m'VE'F-‘tfr: THE EFFECT OF A MAGNETIC FIELD ON THE THERMOPONER 0F SILVER-GOLD ALLOYS By Michae] J. O'Caiiaghan A THESIS Submitted to Michigan State University in partial fulfiiiment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1978 ABSTRACT THE EFFECT OF A MAGNETIC FIELD ON THE THERMOPOWER OF SILVER-GOLD ALLOYS By Michael J. O'Callaghan Four dilute alloys of silver-gold were prepared in the form of long polycrystalline wires. The change in thermoelectric voltage was measured due to the presence of tranverse magnetic fields up to 50 k6. Measurements were made from 4.2°K up to 48°K. From this data the change in thermopower AS was obtained by fitting the voltage vs. temperature data to a curve and differentiating. It was found that the magnetic field enhanced the thermopower and that a peak occurs in A5 at about the same temperature as the peak in the zero field thermopower. The position of the peak in AS shows very little dependence on magnetic field strength. It does, however. move to higher temperatures with increasing gold concentration. The behavior of AS does not correlate well with the quantity wT, where T is the relaxation time for an electron in the alloy and m is the cyclotron frequency of a free electron. ACKNOWLEDGEMENTS I wish to thank Dr. Peter A. Schroeder for suggesting the problem and for his assistance and guidance in conducting this work. Financial support by the National Science Foundation is grate- fully acknowledged. ii Chapter I II III IV V VI Appendix A B References TABLE OF CONTENTS INTRODUCTION THEORY Diffusion Thermopower Phonon Drag Thermopower Magnetic Field Effects Impurities EXPERIMENTAL APPARATUS Cryostat SAMPLE PREPARATION DATA ANALYSIS DISCUSSION OF RESULTS Calibration Data for Resistance Thermometer Computer Program Used to Fit EMF Data to a Polynomial iii Page 47 48 55 LIST OF TABLES Table Page l Data on Ag-Au Alloys 32 2 Calibration Data for Resistance Thermometer 47 3 Coefficients used for Polynomial fit to Resistance Data 48 iv Figure (A) (11 KOCDNON 10 11 12 13 14 15 16 17 18 19 20 21 LIST OF FIGURES Thermocouple schematic Thermoelectric currents Electron-phonon scattering processes The Fermi surface of silver8 Method for directly measuring the change in thermopower Schematic of cryostat sample mount Cryostat-solenoid geometery Electronics Thermometer resistance EMF vs. T Effect of oxygen anneal Resistivity vs. T Variation of AS with concentration (50 k6) AS vs. H Zero field thermopower7 AS/T vs. 12 TAS vs. 72 Change in S of Ag + .011 at. % Au .Change in s of Ag + .013 at. % Au Change in S of Ag + .047 at. % Au Change in S of Ag + .048 at. % Au V Page 11 17 18 19 21 28 3O 34 35 36 38 39 41 42 43 44 45 46 CHAPTER I INTRODUCTION In previous studies of the effects of a magnetic field on the thermopower of a conductor it has been unclear what contribution to the total thermopower was being changed. Theoretical consideration 1 2 are both of diffusion thermopower and phonon drag thermopower compatible with experimentally observed magnetic field dependences. Observing the magnetic field effects on dilute alloys may be of some use in deciding between the two models. Dilute silver-gold alloys have been chosen for the following reasons. (1) They are readily soluble in one another. (2) Extensive measurements have been made of their transport properties. (3) Since the gold is much heavier than silver it is a good scatterer of phonons. Since gold is homovalent with silver, it is a comparatively poor scatterer of electrons. These properties will be useful in differentiating between phonon drag and diffusion effects. The purpose of this thesis is to describe the experimental appa- ratus and methods used to study the effects of magnetic fields on the thermopower of dilute alloys and then to discuss the results of those measurements. CHAPTER II THEORY When a temperature gradient is maintained in a conductor an associated electrical potential gradient is also created. The ratio of the electric field -dV/dx to the temperature gradient dT/dx at a particular temperature defines the thermopower S at that temperature. _ dV/dx dT/dx S = = -dV/dT (2.1) The potential difference between the two ends of the conductor can then be written as: X2 x2 T2 EMF =-jx dV/dx dx = 1x 5 dT/dx dx = j s dT (2.2) 1 1 1 1 Where T1 and T2 are the temperatures at the ends of the conductor. This expression has the advantage that it depends only on the tempera- tures at the ends of the conductor and on the variation of S with temperature. It does not depend on the details of the temperature gradient. In Order to measure the EMF across a conductor leads must be used to make a connection to a voltage measuring device. These leads have a thermopower of their own and will be subjected to temperature 3 differences which will cause them to generate their own thermoelectric voltage which will be in addition to the EMF that is to be measured. This situation is depicted in Figure l. If the conductor has thermo- power SA and the leads have thermopower SL then the EMF actually measured will be: T1 EMF = f dT (2.3) T T2 T T2 T2 SLdT+fT SAdT+fT SLdT = [T (SA-SL)dT = IT SAL 1 2 1 1 where SAL is the effective thermopower of the conductor-lead combi- nation. The EMF measured is actually the difference in the EMFs generated by the two conductors. This is referred to as the Seebeck effect. Since the EMF depends only on T1 and T2 the Seebeck effect is useful for measuring temperature differences electrically. This is the principle of operation for thermocouples. The thermopower of a conductor is commonly broken up into two components, the diffusion thermopower Sd and the phonon drag thermo- power Sg. The total thermopower is just the sum of the two: S = Sd + Sg (2.4) 3 Some theoretical work has been done suggesting that there may be an interference term between Sd and S9 but that it is small enough that the two components may be added linearly to a high degree of accuracy. EI‘L‘F T Fig. 1 Thermocouple schematic Diffusion Thermopower The diffusion thermopower is derived by considering the electrons to be interacting with phonons and randomly distributed scattering centers at local thermal equilibrium. The electrons are considered to have an energy dependent relaxation time and consequently an energy dependent electrical conductivity. The expression for 5d can be shown to be:4 2 2 Sa = 1TskeT (335w) (2'5) Ef where k is Boltzmann's constant, E the electron energy, 0 the electri- cal conductivity, e the electron charge, and Ef is the Fermi energy. This expression is valid for T>eD (Debye temperature) or at low tem- peratures when impurity scattering is dominant; in both cases Sd is proportional to T. The importance of the energy dependent electrical conductivity can be illustrated with the situation depicted in Figure 2. The electron energy distributions should look somewhat as shown in the graph for the two temperature regions. Clearly the electrons above Eo will diffuse towards the low temperature region and the electrons below EO will diffuse to the higher temperature region. If o(E) is sufficiently small for E>Eo then the net current will be towards the low temperature region and the cold end will become electrically positive, resulting in a negative thermopower. 0n the other hand, if o(E) is small for E Neck Bond Belly Bond .// Electron 0" ‘ 8 Fig. 4 The Fermi surface of silver l2 It might be expected that the number of electrons present in the neck regions would increase. Since the neck regions are sites of umklapp scattering, such a redistribution would lead to an enhanced positive thermopower. The diffusion thermopower should also be affected since it depends critically on how many electrons are experiencing what relaxation time. For a given energy electrons can have any of a variety of relaxation times depending on whether they are in the neck or the belly regions. They might be described as having a relaxation time that is a weighted average of the relaxation times in the various regions. The redistri- bution caused by a magnetic field will clearly have an effect on this weighted average and will consequently affect the diffusion thermopower. As the magnetic field increases, the effect it has should also increase. It is useful to consider the quantity wT in this context; w is the cyclotron frequency of the electron and T its relaxation time. If wT<<1 then an electron would barely start its cyclotron orbit before suffering a collision, and the effect of the field should be minimal. When wT>>1 the field should reach peak effectiveness as the electrons are able to make several circuits of their cyclotron orbits before suffering a collision, and the effect of the field would reach satu- ration. Since T decreases with increasing temperature, more intense magnetic fields should be required to reach saturation at higher temperatures. Magneto-resistance reaches a saturation value in compensated materials but does not saturate for uncompensated materials. Thermo- power is expected to differ in that it should saturate for both compensated and uncompensated materials. l3 One approach that seems useful for understanding the effect of a magnetic field is the band model. In this model the thermopower is written as the sum of the individual thermopowers from differing portions of the Fermi surface: 0 0| —' —lo —I S. (2.9) where Oi is the electrical conductivity of the ith band, Si is its thermopower, and 0T is the total electrical conductivity. In the case of silver the relevant bands would be cyclotron orbits lying entirely in the belly regions and those orbits traversing the neck regions. 1 that the It has been shown for a two band uncompensated metal band model predicts that S will reach a saturation value as wT becomes much greater than one. Impurities Impurities in a conductor can have a significant effect on the thermopower. The simplest situation is when the electronic structure of the host and impurity atoms are the same. Electrons would not be greatly scattered by such impurities. The major effect would be the scattering of phonons due to the difference in mass between the host and impurity atoms. The phonon drag thermopower should be strongly affected. If atoms with a different valence or with strong magnetic prop- erties were introduced then electron scattering could be greatly altered. The electron diffusion thermopower could be greatly influenced in this case. l4 Silver and gold have the same outer electron configuration and so have similar electrical properties. For small concentrations of gold in silver the primary effect should be on the phonon drag thermo- power. By studying the change of S in a magnetic field for various concentrations of gold in silver, it may be possible to tell how much of the change is due to phonon drag enhancement and how much to electron diffusion thermopower enhancement. CHAPTER III EXPERIMENTAL APPARATUS There are two commonly used approaches to measuring the thermo- power of a material. They are based on the equivalent relationships: 1 EMF =12 SdT s = -dV/dT (3.1) 1 1 TheSe approaches are referred to as the integral method and the differential method. In the differential method the differential form of the thermo- power relationship is used. If the temperature difference across a sample is sufficiently small the thermopower can be obtained simply from the ratio of the voltage across the sample to the temperature difference. 8y raising or lowering the temperature of the entire sample the thermopower at various temperatures can be found. The experimental apparatus for these measurements is relatively cumbersome but the data analysis is quite simple. The integral method uses the integral form of the thermopower relationship. If one end of the sample is maintained at temperature T1 and V-is recorded for various values of T2, the thermopower is given by S(T2) = -dV/dT2. One difficulty with this method is that for large temperature gradients there may be undesirable heat flow 15 l6 through the sample. This problem may be minimized by using samples in the form of long thin wires. Another difficulty with the integral method is in the data analysis. The discrete data points for V vs. T must be fitted to a curve in order to find dV/dT. The differentiation can be carried out either graphically or numerically. The primary advantage of the integral method is its experimental simplicity. For the measurements presented in this thesis a modification of 6 is used. Since the thermopower the integral method due to Chiang of silver-gold alloys is fairly well established,7 the quantity of interest is the change of thermopower in the presence of a transverse magnetic field. This can be found directly with the arrangement shown in Figure 5. Since the sample is in the form of a long wire it will necessarily be polycrystalline. The measured thermopowers will be an angular average of the thermopower tensor. Cryostat A functional diagram of the cryostat used (constructed by R. Cady) is shown in Figure 6. In order to minimize heat conduction to the sample, the can containing it is evacuated typically to lO'z microns. For the same reason the base supporting the two rods is made of brass. The two rods which support the sample are made of copper in order to minimize any temperature gradients along them. One of the hollow copper support rods is open to the surrounding liquid helium bath, which serves as the constant temperature region for the integral method of measuring the thermopower. The other hollow support rod serves as the variable temperature region. The top of the variable rod is connected to a small can that can be filled 17 EMF T1 zero field T1 £-——-- sample wire T T2 1 central field ¢- solenoid T2 T1 T2 EMF = j S(H,T)dT + f S(O,T)dT = f {S(H,T) - S(H,O)}dT T T T l 2 l _ dEMF = _ Fig. 5 Method for directly measuring the change in thermOpower pump line for He can pump line for cryostat ‘ 1 liquid He can ————L top heater \L /l/ sample wire \ \ / synthane l resistance l- spoo s “—~—1_____rr gradient thermocouple (along rod) bottom \\ ;/,/’ thermometer T control ”—5 l thermocouple P ‘\ vacuum can / heater 7%: variable T rod (hollow) 14 _ _ .._——» .._——- ‘_._:- J __\\*L fixed T rod (hollow) Fig. 6 Schematic of cryostat sample mount 19 liquid helium level L | .1J_1 I L iii or t t o 4 X' y S a AL ' VT ' [: 1111 solenoid .R\g IUJ U, L_ _ _.__ __ ___..__._...L.ll r-— l I l l I l magnetic field Fig. 7 Cryostat-solenoid geometery 20 with liquid helium from the surrounding bath. The temperature of this support rod can then be lowered below that of the surrounding helium bath by maintaining the helium in the can at reduced pressure. This cooling capability was not used for the purposes of this thesis; instead the temperature of the variable rod was raised to various levels between 4.2°K and 50°K. This was accomplished by the use of two electrical heating wires made of Evenohm wire which are positioned at the top and bottom of the rod as shown in Figure 6. The two heaters are used independently. The only temperature that need be measured is that of the variable temperature rod. A difficulty is encountered here in that most types of temperature measuring devices are strongly affected by the presence of a magnetic field. To avoid this problem a carbon-glass resistance thermometer produced by Lake Shore Cryotronics, Inc. was used. It is expected that temperature measurement errors never exceeded one per- cent for the range of magnetic fields used. A diagram of the resistor circuit and for other circuitry is shown in Figure 8. In order to simplify data taking the temperature is controlled electronically. A circuit controls current to the top heater so as to keep the voltage from a thermocouple, mounted between the two supports, constant. This constant is determined by an adjustable voltage reference. By using this control scheme it is very easy to adjust the temperature in small increments simply by changing the value of the reference voltage. The fact that the thermopower of the thermocouple is affected by the presence of a magnetic field is of no importance in this application. A second thermocouple mounted along the length of the variable temperature rod detects any thermal gradient that may exist along 21 constant current supply resistance thermometer 6 voltmeter 10 ohm voltmeter for measuring current current F“ controller heater null detector gradient thermocouple reference voltage Fig. 8 Electronics 22 this rod. The gradient is reduced to negligible proportions by adjusting the current to the bottom heater. By using this method the temperature was kept uniform to within a third of a degree. If care is not taken in the physical placement of the wiring, temperature gradients may be created in the regions that should be isothermal. In order to minimize this problem each wire is wrapped around its support several times before passing to another temperature region. This procedure insures that the anchor points of the wire are at the temperature of their support. All wiring is held in place by a coat of varnish. The two ends of the sample wire are placed next to each other on the constant temperature rod. A twisted pair of copper leads is then connected to the sample with a solder joint and these leads are brought outside of the cryostat to be connected to a voltmeter. Since the two copper wires experience the same thermal environment, they should con- tribute no net voltage to that present across the sample. Consequently the voltage measured across the copper leads outside the cryostat will be essentially that generated by the sample. As with the sample leads, all leads leaving the cryostat are twisted together in pairs to minimize electrical noise pick-up. The length of the support rods is such that the sample passing between the two rods at the top is outside the solenoid. When it again crosses at the bottom it is in the central region of the solenoid. In order that this arrangement yield the desired information it is important that the sample be isothermal as it passes into the region of the solenoid. If it is not isothermal then unwanted thermoelectric voltages will be developed. In addition to the previously mentioned 23 precautions against this used for all wiring, one other thing is done. Heat conduction through the sample is minimized by making the length of the sample joining the two temperature regions as great as is practical. This was done by wrapping the sample several times around a thermally insulating synthane spool before attaching it thermally to the variable temperature rod. The magnet used was an Oxford Instrument Company superconducting solenoid. A calibration table of magnet current vs. field strength was available, so the field strength did not have to be directly measured. To set the desired field all that had to be done was to adjust the solenoid current to the appropriate value. The current supply for the solenoid contains a circuit which automatically adjusts the current to any preset value within its operating range. The current was monitored by measuring the voltage drop across a .OOl ohm resistor that was in series with the solenoid. Once the current had been set the solenoid was shorted out with a length of superconducting wire. The current then remained constant until the short was removed and the power supply reconnected. One difficulty exists with this experimental arrangement. Though the top section of the sample is outside of the solenoid it still experiences the fringe field which exists around the magnet. According 8 the fringe field at the top position of the sample is about to Chiang l/lOO of the central field. Since the largest field used is 50 k6 the largest fringe field will be about .5 k6. To check on the importance of fringe field effects, measurements were made for small central fields. It was found that AS at .5 k6 was negligibly small in comparison to AS for fields of 10 k6 and higher. 24 All voltage measurements were made with digital voltmeters. The thermoelectric voltage from the sample was measured using a Keithley Model 180 digital nanovoltmeter with a resolution of ten nanovolts. CHAPTER IV SAMPLE PREPARATION The samples were in the form of polycrystalline wires. They were prepared from Cominco 6N silver and Cominco 5N gold. The silver came in the form of small pellets. To remove surface contaminants, they were first etched in a l:l solution of ammonium-hydroxide: hydrogen-peroxide and then in nitric acid. The gold also came in the form of small pellets but was etched in aqua regia. As the first step in creating the required very dilute alloys a one atomic percent gold in silver alloy was made. This was diluted with appro- priate amounts of silver to produce the desired concentrations. The gold and silver pellets were placed together in a graphite crucible and heated past their melting points to about l400°K using an induction furnace. The resulting liquid was then agitated by rocking the crucible in order to create a homogeneous mix. The heated alloy was poured from the graphite crucible into a copper mold where it was allowed to cool. This was repeated a second time in order to further insure a homo- geneous mixing of the gold and silver. After cooling, the rod was passed through a rolling mill to decrease its diameter and then through a series of successively smaller tungsten-oxide dies, and eventually diamond dies, until a wire with the diameter of .Ol" had been formed. The annealing of the samples will be discussed with the presentation of the data. The concentrations attempted were .l, .07, .04, and .03 25 26 atomic percent gold in silver. When samples of these alloys were sent to Schwarzkopf Analytical they were reported as being .047, .048, .013, and .011 atomic percent gold in silver respectively. CHAPTER V DATA ANALYSIS The data consist of thermometer resistances and sample EMFs. The first step in the data analysis is to convert the measured carbon- glass resistances to their corresponding temperatures. A table of calibrated values from about 4°K to 85°K was available. It had been made by calibrating the carbon-glass resistor against a germanium resistance thermometer whose calibration was good to :l percent. In order to estimate temperatures at points intermediate to the calibration points a curve was fitted to the R vs. T calibration points. From Figure 9 it is apparent that the £nR vs. EMT plot shows much less curvature than the direct R vs. T plot. A polynomial curve can be fitted to the in plot much easier than the direct plot. The EnT, KnR calibration points were divided into four roughly equal temperature ranges and within each range the points were fit to a sixth degree pblynomial using a computer program. Using these curves another computer program was used to generate a table of R vs. T in 5°mK increments from 4.190°K to 84.945°K. This table was then used to convert the carbon-glass resistance data to temperatures. The calibration data for the carbon-glass thermometer are included in an appendix along with the polynomial coefficients. In order to calculate AS the EMF vs. T curve had to be differ- entiated. The EMF vs. T curve was found by breaking up the data into 27 R ohms 1000 500 28 40 80 Fig. 9 Thermometer resistance 29 at most three temperature ranges and fitting the points within each range to a fourth-degree polynomial using a computer program. Because of slight inhomogeneities in the sample wire and leads, the EMF is not strictly zero when no magnetic field is present. To account for this effect in the data, a zero field set of points for EMF vs. T was taken, and the resultant EMF vs. T curve was subtracted from the curve for non-zero fields. The output from the computer program consisted of calculated EMFs, S, and the coefficients of the polynomials used to fit the EMF vs. T data. The curve fitting was done by the least square method. Plots of some EMF data points along with their polynomial fit are shown in Figure l0. 30 a .m> mam 0H .mme om 0: om um oH 09. on :< & .pm HHo. + m< Na 0H om (ENE siIOAOJotm CHAPTER VI DISCUSSION OF RESULTS One thing that became apparent was that the peak value of AS for the first samples done seemed much too small in comparison with the results of pure silver and silver +.37 at.% gold found by Chiang. The residual resistance ratios were measured and found to be incon- 7 This information is shown. sistent with those of Crisp and Rungis. in Table T. It was thought that perhaps some unwanted impurity might have contaminated the alloys during the mixing process. The only things the alloys came in contact with while they were in the liquid state and most vulnerable to absorbing impurities were the copper mold and graphite crucible. The copper mold seemed the most likely candidate as a possible source of impurity. This was tested by measuring the resistance ratio of pure silver that had been melted in the graphite crucible and comparing it to the resistance ratio of a sample of pure silver that had been poured into the copper mold. The resistance ratios of both samples were essentially the same. With these results it seems unlikely that the copper mold was a source of impurities. Another possibility was that something in the annealing pro- cedure was responsible. It is known that very small amounts of iron can cause large effects in the thermopower of a material.10 One com- mon way of removing these iron impurities is by annealing the alloys 31 32 Table l. Data on Ag-Au Alloys Atomic % Au RRR ASmax(uV/°K) Anneal .011 49.4 .625 argon .Ol3 l02 l.l oxygen .047 45.6 .77 vacuum .048 50.7 .83 argon .000 540 (copper mold) vacuum .000 542 (graphite mold) vacuum .013 87.6 vacuum From Crisp and Rungis .000 700 oxygen .09 50 oxygen .69 7 oyxgen From C. K. Chiang .000 500 2.25 air .37 11.5 .1 ? _ . . _R 273°K RRR-Res1stance Rat1o R 4.2 K 33 in oxygen.7 The previous samples had all been annealed in either vacuum or argon. To test this possibility a .013 at.% alloy was prepared; one sample of the alloy was annealed in vacuum and another in oxygen. The resistance ratio for the sample annealed in oxygen had the higher value. As can be seen from Figure 11 the peak value of AS for the .013 at.% sample annealed in oxygen was significantly greater than that of the .011 at.% sample which was annealed in vacuum. It seems likely then that small amounts of iron impurity were respon- sible for the small AS values. It is common to try interpreting the behavior of AS in terms of wT. w==%%-is the cyclotron frequency of a free electron and T is the relaxation time from the simple conductivity relation 0 = nezt/m. Curves for materials other than those described in this thesis tend to show that the peak in AS moves to higher temperatures as the field strength is increased. 0f the alloys studied in this thesis, only the .011 at.% alloy showed this trend. The trend is slight enough though that its validity may be questioned. If the peak were to occur at some particular value of wT, then it is evident from the graph of R vs. T in Figure 12 that the temperature at which the peak occurs would be expected to move much more than is observed. Another effect which argues against the peak being simply related to wT is the fact that as the concentration of gold in silver increases the peak in AS moves to a higher temperature. This is shown in Figure 13. If the wT dependence were correct then the peak should move to lower temperatures where relaxation times have been shortened by the introduction of impurities. 34 Hamccm :mwhxo Mo pomemm 0: om ON HH .wflm oa ox om _ _ _ Humans Comma o< s .eo “Ho. Hmmccm Cmmhxo :< R .pm mfio. SV /siIOonotm 1o ‘ J nano-ohm cm 35 Ag + .011 at. w Au 20- 10 ” O l l I l 0 10 20 30 40 Fig. 12 Resistivity vs. T microvolts/OK AS 36 Fig. 13 40 °K Variation of A8 with concentration (50 k6) 37 The one aspect in which the wT viewpoint seems appropriate is in its implications for the AS vs. field strength characteristics of the samples for various temperatures. If the thermopower reaches a saturation value, it would seem reasonable for it to be related to NT since this gives some indication of how noticeable the field is to the sample. For very short relaxation times an electron will barely begin to traverse its cyclotron orbit before it is scattered; the magnetic field is not likely to have a very great effect in this situation. At low temperatures only a small field is necessary in order to cause electrons to traverse their cyclotron orbits many times before being scattered. If AS has a saturation value it will likely reach it for fairly small fields at low temperatures. At higher temperatures correspondingly higher fields would be needed to push AS to its satu- ration value. The graphs in Figure 14 seem to suggest this sort of behavior. At the lower temperatures AS seems to be saturating. At the higher temperatures it also seems to be saturating but not to as great an extent. The peak in AS and the phonon drag peak in S(H=0) (Figure 15) both occur in the same temperature region. This suggests that the AS peak may be associated with the phonon drag thermopower. As the concen- tration of gold is increased, the peak in S moves to lower temperatures while the peak in AS moves to higher temperatures. Because of this behavior it seems unlikely that AS is due simply to enhancement of the phonon drag peak. As shown in Figure 13, the peak in AS is greatly reduced by the addition of small amounts of gold. Since the principal effect of the gold should be to scatter phonons, this evidence suggests that the peak in AS may be largely due to phonon drag thermopower. microvolts/OK 38 .2,_ .0 l l 1 Fig. 14 AS vs. H Ag + .011 at. % Au 140' 50 k6 39 Silver-rich Alloys Fig. 15 Zero field thermopower? 4O 3 1 The relations AS = AT + BT and AS = AT + BT' can be used to see how much of the AS is due to phonon drag and how much to diffusion thermopower. By looking at graphs of AS/T at low temperatures and TAS at higher temperatures, the variation of the coefficients A and B may be seen. These graphs are shown in Figures 16 and 17. It is apparent from Figure 17 that A and 8 change proportionately, so if they truly represent diffusion thermopower and phonon drag thermopower then they are both affected equally by the magnetic field. AS is due equally to both diffusion theropower and phonon drag thermopower. In order to be more confident of this conclusion, the experimental curve should be extended well beyond where the phonon drag peak occurs, i.e. up to at least 100°K. microvolts/OK2 AS/T 41 .04 )- 50 k1} Ag + .047 at. % Au ‘02 — 20 k0 /’ \ l I l 50 id} 001+ '— A -+ .(fl48 . a; r' g at / Au .02 " 20 k6 Fig. 16 AS/T vs. T microvolts TAS 42 18‘_' 10 - Ag + .014 at. % AU 14-— A. B 19 Ag + .011 at. % Au 50 RI; 200 Fig. 17 TAS VS. 2000 3< s .eo FPO. + m4 co m e? omeoeu mp .meu 43 vlo as on om 0H 4 a _ _ 0% OH 0% ON 1 nua on ox o: l N. UM on W i I. 3. w. o I O A O T. I .+ S / 0 VA II. 0. 44 2 s so ms. + 2 to m E 8:25 0M 0H 3 om 3 on ox o: ox on O H SV xo/siIOonotm 45 rflllrllo. :< s .oo gee. + me to m as omeoeu om .swa mum on :f \0 8V )1 o/sitozxoatotm :< & .pm wco. + m< mo m cw mmcmcu Fm .mwn 46 vm0 0.3 on ON OH _ _ _ _ ox om ox o: I: N. l 0. I. m. SV )[o/sitoxxoiotm APPENDICES APPENDIX A Table 2. Calibration Data for Resistance Thermometer* Temperature Resistance Temperature Resistance °K Q °K O 4.191 937.2 14.355 95.80 4.192 936.4 15.319 88.755 4.307 894.1 15.810 85.26 4.309 893.8 17.057 78.57 4.322 888.2 18.202 73.305 4.378 862.65 18.618 71.375 4.383 861.9 19.410 68.43 4.557 783.7 19.890 66.78 4.738 701.4 20.630 64.405 4.863 666.8 21.223 62.66 5.183 560.1 21.710 61.315 5.779 437.25 22.600 58.965 6.098 388.8 24.913 54.065 6.230 371.8 27.093 50.325 6.405 351.1 29.641- 46.78 6.800 310.3 31.292 44.845 7.089 286.0 33.224 42.85 7.266 273.05 34.364 41.79 7.563 253.8 36.083 40.32 7.942 231.5 37.799 39.05 8.310 213.75 40.648 37.14 8.743 196.05 43.607 35.43 9.249 178.7 48.326 33.225 10.068 155.95 52.390 31.635 10.731 141.95 57.097 30.06 11.404 129.9 62.460 28.6095 11.755 124.6 70.022 26.9275 12.202 118.35 77.425 25.5918 12.922 109.75 77.927 24.532 13.267 105.95 84.949 24.4695 13.760 101.1 *Carbon Glass Resistor #842, Lake Shore Cryotronics, Inc. 47 Table 3. 3:- II II U 11 II C ll 11 U 11 ll .1889066E+O4 .2064272E+O4 .6727089E+02 .3258694E+02 .1424817E+02 .5773279E+OO .1377278E+02 .2946827E+OO ZMR = A + anT + C(tnT )2 48 + o(2nT)3 + E(2n1 4.191°K to 5.183°K B E .4921020E+O4 -.3339054E+03 4.863°K to 8.310°K B E -.1216509E+O3 .4280011E+O1 7.942°K to 21.223°K B E -.833347OE+01 .4773964E-01 20.630°K to 84.949°K B E -.7123667E+01 .1702847E-O1 'T'IO + F(£nT Coefficients used for Polynomial fit to Resistance Data )5 -.4783237E+O4 .OOOOOOOE+OO .9345597E+02 .OOOOOOOE+OO .2962758E+01 .OOOOOOOE+OO .2040751E+01 .OOOOOOOE+OO APPENDIX B COMPUTER PROGRAM USED TO FIT EMF DATA TO A POLYNOMIAL (PUCPAH MIKI(INPUT HIHFNSION 1(100 l INlI'SER HEAnmc. C C000 DATA DECK “HST C HASECS)QFOIF(S) UITH TVO TRIPLE 7190 CARDS coo .Meis.1.rnF1 o "99 F.nasr1 G-NPlsolotflr.nnsr.HAsr1 —. ‘- U - - ‘- .fi '9‘. £5.” 470—. U A—‘o-fi OWAH ofidu ~c~mc CHM ‘OAP :2 = D We :flwfll Mo 10 ~ - >bfi> >b‘) ~P de dfloa nan . f-PAI- fl dPPZF FPEF ‘fiflmfi andn ;d401 «do - v a- h a no ‘ w «99 ‘W 24 3‘ 49 nnnndnnnnnfi ‘F“‘ “Afi- .3 Jxfixa \ONJfi-CDWJ Q0001) unupomim‘ mush: 11H 1‘] “EMS 1"" I (11114 '1' (K “-11? Llll'l-‘M I‘F‘CR NUHiH’. OF FAT DINA LITEHM N’fCR H'JI’MP OF I‘AT llAlA Tllr F «nwvnfin . :ZFWCDCDCDCD C3'1F1Fl «xxx: «)3 CIZI‘K‘t-413\3.31323 23- Daul>~fifiafi -—« - ‘4 1’? um (10 11c, l" A 111 T T P T A l‘ 50 111 (11an) NC .NPTSQIQFHF) SILEHFCI 00) UPI-- I Oll OF DATA ()1'119 01-] 01' DATA 1 I 0 N15 ROUPS. FloCDLl(5).COL2(DIQCOL3(51qCDLAISI 51 1055 A LEAST SQUARFS FIT TO THT TQYHF DATA T0 FIND [HF r POLYNOMIAL IN T. 1 THIS SURROUTIV FlH'RTl' DEGRE- 1 T L N “P CCCC ”1(1141131oocl-11 L2¢11oicuioo1 Lsiiioitaioo(1611 '4ciio11u1ocilb2i é NPT COL (I)‘T(J)OO(I*3) CDLZ(T)‘FMF(J)OT(J)°O(T-l) 0 I QNPTS 0. =1 1) lo I) D L5 U N n I )L5 0 I C ONT [NH N A U CONT on 0 70 F0" r 60 5 '1 80 100 90 ,”’, ’!:€Defi T141T14l FFFtLEF. 000000 0 0 0 0 9 . :fiUSRhil LLLLLL 000000 CCCCCC voovvvo 000.20. 0 Q 9 0 O Q 2fi¢2n(?? LLIFFLL 000000 CCCCCC , O Q O O 0 121111 LLLLLL 0000 00 CCCCCC ((“(l\ TTilITT. NNMKNNN PfiunhNP PFPPR" Ylvlvlvlvlvl DnKUDPKu LLLLLL LILLLILL AAATAAA CCLCCC HF TURN 'VD 0L5(4) 52 (OHTINU' 1n II”), 12345 x0‘xx Q Q 0 9 0 555.50 LLLLL 00000 CCCCC CCCCC 9 9 9 9 Q .4033 LLLLL 00000 CCCCC CCCCC O 9 Q 9 9 3322? LLLILL 000.00 CCCCC CCCCC O 9 O Q 0 9.1111. [ILLILL 00000 CCCCC ((‘(( RRRRR 000m 0 NNNNN Ill]! "HRH" LLLLL LLLLL AAA AA CCCCC COL](l)0X1-C0L2(1)‘XZ‘COLSII’OXB'COLQTID0X4¢C0L5(1)0X5 RETURN 1N0 DET 53 SUWROUTINF MINOR FIHTWSInN LCLICA FCAL IoJoKvl. VH9” ’0’) no.“ lutl.l\n| 1250 lllL 0000 C(Cr... Z 2 _. .- "UH-Ln: 553$ \l.\l ‘1‘, 353‘. 0“! Z 2 Z .- CGKO (r150. ’0’) Ola/flaw! (((( 12.5.0 LLLL 0000 CCCC : .. ..l. 2 BF \HI. 03310. ,)’) 111.- (((( 12‘s0 LIL-LL 0000 cccr. : : : .— 0‘1" RT TURN rNI) ~mfiinhfi I? ("1:1111151'1’111 VOLT FE “‘ICROVOL ct "FrrALCUl ATV,“ 1Ml CC1‘1F=CALL"|LMID F"F HTPPFCTFH n1 lAlr"thlI\NCr OF TFP '2'11 10 MAC-1‘ 11 211 SUN 'UTI'” ”1'1 1‘“[1QIHH 1(‘0 54 1’UT1'AIIIHCOLPTSQTQFHFQC01F91’AIF’ 0791-01110‘1’0C1‘FF10791’ACF("I 0‘ 1 - IIJTIM lll ADIN Tl'lli‘rTl’lerATUltlF (DICIIFFS 1%. ”F CROVOLTS F‘I'R DEGRET KFLVIN) 1R1r11 unnotulnnlneiiiol:lofll PR1” ‘1 " CH1! =0 1‘ 11011 I‘Tlolli’TS (”PF 1‘. D 012” \':108 0115:!{Hrocofth-Ttllochl-Il THU! CHIGDOUMFHI-CFHFH'? 0.? g': Or !'I:FCEMFH".OFF(J)-DASI(.JH-ltllntd- I) (3 ("h-c .Zlfl-n .‘ 3 Z Ub—ifiw‘ v-n—II.‘-4—f- tad-d". 3‘...) .— h. 'F’z'lLTATl'IN1 1J-II'1COFFCJI-RASF1J) I0T111001J-7) vT1IIoE1‘lF1TIQCEHF oCCEHFoDLTATEP (CHISQI'IPTSI T run Off-‘5', '1' ‘9‘1’ 0 3 .0chch 3.. ”<4 Q q filo-o VéXLOo "’7-411 II‘T'FII O D ‘1 CCFMF0(COFFIJ1-RASI(JIIOTFMPOOCJ- 1) 1‘6.) .- -0...- C. ‘r" _a 0 J19 .1 [A rllPJTIII U PR 1111 "211oTFHI'oCCFIIFoULTATFI‘ 11111“TF"'1“I.O - (1111111111 U - J 1“. orb-«nug... ("O-twat: ’ 11 “TV: DanTAT'P0(J-1ItirfliF1d)-Rh$11d)IOTFMPOtIJ-?I 1'0RM’41(010..' ‘10) 10RVFT(°"090 T‘”“ FNF CFHF .CCFMF DLTATFFO) FORMAT" ‘oFfio‘o5F7.?oFP.3) 10'”“T1°09010X9°SQPI1FHI SQUARIDINPTSI=¢QEI?o 6) F0R"AT1‘3901OVQ‘A0=‘0F17.60‘A12'QEIZ‘oGQ‘A23 ‘0rl?o GQ‘A 2'0E12o69‘A0 1:90'12o51 1”PMAT('I‘Q' T'HP CCFYQF 0LTATFPO) FHRVATCO OoF‘.3o17o?qF8. 3) V'lUH“ "no REFERENCES \1 REFERENCES A. D. Caplin, C. K. Chiang, J. Tracy, P. A. Schroeder, Phys. Stat. Sol. (a) 2g, 497 (1974). F. J. Blatt, C. K. Chiang, L. Smrcka, Phys. Stat. Sol. (a) 24, 621 (1974). F. Napoli, 0. Sherrington, J. Phys. 5, L53 (1971). R. D. Barnard, "Thermoelectricity in Metals and Alloys", Taylor & Francis, Ltd., London (1972). F. J. Koch and R. F. Doezema, Phys. Rev. Lett., 25, 507 (1970). . K. Chiang, Rev. Sci. Inst. 4§, 37 (1974). . S. Crisp and J. Rungis, Philos. Mag. 22, 217 (1970). . K. Chiang, unpublished. J. Blatt, P. A. Schroeder, C. L. Foiles, D. Greig, "Thermo- C R C F electric Power of Metals", Plenum Press, New York (1976). 55