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Electron-phonon scattering angles in copper
Volume 46A, number 1
PFIYSICS LETTERS
19 November 1973
ELECTRON-PHONON SCATTERING ANGLES IN COPPER* D.S. KHATRI** and JR. PEVERLEY Depart ~nent of Physics, The Uat/zo!ic Un ii’ersirs’ oJ A merica, WashingtonD.U. 20017, USA Received 24 September 1973 Electron-phonon scattering rates determined from the linewidths of acoustic open orbit resonances in copper have been analyzed by Monte-Carlo simulation, leading to numerical estimates of the average angle of scatter. The data suggest the dominance of transverse phonons as a scattering agent.
The acoustic open orbit resonance II] is a sharp resonance in the attenuation of ultrasound in a metal which take place when the spatial period of an open electron orbit matches that of the sound wave itself, The width of the resonance line in gauss is proportionalto the rate at which electrons are scattered out of the open orbit by collisions with scattering centers in the lattice. Talaat and Peverley [2] have carried out linewidth studies in copper and have measured the scattering rate due to collisions between electrons and 4. thermal They found a rate whichphonons. is intermediate between theproportional rates to T expected for catastrophic or large angle scattering (T3) and diffusive or small angle scattering(T5). It was concluded that the observed rate is sensitive to. and hence can be used to estimate, the angle of scatter~more precisely the root-mean-square value of the wavevector change suffered by an electron when in collision with a thermal phonon. In this paper we report a Monte-Carlo analysis of the4scattering regime and problem does indeed predict a T average scatenables a which more realistic estimate of the tering angle to be made; to our knowledge the first determination of this parameter. In the model calculation, the open orbit band is taken to be a flat two dimensional strip of width ~k in k-space. on which the electrons execute randoni walks as a result of collisions. The random walks are carried out by computer simulation. Each random walk terminates when an electron is randomized,
* **
36
Work supported by NSF Grant No. GY-3 285 and by ONR Contract No. N00014-67-A-0377-0007. Present address: Department of Physics, Federal City College, Washington D.C. 20005, USA.
either by leaving the strip~ or by progressing more than half the open orbit period in the direction of 2 . The probability distribution for the open orbitT individual collisions is taken to be of the form p(t) = exp( ri) where p(t) is the probability that the electron remains unscattered after time t and r is the mean scattering rate. i.e. l/r is the quasiparticle lifetime. A computer prograni is set up to simulate this distribution and to generate a sequence of times t~.t 2,... between individual collisions. The distribution function for the wavevector change between collisions is taken to be1’3 (C/Ti 2 Q(q) = -~-~-•--~---—-(I) [exp(aq)--lj [1 expLaq)j
where a = hv
5/kT. v~being the phonon velocity in the Dehye approximation. Thus Q(q)dg is the probability that the wavevector change q = k k’ lies in the dcrnent dq = 27rqdq of q-space, C being the constant of normalization. is taken to be randomly. independent the direction of q. Q which is chosen Weofnow define a time T which is the cumulative total of the individual times r~, t 2,... and which represents the time spent by the electron on the open orbit before being randomized in one of the allowed ways. After simulating a sequence of several thousand random walks it is possible. by constructing hisiograms, to investigate the form of the distribution ~ T) of r and we find that to a good approximation P(T) = exp( Rr) where R is now the average rate ai which electrons are scattered 2 This criterion is clearly inexact, thisinmode of The argument justifying this hashowever been given ref. [21. t randomization turns out to be unimportant if the open orbit period is long compared with its width, as in copper. ~ The origin of this expression is discussed in ref. [21.
Volume 46A, number
1
PHYSICS LETTERS
4 giveR = (1.35 ±0.20) X 105T4. copper. This The value of datat r was obtained by orbitally averaging thepoint-by-pointvaluesofDoezemaandKoch [4],. givingr (5.5 ±0.2) X 106T3. The only adjustable parameter in the fitting is the phonon velocity v~. Our results, which must be interpreted as rather
R/r 1.0 135 MHz 0.8
•
19 November 1973
255 MH
0.6
complicated averages, can thus be summarized; = (0.046 ±0.002)T,
(3)
v /
_______________________________________________ 0
0.5
1.0
1.5 qrms/~
2.0
2.5
3.0
k
Fig. 1. Monte-Carlo calcUlation (solid curve) and experimental data (points) for open orbit scattering rates in copper. The dotted lines are asymptotic rates valid in the large-angle and small-angle ~its.
out of the open orbit, i.e. the randomization rate. It follows from this that the open orbit lineshape should be a Lorentzian with half-width proportional to R, since .P(r) rather than p(t) would enter as the damping term in a trajectory calculation for the problem [3]. Noting that in (1) the temperature T is a free parameter, the simulation technique can be used to estimate the temperature dependence ofR. We find it more convenient to use as a free parameter the ratio q~~ 5/~k where q~5is defined by 2. (2) q~ Jq2Q q)dq =(4.l5kT/hu5) The result of this calculation is the smooth curve in fig. 1. It is seen that R is always less, than r because several collisions are sometimes required before an electron is randomized. Since r varies as T3 the asymptote R/r = 1 for q~~/L~k ~ 1 corresponds to catastrophic or large-angle scattering (R T3) and the quadratic region for q~ 5/1.~k ~ 3 corresponds to 5). The diffusiveparabola or small-angle scattering (R2 results T dotted R/r = ~ir2(q~ from 5/~k) limit [2] and the analytical solution in the diffusion is seen to be a good approximation only when ~ 0.2. The computed curve also shows a broad, approximately linear region where locally R —‘ T4. Also shown in fig. 1 are-some experimental points taken on the [111] -directed open orbit in ~‘
5cm/sec. (4) 5=(2.3±0.l)XlO The result (3) can also be expressed as a scattering angle on the equivalent free electron sphere since ~k, the copper neck diameter, is a known fraction of the Fermi radius. From (3) we thus find that ~ subtends (1.00 ±0.04)T degrees of angle at the zone center. The errors quoted above were derived from the spread in values obtained using two samplest4 and two acoustic frequencies, 135 MHz and 255 MHz. it seems not unlikely that larger systematic errors may have arisen on account of the failure of the copper Fermi surface to comply with the restrictions inherent in our model. It is known, for example that the local scattering rate r(k) is highly anisotropic and is highest on the Fermi surface necks, where the open orbit band deviates most drastically from our flat strip model. Nevertheless, we draw two tentative conclusions from the data. The first is that the scattering angle seems large enough so that the method of Doezema and Koch [4], insubtending which the effective electrons occupy a rectangle by 6°at the zone center, probably does count all 4° collisions and hence does give the true rate r(k) averaged only over the rectangle in question. Secondly, the average thermal phonon velocity extracted from the data is much more typical of the transverse branches of the phonon spectrum in copper than the longitudinal branch, for which the velocities are typically 5 X 10~cm/sec. The importance ofsuggested transverseby (strictly quasi-transverse) phonons our data is in accord with a recent theoretical study [5] of the electron-phonon interaction in copper. t~The data were taken on
two copper samples cut from a double having resistivity ratio 20 000. The detailed procedure was a slightly improved version of that described in ref. [2]. 37
Volume 46A. number 1
PHYSICS LETTERS
Theoretical studies of this type should in principle be capable of yielding point-by-point values of q~~ 5’ but this does not appear to have been attempted as yet. We are indebted to Dr. A.F. Clark of NBS, Boulder, Colorado for loaning us the copper boule from which the samples were cut, and to Dr. JR. Leibowitz for the loan of equipment. Computer time was provided by the Catholic Univercity Computer Center.
38
19 November 1973
References Ill E.A. Kaner, V.G. Peschanskii and l.A. Privorotskii, J.E.T.P. 13(1961)147. 12] MI-I. Talaat and JR. Peverley, Phys. Rev., to be published. 13] JR. Peverley, Phys. Rev., to be published. [4] RE. Doesema and J.F. Koch, Phys. Rev. B6 (1972) 207. [51 S.G. Das, Phys. Rev. B7 (1973) 2238.
PFIYSICS LETTERS
19 November 1973
ELECTRON-PHONON SCATTERING ANGLES IN COPPER* D.S. KHATRI** and JR. PEVERLEY Depart ~nent of Physics, The Uat/zo!ic Un ii’ersirs’ oJ A merica, WashingtonD.U. 20017, USA Received 24 September 1973 Electron-phonon scattering rates determined from the linewidths of acoustic open orbit resonances in copper have been analyzed by Monte-Carlo simulation, leading to numerical estimates of the average angle of scatter. The data suggest the dominance of transverse phonons as a scattering agent.
The acoustic open orbit resonance II] is a sharp resonance in the attenuation of ultrasound in a metal which take place when the spatial period of an open electron orbit matches that of the sound wave itself, The width of the resonance line in gauss is proportionalto the rate at which electrons are scattered out of the open orbit by collisions with scattering centers in the lattice. Talaat and Peverley [2] have carried out linewidth studies in copper and have measured the scattering rate due to collisions between electrons and 4. thermal They found a rate whichphonons. is intermediate between theproportional rates to T expected for catastrophic or large angle scattering (T3) and diffusive or small angle scattering(T5). It was concluded that the observed rate is sensitive to. and hence can be used to estimate, the angle of scatter~more precisely the root-mean-square value of the wavevector change suffered by an electron when in collision with a thermal phonon. In this paper we report a Monte-Carlo analysis of the4scattering regime and problem does indeed predict a T average scatenables a which more realistic estimate of the tering angle to be made; to our knowledge the first determination of this parameter. In the model calculation, the open orbit band is taken to be a flat two dimensional strip of width ~k in k-space. on which the electrons execute randoni walks as a result of collisions. The random walks are carried out by computer simulation. Each random walk terminates when an electron is randomized,
* **
36
Work supported by NSF Grant No. GY-3 285 and by ONR Contract No. N00014-67-A-0377-0007. Present address: Department of Physics, Federal City College, Washington D.C. 20005, USA.
either by leaving the strip~ or by progressing more than half the open orbit period in the direction of 2 . The probability distribution for the open orbitT individual collisions is taken to be of the form p(t) = exp( ri) where p(t) is the probability that the electron remains unscattered after time t and r is the mean scattering rate. i.e. l/r is the quasiparticle lifetime. A computer prograni is set up to simulate this distribution and to generate a sequence of times t~.t 2,... between individual collisions. The distribution function for the wavevector change between collisions is taken to be1’3 (C/Ti 2 Q(q) = -~-~-•--~---—-(I) [exp(aq)--lj [1 expLaq)j
where a = hv
5/kT. v~being the phonon velocity in the Dehye approximation. Thus Q(q)dg is the probability that the wavevector change q = k k’ lies in the dcrnent dq = 27rqdq of q-space, C being the constant of normalization. is taken to be randomly. independent the direction of q. Q which is chosen Weofnow define a time T which is the cumulative total of the individual times r~, t 2,... and which represents the time spent by the electron on the open orbit before being randomized in one of the allowed ways. After simulating a sequence of several thousand random walks it is possible. by constructing hisiograms, to investigate the form of the distribution ~ T) of r and we find that to a good approximation P(T) = exp( Rr) where R is now the average rate ai which electrons are scattered 2 This criterion is clearly inexact, thisinmode of The argument justifying this hashowever been given ref. [21. t randomization turns out to be unimportant if the open orbit period is long compared with its width, as in copper. ~ The origin of this expression is discussed in ref. [21.
Volume 46A, number
1
PHYSICS LETTERS
4 giveR = (1.35 ±0.20) X 105T4. copper. This The value of datat r was obtained by orbitally averaging thepoint-by-pointvaluesofDoezemaandKoch [4],. givingr (5.5 ±0.2) X 106T3. The only adjustable parameter in the fitting is the phonon velocity v~. Our results, which must be interpreted as rather
R/r 1.0 135 MHz 0.8
•
19 November 1973
255 MH
0.6
complicated averages, can thus be summarized; = (0.046 ±0.002)T,
(3)
v /
_______________________________________________ 0
0.5
1.0
1.5 qrms/~
2.0
2.5
3.0
k
Fig. 1. Monte-Carlo calcUlation (solid curve) and experimental data (points) for open orbit scattering rates in copper. The dotted lines are asymptotic rates valid in the large-angle and small-angle ~its.
out of the open orbit, i.e. the randomization rate. It follows from this that the open orbit lineshape should be a Lorentzian with half-width proportional to R, since .P(r) rather than p(t) would enter as the damping term in a trajectory calculation for the problem [3]. Noting that in (1) the temperature T is a free parameter, the simulation technique can be used to estimate the temperature dependence ofR. We find it more convenient to use as a free parameter the ratio q~~ 5/~k where q~5is defined by 2. (2) q~ Jq2Q q)dq =(4.l5kT/hu5) The result of this calculation is the smooth curve in fig. 1. It is seen that R is always less, than r because several collisions are sometimes required before an electron is randomized. Since r varies as T3 the asymptote R/r = 1 for q~~/L~k ~ 1 corresponds to catastrophic or large-angle scattering (R T3) and the quadratic region for q~ 5/1.~k ~ 3 corresponds to 5). The diffusiveparabola or small-angle scattering (R2 results T dotted R/r = ~ir2(q~ from 5/~k) limit [2] and the analytical solution in the diffusion is seen to be a good approximation only when ~ 0.2. The computed curve also shows a broad, approximately linear region where locally R —‘ T4. Also shown in fig. 1 are-some experimental points taken on the [111] -directed open orbit in ~‘
5cm/sec. (4) 5=(2.3±0.l)XlO The result (3) can also be expressed as a scattering angle on the equivalent free electron sphere since ~k, the copper neck diameter, is a known fraction of the Fermi radius. From (3) we thus find that ~ subtends (1.00 ±0.04)T degrees of angle at the zone center. The errors quoted above were derived from the spread in values obtained using two samplest4 and two acoustic frequencies, 135 MHz and 255 MHz. it seems not unlikely that larger systematic errors may have arisen on account of the failure of the copper Fermi surface to comply with the restrictions inherent in our model. It is known, for example that the local scattering rate r(k) is highly anisotropic and is highest on the Fermi surface necks, where the open orbit band deviates most drastically from our flat strip model. Nevertheless, we draw two tentative conclusions from the data. The first is that the scattering angle seems large enough so that the method of Doezema and Koch [4], insubtending which the effective electrons occupy a rectangle by 6°at the zone center, probably does count all 4° collisions and hence does give the true rate r(k) averaged only over the rectangle in question. Secondly, the average thermal phonon velocity extracted from the data is much more typical of the transverse branches of the phonon spectrum in copper than the longitudinal branch, for which the velocities are typically 5 X 10~cm/sec. The importance ofsuggested transverseby (strictly quasi-transverse) phonons our data is in accord with a recent theoretical study [5] of the electron-phonon interaction in copper. t~The data were taken on
two copper samples cut from a double having resistivity ratio 20 000. The detailed procedure was a slightly improved version of that described in ref. [2]. 37
Volume 46A. number 1
PHYSICS LETTERS
Theoretical studies of this type should in principle be capable of yielding point-by-point values of q~~ 5’ but this does not appear to have been attempted as yet. We are indebted to Dr. A.F. Clark of NBS, Boulder, Colorado for loaning us the copper boule from which the samples were cut, and to Dr. JR. Leibowitz for the loan of equipment. Computer time was provided by the Catholic Univercity Computer Center.
38
19 November 1973
References Ill E.A. Kaner, V.G. Peschanskii and l.A. Privorotskii, J.E.T.P. 13(1961)147. 12] MI-I. Talaat and JR. Peverley, Phys. Rev., to be published. 13] JR. Peverley, Phys. Rev., to be published. [4] RE. Doesema and J.F. Koch, Phys. Rev. B6 (1972) 207. [51 S.G. Das, Phys. Rev. B7 (1973) 2238.