Possible experimentally observable effects of vertex corrections in superconductors

Transcription

1 PHYSICAL REVIEW B VOLUME 58, NUMBER 21 1 DECEMBER 1998-I Possible experientally observable effects of vertex corrections in superconductors P. Miller and J. K. Freericks Departent of Physics, Georgetown University, Washington, D.C E. J. Nicol Departent of Physics, University of Guelph, Guelph, Ontario, Canada Received 20 May 1998 We calculate the effects of vertex corrections, of nonconstant density of states and of a self-consistently deterined phonon self-energy for the Holstein odel on a three-diensional cubic lattice. We replace vertex corrections with a Coulob pseudopotential C * adjusted to give the sae T c, and repeat the calculations, to see which effects are a distinct feature of vertex corrections. This allows us to deterine directly observable effects of vertex corrections on a variety of therodynaic properties of superconductors. To this end, we eploy conserving approxiations in the local approxiation to calculate the superconducting critical teperatures, isotope coefficients, superconducting gaps, free-energy differences, and therodynaic critical fields for a range of paraeters. We find that the dressed value of is significantly larger than the bare value. While vertex corrections can cause significant changes in all the above quantities even when the bare electronphonon coupling is sall, the changes can usually be well odeled by an appropriate Coulob pseudopotential. The isotope coefficient proves to be the quantity that ost clearly shows effects of vertex corrections that can not be iicked by a C *. S X I. INTRODUCTION The theory of conventional, low-teperature superconductors has been well understood for decades, within BCS theory, 1 and any aterial properties of superconductors have been accurately described within the ore appropriate Migdal-Eliashberg foralis, 2,3 which includes the retardation effects of the electron-phonon interaction in a realistic anner. The success of the foralis arises fro Migdal s theore, 2 which says that vertex corrections can be neglected when the ratio of the phonon energy scale to the electron energy scale is sall such as the value of 10 4 typical of conventional low-teperature superconductors. The physical reason being that the ion oveent is typically too slow to respond to anything but the ean-field potential produced by the fast-oving electrons. In recent years there has been investigation of superconducting aterials 4 10 such as Ba 1 x K x BiO 3,K 3 C 60, and the A15 s, whose phonon energy scale is a larger fraction of their electron energy scale. For such aterials, the standard Migdal-Eliashberg theory ay no longer be valid, either because second-order processes in the electron-phonon coupling so-called vertex corrections can not be neglected, or because structure in the electronic density of states and finite-bandwidth effects becoe significant. 16 Our ai is to identify those experients which can clearly indicate where vertex corrections, or structure in the electronic density of states, are observable in real aterials. In particular, we wish to uncover those effects which can not be iicked by a Coulob pseudopotential C * as this would render the unobservable in practice, because a C * is typically fitted to the experiental data. The ost copelling evidence for the effect of vertex corrections would coe fro a tunneling conductance easureent that provided data for the tunneling conductance out to an energy at least twice that of the axial phonon frequency of the bulk aterial. Then a tunneling inversion could be perfored using only the experiental data out to an energy of the axial bulk phonon energy, and the results for Migdal-Eliashberg theory would be copared to the vertex-corrected theory for the experiental data that was easured in the ultiphonon region, at voltages above the axial bulk phonon energy of the aterial. Unfortunately, such an analysis has only been perfored for lead 12 and in that case, the experiental data was not accurate enough in the ultiphonon region to be able to see if effects of vertex corrections were observable. Siilarly, high energy data fro optical conductivity experients would indicate whether or not vertex corrections are iportant, but there again, the accuracy of the data ight preclude seeing effects of vertex corrections. In order to deterine what effects are unistakenly due to vertex corrections, we fit a Coulob pseudopotential C * to a Migdal-Eliashberg theory so that the superconducting transition teperature T c is the sae for the vertex-corrected theory with no C *) and the conventional Migdal-Eliashberg theory with a C * but no vertex corrections. In the case of dressed phonons, we adjust both the electron-phonon interaction energy and C *, in order to fit the sae values of and T c as the results with vertex corrections. This procedure is exactly the procedure carried out in analyzing experiental data within the conventional theory. We carry out this experiental analysis on a odel syste where we know that vertex corrections have a large effect, in order to see the extent to which the conventional analysis asks their observation. Hence we look for differences in therodynaic quantities between these results and those with vertex corrections. Any quantities which are significantly different /98/58 21 / /$15.00 PRB The Aerican Physical Society

2 PRB 58 POSSIBLE EXPERIMENTALLY OBSERVABLE EFFECTS FIG. 1. The single-spin density of states for noninteracting electrons on a 3D cubic tight-binding lattice. Note the nature of the van Hove singularities, which lead to an abrupt fall in the density of states. They occur at electron fillings of n 0.45 and n The approxiate for used in the calculations is also plotted, and is indistinguishable by eye fro the exact curve. when calculated with a fitted C * instead of vertex corrections pin-point the experients that can give the best indication that a aterial needs to have vertex corrections included in its description. Since this is the initial attept at solving such a proble, we study a siple odel syste, which has a nonconstant electronic density of states Fig. 1, using both Migdal- Eliashberg foralis, and going beyond it, to include vertex corrections. The vertex corrections are second-order diagras where a pair of phonon lines cross, as shown in the self-energies of Fig. 2 b. In both cases, the phonon propagators can be dressed i.e., with a phonon self-energy included or bare, as shown in Fig. 2. We find that dressing the phonons leads to a strong enhanceent of the effective interaction strength, exeplified by a large renoralization in the value of the electron-phonon coupling paraeter. The specific odel we study is the Holstein Hailtonian 17 on a three-diensional 3D cubic lattice. The Hailtonian consists of conduction electrons that hop fro site to site, coupled to haronic, localized Einstein phonons: Ĥ Nˆ i, t ij ĉ i ĉ j 1 j, 2 M 2 2 xˆi i 1 2M i pˆi 2 gxˆi nˆi, 1 i where ĉ i and ĉ i are ferionic operators which create and destroy, respectively, an electron of spin in a single Wannier tight-binding state on the lattice site i, whose total electron occupancy is given by n i ĉ i ĉ i ĉ i ĉ i. The electron hopping is between nearest neighbors only, such that t ij t if i, j are neighboring sites, with t the overlap integral, and t ij 0 otherwise. The phonons, of ass M with displaceent x i and oentu p i are characterized by their FIG. 2. Different fors for the electron self-energy. All diagras include the sae initial Hartree ter, followed by the specific Fock contribution and any other required diagras. All electronic Green s functions are dressed, but phonon lines can be undressed thin or dressed thick as in Fig. 3 b. For bare phonons, a is with no vertex corrections, while b includes the vertexcorrected diagra. When the phonons are dressed, c is without vertex corrections, while d includes the. A factor of 2 appears in front of each electron loop ter, to indicate a su over spins. frequency. The strength of the bare electron-phonon coupling can be easured by the bipolaron binding energy U, where U M. 2 2 The cheical potential is, and is always calculated selfconsistently for a given average nuber of electrons per site n (0 n 2). Particle-hole syetry occurs at half filling, where n 1 and U. We concentrate our work on superconductivity and ignore any possible charge-density wave order that ay occur near half filling. We carry out weak-coupling expansions within the conserving approxiations of Bay and Kadanoff The electron self-energy is given as a functional derivative of the free-energy functional with respect to the electron Green s function G. When dressing the phonons, we aintain a conserving approxiation by careful choice of the phonon self-energy. In this case, there ust be a free-energy functional whose partial derivative with respect to the electron Green s function yields the electron self-energy and whose partial derivative with respect to the phonon Green s function D yields the phonon self-energy. In such cases there always exists a related free-energy functional, which is the skeleton-diagra expansion whose full functional derivative with respect to the electron Green s function yields the electron self-energy. In all cases, the irreducible vertex func- g2

3 P. MILLER, J. K. FREERICKS, AND E. J. NICOL PRB 58 tion is given by the full functional derivative of the electron self-energy with respect to the electron Green s function, i.e., T n, d (i n )/dg(i ). We eploy the local approxiation in our calculations, which eans we neglect the oentu dependence in the self-energy and irreducible vertex functions. The local proble gives an exact solution in the infinite-diensional liit, 11,22 but in this case it is an approxiation, which, as used in Migdal-Eliashberg theory, is expected to give good quantitative results, though of course it eans we can only study s-wave pairing. We use the foralis, described in detail in the next section, to calculate the critical teperature T c and the isotope coefficient by looking at the instability of the noral state. Also, within the superconducting state, we calculate the superconducting gap and the therodynaic critical field H c fro the free-energy difference F S F N. The conserving nature of each approxiation is evidenced fro the fact that the T c deterined by extrapolating the ordered-phase calculations to zero order paraeter agrees withe the T c found fro the Owen-Scalapino ethod 23 in the noral state. We point out in particular those results which deviate fro Migdal-Eliashberg theory, and arise fro vertex corrections or a nonconstant density of states. By including a Coulob pseudopotential, which causes the sae change in T c as vertex corrections, we are able to deonstrate what experients can be used to differentiate between vertex corrections and a Coulob pseudopotential. Section II contains the foralis, describing the particular approxiations we use, and explaining how our calculations are carried out. Section III presents our coputational results, including a nuber of graphs depicting the various quantities that can be deterined experientally for real systes. We ake our conclusions in Sec. IV, which is followed by an Appendix which gives the detailed forula withheld fro Sec. II. FIG. 3. Dyson s equation for the phonon propagators. Thin lines indicate bare Green s functions while thick lines indicate dressed ones. a is the phonon self-energy without vertex corrections, b is with vertex corrections. The factors of 2 coe fro a su over electron spins. while the irreducible vertex function for superconductivity is given by T n, i n F i. Figure 4 shows the Feynan diagras which correspond to the appropriate free-energy functional for each of the four calculations, while we defer the specific forulas to the ap- II. FORMALISM In our calculations, we use four different types of conserving approxiations, which will be expounded below. Four different approxiations are necessary, in order to reveal how dressing the bare phonons affects solutions of the odel, as well as to deonstrate the effects of vertex corrections. Each approxiation includes a specific selfenergy, and hence a specific vertex function, as shown in Figs In short, these can be described as a Migdal- Eliashberg approxiation with a truncated dressing of phonons, b second-order perturbation theory, which contains vertex corrections and a truncated dressing of phonons, c Migdal-Eliashberg theory with dressed phonons also known as the shielded potential approxiation 19, and d vertex-corrected theory with dressed phonons. The calculations are carried out on the iaginary axis, 22 with the Green s functions and self-energies defined at Matsubara frequencies i n (2n 1) it, where T is the teperature. The self-energies are calculated fro the derivative of an appropriate free-energy functional as i n 1 T G i n, i n 1 T F* i n, FIG. 4. Free-energy functional for the conserving approxiations. Equations a and b are with bare phonon propagators, while c and d have dressed phonons. The first ter on the right in each equation is the Hartree ter, which does not change and which is not included in. Its phonon line is always undressed, to avoid double counting. The extra, final diagra in b and d is the vertex-correction ter.

4 PRB 58 POSSIBLE EXPERIMENTALLY OBSERVABLE EFFECTS The phonon propagator D(i ) is defined by 26 FIG. 5. The irreducible vertex-function diagras, for superconductivity. a is with no vertex corrections, while b includes the three vertex-corrected diagras. When the phonons are dressed, c is without vertex corrections, while d includes the. pendix. The functional whose full derivative leads to the electronic self-energy, is equal to when the phonons are bare, but consists of an infinite nuber of skeleton diagras when the phonons are dressed. The precise relationship between and is given in Eq. A18. When calculating properties for the superconducting state, we ust use the Nabu foralis, 24 where the electron Green s function and self-energy are represented by a 2 2 atrix G i n G i n F i n F* i n G i n d i n 3 i n 1, 3 where (0) ( ) is the approxiate for for the noninteracting electron density of states on a 3D cubic lattice as shown in Fig. 1, given by Uhrig, 25 and i n i n i n, * i n * i n Here, the diagonal and off-diagonal Green s functions are defined respectively as G i n 1 d exp i n T N ĉ j ĉ j, j 0 5 F i n 1 d exp i n T N ĉ j ĉ j, j 0 6 where T denotes tie ordering in. The definitions of (i n ) and (i n ) for each calculation are given in the appendix, Eqs. A2 A16, and are represented in Fig. 2. D i 2 d exp i Txˆ N j xˆ j, 7 j 0 such that the bare propagator, with no self-energy, D (0) (i ), is equal to 2 /( 2 2 ). The appropriate Matsubara frequencies in this case, are those which lead to bosonic statistics, such that i 2 it. When we use bare phonons, and have no vertex corrections, the electron self-energy is illustrated in Fig. 2 a. The first ter is the Hartree contribution Un. It can be included by a shift of the cheical potential as it only contributes a constant to the diagonal part of the self-energy (i n ). Such a shift is iplicitly included when appears in a Green s function, so that the Hartree ter is hereafter neglected. It is followed by the Fock ter, then a single-phonon dressing ter, where the phonon line includes a single loop, which is the electron polarizability (0) (i ): i 2T G i G i F i F* i, 8 with the factor of 2 arising fro the suation over spin. The third ter includes dressing of phonons in a truncated anner, and allows us to ake coparison with the coplete second-order approxiation, in Fig. 2 b, where the only difference is the inclusion of the vertex-correction ter. The specific forula for the extra ter in Fig. 2 b, which enters our calculations when we include vertex corrections using bare phonons, is given in Eqs. A7 and A8. The extra contributions to the self-energy coing fro vertex corrections have opposite sign to the Fock ter, near half filling, where the product of two electronic Green s functions is negative the Green s functions are pure iaginary at half filling. Away fro half filling, the Green s functions gain real parts, which eans that near the band edges the product of two Green s functions can be positive, and the vertexcorrection ters then add to the Fock ter. In the calculations with dressed phonons, we have a Dyson s equation for the phonon propagator, as depicted in Fig. 3, D i D i D i i D i, 9 where (i ) is the phonon self-energy. Note that within the conserving approxiations, we ust deterine the phonon self-energy by differentiating the free-energy functional: i 2 T D i. 10 With no vertex corrections, the phonon self-energy is siply given by the electron polarization, as depicted in Fig. 3 a. That is, i U i. 11 In this case, we use the contributions to the electron selfenergy that are shown in Fig. 2 c, whose explicit forula is given in Eq. A11. Note that the ter with the single polarization bubble is issing, as all orders of such necklace

5 P. MILLER, J. K. FREERICKS, AND E. J. NICOL PRB 58 diagras are included in the Fock ter with the dressed phonon propagator. In fact, the approxiation including dressed phonons without vertex corrections is equivalent to the shielded potential approxiation, whereby the infinite series of ring diagras without vertex corrections are included in the skeleton-diagra functional. When we include vertex corrections, as well as dressed phonons, the phonon self-energy gains the extra ter shown in Fig. 3 b. The full expression is given in the Appendix Eq. A16. Note that a fully dressed phonon propagator is included in the phonon self-energy. As shown in Fig. 2 d, the electron self-energy now has the expected extra ter with a crossing of phonon lines. It is alost identical to the extra ter in Fig. 2 b, except of course, now the phonons are dressed. Again, the details of the forula can be found in the Appendix. Our calculations all involve iteration of the Green s functions and self-energies, until a self-consistent solution is reached. We begin with the noninteracting Green s functions set the self-energies to zero, and use it to calculate an initial estiate of the self-energies, according to Eqs. A2 A16. The new self-energies are used to calculate updated Green s functions, according to Eqs. 3 and 9. The procedure is iterated, so that at each step there is an updated self-energy, which includes a fraction of the previous self-energy, the exact fraction variable, dependent upon the progress of the iteration. We stop the process when the change in all the self-energies is less than one part in 10 10, which is typically after tens, but soeties after hundreds of iteration steps. Superconductivity occurs below a critical teperature T c, where the noral state becoes unstable to fluctuations in the pairing potential the Cooper instability. The instability shows itself as a divergence in the pairing susceptibility,n which is given by,n,n T l,l l,, 12 where,n is the irreducible vertex function, to be defined shortly. The bare susceptibility in the superconducting channel for oentu q is defined as q 1 G i N,k G i, k q, 13 k which becoes in the local approxiation for the zerooentu pair : where I G i Z i, 14 Z i I i. 15 The transition teperature T c occurs when the largest eigenvalue of the atrix T (0),l passes through unity. We calculate the highest eigenvalue using the power ethod. The irreducible vertex function,n is given by the functional derivative of the off-diagonal self-energy, with respect to the off-diagonal Green s function T n, i n / F i. 16 As the pairing fluctuations lead to an instability of the noral state, n, ust be calculated in the noral state i.e., in the liit F(i n ) 0]. Figure 5 shows the contributions to the irreducible vertex function for each of the four approxiations. Each diagra is achieved by reoval of one electron Green s function line, fro a self-energy diagra in Fig. 2. The algebraic expressions for each of the four sets of diagras are given in the Appendix. Note that because the calculations are carried out in the noral state, with F 0, rings within the self-energy can not be broken, as the resulting ladder diagras give zero contribution. Copare Fig. 5 a arising fro Fig. 2 a. As we wish to uncover ore than the phase diagra given by the different perturbation approxiations, we ove on to describe how we calculate other properties. In order to find the superconducting gap and therodynaic quantities, calculations are required within the superconducting state, but a siple addition to the previous coputations in the noral state allows us to calculate the isotope coefficient. The isotope coefficient describes how the critical teperature changes with the phonon ass M. It is defined as d ln T c d ln M, 17 so that T c M. The weak-coupling liit of BCS theory, and Migdal-Eliashberg theory with no Coulob repulsion, predict 0.5, which corresponds to T c 1/ M. The phonon frequency changes with ass, according to 1/ M, so both the product M 2 and the interaction energy U reain constant. Hence we calculate the isotope coefficient siply by changing the phonon frequency by 1% corresponding to a typical ass change of 2% between isotopes and coparing the change in critical teperature. To be precise, we copute by 0.5 T new old c T c old old T c new, old 18 so the BCS result is achieved if T c. According to standard ethods, the energy gap in the superconducting state requires a self-consistent calculation within the superconducting state. Note, the order paraeter on the iaginary axis is related to the off-diagonal selfenergy through i n i n /Z i n, 19 where Z(i n ) is the ass-enhanceent paraeter, calculated fro the electronic self-energy (i n ) as given in Eq. 15. The gap itself is found fro the order paraeter on the real axis, at the point where Re ( ). We carry out a Padé analytic continuation 30,31 to obtain the order paraeter on the real axis, and hence the value of the gap. The free energy can be found fro the forula 32 34

6 PRB 58 POSSIBLE EXPERIMENTALLY OBSERVABLE EFFECTS F 2T n 1 2 ln[ 1/detG(i n)] 1 2 Tr (i n)g(i n ) T 2 ln 1/D i i D i n The free-energy functional whose partial differential with respect to the electron or phonon Green s function gives the electron or phonon self-energy is depicted in Fig. 4 and evaluated in Eqs. A2, A6, A11, and A15. We are interested in the free-energy difference between noral and superconducting states at fixed electron filling n in which case the first, Hartree, ter in is neglected as it is a constant Un 2 /2. The final ter, (n 1) cannot be neglected, because the cheical potential can differ considerably between the superconducting and noral state when one includes the effects of nonconstant density of states. We calculate the therodynaic critical field in the superconducting state H c fro the free-energy difference between the superconducting and noral state, according to the forula F S F N 0 H 2 c. The therodynaic field varies with teperature in an alost quadratic anner, so that calculation of the deviation function, which is defined as the difference between H c (T) and the quadratic for H c (0) 1 (T/T c ) 2 gives a sensitive test of changes in therodynaic quantities. In the calculations which include a Coulob repulsion ter U C, we ake the standard siplification 16,27 29 of only including its effects on the off-diagonal self-energy. This siplification is valid, as the noral-state Green s functions in reality include the Coulob repulsion effects, and these change by very little for the diagonal part of the Green s function when superconducting order is present. As our odel does not include the effects of U C on the diagonal Green s functions, it is not strictly the solution of a siple Hailtonian with a U C ter included. 35 However, the siplification does allow us to copare the effects of including a Coulob repulsion versus adding vertex corrections, on the superconducting properties and transition teperature fro a siilar noral state and it is precisely the ethod eployed in analyzing experiental data on real aterials. With these coents understood, the only changes to the calculations that are necessary with the inclusion of a Coulob ter, are that both the off-diagonal self-energy (i n ) and the irreducible vertex function n, gain an extra ter: i n i n U C T F i, 21 n, n, U C. 22 In the coputational calculations, the Matsubara frequency su is cut off after a constant nuber N c of ters. This leads to a renoralization which reduces the Coulob ter 16,36 to a pseudopotential U C * given by U C * U C I G i 1 2TU C Nc 1, 23 where the diagonal Green s function G(i ) is at high frequency, where the self-energies ay be neglected, but ust include the self-consistent cheical potential. In our calculations, with the frequency cutoff on the scale of the bandwidth such that 256 or 512 Matsubara frequencies are used, there is a very sall typically 1 or 2 %) reduction in U C. This is different fro the conventional approach in real aterials because here our energy cutoff is governed by the electronic bandwidth, not soe ultiple of the axiu phonon frequency. To ake contact with the standard foralis, we define a diensionless pseudopotential C * (0) ( ) U C *, where (0) ( ) is the noninteracting electronic density of states at the cheical potential. In calculations at different teperatures, U C is kept fixed, so that C * varies to a sall extent. Finally, we wish to ake clear how, the easure of the electron-phonon coupling strength, is defined in our work. A precise definition is required, because different ethods of calculating lead to different results away fro the weakcoupling liit, especially when the phonons are dressed. Here, is given by U D, 24 where D(0) is the zero-frequency coponent of the dressed phonon propagator, which can be significantly different fro that of the bare propagator D (0) (0). This value of is usually different fro the electronic ass enhanceent paraeter, i.e., Z(0) 1, as the two are only equal in the weakcoupling liit and with 0. The definition of is identical to that coonly calculated fro the electronphonon spectral density, 27 2 F( ), naely, 2 2 F d, 25 0 where 2 F U 1 I D. 26 The real-axis for of the phonon propagator D( ) is calculated fro its iaginary-axis values D(i ) by a Padé analytic continuation. III. RESULTS In choosing the paraeters used to carry out the calculations, a nuber of criterion had to be satisfied. First, we wished to operate outside the weak-coupling regie, so that vertex corrections would not be negligible. The phonon frequency needed to be large copared to conventional lowteperature superconductors, but not so large that it gave no realistic point of contact with those superconductors entioned in the introduction. So we choose t, equal to one twelfth of the bandwidth. We had to ensure the electronphonon coupling strength was not so strong that the ground state would contain bipolarons, 37 aking the perturbation expansion about a Feri liquid state invalid. A axiu cou-

7 P. MILLER, J. K. FREERICKS, AND E. J. NICOL PRB 58 FIG F( ), calculated fro the dressed phonon propagator on the real frequency axis for n 1 and t. Increased coupling leads to a greater downwards shift in frequency fro the initial delta function at t. pling of U 2t, hence a bare 0.5 ensured this. Finally, in order to calculate properties in an achievable tie, while ensuring the iaginary frequency cutoffs were at energies larger than the band-width, the teperature of the calculations could not be too sall, hence T t. The last condition eant that results within the superconducting state were best carried out for as large U and as possible, so that T c would be high. Hence, calculations near the band edges, where the density of states was low, prove to be inaccurate, due to the very low transition teperatures there. Our first result is that dressing the bare phonon propagator leads to considerable renoralization effects. To be specific, the value of doubles fro its bare value of 0.21, to 0.4 after dressing the phonons, using paraeters t and U 1.5t, near half filling. Moreover, at the increased interaction strength of U 2t, is enhanced by a factor of 3 fro the value of 0.28 for undressed phonons to the dressed value of 0.9. Such an enhanceent indicates that the perturbation expansion would be inaccurate at bare coupling strengths lower than ight be naively expected. Figure 6 shows how 2 F( ) is altered fro its bare value, a delta function situated at t, when it is dressed. Note that there is both a shift to lower frequencies as well as a broadening of the spectru. The shift to lower frequencies shows that a Holstein odel with bare phonon frequencies of near 10% of the bandwidth can be required to lead to dressed phonon frequencies at approxiately 5% of the bandwidth. Hence the exact phonon self-energy used is an iportant factor when odeling electron-phonon systes, near the crossover between the weak-coupling and strong-coupling regies. The preceding paragraph explains soe of the draatic differences between the critical teperature (T c ) values for the different approxiations, shown in Fig. 7. In particular, the T c for dressed phonons is arkedly higher than that for bare phonons, which is to be expected as is also higher. Note that the transition teperatures fall rapidly with increasing filling, above a filling of about n 1.5, as the electronic density of states drops significantly in this region. Vertex corrections seriously reduce T c near half filling (n 1). FIG. 7. Transition teperature as a function of filling. The dashed lines indicate vertex corrections are included, while solid lines are without the. The lines with circles have dressed phonons. Both diagras show an enhanced T c by dressing the phonons, while vertex corrections reduce T c near half filling. In all cases, T c falls rapidly above a filling of n 1.55, where the density of states drops rapidly. The two curves with dressed phonons in Fig. 7 show a greater disparity than the two curves with bare phonons in the sae figure, which eans that dressing the phonons, which increases the effective coupling, enhances the effect of vertex corrections. Although it is hard to distinguish the curves due to the low T c near the band edge, above a filling of n 1.75 the vertex corrections do lead to an enhanceent of T c. This result is in agreeent with previous work. 12,14 Note that all results show particle-hole syetry, that is, they are syetric about half-filling. The figures just show half of the band (n 1), neglecting a irror iage below half filling. It is clear that vertex corrections do change T c by a considerable aount, but for any experiental easureent, where the icroscopic paraeters are not known, a Coulob pseudopotential C * can always be fitted to give the sae T c as vertex corrections. Hence we continue with other results, to see where vertex corrections can not siply be iicked by an appropriate C *, which would cause the effects of vertex corrections to be unobservable. So we fit a C * to give the sae value of T c as vertex corrections, and go on to change the unobservable electron-phonon coupling strength U to give the sae as vertex corrections, when phonons are dressed. Hence the effects of vertex corrections can show up as discrepancies over a range of quantities copared to the values obtained with a fitted C * and. Figure 8 is a direct coparison between the effects of vertex corrections and a Coulob pseudopotential C * on the value of the gap paraeter. Our first ethod is to fix the bare electron-phonon coupling and to adjust the value of U C and hence C * in a calculation without vertex corrections, until the sae T c is reached as found in the calculation with vertex corrections shown in Fig. 7 b. U C is then used unchanged, to calculate other properties such as the gap paraeter. The Coulob pseudopotential C * varies with electron filling at T c as shown in the inset. Notice that an artificial value of C * 0 is required when n 1.7, as vertex corrections enhance T c in this region. Near half filling, where C * is positive, and reduces T c as vertex corrections, the gap paraeter is reduced by a saller aount. Hence a Coulob repulsion leads to slightly higher gap ratio than vertex corrections. The second curve, with a lower value of C *, is an alter-

8 PRB 58 POSSIBLE EXPERIMENTALLY OBSERVABLE EFFECTS FIG. 8. A coparison between the effects of vertex corrections and a Coulob pseudopotential, C * on the superconducting gap. The calculations are with dressed phonons, with U 2t and t. The dashed curve is with vertex corrections, while the dotted curves include a C *, whose value changes with filling as shown in the inset, to ensure the two corresponding T c curves are exactly the sae. The dotted curve with triangles indicates a fit to the sae by adjusting U as well as C *. nate approach, where both the dressed value of and T c are fitted to the results with vertex corrections. A siple fit to T c with a fixed bare coupling leads to a higher with a Coulob repulsion than with vertex corrections, because is deterined through the dressed phonon propagator. As a conventional analysis would fit to the experiental data, as well as C *, this second ethod follows the spirit of our paper by trying to fit conventional theory to the vertexcorrected results. In order to give the sae values of, the electron-phonon interaction energy had to be varied, and in fact reduced by 10% at half filling. The result is a lower value of the gap ratio than with only C * fitted, but still a slightly larger value than with vertex corrections alone. Although the agnitude of the gap varies considerably, depending upon the approxiation used, Fig. 9 shows that the gap ratio 2 /kt c varies less arkedly. The gap ratio is greater than 4 in the case of dressed phonons without vertex corrections, which is typical of the strong coupling regie ( 0.5). Note that when the phonons are bare, so the coupling is less strong, the inclusion of vertex corrections, while strongly reducing T c and individually, has little effect on the ratio 2 /kt c. The isotope coefficient has a value of 0.5 in the siplest, BCS, approxiation, and in Migdal-Eliashberg theory with no Coulob repulsion. The reason is that the phonon frequency provides the only cutoff for the coupling between different states, and phonon frequencies are proportional to M 0.5, where M is the ionic ass. Inclusion of a frequencyindependent Coulob repulsion U C leads to a reduction in, as does a finite bandwidth. The reduction in, indicates that the increase in phonon frequency is less effective at increasing T c than otherwise. Higher-frequency phonons reduce the retardation in the electron-electron attraction, so the Coulob repulsion between electrons is less shielded. The FIG. 9. Gap ratio 2 /kt c as a function of filling n. Dressed phonons with circles exhibit strong coupling behavior, by the increased gap ratio. Vertex corrections dashed lines reduce the effective coupling strength in the center of the band, hence the gap ratio is lower in this region. The dotted lines indicate how a Coulob pseudopotential alters the gap ratio when T c is atched to T c with vertex corrections, and where triangles indicate that is also atched, by adjusting U). With dressed phonons, the Coulob repulsion clearly has less of an effect than do vertex corrections, but this is not the case with bare phonons. finite bandwidth eans that the nuber of states coupled together includes a factor independent of phonon frequency, so T c does not increase with as it ight if there were an infinite nuber of electron states extending through all energies. It is known 38,16 that Migdal-Eliashberg theory with a finite bandwidth and including a Coulob repulsion, leads to the allowed range of values for the isotope coefficient, Including a nonconstant density of states 39,40 can in principle lead to any positive value of. The reason being that T c increases because extra electron states near the cheical potential are able to couple together when the phonon frequency increases. If the density of states increases significantly in the region of energy where new states are coupled together, the increase in T c is uch higher than would otherwise be expected, and can be large, even greater than 0.5. The corollary is that if the density of states decreases away fro the cheical potential, also decreases, but never to less than zero, as T c does not go down when the nuber of states coupled together goes up. Figure 10 shows that the inclusion of vertex corrections in the calculations with dressed phonons not only reduces, but can in fact lead to negative values. Indeed, the strongest reduction in by vertex corrections occurs near half filling, and at strong coupling, where T c is coparatively large. By coparison, in all cases the Coulob pseudopotential, which gives the sae reduction in T c as vertex corrections, leads to a uch saller reduction in. In Migdal-Eliashberg theory, a very sall value of requires a very low T c. Hence, any observation of isotope effects which have 0, or a sall with oderate to high T c, iplies that either vertex corrections are involved or soe other echanis outside of Migdal-Eliashberg theory is iportant. Paraagnetic ipurities, 41,42 proxiity effects, 43 anharonicity, 44,45 and an isotopic dependence of the electron density in the conduction band, 42,43 can also lead to a low or negative without requiring a low T c. One or ore of these effects ay be iportant, in those aterials with

9 P. MILLER, J. K. FREERICKS, AND E. J. NICOL PRB 58 FIG. 10. The isotope coefficient plotted against electron filling n for a U 1.5t and b U 2t. Vertex corrections, indicated by dashed lines, reduce. As does dressing the phonons, seen by the lines with circles. b exhibits the unusual feature of 0 for dressed phonons, with vertex corrections included. The dotted curve in b is the result with the Coulob repulsion C * shown in the inset in the previous figure. anoalously low values of, but vertex corrections should also be considered. An iportant effect of dressing the phonons is that a sall increase in the bare phonon frequency does not just result in a constant shift of 2 F( ), through a rescaling of the frequency variable. When calculating the isotope effect, it is coon to assue that a ass substitution siply rescales the frequency, otherwise aintaining the sae for of 2 F( ). However, the phonon self-energy is not independent of frequency, so the agnitude of 2 F( ) at its peak, which is inversely proportional to the iaginary part of the self-energy at that frequency, does not reain constant. In fact, the peak height is reduced by an increase in peak frequency, resulting in a slight reduction in. Hence the increase in T c with bare frequency is less than otherwise expected, reducing the isotope coefficient for dressed phonons. Interestingly, when the phonons are undressed, the Coulob repulsion leads to a very sall increase in. This arises, because the ter with a single polarization bubble the first order ter coing fro dressed phonons is present. The ter acts to reduce, but is less significant at the lower transition teperatures caused by the Coulob pseudopotential. Meanwhile, the increase in * with teperature, which acts to reduce is a uch saller effect. The free-energy difference F F S F N is plotted as a function of filling n in Fig. 11. In the siplest picture, true in the weak-coupling liit, one expects the agnitude of the free-energy difference to be approxiately equal to ( ) 2 /2, representing a nuber of states ( ) each shifted by an average energy of order /2. Although, with ( ) given by Z(0) (0) ( ), the weak-coupling result predicts too high a condensation energy, 28 it does explain the qualitative changes between the different curves of Fig. 11. In fact, the diensionless quantity T 2 c /(8 F) the Soerfeld constant, 2 2 k 2 B (0) ( )Z(0)/3 changes little for these curves. At half filling, with dressed phonons, the value is reduced fro the BCS constant result of to a strong-coupling value of The value with vertex corrections is 0.157, while with the Coulob pseudopotential it is Hence, as with the gap ratio, near half filling, vertex FIG. 11. Condensation energy, F S F N. Circles indicate the phonons are dressed, which leads to a greater condensation energy. The dotted lines indicate dressed phonons with a Coulob pseudopotential, to iic the vertex-corrected T c, the triangles indicating that U is also adjusted to iic the obtained with vertex corrections. The dashed, vertex-corrected curve shows saller condensation energy for n 1.55, though the difference can not be seen with bare phonons. corrections cause quantities to be closer to the weakcoupling values than does a Coulob pseudopotential fitted for the sae T c. The values at a filling of n 1.6 are all closer to the weak-coupling liit, as expected when the density of states falls. The result is for dressed phonons without vertex corrections, changing little to with a Coulob pseudopotential and with vertex corrections. It is worthwhile pointing out that the difference in therodynaic potentials, which is usually calculated as an approxiation to the free-energy difference, leads to very different results away fro half filling. The approxiation is based on the assuption that the cheical potential changes little between noral and superconducting states, but this is not necessarily the case when there is a nonconstant density of states. In fact, there is a particularly large shift fro the noral-state cheical potential N to that in the superconducting state S if N lies near the van Hove singularity, where the noninteracting density of states is falling precipitously. This can be understood, by considering how states above and below the noral-state cheical potential N couple together and create an energy gap. When the density of states is falling rapidly with increasing energy, electronic states fro a larger energy region above N couple to those in a sall region below N. The resulting energy gap is skewed up in energy, so the cheical potential in the superconducting state S, which sits in the iddle of the gap, becoes greater than N. In the irrored exaple below half filling, near n 0.4, S is also pushed away fro half filling, so we find S N there. The therodynaic critical field H c is effectively the square root of the free-energy difference, so shows qualitatively the sae effects. H c is known 27 to vary with teperature in a anner close to the behavior H c (T) H c (0) 1 (T/T c ) 2 which corresponds to the two-fluid odel. The deviation function, plotted in Fig. 12, is the difference be-

10 PRB 58 POSSIBLE EXPERIMENTALLY OBSERVABLE EFFECTS FIG. 12. Critical-field deviation function H c (T)/H c (0) 1 (T/T c ) 2. Solid lines are without, dashed lines are with vertex corrections. The dotted lines with and without triangles include the Coulob pseudopotentials C * of Fig. 8, inset. All results are for dressed phonons. a is at half filling, n 1 while b is at a filling of n 1.6. Note the shift in scale for the second graph. tween the reduced critical field H c (T)/H c (0) and the quadratic fit 1 (T/T c ) 2. The curve without vertex corrections at half filling shows a sall positive deviation, with a axiu of 0.02, typical of strong-coupling superconductors. Interestingly, when vertex corrections are included, the deviation function at half filling becoes negative, showing a iniu of about 0.03, which is ore typical of weak-coupling superconductors BCS theory predicts a iniu of 0.037). This result fits in with those for other properties, deonstrating that vertex corrections reduce the effective coupling strength, near half filling. A Coulob pseudopotential, with the sae power to reduce T c as vertex corrections, also decreases the deviation function, but to a lesser extent than vertex corrections do so. This is still true when the value of is also fitted, by altering U as necessary. The calculations away fro the band center, at n 1.6, lead to a negative deviation for all approxiations. The reduced density of states at this filling leads to weak-coupling behavior. Other therodynaic quantities, which can be derived fro the free-energy data, are affected in siilar ways. That is, vertex corrections reduce the effective coupling strength, to a greater extent than does a Coulob repulsion giving the sae T c. For exaple, vertex corrections reduce the specific heat jup, C at T c, as does a Coulob repulsion to a lesser extent. The following results are obtained by a nuerical differentiation, so are not copletely accurate in theselves, perhaps only to 10% but as uch of the error is systeatic, the trends are reliable. At half-filling, the diensionless quantity, C/ T c is reduced fro the strongcoupling value of 2.44, to 1.63 with vertex corrections and 1.88 with a Coulob pseudopotential the BCS result is Again, notice the typical result that near half filling, vertex corrections lead to a weaker-coupling result than does inclusion of a Coulob pseudopotential. At an electron filling of n 1.6, the results indicate less strong coupling, giving C/ T c 1.66 for dressed phonons, with the value reduced to 1.53 by vertex corrections, and to 1.44 by a Coulob repulsion. IV. CONCLUSIONS We have copleted a nuerical investigation of the effects of vertex corrections, dressing phonons and a nonconstant density of states on the physical properties of strongcoupling superconductors. We solved the Holstein odel, using four distinct perturbation theories, within conserving approxiations. The necessity of incorporating a realistic phonon selfenergy is of considerable iportance to those working with odel Hailtonians. The use of dressed phonons in the Holstein odel, leads to a large renoralization of the paraeters in particular, the value of can be enhanced by a factor of 3, when its bare value of 0.28 would suggest the syste is in the weak-coupling regie. Such an enhanceent of reveals itself in increased T c,,, H c, and a gap ratio (2 /kt c ) greater than 4. The real-frequency data shows that the Einstein spectru a delta function at ) is both broadened and peaked at a lower frequency, when the bare, Einstein phonons are dressed. The nonconstant density of states affects Migdal- Eliashberg results in both expected and unexpected ways. Firstly, all quantities which depend on the density of states as a paraeter within Migdal-Eliashberg theory change in the expected anner as the electron band filling changes. Note that any sharp features in the noral electronic density of states have their effects reduced by the averaging over a large phonon frequency range. Hence the strong fall in T c and is expected at both sall and large fillings (n 0.45 and n 1.55) due to the rapid fall in the density of states in our odel. More subtle, is the result that the cheical potential shifts by a considerable aount between the noral and superconducting states, if it lies near a van Hove singularity. To observe such an effect, the superconductor would have to be coupled to one with a ore constant density of states. We find, in agreeent with previous work, 11,12,14,15 that vertex corrections lead to results that correspond to a reduced effective strength of the electron-phonon coupling near half filling, but an increased coupling strength near the band edges. These effects are exeplified by reductions in critical teperature T c, superconducting gap, isotope coefficient, and therodynaic critical field H c near half filling. As nearly all of these effects can be odeled by an appropriate Coulob pseudopotential C * it akes it extreely difficult for any single experient to reveal that vertex corrections have played a significant role. However, we do find soe trends worth pointing out. First, if there is difficulty in fitting both T c and with a given 2 F( ) and C *, then this is an indication that vertex corrections ay contribute, since they affect the ratio 2 /kt c for fixed T c, reducing it near half filling. Second, if the experientally easured deviation function for the therodynaic critical field lies below the predicted value with a given 2 F( ) and C *] vertex corrections could be iport ant. Thirdly, a theoretical prediction, ignoring vertex corrections, will overestiate the specific heat jup at T c. Finally, a ore striking result, is in the isotope coefficient, which vertex corrections reduce uch ore arkedly than does C *. Indeed, vertex corrections can lead to 0, which C * alone can never do. 16 Hence aterials that have oderate to large T c s, but sall isotope coefficients can still be electronphonon ediated superconductors with vertex corrections included. As the isotope coefficient is the single experiental

11 P. MILLER, J. K. FREERICKS, AND E. J. NICOL PRB 58 quantity affected the ost by vertex corrections, it is iportant to consider aterials which have unexplained by Migdal-Eliashberg theory. Anoalously low, and even negative isotope coefficients have been observed in aterials, such as Ru, 46 -uraniu, 47 PdH, 48 and La 2 x Sr x CuO 4, where vertex corrections are probably not iportant, and other echaniss, such as anharonicity, conduction electron density variations and paraagnetic ipurities play a role. However, a syste such as Rb 3 C 60, 52,53 where the phonon frequency is a sizable fraction of the electron bandwidth, is uch ore likely to have vertex corrections affect the value of. Still, the best way to see the effects of vertex corrections is to directly view their contribution in the ultiphonon region of a tunnel junction, or in the high-energy region of the optical conductivity. So, in order to unequivocally deonstrate the presence of strong vertex corrections in a aterial, ore accurate dynaical easureents at energies beyond the highest phonon frequencies need to be carried out. V. ACKNOWLEDGEMENTS J.K.F. and P.M. would like to thank the Office of Naval Research for support under Grant No. N E.J.N. thanks the Cottrell Scholar Progra of Research Corporation for financial support and was also supported by the Natural Sciences and Engineering Research Council of Canada. We are grateful for useful discussions with J. W. Serene. APPENDIX: CONSERVING APPROXIMATION FORMULA In this appendix, we give the specific forulas for the free-energy functional, self-energy and irreducible vertex functions for each of the four conserving approxiations. Figures 4, 5 are the representations of these equations as Feynan diagras. Hereafter, we eploy the shortened notation G n G(i n ), G n * G( i n ) and siilarly for F n, D, and (0). Note that the difference of two ferionic frequencies leads to a bosonic frequency, i.e., D n D(i i n ) D(i ), where n. The calculations with no vertex corrections and a bare phonon propagator have the free-energy functional bare UT2 2 n, Tr 3 G n Tr 3 G D UT2 2 n, U 2 T 4 Tr 3 G n 3 G D n D 2, A1 where (0) is the electron polarizability, defined in Eq. 8. The inclusion of the 3 atrices, ensures that each pair of off-diagonal Green s functions F n *F corresponding to Cooper pair creation then annihilation, enters the product with a inus sign. Functional differentiation with respect to the diagonal electron Green s function G n and off-diagonal Green s function F n * leads respectively to the diagonal ter in the selfenergy (i n ), bare i n UTn UT U 2 T and the off-diagonal ter (i n ), bare i n UT U 2 T G D n G n F D n F n D n 2, D n 2. A2 A3 We only require the superconducting vertex part, which is given by the derivative: T bare n, i n / F, A4 taken in the liit F 0. Hence with bare phonons and no vertex corrections, the vertex function is that shown in Fig. 5 a : bare n, UD n U 2 n D n 2. A5 There is only one extra ter which coes fro the inclusion of vertex corrections in the free-energy functional. It is the diagra in Fig. 4 b with crossed phonon lines, and is equal to vc U2 T 3 4 n,,l Tr 3 G n 3 G 3 G l 3 G n l D n D l. A6 The total free-energy functional for bare phonons with vertex corrections included is bare vc. The extra ter in the self-energy is found by differentiation of the above ter. For the diagonal and off-diagonal parts, respectively, this leads to vc i n U 2 T 2 G G l G n l F l F n l *,l F n l F l * G l *F F n l * D n D l, A7 vc i n U 2 T 2,l F F l *F n l G l G n l G n l * G l * F l G G n l * D n D l, A8 The extra ter leads to the new self-energy bare vc and bare vc. When the vertex correction ter is added to the selfenergy, three new ters appear in the irreducible vertex function, coing fro each of three Green s functions that can be differentiated. The extra diagras in Fig. 5 b contribute a total of