Towards room-temperature superconductivity
Jacob Szeftel, Nicolas Sandeau, Michel Abou Ghantous, Muhammad El-Saba
aa r X i v : . [ phy s i c s . g e n - ph ] D ec Towards room-temperature superconductivity
Jacob Szeftel , ∗ Nicolas Sandeau , Michel Abou Ghantous , and Muhammad El-Saba ENS Paris-Saclay/LuMIn, 4 avenue des Sciences, 91190 Gif-sur-Yvette, France Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, F-13013 Marseille, France American University of Technology, AUT Halat, Highway, Lebanon and Ain-Shams University, Cairo, Egypt
By taking advantage of a stability criterion established recently, the critical temperature T c isreckoned with help of the microscopic parameters, characterising the normal and superconductingelectrons, namely the independent-electron band structure and a repulsive two-electron force. Theemphasis is laid on the sharp T c dependence upon electron concentration and inter-electron coupling,which might offer a practical route toward higher T c values and help to understand why high- T c compounds exhibit such remarkable properties. PACS numbers: 74.25.Bt,74.25.Jb,74.62.Bf
The BCS theory[1], despite its impressive success, doesnot enable one to predict[2] superconductivity occurringin any metallic compound. Such a drawback ensues froman attractive interaction, assumed to couple electrons to-gether, which is not only at loggerheads with the sign ofthe Coulomb repulsion but in addition leads to question-able conclusions to be discussed below. Therefore thiswork is intended at investigating the T c dependence uponthe parameters, characterising the motion of electronscorrelated together through a repulsive force, within theframework of a two-fluid picture[3] to be recalled below.The conduction electrons comprise bound and inde-pendent electrons, in respective temperature dependentconcentration c s ( T ) , c n ( T ), such that c = c s ( T ) + c n ( T )with c being the total concentration of conduction elec-trons. They are organized, respectively, as a many boundelectron[4] (MBE) state, characterised by its chemicalpotential µ ( c s ), and a Fermi gas[5] of Fermi energy E F ( T, c n ). The Helmholz free energy of independentelectrons per unit volume F n and E F on the one hand,and the eigenenergy per unit volume E s ( c s ) of boundelectrons and µ on the other hand, are related[5, 6], re-spectively, by E F = ∂F n ∂c n and µ = ∂ E s ∂c s . Then a stableequilibrium is conditioned[7] by Gibbs and Duhem’s law E F ( T, c n ( T )) = µ ( c s ( T )) , (1)which expresses[6] that the total free energy F n + E s isminimum provided ∂E F ∂c n + ∂µ∂c s >
0. Noteworthy is that ∂µ∂c s < T = T c E F ( T c , c ) = µ ( c s = 0) = ε b / , (2)with ε b being the energy of a bound electron pair[4]. Notethat Eqs.(1,2) are consistent with the superconductingtransition being of second order[6], whereas it has beenshown[4] to be of first order at T < T c ( ⇒ E F ( T, c − c s ) = µ ( c s )), if the sample is flown through by a finitecurrent. The binding energy[4] of the superconducting state E b ( T < T c ) has been worked out as E b ( T ) = Z T c T ( C s ( u ) − C n ( u )) du , with C s ( T ) , C n ( T ) being the electronic specific heat of asuperconductor, flown through by a vanishing current[4]and that of a degenerate Fermi gas[5]. A stable phase( ⇒ E b >
0) requires C s ( T c ) > C n ( T c ), which can besecured[3] only by fulfilling the following condition ∂E F ∂c n ( T c , c ) = − ∂µ∂c s (0) , ρ ′ ( E F ( T c , c )) > , (3)with ρ ( ǫ ) , ǫ being the independent electron density ofstates and one-electron energy, respectively, and ρ ′ = dρdǫ .Since the remaining analysis relies heavily onEqs.(2,3), explicit expressions are needed for E F ( T c , c ) , ∂E F ∂c n ( T c , c ) , ε b , ∂µ∂c s (0). Because the in-dependent electrons make up a degenerate Fermi gas( ⇒ T << E F /k B with k B being Boltzmann’s constant),applying the Sommerfeld expansion[5] up to T yields E F ( T c , c ) = E F (0 , c ) − ρ ′ ρ ( πk B T c ) ∂E F ∂c n ( T c , c ) = (cid:16) ρ + ρ ′′ ( πk B T c ) (cid:17) − , (4)with ρ = ρ ( E F (0 , c )) , ρ ′ = dρdE F ( E F (0 , c )) , ρ ′′ = d ρdE F ( E F (0 , c )). As for ε b , ∂µ∂c s (0), a truncated HubbardHamiltonian H K , introduced previously[9–11], will beused. The main features of the calculation[4] are sum-marised below for self-containedness.The independent electron motion is described by theHamiltonian H d H d = X k,σ ǫ ( k ) c + k,σ c k,σ .ǫ ( k ) , k are the one-electron energy ( ǫ ( k ) = ǫ ( − k )) and avector of the Brillouin zone, respectively, σ = ± is theelectron spin and the sum over k is to be carried out overthe whole Brillouin zone. Then c + k,σ , c k,σ are creation andannihilation operators on the Bloch state | k, σ i| k, σ i = c + k,σ | i , | i = c k,σ | k, σ i , with | i being the no electron state. The Hamiltonian H K reads then H K = H d + UN X k,k ′ c + k, + c + K − k, − c K − k ′ , − c k ′ , + , with N >> , U > H K =0 , but with U < H K sustains[4] a single bound pair eigenstate, the en-ergy ε b ( K ) of which is obtained by solving1 U = 1 N X k ε b ( K ) − ε ( K, k ) = Z t K − t K ρ K ( ε ) ε b ( K ) − ε dε. (5) ± t K are the upper and lower bounds of the two-electronband, i.e. the maximum and minimum of ε ( K, k ) = ǫ ( k )+ ǫ ( K − k ) over k , whereas ρ K ( ε ) is the correspondingtwo-electron density of states, taken equal to ρ K ( ε ) = 2 πt K s − (cid:18) εt K (cid:19) . The dispersion curves ε b ( K ) are plotted in Fig.1.Though Eq.(5) is identical to the equation yielding theCooper pair energy[12], their respective properties arequite different :• the data in Fig.1 have been calculated with U >
U < U ∂µ∂c s <
0, choosing
U < ∂µ∂c s >
0, which hasbeen shown not to be consistent with persistentcurrents[7], thermal equilibrium[4], the Josephsoneffect[8] and occurence[3] of superconductivity. Asa further consequence of
U > ε b ( K ) shows upin the upper gap of the two-electron band struc-ture ( ⇒ ε b ( K ) > t K ) rather than in the lowergap ( ⇒ ε b ( K = 0) < − t K ) in case of the Cooperpair[12]. Nevertheless the bound pair is thermody-namically stable, because every one-electron stateof energy < E F ( T c , c ), is actually occupied, sothat, due to Pauli’s principle, a bound electronpair of energy ε b ( K ) = 2 E F ( T c , c ), according toEq.(2), cannot decay into two one-electron states ǫ ( k ) < E F , ǫ ( K − k ) < E F ;• a remarkable feature in Fig.1 is that ε b ( K ) → t K for U → t K /
2, so that there is no bound pair for U < t K / Kaπ < . U →
0. This discrepancy results from the three-dimensional Van Hove singularities, showing upat both two-electron band edges ρ K ( ε → ± t K ) ∝ p t K − | ε | , unlike the two-electron density of states,used by Cooper[12] which is constant and thencedisplays no such singularity. Likewise the widthof Cooper’s two-electron band is equal to a Debyephonon energy 2 t K =0 = ω D ≈ meV << E F ≈ eV . Hence the resulting small concentration ofsuperconducting electrons, c s ( T =0) c ≈ ω D E F ≈ . U < ε b / = E F ( T c ), which is typical of a first order transitionbut runs afoul at all measurements, proving con-versely the superconducting transition to be of sec-ond order ( ⇒ ε b / E F ( T c ) in accordance withEq.(2)).The bound pair of energy ε b ( K ) turns, at finite concen-tration c s , into a MBE state, characterised by µ ( c s ). Itsproperties have been calculated thanks to a variationalprocedure[4], displaying several merits with respect tothat used by BCS[1] :• it shows that µ (0) = ε b / | U | → ∞ ;• an analytical expression has been worked out for ∂µ∂c s ( K, c s = 0) as : ∂µ∂c s ( K, c s = 0) = − R t K − t K ρ K ( ε )( ε b ( K ) − ε ) dε (cid:16)R t K − t K ρ K ( ε )( ε b ( K ) − ε ) dε (cid:17) . (6)The T c dependence on c will be discussed by assigningto ρ ( ǫ ) the expression, valid for free electrons ρ ( ǫ ) = η √ ǫ − ǫ b ⇒ c = 23 η ( E F (0 , c ) − ǫ b ) , (7)with η = √ m Vπ ~ , whereas ǫ b , m, V = 17Å stand forthe bottom of the conduction band, electron mass andvolume of the unit-cell, respectively. With help of Eq.(4),Eqs.(2,3) can be recast into a system of two equations E F (0 , c ) − ρ ′ ρ ( πk B T c ) − ε b ( K )2 = 0 (cid:16) ρ + ρ ′′ ( πk B T c ) (cid:17) − + ∂µ∂c s ( K, c s = 0) = 0 , (8)to be solved for the two unknowns c ( T c ) , t K ( T c ) with T c being dealt with as a disposable parameter. (cid:19)(cid:19)(cid:17)(cid:24)(cid:20) (cid:19) (cid:19)(cid:17)(cid:24) (cid:20)(cid:87) (cid:46) (cid:3)(cid:18)(cid:3)(cid:87)(cid:56)(cid:3)(cid:18)(cid:3)(cid:87)(cid:3)(cid:32)(cid:3)(cid:20)(cid:56)(cid:3)(cid:18)(cid:3)(cid:87)(cid:3)(cid:32)(cid:3)(cid:17)(cid:25)(cid:56)(cid:3)(cid:18)(cid:3)(cid:87)(cid:3)(cid:32)(cid:3)(cid:17)(cid:23)(cid:28) (cid:46)(cid:68)(cid:3) (cid:18)(cid:3) (cid:3) (cid:69) (cid:3)(cid:18)(cid:3)(cid:87) FIG. 1. Dispersion curves of t K as a dashed-dotted line andof ε b ( K ) as solid, dashed and dotted lines, associated withvarious U values, respectively; those data have been obtainedwith t K = t cos ( Ka/ t, a are the one-electron band-width and the lattice parameter, respectively.
To that end, starting values are assigned to
U, t K ,which gives access to ε b ( K ) , ∂µ∂c s ( K, c s = 0)) andthence to E F (0 , c ) , ǫ b and finally to c , owing toEqs.(2,3,7). Those values of c , t K are then fed intoEqs.(8) to launch a Newton procedure, yielding the so-lutions c ( T c ) , t K ( T c ). The results are presented in ta-ble I. Since we intend to apply this analysis to high- T c compounds[17], we have focused upon low concentrations c < .
2, which entails, in view of Eqs.(4,7), that (cid:12)(cid:12)(cid:12) ∂µ∂c s (cid:12)(cid:12)(cid:12) takes a high value. This requires in turn ε b ( K ) → t K (see Eq.(6)) and thence[4] U → t K , in agreement with t K U ≈ c , t K are barely sensitive to large variations of T c , i.e. | δc | < − , | δt K | < − for δT c ≈ K . This can beunderstood as follows : taking advantage of Eqs.(2,4,7)results into2 E F (0 , c ) ε b ( K ) − π (cid:18) k B T c ∆( T c ) (cid:19) , which, due to dt K dT c ≈ , ∆( T c ) ≈ eV, T c = 400 K , yieldsindeed δc = c (400 K ) − c (1 K ) ≈ − , in agreementwith the data in table I. Such a result is significant intwo respects, regarding high- T c compounds, for which c can be varied over a wide range :• because of dc dT c ≈
0, the one-electron band structurecan be regarded safely as c independent, whichenhances the usefulness of the above analysis;• the large doping rate up to ≈ . c , which, in view of the ut-most sensitivity of T c with respect to c , will result TABLE I. Solutions c ( T c ) , t K ( T c ) , ∆( T c ) (∆( T c ) = E F (0 , c ( T c )) − ǫ b ) of Eqs.(8); t K , ∆ , U are expressed in eV ,whereas the unit for c is the number of conduction electronsper atomic site. T c ( K ) c t K ∆1 0 . . . . . U = 3 . T c ( K ) c t K ∆1 0 . . . . . U = 1 . T c ( K ) c t K ∆1 0 . . . . . U = 2 . into a heterogeneous sample, consisting in domains,displaying T c varying from 0 up to a few hundredsof K . Thus the observed T c turns out to be theupper bound of a broad distribution of T c values,associated with superconducting regions, the set ofwhich makes up a percolation path throughout thesample. However, if the daunting challenge of mak-ing samples, wherein local c fluctuations would bekept well below 10 − , could be overcome, this mightpave the way to superconductivity at room temper-ature.The T c dependence upon U will be analysed with ρ ( ǫ ) = 4 πt r − (cid:16) − ǫt (cid:17) , where 2 t stands for the one-electron bandwidth. Our pur-pose is to determine the unknowns t K ( E F , T c ) , U ( E F , T c )with E F = E F ( T = 0 , c ) and c = R E F ρ ( ǫ ) dǫ . Tothat end, Eq.(3) will first be solved for t K by replac-ing ∂E F ∂c n ( T c , c ) , ∂µ∂c s (0) by their expressions given byEqs.(4,6), while taking advantage of Eq.(2). Then theobtained t K value is fed into Eq.(5) to determine U . Theresults are presented in Fig.2.It can be noticed that there is no solution for c >.
75, because ∂E F ∂c n ( T c , c ) ≈ ρ ( E F (0 , c )) and ∂µ∂c s (0) > U decrease and increase, respectively, with increasing c , sothat Eq.(3) can no longer be fulfilled eventually. But themost significant feature is that δU is almost insensitiveto large T c variation, except for E F →
0, i.e. for E F closeto the Van Hove singularity, located at the bottom of theband, which has two consequences :• c cannot be varied in most superconducting ma-terials, apart from high- T c compounds, so that U is unlikely to be equal to U ( c ), indicated in Fig.2. (cid:19)(cid:19)(cid:17)(cid:24)(cid:20)(cid:20)(cid:17)(cid:24) (cid:19)(cid:17)(cid:19)(cid:19)(cid:19)(cid:20)(cid:19)(cid:17)(cid:19)(cid:19)(cid:20)(cid:19)(cid:17)(cid:19)(cid:20)(cid:19)(cid:17)(cid:20)(cid:19) (cid:19)(cid:17)(cid:21)(cid:24) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:26)(cid:24) (cid:40) (cid:41) (cid:18)(cid:87) (cid:87) (cid:46) (cid:18) (cid:87) (cid:56)(cid:18)(cid:87)(cid:40) (cid:41) (cid:87) (cid:46) (cid:56)(cid:70) (cid:19) (cid:40) (cid:41) (cid:87) (cid:46) (cid:56) (cid:3) (cid:40) (cid:41) (cid:3)(cid:87) (cid:46) (cid:3)(cid:56) FIG. 2. Plots of E F ( T c , c ) , t K ( T c , c ) , U ( T c , c ) calculated for T c = 1 K and t = 3 eV ; the unit for c is the number of conduc-tion electrons per atomic site; δf with f = E F , t K , U is de-fined as δf = (cid:12)(cid:12)(cid:12) − f (300 K,c ) f (1 K,c ) (cid:12)(cid:12)(cid:12) ; the scale is linear for E F , t K , U but logarithmic for δE F , δt K , δU . Conversely, since high- T c compounds allow for wide c variation, c can be tuned so that U = U ( c );• the only possibility for a non high- T c material toturn superconducting is then offered at the bot-tom of the band, because δU becomes large due to ρ ′ ρ ( E F → ∝ E F in Eq.(4). Such a conclusion,that superconductivity was likely to occur in thevicinity of a Van Hove singularity in low- T c ma-terials, had already been drawn[4] independently,based on magnetostriction data.It will be shown now that ρ ( ǫ ) , ρ K ( ε ) cannot stem fromthe same one-electron band. The proof is by contradic-tion. As a matter of fact ρ ( ǫ ) should read in that case ρ ( ǫ ) = 4 πt r − (cid:16) ǫt (cid:17) . Hence
U > ε b = E F >
0, which implies ρ ′ ( E F ) < ρ ( ǫ ) , ρ K ( ε ),display a sizeable overlap, they should in addition belongto different symmetry classes of the crystal point group,so that superconductivity cannot be observed if there areonly s -like electrons at E F or if the point group reducesto identity. Noteworthy is that those conclusions hadalready been drawn empirically[2]. The critical temperature T c has been calculated forconduction electrons, coupled via a repulsive force,within a model based on conditions, expressed inEqs.(2,3). Superconductivity occurring in conventionalmaterials has been shown to require E F ( T c ) being lo-cated near a Van Hove singularity, whereas a practicalroute towards still higher T c values has been delineatedin high- T c compounds, provided the upper bound of lo-cal c fluctuations can be kept very low. The thermody-namical criterions in Eqs.(2,3) unveil the close interplaybetween independent and bound electrons in giving riseto superconductivity. At last, it should be noted thatEqs.(2,3) could be applied as well to any second ordertransition, involving only conduction electrons, such asferromagnetism or antiferromagnetism. ∗∗