The coherent gradient sensor for film curvature measurements at cryogenic temperature

Coherent Gradient Sensor (CGS) system is presented for measurement of curvatures and nonuniform curvatures changes in filmsubstrate systems at cryogenic temperature. The influences of the interface of refrigerator and itself on the interferograms which are accounting for the temperature effect are successfully eliminated. Based on the measurement technique, the thermal stresses (including the radial stress, circumferential stress and shear stress) of superconducting YBCO thin-film are obtained by the extended Stoney’s formula during the heating process from 30K to 150K. Take the superconducting YBCO thin film as an example, the thermal stresses of which are gained successfully. ©2013 Optical Society of America OCIS codes: (240.0310) Thin films; (120.3180) Interferometry; (120.6780) Temperature; (310.4925) Other properties (stress, chemical, etc.). References and links 1. P. A. Flinn, D. S. Gardner, and W. D. 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Introduction
Thin films deposited on various types of substrates are applied in many technologies, such as microelectronics, optoelectronics, thermal barrier coating technology, and microelectromechanical systems (MEMS), etc. Fabrication of such a film-substrate structure inevitably gives rise to stress in the film due to lattice mismatch, different coefficients of thermal expansion, chemical reactions, or other physical effects.Up to now, there are a few experimental techniques (including scanning laser method [1], multi-beam optical stress sensor [2], coherent gradient sensor [3][4][5][6], and X-ray diffraction [7], etc.) for stress measurement in thin films.Compared with other methods, the coherent gradient sensor (CGS), one type of shear interferometry, has distinguished advantages, including full field, real-time, non-destructive, noncontact, and vibration insensitivity, which is based on the observation of substrate curvature induced by this stress, and is gaining increasingly widespread use as diagnostic procedures [8][9][10].According to the mismatch in thermal expansion coefficient between the film and substrate subjected to a high temperature environment, Dong et al. [11] developed the CGS system to high temperature and presented the analysis expression of the stress based on the Stoney's formula [12] and its expansions [13][14][15][16][17].In addition, CGS system is always used to investigate the deformation of crack tip and facture characteristics in the facture-mechanics [3,[18][19][20], such as the crack tip deformation, stress intensity factor, etc.Some good results are achieved by the CGS system.
However, since the CGS system is applied to measure the curvature and curvature changed in thin film-substrate structures by Rosakis et al. [5] at first, and recently, Liu et al [21] give a theoretical error analysis of the CGS system at low temperature, there are few investigations on the CGS system at the cryogenic temperature.For the superconducting thinfilm systems, which are employed at the low temperature ambient (always explored by a vacuum closed cycle refrigerator with transparent interface), and the thermal stress has remarkable effects on its superconducting characteristics, e.g.critical current density [22].Thus, it is important to measure the thermal stresses of the superconducting thin film system during its cooling process.In this paper, a measurement device including CGS and low temperature Dewar is established firstly, and then a technique to reduce the influences of the interface on the obtained interferogram is presented.In the last, the thermal stresses (including the radial stress, hoop stress and shear stress between the thin film and substrate) of the superconducting YBCO thin-film during temperature increase process are obtained.

Experimental setup and processes
The CGS setup for cryogenic temperature measurement is illustrated in Fig. 1(a).A collimated laser beam passes through a beam splitter and is then directed to the reflecting specimen surface in the Dewar with a transparent window.The reflected beam from the specimen is further reflected by the beam splitter and then passes through two Ronchi gratings, G 1 and G 2 with the same density (40 lines/mm) separated by a distance  .The diffracted beams from the two gratings are converged to interfere using a lens.Either of the ± 1 diffraction orders is filtered by the filtering aperture to obtain the interferogram recorded by a CCD camera.Fig. 1(b) is displayed the actual equipment of the measurement system, in where the number 1 denotes the closed cycle refrigerator (G-M).The detail description of the CGS system will be neglected in this paper.Its process will be referenced by Dong et al. [11] and Liu at el [21].Here, we take emphasis on the technique how to eliminate the influences of interface on the interferogram.At first, in order to remove the effects of the reflected beam of the transparent window on the CGS system, a tilt of the transparent window will be conducted, which is illustrated in Fig. 2 (a).(a) schematic of the lean of transparent windows,  is the angle between the window and the horizontal plane, l denotes the width of the beam splitter, and h is the distance between the project plane of the transparent window and the bottom surface of the beam splitter, and while the thickness of quartz window is neglected, (b) Schematic of the effects of the quartz window on the interferogram, the dotted lines denote the normal of the window's surface.
From Fig. 2 (a), one can see that in order to avoid the reflected light of the transparent window into the CGS system, the angle  between the project plane of the transparent window and the bottom surface of the beam splitter should be satisfied by One can get that the two roots of the Eq. ( 1) are According to the law of refraction, one can see that there is no effect of the quartz window on the light propagation vector easily, which has effect on the location of the interferogram only.Thus, the change of the incidence vector will be decided by the air (its refractive index is equal to 2 n ) and the vacuum (the refractive index is equal to 1 n ).Based on our previous derivation in [21], the propagation vector of the light from the window's surface is satisfied by ) , ( 1) Because the interferogram of the CGS is formed from the propagation vector of the emergent light in air, the influences of the refraction caused by transparent window on the CGS system will be ignored.According to the equation presented by Liu et al. [21], in which    mentioned above into Eq. ( 4) one can obtain () 4 ( , ) , where     where xx  is the curvature in x direction, yy  is the curvature in y direction, and xy  denotes the twist curvature.It should be noted that the way we infer curvature and twist components is by numerical (spatial) differentiation of the surface gradient.Based on the above analysis, in order to obtain the curvature of the specimen' s surface, one can gain the phase distributions in x direction and y direction firstly.Thus, we now turn to how to obtain the phase information from the interferogram fringes.
The interferogram fringes originated from the CGS system can be gave as [23] ( , ) ( , ) ( , )cos ( , ), I x y a x y b x y x y   (10) where the first item includes the intensity information of the background, the second item denotes the vary information of the intensity of the interferogram fringes, in which ( , ) xy  is on behalf of the information of phase distributions, and its change is included the deformation of the specimen.The Eq. ( 10) can be rewritten as where , * c is the complex conjugate of c .Calculated by the Fourier transform, one can get .
By using the band-pass filter, the first item A , the second item C or the third * C should be eliminated.For the residual item C or * C , an inversion of the Fourier transform can be employed, one can obtain It is well know that the fabricated processes of the thin-film/substrate system inevitably introduce the nonuniform nucleation and /or misfit, which can result in serious thermo-stress due to temperature variation.One of the authors(Xue.F.) and his previous associates [15][16][17] had derived an extension of Stonry's formula for a multilayer thin-film/substrate system subjected to nonuniform and nonaxisymmetrical temperature distribution.In their work, they derive relations between the film curvatures temperature, and between the plate system's curvatures and the stresses.These relations featured a "local" part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a "nonlocal" part that reflects the effect of temperature of other points.In this paper, we will use the cylindrical coordinates to present the thermal stresses analysis.Then the nonuniform thin-film stresses form the nonuniform curvatures of the substrates can be expressed as [17]  where s h and f h are the thickness of the substrate and thin film, respectively.
R is the radius of the system,  


. s E is the Young's modulus of the substrate.s  and f  are the Poisson's ratio of the substrate and film, respectively.s  and f  denote the thermal expansion coefficients of the substrate and film, respectively.It should be noted that the thin-film nonuniform stresses are not only dependent on the local curvatures of the substrate, but they also related to the 'nonlocal' curvatures (average curvature).
In summary, in the above paragraphs, we present the CGS system at the cryogenic temperature.The influences of the reflected light from the quartz window's surface on the CGS interferogram fringes have been eliminated successfully.In addition, the effects of the refraction of the quartz window can be ignored by a mathematical derivation.Based on the obtained curvatures of the substrate, the nonuniform thermal stresses of the thin film/substrate system can be gained by the above Eqs.( 14)-( 18).

Substrate curvature measurement
The specimen consists of YBCO film grown by laser ablation on MgO substrate, which is the representative wafer structure widely used in superconducting researches.The thicknesses of the YBCO film and MgO substrate were 200nm and 500μm, respectively; their radius was 10mm.The geometry size agreed with the assumption fs h h R   .The specimen was placed vertically, as shown in Fig. 1(a).The back of the specimen was supported by a stiff frame made by Cu.Moreover, the contact between the specimen and the Cu support is conducted by Indium to make sure the well thermal conduction.The specimen could expand freely subjected to temperature, and there was no additional stresses induced by the boundary condition.As the temperature was elevated from 30K temperature to high temperature (e.g.~150K), the CGS interferograms were recorded by a CCD camera.Fig. 3 shows the interferograms obtained at 30K.The red fringes in Figs.3(a) and 3(c) represent the contour curves of the specimen surface slope in lateral (x direction) and vertical (y direction) directions, respectively.The wrapped phase map is calculated by FFT method and shown in Figs.3(b) and 3(d), respectively.Fig. 4(a) and (b) show the corresponding system curvatures distribution in x and y directions, respectively, while Fig. 4(c) shows the twist curvature distribution.It is obvious that the curvature distribution is nonuniform and thus violates the Stoney's formula assumption.The curvatures in the vicinity of the edge become much greater than those in the other area due to the edge effect.

Nonuniform stresses of the thin film
To calculate the film stresses at cryogenic temperature, we select the room temperature (about 297K) as a reference state.The physical parameters of the system are 123 mm, and 21  mm, respectively.The thermal expansion coefficients of the YBCO thin film and the MgO substrate are displayed in Fig. 5, one can see that with the increase of ambient temperature, their expansion coefficients increase with a close law [25].The thin film stresses for 30K are shown in Fig. 6.Fig. 6(a), 6(b), and 6(c) show the film stresses (radial direction), (circumferential direction), and (shear stress), respectively.Fig. 6(d) and 6(e) show the interfacial shear stresses (radial direction) and (circumferential direction) between the film and the substrate, respectively.The nonuniformity of the film stresses becomes more severe owing to the nonlocal effect shown in Eqs. ( 14)- (18).In addition, the interfacial stresses and with the magnitude of a few Pa are rather smaller compared with the film stresses.To investigate the thermo-stresses of thin film subjected to varied temperature, we conducted the experiment from 30K to 150K with the step of 10K.Then the full-field stresses can be obtained at the different temperatures following the same process as above.The film stresses of the central point in the specimen are selected to illustrate the thermo-stress evolution, as shown in Fig. 7. Due to the different coefficients of thermal expansion and inhomogeneous temperature distribution, the shear stress is formed in the interface, which results in a new stress distribution in the film plane.

Conclusions
The coherent gradient sensing (CGS) is features as full-field nonuniform curvatures measurement and vibration insensitivity, which is used to measure the thin-film/substrate system curvature at cryogenic temperature.Superconducting YBCO thin-film with MgO substrate is used to conduct the proposed CGS method.The stresses including the radial, circumferential and shear are obtained by using this technique.These results provide a fundamental approach to understand the thin-film (especially for superconducting thin-film system) stresses and the feasible measurement method for cryogenic temperature.

Fig. 1 .
Fig. 1.(a) Schematic of the CGS for cryogenic temperature, (b) photo of the measurement system, in which the number 1 denotes the closed cycle refrigerator (G-M).

Fig. 2 .
Fig. 2. (a) schematic of the lean of transparent windows,  is the angle between the window and the horizontal plane, l denotes the width of the beam splitter, and h is the distance the small  and hl  , we can obtain that tan is equal to distributions of the interferogram fringes in x direction and y direction,  denotes the distance between the two Ronchi gratings, p   , , xy ff denote the first-order derivative of the shape functions of the specimen in x direction #189294 -$15.00USD Received 22 Apr 2013; revised 17 Jun 2013; accepted 8 Jul 2013; published 25 Oct 2013 (C) 2013 OSAy direction, respectively.Therefore, the CGS governing equations for cryogenic temperature can be given by   denote the imaginary and real parts of the complex amplitude   , C x y .Subsequently, the phase distribution function   , xy  is fitted by using the Zernike polynomial[24] to obtain the phase distribution ( in-plane stresses of the thin film in the radial and circumferential directions, respectively.between the substrate and thin film in the radial and circumferential directions, respectively.$15.00 USD Received 22 Apr 2013; revised 17 Jun 2013; accepted 8 Jul 2013; published 25 Oct 2013 (C) 2013 OSA

Fig. 6 .
Fig. 6.The nonuniform stresses of thin film measured at 30K: (a) stress rr  in radial

rr
is equal to   at the beginning 30K then increases to 50Gpa (in compression) at 150K,   is equal to 90GPa (in compression) at the same temperature.The multi fluctuations of shear stress r  from tension to compression with the increase of temperature are observed which may result from the nonuniformity and the nonlocal effect.

Fig. 7 .
Fig. 7.The film stresses in radial, circumferential directions (a) and the shear stress (b) at the central point of the specimen vs. temperature.