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Magnetoacoustic determination of deformation potential
Physica B 263—264 (1999) 236—238
Magnetoacoustic determination of deformation potential A.V. Tkach* Institute for Metal Physics, Russian Academy of Sciences, 18, Kovalevskaya St., Ekaterinburg 620219, Russia
Abstract On the basis of ultrasonic measurements in a classically high magnetic field, new information has been obtained about the averaged components of deformation potential on effective orbits in molybdenum and tungsten. In addition, the limiting attenuation of longitudinal waves, propagating along the magnetic field in the direction [0 0 1], has been calculated for Mo by the use of data on strain dependencies of the effective orbits. 1999 Elsevier Science B.V. All rights reserved. Keywords: Deformation potential; Ultrasonic absorption; Magnetoacoustics
At present, geometric characteristics of Fermi surfaces (FS) have been investigated quite well for a number of metals. There is also plenty of data on electron velocities at FSs. However, the deformation potential (DP) remains insufficiently studied. The DP takes into account variations of the electron energy under a lattice deformation. One has to know DP-values at the Fermi Surface to give a description of a physical phenomenon that involves the electron—phonon interaction. However, this information is inaccessible, as a rule. A possible approach, Ref. [1], to this problem consists in using a relationship between the DP (averaged over an effective orbit) and the area change (dA) of the orbit due to a deformation:
d(ln A) m KI "! GH. 2p du A GH * Fax: #7-3432-745344; e-mail: [email protected].
(1)
Here u ,ju /jx is the distortion tensor; m is the GH G H cyclotron mass, and KI is the (averaged) norGH malized deformation potential tensor. Available experimental data on the area variations under deformations have a restricted accuracy of 10—15%, Ref. [2]. A combination of complicated quantum oscillation techniques has to be used in such measurements. We demonstrate here some opportunities of another experimental approach. Let us consider the electronic attenuation C of ultrasound (of the wave vector q), propagating along an n-fold rotation axis (n*3) in a magnetic field H""q""O . If the free X path of electron l is large enough, absorption tends to a saturation level in classically high (ch) magnetic fields far above the edge of cyclotron absorption:
C 2\Np dA \ " N(m KI ) . H dk f
C K X
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 4 6 - 5
(2)
A.V. Tkach / Physica B 263—264 (1999) 236—238
237
Here p-index is equal to 3 for longitudinal waves and 4 for transverse ones, f is the ultrasonic frequency, is the Planck constant, and C denotes the quasi-static elasticity modulus. Summation in Eq. (2) is taken over the effective orbits m at the FS, k being the z-component of the electron wave X vector, and N denotes the orbit multiplicity. KI ,(KI ) if p"3, H XX (3) KI ,(KI )#(KI ) if p"4. H VX WX A geometric characteristic of the FS, (dA/dk), can X be determined with a high accuracy for a number of cases. Hence, the relative error of indirect determination of (m KI ) can be estimated as half of the H C-measurement error. This later one can be as small as 5%. As an example, we consider cubic metals tungsten and molybdenum under q""H""[0 0 1]. Normalized absorption (C /f ) is measured in Ref. [3] in two steps. First, the corresponding value (C /f ) at H"0 is determined, and then the difference is registered D"C /f!C /f. After that, C /f is de termined as C /f!D. Shear waves (Mo and W). Only four (N"4) orbits of type o contribute to a high-field absorption in this case, Ref. [4]. These orbits belong to inclined (with respect to H""[0 0 1]) ellipsoidal FS-sheets. Considering such an orbit at the axis [1 0 1], one has (by the symmetry) KI ,0. WX The ellipsoid semiaxes and the main effective masses can be taken from Refs. [5—7]. Thus, starting from values of (C /f )"42.6$2.2 dB/(cm GHz) in W and 44$7 dB/(cm GHz) in Mo [3], the DP-component "KI " is obtained. It equals 7.5 eV VX for W and 4.6 eV for Mo. We compare these values with those, derived, on the basis of Eq. (1), from data on d(ln A)/du , Refs. VX [8,9]. Fig. 1 gives a comparison of the two methods. It reveals a good consistency, but the method of ultrasonic absorption in a high magnetic field has an advantage of better accuracy as marked by dimensions of the error ellipses. Longitudinal waves (Mo). Eq. (2) can be transformed for longitudinal waves (p"3) to the following form:
C
d(ln A) dA \ " N A . f 4pC du dk K GH X
(4)
Fig. 1. Values of a shear component of deformation potential in Mo and W, obtained by the two methods: +KI , — from VX a high-field ultrasonic absorption, and +KI , — from the quanVX tum oscillation data. The solid line represents equality +KI , "+KI , . VX VX
High-field absorption of longitudinal ultrasound under H""q""[0 0 1] has not been measured in Mo. Thus, we have a good chance to calculate it by using of Eq. (4). Geometric parameters of the effective orbits are evaluated as described in Ref. [3]. Information on areas and area changes under the deformation u is summarized in Ref. [2]. Only XX one of the orbits, q, has not been investigated. The corresponding value d(ln A)/du "!1.9 was obXX tained by digitizing computer-simulated q-contours of Ref. [4] (reproduced in Ref. [2]). The final result of the calculation is C /f"42 dB/(cm GHz). There is an indirect way to check up the calculated value. Taking the experimental data of Ref. [3], one can derive the following empirical relation for Mo and W (longitudinal and transverse waves with H""q along [0 0 1] and [1 1 1]): C /f+ 0.2 ) (Z ) (C /f ), where Z is the atomic number (see Fig. 2). Zero-field absorption of longitudinal waves in Mo, C /f"63 dB/(cm GHz), is known from Ref. [10]. We can plot the calculated ch-absorption (the solid circle in the Fig. 2). Note a satisfactory agreement with the empirical line. Longitudinal waves (¼). Ultrasonic absorption in a high field has been determined in Ref. [3]: C /f"32 dB/(cm GHz) and one can evaluate the
238
A.V. Tkach / Physica B 263—264 (1999) 236—238
Fig. 2. High-field ultrasonic absorption in W and Mo, Ref. [3], as a function of a zero-field absorption. (䢇) Calculated value for longitudinal waves in Mo under H""q""[0 0 1].
corresponding DP-value for the orbit q. Contours of the FS are taken from Ref. [11]. Taking into account high uncertainty of d(ln A)/ du in Ref. [2], one can calculate (dA/dk) with XX X some simplifications. The averaged curvature of octahedron’s cross-sections across the l-orbit is estimated from two contours of Ref. [11]. As for the electron jack, it is formed of six equivalent ‘fingers’, each of them having a 4-fold axis. To calculate (dA/dk) for the orbits q, n, and p, each ‘finger’ is X assumed to form weakly noncircular normal crosssections: R(u, k )"RM (k )#DR(k )cos(4u). Here V V V u is the angle, accounted from the plane (k "0) X and R is the distance from the 4-fold axis. One could perceive these estimates to be an unreasonably rough (and for Mo particularly — because of a more anisotropic shape of a ‘finger’). But first, the assumptions are not obligatory — there are ways to calculate (dA/dk) more precisely, see X Refs. [11,6]. Second, we can carry out a compari-
son with an accurate calculation for Mo, Ref. [6]. For the orbits q and l in Mo, our method gives (dA/dk)"!9 and "!26, respectively, close X enough to the corresponding values of !10.6 and !30 as determined from plots of Ref. [6]. We obtain finally the following values: "d(ln A)/ du ""1.6 and "KI ""1.2 eV on the q-orbit in W. XX XX Relative contributions of different FS-sheets to the total ultrasonic absorption can be estimated also on the basis of Eq. (4). Corresponding values for ¼ are found 31% — for jack, 32% — for ellipsoids, and 37% — for octahedron. This work was supported by the Russian Foundation for Basic Research.
References [1] N.G. Bebenin, N.S. Yartseva, Sol. State Commun. 73 (1990) 579. [2] W. Joss, R. Griessen, E. Fawcett, Landolt—Bo¨rnstein numerical data and functional relationships in science and technology. New Series. Group III, 13(b), Springer, Berlin, 1983. [3] A.V. Tkach, A.B. Rinkevich, Phys. Met. Metallogr. 84 (1997) 621. [4] R. Griessen, M.J.G. Lee, D.J. Stanley, Phys. Rev. B 16 (1977) 4385. [5] R.F. Girvan, A.V. Gold, R.A. Phillips, J. Phys. Chem. Solids 29 (1968) 1458. [6] V.A. Gasparov, M.H. Harutunian, Phys. Stat. Sol. B 93 (1971) 403. [7] K. Dobson, A. Myers, Sol. State Commun. 35 (1980) 27. [8] M.J.G. Lee, J.M. Perz, D.J. Stanley, Phys. Rev. Lett. 37 (1976) 537. [9] J.M. Perz et al., Phys. Rev. B 19 (1979) 4901. [10] C.K. Jones, J.A. Rayne, Phys. Lett. 13 (1964) 282. [11] J.B. Ketterson et al., Phys. Rev. B 11 (1975) 1447.
Magnetoacoustic determination of deformation potential A.V. Tkach* Institute for Metal Physics, Russian Academy of Sciences, 18, Kovalevskaya St., Ekaterinburg 620219, Russia
Abstract On the basis of ultrasonic measurements in a classically high magnetic field, new information has been obtained about the averaged components of deformation potential on effective orbits in molybdenum and tungsten. In addition, the limiting attenuation of longitudinal waves, propagating along the magnetic field in the direction [0 0 1], has been calculated for Mo by the use of data on strain dependencies of the effective orbits. 1999 Elsevier Science B.V. All rights reserved. Keywords: Deformation potential; Ultrasonic absorption; Magnetoacoustics
At present, geometric characteristics of Fermi surfaces (FS) have been investigated quite well for a number of metals. There is also plenty of data on electron velocities at FSs. However, the deformation potential (DP) remains insufficiently studied. The DP takes into account variations of the electron energy under a lattice deformation. One has to know DP-values at the Fermi Surface to give a description of a physical phenomenon that involves the electron—phonon interaction. However, this information is inaccessible, as a rule. A possible approach, Ref. [1], to this problem consists in using a relationship between the DP (averaged over an effective orbit) and the area change (dA) of the orbit due to a deformation:
d(ln A) m KI "! GH. 2p du A GH * Fax: #7-3432-745344; e-mail: [email protected].
(1)
Here u ,ju /jx is the distortion tensor; m is the GH G H cyclotron mass, and KI is the (averaged) norGH malized deformation potential tensor. Available experimental data on the area variations under deformations have a restricted accuracy of 10—15%, Ref. [2]. A combination of complicated quantum oscillation techniques has to be used in such measurements. We demonstrate here some opportunities of another experimental approach. Let us consider the electronic attenuation C of ultrasound (of the wave vector q), propagating along an n-fold rotation axis (n*3) in a magnetic field H""q""O . If the free X path of electron l is large enough, absorption tends to a saturation level in classically high (ch) magnetic fields far above the edge of cyclotron absorption:
C 2\Np dA \ " N(m KI ) . H dk f
C K X
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 4 6 - 5
(2)
A.V. Tkach / Physica B 263—264 (1999) 236—238
237
Here p-index is equal to 3 for longitudinal waves and 4 for transverse ones, f is the ultrasonic frequency, is the Planck constant, and C denotes the quasi-static elasticity modulus. Summation in Eq. (2) is taken over the effective orbits m at the FS, k being the z-component of the electron wave X vector, and N denotes the orbit multiplicity. KI ,(KI ) if p"3, H XX (3) KI ,(KI )#(KI ) if p"4. H VX WX A geometric characteristic of the FS, (dA/dk), can X be determined with a high accuracy for a number of cases. Hence, the relative error of indirect determination of (m KI ) can be estimated as half of the H C-measurement error. This later one can be as small as 5%. As an example, we consider cubic metals tungsten and molybdenum under q""H""[0 0 1]. Normalized absorption (C /f ) is measured in Ref. [3] in two steps. First, the corresponding value (C /f ) at H"0 is determined, and then the difference is registered D"C /f!C /f. After that, C /f is de termined as C /f!D. Shear waves (Mo and W). Only four (N"4) orbits of type o contribute to a high-field absorption in this case, Ref. [4]. These orbits belong to inclined (with respect to H""[0 0 1]) ellipsoidal FS-sheets. Considering such an orbit at the axis [1 0 1], one has (by the symmetry) KI ,0. WX The ellipsoid semiaxes and the main effective masses can be taken from Refs. [5—7]. Thus, starting from values of (C /f )"42.6$2.2 dB/(cm GHz) in W and 44$7 dB/(cm GHz) in Mo [3], the DP-component "KI " is obtained. It equals 7.5 eV VX for W and 4.6 eV for Mo. We compare these values with those, derived, on the basis of Eq. (1), from data on d(ln A)/du , Refs. VX [8,9]. Fig. 1 gives a comparison of the two methods. It reveals a good consistency, but the method of ultrasonic absorption in a high magnetic field has an advantage of better accuracy as marked by dimensions of the error ellipses. Longitudinal waves (Mo). Eq. (2) can be transformed for longitudinal waves (p"3) to the following form:
C
d(ln A) dA \ " N A . f 4pC du dk K GH X
(4)
Fig. 1. Values of a shear component of deformation potential in Mo and W, obtained by the two methods: +KI , — from VX a high-field ultrasonic absorption, and +KI , — from the quanVX tum oscillation data. The solid line represents equality +KI , "+KI , . VX VX
High-field absorption of longitudinal ultrasound under H""q""[0 0 1] has not been measured in Mo. Thus, we have a good chance to calculate it by using of Eq. (4). Geometric parameters of the effective orbits are evaluated as described in Ref. [3]. Information on areas and area changes under the deformation u is summarized in Ref. [2]. Only XX one of the orbits, q, has not been investigated. The corresponding value d(ln A)/du "!1.9 was obXX tained by digitizing computer-simulated q-contours of Ref. [4] (reproduced in Ref. [2]). The final result of the calculation is C /f"42 dB/(cm GHz). There is an indirect way to check up the calculated value. Taking the experimental data of Ref. [3], one can derive the following empirical relation for Mo and W (longitudinal and transverse waves with H""q along [0 0 1] and [1 1 1]): C /f+ 0.2 ) (Z ) (C /f ), where Z is the atomic number (see Fig. 2). Zero-field absorption of longitudinal waves in Mo, C /f"63 dB/(cm GHz), is known from Ref. [10]. We can plot the calculated ch-absorption (the solid circle in the Fig. 2). Note a satisfactory agreement with the empirical line. Longitudinal waves (¼). Ultrasonic absorption in a high field has been determined in Ref. [3]: C /f"32 dB/(cm GHz) and one can evaluate the
238
A.V. Tkach / Physica B 263—264 (1999) 236—238
Fig. 2. High-field ultrasonic absorption in W and Mo, Ref. [3], as a function of a zero-field absorption. (䢇) Calculated value for longitudinal waves in Mo under H""q""[0 0 1].
corresponding DP-value for the orbit q. Contours of the FS are taken from Ref. [11]. Taking into account high uncertainty of d(ln A)/ du in Ref. [2], one can calculate (dA/dk) with XX X some simplifications. The averaged curvature of octahedron’s cross-sections across the l-orbit is estimated from two contours of Ref. [11]. As for the electron jack, it is formed of six equivalent ‘fingers’, each of them having a 4-fold axis. To calculate (dA/dk) for the orbits q, n, and p, each ‘finger’ is X assumed to form weakly noncircular normal crosssections: R(u, k )"RM (k )#DR(k )cos(4u). Here V V V u is the angle, accounted from the plane (k "0) X and R is the distance from the 4-fold axis. One could perceive these estimates to be an unreasonably rough (and for Mo particularly — because of a more anisotropic shape of a ‘finger’). But first, the assumptions are not obligatory — there are ways to calculate (dA/dk) more precisely, see X Refs. [11,6]. Second, we can carry out a compari-
son with an accurate calculation for Mo, Ref. [6]. For the orbits q and l in Mo, our method gives (dA/dk)"!9 and "!26, respectively, close X enough to the corresponding values of !10.6 and !30 as determined from plots of Ref. [6]. We obtain finally the following values: "d(ln A)/ du ""1.6 and "KI ""1.2 eV on the q-orbit in W. XX XX Relative contributions of different FS-sheets to the total ultrasonic absorption can be estimated also on the basis of Eq. (4). Corresponding values for ¼ are found 31% — for jack, 32% — for ellipsoids, and 37% — for octahedron. This work was supported by the Russian Foundation for Basic Research.
References [1] N.G. Bebenin, N.S. Yartseva, Sol. State Commun. 73 (1990) 579. [2] W. Joss, R. Griessen, E. Fawcett, Landolt—Bo¨rnstein numerical data and functional relationships in science and technology. New Series. Group III, 13(b), Springer, Berlin, 1983. [3] A.V. Tkach, A.B. Rinkevich, Phys. Met. Metallogr. 84 (1997) 621. [4] R. Griessen, M.J.G. Lee, D.J. Stanley, Phys. Rev. B 16 (1977) 4385. [5] R.F. Girvan, A.V. Gold, R.A. Phillips, J. Phys. Chem. Solids 29 (1968) 1458. [6] V.A. Gasparov, M.H. Harutunian, Phys. Stat. Sol. B 93 (1971) 403. [7] K. Dobson, A. Myers, Sol. State Commun. 35 (1980) 27. [8] M.J.G. Lee, J.M. Perz, D.J. Stanley, Phys. Rev. Lett. 37 (1976) 537. [9] J.M. Perz et al., Phys. Rev. B 19 (1979) 4901. [10] C.K. Jones, J.A. Rayne, Phys. Lett. 13 (1964) 282. [11] J.B. Ketterson et al., Phys. Rev. B 11 (1975) 1447.