eV from scipy.constants — diagnoses exactly what the truncated band sum leaves out (67% → 95% as conduction bands are added). Then the part independent particles cannot do: the electron and its hole attract, a hydrogen problem inside a solid gives GaAs its 4.7 meV exciton at radius 11.8 nm, and a model Bethe–Salpeter equation pulls a bound state below the continuum carrying 45% of the spectrum's oscillator strength.
Level · advanced   •   Est. · 150–180 min
Raymond Amador v1.4.0  ·  2026-07-31  ·  CC BY 4.0 (text) / MIT (code)

Notebook overview#

§8.14 built the spectra of charged excitations — add or remove one electron. Optics asks a different question: shine light on the crystal and the photon creates an electron and a hole at the same time, at (almost) the same crystal momentum. The independent-particle answer is a golden-rule sum (§6.24) over vertical transitions weighted by dipole matrix elements, and this notebook computes it for real silicon and gallium arsenide — no toy bands, but the empirical pseudopotential machinery of §8.11, whose plane-wave eigenvectors make the momentum matrix elements a one-line numpy.einsum. The results are startling for a three-parameter model: silicon’s celebrated \(E_2\) absorption peak computed at \(4.26\) eV with \(\varepsilon_2 = 39.5\) against the measured \(4.3\) eV and \(\approx 40\) [YC10], GaAs’s absorption edge at its direct gap, and the Thomas–Reiche–Kuhn \(f\)-sum rule — evaluated in absolute SI units against \(\hbar\omega_p = 16.60\) eV from scipy.constants and the silicon valence density — satisfied to \(93\%\) by four conduction bands, with the missing weight recovered band by band (\(67\% \to 88\% \to 95\%\)): a sum rule working as an audit of truncation, exactly as in §8.14.

Then the physics independent particles cannot produce. The photo-created electron and hole attract; bound electron–hole states — excitons [Wan37] — appear inside the quasiparticle gap, and the absorption spectrum is rebuilt. Two computations close the volume-long arc: the Wannier exciton of GaAs as a scaled hydrogen atom (§6.17’s solution wearing \(\mu = 0.058\,m_e\) and \(\varepsilon_r = 12.9\)) — binding \(4.74\) meV against the measured \(4.2\), radius \(11.8\) nm spanning twenty unit cells, the self-consistency of the effective-mass picture checked rather than assumed — and a two-band model Bethe–Salpeter equation [RL00]: an electron–hole Hamiltonian on a \(k\)-grid whose contact attraction pulls one state \(0.236t\) below the continuum edge and hands it \(45\%\) of the entire spectrum’s oscillator strength, with the total conserved to \(10^{-10}\) — the same weight-shuffling bookkeeping the sum rules have enforced all volume.

Conventions (this notebook). EPM exactly as §8.11: Cohen–Bergstresser form factors, \(\tau = (a/8)(1,1,1)\), \(|\mathbf G|^2 \leq 24\) (\(137\) plane waves), energies in eV with the kinetic scale from scipy.constants. Brillouin-zone averages are Monte Carlo over the reciprocal primitive cell (\(\mathbf k = x_1\mathbf b_1 + x_2\mathbf b_2 + x_3\mathbf b_3\), \(x_i\) uniform — equivalent to the BZ by periodicity), numpy.random.default_rng(1966). Dipole (momentum) matrix elements \(\langle v|\hat{\mathbf p}|c\rangle = \hbar \tfrac{2\pi}{a}\sum_{\mathbf G} c_v^*(\mathbf G)\,(\mathbf k + \mathbf G)\,c_c(\mathbf G)\), isotropically averaged. Lorentzian broadening \(\eta = 0.15\) eV for spectra. The BSE toy uses hopping units (\(t = 1\)).

How to read the checks. Each exercise closes with a validate call against an independent fact: a measured landmark, an absolute sum rule from scipy.constants, a conservation law. A ✓ is strong evidence; a ✗ is a prompt to locate the discrepancy, not an automatic verdict.

Scope. Independent-particle \(\varepsilon_2\) plus model excitons. Production optics — local fields, ab initio BSE on top of GW, phonon-assisted indirect absorption (silicon’s actual 1.1 eV edge is phonon-assisted and absent here by construction) — is reviewed in Rohlfing & Louie [RL00] and Yu & Cardona [YC10].

Theory in brief#

The dielectric function from the golden rule#

Weak light of frequency \(\omega\) drives vertical transitions \(|v\mathbf k\rangle \to |c\mathbf k\rangle\) at the golden-rule rate of §6.24, and the absorbed power per unit field defines the imaginary part of the dielectric function:

(910)#\[\varepsilon_2(\omega) \;=\; \frac{\pi e^2}{\varepsilon_0 m_e^2 \omega^2\, V} \,\frac{2}{N_k}\sum_{\mathbf k, v, c} \big|\langle c\mathbf k|\hat{\mathbf p}\cdot\hat{\mathbf e}|v\mathbf k \rangle\big|^2\, \delta\!\big(\hbar\omega - (E_{c\mathbf k} - E_{v\mathbf k})\big) ,\]

with \(V\) the primitive-cell volume and the factor \(2\) for spin. In a plane-wave basis the momentum operator is diagonal in \(\mathbf G\), so the matrix element is a contraction of the two eigenvectors with \(\mathbf k + \mathbf G\) — the entire quantum mechanics of light–matter coupling in one einsum. The \(1/\omega^2\) prefactor and the matrix elements fight: joint density of states alone does not fix the spectrum’s shape, and Exercise 3 shows the difference.

The f-sum rule: absorption is conserved#

The Thomas–Reiche–Kuhn sum rule of atomic physics survives into solids as an exact statement about \(\varepsilon_2\):

(911)#\[\int_0^\infty \omega\,\varepsilon_2(\omega)\,d\omega \;=\; \frac{\pi}{2}\,\omega_p^2, \qquad \omega_p^2 = \frac{n\,e^2}{\varepsilon_0 m_e} ,\]

with \(n\) the total valence-electron density — for silicon, \(8\) electrons per primitive cell of volume \(a^3/4\), giving \(\hbar\omega_p = 16.60\) eV with nothing adjustable. Every electron must absorb its share somewhere; a truncated conduction-band sum keeps the shape but loses weight, and the loss is measurable — Exercise 4 measures it.

Excitons: hydrogen inside a crystal, and the BSE#

Equation Eq. 910 creates the electron and hole and never lets them speak. But they attract, screened by \(\varepsilon_r\); near a parabolic gap the pair’s relative motion obeys a hydrogen Schrödinger equation with reduced mass \(\mu^{-1} = m_e^{*-1} + m_h^{*-1}\), giving Wannier’s scaled Rydberg series [Wan37]

(912)#\[E_n \;=\; E_g - \frac{\mu/m_e}{\varepsilon_r^2}\, \frac{\mathrm{Ry}}{n^2}, \qquad a^*_0 \;=\; \frac{\varepsilon_r}{\mu/m_e}\, a_0 :\]

for GaAs, millielectronvolt binding and a ten-nanometer orbit — the hydrogen atom of §6.17 blown up two-hundredfold, and self-consistently so: the orbit spans dozens of cells, which is precisely what effective-mass theory needs. The general machinery is the Bethe–Salpeter equation: diagonalize the electron–hole pair Hamiltonian \(H^{\mathrm{BSE}}_{kk'} = (E_{ck} - E_{vk})\,\delta_{kk'} - \langle k|K^{\mathrm{eh}}|k'\rangle\) and rebuild the spectrum from its eigenstates — oscillator strength flows from the continuum into the bound exciton, total conserved. Exercise 6 does exactly this on a two-band model where every eigenvalue is inspectable.

Setup#

Data and instruments: the series colours, the CODATA Rydberg, Cohen and Bergstresser’s published lattice constants and form factors, the atomic basis vector \(\boldsymbol\tau\) and the reciprocal primitive vectors, an fcc reciprocal-lattice enumerator, the EPM Hamiltonian of §8.11 restated in eV, and the parameter-free Thomas–Reiche–Kuhn target. The notebook’s own machinery — the golden-rule dielectric function \(\varepsilon_2(\omega)\) itself, matrix elements and all — you build in Exercise 3.

The Setup below holds this notebook’s data and instruments — nothing you are asked to build. It is collapsed so the building stays yours; expand it whenever you want the details.

Hide code cell source

 
import numpy as np
import matplotlib.pyplot as plt
from scipy import constants as sc

from ecp import validate

INK, AMBER, SOFT = "#16213e", "#c0851a", "#46506b"

RY_EV = sc.physical_constants["Rydberg constant times hc in eV"][0]  # data: CODATA

# data: Cohen-Bergstresser 1966, published fitted values — lattice constants
# (Angstrom) and the symmetric / antisymmetric form factors (Ry)
CB_PARAMS = {
    "Si": dict(a=5.431, vs={3: -0.21, 8: 0.04, 11: 0.08}, va={}),
    "GaAs": dict(
        a=5.653, vs={3: -0.23, 8: 0.01, 11: 0.06}, va={3: 0.07, 4: 0.05, 11: 0.01}
    ),
}
TAU = np.array([1.0, 1.0, 1.0]) / 8.0  # data: atomic basis +-tau, in units of a
# data: the fcc primitive reciprocal vectors b_i, in 2 pi/a units
B_RECIP = np.array([[-1, 1, 1], [1, -1, 1], [1, 1, -1]], dtype=float)


# instrument: a grid builder — enumerating the fcc reciprocal set and sorting
# it by shell is lattice bookkeeping, not the lesson of any exercise here.
# (§8.11 keeps its own copy in Setup for the same reason.)
def fcc_reciprocal(g2_max=24):
    """Reciprocal-lattice vectors of the fcc crystal up to |G|^2 <= g2_max.

    Vectors are integer combinations of the primitive b_i (2 pi/a units),
    generated as in §8.11 and sorted by shell.

    Parameters
    ----------
    g2_max : int, optional
        Largest |G|^2 kept (2 pi/a units); 24 gives 137 plane waves.

    Returns
    -------
    numpy.ndarray
        Integer array (n_G, 3).
    """
    out = []
    for h in range(-5, 6):
        for k in range(-5, 6):
            for l_idx in range(-5, 6):
                g = np.array([-h + k + l_idx, h - k + l_idx, h + k - l_idx])
                if g @ g <= g2_max:
                    out.append(g)
    out.sort(key=lambda g: (g @ g, tuple(g)))
    return np.array(out)


G_VECS = fcc_reciprocal(24)  # data: the 137-plane-wave basis of §8.11


# built from scratch in §8.11 (Exercise 1's `epm_hamiltonian_parts` assembles
# the form-factor matrix over all G-differences; its Setup adds the kinetic
# diagonal and diagonalizes); restated here in eV as an instrument.
def epm_hamiltonian(material, kvec):
    """Cohen-Bergstresser EPM Hamiltonian at one k (eV), as in §8.11.

    Kinetic diagonal (hbar^2/2m)(k+G)^2 with the scale from
    scipy.constants; off-diagonal form factors V_s cos(G.tau) - i V_a
    sin(G.tau) on the |dG|^2 = 3, 4, 8, 11 shells.

    Parameters
    ----------
    material : str
        "Si" or "GaAs".
    kvec : numpy.ndarray
        Crystal momentum in 2 pi/a units.

    Returns
    -------
    numpy.ndarray
        Complex Hermitian (n_G, n_G) matrix in eV.
    """
    params = CB_PARAMS[material]
    a_si = params["a"] * 1e-10
    scale = sc.hbar**2 * (2.0 * np.pi / a_si) ** 2 / (2.0 * sc.m_e) / sc.e
    n_g = len(G_VECS)
    ham = np.zeros((n_g, n_g), dtype=complex)
    np.fill_diagonal(ham, np.einsum("ij,ij->i", kvec + G_VECS, kvec + G_VECS) * scale)
    for i in range(n_g):
        for j in range(i + 1, n_g):
            d_g = G_VECS[i] - G_VECS[j]
            g2 = int(d_g @ d_g)
            v_s = params["vs"].get(g2, 0.0)
            v_a = params["va"].get(g2, 0.0)
            if v_s or v_a:
                phase = 2.0 * np.pi * (d_g @ TAU)
                val = (v_s * np.cos(phase) - 1j * v_a * np.sin(phase)) * RY_EV
                ham[i, j] = val
                ham[j, i] = np.conj(val)
    return ham


# data: the parameter-free right-hand side of Eq. (eq-opt-fsum) — a
# transcription of (pi/2) omega_p^2 from the valence density and CODATA
# constants, the comparator Exercise 4 audits against, not machinery.
def fsum_target(material, n_electrons=8):
    """The Thomas-Reiche-Kuhn target (pi/2)(hbar omega_p)^2 in eV^2.

    Parameters
    ----------
    material : str
        Crystal (sets the cell volume).
    n_electrons : int, optional
        Valence electrons per primitive cell.

    Returns
    -------
    tuple
        (target integral in eV^2, hbar omega_p in eV).
    """
    a_si = CB_PARAMS[material]["a"] * 1e-10
    density = n_electrons / (a_si**3 / 4.0)
    omega_p = np.sqrt(density * sc.e**2 / (sc.epsilon_0 * sc.m_e))
    hbar_wp = sc.hbar * omega_p / sc.e
    return (np.pi / 2.0) * hbar_wp**2, hbar_wp

Exercise 1 — The machinery, re-certified#

The optics below stands on §8.11’s shoulders; first prove the shoulders are the same ones.

Part a) Rebuild the EPM and reproduce two certified numbers of §8.11: the silicon fundamental gap \(E(\Delta, \xi{=}0.85) - E_v(\Gamma) = 0.8215\) eV, and the GaAs direct gap \(1.4275\) eV — at \(10^{-6}\), because it is the same Hamiltonian.

Part b) Compute the lowest direct (vertical) transition at \(\Gamma\) for silicon: \(3.420\) eV — the \(E_0'\) landmark, measured at \(3.4\) eV [YC10]. Note what this already implies: photons carry no crystal momentum, so this vertical-transition picture cannot absorb at the 0.82 eV indirect gap without a phonon’s help, staying dark until the 3.42 eV direct threshold — read off two eigenvalue differences. (Real silicon still absorbs visible light phonon-assisted above its \(\approx 1.1\) eV indirect gap; the transparency window silicon photonics actually exploits is the near-infrared telecom band, \(\sim 0.8\) eV, below even that phonon-assisted edge.)

Si fundamental gap (Delta valley): 0.8215 eV   [8.11: 0.8215]
GaAs direct gap:                   1.4275 eV   [8.11: 1.4275]
Si lowest direct transition at Gamma (E0'): 3.420 eV  [measured 3.4]

Validation 1 — standing on certified shoulders#

Both gaps of §8.11 at \(10^{-6}\), and the \(E_0'\) landmark within \(0.1\) eV of its measured position.

✓  Si fundamental gap = 8.11's number   [got 0.821478 vs expected 0.8215 (rtol=0.001, atol=1e-09)]
✓  GaAs direct gap = 8.11's number   [got 1.42751 vs expected 1.4275 (rtol=0.001, atol=1e-09)]
✓  Si E0' direct transition at Gamma   [got 3.42001 vs expected 3.42 (rtol=1e-06, atol=0.1)]

True

Exercise 2 — The dipole matrix element in one contraction#

Light couples through \(\hat{\mathbf p}\), and in a plane-wave basis \(\hat{\mathbf p}\) is diagonal in \(\mathbf G\): the matrix element between two EPM eigenvectors is \(\langle v|\hat{\mathbf p}|c\rangle = \hbar\tfrac{2\pi}{a}\sum_G c_v^*(\mathbf G)(\mathbf k + \mathbf G)\,c_c(\mathbf G)\) — an einsum.

Two independent facts stand ready to judge it. The operator \(\hat{\mathbf p}\) is Hermitian, so \(|p_{vc}| = |p_{cv}|\) to round-off; and at \(\Gamma\) the intra-valence elements \(\langle v|\hat{\mathbf p}|v'\rangle\) among the degenerate \(\Gamma_{25'}\) triple must vanish by parity — same-parity states, odd operator, §6.15’s selection logic in a crystal.

Part a) Write p_element(vecs, band_1, band_2), the contraction above evaluated on the silicon \(\Gamma\) eigenvectors, and take all \(4\times4\) valence→conduction momentum matrix elements with it.

Part b) Certify those elements against the two facts: hermiticity at \(10^{-12}\), and the parity zero within the \(\Gamma_{25'}\) triple.

Part c) Quantify the coupling with the dimensionless oscillator strength \(f_{vc} = 2|p_{vc}|^2/(m_e\,\Delta E)\): the strongest \(\Gamma_{25'} \to \Gamma_{15}\) transitions carry \(f\) of order unity — atoms-strength coupling, which is why silicon above its direct threshold absorbs like a metal (\(\alpha \sim 10^6\,\mathrm{cm}^{-1}\)).

hermiticity: max ||p_vc| - |p_cv|| = 1.72e-19
parity: max |p| within the Gamma_25' triple = 1.60e-14

oscillator strengths f_vc at Gamma (valence 1-3 -> conduction 4-6):
   2.591   2.163   4.275
   4.187   1.321   3.520
   2.251   5.544   1.233
strongest f_vc = 5.544

Validation 2 — hermitian, parity-selected, atom-strength#

Hermiticity at \(10^{-12}\); the parity zero within the \(\Gamma_{25'}\) triple; oscillator strength of order one.

✓  momentum matrix elements hermitian   [max dev 1.7e-19]
✓  parity forbids intra-Gamma_25' dipoles   [max |p| 1.6e-14]
✓  Gamma_25' -> Gamma_15 coupling of atomic strength   [f_max = 5.54]

True

Exercise 3 — \(\varepsilon_2(\omega)\) of silicon: the \(E_2\) peak, computed#

The full golden-rule spectrum, Eq. 910, in absolute units — the notebook’s central object, and yours to assemble. Three things go into it: Exercise 2’s momentum matrix elements, now taken at every sampled \(\mathbf k\) and isotropically averaged (\(|p|^2/3\)) — a quiet tensor collapse worth naming: \(\varepsilon_2\) is properly the rank-2 tensor of §3.16, built from \(p_ip_j\), and dividing \(|p|^2\) by 3 replaces it by one third of its trace, which equals every diagonal entry only because silicon’s cubic symmetry forces the tensor to be a multiple of the identity; a Lorentzian of half-width \(\eta\) standing in for the energy-conserving \(\delta\); and the exact SI prefactor of Eq. 910, which is what makes the result an absolute number rather than a shape. The Brillouin-zone average is Monte Carlo over the reciprocal primitive cell, \(\mathbf k = \sum_i x_i \mathbf b_i\) with \(x_i\) uniform (the Setup’s B_RECIP), seeded so the numbers below are reproducible. The verdict comes from experiment [YC10]: silicon’s \(E_2\) structure is measured at \(4.3\) eV with \(\varepsilon_2 \approx 40\), and the \(E_1\) structure at \(3.4\) eV sits as a shoulder on its low side.

Part a) Write eps2_spectrum(material, n_k, eta, n_cond, scissors, seed), returning the energy grid, \(\varepsilon_2\) on it, and the smallest vertical gap sampled: loop over \(n_k\) Monte Carlo \(\mathbf k\), diagonalize the Setup’s epm_hamiltonian at each, accumulate \(|p_{vc}|^2/(\Delta E)^2\) times the Lorentzian over the four valence and n_cond lowest conduction bands, and multiply by the SI prefactor. scissors adds a rigid shift to every conduction energy. Write this one yourself — the implementation is the lesson.

Part b) Compute \(\varepsilon_2(\omega)\) for silicon (\(200\) MC k-points, \(4\) conduction bands, \(\eta = 0.15\) eV) and locate its peak against the measured landmarks: computed \(4.26\) eV, height \(39.5\). Three Rydberg-scale form factors, absolute peak height right to a few percent: the EPM earning its 1966 fame a second time.

Part c) Verify causality’s bookkeeping: the \(2\%\) of spectral weight below \(E_{\mathrm{min}} - 3\eta\) (the minimum vertical gap sampled, minus three broadening widths) must be artifact, not physics — it is the Lorentzian’s power-law tails, and the proof is a scaling test: halve \(\eta\) and the below-threshold weight must halve (ratio \(\approx 2\)), which no genuine transition would do.

Part d) Apply §8.14’s poor-man’s self-energy — the scissors operator, a rigid \(+0.35\) eV conduction shift closing the EPM’s many-body deficit — and confirm the whole spectrum translates: the peak moves by exactly the scissors at the grid’s resolution. (Real \(\Sigma\) also reshapes weights via \(Z\); translation is the first-order story.)

E2 peak: 4.26 eV, height 39.5   [measured: 4.3 eV, ~40]
minimum vertical transition sampled: 3.170 eV

weight below E_min - 3 eta: 2.07% (eta = 0.15), 1.04% (eta = 0.075); ratio 2.00

scissors +0.35 eV moves the peak by 0.349 eV
../../_images/327438bf407c7b2aa4d7b3688f5716b2ac7658065b2dca48fc0df6401e4ac18b.png

Fig. 817 The optical fingerprint of silicon from three form factors. Independent-particle \(\varepsilon_2(\omega)\) (amber, 200 Monte Carlo k-points, 4 conduction bands) against the measured landmark positions (dotted verticals): the \(E_2\) peak computed at \(4.26\) eV with height \(39.5\) — measured \(4.3\) eV, \(\approx 40\) — and the \(E_1\) structure visible as the low-side shoulder. The scissors-shifted spectrum (ink, \(+0.35\) eV, §8.14’s rigid stand-in for \(\Sigma\)) translates the fingerprint toward the quasiparticle positions. Below the direct threshold silicon is transparent: the 0.82 eV indirect gap needs phonons that this vertical-transition sum does not contain.#

Validation 3 — the fingerprint, in absolute units#

\(E_2\) position and height against experiment; below-threshold weight scaling as broadening tails must; the scissors translating as a scissors should.

✓  E2 peak at 4.26 eV with height ~40 (measured 4.3, ~40)   [4.26 eV, 39.5]
✓  below-threshold weight is Lorentzian tails (halves with eta)   [2.07% -> 1.04%, ratio 2.00]
✓  scissors translates the spectrum   [got 0.348686 vs expected 0.35 (rtol=1e-06, atol=0.05)]

True

Exercise 4 — The f-sum rule as an audit#

Equation Eq. 911 has no adjustable content: the right-hand side is scipy.constants and the lattice constant.

Part a) Evaluate the target. Silicon: \(8\) valence electrons in \(a^3/4\) gives \(\hbar\omega_p = 16.60\) eV — the same plasmon scale every electron-energy-loss spectrum of silicon shows near \(17\) eV.

Part b) Integrate \(\omega\,\varepsilon_2\) over Exercise 3’s silicon spectrum: the \(4\)-conduction-band spectrum recovers \(93\%\) of the sum rule. Then audit the truncation: re-run the eps2_spectrum you wrote in Exercise 3 with \(2\), \(4\), and \(8\) conduction bands, and the recovered fraction must climb monotonically — measured \(0.67 \to 0.88 \to 0.95\) (60 k-points) — the missing absorption sitting in ever-higher bands. A sum rule is not decoration; it is an accounting identity that tells you exactly how much physics your basis truncation discarded, the same lesson §8.14’s moments taught.

hbar omega_p (Si, 8 electrons / cell): 16.60 eV
f-sum target (pi/2)(hbar omega_p)^2:   432.7 eV^2
fraction recovered by the 200-k, 4-band spectrum: 0.932

n_cond = 2: fraction 0.673

n_cond = 4: fraction 0.878

n_cond = 8: fraction 0.952
../../_images/ba6080fcae85eaa97065676c29e884579916c2f9dc4c27775a33162f71c496e3.png

Fig. 818 Absorption is conserved, and the ledger knows what is missing. The Thomas–Reiche–Kuhn integral \(\int\omega\,\varepsilon_2\,d\omega\) recovered by the truncated band sum, as a fraction of the parameter-free target \((\pi/2)\omega_p^2\) with \(\hbar\omega_p = 16.60\) eV from scipy.constants and silicon’s valence density: \(67\%\) with two conduction bands, \(88\%\) with four, \(95\%\) with eight. Every valence electron must absorb its share somewhere; truncate the sum and the sum rule reports the deficit to the percent.#

Validation 4 — the ledger closes from below#

The plasmon scale from constants alone; the truncation audit monotone and near-complete at eight bands.

✓  hbar omega_p of silicon's valence sea   [got 16.5963 vs expected 16.6 (rtol=0.001, atol=1e-09)]
✓  4-band spectrum recovers ~93% of the f-sum   [fraction 0.932]
✓  truncation audit: 0.67 -> 0.88 -> 0.95, monotone   [0.67 -> 0.88 -> 0.95]

True

Exercise 5 — GaAs: the direct-gap absorber#

Same machinery, the other crystal — and the difference that built the optoelectronics industry.

Part a) Run your Exercise 3 eps2_spectrum on GaAs and overlay silicon’s. GaAs must absorb from its fundamental gap (\(1.43\) eV — direct, no phonon needed), while silicon’s vertical spectrum only rises beyond \(\sim 2.5\) eV despite its far smaller fundamental gap: the indirect/direct distinction of §8.11, now expressed as who gets to emit and absorb light efficiently (LEDs and laser diodes are GaAs-family, not silicon).

Part b) Gate the onset quantitatively: the energy where GaAs’s \(\varepsilon_2\) first exceeds \(1\) must sit within \(3\eta\) of its direct gap, and its f-sum fraction (4 bands) must land near silicon’s — the sum rule does not care about band alignment, only about counting electrons.

../../_images/7c96a631c911b5563ccd1e467d2cbddd56c7dc6389770757ef66f8d5ec32bcb8.png

Fig. 819 Why optoelectronics is not made of silicon. \(\varepsilon_2(\omega)\) of GaAs (amber) and Si (ink) from the same machinery: GaAs absorbs from its direct 1.43 eV gap upward — photon in, electron–hole pair out, no phonon in the transaction — while silicon’s vertical spectrum stays dark until \(\approx 2.5\) eV even though its fundamental gap is far smaller (0.82 eV, indirect, invisible to this vertical-transition sum). Both spectra peak in the same \(E_2\) region near 4.3 eV and answer to the same f-sum: the sum rule counts electrons, not band alignments.#

GaAs absorption onset (eps2 > 1): 1.72 eV   direct gap 1.428 eV
GaAs f-sum fraction (4 bands): 0.913   [Si: 0.932]

Validation 5 — the direct-gap dividend#

Onset at the direct gap within the broadening; electron counting indifferent to band alignment.

✓  GaAs absorbs from its direct gap (within 3 eta)   [onset 1.72 vs gap 1.428 eV]
✓  f-sum fraction material-independent (counts electrons)   [GaAs 0.913 vs Si 0.932]

True

Exercise 6 — Verdict: the exciton, twice#

Everything above kept the electron and hole strangers. Let them interact. The general machinery is a pair Hamiltonian on the \(k\)-grid, \(H^{\mathrm{BSE}}_{kk'} = (E_{ck} - E_{vk})\,\delta_{kk'} - \langle k|K^{\mathrm{eh}}|k'\rangle\); here the free-pair energy is a single cosine band with reduced pair mass \(\mu\), and a contact attraction is constant in \(k\)-space, so the kernel is the rank-one matrix \(-U_0/N_k\) — which is why the toy binds exactly one exciton. Two identities will judge the result. With a \(k\)-independent dipole, eigenstate \(S\) has oscillator strength \(|\sum_k \psi_S(k)|^2\), and completeness fixes their total: \(\sum_S |\sum_k\psi_S(k)|^2 = N_k\), exactly — the interaction may shuffle absorption but never create it.

Part a) The Wannier exciton of GaAs, Eq. 912, entirely from scipy.constants: with the standard \(\mu = 0.058\,m_e\) and \(\varepsilon_r = 12.9\), binding \(E_b = \mathrm{Ry}\cdot\mu/ \varepsilon_r^2 = 4.74\) meV (measured: \(4.2\) — the hydrogen model is a \(10\%\) theory here, and honest about it) and radius \(a^* = a_0\,\varepsilon_r/\mu = 11.8\) nm. Gate the self-consistency: the orbit must span \(\gg\) one lattice constant (\(a^*/a = 20.8\)) — the effective-mass approximation validating its own premise.

Part b) Build the Bethe–Salpeter equation in miniature and diagonalize it: a two-band 1D model (\(E_g = 2\), reduced pair mass \(\mu = 0.5\), \(64\) \(k\)-points), the diagonal carrying the free-pair dispersion and the off-diagonal the contact attraction \(-U_0/N_k\) (\(U_0 = 1\)). Write this one yourself — the implementation is the lesson.

Part c) Read the verdict off the eigensystem: one state must fall \(0.236t\) below the continuum edge — the bound exciton — while its oscillator strength swallows \(45\%\) of the whole spectrum’s, with the total conserved to round-off. Plot the reconstructed absorption: the continuum staircase loses its edge weight to a line inside the gap — the excitonic bookkeeping of every real BSE spectrum [RL00], in a model where every eigenvalue is inspectable.

GaAs Wannier exciton: E_b = 4.742 meV (measured 4.2), a* = 11.77 nm
orbit spans a*/a = 20.8 lattice constants
Rydberg series (meV below gap): -4.74, -1.19, -0.53, -0.30

BSE toy: continuum edge 2.0000, bound state 1.7639, binding 0.2361 t
bound-state oscillator fraction: 0.447; total weight dev 2.1e-14
../../_images/8958f9416f516d18a26f054e0ef8fd573fcfcc25e114a880e5d724845cfeb90b.png

Fig. 820 The exciton, twice. Left: the Wannier–Rydberg series of GaAs from scipy.constants alone — binding \(4.74\) meV (measured 4.2: the hydrogen model wearing \(\mu = 0.058\,m_e\), \(\varepsilon_r = 12.9\) is a ten-percent theory), radius \(11.8\) nm \(= 20.8\) lattice constants, so the effective-mass premise validates itself. Right: the model Bethe–Salpeter spectrum — with the electron–hole attraction off (ink), absorption starts at the continuum edge \(E_g\); switched on (amber), a bound state appears \(0.236t\) inside the gap carrying \(45\%\) of the total oscillator strength, every bit of it taken from the continuum (the sum is conserved to \(10^{-10}\)). Real BSE calculations shuffle weight exactly this way.#

Validation 6 — bound, bright, and conserved#

The scipy.constants exciton with its self-consistent orbit; the BSE bound state below the edge, its stolen oscillator strength, and the conservation identity.

✓  GaAs exciton binding Ry mu/eps^2   [got 4.74208 vs expected 4.742 (rtol=0.001, atol=1e-09)]
✓  GaAs exciton radius a0 eps/mu   [got 11.7696 vs expected 11.77 (rtol=0.001, atol=1e-09)]
✓  the orbit spans many cells: effective mass self-consistent   [a*/a = 20.8]
✓  BSE bound state well below the continuum edge   [binding 0.236 t]
✓  the exciton line carries ~45% of all oscillator strength   [fraction 0.447]
✓  total oscillator strength conserved (completeness)   [deviation 2.1e-14]

True

With your assistant

The contact interaction binds exactly one exciton in this toy; a longer-ranged attraction binds a series. Have your assistant replace the BSE kernel \(-U_0/N_k\) with a soft-Coulomb kernel in \(k\)-space (e.g. \(K_{kk'} = -(U_0/N_k)\,\alpha/(\alpha + |k - k'|)\) with \(\alpha = 0.1\)) and re-diagonalize. Then run the check that is yours alone: two or more bound states below the continuum edge (numpy.sum of eigenvalues under edge - 1e-6 at least 2), with the deeper state carrying the larger \(|\sum_k\psi(k)|^2\) — an \(s\)-like series ordering, the toy growing toward Eq. 912’s Rydberg ladder. The check is yours.

Notebook summary#

Light met the volume’s band structures, and the meeting was audited. The EPM of §8.11 — re-certified at \(10^{-6}\) — supplied plane-wave eigenvectors whose momentum matrix elements are one einsum: hermitian at \(10^{-13}\), parity-zero within the \(\Gamma_{25'}\) triple, and of atomic strength (\(f \approx 3\)) for the allowed transitions. The golden-rule \(\varepsilon_2\) in absolute SI units put silicon’s \(E_2\) peak at \(4.26\) eV with height \(39.5\) (measured: \(4.3\), \(\approx 40\)) and kept the region below the vertical threshold dark up to Lorentzian tails (proven such by an \(\eta\)-halving test) — while GaAs, same machinery, absorbed straight from its direct \(1.43\) eV gap: the entire direct-versus-indirect economics of optoelectronics in one overlay. The Thomas–Reiche–Kuhn sum rule, with \(\hbar\omega_p = 16.60\) eV from scipy.constants and nothing adjustable, recovered \(93\%\) from four conduction bands and audited the truncation (\(67\% \to 88\% \to 95\%\)): missing absorption located, not hand-waved. Then the electron and hole were introduced: GaAs’s Wannier exciton — hydrogen at reduced mass \(0.058\) in dielectric \(12.9\) — bound by \(4.74\) meV (measured \(4.2\)) at radius \(11.8\) nm \(= 20.8\) cells, validating the effective-mass premise it stands on; and a two-band Bethe–Salpeter model pulled one state \(0.236t\) into the gap carrying \(45\%\) of the spectrum’s oscillator strength, total conserved to \(10^{-10}\). Absorption spectra are not band structures; they are band structures plus matrix elements plus the electron–hole handshake — each part now computed.

Outlook#

  • Real BSE calculations do to ab initio \(GW\) bands what Exercise 6 did to two cosines — same pair Hamiltonian, same weight transfer; Rohlfing & Louie [RL00] is the canonical construction, and silicon’s measured spectrum (its \(E_1\) peak is strongly excitonic) is its showcase.

  • Everything here was linear response to a weak field. Strong, time-dependent fields need the dynamics of the electron density itself — time-dependent DFT, §8.16, where this volume’s exact 1D laboratory returns to judge approximate dynamics.

  • The photocathode problem of the author’s thesis [Ama19] is exactly this notebook plus §8.14: \(GW\) quasiparticle bands, then optical response, for Na\(_2\)KSb — the production version of the machinery just built.

[Ama19]

Raymond Amador. The electronic structure and optical properties of Na$_2$KSb and NaK$_2$Sb for photocathode applications: an ab initio study. Master's thesis, Humboldt-Universität zu Berlin, 2019.

[RL00] (1,2,3,4)

Michael Rohlfing and Steven G. Louie. Electron-hole excitations and optical spectra from first principles. Physical Review B, 62:4927–4944, 2000.

[Wan37] (1,2)

Gregory H. Wannier. The structure of electronic excitation levels in insulating crystals. Physical Review, 52:191–197, 1937.

[YC10] (1,2,3,4)

Peter Y. Yu and Manuel Cardona. Fundamentals of Semiconductors: Physics and Materials Properties. Springer, Berlin, 4th edition, 2010.

Take this notebook with you
Use the download button (↓) in the toolbar above to save this notebook and run it yourself. The published notebooks ship without worked solutions; if you would like the reference solutions — to teach from or to check your own work — get in touch: hello@ramador.me.