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Magnetoacoustic open-orbit antiresonance in silver
Volume 64A, number 3
PHYSICS LETTERS
26 December 1977
MAGNETOACOUSTIC OPEN-ORBIT ANTIRESONANCE IN SILVER * D.S. KHATRI and A.R. DONFOR Physics Department, University of District of Columbia, Mount Vernon Square Campus, Washington, DC 20005, USA
and G.N. KAMM and AC. EHRLICH Naval Research Laboratory, Washington, DC 20375, USA Received 12 October 1977 Antiresonance (sharp attenuation minima) predicted theoretically by Sievert in 1 967 and the rich harmonics observed experimentally by Khatri and Peverley in 1973 in copper have remained unexplained to a great extent. In this letter, we report our preliminary results on this novel effect (antiresonance) for shear waves propagated along the liii I crystallographic direction in silver.
The interaction between sound waves and the conduction electrons in a metal has been considered in detail by Pippard [1] where the effective force is broken down into two principal parts, i.e., the deformation and the field force. General expressions are derived for the attenuation in a magnetic field, taking into account these two forces, but the open-orbit contribution was not taken into account explicitly. The openorbit resonance predicted theoretically by Kaner, Peschanskii and Privoroskii was observed experimentally in copper by several workers [2, 3]. Kaner et al. recognized the existence of a field force but neglected it in their calculations at an early stage. Sievert [41in 1967 calculated the attenuation for a Fermi surface model consisting of a set of overlapping cylinders, which permits open orbits if a small perturbing potential is assumed. The anitresonance predicted by that model for shear waves was observed experimentally by Khatri and Peverley [51for copper in 1973 but the results remain not fully explained. We report here our preliminary results for magnetoacoustic antiresonance in ultrapure single crystals of silver (residual resistivity ratio, RRR 12000) which have been obtained by propagating shear waves of frequency 165 MHz [11] crystallographic direction with shearalong wave the polarization (~)along [1121. Work supported by the college supported research committee (CSRC) of the University of District of Columbia, Mt. Vernon Square Campus.
322
This novel effect seems to have its origin in the socalled field force (due to the sound wave) on the conduction electrons which is surprising in copper and silver because of the assumed dominance of the deformation force in metals. If a magnetic field is rotated in the (Ill) plane, open orbits are possible when Hues along [112] or equivalent directions. In the k-space the open orbit runs along [110] and in real space it has a component along q. This geometry is depicted in fig. 1. Open orbit resonances can occur whenever the spatial matching condition = (1) q is satisfied, where (V> is mean velocity and wc is cyclotron frequency of the open orbit electrons; n is integer. In the present case eq. (1) reduces to
Hn
=
hck0/(neX),
(2)
where c/c is the electronic charge divided by the velocity of light, X is sound wavelength, and k0 is openorbit repeat distance. Shear waves frequejicy 165 MHz 5 cm/sofwere propagated alongand thevelocity [Ill] 1.64 X i0 crystallographic direction with polarization vector, //[i 121 with the magnetic field in both the[1 121 and two equivalent directions [1211 and [2111. The thickness (1.77 mm) of the sample was taken small enough to avoid the defocussing problem of conical
Volume 64A, number 3
I
PHYSICS LETTERS
26 December 1977
~b4~I65MHZ
0
‘~‘
~
~
0
MAGNETIC FIELDIKILOGAUSS)
Fig. 1. (a) (111) projection of silver Fermi surface showing the principal open orbit. (b) (112) projection of open orbit in k space, drawn to same scale,
internal refraction in which the acoustic energy flux has a component normal to the propagation vector q. The shear wave attenuation versus magnetic field curves are shown in fig. 2a for £ and H in the [1l~] direction and fig. 2b where £ remains in the [1l~]direction but H is rotated 60°to either the [T21] or [21 F] direction. In the former case, strong antiresonances are seen for all values of n except for n = 1, 2, 3. In contrast, very strong resonances are obtained for all values of n in the latter. In agreement with Khatri and Peverley’s Cu results [51,the even number resonances are stronger than the odd number ones. Additional experiments not explicitly shown here were carried out at 75 MHz to explore the behavior for n = 1, 2, 3. It was found that for both experimental arrangements, n = 1, 2, 3 represent resonances. It was necessary to carry out these experiments because at 165 MHz the magnetic field for all these resonances are beyond the range of our electromagnet (n = 1 requires 29 kG.). At sufficiently low fields the odd harmonics die out for both cases and the even harmonics merge imperceptibly into the magnetoacoustic oscillations. This would be expected from the calipering of
Fig. 2. (a) Open-orbit antiresonances for 165 MHz shear propagating along 1111] in silver with the polarization vector E along 11121 and parallel to H, the magnetic field. (b) Openorbit resonances for 165 MHz shear waves propagating along 1111] in silver with the polarization vector along 11121 and Halong either 11211 or [2111 , i.e. at 600 to E.
and resonances are very sensitive to the angle between H and {l 12}. In addition, the antiresonances have been seen to disappear for changes of 2 degrees in the angle between c and {l 1 2}. Gavenda and Chang [6] have explained the source of antiresonance as the vanishing of the component of the resistivity tensorR~1which is responsible for shear wave attenuation when a finite band of electrons move in resonance with the sound wave but confirmation is still lacking. From this and previously given explanations by Khatri and Peverly, we can tentatively condude that in real metals for certain crystallographic symmetry directions resonance or antiresonance occurs depending upon the dominance of either of two types of forces. This is a direct and also the first demonstration in silver of the existence of deformation and field forces which have been seen before in copper [5] and in certain superconductors [7]. A unique feature of silver and copper results is that the contribution of either of the two forces can be
the Fermi-surface belly orbit, which has a dimension very close to one-half of the open-orbit period. It
varied at will by changing the direction of polarization relative to H. Computer simulation and further experi-
must be pointed out here that both the antiresonances
mental work in this direction for other frequencies and 323
Volume 64A, number 3
PHYSICS LETTERS
on high purity single crystals of silver is in progress and will be reported at a later date.
References [11A.B.
324
Pippard, Proc. Roy. Soc. Ser. A257 (1960) 165.
26 December 1977
121
C.W. Burmeister, l).B. Doan and J.D. Gavenda, Phys. Let). 7 (1963) 112. [31J.D. Gavenda and W.R. Cox, Phys. Rev. B6 (1972) 4392. 141 P.R. Sievert, Phys. Rev. 161 (1967) 637. 151 D.S. Khatri and J.R. Peverley, Phys. Rev. Lctt. 30 (1973) 490. [61iD. Gavenda and S.J. Chang, LT-14, Vol. 3(1976)153.
171
JR. Leibowitz, Phys. Rev. 136 (1964) A22.
PHYSICS LETTERS
26 December 1977
MAGNETOACOUSTIC OPEN-ORBIT ANTIRESONANCE IN SILVER * D.S. KHATRI and A.R. DONFOR Physics Department, University of District of Columbia, Mount Vernon Square Campus, Washington, DC 20005, USA
and G.N. KAMM and AC. EHRLICH Naval Research Laboratory, Washington, DC 20375, USA Received 12 October 1977 Antiresonance (sharp attenuation minima) predicted theoretically by Sievert in 1 967 and the rich harmonics observed experimentally by Khatri and Peverley in 1973 in copper have remained unexplained to a great extent. In this letter, we report our preliminary results on this novel effect (antiresonance) for shear waves propagated along the liii I crystallographic direction in silver.
The interaction between sound waves and the conduction electrons in a metal has been considered in detail by Pippard [1] where the effective force is broken down into two principal parts, i.e., the deformation and the field force. General expressions are derived for the attenuation in a magnetic field, taking into account these two forces, but the open-orbit contribution was not taken into account explicitly. The openorbit resonance predicted theoretically by Kaner, Peschanskii and Privoroskii was observed experimentally in copper by several workers [2, 3]. Kaner et al. recognized the existence of a field force but neglected it in their calculations at an early stage. Sievert [41in 1967 calculated the attenuation for a Fermi surface model consisting of a set of overlapping cylinders, which permits open orbits if a small perturbing potential is assumed. The anitresonance predicted by that model for shear waves was observed experimentally by Khatri and Peverley [51for copper in 1973 but the results remain not fully explained. We report here our preliminary results for magnetoacoustic antiresonance in ultrapure single crystals of silver (residual resistivity ratio, RRR 12000) which have been obtained by propagating shear waves of frequency 165 MHz [11] crystallographic direction with shearalong wave the polarization (~)along [1121. Work supported by the college supported research committee (CSRC) of the University of District of Columbia, Mt. Vernon Square Campus.
322
This novel effect seems to have its origin in the socalled field force (due to the sound wave) on the conduction electrons which is surprising in copper and silver because of the assumed dominance of the deformation force in metals. If a magnetic field is rotated in the (Ill) plane, open orbits are possible when Hues along [112] or equivalent directions. In the k-space the open orbit runs along [110] and in real space it has a component along q. This geometry is depicted in fig. 1. Open orbit resonances can occur whenever the spatial matching condition = (1) q is satisfied, where (V> is mean velocity and wc is cyclotron frequency of the open orbit electrons; n is integer. In the present case eq. (1) reduces to
Hn
=
hck0/(neX),
(2)
where c/c is the electronic charge divided by the velocity of light, X is sound wavelength, and k0 is openorbit repeat distance. Shear waves frequejicy 165 MHz 5 cm/sofwere propagated alongand thevelocity [Ill] 1.64 X i0 crystallographic direction with polarization vector, //[i 121 with the magnetic field in both the[1 121 and two equivalent directions [1211 and [2111. The thickness (1.77 mm) of the sample was taken small enough to avoid the defocussing problem of conical
Volume 64A, number 3
I
PHYSICS LETTERS
26 December 1977
~b4~I65MHZ
0
‘~‘
~
~
0
MAGNETIC FIELDIKILOGAUSS)
Fig. 1. (a) (111) projection of silver Fermi surface showing the principal open orbit. (b) (112) projection of open orbit in k space, drawn to same scale,
internal refraction in which the acoustic energy flux has a component normal to the propagation vector q. The shear wave attenuation versus magnetic field curves are shown in fig. 2a for £ and H in the [1l~] direction and fig. 2b where £ remains in the [1l~]direction but H is rotated 60°to either the [T21] or [21 F] direction. In the former case, strong antiresonances are seen for all values of n except for n = 1, 2, 3. In contrast, very strong resonances are obtained for all values of n in the latter. In agreement with Khatri and Peverley’s Cu results [51,the even number resonances are stronger than the odd number ones. Additional experiments not explicitly shown here were carried out at 75 MHz to explore the behavior for n = 1, 2, 3. It was found that for both experimental arrangements, n = 1, 2, 3 represent resonances. It was necessary to carry out these experiments because at 165 MHz the magnetic field for all these resonances are beyond the range of our electromagnet (n = 1 requires 29 kG.). At sufficiently low fields the odd harmonics die out for both cases and the even harmonics merge imperceptibly into the magnetoacoustic oscillations. This would be expected from the calipering of
Fig. 2. (a) Open-orbit antiresonances for 165 MHz shear propagating along 1111] in silver with the polarization vector E along 11121 and parallel to H, the magnetic field. (b) Openorbit resonances for 165 MHz shear waves propagating along 1111] in silver with the polarization vector along 11121 and Halong either 11211 or [2111 , i.e. at 600 to E.
and resonances are very sensitive to the angle between H and {l 12}. In addition, the antiresonances have been seen to disappear for changes of 2 degrees in the angle between c and {l 1 2}. Gavenda and Chang [6] have explained the source of antiresonance as the vanishing of the component of the resistivity tensorR~1which is responsible for shear wave attenuation when a finite band of electrons move in resonance with the sound wave but confirmation is still lacking. From this and previously given explanations by Khatri and Peverly, we can tentatively condude that in real metals for certain crystallographic symmetry directions resonance or antiresonance occurs depending upon the dominance of either of two types of forces. This is a direct and also the first demonstration in silver of the existence of deformation and field forces which have been seen before in copper [5] and in certain superconductors [7]. A unique feature of silver and copper results is that the contribution of either of the two forces can be
the Fermi-surface belly orbit, which has a dimension very close to one-half of the open-orbit period. It
varied at will by changing the direction of polarization relative to H. Computer simulation and further experi-
must be pointed out here that both the antiresonances
mental work in this direction for other frequencies and 323
Volume 64A, number 3
PHYSICS LETTERS
on high purity single crystals of silver is in progress and will be reported at a later date.
References [11A.B.
324
Pippard, Proc. Roy. Soc. Ser. A257 (1960) 165.
26 December 1977
121
C.W. Burmeister, l).B. Doan and J.D. Gavenda, Phys. Let). 7 (1963) 112. [31J.D. Gavenda and W.R. Cox, Phys. Rev. B6 (1972) 4392. 141 P.R. Sievert, Phys. Rev. 161 (1967) 637. 151 D.S. Khatri and J.R. Peverley, Phys. Rev. Lctt. 30 (1973) 490. [61iD. Gavenda and S.J. Chang, LT-14, Vol. 3(1976)153.
171
JR. Leibowitz, Phys. Rev. 136 (1964) A22.