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Planar magnetoresistivity and planar hall effect measurements in nickel single-crystal thin films
Physica
59 (1972) 498-509
PLANAR
o North-Holland Publishing Co.
MAGNETORESISTIVITY
MEASUREMENTS
IN NICKEL T. T. CHEN*
Department
of Electrical
AND
PLANAR
SINGLE-CRYSTAL
HALL THIN
EFFECT FILMS t
and V. A. MARSOCCI
Sciences, State University
Stony Brook,
New
York,
of New
York,
USA
Received 28 June 1971
synopsis The planar magnetoresistivity anisotropy and the planar Hall effect in nickel single-crystal films have been explained successfully by using a phenomenological equation in a tensor form. The anisotropy constants defined for this equation are directly related to the planar measurements and can be used as parameters for making relative comparisons in the study of the magnetoresistivity anisotropy effects. It is demonstrated that these effects in ferromagnetic crystals can be studied by measurements on single-crystal films.
1. I&rod&ion. In ferromagnetic metals both the magnetoresistivity and the Hall resistivity are composed of two distinct parts: the “ordinary” effects and the “extraordinary” or “spontaneous” effects. The ordinary effects are caused by the direct interaction between the magnetic field and the kinetic behaviour of the conduction electrons; this effect is present in all conductive materials. The extraordinary effect, unique to the ferromagnetic state, is contributed by the scattering process between the s-band electrons and the d-band electrons. This effect is related to the spin-orbit interaction of the d electronsi), and is anisotropic and depends on the orientation of the magnetization of the metal. In the case of nickel single crystals, the anisotropic effects for the extraordinary part of both the magnetoresistivity and the Hall resistivity have been reported23 a). The planar Hall resistivity in ferromagnetic polycrystalline films has been extensively studied and has been found to be closely related to the magnetoresistivity anisotropy4). However, the reports of the experimental data taken for single-crystal films5) are few and the results have not been t Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF-AFOSR1116-66. t Present address: RCA Laboratories, Princeton, N. J. 498
MAGNETORESISTIVITY
properly
explained.
AND
In this paper,
HALL
EFFECT
it is proposed
499
IN Ni FILMS
that this planar
Hall
effect is actually contributed by the extraordinary part of the magnetoresistivity only. The anisotropies in both the extraordinary Hall resistivity and the magnetoresistivity, as well as in the planar Hall resistivity, can be described by a generalized phenomenological equation. The possibilities of using the single-crystal films as the media in which to study these anisotropy effects are also discussed. 2. Theory. In a saturated ferromagnetic crystal the extraordinary part of the magnetoresistivity and the Hall resistivity are dependent on the orientation of the magnetization only. These two resistivities may then be included in the resistivity tensor of the crystal. At room temperature, the intrinsic resistivity anisotropy due to the nonspherical Fermi surface structure is negligible as compared with the magnetoresistivity-anisotropy effect. Thus the resistivity tensor, p, can be written as a function of the unit vector, 01,along the direction of the magnetization M. The components of p(a) may then be expanded in a power series, of the CL~‘S, in the form P&4
= +
i
k,l,m...=l
@ii -k akfjak
~klmntpkW%%
+
-b aklijakw
+
ak f
aklmij~x~lolm
. . . ),
(1)
where i, j = 1, 2, 3 and the a’s are expansion coefficients. The above tensor can be further separated into a symmetrical and an antisymmetrical part, p$, given by p$‘(cx) =
lFI
(atj + aklipk4
+
aklrnntpk~l~rn%
+
. - -)
part ply!,
(4
k,l,m...=l
and Pii&)
(akipk
= k 1j ,,
+
akZmi3~kWm
+
. . . ).
(3)
. ..=l
In a cubic-crystal structure, the components in eqs. (2) and (3) can be greatly simplified by arguments of symmetry if the crystalline axes coincide with the axes of the coordinate system. For a fifth-order expansion in the ai’s, the resistivity tensor is given bys) p’S’(oz)
500
T. T. CHEN
0 p@)(B)
=
A@)
V.
A. MARSOCCI
--A2(4
0
A$)
-Al(&)
0
-A&q
A$)
AND
,
(5)
with A$(&) = cua[eo + elaf + ezo$ + es(c&$ + a&~ + a&f)] and i = 1, 2, 3. In the above, the et’s are the extraordinary Hall resistivity-anisotropy constants. For crystals,
such as nickel and nickel-rich
alloys, which have the easy
axes of magnetization along the (11 l> directions, the magnetoresistivity anisotropy along a current direction, b, may be calculated, for the saturated state, by the use of eq. (4). The result is A&% @)/PO z GM, B) -
pollpo= [~~(;(s)~~)- pollpo = Cl(E &Ii” - 5) + Cz(X a$$ - ;) + C3(2 a$@; - ;)
+ 2c4 x wjh% + 2cs 2 w&/%h
(6)
withi,j,k= 1,2,3andC,=C;/pawithr= 1,2,3,4,5. In eq. (6), Ap/pa represents the extraordinary magnetoresistivity with p(&, p) being the value of the resistivity when the crystal is just saturated along the di direction and pa the resistivity measured in a completely demagnetized state. In the present case pa = C’ + $6 + $(G + Cb). Eq. (6) is equivalent to Doring’s equations) except that the anisotropy constants are directly related to the resistivity-tensor elements. In a thin-film sample, when a magnetic field is rotated in the plane of the film, the magnetoresistivity measured along the current direction is defined as the planar magnetoresistivity. There will also exist a voltage produced in the plane of the film but perpendicular to the current direction ; this potential is defined as the planar Hall voltage. In the case of a film grown on the (001) plane, and with the current J at an angle 8 with respect to the < 100) axis and the field l? at an angle 4 with respect to 3, the planar magnetoresistivity anisotropy is given by Ap(& @)/PO = 9Cr + +
SC2
-
+C3
*[(Cl + Cs)s co9 28 + Ci sin2 201s
x sin 2(8 + +) + tan-l ( c1~c2)cos2~]+~Cscos4(~+~). Letting f be a unit vector in the film plane but perpendicular Hall resistivity may be given by PZY(~, &PO = (X)
(7) to B, the planar
*f/PO
= +[Ci cos2 20 + (Cl + Cs)2 sin2 2B]* x sin 2(e + 4) [
tan- r( “Lc2)
tan 281.
(8)
MAGNETORESISTIVITY
In
the
case
AM,
of
AND
HALL
(OOl)--oriented
EFFECT
IN Ni FILMS
single-crystal
films,
501
19= 0 and
&/PO = AP($)/PO = gc, + jjc,
-
;ca + i(cl+
Cs) cos 2$ + &C2 cos 4+ 2
(9)
also pzzl(& B)/Po = pzy(+)lpO = K4sinW
(10)
In the case of (OOl)--oriented single-crystal corresponding equations are A&$)/pa = &Cl + +$C, -
;C, + SC4 cos 24 -
films 8 = x/4 and the
+C2 cos 44
(11)
and pz&$)/pO
=
*(Cl +
C2)sinW
(12)
Hence the anisotropy coefficients Cr, Cs and Cd can be determined by a measurement of the Fourier components of functions Ap(4)/pa and p&$)/p0 in the (OOl)-- or the (OOl)-< 1 IO>-oriented film. The polycrystalline structure may be viewed as a composite structure of various randomly-oriented single-crystal grains. Thus, the galvanomagnetic effects in a polycrystalline structure can be calculated by averaging the effects for a single-crystal structure. If the averaging process is performed on the previous expressions for the planar effects in a polycrystalline film, the results give AP(~)/PO=~C~+~C~-~C~+~.C~+~C~
Here,
+
(gl+&c,
-&c3+&c4
=
&K + &(2Cs -
+&5)cos2+
C3) + $,K cos 2+.
(13)
K has the same definition as in Diiring’s papers) and is expressed as
K = @Cl + $C, -
AC3
+
C4 -
+C5).
(14)
In addition PW(#/PO = &K sin 4.
(15)
These results agree with the previous reports4). When the magnetic field is rotated in a plane perpendicular to the current direction, there will be produced a Hall-voltage component along the 9 direction. The anisotropy appearing in the extraordinary part of the Hall resistivity can be described by the antisymmetric part of the resistivity tensor as shown in eq. (5). In the case of a (OOl)--oriented single-crystal film, if B is rotated in the (100) plane and makes an angle 5 with the axis, the resistivity
502
T. T. CHEN
measured
AND
along the <0 10) direction
?H(<)
=
V. A. MARSOCCI
is described
by
(;‘p^)$
=
-A&q
=
(e +
gel
+ &(es -
+
ye2
+
+e3)
sin 5 -
($er + &;-es -
&es) sin 35‘
es) sin 55.
The extraordinary Hall resistivity anisotropy may then be determined the angular-dependence data taken for a single-crystal sample.
(16) from
3. Experimental results. The planar effects have been measured for three sets of nickel films. Each set consists of one polycrystalline film and two single-crystal films; the three films were fabricated in a single evaporation process to insure that they have the same thickness. The polycrystalline film was deposited directly on a glass substrate, and the two single-crystal films were epitaxially grown on a freshly cleaved and carefully polished (100) surface of a sodium-chloride crystal. These two latter films were oriented one along the (100) axis and the other along the
MAGNETORESISTIVITY
films will cause a negligible
AND HALL EFFECT
or at most
a constant
IN Ni FILMS
effect
503
on the planar
measurements at room temperature. All the previous calculations may then be applied to the present measurements. The angular dependence of Ap(#)/pa and of psY($)/pa were measured when rotating the film inside the magnetic field, on a point-by-point basis, so that the induced voltage due to the rotation of the film is eliminated. Then these data were compared with a continuous function of the form A cos 2(4 + e) + B cos 4$,
(17)
which was generated by an analog computer. By visual alignment of the graphs, the best-fit parameters can be determined. Two typical measurements for a (OOl)-- and a (OOl)-
r
00
1
30°
I
60’=
90°
120”
1500
180”
,
210”
I
240”
1
270” 4
Fig. 1. The planar magnetoresistivity anisotropy and the planar Hall effect in a (OOl)-( 100) oriented nickel single-crystal film (no. 257).
Fig. 2. The planar magnetoresistivity anisotropy and the planar Hall effect in a (OOl)-(100) oriented nickel single-crystal film (no. 258).
Polycrystal
[loo]-(001)
11lOI-(001)
Polycrystal
[loo]-(001)
[l lOI-(001) Polycrystal
1492
1160
1160
1160
1058
1058
239 244
245
246
257
258
from Kaya’s
data
from Doring’s
data
Bulk results calculated
Polycrystal
plate
(OOl)-[loo]-plate
(OOl)-[lOO]-plate Polycrystal plate
[ 1lOI-(001)
1058 259 Bulk results calculated
[loo]-(001)
1492
1492
Orientation
238
(4
Thickness
magnetoresistivity
237
Film no.
Planar
l.lcos4#
1.4 cos 44
1.28 cos 44
12.96 cos 24
25.9
26.6
26.3
1.75 cos 44
13.14 cos 4+ 5.7 cos 24 +
29.0
6 cos 2$ + 2.38 cos 44
12.58
22.65
19.9
12.2
8.4
16
15
8
13.4
8 16.4
19.9
16
15
doe-?
[(~/PO) Wffl
19.7
10.8
23.5
17.3
1.92 cos 44
10.72 cos 2+ -
films
(dpzl/lpo) x IO3
9.8 cos 44
1.5 cos 44
5.90 cos 2fp -
9.32 cos 44
8.9 cos 26, -
5.88 cos 2+ +
10.36 cos 44
11.6cos2+
in nickel-thin
: Anisotropies
1.65 cos 44
VP/PO)x lo3
Planar-effects
and planar Hall resistivity
5.22 cos 24 +
anisotropy
TABLE I
12
17
15
10
16.8
17
11
17
pP-l)
x 1~
MAGNETORESISTIVITY
AND
HALL
EFFECT
Since there is no way to place the thin-film magnetized
IN
Ni FILMS
505
sample in a completely
de-
state
the value of pa is approximated by pa = &[p(# = 90”) + ~(4 = o')l. Th e error which results can be calculated by eqs. (7) and (13) ; in the case of nickel films, it is less than 1Oh. By applying eqs. (9) to (15) to the measured data for the planar magnetoresistivity anisotropy and for the planar Hall effect, the coefficients Ci, Cs, Cq and K are determined. The calculated values are listed in table II. TABLE II Calculated Film no.
anisotropy Cl
coefficients
based on the measured
c2
c3
data in nickel films
c4
c5
K
237
- 0.0006
0.0112
0.02315
0.03385
244
-0.00106
0.01072
0.0188
0.0322
257
-0.00152
0.0136
0.022
0.0375
Bulk data Kaya
-0.007
0.019
-0.070
0.029
0.014
0.0438
Dijring
- 0.0026
0.014
- 0.068
0.0266
0.020
0.0432
Although the constants Ca and Cs also appear in these equations, their values cannot be determined from these planar measurements, since the value of pa is only an approximate one. The results of the measurements of the planar Hall effect in the (OOl)-oriented films happen to represent two special cases. That is, the values of psy(~)/pa show a sin 24 variation for these two cases. Since this variation is also observed for the polycrystalline films, it may be expected that the planar Hall effect in single-crystal films is intrinsically related to the symmetry between the relative orientation of the magnetic field with respect to the current direction. In order to ensure that the crystalline structure is the primary factor which governs the symmetry of the resistivity anisotropy, a single-crystal film deposited on the (001) plane but oriented along an arbitrary direction between the directions were measured. The results are shown in fig. 3 and the corresponding best-fit data are Ap(+)/pa = [8.60 sin 2(+ + 26”) + 3.84 cos 2(+ + 26”) + 2.04 cos 4(+ + 26”)] x 10-s and pZY($)/pO= 9.34 sin 2(4 + 12") x 10-a.
(18) (19)
If the averaged value of the CZ’S in table II are used with 19= 26”, then the results predicted for the film are
Ap($)/po= [8.66 sin 2(rj + 26”) + 3.08 cos 2(4 + 26”) + 1.38 cos 4($ + 26”)] x 10-s
(20)
506
‘I’. T. CHEN
1
I
nickel
V. A. MARSOCCI
Ap, =6.60 sin 2(~+26°)+3.84cos2(~+260~+2.04cos4(~+260$x103 PO
I
IY Fig. 3. The
AND
I
30”
planar
60°
1
I
90’
1200
magnetoresistivity
single-crystal
I
1500
I
180”
anisotropy
and
I
2100 the
film at an angle of 26” with respect
I
1
240”
270° c Hall
planar to the
effect
in a
< 100) axis.
and pZY(~)/pa = 7.98 sin 2(+ + 10.9”) x 10-s.
(21)
These data agree quite well with the measured results given in eqs. (18) and (19). The field dependence of the Hall resistivity has been measured for a (001)~< lOO)-oriented film. The magnetic field was held in the (100) plane making a different angle with respect to the (010) axis. After an appropriate correction is made, for the demagnetization field effect and for the effects
Fig. 4. The
angular
dependence
of the extraordinary crystal
film.
Hall
effect
in a nickel
single-
MAGNETORESISTIVITY
of the planar Hall resistivity, value of the effective magnetizations). Hall resistivity
AND
HALL
EFFECT
the Hall resistivity
IN Ni FILMS
may be plotted
field. This field has the same orientation
The extraordinary PH by the equation
Hall resistivity
RrM
507
versus the
as the internal is related
to the
(22) where Ra is the Hall coefficient and f(C) re p resents the parameter which is measured in a field H ; the field is in the ( 100) plane and makes an angle c with respect to the axis. From the corrected PH([) curves, the angular dependence of RIM was determined as is shown in fig. 4. 4. Discussion. From the results shown in table I and in figs. 1 to 3, it can be seen that the angular dependence of the planar magnetoresistivity anisotropy and the planar Hall resistivity as measured in nickel singlecrystal films agree quite well with the predicted variations. The slight deviations, from the predicted values, as seen in figs. 1 and 2 are due to the fact that the films were not exactly aligned in the appropriate directions. In fig. 3, the orientation of the film was determined by the best-fit curves, hence no such deviation is manifested. In these measurements, only the two-fold symmetry terms (e.g. the cos 2(+ + 0) terms in eqs. (9) and (10) and the four-fold symmetry term (e.g. the cos 4(# + 0) term in eq. (10) are observed; no higher-order symmetry has been found. This indicates that a fourth-order expansion, in terms of the CX~ for $3)(&) as given by eq. (1) is both necessary and sufficient to describe the magnetoresistivity anisotropy in nickel single crystals. This equation, when applied to a polycrystalline structure, indicates that the fourth-order terms are needed to describe the planar Hall effect coefficients. This agrees with recent reports for bulk single-crystals) and polycrystal specimens lo). The planar Hall effect in single-crystal films has been found to be closely related to the magnetoresistivity anisotropy; this correlates with the results obtained for the polycrystalline films. The more generalized measurement, shown in fig. 3, further indicates that the symmetry of the planar Hall effect is determined by the crystalline structure. The present vacuum deposition techniques are still unable to provide for the fabrication of different films with identical physical structures. The data in table I indicate that the variations among the measured values of the coefficients may be larger than 30%, whereas the measurement error is estimated to be less than 2%. These results indicate that the structural imperfections such as dislocations, stacking faults, grain boundaries, etc. do effect the magnetoresistivity-anisotropy measurements. It can be seen from the data in table I that these imperfections reduce the anisotropy effect in thin films as compared with the bulk data.
508
An additional
T. T. CHEN
problem
AND
encountered
V. A. MARSOCCI
in the thin-film
measurements
in-
volves the changes in the susceptibility of the thin-film specimen. It has been observed that, in the polycrystalline films, the value of (1 /pe)[dp(+)/dH] does depend on the relative orientation of the magnetic field with respect to the current direction119 1s). Similar results have been found for the singlecrystal films as shown in table I. As a consequence of this effect all the Ci’s are dependent on the magnitude of the applied magnetic field. However, as indicated by the measured data shown in table I, the differences between the values of (1 /pa)(dp///dH)and (1 /PO)(dp,/dH) are of the order of 1O-s per oersted. The symbols p// and pL indicate the resistivity measured with H parallel to and perpendicular to J, respectively. A 10 kOe change in the value of H will produce a change, in the value of (p,_ - p,,)/po, of the order of 10-d (the values of (pI - p,,)/pa are of the order of 10-s). Hence, the error involved in neglecting this magnetization process is of the order of 10-s. The data, measured by Hiraotaa), for the anisotropy in the extraordinary Hall resistivity were insufficient to experimentally validate the phenomenological equation represented in eq. (5). Also, the present single-crystal thin-film measurement, shown in fig. 4, does not confirm the order of the expansion required for F@(G). If this factor is expanded only to the third order, the data in fig. 4 may be roughly approximated by the expression (11.9 sin 5 + 0.45 sin 35) x lo-11 where c is the angle between the internal magnetization g and the
MAGNETORESISTIVITY
the antisymmetric
AND HALL EFFECT
part of the resistivity
tensor. The calculation
eq. (16) and the results of the measurements
509
IN Ni FILMS
shown in
in the single-crystal
film
demonstrate a very convenient approach for obtaining the anisotropy coefficients related to the extraordinary Hall effects. By employing this approach, all the necessary data can be taken on one single sample, and thus the data points will not be limited by the number of different samples cut from the single-crystal rod with a certain orientation, as is the case in the previous works). Further, the error, induced by the difference in the physical contacts resulting for different sets of Hall probes, can be eliminated since only one pair of contacts is used in the present method. In spite of the technical problems involved with the single-crystal thinfilm measurements, the present study demonstrates that it is possible to deduce magnetoresistivity-anisotropy data, from the thin-film measurements, which are comparable to the bulk data. This suggests that the magnetoresistivity-anisotropy effects in ferromagnetic metals can be studied, at least qualitatively, on the single-crystal thin-film samples. Since the single-crystal films are much easier to fabricate than are the bulk crystals, and the electrical measurements are more conveniently measured for the thin-film specimens it is simpler to employ such specimens to measure the data for the anisotropy effects in ferromagnetic metals. 6. Acknowledgement. The authors wish to express their sincere thanks to Dr. T. C. Li for his helpful technical discussions and for his assistance with the experiments. REFERENCES 1) Thomas, G., Marsocci, V. A. and Lin, P. K., Physica 45 (1969) 407. 2) Doring, W., Ann. Physik 32 (1938) 259. 3) Hiroka, T., Kita, T. and Tatsumato, E., J. Phys. Sot. Japan 22 (1967) 661; Hiroka, T., J. Sci. Hiroshima Univ. Ser. A-II 32 (1968) 153. 4) Vu, D-K. and Kuritsyna, E. F., Soviet Physics-Doklady 10 (1965) 51; Vu, D-K., Bull. Acad. Sci. USSR, Phys. Ser. 29 (1965) 558; Vu, D-K., Soviet Physics-JETP 29 (1968) 407; Battarel, C. P. and Galiner, M., IEEE Trans. Magnetics 4 (1968) 3645, 5 (1969) 18. 5) Marsocci, V. A. and Chen, T. T., J. appl. Phys. 40 (1969) 336 1. 6) Birss, R. R., Symmetry and Magnetism, North-Holland Publ. Comp. (Amsterdam, 1964). 7) Mirchell, E. N., Hankaas, H. B., Bale, H. D. and Steeper, J. B., J. appl. Phys. 35 ( 1964) 2604; Kuwahara, K., Trans. Japan Inst. Metals 6 (1965) 192; Vu, D-K., Phys. Status solidi 26 (1968) 565. 8) Chen, T. T., Ph. D. Thesis, SUNY and Stony Brook (1970). 9) Hirsch, A. A., Kleefield, J. and Frielander, G., Phys. Letters 34A (1971) 253. 10) Yu, M. L. and Chang, J. T. H., J. Phys. Chem. Solids 31 (1970) 1997. 11) Vu, D-K., Fiz. Metallor Metallovedenic 23 (1967) 400. 12) Kuritsyna, E. F. and Valsilyev, Tu. V., Phys. Status solidi 31 (1969) 28 1.
59 (1972) 498-509
PLANAR
o North-Holland Publishing Co.
MAGNETORESISTIVITY
MEASUREMENTS
IN NICKEL T. T. CHEN*
Department
of Electrical
AND
PLANAR
SINGLE-CRYSTAL
HALL THIN
EFFECT FILMS t
and V. A. MARSOCCI
Sciences, State University
Stony Brook,
New
York,
of New
York,
USA
Received 28 June 1971
synopsis The planar magnetoresistivity anisotropy and the planar Hall effect in nickel single-crystal films have been explained successfully by using a phenomenological equation in a tensor form. The anisotropy constants defined for this equation are directly related to the planar measurements and can be used as parameters for making relative comparisons in the study of the magnetoresistivity anisotropy effects. It is demonstrated that these effects in ferromagnetic crystals can be studied by measurements on single-crystal films.
1. I&rod&ion. In ferromagnetic metals both the magnetoresistivity and the Hall resistivity are composed of two distinct parts: the “ordinary” effects and the “extraordinary” or “spontaneous” effects. The ordinary effects are caused by the direct interaction between the magnetic field and the kinetic behaviour of the conduction electrons; this effect is present in all conductive materials. The extraordinary effect, unique to the ferromagnetic state, is contributed by the scattering process between the s-band electrons and the d-band electrons. This effect is related to the spin-orbit interaction of the d electronsi), and is anisotropic and depends on the orientation of the magnetization of the metal. In the case of nickel single crystals, the anisotropic effects for the extraordinary part of both the magnetoresistivity and the Hall resistivity have been reported23 a). The planar Hall resistivity in ferromagnetic polycrystalline films has been extensively studied and has been found to be closely related to the magnetoresistivity anisotropy4). However, the reports of the experimental data taken for single-crystal films5) are few and the results have not been t Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF-AFOSR1116-66. t Present address: RCA Laboratories, Princeton, N. J. 498
MAGNETORESISTIVITY
properly
explained.
AND
In this paper,
HALL
EFFECT
it is proposed
499
IN Ni FILMS
that this planar
Hall
effect is actually contributed by the extraordinary part of the magnetoresistivity only. The anisotropies in both the extraordinary Hall resistivity and the magnetoresistivity, as well as in the planar Hall resistivity, can be described by a generalized phenomenological equation. The possibilities of using the single-crystal films as the media in which to study these anisotropy effects are also discussed. 2. Theory. In a saturated ferromagnetic crystal the extraordinary part of the magnetoresistivity and the Hall resistivity are dependent on the orientation of the magnetization only. These two resistivities may then be included in the resistivity tensor of the crystal. At room temperature, the intrinsic resistivity anisotropy due to the nonspherical Fermi surface structure is negligible as compared with the magnetoresistivity-anisotropy effect. Thus the resistivity tensor, p, can be written as a function of the unit vector, 01,along the direction of the magnetization M. The components of p(a) may then be expanded in a power series, of the CL~‘S, in the form P&4
= +
i
k,l,m...=l
@ii -k akfjak
~klmntpkW%%
+
-b aklijakw
+
ak f
aklmij~x~lolm
. . . ),
(1)
where i, j = 1, 2, 3 and the a’s are expansion coefficients. The above tensor can be further separated into a symmetrical and an antisymmetrical part, p$, given by p$‘(cx) =
lFI
(atj + aklipk4
+
aklrnntpk~l~rn%
+
. - -)
part ply!,
(4
k,l,m...=l
and Pii&)
(akipk
= k 1j ,,
+
akZmi3~kWm
+
. . . ).
(3)
. ..=l
In a cubic-crystal structure, the components in eqs. (2) and (3) can be greatly simplified by arguments of symmetry if the crystalline axes coincide with the axes of the coordinate system. For a fifth-order expansion in the ai’s, the resistivity tensor is given bys) p’S’(oz)
500
T. T. CHEN
0 p@)(B)
=
A@)
V.
A. MARSOCCI
--A2(4
0
A$)
-Al(&)
0
-A&q
A$)
AND
,
(5)
with A$(&) = cua[eo + elaf + ezo$ + es(c&$ + a&~ + a&f)] and i = 1, 2, 3. In the above, the et’s are the extraordinary Hall resistivity-anisotropy constants. For crystals,
such as nickel and nickel-rich
alloys, which have the easy
axes of magnetization along the (11 l> directions, the magnetoresistivity anisotropy along a current direction, b, may be calculated, for the saturated state, by the use of eq. (4). The result is A&% @)/PO z GM, B) -
pollpo= [~~(;(s)~~)- pollpo = Cl(E &Ii” - 5) + Cz(X a$$ - ;) + C3(2 a$@; - ;)
+ 2c4 x wjh% + 2cs 2 w&/%h
(6)
withi,j,k= 1,2,3andC,=C;/pawithr= 1,2,3,4,5. In eq. (6), Ap/pa represents the extraordinary magnetoresistivity with p(&, p) being the value of the resistivity when the crystal is just saturated along the di direction and pa the resistivity measured in a completely demagnetized state. In the present case pa = C’ + $6 + $(G + Cb). Eq. (6) is equivalent to Doring’s equations) except that the anisotropy constants are directly related to the resistivity-tensor elements. In a thin-film sample, when a magnetic field is rotated in the plane of the film, the magnetoresistivity measured along the current direction is defined as the planar magnetoresistivity. There will also exist a voltage produced in the plane of the film but perpendicular to the current direction ; this potential is defined as the planar Hall voltage. In the case of a film grown on the (001) plane, and with the current J at an angle 8 with respect to the < 100) axis and the field l? at an angle 4 with respect to 3, the planar magnetoresistivity anisotropy is given by Ap(& @)/PO = 9Cr + +
SC2
-
+C3
*[(Cl + Cs)s co9 28 + Ci sin2 201s
x sin 2(8 + +) + tan-l ( c1~c2)cos2~]+~Cscos4(~+~). Letting f be a unit vector in the film plane but perpendicular Hall resistivity may be given by PZY(~, &PO = (X)
(7) to B, the planar
*f/PO
= +[Ci cos2 20 + (Cl + Cs)2 sin2 2B]* x sin 2(e + 4) [
tan- r( “Lc2)
tan 281.
(8)
MAGNETORESISTIVITY
In
the
case
AM,
of
AND
HALL
(OOl)-
EFFECT
IN Ni FILMS
single-crystal
films,
501
19= 0 and
&/PO = AP($)/PO = gc, + jjc,
-
;ca + i(cl+
Cs) cos 2$ + &C2 cos 4+ 2
(9)
also pzzl(& B)/Po = pzy(+)lpO = K4sinW
(10)
In the case of (OOl)-
;C, + SC4 cos 24 -
films 8 = x/4 and the
+C2 cos 44
(11)
and pz&$)/pO
=
*(Cl +
C2)sinW
(12)
Hence the anisotropy coefficients Cr, Cs and Cd can be determined by a measurement of the Fourier components of functions Ap(4)/pa and p&$)/p0 in the (OOl)-
Here,
+
(gl+&c,
-&c3+&c4
=
&K + &(2Cs -
+&5)cos2+
C3) + $,K cos 2+.
(13)
K has the same definition as in Diiring’s papers) and is expressed as
K = @Cl + $C, -
AC3
+
C4 -
+C5).
(14)
In addition PW(#/PO = &K sin 4.
(15)
These results agree with the previous reports4). When the magnetic field is rotated in a plane perpendicular to the current direction, there will be produced a Hall-voltage component along the 9 direction. The anisotropy appearing in the extraordinary part of the Hall resistivity can be described by the antisymmetric part of the resistivity tensor as shown in eq. (5). In the case of a (OOl)-
502
T. T. CHEN
measured
AND
along the <0 10) direction
?H(<)
=
V. A. MARSOCCI
is described
by
(;‘p^)$
=
-A&q
=
(e +
gel
+ &(es -
+
ye2
+
+e3)
sin 5 -
($er + &;-es -
&es) sin 35‘
es) sin 55.
The extraordinary Hall resistivity anisotropy may then be determined the angular-dependence data taken for a single-crystal sample.
(16) from
3. Experimental results. The planar effects have been measured for three sets of nickel films. Each set consists of one polycrystalline film and two single-crystal films; the three films were fabricated in a single evaporation process to insure that they have the same thickness. The polycrystalline film was deposited directly on a glass substrate, and the two single-crystal films were epitaxially grown on a freshly cleaved and carefully polished (100) surface of a sodium-chloride crystal. These two latter films were oriented one along the (100) axis and the other along the
MAGNETORESISTIVITY
films will cause a negligible
AND HALL EFFECT
or at most
a constant
IN Ni FILMS
effect
503
on the planar
measurements at room temperature. All the previous calculations may then be applied to the present measurements. The angular dependence of Ap(#)/pa and of psY($)/pa were measured when rotating the film inside the magnetic field, on a point-by-point basis, so that the induced voltage due to the rotation of the film is eliminated. Then these data were compared with a continuous function of the form A cos 2(4 + e) + B cos 4$,
(17)
which was generated by an analog computer. By visual alignment of the graphs, the best-fit parameters can be determined. Two typical measurements for a (OOl)-
r
00
1
30°
I
60’=
90°
120”
1500
180”
,
210”
I
240”
1
270” 4
Fig. 1. The planar magnetoresistivity anisotropy and the planar Hall effect in a (OOl)-( 100) oriented nickel single-crystal film (no. 257).
Fig. 2. The planar magnetoresistivity anisotropy and the planar Hall effect in a (OOl)-(100) oriented nickel single-crystal film (no. 258).
Polycrystal
[loo]-(001)
11lOI-(001)
Polycrystal
[loo]-(001)
[l lOI-(001) Polycrystal
1492
1160
1160
1160
1058
1058
239 244
245
246
257
258
from Kaya’s
data
from Doring’s
data
Bulk results calculated
Polycrystal
plate
(OOl)-[loo]-plate
(OOl)-[lOO]-plate Polycrystal plate
[ 1lOI-(001)
1058 259 Bulk results calculated
[loo]-(001)
1492
1492
Orientation
238
(4
Thickness
magnetoresistivity
237
Film no.
Planar
l.lcos4#
1.4 cos 44
1.28 cos 44
12.96 cos 24
25.9
26.6
26.3
1.75 cos 44
13.14 cos 4+ 5.7 cos 24 +
29.0
6 cos 2$ + 2.38 cos 44
12.58
22.65
19.9
12.2
8.4
16
15
8
13.4
8 16.4
19.9
16
15
doe-?
[(~/PO) Wffl
19.7
10.8
23.5
17.3
1.92 cos 44
10.72 cos 2+ -
films
(dpzl/lpo) x IO3
9.8 cos 44
1.5 cos 44
5.90 cos 2fp -
9.32 cos 44
8.9 cos 26, -
5.88 cos 2+ +
10.36 cos 44
11.6cos2+
in nickel-thin
: Anisotropies
1.65 cos 44
VP/PO)x lo3
Planar-effects
and planar Hall resistivity
5.22 cos 24 +
anisotropy
TABLE I
12
17
15
10
16.8
17
11
17
pP-l)
x 1~
MAGNETORESISTIVITY
AND
HALL
EFFECT
Since there is no way to place the thin-film magnetized
IN
Ni FILMS
505
sample in a completely
de-
state
the value of pa is approximated by pa = &[p(# = 90”) + ~(4 = o')l. Th e error which results can be calculated by eqs. (7) and (13) ; in the case of nickel films, it is less than 1Oh. By applying eqs. (9) to (15) to the measured data for the planar magnetoresistivity anisotropy and for the planar Hall effect, the coefficients Ci, Cs, Cq and K are determined. The calculated values are listed in table II. TABLE II Calculated Film no.
anisotropy Cl
coefficients
based on the measured
c2
c3
data in nickel films
c4
c5
K
237
- 0.0006
0.0112
0.02315
0.03385
244
-0.00106
0.01072
0.0188
0.0322
257
-0.00152
0.0136
0.022
0.0375
Bulk data Kaya
-0.007
0.019
-0.070
0.029
0.014
0.0438
Dijring
- 0.0026
0.014
- 0.068
0.0266
0.020
0.0432
Although the constants Ca and Cs also appear in these equations, their values cannot be determined from these planar measurements, since the value of pa is only an approximate one. The results of the measurements of the planar Hall effect in the (OOl)
(18) (19)
If the averaged value of the CZ’S in table II are used with 19= 26”, then the results predicted for the film are
Ap($)/po= [8.66 sin 2(rj + 26”) + 3.08 cos 2(4 + 26”) + 1.38 cos 4($ + 26”)] x 10-s
(20)
506
‘I’. T. CHEN
1
I
nickel
V. A. MARSOCCI
Ap, =6.60 sin 2(~+26°)+3.84cos2(~+260~+2.04cos4(~+260$x103 PO
I
IY Fig. 3. The
AND
I
30”
planar
60°
1
I
90’
1200
magnetoresistivity
single-crystal
I
1500
I
180”
anisotropy
and
I
2100 the
film at an angle of 26” with respect
I
1
240”
270° c Hall
planar to the
effect
in a
< 100) axis.
and pZY(~)/pa = 7.98 sin 2(+ + 10.9”) x 10-s.
(21)
These data agree quite well with the measured results given in eqs. (18) and (19). The field dependence of the Hall resistivity has been measured for a (001)~< lOO)-oriented film. The magnetic field was held in the (100) plane making a different angle with respect to the (010) axis. After an appropriate correction is made, for the demagnetization field effect and for the effects
Fig. 4. The
angular
dependence
of the extraordinary crystal
film.
Hall
effect
in a nickel
single-
MAGNETORESISTIVITY
of the planar Hall resistivity, value of the effective magnetizations). Hall resistivity
AND
HALL
EFFECT
the Hall resistivity
IN Ni FILMS
may be plotted
field. This field has the same orientation
The extraordinary PH by the equation
Hall resistivity
RrM
507
versus the
as the internal is related
to the
(22) where Ra is the Hall coefficient and f(C) re p resents the parameter which is measured in a field H ; the field is in the ( 100) plane and makes an angle c with respect to the
508
An additional
T. T. CHEN
problem
AND
encountered
V. A. MARSOCCI
in the thin-film
measurements
in-
volves the changes in the susceptibility of the thin-film specimen. It has been observed that, in the polycrystalline films, the value of (1 /pe)[dp(+)/dH] does depend on the relative orientation of the magnetic field with respect to the current direction119 1s). Similar results have been found for the singlecrystal films as shown in table I. As a consequence of this effect all the Ci’s are dependent on the magnitude of the applied magnetic field. However, as indicated by the measured data shown in table I, the differences between the values of (1 /pa)(dp///dH)and (1 /PO)(dp,/dH) are of the order of 1O-s per oersted. The symbols p// and pL indicate the resistivity measured with H parallel to and perpendicular to J, respectively. A 10 kOe change in the value of H will produce a change, in the value of (p,_ - p,,)/po, of the order of 10-d (the values of (pI - p,,)/pa are of the order of 10-s). Hence, the error involved in neglecting this magnetization process is of the order of 10-s. The data, measured by Hiraotaa), for the anisotropy in the extraordinary Hall resistivity were insufficient to experimentally validate the phenomenological equation represented in eq. (5). Also, the present single-crystal thin-film measurement, shown in fig. 4, does not confirm the order of the expansion required for F@(G). If this factor is expanded only to the third order, the data in fig. 4 may be roughly approximated by the expression (11.9 sin 5 + 0.45 sin 35) x lo-11 where c is the angle between the internal magnetization g and the
MAGNETORESISTIVITY
the antisymmetric
AND HALL EFFECT
part of the resistivity
tensor. The calculation
eq. (16) and the results of the measurements
509
IN Ni FILMS
shown in
in the single-crystal
film
demonstrate a very convenient approach for obtaining the anisotropy coefficients related to the extraordinary Hall effects. By employing this approach, all the necessary data can be taken on one single sample, and thus the data points will not be limited by the number of different samples cut from the single-crystal rod with a certain orientation, as is the case in the previous works). Further, the error, induced by the difference in the physical contacts resulting for different sets of Hall probes, can be eliminated since only one pair of contacts is used in the present method. In spite of the technical problems involved with the single-crystal thinfilm measurements, the present study demonstrates that it is possible to deduce magnetoresistivity-anisotropy data, from the thin-film measurements, which are comparable to the bulk data. This suggests that the magnetoresistivity-anisotropy effects in ferromagnetic metals can be studied, at least qualitatively, on the single-crystal thin-film samples. Since the single-crystal films are much easier to fabricate than are the bulk crystals, and the electrical measurements are more conveniently measured for the thin-film specimens it is simpler to employ such specimens to measure the data for the anisotropy effects in ferromagnetic metals. 6. Acknowledgement. The authors wish to express their sincere thanks to Dr. T. C. Li for his helpful technical discussions and for his assistance with the experiments. REFERENCES 1) Thomas, G., Marsocci, V. A. and Lin, P. K., Physica 45 (1969) 407. 2) Doring, W., Ann. Physik 32 (1938) 259. 3) Hiroka, T., Kita, T. and Tatsumato, E., J. Phys. Sot. Japan 22 (1967) 661; Hiroka, T., J. Sci. Hiroshima Univ. Ser. A-II 32 (1968) 153. 4) Vu, D-K. and Kuritsyna, E. F., Soviet Physics-Doklady 10 (1965) 51; Vu, D-K., Bull. Acad. Sci. USSR, Phys. Ser. 29 (1965) 558; Vu, D-K., Soviet Physics-JETP 29 (1968) 407; Battarel, C. P. and Galiner, M., IEEE Trans. Magnetics 4 (1968) 3645, 5 (1969) 18. 5) Marsocci, V. A. and Chen, T. T., J. appl. Phys. 40 (1969) 336 1. 6) Birss, R. R., Symmetry and Magnetism, North-Holland Publ. Comp. (Amsterdam, 1964). 7) Mirchell, E. N., Hankaas, H. B., Bale, H. D. and Steeper, J. B., J. appl. Phys. 35 ( 1964) 2604; Kuwahara, K., Trans. Japan Inst. Metals 6 (1965) 192; Vu, D-K., Phys. Status solidi 26 (1968) 565. 8) Chen, T. T., Ph. D. Thesis, SUNY and Stony Brook (1970). 9) Hirsch, A. A., Kleefield, J. and Frielander, G., Phys. Letters 34A (1971) 253. 10) Yu, M. L. and Chang, J. T. H., J. Phys. Chem. Solids 31 (1970) 1997. 11) Vu, D-K., Fiz. Metallor Metallovedenic 23 (1967) 400. 12) Kuritsyna, E. F. and Valsilyev, Tu. V., Phys. Status solidi 31 (1969) 28 1.