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Focussing of acoustoelectrically amplified phonons in obliquely-cut CdS crystal
Solid State Communications,
Vol. l3,pp. 483—485, 1973.
Pergamon Press.
Printed in Great Britain
FOCUSSING OF ACOUSTOELECFRJCALLY AMPLIFIED PHONONS IN OBLIQUELY.CUT CdS CRYSTAL M. Yamada, M. San’ya, C. Hamaguchi and J. Nakai Department of Electronics, Osaka University, Suita, Osaka 565, Japan (Received 7May 1973 by Y. Toyozawa)
The effect of elastic and acoustoelectric anisotropy on acoustoelectric domain propagation in semiconducting CdS has been studied by Brillouin scattering, in the case where the c-axis of the cr~sta1is obliquely oriented with respect to the electric drift field. In the 45 configuration (the angle between E and c is 450) angular focussing of energy is observed. C-0x5 • ~P
3C
IT IS WELL known that intense phonons are generated through acoustoelectric effect when high voltages are applied to piezoelectric semiconductors.’ The characteristics of acoustoelectrically amplified phonons, especially of shear waves propagating along the [110) direction in GaAs2 and perpendicular to the c-axis in CdS,3 have been extensively studied by means of Brillouin-scattering measurements. In such investigations a weak focussing of phonons was observed, which was caused by virtue of angular dependence of acoustoelectric gain. We report here a strong focussing effect due to elastic and acoustoelectric anisotropy.
CdS
-~
zc
q
~
—,
,
~
2’?
3’?
\~\ 4’?
~o.
60
8’?
/
9~O
/ “ ——
-0
FiG. 1. ~ vs 0,,. Deviation of group-velocity vector from wave vector as a parameter of the angular frequericy of sound wave. ~ ~ is the angular frequency of maximum acoustoelectric gain.
In an elastically anisotropic solid the direction of energy flow associated with a plane wave does not, m general, coincide with the direction of wave vector.4 It follows that even when waves are in a given region of a crystal with a uniform distribution of wave vectors, the energy flow will be enhanced in some direction and decreased in others with respect to the average value. Such an anisotropic effect was studied in detail by Musgrave and Miller,56 and was applied to the case of off-axis domain propagation in ZnO and CdS by Keller7 and in GaAs by Moore et al.8 It is recently pointed out by Hamaguchi et al.9 that the effect of piezoelectric stiffening on anisotropy is very small in GaAs but cannot be neglected in CdS and that a strong
The enhancement (or decrease) can be calculated for any direction in a crystal from the relation between the directions of the phase velocity v~ør the wave vectors q, and the corresponding directions of energy flow or group velocity v~given by = v w(q) = V [qv (q)] (1) S
Q
q
p
It should be noted that v,,(q) stongly depends on q, due to both elastic and acoustoelectric anisotropy. With conventional spherical coordinates, v,,(q) in the hexagonal crystal depends very little on the ~p 1,cornponent but strongly on the 0,, component of q. A computer calculation was carried out of the direction of the group velocity with a large number of wave vectors, which were taken. at 0.05°interval in 0,,. The
phonon focussing occurs in the [110] direction of GaAs. Here we follow their treatment, taking account of the piezoelectric stiffening, and report the phonon focussing effect in CdS. 483
484
PHONON FOCUSSING IN CdS
Vol. 13, No.4
UHP9.45
CdS
.5
‘U __________
I’?
2’?
time (o.usec/div.~
__________
3’?
4’?
5’?
6’?
7’?
8’?
9’?
9;
FIG. 2. The number of group-velocity vectors, v5, lying within intervals of 1°in 0, with a uniform distribution of wave vectors v,,.
FIG. 3. Some typical examples of the acoustoelectric current-oscillation patterns observed in a 45°-cutCdS sample at room temperature. ANODE
result of this computation is summarized in Figs. 1 and 2. In Fig. 1 deviations of ray from wave normal, i~(= 0, 0,,), are plotted as a function of OP for three different frequencies. (See the inset of Fig. I for the definition of the angles used.) The number of group-velocity vectors lying within intervals of 1°in 0, with a uniform angular distribution of wave vectors, and hence the degree of enhancement (or decrease) in that direction is shown in Fig. 2. It is found that the energy is strongly focused in the direction of °g ~ In the limit of ~ -~0,where the acoustoelectric ani7 sotropy vanishes, the result coincides with Keller’s. We examined the focussing of off-axis waves ~ plified in obliquely-cut CdS crystals. The CdS sample used in the experiments had dimensions of 4.8 X 1.83 X 0.91 mm, with its length oriented to the direction at angles of 45°with the c-axis. The sample resistance at room temperature was 580~2and its carrier concentration was I X 1015 cm3. The application of high-voltage pulses of 4-,usec duration gave rise to the current oscillations of square type, similar to those observed in the (c I E) case. Figure 3 shows the current patterns for three different voltages. It is clear from Fig. 3 that a strong saturation of the current occurs. Thus it seems to suggest that the amplified phonons are strongly focussed in the direction of current flow. The saturation value “8 of current density is 35A/cm2, from which the saturation velocity v, may be estimated to be 2.1 X l0~cm/sec with the equation of v, ~ .1~/ne,where e is electron charge and n is carrier concentration. Also, v~may be estimated to be 2.2 X iO~cm/sec from the relation of ~ ~ UT, where 1 is the sample length and T is the current oscillation period. Both estimated values are in good
(mm)
‘.7
—
4
3 —
2
_______
t.O.7~C
V
.~mm1TTTflfl~r~..
O
x.o UHP 9. 45
1.83 (mm)
CATHODE
FIG. 4. Solid delineation of the local peak-intensity of wave 2 GHz, O~= 30’~)in theplane domain as component a parameter(f of=time. agreement with the velocity of off-axis shear waves. Therefore, it may be concluded that off-axis waves strongly interact with the drift electrons. Brillouin-scattering measurements were made in order to investigate the propagation characteristics of the interacting sound waves. The acoustic domain consisting of various frequencies and propagationdirections was found to incline to the direction of current flow and the inclined angle was nearly equal to the angle of maximum acoustoelectric gain (0,, = 30°)expected from the small signal theory. The offaxis angle of amplified acoustic waves was within the range from 15 to 45°.Figure 4 shows the solid delineation of the local peak-intensity of off-axis wave (f = 2GHz, 9,, = 30°)in the domain. We find from
Vol. 13, No.4
PHONON FOCUSSING IN CdS
this figure that the phonon energy is focussed in the same direction as the current flow. Other off-axis waves were also found to be focused in the same direction. These facts may fully explain the current patterns as shown in Fig. 3. The propagation characteristics of off-axis waves were also examined in the 30°-cutsample. The spatial distribution of the domain was normal to the current flow. However, the acoustic energy did not initially propagate in the direction of OP ~ 300 but in the direction of 0, ~45°,and then
485
hit the sample surface and was reflected. As the consequence, the current pattern was the damped oscillation or saturation type rather than the square type. We conclude that off-axis shear waves acousto-
electrically amplified in the obliquely-cut CdS crystal are strongly focused in the direction of 0, ~ 450 and the anti-focusing takes place in the transverse configuration (c IE) due to elastic and acoustoelectric anisotropy.
1.
REFERENCES For a review and other references pertaining to the acoustoelectric effect see MEYER N.I. and JORGENSEN
2.
M.H. Advances in Solid State Physics, p.21, Pergamon, Vieweg (1970). SPEARS D.L.,Phys. Rev. B2, 1931 (1970).
3.
YAMADA M., HAMAGUCHI C., MATSIJMOTO K. and NAKAI J.,Phys. Rev. B7, 2682 (1973).
4.
TAYLOR B., MARIS H.J. and ELBAUM C.,Phys. Rev. Lett. 23,416 (1969).
5.
MUSGRAVE M.J.P., Proc. R. Soc. A 226, 339 (1954); ibid. A226, 356 (1954). MILLER GF. and MUSGRAVE M.JP., Proc. R. Soc. A 236, 352 (1956). KELLERO.,Phys.StatusSolidi(a)8,61 (1971);ibid. (a) 10,581 (1972).
6. 7.
8. 9.
MOORE A.R. and DAVENPORT J.W.,J. App!. Phys. 43,4513 (1972). HAMAGUCHI C., ROSS J.B. and BRAY R., to be published; ROSS J.B., Ph. D. Thesis, Purdue Univ., (1972).
Die Einwirkung der elastischen und akustoelektrischen Anisotropie auf die Fortpflanzung akustoelektrisher Domànen in halbleitendem CdS wurde durch Brillouinstreuung für den Fall untersucht, dali die elektrishe Feldstärke zur hexagonalen Achse geneigt ist. Bei der 45°-Konfiguration(der Winkel zwischen E und c ist 45°)ergeben die Experimente eine Winkelfokussierung der Energie.
Vol. l3,pp. 483—485, 1973.
Pergamon Press.
Printed in Great Britain
FOCUSSING OF ACOUSTOELECFRJCALLY AMPLIFIED PHONONS IN OBLIQUELY.CUT CdS CRYSTAL M. Yamada, M. San’ya, C. Hamaguchi and J. Nakai Department of Electronics, Osaka University, Suita, Osaka 565, Japan (Received 7May 1973 by Y. Toyozawa)
The effect of elastic and acoustoelectric anisotropy on acoustoelectric domain propagation in semiconducting CdS has been studied by Brillouin scattering, in the case where the c-axis of the cr~sta1is obliquely oriented with respect to the electric drift field. In the 45 configuration (the angle between E and c is 450) angular focussing of energy is observed. C-0x5 • ~P
3C
IT IS WELL known that intense phonons are generated through acoustoelectric effect when high voltages are applied to piezoelectric semiconductors.’ The characteristics of acoustoelectrically amplified phonons, especially of shear waves propagating along the [110) direction in GaAs2 and perpendicular to the c-axis in CdS,3 have been extensively studied by means of Brillouin-scattering measurements. In such investigations a weak focussing of phonons was observed, which was caused by virtue of angular dependence of acoustoelectric gain. We report here a strong focussing effect due to elastic and acoustoelectric anisotropy.
CdS
-~
zc
q
~
—,
,
~
2’?
3’?
\~\ 4’?
~o.
60
8’?
/
9~O
/ “ ——
-0
FiG. 1. ~ vs 0,,. Deviation of group-velocity vector from wave vector as a parameter of the angular frequericy of sound wave. ~ ~ is the angular frequency of maximum acoustoelectric gain.
In an elastically anisotropic solid the direction of energy flow associated with a plane wave does not, m general, coincide with the direction of wave vector.4 It follows that even when waves are in a given region of a crystal with a uniform distribution of wave vectors, the energy flow will be enhanced in some direction and decreased in others with respect to the average value. Such an anisotropic effect was studied in detail by Musgrave and Miller,56 and was applied to the case of off-axis domain propagation in ZnO and CdS by Keller7 and in GaAs by Moore et al.8 It is recently pointed out by Hamaguchi et al.9 that the effect of piezoelectric stiffening on anisotropy is very small in GaAs but cannot be neglected in CdS and that a strong
The enhancement (or decrease) can be calculated for any direction in a crystal from the relation between the directions of the phase velocity v~ør the wave vectors q, and the corresponding directions of energy flow or group velocity v~given by = v w(q) = V [qv (q)] (1) S
Q
q
p
It should be noted that v,,(q) stongly depends on q, due to both elastic and acoustoelectric anisotropy. With conventional spherical coordinates, v,,(q) in the hexagonal crystal depends very little on the ~p 1,cornponent but strongly on the 0,, component of q. A computer calculation was carried out of the direction of the group velocity with a large number of wave vectors, which were taken. at 0.05°interval in 0,,. The
phonon focussing occurs in the [110] direction of GaAs. Here we follow their treatment, taking account of the piezoelectric stiffening, and report the phonon focussing effect in CdS. 483
484
PHONON FOCUSSING IN CdS
Vol. 13, No.4
UHP9.45
CdS
.5
‘U __________
I’?
2’?
time (o.usec/div.~
__________
3’?
4’?
5’?
6’?
7’?
8’?
9’?
9;
FIG. 2. The number of group-velocity vectors, v5, lying within intervals of 1°in 0, with a uniform distribution of wave vectors v,,.
FIG. 3. Some typical examples of the acoustoelectric current-oscillation patterns observed in a 45°-cutCdS sample at room temperature. ANODE
result of this computation is summarized in Figs. 1 and 2. In Fig. 1 deviations of ray from wave normal, i~(= 0, 0,,), are plotted as a function of OP for three different frequencies. (See the inset of Fig. I for the definition of the angles used.) The number of group-velocity vectors lying within intervals of 1°in 0, with a uniform angular distribution of wave vectors, and hence the degree of enhancement (or decrease) in that direction is shown in Fig. 2. It is found that the energy is strongly focused in the direction of °g ~ In the limit of ~ -~0,where the acoustoelectric ani7 sotropy vanishes, the result coincides with Keller’s. We examined the focussing of off-axis waves ~ plified in obliquely-cut CdS crystals. The CdS sample used in the experiments had dimensions of 4.8 X 1.83 X 0.91 mm, with its length oriented to the direction at angles of 45°with the c-axis. The sample resistance at room temperature was 580~2and its carrier concentration was I X 1015 cm3. The application of high-voltage pulses of 4-,usec duration gave rise to the current oscillations of square type, similar to those observed in the (c I E) case. Figure 3 shows the current patterns for three different voltages. It is clear from Fig. 3 that a strong saturation of the current occurs. Thus it seems to suggest that the amplified phonons are strongly focussed in the direction of current flow. The saturation value “8 of current density is 35A/cm2, from which the saturation velocity v, may be estimated to be 2.1 X l0~cm/sec with the equation of v, ~ .1~/ne,where e is electron charge and n is carrier concentration. Also, v~may be estimated to be 2.2 X iO~cm/sec from the relation of ~ ~ UT, where 1 is the sample length and T is the current oscillation period. Both estimated values are in good
(mm)
‘.7
—
4
3 —
2
_______
t.O.7~C
V
.~mm1TTTflfl~r~..
O
x.o UHP 9. 45
1.83 (mm)
CATHODE
FIG. 4. Solid delineation of the local peak-intensity of wave 2 GHz, O~= 30’~)in theplane domain as component a parameter(f of=time. agreement with the velocity of off-axis shear waves. Therefore, it may be concluded that off-axis waves strongly interact with the drift electrons. Brillouin-scattering measurements were made in order to investigate the propagation characteristics of the interacting sound waves. The acoustic domain consisting of various frequencies and propagationdirections was found to incline to the direction of current flow and the inclined angle was nearly equal to the angle of maximum acoustoelectric gain (0,, = 30°)expected from the small signal theory. The offaxis angle of amplified acoustic waves was within the range from 15 to 45°.Figure 4 shows the solid delineation of the local peak-intensity of off-axis wave (f = 2GHz, 9,, = 30°)in the domain. We find from
Vol. 13, No.4
PHONON FOCUSSING IN CdS
this figure that the phonon energy is focussed in the same direction as the current flow. Other off-axis waves were also found to be focused in the same direction. These facts may fully explain the current patterns as shown in Fig. 3. The propagation characteristics of off-axis waves were also examined in the 30°-cutsample. The spatial distribution of the domain was normal to the current flow. However, the acoustic energy did not initially propagate in the direction of OP ~ 300 but in the direction of 0, ~45°,and then
485
hit the sample surface and was reflected. As the consequence, the current pattern was the damped oscillation or saturation type rather than the square type. We conclude that off-axis shear waves acousto-
electrically amplified in the obliquely-cut CdS crystal are strongly focused in the direction of 0, ~ 450 and the anti-focusing takes place in the transverse configuration (c IE) due to elastic and acoustoelectric anisotropy.
1.
REFERENCES For a review and other references pertaining to the acoustoelectric effect see MEYER N.I. and JORGENSEN
2.
M.H. Advances in Solid State Physics, p.21, Pergamon, Vieweg (1970). SPEARS D.L.,Phys. Rev. B2, 1931 (1970).
3.
YAMADA M., HAMAGUCHI C., MATSIJMOTO K. and NAKAI J.,Phys. Rev. B7, 2682 (1973).
4.
TAYLOR B., MARIS H.J. and ELBAUM C.,Phys. Rev. Lett. 23,416 (1969).
5.
MUSGRAVE M.J.P., Proc. R. Soc. A 226, 339 (1954); ibid. A226, 356 (1954). MILLER GF. and MUSGRAVE M.JP., Proc. R. Soc. A 236, 352 (1956). KELLERO.,Phys.StatusSolidi(a)8,61 (1971);ibid. (a) 10,581 (1972).
6. 7.
8. 9.
MOORE A.R. and DAVENPORT J.W.,J. App!. Phys. 43,4513 (1972). HAMAGUCHI C., ROSS J.B. and BRAY R., to be published; ROSS J.B., Ph. D. Thesis, Purdue Univ., (1972).
Die Einwirkung der elastischen und akustoelektrischen Anisotropie auf die Fortpflanzung akustoelektrisher Domànen in halbleitendem CdS wurde durch Brillouinstreuung für den Fall untersucht, dali die elektrishe Feldstärke zur hexagonalen Achse geneigt ist. Bei der 45°-Konfiguration(der Winkel zwischen E und c ist 45°)ergeben die Experimente eine Winkelfokussierung der Energie.