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Since its discovery in 1938 [61], magnetic resonance of spin systems has been one of the most important tools in studying properties of matter [62]. Nuclear magnetic resonance (NMR) has been used to elucidate the chemical structure of many molecules—most recently proteins and other complex biomolecules. NMR has also been used in manufacturing control, nondestructive testing and identification of samples, and gave rise to magnetic resonance imaging (MRI), with its own myriad of technical and medical applications. Electron spin resonance (ESR), although less famous than its nuclear cousin, is a powerful tool for studying free radicals and paramagnetic centers and has also found many applications in chemistry, physics, and biology. Finally, ferromagnetic resonance (FMR) has been used to study ferromagnetic materials and devices and thus has played a role in the development of magnetic storage media.

Despite the continuing progress in the sensitivity of magnetic resonance instruments, conventional magnetic resonance essentially remains a bulk technique, i. e., it is normally used to study a sample of macroscopic dimensions. Extending magnetic resonance techniques to the micro- or even nanoscale would open the possibility of studying the magnetic, chemical, and structural properties of individual nanoscale objects or even individual molecules. In 1991, John Sidles proposed to do exactly that by detecting magnetic resonance with microscale mechanical resonators rather than conventional RF coils [63]. The resulting system combined the essential features of MRI and atomic force microscopy (AFM) and therefore acquired the name magnetic resonance force microcopy (MRFM).

M₀

y

z

B₀

F

x

M₀

x

y

z

F

(b)

(a)

B₀

Figure 4.1: Schematics of the longitudinal (a) and transverse (b) MRFM configurations. Both schemes use the dynamics of magnetizationM0 in the external fieldB0. The time-dependent mag-

netization creates a force F on the magnetic tip that affects the motion of the cantilever. In the longitudinal case, the magnetization is modulated with the help of the RF field created by an external coil.

Shortly after the proposal, the first MRFM experiments were performed, detecting magnetic resonance of electron [64] and nuclear [65] spins. The sensitivity of MRFM experiments has been steadily progressing since, improving by a factor of more than one million by now. Recently, Rugar and coworkers demonstrated detection of a single electron spin [8] and nuclear magnetic resonance imaging with 90 nm resolution [9]. Experimenters are now closer than ever to the single-nuclear-spin sensitivity and the holy grail of MRFM—three-dimensional imaging of individual biomolecules and other nanoscale objects with atomic resolution. The progress towards this goal over almost two decades has been reviewed early in Ref. [66] and more recently in Ref. [67].

Fundamentally, MRFM experiments can be done in two different ways [68]: the motion of the mechanical resonator can be coupled to either longitudinal or transverse part of the magnetization of spins (see Figure4.1). The main difference between them stems from the fact that, for spins in a strong magnetic fieldB0, the longitudinal component of the magnetization evolves slowly, whereas

the transverse component performs fast precession at the Larmor frequencyωL=γsB0/~, whereγs

is the gyromagnetic ratio of the spin [62]. Correspondingly, longitudinal MRFM uses low-frequency resonators, typically in the kilohertz range. Transverse MRFM should use resonators that resonate at the Larmor frequency, which is typically in the megahertz range or higher.

experiments, the longitudinal magnetization is typically modulated at the resonance frequency of the cantilever with the help of an external source of electromagnetic field. The resulting time- dependent longitudinal magnetization exerts a periodic force on the magnetic tip of the cantilever, which measurably affects the motion of the resonator.

Experiments using transverse magnetization proved much more difficult. One reason is that transverse MRFM requires resonators that have high resonance frequencies and, at the same, are compliant enough to detect the tiny forces produced by spins. These two requirements conflict with each other because the frequency of a resonator is given by ωR = pkef f/mef f, where kef f

is the effective spring constant and mef f is the effective mass of the resonator. For a resonator of a given mass, the resonance frequency can only be increased by making the resonator stiffer (less compliant). Looking at the same expression another way, we obtainmef f =kef f/ω2

R, which means

that achieving a high resonance frequency and a small spring constant (high compliance) at the same time can only be done by minimizing the effective mass. Since nanomechanical resonators are currently the smallest and lightest mechanical resonators available, they are a natural choice in this situation, but even with nanoscale resonators transverse MRFM remains extremely challenging.

In this chapter I describe our work towards demonstrating a system in which the motion of a nanomechanical resonator is coupled to the transverse rather than longitudinal component of magnetization of nuclear spins.1 _{In the following sections, I analyze in more detail the transverse} MRFM system as well as other physical systems in which such coupling may be realized and describe our attempts to detect the signatures of such coupling in experiment. I conclude the chapter by exploring the theoretical analogy between the coupled spin-resonator system and the quantum optical model of a laser.

1_{I focused on nuclear spins because it is currently possible to fabricate devices that resonate at nuclear Larmor} frequencies, which are typically in the range of 1–100 MHz for an external field of a few Tesla. For electron spin resonance, the Larmor frequencies would be 30 GHz or higher, which far exceeds the frequency range of the currently feasible nanomechanical resonators.

4.1

Mechanisms of coupling between spins and mechanical

motion

One way to make a mechanical resonator interact with transverse magnetization of nuclear spins in a nearby sample is to attach a permanent magnet to a cantilever, as shown in Figure 4.2(a). This is basically the transverse MRFM (T-MRFM) system [68]. Note that in this case the ferromagnet on the cantilever tip creates a constant magnetic field of its own, which can be approximated as the field of a magnetic dipole. When superposed on the uniform external field B0, this field modifies

the total magnetic field seen by the spins and, therefore, their Larmor frequency. As a result, only a certain slice of the sample, known as the sensitive slice, will have the Larmor frequency equal to the resonance frequency of the cantilever. The position of this resonant slice will generally depend on the value of the external fieldB0, and at some values of the field, the resonant slice may be outside

the sample or not exist at all.

In this case of Larmor-frequency resonators, there are two aspects to the interaction between the spins and the resonator. First, the rotating transverse component of nuclear magnetization exerts a magnetostatic force on the ferromagnetic tip, which can drive or otherwise affect cantilever oscil- lations. Secondly, a moving ferromagnet creates an AC magnetic field, oscillating at the frequency of the cantilever motion, inside the nearby sample. At a point characterized by the radius vector~r, this RF field is given byA∂B⊥(~r)

∂x , whereA is the amplitude of cantilever motion and ∂B⊥(~r)

∂x is the

derivative of the transverse component of the ferromagnetic tip’s magnetic field with respect to the direction of the cantilever motionx(see Figure4.2(a)). We can therefore use the cantilever itself to create the RF field that will drive transitions between Zeeman levels of nuclear spins, obviating the need for an RF coil that is used in the longitudinal MRFM experiments.2

In principle, we could still use an external RF coil to control the magnetic spins, similar to the case of longitudinal MRFM, and use the cantilever only to detect the spin dynamics. However, because the RF field created by the coil would oscillate at the Larmor frequency, this field will 2_{This method of producing local RF was used in Ref. [69], where a cantilever with a magnetic tip was used to} create an RF field inside a small volume of a cell containing cesium vapor. This RF field then affected the populations of Zeeman levels of cesium atoms, which was detected optically.

sensitive