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The Electron-Phonon Coupling Constant 

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The Electron-Phonon Coupling Constant  The electron-phonon coupling constant Philip B. Allen Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800 March 17, 2000 Tables of values of the electron-phonon coupling constants and are given for selected tr elements and comp ounds. A brief summary of the theory is also given. I. THEORETICAL INTRODUCTION Grimvall [1] has written a review of electron-phonon e ects in metals. Prominent among these e ects is sup ercon- ductivity. The BCS theory gives a relation T exp1=N 0V for the sup erconducting transition temp erature c D T in terms of the Debye temp erature .Values of T are tabulated in various places, the most complete prior to c D c the high T era b eing ref. [2]. The electron-electron interaction V consists of the attractive electron-phonon-induced c interaction minus the repulsive Coulombinteraction. The notation is used = N 0V 1 eph and the Coulomb repulsion N 0V is called , so that N 0V = , where is a \renormalized" Coulomb c repulsion, reduced in value from to =[1 + ln! =! ]. This suppression of the Coulomb repulsion is a result P D of the fact that the electron-phonon attraction is retarded in time by an amountt 1=! whereas the repulsive D screened Coulombinteraction is retarded byamuch smaller time, t 1=! where ! is the electronic plasma P P frequency. Therefore, is b ounded ab oveby1= ln! =! which for conventional metals should b e 0:2. P D Values of are known to range from 0:10 to 2:0. The same coupling constant app ears in several other physical quantities, such as the electronic sp eci c heat C T whichatlow temp erature equals el 2 2 N 01 + k T: 2 C T = el B 3 This relation enables a value for 1 + to b e extracted if the bare density of states N 0 is known from quasiparticle band theory. This pro ceedure has large uncertainties, so the resulting values of are sub ject to substantial error and are not tabulated here. When the BCS ideas are carefully worked out using the actual electron-phonon interactions Migdal-Eliashb erg theory [3] then a quite complicated but in principle solvable relation o ccurs b etween electron-phonon coupling and T . If the anisotropy of the sup erconducting gap is ignored or washed out by non-magnetic impurity scattering then c 2 theory simpli es and T dep ends on and a single function F which is similar to the phonon-density of states c 2 F . contains an average square electron-phonon matrix element. Quasiparticle tunneling exp eriments in planar tunnel junction geometry [4,5] in principle provide a way of measuring this function, which is related to by Z 1 d 2 =2 F : 3 0 This provides p erhaps the most reliable known values for . The techniques and the data are reviewed in ref. [5]. However, there are still signi cant uncertainties in values of obtained this way, caused by diculties in making good yet partially transparent barriers and ignorance of such details as the transmission co ecients for tunneling, inelastic e ects in the barrier region, etc. For a few metals principally Pb, In, Tl, and alloys of these the barriers 2 seem particularly clean or else the complexities somehow cancel out; the accuracy of the resulting F is well tested through various self-consistency checks. Tunneling in p oint contact geometry gives valuable information ab out 2 F , esp ecially in materials with weaker electron-phonon interactions where the planar junction techniques do not work. However, the absolute values of obtained this way are rather variable and are not listed here. This article was published in Handb o ok of Sup erconductiv ity, edited byC.P.Po ole, Jr. Academic Press, New York, 1999 Ch. 9, Sec. G, pp. 478-483. 1 2 The Migdal-Eliashb erg theory was solved numerically for T as a functional of F and by McMillan [6]. He c tted his results to an approximate formula, generalizing the BCS result T = exp [1= ], c D 1:041 + D T = exp : 4 c 1:45 1 + 0:62 The parameter is assigned a value in the range 0.10-0.15, consistent with tunneling and with theoretical guesses. The choice of is fairly arbitrary, but fortunately its precise value is not to o imp ortant unless T is very low. Most c of our knowledge of values of comes from using Eqn. 4 to extract a value of from measured values of T and . c D Values deduced in this fashion are denoted in this article. Subsequent to McMillan's work, exp eriments and McM further numerical studies [7] worked out the limits of applicability of Eqn. 4. The \prefactor" =1:45 works well D only for elements or materials whose phonon density-of-states is similar in shap e to elements like Nb. The correct [7] prefactor ! =1:20, is not measureable except by exp eriments like tunneling. The de nition of ! is ln ln Z 1 ln 2 2 d F : 5 ! = exp ln 0 Lack of knowledge of ! limits the accuracy of values of , esp ecially in comp ound materials with complicated ln McM phonon disp ersion. When T b ecomes reasonably large T greater than 5 of or 1:2 Eqn. 4 underesti- c c D mates T . Approximate correction factors were given in ref. [7]. Unfortunately, additional parameters are required to c give an accurate formula for T . c Calculations of or are computationally demanding and are not yet under theoretical control. Calculations of are slightly less demanding, are under somewhat b etter theoretical control, and have b een attempted for many years. Prior to 1990, calculations of generally required knowing the phonon frequencies and eigenvectors as input information, and approximating the form of the electron-ion p otential. Results of these calculations are not tabulated here. McMillan [6] and Hop eld [8] p ointed out that one could de ne a simpler quantity, 2 2 N 0hI i = M h! i; 6 Z 1 2 2 2 h! i d F : 7 0 2 The advantage of this is that and hI i are purely \electronic" quantities, requiring no input information ab out phonon frequencies or eigenvectors. Gaspari and Gyor y [9] then invented a simpli ed algorithm for calculating , and many authors have used this. These calculations generally require a \rigid ion approximation" or some similar 2 guess for the p erturbing p otential felt by electrons when an atom has moved. Given , one can guess a value for h! i for example, from and thereby pro duce an estimate for .Values pro duced this way are not tabulated in the D presentchapter. Instead the reader is referred to the following literature: Sigalas and Papaconstantop oulos [10] have given a recent tabulation for d-band elements, and Skriver and coworkers have published calculations for rare earths [11] and lanthanides [12] which are particularly valuable since usually no other estimate of is available. Brorson et al. [13] have extracted measured values of from measured rates of thermal equilibration of hot electrons in various metals, using a theoretical relation [14]. Fairly recently, theory has progressed to the p oint where \ rst- principles" calculations [15] can b e made of phonon disp ersion curves using density functional theory, usually in \lo cal density 2 approximation" LDA for quite complicated systems. It is not to o hard to extend these calculations to give F and ; these values should b e \repro ducible" in the sense that most theorists would agree up on a unique recip e. Such calculations have b een done bySavrasov and Savrasov [16] and by Liu and Quong [17]. The results accord well with other metho ds of nding .Values of obtained this way will b e denoted . LDA The T -dep endence of the electrical resistivity sometimes o ers an accurate wayofevaluating the electron-phonon coupling. In clean metals de ned by a large resistance ratio 300K =T = T + the resistivity is normally c 2 dominated by electron-phonon interactions. Using the standard form =1= = ne =m, the scattering rate h= T at high temp eratures is 2 k T which de nes a coupling constant whichisvery closely related to . Coulomb tr B tr 2 scattering of electrons with each other also contributes, but is smaller by the factor =N 0k T which is usually B 2 10 at ro om temp erature. The derivation of this result dep ends on Blo ch-Boltzmann transp ort theory, whichis closely analogous to Migdal-Eliashb erg theory of sup erconductivity.For b oth theories, the \Migdal theorem" shows that corrections Feynman diagrams with phonon vertex corrections should b e smaller by a factor N 0h! . Both D theories contain anisotropy corrections, which are almost always small. When anisotropy is ignored, sup erconductivity dep ends on the isotropic parameter and resistivity on the isotropic parameter . These two coupling constants tr 0 0 are related to the electron-phonon matrix elements M and the phonon frequencies ! by the formula k;k k k 2 P 0 2 0 0 0 w k; k jM j =h! 0 k;k k k k k k;k P 8 = N 0 w 0 0 w k; k 0 k k k;k where is the quasiparticle energy of an electron constrained by the -function to b e at the Fermi energy, the k 0 0 lab els k; k are short for the electron quantum numb ers wavevector, band index, spin, and w k; k isaweight function to b e sp eci ed.
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