J. Phys. Chem. B 2001, 105, 11251-11255
11251
Resistivity of a Superconductor: A Search for the Origin of Superconductivity R. J. Thorn Chemistry and Material Science DiVisions, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: March 7, 2001; In Final Form: July 10, 2001
The measurements of the variation of the resistivity with temperature in increments of 0.02 K for single crystals of YBa2Cu3O6+x by Pomar et al. (Phys. ReV. B 1996-I, 53, 8245-8248) yield discriminating values and show that the resistivity approaches zero at 0 K asymptotically. These measurements, which can serve as a prototype, have been analyzed in terms of equations, generally applicable for any superconductor, that recognize this fact and that describe the conductivities of the normal state with scattering and of the super state for mixed-valent bipolarons having no scattering. These descriptions, together with a description of the diamagnetic transition as a shift in the chemical equilibrium between polarons and bipolarons, identify a rationale for searching for the origin of superconductivity.
I. Introduction A search through the conventional texts1 and the references in the literature2 reveals that nothing is found on the quantitative description (i.e., equation) for the transition from the normal state to the superconducting state. Perhaps part of the reason for not addressing this topic is that it is commonly stated that the resistance goes to zero at or slightly below the transition. Hence, no theoretical description can be derived. Since the discriminating measurements by Pomar et al.3 of the resistivity of YBa2Cu3O6+x show that dF/dT is not infinite at the transtion and that the resistance can approach 0 at 0 K asymptotically, one has reason for believing that the resistance can be described by a continuous function. Numerous measurements with several superconductors also show that dF/dT is not infinite at the transition. This includes even the measurements made by Onnes:4 in one case, the curve approaches zero asymptotically;5 in another case, the slope at the transition is finite, and the smallest resistance measured is 10-5 ohms.6 Hence, it is reasonable to inquire whether one can derive an equation that describes a transition that is continuous to 0 K. Furthermore, it is commonly accepted that the superconducting state is a paired, diamagnetic state and the normal state is a single, nonmagnetic state. Hence, it is reasonable to assume that the pairs (bipolarons) and the singles (polarons) are in chemical equilibrium and to derive the consequence.
conductivities are listed in Addenda, item A2. These equations apply to any superconductor. III. The Diamagnetic Transition As long ago as 1940, London recognized “the possible simultaneous presence of superconducting and normal electrons” in a superconductor. With the suggestion by Ogg10 that the super state contains electron pairs and the formal use of the pairing concept by Bardeen et al.,11 it is obvious then that pairs (mixedvalent bipolarons) and singles (mixed-valent polarons) (see Addenda, item A1) are assumed to be present simultaneously. But the transition from the normal to the super state must involve a change from the predominant presence of singles to the predominant presence of pairs. Hence, the question is: How does one describe the distribution of the electrons among the singles and pairs during the transition? The answer is: Through the condition for chemical equilibrium between singles (1) and pairs (2) or between mixed-valent polarons and bipolarons:
ln[(q1)2(N1)2/q2N2] ) (-∆Geq/RT) ) 0
in which q1 and q2 are the partition functions for the singles and the pairs respectively, and the N’s are their respective numbers. A complete evaluation of this condition for equilibrium requires the values for the partition functions. The chemists have long known, however, that an equilibrium such as
e2 ) 2e,
II. Conductivities From a study of the measurements in the normal state, Thorn7 found that the variation of the conductivity with temperature is described by an equation for a solid with small Fermi and small gap energies. Using the Hamiltonian for an electron linearly coupled to a harmonic oscillator, Holstein8 found for his small polaron two equations: (1) one with scattering in a semiconductor and (2) one with no change in the phonon number, i.e., no scattering. Following Holstein, Bo¨ttger and Bryksin9 derived the equations for their small polaron on the basis of lattice vibrations, but for a superconductor, only one of these is operative with a frequency, ωo. The equations that describe these
(1)
or p2 ) 2p
(2)
can be described by expressing an equilibrium constant in terms the concentrations, c1and c2, with the total number of species constant:
c2/c12 ) K/cst ) exp(-∆G°/RT)
(3)
The units for c1 and c2 and the value for cst needed not be specified because they cancel out in the final results, c2(T)/c2(0). One can set ∆S°/∆H° equal to 1/Te, where Te is some effective temperature that turns out to be T1/2, the temperature at the half-height of c2(T)/c2(0). The ratio of the concentration
10.1021/jp0108770 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/20/2001
11252 J. Phys. Chem. B, Vol. 105, No. 45, 2001
Thorn
Figure 1. Least squares application of eqs 4, 5, and 8 to the measurements of the real part of the AC diamagnetic susceptibility of a niobium sphere (see ref 13). The symbols, b, represent the measured values. The value at T ) 0 is that predicted for a perfect spherical diamagnetic.
Figure 3. (a) Least squares application of the equations to the measurements (b) of resistivity versus the lower temperatures for single crystal of YBa2Cu3O6+x by Pomar et al. (b) Least squares application of Holstein’s equation with no scattering to the measurements by Pomar et al. in the lower-temperature range. Figure 2. Least squares application of eqs 4, 5, and 8 to the measurements of the real part of the AC diamagnetic susceptibility of a single crystal of YBa2Cu3O6+x. Note that the curvature at the onset is greater than that at the heel for both the measurements and the equations. Data from ref 14.
of pairs at T to that at 0 or, equivalently, the fraction of the solid that is in the super state is:12
c2(T)/c2(0) ) (4 + A - Ax1 + 8/A)/4 ) F(T)
(4)
A ) exp[(∆H°/R)(1/T1/2 - 1/T)]
(5)
(dF(T)/dT)T1/2 ) -∆H°/6R(T1/2)2
(6)
In these equations, ∆H° is the dissociation enthalpy of the pair, and T1/2 is the value of the temperature at F(T) ) 1/2. F(T) is sigmoidally shaped with
lim{F(T)}T)0 ) 1,
and lim{F(T)}T)∞ ) 0
(7)
Thus, the singles (polarons) and pairs (bipolarons) are simultaneously present, except in the limits where only pairs are present at T ) 0 and only singles are present at T ) ∞. F(T) is a measure of the fraction of the solid that is in the super state. Because the super state with the predominant pairs is diamagnetic, one can write that
χ(T)/χ(0) ) -F(T)
(8)
Figures 1 and 2 illustrate the least squares application of these equations to the measurements of the real part of the diamagnetic
susceptibilities of niobium13 and of YBa2Cu3O6+x.14 Notice that the limiting value for the susceptibility of niobium when corrected for the demagnetiztion factor is the theoretical value. IV. Variation of the Conductivity (Resistivity) with Temperature before, through, and after the Transition Because F(T) measures the fraction of the superconductor that is in the super state, its equation is essential to the description of the transition. With the equations that describe the conductivities of normal and super states (Holstein and Bo¨ttger and Bryksin), F(T) will faithfully describe the transition, but without F(T), no transition in the resistivity occurs. The total conductivity can be written as the sum of the normal conductivity and the superconductivity, each weighted by the fractional composition:
σ ) [1 - F(T)]σn + F(T)σs
(9)
At F(T) ) 0, σ ) σn, and only the normal state contributes to the conductivity at T ) ∞. At F(T) ) 1, σ ) σs, and only the super state contributes to the conductivity at T ) 0. For simplicity, one can write just F for F(T). Knowing F(T), one searches for something that describes σs. In the small polaron model of Holstein, one of the components of conductivity is the one that occurs without scattering, and in the modification effected by Bo¨ttger and Bryksin, this component has the nature that defines a bipolaron. These equations have been used in a least squares procedure with the discriminating measurements of the resistivity versus temperature for a single crystal of YBa2Cu3O6+x by Pomar et al. The results shown in Figure 3a describe the precipitous
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Figure 4. Representation of a superconductor consisting of the equilibrium between polarons and bipolarons, F (A, on the left), and the conductivity of the bipolaronic super state, σs (B, on the right). In its derivation, σs contains the zero-point vibrational state with quantum state hν/2 (C). When F is “closed” (F ) 1), F ) 1/σ ) 0.
decrease in the resistivity at the transition and the asymptotic approach to zero at T ) 0. Figure 3b displays the asymptotic approach of the resistivity to zero. The perturbation approach used by Holstein places a lower limit on the temperature for the validity of his equation. For the parameters obtained with the measurements by Pomar et al., the lower limit is near 25 K, where Holstein’s equation attains a maximum value for the normalized conductivity. (The resistivity is nearly zero.) However, this is a mathematical limitation in the perturbation solution of his small polaron. It is not a conceptional limitation. Bo¨ttger and Bryksin’s equation for the conductivity of the Holstein model with no scattering does not contain this limitation. Application of the least squares procedure to the low-temperature data of Pomar et al. yields the same values of F(T) as those from Holstein’s equation shown in Figure 3b. However, their equation yields a value of ∞ for the conductivity at T ) 0, where the quantum state is that for the zero-point vibration. A Model of a Superconductor. The description for a superconductor is illustrated in Figure 4, where the equilibrium between polarons and bipolarons is indicated by the dotted curve, F(T). These two species coexist at all temperatures except at the limits of T ) 0 and T ) ∞, respectively. At the transition, the chemical equilibrium shifts from predominately bipolarons to predominately polarons at the left (A) of the figure. F(T) identifies the underlying mechanism responsible for the superconducting properties. (The negative of F(T) describes the
normalized diamagnetic susceptibility.) F(T) is central to all the measurements, χ(T), F(T), Cp(Τ), and M(H), that identify a superconductor. The conductivity of the superconducting state, σs, (Bo¨ttger and Bryksin) is indicated on the right side (B) of the figure. At T ) 0, σs contains in its derivation the zero-point vibrational state (C), which is the microscopic source of the bipolaronic super current that causes the diamagnetism. At high temperatures, F(T) approaches zero, so it is almost completely “open”, and the normal state contributes the most to the conductivity. At the lower temperatures, F(T) approaches one so that it is mostly “closed”, and the super state contributes the most to the conductivity. Since σ ) F(T)σs + [1 - F(T)]σn (the two are in parallel), at T ) 0, F(T) ) 1 and σ ) σs; at T ) ∞, F(T) ) 1 and σ ) σn. The variation of F(T) ) 1/σ(T) is shown at the bottom of column C. Since F(T;c,T1/2), σs, and σn are functions of T, it is obvious that one equation, F(F,σs,σn), in which F, σs, and σn can in principle be solved for T describes any superconductor. Consequently for any finite T, there is no finite theoretical limit on the transition temperature. Contained in the complete theory is the equation at the bottom of Figure 5 for the transition temperature as a function of the dissociation energy and the slope of F(T) at its half-height, (dF(T)/dT) at T1/2. If the slopes are sufficiently the same for a series of measurements, then T1/2 is a function of ∆H°, as shown in Figure 5 for the cuprates and for the “metals”.
11254 J. Phys. Chem. B, Vol. 105, No. 45, 2001
Thorn The term [1 - F(T)]σn is indeterminate for T < 4 × 10-3, but this is no operational problem because Fn(0) ) 0.
Figure 5. Variation of the transition temperature, T1/2 in F(T), with the dissociation energy of the bipolarn, ∆H°, for the cuprates and some “metals”. The curve labeled BCS 1 is that for the BCS equation with N(0)V as a variable; the curve labled BCS 2 is that for the BCS equation with the pre-exponential and the N(0)V as adjustable parameters for the cuprates.
Summary The measurements that serve to identify a superconductor, χ(T) and F(T), have been unexplained heretofore; no equations describing them have been derived in all the theories of superconductivity. Given a system consisting of singles (polarons) and pairs (bipolarons), it is reasonable to ask what is the consequences of assuming that they are in chemical equilibrium. Thus, through the use of the centralized role of the chemical equilibrium and the Bo¨ttger and Bryskin model of a four-site small polaron, I have found an equation describing, within the precision of experimental error, the precipitous decrease in the resistivity at the transition, the relation between the resistive and diamagnetic transitions, and the origin of superconductivity in the zero point vibration in the small polaron model. It is not possible to establish that the resistance has vanished or disappeared at or slightly below the transition temperature. Furthermore, if the definition of a superconductor is that it does, it is not possible to construct a theory of the superconducting state on the basis of this zero-property. On the other hand, if a superconductor is defined in terms of the precipitous decrease toward zero in the resistance, then it is possible to construct a theory, as shown herein, on the basis of the shift in the chemical equilibrium between predominantly singles (polarons) in the normal state and predominately pairs (bipolarons) in a superconducting state that has a conductivity described by Bo¨ttger and Bryskin’s model of a four site small polaron. In 1950, Bardeen published an article entitled “Zero-Point Vibration and Superconductivity”,15 in which he stated, “A theory of superconductivity which depends on the interaction of the valence electrons with the zero-point vibrations of the crystal lattice is proposed”. In the article, Bardeen did not address the role of conductivity and chemical equilibrium. Besides, he only made a proposal. At that time, Holstein’s model for the conductivity versus temperature of the small polaron in which an electron is linearly coupled to an harmonic oscillator and Bottger and Bryksin’s extension were not published. Now that we know that this model yields the conductivity without scattering that extrapolates to zero at 0 K, the contribution of the zero-point vibrational state can be established. VI. Addenda A1. Mixed-Valent Polarons and Bipolarons. All of the recently discovered superconductors and all the large number discovered by Matthias16 are ones with potentially mixed valences for which equilibria can be written in general as follows (cs and as refer to cation and anion sites):
Figure 6. Predicted curve for normalized resistivity versus temperature for a room-temperature hypothetical cuprate superconductor with the predicted value of the dissociation energy given in Figure 5.
Figure 5 represents the predicted value of ∆H° necessary for a room temperature superconductor in the cuprate series. The data plotted in Figure 5 also show why the transition temperatures for the cuprates are higher than are those for the “metals”; they have a larger dissociation energy of the bipolaron. The relationships shown in the figure elicits the question: Are there any series of superconductors that have curves to the left of that one for the cuprates? Figure 6 represents F versus T for a hypothetical cuprate that has a transition temperature at 298 K, i.e., at room temperature.
(Mcs)m+ ) (Mcs)(m+1)+ + ef
(A1.1)
ef + (Mcs)m+ ) (Mcs)(m-1)+
(A1.2)
(Mcs1)m+ + (Mcs2)m+ ) (Mcs1)(m+1)+ + (Mcs2)(m-1)+ (A1.3) For the superconductor YBa2Cu3O6+x, one writes:
(Cucs)3+ ) (Cucs)2+ + ef
(A1.4)
(Oas)2- ) (Oas)1- + ef
(A1.5)
The energies associated with eqs A1.4 and A1.5 are on the order
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of eV’s, so they are generally observable at room temperature in the visible range. However, their difference
D. ConductiVity and ResistiVity from Normal through the Transition and into the Super State.
(Cucs)3+ + (Oas)2- ) (Cucs)2+ + (Oas)1-
σm ) (F/(2 - F))σs + 2((1 - F)/(2 - F))σn
(A1.6)
has an energy that is significantly below that of room temperature in measurements with the superconductor. These equilibria define the mixed-valent polaron. The existence of the equilibrium (A1.6) means that there exists a configuration such as (Oas1)2-(Cuca1)3+(Oas2)1-(Cucs2)2+ in the solid, so since the oxygen is lighter than the copper, most of the vibration occurs in the oxygen so that during its oscillation an electron on (Oas1)2-can hop to (Cuca1)3+ and then to (Oas2)1and so on, thereby furnishing a mechanism for the conductivity. In the super state, cross-linkages exist between two polarons to form a bipolaron with a closed orbit in which the electrons can circulate with a radius r. This gives rise to the diamagnetic susceptibility (see Seitz17):
χ ) -e2/(6mc2)(ΣaNana〈r2〉)
(A1.7)
A2. Equations Used in the Calculations. A. Fraction of the Phase in the Super State. Diamagnetic susceptibility:
(1 - F)σn is indeterminate at T ) 0, so it can be set equal to 0.
F ) F(T) E. Equations with Tnr or Tsr are used to normalize the property so that the prefactor constants, Cf, Cg, Ch, and Cbb, are canceled out. Parameters k c ) ∆H°/R
T1/2 Ef
A ) exp(c(l/T1/2 - 1/T))
Eg n Tnr
F(T) ) (4 + A - Ax1 + 8/A)/4 ) -χ(T)/χ(0) ) F
Tes
B. ConductiVity in the Normal State.
σfr ) ln((exp(Ef/kTnr) + l)/2) σgr ) ln(exp(-(Eg - Ef)/kTnr) + 1) σnr ) (T(lnr- n))(σfr + σgr) σft ) ln((exp(Ef/kT) +1)/2) σnt ) (T2(l - n))(σft + σgt)
Tr γ g
Boltzmann’s constant, 8.6186e × 10-5 eV/deg energy of dissociation/R of the bipolaron determined from the slope at half-height of the variation of the diamagnetic susceptibility with temperature temperature at the half-height of F(T) Fermi energy (eV) in the Fermi-Dirac distribution function energy (eV) at the bottom of the valence band exponent for phonon or electron scattering normalization constant in the conductivity for the normal state normalization constant for the conductivity for the bipolaron tunneling conductivity reference temperature () hωo/2πk) in the Bo¨ttger and Bryksin theory electron-phonon coupling constant in Holstein’s theory effective coupling constant in the Bo¨ttger and Bryksin theory ) γ2 [1 - cos(Rm′ - Rm)
Acknowledgment. Work at Argonne National Laboratory is performed under contract W-31-109 Eng. 39 of The U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science. References and Notes
σgt ) ln(exp(-(Eg - Ef)/kT) +1) σnt ) (T(1-n))(σft + σgt) σn ) σnt/σnr C. ConductiVity of the Bipolaron State. Holstein:
σsr ) xl/sinh(Tes/2Tsr) exp(-2γ/sinh(Tes/(2Tsr))/Tsr) σst ) xl/sinh(Tes/2T) exp(-2γ/sinh(Tes/(2T))/T) Bo¨ttger and Bryksin:
σsr ) xTes/Tsr exp(St) exp(Tes/Tsr) σst ) xTes/T exp(St) exp(Tes/T) σt ) g/tanh(Tr/T);
Tr ) hωo/2πk
σs ) σst/σsr
(1) Texts: Anderson, 1997; Tinkham, 1996; Poole, Farach, and Creswick, 1995; Kresin and Wolf, 1990; Crisan, 1989; Poole, Datta, and Farach, 1988; Rickayozen, 1965; Blatt (19664); and Schoenberg, 1952. (2) Science Citation Index, Institute for Scientific Information, Philadephia. (3) Pomar, A.; Curra´s, R. S.; Veira, J. A., Vidal, F. Phys. ReV. B 1996I, 53, 8245-8248. (4) Onnes, H. K. Lieden. Commun. 1911, 120b, 122b, and 124b. (5) Kittel, C. Introduction to Solid State Physics; John Wiley & Sons: New York, 1953; p 200, Figure 1.11. (6) Kittel, C. Introduction to Solid State Physics; John Wiley & Sons: New York, 1996; p 334, Figure 1. (7) Thorn, R. J. J. Phys. Chem. Solids 1987, 48, 355-361. Thorn, R. J.; Thorn, C. E. J. Phys. Chem. Solids 1989, 50, 153-161. (8) Holstein, T. Ann Phys. 1950, 8, 325-343, 343-389. (9) Bo¨ttger, H.; Bryksin, V. V. Hopping Conduction in Solids; VCH: Deerfieldbeach, FL, 1985; pp 55-65. (10) Ogg, R. A. Phys. ReV. 1946, 69, 243-244. (11) Bardeen, J.; Cooper, L. N.; Schieffer, J. R. Phys. ReV. 1957, 108, 1175-1204. (12) Thorn, R. J. Chemical Equilibrium Bases for Oxide and Organic Superconductors; John Wiley & Sons: New York, 1996; pp 162-163. (13) Thorn, R. J.; Dudek, J.; Schlueter;. J. A., Wang H. H. Magnetic Susceptilities of κ-(BEDT-TTF)2X. Unpublished manuscript. (14) Dudek, J. Private communication, 1998. (15) Bardeen., J. Phys. ReV. 1950, 79, 167-168. (16) Matthias, B. T. Am. Sci. 1970, 58, 80-83. (17) Seitz, F. Modern Theory of Solids; McGraw-Hill Book Co.: New York, 1940; p 583.
