two-dimensional metals with a cylindrical Fermi surface cannot be applied near the wave vector corresponding to the Kohn singularity. Instead, the Dyson ...
PHYSICAL REVIEW B 77, 014517 共2008兲
Self-consistent theory of phonon renormalization and electron-phonon coupling near a two-dimensional Kohn singularity O. V. Dolgov,1 O. K. Andersen,1 and I. I. Mazin2 1Max-Planck-Institut
für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany Center for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, USA 共Received 3 October 2007; revised manuscript received 12 December 2007; published 23 January 2008兲
2
We show that the usual expression for evaluating electron-phonon coupling and the phonon linewidth in two-dimensional metals with a cylindrical Fermi surface cannot be applied near the wave vector corresponding to the Kohn singularity. Instead, the Dyson equation for phonons has to be solved self-consistently. If a self-consistent procedure is properly followed, there is no divergency in either the coupling constant or the phonon linewidth near the offending wave vectors, in contrast to the standard expression. DOI: 10.1103/PhysRevB.77.014517
PACS number共s兲: 63.20.K⫺, 74.70.Ad
INTRODUCTION
First principles calculations of the phonon spectra and electron-phonon coupling in MgB2 共see Ref. 1 for a review兲 have determined that the interaction mainly responsible for superconductivity in this material is coupling of small-q, high-energy optical phonons of a particular symmetry with approximately parabolic nearly two-dimensional hole bands forming practically perfect circular cylinders occupying only a small fraction of the Brillouin zone. It was realized early enough2 that the case of an ideal two-dimensional 共2D兲 cylinder leads to a divergency in the calculated phonon linewidth at the 2D Kohn singularity, q = 2kF, which presents serious difficulties in calculating the electron-phonon coupling function. One option that was exploited was to use an analytical integration for the wave vectors comparable with or smaller than 2kF,2,3 and a numerical one for the larger vectors. It was also pointed out1,4 that this singularity gets stronger when the Fermi surface gets smaller, while the integrated electron-phonon coupling 共for 2D parabolic bands兲 does not change. While a perfectly cylindrical Fermi surface is an idealized construction, deviations may be quite small, it seems, on the first glance, unphysical that all phonons with 兩q兩 = 2kF have infinite linewidth. Note that the problem is not specific for MgB2: it occurs in doped graphenes such as CaC6 and, YbC6,5 and, in fact, in any 2D material sporting Kohn singularities. In particular, the hypothetical hexagonal LiB, a subject of substantial recent interest, has bands that are even more 2D that those in MgB2 共Ref. 6兲 and the described problem is even more pronounced. This intuition is correct. In this paper, we show that close to a Kohn anomaly, standard formulas for calculating electron-phonon interaction 共EPI兲 become incorrect and new, self-consistent expressions replace them. These expressions have no singularities, and exhibit a much more natural, reasonably smooth, q dependence of the phonon self-energy. To start with, we shall remind the readers of the standard formalism. We first define the retarded phonon Green function  共0兲兴典, D␣共q,t兲 ⬅ − i共t兲具关uq␣共t兲u−q
where 关…兴 is a commutator and 具…典 denotes statistical averaging. The displacement operator uq␣ in the ␣ direction can 1098-0121/2008/77共1兲/014517共7兲
be expressed via the phonon eigenvectors eq and frequencies squared q2 : uq = 兺
冉
冊
1 † eq共aq + a−q 兲. 2M q
共1兲
For simplicity, a primitive lattice with a single kind of ions with a single mass M will be considered below. Also, atomic 共hartree兲 units will be used throughout the paper. In this case, the “bare” phonon Green function has a form D0共q, 兲 =
冋
册
1 1 1 − , 2M q0 − q0 + i␦ + q0 + i␦
共2兲
where q0 is the bare phonon frequency, before accounting for electron-phonon coupling 共screening by electrons兲. Without losing generality, it can be assumed to be q independent, q0 = 0. Correspondingly, the full Green function is D共q, 兲 =
冋
册
1 1 1 − , 2M q − q + i⌫q + q + i⌫q
共3兲
where q is the renormalized 共observable兲 frequency, and ⌫q is damping 共phonon linewidth兲 due to EPI.7 The Dyson equation reads D−1共q, 兲 = D−1 0 共q, 兲 − ⌸共q, 兲,
共4兲
where the polarization operator in the lowest approximation 共as usual, the Migdal theorem8 allows neglecting the vertex corrections兲 along the real frequency axis at T = 0 has a form 共a is the lattice constant in the plane兲 ⌸共q, 兲 = − 2i
冕
冉
冉 冊
q 0 兩gk,k+q 兩2G0 k + , + 2 2
a2d2k d q ⫻G0 k − , − . 2 2 共2兲2 2
冊 共5兲
0 Here, gk,k+q is the bare electron-ion scattering matrix element 共the commonly used EPI matrix element differs in that the potential gradient is replaced by the derivatives with respect to the normal phonon coordinates兲
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©2008 The American Physical Society
PHYSICAL REVIEW B 77, 014517 共2008兲
DOLGOV, ANDERSEN, AND MAZIN
0 gk,k+q =
冕
1 a2
a2
* k+q 共r兲 ⵜ V共r兲k共r兲d2r,
共6兲
where G0共k,兲 =
1
共7兲
− 0共k兲 + i␦ sign共k − kF兲
is the bare electron Green function, and 0共k兲 = 共k2x + k2y − kF2 兲 / 2m. The renormalized phonon frequency and the phonon linewidth ⌫q are determined by the pole of the phonon Green function D共q , 兲 or D−1 0 共q, q
D 共q, q + i⌫q兲 ⬅ −1
+ i⌫q兲 − ⌸共q, q + i⌫q兲 = 0
I. COMPLEX POLARIZATION OPERATOR
Let us consider a model with a cylindrical Fermi surface of radius kF, whose electrons interact with an optical phonon with a bare frequency 0, and a momentum-independent matrix element g0 共we also neglect possible warping of the Fermi-surface cylinder, cf. Ref. 11兲. In this case, the imaginary part of Eq. 共10兲 reads Im ⌸共q, + i␦兲 = − 2兩g0兩2 兺 兵关0共k + q兲兴 − 关0共k兲兴其 k
⫻␦关0共k + q兲 − 0共k兲 − 兴 or
关note that in the experiment, the phonon linewidth is usually defined as the half-width of the peak in Im D共q , 兲, which differs from our definition by terms of the order 共⌫ / 兲4兴. This leads to
q2
=
1 Re ⌸共q, q + i⌫q兲 + ⌫q2 + M
20
Im ⌸共q, 兲 = −
共8兲
=− 1 Im ⌸共q, q + i⌫q兲. 2M q
0 兩2 ⌸共q, + i␦兲 = − 2 兺 兩gk,k+q k
⌸共q,Z兲 =
共10兲
2 appq
and ⌫qapp = −
冋 冉 冊册
2 1 = ⌸共q,0兲 1 + O 20 M F
冏
1 d Im ⌸共q, 兲 2M d
⬅
冏
冑1 − 共q/2k + m/kq兲2
m兩g0兩2 关Re 冑1 − 共Q − ⍀兲2 a 2Q 共13兲
1 P
冕
⬁
−⬁
dE Im ⌸共q,E兲. E−Z
The result is ⌸共q,Z兲 = −
共11兲
m兩g0兩2 a 2Q
冕
dxdy
C
x , − 共Z/4F兲2/Q2 + x2
共14兲
where F = kF2 / 2m, x = kx / kF, y = ky / kF, and N共0兲 = m / 2a2 is the density of states at the Fermi level, per spin. The integration is performed over the range 共x − Q兲2 + y 2 ⬍ 1. The substitution x − Q = r cos and y = r sin leads to
=0
兺 兩g0 兩2␦关0共k + q兲兴␦关0共k兲兴, M k k,k+q
max兵kF,m/q+q/2其
where Q = q / 2kF and ⍀ = / qvF 共vF is the Fermi velocity兲. To find the polarization operator for a complex frequency Z = + i␥, we use the Hilbert transformation
nk − nk+q 0共k + q兲 − 0共k兲 − − i␦
to first order in frequency just above the real axis of the complex frequency . In this case,
2 kF +2m
− Re 冑1 − 共Q + ⍀兲2兴,
共9兲
The next standard step, following Ref. 9, is to expand the polarization operator
冕冑
dk
⫻Re
and ⌫q = −
m兩g0兩2 2 a 2q
冕
共12兲
where the factor 共nk − nk+q兲␦关0共k + q兲 − 0共k兲 − 兴 has been changed to −␦关0共k + q兲兴␦关0共k兲兴. According to Ref. 10, “Except for extremely pathological energy bands, it is an excellent approximation.” Unfortunately, MgB2 and some other recently discovered superconductors are examples where the energy bands are, in some aspects, pathological. Formally, expression 共12兲 for a 2D system is divergent near a Kohn anomaly q → 2kF, and we have ⌫q 艌 q, i.e., phonons are not well defined quasiparticles. To describe electronphonon interaction in these systems, we have to calculate the polarization operator ⌸共q , Z兲 for a complex frequency Z and solve Eqs. 共8兲 and 共9兲.
C
⌸共q,Z兲 = −
冕 冕 1
dxdy ⇒
冋 冉 册
0
m兩g0兩2 Z 2 2− 1− 2a 4FQ2
− 兵Z → − Z其 .
2
rdr
d ,
0
冊冑 冉
1− Q−
Z 4FQ
冊
−2
共15兲
The branches of the square roots are chosen so as to get the correct behavior ⌸ ⬀ q2VF2 / 2 at large frequencies 共 = Re Z兲. For a 2D system for ⌸共q , 兲 on the real frequency axis, one can write:12–15
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SELF-CONSISTENT THEORY OF PHONON…
Re ⌸共q, + i␦兲 = −
m兩g0兩2 关2Q − 共Q − ⍀兲Re 冑1 − 共Q − ⍀兲−2 2 a 2Q
− 共Q + ⍀兲Re 冑1 − 共Q + ⍀兲−2兴, Im ⌸共q, + i␦兲 = −
m兩g0兩2 关Re 冑1 − 共Q − ⍀兲2 2 a 2Q
− Re 冑1 − 共Q + ⍀兲2兴.
共16兲
A three-dimensional 共3D兲 plot of the functions 共16兲 can be found, for instance, in Ref. 16. Note that while the expression for real frequencies 关Eq. 共16兲兴 has been published multiple times, we have not found in the literature the general expression for an arbitrary complex frequency argument 关Eq. 共15兲兴. First, we see that the imaginary part is finite for all values of the wave vector q and vanishes inside the Landaudamping cone q ⬍ / vF 关more exactly, at Q ⬍ 共 / 4F兲 − 共 / 4F兲2兴. It has two maxima: one is rather small, Im ⌸共兲 ⯝ −m兩g0兩2冑 / 4F / a2, at Q ⯝ 1 − 共 / 4F兲, while the other has an antiadiabatical behavior, Im ⌸共兲 ⯝ −m兩g0兩2冑2F / / a2, and occurs at a very low frequency Q ⯝ 共 / 4F兲 + 共 / 4F兲2. However, if we expand the polarization operator at small frequencies, we recover a standard result 共see, e.g., Refs. 2 and 4兲 ⌸app共q, + i␦兲 ⯝ − −
冋
FIG. 1. 共Color online兲 The imaginary and real parts of the normalized polarization operator 2兩⌸共q , = 0兲兩a2 / m兩g0兩2 as a function of the reduced wave vector q / kBZ, for different fillings kF / kBZ. Solid lines represent the exact results, and the dashed lines the approximate solution 关Eq. 共17兲兴. The four different sets correspond, from left to right, to four increasing values of kF / kBZ.
Re ⌸共q,0兲 ⯝ −
m兩g0兩2 i/4F 1+ 共1 − Q兲 2 a Q冑1 − Q2
冑
1−
册
1 共Q − 1兲 , Q2
=−
⫻
1 . 0共k + q兲 − 0共k兲
In this case,
⌸ M 共q,in兲 = −
m兩g0兩2 关1 − 共Q − 1兲冑1 − Q−2兴. a2
Re ⌸共q → 0, 兲 ⬇ −
共18兲
In the opposite limit, Im ⌸共q , 兲 ⬅ 0 for q 艋 m / 冑kF2 − m. For the real part, one can set = 0 in Eq. 共17兲: k
2kF q
1−
2
共19兲
In the opposite limit,
02
Re ⌸共q,0兲 = − 2兩g0兩2 兺 共兩k兩 − kF兲共kF − 兩k + q兩兲
冑 冉 冊册
共17兲
where the imaginary part diverges at q → 0 and q → 2kF. The real part of Eq. 共16兲 practically coincides with the real part of Eq. 共17兲 except inside the Landau-damping region. At → 0 and finite q, we get
m兩g 兩 共1 − Q兲 . Im ⌸app共q, 兲 ⯝ − 4F a2 Q冑1 − Q2
冋
m兩g0兩2 1 − 共q − 2kF兲 a2
冉 冊
m兩g0兩2 qkF 2a2 m
2
.
The momentum dependence of the absolute values of the imaginary 共the upper panel兲 and real parts 共the lower panel兲 of Eqs. 共16兲 共solid lines兲 and 共17兲 共short-dash lines兲 at = 0 = 90 meV as the functions of the reduced wave vector q / kBZ 关kBZ = / a is the Brillouin zone 共BZ兲 vector, or the radius of the Wigner-Seitz cylinder兴 is shown in Fig. 1. Fermi vectors kF / kBZ = 0.075, kF / kBZ = 0.1, kF / kBZ = 0.15, and kF / kBZ = 0.2 correspond to F = 0.15, 0.27, 0.60, and 1.07 eV, respectively. Along the imaginary 共Matsubara兲 axis, the polarization operator has the following form:
冋
册
冑Q4 − Q2 − 共n/4QF兲 + 冑关Q4 − Q2 − 共n/4QF兲兴2 + 共Qn/4F兲2 m兩g0兩2 1+ , 2 冑2Q2 a 014517-3
共20兲
PHYSICAL REVIEW B 77, 014517 共2008兲
DOLGOV, ANDERSEN, AND MAZIN
where n = 2nT . T is temperature and n = 0 , ⫾ 1, ⫾2 , . . ., ⫾⬁. ⌸ M 共q , in → 0兲 coincides with Eq. 共19兲. II. PHONON RENORMALIZATION IN TWO-DIMENSIONAL SYSTEMS
First, let us consider the approximate polarization operator from Eq. 共17兲. Then
qapp = 0冑1 − 2关1 − 共q − 2kF兲冑1 − 共2kF/q兲2兴
共21兲
and, according to Eqs. 共12兲 and 共17兲, ⌫qapp =
m兩g0兩2 20 共1 − Q兲 共2kF − q兲 = , 2 2a MVFq 冑1 − 共q/2kF兲2 4F Q冑1 − Q2 共22兲
where we have introduced, following Ref. 17, an auxiliary coupling constant = N共0兲兩g0兩2 / M 20 共some authors use another dimensionless constant 0 = 2兲. ⍀q = qapp + i⌫qapp gives the renormalized frequency and the damping 关see Eqs. 共11兲 and 共12兲兴. For q 艋 2kF, ⌫qapp
=
qapp
0
1
4F冑1 − 2 Q冑1 − Q2
共23兲
.
This expression diverges in the limits Q → 0 and Q → 1. Turning now to the exact Eq. 共15兲, we observe that in the lowest order in / F ⌸共q → 2kF, 兲 = −
冋
m兩g0兩2 1 − 共1 + i兲 a2
冑 册
. 8F
This leads to
2kF ⯝ 0冑1 − 2 , ⌫2kF =
20 2kF
and ⌫2kF
2kF
=
1 − 2
冑
冑
2kF 8F
,
0冑1 − 2 . 8F
共24兲
This ratio remains finite in the limit Q → 1, although the approximate expression of Eq. 共23兲 diverges for any system with a cylindrical Fermi surface. Of course, the singularity at Q → 0 is also unphysical and, in principle, can be treated in a similar way. The common point is that, in both cases, the well-known popular formula 兩g20兩 ⌫q 兩g20兩 = ␦共k兲␦共k+q兲 = 兺 q M k M共2a兲2
冕
dk 兩v共k兲 ⫻ v共k + q兲兩 共25兲
is not valid near the singularity. We do not consider the Q → 0 case in this paper, but it is worth noting that there are other issues relevant in that limit, but not near the Kohn singularity, such as the Landau threshold and applicability of the Migdal theorem.
FIG. 2. 共Color online兲 The linewidth ⌫q and the renormalized phonon frequency q obtained by using the approximation of Eq. 共17兲 and the exact expression Eq. 共15兲, for the following parameters: the bare phonon frequency 0 = 90 meV and the bare constant of EPI = 1 / 4. The filling corresponds to kF / kBZ = 0.17 and F = 0.2 eV.
The results for the linewidth ⌫q and the renormalized phonon frequency q obtained by using the approximate polarization operator as functions of the reduced wave vector Q = q / 2kF are shown in Fig. 2 by red lines. Parameters are the following: the bare phonon frequency 0 = 90 meV and the bare constant of EPI = 1 / 4. The ratio kF / kBZ is equal to 0.17. It corresponds to F = 0.2 eV. The approximate result agrees with that of Ref. 4 共their Fig. 4兲. The exact result 共black lines兲 has been obtained by a numerical solution of Eqs. 共8兲 and 共9兲 using the polarization operator from Eq. 共15兲. The latter, in contrast to the approximate expression, shows two shoulders in the wave vector dependence of the renormalized frequency q 共see also 兩Re ⌸共q , 0兲兩 in the bottom panel of Fig. 2兲. The first one corresponds to the maximum of 兩Im ⌸共q , 兲兩 共or ⌫q兲 and the second one to the vanishing of these values. III. ELECTRON SELF-ENERGY
The electron self-energy is expressed via the electron and phonon Green functions: ⌺共p兲 = ig0
冕
G共p − k兲D共k兲⌫共p,p − k,k兲
dD+1k , 共2兲D+1
where p = 兵p , 其. It was shown by Migdal8 that one can neglect the vertex corrections ⌫共p , p − k , k兲 ⯝ g0共1 + 冑m / M兲 and that the function G共 , k兲 differs from the bare electron Green function 关Eq. 共7兲兴 only in the narrow interval of momenta 兩k − kF兩 ⱗ ph / VF and frequencies 兩兩 ⱗ ph. Thus, the full electron Green function G共p兲 can be substituted by the corresponding function for noninteracting electrons 关Eq. 共7兲兴. Using Eq. 共3兲, the electron self-energy for T = 0 becomes 共see, e.g., Refs. 18 and 19兲
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SELF-CONSISTENT THEORY OF PHONON…
⌺共k, 兲 = 兺 ␦共k+q兲 q
+
兩g0兩2 2M q
冕 冋 d
册
共兲 − − q + i⌫q
˜ 共0兲 = N q
共− 兲 . − + q − i⌫q
1 兺 ␦共k兲⌺共k, 兲 N共0兲 k
=−
再
兩g0兩2 ⌫2 + 共q − 兲2 1 ␦共k兲␦共k+q兲 ln 2q 兺 兺 N共0兲M k q 4q ⌫q + 共q + 兲2
冋 冉 冊 冉 冊册冎
+ 2i tan−1
q − q + − tan−1 ⌫q ⌫q
˜ 共0兲 = N q
q
兩g0兩2 1 , M ⌫q2 + q2
02
共26兲
=
冒
冋
q2 ,
⌫qapp 1 ␦共 − q兲, = 兺 2N共0兲 q q
冋
册
共30兲
the same function Im ⌸⬘共q , 0兲 ⬅ d Im ⌸共q , 兲 / 兩d兩=0. In a general case, these functions can be different. This result, without using the pole approximation for the phonon Green function, can be trivially obtained in the Matsubara formalism. In this case, one does not need to solve the Dyson equation. In the lowest order in coupling for T = 0, the self-energy has a form
⌺共i,k兲 = − i
共31兲
where in the last equality we have used the approximate Eq. 共12兲. Note that Eq. 共31兲 is a consequence of the fact that the damping ⌫qapp, according to Eq. 共12兲, can be expressed via the nesting function 关Eq. 共27兲兴 and both are determined by
共29兲
册
˜ 共0兲兩g0兩2 1 N 1 ⌫q ⌫q q − . 兺 2q 2 共 − q兲2 + ⌫q2 共 + q兲2 + ⌫q2 2 M q
␦共k兲兩g0兩2␦共k+q兲 1 ␦ 共 − q兲 兺 2q N共0兲M k,q
共28兲
but for ⌫q Ⰷ q, the contribution of strongly damped phonons to total q is suppressed. The result 共26兲 we can get also if we introduce, according to Eq. 共3兲, a generalized Eliashberg function
␦共k兲兩g0兩2␦共k+q兲 1 1 ⌫q ⌫q 兺 2 − 2 2 共 − q兲 + ⌫q 共 + q兲2 + ⌫q2 2q 2 MN共0兲 k,q
The second term in this expression cancels out the nonphysical behavior at low and high frequencies. Otherwise, ⌫ would have been divergent. Equations 共26兲 and 共30兲 are general and valid not only for the 2D systems, where phase space factor 关Eq. 共27兲兴 is divergent. For ⌫q Ⰶ q, we have 2 ␣app 共兲F共兲 =
,
In the weak-damping approximation, we recover the standard formula ˜ 共0兲 兩g 兩 qapp = N q M
where we introduced the phase space function 共sometimes called “nesting function”兲
␣⌫2 共兲F共兲 =
4FQ冑1 − Q2
02 1 ˜ 共0兲 兩g 兩 . q = N q 2 M ⌫q + q2
1 1 兺 兺 ␦共k兲兩g0兩2␦共k+q兲 ⌫2 + 2 MN共0兲 k q q q
˜ 共0兲 =兺N q
共1 − Q兲
which diverges at Q → 0 and Q → 1. Following Ref. 9, we can introduce the “mode ” via the expression = 兺qq. Then
.
The limit = −lim→0 Re ⌺共兲 / is nothing but the standard electron-phonon coupling constant ⌫ =
共27兲
For a 2D cylindrical Fermi surface, we have
This is a trivial generalization of the standard expressions on the finite phonon linewidth case. Let us average the selfenergy over the Fermi surface ⌺共兲 =
1 兺 ␦共k兲␦共k+q兲. N共0兲 k
⫻
where
014517-5
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d共i兲 兺 兩g0兩2DM共i + i,k,k⬘兲 2 k , ⬘
1 , i − k⬘ − ⌺共i,k⬘兲
PHYSICAL REVIEW B 77, 014517 共2008兲
DOLGOV, ANDERSEN, AND MAZIN
D M 共i,k,k⬘兲 = 共1/兲
冕
⬁
d⍀ Im D共⍀ + i␦,k,k⬘兲关共i − ⍀兲−1
0
− 共i + ⍀兲−1兴.
共32兲
We can also average the self-energy over the Fermi surface ⌺共i兲 = 兺k␦共k兲⌺共i , k兲 / N共0兲: ⌺共i兲 =
1 兺 兺 ␦共k兲 N共0兲 k q ⫻兩g0兩2␦共k+q兲
冕
⬁
−⬁
d D M 共q,i兲 2
冕
⬁
−⬁
d . i共 − 兲 −
pling constants and are that they are measures of the renormalization of the phonon frequency from 0 to q 共cf. a discussion for 3D systems in Ref. 20兲. Turning back to the 2D case, according to Eq. 共26兲, we can neglect all divergent contributions near Q = 0 and Q = 1. Elsewhere, we can use Eq. 共29兲. One should keep in mind that the conventional coupling constant is = / 共1 − 2兲. This parameter determines electronic properties 共Fermi velocities, Tc, etc.兲. The other parameter, = / 共1 + 2兲 ⬍ 1 / 2, defines the observable phonon frequency, q = 0冑1 − 2. CONCLUSIONS
d ⬁ 兰−⬁ i共−兲− = −i2
The integral sign共 − 兲 allows us to calculate the physical coupling constant =−
冏
⌺el共i兲 i
冏
= − lim →0
⫻兩g0兩2␦共k+q兲
冕
→0
␣2共兲F共兲 =
1 兺 兺 ␦共k兲 N共0兲 k q
⬅
d D共q,i兲2␦共 − 兲 2
1 =− 兺 兺 ␦共k兲兩g0兩2␦共k+q兲DM共q,i0兲 N共0兲 k q =−
First of all, the standard well-known expression
1 1 . ␦共k兲兩g0兩2␦共k+q兲 −1 兺 兺 N共0兲 k q D M0共i0兲 − ⌸M 共q,i0兲
On the Matsubara axes, for a 2D system, according to Eq. 共20兲, the phase space factor vanishes for q ⬎ 2kF and ⌸ M 共q 艋 2kF , i0兲 = −220 is a constant 关see Eqs. 共20兲 and 共19兲兴. Us2 ing Eqs. 共2兲 and 共32兲, we get D−1 M0共i0兲 = −M 0 and = /共1 − 2兲.
1 ⌫q ␦ 共 − q兲 兺 2N共0兲 q q
␦共k兲兩g0兩2␦共k+q兲 1 ␦ 共 − q兲 兺 兺 2q N共0兲M k q
is valid only in the lowest order in the bare phonon linewidth, ⌫q, which is not an acceptable approximation in the case of strong Kohn singularities, and particularly for a cylindrical Fermi surface. ⌫q in this approximation is not the actual phonon linewidth; as opposed to ⌫q, determined by the oversimplified Eq. 共12兲, the real phonon linewidth does not diverge even for an ideally cylindrical Fermi surface. Second, following Eq. 共33兲 derived above, the more accurate treatment presented here still respects the old sum rules that the total for a cylindrical Fermi surface and parabolic bands do not depend on filling.
共33兲
ACKNOWLEDGMENT
In a 3D case, ⌸ M 共q , i0兲 is a rather complicated function of q and Eq. 共33兲 is only an approximation 共as was probably first mentioned by Fröhlich17兲. The physical meanings of the cou-
I.I.M. would like to thank the Max Planck Society for hospitality during his visit to the Max Planck Institute for Solid State Research, where part of this work was done.
I. I. Mazin and V. P. Antropov, Physica C 385, 49 共2003兲. Kong, O. V. Dolgov, O. Jepsen, and O. K. Andersen, Phys. Rev. B 64, 020501共R兲 共2001兲. 3 A. Y. Liu, I. I. Mazin, and J. Kortus, Phys. Rev. Lett. 87, 087005 共2001兲. 4 J. M. An, S. Y. Savrasov, H. Rosner, and W. E. Pickett, Phys. Rev. B 66, 220502共R兲 共2002兲; W. E. Pickett, J. M. An, H. Rosner, and S. Y. Savrasov, Physica C 387, 117 共2003兲. 5 For a review, see I. I. Mazin, L. Boeri, O. V. Dolgov, A. A. Golubov, G. B. Bachelet, M. Giantomassi, and O. K. Andersen, Physica C 460, 116 共2007兲. 6 A. Y. Liu and I. I. Mazin, Phys. Rev. B 75, 064510 共2007兲; M. Calandra, A. N. Kolmogorov, and S. Curtarolo, ibid. 75, 144506 共2007兲. 7 In publications, one can sometimes find the phonon Green functions which differ from ours in some factors. This ambiguity can be traced down to the definitions of the phonon field operators
关Eq. 共1兲兴. In this case, the corresponding factors appear in the electron-ion matrix element 关Eq. 共6兲兴. However, all physical quantities such as EPI coupling constant, Eliashberg functions, etc., are determined by the unique combination 兩g0兩2D, which is independent of definitions. 8 A. B. Migdal, Sov. Phys. JETP 7, 996 共1958兲. 9 P. B. Allen, in Dynamical Properties of Solids, edited by G. K. Horton and A. A. Maradudin, Vol. 3 共North-Holland, Amsterdam, 1980兲, p. 157. 10 P. B. Allen, Phys. Rev. B 6, 2577 共1972兲. 11 M. Calandra and F. Mauri, Phys. Rev. B 71, 064501 共2005兲. 12 F. Stern, Phys. Rev. Lett. 18, 546 共1967兲. 13 T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 445 共1982兲. 14 Y. Zhang, V. M. Yakovenko, and S. Das Sarma, Phys. Rev. B 71, 115105 共2005兲. 15 A. Isihara and T. Toyoda, Z. Phys. B 23, 389 共1976兲; A. Isihara,
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19 P.
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