Experimental determination of temperature dependence of refractive index of semiconductor materials
Only small group of specialists is familiar with the great potential capabilities of this method. However, they are not in a hurry to transit it on the solution of research and technological tasks into other industrial areas restraining the broad application of laser thermometry .
Method of laser interference thermometry (LIT) used in many operations provides the greatest temperature sensitivity of used signal. The values of physical constants, which characterize their optical properties, must be known for wide range of materials of electronics and optoelectronics. Scientific and technological interest in wide-band gap materials requires the obtainment of temperature dependences of refractive indices of materials at different wavelengths for every new material. LIT allows measuring the transient temperature of plane-parallel plate, and it is the most sensitive method of thermometry of semiconductor and dielectric substrates in microtechnology. This method is most frequently used for the studies of temperature conditions of substrates made of silicon, gallium arsenide, fused quartz and different glasses. The method diagram is given in Fig. 1.
LIT method is based on the fact that transparent or semitransparent plane-parallel plate is Fabry-Perot cavity standard for the probing light beam. In other words, both surfaces of plate play the role of two flat mirrors with the reflection coefficient R0 = (n – 1)2 / (n + 1)2 (n – refractive index), and the optical thickness of plate (nh) (h – geometrical thickness of plate) varies depending on temperature. On every surface the splitting of incident wave into two waves – reflected and transmitted – occurs. The difference of travelpath occurring between two beams of different orders (reflected or transmitted waves) (2nkh, where k = 2 π/λ – wave number, λ – wavelength of probing radiation) results in the occurrence of interference. As a rule, the intensity of reflected radiation is recorded because the contrast of interference in this case is higher than in transmitted light.
In order to measure the temperature, He-Ne laser with low power (5–10 mW) is usually used. Radiation with the wavelength λ = 633 nm is used for the measurement of temperature of wide-band semiconductor crystals (GaP, ZnS, ZnTe etc.). For semiconductor crystals with the width of band gap Eg ≤ 1.5–1.7 eV (Si, Ge, GaAs) radiation at the wavelengths 1.15, 1.52 and 3.39 nm is used.
If the tested sample at the initial moment has the temperature Т 1 and within certain period of time its temperature has changed to the value Т 2, then the variation of interferogram phase will be recorded
Δϕ = 2k [ n(T2) h(T2) – n(T1) h(T1) ].
If the phase shift exceeds 2π (one period of interferogram), the expression Δϕ = 2π (ΔN) can be introduced, where ΔN – number of interference periods. Knowing the dependences n(T) and h(T) the explicit dependence of temperature on the value ΔN can be found.
There is upper limit restricting the measurement of temperature of semiconductor monocrystals. It is stipulated by two reasons. One of them consists in the fact that there is shift of the edge of own absorption into long-wavelength region when heating the crystal. It should be taken into account that the influence of shift of absorption edge will not become apparent until the wavelength of probing light is farther from absorption edge (for example, for this purpose the line of 3.39 µm instead of 1.15 µm can be applied for Si). The second reason consists in the fact that with the increase of concentration of free charge carriers to certain level the semiconductor crystal becomes completely non-transparent within the whole spectral range. For silicon with the thickness of 0.5 mm the upper limit restricting the sensitivity of LIT method is located near 1 000 K at the wavelength 1.8 µm (for other used wavelengths the limit is lower than this temperature). For gallium arsenide the upper limit is located near 1300 K. For wide-band crystalline dielectrics the upper limit of measured temperatures concurs with the melting point; at the temperature higher than melting point the material softening, its viscous flow and plate deformation take place.
RECORDING AND PROCESSING OF INTERFEROGRAM
We have developed the automated device for the measurement of temperature dependence of refractive index n (T) of semiconductor and dielectric materials . The device diagram is shown in Fig. 2.
Helium-neon laser LGN-118–3V (NIIGRP "Plazma") with the generation at the wavelengths 0.633; 1.15 and 3.39 µm is the source of probing light. In experiment the sample is irradiated by one of three lines. The laser radiation power is 5–10 mW. The optical path consists of beam splitter, converging lens with the focal distance of 10 cm and photodetector. The diameter of probing laser beam at the front-face surface of sample is equal to ~0.3 mm, at the same time for the majority of samples high contrast of interference is reached. Studied samples have the shape of plane-parallel plates with the dimensions ~5 Ч 5 mm 2 and thickness h = 0.1–1 mm.
The sample was fixed in aluminum unit with the help of heat conducting glue. The sample temperature is measured by thermocouple brought to the sample backside and glued to its surface near the region, which is probed by laser beam. The time of sample heating from room temperature to Т " 600–700 K is approximately 30 min., the further time of cooling to Т ≈ 350K is about 1.5 hours.
Radiation reflected from the sample with the wavelength λ = 0.633 µm is detected by silicon photodiode FD25K, with λ = 1.15 µm – by germanium photodiode FD-7G, with λ = 3.39 µm – by radiation thermoelement RTN-10G (with germanium aperture). All elements of optical circuit are secured on optical bench, and the capability of adjustment is provided. The connection of temperature-sensitive element and photodetector to computer is performed using two-channel module E-270.
The special program is developed for recording and processing of the data obtained during experiment. Simultaneously, the dependences of sample temperature T(t) and intensity of reflected light I(t) on time are recorded upon heating and cooling of the sample. Then, the time is excluded, and using obtained interferogram I(T) the determination of sought for dependence n(T) is performed. The shift by one interference band corresponds to the variation of optical thickness by wavelength half: D(nh) = l/2.
Correctness of crystal temperature measurement was checked in the following manner. Condition for interferogram extremums obtainment is fulfilled notwithstanding whether the sample temperature increases or decreases. For example, for intensity minimums the condition nh = 0.5 lm is fulfilled, where n and h – refractive index and crystal thickness, l – wavelength, m = 0, 1, 2,... Therefore, the main check of sample temperature measurement correctness consists in the comparison of interferograms I(T) obtained during the heating and cooling. If the temperatures, at which extremums of reflected light are obtained, concur, then the temperature difference between the points of measurement and probing is negligible. If the temperatures of interference extremums upon heating and cooling significantly differ, then the temperature measured by thermocouple differs from the actual sample temperature in the area of probing.
As our measurements have proved, when thermocouple is connected to studied sample at the distance "1 mm from the place of light beam incidence, good concurrency of interferogram extremums on the axis of temperatures is observed – the discrepancy of same-name extremums does not exceed 1оС (Fig. 3).
RESULTS OF EXPERIMENTS
All measurements were started at the room temperature (20°C) and performed up to 400–450°C. Initial data by n(T) and coefficients of thermal expansion of materials was taken from reference books and articles. The dependence h(T) was calculated on the basis of the expression h(T) = h(T1) [ 1 + α(T–T1) ] where the known parameters were inserted.
Temperature dependence n(T) of silicon monocrystal was determined in the paper .
Monocrystal ZnO. Zincite (ZnO) refers to the number of advanced materials for the most progressive technical devices with various intended purposes, which function on the basis of three-dimensional elements and thin films. Combination of anisotropic crystalline structure with wide band gap, luminescent properties, photo-sensibility, radiation resistance, photoconductivity of ZnO are truly unique. Zincite monocrystals can be used in the equipment intended for the control of stress degree of mechanical structures, during the measurements of alternating and quasi-static pressure, in defectoscopy within wide range of temperatures, in production of light guides, gas sensors and UV lasers. Width of ZnO band gap (at Т = 300K) is equal to 3.33 eV. Temperature dependence of ZnO refractive index n (T) was measured and published . The following data was used in calculations: thermal coefficient of linear expansion αt = 2.92 · 10–6 (K–1); refractive index for the wavelength of 0.5 µm with regard to ordinary beam no = 1.989. Thickness of studied crystal was h = 578 µm. Temperature dependence n(T) is given in Fig. 4 .
Gallium arsenide GaAs. Undoped semi-insulating GaAs with high specific resistance (107 Ohm.cm) is used for the production of high-frequency integrated circuits and discrete microelectronic devices. GaAs heavily doped by silicon with the conductivity of n-type is used for the production of light emitting diodes and lasers. Monocrystals of GaAs heavily doped by silicon (1017–1018 cm–3) are widely used in optoelectronics for the production of injection lasers, light emitting diodes and photodiodes, photocathodes, and they represent good material for the generators of SHF oscillations. They are applied for the production of tunnel diodes, which are capable to operate at higher temperatures in comparison with silicon diodes, and at higher frequencies in comparison with germanium diodes. Monocrystals of semi-insulating gallium arsenide doped by chromium are used in infrared optics.
Currently, GaAs is considered in the capacity of the most probable materials for photoelectric systems of solar energy conversion.
PET (photoelectric transducers) based on GaAs have higher theoretical efficiency in comparison with silicon PET because the width of band gap in them practically concurs with the optimal width of band gap for solar energy semiconductor converters (1.4 eV). In silicon PET this parameter is equal to 1.1 eV.
In addition, gallium arsenide is used for the production of lenses and beam splitters, and it is the alternative to ZnSe in optical systems of continuous CO2 lasers with medium and high power.
Dependence n(T) of GaAs was studied using the single crystal with orientation (100), thickness h = 485 nm at the wavelength λ = 1.15 µm. The value of refractive index n = 3.44 (at Т = 300K) was used for calculations; thermal coefficient of linear expansion was αt = 5.82·10–6 K–1 .
On the basis of results of several measurements for the interval (20–400°C) the following data was obtained: n(T) = 3.385335 + 1.26897 · 10–4 T + 1.78814 · 10–7 T 2. The graph n(T) for GaAs is shown in Fig. 5.
Gallium phosphide GaP. Development of electronic components on the basis of gallium phosphide is the part of extensive and branch program of creation of element base of high-temperature electronics or electronics aimed at high-temperature applications. Relevance of generation of such element base is stipulated by the further development of such high-priority areas of current technology as aerospace technologies and matters connected with nuclear reactor safety, deep-hole drilling and solar power engineering, design of robot devices for operation under extreme conditions.
Single-crystal gallium phosphide is the basic material for creation of light emitting diodes with red, red-orange, orange and yellow luminescence applied in large color screens, traffic maintenance equipment and architectural lighting; it is used for the production of optical lenses and laser lenses.
Measurement of gallium phosphide n(T) was performed at two wavelengths – 0.633 µm and 1.15 µm. Width of studied single crystal was h = 313 µm. Initial data for calculations: refractive indices: n = 3.31 (λ = 0.633 µm) and n = 3.13 (λ = 1.15 µm); coefficient of thermal expansion αt = 5.9 · 10–6 K–1.
As a result of these studies, the experimental dependences of photo-electromotive force on temperature were obtained and temperature dependences of refractive index of GaP monocrystal were calculated for two wavelengths:
n(T) = 1,40364 · 10–4 T + 1,12906 · 10–7 T2 + 3,25898
for λ = 0,633 мкм;
n(T) = 1,15634·10–4 T + 4,14194 · 10–8 T2 + 3,09187
for λ = 1,15 мкм.
These dependences are shown graphically in Fig. 6 and 7 respectively.
Zinc selenide ZnSe (CVD). Zinc selenide is used in the capacity of material for the production of optical elements: windows, lenses, mirrors, prisms, beam splitters etc. operating within IR range. Though this material is polycrystalline, it is characterized by homogeneous structure, high transmission in IR region and low internal losses connected with absorption and scattering. Zinc selenide is most often used for the production of components for CO2-lasers (including high-power lasers) and wide-band spectral devices operating within the range of 0.6 to 19 microns. It is also suitable for the production of imaging optics. Zinc selenide has the required set of physical and chemical parameters for the creation of light emitting diodes with moderately blue radiation on its basis.
Measurement of n(T) was performed at two wavelengths – 0.633 µm and 1.15 µm. Thickness of studied monocrystal was h = 2.13 mm. Refractive index at Т = 293K: for λ = 0.633 µm n = 2.59; for λ = 1.15 µm, n = 2.47 . Thermal coefficient of expansion was αt = 7.1 · 10–6 K–1 .
Due to significant thickness of crystal and low heat conductivity (13 W/(m · K)), despite the low speed of heating, minimums for heating and cooling partially did not concur. Therefore, the calculations were carried out on the basis of minimums for heating and minimums for cooling. The result can be compared, and the error can be evaluated.
As a result of a number of experiments, the dependences n (T) were obtained (by minimums for heating):
n(T) = 2,56403 + 7,384033 · 10–5 T + 4,8053767 · 10–8 T2
for λ = 0,633;
n(T) = 2,45322 + 4,78421 · 10–5 T + 2,95636 · 10–8 T2
for λ = 1,15.
Measurement results are given in graphic form in Fig. 8 and 9.
Diamond С. Exclusive hardness of diamond finds its application in industry: it is used for the production of knives, drills, cutters and similar goods. Diamond powder (as waste during the treatment of natural diamond and powder obtained in artificial manner) is used as the abrasive for the production of cutting and grinding blades, rings etc. The development of microelectronics based on diamond substrates is very prospective. There are finished products, which have high thermal and radiation resistance. Also, the use of diamond in the capacity of active element in microelectronics, especially high-current and high-voltage electronics, is prospective due to significant value of breakdown voltage and high heat conductivity.
Measurement of n(T) was performed using the single crystal of natural diamond with the thickness h = 366 µm at the wavelength λ = 0.633 µm. The refractive index of diamond for this wavelength at room temperature was assumed to be n = 2.412 ; coefficient of thermal expansion was αt = 1 · 10–6 K–1. As a result of measurements and calculations, the sought-for dependence n (T) was obtained: n(T) = 2.4117 + (8.82 ± 0.34) · 10–6 T + (2.96 ± 0.10) · 10–8 T2. This dependence is shown graphically in Fig. 10.
PROSPECTIVES OF LIT METHOD
Laser interference thermometry is used in the studies carried out under difficult experimental conditions: upon exposure of charged particle beams on the sample or high-power optical radiation, in non-equilibrium plasma etc. In productivity, interference immunity and accuracy of temperature measurements, the level, which is typical for other optical methods of diagnostics, has been reached. Technological studies using LIT are often carried out in the course of the physical and chemical processes of treatment of metals, semiconductors and dielectrics during the creation of integrated circuits.
In LIT method the temperature sensitivity of recorded signal is by 10–100 times higher than in case of use of thermocouples. Laser radiation has several characteristic features (wavelength, polarization, direction of propagation, modulation of intensity etc.), which make it possible to differentiate it reliably against the background of intense interference. Such measurement characteristics of LIT allow believing that in future the method will be used not only in microtechnology but in other areas as well.
Currently, there is number of problems, which prohibit the spread of LIT, in studies and technological control. With regard to temperature dependences of optical parameters of the majority of materials, there is no data in the literature in sufficiently wide range of wavelengths and temperatures (from cryogenic region to melting point). For many materials data was obtained within narrow range in proximity of room temperature. There is no metrological support of LIT. The experimental comparison of LIT with other methods was not performed. Data on the optical properties of solid bodies given in different papers noticeably differ, and it requires additional experiments and analysis of methodical errors.
Measurement of dn/dT using the method of interferometry with the careful control of sample temperature has significant value for the expansion of application range of laser interference thermometry.