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Experimental study of itinerant Zeeman splitting in platinum group metals
1015
Journal of Magnetism and Magnetic Materials 54-57 (1986~ 1015-1016 EXPERIMENTAL STUDY GROUP METALS L. N O R D B O R G ,
OF ITINERANT
ZEEMAN
SPLITTING
IN PLATINUM
H. O H L S I ~ N a n d P. G U S T A F S S O N
Department of Solid State Physics, Uppsala University, P.O. Box 534, S-751 21 Uppsala, Sweden
The conduction electron Zeeman splitting has been measured by means of the de Haas van Alphen effect. Results are presented for the F-centered electron sheet in Pd, Pt and It, and for the c~-orbit in Pd and Pt. Measurements from the X-pocket in Pd are also reported.
A magnetic field ( B ) applied to a metallic sample gives rise to Landau levels in the conduction electron energy distribution, separated by an energy E L = h e B / m c , where m c is the cyclotron effective mass. In the paramagnetic state the Landau levels are split into spin up and spin down sets of levels (Zeeman splitting). The average of that splitting over cyclotron orbits ( E z), can be studied by the de Haas-van Alphen effect (dHvA), with a resolution of a tenth of a meV. The two sets of levels will interfere with each other according to a cosine factor, cos(k,~R) included in the expression for the amplitude [1]. R is the ratio E z / E L = g c m c / 2 m and k is the harmonic number of the dHvA-frequency. The condition kR = n + 1 / 2 , (n integer) defines a spin splitting zero (SSZ) and can be found as contours on the Fermi surface (FS) of most transition metals. By carefully studying the amplitude variation of the dHvA-signals with angle, it is possible to extract R(O,e~), [2]. The cyclotron effective mass may then be used to obtain the conduction electron cyclotron averaged g-factor (g~), and its variation over the Fermi surface. The experiments were performed using the field modulation technique for fields up to 6.5 T and at a temperature of 0.5 K. Extensive measurements have been performed for the F-centered electron sheet (F6) in palladium and platinum, and also for the a-orbit on the open hole sheet in these metals. The X-centered hole pockets in palladium have been studied and preliminary results from the F6-sheet in iridium are reported.
The F6-sheet (Pd, Pt, Ir). Results for palladium and preliminary results for platinum can be found in refs. [2,3]. In fig. 1 we compare final results for R(0,~b) in the symmetry planes for palladium and platinum. The preliminary results for iridium are also shown. In fig. 2 the SSZ-contours on the F6-sheet in iridium are shown in the irreducible basic 1 / 4 8 : t h of the Brillouin zone. In the shaded area a pronounced low amplitude was found making it very difficult to determine how the different contours are connected with each other. The most probable cause for this low amplitude is a value of the cosine 0304-8853/86/$03.50
factor close to zero in the whole region. With such an interpretation the integers n for the SSZ contours entering this area should be equal. It is then also obvious that the value of R in this angular region must be nearly constant. We are then left with an uncertainty in sign and calibration for the variation of R. We have chosen to display (fig. 1) the alternative with a maximum at [110], which might be caused by the maximum of the cyclotron effective mass at [110], [4]. The entity R is itself a measure of the contribution to the magnetization from those parts of the FS which the orbit is spanning. It is however also interesting to visualize the conduction electron Zeeman splitting through gc- In fig. 3 the
R
Pd
n. +S/Z
nr +I/z
n r -I/z
Pt
n r
+I/z
Ir A ~
n v +l/z
o
ctoo3
3o
eo
(11o)
90/45
~;o~
(1oo)
c1ooJ
Fig. 1. Experimental results for R on the F6-sheet in the symmetry planes for Pd, Pt and Ir.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
L. Nordborg et a L / Zeeman splitting in Pt-group metals
1016
[111]
gc 90
na=23 -----.-.,_.___,
15
na=17
10
na=ll
(d~g) '75
(loo)
(It0) 30 90
15
0
[100]
15
30
a
(de9)
[110]
Fig. 2. SSZ contours (solid lines) in the basic 1/48:th wedge of the Brillouin zone for the F6-sheet in Ir. Measured SSZ's are indicated by.. The SSZ contour around [111] is assigned to the central orbit.
anisotropic gc-factor for p l a t i n u m is shown. The corres p o n d i n g gc-curves for palladium are similar in the m a j o r features. Thus, a great part of the differences in the R graphs is due to differences in the anisotropy of the effective masses.
The e~-orbit (Pd, Pt). M e a s u r e m e n t s of the c~-orbit in p a l l a d i u m a n d p l a t i n u m (the orbit does not exist in iridium) reveal a rather c o n s t a n t gc-factor for any p r o b able choice of n. In fig. 4, gc is shown for palladium, calculated from d a t a on R from ref. [2]. The possible values of the integer n are restricted to odd integers [5].
gc
6
30
[100]
EO
(,lo)
90/45
[1101
30
(,ool
[1001
Fig. 3. The gc factor on the Fo-sheet for Pt in the symmetry planes for different values of n.
15
30
15
[100]
(deg)
Fig. 4. The gc factor on the a-orbit for Pd in the symmetry planes for different values of n. Measured SSZ's are indicated by..
The X-pocket (Pd). A rather small variation of R and gc would be expected on such a small sheet in a very limited region of the Brillouin zone. However, a mapping out of b o t h R and gc show considerable anisotropy (to be presented elsewhere) a n d two different SSZ contours were f o u n d near the " w a i s t " of the ellipsoidal pocket. W e find a variation of gc over the sheet of a b o u t 40%. O n e would expect s p i n - o r b i t coupling to cause an anisotropic g-shift from the free electron value of g. In addition these metals also have a c o n t r i b u t i o n to the g-shift from m a n y b o d y effects which may show anisotropy. There are theoretical works which have dealt with this problem. F o r example, M a c D o n a l d [6] and J a n a k [7] a n d references therein have treated the trends a m o n g transition metals of the effects m e n t i o n e d above. Mueller et al. [8] a n d Jarlborg et al. [9] are examples of calculations where spin orbit interaction a n d m a n y body effects, respectively, are included for palladium a n d platinum. So far, no calculation has b e e n able to explain the different anisotropies found in our results. [1] I.M. Lifshitz and A.M. Kosevitch, Sov. Phys. JETP 2 (1956) 636. [2] H. Ohls6n, P. Gustafsson, L. Nordborg and S.P. H/Srnfeldt, Phys. Rev. B 29 (1984) 3022. [3] P. Gustafsson, H. Ohls6n, L. Nordborg and S.P. H6rnfeldt, Solid State Commun. 45 (1983) 395. [4] S.P. HOrnfeldt, L.R. Windmiller and J.B. Ketterson, Phys. Rev. B 7 (1973) 4349. [5] J. Wolfrat, Thesis (University of Amsterdam, 1984). [6] A.H. MacDonald, J. Phys. F 12 (1982) 2579. [7] J.F. Janak, Phys. Rev. B 16 (1977) 255. [8] F.M. Mueller, AJ. Freeman and D.D. Koelling, J. Appl. Phys. 141 (1970) 1229. [9] T. Jarlborg and A.J. Freeman, Phys. Rev. B 23 (1981) 3577.
Journal of Magnetism and Magnetic Materials 54-57 (1986~ 1015-1016 EXPERIMENTAL STUDY GROUP METALS L. N O R D B O R G ,
OF ITINERANT
ZEEMAN
SPLITTING
IN PLATINUM
H. O H L S I ~ N a n d P. G U S T A F S S O N
Department of Solid State Physics, Uppsala University, P.O. Box 534, S-751 21 Uppsala, Sweden
The conduction electron Zeeman splitting has been measured by means of the de Haas van Alphen effect. Results are presented for the F-centered electron sheet in Pd, Pt and It, and for the c~-orbit in Pd and Pt. Measurements from the X-pocket in Pd are also reported.
A magnetic field ( B ) applied to a metallic sample gives rise to Landau levels in the conduction electron energy distribution, separated by an energy E L = h e B / m c , where m c is the cyclotron effective mass. In the paramagnetic state the Landau levels are split into spin up and spin down sets of levels (Zeeman splitting). The average of that splitting over cyclotron orbits ( E z), can be studied by the de Haas-van Alphen effect (dHvA), with a resolution of a tenth of a meV. The two sets of levels will interfere with each other according to a cosine factor, cos(k,~R) included in the expression for the amplitude [1]. R is the ratio E z / E L = g c m c / 2 m and k is the harmonic number of the dHvA-frequency. The condition kR = n + 1 / 2 , (n integer) defines a spin splitting zero (SSZ) and can be found as contours on the Fermi surface (FS) of most transition metals. By carefully studying the amplitude variation of the dHvA-signals with angle, it is possible to extract R(O,e~), [2]. The cyclotron effective mass may then be used to obtain the conduction electron cyclotron averaged g-factor (g~), and its variation over the Fermi surface. The experiments were performed using the field modulation technique for fields up to 6.5 T and at a temperature of 0.5 K. Extensive measurements have been performed for the F-centered electron sheet (F6) in palladium and platinum, and also for the a-orbit on the open hole sheet in these metals. The X-centered hole pockets in palladium have been studied and preliminary results from the F6-sheet in iridium are reported.
The F6-sheet (Pd, Pt, Ir). Results for palladium and preliminary results for platinum can be found in refs. [2,3]. In fig. 1 we compare final results for R(0,~b) in the symmetry planes for palladium and platinum. The preliminary results for iridium are also shown. In fig. 2 the SSZ-contours on the F6-sheet in iridium are shown in the irreducible basic 1 / 4 8 : t h of the Brillouin zone. In the shaded area a pronounced low amplitude was found making it very difficult to determine how the different contours are connected with each other. The most probable cause for this low amplitude is a value of the cosine 0304-8853/86/$03.50
factor close to zero in the whole region. With such an interpretation the integers n for the SSZ contours entering this area should be equal. It is then also obvious that the value of R in this angular region must be nearly constant. We are then left with an uncertainty in sign and calibration for the variation of R. We have chosen to display (fig. 1) the alternative with a maximum at [110], which might be caused by the maximum of the cyclotron effective mass at [110], [4]. The entity R is itself a measure of the contribution to the magnetization from those parts of the FS which the orbit is spanning. It is however also interesting to visualize the conduction electron Zeeman splitting through gc- In fig. 3 the
R
Pd
n. +S/Z
nr +I/z
n r -I/z
Pt
n r
+I/z
Ir A ~
n v +l/z
o
ctoo3
3o
eo
(11o)
90/45
~;o~
(1oo)
c1ooJ
Fig. 1. Experimental results for R on the F6-sheet in the symmetry planes for Pd, Pt and Ir.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
L. Nordborg et a L / Zeeman splitting in Pt-group metals
1016
[111]
gc 90
na=23 -----.-.,_.___,
15
na=17
10
na=ll
(d~g) '75
(loo)
(It0) 30 90
15
0
[100]
15
30
a
(de9)
[110]
Fig. 2. SSZ contours (solid lines) in the basic 1/48:th wedge of the Brillouin zone for the F6-sheet in Ir. Measured SSZ's are indicated by.. The SSZ contour around [111] is assigned to the central orbit.
anisotropic gc-factor for p l a t i n u m is shown. The corres p o n d i n g gc-curves for palladium are similar in the m a j o r features. Thus, a great part of the differences in the R graphs is due to differences in the anisotropy of the effective masses.
The e~-orbit (Pd, Pt). M e a s u r e m e n t s of the c~-orbit in p a l l a d i u m a n d p l a t i n u m (the orbit does not exist in iridium) reveal a rather c o n s t a n t gc-factor for any p r o b able choice of n. In fig. 4, gc is shown for palladium, calculated from d a t a on R from ref. [2]. The possible values of the integer n are restricted to odd integers [5].
gc
6
30
[100]
EO
(,lo)
90/45
[1101
30
(,ool
[1001
Fig. 3. The gc factor on the Fo-sheet for Pt in the symmetry planes for different values of n.
15
30
15
[100]
(deg)
Fig. 4. The gc factor on the a-orbit for Pd in the symmetry planes for different values of n. Measured SSZ's are indicated by..
The X-pocket (Pd). A rather small variation of R and gc would be expected on such a small sheet in a very limited region of the Brillouin zone. However, a mapping out of b o t h R and gc show considerable anisotropy (to be presented elsewhere) a n d two different SSZ contours were f o u n d near the " w a i s t " of the ellipsoidal pocket. W e find a variation of gc over the sheet of a b o u t 40%. O n e would expect s p i n - o r b i t coupling to cause an anisotropic g-shift from the free electron value of g. In addition these metals also have a c o n t r i b u t i o n to the g-shift from m a n y b o d y effects which may show anisotropy. There are theoretical works which have dealt with this problem. F o r example, M a c D o n a l d [6] and J a n a k [7] a n d references therein have treated the trends a m o n g transition metals of the effects m e n t i o n e d above. Mueller et al. [8] a n d Jarlborg et al. [9] are examples of calculations where spin orbit interaction a n d m a n y body effects, respectively, are included for palladium a n d platinum. So far, no calculation has b e e n able to explain the different anisotropies found in our results. [1] I.M. Lifshitz and A.M. Kosevitch, Sov. Phys. JETP 2 (1956) 636. [2] H. Ohls6n, P. Gustafsson, L. Nordborg and S.P. H/Srnfeldt, Phys. Rev. B 29 (1984) 3022. [3] P. Gustafsson, H. Ohls6n, L. Nordborg and S.P. H6rnfeldt, Solid State Commun. 45 (1983) 395. [4] S.P. HOrnfeldt, L.R. Windmiller and J.B. Ketterson, Phys. Rev. B 7 (1973) 4349. [5] J. Wolfrat, Thesis (University of Amsterdam, 1984). [6] A.H. MacDonald, J. Phys. F 12 (1982) 2579. [7] J.F. Janak, Phys. Rev. B 16 (1977) 255. [8] F.M. Mueller, AJ. Freeman and D.D. Koelling, J. Appl. Phys. 141 (1970) 1229. [9] T. Jarlborg and A.J. Freeman, Phys. Rev. B 23 (1981) 3577.