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Dispersion of the acoustic phonon concerned with C66 of KD2PO4 observed by light scattering
Solid State Communications, Vol. 35, pp. 285—287. Pergamon Press Ltd. 1980. Printed in Great Britain. DISPERSION OF THE ACOUSTIC PHONON CONCERNED WITH C~OF KD2PO4 OBSERVED BY LIGHT SCATTERING H. Tanaka and I. Tatsuzaki Research Institute of Applied Electricity, Hokkaido University, Sapporo 060, Japan (Received 27 March 1980 by Y. Toyozawa) Wave vector dependence of the light scattering spectra of DKDP is investigated for the scattering angles ir/4, ir/2 and 3ir/4. Brillouin shift concerned with an elastic stiffness C66 is obtained from the angular dependence of the quasi-longitudinal and quasi-transverse mode frequencies in the ab plane observed in the VV scattering. The results indicate that the phase velocity associated with C~decreases slightly near Tb. in the case of ir/4 scattering, but does not in ir/2 and 3ir/4 scatterings.
PIEZOELECTRIC KD2PO4(DKDP) undergoes a ferroelectric phase transition at 220 K. The fluctuation in the electric polarization P3 with B2 symmetry increases anomalously near the transition temperature Ttr and couples linearly with the acoustic shear mode x,, even above ~ According to ultrasonic studies [1] the elastic stiffness C~obtained from the sound velocity is identified with C~and decreases to a considerable extent as T—* Ttr. On the other hand, neutron inelastic scattering by Cowley et al. [2] shows that the frequency ,
6328 A light than with 4880 Alight. These results suggest that we can pursue the acoustic phonon peaks up to lower T~,by measuring the spectra of smaller scattering vector q, which are realized by using the smaller scattering angle or the laser of the longer wavelength.
L~
of the acoustic mode at large values of the wave vector
q corresponds C~which The relaxation is almost to, rather rate independent of than theoccurs C~,an fluctuation of temperature. P3 stiffness fromT-~ ultraphonon and30GHzforT-±T~.atIT—Ttrl=1K[1],and sonic measurements frequencies measured isfrom estimated by to Brfflouin be 4the GHz scattering for inT~ changes fact characteristic lie inits this dynamics behavior frequency which region the fast [3, relaxation 4] aselastic Therefore, system regime to slow relaxation one will be observed [5,6] in the wave vector region concerned with Brfflouin scattering. The intermediate behavior can be studied through not only the temperature dependence of the relaxation rate but also the q dependence of the frequency of the acoustic phonon. It is reported first by Reese et al. [3] with 6328 A wavelength light and later by Sawafuji et al. [4] with 4880 A that the acoustic phonon peaks are “washed .
out” by the broad central component associated with the fluctuation ofF3 within several degrees of Ttr as T-÷Ti,. in the x + z(—x + z,y) —x + z scattering geometry. Comparing both experiments with each other, we can see that the temperature T~where the phonon peak merges into the shoulder of the broad central component, is lower for the spectra measured with 285
_______ =
(3 • 2.5
I
-12.0
I
I
I
I
-8.0 —4.0 0 FREQUENCY SHIFT
I
I
4.0 8.0 (GHz)
12.0
Fig. I. jql dependence of the VH spectra with q II [100] in the ac plane. Scattering angles; (A) ir/4, (B) ir/2, (C) 3ir/4. Free spectral range of the Fabry—Perot used is 29.82 GHzatand finesse is about are taken the the temperature T = Ttr70. + These 4.5 K,spectra where Ttr of this sample is 52.5°C.Different gains of detection are denoted by G. —
286
DISPERSION OF THE ACOUSTIC PHONON CONCERNED WITH C~OF KD2PO4 Vol. 35, No.3 I
I
I
30
7/
25~,
15
20
L’
-
N
-
~ 15-
10
0
I 60
50
40
I 30 20 4, (deg.)
I IC
I 0
(b)
-
T’
-
-
-
III 50 40
I I 30 20 4, (deg.)
10
I 0
Fig. 2. q dependence of the quasi-longitudinal L’ and quasi-transverse T’ mode frequencies in the ab plane. Open and closed circles are observed Brillouin shifts in the VV spectra. The solid lines are calculated ones so as to give a good fIt of equation (2) in the text to the observed Brillouin shifts. Scattering angles; (a) ir/4, (b) 3ir/4. Temperature dependences are denoted by the closed circles: ~T = T Tfr = 71 K and open circles; (a) LXT = 1.2 K, (b) L~T= 0.05 K. Ttr of this sample is —52.56 ±0.05°C. —
In this letter we report some preliminary results obtained in ii]4 and 31T/4 scatterings using 4880 A light, in addition to the conventional ir/2 scattering. In the scattering angle of ir/4, the stray light from the surface of a sample often enhances the intensity of Rayleigh line and prevent us from studying the detailed structure of low frequency side of spectra. The cryostat designed specially for the present purpose, however, makes it possible to obtain a small ‘RI’B as seen in Fig. 1, where ‘R and ‘B are intensities of Rayleigh and Brillouin lines, respectively. The experimental set up is essentially the same [7] as that described previously except the cryostat.* Figure 1 shows the typical profile of I~Idependent *
The cryostat will be described elsewhere,
spectra measured in the VH scattering geometry in the ac plane at q II [1001 As expected from the matter mentioned above we can recognize the resonant peaks associated with C~in the ir/4 scattering angle up to about 4.5 K above Ttr(Tw Ttr 4.5 K), which should be compared with T~ Ttr = 10—12 K in the 3ir/4 scattering angle. In this scattering geometry, however, the phonon frequency deduced from ultrasonic measurements lies between the resonant peak and “dip” as is shown by Reese et al. [3] The characteristic behavior of the spectrum, where the phonon system couples linearly with the fluctuation of another system, has been studied theoretically [5, 6] In addition to the development of a broad central component, the extent of softening of a phonon depends on the characteristic parameters z0 in [5] or ~ and ~ in [6] The spectra in .
—
—
.
.
.
Vol. 35, No.3 DISPERSION OF THE ACOUSTIC PHONON CONCERNED WITH C~OF KD2PO4
287
Fig. 1 correspond to, in fact, the display of the q dependence of the characteristic parameter Z~ or In order to obtain further information on the q dependence of the acoustic phonon, measurements are made in the VV scattering in the ab plane. In this scattering geometry we can observe two acoustic modes, the quasi-transverse mode T’ and the quasi-longitudinal mode L’, without the broad central component P3. Frequencies of both modes with wave vector q whose direction cosines are (cos sin 0) are expressed in the following form 2(cos2~)2 2 = (q)2 ~ ~(C~ + C~)±[(C11 C~)
v~(cb= 45°)giveelastic stiffness C11, C6 and (C11 + C~6)/2 in equation (2), they are denoted by v(C11), v(C~)and v[(C11 + C6*~)/2],respectively hereafter. The solid lines in Fig. 2, which are calculated based on the equation (2), show comparatively good agreement with the measured values of v±(~). The essential results of the present q dependent VV scattering study are summarized as follows: As temperature approaches Ti,. both v(C11) and v[(C11 + C6~)!2] become large in the ir/2 and 3ir/4 scatterings. In the ir/4 scattering, on the other hand, v[(C11 + C~)/2] still remains near the value at room temperature although v(C 11) show similar temperature dependence to those in the and 344 scatterings. Therefore, v(C~6) the 2(sin2~)2]1/2} ~ (1) ir/4 ir/2 scattering should show softening to such an inextent + (C12 + C~) that cancels the hardening of v(C 11) in the term of where v~and v_correspond to L’ and T’ modes, respecv[(C11 + C~)/2]at q II [110]. These results strongly tively. The frequency shifts of both Brilouin spectraL’ suggest that there is the dispersion of the phase velocity and T’ are shown in Fig. 2(a) and (b) for the scattering concerned with the x,, shear mode in the close vicinity angle 44 and 3ir/4. Although the spectrum of mode L’ of the Brillouin zone center near Ttr. Detailed results is seen at any angle of~,the peak intensity of mode T’ will be published elsewhere in the near future. tends to zero as q approaches the a axis because of a factor sin q contained inherently in the selection rule of Acknowledgements The authors wish to express their this scattering geometry [4]. In addition, it should be thanks to Dr. M. Tokunaga for his valuable discussions. noted that the broadening of the line width of the mode The present research was supported in part by GrantT’ is observed as the temperature and 0 approach Ti,. in-aid for Scientific Research from Ministry of Education of Japan. and zero, respectively. At q II [110], we can observe only one phonon peak (whose intensity is 1.5—2 times REFERENCES as large as that of each spectrum ofT’ andL’ mode 1. E. Litov & E.A. Uehling,Phys. Rev. Bi, 3713 near q [110]) because the intensity of mode T’ is zero (1970). at q II [110] by the selection rule. Consequently, we 2. R.A. Cowley, W.J.L. Buyers, E.C. Svensson & cannot obtain the precise value of C12 + C~directly. G.L. Paul ,Neutron Inealstic Scattering, Vol. 1, p. 281. International Atomic Energy Agency, Let us assume that C12 + C~= 0 for simplicity. Vienna (1968). Equation (l)is thus rewritten in the form 3. R.L. Reese, Ii. Fritz & H.Z. Cummins,Phys. ~.
~,
~,
—
—
=
;1
(~) q
2
~((C11+ C~)±(C11
—
C~)cos 20}. (2)
Since the Brilouin shift v~(çb= 0), v.(0 = 0) and t him c~ ,-~E 66 ~-~66 —
q-~O
4.
Rev. B7,4l65 M. M.(1973). Tokunaga & I. Tatsuzaki,J. Phys. Soc.Sawafuji, Japan 47, 1860 (1979).
5.
A.P. Levanyuk & A.A. Sobyanin, Soy. Phys. JETP 26,612(1968). Y. Yamada, H. Taketra & D.L. Huber, J. Phys. Soc. Japan.36, 641 (1974). H. Tanaka & 1. Tatsuzaki,J. Phys. Soc. Japan 47, 878 (1979).
6. 7,
PIEZOELECTRIC KD2PO4(DKDP) undergoes a ferroelectric phase transition at 220 K. The fluctuation in the electric polarization P3 with B2 symmetry increases anomalously near the transition temperature Ttr and couples linearly with the acoustic shear mode x,, even above ~ According to ultrasonic studies [1] the elastic stiffness C~obtained from the sound velocity is identified with C~and decreases to a considerable extent as T—* Ttr. On the other hand, neutron inelastic scattering by Cowley et al. [2] shows that the frequency ,
6328 A light than with 4880 Alight. These results suggest that we can pursue the acoustic phonon peaks up to lower T~,by measuring the spectra of smaller scattering vector q, which are realized by using the smaller scattering angle or the laser of the longer wavelength.
L~
of the acoustic mode at large values of the wave vector
q corresponds C~which The relaxation is almost to, rather rate independent of than theoccurs C~,an fluctuation of temperature. P3 stiffness fromT-~ ultraphonon and30GHzforT-±T~.atIT—Ttrl=1K[1],and sonic measurements frequencies measured isfrom estimated by to Brfflouin be 4the GHz scattering for inT~ changes fact characteristic lie inits this dynamics behavior frequency which region the fast [3, relaxation 4] aselastic Therefore, system regime to slow relaxation one will be observed [5,6] in the wave vector region concerned with Brfflouin scattering. The intermediate behavior can be studied through not only the temperature dependence of the relaxation rate but also the q dependence of the frequency of the acoustic phonon. It is reported first by Reese et al. [3] with 6328 A wavelength light and later by Sawafuji et al. [4] with 4880 A that the acoustic phonon peaks are “washed .
out” by the broad central component associated with the fluctuation ofF3 within several degrees of Ttr as T-÷Ti,. in the x + z(—x + z,y) —x + z scattering geometry. Comparing both experiments with each other, we can see that the temperature T~where the phonon peak merges into the shoulder of the broad central component, is lower for the spectra measured with 285
_______ =
(3 • 2.5
I
-12.0
I
I
I
I
-8.0 —4.0 0 FREQUENCY SHIFT
I
I
4.0 8.0 (GHz)
12.0
Fig. I. jql dependence of the VH spectra with q II [100] in the ac plane. Scattering angles; (A) ir/4, (B) ir/2, (C) 3ir/4. Free spectral range of the Fabry—Perot used is 29.82 GHzatand finesse is about are taken the the temperature T = Ttr70. + These 4.5 K,spectra where Ttr of this sample is 52.5°C.Different gains of detection are denoted by G. —
286
DISPERSION OF THE ACOUSTIC PHONON CONCERNED WITH C~OF KD2PO4 Vol. 35, No.3 I
I
I
30
7/
25~,
15
20
L’
-
N
-
~ 15-
10
0
I 60
50
40
I 30 20 4, (deg.)
I IC
I 0
(b)
-
T’
-
-
-
III 50 40
I I 30 20 4, (deg.)
10
I 0
Fig. 2. q dependence of the quasi-longitudinal L’ and quasi-transverse T’ mode frequencies in the ab plane. Open and closed circles are observed Brillouin shifts in the VV spectra. The solid lines are calculated ones so as to give a good fIt of equation (2) in the text to the observed Brillouin shifts. Scattering angles; (a) ir/4, (b) 3ir/4. Temperature dependences are denoted by the closed circles: ~T = T Tfr = 71 K and open circles; (a) LXT = 1.2 K, (b) L~T= 0.05 K. Ttr of this sample is —52.56 ±0.05°C. —
In this letter we report some preliminary results obtained in ii]4 and 31T/4 scatterings using 4880 A light, in addition to the conventional ir/2 scattering. In the scattering angle of ir/4, the stray light from the surface of a sample often enhances the intensity of Rayleigh line and prevent us from studying the detailed structure of low frequency side of spectra. The cryostat designed specially for the present purpose, however, makes it possible to obtain a small ‘RI’B as seen in Fig. 1, where ‘R and ‘B are intensities of Rayleigh and Brillouin lines, respectively. The experimental set up is essentially the same [7] as that described previously except the cryostat.* Figure 1 shows the typical profile of I~Idependent *
The cryostat will be described elsewhere,
spectra measured in the VH scattering geometry in the ac plane at q II [1001 As expected from the matter mentioned above we can recognize the resonant peaks associated with C~in the ir/4 scattering angle up to about 4.5 K above Ttr(Tw Ttr 4.5 K), which should be compared with T~ Ttr = 10—12 K in the 3ir/4 scattering angle. In this scattering geometry, however, the phonon frequency deduced from ultrasonic measurements lies between the resonant peak and “dip” as is shown by Reese et al. [3] The characteristic behavior of the spectrum, where the phonon system couples linearly with the fluctuation of another system, has been studied theoretically [5, 6] In addition to the development of a broad central component, the extent of softening of a phonon depends on the characteristic parameters z0 in [5] or ~ and ~ in [6] The spectra in .
—
—
.
.
.
Vol. 35, No.3 DISPERSION OF THE ACOUSTIC PHONON CONCERNED WITH C~OF KD2PO4
287
Fig. 1 correspond to, in fact, the display of the q dependence of the characteristic parameter Z~ or In order to obtain further information on the q dependence of the acoustic phonon, measurements are made in the VV scattering in the ab plane. In this scattering geometry we can observe two acoustic modes, the quasi-transverse mode T’ and the quasi-longitudinal mode L’, without the broad central component P3. Frequencies of both modes with wave vector q whose direction cosines are (cos sin 0) are expressed in the following form 2(cos2~)2 2 = (q)2 ~ ~(C~ + C~)±[(C11 C~)
v~(cb= 45°)giveelastic stiffness C11, C6 and (C11 + C~6)/2 in equation (2), they are denoted by v(C11), v(C~)and v[(C11 + C6*~)/2],respectively hereafter. The solid lines in Fig. 2, which are calculated based on the equation (2), show comparatively good agreement with the measured values of v±(~). The essential results of the present q dependent VV scattering study are summarized as follows: As temperature approaches Ti,. both v(C11) and v[(C11 + C6~)!2] become large in the ir/2 and 3ir/4 scatterings. In the ir/4 scattering, on the other hand, v[(C11 + C~)/2] still remains near the value at room temperature although v(C 11) show similar temperature dependence to those in the and 344 scatterings. Therefore, v(C~6) the 2(sin2~)2]1/2} ~ (1) ir/4 ir/2 scattering should show softening to such an inextent + (C12 + C~) that cancels the hardening of v(C 11) in the term of where v~and v_correspond to L’ and T’ modes, respecv[(C11 + C~)/2]at q II [110]. These results strongly tively. The frequency shifts of both Brilouin spectraL’ suggest that there is the dispersion of the phase velocity and T’ are shown in Fig. 2(a) and (b) for the scattering concerned with the x,, shear mode in the close vicinity angle 44 and 3ir/4. Although the spectrum of mode L’ of the Brillouin zone center near Ttr. Detailed results is seen at any angle of~,the peak intensity of mode T’ will be published elsewhere in the near future. tends to zero as q approaches the a axis because of a factor sin q contained inherently in the selection rule of Acknowledgements The authors wish to express their this scattering geometry [4]. In addition, it should be thanks to Dr. M. Tokunaga for his valuable discussions. noted that the broadening of the line width of the mode The present research was supported in part by GrantT’ is observed as the temperature and 0 approach Ti,. in-aid for Scientific Research from Ministry of Education of Japan. and zero, respectively. At q II [110], we can observe only one phonon peak (whose intensity is 1.5—2 times REFERENCES as large as that of each spectrum ofT’ andL’ mode 1. E. Litov & E.A. Uehling,Phys. Rev. Bi, 3713 near q [110]) because the intensity of mode T’ is zero (1970). at q II [110] by the selection rule. Consequently, we 2. R.A. Cowley, W.J.L. Buyers, E.C. Svensson & cannot obtain the precise value of C12 + C~directly. G.L. Paul ,Neutron Inealstic Scattering, Vol. 1, p. 281. International Atomic Energy Agency, Let us assume that C12 + C~= 0 for simplicity. Vienna (1968). Equation (l)is thus rewritten in the form 3. R.L. Reese, Ii. Fritz & H.Z. Cummins,Phys. ~.
~,
~,
—
—
=
;1
(~) q
2
~((C11+ C~)±(C11
—
C~)cos 20}. (2)
Since the Brilouin shift v~(çb= 0), v.(0 = 0) and t him c~ ,-~E 66 ~-~66 —
q-~O
4.
Rev. B7,4l65 M. M.(1973). Tokunaga & I. Tatsuzaki,J. Phys. Soc.Sawafuji, Japan 47, 1860 (1979).
5.
A.P. Levanyuk & A.A. Sobyanin, Soy. Phys. JETP 26,612(1968). Y. Yamada, H. Taketra & D.L. Huber, J. Phys. Soc. Japan.36, 641 (1974). H. Tanaka & 1. Tatsuzaki,J. Phys. Soc. Japan 47, 878 (1979).
6. 7,