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Thermopower of cubic transition metals
3-374
TECHNICAL
REFERENCES I. SILVERMAN S. M..J. &em. Phys. 25, 1081 (1956); 1. Geophys. Res. 75 in press. f 1970). 2. DANIEL V., Adr. Phys. (Phil. iwag. Sicppl.) 2 450 (1953). 3. FlNKE H. L., GROSS M. E.. WADDINGTON G. and HUFFMAN H. M., J. Amu. Chem. Sot. 76, 333 (1954).
J. Phys. Chem.
Vol. 3 I.
Solids
NOTES
I’
pp. 2374-2377.
Thermopower of cubic transition metals* 4 Febrmry
WE HAVE measured the thermopower of the transition metals vanadium, niobium, tantalum, molybdenum. tungsten. rhenium. rhodium. and iridium. The measurements were performed on wires of 99.99 per cent purity supplied by Materials Research Corporation;; using an apparatus similar to that of Henry and Schroeder~l]. Except for rhenium which has a hexagonal close packed structure these metals are cubic and the single component of the thermopower tensor can therefore be measured on polycrystalline wires. For rhenium the results will be some kind of average over the two components of the thermopower tensor which applies to hcp metals. The main object is to compare the results for the three groupings (V. Nb, Taf. (MO, Wf, (Rh, It-). within which the elements have similar outer shells of electrons. In Figs. I-4 we show the thermopowers of the metals in the above groupings. The results for Va are similar to those of Potter[2] and McIntosh and Sill[3]. except that we did not observe the small anomaly seen by them at 217°K. The Nb results agree with those of Weinberg and Schultz[4] and Potter [2], but join rather crudely to the high temperature data of Raag and Kowger[5]. Our low *This work Foundation.
was
supported
by the National
0
/
/
50
Fig. 1. Thermopowerof
i 970)
Science
I’
I
l
~~~~,
(Rerrioed
I’
I’
, 100
’
/’
II
Vanadium
‘?~~~’ 150 200 250 300 T (OKI vanadium. niobium. and tantalum.
temperature results for MO and W join neatly onto the results calculated from Lander’s [6] high temperature Thomson coefficient measurements and also, for W, onto the results of Raag and Kowger[5]. For W. the thermopower becomes positive at the lowest temperature in agreement with Trodahl[7]. Measurements on tungsten wire drawn from a zone refined sample gave results - O-02 pV/ “K less than the M.R.C. tungsten for temperatures greater than 40°K. Our rhodium and iridium results differ markedly from those of Potter [2]. The most striking feature of the results is the similarity of the results for the metals within each of the above groupings. Undoubtedly this is related to the similarity of the Fermi surfaces within each grouping. We now briefly consider ways in which the Fermi surface properties may manifest themselves. First of all we assume that the low tempera ture peaks occurring in the vicinity of 19J5. where Bn is the Debye temperature. are caused by phonon drag. The simplest picture to explain the sign of these peaks appears to apply remarkably well to Mg. Zn and Cd@]. In Fig. 5(a), we show a phonon induced transition between two states on an electron sheet of the Fermi surface in the periodically extended zone scheme. For such a transition the electron velocity 2, in the direction of the
2375
NOTES
TECHNICAL
n
0 W - Prsrrnt lnvertigation - Raag ELKowger xw - Lander *W
I
I
i
I
1
1
I
I
f
I
i \ J 1
‘7l -0.5 -
0
1
0
I1
200
I
I
11
400
Fig. 2. Thermopower
600
’
IO I
’
’
20 I
30 I
800 1000 T (OKI
’
40 I
50 II
f
11
1200
of molybdenum and tungsten. I-zone 9999% purity tungsten.
1400
11
1600
refined tungsten: a-
Re
50
100
150 200 T (OKI
Fig. 3. Thermopower
250
of rhenium.
phonon wave vector is increased. This leads Lo a negative contribution to the phonon-drag thermopower[9. lo]. Conversely. the transition across the hole region of the Fermi surface illustrated in Fig. 5(b) leads to a positive contribution to the thermopower. Transitions like that labelled 4’ in Fig. 5 lead to a smaller contribution than q because only a component of the electron velocity is reversed. At the lowest temperature we expect the sign of the phonon drag thermopower to be determined by the nature, hole or
-0
50 I
100 \
Fig. 4. Thermopower
150 \
200 I
1 250
300 I
T f*K) of rhodium and iridium.
electron, of the piece of Fermi surface with the smallest calliper dimensions. There will be relatively few phonons with wave vector large enough to span the larger surfaces. As the temperature increases we expect the contribution of the next largest piece of Fermi surface to begin to predominate. In the vicinity of &,/5 the phonon drag thermopower
TECHNICAL
2376
lb) Fig.
5. Phonon
induced transitions surface.
across
the Fermi
usually begins to decrease because of phononphonon scattering. This highly simplified picture would lead us to expect the observed positive phonon drag thermopower in the vanadium group of metals since the proposed Fermi surface of these metals is made up of hole sheets [ 11, 121. For W and MO the situation is complicated by the presence of hole and electron sheets [1 1. 13. 141. For W the smallest pieces of Fermi surface are hole sheets (at N in Ref. [ 131) with dimensions significantly smaller than the next largest surface which is an electron sheet. This may account for the initial positive thermopower in tungsten and the large negative dip thereafter. In MO analogous pieces of Fermi surface are much closer to being equal. Furthermore MO has an additional small electron sheet (the lenses of Ref. [ 131. It would appear that contributions from these sheets tend to cancel and produce a comparatively small negative contribution only in MO. In rhenium the three smallest sheets of the Fermi surface are hole sheets[l5]. We therefore expect the observed positive contribution to the phonon drag thermopower. When we come to Rh and Ir the above picture does not fit at all. The smallest pieces of Fermi surface are hole sheets and these are significantly smaller than the electron sheets [16. 171. To make any sense of these results we have to resort to the possibility of interband transitions using arguments similar to those used by Fletcher and Greig[181 to explain the positive phonon drag peaks in palladium and platinum. So far we have
NOTES
ignored interband transitions. in part because the 4 vectors associated with these tend to be large in the first two groups of metals considered above. and in part. because frequently any such transition which does occur would be expected to make a comparatively small contribution to the thermopower, since the electron velocity is not reversed as it is in the situations illustrated in Fig. 4. For Rh and Ir the situation is rather like that in Pt and Pa. The q vector connecting diRerent bands in certain directions becomes much smaller than the dimensions of the Fermi surface sheets themselves. Furthermore such transitions are augmented by the high density of states associated with the d-like character of the wave functions over the sheet of the Fermi surface in the second zone. It now becomes even more difficult to predict the sign of the thermopower. At best we can say that the experimental results strongly suggest the importance of interband transitions in Rh and Ir so far as the thermopower is concerned. Finally we note that Colquitt and Fankhauser[l9] have performed a detailed study of the effects of phonon induced and electronelectron interband s-d transitions. They point out that both the small positive peak observed in W at T < 12°K and the results for W and MO above the Debye temperature could be interband electron-electron caused by scattering. Department of Physics, Michigan State University, East Lansing, Mich. 48823. U.S.A.
R. CARTER A. DAVIDSON P. A. SCHROEDER
REFERENCES W. G. and SCHROEDER P. A.. Can. .I. 1. HENRY Phys. 41. 1076 (1963). H. H.. Proc. phys. Sot. Lond. 53. 695. 2. POTTER (1941). 3. MCINTOSH A. R. and SILL L.. .I. Phys. Chem. Solids 24.501 (1963). J. and SCHULTZ C. W.. J. Phys. 4. WEINBERG Chem. Solids 27.474 (1965). 5. RAAG V. and KOWGER H. V.. J. uppl. Phys. 36. 2045 (1965). 6. LANDER J. J., Phys. Rev. 74,479 (1948). H.. R.S.I., 40.648 (1969). 7. TRODAHL
TECHNICAL 8. ROWE V. A. and SCHROEDER P. A.. J. Phys. Chem. Solids 31, 1 (1970). 9. ZIMANJ. M..Adv.Phys.lO. l(1961). 10. BAILYN M.,Phys. Rev. 120.381 (1960). 11. MA~H~ISS L. F.. Phys. Rev. 139. Al893 (1965). 12. FAWCETT E.. REED W. A. and SODEN R. R.. Phys. Rev. 159.533 (1967). 13. SPARLIN D. M. and MARCUS J. A., Phys. Rev. 144.484 (1966).
NOTES 14. 15. 16. 17.
2377
LOUCKS T. L.. Phys. Rev. 143.506 (1966). MATTHEISS L. F.. Phys. Rev. 151.450 (1966). COLERIDGE P. T., Proc. R. Sm., A295.476 (1966). ANDERSON 0. K. and MACKINTOSH A. R.. ~o~~d~tat~ Gommun. 6,285 (1968). 18. FLETCHER R. and GREIG D.. Phil. &fag. 17.21 (1968). 19. COLQUIm C. L. and FANKHAUSER H. R.. (To be published).
TECHNICAL
REFERENCES I. SILVERMAN S. M..J. &em. Phys. 25, 1081 (1956); 1. Geophys. Res. 75 in press. f 1970). 2. DANIEL V., Adr. Phys. (Phil. iwag. Sicppl.) 2 450 (1953). 3. FlNKE H. L., GROSS M. E.. WADDINGTON G. and HUFFMAN H. M., J. Amu. Chem. Sot. 76, 333 (1954).
J. Phys. Chem.
Vol. 3 I.
Solids
NOTES
I’
pp. 2374-2377.
Thermopower of cubic transition metals* 4 Febrmry
WE HAVE measured the thermopower of the transition metals vanadium, niobium, tantalum, molybdenum. tungsten. rhenium. rhodium. and iridium. The measurements were performed on wires of 99.99 per cent purity supplied by Materials Research Corporation;; using an apparatus similar to that of Henry and Schroeder~l]. Except for rhenium which has a hexagonal close packed structure these metals are cubic and the single component of the thermopower tensor can therefore be measured on polycrystalline wires. For rhenium the results will be some kind of average over the two components of the thermopower tensor which applies to hcp metals. The main object is to compare the results for the three groupings (V. Nb, Taf. (MO, Wf, (Rh, It-). within which the elements have similar outer shells of electrons. In Figs. I-4 we show the thermopowers of the metals in the above groupings. The results for Va are similar to those of Potter[2] and McIntosh and Sill[3]. except that we did not observe the small anomaly seen by them at 217°K. The Nb results agree with those of Weinberg and Schultz[4] and Potter [2], but join rather crudely to the high temperature data of Raag and Kowger[5]. Our low *This work Foundation.
was
supported
by the National
0
/
/
50
Fig. 1. Thermopowerof
i 970)
Science
I’
I
l
~~~~,
(Rerrioed
I’
I’
, 100
’
/’
II
Vanadium
‘?~~~’ 150 200 250 300 T (OKI vanadium. niobium. and tantalum.
temperature results for MO and W join neatly onto the results calculated from Lander’s [6] high temperature Thomson coefficient measurements and also, for W, onto the results of Raag and Kowger[5]. For W. the thermopower becomes positive at the lowest temperature in agreement with Trodahl[7]. Measurements on tungsten wire drawn from a zone refined sample gave results - O-02 pV/ “K less than the M.R.C. tungsten for temperatures greater than 40°K. Our rhodium and iridium results differ markedly from those of Potter [2]. The most striking feature of the results is the similarity of the results for the metals within each of the above groupings. Undoubtedly this is related to the similarity of the Fermi surfaces within each grouping. We now briefly consider ways in which the Fermi surface properties may manifest themselves. First of all we assume that the low tempera ture peaks occurring in the vicinity of 19J5. where Bn is the Debye temperature. are caused by phonon drag. The simplest picture to explain the sign of these peaks appears to apply remarkably well to Mg. Zn and Cd@]. In Fig. 5(a), we show a phonon induced transition between two states on an electron sheet of the Fermi surface in the periodically extended zone scheme. For such a transition the electron velocity 2, in the direction of the
2375
NOTES
TECHNICAL
n
0 W - Prsrrnt lnvertigation - Raag ELKowger xw - Lander *W
I
I
i
I
1
1
I
I
f
I
i \ J 1
‘7l -0.5 -
0
1
0
I1
200
I
I
11
400
Fig. 2. Thermopower
600
’
IO I
’
’
20 I
30 I
800 1000 T (OKI
’
40 I
50 II
f
11
1200
of molybdenum and tungsten. I-zone 9999% purity tungsten.
1400
11
1600
refined tungsten: a-
Re
50
100
150 200 T (OKI
Fig. 3. Thermopower
250
of rhenium.
phonon wave vector is increased. This leads Lo a negative contribution to the phonon-drag thermopower[9. lo]. Conversely. the transition across the hole region of the Fermi surface illustrated in Fig. 5(b) leads to a positive contribution to the thermopower. Transitions like that labelled 4’ in Fig. 5 lead to a smaller contribution than q because only a component of the electron velocity is reversed. At the lowest temperature we expect the sign of the phonon drag thermopower to be determined by the nature, hole or
-0
50 I
100 \
Fig. 4. Thermopower
150 \
200 I
1 250
300 I
T f*K) of rhodium and iridium.
electron, of the piece of Fermi surface with the smallest calliper dimensions. There will be relatively few phonons with wave vector large enough to span the larger surfaces. As the temperature increases we expect the contribution of the next largest piece of Fermi surface to begin to predominate. In the vicinity of &,/5 the phonon drag thermopower
TECHNICAL
2376
lb) Fig.
5. Phonon
induced transitions surface.
across
the Fermi
usually begins to decrease because of phononphonon scattering. This highly simplified picture would lead us to expect the observed positive phonon drag thermopower in the vanadium group of metals since the proposed Fermi surface of these metals is made up of hole sheets [ 11, 121. For W and MO the situation is complicated by the presence of hole and electron sheets [1 1. 13. 141. For W the smallest pieces of Fermi surface are hole sheets (at N in Ref. [ 131) with dimensions significantly smaller than the next largest surface which is an electron sheet. This may account for the initial positive thermopower in tungsten and the large negative dip thereafter. In MO analogous pieces of Fermi surface are much closer to being equal. Furthermore MO has an additional small electron sheet (the lenses of Ref. [ 131. It would appear that contributions from these sheets tend to cancel and produce a comparatively small negative contribution only in MO. In rhenium the three smallest sheets of the Fermi surface are hole sheets[l5]. We therefore expect the observed positive contribution to the phonon drag thermopower. When we come to Rh and Ir the above picture does not fit at all. The smallest pieces of Fermi surface are hole sheets and these are significantly smaller than the electron sheets [16. 171. To make any sense of these results we have to resort to the possibility of interband transitions using arguments similar to those used by Fletcher and Greig[181 to explain the positive phonon drag peaks in palladium and platinum. So far we have
NOTES
ignored interband transitions. in part because the 4 vectors associated with these tend to be large in the first two groups of metals considered above. and in part. because frequently any such transition which does occur would be expected to make a comparatively small contribution to the thermopower, since the electron velocity is not reversed as it is in the situations illustrated in Fig. 4. For Rh and Ir the situation is rather like that in Pt and Pa. The q vector connecting diRerent bands in certain directions becomes much smaller than the dimensions of the Fermi surface sheets themselves. Furthermore such transitions are augmented by the high density of states associated with the d-like character of the wave functions over the sheet of the Fermi surface in the second zone. It now becomes even more difficult to predict the sign of the thermopower. At best we can say that the experimental results strongly suggest the importance of interband transitions in Rh and Ir so far as the thermopower is concerned. Finally we note that Colquitt and Fankhauser[l9] have performed a detailed study of the effects of phonon induced and electronelectron interband s-d transitions. They point out that both the small positive peak observed in W at T < 12°K and the results for W and MO above the Debye temperature could be interband electron-electron caused by scattering. Department of Physics, Michigan State University, East Lansing, Mich. 48823. U.S.A.
R. CARTER A. DAVIDSON P. A. SCHROEDER
REFERENCES W. G. and SCHROEDER P. A.. Can. .I. 1. HENRY Phys. 41. 1076 (1963). H. H.. Proc. phys. Sot. Lond. 53. 695. 2. POTTER (1941). 3. MCINTOSH A. R. and SILL L.. .I. Phys. Chem. Solids 24.501 (1963). J. and SCHULTZ C. W.. J. Phys. 4. WEINBERG Chem. Solids 27.474 (1965). 5. RAAG V. and KOWGER H. V.. J. uppl. Phys. 36. 2045 (1965). 6. LANDER J. J., Phys. Rev. 74,479 (1948). H.. R.S.I., 40.648 (1969). 7. TRODAHL
TECHNICAL 8. ROWE V. A. and SCHROEDER P. A.. J. Phys. Chem. Solids 31, 1 (1970). 9. ZIMANJ. M..Adv.Phys.lO. l(1961). 10. BAILYN M.,Phys. Rev. 120.381 (1960). 11. MA~H~ISS L. F.. Phys. Rev. 139. Al893 (1965). 12. FAWCETT E.. REED W. A. and SODEN R. R.. Phys. Rev. 159.533 (1967). 13. SPARLIN D. M. and MARCUS J. A., Phys. Rev. 144.484 (1966).
NOTES 14. 15. 16. 17.
2377
LOUCKS T. L.. Phys. Rev. 143.506 (1966). MATTHEISS L. F.. Phys. Rev. 151.450 (1966). COLERIDGE P. T., Proc. R. Sm., A295.476 (1966). ANDERSON 0. K. and MACKINTOSH A. R.. ~o~~d~tat~ Gommun. 6,285 (1968). 18. FLETCHER R. and GREIG D.. Phil. &fag. 17.21 (1968). 19. COLQUIm C. L. and FANKHAUSER H. R.. (To be published).