Low-temperature physics
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- Elwin Phelps
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1 Low-temperature physics 1. Introduction Superconductors in magnetic fields Early transport measurement results were analyzed with the Drude classical scattering model, which may be viewed as the transport analog to the classical Lorentz dipole models of optical transitions. To observe quantum-mechanical effects in solids we usually have to cool the material, to diminish the disturbances to electron states that arise from lattice vibrations. In the case of superconductors, the effect of these lattice disturbances (different from the virtual phonons coupling electrons into pairs) is effectively removed with the opening of a gap in the electron spectrum at the critical transition temperature TT cc. One may ask what will be the screening properties of such a superconductor. An electric field will not penetrate into a normal metal but magnetic fields will usually do, except for the case of special materials (μμ metals). In contrast, it is observed that an external magnetic field will not penetrate into the interior of a superconductor except for a very small distance. This is due to induced surface supercurrents flowing with no resistance that shield the material from this external perturbation. The superconducting probability current depends on the kinematic momentum (which is not the same as the canonical momentum when a magnetic field is present) or JJ ss = 1 (ψψ 2mmmm s ψψ ss ψψ ss ψψ ss ) eeee ψψ mmmm ss 2 eeee ψψ mmmm ss 2 = eeee ρρ mmmm sswhen there is a uniform macroscopic quantum state inside the material (translational invariance of a macroscopic quantum state). Together with 2 AA = μμ 0 JJ we get 2 AA AA λλ2 = 0 with λλ = mmmm 50 nnnn for ρρ eeρρ ss μμ ss = AA 3. This is the distance a magnetic field will penetrate inside a type-i superconductor (mercury and most of the elemental superconductors, with the exception of niobium and a few others). The Meissner Effect in superconductors is the expulsion of an applied external magnetic field from a material on transition to the superconducting state (a different observation, see Fig. 1(a)). While field screening may be explained by eddy supercurrents induced gradually as the magnet is brought close to the SC and flowing without resistance on the surface of the SC, the expulsion of the external field is not the same as the eddy current screening of classical electrodynamics. The magnet has to move to generate eddy currents, while the magnet is stationary in the Meissner Effect. The field expulsion is a QM effect, and it came as a surprise when first observed. In practice, the high TT cc superconductor Bi 2 Sr 2 Ca 2 Cu 3 O 10 or BSCCO-2223 we use is type-ii. The magnetic field penetrates in vortices (at lower critical field HH cc1, on the order of 0.1 TT and Page 1
2 less), which are first aligned in a regular lattice, and completely remove the diamagnetism at a much higher upper critical field HH cc2 [measured or estimated at ss of TT]. A printout in the lab [Hellman et al., J Appl. Phys. 63, 447 (1988)] describes a similar experiment with YBa 2 Cu 3 O 7 ( YBCO ), another type-ii high TT cc superconductor. N N 4ππππ SSSS Type-I (b) SC N BB eeeeee Type-II HH eeeeee HH ccc HH cc HH ccc SC BB eeeeee SC BB eeeeee (a) II + VV + VV II (c) Fig. 1: (a) Field screening for zero-field cooling (cooling first, applying field next) compared to field expulsion for field cooling (applying field first, cooling next) in type-i superconductors. The Meissner Effect relates to the 2 nd case. (b) Magnetization of a type-i and type-ii superconductor ( 4ππππ MM in SI) [the numerical values for fields vary widely between materials]. (c) The geometry of a 4-probe resistivity measurement. 2. Experimental setup Safety: BSCCO is a skin and eye irritant. Avoid creation of dust (the material is brittle and will chip at the edges). Work in the area of the brown paper towel. Use gloves. Remove them when done without touching their outer surface. Dispose of the used gloves, the brown paper towel, and the white wipes used to remove moisture from the sample in a disposal plastic bag and outside in the hallway bins. Wash hands at the end of experiments. Safety: Cryogenics is in itself an interesting and widely-used technology, with cooling done in a variety of ways. We use liquid nitrogen. Cooling with liquid nitrogen has two hazards: an obvious one, which can hurt badly, but usually does not kill, and a less obvious one, which kills. Use the large thick cryo-gloves when raising and lowering the transfer line. Do not take the transfer line fully out of the Dewar. Lower it when transferring liquid and raise it about 1.5 feet when done (optional). Page 2
3 Danger: in addition, to the hazard of low temperatures, liquid nitrogen is an asphyxiation hazard, as you can see from the papers posted in the lab [ 8 deaths/year in US, with 1 death/ 2 years in a laboratory]. The liquid nitrogen expansion ratio is 1:695 (1 liter of liquid evaporates into 695 liters of gas at STP). The volume of the room is 2500 cubic feet. Replacing ½ of the air with pure nitrogen requires 1.8 cubic feet of liquid ( 50 LL) and would lower the oxygen concentration from 21 % to 1 21% = 10.5%. This would be a deadly environment. There is some ventilation in the 2 room, but it can fail without warning, we are using small quantities of liquid, but if we forget the needle valve open, the liquid volume is larger. This hazard is particularly of concern in small rooms located below grade, as is 024-LL. Read the highlighted lines on the board. To increase the air flow and minimize any possibility of a hazardous situation: (1) The lab door must be wide-open at all times during the experiment (2) Open the door to the lab across the hallway (key is next to note) (3) Prop open and post the door to the staircase (the 1 kg brass weight is a good door prop) (4) Roll the Dewar out of the room at the end of experiments Be alert! Because of this hazard, this experiment should be done only during the 9-7 interval during the week. 3. Experiments First, you will observe the levitation that arises from the magnetic field screening the superconductor s interior with zero-field cooling and correlate the observations with the resistivity transition. Then, you will observe the levitation due to field expulsion with fieldcooling (the Meissner Effect) Zero-field cooling and levitation from field screening There are three Ni-plated NdBFe neodymium permanent magnets on the cover of the power supply. Please do not lose them. Black tape marks the magnetic poles of the cubic magnet. Hint: the cubic magnet is much easier to pick up later on, if you put this magnet on the cover with its moment horizontal. Measure HH mmmmmmmmmmmm with the Hall Effect sensor (may zero the needle in the middle of the display to see the sign change as the sensor face orientation is reversed). Measure the approximate variation of the field for the cube along its moment and across it (Fig. 2) [I got 700 G touching the side, 300 G with 1 mm spacing, giving 400GG gradient]. mmmm Check that no moment can be detected for the superconductor in its normal state, Place the SC disk in a glass beaker at the bottom of the open Dewar. Please use the foam to prevent scratches to the Dewar coating. Roll the large Dewar in, open the needle valve and cool to the lowest temperature. BSCCO has a transition temperature of TT cc 110 KK (subtract 20 KK from the lowest Page 3
4 temperature reading, and 0 KK from the reading at room temperature, with interpolation inbetween). Take the SC out and put it on the piece of white foam on the table. Check the material does not have a detectable moment in the superconducting state. The material is a perfect diamagnet below the transition and has no moment of its own (except for a small amount of trapped flux lines, giving a field of a few tens of GG). The magnetic properties of the SC arise entirely in response to an applied magnetic field. Place the cubic and cylindrical magnets on the superconducting disk. The interaction energy depends on the magnet and SC shapes and relative orientation. With the SC sample being fixed in position, and given shapes, the magnet will rotate and settle in a position that minimizes this interaction energy. The cylinder will flip on its side with the moment horizontal. Similarly, the cube will have its moment approximately horizontal or slightly tilted. This is because the normal component of the field is continuous, as shielding surface super-currents will not introduce a field along this direction. Therefore, an orientation with the magnet moment vertical will necessarily introduce a lot of vortices in the superconductor (the magnet has a field > HH cc1 a few mm away, so it introduces vortices into BSCCO). This costs energy and the configuration is not favored. Observe how the cubic magnet moves to the SC disk sides when its moment is aligned vertically. In contrast, its position will be relatively stable when the moment is oriented inplane. This orientation has lower energy. In addition, pinning of relatively few vortices induced inside the SC prevent sideways sliding. Use the plastic tweezers to move the magnet back toward the center, if it drifts off toward the edge. Orientation Of the Hall Effect Sensor Fig. 2: (a) Magnetic field lines for the cube and sketch of the sensor position. (b) Flux lines entering the superconductor The superconducting transition in resistivity measurements You will use a 4-probe geometry [Fig. 1(c)]. Connect the red wires to the power supply outputs (top pair, not the lower pair) Page 4
5 Connect the blue and yellow wires to the first two inputs in the DAQ card (to ground and to AI-0 input). Connect the thermocouple. Turn on the power supply. The voltage that triggers the current limit light is a few volts. Verify that the light turns on at 3 VV. This checks that the current is indeed passing through the sample and the current connections are good. Then, lower the voltage until the light turns off. Select NI-MAX icon, arrow in Devices and Interfaces, USB-6002, and Test Panel Set channel Dev1/ai0, mode on demand, RSE input configuration, and click Start. You can change the horizontal range by changing the numbers directly on the plot. Verify that a comparable excursion in the detected voltage is observed as you vary the power supply voltage. This checks that the voltage connections are good. Cool the SC, put the magnet on top and correlate drop of the magnet on the surface, which signals the disappearance of diamagnetism as the sample warms up, with a jump in the measured voltage. When done with this section, gently heat the superconducting sample in the brass casing with the heat gun (use the middle setting and keep at least 2-3 inches between the heat gun muzzle and the sample). It will take 2-3 minutes of constant heating to melt the condensation. Wipe off all resulting moisture carefully with white paper towels (put these in a plastic bag for disposal). Return the sample with the wires neatly coiled into the jar with desiccant (Hint: coil wires outside first before lowering the sample into the jar). Lock the lid into place Field-cooling and levitation from field expulsion (the Meissner Effect) You observed magnetic levitation, when first cooling and then applying the magnetic field ( zero-field cooling ). The Meissner Effect is the name given to field expulsion, not filed screening. The magnetic field is applied first and will enter into a material in its normal (not superconducting) state. The material is cooled next ( field-cooling ). The magnetic field is unexpectedly expelled from within the material at the superconducting transition temperature. The field re-enters on warming back up. This is illustrated in the right column of Fig. 1(a). It is possible to demonstrate levitation following this different experimental procedure. Place the other SC disk, without a brass casing, on the 4-magnet base in the beaker. This is more fragile sample. Please wipe any fragments that you see and place them in the plastic bag for disposal at the end. Pour liquid nitrogen on it, to cool it in a preliminary step (the beaker should be on the white foam outside the open Dewar for better visibility). Then, placing the end of the transfer line Page 5
6 a few mm above the SC disk will gradually (may take a few minutes) cool it below TT cc. You should then see the SC disk slightly levitating above the 4-magnet base. It may then slide sideways off the base. This shows that a levitation force is present and, just as in the 1 st case, the field is zero inside the superconductor. For this SC sample, just put it back in its plastic cover and in the jar quickly, without heating and wiping moisture, as this will produce more fragments and dust. When done: Turn off the gaussmeter and the power supply. Verify the Dewar needle valve is in the closed position (do not overtighten). Raise the transfer line about 1.5 feet (optional). Move the Dewar to the hallway and secure the transfer line with the yellow rope. 4. Conclusion Meissner Effect can be used to measure the dependence of the field HH cc on temperature. It is an example of the insight that can be obtained into a phase transition, without a microscopic model, as it gives a convenient way of measuring the thermodynamic potentials. For instance, for type-i superconductors, the results of HH cc (TT) can be applied to show that the transition is 2 nd order at BB = 0 and 1 st order at BB 0 and to predict the dependence of specific heat cc VV (TT) on temperature [see the spring 2017 Thermal Physics qualifying problems]. In addition, all microscopic models must be able to explain the Meissner Effect. Appendix Estimate of the levitation force The levitation force for type-ii is FF = U = dddd (MM BB ), where UU is the magnetic interaction energy. The integral is over the volume of superconductor, and MM is the induced magnetization in the superconductor [Fig. 1(b)] in response to the applied magnetic field. Because both BB and MM depend on the magnet and superconductor shapes (demagnetization fields) and relative orientation, calculations usually require numerically obtaining the fields and then integrating. The order of magnitude can be estimated for a constant derivative and magnetization, asff zz VVMM zz BB zz. The magnetic field and its derivative at the superconductor position (neglecting the SC demagnetization factor) measured with the gaussmeter should be 400 GG 300 GG and. Then, FF 3 mmmm ππ (2 mmmm mmmm)2 400 GG 300GG 5 mmmm 10 3 NN. The 0.92 gg cube weight FF ww NN is of the same order of magnitude. This estimate shows why this force can be used in niche applications, but can never levitate a train. The article printout has an alternative calculation for type-ii, which considers the energy of the vortices. Page 6
7 Name Phys-602 Quantum Mechanics Laboratory I The Meissner Effect lab report Date Page 7
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