ORGANIC SUPERCONDUCTIVITY

Denis Jerome

► To cite this version:

Denis Jerome. ORGANIC SUPERCONDUCTIVITY. J. Phys.Soc. Japan, Suppl, 1980. hal-05030509

HAL Id: hal-05030509

https://hal.science/hal-05030509v1

Submitted on 11 Apr 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
ORGANIC SUPERCONDUCTIVITY

D. Jérôme

Laboratoire de Physique des Solides
Université Paris-Sud 91405 Orsay (France)

We present the experimental evidences for the existence of a superconducting state in the Quasi One Dimensional organic conductor·(TMTSF)₂ PF₆. Superconductivity occuring at 1 K under 12 kbar is characterized by a zero resistance diamagnetic state. The anisotropy of the upper critical field of this type II superconductor is consistent with the band structure anisotropy. We present evidences for the existence of large superconducting precursor effects giving rise to a dominant paraconductive contribution below 40 K. We also discuss the anomalously large pressure dependence of Tₛ which drops to 0.19 K under 24 kbar in terms of the current theories.

The aim of this survey is to present the basic results related to the discovery of organic superconductivity in the conducting salt : di (Δ²,² - bi - 4,5 - dimethyl - 1,3 - diselenole) - hexafluorophosphate , (TMTSF)₂ PF₆. [1,2]

I. The superconducting state of (TMTSF)₂ PF₆

The triclinic structure [1] of (TMTSF)₂ PF₆ consists of nearly uniform stacks of TMTSF molecules ordered in sheets separated by anion (PF₆⁻) sheets, Fig.(1). As a consequence of a slight dimerization of the TMTSF molecules observed at room temperature the electron energy band created by the carrier delocalization becomes half-filled.

Good quality crystals have been grown by an electrochemical
technique electrolysing a solution of highly purified TMTSF in
CH₂Cl₂ containing a large excess of PF₆ at very low current.
As the band filling of this conductor is determined by the 2:1
stoichiometry, it cannot be changed by pressure induced band broad-
dening in contrast to the related two-chain materials, (i.e TTF-
TCNQ [3]). Ambient pressure studies [1] have shown that the low
temperature conductivity exceeds α 10⁵ (Ωcm)⁻¹ at 20 K prior to a
sharp transition towards a semiconducting state at α 15 K. The semi-
conducting state is suppressed by the application of a hydrostatic
pressure of 9 kbar, [2], Fig.(2) and thus the resistance exhibits
a metallic-like behaviour down to the liquid helium temperature,
Fig.(3). The longitudinal conductivity measured on a large number
of single crystals by the 4-probe technique amounts to 400 ± 100
(Ωcm)⁻¹ under ambient conditions and increases at a rate of 23 %
kbar⁻¹under pressure [4]. Below 1.2 K a clamped copper-beryllium
pressure vessel cooled by a He³-He⁴ dilution refrigerator has been
used for the study of (i) the resistivity [2] and (ii) the AC sus-
ceptibility (v = 68 Hz) [5].
<figure><img src="image_2.png" /><figcaption>Fig. (1) A schematic side view in perspective of $(TMTSF)_2PF_6$</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. (2) The pressure dependence of the Metal to Insulator transition temperature</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig.(3) The temperature dependence of the longitudinal resistivity of (TMTSF)₂PF₆ under 12 kbar</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig.(4) Observation of a superconducting state in (TMTSF)₂PF₆ at 12 kbar by resistive measurements on two samples (left), by inductive method, AC susceptibility (right)</figcaption></figure>

<figure><img src="image_6.png" /></figure>

Figure (4) shows that the longitudinal resistivity of (TMTSF)₂PF₆ drops sharply to a zero value at a temperature which is sample independent (Tₛ = 0.9 K, for P = 12 kbar). Moreover, the AC susceptibility becomes strongly diamagnetic below 0.9 K. The comparison between the χ-responses of (TMTSF)₂PF₆ crystals and of a niobium marker specimen which becomes superconducting below 9.5 K under 12 kbar shows that the diamagnetic signal of the organic sample corresponds to about 1/3 of the signal expected for the entire flux expulsion. Figure(4) brings the proofs that a superconducting state with zero resistivity and quasi complete flux expulsion is observed below 0.9 K in earth magnetic field under 12 kbar. For the following reasons we can rule out the possibility of superconductivity originating in filaments or metal inclusions : (TMTSF)₂PF₆ single crystals have roughly the same cross sections, namely
<figure><img src="image_7.png" /><figcaption>Fig. (6) Parallel and perpendicular upper critical fields obtained from susceptibility</figcaption></figure>

(where $\phi_O$ is the flux quantum), one obtains from Fig. (6), $\xi_\perp(0) = 250$ Å and $\xi_{\parallel}(0) = 3750$ Å. Since $\xi_{\parallel}(0)$ is considerably larger than a reasonable electron mean free path, $(TMTSF)_2 PF_6$ is to be considered as a "dirty limit" type II superconductor. The theory of Josephson-coupled chains [9] shows that for a coherence length anisotropy of about 15 a misalignment of only a few degrees may easily reduce $H_{c2}^\perp$ by a factor larger than two [8]. Therefore, the anisotropy of 15 should be considered as a lower limit on the actual value, which may be larger by a factor two or more. Consequently $(TMTSF)_2 PF_6$ is a strongly anisotropic superconductor and the Fermi surface anisotropy, namely the ratio of the longitudinal to transverse overlap integrals derived from the superconducting properties $t_{\parallel}/t_\perp = \xi_{\parallel}(0)/\xi_\perp(0) = 15$ compares favourably with the value derived from the ambient conditions conductivity anisotropy, $(t_{\parallel}/t_\perp)^2 \propto \sigma_{\parallel}/\sigma_\perp = 1$ (for diffusive transverse conductivity) [1, 4, 11].

## II. High temperatures precursors of superconductivity

Besides the existence of a 3-D ordered superconducting state in $(TMTSF)_2 PF_6$ an other highlight of the electronic properties is given by the occurrence of superconducting fluctuations below temperatures as high as $\lambda \approx 40$ times $T_s$. The picture of strong fluctuations in this anisotropic conductor is supported by several unusual features in the transport properties of the normal state at 12 kbar [4].

(i) below about 40 K, $\sigma_{\parallel}$ becomes larger than $10^4$ (Ωcm)$^{-1}$ and reaches $10^5$ (Ωcm)$^{-1}$ at helium temperature, an order of magnitude bigger than the figure obtained in other organic conductors [12].

(ii) $\sigma_{\parallel}$ is still strongly T-dependent below 4.2 K in a temperature domain where the conductivity of regular metals is limited by the lattice defects and the impurities.

(iii) $\sigma_{\parallel}/\sigma_\perp$ (with $\sigma_\perp$ measured along the Se-Se direction) remains constant down to 50 K and decreases by a factor 3 on further cooling, Fig. (7).

(iv) an enourmous magnetoresistance is observed up to 30 K when a field of 55 kOe is applied perpendicular to the chain axis, Fig. (8). $(TMTSF)_2 PF_6$ is a single-chain conductor. Therefore, the interpretation of the conductivity in terms of a semi-metallic band model leading to coherent transverse motion at low temperature cannot hold [13]. Unless one is ready to accept the possibility of electron mean free paths as large as 700 intermolecular spacings at 1.2 K which are hard to reconcile with the extreme sensitivity of the low temperature resistivity to impurities [14], the transport properties of
<figure><img src="image_8.png" /><figcaption>Fig.(7) Conductivity anisotropy of a single sample of $(TMTSF)_2 PF_6$</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig.(8) Behaviour of the $(TMTSF)_2 PF_6$ resistance at 12 kbar under a magnetic field perpendicular to the chain axis</figcaption></figure>

$(TMTSF)_2 PF_6$ cannot be understood by a single-particle conductivity scheme below 40 K. Instead, it has been shown that the unusual behaviour of the conduction can be fairly well explained by superconducting fluctuation effects dominant in a very wide temperature domain above $T_C$ [4].

The influence of such fluctuations on conductivity and susceptibility of a quasi 1-D conductor has been calculated using a Ginzburg-Landau theory [4,15]. The total conductivity is the summation of two terms (i) the paraconductivity $\sigma'$ and (ii) the single particle conductivity $\sigma_o$ which usually at low temperatures is limited by the impurities and becomes independent of the temperature. The derivation of the paraconductivity contribution leads to

$$ \sigma'_{//} = \left(\frac{V_F}{2t\downarrow a}\right)^2 \frac{e^2ma'}{16} \xi_{//}(T) \left[1 - \frac{a}{(a^2 + 4\pi\xi_{\downarrow}^2)^{1/2}}\right], \quad (2a) $$

$$ \sigma'_{\perp} = \frac{e^2ma'}{16} \xi_{//}(T) \left[ 1 - \frac{3a}{2(a^2 + 4\pi\xi_{\perp}^2)^{1/2}} + \frac{a^3}{2(a^2 + 4\pi\xi_{\perp}^2)^{1/2}} \right], \quad (2b) $$

for the longitudinal and transverse components, respectively, where $V_F$ is the Fermi velocity, a the interchain distance, m the carrier mass, and $\alpha'$ a constant depending on the single-particle scattering time.

Near $T_C$, $\xi_{\perp}$ is much larger than a (3-D limit) and $\sigma'_{//}$ is proportional to $(T-T_C)^{-1/2}$ (the typical square root divergence of 3-D superconductors, but $\sigma'_{//}$ is enhanced over its value in an isotropic superconductor by the factor $(t_{//}/t_{\perp})^2$ which may be very large in organic conductors due to the small interchain tunneling integral.
At T sufficiently above T_c, ξ_⊥ becomes much smaller than a (1-D limit) eqs 2 lead to

$$ \sigma_{//}^{1-D} \sim (T - T_c^o)^{-3/2} \qquad (3a) $$

and

$$ \sigma_{\perp}^{1-D} \sim (T - T_c^o)^{-5/2} \qquad (3b) $$

where T_c^o is a mean-field transition temperature. A consequence of eqs.(3) is that σ_{//}/σ_{⊥} becomes T-dependent in the 1-D limit. Moreover, when compared to the paraconductivity of an isotropic superconductor, σ_{//}^{1-D} is enhanced by the factor (E/(O))² which can easily reach values as large as 10⁵ or so.

In an attempt to see whether the large value of σ_{//} and its strong temperature dependence can be related to the paraconductivity theory proposed above it is convenient to deal with the resistivity, which according to (2a) becomes

$$ \rho_{//}^{1-D} = 5.10^{-3} \left(\frac{T}{T_F}\right)^{1/2} \left(\ln\frac{T}{T_c^o}\right)^{3/2} (\Omega.cm) \qquad (4) $$

in the 1-D region. We have set a = 4 Å and θ is the life time broadening of the Fermi surface.

Eq.(4) is expected to account for the temperature dependence of the resistivity as long as the paraconductivity dominates over the single-particle conductivity.

Figure (9) shows the calculated resistivity given by eq.(4) (with T_c^o = 5.6 K and θ/T_F = 0.2) up to 40 K, curve a. The theory leads to twice too much resistivity at 40 K. The agreement with the experimental data can be significantly improved by plotting ρ = (σ_{//}^{1-D} + σ_{⊥})^{-1} instead of eq.(4), and taking a T-independent single particle contribution σ_o = 4000(Ωcm)^-1 below 40 K, curve b.

In some other samples (probably more dirty), the experimental T-dependence below 40 K follows even more closely the theoretical behaviour of Fig.(9), curve b.

Fig.(9) The longitudinal resistivity of (TMTSF)₂PF₆ (dots) and the theoretical results without (a) and with (b) inclusion of residual resistance

As discussed above, the decrease of the conductivity anisotropy below 50 K is in agreement with the theory and marks the onset of large superconducting fluctuations.

Let us now discuss qualitatively the enormous magnetoresistance observed up to 30 K, Fig.(8). The minimum of resistance under magnetic field around 20 K, corresponds to σ = 1.8 x 10⁴ (Ωcm)⁻¹. We think as very plausible that this magnetoresistance effect corresponds to the destruction of the superconducting fluctuations under magnetic field. Actually the surprising question raised by this magnetoresistance is that a field of 55 kOe does not seem to remove completely the whole paraconductive contribution. If it were the case,

<figure><img src="image_10.png" /></figure>
first the conductivity should drop to the value of the residual conductivity, namely $10^4 (\Omega cm)^{-1}$ or so, and secondly a saturation of the magnetoresistance should have been observed under magnetic field (which is not the case up to 55 kOe). However, we think that the apparent stability of the fluctuations against the application of a magnetic field can qualitatively be understood by the following argument. The superconducting fluctuations are suppressed by the creation under magnetic field of diamagnetic Meissner currents.

When the field is applied in a perpendicular configuration, the currents must also travel interchain in order to form closed loops. Since the interchain coupling ($t^2_{\perp}/t_{//}$) is small in the anisotropic conductor (of the order of $1/K$ or so) large magnetic fields are needed to generate the Meissner currents.

This situation becomes even more extreme for magnetic fields applied along the chain-axis since in this case the Meissner currents cannot benefit at all from the large longitudinal conductivity.

This picture is exemplified by the behaviour of the magnetoresistance in the parent compound [16] TMTSF-DMTCNQ for which superconducting fluctuations were claimed to explain the large conductivity at low temperature above 10 kbar [17]. Figure (10)

<figure><img src="image_11.png" /><figcaption>Fig. (10) Magnetoresistance in TMTSF-DMTCNQ at 13 kbar with the magnetic field applied $\perp$ (left) and $//$(right) to the chain axis</figcaption></figure>
shows that a 75 kOe field induces an enormous magnetoresistance when applied perpendicular to the chain-axis, whereas it can hardly affect the large conductivity if it is aligned with the large conductivity direction [16].

Attempts to detect an effect of the fluctuations above $T_c$ on the susceptibility of (TMTSF)₂PF₆ have not given significant results [8]. From the sensitivity of the experimental set-up, we found that the fluctuation diamagnetism is smaller than, say $10^{-4}$ times the diamagnetism corresponding to a Meissner effect, namely $1/4 \pi$. However, these results are in agreement with the fact that fluctuation diamagnetism above $T_c$ is not enhanced in a quasi 1-D conductor by dimensionality effects [4,10]. This result may be understood noting that the circular Meissner currents which are at the origin of the diamagnetism have to pass between adjacent chains. Due to the weak coupling the currents between chains are strongly suppressed. This effect compensates the large amplitude of the fluctuations. The only experimental situation allowing "some" fluctuation diamagnetism is for temperatures between $T^o$ and the actual $T_c$, whenever $\xi''(T)$ becomes large. This may be the reason for the observation of a strong diamagnetism by the Faraday technique at low temperature in TMTSF-DMTCNQ under pressure [18].

### III. Superconducting versus Dielectric Instability

Russian theoreticians [19] in 1966 have first raised the point that the proper understanding of the Physics of 1-D conductors requires the simultaneous summation of two kinds of diverging diagrams (i) the electron-hole diagrams and (ii) the electron-electron diagrams giving rise to charge density (or spin density) wave instability and superconducting instability respectively at finite temperature in actually quasi 1-D conductors. Later on, a renormalization group technique has been applied to the 1-D electron gas problem [20]. Using an attractive (electron-phonon mediated) electron-electron interaction the Jerusalem group [21] concluded that when the electron-phonon parameter is reduced, the superconducting instability can overwhelm the dielectric one. In a preliminary attempt to understand the origin of the superconducting state of (TMTSF)₂PF₆ we have looked at the volume dependence of the two instabilities.

The results are reported on Fig.(11), which shows the dependence of $T_c$ ($T_{M-I}$ and $T_s$) versus the decrease of the intrachain intermolecular distance. In this figure we have used the compressibility of the parent compound TTF-TCNQ [22] which happens to be very close to the compressibility of its selenium analogue TSF-TCNQ [23]. The salient results of Fig.(11) are (i) the anomalously large pressure coefficient of $T_s$, namely

$\frac{\partial \ln T_s}{\partial \ln b} \sim 300$ or $\frac{\partial \ln T_s}{\partial \ln V} \sim 150$, as given by the 12 and 24 kbar experiments and (ii) the very likely evidence of a maximum of $T_s$ around, say 9.5 kbar above 1 K followed by a very sharp dip of the transition temperatures occuring between 9.5 and 9 kbar (with our pressure scale). We do not want to discuss in great details in this article the implications of Fig.(11) (the subject of a forthcoming paper [24]) but we shall briefly mention several comments and suggestions.
<figure><img src="image_12.png" /><figcaption>Fig.(11) Dependence of $T_{M-I}$ and $T_s$ in $(TMTSF)_2 PF_6$ on the longitudinal intermolecular distance up to 24 kbar: The value of $T_s$ at 24 kbar is 0.9K. The open circle at ambient pressure is taken from reference[1]</figcaption></figure>

treatments [30,31] a maximum of $T_s$ is expected to occur very close to the border line ($g_1 = 2g_2$) separating the dielectric domain, $g_1 < 2g_2$ from the superconducting domain $g_1 > 2g_2$.

There are several interesting problems raised by the previous model, (i) the superconducting state could be of a triplet nature and (ii) the dielectric state may be due to a spin density wave instability which for large $g_1$ parameters becomes more strongly divergent than the CDW instability.

Obviously, intramolecular modes which have often very high frequencies are also favourable for a superconducting transition as pointed out by Horovitz [32]. Furthermore it is well known that in organic conductors intramolecular modes of high frequencies may be strongly coupled to the conduction electrons [33,35].

## IV. Concluding remarks

The discovery of the superconductivity in the organic state seems to have opened new fields of research. The mere existence of a superconducting state brings strong supports to the previous predictions of 1-D theories and confirms the early ideas of Little [36].

Moreover, we feel confident that superconductivity of $(TMTSF)_2 PF_6$ corresponds to a basic one dimensional divergence in the electron-electron channel even at temperatures much larger than $T_s$ [37]. The existence of a 3-D ordered state at 1 K is a result of

The decrease of $T_s$ under pressure is too large to be explained by the band broadening (decrease of the Fermi level density of states) occurring between 11 and 24 kbar. In a tight binding model for the electron acoustic phonon interaction, the pressure should increase (and not decrease) the coupling constant [25]. Finally, we have no reasons to believe that the transverse tunneling integral is severely affected by pressure. This statement is based on the measurement of the electron escape time by NMR and on conductivity data under pressure in TTF-TCNQ [26,27]. Furthermore we have checked that the room temperature conductivity anisotropy $(t_{\parallel}/t_{\perp})^2$ is hardly increased up to 12 kbar in $(TMTSF)_2 PF_6$ [4]. Consequently we believe that the reason for the superconductivity of $(TMTSF)_2 PF_6$ lies in the existence of peculiar values of repulsive electron-electron interactions ($g_1$, backward scattering and $g_2$, forward scattering) satisfying the relation $g_1-2g_2 > 0$ above 9 kbar [28]. We suggest that pressure induce a change in the 1-D electron-electron parameters $g_1$ and $g_2$ [29] and therefore following recent theoretical
the weak interchain Josephson coupling but the existence of a superconducting 1-D divergence must be attributed to the values of the intrachain electron-electron interactions.

This picture is supported by the paraconductivity model which gives a very convincing (even quantitative) explanation of the unusual transport properties at low temperatures.

### Acknowledgments

This survey represents the "digest" form of more extended pieces of works performed in close cooperation with K. Bechgaard, M. Ribault and H.J. Schulz.

I wish to acknowledge also the help of my Orsay co-workers A. Andrieux, G. Benedek, A. Mazaud and C. Weyl. I have benefited from useful discussions with J. Friedel on several theoretical aspects and with J.P. Pouget on structural problems.

### References

1) K. Bechgaard, C.S. Jacobsen, K. Mortensen, H.J. Pedersen and N. Thorup: Solid State Comm. 33, 1119, 1980.

2) D. Jérome, A. Mazaud, M. Ribault and K. Bechgaard: J. Physique Lett. 41, L-95, 1980.

3) R.H. Friend, M. Miljak and D. Jérome: Phys Rev Lett 40, 1048, 1978.

4) H.J. Schulz, D. Jérome, A. Mazaud, M. Ribault and K. Bechgaard: Phys Rev B (submitted 1980).

5) M. Ribault, G. Benedek, D. Jérome and K. Bechgaard: J. Physique Lett. 41, L-397, 1980.

6) R.I. Civiak, C. Elbaum, L.F. Nichols, H.I. Kao and M.M. Labes: Phys. Rev. B 14, 5413, 1976.

7) D. Saint James, E.J. Thomas and G. Sarma: Type II Superconductivity, Pergamon Press, 1969.

8) D. Mailly: unpublished report, Orsay, 1980.

9) L.A. Turkevich and R.A. Klemm: Phys Rev B, 19, 2520, 1979.

10) D. Jérome and H.J. Schulz: Proceedings of the Conference on Physics in One Dimension, Fribourg 1980. J. Bernasconi and T. Schneider editors, Springer, 1980.

11) C.S. Jacobsen, K. Mortensen, M. Weger and K. Bechgaard preprint 1980.

12) See for example, The Physics and Chemistry of Low Dimensional Solids, L. Alcacer editor, D. Reidel Publishing Company, 1980.

13) M. Weger: Solid State Comm. 19, 1149, 1976.

14) S. Bouffard: to be published.

15) H.J. Schulz: to be published.

16) A. Andrieux, P.M. Chaikin, C. Duroure, D. Jérome, C. Weyl, K. Bechgaard and J.R. Andersen: J. Physique 40, 1199, 1979.

17) A. Andrieux, C. Duroure, D. Jérome and K. Bechgaard: J. Physique Lett 40, L-381, 1979.

18) V. Hardebusch, W. Gerhardt, J.S. Schilling, K. Bechgaard, M. Weger, M. Miljak, J.R. Cooper: Solid State Comm 32, 1151, 1979.
19) Y. Byschkov, L.P. Gorkov, I.E. Dzyalosinskii: Sov. Phys JETP, 23, 489, 1966.

20) N. Menyhard and J. Solyom. J. Low Temp: Phys. 12, 529, 1973.

21) H. Gutfreund, B. Horovitz and M. Weger: J. Phys. C.7, 383, 1974.

22) D. Debray, R. Millet, D. Jérome, S. Barisic, L. Giral and J.M. Fabre: J. Physique Lett. 38, L-227, 1977.

23) J.F. Thomas and D. Jérome: Solid State Comm. 1980 (in press).

24) M. Ribault, H.J. Schulz, A. Mazaud, D. Jérome and K. Bechgaard (to be published).

25) S. Barisic: Phys Rev B 5, 932 (and) 941, 1972.

26) G. Soda, D. Jérome, M. Weger, J. Alizon, J. Gallice, H. Robert, J.M. Fabre, L. Giral: J. Physique 38, 931, 1977.

27) J.R. Cooper, D. Jérome, S. Etemad and E.M. Engler. Solid State Comm. 22, 257, 1977.

28) S. Barisic and S. Brazovskii: Proc. EPS Conference Antwerpen 1980.

29) The possibility of a pressure effect on the intrachain parameters has been already proposed by D. Jérome and L.G. Caron in Quasi-One-Dimensional Conductors, Lecture Notes in Physics 95, Springer Verlag 1979.

30) V.N. Prigodin and Yu. A. Firsov: JETP Lett, 25, 79, 1977 and JETP 49, 813, 1979.

31) Y. Suzumura and H. Fukuyama: J. Low Temp. Phys. 31, 273, 1978.

32) B. Horovitz: Solid State Comm. 18, 445, 1976.

33) M.J. Rice, C.B. Duke and N.O. Lipari: Solid State Comm. 17, 1089, 1975.

34) M.J. Rice and N.O. Lipari: Phys Rev Lett, 38, 437, 1977.

35) B. Horovitz, H. Gutfreund and M. Weger: Phys Rev B-17, 2796, 1978.

36) W.A. Little: Phys Rev 134, A 1416, 1964.

37) The divergence of the One-Dimensional Superconducting response instead of the Peierls response is also supported by the absence of structural precursor effects in (TMTSF)₂ PF₆, (J.P. Pouget, private communication and to be published).