We show that charge carrier mobilities can be measured by reflection time resolved THz spectroscopy (R-TRTS) even for thin films on metal contacts, such as polycrystalline Cu2SnZnSe4 grown on molybdenum. In the measurement a reduced THz reflection upon photo-excitation is observed in contrast to increased THz reflection commonly observed on insulating substrates, and which excludes standard analytic R-TRTS analyses. Instead, a numerical transfer matrix method is used to model the THz reflection from which we derive carrier mobilities of 100 cm2/Vs consistent with literature. We show that R-TRTS on metal substrates is ~100x less sensitive compared to measurements on insulating substrates. These sensitivity of these R-TRTS measurements can be increased by using lower substrate refractive indices, lower substrate conductivities, thicker sample layers or higher THz probe frequencies.
© 2017 Optical Society of America
Charge carrier mobility and lifetime are key properties of semiconductor materials. However, often semiconductors are grown on conductive substrates, which prohibits mobility measurements by Hall-effect or transmission mode time resolved THz spectroscopy. For such samples reflection mode time resolved THz spectroscopy (R-TRTS) is a promising alternative. R-TRTS has been applied previously to retrieve the mobility of perovskite single crystals , polymer-fullerene films  or of nano-crystalline silicon  and GaAs crystals . However, it has so far never been applied to highly conductive substrates like metals or transparent conductive oxides which are used in many commercial applications such as photovoltaics, light emitting diodes and photo detectors. In this work we will focus on THz measurements of semiconductors grown on highly conductive substrates and discuss the challenges arising from the analysis of such experiments. Once retrieved, the THz mobility can be used on the one hand to extrapolate the DC charge carrier mobility and on the other hand to elucidate the nature of charge carriers by analyzing the dependence of the mobility on the THz-frequencies [5–10]. From the specific spectral signature, free electrons and holes can be distinguished from localized electrons and holes as well as from excitons or polarons. Free carriers exhibit a Drude-like mobility spectrum from which fundamental properties such as the momentum relaxation time as well as the effective carrier mass can be obtained . Localized charge carriers often exhibit a Drude-Smith-like mobility spectrum which can be used to study charge carrier localization in nanoparticles [6,7] or polycrystalline grain boundaries . Excitons  and polarons [10,11] exhibit Lorentz resonances in the mobility spectrum, from which their binding energies can be deduced. Further, TRTS not just offers the possibility to determine the kind of charge carriers transport in the sample it also can probe their dynamics as a function of pump-probe delay with a resolution of ~100 fs .
In this work we will first describe R-TRTS and its analytical and numerical analysis. Then differences in the substrate refractive indices for insulating and conductive substrates will be evaluated. Finally, the impact of the substrates on the signal amplitude as well as the derived mobility is explained on the basis of a comparison between Cu2SnZnSe4 thin film on Mo and on glass.
2. R-TRTS setup and analysis
Transmission TRTS setups [13,14] and static time domain terahertz spectroscopy setups in reflection mode [15,16] are well known. The convertible transmission and reflection TRTS setup used in this work (Fig. 1) is a combination of both types and its reflection mode is similar to the reflection OPTP setup described in [1–3].
A laser pulse with 800 nm central wavelength, 70 fs pulse width and 150 kHz repetition rate from an amplified Ti-Sapphire system is split into three beams which are used for THz generation, THz detection and the optical pumping of the sample. The THz pulse is generated by optical rectification in a ZnTe crystal. It is guided by parabolic mirrors through a silicon beam splitter (BS) onto the sample. The reflected THz Pulse is detected by electro-optical sampling in a second ZnTe crystal. By removing the beam splitter (BS) and the mirror (M) in Fig. 1 the setup can be converted into a transmission TRTS setup. Both pump and THz beam are chopped (CH) at individual frequencies which allows to record the electric field of the THz pulse E and its pump-induced change ΔE with a lock-in amplifier simultaneously. The delay line (DL) of the optical pump beam defines the optical-pump THz-probe delay and the delay line in the THz generation scans the THz pulse vs. the sampling pulse which resolves the THz fields E(t) and ΔE(t) in time.
The Fourier transformed relative change of the reflected THz pulse ΔR/R(ω) = (Rex-R)/R is related to the conductivity Δσ(ω) of the pump induced charge carriers in the sample. Rex and R are the reflected THz fields of the excited and unexcited samples. Under known excitation conditions the complex charge carrier mobility µ(ω) in the THz regime can be retrieved. For this analysis analytic approximations or a numerical methods can be used. Analytic approximations for R-TRTS are used in [1–4].
Equation (1) was originally derived for a thick wafer sample with refractive index n were the internal reflections yield separated pulses which can be cut out by a time window. It retrieves the mobility µ as a function of the angular frequency ω of the THz probe pulse. ΔN and Δσ are the photo-induced charge carrier concentration and conductivity at the surface of the sample and the carrier excitation follows the Lambert-Beer law with absorption coefficient α. The second factor in Eq. (1) vanishes for absorption lengths 1/α much smaller than the shortest wavelength present in the THz pulse which is used in [1–3] and the third factor reduces to ΔR/R for small induced reflection changes ΔR which is used in . Both simplifications together yield the so called thin film approximation (last part of Eq. (1). Although only strictly valid for thick bulk samples Eq. (1) was used to analyze thin film sample geometries on insulating substrates [2,3]. Therefore, in this work we will investigate if Eq. (1) can also be used as a first order approximation for semiconductor thin films on highly conductive substrates. In reference  more general expressions of form ΔR(Δσ) are given which have to be solved numerically for Δσ. They are not further regarded here as the transfer matrix method is our numerical method of choice to account for the multiple internal reflections of the THz probe pulse in the thin film sample.
The numerical transfer matrix method has the advantage that it can handle reflection as well as transmission TRTS signals for a wide variety of different sample geometries. It models the reflection of a THz pulse at the photo-excited sample  and optimizes an assumed mobility until the modeled reflection fits the measured reflection . The reflection is calculated using the transfer matrix method which is described in literature  and was applied to TRTS analysis previously [4,20,21]. It connects the electric THz fields in front of and behind the studied layers and can handle multi-layer systems which are represented by the product of the matrices of the individual layers. Thereby it accounts for all internal reflections of the multi-layer systems. The transfer method can also model an inhomogeneous carrier concentration such as a Lambert-Beer like carrier profile present in the sample directly after photoexcitation. To do so, the inhomogeneously excited layer is cut into a series of homogeneous sublayers with a stairway-like decrease of carrier concentration. Therefore, the analysis utilizing the transfer matrix method can be regarded to give more accurate results than the approximation in Eq. (1).
3. Input sample properties for R-TRTS analysis
Photovoltaic thin films are usually deposited on conductive contacts and are one of the possible applications of R-TRTS. Cu2SnZnSe4, in particular, attracts great interest as thin film photovoltaic material that reaches efficiencies up to 12.7% . Therefore, to demonstrate the R-TRTS of a semiconductor thin film on a metal substrate, we chose a 2.7 µm thick Cu2SnZnSe4 thin film on a 700 nm Mo coated glass substrate. The Cu2SnZnSe4 thin film was deposited by sequential sputtering of metal precursors and subsequent selenization . For comparison with a sample that has a low conductive substrate we also measured a similar Cu2SnZnSe4 thin films on glass. To probe the limits of the approximations assumed for deriving Eq. (1) and to show the broad applicability of R-TRTS in combination with the transfer matrix analysis we also probed a 0.55 mm thick compensated InP wafer in the same setup. To analyze R-TRTS measurements by means of the transfer matrix analysis the excited charge carrier profile as well as the refractive indices and thicknesses of all sample layers have to be known. The carrier profile after excitation follows the Lambert-Beer law with absorption coefficients of 4.4*104 cm−1 for Cu2SnZnSe4  and 2*104 cm−1 for InP  at 800 nm pump wavelength. The refractive indices in the THz regime for the samples together with other frequently used substrates are shown in Fig. 2. They were measured by transmission time domain THz spectroscopy (TDTS) [15,16]. As transmission TDTS requires samples on transparent substrates and no such Cu2SnZnSe4 sample was available we used a 1 mm thick Cu2SnZnS4 pellet . It has the same kesterite structure and is expected to have similar properties as the Se based counterpart in which the S atoms are replaced by Se atoms. The other refractive indices in Fig. 2 were measured for a sputtered 20 nm Mo thin film on quartz, a 2 mm epoxy film and a 1 mm glass slide.
The low conductivity materials InP, Cu2ZnSnS4, AlO3 and epoxy exhibit almost frequency independent and purely real refractive indices of 3.6; 3.35; 2.8 and 1.65, respectively, due to the lack of free carrier or phonon absorption in the lower THz region. In contrast, the refractive index of Mo is strongly frequency dependent and has a very large amplitude of ~100 at 1 THz with equal real and imaginary parts. It is described by Eq. (2) in which the free carrier contribution with conductivity σ dominates over the static refractive index nst.
If the refractive index of Mo is modelled by Eq. (2) a conductivity of 6.2 ± 0.5 *105 Ω−1m−1 is derived. On the other hand Eq. (2) can be used to estimate the refractive index of a metal substrate if its conductivity is known. A conductivity measurement of the substrate may be more convenient than a TDTS measurement of its refractive index.
4. Origin and amplitude of R-TRTS signals
The difference of one order of magnitude in the refractive indices for insulating and metal substrates will affect the THz reflection in the R-TRTS measurement. First, this will be shown experimentally by comparing a Cu2SnZnSe4 thin film on Mo and on glass. Second, the different R-TRTS signals will be numerically reproduced and modeled as function of the substrate refractive index. Finally, a qualitative explanation will be is given.
The R-TRTS measurements of Cu2SnZnSe4 on Mo and glass substrates are shown Fig. 3. They were performed 5 ps after pump excitation to prevent carrier diffusion which preserves the initial Lambert-Beer like carrier profile. This R-TRTS measurements show a decreased THz probe reflection for Cu2SnZnSe4 thin film on Mo and an increased THz probe reflection for the Cu2SnZnSe4 thin film on glass which indicates a fundamental difference in the R-TRTS measurement due to the different substrates. Further, although the same pump intensity of 0.15 W/cm2 (4*1012 photon/cm2 at 150 kHz) was used for both R-TRTS measurements the signal amplitude for the Cu2SnZnSe4 thin film on Mo is 1-2 order of magnitude lower (magnified by a factor of 20 in Fig. 3) and further decreases for lower THz probe frequencies.
To find the reason why the pump induced change in THz reflection ΔR/R is negative and relatively small for the metal substrate we simulated the R-TRTS signal ΔR/R by the transfer matrix method as a function of the substrate refractive index nsubstrate (Fig. 4).
The modelling shows lower R-TRTS signals |ΔR/R| for substrates with higher refractive indices. This dependency on the refractive index is also roughly predicted by the analytic thin film approximation (Eq. (1). However Eq. (1) agrees with the transfer matrix approach only for substrates with roughly the same refractive index as the overlaying thin film (2.7 in the simulation). This case corresponds to a wafer sample which is its own substrate and for which Eq. (1) was originally derived. The thin film approximation in Eq. (1) should therefore only be used in the region indicated in Fig. 4. This Figure also resolves the real and imaginary contributions to ΔR/R. A transition from an increased reflection (real part of ΔR/R >0) to a decreased reflection (real part of ΔR/R <0) as well as a transition from later arrival (imaginary part of ΔR/R >0) to an earlier arrival of the reflected THz pulse (imaginary part of ΔR/R >0) is found. Therefore, the different refractive indices of glass (n~2.5) and Mo (n~75 + 75i at 1 THz) are identified as the reason for the lower amplitude and negative sign of the ΔR/R signal on Mo substrates observed in Fig. 3. The even lower ΔR/R signal at lower THz frequencies for Cu2SnZnSe4 thin film on Mo as seen in Fig. 3 is partially an effect of the Mo refractive index which increases for lower frequencies (Fig. 2) and therefore reduces the R-TRTS signal according to its dependency on the substrate refractive index (Fig. 4). Additionally, the thin film geometry of the sample induces a lower ΔR/R signals for lower THz frequencies (not shown here).
The transition from a strong increase to a weak decrease in THz reflection upon photoexcitation indicates a change in the dominating effect depending on the substrate refractive index. In general, the photoexcitation induces charge carriers which absorb THz radiation depending on their mobility. But also the real part of the refractive index changes via the Kramers-Krönig relation when the absorption is changed. The increased THz reflection for samples on glass and other insulating substrates is an effect of the photo-increased refractive index which increases the THz reflection at the air to semiconductor interface and at the graded interface of excited semiconductor to unexcited semiconductor . When the substrate refractive index reaches high values this contribution becomes less effective because the reflection of the unexcited sample already approaches unity. In the extreme case of a total reflection at the semiconductor to substrate interface an increased reflection at the air to semiconductor interface wouldn’t influence the amplitude of the overall reflection anymore. It would only shift the reflected THz pulse slightly to earlier times. The Cu2SnZnSe4 thin film on Mo is close to such a case. In this case the reflection of the THz probe is found by experiment (Fig. 3) and modelling (Fig. 4) to decreases upon photoexcitation and to shift to earlier times. This can be attributed to the absorption of the internal THz reflections which contribute to the overall reflection in the thin film geometry. Further, a delay of this internal reflections by an increased refractive index shifts their interference condition away from the constructive case and also decreases the overall THz reflection.
To give a guideline as to which thin films samples on conductive substrates can be measured by R-TRTS the signal |ΔR/R| (f = 1 THz) was simulated by the transfer matrix method for a semiconductor thin film (n = 2.7) on a typical metal substrate with a refractive index of 75 + 75i. With decreasing layer thickness d and mobility µ the R-TRTS signal decreases and typical empirical noise levels for R-TRTS measurements of 1 minute and of 1 day are shown in Fig. 5.
The measurements shown in Fig. 3 for example were completed within three minutes which is much shorter than for example mobility measurements by Hall effect where the sample has to be electrically contacted. If higher R-TRTS signals are desired higher THz probe frequencies or higher pump intensities than assumed for the simulation in Fig. 5 can be used. However, heating of the sample by the pump beam should be prevented and the pump spot size should always be large compared to the longest wave length in the THz probe pulse (we use 0.3 cm FWHM).
5. Retrieved charge carrier mobilities
Finally it is to be verified, that R-TRTS measurements and the transfer matrix analysis retrieve the same charge carrier mobilities for thin films on insulating and metal substrates and that the extracted mobilities for wafers are in line with Eq. (1).
Indeed, the mobilities derived by the numerical transfer matrix method agree closely for the Cu2SnZnSe4 thin film on glass and on Mo as shown in Fig. 6. It is highlighted that the corresponding measured signals ΔR/R in Fig. 3 were completely different in amplitude and frequency dependency but the retrieved charge carrier mobilities are very similar. The derived frequency dependent mobility can be extrapolated to a DC-value of ~100cm2/Vs which is also in agreement with DC mobilities of 70-100 cm2/Vs reported from transmission TRTS measurements of Cu2SnZnSe4 on transparent substrates [27,28]. Further, the transfer matrix derived mobility shows a Drude-Smith like frequency dependency  with an increasing real part and a negative imaginary part which indicates carrier localization and was previously reported for Cu2SnZnSe4 in . This demonstrates the applicability of R-TRTS in combination with the numerical transfer matrix analysis to thin film semiconductors on metal substrates. In contrast, the mobility calculate analytically by Eq. (1) for the thin film on Mo yields unphysical negative mobilities of −3000-10000i cm2/Vs at 1 THz and shows again that Eq. (1) is not suited for this thin film geometry as already concluded from SEC IV. The minor differences between Mo and glass substrate in Fig. 6 for frequencies above 1.5 THz can be attributed to the uncertainty in the refractive index of the used glass substrate in this region (Fig. 2) which was set to 2.55 for the analysis. Another major contribution to the uncertainty is the surface roughness which is not regarded here to minimize the number of parameter. Additionally the Cu2ZnSnSe4 growth on Mo and glass substrate may be different and cause real differences in the charge carrier mobilities of these samples.
For an InP wafer the carrier mobilities derived by the transfer matrix method and the approximation of Eq. (1) agree with each other as shown in Fig. 7. It is found that the deviations are within 5%, which is equal to the reported deviation between transfer matrix method and the slab-like approximation for transmission time resolved THz spectroscopy (T-TRTS) . The mobility of InP is well described by the Drude formula µ = eτ/meff/(1 + iωτ) with an effective mass meff of 0.095 and a momentum relaxation time τ of 117 fs which proves a non-localized carrier transport and is in line with former transmission TRTS measurements on InP . In summary we have shown that R-TRTS in combination with the numerical transfer matrix analysis is well-suited for charge carrier mobility measurements of thin films on insulating and metal substrates as well as of wafer samples.
We have demonstrated R-TRTS measurements and mobility analysis for a 2.7 µm thick Cu2SnZnSe4 thin film on Mo and verified the validity of the obtained result with measurements of a Cu2SnZnSe4 thin film grown on a glass substrate. The mobilities of ~100 cm2V−1s−1 retrieved by a numerical transfer matrix method agree better than 10% with each other and with literature values. Previously reported analytical thin film approximations are shown to apply only to thin films on substrates with similar refractive indices which is not fulfilled for a semiconductor thin film on metal. Especially the observed reduced THz reflection upon photoexcitation is beyond the thin film approximation. This reduced THz reflection is a result of the absorption and delay of internal THz reflections by photoexcited charge carriers and can be modeled numerically to derive the charge carrier mobility. The modelling and the derived mobility depend also on other sample properties which are the main sources of uncertainty of the method and set the resolution limit of R-TRTS. In general the R-TRTS signal is shown to decrease for higher substrate refractive indices, higher substrate conductivities, thinner sample layers, lower pump intensities and lower THz probe frequencies. Also the thin film refractive index, surface roughness and carrier distributions have some impact. In particular the refractive index of a conductive substrate in the THz regime cannot be estimated by literature values as it is governed by its individual conductivity and becomes complex valued as well as strongly frequency dependent. A metal as substrate reduces the signal amplitude by 1-2 orders of magnitude compared to insulation substrates and limits R-TRTS resolution typically to film thicknesses and mobility combinations of 100 nm and 50 cm2/Vs or 5 µm and 0.1 cm2/Vs.
Following the results of this work, charge carrier mobilities of semiconductor thin films grown on conductive substrates as in photovoltaics devices, light emitting diodes or photon detectors can be retrieved with high throughput by the contactless R-TRTS.
Helmholtz Association Initiative and Network Fund (HNSEI Project SO-075).
The authors thank Anna Ritscher for providing the Cu2ZnSnS4 pellet, Karsten Harbauer for providing the sputtered Mo thin film on quartz and Alex Redinger for the Cu2ZnSnSe4 thin film on Mo and glass.
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