# Fundamentals of carrier diffusion waves in electronic solids

Andreas Mandelis¹,²*, Qiming Sun¹,², Alexander Melnikov¹

¹Center for Advanced Diffusion-Wave and Photoacoustic Technologies (CADIPT),
University of Toronto, Toronto, Canada

²School of Optoelectronic Information, University of Electronic Science and Technology of China,
Chengdu, China

*mandelis@mie.utoronto.ca

Photocarriers in semiconductors excited by modulated laser sources give rise to charge diffusion waves that can be used to study and characterize the electronic transport properties of materials and devices. In this talk the concept of carrier diffusion waves (CDW) will be introduced for continuous-band semiconductors (e.g. Si); and of hopping diffusion waves in nanolayers (e.g. colloidal quantum dot (CQD) excitonic ensembles). The fundamental transport equations describing these coherent oscillations will be presented with focus on the key CDW parameters: wavenumber, diffusivity, drift velocity, mobility, diffusion length and surface/interface diffusion currents determined by recombination velocities and flux transfer coefficients [1,2].

*In continuous-band materials* CDW transport is limited by band-to-band recombination and band-to-defect state decays with relaxation times controlled by bandgap defect and impurity state trap densities interacting with the free carrier bandedges through thermal emission and capture rates. The CDW concept has given rise to new material and device diagnostic methodologies, specifically photocarrier radiometry (PCR) [3] and its imaging analog, lock-in carrierography (LIC) [4]. Examples of PCR and LIC applications to the measurement and imaging of diffusion-wave parameters in micro- and optoelectronic materials and devices will be discussed.

*In discrete-particle nanolayers* (CQD ensembles, in particular), very recent formulations of the hopping CDW theory leading to physical insights into diffusion and drift mechanisms in band-energy-equivalent structures consisting of CQD will be presented with focus on how excitonic and/or dissociated carrier hopping diffusion determines anomalous S-shaped current-voltage characteristics of state-of-the-art nanolayered solar cells. The effects of the presence of electrodes in device quality will be identified by means of ultra-high-frequency LIC images revealing compromises in carrier diffusivities and diffusion lengths at the contract-nanolayer interface.

## References

[1] A. Mandelis: *Diffusion-Wave Fields*, Springer, New York 2001, Chap. 9.

[2] A. Mandelis, L. Hu, J. Wang: *Quantitative measurements of charge carrier hopping transport properties in depleted heterojunction PbS colloidal quantum dot solar cells from temperature dependent current-voltage characteristics*. RSC Adv. **6**, 93180 (2016).

[3] A. Mandelis, J. Batista, D. Shaughnessy, *Infrared photo-carrier radiometry of semiconductors: Physical principles, quantitative depth profilometry and scanning imaging of deep sub-surface electronic defects*. Phys. Rev. B **67**, 205208 (2003).

[4] A. Melnikov, A. Mandelis, J. Tolev, P. Chen, S. Huq, *Infrared lock-in carrierography (photocarrier radiometric imaging) of Si solar cells*. J. Appl. Phys. **107**, 114513 (2010).