Design of planar microcoil-based NMR probe ensuring high SNR

NMR analyses at micro-scale, such as NMR spectroscopy, can be a very useful tool to determine the composition of very minute volumes of samples. Micro-NMR systems reported previously have been realized with several types of coils, such as Helmholtz,^1–3 solenoidal,^4–6 planar^7–12 or even micro-strip lines.¹³ These systems operate at different Larmor frequencies with values varying in a wide range, from 63.88 MHz when the externally applied DC magnetic field (MF) has the magnetic field strength (MFS, i.e. magnetic flux density) B₀ = 1.5 T,¹⁴ to as high as 500 MHz when B₀ = 11 T.⁸ 

The first generic expression for the signal-to-noise ratio (SNR) for an NMR experiment was formulated almost 40 years ago by D. Hoult et al.:¹⁵ 

where K is the inhomogeneity factor of the MF generated by the coil, $η$ is the filling factor which indicates the extent of coupling of the generated magnetic energy of the coil with the sample, B₁ is the time-varying MFS which lies in the plane perpendicular to the direction of B₀, i is the current generating the RF MF of strength B₁, $ω0$ is the Larmor frequency, M₀ is the strength of the magnetization vector which depends on B₀ and the sample under investigation, and V_s is the sample volume. Equation 1 indicates the parameters involved in the calculation or optimization of SNR, but it may need to be modified (or some of its components recalculated) depending on the specifics of the used system.

Here we are mainly interested in estimating the SNR of miniaturized micro-scale NMR probes for which the sample noise is not significant owing to its small volume (nL - μL). This paper focuses on the theoretical study of various factors affecting SNR of such a microNMR probe that consists of a planar microcoil with conductors buried in its substrate, and an adjacent microfluidic structure with which the sample is manipulated and/or placed in the vicinity of the microcoil. Most importantly, our paper defines a microcoil performance factor P that enables one to optimize the SNR performance of the microNMR probe, independently of the external setup and the proton density of the measured sample. Additionally, the paper details the analysis and calculation of the main components of P, namely the filling factor $η$ and the inhomogeneity factor K. The factor P can help the design of planar microcoil-based microNMR probes with optimal sample volume which will provide an optimal performance independent of the external setup used for the NMR experiment(s). We focus mainly on planar microcoils, since they offer the easiest fabrication for smart lab-on-a-chip (LOC) microsystems, and are already widely used in many RF applications. However, a comparison with a conventional solenoidal coil is also given for a better understanding of the performance of planar microcoil-based microNMR probes.

The literature review summarised in the  Appendix indicates clearly that it is very difficult to compare the performance of all the various types of coils used in different NMR/MRI probes and especially to understand which are the key parameters influencing their SNR. Therefore, P could serve as a quantitative performance indicator that would aid the design and optimization of microNMR probes. This is necessary because it is very difficult to make a one-to-one comparison for different setups even when the SNR values are reported since in each case the experiments were carried out with a different set-up, different strengths of DC magnetic flux density and different bio-chemical samples having different proton densities.

Additionally, until now there has been very little, if any, discussion regarding the variation and quantification of the magnetic flux density homogeneity within the sample. Moreover, the filling factor $η$ −an essential component of Equation 1− does not appear to have been calculated explicitly in any previously published paper nor has it been shown how it depends on the sample volume or on its placement with respect to the microcoil. Previously, a simplified parameter, the product of $η$ and coil quality factor Q, was suggested by Darrasse et al.¹⁶ to analyze the coil performance with respect to the sample placement. However, although the Q factor indicates the magnetic performance of a coil, it is related to the total MF generated everywhere around the coil, while in the case of NMR we are interested only in the field inside the sample (whose volume is strictly determined by several factors). Hence, the $η⋅Q$ product may not be a useful parameter to evaluate a microNMR probe’s performance.

For the case considered here, when the sample noise is quite negligible compared to the thermal noise from the microcoil, Equation 1 reduces to

where T_c is the temperature of the microcoil, R the resistance of the microcoil, $Δf$ the bandwidth of the pre-amplifier, k is the Boltzmann’s constant, and B₁ is the average value of the magnetic flux density generated by the coil.

Equations 1 and 2 show clearly that the SNR is improved by maximizing the numerator value. Since both the magnetization vector M₀ and the Larmor frequency $ω0$ are directly proportional to B₀, they can both be enhanced by boosting B₀ and consequently improve the SNR. This explains the adoption of increasingly stronger superconducting (electro)magnets in NMR/MRI machines in the last decade.

Unfortunately, the choice of increasing B₀ is not always available to the user: once set in place, the value of B₀ is a given, dictated by the characteristics of the NMR/MRI machine available to the researcher and cannot be further increased. Therefore, another way of improving the SNR is to optimize the other remaining parameters in the numerator: the MF inhomogeneity factor K, the filling factor $η$⁠, the average value of B₁ within the sample volume V_s, and the sample volume itself. This paper examines all these factors and their influence on the final value of the P factor, and, hence, on SNR.

The first term $(M0Vsω0)$ in Equation 2 is a sample-dependent term and indicates the number of proton spins of the particular element of interest to be detected within the total sample volume at the DC MF B₀ and at the Larmor frequency $ω0$⁠. It can be shown that this term’s value depends on the number of spins present in the sample under investigation and on the gyro-magnetic ratio (γ) of the element in whose NMR spectra one is interested. The second term in the right hand side of Equation 2 represents solely the coil performance in an NMR experiment, since the numerator is proportional to the spatial homogeneity and strength of the MF generated by the coil, and to the filling factor $η$ which represents the time-varying MF’s effective coupling with the sample, while the denominator is the thermal noise due to the coil resistance. Therefore, this second term in Equation 2 is independent of both the proton spin density γ and B₀ and hence can be defined as a “performance factor,” P characterizing only the micro-probe itself. Equation 2 can thus be re-written as

Consequently, P can be used to calculate and compare the performance of various microNMR probes by quantitatively assessing a probe’s efficacy independent of proton spin density and external DC MFS. However, although the factor P does not explicitly include the sample volume term V_s, all of it’s numerator parameters do depend on V_s. Thus, V_s indirectly influences P and hence the determination of an optimal sample volume, or at least studying how P varies with the volume of the sample, is extremely important and is to be determined foremost.

This paper first discusses the mean value of the concerned MF and defines the inhomogeneity factor K inside the sample volume in section III. This is followed by a novel approach of calculating the filling factor $η$ from the data obtained by electromagnetic simulations using the software package Computer Simulation Technology (CST) in section IV. After this, in order to compare and verify the effect of our P factor, in section V we calculate the P factor of the same planar microcoil and sample volume combination as mentioned by Armenean et al.¹⁷ but using our approach of calculating the MFS, K & $η$ from electromagnetic simulations as discussed in the previous two sections, followed by the calculation of SNR from these calculated P factors and comparing them with the experimental value of SNR reported by Armenean et al.¹⁷ Next, section VI introduces the geometrical parameters of different planar microcoils and structural variants of our microNMR probes. This includes the chosen dimensions of the different microcoil variants in subsection VI A, followed by the determination of an optimal sample volume for each microcoil specific probe variant according to an angle criteria meant for determining the dominant component of the generated MF in subsection VI B. In section VII the value of each of the components of the P factor (as well as that of the P factor itself) is calculated for our probe variants with different aspect ratios, conductor cross-section, number of turns and sample volumes using the approaches given in sections III and IV. This section highlights the probe performances for both the maximum optimal sample volume (subsection VII A) and the minimum common sample volume for all the probe variants (subsection VII B). Subsection VII C then summarizes the outcome of P factor analysis and concludes which parameters should be considered for an optimal probe design. Finally, section VIII compares the performance of our microcoil with a corresponding solenoid structure.

The three most important factors influencing both P and SNR are the spatial homogeneity of the MF reflected in the parameter K (which takes into account the inhomogeneity of the time-varying field within the sample volume), the MFS (which can be represented by the mean value of its dominant component within the sample volume B_z−mean,), and the filling factor $η$⁠. The magnitude of B_z within the sample volume depends on the microcoil dimensions and number of turns, and also on the distance of the sample from the coil (but which is fixed in our case, for all microcoil variants). The inhomogeneity factor K shows how much the MF varies i.e. how inhomogeneous it is compared to (as a fraction of) the mean value of its strength. Previously, K has been discussed only qualitatively by Hoult et al.¹⁵ We define the inhomogeneity factor K as

where $σ$ is the standard deviation of the MFS within the sample volume. For an ideal MF that is perfectly homogeneous within the sample volume, $σ=0$ and K = 1. On the other hand, the smallest possible value of K is 0.

Hence, Equation 4 is valid as long as $σ$ is smaller than the mean value of the MFS B_z−mean, or else $K≤0$⁠, resulting in impractical negative values of P. The values of the standard deviation σ and of B_z−mean for all probe variants presented in this work were extracted from electromagnetic simulations using CST for the useful maximum sample volumes.

The filling factor $η$ determines the extent of coupling between B₁ and the sample volume that will induce the sample response. It is defined as¹⁵ 

Thus, $η$ is the ratio of the magnetic energy contained within the sample volume to the magnetic energy in all space:

The denominator of Equation 6 essentially represents the inductance of the coil, which for a unit current can be expressed as

which can be rearranged as

If B₁ is perfectly spatially homogeneous over the sample volume, then the numerator of Equation 6 can be simplified as the product of $B12$ and the volume of the sample V_s. In reality B₁ always has a spatial variation in all three directions x, y, and z. Also since B_z is the dominant MF component for a planar spiral microcoil placed in the xy plane, Equation 8 can be further written as:

Hence, the analytical expression for the filling factor $η$ for a sample holder with the length aligned along the z axis is:

Therefore, in order to evaluate the filling factor $η$ analytically, it is necessary to first deduce the expression of B_z(x, y, z) that gives the MFS variation along the x, y & z axes. Alternatively, both the energy stored in the sample and that over the entire space can be extracted from electromagnetic simulations of each microcoil and sample placement, and then used to evaluate $η$ according to Equation 6.

The approach described above for determining the filling factor $η$ based on electromagnetic simulations must first be verified, especially since this method has not been reported previously in the literature. For this purpose, we use as a reference the approximate closed form expression of the filling factor $η$ for a sample kept inside the core of a solenoid¹⁵ 

where V_s is the sample volume and V_c is the cylindrical volume occupied by the solenoidal coil. Considering a generic solenoidal coil with the sample in its core along the z axis and the corresponding values of V_s and V_c of 0.021 pL and 0.15 pL, respectively, the filling factor value results as $η=0.07$ using Equation 11.

Next, we also simulated in CST the 3D model of the same solenoidal coil and a value of 0.06 was obtained for the filling factor $η$⁠. This result is close to the analytically calculated value of 0.07 with a variation of only 14.28%. This demonstrates the validity of our approach to calculate the filling factor $η$ using energy extraction from simulations.

In order to verify the P factor defined by us and the approach described in the previous 2 sections to calculate each terms of the P factor we calculate the P factor for a planar coil reported by Armenean et al.¹⁷ Armenean et al.¹⁷ clearly mention all the parameters and the measurement setup details which can be used for theoretical calculation of SNR using Equation 1. We built a 3D model of the same planar coil immersed in 5 μL of water sample, exactly as described in Armenean et al.¹⁷ The 3D model is shown in FIG 1.

We calculated the average value of MFS within the sample volume, the K factor and $η$ as well as the resistance R, followed by the calculation of the P factor. Next, we used this P factor to calculate the SNR according to the measurement parameters and, the volume of the sample and the resulting values are given in Table I along with the measured value reported by Armenean et al.¹⁷ However, the SNR equation cited in Armenean et al¹⁷ does not consider the effects of K and $η$ which are essential in the calculation of SNR. In order to have a direct one-to-one comparison with Armenean’s system, the P value thus calculated by us is also devoid of K and $η$ values.

While the two SNR values have the same order of magnitude the values differ by a factor of $≈$ 3.5. This difference mostly stems from the fact that the investigated zone of sample around the microcoil mentioned by Armenean et al¹⁷ is an approximate volume of $5μL$ surrounding the microcoil immersed in a larger volume of 0.33 cm³ water. This uncertainty in actual volume will result in change in the mean value of MFS inside the sample volume and also the number of protons present under the influence of the magnetic field. In summary, the P factor proposed here appears to predict SNR value reasonably well.

The probe structure considered here for analysis consists of a planar spiral microcoil with a rectangular cross section conductor embedded in the substrate, with the conductor aspect ratio (CAR) defined as the (conductor thickness, or height)/(conductor width) ratio. As shown in FIG 2, the sample holder is considered to be placed right on top of the microcoil. The exact distance between the sample holder and the microcoil, ideally as small as possible, is dictated by microfabrication limitations, e.g. the minimal thickness of the material used which to realize the sample holder. In our simulations, a value of $2μm$ was considered for the material thickness/distance of the sample holder from the microcoil/chip surface. The 3D CST model built and used here considers the coil to be in the xy plane.

Several microcoil variants, with different number of turns and conductor aspect ratios (CARs), were analyzed. The details of all the microcoil variants are given in Table II. These dimensions are dictated by the minimal photolithographic feature size and cost of fabrications. The microcoils with a conductor having a depth of $8μm$ and a trace width of $25μm$ will have a CAR value approximated as 1:3. In this study, the trace width is considered fixed at $25μm$ and the depth (or height) is varied to achieve different CAR values.

Since the microcoil is located in the xy plane, the dominant component of its MF is along the z axis, B_z. For NMR analysis, if we consider the external DC MF B₀ to be oriented along the x axis (hence perpendicular to B_z), then in order to tip the M₀ vector within the zy plane only one of the B_z or B_y components must be dominant. Although for a planar microcoil B_z is usually considered dominant within the entire sample volume, we need to carefully check that this is indeed correct and that there are no areas where B_y is stronger. In order to determine this, we considered the angle $θ$ between B and B_y along the x and y axes for different values of z. B_z is considered stronger than B_y in the regions where $θ≥75o$⁠. The angle criterion of $θ≥75o$ ensures that the component B_y can be neglected within the sample volume. The height of the sample along z axis was restricted to $10μm$ since the magnitude of B_z along the z axis falls as $≈$ $1z$⁠, as given by Equation 12 obtained through modification of a similar relation given in literature.¹⁸ 

where $e=Nw+(N−1)s$⁠, I is the current and a₀ is the internal radius of the coil.

FIG 3 shows one set of such angle variation plots for a microcoil with 5.5 turns and 3 different CAR values. The extent along the x and y axes where the angle is below 75⁰ was estimated from these plots, thus indicating the largest region in which the sample could be situated. The same procedure was repeated for the microcoils with 3.5 and 15.5 turns and all the different conductor CAR values to determine the maximum extent of the sample volume for which $θ≥75o$ was determined and the resulting values are tabulated in Table III. In all cases it was considered that the height (i.e. thickness) of the sample is held constant at a value of $10μm$ along z axis.

As can be seen from Table III, the sample volume extent along the x axis increases with the number of turns while the change is insignificant along the y axis, and the CAR value also has little to no effect. The maximum permitted sample volume resulting from these plots for each microcoil variant was then used to determine the average values of B_z, the inhomogeneity factor K and the filling factor $η$⁠, with the results shown in the next section.

Using the maximum permitted sample volume and B_z−mean, K and $η$⁠, the P factor values have been calculated for different microcoils and these results are tabulated and discussed in the following sub-sections.

First we consider the maximum sample volume determined for each microcoil variant using the angle condition $θ≥75o$⁠, as described in section VI B. The resulting values are tabulated in Table IV along with the total resistance of each microcoil.

As can be seen from Table IV for a microcoil with a given number of turns, the B_z−mean value increases with decreasing CAR value, with the largest variation for the smallest microcoil (3.5 turns). This happens since more of the MF is distributed within the sample placed above the coils as the CAR value decreases (or the conductor height decreases), i.e. the microcoil conductors, and, therefore, the MF-generating currents are closer to the chip surface and hence to the sample. This phenomenon also affects the filling factor $η$⁠. Since the microcoil CAR value has no effect over the magnetic energy over the entire space but does influence (as mentioned above) the magnetic energy within the sample volume, the microcoils with same number of turns have lower filling factors as the CAR value increases due to lowering of B_z−mean values within the sample volume. However, increasing the CAR value reduces the resistance for a given number of turns and constant conductor width since it results in an increased conductor cross-sectional surface area through which the current flows.

As the number of turns increases from 3.5 to 5.5 the increase of the mean MFS within the sample volume B_z−mean is larger than the increase of the magnetic energy over the entire space and, consequently, the filling factor $η$ also increases. However, for the bigger microcoils, the inhomogeneity factor K suffers due to an increased variation of the MF distribution across a larger microcoil (and hence within the sample volume as well), and so does the resistance which increases due to an increased conductor length.

Contrary to expectations, the results deteriorate when the number of turns is further increased to 15.5 turns. Although the larger number of turns allows a larger sample volume, the MFS B_z−mean is quite low compared to microcoils with less turns and this reduces the filling factor $η$ while the microcoil resistance resistance. Consequently, the performance of the microcoil with 15.5 turns is quite poor in all aspects and is reflected in a small P value as shown in the last column of Table IV.

The previous section discussed the performance of the microcoils when the maximum permitted volume was used. Now, in order to get an idea about the effect of the sample volume on the P factor values, we selected the smallest volume for which the angle condition $θ≥75o$ is simultaneously satisfied by all microcoil variants. This volume is the volume corresponding to the largest permitted volume for the 3.5 turn coil since this volume is lower than the largest volume allowed for the 5.5 and 15.5 turn coil variants. Then we recalculated the values of all the parameters (B_z−mean, K, $η$⁠) and, of course, that of the P factor, and the results are given in Table V.

In Table V the first 3 rows are the same as in Table IV since the lowest common volume is the same as the largest allowed volume for the 3.5 turn coil. However, for the rest of the coil variants the MFS, as well as the homogeneity of the MF have improved since the volume, considered now is lower than the maximum allowed volume while on the other hand filling factor $η$ has reduced for the same reason. Another conclusion which can be drawn from Table V is that the effect of an increased microcoil CAR for a particular number of turns is still the same as before: it reduces the B_z−mean, which in turn affects the values of both K and $η$⁠.

The values of the performance factor P, calculated using the values of its components are listed in Tables IV & V and are plotted in FIG 4 for a better comparison. Fig. 4 can thus be used to draw some useful conclusions for the design of microcoil-based NMR probes.

• The sample volume reduction from the maximum allowed volume to lowest common volume has improved both B_z−mean and K, as reflected by the K factor values closer to 1 for the larger number of turns. However, the opposite is true for the filling factor $η$⁠: its value reduced considerably since the minimal sample volume is very low, especially for the microcoil with 15.5 turns. Thus, although this latter microcoil has the strongest B_z−mean and better MF homogeneity within the sample, its smaller filling factor and higher resistance reduce the value of P.

• The microcoil’s CAR has a major effect on resistance. In terms of magnetic performance, however, its increase only reduces the MFS within the sample which in turn lowers the filling factor value. Additionally, conductors with high aspect ratios are more difficult to fabricate/fill in using copper electroplating.

• Increasing the number of turns initially, i.e. for a relatively small number of turns of, e.g., up to 5…6, enhances the MFS inside the sample and also may be necessary to accommodate a minimal sample volume. This is because a too small sample volume, although characterized both by a large B_z−mean and a high K value, would have a very small filling factor that would reduce drastically the value of the entire performance factor P.

• However, increasing excessively the number of turns of the planar microcoil would increase the maximum permitted sample volume (i.e. for which the angle condition $θ≥75o$ is satisfied). Although having the largest possible sample volume may seem desirable to maximize the SNR (see Equations 1 & 2), in fact it would have only negative effects in this case. The average MFS would be reduced within the larger sample volume, hence the filling factor $η$ decreases, and the MF homogeneity worsens too. Additionally, the microcoil resistance also rapidly increases with the number of turns and contributes considerably to the rapid reduction of the performance factor value. Consequently, an optimum compromise has to be found (also see next point), with the smallest number of turns that would accommodate the required sample volume (considering a sample holder of fixed height and as close as possible to the microcoil surface) but which would still generate the strongest MF without increasing R.

• Our data obtained for planar microcoils variants with 3 different CARs (1:3, 1:1 and 3:1) and with 3 different numbers of turns (3.5, 5.5 and 15.5) indicate that the optimum performance factor value is obtained for an aspect ratio of 1:1 and 5.5 turns. 5.5 turns seem to be a compromise between low and high number of turns and so does a 1:1 CAR where the field B_z−mean is stronger than that of the 3:1 variant and the resistance is less than that of the 1:1 variant.

Since the solenoid is one of the most conventional structure used for MicroNMR, we simulated a set of solenoidal coils whose dimensions were chosen such that the cross sectional area of their wires are same as those of our microcoils with the different CAR values of 1:3, 1:1 and 3:1. Thus, from the Equation

where r_o is the radius of the solenoid wire, $h/w=CAR&w=25μm$ we get 3 values of r₀ corresponding to each CAR. The inter-turn spacing is taken to be the same as for our microcoils, with a total number of turns of 5. The radius of the core of the solenoid is also taken the same as the radius of the innermost turn of the microcoils i.e. $25μm$⁠. The sample is placed inside the core of the coil and hence its volume is that of the cylinder the solenoidal coil encompasses. This model is shown in FIG 5. Table VI tabulates the different component values for the solenoidal coils along with their calculated P factor values. As can be seen, the P factor value falls as the cross-sectional area of the solenoidal coil increases. The reduction in mean value of B_z (we considered the solenoid to be oriented along the z axis and hence the dominant field is B_z) when increasing the wire cross-section seems to be the main culprit for the reduction of the P factor despite the fact that the resistance decreases. The reduction in B_z value is due to the fact that the length, and hence the volume of the cylinder encompassed by the solenoid increases with the cross-sectional area of the solenoid wire, ultimately reducing the mean value of B_z and also the filling factor.

The values of P for the solenoidal coil appear much larger than those for planar microcoils in Tables IV and V. But it is difficult to miniaturize solenoidal coils. For the cross-sectional area equivalent to a CAR of 3:1 the solenoidal coil’s diameter is about $48μm$⁠, which is extremely small for a 3D solenoidal coil to be. Hence, in reality, in order to be used wound around capillary tubes of practical sizes solenoidal coils must have much larger diameters which will reduce the P factors significantly. This suggests that the comparatively bigger solenoids used conventionally will have a lower P factor compared to planar microNMR coils unless other dimensional parameters are varied to nullify the effect.

This paper has explored the effect of various design parameters on the performance of probes for microNMR analysis using spiral microcoils whose conductor is buried in the substrate and with a cylindrical sample holder of fixed height, situated just above the microcoil surface, but whose extent (radius) may be varied to accommodate various sample volumes. In order to be able to compare micro-probes that may be operated in different NMR experiments that use very different operating parameters, we defined and used a sample- and setup-independent performance factor P. Its dependence on three main parameters has been investigated: the size of the microcoil (i.e. number of turns N), the aspect ratio of the microcoil’s conductor cross-section and the volume of the sample. By comparing the values of P obtained for several microcoil variants, each with different values for the above mentioned parameters, we show that the best performance can be obtained using microcoils with a small conductor aspect ratio (i.e. flatter conductors), an optimum number of turns and a restricted sample volume. A larger number of turns increases the maximum sample permitted volume but decreases both the MF homogeneity K and its mean strength B_z−mean, resulting in a decreased filling factor $η$⁠. Higher conductor aspect ratios (i.e. increasing the conductor thickness at constant width) reduce the resistance for a given number of turns but this advantage is negated by decreased values of B_z−mean, K and $η$⁠, ultimately resulting in a much lower value of the performance factor. Also a comparison between corresponding solenoidal coil and planar microcoil indicates that for the SNR performance of conventional solenoids is most likely to be worse due to the much larger dimensions which affect negatively their P factor values. This work provides guidelines for designing microcoil geometry - sample volume combination to achieve high SNR under any external setup.

This work has been supported by Office for space technology and industry (OSTIn), Singapore, under the project S14-1126-NRF OSTIn-SRP.

All the previously published papers about microNMR probes can be divided into 3 categories, depending on the MF-generating micro-structures: solenoidal coils, microstrip line and planar microcoils.

In the case of 3-dimensional (3D) coils of relatively large dimensions, such as Helmholtz and solenoidal coils, samples of large sizes and/or volumes¹ are generally used which makes both the sample noise and coil noise significant. One such example is a solenoidal coil with inner radius of 37.5 $μm$ and 5 turns for which an SNR of 13.6 was reported.⁴ Another miniaturized 3D solenoid-like coil, fabricated by Badilita et al.,⁵ exhibited a high Q factor of 46 at 400 MHz and has been successfully used for MRI.

Microstrip lines have also been used for MRI microscopy as an alternative to coils: a micro-strip line generated the RF magnetic field and a separate coil was used to pick up the sample signal with a SNR between 4.5 to 5.2 for microstrip lines of widths ranging from 500 $μm$ to 2000 $μm$⁠.¹³ Among the planar microcoils used for NMR experiments,^8–11,19,20 the highest average SNR per acquisition of 60 was achieved with a circular spiral microcoil made of Cu, having an outer radius of 1000 $μm$⁠, 4 turns and a sample volume of 470 nL located 50 $μm$ above the coil center. Other reported planar architectures include stacked multiple planar microcoils. One such realization comprised 4 square spiral microcoils made of Au, having 5 turns each, stacked on a $4×4$ mm² area, and fabricated on a 5 $μm$ thick polymide substrate. The assembly exhibited a Q factor of 24.3, resistance of 9.5 $Ω$ and an SNR of 7 at 17.1 MHz when used with a natural rubber sample.¹² 

It can be seen from the data in Table VII above that it is very difficult to compare the performance of all these various types of coils and especially to understand which are the key parameters influencing their SNR (particularly for a new researcher) given the great diversity of microcoils and large number of papers reporting such devices, of which only some may indicate the obtained SNR. Two very good reviews by A.G. Webb^21,22 discuss the different problems associated with various microcoil types used for NMR. Solenoids produce a uniform magnetic flux density concentrated in the core of the coil; however, the dimensions of the solenoidal ‘microcoils’ (which actually are minicoils given their large dimensions, typically on the order of mm when realized with a normal wire of circular cross-section) are usually limited by the constraints imposed by the NMR equipment. In contrast, for on-chip microcoil based realizations, the dimensions are dictated by the fabrication process. Additionally, solenoidal coils also feature a susceptibility mismatch between air, coil and the sample when the sample under study is placed within the core of the solenoid. To solve this latter problem, susceptibility matching fluids may need to be used.