What can transient absorption spectroscopy reveal about the trap distribution in a semiconductor?

Highlights

• Only a very small trap energy window around the Fermi level is actually probed in the experiment.

• Vastly different trap distributions give rise to virtually identical kinetic traces.

• Most of the trap distributions lead to kinetic traces very well fitted by either a single or a double-exponential.

• There is an approximate scaling between kinetic traces obtained at nearby temperatures that have not been explored experimentally.

• An upper limit for the average trap capture cross section can also be found when the kinetic trace is mono-exponential.

Abstract

We use a master equation to model the electronic relaxation process following the absorption of a pump pulse. The relaxation process is assumed to be dominated by trapping of conduction band electrons in shallow trap levels. We solve the master equation and obtain an expression for the experimentally accessible kinetic trace of the conduction band electronic population in terms of the trap energies and the equilibrium Fermi level perturbed by the pump pulse. The expression reveals the small window of trap energies that actually contributes to the signal and, investigating different trap distributions, we argue that basically the kinetic trace will be very well fitted either by a single-exponential, when all traps lie below the Fermi level, or by a double-exponential (and, naturally, even better by a triple-exponential) when some of the traps are above the Fermi level. The rates obtained do not allow one to reconstruct the trap distribution, but the equilibrium Fermi level and, in the case of single-exponential decays, an upper limit for the capture cross-section may be found.

Introduction

Shallow traps directly affect the transport of charge carriers, and hence the performance, of many semiconductor based devices, see for instance the cases of TiO₂-based photocatalysts [1], [2], [3], [4], [5], [6], [7] and dye sensitized solar cells [8], [9], [10], [11], [12]. It is essential to determine the spatial and energetic distribution of traps in order to access how a certain sample growth procedure or an after-growth treatment affect the device performance.

Transient absorption spectroscopy (TAS) [13] is a widely used method to investigate the trapping dynamics of electrons and holes [8], [14], [15], [16], [17], [18], [19], [20], [21] in semiconductors. The temporal resolution for this technique is of the order of 100 fs, providing information about the electronic relaxation in an ultrafast time scale.

In this experiment an initial pump pulse creates transient excited species which are analyzed employing a white light probe beam to study the absorption at different wavelengths (UV–vis–NIR) as a function of time. If at least one of the wavelengths can be associated with the conduction band (CB) absorption, the corresponding kinetic trace reveals the relaxation of the CB population. In the supplementary material we comment on the main processes involved following the pump pulse.

It is fairly common for the kinetic traces in TAS experiments to be fitted by sums of up to three exponentials [20], [22], [23], [24], [25], suggesting the presence of discrete trap levels without attempting to pinpoint their energy or addressing the role of the disturbed Fermi level. To properly interpret the TAS kinetic traces a thorough theoretical description of the electronic relaxation process in such experiment is needed, one that clearly establishes what exactly the CB kinetic trace reveal about the trap energetic distribution. Our work addresses this fundamental question.

We model the relaxation between the CB and the shallow traps with a master equation [26], assuming that the pump pulse is of such low intensity that: (i) we can neglect any second-order processes like direct electron-hole recombination and (ii) the relaxation can be treated in the near-equilibrium approximation. As argued in Refs. [18], [27], the weak-excitation regime is the relevant one when using TAS to study photocatalysis.

We solve the master equation and obtain an integral equation relating the CB kinetic trace to the trap distribution and to the equilibrium Fermi level. The analysis of this expression reveals that: (i) at room temperature, the traps that actually contribute to the kinetic trace have energies in a range of 260 meV centered around the equilibrium Fermi level, (ii) very different trap distributions lead to nearly identical kinetic traces, (iii) trap distributions entirely concentrated below the Fermi level give rise to kinetic traces that can be perfectly fitted by single-exponentials, whereas other types of distributions produce kinetic traces almost perfectly fitted by double-exponentials (and certainly not by single-exponentials), (iv) in the first case the decay rate reveals the Fermi level, not any feature of the trap distribution, and provides an upper limit for the average trap capture cross-section; in the second case the two decay rates inform nothing about the trap distribution but the Fermi level can still be obtained exploiting an approximate scaling between kinetic traces on nearby temperatures. In both cases it is essential to measure the kinetic trace at more than a single temperature.