# Possible observation of a Luttinger liquid behaviour from photoemission spectroscopy of one dimensional organic conductors

Denis Jerome, B. Dardel, Marco Grioni, Johannes Voit, Y. Baer, Paul Weibel, Daniel Malterre

► To cite this version:

Denis Jerome, B. Dardel, Marco Grioni, Johannes Voit, Y. Baer, et al.. Possible observation of a Luttinger liquid behaviour from photoemission spectroscopy of one dimensional organic conductors. EPL - Europhysics Letters, 1993. hal-04989933

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Distributed under a Creative Commons Attribution 4.0 International License
*Europhys. Lett.*, **24** (8), pp. 687-692 (1993)

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# Possible Observation of a Luttinger-Liquid Behaviour from Photoemission Spectroscopy of One-Dimensional Organic Conductors.

B. DARDEL(*), D. MALTERRE(*), M. GRIONI(*), P. WEIBEL(*), Y. BAER(*)

J. VOIT(**) and D. JÉRÔME(***)

(*) *Institut de Physique, Université de Neuchâtel - CH-2000 Neuchâtel, Switzerland*

(**) *Institut Laue-Langevin - BP 156, 38042 Grenoble, France*

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

(received 22 June 1993; accepted in final form 22 October 1993)

PACS. 79.60C - Clean metals.

PACS. 72.15N - Collective modes; low-dimensional conductors (inc. synthetic metals).

PACS. 71.20H - Organic compounds, polymers, and complex systems.

**Abstract.** - High-resolution photoemission experiments have been carried out on one-dimensional organic conductors in their normal state. In contrast to usual metals, the spectral function vanishes at the Fermi level and no Fermi edge is detectable. These results are consistent with a Luttinger description and an exponent $\alpha$ of the spectral function at the Fermi level larger than 1 is found. This result corroborates recent nuclear-magnetic-resonance experiments. Such a large correlation exponent shows that the 1D Hubbard model is not appropriate and strong long-range interactions must be assumed to play an important role in these materials. The compatibility with other correlation functions is discussed.

In a recent letter, we have presented an intriguing spectroscopic behaviour in a particular class of metals: the photoemission spectrum of charge density wave quasi-one-dimensional systems does not exhibit the characteristic Fermi edge in their metallic phase [1]. We have discussed this striking result in terms of the singular behaviours prevailing in one-dimensional systems and we have identified three possible causes of the observed reduction of the spectral weight at $E_F$: one-dimensional (1D) fluctuations, correlations and phonon effects [1,2]. First, the thermodynamical 1D fluctuations are very large and suppress any transition at finite temperature. Thus, the finite transition temperature observed in real quasi-1D materials results from a residual transverse coupling between chains but the one-dimensionality is reflected by the presence of large critical fluctuations extending well above the actual transition temperature. These fluctuations induce a pseudogap in the metallic phase that significantly reduces the spectral weight just below the Fermi level [3, 4]. Secondly, singular correlation effects are known to occur in one dimension [5, 6]. Although 3D strongly correlated systems are well described in the framework of Fermi liquids (FL),
correlations destroy this picture in 1D. The adiabatic continuity is no longer valid and low-energy excitations cannot be represented by quasi-particles (all the degrees of freedom are collective and the excitations are bosonlike). Moreover, the momentum distribution function does not exhibit the emblematic discontinuity of Fermi liquids at the Fermi momentum ($k_F$) and the electronic spectral function vanishes at the Fermi level [7]. Finally, the polaronic effects have been considered. Photoemission of an electron could be accompanied by the excitation of molecular vibration modes. Thus, the system left behind could be in several excited phononic states leading to a transfer of spectral weight from the vicinity of $E_F$ to higher binding energies [8,9].

In inorganic systems, the relative importance of these different mechanisms cannot be discriminated and it is possible that their physical properties result from a complex interplay of them. It is therefore important to examine materials where one of these mechanisms dominates. In this paper, we present a photoemission study of a series of quasi-one-dimensional organic compounds (the Bechgaard salts: tetramethyltetraselenafulvalene (TMTSF)₂X with X = ClO₄, PF₆ etc.). These materials generally behave as good metallic conductors and exhibit at low temperature (below T < 12 K) various phase transitions resulting from competing ground states (superconducting, charge or spin density waves, spin Peierls, etc.). A large amount of experimental data shows that the electronic correlations are at the origin of richness of the phase diagram in these materials [10,11]. Thus, in contrast to the previously studied inorganic materials where electron-phonon interactions are important, electronic correlations play the dominant role in the Bechgaard salts. Moreover, critical fluctuations associated with the low-temperature transition are expected to be small in the metallic temperature range. In this letter, we show that photoemission spectra exhibit strong deviations from the Fermi-liquid picture and are more consistent with a Luttinger-liquid character of these 1D metals.

The Bechgaard salts consist of a zig-zag stacking of TMTSF planar molecules along regular chains. A detailed description of their structure can be found in ref. [12]. These materials exhibit highly anisotropic transport and optical properties revealing a quasi-1D character. In particular, no plasma edge in the polarized-light reflectance spectrum is observed at room temperature and down to 100 K for light polarized along the transverse *b* direction [13]. Only around 25 K, very close to the SDW transition, a rather well-defined edge near 1000 cm⁻¹ is observed (as compared to the edge at 10⁴ cm⁻¹ for a longitudinal polarization). In addition, the conductivity anisotropy is temperature independent down to about 40 K, as expected for an incoherent transverse conduction [10]. Samples, mounted on a He closed-cycle refrigerator, were cleaned by cleavage in a vacuum of 1·10⁻¹⁰ Torr. Our spectrometer, equipped with a helium discharge lamp producing very narrow photon lines (HeI *h*ν = 21.2 eV and HeII *h*ν = 40.8 eV), has a total energy resolution better than 20 meV and an angular resolution of about ± 3°. The position of the Fermi energy (*E*<sub>F</sub>) is determined by measuring the Fermi edge at low temperature on an adjacent Cu reference in electrical contact with the sample. The uncertainty of this determination is estimated to be less than 3 meV.

In fig. 1, we present the HeII photoemission spectrum of (TMTSF)₂PF₆ in the metallic state (T = 50 K). We have chosen to measure at a temperature sufficiently low to limit the Fermi-function broadening and still large enough so that the interchain coupling can be neglected. The spectrum is dominated by broad structures with the shallowest one near *E* = -1.2 eV. No angle dependence has been detected and a very weak spectral weight just below the Fermi level is found for all emission angles. Our poor angular resolution (3°) corresponds to an uncertainty of the photoelectron momentum in the chain direction of about Δ*k* = 0.17 Å⁻¹ (for *h*ν = 40.8 eV at normal emission) which is a significant amount (about one quarter) of the first Brillouin zone (0.431 Å⁻¹). Thus, the spectrum of fig. 1 represents a
partial momentum integration. A close-up of the region near the Fermi level clearly shows that no Fermi edge is detectable (inset of fig. 1) as previously observed in the inorganic quasi-1D materials ($K_{0.3}MoO_3$ and ($TaSe_4)_2I$) [1]. In the present case, the fluctuation and polaronic effects are unlikely to be responsible for a vanishing intensity at $E_F$. Near 12 K the Bechgaard salts undergo a metal-insulator transition toward a spin density wave state and experiments lead to a single-particle gap $\Delta$ which is close to the BCS value $2\Delta = 3.5k_B T_{SDW}$ [14], in contrast to CDW systems, like $K_{0.3}MoO_3$, where $\Delta$ is usually significantly larger than the BCS value. Thus, as $T_{SDW}$ is very close to the mean-field transition temperature, the SDW transition in ($TMTSF)_2PF_6$ has a strong 3D character and the corresponding fluctuations take place close to the critical temperature as also indicated by the narrow region of 3D antiferromagnetic fluctuations ($\Delta T/T_c \approx 0.1$) detected by NMR [15]. Owing to the small value of the transition temperature, these fluctuations would induce a very narrow pseudogap (a few meV width). Recent optical measurements of ($TMTSF)_2PF_6$ [16] seem to rule out the existence of such a pseudogap in the optical spectrum above the SDW transition and show no evidence of a narrow resonance at zero frequency as implied by other studies [17]. This contrasts with the situation encountered in CDW systems like $K_{0.3}MoO_3$ where a large pseudogap was found above the CDW transition [18]. Polaronic effects, on the other hand, reduce the quasi-particle weight by a factor directly related to the mass of the polaron so that, as no spectral weight is observed near $E_F$, the effective mass would be huge and should be reflected in transport properties. As no evidence of a large polaronic enhancement of the effective mass is found in resistivity and optical measurements

<figure><img src="image_3.png" /><figcaption>Fig. 1</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2</figcaption></figure>

Fig. 1. – HeII photoemission spectrum of ($TMTSF)_2PF_6$ at $T = 50$ K. In the insert, the HeI spectrum near the Fermi level clearly shows with better statistics that no Fermi edge is detectable and that the correlation exponent is larger than 1. Similar spectra are obtained in other Bechgaard salts like ($TMTSF)_2ClO_4$ and ($TMTSF)_2AsF_6$

Fig. 2. – Calculated spectral function of the Luttinger model for various correlation exponents $\alpha$ (— $\alpha = 0.125, \cdots, \alpha = 0.5$, --- $\alpha = 1, \cdots, \alpha = 1.5$, - - - $\alpha = 2$). We have assumed $g_{4\parallel} = 0$ and $g_{2\parallel} = g_{4\perp} = g_{2\perp}$ implying specific relations between $\alpha$ and the renormalized charge and spin velocities $v_z$ and $v_\sigma$ (cf. ref. [23]).
of the Bechgaard salts, polaronic effects cannot lead to a strong reduction of the quasi-particle feature near $E_F$.

Since in the Bechgaard salts fluctuations associated with the SDW transition and polaronic effects cannot explain the absence of a Fermi edge in the photoemission spectrum, we now have to investigate if the electronic correlations can be responsible for this peculiar observation in 1D electron systems. Tomonaga [5] and Luttinger [6] have proposed a simple model of interacting 1D electrons with a linearized dispersion and two coupling constants ($g_2$ and $g_4$) parametrizing the small-q two-body scattering amplitudes. The salient features of this model are i) the absence of quasi-particles—all low-energy excitations are collective bosonic modes; ii) charge-spin separation (the charge and spin modes propagate with different velocities); iii) all correlation functions can be computed exactly and decay with non-universal power laws. Haldane [19] has suggested that this low-energy structure is universal to a wide class of 1D systems and proposed that Fermi liquids should be replaced by Luttinger liquids in 1D. There is also indirect supporting evidence from NMR for the possible relevance of this picture [15].

The spectral properties of the Luttinger liquid are strongly influenced by electronic correlations and exhibit dramatic differences from the familiar Fermi liquids. Fermi liquids are characterized by a finite jump of the momentum distribution function $n(k)$ at the Fermi momentum ($k_F$) reflecting quasi-particle excitations, whereas $n(k)$ is continuous at $k_F$ in interacting 1D systems (Luttinger liquids) [20-22]:

$$n(k) = n(k_F) + C|k - k_F|^\alpha \operatorname{sgn}(k - k_F) + D|k - k_F|, \quad (1)$$

where $\alpha$ depends on the model parameters (g-coupling strengths), C and D are two constants. The Fermi-liquid discontinuity is recovered in the non-interacting limit $\alpha = 0$. For $\alpha > 1$, the last term dominates and a linear variation near $E_F$ is expected. The absence of quasi-particle excitations is also reflected on the form of the spectral function. The momentum-integrated spectral function $\rho(\omega)$ consists entirely of an incoherent background, and it vanishes at the Fermi energy, according to the asymptotic power law [7, 22-24]

$$\rho(\omega) \approx \omega^{\alpha}. \quad (2)$$

As in the X-ray edge problem [25], this singular behaviour results from the fact that, because of interactions, a single-particle excitation is dressed with an infinite number of particle-hole pairs.

In fig. 2, we report the calculated momentum-integrated photoemission spectral function for several values of $\alpha$ at zero temperature. In the non-interacting limit ($\alpha = 0$), the spectral density would be a constant. Correlation effects lead to a significant decrease in the spectral-weight near $E_F$ and a broad maximum appears around ($\omega\Lambda/v_F$) = 1 in order to obey the spectral weight sum rule ($v_F$ is the Fermi velocity and $\Lambda$ is a momentum transfer cut-off in the interactions [23]). For $\alpha < 1$, the spectral function exhibits an infinite slope at $E_F$ and will be hardly distinguishable from a true Fermi edge experimentally. The slope is zero for $\alpha > 1$ which compares favourably with fig. 1. For large $\alpha$, spectral weight near $E$ is very low and the momentum-resolved spectral functions do not exhibit any significant dispersion [23]. An identification of the maximum with broad feature at $E = -1.2$ eV would be abusive: the approximation of a linear dispersion is certainly questionable on this energy scale (the bandwidth is estimated to be of order of 1 eV [10] in these materials). Moreover, the -1.2 eV feature could also reflect low-lying orbitals of the TMTSF molecules [26]. However, the most important result is the absence of Fermi edge and photoemission weight near $E_F$, which clearly indicates that the band states on the Fermi surface do not contribute near $E_F$. This
behaviour suggests that the usual Fermi-liquid picture is no longer valid in these 1D metals.

As shown on the low-energy part of the calculated and experimental spectra (inset of fig. 1 and 2), consistency of the experiments with a Luttinger-liquid picture would require a surprisingly large correlation parameter $\alpha > 1$. It is interesting to notice that such a large $\alpha$ value is incompatible with the standard Hubbard model which yields a maximum value $\alpha = 1/8$ (for infinite $U$). Then strong interactions of longer range, like inter-site electronic interactions in the extended Hubbard model, are required to increase $\alpha$. But, is $\alpha > 1$ compatible with the other experimental quantities? The exponents of the various correlation functions of the Luttinger model can all be expressed as a function of the charge correlation exponent $K_c$ [27]. For the spectral function (2), this relation is $\alpha = (K_c + K_c^{-1} - 2)/4$. Two other exponents are particularly interesting: the spin density wave exponent $\gamma = 1 - K_c$ and the $4k_F$ CDW exponent ($\delta = 2 - 4K_c$). $\gamma$ determines the temperature dependence of the $q = 2k_F$ contribution to the nuclear spin-lattice relaxation rate $T^{-1}(2k_F) = T(T/E_F)^{-\gamma}$ [28]. Recent NMR measurements yield $\gamma = 0.85$ implying $K_c = 0.15$ and $\alpha = 1.25$ [29] in good agreement with our spectroscopic estimate. Therefore, two independent measurements, photoemission and NMR, support a low value of the charge correlation exponent $K_c$ (large value of $\alpha$). For $\alpha > 1$, strong $4k_F$ CDW fluctuations are expected ($\delta > \gamma$). However, no $4k_F$ diffuse signal in X-ray diffraction experiments is observed in $(TMTSF)_2PF_6$ [30]. This is puzzling, but it is neither observed in $(TMTTF)_2PF_6$, where charge localization below 250 K is clearly related with large $4k_F$ CDW fluctuations. The absence of a $4k_F$ diffuse signal therefore does not imply a small $\alpha$. Moreover, while there is no clear signature of charge localization in the d.c.-transport properties, infrared data even at room temperature apparently require a sizeable degree of charge localization [31]. Obviously, due to a more pronounced dimerization, charge localization and $4k_F$ CDW fluctuations are more important in $(TMTTF)_2PF_6$. This is fully consistent with the estimate of the SDW exponent from NMR relaxation $\gamma = 1$ (and then $K_c = 0$) just below the localization temperature [29] and a larger optical gap: in going from $(TMTTF)_2PF_6$ to $(TMTSF)_2PF_6$, $K_c$ increases leading to a decrease of the $4k_F$ fluctuations and, correlatively, a decrease of the charge localization temperature which could be smaller than the SDW transition temperature.

To summarize, high-resolution photoemission experiments on the metallic phase of the organic compound $(TMTSF)_2PF_6$ show that the spectral function does not exhibit the characteristic Fermi edge of Fermi liquids. A similar behaviour was previously found in inorganic CDW quasi-1D compounds [1] but in these latter systems electron-phonon interactions and fluctuations are important and cannot be *a priori* neglected. As the physical properties of the Bechgaard salts are dominated by electronic correlations, we can conclude that the striking spectroscopic behaviour is the consequence of 1D electronic correlations. Our measurements unambiguously show the absence of a quasi-particle signature and we propose that these 1D correlated materials, in their normal metallic state, are Luttinger liquids with a correlation parameter $\alpha > 1$. This large correlation parameter suggests the importance of long-range interactions in these materials since a simple Hubbard model cannot yield an $\alpha$ value larger than 1/8. This speculative idea is corroborated by very recent NMR measurements whose analysis leads to $\alpha = 1.25$.

***

We thank Drs. P. BATAIL, P. AUBAN-SENZIER and C. LENOIR for kindly providing us with the samples and we acknowledge Prof. C. BOURBONNAIS, J. P. POUGET and F. MILA for fruitful discussions and Prof. K. SCHÖNHAMMER for sending us the preprint of a paper reporting similar calculations before publication. This work was supported by the Fonds National Suisse de la Recherche Scientifique.
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