2.1 - Terahertz Generation
There was a lot of methods can generated terahertz such as irradiation of photoconductive antennas, semiconductor surfaces, or quantum structures with femtosecond optical pulses laser. The most common mechanism are surge current and transient polarization which acts as broadband terahertz sources. In this work, the terahertz waves are generated via surge current, therefore we will briefly introduce this mechanism.
Surge-Current Model
The basic mechanism of surge current begins with an ultrashort pulse laser which creates electron-hole pairs in semiconductors. Then the carriers accelerate in the external or internal electrical field to form a transient current which radiate THz waves. In the far field approximation, the field amplitude of the radiation is proportional to the time derivative of the photo-current.
From the Maxwell’s equation[20]：
(Faraday’s Law) (2-1)
(Gauss’s Law) (2-2)
(Ampere’s Law) (2-3)
(2-4) From Equation (2-1), (2-2), (2-3) and (2-4), we get
(2-5)
(2-6)
7 We got two inhomogeneous wave equations (2-10 and 2-11) in terms of magnetic potential and electric potential V that is the current density and is the charge density. These equations describe the propagation of the electromagnetic disturbances.
From equation (2-3), the continuity equation of free carriers is obtained.
electric field. Therefore, from Eq (2-6) we have
The solution of the vector potential in Eq (2-10) leads into the expression for the time-dependent radiated electric field E( ,t) at a displacement from the center of
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current in emitter evaluated at the retard time, and is the increment of surface area at displacement from the center of the emitter. In the far field and assuming is a constant at all points in the emitter, the radiated field can be written as
where b is a constant, and the radiated field proportional to the time derivative of photo-current. In general, the semiconductors with a wide bandgap, such as GaAs (Eg
= 1.43eV) and InP (Eg = 1.34eV) which by surface depletion field[21] to generate terahertz wave. However, the semiconductors with small bandgaps and small effective mass, such as InAs and InSb, radiate terahertz via the photo-Dember effect.[22-23]
2.2 - Terahertz Detection
The most common methods for THz detection are photoconductive sampling and free space electric-optic sampling (FS-EOS). Both of them are coherent detection which can get both amplitude and phase information by scanning THz time-domain waveforms. They have high signal to noise ratio (SNR) in comparison with bolometer.
In our experimental setup, free space electric-optic sampling mechanism was used for our detection. The properties are described below.
2.2.1 - Free space Electro-Optic Sampling (FS-EOS)
The EO sampling method is based on a birefringence in an EO crystal induced by the incident radiation, or Pockels effect. Under the external field, the phase retardation induced in the EO crystal can be expressed as[24]
Where d is the thickness of the crystal, n is the refractive index of the crystal at the
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wavelength of the near IR probe beam, is the probe beam wavelength, is the electric-optic coefficient, and is the electric field of THz waves. The detector of EO sampling has broad bandwidth spectrum and is easy to implement. There are many different materials for EO sampling such as ZnTe, GaP, GaSe, and InP. Among all crystals, ZnTe is commonly used for EO sampling of the THz pulse because ZnTe crystal has relatively large electric-optic coefficient and good group velocity match.
The linearly polarized laser probe beam co-propagates inside the crystal with the THz beam and its phase is modulated by the refractive index change induced by the electric field of the THz pulse as shown in Fig. 2-4.
Fig. 2-1 Schematic figure of Electro-Optic Sampling
The principle can be explained as follow. Suppose z is the probe beam propagating direction, and x and y are the crystal axes of the EO crystal, respectively. When an electric field is applied to the EO crystal, the birefringence axes are induced by electric field x’ and y’ at an angle of with respect to x and y axes. If the input probe beam is polarized along x with amplitude, then the output beam can be expressed by
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Where is the phase difference between the x’ and y’ polarizations, including dynamic term caused by THz field and static term, .
The intensity of probe beam in x and y direction can be expressed by
where is the input intensity and Ix+Iy=Io is the total energy. A Wollaston prism is use to extract the and independently. A quarter-wave plate for balance detection that the static term is often set at . It’s operating at the linear region to avoid distortion. In the most cases of EO sampling, is much smaller than 1, so we can measure the signal and by a balance detector.
The signal difference that is proportional to the phase change
is
induced by THz field. Thus, we can obtain the total THz time-domain waveforms by measuring the signal difference via the balance detector as a function of delay time between the THz pulse and the laser probe pulse.11
2.3 - Time-Domain Measurement Technique
Pump-and-probe technique can be used to obtain the time-domain information on ultrafast phenomena. The general principles of pump-probe method need two ultrashort pulses: one is called “pump” beam which excites samples and the other one is called “probe” beam which can detect the temporal evolution of the perturbed sample. The time-resolved study of the photoexcited samples can be obtained by adjusting the time between pump and probe beamsThe single-cycle waveform of terahertz field can be obtained by this pump-and-probe method.
Fig. 2-2 The typical scheme of pump-probe experiment
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2.4 - Models and Analysis Methods
The THz time-domain spectroscopic (THz-TDS), both of frequency dependent amplitude and phase information can be obtained by Fast Fourier Transform (FFT).
We can use these information to calculate the material’s frequency dependent optical constants of samples that including complex refractive index and conductivity without via Kramers-Kronig analysis. The information extraction process of time-domain spectroscopy is expressed as shown in Fig. 2-5. In this section, we will discuss the electromagnetic theories and analysis methods for two kinds of samples that are thick samples and thin samples, respectively.
Fig. 2-3 THz-TDS measurement and the parameter extraction process
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2.4.1 - Thick Samples
In general, the samples of thickness are hundred microns (typically 100m), the temporal terahertz reflection signal in time domain is clearly separated which caused by multiple reflections in the sample. We can reduce the reflection signal to simplify the analysis as show in Fig. 2.6. The schematic figure of the electromagnetic model is shown as Fig. 2.7, assuming normal and plane wave incidence. We assumed is the incident terahertz field, is the reference field, is the transmitted through the sample with thickness d, and c is speed of light in free space.
and are refractive index for air and measured thick sample.
The reference and signal field can be expressed by
Where and are transmission coefficient that is air (medium 1) to sample (medium 2) and sample to air, respectively. The formula can be expressed by
From above equations, the complex transmission coefficient of the sample can be expressed by complex refractive index can be obtained as
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Fig. 2-4 Terahertz time-domain waveform transmitted through substrate of thick silicon and the reflection signal is clearly separated from the main signal
Fig. 2-5 Diagram of the electromagnetic model of thick sample
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2.4.2 - Thin Samples
In general, the thickness of the samples was several micron meters or even only several hundred nanometers. So the reflection signal couldn’t be clearly separated in time-domain as shown in Fig. 2.8. The Fig. 2.9 illustrate with a schematic of the interaction of the THz pulse with sample. We assumed , , and be the incident THz field, the reference field through substrate with thickness d’ (substrate only), signal field transmitted through the thin film with thickness d and the signal field transmitted through both the thin film and substrate, respectively. In the formula, c is speed of light in free space. , transmission and reflection coefficients which can be expressed by
16 The theoretical complex transmittance can be determined by
The transmission and reflection coefficients can be expressed by
we obtained the experimental complex transmittance from THz-TDS data but there is no exact solution for the complex refractive index. For this goal, we use a numerical method and define an error function that can be expressed
If there exists a complex refractive index which makes the error function closest to zero at a part of frequency , we can extract the values at each frequency by use of mathematical program. We set a range of values for calculation and extract the best one that can makes the error function almost zero within them for each frequency to have the complex refractive index values.
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Fig. 2-6 Terahertz time-domain waveform transmitted through thin samples and the main signal is mixed with multiple reflection signals
Fig. 2-7 Diagram of the electromagnetic model of thin sample
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2.5 - Determination of Optical Conductivity and Mobility from Drude Model and Drude Smith Model
In this section, we will discuss the optical conductivity and introduce Drude model and Drude-Smith model. From numerical methods discussed above, we have the frequency dependent complex refractive index of thin films. We can use the relation to obtain the frequency dependent complex optical conductivity. We start from Maxwell equation first of all. Assuming a simple conducting medium with flowing current, , and the formula can be expressed by
Where is the contribution of the bound electrons and is the effective dielectric constant. We can get by
and therefore the complex conductivity can be obtained from
The Drude model[25] was developed in the 1900s by Paul Drude that can be describe the transport properties of electrons in materials (especially metals).
Consider the microscopic behavior of electrons in a solid as classical point electronic charges subject to random collisions that affect by an applied field. The classical simple Drude model that conductivity can be expressed by
19 is the electronic charge, is the effective carrier mass and is carrier relaxation time. The DC conductivity is given by , where is carrier mobility.
The simple Drude model indicates that the velocity of carriers is damped with a time constant and is randomized following each collision event. Many semiconductors in the terahertz region have been conform the simple Drude model, but still some materials, such as nanostructured materials cannot explain by simple Drude model.[26]
Recently, Smith introduced a modified Drude model [27], which can explain the deviations from the simple Drude model, especially for the negative values of imaginary part of conductivity. Smith proposed the complex conductivity in the Drude-Smith model is given by
Where c is a parameter describing fraction of the electron’s original velocity after scattering and varied between -1 and 1. In the simple Drude model, the carrier is randomized after each scattering event, but in the Drude-Smith model, carriers retain a fraction c of their initial velocity. Mainly, c = 0 corresponds to the simple Drude conductivity and c = -1 means that carrier undergoes complete backscattering. The Drude-Smith model predicts a DC conductivity of and thus the reduced macroscopic DC mobility is given by .
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