Pair formation temperature in jelliumlike two-dimensional electron gases

I. Nagy,1,2 N. Zabala,3,2,4 and P. M. Echenique5,2,4 1Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, H-1521 Budapest, Hungary 2Donostia International Physics Center (DIPC), P. Manuel de Lardizabal 4, 20018 San Sebastián, Spain 3Elektrizitatea eta Elektronika Saila, Zientzia eta Teknologika Fakultatea, UPV/EHU, 644 Postakutxatila, 48080 Bilbao, Spain 4Centro de Física de Materiales CFM, Centro Mixto CSIC-UPV/EHU, 20018 San Sebastián, Spain 5Departamento de Fisica de Materiales, Facultad de Quimica, UPV/EHU, Universidad del Pais Vasco, Apartado 1072, 20018 San Sebastián, Spain Received 4 June 2009; revised manuscript received 13 August 2009; published 16 September 2009

There are clear experimental indications 1 that cuprate superconductors can exhibit significant pairing correlation for a range of temperatures that extends above the highest measured superconducting T c . The anomalous normal state of these materials is exemplified by their pseudogap in the excitation spectra, the proper characterization of which is a strong motivation behind various more recent experimental works [2][3][4][5] also. The problem of pair formation is an important issue to a successful theory in our understanding of hightemperature superconductivity in cuprates. A very recent observation 6 on the onset of a pseudogap in a nearly optimally doped pnictide superconductor ͑T c Ӎ 50 K͒ gives an additional strong motivation to the present attempt on pairing since the experimental onset temperature ͑about 180 K͒ is in the range of 100-250 K, estimated experimentally 1-5 for high-temperature cuprate superconductors. These data suggest us a search for clarification of common origin.
On the theoretical side of the pairing problem in jelliumlike systems, we should mention the earlier systematic studies 7,8 of the group of Jankó. Their method is based on the so-called pair-correlation approximation 9 of Kadanoff and Martin for superconductivity, with a built-in attractive interaction ͑to a conventional s-type channel͒ in momentum space. In brief, the key element of this approximation is the many-body scattering t matrix. This is expressed selfconsistently in their framework in terms of the built-in attractive interaction in momentum space and the pair susceptibility. Then, a resonant ͑above T c ͒ pair scattering is characterized 7 by the condition that the real part of the inverse t matrix is zero.
Following the lead of a more recent work 10 by Galitski, where a built-in l wave ͑l Ͼ 1͒ pairing was applied to analyze experimental 3 predictions, we shall use the T p notation for this thermodynamical quantity in the present work on pair ͑p͒ formation in a two-dimensional ͑2D͒ fermionic system. Galitski discussed, within the framework of a disordered BCS theory, the fluctuational aspects of the local 3 pairing temperature above T c and deduced a dome-shaped form for T p as a function of doping.
In the present study on the onset temperature ͑T p ͒ for pairing in 2D electron gases we combine a well-known conventional many-body approximation [11][12][13] for the pairing vertex ͑⌫ ͒ in momentum space and the modeling of the input effective interparticle interaction using real-space constraints. Such a transparent combination is motivated, apart from its intrinsic theoretical interest, by the experimentally established local pairing picture 3 and intricate duality 14 between real and momentum spaces. The obtained estimations for the onset temperature for pairing are in the abovementioned experimental range. Thus, the underlying treatment could provide an important contribution to our general understanding of a phenomenon which appears in the normal state of different unconventional superconducting materials.
According to the many-body theory [11][12][13] in momentum space, in the temperature-dependent formalism of the twoparticle Green's function for a normal Fermi system, the Cooper ͑opposite momenta͒ channel is characterized by zero total momentum at the Fermi surface. This implies a vanishing excitation energy ͑͒ for quasiparticle-quasiparticle scattering. In this formalism the signal for pair formation appears ͑at T p ͒ as a pole of the total pairing vertex function. In order to solve the Cooper problem one has to expand the irreducible interaction ͑⌫͒ in the eigenfunctions of the angular momentum 11 at the Fermi surface with zero total momentum.
Under this procedure, the integral equation for the pairing ͑⌫ ͒ vertex decouples [11][12][13] to a set of algebraic equations for its partial ͑l͒ component ⌫ l . In each equation, a partial component of ⌫ l is coupled to a component of ⌫ l as given ͑Hartree atomic units are used͒ below: In this equation = ͑1 / 2͒ln͑E F / k B T͒ at vanishing excitation energy at the Fermi surface 13 of the 2D system. The decoupling in the Cooper channel has a profound consequence, as was emphasized 11 by Lifshitz and Pitaevski. The liquid state is unstable against pairing if there is an attraction even for a single ⌫ l , at which the denominator of the above equation becomes zero. If several ⌫ l Ͻ 0, the pair formation occurs 11 at that temperature ͑denoted here by T p ͒ which corresponds to the largest ͉⌫ l ͉ value, i.e., the strongest attraction. Introducing the ⌫ l = ͑1 / 2k F ͒␥ l notation, this T p is determined ͑for a ␥ l Ͻ 0͒ from since E F =1/ r s 2 and k F = ͱ 2 / r s in 2D. With a fixed ͉␥ l ͉ there is ͑for ␥ l Ͻ 0͒ an optimal ͑o͒ density parameter ͓r s ͑o͒ =2 ͱ 2 / ͉␥ l ͉͔ at which the pair formation temperature T p has its maximum ͑m͒ value ͑in a.u.͒ of Thus, we get T p ͑m͒ = 540͉͑␥ l ͉͒ 2 in thermodynamical units, K. The l-dependent partial component is where V eff ͑q͒ is an effective ͑input͒ interaction in 2D which models the quasiparticle-quasiparticle irreducible vertex ͑⌫͒ in the Cooper channel. Notice that according to Mermin's theorem, for the ͑ / q͒ → 0 limit this vertex is zero 11,15 on the Fermi surface at q → 0. The practical attempt in the present work rests, partly, on our previous 16,17 experience of modeling interparticle interactions in 2D jellium systems. In brief, we modeled the normalized holes surrounding moving electrons using realspace arguments such as the physically limited magnitude of charge depletion at a repulsive-particle position. Here, the following three models for an instantaneous hole are investigated. The Gaussian ͑G͒ distribution with ⌬n G ͑q͒ = exp͑−q 2 / 2␤ 2 ͒ Fourier transform. The more extended hydrogenic ͑H͒ distribution for which we get ⌬n H ͑q͒ = ␤ 3 / ͑␤ 2 + q 2 ͒ 3/2 . Finally, a very extended powerlike ͑P͒ distribution is introduced, which also has a finite value at r = 0. In this case one has ⌬n P ͑q͒ = exp͑−q / ␤͒. In our comparative study we fix ␤ = k F = ͱ 2 / r s ; i.e., we apply the complete depletion at r = 0. This is supported by recent Monte Carlo data 18 on the paircorrelation function at contact. Next, by using a symmetry argument 17,19 for real-space screening of two ͑equivalent͒ charges in their many-body environment, and the rules of convolution, we write for the quasiparticle-quasiparticle effective interaction in wave vector ͑q͒ space where V͑q͒ =2 / q in 2D for Coulomb potential. For all interaction V eff ͑i͒ ͑q =0͒ = 0, i.e., the properly weighted sum of scattering phase shifts 17,20 is zero. Thus, our effective interactions are well constrained considering the important 11,15 limit value of a renormalized scattering vertex at q → 0 on the Fermi surface.
For the details on the real-space V eff ͑R͒ forms we refer the interested reader to earlier 16,17 works, in the Gaussian and hydrogenic cases; R is the interparticle relative distance. In the powerlike case one gets via standard inverse Fourier-Hankel transformation

͑9͒
This becomes attractive only at high R values. A convenient representation of the key quantity ͑␥ l ͒ to the above Eq. ͑2͒ is as follows:

͑10͒
This expression is used in our practical estimations based on three models with ͑common͒ complete depletion. Due to this physical constraint on normalized holes, the calculated ␥ l ͑i͒ become pure numbers. For completeness, we would like to emphasize at this point of evaluation that with a simple Thomas-Fermi ͑TF͒type model ͓⌬n TF ͑q͒ =2/ ͑2+q͔͒ we get via Eqs. ͑4͒ and ͑8͒ in the low density limit. This becomes negative only for even l, with a maximum in ͉␥ l ͑TF͒ ͉ for l = 2. This particular conclusion fits to the statement of Galitski and Das Sarma, who discussed 21 a retarded interaction to pairing in the d channel. We refer, at this point of discussing an effective interaction, to the particularly clear work 22 of Anderson on the possible ways of eliminating the elementary repulsive force between electrons. The high l scaling, i.e., log͑T p ͒ϳ͓−͑2l͒ 2 ͔ obtained from Eq. ͑2͒, is in harmony with the 2D result 12 of Chubukov. The difference in comparison with the pioneering 23 three-dimensional ͑3D͒ case treated by Kohn and Luttinger is notable since in 3D one gets a log͑T p ͒ϳ͓−͑2l͒ 4 ͔ dependence for high l.
It is clear from Eq. ͑2͒ with Eq. ͑11͒ that a TF modeling results in an exponentially decreasing ͑T p / E F ͒ as a function of r s at low densities. This r s dependence of the exponent is opposite to the one obtained in our study where the relevant ␥ l is a number. In our modeling of an interparticle interaction between system particles, the effective range is determined by the extension ͓ϰ͑1 / k F ͒ϳr s ͔ of a normalized comoving hole. We stress that such a scaling in screening is the only one, according to detailed scattering 24,25 calculations, which can yield a perfect agreement with an exact limiting behavior in r s of the pair-correlation function at contact.
We performed the numerical calculations, using Eq. ͑10͒, with the inputs based on Eqs. ͑5͒-͑7͒ and the prefixed ␤ = k F = ͱ 2 / r s . For the Gaussian model 17 16,17 character due to strong dynamical correlation, are in the experimental range. The powerlike model, due to its extension in real space, results in values which are out of this range.
In conclusion, motivated by experimental indications on the values of pair formation temperatures in the normal state of different unconventional superconductor materials, a model calculation is performed within the framework of the Bethe-Salpeter ladder method in order to point out a possible common origin to a proper phenomenology. We imple-mented the many-body approximation by physically constrained inputs.
The applied constraints are based on simple but fundamental aspects of modeling in real and momentum spaces simultaneously. The theoretical pair formation temperatures are in the experimentally established range. Our results show that it is the extension of comoving holes around electrons in their many-body environment which influences the quantitative prediction for pairing.