A numerical calculation of the electronic specific heat for the compound Sr 2 RuO 4 below its superconducting transition temperature
Pedro Contreras, Jose Burgos, Ender Ochoa, Daniel Uzcategui, Rafael Almeida
AA numerical calculation of the electronic specific heat for thecompound Sr RuO below its superconducting transitiontemperature Pedro Contreras , , Jos´e Burgos , Ender Ochoa , Daniel Uzcategui Departamento de F´ısica, Universidad de Los Andes, M´erida 5101, Venezuela Centro de F´ısica Fundamental ULA
Rafael Almeida Departamento de Qu´ımica, Universidad de Los Andes, M´erida 5101, Venezuela (Dated: December 18, 2018)In this work, a numerical study of the superconducting specific heat of the uncon-ventional multiband superconductor Strontium Ruthenate, Sr RuO , is performed.Two band gaps models are employed, and the results rendered for each of them arecompared. One of the models, previously proposed by one of the authors to explainthe experimental temperature behavior of the ultrasound attenuation, considers twogaps with point nodes of different magnitude on different gap surface sheets, while theother one is an isotropic and line node model, reported in the literature for describ-ing quantitatively experimental specific heat data. The Sr RuO superconductingdensity of states DOS is computed by employing these two models and then, a de-tailed numerical study of the electronic specific heat, that includes the contributionfrom the different Fermi sheets, is carried out. It is found that the calculated pointnode model specific heat temperature behavior shows an excellent agreement withthe existent Sr RuO experimental data at zero field, particularly, it is obtainedthat the observed specific heat jump at T c is precisely reproduced. Also, it is foundthat the sum of the contributions from the different bands fits quantitatively themeasured specific heat data. The results in this work evidence that the Sr RuO superconducting states are of unconventional nature, corresponding to those of apoint node superconductor, and show the importance of taking into account themultiband nature of the material when calculating thermodynamic superconductingquantities. a r X i v : . [ c ond - m a t . s up r- c on ] D ec Keywords:
Electronic specific heat; unconventional superconductors; gap struc-ture; superconducting density of states; point nodes; line nodes.
PACS numbers: 74.20.Rp; 74.70.Pq, 74.25.Bt
I. INTRODUCTION
The strontium ruthenate ( Sr RuO ) is a multiband superconductor with Fermi surfacecomposed of three sheets ( α , β and γ sheets). Sr RuO has a body centered tetragonalstructure with a layered square-lattice similar to that of many high temperature copper-oxide superconductors [1] and its normal state displays Fermi liquid behavior [2]. For puresamples, its critical temperature, T c , is approximately 1.5 K, and is found that T c variesstrongly with non magnetic impurity concentration. It has been proposed that Sr RuO isan unconventional superconductor having some kind of nodes in the superconducting gap[1]. Thus, a number of theoretical works [3–5] have predicted the existence of linear nodeson two of the three Fermi surface sheets ( α , β sheets), while other ones have proposed thatthe γ sheet is nodeless [4, 5]. The predictions in these works agree with the results obtainedfrom measurements of specific heat C ( T ) [6, 7], electronic heat transport κ ( T ) [8], and depthpenetration λ ( T ) [10]. In electronic thermal conductivity and specific heat experiments, thethree sheets have similar contributions to the κ ( T ) and C ( T ) results, i.e. they have anintegral effect. Because of this, from these experiments is very difficult to discern if theorder parameter in each of the Fermi sheets has similar nodal structure. In contrast, fromSr RuO sound attenuation experiments [11] is possible to distinguish the nodal structureof the γ sheet from those of the α and the β sheets. Moreover, experiments on Sr RuO ultrasound nodal activity α ( T ), measured below T c , have yielded the anisotropy inherentto the k -dependence of electron-phonon interaction [12]. The results have showed thatthe γ sheet dominates the ultrasound attenuation α ( T ) for the L [100], L [110] and T [110]sound modes, and that below T c , these three modes exhibit comparable temperature powerlaw behavior. These results lead to think that the γ sheet and the experimentally measuredultrasound nodal activity should have nodal structure alike, which is similar to that displayedby the other two sheets, conclusion that contradicts the proposition of a nodeless γ sheet.Additionally, according to [13], the symmetry of the gap structure is believed to be a timereversal broken state, with the symmetry transforming as the two dimensional irreduciblerepresentation E u of the tetragonal point group D h .An extensive experimental investigation of the electronic specific heat C ( T ) for the un-conventional superconductor Sr RuO has been performed in a series of experiments byMaeno and Collaborators [6, 7, 9]. Through these experiments, they look to elucidate thegap structure of this material by means of electronic specific heat measurements. Amongthese experiments, the Nishizaki and collaborators specific heat measurements [9] on cleansamples of Sr RuO and under zero magnetic field, showed a remarkable near-linear be-havior of C s ( T ) /T at low temperatures. This result provides evidence, supporting the ideathat Sr RuO is an unconventional superconductor with some kind of line of nodes in itsorder parameter. They pointed out that the measurements results were no consistent witha single band isotropic model with triplet order parameter, d z ( k ) = ∆( T )( k x + ik y ). Asa consequence of this, any fittings performed with a single line node order parameter, orwith multiband gaps having same order parameter on each band, will not agree with theexperimental results. On the other hand, several theoretical works [3, 15, 16] have proposedmodels for calculating zero field specific heat, and their results have been able to successfullyfit the experimental data. However, due to its relevance to our work, we will only refer tothe calculation by Zhitomirsky and Rice [3], which uses which uses a Sr RuO nodeless γ sheet superconductivity tight binding microscopic model. These authors performed a threeparameter fitting to specific heat experimental results [9], reproducing well the experimentalcurve, but only rendering an approximate adjustment to the observed specific heat jump atT c . Their calculations, that employed a lines of nodes model, yield a jump larger than thatexpected if a multiband model would have been used.Recently, one of us has proposed a model based on symmetry considerations [12], which isable to explain the experimental temperature behavior of the ultrasound attenuation for the L [100], L [110], and T [110] sound modes [11]. According to this model, the γ sheet shouldhave well-defined point nodes, and the α and β sheets could have also point nodes, but anorder of magnitude smaller than those of the γ band, and which resembles line of nodes withvery small gap. In this work, this anisotropic model will be applied to calculate the specificheat will, aiming to improve the calculated value of the specific heat at T c . Summarizing,at this point there is a considerable consensus regarding the Sr RuO unconventional super-conducting behavior [1, 2, 4], about the symmetry of the superconducting gap [13], and, alsoabout the multiband nature of the superconducting state; nevertheless, certainly, yet thereis no agreement regarding the nodal structure of the superconducting gap on the differentsheets of the Fermi surface. Within this context, taking into account the suitability shownby the gap model introduced in [12] to interpreting the electronic heat transport resultsbelow the transition temperature [17], in this article we will apply this model to the studyof the electronic specific heat of Sr RuO . II. THE SUPERCONDUCTING GAP STRUCTURE MODEL
As was mentioned before, here the gap model proposed in reference [12] will be extendedto study the electronic specific heat. This model assumes a superconducting order parameterbased on symmetry considerations, where the gap ∆ ik ( T ) is given by:∆ ik = ( d i ( k ) · d i, ∗ ( k )) ∆ i ( T ) , (1)here ∆ i ( T ) is taken as ∆ i (cid:112) − ( T /T c ) , where ∆ i is an adjustable parameter from experi-mental data [18]. Before continuing it is important to point out that the particular choice for∆ i ( T ) does not seem to affect the final results. Thus, in this expression we have employed( T /T c ) instead of ( T /T c ) with no effect on the fitting of the experimental specific heat.The functions d i ( k ) are the vector order parameters for the i -Fermi sheets, transforming ac-cording to the two dimensional irreducible representation E u of the tetragonal point group D h . The form of this function is [4], d i ( k ) = e z [ d ix ( k ) + i d iy ( k )] . (2)Here d ix and d iy are real functions given by: d ix ( k ) = δ i sin( k x a ) + sin( k x a k y a k z c , (3)and d iy ( k ) = δ i sin( k y a ) + cos( k x a k y a k z c . (4)In reference [12], the factors δ i were obtained by fitting the experimental data obtained fromultrasound attenuation measurements [11].For the γ band nodal structure, this model predicts eight symmetry-related k-nodes lyingon the symmetry equivalent 100 planes (see Fig. 1), and also eight symmetry-related nodes in FIG. 1: The black dots show the positions of the point nodes in the superconducting gap on the β and γ Fermi surface sheets in Sr RuO , as determined by Eqs. 3 and 4. Each solid circlerepresents two nodes, at positions ± k z . the 110 planes. Similarly, for the α and β sheets, the nodal structure of the order parameteryields eight symmetry-related k -nodes, lying on the symmetry equivalent 100 planes (seealso Fig. 1), and also eight symmetry-related nodes in 110 planes. All these point nodes are”accidental” in the sense that they are not required by symmetry; instead, they only exist forcertain range of the values of δ γ and δ β/α material parameters [12]. The 3-dimensional resultsfor the superconducting band structure of the α , β and γ Fermi surface sheets, resulting byapplying Eqs. 1, 2, 3, and 4 for different values of the parameter δ i , are displayed in Fig. 2[14].The gap structure of [3] can be described as follows: for the gamma sheet it has the form d γz ( k ) ∝ c γ (sin k x + i sin k y ) and for the β and α sheets d β/αz ( k ) ∝ c β/α (sin k x / k y / i cos k x / k y /
2) cos( k z c/ γ and c β/α are temperature dependent quan-tities which fix the values for the maximum gaps ∆ γ = c γ ( T = 0) and ∆ β/α = c β/α ( T = 0).The order parameter for the γ sheet in this model is nodeless, but the order parameter forthe β/α sheets has horizontal line nodes located at k z c = π/ FIG. 2: Three dimensional position of the point nodes in the superconducting gap on the α and β and γ Fermi surface sheets in the compound Sr RuO as determined by Eqs. 1, 2, 3, and 4 fordifferent values of the material parameter δ i . Each solid cone represents two nodes, at positions ± k z [14]. III. SUPERCONDUCTING DENSITY OF STATES FOR A TWO GAP MODELWITH POINT NODES OF DIFFERENT MAGNITUDE
In this section, the results for the superconducting density of states are presented. Twomodels are employed; the first one considers two gaps with point nodes of different magnitudeon different sheets of the Fermi surface [12], while the second one, the Zhitomirsky and Ricemodel [3], assumes horizontal line nodes. In unconventional superconductors, the orderparameter goes to zero at some parts of the Fermi surface. Due to this fact, the density ofstates at very low energy arises from the vicinity where the nodes of the order parametersare located. Well known examples of this are the high temperature superconductors. Ingeneral, line nodes and point nodes give a density of states that varies, at the low energylimit, as (cid:15) and (cid:15) respectively [19]. Besides the nodes in the order parameter, scatteringfrom non-magnetic impurities also influences the calculation of the low energy density ofstates [19, 20]. This scattering mechanism leads to the lowering of T c ; and therefore, tothe suppression of the superconducting state. In general, for temperatures much smallerthan T c , the effect of very low concentrations nonmagnetic impurities can be neglected. Itis found that only for very low temperatures, the effect of impurities becomes important forthe so called unitary limit. However, for clean samples this effect can be neglected [19, 20].For the calculations of the density of states carried out in this work, we consider the tight-binding approximation to be valid. Hence, in the performed calculations we neglect any self-consistency, only the tight-binding structure of the normal state energy is considered and theorder parameters are taken into account. Following the general approach, the unconventionalsuperconductor Fermi surface-averaged density of states (DOS) can be calculated using theequation, N i ( (cid:15) ) = N i Re [ g i ( (cid:15) ) ] . (5)Here, the i label denotes the conduction bands α/β and γ . The quantity N i is the normalmetal DOS at the Fermi level. For the case of a multiband superconductor, the following g i ( (cid:15) ) function is employed [14, 21], g i ( (cid:15) ) = (cid:42) (cid:15) (cid:112) (cid:15) − | ∆ ik | (cid:43) i FS , (6)where (cid:104)· · · (cid:105) i FS denotes the average over the i th Fermi sheet. The numerical parametersinvolved in the tight-binding normal state energy are also used for calculating the super-conducting DOS. Their values are determined from the band structure expression of theFermi velocity, and correspond to those in the Haas-Van Alfen and ARPES experiments[2, 22]: ( E − E F , t, t (cid:48) ) = ( − . , − . , − . i is the zero amplitude gap parameter for the i th Fermi sheet.The normalized total density of states can be written in a dimensionless form as N ( (cid:15) ) N = p γ N γ (cid:16) (cid:15) ∆ γ (cid:17) + p αβ N α/β (cid:16) (cid:15) ∆ α/β (cid:17) . (7)Here p αβ and p γ are the fractions of the normal-state density of states in the normalmetal associated with the α , β and γ bands, respectively. These two quantities are relatedby the sum rule: p γ + 2 p αβ = 1. For our calculations, the experimentally determined values[2], p α + β = 0 .
42 and p γ = 0 .
58, are used. The results obtained for the Fermi averaged andnormalized total DOS, together with those for each Fermi surface sheet, all at T = 0 K, aredisplayed in Fig. 3. The results in Fig. 3(a) correspond to the point node model [12], whilethose in 3(b) are obtained by employing the Zhitomirsky and Rice lines of nodes model [3].For the point node model, it can be seen that the lines of very small point gaps on the α and β sheets dominate the low energy behavior of the density of states. For these cases, thedensity of states increases faster than the density of states corresponding to the point gaps ofthe γ sheet. This result agrees with that previously reported in [12]. The parameters ∆ β/α and ∆ γ are determined in [8, 17] by fitting the thermal conductivity data, and their valueswill be also used for fitting C s ( T ). From the lines of nodes model, figure 3(b), one noticesthat the horizontal gaps on the α and β sheets dominate the low energy behavior of thedensity of states, while the density of states of the γ band opens a gap below certain criticalvalue due to its nodeless nature of this model. For both cases, it is found that as more linesof nodes or point of nodes are added to the gap functions, the density of states inside thegap increases faster [12]. Also it is obtained that the density of states increases faster thanlinearly inside the superconducting region. In addition, both models present DOS van-Hovesingularities, which are due to the tight-binding structure of Sr RuO . These singularitiesare responsible for the two coherence peaks observed in the total density of states. IV. ELECTRONIC SUPERCONDUCTING SPECIFIC HEAT IN AMULTIBAND SR RUO For an unconventional single band superconductor, the electronic specific heat low tem-perature behavior is expected to vary as T , for the horizontal lines of nodes (zeros) and as T for the point of nodes models [19]. For multiband superconductors, at T smaller thanT c , the values of the specific heat show a strong dependence on the order parameters ∆ i .However, due to the absence of a self-consistent evaluation of ∆ i ( T ) in the tight-bindingmethod employed here, and since at temperatures close to T c , the numerical value of thederivatives d ∆ i ( T ) dT are relevant, at first sight, a good agreement with the observed electronicspecific heat behaviors may not be anticipated. Nevertheless, the order parameters havea strong dependence on the anisotropic effects [14, 20, 22]. Thus, in this section we willexplore if, through calculations that incorporate anisotropic effects, is possible to overcomethe tight-binding method initial handicap, to provide a good description of the specific heattemperature behavior. Before continuing it is important to point out that, as was done in FIG. 3: Normalized total and partial superconducting density of states N ( (cid:15) ) for the multibandmodels with accidental point nodes of [17] panel (a), and the horizontal lines model of [3] panel(b). the thermal conductivity case [17], for comparing the calculation results with the specificheat experimental data, the sum of the contributions from all the band sheets has to beconsidered. The expression used in this work to calculate the electronic specific heat foran unconventional multiband superconductor is an extension of that developed in [20, 23],obtained through a single band unconventional superconductor formalism. Here, a moregeneral expression for C s ( T ), that takes into account the anisotropic and multiband effectsof an unconventional superconductor is used: C ( T ) = 2 T (cid:88) i (cid:90) d(cid:15) (cid:16) − ∂f∂(cid:15) (cid:17) F i ( (cid:15) ) , (8)the Fermi surface averaged function F i ( (cid:15) ) appearing in this expression is given by: F i ( (cid:15) ) = p i (cid:28) N i ( (cid:15), k ) (cid:104) (cid:15) − T d | ∆ ik | d T (cid:105)(cid:29) i FS ; (9)where p i denotes the partial density of states in the band i , N i ( (cid:15), k )) is the expression for themomentum dependence of the DOS and the quantity N i is the DOS at the Fermi level. Theonly input parameters required for the specific heat temperature behavior fitting are the zero0temperature energy gaps ∆ γ and ∆ α/β . Here we use the values calculated in [8, 17] for fittingthe thermal conductivity data. For the models employed here, the accidental point nodeand the horizontal line node models, fig. 4 shows the theoretical results for the temperaturedependence of the normalized electronic specific heat, calculated from Eq.8 together withthe experimental results reported in [9]. For the point node model [12, 14, 17], the lowerpanel displays of fig. 4 shows an excellent fitting of the experimental data.One point which shows excellent agreement is the size of the specific heat jump at T c ,∆ C/C n (cid:39) C/C n (cid:39) C/C n = 0.70. Thisis a remarkable result since, as mentioned before, only anisotropic arguments are considered,and also because only two parameters, obtained from the literature, corresponding to fittinga different thermal conductivity experiment, are used. The contributions of each of the threeband sheets are displayed in the figure, from there is observed that at low temperatures,close to T=0, the anisotropy of the gap in the α and β bands dominates the behavior, whileat higher temperatures, the electronic specific heat behavior is dominated by the γ band.The results in the lower panel of fig. 4 show that the horizontal line node model also providea good fitting; however, a worse adjustment to the experimental data is obtained as Tapproaches T c , and, as reported in the literature [2, 3, 18] , from this model, the theoreticalcalculations only yield a modest adjustment to the observed specific heat jump at T c i.e.this jump seems to be better reproduced by our point nodes model. It is interesting to pointout that [6, 9]reported that the γ -band is the one responsible for the largest contribution tothe total density of states; thus, as a consequence of this, Fig. 4 also shows that, for bothmodels, the γ -band dominates the contribution to the specific heat. V. CONCLUSIONS
In this work, two different models for the gap structures characterizing the α , β and γ Fermi surface sheets are employed to calculate the Sr RuO superconducting electronicspecific heat C s ( T ). One of them considers two gaps with point nodes of different magnitudeon different sheets, while the other one assumes horizontal line nodes. Through the firstmodel, it is found that the calculated temperature behavior of C s ( T ) shows an excellentagreement with the existent Sr RuO experimental data at zero field [6, 7, 9], particularly,1 FIG. 4: Normalized electronic specific heat C s ( T ) /T for the multiband models: Lower paneldisplays the results yielded by the point node model [12, 17], and the upper panel exhibits thoseobtained from the horizontal lines node model [3]. The points correspond to the experimental dataof [9] . the observed specific heat jump at T c is better reproduced by the point nodes model. Theresults obtained here seem to confirm that the Sr RuO superconducting state correspondsto that of a point node unconventional superconductor. Acknowledgments