Temperature-linear Resistivity in Twisted Double Bilayer Graphene

We report an experimental study of carrier density (n), displacement field (D) and twist angle ({\theta}) dependence of temperature (T)-linear resistivity in twisted double bilayer graphene (TDBG). For a large twist angle ({\theta}>1.5{\deg}) where correlated insulating states are absent, we observe a T-linear resistivity (with the slope of the order ~10{\Omega}/K) over a wide range of carrier density and its slope decreases with increasing of n, in agreement with acoustic phonon scattering model semi-quantitatively. The slope of T-linear resistivity is non-monotonically dependent on the displacement field with a single peak structure. For device with {\theta}~1.23{\deg} at which correlated states emerge, the slope of T-linear resistivity is found maximum (~100{\Omega}/K) at the boundary of the halo structure where phase transition occurs, with signatures of continuous phase transition, Planckian dissipation, and the diverging effective mass; these observations are in line with quantum critical behaviors, which might be due to the symmetry-breaking instability at the critical points. Our results shed new light on correlated physics in TDBG and other twisted moir\'e systems.

processes that are rooted in quantum criticality [2][3][4] (Fig. 1b). The latter strange metal behaviors are observed in various correlated systems, for instance cuprates [5], iron-based compounds [6,7], heavy fermion systems [8,9], Kondo lattices [10], and frustrated lattices [11,12], where quantum critical points are found with the instabilities of order parameters. So far, most previous studies require multiple samples to obtain a single phase diagram, and improved in-situ sample tunability is demanded.
Moiré superlattices have emerged as a flat band system [13] for realizing a variety of interaction-driven quantum phases [14][15][16][17][18][19]. Large temperature linear (T-linear) resistivity has been observed in twisted bilayer graphene (TBG) [20,21], which bears a lot of similarity with that in optimally doped cuprates [22]. While the experiments in ref.20 support the electron-phonon scatterings [23,24], those in ref. 21 raise the possibility of strange metal behavior with near Planckian dissipation in magic angle TBG, leaving the origin of the T-linear resistivity in TBG still elusive. In twisted double bilayer graphene (TDBG), observations of halo structures with field-tunable symmetry-breaking correlated states, as well as T-linear resistivity, have been reported [25][26][27][28]. The phase transitions from normal metallic states to correlated states occur at the boundaries of the halo structure in TDBG. Compared to TBG, TDBG is advantageous in tuning electron interactions insitu by displacement field. The displacement field changes the flat bandwidth W and superlattice gap Δ in the band structure of TDBG [29], and thus acts as an extra parameter to control the relative strength of electron interactions to kinetic energy. The previous reports mainly focus on the nonlinear regime where ρ rapidly drops at low T, and the linear behaviors at high T are observed yet barely explored, demanding an in-depth and systematic exploration.
In this work, we systematically study T-linear resistivity in TDBG. We found T-linear resistivity in devices with twist angles from 1.23° to 1.91°. Firstly, we study the devices with large twist angles (θ >1.5°) where correlated insulating states are absent, and find that acoustic phonon scatterings can fully account for the T-linear resistivity. Meanwhile, we also demonstrate a displacement field tunable electron-phonon interaction. Secondly, in devices with ~1.23°, we reveal the features beyond the phonon model, and discuss the possibility of quantum criticality.
The TDBG samples are prepared by 'cut and stack' technique [30][31][32][33], and the devices are fabricated with a dual gate Hall bar geometry in bubble-free region of the samples. The dual gate geometry enables independent tuning of carrier density (n) and electric displacement field (D), via We first focus on 1.55° device in Fig. 2 where correlated states are absent at T=1.8K. Fig. 2b shows the ρ(T) curves for fillings v from 0.5 to 2.5 at D=0 V/nm. T-linear resistivity with ρ ∝ A1T is observed at T>T*, indicated by the yellow dashed lines in Fig. 2b. Here, A1 is the linear slope and T* is the characteristic temperature that separates linear regime at T>T* and nonlinear at T<T*. The A1 is plotted against n in Fig. 2c with values of about 10-30 Ω/K, and it decreases as n increases; the onset temperature T* can be obtained in a quadratic plot ρ−T 2 (see Supplementary Material Fig. S3) [34], and it increases with n as shown in Fig. 2d.
Our experimental data are captured quantitatively by electron-phonon scattering model in TDBG [35]. In this model, acoustic phonon scattering [36][37][38][39] is enhanced due to a reduced Fermi velocity in TDBG moiré bands, and it predicts a crossover from T at high temperature to T 4 at low temperature. If electron-electron Umklapp scattering [40] dominates over phonon scattering at low temperature, the nonlinear term could become T 2 , as shown by the black dashed lines in Fig. 2b. In the linear regime, the slope A1 resulted from acoustic phonon scattering [35] is given by  [20,35], and = (ℏ /2)/ * with effective mass * ≈ 0.15me calculated from band structure in continuum minimal model [13,41]. The calculated A1 and the measured value in Fig. 2c are in good agreement. Moreover, the measured T* matches well with 1/4 of Bloch-Grüneisen temperature in Fig. 2d, which further supports the phonon model [35].
Here is defined as 2ℏ / , where = /2 is the effective Fermi wave vector.
We also demonstrate that the electron-phonon scatterings are tunable by D field. Fig. 2e shows the ρ(T) curves at v = 2 for different field D from 0 to -0.7V/nm in the 1.55° device, with the linear behaviors well preserved. Fig. 2f presents the curves of extracted A1 as a function of D at different v, which shows a non-monotonical dependence with peaks at finite D. Since ∝ * at a fixed v, the D-dependent A1 suggests a field tunable moiré band dispersion in TDBG [42]. As indicated by the arrow in Fig. 2f, the peak position in D shifts with v, demonstrating the electron filling effect on band structure [26,43].
Next, we turn to the 1.23° device with correlated insulating states and concomitant halo structure in moiré conduction band. T-linear resistivity with ρ~A1T is observed at v=2 for T > T* in transitions between metallic phases with broken symmetry and those without [28]. Moreover, these A1 peaks are found following the boundary of the halo structure in Fig. 4d, implying the correlation between enhancement of A1 and phase transitions. From the phonon model, the increase of A1 at the halo boundary suggests the enhancement of effective mass.
To better reveal the effective mass, we analyze the temperature dependence of resistivity at T<T*. For the metals outside the halo structure, we observe signatures of quadratic resistivity ρ~A2T 2 at T < T*, as shown in Fig. 3a. The D dependent T* and A2 are plotted in Fig. 3d-e. We find a decreasing T* and an increasing A2 when approaching the critical displacement field. The quadratic power law at low temperature indicates Landau Fermi liquid behavior, in which ∝ / ∝ * / with α and β being the exponents that depend on the details of band structure [44]. It is worth noting that the A2(D) also qualitatively agrees with the A1(D) in the phonon scatterings model, since ∝ 1/ ∝ * / . The increased m* at the halo boundary is also revealed in the carrier density dependent A2(n) at a fixed D (Fig. 4c). Such quadratic dependence of resistivity is also well reproduced from another TDBG device with θ =1.28° in Fig.5a and 5b, and the corresponding field dependent A2 also tends to diverge at the halo boundary in Fig. 5c, similar to those observations in  [45,46], which we discuss in the following.
Firstly, the phase transitions at the critical points of the halo structure are continuous in the temperature range we studied, indicated by the vanishing thermal activation gap (Fig. 3f). Secondly, the decreasing T* and the increasing A2 when approaching the critical displacement field outside the halo structure in Fig. 3d and Fig. 4b are also signatures of the quantum critical behaviors [45]. Lastly, we find the large T-linear resistivity falls into the Planckian dissipation regime where the scattering rate is proportional to temperature, ℏ = and the coefficient C approaches O(1). In graphene moiré system [21], the number C can be extracted from T-linear resistivity slope A1 by = ℏ / * . In our cases, we obtain the C~1.8 for the A1 peaks at v=2 in Fig. 3c, by taking m*=0.3me, n=1.75×10 12 /cm 2 , A1=146Ω/K. All these facts suggest the possible existence of quantum criticality at the boundaries of halo structure.
At continuous phase transition critical points, quantum fluctuations associated with symmetrybreaking order parameters can be significant. In TDBG, the boundary of the halo structure separate states with different symmetry, which is likely to generate quantum fluctuations. More specifically, we suspect it is the spin fluctuations [47] that contribute to the T-linear resistivity. While spin-up electrons and spin-down electrons are equally filled outside the halo, one is more favored than the other inside the halo structure [26]. However, more experimental and theoretical investigations are required for a better understanding.
In conclusion, we systematically investigate the carrier density, displacement field and twist angle dependences of T-linear resistivity in TDBG. We demonstrate a dominant role played by acoustic phonons when correlated states are absent at >1.5°. The T-linear resistivity has a nonmonotonic displacement field dependence with a single A1 peak, revealing the field-tunable electron phonon interaction in TDBG. Moreover, we observe a "M"-shaped two peak structure in the presence of correlated states at ~1.23°. These peaks are found located at the halo boundary, separating states with different symmetries. We propose the possible existence of quantum criticality, supported by the evidences of continuous phase transition, vanishing T*, Planckian dissipation, and the diverging effective mass at the critical points. Our observations establish a close link between high temperature T-linear resistivity to low temperature ground states, and hopefully inspire more works about the nature of quantum criticality and ground states instability in TDBG [48]. Similar phenomena may also be expected in other field-tunable systems such as twisted monolayer-bilayer graphene [49][50][51] and ABC trilayer graphene/hBN moiré system [52].       Figure S1. Linear fit procedures. In the correlated regime, the linear range is limited and the linear fit is done at a temperature above the phase transitions. Firstly, we identify the critical temperature T* that separates correlated states and normal metal inside the halo structure. As shown in Fig. 3d and 4c, T* is also a good indicator as the onset temperature for the T-linear resistivity, i.e. ρ(T) shows a good linearity at T>T*. Then, as the field is tuned away from the center of the halo regime, T* gradually gets smaller and smaller, and it eventually approaches our base temperature at the boundary of halo structure. Once across the boundary, T* separate the regimes between the T-linear and quadratic power law, and it gradually grows when D is tuned further away from the halo regime. Others. For the field dependent A1 at a filling close to v=2, the peaks can be identified directly from the "M" shaped plots. For the field dependent A1 at a filling away from v=2, one peak is prominent while the other is very faint, giving a shoulder like structure. We identify the faint one by subtracting a linear background of the shoulder feature. The linear background is defined by linking the two side-turning-points of the shoulder.

Figures and Figure captions
Note A: Supplementary information for electron-phonon interaction in TDBG. Figure S1. Carrier density scaling of A1.The A1 scales linearly with 1/√ in (b), while scales with 1/ at D=0V/nm in 1.55° device in (a). The different scaling of 1/n or 1/√ may root in field tunable band structure in TDBG. For ~1.3° device, (c) at D=0V/nm, the A1 is linear in 1/n as well in both CB and VB. Here, n is relative value to full filling of moiré conduction (+ns) and valence (-ns) band, respectively. At D=-0.39V/nm. (d) A1 obtained from 1.33° device at the similar displacement field shows 1/√ scaling of A1. The scaling other than 1/n rule suggests nonparabolic band at finite displacement field of TDBG.