extraordinary hall effect in magnetic films

Journal of Magnetism and Magnetic Materials 242–245 (2002) 90–97

Extraordinary Hall effect in magnetic films

A. Gerbera,*, A. Milnera, M. Karpovskya, B. Lemkeb, H.-U. Habermeierb,J. Tuaillon-Combesc, M. N!egrierc, O. Boisronc, P. M!elinonc, A. Perezc

aRaymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University,

Ramat Aviv 69978, Tel Aviv, IsraelbMax-Planck-Institut f .ur Festk .orperforschung, Heisenbergstrabe, 1 D-70569 Stuttgart, Germany

cD!epartement de Physique des Mat!eriaux, Universit!e Claude Bernard Lyon 1, F-69622 Villeurbanne, France

Abstract

We review some of the recent developments in the studies of the extraordinary Hall effect in magnetic films. Three

major topics are discussed: (1) physics and characterization of the effect in heterogeneous and geometrically confined

magnetic systems, (2) use of the effect as a tool for the study of ultra-thin films and nanoscopic magnetic

objects, and (3) potential technical applications for magnetic sensors and memory devices.r 2002 Elsevier Science B.V.

All rights reserved.

Keywords: Extraordinary Hall effect; Magnetic films; Heterogeneous ferromagnets; Magnetic nanoparticles

1. Introduction

One needs solid arguments to promote a comeback of

a subject known and studied for decades. This paper is

an effort to provide for such a phenomenon known as

the anomalous or extraordinary Hall effect. There are at

least three major reasons for the revision: (1) the effect

has not been systematically studied in heterogeneous

and geometrically confined magnetic systems; (2) it can

be used as a simple and effective tool for the study of

thin films and nanoscopic magnetic objects, and (3) use

of the effect can develop into a new generation of

magnetic sensors and memory devices. Each of these

subjects is good enough to study.

The Hall effect in magnetic materials is commonly

described by the phenomenological equation [1]

rH ¼ R0B þ Rem0M; ð1Þ

where rH is the Hall resistivity, B the magnetic

induction, and M the magnetization. R0 is the ordinary

Hall coefficient and is related to the Lorentz force acting

on moving charge carriers. Re the extraordinary Hall

coefficient is associated with a break of the right–left

symmetry during the spin–orbit scattering in magnetic

materials and can be much larger than R0: In these cases,the Hall voltage can serve as a direct measurement of

magnetization. Typical magnetic-field dependence of the

Hall resistance RH (rH RHt; where t is the thickness of

the film) of 130 nm thick Ni films is plotted in Fig. 1,

together with its magnetization measured by SQUID.

The scales are normalized for the clarity of presentation.

Both curves are practically identical which illustrates an

intimate link between the extraordinary Hall effect and

magnetization.

In bulk magnets it has been established, both

experimentally and theoretically, that there is a direct

correlation between the extraordinary Hall coefficient

and longitudinal resistivity in the form Reprn; where n

depends on the predominant scattering mechanism

involved: n ¼ 1 for the skew scattering and n ¼ 2 forthe side jump [2,3]. Superposition of two contributions is

usually presented as Re ¼ arþ br2; where a and b are

coefficients corresponding to the skew scattering and

side jump, respectively. The skew scattering is assumed

to be dominant in low-resistivity systems and the only

bulk materials were n ¼ 1 has been observed are low-resistivity dilute alloys at low temperatures [4]. The rest

*Corresponding author.

E-mail address: [email protected] (A. Gerber).

0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 2 0 7 - 0

of the previously studied homogeneous ferromagnets

with relatively high resistivity demonstrated nD2 andthe side jump mechanism was claimed to be dominant. It

is important to mention that in bulk ferromagnets the

resistivity was varied either by temperature or by modest

doping, low enough to avoid significant changes in the

band structure of the material.

Remarkably different results have been reported

recently in heterogeneous ferromagnetic systems, mul-

ti-layers and granular mixtures. To mention only some

of them: n ¼ 2:6 was found for electron-beam evapo-

rated Fe/Cr multilayers [5] and n as high as 3.7 was

reported for heterogeneous giant magnetoresistance

films of Co–Ag [6]. Parameters additional to those

present in the bulk seem to affect the phenomenon in

spatially inhomogeneous systems. Roughness of the

interfaces, for example, has been found [7] to modify the

relationship between the extraordinary Hall coefficient

and longitudinal resistivity of Fe/Cr multilayers. The

extraordinary Hall coefficient appears to be field

dependent in several systems [5,8]. Its polarity in

granular Co–Ag mixtures has been found [9] to change

from negative in thick (2000 (A) to positive in thin

(100 (A) films. The latter was interpreted as an evidence

for a competition between the bulk and surface

scattering contributions.

The theory of the effect in systems affected by

geometrical constrains is far from being complete. Since

the first attempts [10] to adapt the bulk model [11] to the

surface scattering in ferromagnetic films, more works

have been recently devoted to the heterogeneous

systems. Controversy remains regarding the question

as to which of the two mechanisms, side jump [12] or

skew scattering [13–15], is dominant in heterogeneous

systems. Common to the two approaches is the

conclusion that the traditionally used scaling relation

between the extraordinary Hall resistivity and long-

itudinal resistivity is probably not valid. Moreover,

unusual scaling power values, including Repr3:8; havebeen calculated for the skew scattering involving not one

but few grains [15].

Description of the extraordinary Hall effect in terms

of the total resistivity might be among the reasons for

the existing controversy. The total resistivity is a

combination of several scattering processes; contribu-

tion of each of them to the extraordinary Hall effect

must not be the same. Our first task will be the search

for proper characteristic parameters, clarification of

different scattering components and determination of

their contributions to the effect.

2. Contribution of the surface scattering

The system of choice to establish the effect of the

surface scattering on the extraordinary Hall coefficient

are thin films with an electronic mean free path of the

order of or shorter than their thickness. Following the

original Fuchs size-effect model [16], external surfaces

impose a boundary condition on the electron-distribu-

tion function, which enhances the intrinsic, thickness-

independent bulk resistivity rb to a thickness-dependenttotal resistivity r. Assuming that the total resistivity

follows the Matthiesen rule and is a simple superposition

of several scattering contributions, the surface scattering

term rss can be extracted explicitly as rss ¼ r rb: In asimilar way, the contribution of the surface scattering to

the extraordinary Hall resistivity can be found as rHss ¼rH rHb; where rH and rHb are the extraordinary Hallresistivity of a given film and the bulk, respectively. For

the following discussion, rH is defined as the saturatedHall resistivity by extrapolating the high field linear

slope of rH ðB0Þ to zero field (see Fig. 1).Fig. 2 presents both the longitudinal r and the Hall

resistivity rH of several Ni films plotted as a function oftheir thickness. As expected, resistivity is constant in

thick films and is strongly enhanced in thin ones.

Resistivity of the 5 nm thick film is about an order of

magnitude higher than that of the 100 nm thick film.

Qualitative behavior of the Hall resistivity is similar:

constant in thick and strongly enhanced in thin films.

Standard analysis of the data in the form rH ¼arþ br2; is shown in Fig. 3, where rH=r are plotted as afunction of r. Each symbol here represents a film of a

given thickness measured at 4.2, 77 and 294K. Quite

limited information can be extracted from this plot,

which might mean that the total resistivity is not a good

characterization parameter.

Fig. 4, on the other hand, presents rHss as a functionof rss for three sets of samples: (a) Ni films deposited inhigh-vacuum conditions, (b) Ni films deposited in

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

-400

-200

0

200

400

B0, T

M, em

u/c

m3

-6

-4

-2

0

2

4

6Ni 130 nm

T = 77 K

RH

, m

Ω

Fig. 1. Hall resistance (solid circles, right-hand axis) and

magnetization (open circles, left-hand axis) of 130 nm thick

Ni film measured at 77K with magnetic field B0 perpendicular

to the film plane.

A. Gerber et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 90–97 91

medium-vacuum conditions (vacuum of the order of

105–106 Torr), and (c) co-deposited Ni–SiO2 granular

samples of about 200 nm thick. Longitudinal resistivity

of the series (b) is about three times higher than that of

(a); room temperature resistivity of the 100 nm-thick

film of this series is about 35 mO cm. Enhancement ofresistivity of Ni–SiO2 samples with increasing SiO2content is mainly related to the scattering on interfaces

of the insulating inclusions. Concentration of SiO2 in

these samples is lower than 25% and the films are far

from the metal–insulator percolation threshold. The

interface contribution to the extraordinary Hall effect

and longitudinal resistivity can be extracted following

the same procedure as for chemically uniform thin Ni

films. The data are summarized for all three systems:

high-quality thin Ni films, ‘‘low’’-quality Ni films and

Ni–SiO2 mixtures at three temperatures: 4.2, 77 and

294K. All the results collapse on temperature-indepen-

dent straight lines, single for each series. The slopes are

different but the behavior is general.

Magnetization of Ni films thicker than 5 nm can be

considered as thickness-independent in the saturated

state [17]. The surface scattering part of the extraordin-

ary Hall coefficient Ress is, therefore, proportional to the

experimentally measured saturated Hall resistivity rHss;and depends linearly on the surface scattering resistivity

rss:Ressprss:In the same content, one should mention recent

studies of the extraordinary Hall effect in ferromagnet–

insulator mixtures in the vicinity of the percolation

threshold [18–20], where surface scattering should be

important. An enhancement of the extraordinary Hall

resistivity by three to four orders of magnitude above

the bulk Ni value has been found in Ni–SiO2 [18], the

ratio claimed to be much larger than that predicted by

the percolation theory. The effect has been also observed

in the tunneling conduction regime below the percola-

tion threshold in Fe–SiO2 [19] with a low scaling

exponent rHpr0:5:

3. Granular ferromagnets

At least two puzzling properties of the extraordinary

Hall effect have been discovered so far in granular

metallic ferromagnets demonstrating the giant magne-

toresistance: (a) anomalously high scaling exponent

0 20 40 60 80 1000

10

20

30

40

thickness, nm

ρ , µ

Ω cm

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ρH , µΩ

cm

Fig. 2. Resistivity and the saturated extraordinary Hall resis-

tivity of Ni films as a function of their thickness. T ¼ 294K.

0 5 10 15 20 25 30 35 40 450.000

0.005

0.010

0.015

5 nm 6 7 8 10 50 100

ρ H/ρ

ρ, µΩ cm

Fig. 3. The saturated extraordinary Hall resistivity of Ni

samples as a function of the total resistivity plotted as rH=rversus r: Three data points per sample correspond to threetemperatures 294, 77 and 4.2K.

0 25 50 75 100 125

0.0

0.1

0.2

0.3

0.4

0.5

0.6

c

ba

ρ Hss, µ

Ω cm

ρSS

, µΩ cm

Fig. 4. The surface scattering component of the extraordinary

Hall resistivity as a function of the respective resistivity term for

(a) high-quality thin Ni films, (b) ‘‘low’’-quality Ni films, and

(c) Ni–SiO2 mixtures. Group (a) is shown for three tempera-

tures: 294, 77 and 4.2K. Groups (b) and (c) are shown for 294

and 77K. Straight lines are guides for the eyes.

A. Gerber et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 90–9792

n ¼ 3:7 (rHpr3:7) in Co–Ag mixture [6], and (b) a non-monotonic field dependence of the extraordinary Hall

resistivity in granular [8] and multilayer [5] systems. In

Ref. [6], a comparison has been made among several

Co–Ag samples with the same content of Co annealed at

different conditions. Annealing affected both longitudi-

nal and Hall resistivity and n ¼ 3:7 has been obtainedfrom a direct plot of rH versus r. It is mentioned in the

paper that a typical size of Co particles in different

samples ranged from about 2 to 13 nm depending on the

annealing temperature. What has been missed is that the

total number of Co particles (N) decreases with their

diameter (D) as Np1=D3 and is of the order of 300 for

this change of sizes. It is likely that the observed

reduction of the Hall resistivity by two orders of

magnitude is simply related to a decrease in density of

scattering centers and not to the change of total

resistivity. Such correlation has been found in our

preliminary measurements of Co–Pt mixtures.

Another feature of interest is the field dependence of

Re: In most experiments with ordinary magnetic

materials, the Hall resistivity can be described quite

well by Eq. (1) with a constant field-independent

coefficient Re: This description fails in systems demon-strating the giant magnetoresistance effect (GMR) [5,8].

We show in Fig. 5 the Hall resistance of the Co–Ag

granular film with 25% volume of Co measured as a

function of magnetic field at 20, 4.2 and 1.5K. Each

isothermal curve is non-monotonic. Pronounced peaks

are developed at low temperatures at about 3 kG. Eq. (1)

cannot describe this feature using a field independent Re:We can, nevertheless offer a simple phenomenological

model that mimics the general form of rH: Let us modifyEq. (1) as

rH ¼ R0B þ ðRe;sat þ Re;GMRÞm0M;

where Re;sat is a field-independent extraordinary Hall

coefficient corresponding to a magnetically saturated

state at high applied field. Re;GMR is a coefficient

corresponding to the magnetically disordered state and

is related to an additional ‘‘GMR’’ component of

resistivity: rGMR ¼ r rsat; where rsat is the saturatedresistivity at high fields. Following the standard

presentation of the extraordinary Hall effect as a sum

of the skew scattering and side jump components, let us

assume that Re;GMR is given by

Re;GMR ¼ arGMR þ br2GMR ¼ aðr rsatÞ þ bðr rsatÞ2:

Re;GMR reduces to zero in the magnetically saturated

state and Re;sat determines the high-field Hall resistivity.

In our experiments, magnetization has been measured

up to 5T and we use the data at this field to extract Re;satas Re;sat ¼ ½rHð5 TÞ 5R0=m0Mð5 TÞ: Fit to the experi-mentally measured Hall resistivity at 4.2K is calculated

in the following form:

rH ¼R0B þrHð5 TÞ 5R0

Mð5 TÞM

þ ½aðr rsatÞ þ bðr rsatÞ2m0M ð2Þ

with a and b as fitting parameters, and is plotted inFig. 6. Agreement between two curves is notable for

a=b ¼ 2: Reasonably good fit can also be obtained witha ‘‘side jump’’ term only, i.e. with a ¼ 0: Two remarksshould be mentioned: (i) no peak in rHðBÞ can be

reproduced by this model if rGMR is replaced by thetotal resistivity r; (ii) no peak in rHðBÞ has beenobserved in granular ferromagnets Co–Pt that do not

demonstrate GMR. The model is successful in reprodu-

cing the non-monotonic character of the Hall voltage.

However, it does not provide a microscopic mechanism

of the effect and does not explain the reversal of polarity

of the peak and the zero field slope within a very short

temperature interval between 4.2 and 1.5K (Fig. 5). At

-4 -3 -2 -1 0 1 2 3 4

-5.0

-2.5

0.0

2.5

5.0

1.5 K

4.2 K

20 K

RH

, m

B0, T

Fig. 5. Hall resistance of the Co25Ag75 granular sample as a

function of applied magnetic field at 20, 4.2 and 1.5K.

0 1 2-4x10-8

-2x10-8

0

2x10-8

4x10-8

ρ H,

Ωcm

B0, T

Fig. 6. Hall resistivity of the Co30Ag70 granular sample as a

function of applied magnetic field. Open circles indicate the

measured data, solid line is a fit calculated with Eq. (2).


higher temperatures, the low field Hall signal is even

more complicated, although the magnitude of the peak

decreases. The latter is interesting by itself, since neither

magnetization nor magnetoresistance change signifi-

cantly within this temperature range. No anomaly is

found in the saturated high field Hall resistance. Rever-

sal of the peak polarity with temperature is not a general

property, it was not observed in e.g. Co30Ag70 sample.

4. Study of thin films and nanoscopic magnetic object

Correlation between the extraordinary Hall effect and

magnetization enables the use of the Hall voltage

measurement for the study of magnetic properties of

materials in conditions hardly accessible by other

techniques. Already at 50s, the effect has been used to

detect the details of the magnetization processes [21] and

magnetism of very thin films [22]. Recently, the

technique has been used in a number of occasions: to

map magnetization reversal and the perpendicular

magnetic anisotropy in e.g. Co–Cr and Co–Cr–Ta thin

films under the canted magnetic fields [23], to probe the

influence of the surface-induced anisotropy on the

impurity spin magnetization in thin Au–Fe films [24]

and to detect an onset of ferromagnetism in a thin-film

semiconducting alloy under applied electric field [25].

Apart from continuous magnetic films, the Hall

technique is also able to detect magnetic properties of

individual nano-particles and their arrays [26]. It was

mentioned earlier that the extraordinary Hall effect has

been observed in granular ferromagnets with a content

of magnetic component below the percolation threshold.

This observation is crucial, since it indicates the

sensitivity of the macroscopic Hall voltage to the local

magnetic moments of individual nano-particles, in

contrast with the macroscopic magnetization of bulk

or thin films.

The technique has been tested using a series of Co

nano-particle arrays embedded in a thin Pt matrix.

Crystalline Co clusters in FCC-phase with a narrow

distribution of diameters about 3 nm were produced by

the low-energy clusters beam deposition (LECBD)

technique [27,28]. Under-layer and over-layer Pt films

of 5 and 15 nm, respectively, were deposited from an

electron-gun evaporator mounted in the same deposition

chamber. The mean thickness of Co clusters varied

between 0.01 and 1.1 nm. An average distance between

centers of neighbor spherical Co clusters with diameter d

can be estimated as about d3=2=ð2tÞ1=2; where t is the

mean thickness of the deposited magnetic material. For

d ¼ 3 nm and t ¼ 0:01 nm, the array is strongly dilutedwith an average intergranular distance of about 37 nm,

more than an order of magnitude larger than the size of

each grain. The Hall voltage measured in this sample at

different temperatures is shown in Fig. 7. Scattering by

magnetic grains embedded in a conducting matrix

generates the extraordinary Hall voltage proportional

to magnetic moment of individual magnetic clusters

averaged over the volume of the sample. In samples

prepared by the LECBD technique, the size distribution

of clusters is narrow and the signal is proportional to a

mean magnetization of a single Co particle. At 77K, the

system is superparamagnetic and the signal is reversible.

Hysteresis develops below a well-defined temperature,

which can be identified as the blocking temperature Tbof Co clusters. Tb can be found by extrapolating the

coercive field or the width of the hysteresis to zero. More

precisely, it can be found by monitoring the remnant

Hall signal as a function of temperature, as plotted in

Fig. 8, for a sample with a mean Co thickness of 0.4 nm

(intergranular distance of about 6 nm). The sample has

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0 Pt(11nm)/Co(0.01nm)/Pt(5nm)

4K

10K

77K

RH

, a.

u.

B0,T

Fig. 7. Hall resistance of the Pt/Co/Pt sample with 0.01 nm

mean thickness of Co clusters as a function of applied magnetic

field measured at 4.2, 10 and 77K. Arrows indicate the

direction of the field sweep.

10 20 30 40

0

5

10

15

20

RH

(0),

mΩ

T, K

Fig. 8. Remnant Hall resistance of the Pt/Co/Pt sample with

0.4 nm mean thickness of Co clusters as a function of

temperature.


been magnetized at 4.2K and slowly heated at zero

external field. The remnant Hall signal decreases

monotonically and approaches zero at about 25K,

which indicates the blocking temperature of Co clusters

in this sample. Determination of Tb in this experiment is

equivalent to the zero-field versus field-cooled magneti-

zation measurement. Magnetic anisotropy energy den-

sity C of a nano-particle can be calculated using e.g.

CVE25kBTb; where V is the particle volume, and kB is

the Boltzmann constant. For Tb ¼ 30K, we calculateCE7 106 erg/cm3 which is close to the value found in3D films [29]. The absolute value of the particle’s

magnetic moment can be found from the fit of the

measured Hall signal by the Langevin function at

temperatures above Tb; as shown in Fig. 9. Perfect fitof the 0.01 nm thick Co sample is found for

M ¼ 4:6 1017 erg/G, which corresponds to about

3 103 atoms of Co with a bulk value of atomic

magnetic moment (1.7 mB).The very use of the Hall resistivity for the study of

magnetic properties is attractive. Standard magneto-

metric techniques, like VSM, SQUID, Hall probe and

optical, detect magnetization of a sample by an external

measurement apparatus. In the present approach,

magnetization is revealed by an intrinsic transport

property of the material itself. The technique is

remarkable due to its simplicity; it is unlimited by field

and temperature constrains and is well adapted for the

study of ultrathin magnetic films and arrays of diluted

nano-particles.

5. Technical applications

So far, the extraordinary Hall effect has been almost

entirely ignored as a possible tool for technical applica-

tions in field sensors and memory devices. Time is

probably ripe to change this attitude. The basic

requirements for competitive magnetic field sensors

are: (a) high sensitivity; (b) linear field response; and

(3) absence of hysteresis in field dependence. Fig. 10

presents the Hall resistance of a 10 nm thick nickel film

(out of series (b), Fig. 3) as a function of an applied

magnetic field. One can mention at least three advan-

tages of this film as compared with bulk material and

thick films (see Fig. 1): (a) improved linearity of the Hall

signal versus magnetic field; (b) improved

sensitivityFdRH=dBE10O/T versus 2 102O/T in

130 nm thick film, and (c) hysteresis width less than

5G at room temperature. The magnitude of the

extraordinary Hall coefficient and, therefore, sensitivity

to magnetic field can be enhanced in a number of ways.

One of them is by decreasing the film thickness and

enhancing the surface scattering. A typical enhancement

of the field sensitivity in the low thickness limit of

chemically pure Ni films (series (a) Fig. 3) is shown in

Fig. 11. The data presented here are limited for uniform

continuous films and the thinnest sample shown is 7 nm

thick. Since the surface scattering resistivity diverges in

the zero-thickness limit, the sensitivity can be dramati-

cally increased in ultra-thin continuous films.

Combination of the Hall effect measurement techni-

que and use of films with perpendicular magnetic

anisotropy can be applied for memory devices [30].

The Hall resistance measured in a 6 nm thick Ni film is

plotted in Fig. 12. The signal is sharp with an almost

rectangular hysteresis. The difference between the up

and down magnetized states is about 2O, the width ofthe hysteresis is 300G and the magnetization reverse

slope is about 50O/T.Field sensitivity of the order of 10O/T in the simplest

Ni films justifies more work to be done. Materials with

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Mfit=4.6×10-20Am2

Pt(11nm)/Co(0.01nm)/Pt(5nm)

T=77 K

RH

, a.

u.

B0, T

Fig. 9. Measured Hall resistance of the Pt/Co/Pt sample with

0.01 nm mean thickness of Co clusters as a function of the

applied magnetic field at 77K (symbols). Solid line is the

Langevin function calculated with Mfit ¼ 4:6 1020 Am2.

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

-0.50

-0.25

0.00

0.25

0.50 Ni 10nm

T=290 K

dRH 0/dB 10Ω/T

RH

, Ω

B0, T

∼∼

Fig. 10. Hall resistance of the 10 nm thick Ni film (out of series

(b), Fig. 4) as a function of applied field at 290K. Field

sensitivity is about 10O/T.


high extraordinary Hall response and low saturation

fields should be tried for the field sensing applications.

Films with perpendicular magnetic anisotropy should be

explored as the potential memory units.

6. Spin Hall effect

Finally, one should mention a particularly interesting

phenomenon closely related to the one discussed ear-

lierFspin Hall effect [31,32]. Asymmetric scattering of

spin-up versus spin-down electrons due to the spin–orbit

interaction, which is responsible for the anomalous Hall

effect, operates as a spin separation pump. This spin

Hall effect is predicted also in the cases with zero net

magnetization, e.g. in paramagnetic metals or doped

semiconductors. A variety of applications can be

anticipated. Use of the effect as an experimental tool

to measure spin current [33] is only one of them.

Acknowledgements

This research has been supported in part by AFIRST,

Franco–Israeli research program on nanotechnology,

grant No. 9841.

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0 20 40 60 80 100

0

1

2

3

4

5

Ni

T=290 K

dR

H/d

B0, Ω

/T

thickness, nm

Fig. 11. Field sensitivity of Ni films (series (a) Fig. 4) as a

function of their thickness.

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

-1.0

-0.5

0.0

0.5

1.0

Ni, 6 nm, 4.2 K

RH

, Ω

B0, T

Fig. 12. Hall resistance of the 6 nm thick Ni film as a function

of applied field at 4.2KFprototype of the memory unit.


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extraordinary hall effect in magnetic films

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