1. Introduction

Terahertz (THz) time-domain spectroscopy (TDS) has developed towards an alternative technique for characterizing electrical transport in condensed matter. Many studies on semiconductors demonstrate excellent agreement with the Drude model and characteristic properties such as the complex conductivity have been deduced [1–5]. Recent reports on organic semiconductors [6], low doped inorganic semiconductors [7], and nanoparticles [8] reported deviations from a pure Drude behavior. Experimental data were reproduced with excellent precision using modified Drude models such as the Drude-Smith model [9], the Cole-Davidson model [10], or by incorporating the Maxwell Garnett theory [11].

All Drude models have in common that scattering is described by the momentum relaxation time τ which makes this time constant one of the most fundamental quantities that control charge transport. Deducing τ from optical data, however, demands knowledge of further parameters, such as of the carriers' density N, the effective mass m*, the background permittivity ε _∞, and the thickness d of the sample [1,12,13]. The thickness d can be deduced in time-resolved experiments when the optical thickness exceeds the wavelength of the radiation [14–16]. However, the progress in material science provides many novel materials, which are extremely thin and have unknown charge carrier densities and effective masses. Besides THz TDS, far-infrared as well as microwave techniques are used for characterizing such modern materials [17,18]. But these approaches require precise knowledge of material parameters or use fitting procedures.

In this work, we present an optical method for probing the Drude relaxation time without using any further parameters. In our technique we modulate the optical properties of the sample and record the differential response signal at THz frequencies. We will show that the phase of this signal directly provides the relaxation time. It is worth pointing out, that our method is not restricted to the THz band. The method can be adapted to any spectroscopic technique, which provides the optical phase of transmitted or reflected radiation.

Small modulations of the sample's optical properties can be achieved most conveniently by electromodulation techniques similar to the method described by Allen et al. [19]. The fact that the modulation of the charge carrier density resembles the operation of a field-effect transistor illustrates the application potential of the technique. Currently, numerous novel materials such as nanostructures or organic semiconductors have been developed for similar devices. Many of these materials, however, are inhomogeneous, include a high density of traps and grain boundaries, or have a broadened density of electronic states. Dispersive transport and hopping dominate electrical properties and mask the fundamental carrier dynamics on microscopic scales [20,21]. In such cases THz modulation spectroscopy may provide further insight into the fundamental carrier relaxation dynamics.

2. Method

The concept of our method is illustrated in Fig. 1a ). We study a thin sheet of a semiconductor, which has the thickness d and contains mobile carriers. Within the part Δd the charge carrier density can be modulated, e.g. the density can be reduced to zero. A fraction of an incident THz electric field E _inc is transmitted through the doped layer, and generates a current density j which is proportional to the complex conductivity σ . In the following, it is convenient to discuss differential quantities such as the differential field ΔE = E ₁ -E ₀ where E ₁ and E ₀ are the transmitted electric fields in the case of an interaction across the entire layer and in the case the layer thickness is reduced by Δd, respectively.

 

Fig. 1 a) Schematic diagram of a thin metallic layer of thickness d embedded in a dielectric with a refractive index n. The differential current Δj driven by the incident field E _inc causes a differential magnetic field ΔH which in turn causes the differential field ΔE measured by THz-TDS. b) Complex diagram illustrating the relation of the phase angle φ with the measured complex fields E _1y, E _0y, and ΔE _y.

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The differential current density Δj can be described by a change of the thickness

In layers which are thin compared with the wavelength ($d\ll \lambda$), the x-components of ΔH dominate the above equation whereas the z-components can be neglected. We also consider that Δd is smaller than the skin depth. In consequence, the modulated current density Δj does not change significantly across the depleted layer. The magnetic field ΔH causes a differential electrical field given by ΔE = Z ₀ΔH/ n, where Z ₀ is the impedance of vacuum and n is the refractive index of the surrounding medium. Figure 1b) displays the fields in the complex plane. Using E _1y as reference provides:

At this point we assume the conductivity of the original Drude model:

Thus, the momentum relaxation time can be directly deduced from φ without the use of further parameters. This includes the change of the layer thickness Δd. Alternatively, the amplitude or intensity of the signal S can be analyzed in case that ωτ ≈1 [24]. In the Hagen-Rubens range (ωτ$\ll 1$), however, the spectra become structureless because the amplitude is proportional to 1/(1 + ω ² τ ²)^1/2. In contrast, the tangent of the phase angle is proportional to τ, which illustrates the strength of our technique when characterizing materials in which ultrafast scattering occurs.

3. Experimental

We tested the method on a GaAs structure grown by molecular beam epitaxy (MBE). It comprises an n-doped layer grown on a semi-insulating GaAs substrate. The layer is doped at 2·10¹⁶ cm⁻³ and has a thickness of 2 µm. For electronic control we fabricated a Schottky contact on top of the structure by depositing a 10 nm thick film of Cr. The second terminal is an ohmic AuGe contact. The doped layer in the GaAs can be partially depleted from electrons by the application of a negative bias V to the Schottky contact. The width of the depletion region follows a square root dependence on V [25], from which the modulated depth Δd can be calculated. In our experiments Δd is of the order of 100 nm. This is much smaller than the skin depth of the electron gas, which is about 20 μm at THz frequencies. Thus, the assumptions made in section 2 are justified. They can be applied to many other devices where conductive sheets are modulated, as for instance, in field-effect transistors.

In our experiments we apply time-resolved THz transmission spectroscopy. The sample is mounted in a cryostat, which allows for low-temperature measurements. A titanium-sapphire laser delivers pulses of 80 fs duration at 780 nm center wavelength and at a repetition rate of 80 MHz. About 700 mW of the laser power is used for the photoexcitation of an interdigitated THz emitter [26,27]. Parabolic mirror optics with NA = 0.3 focus the THz radiation onto the GaAs structure. The transmitted radiation is time-resolved by standard electro-optic sampling in a [110] ZnTe crystal of 1 mm thickness [28]. The setup has a bandwidth of 2.8 THz and the signal to noise ratio is about 10⁵ Hz^1/2. We measured the field strength of the THz pulses using the procedure described in Ref [29]. and deduced a peak field of 70 V/cm. The curve in Fig. 2a ) shows a THz pulse transmitted through the GaAs structure, whereas Fig. 2b) displays the differential signal, which we obtained by modulating the structure with a bias of −5 V. This corresponds to a change of the depletion width of about 400 nm.

 

Fig. 2 a) Time-resolved few-cycle THz pulse after transmission through the metal-semiconductor structure. b) Differential signal obtained by switching the GaAs between equilibrium and partial depletion. The modulation bias of −5 V increased the depletion zone by 400 nm.

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4. Results and discussion

In order to prove the concept of our technique we measured the differential transmission through the GaAs structure at room temperature. The electron gas was modulated by switching the bias to the Schottky contact between 0 V and −1 V. This corresponds to a modulated layer thickness of 120 nm or to a modulated sheet density of n_{2 D} = 2.4· 10¹¹ cm⁻². We recorded the THz signals E _1y and E _0y [30]. After Fourier transform of the differential THz signal ΔE_y = E _1y-E _0y, the optical phase was obtained as described above. The data depicted in Fig. 3 show the expected dependence of the phase on the frequency. Using a relaxation time of 198 ± 2 fs provides an excellent agreement with Eq. (5). The linear behavior of tan φ confirms the validity of the original Drude model in our case. We do not observe deviations from the Drude model as reported for poly(3-hexylthiophene) [6], low doped silicon [7], and nanoparticles [8].

 

Fig. 3 Frequency dependence of the tangent of the phase angle φ obtained on the GaAs structure at room temperature (symbols). The solid line shows a calculation for τ = 198 fs. The dashed line indicates ωτ = 1.

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In section 2 we assumed that only the lateral components of ΔH contribute to Ampere's law whereas the perpendicular components can be neglected. We also assumed that the exciting THz field is virtually constant across the modulated layer. We validate these assumptions by measuring the THz signals in dependence on the modulation depth Δd by increasing the modulation bias V. As displayed in Fig. 4a ) the measured phase shows no dependence on V, which proves the validity of the assumptions. In contrast, the amplitude plotted in Fig. 4b) follows a square root dependence as expected for the increase of the depletion width. The calculation, however, shows deviations for higher voltages. We attribute these differences to imperfections of the Schottky contact and to inhomogeneities of the doping profile. This illustrates the difficulties that arise in deducing transport properties from amplitude data and highlights the reliability of phase measurements.

 

Fig. 4 a) Dependence of the phase angle φ on the modulation bias for room temperature. The data show the phase for frequencies of 1 and 2 THz, respectively. b) Dependence of the amplitude of the electromodulation signal on the modulation bias at 1 and at 2 THz, respectively. The solid lines depict calculations that take into account the depletion underneath the Schottky contacts with increasing bias.

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In the case of our well-defined MBE grown structure, parameters such as the doping density and the effective carrier mass are well known. Thus in this particular case, the momentum relaxation time can be extracted from four-point measurements [31]. Figure 5 shows temperature resolved data. Electrical and phase measurement differ by about 3%, which illustrates that our technique provides precise data. At low temperatures, however, freeze-out of the donors may lead to a reduced electrical conductivity, which explains the differences at about 100 K.

 

Fig. 5 Temperature dependence of the momentum relaxation time τ as extracted from THz phase measurements (symbols). The solid line shows the result of the electrical four-point characterization.

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We emphasize that our approach is not advantageous in the case of homogeneous semiconductors with well known properties. In this case electrical characterization certainly is less elaborate. But many modern materials exhibit unknown charge carrier densities and effective masses. Additionally, many new materials are inhomogeneous. In such systems the mesoscopic morphology may limit electrical transport and in consequence extremely low mobilities are observed by electrical characterization. Examples are polycrystalline semiconductors, percolated networks of carbon nanotubes, and many organic substances where hopping transport is the most limiting factor. As mentioned in the introduction, such systems can be well described by modified Drude models. For the technique described in this work, we so far assumed the original Drude conductivity. It is worth pointing out that similar results as Eq. (5) can be achieved for the Drude-Smith approach [9] and the Cole-Davidson model [10]. This indicates that the presented method of phase analysis can be applied in a much broader framework and may open up new opportunities for characterizing charge transport in modern materials.

5. Conclusions

We developed a technique for directly measuring the Drude momentum relaxation time τ of charge carriers using the optical phase of a differential THz signal. The modulation of the structure under investigation provides a phase signal, which is proportional to τ. In contrast to other spectroscopic approaches, our technique is advantageous particularly in the Hagen-Rubens range. Thus, we are confident that our technique will provide new insights into many modern materials where scattering occurs on ultrafast time scales and electronic properties are not directly accessible by electronic means.