PHONC'N DRAG THERMOWWER IN CADMSUM, ZWC AND MAGNESESM Thesis for the Degree of Ph. D. MICHIGAN S‘fATE UNNERSiTY Vi CENT ALAN ROWE 195? THESlS This is to certify that the thesis entitled l’liUT-{LJN D'L’xG Slitlifl‘UBU "1’ i n CADMILM, ZINC and MAGNESIUM presented by Vincent Alan Rowe has been accepted towards fulfillment of the requirements for Ph . l) . degree mm M Major professor 0—169 ABSTRACT PHONON DRAG THERMOPOWER IN CADMIUM, ZINC AND MAGNESIUM by Vincent Alan Rowe AnisotrOpic thermopowers and thermal conductivities of single crystals of the three Group II metals Cd, Zn, and Mg have been measured from 1.250K to room temperature. An attempt has been made to correlate the phonon drag thermo- power in these metals with the detailed tOpologies of their Fermi surfaces. PHONON DRAG THERMOPOWER IN CADMIUM, ZINC AND MAGNESIUM by ‘Vincent Alan Rowe A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1967 @qghog '3..2l—é>3 Acknowledgments The assistance and encouragement of Professor P. A. Schroeder throughout all phases of this investigation is gratefully acknowledged. I am indebted to Professor F. J. Blatt for many illuminating discussions. The work was supported by the National Science Foundation. 11 TABLE OF CONTENTS Introduction Theory Thermolectric Power - Thermodynamics The Diffusion Term Phonon Drag The Bailyn Equation A Working Hypothesis Thermal Conductivity Sample Preparation A. B. C. D. Generalities The Growth of Cadmium Crystals Orientation and Shaping Sample Purity Apparatus and Experimental Technique A. B. C. D. Objectives The Cryostat Electronics Method of Measurement Results and Discussion A. B. Data Reduction Results 111 6. C. D. E. F. G. TABLE OF CONTENTS, Continued Accuracy Fermi Surfaces Phonons An Order of Magnitude Calculation The Working Hypothesis Comparison With Experimental Results Conclusions iv List of Figures Figure Page Description 1-1 2b ThermOpower of polycrystalline Cd 2-1 5 Thermoelectric Circuit 2-2 7 Angular Dependence of S 2-3 18 Some Scattering Events 2-4 20 Phonon Distribution 3-1 28 Spark Tool 3-2 29 Sample Schematic 4-1 34 The Sample Helder 4-2 37 Vacuum System 4-) 45 Block Diagram of Electronics 4-4 50 Level Indicator Circuit 5-1 57 Thermopower of Cd, o-3oo°K 5-2 58 Thermopower of Zn, O-BOOOK 5-3 60 Thermopower of Mg, 0-3000K 5-4 61 ThermOpower of Polycrystalline Mg 5-5 62 ThermOpower of Mg, 0 - 150K 5-6 64 Thermoconductivity of Cd 5-7 66 Thermoconductivity of Zn 5-8 67 Thermoconductivity of Mg 5-9 70 O.P.W. Model for Fermi surface of Mg 5-10 73 Scattering across Needles 5-11 81 Parallel Components, 0 - 200K 5-12 82 Perpendicular Components, 0 - 20°K 4-471 102 Heater Control Circuit 4-4-2 104 Power Supply Table 5-1 5—2 u-2-1 u-u-1 Page 30 76 77 94 103 List of vi Tables Description Physical Details of the Samples Some Parameters of Cd, Zn, Mg Fermi Surface Calipers and Equivalent Temperatures R(T) for Manganin Circuit Parameters for Heater Control Appendix 4-1 4-2 4-4 Appendices Page Description 92 Magnesium Solder 93 Heaters 95 Thermocouples 101 Heater Controls vii 1. Introduction In 1963 Jan and Pearson(1) published the results of their investigation of the thermOpower of intermetallic AuSn. Finding considerable anisotropy in the data taken on crystals of differing orientations, they attempted to deduce a topology for the Fermi surface of AuSn consistent with their results. We felt, however, with the present limited understand- ing of phonon drag thermOpower that it would be more logical to work with metals with known Fermi surfaces and try to correlate detailed features of these surfaces with their observed phonon drag thermOpowers. This project was there- fore commenced with a view to testing the hypothesis that various sheets of the Fermi surface would make discernible contributions to the phonon drag thermopower. We esta- blished four criteria for the choice of metals on which to take these phonon drag thermopower measurements: (1) the Fermi surfaces should be well established, (2) more inform- ation would accrue if the materials studied were not cubic --thus resulting in an anisotropic thermOpower, (3) oriented single crystals of fairly high purity are required if advantage of number (2) is to be taken and if the results are to clearly reflect the thermOpower of the pure metals, (4) metals sufficiently alike are needed to facilitate com-. parison of the results. The three group II metals, Cadmium, Zinc, and Magnesium satisfy these criteria. Their Fermi surfaces are quite similar and well known. All have hexagonal symmetry and thus will have 2 independent thermoelectric coefficients. With the possible exception of Mg, all are fairly easy to obtain as oriented single crystals. Measurements on poly- crystalline Cd wire had already been carried out in our laboratory. Figure 1-1 shows the interesting temperature dependence diSplayed by Cd in these results. At the outset, we had hOped to be able to correlate the phonon drag thermopower with Fermi surface geometry, expecting the diffusion term to be a simple monotonically increasing function of temperature whose sign is determined at room temperature. As it turns out, one cannot be certain of the behavior of a diffusion thermopower arising from many sheeted Fermi surfaces such as those of the present metals. Very few attempts to directly correlate transport prOperties with Fermi surface topology have been made to date. We mention work by Klemens(2) on the thermOpower and resistivity of tin at low temperatures, Bailyn and Dugdale(3) on c0pper, and Ziman on noble meta1s(u). We should point out that Gruneisen and Goens(5) measured the thermopowers of Cd and Zn from 200K to room temperature in 1926. Hewever, their data were taken with c0pper as the reference metal which has been shown to have an anomalous thermOpower at low temperatures. 2b 2s 2 2.4“ _ ca 2.2 s 2 I. I. I! (IA/PK) I.‘ I. .a' £1 4‘ .2‘ 724’ T I I T ' 40 30 I20 It?) 200 240 zeos T PK) Fig. 1-1 IO We have also measured the thermal conductivity of each sample. For various reasons, however, this data was not found useful for the present purposes. 2. Theory A. Thermoelectric Power - Thermodynamics The theory of irreversible thermodynamics leads to the following equation:(6) _ 1 . - V+ — - "e- [a VT + Vu] 2‘1 where d is the sealer electric potential, e the electronic charge, E the second rank thermoelectric tensor, T the abso- lute temperature and u the chemical potential; All the materials with which we are concerned being of hexagonal symmetry, the tensor takesthe forml 7 ) "M u (3 to C) L 2-11 where u refers to the hexad axis, and.i.refers to the basal plane. This implies that the various thermoelectric preper- ties will be isotrOpic only in the basal plane. Because of the basal plane isotropy, the problem may be reduced to 2 dimensions:/ \ r ’ \ ’ I f I x 1 . I I= .II II .T I + I... >> at 0 2 63.— a? IBETiI \ ’ \ I K J \ / 5 Then, in the basal plane; for example: 89 _ 1 3T l Bu _ .x I E 215x + '5 5x 2 IV To see the significance of equation 2-IV, consider the schematic circuit shown in Figure 2-1: T T+; if}; J[ - lfpzpb dT To T+6T To To 0 0 Similarly, the second term on the right: T+6T 6T f 21 dT = [21. dT T 0 Equation 2-V reduces, therefore, to: ¢+5¢ 6T [M = [-é-(zl- sz)dT ch 0 or for small 6T _ 1 mb _ -é- - 2%) 6T or: 6 _ 1 6T " 5(3I- 2%) Z or setting _i 1 e 6 Si .. 75% + st 2-VI Thus, Zlis seen to be the usual absolute thermopower divided by the electric charge. The case for 2” follows along the lines of the above treatment. Angular dependence of S: These considerationst not allow for the case where the sample axis lies not along a symmetry direction but in some arbitrary direction relative to the principle axes. 2-2 shows such an example. X Xi ? <3) 0 Y Figure 2-2 Figure The coordinate transformation that expresses x, y in terms of xi, x“ is: x cose -sin9 xL y sine cosG \xm Hence the thermoelectric tensor will have the following form in the (x,y) coordinate system: / \ \ / cosO -sin9 31 O 0089 sine é' (Kay) ll V A sine cose O 2” -sin9 cosO / I \ 2L c0329 + Z" sinze (2l - Z“) cosO sine ( (zl - zll ) cose sine 2L sin29 + z" c0329 \ In this instance 2-I becomes: ”e :— as X X < = $- .Z'(x,y)< + a > 2-VII gi 8T Bu Ky \337 \337) If we assume that with the heat current flowing along x, 92- 0 Then: 5y a __ 1 2 2 5T 1 Bu 2-VIII 3% — -e-{z-LCOS 9 + ZHSin 9} 63(- + E- 63(- 9 From 2-VIII we then may deduce the thermoelectric power, bearing in mind the more specific results obtained in 2-VI. 3(9) = SLCOSZG + SnsinEG 2-IX Hewever, it is unrealistic to presume that in an anisotrOpic material thermal gradients will not persist in directions not parallel to the heat flow. Such gradients do in fact exist whenever the thermal conductivity is anisotropic. In our case the thermal conductivity tensor has the same form as the thermoelectric tensor and so is aniso- trOpic outside the basal plane. Kohler( 9) shows that in this instance the thermOpower as a function of angle as defined in Figure 2-2 is: 29 +.;£ 1 cos 8 n H 2 2 2 ~90329 sin 9 8(9) = Slcos 9 + Snsin e + 29 2-X A1 and A“ are the two independent elements of the thermal conductivity tensor. In general the additive term is but a small correction. This is not to say that equation 2-x has not had ample experimental verification(lo) The expression given in 2-IX is often referred to as the isothermal thermopower, whereas that in 2-X is called the adiabatic thermopower. 10 B. The Diffusion Term In non-magnetic materials, the thermOpower may be Split(ll) into two terms, a diffusion term Sd and a phonon drag term 88: =tota1 = 3d +’§g 2'XI Briefly, the diffusion part arises from the dynamic balance established between the two driving forces in the Boltzmann equation, the temperature gradient VT and the electric field E, The rate of change of this electric field with respect to temperature is proportional to the diffusion thermopower. An expression originally derived by Mott(12) is pre- sented here(13) in a form applicable to anisotropic metals. The derivation depends on the Boltzmann equation in the relaxation time approximation. 4 _ _ w2k2T Baz (E) (8d)13 - W pi£(+)Er (Sd)1J, pifi’ Old indicate tensor components of the diffusion thermOpower, the electrical resistivity, and the electrical conductivity. k is Boltzmann's constant, T the absolute temperature and e the electronic charge. E is the energy and Ef the fermi energy. Note that the repeated index 1 is summed over the applicable range. For cubic metals the tensors degenerate into sealers. 11 For a free electron model this equation then predicts a negative thermOpower with a linear dependence on the tempera- ture. This behavior is rarely observed because of the com- plexity of the bracketed term even in the simple monovalent alkali and noble metals. For an anisotrOpic metal with a Fermi surface of many sheets the evaluation of the bracketed term becomes prohibitively difficult. We, therefore, do not intend to consider this term in detail. In undertaking this work, however, we assumed in analogy with the group 1 metals that the thermOpower would be a monotonically increasing or decreasing function of T, the sign of which could be deter- mined at high temperatures where the phonon drag contribu- tion is expected to become small. Hewever, recently Colquitt(14) has performed theory calculations which suggest that it may be possible to obtain maxima and minima in the low temperature thermOpower of ametal which has a Fermi surface of many sheets. C. Phonon Drag The second term in the right hand side of equation 2-XI represents a contribution to the thermopower that results from interactions between non-equilibrium phonon distribu- tion and conduction electrons. Phonon-electron collisions result' in net changes in the velocity of the electrons involved, thus altering the electric field set up by the diffusion process. We believe that effects of this sort play _<_9 the dominant role at low temperatures ( .Ig, say) giving rise 12 to complicated temperature dependence. Phonon drag as a possible mechanism was first suggested by Gurevich(15) in 1945. MacDonald(16) has considered the problem in a simple way and arrives at the following expressions: 2-XIII where CD is the lattice specific heat at constant pressure, N the number of changes/inner volume, q the change of the carriers, 1/T2 the'rate of momentum transfer to the free carriers", and 1/‘r1 the "rate of transfer for all other mechanisms (phonon-phonon, phonon-impurity, etc.)". MacDonald analyzes this equation into two regions, above OD and below 9D. His result is: k g anl $., T > e 2-XIV 3 200k T , Here k is the Boltzmann constant, n the number of free changes/atom, A the lattice thermal conductivity, and 9D the Debye temperature. These results depend on the free electron model, a relaxation time Boltzmann equation. Using a variational method, Ziman(17) arrives at a result involving Specific reference to the scattering pro- cesses through (scattering Operators). 13 Ziman's expressions are: . -P 1 1L g q n P11 D 4 2-XV s = 5.1.1. 1m T ”PIL T<9 g q n 5 9 PLL D The PiJ's are Ziman's "scattering Operators" and have the following form(18) 1 k' P = m (Ch—{- "' (bk-t) «Eadl‘. d9 d5. k' _ ”P11; = TICTfffl¢E '¢-1£I)(l>a 2.123 (3.12 d9. d5: 2-XVI P III/Elf E'dkd dk' LL“ E q 9153 _9._ The d's are the variational trial functions and the 9's are the equilibrium transition rates for the scattering processes envisioned. Simple assumptions regardingthe Pij's lead one back to expressions like equations 2-XIV. What is observed in experiment, however, seldom looks much like anything one can calculate from these equations. For example, the negative bump at low temperatures in Pb is attributed to phonon drag, whereas 2-XIV gives positive phonon drag componentl. The next approximation involves looking a little harder at equations 2-XVI, and using more realistic trial functions. l4 Ziman( 19)now inserts for the d's the following expression: 4 E, m 2-XVII T k A9 = .4: .9 '.E T L Te and TL are average relaxation times construed to make the old kinetic formulae: _ nN 2 Ge — F 8 Te and l 2 K a ._ L 3 CL uL TL give the actual ideal electrical and thermal conductivities. The new T'S are the true k and 3 dependent quantities presumably required in any detailed theory. The result of inserting 2-XVII into 2-XVI is, of course, calamitous. There is, however, something to save the crew's drowning. Note that P1L will contain a factor not unlike the following: gfikl = Tg-g . (Tk"¥g' - TE :5) 2-XVIII We note that such a factor could cause the resultant thermo- power to be of either sign depending on the relevant band structure and fermi surface geometry. We shall return to 15 this again. The above results, although complicated, are not yet detailed enough to describe a real metal. They do not take account of phonon diSpersion or of the complex band structure and fermi surface tepology found in the oscillatory experiments. 16 D. The Bailyn Equation 9 Bailyn(20’21’2“) has derived an expression for the phonon drag thermopower which does allow for these facts. 1 dN [35] m 0,1 . . I I 83 = TIE—dfilrr Z dT ‘7‘” Z “(Jg’y’k ’2 )x ' £13 1512'” 2-XIX X [id/31$) - BMW] From the left, the various terms are: e is the electronic charge _ 1 1219:: ($1,... - mg W N is the number of atoms in the sample, A0 the atomic volume, gth A(£§) is the area of the sheet of the fermi surface, and v(£g) is the fermi velocity on the Zth sheet. This expression becomes %: in the effective mass approximation. NOJ is the Bose distribution function. Vm is the usual sound velocity which depends on the direction and magnitude of q and its polarization, J. A given phonon, 3, may scatter from other phonons, from impurities and so forth, as well as inducing transitions of the form (13, 5!, k'l'). “(39}.E£’.E'£') represents the probability that the phonon 13 will induce the event (13, kt, k'fi') relative to all the other scattering events 17 open to 13. The interested reader is referred to reference (20,21) for explicit expressions. Finally, 3(flk) is the velocity of the electron on the 2th sheet. Unfortunately, even this detailed expression will not serve for non-cubic materials. The factor of-% in equation 2-XIX was introduced as an average over 3 cubic directions. Shortcomings aside, the salient point is the emergence of a factor similar in form to 2-XVIII. SJ "w = W» ° [3(a) - 2021139] (on) 2-XX = §(SLJ) ° [10%) - KNEW] (a) This must be summed for all 3k, £k' at a given q_and polarization J. If the q and direction dependence of S,the sound velocity,may be ignored, then: nkk. = 2% ' [1025)- 1(£‘1<_’)](u) It is this factor that fixes the sign of the resultant phonon drag component. The difficulty is that all such com- ponents must be summed up weighted by their relative proba- bilities. Some generalities emerge, nevertheless. Consider the event pictured in Figure 2-3. 18 Figure 2-3 The left hand drawing represents a transition crossing a filled region in k-space. Then: -2qv a'QL-z') But the right hand drawing which represents a transition across an empty region gives: 3°(1-z') = 2qv Hence, dispersion ignored, transitions that cross filled regions contribute negative n's while those that cross unfilled regions contribute positive n's. Which sign wins out depends on the a's, and on the weighting given the g}s in equation 2-XIX by the temperature derivative of the Bose distribution. 19 E. A Working_Hypothesis In the absence of detailed knowledge of all the factors that appear in equation 2-XIX, we appeal to a simple minded argument to gain some insight into the temperature depen- dence of the observed thermopower. If one adepts a simple Debye spectrum for the lattice waves, then the number of phonons with frequency v is: 2 N(v) = XCESStAV 2-XXI eET - l Differentiating 2-XXI with respect to v and setting the derivative to zero yields: Graphical solution gives x e:1.6l; that is: hvc a: 1.61 kT 2-XXII A plot of 2-XXI shows a function (figure 2-4) highly peaked at the frequency predicted by 2-XXII. What this implies is that we may define a "dominant" phonon frequency at each temperature, realizing that the Debye Spectrum will obtain only at very low temperatures (say 0 - GD/lOOOK). Add to this the fact that the shortest wave length that is (23) I allowed to propagate in a Debye solid is (4%,? times 21 the lattice spacing: = 1.6l2a, which implies: hvmax = k9D or since: _ .E;. Vmax — xmin and = 2r _ 2w qmax imin “ T1. 12a- Then: 2W _ h” T1. 12a " ken 2-XXIII Similarly 2-XXII is: hvc = huqc = 1.61 kT 2-XXIV Dividing 2-XXIV by 2-XXIII, we get: T = T 2"XXV This says that as the temperature is raised phonons of larger and larger q-vector become available in large numbers. Looking at it in another way, a given transition involving a phonon wave vector q, will have a high probability of occurring at a temperature given by 2-XXV. 22 We have used an extremely crude model for our phonons, and therefore must not expect much in the way of correlation with experimental results. Furthermore, the phonon Spectrum is not delta-function like at go, but rather contains all q's up to qc and many above (up to the cut off necessitated by GD). Hence, one would expect from such a model a monotonically increasing (or decreasing contribution from each piece of the Fermi surface somewhere in the vicinity of the temperature predicted by 2-XXV. Further smearing out of such an effect may be envisioned from the transition (3';.v",.1"') in figure 2-3, left. We expect the large angle scattering to give the largest contribution, but there are many such small angle events possible even with vanishingly small q-vectors. At some point, however, phonon-phonon interactions will come into play and tend to reduce a in 2-XIX to the point where phonon drag effects will essentially disappear. 23 F. Thermal Conductivity Thermodynamics The thermal conductivity tensor is defined in the following way: 2=£'VT 2-XX‘VI where P is the power or heat energy per unit time trans- mitted through the sample, fi'is the thermal conductivity tensor, and T is the absolute temperature. In the hexagonal system, £_has the form: (KL 0 O i = (0 ”J- 0) o o It“) where 1 refers to.L or H . Temperature Dependence The Free electron - Debye Spectrum result for the ther- mal resistivity at low temperatures (W = l/K) 13(24): 2 W = + BT 2-XXVIII I-3I:> 2a where %-is the contribution due to impurity scattering, and the BT2 term results from phonon-electron scattering. Equation 2-XXVIII is the electronic part of the thermal conductivity and is the dominant factor in pure metals. It predicts a sharp rise in the thermal conductivity at lowest temperatures and then a gradual fall off at higher tempera- tures. At higher temperatures 2-XXVIII predicts a rise in K, but more complicated treatments allowing for U-processes can give a temperature independent conductivity. From another point of view, if the Wiedemann-Franz law is valid, then K will go like the electrical resistivity over T. Since the resistivity goes like T, K is constant. 25 3. Sample Preparation A. Generalities Preliminary eXperimentS indicated that sample dimensions on the order of three inches length by 0.1 inch diameter would yield thermal voltages of sufficient size to give reli- able measurements of the thermopower. At the same time, it was heped that this geometry would suffice for measuring thermal conductivities, at least in cases where this para- meter is not inordinately large. Most of the samples were cut from commercially obtained single crystal rods. Initially, attempts were made to grow large Single crystals of Cadmium in a Bridgmann furnace. We found that while relatively thin, long crystals of random orientation were easily grown, attempts to increase the diameter above 1/8 inch resulted in multicrystalline blocks. Since it is necessary to have samples that are oriented in specific symmetry directions relative to the sample axis, it becomes necessary either to seed small crystals in the desired orientation or to cut the sample of the required orientation from a large block. Seeding in a vertical Bridgmann furnace is at its best a black art and large mono- crystals seemed out of the question. The commercial firm* from which one of the zinc crystals was obtained ran into the same troubles with Cadmium we encountered. *Alfa Inorganics, 6 Congress Street, Beverly, Massachusetts 26 We anticipated the same sort of difficulties with Zinc and Magnesium. In addition, Magnesium attacks glass. Since crucibles containing the stock material are usually sealed (off under vacuum in glass, this difficulty becomes prohibi- tive. In the end, however, we did come upon a method of grow- ing Cadmium crystals. The method employed is discussed in section 3-B below. B. The Growth of Cadmium Crystals Both Cadmium crystals were grown in our laboratories using a horizontal zone technique develOped by J. C. Abele. Heat to produce the zone was provided by a movie projection bulb mounted at one focal point in an elliptical reflector.* The sample is then located at the other focus of the ellipse, resulting in a highly concentrated and confined Spot of radient energy. The molten zone was easily controlled by adjusting the lamp voltage with a variac. Crystals were grown in this apparatus in two steps. Pellets of "69" Cadmium** were first etched in a solution consisting of 3 parts nitric acid and 1 part glycerine and rinsed in double distilled water. A boat was made of SpectrOSCOpic carbon rod and etched in aqua regia. After rinsing in distilled water the boat was baked and then out- gassed under vacuum. The Shot was placed in this boat and *- Source: Materials Research Corp., Route 303, Orangeburg, N.Y. **Source: Cominco American, Inc., Spokane, Washington 27 the combination was outgassed, backfilled with ~'l0.0 cm of hydrogen to produce a reducing atomosphere, and sealed off in cleaned pyrex tubing. The Shot were then melted together in the horizontal zone apparatus and the resulting bar removed from the pyrex and etched again to remove the surface Slag. An inch or so was removed from one end to make room for the seed and the bar re-etched. Having prepared the seed (see below), it was placed in the boat together with the rod. This was again sealed off in a hydrogen atomOSphere and positioned on the cart. The lamp was adjusted to produce a small zone (%-- 1 inch) in the rod at some distance from the seed. By moving the cart the zone was inched up to the seed and finally advanced part way into it. The cart drive was reversed and the zone allowed to proceed to the other end of the boat. In this way the entire rod was caused to grow in a Single crystal oriented in the direction of the seed. The end product was about 3/8" in diameter and 5" long. The Seed The seed was produced in the same way, but in this case a molten zone was allowed to proceed Slowly down the bar by pulling the pyrex tube past the lamp in a cart by means of a variable Speed motor. By chance the first such attempt produced a single crystal only a few degrees from the basal plane. A two inch section was Spark cut from the bar, oriented by means of a back reflection Laue camera, and spark cut along a direction in the basal plane with a 28 tool made of 3/16 inch thinwall stainless tubing. This technique is highbyrecommended as a source of excellent quality Single crystals of materials with melting points below lOOOOC. At first glance, however, one might think that low melting point metals like Cadmium that have high vapor pressures would tend to plate out on the glass tube, thus preventing light from entering the sample container. In actual practice the lamp merely vaporizes the plated metal and all proceeds as desired. C. Orientation and Shaping The single crystal rods were oriented by the Laue method and transferred to a Servomet Spark cutter. This procedure was capable of producing Specimens with orientations within 1 to 3 degrees of the desired direction. Long, thin, cylindrical samples were cut (with the ex- ception of Zn" for which a long tube was used) with a tool like that shown schematically in figure 3-1. Figure 3-1 29 The actual cutting was done by a small length of 1/8" x .010 wall stainless steel tubing. This arrangement gave the most uniform cross sections and took the least amount of cutting time. The resultant physical dimensions are given in Table 3-1 with reference to Figure 3-2 below. N I 1 I l | A. 7 t Figure 3-2 A is the average cross-sectional area as determined by actual micrometer measurements and by weight. The later procedure consisted of weighing the sample, measuring its overall length (B in Figure 3-2 and Table 3-1) and calculating the area from the known densities of the metals used. In order to ensure that no significant temperature gradienhsappeared at the differential thermocouple - sample interface, holes were spark cut using 18 gauge copper wire Hum wanes nmm. : «em. a om. m mono. mm :9... 025.2 m: an.: un.: mm.m nwdo. :m ems: cosmg< mH.: mn.: mm.m mmfio. oooqma emme .H.o.m cu moazmeOCH mmm. : mmm. : 8n. K. 8:0. com. m __ mm: 32 mmo.m mmo.m mmo.m Homo. ooo.:: zmme .D.m.z no mm.: mm.: om.m nmgo. ooo.zm emme .D.m.= AEov “Eov .AEov A Eov a o m m a mZOHmzman .m.m.m Hannah condom 30 II. III II .III III. ll 31 as the tool, and the thermocouples firmly varnished inside after having been insulated with cigarette paper. The distance between these holes is shown as "D". Also included in these tabulations is the distance between the two soldered potential probes, C. D. Sample Purity Included in Table 3-1 are the sources and purities of the crystals used. In thiscase purity refers to the quoted purity of the starting materials. The firms mentioned are Alfa Inorganics, 6 Congress Street, Beverly, Massachusetts; Research Crystals, Incorporated, Richmond, Virginia; and Aremco Products, Briarcliff Manor, New YOrk. The entries, "M.S.U." refer to crystals grown in our laboratories. We also measured the Residual Resistivity Ratios of all the samples. This term is defined below: p2970K p4.20K R.R.R. = and is a measure of the purity and state of anneal of a given metal, ie. the higher R.R.R., the more perfect the metal crystal. Since the thermopower of a pure metal is extremely sensitive to impurity content, the necessity of having clean samples is apparent. We now feel that considerably better samples could have been grown in the cases of Zn_L and both 32 Magnesiums using the horizontal lamp technique used for the Cd crystals. Whether or not Significant, it is interesting to note that in each case the perpendicular sample had a lower R.R.R. than its parallel partner. 53 4. Apparatus and Experimental Technique A. Objectives In order to measure thermopower and thermal conductivity on single crystals, one must introduce and measure a thermal gradient along the sample, measure the power required to pro- duce each gradient, and finally measure the electrical poten- tial difference generated by the gradient. To obtain these coefficients as functions of temperature, it is necessary to have some scheme of changing and accurately monitoring the sample temperature. How these requirements were met is the tOpic of this section. B. The Cryostat The cryostat used in these measurements consists of two concentric cylindrical vacuum cans, suspended by cupro nickel tubes from a brass flange at the dewar head, with an arrange- ment for controlling the pressure in each one independent of the other (Fig. 4-1). The inner can is always evacuated to pressures of the order of 5 x 10'6mm of Hg, thus eliminating the possible source of error in K due to convection heat exchange and, at the same time, minimizing the heat required to maintain the necessary temperature gradient for thermo- power measurements. The space between the two cans may be held at any convenient pressure ranging from hard vacuum (5 x lO-6mm) when the inner can is maintained at a high temperature relative to the working bath, to l atmOSphere during precool. This arrangement provides a Simple means of obtaining a controlled heat leak between the sample holder and the bath. It is therefore possible to work at ll!" iiililfillf [It 'lllrl'lll Ill .l'.ll‘ " ‘I' 34 J W I ‘ r h---L- H B eM Fir Fig. 4-1 55 temperatures near the bath, an otherwise difficult feat in view of the rather large quantities of heat involved in pro- ducing sizeable temperature gradients in samples of high thermal conductivity. The Vacuum System: The vacuum system is shown schematically in Figure 4-2. The left hand side of the drawing displays the high vacuum section which serves the sample holder. The method for intro- ducing controlled quantities of He gas into the Space between the two vacuum cans for purposes of heat exchange is Shown at the t0p left. The ballast tank remains at approximately 1 atmosphere absolute and He at this pressure is admitted by means of a 3-way vacuum ball valve into a Short stub. The gas in this stub expands into the cryostat to a final pressure of ~25 microns. Repeating the procedure builds up the exchange gas pressure. When less is desired, some may be pumped away with the diffusion pump. In this fashion the heat leak may be varied over quite a range of values. This is important between 5 - 300K where the sample conductivity becomes high, requiring large quantities of heat to produce a measurable temperature gradient. This heat must go some- where (the bath) or the sample will heat as the gradient is raised. At higher temperatures the heat capacity of the inner can as well as radiative heat transfer become large enough to accomodate the impressed heat. In addition, the cupro nickel tubes contribute to the required heat leak. 36 The situation is somewhat similar in the range from 78 - 850K with LN2 as cryogen. Generally 200-300 u of He* gas is required to hold 780K in the course of a measurement. Even with several hundred microns of exchange gas present, reliable data below 6 0r 70K was hard to collect. A rather larger heat leak in the form of a wire, perhaps, would be required to take measurements in this region. It was partly for this reason that it was decided to attempt measurements at 4.2OK and below. To achieve this, the outer can was removed, thus placing the OFHC**copper flange of the inner can directly in contact with the He bath. The vacuum arrangements for achieving reduced temperatures are Shown on the R.R.S. of Figure 4-2. A Walker manostat is used to control the pressure over the He bath. The vapor pressure of the bath is read on a mercury manometer, and at lowest pressures, a McCleod gage. These pressures were converted to temperatures by the 1958 He temperature scale. During the course of these measurements, a gaseous He recovery system was installed in our laboratories. Hence provisions are shown for returning boil-off to the central collection bag. Sample Helder: Figure 4-1Ideta11s the sample holder utilized. The two vacuum cans are sealed with pure lead wire wrapped in a * The thermal conductivity of helium is not appreciably larger at STP than at ~300u ** OFHC: Oxygen free high conductivity 37 Niamgm U028: L .r mmom .8. me u>00om 37 NIV..0_n_ CONGO: .T L doom .8. me 9603. 38 single turn around the protruding lip on the upper flanges, twisted tight and squashed by the screws Shown. The lead used was 41 mill. for the outer can and 31 mill. for the inner can. Both are smeared lightly with Silicone grease before installation. This apparently helps the lead flow into nicks and faults, thus insuring a good seal. The outer can is constructed entirely of Brass and soldered together with Eutectic Solder which fuses at about 900°F. This was used so that further connections (for example, to the tube carrying the epoxy seal) could be made with soft solder. The inner can is made of copper and assembled with the same Eutectic Solder. In particular, as previously men- tioned, the flanges are made of OFHC copper. This was done to insure rapid heat transfer within the various parts of the can so that equilibrium could be quickly attained. Further- more, when Operating below 4.2°K, high conductivity Cu was deemed essential to keeping the sample's cold end at the bath temperature. The binding post (B in Figure 4-1), similarly constructed of OFHC Cu, serves two purposes. It provides a convenient place to tie down fragile wires and, more important, it helps make sure that the various probe wires are near the temperature of the sample. This helps eliminate the chance of large temperature gradients being deve10ped across solder joints and glue joints 0n the sample which might otherwise lead to measurement errors. The binding post was first 39 wrapped with a layer of cigarette paper and then varnished with G.E. 7031. All probe wires were then wrapped with several turns and varnished in place. This afforded good electrical isolation and, at the same time, seems to have produced the required heat Sink. The samples were mounted by soldering them with indium into the Cu flange (see Appendix 4-1). This method was decided upon as affording the best thermal contact to the can. Indium melts at a suffic- iently‘low temperature (156.400) to allow it to be used as a solder for the present samples and, since it can be obtained in a pure state, provides a high thermal conduc- tance to the flange. Mounting was accomplished bytflxuing the sample with In, heating the entire inner flange to the melting point of In, and then plunging the sample into the pot of now molten metal contained in the flange. The procedure outlined in Appendix 4-1 was necessitated in the case of Mg, since ordinary soldering techniques are inapplicable. A Small heater was wound on an OFHC Cu cylinder approximately 3/8" long and 1/4" diameter over a layer of cigarette paper cemented in place with G.E. 7031 varnish. This heater was in turn soldered with In to the hot end of the sample, except in the case of Mg. Here cigarette paper was cemented around the hot end of the sample itself with thinned Duco cement and a heater wound directly on this base. Details on all heaters can be found in Appendix 4-2. 4O Probes: Measurement probes were attached in the following fashion. As outlined in Section 3, small holes were Spark cut in the samples. Into these were placed the AT thermocouple probes which were first wrapped with cigarette paper and then cemented in. G.E. 7031 was used for all but Mg where acetone thinned Duco cement was employed. Spark cut holes were used in order to accomplish a low thermal resistance from sample to probe. Temperature gra- dients here would, of course, be detrimental to the measure- ments. Thinned Duco cement has the advantage that it dries considerably faster than G.E. 7031 varnish and thus allows a reduction in the time lapse between sample mounting and apparatus seal up. The thermal resistance of this cement appears to be low enough to suit the present purposes. The thermocouple measuring the average temperature of the sample was wrapped with cigarette paper and glued to a Spot on the sample approximately midway between the two AT probes ("M" in figure 4-1). Since the T probe remains at roughly the temperature of the can and since it is firmly anchored mechanically and thermally to the binding post, it is not necessary to go to such great extremes in keeping down the thermal resistance of the sample-probe junction. Specifics and calibrations of the Gold-Iron vrs. Chromel thermocouples employed are recorded in Appendix 4-3. The thermal emf. probes were cut from "69" 10 mill. lead wire supplied by Cominco Products, Inc., Spokane, 41 Washington. In all cases these probes were soldered in place with Wood's metal, using Sta-Clean flux. These joints were kept as close as possible to the Spark cut holes that contained the AT probes. See Table 3—1 and figure 3-1 in Section 3. Epoxy Seal: An epoxy seal (S in figure 4-2) was employed to remove the thermocouple leads from the vacuum space into the bath, while the lead wires were encased in teflon tubing and taken directly up the inner can's pumping line where they, too, left the vacuum via an epoxy seal. The epoxy seal that Sits at0p the outer can and contacts the liquid cryogen was itself the object of considerable effort. Many diverse methods were tried before a reliable scheme was developed. Several authorsl25’26) have described various approaches to the feed-thru problem. Most of these were found unsatisfactory for one reason or another, usually unreliability in cycling to LHe temperatures. Wheatly's method(26 ) was acceptable in this respect, but seals constructed after his recipe invariably caused the fragile Gold-Iron wire to sever somewhere inside the seal. The method adopted is similar to that described by Balain and Bergeran (25) A Small brass button was turned to fit snugly into 3/8" O.D., .010" wall stainless tubing. It was then soft soldered into a 1.3" length of the above mentioned stainless steel tubing. Finally, a 0.1" hole was drilled in the button. 42 The whole piece was rinsed in methanol and the thermocouple wires were passed through, leaving sufficient length extend- ing from each end. The chromel being already enameled, no further insulation was necessary. The Gold-Iron was bare and it was left so. At this point a small batch of Stycast 2850 GT mixed with 5% by weight of catalyst No. 9* was prepared. This viscous mixture was then filled into the previously drilled hole and the result allowed to cure for 48 hours. An earlier attempt with the recommended 7% catalyst N0. 11 baked at 150°C cracked after a short period of time. The seal described has been in use for several months and has withstood thermal cycling from 1.25°K to room temperature in the course of all the measurements that were taken on this piece of apparatus. It is vacuum tight (< 2 x 10'6mm) and provides extremely good electrical insulation from ground. Once prepared, the wires were protected with teflon tubing and the end opposite the seal was soldered into a 3/8" 00pper coupling to join the seal to the seal line (S in figure 4-1). During soldering the seal was protected with wet tissue paper. *Stycast supplied by Emerson and Cuming, Inc., Canton, Massachusetts. 43 C. Electronics This section sets forth the circuitry and allied elec- tronics used in the present measurements. Electrical Leads: It is necessary to bring Signal leads out of the cryostat in a manner that minimizes pickup and stray thermoelectric potentials. This was accomplished by the use of extensive electrical Shielding throughout and heavy heat sinking and thermal shielding, especially where joints were required. AS mentioned previously, the lead potential probe wires were run up the inner can's pumping line. They were removed from the pumping line by means of an epoxy seal at the dewar header (H in figure 44;) and joined to copper wires by squashing the prepared Pb and Cu wires between pure copper washers. The connections and seal were encased in a heavy brass can supported above the dewar flange to prevent frosting during transfer of liquid helium. Another seal and can assembly removes cOpper wires that have been soldered to the thermocouple wires on a terminal strip located near the low temperature seal just above the sample holder. All these cOpper wires pass through one shielded braid to an aluminum box located near the instruments. This box contains a heavy terminal block constructed as follows: A 6" x 4" x l" Slab of lucite was attached inside a 7" x 5" x 3" aluminum minibox. Twelve l" x l" x fir c0pper slugs were formed and drilled to make square "washers ". These were ll secured in pairs to the lucite base with 4' x 20" nylon bolts. 44 Cryostat leads are joined in this box to shielded copper wires that go directly to the various measuring instruments. The foregoing precautions were undertaken to minimize thermal voltages in the leads. A modicum of success was achieved in this regard -- ambient thermals being on the order of tenths of microvolts and relatively stable. Measuring instruments: Figure 4-3 shows a block diagram of the circuitry involved. From the left, the abbreviations employed stand for the following: K3 is a Leeds and Northrup type K-3 potentiometer. L&N N.D. is a Leeds and N0rthrup model 9834 null detector. K149 refers to a Keithley model 149 milli- microvoltmeter; K147 is a Keithley model 147 nanovolt null detector, M2D2 is a Moseley model 2D2 x-y recorder, and VTVM is a Hewlett-Packard model 410BR vacuum tube voltmeter. The boxes labelled "heater control" and "power supply" are described in‘detail in Appendix 4-4, and the heaters in Appendix 4-2. 7 The vertical string of instruments on the left was used to measure and control the average sample temperature. Out- put from the Gold Iron - Chromel thermocouple, whose reference is the working cryogen (liquid He or nitrogen),feeds the K3 potentiometer. Voltage unbalance between the actual thermo- couple voltage and the setting on the potentiometer is detected and amplified by the model 9834 null detector. This imbalance is used to control the heater marked T-control via the heater control circuit described in Appendix 4-4. 45 z>e> E: 5.50 I A NONE _o._+cou Loewe: 91 v. mi v. 4. >4 C80... Te, :2“. E L050: l, A Emmom .3301 - 2828.: .55» I ,1 .220 c 22.00: My. ESP 46 This arrangement allowed voltage discrimination on the order of .5 uv which corresponds to .05 K0 at 4.2 and .02°x at room temperature. The controller generally held the sample temperature to within .1 KO over the range from 4 - 3000K. The second string of blocks in figure 4-3 consists of an adjustable current source and a power heater. This com- bination was used to provide the gross quantities of heat required for quick temperature changes and for holding temperatures well above the bath, eSpecially in the case of 1N2 with the sample can at 3000K. In this instance rather larger quantities of heat than can conveniently be supplied by a sensitive controller were needed to maintain the high difference in temperature between the bath and the sample can. The third string of instruments utilizes a Keithley model 149 millimicrovoltmeter to measure the output of the differential thermocouple. The output from this Keithley drives a second heater control,as described in Appendix 4-4, and the x-axis of the Moseley x-y recorder. For runs above 40K, the K149 was used in its lOuV f.s. setting, thus giving accuracies on the order of .2uV which translates to .02K° at 4° and .01x0 at 3000K. Since the largest temperature gradient employed was about 0.5 KO, this corre- sponds to 2 - 4% accuracy in the measurement of AT. Below 4° K149 was used on either the 3uV or the luv f.s. setting. Hewever, ambient noise cancels out any gain in relative accuracy. The Keithely model 147 nanovolt null detector was used 47 to measure and amplify the signal from the lead potential probes. The output from this instrument drives the y-axis of the Moseley recorder. The Keithley 147 was used on various ranges depending on the relative size of the thermoelectric voltage. Ranges utilized were between .03 and 10 uv f.s. Accuracy in this leg can be expected to be 2-4%. This leaves only the Hewlett Packard VTVM. It was used to measure the voltage drop across the AT heater, thus giving a measure of the power feed through the sample. Estimated accuracy is 3 - 5%. 48 D. Method pf Measurement Data were taken in three separate steps, correSponding to three temperature ranges. The first step covered the range from room temperature (~3OOOK) to the boiling point of liquid nitrogen (770K). With both vacuum cans evacuated to roughly 2 x 10-6mm of Hg, liquid nitrogen was transferred into the inner dewar. As previously noted, considerable power was required to hold the inner can at 3000K. This power was supplied by the heaters marked (T) and (T-control) in figure 4-3. The potentiometer marked (K3) in figure 4-3 was set at the vol- tage correSponding to the desired temperature as read on the thermocouple calibration curve. In this manner the tempera- ture controller provided vernier corrections to the power required to hold the desired temperature. When the average sample temperature had stabilized, measurements were begun. With no current through the AT- Heater, and the remainder of the electronics activated, the point plotting mechanism of the Moseley x-y recorder was activated, thus plotting a point mark on 8%IX ll graph paper. This done, the AT heater control was advanced so as to increase the temperature gradient along the sample. Sufficient time was allowed for dynamic equilibrium to set in, and a second point was plotted. This was repeated at least four times. At each point the voltage across the AT heater was read and recorded. The thermal emf vrs. tempera- ture difference points lie approximately on a straight line. 49 The final temperature difference varied from .5KO at high temperatures to .1K0 at lowest temperatures. When these measurements were completed, the K3 was set for the next, usually lower, temperature (say, 2900K), causing the control heater to cut off; the T-heater was turned off, and lOOu of He gas was admitted into the outer can. Exchange between the 2 cans then took over, rapidly cooling the inner can to the next point of interest. Somewhat before the next set point was reached the T-heater was turned back on and the exchange gas pumped away. After equilibrium was reestablished the procedure indicated above was repeated. In this manner the range from 300 to 780K was covered in 5 - 10K° steps. The second step proceeded similarly to the first with the exception that liquid helium was utilized in order to cover the region from 80 - 60K. The inner can was precooled to ~8O°K along with the rest of the apparatus by admitting exchange gas into the outer can during the precool stage prior to transferring helium. The temperature of the appar- atus was monitored during precool by measuring the current passed by a 1N 34A Germanium diode selected for high temper- ature coefficient. Once calibrated, this device could be depended upon to indicate temperatures accurately to within a few degrees at 1000K. The circuit particulars are shown in figure 444. Included in the same package is a bridge circuit used in conjunction with 1000 1/8 watt carbon resistors to sense the He liquid level. Similar devices are in wide . I'll...“ 1 {‘1 lllllll 50 m: .88 van .9“. DUE :0 ocean“; 0:03.... 0:31— , Massive..- score st. to... <8 o eL me e e e o I. . at «was . . .e. - c o .. ~ . I... s— 4109 OI. .fiJC - + *m a \- . q f 0 s M . . l l I . - ~ ~ ‘ OIL 4. . l . - w Q. . .5: HI. a H... . 5W. l _____. I. q >m.¢ 51 spread use. The prototype of this particular instrument was designed by Dr. D. J. Sellmeyer. AS in step one, measurements were initiated at an elevated temperature relative to the bath. Hewever, upon reading 300K the specific heat of the inner can and the var- ious exchange mechanisms (excluding gaseous transfer) were insufficient to hold the sample temperature reasonably constant during the incrementing of the sample temperature gradient. At this point a new procedure was adopted. Some lOOu of He exchange gas was admitted to the outer can and the inner can allowed to cool to 4.2OK or so. Then by using the controller and gradually pumping the exchange gas as the set point was raised, the exchange could be carefully con- trolled and stable temperatures easily maintained. Overall sample temperatures in this mode could be held to something like .OloK during an entire measurement sequence. The upper limit under these circumstances was about 400K and the lower limit 6 - 80K, depending on the sample's thermal conductivity. This has been discussed in the cryostat section. The last step covers the range from 4.20K to 1.2OK. In this case the outer can was removed and the rest submerged in liquid helium. The overall temperature was then controlled by pumping on the He hath. Needless to say, the thermo-emf Shrinks rapidly with temperature, necessitating a somewhat different approach. In these low temperature runs a great number of points were printed representing repeated excursions of AT. This resulted in a large quantity of points from which 52 the required SlOpe was extracted, even though the information was masked by noise and drift. Because of the rather small voltages involved (.01 — .l microvolts) the uncertainties are considerably larger in these low temperature data. 53 5. Results and Discussion A. Data Reduction We describe in this section the methods used to analyze the raw data taken into meaningful thermopowers and thermal conductivities. Thermopower: The raw data consists of plots from the Moseley x-y recorder of the sample emf vrs. the emf from the Au-Fe thermocouple measuring the temperature difference along the sample. The SlOpe of the best straight line drawn through these points is pr0portional to the therm0power. This scheme eliminates stray constant thermal emfs generated in the lead wires from the cryostat. The analysis is Similar to that described below for the thermal conduct- ivity. However, scaling factors for the various amplifiers used and the temperature dependent sensitivity of the differential thermocouple must be taken into account. This can all be eXpressed by a simple equation: El. = (scale factor) '(%¥j '(SlOpG) where.%% is the measured thermopower, "scale factor" is the ratio of Y amplification to that of X amplication in the various instruments,-%¥ is the sensitivity of the Gold-Iron vrs. Chromel differential thermocouple*, and "s10pe" is the 1See Appendix 4-3 54 measured s10pe 0f the best fit line through the data points. Now equation 2-VI states that we must add to the measured therm0power the thermopower of pure lead. The lead data used in this analysis is that of Christian, et al.(27 ) A program was written to perform the above calculations on a CDC-3600 computer, using least squares to find the s10pes. It did not produce significant improvements over the simple hand calculations in most instances and was abandoned. Thermal Conductivity Equation 2-XXVI says that we must measure the tempera- ture gradient produced by a given heat flux. The latter is given by: where P is the power flowing, V the voltage drop across the heater, and R(T) the temperature dependent resistance of the heater. Then by 2-XXVI V2 R(T) ll 3’; «I» AT 5-I AT is not measured directly. The measured quantity is the voltage dr0p across the differential thermocouple, VT C , together with a thermal voltage associated with the wires com- ing from the cryostat. This thermal is presumed to be 55 constant: ie.: Vobs. VT.C. + const. or VT.C. = Vobs. - const. but 1 ( ) 1 AT = V = V - const. T'C° dV/dT °bS° dV/dT Substitute into 5-I: _ RA . 1 v2 _ x ‘1'(Vobs.) (577EE)+ const. Hence, if V2 is plotted against Vobs one should get a straight line whose slope is related to the thermal conduct- ivity as follows: _ dV E K — (lepe)(scale factor)(aT)(fiK) 5-11 V and V are measured at the same time as the thermopower obs. measurements are taken. AT is, of course, always <