JOURNAL OF RESE ARC H of th e National Bureau of Standards - A. Physics and Chemistry Val. 74A, No.3, May-June 1970 Density of States I nformation from Low Temperature Specific Heat Measurements* Paul A. Beck and Helmut Claus University of Illinois, Urbana (October 10, 1969) The c a lcul ati on of one -electron d ensit y of s tate va lues from the coeffi cient y of the te rm of the low te mperature specifi c heat lin ear in te mperature is compli cated by many- body effects. In parti c ul ar, the electron-p honon inte raction may enhance the measured y as muc h as tw ofo ld. The e nha nce me nt fa ctor can be eva luat ed in the case of supe rconducting metals and a ll oys. In the presence of magneti c mo­ ments, add it ional complicati ons arise. A magneti c contribution to the measured y was ide ntifi e d in the case of dilute all oys and a lso of concentrated a lJ oys wh e re parasiti c antife rromagnetis m is s upe rim­ posed on a n over-a ll fe rromagneti c orde r. No me thod has as ye t bee n de vised to e valu ate this magne ti c part of y. T he separati on of the te mpera ture- li near term of the s pec ifi c heat may itself be co mpli cated by the a ppearance of a s pecific heat a no ma ly due to magneti c cluste rs in s upe rpa ramagneti c or we ak ly ferromagneti c a ll oys. Key words: All oys; de nsity of states; low te mperature specifi c heat; magnetic s pecific heat; many­ body effects; superconductivity. 1. Introduction factor determined by th e uni ts used for y, N(Ep) and k. Unfortunately, in a very large majority of cases, the In the Sommerfeld-Bethe theor y of metals the elec­ simple procedure just described cannot be used , or at tronic specific heat at low temperatures is linear in tem­ least it does not give re liable results. Many-body effects perature in first order approximation. The lattice and, in some alloys, magne ti c effects may make the specific heat in the low temperature approximation .is determination of N(Ev} from low te mperature specific proportional to T3 so that, in the absence of other con­ heat data more complicated than implied by eqs (1) and tributions, the total specific heat (2), or even impossible at the present state of the art. C=yT+{3T3 (1) 2. Many-body Effects If C is known as a function of T, the two terms can be In recent years it has become known that many-body separated by making use of the linear variation of CIT effects, in particular the electron-phonon interaction, with rz and by extrapolating to T = 0. The intercept of require renormalization of the effective mass of the the extrapolated li ne with the ordinate axis gives the electrons at the Fermi surface. This increases the mea­ temperature coefficient of the electronic s pecific heat sured electronic specific heat coefficient over the one­ y. In the sim plest case, y is proportional to the elec­ electron "band structure" value by the enhancement tronic density of states at the Fermi s urface,N(Ep); factor (l + A). For Na, Al and Pb, it was possible to determine the value of this factor [1J , by comparing the (2) "band structure electronic s pecifi c heat," calculated from the known band structure and the topography of where k is the Boltzman constant and ex is a numerical the Fermi surface, with the measured electronic specific heat. These values: 1.25,1.45 and 2.00, respec­ ,.. An in vit ed pa per presented al the 3rd Materials Resp.a rch Symposium , Electronic Density of Slat.es, November 3·6. 1. 969, Caith ersburg, Md. tively, were found to agree quite well with the e nhance- 449 me nt factors calculated from band structure, Fermi tron "band structure" situation. Hence, the lower, surface topography and phonon dispersion curves, on " band structure density of states" values must be used the basis of the electron-phonon interaction [1] . Unfor­ in determining the band width, for instance, rather than tunately, for most other metals calculations of this sort the density of states enhanced by electron-phonon in­ cannot be made at present since at least some of the teraction, as obtained from low temperature specific required data are not yet available. For superconduct­ heat measurements. Using the electron-phonon ing metals the electron-phonon coupling constant 'A has coupling constant 'A, for instance the values given in been recently calculated by McMillan [2], using the table I, the "band structure density of states" at the following equation which he derived from the strong Fermi level N(Ep) can be calculated from the experi­ coupling theory: mentally determined value of the low temperature specific heat coefficient y' as follows: () [1.04(1 + 'A) ] (3) T C =1.45 exp- 'A-p,*(1+0.62'A) , (4) where Tc is the superconducting transition temperature and () is the Debye temperature. The electron·electron For most of the nonsuperconducting metals and alloys interaction constant p,* was assumed to have a value of the value of 'A is at present unknown and, as a result, 0.13 for all transition metals. The values of 'A calculated the "band structure density of states" cannot be calcu­ by McMillan [2] for superconducting metals are given lated from the low temperature specific heat. in table I. As seen in figure 1, the experimental electronic specific heat coefficient y' for the b.c.c. 3d-transition TABLE I. The Electron·Phonon Interaction Coefficient A and metals and their alloys as a function of electron concen­ " Band Structure" Density of States· N(Erl for Superconducting tration [3] shows prominent maxima and minima in the Metals.' range of eta from 4 to 9. Since in the region of the Tc N(El'l minima and of the second maximum the alloys are not Element e A oK oK 1 eV-latom- superconducting, the "band structure density of states" cannot be calculated at present. Thus, the in­ Be ........... ........... .026 1390 .23 .032 AI.. .... ............. ... LI6 428 .38 .208 teresting question whether the prominent features of Zn ...................... .85 309 .38 .098 these curves are due to changes in the electron-phonon Ga ..................... l.08 325 .40 .091 enhancement factor upon alloying, or indeed these fea­ Cd ...... .. ............. .52 209 .38 .106 tures are characteristic of the electronic band structure In ......... ............. 3.40 ll2 .69 .212 Sn ............. ......... 3.72 200 .60 .238 of the transition metals concerned, cannot be answered Hg ..................... 4.16 72 l.00 .146 TI... .. ................ 2.38 79 .71 .182 Pb ..................... 7.19 105 LI2 .276 40 9 Ti ............... ....... .39 425 .38 .51 V ................... .... 5.30 399 .60 1.31 35 ~ Zr ...................... .55 290 .41 .42 Nb ....... .............. 9.22 277 .82 .91 1;: 30 Mo .................... .92 460 .41 .28 Q Ru ... ............ ...... .49 550 .38 .46 T ~ 25 J Hf .. .................... .09 252 .34 .34 i '"""'" '\ Ta ..................... 4.48 258 .65 .77 1 ;;l 20 W ...... .. .. ..... .... ... .012 390. .28 .15 u Re ............ ......... l.69 415 .46 .33 'Q ~ .5 15 1\ \ Os .. .......... .. ....... .65 500 .39 .35 ..... 'G • Ir ........... ........... , .14 420 .34 .51 \ ..... 1\.. 10 ....,.. ! "- II 5 1\ \ It is now clear that the enhancement factor (1 + 'A) of lICf the electronic specific heat due to the electron-phonon interaction can be as high as 2, or more. This interac­ Ti V Cr (Mn) Ft Co 4 5 6 T 8 9 tion affects only the electrons whose kinetic energy is FIGURE l. Coefficient y' or y" of the low temperature specific heat close to the Fermi energy. The density of states at lower term linear in temperature vs electron concentration e/afor b.c.c. levels, that is for most of the electrons in the metallic alloys of3 d transition metals [3] . Points marked by filled squares band, may be assumed to correspond to the one-elec- represent data for close-packed structures. 450 with certainty. H owever, the work of McMillan [2] al· lows the conclusion that the electron-phonon coupling constant (and, thus, the enhancement factor) de pends ~ :;-1- I 7 primarily o n the phonon frequencies, rather than on th e ~:; ~ electronic properties. Since the elastic constants and, therefore, the phonon frequencies are not known to undergo drastic c hanges with the composition in suc h !,. W1ffiH5115 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 Q T'in DEG' solid solution alloys composed of me tals near one FIGU RE 2. (C-Al/T vs T2 (where C-A is low temperatlire specific heat a nothe r in the same row of the periodic table, it may be less magnetic cluster contribution, see eq (5)) for alloy Nio.48 CUo.52 conclude d with a reasonable degree of probability that cooled without a magnetic field (top graph), cooled in 14 kOeji.eld the prominent features mentioned of the y' vers us ela from 300 to 4.2 K with the field turned off during the measurements c urve of figure 1 are in fact resulting from correspond­ (graph m) andfield-cooled with the field on during m.easurements ing variations in the "band struc ture density of states," (graphm-m) [8]. even tho ugh the relative magnitude of th e various minima and maxima may be appreciably altered by the The occurrence in the same alloys of "exchange gradual c hanges in the coupling constant with composi- anisotropy" a nd of a magneti c contribution to the low ti on.