Dynamic nuclear polarization in a magnetic resonance force microscope experiment

We report achieving enhanced nuclear magnetization in a magnetic resonance force microscope experiment at 0.6 tesla and 4.2 kelvin using the dynamic nuclear polarization (DNP) effect. In our experiments a microwire coplanar waveguide delivered radiowaves to excite nuclear spins and microwaves to excite electron spins in a 250 nm thick nitroxide-doped polystyrene sample. Both electron and proton spin resonance were observed as a change in the mechanical resonance frequency of a nearby cantilever having a micron-scale nickel tip. NMR signal, not observable from Curie-law magnetization at 0.6 tesla, became observable when microwave irradiation was applied to saturate the electron spins. The resulting NMR signal's size, buildup time, dependence on microwave power, and dependence on irradiation frequency was consistent with a transfer of magnetization from electron spins to nuclear spins. Due to the presence of an inhomogenous magnetic field introduced by the cantilever's magnetic tip, the electron spins in the sample were saturated in a microwave-resonant slice 10's of nm thick. The spatial distribution of the nuclear polarization enhancement factor $\epsilon$ was mapped by varying the frequency of the applied radiowaves. The observed enhancement factor was zero for spins in the center of the resonant slice, was $\epsilon = +10$ to $+20$ for spins proximal to the magnet, and was $\epsilon = -10$ to $-20$ for spins distal to the magnet. We show that this bipolar nuclear magnetization profile is consistent with cross-effect DNP in a $\sim \! 10^{5} \: \mathrm{T} \: \mathrm{m}^{-1}$ magnetic field gradient. Potential challenges associated with generating and using DNP-enhanced nuclear magnetization in a nanometer-resolution magnetic resonance imaging experiment are elucidated and discussed.

We report achieving enhanced nuclear magnetization in a magnetic resonance force microscope experiment at 0.6 tesla and 4.2 kelvin using the dynamic nuclear polarization (DNP) effect. In our experiments a microwire coplanar waveguide delivered radiowaves to excite nuclear spins and microwaves to excite electron spins in a 250 nm thick nitroxide-doped polystyrene sample. Both electron and proton spin resonance were observed as a change in the mechanical resonance frequency of a nearby cantilever having a micron-scale nickel tip. NMR signal, not observable from Curielaw magnetization at 0.6 T, became observable when microwave irradiation was applied to saturate the electron spins. The resulting NMR signal's size, buildup time, dependence on microwave power, and dependence on irradiation frequency was consistent with a transfer of magnetization from electron spins to nuclear spins. Due to the presence of an inhomogenous magnetic field introduced by the cantilever's magnetic tip, the electron spins in the sample were saturated in a microwave-resonant slice 10's of nm thick. The spatial distribution of the nuclear polarization enhancement factor was mapped by varying the frequency of the * née Longenecker. Present address: Quantum Information and Integrated Nanosystems Group, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02420-9108, USA † Present address: Department of Chemistry, Lehigh University, Bethlehem, Pennsylvania 18015, USA ‡ Present address: State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China applied radiowaves. The observed enhancement factor was zero for spins in the center of the resonant slice, was = +10 to +20 for spins proximal to the magnet, and was = −10 to −20 for spins distal to the magnet. We show that this bipolar nuclear magnetization profile is consistent with cross-effect DNP in a ∼ 10 5 T m −1 magnetic field gradient. Potential challenges associated with generating and using DNP-enhanced nuclear magnetization in a nanometer-resolution magnetic resonance imaging experiment are elucidated and discussed.

I. Introduction
Magnetic resonance force microscopy (MRFM) is a highly sensitive method for detecting and imaging magnetic resonance [1,2]. The highest-resolution MRFM imaging experiment to date achieved a spatial resolution of 4 to 10 nm [3], on the verge of what is necessary to study individual macromolecular complexes, but this experiment required that the sample be affixed to a high compliance microcantilever. Proton magnetic resonance was observed in a polymer film at comparable sensitivity in a scanned-probe experiment employing a magnet-tipped cantilever [4], suggesting the possibility of performing a nanometer-resolution magnetic resonance imaging (nano-MRI) experiment on an asfabricated device or a flash-frozen biological sample. Remarkably, these experiments detected magnetic resonance as a modulation of statistical fluctuations in the sample's proton magnetization. In a small spin ensemble, these random-sign, statistical "spin noise" fluctuations in magnetization greatly exceed the thermal equilibrium, Curie-law magnetization that one usually observes in a magnetic resonance experiment. Here we use dynamic nuclear polarization (DNP) to create Typeset by REVT E X arXiv:1601.07253v2 [cond-mat.mes-hall] 18 Feb 2016 hyperthermal nuclear spin magnetization in an MRFM experiment, with the goal of pushing the experiment out of the spin-noise limit.
The ultimate goal of this work is to increase the signal-to-noise ratio (SNR) of the MRFM experiment. The SNR enhancement achievable with DNP in an inductively-detected magnetic resonance experiment is determined primarily by the ratio of the hyperpolarized magnetization to the thermally-polarized magnetization. Assessing the SNR achievable with DNP in an MRFM experiment is not so simple. Because of the small spin ensembles observed in an MRFM experiment, at low polarization it is preferable to detect magnetization fluctuations, while at high polarization detecting the average magnetization gives higher SNR. To assess the DNP gain in MRFM, one therefore needs to compare the SNR of two very different experiments.
Before continuing, it is helpful to consider the ratio of magnetization fluctuations to the thermal-average magnetization. The thermal Curie-law magnetization is where N s is the number of spins in the sample, assumed here to be protons; µ p = 1.41 × 10 −26 J T −1 is the proton magnetic moment; and p therm = tanh (µ p B 0 /k B T 0 ) is the thermal spin polarization, with B 0 the external magnetic field, T 0 the temperature, and k B Boltzmann's constant.
There is a statistical uncertainty in the spin magnetization whose root-mean-square variation is given by At thermal equilibrium, the probability of measuring a certain magnetization is described by a Gaussian distribution whose mean is given by Eq. 1 and whose standard deviation is given by Eq. 2 ( Fig. 1(a)). In a large ensemble the Curie-law magnetization exceeds the rootmean-square variation, while in a small ensemble the root-mean-square variation exceeds the Curie-law magnetization. In most experiments p therm 1, and in this limit the crossover from large-ensemble to smallensemble behavior occurs when N s ≤ 1/p 2 therm . Dynamic nuclear polarization increases the spin polarization to p = p therm (3) and increases the magnetization to µ z = p N s µ p , with an enhancement factor that lies between −660 and +660 for protons. The time-averaged net magnetization now exceeds the root-mean-square magnetization fluctuations when By increasing p, DNP allows us to study smaller ensembles of spins while remaining in a regime where the Detecting magnetic resonance as a force acting on the cantilever: cantilever, resonant slice of magnetization, and tip-field gradient. Because of the symmetry of the gradient, the net force on the cantilever from uniformly polarized spins is zero. (c) Detecting magnetic resonance as a cantilever frequency shift: cantilever, resonant slice of magnetization, and the tip-field second derivative.
average magnetization dominates over the magnetization fluctuations. Now let us assess the potential signal-to-noise ratio achievable in a magnetic resonance force microscope experiment enhanced by DNP. Degen and coworkers carefully considered the SNR for detecting magnetization fluctuations from N s protons in an MRFM experiment [5]. Fluctuating proton magnetization gives rise to a stochastic force acting on the cantilever whose variance is where G tip zx = ∂B tip z /∂x is the field gradient from the magnetic tip at the location of the spins. In the above equation we have written the variance in terms of the magnitude of force from a single proton, The challenge is to perform enough measurements to determine this variance with certainty. In the MRFM experiment, radiofrequency (rf) waves are applied to cyclicly invert the proton spins twice per cantilever period, creating a cantilever-resonant spin force. In the presence of the rf, the magnetization fluctuations exhibit a correlation time τ m that is typically shorter than T 1 , the spin-lattice relaxation time, but longer than T 1ρ , the spin-lattice relaxation time in the rotating frame. Degen and coworkers showed that the magnetization fluctuations could be detected with improved SNR by applying rf pulses every t seconds to actively randomize the sample magnetization. Ideally, t < τ m , allowing the acquisition of more independent measurements of the sample's magnetization than without the periodic randomization. Decrease the measurement time too much, however, and cantilever mean-square force fluctuations begin to obscure the spin signal. There is an optimal reset time given by [5] t opt r with the power spectrum of environmental force fluctuations acting on the cantilever and Γ = k c /(2π f c Q) the cantilever dissipation constant at a temperature T 0 . The SNR for detecting magnetization fluctuations may be written as [5] with T acq the total signal-acquisition time, Here t is the measurement time and α 1 is a unitless constant that depends weakly on the ratio r of the measurement time to the optimal reset time. With t adjusted to be optimal, r = 1 and α 1 = 0.38. Now let us consider the SNR for detecting the average magnetization in a magnetic resonance force microscope (MRFM) experiment. For simplicity, let each experiment begin with the average magnetization equal to zero and let the sample polarize for a time T before detecting. The SNR for this polarized-spin experiment is derived in the Appendix. The result is The measurement is detector-noise limited when P δF 4 t σ 2 spin , or equivalently t t opt r . In this limit, for t T , SNR 2 is maximized by setting T ≈ 1.256T 1 and t ≈ 0.628 τ m . In the extreme case that τ m → T 1 , SNR 2 is maximized by setting T ≈ 1.655T 1 and t ≈ 0.461 T 1 . In the detector-noise limit, the signal-to-noise ratio for the polarized-spin experiment may be summarized as Our motivation for pursuing DNP in an MRFM experiment is revealed by comparing the SNR in the polarized-spin experiment, Eq. 10, to the SNR in the unpolarized-spin experiment, Eq. 8. An analytical result for this ratio can be obtained in two limiting cases. Above we considered Eq. 10 in the detector-noise limit. The measurement is spin-noise limited when P δF 4 t σ 2 spin , or equivalently t t opt r . For t T , SNR 2 is now maximized by setting T ≈ 1.256 T 1 , and keeping t τ m .
The signal to noise ratio in these two limits is We see that in both cases the SNR of the polarizedspin experiment is larger than that of the unpolarizedspin experiment by p √ N s times a numerical factor. The potentially significant numerical factor depends on the relaxation times of the sample and the measurement sensitivity expressed in terms of Degen's optimal reset time t opt r . In the limit of slow modulation τ m approaches T 1 and SNR 2A /SNR 1 ≥ 1 when p √ N s ≥ 1 -precisely the spin-noise-avoidance criterion introduced in Eq. 4.
In addition to improving SNR, there are a number of other reasons for wanting to detect signal from well-polarized sample spins in an MRFM experiment. Imparting the nanoscale ensemble of spins detected in an MRFM experiment with a non-zero net spin polarization will facilitate the detection of dilute species via polarization transfer [6] and double resonance [7]. Moreover, the ability to create and detect a net spin polarization in a nanometer-scale sample is expected to increase the resolution of imaging experiments. In the virus imaging experiment of Ref. 3, Degen and coworkers collected a force-noise map while slowly scanning the sample with respect to the magnet. They subsequently applied a time-consuming, iterative nonlinear deconvolution to the force-noise map to reconstruct an image of the sample's spin density. Nichol and Budakian showed that a spin-density map could instead be obtained by evolving spin fluctuations in a pulsed magnetic field gradient; they built up a multidimensional correlation function through signal averaging and applied a Fourier transform (FT) to obtain an image [8]. While this FT approach has many advantages including rapid and essentially linear image reconstruction and a favorable signal-to-noise ratio, obtaining an FT image in the spin-noise limit requires an inordinate amount of signal averaging. The experiments described below were motivated by our conclusion that the time required to perform an FT-MRFM imaging experiment on a nanoscale ensemble of spins could be decreased significantly by using DNP to create a magnetic resonance signal with a well-defined sign.
In an inductively-detected magnetic resonance experiment, the Curie-law magnetization and the DNPenhanced magnetization are both detected as a Faradaylaw voltage. In an MRFM experiment, detecting DNPenhanced nuclear magnetization requires some thought. The most sensitive MRFM experiments to date [3,4,9,10] have detected resonance-induced modulations of longitudinal spin magnetization as a change in the force acting on a cantilever, Here z is the direction of the applied magnetic field, G tip zx = ∂B tip z /∂x is the derivative of the tip magnetic field in the direction x of the cantilever motion, r k is the location of the k th spin, and the sum is over all spins in resonance. To avoid snap-in to contact, the cantilever is operated in the "hang-down" geometry [11]. Due to the symmetry of the gradient, Fig. 1(b), the net force from uniformly polarized spins is zero in this geometry. 1 The gradient in the spin force shifts the resonance frequency of the cantilever by an amount where f c is the cantilever frequency, k c is the cantilever spring constant, G tip zxx = ∂ 2 B tip z /∂x 2 , and the sum is over all spins in resonance. As can be seen in Fig. 1(c), the second derivative of the tip field G tip zxx is a symmetric function of x. Consequently, the frequency shift given by Eq. 14 is sensitive to the average, net magnetization [12][13][14] as well as magnetization fluctuations [15]. In the experiments detailed below, we detect changes in sample magnetization due to magnetic resonance and DNP as a shift in the resonance frequency of a magnettipped cantilever.
There are only a few examples of observing hyperthermal spin polarization in an MRFM experiment. Thurber, Smith, and coworkers used optical-pumping DNP to increase the nuclear spin magnetization 12-fold in a gallium arsenide sample affixed to a cantilever in a magnetic resonance force microscope experiment [16,17]. Optical pumping is challenging to implement in an 1 Spin fluctuations create a temporary left/right imbalance in magnetization observable as a force fluctuation in the experiment of Fig. 1 MRFM experiment because of heating, and the optical nuclear polarization mechanism is restricted to semiconducting samples. Chen, Marohn, and coworkers observed a long-lived shift in the frequency of a magnettipped cantilever in an ESR-MRFM experiment carried out on a nitroxide-doped perdeuterated polystyrene film [18,19]. The size and buildup time of the frequency shift signal led them to hypothesize that it arose from a DNP enhancement of 2 H magnetization; they were unable to definitively prove this hypothesis, however, because they were not able to apply the radiowaves required to flip the 2 H nuclear spins.
Below we use DNP to create enhanced 1 H magnetization in a magnet-on-cantilever MRFM experiment carried out on a nitroxide-doped polystyrene film at 4.2 kelvin. The experiment is sketched in Fig. 2. In these experiments, we can simultaneously apply both microwaves and radiowaves to the sample and thereby verify that DNP-enhanced magnetization has been created, can measure the background Curie-law signal and therefore quantify the enhancement, and can vary the rf center frequency to probe the spatial distribution of the enhanced nuclear magnetization. There are only a few examples of observing DNP at liquid helium temperatures or in the TEMPAMINE/polystyrene system studied here [20][21][22]. We know of no precedent for observing DNP in the large magnetic field gradient present in our MRFM experiment.

II. Materials and Methods
A. Sample.

B. Cantilever.
Magnetic resonance was detected with a customfabricated attonewton-sensitivity silicon cantilever, as described in Refs. 23 and 24. A radius r = 3.5µm nickel sphere (saturation magnetization µ 0 M = 0.6 T) was manually affixed to the leading edge of the cantilever with epoxy.
Cantilever displacement was observed with a temperature-tuned fiber optic interferometer [25] (wavelength λ = 1310 nm). The power spectrum of cantilever displacement fluctuations was recorded and a spring constant k c computed from the integrated fluctuations and the known temperature using the equipartition theorem [26]. The cantilever frequency f c and ringdown time τ c were determined by exciting the cantilever at resonance and measuring the decay of the induced cantilever oscillation; the cantilever quality factor was computed using Q = π τ c f c . A dissipation constant was calculated using Γ = k c /(2π f c Q) and a power spectral density of thermomechanical force fluctuations computed using P δF = 4 Γ k b T 0 .
At a temperature of T 0 = 4.2 K, a pressure of P = 5 × 10 −6 mbar, and an applied magnetic field B 0 of zero, the cantilever had an apparent spring constant of k c = 1.0 mN m −1 , a resonance frequency f c = 3500 Hz, and a quality factor of Q = 5 × 10 4 . With the magnetic field applied parallel to the long axis of the cantilever, the cantilever's quality factor decreased to Q = 1.0 × 10 4 at B 0 = 1T and to 1.6×10 3 at B 0 = 6T. The cantilever was positioned over the center of the coplanar waveguide in the "hang-down" geometry ( Fig. 2); the cantilever was brought near the sample surface with its long axis parallel to the surface normalẑ, an external magnetic field was applied along the z direction, and the cantilever oscillated in the x direction. At T 0 = 4.2 K, over the copper centerline of the coplanar waveguide, with a tipsample separation of h = 1500 nm, the power spectral density of cantilever force fluctuations ranged from P δF = 1300aN 2 Hz −1 at B 0 = 1T to P δF = 8100aN 2 Hz −1 at B 0 = 6 T.
To continuously measure the cantilever frequency, the cantilever was driven into self oscillation by making it part of an analog positive feedback loop [27], as follows: the cantilever displacement signal was measured, phase shifted by ninety degrees, amplitude limited, and fed to a piezoelectric element located below the cantilever mount. The feedback gain was adjusted to achieve a zero-to-peak cantilever amplitude of x 0p = 100 nm during the magnetic resonance experiments described below. The cantilever displacement-versus-time signal was digitized and the instantaneous cantilever frequency determined using a software frequency demodulator [28].

C. Coplanar waveguide.
Poggio and coworkers showed that nuclear spin transitions in an MRFM experiment could be excited efficiently with a transverse magnetic field produced by passing a radiofrequency (rf) current through a microwire [29]. By integrating the microwire into a coplanar waveguide [30] we are able excite the sample at frequencies up to 20 GHz for ESR experiments. The long axis of our waveguide's center line was oriented parallel to the y axis in Fig. 2.
The coplanar waveguide (CPW) was fabricated in two sections. The first section served to couple microwaves (MW) from a semi-rigid coaxial cable with an SMA connector to a CPW, made of copper plated onto an Arlon substrate. The second section consisted of a copper CPW microfabricated on a high resistivity silicon substrate; this CPW tapered down to a 500 µm long, 10 µm wide, and 0.2 µm thick copper wire flanked on either side by ground plane. The two sections were brought to within approximately 200 µm of each other and their center lines and ground planes were connected via multiple wire bonds. Transmission losses were low at frequencies ≤ 5 GHz and at certain frequencies between 5 and 20 GHzpresumably line resonances. For electron spin resonance experiments, the irradiation frequency was set to one of these line resonances.

D. Probe and nanopositioning.
Experiments were performed at T 0 = 4.2 K, nominal P = 5× 10 −6 mbar, and at external fields from B 0 = 0.6 to 6 T using a custom-built magnetic resonance force microscope. The CPW and sample were affixed to a stationary cooling block while the cantilever and associated driving piezo and optical fiber were mounted on a custom-built scanner. Coarse x, y and z scanning was achieved using custom-built Pan-style walkers, [31] while fine motion was achieved with a piezo-tube actuator. Three fiber-optic interferometers were used to observe the motion of the cantilever holder relative to the sample and CPW. The Pan walkers were used to position the cantilever over the centerline of the CPW. Alignment of the cantilever to the CPW was registered by observing a small shift in the cantilever frequency fortuitously present when the cantilever was located at the edge of the CPW's center line or ground plane. The cantilever was brought into contact with the sample surface using a combination of coarse and fine motion; the h = 0 location was determined by gently touching the cantilever to the surface while looking for the cantilever to stop oscillating and undergo a small buckling motion.

Nuclear magnetic resonance.
Nuclear magnetic resonance (NMR) signal from 1 H Curie-law magnetization was detected at fields between 4 and 6 tesla. Frequency-modulated sine and cosine waves were generated at 27 MHz using a National Instruments PXI-5421 arbitrary waveform generator and up-converted to a final rf frequency f rf between 170 and 260 MHz using single-sideband mixing. The resulting frequency-modulated radiowaves were amplified at room temperature (Kalumus Model 320CP-CE) and delivered to the CPW at 4.2 K through a combination of flexible and semi-rigid coaxial cables equipped with SMA connectors. An adiabatic rapid passage (ARP) through resonance (linear sweep; width ∆f rf = 1 MHz, except where noted) was used to invert 1 H magnetization. Each passage lasted between one and ten cantilever cycles (T c ) and was triggered to start when the cantilever was at maximum displacement. The FMmodulated rf inverted the sample's 1 H magnetization in a region of the sample -a "resonant slice" -whose location and size was determined by the static field B 0 , tip field B tip (e.g., tip-sample separation h), rf center frequency f rf , and ∆f rf . The inversion of the sample's Curie-law 1 H magnetization was detected as a dc shift of the cantilever's resonance frequency [12].

Electron spin resonance.
Following Moore et al. [13], electron spin resonance (ESR) from Curie-law electron-spin magnetization was detected near 0.6 T. Amplitude-modulated 18.5 GHz microwave irradiation (Anritsu-Wiltron source, model 6814B; American Microwave Corporation switch, model SWN-218-2DT, options 912 and B05HS20NS; Narda Microwave amplifier, model DBP-0618N830) was delivered to the CPW through a second coaxial cable. Microwave delivery was timed to start when the cantilever was at its maximum displacement. Microwave irradiation was applied for one cantilever cycle, during which time the cantilever motion swept out a region of partially saturated electron spin magnetization in the sample. A 1 T c interval of irradiation was followed by a 2 T c interval during which no MW irradiation was applied to avoid sample heating. As in Ref. 13, this on/off modulation sequence was interspersed with intervals of no irradiation in order to impose a squarewave modulation on the spin-induced cantilever frequency shift. The modulation frequency f mod was set to between 4 and 20 Hz to avoid 1/f frequency noise from sample dielectric fluctuations and ∝ f 2 frequency noise arising from white voltage noise in the interferometer circuitry [33]. With f mod so chosen, the cantilever frequency noise in the ESR experiment was close to the thermomechanical limit. The spin-induced frequency shift was obtained from the frequency-demodulator output using a software lock-in detector.
The electron spin magnetization was measured for various microwave powers P and fit to the following equation to obtain a value for the coil constant c p of the coplanar waveguide: where δf peak c is the maximum cantilever frequency shift and the saturation parameter S is given by with γ e = 28 GHz T −1 the gyromagnetic ratio of the electron, T 1 the electron spin-lattice relaxation time, and T 2 the echo decay time. The coil constant was determined assuming [13] T 1 = 1.3 ms and T 2 = 450 ns.

F. Dynamic nuclear polarization.
Dynamic nuclear polarization experiments were performed at 0.6 tesla.
Microwaves and radiofrequency waves were applied to the CPW simultaneously through two separate SMA connections on either end of the waveguide. An rf isolator was used to keep transmitted rf from damaging the microwave amplifier and a low-pass filter and attenuator eliminated transmitted microwaves before reaching the rf amplifier. The cantilever's resonance frequency was recorded continuously during each DNP experiment. To create DNP-enhanced nuclear magnetization, electron spins in the sample were saturated by applying microwave irradiation starting at time t = 0; while on, the microwaves were continuously modulated in a T c -on: 2 T c -off sequence. Subsequently, the cantilever frequency shifted because of microwave heating, saturation of electron-spin magnetization, and buildup of nuclear-spin magnetization. To infer the change in cantilever frequency arising from nuclear magnetization, with the microwaves still on a single ARP was applied at time t = τ to invert the 1 H magnetization as described previously in Sec. II E 1. The resulting frequency shift δf c was fit to where δf max c is the maximum, steady-state frequency shift due to microwave-enhanced 1 H magnetization and τ buildup is the time constant associated with the enhancement.
Real-time measurements of the spin relaxation time were performed following Alexson and coworkers [14]. After an interval of DNP, an ARP was applied to invert the 1 H magnetization. With the microwaves still on, the cantilever frequency shift was recorded with a commercial frequency counter (Stanford SR620) as a function of time t and fit to δf c (t) = δf initial c e −t/T eff 1 (18) to obtain δf initial c , the initial frequency shift due to microwave-enhanced 1 H magnetization, and T eff 1 , an effective 1 H spin-lattice relaxation time with the microwaves on. The cantilever frequency shift was calculated using Eq. 14 and measured values for f c and k c . The sum over spins in resonance in Eq. 14 was evaluated numerically, approximating the sample as a collection of independent spin 1/2 particles. Equation 14 is valid when the cantilever amplitude, x 0p = 0.1 µm, is small compared to the distance r + h = 5.0 µm between the center of the magnetic sphere and the sample, which is the case here.
The tip magnetic field component B tip z and second derivative G tip zxx were calculated using analytical formulas for a uniformly magnetized sphere. The sample was modeled as a box having dimensions ∆x = 10 µm (the width of the CPW center line), ∆y = 30 µm, and ∆z = 0.25µm. At each point in the sample, the change in electron or nuclear Curie-law magnetization ∆µ z,k due to either inversion or saturation was computed using the Bloch equations.

Nuclear magnetic resonance.
The sample box was approximated using N x = 250, N y = 750, and N z = 13 grid points. The simulations employed a proton density of ρ p = 49 spins nm −3 and a proton magnetic moment of µ p = 1.4106 × 10 −26 J T −1 . The magnitude of the (assumed perfectly homogeneous) transverse oscillating magnetic field was taken to be B 1 = 2.5 mT. This number was obtained from electromagnetic simulations of the CPW (Sonnet Software, Inc.) carried out using the measured input power of 200 mW. The Bloch equations were used to compute ∆µ z,k under the assumption that spin-locked magnetization followed the effective field in the rotating frame adiabatically. The only free parameter in each simulation was a lateral x offset between the center of the magnetic sphere and the center of the CPW; this parameter was adjusted to achieve improved agreement between the measured and calculated frequency shift versus magnetic field curves.

Electron spin resonance.
The sample box was approximated using N x = 400, N y = 1200, and N z = 13 grid points. It was assumed that the magnetic sphere was positioned directly over the center of the CPW. The simulations employed an electron-spin density of ρ e = 2.41 × 10 −2 spins nm −3 , an electron magnetic moment of µ e = −928.4×10 −26 JT −1 , T 1 = 1.3 ms, T 2 = 450 ns, and B 1 = 1.3 µT. The B 1 was taken from a Sonnet simulation of the CPW operating at 18.5 GHz; the input power used in the simulation was computed from the estimated experimental input power and the measured transmission losses.

A. Nuclear magnetic resonance at high field.
Nuclear magnetic resonance signal is shown in Fig. 3. In Fig. 3(a,b) we show representative plots of cantilever frequency shift versus time acquired at six different combinations of rf center frequency and magnetic field.
The spin-inverting ARP sweep was applied at t = 2.0 s in each plot. The sweep induced a transient positive frequency shift whether or not the rf was in resonance with sample spins. When the sweep was in resonance with sample spins it induced a long-lived shift in the cantilever frequency. This long-lived shift -the mechanically-detected nuclear spin signal -is highlighted in gray in Fig. 3(a,b). A spin signal was calculated by subtracting the cantilever frequency before and after the ARP sweep, with the "after" time window adjusted to reject the spurious rf-induced frequencyshift transient.
In Fig. 3(c,d,e) we plot the resulting spin signal acquired as a function magnetic field B 0 at three different rf center frequencies. The spin signal is either positive or negative, depending on the exact rf center frequency and field [12]. In Fig. 3(f) we show the calculated spatial distribution of the change in spin polarization induced by the ARP at two B 0 values in the experiment of Fig. 3(e). At 4.01 T, the spin signal is dominated by "bulk spins" far away from the magnet where G tip zxx is positive. At 3.91 T, in contrast, the spin signal is dominated by "local spins" close to the magnet where G tip zxx is negative. The calculated signal is shown as solid lines in Fig. 3(c,d,e). Both the absolute size of the nuclear spin signal and its complicated dependence on field are in excellent agreement with simulations and consistent with our observing Curie-law magnetization from 1 H spins at T 0 = 4.2 K.

B. Electron-spin resonance at low field.
The applied magnetic field was reduced to near 0.6 T and electron-spin resonance signal was acquired as a function of magnetic field as described in the Methods section (see Fig. 4). The observed frequency shift versus magnetic field had a lineshape similar to that seen in the NMR case but with larger deviations from the calculated signal.
No signal was observed at high field ( Fig. 4(a); region I), as expected. As the field was lowered, large microwave-induced shifts in cantilever frequency were seen at certain magnetic fields ( Fig. 4(a); grey shaded region II). The magnitude of these signals depended linearly on microwave power (Fig. 4(b); red line) and their associated frequency-shift power spectra exhibited large, low-frequency fluctuations (Fig. 4(c); right); we tentatively attribute the region II frequency shifts to spurious excitation of ferromagnetic resonances in the cantilever tip. As the field was lowered further, sample spins came into resonance and we observed the expected spin-induced changes in cantilever frequency. The size and lineshape of the region-III signal agreed well with the simulated force-gradient ESR signal ( Fig. 4(a); grey line). The region-III frequency shifts had a dependence on microwave power consistent with saturation of an electron-spin magnetic resonance signal (Fig. 4(b); blue line). When the field was lowered further, to below 580 mT (region IV; Fig. 4(a)), large variations in the cantilever frequency shift versus (c) Cantilever frequency-shift power spectrum for the field set to be on resonance (center) and off resonance (left and right). Note the logarithmic y axis. A modulated cantilever-frequency signal is apparent in each plot as a large peak at f = f mod and its harmonics. The noise floor is orders of magnitude higher with the field set off resonance to Ba and Bc. Experimental parameters: tipsample separation h = 1500 nm and microwave frequency f MW = 18.5 GHz.
field signal were observed. There variations arose, we hypothesize, from either ferromagnetic resonances or magnetization fluctuations in the cantilever's magnetic tip; below B 0 = 580 mT, the magnetization may not be fully saturated at all locations in the magnet given that the saturation magnetization for nickel is µ 0 M = 600 mT.
In summary, while spurious frequency-shift signals were present in regions II and IV of the Fig. 4 electronspin resonance signal, a microwave-induced spin signal could be observed in field region III whose dependence on magnetic field and microwave power was consistent with magnetic resonance signal from Curie-law electron spin magnetization at T 0 = 4.2 K. The above experiments demonstrate our ability to excite nuclear spins at high field and electron spins at low field using a single coplanar waveguide. We detected magnetic resonance in both experiments as a change in the mechanical resonance frequency of a magnet-tipped cantilever. We next looked for evidence that hyperthermal nuclear magnetization could be created in our microscope at low field via the dynamic nuclear polarization effect. We emphasize that the same sample, waveguide, and magnet-tipped cantilever was employed in all three experiments.
Microwaves and radiowaves were applied as indicated in the timing diagram of Fig. 5. To demonstrate DNP, cantilever frequency was recorded as a function of time in experiments employing combinations of onand off-resonance microwaves and rf. A frequency shift due to nuclear magnetization was observed following a period of on-resonance microwave irradiation. This microwave-enhanced nuclear-spin signal is shown shaded in blue in Fig. 5. No such nuclear-spin signal was observed when either the microwaves or the radiowaves were applied off resonance. Applica-tion of microwaves led to a decrease in cantilever frequency at times 0 < t ≤ 15 s; because it was present when both on-and off-resonance microwave irradiation was applied, we attribute this decrease to a heatingrelated artifact. Significantly, no nuclear spin signal could be detected at B 0 = 0.6 T without first applying microwaves. This is expected. The Curie-law nuclearspin signal apparent at B 0 = 6 T in Fig. 3 scales linearly with B 0 ; we would predict the Curie-law signal at B 0 = 0.6 T to fall below the cantilever frequency noise floor and therefore be undetectable. Comparing the nuclear spin signal observed following microwave irradiation at B 0 = 0.6 T to the nuclear spin signal extrapolated from the B 0 = 6 T experiments, we can nevertheless estimate that the applied microwaves are inducing a hyperthermal nuclear magnetization enhanced by a factor of between = 10 and 20.
The dependence of the microwave-enhanced nuclear spin signal on microwave power is shown in Fig. 6(a). Both the nuclear-spin signal and the electron-spin signal plateau at the same microwave power, suggesting that the Fig. 5 nuclear-spin signal originates in the electronspin magnetization. The dependence of the nuclearspin signal on microwave irradiation time is shown in Fig. 6(b); the measured buildup time is τ buildup = 12.7 ± 3.3 s. Ideally the DNP buildup time should be compared to the nuclear spin's spin-lattice relaxation time, T 1 . The 1 H spin-lattice relaxation time in the absence of microwave irradiation was difficult to measure at B 0 = 0.6 T because of the large transient change in cantilever frequency created by turning off the microwave irradiation. An effective 1 H T 1 could be measured in the presence of resonant microwave irradiation by observing the cantilever frequency shift in real time [14,32] after the application of a single ARP sweep (Fig. 6(c)). The resulting cantilever frequency transient was well described by a single exponential decay having an effective spin-lattice relaxation time of T eff 1 = 14.3 ± 1.0 s. This value is in reasonable agreement with the 1 H spin-lattice relaxation time T 1 = 30.8 ± 0.9 s measured at B 0 = 5 T (Supporting Information). The factor-oftwo agreement between these estimated 1 H spin-lattice relaxation times and τ buildup is consistent with buildup of nuclear magnetization via a DNP effect.
The reader will have noticed that, in the experiments of Fig. 5 and 6, the rf was not exactly in resonance with nuclear spins in the center of the resonant slice defined by the applied microwaves. 2 If the ARP was adjusted to flip nuclear spins in a region exactly centered on the resonant slice, Fig. 7(a), then there was no observable frequency shift, implying a net nuclear-spin enhancement of zero. Further experiments were carried out in which resonant microwave irradiation was applied at t = 0 s to initiate DNP, one or two ARP sweeps were applied subsequently to generate a frequency shift proportional to the nuclear polarization, and the center frequency of the ARP sweep was varied to selectively invert nuclear spins that were nearer to or further from the magnet tip than were the resonant electron spins. The results of these experiments are shown in Fig. 7(b-d). These results are consistent with the nuclear spins proximal to the magnet tip having a positive and distal spins having a negative .
The associated spatial distribution of enhanced nuclear magnetization is sketched in Fig. 8(a). To map out this distribution, the ARP sweep width was reduced to ∆f rf = 0.3 MHz, the rf center frequency varied systematically about f rf = 28 MHz, and the above experiments were repeated. The B 0 field was varied to examine the nuclear enhancement about both a "local" resonant slice, Fig. 8(b), and a "bulk" resonant slice, Fig. 8(c). Considering that G tip zxx is negative in the local slice and positive in the bulk slice, the Fig. 8(b,c) data is consistent > 0 for proximal spins and < 0 for distal spins in both slices.

IV. Discussion
In summary, the dependence of the cantilever frequency shift seen in Figs. 5, 6, 7, and 8 on 1. the longitudinal magnetic field; 2. the frequency and timing of the applied rf; and 3. the microwaves' frequency, timing, and intensity is consistent with signal arising from DNP-enhanced nuclear magnetization interacting with a magnet-tipped cantilever. A number of DNP mechanisms [34,35] are possible in TEMPAMINE-doped polystyrene, including the solid effect, thermal mixing, the cross effect, and the recently proposed separative magnetization transport (SMT) mechanism [36][37][38]. The spatial distribution of the DNP enhancement in Fig. 8 is inconsistent with the SMT mechanism alone. The relative values of line widths and resonance frequencies govern which of the established DNP mechanisms is active in a sample at a given field. The homogeneous linewidth of the ESR spectrum in our sample is δ = 1/(γ e T 2e ) = 2 MHz. At B 0 = 0.6 T, the inhomogeneous linewidth of the ESR spectrum is approximately ∆ ∼ 77 MHz (due primarily to hyperfine anisotropy, ∼ 77 MHz, and not g-factor anisotropy, ∼ 61 MHz). The nuclear Larmor frequency is ω = 28 MHz here. These line widths and frequencies satisfy [39] ∆ > ω > δ, making cross-effect the most likely DNP mechanism at play in our experiment. We expect the large magnetic field gradient present in our experiment to have a number of effects on the sample's nuclear and electron spins that we will now consider. Relevant properties of our sample's 1 H and electron spins are summarized in Table I.
The presence of the cantilever tip's magnetic field gradient qualitatively explains why cross-effect DNP would produce the enhancement profile shown in Fig. 8. In a homogeneous magnetic field, an ensemble of randomly oriented nitroxide molecules will absorb microwaves across a distribution of frequencies ν because of the anisotropy in the nitroxide electron's hyperfine coupling and g tensor. In our experiment, each nitroxide's resonance frequency is shifted by an additional amount inversely proportional to its distance from the cantilever's magnetic tip. The observed polarization profile can be understood by considering both these effects together (Fig. 9). The observed profile is rationalized by realizing that applied microwaves simultaneously excite distal molecules at the high-ν end of the absorption profile a(ν), yielding < 0 proton polarization, and proximate molecules at the low-ν end of the a(ν) profile, yielding > 0 proton polarization.
The picture of Fig. 9 predicts that the nuclear spin enhancement should be limited to distances away from the central resonance slice where the electron Larmor frequency is shifted by no more than ±ω due to the magnetic field gradient. The electron spins directly below the magnetic tip experience a gradient of G tip zz = −2µ 0 M sat a 3 (h + a) −4 . With µ 0 M sat = 600 mT for nickel, a tip radius of a = 3500 nm, and a tip-sample separation of h = 1500 nm, the gradient is estimated to be G tip zz = 0.082 mT nm −1 . The peak enhancement is therefore pre- dicted to occur at a distance ∆z = ω/(γ e G tip zz ) = 12 nm to either side of the resonant slice. The nuclear spin polarization in Fig. 8(b,c) is observed to peak at a distance approximately 0.25 MHz/(γ p G tip zz ) = 72 nm from the center of the resonant slice, a distance that is good agreement with the calculated proton spin-diffusion length D = √ DT 1 = 80 nm; see Table I. Based on the calculated ∆z, we would expect spin diffusion during DNP to lead to a "blurring" of the enhanced nuclear magnetization in our experiment. This blurring of the bipolar magnetization profile by spin diffusion may account for the modest 1.5 to 3.0 % efficiency of DNP seen here.
The large magnetic field gradient present in an MRFM experiment has been shown to affect both electron [40] and nuclear [42] spin diffusion. For each kind of spin, there are two effects to consider. Eberhardt and coworkers [42] studied nuclear spin diffusion and showed that spin diffusion was impeded when the gradient was large enough to be felt by diffusing spin polarization during the spin polarization's lifetime T 1 . They concluded that the onset of reduced spin diffusion should occur at a critical gradient of where B L (in our notation) is the mean-square "local field" produced by dipolar interactions with the other spins in the sample. Budakian et al. [40], in their MRFM study of electron spin-lattice relaxation, argued that spin diffusion should be quenched entirely when the resonance frequency difference at different lattice sites is larger than the spin-spin dipolar coupling. According to this argument, the onset of spin diffusion quenching should occur at a critical gradient of approximately with a the lattice spacing.
In Table I we provide estimates of G crit 1 and G crit 2 for both the 1 H and electron spins in our sample. For electron spins, we estimate that G tip zz ≥ G crit 2 > G crit 1 ; electron spin diffusion should be quenched or at least strongly affected by the magnetic field gradient present in our experiment. For the 1 H spins, on the other hand, G crit 2 > G tip zz > G crit 1 ; we should be in a regime where nuclear spin diffusion is impeded, but not quenched, by the tip's magnetic field gradient. The localized polarization profile in Fig. 8(b) is consistent with the electron spin diffusion length being negligibly small but the nuclear spin diffusion length being close to the unperturbed bulk value.

Sensitivity and resolution
The absolute proton spin polarization achieved here is far less than unity but nevertheless significant. At The sensitivity and imaging resolution achievable in a small-magnet tip MRFM experiment with hyperpolarized magnetization is estimated using Eqs.11a and 11b in Table II. The table considers  We take the base electron polarization to be 0.1, consistent with our present operating conditions of 18GHz and 4.2 K. Assuming a DNP efficiency of 10 percent (e.g. = 66) gives p = 0.01. We calculate the SNR in two cases. The Eq. 11a calculation assumes T 1 = 12 s (vide supra) and τ m = 0.020 s [3] while the Eq. 11b calculation assumes τ m → T 1 . In Expt. B we assume a smallerradius tip having a gradient improved by a factor of two, In the virus imaging experiment of Degen and co-workers [3] the peak lateral gradient was G tip zx = 4 mT nm −1 ; the peak spin variance signal, σ 2 spin = 600 aN 2 , arose from approximately N s = σ 2 spin /(µ 2 p (G tip zx ) 2 ) = 1.9 × 10 5 protons in resonance. The associated volume of spins in resonance is V s = 3800 nm 3 (assuming a proton density of 50 spins/nm 3 ). The computed spin density at each sample point in the Ref. 3 experiment contains contributions from signal measured at many sample locations; due to this signal averaging, the imaging resolution is smaller than V s by nearly an order of magnitude, 4 nm × 10 nm × 10 nm = 400 nm 3 . Let us take the N s used to compute the signal-to-noise ratios in Table II and divide by ten to account for the improvement in the signal-to-noise ratio expected from the imagereconstruction step. The implied imaging resolution ranges from slightly worse than (6 nm) 3 in the τ m T 1 case of Expt. A to much better than (2 nm) 3 in the τ m → T 1 case of Expt. B. In both experiments, the SNR is improved 25 fold by pushing τ m to the T 1 limit.
Creating, detecting, and imaging DNP-enhanced nuclear magnetization in a small-tip nanometerresolution MRFM experiment will be challenging. We can draw on the above analysis and prior work to enumerate the key expected challenges.

Polarization
The cross-effect DNP mechanism is a three-spin process requiring two electrons in close proximity having a difference in Larmor frequencies equal to the Larmor frequency ω of the adjacent nuclear spins. Spin diffusion is needed to share the nuclear spin polarization created near the paramagnetic site with spins many nanometers away; to achieve homogeneous magnetization in a single macromolecule, it will therefore be important to keep the gradient below G crit 2 = 0.7 mT nm −1 . The gradient could impede CE-DNP in a more fundamental way as well, by creating a difference in Larmor frequency for adjacent electrons that could exceed ω = γ p B 0 and therefore shut down the cross-effect mecha-nism. The associated critical gradient is where a is the distance between paramagnetic molecules. For a paramagnetic dopant concentration of 40 mM, the critical gradient is G crit CE-DNP = 0.29 mT nm −1 at B 0 = 0.6 T (f MW = 17 GHz), rising to G crit CE-DNP = 2.9 mT nm −1 at B 0 = 6 T (f MW = 170 GHz).

Imaging
Harnessing DNP-enhanced nuclear magnetization for nanometer-scale imaging, on the other hand, will require reducing or eliminating spin diffusion. Assuming one-dimensional diffusion for simplicity, spin diffusion will broaden a σ(0) = 1nm wide Gaussian distribution of magnetization to a width of σ(t) = 2 nm in only t = 1.5σ 2 /D = 3.3 ms. This calculation indicates that nanometer-scale spin-magnetization gradients created during Fourier image encoding [8,43] in an MRFM experiment will be erased rapidly by spin diffusion. To avoid this problem, Kempf and Marohn have proposed applying rf pulses in synchrony with cantilever motion to enable FT-image encoding while eliminating spin evolution from dipolar couplings [43]. Alternatively, an image can be acquired by collecting signal while slowly scanning the cantilever [3]. To image a polarized-spin sample in this way, it will be important to keep magnetization from transferring between adjacent slices by, for example, detecting in a gradient larger than G crit 2 to suppress spin diffusion. Detection For simplicity, here we used the dc CERMIT effect to observe the polarized nuclear spins. The signalto-noise ratio in our experiments was limited by surface frequency fluctuations and not thermomechanical noise. It is likely that surface frequency noise will be an even more pronounced problem in a small-tip experiment [4,33,44]. In future experiments, optimizing the signal-to-noise ratio will require balancing two competing demands. To achieve long τ m , the nuclear magnetization should be modulated slowly. To evade surface frequency noise, the nuclear magnetization should be modulated rapidly; at fast modulation frequencies parametric upconversion may be necessary to evade detector noise [45]. It is unknown how well this upconversion scheme will work in practice in a high surface noise environment.
In an MRFM experiment, the per-spin sensitivity is proportional to the magnetic field gradient. The gradient in the experiment of Ref. 4 was 5.5 mT nm −1 , enabling 500 µ p sensitivity but already larger than both G crit 2 and G crit CE-DNP ; the gradient requirements for achieving optimized polarization, imaging, and highsensitivity detection are seemingly incompatible. Eberhardt, Meier, and coworkers encountered a similar problem when trying to perform NMR spectroscopy measurements in a sample-on-cantilever MRFM experiment [46]. They obtained high-resolution spectra by cycling their millimeter-scale gradient source away from the sample temporarily to allow for a period of spin evolution under the chemical shift in a homogeneous field. Shuttling the 200 nm wide tip of Ref. 4 laterally by 500 to 1000 nm on the timescale of T 1 (many seconds at cryogenic temperatures) appears feasible and would enable the independent optimization of the gradient during periods of polarization, image encoding, and spin detection.

V. Conclusion
Many magnetic resonance experiments have been shown to be compatible with magnetic resonance force microscopy. These experiments include the Rabi nutation experiment [47,48], spin echoes [49], dipolar spectroscopy [50,51], cross polarization [52], indirect observation of low gamma nuclei via cross depolarization [6], two-dimensional spectroscopy [46], and nuclear double resonance [53]. Here we have used the widely applicable cross-effect DNP mechanism to create hyperthermal nuclear spin polarization in a thin-film polymer sample in a magnet-on-cantilever MRFM experiment. If the challenges discussed above can be addressed, using DNP to create hyperthermal spin polarization in an MRFM experiment offers many exciting possibilities for increasing the technique's sensitivity in both imaging and double-resonance experiments.
Here we calculate the signal-to-noise ratio for detecting the average magnetization in a magnetic resonance force microscope (MRFM) experiment. As described in the text, we let each experiment begin with an average magnetization equal to zero. After waiting a time T for the sample to polarize, there is now a spin force acting on the cantilever whose magnitude is At a subsequent time t the force acting on the cantilever is F s (t) = F init s e −t/τm . The force signal is multiplied by a matched filter e −t/τm , integrated, and the resulting integral is divided by the observation time t to obtain the following estimate for the magnitude of the mean spinforce signal: There are two contributions to the force noise acting on the cantilever: the environmental force noise, whose power spectrum is given by Eq. 7, and the force noise arising from magnetization fluctuations, whose variance is given by Eq. 5. The two noise sources are uncorrelated and we may therefore add their variances to obtain the total force-noise variance observed during one measurement period: σ 2 F, 1 = P δF /(4t) + σ 2 spin , where b = 1/(4t) is the noise-equivalent bandwidth of the matched exponential filter. The measurement is repeated N avg = T acq /(T + t) times and the resulting signals are averaged together. As a result of the averaging, the noise variance will be reduced by a factor of 1/N avg to σ 2 F, Navg = σ 2 F, 1 /N avg . The resulting signal-tonoise expression SNR 2 = F avg s /σ 2 F, Navg is given by Eq. 10.
As described in the text, we can calculate approximate expressions for the SNR in the detector-noise limit and in the spin-noise limit. These two cases can be summa-rized as detector-noise limited: t t opt r spin-noise limited: t t opt r For t T (or, equivalently, when τ m T 1 ), the detector-limited SNR is maximized by setting T ≈ 1.256 T 1 and t ≈ 0.628 τ m giving The spin-noise limited SNR is maximized by setting T ≈ 1.256 T 1 and keeping t τ m to give an SNR expression

Supporting Information A. Frequency noise and equivalent force noise
In the experiments described in the manuscript, electron-spin resonance and nuclear magnetic resonance were registered as a change in the mechanical resonance frequency of a cantilever. Thermo-mechanical position fluctuations place a fundamental limit on how small a FIG. S1: Power spectrum of cantilever frequency fluctuations P δfc versus offset frequency f at B0 = 0.655 T (red line) and B0 = 6.0 T (blue line). The P δfc data above f ≥ 10 2 Hz was fit to Eq. S.1 with xrms = 69 nm to obtain P det δx = 3.6 × 10 −6 nm 2 Hz −1 (dotted black line), the power spectrum of detector noise expressed in units of equivalent position noise. The right-hand axis is P δfc rewritten in terms of an equivalent force fluctuation using Eq. S. 10  cantilever frequency shift can be measured in a given averaging time [27,28,33,54]. In this section we present cantilever frequency-fluctuation power spectra and use these spectra to assess how close the experiments in the manuscript were to operating at the thermo-mechanical limit.
A power spectrum of cantilever frequency fluctuations P δfc (f ) was collected at B 0 = 0.655T and B 0 = 6T, in vacuum, at 4.2 K (see Fig. S1). Apparent in the spectrum are ∝ 1/f dielectric fluctuations at low offset frequency f [44] and detector noise ∝ f 2 at high f [33]. The detector-noise contribution to the cantilever frequencynoise power spectrum is [28,33] P det where x rms is the root-mean-square cantilever amplitude and P det δx is the power spectrum of detector noise expressed in units of equivalent position noise. The high-f data in Fig. S1 was fit to Eq. S.1 to obtain P det δx = 3.6 × 10 −6 nm 2 Hz −1 .
The cantilever frequency fluctuations can be analyzed, as follows, to obtain a power spectrum of equivalent force fluctuations. Fluctuating forces acting on the cantilever lead to fluctuations in the cantilever position whose power spectrum is given by where P δF (f ) is the power spectrum of force fluctuations and k c , f c , and Q are the cantilever spring constant, resonance frequency, and quality factor, respectively. These fluctuations in cantilever position contribute noise to the measured cantilever frequency. The resulting power spectrum of induced frequency fluctuations is given by We could at this point substitute Eq. S.2 into Eq. S.3 and obtain a relation between P δfc to P δF . Before doing so, it is helpful to examine which simplifies to where in going from Eq. S.4 to Eq. S. 5 we have used that f f c . Substituting Eq. S.5 into Eq. S.3 yields where we have used that P δF (f ) is an even function of f to write P δF (f ± f c ) → P δF (f c ± f ). This expression may be simplified further by defining the average power spectrum of force fluctuations at an offset frequency f below and f above the cantilever frequency. Substituting Eq. S.7 into Eq. S. 6 gives . (S.8) The last term in Eq. S.8 becomes 1 in the limit that f f c /(2Q), that is, when f is larger than the width of the oscillator resonance in cycles s −1 . In this limit, Solving for P avg δF we obtain If the only source of frequency noise was the underlying force noise, then we could use Eq. S.10 to calculate the fluctuating forces driving the cantilever from the measured power spectrum of cantilever frequency fluctuations. In practice, however, P δfc (f ) contains additional contributions from surface noise and detector noise. Applying Eq. S.10 to the measured frequency fluctuations we obtain an equivalent or effective power spectrum of force noise. In this case Eq. S.10 can be interpreted as the power spectrum of force fluctuations that, when applied to the cantilever, would yield frequency fluctuations having the observed power spectrum P δfc . The P avg δF calculated in this way is shown as the right-hand y axis in Fig. S1.
For comparison, we can plot the power spectrum of thermo-mechanical force fluctuations P therm δF . This power spectrum is independent of frequency. In terms of measured parameters, where k B is Boltzmann's constant, T 0 is temperature, and τ c is the cantilever ringdown time. Comparing the observed P avg δF data in Fig. S1   we see that the equivalent force noise at B 0 = 6.0 T was near the thermo-mechanical limit at offset frequencies 20 Hz < f < 50 Hz while at B 0 = 0.655 T the equivalent force noise was never better than 10× the thermomechanical limit.

B. Effect of cantilever motion on the DNP signal
During the NMR, ESR, and DNP experiments described in the manuscript, the cantilever was oscillated at its resonance frequency.
According to Eq. S.9, cantilever root-mean-square frequency noise is inversely proportional to cantilever amplitude. Moreover, in the ESR experiment some cantilever motion is required to scan the resonant slice through the sample and bring a measurably large number of electron spins into resonance.
The microwave induced 1 H spin signal was found to be independent of the peak-to-peak displacement of the cantilever as seen in Fig. S2. This observation is consistent with the manuscript's finding that the nuclear spins are polarized in a thin region on the proximal and distal sides of the resonant slice. Oscillating the cantilever in the x direction causes a lateral blurring of the resonant slice but does not, to first order, change location or thickness of the slice in the z direction. Consequently, increasing the oscillation amplitude of the cantilever should not cause any cancelation of the DNP enhancement.

C. Coplanar waveguide details
The coplanar waveguide consisted of two sectionsa copper CPW on an Arlon substrate and a copper CPW fabricated on high resistivity silicon.
Arlon section -The CPW-on-Arlon section was purchased from PCB Fab Express with precut holes to facilitate making a connection to an SMA coaxial connector. The Arlon substrate had a thickness of H = 2540µm and a (specified) relative permittivity of r = 9.8. The waveguide was made of 35 µm thick copper. The waveguide's center line was w = 457 µm wide and the gap to the flanking ground plane was s = 228.6 µm wide on each side. A 10 mm by 2 mm section was removed from the center of the Arlon substrate to accommodate the CPWon-Si section described below.
Silicon section -The CPW-on-Si section of the waveguide was microfabricated at the Cornell Nanoscale Science and Technology Facility. The substrate was made of high-resistivity silicon, had a thickness of H = 500 µm, and had a (specified) relative permittivity of r = 11.8. The waveguide was made of 0.2 µm thick copper. The waveguide's outer center line was w = 480 µm wide and the gap to the flanking ground plane was s = 230 µm on each side. This section tapered, over a distance of 450 µm, to a narrower waveguide; the w/s ratio was maintained in the tapered region. The narrower, "microwire" section of coplanar waveguide was L = 500 µm long, w = 10 µm wide, and had an s = 6 µm gap. The dimensions of the 10 mm by 2 mm hole in the CPW-on-Arlon section were precisely measured, and the CPW-on-Si was cut using a dicing saw to fit into the hole leaving less than a 200 µm gap between the two sections.
Connections -The two CPWs were connected via wire bonds. Three wire bonds were used to connect the center line and three wire bonds were used to connect each of the flanking ground planes.
S parameters -See Table S1 for measured and calculated scattering parameters.

D. Adiabaticity of nuclear spin inversions
The CPW described above was designed to deliver broadband irradiation. Electromagnetic simulations (Sonnet Software, Inc.) predicted a transverse magnetic field strength B 1 of 2.5 mT with only 200 mW of input power at frequencies below 5 GHz where simulated and measured scattering parameters agreed within 1 dB. To invert the nuclear magnetization reversibly, the nuclear spin magnetization must stay aligned with the effective field in the rotating frame during an adiabatic rapid passage through resonance. Maintaining this alignment requires a B 1 large enough to meet the adiabatic condition, According to Eq. S.12, a transverse magnetic field of strength B 1 = 2.5 mT should meet the adiabatic condition during a ∆f rf = 1 MHz sweep as long as the sweep duration ∆t rf was ≥ 0.014 ms.
Harrell et al. provide guidelines that allow us to further quantify how efficiently we are inverting nuclear spins [55]. Considering a spin-1/2 system and a finite radiofrequency sweep rate, one can calculate the probability that spins undergo a diabatic transition rather than an adiabatic transition using Under our experimental conditions, we calculate a 10% likelihood of a diabatic transition with a ∆f rf = 1 MHz sweep lasting ∆t rf = 0.021 ms. Equation S.13 is valid in the limit that ∆f rf ≥ 5γB 1 . This condition sets a lower limit on the width of an ARP frequency sweep necessary to prevent projection losses -losses incurred from projecting the magnetization on the effective field when the rf is turned on. For B 1 = 2.5 mT, the 1 MHz wide frequency sweep used throughout these experiments should be adequate. The ∆f rf = 0.3 MHz ARP sweeps used to map the DNP enhancement, however, do not strictly satisfy this condition. The signal from the 0.3 MHz sweeps was likely affected by (modest) projection losses.
The applied sweeps in our experiments were 0.28 to 2.8 ms in duration -sufficient, we predicted, to meet the adiabatic condition with negligible diabatic transitions and projection losses. After applying consecutive adiabatic rapid passage sweeps through resonance, however, we did not observe a complete return of the cantilever resonance frequency to its initial value (see the experiments and data presented in Fig. S3). Figure S3(a) shows the percent return of the cantilever frequency to its initial value following two identical ARP sweeps with a two second delay between them. The fidelity of the inversions is poor. Three hypotheses were developed to explain the observation: 1. the oscillating field was not as strong as predicted; thus, we were not meeting the adiabatic condition; 2. a short T 1ρ was causing a loss of magnetization during the rf sweep; and 3. spin diffusion was moving polarized spins out of the resonant slice during the delay before the second rf sweep was applied to re-invert the spins.
As the duration of the sweep was shortened, the percent return improved, indicating that we were likely meeting the adiabatic condition but were possibly losing magnetization due to a short T 1ρ . Fig. S3(b) shows that the percent return improves as the inter-sweep delay is decreased. Fig. S3(c) shows a real-time measurement of the 1 H spin-lattice relaxation time; the measured T = 30.9 ± 0.9 s is considerably longer than the fewsecond lifetime of the inverted magnetization apparent in Fig. S3(b). Taken together, these findings support the hypothesis that spin diffusion is carrying the inverted spin polarization away from the resonant slice on the time scale of just a few seconds.

VI. Absolute nuclear spin polarization
The experiment described in this manuscript is the first time that microwave-assisted DNP has been definitively demonstrated in an MRFM experiment. A 10 to 20-fold 1 H polarization enhancement was achieved. Figure S4 summarizes the absolute polarization and buildup time achieved in inductively-detected DNP experiments carried out at temperatures ranging from T 0 = 4.2 K to T 0 = 7 K. The significantly greater enhancements achieved in these previous DNP experiments often came at the expense of long polarization buildup times. The buildup time of τ ∼ 13 s seen in this experiment is favorably low compared to prior inductively-detected low temperature DNP experiments. Implementing more optimized polarizing agents [21,22,[56][57][58], freezing the sample in a partially deuterated glass-forming solvent matrix, and operating at higher fields should lead to significantly greater absolute 1 H polarization in the MRFM experiment.