E18 · Electron excitations

Cathodoluminescence: Where Light Is Born

Does a dark spot mean the beam generated fewer carriers? Not necessarily. The light you see depends on how many carriers the beam creates, how far they diffuse, how many recombine without giving off light, and how much light is collected. At a dark spot the beam may have created carriers as usual, only for them to be lost to non-radiative recombination, for example at a defect.

Scan the beam across a synthetic semiconductor and follow how carriers diffuse, emit light or are captured without emitting. Then select a region of the recorded cube of spectra and compare the light in fixed wavelength windows with the total brightness.

Recorded region photons $\text{—}$

Recorded defect-window photons $\text{—}$

Defect-window share of counts $\text{—}$

Model mean residence time $\text{—}$

More measurements and target

Assumed generation footprint FWHM $\text{—}$

Specified band emission wavelength $\text{—}$

Raster acquisition time $\text{—}$

Selected stored pixels $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Landing energy

E_0

5\,\mathrm{keV}

Dwell per scan pixel

t_d

0.005\,\mathrm{s}

Specified temperature

T

100\,\mathrm{K}

Nominal bulk diffusion length

L_0

0.15\,\mathrm{\mu m}

Advanced settings

Incident beam current

I

10\,\mathrm{pA}

Extra nonradiative rate in domain

\Gamma_q

1\,\mathrm{\frac{1}{ns}}

Optical collection solid angle

\Omega

0.8\,\mathrm{sr}

Region center: horizontal

x

0.8\,\mathrm{\mu m}

Region center: vertical

y

0\,\mathrm{\mu m}

Region radius

r

0.3\,\mathrm{\mu m}

Learning goal and model limits

Make a prediction, then test it

Does a dark spot mean the beam generated fewer carriers?

Target: collect at least 500 defect-window photons from a region that lies entirely inside the specified domain. Keep the temperature at or below 150 K, the nominal diffusion length at or below 0.2 micrometers, the landing energy at or below 8 keV and the extra nonradiative rate at or below 2 per ns.

Model assumptions

This linear, steady-state teaching model treats the specimen as a projected sheet, solved on a 4 micrometer square grid of 20 by 20 cells with reflecting edges. A specified circular defect domain adds its own radiative and nonradiative rates. All material energies, lifetimes, yields and their temperature dependence are assumed and do not describe any real semiconductor. The generation footprint is a normalized Gaussian that widens with landing energy; it is not an electron transport calculation. A finite-volume diffusion equation makes the radiative and nonradiative channels together account for every generated carrier. Emission lines are Gaussians in energy, integrated over the boundaries of each wavelength bin. Photon collection assumes isotropic escape, 40 percent efficiency and no reabsorption. Each pixel stores a seeded count spectrum, with a normal approximation above 40 expected counts. A fixed wavelength window can miss a peak that shifts. Each map has its own log color scale, so the colors are not absolute concentrations. Regions and profiles reuse the acquired counts. The lifetime readout is a forward-model diagnostic, not a time-resolved measurement. Saturation, charging, beam damage, optical interference, transport in depth and calibrated inversion of the diffusion length are left out. The specimen and carrier paths in 3D are enlarged and schematic.

Physics tutorial

Cathodoluminescence: Where Light Is Born

BackgroundYou scan the beam across a synthetic semiconductor and follow how carriers diffuse, emit light or are captured without emitting. Then you select a region of the recorded cube of spectra and compare the light in fixed wavelength windows with the total brightness.

Why it mattersThe goal is to collect at least 500 photons in the defect window from a region that lies entirely inside the specified domain. Keep the temperature at or below 150 K, the nominal diffusion length at or below 0.2 micrometers, the landing energy at or below 8 keV and the extra nonradiative rate at or below 2 per ns.

Start with the essentials

Focus question: Does a dark spot mean the beam generated fewer carriers?
One-sentence intuition: How bright a pixel looks depends on how many carriers the beam creates, how far they travel, which recombination channel wins and how much light is collected. A photon map records where the beam excited the sample, not where each carrier actually emitted.

Core mathematical model

Steady-state carrier continuity

-D\nabla^2n+(\Gamma_b+\Gamma_d+\Gamma_{nr})n=G

In steady state, every generated carrier ends up in one of the competing recombination channels.

Expected photons in each wavelength bin

\mu_j=\frac{\eta\Omega}{4\pi}\int_{\lambda_j^-}^{\lambda_j^+}S_E\!\left(\frac{hc}{\lambda}\right)\frac{hc}{\lambda^2}\,d\lambda

The emission lines are defined in energy. Integrating over each wavelength bin, with the Jacobian for the change from energy to wavelength, gives the expected photon count in that bin.

Nominal diffusion length

L=\sqrt{D/\Gamma_{bulk}}

The nominal diffusion length sets the diffusivity through a reference lifetime of one nanosecond. The actual competing rates are different inside the domain.

Common difficulties

Brighter does not mean more local

Typical misconceptionA brighter or cleaner map means the measurement is more local.

Better mental modelHow bright a pixel looks depends on how many carriers the beam creates, how far they travel, which recombination channel wins and how much light is collected. A photon map records where the beam excited the sample, not where each carrier actually emitted.

Run the experiment

01
Predict why the domain looks dark
Compare Local domain with usable counts and Nonradiative capture at the same beam current.
What to observe: The beam creates the same number of carriers, but fewer of them recombine by emitting light.
02
Track carrier migration
Increase the nominal diffusion length and compare the known domain with the recorded central profile.
What to observe: Counts are filed under the position where the beam excited the sample, and carriers can travel before they emit.
03
Work with the stored spectra
Move the selected region onto the matrix and back, then reach the target. Before you interpret a fixed-window map, try heating the sample.
What to observe: Selecting a region does not acquire new photons. When the peak shifts, the fixed-window map changes.