”Small is beautiful” in NMR

Jan G. Korvink, Neil MacKinnon, Vlad Badilita, Mazin Jouda

Institute of Microstructure Technology, Karlsruhe Institute of Technology, P.O.Box 3640, 76021 Karlsruhe, Germany

Abstract

In this prospective paper we consider the opportunities and challenges of miniaturized nuclear magneticresonance. As the title suggests, (irreverently borrowing from E.F. Schumacher’s famous book), minia-turized NMR will feature a few small windows of opportunity for the analyst. We look at what these are,speculate on some open opportunities, but also comment on the challenges to progress.

1. Introduction

The miniaturization of devices belongs to a re-search field known as MEMS, short for micro-electro-mechanical systems. The field was bornout of a 1950’s talk by Richard Feynman, ”There’splenty of room at the bottom” that inspired en-gineering scientists of the time and still does tothis day. The MEMS community collectively de-velops microscopically sized detectors and actu-ators, combined with electronic systems, some-times integrated with complex microfluidics for han-dling minute amounts of sample called Lab-on-a-Chip (LOC) systems. All sorts of applicationsare targeted, covering biology, chemistry, and evenchemical engineering, in addition to those knownfrom consumer products such as the instrumentsfound in smart devices. During the time the firstauthor was a PhD student (early 1990’s), our pro-fessors were seeking the ”killer applications” forMEMS, hoping that their field would establish it-self before running out of steam. Their wishwas granted: automobiles and smart phones cre-ated the necessary initial market pull, with theproductive research field giving rise to billions ofdollars worth of business over the course of afew decades. Miniaturized detector systems areclearly here to stay.

MEMS systems are currently revolutionizingmeasurement science, yielding more sensitivityand information, with less effort, in the smallest ofspace. Thus electron microscopy, atomic force mi-croscopy, plasmonics, optical spectroscopy, and soon, all benefit significantly from MEMS technology,

Figure 1: The entire NMR system is becoming available in morecompact format, facilitated by the shrinking of electronics, mag-nets, and resonators. However, all but the large laboratoryNMR systems have a field strength on the order of 1 Tesla. Onthe other hand, compact palmtop and chip-scale NMR systemshave yet to make it to generally available products.

opening up fantastic new opportunities. It is there-fore well worth considering whether NMR and MRIcould equally benefit from this approach. For thiswe need to look more deeply into the crystal ball,to see whether we can find pathways that lead tosome ”killer applications”.

2. Why bother with small?

Since the mid 1990’s, numerous publications[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] have reported

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on efforts to miniaturize NMR. Many of these re-ports are truly impressive, either because of the in-creased NMR performance, or in the sophisticationof the technical systems. To date, very few of thesedevelopments have made it into products (see Fig-ure 1), so that accessibility is limited to only a fewpractitioners and their associates. Overall, such aprogression is not unusual for any field of endeav-our, but of course there comes a time when it isprudent to ask: ”why bother?” Once the novelty of”because we can” is over, one might hope for ei-ther more science, or more applications, in orderto justify further exploration and adoption.

The core of an NMR system is the detector. Inan elegant paper, John Sidles showed that thesignal-to-noise ratio (SNR) of a miniaturized LCresonator and a magnetic resonance force micro-scope (MRFM) are of the same order of magni-tude, even though the techniques have very differ-ent absolute spin sensitivities [13]. Sidles’ papershowed also that it is possible for magnetic res-onance force microscopy to outperform inductiveNMR, provided that a proper mechanical resonatorwith small mass, high resonance frequency, andhigh quality factor is used. However, the complex-ity, particularly with liquid-state samples, and lim-ited applicability of this technique, together with ac-cess to the toolbox of sophisticated modern NMRexperiments, are the reasons why inductive detec-tion is, and is most likely to stay, overwhelminglyused. Since this covers our length-scale of inter-est, we will therefore restrict our attention to in-ductive measurements using coils (also coveringstriplines), knowing that other approaches such asthe use of nitrogen vacancies in diamond, or can-tilever probes, may offer slightly better sensitivitiesfor specific situations.

In the NMR experiment, the signal-to-noise ratio(SNR) per sample volume, measured at the termi-nals of the RF coil, can be described by [14]

SNR

Vs=KNγh2I(I + 1)/(3kBTs) · ω2

0 ·B1√4kBTc∆f ·

√Rnoise

, (1)

where K is a field inhomogeneity factor, N is thenumber of resonant spins per unit volume, γ is thegyromagnetic ratio, ω0 is the Larmor frequency, h isthe reduced Planck’s constant, I is the spin quan-tum number, kB is the Boltzmann’s constant, Tsis the sample’s temperature, Tc is the coil’s tem-perature, ∆f is the measurement bandwidth, B1 isthe magnetic flux density per unit current flowing inthe coil, and Rnoise is the AC resistance including

both the sample and the coil losses. In microcoils,sample losses can be neglected since the coil isthe major source of noise. This assumption is validonly for non-conductive samples, or samples withvery low concentration of salt, in which case theeffective resistance that represents sample lossesis too small and thus negligible. However, if thesalt concentration in a sample increases, then theinductive losses within the sample increase andas a result, the overall coil sensitivity decreases[15]. This effect becomes more pronounced withcoils that have low noise contribution, such as thecryoprobes [16]. Thus, defining a size of the coilbelow which sample losses can be neglected, de-pends mainly on the noise performance of the coil,as well as the conductivity of the sample.

Neglecting sample losses for simplicity, the SNRper sample volume of a microcoil is then propor-tional to the square of the frequency ω2

0 , and thecoil sensitivity B1/

√Rcoil:

SNR

Vs∝ ω2

0 ·B1√Rcoil

. (2)

Rcoil is generally dependent on the coil geometry,the wire used in its construction, and by exten-sion, the frequency of operation. We will considera solenoid as an example to demonstrate the ben-efits of miniaturization when accounting for prox-imity and skin effects, although the discussion canbe generalized to any coil geometry. The mag-netic flux density per unit current of a single layersolenoid of multiple windings n � 1, with radiusrcoil, and length l, can be calculated along theaxis of the coil by the direct application of the Biot-Savart law [17]:

B1 =µsn

rcoil√

4 + (l/rcoil)2, (3)

where µs is the sample’s permeability. The ACresistance of the solenoid, assuming the radiusof the wire to be significantly larger than the skindepth (rwire � δ), can be approximated as [18, 19,15]:

Rcoil ≈3ρn2rcoilξ

lδ, (4)

where ρ is the resistivity of the coil material, ξ isa proximity factor that depends on the number ofcoil layers and windings, and δ =

√2ρ/(µwireω0) is

the (frequency dependent) skin depth. SubstitutingEquations 3 and 4 into Equation 2, and assuming

2

that the number of windings and the ratio l/rcoil aremaintained constant, results in:

SNR

Vs∝ ω

7/40

rcoil. (5)

Thus, if the coil radius is reduced by a factor of2, then the coil sensitivity as well as the SNR persample volume doubles, whereas the excitationpower required for a certain flip angle decreasesby a factor of 4.

It is therefore instructive to briefly consider thescaling laws for detectors. We assume a flat circu-lar coil (saddle or solenoidal coils will behave simi-larly) with diameter D and wire diameter d (impor-tant for the resistance), and a linear miniaturizationscaling factor 0 < αL ≤ 1 so that Dnew = αL ·Dold:

• Volume and signal: The sample volume andhence NMR signal strength scales as αV =α3L. One clearly accesses a rapidly decreas-

ing number of spins by going down in size.

• Radiofrequency resistance: Since R =ρl/(πdδ), with ρ the resistivity, l the wirelength, and δ the skin depth, the coil resis-tance is a constant under scaling as long asδ � d.

• Bandwidth and excitation field strength:For a given current I, the B1-field scales asαB = α−1

L , as does the excitation bandwidth.See below for more detail.

• Power: The RF power required to drive agiven current scales as αP = α2

L, as long assufficient skin depth in the conductor wires isretained, i.e., δ � d.

• Signal-to-noise ratio (SNR): The SNRscales as αSNR = α−1

L [20], provided that theskin depth is significantly smaller than the wirediameter, δ � d.

• MAS rotation speed: For a given mate-rial and geometry, the achievable rotation fre-quency scales with the inverse of the diame-ter: αMAS = α−1

L .

Miniaturization thus brings along some interest-ing general advantages, such as more excitationbandwidth, less power for a given bandwidth, andpotentially, considerably more SNR. It also offers afew further features that are potentially useful forthe practitioner:

• Smaller compartments, for example in bi-ology. The size of the smallest detectablevoxel determines the kind of system for whichNMR can provide distinguishable data, whichis useful especially in biological systems.

• Naturally small samples. Cells, cell clus-ters, or tiny organisms such as C. elegans,are ideal small systems for metabolomic stud-ies. Tissue biopsies or spinal extract from ail-ing patients yield unique information on dis-ease progression, but must be kept as smallas possible to minimise patient discomfort.The bacterial production of bio-molecules istime-consuming, so that smaller samples area boon because they are quicker to establish.Thin films and fibers present functionality de-pendent on the arrangement of small quanti-ties of material. Also, lab-scale chemical syn-thesis often results in tiny, yet valuable sam-ples.

• Tight spaces: Arteries and other biologicalducts need MRI at the highest spatial resolu-tion, but have little extra space to maneuver,requiring highly miniaturized solutions. Thesame is true for samples inside high pressurediamond anvils.

• Less analysis time. Better SNR naturallyleads to quadratically shorter measurementtimes.

• Improved excitation. A consequence of thethe reciprocity principle ensures that smallresonators, equipped with lower inductancethan their macroscopic counterparts, createhigher fields per unit current, and thus excitespins over a broader range of frequencies.

• Better use of magnetization real estate.Miniaturization offers the opportunity to ad-vantageously parallelize NMR, by placingmany NMR experiments side-by-side. Thiswill not only reduce the cost-of-ownership, butwill also lead to dramatically more SNR, andmore freedom in experimentation.

• More efficient signal processing. NMRminiaturization meets and benefits from thewell-established field of CMOS technology.When NMR signal processing chains are im-plemented in CMOS chips, all of the advan-tages of integration come to play: higher pro-

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cessing speed, less power, channel multiplex-ing, shorter cables reducing losses, to namethe most obvious.

• Ubiquity. The level of expertise needed forNMR is currently extraordinarily high. Minia-turized systems can help drive NMR into novelapplication areas and simplify modes of us-age.

• Hyphenation and correlation. It is becom-ing increasingly easy to bring extra technologyinto the bore of the magnet, including novel”excitors”, ”detectors”, and ”influencors”. Suchtiny MEMS systems open up the door for newkinds of correlative measurements, expandingthe conclusions we can draw from an NMR ex-periment.

• Complex sample handling. Microfluidic sys-tems that are compatible with small detectorplatforms open the door to the lab-on-a-chipphilosophy of sample management.

• More efficient hyperpolarization: At the mi-croscale, less transport distance between hy-perpolarization and detector leads to less sig-nal loss [21].

• Faster MAS. In magic angle sample spin-ning, smaller samples would yield faster spin-ning. The current rotation speed limit forMAS is 150 kHz, and for reasons of materialstrength requires rotors of around 0.5mm di-ameter, close to the limit of conventional man-ufacturing precision. Rotor geometry has re-cently been shown to be a degree of freedom[22]. However, there are also potential chal-lenges associated with MAS miniaturization:whereas spheres appear to be more stablethan cylinders, the wall thickness requirementof small rotors depends critically on the ulti-mate strength to density ratio of the wall ma-terial, and adding more wall thickness is at thecost of sample volume.

Before proceeding with a discussion on what tominiaturize, it is also instructive to consider thechoices and compromises we have to make. Whenan NMR signal is detected from a volume of mat-ter, we either try to maintain conditions as constantas possible over the volume, thereby treating theentire sample as an ”ensemble” over which we av-erage, or we use some scheme to select a sub-

set of the sample, for example by applying a gradi-ent field that locally varies the magnetic resonancecondition, as is routine in MRI. When a detectoris miniaturized with respect to the sample, both itsuseful interaction volume and its penetration depthdecrease, which has consequences for the spatiallocalization of spins, and for the number of spins inthe observed ensemble. Furthermore, there is nopoint in using a detector beyond its intrinsic sen-sitivity limit, which fundamentally restricts the en-semble size. This in turn limits the concentration ofdetectable spins, for example in biological systems[23].

The ideal choice for a particular detector type(microcoil, stripline, magnetic resonance force mi-croscope, NV centre, etc.) is to be well matchedto the geometry of the sample and to the kind ofquestions we wish to answer. The most accom-modating sample to deal with is a liquid, fitting intoany space available. The least accommodating isa live cell, being mechanically fragile, and requiringa cosy environment and life-sustaining consum-ables, including the management of waste prod-ucts [24].

3. What can we miniaturize?

Both NMR and MRI are instrumentation-intensive activities. Apart from solid state magicangle spinning and benchtop NMR magnets,miniaturization is certainly not yet (2019) part ofthe established commercial equipment roadmap,as was sketched in Figure 1. Desirable NMR andMRI magnets are very large and sensitive instru-ments, and so are well-equipped consoles (ourconsole includes gradient amplifiers, an MAS con-trol unit, and 6 RF channels, running over three19” cabinets, each 1.5 m high and 0.5 m wide).Therefore we might first consider what miniatur-ization could offer for the existing equipment base,and what the consequences could be.

The NMR magnet is probably the most obviouscandidate for miniaturization. A number of ven-dors have demonstrated impressive tabletop sizedpermanent magnet systems, with magnetic fieldsreaching up to 2T, which is probably near to theupper limit in field strength for classical Halbachdesign magnets. Already now, these benchtopsystems are opening up NMR to a much larger andbroader community. Two aspects will almost cer-tainly be game changers. For one, the price must

4

Figure 2: A compact NMR system could be achieved by a reg-ular shrinking of each of its parts. Only the miniaturized super-conducting high field magnet is not yet on the market.

still come down – at a target of 20 ke most labora-tories could probably afford a magnet. For two, thefield must go up. Ever since the advent of closedcycle cryogen free magnet technology, and high-Tcmagnetic wire, there is no ”engineering” reason leftwhy we cannot achieve a 23T shielded benchtopsystem (the vision is to have this magnet proppedup next to a laptop in the office, as tiny as indi-cated in Figure 2)[25, 26]. Of course this is a greatidea even at half of that field strength; each studentin the lab would have their own instrument, andproductivity would go up tremendously. But morethan that, we would be able to run different kindsof experiments, and industry could use high fieldNMR for new purposes. Because the bore wouldbe much smaller, we would have to use smallerdetectors and would need less sample, a boon formany applications.

The next candidate for miniaturization is thespectrometer console. Consider this analogy fromelectronics: when the first author was a PhD stu-dent at the ETH-Zurich, the CRAY-XMP/28 super-computer was used for his thesis calculations1. Itwas a huge box of electronics, with a water coolingroom in the basement to allow the circuits to workat full power. His current smartphone has morememory, and more signal processing power, thanthe Cray-XMP of 1990. And it is air cooled, runningoff a battery. This radical miniaturization mindset

1The CRAY-XMP/28 computer had 2 processors, 64 MB ofRAM, and a clock speed of 100 MHz. It cost about 15 milliondollars. Today, a raspberry pie has 512 MB of RAM, runs at 700MHz, and costs 38 dollars.

has not yet come to NMR consoles, apart from anumber of pioneering research papers [27, 9, 28].The immediately accessible technology is perhapsnot the smart phone, but rather the so-called Soft-ware Defined Radio (SDR) platform [29], which iscapable of running pulsed NMR experiments. Bydefining the operating principles using a softwarespecification, a range of SDR hardware platformshave emerged, on which Rx-Tx experiments canbe performed by portable software, over frequen-cies ranging from 20MHz to 5.4GHz. These cir-cuit boards cost around 250 e a piece, making afour channel console about 1000 e in hardware in-vestment! Indeed, there is no ”engineering” reasonleft why the SDR hardware cannot be implementedinto a single CMOS chip, even in a multi-channelformat, achieving an overall console size similar tothat of a smart phone [9].

Gradient systems also have much to offer frombeing smaller. By reducing the distance betweentwo parallel wires carrying currents in opposite di-rection, the gradient field strength goes up propor-tionally. Numerous advances in topology optimiza-tion have made it possible to design microgradi-ent coils that produce high gradient strengths atthe lowest possible power dissipation [30]. As thecoils get smaller, so their inductance reduces too,making it possible to switch the gradients muchfaster than their macroscopic counterparts. In turn,high precision current sources are required to en-sure precise current-time profiles. Because of thesmaller size, also gradient coil power consumptionbenefits from miniaturization.

Dynamic nuclear polarization is an equipment-heavy discipline – a practitioner effectively needsa second large magnet, one major barrier towidespread use. Much power is lost between agyrotron and the sample, so one approach couldbe to optimize power transfer between source andresonator, and here miniaturization technologiescould certainly help to improve waveguides andresonators. Solid state sources based on diodes,have emerged as viable miniaturization candidateschallenging the use of gyrotrons, but achieving suf-ficient power levels remains an issue. One col-league has started to use solid state oscillators inCMOS to generate exceptionally high frequencies[9], since CMOS enables exquisite signal controlso that this approach is certainly challenging theuse of diodes, but CMOS also suffers from lowpower output. So what if we miniaturize the gy-rotron itself? It will require a concentrated effort,

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Figure 3: Fully integrated chip-based NMR and MRI sys-tems could open up unique applications, from instrumentedcatheters, to wearable health monitoring systems, point-of carediagnosis, lifestyle assistants, or augmented labware.

since highly precise 3D micromanufacturing will berequired for the miniaturized microwave resonator.

Miniaturization through MEMS technologiesmake it possible to mass produce technical sys-tems, achieving high precision and robustness ina tiny package, and at a low unit cost. As a re-sult, the patterns of using the systems dramaticallychange when this becomes possible, because weneed to be less careful, an approach which has al-ways benefited research. Another consequence ofminiaturization and cost reduction is the ability toparallelize the application. It is therefore interest-ing to see whether this approach could also ben-efit NMR, by enabling novel approaches, and newexperiments.

4. Novel applications - the ’killer apps’?

This suggestion is not just speculative. Withtransistor gates currently at around 10 nm, agileCMOS circuits with billions of transistors capableof gigahertz frequency operation and ultimately lowpower consumption are routinely manufacturablewithin a single digit cubic millimetre chip size. Thetrend in miniaturization is currently (2019) still un-derway. Further ideas that arise from miniaturisa-tion could be (also see Figure 3):

• Disposable sensors. Current NMR detec-tors are exquisite instruments, much like ex-pensive watches, since they are crafted byhand. Miniaturization benefits from precisionand automatic parallel manufacturing, so thatsensors can become disposable (or perhapsact as sample containers for lengthy storageof rare specimens or forensic evidence).

• Implantable sensors and long term mi-croscopy. For the extended monitoring ofdisease progression over time, tiny inductivelycoupled NMR sensors can be implanted closeto the area of interest, offering high enoughSNR.

• NMR in a coin; NMR in a pencil. We shouldalso consider those applications that becomepossible when NMR is radically miniaturized,i.e., when the entire system is packed into thespace of a pencil tip without loss of analysisquality. Such a simple and feasible idea wouldopen up NMR for a vast range of new modal-ities, perhaps not all of them for scientific ap-plications, but then NMR will perhaps becomeubiquitous.

• Integration of NMR into production pro-cesses. Considering the importance of NMRin the field of materials science, and its spe-cific sensing ability, it is remarkable that themethod has not yet become a mainstreamonline control technique in the production ofchemicals. Ubiquitous NMR could dramati-cally change the fields of chemical production(and of course pharamceuticals, cosmetics,energy technology, etc.), if engineering chal-lenges such as compactness and temperaturecontrol can be overcome.

• In situ NMR; in vitro metabolomics. Minia-turized NMR detectors easily fit inside theequipment of other experiments, making itpossible to study dynamic phenomena as theyhappen, localized within very small compart-ments, see Figure 4. This generic idea holdsmany opportunities for materials science andbiology. Especially in metabolomics, NMRmay one day be able to deliver metabolomicrates within a single cell. Even now,miniaturization is enabling the monitoring ofmetabolomic waste products downstream ofcells and small organisms.

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Figure 4: Miniaturized NMR would open the opportunity toperform simultaneous measurements across different methods,whilst selectively varying the experimental conditions (excita-tions) for a single sample. The acquired data cubes (or hy-percubes) would allow the extraction of correlations across alldimensions, to yield more comprehensive information on thematerial or biological sample, thereby directly informing the cre-ation of digital material twins.

• Correlative materials characterization. Tak-ing the in situ and in operando idea one stepfurther, the ability to perform a double (triple,...) measurement at the same sample lo-cality and time, provides the opportunity tocorrelate physical effects and gain additionalinsight into material behaviour, for examplealong coordinates that are otherwise not ac-cessible to NMR. Additionally, these coordi-nates (light, plasmons, ...) have their own exci-tation modalities, which of course can be suit-ably combined to explore coupled pathways.All this is known, of course, but miniaturiza-tion renders these kind of experiments mucheasier to realize, and may provide improvedcontrol over the experiments.

• Opportunities for autonomous operationand discovery. By transforming a miniatur-ized NMR detector into a complete systemequipped with additional detectors, as wellas advanced computational capabilities, dataexploration and data completion approachesmore familiar from the world of automomoussystems becomes possible. Arrangements ofsuch detectors into cooperating arrays openup the opportunity to more rapidly explore theavailable information space and complete thespectroscopic picture of a considered sample.

5. Challenges ahead

NMR microscopy suffers from a ”measurementgap” challenge. One of the exceptional features ofNMR is its non-invasiveness, but all current effortsin driving up detection sensitivity is challenging thisnotion.

For example, the most sensitive detectors avail-able are the SQUID, the nitrogen-vacancy (NV)centre, and the magnetic resonance force micro-scope (the MRFM is essentially an atomic forcemicroscope). SQUIDs, operated at cryogenic tem-peratures, must be well-insulated from any livematter such as cells, which is hard to do when thesystems are miniaturized and brought into closeproximity, requiring additional spin-detector dis-tance due to insulation. NV-centres in diamond,and the MRFM microbeam tip, need to be nano-metrically close to target spins, so that these sys-tems cannot operate noninvasively for example inthe context of single cell or small organism work.At the other end of the gap we have miniaturized in-ductive detectors, such as the microcoil or stripline,both of which are sensitive across the diameter ofa single cell but require at least 1013 room tem-perature spins at 11.74 T for a detectable signal[18, 19, 1, 31, 2, 6, 32].

Taken together, therefore, spatial resolutions be-tween 10 nm and 10 µm are not adequately cov-ered by sensitive enough non-invasive detectors –those are three invisible orders of magnitude in dis-tance! The problem is caused by the rapid decay1/r3 over distance r of the dipole strength, whichdecreases by a factor of 109 when going from 10nm to 10 µm. At the moment, no principle is in sightfor a ”measurement gap” NMR micro-detector thatis sensitive to the small signals produced by typi-cal levels of thermal equilibrium spin polarization indiluted samples.

From the current perspective, a concerted effortis required. All available SNR must be comman-deered of course, but we also have to open upnovel hyperpolarization pathways into small organ-isms or cells, to raise NMR contrast in these verysmall compartment systems in a gentle mannerwithout disturbing the underlying biological func-tion. Ideally, however, novel detector principlesmust be devised that reach beyond current induc-tive detection sensitivity to bridge the ”measure-ment gap”.

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6. Acknowledgements

We sincerely acknowledge our collaboratorswith whom we often speculate about the fu-ture of miniaturized NMR. These include fromour own group: Jurgen Brandner, Lorenzo Bor-donali, Pedro Silva, and Dario Mager. Out-side of our group we sincerely acknowledge:Jens Anders, Bernhard Blumich, Michael Bock,Jurgen Hennig, Marcel Utz, and Ulrike Wallrabe.We acknowledge our generous funders, includ-ing the European Research Council under Ad-vance Grant number 290586 NMCEL, the Eu-ropean Union’s Future and Emerging Technolo-gies Framework (H2020-FETOPEN-1-2016-2017-737043-TISuMR), and the DFG project screeMR(DFG KO 1883/29-1) and Bio-PRICE (DFG BA4275/4-1, DFG MA 6653/1-1).

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