Optical transmission measurement is a powerful tools to explore the physics mechanism of material properties. There are several advantages: (i) the electromagnetic (EM) wave have a proper penetrating depth depending on the wavelength and different samples. Therefore, it can easily provide the information from the deep bulk excluding the surface signal; (ii) the optical components are well developed so that we can easily acquire a focused light beam (~ 1 μm) to measure a small-sized sample. Another useful optical component is fiber. It makes optical spectrometers in conjunction with some other equipment, for example high magnetic field magnet; (iii) optical technique can be used in solid, liquid and gas samples. The limitation of sample is not very strict; and (iv) the obtained optical responses come from different physics mechanism. It can be controlled by varied frequency of incident light.
The infrared spectroscopy probes different response of material in the different frequency region. In the far-infrared (FIR) region (10 ~ 100 cm-1), the incident wave interacts with free carriers so that we can study the properties of free electrons, for example, the scattering rate. In the range between 100 and 1000 cm-1, the optical response shows the information of lattice phonons. Parameters of the optical-active phonon can be obtained. These information are useful for studying the spin-phonon coupling. When the incident wave energy is between 1000 and 100000 cm-1, the optical spectrum shows the electronic transitions. Usually, in the mid-infrared
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(MIR) region (1000 ~ 10000 cm-1), the incident light energy is relatively small in the case of oxides; therefore, only intraband transition can be triggered. Naturally the higher energy can trigger the interband transition of valence electrons, which can be contributive to understand the band structure around Fermi surface. By using a Fourier transform infrared spectrometer (FTIR), we can measure optical spectrum from 30 cm-1 to 60000 cm-1.
In this study, the samples are thin films. Therefore, using transmittance spectrum as a base to analyze the optical properties, such as dielectric function of optical conductivity, is a practical method. In order to obtain the absolute transmission, we used hole as reference in FTIR spectrometer. The as-measured transmittance Texp can express as,
Texp = Tsample / Thole. (3.1.1) To understand the properties of solid by analyzing optical spectra, we should begin with Maxwell’s equations. The microscopic form of Maxwell’s equations is
4
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electrons), so we can treat solid as continuous system. In this case, an averaged macroscopic form of Maxwell’s equations is more suitable. That is
4
Consider only linear response, using the linear approximation:
: properties of materials. Substitute Eq. (3.1.6) into Eq. (3.1.4), now the Maxwell’s equations are involved with the properties of the materials,
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In our measuring, there is no external charge and current, and then we can reduce the Maxwell’s equations into single equation: wave is a single plane wave and traveling in an isotropic medium, assume the electronic field has the form,EE e0 i q r(ˆ t). Substitute electronic field in Eq (3.1.8), resulting in a dispersion
then, the form of plane wave changes into:
( ) ( )
0
k r i n r t
c c
the attenuation of a traveling wave wave amplitude with phase velocitycn with distance
44 The absorption coefficient (α) represents the fractional decrease of wave intensity with penetrating distance:
Moreover, we can define a complex dielectric function (ε):
2
Combine Eq. (3.1.10) and Eq. (3.1.13) we obtain the relation among complex dielectric function and complex refractive index:
Here, these optical parameters, 1, 2,nandk are strongly influenced by electronic structure of solids. In general, we measured transmission spectrum and calculated the optical parameters, especially dielectric function.
Now we consider a light incident to a single interface of two medium whose refractive index are n and i n . Fig. 3.1 is a simple sketch graph. Again, we assume the electronic field of t light has the plane wave form and consider the boundary conditions of Maxwell’s equations.
Then, the relation among electric field of incident, reflect and transmit light can be obtain.
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They are usually represented as Fresnel coefficient which includes reflectance coefficient and transmittance coefficient. But, the vibration direction of electronic field will influence the solution when we apply the boundary conditions. We need to separate incident electronic field into two components: (i) The electronic field is perpendicular to the plane of incidence, called transverse electronic field mode (TE mode); (ii) The electronic field is parallel to the plane of incidence, called transverse magnetic field mode (TM mode). Moreover, we consider the nonmagnetic materials. Then we get two set of solution:
0
The reflectance is defined as the intensity ratio of the reflective and incident beam. Besides, the
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intensity is square of the electronic field amplitude. Then, the relation of the reflectance and refractive index can be written as
* 2 An extended complex refractive index is suitable in general case. So Eq. (3.1.20) is changed into a general form, reflectance as linear response of the optical system. In this case, the Kramers-Kronig relations, who connect the real and image part of complex optical response are valid. Consider a complex optical parameter (N, ... ) whose general form is
ˆ( ) ( ) ( )
f f if (3.1.22) The general form of Kramers-Kronig relations are
2 2
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where ρ indicates the Cauchy principal value. It is clear to see that we can use Kramers-Kronig relations to obtain image part of optical parameter spectrum by integrate the real part spectrum, and vice versa.
Using the Kramers-Kronig relations in our analysis, the reflectance coefficient is a good starting place, since it is also a complex optical response. Apply all assumption which is mentioned in Eqs (3.1.16) and (3.1.17), the reflectance coefficient can be written as
1 [ ( ) ( )] ( ) know the relation between reflectance and reflectance coefficient from Eq. (3.1.19),
ln ( ) ln R( ) (3.1.26)
48 indicating the relation between N and .
It should be noted that the integral interval of Kramers-Kronig relations is up to infinity, which is impossible in practical. Besides, in much higher frequency range, the correlation function of EM wave and solids would be different, resulting in the Kramers-Kronig relations become invalid. Therefore, when we use the Kramers-Kronig relations, a reasonable and suitable extrapolation for each compound should be applied. Typically, the high-frequency extrapolations were done by using weak power law dependence, R ~ ω-s with s ~ 1 – 2. The low-frequency extrapolations are strongly dependent on sample properties. For the insulating sample, we could use the reflectivity of end point to perform a constantly extrapolation down to DC, usually. For the metallic conducting compounds, we usually use a Hagen-Rubens relation, R = 1- Aω1/2, to extrapolate low-frequency reflectance.