1-1 Introduction of THz
Terahertz (THz) radiation lies in the frequency gap between the infrared and microwaves(see Fig. 1-1-1), typically is referred to as the frequencies from 100 GHz to 30 THz. 1 THz is equivalent to 33.33cm-1 (wave numbers), 4.1 meV photon energy, 300μm wavelength. Before 1980s people don’t know much about THz because the generation and detection technologies are not well established, but since the development of femtosecond laser, THz has been intensely studies. At middle 1980s, Auston [1] successfully used photoconductive dipole antenna to generate and detect coherently THz radiation in time domain, and this technology is called THz time-domain spectroscopy (THz-TDS). After Auston’s research many other generation methods have been developed including optical rectification [2], surge current in semiconductor surface [3], quantum cascade laser [4]…, and in 1995 XC-Zhang et al[5] successfully used ZnTe crystal to detect THz radiation by free-space electro-optic sampling that highly increased the detection bandwidth and signal to noise ratio.
Fig.1-1-1 The spectral range of electromagnetic waves (http://www.rpi.edu/terahertz/about_us.html)
Terahertz has much smaller photon energy (4.1 meV) compared to X-ray and therefore this kind of non-destruction measurement can be used for biology and medical sciences [6]. Image and tomography [7] of THz have also been studied and can be applied to homeland security.
For semiconductors measurement, conventional four point probe and Hall effect measurement can measure the characteristics including mobility, carrier concentration and resistivity of the semiconductor materials by direct sample contact. All these electrical measurements measure only DC value of the sample but some characteristics, including refractive index, conductivity, are frequency-dependent. For some semiconductors with high resistivity and low concentration, the electrical properties are difficult to be measured by simple direct contact because at the metal-semiconductors interface, the Schottky barrier may disturb the measurement value. Therefore, THz-TDS with advantages of non-contact and frequency-dependant measure is desirable for semiconductors characterization. In 1990 D. Grischkowsky et al [8] successfully measured optical properties including refractive index and
3
GaAs, other semiconductors such as silicon [9] have also been widely studied.
Recently, many nanostructured semiconductors like InP-nanoparticle [10], ZnO-nanowire [11], Si-nanoparticle [12] have been studied using THz-TDS technology and the particular conduction behavior have been observed. In comparison with conventional far-IR source and detector, THz-TDS is a coherent technology that means both amplitude and phase information can be obtained. Both absorption coefficient and refractive index could be extracted without use of Kramers-Kronig relation that simplifies the analysis process.
1-2 Transparent Conductive Oxides (TCOs)
Basic introduction of PV Solar Cell
Nowadays, solar cells have become essential devices because of the energy crisis.
The major and subsequence field of studying solar modules become the top topic of the Green-energy because of the shortage of petroleum and coal which are the powerful and useful energy sources creating a successfully prosperous industry world in nineteenth and twentieth century.
The quest for low-cost, high-effective, sustainable energy source is one of the prior concerns of the industrial society at the present age. Photovoltaic solar energy is the major part in this quest. The main reason is related to their manufacturing cost, to their market price. This kind of thin film photovoltaic solar cell can be deposited on low-cost large-area substrates. Various semiconductors, compound semiconductors, have been invested for thin film solar cell such as CdTe and Cu (In, Ga)Se2 (CIGS) have attracted much attention in the past research, because of relatively high laboratory efficiencies. For the low cost purpose, the thin film silicon solar cell manufacturing is the best choice.
TCOs of PV Solar Modules
Thin film silicon solar cells, consisting of amorphous and microcrystalline silicon which have the relatively low value of absorption coefficient, need elaborate light-trapping schemes in order to absorb a sufficient part of incoming solar spectrum, within reasonable thickness to ensure satisfactory efficiencies. In a solar cell, to increase the photo-generation and short circuit current density, all reflection and absorption losses have to be minimized. It means as below:
(a.) AR coating might be used on the solar module where light enters into it.
(b.) Back reflectors necessarily have as little absorption as possible
(c.) The substrate and the front TCO layer must have high transparency at 300 to 1200 nm, the solar spectral range.
(d.) TCO layers and the layers which do not contribute to photo-generation and collection, should be kept as thin as possible and have very low absorption coefficient in the acting spectral range.
Fig. 1-2-1 and 1-2-2 show the diagrams, that light propagate in the solar cell. With the rough surface of the TCO layers, the sunlight is widely scattered and improving the absorption at the silicon layers which transverse the light energy to the electrical energy. And in the diagram, it also shows the benefit of the back reflector, which reflects the light to confine the propagating light. [13][14]
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Fig. 1-2-1 The p-i-n type thin film silicon solar cell with light incident
Fig. 1-2-2 Rough surface of TCO and back reflector to increase the light absorption
TCO layers are used as a front electrode and as part of the back side reflector in the solar module. When it applied at the front side, there are few properties should followed:
(a.) High transparency ( in the spectral rang where solar cell is operating).
(b.) High conductivity
(c.) Strong scattering of the incoming light into Si absorber layer.
(d.) To be favorable physical-chemical properties for the growth of the silicon.
ITO Film from NDL:
ITO thin films with thickness as followed, 961.6nm, 483.0nm, and 188.9nm, are prepared by NDL with the DC reactive magnetron sputtering. The growth conditions are shown below:
1. vacuum pressure: 6 m-torr 2. sputtering power: 300W
3. growth on fused silica, the substrate 4. substrate temperature: 250˚C
5. with O2 and Ar
By using the commercial machine, Ulvac DC sputter, with target compositions are 90 wt% In2O3 +10 wt% SnO2 and purity of it is 99.99 wt% manufactures the ITO thin films.
7
The morphology of the ITO thin film: (the surface roughness and the grain) Fig. 1-2-3 ~ 1-2-5 are shown in the SEM photography.
Fig. 1-2-3 Film with thickness 961.6(nm)
Fig. 1-2-4 Film with thickness 483.0(nm)
Fig. 1-2-5 Film with thickness 188.9(nm)
The Fig.1-2-3 to 1-2-5 show the top views of our ITO film samples. The magnification is ×50,000. The pictures with a scale 100nm make us find out the grain size of each sample. Therefore, we have the grain size, 30nm, 20nm, 10nm for the sample with thickness, 961.6nm, 483.0nm, 188.9nm, respectively.
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ITO Nanocolumn from Prof. Yu
ITO nanoculumn with thickness 1µm is grown under the following conditions:
1. deposition angle: α~70˚
2. vacuum pressure: 10-4 torr 3. growth on Si cell
4. substrate temperature: 240˚C 5. with N2
Using the commercial target compositions are 95 wt% In2O3 + 5 wt% SnO2.
The chamber diagrams are shown below: Fig.1-2-6 and Fig.1-2-7
Fig. 1-2-6 The picture of the chamber
Fig. 1-2-7 A cutaway view of the chamber
Fig. 1-2-8 SEM image of ITO nanocolumns deposited with obliquely incident with
Vacuum
pump
Normal Line
ITO
φ
C H A M B E R
Holder θ
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1-3 Motivation
The THz region is ideal for probing semiconductors because the frequency range closely matches typical carrier scattering rates of 1012 to 1014 (Hz).
Therefore, we introduce the optical characterization techniques including FTIR and THz-TDS to measure our samples, ITO thin films and ITO nanocolumn, prepared by NDL and Prof. Yu’s group, respectively. It is a non-contact probe method that could avoid destroying our sample surface. By optical analysis with Drude and Drude Smith model, we also could get the electric properties, such as DC mobility, DC conductivity and carrier concentration.
We will introduce the effective medium theory to the nanostructure sample. In the scale around the atom site, the localized field cannot be ignored. And it is a very interesting topic for the physical phenomenon.
Chapter 2 : Theoretical and experimental methods
2-1 Analysis Models
Macroscopic Fields and Maxwell’s Equations:
The electromagnetic state of matter at a given point is described by four quantities:
(1). the volume density of electric charge ρ
(2). the volume density of electric dipoles, called the polarization P
(3). the volume of density of magnetic dipoles, called the magnetization M (4). the electric current per unit area, called the current density J
All of these quantities are considered to be macroscopically averaged to smooth out the microscopic variations due to the atomic makeup of all matter. The relations of the macroscopically averaged field E and H are described as following by Maxwell equations:
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D=ε0E+P= Eε , called the electric displacement --- (2-1-5)
0( )= H
B=μ H+M μ , called the magnetic induction --- (2-1-6)
In our case, we separate bound charge and bound current from free charge and free current. This separation is more useful for calculations involving dielectric materials.
z E= B
Propagation of Light in Isotropic Dielectrics: [16]
The electrons are permanently bound to the atoms comprising the medium of a non-conducting, isotropic medium. Assume that each electron, of charge -e, has a displaced distance r from its equilibrium position. The macroscopic polarization P of medium is given by P = -Ner, where N(tilted) is the number of electrons per unit volume. Apply a static electric field E, the restoring force of electron is –eE=κr. If the electric field varies with time, the motion of equation with the damping term is
. ..
Consequently, the polarization is given by
where ω0 is the effective resonance frequency of bound electrons.
To show how the polarization affects the propagation of the light, we return the the general wave equation derived from Maxwell’s equation.
2 2
In the above wave equation, the polarization term is important for non-conductive medium. On the other hand, the conduction term dominates for the metals.
Considering ▽•E=0 and equation 2-1-13, we have
2 2
To seek a solution of the form
( )
0 i z t . E E e Κ −ω
⎡ = ⎤
⎣ ⎦
It’s called homogeneous plane harmonic waves.
The possible solution is provided
2 2 2
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Propagation of Light in Conducting Media: (free electron gas) [16]
S
ince the conduction electrons are not bound, there is no elastic force involved.We only consider the electron scattering inside the motion of equation as the form:
d v 1
m m v eE
dt + τ− = −
( mτ-1 is the frictional dissipation constant ) --- (2-1-16)
The current density is J=-Nev, N is the number of conduction electrons per unit volume. The equation can be derived as followed:
1 2
d J Ne
J E
dt +τ− = m --- (2-1-17) (1.) Homogeneous solution:
1 0,
From above equation we get the current density as the form:
0 ,
Here we also introduce J=σE in our assumption. We have ( )= 0 ;
1 i σ ω σ
ωτ
− (It’s the well-known Drude formula.) --- (2-1-19)
We also introduced the result to the general wave equation, and the equation is reduced to
--- (2-1-20)
We also take the simple homogeneous plane-wave solution as our trial solution.
( )
0 i z t . E E e Κ −ω
⎡ = ⎤
⎣ ⎦
It is easily found that K must satisfy the relation
2 2 0 0
In the condition of very low frequency, the formula is reduces to the approximate formula
2
0 0, iωμ σ
Κ ≈ Κ ≈ iωμ σ0 0 = +(1 )i ωμ σ0 0 / 2; --- (2-1-22)
In the low frequency case, the real and imaginary parts of K are equal.
(K=k+iα), k≈ ≈α ωμ σ0 0/ 2. --- (2-1-23)
Similarly, the real and imaginary parts of N are equal too.
(N=n+iκ), n≈ ≈κ σ0/ 2ωε0 . --- (2-1-24)
Without the assumption, we drive the complex refraction index from the relation of K.
2
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Drude Smith Model: [17]
While THz is widely used in Chemistry research and measurements of the conductivity of semiconductor, the flexibility Drude model have been invested in THz region. Better fits model are in the form of modified Lorentzians
0
where the exponents 1-αand βare treated as disposable parameters. With α=0 and β=1, we have the Drude result. With β=1 and α setting as variables is called the Cole-Cole(CC) model. With α=0 and varying the value of β, it is called the Cole-Davidson(CD) model. Some well-fit results were published in the past literature, such as:
On doped Si, Grischkowsky et al. report success with CD model and in transient photoconductivity measurements on GaAs, Schmuttenmaer et al. report success with keeping both α and β as varying parameters.
The formula with Lorentzian form requires that the frequency dependent conductivity should have the maximum at the zero frequency and then fall off.
Departure phenomena have been observed, and we have to concern with those materials in which σ(ω) displays a minimum at zero frequency and a transfer of oscillator strength to higher frequencies in the form of an impulse response.
Let us introduced a simple impulse response approach to the optical conductivity, we have the initial current decays exponentially to its equilibrium value with a relaxation time τ as the form
( )
and
It is the form of Drude model. Then let us assume that an electron experiences collisions that are randomly distributed in time but with an average time interval τ between collision events. We have the resulting current response
( )
This generalized Drude formula is called Drude Smith model following the name of the inventor, Smith. The coefficient cn represents the fraction of the electron’s original velocity that is retained after the nth collision. In order to simplify our calculation, we assume that the persistence of velocity is retained for only one collision (cn=0, for n>1). We have the current response
( )
For elastic collisions, c would be <cosθ> where θ is the scattering angle. The most important character of c is described as followed: When c is a negative value, it implies a predominance of backscattering.
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2-2 The Method of Measurement and Data Analysis
Thin Film Sample (Transmission Type THz-TDS)
Considering a thin film with a thickness below 150µm, the time delay of it will be smaller than 1ps which is the duration of THz pulse. We cannot distinguish the second reflection from the main pulse. Therefore we introduce the multi-beam interference in our analysis as that to extract the information from the THz waveform by Fourier transform method (time domain to frequency domain). And then, we take the refraction index of substrate as a constant value to simplify the analysis method. (In practical, we use low absorption and constant refraction index material in THz range.)
Theoretical transmission formula
E0(ω) is the incident THz field, Eref(ω) is the reference electric field gone through both the substrate ( the thickness is “D” ) and a bunch of air with thickness
“d”, and Esig(ω) is the signal electric field transmitted through both the thin film ( the thickness is “d” ) and the substrate ( the thickness is “D” ). We could write down the reference and signal field in the form of E0:
* 3
Fig. 2-2-1 The diagram of the reference
Fig. 2-2-2 The diagram of the sample
Fig. 2-2-1 and 2-2-2 are showing the basic diagrams of the light propagating through the sample and the reference. The concept will be applied to the THz-TDS analysis with gathering both the reference and signal data.
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Where q is the number of multiple reflections, and t12, t13, t31, t23, r21, and r23 are Fresnel amplitude transmission and reflection coefficients in the condition of normal incidence which can be expressed by
2 i
By assuming the number of reflection is infinite (q → ∞), Efilm(ω) could be
simplified as
From the equation 2-2-1 and 2-2-2 by insert 2-2-5 into them, the theoretical complex transmittance can be given by
2 The contents mentioned above where n1 is the refraction index of the air, n2is the complex refraction index of the thin film, and n3 is the refraction index of the substrate.
Refraction index extraction
Experimental data is collected from transmission type THz-TDS system with PC antenna as emitter and receiver. By means of the Fourier transform, the original time domain data is to be the frequency domain data. Therefore we have the experimental amplitude transmission in frequency domain by dividing the two spectrums, here we
given it by texp( , , )ω n2 κ2
.
In order to extract the parameters of the experimental data, we defined error function as below:
exp( , , )i 2 2 the( , , )i 2 2 ( , , );i 2 2
t ω n κ −t ω n κ =Error ω n κ
First, given a periodic set of n2 and κ2, we set { n2, 0, 200, 0.1 }, { κ2, 0, 200, 0.1 }. (It means that an interval from 0 to 200 with spacing 0.1.) Then we can get a 2D-matrics of various n2 and κ2 at each ωi. With the sorting program, we find out the local minimum of the 2D sets. Gathering all of the local minimums at each ωi, we make a new set which is the refraction index extracting from the experimental data.
Fig.2-2-3 The flow chart of the n κ value extraction
Figure 2-2-3 demonstrates the flow chart of extracting the refractive index of the sample. First, we have the time domain waveforms, reference and signal, from the THz-TDS system. With the fast Fourier transform, we have the frequency domain spectrum. Then, we could get the experimental transmittance. Following, to calculate
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experimental data. By calculating from computer program, we could extract the frequency dependent refractive index.
Optical conductivity
From above calculations, we have the complex refractive index, and we can use it to obtain the complex conductivity of a conductor. First, we start from the Maxwell equation assuming a simple conducting medium with a flowing current, J=σE, and the formula can be expressed by:
where ε∞is the contribution of the bound electrons and ε is the effective dielectric constant. We can have ε from the refractive index via the relation of ε=(εr+iεi)=(n+iκ)2 and therefore the complex conductivity can be obtained from Eq.
0
The conduction of electrons in simple metals can be describe by a classical simple Drude model [18] which treat the free carriers in a solid as classical point charges
subject to random collisions denoted as
The plasma frequency is defined by ωp2 = Ne2/(εm*) where N is carrier concentration, e is the electronic charge and m* is the effective carrier mass, and τ is the carrier
relaxation time. The DC conductivity is given by σDC = eNµ, where µ = eτ/m* is the carrier mobility. The simple Drude model indicates that the velocity of carriers is damped with a time constant τ and is randomized following each collision event. The conduction properties of many semiconductors in the terahertz region have been justified to follow the simple Drude model, but some nanostructured materials show deviations from it. Recently, Smith proposed a modified Drude model, which can explain the deviations from the simple Drude model for the nanostuctured materials, particularly the negative values of imaginary part of conductivity. The complex conductivity in the Drude-Smith model [17] is given by
2 2
where c is a parameter describing fraction of the electron’s original velocity after scattering and vary between -1 and 0. In the simple Drude model, the momentum of
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carriers retain a fraction, c, of their initial velocity. In particular, c = 0 corresponds to the simple Drude conductivity
a
nd c = –1 means that carrier undergoes complete backscattering. The Drude-Smith model predicts a DC conductivity of σ = eNµ(1+c) and thus the reduced macroscopic DC mobility is given by µm = (1+c)µ. [17]Fig.2-2-4 The flow chart of the conductivity value extraction
Figure 2-2-4 demonstrates the flow chart of the conductivity value extraction. It follows the steps as below. To get the experimental conductivity from the refractive index, we use the relations between the refractive index to the permittivity and the conductivity to the permittivity. The relations are derived from Maxwell equations.
And then, we introduce Drude free electron model into the fitting process. With the experimental data and theoretical formula, we also define an error function to extract our parameters, ωp and τ.
2-3 Effective Medium Approximation:
If the body is neutral, the contribution to the average field may be expressed in terms of the sum of the fields of atomic dipoles. We define the average electric field E (r0) as the average field over the volume of the crystal cell that contains the lattice point r0:
where e(r) is the microscopic electric field at the point r. Vc is the total volume of the body. We called E the macroscopic electric field.
Depolarization Field: [19]
The geometry in many of our problems is such that the polarization is uniform within the body, and then the only contributions to the macroscopic field are from E0
and E1:
0 1;
E=E +E --- (2-3-2)
Here E0 is the applied field and E1 is the field due to the uniform polarization. The field E1 is called the depolarization field, for within the body it tends to oppose the applied E0 as in Fig. 2-3-1.
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Fig.2-3-1 Depolarization field E1 tends to oppose the applied field E0.
If Px, Py, Pz are the components of the polarization P referred to the principal axes
If Px, Py, Pz are the components of the polarization P referred to the principal axes