Emergence of pseudogap from short-range spin-correlations in electron doped cuprates

F. Boschini†,1, 2, ∗ M. Zonno, 2, † E. Razzoli, 2 R. P. Day, 2 M. Michiardi, 2, 3 B. Zwartsenberg, 2 P. Nigge, 2 M. Schneider, 2 E. H. da Silva Neto, A. Erb, S. Zhdanovich, 2 A. K. Mills, 2 G. Levy, 2 C. Giannetti, 7 D. J. Jones, 2 and A. Damascelli 2, ‡ Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada Department of Physics & Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, Dresden 01187, Germany Department of Physics, University of California, Davis, CA 95616, USA Walther-Meißner-Institute for Low Temperature Research, Garching, 85748, Germany Department of Mathematics and Physics, Università Cattolica del Sacro Cuore, Brescia, BS I-25121, Italy Interdisciplinary Laboratories for Advanced Materials Physics (ILAMP), Università Cattolica del Sacro Cuore, Brescia I-25121, Italy


INTRODUCTION
In the presence of strong correlations, the interactions within and among various degrees of freedom often obfuscate the microscopic origin of exotic electronic phenomena [1][2][3]. As a most prominent example, the interplay between intertwined orders [4] continues to preclude a thorough understanding of the pseudogap phenomenon, an enigmatic state of correlated matter by now notorious from systems as diverse as unconventional superconductors [5][6][7], dichalcogenides [8,9], and ultracold atoms [10][11][12]. Broadly speaking, the pseudogap (PG) in the condensed matter is associated with a partial suppression of the electronic spectral weight in the vicinity of the Fermi level (ω=0), and evidence for the PG has been widely reported [13]. This behavior may be anticipated in the presence of long-range (or mesoscopic) order, e.g. spin-or charge-order: the loss of spectral weight in particular momentum-energy regions would be a simple consequence of the avoided crossings in the symmetry-reduced bandstructure [8,9,14,15]. However, this argument is unsatisfactory in the presence of strong electronic correlations and short-range order. Copper-oxide hightemperature superconductors are a paradigmatic example where the origin of the PG is still debated and, for both hole-and electron-doped cuprates, a clear answer has yet to emerge [1,[5][6][7][14][15][16][17][18].
For the electron-doped cuprates, the PG is believed to bear some relation to the antiferro-magnetic (AF) order which itself extends over a wide doping range. As illustrated in Fig. 1A, the PG is stable above the entire AF and superconducting (SC) domes, with its onset temperature indicated by T* [18][19][20][21]. Scattering experiments on electron-doped cuprates have shown that the long-range AF order disappears when entering the narrow SC dome [22,23], and that the commonly reported charge-order in cuprates does not exhibit a clear connection to the AF order [24,25], although a coupling to dynamic magnetic correlations has been recently shown [26]. In addition, 3D collective charge modes, which may play a substantial role in mediating high-temperature superconductivity, have been reported [27]. Several studies have discussed the connection between AF order and the PG in electron-doped cuprates [18-21, 23, 28-32]. In particular, T* has been proposed to be a temperature crossover for which the instantaneous spin-correlation length (ξ spin ) is comparable to the quasiparticle de where v F is the Fermi velocity) [23,33]. However, these considerations seem to fail close to optimal-doping, and a momentum-resolved study unveiling explicitly the relation between the PG spectral features and short-range AF correlations in electron doped cuprates is still missing.
We note that for dopings where long-range AF order disappears, i.e. when the SC dome arises, only short-range spin-fluctuations (ξ spin 50 a, where a is the unit cell size) are detected by inelastic neutron scattering [23] (see Fig. 1A). In this doping range, two distinct and seemingly unrelated temperatures can be identified: T* and T ξ . While T* has been measured by spectroscopic and transport probes as the onset of the PG [19][20][21], T ξ defines the temperature above which the low-temperature, already short-range, ξ spin starts to decrease with a hyperbolic behavior [23] (see Fig. 1A). To elucidate the relationship between these temperatures, and by extension the associated PG and short-range AF correlations, we have performed a time-and angle-resolved photoemission spectroscopy (TR-ARPES) study of optimally doped Nd 2-x Ce x CuO 4 (NCCO) (x=0.15, yellow arrow in Fig. 1A), which is characterized by ξ spin ≈20 a for T<T ξ [23].
TR-ARPES allows one to circumvent many of the challenges of a detailed temperaturedependent ARPES study such as surface degradation as well as coarse and uncorrelated sampling. As in standard pump-probe spectroscopy, a near-infrared pump pulse is used to perturb the system, with its relaxation studied by varying the temporal delay of a subsequent UV probe pulse. Due to the strong coupling of excited quasiparticles to underlying excitations, the energy released by the pump excitation is rapidly (≈100 fs) shared among all the degrees of freedom [34][35][36][37][38][39]. After this initial relaxation, an effective electronic temperature T e may be defined at each point in time, allowing a temperature-dependent scan to be performed continuously and with remarkable detail [40]. By applying this transient approach, we demonstrate here a direct connection between momentum-resolved spectroscopic features of the PG and short-range ξ spin (T e ), as extracted from inelastic neutron scattering data reported in Ref. [23]. In particular, we identify T* as a temperature crossover where the spin-fluctuation-induced spectral broadening exceeds the PG amplitude significantly, establishing the PG as a precursor of the underlying AF order. commonly referred to as the hot-spot (HS), and coincides with the location where an AFdriven PG is expected to be particle-hole symmetric [18]. In a mean field description, the commensurate q=(π,π) folding of the Fermi surface is driven by a strong quasi-2D AF order in the copper-oxygen plane [18]. The Green's function can then be written as [15,32,33]:

Fermi surface mapping and modeling
where k is the bare energy dispersion, ∆ PG the AF-driven pseudogap spectroscopic amplitude, which is determined by the local Coulomb interaction and spin susceptibility [33], and Γ a quasiparticle broadening term. For simplicity, we do not distinguish the two Γ terms in Eq. 1, even if they may be associated with different effects. The first term is related to quasiparticle spectral broadening and the second to pseudogap filling via reduction of ξ spin [31,33]. Using Eq. 1 we can calculate the spectral function A(k, ω) = − 1 π Im[G(k, ω)] [42] and compute the Fermi surface (Fig. 1C), finding that it agrees well with our experimental data and previous ARPES mapping studies [28,29]. We used ∆ PG =85 meV for simulation purposes, in agreement with previous optical and ARPES studies [19-21, 28, 29, 43] (details in Supplementary Materials).

Tracking of the pseudogap spectral weight in an ultrafast fashion
Before moving to a detailed analysis of our TR-ARPES data, we provide a simple picture to illustrate the general concepts which underlie our experimental strategy. Motivated by previous model calculations, we assume particle-hole symmetry at the HS [15,[31][32][33]: Figure 1D sketches how the temperature evolution of the PG would influence the photoemission intensity. In the scenario where the PG arises from AF correlations, for T e <T ξ a well established ξ spin supports a robust PG (black solid line, left panel). When the system is driven above T ξ , we would expect to observe a suppression of ξ spin followed by a filling of the PG (red solid line, left panel). In Fig. 1D, top-right, we depict the temperature evolution of the photoemission intensity in the framework of this model. The photoemission intensity, given experimentally by the energy distribution curves (EDCs) integrated along the momentum direction through HS, is proportional to the local density of states (DOS) modulated by the electronic distribution (f ). By computing the difference between the photoemission intensity for T e >T ξ and its counterpart for T e <T ξ , it is evident that a filling of the PG may lead to an increase of the photoemission intensity for negative binding energies (Fig. 1D, bottom-right). As shown later in more detail in Eq. 2, this differential curve is proportional to the experimental differential momentum-integrated EDCs (dEDCs). We emphasize that the increase of the photoemission intensity for negative binding energies is associated directly with a transient modification of the DOS, as a pure thermal broadening would suppress (increase) photoemission intensity symmetrically for all ω <0 meV (ω >0 meV), independently of the explored momentum region.

Intimate relation of the pseudogap to the spin-correlation length
Having defined the framework for our investigation, we now present our experimental results. By introducing thermal excitations via optical pumping, we investigate the temperature-dependent modification of the low-energy DOS at the HS in optimally-doped NCCO. Figure Fig. 1D, as well as experimental dEDCs at the HS which confirm the anticipated modification of the DOS (see Fig. 3B), with an enhancement of I HS ω=−50 in both the low (LF) and high fluence (HF) regimes. However, we note that the temporal response of I HS ω=−50 is dependent on the pump fluence. While the enhancement of I HS ω=−50 recovers exponentially within 2 ps for the LF regime, in the HF regime I HS ω=−50 saturates for approximately 2 ps and does not recover within the domain of pump-probe delay studied.
In order to establish a connection between the TR-ARPES phenomenology and the PG, as well as the reported T* and T ξ , we must convert the measured time dependence of our data to an effective temperature evolution [40]. This is done by fitting the Fermi edge width of the momentum-integrated EDCs along the near-nodal direction (ϕ ≈39 o , green solid line in Fig. 1B), establishing an effective electronic temperature T e for each time delay. We note that T e exceeds T ξ ≈75 K for approximately ∼2 ps (LF) and more than 8 ps (HF), as shown in Fig. 2B. The time frame for which T e >T ξ matches well the observed timescale of the enhancement of I HS ω=−50 in both LF and HF regimes ( Fig. 2A). This suggests that the increased intensity at the HS in the occupied states may be related to the modification of the PG as a consequence of the sudden suppression of the spin-correlation length for T e >T ξ . We confirm this hypothesis by plotting I HS ω=−50 directly as a function of T e in Fig. 2C. A substantial enhancement of I HS ω=−50 is observed only for T e >T ξ . Recognizing a resemblance between the temperature dependence of I HS ω=−50 and ξ −1 spin reported in Ref. [23], we have superimposed the latter as a green line and shadow in Fig. 2C, with appropriate offset and scaling. For temperatures T e >120 K, I HS ω=−50 (T e ) is found to saturate, marking a departure from ξ −1 spin . This saturation point is in very good agreement with the T* reported by other experimental probes [19][20][21]. Note that the deviation of I HS ω=−50 (T e >T*) from ξ −1 spin is not driven by a phase transition, but is rather a consequence of the PG filling, as we will discuss in the following.

Filling of the pseudogap driven by spin-fluctuations
To further clarify the origin of T*, we now present a comprehensive analysis and modeling of the photoinduced thermal modification of the PG for both LF and HF pump regimes.
The spectral features of NCCO are inherently broad, precluding the sort of detailed analysis of the transient spectral function which has been demonstrated for hole-doped cuprates [44].
Following Ref. [45], we focus our analysis on the temporal evolution of dEDCs, defined as: where DOS 0 (ω) and f 0 (ω) are the unperturbed quantities (see Supplementary Materials).
Note that the photoemission matrix-elements, not included in Eq. 2, represent a simple renormalization factor. The left column of Fig. 3A displays experimental dEDCs as a function of the pump-probe delay (τ ) and binding energy (ω) for the near-nodal cut (top panel, LF), and the HS (middle and bottom panels for LF and HF, respectively). The TR-ARPES data reported here can be simulated remarkably well using the simple model of Eq. 1 (see right column in Fig. 3A). The experimental dEDCs are reproduced through a substantial increase of the broadening term Γ alone, which phenomenologically describes the filling of the PG due to the reduction of the spin-correlation length [31,46]. Note that, Γ(T e ) ∝ ξ −1 spin (T e ) in Eq. 1 is assumed for simulation purposes, in agreement with theoretical predictions of Vilk and Tremblay [33]. A full closure of the gap fails to reproduce our TR-ARPES data (details in the Supplementary Materials). The filling, instead of closure, of the PG is in agreement with scattering studies [22,23], which show that while the spin-correlation length clearly decreases for T e >T ξ , the spectral weight associated with magnetic excitations displays a much weaker temperature dependence.
In further support of our interpretation of the HS data, we compare single experimental and simulated dEDCs along the near-nodal direction and HS for τ =+0.6 ps in Fig. 3B. Along the near-nodal direction we find an almost symmetric transient population/depletion (increase/decrease of the photoemission intensity for ω >/<0 across the Fermi level), characteristic of a mere thermal broadening effect. In contrast to this, at the HS, we observe instead the increment of the intensity for ω ≈ −50 meV and a modest (null) depletion signal for ω ≈ −20 meV for LF (HF), in good agreement with the model shown in Fig. 1D and data in Fig. 2. Finally, we plot in Fig. 3C the simulated analog to Fig. 2C, noting a good agreement between the two figures. In particular, assuming the direct relationship between the filling of the PG and ξ spin (predicted for 2D spin-fluctuations [33]), we find that the simulated filling of the PG saturates for temperatures T≈T* when the broadening Γ(T*)≈ 2∆ PG =170 meV.
We note that this empirical observation agrees well with recent theoretical investigations of the vanishing of the PG in the electron-doped cuprate Pr 1.3-x La 0.7 Ce x CuO 4 [47].

DISCUSSION
Our findings indicate that T* is a temperature crossover driven by spin-fluctuationinduced spectral broadening, which is associated with the weakening of the short-range AF correlations and incipient (π,π)-folding [31,46]. In addition, we note that at T≈120 K, which matches well with T* as extracted from transport and spectroscopic probes [19][20][21], ξ spin is comparable to λ B [ξ spin (T*)≈10-15 a and λ B (T*)≈15 a, see Supplementary Materials]. This suggests that the spin-fluctuation-induced spectral broadening can be discussed in terms of the loss of quasiparticle integrity, as proposed in Refs. [23,33], even when only short-range AF correlations are present. Note that these results prove that even underlying orders with correlation lengths comparable to λ B can dramatically affect the topology of the Fermi surface and associated transport properties [18-21, 28-31, 43, 48, 49].
In conclusion, we have performed a detailed TR-ARPES study at the hot-spot of the optimally-doped NCCO (x=0.15) electron-doped cuprate and demonstrated that the temperature dependence of the low-energy DOS is closely related to the spin-correlation length ξ spin . In particular, we identified three different temperature regimes for the PG, moving from low to high temperature: i) for T<T ξ , the PG spectral features do not change; ii) T ξ <T<T*, in which the PG begins to fill alongside with the reduction of ξ spin ; iii) T>T*, where the PG is completely filled-up. Our results show that the frequently reported onset temperature T* does not represent a thermodynamic phase transition, i.e. a sudden quenching of a well-defined order parameter; rather, T* is representative of an energy crossover driven by the shortening of the AF correlation length, manifest spectroscopically as a filling of the PG. This suggests that the energy scale associated with the PG survives for temperatures well above T* and bears a relation to the underlying Mott physics. Therefore, while the PG amplitude is mainly driven by local interactions, PG spectral features are washed out when the correlation length of spin-fluctuations is of the order of the quasiparticle de Broglie wavelength λ B . In addition, we note that our momentum-resolved study demonstrates how short-range correlations may play a significant role in defining the Fermi surface topology in complex materials.

Experimental design
Our TR-ARPES setup exploits the classic pump-probe scheme. A 6.2 eV probe beam is generated by fourth-harmonic generation of the fundamental wavelength (800 nm, 1.55 eV) of a Ti:sapphire laser (Vitesse Duo and RegA 9000 by Coherent, 250 kHz repetition rate).  scattering studies for x=0.145 [23], appropriately scaled and offset (we use the x=0.145 data from [23] as the closest to the doping x=0.15 studied here). T ξ marks where ξ spin starts to change as the temperature increases (see Fig. 1A), while we identify T* as the temperature at which the PG is completely filled (in agreement with the saturation of the photoemission intensity at the HS for ω=-50±10 meV for HF). T ξ for x=0.145 is indicated by the white filled diamond in Fig. 1A (from Ref. [23]), while for this explored doping (x=0.15) we estimate T ξ ≈75 K as the temperature above which the photoemission intensity at the HS starts to change (white open diamond in Fig. 1A).  Fig. 2C, and assuming Γ(T e ) = C · ξ −1 spin (T e ) (C ≈1.7 a eV, where a is the unit cell size, details in Supplementary Materials), as in Ref. [33].
(B), Experimental (upper panel) and simulated (bottom panel) dEDCs along the near-nodal direction (green line, LF), and at the hot-spot (black and red lines for LF and HF, respectively), at τ =+0.6 ps. (C), Simulated photoemission intensity at the HS, ω=-44±10 meV, as a function of the electronic temperature (black and red markers for LF and HF, respectively). The green line is the inverse of ξ spin , obtained from Ref. [23] as discussed in Fig. 2C.