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De Haas-van Alphen effect and Fermi surface of PdSb and other NiAs structure compounds
Physica 95B (1978) 183-189 0 North-Holland Publishing Company
DE HAAS-VAN ALPHEN EFFECT AND FERMI SURFACE OF PdSb AND OTHER NiAs STRUCTURE COMPOUNDS M. A. C. DEVILLERS, N. J. COENEN and A. R. DE VROOMEN l?vsisch Laboratorium, Universiteit van Nijmegen, Toernooiveld, Nijmegen, The Netherlands Received 10 March 1978
High-field de Haas-van Alphen measurements in PdSb are reported and a semiquantitative comparison is made with a recent bandstructure calculation. The interpretation of the de Haas-van Alphen branches is given extra support by a Fourier series description of some of the sheets. On the whole the Fermi surface is rather free-electron like despite the presence of the Pd d-bands near the Fermi energy. A comparison with the Fermi surfaces of AuSn and PtSn is made, from which the general features are expected to hold for all NiAs-structure compounds which are isoelectronic with either PdSb or PtSn.
We assign B1 to the neck of the humbugs because it has lowest effective mass and the band structure indicates that those orbits have an effective mass about three times lower than the/~ and v-orbits. B, and B, determine the humbug (Coenen, model 2, 123), B2 being belly oscillations around H. B3 starting where the second harmonic of B, stops is assigned to an orbit across zwo bellies of the humbug. Coenen et al. [l] interpolated their C-branches with the aid of a Fourier expansion. This model has got further confirmation by our measurements, in particular the C, branch is now found to vary continuously across the [ 11 TO] -direction and to lie close to the predicted values. A,, A,, 4, A7 and Ag are assigned to the multiply connected surface, centered at A in the Brillouin zone, A,, AZ, & and A7 being v, S, A and J/-orbits, respectively (compare with fig. 4). A8 exists over the range 10” to 30” from the [ lOTO] -direction and most likely belongs to an orbit around two horizontal arms and one vertical neck of the A-surface: an estimation from Coenen’s interpolated model yields a dHvA frequency of about 5 X lo3 T at a field direction 20’ away from the [ lOiO] -direction. AB is thought to be magnetic break down orbit (fig. 3) between the A and B surfaces along LH near L, where spin-orbit interaction only weakly splits the bands. At the same point magnetic breakdown causes
Recently [l] we reported de Haas-van Alphen (dHvA) experiments on the metallic compound PdSb. The measurements were performed in a 3.5 T iron core magnet. Lacking a realistic band-structure calculation, the interpretation was guided by the nearly free electron model; a symmetrized plane wave interpolation describing the sheets of the Fermi surface appeared to be of good use. As the interpretation of the experiments did not appear to be conclusive we searched for additional information with the aid of a 20 T pulsed magnetic field [2] (fig. l), a 8 T superconducting solenoid, and a 15 T Bitter magnet of the Nijmegen High Magnetic Field Laboratory (fig. 2, table I). Moreover an APW bandstructure calculation by Myron and Mueller [3] has become available now, which appears accurate enough to serve as a semi-quantitative guidance for an identification of our dHvA branches. In general the various surfaces, proposed in [l] (compare with fig. 4) are confirmed, i.e. the M-centered “double beret” corresponding to the C-branches, the “humbugs” along KH, and a multiple connected surface with necks centered at P and at L, respectively. The main difference is that there is a fourth surface centered at r, to which we attribute the D ‘oscillations, and that the identification of A and B of [l] should be interchanged. 183
M. A. C. Devillers et al.fDe Haas-van Alphen effect of PdSb
184
Fig. 1. De Haas-van Alphen frequencies in PdSb using a 3.5 T iron core magnet (circles) and a 20 T pulsed field (crosses) respectively (after Coenen [ 21). The solid lines are from a symmetrized plane wave interpolation of the data for the M-centered surface (C-branches), of the H-centered surface (B-branches) and of the L-centered necks (A-surface) respectively (after Coenen [ 21).
[iaio]
[iiio]
[ooot]
Flll
lo2
’
go
I
I
I
I
I
I
I
I
eo
70
60
50
&II
30
20
10
I 0
I
,
10
20
1
I
I
I
I
I
30
LO
50
60
70
80
I 90
Fig. 2. De Haas-van Alphen frequencies using an 8 T superconducting magnet and by a 15 T Bitter magnet. The experiments from [ 1 l?O] to 62.5” from [OOOl] have been obtained with the 15 T Bitter magnet. The experimental errors are 0.7% for the superconducting magnet results and 1% for the Bitter magnet results, respectively.
M. A. C. Devillers et al./De &as-van Alphen effect of PdSb
185
Table I dHvA frequencies along symmetry directions Field direction
Branch
Orbit
Frequency (10’T) Experiment
[OOOl]
BI f’, Cl
Al 9
2 [ 11~0]
A2 C2
6 E
Coenen [2]
This work
3.11 5.05 18.3 22.7
f 0.06 f 0.10 f 0.3 ?z0.7
72.0 77.0
i 1.5 i 1.5
3.15 5.04 18.2 22.7 39.0 71.7 17.2
f f f f f f f
0.02 0.05 0.1 0.1 0.4 0.5 0.5
4.0 4.9 16.9 11.5 35.7 75.1 75.1
2.32 15.8 27.0 42.3 44.7 58.4 61.0 99.3
f 0.03 i0.2 f 0.3 f 1.0 f 1.0 f 1.0 f 1.0 f 1.0
1.3 18.9
1.98 4.81 15.8 18.9
i i i i
2.27 * 0.05 15.8 f 0.3
Cl
D4 D*l D6 Dd
[ioio]
A7
*
2
6
C2
El
Cl
e2 9*
JJS
Fig. 3. Magnetic breakdown between bellies of the B-pieces via L-centered necks of the A-pieces. From his symmetrized plane wave interpolation for the B-piece Coenen [2] calculated the magnetic breakdown frequency to be 60 T which compares well with the experimental value (57 f 2) T.
Theory [3]
1.95 f 4.8 i 15.7 f 19.0 f 37.7 *
0.04 0.1 0.3 0.4 0.8
0.04 0.05 0.3 0.4
38.3
98.6 1.1 17.6 20.3 30.0
zone6 l&C] Fig. 4. Sketch of the Fermi surface of AuSn (after Edwards et al. [4] ). Zone 4 has been revised according to the bandstructure calculation of Myron and Mueller [ 31 for PdSb. The drawings are not precisely to scale with the dHvA data.
M. A. C. Devillers et aI./De Hans-van Alphen effect of PdSb
186
the frequency B2-A2 at field directions about 45’ away from the [OOOl ] -direction towards the [ lOi direction. DI and D, are assigned to central cross sections of the D-surface along [OOOl] , and D4 and D, are attributed to cross sections along the other two symmetry directions. The splitting in D4 is of some interest. Although the band-structure calculation of Myron [3] does not give the Fermi surface around the u-orbit in enough detail, it is not unlikely that around the u-orbit the cross section has a shape as sketched in fig. 5a (compare also with fig. 4, zone 4, of Edwards et al. [4] ). Then the dHvA-frequency will behave like as indicated in fig. 5b. So a small misalignment of the rotation plane of the sample with respect to the field direction would explain this splitting. This same sharp structure would explain the separation of D4 and D3. The weak signals of D3, D4, D& CI, A, and A8 were observed with the 15 T Bitter magnet only. In table II we compare the experimentally observed
Fig. 5. a) Sketch of the (I (D+rbit in the fourth zone. Such a shape gives rise to a splitting of D4 in the (0001) rotation plane, sketched in fig. b. Table II Cut-off angles of the D-oscillations Rotation plane
Branch
Experiment
Theory
[OOOl]-[1120]
D1
46“ 50” 67”
< 48” 6 51° > 55”
38” 54” 60”
< 50” G 54” > 58”
"2 D3
[oool]+[ioio]
Dt J'2
D5
cut-off angles with the theoretical ones. We note that none of the former is at variance with theory. BCI, BC, and BC3 have frequencies of the form nB2 + mC1,2 and are found at directions where both 82 and CI,, have large amplitudes, therefore we interpret those as being caused by magnetic interaction. Also the. branches DC and DC1 can be interpreted-as magnetic interaction induced frequencies 04 + C, and Di + C’,. This completes the interpretation of all observed dHvA branches in PdSb. The proposed Fermi surface for PdSb is shown in fig. 6. As PdSb is a compensated metal, the volume of the electron surfaces has to equal the volume of the hole surfaces. With the aid of his symmetrized plane wave expansion Coenen [2] calculated the volume fractions of the A, B and C surfaces. As B and C are assigned now to be electron surfaces and A and D to be hole surfaces, we can deduct a value for the volume fraction of the D-surface (table III). Using those volume fractions of PdSb as reference values we can estimate the corresponding values for AuSn and PtSn by scaling with measured dHvA frequencies, assuming that the assignments as in table IV are correct and, eventually, after correcting for differences in the lattice constants (table V). The constraint of compensation is well satisfied for AuSn, thus reinforming the validity of the interpretation of the various dHvA branches. For PtSn the constraint of compensation is violated. However, if we assume for the 4th zone of PtSn a shape qualitatively similar to the one of PdSb, rather than the model proposed by Cathey et al. [5], which implies the interchange of the u and pbranch, we also get compensation for PtSn (fig. 6, table III). From the comparison between PdSb, AuSn and PtSn we deduce that the iso-electronic compounds PdSb and AuSn have very similar Fermi surfaces, both topologically and quantitatively; the slightly smaller values for the dHvA frequencies in AuSn compared with PdSb (typically 15%) are completely due to the 6% larger lattice constant II of AuSn. The volume fractions of the surfaces in AuSn are very close to those in PdSb however. Going from PdSb and PtSn, the latter having 28 electrons/cell in the highest s,p,d-bands compared with 30 electrons/cell for the former, it appears (table III), that the two less electrons in PtSn are accounted for by creating two additional holes in the fourth and fifth bands around F, rather than by emptying the sixth zone as might be expected in a rigid band model. Using the band structure for
187
M. A. C. Devillerset al./De Haas-van Alphen effect of PdSb Table III Number of electrons per zone Zone 3
r
PdSb AuSn PtSn5) PtSn6)
2.00 2.00 1.944) 1.78
Zone 4 Zone 5 Zone 6 (D) (A) (B)
(C)
1.923) 1.944) 0.84) 0.9,
0.1201) 6.00 0.0g4) 5.98 0.154) 4.0 0.00 3. 0
l.5S2) l.5S4) 0.74) 0.27
Total
0.406l) 0.404) 0.454) 0.01,
From Coenen [ 1,2] ; from Coenen [ 1, 21 after correction (see text); from compensation criterium; PdSb-derived Fermi surface after dHvA resealing including lattice constant corrections; 5) PdSb derived Fermi surface (nonrigid band model); 6) rigid band model derived Fermi surface (see text).
1) 2) 3) 4)
Table IV Comparison dHvA frequencies along symmetry directions
[ OOOl] BI J'l Cl Al D2 B2 A6
8 Ic P V
0 7 h @ e
PdSb’)
AuSn2)
PtSn3)
3.15 5.04 18.2 22.7 39.0 71.7 77.2
0.711 1.65 14.9 15.4
22 103 25 120 69 76 78 21 3.3
61.4 69.6
[Ill01 Fig. 6. a) Fermi surface cross sections of PdSb. The A, B and C pieces are from a symmetrized plane wave interpolation to the dHvA measurements (Coenen [ 21) where it should be noted that the A-cross section around r has been revised. The D-cross section is taken from the band-structure calculation of Myron and Mueller [ 31 after a slight modification to fit the dHvA measurements. The zones are indicated by the numbers. b) Our proposed Fermi surface of PtSn (nonrigid band model). c) Rigid band model derived Fermi surface of PtSn. The greek letters indicate the assignment of the dHvA frequencies of Cathey et al. [ 5 1.
PdSb of Myron and Mueller [3] as a rigid band model for PtSn yields a Fermi surface in which also most of the dHvA data of Cathey et al. [S] could be identified qualitatively (fig. 6~). The main setback of this otherwise simpler Fermi surface is, that in using the alterna-
A2 c2 Cl D4 Al
6
E ;*,
2.32 15.8 42.3 27.0
J, 0 e
99.3
6
1.98 4.81 15.8 18.9 37.7
2.20 14.0
23
35 20 5.8
[ioio] A2 A5 c2 Cl DS
El E2
+* X @ e
1.86 13.1
18
45 21 5.7
1) This work; 2) Ref. 4; 3) ref. 5; 4) as in table I. and after ref. 5.
M. A. C. Devillers et al./De Haas-van Alphen effect of PdSb
188 Table V
Non-magnetic compounds with NiAs structure Molecular valency
Metal
a
C
da
5
AuSn PdSb PtSb PtBi NiAs NiSb NiBi CoSe RhTe
4.3218 4.018 4.138 4.324 3.617 3.924 4.018 3.6294 3.99
5.5230 5.593 5.483 5.501 5.038 5.142 5.36 5.3006 5.66
1.278 1.374 1.325 1.272 1.393 1.311 1.314 1.460 1.419
PtSn PtPb RhBi IrSb
4.111 4.259 4.075 3.978
5.439 5.467 5.669 5.521
1.323 1.2837 1.391 1.388
3
IrSn IrPb
3.988 3.993
5.567 5.566
1.396 1.394
6
PdTe
4.1521
5.6719
1.366
tive assignment of the dHvA branches the compensation criterium is ‘violated severely: the zones 3 to 6 contain only (3.0 f 0.1) electrons instead of the required 4 electrons (table III). Or stated in other words the (1.3 + 0.1) holes in the third and fourth zones of both the rigid band model derived Fermi surface and the nonrigid band model derived Fermi surface are “compensated” by (0.28 + 0.03) electrons and (1.3 + 0.1) electrons in zone five and six of those two models respectively. This implies that the Fermi surface derived from the rigid band model should be rejected. From the above we note that the sixth zone remains remarkably constant through the above three NiAsstructure compounds. In view of this tendency towards enlarging the hole surfaces around I’ and comparing with the band structure of PdSb it seems most likely now to attribute the $-orbits and e-orbits in PtSn to third zone hole surfaces around A and I?, respectively (fig. 6; compare Myron and Mueller [3] ). The alternative assignment 13at A and @at I’ is not ruled out however. The picture that arises is that the Fermi surfaces of the above N&-structures are mainly determined by:
a) the nearly free electron bands for the total of s, pelectrons; so it is convenient to characterize the compounds by their total molecular (s,p) valency; b) the influence of the d-bands which modifies the extent to which the hole bands at r overlap with the H-centered electron bands; c) a “constant sixth zone hypothesis” as suggested for the five and four-valent compounds. With some confidence we can extrapolate the above findings to other iso-electronic NiAs-structures (table V). Thus PtBi and PtSb are expected to have Fermi surfaces nearly identical to that of PdSb, with dHvA frequencies deviating only 10 to 20% from those in PdSb, the latter mainly due to lattice constant deviations. Similarly NiSb and NiBi will show strong resemblance with the PdSb-Fermi surface. For NiAs the picture might be altered significantly by the increased electron density (30%) which could alter the relative positions of the P-bands with respect to the H-bands. Also CoSe and RhTe might deviate from PdSb. In the group of 4-valent NiAs-compounds we expect PtSn and PtPb to have identical Fermi surfaces. For RhBi and IrSb, being VII B-V compounds rather than VIII B-IV compounds like PtSn and PtPb, the above extrapolation might be less correct. We note a tendency for Fermi surface nesting of the fifth and sixth zone around H. This Fermi surface nesting around the hexagonal plane of the Brillouin zone tends to phase instabilities; on the other hand stronger spin-orbit coupling enlarges the band splittings in the hexagonal Brillouin zone plane, so this is presumably the reason why there is a trend for only the heaviest metals to stabilize in the NiAs structure. For the 3-valent NiAs-compounds IrSn and IrPb the “constant sixth zone hypothesis” would imply a close nesting of the 3rd to 6th zones around H with about half an electron in each zone (the (n - 1)th band contains at least as much electrons as the nth band). Only large spin-orbit splitting could stabilize such a Fermi surface, which might be the reason why only the heaviest compounds IrSn and IrPb stabilize in the NiAs-structure. Likely there exist d-band holes in those 3-valent NiAs-structures too. To solve those problems more experimental and theoretical work should be done. For the 6-valent compound PdTe one may expect the nearly free electron model to hold even better
M. A. C. Devillers et al./De Hans-van Alphen effect of PdSb
than for PdSb, with surfaces in zone 5 to zone 8 (compare with table III, where one has to accomodate two more electrons in zone 4 and higher zones).
189
111 Coenen, N. J., Morsing, C. A. J., Reijers, A. P. J. M. and Schreurs, L. W. M., Sol. State Comm. 16 (1975) 557.
121 Coenen, N. J. M., thesis, Nijmegen, 1976. Acknowledgements We thank ir. L. Schreurs for the sample preparations, mr. W. Hooghof for the development of the datasampling and data-analysis programs, the technical staff of the Nijmegen High Magnetic Field Laboratory for their nightly assistance, and dr. H. W. Myron and Prof. F. M. Mueller for communicating the results of their band structure calculation prior to publication.
131 Myron, H. W. and Mueller, F. M., Phys. Rev. B I7 (l-978) 1828. 141 Edwards, G. J., Springford, M. and Saito, Y., J. Phys. Chem. Solids 30 (1969) 2527. 151 Cathey, W. N., Coleridge, P. T. and Jan, J. P., Can. J. Phys. 48 (1970) 1151.
DE HAAS-VAN ALPHEN EFFECT AND FERMI SURFACE OF PdSb AND OTHER NiAs STRUCTURE COMPOUNDS M. A. C. DEVILLERS, N. J. COENEN and A. R. DE VROOMEN l?vsisch Laboratorium, Universiteit van Nijmegen, Toernooiveld, Nijmegen, The Netherlands Received 10 March 1978
High-field de Haas-van Alphen measurements in PdSb are reported and a semiquantitative comparison is made with a recent bandstructure calculation. The interpretation of the de Haas-van Alphen branches is given extra support by a Fourier series description of some of the sheets. On the whole the Fermi surface is rather free-electron like despite the presence of the Pd d-bands near the Fermi energy. A comparison with the Fermi surfaces of AuSn and PtSn is made, from which the general features are expected to hold for all NiAs-structure compounds which are isoelectronic with either PdSb or PtSn.
We assign B1 to the neck of the humbugs because it has lowest effective mass and the band structure indicates that those orbits have an effective mass about three times lower than the/~ and v-orbits. B, and B, determine the humbug (Coenen, model 2, 123), B2 being belly oscillations around H. B3 starting where the second harmonic of B, stops is assigned to an orbit across zwo bellies of the humbug. Coenen et al. [l] interpolated their C-branches with the aid of a Fourier expansion. This model has got further confirmation by our measurements, in particular the C, branch is now found to vary continuously across the [ 11 TO] -direction and to lie close to the predicted values. A,, A,, 4, A7 and Ag are assigned to the multiply connected surface, centered at A in the Brillouin zone, A,, AZ, & and A7 being v, S, A and J/-orbits, respectively (compare with fig. 4). A8 exists over the range 10” to 30” from the [ lOTO] -direction and most likely belongs to an orbit around two horizontal arms and one vertical neck of the A-surface: an estimation from Coenen’s interpolated model yields a dHvA frequency of about 5 X lo3 T at a field direction 20’ away from the [ lOiO] -direction. AB is thought to be magnetic break down orbit (fig. 3) between the A and B surfaces along LH near L, where spin-orbit interaction only weakly splits the bands. At the same point magnetic breakdown causes
Recently [l] we reported de Haas-van Alphen (dHvA) experiments on the metallic compound PdSb. The measurements were performed in a 3.5 T iron core magnet. Lacking a realistic band-structure calculation, the interpretation was guided by the nearly free electron model; a symmetrized plane wave interpolation describing the sheets of the Fermi surface appeared to be of good use. As the interpretation of the experiments did not appear to be conclusive we searched for additional information with the aid of a 20 T pulsed magnetic field [2] (fig. l), a 8 T superconducting solenoid, and a 15 T Bitter magnet of the Nijmegen High Magnetic Field Laboratory (fig. 2, table I). Moreover an APW bandstructure calculation by Myron and Mueller [3] has become available now, which appears accurate enough to serve as a semi-quantitative guidance for an identification of our dHvA branches. In general the various surfaces, proposed in [l] (compare with fig. 4) are confirmed, i.e. the M-centered “double beret” corresponding to the C-branches, the “humbugs” along KH, and a multiple connected surface with necks centered at P and at L, respectively. The main difference is that there is a fourth surface centered at r, to which we attribute the D ‘oscillations, and that the identification of A and B of [l] should be interchanged. 183
M. A. C. Devillers et al.fDe Haas-van Alphen effect of PdSb
184
Fig. 1. De Haas-van Alphen frequencies in PdSb using a 3.5 T iron core magnet (circles) and a 20 T pulsed field (crosses) respectively (after Coenen [ 21). The solid lines are from a symmetrized plane wave interpolation of the data for the M-centered surface (C-branches), of the H-centered surface (B-branches) and of the L-centered necks (A-surface) respectively (after Coenen [ 21).
[iaio]
[iiio]
[ooot]
Flll
lo2
’
go
I
I
I
I
I
I
I
I
eo
70
60
50
&II
30
20
10
I 0
I
,
10
20
1
I
I
I
I
I
30
LO
50
60
70
80
I 90
Fig. 2. De Haas-van Alphen frequencies using an 8 T superconducting magnet and by a 15 T Bitter magnet. The experiments from [ 1 l?O] to 62.5” from [OOOl] have been obtained with the 15 T Bitter magnet. The experimental errors are 0.7% for the superconducting magnet results and 1% for the Bitter magnet results, respectively.
M. A. C. Devillers et al./De &as-van Alphen effect of PdSb
185
Table I dHvA frequencies along symmetry directions Field direction
Branch
Orbit
Frequency (10’T) Experiment
[OOOl]
BI f’, Cl
Al 9
2 [ 11~0]
A2 C2
6 E
Coenen [2]
This work
3.11 5.05 18.3 22.7
f 0.06 f 0.10 f 0.3 ?z0.7
72.0 77.0
i 1.5 i 1.5
3.15 5.04 18.2 22.7 39.0 71.7 17.2
f f f f f f f
0.02 0.05 0.1 0.1 0.4 0.5 0.5
4.0 4.9 16.9 11.5 35.7 75.1 75.1
2.32 15.8 27.0 42.3 44.7 58.4 61.0 99.3
f 0.03 i0.2 f 0.3 f 1.0 f 1.0 f 1.0 f 1.0 f 1.0
1.3 18.9
1.98 4.81 15.8 18.9
i i i i
2.27 * 0.05 15.8 f 0.3
Cl
D4 D*l D6 Dd
[ioio]
A7
*
2
6
C2
El
Cl
e2 9*
JJS
Fig. 3. Magnetic breakdown between bellies of the B-pieces via L-centered necks of the A-pieces. From his symmetrized plane wave interpolation for the B-piece Coenen [2] calculated the magnetic breakdown frequency to be 60 T which compares well with the experimental value (57 f 2) T.
Theory [3]
1.95 f 4.8 i 15.7 f 19.0 f 37.7 *
0.04 0.1 0.3 0.4 0.8
0.04 0.05 0.3 0.4
38.3
98.6 1.1 17.6 20.3 30.0
zone6 l&C] Fig. 4. Sketch of the Fermi surface of AuSn (after Edwards et al. [4] ). Zone 4 has been revised according to the bandstructure calculation of Myron and Mueller [ 31 for PdSb. The drawings are not precisely to scale with the dHvA data.
M. A. C. Devillers et aI./De Hans-van Alphen effect of PdSb
186
the frequency B2-A2 at field directions about 45’ away from the [OOOl ] -direction towards the [ lOi direction. DI and D, are assigned to central cross sections of the D-surface along [OOOl] , and D4 and D, are attributed to cross sections along the other two symmetry directions. The splitting in D4 is of some interest. Although the band-structure calculation of Myron [3] does not give the Fermi surface around the u-orbit in enough detail, it is not unlikely that around the u-orbit the cross section has a shape as sketched in fig. 5a (compare also with fig. 4, zone 4, of Edwards et al. [4] ). Then the dHvA-frequency will behave like as indicated in fig. 5b. So a small misalignment of the rotation plane of the sample with respect to the field direction would explain this splitting. This same sharp structure would explain the separation of D4 and D3. The weak signals of D3, D4, D& CI, A, and A8 were observed with the 15 T Bitter magnet only. In table II we compare the experimentally observed
Fig. 5. a) Sketch of the (I (D+rbit in the fourth zone. Such a shape gives rise to a splitting of D4 in the (0001) rotation plane, sketched in fig. b. Table II Cut-off angles of the D-oscillations Rotation plane
Branch
Experiment
Theory
[OOOl]-[1120]
D1
46“ 50” 67”
< 48” 6 51° > 55”
38” 54” 60”
< 50” G 54” > 58”
"2 D3
[oool]+[ioio]
Dt J'2
D5
cut-off angles with the theoretical ones. We note that none of the former is at variance with theory. BCI, BC, and BC3 have frequencies of the form nB2 + mC1,2 and are found at directions where both 82 and CI,, have large amplitudes, therefore we interpret those as being caused by magnetic interaction. Also the. branches DC and DC1 can be interpreted-as magnetic interaction induced frequencies 04 + C, and Di + C’,. This completes the interpretation of all observed dHvA branches in PdSb. The proposed Fermi surface for PdSb is shown in fig. 6. As PdSb is a compensated metal, the volume of the electron surfaces has to equal the volume of the hole surfaces. With the aid of his symmetrized plane wave expansion Coenen [2] calculated the volume fractions of the A, B and C surfaces. As B and C are assigned now to be electron surfaces and A and D to be hole surfaces, we can deduct a value for the volume fraction of the D-surface (table III). Using those volume fractions of PdSb as reference values we can estimate the corresponding values for AuSn and PtSn by scaling with measured dHvA frequencies, assuming that the assignments as in table IV are correct and, eventually, after correcting for differences in the lattice constants (table V). The constraint of compensation is well satisfied for AuSn, thus reinforming the validity of the interpretation of the various dHvA branches. For PtSn the constraint of compensation is violated. However, if we assume for the 4th zone of PtSn a shape qualitatively similar to the one of PdSb, rather than the model proposed by Cathey et al. [5], which implies the interchange of the u and pbranch, we also get compensation for PtSn (fig. 6, table III). From the comparison between PdSb, AuSn and PtSn we deduce that the iso-electronic compounds PdSb and AuSn have very similar Fermi surfaces, both topologically and quantitatively; the slightly smaller values for the dHvA frequencies in AuSn compared with PdSb (typically 15%) are completely due to the 6% larger lattice constant II of AuSn. The volume fractions of the surfaces in AuSn are very close to those in PdSb however. Going from PdSb and PtSn, the latter having 28 electrons/cell in the highest s,p,d-bands compared with 30 electrons/cell for the former, it appears (table III), that the two less electrons in PtSn are accounted for by creating two additional holes in the fourth and fifth bands around F, rather than by emptying the sixth zone as might be expected in a rigid band model. Using the band structure for
187
M. A. C. Devillerset al./De Haas-van Alphen effect of PdSb Table III Number of electrons per zone Zone 3
r
PdSb AuSn PtSn5) PtSn6)
2.00 2.00 1.944) 1.78
Zone 4 Zone 5 Zone 6 (D) (A) (B)
(C)
1.923) 1.944) 0.84) 0.9,
0.1201) 6.00 0.0g4) 5.98 0.154) 4.0 0.00 3. 0
l.5S2) l.5S4) 0.74) 0.27
Total
0.406l) 0.404) 0.454) 0.01,
From Coenen [ 1,2] ; from Coenen [ 1, 21 after correction (see text); from compensation criterium; PdSb-derived Fermi surface after dHvA resealing including lattice constant corrections; 5) PdSb derived Fermi surface (nonrigid band model); 6) rigid band model derived Fermi surface (see text).
1) 2) 3) 4)
Table IV Comparison dHvA frequencies along symmetry directions
[ OOOl] BI J'l Cl Al D2 B2 A6
8 Ic P V
0 7 h @ e
PdSb’)
AuSn2)
PtSn3)
3.15 5.04 18.2 22.7 39.0 71.7 77.2
0.711 1.65 14.9 15.4
22 103 25 120 69 76 78 21 3.3
61.4 69.6
[Ill01 Fig. 6. a) Fermi surface cross sections of PdSb. The A, B and C pieces are from a symmetrized plane wave interpolation to the dHvA measurements (Coenen [ 21) where it should be noted that the A-cross section around r has been revised. The D-cross section is taken from the band-structure calculation of Myron and Mueller [ 31 after a slight modification to fit the dHvA measurements. The zones are indicated by the numbers. b) Our proposed Fermi surface of PtSn (nonrigid band model). c) Rigid band model derived Fermi surface of PtSn. The greek letters indicate the assignment of the dHvA frequencies of Cathey et al. [ 5 1.
PdSb of Myron and Mueller [3] as a rigid band model for PtSn yields a Fermi surface in which also most of the dHvA data of Cathey et al. [S] could be identified qualitatively (fig. 6~). The main setback of this otherwise simpler Fermi surface is, that in using the alterna-
A2 c2 Cl D4 Al
6
E ;*,
2.32 15.8 42.3 27.0
J, 0 e
99.3
6
1.98 4.81 15.8 18.9 37.7
2.20 14.0
23
35 20 5.8
[ioio] A2 A5 c2 Cl DS
El E2
+* X @ e
1.86 13.1
18
45 21 5.7
1) This work; 2) Ref. 4; 3) ref. 5; 4) as in table I. and after ref. 5.
M. A. C. Devillers et al./De Haas-van Alphen effect of PdSb
188 Table V
Non-magnetic compounds with NiAs structure Molecular valency
Metal
a
C
da
5
AuSn PdSb PtSb PtBi NiAs NiSb NiBi CoSe RhTe
4.3218 4.018 4.138 4.324 3.617 3.924 4.018 3.6294 3.99
5.5230 5.593 5.483 5.501 5.038 5.142 5.36 5.3006 5.66
1.278 1.374 1.325 1.272 1.393 1.311 1.314 1.460 1.419
PtSn PtPb RhBi IrSb
4.111 4.259 4.075 3.978
5.439 5.467 5.669 5.521
1.323 1.2837 1.391 1.388
3
IrSn IrPb
3.988 3.993
5.567 5.566
1.396 1.394
6
PdTe
4.1521
5.6719
1.366
tive assignment of the dHvA branches the compensation criterium is ‘violated severely: the zones 3 to 6 contain only (3.0 f 0.1) electrons instead of the required 4 electrons (table III). Or stated in other words the (1.3 + 0.1) holes in the third and fourth zones of both the rigid band model derived Fermi surface and the nonrigid band model derived Fermi surface are “compensated” by (0.28 + 0.03) electrons and (1.3 + 0.1) electrons in zone five and six of those two models respectively. This implies that the Fermi surface derived from the rigid band model should be rejected. From the above we note that the sixth zone remains remarkably constant through the above three NiAsstructure compounds. In view of this tendency towards enlarging the hole surfaces around I’ and comparing with the band structure of PdSb it seems most likely now to attribute the $-orbits and e-orbits in PtSn to third zone hole surfaces around A and I?, respectively (fig. 6; compare Myron and Mueller [3] ). The alternative assignment 13at A and @at I’ is not ruled out however. The picture that arises is that the Fermi surfaces of the above N&-structures are mainly determined by:
a) the nearly free electron bands for the total of s, pelectrons; so it is convenient to characterize the compounds by their total molecular (s,p) valency; b) the influence of the d-bands which modifies the extent to which the hole bands at r overlap with the H-centered electron bands; c) a “constant sixth zone hypothesis” as suggested for the five and four-valent compounds. With some confidence we can extrapolate the above findings to other iso-electronic NiAs-structures (table V). Thus PtBi and PtSb are expected to have Fermi surfaces nearly identical to that of PdSb, with dHvA frequencies deviating only 10 to 20% from those in PdSb, the latter mainly due to lattice constant deviations. Similarly NiSb and NiBi will show strong resemblance with the PdSb-Fermi surface. For NiAs the picture might be altered significantly by the increased electron density (30%) which could alter the relative positions of the P-bands with respect to the H-bands. Also CoSe and RhTe might deviate from PdSb. In the group of 4-valent NiAs-compounds we expect PtSn and PtPb to have identical Fermi surfaces. For RhBi and IrSb, being VII B-V compounds rather than VIII B-IV compounds like PtSn and PtPb, the above extrapolation might be less correct. We note a tendency for Fermi surface nesting of the fifth and sixth zone around H. This Fermi surface nesting around the hexagonal plane of the Brillouin zone tends to phase instabilities; on the other hand stronger spin-orbit coupling enlarges the band splittings in the hexagonal Brillouin zone plane, so this is presumably the reason why there is a trend for only the heaviest metals to stabilize in the NiAs structure. For the 3-valent NiAs-compounds IrSn and IrPb the “constant sixth zone hypothesis” would imply a close nesting of the 3rd to 6th zones around H with about half an electron in each zone (the (n - 1)th band contains at least as much electrons as the nth band). Only large spin-orbit splitting could stabilize such a Fermi surface, which might be the reason why only the heaviest compounds IrSn and IrPb stabilize in the NiAs-structure. Likely there exist d-band holes in those 3-valent NiAs-structures too. To solve those problems more experimental and theoretical work should be done. For the 6-valent compound PdTe one may expect the nearly free electron model to hold even better
M. A. C. Devillers et al./De Hans-van Alphen effect of PdSb
than for PdSb, with surfaces in zone 5 to zone 8 (compare with table III, where one has to accomodate two more electrons in zone 4 and higher zones).
189
111 Coenen, N. J., Morsing, C. A. J., Reijers, A. P. J. M. and Schreurs, L. W. M., Sol. State Comm. 16 (1975) 557.
121 Coenen, N. J. M., thesis, Nijmegen, 1976. Acknowledgements We thank ir. L. Schreurs for the sample preparations, mr. W. Hooghof for the development of the datasampling and data-analysis programs, the technical staff of the Nijmegen High Magnetic Field Laboratory for their nightly assistance, and dr. H. W. Myron and Prof. F. M. Mueller for communicating the results of their band structure calculation prior to publication.
131 Myron, H. W. and Mueller, F. M., Phys. Rev. B I7 (l-978) 1828. 141 Edwards, G. J., Springford, M. and Saito, Y., J. Phys. Chem. Solids 30 (1969) 2527. 151 Cathey, W. N., Coleridge, P. T. and Jan, J. P., Can. J. Phys. 48 (1970) 1151.