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dHvA effect in Hg3-δSbF6
0038-1098181/241203-03502.00/0
Solid State Communications, Vol. 38, pp. 1203-I 205. Pergamon Press Ltd. 1981. Printed in Great Britain. dHvA EFFECT IN Hga_~SbF6* E. Batalla and W.R. Datars
Department of Physics, McMaster University, Hamilton, Canada LSS 4M1 and D. Chattier and R.J. Gillespie Department o f Chemistry, McMaster University, Hamilton, Canada LSS 4M1
(Received 15 January 1981 by M.F. Collins) The de Haas-van Alphen effect has been observed in the linear-chain compound Hga_sSbF6. Cylindrical Fermi surface cross-sections and the cyclotron masses are accounted for with a one-dimensional energy dispersion relation for the electrons in the chains and the perturbating potential caused by the incommensurability of the chains with respect to the host lattice. Hg3_6SbF 6 IS A COMPOUND that contains linear chains of mercury along two mutually perpendicular directions [1] and, like the similar compound Hg3_sAsF~, exhibits anisotropic electrical [2] and optical [3] properties. The mercury chains occupy channels of the tetragonal lattice formed by the antimony hexafluoride anions (a = 7.710(2) A, c = 12.638(8)A)and the distance between mercury atoms along the chains is 2.65 +--0.03 A [4]. Razavi et al. [5] have obtained extremal cross-sections of the Fermi surface of Hg3_, As F6 from observations of the de Haas-van Alphen (dHvA) effect and concluded that it consists of cylindrical pieces oriented along the c-axis. Using a coupled one-dimensional model for the electrons and the incommensurability of the chains with respect to the tetragonal lattice, they were able to derive values of minimum dHvA frequencies of the cylinders that are in good agreement with the experimental values with a value of G of 0.210. In this paper, we report that the Fermi surface of Hg3_6SbF~ is also cylindrical and that the same theoretical model can be used to determine the cross-sectional areas of the Fermi surface. Samples of Hg3_6SbF 6 were grown by direct oxidation of elemental mercury by SbFs. Samples with typical dimensions 1 x 1 x 0.2 mm 3 were placed in the detection coil of the field modulation technique with the c-axis along the coil's axis which was parallel to the magnetic field produced by a 5.5 T superconducting magnet. The sample was rotated about zn axis perpendicular to the c-axis to study the Fermi surface for * Research supported by the Natural Sciences and Engineering Research Council of Canada.
50
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~.
/'/
20
o
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7 5
t
r
~
t ~
t
l
~
T
l
o
t
~ 30
6o
Angle, degrees Fig. 1. dHvA frequencies for magnetic field directions up to 40 ° away from the c-axis. The solid lines are a fit to cylindrical Fermi surface pieces parallel to the c-axis. different magnetic field directions in an (hk0) plane. The dHvA frequencies contained in the oscillatory part of the magnetization were obtained by fast-Fourier analysis. The results from four samples of Hg3_6SbF6 indicated that each sample contained more than one 1203
dHvA EFFECT IN Hgs_&bF,
1204
Vol. 38, No. 1.2
Table 1. dHvA frequencies and cyclotron masses of Hg,_sSb F, with the magnetic field along the c axis for a cylindn’cai fit to experimental data and for our model calculation Orbit
of cf 6’ c! E
Experimental
Calculated
F(T)
m”lme
F(T)
m*lme
345 770 975 1680 2580 2700
0.15 +0.02 0.18 f 0.03 0.32 f 0.01
355 773 895 1680 2587 2710
0.124 0.197 0.197 0.269 0.342 0.342
Table 2. Cross-sectional areas of the Fermi surface of Hg,_sSbFb in the (00 1) plane and probabilities of observing these orbits Orbit
Area
Probability
ff* 01
(1 - 46)‘(n/a)* (1 - 46)(n/a)2 (1 - 26)2(n/a)’
(1 -q)2q4 (1 -q)2q5 q8 (1 -q)2q4 (1 -q)2q” qa
Y 6 P E
(n/a) 2 (1 + 46)(77/a)2 (1 + 26)‘(n/a)’
single crystal with different orientations of the crystal axes with respect to the applied magnetic field. The frequencies corresponding to a major single crystal isolated in the data for one sample consisted of six fundamental frequencies plus overtones and combinations. A semi-log plot of the fundamental frequencies as a function of angle is shown in Fig. 1. The direction of the minimum frequencies is taken to be the c-axis. The solid lines on this graph show the angular dependence of frequencies that would arise from cylindrical pieces of Fermi surface. It can be seen that the six fundamental frequencies have this angular dependence and the Fermi surface consists of cylindrical pieces. The cyclotron masses for three of these frequencies were obtained from the temperature dependence of the dHvA amplitude in a second sample. These are shown in Table 1 together with the minimum values of the six fundamental frequencies. The model of the Fermi surface of Hg,_sAsFB is used to interpret our results for Hg,_sSbF,. The electrons are assumed to be bound to individual chains with free-electron motion only along the direction of the chains. This implies that the electron energy depends only on k.,(k,), the component of the wavevector along the x(y) chain. The Fermi wavevector, kF, for chains going in one direction is determined by tilling electron states with electrons coming from the valence electrons
Fig. 2. (001) plane of the Fermi surface in the extended zone scheme. I‘, X and 2 are symmetry points of the body centered tetragonal Brillouin zone. The hatched area contains the occupied electron states. The darker lines represent the electron orbit y and the hole orbit E.
Fig. 3. Overlap of the Fermi surface displaced by (2n/a)(6,6, 0) with the original Fermi surface. The primed symmetry points correspond to the displaced Fermi surface. The darker lines are the contours of the most probable orbits created by this translation. of each mercury atom after one electron for every (3 - 6) mercury atom is used for the bonding between the chains and the antimony hexafluoride anions. In the Brillouin zone for the body-centered tetragonal lattice of
Vol. 38, blo. 12
dHvA EFFECT IN Hg3_aSbF 6
the anions, this gives k s = rt ( 0 . 5 - - 5 ) a
(1)
1205
The cyclotron mass for the orbits, assuming a free electron dispersion for the energy as a function of the wavevector along the chain, is 2
in the third zone. The Fermi surface then consists of four plane surfaces at -+k r along the k x and ky directions. A crosssection of this Fermi surface in the (1301) plane of the reciprocal lattice is shown in Fig. 2. The gap that removes the degeneracy at the crossing of the surfaces is a consequence of an interaction between perpendicular chains. Such an interaction has been observed to cause ordering of mercury chains in Hg3_6AsF 6 at 120 K [6]. This ordering transition has not yet been investigated in Hg3_6SbF 6 but from the similarity of the two compounds, it is quite likely that it occurs in this compound. This transition adds an extra periodicity to this system because the two sets of chains order so as to have common reciprocal points at (3 +- 6, 3 -+ 6, 0). As in the case of spin density waves in Cr [7], the effect of the perturbating potential associated with the ordering is obtained by displacing the whole Fermi surface by a vector equal to the difference between the wavevector of the perturbation and the nearest wavevector of the unperturbed lattice. Energy gaps are opened at intersections of the displaced and non-displaced Fermi surface. If the breakdown probability across this gap is q, we can calculate the probability of observing various types of orbits in this system. The most probable orbits are shown in Fig. 3 and their areas and probabilities of occurrence are listed in Table 2. Since the incommensurability parameter has not been measured at low temperature in Hg3_6SbF6, we use it as a parameter to calculate the areas in Table 2. There is excellent agreement between calculated and measured frequencies in Table 1 for all orbits except the 3' orbit for which there is an 8% difference for a 8 of 0.135. This value is reasonable since 8 is 0.08 at room-temperature and is expected to be at least 0.12 at low temperatures from thermal contraction of the lattice [6].
m* =
n"
mo
(2)
where A is the area of the orbit and k F is the Fermi wavevector in the extended zone scheme. The theoretical predictions of equation (2) are compared with the experimental values in Table 1. The theoretical masses are larger than the experimental values. In conclusion, we have shown that the Fermi surface of Hg3_sSbF6 consists of cylindrical sections and that the observed dHvA frequencies and cyclotron masses can be accounted for using the Fermi surface model suggested for Hg3_~AsF e even though the incommensurability parameter 6 is more than 50% larger in the arsenic compound (6 = 0.210) than in the antimony compound (6 = 0.135).
REFERENCES 1. 2. 3. 4. 5. 6. 7.
B.D. Cutforth, Unpublished Ph.D. thesis, McMaster University (1975). B.D. Cutforth, W.R. Datars, A. van Schyndel & R.J. Gillespie, Solid State Commun. 21,377 (1977). E.S. Koteles, W.R. Datars, B.D. Cutforth & R.J. Gillespie, Solid State Commun. 20, 1129 (1976). Z. Tun & I.D. Brown (private communications). F.S. Razavi, W.R. Datars, D. Chartier & R.J. Gillespie, Phys. Rev. Lett. 42, 1182 (1979). J.M. Hastings, J.P. Pouget, G. Shirane, A.J. Heeger, N.D. Miro & A.G. MacDiarmid, Phy~ Rev. Lett. 39, 1484 (1977). W.M. Lomer, Proc. Int. Conf. Magnetism, Nottingham, 1964, p. 127. The Institute of Physics and the Physical Society, London (1965).
Solid State Communications, Vol. 38, pp. 1203-I 205. Pergamon Press Ltd. 1981. Printed in Great Britain. dHvA EFFECT IN Hga_~SbF6* E. Batalla and W.R. Datars
Department of Physics, McMaster University, Hamilton, Canada LSS 4M1 and D. Chattier and R.J. Gillespie Department o f Chemistry, McMaster University, Hamilton, Canada LSS 4M1
(Received 15 January 1981 by M.F. Collins) The de Haas-van Alphen effect has been observed in the linear-chain compound Hga_sSbF6. Cylindrical Fermi surface cross-sections and the cyclotron masses are accounted for with a one-dimensional energy dispersion relation for the electrons in the chains and the perturbating potential caused by the incommensurability of the chains with respect to the host lattice. Hg3_6SbF 6 IS A COMPOUND that contains linear chains of mercury along two mutually perpendicular directions [1] and, like the similar compound Hg3_sAsF~, exhibits anisotropic electrical [2] and optical [3] properties. The mercury chains occupy channels of the tetragonal lattice formed by the antimony hexafluoride anions (a = 7.710(2) A, c = 12.638(8)A)and the distance between mercury atoms along the chains is 2.65 +--0.03 A [4]. Razavi et al. [5] have obtained extremal cross-sections of the Fermi surface of Hg3_, As F6 from observations of the de Haas-van Alphen (dHvA) effect and concluded that it consists of cylindrical pieces oriented along the c-axis. Using a coupled one-dimensional model for the electrons and the incommensurability of the chains with respect to the tetragonal lattice, they were able to derive values of minimum dHvA frequencies of the cylinders that are in good agreement with the experimental values with a value of G of 0.210. In this paper, we report that the Fermi surface of Hg3_6SbF~ is also cylindrical and that the same theoretical model can be used to determine the cross-sectional areas of the Fermi surface. Samples of Hg3_6SbF 6 were grown by direct oxidation of elemental mercury by SbFs. Samples with typical dimensions 1 x 1 x 0.2 mm 3 were placed in the detection coil of the field modulation technique with the c-axis along the coil's axis which was parallel to the magnetic field produced by a 5.5 T superconducting magnet. The sample was rotated about zn axis perpendicular to the c-axis to study the Fermi surface for * Research supported by the Natural Sciences and Engineering Research Council of Canada.
50
~
/ e/
e%%e
~.
/'/
20
o
S /
7 5
t
r
~
t ~
t
l
~
T
l
o
t
~ 30
6o
Angle, degrees Fig. 1. dHvA frequencies for magnetic field directions up to 40 ° away from the c-axis. The solid lines are a fit to cylindrical Fermi surface pieces parallel to the c-axis. different magnetic field directions in an (hk0) plane. The dHvA frequencies contained in the oscillatory part of the magnetization were obtained by fast-Fourier analysis. The results from four samples of Hg3_6SbF6 indicated that each sample contained more than one 1203
dHvA EFFECT IN Hgs_&bF,
1204
Vol. 38, No. 1.2
Table 1. dHvA frequencies and cyclotron masses of Hg,_sSb F, with the magnetic field along the c axis for a cylindn’cai fit to experimental data and for our model calculation Orbit
of cf 6’ c! E
Experimental
Calculated
F(T)
m”lme
F(T)
m*lme
345 770 975 1680 2580 2700
0.15 +0.02 0.18 f 0.03 0.32 f 0.01
355 773 895 1680 2587 2710
0.124 0.197 0.197 0.269 0.342 0.342
Table 2. Cross-sectional areas of the Fermi surface of Hg,_sSbFb in the (00 1) plane and probabilities of observing these orbits Orbit
Area
Probability
ff* 01
(1 - 46)‘(n/a)* (1 - 46)(n/a)2 (1 - 26)2(n/a)’
(1 -q)2q4 (1 -q)2q5 q8 (1 -q)2q4 (1 -q)2q” qa
Y 6 P E
(n/a) 2 (1 + 46)(77/a)2 (1 + 26)‘(n/a)’
single crystal with different orientations of the crystal axes with respect to the applied magnetic field. The frequencies corresponding to a major single crystal isolated in the data for one sample consisted of six fundamental frequencies plus overtones and combinations. A semi-log plot of the fundamental frequencies as a function of angle is shown in Fig. 1. The direction of the minimum frequencies is taken to be the c-axis. The solid lines on this graph show the angular dependence of frequencies that would arise from cylindrical pieces of Fermi surface. It can be seen that the six fundamental frequencies have this angular dependence and the Fermi surface consists of cylindrical pieces. The cyclotron masses for three of these frequencies were obtained from the temperature dependence of the dHvA amplitude in a second sample. These are shown in Table 1 together with the minimum values of the six fundamental frequencies. The model of the Fermi surface of Hg,_sAsFB is used to interpret our results for Hg,_sSbF,. The electrons are assumed to be bound to individual chains with free-electron motion only along the direction of the chains. This implies that the electron energy depends only on k.,(k,), the component of the wavevector along the x(y) chain. The Fermi wavevector, kF, for chains going in one direction is determined by tilling electron states with electrons coming from the valence electrons
Fig. 2. (001) plane of the Fermi surface in the extended zone scheme. I‘, X and 2 are symmetry points of the body centered tetragonal Brillouin zone. The hatched area contains the occupied electron states. The darker lines represent the electron orbit y and the hole orbit E.
Fig. 3. Overlap of the Fermi surface displaced by (2n/a)(6,6, 0) with the original Fermi surface. The primed symmetry points correspond to the displaced Fermi surface. The darker lines are the contours of the most probable orbits created by this translation. of each mercury atom after one electron for every (3 - 6) mercury atom is used for the bonding between the chains and the antimony hexafluoride anions. In the Brillouin zone for the body-centered tetragonal lattice of
Vol. 38, blo. 12
dHvA EFFECT IN Hg3_aSbF 6
the anions, this gives k s = rt ( 0 . 5 - - 5 ) a
(1)
1205
The cyclotron mass for the orbits, assuming a free electron dispersion for the energy as a function of the wavevector along the chain, is 2
in the third zone. The Fermi surface then consists of four plane surfaces at -+k r along the k x and ky directions. A crosssection of this Fermi surface in the (1301) plane of the reciprocal lattice is shown in Fig. 2. The gap that removes the degeneracy at the crossing of the surfaces is a consequence of an interaction between perpendicular chains. Such an interaction has been observed to cause ordering of mercury chains in Hg3_6AsF 6 at 120 K [6]. This ordering transition has not yet been investigated in Hg3_6SbF 6 but from the similarity of the two compounds, it is quite likely that it occurs in this compound. This transition adds an extra periodicity to this system because the two sets of chains order so as to have common reciprocal points at (3 +- 6, 3 -+ 6, 0). As in the case of spin density waves in Cr [7], the effect of the perturbating potential associated with the ordering is obtained by displacing the whole Fermi surface by a vector equal to the difference between the wavevector of the perturbation and the nearest wavevector of the unperturbed lattice. Energy gaps are opened at intersections of the displaced and non-displaced Fermi surface. If the breakdown probability across this gap is q, we can calculate the probability of observing various types of orbits in this system. The most probable orbits are shown in Fig. 3 and their areas and probabilities of occurrence are listed in Table 2. Since the incommensurability parameter has not been measured at low temperature in Hg3_6SbF6, we use it as a parameter to calculate the areas in Table 2. There is excellent agreement between calculated and measured frequencies in Table 1 for all orbits except the 3' orbit for which there is an 8% difference for a 8 of 0.135. This value is reasonable since 8 is 0.08 at room-temperature and is expected to be at least 0.12 at low temperatures from thermal contraction of the lattice [6].
m* =
n"
mo
(2)
where A is the area of the orbit and k F is the Fermi wavevector in the extended zone scheme. The theoretical predictions of equation (2) are compared with the experimental values in Table 1. The theoretical masses are larger than the experimental values. In conclusion, we have shown that the Fermi surface of Hg3_sSbF6 consists of cylindrical sections and that the observed dHvA frequencies and cyclotron masses can be accounted for using the Fermi surface model suggested for Hg3_~AsF e even though the incommensurability parameter 6 is more than 50% larger in the arsenic compound (6 = 0.210) than in the antimony compound (6 = 0.135).
REFERENCES 1. 2. 3. 4. 5. 6. 7.
B.D. Cutforth, Unpublished Ph.D. thesis, McMaster University (1975). B.D. Cutforth, W.R. Datars, A. van Schyndel & R.J. Gillespie, Solid State Commun. 21,377 (1977). E.S. Koteles, W.R. Datars, B.D. Cutforth & R.J. Gillespie, Solid State Commun. 20, 1129 (1976). Z. Tun & I.D. Brown (private communications). F.S. Razavi, W.R. Datars, D. Chartier & R.J. Gillespie, Phys. Rev. Lett. 42, 1182 (1979). J.M. Hastings, J.P. Pouget, G. Shirane, A.J. Heeger, N.D. Miro & A.G. MacDiarmid, Phy~ Rev. Lett. 39, 1484 (1977). W.M. Lomer, Proc. Int. Conf. Magnetism, Nottingham, 1964, p. 127. The Institute of Physics and the Physical Society, London (1965).