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Phonon effects in the motion of positrons in metals
PHYSICS
Volume 24A, number 8
LETTERS
alloys Kg is negligible and Kmeas = 1 We, so that meas - Wpt$“) = (W” - Wzure) + Aw. There is (w
For the belly region of Fermi surface, small angle scattering processes hardly contribute to the electrical resistivity but can change electrons from ‘hot’ to ‘cold’. They are therefore much more effective in reducing th thermal than the p’ >>T&, . In the electrical current, and so 7pb neck regions, on the other hand, the same phonon wave vector can scatter an electron through a much lar r angle so that there is less differIt follws that 7ph/7 ph ’’ ence in C n and 7&. Wn Wb ph P > ?pn $,h , and so CYWb CY~in accordance with our experimental observation that yw > yp.
clear evidence from fig. 1 for the existence of these deviations. In NiCu, phonon scattering due to mass difference effects is much smaller than in NiPd. On the other hand, the difference in valency between the two elements leads to an increase in WO so that the ratio K~/K~ is greater than in NiPd and ~~ can no longer be neglected. When a direct separation of K and Ke is made in NiCu without taking account of 1 w, the lattice component is found to be negative over the range of temperatures in which AW should be greatest. Sensible values of K can only be obtained when Ke is corrected to 1.I?elude Aw. However, the value of yw required to make this correction is -0.6, which is about 2Yp. The electrical and thermal conductivities originating from any area AY of the Fermi surface both depend on the product Arvrrr, where vy is the velocity of electrons at A,, and 7y is their relaxation time [7]. The anisotropy factors in eq. (1) are therefore Of the form AnvnTn/AbvbTb, where the subscripts n and b refer to ‘neck’ and ‘belly’ respectively. Since the ratio A,v,/Abvb (= C) is the same for p and W,, the four aniso&-oby factors we are concerned with are, (YP = = c$/r;;, ‘yw = C$;/T&, @ = CT&, /;b ,
We are indebted to Professor J. S. Dugdale for leading numerous discussions on the subject of anisotropic relaxation times, and we whould like to thank SRC for financial assistance and for a maintenance grant to T. F.
References 1. N.F.Mott, Proc.Roy.Soc.Al53 (1936) 699; Al56 (1936) 368, 2. A. I.Schindler, R. J.Smith and E. I. Salkovitz, J. Phys. Chem.Solids 1 (1956) 39. 3. A. Hasegawa, S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 20 (1965) 1865. 4. L. Hodges, H. Ehrenreich and N. D. Lang, Phys.Rev. 152 (1966) 505. 5. J.M: Zim& Phys.Rev. 121 (1961) 1320. 6. Z.S. Basinski and J.S. Dugdale, Phgs.Rev. , to be published. 7. J. M. Ziman, Principles of theory of solids (Cambridge University Press, 1964) p. 196.
and pw = C’r”wnj.T&b. The validity of the Wiedemann-Franz law at low temperatures implies that the relaxation times for impurity scattering are the same in p and We, so that pp = pw. However, this will not be so for low temperature phonon scattering. **
PHONON
EFFECTS
IN
THE
10 April 1967
* * *
MOTION
OF
POSITRONS
IN
METALS
H. J. MIKESKA Physik Department,
Technische
Hochschule,
Miinchen,
Deutschland
Received 10 March 1967
The positron-phonon interaction is shown to lead to an important tail in the positron momentum distribution. The tail can account for at least part of the large so-called effective mass of positrons in sodium.
Recent experiments on the annihilation of positrons in sodium [l] have shown a thermal smearing of the Fermi-cutoff which leads to an effective 402
mass rn: = (1.9 f 0.4)m of the positron (bare mass m) when fitted by a Gaussian momentum distribution for the positron. Calculations of the positron
Volume
24A,
number
8
PHYSICS
self energy due to electron-positron interaction [2] and to band effects [3] give rn: z 1.25m. We have estimated the contribution of positron-phonon interaction (using Migdal’s model of the electron-phonon interaction [4]) and found an increase of 4% in the effective mass. The aim of ,this paper is to show that the remaining discrepancy can be understood by considering the momentum distribution of the positron in detail. We express the probability for two Y-quanta to emerge with momentum kZ in z-direction (which is the quantity determined experimentally) by a convolution of the momentum distribution of the electron n(p) (taken as the free electron Fermi distribution) and that the positron m(p) P(k,)
cc Sd&d$
sd3p n(P) m&-p)
.
0)
This corresponds to approximating the electronpositron Green function by a product of the electron and the positron propagator. This simplification has been shown to be correct [5] apart from a factor of about 10 which describes the increased electron density surrounding the positron due to Coulomb attraction, but is fortunately nearly momentum independent. Some account of the neglected correlations is given below. In the following we consider the dressing of the positron due to positron-phonon interaction. Our essential result is that this interaction leads to a tail ap-4 in the momentum distribution which dominates the Gaussian part for p > 0.2 kD (Debye momentum kD). On the other hand higher momenta are important in eq. (1) because of the integration over two components of the momentum. We define the effective mass for a nonGaussian momentum distribution as in [6], i. e. comparing the momentum distribution to a Gaussian one giving the same value P(kf) (Fermi momentum kf). The momentum distribution is expressed by the positron self energy kf(p, w) + iP(p, w)
m(p) = =
m
10 April
LETTERS
1967
are well separated for p >> m:c, the first one leading to a Gaussian part in m(p), the second one to a p-4 part. We calculate P by applying Migdal’s model for the electron-phonon interaction to the positronphonon interaction and obtain for 2T >>pc m(p) = ep’T
zp exp[-P2/2m*,T)
1+
1
(3)
The deviation of the renormalization factor Zp from 1 can be neglected. For 2T <
++3-ad@&Qw w;::;:r2(p I -E(p,
-m
, w)
~_ris the chemical potential of positrons, -p/T >> 1, and E(p, w) =p2/2m +M(p, w). In the following we neglect the retardation of E(p, o), using E(p, w) = Ep =p2/2m;f. We consider the contribution to I from the interaction with phonons having a linear spectrum w(k) = ck (k =SkD, no Umklapp-processes). From energy and momentum conservation follows that r(p, w) is nonvanishing for w + j.~ > A@ - $m]Sc2. There are two important contributions to the integral in (2), w = -cc +Ep and w * -EL. The two contributions
I wish to thank Prof. W. Brenig, who called my attention to this problem, for interesting discussions.
403
Volume
24A,
number 8
References 1. A. T. Stewart,
PHYSICS
J. B. Shand and S. M.Kim, Proc. Sot. 88 (1966) 1001. 2. D.R.Hamann, Phys. Rev. 146 (1966) 277. 3. C.K.Majumdar, Phys.Rev. 149 (1966) 406.
LETTERS
Phys.
10 April
1967
4. A. B. Migdal, Sov.Phys. JETP 7 (1958) 996. 5. J. P. Carbotte, S.Kahana, Phys. Rev. 139 (1965) A213. 6. C.K.Majumdar, Phys.Rev. 140 (1965) A237.
*****
ELECTRICAL
I. Physikalisches
PROPERTIES
OF
THE
SYSTEM
Cd,Srl_xO
M. VOLKMANN and W. WAIDELICH Institut der Technischen Hochschule Darmstadt, Received
6 March
Gemzany
1967
Measurements of electrical conductivity, Hall effect and thermoelectric power were made at sintered samples of CdxSrI_xO from -160 to +50°C. The temperature dependence of the mobility is interpreted as due to scattering by ionized impurities.
Previous work about the electrical and optical properties of Cd0 mostly deals with thin layers with electron concentrations from 1016 to some 1020 cmw3 [l]. The electrical properties of thin layers probably are influenced by the grain structure of these films. Even several investigations were made with sintered samples [2] and monocrystals [3]. Though these studies ppssibly better relate to the bulk properties, other difficulties arise: reproducable preparation was possible only in the concentration range between 1018 and 2 x 1020 cm-3. All experiments to lower the carrier concentration by doping with impurities at concentrations less than 10% failed. Considering this situation it seems of some interest to investigate the properties of Cd0 mixed crystals. For this purpose the systems CdfirI _& [4], Cd&al _& and probably CdxMnl _%O [5] may be used. Starting with CdO, the properties of which are well known by the investigations mentioned before, we prepared the system Cd&l_&0 by heating mixtures of Cd0 and SrO powder at llOO°C for about 4 hours [4]. The lattice constant, determined by X-ray diffraction, varies nearly linear from 4.69 A for Cd0 to 5.15 R for SrO. The electrical measurements were made at samples, pressed with about lo4 kp/cm2 and after that sintered at 8000C over four hours. From these sam les we cut pieces of about 5 x IO x conductivity, Hall efx0.5 mm B. The electrical fect and thermoelectric power was investigated between -160 and +50°C. The room temperature 404
100
2003aO 512)‘K lempemfure ----t
Fig. 1. Hall mobility versus absolute temperature. The parameter x corresponds to the Cd/Sr ratio, x = 1 relates to a polycrystalline Cd0 sample grown from the vapour phase.
conductivity varies from 3.8 X lo2 51-l cm-l for CdO.gSrO.10 to 3.1 x lo-11 52-l cm-l for CdO.6SrO.40. The logarithm of the conductivity was found to be proportional to the reciprocal of the absolute temperature. The activation energy was about 0 eV for CdO.gSrO.10 and 0.8 eV for
Volume 24A, number 8
LETTERS
alloys Kg is negligible and Kmeas = 1 We, so that meas - Wpt$“) = (W” - Wzure) + Aw. There is (w
For the belly region of Fermi surface, small angle scattering processes hardly contribute to the electrical resistivity but can change electrons from ‘hot’ to ‘cold’. They are therefore much more effective in reducing th thermal than the p’ >>T&, . In the electrical current, and so 7pb neck regions, on the other hand, the same phonon wave vector can scatter an electron through a much lar r angle so that there is less differIt follws that 7ph/7 ph ’’ ence in C n and 7&. Wn Wb ph P > ?pn $,h , and so CYWb CY~in accordance with our experimental observation that yw > yp.
clear evidence from fig. 1 for the existence of these deviations. In NiCu, phonon scattering due to mass difference effects is much smaller than in NiPd. On the other hand, the difference in valency between the two elements leads to an increase in WO so that the ratio K~/K~ is greater than in NiPd and ~~ can no longer be neglected. When a direct separation of K and Ke is made in NiCu without taking account of 1 w, the lattice component is found to be negative over the range of temperatures in which AW should be greatest. Sensible values of K can only be obtained when Ke is corrected to 1.I?elude Aw. However, the value of yw required to make this correction is -0.6, which is about 2Yp. The electrical and thermal conductivities originating from any area AY of the Fermi surface both depend on the product Arvrrr, where vy is the velocity of electrons at A,, and 7y is their relaxation time [7]. The anisotropy factors in eq. (1) are therefore Of the form AnvnTn/AbvbTb, where the subscripts n and b refer to ‘neck’ and ‘belly’ respectively. Since the ratio A,v,/Abvb (= C) is the same for p and W,, the four aniso&-oby factors we are concerned with are, (YP = = c$/r;;, ‘yw = C$;/T&, @ = CT&, /;b ,
We are indebted to Professor J. S. Dugdale for leading numerous discussions on the subject of anisotropic relaxation times, and we whould like to thank SRC for financial assistance and for a maintenance grant to T. F.
References 1. N.F.Mott, Proc.Roy.Soc.Al53 (1936) 699; Al56 (1936) 368, 2. A. I.Schindler, R. J.Smith and E. I. Salkovitz, J. Phys. Chem.Solids 1 (1956) 39. 3. A. Hasegawa, S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 20 (1965) 1865. 4. L. Hodges, H. Ehrenreich and N. D. Lang, Phys.Rev. 152 (1966) 505. 5. J.M: Zim& Phys.Rev. 121 (1961) 1320. 6. Z.S. Basinski and J.S. Dugdale, Phgs.Rev. , to be published. 7. J. M. Ziman, Principles of theory of solids (Cambridge University Press, 1964) p. 196.
and pw = C’r”wnj.T&b. The validity of the Wiedemann-Franz law at low temperatures implies that the relaxation times for impurity scattering are the same in p and We, so that pp = pw. However, this will not be so for low temperature phonon scattering. **
PHONON
EFFECTS
IN
THE
10 April 1967
* * *
MOTION
OF
POSITRONS
IN
METALS
H. J. MIKESKA Physik Department,
Technische
Hochschule,
Miinchen,
Deutschland
Received 10 March 1967
The positron-phonon interaction is shown to lead to an important tail in the positron momentum distribution. The tail can account for at least part of the large so-called effective mass of positrons in sodium.
Recent experiments on the annihilation of positrons in sodium [l] have shown a thermal smearing of the Fermi-cutoff which leads to an effective 402
mass rn: = (1.9 f 0.4)m of the positron (bare mass m) when fitted by a Gaussian momentum distribution for the positron. Calculations of the positron
Volume
24A,
number
8
PHYSICS
self energy due to electron-positron interaction [2] and to band effects [3] give rn: z 1.25m. We have estimated the contribution of positron-phonon interaction (using Migdal’s model of the electron-phonon interaction [4]) and found an increase of 4% in the effective mass. The aim of ,this paper is to show that the remaining discrepancy can be understood by considering the momentum distribution of the positron in detail. We express the probability for two Y-quanta to emerge with momentum kZ in z-direction (which is the quantity determined experimentally) by a convolution of the momentum distribution of the electron n(p) (taken as the free electron Fermi distribution) and that the positron m(p) P(k,)
cc Sd&d$
sd3p n(P) m&-p)
.
0)
This corresponds to approximating the electronpositron Green function by a product of the electron and the positron propagator. This simplification has been shown to be correct [5] apart from a factor of about 10 which describes the increased electron density surrounding the positron due to Coulomb attraction, but is fortunately nearly momentum independent. Some account of the neglected correlations is given below. In the following we consider the dressing of the positron due to positron-phonon interaction. Our essential result is that this interaction leads to a tail ap-4 in the momentum distribution which dominates the Gaussian part for p > 0.2 kD (Debye momentum kD). On the other hand higher momenta are important in eq. (1) because of the integration over two components of the momentum. We define the effective mass for a nonGaussian momentum distribution as in [6], i. e. comparing the momentum distribution to a Gaussian one giving the same value P(kf) (Fermi momentum kf). The momentum distribution is expressed by the positron self energy kf(p, w) + iP(p, w)
m(p) = =
m
10 April
LETTERS
1967
are well separated for p >> m:c, the first one leading to a Gaussian part in m(p), the second one to a p-4 part. We calculate P by applying Migdal’s model for the electron-phonon interaction to the positronphonon interaction and obtain for 2T >>pc m(p) = ep’T
zp exp[-P2/2m*,T)
1+
1
(3)
The deviation of the renormalization factor Zp from 1 can be neglected. For 2T <
++3-ad@&Qw w;::;:r2(p I -E(p,
-m
, w)
~_ris the chemical potential of positrons, -p/T >> 1, and E(p, w) =p2/2m +M(p, w). In the following we neglect the retardation of E(p, o), using E(p, w) = Ep =p2/2m;f. We consider the contribution to I from the interaction with phonons having a linear spectrum w(k) = ck (k =SkD, no Umklapp-processes). From energy and momentum conservation follows that r(p, w) is nonvanishing for w + j.~ > A@ - $m]Sc2. There are two important contributions to the integral in (2), w = -cc +Ep and w * -EL. The two contributions
I wish to thank Prof. W. Brenig, who called my attention to this problem, for interesting discussions.
403
Volume
24A,
number 8
References 1. A. T. Stewart,
PHYSICS
J. B. Shand and S. M.Kim, Proc. Sot. 88 (1966) 1001. 2. D.R.Hamann, Phys. Rev. 146 (1966) 277. 3. C.K.Majumdar, Phys.Rev. 149 (1966) 406.
LETTERS
Phys.
10 April
1967
4. A. B. Migdal, Sov.Phys. JETP 7 (1958) 996. 5. J. P. Carbotte, S.Kahana, Phys. Rev. 139 (1965) A213. 6. C.K.Majumdar, Phys.Rev. 140 (1965) A237.
*****
ELECTRICAL
I. Physikalisches
PROPERTIES
OF
THE
SYSTEM
Cd,Srl_xO
M. VOLKMANN and W. WAIDELICH Institut der Technischen Hochschule Darmstadt, Received
6 March
Gemzany
1967
Measurements of electrical conductivity, Hall effect and thermoelectric power were made at sintered samples of CdxSrI_xO from -160 to +50°C. The temperature dependence of the mobility is interpreted as due to scattering by ionized impurities.
Previous work about the electrical and optical properties of Cd0 mostly deals with thin layers with electron concentrations from 1016 to some 1020 cmw3 [l]. The electrical properties of thin layers probably are influenced by the grain structure of these films. Even several investigations were made with sintered samples [2] and monocrystals [3]. Though these studies ppssibly better relate to the bulk properties, other difficulties arise: reproducable preparation was possible only in the concentration range between 1018 and 2 x 1020 cm-3. All experiments to lower the carrier concentration by doping with impurities at concentrations less than 10% failed. Considering this situation it seems of some interest to investigate the properties of Cd0 mixed crystals. For this purpose the systems CdfirI _& [4], Cd&al _& and probably CdxMnl _%O [5] may be used. Starting with CdO, the properties of which are well known by the investigations mentioned before, we prepared the system Cd&l_&0 by heating mixtures of Cd0 and SrO powder at llOO°C for about 4 hours [4]. The lattice constant, determined by X-ray diffraction, varies nearly linear from 4.69 A for Cd0 to 5.15 R for SrO. The electrical measurements were made at samples, pressed with about lo4 kp/cm2 and after that sintered at 8000C over four hours. From these sam les we cut pieces of about 5 x IO x conductivity, Hall efx0.5 mm B. The electrical fect and thermoelectric power was investigated between -160 and +50°C. The room temperature 404
100
2003aO 512)‘K lempemfure ----t
Fig. 1. Hall mobility versus absolute temperature. The parameter x corresponds to the Cd/Sr ratio, x = 1 relates to a polycrystalline Cd0 sample grown from the vapour phase.
conductivity varies from 3.8 X lo2 51-l cm-l for CdO.gSrO.10 to 3.1 x lo-11 52-l cm-l for CdO.6SrO.40. The logarithm of the conductivity was found to be proportional to the reciprocal of the absolute temperature. The activation energy was about 0 eV for CdO.gSrO.10 and 0.8 eV for