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Introduction to AC Susceptibility QuantumDesign Introduction to: AC Susceptibility AC Magnetic Measurements Dinesh Martien Introduction AC magnetic measurements, in which an AC field is applied to a sample and the resulting AC moment is measured, are an important tool for characterizing many materials. Because the induced sample moment is time-dependent, AC measurements yield information about magnetization dynamics which are not AC Magnetometry obtained in DC measurements, where the sample moment is constant during the measurement time. This application note In AC magnetic measurements, a small AC drive magnetic field will briefly describe how AC magnetic measurements are is superimposed on the DC field, causing a time-dependent performed, discuss the meaning of the data that come out of moment in the sample. The field of the time-dependent an AC measurement, and show some measurement examples. moment induces a current in the pickup coils, allowing meas- urement without sample motion. The detection circuitry is configured to detect only in a narrow frequency band, normally DC Magnetometry at the fundamental frequency (that of the AC drive field). DC magnetic measurements determine the equilibrium value In order to understand what is measured in AC magnetometry, of the magnetization in a sample. The sample is magnetized first consider very low frequencies, where the measurement is by a constant magnetic field and the magnetic moment of the most similar to DC magnetometry. In this case, the magnetic sample is measured, producing a DC magnetization curve moment of the sample follows the M(H) curve that would be M(H) . The moment is measured by force, torque or induc- measured in a DC experiment. As long as the AC field is small, . ω tion techniques, the last being the most common in modern the induced AC moment is MAC = (dM/dH) HAC sin( t) ω instruments. Inductive measurements are performed by where HAC is the amplitude of the driving field, is the moving the sample relative to a set of pickup coils, either by driving frequency, and χ = dM/dH is the slope of the M(H) vibration or one-shot extraction. In conventional inductive curve, called the susceptibility. The susceptibility is the quant- magnetometers, one measures the voltage induced by the ity of interest in AC magnetometry. moving magnetic moment of the sample in a set of copper pickup coils. A much more sensitive technique uses a set of As the DC applied magnetic field is changed, different parts of superconducting pickup coils and a SQUID to measure the the M(H) curve are accessed, giving a different susceptibility. current induced in superconducting pickup coils, yielding One advantage of the AC measurement is already evident: the high sensitivity that is independent of sample speed during measurement is very sensitive to small changes in M(H) . extraction. Inductive magnetometers can also be used to Since the AC measurement is sensitive to the slope of M(H) perform AC magnetic measurements. and not to the absolute value, small magnetic shifts can be detected even when the absolute moment is large. Introduction to: AC Susceptibility At higher frequencies than those considered above, the AC important for spin-glasses, because the freezing temperature moment of the sample does not follow along the DC magnet- cannot be extracted from specific heat.1 Furthermore, the ization curve due to dynamic effects in the sample. For this location of the cusp is dependent on the frequency of the AC reason, the AC susceptibility is often known as the dynamic susceptibility measurement, a feature that is not present in susceptibility. In this higher frequency case, the magnetization other magnetic systems and therefore confirms the spin-glass of the sample may lag behind the drive field, an effect that is phase. Both of these features are evident in AC susceptibility detected by the magnetometer circuitry. Thus, the AC mag- data for Cu1-xMnx as shown in Fig. 1. netic susceptibility measurement yields two quantities: the magnitude of the susceptibility, χ , and the phase shift, ϕ (relative to the drive signal). Alternately, one can think of the 0.3 susceptibility as having an in-phase, or real, component χ' 0.335 1% χ' and an out-of-phase, or imaginary, component χ" . The two Max representations are related by 0.330 0.2 (emu/mole (emu/mole Mn) ' 2 2 χ χ' = χ cos ϕ χ = χ' + χ" 0.325 (emu/mole (emu/mole Mn) χ χ ϕ ϕ χ / χ ' " = sin = arctan( " ' ) χ 0.1 0.320 9.00 9.50 10.00 In the limit of low frequency where AC measurement is most Tf Temperature (K) similar to a DC measurement, the real component χ' is just 0.0 0 50 100 150 the slope of the M(H) curve discussed above. The imaginary Temperature (K) component, χ" , indicates dissipative processes in the sample. In conductive samples, the dissipation is due to eddy currents. Figure 1. AC susceptibility of CuMn (1 at% Mn) showing the Relaxation and irreversibility in spin-glasses give rise to a non- cusp at the freezing temperature. The inset shows the frequency zero χ" . In ferromagnets, a nonzero imaginary susceptibility dependence of the cusp from 2.6 Hz (triangles) to 1.33 kHz 2 can indicate irreversible domain wall movement or absorption (squares). Figure reprinted with permission. due to a permanent moment. Also, both χ' and χ" are very sensitive to thermodynamic phase changes, and are often used The irreversibility in spin-glasses leads to a nonzero out-of- to measure transition temperatures. AC magnetometry allows phase component, χ" , below the spin-glass freezing tempera- one to probe all of these interesting phenomena. Typical meas- ture. Because spin-glasses have unique magnetization dynam- urements to access this information are χ vs. temperature, χ ics, many interesting effects are observed in the susceptibility vs. driving frequency, χ vs. DC field bias, χ vs. AC field behavior. For example, Jonsson, et al. showed how χ" has a amplitude, and harmonic measurements. Some of these will "memory" for temperature treatment.3 They cooled a sample be discussed in the examples below. of Ag89Mn11, paused at 23 K, and then continued cooling. Upon warming, χ" showed a dip at 23 K, indicating a memory of the pause. By examining the behavior of χ" during various Measurement Examples temperature treatments, these workers probed the dynamics of the Ag89Mn11. They developed a detailed picture of the dy- SPIN-GLASS. namics in terms of spin-glass domains and droplet excitations. Spin-glass behavior is usually characterized by AC susceptibility. In a spin-glass, magnetic spins experience random interactions SUPERPARAMAGNETISM. with other magnetic spins, resulting in a state that is highly AC susceptibility measurements are an important tool in the irreversible and metastable. This spin-glass state is realized characterization of small ferromagnetic particles which exhibit below the freezing temperature, and the system is paramagnetic superparamagnetism, the theory of which was originally above this temperature. The most studied spin-glass systems explained by Néel and Brown.4,5 In this theory, the particles are dilute alloys of paramagnets or ferromagnets in nonmag- exhibit single-domain ferromagnetic behavior below the block- netic metals, typified by Cu1-xMnx. ing temperature, TB , and are superparamagnetic above TB . In the superparamagnetic state, the moment of each particle The freezing temperature is determined by measuring χ' vs. freely rotates, so a collection of particles acts like a paramagnet temperature, a curve which reveals a cusp at the freezing tem- where the constituent moments are ferromagnetic particles perature. The AC susceptibility measurement is particularly (rather than atomic moments as in a normal paramagnet). QuantumDesign In the Néel-Brown theory, the particles are assumed to be non- The scaling behaviors of χ' and χ" provide additional inform- interacting and the blocking temperature is given by ation about particle interactions and the distribution of particle sizes. For noninteracting particles, χ' vs. T curves for various ∆E T = , particle concentrations are identical when properly normalized. B ln(τ/τ )k 0 B Deviation from this behavior indicates that interparticle inter- where ∆E is the energy barrier to magnetization reversal in actions are important. As shown in Fig. 2, Luis, et al. applied τ τ a single particle, is the measurement time, 0 is called the this method to ferritin particles and found no evidence for 8 χ attempt frequency, and kB is the Boltzmann constant. The interactions. The same experimenters plotted " vs. a scaling measurement time is typically 1-100 sec for DC measurements, variable that depends on the measurement frequency to find and is the inverse of the measurement frequency for AC meas- that the particle sizes follow a gamma-function distribution. urements. The utility of AC susceptibility for superparamag- netism stems from the ability to probe different values of τ by MAGNETIC PHASE TRANSITIONS. varying the measurement frequency. The dynamic susceptibility is also a powerful tool for examin- ing the nature of magnetic phase transitions, such as ferro- Above the blocking temperature, χ" is small and χ' follows magnetic transitions. Typically, χ diverges at the critical tem- the Curie law χ' ∝T -1, as expected for paramagnetic behavior. perature of a ferromagnetic phase transition. Critical exponents From the slope of 1/ χ' vs. T, one obtains the volume of the characterize the nature of the divergence as a function of magnetic particles (assuming monodisperse particle size).6 In temperature and DC applied field. Determination of these a recent experiment, Lee and coworkers used this method to critical exponents allows one to distinguish between various determine the size of Fe platelets in Cu/Fe multilayers.7 Futher- models of magnetic interactions, such as the 3-d Heisenberg, more, the imaginary component of the susceptibility peaks at X-Y, or Ising model.
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