Minority and Majority Charge Carrier Mobility in Cu2ZnSnSe4 revealed by Terahertz Spectroscopy

The mobilities of electrons and holes determine the applicability of any semiconductor, but their individual measurement remains a major challenge. Here, we show that time-resolved terahertz spectroscopy (TRTS) can distinguish the mobilities of minority and majority charge carriers independently of the doping-type and without electrical contacts. To this end, we combine the well-established determination of the sum of electron and hole mobilities from photo-induced THz absorption spectra with mobility-dependent ambipolar modeling of TRTS transients. The method is demonstrated on a polycrystalline Cu2ZnSnSe4 thin film and reveals a minority (electron) mobility of 128 cm2/V-s and a majority (hole) carrier mobility of 7 cm2/V-s in the vertical transport direction relevant for light emitting, photovoltaic and solar water splitting devices. Additionally, the TRTS analysis yields an effective bulk carrier lifetime of 4.4 ns, a surface recombination velocity of 6 * 104 cm/s and a doping concentration of ca. 1016 cm−3, thus offering the potential for contactless screen novel optoelectronic materials.


Calibration of TRTS transients
The presented method relies on the accurate measurement of the transients and TRTS uses a mechanical delay of the pump beam to realize time resolution. However, this mechanical delay can result in a change of pump spot size or the overlap of THz probe and optical pump pulse which obscures the original transient. To account for these effects we calibrated TRTS-transient on a Si wafer sample with a µs lifetime. Fig. S1

Uncertainty estimation
The uncertainties in the modelled properties Dam, S and Tb are estimated from the uncertainties of ±10% in the input parameter which are the absorption coefficient at 400 nm α400nm and 800 nm α800nm and the thin film thickness d. Table S1 shows that a change of ±10 % in the absorption coefficients causes a relative large change in the derived Dam and S of up to 38 %. The combined uncertainties of S and Dam are ca. ±50 %.
The uncertainty in the hole mobility µh is dominated by the uncertainty in Dam of ±50 %. The uncertainty in the election mobility µe is dominated from the uncertainty in the sum mobility µe+h of ca. 25 %.
Supplementary Tab. S1: Uncertainty of the properties derived by the TRTS-transient modelling.

Estimate of doping from injection dependent transients
Modelling injection dependent transients at intermediate injection levels Δn ≈ p0 can in principle yield the doping concentration p0 which is a key property of semiconductors, and its contactless derivation is very desirable.
However, modelling the injection dependence is complex and requires combining the continuity equations  Fig.3 now shown with a common global modelling (dotted line) for a) and b) based on equations (3,4,5,S1,S2) and labelled with the initially excited peak carrier concentration at the sample surface Δnpeak. The TRTS-transients start to differ from the high injection behaviour when they have decayed to ca. 10 16 cm -3 which is an estimate for the doping concentration.

Fig. S2: Global injection-dependent modelling of TRTS-transients. Same injection dependent TRTS-transients as in
A qualitative agreement to the measured TRTS transients is found for p0 = 10 16 cm -3 , Sh = 7×10 4 cm/s, Se = 3×10 5 cm/s, = 5 ns, µe = 129 cm 2 /Vs, µh = 6 cm 2 /Vs and pt = pt' = ℎ = 0 as shown by the dotted lines in Fig.S2. However, due the large amount of 9 parameters and the limited computation power we cannot exclude that another parameter combination reproduces the measured transients with a smaller the standard error than this parameter set. Additionally a more advanced model should include a distribution of tail states instead of a single discrete defect level and also trapping effects which have been shown to increases the effective bulk lifetime τB particularly in kesterites [2]. On the flipside, these more advanced models require even more parameters and modelling may become unambiguous.
Still the general trend of the injection dependence is clear and exhibits the transition from injectionindependent transients in high injection to the injections dependence at Δn ≈ p0 ≈ 10 16 cm -3 . These general trend described by equations (5,S1,S2) is caused by the transition from the dominance of the minority carriers in low injection to the impact of both carrier types in high injection. In the case of the p-type kesterite sample these dynamics are given by Dam ~ µe, S = Se and = in low injection and Dam ~ 2µh, S -1 = Se -1 + Sh -1 and = + ℎ in high injection. For the estimation of p0 it is not relevant if the change in the transients is due to a change in diffusion, effective bulk life time or surface recombination velocity. It is only relevant that all these changes take place at Δn ≈ p0 ≈ 10 16 cm -3 . Therefore, this value maybe taken as a rough estimate of the doping and is in line with the value derived from capacitance-voltage measurements on the completed solar cell.