Carrier Multiplication in Semiconductors

• Carrier Multiplication (CM) is a process where excess photon energy creates additional electron-hole pairs via impact ionization or multiple exciton generation.
• CM efficiency is influenced by material parameters such as band gap, excitation density, phonon cooling, and doping, with notable behaviors in graphene, Bi2Se3, and quantum dots.
• Advanced techniques like trARPES, transient absorption, and photocurrent measurements enable detailed observation of CM dynamics and quantification of quantum yields in various semiconductor platforms.

Carrier multiplication (CM) is a nonequilibrium many-body process in which the absorption of a single high-energy photon generates more than one electron-hole pair. In bulk-semiconductor language it is usually identified with impact ionization, the inverse of Auger recombination; in quantum-confined systems it is often discussed as multiple exciton generation. The central physical idea is that photon excess energy above the band gap is redirected into additional electronic excitations rather than being dissipated as heat through phonons. Because of that redirection, CM is repeatedly invoked in the context of photovoltaics, photodetection, and ultrafast carrier dynamics beyond the single-exciton limit (Herb et al., 2024, Maiti et al., 2020).

1. Definition and microscopic mechanism

CM denotes an increase of carrier number beyond the population created directly by optical absorption. A useful operational definition employed in graphene is

$$\mathrm{CM}(t)=\frac{\mathcal{N}(t)}{\mathcal{N}_{\mathrm{opt}}(t)},$$

where $\mathcal{N}(t)$ is the total carrier density and $\mathcal{N}_{\mathrm{opt}}(t)$ is the density expected from optical excitation alone; $\mathrm{CM}>1$ indicates net multiplication (Malic et al., 2016). In nanocrystal language the same physics is հաճախ cast in terms of quantum efficiency,

$QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$

with $N_x$ and $N_{xx}$ the exciton and biexciton populations, so that $QE>1$ signals multiexciton production (Velizhanin et al., 2010).

Microscopically, the elementary pair-number-increasing channel is impact ionization or impact excitation: a hot carrier loses part of its kinetic energy while promoting another electron across the gap. Its inverse is Auger recombination, in which an electron-hole pair annihilates and transfers its energy to another carrier. This distinction is essential because ordinary hot-carrier cooling only redistributes energy into phonons, whereas CM changes the number of mobile carriers. In Bi$_2$Se$_3$, for example, the defining hallmark is that the occupied conduction-band population and the valence-band hole population continue to grow after the pump pulse has ended, which cannot be explained by direct optical excitation alone (Herb et al., 2024).

The nonequilibrium asymmetry between impact excitation and Auger recombination is often governed by phase space and Pauli blocking. In graphene, the weak-excitation argument is especially transparent: $\mathcal{N}(t)$0 Near the Dirac point immediately after optical excitation, $\mathcal{N}(t)$1 and $\mathcal{N}(t)$2, so impact ionization is strongly favored while Auger recombination is blocked. This transient imbalance is the core microscopic origin of large predicted CM in graphene (Winzer et al., 2010).

A recurrent terminological distinction is that CM counts new electron-hole pairs across the band gap or Dirac point, whereas “hot-carrier multiplication” can instead count carriers above the Fermi level in doped systems. The latter becomes important in doped graphene, where intraband Auger processes around $\mathcal{N}(t)$3 can increase the number of hot carriers even when ordinary interband CM is reduced (Malic et al., 2016).

2. Energetic threshold and competition with cooling

The minimal energy-conservation criterion for one extra pair is $\mathcal{N}(t)$4. In the ideal staircase picture, $\mathcal{N}(t)$5 below $\mathcal{N}(t)$6, $\mathcal{N}(t)$7 between $\mathcal{N}(t)$8 and $\mathcal{N}(t)$9, and so forth. Real materials often deviate from that ideal because of unfavorable band structure, selection rules, phase-space restrictions, and rapid phonon-mediated cooling. The CM threshold is therefore a central metric, usually expressed as a multiple $\mathcal{N}_{\mathrm{opt}}(t)$0 (Maiti et al., 2020).

The literature summarized here makes clear that thresholds near $\mathcal{N}_{\mathrm{opt}}(t)$1 are exceptional rather than generic. In many conventional bulk semiconductors the threshold is much higher; the review on emerging photovoltaic materials cites about $\mathcal{N}_{\mathrm{opt}}(t)$2 as a typical bulk threshold in the simple symmetric-band picture, while quantum dots with near-equal electron and hole effective masses often sit closer to $\mathcal{N}_{\mathrm{opt}}(t)$3. Near-minimal thresholds are associated with asymmetric optical excitation from a deeper valence band or into a higher conduction band, together with low exciton binding energy and sufficient mobility for free-carrier extraction (Maiti et al., 2020).

CM is always a race between carrier-carrier scattering and energy loss to phonons. That competition is material specific. Bi$\mathcal{N}_{\mathrm{opt}}(t)$4Se$\mathcal{N}_{\mathrm{opt}}(t)$5 is presented as favorable because its gap is only about $\mathcal{N}_{\mathrm{opt}}(t)$6 meV while the pump photon energy is $\mathcal{N}_{\mathrm{opt}}(t)$7 eV, leaving large excess energy, and because its phonon frequencies are below about $\mathcal{N}_{\mathrm{opt}}(t)$8 meV, which suppresses rapid lattice dissipation (Herb et al., 2024). In graphene, the same competition is controlled strongly by excitation density: microscopic calculations predicted CM of approximately $\mathcal{N}_{\mathrm{opt}}(t)$9 for weak excitation around $\mathrm{CM}>1$0, but only about $\mathrm{CM}>1$1 around $\mathrm{CM}>1$2, because the nonequilibrium asymmetry favoring impact ionization collapses much faster at high density (Winzer et al., 2010).

Phase space can also be tuned by doping. In graphene, stronger $\mathrm{CM}>1$3-doping increases the density of states and the number of scattering partners near the Fermi level, enabling larger CM and faster phonon-mediated cooling. Time- and angle-resolved photoemission found peak CM around $\mathrm{CM}>1$4 in the more strongly $\mathrm{CM}>1$5-doped sample, while the less strongly $\mathrm{CM}>1$6-doped sample only briefly exceeded unity (Johannsen et al., 2016). This does not imply a fundamental asymmetry between electron and hole doping; the reported interpretation is that the decisive variable is the magnitude of doping and hence the available scattering phase space.

3. Experimental observables and measurement strategies

CM has been probed with a wide range of techniques, but not all observables are equally direct. Time- and angle-resolved photoemission spectroscopy (trARPES) is especially powerful because it separately tracks occupied conduction-band populations, depleted valence-band populations, transient band shifts, and hot-carrier distributions in momentum-resolved form. In Bi$\mathrm{CM}>1$7Se$\mathrm{CM}>1$8, trARPES showed that after $\mathrm{CM}>1$9 eV excitation the number of electrons in the conduction band and the number of holes in the valence band continued changing until about $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$0 fs, even though the pump-probe cross-correlation was only about $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$1 fs. From amplitudes at $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$2 fs and at the delayed extrema, the authors estimated a CM factor around $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$3, while explicitly noting that surface photovoltage makes this an empirical estimate rather than a fully intrinsic branching ratio (Herb et al., 2024).

In graphene, trARPES and related temperature-extraction procedures have provided both direct and indirect CM diagnostics. One key signature is the simultaneous increase in the number of conduction-band carriers and decrease in their average kinetic energy within about $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$4 fs, which is the expected fingerprint of impact excitation. In doped graphene, extracted hot-carrier densities and temperatures were used to define

$QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$5

with $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$6 the directly injected carrier density, yielding peak CM of approximately $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$7 in strongly $QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$8-doped graphene (Johannsen et al., 2016).

Transient absorption, transient conductivity, and optical gain spectroscopy are the dominant tools in nanocrystals and layered semiconductors. In 2H-MoTe$QE=\frac{2N_{xx}+N_x}{N_{xx}+N_x},$9 and 2H-WSe$N_x$0, transient absorption in photo-induced absorption geometry was used to calibrate an intraband cross section and infer a quantum yield $N_x$1, producing reported CM efficiencies of about $N_x$2 in MoTe$N_x$3 and $N_x$4 in WSe$N_x$5, with onset near $N_x$6 (Kim et al., 2018). In perovskite FAPbI$N_x$7/NdF$N_x$8 nanocrystals, single-particle time-resolved photoluminescence and biexponential fits were used instead; under $N_x$9 nm excitation at $N_{xx}$0, the average CM efficiency over $N_{xx}$1 single nanocrystals was reported as $N_{xx}$2 (Zhang et al., 6 Apr 2026).

Photocurrent-based measurements aim to bypass some ambiguities of purely optical inference. Dual-gated MoTe$N_{xx}$3 lateral $N_{xx}$4-$N_{xx}$5 homojunctions on quartz were used to extract external and internal quantum efficiencies,

$N_{xx}$6

with $N_{xx}$7. The reported IQE increased from about $N_{xx}$8 to about $N_{xx}$9 as the photon energy was raised from $QE>1$0 to $QE>1$1, corresponding to a normalized quantum yield rising from roughly $QE>1$2 to roughly $QE>1$3, which was interpreted as direct photocurrent evidence for CM (Kim et al., 2020).

4. Material platforms and representative regimes

CM has been reported or modeled in a broad set of materials, but the controlling physics differs sharply among them. Gapless Dirac materials emphasize Pauli-blocking asymmetry and rapid Coulomb scattering; narrow-gap van der Waals semiconductors emphasize low threshold and weak cooling; nanocrystals emphasize confinement, state counting, and biexciton kinetics; transport devices emphasize whether multiplied carriers can actually be harvested before they cool or recombine.

System	Representative CM behavior	Source
Bi$QE>1$4Se$QE>1$5	Estimated factor $QE>1$6 after $QE>1$7 eV excitation; carrier-pair maximum at $QE>1$8 fs	(Herb et al., 2024)
Strongly $QE>1$9-doped graphene	Peak CM $_2$0; faster cooling than less strongly $_2$1-doped graphene	(Johannsen et al., 2016)
2H-MoTe$_2$2 / 2H-WSe$_2$3	Step-like onset near $_2$4; reported CM efficiencies about $_2$5 and $_2$6 by PIA	(Kim et al., 2018)
Landau-quantized graphene	Predicted $_2$7 at $_2$8 T, $_2$9 meV pump, $_3$0	(Wendler et al., 2014)
FAPbI$_3$1/NdF$_3$2 nanocrystals	Average CM efficiency $_3$3 at $_3$4 nm ($_3$5)	(Zhang et al., 6 Apr 2026)
Si$_3$6H$_3$7 vs Si$_3$8(OH)$_3$9	Gaps $\mathcal{N}(t)$00 eV and $\mathcal{N}(t)$01 eV; OH lowers threshold, H gives more efficient CM on a relative scale	(Marri et al., 2018)
CNT photodiodes	Field-assisted impact excitation can drive efficiency toward $\mathcal{N}(t)$02 near threshold, with strong temperature and length dependence	(Baer et al., 2010)

Graphene is the canonical gapless CM platform. Microscopic calculations predicted CM around $\mathcal{N}(t)$03 under weak excitation and established impact ionization as the dominant early-time channel (Winzer et al., 2010). Later reviews integrated theory and experiment, reporting Coulomb-only CM of $\mathcal{N}(t)$04, reduction to about $\mathcal{N}(t)$05 when phonons are included, suppression of ordinary interband CM with doping above about $\mathcal{N}(t)$06 meV, and growth of hot-carrier multiplication up to about $\mathcal{N}(t)$07 around $\mathcal{N}(t)$08 meV (Malic et al., 2016). Under Landau quantization, the discrete $\mathcal{N}(t)$09-scaled Landau levels preserve selected resonant Auger channels while providing an extraction-relevant gap; the predicted CM of about $\mathcal{N}(t)$10 in the low-Landau-level pumping scheme is smaller than in zero-field graphene but conceptually important because it addresses graphene’s gapless-extraction problem (Wendler et al., 2014).

Van der Waals semiconductors provide the clearest near-threshold CM examples. For MoTe$\mathcal{N}(t)$11 and WSe$\mathcal{N}(t)$12, the reported onset at approximately $\mathcal{N}(t)$13 and near-ideal slope above threshold were attributed to strong Coulomb interactions in quasi-two-dimensional layers, relatively weak phonon losses, and multiple electron pockets in momentum space (Kim et al., 2018). Bi$\mathcal{N}(t)$14Se$\mathcal{N}(t)$15 adds a topological-insulator variant: it combines a small gap with low phonon energies and exhibits directly measured delayed carrier buildup after the pump pulse (Herb et al., 2024).

In silicon and lead-chalcogenide nanocrystals, the central issue is the balance between enhanced Coulomb interactions and reduced multiexciton density of states. First-principles work on Si nanocrystals showed ultrafast CM lifetimes and a strong passivation dependence: OH termination lowers the threshold by reducing the gap from $\mathcal{N}(t)$16 eV to $\mathcal{N}(t)$17 eV, but H termination yields more efficient CM when compared on a relative excess-energy scale (Marri et al., 2018). For PbSe and PbS, interband exciton-scattering calculations concluded that direct biexciton photogeneration is only a minor part of QE and that multiple impact-ionization events during phonon-assisted relaxation dominate total QE. On an absolute photon-energy scale, bulk outperforms nanocrystals because the reduction in biexciton DOS from confinement outweighs the modest Coulomb enhancement; on the normalized $\mathcal{N}(t)$18 scale, nanocrystals regain an advantage relevant to photovoltaics (Velizhanin et al., 2011).

5. Device relevance and thermodynamic context

The appeal of CM is straightforward: if high-energy photons yield more than one collected pair, photocurrent can rise beyond the single-exciton limit. Reviews of CM-enabled photovoltaics cite an ideal single-junction efficiency increase from about $\mathcal{N}(t)$19 to about $\mathcal{N}(t)$20 when the absorber gap lies in the rough range $\mathcal{N}(t)$21–$\mathcal{N}(t)$22 eV and the threshold approaches $\mathcal{N}(t)$23 (Maiti et al., 2020). That motivates the emphasis on narrow-gap systems such as PbSe networks, mixed Sn/Pb halide perovskites, and MoTe$\mathcal{N}(t)$24, where both threshold and carrier extraction appear more favorable than in earlier CM platforms (Maiti et al., 2020).

Detailed-balance analyses, however, impose important constraints. A thermodynamic comparison between carrier multiplication and external down-conversion showed that the two are not equivalent: down-conversion gains an additional open-circuit-voltage term,

$\mathcal{N}(t)$25

whereas in the corresponding CM model the enhancement cancels out of the open-circuit balance, yielding $\mathcal{N}(t)$26. The implication is an entropic penalty for performing the spectral splitting internally rather than before the cell (Abrams et al., 2011).

A newer TMD-specific detailed-balance framework reaches a related conclusion in the combined CM/hot-carrier setting. There CM is represented by an energy-dependent multiplicity function $\mathcal{N}(t)$27, with a piecewise-linear Beard-type model and an optimistic upper-bound $\mathcal{N}(t)$28. The framework concludes that CM and hot-carrier extraction draw on the same above-gap energy reservoir; therefore CM does not raise the reversible hot-carrier limit. Instead, CM can help only under finite cooling leakage by shifting excess-energy utilization from a cooling-sensitive voltage channel into current (Lee, 1 May 2026). Under one-sun AM1.5G illumination, this logic is devastating for wide-gap monolayer TMDs: for monolayer WSe$\mathcal{N}(t)$29 with $\mathcal{N}(t)$30 eV, only about $\mathcal{N}(t)$31 of above-gap photons satisfy $\mathcal{N}(t)$32, implying only about $\mathcal{N}(t)$33 idealized short-circuit-current gain before nonidealities (Lee, 1 May 2026).

Transport devices reveal a different CM role. In graphene field-effect transistors, impact ionization can become relevant at moderate fields because the band gap vanishes. A self-consistent rate model with

$\mathcal{N}(t)$34

produced a drain-current “up-kick” at high bias and showed that carrier generation redistributes the electric field and reduces the velocity increase near the drain (Pirro et al., 2012). This is CM as field-induced pair generation rather than photon-to-multiple-pair conversion, but it underscores the broader point that pair-number-changing Auger physics can influence both optoelectronic and transport observables.

6. Ambiguities, controversies, and open problems

The CM literature is technically mature but still contains major interpretive cautions. One recurring issue is whether extra biexcitons are generated promptly during optical excitation or later during relaxation. Interband exciton-scattering calculations for PbSe concluded that the usual photogeneration pathways exhibit complete lack of interference, allowing biexciton photogeneration to be reinterpreted as a single impact-ionization event on the dephasing timescale. In that treatment, prompt photogeneration contributed only about $\mathcal{N}(t)$35 of total QE, while multiple impact-ionization events during phonon-induced relaxation dominated the final yield (Velizhanin et al., 2010). The later PbSe/PbS study retained the same conclusion and further showed that confinement lowers QE on the absolute photon-energy scale because the reduction in biexciton DOS overwhelms the weak Coulomb enhancement (Velizhanin et al., 2011).

A second issue is that indirect observables can be contaminated by unrelated ultrafast effects. In Bi$\mathcal{N}(t)$36Se$\mathcal{N}(t)$37, the authors devoted substantial effort to separating CM from transient surface photovoltage. The key argument was that surface photovoltage develops on the pump-probe cross-correlation timescale and cannot explain carrier-population extrema delayed to $\mathcal{N}(t)$38 fs; moreover, it cannot explain the continued increase in valence-band holes. Even so, the extracted CM factor remained an estimate because the conduction-band signal probably overestimates the true multiplication while the valence-band signal underestimates it (Herb et al., 2024).

A third ambiguity concerns whether device-level gain should be conflated with intrinsic ultrafast CM. Moiré graphene superlattices with $\mathcal{N}(t)$39 twist were reported to yield device-level hot-carrier multiplication gains of $\mathcal{N}(t)$40, far above the “below 5” practical benchmark cited for most 2D photodetectors, and an overall gain of $\mathcal{N}(t)$41 when combined with silicon avalanche multiplication. Yet the same work states that the intrinsic spectroscopy-derived CM process peaks around $\mathcal{N}(t)$42 fs and approaches zero after about $\mathcal{N}(t)$43 ps. The much larger gain therefore depends not only on Auger multiplication but also on a hot-phonon bottleneck, long accumulation times, and secondary ballistic avalanche in silicon (Du et al., 10 Mar 2026). This suggests that large detector gain and large intrinsic CM should not be treated as interchangeable quantities.

Finally, even when CM is well established microscopically, extraction remains the practical bottleneck. In strongly $\mathcal{N}(t)$44-doped graphene the multiplied hot carriers persist only for about $\mathcal{N}(t)$45 fs before cooling dominates (Johannsen et al., 2016). In ordinary graphene more generally, the lack of a band gap makes photovoltaic harvesting difficult, which is one reason Landau quantization was proposed as a route to preserve Auger-enabled multiplication while creating an extraction-relevant gap (Wendler et al., 2014). A plausible implication is that future progress will depend less on demonstrating CM per se than on jointly optimizing threshold, cooling suppression, and ultrafast extraction in architectures where multiplied carriers remain distinguishable from heating, charging, and secondary gain.