as e-doped NbSe₂. The matching between the two band structures is even more accentuated in the unfolded Brillouin zone representation. On figure 3.17 **a**, the darker the dot, the higher the spectral weight of the band.

In figure 3.17 **a**, we see that at the energy range in which one should see the La bands, namely between 0, 4 eV and 1 eV (see figure 3.16 **a**), the bands are made of light grey dots, meaning small spectral weight contribution. This is confirmed in 3.17 **b** in which we can see that the La contribution (red dashed line) to the total density of states around Fermi level is negligible with respect to the Nb contribution (blue dashed line), which further comforts our hypothesis of $(LaSe)_{1,14}(NbSe_2)_2$ behaving as electron doped monolayer NbSe₂.

Since a strong Ising spin-orbit coupling is present in monolayer NbSe₂, it is very important to emphasize that a good correspondence between the two band structures also implies the presence of such a spin-orbit coupling in $(LaSe)_{1,14}(NbSe_2)_2$. The splitting due to this spin-orbit coupling is indeed visible, particularly in the unfolded representation of the band structure of $(LaSe)_{1,14}(NbSe_2)_2$ (see the splitting around *K* in figure 3.17 **a**). Contrary to bulk 2H-NbSe₂ in which no such effect was observed, theoretical calculations show here that $(LaSe)_{1,14}(NbSe_2)_2$, although being a bulk material, actually possesses a quasi-2D physics. This will be of the utmost importance when we will present the results obtained in the superconducting phase of $(LaSe)_{1,14}(NbSe_2)_2$ in 3.6.

A ±1 V dI/dV spectrum was acquired and is superimposed on top of the density of states of $(LaSe)_{1,14}(NbSe_2)_2$ for comparison (green full line in figure 3.17 **b**). All the visible peaks can be related to positions in energy at which the group velocities of the bands are zero and thus at which a lot of spectral weight is expected in the density of states. For positive energies, there is a whole energy span for which there are almost no states. One thus expects the conductance to be close to zero within this range which is also supported by a strong depletion visible in the experiment. Also, around -0,8 eV, we see a gap in the expected density of states. In the experimental data, a small depletion of density of states seems to reflect this feature. The gap is however not seen, and we attribute this to the fact that, as the bias voltage increases, one starts to probe the bulk material rather than the surface. The density of states observed experimentally is in an overall good agreement with the DFT one, enforcing the validity of our first principles calculations.

The following paragraphs are dedicated to the comparison between the theoretical QPIs patterns and the experimental ones. In order to do so, we will base our discussion on figure 3.18⁴. In figure 3.18 **a**, is represented the Fermi surface of e-doped NbSe₂ with two nearly superimposed spin-polarized pockets around $\Gamma$ and two split spin-polarized pockets around $K/K'$ (due to Ising spin-orbit coupling). In figure 3.18 **b** is plotted the expected QPIs patterns (spin-selective joint DOS) at 0-energy for e-doped NbSe₂. The tiny black dots are located at Bragg peaks positions. Figure 3.18 **d** depicts the Fourier transform modulus of $dI/dV$ map at 0 energy<fnref n="5" />. In the following, we will see if it is possible to interpret our STS data in terms of QPIs generated by nesting properties of the Fermi surface of e-doped NbSe₂. Calculating the expected QPIs patterns for $(LaSe)_{1,14}(NbSe_2)_2$ would be too costly numerically speaking.

**Remark :** According to what was said in section 2.3 dedicated to the theoretical description of QPIs, only scattering channels which conserve the spin are to be considered. The wave functions of an incoming electron and a scattered electron with opposite spin are, indeed, orthogonal and thus, cannot interfere and lead to standing-wave patterns. The QPIs expected patterns are thus given by the calculation of the **spin-selective** joint DOS.

A few scattering wave vectors which connect efficiently (good nesting) different parts of the

<footnote n="4">⁴For all panels of figure 3.18, the misfit direction is along the vertical axis.</footnote>

<footnote n="5">⁵400 × 400 *I*(*V*) spectra were acquired in a 30nm×30nm area, following the grid spectroscopy procedure described in figure 2.7 of section 2.2.4.</footnote>
Fermi surface were drawn on figure 3.18 a. Q1 links the hole pocket around $\Gamma$ to itself. It gives the big red circle in figure 3.18 b. Other circles with same radius can be observed around all six Bragg peaks. Their corresponding wave vectors link the pocket around $\Gamma$ in 1BZ to the one in 2BZ. Due to Ising spin-orbit coupling present in monolayer NbSe$_2$ (in which inversion symmetry is broken), a spin up/spin down band splitting occurs around $K/K'$ point. This splitting also exists around $\Gamma$ (not along $\Gamma M$ though) but is negligible. Q2 is a scattering wave vector linking the hole pocket around $\Gamma$ and the $K/K'$ inner pocket. It gives rise in QPIs patterns to arcs which are displayed in green in figure 3.18 b. Again, scattering processes between 1BZ and 2BZ also lead to similar arcs. One is highlighted in green at the bottom of figure 3.18 b. Q3 is a scattering wave vector linking the hole pocket around $\Gamma$ and the $K/K'$ outer pocket. It gives rise in QPIs patterns to arcs which are displayed in blue in figure 3.18 b.

<figure><img src="image_43.png" /><figcaption>Figure 3.18: a. Fermi surface of e-doped monolayer NbSe$_2$. b. Spin-selective joint DOS of the Fermi surface of e-doped monolayer NbSe$_2$ reflecting expected QPIs patterns. c. Modulus square Fourier transform of a STM image taken at -0.25 meV bias voltage. d. Modulus square Fourier transform of a STS dI/dV map at 0 energy.</figcaption></figure>

All the expected QPIs patterns that were presented so far can be seen in 3.18 d and at the right positions in reciprocal space. The only spectroscopic signal that cannot be simply interpreted with the Joint DOS of e-doped NbSe$_2$ Fermi surface is the intense halo around $\Gamma$. A study which is not detailed here but that was done by the theoreticians of our group showed that this signal cannot be explained by QPIs for the La band crossing Fermi level. In figure 3.16 c, it is possible to see that the S and Y points of the approximate commensurate Brillouin
zone, which are close to $\Gamma$, are repeated many times throughout the image. Several reduced Brillouin zones are indeed contained within the hexagonal first Brillouin zone, meaning that strong signal can potentially be sent to $\Gamma$ thanks to the incommensurability. This is only an hypothesis but it could also explain the unexpectedly large widths of the Bragg peaks.

The spin-splitting due to strong Ising spin-orbit coupling predicted theoretically is also present. In figure 3.18 **d**, this last statement is highlighted by the splitting between the two arc in the misfit direction (with blue and green arcs as guide for the eye). In our knowledge, it is the first time that this Ising spin-orbit coupling splitting is seen *via* STS, regardless of the compound.

Q4 is a scattering wave vector linking the inner pockets around $K/K'$ points. It conduced to the presence of a circle around $\Gamma$ highlighted by a pink arc in figure 3.18 **b**. Figure 3.18 **c** is the modulus square of the Fourier transform of a topographic STM image taken at -25 meV. Figure ??c also clearly displays the Q1 contribution, but also the Q4 contribution with the smaller radius signal around $\Gamma$, which further comforts the rigidly shifted monolayer $NbSe_2$'s band structure scenario.

**Partial conclusion :** In this study, we demonstrated that due to charge transfer occurring between La and Nb atoms, the system behaves as electron-doped monolayer $NbSe_2$ (rigid band structure shift) with a $5, 8 \cdot 10^{14}$ electron per cm$^2$ increase in carrier density (60% of intrinsic density) and chemical potential shift of +0,3 eV. In figure 3.19, is represented the band structure of monolayer $NbSe_2$. On top of this band structure is superimposed a grey band delimiting the accessible doping range achieved in ref. [87] by mean of ionic gating. The value of electron-doping reached in this study was the best achieved in similar systems so far, but it is more that four times lower than ours (see orange line in figure 3.19). Our study opens the possibility to use misfit exotic TMD heterostructures to thoroughly tune doping to unprecedented values and thus considerably modify the electronic properties of quasi-2D $NbSe_2$.

<figure><img src="image_44.png" /></figure>

Figure 3.19: $(LaSe)_{1,14}(NbSe_2)_2$ acts as a electron-doping device for $NbSe_2$. The Fermi level position is shifted of 0,3 eV. The Fermi surfaces of monolayer $NbSe_2$ and e-doped $NbSe_2$ are represented next to each other, hence illustrating the possible electronic changes provoked by strong electron doping.
### 3.3.4 2 × 2 charge modulation order

Here, we will focus on the 2 × 2 CDW in $(LaSe)_{1,14}(NbSe_2)_2$ that was previously mentioned in the text. First, we will look at the temperature dependence of this charge order and second, we will see if it can be predicted through electron-phonon dispersion DFT calculations performed on electron-doped monolayer NbSe$_2$.

For now, we only stressed the 2 × 2 CDW signature in reciprocal space, but it can also be seen in real space. Figure 3.20 is a topographic map taken at 4, 2 K. On this image, we can see that, locally, one every two atoms appears brighter. White circle were superimposed on the image as a guide for the eye. In figure 3.21 **a** is represented a STM image acquired at 25 meV and 2 K. One every two atomic columns appears darker, and this, in two directions as suggested by the guide-for-the-eye white lines. The Fourier transform modulus on this image is displayed in figure 3.21 **b**, where the four CDW spots are clearly visible. On the right, are plotted modulus Fourier transforms of STM images taken at 77 K, 100 K and 300 K, respectively. Only a part of the full images is represented to emphasize on the temperature evolution of CDW spots only. At 77 K and 100 K the CDW spots can still be seen. Nevertheless, at room temperature the CDW is no longer present. By performing STM imaging while increasing slowly the temperature, the CDW survives up to 105 K, temperature which we evaluated as being the CDW critical temperature. This critical temperature is above the 3 × 3 CDW critical temperature observed in monolayer NbSe$_2$ in ref.[94] (between 25 K and 45 K), but beneath the 3 × 3 CDW critical temperature of ref. [88] (around 145 K).

<figure><img src="image_45.png" /><figcaption>Figure 3.20: Topography map of $(LaSe)_{1,14}(NbSe_2)_2$ taken at 4, 2 K displaying a 2 × 2 CDW order in real space. Locally, every other atom appears brighter. The white circles are a guide for the eye.</figcaption></figure>
<figure><img src="image_46.png" /><figcaption>Figure 3.21: a. Topography image of (LaSe)₁,₁₄(NbSe₂)₂ displaying a 2 × 2 CDW in real space.
b. Fourier transforms of topography maps taken at 2 K, 77 K, 100 K and 300 K, respectively.
The transition between CDW and normal phase occurs around 100 K.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Figure 3.22: Phonon dispersions for e-doped monolayer NbSe₂ with no doping (black curves), 0,4 e⁻/Nb doping (green curves) and 0,6 e⁻/Nb doping (red curves). The correct amount of strain necessary to reproduce the one of (LaSe)₁,₁₄(NbSe₂)₂ was taken into account in the DFT calculations.</figcaption></figure>

It is interesting to see whether the 2 × 2 CDW naturally appears in e-doped monolayer NbSe₂. The NbSe₂ monolayer considered in this study has two particularities. The first one is the compression along the misfit direction. The theoreticians from our group observed a
tendency to switch from a 3 × 3 CDW to a 2 × 2 CDW with increasing compression but this
was for distortions far much substantial than the one (LaSe)₁,₁₄(NbSe₂)₂ is dealing with. Strain
is, hence, not likely to be responsible for the 2 × 2 CDW order in our compound. The second
particularity is electron-doping.

In figure 3.22 are represented phonon dispersions for 0 charge transfer, 0,4 electrons per Nb atom charge transfer and 0,6 electrons per Nb atom charge transfer<fnref n="6" />. For 0 charge transfer, the well-known phonon softening (imaginary frequency) mode lying at about 2/3[ΓM] and responsible for the 3 × 3 CDW order (see [95]) is quite visible. For 0,4 e⁻/Nb doping the phonon softening mode move from 2/3[ΓM] to [ΓM], or so to say, a 2 × 2 CDW naturally appears with increasing electron doping. In our experiment, given our QPIs results, our doping level should be around 0,6 e⁻/Nb charge transfer. Unfortunately, the phonon softening does not survive up to our experimental doping value. It seems like the calculations do not show any phonon softening for the doping (or the evaluated charge transfer) expected for our compound.

**Partial conclusion :** This study can mean two things : either our model is wrong or incomplete, or the 2 × 2 CDW has another origin differing from regular electron-phonon coupling. Our e-doped monolayer NbSe₂ model might not be correct regarding the 2 × 2 CDW. It is probably wrong to neglect the iono-covalent interaction between NbSe₂ and LaSe layers, interaction which probably induces modifications in the phonon spectrum and thus shift the doping levels at which the 2 × 2 CDW survives. At least this simple model grabs the tendency of the system to switch from 3 × 3 to 2 × 2 charge modulations with electron-doping. However, it is important to note that numerical simulations of STM topography maps of our compound (not shown here but performed by M. Calandra, C. Tresca and M. Campetella) do not show such 2 × 2 modulations. Another possible interpretation would be that the presence of La vacancies (invisible to our probe because lying beneath the NbSe₂ top layer) tend to locally change the chemical potential (the local electron doping is lower) and thus locally tune the system into a 2 × 2 charge modulation phase. This last hypothesis could also explain the "blurry" nature of our 2 × 2 CDW.

## 3.4 (LaSe)₁,₁₄(NbSe₂)₂ : an Ising superconductor

As we have seen in the first chapter (see section 1.1), when a BCS superconductor is submitted
to an external magnetic field, counter currents will start to flow contributing to globally expel
this field. We speak of orbital effects, since it is the motion of Cooper pairs which is at the
origin of the suppression of the magnetic field. Let us now consider a superconducting plane
in presence of a magnetic field taken parallel to this plane. Because of the two-dimensional
nature of the system, the motion of the electrons is limited in the direction perpendicular to the
superconducting plane and the orbital effects are thus considerably limited. This does not mean
that a 2D superconductor can support any arbitrarily strong magnetic field. In a conventional
superconductor, the electrons forming a Cooper pair have opposite spins. When submitted
to a magnetic field, the interaction of one of the spins of the Cooper pair with the magnetic
field is energetically unfavorable, potentially leading to Cooper pair breaking<fnref n="7" />. Let us call $H_c$
the value of the magnetic field for which the interaction with the magnetic field overcomes the
pairing energy. Above $H_c$, superconductivity is killed and one has : $H_c = 1,85 T_{SC}$ (where $H_c$
is expressed in tesla and $T_{SC}$ in kelvin) [19]. This is called the Pauli paramagnetic limit.

In few layers NbSe₂, in-plane upper critical fields are several times bigger than the Pauli
paramagnetic limit [33, 34, 35]. This is due to the competition between the external in-plane

<footnote n="6">⁶This value corresponds to the charge transfer effectively occurring in our compound.</footnote>

<footnote n="7">⁷Such pair breaking occurs because of the Zeeman splitting between the states with opposite spins of the electrons of the Cooper pair. It becomes efficient when $g\mu_B H$ (with g the Landé factor), the energy difference between the two states, becomes of the order of $2\Delta$ (with $\Delta$ the superconducting gap).</footnote>
magnetic field and the strong Ising spin-orbit coupling which tends to lock the spins of the electrons in the out-of-plane direction<fnref n="8" />. Very high in-plane upper critical fields are one of the signatures of Ising superconductivity.

As shown in ref. [89], $(LaSe)_{1,14}(NbSe_2)_2$ is superconducting under 5,3 K (see figure 3.23 **a**). It was shown one year later that $(LaSe)_{1,14}(NbSe_2)_2$ possesses a substantial in-plane upper critical field (more than 25 T by extrapolating the resistivity vs magnetic field curve of figure 3.23 **c**, while $H_{c,Pauli} \simeq 10$ T) implying the presence of a strong Ising spin-orbit coupling [9]. This is further corroborated by the DFT calculations presented in the previous section.

<figure><img src="image_48.png" /><figcaption>Figure 3.23: **a.** Resistivity vs temperature in $(LaSe)_{1,14}(NbSe_2)_2$ showing superconducting behaviour under 5,3 K (from [89]). **b.** Resistivity vs perpendicular magnetic field in $(LaSe)_{1,14}(NbSe_2)_2$ (from [9]). **c.** Resistivity vs parallel magnetic field dependence of $(LaSe)_{1,14}(NbSe_2)_2$ (from [9]). The different plots obtained in **b** and **c** were taken at temperatures going from 4,2 K to 2 K.</figcaption></figure>

## 3.5 Topological superconductivity in monolayer $NbSe_2$

In this section, we will show that some theoretical studies predict the existence of unconventional superconductivity and possibly topological superconductivity in monolayer transition metal dichalcogenides. The conclusion of section 3.3 was that $(LaSe)_{1,14}(NbSe_2)_2$ was behaving up to some extent as an electron-doped monolayer $NbSe_2$. Because of this, we think that $(LaSe)_{1,14}(NbSe_2)_2$ could possibly host unconventional superconductivity.

In ref. [97], it was shown that dichalcogenides monolayer $NbSe_2$ and $TaS_2$ should enter a nodal superconducting topological regime when submitted to in-plane magnetic fields higher than the Pauli paramagnetic limit field but smaller than the critical field $H_{c2}$. This nodal topological phase is a consequence of the strong Ising spin-orbit coupling present in these materials. As it was already detailed in section 1.3.3, the locking of the electrons' spins in the out of plane direction does not occur along $\Gamma M$ lines. The in-plane magnetic field first shifts the positions in reciprocal space of spin up and spin down bands along the $\Gamma M$ lines in a way illustrated in figure 3.24 **a**. Second, since the applied in-plane magnetic field is higher that the Pauli limit field, superconductivity is killed along those lines, and the gap closes at the green and yellow dots (referred to as "nodal points") positions of figure 3.24 **a**, leading to nocal superconductivity.

<footnote n="8">⁸In order to suppress superconductivity, the Zeeman effect of an external magnetic field has to be stronger than the effective spin-orbit field, which can be huge (around 100 T in $MoS_2$ [36, 96]).</footnote>
Wen-Yu He et al. then demonstrated that Majorana flat bands should link each couple of nodal points, as represented in figure 3.24 **b**. The topological nodal feature can further be understood by noticing that those Majorana zero-bias states only develop along specific edges oriented in $\Gamma M$ directions ($k_y$ direction in figure 3.24).

<figure><img src="image_49.png" /><figcaption>Figure 3.24: **a.** The in-plane magnetic field induces a shift in the Fermi surface between the spin up and spin down band along the $\Gamma M$ lines. **b.** Majorana zero modes developing between couples of nodal points (taken from [97]).</figcaption></figure>

In ref. [8], D. Shaffer et al. assumed the most general symmetry-authorized Hamiltonian describing pairing interactions in monolayer NbSe$_2$ (see figure 3.25). Similarly to ref. [97], they considered Ising spin-orbit coupling but they additionally investigated the effect of Rashba spin-orbit coupling, which is also likely not to be negligible in monolayer NbSe$_2$ (which has to be grown on a substrate) as well as in $(\text{LaSe})_{1.14}(\text{NbSe}_2)_2$. Interestingly, the first noticeable effect of Rashba spin-orbit coupling is to lift the degeneracy between spin up and spin down bands along the $\Gamma M$ direction. The nodal topological phase of ref. [97] however remains reachable by applying in-plane magnetic field in the $\Gamma K/\Gamma K'$ directions. The Fermi surface then looks like the one represented in figure 3.25.

<figure><img src="image_50.png" /><figcaption>Figure 3.25: Fermi surface of monolayer NbSe$_2$ in presence of Rashba SOC. The different pairing interactions are represented by dotted arrows with diffrent colors. (taken from [8])</figcaption></figure>

Their interaction Hamiltonian had the following form :
$$ H_{int} = \frac{g_1}{2} d_{\Gamma s}^{\dagger} d_{\Gamma s'}^{\dagger} d_{\Gamma s} d_{\Gamma s'} + \frac{g_2}{2} d_{Ks}^{\dagger} d_{-Ks'}^{\dagger} d_{Ks} d_{-Ks'} + \frac{g_3}{2} d_{Ks}^{\dagger} d_{-Ks'}^{\dagger} d_{-Ks} d_{Ks'} + \frac{g_4}{2} d_{\pm Ks}^{\dagger} d_{\mp Ks'}^{\dagger} d_{\Gamma s'} d_{\Gamma s} + \text{h.c.}, \quad (3.3) $$

For simplicity the momentum indices were suppressed. Here, $s, s'$ are spin indices and $d_{Xs}^{\dagger}$ ($d_{Xs}$) is the creation (annihilation) fermionic operator of an electron of spin $s$ in the $X$ band. The different interactions are schematically represented in figure 3.25. It would seem that a superconducting instability becomes possible provided that either $g_3$ or $g_4$ inter-pocket interactions overcome intra-pocket interactions $g_1$ or $g_2$. The article then exposes the topological phase diagrams of monolayer NbSe₂ as a function of the Rashba spin-orbit coupling and the in-plane magnetic field oriented along the $\Gamma K$ direction depending on which interaction between $g_3$ or $g_4$ is dominant.

Their conclusions are summarized in the phase diagrams shown in figure 3.26.

<figure><img src="image_51.png" /><figcaption>Figure 3.26: On the left is displayed the topological phase diagram of monolayer NbSe₂ as a function of Rashba SOC and $\Gamma K$ oriented magnetic field when the $g_4$ interaction is dominant. On the right is displayed the topological phase diagram of monolayer NbSe₂ as a function of Rashba SOC and $\Gamma K$ oriented magnetic field when the $g_3$ interaction is dominant. (taken from [8])</figcaption></figure>

When $g_4$ is dominant, at zero magnetic field and any value of Rashba spin-orbit coupling, the system is in a s-wave topologically trivial superconducting phase. Nevertheless, by increasing the value of the $\Gamma K$ oriented magnetic field, the system can enter a nodal topological regime similar to the one of ref. [97].

For now, we only focused on cases where in-plane magnetic fields are needed to turn 2D-like transition metal dichalcogenides into nodal topological phases. This approach was thus not feasible in our case since the experiments at 350 mK presented in the next section were performed on the M3 set-up presented in section 2.4, on which only **out-of-plane** magnetic field up to 7 T can be applied.

The case in which the $g_3$ interaction dominates is very interesting because it allows for topological phases to emerge without any magnetic field. At zero magnetic field and Rashba spin-orbit coupling, the system naturally is in a f-wave triplet phase. Although being topologically trivial, the associated superconducting state is nonetheless unconventional. It is also
important to emphasize on the fact that two gaps are expected, one around $\Gamma$ and one around $K/K'$. The $K/K'$ gap is found to be momentum independent<fnref n="9" /> while the $\Gamma$ gap goes as $\cos(3\theta)$ ($\theta$ is the angle made by the momentum vector and is defined in figure 3.25), and therefore closes for $\theta$ equal to any odd multiple of $\frac{\pi}{6}$. The nodal feature of the $\Gamma$ gap can be removed by including small s-wave correlations authorized by the symmetry which prevent it from going to zero.

When the value of the Rashba spin-orbit coupling becomes high enough, the system enters a $p \pm ip$ topologically non-trivial triplet superconducting state exhibiting gapless chiral edge modes reminiscent of the physics presented in section 1.4.3.

In ref. [98], Y. Hsu et al. also predicted the possible existence of topological phases in monolayer transition metal dichalcogenides. The study mainly focuses on compounds such as hole-doped monolayer $MoS_2$ [99] which do not present the hole bands around $\Gamma$, but most of the results can also be applied in the case of monolayer $NbSe_2$ since most of the arguments are symmetry-based and are a fortiori also valid for monolayer $NbSe_2$.

<figure><img src="image_52.png" /><figcaption>Figure 3.27: **a.** Inter-pocket pairing between $K$ and $K'$ hole-like pockets. **b.** Intra-pocket pairing within a $K$ ($K'$) pocket. **c.** and **d.** The absence of inversion symmetry allows for mixing between even and odd parity order parameters. The $C_{3v}$ symmetry of the material authorizes s-wave to mix with f-wave and ($p \pm ip$) to mix with ($d \mp id$) (taken from [98]).</figcaption></figure>

So far, the considered pairing interactions were only in favor of Cooper pairs with zero total momenta. One of the originalities of ref. [98] is the consideration of intra-pocket pairing within a $K$ ($K'$) pocket. Such a pairing is sketched in figure 3.27 **b**. The two electrons of the Cooper pair are at opposite locations in reciprocal space with respect to $K$ ($K'$) but not with respect to $\Gamma$ leading to non-zero center of mass momenta for the Cooper pairs. Such kind of pairing is called Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) and provokes spatial modulations either of the superconducting phase or of the superconducting gap amplitude across the sample [100]. In the case of monolayer $NbSe_2$, given the Ising splitting around $K/K'$ points, it also favors

<footnote n="9">⁹More precisely, its magnitude is momentum independent but the gap changes sign as going from a $K$ point to a $K'$ point.</footnote>
superconducting triplet correlations. However, it is important to understand that this intra-
pocket pairing becomes less and less probable as the trigonal warping of the *K* pocket becomes
more and more pronounced (illustrated in figure 3.28)

<figure><img src="image_53.png" /></figure>

Figure 3.28: Illustration of the fact that a strong trigonal warping is less favorable for FFLO pairing than a nearly circular pocket around $K/K'$. On the left, because of strong trigonal warping, the Cooper pair is formed by two electrons which do not have opposite momenta $q'_1$ and $q'_2$ with respect to $K'$. The pairing is thus considerably reduced (illustrated by the light grey dotted ellipse). On the contrary, on the right, no trigonal warping is present and the two electrons have opposite momenta $q_1$ and $q_2$ with respect to $K$. The pairing is, hence, much more favorable (illustrated by the full black ellipse) than is the trigonal warped case. In both cases, the Cooper pairs have non-zero center-of-mass momenta (FFLO pairing) as represented by the $q_{\Gamma K'} + Q$ and $q_{\Gamma K}$ vectors, respectively.

On the left of figure 3.28, is represented a pocket with a strong trigonal warping. In this case, the electrons of the Cooper pair do not have opposite momenta with respect to $K'$. This situation is thus less favorable than the weak trigonal warping case represented on the right for which the electrons of the Cooper pair do not have opposite momenta with respect to $K$. In both cases, the Cooper pair has a total momentum different from zero.

We recall that as monolayer NbSe₂ is electron-doped, the trigonal warping is less pronounced
around $K/K'$ points at Fermi level. This means that $(LaSe)_{1,14}(NbSe_2)_2$ is potentially a good
platform for FFLO superconducting pairing.

**3.6** $(LaSe)_{1,14}(NbSe_2)_2$ **in the superconducting phase**

**3.6.1 Measurement of the superconducting critical temperature**

Before performing STS measurements on our samples, it was mandatory to priorly check whether
they indeed were superconducting. In order to do so, we performed magnetometry experiments
using a SQUID device in the Mesures Physique à très Basses Températures (MPBT) platform
at Sorbonne university. The samples were 0-field cooled down to 1,8 K after what they were
submitted to a perpendicular magnetic field of 10 Oe. The value of the magnetic field was chosen
so that the diamagnetic response of the system was maximized. The experiments were carried
out on two samples of the same batch referred to as batch#2-sample3 and batch#2-sample5,
respectively. The obtained magnetic moment *vs* temperature curves are represented in figure 3.29 a.

<figure><img src="image_54.png" /><figcaption>Figure 3.29: **a.** Magnetic moment *vs* temperature dependence for two (LaSe)₁,₁₄(NbSe₂)₂ samples. The samples were 0-field cooled down and submitted to an external magnetic field of 10 Oe. **b.** Resistivity *vs* temperature transport measurements carried out by our collaborators in Slovakia.</figcaption></figure>

Batch#2-sample3 exhibited a superconducting critical temperature approximatively equal to 3.6 K whereas the batch#2-sample5 (which is the one on which the QPIs measurements presented in section 3.3 were done) exhibited a slightly higher superconducting critical temperature approximatively equal to 4 K. Complementary transport measurements were also carried out by our collaborators of the Center of Low Temperature Physics of Kosice and revealed a superconducting critical temperature of ~4.3 K for samples of the same batch as ours. Surprisingly, our combined results present critical temperatures inferior to *T*<sub>SC</sub> = 5.3 K evaluated
in $(LaSe)_{1,14}(NbSe_2)_2$ during the 1990's [89]. A re-investigation of the old data showed that such a $T_{SC}$ could be found in one specific part of an old sample, the remaining of the sample rather exhibiting superconducting critical temperatures comparable to ours, pointing out that the superconducting critical temperature is extremely sensitive to disorder and growth conditions. The magnetometry and transport measurements combined with the observed Ising superconductivity reported by the old data comforts the bulk superconducting nature of our $(LaSe)_{1,14}(NbSe_2)_2$ samples.

3.6.2 Measured differential conductance in the superconducting phase

<figure><img src="image_55.png" /><figcaption>Figure 3.30: a. Collection of $dI/dV$ spectra taken on several areas of two different samples. b. Comparison between a typical experimental $dI/dV$ spectrum (blue) and the expected $dI/dV$ spectrum in the case of a BCS superconductor presenting a superconducting critical temperature equal to 4 K.</figcaption></figure>
After this preliminary verification, the samples were prepared in the same way as shown in section 2.4.3 and were cooled down to 350 mK using $^3$He condensation process (see section 2.4). In figure 3.30 **a** is represented a collection of averaged $dI/dV$ spectra taken at many places of two samples of two different growth batches. The value of the superconducting gap does not change significantly with both location and sample but we do observe different filling of the gap depending on the approaching area. In-gap states fill the superconducting gap up to more than 50%. It is not rare to see gap filling comprised between 70% and 80%. We emphasize on the fact that such spectra were observed in all explored areas of two different samples (among which one clearly exhibited superconductivity in SQUID) of two different batches. On figure 3.30 **b** is represented (blue line) a typical $dI/dV$ spectrum obtained at the surface of $(LaSe)_{1.14}(NbSe_2)_2$ at 350 mK, and thus, well under the superconducting critical temperature. The expected $dI/dV$ spectra at 350 mK in the case of conventional superconductors of critical temperature equal to 4 K (black) is superimposed on figure 3.30 **b**. This $T_{SC}$ was chosen to match the one measured in SQUID. First, the measured gap is equal to $\Delta_{SC} = 0.28$ meV. This value is more than twice as small as the BCS expectation for our samples (0.6 meV for $T_{SC} = 4$ K). Second, at zero bias, the conductance does not fall to zero and a hard gap is never seen. One can notice that the experimental $dI/dV$ spectrum differs greatly from the one of a conventional BCS superconductor.

### 3.6.3 Exclusion of a proximity effect between the bulk and the surface

<figure><img src="image_56.png" /></figure>

Figure 3.31: Proximity effect between a normal surface and a superconducting layer with of superconducting gap equal to 0.6 meV. The proximity gap is imposed to be equal to 0.28 meV. Changing the thickness of the superconducting layer does not change much the proximity spectrum. This model does not explain our experimental features meaning that proximity effect is not at the origin of our observed gap.

One could first think that such behaviour is the result of a superconducting proximity effect between the bulk of the sample (demonstrated to be superconducting *via* SQUID and transport measurements) and the surface, should it act as a normal metal. Proximity effect can indeed change both the gap value and its shape and would be a natural explanation of what we observe.
This last hypothesis was removed by numerically considering a system with the following geometry : a superficial normal layer, of width $L_1$, on top of a superconducting layer with varying thickness $L_2$ (this is schematically illustrated in figure 3.31). By using Usadel formalism and by imposing a bulk gap of 0,6 meV<fnref n="10" /> and a proximity gap of 0,28 meV (our measured gap at the surface), we looked at the expected density of states at 350 mK for geometries in which $L_2 = L_1$, $L_2 = 2L_1$ and $L_2 = 10L_1$, respectively. The results are displayed in figure 3.31. The main conclusion of this short study is the fact that no matter the considered superconducting layer thickness, the experimental filled-V-shaped gap is not reproduced. This means that we must look for our measured gap's origin somewhere else.

### 3.6.4 Other signatures of superconductivity

Figure 3.32 presents other experimental results further attesting the superconducting nature of the samples. Figure 3.32 a represents the evolution of the $dI/dV$ spectra with temperature from 350 mK to 2,2 K clearly highlighting the disappearance of superconductivity with temperature.

The appearance of vortices in the presence of an external magnetic field is another signature of superconductivity which can be probed by performing grid spectroscopy. However, we never succeeded in observing vortices. Our guess is that the vortices are not easily pinned and are thus constantly moving, making their observation impossible. This absence of pinning of the vortices is probably due to the quasi-2D nature of $(LaSe)_{1,14}(NbSe_2)_2$. Here, the $NbSe_2$ planes of $(LaSe)_{1,14}(NbSe_2)_2$ are more decoupled than they are in bulk 2H-Nb$Se_2$ in which the vortices tend to align from one $NbSe_2$ plane to the next consequently stabilizing their position/pinning [101]. It does not mean that it is impossible to find signatures of vortices in the material. Figure 3.32 b represents the bottom of the gap as a function of increasing perpendicular magnetic field $H$. In order to obtain the curve we used the "lazy fisherman" technique [102]. It basically consists in continuously taking $I(V)$ spectra at a given position while the magnetic field increases. This method is called the "lazy fisherman" method because, sometimes, a moving vortex can flow under the tip's position and be "caught" just as a fish can be caught by a fishing rod fixed to the shore. It is in this way that we interpret the sudden filling of the gap observed in figure 3.32 b around 70 mT which is probably the signature of a moving vortex that was passing by.

<figure><img src="image_57.png" /><figcaption>Figure 3.32: a. Evolution of $dI/dV$ spectra with temperatures going from 350 mK to 2,2 K. b. Bottom of the gap as a function of perpendicular applied magnetic field.</figcaption></figure>

<footnote n="10">¹⁰Again supposing a superconducting critical temperature of 4 K.</footnote>
### 3.6.5 QPIs in the superconducting phase

In order to further investigate the potential unconventional nature of the order parameter, we performed quasiparticle interferences measurements but this time in the superconducting phase. The experiments, which are presented here, were carried out on two different samples of different batches and with different Pt/Ir tips. The QPIs obtained at 0, 2 meV for the first and the second samples are represented in figure 3.33 c and d, and their corresponding mean $dI/dV$ spectra in figure 3.33 e and f, respectively. The chosen energy (0, 2 meV) is inferior to the gap magnitude and yet, we can still attribute the observed QPIs patterns to scattering processes already encountered in the normal phase. The circle around $\Gamma$ and of radius $Q_1$, where $Q_1$ is the scattering wave-vector linking states of the $\Gamma$ pocket to itself, is still visible, and this, in both experiments. The $Q_4$ scattering process linking inner $K$ pockets to themselves is also observed but only for the first experiment, probably because the two tips did not favor the exact same tunnelling channels.

<figure><img src="image_58.png" /><figcaption>Figure 3.33: a. Fermi surface of electron-doped monolayer NbSe$_2$. Scattering processes between $\Gamma$ pocket to $\Gamma$ pocket are labelled by $Q_1$ wave vector. Scattering processes linking inner $K/K'$ pocket to itself are labelled by $Q_4$ wave vector. b. Expected QPIs patterns at Fermi energy for electron-doped monolayer NbSe$_2$. c. QPIs obtained at 0, 2 meV for the first sample. d. QPIs obtained at 0, 2 meV for the second sample. e. Averaged $dI/dV$ spectrum of the first sample. f. Averaged $dI/dV$ spectrum of the second sample.</figcaption></figure>

One can notice that the averaged $dI/dV$ spectrum of the second sample (figure 3.33 f) has a different shape than the one of the first sample and the ones of figure 3.31 b. We recall that the differential conductance $dI/dV$ can be assimilated to the local density of states of the sample only if the density of states of the tip in constant. It is important to prove that the measured $dI/dV$ spectra of the second sample (typically the one of figure 3.33 f), reflect the superconducting nature of the system and not something else.

Before performing spectroscopy in the second sample, other measurements were done using the same tip and in particular, the one from which the spectra displayed in figure 3.34 a were extracted. This other measurement was also taken at 350 mK. At the beginning of this other spectroscopic grid, the spectra had the shape of the red $dI/dV$ curve characteristic of the filled superconducting gaps observed many times. By the end of the map, the tip might have
interacted with the surface and changed to give the blue $dI/dV$ curve of figure 3.34 a.

This change can simply be illustrated with the help of figure 3.34 b in which the dotted red curve represents the density of states of the superconducting sample and the dotted green curve represents an hypothetical density of state for the tip. Here, the tip presents a small depletion of density of states around Fermi level. By convoluting the density of states of the sample and this hypothetical tip's density of states one obtains (at 350 mK) the blue curve, which reproduces rather satisfyingly the experimental data. One can clearly see elbows at the gap positions in energy in both numerical calculation and experiment. The shape of the density of states within $[-\Delta_{SC}, +\Delta_{SC}]$ is almost not changed meaning that in this energy range our data can really be seen as a collection of local density of state maps. Moreover, this additional tip background is not likely to either add or kill any information because it does not depend on space contrary to QPIs patterns. Deconvoluting the experimental data by the supposed density of states of the tip (which was not done here) however remains the best and cleanest way of treating the data. One last argument in favor of the validity of the second experiment's results is the fact that spectroscopy performed in the first sample (for which the tip had no apparent background) leads to the same conclusions. Exploiting the second sample measurement has two purposes : first it confirms the reproducibility of the results and second it is more spatially resolved than the first experiment, leading to also better resolved QPIs patterns.

<figure><img src="image_59.png" /><figcaption>Figure 3.34: **a.** Spectroscopic grid experiment performed with the same tip as in the second sample. We see that during data acquisition, the typical $dI/dV$ spectra changed from red to blue curve. **b.** Interpretation of the shape of the $dI/dV$ spectra in the second sample. The measured signal (blue) is related to the convolution between the density of states of the sample (red) and a depleted tip’s density of states (green).</figcaption></figure>

We mentioned in the previous paragraph that the results on first and second samples are consistent with one another. We will now present those results.

In figure 3.35 c, d and f are represented the Fourier transforms of the local density of states maps obtained at -0.25 meV, 0 meV and +0.25 meV, respectively. What is striking is the great similarity between the three QPIs patterns. This observation can be extended for any energies within $[-\Delta_{SC}, +\Delta_{SC}]$. The only recognizable QPIs feature is the one resulting from $\Gamma$ to $\Gamma$ scattering (see figure 3.33).
<figure><img src="image_60.png" /><figcaption>Figure 3.35: **a.** Averaged $dI/dV$ spectrum of the second experiment. **b.** Fermi level $dI/dV$ map. **c.** Fourier transform of the $dI/dV$ map at −0, 25 meV. **d.** Fourier transform of the $dI/dV$ map at Fermi level. **e.** Fourier transform of the $dI/dV$ map at +0, 25 meV.</figcaption></figure>

Two main observations can be extracted from this QPIs study in the superconducting phase. Firstly, the QPIs patterns are not different from the ones observed in the normal phase. Secondly, the QPIs patterns are non-dispersive as they do not depend on the considered energy.

In ref. [103], Brian B. Zhou et al. performed Fourier transform STM on superconducting heavy fermion compound CeCoIn$_5$. They exhibited the d-wave nature of the superconducting order parameter by observing **dispersive QPIs patterns** at the energy scales of superconductivity. Such a dispersion is actually expected whenever the order parameter presents nodes, as it was the case there. If one is dealing with a nodal superconductor with gap magnitude $\Delta_{nodal}$, then, partly hole and partly electron low energy excitations appear at energies inferior in absolute value to $\Delta_{nodal}$. Scattering between those states, which are dispersive, can eventually lead to QPIs in presence of disorder.

We think that the absence of dispersion in our QPIs calls for a nodeless order parameter. The states filling the superconducting gap should then be seen as a superposition of many Shiba-like states<fnref n="11" />. Those Shiba states being a mixture of electrons and holes are then likely to scatter on the misfit potential thus leading to QPIs patterns reflecting the normal phase energy contours rather than the symmetry of the order parameter. In ref. [105], Gerbold C. Menard et al. deposited Co magnetic impurities on a conventional superconducting monolayer of Pb, leading to the apparition of in-gap Shiba-states. Around those magnetic impurities, QPIs patterns mirroring the Fermi surface of the Pb monolayer in the normal phase were observed. This last example shows that the QPIs obtained in the superconducting phase can sometimes reflect the Fermi surface in the **normal phase**.

This does not mean that no information can be extracted through QPIs performed on a nodeless superconductor. In our case, the sensitivity to non-magnetic disorder definitely implies

<footnote n="11">¹¹Yu-Shiba-Rusinov states are electron-hole excitations which appear when one introduces magnetic impurities in a superconductor (see ref. [104]). In STS, they manifest as pairs of in-gap states with opposite energies. Similar states can appear in unconventional superconductors in presence of **non-magnetic** impurities.</footnote>
unconventional superconductivity. Also, if the system is in a topologically non-trivial phase, one should see topological edge states at the interface with a region with different topology.

<figure><img src="image_61.png" /><figcaption>Figure 3.36: **a.** Trilayer Pb islands grown on top of $(LaSe)_{1,14}(NbSe_2)_2$. **b.** Zoomed topography map taken far from the islands. The visible balls are Pb atoms. After counting, there is a 5% coverage. **c.** Averaged $dI/dV$ spectra obtained in an area similar to **b**.</figcaption></figure>

Atomic steps naturally define an interface between the material and the vacuum. If the system is already topological (recalling that vacuum is equivalent to a topologically trivial medium) then one should observe edge states along such steps. Unfortunately, after countless tries we never were able to find any steps small enough not to damage our tip during scanning regulation. We thus opted for another solution: depositing enough Pb to grow conventional superconducting Pb islands which could potentially define better interfaces with $(LaSe)_{1,14}(NbSe_2)_2^{12}$.

After our first Pb deposition, we grew triangular-shaped trilayer Pb islands of typically 30 nm lateral extension in a way explained in subsection 2.4.3 (see figure 3.36 **a**). The obtained Pb islands were however too small to develop superconductivity [106]. Away from the islands, in areas like the one encircled in green on figure 3.36 **a**, one can observe Pb atoms distributed randomly across the surface (see figure 3.36 **b**). After counting, we evaluated a 5% Pb coverage in such areas, meaning that there is 1 Pb atom for every 20 Nb atoms of the superficial $NbSe_2$ monolayer.

On figure 3.36 **c**, is represented the averaged $dI/dV$ spectrum in an area similar to the one of figure 3.36 **b**. This spectrum is completely flat and does not present any of the features encountered in previous experiments. This means that a 5% coverage of non magnetic Pb impurities is enough to kill superconductivity which is, according to us, a clear signature of the unconventional nature of superconductivity in $(LaSe)_{1,14}(NbSe_2)_2$. Nevertheless, one needs to be cautious, for this experiment was only performed once on few areas of one same sample (which nonetheless exhibited clear superconducting signatures before Pb evaporation). Moreover, some areas of the samples presented significant filling of the gap up to more than 90%. It is not impossible that our tip landed in such areas. To further study the effect of non-magnetic disorder on superconductivity we are planning on performing spectroscopy measurements on

<footnote n="12">¹²One way of seeing the unconventional nature of the superconducting order parameter (gap sign changes) is to use well-controlled defects.</footnote>
samples on which disorder was introduced by electron irradiation.

### 3.6.6 Interpretation of the small value of the superconducting gap in terms of tunnelling selectivity

For now, we mostly commented on the filling of the superconducting gap but not much on its small value $\Delta_{SC} = 0, 28$ meV. We recall that if $(LaSe)_{1,14}(NbSe_2)_2$ was a BCS superconductor, its superconducting gap should be approximatively equal to 0, 6 meV. The potential unconventional nature of the superconductivity in $(LaSe)_{1,14}(NbSe_2)_2$ is not sufficient to fully understand this difference. In the following, we propose an interpretation able to explain the smallness of the measured gap $\Delta_{SC}$.

With all this in mind, we can now give a possible interpretation for the observed superconducting gap of 0, 28 meV in $(LaSe)_{1,14}(NbSe_2)_2$. Thanks to the theoretical predictions of ref. [8], one can expect to have two unconventional order parameters: one developing around $\Gamma$ and the other one around $K/K'$. Since the electrons of the $K/K'$ pockets possess higher momenta $\vec{k}_{\parallel}$ compared to the ones of the $\Gamma$ pocket, it is possible that our tip-sample tunnelling geometry was mostly sensitive to the $\Gamma$ gap and only to a limited degree to the $K/K'$ gap (see subsection 2.2.2). The fact that such tunnelling selectivity was also observed in parent compound $2H-NbSe_2$ further supports our hypothesis [107]. Nevertheless, this assertion needs to be proven experimentally. One way of verifying this hypothesis would be to carry out spectroscopy measurements with a different tip-sample geometry. For example, one can imagine to approach a Au surface with a $(LaSe)_{1,14}(NbSe_2)_2$ sample glued to a tip holder in a configuration such that the $NbSe_2$ layers are perpendicular to the Au surface. In this configuration the tunnelling selectivity might greatly differ and it could in principle be possible to look for previously "invisible" spectroscopic signatures. We are currently planning on performing this experiment.

## 3.7 Example of another misfit compound

This short subsection is meant to illustrate the fact that it is possible to exploit other, but similar, heterostructures to study the physics of few layers TMDs. Here, we consider the example of $(LaSe)_{1,14}(NbSe_2)$. This compound in another misfit superconducting compound of the family of $(LaSe)_{1,14}(NbSe_2)_2$. Both materials are very similar. $(LaSe)_{1,14}(NbSe_2)$ consists in alternation of LaSe bilayers and of $NbSe_2$ **monolayers**. The two kind of layers are also commensurate in one direction and incommensurate in the perpendicular one with an approximate commensurate ratio also of $7/4$ (just as $(LaSe)_{1,14}(NbSe_2)_2$). In the end, the main structural difference between the two compounds is the number of $NbSe_2$ layers sandwiched between LaSe bilayers.

Since in $(LaSe)_{1,14}(NbSe_2)$, only one monolayer of $NbSe_2$ is sandwiched between LaSe bilayers, $NbSe_2$ is potentially twice as electron doped as $(LaSe)_{1,14}(NbSe_2)_2$. Taking into account the tendency of $(LaSe)_{1,14}(NbSe_2)_2$ to behave as e-doped monolayer $NbSe_2$, one can easily imagine than $(LaSe)_{1,14}(NbSe_2)$ will behave as monolayer $NbSe_2$ but with a different doping value. When cleaving $(LaSe)_{1,14}(NbSe_2)$, the top cleaved surface is either a LaSe terminated surface or a $NbSe_2$ terminated surface with a $1/2$ probability. This is due to the fact that no adjacent layers are van der Waals bonded.

In figure 3.37 a, is displayed a STM image (taken at 2 mV) of $(LaSe)_{1,14}(NbSe_2)$ surface. The modulus of the Fourier transform of the STM map of figure 3.37 a is represented in figure 3.37 b. Those images were obtained by collaborators of the Center of Low Temperature Physics of Kosice, Slovakia. The topography image presents $NbSe_2$ terminated surface. A $2 \times 2$ charge modulation, just as the one observed at the surface of $(LaSe)_{1,14}(NbSe_2)_2$, can be seen. This later one is also seen in the Fourier transform, lying at half the Bragg peaks. What is striking in this Fourier transform image is the similarities it shares with its $(LaSe)_{1,14}(NbSe_2)_2$ homologous. The incommensurability peaks and replica are all present as well as the "Q1"
generated QPIs circle around Γ sitting at the exact same position in reciprocal space as the one seen in (LaSe)₁,₁₄(NbSe₂)₂. This might seem a bit unsettling given the arguments on the doping that were just exposed, but one has to remember that this NbSe₂ terminated surface does not behave as the bulk. For bulk (LaSe)₁,₁₄(NbSe₂), a monolayer NbSe₂ receives electrons from its two adjacent LaSe layers. For the surface, **only one** LaSe layer is likely to give electrons to the superficial NbSe₂ layer. The surface of (LaSe)₁,₁₄(NbSe₂)₂ thus behaves as the bulk of (LaSe)₁,₁₄(NbSe₂)₂. Luckily for us, the surface of (LaSe)₁,₁₄(NbSe₂)₂ shares the same doping level as bulk (LaSe)₁,₁₄(NbSe₂)₂. The critical superconducting temperatures and, *a fortiori*, the superconducting properties of (LaSe)₁,₁₄(NbSe₂) and (LaSe)₁,₁₄(NbSe₂)₂ differ<fnref n="13" />. Studying (LaSe)₁,₁₄(NbSe₂)'s surface potentially means studying monolayer NbSe₂ in the same doping regime as in (LaSe)₁,₁₄(NbSe₂)₂ but with different superconducting properties.

<figure><img src="image_62.png" /><figcaption>Figure 3.37: **a.** Topography STM image of parent misfit compound (LaSe)₁,₁₄(NbSe₂). **b.** Modulus of the Fourier transform of **a**.</figcaption></figure>

## 3.8 Conclusion

In this chapter, we have seen (by mean of QPIs experiments) that the band structure of (LaSe)₁,₁₄(NbSe₂)₂ is very similar to the one of monolayer NbSe₂, but rigidly shifted of approximately 0, 3 eV towards lower energies. In particular, the comparison between our STS measurements and DFT calculations showed the presence of a strong Ising spin-orbit coupling in (LaSe)₁,₁₄(NbSe₂)₂. Our work opens the possibility of using bulk misfit heterostructures (such as (LaSe)₁,₁₄(NbSe₂)₂) to investigate the physics of few layers transition metal dichalcogenides at electron-doping levels never achieved in the literature. More specifically, one can imagine to either change the rock-salt layer (by replacing LaSe with PbSe for example) or the number of NbSe₂ layers to effectively change the doping, and consequently, the electronic properties of the misfit heterostructure.

In our doping regime, we observed different electronic features than in monolayer NbSe₂. First, we highlighted the presence of a 2 × 2 charge modulation in place of the 3 × 3 CDW observed in both bulk and monolayer NbSe₂. DFT calculations exhibited the tendency of

<footnote n="13">¹³In ref. [9], it is shown that the superconducting critical temperature of (LaSe)₁,₁₄(NbSe₂) is equal to 2, 2 K. We recall that (LaSe)₁,₁₄(NbSe₂)₂ has a superconducting critical temperature somewhere between 4 K and 5, 3 K.</footnote>
monolayer NbSe₂ to undergo a electron-doping-driven phase transition from a 3 × 3 charge order to a 2 × 2 charge order. Nevertheless, such a transition should occur at lower electron doping realizations than ours, which either means that the 2 × 2 charge modulation has a different origin (possibly correlations which were not taken into account in the calculations) or that the charge transfer (and *a fortiori* the doping) locally varies because of invisible-to-our-probe La vacancies, thus allowing the system to, locally, enter a 2 × 2 charge order. Second, we observed signatures of nodeless unconventional superconductivity, in qualitative agreement with theoretical works which predict triplet superconducting order parameters (p-wave or f-wave) and a possible route towards topological superconductivity. In order to confirm the unconventional nature of the order parameter we plan on performing several additional experiments such as specific heat measurements, quasiparticle interferences measurements after introduction of defects in a controlled way by mean of Ar sputtering or electron irradiation and finally, performing spectroscopy in another tip-sample configuration to exhibit eventual tunnelling selectivity effects.
# Chapter 4

## Inhomogeneous insulating to pseudo-metallic transition in a doped Mott insulator

The Iridate compound $Sr_2IrO_4$ was originally studied because it shares a common structure with the cuprate $La_2CuO_4$. This material was thus likely to provide new physical insight as to the intriguing nature of the pseudo-gap phase and the origin of superconductivity in high-$T_c$ superconductors [108, 109]. However, as a spin-orbit induced Mott insulator, $Sr_2IrO_4$ also presents a different yet equally interesting kind of physics. Moreover, recent theoretical works suggest that doping should turn the material into a topological superconductor.

In this chapter, we present a STM/STS study performed on electron-doped $Sr_2IrO_4$. We first show that, as the doping increases, the system undergoes an inhomogeneous phase transition from a Mott insulating phase to a pseudo-metallic one. Second, we also exhibit the appearance of orientational order in the organization of the pseudo-metallic areas. Finally, we present spectroscopy performed on pure $Sr_2IrO_4$. This study is the only one, in our knowledge, which combines measurements in both pristine and doped compound. As for the question of topological superconductivity, further studies are yet to be done.

The first section presents one of our motivations, namely the theoretical prediction of topological superconductivity in doped $Sr_2IrO_4$. The second section consists in a short introduction to Mott physics. The third section explains the spin-orbit origin of the Mott insulating behaviour of $Sr_2IrO_4$. Sections 4, 5 and 6 are dedicated to the presentation and discussion of the STM/STS results obtained during the thesis.

### 4.1 Predictions of topological superconductivity in doped $Sr_2IrO_4$

The crystal structure of $Sr_2IrO_4$ is represented in figure 4.1 and is close to being a perfect perovskite layered structure. The material is made out of alternations of $IrO_2$ layers and $SrO$ bilayers. All Ir atoms lie at the centers of oxygen octahedra. An $IrO_2$ plane of a perfect perovskite structure is represented in figure 4.2 a. Nevertheless, the structure of $Sr_2IrO_4$ is slightly deviated from a perfect perovskite structure and rather looks like the one of figure 4.2 b. In the actual compound, the oxygen octahedra are rotated of $\theta \simeq 12^\circ$, resulting in an expanded unit cell [110].

One of the consequences of this tilting is that all the O atoms within an $IrO_2$ plane are no longer inversion centers for the structure, as they were in the perfect perovskite scenario. This implies that the electron hopping between two neighbouring Ir atoms becomes spin-dependent, thus making the hopping Hamiltonian very similar to a Rashba spin-orbit coupling Hamiltonian. If $Sr_2IrO_4$ can be made superconducting, then, according to what has already been said in
section 1.3, the superconducting order parameter necessarily mixes both spin-singlet and spin-
triplet contributions. It was predicted that Sr₂IrO₄ could be turned into a superconductor by
mean of electron or hole doping [108, 111]. In 2015, Y. J. Yan *et al.* performed a STM/STS
study in which they deposited potassium on top of Sr₂IrO₄, consequently electron-doping it,
and observed what was interpreted as a superconducting V-shaped low-energy gap [112]. In
2017, it was shown that hole-doped Sr₂IrO₄ should theoretically enter a topological regime [10].

<figure><img src="image_63.png" /><figcaption>Figure 4.1: Crystal structure of Sr₂IrO₄.</figcaption></figure>

<figure><img src="image_64.png" /><figcaption>Figure 4.2: **a.** Top view of an IrO<sub>2</sub> plane if the structure was perfectly perovskite-like. **b.** Top view of an IrO<sub>2</sub> plane of Sr<sub>2</sub>IrO<sub>4</sub>. Contrary to **a**, the oxygen atoms are no longer inversion centers.</figcaption></figure>
## 4.2 Mott insulators

Band theory was developed in the 1930s and describes the states of electrons in solid materials. In the same way that there are a collection of accessible electronic levels in an individual atom, which can be either empty or occupied, band theory predicts that not all energies are available for the electrons in a solid but only energies within given ranges. When atoms are brought together in a periodic fashion in order to form a solid, the atomic orbitals will overlap and give rise to a set of states really close from one another. Those sets are called “bands”. The unavailable energy ranges separating two consecutive bands are called “forbidden bands”. In metals, the position of the chemical potential lies within a band, such that low energy excitations are possible and a charge current is likely to appear, contrary to insulators, where it lies within a forbidden band.

Within the context of band theory, only single electronic states are considered because the interaction between electrons is neglected<fnref n="1" />. This approach was very fruitful at the time since it allowed to describe the electronic properties of many materials. Nonetheless, some compounds presenting intriguing insulating behavior, when expected to be metals within the framework of band theory, were discovered. In order to interpret such physics, one needs to take into account correlations, or so to say, Coulomb interaction between electrons in this specific case.

A famous example of correlated system is the Mott insulator. In this section, we will present a short introduction to Mott Physics through the so-called Hubbard model. The calculations performed in the section are inspired by ref. [114].

Let us consider atoms sitting on an infinite periodic lattice with one electron per site and an inter-atomic distance $a$. For simplicity, all equations will be derived on a square lattice. The Hubbard model simply consists in considering both hopping to first neighbor and on-site Coulomb repulsion. The Hubbard Hamiltonian is the following :

$$H = H_{kin} + H_{int} = - \sum_{<i,j>,\sigma} (t_{ij}c_{i,\sigma}^{\dagger}c_{j,\sigma} + \text{h.c.}) + U \sum_{i} n_{i,\uparrow}n_{i,\downarrow}, \quad (4.1)$$

where $c_\alpha^\dagger$ is the electronic creation operator on the $\alpha$ site and $c_\beta$ is the electronic annihilation operator on the $\beta$ site. As fermionic operators, one has the anti-commutation relations $\{c_\alpha^\dagger, c_\beta^\dagger\} = \{c_\alpha, c_\beta\} = 0$ and $\{c_\alpha^\dagger, c_\beta\} = \delta_{\alpha\beta}$. The $t_{ij}$ parameter is the hopping amplitude from site $i$ to $j$. For simplicity, we will take $t_{ij}$ constant and equal to $t$. $U$ represents the on-site Coulomb repulsion, thus $U > 0$. The term $n_{\alpha,\sigma} = c_{\alpha,\sigma}^\dagger c_{\alpha,\sigma}$ is the number operator giving the number of electrons of spin $\sigma$ on site $\alpha$.

The first term is the kinetic part of the Hamiltonian. Supposing $U = 0$, it is possible to diagonalize $H_{kin}$ in k-space. Then one can compute the following dispersion : $\epsilon(\vec{k}) = -2t(\cos(k_x a) + \cos(k_y a))$. Because there is only one electron per site, the band is half-filled and the Hubbard Hamiltonian at $U = 0$ depicts a metallic behavior.

Now, we will turn on a finite Coulomb repulsion $U \neq 0$. An analytic solution of the Hubbard model on the square lattice is still not known such that one should get physical insight through simpler considerations. In the following, we will present one way of intuiting how the Hubbard model allows the opening of a gap when band theory predicts a metallic behavior.

For simplicity, let us first focus on the two-site Hubbard Hamiltonian (the sites are called respectively 1 and 2) :

$$H_{2 \text{ sites}} = -t(c_{1\uparrow}^{\dagger}c_{2\uparrow} + c_{2\uparrow}^{\dagger}c_{1\uparrow} + c_{1\downarrow}^{\dagger}c_{2\downarrow} + c_{2\downarrow}^{\dagger}c_{1\downarrow}) + U(n_{1\uparrow}n_{1\downarrow} + n_{2\uparrow}n_{2\downarrow}), \quad (4.2)$$

In the Foch basis $|n_{1\uparrow}, n_{1\downarrow}, n_{2\uparrow}, n_{2\downarrow}\rangle$, because both spin and number of particles are conserved, one can consider several state subspaces completely independent from each other. The first space corresponds to the case where both sites 1 and 2 are empty and is only formed by $|0, 0, 0, 0\rangle$.

<footnote n="1">¹The Fermi liquid theory justifies this approximation [113].</footnote>
The energy of this state is equal to 0. The second subspace is composed of one particle states of spin up : $|1,0,0,0\rangle$ and $|0,0,1,0\rangle$. The expression of $H_2$ sites on the $(|1,0,0,0\rangle, |0,0,1,0\rangle)$ basis for this subspace is $\begin{pmatrix} 0 & -t \\ -t & 0 \end{pmatrix}$, such that the eigenvalues of the associated eigenstates are equal to $\pm t$, signifying that electrons can freely hop from one site to the other. The third subspace is composed of one particle states of spin down. Because of the symmetry of the model with respect to spin polarization, the associated eigenvalues are also equal to $\pm t$. The fourth (fifth) subspace includes two-particle states with spin up (down) electrons and is composed of $|1,0,1,0\rangle$ ($|0,1,0,1\rangle$). The eigenvalues associated to both fourth and fifth subspaces are equal to $0^2$. The sixth subspace contains two-particle states where both electrons have opposite spins. It includes $|1,1,0,0\rangle$, $|0,0,1,1\rangle$, $|1,0,0,1\rangle$, and $|0,1,1,0\rangle$. The expression of the two-site Hubbard

Hamiltonian on the $(|1,1,0,0\rangle, |0,0,1,1\rangle, |1,0,0,1\rangle, |0,1,1,0\rangle)$ basis is $\begin{pmatrix} U & 0 & -t & -t \\ 0 & U & -t & -t \\ -t & -t & 0 & 0 \\ -t & -t & 0 & 0 \end{pmatrix}$.

After calculation one finds the following four eigenvalues: 0, U and $\frac{U}{2} \pm \sqrt{4t^2 + \frac{U^2}{4}}$. The seventh (eighth) subspace touches the three-particle states $|1,1,1,0\rangle$ and $|1,0,1,1\rangle$ ($|1,1,0,1\rangle$ and $|0,1,1,1\rangle$). On the $(|1,1,1,0\rangle, |1,0,1,1\rangle)$ or the $(|1,1,0,1\rangle, |0,1,1,1\rangle)$ basis, the expression of $H_2$ sites is $\begin{pmatrix} U & -t \\ -t & U \end{pmatrix}$ such that the eigenenergies are equal to $U \pm t$ for both seventh and eighth subspaces. The ninth and final subspace is the four-particle state $|1,1,1,1\rangle$ when all sites are doubly occupied. The eigenvalue of this state is equal to $2U$.

Let us not forget that we were originally interested in a half-filled band for the full Hubbard model. In the two-site Hubbard model, a maximum of four electrons can occupy the available states. Thus, in order to be the closest to the physical situation of interest, we will only focus on fourth, fifth and sixth subspaces which are exclusively composed of two-particle states. We recall

that the eigenvalues of this new subspace of dimension 6 can be equal to 0, U or $\frac{U}{2} \pm \sqrt{4t^2 + \frac{U^2}{4}}$. In the limit $U \gg t$, a set of four states at 0 energy is separated by U from an higher-energy set of two states<fnref n="3" />. One qualitatively understands that by going from the two-site system to the infinite lattice while staying in the large-U limit, the two sets of states will become two bands separated by a gap roughly equal to U. The lower-energy collection of states is called the “lower Hubbard band” (LHB) whereas the higher-energy collection of states is called the “upper Hubbard band” (UHB).

Let us imagine a lattice with one electron per site and antiferromagnetic ordering such that any electron with a given spin polarity is surrounded by nearest neighbor electrons with opposite spin polarity. If $U = 0$, then an electron can easily hop to its neighboring site, such that a charge current can flow : the system is metallic. Now, if U is turned on, it will be energetically unfavorable for an electron to jump to its nearby site unless its kinetic energy (represented by $t$) is sufficiently large to overcome Coulomb repulsion. If correlations are high enough, then the electrons are somehow bound to their sites, no electrical current is possible : the system becomes insulating. We can see that the ratio $\frac{U}{t}$ is at play in deciding the metallic or insulating nature of the materials. It will not be derived here but one can prove that the LHB of the two-site Hubbard model has exactly the same eigenspectrum as the two-site Heisenberg

<footnote n="2">²Because of Pauli's exclusion principle, no hopping is possible, and thus, the on-site Coulomb interaction does not contribute.</footnote>

<footnote n="3">³It is important to keep in mind that those states are not single electronic states but many particles states. One cannot think in terms of single electron occupations of two out of the four lower energy states, and thus that two degenerate states are available. The excitations of the system are also many particles excitations and no longer single quasiparticle excitations.</footnote>
Hamiltonian $H_{Heisenberg} = J\vec{S}_1 \cdot \vec{S}_2$ with an antiferromagnetic coupling $J = \frac{4t^2}{U}$ and $\vec{S}_i$ a 1/2 spin on site i (this is valid only when $U \gg t$ [29]). This suggests that a Mott insulator described by the Hubbard model possesses an antiferromagnetic order. Given this last statement one can now work out a hand-waving explanation of how insulating behavior arises in such materials.

It is possible to estimate a value of the ratio $\frac{U}{t}$ at which the metal-insulator transition (MIT) occurs by estimating the energy associated to the two ground states of the system in the two extremal limits $U = 0$ and $U \to +\infty$ of the model which are respectively the Fermi sea $|FS\rangle^4$ and the localized one-particle per atomic site state $|I\rangle = \prod_{i=1}^N c_{i,\sigma_i}^\dagger |0\rangle$, where $N$ is the total number of sites and $\sigma_i$ the spin of the electron on site $i$. The energy of the Fermi sea ground state is:

$$E_{FS} = \langle FS | H | FS \rangle = \langle FS | H_{kin} | FS \rangle + \langle FS | H_{int} | FS \rangle, \quad (4.3)$$

Considering a half-filled band of width $2W$, with the position of the Fermi level at 0 energy, one has the following expression for the kinetic part of the Fermi sea ground state energy:

$$\langle FS | H_{kin} | FS \rangle = \int_{-W}^{0} \rho(\varepsilon) d\varepsilon, \quad (4.4)$$

where $\varepsilon$ is the energy and $\rho(\varepsilon)$ the density of states. The spin degeneracy is taken into account in $\rho(\varepsilon)$. If one now assumes a constant density of states $\rho = \frac{2N}{2W}$ one gets:

$$\frac{\langle FS | H_{kin} | FS \rangle}{N} = -\frac{W}{2}, \quad (4.5)$$

The interacting part of the Fermi sea ground state energy is:

$$\langle FS | H_{int} | FS \rangle = \langle FS | U \sum_{i} n_{i,\uparrow} n_{i,\downarrow} | FS \rangle = U \sum_{i} \langle FS | n_{i,\uparrow} n_{i,\downarrow} | FS \rangle, \quad (4.6)$$

$$\langle FS | H_{int} | FS \rangle = U \sum_{i} \langle FS | n_{i,\uparrow} | FS \rangle \langle FS | n_{i,\downarrow} | FS \rangle = U \sum_{i} \frac{1}{2} \times \frac{1}{2} = \frac{UN}{4}, \quad (4.7)$$

$$\frac{\langle FS | H_{int} | FS \rangle}{N} = \frac{U}{4}, \quad (4.8)$$

The total energy of the Fermi sea brought back to one site is:

$$\frac{E_{FS}}{N} = -\frac{W}{2} + \frac{U}{4}, \quad (4.9)$$

We can also compute the energy of the ground state for infinite $U$ limit:

$$E_I = \langle I | H | I \rangle = \langle I | H_{kin} | I \rangle + \langle I | H_{int} | I \rangle, \quad (4.10)$$

Because in this case an electron is bounded to its atomic site, the kinetic part of $E_I$ is equal to 0. This can easily be computed given the expression of $H_{kin}$. Moreover, by definition of $|I\rangle$, no double occupancy is possible such that $\langle I | H_{int} | I \rangle = 0$. In the end:

$$\frac{E_I}{N} = 0, \quad (4.11)$$

One can notice that if $U < 2W$ then the delocalized ground state $|FS\rangle$ is more favorable than $|I\rangle$. This observation is consistent with the fact that when the kinetic energy of the electrons is big enough, they tend to delocalize such that the system becomes metallic. The bandwidth $2W$ is indeed directly proportional to the transfer integral $t$. The approach which was just exposed

<footnote n="4">⁴In this picture, the electron are delocalized and labelled by crystalline momenta.</footnote>
allows to highlight the existence of a critical value $U \approx 2W$ (although very dependent on the band structure) of the on-site Coulomb repulsion above which one switches from metallic to an insulating state.

<figure><img src="image_65.png" /><figcaption>Figure 4.3: a. Cartoon of a band insulator. Every site is doubly occupied and no hopping is possible due to Pauli exclusion principle. b. Cartoon of a metal in band theory. Every site is singly occupied and electrons can hop from one site to the next potentially carrying current. c. Cartoon of a Mott insulator. Every site is singly occupied but the strong on-site Coulomb repulsion prevents hopping. d. Cartoon of a doped Mott insulator. Here it is electron doped. The Coulomb energy $U$ has already been paid: a current can flow.</figcaption></figure>

So far, we only discussed Mott Physics at half-filling. It may be interesting to look at what happens when the chemical potential lies above or under half-filling. Experimentally, a way of accessing such a situation is to dope either with electrons or hole a Mott insulator. If the undoped material is insulating, which implies $U > 2W$, then a ground state with one electron localized on each site is more likely to describe faithfully the system. If now the system is electron-doped, then a few sites will be doubly occupied, which means that some pairs of electrons will whatever happens be subject to Coulomb repulsion. For electrons sitting on a doubly occupied sites, hopping to singly-occupied nearest neighbor sites won't cost any additional energy such that delocalization becomes favorable. The reasoning is similar in the case of hole-doping where electrons can hop on many empty sites causing again delocalization. This way, electron or hole doping will reduce the size of the Mott gap, closing it for large enough dopant concentrations eventually turning the system into a metal. Moreover, a quasiparticle peak at Fermi level is expected to appear [115]. Through this very naive picture, one understands that introducing doping into a Mott insulator is a way of tuning the $U/t$ ratio. For a given
Mott material, the values of both *U* and *t* can often be extracted by comparing the data with theory, however, the Hubbard model is most of the time insufficient for catching all the physics. In the next part, we will see an example of compound in which the spin-orbit interaction has to be taken into account in order to explain the observed Mott features.

## 4.3 Spin-orbit induced Mott insulator $Sr_2IrO_4$

The Mott insulator behavior of 5d Transition Metal Oxide iridate compound $Sr_2IrO_4$ cannot be explained with the simple model presented previously. Contrary to localized 3d electrons which can lead to Mott physics [116], 5d electrons are more spatially extended such that the overlap of neighboring orbitals should increase, and thus, *t* also increases. Because of this last observation, iridium oxides were expected to be metallic, and yet, many were actually found to be insulating [117]. This last statement is illustrated in figure 4.4 in which one can see the comparison between that the spatial extents of the radial part of the wave functions of 3d, 4d and 5d electrons.

<figure><img src="image_66.png" /><figcaption>Figure 4.4: Comparison between the spatial extents of the radial part of the wave functions of 3d, 4d and 5d electrons. 5d are much more extended spatially which is not in favor of a Mott insulating behaviour (image taken from ref. [118]).</figcaption></figure>

In order to understand the origin of the insulating properties of $Sr_2IrO_4$, one has to have a look at its crystal structure, represented on figure 4.1. It has a layered perovskite structure consisting in alternation of $IrO_2$ and $SrO$ layers. Each Ir atom has a 5d⁵ electronic configuration and is located at the center of an octahedron (which 6 apexes are oxygen atoms), such that octahedral crystal field lifts the degeneracy of the 5d orbitals between $t_{2g}$ and higher in energy $e_g$ sets of states. In addition to this effect, Ir has a substantial atomic number (Z=77) which implies that the 5d electrons are subject to strong spin-orbit coupling. This spin-orbit coupling splits the $t_{2g}$ states between a $J_{eff}=3/2$ completely filled band and a $J_{eff}=1/2$ half-filled band [29]. The correlations then suffice to split the $J_{eff}=1/2$ band into a lower Hubbard band (LHB) and an upper Hubbard band (UHB) separated by a gap, turning the system into a Mott insulator.
<figure><img src="image_67.png" /></figure>

Figure 4.5: **a.** Ir atom at the center of an oxygen octahedron. **b.** Its electronic configuration is 5d⁵. The crystal field lifts the degeneracy between t₂g and e_g stets of states. **c.** The spin-orbit lifts the degeneracy within t₂g levels in a Jₐff=3/2 completely filled band and a Jₐff=1/2 half-filled band with reduced band width allowing the system to turn into a Mott insulating phase (image taken from ref. [29]).

4.4 Metal to insulator transition in (Sr$_{(1-x)}$La$_{x}$)$_{2}$IrO$_{4}$

Pure Sr₂IrO₄ has already been intensively studied by many methods such as angle-resolved photoemission spectroscopy (ARPES) [119, 120], optical conductivity [121, 122, 119], resonant inelastic X-ray scattering (RIXS) [123] and Scanning Tunnelling Microscopy/Spectroscopy (STM/STS) [124, 125]. The measured values of the gap vary greatly from one experiment to the other but were partly reconciled by taking account tip-induced band bending effects in STM studies on lightly doped Sr₂IrO₄ samples [126].

The Metal-Insulator Transition (MIT) in doped Sr<sub>2</sub>IrO<sub>4</sub> is actively investigated [127, 128, 129] but many questions remain unanswered. There are several ways to turn a Mott insulator into a metal such as applying pressure [130, 131], gating it by deposing another material on top of it [132, 133] or even doping the system[134, 135, 136]. In electron doped compound (Sr<sub>(1-x)</sub>La<sub>x</sub>)<sub>2</sub>IrO<sub>4</sub>, only few experiments were performed. The ARPES ones agree as to where in First Brillouin Zone the Mott gap should close for high enough dopants concentrations but not on the position of the LHB with respect to Fermi level for lightly doped samples [127, 128]. This last remark suggests to have a deeper look at the passage from pure Sr<sub>2</sub>IrO<sub>4</sub> to lightly doped (Sr<sub>(1-x)</sub>La<sub>x</sub>)<sub>2</sub>IrO<sub>4</sub>. Because only occupied states can be observed in photoemission, the gap cannot be accessed which makes the interpretation of the results delicate. STS measurements allow to measure the local density of states (LDOS) both beneath and above Fermi level which makes it the perfect tool to probe the spatial distribution of the Mott gap in those materials. It was in this perspective that a recent STM/STS study was performed in (Sr<sub>(1-x)</sub>La<sub>x</sub>)<sub>2</sub>IrO<sub>4</sub> [129], revealing a spatially inhomogeneous doping driven MIT as the Mott gap varies at nanometer scale. However, no STM/STS experiments as yet shown a full analysis of MIT with doping
including pure $\text{Sr}_2\text{IrO}_4$.

4.4.1 Topographic measurements

In this section, we present scanning tunnelling microscopy (STM) measurements which were
carried out on pure $\text{Sr}_2\text{IrO}_4$ as well as on $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$ samples with varying La doping
concentrations ($x = 1,5\%$, $x = 2,3\%$ and $x = 6\%$). The doping concentrations $x$ were deter-
mined via EDX studies. Given the great variety of contradictory results in iridate compounds
systems, it is also very important to emphasize that all the observed crystals were grown in
the same conditions using the same self-flux method. The samples were cooled down to liquid
nitrogen temperature and cleaved in ultrahigh vacuum under a pressure of approximatively
$8 \cdot 10^{-11}$ mbar directly before being loaded into the STM head, also thermalized at 77 K. Cleav-
ing samples at low enough temperature is mandatory in order to obtain clean surfaces. Samples
cleaved at room temperature are indeed subject to unwanted surface reorganization of Sr and/or
O atoms [137].

<figure><img src="image_68.png" /><figcaption>Figure 4.6: **a.** Topography map of the surface of $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$ with $x = 1,5\%$. **b.** Topography map of the surface of $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$ with $x = 2,3\%$. **c.** Topography map of the surface of $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$ with $x = 6\%$. **d.** Fourier transform of a topography image of $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$ with $x = 1,5\%$. The blue, yellow and red circles highlight vacancies, La dopants and adatoms, respectively.</figcaption></figure>
The crystal structure of Sr₂IrO₄ is represented on figure 4.1. All iridium atoms are at the centers of oxygen octahedra. Those octahedra organize on a square lattice of lattice vectors **a** and **b** with || **a** ||=|| **b** ||= 3,9 Å and **a** ⊥ **b**. Within an iridium plane, each oxygen atom is shared between two neighboring iridium. As a consequence, there are twice as much oxygen than iridium atoms in such a plane meaning it has a IrO₂ stoichiometry.

The IrO₂ planes are sandwiched between SrO bilayers. The Sr atoms of a given layer form also a square lattice of basis vectors **a** and **b** where each Sr is shifted of 1/2(**a**+**b**) with respect to its adjacent octahedra lattice nodes. The top most (or bottom most) oxygen atom of each octahedron indeed lies at the center of an on top (underlying) Sr square.

After cleaving, the sample terminates on a SrO layer [129, 125]. Figure 4.6 **a**, **b** and **c** represent the topography maps of 1,5%, 2,3%, and 6% samples, respectively. The topographic data show a square lattice of lattice parameter *a* = 3,9 Å. This value corresponds to the distance between two neighboring Sr atoms within a SrO layer. This can also be seen by looking at the Fourier transform of figure 4.6 **d** where a reciprocal square lattice is clearly distinguishable. It is important to emphasize that the oxygen atoms cannot be seen by STM such that the visible square lattice is composed out of Sr atoms [125].

Many defects cover the surfaces of our samples. It is possible to distinguish both vacancies and adatoms encircled respectively in blue and green on figure 4.6 **b** and **d**. Those are not due to improper growth processes of our crystals but rather to cleaving [137]. Some Sr/La atoms are indeed taken off either the newly obtained SrO layer or the thrown away one.

Adatoms and vacancies do not constitute the majority of defects. Most of them actually present a square-shaped modulation. Such kinds of defects are highlighted with yellow circles on 1,5%, 2,3% and 6% maps. One can notice that all those square-defects sit at nodes of the Sr square lattice. This is no surprise since such features are commonly attributed to La dopants in substitution to Sr atoms [129]. Thus, by spotting the square-shaped defects on a given topography map, it is possible to access both the number of dopants (and by extension to compute the dopant concentration), and their spatial distribution. Such an approach was already performed in other studies [129]. The dopant concentrations were evaluated in this way for 1,5%, 2,3% and 6% samples and are in good agreement with the measured EDX values.

The considered 1,5% to 6% doping concentrations values are not arbitrary. It was indeed already observed that it is not possible to grow (Sr$_{(1-x)}$La$_x$)$_2$IrO$_4$ with $x$ exceeding 6% because of the constraints induced by the substitutions of Sr atom by La dopants [129, 127]. Our study thus covers the full accessible La doping concentration range in doped Mott insulator (Sr$_{(1-x)}$La$_x$)$_2$IrO$_4$.

### 4.4.2 Inhomogeneous transition from insulating to pseudo metallic phase induced by La doping

The IrO₂ layers are the ones bearing the Mott physics of interest since the paradigmatic J<sub>eff</sub>=1/2 ; 3/2 band picture is derived from the crystal field lifting of degeneracy of Ir atomic orbitals. It was already shown that there is a mixing between the Sr and IrO₂ [138] bands such that the observed LDOS on the SrO layer terminated surface should also present Mott physics features [125].

In order to access to the LDOS at different energies within a range [-1 eV,+1 eV], spectroscopic *I*(*V*) maps were acquired, consisting in taking individual *I*(*V*) spectra at many positions determined by a predefined spatial grid (see section 2.2). The differential conductance *dI/dV* maps (derived afterwards) at a given *V* can directly be related to the LDOS of the sample at the corresponding energy. Performing LDOS experiments at 77 K allowed us to exhibit spatial variations of both Mott gap and Fermi level position. The study was realized with an Omicron LT STM using Pt80%/Ir20% tips. Prior to each experiment, the DOS of the tip was verified on Pt and HOPG samples and checked to be featureless (see subsection 2.4.3). The topography
maps were taken in constant current mode and the spectra were taken at constant tip-sample distance for a given tunnelling set-point (mainly around $U = -1$ V, $I = 100$ pA).

The first part of this study focuses on describing the transition from a Mott insulating state to a (pseudo) metallic one as the doping concentration $x$ goes up. The discussion is built on spatial representations referred to as “gap maps”. They simply represent the magnitude of the Mott gap as a function of space. The gaps are extracted from each spectrum/point of each spectroscopic map using a method illustrated on figure 4.7. In all $I(V)$ spectra, there is a whole energy range in which the current is equal to zero. This range is roughly equal to the insulating gap $\Delta$. In order to extract $\Delta$, we looked at the energy coordinates at which the crossing between the $I(V)$ spectrum and horizontal lines lying at $\pm 1\%$ of the set-point current $\|I_{\text{setpoint}}\|$ occur. Those two energies are called $E_{\text{min}}$ and $E_{\text{max}}$, respectively. In the end: $\Delta = E_{\text{max}} - E_{\text{min}}$.

It is important to keep in mind that it is often delicate to extract the value of a gap. Here, we do not have any model which could give us an analytical expression of the expected differential conductance, and thus, allow us to efficiently fit our data. The method used here to extract gap is one out of many, and we do not pretend that such gap estimations are absolute. However, this process is good enough for what we intend to do, namely, looking at the spatial dependence of the gap in $(\text{Sr}_{(1-x)\text{La}_x})_2\text{IrO}_4$ for several doping values.

On figure 4.8 is shown the evolution of the gap maps with doping concentrations $x = 1, 5\%$, $x = 2, 3\%$, $x = 6\%$. The selected areas have more or less the same size (typically 20nm×20nm). Those data are referred as large-scale experiments. All the gap maps share the same black and white color bar. The smaller the gaps the whiter they appear. The corresponding gap distributions are superimposed on figure 4.8 d, e and f.

<figure><img src="image_69.png" /></figure>

Figure 4.7: Illustration of the way the gaps were extracted from $I(V)$ spectra. The blue curve is a typical $I(V)$ spectrum and is normalized with respect to the set-point current. There is a whole range of biases voltages for which the spectrum is flat and for which the derivative and a *fortiori* the density of states will be equal to zero. The idea is to estimate this range which will be assimilated to the gap. In order to do so, we looked at the intersections between the spectrum and the horizontal lines $y = \pm 1\% \|I_{\text{setpoint}}\|$. The $V$ coordinates at which those intersection occur are respectively called $E_{\text{max}}$ (for $y = +1\% \|I_{\text{setpoint}}\|$) and $E_{\text{min}}$ (for $y = -1\% \|I_{\text{setpoint}}\|$). The gap is finally defined as $\Delta = E_{\text{max}} - E_{\text{min}}$.
<figure><img src="image_70.png" /><figcaption>Figure 4.8: **a.** Gap map of the *x* = 1, 5% sample. **b.** Gap map of the *x* = 2, 3% sample. **c.** Gap map of the *x* = 6% sample. **d.** Gap distribution for the *x* = 1, 5% sample. **e.** Gap distribution for the *x* = 2, 3% sample. **f.** Gap distribution for the *x* = 6% sample.</figcaption></figure>
<figure><img src="image_71.png" /><figcaption>Figure 4.9: Two typical averaged $dI/dV$ spectra taken in the 1,5% sample on a bubble and on the background, respectively.</figcaption></figure>

First of all, the spatial distribution of the gaps is very inhomogeneous, and this, for all doping concentrations. The magnitude of the gap evolves on energy ranges varying from one sample to another but typically going from 100 meV to 750 meV. This is also emphasized by the finite widths of the gap distributions. Those substantial gap variations occur at nanometer scale. The smaller-gap areas are disk-shaped such that we refer to them as “bubbles”. Such small-gap bubbles are highlighted with green circles on 1,5%, 2,3%, and 6% gap maps. The bubbles have a diameter approximatively equal to 2,5 nm. Each gap map thus consists in many small-gap bubbles (white) distributed across a more insulating background (black). Two typical averaged $dI/dV$ spectra taken in the 1,5% gap map on a bubble and on the background respectively are represented in figure 4.9 and illustrate the gap inhomogeneities throughout the samples. Secondly, the gap maps are whiter and whiter as the doping concentration $x$ increases implying that the samples become less and less insulating. This is consistent with a Mott insulator to metal transition driven by doping.

An inhomogeneous MIT in $(\text{Sr}_{(1-x)\text{La}_x}\text{Iro}_4)^2$ was also observed in the work of Battisti et al. [129] but their results/conclusions differ from ours. They report an abrupt transition from an homogeneous insulating phase for low doping concentrations (2,1% to 3,7%), to a mixed inhomogeneous phase where both pseudo metallic and insulating puddles cohabit for doping concentrations higher than 4,8%. It is important to point out that in our study the inhomogeneities start emerging even for lightly doped samples with $x$ going as low as 1,5%. There are two possible explanations for such discrepancies between our study and the one of Battisti et al.. First, even though both our crystal growth methods are similar, they are not identical. Moreover, the used sources were also different. Maybe the physics of $(\text{Sr}_{(1-x)\text{La}_x}\text{Iro}_4)^2$ is very sensitive to the growth procedure, which would also explain the great variety of results reported in the literature. Second, in the last figure of ref. [129], spatial color plots representations equivalent to our gap maps (their way of extracting the gap is different) are compared for samples with different doping levels (namely 2,1%, 2,3%, 3,7%, 4,8%, 5,2% and 5,5%). The color plots for 2,1%, 2,3% and 3,7% samples appear perfectly uniform, meaning that the images were somehow saturated and that spatial inhomogeneities were maybe killed.
<figure><img src="image_72.png" /><figcaption>Figure 4.10: **a.** Middle-gap map of the *x* = 1, 5% sample. **b.** Middle-gap of the *x* = 2, 3% sample. **c.** Middle-gap of the *x* = 6% sample. **d.** Middle-gap distribution for the *x* = 1, 5% sample. **e.** Middle-gap distribution for the *x* = 2, 3% sample. **f.** Middle-gap distribution for the *x* = 6% sample.</figcaption></figure>
<figure><img src="image_73.png" /><figcaption>Figure 4.11: Two typical averaged dI/dV spectra taken respectively on a positive-valued middle-gap area and on a negative-valued middle-gap area in the 1,5% sample.</figcaption></figure>

In doped semiconductors, the presence of dopants leads to a chemical potential shift towards conduction or valence band depending to whether the semiconductor is n or d doped. In doped Mott insulators, dopants also induce displacements of the Fermi level towards LHB or UHB. However, contrary to semiconductors, a reduction of the gap (and a modification of the band structure) is likely to accompany this shift. In my work I tried to highlight such effects by building up different spatial representations referred to as “middle-gap maps”. Let $E_{min}(\mathbf{r})$ and $E_{max}(\mathbf{r})$ the energies of respectively the last occupied state and the first unoccupied state at a given position $\mathbf{r}$. A middle-gap map represents $M = (E_{max}+E_{min})/2$ as a function of space (the Mott gap is $\Delta = E_{max} - E_{min}$). It represents the position in energy of the "center" of the gap with respect to Fermi level. Only the combined information of both gap and middle-gap maps is likely to give without any ambiguity the shape of the LDOS as a function of space.

On figure 4.10 is shown the evolution of the middle-gap maps with doping concentrations $x = 1,5\%$, $x = 2,3\%$, $x = 6\%$. The selected areas are the same than in figure 4.8. All the middle-gap maps share the same red/white/blue color bar. The more negative (positive) the middle-gaps the redder (bluer) they appear. The corresponding middle-gap distributions are superimposed on figure 4.10 d, e and f.

Similarly to the Mott gap, the energy position of the middle of the gap is inhomogeneous in space as it can vary typically from -100 meV to 100 meV. For the 1,5% sample, about half of the surface has negative middle-gaps values when the other half has positive middle-gaps values. Two typical averaged dI/dV spectra taken on a positive-valued middle-gap area and on a negative-valued middle-gap area, respectively, in the 1,5% middle-gap map are represented in figure 4.11. The two spectra were especially picked up to show that some areas may present the same Mott gap while having different Fermi level positions.

Scattered white zones (gap center at Fermi level) can also be seen. Such a zone is encircled in green on figure 4.10 a. Most of the time, such areas correspond to small gap bubbles. This apparent correlation can be simply explained. In Mott physics, correlations between electrons around Fermi level open a gap. This implies that the Fermi level should lie within the gap. When the gap becomes smaller, as in a small-gap bubble, the middle of the gap can only vary within a limited energy range such that it is more likely to be close to 0.
As the doping concentration increases, the middle-gaps are more and more negative (red). The increasing asymmetry with respect to 0-energy of the middle-gap distributions in figure 4.10 *d*, *e* and *f* comforts this last observation.

The short-range inhomogeneous character of both Mott gap and position of the middle of the gap can be further accentuated by investigating the variation of spectra at nanometer scale. In figure 4.12 are shown the different kinds of dI/dV spectra encountered in *x* = 1, 5%, *x* = 2, 3% and *x* = 6% samples. Those are extracted from different spectroscopic grid measurements taken on zooms of the areas of figure 4.8/4.10. The color of a given spectrum corresponds to the magnitude of its gap. Each spectrum is obtained after averaging of a collection of similar spectra sharing the same general features enclosed in specific areas on the map. The number of spectra per unit of surface is higher than in the previous set of experiments (more than 50 times). This second set of spectroscopy measurements is more spatially resolved regarding the typical scale of both atomic variations and inhomogeneities such as La modulations. These measurements are referred to as small-scale experiments. In the insets of each collection of spectra are represented the corresponding gap maps. The presented collections of spectra are assumed to be representative of the samples.

Many of the remarks that were made on the large-scale experiments can also be made by analysing the spectra of figure 4.12.

First, all samples present spectra with significantly different magnitudes of the gap. The spectra with reduced gaps (red) are characteristic of the bubbles and the ones with higher gaps (blue) are characteristic of the insulating background. The inhomogeneity of the samples is thus again visible in small-scale experiments. The magnitude of the gap for the small-gap spectra reduces as doping concentration increases, which is consistent with previous observations. However, another piece of information arises from this second study. For the 1,5% sample, there are no states at zero energy for the spectra taken on the bubbles. They thus have a well-defined gap and the bubbles remain insulating. On the contrary, for the 6% sample, the bubble-type spectra show some density of states at zero energy. This means that for high-enough doping concentrations the bubbles become pseudo-metallic. However, the magnitude of the gap of the insulating background-type spectra remains the same with doping (typically 650 ± 50 meV). The pseudo-metallic aspect of the Mott transition areas at high doping was also observed by Battisti et al. in their study. These last statements can all be summed up in the following way: doping the system induces the apparition of small gap/pseudo-metallic bubbles on top of an insulating background; the insulating background gaps remains constant with doping while the magnitude of the bubble (pseudo-)gaps reduce as the number of dopants increases. In this system, the presence of dopants does not induce a smooth variation of the ratio U/t as additional electric charges are not delocalized throughout the sample but rather an inhomogeneous MIT.

Second, for low doping concentrations (1,5%, 2,3%), the position of the middle of the gap greatly varies from negative values to positive values while for the 6% sample, it mainly stays negative. Moreover, the middle of the gap lies around 0-energy for small gap/pseudo-metallic spectra. Again, this is in agreement with large-scale experiments data.
<figure><img src="image_74.png" /><figcaption>Figure 4.12: **a.** Small-scale gap map of the *x* = 1,5% sample. **b.** Small-scale gap map of the *x* = 2,3% sample. **c.** Small-scale gap map of the *x* = 6% sample. **d.** Typical spectra encountered in **a** area. **e.** Typical spectra encountered in **b** area. **f.** Typical spectra encountered in **c** area.</figcaption></figure>
The question that immediately arises is whether there is or not a correlation between the positions of the dopants and the positions of the pseudo-metallic bubbles. In their article, Battisti *et al.* suggest that there is an equivalence between the locations of the La dopants and of the pseudo-metallic puddle areas. Our conclusion is partly in agreement with theirs. When doing the cross correlations between the gap maps and the position of the dopants maps, one finds a 33% matching for the 1,5% sample and a bit less for higher doping samples. These values mean there exist a certain degree of correlations between the locations of La atoms and the positions of metallic bubbles since by generating a random spatial distribution with an equal number of dopants one rarely reaches values above 10%. A rough estimation by the eye would indeed suggest correlations because for large-scale images a great number of dopants are localized in the pseudo metallic areas. Yet, it is also very common for dopants to lie within the insulating background. Figure 4.13 displays on the left a dopants concentration map of the same 1,5% large-scale area than in figure 4.8 a (displayed on the right). In order to obtain the dopants concentration map, the positions of the dopants were targeted, after what a map was created with signal peaked at those positions. A Gaussian filter was applied to this map to give a spatial extension to the dopants. The width of the Gaussian in the filtering process was fixed to maximize the cross-correlations between the gap map and the dopants concentration map. For 1,5% large-scale images, one can clearly see dopants sitting in the very middle of large-gap areas as suggested by pink circles in figure 4.13. The fact that one does not observe a one to one correspondence can naively be understood in the following way: in STM, only the SrO layer is visible, nevertheless, Mott physics emanates from the electrons of $IrO_2$ layers. The presence of La atoms substituting the Sr atoms in the neighboring layers modifies the physical properties of the system such that potentially half of what causes the MIT is invisible to our probe. With this last statement in mind, it would then be odd to find perfect matching between the positions of the dopants at the surface and of the pseudo-metallic bubbles. An additional way of interpreting this mismatch will be discussed in section 4.5.

<figure><img src="image_75.png" /><figcaption>Figure 4.13: On the left is displayed a dopants concentration map. The hotter the color, the more dopants are present. On the right is represented a gap map of the same area. These data correspond to the 1,5% case.</figcaption></figure>
It also looks like there is a correlation between the locations of pseudo-metallic bubbles and the 0-valued middle gap areas (in white). In Mott physics, a gap opens because of correlations between electrons at Fermi level, which implies that the Fermi level should lie within the gap. When the gap becomes smaller, as in a pseudo-metallic bubble, the middle of the gap can only vary within a limited energy range such that it is more likely to be close to 0. Given the discussion of the previous paragraph, this suggests a correspondence between the locations of La atoms and the negative valued middle-gap areas. One can thus conjecture the following statement : the introduction of dopants locally shifts the chemical potential towards lower energies because new electric charges are brought in the material ; as more and more dopants are present, the chemical potential starts to homogenize across the sample and the Fermi level position locks right beneath the UHB. Here, our observations are partly in agreement with the Battisti et al. study. They indeed do not see any spatial chemical potential shifts in lightly doped samples (2,2%). Yet, in the case of highly doped samples (5,5%), they also observe a locking of the Fermi level position close to the UHB, which is in agreement with electron doping (seen also in La-doped bilayer iridate Sr$_3$Ir$_2$O$_7$ [139] and which is the opposite of what is observed in hole-doped cuprates [140]).

4.5 Orientational emergent order in $(\text{Sr}_{(1-x)}\text{La}_x)_2\text{IrO}_4$

Up to now, no comment was made on the distribution and number of pseudo-metallic bubbles
and their evolution with doping concentration *x*. A good way of accessing these information
is to compute the autocorrelation maps of gap maps. The autocorrelation of a given map
is the 2D-autoconvolution of that map. It roughly consists in sliding a map on top of itself
for many displacement vectors and see for which the obtained translated maps coincide in
the best way with the original one. Our convention is that the center of the autocorrelation
map corresponds to no displacement at all such that both original and translated maps match
perfectly. Autocorrelation functions are a powerful tool for exhibiting order and periodicity in
a system.

In figure 4.14, are represented the autocorrelation maps of the large-scale gap maps of figure 4.8. By construction, at the center of every autocorrelation map, the value is equal to 1. The directions of the Sr atomic square lattice were drawn as black dotted arrows on top of each autocorrelation map.

<figure><img src="image_76.png" /><figcaption>Figure 4.14: **a.** Autocorrelation map of the gap map of figure 4.8 **b.** Autocorrelation map of the gap map of figure 4.8 **c.** Autocorrelation map of the gap map of figure 4.8 **d.** The black arrows represent the atomic directions of the Sr lattice.</figcaption></figure>

The first noticeable result one gets by looking at the autocorrelation maps is that the widths
of the central spots are more or less constant with doping. The width of a central spot can directly be related to the typical size of the smallest objects in a gap map, namely the pseudo-metallic bubbles. At 20% the height of the central autocorrelation peaks, one can then evaluate the diameter of a bubble: $D_{bubble} = 2,5 \pm 0,5$ nm, which is consistent with both large-scale and small-scale experiments. The second observation one can make is that the autocorrelation patterns seem to organize along the atomic directions. It is visible for all doping concentrations but is particularly remarkable in the 1,5% case where one can even distinguish the development of a rectangular lattice fading as one goes away from the center.

In the 1,5% case, the first neighboring spot to the center lies at $L = 5 \pm 1$ nm from the central peak. This last piece of information suggests that, for this level of doping, the pseudo-metallic bubbles do not distribute in a glassy way, but rather tend to organize along the surface atomic directions forming a square-like lattice of lattice parameter $L$. Thus, the locations of pseudo-metallic bubbles do not reflect the random distribution of dopants, which consolidates what was already told in the previous part. Further studies would need to be performed to see whether the formation of a lattice is typical of the lightly doped regime.

Our study highlights what we interpret as being the emergence of a short-range orientational order in the gap spatial distribution. If the appearance of a lattice is not necessarily obvious for all doping, one can however claim that the pseudo-metallic bubbles preferentially arrange along the Sr lattice atomic directions.

A similar order, manifesting as pseudo-gap stripes, was also observed in the study of Battisti *et al.* [129] and in lightly hole-doped cuprate parent compounds [141, 142, 143, 144, 140]. It would seem that the appearance of such orders is representative of the inhomogeneous insulator to metal transition in lightly doped Mott insulators.

## 4.6 Pure sample case

As it was already mentioned, the different studies that were carried out so far on pure $Sr_2IrO_4$ are not in agreement regarding the value of the Mott gap. Gaps as low as 0, 1 eV have been reported in transport [145] and optical conductivity [119] measurements. In angle-resolved photoemission spectroscopy experiments, the maximum of the valence band was measured around -0,2 eV, implying a Mott gap at least equal to 0,2 eV [119, 120, 127, 128]. In the first STM/STS study performed on pure $Sr_2IrO_4$, the value of the gap was evaluated to be equal to ~ 0,15 eV [124]. However, even down to 4, 2 K, the spectra did not exhibit a hard gap but rather a strong depletion of the density of states around Fermi level. Two other STM/STS studies were published one year later. In the first study, again, no hard gap was observed, which the authors interpreted as gap filling due to inelastic magnons features [146]. In the second study, J. Dai *et al.* reported a substantial Mott gap of 620 meV [125]. Nevertheless, the measured samples presented many defects and local spectroscopy measurements performed on top of these defects exhibited drastic changes of the local density of states. In their study on Mott insulator to metal transition in $(Sr_{1-x}La_x)_2IrO_4$, Battisti *et al.* did not pursue any measurements on pure $Sr_2IrO_4$ [129]. According to their complementary article (see ref. [126]), the reason was that the quality of the samples was such that the tip would crash during the approach, consequently preventing any STM experiment.

One of the strong points of the STM/STS work presented in this chapter is the additional experiments performed on **pure** $Sr_2IrO_4$ samples. The chemical composition of the observed $Sr_2IrO_4$ samples was checked beforehand by mean of EDX. No trace of any of the usual elements used in the doping process was found (namely La and Rh). Besides, the measured stoichiometric ratio corresponds to the one of pristine $Sr_2IrO_4$. The experiments were carried out in the exact same conditions as for the $(Sr_{1-x}La_x)_2IrO_4$ samples.

Figure 4.15 **a** represents a typical topography map of pure $Sr_2IrO_4$. One can clearly see the Sr square lattice of lattice parameter 3,9 Å. The absence of the square-shaped defects
associated to La atoms further supports the purity of the samples. Nonetheless, similarly to all previous STM studies performed on pure $Sr_2IrO_4$, some defects are visible on the surface. In our case, they appear as "dark spots" on the topography map. The nature of these last defects is not known for, most of the time, no Sr atom is missing. Dark patches with similar appearance and interpreted as oxygen-related defects were reported in ref. [125]. On top of such dark patches, J. Dai et al. saw a significant reduction of the Mott gap. Such a behaviour was not observed in the case of our "dark spots" defects and their nature is still unclear, their origin potentially lying deeper within the structure.

In addition to those defects, one can also see bright density of states puddles encircled in blue on figure 4.15 a. Their nature is also unknown. An averaged $dI/dV$ spectrum taken on such a bright area is represented in blue in 4.15 b. The local Mott gap was determined using the same method as the one of ref. [125], namely linearly extrapolating both LHB and UHB as illustrated by the blue dotted lines in 4.15 b. By doing so, one finds a value for the Mott gap equal to approximately 260 meV. Even if the method to extract the gap was different, we recall that in $(Sr_{1-x}La_x)_2IrO_4$, insulating areas could display Mott gaps bigger than 500 meV.

For now, we only commented on spectroscopic features on top of defects. By performing spectroscopy far from such "dark or bright spots" one should recover the density of state of pristine $Sr_2IrO_4$. Such an averaged $dI/dV$ spectrum was taken in a free-of-defects area like the one encircled in green on figure 4.15 a and is represented in green in figure 4.15 b.

<figure><img src="image_77.png" /><figcaption>Figure 4.15: **a.** Typical topographic map of the surface of pure $Sr_2IrO_4$. **b.** $dI/dV$ spectra taken on top of a "bright" density of states puddle (blue curve) and away from any apparent defect (green curve), respectively.</figcaption></figure>

Surprisingly, by applying the same gap-determining method as for the blue curve, one finds a Mott gap of pure $Sr_2IrO_4$ approximately equal to 80 meV, or so to say much smaller than the one of any insulating area of $(Sr_{1-x}La_x)_2IrO_4$. This value is actually much closer to the one reported in transport [145] and optical conductivity [119] than the ones reported in both ARPES and STS.

Our study of $(Sr_{1-x}La_x)_2IrO_4$ teaches us that increasing the electron doping level drives a transition from a Mott insulating phase to a pseudo-metallic phase. However, this last statement seems to be true only when the system is already electron-doped. By going from $x = 0\%$ to $x = 1,5\%$ the system indeed at first becomes more insulating than it was. It is only when the doping level further increases that the Mott insulator to metal transition seems to pursue in
the expected direction.

We will now propose possible reconciliation between our data and the literature. In ARPES measurements, the position of the maximum of the valence band with respect to Fermi level can potentially be explained by the fact that photoemission is an integrative method, which by definition will average all the different contributions found in the sample. Our extrapolated Mott gap on top of bright areas, is actually large enough for this interpretation to be plausible. Moreover, the differential conductance spectra that were observed in STM/STS studies of refs. [124, 146] are also compatible with ours. Also, it remains purely speculative but, from our study, it looks like introducing disorder tends to first increase the value of the Mott gap<fnref n="5" />. The large gaps observed in ref. [125] could possibly be explained by the presence of many superficial defects.

## 4.7 Conclusion

In this chapter, we have observed by mean of grid spectroscopy a spatially inhomogeneous Mott insulator to pseudo-metallic phase transition driven by electron doping in $(Sr_{1-x}La_x)_2IrO_4$ (with x the La concentration). This result further justifies the use of a local probe and that integrative methods like ARPES are not sufficient to study the Mott physics in this material. Contrary to previous work reported in the literature, our study exhibits the inhomogeneous character of the transition down to the lightly doped regime. We also observed the appearance of a short-range orientational order in the pseudo-metallic puddles distribution, in qualitative agreement with similar orders observed in lightly hole-doped cuprates parent compounds. Moreover, our study is the first one which includes STM/STS measurements on pure $Sr_2IrO_4$. The smallness of the Mott gap exhibited in our pristine compound allowed us to propose a reconciliation between the great variety of results reported in the literature. More particularly, our study suggests a significant dependence of the physical properties of $(Sr_{1-x}La_x)_2IrO_4$ on the crystal growth processes and/or on disorder. Finally, signatures of superconductivity were observed in a recent experimental study in which potassium was deposited at the surface of pristine $Sr_2IrO_4$ (which would be an alternative way to electron-dope the system). We plan on carrying out similar studies since, according to literature, electron or hole doped $Sr_2IrO_4$ could host topological superconductivity.

<footnote n="5">⁵The origin of this behaviour still as to be determined.</footnote>
Conclusion

First have shown that superconducting misfit transition metal dichalcogenide compound (LaSe)₁,₁₄(NbSe₂)₂ possesses a band structure similar to the one of 0, 3 eV electron-doped monolayer NbSe₂. Moreover, we highlighted the unconventional nature of the superconducting order parameter in this misfit compound.

In view of our study, one can imagine to use bulk misfit heterostructures such as (LaSe)₁,₁₄(NbSe₂)₂
in order to investigate the physics of monolayer transition metal dichalcogenides. The generic
formula of misfit heterostructures of the same family is ([MX]₁+δ)ₘ(TX₂)ₙ, where M is either
Sn, Pb, Bi, Sb or any lanthanide element, T is a group 4 (such as Ti) or a group 5 (such as Nb)
transition metal and X is a either S or Se chalcogen element. m and n represent the number
of rock-salt and transition metal dichalcogenide layers, respectively. Finally, δ represents the
degree of structural mismatch between MX and TX₂ layers.

There would be several ways on tuning the doping level in such heterostructures. For
simplicity, we will base our discussion on comparison with our compound. Let us imagine
that we change the LaSe layers for PbSe layers. The main difference resides in the fact that
the PbSe layer neither gives or accepts any electrons (the same can be said of SnX layers).
In this configuration, we can thus imagine that the newly obtained compound should behave
as regular undoped monolayer NbSe₂ (it has been shown in ref. [9] that this compound is
superconducting under 3,64 K). Such a behaviour was actually observed in (SnS)₁,₁₇(NbS₂),
where ARPES measurements exhibited monolayer NbS₂-dominated Fermi level spectroscopic
features [147]. One could also imagine to substitute a fraction of the La atoms with Pb atoms.
Depending on the degree of substitution, the whole [0 eV, +0,3 eV] electron doping interval of
monolayer NbSe₂ could theoretically be explored, and one could potentially perform a study
of the evolution of both charge density wave order and superconductivity with continuous
doping values. Also, reducing *n* (which is equal to 2 in our compound) down to 1 should
theoretically double the number of additional carriers brought to each NbSe₂ layer. In this way,
even higher doping value could in principle be reached. The transition metal dichalcogenide
monolayer of interest could also be for MoS₂ or NbS₂ just as an example (in ref. [147], similar
conclusions were made in a misfit compound based on NbS₂ instead of NbSe₂). As it was
already mentioned in the main text, one possible route towards topological superconductivity
consists in inducing superconductivity in topological insulators [64][7]; It has been shown very
recently that single-layer 1T'-WSe₂ is a topological insulator exhibiting topological edge states
[148]. By replacing NbSe₂ by 1T'-WSe₂, it might be that the electron doping is such that
1T'-WSe₂ layers become superconducting, hence potentially turning the whole system into a
topological superconductor. Moreover, even though we exhibited signatures of unconventional
superconductivity in (LaSe)₁,₁₄(NbSe₂)₂, it does not mean that it is automatically topologically
non trivial. One could think of replacing non magnetic La by magnetic lanthanide elements
such as Eu or Dy. Then, it is possible that the underlying magnetic lattice acts as an external
magnetic field potentially turning the system into a topologically non trivial regime.

Second, we have exhibited the spatially inhomogeneous character of the Mott insulator to
pseudo-metallic phase transition in electron doped Sr₂IrO₄. In view of our study, it appears
that integrative methods such as ARPES, although very powerful, are not sufficient to entirely
address the Mott physics in iridium oxyde compounds and that the use of a local probe such
as the STM is of the atmost importance. Moreover, the discrepancies between our study and
previous STM/STS works reported in the literature suggests that the physical properties of
interest are highly sensitive to the crystal growth processes and also, potentially to disorder.
Those questions have to be further investigated. For example, it would be interesting to study
the effect of disorder on the physics of the system by introducing defects in a controlled way
using electron irradiation. Also, as it was already mentioned, it would also be interesting to
deposit potassium at the surface of pristine Sr₂IrO₄ to see whether it is possible or not to turn
the system into a superconducting (topological) phase. Performing the same kind of study on
other iridium oxide compounds of the same family such as Sr₃Ir₂O₇ [139] would also be a good
idea.
# Appendices

## .1 Computation of the tunneling conductance.

This part can be inserted in part 1.4.1 where a reference to this appendix was made.

For simplicity reasons and because the intensity of the tunneling current is low, we will consider that the number of electrons in both systems 1 and 2 remains conserved such that $[\hat{H}_{\alpha}, \hat{N}_{\beta}] = 0$ with $\alpha, \beta = 1, 2$.

$$ \begin{gathered} \frac{d\hat{N}_1}{dt} = i[\hat{H}, \hat{N}_1](t) = ie^{i\hat{H}t}[\hat{H}, \hat{N}_1]e^{-i\hat{H}t}, \\ \frac{d\hat{N}_1}{dt} = ie^{i\hat{H}t}([\hat{H}_1, \hat{N}_1] + [\hat{H}_2, \hat{N}_1] + [\hat{H}_T, \hat{N}_1])e^{-i\hat{H}t}, \\ [\hat{H}_T, \hat{N}_1] = \sum_{kk', \sigma} \sum_{k'', \sigma'} T_{kk'} \underbrace{[c_{1k\sigma}^{\dagger}c_{2k'\sigma}, c_{1k''\sigma'}^{\dagger}c_{1k''\sigma}]}_{-\delta_{kk'}\delta_{\sigma\sigma'}c_{1k''\sigma'}^{\dagger}c_{2k'\sigma}}, + \sum_{kk', \sigma} \sum_{k'', \sigma'} T_{kk'}^{*} \underbrace{c_{2k'\sigma}^{\dagger}c_{1k\sigma}, c_{1k''\sigma'}^{\dagger}c_{1k''\sigma}]}_{\delta_{kk'}\delta_{\sigma\sigma'}c_{2k'\sigma}^{\dagger}c_{1k''\sigma'}}, \\ [\hat{H}_T, \hat{N}_1] = T^{\dagger}(0) - T(0), \end{gathered} $$

Thus one gets equation 12:

$$ \hat{I}_p = i(T(t) - T^{\dagger}(t)). \qquad (12) $$

Let us derive the induced current using linear response theory. Indeed, one can treat $H_T$ perturbatively because the probability of tunneling exponentially decreases with the distance between the two conducting materials (see 1.1).

**Reminder (Kubo formula):** Let us consider a system described by the Hamiltonian $\hat{H}$, and $\hat{A}$ an observable. We now add a perturbation $\hat{V}(t)$ in the system so that the Hamiltonian of the system is $\hat{H} + \hat{V}(t)$. $\hat{V}(t)$ can be written as $\hat{V}(t) = \sum_{\alpha} B_{\alpha} f_{\alpha}(t)$, where $f_{\alpha}(t)$ is a field and $B_{\alpha}$ the conjugate variable of the field. If we are in linear response theory, then one can derive the Kubo Formula:

$$ \langle \hat{A}(t) \rangle - \langle \hat{A} \rangle_0 = \sum_{\alpha} \int_{-\infty}^{+\infty} dt' \chi_{AB\alpha}^{R}(t-t') f_{\alpha}(t') + o(f^2), \qquad (13) $$

$$ \chi_{AB\alpha}^{R}(t-t') = -i\Theta(t-t')\langle[\hat{A}(t), \hat{B}_{\alpha}(t')]\rangle_{0}, \qquad (14) $$

Where $\chi_{AB\alpha}^R$ is the susceptibility and $\langle...\rangle_0$ is the mean value taken on the unperturbed system.

In our case, $\hat{V}(t) = \underbrace{\hat{H}_T(t)}_{B} \underbrace{\Theta(t)}_{f}$ and $\langle \hat{I}_p \rangle_0 = 0$. All this leads to equation 15:
$$ \langle \hat{I}_p(t) \rangle = \int_{-\infty}^{+\infty} dt' \chi^R(t-t') \Theta(t'), \qquad (15) $$

$$ \chi^R(t-t') = -i\Theta(t-t')\langle[\hat{I}_p(t), \hat{H}_T(t')]\rangle_0, \qquad (16) $$

Let us compute $\chi^R(t-t')$:

$$ \chi^R(t-t') = \Theta(t-t')\langle[(T-T^\dagger)(t), (T+T^\dagger)(t')]\rangle_0, $$

But $\langle[T(t), T(t')]\rangle_0 = \langle[T^\dagger(t), T^\dagger(t')]\rangle_0 = 0$ because of Wick's theorem (the mean value is taken for the unperturbed system).

Indeed, because of Wick's theorem, only terms like $\langle c_\alpha c_\alpha \rangle_0$ or $\langle c_\alpha^\dagger c_\alpha^\dagger \rangle_0$, which give 0, intervene in the computation.

$$ \chi^R(t-t') = \Theta(t-t')\langle[T(t), T^\dagger(t')]\underbrace{-[T^\dagger(t), T(t')]}_{[T(t), T^\dagger(t')]^\dagger} \rangle_0, $$

$$ \chi^R(t-t') = 2\Re[\Theta(t-t')\langle[T(t), T^\dagger(t')]\rangle], $$

$$ \chi^R(t-t') = 2\Re[\Theta(t-t') \sum_{kk',\sigma} \sum_{k''k'''',\sigma'} T_{kk'} T_{k''k''''}^* (\langle c_{1k,\sigma}^\dagger(t) c_{2k',\sigma}(t) c_{2k'''',\sigma'}^\dagger(t') c_{1k'',\sigma'}(t') \rangle_0 \\ - \langle c_{2k'''',\sigma'}^\dagger(t') c_{1k'',\sigma'}(t') c_{1k,\sigma}^\dagger(t) c_{2k',\sigma}(t) \rangle_0)], $$

Here again, also because of Wick's theorem:

$$ \left\{ \begin{aligned} \langle c_{1k,\sigma}^\dagger(t) c_{2k',\sigma}(t) c_{2k'''',\sigma'}^\dagger(t') c_{1k'',\sigma'}(t') \rangle_0 &= \langle c_{1k,\sigma}^\dagger(t) c_{1k'',\sigma'}(t') \rangle_0 \langle c_{2k',\sigma}(t) c_{2k'''',\sigma'}^\dagger(t') \rangle_0, \\ \langle c_{2k'''',\sigma'}^\dagger(t') c_{1k'',\sigma'}(t') c_{1k,\sigma}^\dagger(t) c_{2k',\sigma}(t) \rangle_0 &= \langle c_{2k'''',\sigma'}^\dagger(t') c_{2k',\sigma}(t) \rangle_0 \langle c_{1k'',\sigma'}(t') c_{1k,\sigma}^\dagger(t) \rangle_0. \end{aligned} \right. $$

The greater and lesser Greens functions are defined as follows:

$$ \left\{ \begin{aligned} G_{ab}^{>} (t, t') &= -i \langle c_a(t) c_b^\dagger (t') \rangle_0, \\ G_{ab}^{<} (t, t') &= i \langle c_b^\dagger (t') c_a(t) \rangle_0, \end{aligned} \right. \qquad (17) $$

Moreover, one has the following relations:

$$ \left\{ \begin{aligned} G_{ab}^{>} (\omega) &= -2i\pi(1 - n_F(\omega - \mu))A_{ab}(\omega - \mu), \\ G_{ab}^{<} (\omega) &= 2i\pi n_F(\omega - \mu)A_{ab}(\omega - \mu), \end{aligned} \right. \qquad (18) $$

Where $n_F$ is the Fermi-Dirac distribution and $A_{ab}$ the spectral function.

In our case, assuming the lesser and greater Green's functions are diagonal in $k$, one gets:

$$ \langle \hat{I}_p \rangle = \int_{-\infty}^{0} 2\Re[\sum_{kk',\sigma\sigma'} |T_{kk'}|^2 (G_{1k,\sigma}^{<}(t') G_{2k',\sigma'}^{>}(t') - G_{1k,\sigma}^{>} (t') G_{2k',\sigma'}^{<}(t'))], $$

$$ \langle \hat{I}_p \rangle = \int_{-\infty}^{0} 2\Re[\sum_{kk',\sigma\sigma'} |T_{kk'}|^2 \frac{1}{(2\pi)^2} \int d\omega d\omega' e^{-i(\omega-\omega')t'} (G_{1k,\sigma}^{<}(\omega) G_{2k',\sigma'}^{>}(\omega') - G_{1k,\sigma}^{>} (\omega) G_{2k',\sigma'}^{<}(\omega'))], $$

$$ \langle \hat{I}_p \rangle = \int_{-\infty}^{0} 2\Re[\sum_{kk',\sigma\sigma'} |T_{kk'}|^2 \frac{1}{(2\pi)^2} \int d\omega d\omega' e^{-i(\omega-\omega')t'} (G_{1k,\sigma}^{<}(\omega) G_{2k',\sigma'}^{>}(\omega') - G_{1k,\sigma}^{>} (\omega) G_{2k',\sigma'}^{<}(\omega'))], $$
Because $\int_{-\infty}^{+\infty} dt e^{i\alpha t} = \int_{-\infty}^{0} dt 2\Re e^{i\alpha t}$ and because $\int_{-\infty}^{+\infty} \frac{dt}{2\pi} e^{-i(\omega-\omega')t} = \delta(\omega - \omega')$ one gets:

$$ \langle \hat{I}_p \rangle = \frac{1}{(2\pi)^2} \int_{-\infty}^{+\infty} d\omega \sum_{kk',\sigma\sigma'} |T_{kk'}|^2 (G_{1k,\sigma}^{<}(\omega)G_{2k',\sigma'}^{>}(\omega) - G_{1k,\sigma}^{>}(\omega)G_{2k',\sigma'}^{<}(\omega)), $$

Using relation 18:

$$ \langle \hat{I}_p \rangle = 2\pi \sum_{kk',\sigma\sigma'} |T_{kk'}|^2 \int_{-\infty}^{+\infty} d\omega A_{1k,\sigma}(\omega - (\mu + eV_1)A_{2k',\sigma'}(\omega - (\mu + eV_2)[n_F(\omega - (\mu + eV_1)) - n_F(\omega - (\mu + eV_2))], $$

$$ \langle \hat{I}_p \rangle = 2\pi \sum_{kk',\sigma\sigma'} |T_{kk'}|^2 \int_{-\infty}^{+\infty} d\omega A_{1k,\sigma}(\omega) A_{2k',\sigma'}(\omega + eV)[n_F(\omega) - n_F(\omega + eV)], $$

One finally gets equation 19:

$$ \hat{I} = -e2\pi \sum_{kk',\sigma\sigma'} |T_{kk'}|^2 \int_{-\infty}^{+\infty} d\omega A_{1k,\sigma}(\omega) A_{2k',\sigma'}(\omega + eV)[n_F(\omega) - n_F(\omega + eV)], \quad (19) $$

Now, if we assume $|T_{kk'}|$ to be constant equal to $|T|$ (which is often the case if the fermi surface is not too anisotropic), then we get:

$$ \hat{I} = -e2\pi|T|^2 \int_{-\infty}^{+\infty} d\omega n_1(\omega)n_2(\omega+eV)[n_F(\omega) - n_F(\omega+eV)]. \quad (20) $$

Where $n_i = \sum_{k,\sigma} A_{ik,\sigma}$ is the density of states of conductor $i = 1$ or 2.

Let us now suppose that conductor 2 is a normal metal. Therefore, its density of states can be considered as constant and equal to $n_0$ because only electrons close to Fermi level participate to the dynamics of the whole system. The conductance of the system is then:

$$ \frac{d\hat{I}}{dV} = e^2 2\pi |T|^2 n_0 \int_{-\infty}^{+\infty} d\omega n_1(\omega) \frac{dn_F}{d\omega}(\omega + eV). \quad (21) $$
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