PHYSICAL REVIEW B 98, 220405(R) (2018)

Rapid Communications

Corbino magnetoresistance in ferromagnetic layers: Two representative

examples Ni81 Fe19 and Co83 Gd17

B. Madon and J.-E. Wegrowe*

Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France

M. Hehn, F. Montaigne, and D. Lacour

Institut Jean Lamour, UMR 7198, CNRS, Université de Lorraine, Vandoeuvre les Nancy, France

(Received 21 July 2018; revised manuscript received 26 November 2018; published 12 December 2018)

The magnetoresistance of Ni81 Fe19 and Co83 Gd17 ferromagnetic thin films is measured in Corbino disk
geometry, and compared to the magnetoresistance of the same films measured in the Hall-bar geometry. The
symmetry of the magnetoresistance profiles is drastically modified by changing the geometry of the sample, i.e.,
by changing the boundary conditions. These properties are explained in a simple model, showing that the Corbino
magnetoresistance is defined by the potentiostatic boundary conditions while the Hall-bar magnetoresistance is
defined by galvanostatic boundary conditions.

DOI: 10.1103/PhysRevB.98.220405

The Hall effect was first measured in 1879 by Hall [1] and spin-dependent potentials, and has paved the way to
by applying a magnetic field to a conducting slab contacted the realization of new electronic devices. Recently, various
to an electric generator at the extremities. Later on, Corbino developments about the spin-Hall effects (anomalous Hall
[2] found a similar effect by applying a magnetic field on a effect, spin-Hall effect, spin-pumping effect, spin-Seebeck
disk geometry with two concentric electrodes. Quickly the effects, etc. [13,14]) tend to show that the usual Hall-bar
question arose on whether the effect measured by Corbino conditions with spin relaxation could be turned into “Corbino-
(the so-called Corbino effect) in a disk and by Hall in a bar like boundary conditions,” in the sense that the electric charge
have the same origin. In 1914, Adams and Chapman measured accumulation drops to zero at the edges and a pure spin current
the Corbino effect in many different metals [3] by using an can be generated instead of a Hall voltage [11].
oscillating current flowing from the center of the disk to its In this context, the goal of this Rapid Communication
outer. Adams concluded in 1915 that “the Corbino effect is to study NiFe and GdCo ferromagnetic Corbino disks
is, essentially, the same as the Hall effect” [4]. However, and Hall bars, in order to understand the behavior of the
the question remains about the exact meaning of the adverb magnetoresistance [13,15–17] when the boundary conditions
“essentially.” are switched (by changing the geometry) from spin current
In the 1950’s, the Hall effect in the Corbino geometry was to spin-dependent voltage. The alloys Ni81 Fe19 and Co83 Gd17
studied for its practical applications. The magnetoresistance are chosen for their maximum contribution to the anisotropic
of InSb slabs was shown to depend strongly on the shape of magnetoresistance and the anomalous Hall magnetoresistance
the samples [5]. The reason is that near the current injection (that defines the anomalous Hall angle), respectively.
edge, the Hall electric field is shorted and a transverse electric First, we will present our measures of Corbino magne-
current appears which causes an increase of the resistance toresistance performed on NiFe and CoGd rings. The re-
as in the Corbino geometry [6–9]. Accordingly, one can see sults are then analyzed in the framework of the generalized
the Corbino geometry as the extreme scenario where the Hall Ohm’s law by defining the Corbino magnetotransport coeffi-
electric field is zero everywhere and a Hall current is flowing, cients C as a function of the usual Hall-bar coefficients [see
or, in other terms, one can view the Corbino disk as a Hall Eq. (12) below]. The consistency of the proposed explanation
bar in which the electrostatic charge accumulation is reduced is checked independently, by measuring the magnetotransport
to zero everywhere. The system cannot generate a voltage coefficients of the Hall bar.
between the edges so that a Hall current is flowing and the The samples studied are 20-nm-thick layers of Ni81 Fe19
Joule heating is higher than in the Hall bar for an equivalent and Co83 Gd17 sputtered on glass substrates. The magnetic
volume [10–12]. The mechanism responsible for both the layers are sandwiched between 5-nm-thick Ta buffers and
Hall effect and the Corbino effect is indeed the same, but the 3-nm-thick Pt caps. The magnetic properties of the thin layers
Corbino disk is a device that is more constrained than the Hall have been previously studied [18] (see Supplemental Material
bar, due to the change of the boundary conditions. [19]). The sample magnetization is uniform for quasistatic
At the turn of the last century, the emergence of spintronics states, although nonuniform states could take place at low
has shown the possibility of exploiting spin-polarized currents magnetic fields (this regime is, however, not considered in
the present study). The NiFe is textured with small uniaxial
anisotropy lying in the sample plane. The coercivity field
*
jean-eric.wegrowe@polytechnique.edu in the in-plane geometry is of the order of 1 mT. The out-

2469-9950/2018/98(22)/220405(5) 220405-1 ©2018 American Physical Society

MADON, WEGROWE, HEHN, MONTAIGNE, AND LACOUR PHYSICAL REVIEW B 98, 220405(R) (2018)

FIG. 2. (a) Schematic view of the Hall-bar devices. Resistance

FIG. 1. (a) Schematic view of the Corbino geometry, with the measurements vs applied field for the NiFe device in the (b) lon-
three unit vectors ur , uφ , ux confined in the plane. (b) and (c) Corbino gitudinal and (c) transverse geometries. Resistance measurements
magnetoresistances for the CoGd and NiFe samples measured as vs applied field for the CoGd device in the (d) longitudinal and
a function of the out-of-plane magnetic field. The gray lines are a (d) transverse geometries. The two insets are zooms (magnification
quadratic fitting of the data. 100×).

of-plane shape anisotropy corresponds to a field of about to ferromagnetic conductors. The resistivity tensor ρ̄¯ contains
1 T, defined by the magnetization at saturation Ms (NiFe) ≈ both anisotropic terms and anomalous Hall terms [15,17]
800 emu cm−3 . The CoGd has a comparable magnetization, (see Supplemental Material [19]). In the Hall-bar geometry,
with a similar shape anisotropy. The structure of CoGd is a current is imposed along the x axis and the device is in a
amorphous and it is not textured. Owing to a standard two-step galvanostatic mode: We then use the resistivity representation
UV lithographic process, they have been patterned in Corbino of the Ohm’s law,
rings with an inner and outer radius respectively equal to 7
E = ρ J + ρ(m
 · J)m  × J,
 + ρAH m (1)
and 2 mm. Gold contact pads are formed in the second step.
Figure 1(a) presents a sketch of the obtained devices. where m  is the magnetization normalized to unity and E is
The measured magnetoresistance of NiFe and CoGd the electric field. The first coefficient ρ on the right-hand side
Corbino rings is presented respectively in Figs. 1(b) and 1(c). is the isotropic resistivity and it accounts for isotropic trans-
The voltage is measured as a function of the current with a port. The second term is the anisotropic magnetoresistance
current swept from −1 to +1 mA. The resistance is deduced described by the the coefficient ρ/ρ, where ρ = ρ − ρ⊥
by fitting the I (V ) curve in order to eliminate thermoelec- is the difference between the resistivity measured with a
tric contributions. In both materials, an increasing magnetic current parallel to the magnetization ρ and the resistivity
field leads to parabolic magnetoresistance. The nonparabolic measured with a current perpendicular to the magnetization
point at zero magnetic field in Fig. 1(c) corresponds to the ρ⊥ (ρ/ρ is positive). The last term is the anomalous Hall
variation observed in Fig. 2(b) and is due to the small in- magnetoresistance described by the anomalous Hall resistivity
plane anisotropy for the NiFe sample [19]. Note that while ρAH (as discussed in the Supplemental Material [19], the
a negative Corbino magnetoresistance is measured for NiFe coefficient ρAH plays the same role as the Hall resistivity
devices, it is positive in the CoGd case. It is also remarkable [13,20,21]).
that the amplitude of the NiFe Corbino signal is one order of In the following, we describe the magnetization m  in
magnitude higher than the one measured in the CoGd case. Cartesian coordinates m  = mx ux + my uy + mz uz with the set
To shed light on these Corbino magnetoresistance profiles of unit vectors { ux , uy , uz } in the case of Hall bars, while it
and the drastic differences between NiFe and CoGd, it is nec- is described in cylindrical coordinates m  = mr ur + mφ uφ +
essary to have a look at the Ohm’s law E = ρ̄¯ J generalized mz uz with the set of unit vectors { ur , uφ , uz } in the case of

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CORBINO MAGNETORESISTANCE IN FERROMAGNETIC … PHYSICAL REVIEW B 98, 220405(R) (2018)

Corbino disks [see Fig. 1(a)]. In this last case, the boundary of the Hall bar are contacted together in the Corbino disk). As
conditions are such that the potential V0 is imposed in the a consequence, the Poisson equation reduces to the Laplace
inner disk while the outer disk is set at the ground. The device equation ∇ 2 V = 0, which leads to the following form of the
is then in a potentiostatic mode. The resistance is given by electric potential,
the ratio V0 /I , where the current is the integral of the radial  
current densities over the angle φ and the thickness t of the ln (r/rin )
V (r ) = V0 1 − , (5)
disk, ln (rout /rin )
 t  2π
I (r ) = Jr (r, φ)r dφ dz. (2) where rin is the inner disk radius, and rout is the outer disk
z=0 φ=0 radius. This leads to the planar component of the electric field,
In this potentiostatic mode, it is convenient to inverse Eq. (1) V0 1
in order to use the conductivity representation [8]. The Ohm’s E = −∇V
 = ur . (6)
law now reads J = σ E,  where the conductivity tensor σ = ln(rout /rin ) r
(ρ)−1 is the inverse of the resistivity tensor ρ used in Eq. (1). Furthermore, an out-of plane electric field Ez is introduced
We then have the current as a function of the electric field, in order to take into account the thickness of the sample.
J = σ E + σ (m  m
 · E)  − σAH m 
 × E, (3) Since the difference rout − rin between the radius of the inner
disk and the outer disk is much larger than the thickness t of
where the three conductivity coefficients are related to the the layer, we assume that Ez is uniform and the out-of-plane
three resistivity coefficients by the relations component of the current is zero. Consequently, the electric
ρ 1 ρ field and electric current read
σ = , σ = − 2 ,
ρ 2 + ρAH
2 ρ + ρ ρ + ρAH
2 ⎛ V0 1 ⎞ ⎛ ⎞
ln(rout /rin ) r Jr
σAH = 2
ρAH
. (4) E = ⎝ 0 ⎠ and J = ⎝Jφ ⎠ . (7)
ρ + ρAH 2
Ez 0
r,φ,z r,φ,z
As shown in the following, this simple inversion between
Eqs. (1) and (3) is responsible for the considerable change ob- Introducing relations (7) in Eq. (3), we obtain a system
served in the magnetoresistance profiles between the Corbino of three equations with the three unknowns {Jr , Jφ , Ez }. We
disk (Fig. 1) and the Hall bar (Fig. 2). set the x axis as the direction of the in-plane component of
The striking specificity of the Corbino geometry is the the magnetization. Using m2r + m2φ + m2z = 1, and the relation
absence of charge accumulation (since the two opposite edges m2φ = (1 − m2z ) sin2 φ, we obtain

σ 2 + σ σ + sin2 φ σAH
2
− σ σ 1 − m2z V0 1
Jr (r, φ) = . (8)
σ + σ mz2 ln(rout /rin ) r
Integrating according to Eq. (2) and dividing by the current gives the Corbino resistance,
V0 ln(rout /rin ) σ + σ 1 − m2z
RCor = = . (9)
I πt 2σ 2 + 2σ σ + m2z σAH
2
− σ σ
To obtain RCor as a function of the resistivity coefficients ρ, ρ, and ρAH that are usually measured, we insert Eqs. (4) into
Eq. (9) and we obtain
ln(rout /rin ) 1 + ρ/ρ 1 − m2z + ρAH
2
m2z /ρ 2
RCor (mz ) = ρ . (10)
2π t 1 + ρ/(2ρ) 1 − m2z

Considering that ρ/ρ  1 (this is the case for the ma- prefactor on the right-hand side of Eq. (11)—the Corbino
terials used in this study) the magnetoresistance defined as coefficient—which is defined by
RCor (mz )/RCor ≡ [RCor (mz ) − RCor (0)]/RCor (0) can finally
 2
be expressed as ρAH ρ
C= − , (12)
 2 ρ 2ρ
RCor (mz ) ρAH ρ
= − m2z . (11)
RCor ρ 2ρ is the difference between the square of the anomalous Hall
magnetoresistance and half the anisotropic magnetoresis-
Accordingly, for ferromagnetic devices patterned in the tance. The positive parabolic profile then corresponds to a
Corbino geometry we expect a Corbino magnetoresistance dominance of the anomalous Hall magnetoresistance, while
proportional to m2z . This is qualitatively in agreement with the negative parabolic profile corresponds to a dominance of
the trend measured in Figs. 1(b) and 1(c). Quantitatively, the the anisotropic magnetoresistance.

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MADON, WEGROWE, HEHN, MONTAIGNE, AND LACOUR PHYSICAL REVIEW B 98, 220405(R) (2018)

In order to verify the validity of Eq. (11), it suffices to TABLE I. Comparison between the measured Corbino magne-
measure independently the terms (ρAH /ρ)2 and ρ/2ρ on toresistance and the Corbino coefficients C defined in Eq. (11) from
the Hall bar. The results measured on the Hall-bar devices the Hall-bar measurements.
patterned from the previous NiFe and GdCo multilayers are
reported in Fig. 2. As expected [16], it can be seen that the Coefficients NiFe CoGd
magnetoresistance is dominated by the anisotropic magne- ρ/ρ [from Figs. 2(b) and 2(d)] 1.3 × 10−2 8.5 × 10−5
toresistance RAMR /RAMR for the NiFe sample in Fig. 2(b) RAH [from Figs. 2(c) and 2(e)] 4 × 10−3  8 × 10−1 
(transition metal ferromagnet), while it is dominated by the (ρAH /ρ )2 5 × 10−7 2.35 × 10−4
anomalous Hall effect (ρAH /ρ)2 for the GdCo rare earth– Corbino coeficient C −6.5 × 10−3 1.5 × 10−4
transition metal alloys in Fig. 2(e). Figure 2(c) shows that [calculated from Eq. (11)]
the anomalous Hall effect is negligible in NiFe and Fig. 2(d) Corbino magnetoresistance −6.7 × 10−3 3.5 × 10−4
shows that the anisotropic Hall effect is negligible in GdCo. (measured from Fig. 1)
The insets in both figures are zooms with a magnification of
100×. Note that a small contribution of the AMR can also be
present in the inset of Fig. 2(c) due to a slight misalignment
of the electrodes (see Supplemental Material for a detailed
study [19]). contacts of Pt (higher ρAH measured in the Hall bar). Indeed,
More qualitatively, we deduce from Eq. (1) that, with ho- this problem is inherent in Hall-cross devices [8], and is one
mogeneous magnetization and current density, the anisotropic of the main motivations for the use of Corbino devices. In
magnetoresistance (AMR), which is by definition measured the last case, the two electrodes follow the radial geometry by
longitudinally to the current (along x), reads imposing the radial voltage, in contrast to the former case, for
L which the electrodes break to translational invariance along
RAMR = ρ + ρm2x , (13) the longitudinal axis. We observe nevertheless a good order of
wt magnitude in the application of Eq. (12).
where t is the thickness, L is the length, and w is the width In conclusion, the magnetoresistance of Corbino disks of
of the Hall bar. Close to the magnetic saturation (at Bz = 1 T) NiFe and GdCo has been measured as a function of mag-
the magnetization is almost aligned with the magnetic field, netization direction. This observed Corbino magnetoresis-
i.e., mx ≈ 0. At zero field the magnetization lies in the plane tance is interpreted in the framework of a phenomenological
of the sample, aligned with the current direction (mx = 1). model, that allows the Corbino parameters to be expressed
Consequently, the AMR amplitude RAMR /RAMR at the sat- as a function of the usual Hall-bar parameters, which are
uration field is equal to ρ/ρ. From Figs. 2(b) and 2(d), the isotropic resistivity ρ, the anisotropic resistivity ρ/ρ,
we can deduce the values of the parameters ρ NiFe /ρ NiFe ≈ and the anomalous Hall resistivity ρAH . The typical negative
1.3 × 10−2 and ρ CoGd /ρ CoGd ≈ 8.5 × 10−5 . (NiFe) or positive (GdCo) quadratic Corbino magnetoresis-
On the other hand, RAH ≡ Ey l/I is the anomalous Hall tance observed with respect to the external magnetic field is
resistance defined by the voltage measured transversely explained by the competition between anisotropic magnetore-
(along y) divided by the current I = Jx wt injected along x. sistance terms (negative contribution) and the anomalous Hall
The expression of Ey is deduced from Eq. (1), magnetoresistance (positive contribution).
  This Rapid Communication shows that the magnetoresis-
1 ρ ρAH tance profiles between the Hall bar and the Corbino disks are
RAH = mx my + mz . (14)
t ρ ρ due to the absence of Hall current in the former geometry, and
the presence of the orthoradial spin-dependent Hall current
The right-hand side of Eq. (14) defines the planar Hall mag- in the latter geometry. This difference changed drastically the
netoresistance (first term) and the anomalous Hall magne- symmetry of the magnetoresistance, from negative to positive
toresistance (last term). At saturation, the magnetization is magnetoresistance observed between Figs. 1(c) and 2(b) for
aligned along the z direction, and we can rewrite Eq. (14) NiFe, and from even to odd magnetoresistance observed be-
as RRAH ≈ ρρAH Lω , or ρρAH = RAH
ρ
t
. Since the resistivity ρ of the tween Figs. 1(b) and 2(e) for GdCo. Similar effects are also
CoGd alloy is of the order of 120 μ cm and t = 23 nm, the expected in the context of the spin-Hall effect (e.g., using
measurement of the Hall bar then gives ρAH /ρ ≈ 1.6 × 10−2 . nonferromagnetic layers) while playing with the electrical
The results are summarized in Table I. properties of the interfaces, with potential applications in
The comparison is excellent in the case of the NiFe spin-to-charge conversion devices.
(−6.5 × 10−3 instead of −6.7 × 10−3 ), with a largely dom-
inant contribution of the anisotropic magnetoresistance (the Financial funding RTRA Triangle de la physique Projects
anomalous Hall contribution is negligible). On the other hand, DEFIT No. 2009-075T and DECELER No. 2011-085T, the
the value of the anomalous Hall coefficient ρAH /ρ of GdCo is DGA project “Spin” CNV 2146, the FEDER, La région
underestimated in the Hall-bar geometry (1.5 × 10−4 instead Lorraine, Le grand Nancy, ICEEL, and the ANR-12-ASTR-
of 3.5 × 10−4 ). This discrepancy can be understood by the 0023 Trinidad are greatly acknowledged. This work was also
3-nm Pt cap layer (lower ρ measured in the Hall bar) and by supported by the French PIA project Lorraine Université d’
the strong perturbation of the current lines due to the lateral Excellence ANR-15-IDEX-04-LUE.

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CORBINO MAGNETORESISTANCE IN FERROMAGNETIC … PHYSICAL REVIEW B 98, 220405(R) (2018)

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