Michael Gedion
email: michael.gedion@ies.univ-montp2.fr

Frédéric Wrobel

Frédéric Saigné

Natural radioactivity consideration for high-κ dielectrics and metal gates choice in nanoelectronic devices

In order to face downscaling, new chemical elements are used and suggested for the semiconductor industry. However, some of these elements have natural radioactive isotopes, which may cause reliability issues in nanoelectronic devices by triggering soft errors. In this paper, we focus on high-κ dielectric materials and metal gates. We show that beside physical, chemical and mechanical properties of high-κ dielectrics and metal gates, natural radioactivity is also a crucial property to be considered in order to select suitable materials. Using samarium in gate oxides and platinum in electrodes turns out to be a crucial issue for ground level applications.

Alpha emitting nuclei, Soft Errors, High-κ dielectrics, Metal gates

Introduction

Alpha particles are the major reliability issues in electronic devices causing soft errors at ground level [START_REF] Autran | Altitude and underground real Time SER characterization of CMOS 65 nm SRAM[END_REF]. They are emitted from natural disintegrations of radioactive nuclei called alpha-emitters. Once emitted, alpha particles lose their energy by ionizing atoms of the medium in which they travel, resulting in charges generation, which are then collected by sensitive nodes. The collected charge may induce a soft error by changing a bit from the state "1" to the state "0" or vice versa [START_REF] May | Alpha particle induced Soft Error in dynamic memories[END_REF]. Sources of soft errors induced by alpha particles are classified into two categories. The first source corresponds to common impurities ( 238 U, 232 Th and their relative daughters) that are naturally present in refined materials or incorporated in devices during their processing (e.g. residual left behind etching with acid solutions). On the other hand, the second source corresponds to elements used in devices, naturally containing an amount of isotopes emitting alpha particles. These elements are introduced in the semiconductor industry in order to face the scaling of electronic devices ignoring sometimes their ability to emit alpha particles (e.g. HfO 2 as high-κ dielectric containing 174 Hf , which is a natural alpha emitting isotope (see Table 1). This paper focuses on materials used and suggested for present and future nodes, containing alpha emitting isotopes. We evaluated the materials contribution to Soft Error Rate (SER) in order to evaluate their risk. [START_REF] Osten | Introduction crystalline rare-earth oxides into Si technologies[END_REF][START_REF] Yue | Gd 2 O 3 high-κ gate dielectrics deposited by Magnetron Sputtering Sympo[END_REF][START_REF] Dimoulas | Advanced Gate Stacks for High-Mobility Semiconductors[END_REF], GdScO 3 [START_REF] Lopes | Amorphous ternary rare-earth gate oxides for future integration in[END_REF][START_REF] Wagner | Gadolinium scandate thin films as an alternative gate dielectric prepared by electron beam evaporation App[END_REF][START_REF] Roeckerath | Gadolinium scandate as an alternative gate dielectric in field effect transistors on conventional and strained silicon App[END_REF], SmScO 3 [START_REF] Ozben | SmScO 3 thin films as an alternative gate dielectrics App[END_REF], and Nd 2 O 3 [START_REF] Pan | Structural and electrical properties of neodymium oxide high-κ gate dielectrics App[END_REF] have been suggested as favorite candidates for future MOS technologies. Moreover, traditional poly-silicon gate electrodes do not satisfy the desired requirements because of the poly-silicon depletion effect (boron penetration in Si) [START_REF] Robert | Application of high-κ gate dielectrics and metal gate electrodes to enable silicon and non silicon logic nanotechnology[END_REF]. To overcome this problem, fully silicided (FUSI) and fully germinide (FUGE) metal electrodes have been developed due to their CMOS compatibility [START_REF] Doering | Handbook of Semiconductor Manufacturing Technology[END_REF]. An alloy of nickel and platinum (Ni/Pt) [START_REF] Doering | Handbook of Semiconductor Manufacturing Technology[END_REF]22], PtSi [START_REF] Kadoshima | Silicon-atom induced fermi-level pinning of fully silided platinum gates on HfO 2 dielectrics[END_REF], and Pt 3 Ge 2 [START_REF] Tsuchiya | Material characterization of metal germanide gate electrodes formed by fully germanided gate process Jap[END_REF] are proposed as gate electrodes for P-MOSFETs. On the other hand, by alloying nickel and gadolinium, Ni(70)Gd(30) has been proposed as metal electrode for N-MOSFETs [START_REF] Lee | Characteristics of Ni/Gd FUSI for NMOS gate electrode applications[END_REF]. These propositions for both gate oxides and metal electrodes rely only on physical, chemical properties and process compatibility, but the radioactivity aspect of the new chemical elements have not been touched on.

Alpha Particle induced Soft Error Rate calculation

Gadolinium (Gd), samarium (Sm), neodymium (Nd), and platinum (Pt) have natural alpha emitter isotopes (see Table 1 All these oxides have a dielectric constant between 10 and 25. In this paper, an average dielectric constant of 20 is taken in order to estimate the order of magnitude of the gate oxide thickness. Using equation ( 1), the thickness of the high-κ insulator required to form a gate oxide of 9 Å EOT is 46 Å.

t ox = t eq × k x k SiO 2 [12]. (1)
Where, t eq , and t ox are the thicknesses of the equivalent oxide and the high-κ dielectric respectively. k x and k SiO 2 are dielectric constants of the high-κ dielectric, and SiO 2 , respectively. Hence, the volume of the gate dielectric is then 7,452 nm 3 and that of the metal electrode is 42,120 nm 3 . By using the basic molar mass calculation, we developed equation 2 which determines the number of radioactive atoms N 0 present in these volumes.

N 0 = ρV M × N A × a 100 × n. ( 2
)
Where ρ is the density of the compound, V and M are the volume and the molar mass, respectively, of the gate oxide or the metal electrode. N A is the Avogadro number, a and n are the percentage of natural abundance and the number of radioactive atoms in the molecule, respectivily. Using equation ( 2), a 32 nm half-pitch gate made by HfO 2 contains 334 atoms of 174 Hf, which are alpha emitters. A unit SRAM memory cell contains six MOSFETs (Metal Oxide Semiconductor Field Effect Transistor). In this paper, we take a 6T SRAM, thus the cell is made of six gate oxides. Assuming that the entire six transistors have the same size, a 1 MBit memory cell encloses 2.01 × 10 9 atoms of radioactive 174 Hf. The number of disintegrations involved by these radioactive atoms is obtained by using the exponential radioactive decay law [START_REF] Wrobel | Effects of atmospheric neutrons and natural contamination on advanced microelectronic memories App[END_REF].

N = N 0 (1 -e -λT ). ( 3
)
Where N , N 0 , and λ are the number of emitted alpha particles, the above calculated radioactive atoms, and the decay constant respectively. T is defined as 10 9 hours so that results can be directly compared to Failure in Time (FIT), which represents the number of failures during 1 billion hours. Thus, a 1 MBit memory cell, made of HfO 2 gates, undergoes 0.08 disintegrations/MBit/10 9 hours. Supposing that, in the worst case, every emitted alpha particle gives rise to a soft error, a one MBit SRAM memory made by HfO 2 has a failure rate of 0.08 FIT/MBit. This evaluation is consitant according to [START_REF] Wrobel | Hafnium and uranium contributions to Soft Error Rate at ground level[END_REF]. Note that this failure rate only concerns failures caused by alpha particles emitted from the gate oxide. The value of 0.08 FIT/Mbit has to be compared with the acceptable SER. From the ITRS road map [22], the targeted SER is between 1000 and 2000 FIT/MBit for 32 nm based SRAM memories. Consequently, a one MBit SRAM memory cell has a very low SER induced by the HfO 2 gates. Therefore, Hf based gate oxides cannot be a concern for device failures induced by alpha particles. Using the same reasoning, table 2 summaries the failure rate caused by each material containing alpha emitters. GdScO 3 , Gd 2 O 3 and Nd 2 O 3 as gate oxides in the worst case, lead respectively to only 3.3, 4.8, and 9.6 FIT/MBit respectively. In comparison to the targeted SER, we conclude that gate oxides made of gadolinium and neodymium based compounds should not be a problem (in terms of soft error induced by alpha particles), even though the latter are alpha emitters.

The case of samarium (used in SmScO 3 ) is different. Results show that a 1 MBit SRAM memory cell using Sm based gate dielectrics can have, in the worst case, a rate of 7.9×10 3 FIT/MBit. This high rate is explained by a high disintegration rate in comparison to 147 Sm, because it has a shorter half-life than 148 Sm. Hence, Sm based materials may lead to an unacceptable level of SER. However, this evaluation remains in the worst case. In fact, not all disintegrations result in soft errors. In order to estimate definitely the influence on SER, a Monte Carlo based simulations would have to be performed on the device of interest.

For metal electrodes, the SER is calculated by assuming that the 6T SRAM memory is composed of two N-MOSFETs and two P-MOSFETs to store data and two N- The contribution of P-MOSFETs to SER is higher than the N-MOSFETs, this is due to platinum high disintegration rate. Whereas, the N-MOSFET composed of gadolinum will result in a low alpha particle emission rate because gadolinium has a low disintegration rate. We have to note if platinum is used in N-MOSFETs, the SER level will increase.

Conclusion

To summarize, we have evaluated the SER in the worst-case approach. Gd, Hf, and Nd based materials do not play a major role in triggering soft errors. This is due to their high half-life resulting in a low disintegration rate. These elements may be used in gate oxides without taking account their contributions to soft errors. Conclusion must be, however revised, if the quantity of these atoms based materials introduced in devices is significantly increased. In literature, samarium is suggested to be used in gate oxide compound (SmScO 3 ) [START_REF] Ozben | SmScO 3 thin films as an alternative gate dielectrics App[END_REF]. However, in the worst-case calculation, we show a very high contribution to SER. Hence, we do not recommend using this material in nanoelectronic devices. However, if its introduction in devices is essential, its contribution to SER must be first evaluated precisely (e.g. Monte Carlo method). Otherwise, samarium isotopes especially 147 Sm must be purified from the samarium based material.

Separating samarium isotopes can be achieved by chemical means either by crow ether [START_REF] Fujii | Separation of gadolinium and samarium isotopes by crown etherTechnology reports of the[END_REF] or by acetate/amalgam separation system [START_REF] Dembinski | Isotopes effects of samarium and ytterbium in the acetate/amalgam separation system[END_REF]. Finally, as far as platinum is used, it is also a potential source of soft errors induced by alpha particle. Even if its activity is much lower than that of samarium, its contribution to soft errors must be taken into account because of its proximity with the sensitive zones.

Table 1 .

 1 List of radioactive isotopes contained in new elements used or suggested for the semiconductor industry[START_REF] Wrobel | Effects of atmospheric neutrons and natural contamination on advanced microelectronic memories App[END_REF]. , suggested rare earth based oxides as future gate dielectrics[START_REF] Leskela | Rare-earth oxide films for gate dielectrics in microelectronics[END_REF][START_REF] Leskela | Rare-earth oxide thin films as gate oxides in MOSFET transistors[END_REF][START_REF] Scarel | Rare earth oxide yhin Films : growth, characterization, and applications[END_REF][START_REF] Osten | Introduction crystalline rare-earth oxides into Si technologies[END_REF][START_REF] Yue | Gd 2 O 3 high-κ gate dielectrics deposited by Magnetron Sputtering Sympo[END_REF][START_REF] Dimoulas | Advanced Gate Stacks for High-Mobility Semiconductors[END_REF][START_REF] Lopes | Amorphous ternary rare-earth gate oxides for future integration in[END_REF][START_REF] Wagner | Gadolinium scandate thin films as an alternative gate dielectric prepared by electron beam evaporation App[END_REF][START_REF] Roeckerath | Gadolinium scandate as an alternative gate dielectric in field effect transistors on conventional and strained silicon App[END_REF][START_REF] Ozben | SmScO 3 thin films as an alternative gate dielectrics App[END_REF][START_REF] Pan | Structural and electrical properties of neodymium oxide high-κ gate dielectrics App[END_REF]. Among these potential rare earth based gate dielectrics, Gd 2 O 3

	Alpha emitter isotopes	Half life (years)	Natural abundance(%)
	174 Hf	2.0 × 10 15	0.162
	152 Gd	1.08 × 10 14	0.20
	144 Nd	2.29 × 10 15	23.8
	148 Sm	7.0 × 10 15	11.3
	147 Sm	1.06 × 10 11	15.0
	190 Pt	6.5 × 10 11	0.01
	2. New chemical elements for gates oxides and metal electrodes
	Scaling of gate oxides made that recent technologies nodes have an oxide thickness
	around 10 Å. At this range, quantum tunneling leads to unacceptable leakage current
	and unexpected values of key parameters affecting device operations. Thus, it appears
	likely that dielectrics other than SiO 2 are required for new CMOS technologies. Solutions
	to this problem suggest employing high-κ materials for gate oxides [4]. High-κ
	insulators are materials with higher permittivity, which means they are sufficiently
	thicker preventing quantum tunneling. For example, hafnium based oxides has attracted
	much attention for the last decade. As a result, Intel integrated these oxides in
	its 45 nm MOS devices [5]. Even though the approval, hafnium based gate oxides

Table 2 .

 2 Summary of SER of different compounds studiedMOSFETs access transistors. For metal electrodes composed of Ni(70)Gd(30), the SER is negligible (6.5 FIT/MBit). This type of electrode should not be a problem in SRAMs. Metal electrodes of P-MOSFETs, PtSi and Pt 3 Ge 2 , give 34.2 FIT/MBit and 38.0 FIT/MBit, respectively. These values may appear lower than the recommendations of ITRS but they are not negligible since metal electrodes are near sensitive zones.

	Compound	Used as	SER (FIT/MBit)
	HfO 2	Gate Oxide	0.08
	GdScO 3	Gate Oxide	3.3
	Gd 2 O 3	Gate Oxide	4.8
	Nd 2 O 3	Gate Oxide	9.6
	SmScO 3	Gate Oxide	7.9 × 10 3
	PtSi	P-MOSFET metal electrode	34.2
	Pt 3 Ge 2	P-MOSFET metal electrode	38.0
	Ni 70 Gd 30	N-MOSFET metal electrode	6.5


Denise Erb

Gregor Nowak

Kurt Westerholt

Hartmut Zabel

Thin films of the Heusler alloys Cu2MnAl and Co2MnSi: recovery of ferromagnetism via solid-state crystallization from the x-ray amorphous state

Keywords: numbers: 75.70.i, 76.50.g, 78.20.Ls, 78.66.Bz Heusler compounds, magnetic thin films, amorphous metals 1

X-ray amorphous thin films of the Heusler alloys Cu 2 MnAl and Co 2 MnSi have been prepared by magnetron sputter deposition at room temperature. In the amorphous state the Cu 2 MnAl phase is non-ferromagnetic; Co 2 MnSi is weakly ferromagnetic with a ferromagnetic Curie temperature of 170 K. By solid-state crystallization at high temperatures strong ferromagnetic order and high Curie temperatures are established in both alloys. The saturation magnetization of the Co 2 MnSi alloy reaches 5.1 µ B /f.u. at 4 K, corresponding to 100% of the theoretical value; for Cu 2 MnAl we obtain 2.8 µ B /f.u. at 4 K, which corresponds to 87.5 % of the theoretical value. In samples of the Co 2 MnSi phase with optimum saturation magnetization Bragg reflections, indicating a long-range chemical order, are missing, whereas for the Cu 2 MnAl phase Bragg reflections confirm epitaxial quality and long-range L2 1 order.

I. INTRODUCTION

Heusler alloys are ternary intermetallic compounds of the generic composition X 2 YZ.

They have attracted considerable interest since their discovery about 100 years ago [START_REF] Heusler | [END_REF], mainly because of their versatile magnetic properties. Heusler compounds essentially exist in three crystallographically different structural variants: The ordered L2 1 structure with a cubic unit cell composed of four interpenetrating fcc lattices, each occupied solely by X-, Yor Z-atoms; the B2 structure with the Y-and Z-atoms randomly occupying the Y-and Zsublattices; and the A2 structure with all four sublattices occupied at random by X-, Y-and Z-atoms, corresponding to a simple bcc lattice with random site occupancy. The magnetic properties of Heusler alloys depend sensitively on the chemical order, with the perfect L2 1 structure usually possessing the largest magnetic moment [2], [START_REF] Potter | Proc. Phys. Soc. London[END_REF], [START_REF] Bradley | [END_REF], [START_REF] Webster | Heusler Alloys[END_REF].

Here we report on the Heusler alloys Cu 2 MnAl and Co 2 MnSi: Cu 2 MnAl, the original Heusler phase described by Heusler [START_REF] Heusler | [END_REF], may in many respects serve as a valuable model system for the complex physics of Heusler alloys, although it does not belong to the class of half metals [START_REF] Kübler | [END_REF]. The half-metallic phase Co 2 MnSi is established as one of the most important Heusler alloys for applications. Given the highly ordered L2 1 structure, both Heusler alloys are strong ferromagnets with magnetic moments and ferromagnetic Curie temperatures of 3.2 μ B /f.u. [7] and 603 K [8] for Cu 2 MnAl, and 5.07 μ B /f.u. and 985 K for Co 2 MnSi [9]. Ferromagnetic Heusler alloys experienced a striking upsurge in scientific and technological interest after the discovery of half-metallicity in some of these compounds in the early 1980s [10]: Half-metallic ferromagnets -i.e. ferromagnets with full spin polarization at the Fermi level -are promising materials for applications in spintronics [11], [12]. They can be employed as spin filters [13], in spin injection/detection devices [14], [15], or in magnetic tunneling junctions (MTJs) [16]. Apart from a few oxidic compounds and Mn-doped semiconductors [17], [18], [19], the class of Heusler alloys provides the only metallic alloy system with halfmetallicity. Research on integrating half-metallic Heusler alloys into GMR elements is in progress [20], Co 2 MnSi has been successfully employed in MTJs [21], [22], [23], and fully epitaxial MTJs utilizing Co-based Heusler alloys have been produced recently [24], [25]. A topical review on magnetic properties of Co-based Heusler alloys can be found in [26].

The magnetic and half-metallic properties of Heusler alloys depend crucially on the atomic order [27], [28], [29], [30], [31], [32], as mentioned above: True half-metallicity with a spin po-larization of 100% is theoretically predicted only for perfect L2 1 structure. Therefore, the preparation of Heusler alloy thin films with L2 1 crystal structure is of utmost importance for optimizing the performance of Heusler alloys in spintronic devices. Perfect site order can usually best be achieved by preparing films at low growth rates and high substrate temperatures. However, high preparation temperatures are often prohibited for devices by the requirement of smooth surfaces, sharp interfaces and minimized interdiffusion. The standard preparation procedure for MTJ devices with Heusler alloy electrodes provides an illustrative example: The samples, which typically comprise seven different functional layers, are prepared by sputtering all layers in one run at room temperature, since this is the only way to obtain high quality MgO or Al 2 O 3 tunnel barriers. In the as-prepared state the Heusler alloy electrodes in these junctions exist in A2 order rather than in L2 1 structure.

A high-temperature post-annealing procedure of the complete sample stack then establishes the magnetic moment and the high degree of spin polarization of the Heusler alloy electrodes [33]. During this annealing procedure chemical ordering or even crystallization of the Heusler alloy layers takes place. Only very few studies of these processes have been published until now [34], [35].

In the experimental section below we will report on how the ferromagnetism in thin films of the Heusler phases Cu 2 MnAl and Co 2 MnSi evolves gradually during annealing from the as-prepared, weakly magnetic state until the maximum magnetic moment is obtained.

Our main objective is to describe the correlation between long-range structural order and magnetic order, and to elucidate the kinetics of the processes on the atomic level which establish this order.

II. PREPARATION AND EXPERIMENTAL

Thin films of the Heusler phases Cu 2 MnAl and Co 2 MnSi of nominally 50 nm thickness were deposited on MgO (100) and Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) substrates by RF magnetron sputter deposition at room temperature from stoichiometric targets. The target diameter was 50 mm, with the distance between substrate holder and target being 80 mm. The base pressure in the UHV system was 5×10 -9 mbar, and we used Ar at a pressure of 5×10 -3 mbar as sputter gas. We deposited the Heusler alloy layers at very low rates of 0.003 nm/s and 0.004 nm/s for Cu and Co, by reducing the RF power to 25 W or 50 W, respectively. These low rates are essential for establishing the amorphous as-prepared state. All samples were capped by 3 nm of Al, which oxidizes in air forming a protective Al 2 O 3 layer stable up to high temperatures.

The stoichiometry of the films was checked by Rutherford backscattering (RBS), and found to match the nominal composition 2:1:1 within the resolution of the RBS method of about 2 at%. MgO (100) and Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) substrates induce different characteristic textures in the Heusler alloy film during its crystallization. For Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) substrates one expects the (110) planes of the Heusler alloy film to be parallel to the film surface [36]; for MgO (100) the preferred orientation of the Heusler alloy film has the (100) planes parallel to the MgO (100) surface. Thus, the fundamental (220) Bragg reflection, which is already present in the chemically completely disordered A2-type Heusler alloy structure, can readily be observed in the films grown on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20). For studying the (200) and (400) Bragg reflections, characterizing the B2-type of chemical long-range order, films grown on MgO (100) are best suited.

X-ray reflectometry (XRR) and diffractometry (XRD) were conducted for determining the film thickness, as well as for investigating the crystal structure of the samples, and the epitaxial relation between film and substrate. Conventional in-house fixed-anode and rotating-anode two-axes diffractometers as well as the (6+2)-circle diffractometer at the synchrotron-based beamline W1.1 at HASYLAB (DESY) were employed for these measurements. All these facilities provide radiation with a wavelength of λ = 0.154 nm, which belongs to the characteristic Cu-K α x-ray emission line. The magnetic properties of the samples were characterized by a vibrating sample magnetometer (VSM), integrated in a physical properties measurement system (PPMS, by Quantum Design). The VSM is equipped with an UHV oven (temperature range: from room temperature to 1000 K), and thus allows for an in-situ observation of the development of the magnetic moment during thermal treatment.

Samples designated for structural characterization were annealed ex-situ in the heating stage of the UHV sputter deposition system, whereas samples appointed for magnetic characterization underwent in-situ thermal treatment in the VSM while measuring the sample magnetization. Since the vacuum conditions in the UHV sputter deposition system are considerably better than in the VSM, long-time annealing was only conducted in the UHV sputter deposition system.

III. RESULTS AND DISCUSSION

In Fig. 1 we show examples of x-ray reflectivity scans of Cu 2 MnAl and Co 2 MnSi thin films prepared at room temperature. The period of the well-defined total thickness oscillations corresponds to a total film thickness of 53 nm, in good agreement with the nominal thickness including the capping layer.

Interestingly, in the as-prepared sample state we could not detect any Bragg reflections in the range of high angles -irrespective of the substrate type, and even when employing the high intensity of a synchrotron radiation source (see black curves in Fig. 9, Fig. 7(a)-(b)).

This also holds true for the fundamental Bragg reflection (220), which already exists for Heusler alloys in the chemically completely disordered A2 structure. Thus, the chemical structure of the Heusler alloy films in the as-prepared state is amorphous, or -to be more precise -x-ray amorphous; meaning that the structural coherence length is too small to yield Bragg reflections detectable by standard XRD.

A. Development of Magnetic Order

In Fig. 2 we depict low-temperature magnetization measurements of a Cu 2 MnAl film and a Co 2 MnSi film in the as-prepared state. These films were prepared on MgO (100), but virtually the same curves are obtained for films deposited on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20). For Cu 2 MnAl we find a weakly temperature-dependent magnetization with a magnetic moment of about 0.02 μ B /f.u. (see Fig. 2(a)). At 250 K the field-cooled (FC) and the zero-field-cooled (ZFC) magnetization curve begin to deviate slightly from each other, indicating the onset of magnetic irreversibility. Such a behavior is characteristic for a spin glass order with a freezing temperature of about 250 K. Spin glass order in bulk Cu 2 MnAl with A2 or B2 order has already been reported [37], [38]. Thus, in this respect a x-ray amorphous thin film of Cu 2 MnAl behaves similar to disordered bulk material.

As mentioned in the introduction, Cu 2 MnAl in the fully ordered L2 1 state is a strong ferromagnet with a ferromagnetic Curie temperature of 603 K [8] and a magnetic moment of 3.2 μ B at room temperature [7]. The ferromagnetism of Cu 2 MnAl results from an indi- rect type of exchange interaction between the Mn-d orbitals, mediated by intermediate Al orbitals [START_REF] Kübler | [END_REF]. In the unit cell of the perfectly ordered L2 1 state there are no Mn nearest neighbors, whose strong antiferromagnetic exchange coupling would reduce the total magnetic moment per unit cell. In the chemically disordered A2 structure, however, nearest neighbor Mn atoms exist with a certain probability, and the strong antiferromagnetic Mn-Mn direct exchange interaction becomes relevant. The competition between the ferromagnetic and the antiferromagnetic exchange interactions in combination with random atomic disorder gives rise to magnetic frustration and spin glass type of order [39], [40].

The magnetic behavior of the Co 2 MnSi film in the as-prepared state is significantly different from that of the Cu 2 MnAl film (see Fig. 2(b)): We observe a ferromagnetic type of M(T)-curve with a ferromagnetic Curie temperature of about 170 K, and typical ferromagnetic hysteresis loops below this temperature. The saturation magnetic moment reaches 0.6 μ B /f.u. at low temperatures. In addition, below a temperature of about 31 K, a strong magnetic irreversibility sets in -as evidenced by the sudden deviation of the FC and ZFC magnetization curves at 31 K. This behavior is typical for re-entrant ferromagnetism [41].

In comparison, Co 2 MnSi in the fully ordered L2 1 state has a ferromagnetic Curie temperature of 985 K -which is actually one of the highest Curie temperatures ever detected within the family of Heusler alloys [START_REF] Webster | Heusler Alloys[END_REF] -and a magnetic moment of 5.07 μ B /f.u. [9]. Thus, Curie temperature and magnetic moment of the as-prepared Co 2 MnSi film are drastically reduced as compared to the values in the chemically ordered L2 1 state. This observation can qualitatively be explained following the same lines of argument as for the Cu 2 MnAl phase:

In the x-ray amorphous state with a short-range A2-type of atomic order strong antiferromagnetic interactions between nearest neighbor Mn atoms compete with ferromagnetic Co-Mn and Co-Co interactions [42], thus the average total magnetic moment per formula unit is diminished. In contrast to the situation in Cu 2 MnAl, there are two magnetic atoms in the unit cell of Co 2 MnSi: Mn contributing about 3 μ B and Co contributing about 1 μ B per atom [43]. Since the majority of the magnetic interactions in the unit cell are ferromagnetic, a finite ferromagnetic magnetization remains even in the case of complete chemical disorder.

However, the magnetic ground state is not purely ferromagnetic, but random spin disorder coexists with ferromagnetic order: Such a situation is characteristic for the re-entrant ferromagnetism observed in the vicinity of the ferromagnetic and spin glass phase boundaries of systems with competing magnetic interactions [41].

In Fig. 3 we show high-temperature magnetization measurements recorded in-situ during thermal treatment of Cu 2 MnAl and Co 2 MnSi thin films. The thermal treatment comprises two cycles for Cu 2 MnAl (see Fig. 3(a): the black curve corresponds to first cycle, the red curve corresponds to the second cycle), and three cycles for Co 2 MnSi (see Fig. 3(b): the black curve corresponds to the first cycle, the red curve to the second cycle, and the green curve corresponds to the third cycle). In each cycle the sample was heated up from room temperature to a specific annealing temperature at a constant rate of 5 K/min. After annealing the films for one hour per cycle, the samples were cooled back down to room temperature at a rate of -5 K/min. The films for the measurements shown in Fig. 3 have been grown on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20), but films grown on MgO (100) show virtually identical behavior.

For Cu 2 MnAl the initial magnetization at room temperature is very small, because the sample is non-ferromagnetic in the as-prepared state (see Fig. 3(a)). The magnetization increases rapidly for temperatures above 480 K, and quickly reaches typical ferromagnetic magnetization values of about 1 μ B /f.u. The formation of the ferromagnetic state out of the x-ray amorphous as-prepared state is accomplished by an irreversible solid-state crystallization and chemical ordering process. Upon cooling we observe a typical ferromagnetic magnetization curve, which extrapolates to a Curie temperature of about 600 K and to a ferromagnetic saturation moment of 2.5 μ B /f.u. at room temperature.

For Co 2 MnSi the magnetization curve recorded during heating the film from the asprepared state up to 660 K (black curve in Fig. 3(b)) principally exhibits a behavior very similar to that observed for Cu 2 MnAl. However, the irreversible transition to the ferromagnetic state is shifted to higher temperatures. The onset of ferromagnetic order is observed at 640 K only -as compared to 480 K for the Cu 2 MnAl phase. Thus, the thermal activation energy involved in the crystallization process is definitely higher in Co 2 MnSi than in Cu 2 MnAl. After annealing the sample at 660 K for 1 h, we observe a typical ferromagnetic M(T)-curve upon cooling to room temperature. The recorded magnetization curve is compatible with a ferromagnetic Curie temperature well above 900 K and with a saturation magnetic moment of 3.2 μ B /f.u. at room temperature.

For the observation of the irreversible structural changes via the in-situ measurement of the magnetization shown in Fig. 3 we have applied a magnetic field of H = 1 kOe. analogous measurements in fields of H = 200 Oe and H = 50 kOe, respectively, yielded virtually the same temperatures for the onset of the irreversible structural changes. Thus, the applied magnetic field has no direct influence on this development.

In Fig. 4 we depict a series of magnetization measurements, in which the samples are subsequentially heated to higher and higher temperatures and quenched to room temperature while applying a constant external field of H = 1kOe. Thereby we demonstrate, that the saturation magnetization of both Heusler phases can be tuned to any value between zero and full saturation magnetic moment. Fig. 5 shows the corresponding magnetic hysteresis loops measured at room temperature after quenching from a specific high temperature. The hysteresis loops have a nearly rectangular shape with small coercive fields of about H c ≈ 10 Oe for Cu 2 MnAl and H c ≈ 50 Oe for Co 2 MnSi, essentially independent of the degree of magnetic order established in the sample. These coercive forces seem exceptionally small as compared to those in thin films of the same Heusler phases prepared at high temperatures [44], and indicate an excellent magnetic homogeneity of the films. We have also tested systematically the maximum magnetic moment which we can achieve by long-time annealing treatment of the samples at high temperatures: For the Cu 2 MnAl phase we could not improve the maximum magnetization, which was already obtained after annealing at 560 K and at 580 K for one hour each. The maximum saturation magnetic moment was 2.5 μ B /f.u at room temperature and 2.8 μ B /f.u at 4 K, which corresponds to 87.5 % of the value for bulk Cu 2 MnAl [7]. Prolonging the annealing time or increasing the annealing temperature above 600 K is counterproductive, i.e. it lowers the maximum saturation magnetic moment.

For the Co 2 MnSi phase we could increase the maximum saturation magnetic moment significantly by annealing the films at 700 K for one hour; the corresponding magnetization curve has been included in Fig. 3(b) (green curve). The saturation magnetic moment, which we achieved for Co 2 MnSi after this heat treatment, was 4.9 μ B at room temperature and 5.1 μ B /f.u. at 4 K, which corresponds to the maximum magnetic moment expected theoretically for this phase in the perfectly ordered L2 1 structure [9].

B. Thermal Activation Energies

The irreversible change of the magnetization of the Cu 2 MnAl and Co 2 MnSi thin films at high temperatures is caused by atomic nearest neighbor hopping processes in the Heusler compound unit cell. Therefore, it appears sensible to assume that a proportionality exists between the initial change of the ferromagnetic magnetization M(T ) and the probability for nearest neighbor hopping p(T ). With the reasonable assumption of an Arrhenius law for the temperature dependence of the probability for nearest neighbor hopping processes

p(T ) = p 0 exp(-E a /k B T ) we thus can assume that M(T, t) ∝ t • exp(-E a /k B T )
with the heating time t and the thermal activation energy E a . In Fig. 6 we show corresponding Arrhenius plots for Cu 2 MnAl and Co 2 MnSi, deduced by numerical differentiation of magnetization data shown in Fig. 4. From the straight line fits in Fig. 6 we derive thermal activation energies E a of 2.2 eV and 3.0 eV for Cu 2 MnAl and Co 2 MnSi, respectively, which are reasonable values for nearest neighbor hopping processes in metals.

C. Development of Chemical Order in Co 2 MnSi

As stated at the beginning of this section, there are no detectable Bragg peaks in the XRD patterns of samples in the as-prepared state. This observation has led us to the conclusion, that before thermal treatment the samples are in an x-ray amorphous state, i.e. the coherence length of their structural order is too small to yield detectable Bragg reflections. After annealing Co 2 MnSi films for an accumulated time of three hours at temperatures ranging from 660 K to 700 K (see Fig. 3(b)), the XRD patterns still show no Bragg reflections, although the sample is already ferromagnetic with a ferromagnetic saturation moment close to the theoretical limit. As discussed above, a reduction of the number of antiferromagnetically coupled Mn nearest neighbors is sufficient for the occurrence of ferromagnetism with large saturation moments in Heusler compounds. This can be achieved by some re-arrangement of the site occupancy within the unit cell via nearest neighbor hopping processes.

Eventually, after further annealing the sample for 10 h at 730 K, a (220) Bragg reflection indicating long-range A2 order could be detected for the Co 2 MnSi film grown on Al 2 O 3 (11-20) (see Fig. 7(a)). However, there are two additional peaks at 2Θ = 44.2 • and at 42.3 • , respectively, which do not belong to the Heusler phase, but suggest a decomposition of the film. This is corroborated by the change of the ferromagnetic hysteresis loops after this high- temperature annealing, as depicted in Fig. 8(a): The saturation magnetization is definitely lowered, and the coercive field is clearly increased as compared to the sample state which is obtained after annealing at lower temperatures (see Fig. 3(b)). We did not try to identify the impurity phases formed during the decomposition at 730 K. The precipitation of fcc Co from the Co 2 MnSi Heusler phase at high temperatures has been suggested in literature [45].

Given a long-range L2 1 structural order, the (200) and (400) reflections are expected to be displayed in out-of-plane Bragg scans of Co 2 MnSi films deposited on MgO (100). These reflections remain absent, even after annealing at 730 K for 10 h (see Fig. 7(b)). Thus, only A2-type of long-range structural order has been formed; the coherence length of B2-and L2 1type of long-range order still does not exceed the sensitivity limit of standard x-ray Bragg diffraction. In Fig. 7(c)-(d) we depict Bragg scans for Co 2 MnSi films grown on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and on MgO (100) at a substrate temperature of 770 K for comparison. For these films we observe the (220) Bragg reflection on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20), as well as the ( 200) and (400) Bragg reflections on MgO (100). However, concerning the magnetic properties of these films, there is not much of a difference due to the alternative preparation conditions; the films prepared via the amorphous state even exhibit a slightly larger saturation magnetization and smaller For Cu 2 MnAl deposited on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and annealed for 1 h at 580 K we observe a (220) Bragg reflection at an angle of 2Θ = 44 • in the out-of-plane Bragg scan, consistent with the expected [110] texture for the Heusler alloy growth on Al 2 O 3 (11)(12)(13)(14)(15)(16)(17)(18)(19)(20). The intensity of the (220) peak is lower than that observed for the (200) and (400) peaks in Fig. 9(a

IV. SUMMARY AND CONCLUSIONS

We have shown in this investigation, that thin films of the Heusler alloys Cu 2 MnAl and Co 2 MnSi grown by magnetron sputtering at room temperature and at very low deposition rates are x-ray amorphous and non-ferromagnetic. Annealing of the as-prepared state establishes strong ferromagnetism via solid state crystallization and atomic ordering. Atomic nearest neighbor hopping processes within the unit cell of the disordered Heusler phases create a short-range chemical order, which is the prerequisite for the ferromagnetic state.

For both Heusler alloys we obtained large ferromagnetic saturation magnetic moments; the hysteresis curves are of a square type with very low coercive field values.

The degree of atomic long-range order was probed by XRD before and after heat treatment, revealing for the Co 2 MnSi phase that even the sample state with the highest saturation magnetic moment yields no detectable Bragg reflections. This Heusler phase provides further evidence, that chemical long-range order is not required for strong ferromagnetismchemical short-range order is sufficient. The prototype for this situation are amorphous ferromagnetic metals such as NiP [46] or Fe 80 B 20 [47]. Our experimental data allow no infer- ences on the details of the atomic short-range order and electronic structure of the samples. Future research will provide the necessary information and will suggest explanations for the large saturation magnetic moments which can be observed in amorphous Heusler alloys.

In contrast, for Cu 2 MnAl films grown on MgO (100) we achieved epitaxial film quality from the x-ray amorphous as-prepared state by short-time annealing at a moderate temperature of 580 K. The epitaxial quality of these films is comparable to that which we have obtained before by growing Cu 2 MnAl films on MgO (100) at high substrate temperatures [48]. It is common belief that high substrate temperatures are necessary for single crystalline thin film growth, since high surface mobility of the atoms during the growth process is essential. Thus, it seems remarkable that for the Cu 2 MnAl phase an alternative route via solid state crystallization from an x-ray amorphous phase also leads to single crystallinity and epitaxy.

The crystallization and atomic ordering processes for Cu 2 MnAl and Co 2 MnSi films are significantly different. For the Co 2 MnSi phase it is easy to establish a high magnetic moment, but very difficult to achieve any type of crystalline long-range order. Actually, an annealing temperature of 730 K was required before we could observe at least the fundamental (220) Bragg reflection. However, the Co 2 MnSi films decompose under these conditions. In contrast, long-range structural order is readily established in Cu 2 MnAl thin films. The main reason for this striking difference is the fact, that the activation energy in Co 2 MnSi is much larger than in Cu 2 MnAl. We assume that due to the covalent character of the Si metal bonds the Si atoms in the Co 2 MnSi unit cell are nearly immobile at the temperatures applicable, so that the system can hardly remove the frozen-in atomic disorder on a long-range scale. As a compromise, only chemical short-range order is formed during annealing. However, this is sufficient to establish a magnetic saturation moment reaching that predicted by theory for perfectly ordered L2 1 symmetry.

Figure 1 :

 1 Figure 1: X-ray reflectivity scans of thin films of Cu 2 MnAl and Co 2 MnSi, deposited on Al 2 O 3 (11-20) and MgO (100) and capped by Al. The curves have been shifted along the y-axis for clarity.

Figure 2 :

 2 Figure 2: Field-cooled and 0-field-cooled magnetic moment versus temperature of Cu 2 MnAl (a) and Co 2 MnSi (b) thins films grown on MgO (100) in the as-prepared state, measured in an external field of H = 200 Oe. The arrows indicate the direction of the temperature variation.

Figure 3 :

 3 Figure 3: Magnetic moment versus temperature for (a) Cu 2 MnAl and (b) Co 2 MnSi thin films grown on Al 2 O 3 (11-20), recorded in-situ during thermal cycling (see main text) in an external field of H = 1kOe. The direction of the arrows indicates the direction of the temperature variation; the temperature was kept constant for one hour at the maximum temperature during each cycle.

Figure 4 :

 4 Figure 4: Magnetic moment versus temperature for (a) Cu 2 MnAl and (b) Co 2 MnSi thin films grown on Al 2 O 3 (11-20) during sequential annealing (see main text) in an external field of H = 1 kOe. During each step of the sequence the samples were heated up to the respective maximum temperature at a rate of 5K/min and rapidly quenched to room temperature. Only the magnetic moment measured during the heating branch is shown.

Figure 5 :

 5 Figure 5: Magnetic hysteresis loops of (a) Cu 2 MnAl and (b) Co 2 MnSi thins films grown on Al 2 O 3 (11-20), measured at room temperature after quenching from the temperatures given in the figureduring the sequential annealing procedure described in the main text (see also Fig.4).

Figure 6 :

 6 Figure 6: Logarithm of the absolute value of the temporal derivative of the magnetization versus the reciprocal temperature for (a) Cu 2 MnAl and (b) Co 2 MnSi grown on MgO (100). The slope of the straight line defines the thermal activation energy.

Figure 7 :

 7 Figure 7: Out-of-plane x-ray diffraction patterns for Co 2 MnSi thin films: (a)-(b) deposited at room temperature and annealed for 10 h at 730 K, recorded at HASYLAB and at a rotating anode inhouse facility, respectively; (c)-(d) deposited at 770 K with an Au capping layer of 5 nm thickness, recorded at a fixed anode in-house facility. Panels on the left hand side show films deposited on Al 2 O 3 (11-20); panels on the right hand side show films deposited on MgO (100).

  coercive field (see Fig. 8(b)). D. Development of Structural Long-Range Order in Cu 2 MnAl The results of our XRD studies on Cu 2 MnAl differ significantly from those discussed above for the Co 2 MnSi films in several respects: After annealing Cu 2 MnAl films for 1 h at a moderate temperature of 580 K, the out-of-plane XRD patterns already show the characteristic Bragg reflections for this phase (see Fig. 9(a)-(b)). For the film prepared on MgO

Figure 8 :

 8 Figure 8: Hysteresis curves of Co 2 MnSi thin films (a) annealed from the amorphous as-prepared state for 3 h at 660 to 700 K (black curve) and additionally annealed for 10 h at 730 K (red curve), (b) annealed from the amorphous as-prepared state for 3 h at 660 K to 700 K (black curve) and deposited at 770 K without subsequent annealing (red curve).

(

  100) we observe well resolved (200) and (400) Bragg reflections, indicating the expected texture for [100] growth on MgO (100). The (200) superstructure reflection originates from a substantial degree of long-range B2-type of chemical order (i.e. separation of Mn atoms and Al atoms on their respective sublattices). From the full width at half maximum (FWHM) of these Bragg peaks we can derive the structural coherence length ξ by applying the Scherrer formula; we obtain a value of about ξ = 25 nm, corresponding to 50% of the total film thickness. The rocking width of the (400) peak correlates with a mosaic spread of 0.45 • for the crystallites within the plane. Both parameters indicate a high-quality textured growth on MgO (100).

Figure 9 :

 9 Figure 9: Out-of-plane x-ray diffraction patterns of Cu 2 MnAl thin films deposited on (a) Al 2 O 3 (11-20) and (b) MgO (100), in the as-prepared state (black curves) and after annealing for 1 h at 560 K (red curves). Recorded at a rotating anode in-house facility.

  ), although the (220) peak is the fundamental Bragg reflection for the Heusler compounds, and therefore is expected to have the highest intensity. Moreover, the mosaic spread of more than 2 • for Cu 2 MnAl grown on Al 2 O 3 (11-20) is significantly larger than for Cu 2 MnAl on MgO (100); thus the structural quality of the Cu 2 MnAl phase when grown on Al 2 O 3 (11-20) is clearly lower than when grown on MgO (100). Although after crystallization and thermally induced chemical order the resulting structural qualities of the Heusler phases deposited on MgO (100) and on Al 2 O 3 (11-20) differ markedly, the ferromagnetic properties are very similar. The Cu 2 MnAl films deposited on MgO (100) even have epitaxial film quality after annealing at 580 K for 1 h, as proven by the in-plane azimuthal scans in Fig. 10, which were recorded employing synchrotron radiation. The in-plane 360 • azimuthal scans on the Heusler (400) Bragg reflection and the MgO (200) Bragg reflection demonstrate the perfect four-fold symmetry of the crystal structure of this film, with the Heusler (400) Bragg peaks shifted by 45 • from the MgO (200) Bragg peaks. This corresponds to an in-plane epitaxial relation between substrate and film with the [100] direction of the Heusler phase rotated by 45 • from the [100] direction of the MgO substrate. For this film we also conducted an in-plane 360 • azimuthal scan on the asymmetric (111) Bragg reflection, which is tilted in the scattering plane by 26.14 • with respect to the sample surface. This reflection unambiguously identifies the L2 1 -type of chemical order in Heusler alloys. The results presented in Fig. 10(b) again reflect the perfect cubic symmetry of the film. Moreover, they prove that the film possesses an appreciable degree of long-range L2 1 order.

Figure 10 :

 10 Figure 10: In-plane 360 • azimuthal scans on the (400) Bragg-reflection (a) and on the asymmetric (111) reflection (b) of the Cu 2 MnAl thin film from Fig. 9(b). Recorded at HASYLAB.

Acknowledgments

We wish to acknowledge financial support of this work within SFB 491 by the DFG. We thank P. Stauche and S. Erdt-Böhm for technical assistance with the sputter deposition, as well as Dr. H. J. Becker and Dr. D. Rogalla for conducting the RBS measurements. We


R R A Syms
email: r.syms@imperial.ac.uk

L Solymar

I R Young

Broad-band Coupling Transducers for Magneto-Inductive Cables

Keywords: Metamaterial, periodic structure, magneto-inductive wave PACS numbers 41.20.-q Applied classical electromagnetism 81.05.Zx New materials: theory, design and fabrication 84.30.Bv Circuit theory

A broadband resonant transducer capable of low-loss coupling between magnetoinductive waveguides and a real impedance is introduced. The transducer is an L-C circuit resonating at the resonant frequency of the elements forming the guide. However, the values of the components in the transducer are different, and chosen to obtain two separate nulls in reflection so that low reflection is obtained over a wide spectral range.

The transducer can be incorporated into the MI waveguide itself, allowing a connection between a magneto-inductive cable and a conventional system to be made as a simple splice. The design is confirmed using two metre lengths of low-loss thin-film magnetoinductive cables formed using copper-clad polyimide and operating near 100 MHz frequency.

Introduction

Magneto-inductive (MI) waveguides are periodic low-frequency electrical structures, formed by magnetically coupling a set of lumped-element L-C circuits [START_REF] Shamonina | Magneto-inductive waveguide[END_REF]. The properties of linear arrangements and 2D and 3D MI arrays have been extensively studied [START_REF] Shamonina | Magnetoinductive waves in one, two and three dimensions[END_REF][START_REF] Wiltshire | Dispersion characteristics of magneto-inductive waves: comparison between theory and experiment[END_REF][START_REF] Shamonina | Magneto-inductive waves supported by metamaterial elements: components for a one-dimensional waveguide[END_REF][START_REF] Maslovski | Artificial magnetic materials based on the new magnetic particle: metasolenoid[END_REF][START_REF] Jylhä | Traveling waves along the metasolenoid[END_REF].

Propagation losses have been reduced [START_REF] Syms | Low loss magneto-inductive waveguides[END_REF], and the effects of non-nearest neighbour interactions [START_REF] Syms | Higher order interactions in magneto-inductive waveguides[END_REF], bi-periodicity [START_REF] Radkovskaya | Experimental study of a bi-periodic magnetoinductive waveguide: comparison with theory[END_REF], coupling to electromagnetic waves [START_REF] Syms | A theory of metamaterials based on periodically loaded transmission lines: interaction between magneto-inductive and electro-magnetic waves[END_REF] and retardation [START_REF] Radkovskaya | An experimental study of the properties of magnetoinductive waves in the presence of retardation[END_REF] have all been considered. Although the first demonstrations used simple resonant loops, MI waveguides have also been formed as planar structures [START_REF] Freire | Planar magnetoinductive wave transducers: Theory and applications[END_REF] and thin-film cables [START_REF] Syms | Thin-film magneto-inductive cable[END_REF]. The latter have potential applications as patient-safe cable in magnetic resonance imaging [START_REF] Syms | Periodic analysis of MR-safe transmission lines[END_REF][START_REF] Weiss | Transmission line for improved RF safety of interventional devices[END_REF]. Many MI devices have been proposed or demonstrated, including magnetic flux concentrators [START_REF] Wiltshire | Resonant magnetic concentrators[END_REF], delay lines [START_REF] Freire | Planar magnetoinductive wave transducers: Theory and applications[END_REF], filters [START_REF] Nefedov | On potential applications of metamaterials for the design of broadband phase shifters[END_REF], directional couplers [START_REF] Shamonina | Magneto-inductive waves supported by metamaterial elements: components for a one-dimensional waveguide[END_REF][START_REF] Radkovskaya | Transmission properties of two shifted magnetoinductive waveguides[END_REF], splitters [START_REF] Syms | Magneto-inductive waveguide devices[END_REF], lenses for near-field imaging [START_REF] Freire | Planar magnetoinductive lens for three-dimensional subwavelength imaging[END_REF][START_REF] Sydoruk | Mechanism of subwavelength imaging with bilayered magnetic metamaterials: theory and experiment[END_REF][START_REF] Freire | Experimental demonstration of a µ = -1 metamaterial lens for magnetic resonance imaging[END_REF] and detectors for magnetic resonance imaging [START_REF] Solymar | Rotational resonance of magnetoinductive waves: basic concept and application to nuclear magnetic resonance[END_REF]. Parametric amplification has been considered as a method of reducing propagation losses [START_REF] Sydoruk | Parametric amplification in coupled magnetoinductive waveguides[END_REF][START_REF] Syms | Three-frequency parametric amplification in magneto-inductive ring resonators[END_REF]. Despite this effort, losses per metre have historically been high (150 dB/m in [START_REF] Shamonina | Magneto-inductive waveguide[END_REF] and 50 dB/m in [START_REF] Syms | Low loss magneto-inductive waveguides[END_REF]), propagation distances have been limited, and end-reflections have mainly been ignored. However, the demonstration of losses of ≈ 2.5 dB/m in thin-film cables [START_REF] Syms | Thin-film magneto-inductive cable[END_REF] has re-affirmed the need for an effective transducer for coupling to a conventional transmission line or for termination. Without such a transducer, it will be difficult to construct any high-performance MI system. Although multi-element absorbers have been proposed [START_REF] Syms | Absorbing terminations for magnetoinductive waveguides[END_REF], these do not solve the coupling problem. In this paper, we consider possibilities for inductive coupling to a MI waveguide, and demonstrate a simple broadband transducer that can be constructed from passive components. The theory of matching is described in Section 2, transducer optimisation is explored in more detail in Section 3, experimental verification is presented in Section 4, and conclusions are drawn in Section 5.

Matching to magneto-inductive waveguides

We first consider the principle of matching a magneto-inductive waveguide to a real load.

Figure 1a shows the equivalent circuit of an ideal, loss-less guide, formed from a set of L-C resonators coupled to their nearest neighbours by a mutual inductance M. Away from any termination (n < 0) the equation governing the current I n in the n th element at angular frequency ω can be found from Kirchhoff's voltage law as [START_REF] Shamonina | Magneto-inductive waveguide[END_REF]:

{jωL + 1/jωC}I n + jωM{I n-1 + I n+1 } = 0 (1)
A solution can be found by assuming that I n is the travelling wave I n = I 0 exp(±jnka), where I 0 is the wave amplitude, ka is the phase shift per element, k is the propagation constant and a is the period. Substitution into Equation 1 yields the well-known dispersion equation:

(

1 -ω 0 2 /ω 2 ) + κ cos(ka) = 0 (2)
Here ω 0 2 = 1/LC is the angular resonant frequency and κ = 2M/L the coupling coefficient.

For positive κ, propagation can only take place for ω/ω 0 between 1/√(1 + κ) and 1/√(1κ). The effect of loss may be modelled by repeating the analysis, assuming the presence of a resistor R in each resonant loop [START_REF] Shamonina | Magneto-inductive waveguide[END_REF]. If this is done, the dispersion equation is modified to (1 -ω 0 2 /ω 2 -j/Q) + κ cos(ka) = 0, where Q = ωL/R is the Q-factor. Assuming a complex-valued propagation constant ka = k'a -jk''a, and further assuming that k''a is small, it can be shown that k''a = 1/{κQ sin(k'a)}. Losses are lowest at mid-band, and strong coupling and a high Q-factor are required for low loss.

In [START_REF] Shamonina | Magnetoinductive waves in one, two and three dimensions[END_REF], it was shown that a non-reflective termination is formed by inserting an impedance Z 0 into the last (0 th ) element of the waveguide, where Z 0 is given by:

Z 0 = jωM exp(-jka) (3)
At mid-band, when ω = ω 0 and ka = π/2, Z 0 reduces to the real value Z 0M = ω 0 M. Thus, in principle it should be possible to couple a magneto-inductive waveguide to a conventional transmission line (which has real impedance) if M is appropriately chosen.

Unfortunately, Z 0 is complex away from the band centre, and moreover as expressed here is a function of both ka and ω. Consequently, Equation 3 has so far represented a mathematical contrivance, rather than an element that can be realised. Further development of MI systems clearly requires simple circuits that can approximate this impedance.

In a search for suitable circuits we consider the termination in Figure 1b. Here the final element of a MI waveguide is coupled via a mutual inductance M' to a loop containing an inductance L', a capacitance C' and a real load R L , which will typically represent a 50 Ω system. The loop is resonant at an angular frequency ω 0 ' = 1/L'C', but may be considered non-resonant when ω 0 ' is zero. For the final elements, the circuit equations are:

{jωL + 1/jωC}I 0 + jωMI -1 + jωM'I L = 0 {R L + j(ωL' -1/ωC')}I L + jωM'I 0 = 0 (4)

Combining Equations 4, we obtain for the 0 th element:

{jωL + 1/jωC + Z L }I 0 + jωMI -1 = 0 (5)
Here Z L is a load that has effectively been inserted into the 0 th element, given by:

Z L = ω 2 M' 2 /{R L + jωL'(1 -ω 0 ' 2 /ω 2 )} (6)
We may evaluate the performance of Z L as a termination by considering the reflection of current waves as shown in Figure 1c. Assuming solutions of Equations 1 and 5 as the sum of incident and reflected waves, i.e. as I n = I I exp(-jnka) + I R exp(+jka), substituting into Equation 5 and using Equation 2we can obtain the reflection coefficient Γ = I R /I I as:

Γ = -{Z L -Z 0 } / {Z L + Z 0 *} ( 7
)
Equation 7 is similar to the reflection coefficient for current waves obtained when a conventional transmission line is terminated with a load. However, due to the presence of a complex conjugate term, it is clearly not identical.

Since the current in element zero, which passes through the effective load, is I 0 = I I + I R , we can also define a transmission coefficient as T = I 0 /I I = (1 + Γ), or:

T = 2Re(Z 0 )/{Z L + Z 0 *} ( 8
)
Equation 8 is again similar, but not identical to the conventional transmission coefficient, due to the presence of a real operator. It is simple to show that Γ and T satisfy the power conservation relation:

ΓΓ* + TT* Re(Z L ) / Re(Z 0 ) = 1 (9)
Once again, Equation 9is similar to the corresponding result for real-valued systems.

Different terminations can be compared by plotting the scattering parameter S 11 in dB, as:

S 11 ≈ 10 log 10 { Γ 2 } (10)
Equation 7 implies that Z L should be chosen to approximate Z 0 as far as possible. We now compare a number of possibilities, assuming for simplicity that Z 0M = R L and that M' = M.

Figure 2 shows the frequency variation of S 11 obtained in a non-resonant transducer for different values of L'/L, calculated assuming the typical coupling coefficient κ = 0.6.

Propagation can take place over the band 0.79 ω/ω 0 1.58, and S 11 rises to 0 dB at the band edges. Within the band, S 11 reduces somewhat. However, when L'/L = 1, the reflection coefficient is generally high and the transducer is correspondingly ineffective.

As L'/L reduces, a deeper and deeper minimum in S 11 develops, and gradually shifts towards ω/ω 0 = 1. These results imply that the performance of the transducer improves as L'/L reduces, and that the best result is obtained when only a small reactance is inserted into the final loop. However, the best that can be achieved still only represents a narrowband impedance match. Furthermore, the minimum in S 11 is relatively high (-35 dB), and even this result is only obtained for very small values of L'/L, when it may be difficult to maintain M'/M = 1.

The inserted reactance can clearly be cancelled more effectively if the transducer is made resonant. Figure 3 shows the corresponding variation of S 11 obtained for a resonant transducer, calculated assuming that κ = 0.6 and that L' and C' are chosen so that ω 0 ' = ω 0 . Results are again shown for different values of L'/L. The most obvious choice of L'/L = 1 provides a complete null in reflectivity when ω/ω 0 = 1. However, this result again only represents a narrow-band match. In contrast, the less obvious choice of L'/L = 0.5 provides two nulls in reflectivity, the first again being at ω/ω 0 =1. Because the nulls are widely separated, low reflectivity is obtained over a wide band. For example, S 11 is less than -20 dB for 65% of the pass-band, and less than -30 dB for 45%. This form of transducer is extremely effective. Importantly, it can be realised very conveniently as shown in Figure 1d. Here each resonant element in a MI waveguide is now formed from two separate inductors of value L/2, and two capacitors of value 2C. If half the final element is simply removed, the remainder may then be connected directly to a resistive load. This modification allows MI waveguides of arbitrary length to be terminated without the need for additional components. To realise such an arrangement, all that is required is to stagger the inductors and capacitors in each unit cell, so that the cell may be exactly bisected using a transverse cut.

Transducer optimisation

We now consider the broadband resonant transducer in more detail. To understand how it achieves its effect, we introduce the normalised characteristic impedance Z 0N = Z 0 /ω 0 M, obtained from Equation 3 as:

Z 0N = w/{sin(ka) -j cos(ka)} (11)
Here w = ω/ω 0 is a normalised frequency. Using the dispersion equation ( 2), Z 0N may be written alternatively as:

Z 0N = w/{√[1 -(1 -1/w 2 ) 2 /κ 2 ] + j(1 -1/w 2 )/κ} (12)
In this form, Z 0N is clearly a function only of w. Consequently it may be compared directly with the corresponding normalised load impedance Z LN /ω 0 M, which may be written as:

Z LN = w/{ρ/wµ 2 + j(2λ/κµ 2 )(1 -η 2 /w 2 )} (13)
Here we have introduced four normalised variables. The first, ρ = R L /ω 0 M, is the ratio of the R L to the mid-band impedance of the MI waveguide. The second, λ = L'/L, is the ratio of the self-inductances in the transducer and the guide, the third, µ = M'/M, is the ratio of mutual inductances, and the fourth, η = ω 0 '/ω 0 , is the ratio of resonant frequencies.

Clearly, reflectivity will be low if the real and imaginary parts of Z 0N and Z LN are similar, Considering now the real parts, and assuming that the transducer is correctly resonant, Re(Y 0N ) can be made equal to Re(Y LN ) if:

or
ρ/wµ 2 = √[1 -(1 -1/w 2 ) 2 /κ 2 ] ( 14
)
Equation 14 is actually a quadratic equation, which can be expanded as:

w 4 {κ 2 -1} + w 2 {2 -α 2 κ 2 } -1 = 0 (15)
Here, α = ρ/µ 2 . When α = 1, a condition that can be achieved by taking M' = M and ω 0 M = R L , Equation 15 has the simple solutions of w = 1 and w = 1/√(1 -κ 2 ). Both lie in the propagating band. These results imply that if ω 0 M = R L , and if the transducer is made resonant at ω 0 using an inductance L' = L/2 (which requires a capacitance C' = 2C), the imaginary parts of Y L and Y 0 can be made equal across the band and the real parts at two discrete frequencies. Consequently, Z 0 and Z L can be equalised at the same two frequencies, and at these points there can be no reflection. These results are confirmed in Figure 4, which shows the frequency variations of the real and imaginary parts of Z 0N and Z LN , calculated assuming κ = 0.6. Matching is achieved when w = ω/ω 0 = 1 and ω/ω 0 = 1/√(1 -0.6 2 ) = 1.25, the points at which nulls in reflectivity are seen in Figure 3.

Broadband operation then follows from the existence of these two separate nulls.

A broadband transducer can still be constructed if α 1 (for example, if M' = M but ω 0 M R L ). Figure 5 shows the frequency variation of the scattering parameter S11 for resonant transducers, calculated assuming κ = 0.6, and L'/L = 0.5 and assuming different values of α. For α < 1, there are again two nulls in reflectivity, which move further apart as α reduces. For α > 1, there is only a single minimum, not a null. However, in each of the cases shown, the return is generally low over a wide spectral range. These results suggest that the broadband resonant transducer will give reasonable performance even when the mid-band impedance of the MI waveguide is slightly mismatched from RL.

Finally, it is simple to show that the two nulls in reflectivity just merge together when the roots of Equation 15are repeated. This occurs when

(2 -α 2 κ 2 ) 2 = 4(1 -κ 2 ) ( 16
)
Or when:

α 4 κ 2 -4α 2 + 4 = 0 ( 17
)
Equation 17 has the solutions α = (√2/κ){1 ± √ (1 -κ 2 )]} 1/2 . For κ = 0.6, for example, the solutions are α = 3.162 and α = 1.054. The latter value is midway between α = 1 and α = 1.1, and generates a variation in reflectivity with a single null.

Experimental verification

Experimental confirmation of the theory of the previous section was provided using thinfilm magneto-inductive cables formed by double-sided patterning of copper-clad polyimide, as described in [START_REF] Syms | Thin-film magneto-inductive cable[END_REF]. Figure 6a The photomasks contained a set of MI waveguides with different parameters. The overall width and length were taken as w = 4.7 mm and a = 100 mm throughout, so that a twometre length contained 19 resonant elements. The track width t, and the small gaps g C and g L between capacitor plates and between plates and tracks were all taken as 0.5 mm. The main variables were the capacitor and inductor lengths d C and d L , which were varied to obtain different properties. Of particular importance were the inductance L, the capacitance C, the mutual inductance M and the Q-factor of the resonant elements. These parameters determine the resonant frequency f 0 = ω 0 /2π, the coupling coefficient κ = 2M/L, the mid-band impedance Z 0M = ω 0 M and an imaginary part of the propagation constant.

Electrical performance was evaluated using an Agilent E5061A Electronic Network Analyser (ENA). The inductance was determined by making the transducers resonant at low frequency with a known capacitor and measuring the resonant frequency with a weak inductive probe. The remaining parameters were estimated by attaching SMA-type endlaunch connectors as shown in Figure 6c, measuring transmission and reflection data, and fitting the data to a theoretical model. Recently, it has been shown that flexible magnetoinductive cable is very tolerant to bends, due to the extreme stability of the mutual inductance between adjacent elements [START_REF] Syms | Bends in magneto-inductive waveguides[END_REF]. Despite this, small reflections tend to mask the appearance of predicted nulls in S 11 . The smallest overall reflections were therefore obtained with the MI waveguide held straight, using an additional co-axial cable to return the transmitted signal to the ENA. The terminations were made resonant using surface mount capacitors, using the optimum capacitance value of 2 x 11.6 ≈ 23 pF. Electrical performance was then re-measured to give the results shown in Figure 8. Here, peak transmission has now increased to -5.2 dB, and the oscillations due to multiple reflections have largely disappeared. The return has significantly reduced, and S 11 is below -25 dB for much of the band. The data are again compared with theory, this time assuming a resonant termination, and good agreement is again obtained. However, the predicted nulls in reflection cannot be seen in the experiment, presumably due to other small reflections from connections. These results confirm the improvement in performance offered by the optimum resonant termination.

Conclusions

A simple inductive transducer for coupling a magneto-inductive waveguide to a real load has been introduced. A theory of reflection from lumped-element coupling transducers has been developed. It has been shown that zero reflection can be obtained at a single frequency if the (real-valued) mid-band impedance of the MI waveguide matches the load, and if the transducer is resonant at the same frequency as the resonant elements forming the guide. If in addition the inductance of the transducer is half that used in the resonant elements and the capacitance is correspondingly double, matching to the load can also be achieved at a second frequency. Since zero reflection is now obtained at two separate frequencies, low reflectivity can be obtained over a broad spectral range. The theory has been compared with experimental results obtained from thin-film magnetoinductive cable. Excellent agreement has been obtained, and the improvement in performance offered by the optimised resonant transducer has been confirmed by comparison with a non-resonant equivalent.

Other more complicated transducer designs can doubtless be developed to achieve improved broadband performance. However, the simple design presented here has the important advantage that a transducer with exactly the required properties can be obtained from the waveguide itself, if the resonant elements are formed using pairs of inductors and capacitors in series, rather than single components. As a result, a connection between a MI waveguide and a system with real impedance may be obtained simply by splicing, exactly as is done for other cable types such as co-axial cable. The demonstration of a Figures

  alternatively if the real and imaginary parts of the normalised admittances Y 0N = 1/Z 0N and Y LN = 1/Z LN correspond. Considering first the imaginary parts, Im(Y 0N ) can be made equal to Im(Y LN ) for all ω if η = 1 and λ = µ 2 /2. If µ = 1, as previously assumed, a complete match in admittance can therefore be obtained if the transducer is resonant at ω 0 and the inductance L' of the transducer is half that of the resonant elements forming the guide.

  shows the physical arrangement and key dimensions, and Figure 6b a short section of cable. Each resonant element is formed from two inductors and two capacitors. The inductors are single-turn loops of inductance L/2, located on either side of a thin substrate, while the capacitors are parallel-plate components of capacitance 2C, which use the substrate as a dielectric interlayer. This arrangement approximates that of Figure 1d, although it does not allow a resonant termination to be formed by cutting the cable. Instead, it provides non-resonant terminations of inductance L' = L/2, which may be made resonant with additional capacitors. Since the mutual inductances M and M' are the same, this arrangement has normalised parameter values λ = 0.5 and µ = 1.Cables were fabricated in two metre lengths by the UK company Clarydon (Willenhall, West Midlands). The base material consisted of 25 µm thick Kapton ® carrying a 35 µm thick layer of copper on either side. The copper was patterned by step-and-repeat lithographic exposure to a pair of one metre long photomasks, followed by wet etching.

18 )S 21 ≈

 1821 The theoretical model includes propagation loss but ignores multiple reflections. In this case, the scattering parameters S 11 and S 21 for an N-element MI waveguide connected to a source with real output impedance R L and a similar load are given approximately by: S 11 ≈ 10 log 10 { Γ 2 } (10 log 10 {(1 -Γ 2 ) exp(-2Nk''a) (1 -Γ 2 )}[START_REF] Syms | Magneto-inductive waveguide devices[END_REF]

Figure 7

 7 Figure 7 shows a comparison between experimental measurement of the frequency variations of S 11 and S 21 and the theory above, for a cable with d C = 10 mm and d L ≈ 90

1 . 2 . 3 . 4 . 5 . 7 . 8 .

 1234578 Magneto-inductive waveguide, with a) ideal termination, b) real termination c) equivalent termination and d) self-termination. Frequency variation of the scattering parameter S11 for non-resonant transducers calculated assuming M' = M, κ = 0.6 and different values of L'/L. Frequency variation of the scattering parameter S11 for resonant transducers, calculated assuming M' = M, κ = 0.6, ω 0 ' = ω 0 and different values of L'/L. Frequency variation of the real and imaginary parts of the normalised impedance Z 0N = Z 0 /ω 0 M (thin lines) and Z LN = Z L /ω 0 M (thick lines), calculated assuming M' = M, κ = 0.6, ω 0 ' = ω 0 and L' = L/2. Frequency variation of the scattering parameter S11 for resonant transducers, calculated assuming M' = M, κ = 0.6, L'/L = 0.5 and different values of α. 6. a) Arrangement and b) experimental realisation of thin-film magneto-inductive cable; c) arrangement for resonant connection to conventional coaxial cable. Frequency variation of S 11 and S 21 for thin-film magneto-inductive cable, with nonresonant transducers. Thick lines are experimental data and thin lines are theory. Frequency variation of S 11 and S 21 for thin-film magneto-inductive cable, with resonant transducers. Thick lines are experimental data and thin lines are theory.

simple arrangement for coupling should greatly increase potential applications for MI waveguides.