Water vapor absorption in terahertz frequency region

The absorption of the terahertz radiation by air has already been mentioned in the previous works (there existed the deep absorption in the frequency spectrum of the terahertz pulses). The THz-TDS can be effectively applied to characterize gases that have significant absorption in terahertz frequency region [40,91]. As shown in Fig.

5-4, it is apparent that there exists sharp absorption of the terahertz radiation by water vapor contained in the laboratory air. Hall et al. attributed the deep absorption to H2O molecule rotation transitions [91,92]. In order to suppress the humidity in the laboratory air, the THz-TDS system is put into a Plexiglas box and purged with dry nitrogen flow [as shown in Fig. 5-6]. Figs 5-4 and 5-5 display the influence of the water vapor absorption on the waveforms and spectra of terahertz pulses. The

-15 -12 -9 -6 -3 0 3 6 9 12 15 -1.0x10^-5

-5.0x10^-6 0.0 5.0x10^-6 1.0x10^-5 1.5x10^-5 2.0x10^-5

TH z electric field (arb. units)

Time Delay (ps)

FIG. 5-5. Illustration the influence of the water vapor absorption on the waveform of terahertz temporal profile (red line). After purged with dry nitrogen flow, the time profile displays no oscillation tail on the waveform (dark line).

oscillation tail in the temporal time profile apparently is caused by the water vapor absorption.

In general, the abrupt absorption at several frequencies will result in large errors in the estimation of sample’s intrinsic characteristics. It is essential that the influence of water vapor is minimized in order to obtain more precise information using the THz-TDS system. Table 5.1 shows the measured frequencies and wave numbers of the absorption lines compared to the results reported by Hall et al. [92].

The corresponding absorption lines located at 0.56 THz、0.76 THz、1.1 THz、1.4 THz and 1.7 THz are in good agreement with our data.

Table 5.1 Water vapor absorption lines in the terahertz range

Line # 1 2 3 4 5

f [THz] measured 0.56 0.76 1.10 1.17 1.42

f [cm^-1] measured 18.6 25.3 36.6 39 47.3

f [cm^-1] Hall et al. [92] 18.58 25.09 36.59 40.36 48.33

Line # 6 7 8 9 10

f [THz] measured 1.71 1.78 1.88 2.17 2.27

f [cm^-1] measured 57 59.3 62.6 72.3 75.6

f [cm^-1] Hall et al. [92] 57.29 59.68 62.24 72.2 75.52 FIG. 5-6 Display the experimental setup of terahertz time-domain spectroscopy in our laboratory. The system is put into a Plexiglas box and purged with dry nitrogen flow to suppress the humidity in the laboratory air.

-10 -5 0 5 10 15 20 -1.0x10^-5

0.0 1.0x10^-5 2.0x10^-5

E lectric F ield (arb. units)

Time Delay (ps)

air (reference signal)

NdGaO₃ substrate (sample's signal) T=300 K

multiple reflection SI-GaAs

signal cut off point

∆t

FIG. 5-7. Demonstration the terahertz time-domain spectra obtained for radiation through vacuum and with NdGaO3 substrate (red line).

5.3.4 Index of refraction of NdGaO3 substrate in terahertz region

This section presents the THz-TDS measurements of the dielectric substrate NdGaO3. Employing the method discussed previously, one can take the sample’s signal (substrate) and the reference signal (air) to determine the index of refraction of the sample quite accurately. In the case of a homogeneous sample the complex refraction index of substraten_s*=n+ikcan be then converted to complex dielectric function ε*=ε_r +iε_i using the relation n_s*= ε*.

Figure 5-7 demonstrates the time-domain spectra for radiations through vacuum and NdGaO3 substrate, respectively. The NdGaO3 substrate causes a time delay relative to vacuum by ∆t =(n_s −1)d/c. Where d is substrate’s thickness and c is the speed of light. The minor secondary signal appearing after about 15 ps delay of

the main terahertz pulse is believed to arise from the substrate (SI-GaAs) multiple reflections. After Fourier transforming the pulse with longer time scanning (terahertz signals including the part of multiple reflection), the effect of the multiple reflection in the GaAs, as shown in the Fig. 5-8(a), is seen in the spectrum. The periodic oscillation hinders the data processing in obtaining the material’s intrinsic characteristics. Hence, by losing some resolution, we choose a time scanning range which avoids the effect of the multiple reflection in the terahertz temporal profile. In Fig. 5-7, we collect the sampling range of time delay between -10 ps to 14 ps (Fig.

5-8(a) shown the Fourier transforms without the effect of multiple reflection). Then, according to Eq. (5-1) and by numerically solving Eq. (5-2), the amplitude of transmittance and the index of refraction of NdGaO3 substrate are plotted as a function of frequency. In Fig. 5-8(b), where we plot transmission data taken on a bare 0.5-mm NdGaO3 substrate at 300 K. The real part of the index of refraction n_s as a function of frequency is plotted in Fig. 5-9. The imaginary part, being nearly zero at all frequencies, is also plotted in Fig. 5-9. At 60 K, n varies slightly from 4.62 for low frequencies up to 4.7 for higher frequencies. The frequency dependence of the index of refraction remains approximately the same with temperature, displaying the linear dispersion. The value of index of refraction of substrate, for example at 0.5 THz, will vary with temperature from 4.65 at 60 K to 4.75 at 276 K. This variation must be taken into account in the data processing. Figure 5-10 shows the complex dielectric function as a function of frequencies for NdGaO3 substrate. In this case, k~0 leads to εi~0 and ε_r~n² over the whole range of frequencies and temperatures. Similarly, the frequency dependence of the real part of the dielectric function remains approximately the same over the entire temperature range studied. The value ε_r varies from 21.5 for low frequencies up to about 22.4 for higher frequencies at 60 K, while changes with temperature from 21.6 [T=60 K] to 22.6 [T=276 K] at 0.5 THz.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

2.0x10^-4 4.0x10^-4 6.0x10^-4

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.0

0.2 0.4 0.6 0.8 1.0

Amplitude Transmittance

Frequency (THz)

Frequency (THz)

Amplitude

signal with multiple reflection remove the effect of multiple reflection

(a)

(b)

FIG. 5-8. (a) corresponding frequency spectrum by Fourier transform for radiation through vacuum. The multiple internal reflection of the SI-GaAs substrate is presented in the frequency spectra. The sampling range in time profile is collected (denoted in Fig. 5-7) to avoid the effect of the multiple reflection and shown in the frequency-domain (red line); (b) frequency-dependent transmission spectra of the NdGaO3 substrate at 300 K.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

FIG. 5-9. Calculated index of refraction of NdGaO3 substrate at various temperatures. .

FIG. 5-10 Dielectric function of NdGaO3 substrate versus frequency at various temperatures.

5.4 Optical conductivity in superconducting YBa

₂

Cu

₃

O

_7-δ

thin films

5.4.1 Motivation

The measurements of the low-frequency electrodynamics in high-temperature superconductors has been very important. A detailed review in microwave spectroscopy measurements has been given by Bonn and Hardy [93]. Recently, microstrip ring resonators made of double-side YBCO films were fabricated to investigate the temperature- and frequency-dependent penetration depth and microwave conductivity and furthermore to enlighten the properties of quasiparticles at microwave frequencies [94]. The optical conductivity σ^*(ω,T) in superconducting thin films at terahertz and submillimeter regime were carried out straight by using the newly developed technique of coherent terahertz time-domain spectroscopy [5,17,18,50,95-102]. However, the temperature dependence of the complex index of refraction n^*(ω,T) at terahertz frequencies is still not well characterized and it seems important to determine the spectral shape of the terahertz transmission temporal profile in the time-domain. Meanwhile, it seems that similar pulse reshaping phenomena have occurred in transmission spectra between the terahertz generation in FSEOS and the terahertz transmission in THz-TDS measurements by using YBCO thin films as sample. In addition, the method of the THz-TDS can also provide the necessary frequency-dependent information [103]. In particular, we are able to obtain the complex optical conductivity of the YBCO film and use it to directly extract the London penetration depth and the quasiparticle scattering rate with the aid of the two-fluid model.

5.4.2 Electrodynamics and the two-fluid model

Terahertz measurements can be satisfactorily analyzed by using a phenomenological two-fluid model that treats the conductivity of the superconductor as composed of two parts: one that is due to normal carriers whose motion is governed by the Drude equation and one that is due to superconducting carriers whose motion is determined by the London equation. Here, we assume that the carrier response to an applied field is local. The current flow at a point is proportional to the field at the same point satisfiedJ =σE.

In the two-fluid model, each carrier type has its own frequency-dependent conductivity and that the total conductivity is given by the sum of the normal and superconducting part, where the subscripts n and s refer to normal and superconducting, respectively. The temperature T sets the relative ratio of the normal to superconducting carrier densities,

) (T

n_n and n_s(T), respectively.

The normal conductivity is determined by the frequency-dependent Drude expression,

The expression for the frequency-dependent conductivity of the superconducting carriers is given,

)

We note that the conductivity of the superconducting carriers is pure imaginary. One can write the conductivity in terms of the temperature-dependent London penetration depth,

2

where µ_o is the magnetic permittivity of free space. Physically, λ_L represents the distance that an electromagnetic field may penetrate into a superconductor.

The full expression for the temperature-dependent conductivity, thus, can be written as momentum relaxation time. The real part of the complex conductivity then can be expressed as follow,

The quasiparticle scattering rate 1/τ(T) can be calculated directly by using Eq. (5-12) to fit the real part of the conductivity σ₁(ω,T) with frequencies (as called full Drude fit). Besides, the penetration depth λ_L(T), based on the circumstance of

2 1

2τ <

ω , can be extracted from fitting the frequency-dependent imaginary part of complex conductivity and λ_L(0) (or the plasma frequency ω_p) is obtained by the extrapolation of σ₂ to T=0 (ω_p =c/λ_L(0)).

5.4.3 Terahertz transmission spectra

We have investigated the terahertz temporal profile and the frequency-dependent transmission spectra of the radiation of several YBCO thin films at various temperatures. As discussed in section 5.3.1, the terahertz signals of the sample A with higher film’s quality was measured with exposing the THz-TDS system to the air (the humidity is about 42%) and sample B was measured with the system installed in a Plexiglas box and purged with dry nitrogen flow (the humidity is about ~10%).

Figure 5-11 demonstrates the time-domain spectra obtained for radiation through NdGaO3 substrate and YBCO thin film deposited on NdGaO3 substrate. The transmitted electric field of the terahertz pulse is determined by the response of the film and the dielectric properties of the substrate. The transmitted terahertz pulses are almost unchanged in amplitude and phase in whole measured temperature range [as shown in the Fig. 5-11(a)] showing that this substrate can be treated as a ideal material for our purposes. In Fig. 5-11(b), the transmitted terahertz pulses of YBCO thin film in normal state (T ≥90 K) have the same waveform of radiation as like NdGaO3 substrate data, excepting that the transmitted signal is reduced owing to the film’s reflection and absorption. In superconducting state, however, besides a further decrease in amplitude, there is a radical change in pulse shape, implying a dramatic phase shift. The pulse reshaping effect can be regarded as a rapid decrease in the real part of the index of refraction of the YBCO films which will be discussed in the next section.

The changes of the transmitted field can be seen more clearly in the frequency-domain transmission by performing a Fourier transform on the time-domain data. The frequency-dependent amplitude transmittance of the YBCO thin film in the normal and superconducting states was shown in the Fig. 5-12. Above

Tc, the transmitted pulse is almost undispersed, as shown by the date taken at 300 K and 90 K, suggesting the metallic behavior (the intrinsic impedance is dominated by the resistance in normal state). The transmission, below Tc, abruptly drops implying a dominant response of the imaginary part of the conductivity. The overall transmission level falls with decreasing temperature as the density of superconducting carriers (Cooper pairs) increases. These paired electrons that generates a surface supercurrent and strongly screens the external electromagnetic field.

The time-domain transmission spectra for temperatures ranging from 6 K to 300 K in sample B were shown in Fig. 5-13. The dramatic change of the pulse shape, below Tc, can also be found. Figure 5-14 display the corresponding amplitude transmittance as a function of frequency for various temperatures. At room temperature, the frequency response is flat, consistent with metallic conductivity. The amplitude of the transmittance decreases slowly with decreasing temperature until Tc.

Below Tc, there is a distinct drop in the transmission when the samples become superconducting and the Cooper pair density builds up. The noteworthy feature is that the transmission at low frequencies is suppressed more than at high frequencies, suggesting the film acts like a high-pass filter [18].

-6 -4 -2 0 2 4 6 8 10

FIG. 5-11 Transmission time-domain spectra obtained for radiation through (a) NdGaO3 substrate and (b) YBCO thin film deposited on NdGaO3 substrate.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FIG. 5-12 Frequency-dependent amplitude transmittance of the YBCO thin film (sample A) in the normal and superconducting states.

-6 -4 -2 0 2 4 6 8 10 12

FIG. 5-13 Time-domain transmission spectra for temperatures ranging from 6 K to 300 K in sample B.

FIG. 5-14 Corresponding amplitude transmittance as a function of frequency for various temperatures of YBCO thin film.

0 2 4 6 8 10 12 14 16 T=70 K Measured E_o(t)

Time Delay (ps)

(c)

Calculated E_o(t)

Electric Field (arb. units)

(b)

T=90 K Input E_i(t) (a)

5.4.4 The pulse reshaping effect in terahertz temporal profile

In section 5.4.3, we have seen the pulse reshaping effect in sample A, especially the dramatic phase shift in pulse shape. In fact, the pulse reshaping in the measured terahertz electric field caused by the kinetic inductance of the superconducting charge carriers is identified. The kinetic inductance of charge carriers is neglected since the intrinsic impedance is usually dominated by the resistance. In a superconducting state, however, the kinetic inductance becomes more significance and must be considered as an important parameter. Figure 5-15 illustrated the pulse reshaping results in sample A. Details of the phenomenon of kinetic inductance induced pulse reshaping effect has been reported in Chapter 4, and the duplicate circumstance is present in this section.

FIG. 5-15 (a) Measured (output) terahertz pulse

) (t

E_o at 90 K; (b) original terahertz pulse

) (t

E_i obtained by Eq.

(4-7) using the data in (a); (c) measured terahertz pulse at 70 K.

5.4.5 Optical constant and dielectric function

Terahertz time-domain spectroscopy proved a powerful probe for the experimental study of electromagnetic properties of high temperature superconducting thin films at THz-frequencies in basic and applied research. The determination of the index of refraction and dielectric function of superconducting thin films is using the THz-TDS measurements by numerically solving the observed complex transmittance without taking up Kramers-Kronig analysis. In our analysis, the complex index of refraction of the superconducting thin film n_f*=n+ik is determined by numerically minimizing the difference between the experimental data and the theoretical one on the right side of Eq. (5-6) at each frequency [44]. Then the dielectric function *ε and the optical conductivity σ* can be carried out through the relationships

2 1

*2 ε* ε iε

n_f = = + , Eq. (5-13)

*

* σ₁ σ₂ ωε₀ε

σ = +i =−i , Eq. (5-14) whereε₁ =n² −k² and ε₂ =2nk represent the real part and imaginary part of the dielectric function, respectively. ω =2πf is the angular frequency and ε_o is the dielectric function in vacuum space.

The real and imaginary part of the index of refraction of the YBCO thin film (sample B) as a function of temperature at various frequencies is displayed in Fig.

5-16 and Fig. 5-17. With decreasing temperature, the real part of the index of refraction decrease intensively below Tc demonstrated the circumstance of the dramatic change in phase shift in terahertz transmission temporal profile in time-domain. Further, it offers a clear manifestation in study the phase shift of the pulses of terahertz radiation that photogenerated from current-biased superconducting YBCO thin films itself mentioned in Fig. 4-11. Besides, in Fig. 5-17, the imaginary

part of the index of refraction strongly arises as cooled down the measured temperature indicating the strong absorption in YBCO thin films itself in terahertz frequencies. We found that the performances regarding to the peak amplitude of terahertz signals rapidly increases with increasing temperature which observed in Figs 4-11 and 5-13. Hence, the variety of the complex index of refraction with temperature and frequency will result in the differences in the output terahertz time-domain spectra. Whatever the investigation of the terahertz transmission in THz-TDS analysis or the terahertz generation in current-biased YBCO films by using the FSEOS detection, determination of complex index of refraction of terahertz broadband of films itself is significant.

According to the Eq. (5-13), one can calculate directly the dielectric function of YBCO thin films. The real part and imaginary part of the dielectric function of the YBCO thin film (sample B) as a function of temperature and frequency is displayed in Fig. 5-18 and Fig. 5-19. An increase in the real part of *ε from 95 K down to 6.3 K by a factor of about 20 at 0.25 THz is observed, and the imaginary part of *ε present the large broad peak in whole measured terahertz range. There, the tendency for the peak is to become smaller and to be shifted slightly to higher temperatures for increasing frequency.

0 10 20 30 40 50 60 70 80 90 290 300 0

100 200 300 400 500 600

k

0.25 THz 0.50 THz 0.75 THz 1.25 THz

Imaginary Part of Index of Refraction

Temperature (K)

FIG. 5-17 The imaginary part of the index of refraction of the YBCO thin film (sample B) as a function of temperature at various frequencies.

0 10 20 30 40 50 60 70 80 90 290 300

0 20 40 60 80 100 120

n

0.25 THz 0.50 THz 0.75 THz 1.25 THz

Real part of Index of Refraction

Temperature (K)

FIG. 5-16 The real part of the index of refraction of the YBCO thin film (sample B) as a function of temperature at various frequencies. The solid line is drawn to indicate the trend.

0 10 20 30 40 50 60 70 80 90 290 300 0.0

5.0x10⁴ 1.0x10⁵ 1.5x10⁵ 2.0x10⁵ 2.5x10⁵ 3.0x10⁵ 3.5x10⁵ 4.0x10⁵

0.25 THz 0.50 THz 0.75 THz 1.00 THz 1.25 THz

Real Part of Dielectric Constant

Temperature (K)

FIG. 5-18 The real part of the dielectric function of the YBCO thin film (sample B).

0 10 20 30 40 50 60 70 80 90 290 300 0

1x10⁴ 2x10⁴ 3x10⁴ 4x10⁴ 5x10⁴

0.25 THz 0.50 THz 0.75 THz 1.00 THz 1.25 THz

Imaginary part of Dielectric Constant

Temperature (K)

FIG. 5-19 The imaginary part of the dielectric function of the YBCO thin film (sample B) as a function of temperature at various frequencies. The solid line is drawn to indicate the trend.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

FIG. 5-20 The frequency spectra of the real part of the complex conductivity for different temperatures.

5.4.6 Frequency and temperature dependence of complex optical conductivity spectra Now, based on the expression of the Eq. (5-14), the frequency and temperature dependence of complex optical conductivity can be obtained in sample B. Figs 5-20 and 5-21 display the frequency spectra of the real part and imaginary part of the complex conductivity for different temperatures. The real part of the optical conductivity depends weakly on frequency in normal state as data shown at 300 K and 95 K. However, as the temperature fall down below Tc, a frequency dependence of

)

1(ω

σ is observed, which becomes more pronounced as the temperature is lowered further. The imaginary conductivity σ₂(ω) below Tc followed ~1/ω dependence [indicated in Eq. (5-9)].

The temperature dependence of the real component of complex conductivity is shown in Fig. 5-22. All results at different frequencies display the similar behavior.

There present the large broad peak in whole measured terahertz range, where the peak

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FIG. 5-21 Frequency spectra of the imaginary part of the complex conductivity for different temperatures.

is to become smaller and to be shifted slightly to higher temperatures for increasing frequency as described in the imaginary part of dielectric function. The temperature-dependent imaginary conductivity as shown in Fig. 5-23 displayed the sharp rise below Tc and then the whole complex conductivity is dominated by σ₂ at lower temperature measurements, manifesting the electrodynamics in superconductivity (The starting point of the sharp rise will be close to the superconducting transition temperature).

The temperature dependence of σ₁(T) is completely determined by the temperature dependence of the number of normal carriers and the scattering rate 1/τ as depicted in the Eq. (5-12). The phenomenon of existence of the broad peak in

)

1(T

σ is already present in microwave range [93,104,105]. The general trend is for the peak to become smaller and move to higher temperature as the frequency is increased. At terahertz frequencies, the same behavior of the peak is observed in our

0 10 20 30 40 50 60 70 80 90 290 300

FIG. 5-22 The temperature dependence of the real component of complex conductivity. The solid line is drawn to indicate the trend.

0 10 20 30 40 50 60 70 80 90 290 300

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

results and other groups [17,67,95,96,106]. Within the two-fluid model used here to extract the normal conductivity, the occurrence of the broad peak is attributed to a competition between an increase in the normal carrier relaxation time and a decrease in the number of normal carriers with decreasing temperature below Tc. The detailed origin of this peak remains a point of intense discussion, although rough arguments have been invoked to explain its existence. Measurements by Nuss et al. have

results and other groups [17,67,95,96,106]. Within the two-fluid model used here to extract the normal conductivity, the occurrence of the broad peak is attributed to a competition between an increase in the normal carrier relaxation time and a decrease in the number of normal carriers with decreasing temperature below Tc. The detailed origin of this peak remains a point of intense discussion, although rough arguments have been invoked to explain its existence. Measurements by Nuss et al. have

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