The Thermal Conductivity of Metals at Low Temperatures* K. MENDELSSOHN AND H. M. ROSENBERG The Clarendon Laboralory, O x f d , England
I. Theoretical Outline 1. Introduction ...,.... .............................................. 2. The Electronic Thermal Reaistivity.. ................................ 3. The Validity of the Wiedemann-Franz Law........................... 4. The Lattice Thermal Reaistivity. ................................... 5. The Separation of the Lattice and Electronic Conductivities.. .......... 6. Superconductors .................................................. 11. Discussion and Interpretation of the Experimental Reaults 7. Pure Metals.. .................................................... 8. Alloys.. ......................................................... 9. Superconductors.................................................. 111. The Use of Heat Conduction Data in Research on Metal Imperfections 10. Introduction.. ................................................... 11. The Detection of Dislocations in Deformed Alloys. .................... 12. The Detection of Defects in Superconductors. ........................ IV. Experimental Techniques 13. Introduction ...................................................... 14. Thermometry ..................................................... 15. Types of CryoAtat................................................
223 224 229 230 234 236 239 244 246 256 257 261 264 264 271
1. Theoretical Outline
Heat can be transported in a solid by two different mechanisms. In all solids there is the possibility of heat being conducted by the lattice vibrations (or “phonons”); however, if the solid also has a reasonable electrical conductivity, and hence has a supply of “free” electrons, the electrons can also act as carriers of heat. Experiment shows that in reasonably pure metals nearly all the heat is, in fact, carried by the electrons. In impure metals, alloys, and semiconductors, however, an appreciable proportion of the thermal conductivity is due to lattice conduction. In dielectric crystals, of course, all the heat is transported by the phonons. I n order to understand the behavior of heat conduction a t low temperatures, we must study the way in which the two carriers of heat, the electrons and the phonons, may be scattered in their passage through the substance,
* Submitted for publication in June,
K. MENDELSSOHN AND H. M. ROSENBERQ
for if there were no scattering we would observe no temperature gradient along the specimen; i.e., it would have an infinite thermal conductivity. In studying thermal conductivity K , it is always useful to recall the kinetic theory formula for heat conduction
K = Aclv
where c is the specific heat of the heat carriers per unit volume, v is their velocity, 1 is their mean free path (i.e., a measure of the distance they can travel between collisions), and A is a constant which is usually equal to *. For both electrons and phonons we assume that v is constant. The absolute magnitude and the temperature dependence of the thermal conductivity are therefore determined by the variations of c and 1. The variation of c is well known. For electrons c is proportional to T,whereas for phonons it is proportional to T3a t very low temperatures and is constant a t high temperatures. In this article we shall not consider explicitly the case a t intermediate temperatures, but we can always use the Debye or some more detailed theory if it is required. The main problem is in the calculation of 1. We should first observe that there may be several different scattering mechanisms which are effective a t the same time. It is usually assumed that we can treat each mechariism separately, so that each gives a contribution t o the thermal resistance and that we can then add the thermal resistances to obtain the total resistivity. The approximate validity of this assumption has been verified in the case of electrical resistivity; it forms the basis of Matthiessen’s rule. Hence the total electronic thermal resistivity We, will be of the form We = W1 W2 W3 where W1, W2, etc., are the resistivities arising from the scattering mechanisms 1, 2, * * . There will be a similar expression for the total phonon resistivity W,. The total heat transport through the metal, however, will be the sum of the heat carried by the electrons and the phonons; hence the total heat conductivity is the sum of the conductivities of the phonons and the electrons, i.e.,
where K , = 1/We and K , = l/W,. We shall now consider the various types of scattering mechanism and their effect on We and W,. 2. THEELECTRONIC THERMAL RESISTIVITY Bloch’s treatment of the case of an electron in a region of a periodically varying potential showed that it would not be scattered unless there was a M. Blackman, in “Handbuch der Physik,” Vol. 7, Springer, Berlin, 1956.
THERMAL CONDUCTIVITY OF METALS
departure from perfect periodicity. There are two important ways in which this might occur. The ideal crystal lattice might be upset by distortions of various kinds, e.g., vacancies, interstitials, or dislocations, or by impurity atoms. We shall call this impurity scattering. It gives rise to a thermal resistivity which is usually denoted by Wo.An otherwise perfect crystal lattice will also be disturbed by the thermal vibrations of the atoms characteristic of the temperature of the crystal. These will give rise to a thermal resistivity denoted by Wi. This is sometimes called the “ideal” resistivity because it is characteristic of the perfect specimen of 100% purity. Let us now consider how the two types of scattering, that arising from the impurities and that from the thermal vibrations, depend on the temperature. In the case of impurity scattering there is no reason why the mean free path of the electrons should change with the temperature, hence 1 in Eq. (1.1) will be a constant and Wo will be proportional to l/c, i.e., WOwill be of the form PIT. The more impure the sample, whether by the addition of chemical impurities or by upsetting the crystal lattice, e.g., by deformation or by irradiation, the smaller I will be and hence the larger 8. This scattering of electrons by impurities gives rise to the constant residual electrical resistivity Ro a t low temperatures, and is connected with Wo by the WiedemannPranz law
Ro/(WoT) = L. Here L is the Lorenz constant, which is equal to 2.45 X pressed in ohm cm and Wo in watt-’ cm deg. Hence
if Ro is ex-
The numerical value of 1 for a given impurity cannot be computed with any certainty. Chemical impurities give a scattering which is approximately proportional to the square of the difference in valency between the impurity atoms and the atoms of the matrk2S3This variation does not take into account any change in the Fermi surface of the metal which may arise when impurity atoms are present and which may also affect the heat transport. Up to a few per cent of impurity /3 is approximately proportional t o s(1- s), where x is the impurity concentration, but no general rule can be formulated for larger amounts. The scattering of electrons by lattice defects has been calculated by 1
J. 0. Linde, Ann. Physik  10, 52 (1931); 14,353 (1932); 16, 219 (1932). N. F. Mott, Proc. Cambridge Phil. Soc. 31, 281 (1936). M. H. Cohen and V. Heine, Advances in Phys. 7,395 (1958).
E. MENDELSSOHN AND H. M. ROSENBERG
several workers.6* There is some measure of agreement that the electrical resistance is in the vicinity of 1.0 X lo-’ ohm cm per atomic per cent of vacancies, which corresponds to B = 40 watt-’ cm deg2. That for an interstitial atom is in the same range. Calculations of the scattering of electrons by dislocations are far less certain because the geometry and electron configuration of the dislocation core are not known with any precision. The values range from 4 X lo-’’ ohm cm per unit density of dislocations (i.e., /3 = 1.8 X watt-’ cm deg’) if the core scattering is neglected,’ to a value about ten times greater if the effect of the core is taken into account in a way which possibly overestimates it.8 There is increasing evidence o. lo,l1 that a considerable amount of electronic scattering may be a result of the presence of stacking faults in the crystal rather than of the dislocations associated with them. Since the density of stacking faults can vary widely from one metal or alloy to another, it is difficult to give any guidance on the magnitude of this effect. In many cmw, however, the stacking fault scattering may be several times greater than that arising from dislocations. We must now consider the term Wi which is due to the scattering of the electrons by the thermal vibrations of the crystal lattice. There is no simple method of dealing with this term. Although Wi can be given a simple analytic form at high and at very low temperatures, there is no general straightforward expression covering the whole temperature range. Qualitatively, it can be appreciated that the amplitude of the lattice vibrations decreases as one goes to lower temperatures; hence their scattering effect becomes smaller. The thermal resistance will therefore decrease as the temperature is decreased. A full treatment of the problem however can only be obtained by solving the Boltzmann transport equation. This leads to a complicated integral equation which has only been solved for the case of quasi-free electrons. Details and reviews of various approximate solutions have been given in several publications 12-16 to which the reader is referred for further information.
‘P.Jongenburger, Appl. Sci. Research B3, 237 (1953).
OF. J. Blatt, Phys. Rev. 99, 1708 (1955). S.C. Hunter and F. N. H. Nabam, Proc. Roy. Soc. A220,542 (1953). W. A. Harrison, Phys. and Chem. Solids 6,44 (1958). J. W. Christian and J. Spreadborough, Phil. Mag.  1, 1069 (1956). loA. Seeger, Can. J . Phys. 34, 1219 (1956). l1 J. N. Lomer and H. M. Rosenberg, Phil. Mag.  4,467 (1959). 11 J. L. Olsen and H. M. Rosenberg, Aduancea in Phys. 2,28 (1953). P. G. Klemens, in “Handbuch der Physik,” Vol. 14, p. 198. Springer, Berlin, 1956. l4 A. H. Wilson, “Theory of Metals,” 2nd ed. Cambridge Univ. Press, London and New York, 1953. 16 J. M. Ziman, “Electrons and Phonons.” Clarendon Preaa, Oxford, 1960. @
THERMAL CONDUCTIVITY OF METALS
The general result of these calculations shows that a t low temperatures
(T < 8/10) where 8 is the Debye characteristic temperature) the thermal
resistance Wi is of the form aT2, where a! is given by
Here N is the number of “free” electrons per atom, K, is the limiting thermal conductivity a t high temperatures, and G is a numerical factor which has a value of about 70, depending on which approximation is used for the solution of the Boltzmann equation. It is unfortunate, as Eq. (2.3) shows, that a! can only be calculated with the aid of other experimental data, in particular K, and 8. A constant value of 8 implies a simple Debye spectrum, but deviations from this model are most apparent a t low temperatures. K, appears in Eq. (2.3) because it is not possible to calculate the true interaction function of the electrons and the lattice vibrations. A substitution involving K , can be made for this function, however, since K , is also dependent on the interaction between the electrons and the phonons. Nevertheless, if N is taken as unity and experimental values of 8 and K , are used, the calculated value of a! appears to be about four times greater than that obtained from actual measurements of the thermal conductivity a t low temperatures.lB This substitution for the interaction function is based on the assumption that the function is the same both a t high and low temperatures. It is unlikely that this postulate is correct. At high temperatures there is a much greater probability that an umklapp process will occur. In this form of interaction, which was first suggested by Peierls,” momentum is not conserved among the interaction components themselves, but some is transferred to the lattice as a whole. Such processes enhance the scattering of the electrons and increase the value of the interaction function a t high temperatures. Thus the use of an expression which involves this function a t low temperatures will lead t o an overestimate of the low-temperature scattering; i.e., the calculated value of a! in Eq. (2.3) will be too large. This is what has been observed. It has also been suggested I8 that Eq. (2.3) should be modified so that the parameter 8 refers to the longitudinal lattice vibrations, for the Bloch theory involves the assumption that the electrons are scattered only K. H u h , Prm. Roy. SOC.A m , 98 (1960). R. Peierle, Ann. P h y ~ k 3, 1055 (1929); “Quantum Theory of Solids.” Clsrendon
Press, Oxford, 1955. M.Blackman, Proc. Phys. SOC.AM,681 (1951).
K. MENDELSSOHN AND H. M. ROSENBERG
by such vibrations. Hence a special value of 8 should be used, say 8 L . I n contrast the value of 8 for the specific heat takes account of lattice vibrations of all polarizations. Since the value of Or, is about 1.5 times greater than the specific heat 8, the use of the former would also help to reduce the value of a. Ziman has developed a theory which takes into account both the umklapp processes and the fact that only longitudinal phonons should be lsolg
FIG.1. Diagrammatic curvea showing the dependence of the electronic thermal conductivity K , on temperature and purity. The topmost curve is for a high-purity specimen and the bottom curve for an impure specimen.
considered. His numerical calculations for sodium are in quite good agreement with the experimental results of Berman and MacDonald.20His calculations also show that a maximum in Wi which the earlier theories had suggested ought to exist at a temperature of about 8/5, and which had never been observed, is almost entirely absent. Nevertheless the T 2 temperature dependence of Wi still holds at low temperatures even though the expression for a in Eq. (2.3) should be reduced by a factor of about four to be in accord with experiments on many metals. Thus the total electronic thermal resistance of a metal is of the form
J. M. Ziman, PTOC. Roy. Soc. A226,436 (1954). R.Berman and D. K. C. MacDonald, Proc. Roy. Soc. A209,368 (1951).
THERMAL CONDUCTIVITY OF METALS
This expression should apply when T/8 is small, which is usually taken to be the case for values below 8/10. This is not the only criterion for the validity of (2.4), however. If the simple theory is to apply we must also be in the region of temperature where the specific heat varies as T3. This does not occur until T is as low as 8/50 or 8/100' in many metals. Thus Wi will not be strictly proportional to T2until one is at these temperatures. The preceding discussion shows that a i s the more important parameter in Eq. (2.4) because it should be a constant for a particular metal, whereas P, the impurity coefficient, depends on the particular sample of metal which is being measured. Providing that /3 is fairly small, it can be seen that there will be a minimum in the value of We a t low temperatures; i.e., there will be a maximum in the electronic thermal conductivity, Ke. The purer the sample of the metal the smalkr /3 will be so that the maximum in K , will be higher and will occur at a lower temperature. Typical conductivity curves are shown in Fig. 1. 3. THEVALIDITYOF
It should be noted that, whereas the Wiedemann-Frans law [Eq. (2.1)] holds for the relationship between the residual electrical resistivity Ro and the impurity thermal resistance Wo,there is no such simple connection between the temperature dependent electrical resistivity Ri and the thermal resistivity Wi. The quantity Ri/(WiT) is less than the Lorens number a t low temperatures and approaches the simple theoretical value once again only a t higher temperatures. This means that Wi increases more rapidly than Ri a t low temperatures; that is, the mean free path of the electrons is less for thermal resistivity than for electrical resistivity. At first sight this seems rather strange, since the same electrons are being scattered in both cases. The reason for the difference is that the electrons interact with lowenergy phonons a t low temperatures and hence are scattered only through small angles. Since the electronic charge is unaltered by the scattering, the flow of current in a given direction is not changed very much. If, on the other hand, we are concerned with the transport of heat, the interaction with a phonon, even though it results in small-angle scattering, still changes the electron energy by kT.The temperature of the electron gas is reduced and less heat is transported along the specimen. Hence Wi is increased much more effectively than Ri. At high temperatures where the energy interchange is of the magnitude k0, which is small compared with kT,the main scattering mechanism for both Ri and Wi produces a change in direction through a much larger angle. Under these circumstances the WiedemannFranz law is valid once more. Typical curves for the variation of L with temperature are shown in Fig. 2.
AND H. M. ROSENBERG
FIQ.2. Diagrammatic curves showing the dependence of the Lorenz number L on temperature and purity. The bottom curve is for an ideally pure specimen (with zero reddual electrical reaistivity) and the topmost curve for an impure specimen. If the impure specimen haa an appreciable lattice conductivity (see Section 4), the value of L watt ohm deg-2. might rise above its theoretical limiting value of 2.45 X
4. THELATTICETHERMAL RESISTIVITY
An earlier article in this series 21 dealt in some detail with the theory of heat transport by the phonons so we shall give here only a brief outline of the aspects which are important when dealing with metals and alloys. Whereas the mean free path 1 for scattering by defects is independent of the temperature for electron heat transport, this is not the case when one considers the phonon conductivity. This is a result of the fact that the spectrum of the phonon wavelengths which are excited is dependent on the temperature. Only long waves are present a t low temperatures whereas shorter wavelength phonons are excited a t higher temperatures. The amount of scattering will depend on the size of the defect relative to the dominant phonon wavelength a t the temperature we are considering. Hence the amount of scattering by a particular defect is very dependent on temperature. The most important scattering mechanisms for phonons, in the order of increasing effectiveness as the temperature is increased, are as follows: 1, the scattering of the waves a t the specimen boundaries which occurs a t the lowest temperatures; 2, the scattering by electrons; 3, scattering by dislocations; 4, scattering by point defects; 5, and lastly, the scattering of the phonon waves by interaction with one another (umklapp processes). In annealed metals and alloys a t low temperatures, the main contribution 11
P. G. Klemens, Solid State Phye. 7, 1 (1958).
THERMAL CONDUCTIVITY OF METALS
to the total phonon resistivity, W,, arises from the scattering by the electo be of the form ET-2. trons. This resistivity Wge can be shown 13,15,21,n A simple argument can be used to derive this behavior from the kinetic theory Eq. (1.1). The mean free path 1 of the phonons is proportional to the number of electrons with which they can interact. The only electrons with which this is possible are those whose energy lies within kT of the Fermi surface Eo. The proportion of these electrons to the total number present is of the order kT/Eo. Hence the number of electrons available for scattering is proportional to T and therefore 1 varies as T-'. If one assumes that the phonon velocity v is constant and that the specific heat is proportional to T 3 , W,, will be proportional to T-2. The calculation of the constant of proportionality E involves the problem we have already encountered in discussing the scattering of electrons by phonons in the calculation of W,-that is, the uncertainty of the form of the interaction constant between the electrons and the phonons. Makinson tried to circumvent this uncertainty by using the limiting high-temperature value of the electronic thermal conductivity, K,, in his expression for E . As has already been remarked, however, such a substitution does not take account of the umklapp processes which will be more operative a t high avoided this source of error by using than a t low temperatures. Klemens the low-temperature electronic thermal resistance Wi as the experimental quantity which contained the interaction constant. His expression for Wge is W,, = 3.2 X lo"[email protected]
"T-~ = ET-2 where a is defined by the relation Wi = aT2 as in Eq. (2.3). Calculations of W,, have been made by Ziman 28 in cases in which there is only a very small number of electrons which have a nondegenerate distribution (such as occurs in semiconductors). In the limit of a degenerate distribution, his solution is equivalent to that in Eq. (4.1). Apart from Wge the other contributions to the phonon thermal resistivity can apply equally well to nonmetals. At higher temperatures, the main contributions to W, arise from umklapp processes and from point defect scattering. The scattering by impurities and defects can be treated by considering the analogous example of the scattering of waves by particles which are smaller than the wavelength (Rayleigh scattering). The mean free path of the waves for such scattering is proportional to the fourth power of the wavelength. If we assume that the dominant lattice wavelength is inversely proportional to the temperature, 1 is approximately proportional to T-4. This, of course, is very much an oversimplification. We should integrate the
R. E.B. Makinson, Proc. Cambridge Phil. Soc. 34,474 (1938).
** J. M.Zimn, Phil. Mag. [S] 1, 191 (1966).
K. MENDELSSOHN AND H. M. ROSENBERG
scattering over the whole phonon spectrum. Using Eq. (1.1) we see that, in the region where the specific heat is proportional to T3, K will be proportional to T-'; i.e., the thermal resistance of the phonons due to point defects and individual impurity atoms, Wp, should be of the form P T . At higher temperatures Wp tends to become more nearly constant because (a) the specific heat is not so temperaturedependent and (b) Rayleigh scattering no longer holds as the phonon wavelength becomes shorter, the scattering tending to become independent of the wavelength. The umklapp processes at low temperatures introduce a thermal resistance WUwhich is exponential
Wu = gTne-e/mT
in which g, m, and n are constants. Wu is very small; its contribution is If *a ~specimen observed only in crystals which are isotopically p ~ r e . 2 ~ ~,~~ contains the usual proportion of naturally occurring isotopes, the less abundant isotopes act as impurity atoms of different mass when compared with the most abundant isotope. This introduces an extra contribution to W p which tends to overshadow the effect of WU. In spite of such increases in Wp, it tends to become rather small a t lower temperatures, as has already been mentioned, since it is proportional to the temperature. The mean free path of the. phonons will therefore increase as the temperature is reduced until a t some stage it becomes comparable with the dimensions of the specimen itself. When this occurs it is obvious that it cannot increase any further. Thus below a certain temperature the value of 1 becomes constant and is comparable to the smallest dimension of the specimen (i.e., the diameter). Using Eq. (1.1) we see that this boundary scattering leads to a thermal resistance WB which is proportional to T-3 and is inversely proportional to the specimen diameter. If the sample is a polycrystal, 2 will be of the order of the linear dimensions of the crystallites. The last important scattering mechanism which must be dealt with is the scattering of phonons by dislocations, which gives a resistance w&s.The following simplified treatment is based on that given by Z i m a ~ ~The . ~ ?scattering of phonons by a dislocation can be divided into two parts, that due to the misplaced atoms at the core of the dislocation and that caused by the elastic strain around it. At low temperatures the phonon wavelength is long compared with the lattice spacing and the scattering by the dislocation core can be considered as Rayleigh scattering by a long cylinder. This gives a R. Bennan, E. L. Foster, and J. M. Ziman, PTW.Roy. SOC.A237,344 (1956). G . A. Slack, Phys. Rev. 106,829 (1957). 26T. H. Geballe and G . W. Hull, Proc. 6th Intern. conf.on Low TempeTntiLre Phys. and Chem. p. 380 (1958). '7 J. M. Ziman, Nuow cimento [lo]7, Suppl. 2, p. 353 (1958). 24
THERMAL CONDUCTIVITY O F METALS
mean free path proportional to (where X is the phonon wavelength). If we use “dominant wavelength” reasoning similar to that used earlier in discussing the scattering of phonons by point defects, we are led to a temperature-independent thermal resistance. The absolute magnitude of this resistance is very small compared with that due to the scattering of the phonons by the region of elastic strain around the dislocation. This strain field extends a long way from the dislocation and has considerable influence in scattering of phonons even a t large distances T from the center of the dislocation which are greater than k. The elastic strain at any point is of the order of b/r, where b, which is
FIG.3. The sdttering of a phonon by the strain field of a dislocation. The angle of scattering, 6,is of the order of rb/ro, where ro is the distance of closest approach of the phonon to the core of the dislocation.
about an atomic spacing, is the magnitude of the Burgers vector of the dislocation. Since the elastic potential energy of the lattice is not strictly harmonic, the elastic constants of the strained lattice are different from those in the unstrained lattice, i.e., the velocity of sound v will be changed. Hence the phonons will be refracted when they pass through the region around the dislocation. If the velocity of sound in the unstrained lattice is vo, it becomes v = vo(1 f yb/r)
in a region where the lattice strain is f b / r . Here y is the Griineisen constant derived from thermal expansion data and is a measure of the anharmonicity. It can be shown that a phonon whose closest distance of approach to the center 0 of the dislocation is ro (Fig. 3) will be scattered through an angle 4 of about yb/ro. If the incident phonon flux is distributed homogeneously and is traveling in the direction indicated by the arrow in Fig. 3, the amount which will pass within a distance T and r dr from 0 can be written ~ 1 9F dr. The amount which is scattered a t right angles to its original direction will be F dr (1 - cos 4). Since c#, is small this is equal t o F + ( y b / ~dr. ) ~ Hence the fraction of the total flux which is scattered is 3y2b2/rowhere ro is the shortest distance from 0 for which this treatment is valid, i.e., ro = A. Thus the
E. MENDELSSOHN AND H. M. ROSENBERG
scattering is proportional to y2b2/X. Hence in the region where the specific heat is porportional to T3,we get, using (1.1)) Klemens 21 suggests that the most satisfactory expression for the calculation of D for a random array of dislocations is
where h is Planck’s constant, v is the velocity of the phonons, k is Boltzmann’s constant, and N is the density of dislocations per unit area. It will be noted by comparison with Eq. (4.1)) that the temperature dependence of wdb is the same as that of Wge. Hence a T-2 lattice thermal resistance might be the result of a combination of the two scattering processes. To summarize the behavior of the total phonon resistivity W,, we can write Wg = w g e Wl? WP W U Wdis (4.6)
+ + + +
W g = ETd2
+ BT-3 + P T + gTne4lmT + D T - ~
where the subscripts in Eq. (4.6) refer to the scattering of phonons by electrons, specimen or grain boundaries, point defects and isotopes, umklapp processes, and dislocations, respectively. In ordinary metals and alloys, Wgeand Wdb are the only resistivities we need consider, although WBcan become important in superconductors (see Section 6). In nonmetallic crystals all except Wge may be present, although in practice WBis the overriding resistivity a t the lowest temperatures whereas a t higher temperatures W p or WUis the determining factor. If the specimen is isotopically, chemically, and physically pure, W Uwill determine the main resistivity; otherwise Wp will probably do so. In semiconductors, the behavior depends very much on the effective number of carriers which are present. If this number is large, W,, will be very effective and there may also be an appreciable electronic thermal conductivity K,.If the number of carriers is very small, the semiconductor tends to behave like a dielectric crystal. Quite complicated phenomena can occur in between these two cases. 5. THESEPARATION OF THE LATTICE AND ELECTRON CONDUCTIVITIES
A full discussion of this subject has been given by Klemens.21We shall limit ourselves to the methods which have been used satisfactorily in experiments. From the practical point of view it is not possible, in general, to measure the thermal conductivity of a specimen with an accuracy greater
THERbUL CONDUCTIVITY OF METALS
than 1%. It is clear, therefore, that, if Ke or K , is very small compared with the total conductivity K , it will not be possible to determine the smaller component. In practice this means that it is not possible to determine K , for a pure metal, since nearly all the heat is transported by the electrons. The exception to this occurs for a metal in the superconducting state where, as will be described in Section 6, nearly all the heat is carried by the phonons a t very low temperatures. Hence the heat transport which is measured is due to lattice conduction. It should be noted, however, that this value of K , is peculiar to the superconducting state and is not the same as that appertaining to the normal state in which the electrons can interact with the phonons and so reduce K,. I n metals, K , is measured by adding impurity atoms to the parent metal. The addition of such atoms decreases the electronic thermal conductivity K , because of the impurity Scattering effect. This is most pronounced a t low temperatures [Eq. (2.4)]. The effect of impurity atoms on K,, however, is rather diflerent. Since they are point defects, they tend to scatter only the shorter lattice waves. Thus the low-temperature phonon conductivity is unaffected (see Section 4). Hence the addition of impurities to a metal will make K , a much larger fraction of the total conductivity a t low temperatures than it is in the pure metal. There still remains the problem of separating K into K , and K,. To do this one must make assumptions regarding the dominant scattering processes which limit the two conductivities. It is obvious that, if one had to take into account all the scattering mechanisms at once, it would be impossible to achieve a satisfactory analysis. If we are dealing with an unstrained impure metal we assume that K , is limited by impurity scattering in the helium region, that is, Ke = l/Wo. This can be checked by ensuring that the electrical resistivity is constant in this region, i.e., that we are measuring Ro. By using Eq. (2.2) we can calculate Wo.Since K , = 1/Wo in this case, we can find K,, which is equal to K - K,. Although this method can be used, it is also quite common to assume that K , is limited by the scattering of phonons by electrons, that is, K , = l/W,,. Thus we find from Eqs. (1.2), (2.4)) and (4.7) that the total conductivity is of the form
K = T2/E
Hence a graph giving K I T against T should be a straight line having a slope 1/E and an intercept on the K I T axis of l/@.The low-temperature values of both K , and K , may be found in this way. This method is used only for the determination of K , in practice because K , can be found from Ro. Hence l/@can be calculated independently and is used on the graph as the actual intercept. It should be noted that the value of K , which is obtained in this
K. MENDELSSOHN AND H. M. ROSENBERQ
way is significant only for the impure sample which has been measured. It need not necessarily be equal to K , for the pure metal. There are two main reasons for this. First, the addition of impurities might affect the electron distribution and hence the amount of scattering w d d be changed. Secondly, the addition of the impurity atoms will alter the phonon spectrum which will also change the value of K,. This can be mimimized by using an impurity atom whose atomic weight and radius is similar to that of the parent metal. The simple expression for K , cannot be used if we wish to find K , at higher temperatures. We must also take into account the aT2 term of Eq. (2.4). This can only be done by using the value of a found for the pure metal since values of a for alloys cannot be measured. This may not be very accurate because a will also be affected byzhanges in the electron distribution and the phonon spectrum when impurities are added. Providing the temperature is not too high, however, this term is rather small. Hence any inaccuracies in a will not he very important. 6. SUPERCONDUCTORS Almost immediately after the discovery of electrical superconductivity in 1911, Kamerlingh Onnes initiated experiments to determine the effect of the phenomenon on'the heat transport. The first results were obtained in the following year but were not published until two years later 28 because they were quite unexpected. Instead of thermal superconductivity, it was found that the heat conduction in the superconductive state was lower than in the normal one. Work between the wars in Leiden and Oxford confirmed this result for pure metals; however, it was observed that the position is reversed in certain alloys. In all these measurements, the heat conduction Knin the normal state, which is obtained by carrying out the observations in a magnetic field larger than that necessary to quench superconductivity, appeared as a continuous function of temperature. At the electrical trsznsition temperature T,, the superconductive heat conductivity K , breaks away from K,. The departure may be sudden or gradual (cf. Fig. 7) but no discontinuity in K is observed at T , even though this point has been carefully investigated.16 The fall of K , below K n in the case of a pure metal is in good agreement with the general phenomenological pattern which is now accepted for the superconductive state. As the metal is cooled below T,, the electronic contribution t o the entropy decreases more rapidly with temperature than the linear function (S = r T ) characteristic of the normal state. The thermo-
** H. Kamerlingh Onnm and H. G . Holst, Communs. Kamerlingh Onnes TAb. Univ. Leiden 142c (1914).
THERMAL CONDUCTIVITY OF METALS
dynamic and electrodynamic behavior of superconductors has led to a working hypothesis which has been remarkably successful in a rough interpretation of the observed effects, although it cannot have any physical significance in its crude form. This is the so-called “two-fluid model” in which the electron fluid is regarded as a completely interpenetrating mixture of a normal and a superconductive constituent. Different thermodynamic and electrodynamic properties are assigned to these two fluids and it is assumed that at T, the fractional concentration of the normal fluid is 1 and that it decreases monotonically to zero at T = OOK. It has been shown experimentally that the decrease in total entropy is a result of the growth of the superconductive concentration and that, in fact, the entropy of this constituent is zero at all temperatures. This conclusion led to the postulate that there is an energy gap in the electron spectrum of the metal 28 which is roughly coincident with the Fermi energy. Recent theoretical 30 as well as experimental *I developments seem to favor such a model. The energy gap confers on the metal an aspect which is not too different from that of a dielectric crystal. Owing to the small size of the gap, however, this behavior cannot become apparent except at very low temperatures. A comparison with a dielectric solid may appear strange in view of the infinite electrical conductivity; however, it is remarkable that this feature of superconductivity becomes very convincing in the heat conduction. Since the superconductive constituent has zero entropy, it cannot contribute to the heat transport; therefore KOrepresents a progressively smaller fraction of K nas the temperature is lowered. The thermal conduction of the superconductive metal can be calculated by making plausible assumptions concerning both the rate at which the normal constituent vanishes below T , and its change in heat conduction as it becomes diluted. The temperature function is usually given for KO/& and was first computed by Koppe 32 on the basis of Heisenberg’s theory. A very similar function has recently been obtained from the Bardeen-Cooper-Schrieffer theory.33The temperature dependence of KO/& is given in Fig. 4. It may appear surprising that the two theories, based as they are on very different interaction mechanisms, should yield closely similar results. However, it must be remembered that the temperature variation of the free energy which determines the other temperature functions is given by the experimentally obtained values of the critical field. The scattering suffered by the normal electron fluid, much like that in a a3 J. G . Daunt and K. Mendeleaohn, Proc. Roy. Soc. A188, 226 (1946). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). *1P. L. Richards and M. Tinkham, Phys. Reu. Letters 1, 318 (1958). a* W. Heisenberg, 2. Naturforsch. Sa, 65 (1948). SJ J. Bardeen, G . Richayzen, and L. Tewordt, Phys. Rev. 113, 982 (1959).
K. MENDELSSOHN AND H. M. ROSENBERG
FIG.4. The semiempirical function for KJK, against the reduced temperature.
nonsuperconductive metal, occurs as a result of interaction with either phonons or impurities as indicated in Eq. (2.4). Since the temperature-dependent (ideal) electrical resistance is small in comparison with residual resistivity in the range of temperatures where superconductivity occurs, impurity scattering has to be considered primarily. As will be seen in Part 11, reasonable agreement between theory and experiment obtains for this case, whereas the scattering of electrons by phonons, such as is observed in the superconductive state of very pure specimens of lead and mercury, clearly requires further assumptions. As is shown in Fig. 4,K , should become extremely small a t sufficiently low temperatures. However, only the electronic conductivity has been considered in these calculations. This is indeed by far the dominant mechanism for a pure metal in the normal state, but it should be remembered that this is only the case because phonon conduction in metals is inhibited owing to the extremely effective scattering of phonons by the free electrons. The removal, on cooling, of a progressively increasing fraction of the electron fluid from the thermal distribution in the superconductive state not only decreases the heat conduction by electrons but also decreases the scattering
THERMAL CONDUCTIVITY OF METALS
of phonons by them. Accordingly, phonon conduction in the metal will become the dominant process a t sufficiently low temperatures. Its behavior will be closely analogous to that of a dielectric crystal. Thus, a t some temperature below T,,Ka/Kn must become larger than the function of Fig. 4. So far no theoretical attempt has been made to determine this temperature ; however, experimental data suggest that it will be below O.4Tcin the case of a pure metal. At sufficiently low temperatures, the thermal conduction should become proportional to T3 in the case of good single crystals, the factor of proportionality depending on the diameter of the specimen since the only relevant process will be scatter of phonons on the walls of the specimen. Thus a size effect in the heat conductivity of a superconductor is to be expected. Moreover, KO,which is chiefly limited by scatter of electrons on point imperfections a t higher values of T/T,must become less sensitive to these at lower temperatures, since their scattering cross section is small for phonons. Instead, K, must be strongly influenced by extended lattice faults. The relative importance of conduction by phonons will be shifted to higher reduced temperatures as the impurity content of the specimen rises. For very impure specimens which have a low electronic conduction in the normal state, one can expect values of K , which approach or even exceed those of Kn. The explanation of the inversion of the Ka/Kn ratio in the case of alloys is thus provided by a combination of depressed electron and enhanced phonon conduction. The correctness of this model is borne out, at least qualitatively, by experiments.36 Even so, it must be regarded as surprising that, in some cases, for example in a lead alloy with 10% bismuth, a rapid rise of K , occurs already a t T,. The opposite case is given by a pure metal with an intrinsically high electrical conductivity, such as aluminum, in which the residual heat conduction due to the normal electrons will remain the dominant factor in K , even a t the lowest temperatures in spite of an enhanced phonon conductivity. In such cases one should expect a close adherence t o the semiempirical formula particularly when, as in aluminum, T,is well below the maximum in the thermal conductivity. II. Discussion and Interpretation of the Experimental Results
7. PUREMETALS In recent years the thermal conductivity of most metals has been measured over a wide range of temperature. For details of the experimental data
*' H. B. G . Casimir, Physicu 6,495 (1938).
K. Mendekaohn and J. L. Olsen, Prm. Phys. Soc. A8S, 1182 (1960).
K. MENDELSSOHN AND H. M. ROSENBERG
on various metals the reader is referred to the review papers by Klemens la and by Powell and Blanpied.36The latter has most of the data in the form of graphs and includes information on alloys and nonmetals. Qualitatively, the thermal conductivity of pure annealed metals agrees with the theoretical description which has been outlined in the previous section. All the heat is transported by the electrons, no contribution by the
I I I 40 60 Temperature ( O K )
PIQ.5. The thermal conductivity of two samples of lithium as a function of temnperature.
phonons being detectable experimentally. For a given sample the conductivity increases linearly from 0°K to a maximum which occurs at a temperature of about B/lO°K and then decreases, becoming constant a t higher temperatures. The purer the sample the higher is the conductivity maximum and the lower the temperature at which it occurs. Typical curves are shown for two samples of lithium (Fig. 5 ) of differing purity. It will be noted that at higher temperatures the conductivity is the same for both specimens. This is the region in which the dominant scattering agents are the thermal vibrations which, to a first approximation, have an effect independent of the impurity content. The conductivity curves for all high-purity metals have this form. The conductivity a t the maximum of the curve can often 36
R. L. Powell and W. A. Blanpied, Thermal Conductivity of Metals and Alloys at Low Temperatures, A review of the literature, Nail. Bur. Standards Circ. 666 (1954).
THERMAL CONDUCTIVITY O F METALS
be very high indeed, that is, of the order of 100 watt/(cm deg) or more [the conductivity of copper a t room temperature is about 4 watt/(cm deg)]. The alkali metals are, of course, the ones for which one would expect any free electron theory to apply most successfully. Unfortunately, both lithium and sodium undergo a martensitic transformation a t low temperatures in which certain regions of the specimen are transformed to another crystal structure. The amount of the transformation is considerable, being about 45q10 in sodium 37 and between 80 and 100% in lithium.38Since the vibrational spectra of the phases are different from one another,39this can lead to a complicated conductivity behavior. In actual fact the general form of the curves 20,40,41 is similar to those for all other metals, but it is clear that a detailed, qualitative study might lead t o difficulties in interpretation. I n spite of this, Ziman l9 as has already been mentioned, has made a calculation for the heat conductivity of sodium which does give good agreement with experiment. Unfortunately his theory has not yet been applied to the other alkali metals. Bailyn 42 has also developed similar work. Apart from this, no satisfactory quantitative calculation for the heat conductivity has been made. Although K , is represented with some degree of accuracy by Eq. (2.4) in most cases, calculations of the value of a obtained using expressions such as (2.3)give values which are about four times higher than the experimental ones. Some remons for this discrepancy have already been given in Section 2. While Wi is usually of the form aT2, it is found that the power of T is slightly greater than 2 in some cases, 43 for example copper and silver. It is not clear whether this is the effect of a complicated electronic band structure or whether it might be caused by a deviation from the simple Debye spectrum for the phonons, as is discussed below. Even the data for the transition metals, which have complicated overlapping bands, are in rough agreement with the simple theory. However, for tungsten, rhenium, osmium, rhodium, and iridium, the power of T in the expression for Wi was greater than 2.5.44It was suggested that this behavior might be the result of an exponential decrease in the probability of s,d transitions as the temperature is reduced. In the ferromagnetic metals, iron, cobalt, and nickel, it was found 44 that Wi did not become constant a t higher temperatures as it does in other metals but that it continued to increase. n D. Hull and H. M. Rosenberg, Phys. Rev. Letters 2,205 (1959).
D.Hull and H. M. Rosenberg, Cryogenics, 1,27 (1960) J. S. Dugdale and D. Gugan, Proc. Roy. Soc. A264, 184 (1960). 40 D.K.C. MacDonald, G. K. White, and S. B. Woods, Proc. Roy. Soc. A!236,358 (1956).
41H. M. Rosenberg, Phil. Mag. IS] 1, 738 (1956). M. Bailyn, Proc. 6th Intern. C m f . on LOLO Temperature Phys. and Chem. p. 373 (1958). 4 G. K.White, Proc. Phys. Soc. A66,844 (1953);Australian J . Phyu. 6,397 (1953). u G. K.White and S. B. Woods, Phil. Trans. Roy. Soc. A261,273 (1959).
K. MENDELSSOHN AND H. M. ROSENBERG
Although one presumes that this is connected with the electronic configuration of a ferromagnetic, there is no obvious indication of the nature of the connection. One rather general result for a has been obtained from Eq. (2.3). Assuming the form of this expression is correct, even though the numerical values of G and N are uncertain, one might expect that metals with similar electron configurations would have the same values of G and N and hence that the value of aKWB2would be the same for those metals. Rosenberg46has analyzed experimental results and has shown that the values for aKw02are the same for metals of a given chemical group. A similar type of analysis has been made by Klernen~,'~ and MacDonald et aZ.4O show that it holds for their results on the alkali metals. The value of CY is usually obtained from the experimental data by plotting a graph of WeT against T3. Inspection of Eq. (2.4) shows that this should be a straight line with a slope of a. A reasonably straight line is obtained for most metals, but there are many cases where it has a distinct curvature even at very low temperatures. In some cases (e.g., the impure mercury specimens of Hulm 16), the curve is convex towards the T3 axis. Olsen and Rosenberg la have shown that this is consistent with the theory W eis calculated as a of conductivity developed by S ~ n d h e i m e rin , ~which ~ whole and is not subdivided into parts Wi and Wo initially, i.e., the validity of Mattheissen's law is not assumed. Such a calculation shows that the deviation of the WT against T3 curves from linearity becomes more pronounced at low temperatures the more impure the sample. Nonlinear behavior is sometimes observed in quite pure samples. In this case, however, the curves are concave, away from the T3 axis. This deviation cannot be explained by an impurity effect. The explanation of the behavior lies in the fact that the simple theory is based on the assumption that the metal has a Debye spectrum of lattice vibrations and that the specific heat is proportional to T3 in the temperature range. In actual fact this is not true. As Blackman has shown,' a true T3behavior should not set in until the specimen is below a temperature of 8/50 or 8/lOO, and not 6/10 as the simple Debye theory predicts. At higher temperatures, greater than 8/50, the specific heat rises above the Debye value, i.e., the effective value of O decreases. Since a linear WT versus T3 curve implies a constant value of a,a changing value of 8 is reflected in a variation in a,i.e., the curve is no longer linear. As 8 decreases, a increases [Eq. (2.3)] and the curve is concave away from the T3 axis, as is observed experimentally. This hypothesis has been tested 47 by experiments on two metals, zinc and cadmium, which 46 H. M. Rosenberg, Phil. Trans. Roy. Soc. AM?, 441 (1955). W E . H.Sondheimer, Proc. Roy. Soc. A203,75 (1950). 47 H.M. Rosenberg, Phil. Mag. [S]4,541 (1957).
THERMAL CONDUCTIVITY OF METALS
exhibit a particularly large change in the value of 8 a t low temperatures. The results for cadmium are shown in Fig. 6. It will be seen that a very marked change in slope, by a factor of three, occurs in the region where the value of 8 changes. It should of course be noted in this connection that one should not expect quantitative agreement between the changes in 0 deduced from specific heat measurements and the variations in a which are obtained. 8 is only inserted into the expression for a as a convenient way of
FIG.6. A plot of WT against T*for cadmium, showing the change in slope of the curve which occurs at approximately the same temperature &B the rapid decreaae in the value of e (see inset). [H. M. Rosenberg, Phil. Mag. [S] 2, 641 (1957).1
averaging over the phonon spectrum. It is not introduced to provide a measure of the specific heat. Since the averaging process is different for the specific heat and for the thermal conductivity, the effect on a of a slight change in 8 should not be taken too seriously. The quite marked changes in 8 which are observed in some metals must however be reflected in a change in a, as has been verified in the experiments on cadmium and zinc. We have stated that all the heat transport in pure metals can be accounted for by electronic conduction. This is not quite true because there are some metals in which the value of Wo calculated from the residual resistance Ro with Eq. (2.1) is higher than the experimental value even though the curve for K is apparently linear a t low temperatures, where W , is dominant. Thus the measured conductivity is larger than it ought t o be. In view of the reliability of the Wiedemann-Frans law in calculating Wo, it seems reasonable to suppose that the small amount of extra conduction originates in the phonons. The metals in which this occurs are those which have a very low value of K-titanium, zirconium, hafnium, niobium, chromium, vana-
K. MENUELSSOHN AND H. M. ROSENBMRG
dium, and m a n g a n e ~ e . These ~ . ~ ~ metals are very difficult to prepare in a high state of purity. Many of them, even when as pure as it has been possible to prepare them so far, are physically very hard, which indicates that some impurities may still be present. Thus it is very likely that their low conductivity and the appearance of a recognizable component K , is not an intrinsic property of the metal, but is an impurity effect similar to that discussed in the next section. a. Other Experiments on Pure Metals
For completeness, we may mention other types of heat conductivity experiments which have been made; however, no details are given. Although the results aye interesting, no satisfactory interpretation has been provided because of the complexity of the theory. The object of these experiments is to determine whether anomalies which have been detected in the electrical conductivity are reflected in the thermal conductivity. As might be expected, similar effects are generally found. Several workers 48,49,60 have observed deviations in the W T versus T3 plot which are associated with the minimum in the electrical resistivity found a t low temperatures in some metals. Others have measured the decrease in the thermal conductivity which occurs when a magnetic field is applied to a specimen. This effect can be very large 61 in some cases. Moreover oscillatory effects akin to the de Haas-van Alphen effect (the oscillatory variation of magnetic susceptibility with changing field) have been observed 62 at high fields. The reader is referred to earlier review articles 12,13-63 for further details. Experiments have been made on single crystals in which the anisotropy of the conductivity along different crystal axes has been measured. The most outstanding case studied is that of gallium4Swhich is very anisotropic. Both zinc and cadmium also exhibit some anisotropy.
8. ALLOYS The main effect of the addition of another metal is to increase the impurity scattering (i.e., W o ) .If sufficient alloying material is present, K , will be so small that it is comparable in magnitude with the lattice conductivity K , . As has been explained in Section 4,this is unaffected by impurity scatW. R. G. Kemp, A. K. Sreedhar, and G . K. White, Proc. Phys. Soc. A66, 1077 (1953). H. M. Rosenberg, Phil. Mag. [71 46, 73 (1954). 60 D. A. Spohr and R. T. Webber, Conf. phys. dea basses temps., Pan's p. 453 (1955); Phys. Rev. 96,602 (1954). 61 K. Mendelssohn and H. M. Rosenberg, Proc. Roy. SOC. A218, 190 (1953). * M. C. Steele and J. Babiskin, Phys. Rev. BS, 359 (1955). H. M. Rosenberg, Progr. in MdaE Phys. 7,339 (1968).
THERMAL CONDUCTIVITY OF METALS
tering at low temperatures. In this case the conductivity K must be considered to be the result of the addition of contributions froin both K, and K,. It is difficult to decide when K, must be taken into account in the interpretation of K since the reduction in K, which is necessary to make K, appear is dependent on the type of impurity atom introduced. To provide an example, it has been found 54 that the effect of K, becomes evident at an impurity-concentration of about 1% zinc when zinc is added to copper even though K, cannot be estimated with much accuracy. At 5% of zinc, about one-half of the total conductivity can be ascribed to lattice conduction. If, however, the valency difference between solute and solvent were greater, K, would be observable at lower impurity concentrations. In the region where Ke is dominated by WO (i.e., Ke G l/Wo), the electrical resistivity of the specimen is constant, having the “residual” value Ro. Early analyses ss-s6 were carried out in which Ro was measured and Ke was calculated with the use of the Weidemann-Franz law [Eq. (2.1)1. Thus K, was found from the expression
When this procedure was used it waa found that K , was proportional to T 2 up to about 20’K. This result indicated that K, was limited by the scattering of phonons by the free electrons, as has been mentioned in Section 4. The T 2 dependence of K, has usually been assumed in later work, and the results have been analyzed by plotting a graph of K I T against T as described in Section 5. The reciprocal of the slope of the straight line so obtained gives the value of the constant E in Eq. (4.1). The results found in this way are in qualitative agreement with the theory. The reader is referred to an earlier article in this series for a fuller discussion of the lattice heat conduction of metals. It should be pointed out that the value of K, which is calculated from the experimental data in the way described here is, of course, the value for the alloy and not the pure parent metal. The value of N and of the interaction constant may change as impurity atoms are added. The lattice conductivity for the pure metal can be estimated by measuring K, as a function of impurity content, and then extrapolating to zero impurity. So far we have only considered the behavior of K, at the lowest temperatures. When the temperature is increased, other scattering mechanisms take over, in particular those associated with point defect scattering W P . The latter will tend to stop the original T2 increase in K, which then passes through a maximum and decreases. It is difficult to estimate K , at higher J. N. Lomer, D.Phi1. Thesis, Oxford University (1958). J. K. Hulm, Proc. Phy8. Soc. B64,207 (1951). 56 R. Beman, Phil. Mag. 42, 642 (19511.
K. MENDELSSOHN AND H. M. ROSENBERQ
temperatures, however, for K , cannot be calculated with sufficient accuracy in this region. The behavior of K , in alloys which have a very high electrical resistivity R is interesting. If R is large, the mean free path of electrons is short. Since the phonon wavelength can be a hundred atomic distances or more a t low temperatures, this means that the phonons are scattered by electrons whose mean free path is shorter than the phonon wavelength. Under these circumstances the ordinary phonon-electron scattering theory is not valid. It can be shown 67 that the phonon scattering is proportional t o X-' and to the mean free path of the electrons. This latter quantity is constant a t low temperatures, particularly for the alloys in which this mechanism operates. It follows from Eq. (1.1) that K , is proportional to T.Such behavior has been observed in copper-nickel and silver-antimony alloys 68 which have particularly high values of Ro. It will be recalled that K , is proportional to T2in the normal case of scattering of phonons by electrons. At higher temperatures, of course, where the dominant lattice wavelength becomes shorter, the more usual T2term tends to become dominant once again.
9. SUPERCONDUCTORS The temperature dependence of the thermal conductivity in the normal and in the superconductive state, K , and K , respectively, are shown in Fig. 7 for typical samples of lead and tin. The difference in behavior, indicated in Section 6 is clearly apparent, the K , curve departing from Kn abruptly in the case of lead and gradually in the case of tin. It can also be seen that the transition temperature T,for lead is above the maximum in K nand below it in tin. This means that the scatter of the normal electrons a t the onset of superconductivity is mainly by phonons in lead and by impurities in tin. In both cases, however, K , is lower than K n for the whole range of superconductivity. It was mentioned in Section 6 that one should expect heat to be carried by phonons rather than by electrons a t low temperatures, where the concentration of normal electrons must become vanishingly small. This phonon conduction will be enhanced by the reduction of the scattering which the phonons experience by encounters with normal electrons. Evidence for this effect can be obtained either by observing the temperature dependence of K , a t sufficiently low temperatures or by introducing agents into the specimen which will scatter phonons selectively. Both types of experiment have been carried out, and have shown beyond doubt that a superconductive metal exhibits behavior identical with that of a dielectric crystal, 67 68
A. B. Pippard, Phil. Mag.  46, 1104 (1955). J. E. Zimmerman, Proc. 6th Intern. Cmj. on Loto Temperalure Phys. and Chem. p. 392 (1968); Phys. and Chem. of Solids 11,299 (1969).
THERMAL CONDUCTIVITY OF METALS
as far as the thermal conduction is concerned, when sufficiently near to absolute zero. In the dimensional Eq. (l.l),u is temperature-independent and c is proportional to T3. Hence K will vary with T3 since 1 is constant. As was pointed out by Casimir," scatter of phonons only occurs a t the geometric surface of the specimen in a perfect dielectric crystal a t low enough temperatures. This means that 2 is constant and is determined by the specimen
FIQ.7. The heat conductivity of (a) tin and (b) lead in the normal and superconduct-
ing s t a h .
diameter. Thus the magnitude of K is dependent on size alone. This type of behavior has indeed been observed," in dielectric crystals, the heat conductivity being proportional to T3and of the predicted magnitude. Clear evidence for a similar behavior was first found in superconductors in measurements of K , for lead 6o a t temperatures below 1°K. A typical case is shown in the logarithmic diagram of Fig. 8. Similar results have since been obtained for tin, indium, and niobium. While the graph of Fig. 8 shows it was found that the numerical value was a definite variation of K, with T3, rather smaller in this case than would be expected on the basis of Casimir's treatment which, for a cylindrical specimen, predicts 1 d, the specimen diameter. Similar deviations are found in other cases. A study of the effect of strain on K,, to be discussed in Section 12, has made it clear that even a very small degree of cold work, such as is produced by slight vibration of u,
R. Berman, F. E. Simon and J. M. Ziman, Proc. Roy. Soc. A2N, 171 (1953). K. Mendelssohn and C. A. Renton, Proc. Roy. Soc. A230, 157 (1955).
K. MENDELSSOHN AND € M. I.ROBENBERG IX
FIG. 8. A logarithmic plot sliowing the proportionality of K, at low temperatures with Ta for lead and thallium.
the cryostat, can raise the numerical value of K, substantially. This difficulty was not appreciated in most of the researches quoted in the foregoing. More careful work carried out recently by Graham 61 has shown that, a t least in the case of tin, values for K, can be obtained which are sufficiently close to the value predicted by Casimir t o make scattering by the specimen boundaries the dominant process. Even so, however, the tendency of K, to fall short of this value deserves attention. It is convenient to represent the situation by introducing several thermal resistances which are characterized by different mean free paths and which are combined additively. Thus, denoting the mean free path derived from observation by lo, we write
l/lO = (I/’$
in which is a mean free path characteristic of an additional resistance in the bulk material. The circumstance that no metal specimen has been found so far in which this additional internal resistance is zero whereas it is of considerable magnitude in most may be due to the fact that the metals 61
G . M. Graham, Proc. Roy. Soc. M48, 522 (1958).
THERMAL CONDUCTIVITY OF METALS
are much more plastic than the dielectric crystals investigated. This is a field in which further research is indicated, particularly because the features associated with the internal scattering appear to be complex. Although the temperature dependence of K, is cubic in many cases, both higher and lower powers of T have been observed. Moreover, it has been found that even a t the lowest temperatures, a single crystal of the given material will exhibit a thermal conductivity proportional to T 2even though IL polycrystalline sample may show a T3 dependence. This suggests that the internal scattering centers, which are most probably dislocations, may have a more profound effect than the crystal boundaries. A more convincing proof of the phonon nature of K , a t low temperatures is provided by work in which selected scattering mechanisms are introduced into the specimen. Observation of the different effects of changes on K, and Kn allows a more unambiguous assessment of the nature of the energy transport in each case. Assuming that heat is entirely carried by phonons in the superconductor, a t low enough temperature, we may expect the following behavior of K, and Kn in the same specimen: (1) K. should be insensitive to point imperfections, i.e., to the amount or nature of the impurity, whereas K , should be much reduced by small amounts of impurity and should be sensitive t o its nature. (2) K, should be reduced by large scale imperfections (e.g., grain boundaries) such as are introduced by plastic deformation and recrystallization, whereas K , should not be greatly influenced by them. Providing that the specimen is pure enough, dislocations should reduce both K , and K n . In many cases however K n is already so small because of impurities, that the only observable effect is the reduction in K,. (3) K , should be reduced when the specimen diameter becomes small compared with the phonon mean free path in the bulk material, whereas K , should not. Experiments designed t o test these predictions have been carried out on lead samples.s2The results leave no doubt concerning the phonon nature of the heat transport. Figure 9a shows K , and K , for two single crystals, one of pure lead and one of an alloy containing 0.7% bismuth. The conductivities in the normal state at 1°K differ by a factor one hundred and there is still a wide divergence even in the superconductive state a t 4°K. The conductivity a t this temperature evidently is still mainly electronic, but the two K , curves merge below 2°K. Numerically very similar values of K , were obtained a t 1°K with specimens of lead containing the same amount of tin or thallium. On the other hand, when a sample of lead containing 0.6% of thallium a H. Montgomery, Proc. Roy. Xoc. A244,85 (1958).
IC. MENDELSSOHN AND H. M. ROSENBERG
was measured before and after being severely strained by bending, the normal state conductivity was found to be completely unchanged; however, K , was found to be reduced to one-sixth of the original value (Fig. 9b). It seems probable that the dislocations introduced into the sample by straining are very effective in scattering phonons but that the electronic conduction is unaffected by them because of the thallium impurity.
Bi 1 %
no. 9. The effect of (a) impurity and (b) strain on K, and K,, in pure lead and in l e d alloys. (b) Effect of strain on the conductivity of the specimen PbTlO.S%.
It is more difficult to demonstrate the effect of specimen size on K , clearly. Simple comparison of the values obtained on a thick rod and a thin wire of the eame material is not too convincing since it is almost impossible to avoid straining the latter. Thus, it would be difficult to distinguish with certainty between reduction in K , due to limitation of the phonon mean free path resulting from geometrical boundaries or from dislocations. However, an experiment has been carried out 0z on a lead foil 0.07 mm thick which was made mechanically stable by being rolled into a scroll. Since the phonon mean free path I?b of the material was of the order of 0.5 mm, a size effect should have been noticeable in the foil. Indeed the heat conduction of the foil a t 1°K was found to be five times smaller than that of a bulk
THERMAL CONDUCTIVITY OF METALB
specimen of the same material whereas K , is essentially the same in both cases. Moreover, the temperature function of K, for the scroll approaches T3, which is to be expected for boundary scattering. The effect of enhanced phonon conduction can be expected to be most
FIG.10. K , / K , for tantalum and niobium single crystals.
pronounced in metals having low intrinsic electrical conductivity, such as the transition metals. The most striking example of this kind has, indeed, been observed in tantalum BB which was recently measured down to 0.2”K. The result of these investigations shows that K./Kn is even larger than 1 in the neighborhood of 1°K. Plotting the data in a form which permits comparison with the semiempirical function (Fig. lo), it is clear that the latter represents the data remarkably well down to about 0.4Tc.Below this WP. Rowell, D.Phi1. The&, Oxford University (1968); K. Mendelssohn, PhySica M, Suppl. 53 (1958).
K. MENDELSSOHN AND H. M. ROSENBERG
temperature, there is an enormous rise in the phonon conduction which at 0.2T, assumes a value about a hundred times larger than the electronic contribution. Data on niobium, included in the figure, and on vanadium, present a similar pattern. Connolly has used the results for tantalum to separate the phonon contribution K,, from the electronic part Kes. This analysis
l h . 11. K , for pure aluminum and an alloy with 1.7 atomic per cent Cu.
shows that a t a temperature T c / 4 ,the phonon conductivity is still rising with falling temperature even though the ratio Kg8lKe.g approaches lo2, indicating the strong scatter of phonons by the normal conduction electrons. Comparison of the numerical data also makes it clear that the maximum in K , will be less pronounced or will disappear for metals with better intrinsic electron conduction. A typical case in which the latter behavior is observed is presented by aluminum, which has been investigated both by Satterthwaite 66 and Con(6
A. Connolly, D.Phil. Thesis, Oxford University (1960). C. B. Satterthwaite, Phy.9. Rev. to be published, see also N. V. Zawaritskii, J.E.T.P. (USSR) 83, 1085 (1957).
THERMAL CONDUCTIVITY OF METALS
nolly whose results are in good agreement. Here the data agree well with the semiempirical function over the entire range down to the lowest temperature. Connolly's results are given in Fig. 11, in which observations of K , are plotted for pure aluminum and for an alloy with 1.7 atomic per cent copper. The pure metal follows a line corresponding t o the B.C.S. function with an energy gap slightly smaller than that predicted by the theory. At temperatures below TJ3, however, the alloy shows a deviation to higher values, indicative of a phonon component. It was mentioned in Section 6 that the semiempirical formula breaks down in those cases in which scattering of electrons in the normal state is not caused by impurities but by phonons. In fact, the observed values for K , / K n for a metal like lead, which has a low characteristic temperature and a relatively high transition point (7.2"K), do not follow the theoretical function for any temperature region. As is evident from Fig. 7, however, K , exhibits a maximum at 0.5T, which is not unlike those shown by tantalum and niobium a t much lower reduced temperatures. The question therefore arises whether the maximum in lead is different in nature from the enhanced phonon conduction found in the transition metals. Fortunately, the sensitivity of the phonon conduction to dislocations allows this problem to be decided experiment8ally.66 When the pure single crystal of lead of Fig. 7 was strained a t helium temperatures, it was found that K , was indeed reduced. However, this reduction took place only a t temperatures well below the maximum, i.e., in the reduced temperature region found in tantalum and niobium. The maximum in lead was quite unaffected whereas it was drastically reduced by similar treatment in the transition metals, as will be discussed in the following section. A much better idea of the situation in the case of lead can be obtained from the plot of K , / K n shown in Fig. 12. This shows that a t temperatures above -0.4Tc the experimental values are lower than those given by the semiempirical function, whereas they are higher below that temperature. Introducing dislocations by strain does not affect the higher temperature region of K , at all. However, K , now follows the theoretical function remarkably well below -0.4T,. Thus it appears that the failure to obey the semiempirical formula arises from two quite different reasons. At high temperatures, where phonon scatter of electrons is predominant, the theoretical understanding is not yet sufficient whereas phonon conduction becomes predominant in the superconductor below 0.4Tc. Once phonon conduction is drastically reduced by scattering on dislocations, the semiempirical formula is obeyed well. Hence the maximum of I<, in lead is entirely electronic in nature and is clearly connected with the maximum in K,. UA. Cdverley, K. Mendelseoh, and P. M. Rowell, Cryogenics, 2, 26 (1961).
K. MENDELSSOHN AND H. M. ROBENBERG
0.5 0.6 a7 OB
FIG.12. The effect of strain on the K , / K , function of lead.
Finally, mention must be made of the curious behavior of the thermal conduction, f i s t noticed by Mendelssohn and 01seqB7when superconductivity is destroyed in a cylinder by a transversely applied magnetic field. The thermal resistance W remains constant until the tangential component a t the metal surface reaches the critical field. Then, instead of decreasing monotonically to the lower value found in the normal state, W rises, sometimes quite appreciably, until the value characteristic of the normal state finally is reached. The effect is very pronounced in dilute alloys but also occurs in pure metals. It is absent in heavily alloyed specimens. Moreover, there is a strong but not simple dependence on temperature. It is known from observation of magnetic powder patterns and other experimental techniques that, in the intermediate state, the specimen splits up into a series of normal and superconductive laminae which stand normal to the axis of the cylinder. Various scattering mechanisms have been assumed to act at the boundaries of these laminae to provide an explanation of the curious thermal conduction in the intermediate state. The situation has now been greatly clarified by a systematic investigation of the alloy series Sn-In.68 The induced magnetic moment and the dependence of the electrical and thermal resistance on the field strength were measured on these speci0
K. Mendelssohn and J. L. Olsen, Proc. Phys. SOC.A M , 2 (1950). K. Mendelssohn and C. A. Shiffman, Proc. Roy. Soc. A266,199 (1960).
THERMAL CONDUCTIVITY OF METALS
mens. A typical set of results for an almost perfect single crystal containing 2.8% indium is given in Fig. 13 for two temperatures. Whereas the fraction of trapped flux is largely temperature-independent, the maximum in the thermal resistance is pronounced a t 2.03"K and practically nonexistent a t 2.65"K. From this it can be deduced that the structure of the intermediate state is much the same at different temperatures and that the relevant scattering mechanisms may be different. More detailed analysis does indeed suggest that the phase boundaries scatter both electrons and phonons but that the temperature dependence is different for the two mechanisms.
.-0- 1.0 d zV
u V 0 c
FIQ.13. Magnetic moment M,electrical resistance ratio p/po, and thermal resistivity ratio W of a tin cylinder with 2.8% indium during transveme destruction of superconductivity by a magnetic field. (a) 2.03"K transverse field; (b) 2.65'K transverse field.
K. MENDELSSOHN AND H. M. ROSENBERQ
111. The Use of Heat Conduction Data in Research on Metal Imperfections
The study of imperfections in metals has achieved very great importance today. In developing theories of work hardening, information concerning the multiplication of dislocations and point defects during deformation is necessary in order that mechanisms of plastic deformation can be formulated. In studies of irradiation damage it is obviously essential that we should know as-much as possible about the type of damage that is introduced. Both of these fields can be investigated with the aid of measurements of electrical resistivity because the defects produce an increase in resistivity which can be studied during deformation or irradiation. Many experiments have been made on these lines with particular emphasis on the way in which the resistivity anneals as the specimen temperature is raised. The disadvantage of such measurements is that, for a certain change in resistivity, it is very difficult to separate out the contributions which arise from specific types of damage. In particular there is considerable uncertainty in separating the effects of dislocations from those of point defects. Even though the resistivity arising from each type of scattering is essentially temperatureindependent and even though various annealing effects occur which can be ascribed to certain defects from knowledge of the temperature a t which the annealing takes place, it is very difficult to analyze the results in practice. Even if a satisfactory separation of the resistivity into various components can be made, there is still the problem of calculating the actual number of defects which are present, i.e., we need to know the resistivity associated with a single dislocation or a point defect. The resistivity of point defects is fairly well established in certain cases; however, the resistivity of dislocations is still not known with any confidence because of uncertainties concerning the electron configuration a t the core of the dislocation (Section 2). To overcome the disadvantages of electrical measurements, some heat conduction experiments have been made as part of the study of defects in metals. We should make it clear at the outset that the experiments have not been concerned with the electronic thermal conductivity K,, which will of course follow the behavior of the electrical resistivity and hence will be subject t o all the uncertainties of interpretation which have already been discussed. Rather we should expect measurements of the lattice conductivity K , to provide additional information. There are two reasons for this. First, as has been shown in Sections 2 and 4,the thermal resistance of dislocations has a different temperature dependence (T-2) from that of point
THERMAL CONDUCTIVITY O F METALS
defects (T). This means that at low temperatures only dislocations coiitribUte to the extra thermal resistance. Second the scattering of phonons by defects can be calculated with far more confidence. The detailed configuration, particularly a t the core of a dislocation, is not required since a t low temperatures only long phonon wavelengths are involved. The problem which remains is to measure the lattice conductivity of a metal. There are two possible methods. Either the conductivity of an alloy can be measured and K , can then be estimated as in Section 5 , or the conductivity of a pure metal in the superconducting state can be determined a t very low temperatures. As has been described in Section 6 , K , is then due entirely to phonon conduction. We shall deal separately with experiments which have relied on each of these methods. 11. THEDETECTION OF DISLOCATIONS IN DEFORMED ALLOYS
The determination of K , from the total conductivity haa been dealt with in Sections 5 and 8. It was shown that at low temperatures K , is proportional to T2and that this dependence is produced by the scattering of phonons by both electrons and by dislocations [Eqs. (4.1) and (4.4)]. Some early experiments were made on a copper-nickel 69 alloy containing 10% Ni and on silver-cadmium and silver-palladium al10ys.~OThese showed that K , is proportional to T2a t low temperatures both when the specimens have been colddrawn and after they have been annealed, although in the latter case the absolute value of K , is appreciably higher. The decrease in conductivity associated with the colddrawing was ascribed 'O to dislocations introduced during the deformation. Although the number of dislocations as calculated from the change in conductivity was higher than would have been expected, there seemed to be no doubt that the change in K , was indeed the result of dislocations. Similar experiments on copper-zinc alloys 71 bore this out, although once again the dislocation density waa apparently rather high. A later series of experiments was made on copper-zinc alloys.ll Both single and polycrystals containing 7, 15, and 30% zinc were extended in stages by small amounts, the heat conduction in the range 2 to 4.2"K being measured after each successive deformation. After the specimens broke they were deformed further by drawing down one of the broken halves through dies. The conductivity was measured again a t various stages. K , was found to decrease after each deformation although it remained proportional to T2. Figure 14 shows how the conductivity changes. It shows plots of K I T 70
I. Estermann and J. E. Zimmerman, J . Appl. Phys. 28, 578 (1952). W. R. G. Kemp, P. G . Klemens, A. K. Sreedhar, and G . K. White, Proc. Roy. SOC. A2!?3,480 (1966). W.R.G.Kemp, P. G. Klemens, R. J. Tainah, and G . K. White, A& Met. 6,303 (1957).
K. MENDELSSOHN AND H. M. ROBENBERG
FIG 14. Graphs of K/T against T for Cu-7% Zn showing the marked decrease in slope (i.e., the decrease in lattice conductivity K I ) when an annealed specimen is progressively strained. [J. N. Lomer and H. M. Rosenberg, Phil. Mag. [S]4,467 (1959).]
against T (the slope of which should provide a measure of K , [Eq. (5.1)]) for an annealed specimen and after successive deformations. The decrease in slope (i.e., of K,) is quite marked. The dislocation density was calculated from these changes in slope with the aid of Eq. (4.5). The results of these experiments are shown in Figs. 15 and 16. Several interesting facts emerge. For small to moderate extensions (Fig. 15) the dislocation density is independent of the zinc content, but it rises rapidly for polycrystals and very much less rapidly for single crystals oriented for single slip. For larger extensions (Fig. 16) the dislocations stop multiplying a t a value which is very strongly dependent on the zinc content. These results are in good qualitative agreement with modern theories of work-hardening. The initial slow rise in the dislocation density in single crystals corresponds to the so-called stage 1 of the stress-strain curve, where dislocations can move large distances with little obstruction. The rapid rise in the polycrystals corresponds to the linear section (stage 2) of the stress-strain curve where a large amount of dislocation multiplication is expected. The saturation region corresponds to the place where dislocations can cross slip on to neighboring planes in order to avoid obstacles (stage 3) and little extra multiplication is necessary for further deformation. The energy for cross slip is very dependent on impurity concentration; this is borne out by these heat conduction measurements. A more detailed discussion of the results would be out of place in this paper but sufficient has been given to show that essential information about dislocation multiplication can be obtained from such experiments.
THERMAL CONDUCTIVITY OF METALS
0 30% ZINC X
I 5 % ZINC polycrystols
FIG.15. Plots of dislocation density N against strain, for small and moderate strains, aa derived from heat conduction measurements on copper-zinc alloys. The upper and lower curve are for poly- and single crystah respectively. [J.N. Lomer and H. M. Rosenberg, Phil. Mag.[Sl 4,467 (19591.1
FIG.16. Plots of dislocation density N against strain for large strain aa derived from heat conductivity measurements on copper-zinc alloys. The saturation of the value of N coincides with the onset of stage 3 in the strw-atrain CUNW. [J. N. Lomer and H. M. Rosenberg, Phil. Mag. [81 4,467 (19591.1
K. MENDELSSOHN AND H. M. ROSENBERG
Some doubts had been raised in earlier work about the absolute magnitude of the dislocation density calculated from K , since the values seemed to be too high when compared with, say, dislocation densities derived from the interpretation of X-ray data. Some experiments have been made to check the situation l1 by an independent counting method. The heat conduction of two thin brass strips was measured before and after deformation and the density of the dislocations thereby introduced was calculated by the decrease in K,. The strips were then electro polished until very thin and they were then examined by transmission electron micr0scopy.7~This technique shows the dislocations directly so that they can be counted. It was found that Eq. (4.5) overestimated the dislocation density by a factor of about six. For this reason the ordinates in Figs. 15 and 16 have been reduced by this factor so that they are more nearly in accord with the actual number of dislocations which are present. This does not mean of course that the numerical values are accurate, because the electron microscope technique samples only a very small portion of the specimen which might not be typical of the whole. Other independent experiments, however, also suggest that Eq. (4.5)overestimates the dislocation density. For example the change in density has been used to estimate the number of dislocations in a specimen after deformation and this has been compared with the change in K,.7aThe dislocation density has also been determined by etch pit techniques on deformed specimens of lithium fluoride and this has been compared with the density calculated from heat conduction data.s1In both these investigations it was found that Eq. (4.5)gave too high a dislocation density. It is not clear theoretically why this should be so. Nevertheless even though there might still be some question about the accuracy of the absolute values obtained, there is no doubt that the method based on heat conduction is very reliable in determining relative changes in dislocation densities. In experiments of the type just described, the value of K , for the annealed specimens is assumed to be that which is limited by phonon-electron scattering. In some alloys, however, K , has what is apparently too low a value 74-76 even in annealed specimens. It has been suggested 21 that this might be a consequence of the fact that considerable numbers of dislocations are anchored by the impurity atoms in such alloys and that these dislocations reduce K , to the observed values. Since, however, the numbers of dislocations which would be required to account for this are about 10% 72 P. B. Hirsch, R. W. Horne, and M. J. Whelm, Phil. Mug.  1, 677 (1956). W. R. G. Kemp, P. G. Klemens, and R. J. Tainsh, Proc. Kamerlingh Onnea Conf., Physicu 24, Suppl., S170 (1958). 7' W. R. G. Kemp, P. G. Klemem, and R. J. Tainsh, Austrulian J . Phys. 10,454 (1957). 76 J. A. Birch, W. R. G. Kemp, and P. G. Klemens, PTOC. Phys. SOC.71,843 (1958). g1 R. L. Sproull, M. Moss, and H. Weinstock, J . A p p l . Phys. 30, 334 (1959). 7*
THERMAL CONDUCTIVITY O F METALS
of the iiumber introduced by heavy deformation, and traiisrnission electron microscopy experiments show no evidence for their presence, it is felt that there must be some other explanation for the low values of K,. Although most of the work on deformed alloys has been concerned with the detection of dislocations from the measurement of K , a t helium temperatures, some experiments have also been made at higher temperatures 71,73 where the presence of point defects will influence K,. Some evidence has been put forward on the basis of such experiments which indicates the production of vacancy clusters when deformed copper-zinc alloys are annealed. In view of the uncertainties involved in estimating K , at higher temperatures (Section 5 ) one cannot place as much reliability on these experiments as on those which are concerned with dislocations. Another investigation has been concerned with the multiplication of dislocations as a copper-zinc alloy is fatigued. Heat conduction experiments showed l1 that the dislocations multiply very rapidly during the first 200 cycles of fatigue; thereafter their number remains practically constant. 12. THEDETECTION OF DEFECTSIN SUPERCONDUCTORS
The predominance of phonon conduction in K , at low temperatures was demonstrated in Section 9 by its sensitivity to strain and its independence on point imperfections. It is clear that this feature can be turned to advantage to provide a distinction between these two different types of lattice faults. Investigations of this nature have the added advantage that, in the same specimen, the scattering of phonons and (by the simple application of a magnetic field) that of electrons can be studied without even warming the sample to room temperature. In most cases the magnetic fields required to destroy superconductivity are less than lo00 oersteds. The accompanying magneto-resistive effects occurring in the normal state are small. Those which exist can usually be taken into account by carrying out comparison experiments above T,. Systematic experiments on the effect of strain and impurity were made . ~ ~clearest results were obtained by by Montgomery 62 and by R ~ w e l lThe the second author who subjected pure lead and a lead alloy to controlled bending a t helium temperatures, and measured the heat conductivities in the normal and in the superconductive state before and after the introduction of strain. Another feature of this work is a study of annealing effects at different temperatures between that of liquid helium and room temperature. Order of magnitude agreement was obtaitied between the density of dislocations derived from the measurements of heat conductivity and 76P.
M. Rowell, Proc. Roy. Soe. A164, 542 (1960).
K. m N D E L S S O H N AND H. Y. ROSENBERG
those predicted on the basis of the strain introduced. More spectacular than the results on lead are those obtained on a niobium rod which WM originally in single crystal form and which was subsequently stretched in steps until it ruptured. Except for the state of highest strain, there is no sign of any effect on Kn. Even after fracture K , changed only by a few per cent. On the other hand, the effect on K , is far-reaching. In its undisturbed condition, the niobium sample showed a very pronounced maximum in K , a t temperatures below 0.4TC.With successive stretching, this maximum was largely removed. STRAIN
Fro. 17. Diagram showing the effect of impurity and strain on the thermal conductivity in the normal and superconductive states.
Applying these results to specimens of other transition metals, particularly to vanadium, it has been possible to use the data obtained on the heat conductivity in the superconductive and normal states of the metal for guiding the manufacturing process of subsequent specimens. The behavior of the transition metals can indeed serve as a simple model to explain qualitatively the way in which a distinction between different types of lattice faults can be obtained with this method. The effect of (a) point imperfections and (b) larger scale defects is shown diagrammatically in Fig. 17. Point imperfections, which scatter electrons, will reduce both the conductivity in the normal state and K , a t high reduced temperatures where the heat transport is still by electrons. The maximum in K , a t low temperatures due t o phonon conduction is not affected. Since K n is drastically reduced, K , may now exceed K n in this temperature region. At still lower temperatures this relation is again reversed and K J K n becomes smaller than unity. Extended lattice faults, such as dislocations, introduced by strain appear to have little or no effect on the electronic part of the conduction mechanism; therefore Kn as well as the high-temperature part of
THERMAL CONDUCTIVITY O F METALS
K , will not change materially. In the low-temperature region, on the other hand, where heat is carried by phonons, K , is decreased. The phonons are scattered by the dislocations; with increasing strain the ratio Ka/Kn becomes progressively smaller. Another type of damage to metals which can be investigated with this
K (watt cm-' deg-' 1
FIQ.18. Thermal conductivity of a niobium single crystal in the normal and superconductive statea before (-) and after (- - -) neutron irradiation.
method is that introduced by radiation in a nuclear reactor. This type of damage is by no means fully understood as yet and a method which is capable of separating small- and large-scale lattice disturbances is useful in elucidating the processes involved. The fist set of results concerning the application of low-temperature heat conduction to radiation damage has now been obtained on niobium.I1 The thermal conductivity of a single crystal rod of this metal was measured first in the undamaged state and again after it had been subjected to neutron irradiation at room temperature. The result is given in Fig. 18. Both Kn and K , are reduced by irradiation. ASK. D. Chaudhuri, K. Mendelssohn, and M. W. Thompson, Cryogenics, 1, 47 (1960).
K. MENDELSSOHN AND H. M. ROSENBERG
suming that both interstitials and vacancies have been produced by irradiation, we have reason to believe that, although the former may have migrated a t the temperature of irradiation, the temperature was never high enough to cause migration of vacancies in niobium. It is thus tempting to regard the decrease in K , as due to the vacancies produced directly by irradiation and to assume that a more complicated process is responsible for the drop in K,. From other evidence it has been postulated that, during migration, the interstitials tend to form small dislocation loops or to condense on existing dislocations causing jogs to appear on originally straight lines. Making the reasonable assumption that about 10% of the interstitials caused by irradiation condense in this way and that the loops or jogs are 100 A in linear dimensions, an increase of the order of log dislocations per cm2 was to be expected for the irradiated niobium sample. Using Klemens’ formula (4.5) to determine the change in K with the number of dislocation lines, analysis of the observed change in K . yields 3 X lo9 lines per cm2, which is in surprisingly good agreement. It is evident that further work alone can show whether the agreement in this case is fortuitous or not. However, the method seems to show good prospects for the investigation of irradiation damage. IV. Experimental Techniques
Nearly all workers in the field have used a miniature version of the classic Searle’s bar experiment which is frequently used as a class demonstration. The specimen, in the form of a rod, usually between 5 and 10 cm long, and a few mm in diameter is mounted in a high vacuum with one end in good thermal contact with a heat sink-usually a bath of liquid helium or a paramagnetic salt. A heater is fixed a t the other end of the specimen. This consists of a few hundred ohms of fine resistance wire. In some cases it is wound and cemented directly on to the specimen whereas in others it is mounted on a former and this is soldered, if possible, or clamped to the end of the specimen. Current and potential leads are attached to each end of the heater in order to measure Q the power generated. Two thermometers, a few cm apart, are fixed to the specimen, so that the temperature gradient 6T/6x may be measured. Then the thermal conductivity K = Q 6x/(A 6T), where A is the cross-sectional area of the specimen. 14. THERMOMETRY
The choice of a thermometer depends on the temperature range to be covered and the idiosyncrasies of the experimenter. Three main types have
THERMAL CONDUCTIVITY O F METALS
been used : helium gas thermometers, carbon resistance thermometers, and thermocouples.
a. Helium Gas Themmeters Helium gas thermometers may be used a t all temperatures down to 2°K. They are usually in the form of small copper cylinders with a volume
FIG. 19. A typical arrangement for the measurement of temperature difference for heat conduction work. A gas thermometer is connected to each of the two leftrhand l i i b s and the temperature difference is indicated by a difference in the oil levels. The absolute pressure is measured by the height of the rightrhand limb which is kept continuously pumped. The bellows is used to adjust the levels so as to achieve constant volume conditions. [H. M. Rosenberg, Phil. Trans. Roy. Soc. A247, 441 (1955).]
of a few cc. They are filled with helium gas to a pressure of, say, 5 cm of mercury a t 4.2"K,via thin-walled (0.1-mm) German silver, Inconel, or stainless steel capillaries which have a diameter of 0.5 to 1 mm. The temperature difference between the two thermometers gives rise to a corresponding pressure difference which can be measured quite accurately by connecting the two capillaries to either side of a U tube containing oil. The absolute pressure in each thermometer can be measured by connecting a third vertical tube, which is kept pumped, to the bottom of the U tube. The difference in levels between the meniscus in each arm of the U tube and that in the third tube gives the pressure in each thermometer. Some type of bellows device is connected to the system so that by adjusting the oil levels, the thermometers may be used under constant volume conditions. A typical arrangement is shown in Fig. 19. Helium gas thermometers have the advantage that, once the dead vol-
K. MENDELSBOHN AND H. M. ROSENBERG
ume correction has been calculated, they need only one calibration point. If po and TOare the calibration pressure and temperature respectively, the temperature T corresponding to a pressure p is given by:
T = PTo/(Po The correction term F is given by
(14.2) E(To/T')(Po - PI where B is the ratio of the external volume of the thermometer system to the internal volume, and T' is the mean temperature of the external volume. It can be seen that F is usually quite small, for T' is approximately room temperature whereas To is usually 4.2 or 20°K and E can be made considerably less than unity. The value of E may be found by measuring the thermometer pressures at two known temperatures, e.g., in baths of liquid hydrogen and helium boiling under atmospheric pressure. When a thermal conductivity measurement is being taken, the mean pressure p , of the two thermometers is used to calculate the mean temperature T , of the specimen by using Eq. (14.1). If 6p is the pressure difference between the two thermometers, the temperature difference 6T is given by
6T = T , 6p/p.
Corrections for nonideality of the gas are very small. They need only be taken into account below about 2.5"K, and even then only for precision work. Gas thermometers are unaffected by magnetic fields, and may be used up t o 90°K and above. The main disadvantage in the liquid helium region is that they cannot be employed below 2°K because the filling pressures used are too small for readings of pressure difference to be made with any accuracy as a consequence of the low vapor pressure of helium. They tend to be rather sluggish in reaching equilibrium, although this is much more noticeable at 20°K and above where of course all equilibrium times start to increase. Further details concerning the use of the thermometers are given by Rosenberg.'s
b. Carbon Resistance Therrnmters Carbon resistance thermometers are widely used, particularly if measurements are to be taken down to 1°K and below. There are two main types. Ordinary composition.radio resistors which have a room-temperature resistivity of about 30 to 100 ohms, have a resistivity in the liquid helium range of a few thousand ohms. If a batch of about a dozen resistors is tested, several will be found to have satisfactory characteristics in the liquid helium range. When such resistors are being selected they should be tested throughout the temperature range to be covered. Dipping them into liquid
THERMAL CONDUCTIVITY OF METALS
helium at 4.2"K does not always provide a satisfactory test as it does not necessarily give a good indication of the resistivity at lower temperatures. The resistors from some manufacturers are found to be much more suitable than those from others. Another type of carbon resistance thermometer is home-made and consists of a layer of carbon paint, either applied directly [e.g., see Fig. 22, inset 11 to the specimen or on to a small former. These paint thermometers have a very short equilibrium time and they have been found to be particularly useful a t temperatures below 1°K, although even some radio resistors are quite satisfactory down to 0.1"K. The main disadvantage of carbon thermometers is that they need to be calibrated carefully. This must be done throughout the temperature range, either against the helium vapor pressure, or, below 1"K, against the susceptibility of a paramagnetic salt of known characteristics. They may be calibrated against the vapor pressure of liquid He3 down to 0.3"K. The calibration usually changes slightly when they have been warmed up to room temperature and then recooled, although their reproducibility can be improved if no water vapor is allowed to condense on them when they are warming up. Fields of several thousand gauss do not affect the calibration although Douthett and Friedberg 78 had t o take the magnetoresistance into account at 10 kgauss. Because it is practically impossible to obtain two carbon thermometers which are perfectly matched, an accurate determination of the temperature difference between two thermometers presents some difEculties. One satisfactory method has been used by Montgomery 78 in the range 1 to 4.2"K. He first selected two thermometers whose resistances were as nearly as possible the same over this temperature range and calibrated each of them against the helium vapor pressure. He found that the variation of the resistance R of each thermometer with the temperature was of the form:
- Ro) = C
where Ro was about 25 ohms. C was approximately proportional to the temperature between 2.5 and 4°K. A simplified diagram of the circuit used is shown in Fig. 20. It is really a double Wheatstone bridge. The two thermometers are Rh at the free (heated) end of the specimen and R, at the fixed (cold) end. The procedure was first to use the top two arms of the bridge and to balance R, and X , against the two 1OOO-ohm resistors by adjusting X,. The two 1000-ohm resistors were then disconnected from the circuit and the lowest arm conD. Douthett and s. A. Friedberg, Camegie Inst. Technol. ONR Research Contract NONR-760(05), Tech. Rept. No. 3 (1958). 79 H. Montgomery, D.Phi1. The&, Oxford University (1956): Proc. Rov. Soc. A244. 78
K. MENDELSSOHN AND H. M. ROSENBERG
R h and x h was balanced x h
against R, and X,, by adjusting x h . Then
= X c(
W R J*
This method tends to counteract any inaccuracies which might arise as a result of a slight temperature change of the specimen during the measurements. If such a change did occur, R h and R, would change by about the same amount and their ratio would remain approximately constant. Thus X h is the value which R h had when R, was being measured.
FIO.20. A simplified diagram of the circuit used by Montgomery [D. Phil. Thesis, Oxford University (1956); Proc. Roy. Soc. A244, 85 (1968)l to measure temperature difference with radio resistance thermometers.
Using this circuit, Montgomery calibrated each thermometer against the helium vapor pressure. He then plotted three graphs: (a) (b) (c)
against T [C as defined by Eq. (14.4)] against C h ( c h - C,) a t constant temperature against Cc
where c h and C , are defined by Eq. (14.4). For a thermal conductivity measurement, current was passed through the specimen heater and C, and c h were determined using the bridge circuit already described. With the aid of graph (c) the quantity AC given by
- Cc) - ( c h
- Cc)oonat temp
was calculated. The mean temperature T,,, of the specimen then corresponds to a value of C, say C,,,,where
From this value of C, T,,, may be found with the aid of graph (a). We may then find dT/dC,,, appropriate to C,,, with the use of graph (b) and hence
THERMAL CONDUCTIVITY OF METATAS
6T = (dT/dC)c, AC.
With this method 6T could be determined to degree or less and T,,, to about degree. Graphs (a) and (c) can be drawn directly from the bridge readings, but graph (b) is more difficult to determine accurately. Montgomery fitted an empirical function to C and then differentiated it. Graphs (a) and (c) were redrawn for each run. As soon as graph (a) had changed by 1%, graph (b) was redrawn.
FIQ.21. The circuit used by J. L. Olsen and C. A. Renton [Phil. Mag.  43, 946 (1952)l and by K.Mendelssohn and C. A. Renton [Proc. Roy. Soc. A230, 157 (1955)l for the measurement of temperature difference below 1°K using carbon paint thermometers.
The electrical circuit which actually was used compensated for the resistance of the leads. It also took into account the fact that Ro was 25 ohms, so that the resistances X could be made to give the values of C directly. This is by no means the only method of determining the temperature difference when carbon resistance thermometers are used. In their experiments below 1"K, Olsen and Renton *O and Mendelssohn and Renton 6o used an off-balance bridge method in which the off-balance potential gave a direct indication of the temperature difference. This method, although perhaps not as accurate as that used by Montgomery, enables the temperature gradient to be calculated with far less labor. A simplified circuit is shown in Fig. 21. The carbon paint thermometer a t the cold end of the specimen is connected between A and B and that a t the heated end between A and D. BC is made 200 ohm. Just before demagnetino J. L. Olsen and C. A. Renton, Phil. Mag.  43,946 (1952).
K. MENDELSSOHN AND H. M. ROSENBERQ
zation the bridge is balanced by adjusting CD. If the temperature dependence of both thermometers is the same, the galvanometer deflection A0 should remain zero after demagnetization. In practice this does not occur and readings of A0 and of VBC,the potential across BC, are taken as a function of magnetic temperature. When heat is supplied to one end of the specimen for a conductivity measurement, A changes to AH. In these experiments it was found that VBCis approximately proportional to T to within the limits of experimental error and also that dAo/dT << dVBc/dT. These results enable one to express the temperature difference 6T in the simple form 6T = (A0 - A H ) dvBc/dT. (14.9)
An ac bridge circuit has been used by Fairbank and Wilks 81 and some workers (e.g., Douthett and Friedberg 78) have measured the resistance of their thermometers with a potentiometer. Although carbon resistors are most useful below 4.2"K, they can be used above this temperature. However, most of them are found to be too insensitive for the accurate determination of temperature differences above about 20°K. Other semiconductors besides carbon may be used as thermometers. Germanium possessing small amounts of added impurity has a temperaturedependent resistivity. Specimens can be used as thermometers which are very stable in their characteristics.e86 Since they can be made from single crystals, their heat conduction (and hence their thermal diffusivity) is high and so they come t o equilibrium very quickly. Unfortunately the amount of impurity which is required can be critical. Thus very few workers have used them because of the difficulty in obtaining suitable samples. Although most workers have used carbon resistors for work below 1"K, in some early experiments 87.88 the ends of the specimen were fixed into pills of paramagnetic salt and the temperature difference across the whole specimen was determined by susceptibility measurements on the pills at either end. This method can give rise to considerable inaccuracies, in view of the unknown thermal resistance between the salt pills and the specimen. H. Fairbank and J. Wilks, Proc. Roy. Soc. A231,545 (1955).The theory of this bridge is given by L. J. Challis, D.Phi1. The&, Oxford University (1957). 8 ) T. H. Geballe, F. J. Morin, and J. P. Maita, Coqf. Phys. des Basses Temps., Pan's p. 425 *I
J. E. Kunzler, T. H. Geballe, and G. W. Hull, Rev. Sn'. Inatr. 26,96 (1957).
I. Estermann, Phys. Rev. 78, 83 (1950). A. Friedberg, Phys. Rm. 82, 764 (1951). 6 0 1 . Estermann, 5. A. Friedberg, and J. E. Goldman, Phys. Rev. 87, 582 (1952). 87 J. G. Daunt and C. V. Heer, Phys. Rev. 76,854 (1949). ' 8 B. B. Goodman, Proc. Phya. Soc. A86,217 (1953). 84
THERMAL CONDUCTIVITY OF METALS
In principle, a thermocouple is the ideal way of measuring the temperature difference between two points. For most pairs of metals, however, the thermal emf is so small a t low temperatures that only a few workers have attempted to use them. Geballe and Hull 88 have used copper/constantan junctions down to about 20°K and Powell et al?O have used gold-cobalt/copper junctions a t 4.2"K and above. These are much more sensitive than copper/constantan thermocouples a t liquid helium temperatures. 15. TYPESOF CRYOSTAT Among the most interesting features of the thermal conductivity are its temperature dependence and the way in which this dependence changes in different temperature ranges. Since the interesting range can vary widely, depending on the material, several different techniques have had to be adopted . a. Below 1 "K
For measurements below 1°K the usual methods of adiabatic demagnetization are used. The specimen is thermally anchored to a copper block which is itself connected to copper vanes which are pressed into a paramagnetic salt. The salt is demagnetized and measurements are taken a t appropriate temperatures during the warming-up period. The choice of paramagnetic salt is determined by the temperature range to be studied, for the salt should have a specific heat anomaly in this range in order that the warm-up time will be sufficiently slow for satisfactory measurements to be taken. I n addition a salt should be chosen whose magnetic properties have been thoroughly investigated because the susceptibility of the salt is used to calibrate the thermometers. For measurements down to about 0.2"K Mendelssohn and Renton 6o used manganese ammonium sulfate. If experiments are to be made on superconductors, it is important that the specimen be shielded from the magnetic field which is applied to the paramagnetic salt. Otherwise some magnetic flux might be trapped in the specimen on demagnetization and misleading results will be obtained. A typical experimental arrangement for a superconducting specimen is shown in Fig. 22. A helium-3 cryostat 66 may also be used for measurements down to -0.3"K. T. H. Geballe and G . W. Hull, Phys. Rev. 110, 773 (1958); Conf. Phys. dea Basses Temps., Paris p. 460 (1955). W R. L. Powell, W. M. Rogers, and D. 0. coffin, J . Resmrch Natl. Bur. Shndurda U.S. 69,349 (1957); R. L. Powell,H. M. Roder, and W. M. Rogers, J . App2. Phya. 28,1282
K. MENDELSSOHN AND H. M. ROSENBERO
PIQ.22. AII apparatus used for the measurement of the heat conductivity of superconductors below 1°K [K. Mendelssohn and C. A. Renton, Proc. Roy. Soc. Aa30, 157 (1955)l. Note the magnetic shielding,S, of the specimen J , which is necemary to prevent it from entering the normal state during the process of magnetic cooling. It is mounted 13 cm away from the paramagnetic salt H a t the end of a high conductivity copper rod I . When a field of 5 koersteds was applied to H , the field a t J was not greater than 10 oersteds. A and C are liquid helium baths a t 4.2 and 1°K respectively. Inset (1) shows how the carbon thermometers were made by painting “Dag” colloid over copper wires which were wound around the specimen. Inset (2) shows the detail of the specimen mounting.
b. 1 to 4.2”K
Readings between 1 and 42°K are made in the usual way, by pumping the liquid helium bath to the appropriate pressure. As has already been remarked, gas thermometers can only be used in this range down to 2°K. Below this temperature carbon resistance thermometers must be used.
Above 4 . P K The design of a cryostat for this range is dictated much more by local conditions and tradition than is true for measurements below 4.2%. If no liquid hydrogen is available, readings above 42°K must be made by attaching an auxiliary heater to the fixed end of the specimen and adjusting the current through the heater until the specimen is a t the temperature at c.
THERMAL CONDUCTIVITY O F METALS
which a measurement is required. This method can also be used to bridge the gap between 4.2 and 10°K which occurs even when liquid hydrogen is available since 10°K is about the lowest temperature to which hydrogen can be pumped. An auxiliary heater must also be used above 20°K until temperatures which are attainable with liquid nitrogen or oxygen are reached. Powell et d.B0have obtained readings above 4.2"K by using a very long specimen with several thermocouples spaced along it. The current input to the heater was adjusted so that several readings a t different temperatures could be taken at the same time. Whereas this method does have the disadvantage that different parts of the specimen are being used for the various experimental points, it is a useful method of making measurements above 4.2"K providing that homogeneity of the specimen can be ensured. In addition to the methods which have just been described, we should emphasize the usefulness of the Simon expansion helium liquefier for work in which measurements are required above 4.2"K. The whole temperature range from 90 to 2°K can be covered in the liquefaction process and measurements in what are sometimes considered to be difficult temperature regions are very much simplified. The main requirement is a supply of liquid hydrogen. Readings in the range from 90 to 20°K can be taken when the apparatus is being cooled with liquid hydrogen. The helium exchange gas can be pumped out a t any intermediate temperature. Providing the highpressure chamber of the liquefier is filled with helium gas a t about 120 atmospheres, this gas will act as a sufficient thermal anchor that the temperature will remain practically constant during the time necessary for a reading to be taken. Measurements between 20 and 10°K may be taken by interrupting the pumping of the liquid hydrogen bath a t any point. The final set of readings down to 4.2"K may be taken during the expansion of the high-pressure helium gas. In the normal liquefaction procedure, the helium is expanded from 120 atmospheres t o 1 atmosphere over a period of a few minutes. If however the expansion is stopped a t intermediate pressures, temperatures between 10 and 4.2"K can be achieved. They can be maintained constant by controlling the flow of gas so that a very slow expansion takes place which just counteracts the warming-up which would otherwise occur.
d . Other Experimentul Points The specimen is usually mounted vertically, because this usually enables an apparatus of small diameter to be used, which is often an over-riding requirement in the design of a cryostat and Dewar assembly. In some cases, however, a horizontal mounting is more useful, particularly if measurements are to be made in both transverse and longitudinal fields, as is often the case with superconductors. When the specimen is mounted horizontally,
K. MENDELSSOHN AND H. M. ROSENBERQ
both field directions can be produced with the same horse-shoe type of magnet by rotating it through 90" about the vertical axis of the apparatus. If the specimen is vertical, a solenoid must be used for the longitudinal field and a horse-shoe magnet for the transverse field. In most experimental arrangements, one end of the specimen, which is kept permanently in a vacuum, is in contact with the heat sink. The thermometers, which are attached to the specimen, are calibrated against the temperature of the heat sink and one assumes that they are a t the same temperature as the sink. This is, of course, accurate only if there is no heat input to the specimen along the leads and capillaries. While this is usually very small it can cause an appreciable error in calibration (a) if the specimen has a low conductivity, or (b) if there is a bad thermal contact to the heat sink, as sometimes occurs if the specimen cannot be soldered. In either of these circumstances, it is advisable to design the apparatus so that helium exchange gas can be admitted to the experimental chamber while the thermometers are being calibrated, in order to avoid temperature gradients. Care must be taken to ensure that the chamber is properly evacuated before taking a heat conductivity measurement. e. Heat Losses by Conduction and Radiation
The importance of any inaccuracy arising from radiation or conduction depends to a considerable extent on the absolute value of the thermal conductivity of the specimen. Most workers tend to ignore these losses at temperatures below 90°K. This is quite justifiable in the case of radiation losses, although they might introduce an error of one or two per cent at 90°K) particularly if gas thermometers, which have a relatively large surface area, are used. Conduction through the residual gas in the experimental chamber can be appreciable, unless care is taken. It is not possible to give specific recmmendations, because so much depends on the details of the experiment, but the following points should be watched. (a) Any slight deterioration in the vacuum at higher temperatures (above liquid hydrogen temperature) should be noted and cured. (b) If helium exchange gas has been introduced into the experimental chamber at liquid helium temperatures, the chamber should be pumped for a considerable time (of the order of half an hour) before measurements are taken. (c) While very small leaks can usually be tolerated at liquid helium temperatures, because the conductivity of the gas is so low compared with that of the specimen, with some materials for which the heat conductivity can be very small at liquid helium temperatures watt cm-' deg-' or less) it is essential that a good vacuum be maintained (i.e.10" to lo-' mm Hg).