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Fermi surface changes in dilute alloys of magnesium in cadmium
Volume 47A, number 2
PHYSICS LETTERS
11 March 1974
FERMI SURFACE CHANGES IN DILUTE ALLOYS OF MAGNESIUM IN CADMIUM* JR. BOYD*, JR. RESCHLY and W.L. GORDON Department of Physics, Case Western Reserve University, Cleveland, Ohio 441 06, USA Received 24 January 1974 The addition of Mg in concentrations up to 2 at.% is found to reduce the cross section of the Cd Fermi surface near the point H such that ~ lnA/ö(c/a) is 4.8 ±0.2 compared with the uniaxial compression result of 4.
The addition of Mg to Cd produces a strong decrease in axial ratio with no change in the number of electrons per atom. Thus, a direct comparison of Fermi surface changes with those observed in either uniaxial strain or pressure studies can provide an indication of the changes in Cd band structure due to the Mg potential. Results on two related families of de Haas—van Alphen oscillations are presented here: the a orbits on the “cap” and the 13 orbits on the “undulating cylinder” in the notation of Tsui and Stark [1]. Single crystal samples containing up to 2.3 at.% Mg were prepared from 6—9’s materials by chill casting and zone levelling. Experimental samples were cut with an acid string saw and their composition determined by chemical analysis of adjacent sections. The dHvA frequencies were obtained with an accuracy of approximately 0.1% using a null deflection torsion balance of the Condon type in conjuction with a rotatable 1 5” Varian electromagnet providing fields up to 36 kG. The magnitude and angular variation of both a and 13 frequencies in pure Cd agreed well with the results of ref. [1]for magnetic field in the (1010) and (1120) planes. In each of these alloys, the angular variation of frequency was simply related to that of Cd by a scale factor for B within 30°of [0001]. Thus the frequency at [0001]was used to represent each composition and the linear dependence of this area change on concentration is shown in fig. 1. Some evi*
Supported initially by the US Army Research Office, Dur-
*
ham, and now by the National Science Foundation. Present address: Department of Physics, Southern Methodist University, Dallas, Texas 75222, USA.
ATOMIC % Mg
oO
.5
1.0
I 5
2.0
25
a ORBITS
-2
ORBIT S
~
-s Fig. 1. Fractional change in extremal cross section areas of the Fermi surface in the AHL plane for Cd Mg alloys versus Mg concentration. The upper solid line shows the remapped free electron prediction as described in the text.
dence of the 71/3 frequency, ref. [1], was present in the low concentration alloys but uncertainties in frequency determinations obscured any change with alloying. Tsui and Stark [1]have pointed out that the separation between a and j3 frequencies for B near [0001] is due to the lifting of a degeneracy in the AHL plane of the Brillouin zone by spin-orbit interaction. Since there is no detectable difference between the concentration dependence of these two frequencies, we conclude that Mg does not perturb the Cd spin-orbit interaction significantly.
The remapped-free electron cross section for the cap in the AHL plane is simply approximated as a tn137
Volume 47A. number 2
PHYSICS LETTERS
angle with area A
=
Cyclotron masses for B along 100011 are 0.16 ± 2[~~~j2
3~3(2~/a)
with = [(\/~a/7rc)2/3—a2!4c2]112. The change of this area with a and c appropriate to each alloy composition [2] is shown by the upper solid line in fig. I. A more meaningful comparison is provided by uniaxial compression [3] and pressure [4] results on the a and 13 orbits. The former yields a value of 4.2 for a In A/a (c/a) and the latter a value of approximately 4 (based on the large anisotropy in linear compressibilities in Cd) while alloying with Mg produces a value of4.8 ±0.2. Thus, assuming a separable dependence of cross secion on lattice parameter and on the difference between solute and solvent potential. we can attribute ‘~ 80% of the change of cross section to lattice parameters. A proper band structure calculation ~s needed to extend this interpretation. It is interesting to note, however, that the atomic volumes of Cd and Mg are similar and that only a very small change of atomic volume occurs up to 25 at.% Mg in spite ~f the large change in c/a.
138
II March 1974
0.01 for both a and 13 orbits. Dingle temperatures of approximately 4°Kper at.% were obse~edfor both a and 13 orbits up to 30°from [00011 hut were not checked over a wider angular range. We are grateful to Professor R.W. Stark for helpful and stimulating discussions on this work.
References Ill D.C. Tsui and R.W. Stark, Phys. Rev. Lett. 16(1966) 19. 121 Room temperature values were obtained from W.B. Pearson, Handbook of lattice spacing and structure of metals and alloys (Pergamon Press, London, 1964) and corrected for thermal contraction obtained from three terminal capacitance measurements on two representative alloys. 13] D. Gamble and B.R. Watts, J. Phys. F Metal Phys. 3 (1973) 98.
14] i.E. Schirber and W.J. O’Sullivan, Proc. Eleventh Intern. Conf. on Low temperature physics, St. Andrews (1968), P• 1141.
PHYSICS LETTERS
11 March 1974
FERMI SURFACE CHANGES IN DILUTE ALLOYS OF MAGNESIUM IN CADMIUM* JR. BOYD*, JR. RESCHLY and W.L. GORDON Department of Physics, Case Western Reserve University, Cleveland, Ohio 441 06, USA Received 24 January 1974 The addition of Mg in concentrations up to 2 at.% is found to reduce the cross section of the Cd Fermi surface near the point H such that ~ lnA/ö(c/a) is 4.8 ±0.2 compared with the uniaxial compression result of 4.
The addition of Mg to Cd produces a strong decrease in axial ratio with no change in the number of electrons per atom. Thus, a direct comparison of Fermi surface changes with those observed in either uniaxial strain or pressure studies can provide an indication of the changes in Cd band structure due to the Mg potential. Results on two related families of de Haas—van Alphen oscillations are presented here: the a orbits on the “cap” and the 13 orbits on the “undulating cylinder” in the notation of Tsui and Stark [1]. Single crystal samples containing up to 2.3 at.% Mg were prepared from 6—9’s materials by chill casting and zone levelling. Experimental samples were cut with an acid string saw and their composition determined by chemical analysis of adjacent sections. The dHvA frequencies were obtained with an accuracy of approximately 0.1% using a null deflection torsion balance of the Condon type in conjuction with a rotatable 1 5” Varian electromagnet providing fields up to 36 kG. The magnitude and angular variation of both a and 13 frequencies in pure Cd agreed well with the results of ref. [1]for magnetic field in the (1010) and (1120) planes. In each of these alloys, the angular variation of frequency was simply related to that of Cd by a scale factor for B within 30°of [0001]. Thus the frequency at [0001]was used to represent each composition and the linear dependence of this area change on concentration is shown in fig. 1. Some evi*
Supported initially by the US Army Research Office, Dur-
*
ham, and now by the National Science Foundation. Present address: Department of Physics, Southern Methodist University, Dallas, Texas 75222, USA.
ATOMIC % Mg
oO
.5
1.0
I 5
2.0
25
a ORBITS
-2
ORBIT S
~
-s Fig. 1. Fractional change in extremal cross section areas of the Fermi surface in the AHL plane for Cd Mg alloys versus Mg concentration. The upper solid line shows the remapped free electron prediction as described in the text.
dence of the 71/3 frequency, ref. [1], was present in the low concentration alloys but uncertainties in frequency determinations obscured any change with alloying. Tsui and Stark [1]have pointed out that the separation between a and j3 frequencies for B near [0001] is due to the lifting of a degeneracy in the AHL plane of the Brillouin zone by spin-orbit interaction. Since there is no detectable difference between the concentration dependence of these two frequencies, we conclude that Mg does not perturb the Cd spin-orbit interaction significantly.
The remapped-free electron cross section for the cap in the AHL plane is simply approximated as a tn137
Volume 47A. number 2
PHYSICS LETTERS
angle with area A
=
Cyclotron masses for B along 100011 are 0.16 ± 2[~~~j2
3~3(2~/a)
with = [(\/~a/7rc)2/3—a2!4c2]112. The change of this area with a and c appropriate to each alloy composition [2] is shown by the upper solid line in fig. I. A more meaningful comparison is provided by uniaxial compression [3] and pressure [4] results on the a and 13 orbits. The former yields a value of 4.2 for a In A/a (c/a) and the latter a value of approximately 4 (based on the large anisotropy in linear compressibilities in Cd) while alloying with Mg produces a value of4.8 ±0.2. Thus, assuming a separable dependence of cross secion on lattice parameter and on the difference between solute and solvent potential. we can attribute ‘~ 80% of the change of cross section to lattice parameters. A proper band structure calculation ~s needed to extend this interpretation. It is interesting to note, however, that the atomic volumes of Cd and Mg are similar and that only a very small change of atomic volume occurs up to 25 at.% Mg in spite ~f the large change in c/a.
138
II March 1974
0.01 for both a and 13 orbits. Dingle temperatures of approximately 4°Kper at.% were obse~edfor both a and 13 orbits up to 30°from [00011 hut were not checked over a wider angular range. We are grateful to Professor R.W. Stark for helpful and stimulating discussions on this work.
References Ill D.C. Tsui and R.W. Stark, Phys. Rev. Lett. 16(1966) 19. 121 Room temperature values were obtained from W.B. Pearson, Handbook of lattice spacing and structure of metals and alloys (Pergamon Press, London, 1964) and corrected for thermal contraction obtained from three terminal capacitance measurements on two representative alloys. 13] D. Gamble and B.R. Watts, J. Phys. F Metal Phys. 3 (1973) 98.
14] i.E. Schirber and W.J. O’Sullivan, Proc. Eleventh Intern. Conf. on Low temperature physics, St. Andrews (1968), P• 1141.