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Empirical correlation between the superconducting Tc and the Hall constant
Volume 70A, number 2
PHYSICS LETTERS
19 February 1979
EMPIRICAL CORRELATION BETWEEN THE SUPERCONDUCTING T~ AND THE HALL CONSTANT* J.O. UNDE and Osten RAPP Department of Solid StatePhysics, Royal Institute of Technology, S-100 44 Stockholm 70, Sweden
Received 2 November 1978
It is found that the superconducting transition temperature and the galvano—magnetic Hall constant at room temperature are correlated in a large number of non-transition elements and alloys.
In the present letter an extensive correlation between the superconducting transition temperature, T~, and the galvano—magnetic Hall constant R is reported. Such a correlation was found a couple of years ago for
Superconductivity has recently been discovered in several noble metal based alloy systems [2,3]. In all cases where results for a number of alloys in the same alloy system were reported it was found that T0 in-
the non-transition elements by one of us [1]. It is seen from fig. 1 that for these elements a low value of IRI favours a relatively high value of T~.We now show that this correlation, in itself not widely known, applies also to a large number of non-transition metal alloy systems.
creases with increasing solute concentration. Transitions were observable down to 7 mK in those investigations corresponding to a lower limit of solute concentration of about 3 at%. Comprehensive measurements of the Hall coefficient at room temperature for noble metal alloys are available in the literature [4,5]. Neglecting an initial anisotropy effect, which in several alloy systems causes an initial increase of RI with a maximum at about 1 at% of solute, the overall behaviour of RI in these systems is a linear decrease with
Work supported in part by Naturvetenskapliga Forsknings-
r~det.
increasing solute concentration. Fig. 2 shows a plot of T~versus R for the six Aubased alloy systems where both Tc and R have been measured. For AuZn alloys the evaluation required extrapolation of the Hall effect data over a few at% of solute concentration. For comparison a few other alloy systems were similarly evaluated. In some cases was measured for a few samples of the same composition. The separate values then obtained have all been plotted
TIK) C
51 /
~T1
Ga.. -60000-50
-25
T)~..05
.09
.011
0
J~.~11~
P5
Au
~ .152
R. 1O”(m~/As) Fig. 1. Correlationbetween room-temperature R and
Tc for non-transition metal elements. T~denotes normal conductivity down to the temperature given (from ref. [11).
in the figure. The general character of the correlation between R and Tc is the same as that in fig. 1. l’his result remains valid also when data from the literature for Ag and Cu based alloys are included in a diagram like fig. 2 although the values are then more scattered. The change of the electron—phonon interaction A with alloying can be estimated from the temperature derivative of resistivity, dp/dT, above room tempera147
Volume 70A, number 2
o •
Au Ga
4
AuA(
:
£
~‘
PHYSICS LETTERS
ranges. A more definite statement may therefore have to await additional measurements on CuMg alloys. This correlation between R and T~is found also in several alloy systems from other parts of the periodic •
~“
0.2
table. Oneofsuch is PbBimetal which has system. the highest T~value any example non-transition alloy
o
Measurements of the Hall constant in this alloy system
o
have been reported Bi concentrations up towith 15 at% [9] and show that Rfor decreases monotonously the addition of Bi in agreement with the general correlation between R and T0 and the observed increase of T~with
4 V
01
Bi concentration [10]. We have extended the Hall effect measurements Bi concentrations [11] and have found thattoRhigher continues to decrease with in-
O
0.05
____________________________ -7 -6 -5 -‘ “~ 3A si RI” ~m Fig. 2. Tc versus R for Au based alloys. T~data are from refs. [2,3], R data are from ref. [5]. Open symbols: linearly interpolated Hall constants. Closed symbols: linearly extrapolated
ture for noble metal alloys [6,7]. Resistivity data are available for a large number of noble metal systems [8]. From these data further support is found for the correlation between R and Tc in a number of alloy systems where T~is not directly measurable. There are in all 27 alloy systems for which data on the concentration dependence of dp/dT [8] as well as R [4,5] are available. In 25 of these systems the electron—phonon coupling, as inferred from the data of dp/dT, increases on alloying while RI decreases in agreement with the correlation between R and T0. AuMg is particularly interesting because it is the only alloy system in refs. [4] and [5] where RI increases with solute concentration. From resistivity data it is inferred that A decreases with solute concentration for AuMg [7,8]. The correlation between R and T~is thus again confirmed also in this anomalous case. Finally, for the remaining alloy system, CuMg, resistivity data indicate a decreasing A while RI also decreases. It would thus seem that the correlation between R and T~fails in this case. The effects are small, however, Compared to the other anomalous system, AuMg, resistivity as well as Hall constant data for CuMg are restricted to smaller or much smaller concentration
148
19 February 1979
creasing Bi content at least up to below about 30 at% which gives further support to this correlation between RandT~. It should be noted that the correlation discussed
here is more general than the prevailing trend for nontransition metals in the periodic table that T~increases ..
and RI decreases with increasing number of valence electrons per atom, e/a. AuMg, which was discussed above, is one example of an alloy system where an electron per atom rule fails but the correlation between
R and T~is valid. Furthermore, in an iso-electronic alloy system, such as InTl, an electron per atom rule cannot be applied to predict the variation of T~and R. It is observed, however, that T~decreases [12] andR increases [13] with increased proportion of Tl which is again in agreement with the general correlation between R and T 0. A different example is provided by the unusual superconducting properties of Be. In the crystalline hcp phase T0 is 0.026 K to be compared with the thickness dependent T0’s well in excess of 9 K that have been observed [14,15] in thin Be-films condensed from the vapor on cold substrates. The low temperature phase is unstable at higher temperatures and transforms irreversibly, upon heating, to the hcp-low-T~phase. The resistance and the Hall constant have been measured simultaneously as a function of temperature on a highT~quench-condensed Be film [16]. It was found that the resistance decreased rapidly at about 50 K marking the disappearance of the high-Ta phase. This change was accompanied by a rapid increase of R in the same temperature interval. These observations are in line with the correlation presently discussed and give further support for its general applicability.
Volume 70A, number 2
PHYSICS LETTERS
The very high numerical value of R of Bi reflects the small number of carriers in this element. With the application of pressure, however, important changes in the electronic structure take place as apparent e.g. from the observation of superconductivity in most of the several high pressure phases of Bi [17]. From measurements of R under pressure Vaisnys and Kirk [18] found a value for Bill of about 1% of the low pressure value while R in phases III and IV could not be distinguished from zero within their experimental 3/As accuracy. Later a value of about +20 X 10—11 m was reported [19] for Bi III at 35 kbar and room temperature. The difference between the detailed pressure results of refs. [18] and [19] indicate the possibility of considerable experimental difficulties. Nevertheless, although the correlation of fig. 1 may not be valid for
Bi under pressure, there is strong qualitative support for the general trend. With the occurrence of superconductivity in Bi under pressure IRI decreases by several orders of magnitude.
On the other hand the Tc’s of Pb and PbBi alloys are only slightly affected by quench condensing [20] whereas R of Pb is strongly affected and will approach the free electron value [21] as expected for the liquid like structure of amorphous metals. These observations illustrate that it is not excluded to find examples where the correlation between R and Tc apparently breaks down. In conclusion we find nevertheless that the experimental evidence for a correlation between R and Tc is quite impressive. To our knowledge such a correlation has not hitherto been deduced from observed data and its connection with basic theory of superconductivity would merit further investigation. We have benefited from useful discussions with T. Claeson and R.E. Glover and stimulating correspond-
19 February 1979
References [1] J.O. Linde, Trans. Royal Inst. Technology FYS-5027 (Stockholm, 1975), unpublished. [2] R.F. Hoyt and A.C. Mota, Solid State Commun. 18 (1976) 139. [3] A.C. Mota and R.F. Hoyt, Solid State Commun. 20 (1976) 1025. [4] W. Köster and H.-P. Rave, Z. Metallkunde 55 (1964) [5] Köster and Hank, Z. Metallkunde 56 LT14, (1965)Vol. 846.3, [6] W. G. in:J.and Low physics eds.Grimvall, M. Krusius M.temperature Vuorio (North-Holland, Amsterdam, 1975), p. 126. [7] 6. Rapp, Phys. Lett. 64A (1977) 75. [8] J.O. Linde, Elektrische Widerstandseigenschaften der verdünnten Legierungen desLund. Kupfers, Silbers und Goldes (1939), Thesis, Gleerup, [9] K. Takano and T. Sako, J. Phys. Soc. Japan 20 (1965) 2013. [10] R.C. Dynes and J.M. Rowell, Phys. Rev. B1 1(1975) 1884. [11] J.O. Linde, 6. Rapp and(Stockholm, A. Saenz, Trans. Inst. Technology FYS-5045 1978),Royal unpublished. [12] M.F. Merriam, J. Hagen and H.L. Luo, Phys. Rev. 154
(1967) 424. [13] C.M. Hurd, The Hall effect of metals and alloys (Plenum, New York, 1972) p. 314. [14] R.E. III,519. S. Moser and F. Baumann, J. Low Temp. Phys. Glover 5 (1971) [15] C.G. Granqvist and T. Claeson, Phys. Lett. 47A (1974) [16] K. Yoshihiro and R.E. Glover III, in: Low temperature physics LT1 3, Vol. 3, eds. K.D. Timmerhaus, W.J. O’Sullivan and E.F. Hammel (Plenum, New York, 1974) p. 547. [17] B.W. Roberts, J. Phys. Chem. Ref. Data 5 (1976) 619. [18] J.R. Vaisnys and R.S. Kirk, J. Appl. Phys. 38(1967) 4335. [191 W.B. Holzapfel and713. D. Severin, High Temp. — High Pressures 1(1969) [20] J. Petersen, Z. Phys. B24 (1976) 283. [21] R. Koepke, Z. Phys. 264 (1973) 155.
ence with J. Friedel.
149
PHYSICS LETTERS
19 February 1979
EMPIRICAL CORRELATION BETWEEN THE SUPERCONDUCTING T~ AND THE HALL CONSTANT* J.O. UNDE and Osten RAPP Department of Solid StatePhysics, Royal Institute of Technology, S-100 44 Stockholm 70, Sweden
Received 2 November 1978
It is found that the superconducting transition temperature and the galvano—magnetic Hall constant at room temperature are correlated in a large number of non-transition elements and alloys.
In the present letter an extensive correlation between the superconducting transition temperature, T~, and the galvano—magnetic Hall constant R is reported. Such a correlation was found a couple of years ago for
Superconductivity has recently been discovered in several noble metal based alloy systems [2,3]. In all cases where results for a number of alloys in the same alloy system were reported it was found that T0 in-
the non-transition elements by one of us [1]. It is seen from fig. 1 that for these elements a low value of IRI favours a relatively high value of T~.We now show that this correlation, in itself not widely known, applies also to a large number of non-transition metal alloy systems.
creases with increasing solute concentration. Transitions were observable down to 7 mK in those investigations corresponding to a lower limit of solute concentration of about 3 at%. Comprehensive measurements of the Hall coefficient at room temperature for noble metal alloys are available in the literature [4,5]. Neglecting an initial anisotropy effect, which in several alloy systems causes an initial increase of RI with a maximum at about 1 at% of solute, the overall behaviour of RI in these systems is a linear decrease with
Work supported in part by Naturvetenskapliga Forsknings-
r~det.
increasing solute concentration. Fig. 2 shows a plot of T~versus R for the six Aubased alloy systems where both Tc and R have been measured. For AuZn alloys the evaluation required extrapolation of the Hall effect data over a few at% of solute concentration. For comparison a few other alloy systems were similarly evaluated. In some cases was measured for a few samples of the same composition. The separate values then obtained have all been plotted
TIK) C
51 /
~T1
Ga.. -60000-50
-25
T)~..05
.09
.011
0
J~.~11~
P5
Au
~ .152
R. 1O”(m~/As) Fig. 1. Correlationbetween room-temperature R and
Tc for non-transition metal elements. T~denotes normal conductivity down to the temperature given (from ref. [11).
in the figure. The general character of the correlation between R and Tc is the same as that in fig. 1. l’his result remains valid also when data from the literature for Ag and Cu based alloys are included in a diagram like fig. 2 although the values are then more scattered. The change of the electron—phonon interaction A with alloying can be estimated from the temperature derivative of resistivity, dp/dT, above room tempera147
Volume 70A, number 2
o •
Au Ga
4
AuA(
:
£
~‘
PHYSICS LETTERS
ranges. A more definite statement may therefore have to await additional measurements on CuMg alloys. This correlation between R and T~is found also in several alloy systems from other parts of the periodic •
~“
0.2
table. Oneofsuch is PbBimetal which has system. the highest T~value any example non-transition alloy
o
Measurements of the Hall constant in this alloy system
o
have been reported Bi concentrations up towith 15 at% [9] and show that Rfor decreases monotonously the addition of Bi in agreement with the general correlation between R and T0 and the observed increase of T~with
4 V
01
Bi concentration [10]. We have extended the Hall effect measurements Bi concentrations [11] and have found thattoRhigher continues to decrease with in-
O
0.05
____________________________ -7 -6 -5 -‘ “~ 3A si RI” ~m Fig. 2. Tc versus R for Au based alloys. T~data are from refs. [2,3], R data are from ref. [5]. Open symbols: linearly interpolated Hall constants. Closed symbols: linearly extrapolated
ture for noble metal alloys [6,7]. Resistivity data are available for a large number of noble metal systems [8]. From these data further support is found for the correlation between R and Tc in a number of alloy systems where T~is not directly measurable. There are in all 27 alloy systems for which data on the concentration dependence of dp/dT [8] as well as R [4,5] are available. In 25 of these systems the electron—phonon coupling, as inferred from the data of dp/dT, increases on alloying while RI decreases in agreement with the correlation between R and T0. AuMg is particularly interesting because it is the only alloy system in refs. [4] and [5] where RI increases with solute concentration. From resistivity data it is inferred that A decreases with solute concentration for AuMg [7,8]. The correlation between R and T~is thus again confirmed also in this anomalous case. Finally, for the remaining alloy system, CuMg, resistivity data indicate a decreasing A while RI also decreases. It would thus seem that the correlation between R and T~fails in this case. The effects are small, however, Compared to the other anomalous system, AuMg, resistivity as well as Hall constant data for CuMg are restricted to smaller or much smaller concentration
148
19 February 1979
creasing Bi content at least up to below about 30 at% which gives further support to this correlation between RandT~. It should be noted that the correlation discussed
here is more general than the prevailing trend for nontransition metals in the periodic table that T~increases ..
and RI decreases with increasing number of valence electrons per atom, e/a. AuMg, which was discussed above, is one example of an alloy system where an electron per atom rule fails but the correlation between
R and T~is valid. Furthermore, in an iso-electronic alloy system, such as InTl, an electron per atom rule cannot be applied to predict the variation of T~and R. It is observed, however, that T~decreases [12] andR increases [13] with increased proportion of Tl which is again in agreement with the general correlation between R and T 0. A different example is provided by the unusual superconducting properties of Be. In the crystalline hcp phase T0 is 0.026 K to be compared with the thickness dependent T0’s well in excess of 9 K that have been observed [14,15] in thin Be-films condensed from the vapor on cold substrates. The low temperature phase is unstable at higher temperatures and transforms irreversibly, upon heating, to the hcp-low-T~phase. The resistance and the Hall constant have been measured simultaneously as a function of temperature on a highT~quench-condensed Be film [16]. It was found that the resistance decreased rapidly at about 50 K marking the disappearance of the high-Ta phase. This change was accompanied by a rapid increase of R in the same temperature interval. These observations are in line with the correlation presently discussed and give further support for its general applicability.
Volume 70A, number 2
PHYSICS LETTERS
The very high numerical value of R of Bi reflects the small number of carriers in this element. With the application of pressure, however, important changes in the electronic structure take place as apparent e.g. from the observation of superconductivity in most of the several high pressure phases of Bi [17]. From measurements of R under pressure Vaisnys and Kirk [18] found a value for Bill of about 1% of the low pressure value while R in phases III and IV could not be distinguished from zero within their experimental 3/As accuracy. Later a value of about +20 X 10—11 m was reported [19] for Bi III at 35 kbar and room temperature. The difference between the detailed pressure results of refs. [18] and [19] indicate the possibility of considerable experimental difficulties. Nevertheless, although the correlation of fig. 1 may not be valid for
Bi under pressure, there is strong qualitative support for the general trend. With the occurrence of superconductivity in Bi under pressure IRI decreases by several orders of magnitude.
On the other hand the Tc’s of Pb and PbBi alloys are only slightly affected by quench condensing [20] whereas R of Pb is strongly affected and will approach the free electron value [21] as expected for the liquid like structure of amorphous metals. These observations illustrate that it is not excluded to find examples where the correlation between R and Tc apparently breaks down. In conclusion we find nevertheless that the experimental evidence for a correlation between R and Tc is quite impressive. To our knowledge such a correlation has not hitherto been deduced from observed data and its connection with basic theory of superconductivity would merit further investigation. We have benefited from useful discussions with T. Claeson and R.E. Glover and stimulating correspond-
19 February 1979
References [1] J.O. Linde, Trans. Royal Inst. Technology FYS-5027 (Stockholm, 1975), unpublished. [2] R.F. Hoyt and A.C. Mota, Solid State Commun. 18 (1976) 139. [3] A.C. Mota and R.F. Hoyt, Solid State Commun. 20 (1976) 1025. [4] W. Köster and H.-P. Rave, Z. Metallkunde 55 (1964) [5] Köster and Hank, Z. Metallkunde 56 LT14, (1965)Vol. 846.3, [6] W. G. in:J.and Low physics eds.Grimvall, M. Krusius M.temperature Vuorio (North-Holland, Amsterdam, 1975), p. 126. [7] 6. Rapp, Phys. Lett. 64A (1977) 75. [8] J.O. Linde, Elektrische Widerstandseigenschaften der verdünnten Legierungen desLund. Kupfers, Silbers und Goldes (1939), Thesis, Gleerup, [9] K. Takano and T. Sako, J. Phys. Soc. Japan 20 (1965) 2013. [10] R.C. Dynes and J.M. Rowell, Phys. Rev. B1 1(1975) 1884. [11] J.O. Linde, 6. Rapp and(Stockholm, A. Saenz, Trans. Inst. Technology FYS-5045 1978),Royal unpublished. [12] M.F. Merriam, J. Hagen and H.L. Luo, Phys. Rev. 154
(1967) 424. [13] C.M. Hurd, The Hall effect of metals and alloys (Plenum, New York, 1972) p. 314. [14] R.E. III,519. S. Moser and F. Baumann, J. Low Temp. Phys. Glover 5 (1971) [15] C.G. Granqvist and T. Claeson, Phys. Lett. 47A (1974) [16] K. Yoshihiro and R.E. Glover III, in: Low temperature physics LT1 3, Vol. 3, eds. K.D. Timmerhaus, W.J. O’Sullivan and E.F. Hammel (Plenum, New York, 1974) p. 547. [17] B.W. Roberts, J. Phys. Chem. Ref. Data 5 (1976) 619. [18] J.R. Vaisnys and R.S. Kirk, J. Appl. Phys. 38(1967) 4335. [191 W.B. Holzapfel and713. D. Severin, High Temp. — High Pressures 1(1969) [20] J. Petersen, Z. Phys. B24 (1976) 283. [21] R. Koepke, Z. Phys. 264 (1973) 155.
ence with J. Friedel.
149