* vavg = _v _d,avg = -_eE τ_/m
</WRAP> | (1) |
where τ is the average time between scatters, also called the mean free time.
When there is a net velocity, there is a net current I flowing through the material. If we have a density of n electrons per unit volume and a cross sectional area of the material A, then the current density is given by
| j = I/A = -nev = ne_2E τ_/m. | (2) |
If we rearrange this, we find the fundamental form of Ohm’s law,
| E = ρj, | (3) |
where ρ = m/ne2τ is the resistivity of the material. Ohm’s law may be more commonly presented in the form V = IR, but the resistance R of an object depends on the shape and size of that object, whereas the resistivity is an intrinsic property of a material. For a wire of uniform density, the resistance is related to the resistivity by
| R = ρL/A | (4) |
==== 2.2.2 Sources of electron scatter ====
So far, we have not yet said what causes the electron to scatter. In most situations, it is appropriate to assume the electrons do not interact with each other or with the positive atom cores, and to approximate the discrete atom cores as producing a uniform background electric potential. In such a case, an electron will move along without changing speed or direction. Electrons instead scatter off things which break the periodicity of the potential such as impurities or vibrations in the lattice. A dislocation caused by a defect or a vibration will produce a local perturbation in the electric field which will cause an electron to scatter. Such a scatter decreases the average drift velocity and increases the material’s resistance.
We can neatly separate the resistivity into contributions from different scattering processes as
| { ${/download/attachments/201099372/eqn_5.png?version=1&modificationDate=1547062196000&api=v2}$ | (5) |
Because the concentration of impurities in a sample is fixed, the intrinsic resistivity due to impurities should not vary with temperature. However, the number of scattering events due to lattice vibrations in a sample is zero when T = 0 and increases with temperature. If we neglect all other scattering effects, we therefore have
| ρ(T = 0) = _ρ_i | (6) |
and
| _ρ_L(T) = ρ(T) - _ρ_i, | (7) |
where ρ__i is the resistivity due to impurities and ρ__L is the resistivity due to lattice vibrations.
Let us look more closely at lattice vibrations and their effect on the temperature dependence of the resistivity.
==== 2.2.3 Lattice vibrations and the Debye model ====
At zero temperature, the atoms in a solid possess no thermal energy and their positions remain fixed. However, as the temperature of the solid increases, the atoms acquire kinetic energy and begin to vibrate around their equilibrium positions. These atoms do not vibrate independently, though. They instead behave as though they were connected by springs and any vibration can be thought of as a sum of normal modes of this coupled system.
The geometry of the crystal puts constraints on which frequencies are allowed. There are, for example, more high frequency modes – which are modes of short wavelength involving only a few atoms – than low frequency modes – which are modes of long wavelength involving many atoms. In 1912, Peter Debye proposed a model to quantify this phenomenon where the density of states increased quadratically with frequency. This model also included a high frequency cutoff, ωmax, called the Debye frequency above which no further modes exist. This upper cutoff can also be expressed in terms of the Debye temperature _Θ_D,
| { ${/download/attachments/201099372/eqn_8.png?version=1&modificationDate=1547062196000&api=v2}$ | (8) |
which can be computed from physical properties of the material. For convenience we list tabulated values of _Θ_D below in Table 1 and show a comparison of the Debye model density of lattice vibration normal modes and with a realistic density of states in Fig 3.
|
*Element
</WRAP> |
*_Θ_D (K)
</WRAP> |
| Copper | 343 |
| Niobium | 275 |
| Vanadium | 380 |
| Tantalum | 240 |
Table 1 : Debye temperatures for the metals used in this experiment. (Source: Kittel).
{ ${/download/attachments/201099372/fig_3.png?version=1&modificationDate=1547062196000&api=v2}$\\
Figure 3 : (a) Debye model density of states. (b) A realistic density of states. Note that while the details differ, both have similar behaviors at small energy and both drop to zero at large energy. (After Kittel Fig. 5-14, page 113.)
The thermal energy of the crystal is bound up in vibrations of these modes. The energy contained in a particular mode is given by the quantum harmonic oscillator energy,
| { ${/download/attachments/201099372/eqn_9.png?version=1&modificationDate=1547062196000&api=v2}$ | (9) |
where ω is the frequency of that mode and n describes the quantum energy state. A vibrational mode can only gain or lose energy in discrete amounts, and these quanta of heat energy are called phonons. Therefore, a mode in the _n_th energy state is occupied by n phonons, each with energy { ${/download/attachments/201099372/E_p.png?version=1&modificationDate=1547062196000&api=v2}$.
==== 2.2.4 Temperature dependence of lattice vibration-electron scattering contributions ====
Let us first consider a solid at temperatures well above the Debye temperature, _T » Θ_D. In this limit, electrons possess enough thermal energy to excite all the modes of the system up to the cutoff frequency. Thus, as more thermal energy is added above the Debye energy, no additional modes can appear, but the energy of each individual mode increases. If we recall that the amplitude of a wave is related to energy as { ${/download/attachments/201099372/EproptoA2.png?version=1&modificationDate=1547062196000&api=v2}$, we see that a lattice vibration carves out a cross-sectional area σ which is proportional to temperature:
| { ${/download/attachments/201099372/eqn_10.png?version=1&modificationDate=1547062196000&api=v2}$ | (10 |
As the cross-sectional area goes up, the average time between scattering decreases and the resistivity increases. We can model this as
| { ${/download/attachments/201099372/eqn_11.png?version=1&modificationDate=1547062196000&api=v2}$ | (11) |
Combining this with the intrinsic resistivity, the total resistivity can be rewritten in the more common form
| ρ = _ρ_0[1 + α(T - _T_0)]. | (12) |
Here, the resistivity at temperature T is related to two constants measured at a reference temperature T0: α is called the temperature coefficient of resistance for a particular metal at temperature T0 and ρ__0 = ρ_(T = T0)._
At lower temperatures (T « ΘD), the electrons possess thermal energy only sufficient to excite those lattice vibration modes with energy of order E ≈ kBT (or, equivalently, with frequencies { ${/download/attachments/201099372/omega.png?version=1&modificationDate=1547062196000&api=v2}$). In this region, adding thermal energy does not go to increasing the energy of existing modes, but to exciting new modes. The resistivity in this low temperature region grows faster than linearly, but its behavior is complicated. For that reason, we will not attempt to quantitatively model the lattice contributions in this experiment.
==== 2.2.5 Superconductivity ====
At very low temperatures, some metals undergo a transition from conductor (with non-zero resistivity) to superconductor (with zero resistivity). Such a transition is due to the formation of so-called Cooper pairs.
To help understand Cooper pairs, consider a conduction electron which is moving freely in a solid. When the electron passes near an ion with positive charge, there is an attractive Coulomb force causing the ion to displace slightly. This deformation alters the charge density and in turn attracts more electrons. The effect is very weak, (we in fact said it was negligible above in Section 3.2.2), and at high temperatures just thermal energy is enough to overcome any attraction. However, in some metals at low temperatures – typically less than 10 K – this effect can become appreciable; two electrons with opposite spins, but similar energies can exchange energy with each other through the exchange of phonons and form a bound pair.
The superconducting transition occurs when all conduction electrons simultaneously form such pairs. Whereas free electrons are fermions which must obey the Pauli exclusion principle, Cooper pairs behave effectively like bosons and can condense into a single low energy quantum ground state. This ground state wavefunction extends over the whole system and correlates all the conduction electrons together. For this reason, the force is long-ranged even though it is weak; the distance between paired electrons can be much greater than the lattice spacing.
Importantly, energy is no longer exchanged between the lattice and electrons, but between the lattice and the condensate as a whole. Whereas a collision previously might have provided sufficient energy to break up a single Cooper pair, it now would need to provide enough energy to break up all the Cooper pairs to have any effect. Hence, below the superconducting temperature, no scattering occurs at all and the resistivity goes to zero.
===== 2.3 Semiconductors =====
In semiconductors, the resistivity also depends on temperature, but in a very different manner from metals. Whereas resistivity increases with temperature in metals, it decreases in semiconductors over a wide range of temperatures.
To understand why, recall the band structure of a semiconductor. At zero temperature, all electrons have energies in the valence band and none in the conduction band. The bands are separated by a small band gap and at T = 0, no electrons can cross the gap into the conduction band. But as T increases, more electrons gain thermal energy and move into the conduction band, leaving holes in the valence band. When an electric field is applied, not only do these free electrons move to produce a current, but the holes left behind also move. These holes move in the opposite direction from the electrons, but since holes also carry the opposite charge sign, the current from the two processes is in the same direction. Pure solids of semiconducting material with an equal number of holes and electrons are called intrinsic semiconductors.
However, it is common to intentionally dope a semiconducting material with a small fraction of a different material. The doping material is typically picked so that it has either one more or one fewer electron per atom than the base semiconductor. This extra electron (or hole) remains in the valence band and is localized, but it acts as a donor (or acceptor) thereby making the motion of the other electrons (or holes) easier. The extra electron of a donor will have an energy within the gap, but slightly below the conduction band. Similarly, the extra hole of an acceptor will have energy within the gap, but slightly above the valence band. Such doped semiconductors are called extrinsic semiconductors and are further broken down into n-type and p-type depending on whether they are donors of an extra negative charge (electrons) or acceptors of an extra positive charge (holes), respectively.
{ ${/download/attachments/201099372/fig_4.png?version=1&modificationDate=1547062196000&api=v2}$\\
Figure 4 : Band structure of n-type and p-type extrinsic semiconductors.
At high temperature, intrinsic and extrinsic semiconductors behave identically. The dominant effect in this regime is that as the temperature increases, more electrons are able to cross the semiconductor gap and participate in the conduction; the resistivity decreases exponentially as the temperature rises,
| { ${/download/attachments/201099372/eqn_13.png?version=1&modificationDate=1547062196000&api=v2}$ | (13) |
At lower temperatures, however, extrinsic semiconductors encounter a second effect. In order to help with conduction, the donor or acceptor has to be “activated” by receiving enough energy EA to liberate its electron into the conduction band (or its hole into the valence band). This behavior is similar to that outlined above, but with EA replacing EG:
| { ${/download/attachments/201099372/eqn_14.png?version=1&modificationDate=1547062196000&api=v2}$ | (14) |
At even lower temperatures, quantum mechanical tunneling of electrons or holes between doping atoms can occur and the temperature dependence changes to
| { ${/download/attachments/201099372/eqn_15.png?version=1&modificationDate=1547062196000&api=v2}$ | (15) |
This process is called Mott variable range-hopping.
In this experiment, we use germanium (Ge) doped with a small concentration of gallium (Ga); the gallium acceptors have a density of about 5 x 1015 cm-3. The energy gap for germanium is about _E_G = 0.67 eV, while the activation energy for the gallium is about _E_A = 0.01 eV. (See Chapter 11 of Kittel.)
====== 3 Procedure ======
—-
===== 3.1 Equipment =====
The sample holder contains four metal wire samples, one germanium semiconductor crystal, and a diode thermometer. The dimensions of the cylindrical samples are given with 1% uncertainty in Table 2.
The wires from the samples and thermometer run up through a thin walled stainless steel tube to a breakout box at the top of the cryostat where they connect with a ribbon cable. At the other end of the ribbon cable, the connections from the samples go into a scanning card in the Keithley Scanning digital multimeter (DMM). The leads from the diode thermometer go to the Lakeshore Cryonics temperature readout from which an output voltage, which is proportional to the temperature, is connected to the scanning DMM.
We determine the resistance of the each sample by measuring the current through and the voltage across that sample. However, if we measure the voltage near the current source, we would also be measuring the voltage drop along the wire leads, see Fig. 5(a). Therefore, we use the 4-terminal method shown in Fig. 5(b). The Keithley scanning multi-meter measures the voltage while supplying a constant current of 1.0 mA for this purpose. Note that the digital multi-meter, used as a voltmeter, has a very large impedance and therefore there is almost no current in the inner loop of Fig. 5(b).
The 4-terminal measurement technique applied to our samples is shown in Fig. 6.
|
*Sample
</WRAP> |
*Diameter (cm)
</WRAP> |
*Length (cm)
</WRAP> |
| Cu | 0.0077 | 600 |
| Nb | 0.013 | 150 |
| V | 0.0104 | 52.5 |
| Ta | 0.038 | 51.5 |
| Ge | 0.30 | 0.20 |
Table 2: Sample dimensions (to 1% uncertainty)
{ ${/download/attachments/201099372/fig_5.png?version=1&modificationDate=1547062196000&api=v2}$
Figure 5 : (a) A two-terminal measurement where voltage is measured near the current source such that V = IRSample + IRLeads. (b) A four-terminal measurement where voltage is measured near the sample such that V = IRSample.
{ ${/download/attachments/201099372/fig_6.png?version=1&modificationDate=1547062196000&api=v2}$
Figure 6 : Sample wiring
The resistance of the germanium semiconductor becomes so high at low temperatures that it cannot be made part of the circuit for the metal samples. To measure the resistance of the Ge sample, the multimeter is directly connected across it and switched to its ohmmeter function.
To measure temperature, the Lakeshore thermometer measures the voltage drop across a silicon diode which is supplied with a constant current of 10 mA. The Lakeshore device compares this voltage drop with an internal calibration and displays the temperature in Kelvin. Note that the conversion from voltage to temperature is done slightly different on the device display versus in software; the reading computed in software is more accurate while the display is approximated in such a way as to introduce systematic error of up to ±0.5 K. For this reason, use the display temperature as a rough measure only. The operation of the diode thermometer is briefly explained under Fig. 3 of the Specific Heat manual.
Fig. 7 shows the cryostat, samples, and the silicon diode thermometer in the copper can which provides a uniform temperature environment.
{ ${/download/attachments/201099372/fig_7.png?version=1&modificationDate=1547062195000&api=v2}$\\
Figure 7 : Cryostat and sample chamber
===== 3.2 Preliminary questions =====
Before we begin cooling the apparatus, it is useful to answer a few preliminary questions in your notebook. These questions will help you better understand what to expect as you collect data.
> NOTEBOOK:
>
> Question 1 : Look up the room temperature resistivities for copper, niobium, vanadium and tantalum. Using Table 2, compute the expected room temperature resistances of the four samples.
>
> Question 2: Is the uncertainty in the geometry of the samples a statistical or systematic source of error?
>
> Question 3: The dimensions of a material will expand or contract under temperature changes. Using the fact that copper's coefficient of thermal expansion (i.e. the fractional change in length per unit temperature increase) is 16.6 x 10-6 K-1, show whether or not this source of systematic error is negligible.
>
> Question 4 : Based on the theory section, what features should the resistivity versus temperature data show? Where is the curve linear? Where is it constant? Where is it exponential? Sketch (without using numbers) your expectations for a normal metal, a superconducting metal and a semiconductor.
>
> Question 5 : Based on the equipment description, what do you expect to be the main sources of error in temperature? In resistivity? What uncertainties will need to be propagated?
>
> Question 6 : Your measured temperatures will span a large range from 300 K down to 4 K. Will your relative uncertainty be largest at the low end or the high end of this range?
>
> Question 7 : How will you test the reproducibility of your temperature measurements? In other words, how can you estimate the statistical uncertainty in resistance and temperature?
===== 3.3 Preliminary measurements =====
While the apparatus is at room temperature, measure all resistances, measure temperature, and familiarize yourself with all instruments. Instructions for how to use the data acquisition (DAQ) software is given in the next section.
> NOTEBOOK : Record the resistance for each sample several times at room temperature. Because the temperature is stable, the spread in these data points will give you an estimate of the statistical uncertainty in your measurements of R and _T _that can be used elsewhere (where the temperature is not stable enough for multiple readings).
A note on uncertainties: You will need to estimate uncertainties on all the quantities you measure. As mentioned above, the temperature is, in general, not stable and so you cannot make multiple measurements of exactly the same temperature. The exceptions to this are the two endpoints – room temperature on the high end and 4.2 K at the low end. At these two extremes, you can estimate fractional statistical uncertainties based on repeated measurements. (You may also want to consider additional sources of uncertainty.)
==== 3.3.1 Data acquisition (DAQ) software ====
The Keithley Scanning DMM is controlled by a LabView application (Resistivity_DAQ) and performs the resistance and voltage measurements which can then be saved to a file. A screenshot of the software interface is shown in Fig. 8.
{ ${/download/attachments/201099372/image2016-3-30%208%3A52%3A39.png?version=1&modificationDate=1547062195000&api=v2}$
Figure 8 : Screenshot of the Electrical Resistivity DAQ software.
To begin, do the following:
- Make sure that the Keithley 2700 DMM is turned on and connected via a serial cable to the computer. If there is any doubt about what mode the Keithley DMM is in, simply turning it off and back on will reset it to the default (correct) mode. - On the computer desktop is a shortcut for the application Restivity_DAQ. Double-click it to open the application.
The upper left portion of the interface allows the user to perform a measurement of all five samples with temperatures by clicking on the Initiate Scan button. You will hear clicking sounds from relays in the Keithley as the instrument scans through all of the samples. As each sample measurement is completed, the corresponding resistance and temperature values are displayed. The measured values are also stored in a table and shown graphically.
In the upper middle portion of the interface are controls for performing a measurement on a single sample. The sample to be measured is selected from the drop down menu labeled Select Sample. Clicking on the Initiate button will initiate the measurement. The measured values are displayed and added to the table and graph.
In the upper right portion of the interface are buttons to Save the data stored in the table to a text file on disk. The _Read _button allows the user to read previously saved data into the application. Exit quits the program.
The bottom portion of the interface contains a graph of the resistance versus temperature data collected. At the top of the graph is a legend showing the point type and color for each sample. Clicking the colored radio button on the right-hand side of the legend toggles the plot for the associated sample on and off. It is recommended that the plot of the Ge data be turned off when viewing the metal samples (and vice versa) as the resistances of semiconductors and metals vary differently with temperature. Clicking on the black box showing the point and line style brings up options for how the data are plotted
A note on thermoelectric potentials: The leads and solder connections of your electrical circuit are at greatly different temperatures during your experiment. Consequently thermoelectric voltages appear in addition to the IR voltage drop across the metal sample you wish to measure. This is a serious problem when the IR voltage is small. The LabView software used in this experiment will subtract the thermoelectric voltage from your measured voltage: IR = V(I) - V(0), where V(I) and V(0) are the voltages measured with and without current I. V(0) is the thermoelectric voltage.
===== 3.4 Preparing the cryostat for cool down =====
Cryostats are designed to insulate against the three forms of heat transfer: conduction, convection and radiation. In our cryostat, glass is chosen since it is a poor thermal conductor. Two layers of vacuum help reduce convection. In many glass cryostats the glass is coated with aluminum to reflect radiation. However, we have chosen not to aluminize our cryostats so that you can see inside. Thus, we have a bit of radiative warming which will evaporate away the liquid helium within a few hours.
==== 3.4.1 Pump-able vacuum jacket preparation ====
The cryostat is made of Pyrex glass through which He atoms can slowly diffuse, reducing the insulating property of the vacuum jacket over time. You must reduce the amount of residual He gas in the jacket to restore the insulation before adding liquid helium. To do so, you will evacuate the jacket, backfill with air (in which there is very little He), and re-evacuate. You will then repeat this process three times. Any remaining air will freeze onto the glass surfaces, reducing the ultimate pressure to an acceptable level.
Start with all valves closed (knobs clockwise, flip handles down).
A: Turn on the pump to the vacuum jacket. Fully open valve V2.
B: Evacuate the jacket by opening valve V4 (flip up to open). You should hear a hiss and observe the pressure gauge reading drop. Continue to pump for about 30 seconds.
C: Close V4 and open V5 to allow air into the jacket.
D: Repeat steps B and C.
E: Repeat step B, this time allowing the pump to run for several minutes to establish a good vacuum.
F: Close V2 firmly, but do not force it. Turn off the pump and vent it by opening V5.
==== 3.4.2 Liquid helium chamber preparation ====
The liquid helium chamber is likely to contain air since its last use. You should replace this air with He gas so that there are no contaminants when you introduce liquid He into the chamber. For this procedure you will use a pump different from the one used for pumping out the vacuum jacket because the jacket should not be contaminated with He.
Pumpout
Again, start with all valves closed. Identify the piping to the He chamber. Attach the hose from the large pump under the lab bench to the port of V1. Turn on the large pump and then open valve V1 slowly to pump out the He chamber. Note the movement of the pressure gauge. After about 10 seconds of pumping, close V1 and turn off the large pump. Pull off its hose from V1 to vent the large pump.
Backfill with He gas
Backfill the He chamber with He gas. To do so, run some He gas through the hose from the He compressed gas cylinder to flush out air, then attach the He gas hose to the input controlled by V3. Open V3 to fill the chamber with He gas up to atmospheric pressure, a reading of 0 on the pressure gauge.
Leave V3 connected to He gas supply, since it will be necessary to add He gas in order to keep the He chamber at atmospheric pressure while the cryostat cools down.
===== 3.5 Taking data from 300 K to 100 K =====
Prepare to take measurements as a function of T as the cryostat is cooled by liquid nitrogen. This will take about 45 minutes, and you should try to capture as many data points as practical over the range from 300 K to about 100 K where the temperature will stabilize.
Pour liquid nitrogen (LN2) into the space between the two double-walled containers and cap the opening with a cloth to slow evaporation. During the experiment it will be necessary to add LN2 as it evaporates. Note that while surrounded by LN2 alone, your samples will never reach the temperature of boiling LN2, because some heat is constantly being supplied by conduction down the wires and by radiation into the cryostat.
> NOTEBOOK : Collect data as the temperature falls. Consider ways to minimize uncertainties in T, since T is changing with time.
After 45 minutes the samples should be close to 100 K and you should transfer liquid He into its chamber.
===== 3.6 Liquid helium temperature range =====
Liquid He is an expensive, nonrenewable resource and filling the cryostat is a delicate procedure. Do not attempt the transfer yourself. Ask the lab staff for help.
A: Position the liquid He storage vessel near the cryostat.
B: Take down the transfer tube and purge the air from the transfer tube by pushing He gas through it. If you fail to do that, the air will freeze and may block the flow of liquid He.
C: Open the transfer ports of the storage vessel (dewar) and of the cryostat, and insert the transfer tube with the longer part going into the storage vessel.
D: Now remove the He gas line from V3 and leave V1 open as an escape path for the evaporating He. Use the He gas line to pressurize the He storage vessel. This will push the liquid He from the storage vessel into your cryostat.
E: Use a light to look for liquid appearing at the end of the transfer tube in the cryostat. Helium will rapidly boil off (creating a loud hiss out of port V1) as the cryostat fills and the samples cool from 100 K down to 4.2 K. Stop the LHe flow by pulling the transfer tube adaptor out of the storage vessel port. One would like the cryostat to be about half full. If the helium level rapidly falls, it may be necessary to reinsert the transfer tube and repeat.
F: When the helium level has stabilized, remove the transfer tube, which tends to freeze to the O-ring of the cryostat port.
G: Close the two transfer ports and hang the transfer tube on its rack. Leave V1 open.
> NOTEBOOK : Take resistivity measurements while the sample holder is immersed in liquid He after the temperature stabilizes. As the temperature is constant, repeat each reading several times to establish the spread (statistical uncertainty) at this extreme.
When the sample is submerged in the liquid helium, the temperature will be a constant 4.2 K. To take readings above this temperature, you must raise the sample holder above the top level of liquid He into the gaseous region. The temperature that you find will vary according to the height above the liquid level. As the liquid evaporates, higher and higher temperatures will become available at the top of the gas column.
<blockquote>
NOTEBOOK : Collect data points by raising and lowering the sample holder to change the temperature. Consider the following advice:
</HTML>
* Collect data at lower temperatures first. If your samples rise to a high temperature (say > 50 K) and you attempt to reinsert the holder back into the liquid, you risk evaporating all of of the remaining liquid helium.
* For the metals, you should aim to have 30 or more data points between 4.2 K and 100 K. Whenever you get to a superconducting transition, measure repeatedly on either side of this point in order to accurately determine the transition temperature.
* For germanium, the resistivity changes rapidly below T = 10 K. In order to have sufficient data for the measurement of the activation energy Ea, take at least 20 data points between 6 K and 10 K.
</blockquote></HTML>
===== 3.7 Superfluid helium (if liquid remains) =====
At even lower temperatures, liquid helium can undergo a transition from a normal fluid to a superfluid. If liquid helium remains at the end of your resistivity measurements, we can observe this transition by reducing the vapor pressure in the helium chamber, thereby lowering the temperature.
A: Move the sample holder to its highest position in the cryostat, and close V3 to stop the flow of He gas.
B: Attach the larger vacuum pump to V1 and turn it on. This will reduce the pressure above the liquid He. Faster moving helium particles will evaporate off the surface of the liquid, leaving behind slower moving particles; thus, the temperature is reduced.
C: If the temperature can be lowered enough, the rapidly boiling helium will suddenly become still. This is superfluid transition.
D: Quickly note the vapor pressure and temperature at the transition point. These can be compared with literature values.
E: Switch off the big pump and close V1.
===== 3.8 Shutdown =====
When you are finished, leave the cryostat valves closed. Any excess He will blow out of the safety valve.
====== 4 Analysis ======
—-
For the REPORT , complete all of the following analysis and answer all of the questions therein.
===== 4.1 Metals =====
For the metals, plot the resistivity ρ against temperature T (in Kelvin). Compare ρ at room temperature with literature values. The room temperature resistivity values of the metals might disagree with literature values because you are measuring metal wires – which are polycrystalline and which contain some impurities – instead of single crystals of high purity.
> QUESTION: Do we expect the addition of impurities to increase or decrease the resistivity?
Determine the superconducting transition temperatures _T_c of V, Ta, and Nb and compare with the literature values.
> QUESTION: Is each superconducting transition sharp, or does it occur over a range of temperatures? How can you estimate an uncertainty for _T_C?
For each metal you studied, fit the high-temperature resistivity to the form of Eq. (12), ρ = ρ__0[1 + α(_T - T_0)]. (Let ρ__0 and α be fit parameters, but fix T0 = 293 K (i.e. room temperature).) Be careful to include only the part of the data which is linear. You may need to try fitting the data multiple times – each time cutting at a different temperature – in order to find the best fit region for each sample. While the theory predicts linear behavior only above the Debye temperature, you may find the fit works well below this point.
From this fit, identify the resistivity coefficient near room temperature for each metal sample. A literature value for the temperature coefficient of copper is widely available. Compare this to your result. Literature values are not as readily available for the other metals, but comment on how these compare to the value you find for copper.
For the low-temperature resistivity of metals, (at least where _T > T_C), the contribution due to lattice vibrations decreases with T. Identify the intrinsic resistivity due to impurities _ρ_i as the y-intercept of your curves for each metal. You may need to extrapolate (by eye) your curve to T = 0 or otherwise determine the constant value to which the resistivity is trending.
===== 4.2 Semiconductor =====
For the germanium semiconductor, plot Ln(ρ) against 1/T. You should notice a number of regions with different behaviors. Identify each of the regimes discussed in the theory section and comment on the qualitative properties of the plot results.
> QUESTION: Your sample of germanium doped with gallium is a p-type extrinsic semiconductor. Qualitatively, how would the results differ if this had been a pure germanium (intrinsic) sample? What if it had been germanium doped with a donor impurity (such as phosphorus) to form an n-type extrinsic semiconductor?
>
> QUESTION: Is the resistivity monotonic over the full temperature range or does the slope of the resistivity change sign at any point? What physical process is dominating when the sign of the slope changes?
In the temperature range of about 6.5 K to 9 K, fit the data to the form Ln(ρ) = A/T + B and use this form to estimate the activation energy. Compare to the literature value of _E_A = 0.01 eV. One should, in principle, also be able to fit a similar straight line form in the “intrinsic” range at higher temperatures in order to estimate the band gap energy, but complicating effects lead to a poor estimate. Both temperature-dependent charge carrier mobility and temperature-dependent lattice vibration scattering change the shape from the model prediction in this region.
====== Rubric ======
—-
When writing your report, consult the rubric and notes below for the appropriate quarter**.
Spring quarter
| Description | Percent |
| Abstract | 5 |
| Summarize the motivation, methods, results, and conclusions of your experiment. Include numerical results (with uncertainties). | |
| Introduction | 10 |
| Provide an introduction that describes the experiment in sufficient detail to give context for the reader to understand your subsequent analysis and discussion of the results. You may touch on motivation, theory, apparatus, experimental method and/or procedure. The introduction has a strict limit of no more than 2 pages of text (figures not included). You may lose points for exceeding this limit. | |
| Metal Resistivity | 40 |
| Provide ρ vs. T plots for each metal sample. For at least one metal, identify and qualitatively describe the relevant features in the resistivity plot. Fit the linear region of the high temperature metal resistivities to the model described in the theory section and provide values of the resistivity coefficients at room temperature. Identify the intrinsic resistivity due to impurities for each metal. Identify the superconducting transition temperature _T_C (for each metal in which a transition is observed) and estimate uncertainties. Discuss how you estimated and propagated uncertainties for all measured and calculated values. Answer any additional questions from the Analysis section not addressed specifically above. | |
| Semiconductor Resistivity | 20 |
| Provide a plot of Ln(ρ) vs. 1/T for germanium. Identify and qualitatively describe the relevant features in the semiconductor resistivity plot. In the temperature range of about 6.5 K to 9 K, fit the data to the form Ln(ρ) = A/T + B and use this form to estimate the activation energy. Answer any additional questions from the Analysis section not addressed specifically above. | |
| Discussion and Conclusions | 15 |
| Discuss your observations and conclusions from the experiment. Compare quantitative results to literature values when possible or to physical expectations when values are unavailable. Discuss uncertainty contributions (both statistical and systematic) and name any unaccounted for sources of error. | |
| Style | 10 |
| The report should conform to PHYS 211 style expectations including appropriate plot formatting, use of figure captions, numbering (and in-text referencing) of figures and tables, numbering (and in-text referencing) of equations, and proper citation of sources (including the wiki). The report should read as a narrative, not a fact dump, with complete sentences and and a logical report structure. You will be evaluated on the strength of the overall arguments and the clarity of writing. | |
Sample Resistances at Start of Spring 2018.
| Sample | T = 291K 2Ω | T =83.5K 4Ω |
| 1 | 33.4 | 2.82 |
| 2 | 41.4 | 3.25 |
| 3 | 19.8 | 0.22 |
| 4 | 38.6 | 5.44 |
| 5 | 24.2 |
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