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Possible explanation of unstable superconducting phase in KxC60 with Tc=21 K
Solid State Sciences 3 (2001) 531–537 www.elsevier.com/locate/ssscie
Possible explanation of unstable superconducting phase in Kx C60 with Tc = 21 K Jan Stankowski a , Tadeusz Luty b , Wojciech Kempi´nski a , Lidia Piekara-Sady a a Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna´ n, Poland b Institute of Physical and Theoretical Chemistry, Wrocław Technical University, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland
Received 2 May 2000; revised 29 December 2000; accepted 23 January 2001
Abstract EPR and MMMA methods revealed two K3 C60 phases, the unstable one with Tc = 21.0 K containing C1− 60 centers and a stable ions. The metastable phase appears only in the first one with Tc = 18.0 K constituting of C60 molecules in the form of C3− 60 stage of potassium intercalation of fullerene at the C60 /K3 C60 phase boundary. This phase is being transformed to a stable one on further doping to K3 C60 . The appearance of a metastable phase is related to the instability of the fullerene C60 fcc structure during the process of octahedral sites being filled by potassium ions. The critical temperature Tc shift upwards from 18.5 to 3− 1+ 1− 1− 21.0 K is ascribed to the possible process of potassium transient back charge transfer valency, i.e. K1+ 3 C60 ↔ K2 K C60 1+ + K1− + C1− ], which could enlarge the unit cell necessary to shift a superconducting critical temerature [3K1+ + C3− 60 ⇔ 2K 60 according to Tc on volume dependence. 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
0. Introduction Fullerene C60 is an electron acceptor molecule. The minimum energy of the Cv60 ion is achieved for the valency state of v = −1.5, whereas zero energy corresponds both to neutral C60 and C3− 60 ion [1]. Assuming an ionic model, the energy of the charged state of a fullerene has been correlated to the spectroscopic g-value, experimentaly determined from EPR data [2–9], related to the spinorbit interaction of the conducting electron on each carbon atom along its equatorial trajectory on the C60 surface. This correlation shows that the interaction in the Cv60 state of the fullerene molecule considered as conducting mesoscopic object is satisfactorily described [10]. In the C60 fcc structure there are spacious octahedral (2R = 0.412 nm) and smaller tetrahedral sites (2R = 0.224 nm), which become occupied during the alkali E-mail address: [email protected] (J. Stankowski).
metal intercalation process. Intercalation of potassium into the fullerene C60 can be divided into three consecutive stages: 1. C1+ 60 centers, present as a defect in even freshly prepared and degassed polycrystalline sample, disappear after charge recombination, i.e. conducting electron trapped on the C60 molecule during the diffusion of potassium atoms (or ions) into fullerene fcc lattice. 3− 2. C1− 60 and C60 centers appear as a result of subsenquently trapped electrons on fullerene molecule: K1+ ions start to occupy octahedral sites of the fcc fullerene structure which they are very mobile, then K1+ tightly occupy much smaller tetrahedral sites; the K1− transient state can appear at the initiation of intercalation, when C1− 60 is observed as the nonequilibrium charged state in Kx C− 60 system. 3− 3. Only C60 centers exist: K1+ ions occupy both octaand tetrahedral sites in fully doped superconducting K3 C60 , when the whole crystal has reached charge compensated state.
1293-2558/01/$ – see front matter 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 1 ) 0 1 1 5 7 - 8
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At low doping levels of polycrystalline C60 , during each heating period of the potassium and fullerene mixture, K1+ ions propagate into the granular sample and into granules. There exists a phase boundary between pure phase of the C60 and completely synthesized superconducting K3 C60 . At this phase boundary the solid state reaction takes place. If the process is stopped at the stage 2, redistribution of potassium, accompanied by fur3− ther reduction of C1− 60 to C60 as a result of phase separation of K1 C60 to α-C60 and K3 C60 can be observed at room temperature by EPR. Measurement of the EPR integral intensity is a measure of the reaction progress, and this kinetics of potassium diffusion has been already described [11]. Magnetically Modulated Microwave Absorption (MMMA) has been used to detect superconductivity in potassium intercalated fullerene [11,12]. Observation of Josephson hysteresis loops served as an evidence for the existence of the superconducting state.
1. EPR of potassium-doped fullerene C60 It has been shown that even though there is a broad range of g-factor values for a given paramagnetic fullerene ions [2–9]. The g-factor for a given charged state differs in value sufficiently to enable identification of the fullerene center solely on the basis of the resonant field of EPR [10]. Thus, EPR can distinguish between paramagnetic centers generated in the process of intercalation if the linewidths are sufficiently narrow (0.5 mT). In the case of potassium intercalation, although at room temperature the signals overlap due to the small g value difference and substantial linewidths (Fig. 1), the temperature de3− pendence of the individual linewidths of C1− 60 and C60 separates these signals at lower temperatures.
2. Superconductivity in potassium doped Kx C60 If the process of intercalation at the very beginning (5 min of heating fullerene and potassium mixture at 500 K) is abruptly stopped (by quenching the sample), we found that the superconducting phase has already been formed. The critical temperature of the transition from the normal to the superconducting state (Tc = 21.0 K, Fig. 2(a)) is found, however, to be higher than that of the stable K3 C60 phase [13,14]. The C1+ 60 EPR line in Fig. 2(a) is due to defect centers in polycrystalline fullerene samples (before stage 1 of the intercalation
3− Fig. 1. The X-band EPR signals of C1− 60 and C60 become separated below room temperature due to the line narrowing.
process has not been completely formed). On further doping, one can find two coexisting superconducting phases (Fig. 2(b)), one with Tc = 21.0 K and that of (0) K3 C60 , i.e. Tc = 18.0 K. These two superconducting phases can be observed only when the EPR spectrum 3− consists of both C1− 60 and C60 signals (Fig. 2(b)). These two lines originate from K1 C60 and α-C60 (Kx C60 for 3− x ≈ 0.1) phases (C1− 60 ), and from the K3 C60 phase (C60 ). Superconductivity in both phases is evidenced (Fig. 3) (1) by Josephson hysteresis loops at Tc = 21.0 K (unstable (2) (1) (2) phase, i.e. Tc < T < Tc ) and at Tc = 18.5 K (stable (2) phase, i.e. T = Tc ). 3. Superconductivity in K3 C60 at T c = 21 K It has been already known that for potassium-doped fullerenes, the K3 C60 is the only superconducting phase (0) with Tc = 18–19 K. The existence of an unstable superconducting phase with Tc = 21.0 K [13,14], appearing in the course of the doping process, must be related (0) to an upward shift of Tc in K3 C60 . On looking for a (0) possible mechanism leading to explain Tc > Tc one has to seek for enlarging the lattice constant of crystal nuclei of K3 C60 in the matrix which have a lattice constant
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(a)
(a)
(b) (b) (1)
(0)
Fig. 2. Superconducting phase with Tc > Tc exists only when K1 C60 is present in the sample as evidenced by C1− 60 EPR signal in X-band: (a) EPR and MMMA signal at a very low doping level; C1+ 60 signal of polycrystalline C60 is still present: the stage 1 of the intercalation process has not been completed yet; (b) EPR and MMMA signal at slightly higher doping level (stage 3 has already been started).
ahost > aK3 C60 . K1 C60 and C60 do not satisfy this relation, both having the lattice constants smaller than that of K3 C60 . Taking into account the fact that the supercon(0) ducting phase with Tc > Tc exists only when K1+ C1− 60 is present in the sample (K1 C60 or α-C60 and even pristine C60 at a very low doping level), the phenomenon has to be a transient feature. This unstable phase appears only if the material exhibits the EPR line of C1− 60 (Fig. 2). This implies the presence of α-C60 and/or K1 C60 phases. The K1 C60 phase can be dimerized by fast cooling [15]. It has been already proposed that the strains at the boundary
Fig. 3. Typical Josephson hysteresis loops in Kx C60 at T = 21.0 K (2) (1) (unstable phase, i.e. Tc < T < Tc ) and at T = 18.5 K (stable (2) superconducting phase, i.e. T < Tc ).
in the “onion-like” structure of K1 C60 and K3 C60 phases within a granule could give a negative stress and shift Tc to higher temperature. In this model another mechanism of shifting Tc , also based on the assumption of a BCS-like dependence of Tc on the lattice constant, will be presented. We will be discussing the process of K-intercalation at a very low doping level. The C60 molecule is an electron acceptor. Theoretical calculations have shown that the energy of the fullerene molecule versus the negatively charged state 1.5− [1]. This of the molecule has a minimum for C60 1− 2− implies that C60 and C60 have lower energy than neutral C60 , which has energy almost identical to C3− 60 . This 3− conclusion is very important since C60 appears to be
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“transparent” for electrons jumping between potassium ions in K3 C60 lattice. The contact of potassium vapor with a fullerene induces the flow of electrons from the conduction band to appropriate states in the fullerene molecules. C1− 60 ions are then formed, having energy lower than C60 molecules by 2.7 eV. Heating potassium and fullerene results in potassium diffusion to octahedral (Roct = 0.206 nm) and tetrahedral (Rtetr = 0.112 nm) sites. In tetrahedral sites K1+ ions, rK+ = 0.133 nm, are tightly locked. The fcc structure is preserved because these voids are filled. The larger octahedral sites (the difference between octa- and tetrahedral site amounts to Roct − Rtetr = 0.073 nm) prevent the K1+ ion from having a central position at the octahedral site on the intercalation front. There is also no charge equilibrium at the phase boundary between K3 C60 and C60 . Thus, the octahedral void becomes distorted, moving the C3− 60 ion towards K1+ . It is conceivable to presume that, at the beginning of doping, the back charge transfer of two electrons from C3− 60 to the conduction band can take place. The only acceptor available is K1+ ion, which can accept one or, even more probably, two electrons to become K1− ion and to form the transient state of 1− 1+ 1+ (C1− 60 K K K ). Ionization energy of the potassium atom is 4.34 eV. The neutral atom can make the stable negative ion K1− with bonding energy of extra electron 0.5 eV [17]. Total energy of two electrons in K1− is close to the excitation energy of fullerene molecule to C1+ 60 ionic state [1]. Such charge transfer can restore fcc structure because the radius rK1− = 0.231 nm of K1− is much larger, than that of K1+ , rK1+ = 0.133 nm. The K1− potassium state can appear in a specific situation as a result of the K1+ potassium ion separation. In liquid state, NMR evidenced K1− formation when the K1+ complex with 18-crown 6-molecule had been formed [16,18] and free electrons could be trapped by weak bonded potassium outside of the complex. In the case of potassium doping of a fullerene, K1+ is locked in tetrahedral site whereas the K1+ ion may form K1− at the front of intercalation in previously asymmetrically occupied octahedral voids of the fullerene lattice. Hence, the phase boundary of K3 C60 can be moved towards the not intercalated C60 lattice. Large difference of the activation energy for diffusive electrical transport of the K1+ ions through the C60 lattice was observed [19]. The activation energy for diffusion to octahedral sites is ∆octa = 0.572 eV and for tetrahedral sites is two times larger (∆tetr = 1.02 eV). Similar difference was obtained in NMR relaxation eperiments.
The mechanism is illustrated in Fig. 4. The potassium ion distribution determines the front of intercalation between K3 C60 and C60 . Both of the phases are fcc (Fig. 4(a)). As the potassium ion moves the front boundary, exhibiting unstability of the structure, this results in the distortion of the octahedral site. Thus, one C3− 60 ion is pushed out from its stable position (Fig. 4(b)) and becomes C1− 60 , the more stable fullerene anion. The charge 1+ is transferred from fullerene C3− 60 to potassium ion K 1− resulting in much larger K ion, so the octahedral site becomes tightly filled (Fig. 4(c)). Therefore, the fcc structure is restored at the phase boundary, and the phase boundary is shifted. This mechanism of C1− 60 formation accompanying K1− ion generation is possible until potassium ions have diffused towards the whole interior of the fullerene grain. A non-stable phase with octahedral sites occupied either by K0 or K1− ions would have a larger lattice constant than K3 C60 . Therefore, the presence of an unstable phase with a larger lattice constant can induce an internal pressure in the K3 C60 lattice, which shifts the critical temperature Tc of K3 C60 upwards. The unstable phase reveals itself as a discrete state because it is ob(1) served via a distinct critical temperature Tc = 21.0 K [13,14]. If all octahedral sites had been occupied by K0 or K1− ions, the Tc would have been shifted to Tcmax = 40 K according to Tc linear dependence on the unit cell dimension [20]. 4. Simple model We consider the solid-state reaction of C60 having fcc crystal structure and metallic potassium vapor diffusing into that structure C60 + 3K ⇒ K3 C60
(1)
At the reaction front, a transient state appears as a result of back two-electron transfer: 1+ + K1− + C1− 3K1+ + C3− 60 ⇒ 2K 60
(2)
which is reflected in the EPR spectra as strong C1− 60 signal. The system may be considered as a solid solution of potassium in C60 matrix. The dopant at the reaction front may be either in normal (K1+ ) or excited (K1− ) state. The thermodynamics of the system can be described in the framework of regular solid solution theory [21]. First, let us consider pure compounds corresponding to normal (n) and excited (e) states, i.e. (3K1+ + C3− 60 ) 1− 1+ 1− and (2K + K + C60 ), respectively. These two
J. Stankowski et al. / Solid State Sciences 3 (2001) 531–537
(a)
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(b)
(c) Fig. 4. A cartoon of the potassium doping process of C60 fullerene lattice. (a) The phase boundary between K3 C60 and C60 fcc structures. (b) Transient state at the phase boundary: distortion of fcc structure results from K1+ ion asymmetrical position in octahedral void, thus, C3− 60 is shifted. The process 1+ may occur to form C1− and K1− . (c) Two electrons are transferred back from K1− to C1− : the phase of two electrons transfer from C3− 60 to K 60 60 boundary between the perfect fcc K3 C60 and C60 is shifted towards C60 lattice.
states can be described by adiabatic potentials in terms of a coordinate related to the radii of K1+ and K1− ions (Fig. 5). Chemical potential difference for the pure “compounds”, µ(ne) = µ(e) − µ(n) , is defined by the energy difference between these states, and electronic and vibrational contributions. For an ideal solution (no interaction with the matrix and between species) the
degree of the transformation (2) is given by a standard formula: 1 , c(T ) = (3) 1 + [g (n) /g (e) ] exp(µ(ne) /kT ) where g (i) is degeneracy of ith state. Now, it should be taken into account that the reaction takes place in a solid and that the reacting species
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volume change Vne , due to the K1+ · · · ⇒ · · · K1− transformation is related to the difference in volume of the “hard spheres” (potassium ions), i.e. vne = vK− − vK+ , by the relation Vne = γ0 vne ,
(8)
and it is the so-called “image-pressure” approach. This change in volume leads to the internal chemical pressure, which drives the reaction front and which is negative of the pressure defined as ∂f p=− (9) , ∂Vne T Fig. 5. Free energy versus distortion Qi for stable state K3 C60 and 1+ excited state 2K1+ K1− C1− 60 at the front of a solid state reaction 3K 1+ + K1− + C1− . + C3− 60 ⇒ 2K 60
can interact (via matrix). We adopt a simple model of the “hard spheres” embedded in an isotropic matrix (pure C60 ). The elastic effects will be considered as the most important, neglecting the polarization etc., due to the electron transfer. The model is similar to that of Spiering [22], which is a “minimum version” of a more general one [23]. The free energy per potassium ion of the solid solution, where x is the fraction of potassium embedded into C60 , and c is the fraction of potassium ions, transformed into K1− state, is given by f (x, c, T ) = cµ(ne) − T Smix (c) + x cδ − c2 Γ . (4) First two terms describe an ideal solution with Smix = −k c ln c + (1 − c) ln(1 − c) .
(5)
The third term describes self-deformation effect, i.e. the energy due to the transformation in a solid matrix with δ = 2qΓ
and q =
Vno , Vne
(6)
where Vno = Vn − V0 and Vne = Ve − Vn , where V0 is the unit cell volume of the matrix, Vn and Ve is the unit cell volume of normal and excited state, respectively. The fourth term stands for the interaction (mediated by the elastic matrix) between transforming sites. If the elastic modulus K, and Poisson ratio σ , characterize the elastic medium (C60 ) where the transformation takes place, the interaction parameters are given as [24] 1 Vne 1 Γ = K γ0 − (7) 2 γ0 Vc with γ0 = 3(1 − σ )/(1 + σ ) known as Eshelby constant, whereas Vc is the volume per transforming site. The
and p = p0 + p ∂µne = −c ∂Vne T γ0 − 1 1 Von − cVne . − xcK γ0 Vc
(10)
The first part is the pressure due to the “independently transforming sites”, the result for an ideal solution. The other contributions are due to the self-deformation effect (linear xc term) and to the interaction between transforming sites (quadratic xc2 term). This chemical pressure leads to the change in Tc of the transition from normal to superconducting phase according to Fleming [20]: ∂Tc = [−0.63, −0.78] K kbar−1 . (11) ∂p V Thus, Tc = [−0.63, −0.78] K kbar−1 × p. In conclusion: the correlation of Tc to the interaction between transforming sites (∝ cx 2 ) could be expected to be important for a large doping level and a higher degree of K1+ · · · ⇒ · · · K1− transformation. An increase in the lattice constant by an internal “chemical” pressure gives an increase of Tc according to Fleming universality rule [20]. At low doping level, the lattice constant of the superconducting phase corresponding to the shift of Tc observed in the experiment, can be expressed as a = a (n)(1 + cA),
(12)
where A describes the difference between normal and excited (hypothetical) phase lattice constant: A = (a (e) − a (n))/a (n) = 0.003, and c is the degree of the back charge transfer, i.e. it can express the percentage of the excited state. Thus, to shift Tc of K3 C60 by 2.5 K, the
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presence of 10% of the excited phase is required at the reaction front.
5. Conclusions The chemical equilibrium in the octahedral site in the fcc K3 C60 structure at the reaction front has been considered. The transfer of two electrons from C3− 60 to form C1− can be observed by the EPR only during the 60 doping process. This transfer gives higher energy state than K3 C60 . Two electrons are transferred between two spherical Cv60 molecules pointing to the local pairing mechanism. Superconductivity in K3 C60 could be related to the interaction between electrons localized on the neighboring fullerene ions as the very short correlation length, ξ = 2.6 nm, corresponds to two neighboring fullerene molecules exchanging pair of electrons. This two-electron transfer process has been observed only at the phase boundary during the formation of the superconducting phase K3 C60 . Acknowledgements We would like to thank Prof. F. Rozpłoch for stimulating discussions. This work was partially supported by KBN grant 5 P03B 06120.
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[3] P.J. Krusic, E. Wasserman, B.A. Parkinson, B. Malone, E.R. Holler Jr., P.J. Keizer, I.R. Motrton, K.F. Preston, J. Am. Chem. Soc. 113 (1991) 6274. [4] D. Dubois, M.T. Janes, K.M. Kadish, J. Am. Chem. Soc. 114 (1992) 6446. [5] A.J. Schell-Sorokin, F. Mehran, G. Eaton, S.S. Eaton, A. Viehbeck, T.R. O’Tole, C.A. Braun, Chem. Phys. Lett. 195 (1992) 225. [6] K. Tanikagi, M. Kosaka, T. Manako, Y. Kubo, I. Hirosawa, K. Uchida, K. Prassides, Chem. Phys. Lett. 240 (1995) 627. [7] S.S. Eaton, G.R. Eaton, Appl. Magn. Reson. 11 (1996) 155. [8] L. Piekara-Sady, A.V. Il’yasov, V. Moroyov, J. Stankowski, W. Kempi´nski, Z. Trybuła, Appl. Magn. Reson. 9 (1995) 367. [9] J. Stankowski, L. Piekara-Sady, W. Kempi´nski, O. Huminiecki, P.B. Sczaniecki, Full. Sci. Technol. 5 (1997) 1203. [10] J. Stankowski, L. Piekara-Sady, W. Kempi´nski, Appl. Magn. Reson. 19 (2000) 539. [11] W. Kempi´nski, L. Piekara-Sady, J. Stankowski, in: Molecular Nanostructures, World Scientific, 1998, p. 249. [12] J. Stankowski, B. Czy˙zak, J. Martinek, Phys. Rev. B 42 (1990) 13. [13] W. Kempi´nski, J. Stankowski, Acta Phys. Polon. A 88 (1995) 549. [14] W. Kempi´nski, J. Stankowski, Solid State Commun. 97 (1996) 1079. [15] Q. Zhu, D.E. Cox, J.E. Fisher, Phys. Rev. Lett. B 51 (1995) 3966. [16] M. Sokoł, Z. Grobelny, Z. Jedli´nski, Magn. Res. Chem. 28 (1990) 934. [17] H. Massey, Atomic and Molecular Collisions, Taylor & Francis, 1979, Polish translation: PWN, 1982, pp. 85, 90. [18] M. Sokoł, J. Grobelny, Z. Grobelny, Z.J. Jedi´nski, Spectrochim. Acta A 47 (1991) 1547. [19] L. Firlej, Cond. Matter News 8 (2000) 22. [20] R.M. Fleming, A.P. Ramirez, M.J. Rosseinsky, D.W. Murphy, R.C. Haddon, S.M. Zahurak, A.V. Makhija, Nature 352 (1991) 787. [21] C.P. Slichter, H.G. Drickamer, J. Chem. Phys. 56 (1997) 2142. [22] H. Spiering, E. Meissner, H. Koppen, E.W. Muller, P. Gutlich, Chem. Phys. 68 (1982) 65. [23] T. Luty, C.J. Eckhardt, J. Am. Chem. Soc. 117 (1995) 2441. [24] M. Willembachen, H. Spiering, J. Phys. C: Solid State Phys. 21 (1988) 1423.
Possible explanation of unstable superconducting phase in Kx C60 with Tc = 21 K Jan Stankowski a , Tadeusz Luty b , Wojciech Kempi´nski a , Lidia Piekara-Sady a a Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna´ n, Poland b Institute of Physical and Theoretical Chemistry, Wrocław Technical University, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland
Received 2 May 2000; revised 29 December 2000; accepted 23 January 2001
Abstract EPR and MMMA methods revealed two K3 C60 phases, the unstable one with Tc = 21.0 K containing C1− 60 centers and a stable ions. The metastable phase appears only in the first one with Tc = 18.0 K constituting of C60 molecules in the form of C3− 60 stage of potassium intercalation of fullerene at the C60 /K3 C60 phase boundary. This phase is being transformed to a stable one on further doping to K3 C60 . The appearance of a metastable phase is related to the instability of the fullerene C60 fcc structure during the process of octahedral sites being filled by potassium ions. The critical temperature Tc shift upwards from 18.5 to 3− 1+ 1− 1− 21.0 K is ascribed to the possible process of potassium transient back charge transfer valency, i.e. K1+ 3 C60 ↔ K2 K C60 1+ + K1− + C1− ], which could enlarge the unit cell necessary to shift a superconducting critical temerature [3K1+ + C3− 60 ⇔ 2K 60 according to Tc on volume dependence. 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
0. Introduction Fullerene C60 is an electron acceptor molecule. The minimum energy of the Cv60 ion is achieved for the valency state of v = −1.5, whereas zero energy corresponds both to neutral C60 and C3− 60 ion [1]. Assuming an ionic model, the energy of the charged state of a fullerene has been correlated to the spectroscopic g-value, experimentaly determined from EPR data [2–9], related to the spinorbit interaction of the conducting electron on each carbon atom along its equatorial trajectory on the C60 surface. This correlation shows that the interaction in the Cv60 state of the fullerene molecule considered as conducting mesoscopic object is satisfactorily described [10]. In the C60 fcc structure there are spacious octahedral (2R = 0.412 nm) and smaller tetrahedral sites (2R = 0.224 nm), which become occupied during the alkali E-mail address: [email protected] (J. Stankowski).
metal intercalation process. Intercalation of potassium into the fullerene C60 can be divided into three consecutive stages: 1. C1+ 60 centers, present as a defect in even freshly prepared and degassed polycrystalline sample, disappear after charge recombination, i.e. conducting electron trapped on the C60 molecule during the diffusion of potassium atoms (or ions) into fullerene fcc lattice. 3− 2. C1− 60 and C60 centers appear as a result of subsenquently trapped electrons on fullerene molecule: K1+ ions start to occupy octahedral sites of the fcc fullerene structure which they are very mobile, then K1+ tightly occupy much smaller tetrahedral sites; the K1− transient state can appear at the initiation of intercalation, when C1− 60 is observed as the nonequilibrium charged state in Kx C− 60 system. 3− 3. Only C60 centers exist: K1+ ions occupy both octaand tetrahedral sites in fully doped superconducting K3 C60 , when the whole crystal has reached charge compensated state.
1293-2558/01/$ – see front matter 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 1 ) 0 1 1 5 7 - 8
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At low doping levels of polycrystalline C60 , during each heating period of the potassium and fullerene mixture, K1+ ions propagate into the granular sample and into granules. There exists a phase boundary between pure phase of the C60 and completely synthesized superconducting K3 C60 . At this phase boundary the solid state reaction takes place. If the process is stopped at the stage 2, redistribution of potassium, accompanied by fur3− ther reduction of C1− 60 to C60 as a result of phase separation of K1 C60 to α-C60 and K3 C60 can be observed at room temperature by EPR. Measurement of the EPR integral intensity is a measure of the reaction progress, and this kinetics of potassium diffusion has been already described [11]. Magnetically Modulated Microwave Absorption (MMMA) has been used to detect superconductivity in potassium intercalated fullerene [11,12]. Observation of Josephson hysteresis loops served as an evidence for the existence of the superconducting state.
1. EPR of potassium-doped fullerene C60 It has been shown that even though there is a broad range of g-factor values for a given paramagnetic fullerene ions [2–9]. The g-factor for a given charged state differs in value sufficiently to enable identification of the fullerene center solely on the basis of the resonant field of EPR [10]. Thus, EPR can distinguish between paramagnetic centers generated in the process of intercalation if the linewidths are sufficiently narrow (0.5 mT). In the case of potassium intercalation, although at room temperature the signals overlap due to the small g value difference and substantial linewidths (Fig. 1), the temperature de3− pendence of the individual linewidths of C1− 60 and C60 separates these signals at lower temperatures.
2. Superconductivity in potassium doped Kx C60 If the process of intercalation at the very beginning (5 min of heating fullerene and potassium mixture at 500 K) is abruptly stopped (by quenching the sample), we found that the superconducting phase has already been formed. The critical temperature of the transition from the normal to the superconducting state (Tc = 21.0 K, Fig. 2(a)) is found, however, to be higher than that of the stable K3 C60 phase [13,14]. The C1+ 60 EPR line in Fig. 2(a) is due to defect centers in polycrystalline fullerene samples (before stage 1 of the intercalation
3− Fig. 1. The X-band EPR signals of C1− 60 and C60 become separated below room temperature due to the line narrowing.
process has not been completely formed). On further doping, one can find two coexisting superconducting phases (Fig. 2(b)), one with Tc = 21.0 K and that of (0) K3 C60 , i.e. Tc = 18.0 K. These two superconducting phases can be observed only when the EPR spectrum 3− consists of both C1− 60 and C60 signals (Fig. 2(b)). These two lines originate from K1 C60 and α-C60 (Kx C60 for 3− x ≈ 0.1) phases (C1− 60 ), and from the K3 C60 phase (C60 ). Superconductivity in both phases is evidenced (Fig. 3) (1) by Josephson hysteresis loops at Tc = 21.0 K (unstable (2) (1) (2) phase, i.e. Tc < T < Tc ) and at Tc = 18.5 K (stable (2) phase, i.e. T = Tc ). 3. Superconductivity in K3 C60 at T c = 21 K It has been already known that for potassium-doped fullerenes, the K3 C60 is the only superconducting phase (0) with Tc = 18–19 K. The existence of an unstable superconducting phase with Tc = 21.0 K [13,14], appearing in the course of the doping process, must be related (0) to an upward shift of Tc in K3 C60 . On looking for a (0) possible mechanism leading to explain Tc > Tc one has to seek for enlarging the lattice constant of crystal nuclei of K3 C60 in the matrix which have a lattice constant
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(a)
(a)
(b) (b) (1)
(0)
Fig. 2. Superconducting phase with Tc > Tc exists only when K1 C60 is present in the sample as evidenced by C1− 60 EPR signal in X-band: (a) EPR and MMMA signal at a very low doping level; C1+ 60 signal of polycrystalline C60 is still present: the stage 1 of the intercalation process has not been completed yet; (b) EPR and MMMA signal at slightly higher doping level (stage 3 has already been started).
ahost > aK3 C60 . K1 C60 and C60 do not satisfy this relation, both having the lattice constants smaller than that of K3 C60 . Taking into account the fact that the supercon(0) ducting phase with Tc > Tc exists only when K1+ C1− 60 is present in the sample (K1 C60 or α-C60 and even pristine C60 at a very low doping level), the phenomenon has to be a transient feature. This unstable phase appears only if the material exhibits the EPR line of C1− 60 (Fig. 2). This implies the presence of α-C60 and/or K1 C60 phases. The K1 C60 phase can be dimerized by fast cooling [15]. It has been already proposed that the strains at the boundary
Fig. 3. Typical Josephson hysteresis loops in Kx C60 at T = 21.0 K (2) (1) (unstable phase, i.e. Tc < T < Tc ) and at T = 18.5 K (stable (2) superconducting phase, i.e. T < Tc ).
in the “onion-like” structure of K1 C60 and K3 C60 phases within a granule could give a negative stress and shift Tc to higher temperature. In this model another mechanism of shifting Tc , also based on the assumption of a BCS-like dependence of Tc on the lattice constant, will be presented. We will be discussing the process of K-intercalation at a very low doping level. The C60 molecule is an electron acceptor. Theoretical calculations have shown that the energy of the fullerene molecule versus the negatively charged state 1.5− [1]. This of the molecule has a minimum for C60 1− 2− implies that C60 and C60 have lower energy than neutral C60 , which has energy almost identical to C3− 60 . This 3− conclusion is very important since C60 appears to be
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“transparent” for electrons jumping between potassium ions in K3 C60 lattice. The contact of potassium vapor with a fullerene induces the flow of electrons from the conduction band to appropriate states in the fullerene molecules. C1− 60 ions are then formed, having energy lower than C60 molecules by 2.7 eV. Heating potassium and fullerene results in potassium diffusion to octahedral (Roct = 0.206 nm) and tetrahedral (Rtetr = 0.112 nm) sites. In tetrahedral sites K1+ ions, rK+ = 0.133 nm, are tightly locked. The fcc structure is preserved because these voids are filled. The larger octahedral sites (the difference between octa- and tetrahedral site amounts to Roct − Rtetr = 0.073 nm) prevent the K1+ ion from having a central position at the octahedral site on the intercalation front. There is also no charge equilibrium at the phase boundary between K3 C60 and C60 . Thus, the octahedral void becomes distorted, moving the C3− 60 ion towards K1+ . It is conceivable to presume that, at the beginning of doping, the back charge transfer of two electrons from C3− 60 to the conduction band can take place. The only acceptor available is K1+ ion, which can accept one or, even more probably, two electrons to become K1− ion and to form the transient state of 1− 1+ 1+ (C1− 60 K K K ). Ionization energy of the potassium atom is 4.34 eV. The neutral atom can make the stable negative ion K1− with bonding energy of extra electron 0.5 eV [17]. Total energy of two electrons in K1− is close to the excitation energy of fullerene molecule to C1+ 60 ionic state [1]. Such charge transfer can restore fcc structure because the radius rK1− = 0.231 nm of K1− is much larger, than that of K1+ , rK1+ = 0.133 nm. The K1− potassium state can appear in a specific situation as a result of the K1+ potassium ion separation. In liquid state, NMR evidenced K1− formation when the K1+ complex with 18-crown 6-molecule had been formed [16,18] and free electrons could be trapped by weak bonded potassium outside of the complex. In the case of potassium doping of a fullerene, K1+ is locked in tetrahedral site whereas the K1+ ion may form K1− at the front of intercalation in previously asymmetrically occupied octahedral voids of the fullerene lattice. Hence, the phase boundary of K3 C60 can be moved towards the not intercalated C60 lattice. Large difference of the activation energy for diffusive electrical transport of the K1+ ions through the C60 lattice was observed [19]. The activation energy for diffusion to octahedral sites is ∆octa = 0.572 eV and for tetrahedral sites is two times larger (∆tetr = 1.02 eV). Similar difference was obtained in NMR relaxation eperiments.
The mechanism is illustrated in Fig. 4. The potassium ion distribution determines the front of intercalation between K3 C60 and C60 . Both of the phases are fcc (Fig. 4(a)). As the potassium ion moves the front boundary, exhibiting unstability of the structure, this results in the distortion of the octahedral site. Thus, one C3− 60 ion is pushed out from its stable position (Fig. 4(b)) and becomes C1− 60 , the more stable fullerene anion. The charge 1+ is transferred from fullerene C3− 60 to potassium ion K 1− resulting in much larger K ion, so the octahedral site becomes tightly filled (Fig. 4(c)). Therefore, the fcc structure is restored at the phase boundary, and the phase boundary is shifted. This mechanism of C1− 60 formation accompanying K1− ion generation is possible until potassium ions have diffused towards the whole interior of the fullerene grain. A non-stable phase with octahedral sites occupied either by K0 or K1− ions would have a larger lattice constant than K3 C60 . Therefore, the presence of an unstable phase with a larger lattice constant can induce an internal pressure in the K3 C60 lattice, which shifts the critical temperature Tc of K3 C60 upwards. The unstable phase reveals itself as a discrete state because it is ob(1) served via a distinct critical temperature Tc = 21.0 K [13,14]. If all octahedral sites had been occupied by K0 or K1− ions, the Tc would have been shifted to Tcmax = 40 K according to Tc linear dependence on the unit cell dimension [20]. 4. Simple model We consider the solid-state reaction of C60 having fcc crystal structure and metallic potassium vapor diffusing into that structure C60 + 3K ⇒ K3 C60
(1)
At the reaction front, a transient state appears as a result of back two-electron transfer: 1+ + K1− + C1− 3K1+ + C3− 60 ⇒ 2K 60
(2)
which is reflected in the EPR spectra as strong C1− 60 signal. The system may be considered as a solid solution of potassium in C60 matrix. The dopant at the reaction front may be either in normal (K1+ ) or excited (K1− ) state. The thermodynamics of the system can be described in the framework of regular solid solution theory [21]. First, let us consider pure compounds corresponding to normal (n) and excited (e) states, i.e. (3K1+ + C3− 60 ) 1− 1+ 1− and (2K + K + C60 ), respectively. These two
J. Stankowski et al. / Solid State Sciences 3 (2001) 531–537
(a)
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(b)
(c) Fig. 4. A cartoon of the potassium doping process of C60 fullerene lattice. (a) The phase boundary between K3 C60 and C60 fcc structures. (b) Transient state at the phase boundary: distortion of fcc structure results from K1+ ion asymmetrical position in octahedral void, thus, C3− 60 is shifted. The process 1+ may occur to form C1− and K1− . (c) Two electrons are transferred back from K1− to C1− : the phase of two electrons transfer from C3− 60 to K 60 60 boundary between the perfect fcc K3 C60 and C60 is shifted towards C60 lattice.
states can be described by adiabatic potentials in terms of a coordinate related to the radii of K1+ and K1− ions (Fig. 5). Chemical potential difference for the pure “compounds”, µ(ne) = µ(e) − µ(n) , is defined by the energy difference between these states, and electronic and vibrational contributions. For an ideal solution (no interaction with the matrix and between species) the
degree of the transformation (2) is given by a standard formula: 1 , c(T ) = (3) 1 + [g (n) /g (e) ] exp(µ(ne) /kT ) where g (i) is degeneracy of ith state. Now, it should be taken into account that the reaction takes place in a solid and that the reacting species
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volume change Vne , due to the K1+ · · · ⇒ · · · K1− transformation is related to the difference in volume of the “hard spheres” (potassium ions), i.e. vne = vK− − vK+ , by the relation Vne = γ0 vne ,
(8)
and it is the so-called “image-pressure” approach. This change in volume leads to the internal chemical pressure, which drives the reaction front and which is negative of the pressure defined as ∂f p=− (9) , ∂Vne T Fig. 5. Free energy versus distortion Qi for stable state K3 C60 and 1+ excited state 2K1+ K1− C1− 60 at the front of a solid state reaction 3K 1+ + K1− + C1− . + C3− 60 ⇒ 2K 60
can interact (via matrix). We adopt a simple model of the “hard spheres” embedded in an isotropic matrix (pure C60 ). The elastic effects will be considered as the most important, neglecting the polarization etc., due to the electron transfer. The model is similar to that of Spiering [22], which is a “minimum version” of a more general one [23]. The free energy per potassium ion of the solid solution, where x is the fraction of potassium embedded into C60 , and c is the fraction of potassium ions, transformed into K1− state, is given by f (x, c, T ) = cµ(ne) − T Smix (c) + x cδ − c2 Γ . (4) First two terms describe an ideal solution with Smix = −k c ln c + (1 − c) ln(1 − c) .
(5)
The third term describes self-deformation effect, i.e. the energy due to the transformation in a solid matrix with δ = 2qΓ
and q =
Vno , Vne
(6)
where Vno = Vn − V0 and Vne = Ve − Vn , where V0 is the unit cell volume of the matrix, Vn and Ve is the unit cell volume of normal and excited state, respectively. The fourth term stands for the interaction (mediated by the elastic matrix) between transforming sites. If the elastic modulus K, and Poisson ratio σ , characterize the elastic medium (C60 ) where the transformation takes place, the interaction parameters are given as [24] 1 Vne 1 Γ = K γ0 − (7) 2 γ0 Vc with γ0 = 3(1 − σ )/(1 + σ ) known as Eshelby constant, whereas Vc is the volume per transforming site. The
and p = p0 + p ∂µne = −c ∂Vne T γ0 − 1 1 Von − cVne . − xcK γ0 Vc
(10)
The first part is the pressure due to the “independently transforming sites”, the result for an ideal solution. The other contributions are due to the self-deformation effect (linear xc term) and to the interaction between transforming sites (quadratic xc2 term). This chemical pressure leads to the change in Tc of the transition from normal to superconducting phase according to Fleming [20]: ∂Tc = [−0.63, −0.78] K kbar−1 . (11) ∂p V Thus, Tc = [−0.63, −0.78] K kbar−1 × p. In conclusion: the correlation of Tc to the interaction between transforming sites (∝ cx 2 ) could be expected to be important for a large doping level and a higher degree of K1+ · · · ⇒ · · · K1− transformation. An increase in the lattice constant by an internal “chemical” pressure gives an increase of Tc according to Fleming universality rule [20]. At low doping level, the lattice constant of the superconducting phase corresponding to the shift of Tc observed in the experiment, can be expressed as a = a (n)(1 + cA),
(12)
where A describes the difference between normal and excited (hypothetical) phase lattice constant: A = (a (e) − a (n))/a (n) = 0.003, and c is the degree of the back charge transfer, i.e. it can express the percentage of the excited state. Thus, to shift Tc of K3 C60 by 2.5 K, the
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presence of 10% of the excited phase is required at the reaction front.
5. Conclusions The chemical equilibrium in the octahedral site in the fcc K3 C60 structure at the reaction front has been considered. The transfer of two electrons from C3− 60 to form C1− can be observed by the EPR only during the 60 doping process. This transfer gives higher energy state than K3 C60 . Two electrons are transferred between two spherical Cv60 molecules pointing to the local pairing mechanism. Superconductivity in K3 C60 could be related to the interaction between electrons localized on the neighboring fullerene ions as the very short correlation length, ξ = 2.6 nm, corresponds to two neighboring fullerene molecules exchanging pair of electrons. This two-electron transfer process has been observed only at the phase boundary during the formation of the superconducting phase K3 C60 . Acknowledgements We would like to thank Prof. F. Rozpłoch for stimulating discussions. This work was partially supported by KBN grant 5 P03B 06120.
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