Anisotropic magnetoresistance effect of nickel nanowires

 Communications in Physics, Vol.31, No. 2 (2021), pp. 227-234
 DOI:10.15625/0868-3166/15674
 ANISOTROPIC MAGNETORESISTANCE EFFECT OF NICKEL NANOWIRES
 PHAM DO CHUNG1,†, NGO VAN THANH2 AND JEAN-ERIC WEGROWE3
1Hanoi National University of Education,
 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Centre for Informatics and Computing,
 Vietnam Academy of Science and Technology,
 18 Hoang Quoc Viet, Nghia Do, Cau Giay, Hanoi, Vietnam
3LSI, Ecole Polytechnique,
 CEA/DRF/IRAMIS, CNRS, Institut Polytechnique de Paris,
 F-91128 Palaiseau, France
 E-mail: †chungpd@hnue.edu.vn
 Received 10 November 2020
 Accepted for publication 15 December 2020
 Published 15 January 2020
 Abstract. In this work, we study the magnetic properties of nickel nanowires by measuring their
 anisotropic magnetoresistance at room temperature. The single nickel nanowire is grown by elec-
 trodeposition in a polymer membrane (Polycarbonate). We measure the anisotropic magnetore-
 sistance effect of nickel nanowires for the various values of the magnitudes and orientations of an
 external magnetic field. The results clearly show the existence the anisotropic magnetoresistance
 effect in the nickel nanowires. Besides, the experimental data are best fit to the analytical calcula-
 tions using the Stoner-Wohlfarth model for the magnetization of the wires.
 Keywords: AMR, single domain, magnetic nanowire.
 Classification numbers: 75.47.De, 81.07.Gf, 75.30.Gw, 85.75.-d.
 ©2021 Vietnam Academy of Science and Technology
228 ANISOTROPIC MAGNETORESISTANCE EFFECT OF NICKEL NANOWIRES
I. INTRODUCTION
 Spintronics is one of the prominent fields in finding out a new phenomenon that was dis-
covered in the past few decades. In solid-state devices, ones have found the dependence of the
electron transport on the spin which is an intrinsic property of electron. So, the concept of spin-
tronics relates to spin-charge coupling in metallic materials. The performance of spintronic devices
is controllable therefore it provides a larger diversity of functionality. Spintronics has become the
new science of computers and memory chips in the good advantages to reduce their power con-
sumption, to increase their memory and processing capabilities [1, 2].
 Giant magnetoresistance (GMR) is one of typical phenomena in spintronics that is a quan-
tum effect in essence. GMR effect was found in the multilayer materials with the electron scat-
tering depended on the spin orientation. The tunnelling magnetoresistance (TMR) effect was ob-
served in a tunnel junction composed of two ferromagnetic layers and a thin insulator one [3–5].
Besides, the anisotropic magnetoresistance (AMR) effect is a most interesting phenomenon in
ferromagnetic materials. The electrical resistance of materials depends on the relative angle be-
tween the direction of magnetization and the one of electric current [6–9]. The AMR effect
occurs not only in bulk materials [10–12] but also in nanomaterials such as the ferromagnetic
nanowires (NWs) [11–13]. The magnetoresistance of NWs is easily in control of the shape and
the size of wires for making the magnetic devices such as memory, sensor, signal processing,
logic [10–12, 14].
 Recently, Nickel NWs materials have been extensively studied in both experimental mea-
surements and theoretical calculations. Because the susceptibility is very small in Ni NWs com-
paring to in other ferromagnetic materials, the activity of these devices could be controlled by a
weak external magnetic field for saving the power consumption. However, the difficulty in exper-
iments is that the magnetization of a single nickel nanowire is extremely small (less than 10 emu
in the nano-scale of size) [7,8,12,15]. So, ones have to use the micro-SQUID susceptor-meters or
the magnetic force microscopy (MFM) probes which are the high cost equipment [7, 8, 16].
 In this paper, we presented the experimental measurement of the resistance of the nickel
nanowires at room temperature. The samples were made by the electrodeposition technique with
the size of 10 nm, 30 nm and 100 nm. We investigated the AMR effect by varying the applied mag-
netic field on both magnitude and direction. The results were confirmed by fitting the experimental
data to the theoretical calculations based on the Stoner-Wohlfarth model.
II. EXPERIMENT
II.1. Materials
 In this study, the electrodeposition technique is used to produce the nickel nanowires that
is one of common methods to prepare the nanostructured samples [8, 10, 15, 17]. The structure of
polycarbonate (PC) membrane filter is shown in Fig. 1, the membranes which have the cylindrical
pores inside with the diameter in the range of 10 nm to 100 nm , are about 6 µm of thickness.
The two gold layers at surfaces are deposited by the sputtering technique to prevent the blocking
of the pores. The below layer is a solid base of 200 nm thickness, and the upper one of 35 nm
has the pores coinciding exactly with the cylindrical ones of the membrane. The PC membrane is
soaked in the aqueous electrolytic solution composed of NiSO4 and H3BO4 at pH = 6.2. During
the electrochemical action, Ni nanowires is growing up in the cylindrical pores of the membrane,
 CHUNG PHAM DO et al. 229
when one of them is filled, the current through the two electrodes (gold layers) is soaring from
zero to high intensity and then we can terminate the sample production process [8, 18, 19].
 Fig. 1. The structure of polycarbonate membrane filter.
 Fig. 2. (a) SEM image of Ni NWs, (b) SEM image of a single NW with the diameter of
 30 nm, (c) the EDS spectrum.
230 ANISOTROPIC MAGNETORESISTANCE EFFECT OF NICKEL NANOWIRES
 We obtained here the samples notated by PC10, PC30 and PC100 for the Ni NWs with
the diameters of 10 nm, 30 nm and 100 nm, respectively. SEM images of PC30 are shown in the
figures 2a and 2b (taken by FESEM equipment named Hitachi S-4800 at LSI, Ecole Polytechnique,
French). Fig. 2c is the energy dispersive X-ray spectroscopy (EDS) spectrum of nanowire which
shows the contributions of C (11%), O (16%), S (1%) and Ni (72%).
II.2. Experimental methods
 We designed an experimental measuring system with the components are shown in Fig. 3a.
The sample was mounted in a self-constructed holder connecting to the amperemeter and volt-
meter. The holder was put in the Faraday cage to reduces the electromagnetic noise. We used the
Keithley 6221 device to provide a fixed DC current I < 100 µA across the nanowire axis in z di-
rection. The Keithley 2182A nanovoltmeter device was used for measuring voltages between the
ends of nanowire. Note that, this device has just 15 nV noise at 1s response time with an external
circuit of 1 kΩ resistance.
 Fig. 3. (a) diagram of the experimental measuring system, (b) the angles between the
 magnetic field and the z direction.
 In this experiment, we applied an external magnetic field H in a direction which made
an angle θH with the nanowire axis (see in Fig. 3b). This field was provided by the Bouhnik
electromagnet and the current was supplied by a ±50 A generator. We measured the voltage
across the nanowire at various values of H and θH , then the resistance R of the nanowire was
easily derived from Ohm’s law. We used here the expression of AMR ratio [1, 5–7, 18, 19] as
 ∆R R − R
 = min (1)
 R Rmin
 ◦
with Rmin is the minimum resistance of the nanowire measured at θH = 90 where the applied field
is perpendicular to the wire axis (H ⊥ z in Fig. 3b).
 CHUNG PHAM DO et al. 231
 For a macroscopic scale sample, the nickel NWs can be considered as a single domain so
that their magnetization is almost uniform along, i.e., m = mseu, in which ms is their saturation
magnetization and eu is the unit vector of modulus ms. Let denote θ the angle between m and the
z axis obtained by minimising the ferromagnetic free energy in the Stoner-Wohlfarth model
 2
 E(θ) = −Hms cos(θH − θ) − Hdms cos θ (2)
where Hd is the demagnetising field. The relation between the estimated resistance and the mag-
netization is given by
 2
 R(m) = Rmin + δRcos θ (3)
In which, δR = Rmax − Rmin is the maximum variation of resistance due to AMR, Rmax is the
 ◦
maximum resistance of the nanowire measured at the applied field θH = 0 .
 We defined a similar formula as the equation (1) for the AMR ratio of estimated resistance:
 ∆R R(m) − R
 = min
 R R
 min (4)
 δR
 = cos2 θ,
 Rmin
where θ is obtained by solving the equation
 ∂E(θ)
 = 0. (5)
 ∂θ
III. RESULTS AND DISCUSSIONS
 In this section, we show the measured data and the analytical results for three samples
PC10, PC30 and PC100. The external magnetic field H is varying from −14 kOe to +14 kOe with
 ◦ ◦ ◦ ◦ ◦
some angles θH = 0 , 30 , 60 , 65 and 90 .
 At the first stage of analytical calculations, we have to find the Rmax and Rmin from the
 ◦ ◦
measured data for each sample at θH = 0 and 90 . Note that, the demagnetising field Hd has
not been known yet but it can be calculated by using the self-consistent method. First step, we
choose an initial value of demagnetising field Hd, calculate the angle θ from Eq. (5) and then fit
the Eq. (3) to the experimental data of R with an error ε. Second one, we put the angle θ back
 −3
to the equation (5) to obtain the new value of Hd and return to first step until the ε < 10 . We
have Hd after the self-consistent process completes. In the all of figures, the measured data are
presented by the dots and the solid lines are the plots of equation (4).
 ◦ ◦
 Figures 4(a)-4(c) show the percentage of the magnetoresistance ∆R/Rmin for θH = 0 , 30 ,
60◦, 65◦ and 90◦. At zero field, the measured resistance of Ni NWs is about 800, 300 and 100 for
PC10, PC30 and PC100, respectively. The AMR ratio is about 1.48%, 2.45% and 2.71% those
decrease with increasing the magnetic orientation at the magnitude H > 0. The magnetization of
 ◦
Ni NWs tends to the saturation state at θH = 90 and the magnitude H > 5 kOe. The calculated
curves are excellent fitting with the measured data (dotted lines) with Hd ≈ 2.6 kOe. The results
clearly show the existence of AMR effect where the magnetoresistance strongly depends on both
the magnitude and orientation of external field.
 Fig. 4 also shows the single jump in the magnetoresistance through the varying field. It
means that the magnetic anisotropy of the Ni NW is uniquely due to the shape anisotropy (the
dipolar field or the demagnetizing field). The average of all magnetic anisotropies approximates
232 ANISOTROPIC MAGNETORESISTANCE EFFECT OF NICKEL NANOWIRES
to zero because the Ni NW is composed of the nanocrystallites with random orientations. The
magnetic hysteresis loop confirms our samples to be the single-domain magnets [12, 14, 15] and
the valid value of Hd.
 Fig. 4. The dependence of the ∆R/Rmin on the magnetic field H with various angles θH
 for (a) PC10, (b) PC30 and (c) PC100 Ni NWs.
 Theoretically, we plot in Fig. 5a the magnetoresistance (from Eq. (4)) for Hd = 2.6 kOe and
 ◦
θH = 65 , where the direction of applied field is parallel to the easy axis of magnetization of nickel
material. In the range of ±HC ≈ 1.5 kOe, we obtained the two values of the magnetoresistance
 2
which correspond to the two solutions of equation (5) with variable H. The ratio mz/ms = cos θ
clearly shows the hysteresis loop in Fig. 5b. Note that, the rectangle form of the hysteresis loop
for H = 0 is a signature of the single-domain magnets.
 CHUNG PHAM DO et al. 233
 Fig. 5. The ∆R/Rmin(a) and the hysteresis loop (b) of the Ni nanowire versus the applied
 ◦ ◦
 field with θH = 65 (blue) and θH = 0 (black).
IV. CONCLUSION
 We studied in this paper the magnetoresistance effects in the nickel nanowire of sizes 10
nm, 30 nm and 100 nm. The results show that the nickel nanowires are the single-domain magnets
with a unique demagnetizing field. The anisotropy of Ni NWs is as a result of the shape anisotropy
only.
 The experimental data are verified by analytical calculations based on the Stoner-Wohlfarth
model. So, the demagnetizing field in the Ni NWs could be calculated by fitting the measured
data to the expression of resistance. The combination of experiment and theoretical methods gives
us an efficient approach to study the effects of anisotropy magnetoresistance in the ferromagnetic
nano materials with the high precision and the low cost.
ACKNOWLEDGMENT
 This work was supported by the NAFOSTED (National Foundation for Science and Tech-
nology Development, Vietnam), Grant No. 103.01-2017.62.
REFERENCES
 [1]I. Zutiˇ c,´ J. Fabian and S. D. Sarma, Reviews of Modern Physics 76 (2004) 323.
 [2] S. A. Wolf, Science 294 (2001) 1488.
 [3] A. Fert and I. A. Campbell, Physical Review Letters 21 (1968) 1190.
 [4] A. Fert and S.-F. Lee, Physical Review B 53 (1996) 6554.
 [5] A. Fert and H. Jaffres,` Physical Review B 64 (2001) 184420.
 [6] T. McGuire and R. Potter, IEEE Transactions on Magnetics 11 (1975) 1018.
 [7] J.-E. Wegrowe, S. Gilbert, D. Kelly, B. Doudin and J.-P. Ansermet, IEEE Transactions on Magnetics 34 (1998)
 903.
 [8] J.-E. Wegrowe, D. Kelly, A. Franck, S. E. Gilbert and J.-P. Ansermet, Physical Review Letters 82 (1999) 3681.
 [9] M. I. Hossain, M. Maksud, A. Subramanian, J. Atulasimha and S. Bandyopadhyay, Journal of Physics D: Applied
 Physics 49 (2016) 455003.
234 ANISOTROPIC MAGNETORESISTANCE EFFECT OF NICKEL NANOWIRES
[10] A. Fert and L. Piraux, Journal of Magnetism and Magnetic Materials 200 (1999) 338.
[11] L. Piraux, Applied Sciences 10 (2020) 1832.
[12] S. N. Kozlov, O. V. Skryabina, S. V. Egorov, I. A. Golovchanskiy, A. A. Klimenko, K. S. Napolskii and V. S.
 Stolyarov, Journal of Applied Physics 125 (2019) 063902.
[13] T. da Camaraˆ Santa Clara Gomes, N. Marchal, F. A. Araujo and L. Piraux, Applied Physics Letters 115 (2019)
 242402.
[14] O. V. Skryabina, S. N. Kozlov, S. V. Egorov, A. A. Klimenko, V. V. Ryazanov, S. V. Bakurskiy, M. Y. Kupriyanov,
 N. V. Klenov, I. I. Soloviev, A. A. Golubov, K. S. Napolskii, I. A. Golovchanskiy, D. Roditchev and V. S. Stolyarov,
 Scientific Reports 9 (2019) 14470.
[15] F. Zighem, D. Faurie, M. Belmeguenai, A. Garcia-Sanchez, P. Lupo and A. O. Adeyeye, Applied Physics Letters
 111 (2017) 052408.
[16] W. Wernsdorfer, B. Doudin, D. Mailly, K. Hasselbach, A. Benoit, J. Meier, J. P. Ansermet and B. Barbara, Physical
 Review Letters 77 (1996) 1873.
[17] M. Stanoˇ and O. Fruchart, Magnetic nanowires and nanotubes, Handbook of Magnetic Materials, Elsevier, 2018,
 pp. 155–267.
[18] D.-C. Pham, N. Biziere, G. Melilli, R. Pajon, D. Lacour, L. Bouvot, M. Tabellout, D. Lairez, H.-J. Drouhin, M.-C.
 Clochard and J.-E. Wegrowe, Materials Research Express 1 (2014) 045017.
[19] G. Melilli, B. Madon, J.-E. Wegrowe and M.-C. Clochard, Nuclear Instruments and Methods in Physics Research
 Section B: Beam Interactions with Materials and Atoms 365 (2015) 675.