Dressel.Gruner.Electrodynamics of Solids.2003

	278	11 Measurement configurations
		E
	Ei	Plane of	Er
		incidence
	E	ψi	Surface
			ψr
	N'

ψt

N

Et

Fig. 11.5. Reflection and transmission of a light beam at a surface. The plane of incidence contains the incoming beam Ei, the outgoing beam Er, and the transmitted beam Et. The ratio of the angles of incidence ψi and transmission ψt at a single interface between two

media is characterized by Nˆ and Nˆ . The amplitudes of the reflected and transmitted fields depend on the polarization with respect to the plane of incidence, E and E , and are given by Fresnel’s equations (2.4.3).

optical properties of solids, such as frustrated and attenuated total reflectance, are not presented here, but are discussed in handbooks, for example [Pal85].

11.1.4 Ellipsometry

Ellipsometry or polarimetry measures the polarization of an electromagnetic wave in order to obtain information about an optical system which modifies the polarization; in general it is used for reflection measurements. While most of the early work was concentrated in the visible spectral range, recent advances have made ellipsometric investigations possible from the far-infrared to the ultraviolet. In contrast to standard reflectivity studies which only record the power reflectance, two independent parameters are measured, thus allowing a direct evaluation of the complex optical constants. Furthermore, as the magnitude of the reflected light does not enter the analysis, ellipsometric studies are not sensitive to surface roughness and do not require reference measurements. Several reviews of this topic and a selection of important papers can be found in [Azz87, Ros90, Tom93].

As discussed in Section 2.4.1, an electromagnetic wave reflected at a surface can be decomposed into a wave with the polarization lying in the plane of incidence (subscript ) and a part which is polarized perpendicular to the plane of incidence (subscript ) according to Fig. 11.5. The two components of the electric field

11.1 Single-path methods

279

generally experience a different attenuation and phase shift upon reflection even for isotropic media, and hence the state of polarization changes as described by Fresnel’s equations (2.4.7d) and (2.4.7b). Ellipsometry is a measurement of the complex ratio of both reflection coefficients:

	ρ	rˆ	=	tan	θ	}	exp	i	}	.	(11.1.2)
		rˆ
	ˆ =			{			{

The two angles which have to be determined are related to the amplitude ratio θ and the phase = φr − φr , where φr and φr are the phase shifts induced by the sample upon the reflection. These coefficients depend upon the angle of incidence ψi, upon the material properties (the complex conductivity σˆ or dielectric constantˆ), and typically also upon the frequency ω. As expected, for normal incidence the ellipsometric effect disappears because R = |rˆ |2 = |rˆ |2 = R .

It is important to note that ellipsometers measure θ and , and not the optical constants: to evaluate these, models are used with certain assumptions. With the simplest model (two semi-infinite dielectric materials with an abrupt discontinuity in Nˆ at the interface as shown in Fig. 11.5), both the real and imaginary components of the dielectric constant are determined using the following:

1

2

=

{

i}

+

{

}(1 + sin 2{θ } cos{ })2

sin2

ψ

1

tan2

ψi (cos2{2θ } − sin2

{ } sin2{2θ })

(11.1.3)

=

sin2

{

ψ

i}

tan2{ψi} sin{4θ } sin{ }

.

(11.1.4)

(1 + sin{2θ } cos{ })2

Two configurations are often used to obtain the ellipsometric angles. The nulling technique determines the angle of the analyzer for which the reflected signal vanishes; while in the case of the photometric technique the transmitted signal is measured as a function of the polarization angle. The scheme of a null ellipsometer is displayed in Fig. 11.6: the linearly polarized light is reflected from the surface of the sample in an angle larger than 60◦ and becomes elliptically polarized; it is then converted back to a linearly polarized beam by a suitable rotation of the quarter wave (λ/4) plate. The intensity of the light at the detector is finally minimized by rotating the analyzer until it is perpendicular to the axis of polarization. The rotation angles of both the λ/4 plate and the analyzer allow the determination of θ and . In other words, in general the transmitted light is elliptically polarized so that the major and minor axes are aligned with the directions of fast and slow propagation. The angle γ of the major axis is adjusted by rotating the λ/4 plate; by rotating the polarizer in front of the λ/4 plate, we account for the eccentricity e = b/a – the ratio of the minor axis b to the major axis a.

In general no attempt is made to reconvert the elliptically polarized light to

280	11 Measurement configurations

Source

Detector

		Linearly	Null
	Linearly	polarized
		Elliptically	Analyzer
	polarized	polarized
	Polarizer
	ψi	Quarter
		wave plate
	Sample

Fig. 11.6. Schematic arrangement of a null ellipsometer. A monochromatic light (polarized 45◦ out of the plane of incidence) hits the sample at an angle ψi. The reflected light is elliptically polarized; by passing through a λ/4 (quarter wave) plate it is transformed back to linear polarization. A second polarizer determines the polarization. The angular settings of the quarter wave plate and the analyzer are used to determine the phase shift and attenuation ratio θ (after [Tom93]).

linearly polarized light; instead the first polarizer is set at a fixed value – usually 45◦ with respect to the plane of incidence – and the reflected intensity of the light is measured as a function of the analyzer angle (so-called rotating analyzer ellipsometer). The intensity as a function of analyzer angle is given by the expression

I (α) = I0 + I1 cos{2α} + I2 sin{2α} ,

where I1 and I2 depend on both γ and e; here α is the position of the analyzer. The coefficients I1 and I2 can be found from the measured data since the azimuth is

π

− arctan

I1

2γ =

(11.1.5)

2

I2

and the eccentricity is

e =

I0

+

I12

I22

1/2

1/2

I0

I 2

+ I 2

1/2

.

(11.1.6)

−

1

+ 2

Thus, as with the nulling technique, we determine θ and from the measured parameters; the material properties are finally evaluated from these values.

Ellipsometric investigations which cover a wide spectral range are cumbersome because the experiments are performed in the frequency domain; in the infrared range Fourier transform ellipsometers have become available [Ros90]. At each analyzer position an interferogram is recorded and a Fourier transform of the

11.2 Interferometric techniques

281

interferogram generates the frequency spectra of the reflected light at this position of the analyzer. The measurements are then repeated for the other analyzer position, and thus for a certain frequency the intensity is obtained as a function of angle. Measurements with these techniques have recently been made in the far-infrared frequency range using synchrotrons as highly polarized light sources [Kir97].

11.2 Interferometric techniques

In order to obtain the complex electrodynamic response of the material of interest, single-path configurations – as just described – have to record the absolute values of four different quantities; for example, magnitude and phase of the signal before the interaction and magnitude and phase after the interaction of the light with the sample. If the variations caused by the material are small, it is advantageous not to probe the signal itself, but to compare it with a (well characterized) reference and measure only the difference. The basic idea of interferometric measurements is therefore to compare the parameters of interest with known parameters, and to analyze the difference; this technique is often called the bridge method.

An interferometer splits the monochromatic radiation coming from one source into two different paths, with the sample being introduced in one arm and a reference into the other (no reference is needed for transmission measurements); the radiation is eventually combined and guided to a detector. If the coherence of the source (introduced in Section 8.3) is larger than the path difference, the recombined beam shows interference. From the two parameters, the phase difference and the attenuation caused by the sample, the material properties (e.g. complex conductivity σˆ ) are evaluated.

Since monochromatic radiation is required, interferometric techniques can only be utilized in the frequency domain. In the following we discuss three different setups which cover the spectral range where circuits, transmission lines, and optical arrangements are utilized.

11.2.1 Radio frequency bridge methods

The arrangement of a low frequency bridge is best explained by the Wheatstone bridge (Fig. 11.7). An element, the electrical properties of which we intend to measure and which is described by a complex impedance Zˆ 1, is inserted into a network of known impedances Zˆ 2 and Zˆ 3. Two points in the network are connected to an alternating current source, while a detecting instrument bridges the other two points. The fourth impedance Zˆ 4 is then adjusted until the two bridged points are at the same potential and phase – leading to a null reading at the detector

282	11 Measurement configurations

Z4 Z1

~

Detector

Z3 Z2

Fig. 11.7. Wheatstone bridge circuit with four impedances. The impedance Zˆ 1 is determined by adjusting Zˆ 4 in such a way that no signal is recorded at the detector. In the simplest case the two elements Zˆ 2 and Zˆ 3 are equal.

[Hag71]. The experimental determination of the electromagnetic properties is then reduced to the measurement of the values of an impedance Zˆ 4. In the simplest case the Wheatstone bridge contains only resistors, but the general layout also works for complex impedances, consisting of ohmic and capacitive contributions, for example. Then two parameters have to be adjusted to bring the detector signal to zero, corresponding to the phase and to the amplitude of the signal. Knowing the geometry of the specimen, and from the values of the resistance and the capacitance, the complex conductivity of the material can be calculated. Bridge configurations as depicted in Fig. 11.7 are available in a wide frequency range up to a few hundred megahertz. The sample is either placed in a capacitor or has wires attached (Fig. 11.1).

11.2.2 Transmission line bridge methods

A large variety of waveguide arrangements have been developed which follow the principles of bridge techniques [Gru98] to measure the complex conductivity of a material in the microwave and millimeter wave range (10–200 GHz). In general the beam is split into two arms with the sample placed in one and the second serving as a reference; both beams are finally recombined and the interference is observed. Up to approximately 50 GHz coaxial components are also employed for similar bridge configurations.

The technique is used for transmission and for reflection measurements. Since the former arrangement is similar to the Mach–Zehnder interferometer discussed in Section 11.2.3, we only treat the reflection bridge here. This configuration

	11.2 Interferometric techniques			283
	Short
	Reference arm
Frequency		Phase shifter
counter	AS	φS
	Attenuator			Lock-in
	Modulator		Isolator	Ampl.
Source	Isolator		Detector

Measurement arm

Sample

Fig. 11.8. Diagram of the millimeter wave impedance bridge for reflection measurements (after [Sri85]). First the bridge is nulled by terminating the measurement arm with a metal

of known Zˆ S. Then, with the sample replacing the metal, the interferometer is readjusted by changing the phase shift φS and attenuation AS. From these two readings, φS and AS, the load impedance terminating the transmission line and eventually the complex conductivity of sample are determined.

measures the complex reflection coefficient (or the scattering parameter Sˆ) of the sample, which is written in the form

rˆ = −10AS/20 exp{iφS} ,

(11.2.1)

where AS is the change of the attenuation given in decibels and φS is the change in the phase. Once this quantity is known, using Eq. (9.2.4) the impedance of the sample is extracted, and from that the complex conductivity can be evaluated. The arrangement of a microwave reflection bridge [Joo94, Kim88, Sri85] is shown in Fig. 11.8. The sample either terminates the measurement arm (in this case the specimen has to be metallic or at least thicker than the skin depth – Zˆ L then simplifies to the surface impedance Zˆ S of the material); or the sample is placed at a distance from a short end of the transmission line, where the electric field is at a maximum. This is the case for = (2n + 1)λ/4 (with n = 0, 1, 2, . . .) and thus the position depends on the particular frequency used (Fig. 11.2); for other frequencies appropriate transformations have to be made [Ram94].

As an example of a measurement performed by a millimeter wave impedance bridge operating at 109 GHz, we present results from TaSe3 [Sri85]. The phase

284		11 Measurement configurations
	8	(a)
				0
(dB)	6				Phase
				−25
S
A	4
Attenuation					φ shift
				−50
	2				S
				−75
	0
100		(b)
K)
10− 1				109 GHz
				dc
Resistivity (300(T)ρ/
ρ
	− 2			TaSe3
10		100	200	300
	0
		Temperature T (K)

Fig. 11.9. (a) Attenuation (open diamonds) and phase shift (full circles) measured as a function of temperature on a sample of TaSe3 with a millimeter wave bridge at 109 GHz.

(b) From the data shown in (a) the temperature dependent resistivity of TaSe3 is obtained at 109 GHz and compared with dc measurements. The sample was a thin needle with an approximate cross-section of 1 µm × 1 µm. At room temperature, the dc resistivity is 500 µ cm and the microwave results were normalized to this value (after [Sri85]).

shift and the attenuation due to the needle shaped crystal placed in the maximum of the electric field are shown in Fig. 11.9a as a function of temperature. For this configuration the load impedance and from that the complex conductivity can be evaluated; the calculated resistivity ρ(T ) is displayed in Fig. 11.9b. The solid line is the four-probe dc resistivity measured – the experiments provide evidence that the conductivity is independent of frequency in the measured spectral range.

11.2 Interferometric techniques		285
	Wire-grid
Mirror 2	mirror	Analyzer

Detector

			Sample
Source	Polarizer	Beam-	Mirror 1
		splitter

Fig. 11.10. Mach–Zehnder type interferometer used for quasi-optical transmission measurements in the submillimeter wave range. The coherent radiation is split by wire grids. The length of the reference arm can be adjusted for destructive interference by moving mirror 2. After the sample is introduced, the interferometer is readjusted in order to obtain

φt.

11.2.3 Mach–Zehnder interferometer

The Mach–Zehnder interferometer is arranged following the outline of a transmission bridge; it is common in the optical range (the frequency range from millimeter waves up to the visible spectrum) where the electromagnetic waves propagate in free space. The monochromatic beam is split into two paths which are finally recombined. The sample is placed in one arm, and the changes in phase and the attenuation are measured by compensation; this procedure is repeated at each frequency. Provided the sample thickness is known, the refractive index n of the material is evaluated from the change in phase. The absorbed power is measured by a transmission measurement in a single-bounce configuration as discussed in Section 11.1.3, for instance. From the absorption coefficient α of the sample, the extinction coefficient k is determined using Eq. (2.3.18). Other material parameters, such as the complex conductivity σˆ or dielectric constant ˆ, are evaluated from n and k using the relations given in Section 2.3.

Fig. 11.10 shows the outline of a Mach–Zehnder type interferometer developed for the submillimeter wave region, i.e. from 2 cm−1 to 50 cm−1, based on backward wave oscillators as tunable and coherent sources [Koz98]. As an example of a measurement conducted using this interferometer, the transmission TF(ω) and phase shift φt(ω) obtained on a semiconducting TlGaSe2 sample (thickness 0.1 mm) is displayed in Fig. 10.1a and b. For each frequency, TF and φt are obtained by separate measurements. From these two quantities both components of

286	11 Measurement configurations

the complex dielectric constant ˆ(ω) are determined and are depicted in Fig. 10.1c and d. The close relation between the phase shift φt(ω) and the dielectric constant1(ω) or the refractive index n(ω) ≈ [ 1(ω)]1/2 is clearly seen.

In contrast to resonant techniques, which are limited to a narrow range of frequency, interferometric methods can in general be used in a broad frequency range. The most significant advantage of interferometric arrangements compared to single-path methods is the possibility – in addition to the attenuation of the radiation – of determining the phase shift introduced by the sample. Furthermore, the method has increased sensitivity by directly comparing the electromagnetic wave with a reference wave in a phase sensitive way. The interferometric method can be combined with resonant techniques to enhance the sensitivity further.

Fig. 11.11 displays the results of a transmission experiment [Pro98] performed on a metal film on a substrate. The niobium film (thickness 150 A)˚ was deposited on a 0.45 mm thick sapphire substrate, which acts as a Fabry–Perot resonator due to multireflection. The transmission through this arrangement is measured by a Mach–Zehnder interferometer in order to also determine the phase shift. In Fig. 11.11a the transmission TF through this composite sample is shown as a function of frequency, and the phase shift is displayed in Fig. 11.11b. As the temperature decreases below the superconducting transition Tc = 8.3 K, the transmitted power and phase shift are modified significantly. Since the properties of the dielectric substrate do not vary in this range of frequency and temperature, the changes observed are due to the electrodynamic properties of the superconductor. The change of the electrodynamic properties at the superconducting transition strongly changes the transmission and the phase of the composite resonator [Pro98]. The data can be used to calculate directly the real and imaginary parts of the complex conductivity σˆ (ω). The results, together with the theoretical prediction by the Mattis and Bardeen formulas (7.4.20), are displayed in Fig. 14.5.

11.3 Resonant techniques

Resonant methods utilize multiple reflection to increase the interaction of the electromagnetic radiation with the material under investigation. The fundamental concept of resonant structures were discussed in Section 9.3 where the technical aspects of these measurement configurations are summarized. The quality factor Q of a resonant structure – as defined in Section 9.3.1 – indicates the number of times the wave bounces back and forth in the resonator, and, roughly speaking, the sensitivity of the measurement by a resonant technique is Q times better than the equivalent non-resonant method. The increase in sensitivity is at the expense of the bandwidth – the major drawback of resonant techniques. In general the applicability of resonant structures is limited to a single frequency, and only in

11.3 Resonant techniques

	10− 2
			Nb film
			on sapphire
F	10	− 3
T			9 K
Transmission
	10	− 4	6 K

)	10− 5
−1
/ cm		9 K
(radfrequency/	0.6
	0.8
shiftPhase		6 K
	0.40	10	20
			Frequency ν (cm−1)

287

(a)

(b)

30

Fig. 11.11. Frequency dependent transmission TF(ω) and phase shift φ (ω) spectra of a niobium film on a sapphire substrate (0.45 mm) at two temperatures above and below the superconducting transition Tc = 8.3 K (after [Pro98]).

some cases are higher harmonics or different modes utilized. Resonant techniques are widely used in the gigahertz range (microwaves up to submillimeter waves) where non-resonant measurement techniques lack sensitivity. The experiments are usually performed in the frequency domain but the resonator can also be excited by a short pulse, and then the timely decay of the signal is observed. This so-called cavity ring-down method is preferable for extremely large Q factors.