Bronson Philippa

Photocarrier drift distance in organic solar cells and photodetectors

Light harvesting systems based upon disordered materials are not only widespread in nature, but are also increasingly prevalent in solar cells and photodetectors. Examples include organic semiconductors, which typically possess low charge carrier mobilities and Langevin-type recombination dynamics – both of which negatively impact the device performance. It is accepted wisdom that the “drift distance” (i.e., the distance a photocarrier drifts before recombination) is defined by the mobility-lifetime product in solar cells. We demonstrate that this traditional figure of merit is inadequate for describing the charge transport physics of organic light harvesting systems. It is experimentally shown that the onset of the photocarrier recombination is determined by the electrode charge and we propose the mobility-recombination coefficient product as an alternative figure of merit. The implications of these findings are relevant to a wide range of light harvesting systems and will necessitate a rethink of the critical parameters of charge transport.

The impact of hot charge carrier mobility on photocurrent losses in polymer-based solar cells

A typical signature of charge extraction in disordered organic systems is dispersive transport, which implies a distribution of charge carrier mobilities that negatively impact on device performance. Dispersive transport has been commonly understood to originate from a time-dependent mobility of hot charge carriers that reduces as excess energy is lost during relaxation in the density of states. In contrast, we show via photon energy, electric field and film thickness independence of carrier mobilities that the dispersive photocurrent in organic solar cells originates not from the loss of excess energy during hot carrier thermalization, but rather from the loss of carrier density to trap states during transport. Our results emphasize that further efforts should be directed to minimizing the density of trap states, rather than controlling energetic relaxation of hot carriers within the density of states.

Advantage of suppressed non-Langevin recombination in low mobility organic solar cells

Photovoltaic performance in relation to charge transport is studied in efficient (7.6%) organic solar cells (PTB7:PC71BM). Both electron and hole mobilities are experimentally measured in efficient solar cells using the resistance dependent photovoltage technique, while the inapplicability of classical techniques, such as space charge limited current and photogenerated charge extraction by linearly increasing voltage is discussed. Limits in the short-circuit current originate from optical losses, while charge transport is shown not to be a limiting process. Efficient charge extraction without recombination can be achieved with a mobility of charge carriers much lower than previously expected. The presence of dispersive transport with strongly distributed mobilities in high efficiency solar cells is demonstrated. Reduced non-Langevin recombination is shown to be beneficial for solar cells with imbalanced, low, and dispersive electron and hole mobilities.

Analytic solution of the fractional advection-diffusion equation for the time-of-flight experiment in a finite geometry.

A general analytic solution to the fractional advection diffusion equation is obtained in plane parallel geometry. The result is an infinite series of spatial Fourier modes which decay according to the Mittag-Leffler function, which is cast into a simple closed-form expression in Laplace space using the Poisson summation theorem. An analytic expression for the current measured in a time-of-flight experiment is derived, and the sum of the slopes of the two respective time regimes on logarithmic axes is demonstrated to be -2, in agreement with the well-known result for a continuous time random-walk model. The sensitivity of current and particle number density to the variation of experimentally controlled parameters is investigated in general, and the results applied to analyze selected experimental data.

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

The solution of a Caputo time fractional diffusion equation of order 0 < α < 1 is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an N-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from O ( N 2 ) to O ( N α ) , given a precomputation of O ( N 1 + α ln ? N ) . The mapping is applied successfully to the least squares fitting of a fractional advection-diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes. A mapping between fractional diffusion and normal diffusion is explored numerically.An efficient solution method for fractional diffusion equations is proposed.A benchmark fractional-order model fit for Scher-Montroll transport is presented.

Solution of a generalized Boltzmann's equation for nonequilibrium charged-particle transport via localized and delocalized states.

We present a general phase-space kinetic model for charged-particle transport through combined localized and delocalized states, capable of describing scattering collisions, trapping, detrapping, and losses. The model is described by a generalized Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. The velocity of the center of mass and the diffusivity about it are determined analytically, together with the flux transport coefficients. Transient negative values of the free particle center-of-mass transport coefficients can be observed due to the trapping to, and detrapping from, localized states. A Chapman-Enskog-type perturbative solution technique is applied, confirming the analytical results and highlighting the emergence of a density gradient representation in the weak-gradient hydrodynamic regime. A generalized diffusion equation with a unique global time operator is shown to arise, reducing to the standard diffusion equation and a Caputo fractional diffusion equation in the normal and dispersive limits. A subordination transformation is used to solve the generalized diffusion equation by mapping from the solution of a corresponding standard diffusion equation.

Generalized phase-space kinetic and diffusion equations for classical and dispersive transport

We formulate and solve a physically-based, phase space kinetic equation for transport in the presence of trapping. Trapping is incorporated through a waiting time distribution function. From the phase-space analysis, we obtain a generalized diffusion equation in configuration space. We analyse the impact of the waiting time distribution, and give examples that lead to dispersive or non-dispersive transport. With an appropriate choice of the waiting time distribution, our model is related to fractional diffusion in the sense that fractional equations can be obtained in the limit of long times. Finally, we demonstrate the application of this theory to disordered semiconductors.

Third-order transport coefficients for localised and delocalised charged-particle transport

We derive third-order transport coefficients of skewness for a phase-space kinetic model that considers the processes of scattering collisions, trapping, detrapping and recombination losses. The resulting expression for the skewness tensor provides an extension to Fick’s law which is in turn applied to yield a corresponding generalised advection-diffusion-skewness equation. A physical interpretation of trap-induced skewness is presented and used to describe an observed negative skewness due to traps. A relationship between skewness, diffusion, mobility and temperature is formed by analogy with Einstein’s relation. Fractional transport is explored and its effects on the flux transport coefficients are also outlined.

Generalized balance equations for charged particle transport via localized and delocalized states: mobility, generalized Einstein relations, and fractional transport

A generalized phase-space kinetic Boltzmann equation for highly nonequilibrium charged particle transport via localized and delocalized states is used to develop continuity, momentum, and energy balance equations, accounting explicitly for scattering, trapping and detrapping, and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on mobility and diffusivity. Generalized Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field dependence of the mobility. Interesting phenomena such as negative differential conductivity and recombination heating and cooling are shown to arise from recombination loss processes and the localized and delocalized nature of transport. Fractional transport emerges naturally within this framework through the appropriate choice of divergent mean waiting time distributions for localized states, and fractional generalizations of the GER and mobility are presented. Signature impacts on time-of-flight current transients of recombination loss processes via both localized and delocalized states are presented.

A route to high gain photodetectors through suppressed recombination in disordered films

Secondary photocurrents offer an alternative mechanism to photomultiplier tubes and avalanche diodes for making high gain photodetectors that are able to operate even at extremely low light conditions. While in the past secondary currents were studied mainly in ordered crystalline semiconductors, disordered systems offer some key advantages such as a potentially lower leakage current and typically longer photocarrier lifetimes due to trapping. In this work, we use numerical simulations to identify the critical device and material parameters required to achieve high photocurrent and gain in steady state. We find that imbalanced mobilities and suppressed, non-Langevin-type charge carrier recombination will produce the highest gain. A low light intensity, strong electric field, and a large single carrier space charge limited current are also beneficial for reaching high gains. These results would be useful for practical photodetector fabrication when aiming to maximize the gain.