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Ultrasonic attenuation in copper at low temperatures
Volume 100A, number 1
PHYSICS LETTERS
2 January 1984
ULTRASONIC ATTENUATION IN COPPER AT LOW TEMPERATURES S. SATHISH and O.N. AWASTHI 1
Physics Department, Regional Collegeof Education, Mysore 5 70 006, India Received 28 April 1981 Revised manuscript received 14 November 1983
Ultrasonic attenuation in copper at low temperatures has been analysed in terms of electron-electron and electronphonon interactions. The contribution due to electron-electron interactions has been evaluated using a simplified spherical Fermi-surface model with an isotropic Umklapp transition probability. We observe that the contribution to the attenuation constant (c0 due to electron-electron scattering in copper at 3 K is about 10% of the electron-phonon contribution. We have also considered the viscous attenuation and the induction attenuation mechanisms simultaneously in copper as suggested by Orlov and find that our results improve the existing theoretical data considerably.
The measurements of ultrasonic attenuation in copper at low temperatures (below 20 K) provide significant information about the electron-electron and e l e c t r o n - p h o n o n interactions. The effect of these interactions has been extensively investigated both theoretically and experimentally. Kolouch and McCarthy [1], Wang et al. [2], and Macfarlane and Rayne [3] have measured the attenuation of longitudinal ultrasonic waves in single crystals of pure copper as a function of frequency and temperature. These experimental results follow the frequency and temperature dependence predicted by the free electron theory of Pippard. But Kolouch and McCarthy [1] observe that the theoretically predicted attenuation should be multiplied by an average of 1.7 to match with the experimentally measured attenuation. The authors [ 1 - 3 ] have tried to explain the discrepancies observed but these are not so satisfactory. According to Pippard's [4] theoretical formulation the attenuation constant, ct, for longitudinal ultrasonic waves is given as
a= hq2
f
D2lds.
(1)
4~'3 VS Fermi surface 1 Present address: Department of Physics, University of Ibadan, Ibadan, Nigeria. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Eq. (1) for a free electron metal reduces to
a= 4 nm~2V2 z/pV3s,
(2)
where n is the electron density, m is the electron mass, p the density of the metal, VS the ultrasonic velocity, r the relaxation time, l the mean free path, VF the Fermi velocity of the electrons, q the wave number and co the angular frequency of the ultrasonic wave. The analysis of experimental results in copper shows that in the limit ofql ~ 1, the attenuation constant associated with the electron-phonon interaction is a function of the crystalline orientation. For values o f ql '~ 1, however, the attenuation is found to be independent of the crystalline orientation. We shall confine ourselves here for the discussion of ultrasonic attenuation in copper in the local limit (ql 1), since in this case the application of Pippard's free electron model is feasible. In early comparisons with experiments the r which appears in expression (2) for a was generally assumed to be the same as encountered in the electrical resistivity caused by electron-phonon interaction. Bhatia and Moore [5], however, have shown that the simultaneous application of the electron transport equation, Maxwell's equations and the equations o f the elasticity theory leads to the conclusion that the relaxation times appearing in the ex39
Volume 100A, number 1
PHYSICS LETTERS
pressions for the electrical conductivity, o, and the ultrasonic attenuation constant, or, differ from one another. In fact r for o contains the first Legendre polynomial of the cosine of the angle between the orientation of the field and the electron momentum, whereas in the latter case it contains the second Legendre polynomial of the same argument. Rice and Sham [6] using the pseudopotential method for the evaluation of the ultrasonic attenuation constant for potassium due to the electron-phonon normal scattering process in the local limit (ql ~ 1) have shown that their calculated values for r are only 50% of the experimental ones. Ekin et al. [7] considered the role of the electron-phonon Umklapp scattering processes als9 for the ultrasonic attenuation (a) and the electrical conductivity of potassium at low temperatures. These workers have concluded that the electron-phonon Umklapp scattering processes contribute very little for the attenuation but play a significant role for the electrical conductivity. Recently Awasthi and Sathish [8] have included the effect of the electron-electron Umklapp scattering process on the ultrasonic attenuation in potassium at low temperatures. Their results show that the contribution of the electron-electron Umklapp scattering process to the ultrasonic attenuation in potassium is only about 5% of the electron-phonon value as evaluated by Ekin et al. [7] at 3 K. In the present paper we have tried to include the effect of the electron-electron Umklapp interactions to the low temperature ultrasonic attenuation in copper. In potassium the Fermi surface does not touch the Brillouin zone and hence the electron-phonon Umklapp scattering contribution to the attenuation is quite small. But in the case of copper the Fermi surface touches the Brillouin zone along the (111) direction. So, this electron-phonon Umklapp scattering contribution to the attenuation may be significant. Following Ziman [9] and Awasthi and Sathish [8, 10] the relaxation time for the electron-electron Umklapp scattering processes can be evaluated using the Born approximation for estimating the scattering cross section of one electron for another. Neglecting the band structure effects for simplicity and using the spherical Fermi surface model with an isotropic transition probability, we obtain the following expression for the relaxation time (ree) in copper:
40
2 January 1984
rTe1 = 1.73 X
IO-41(rIZG2/rsVFE3 ) T2s -1
(forg ~ 2kv),
(3)
where Z is the coordination number of the reciprocal lattice; r s = ro/a H (all is the Bohr radius and r0 is the radius of a sphere that contains one electron);E F is the Fermi energy; kF, Fermi wave vector ;g, reciprocal lattice vector; G, overlap integral or the interference factor for the incident and scattered electron wavefunctions inside the Wigner-Seitz cell of the lattice. The expression for G can be written as [8] :
G = ~2-1 f
U*kl(r ) exp(ig.r) d3r,
(4)
cell where g2 is the atomic volume and g = k 1 + k 2 + k 3 + k4. The value of G can be evaluated using the WignerSeitz method [ 11 ]. This gives us the wavefunction for the ground state of a metal, i.e., for k = 0. Denoting this wavefunction by ~0(r) a fair approximation to the wavefunctions of higher states will be within one atomic sphere ~k(r) = exp(ik'r) ~0(r) provided k lies within the first Brillouin zone, not too close to its boundaries. Using this approximation expression (4) can be written as G ~ f [~k0(r)l2 exp(ig-r) d3r. cell
(5)
Results and discussion. To estimate r~e1 we have evaluated the overlap integral G numerically using expression (5). Values of ff0(r) for copper have been taken from the literature [12]. We estimate G = 0.065 and r~e1 = 0.85 X 105 T 2 s-1 for copper with g ~ 2k F. Assuming that the relaxation times for the ultrasonic attenuation due to different scattering processes can be combined in a similar manner as for the electrical resistivities, we can write r - 1 = r~-I + %1 + repl,
(6)
where r is the effective relaxation time responsible for the total attenuation. All other symbols have their usual meanings.
Volume 100A, number 1
PHYSICS LETTERS
The attenuation constant a can therefore be expressed as t~- 1 = Oq- 1 + Otee1 + 0%1 .
(7)
Our calculations for a in copper show that elect r o n - e l e c t r o n Umklapp scattering processes contribute about 10% of the e l e c t r o n - p h o n o n part around 3K. It has been observed [13,14] that electron-electron scattering in copper is dominant below 1.5 K. If one measures carefully the electronic attenuation at much lower temperatures ( < 1.3 K) and lower frequencies, it may be possible to observe the T 2 dependence for a due to e l e c t r o n - e l e c t r o n Umklapp scattering. We have also examined the suggestion of Orlov [15] that one should consider the viscous attenuation mechanism and the induction attenuation mechanism simultaneously. He has shown in the long wavelength limit (ql "~ 1) that both the above mechanisms make comparable contributions to the attenuation. The expression for ultrasonic attenuation for longitudinal waves due to Orlov's theory can be expressed as
a= 4 (mn6o2V2 r/pV3) X [1
+(3X2/rnV2)(¼X2/mV 2 - 1)],
(8)
where ~2 = p2/3m2 is calculated by the deformation potential method. The first term in eq. (8) is due to induction attenuation mechanism. The second and third terms are due to viscous attenuation mechanism. To estimate the contribution of viscous mechanism to a, we have used the results o f Hunter an Nabarro [16]. Using m/m 2 = - 1 . 1 1 6 in eq. (8), we get
a=2.4×(~nmw2V2/pV3)
(for q l ¢ 1).
We thus find that the theoretical values o f ct for
(9)
2 January 1984
copper are enhanced by 140% bringing them nearer to the experimental values, though slightly on the higher side. The remaining discrepancy in the experimental and theoretical values o f a may be attributed to the anisotropy in transition rate and nonspherical nature o f the Fermi surface o f copper. The authors wish to thank UGC, T I F R , NCERT and the University o f Mysore for financial and other assistance. They also thank a referee for some of his useful comments which have really helped in our work.
References [1] R.J. Kolouch and K.A. McCarthy, Phys. Rev. 139 (1965) A700. [2] E.Y. Wang, R.J. Kolouch and K.A. McCarthy, Phys. Rev. 175 (1968) 723. [3] R.E. Macfarlane and J. Rayne, Phys. Rev. 162 (1967) 532. [4] A.B. Pippard, Proc. R. Soc. A257 (1960) 145. [5] A.B. Bhatia and R.A. Moore, Phys. Rev. 121 (1961) 1075. [61 T.M. Rice and L.J. Sham, Phys. Rev. B1 (1970) 4546. [7] P.N. Trmfomenkoff and J.W. Ekin, Phys. Rev. B4 (1971) 2392. [8] O.N. Awasthi and S. Sathish, Phys. Lett. 77A (1980) 356. [9] J.M, Ziman, Electrons and phonons (Oxford, 1960) p. 415. [10] O.N. Awasthi and S. Sathish, Phys. Lett. 83A (1981) 283. [11 ] E.P. Wigner and F. Seitz, Phys. Rev. 43 (1933) 804; 46 (1934) 509. [12] K. Fuchs and H.H. Wells, Proc. R. Soc. 151 (1935) 585. [13] M. Khoshnevisan, W. Pratt Jr., P.A. Scttroeder and S.D. Steenwyk, Phys. Rev. B19 (1979) 3873. [14] S.D. Steenwyk, J.A. Rowlands and P.A. Schroeder, J. Phys. F l l (1981) 1623. [15] V.G. Orlov, Soy. Phys. Solid State 18 (1976) 539. [16] S.C. Hunter and F.R.N. Nabarro, Proc. R. Soc. A220 (1953) 542.
41
PHYSICS LETTERS
2 January 1984
ULTRASONIC ATTENUATION IN COPPER AT LOW TEMPERATURES S. SATHISH and O.N. AWASTHI 1
Physics Department, Regional Collegeof Education, Mysore 5 70 006, India Received 28 April 1981 Revised manuscript received 14 November 1983
Ultrasonic attenuation in copper at low temperatures has been analysed in terms of electron-electron and electronphonon interactions. The contribution due to electron-electron interactions has been evaluated using a simplified spherical Fermi-surface model with an isotropic Umklapp transition probability. We observe that the contribution to the attenuation constant (c0 due to electron-electron scattering in copper at 3 K is about 10% of the electron-phonon contribution. We have also considered the viscous attenuation and the induction attenuation mechanisms simultaneously in copper as suggested by Orlov and find that our results improve the existing theoretical data considerably.
The measurements of ultrasonic attenuation in copper at low temperatures (below 20 K) provide significant information about the electron-electron and e l e c t r o n - p h o n o n interactions. The effect of these interactions has been extensively investigated both theoretically and experimentally. Kolouch and McCarthy [1], Wang et al. [2], and Macfarlane and Rayne [3] have measured the attenuation of longitudinal ultrasonic waves in single crystals of pure copper as a function of frequency and temperature. These experimental results follow the frequency and temperature dependence predicted by the free electron theory of Pippard. But Kolouch and McCarthy [1] observe that the theoretically predicted attenuation should be multiplied by an average of 1.7 to match with the experimentally measured attenuation. The authors [ 1 - 3 ] have tried to explain the discrepancies observed but these are not so satisfactory. According to Pippard's [4] theoretical formulation the attenuation constant, ct, for longitudinal ultrasonic waves is given as
a= hq2
f
D2lds.
(1)
4~'3 VS Fermi surface 1 Present address: Department of Physics, University of Ibadan, Ibadan, Nigeria. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Eq. (1) for a free electron metal reduces to
a= 4 nm~2V2 z/pV3s,
(2)
where n is the electron density, m is the electron mass, p the density of the metal, VS the ultrasonic velocity, r the relaxation time, l the mean free path, VF the Fermi velocity of the electrons, q the wave number and co the angular frequency of the ultrasonic wave. The analysis of experimental results in copper shows that in the limit ofql ~ 1, the attenuation constant associated with the electron-phonon interaction is a function of the crystalline orientation. For values o f ql '~ 1, however, the attenuation is found to be independent of the crystalline orientation. We shall confine ourselves here for the discussion of ultrasonic attenuation in copper in the local limit (ql 1), since in this case the application of Pippard's free electron model is feasible. In early comparisons with experiments the r which appears in expression (2) for a was generally assumed to be the same as encountered in the electrical resistivity caused by electron-phonon interaction. Bhatia and Moore [5], however, have shown that the simultaneous application of the electron transport equation, Maxwell's equations and the equations o f the elasticity theory leads to the conclusion that the relaxation times appearing in the ex39
Volume 100A, number 1
PHYSICS LETTERS
pressions for the electrical conductivity, o, and the ultrasonic attenuation constant, or, differ from one another. In fact r for o contains the first Legendre polynomial of the cosine of the angle between the orientation of the field and the electron momentum, whereas in the latter case it contains the second Legendre polynomial of the same argument. Rice and Sham [6] using the pseudopotential method for the evaluation of the ultrasonic attenuation constant for potassium due to the electron-phonon normal scattering process in the local limit (ql ~ 1) have shown that their calculated values for r are only 50% of the experimental ones. Ekin et al. [7] considered the role of the electron-phonon Umklapp scattering processes als9 for the ultrasonic attenuation (a) and the electrical conductivity of potassium at low temperatures. These workers have concluded that the electron-phonon Umklapp scattering processes contribute very little for the attenuation but play a significant role for the electrical conductivity. Recently Awasthi and Sathish [8] have included the effect of the electron-electron Umklapp scattering process on the ultrasonic attenuation in potassium at low temperatures. Their results show that the contribution of the electron-electron Umklapp scattering process to the ultrasonic attenuation in potassium is only about 5% of the electron-phonon value as evaluated by Ekin et al. [7] at 3 K. In the present paper we have tried to include the effect of the electron-electron Umklapp interactions to the low temperature ultrasonic attenuation in copper. In potassium the Fermi surface does not touch the Brillouin zone and hence the electron-phonon Umklapp scattering contribution to the attenuation is quite small. But in the case of copper the Fermi surface touches the Brillouin zone along the (111) direction. So, this electron-phonon Umklapp scattering contribution to the attenuation may be significant. Following Ziman [9] and Awasthi and Sathish [8, 10] the relaxation time for the electron-electron Umklapp scattering processes can be evaluated using the Born approximation for estimating the scattering cross section of one electron for another. Neglecting the band structure effects for simplicity and using the spherical Fermi surface model with an isotropic transition probability, we obtain the following expression for the relaxation time (ree) in copper:
40
2 January 1984
rTe1 = 1.73 X
IO-41(rIZG2/rsVFE3 ) T2s -1
(forg ~ 2kv),
(3)
where Z is the coordination number of the reciprocal lattice; r s = ro/a H (all is the Bohr radius and r0 is the radius of a sphere that contains one electron);E F is the Fermi energy; kF, Fermi wave vector ;g, reciprocal lattice vector; G, overlap integral or the interference factor for the incident and scattered electron wavefunctions inside the Wigner-Seitz cell of the lattice. The expression for G can be written as [8] :
G = ~2-1 f
U*kl(r ) exp(ig.r) d3r,
(4)
cell where g2 is the atomic volume and g = k 1 + k 2 + k 3 + k4. The value of G can be evaluated using the WignerSeitz method [ 11 ]. This gives us the wavefunction for the ground state of a metal, i.e., for k = 0. Denoting this wavefunction by ~0(r) a fair approximation to the wavefunctions of higher states will be within one atomic sphere ~k(r) = exp(ik'r) ~0(r) provided k lies within the first Brillouin zone, not too close to its boundaries. Using this approximation expression (4) can be written as G ~ f [~k0(r)l2 exp(ig-r) d3r. cell
(5)
Results and discussion. To estimate r~e1 we have evaluated the overlap integral G numerically using expression (5). Values of ff0(r) for copper have been taken from the literature [12]. We estimate G = 0.065 and r~e1 = 0.85 X 105 T 2 s-1 for copper with g ~ 2k F. Assuming that the relaxation times for the ultrasonic attenuation due to different scattering processes can be combined in a similar manner as for the electrical resistivities, we can write r - 1 = r~-I + %1 + repl,
(6)
where r is the effective relaxation time responsible for the total attenuation. All other symbols have their usual meanings.
Volume 100A, number 1
PHYSICS LETTERS
The attenuation constant a can therefore be expressed as t~- 1 = Oq- 1 + Otee1 + 0%1 .
(7)
Our calculations for a in copper show that elect r o n - e l e c t r o n Umklapp scattering processes contribute about 10% of the e l e c t r o n - p h o n o n part around 3K. It has been observed [13,14] that electron-electron scattering in copper is dominant below 1.5 K. If one measures carefully the electronic attenuation at much lower temperatures ( < 1.3 K) and lower frequencies, it may be possible to observe the T 2 dependence for a due to e l e c t r o n - e l e c t r o n Umklapp scattering. We have also examined the suggestion of Orlov [15] that one should consider the viscous attenuation mechanism and the induction attenuation mechanism simultaneously. He has shown in the long wavelength limit (ql "~ 1) that both the above mechanisms make comparable contributions to the attenuation. The expression for ultrasonic attenuation for longitudinal waves due to Orlov's theory can be expressed as
a= 4 (mn6o2V2 r/pV3) X [1
+(3X2/rnV2)(¼X2/mV 2 - 1)],
(8)
where ~2 = p2/3m2 is calculated by the deformation potential method. The first term in eq. (8) is due to induction attenuation mechanism. The second and third terms are due to viscous attenuation mechanism. To estimate the contribution of viscous mechanism to a, we have used the results o f Hunter an Nabarro [16]. Using m/m 2 = - 1 . 1 1 6 in eq. (8), we get
a=2.4×(~nmw2V2/pV3)
(for q l ¢ 1).
We thus find that the theoretical values o f ct for
(9)
2 January 1984
copper are enhanced by 140% bringing them nearer to the experimental values, though slightly on the higher side. The remaining discrepancy in the experimental and theoretical values o f a may be attributed to the anisotropy in transition rate and nonspherical nature o f the Fermi surface o f copper. The authors wish to thank UGC, T I F R , NCERT and the University o f Mysore for financial and other assistance. They also thank a referee for some of his useful comments which have really helped in our work.
References [1] R.J. Kolouch and K.A. McCarthy, Phys. Rev. 139 (1965) A700. [2] E.Y. Wang, R.J. Kolouch and K.A. McCarthy, Phys. Rev. 175 (1968) 723. [3] R.E. Macfarlane and J. Rayne, Phys. Rev. 162 (1967) 532. [4] A.B. Pippard, Proc. R. Soc. A257 (1960) 145. [5] A.B. Bhatia and R.A. Moore, Phys. Rev. 121 (1961) 1075. [61 T.M. Rice and L.J. Sham, Phys. Rev. B1 (1970) 4546. [7] P.N. Trmfomenkoff and J.W. Ekin, Phys. Rev. B4 (1971) 2392. [8] O.N. Awasthi and S. Sathish, Phys. Lett. 77A (1980) 356. [9] J.M, Ziman, Electrons and phonons (Oxford, 1960) p. 415. [10] O.N. Awasthi and S. Sathish, Phys. Lett. 83A (1981) 283. [11 ] E.P. Wigner and F. Seitz, Phys. Rev. 43 (1933) 804; 46 (1934) 509. [12] K. Fuchs and H.H. Wells, Proc. R. Soc. 151 (1935) 585. [13] M. Khoshnevisan, W. Pratt Jr., P.A. Scttroeder and S.D. Steenwyk, Phys. Rev. B19 (1979) 3873. [14] S.D. Steenwyk, J.A. Rowlands and P.A. Schroeder, J. Phys. F l l (1981) 1623. [15] V.G. Orlov, Soy. Phys. Solid State 18 (1976) 539. [16] S.C. Hunter and F.R.N. Nabarro, Proc. R. Soc. A220 (1953) 542.
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