Application of surface coil in nuclear magnetic resonance studies of physical properties of quasi-2D materials

The nuclear magnetic resonance (NMR) technique is a very powerful scientific tool, both in medical imaging and basic science. More precisely, NMR is a bulk microscopic probe of magnetism that can be used to effectively determine spatial variations of local magnetic properties in matter. Therefore, additionally to being a valuable tool for studying ordered magnetic states, it can be used to access real space features in spatially inhomogeneous states and those characterized by short range magnetic orders. Furthermore, the element site-specific nature of our probe permits the separation of the itinerant from local electronic properties. The applicability of NMR over a broad range of magnetic fields up to 45 T¹ allows an unparalleled exploration of phase diagrams. However, the problem is that detection of the NMR signal typically requires an order of 10²⁰ spins. For this reason, the application of NMR to study properties of nano-structures, 2D materials, and devices is precluded.

Inherently low sensitivity of NMR can be improved by enhancing the difference of Boltzmann population of the nuclear Zeeman energy levels by either lowering temperature and/or increasing the applied magnetic field. However, we often require knowledge of physical properties of matter as a function of temperature and/or applied field. Thus, the sensitivity has to be improved by varying some other parameter to probe the intrinsic physical properties. The signal-to-noise ratio (SNR) in an NMR experiment is also controlled by the filling fraction, which is the ratio of the volume of the effective radio frequency (RF) field to the sample volume.² Consequently, by increasing the filling factor of the solenoid coil, the sensitivity can be improved, as illustrated in Fig. 1. Nonetheless, this strategy does not work for volume limited samples and/or 2D thin-films. In such cases, the application of surface micro-coils can be beneficial.^3–5 This is because the effective RF field strength rapidly decays away from the surface of the coil.³ Since the effective RF field is confined to a region adjacent to the coil, the filling factor, and thus SNR, is maximized for 2D-like samples or for probing sizable surface area to a limited depth. In this paper, we investigate the performance of surface coils for NMR studies of quasi-2D materials. We compare the reception sensitivity of different geometries of surface coils and identify one which is optimal for a given application. We find that the optimal sensitivity for an applied field parallel to the coil plane is obtained by placing the sample into the cavity of the spiral surface coil. Furthermore, we identify meander-line surface coils as best suited for the studies of intrinsic anisotropic properties. We emphasize that in addition to the gain in sensitivity for such samples the application of the surface coils offers ready access to the sample. This is important if other external parameters, such as bias gate voltages and/or applied strain, are to be varied.

All surface coils produce an RF field whose magnitude decays away from the coil in the direction perpendicular to its plane. The inhomogeneity of the RF excitation field is one of the major challenges in the application of surface coils to imaging and high resolution NMR spectroscopy.⁶ However, for the following reasons, this is not an issue in the NMR study of fundamental properties of quantum materials. Often intrinsic magnetic and/or electronic inhomogeneities and textures far exceed the inhomogeneity of the excitation field generated by the surface coil.^7–10 The penetration of the RF field is spatially varying due to electronic and superconducting current shielding effects in metallic and superconducting samples, respectively.¹¹ As far as applications to condensed matter physics examined here, the main advantage for using surface coils is increased sensitivity for resonance studies of devices and quasi-2D materials, tuning of the probing depth of sizable surface area of the sample, and ready access to the sample.

The paper is organized as follows. We discuss basic principles and assumptions necessary for the intuitive understanding of the NMR performance of different surface coils in Sec. II. In Sec. III, we describe experimental details about our NMR setup, coil fabrication, and the samples. Measurements of the NMR spectra acquired by separate surface coils and for the different orientations of the magnetic field are presented in Sec. IV.

The optimal choice of surface coil geometry relies on the specifics of a particular application. We will discuss the two geometries illustrated in Fig. 2. They differ in uniformity of the in-plane RF fields. These are meander-line and Archimedean spiral coils. As shown in Fig. 2(a), the meander-line coil is a serpentine array of parallel conductors of mutual separation a. The Archimedean spiral is defined as r = sθ in polar coordinates, where s is the spiral constant. Coils of spiral geometry induce more uniform in-plane RF fields and thus represent a more appropriate geometry for applications requiring uniform excitations. On the other hand, the meander-line coils are better suited for the investigation of intrinsic magnetic anisotropies, since they minimize artifacts associated with the effective RF field anisotropy, as we demonstrate in Sec. IV C.

The excitation field pattern of the surface coil is fully described by the spatial distribution of the magnetic field (B₁) produced by unit current through the coil and can be calculated from the Biot-Savart law.¹² For a given surface coil, excitation and reception patterns are identical.⁶ Therefore by calculating the spatial distribution of the B₁ magnetic field, one can also obtain information about the reception sensitivity of the coil. In materials such as metals, where RF field penetration is spatially non-uniform, calculation of the spatial distribution of B₁ does not necessarily give precise reception sensitivity. Thus, to compare true reception sensitivity of different coils in our study, we have independently optimized excitation pulses for each coil and applied field orientation.

The spatial distribution of the B₁ magnetic field has been calculated for both spiral⁴ and meander-line^3,12 coils. In what follows, we give a brief overview of the main results of these calculations. The distribution of the magnitude of B₁ calculated near a conductor of a meander-line coil, defining the x–y plane, in the near field region, for z/a $≪$ 1, is given by

for $−a/2≤x≤a/2$⁠. Here z is the distance away from the coil plane, and the coil strips are considered as infinitely thin but of finite width. The magnitude of B₁ is the quantity of interest in nuclear quadrupole resonance (NQR) detection, while the individual components of B₁ perpendicular to the applied field define excitation/detection pattern in NMR. The important finding is that the RF magnetic field has the periodicity of the meander-line itself, in planes parallel to the surface of a meander-line coil. However, its strength at a distance z away from the coil is given by exp(−πz/a). Therefore, the effective RF field is confined to a region adjacent to the coil and its penetration depth is determined only by the spacing a and not by the overall size of the coil. Since the signal-to-noise ratio in an NMR experiment is proportional to the filling fraction, the meander-line coils are ideally suited for probing thin 2D-like samples or sizable surface area to a limited depth.^3,6 Consequently, by adjusting only the spacing a, one can control the SNR and sensitivity depth.

For spiral coils, the on-axis field decays more slowly at a distance z away from the coil plane than that for the meander-line coil. In the limit of spacing between conducting traces (Δ) going to zero, the on-axis field reduces to the expression for a single loop RF coil carrying a current I in the AC conductor limit,^4,6

Evidently, the RF field and consequently SNR decrease with increasing axial distance from the coil’s center. Total B₁ field can be increased by adding turns to the spiral. Each additional turn strengthens total B₁ field by superposition. However, as additional turns add to the total resistance to the coil, there is an optimal number of turns for each coil configuration that maximizes SNR.⁴ For our designs, we did not find decreasing reception sensitivity for the coils with up to 7 turns.

The measurements were done using a high homogeneity superconducting magnet with a field strength of 7 T. Data were taken at room temperature. The NMR data were recorded using a state-of-the-art laboratory-made NMR spectrometer. The coils were mounted on a homemade broadband NMR probe, constructed based on the design described in Ref. 13. Variable capacitors were used to tune each coil to the desired resonance frequency and assure that the circuit is matched to 50 Ω. Thus, the Q factor was comparable for each coil used in this study. NMR absorption spectra were obtained from the Fourier transform of the spin-echo. We used a standard spin echo sequence (π/2 − τ − π), with pulses independently optimized for each coil and applied field orientation. This was done to assure that only reception sensitivity is compared in each case. Because the samples are highly inhomogeneous, only a single line was observed for ¹¹⁵In NMR. That is, we did not observe nine distinct quadrupole satellite lines expected for ¹¹⁵In with nuclear spin I = 9/2 in a non-cubic local environment.

We used CST Microwave Studio, a popular commercial tool for 3D electromagnetic (EM) simulation of microwave frequency and RF, to build 3D models of the various surface coils. This software generates the 3D field distribution using a discretized solution of the integral formulation of Maxwell’s equation. The calculated field distribution depends on the exact boundary conditions used. We used open boundary conditions for the spatial part and in time domain by a so-called “transient finite difference time domain” approach. These conditions produce results that are sufficiently accurate for magnetic resonance imaging (MRI) applications. However, we used the software just as a rough guide for the field distribution and used our measurements to identify an ideal coil geometry for a particular application. More importantly, the CST Microwave Studio output file is used by CircuitCAM software to build the design for machining the coil. CircuitCAM outputs the file in .lmd format [common printed circuit (PC) board format] which is then loaded to BoardMaster software to run computer numerical control (CNC) machines for physical coil fabrication. We used a 300 μm thick PC board with a Cu conducting layer of a thickness of 25 μm to machine the coils of desired geometry. Coils were fabricated at the NMR facility, Division of Structural and Synthetic Biology Centre for Life Science Technologies, RIKEN Yokohama Campus, Japan, led by Professor Maeda.

We also fabricated meander-line micro-coils with 100 nm spacing between the conductors using lithographic techniques. The gold conducting leads were wire bound and used to connect to the NMR tank circuit. We were able to send sufficient power to detect an NMR echo signal without burning the coil with the RF power. These type of coils can be very beneficial for the NMR studies of nano-devices.

For this study, we used 400 μm thick InP substrates as our test samples made at the IBM research center in the group of Dr. D. Sadana. The InP substrates are semi-insulating films doped with Fe, with room temperature resistance exceeding 1 MΩ. Such samples were chosen for the following reasons. Both ¹¹⁵In and ³¹P nuclei provide good NMR sensitivity. These substrates can be easily cut to the desired shape, allowing us to perform a comprehensive set of desired performance tests. The thickness of the substrates was selected to be comparable to the spacing between neighboring Cu conductors in the surface coil. For meander-line coils, this assured that the effective RF field would penetrate the entire thickness of the sample. To eliminate artifacts associated with the skin-depth and induced surface currents, Fe doped insulating InP samples were taken.

In a typical NMR experiment, a sample is placed in a solenoid coil, as described earlier. Such coil provides very low filling fraction, and thus low SNR, for flat 2D-like samples. This can be easily resolved in part by using a flat solenoid, as depicted in Fig. 1(b). As a matter of fact, we found that the SNR can be improved by a factor of two, as shown in Fig. 3. However, significant improvement of reception sensitivity by a factor of six can be achieved when a surface coil is employed as compared to the solenoid coil. In Fig. 3, we plot magnitude of the ¹¹⁵In NMR spectra for the same square sample acquired using three different coils, as denoted. The results clearly demonstrate the advantage of the surface coil. Here, we used the spiral surface coil, with the coil plane oriented parallel to the applied field and the sample covering significant area of the coil. We emphasize that the observed sensitivity enhancement comes exclusively from the improvement of the effective filling fraction, defined in this case as the ratio of the sample thickness to the effective RF field decay length, for the thin sample placed on the surface coil.

Our next step is to determine the optimal geometry of the surface coil for a given sample size and the appropriate variation of the external parameters, such as the applied magnetic field. This is particularly important when specific anisotropic and/or inhomogeneous quantities are to be investigated.

In this section, we will examine the performance of different surface coils with the plane oriented parallel to the applied field. We first consider spiral coils. As discussed in Sec. II, the strongest effective RF field is induced at the center of the spiral and quickly decays as one moves away from the plane of the coil. When the applied field is oriented parallel to the surface of the coil, this component of the effective RF field is responsible for the spin flip since it is perpendicular to the applied field. This also implies that the most sensitive reception region is in the center cavity of the spiral coil. Therefore, for this applied field orientation, we expect that placing the sample in the center cavity of the spiral should provide the best reception sensitivity and consequently SNR.

To test this hypothesis, we used the same spiral coil to obtain the NMR signal from two samples of different sizes, as illustrated in Figs. 4(a) and 4(b). One sample is such that it can be entirely placed into the center cavity of the spiral, while the other covers most of the area of the same coil. In Fig. 4(c), we plot the magnitude of the ¹¹⁵In NMR signal per nucleus acquired by the same spiral coil for samples of different sizes, placed as shown in the figure. For these particular geometries, we find that the magnitude of the signal for the case when the sample is in the cavity exceeds by more than a factor of three than that when the sample covers the coil. This result confirms the hypothesis that highest sensitivity can be obtained by placing the sample in the center cavity of the spiral.

Therefore, for this applied field orientation and given sample size, the best sensitivity can be achieved by designing the spiral coils such that the entire sample can be placed into the center cavity. The sensitivity can be further improved by increasing the effective RF field by adding turns to the spiral. The requirement that the tank circuit must tune to the resonance frequency of a particular nuclear species imposes an upper bound on the total number of coil turns. Therefore, high frequency applications of the spiral coil with the sample placed in the center cavity might not offer significant gain in sensitivity as compared to other coil geometries such as meander-line.

Next, we compare the signal magnitude obtained using meander-line and spiral surface coils, as depicted in Figs. 5(a) and 5(b). The comparison is performed at two separate frequencies, differing by nearly a factor of two, corresponding to ¹¹⁵In and ³¹P resonances. Two spiral coils with the same size of the center cavity are employed to achieve the required resonance frequencies. In both cases, the sample is positioned in the center cavity of the coil. The spiral with less turns is used for higher frequency ³¹P NMR, as displayed in Figs. 5(c) and 5(d).

In Fig. 5(c), we plot the magnitude of ¹¹⁵In NMR signals acquired by different surface coils. Two traces are acquired from the same sample, as depicted: one from the sample being placed in the cavity of the spiral coil and the other on the meander-line. The signal detected by a dense seven turn spiral is nearly 16 times stronger than that detected by the meander-line coil. As previously discussed, decreasing the number of turns of the spiral lowers the effective RF field in the center and thus the sensitivity. We demonstrate this effect by comparing the signal detected from the same sample by a three turn spiral and the meander-line. As shown in Fig. 5(d), the signal detected by a three turn spiral is only 2 times stronger than that detected by the meander-line coil. However, signal detected by the dense 7-turn spiral is nearly 8 times that detected by that with three turns.

We now comment on the applications of surface coils in studies of metallic samples with a finite skin depth, i.e., a characteristic length of penetration of the AC current. As described earlier, in an applied field oriented parallel to the coil plane, the effective RF field responsible for the spin flip quickly decays away from the plane of the coil. This indicates that the effective RF field is confined to a region adjacent to the coil and its penetration into the sample is finite. Therefore, the filling fraction, and thus SNR, is determined by the ratio of the decay length of the effective RF field and the total sample thickness. For the meander-line coil, the decay length is determined only by the spacing between the adjacent parallel conductors of the coil, while it is the opening radius of the spiral that controls the decay length for the spiral coil, as described in Sec. II. In either case, the decay length is not controlled by the overall size of the coil. The finite skin depth determines the effective penetration depth for the RF field in metallic samples. Therefore, the optimal filling fraction, and SNR, is obtained when the decay length matches the skin depth. That is, a meander-line coil with a spacing between conducting lines that matches the skin depth provides optimal filling fraction and thus sensitivity. For a spiral coil, the opening radius should match the skin depth to optimize sensitivity. This theoretical prediction was tested on a metallic sample with a skin depth of the order of 100 μm at 100 MHz. The platelet-type sample, of thickness ≈5 mm, was separated from the surface of the coil by ≈250 μm thick insulating Kapton tape. A 30% stronger signal was detected by the meander-line coil, with a ≈ 1 mm, and then by the flat solenoid. Therefore, for platelet-type samples, it becomes advantageous to use surface coils over the conventional solenoids when the overall thickness of the sample (for non-conducting materials), and/or the RF skin depth (for conducting materials), is of the order of the spacing between conducting lines for a meander-line coil and/or the opening radius for a spiral coil. We note that for the practical purposes described here, the penetration depth in a superconducting material plays the role of the skin depth in a metal. Since the superconducting penetration depth is typically much shorter than the skin depth, the meander-line coils with a sub-millimeter spacing between conductors should be used to investigate superconducting properties. For spiral coils, the opening radius should be made as small as possible to optimize the sensitivity in the superconducting phase.

In what follows, we will examine the performance of surface coils when their plane is oriented perpendicular to the applied field. This is an important case, as this geometry is required in the study of physical phenomena such as vortices in superconductors and quantization of 2D electron gas. In Sec. IV A, we established that placing the sample in the center cavity of the spiral coil provides the best reception sensitivity because the strongest effective RF field is induced at the center of the spiral. However, for the applied field oriented perpendicular to the plane of the coil, this RF field, being aligned with the applied field, cannot induce spin flips. Therefore, no NMR signal can be detected in this geometry.

For this applied field orientation, the NMR signal is generated by the effective in-plane RF fields. We compare the NMR signal acquired on the same sample by three different coil geometries that produce such RF fields. In Fig. 6, we plot the magnitude of ¹¹⁵In NMR signals acquired by meander-line, square, and spiral surface coils, as denoted. In all three cases, the sample covers a significant area of the coil. For this insulating sample, the signal acquired by the meander-line coil is the weakest.

To study intrinsic anisotropies, it is crucial to investigate sample properties as a function of the orientation of the applied magnetic field with all other parameters being fixed. It is important to compare the relative magnitude of the signals acquired with the plane of the coil oriented parallel and perpendicular to the applied field. Next, we present such comparison for three different coil designs described earlier.

Relative magnitude of the ¹¹⁵In signals obtained using meander-line, square, and spiral surface coils with the plane of the coils oriented parallel and perpendicular to the applied field is plotted in Figs. 7–9, respectively. The smallest variation in the detected magnitude for two field orientations is observed for signals acquired by a meander-line coil. That is, when the plane of the meander-line coil is perpendicular to the applied field, a signal 1.8 times stronger is detected. On the other hand, the signal for a spiral coil varies by a factor of three for two different field orientations. Therefore, meander-line surface coils are best suited for the studies of intrinsic anisotropic properties as their use minimizes artifacts associated with the effective RF field anisotropy.

Next, we comment on the applications of surface coils in studies of samples with a finite skin depth in an applied field oriented perpendicular to the coil plane. As described earlier, the effective in-plane RF fields are responsible for the spin flip in this field orientation. These RF fields have the periodicity of the coil and are confined to a region adjacent to the coil. Therefore, the filling fraction, and thus SNR, is optimal when the spacing between the conductors matches the skin depth, as was the case for the field applied parallel to the coil plane.

Lastly, we discuss the application of surface coils to investigate numerous exotic properties in layered organic conductors. This class of materials exhibits rich physics (see, for example, Refs. 14–17). Among this, their superconducting properties are highly anisotropic and unique. For example, one can use NMR to investigate the dynamics of Josephson vortices and/or properties of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, the spatially modulated superconducting phase that emerges in fermionic systems with population imbalance between pairing species. In both cases, one is required to align the applied magnetic field with high precision along one particular axis of the crystal. This poses a problem for an NMR experiment, where the sample is very small (e.g., 0.04 × 0.1 × 0.5 mm) and sometimes needle shaped.¹⁷ In this case, physics demands that the magnetic field be oriented along the needle direction. One would have to make an extremely thin solenoid of one or few turns to optimize the filling fraction. However, such a sample can be readily mounted into the central cavity of a surface coil. Ideally, one should use a square surface coil with a cavity length and width matching those of the sample. In this case, the coil plane would be parallel to the applied field.

We investigate the performance of surface coils for NMR studies of quasi-2D materials. That is, we compared the reception sensitivity of surface micro-coils of spiral and meander-line geometries. The optimal geometry of the surface coil for a given application and the direction of the applied field are identified. In this study, we were not concerned with the homogeneity of the effective RF field induced by the coil since the phenomena in condensed matter systems often produce intrinsic inhomogeneity exceeding by far those associated by the RF field.

For the magnetic field applied parallel to the coil plane, the best sensitivity can be achieved by employing the spiral coil designed so that the entire sample can be placed in the center cavity of the coil. However, this geometry will yield no signal for the magnetic field applied perpendicular to the coil plane. In this case, the best sensitivity can be achieved by employing the spiral coil designed so that the sample covers a significant area of the coil.

Meander-line surface coils are best suited for the studies of intrinsic anisotropic properties since their use minimizes artifacts associated with the effective RF field anisotropy. Furthermore, meander-line geometry is well suited for the study of samples in which RF penetration is spatially inhomogeneous, such as metals and superconductors. In this case, optimal sensitivity is obtained if the spacing between conducting lines matches the RF penetration depth. For an applied field parallel to the coil plane in platelet-type samples, it becomes advantageous to use surface coils over conventional solenoids when the overall thickness of the sample (for non-conducting materials), and/or the RF skin depth (for conducting materials), is of the order of the spacing between conducting lines for a meander-line coil and/or the opening radius for a spiral coil. Finally, in addition to the gain in sensitivity for 2D-like samples, application of the surface coils offers ready access to the sample, which can be crucial in allowing in situ variation of parameters, such as bias gate voltage.

We would like to thank Dr. Devendra K. Sadana, IBM—T. J. Watson Research Center, for providing InP thin substrate samples for our work and Professor Hideaki Maeda, RIKEN Yokohama Campus, for hosting Lu Lu and making coil fabrication facility available to us. We acknowledge guidance for software use from Dr. Yoshinori Yanagisawa and Dr. Masato Takahashi. This research was supported in part by the National Science Foundation under Grant No. DMR-1608760.