The Electron-Phonon Coupling Constant

Home , Coupling (physics)

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

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

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

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

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

N 01 + k T: 2 C T =

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

theory simpli es and T dep ends on and a single function F which is similar to the phonon-density of states

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

=2 F : 3

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

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

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

The Migdal-Eliashb erg theory was solved numerically for T as a functional of F and by McMillan [6]. He

tted his results to an approximate formula, generalizing the BCS result T = exp [1= ],

c D

1:041 +

T = exp : 4

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

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

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

ln 2

d F : 5 ! = exp

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

give an accurate formula for T .

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

2 2

h! i d F : 7

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

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

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

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

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

scattering of electrons with each other also contributes, but is smaller by the factor =N 0k T which is usually

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

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

0 0

are related to the electron-phonon matrix elements M and the phonon frequencies ! by the formula

k;k k k 2

0 2

0 0 0

w k; k jM j =h!

k;k k k k k

k;k

8 = N 0

w k; k

k k

k;k

where is the quasiparticle energy of an electron constrained by the -function to b e at the Fermi energy, the

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. The sup erconducting uses w = 1 for the weight function, while the transp ort uses

0 2

w k; k =v v , where hv is the group velo city @ =@ k . A review of the theory and the formulas is given

kx k x k;x k x

in ref. [18]. Simultaneous calculations [16,19] of these two parameters are available for a numb er of metals, shown

in Table I. From this it is known that the di erence is typically no more than 15, and never as much as a factor

of 2. In principle there are separate coupling constants and in hexagonal or tetragonal metals, but the

tr;xx tr;zz

anisotropyof is b elieved to b e quite small. The explanation for the observed similaritybetween and must

tr tr

2 2

0 0

j . The same lackof b e that the weight factor v v correlates only weakly with the matrix element jM

kx k x k;k

correlation will also guarantee that the anisotropy of the resistivity tensor in non-cubic materials will derive primarily

from the anistropy of the inverse e ective mass tensor n=m rather than from anisotropyin .

To extract a value of from T data, the safest pro ceedure is a three parameter t to the data using the

Blo ch-Gruneisen formula,

2 4

2 k T=h

h =k T d

tr B B

= + : 9

BG 0

n=me ! sinhh =2k T

D B

At high T the factor [ ] b ecomes 4 and thus the integral b ecomes 1. The three tting parameters are , ! ,

0 D

and the ratio =n=m. This equation assumes either an isotropic p olycrystalline sample or else cubic symmetry,

and also assumes that the phonon disp ersion is adequately represented by a Debye mo del. It is easy to generalize

to anisotropic or non-Debye cases, at the cost of further tting parameters. For example, the case of completely

4 2

general phonon disp ersion is handled simply by replacing the factor 2 =! by a function F . Fortunately,

tr D

except at low T , the form of is not very sensitive to the form of F . One exception is ReO where the

BG 3

acoustic vibrations are mainly Re-like and low in energy, while the optic vibrations are mainly O-like, and very high

in energy.For that material, it is adequate [20] to represent F by a sum of a Debye and an Einstein piece,

2 =! + ! =2 ! , where is + . The remaining problem is that the tensor parameter n=m

D D E E E tr D E

must also b e known in order to get a value for , and there is no rm exp erimental metho d. The theoretical formula

= v v : 10

k; k; k

For cubic symmetry, the tensor is a scalar, n=m =n=m . It is usually not useful or even p ossible to make

separate de nitions of a scalar n b ecause it is unclear how many of the valence electrons should b e counted or of

a tensor or scalar m or 1=m. Another way to write Eqn. 10 is n=m = N 0hv v i where the angular brackets

2 2

denote a Fermi surface average. One can also de ne a \Drude plasma frequency" =4e n=m whichgoverns

b oth dc and ac conductivity. Unfortunately it is not p ossible to get reliable values of the Drude plasma frequency

from optical exp eriments, for a variety of reasons, outlined by Hop eld [21]. Surprisingly, it seems that LDA band

theory gives for many metals very reliable values of n=m which yield therefore go o d values of . In quite a few

metals, this provides the b est available estimate of .

There are several other ways of getting values of parameters related to . Quasiparticles near the Fermi surface

have energies and lifetimes given by p oles of the Green's function or zeros of G k; ! for small j! j in the complex

! -plane,

G k; != ! k; !

=1+ T ! + i=2 !; T 11

k k k

where T is de ned as @ k; !=@ ! j , =2 is de ned as k; !, and the zero of energy is shifted so

k 1 ! =0 k k 2

that k; ! = 0 is absorb ed and disapp ears. Then T is the mass enhancement parameter for the quasiparticle

1 k

state k , and is the average of T =0over the Fermi surface. The sp ectral weight function, -ImG k; ! then

has a Lorentzian resonance centered at the renormalized quasiparticle energy E = =1 + T with width equal

k k k

to the lifetime broadening 1= E ;T. If electron-phonon interactions dominate, then at T , where T 0,

k k D k

h= E ;T= 2 T =0k T . This also assumes that the state k b eing prob ed has a small enough j j so that

k k k B k

jE jk .

k B D 3

Various resonance metho ds can measure either E T and 1= T . As an example, cyclotron resonance can do

k k

this for extremal orbits on the Fermi surface, but generally only at low temp eratures where E = =1 + T = 0.

k k k

Under these conditions, 1= T is small and do es not contain complete information ab out ; an orbit-averaged

k k

value of 1 + can b e extracted using theoretical values of the unrenormalized quasiparticle band structure .In

principle, photo emission sp ectroscopy could measure 1= T at high T , and thus directly measure with no need

k k

for theoretical input. The principal diculty is that the p erp endicular comp onent k of the wavevector k can not

b e directly measured; a measured sp ectrum is a sup erp osition of sp ectra for a range of values of k . One case where

this is avoided is for a surface state. If there is no bulk state for some range of energy and some k , then a surface

state may o ccur in that energy range with a sharp two-dimensional k-vector. Then high-resolution photo emission can

measure 1= and thus for that state.

k k

I I. COMMENTARYON VALUES

Tables I I through V contain values of for selected materials. I have tabulated only those cases where it seems

likely that the value will not change greatly with time. Therefore, manyinteresting material are omitted, such as the

cuprate, bismuthate, and fullerene sup erconductors. Esp ecially in the cuprates, prop erties of quasiparticles, including

even whether they exist and what spin and charge they carry, are still mysterious. Electron-phonon coupling seems

to a ect some prop erties of cuprates, yet seems to b e missing from other prop erties. My opinion is that Migdal-

Eliashb erg theory and Eqn. 4 do not apply to cuprates, but probably do apply to fullerenes and to the BaBiO

family of sup erconductors. Values of of order 1 probably apply to the latter two families, but rm numb ers are

hard to sp ot.

Table I I contains elements which are sup erconducting in crystalline form at atmospheric pressure, including two

cases Ga and La where there is information ab out a metastable phase. The entries are taken from previously cited

sources plus refs. [22]- [33]. The sup erconducting elements provide the b est opp ortunity to compare values obtained

by di erent metho ds. In particular, for Pb, V, Nb, and Ta there are transp ort, tunneling, and LDAvalues in addition

to values from the McMillan equation, Eq. 4. In all cases, McMillan values are less than all other values, strongly

suggesting that the McMillan equation underestimates . There are two main causes of this: 1 the McMillan

prefactor =1:45 is often larger than the correct prefactor ! =1:2, although although probably not bymuch for the

D ln

elements V, Nb, and Ta b ecause the McMillan equation was based on the phonon sp ectrum of Nb; 2 knowledge of

is still primitive; it is assumed to b e 0:10 0:13 but Table I I suggests that it is often larger. In particular, in V it is

susp ected that spin- uctuation e ects may increase ab ove the \renormalized" value =[1 + ln! =! ] b ecause

P D

the characteristic spin- uctuation frequencies are lower than ! .For very strong-coupling materials like Pb, the

McMillan equation should overestimate b ecause in this regime the McMillan equation is known to underestimate

T .However, in Pb the other causes must dominate since is an underestimate. I b elieve that theoretical LDA

c McM

values are now more reliable than McMillan values. Tunneling values are excellent for the \simple" metals -Ga, In,

Sn, Hg, Tl, and Pb, but require more complicated junction preparation metho ds and more complicated theoretical

analyses for d-band and exotic materials, which degrades the b elievability of the numb ers. For most elements I think

the transp ort values are go o d, but Tl and Re mayhave to o high values of :

Table I I I shows values for four elements where tunneling exp eriments [34,35] have b een done on sup erconducting

amorphous phases. Bi is semimetallic and not sup erconducting in its crystalline phase. The divergent exp erimental

values of may re ect di erences b etween samples of amorphous Bi made by di erent pro ceedures, or may instead

re ect diculties in accurate measurement of tunneling characteristics, esp ecially at bias voltages near the sup ercon-

ducting gap where can have imp ortant contributions due to soft vibrations which are hard to measure accurately.

In Ga and Sn but not in Pb the amorphous phase has enhanced values of T . In all three elements, is enhanced.

The reason is the softening of the vibrational sp ectrum in amorphous phases, which raises more than T .

Table IV gives values for crystalline elements which are non-sup erconducting, using information from previously

cited sources plus refs. [36]- [38]. Values from transp ort are known for many elements and app ear consistent with

other values when available. It would b e very interesting to have reliable values for ferromagnetic elements. However,

one exp ects that there should b e quite di erentvalues of for the up and the down spin sp ecies, but twovalues

and cannot b e extracted indep endently from transp ort measurements. Therefore wemust wait for theoretical

values and measurements by spin-sensitive techniques. As an example, Gd is ferromagnetic with a Curie temp erature

292K. Photo emission b elow this temp erature has seen a surface state carrying the ma jority" spin, and its value

is listed.

Table V lists most of the intermetallic comp ounds where there is reliable information [39]- [50]. Evaluating for

comp ounds is often risky. There are two main diculties: 1 the measured T is sometimes sample dep endent,

esp ecially in oxide materials where p olycrystalline bulk, single crystal, and thin lm samples give di erent results; it 4

is hard to know the \true" T , but usually safe to assume that the smallest values are the b est; 2 it is imp ortant

to b e sure that the transp ort mean free path is at least 10A; otherwise, wavevector k is not a go o d quantum numb er,

Blo ch-Boltzmann theory is not applicable, and is ill-de ned. In Eqn. 8, the group velo cityhv = @ =@ k

tr k;x x

is only well de ned when wavevector k is a go o d quantum numb er. The sup erconducting ,however, is still well

de ned from Eqn. 8 provided the quantum numb ers k; k are reinterpreted as lab els for the exact eigenstates of the

disordered material.

RuO is an interesting case where the transp ort value of suggests the p ossible o ccurrence of sup erconductivityat

not to o low a temp erature, and exp erimentmay not have tested b elow helium temp erature. The layered intermetallic

b oro carbides have gotten a lot of attention recently. In the case of LuNi B C, McMillan and transp ort values agree

2 2

which suggests as do es other evidence conventional sup erconductivity. For the related material La Ni B N ,

3 2 2 3

transp ort and McMillan values badly disagree. My guess is that the transp ort value maychange with b etter data, but

usually the transp ort value go es down in time as samples improve, rather than up in time. An alternate p ossibilityis

the o ccurrence of unconventional sup erconductivity. A similar situation holds for cuprates, where transp ort values of

of order 1 are found using LDA bands, whereas to account for T near 100K, one needs of order 3 or higher. There are

clear signs of unconventional b ehavior, whichinmy mind invalidates the transp ort analysis based on Blo ch-Boltzmann

theory. Another interesting intermetallic, Sr RuO , is also not listed. Nice-lo oking resistivity measurements are now

2 4

b eing published, but do not accord well with the Blo ch-Gruneisen formula, which is another sign of unconventional

b ehavior. It is not clear whether the concept of an electron-phonon value can b e retained in such cases. In most

materials where the confusion is large, I have not tabulated any values, but La Ni B N is included as a sign of

3 2 2 3

the p otential hazards.

ACKNOWLEDGMENTS

I thank R. C. Dynes and W. E. Pickett for help with the manuscript. This work was supp orted in part by NSF

grant no. DMR-9725037.

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TABLE I. Comparison of theoretical values of and

Metal

theor tr;theor

Al 0.44 0.37

Pb 1.68 1.19

V 1.19 1.15

Nb 1.26 1.17

Nb 1.12 1.07

Ta 0.86 0.83

Ta 0.88 0.57

Mo 0.42 0.35

Cu 0.14 0.13

Cu 0.111 0.116

Pd 0.35 0.43

Pd 0.41 0.46

Full LDA theory, Ref. [16].

LDA energy bands, rigid ion approximation, exp erimental

phonons, Ref. [19]. 6

TABLE I I. Values of for sup erconducting crystalline elements

Metal T K K

c D McM tr tun LDA

Be 0.026 1390 0 .23

b cd

Al 1.16 428 0 .38 0.39 0 .44

Zn 0.85 309 0 .38 0.46

-Ga 1.08 325 0 .40

-Ga 5.9 0.97

Cd 0.52 209 0 .38 0.37

In 3.40 112 0 .69 0.834

Sn 3.72 200 0 .60 0.72

Hg 4.16 72 1 .00 1.60

e f

Tl 2.38 79 0 .71 1.11 0.78

b j d c

Pb 7.19 105 1 .12 1.48 1.55 1.20 ,1.68

Ti 0.39 425 0 .38 0.50

b l c

V 5.30 399 0 .60 1.09 0.83 1 .19

Zr 0.55 290 0 .41 0.55

b m c

Nb 9.22 277 0 .82 1.06 1.05 1 .26

b c

Mo 0.92 460 0 .41 0.32 0 .42

Ru 0.49 550 0 .38 0.45

Hf 0.09 252 0 .34 0.42

b n o c

Ta 4.48 258 0 .65 0.87 0.69 , 0.73 0 .86

W 0.012 390 0 .28 0.26

Re 1.69 415 0 .46 0.76

Os 0.65 500 0 .39 0.54

Ir 0.14 420 0 .34 0.50

p q

-La 4.88 151 0 .81 0.77

-La 6.00 139 0 .93

b b

Th 1.38 165 0 .56 0.52

Unless otherwise noted, from Ref. [6].

Ref. [22].

Ref. [16].

Ref. [17].

Ref. [23].

Ref. [24].

Ref. [25].

Ref. [26].

Ref. [27].

Ref. [28].

Ref. [18].

Ref. [29].

Ref. [30].

Ref. [31].

Ref. [32].

from Eqn. 4 using data from Ref. [2].

Diculties with the junction required ad ho c mo di cations in the analysis, Ref. [33].

TABLE I I I. Values of for sup erconducting amorphous

elements

Metal T

c tun

Ga 8 .56 2 .25

Sn 4.5 0 .84

Pb 7.2 1 .91

a b

Bi 6.1 2.46 ,1.84

Ref. [34].

Ref. [35]. 7

TABLE IV. Values of for non-sup erconducti ng crys-

talline elements

Metal

LDA other tr

Li 0 .35 0.45 { 0.51

Na 0 .14 0.24

K 0 .11

Rb 0 .15

Cs 0 .16

d e

Cu 0 .13 0.14 0.140.02

Ag 0 .12

Au 0 .15 0.2

Mg 0 .20

Ca 0 .05

Ba 0 .27

Sc 0 .51

Y 0 .62

Pd 0 .47 0.35

Pt 0 .66

Gd 0.6

Unless otherwise noted, from Ref. [22].

Ref. [17].

Surface quantum well state, Ref. [36].

Ref. [16].

Surface state on Cu111, Ref. [37].

Extrap olation from sup erconductin g alloys, Ref. [38].

Ref. [23].

h 2

Surface state with quantum numb ers 5dz ; " on Gd0001,

P. D. Johnson, private communication. 8

TABLE V. Values of for crystalline comp ounds and or-

dered intermetallics

Metal T

c tun tr McM

In Bi 5.6 1.40

Bi Tl 6.4 1.63

Tl Sb 5.2 1.43

7 2

V Si 17 .1 0.89

Nb Al 18 .5 1.7

Nb Sn 17 .8 1.75

Nb Ge 20 1.7

NbN 16 .0 1.46

e e

NbO 1.4 0.51 0.41

ReO <0.02 0.35

RuO <4.2 0.5 0.1

CoSi 1 .22 0.44

Pd Si 0.15 { 0.20

h i h

LuNi B C 16 .1 0.9 , 0.8 1.0

2 2

j j

La Ni B N 12 .25 0.29 0.86

3 2 2 3

Ref. [39].

Ref. [40].

Ref. [41].

Ref. [30,42].

Ref. [43].

Ref. [20].

Ref. [44].

Using data from Ref. [45] and theory from Refs. [46] and

[47].

Ref. [48].

Using data from Ref. [49] and theory from Ref. [50]. 9