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. 2020 May 13;53(Pt 3):734–740. doi: 10.1107/S1600576720005154

Measurement of single-crystal piezo modulus by the method of diffraction of synchrotron radiation at angles near π

P V Gureva a,*, N V Marchenkov a,b, A N Artemev a, N A Artemiev c, A D Belyaev a, A A Demkiv a, V A Shishkov b
PMCID: PMC7312137  PMID: 32684888

The diffraction response of a single crystal to an electric field is measured by X-ray diffraction at angles close to π. Such schemes allow one to determine with high (∼10−5–10−6) accuracy the relative changes in the lattice constant.

Keywords: crystal structure, lattice constant variation, diffraction at angles near π, synchrotron radiation, piezoelectric crystals, piezoelectric modulus

Abstract

This article presents measurements of the piezoelectric modulus d 11 of a single crystal of lanthanum gallium silicate (LGS, La3Ga5SiO14). The piezoelectric modulus was measured by X-ray diffraction at angles close to backscattering. Experiments in such schemes are very sensitive to relative changes in the lattice constant in crystals caused by external influences (constant or alternating electric field, mechanical load, temperature change etc.). The development opportunity of the technique is shown, its applicability is evaluated and results of measurement of the LGS single-crystal piezo modulus by the method of diffraction of synchrotron radiation at angles near π are discussed.

1. Introduction  

Piezoelectric crystals of the langasite family are widely used in various fields of electronics. The combination of a zero coefficient of thermal expansion with a high piezoelectric effect and the absence of a phase transition up to the melting point makes these crystals suitable for use under extreme conditions (Iwataki et al., 2001 ▸; Masatochi et al., 1999 ▸; Malocha, 2000 ▸).

Measurement of the characteristics of these crystals, including the piezoelectric effect, is an important step in the development of new devices.

The advantages of using X-ray diffraction methods to study the effect of external influences on the crystal are their nondestructive nature, spatial locality, determined by the cross section of the X-ray beam, and high sensitivity to deformation of the crystal structure. Owing to the similarity of the wavelength and the interatomic distances, the crystal is a natural diffraction grating for electromagnetic radiation of the X-ray range. When irradiated by an X-ray beam, the crystal forms a well defined diffraction pattern. Tiny changes (fractions of a percent) in the spatial configuration of atoms in a crystal lead to a significant change in this diffraction pattern. This makes it possible to observe the slightest deformation of the sample under investigation.

X-ray research methods were naturally improved with the use of synchrotron radiation. For example, X-ray diffraction of synchrotron radiation (at HASYLAB, Hamburg) was used by Gorfman et al. (2007 ▸) to study the piezoelectric properties of a BiB3O6 bismuth single crystal.

Methods of high-resolution X-ray diffractometry and topography were applied by Roshchupkin et al. (2008 ▸) (BESSY, Berlin) to study acoustic properties of lanthanum gallium tantalate crystals (LGT, langatate, La3Ga5.5Ta0.5O14) and polar properties of ferroelectric crystals of lithium niobate and lithium tantalate (LiNbO3 and LiTaO3) (Roshchupkin et al., 2013 ▸).

Internal deformations in piezoelectric BiB3O6 crystals caused by an electric field were studied by Schmidt et al. (2008 ▸) (HASYLAB, Hamburg).

The response of piezoelectric BiB3O6 and Li2SO4·H2O crystals to rapid changes in an applied electric field was studied by the method of synchrotron radiation diffraction (ESRF, Grenoble) with time resolution (Gorfman et al., 2010 ▸).

Among the large number of X-ray diffraction methods, one can distinguish the diffraction method at 2θ angles of the order of π (Shvydko, 2004 ▸; Lider, 2012 ▸).

The use of laboratory X-ray sources with their fixed set of emitted wavelengths significantly limits the range of problems that can be solved by this method (Freund & Schneider, 1972 ▸). Therefore, the use of synchrotron radiation, which allows one to smoothly change the radiation energy, is preferable.

Contrary to laboratory X-ray sources, the use of radiation emitted by high-brightness synchrotron sources allows the spatial resolution of this method to be improved and makes it possible to tune the wavelength in a wide range with high accuracy. The ability to control the wavelength allows a diffraction geometry in which the Bragg angle approaches π/2 and the diffraction angle approaches π. The key feature is a sharp increase in the sensitivity of the diffraction pattern to the slightest changes in the crystal structure. This allows the relative change in the lattice constant caused, for example, by an electric field or thermal load to be determined with high accuracy.

After the first work in this field (Köhra & Matsushita, 1972 ▸), and the subsequent steps in this direction (Bruommer et al., 1979 ▸; Caticha & Caticha-Ellis, 1982 ▸), a number of reports appeared where diffraction in the backscattering geometry was studied.

Graeff & Materlik (1982 ▸) showed that the energy resolution can reach the millielectron volt range, and Verbeni et al. (1996 ▸) reported the development of an X-ray monochromator with a resolution of 2 × 10−8.

Colella & Luccio (1984 ▸) proposed a new type of X-ray resonator, and Kikuta et al. (1998 ▸) achieved a resolution of a fraction of a millielectron volt.

Despite the large number of articles in which the measured piezoelectric parameters are presented, techniques to obtain more accurate measurements of such parameters, including those with high locality across the crystal, are still in high demand.

Kaurova et al. (2014 ▸) showed that the crystal lattice parameters vary from sample to sample cut from one growth boule of a langatate crystal (LGT). Such measurements are important for improving the processes of piezoelectric single-crystal growth.

Compared to the single-crystal geometry (Bond, 1960 ▸), X-ray diffractometry in the double-crystal geometry (Freund & Schneider, 1972 ▸) minimizes the influence of spectral bandwidth of a probe beam on the accuracy of the method. This scheme forms the basis of modern multi-crystal diffractometers and allows measuring diffraction curves with high resolution (Roshchupkin et al., 2004 ▸). Great opportunities are provided by the well developed but very difficult to implement method of registering diffracted beams at angles close to or even equal to backscattering. This method is characterized by high sensitivity to deformation of the unit cell at the level Δd/d ≃ 10−6–10−7 (Lider, 2012 ▸).

As was indicated above, the method of detecting diffracted beams at angles close to backscattering, that is 2θ ≃ π, has been known and studied since the beginning of the 1970s. Note that the application of this method was mainly associated with obtaining and using high energy resolution.

In this article, we use another aspect of this method: the high sensitivity of such diffraction to small relative changes in the crystal lattice parameters due to external influences on the crystal.

In our previous article (Artemiev et al., 2013 ▸), we presented advantages of the method of diffraction at large angles for polycrystalline samples in the Debye–Scherrer geometry. In this article we show the power of this technique for studying single-crystal samples.

2. Method of diffraction at angles near π  

The use of X-ray diffractometry in the backscattering geometry can be applied, for example, for studying the piezoelectric effect, when the characteristic crystal deformations are of the order of Δd/d ≃ 10−5. The task is to precisely determine the relative change in the lattice parameter from the angular displacement of the diffraction reflection curve in accordance with the Wulf–Bragg law. Such changes can be driven by an external electric field. This approach allows one to determine quantitatively and locally the piezoelectric modulus of the crystal, and to obtain this information by a direct method, based on deformation of the crystal at the unit-cell level. Estimations of the sensitivity of this method using the example of determination of the piezoelectric constants of a single crystal of lanthanum gallium silicate (LGS, langasite, La3Ga5SiO14) are shown below. The piezoelectric constants of an LGT crystal have been previously determined by high-resolution diffractometry (Blagov et al., 2013 ▸). This type of piezoelectric crystal is characterized by several remarkable properties, including the high value of the electromechanical coupling coefficient. Such crystals are often used in extreme conditions (Iwataki et al., 2001 ▸; Masatochi et al., 1999 ▸; Malocha, 2000 ▸) – at high temperatures and under strong electromagnetic fields. However, the behavior of their structural parameters and functional characteristics under extreme conditions are still poorly understood.

Differentiating the Wulf–Bragg equation at a fixed wavelength λ, we obtain an equation relating the angular shift of the diffraction curve to a change in the interplanar distance of the corresponding system of atomic planes:

2.

where the index s refers to the sample under study, θs is the Bragg angle of the sample and Δθs is the angular shift of the rocking curve (RC) with a change in the lattice constant Δd/d of the sample. Fig. 1 ▸ shows the relation between the angular shift of the RC and the Bragg angle θs, as we approach the backscattering geometry, for various fixed values of the relative lattice deformation Δd/d of the sample under study.

Figure 1.

Figure 1

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The Bragg angle shift (Δθs) at large Bragg angles for different lattice changes. (1) Δd/d = 10−6, (2) Δd/d = 10−5 and (3) Δd/d = 10−4.

Note that the sensitivity of the angular position of the diffraction curve to a change in the interplanar distance, as well as the width of the rocking curve, increases when approaching diffraction angles close to π.

To calculate the broadening of the rocking curve of a crystal, it is necessary to define the parameters of the probe beam incident on the sample after the monochromator. By differentiating the Wulf–Bragg equation at a fixed interplanar distance, and dividing the result by the original equation, we obtain the relation between the spectral width of the monochromated beam and the angular divergence of the beam after reflection from the monochromator:

2.

where the index m refers to the monochromator, Δλm is the spectral width of the radiation after the monochromator, λm is the wavelength corresponding to the center of the spectral bandwidth of radiation from the monochromator, Δθm is the angular divergence of the beam incident on (and diffracted from) the monochromator in the plane of diffraction and θm is the Bragg angle of the monochromator.

The FWHM of the double-crystal rocking curve of the monochromator and the sample ωe is determined by equation (3):

2.

where ωm, ωs are the intrinsic widths of the Bragg reflections of the monochromator and the sample, θm, θs are the Bragg angles of the monochromator and the sample, and Inline graphic is the relative width of the energy spectrum of a beam from the monochromator, defined by equation (2). The value ωe was calculated for the experimental design used in this work (see Section 4). The divergence of the beam incident on the channel-cut monochromator was determined as

2.

where sourcev is the vertical size of the X-ray source, slitv is the vertical slit size and L is the distance from the source to the slit. The values of ωm and ωs were calculated using a web calculator (http://xrayd.ru/polarizability/; Atknin et al., 2018 ▸). The spectral width Inline graphic of the beam from the monochromator makes the main contribution to ωe.

Fig. 2 ▸ shows the dependence of the rocking curve width of the sample on the Bragg angle θs as it approaches the backscattering geometry. The calculations were made for different monochromators, in accordance with equation (3). From Fig. 2 ▸ it is seen that, for diffraction angles close to π, the rocking curve widths become very large – up to thousands of arcseconds.

Figure 2.

Figure 2

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The width of the rocking curve as a function of diffraction angle for Δd/d = 10−5 when using different monochromators. Curve 1 corresponds to monochromator Si(220), curve 2 to Si(440) and curve 3 to Si(660).

The diffraction method for determining the piezoelectric modulus is based on measuring the shift of the diffraction peak depending on the relative change in the lattice constant of the crystal. This change is caused by an electric field applied to the piezoelectric crystal.

Determination of the position of the diffraction peak is the most important result of the measurement. The accuracy of measurement of the position of the diffraction peak is determined by a number of parameters, including its width.

Therefore, it is of interest to evaluate the ratio of the shift of the diffraction curve to its width as a function of diffraction angle when using various monochromators (Fig. 3 ▸).

Figure 3.

Figure 3

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The ratio of the rocking curve shift Δθs to the width of the rocking curve ωe for Δd/d = 10−5 as a function of Bragg angle, when using different monochromators. Curve 1 corresponds to monochromator Si(220), curve 2 to Si(440) and curve 3 to Si(660).

The value of the relative change in the crystal lattice parameter Δd/d = 10−5 is assumed for all the curves.

From Fig. 3 ▸ it is seen that, with an increase of the order of diffraction from the monochromator, the ratio increases, which allows us to obtain better accuracy. We also note that, with increasing diffraction angle, the ratio changes slightly.

3. Evaluation of the method  

When evaluating the capabilities of the large-angle diffraction method, the characteristic value of the relative deformation resulting from the inverse piezoelectric effect Δd/d = 10−5 was used. In this case, Fig. 1 ▸ shows that the angular shift of the center of the rocking curve at Bragg angles close to 90° noticeably exceeds hundreds of arcseconds (curve 3) for a sample in an electric field. This makes it possible to measure this shift with very high accuracy. The calculation shows that at such diffraction angles, mainly owing to the dispersion (determined by the difference in the Bragg angles of the sample and the monochromator), the corresponding rocking curve width (FWHM, Fig. 2 ▸) can also reach several hundred arcseconds.

A way to reduce the rocking curve width of the sample is to reduce the difference in the Bragg angles of the monochromator and the sample by selecting the reflection from the monochromator with the Bragg angle as close as possible to the Bragg angle of the sample.

Another way is to reduce the angular divergence of the ‘white’ beam Δθm incident on the monochromator.

The range of Bragg angles of the monochromator is limited by the geometry of the experiment and the interplanar distances available for the monochromator crystal, as well as the value of the wavelength of the monochromated radiation that is ‘convenient’ for the experiment. In its turn, the wavelength is determined by the sample crystal and the choice of the largest value of the diffraction angle (2θ).

Note that decreasing the rocking curve width decreases the statistical error of determining the center of this curve. Therefore, it is important to choose an experimental scheme that provides the maximal ratio of the angular shift of the diffraction peak to the width of the corresponding rocking curve as a result of the inverse piezoelectric effect. The larger this ratio, the more accurate a result can be obtained.

Fig. 3 ▸ shows how the ratio of the shift of the rocking curve to its width changes with scattering angle. The following values of the piezoelectric crystal parameters were used: LGS reflection 770, relative variation of the lattice constant of the piezoelectric crystal Δd/d = 10−5. It is seen that at Bragg angles of the sample close to 90° the ratio of the shift of the rocking curve to its width is about 0.4. This makes it possible to increase the measurement accuracy or reduce the time of the measurement of the piezoelectric modulus.

The tensile/compression piezoelectric modulus was calculated using the following equation:

3.

where t is the thickness of the sample and U is the magnitude of the applied voltage.

4. Experimental setup  

Fig. 4 ▸ shows the optical setup for measurements of piezoelectric modulus by X-ray diffraction at angles near π, implemented on beamline 6.2 of the Kurchatov synchrotron radiation source.

Figure 4.

Figure 4

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X-ray optical scheme of a synchrotron radiation spectrometer for measurements of piezoelectric modulus by the method of diffraction of X-rays at angles near π. SR – synchrotron radiation source; slit 1 × 1 mm, monochromator Si(111), monochromator Si(440); detector – photoelectron multiplier (NaI); sample – LGS(770).

The synchrotron radiation (SR) beam collimated by a slit of 1 × 1 mm falls on the preliminary Si(111) monochromator located 22 m from the source. This monochromator with a horizontal diffraction plane is used to cut off unnecessary parts of the SR spectrum, which would otherwise provide excessive heat load on the main monochromator and diffract from the main monochromator as harmonics. The channel-cut Si(440) main monochromator and the single-crystal sample have vertical diffraction planes.

A photoelectron multiplier (NaI) in counting mode was used as detector. The detector was mounted on the 2θ axis of the diffractometer, allowing a range of measurements of the diffraction angle from zero up to 174°. To measure larger diffraction angles up to 179° 30′, the detector was mounted on an arm that extended the distance between the sample and the detector up to 2 m.

To calibrate the wavelength, the positions of the Si(111), Si(333), Si(444) and Si(555) diffraction orders were measured with the help of a reference Si(111) single crystal installed on the diffractometer θ axis. Using angular distances between these positions we can accurately measure the wavelength of the probe beam (X-ray Data Booklet, 2009 ▸).

To measure the inverse piezoelectric effect, X cuts of the langasite piezoelectric crystals were used. The crystal size was 8 × 8 mm and the thickness 0.634 ± 0.001 mm. The crystals were grown by the Czochralski method by the Fomos Materials Company (http://newpiezo.com/). Silver electrodes (70 nm thick) with a chromium sublayer (30 nm) for better adhesion were deposited on opposite crystal surfaces.

Fig. 5 ▸ shows three of the 33 measured rocking curves. The solid lines represent five-parameter Lorentz function fits. Two of the five parameters are the background and the slope of the background to the X axis. The slope parameter determines the asymmetry of the peak. The contribution of this parameter to the width is small.

Figure 5.

Figure 5

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An example of the shifts of LGS(770) rocking curves. Red, blue and black dots are experimental points for zero, positive and negative voltage, applied to the crystal, respectively. Every rocking curve consists of 3000 experimental points. The solid lines represent five-parameter Lorentz function fits.

At the selected wavelength, the double diffraction angle for LGS(770) was 2θ = 178.23°. Fig. 5 ▸ shows LGS(770) rocking curves with and without an electric field [+997.7 ± 0.4 V and −999.3 ± 0.4 V; the signs (+) and (−) indicate the polarity of the applied voltage]. Each rocking curve consists of 3000 experimental points. The solid lines represent five-parameter Lorentz function fits considering linearly changing background. The obtained widths and shifts of the rocking curves are in good agreement with the calculations discussed in Section 5. The angular shifts of the rocking curves are approximately 250′′ when the voltage changes from negative to positive values.

Measurements of the rocking curves were performed in the following order of application of the external electric field: zero, negative, zero and positive field. This sequence of measurements was repeated. To allow for possible relaxation of the crystal a 2 min pause was maintained each time after switching the electric field. The duration of measurement of one rocking curve was about 7 min.

Thirty-three rocking curves were measured. In Fig. 6 ▸, the centers of gravity of the rocking curves of the LGS crystal (770) are shown as a function of time of the measurement. Triangles correspond to negative voltage on the crystal, circles to zero and squares to positive voltage.

Figure 6.

Figure 6

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Centers of gravity of LGS(770) rocking curves. X axis – time from start of experiment; Y axis – rocking curve gravity centers. Triangles correspond to negative voltage on the crystal, circles to zero and squares to positive voltage.

The time trend towards larger angles is clearly visible for the centers of gravity of all RCs measured with a zero, positive and negative field applied. Most probably, this shift is due to temperature drift of the entire beamline, including the SR source, the experimental setup and the sample.

5. Data processing  

To exclude the influence of the temperature drift on the accuracy of the measurements, the shift of the rocking curve due to the application of an electric field to the sample was calculated as follows. The positions of two adjacent zero-field rocking curves were averaged and subtracted from the position of the rocking curve measured with the field.

The average value of the shift of the rocking curve when applying a voltage of +997.7 V was 118.5′′. The shift at a voltage of −999.3 V was 121.2′′. (See error analysis below.) These shifts correspond to the change in the interplanar spacing Δd/d = 1.04 × 10−5. The piezoelectric modulus was calculated using equation (5) for 11 rocking curves corresponding to the positive voltage and the same number of curves corresponding to the negative voltage. |d 11| = 6.32 × 10−12 C N−1 at +997.7 V and |d 11| = 6.53 × 10−12 C N−1 at −999.3 V (Table 1 ▸).

Table 1. Results of the measurements of the langasite (770) crystal and the values of the obtained piezoelectric modulus d 11 .

  2θ (°)   2θ (°)  
Reflection (hkl) 0 V −999.3 V d 11 × 10−12 (C N−1) 0 V +997.7 V d 11 × 10−12 (C N−1)
770 88.94 88.97 6.53 ± 0.08 88.93 88.90 6.32 ± 0.08
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6. Error analysis  

A number of errors affect the accuracy of determining the value of the piezoelectric modulus d 11. Measurement errors can be divided into systematic errors and random errors. The random errors include

(1) the statistical nature of the photon flux on the detector,

(2) the temperature instability of the elements of the storage ring, the spectrometer and the sample.

The systematic errors include

(1) the influence of the finite angle between the surfaces of the crystal where electrodes are attached,

(2) the error of measurement of the sample thickness,

(3) the finite unknown angle between the Bragg planes and the surface of the sample,

(4) the error of measurement of the voltage applied to the sample,

(5) the error in measurement of the angular shifts of the rocking curves.

The statistical nature of the photon flux on the detector. The values of the rocking curve shift (118.5 and 121.2′′) obtained as a result of processing the data shown in Fig. 6 ▸ were considered as a set of independent measurements. A standard statistical processing was applied – determination of the r.m.s. values and confidence interval (±1.4′′).

The calculation of the absolute error was carried out according to formula (6), since (n < 30):

6.

where tαn is the Student coefficient, Shn is the mean square error of Sh, Shn is the value of the mean square error of the shift of the rocking curve (from probability theory; Fisher, 1934 ▸, p. 118) and n is the number of measurements. The absolute error of the shift of a rocking curve was 1.4′′ with a confidence probability of α = 68% (Fisher, 1934 ▸).

Temperature instability. The temperature instability of the elements of the synchrotron radiation storage ring, the spectrometer and the sample were taken into account by measuring the position of the rocking curve at zero field before and after each measurement with the field. Here we assume linear dependence of the rocking curve positions on this temperature instability on the timescale needed to measure three consecutive rocking curves – without the field, with the field and without the field.

Sample thickness. The thickness of the crystal is directly included in equation (5) for piezo modulus d 11. The thickness was measured at ten points along the surface of the crystal using a dial indicator with a graduation of 1 µm. Calibration of the indicator was carried out with gauge blocks (Johansson gauges). The average crystal thickness was 0.634 ± 0.001 mm. The relative error in determining the thickness, considering the incomplete parallelism of the crystal surfaces to each other, is estimated as 1.6 × 10−3.

The angle between the Bragg planes and the surface of the crystal. The applied electric field is orthogonal to the surfaces of the crystal, while the reaction of the crystal is determined by the component of this vector that is orthogonal to the Bragg planes. The crystal supplier (Fomos) estimates the considered angle to be 5′ (1.44 × 10−3). The corresponding relative correction factor is 1 × 10−6.

Electric field. The voltage applied to the electrodes deposited on the crystal surfaces was measured with a reference instrument: digital multimeter ARRA-505, voltmeter C511, Testo 622. The voltage values were −997.7 ± 0.4 V  and +999.3 ± 0.4 V; the relative error of the voltage value is 0.4 × 10−3.

Shift angle of the rocking curves. The systematic error in measuring the value of angular shift of the rocking curves can be determined by the accumulated error of the diffractometer and the correction for the refraction of X-rays in the crystal. These corrections in our case are less than 3 × 10−4.

It is clearly seen that the main contribution to the overall error is made by the statistical component.

The total relative measurement error of the piezoelectric modulus d 11 is the quadratic sum of the relative errors of the parameters considered above and is 1.1%.

The obtained values of the modulus are in good agreement with previously published data (Schreuer, 2002 ▸; Bohm et al., 2000 ▸; Weihnacht et al., 2012 ▸; Irzhak & Roshchupkin, 2018 ▸).

Note that the differences in the d 11 values presented in this article for the positive and negative voltages are not statistically proven.

7. Conclusion  

A method for precise measurement of piezoelectric modulus is proposed. An X-ray optical scheme for diffraction by angles close to backscattering is considered. Evaluation of the sensitivity of this method to variation of crystal lattice parameters due to external influences is performed. We show that, in the backscattering geometry, when the diffraction angle approaches π the shift of the rocking curve due to an external field increases by orders of magnitude compared with the forward-scattering geometry measurements.

Accurate measurement of very small shifts in forward-scattering geometry requires ultra-precise, very expensive equipment. It is much simpler and more accurate to measure large rocking curve shifts. Measuring large shifts seems to be a very convenient way to make quantitative assessments of the reaction of the crystal to an external electric field.

A synchrotron radiation spectrometer for measurements of crystal rocking curves at diffraction angles up to 179.5° has been developed. A langasite single crystal (La3Ga5SiO14) was studied and the values of the piezo modulus were precisely determined as |d 11| = (6.32 ± 0.01) × 10−12 C N−1 and |d 11| = (6.53 ± 0.01) × 10−12 C N−1 for different orientations of electric field. A number of factors affecting possible errors are analyzed.

Acknowledgments

The authors are very grateful to A. E. Blagov and V. G. Kohn for problem statement and useful discussions.

Funding Statement

This work was funded by Russian Foundation for Basic Research grant 19-32-90136.

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