negative resistance contribution of a domain-wall structure in a constricted geometry

Negative resistance contribution of a domain-wall structurein a constricted geometry

S. J. C. H. Theeuwen,a) J. Caro, K. I. Schreurs, R. P. van Gorkom, K. P. Wellock,N. N. Gribov, and S. RadelaarDepartment of Applied Physics and DIMES, Delft University of Technology, Lorentzweg 1,2628 CJ Delft, The Netherlands

R. M. Jungblut, W. Oepts, and R. CoehoornPhilips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands

V. I. KozubA.F. Ioffe Physico-technical Institute. St. Petersburg 194021, Russia

~Received 12 July 2000; accepted for publication 3 January 2001!

We study the magnetoresistance ~MR! of Py/Py, Co/Py, Co/Co, Ni/Ni, and Co/Cu point contacts

(Py5permalloy5Ni80Fe20). These devices are narrow constrictions or channels ~diameter, length'30 nm! between two thin film electrodes. Due to the small size of the constriction, which is

comparable to a bulk domain-wall ~DW! thickness, a DW can be caught in it. For almost all material

combinations studied we find that low resistance contacts show an MR minimum at zero field

(H50) of magnitude 0.4%–1.3%, for temperatures between 1.5 and 293 K. The minimum occurs

for all field orientations with respect to the channel axis. When the contact resistance increases

beyond the value set by a diameter-to-length ratio for the channel of about unity, the resistance

minima at H50 evolve into a maximum/minimum combination as expected for a predominant

anisotropic magnetoresistance ~AMR! effect. We use micromagnetic calculations based on

magnetostatic and exchange interactions to obtain the magnetization in the constriction. These

calculations predict that, due to the finite channel length, there are two partial DWs at either side of

the channel. For high resistance contacts this agrees with the observed AMR, which results from

scattering in the homogeneously magnetized material in the channel. The MR minimum for low

resistance contacts arises from the DWs, which cause a resistance decrease. We attribute this

decrease to a change of spin-dependent diffuse scattering at the constriction boundary due to the

DWs. © 2001 American Institute of Physics. @DOI: 10.1063/1.1351547#

I. INTRODUCTION

Presently, there is a strong interest in the contribution of

domain walls ~DWs! to the resistance of ferromagnetic ~FM!structures operating in the diffusive transport regime. A DW

is the nanoscale transition region between two domains

within which the magnetization changes direction. The resis-

tivity of a DW differs from that of uniformly magnetized

material, i.e., the domains. This is of fundamental interest,

but also of importance for practical magnetoresistive FM

micro- and nanostructures. Measurements of Co and Ni films

with multiple DWs indicate a resistance increase due to the

DWs in the range 0.5%–5%,1,2 but for Fe wires a resistance

decrease of 1% is reported.3 Typically, in these experiments

an external magnetic field is swept from a high value, where

the film is in a single domain state, through zero, where DWs

are present. Thus, the DW resistance can be extracted from

the magnetoresistance ~MR!. This DW MR ~DWMR! has amaximum ~resistance increase due to walls! or a minimum

~resistance decrease due to walls! around zero field. How-

ever, a complication of the analysis is that the DWMR is

always accompanied by the anisotropic magnetoresistance

~AMR!,4 which arises from the spin-orbit interaction, and for

high purity films by the ordinary magnetoresistance ~OMR!,5

as recently discussed for Fe and Co films.3,6

The various theories of the DW resistance do not un-

equivocally predict the sign of the effect. In a semiclassical

two-band model a resistance increase is calculated, which

depends on the spin-dependent scattering lifetimes.7,8 How-

ever, at low temperature DW-specific electron scattering can

also lead to a resistance decrease compared to the single-

domain state due to destruction of weak localization.9 A cal-

culation by one of us10 shows that the sign of the DW resis-

tance can depend on the scattering properties of impurities in

the material.

Recently, we presented a new approach of the DW prob-

lem by measuring the resistance contribution of a single DW

in Py point contacts (Py5Permalloy5Ni80Fe20).11 These

contacts are constrictions with a diameter of a few tens of

nanometers between a pair of three-dimensional electrodes

@Fig. 1~a!#. The constriction length ~'30 nm! is comparable

to the DW thickness, so that one expects that a DW is caught

in the constriction for nonparallel magnetized electrodes.

Therefore, these contacts, which are so small that their resis-

tance is dominated by the constriction, are expected to show

a DWMR extremum between the coercive fields of the elec-

a!Author to whom correspondence should be addressed; electronic mail:

[email protected]

JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 8 15 APRIL 2001

44420021-8979/2001/89(8)/4442/12/$18.00 © 2001 American Institute of Physics

Downloaded 12 Nov 2003 to 145.94.80.165. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

trodes. As exemplified in Fig. 1~b! for an 8.0 V Py/Py con-

tact at T54.2 K, the MR of the Py contacts almost system-

atically shows a weak minimum ~'1%! for different anglesu between the magnetic field and the constriction axis. How-

ever, the minimum occurs close to H50, instead of between

the coercive fields. Further, the minimum persists up to room

temperature, where it is virtually unaffected. The resistance

level at 150 mT of the Py/Py contacts follows the cos2 ubehavior of the AMR, while its correlation with a current

direction along the constriction axis indicates that this AMR

comes from the constriction.

One may be inclined to attribute the observed minima to

a general positive MR effect, such as weak localization ~WL!

or the OMR. WL, a quantum interference correction to the

classical resistance, leads to a resistance minimum for metals

with strong spin-orbit scattering ~antilocalization!.12 Electron

paths contributing to WL approximately span the constric-

tion ~path size'2a'30 nm!. This implies a field scale of

magnitude m0Hc'(h/e)/(2a)2'5 T, much larger than 50

mT @Fig. 1~b!#. This excludes WL as the origin of the

minima. The persistence of the minima up to room tempera-

ture, where WL is usually completely suppressed, is another

counterargument, which simultaneously indicates that the

DW-induced destruction of WL of Ref. 9 cannot play a role

either. The OMR is the resistance increase with field due to

bending of the electron trajectories by the Lorentz force.5 In

the field range of Fig. 1~b! ~'150 mT! the OMR is not

expected to saturate, contrary to our finding, while its esti-

mated magnitude at 1.1 T ~1.1 T is the internal field;

50mT11.1 T'1.1 T! is Dr/r'(vct)2'431026, much

smaller than observed. Here vc5eB/m is the cyclotron fre-

quency and t5l/vF'10214 s is the scatter time of the elec-

trons. l is estimated from the rl product, which for many

metals is '1 fVm2. So, OMR is excluded as well.

From this exclusion it follows that the only remaining

effects are the DWMR and the AMR, both intrinsic to mag-

netic materials. The field scale of the minima, which is close

to the coercive field of the films, and their magnitude indeed

point to DWMR and AMR.

To find the magnetization structure of the Py/Py contacts

and to estimate their AMR and DWMR we performed mi-

cromagnetic calculations.11,13 The magnetization in the chan-

nel is rather homogeneous and at either side of the channel

there is a gradual partial DW with a Neel-type spin rotation.

This is seen in the simulation result in Fig. 1~c!, which fur-

ther shows that partial DWs also occur for parallel electrode

magnetizations. This indicates that apart from the electrode

magnetizations, the constricted geometry is a major factor in

determining the magnetization structure. The AMR and

DWMR estimated for the calculated evolution of the magne-

tization with field are of comparable magnitude.11,13 The

AMR shows a resistance minimum at H50 for the field

parallel to the constriction axis and a maximum for the field

perpendicular to this axis. The DWMR shows only maxima,

which is simply due to the use of a resistivity increase due to

a DW.7 These estimated AMR and DWMR thus disagree

with the measured MR minima for all angles between the

applied field and the constriction axis. This may indicate that

the real magnetization structure differs from the one calcu-

lated, which is possible in view of limitations of the calcu-

lations ~neglect of magnetostrictive and magnetocrystalline

anisotropy energies!. However, on the other hand, the dis-

agreement may indicate that the DWs in the Py contacts

actually cause a resistance decrease.

To further investigate the magnetization structure of FM

contacts and to disentangle AMR and DWMR contributions

FIG. 1. ~a! A schematic cross section of a point contact, showing the insu-

lating membrane and the two ferromagnetic ~FM! electrodes, which are

connected by a narrow channel ~length530 nm, diameter510–50 nm!.Several current paths are shown. Most of an applied voltage drops within the

spherical region of high current density indicated by the dashed circle. ~b!MR curves of an 8.0 V Py/Py contact for different angles u between the

magnetic field H and the constriction axis at T54.2 K. The curves are offset

for clarity, retaining the angular ordering at high fields. The thickness of the

Py electrodes is 200 and 100 nm, respectively. ~c! The calculated magneti-zation structure at zero field using micromagnetic simulations for a Co/Co

contact ~see Ref. 13!. The dimensions are in nanometers.

4443J. Appl. Phys., Vol. 89, No. 8, 15 April 2001 Theeuwen et al.


to the magnetoresistance, we extended our preliminary ex-

periments with additional measurements on Py/Py contacts

and measurements on Co/Py, Co/Co, Ni/Ni, and Co/Cu con-

tacts ~A/B denotes the material combination used for the

electrodes!. Tools are material variation, which implies a dif-ferent exchange stiffness, saturation magnetization, and mag-

netocrystalline anisotropy, and contact-diameter variation.

These variations are expected to affect the magnetization

structure and the different contributions to the total resis-

tance.

This article is organized as follows. In Sec. II we de-

scribe the fabrication of the point contacts and the magnetic

properties of the electrodes. In Sec. III we show that low

resistance ~wide channel! Co/Py, Co/Co, and Co/Cu contactsexhibit MR minima similar to those of the Py/Py contacts.

The results for high resistance ~narrow channel! Py/Py, Co/Py, and Co/Co contacts are presented in Sec. IV. The MR of

these contacts tends to develop characteristics of the AMR of

the channel between the electrodes. This is accompanied by

device-specific step-like transitions in the MR. Section V is

about the MR of Ni/Ni contacts, which with increasing tem-

perature shows a continuous transition from AMR dominated

to DWMR dominated behavior. In Sec. VI we discuss the

magnetization structure. For both low and high resistance

contacts this is characterized by a homogeneous magnetiza-

tion in the channel and partial DWs at either side of the

channel. For low resistance contacts a resistance decrease

due to a DW then gives a MR minimum, while for high

resistance contacts the AMR of the channel dominates. Fur-

ther, the crossover of the MR of Ni/Ni contacts is discussed

in terms of a temperature-dependent magnetization configu-

ration. In Sec. VII we suggest that DW-specific boundary

scattering is the origin of the negative DW resistance and in

Sec. VIII we summarize the conclusions.

II. EXPERIMENT

The point contacts are made using the membrane

technique,14 which comprises the etching of a circular nano-

hole in a 30 nm thick silicon–nitride membrane and deposi-

tion of a polycrystalline FM electrode on either side of the

membrane @Fig. 1~a!#. Deposition is performed in an

ultrahigh-vacuum molecular beam epitaxy ~MBE! system,

equipped with e guns, or in a high-vacuum magnetron sput-

tering system. The electrode films are deposited succes-

sively, using in situ sample rotation. Thus, the hole is filled

with FM material, so that a constriction or narrow channel is

formed between the electrodes. Initially, we use electrodes

with different coercivities Hc ,A and Hc ,B , so that the mag-

netizations of the electrodes are antiparallel between Hc ,A

and Hc ,B . The antiparallel alignment forces a DW structure

to exist in the constriction. Different coercive fields result

from different electrode material or thickness. We study

MBE-grown Co/Py, Py/Py, and Co/Co contacts and sput-

tered Co/Co, Ni/Ni, and Co/Cu contacts, where the first-

noted material is deposited first and the second-noted mate-

rial is deposited last. Electron microscopy of point contacts

formed by evaporation15 indicates that the nanohole is closed

during deposition of the first electrode, while the channel is

filled with the metal of the second electrode during the sec-

ond deposition. The channel of MBE fabricated Co/Py con-

tacts thus consist of Py. The channel of sputtered Co/Cu

contacts, however, is approximately half filled with Co and

half filled with Cu. The MR of the contacts is measured in

the temperature range 1.5–293 K, in a flow cryostat

equipped with a superconducting magnet and in situ sample

rotation, and using an ac resistance bridge.

The electrodes of the contacts are characterized by MR

measurements at 4.2 K, the magneto-optical Kerr effect

~MOKE! at 300 K, the Bitter method, and x-ray diffraction.

In Fig. 2 we show, for different field orientations, typical MR

curves of a rectangular 300 nm thick sputtered Co electrode,

bonded in a Van der Pauw geometry. The MR is the AMR

and has the usual characteristics of a polycrystalline multi-

domain film: a minimum for the longitudinal geometry and

maxima for the perpendicular and transverse geometry. The

perpendicular curve saturates at a much higher field than the

in-plane curves. This identifies the film plane as the easy

plane and the direction perpendicular to the film as the hard

axis. Similar AMR curves for the Py and Ni electrodes lead

to the same identification, which for each electrode type was

confirmed by the MOKE data. Although a Van der Pauw

structure is not ideal for AMR measurements, we use the

curves to estimate the AMR ratio, yielding 3%, 4%, and 3%

for the Py, Co, and Ni electrodes, respectively. The AMR

ratio equals DR/R0 , where DR is the increase of the satu-

rated resistance when changing from the transverse ~or per-pendicular! geometry to the longitudinal geometry and R0

5R(H50). The electrode resistivities r4.2K , their residual

FIG. 2. MR curves of a 300 nm MBE grown Co film at T54.2 K for several

field orientations at high fields ~a! and at low fields ~b!. The AMR ratio is

about 4%.

4444 J. Appl. Phys., Vol. 89, No. 8, 15 April 2001 Theeuwen et al.


resistance ratios (RRR)5R300K /R4.2K and coercive fields, all

extracted from the MR data, are listed in Table I, along with

other characteristics of the contacts discussed in detail in this

article.

X-ray diffraction shows that Ni and Py films have the fcc

structure, while Co has a mixed hcp/fcc structure. The Ni

films are strongly $111% textured, while the Py and Co filmsare only weakly textured. The position of the Bragg peaks

for the Ni indicates a tensile stress at room temperature. The

average grain size of the Ni, determined from the width of

the Bragg peaks, is 32 nm, which is ~much! larger than the

grain size of the sputtered Co ~17 nm!, the MBE-grown Co

~15 nm!, and the Py ~9 nm!. Further, the domains made vis-ible with Bitter fluid at H50 are about 10 mm in the Py films

and 1 mm in the Co films, both much larger than the con-

striction diameter ~10–50 nm!.The electronic transport through the point contacts is dif-

fusive or quasiballistic, as deduced from the smooth

d2I/dV2(V) of the current–voltage characteristics.16 The

point-contact resistance is given by the sum of the diffusive

Maxwell resistance17 RM5r/2a ~2a5d is the constriction

diameter!, the channel resistance, and the series resistance ofthe electrodes, leading to18

R5~rA1rB!/4a1rBt/pa211.16Rsq,A11.16Rsq,B . ~1!

Here rA and rB are the resistivities of the first-deposited and

second-deposited electrode, Rsq,A and Rsq,B are the sheet re-

sistances of these electrodes, and t530 nm is the channel

length. The resistance of the fabricated contacts ranges from

about 0.4 to 55 V. Using Eq. ~1! and the resistivities in TableI, this resistance range is converted to a range of constriction

diameters, yielding 9 nm,2a,60 nm.

The MR of the contacts is independent of bias voltage in

the range 10 mV–100 mV and almost temperature indepen-

dent in the range 1.5–293 K. An exception are the Ni/Ni

contacts, which show a very interesting temperature-

dependent MR.

III. LOW-RESISTANCE CoÕPy, PyÕPy, CoÕCo,AND CoÕCu CONTACTS: MR MINIMA

We find that low resistance and high resistance contacts

have a different MR, the transition being determined by a

diameter-to-length ratio for the channel of about unity.

Therefore, we discuss the two regimes separately.

To determine whether the absence of a DW signal be-

tween Hc ,A and Hc ,B for the Py/Py contacts of Ref. 11 is due

to an ill definition of a field range for antiparallel electrode

magnetizations, involving coercivities not far enough apart,

we shift to Co/Py contacts. For these contacts the coercivities

are m0Hc ,Co515mT and m0Hc ,Py52 mT, certainly far

enough apart for nonparallel alignment to exist between

them. Since rPy /rCo53.5, the Py channel dominates the re-

sistance, so that conditions are fulfilled to measure a DWMR

between Hc ,Co and Hc ,Py .

In Fig. 3~a! we plot the MR at 4.2 K of a 5.5 V Co/Py

contact, representative of low-resistance Co/Py contacts (R

,15V), for different orientations of the applied field. The

curves resemble those in Fig. 1~b!. In particular, the Co/Py

device also shows minima around H50, with a field scale,

amplitude, and hysteresis very similar to those of the mini-

mum of Py/Py. This holds for u590°, which in view of the

easy plane of the electrodes defines the obvious geometry to

measure a DWMR, but for the other angles as well. We

conclude that a nonparallel alignment, expected between 2

and 15 mT from the coercivities of the bulk electrodes, does

not play a role. In Fig. 3~a!, in some minima, a small featureis discernible. This, however, is not a DW signal, but a re-

sidual AMR signal from Rsq,Py @see Eq. ~1!#, since the featuregrows when the four-point measurement is changed to a

TABLE I. Characteristics of the contacts shown in the figures. Numbers under the electrode materials give the

electrode thickness in nanometers. Resistivities r1 , r2 were determined with resistance strips patterned in the

electrode films. 2a , DR , RRR, and Hc ,1 ,Hc ,2 are the diameter as determined from the resistance R, the range

of the u-dependent resistance change around H50, the residual resistance ratio, and the coercivities of the films.

Figure

Electrodes

~nm!R

~V!

r1 ,r2

~mV cm!

2a

~nm!

DR

~mV! RRR

m0•Hc ,1 ,Hc ,2

~mT!

1~b! Py/Py 8.0 14 45 40–100 1.5, 1.5 7, 2

200/100

3~a!, 3~b! Co/Py 5.5 4, 14 50 20–70 2, 1.5 15, 2

185/100

4~a!, 4~b! Co/Co 1.4 4 60 0.9–3.8 2, 2 12, 12

300/300

5 Co/Cua 0.7 1.6, 0.5 48 0.2–0.5 4, 5 10,—

400/200

6~a! Co/Py 19.1 4, 14 20 40–70 2, 1.5 15, 2

185/100

6~b! Py/Py 51.5 14 12 40–80 1.5, 1.5 7, 2

200/100

6~c! Co/Coa 9.2 1.6 9 25–60 4, 4 10, 10

400/400

7 Ni/Nia 0.4 1.1 55 2–4 8, 8 5, 5

500/300

aThese electrodes have been sputter deposited, while the other electrodes are MBE deposited.



three-point measurement with only one wire bonded to the

Py electrode. In the latter case an extra series resistance of

the Py electrode is added with a larger net AMR signal and

with a position equal to the u-dependent coercivity of the

electrode. The small features close to H50 for Py/Py @Fig.1~b!# also are a residual AMR from Rsq,Py .

The magnitude of the minimum in Fig. 3~a! is in the

range 20–100 mV, corresponding to 0.4%–1.3% ~see also

Table I!. The width of the minimum is independent of u.Since characteristic fields of the electrodes ~coercivity, satu-ration field! do depend on u, the minimum must be generated

in the constriction region and thus must be determined by

magnetic characteristics of the material therein. This was al-

ready apparent from the micromagnetic calculations. Con-

trary to Py/Py contacts, the u-dependent MR at 150 mT of

Co/Py contacts is device dependent and does not follow a

cos2 u behavior. Around m0H5120mT the 45° curve of the

Co/Py contact has an extra, hysteretic feature. As for Py/Py,

the MR at 293 K is almost identical to the curves at 4.2 K,

albeit that the width and amplitude of the minimum for 293

K are slightly smaller.

The typical high-field behavior is shown in Fig. 3~b!, foru50° and for the 5.5 V Co/Py contact. On this scale the

minimum is a spike, followed by a negative MR. The elec-

trodes also show this negative MR, which is attributed to the

magnetization increase by the applied field above its sponta-

neous value.4 At 10 T the MR is still negative. The OMR is

thus small, in agreement with Dr/r'(vct)2'331024.

As a next step we measure Co/Co contacts with elec-

trodes of equal thickness, i.e., equal coercivity, for different

angles u and w, where w is the angle of the field in the

electrode plane. Curves for a 1.4 V contact are given in Fig.

4~a!. They resemble those of Figs. 1~b! and 3~a!, having a

similar relative magnitude of the minimum. However, they

have a much larger field scale, since Co is a hard and Py a

soft magnet. The similarity with Figs. 1~b! and 3~a! confirmsthe irrelevance of different electrode coercivities. As found

for Co/Py contacts, the angular behavior of Co/Co contacts

outside the minimum deviates from an AMR-type cos2 u. Forsome Co/Py and Co/Co devices we find an onset of ‘‘fre-

quency’’ doubling in the u dependence of the MR @this ten-dency is also present in Fig. 3~a!#. This is reminiscent of thefrequency doubling found by Doring for the AMR of Ni

single crystals.19 For point contacts this complicated AMR

effect may be caused by only a few grains in the constriction.

Above 5 T ~not shown! the u dependence of the MR tends to

an OMR-type sin2 u. In Fig. 4~b! we see, unexpectedly, a

clear w dependence, the minimum being very weak for w50°. Apparently, the constriction axis is not a symmetry

axis of the magnetization.

Finally, we measure Co/Cu contacts, made by sputter

deposition. This results in a Co/Cu interface positioned in-

side the channel, let us assume halfway. Thus, the MR is

expected to come from one Co region of three-dimensional

current spreading and from half a Co channel, between

FIG. 3. ~a! MR curves of a 5.5 V Co/Py contact for different angles ubetween the magnetic field and the constriction axis at T54.2 K. The curves

are offset for clarity, retaining the angular ordering at high fields, but losing

the excellent reproducibility at H50. ~b! Typical high field behavior for the

5.5 V Co/Py contact for u50°.

FIG. 4. ~a!, ~b! MR of an 1.4 V Co/Co contact for different angles u and w,respectively. w is the field orientation in the plane of the electrodes. The

curves are offset for clarity, retaining the angular ordering at high fields.

T54.2 K.



which a DW structure exists. The MR of these contacts, ex-

emplified in Fig. 5 for a 0.7 V device, shows minima, just as

for Co/Co. The magnitude of the MR is smaller ~by about afactor 2! than for Co/Co. This agrees with the smaller resis-tance contribution of FM material ~about half a FM point

contact is measured!. The field scale of the minima is the

same as for Co/Co. This indicates that the minima do not

involve the magnetic properties of two electrodes.

In summary, the low resistance FM homo- and hetero-

contacts of this section show a MR minimum around H50

for all field orientations, and a regular or more complicated

AMR in the saturation range. The minimum must arise from

a combination of the MR of a geometry-determined DW at

the constriction and AMR.

IV. HIGH-RESISTANCE CoÕPy, PyÕPy, AND CoÕCoCONTACTS: MR MAXIMUMÕMINIMUM COMBINATION

With increasing contact resistance the MR evolves from

systematic minima to another behavior. This change is ac-

companied by device-specific step-like transitions in the MR.

In Figs. 6~a! and 6~b! we demonstrate this, for a 51.5 VPy/Py contact and a 19.1 V Co/Py contact, respectively. The

minima occur at a higher field than before. More subtle is the

weak but discernible maximum in the u590° curves close to

H50 ~indication ‘‘max’’!, which occurs for several high re-sistance contacts. This maximum is a precursor of the usu-

ally observed AMR maximum in the perpendicular geom-

etry. The AMR comes from the channel, which carries an

axial current and dominates the total resistance ~Py/Py:Rch /R tot50.72; Co/Py: Rch /R tot50.70!. An AMR minimum

is then expected for u50°. This indeed is present, but it is

not clear which part is AMR and which part corresponds to

the minimum already seen for low resistances. The different

shape and position of the minimum probably indicate a sub-

stantial admixture of AMR. For the curves in Fig. 6 the udependence of the resistance level at 130 mT follows the

AMR-type cos2 u behavior.

The AMR of the channel is clearly developed for the

parallel and the perpendicular curve of the 9.2 V Co/Co con-

tact (Rch /R tot50.82), plotted in Fig. 6~c!. The curves have a

clear minimum and maximum, respectively, and step-like

transitions ~indicated by ‘‘step’’!. Steps are also present in

Figs. 6~a! and 6~c!. The nonzero longitudinal AMR indicates

that at H50 the magnetization is tilted with respect to the

constriction axis or is nonuniform. The observed minimum/

maximum combination agrees with a tilt angle of 55°. This is

similar to the magnetization of Co nanowires of a different

geometry,20 which is also tilted. The w depen-

FIG. 5. MR of a 0.7 V Co/Cu contact at T54.2 K for different angles u.The curves are offset for clarity.

FIG. 6. MR curves of a 19.1 V Co/Py contact ~a! and a 51.5 V Py/Py

contact ~b! for different angles u between the magnetic field and the con-

striction axis at T54.2 K. The curves are offset for clarity. The arrows

indicate the field-sweep direction. Many Barkhausen jumps in the resistance

can be distinguished ~indicated by ‘‘step’’!. The subtle MR maximum is

indicated by ‘‘max.’’ ~c! MR of a 9.2 V Co/Co contact at T54.2 K. The

MR behavior is explained with the AMR effect of the channel.



dence of the MR for Co/Co in Fig. 4~c! already indicated a

tilted magnetization or absence of axial symmetry. The AMR

ratio of the contact is 0.5%, smaller than for the electrodes.

The device-specific steps resemble Barkhausen jumps of the

magnetization, which arise from sudden motion of DWs

when they break away from pinning sites. We suggest a

similar origin of the steps: sudden motion or rotation of

pinned parts of the magnetization pattern in the constriction.

The steps, which are reproduced for different field sweeps,

are resolved because the constriction region is so small that it

contains only a few active pinning centers.

In summary, with increasing resistance the MR of the

high resistance FM homo- and heterocontacts of this section

evolves to the AMR of the channel. This evolution goes

along with the occurrence of Barkhausen-type steps in the

MR, as expected for probing of a small FM volume.

V. LOW AND HIGH RESISTANCE NiÕNi CONTACTS:EVOLUTION WITH TEMPERATURE FROM MRMINIMA TO MR MAXIMUMÕMINIMUM COMBINATION

Ni/Ni contacts behave differently. This is illustrated in

Fig. 7, where we show the MR of a 0.44 V Ni/Ni contact for

T54.2, 120, and 160 K. At 4.2 K the MR resembles that of

Co/Co in Fig. 6~c!, suggesting it is the AMR of the channel.

With increasing temperature the MR gradually evolves. At

120 K the u50° curve still has a minimum, but the maxi-

mum for u545° has turned into a minimum and the rela-

tively strong maximum for u590° has turned into a weak

feature. This tendency to minima continues upon further

temperature increase, yielding systematic minima for 160 K,

the highest measurement temperature for these contacts. The

same results are found when the temperature range is tra-

versed in the opposite direction. Apparently, for Ni/Ni tem-

perature variation enables continuous interpolation between

the systematic minima of low resistances and the AMR-type

behavior of high resistances. Further, for the largest diameter

~'100 nm! we no longer find AMR behavior at T54.2 K,

but we again find only minima.

The resistance level at m0H53 T follows an AMR-type

cos2 u behavior ~inset of Fig. 8!. For the device of Fig. 7 theAMR amplitude (R i2R') at m0H53 T is about constant in

the range 4.2–160 K. Further, the slope of the MR outside

the central region changes from positive at 4.2 K to negative

at 160 K, the sign reversal occurring at '120 K. Such a

reversal is usually observed for ferromagnets of high purity

and results from the competition of the OMR and the resis-

tivity decrease resulting from the magnetization increase

above its spontaneous value.4 The OMR of the Ni/Ni con-

tacts is pronounced, as the Ni has RRR58, a relatively high

value for a FM film and the highest value in Table I.

In Fig. 8 we plot the AMR amplitude at m0H53 T of the

Ni/Ni contacts as a function of contact diameter @calculatedwith Eq. ~1!#, for T54.2 K. There is a clear increase of the

amplitude with decreasing diameter. For small contacts the

resistance of the channel, with its axial current direction,

exceeds the resistance of the current-spreading regions. The

AMR of the latter regions is reduced due to averaging over

different current directions, so that small contacts, with a

dominating channel resistance, have a larger AMR. The

FIG. 7. MR curves of an 0.44 V Ni/Ni contact for different angles u be-

tween the magnetic field and the constriction axis at T54.2 K ~a!, at T

5120 K ~b!, and at T5160 K ~c!. The curves ~a! and ~b! are offset for

clarity, retaining the angular ordering at high fields. The point contact resis-

tance is 0.44 V at T54.2 K, 0.69 V at T5120 K, and 0.83 V at T

5160 K. For the temperature T5120 K the 90° curve is almost featureless.

FIG. 8. The diameter dependence of the AMR amplitude for the Ni/Ni

contacts at T54.2 K. The inset shows the cos2 u dependence of the AMR at

saturation.



AMR ratio of the Ni/Ni contacts is typically 0.5%, a factor of

6 smaller than that of the Ni electrodes. The AMR ratio of

the smallest contacts is smaller than 0.5%. We attribute this

to boundary scattering in the narrow channel.21

VI. RELATION BETWEEN THE MR AND THEMAGNETIZATION STRUCTURE IN THECONSTRICTION

We have presented the MR of FM point contacts, de-

signed to catch a DW in the constriction. These nanometer-

size devices belong to the smallest FM structures for which

MR data are known. The most striking result is that for low

resistances almost all material combinations studied show an

MR minimum around H50, irrespective of the orientation of

the applied field. This intriguing MR must arise from the

magnetization structure of the constriction region and its

evolution with field or temperature, the latter being relevant

for Ni/Ni. To make an educated guess for the magnetization

and its evolution, we use the results of our micromagnetic

calculations11,13 as a guide.

A. Two partial domain walls in low resistance PyÕPy,CoÕPy, and CoÕCo contacts

The calculated magnetization at H50 for low-resistance

contacts @Fig. 1~b!# is characterized by a homogeneous chan-nel and a partial DW at either side of the channel. The walls

disappear with increasing field, as seen in Figs. 9~a!–9~c!,where three evolution stages are depicted for Co/Co.13 For

u590° the DWs disappear as a result of rotation of the chan-

nel magnetization, whereas for u50° ~not shown! this hap-pens as a result of rotation of the electrode magnetization. If

a DW gives a resistance decrease, as suggested in Sec. I, this

leads to MR minima at H50 for different field orientations,

provided the AMR of the central region is small.

The amplitude of AMR ~the regular or the more compli-cated type! at high field in general is 2–3 times weaker thanthe minima. Thus, the AMR must indeed make a secondary

contribution to the minima. Further, for the various contacts

the largest minimum occurs for u50°. This agrees with the

fact that the AMR minimum at H50 of regular thin film

samples usually is largest in the longitudinal geometry. Ap-

parently, for u50° the AMR gives the strongest enhance-

ment the point-contact MR minimum. In this picture varia-

tion of u gradually changes enhancement into partial

suppression of the minimum for a certain range of u values.

This qualitative decomposition of the MR minima into

DWMR minima and weaker AMR minima/maxima is con-

sistent with the magnetization structure and evolution of Fig.

9 and related ones for other field orientations. Therefore, we

adopt these as being generic for low resistance contacts

~diameter-to-length ratio of the channel .1!, and the DWMR

as the origin of the minima ~to Co/Cu contacts half of the

magnetization structure applies!. Since the magnitude of theminimum depends only weakly on the contact material, in

spite of strong DW thickness variation for bulk samples, the

magnetization structure of the different types of contacts

must be mainly geometrically determined. This leaves us

with the mechanism of the resistance decrease due to a DW.

This is discussed in Sec. VII.

B. Channel with homogeneous magnetization of highresistance PyÕPy, CoÕPy, and CoÕCo contacts

The magnetization structure of high resistance contacts,

which have a narrower channel, was not calculated in Refs.

11 and 13. However, due to shape anisotropy the tendency to

show a homogeneous channel magnetization at H50 will be

stronger for narrower channels, while partial DWs in the

current spreading regions will remain. So, the structure

should be similar to that of Fig. 9. Since the channel domi-

nates the resistance, the MR is dominated by the AMR of the

channel, while the DWMR will be negligible. This agrees

with the measurements.

The measured MR, however, shows a nonzero longitu-

dinal AMR, implying a nonaxial preferential direction in the

channel. This means that the channel magnetization in Fig. 9

should be adjusted with a tilt. This tilt may also apply to low

resistance contacts. The origin of the preferential direction is

unknown. Candidates are crystalline anisotropy and a

fabrication-induced anisotropy, e.g., involving a noncircular

hole or nonideal filling of the channel. These effects are not

taken into account in the calculations, but will certainly have

an effect.

FIG. 9. ~a!–~c! Three stages of the calculated magnetization evolution for a

Co/Co contact with 2a525 nm for u590°. The length scales are in

nanometers.



C. Temperature-dependent position of domain wallsin NiÕNi contacts

The AMR of Ni/Ni contacts is temperature independent

in the range 1.5–160 K ~Sec. V!, and the DWMR is not

expected to be stronger at high temperatures. Therefore, the

temperature-dependent MR of Ni/Ni contacts must come

from a temperature-dependent magnetization. We associate

the MR minima of Ni/Ni contacts at 160 K with the magne-

tization of low resistance contacts ~Fig. 9! and the AMR

behavior at 4.2 K with the magnetization of high resistance

contacts. The magnetization at intermediate temperature is

intermediate between the extremes.

The mechanism of the temperature-dependent magneti-

zation follows from the total free energy of the system,

which is22

FT5FH1Fms1Fe1FK1Fs . ~2!

Here FH is the Zeeman energy, Fms the magnetostatic en-

ergy, Fe the exchange energy, FK the crystalline anisotropy

energy, and Fs the magnetostrictive energy. Our

calculations11,13 do not include FK and Fs , which are just

the temperature-dependent terms. Therefore, we use simple

micromagnetic considerations to find out whether a

temperature-dependent change of these energies can induce a

magnetization change. We first assume that there is no mag-

netic interaction between electrodes and channel, so that

simple arguments for separate regions apply.

In the thin disk limit22 the magnetostatic energy density

of the electrodes is Fmsel /V51/2m0M s

2 sin2 g5Kms sin2 g,

where g is the angle between the magnetization and the elec-

trode plane. This shape anisotropy of magnitude Kms

5140 kJ/m3 strongly favors an in-plane magnetization. The

aspect ratio of the channel, on the contrary, is close to unity,

so its approximately isotropic magnetostatic energy does not

clearly define a preferential direction.

The crystalline anisotropy of Ni is such that the cubic

^111& directions are the easy axes.23 The electrodes are $111%textured ~Sec. II!. Further, we assume that the growth of Ni

in the channel is seeded, so that the ^111& grain orientation ofthe first-deposited electrode is transferred to the channel.

Both in the channel and the electrodes the crystalline anisot-

ropy thus favors a magnetization along the ^111& direction,i.e., parallel to the constriction axis. This tendency increases

when cooling from 160 to 4.2 K. It can be shown that24

DFK /V5FK(4.2 K)/V2FK(160K)/V'225 kJ/m3111 kJ /

m35214 kJ/m3.

The magnetostrictive energy depends on the magneto-

strictive constant l ~we use22 l523431026 for Ni! and thestress s(T) in the Ni, which makes an angle j with the

magnetization. Because of the huge thickness ratio of the Si

substrate and the SiN membrane, the stress in the Ni elec-

trodes is due to differential thermal contraction of Ni and Si.

The thermal-contraction-induced change of the magnetostric-

tive anisotropy constant between 160 and 4.2 K then is25

DKs529 kJ/m3. So, upon cooling the magnetostrictive en-

ergy favors an out-of-plane electrode magnetization, i.e., par-

allel to the constriction axis. Due to the channel’s aspect

ratio ~'1!, the in-plane electrode stress penetrates the chan-nel without much decay. For the channel it then follows that

the temperature-dependent stress is approximately isotropic,

so that no effect on the magnetization is expected.

In the electrodes the increased magnetocrystalline and

magnetostrictive anisotropies at low temperature are over-

ruled by the stronger shape anisotropy. In the channel, on the

contrary, the increase of the magnetocrystalline anisotropy is

important, since competing anisotropies are negligible.

When the interaction between electrodes and channel is

turned on, the magnetostatic interaction effect of the elec-

trodes on the channel, which already induces a homogeneous

channel magnetization and two partial DWs in the absence of

magnetocrystalline anisotropy @see Fig. 9~a!#, is augmentedby the crystalline anisotropy of the channel. The effect of

taking into account the increase of FK in the micromagnetic

calculations ~as occurs upon cooling! will be an outward dis-placement of the DWs. This is equivalent to extending the

homogeneous channel outside the geometrical channel. More

remote DWs make a weaker contribution to the MR, so one

expects that in the range of 160–4.2 K a transition occurs

from MR minima to AMR of a kind of elongated channel.

This is what we observe.

Py/Py, Co/Py, and Co/Co contacts do not show a transi-

tion from MR minima to AMR at low temperature, indicat-

ing that the above mechanism is absent in these contacts.

This is due to the weaker texture and smaller grain size of Py

and Co. These lead to a reduced effective crystalline anisot-

ropy, as in the random anisotropy model,26 and thus to a

reduction of the driving force of the transition, apparently to

a level low enough to prevent it.

VII. MECHANISM OF THE RESISTANCE DECREASEDUE TO A DOMAIN WALL

As indicated in Sec. VI, the MR minima must arise from

a resistance decrease due to a DW structure. We now discuss

the mechanism of such a decrease. Electron transport in the

contacts is diffusive or intermediate between diffusive and

ballistic for each type, i.e., irrespective the magnetic material

used. This common property suggests a role of bulk scatter-

ing in the constriction. A mechanism of spin-dependent bulk

scattering in a DW, which can cause MR minima, is pro-

posed in Ref. 10. In the two-band Stoner model this mecha-

nism gives the following resistivity change due to a DW:10

drDW'r02e2

mdn~t12t2!5r0

2e2

mdnt2S t1

t2

21 D . ~3!

Here drDW is the resistivity change of DW material com-

pared to r0 , the resistivity of single-domain material, e is the

electron charge, m is the electron mass, and dn5dn2

52dn1(dn.0). dn65n62n06 is the change of the den-

sity of majority ~minority!-spin electrons resulting from the

smaller exchange splitting of the majority and minority

bands in a domain wall.

Equation ~3! leads to a resistance decrease if scattering

in each type of contact obeys t1 /t2,1, i.e., the scatter-

relaxation time of majority-spin electrons is shorter than that

of minority-spin electrons. The MR minimum occurs for the

different materials ~except Ni! for all temperatures. If Eq. ~3!describes the minima, then t1 /t2,1 should hold for all



temperatures as well. For values of t1 /t2 we rely on values

of the ratio r2 /r1(t1 /t2'r2 /r1) extracted from giant

magnetoresistance measurements of magnetic multilayers

based on Co and Py. This is legitimate since the high-

temperature resistivity of our contacts, which have RRR

52–3, is not dominated by impurity scattering, similar to

the Co and Py layers of the multilayers. The ratio r2 /r1 of

the multilayers is almost temperature independent, and is in

the range of 1.5–3,27 so that t1 /t2 exceeds unity. This

holds for the temperature-dependent scattering in our con-

tacts and, in view of the constant sign of the DWMR, for

impurity scattering as well. However, the inequality t1 /t2

.1 corresponds to MR maxima. Therefore, the bulk scatter-

ing mechanism cannot be the origin of the minima.

Another common feature of the contacts is that their

channel has a boundary. Boundary scattering may contribute

to the resistance if l6/2a , the ratio of the mean free path to

the contact diameter, is on the order of unity or larger. From

r6l6'1 fVm2 and the resistivities and diameters in Table I

we estimate that l6/2a'0.2– 2, which justifies a consider-

ation of boundary or surface scattering and its role in the DW

resistance. According to Soffer,28 surface scattering depends

on the electron wavelength and the surface roughness. For

ferromagnets this implies a spin-dependent secularity param-

eter p(k6)5exp(24b2 k6

2 cos2 u), which is the probability

that incident electrons are specularly reflected from the sur-

face. Here b is the surface-roughness amplitude, k6 the wave

vector, and u the angle of incidence, measured with respect

to the surface normal. f (k6)[12p(k6) is the probability of

diffuse scattering, which contributes to the resistance, con-

trary to specular scattering. In general k2,k1 , so that f2

, f1@ f6[ f (k6)# . Thus, majority-spin electrons scatter

more diffusely at a boundary than minority-spin electrons. In

a DW a small decrease of the density of majorities and an

equally small increase of the density of minorities occur.10

This identifies boundary scattering as a possible origin of the

minima.

To estimate the effect of boundary scattering, we extend

the approach of Ref. 10 by taking into account this scattering

according to Matthiessen’s rule: 1/leff6

51/l61 f6/2a . Allow-

ing for both dn6 and d f6 this leads to

drDW

r0

52

dsDW

s0

'2

t2dn2

11

l2 f2

2a

1

t1dn1

11

l1 f1

2a

2

l2

2at2n2d f2

S 11

l2 f2

2aD 2

2

l1

2at1n1d f1

S 11

l1 f1

2aD 2

t2n02

11

l02 f 0

2

2a

1

t1n01

11

l01 f 0

1

2a

. ~4!

With the assumption dn6'3n06dk6 /k0

6 ~free-electron approximation; dk65kDW6

2k06! we can write, d f6'7

83(bk0

6)2(1

2 f6)dn/n06 so that Eq. ~4! is rewritten as

drDW

r0

'dn

n02

•

1

11bg31L f 0

2~bgF1bg3!H b211L f 0

2~b2bgF !

1

83~bk0

2!2L f 02@~b2g3F21 !12L f 0

2~b2g3F2bgF !1~L f 02!2~b2g3F2b2g2F2!#

11L f 02

1bgFL f 02

1bgF~L f 02!2

2

83~bk0

2!2L@~b2g321 !12L f 0

2~b2g32bgF !1~L f 0

2!2~b2g32b2g2F2!#

11L f 02

1bgFL f 02

1bgF~L f 02!2

J . ~5!

Here we have introduced the dimensionless parameters b

[t1 /t2 , g[k01/k0

2(g.1), F[ f 01/ f 0

2(F.1), and L

[l02/2a . The first term in the outer parentheses is the bulk

term (b21) of Eq. ~3!. This term is positive in our case, as

discussed above. The other three terms come from boundary

scattering. Of these L f 02(b2bgF) is always negative,

while the sum of the other two terms makes a positive resis-

tivity contribution for large f 06 and a negative one for small

f 06 . Boundary scattering gives a resistivity decrease if the

sum of the boundary related terms is more negative than the

~positive! bulk term.

To calculate drDW /r0 as a function of b we have to

input the parameters dn , b, k02 , g and L @see Eq. ~5!#. The

expression for dn/n02 is obtained from Ref. 10.29 We have

no data on the boundary roughness b of the contacts. How-

ever, drDW /r0 , when plotted as a function of b for a rea-

sonable choice of the other parameters, shows a narrow mini-

mum around b'0.03 nm, while for values b.0.1 nm this

function is rather flat. It would be accidental if all the con-

tacts have b'0.03 nm, a value which is also somewhat

small. This leads us to the flat region of the function, where

we choose b'0.3 nm, an interatomic distance, as the obvi-



ous value. We further choose k02

510 nm21 and vary g in the

range 1.1–1.5. Finally, we omit the u dependence in f 06 by

taking cos2u50.5. This parameter choice leads to f 06'1,

which means strong diffuse boundary scattering. To relate

drDW /r0 to the experiment we use the relation dR/R'0.5drDW /r , since the DWs are situated at the edge of the

channel.

In Fig. 10 we plot drDW /r0 versus b, for g51.5. For

g51.1, the free electron value for D0'0.1EF ~D0

5exchange splitting; EF5Fermi energy! the evolution of theboundary scattering with the parameter L is similar but some

less pronounced. The panels show separate curves for the

bulk term and the boundary terms of Eq. ~5!, and for the sumof these. Panel ~a! is for L50.1, so that boundary scattering

is very weak. Indeed, since the negative contribution of

boundary scattering is very small compared to the bulk con-

tribution, the sum curve is almost the same as the bulk curve.

Panel ~b! is for the realistic value L51. The curve for the

boundary term gives drDW /r0 between 20.1% and 20.3%

for the whole b range, which is the right order of magnitude.

The total curve has now clearly shifted downward. For ex-

ample, drDW /r0 is between 20.2% and 20.05% for b in the

range 1–1.7, which overlaps the range where we expect b.Clearly, if in Eq. ~5! boundary scattering were emphasized

compared to bulk scattering due to some mechanism, then

the total curve would be negative for a wider range of posi-

tive b values. In diffusive contacts such a mechanism exits,

viz. a peaked current density at the boundaries at the en-

trance and exit of the constriction. Our contacts operate in or

close to the diffusive regime and have partial domain walls

near the entrance and exit of the constriction, so that bound-

ary scattering due to a domain wall indeed will be empha-

sized. A possible stronger spin asymmetry of boundary scat-

tering than given by the Soffer model will further emphasize

boundary scattering. To simulate these effects we have en-

hanced boundary scattering in Eq. ~5! by a factor of 2, re-

sulting in the curves of panel ~c!. The sum curve is now

negative for the whole b range having an average amplitude

of 20.3% in the range 1.5–3, which comes down to 20.15%

after correction for the volume fraction of DW material in

the constriction.

These results suggest that a change of spin-dependent

boundary scattering due to the domain wall is the mechanism

for the observed resistance decrease. We expect that the

mechanism of boundary scattering is not only involved in the

DW resistance of our small point contacts, but also in the

DW resistance of large-area ferromagnetic thin films with a

long mean free path.

VIII. CONCLUSIONS

In conclusion, the magnetoresistance measurements of

Py/Py, Co/Py, Co/Co, Ni/Ni, and Co/Cu point contacts re-

ported here confirm our preliminary results for Py/Py con-

tacts. In particular, all low resistance contacts show an MR

minimum at zero magnetic field, with a magnitude in the

range of 0.4%–1.3%, irrespective of the orientation of the

external field. Further, by extending the measurements to

high resistance values we find a crossover from MR minima

to the minimum/maximum combination of the AMR. This

AMR comes from the narrow channel between the elec-

trodes, which for a diameter-to-length ratio of the channel

smaller than unity dominates the resistance.

To explain the MR minima we combine their relative

independence of the contact material with the magnitude of

the AMR at high field and with the magnetization in the

constriction derived from the micromagnetic simulations. As

a result, we attribute the minima to a resistance decrease due

to geometry-determined partial DWs at either side of the

channel. The weak dependence of the magnitude of the mini-

mum on the field direction comes from a residual AMR of

the channel and the current spreading regions. To find the

mechanism of the resistance decrease, we analyze the influ-

ence of the modified band structure in a DW on the resistiv-

ity for bulk scattering and for boundary scattering. We find

that for the ratio of spin-dependent scattering times appli-

cable to our point contacts the required negative resistance

contribution can only arise from boundary scattering, which

in addition gives the right order of magnitude for a proper

choice of parameters. We take this suggestive result as a sign

that boundary scattering indeed is the mechanism of the

DWMR minima.

ACKNOWLEDGMENTS

The authors are pleased to acknowledge C. M. Schep for

his suggestion to study the domain-wall problem with point

contacts, B. J. Hickey for providing the MOKE facilities, and

G. E. W. Bauer and T. M. Klapwijk for stimulating discus-

FIG. 10. Relative resistivity change of domain-wall ~DW! material com-pared to single-domain material due to the reduced magnetization of a DW,

for boundary scattering, bulk scattering, and the sum of both @according to

Eq. ~5!# as a function of the ratio of relaxation times. The ratio of Fermi

wave vectors g51.5 and the boundary roughness b50.3 nm: ~a! L50.1, ~b!

L51, ~c! L51, and enhancement of the boundary term by a factor of 2.



sions. This work is part of the research program of the

‘‘Stichting Fundamenteel Onderzoek der Materie ~FOM!,’’which is financially supported by the ‘‘Nederlandse Organi-

satie voor Wetenschappelijk Onderzoek ~NWO!.’’

1M. Viret, D. Vignoles, D. Cole, J. M. D. Coey, W. Allen, D. S. Daniel,

and J. F. Gregg, Phys. Rev. B 53, 8464 ~1996!.2 J. F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S. M. Thompson,

and J. M. D. Coey, Phys. Rev. Lett. 77, 1580 ~1996!.3U. Ruediger, J. Yu, S. Zhang, A. D. Kent, and S. S. P. Parkin, Phys. Rev.

Lett. 80, 5639 ~1998!.4T. R. McGuire and R. I. Potter, IEEE Trans. Magn. MAG-11, 95193

~1975!.5 J. P. Jan, in Solid State Physics, edited by F. Seitz and D. Turbull ~Aca-demic, New York, 1957!, Vol. 5, pp. 1–96.

6U. Ruediger, J. Yu, L. Thomas, S. S. P. Parkin, and A. D. Kent, Phys.

Rev. B 59, 11914 ~1999!.7P. M. Levy and S. Zhang, Phys. Rev. Lett. 79, 5110 ~1997!.8A. Brataas, G. Tatara, and G. E. W. Bauer, Phys. Rev. B 60, 3406 ~1999!.9G. Tatara and H. Fukuyama, Phys. Rev. Lett. 78, 3773 ~1997!.10R. P. van Gorkom, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 83,

4401 ~1999!.11R. P. van Gorkom, J. Caro, S. J. C. H. Theeuwen, K. P. Wellock, N. N.

Gribov, and S. Radelaar, Appl. Phys. Lett. 74, 422 ~1999!.12G. Bergmann, Phys. Rep. 107, 1 ~1984!.13R. P. van Gorkom, S. J. C. H. Theeuwen, K. P. Wellock, N. N. Gribov, J.

Caro, and S. Radelaar, J. Appl. Phys. 85, 6196 ~1999!.14P. A. M. Holweg, J. Caro, A. H. Verbruggen, and S. Radelaar, Microelec-

tron. Eng. 11, 27 ~1990!.

15N. N. Gribov, J. Caro, and S. Radelaar, Physica B 218, 97 ~1996!.16P. A. M. Holweg, J. Caro, A. H. Verbruggen, and S. Radelaar, Phys. Rev.

B 45, 9311 ~1992!.17 J. C. Maxwell, A Treatise on Electricity and Magnetism ~Clarendon, Ox-

ford, 1904!.18K. P. Wellock, S. J. C. H. Theeuwen, J. Caro, N. N. Gribov, R. P. van

Gorkom, S. Radelaar, F. D. Tichelaar, B. J. Hickey, and C. H. Marrows,

Phys. Rev. B 60, 10291 ~1999!.19W. Doring, Ann. Phys. ~Leipzig! 32, 259 ~1938!.20 J.-E. Wegrowe, D. Kelly, A. Franck, S. E. Gilbert, and J.-Ph. Ansermet,

Phys. Rev. Lett. 82, 3681 ~1999!.21Th. G. S. M. Rijks, R. Coehoorn, M. J. M. de Jong, and W. J. M. de Jonge,

Phys. Rev. B 51, 283 ~1995!.22A. H. Morrish, The Physical Principles of Magnetism ~Wiley, New York,

1965!.23 J. Smit and H. P. J. Wijn, Ferrites ~Philips Gloeilampenfabrieken, Eind-

hoven, 1959!, pp. 46–47.24R. M. Bozort, Ferromagnetism ~van Nostrand, Toronto, 1951!, p. 569.25We use Ds'CNi(aSi2aNi)DT . Here CNi is the elastic constant of Ni and

aSi ,aNi are the temperature-dependent thermal expansion coefficients of

Si and Ni, for which we use the values in Landolt Bornstein, Band II/1

~Springer, Berlin, 1971!, pp. 401–422.26 J. F. Loffler, J. P. Meier, B. Doudin, J.-P. Ansermet, and W. Wagner,

Phys. Rev. B 57, 2915 ~1998!.27M. A. M. Gijs and G. E. W. Bauer, Adv. Phys. 46, 285 ~1997!, see Tables

2 and 4.28S. B. Soffer, J. Appl. Phys. 38, 1710 ~1967!.29dn/n0

2 is obtained by combining Eqs. ~2! and ~14! of Ref. 10, and using

the n02—dependence of D0 .



negative resistance contribution of a domain-wall structure in a constricted geometry

Documents

dw resistance

minimum resistance

resistance decrease

maximum resistance

resistance minima

contact resistance

low resistance contacts

high resistance contacts