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Electron interference oscillations in the transverse magnetoresistance of antiferromagnetic chromium
ELECTRON INTERFERENCE OSCILLATIONS IN THE TRANSVERSE MAGNETORESISTANCE OF ANTIFERROMAGNETIC CHROMIUM
E. FAWCETT*, F. W. HOLROYD t University of Toronto, Canada
and R. REIFENBERGER Purdue University, USA
Magnetic breakdown at energy gaps on the Fermi surface of antiferromagnetic Cr resulting from interaction with the incommensurate spin density wave causes quantum interference oscillations in the transverse magnetoresistance. Their existence and harmonic content is used to determine the size of the gaps, with results inconsistent with a perturbation theory.
Since Overhauser [1] first predicted the existence of spin density waves (SDW) due to the exchange interaction between itinerant electrons, and Lomer [2] pointed out that the Fermi surface of Cr exhibits nesting electron and hole sheets favouring the observed incommensuracy of the SDW with the lattice, there has been keen interest in the electronic energy band structure of this metal. Lomer [2] developed further his model for the Fermi surface and predicted that energy gaps would appear in the band structure of antiferromagnetic Cr where the energy ¢(k) as a function of the electron wavevector k satisfies the relation, c ( k ) = ¢(k + G +-- n Q ) ,
(1)
G being any reciprocal lattice vector and n an integer corresponding to the order of the energy gap resulting from interaction with the SDW of wavevector Q. Graebner and Marcus [3] employed the de Haas -van Alphen (dHvA) effect to show that Lomer's model explains some features of the rich dHvA spectrum of Cr. The uniaxial stress-dependence of the Fermi surface of Cr [4] provided further confirmation of Lomer's model, with the assumption that the magnetic breakdown fields Hn characterising first, second and third order energy gaps [n = 1, 2 and 3 in eqn. (1)] are all in the range from 1-10 T of the dHvA measurements. However, Falicov and Zuckermann [5], using an effective one-electron Hamiltonian to include many-body interactions in the band structure, found by a perturbative method that H l / H 2 ~, H z / H 3 ~-~ 103 to 104, as indicated in table 1. *Supported by the National Research Council of Canada. tNow with General Electric Company, Schenectady, USA
H n may be measured by employing the electron interference effect, which produces oscillations periodic in H -~ in the transverse magnetoresistance for current perpendicular to open orbits produced by magnetic breakdown. Stark and Friedberg [6] pointed out that the amplitude of electron interference oscillations is normally only weakly dependent on temperature, unlike that of dHvA oscillations, and that this enables one to measure the quantum lifetime of the electron state [7]. Arko, Marcus and Reed [8] demonstrated the existence of open orbits in Cr and observed magnetoresistance oscillations whose frequency was distinctly different from that of neighbouring branches of the dHvA spectrum. They remarked the weak temperature-dependence of the amplitude of the oscillations, but were unaware at that time (1969, cf. the date 1971 of ref. [6]) of the significance of this result. We have measured the transverse magnetoresistance oscillations of a single-Q Cr crystal and have confirmed the results of Arko, Marcus and Reed [8]. The frequency of one prominent branch of magnetoresistance oscillations corresponds to an area ~. formed on the Fermi surface chain which is developed on applying eq. (1) to the hole ellipsoids on the face of the Brillouin zone of paramagnetic Cr (see figs. 19 and 20 of ref. [3]). We emphasize that ~. is an interference area and not a dHvA orbit, so that the contributions to d ~ / d ¢ from the different hole-like segments effectively cancel, giving only a very weak temperature-dependence due to thermal smearing of the Fermi surface. Furthermore, the interference oscillations have a rich harmonic content, whose amplitude ratios show qualitatively that H, must have values in the range given in table 1. For an average Fermi veloc-
Journal of Magnetism and Magnetic Materials 15-18 (1980) 901-902 ©North Holland
901
E. Fawcett et al./ Electron interference oscillations in antiferromagnetic Cr
902
TABLE 1 Magnetic breakdown fields H, and SDW energy gaps An in antiferromagnetic chromium Hn: theory (T) (ref. [5])
H,: experiment (T) (present work)
An (eV)
102 4 x 10-2 1.6 × 10-5
5- ! 0 0.5-1.5 ~10
0.05-0.07 0.015-0.027 ~>0.07
Hi H2 H3
ity, 7.5 × 105 m s - ~ , for the h o l e ellipsoids [9], these c o r r e s p o n d to the given values A, of the e n e r g y gaps. I n o r d e r to d e t e r m i n e the v a r i a t i o n in f r e q u e n c y of ~ with m a g n e t angle, we c o m p u t e d its a r e a using the d i m e n s i o n s of the h o l e ellipsoid given b y G r a e b n e r a n d M a r c u s [3]. Since H 1, H 2 a n d H 3 all
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x
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lie in the range of m a g n e t i c fields used in the d H v A m e a s u r e m e n t s o n Cr [3, 4], we also calcul a t e d the a n g u l a r v a r i a t i o n for a l l the d H v A orbits to b e f o u n d on the ellipsoid c h a i n s w h e n the possibility of either m a g n e t i c b r e a k d o w n or Bragg reflection at each e n e r g y g a p is i n c l u d e d . T h e results are strikingly similar to the d H v A spectrum, as s h o w n in fig. 1. T h e b r a n c h e s ~r, p a n d ~ ( G r a e b n e r a n d M a r c u s ' [3] n o t a t i o n ) p r o b a b l y c o r r e s p o n d to the F e r m i surface chains f o r m e d b y a p p l y i n g eq. (1) to electron pockets, b u t w e d i d not c o m p u t e t h e m since the a b s e n c e of a c e n t r e of s y m m e t r y for the e l e c t r o n p o c k e t m a k e s it c o n s i d e r a b l y m o r e difficult to calculate the d H v A o r b i t s for these chains. H o w e v e r , it a p p e a r s t h a t the c h a i n m o d e l c a n a c c o u n t for the g r e a t e r p a r t of the rich d H v A s p e c t r u m of a n t i f e r r o m a g n e t i c c h r o m i u m , as well as for the i n t e r f e r e n c e o s c i l l a t i o n s . F u r t h e r m e a s u r e m e n t s of the latter will p r o v i d e a d e t a i l e d picture of the v a r i a t i o n over the F e r m i surface of the energy gap a s s o c i a t e d with the S D W . W e h o p e that this will e n c o u r a g e efforts to p e r f o r m a selfc o n s i s t e n t c a l c u l a t i o n of the e l e c t r o n i c energy b a n d structure of a n t i f e r r o m a g n e t i c Cr. A m o r e c o m plete r e p o r t of the p r e s e n t w o r k is given in ref. [10].
[,ool ANGLE
I,~o~
~o,o2
[,2,] t,,,;
t,o,]
Fig. I. Upper: d H v A spectrum of single-Q antiB~oma~¢tic Cr, with frequency branches corresponding to hole ellipsoid chains shown by heavier dots (ref. [3] with additional c branches from ref. [4D; lower: calculated dHvA frequencies for the hole ellipsoid chain model. In each diagram the branch attributed to the interference area ~. is so labelled (magneto-resistance oscillation data partially from ref. [8]).
[1] A. W. Overhauser, Phys. Rev. Lett. 4 (1960) 462; Phys. Rev. 128 (1962) 1437. [2] W. M. Lomer, Proc. Phys. Soc. (London) 80 (1962) 489; in Proc. of Intern. Conf. on Magnetism, Nottingham 1964 (Institute of Physics and the Physical Society, London, 1965) p. 127. [3] J. E. Graebner and J. A. Marcus, Phys. Rev. 175 (1968) 659. [4] E. Fawcett, R. Griessen and D. J. Stanley, J. Low Temp. Phys. 25 (1976) 771. [5] L. M. Faiicov and M. J. Zuckermann, Phys. Rev. 160 (1967) 372. [6] R. W. Stark and C. B. Friedberg, Phys. Rev. Lett. 26 (1971) 556; J. Low Temp. Phys. 14 (1974) 111. [7] R. Reifenberger, J. Low Temp. Phys. 26 (1977) 827. [8] A. J. Arko, J. A. Marcus and W. A. Reed, Phys. Rev. 176 (1968) 671; 185 (1969) 901. [9] J. Path and J. Callaway, Phys. Rev. B8 (1973) 5398. [10] R. Reifenberger, F. W. Holroyd and E. Fawcett, J. Low Temp. Phys., to be published.
E. FAWCETT*, F. W. HOLROYD t University of Toronto, Canada
and R. REIFENBERGER Purdue University, USA
Magnetic breakdown at energy gaps on the Fermi surface of antiferromagnetic Cr resulting from interaction with the incommensurate spin density wave causes quantum interference oscillations in the transverse magnetoresistance. Their existence and harmonic content is used to determine the size of the gaps, with results inconsistent with a perturbation theory.
Since Overhauser [1] first predicted the existence of spin density waves (SDW) due to the exchange interaction between itinerant electrons, and Lomer [2] pointed out that the Fermi surface of Cr exhibits nesting electron and hole sheets favouring the observed incommensuracy of the SDW with the lattice, there has been keen interest in the electronic energy band structure of this metal. Lomer [2] developed further his model for the Fermi surface and predicted that energy gaps would appear in the band structure of antiferromagnetic Cr where the energy ¢(k) as a function of the electron wavevector k satisfies the relation, c ( k ) = ¢(k + G +-- n Q ) ,
(1)
G being any reciprocal lattice vector and n an integer corresponding to the order of the energy gap resulting from interaction with the SDW of wavevector Q. Graebner and Marcus [3] employed the de Haas -van Alphen (dHvA) effect to show that Lomer's model explains some features of the rich dHvA spectrum of Cr. The uniaxial stress-dependence of the Fermi surface of Cr [4] provided further confirmation of Lomer's model, with the assumption that the magnetic breakdown fields Hn characterising first, second and third order energy gaps [n = 1, 2 and 3 in eqn. (1)] are all in the range from 1-10 T of the dHvA measurements. However, Falicov and Zuckermann [5], using an effective one-electron Hamiltonian to include many-body interactions in the band structure, found by a perturbative method that H l / H 2 ~, H z / H 3 ~-~ 103 to 104, as indicated in table 1. *Supported by the National Research Council of Canada. tNow with General Electric Company, Schenectady, USA
H n may be measured by employing the electron interference effect, which produces oscillations periodic in H -~ in the transverse magnetoresistance for current perpendicular to open orbits produced by magnetic breakdown. Stark and Friedberg [6] pointed out that the amplitude of electron interference oscillations is normally only weakly dependent on temperature, unlike that of dHvA oscillations, and that this enables one to measure the quantum lifetime of the electron state [7]. Arko, Marcus and Reed [8] demonstrated the existence of open orbits in Cr and observed magnetoresistance oscillations whose frequency was distinctly different from that of neighbouring branches of the dHvA spectrum. They remarked the weak temperature-dependence of the amplitude of the oscillations, but were unaware at that time (1969, cf. the date 1971 of ref. [6]) of the significance of this result. We have measured the transverse magnetoresistance oscillations of a single-Q Cr crystal and have confirmed the results of Arko, Marcus and Reed [8]. The frequency of one prominent branch of magnetoresistance oscillations corresponds to an area ~. formed on the Fermi surface chain which is developed on applying eq. (1) to the hole ellipsoids on the face of the Brillouin zone of paramagnetic Cr (see figs. 19 and 20 of ref. [3]). We emphasize that ~. is an interference area and not a dHvA orbit, so that the contributions to d ~ / d ¢ from the different hole-like segments effectively cancel, giving only a very weak temperature-dependence due to thermal smearing of the Fermi surface. Furthermore, the interference oscillations have a rich harmonic content, whose amplitude ratios show qualitatively that H, must have values in the range given in table 1. For an average Fermi veloc-
Journal of Magnetism and Magnetic Materials 15-18 (1980) 901-902 ©North Holland
901
E. Fawcett et al./ Electron interference oscillations in antiferromagnetic Cr
902
TABLE 1 Magnetic breakdown fields H, and SDW energy gaps An in antiferromagnetic chromium Hn: theory (T) (ref. [5])
H,: experiment (T) (present work)
An (eV)
102 4 x 10-2 1.6 × 10-5
5- ! 0 0.5-1.5 ~10
0.05-0.07 0.015-0.027 ~>0.07
Hi H2 H3
ity, 7.5 × 105 m s - ~ , for the h o l e ellipsoids [9], these c o r r e s p o n d to the given values A, of the e n e r g y gaps. I n o r d e r to d e t e r m i n e the v a r i a t i o n in f r e q u e n c y of ~ with m a g n e t angle, we c o m p u t e d its a r e a using the d i m e n s i o n s of the h o l e ellipsoid given b y G r a e b n e r a n d M a r c u s [3]. Since H 1, H 2 a n d H 3 all
i,,21~\ [,ool 2~i
i
/'
,~ 0,.0% B .. "b
A
_"-
I ...........
.. C ;
"-
-
-
--
"~
References
........
L..z_ • [::.-
x
15
[,i~l
D
-: :'=!-¢, -
E
[,e]
le,,]
[,,o]
[,,q [,,z2
Ioo,] bo,} Q[ MAGNET
lie in the range of m a g n e t i c fields used in the d H v A m e a s u r e m e n t s o n Cr [3, 4], we also calcul a t e d the a n g u l a r v a r i a t i o n for a l l the d H v A orbits to b e f o u n d on the ellipsoid c h a i n s w h e n the possibility of either m a g n e t i c b r e a k d o w n or Bragg reflection at each e n e r g y g a p is i n c l u d e d . T h e results are strikingly similar to the d H v A spectrum, as s h o w n in fig. 1. T h e b r a n c h e s ~r, p a n d ~ ( G r a e b n e r a n d M a r c u s ' [3] n o t a t i o n ) p r o b a b l y c o r r e s p o n d to the F e r m i surface chains f o r m e d b y a p p l y i n g eq. (1) to electron pockets, b u t w e d i d not c o m p u t e t h e m since the a b s e n c e of a c e n t r e of s y m m e t r y for the e l e c t r o n p o c k e t m a k e s it c o n s i d e r a b l y m o r e difficult to calculate the d H v A o r b i t s for these chains. H o w e v e r , it a p p e a r s t h a t the c h a i n m o d e l c a n a c c o u n t for the g r e a t e r p a r t of the rich d H v A s p e c t r u m of a n t i f e r r o m a g n e t i c c h r o m i u m , as well as for the i n t e r f e r e n c e o s c i l l a t i o n s . F u r t h e r m e a s u r e m e n t s of the latter will p r o v i d e a d e t a i l e d picture of the v a r i a t i o n over the F e r m i surface of the energy gap a s s o c i a t e d with the S D W . W e h o p e that this will e n c o u r a g e efforts to p e r f o r m a selfc o n s i s t e n t c a l c u l a t i o n of the e l e c t r o n i c energy b a n d structure of a n t i f e r r o m a g n e t i c Cr. A m o r e c o m plete r e p o r t of the p r e s e n t w o r k is given in ref. [10].
[,ool ANGLE
I,~o~
~o,o2
[,2,] t,,,;
t,o,]
Fig. I. Upper: d H v A spectrum of single-Q antiB~oma~¢tic Cr, with frequency branches corresponding to hole ellipsoid chains shown by heavier dots (ref. [3] with additional c branches from ref. [4D; lower: calculated dHvA frequencies for the hole ellipsoid chain model. In each diagram the branch attributed to the interference area ~. is so labelled (magneto-resistance oscillation data partially from ref. [8]).
[1] A. W. Overhauser, Phys. Rev. Lett. 4 (1960) 462; Phys. Rev. 128 (1962) 1437. [2] W. M. Lomer, Proc. Phys. Soc. (London) 80 (1962) 489; in Proc. of Intern. Conf. on Magnetism, Nottingham 1964 (Institute of Physics and the Physical Society, London, 1965) p. 127. [3] J. E. Graebner and J. A. Marcus, Phys. Rev. 175 (1968) 659. [4] E. Fawcett, R. Griessen and D. J. Stanley, J. Low Temp. Phys. 25 (1976) 771. [5] L. M. Faiicov and M. J. Zuckermann, Phys. Rev. 160 (1967) 372. [6] R. W. Stark and C. B. Friedberg, Phys. Rev. Lett. 26 (1971) 556; J. Low Temp. Phys. 14 (1974) 111. [7] R. Reifenberger, J. Low Temp. Phys. 26 (1977) 827. [8] A. J. Arko, J. A. Marcus and W. A. Reed, Phys. Rev. 176 (1968) 671; 185 (1969) 901. [9] J. Path and J. Callaway, Phys. Rev. B8 (1973) 5398. [10] R. Reifenberger, F. W. Holroyd and E. Fawcett, J. Low Temp. Phys., to be published.