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Anisotropy of planar Hall effect in a single crystal of nickel
Volume 34A, number 5
ANISOTROPY
PHYSICS LETTERS
OF PLANAR
HALL EFFECT
IN A SINGLE
22 March 1971
CRYSTAL
OF NICKEL
A. A. HIRSCH, J. KLEEFELD and G. FRIEDLANDER Department of Physics, Technion, Israel institute of Technology, Haifa, israel
Received 20 February 1971
The planar Hall effect was investigated in a single crystal of Ni at 300°Kwhen the electric current was passed in various directions through the crystal. The planar Hall coefficient is anisotropic in a manner similar to that of the resistivity anisotropy of the crystal
In the last few years considerable interest has developed in the planar Hall effect (PHE) in ferromagnetic polycrystalline films [1-2]. One specifies the PHE by three coplanar vectors: the applied magnetic field (H), the current density U) and the planar Hall field (E) arising in the direction perpendicular to j. One assumes j to be along the applied electric field. The PHE in a single domain film below magnetization saturation is normally described as a two-dimensional phenomenon implied by the rotation of the spontaneous magnetization (M 5) in the film plane. The planar Hall field in such a film is given by E
Pj cos 6 sine
(1)
where 9 is the angle between j and M5. At saturation 0 becomes equal to the angle ~ between j and H. Vu Dinh Ky [3] has shown experimentally that the planar Hall coefficient P in a polycrystalline ferromagnetic film is equal to the resistivity anisotropy ~p5 = p51~-p5~(p511 and p5~ are the resjstivities at saturation when H is parallel and perpendicular to j respectively), Recently [4] when studying in our laboratory the PHE in epitaxial films of Ni grown on rocksalt and exhibiting single crystal properties, we found that P is strongly anisotropic. The PHE anisotropy in ferromagnetic single crystals has not yet been investigated. Moreover, only few reports are available on the anisotropy of the normal Hall effect in crystals [5]. In the present work an attempt was made to study the PHE in a single crystal of Ni. The planar Hall field measurements were accompanied by magnetoresistance measurements in order to find the correlation between p and i~p5 which depends on the direction of j in respect to the crystallographic axes. A circular disk
19 mm in diameter and 0.27 mm thick was carefully cut from a large cylindrical single crystal. The crystal orientation of the disk was checked by X-rays and the direction cosines of H and j in respect to the tetragonal axes were computed from the stereographic projection. The angle O between the applied electric and magnetic fields was changed by rotating the disk around its axis. The [110] axis of the crystal was parallel to the disk plane, and the [110] axis formed an angle of 22.8°with the plane. Current direction is given by the polar angle ~Jibetween j and the fixed [110] axis. The sample holder design made it possible to change the position of the current and Hall voltage electrodes on the disk surface to obtain various angles iI.’. To make the current distribution uniform, current electrodes were made 14 mm wide, with 13 mm between them. The Hall voltage was measured between probes placed 14 mm from each other. Series of PHE and magnetoresistance loops were traced with low frequency (0.005 Hz) triangular waveform fields having a drive amplitude of 2 kOe. Such low fields were sufficient to reach saturation because of the small and constant demagnetization factor of the disk. In fig. 1 are given experimental data at 300°K for the saturation values of the planar Hall field as a function of the angle 6 with ~,1ias parameter. The dashed lines were drawn assuming E to be proportional to sin 6 cos 6 as given by eq. (1). Fig. 2 shows a comparison between the reduced values of planar Hall coefficient P’p0 and the crystal resistivity anisotropy ~p5 p0 which varies with the current direction. We computed ~p5 ~ by using the Döring [6] five-coefficient expression for the fractional change in resistivity of a cubic crystal. The full circles correspond to data computed by using the resistivity 253
Volume 34A. numberS 10
PHYSICS LETTERS
22 March
,j, ~ •
8
1 ~
6
Resistivity coefficients ~fr°20°
\
0.030
•_~4.__~_~.7 ~
~
DOrung Pt’esent data
...
0.025
A
~
I,
,~
\
P/\
.1/7’ ‘I, /
2
/
q.’-120
~
~
1971
:-:°=i.~~,~\/\ ~, 45°
0.020
~
~
AA’~
°‘~k~
LA
I
\
~ I ~ I
~I
A
I A’,
.‘
f
~
‘~
IA//
II
‘~
,
I
I
A A’
~1
0.015
‘It
>
A
0.010
//
1I‘A
‘A/
A
~
/0//
4
\~•‘.. •i”
-6
°‘;—-;°-“
0/
‘~ ~..2
°/ //~
0°
60°
1800
Fig. 2. Hall Comparison between planar coefficient P/p the reduced values of the 0 and the resistivity aniso—
,v/ ,~
tropy.
-8 ~ -I0~_____________________ I 0° 30° 60° 90° 20° 50°
120°
‘4,
/~i
i/I
1800
~~5/~0
6 ~ir and with the anisotropy. coefficient PFourth replaced by the crystal resistivity order effects should be considered in formulating a the-
Fig. 1. Planar Hall field at saturation as a function of the angle 6 between the applied electric and magnetic fields. The parameter 1// is the polar angle between the current and the fixed 11101 axis. coefficient values given by Döring [6]. With the resistivity coefficients derived from the present magnetore sistance measurements we computed the points shown by open circles. As can be seen the plots of both P p0 and exhibit the same anisotropic character. The displacement of the plots along the ordinate seems to be a resuit of the difference between the planar Hall voltage measured on a finite sample and the true planar Hall voltage in an infinitely long sample. As analysis of the PHE in a cubic crystal based on symmetry considerations brought us to the conclusion that eq. (1), originally derived for a polycrystalline medium, describes relatively well also the phenomenon in the crystal when
254
ory of PHE in cubic crystals. As reported [7] P in polycrystalline Ni depends on M~ The authors wish to thank Mr. I. Zviely, of the Physics Laboratory, Technion for his helpful technical assistance.
References [1] Vu Dinh Ky, Phys.Stat.Sol. 26 (1968) 565.
[2] Vu Dinh Ky, Izv.Akad.Nauk. ser. fiz. (USSR) 29 (1965) 576. [3] Vu Dinh Ky and Ye. F.Kuritsyna, Dokl.Akad.Nauk. (USSR) 160 (1965) 77. [4] A. A. Hirsch and J. Kleefeld, to be presented at the Intermag Conference to be held in Denver, Colorado, USA. April 1971 [5] T.Hiraoka and T.Kitai, J.Phys.Soc.Japan 22 (1967) 661. [6] W.Doring. Ann.Phys. 32 (1938) 259. [7] M.L.Yu and J.T.H.Chang, J.Phys.Chem.Solids 31 (1970) 1997.
ANISOTROPY
PHYSICS LETTERS
OF PLANAR
HALL EFFECT
IN A SINGLE
22 March 1971
CRYSTAL
OF NICKEL
A. A. HIRSCH, J. KLEEFELD and G. FRIEDLANDER Department of Physics, Technion, Israel institute of Technology, Haifa, israel
Received 20 February 1971
The planar Hall effect was investigated in a single crystal of Ni at 300°Kwhen the electric current was passed in various directions through the crystal. The planar Hall coefficient is anisotropic in a manner similar to that of the resistivity anisotropy of the crystal
In the last few years considerable interest has developed in the planar Hall effect (PHE) in ferromagnetic polycrystalline films [1-2]. One specifies the PHE by three coplanar vectors: the applied magnetic field (H), the current density U) and the planar Hall field (E) arising in the direction perpendicular to j. One assumes j to be along the applied electric field. The PHE in a single domain film below magnetization saturation is normally described as a two-dimensional phenomenon implied by the rotation of the spontaneous magnetization (M 5) in the film plane. The planar Hall field in such a film is given by E
Pj cos 6 sine
(1)
where 9 is the angle between j and M5. At saturation 0 becomes equal to the angle ~ between j and H. Vu Dinh Ky [3] has shown experimentally that the planar Hall coefficient P in a polycrystalline ferromagnetic film is equal to the resistivity anisotropy ~p5 = p51~-p5~(p511 and p5~ are the resjstivities at saturation when H is parallel and perpendicular to j respectively), Recently [4] when studying in our laboratory the PHE in epitaxial films of Ni grown on rocksalt and exhibiting single crystal properties, we found that P is strongly anisotropic. The PHE anisotropy in ferromagnetic single crystals has not yet been investigated. Moreover, only few reports are available on the anisotropy of the normal Hall effect in crystals [5]. In the present work an attempt was made to study the PHE in a single crystal of Ni. The planar Hall field measurements were accompanied by magnetoresistance measurements in order to find the correlation between p and i~p5 which depends on the direction of j in respect to the crystallographic axes. A circular disk
19 mm in diameter and 0.27 mm thick was carefully cut from a large cylindrical single crystal. The crystal orientation of the disk was checked by X-rays and the direction cosines of H and j in respect to the tetragonal axes were computed from the stereographic projection. The angle O between the applied electric and magnetic fields was changed by rotating the disk around its axis. The [110] axis of the crystal was parallel to the disk plane, and the [110] axis formed an angle of 22.8°with the plane. Current direction is given by the polar angle ~Jibetween j and the fixed [110] axis. The sample holder design made it possible to change the position of the current and Hall voltage electrodes on the disk surface to obtain various angles iI.’. To make the current distribution uniform, current electrodes were made 14 mm wide, with 13 mm between them. The Hall voltage was measured between probes placed 14 mm from each other. Series of PHE and magnetoresistance loops were traced with low frequency (0.005 Hz) triangular waveform fields having a drive amplitude of 2 kOe. Such low fields were sufficient to reach saturation because of the small and constant demagnetization factor of the disk. In fig. 1 are given experimental data at 300°K for the saturation values of the planar Hall field as a function of the angle 6 with ~,1ias parameter. The dashed lines were drawn assuming E to be proportional to sin 6 cos 6 as given by eq. (1). Fig. 2 shows a comparison between the reduced values of planar Hall coefficient P’p0 and the crystal resistivity anisotropy ~p5 p0 which varies with the current direction. We computed ~p5 ~ by using the Döring [6] five-coefficient expression for the fractional change in resistivity of a cubic crystal. The full circles correspond to data computed by using the resistivity 253
Volume 34A. numberS 10
PHYSICS LETTERS
22 March
,j, ~ •
8
1 ~
6
Resistivity coefficients ~fr°20°
\
0.030
•_~4.__~_~.7 ~
~
DOrung Pt’esent data
...
0.025
A
~
I,
,~
\
P/\
.1/7’ ‘I, /
2
/
q.’-120
~
~
1971
:-:°=i.~~,~\/\ ~, 45°
0.020
~
~
AA’~
°‘~k~
LA
I
\
~ I ~ I
~I
A
I A’,
.‘
f
~
‘~
IA//
II
‘~
,
I
I
A A’
~1
0.015
‘It
>
A
0.010
//
1I‘A
‘A/
A
~
/0//
4
\~•‘.. •i”
-6
°‘;—-;°-“
0/
‘~ ~..2
°/ //~
0°
60°
1800
Fig. 2. Hall Comparison between planar coefficient P/p the reduced values of the 0 and the resistivity aniso—
,v/ ,~
tropy.
-8 ~ -I0~_____________________ I 0° 30° 60° 90° 20° 50°
120°
‘4,
/~i
i/I
1800
~~5/~0
6 ~ir and with the anisotropy. coefficient PFourth replaced by the crystal resistivity order effects should be considered in formulating a the-
Fig. 1. Planar Hall field at saturation as a function of the angle 6 between the applied electric and magnetic fields. The parameter 1// is the polar angle between the current and the fixed 11101 axis. coefficient values given by Döring [6]. With the resistivity coefficients derived from the present magnetore sistance measurements we computed the points shown by open circles. As can be seen the plots of both P p0 and exhibit the same anisotropic character. The displacement of the plots along the ordinate seems to be a resuit of the difference between the planar Hall voltage measured on a finite sample and the true planar Hall voltage in an infinitely long sample. As analysis of the PHE in a cubic crystal based on symmetry considerations brought us to the conclusion that eq. (1), originally derived for a polycrystalline medium, describes relatively well also the phenomenon in the crystal when
254
ory of PHE in cubic crystals. As reported [7] P in polycrystalline Ni depends on M~ The authors wish to thank Mr. I. Zviely, of the Physics Laboratory, Technion for his helpful technical assistance.
References [1] Vu Dinh Ky, Phys.Stat.Sol. 26 (1968) 565.
[2] Vu Dinh Ky, Izv.Akad.Nauk. ser. fiz. (USSR) 29 (1965) 576. [3] Vu Dinh Ky and Ye. F.Kuritsyna, Dokl.Akad.Nauk. (USSR) 160 (1965) 77. [4] A. A. Hirsch and J. Kleefeld, to be presented at the Intermag Conference to be held in Denver, Colorado, USA. April 1971 [5] T.Hiraoka and T.Kitai, J.Phys.Soc.Japan 22 (1967) 661. [6] W.Doring. Ann.Phys. 32 (1938) 259. [7] M.L.Yu and J.T.H.Chang, J.Phys.Chem.Solids 31 (1970) 1997.