# Superconducting and spin-density wave phases probed by scanning tunneling spectroscopy in the organic conductor (TMTSF)₂ClO₄

Mohammadmehdi Torkzadeh, Pascale Senzier, Claude Bourbonnais, Abdelouahab Sedeki, Cécile Méziere, Marie Hervé, Francois Debontridder, Pascal David, Tristan Cren, Claire Marrache-Kikuchi, et al.

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Mohammadmehdi Torkzadeh, Pascale Senzier, Claude Bourbonnais, Abdelouahab Sedeki, Cécile Méziere, et al. Superconducting and spin-density wave phases probed by scanning tunneling spectroscopy in the organic conductor (TMTSF)₂ClO₄. Physical Review B, 2025, 112 (4), pp.045107. 10.1103/ cs8w-ttxs. hal-05153939

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Distributed under a Creative Commons Attribution 4.0 International License
Superconducting and spin density wave phases probed by scanning tunneling spectroscopy
in the organic conductor (TMTSF)₂ClO₄

Mohammadmehdi Torkzadeh,¹ Pascale Senzier,² Claude Bourbonnais,³,*, Abdelouahab Sedeki,⁵

Cécile Méziere,⁵ Marie Hervé,¹ Francois Debontridder,¹ Pascal David,¹ Tristan Cren,¹

Claire Marrache-Kikuchi,⁶ Denis Jerome,² and Christophe Brun,¹,†

¹*Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252 Paris, France*

²Laboratoire de Physique des Solides (UMR 8502) - Université Paris-Saclay, F-91405 Orsay, France

³Regroupement Québécois sur les Matériaux de Pointe and Institut Quantique, Département de physique,
Université de Sherbrooke, Sherbrooke, Québec, Canada J1K-2R1

⁴Université Moulay Tahar de Saida, B.P. 138 cité ENNASR 20000, Saida, Algeria

⁵Univ Angers, CNRS, MOLTECH-Anjou, SFR MATRIX, F-49000 Angers, France

⁶Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

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(Received 20 February 2025; revised 30 May 2025; accepted 10 June 2025; published 7 July 2025)

By scanning tunneling microscopy (STM) we have probed the local quasiparticle density of states (DOS) of the Bechgaard salt organic superconductor (TMTSF)₂ClO₄ in slowly cooled single crystals cleaved under ultrahigh vacuum conditions. In well STM imaged crystallographic surface planes, the local DOS has been probed for different surface areas at temperatures above and below the critical temperature of superconducting or insulating spin density wave states. While a rather homogeneous superconducting state is expected in the bulk from previous studies, depending on the degree of disorder introduced by cleavage in the anion lattice, an inhomogeneous granular state is predominantly observed at the surface. A pronounced linear V-shape profile of the local DOS is observed from intermediate to the lowest-energy scale in the less disordered superconducting surface areas. This supports the existence of an unconventional *d*-wave-like order parameter with nodes at low energy, which is preceded by more energetic fluctuations attributed to quantum criticality of the material. At higher energy disorder combined to correlations deplete further the DOS. By contrast, a nonlinear *U* shape characterizes the local low-energy DOS profile for the more disordered and insulating surface areas of the spin density wave state. The experimental results are compared quantitatively with those predicted by the renormalization group theory of the quasi-one-dimensional electron gas model and its description of the superconducting and spin density wave states that are interlinked by quantum criticality in the Bechgaard salts.

DOI: 10.1103/cs8w-ttxs

# I. INTRODUCTION

Among the various families of unconventional superconductors discovered over the last four decades or so, organic materials occupy a noticeable place. This is well exemplified by the first organic superconductors to be synthesized, the Bechgaard salts [$(TMTSF)_2X$] series. These strongly anisotropic, quasi-one dimensional (quasi-1D), conductors are distinctive by the emergence of superconductivity on the edge of an insulating antiferromagnetic spin density wave (SDW) state when pressure is tuned either hydrostatically or chemically from anion X substitution [1–3]. This sequence is now known as a classical example of pressure-tuned antiferromagnetic quantum critical point (QCP) made unstable by the development of a superconducting dome.

A major effort in the quest of understanding these materials has resided in explaining how both quantum criticality and superconductivity are reciprocally conditioned. To this end, most sophisticated experiments have been conducted on $(TMTSF)_2ClO_4$, the only member of the $(TMTSF)_2X$ series

displaying at ambient pressure a superconducting state below $T_c = 1.2$ K, which is located in the immediate vicinity of the QCP. Anomalous-featured order parameter for superconductivity is illustrated in various situations. These are for instance given by the detrimental effect of nonmagnetic impurities on $T_c$ [4,5], the power-law temperature dependence of the nuclear magnetic resonance (NMR) relaxation rate [6,7] and of specific heat below $T_c$ [8], and the Fermi surface gap profiling by angle-resolved field-dependent specific heat and its $\sqrt{H}$ field profile in the vortex state [8]. Combined with the suppression of the NMR Knight shift in the superconducting state at low field [7], these findings support the existence of a singlet order parameter for superconductivity with a nodal structure on the Fermi surface, probably *d* wave. Other experiments like former specific heat measurements, thermal conductivity, and $\mu$-sr, however, do not accord with this view and rather support the existence of a nodeless gap along the Fermi surface [9–11].

The hallmarks of quantum criticality fanning out through the metallic state are found in different conditions. This is the case of the linear temperature dependence of electrical resistivity [12,13]. Also known as Planckian dissipation [14,15], it is present over almost two decades in temperature around $T_c$ in

<footnote>*Contact author: Claude.Bourbonnais@usherbrooke.ca</footnote>

<footnote>†Contact author: christophe.brun@sorbonne-universite.fr</footnote>
the Bechgaard salts. Another imprint comes from NMR spin-lattice relaxation rate. Different sets of experiments reveal the existence of pronounced antiferromagnetic spin fluctuations, extending from the lowest temperature to far beyond $T_c$ in the metallic state and whose amplitude is tuned by the pressure distance from the QCP [2,7,16–19].

None of these observations address directly the spectroscopic properties of quasiparticles in the quantum critical metallic domain and how these modify when entering in the superconducting phase. It is the main motivation of this work to address these important issues from the angle of tunneling spectroscopy that can probe the quasiparticle density of states in the metallic, superconducting, and insulating phases [20].

Several attempts to this end have been performed in the past using different tunneling setups. Early experiments used a n-doped GaSb Schottky barrier evaporated onto a (TMTSF)₂PF₆ single crystal down to 50 mK under a pressure of 11 kbar [21]. It provided evidence of a well-developed gap 2Δ ≈ 3.6 meV at the Fermi level, interpreted as a superconducting gap, persisting above $T_c$ like a pseudogap structure. Subsequent (TMTSF)₂ClO₄ studies at ambient pressure revealed similar results for superconductivity and evidenced also the role of the cooling speed on ClO₄ anion disorder leading to a shift on the pressure axis and to the onset of an insulating SDW state. An estimation of the SDW gap of 2Δ<sub>SDW</sub> ≈ 6 meV was given [22]. Further planar tunneling superconductor-insulator-superconductor junction (SIS') experiment using (TMTSF)₂ClO₄-amorphous Si-Pb junctions, revealed a much smaller superconducting energy gap in the ordered state of 2Δ ≈ 0.8 meV and an SDW gap of 2Δ<sub>SDW</sub> ≈ 3.0 meV for the anion disordered state [23]. In (TMTSF)₂PF₆ at ambient pressure, a scanning tunneling spectroscopy study of the as-grown a-b surface reported a SDW gap 2Δ<sub>SDW</sub> ≈ 4.6 meV below the ordering temperature $T_{SDW} \simeq 12$ K, and preceded over a broad range of temperature by a pseudogap or dip in the density of states [24].

Given the diversity of these results, it turned out to be important to have a reliable spectroscopic tunneling method to perform a direct determination of the local quasiparticle density of states (LDOS) in the superconducting, metallic, and SDW phases. The method we have used is very low-temperature scanning tunneling microscopy and spectroscopy (STM and STS) after having cleaved the (TMTSF)₂ClO₄ sample under ultrahigh vacuum. This technique has been widely used on inorganic conventional and unconventional superconducting materials [25], including layered organic superconductors [26,27]; it also has proven to achieve molecular resolution when applied to quasi-1D organic charge transfer compounds [28].

In this work, we could for the first time succeed to image the elementary (a,b) planes of (TMTSF)₂ClO₄ resulting from the *in situ* cleavage and perform STS on these planes. The *dI/dV* tunneling conductance measurements performed at *T* = 300 mK revealed various types of regions that are consistent with either superconducting or SDW phases depending on disorder conditions. Due to the cleavage process inducing disorder in the ClO₄ lattice sites, superconducting regions are rarely found in the surface plane. Instead, an inhomogeneous granular state is predominantly observed at the surface. In rare surface superconducting regions, the quasiparticle exci-

tation spectrum in the meV energy range reveals a strongly V-shaped DOS presenting a finite residual DOS at the Fermi level. The shape of the local excitation spectrum measured by STS is consistent with an order parameter having a nodal *d*-wave structure rather than nodeless *s* wave. In contrast, SDW puddles reveal U-shaped excitation spectra consistent with a nodeless energy gap having larger values than superconducting regions. In the energy range above the superconducting gap and above $T_c$, the V-shaped depletion of the quasiparticle DOS persists up to intermediate energy where disorder and correlations govern the depletion of the DOS. In the few meV range, we show that our results are rather congruent with the predictions of the renormalization group approach to the quasi-1D electron gas model for the SDW to *d*-wave SC sequence of ground states exhibited across a QCP [29].

## II. EXPERIMENTAL SETUP

The STM and STS experiments were carried out in an ultrahigh vacuum setup having a base pressure in the low 10⁻¹¹ mbar range. (TMTSF)₂ClO₄ single crystals of about several mm lengths were selected and glued on stainless steel sample holders using silver epoxy. In a second step, above the free sample surface, cleavers were glued using silver epoxy. The (TMTSF)₂ClO₄ single crystals were cleaved at room temperature inside the STM chamber and introduced into the cold STM head at a temperature of about 100 K (please see Appendix C for more details regarding the sample preparation and our efforts to minimize internal stress effects). This preparation procedure is reliable and was used by one of us on another very fragile needlelike inorganic quasi-1D material: NbSe₃ [30].

The (TMTSF)₂ClO₄ samples were further cooled down to helium temperature. Between 45 and 4.2 K the samples were slowly cooled down to maintain a cooling speed of about 1.5 K per hour in order to ensure a very good structural ordering of the ClO₄ ions and favor the bulk development of the superconducting phase. The STM and STS experiments were carried out at a base temperature of 300 mK using a homemade apparatus based on a ³He single-shot cryostat [31–33]. Mechanically cut PtIr tips were used. The differential tunneling conductance *dI/dV* spectra were obtained from the numerical derivative of single *I(V)* curves. The individual spectra were further convolved with a Gaussian filter compatible with the thermal broadening at 300 mK.

## III. RESULTS

Figure 1 shows the result of a nicely cleaved (TMTSF)₂ClO₄ sample. Figure 1(a) shows a 400 × 330 nm² STM topography measured at *T* = 300 mK emphasizing several (a,b) crystallographic planes separated by elementary steps oriented along the *c* direction. This is corroborated by the *Z(Y)* profile measured along the vertical dashed line and presented in Fig. 1(b). This profile shows flat sections of molecular (a,b) planes separated by an average *c* parameter of about *c* ≈ 1.49 nm as measured by STM locally in this particular area. It is consistent with the bulk reported value of 1.33 nm but presents an increase of about 10%. This is a general observation that we made on other areas or samples.
<figure><img src="image_3.png" /><figcaption>FIG. 1. Measurements performed at $T = 300$ mK. (a) Raw color-coded constant current $Z(x, y)$ STM topography of a $(TMTSF)_2ClO_4$ single-crystal cleaved under ultrahigh vacuum. The scanned area is $400 \times 330$ nm$^2$ measured with $I = 50$ pA and $V_{bias} = -0.1$ V. Several crystallographic (a,b) planes are seen. The color bar of $Z(x, y)$ expressed in nm is shown on the right. (b) $Z(Y)$ profile of the STM tip acquired along the vertical dashed line shown in image (a). Both displacements $Z$ and $Y$ are in nm. The height variations are consistent with the presence of elementary crystallographic steps along $c$. (c) STM topography of a smaller area (of about $15 \times 15$ nm$^2$), where molecular resolution could be obtained emphasizing the quasi-1D chains lying along **a**. The **b** direction is also indicated. The color bar of $Z(x, y)$ expressed in pm shows a corrugation less than 0.1 nm. This image was Fourier filtered to reduce the signal-to-noise ratio and highlight the molecular resolution.</figcaption></figure>

Additionally, on rare occasions, we have been able to obtain a clear molecular resolution on the (a,b) planes, as shown in Fig. 1(c). Here, we clearly see the quasi-1D chains lying along $a$. Note that a tip change occurs in the bottom part of the image. Our results emphasize that the electronic properties of $(TMTSF)_2ClO_4$ samples are accessible to STM and STS measurements if care is taken in the sample mounting and cleavage process. Based on [34] by Alemany et al., we expect that the LDOS as measured by STM should be dominated by the highest occupied molecular orbital (HOMO) orbitals of the TMTSF molecules, with almost no weight on the $ClO_4$ ions.

Taking advantage of being able to scan crystallographic surface planes, we investigated the low-energy electronic properties of $(TMTSF)_2ClO_4$ at various locations. The bulk critical temperature $T_c \simeq 1.2$ K was precisely determined from *ex situ* electronic transport measurements on samples taken from the same batch, revealing a critical field $H_{C_2-c^*} \approx 0.2$ T along $c^*$, in good agreement with the literature (see Appendix A). Overall, our experimental results carried out on four different samples from the same batch all revealed inhomogeneous electronic properties at the surface, as measured by the STM-STS technique. This is in strong contrast to what can be expected from the $T_c$ and bulk homogeneousness of the electronic properties as probed using macroscopic techniques for a slow sample cooling speed [3,35].

In this work, we ascribe the electronic inhomogeneities observed at the surface to a structural disorder induced in the anions sublattice. Indeed the $ClO_4$ ions are situated in-between molecular (a,b) planes, as can be seen in Fig. 2 (see also the Fig. 9 in the Appendix section for another view of the cristallographic structure). As the natural cleavage plane parallel to (a,b) is likely to pass in-between the molecular (a,b) planes, i.e., through the $ClO_4$ ions, the cleavage process is expected to statistically remove a large part of these anions. Such an effect was already reported earlier by one of us in another well-known quasi-

1D inorganic material, called the blue bronze, developing a charge density wave ground state where the ions are located in-between the molybdenum oxides planes [36,37]. In $(TMTSF)_2ClO_4$ samples cleaved under ultrahigh vacuum we could identify at $T = 300$ mK different phases at the surface, including superconducting regions and SDW areas. In addition, other regions presenting correlated metallic states were also observed. We present below the characteristic tunneling spectra measured for each type of region.

Figure 3 shows differential tunneling conductance measurements performed both below $T_c$ in the superconducting state and above $T_c$. Figure 3(a) shows representative $dI/dV(E = eV_{bias})$ spectra measured at $T = 300$ mK (blue spectrum) and at $T = 2.13$ K (yellow spectrum), in zero-field-cooled sample, in a $8 \times 8$ nm$^2$ region located in the lower terrace of the sample area seen in Fig. 1(a)

<figure><img src="image_4.png" /><figcaption>FIG. 2. Crystal structure of $(TMTSF)_2ClO_4$ viewed along the $a$ direction. Note that the final $ClO_4$ anions' order in the crystal structure depends on the cooling procedure: they can be either disordered or ordered with a doubling of the $b$ period (see text).</figcaption></figure>
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FIG. 3. Tunneling spectroscopic properties of (TMTSF)₂ClO₄ cleaved under ultrahigh vacuum and acquired by scanning tunneling spectroscopy in the area shown on the lower terrace of Fig. 1(a). Differential tunneling conductance dI/dV (E = eVbias) spectra measured in the energy range [-4; +4] mV with the following set point for STS: I = 300 pA and Vbias = -9 mV. Each presented spectrum is the average of a 60 × 60 STS grid measured at T = 300 mK (blue spectrum) or T = 2.13 K (yellow spectrum) over the same area of 8 × 8 nm². (b) Magnetic field dependence of the dI/dV (E = eVbias) spectra at T = 300 mK with B applied along c* between 0 and 0.3 T (blue: 0 T, purple: 0.15 T, red: 0.3 T). The magnetic field dependence was probed in another sample area than the one probed in (a).

indicated by a white square. The comparison between these two spectra enables to associate the internal dip structure of the 300-mK spectrum to the superconducting local density of states (LDOS). This is confirmed by Fig. 3(b) presenting the magnetic field dependence of the dI/dV spectra at T = 300 mK for a magnetic field applied along c* up to 0.3 T. A continuous increase of the conductance around E<sub>F</sub> is observed with increasing magnetic field. A maximum of conductance is achieved around 0.25-0.3 T that we attribute to the surface critical field along c* H<sub>C2S</sub>. This is in good quantitative agreement with bulk H<sub>C2</sub> ≈ 0.2 T measurements [8] as well as our own electrical resistivity measurements along c* (see Appendix A). We provide in this Appendix other dI/dV spectra showing similar superconducting characteristics but emphasizing various gap filling, measured on other areas or samples (see Appendix F).

Figure 4 presents a typical tunneling spectrum, acquired in the same superconducting region as the low-energy spectra presented in Fig. 3(a), but over a much larger energy range of about 50 times the superconducting energy gap. It is seen that the characteristic spectrum is strongly V shaped up to more than 40 meV.

Figure 5 presents an interesting situation of local inhomogeneous electronic properties. It features a 40 × 13 nm² region where a spin density wave puddle of size 15–20 nm [blue (darker) area in (b)] is comprised between two superconducting areas [red-yellow-green (brighter area) in (b)]. The local tunneling spectra [blue and red (lighter colors) in (c)] corresponding to the superconducting areas are similar to the one presented in Fig. 3(a) but reach zero conductance very close to zero energy. In contrast, the middle area presents a strongly U-shaped spectrum (black curve) with a concavity that is opposite to the one of the superconducting spectra. Moreover, this U-shaped spectrum presents a larger energy gap and a lower conductance around the Fermi level reaching zero at E<sub>F</sub>. In order to further support this result, we heuristically

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FIG. 4. Tunneling spectroscopic properties of (TMTSF)₂ClO₄ acquired by scanning tunneling spectroscopy in the area shown on the lower terrace of Fig. 1(a), where the superconducting spectra shown in Fig. 3(a) were also acquired. Differential tunneling conductance dI/dV (E = eVbias) spectra measured in the energy range [-40; +40] meV with the set point: I = 300 pA and Vbias = -40 mV. The presented spectrum (continuous thicker curve) is the average of a 60 × 60 STS grid measured at T = 2.13 K over a 8 × 8 nm² area. The same normalization procedure as in Fig. 3(a) is applied. Comparison with the Anderson-Hubbard model (dashed thinner curve), modeling the influence of disorder on the DOS at high energy (see text).

fitted all dI/dV spectra acquired over a 92 × 29 grid in this area, assuming a power-law energy dependence in the form dI/dV(E) ∝ E^[η]. The extracted exponent map η(x, y) is presented in Fig. 5(a). It anticorrelates with the conductance map of Fig. 5(b): the larger the local exponent η(x, y) (SDW puddle), the smaller the local conductance around -2 meV. Our result is in line with the reported SDW energy gap of about 3 meV [23], 4.6 meV [24], up to 6 meV [22]. Further analysis of the experimental local exponent variations shown in Fig. 5(a) enables to extract a superconducting and SDW coherence length of about 5 nm along the chain direction, implying a low mean-free path of about 1 nm. As we will see below, these low values with respect to the bulk clean limit ones are consistent with the high broadening parameter required to fit the single-particle spectra. We have found numerous SDW puddles at the surface (additional examples are presented in Appendix G).

In addition to the superconducting and SDW puddles we have also found local regions where the tunneling spectra present only a correlated V-shaped background at large energy but without a clear signature of superconducting dip or SDW gap at low energy. Such regions could correspond to puddles of weakened or destroyed superconductivity having additional in-gap filled states, as for instance proposed in a variable-cooling transport study to describe a granular superconducting network [35]. Examples of this case are shown in Appendix H. Finally, we have also encountered several locally insulating planes at the surface, where it was not possible to perform STS of the low-energy spectrum. In such cases, the STM tip dug several nanometers or tens of nanometers into the interior of the material in order to
<figure><img src="image_7.png" /><figcaption>FIG. 5. Measurements done at T = 300 mK. (a) Spatial map presenting the exponent η fitted from individual $dI/dV(E)$ spectra assuming the energy dependence $dI/dV(E) \propto E^\eta$ acquired on a 92 × 29 grid on a 40 × 13 nm² area. The scale bar is indicated. (b) Differential tunneling conductance $dI/dV(E)$ map plotted at -2 meV, measured over the same 13 × 40 nm² area, extracted from a full $I(V)$ grid in the energy range [-10; +10] mV with the set point for STS I = 300 pA and $V_{bias}$ = -9 mV. Three different neighboring regions are seen: left superconducting [red-yellow-green (brighter)], middle spin density wave [blue (darker) region], right superconducting [red-yellow-green (brighter)]. (c) Representative $dI/dV(E)$ spectra averaged locally in each left, middle, right area of the conductance map presented in (b). Each spectrum is the average of about 50 single spectra.</figcaption></figure>

find low-energy conducting planes satisfying a set point such as I = 200 to 300 pA and $V_{bias}$ = -5 to -9 mV. We interpret such situations as damaged or defected surface parts of the sample where some regions of the surface crystallographic planes are electrically badly connected to bulk parts, preventing the establishment of a tunneling current at low voltages. An example of such a situation is provided in Appendix E, where the area could be imaged using a larger bias voltage before and after the tip dug into the surface while performing low-energy STS.

We summarize as follows the general observations that could be inferred from our numerous investigations: (i) Superconducting regions are fewer at the surface plane than expected in the bulk. (ii) Superconducting coherence peaks are strongly smeared out. (iii) In the superconducting regions, the zero-bias conductance, i.e., the LDOS at the Fermi energy, is most often finite and strongly fluctuates from one location to another. (iv) In the SDW regions, the single-particle excitation spectra are U shaped and reach zero at the Fermi energy.

(v) The normal state presents a strong V-shape background extending to energies at least two decades above the superconducting energy gap.

## IV. THEORY AND DISCUSSION

From electronic band structure calculations, a strongly V-shaped LDOS is not expected around $E_F$ [34]. The existence of a quasi-1D band structure could lead to a $k_\parallel$ tunneling dependence but not as strong as to induce such a fast increase of conductance as observed in our case over only 40 meV assuming a reasonable sample work function of at least one eV. A more convincing explanation is that we measure an intrinsic V-shaped LDOS due to the strongly correlated quasi-1D character of the LDOS combined to a rather large surface disorder associated to an inhomogeneous granular state. This emerges by first noting that the pronounced background depletion of LDOS seen up to at least 40 meV in Fig. 4 is not uncommon in disordered materials [38–41], including layered organic superconductors [26,27]. These are characterized by sizable electron-electron interactions that can couple to disorder and then alter the density of states over relatively large energy scales. The perturbative approach of Altshuler and Aronov to this form of coupling is well known to produce such a suppression of the tunneling density of states [42]. Analogous features can also be found from a numerical treatment of the disordered Anderson-Hubbard model [43,44]. These conditions are likely to be fulfilled for a compound as (TMTSF)₂ClO₄ in which quantum criticality linking superconductivity to a correlated antiferromagnetic insulating state is occurring in the presence of nonmagnetic defects of the anion ordering. As shown below, the coupling of electron-electron interaction to randomness is responsible for the shaping of LDOS in energy down to 10 meV or so.

We follow the numerical approach of Ref. [43] to the Anderson-Hubbard model with the expression for the DOS in two spatial dimensions,

$$N_{AH}(E) = c \exp[-\alpha \ln^2(\sqrt{2}|E|/E_F)], \quad (1)$$

where c and α are positive constants and $E_F$ is the Fermi energy which is typically $E_F \simeq 0.26$ eV for a quasi-1D metal like (TMTSF)₂ClO₄ [1]. Using a value of α ≃ 0.3, the above expression fits fairly well the data of Fig. 4 over the whole range of intermediate energy. The value of α is comparable to the one found in layered organic superconductors showing a similar DOS depletion at intermediate energy [27]. However, the data do not evolve to the formation of a soft Hubbard gap as predicted to take place by the model around the Fermi level when the disorder-free electron system is a correlated insulator with a sharp gap. Deviations found below $E_0 \sim 10$ meV come from the onset of coherent quasiparticle states with a finite DOS on which our model developed below is based.

Below $E_0$, the measured energy dependence of the STS spectra of Figs. 3 and 5, associated with superconducting regions, is linearlike all the way down to the energy gap region. There, the LDOS is most often finite at zero E as in Fig. 3 (see also Fig. 12 of Appendix F) which is in agreement with macroscopic specific heat measurements [8,45], but can also reach zero locally as in Fig. 5. This is not consistent with an s-wave nodeless order parameter but reminds what
is observed in d-wave cuprate superconductors by STS [25]. Furthermore, nonmagnetic disorder in the $ClO_4$ anions lattice can be expected to create local defects, as a result of the cleaving process. A more natural explanation compatible with point group symmetry would then be a d-wave symmetry where line nodes would give a linear energy-dependent DOS in the superconducting energy gap, consistent with the angular and the temperature dependence of the specific heat [8,45]. Moreover for a d-wave symmetry nonmagnetic point defects behave as pair breakers, leading to additional gap filling, as predicted for a d-wave superconductor with impurity scattering in the unitary limit [46,47] and often observed experimentally.

In the interpretation of the whole low-energy part of the STS spectrum below $E_0$, we shall adopt the view that SC and SDW states of the Bechgaard salts are the result of repulsive interactions alone, as approached by the theory of quasi-1D electron gas model. This model is based upon an electron spectrum with strongly anisotropic nearest- and next-to-nearest-neighbor transfer integrals. These lead to an open Fermi surface on which a set of electron-electron scattering amplitudes or coupling constants $g_i$ can be defined. We shall make use of the extensive studies of renormalization group (RG) method for this model [19,29,48] (see also Appendix J). At the one-loop level, for instance, these give a very reasonable description of the actual sequence of instabilities of the metallic state against SDW and superconductivity (SC) (d-wave) orders as the antinesting, namely, the interchain next-to-nearest-neighbor hopping parameter $t'_\perp$ is tuned to simulate the influence of pressure. This occurs together with a dome of SDW quantum critical fluctuations that permeate deeply the metallic phase and whose amplitude scales with the one of $T_c$ under pressure.

At the two-loop level of the RG, the one-electron spectral properties of the quantum critical domain of the model can be obtained. This the case of the energy profile of the DOS (normalized by the bare part $N_0$), $\bar{N}_n(E) = -\frac{2}{N_0\pi} \frac{1}{V} \sum_k \text{Im} G(k, E)$, which is linked to the coherent part of the one-particle retarded Green function $G$, as a function of wave vector $k$ and energy distance $E$ from the Fermi level. Its renormalization is governed by the diagrammatic RG equation $O(g_i^2)$ for the self-energy [29], as portrayed in Fig. 16 of Appendix J. The local DOS probed by single-particle tunneling, $\bar{N}_n(E) = \langle z(k_\perp, E) \rangle_{k_\perp}$, is expressed in terms of the energy-dependent quasiparticle weight $z$ averaged over the transverse wave vector $k_\perp$. In the metallic phase for $T > T_{c,SDW}$ calculations predict a drop of $\bar{N}_n(E)$ upon decreasing $|E|$ that is well described by a power law

$$ \bar{N}_n(E) \propto (\text{Max}\{\pi k_B T, |E|\})^\eta, \quad E < 2t'_\perp \quad (2) $$

for low energy up to the scale of antinesting $2t'_\perp$ of the order of 5 meV for a quasi-1D compound like $(TMTSF)_2ClO_4$ (see Appendix J). This scale marks the onset of an interplay between SDW and d-wave pairing fluctuations that initiates the low-energy quantum critical domain. At higher energy for $E > 2t'_\perp$, the calculations show a transient weaker energy dependence of $\bar{N}_n(E)$ (see for instance Fig. 6). The latter evolves towards the higher-energy scale of the nearest-neighbor interchain hopping $t'_\perp \sim 10r'_\perp \sim E_0$, located in turn at the onset of randomness scale for the DOS. In the low-energy sector of interest, the exponent $\eta(t'_\perp)$ is nonuniversal being $t'_\perp$ or

<figure><img src="image_8.png" /><figcaption>FIG. 6. Comparison of the renormalization group prediction (dashed dotted lines) and STM experiments for the density of states at (a) $T = 300$ mK in the SC state for the two regions shown in Figs. 3 and 5, and (b) $T = 2$ K in the metallic state in the region shown in Fig. 3(a).</figcaption></figure>

pressure dependent. Close to the QCP where SC emerges on the brink of SDW, $t'_\perp \simeq t'^*_\perp$ and $\eta(t'^*_\perp) \simeq 1$ with $\bar{N}_n(E)$ exhibiting a linear energy profile. On the SDW side of the phase diagram, at lower $t'_\perp < t'^*_\perp$, $\eta(t'^*_\perp) > 1$ increases above unity and $\bar{N}_n(E)$ shows a concave U-shape behavior in energy for temperature $T > T_{SDW}$ in the metallic state. By contrast, moving away from the QCP on the SC side, $\eta(t'^*_\perp)$ decays below unity, a drop that correlates with the one of $T_c$ with $t'_\perp$. The influence of nonmagnetic impurity scattering on $\bar{N}_n(E)$ is here neglected since its impact on the strength of electron correlations of the quantum critical domain can be considered to be small [49].

At sufficiently low $|E|$, the metallic $\bar{N}_n(E)$ will be connected to the suppression of the DOS following the opening of a gap $\Delta_\mu$ from either $\mu = \text{SC}$ or SDW long-range order. The normalized DOS will take the form

$$ \bar{N}(E) = \bar{N}_n(\Delta_\mu, |E|) \left\langle \text{Re} \left\{ \frac{|E + i\Gamma|}{\sqrt{(E + i\Gamma)^2 - \Delta_\mu^2(k_\perp)}} \right\} \right\rangle_{k_\perp}, \quad (3) $$

where $\bar{N}_n(\Delta_\mu, |E|) \propto (\text{Max}\{\Delta_\mu, |E|\})^\eta$. The $k_\perp$ average in brackets is a normalized BCS expression for the DOS of the quasi-1D electron gas in the presence of a gap $\Delta_\mu$ and the broadening $\Gamma$ due to impurity scattering assumed to be pair breaking for both orderings [47,50].

The above expression for the DOS enters in the differential tunneling conductance at temperature $T$:

$$ \frac{dI}{dV}(E) = G_{nn} \int_{-\infty}^{+\infty} \bar{N}(\omega) \left[ -\frac{\partial f(\omega + eV)}{\partial (eV)} \right] d\omega + G_0, \quad (4) $$

where $E = eV$, $f(\omega)$ is the Fermi distribution, $G_{nn}$ and $G_0$ are adjustable constants for the normal conductance and the finite LDOS at the Fermi level.

To fit the above expressions with the low-energy STM data at $T = 300$ mK in the SC state, one substitutes the $k_\perp$-dependent, d-wave, superconducting gap $\Delta_{SC}(k_\perp) = \Delta_{SC} \cos k_\perp$ for an open quasi-1D Fermi surface, together with the RG results for $N_n(E)$ obtained from typical figures for the couplings $g_i$ and band parameters for the Bechgaard salts [29] (see Appendix J). These lead to a calculated critical temperature $T_c \simeq 1.1$ K at the value of antinesting pressure parameter
<figure><img src="image_9.png" /><figcaption>FIG. 7. Comparison of the normal state renormalization group prediction (dashed line) and STM experiments of Fig. 5 for the density of states in the SDW state at T = 300 mK. The theoretical prediction including in addition the SDW gap in the DOS is shown in Fig. 19 of Appendix K.</figcaption></figure>

$t'_{\perp}/k_B = 26$ K, namely, close to the QCP, a value of the magnitude found experimentally in the bulk. The impact of impurity scattering on the ordering will be incorporated through the broadening factor $\Gamma$. The calculated energy profile of $dI/dV$ is shown in Fig. 6(a) with the STM data at T = 300 mK for the data of two different SC areas shown in Figs. 3 and 5. The best fit is obtained for $\Gamma/\Delta_{sc} = 0.6$, $\Delta_{sc} \simeq 0.76$ meV, and a ratio $\Delta_{sc}/k_B T_c = 8$ that exceeds by a factor about 4 the BCS value for the SC state, a ratio known to be partly enhanced by impurity scattering [47]. The prediction gives a fair description of the differential conductance extending from the Fermi level up to $|E| \sim 5$ meV ($\gg \Delta_{sc}$). At $|E| < \Delta_{sc}$, this is consistent with a d-wave gap having nodes on the Fermi surface which leads to a linear dependence of the DOS on energy. A sizable broadening factor $\Gamma$ accounting for the enhanced impurity scattering at the surface is responsible for the suppression of the coherence peaks. This behavior then joins the E-linear DOS caused by quantum fluctuations spreading over a sizable energy interval well above the gap in accord with the RG solution close to the QCP.

In the normal phase above $T_c$, the $T = 2$ K data of the differential conductance of Fig. 3(a) and reproduced in Fig. 6(b) show a significant filling of the DOS expected by temperature effects close to the Fermi level. In that case Eq. (K2) reduces to the normal state expression $(dI/dV)(E) \propto \bar{N}_n(E)$, where $\bar{N}_n(E)$ is given by the RG results (2) computed at $T = 2$ K with the same parameters that led to the $T_c$ discussed above [29] (see Appendix J). $\bar{N}_n(E)$ is $E$ linear ($\eta \simeq 1$) outside the thermal broadening region at $|E| > \pi k_B T$ up to the beginning of the weaker energy transient dependence taking place from 5 meV or so. This behavior is congruent with the data of Fig. 6(b) obtained up to 4 meV.

We now consider the results of the more disordered region where the system turns to be a SDW insulator at $T = 300$ mK, as displayed in Fig. 5 and reproduced in Fig. 7. In the phase diagram this would correspond to a negative shift along the pressure axis towards the SDW region. According to the figures, the data show a noticeable concavity or a U-shape in energy for the DOS. This can be

simulated theoretically by a downward shift of antinesting at $t'_{\perp}/k_B \simeq 25.4$ K in the phase diagram of the RG solution. An instability of the metallic state against SDW order then occurs at the critical temperature $T_{SDW} \simeq 5$ K, a magnitude typically found in quenched $ClO_4$ samples [51,52]. In this SDW regime, the single-particle gap $\Delta_{SDW}(k_{\perp}) \approx \Delta_{SDW}$ presents no nodes and can be taken as essentially momentum independent along the Fermi surface. From the normal state fit in Fig. 7 obtained using $\Delta_{SDW}/k_B T_{SDW} = 3.6$ and $\Gamma/\Delta_{SDW} = 0.6$ in Eqs. (K1) and (K2), no coherence peaks for the DOS at $|E| \sim \Delta_{SDW} \simeq 1.55$ meV are found but a small kink is visible between $\pm 2$ and $\pm 3$ meV associated with a change in the energy dependence similar to the simulated curve shown in the bottom panel of Fig. 19 (see Appendix K) taking into account the smeared SDW gap. This is followed by a U-shape E profile below the SDW gap scale, making scarcely visible the connection with the normal part (2) characterized by an exponent $\eta \simeq 1.3$ and a similar positive curvature in energy up to $2t'_{\perp}$.

It is worth noticing that the spatial map of the exponent $\eta$ displayed in Fig. 5(a) shows experimental values between 0.75 for superconducting regions and 5 for SDW insulating ones. Such a range of $\eta$ values is compatible with the ones found theoretically when the RG solution is carried out at slightly weaker or stronger $t'_{\perp}/k_B$ around 26 K. When larger $\eta > 1$ values are encountered, stronger insulating SDW gaps take place (see Figs. 17 and 18 of Appendix K) which is also observed experimentally (see Fig. 13 of Appendix G). However, it cannot be excluded that in these strongly insulating conditions the predictions of the disordered Anderson-Hubbard model can lead to results comparable to those of $\bar{N}(E)$ in Eq. (K1) for the DOS at low energy.

V. CONCLUSION

Since the discovery of the Bechgaard salts as the first series of organic superconductors, the nature of the superconducting order parameter in its prototype compound, $(TMTSF)_2ClO_4$, has long framed a major part of the debate surrounding superconductivity in the series. Revealing the intricate link of superconductivity with bordering magnetic spin density wave and quantum critical metallic phases has become another determinant issue regarding these materials.

The experimental and theoretical insights brought out by this work have direct bearing upon these matters. Of particular note is the linearlike or convex low-energy dependence of the local excitation spectra that has been probed by scanning tunneling spectroscopy. The linear profile appears to be akin to the Planckian linear temperature dependence of resistivity observed close to the QCP. The E-linear density of states is consistent with nodal $d$ wave rather than nodeless s-wave symmetry for the order parameter in the superconducting phase. By contrast, the concave energy dependence of the local excitation spectra, found in more disordered surface conditions, matches very well with the nodeless order parameter expected in an insulating spin density wave phase.

Our experimental results find a quantitative support in the renormalization group approach to the quantum critical precursors of both magnetic and superconducting orderings. This manifests itself through the prediction of a power-law energy dependence $E^\eta$ for the density of states at low
energy. The exponent $\eta$ that characterizes metallic quantum fluctuations varies in relation with the nature of order in the low-temperature phase. For superconductivity, $\eta \lesssim 1$ whereas $1 < \eta < 5$ for spin density wave, in definite connection with the surface mapping of tunneling conductance obtained in our experiments. This overall agreement between theory and experiments gives strong support for instabilities in the organic (TMTSF)₂ClO₄ crystals as being primarily mediated by electron-electron repulsion combined to quantum criticality, resulting in nodal *d*-wave superconductivity and spin density wave phases. Both phases are found to coexist in different regions of the surface, as a result of large anions disorder induced by the cleavage process.

## ACKNOWLEDGMENTS

We thank C. Pasquier, N. Dupuis, and S. Yonezawa for fruitful discussion. We would like to thank M. Allain for helping us produce the images of the structure of (TMTSF)₂ClO₄.

## DATA AVAILABILITY

The data that support the findings of this article are not publicly available upon publication because it is not technically feasible and/or the cost of preparing, depositing, and hosting the data would be prohibitive within the terms of this research project. The data are available from the authors upon reasonable request.

## APPENDIX A: ELECTRICAL RESISTIVITY MEASUREMENTS

One (TMTSF)₂ClO₄ single crystal from the same batch as the one used for STM and STS experiments was selected for electrical transport measurements. The temperature dependence of the transverse resistivity $\rho_c$ has been measured in a Quantum Design physical property measurement system (PPMS) using an adiabatic demagnetization refrigerator adapted to the PPMS setup. The minimum temperature reached was 0.3 K at $H = 0$ and 0.6 K at $H = 0.2$ T. The resistance was measured applying a dc current of 50 µA. The cooling speed was decreased down to 0.01 K/min between 26 and 20 K in order to perform the ordering of the ClO₄ anions.

The resistivity shows a very large decrease when lowering the temperature between 300 K and about 2 K, as seen in the top left inset of Fig. 8. The overall behavior attests to very high sample quality, the residual resistivity ratio being larger than 340, in agreement with previous study [35]. The superconducting transition develops below 1.4 K with a midpoint $T_c$ of about 1.26 K. The critical magnetic field along the *c* direction is about 0.2 T (green curve).

As shown on Fig. 8, the inelastic part of the resistivity follows somewhat accurately around the anion ordering temperature up to about 50 K a polynomial law such as $\rho_c(T) \sim AT + BT^2$ displayed by the red dashed line on this figure. A temperature drop $\Delta\rho_c = 0.27$ Ω cm occurs between 25 and 20 K due to the anion ordering (see down right inset). However, the experimental behavior departs significantly from the polynomial law below 5 K and is not affected by a magnetic field (see the green curve on the figure). Such

<figure><img src="image_10.png" /><figcaption>FIG. 8. Temperature dependence of $\rho_c$, i.e., electrical resistivity along the *c* axis, for a very slowly cooled sample of (TMTSF)₂ClO₄ measured between $T = 300$ and 300 mK. The upper left inset displays the behavior starting from 32.3 Ω cm at room temperature. The lower right inset shows in more detail the behavior below 50 K with a drop in resistivity at 24.3 K due to the ordering of ClO₄ anions. Blue curves represent experimental measurements under zero magnetic field. The green curve represents experimental measurement under 0.2 T applied magnetic field the *c* direction. The red curves (continuous and dashed lines) are fits according to the indicated formulas (see text).</figcaption></figure>

a behavior is suggestive of a paraconductive contribution to the conductivity. The downturn of the resistivity below 5 K already reported in (TMTSF)₂ClO₄ [1,53], has actually been attributed to a SDW paraconductive contribution given by $\Delta\sigma = \frac{\rho_n-\rho_{ex}}{\rho_n\rho_{ex}}$, where $\rho_{ex}$ is the measured resistivity (blue dots on Fig. 8) while $\rho_n$ is the behavior of the inelastic resistivity (red line) derived from fitting the experimental transport to a polynomial law between 6 and 16 K. In addition, we may notice the good quantitative agreement for the paraconductive contribution between the data on Fig. 8 and those reported in Ref. [1].

## APPENDIX B: CRYSTAL STRUCTURE OF (TMTSF)₂ClO₄

We provide here an additional view of the crystal structure unit cell of (TMTSF)₂ClO₄ enabling seeing better the molecular stacking along the *a* direction.

## APPENDIX C: SAMPLE PREPARATION-LIMITING INTERNAL STRESS

(TMTSF)₂ClO₄ single crystals are very fragile needlelike materials that are notoriously difficult to manipulate and prepare for STM and STS experiments. We provide here additional pieces of information about the sample preparation procedure we have followed in order to minimize internal stress effects induced by the mismatch of thermal expansion coefficients of the various materials used to hold the (TMTSF)₂ClO₄ sample and cleave it.
<figure><img src="image_11.png" /><figcaption>FIG. 9. Unit cell of the crystal structure of (TMTSF)₂ClO₄.</figcaption></figure>

We took great care in gluing our samples. In particular, in a first step, we glued the sample on a stainless steel sample holder using a silver epoxy polymerizing at room temperature to avoid any additional stress induced by sample heating and also to avoid chemical diffusion. In a second step, we tried to glue the cleaver by dropping glue only on the sample surface, not clasping the edges. Then the cleavage process occurred at room temperature in the STM chamber before inserting the sample holder into the cold STM head. We believe that this way of doing might help avoiding clamping effect induced by different thermal coefficients of the glue, sample, sample holder, and cleaver. In particular, we verified optically in the STM chamber that the cleavage exposes a large part of the sample above the glue level along the *c*\* direction.

Thus, upon cooling, the part of the sample embedded deeply in the glue should indeed be subject to a large internal stress but is not the one that is measured. On the contrary, the cleaved sample surface is connected to a sample part that is held well above the glue level along *c\***, enabling in principle the underlying stress in the bottom part of the sample to relax in the transverse direction. Nevertheless, despite all the care taken, an additional internal stress induced by the various thermal expansion coefficients of the different materials used could appear and induce additional disorder in the cleaved (TMTSF)₂ClO₄ sample affecting in the end its surface electronic properties. This said, our experience with another very fragile inorganic quasi-1D needlelike compound NbSe₃ showed us that this preparation procedure in principle works and enables in this latter case to recover at the surface electronic properties similar to the bulk ones [30].

## APPENDIX D: SAMPLE COOLING CURVE FOR STM AND STS MEASUREMENTS

Below 45 K, the (TMTSF)₂ClO₄ samples used in this study were cooled down very slowly at cooling rates between 1 and 2 K per hour. This is presented in Fig. 10. This ensures that the structural ordering of the anions is best realized in the

<figure><img src="image_12.png" /><figcaption>FIG. 10. Cooling speed followed for all four (TMTSF)₂ClO₄ studied samples by STM and STS experiments in this study.</figcaption></figure>

material, leading to the largest superconducting bulk fraction below the superconducting critical temperature [1].

## APPENDIX E: ADDITIONNAL STS MEASUREMENTS IMPLYING TIP DIGGING

In nicely cleaved (TMTSF)₂ClO₄ areas, it was usually possible to scan the STM tip using constant current conditions with closed feedback loop using *I* = 10 to 50 pA and *V* = -0.1 V. This corresponds to the case of Fig. 1(a) of the main text and to Fig. 11. Indeed, nice and wide molecular terraces are seen in both cases, separated by single or multiple lattice constant along the *c* axis. Nevertheless, it was not always possible to go from these scanning conditions corresponding to a 2–10 GΩ tunneling resistance to much lower tunneling resistance, i.e., about 10 MΩ needed to measure the superconducting excitation spectrum with enough signal-to-noise ratio. In some sample areas, when changing the scanning conditions to acquire a superconducting excitation spectrum, the STM tip dug several nm or even tens of nm below the originally flat sample surface. Such a behavior is exemplified in Fig. 11, the *Z(X)* tip profile enabling to appreciate the depth (about 25 nm) and width (about 50 nm) of the hole created by the tip. In particular, it can be noticed that the depth of this hole is about twice larger than the original tip height variation along this profile (about 10 nm) before the hole was made.

## APPENDIX F: ADDITIONAL STS MEASUREMENTS IN SUPERCONDUCTING REGIONS

In order to be complete, we present here the entire diversity of single excitation spectra corresponding to superconducting regions measured on various areas and different (TMTSF)₂ClO₄ samples from the same batch. Figure 12 presents these additional STS *dI/dV* spectra.

It is seen that the higher-energy background, i.e., above ≈4 meV is very similar among these different spectra. Nevertheless, the low-energy part, corresponding to the
<figure><img src="image_13.png" /><figcaption>FIG. 11. (a) Raw color-coded constant current Z(x, y) STM topography of a (TMTSF)₂ClO₄ area of 550 × 550 nm² measured with *I* = 50 pA and *V* = -0.1 V. The color bar is shown on the right and varies between 0 and about 40 nm. Numerous molecular terraces are seen corresponding to flat areas of uniform color. The top black feature crossed by the continuous almost vertical line is a hole created by the STM tip into the surface after attempting measuring the superconducting excitation spectrum with a set point *I* = 200 pA and *V* = -5 mV. (b) The STM tip profile Z(X) measured in nm along the continuous black line shown in (a) enables to quantify how deep the dip dug into the surface, i.e., about 25 nm, while the sample surface was originally a flat molecular plane at this location. In addition, this profile enables to see the regular spacing in *z* variations corresponding to single or multiple molecular planes of (TMTSF)₂ClO₄ along the *c* direction.</figcaption></figure>

superconducting excitation spectrum, varies considerably. In
particular the gap filling varies considerably from ones reach-
ing zero exactly around the Fermi level (black spectrum in

<figure><img src="image_14.png" /><figcaption>FIG. 12. This figure summarizes representative superconducting excitation spectra measured at different locations and for different (TMTSF)₂ClO₄ samples from the same batch. For each spectrum it is indicated which experimental conditions were encountered in the corresponding location. In particular, it is written whether the excitation spectrum could be measured at the top molecular surface plane (it is then written “no digging”) or not (it is then written “digging”). In addition, the term “rough” or “flat” is indicated to qualify the local surface roughness measured while scanning with typical conditions such as in Fig. 11 (see text). The term “flat” means that the local tip height variations were below 0.1 nm [such as in Fig. 1(c) of the main text]. The term “rough” means that the local tip height variations ranged from several angstroms to about 1 nm. The spectrum denoted “reconstructed” was measured in an area modified by the tip after it dug into the surface.</figcaption></figure>

Fig. 12 or in Fig. 5 from the main text), very large gap filling
like the m164 or m192 spectra. However, these spectra also
share an almost linear low-energy behavior, in agreement with
the other representative spectra presented in the main text.
As the spectra presented in the main text, the superconduct-
ing coherence peaks are smeared out due to disorder, the
presence of a strong V-shaped background and temperature
broadening.

In addition, we also presented in this figure the various
cases encountered in the experiment, as described in the

<figure><img src="image_15.png" /><figcaption>FIG. 13. This figure summarizes $dI/dV$ excitation spectra associated with representative spin density wave spectra measured at different locations and for different (TMTSF)₂ClO₄ samples from the same batch at *T* = 300 mK. With respect to the terminology introduced in Fig. 12, all these spectra were acquired in rough regions. The typical set point for spectroscopy was *I* = 200 pA to *I* = 300 pA and *V* = -9 mV.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>FIG. 14. Left panel: $dI/dV(E)$ map measured at -20 meV over a 60 × 60 nm² area. The color bar is expressed in pS and is located on the right of the map. The STM topography (not shown here), acquired while performing this spectroscopy grid, was measured using $I = 50$ pA and $V_{bias} = -70$ mV. The set point for each 75 × 75 STS spectroscopy was $I = 200$ pA and $V_{bias} = -70$ mV, from which single $I(V)$ curves were acquired between [-70; +70] mV a each point of the 75 × 75 grid. Right panel: selected $dI/dV(E)$ spectra. Each colored spectrum represents an average inside a circle of comparable size comprising about 150 individual spectra and is indicated on the left $dI/dV(E)$ map using a corresponding color. The various observed high-energy backgrounds range from almost linear to convex.</figcaption></figure>

section entitled “additionnal STS measurements in superconducting regions” above. The various cases are described in detail in the caption of the figure.

## APPENDIX G: ADDITIONAL STS MEASUREMENTS IN SDW REGIONS

Figure 13 presents representative $dI/dV$ differential conductance spectra corresponding to spin density wave excitation spectra. The various spectra were acquired at different locations and for different (TMTSF)₂ClO₄ samples from the same batch at $T = 300$ mK.

## APPENDIX H: ADDITIONAL STS MEASUREMENTS SHOWING THE LOCAL VARIATIONS OF THE HIGH-ENERGY BACKGROUND

We also performed local measurements to estimate the local variations of the high-energy background encountered at the sample surface. To this end, full $I(V)$ maps were acquired at several areas on various samples between typically [-60; 60] mV. One example is shown in Fig. 14. Local variations of the $dI/dV(E)$ background are measured across this area, exemplified by the selected spectra. The red spectrum, acquired in the top left part of the area, presents characteristics very similar to the one presented in Fig. 4 of the main text, i.e., a close to linear energy dependence. The other spectra present an energy dependence closer to a power-law behavior with an exponent smaller than one. The microscopic origins of this feature are discussed in details in the main text. The local variations observed here at the nanometer scale are likely to be linked to structural disorder in the anion lattice.

## APPENDIX I: MODEL OF DISORDERED DOS BACKGROUND

The DOS in the intermediate energy range is governed by disorder coupled to electron-electron interaction. Following Shinaoka and Imada [43,44], it can be modeled by the Anderson-Hubbard model which consists of tight-binding electrons interacting through a local Hubbard term $U$ between electrons of opposite spins and one-body term of electrons that scatter on a spatially distributed random site potential $V$. From the numerical solution for site-dependent Hartree-Fock equations, the energy profile of the DOS can be shown to scale according to the following expression in two dimensions:

$$N_{AH}(E) = c \exp[-\alpha \ln^2(E/t)], \quad (I1)$$

where $c$ and $\alpha$ are nonuniversal constants. Here the energy difference $E$ is the energy difference from the Fermi energy

<figure><img src="image_17.png" /><figcaption>FIG. 15. Calculated RG phase diagram for the quasi-one-dimensional electron gas model of the Bechgaard salts.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>FIG. 16. One-particle self-energy RG flow equation at the two-loop $O(g^2)$ level. The open (full) diamond stands for normal (umklapp) electron-electron scattering vertex. Adapted from Ref. [29].</figcaption></figure>

$E_F$. For quasi-1D weakly dimerized metallic chains at quarter-filling like the Bechgaard salts, the normalization hopping energy $t$ is connected to the Fermi energy by the relation $E_F = \sqrt{2t}$. The above expression is used to fit the STM data at intermediate energy.

## APPENDIX J: THE DENSITY OF STATES AND THE QUASI-1D ELECTRON GAS MODEL

The quasi-1D electron gas model for a linear array of $N_⊥$ weakly coupled chains consists of a noninteracting Hamiltonian with the one-electron spectrum

$$E_p(\vec{k}) = v_F(pk - k_F) - 2t_{\perp} \cos k_{\perp} - 2t'_{\perp} \cos 2k_{\perp}, \quad (J1)$$

where $v_F$ is the Fermi velocity of right ($p = +$) and left ($p = -$) moving electrons along the chains ($\hbar = 1$); $t_⊥$ and $t'_⊥$ are the transverse hopping between nearest- and next-to-nearest-neighbor chains, respectively (interchain distance $d_⊥ = 1$). The hopping parameter $t'_⊥$, also called antinesting, models deviations from perfect nesting conditions at wave vector $\vec{Q}_0 = (2k_F, \pi)$ for an open Fermi surface, namely,

$$E_+(\vec{k} + \vec{Q}_0) = -E_-(\vec{k}) - 4t'_\perp \cos 2k_\perp. \quad (J2)$$

Regarding interactions of the model these are commonly defined by three main scattering amplitudes on the Fermi surface, that is, the backward and forward normal scattering processes $g_{1,2}(k'_{⊥1}, k'_{⊥2}; k_{⊥1}, k_{⊥2})$ for large ($\sim 2k_F$) and small ($\sim 0$) momentum transfer along the chains and $g_3(k'_{⊥1}, k'_{⊥2}; k_{⊥1}, k_{⊥2})$ for umklapp scattering that transfers

<figure><img src="image_19.png" /><figcaption>FIG. 17. Energy dependence of the normal state single-particle density of states for the quasi-1D electron gas model as calculated by the RG method for different antinesting $t'_⊥$ values of the phase diagram of Fig. 15. The temperature is set above but close the critical point $T \gtrsim T_{SDW,c}$.</figcaption></figure>

two electrons from one side ($p$) of the Fermi surface to the other ($-p$). By lowering energy, the couplings acquire a momentum dependence along the Fermi surface parametrized by ingoing ($k_{⊥1,2}$) and outgoing ($k'_{⊥1,2}$) transverse wave vectors.

The possible instabilities of the normal state against the formation of either an ordered SDW or a SC state at low energy can be obtained by the renormalization group method [19,29,48]. At the one-loop level, the method proceeds to the successive integration of electron degrees of freedom as a function of energy distance $E$ from the Fermi surface. It allows to sum electron-hole (e-h) closed and vertex corrections diagrams with interfering electron-electron (e-e) ladder diagrams. This yields the flow equations of the normalized scattering amplitudes [29], which can be written in the schematic form

$$\begin{align}
E \partial_E g_{i=1,2}(\{k_\perp\}) &= \sum_{\mathcal{P}_{nn'}} \left\{ \epsilon_{C,i}^{n,n'} \langle g_n g_{n'} E \partial_E \mathcal{L}_C \rangle_{k_\perp} \right. \nonumber \\
&\qquad \left. + \epsilon_{P,i}^{n,n'} \langle g_n g_{n'} E \partial_E \mathcal{L}_P \rangle_{k_\perp} \right\}, \nonumber \\
E \partial_E g_3(\{k_\perp\}) &= \sum_{\mathcal{P}_{3n}} \epsilon_{P,3}^{3,n} \langle g_3 g_n E \partial_E \mathcal{L}_P \rangle_{k_\perp}, \tag{J3}
\end{align}$$

where the $k_⊥$ dependence of convolution products, constrained by momentum conservation, has been masked for simplicity; here $\langle \dots \rangle_{k_⊥}$ stands for an average over $k_⊥$. The e-h ($P$) and e-e ($C$) loops at finite $T$ are given by

$$\begin{align}
E \partial_E \mathcal{L}_{P,C}(k_\perp, q_{P,C}) = & - \sum_{\nu=\pm} \theta(|E + \nu A_{P,C}| - E) \nonumber \\
& \times \frac{1}{2} \left( \tanh \frac{E + \nu A_{P,C}}{2k_B T} + \tanh \frac{E}{2k_B T} \right) \nonumber \\
& \times \frac{E}{4E + \nu A_{C,P}}, \tag{J4}
\end{align}$$

which are evaluated at $q_p = k_{\perp 1} - k'_{\perp 1}$ and $q_C = k_{\perp 1} + k_{\perp 2}$, respectively; $A_{P,C}(k_\perp, q_{P,C}) = -E_+(k_F, k_\perp) \mp E_-(k_F, k_\perp + q_{P,C})$ and $\theta(x)$ is the step function. For a given $g_i$, the related sum in (J3) collects all possibilities $\mathcal{P}_{nn'}$ of diagrams with normal scattering $g_{n=1,2}$ ($g'_{n=1,2}$) and those $\mathcal{P}_{3,n}$ involving umklapp scattering. The coefficients $\epsilon_{C,P,i}^{nn'}$ fix the sign and spin multiplicity of each loop contribution; it differs for closed loops ($\epsilon_{P,i}^{nn'} = -2$), vertex corrections ($\epsilon_{P,i}^{nn'} = 1$), and ladder ($\epsilon_{C,i}^{nn'} = -\epsilon_{P,i}^{nn'} = -1$) diagrams.

Substituting in these flow equations initial parameters that reasonably describe the Bechgaard salts, the instabilities of the metallic state can then be followed as function of pressure simulated by $t'_⊥$. For band parameters, one has typically $v_F k_F = E_F \simeq k_B 3000$ K for the high (Fermi) energy cutoff and $t_⊥/k_B = 200$ K for the smaller transverse hopping amplitude. As for the couplings, different experiments can fix their initial normalized values at $E = E_F$, with $g_1 \simeq 0.3$ and
<figure><img src="image_20.png" /><figcaption>FIG. 18. Variation of the power-law exponent for the calculated single-particle density of states in the antinesting low-energy range ($E < E_0^*$) and for different $t'_\perp$.</figcaption></figure>

$g_2 \approx 0.6$ and $g_3 \approx 0.03$ for the small, but essential, half-filling umklapp amplitude, a consequence of the weak dimerization of molecular chains in the Bechgaard salts.

Following the integration of (J3) down to $E \to 0$ at the Fermi surface, two possible instabilities can occur as a function of $t'_\perp$ at a given critical temperature scale. The first is against the formation of a SDW state at the ordering temperature $T_{SDW}$ which is signaled by a singularity in the combination of couplings $(g_2 + g_3)_{\{k_\perp, \vec{Q}_0\}}$, for $k_\perp$ along the Fermi surface and at the momentum transfer $\vec{Q}_0$ of the SDW modulation. Increasing $t'_\perp$, $T_{SDW}$ undergoes a monotonic decrease until a critical $t'^*_\perp$ is approached from below and where a sharp drop of $T_{SDW}$ points towards a quantum critical point (QCP). However, the contact with the QCP at zero temperature is prevented by another singularity at temperature $T_c$, occurring this time in the combination $(g_1 + g_2)_{\{k_\perp, \vec{Q}_c\}}$ at zero pair momentum $\vec{Q}_c = 0$ for ingoing and outgoing electrons. The combination of couplings is cosine modulated in $k_\perp$ space. This signals an instability towards the formation of a d-wave superconducting gap $\Delta(k_\perp) = \Delta \cos k_\perp$ along the Fermi surface below $T_c$. Moving away from the QCP as $t'_\perp$ is increased, $T_c$ declines progressively. This characteristic sequence of instabilities as a function of the pressure parameter $t'_\perp$ is summarized in the phase diagram of Fig. 15.

The impact of the above one-loop scattering amplitudes on quasiparticle spectral properties of the model can be obtained from the calculation of one-particle self-energy. The corresponding flow equation derived at the two-loop level in Ref. [29] is shown in the diagrammatic form in Fig. 16. From that work, the quasiparticle weight $z(k_\perp, E)$ located at $\vec{k}_F(k_\perp)$ for p electrons on the Fermi surface and energy distance E can be written in the schematic form

$$E \partial_E \ln z(k_{\perp}, E) = \int \frac{d^2 k'_{\perp}}{(2\pi)^2} \{(g_2 g_1 - g_2 g_2 - g_1 g_1) I_{n,E} - g_3 g_3 I_{3,E}\}, \quad (J5)$$

<figure><img src="image_21.png" /><figcaption>FIG. 19. Low-energy dependence of the calculated differential conductance [Eq. (K2)] at $T \ll T_\mu$ in (top) the ordered SC ($t'_\perp/k_B = 26$ K, $T_c \approx 1.1$ K) and (bottom) SDW ($t'_\perp/k_B = 25.4$ K, $T_{SDW} \approx 5$ K) phases at different normalized scattering rate parameters $\Gamma/\Delta$.</figcaption></figure>

in which the transverse momentum dependence and diagram permutations have been masked for simplicity. Here the transverse momentum and energy-dependent expressions of the two-loop diagrams of normal and umklapp vertices are of the form

$$I_{n(3),E} \sim \frac{1}{2} \frac{T^2}{L^2} (\pi v_F)^2 \partial_E \sum_{\{\omega\}} \int_E^{E_F} dE' G^0_{-p,E} G^0_{-p,E'} G^0_{p,E'}$$

where $G^0_{p,E} = [i\omega_n - E_p(\vec{k})]^{-1}$ is the bare electron propagator for the p branch.

Single-particle tunneling along the $c^*$ direction probed by STM experiments couples to local density of states derived from the quasiparticle weight of each chain of the conducting ab plane, namely, $N_n(E) = N_0 \langle z(k_\perp, E) \rangle_{k_\perp}$, where $N_0 = 2(\pi v_F)^{-1}$ is the total bare density of states for both spin orientations.
Inserting the results for the scattering amplitudes (J3) in the flow equation (J5) for self-energy, one finds in Fig. 17 the energy profile of density of states of the metallic phase; it is computed at different antinesting $t'_\perp$ and for a temperature $T$ above but close to the critical $T_{c,SDW}$. The results display a characteristic dip at the Fermi level with a depth and energy dependence that vary with $t'_\perp$. According to (J2), the latter indeed introduces an energy scale at $E_0^* \sim 2t'_\perp$ (~5 meV for the model parameters used in Fig. 17) below which the electron-hole pairing responsible for SDW correlations becomes $k_\perp$ dependent and then weakened. It follows from (J3) an interplay between SDW and d-wave SC correlations at $E < E_0^*$, which is responsible for a dip in $N_n(E)$ that is well described by a power law in energy:

$$ \bar{N}_n(E) \approx C(\text{Max}\{\pi k_B T, |E|\})^\eta, \quad (E < E_0^* \sim 2t'_\perp). \quad (J6) $$

which is cutoff by the temperature at sufficiently low energy. The exponent $\eta$ and constant $C$ are $t'_\perp$ dependent as shown in Fig. 18. At the left of the QCP, on the SDW side of the phase diagram, $\eta$ grows above unity which is compatible with a concave U shape of $N_n(E)$ with energy. Close to the QCP on the SC side, $N_n(E)$ displays a linear V-shape energy dependence with $\eta \simeq 1$ which transforms into a convex variation as $\eta$ progressively decreases below unity when $t'_\perp$ is made larger.

## APPENDIX K: CALCULATED TUNNELING CONDUCTANCE IN THE ORDERED STATE

The above perturbative renormalization group results for the energy-dependent density of states are restricted to the normal metallic state. The impact of the entry into the ordered phase at low energy will be accounted for by completing the expression of the density of states by a BCS form for $|E| < \Delta_\mu$ and in the low-temperature limit [47,50]. One shall

consider the expression

$$ \bar{N}(E) = \bar{N}_n(\Delta_\mu, |E|) \left\langle \text{Re} \left\{ \frac{|E + i\Gamma|}{\sqrt{(E + i\Gamma)^2 - \Delta_\mu^2(k_\perp)}} \right\} \right\rangle_{k_\perp} \quad (K1) $$

where $\bar{N}_n(\Delta_\mu, |E|) = C(\text{Max}\{\Delta_\mu, |E|\})^\eta$ is the normal part. Here the gap $\Delta_\mu(k_\perp)$ is given by $\Delta_{sc}(k_\perp) = \Delta \cos k_\perp$ for a quasi-1D d-wave superconductor and by the constant $\Delta_{sdw}(k_\perp) = \Delta$ in the insulating SDW case; $\Gamma$ stands for the impurity scattering rate that broadens the coherence peaks for both orderings.

The above density of states takes place in the expression of the differential conductance

$$ \frac{dI}{dV}(E) = G_{nn} \int_{-\infty}^{+\infty} \bar{N}(\omega) \left[ -\frac{\partial f(\omega + eV)}{\partial (eV)} \right] d\omega, \quad (K2) $$

where $E = eV$, $f(\omega)$ is the Fermi distribution and $G_{nn}$ is adjustable constant for the amplitude of normal conductance. The low-energy dependence of $dI/dV$ is illustrated in Fig. 19 for two typical values of pressure $t'_\perp$ in the SC and SDW domains of the phase diagram. In the SC case, one sees the suppression of the coherence peaks in the density of states by the impact of the scattering rate. The comparison with the experimentally measured spectra supports high scattering rates occurring at the sample surface, most probably linked to the cleavage process and disorder in the $ClO_4$ ion sublattice. In the SDW case where the reduction of the normal part $\bar{N}_n$ is relatively more pronounced, this effect makes barely visible the impact of the ordered phase on the density of states. Nevertheless, a close look at Figs. 6(a) and 7 of the main text shows small kink features located around 1 and 2.5 meV, respectively, that are consistent with the theoretically expected superconducting and SDW peaks broadened by disorder shown in Fig. 19.

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