Magnetism of materials 1. Introduction Magnetism and quantum mechanics In the previous experiment, you witnessed a very special case of a diamagnetic material with magnetic susceptibility χχ = 1 (usually this parameter is in the range 10 5 10 6 ). The effect is related to the orbital part of the state, not its spin. In contrast, because the atomic orbital angular momentum is quenched in transition metal materials (almost zero), their ferromagnetism is related to the spin of the electron, a quantum concept without a classical counterpart. The magnetic moment of a material quantifies its ferromagnetism. In practice, this quantum operator is replaced with its expectation value and with a classical vector in macroscopic materials. It gives for instance the force between two magnets. In principle, quantum methods in atomic magnetism can be extended to models of magnetism in materials at zero TT. For instance, the exchange interactions within atoms give the ferromagnetism interaction constant JJ between neighboring atoms. Non-zero TT makes matters more complicated. Except for a few cases (1D Ising, 2D Ising), an exact solution of the problem of magnetism in materials at finite TT has not been found (for instance, for the Hubbard model (see Appendix)). Near TT = 0 we can consider ferromagnetism by adding a type of spin excitation (spin density wave), following classical Boltzmann statistics, which will give the Bloch s 3/2 law for the magnetization dependence MM(TT) on temperature. At higher TT, these spin density waves will scatter off each other and an approximation is needed. Usually, these quantum models are then replaced with a mean field theory (MFT) approximation with TT dependent parameters. Several interactions are included in MFT models. The spin-orbit interaction links the spins to a crystalline direction and gives the magneto-crystalline anisotropy KK(TT), the long-range magnetic dipole interactions give the shape energy, and we may have interactions with an external field. Although the basic equations may be written down, solutions for the magnetization MM(TT) can be very complicated. Many important and useful effects at finite TT (for instance, the existence and types of magnetic domain walls) are well described in these models. In this experiment, you will quantify the magnetic moment of a material and its dependence on the external field direction (the magnetic hysteresis loops). 2. Experimental setup In this experiment you will investigate one such effect: the dependence of a material s magnetization curves on the applied external field direction. The offset gives the magnetic hysteresis loop of a material. The opening of this loop may have many different origins because Page 1
the several small interactions above may all contribute. Even though it is difficult to be quantitative, this is a useful and widely-used diagnostic tool of the overall quality of a magnetic sample. Measuring magnetization can be done in several different ways. Two common methods are optical (applying the magneto-optical Kerr and Faraday Effects) and inductive (measuring a current induced in a coil). The experiment uses a vibrating sample magnetometer (VSM). An inverted VSM, in which the current is modulated and the force on the sample is measured is called an Alternating Gradient Magnetometer (AGM). Another, more sensitive method is SQUID, which developed from VSM with a special detection method, applies magnetic flux quantization. The VSM is sufficient for our purposes. It applies a modulation technique, periodically varying the sample position and detecting the induced current in the pick-up coils at the oscillation frequency only with a lock-in amplifier. This greatly improves the signal/noise ratio and is used in many experiments. MM HH cc MM ss Horizontal sample inside plastic straw Oscillation axis aa aa cc HH eeeeee HH aaaaaaaaaaaaaa Pairs of pick-up coils HH DDDD (a) (b) (c) Fig. 1: (a) Two offset magnetization curves make a hysteresis loop with a coercive field HH cc and saturation magnetization MM ss. Real loops will have more rounded edges. (b) The variation of flux through the pick-up coils gives a small periodic current. Its Fourier component at the frequency of the modulation is the signal. This complex number can be viewed as two real numbers (the in-phase and in-quadrature components). (c) A prolate ellipsoid and the HH DDDD field (see Appendix). 3. Experiments 3.1 Preparing the samples Paper clips are slightly ferromagnetic, with a magnetization sufficiently large to show good hysteresis loops with our VSM. Cut two pieces a few mm long, which should fit horizontally within a plastic straw. Their lengths should be close (see appendix for NN DDDD ). Page 2
This shape and orientation with respect to the applied field are preferred because the effect of demagnetizing fields is minimal (Appendix). Anneal one piece with the heat gun for a few minutes Magnetize both pieces for a few minutes, one while still hot, one at room temperature Next, measure the magnetization curves for these two samples. 3.2 Measuring the magnetization curves Switch the console power to On and wait 15 minutes Verify that Signal input is on Turn on the magnet cooling water. Press Power button on the magnet controller. The red button should be lighted, indicating adequate water flow. Press DC on. Password is DAQ. Open USB-6001 card test panel in NI MAX/Devices and Interfaces/NI USB-6001/Test Panel. Choose Analog output and enter a voltage of 1 V. The controller should go to 10 % of maximum current. Then, enter 0 V to bring it back. Load the first sample with its axis along the applied field direction (watch a small rubber O-ring at the top of the rod). Set Meter to Moment to start the oscillation and check that it oscillates. Open final.vi, select Sample rate = 1, samples to read = 1. The Write to measurement sub-vi should have one header only and one column only. Choose Analog Output (AO) Sinewave, 5 V amplitude, and rate =1. The frequency should change to f=10-3 Hz. Start Analog Input (AI) to collect data, then start AO Stop AO when it completes one half-period (up to 9V, then down to 0V) because of reversal switch issues. Stop the AI (stop the data acquisition or it will continue indefinitely, increasing the data file size). Set the voltage to 0 V in the AO DC mode Save the measurements in txt files Convert the time axis to applied magnetic fields using the applied AO sinewave Repeat for the other sample When done, press DC off, turn the water off, and press the Power button. Set Meter to Standby to turn the oscillation off and turn off the VSM console power Email for further analysis with Origin software. The plot of magnetization vs. applied field should resemble the part in the upper-right quadrant of Fig. 1(a). 3.3 Analysis Compare the coercive fields and the saturation magnetizations for the two samples Discuss any observed differences and their possible origins Page 3
4. Conclusion MFT can be applied in classical physics as well and the macrospin we measure can be given a classical analog. However, most of a ferromagnetic material magnetization is due to adding atomic spins, tracing its origin to the atomic quantum spin SS. Experiments on smaller samples or at faster timescales cannot be described with MFT models and require quantum-mechanical methods. The magnetism of materials is a dynamic scientific and technological research field. For instance, spin currents, stable radio-frequency sources from magnetization precession, ultrafast demagnetization and all-optical switching, are only some recent developments. This is the focus of the research in NS-005 and NS-123 laboratories. 5. Appendix Demagnetizing fields distort the results of a magnetization curve measurement. They depend on the geometry of the sample and its orientation with respect to the applied external field. The demagnetizing field always points opposite the applied field and is proportional to the sample magnetization. It is customary to define a demagnetizing factor from HH DDDD = MMNN DDDD. It can be shown that NN DDDD = 4ππ 3 for a sphere. Closed-form forms also exist for NN DDDD for an ellipsoid with two axes equal, as well as for rectangular objects. For instance, for an ellipsoid oriented as shown in the figure, we have (in CGS units) NN DDDD = 4ππ mm 2 1 mm mm 2 1 ln mm + mm 2 1 1 where mm = cc aa. For mm = 5, NN DDDD = 4ππ (0.056), or the measurement will be off by 5%. For arbitrary orientations and shapes, NN DDDD must be calculated numerically. These factors must be considered when testing and calibrating magnets. Exercise: approximate the sample with the prolate ellipsoid and calculate NN DDDD. This shows why we chose this shape and oriented it along the field. The effect of HH DDDD on the measurement can be accounted for by replacing the applied field with the true field HH tttttttt = HH aaaaaaaaaaaaaa HH DDDD. This re-scales the horizontal axis of the hysteresis loops. The Hubbard model applies real space 2 nd quantization of the electron field on a discrete lattice. It is the real-space analog of quantizing EM fields in momentum-space modes. In the Hubbard model, we define operators cc ii, cc ii +, which create or annihilate a particle at a specific Page 4
site ii (compare to the light, where the operators aa kk, aa kk + create and annihilate a photon in a mode with a specific momentum kk). Just as for light, to insure the permutation symmetry of the quantum states, the operators cc ii, cc ii + are required to satisfy specific commutation (integer spin) or anti-commutation (halfinteger spin) relations. Then, HH = tt iiii cc + iiiiii iiii cc jjjj + UU ii nn ii nn ii. The 1 st term is the kinetic energy of hopping from site to site, which depends on TT. The 2 nd term is a simplified interaction potential between two electrons at the same site. It depends on the electron spin (the arrows), just as it does in single atoms. The Hubbard model is useful in analyzing measurements in superconducting samples. For our samples, a simpler and effective model applies a spin-split band structure (the Stoner model of ferromagnetism). Page 5
Name Phys-602 Quantum Mechanics Laboratory I Magnetism of materials lab report Date Page 6