C- Factor:
4.3 Performing MRFM in Practise
4.3.5 Further Applied MRFM Methods
The capabilities of MRFM extend the ability of measuring spin noise and produce spatial images of the nuclear spin density. In principle most of the spin manipulation techniques developed and well-tried by the NMR community could possibly be adapted to MRFM. Even tough some would need modifications in the setup as e.g. a magnetic field gradient that can be switched off or a second RF-field perpendicular to ~B1. Three applied MRFM
methods, which have been examined during this work are discussed in the following. MANIPULATION THESPINNOISE’SDISTRIBUTION: The spin noise responsible for the statistical polarisation of an ensemble naturally follows a distribution determined by statistical mechanics. By observing the trace of the ensembles magnetisation Mz
4.3 Performing MRFM in Practise 79
manipulated [61,103]. In classical MRFM the spins are inverted twice per cantilever oscillation by a pair of two ARP pulses forming a 2π pulse. By applying only a π pulse within the oscillation period TL0, i.e. one ARP pulse with tp = TL0 instead of two with
tp = 1/2TL0, the sign of the spin fluctuation is inverted. If this is done each time Mz
exceeds or falls below a certain threshold, the natural Gaussian distribution of Mz is
narrowed and σMz is reduced. Similarly, the distribution can be shifted yielding the
averaged magnetisation Mz 6= 0, or it can be broadened.
The method finds application e.g. in reducing dephasing in quantum dots [157].
STORINGSPIN FLUCTUATIONS: By interrupting the continuous spin inversions statis- tical polarisations can be stored and recaptured for several seconds [61,103]. This is possible since the spin-lattice relaxation time T1 is bigger than the spin-lattice relaxation
time in the rotating frame T1ρ and the ensemble spin correlation time τm. As for the
manipulation of the noise’s distribution above, the trace of Mz is observed in real time.
Now, the ARP pulses are interrupted when a bigger polarisation than the statistical aver- age is measured. When restarted after a certain time tstorage, whereas T1ρ< tstorage< T1,
the polarisation is found to be still as high as before. It is only11 reduced by the - in
semiconductors - slow T1 decay. The ability to manipulate and store polarisations of
small spin ensembles is important for future development of MRFM and possibly for the implementation of solid-state spin qbits.
This technique builds the base for the experiment presented in chapter5and in the publication by Herzog et al. [104].
DOUBLE RESONANCE: Polarisations of different isotopes (i) in a sample can be transferred by tuning the particular Larmor frequencies12ωLarmor,i(t) = γi| ~B0+ ~B1,i(t)|
to be the same when the spins are in the x, y-plane of the Bloch sphere during the ARP. In this situation, ~B0 is compensated by the AC-component of ~B1,i(t), i.e. ~ω0,i = ~ωRF,i
and ~Beff,i = ~B1,i. The thereby established criteria is known as the Hartmann-Hahn
condition: γIB1,I = γSB1,S, whereas the indices I and S denote the two isotopes. By
bringing the two spin species into contact, the polarisations equilibrate in analogy to a thermal bath, and a possibly higher polarisation of one of the isotopes enhances the other.
The technique is called double resonance or cross-polarisation and was already pointed out by Bloch in the earliest reflections about magnetic resonance [2]. It is widely used in NMR to enhance signals of isotopes with low γ and low abundance (e.g.13C or
15N) and is part of most of the advanced NMR pulse sequences [158–161].
11The measurement observable ∆x is additionally reduced by a noise contribution inherent to the
measurement process.
12The pulse field ~B
1,i(t) has only an effect on the spins and adds up to ωLarmor,i(t) if its frequency ωRF,i(t)
Double resonance can not only be used for thermal or other kind of polarisations, it can also be applied to statistical polarisations in MRFM. Poggio et al. demonstrated the reduction of the statistical polarisation of one spin species due to this effect [98].
In principle, double resonance can also be combined with the technique of storing and recapturing statistical fluctuations. A high statistical spin polarisation of one isotope will thereby be transferred to another isotope and the enhanced polarisation will be measured there. An experiment with KPF6with the focus on doing a transfer between
fluorine and phosphorus has been initiated, but due to technical reasons13no successful
measurements could be made.
13RF reflections in the micro-wire causing the power of B
5 Thermal versus Statistical Polarisation
“
Das Leben ist werth gelebt werden, sagt die Kunst, die schönste Verführerin;das Leben ist werth, erkannt zu werden, sagt die Wissenschaft.
”
Friedrich Nietzsche, Homer und die klassische Philologie,Antrittsvorlesung in Basel, 1869 This chapter reflects and extends the experiments presented in the article
Boundary between the thermal and statistical polarisation regimes in a nuclear spin ensemble
by B. E. Herzog, D. Cadeddu, F. Xue, P. Peddibhotla, and M. Poggio published in Applied Physics Letters 105, 043112 (2014) [104].
5.1 Introduction
Nanometre-scale spin ensembles differ from larger ensembles in that random fluctuations in the total polarisation – also known as spin noise – exceed the normally dominant mean thermal polarisation. This characteristic imposes important differences between nano- MRI and conventional MRI protocols. In the former technique, statistical fluctuations are usually measured, whereas in the latter the signal is based on the thermal polarisation [2,
97,103]. Here, the nuclear polarisation of nanometre-scale volumes of19F spins using MRFM is studied. Thereby, the focus lies on the transition between the regimes in which thermal and statistical polarisation dominate.
The thermal polarisation – also known as Boltzmann polarisation – results from the alignment of nuclear magnetisation under thermal equilibrium along a magnetic field. The statistical polarisation, on the other hand, arises from the incomplete cancellation of magnetic moments within the ensemble. Depending on the number N of spins and the time it takes to reach thermal equilibrium, i.e. the longitudinal relaxation time T1,
either one or the other type of polarisation is more advantageous to measure. In cases where both, the statistical and the thermal polarisation, can be measured simultaneously,1 this method enables a simple manner to determine N and subsequently the ensemble’s
1Here, simultaneously means within the same measurement sequence, but not at the very same time.
volume. Furthermore it is much less flawed than the usual method, based only on the statistical polarisation (eq. (4.9) in section4.1.5).
The chapter gives at first an explanation of the origin and characteristics of the two types of polarisation. Subsequently the experimental details are carried out, followed by the presentation of the results and their discussion.