# Interaction driven giant thermopower in magic-angle twisted bilayer graphene

Arup Kumar Paul1111equally contributed Ayan Ghosh1222equally contributed
Souvik Chakraborty1333equally contributed Ujjal Roy1 Ranit Dutta1 K. Watanabe2
T. Taniguchi2 Animesh Panda1 Adhip Agarwala3 Subroto Mukerjee1 Sumilan
Banerjee1 and Anindya<EMAIL_ADDRESS>
Department of Physics,Indian Institute of Science, Bangalore, 560012, India.

National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.

Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38,
01187 Dresden, Germany.

Magic-angle twisted bilayer graphene (MtBLG) has proven to be an extremely
promising new platform to realize and study a host of emergent quantum phases
arising from the strong correlations in its narrow bandwidth flat band. In
this regard, thermal transport phenomena like thermopower, in addition to
being coveted technologically, is also sensitive to the particle-hole (PH)
asymmetry, making it a crucial tool to probe the underlying electronic
structure of this material. We have carried out thermopower measurements of
MtBLG as a function of carrier density, temperature and magnetic field, and
report the observation of an unusually large thermopower reaching up to a
value as high as $\sim\bf{100\mu V/K}$ at a low temperature of 1K.
Surprisingly, our observed thermopower exhibiting peak-like features in close
correspondence to the resistance peaks around the integer Moire fillings,
including the Dirac Point, violates the Mott formula. We show that the large
thermopower peaks and their associated behaviour arise from the emergent
highly PH asymmetric electronic structure due to the cascade of Dirac
revivals. Furthermore, the thermopower shows an anomalous peak around the
superconducting transition on the hole side and points towards the possible
role of enhanced superconducting fluctuations in MtBLG.

Introduction. Interactions in many body systems lead to various complex
emergent quantum phenomena like superconductivity, magnetism and correlated
insulating phases. Understanding these many body quantum phenomena and
utilizing their various applicability remains a key focus of condensed matter
research. For this reason, MtBLG is a promising material with its flat band
[1, 2, 3] induced plethora of exotic states like correlated insulator [2, 4,
5, 6, 7], superconductivity [3, 4, 7, 8, 6, 9], ferromagnetism [10], Chern
insulator [5, 11, 12, 13, 14, 15], quantum anomalous Hall effect [16],
nematicity [17, 18] and Pomeranchuk effect [19, 20]. The discovery of these
emergent quantum phases together with its easy tunability using a variety of
experimental knobs makes MtBLG an unprecedented platform to probe the role of
interactions in its unique electronic band-structure and further the search of
novel electronic properties with technological applicability. In this
direction, primarily electrical transport and local spectroscopic measurements
have been utilised to probe and study the nature of the various symmetry-
breaking electronic states [21, 22, 23, 24, 25]. Notably, recent measurements
of local compressibility [25] and scanning tunneling microscopy [24] have
revealed that the Fermi surface of MtBLG is highly malleable and undergoes
interaction-driven quantum phase transitions at integer fillings of Moire
lattice. The key finding is the resetting of the Fermi surface with strongly
PH asymmetric density of states (DOS) around the integer fillings via a
cascade of Dirac revival transitions [24, 25]. However, their unambiguous
signatures in global transport measurement are still lacking. Some signature
is observed in Hall measurement [5, 9], where the Hall carrier density
suddenly resets from finite value to zero without changing its sign at the
integer fillings, but, the nature and degree of PH asymmetry of the electronic
structure at the transition points remain unexplored.

Figure 1: Thermopower measurement set-up and device response. a, Set-up of
devices. Passing a current, $I_{h}$ through the heater creates the temperature
gradient across the device, where the colder end was directly bonded to the
cold ground (c.g). The gate voltage, $V_{bg}$ controls the carrier density
($n$) of the device. The relay switches between the low-frequency and high-
frequency measurement schemes. Low frequency, $2\omega$ method was used to
measure the thermoelectric voltage ($V_{2\omega}$) at $\sim 13$ Hz using
standard lock-in technique. High frequency ($\sim 720kHz$) thermal noise
($S_{V}$) measurement consisting LC resonant tank circuit and cryo-amplifier
(CA) was used to measure the temperature difference ($\Delta T$) across the
device as $S_{V}=2k_{B}\Delta TR$, where $R$ is the resistance of the device.
b, Resistance versus filling fraction ($n/n_{s}$) as a function of increasing
temperatures, where $n_{s}$ is the carrier density required to fill the flat
band. The top axis shows in terms of numbers of electrons ($\nu$) per Moire
lattice. The resistance peaks at the positive integer fillings ($\nu$) are
visible at lower temperatures. At the hole side no such peaks are observed
except at full filling, and bellow $\sim 500mK$ resistance drops to $\sim
1.8k\Omega$, within $n/n_{s}-0.5$ to $-0.75$, which shows emergence of
superconductivity. c, Measured $V_{2\omega}$ (upper panel) and $\Delta T$
(lower panel) as function of $I_{h}$ near the Dirac point at $1K$. d
$V_{2\omega}$ as function of $\Delta T$ at $1K$, for different $n/n_{s}$,
showing linear response regime. The slope of each curve gives the value of
$S$. e, Differential resistance ($dV/dI$) versus bias current ($I_{sd}$) as
function of temperature, at $n/n_{s}\sim-0.67$, where critical current
($I_{c}$) is maximum. The dark region corresponds to the superconducting
region. The green solid line shows the theoretically generated $I_{c}$ using
the BCS theory, $I_{c}(T)=I_{c}(0)(1-T/T_{c})^{2}$, where the $I_{c}(0)$ is
the experimentally measured value at $T=20mK$. f, $dV/dI$ with increasing
perpendicular magnetic field at $n/n_{s}\sim-0.67$ and at $\sim 20mK$. The
dark black region corresponds to the superconducting region, which is killed
at $B_{\perp}=0.1T$.

In this context, thermopower or the Seebeck effect is a unique tool to probe
the PH asymmetry of the electronic structure of MtBLG. Compared to electrical
transport, it is relatively non-invasive as an open circuit voltage ($\Delta
V$) is measured across the sample in the presence of a small temperature
gradient ($\Delta T$) relative to the sample temperature. In the linear
regime, using semi-classical Boltzmann transport theory and assuming energy
independent scattering time, the Seebeck coefficient ($S=-\Delta V/\Delta T$)
can be written as
$S=-(k_{\mathrm{B}}/Te)[\int(\epsilon-\mu)g(\varepsilon)(-df/d\varepsilon)d\varepsilon]/[\int
g(\varepsilon)(-df/d\varepsilon)d\varepsilon]$, where $e$, $T$, $\mu$,
$g(\varepsilon)$ and $-df/d\varepsilon$ are respectively the electronic
charge, temperature, chemical potential, DOS and derivative of Fermi function.
It can be seen that the numerator is an odd function due to the
($\epsilon-\mu$) term, and thus, the sign and magnitude of $S$ depend on
nature and extent of asymmetry of the DOS around the chemical potential. As a
result, $S$ is a highly sensitive probe to study the electronic structure
around the transition points of MtBLG. Moreover, MtBLG with superconducting
dome around half filling analogous of high-$T_{c}$ cuprate superconductors is
an ideal playground to study the thermopower response as it has been employed
to study the superconducting fluctuations in cuprates[26].

Motivated by these, we have extensively explored the thermopower response of
MtBLG and non magic-angle tBLG devices. Unlike previous works involving
graphene and tBLG [27, 28, 29, 30, 31, 32, 33, 34], we have utilized Johnson
noise thermometry [35, 36, 37, 38] to directly measure the temperature
gradient across the MtBLG device and accurately determine $S$ across a
temperature ranging from $100mK$ to $10K$. Our measurements reveal intricate
dependence of $S$ on carrier density ($\nu$), temperature ($T$), and magnetic
field ($B$). Our key observations are following: i) The measured thermopower
at low temperatures deviates completely from the expected zero-crossings
following the semi-classical Mott formula [39]. Instead, the thermopower
exhibits peak-like features at all positive integer fillings including the
Dirac point. ii) We observe a non-monotonic temperature dependence of the
thermopower. The thermopower reaches a record high value of $\sim 100\mu V/K$
at $1K$ for half filling of the conduction band. iii) We also observe
unusually large peaks in $S$ $\sim-(10-15)\mu V/K$ at sub-Kelvin temperatures
around the superconducting transition tracing the superconducting dome in the
hole side. We explain the first two results qualitatively using a simple model
within self-consistent Hartree-Fock (HF) approximations showing emergent
highly PH asymmetric DOS at integer fillings. Furthermore, we attribute the
anomalous peaks around the $T_{c}$ to enhanced superconducting fluctuations
due to inherent PH asymmetry in MtBLG. Our work highlights the ability of
thermopower to independently provide unique insights into the novel quantum
phenomena observed in MtBLG and opens a new route to achieve high
thermoelectric cooling devices and generators at cryogenic temperatures.

Set-up and device response. Figure 1a shows the schematic of the device and
the measurement setup for thermopower measurement. The devices consist of hBN
encapsulated twisted bilayer graphene (tBLG) on a $Si/SiO_{2}$ substrate. The
details are described in method and supplementary information (SI-1). For the
thermopower measurement, an isolated gold heater line, as shown in Fig. 1a, is
placed parallel to one side of the tBLG. To determine the thermopower or
Seebeck coefficient ($S$), one needs to measure the generated thermoelectric
voltage and the temperature difference ($\Delta T=T_{h}-T_{c}$). We have
utilized well established $2\omega$ lock-in technique [27, 28, 29, 30, 31, 32,
33] for measuring the thermoelectric voltage ($V_{2\omega}$) at $\omega\sim
13Hz$. To measure $\Delta T$, we have utilized Johnson noise thermometry [35,
36, 37, 38]. The details of the noise thermometry setup can be found in our
earlier works [35, 40] and shown in SI-3. The excess thermal noise,
$S_{V}=2k_{B}\Delta TR$ measured across the sample is used to determine the
$\Delta T$ (see SI-6 and SI-7), where $k_{B}$ is the Boltzmann constant and
$R$ is the resistance of the device. Fig. 1c shows the measured $V_{2\omega}$
and the $\Delta T$ as a function of the heater current at a bath temperature
($T$) of 1K. In Fig. 1d, we plot $V_{2\omega}$ with $\Delta T$ for MtBLG at
different carrier densities ($n$). The linearity of the plots in Fig. 1d
suggests that we are in the linear regime, and the slope of each curve gives
the $S$ for a given $n$. We have measured the $S$ from $100mK$ to $10K$ in the
linear regime by adjusting the heater current such that the $\Delta T$ always
remains much smaller than $T$ (SI-7). We have used three devices with twist
angles of $\sim 0.26^{0}$, $1.05^{0}$ $1.86^{0}$.

The gate-dependent resistance ($R$) of MtBLG for different temperatures is
shown in Fig. 1b. Here the gate voltage is replaced with an equivalent Moire
filling factor, $\nu=4n/n_{s}$, where $n$ is the carrier density induced by
the gate voltage and $n_{s}$ is the carrier density required to full-fill the
flat band (4 electrons/holes per Moire unit cell). As can be seen from the $R$
versus $\nu$ response, multiple resistance peaks appear at positive integer
fillings, including the Dirac point, and these peak features survive up to
$\sim$50K and above. On the contrary, for negative filling, we see the
prominent resistance peak at $\nu=-4$, and between $\nu=-2$ and $\nu=-3$, the
resistance drops below $600mK$ and saturates like a plateau at $\sim
1.8k\Omega$ showing the emergence of superconductivity. The resistance value
at the full-filling ($\nu=\pm 4$) continuously decreases with increasing
temperature. On the other hand, resistance value at $\nu=0$ and $2$ decreases
with increasing temperature up to $\sim 10K$ and then increases linearly,
showing metallic nature (SI-10 and SI-11). Fig. 1e and 1f plot the evolution
of differential resistance ($\frac{dV}{dI}$) versus bias current ($I_{sd}$)
response with temperature and perpendicular magnetic field, respectively, at
$\nu\sim-2.5$, and confirms the existence of the superconductivity though the
resistance is measured in two-probe geometry (Method and SI-12).

Figure 2: Thermopower response at integer Moire fillings. a, Measured
thermopower of MtBLG with carrier fillings at $10K$. The dashed horizontal
line corresponds to zero thermopower and vertical dashed lines correspond to
the integer number of electron fillings of the Moire lattice. b, Measured
thermopower at several temperatures from $0.2K$ to $7K$. For the clarity the
data are shifted by $20\mu V/K$ along the y-axis. The dashed horizontal lines
are the zero thermopower for the corresponding temperatures. c, Temperature
dependence of $S$ for the flat band in log-log scale at
$n/n_{s}=0,0.25,0.5,0.75$, and d, for dispersive bands at hole-side
($n/n_{s}=-1.31$) and electron-side ($n/n_{s}=1.47$). e, $S$ with
$B_{\parallel}$ for a different thermal cycle at $2K$. f, Comparison of $S$ at
half-filling with temperature between zero and $B_{\perp}$ = $3.5T$. g,
Derivative of resistance of MtBLG with carrier density at $0.2K$ according to
Mott formula. The dashed horizontal line correspond to zero derivative line.
The clear sign changes at $\nu=0,1,2$ are seen but absent in the $S$ data in
Fig. 2b. h, Measured $S$ and derivative of resistance for $0.26^{0}$ non-magic
angle tBLG at $1K$. The qualitative agreement of sign changes between them can
be seen.

Band reconstruction of MtBLG probed by thermopower. The Fig. 2a and 2b show
the measured thermopower versus $\nu$ at several temperatures, from $200mK$ to
$10K$ for MtBLG. At $10K$ (Fig. 2a), thermopower has approximate mirror
symmetry for both conduction and valence band albeit with opposite signs. It
can be seen that the thermopower changes its sign at the Dirac point, at flat
band full-filling ($\nu\sim\pm 4$) and around $\nu\sim\pm 1$. The sign of the
thermopower depends on the type of the carriers, positive for hole-like and
negative for electron-like carriers, and its magnitude goes to zero at the
symmetric points of the electronic structure as described by the semi-
classical equation. Like, at the Dirac point, the density of states (DOS) goes
to zero symmetrically from both the conduction and valence band. Similarly, at
the band full-filling with the energy gap between the flat and higher energy-
dispersive bands, the $S$ is expected to change the sign. One more sign change
is expected at the middle of the conduction or valence band as the single-
particle DOS of the flat band reaches a maximum (van-hove singularity - VHS)
around $\nu=\pm 2$. If the DOS is symmetric around the maxima, one would
expect a sign change in $S$ exactly at $\nu=\pm 2$. However, the inherent
asymmetry of the DOS, which is complex for MtBLG, of the conduction band or
valence band can give rise to the sign change shifted from $\nu=\pm 2$.

As we decrease the temperature below $10K$, the apparent asymmetry of the $S$
(Fig. 2b) between the conduction and valence band grows similar to the
asymmetry observed in the resistance data in Fig. 1b. Most importantly, the
thermopower exhibits a positive peak around $\nu\sim 2$, and its magnitude
increases rapidly with decreasing temperature and reaches a maximum value of
$\sim 95-100\mu V/K$ at $\sim 1K$, followed by a decrement of the magnitude
with a further reduction of the temperature. Similar, positive peaks are also
seen around $\nu\sim 1,3$ at $\sim 2K$ and at the Dirac point below $1K$. The
observed positive peak in thermopower at the positive integer fillings,
including the Dirac point, is quite striking. Any energy gap $\gtrsim
k_{\mathrm{B}}T$ either from the single-particle band structure or induced by
electronic interactions will give a sign change of $S$. In particular, one
would expect $S$ to go to zero at the resistance maxima, i.e., at the integer
fillings as the Mott formula [39]
$S=(\pi^{2}k_{B}T/3e)(d\ln(R)/dn)g(\epsilon)$ gives zero at those points and
shown in Fig. 2g for $0.2K$. Thus, one can see a complete violation of the
Mott formula for MtBLG. The violation persists even up to $10K$, as shown in
SI-13. On the contrary, for non magic-angle tBLG devices, the measured sign of
$S$ and the Mott formula matches well, as shown in Fig. 2h for $\sim 0.26^{0}$
(SI-Fig. 15b for $1.86^{0}$).

The recurring thermopower peaks (Fig. 2b) at integer fillings with a positive
sign (which usually occurs for hole-like carriers) suggest, at least within an
effective single-particle picture, repeated restructuring of the Fermi surface
at integer fillings such that overall hole-like carriers are dominant. The
pliable Fermi surfaces due to interactions around the integer fillings have
been reported in MtBLG and Stoner like transitions [24, 25, 5, 9] have been
observed experimentally. The key features of these transitions are – a
Lifshitz transition followed by a Dirac revival, which essentially gives rise
to large asymmetric DOS around the transition point such that, for $\nu>0$,
from one side (left side of the transition), the DOS rapidly drops whereas
other side (right side of the transition) the DOS increases gradually, similar
to a sawtooth. Such asymmetric DOS can give rise to peak in $S$ with large
value around the transition point as discussed in the theoretical section. It
can be seen in Fig. 2h and SI-13 that for non magic-angle tBLG devices, we do
not observe any thermopower peaks, and the measured $S$ is around $\sim 1\mu
V/K$ at $\sim 1K$ as expected for graphene-based devices at such low
temperatures [27, 28, 29].

The temperature dependence of $S$ for different integer fillings, including
Dirac point, is shown in Fig. 2c. The common key feature is the non-monotonic
temperature dependence of $S$ with a maxima at a certain temperature, which
depends on the fillings. For example, at $\nu\sim 2$ and $\sim 3$ the peak
appears around $\sim 1K$ whereas it is $\sim 0.3K$ for $\nu\sim 1$ and the
Dirac point. The deviation from the linear $T$ dependence of $S$ again
suggests the strong violation of Mott’s formula [39] for the flat band of
MtBLG. However, it can be seen that for the dispersive bands ($n/n_{s}\sim
1.47$ and $n/n_{s}\sim-1.31$) the $S$ increases almost linearly with $T$ (Fig.
2d), consistent with the Mott formula. Furthermore, the response of $S$ with
in-plane magnetic field ($B_{\parallel}$) underlies the nature of the ground
states at different integer fillings. As can be seen in Fig. 2e (for a
different thermal cycle as shown in SI-14) the thermopower peaks increase with
$B_{\parallel}$ at $\nu\sim 1$ , but decreases at $\nu\sim 2$. These
observations are consistent with the cascade of Dirac revival picture in Ref
[25], where the emergence of flavored symmetry breaking in MtBLG with
polarized ground state at $\nu\sim 1$ strengthens the transition, and thus
make it more PH asymmetric DOS resulting in higher $S$. At $\nu\sim 2$ the
value of $S$ decreases with both $B_{\parallel}$ and $B_{\perp}$, as shown as
a function of $T$ in Fig. 2f. It can be noticed that the peak position of the
$S$ shifted to lower temperature $\sim 0.6K$ at $B_{\perp}=3.5T$ with a value
of $\sim 70\mu V/K$, thus emphasizing its tunability for use as thermoelectric
cooling devices at cryogenic temperatures. It should be noted that the $S$ of
the dispersive band and non magic-angle tBLG devices remain insensitive to
$B_{\parallel}$ (see SI-14).

Figure 3: Thermopower across the superconducting transition. a, 2D colourmap
of resistance as a function of temperature and carrier filling for the hole
side flat band of MtBLG. The dark blue portion corresponds to the
superconducting region around the weaker Mott peak at $n/n{s}\sim-0.55$. The
open circles are the $T_{c}$ at different carrier densities as determined from
the differential resistance versus critical current plot as a function of
temperature as shown in Fig. 1e. b, 2D colourmap of measured thermopower as a
function of temperature and carrier filling. The blue dark portions are the
negative thermopower peak as shown as the cut lines in Fig. 3d for the carrier
densities marked by the white vertical arrows. The regions $I$ and $II$
correspond to two different dome like portions, where region $II$ matches well
with the superconducting dome seen in Fig. 3a as shown by the white dashed
line, which is the trace of $T_{c}$ as shown in Fig. 3a. The other white
dashed line enclosing the regions $I$ is the guiding line. c, The measured
thermopower at a $B_{\perp}=0.1T$ without peak like features in $S$. d, Open
circles with the connected lines in different panels are the cut lines of
measured $S$ as a function of temperature from Fig. 3b at
$n/n_{s}=-0.56,-0.65,-0.74$ and $-0.8$. The solid lines are corresponding cut
lines of $dR/dT$ from Fig. 3a. e, Thermopower cut lines for $B_{\perp}=0.1T$
at $n/n_{s}=-0.65.$ and $-0.8$.

Anomalous thermopower response around the superconducting dome. As shown in
Fig. 2b, there are no thermopower peaks for the valence flat band of the MtBLG
device in the temperature range of $2-10K$. This is consistent with the weaker
cascaded transitions observed for the hole side in Ref [25]. It was shown that
a larger twist angle $\sim 1.1^{0}$ is required to observe the sawtooth
behavior in the DOS for the hole side. However, for the MtBLG device, some
peak-like features are developed within $\nu\sim-2$ to $-3.5$ below $2K$. The
number of peaks and their positions in filling changes with decreasing
temperature; one weak negative peak is seen at $2K$, whereas three prominent
negative peaks are observed at $0.2K$. Fig. 3a shows the two-dimensional (2D)
color map of the resistance with temperature and fillings. The darker blue
region corresponds to the superconducting phase seen close to a weak
resistance peak at $n/n_{s}\sim-0.55$ (details in SI-12). The open circles
denote superconducting transition temperature $T_{c}$ as a function of
filling, obtained from critical current measurement shown in Fig. 1e and
SI-12.

Fig. 3b shows the corresponding 2D color map of $S$ at zero magnetic fields.
Here the darker blue shaded ribbons enclosing dome-like structures correspond
to negative thermopower peaks. A closer view shows two inter-penetrating dome-
like structures marked by the dashed white lines and labeled as $I$ and $II$
in Fig.3b. The dashed white line enclosing the region $II$ is the trace of
$T_{c}$ as shown in Fig.3a, whereas the white dashed line enclosing the region
$I$ is a guide to the eye to follow the locus of the broad negative
thermopower peaks. As evident from Fig. 3a and 3b, there are clear
resemblances between the peak position in $S$ enclosing region $II$ and
$T_{c}(n)$ dome. This can be further ascertained from Fig. 3d, where the $S$
from Fig. 3b and $dR/dT$ for Fig. 3a are plotted as a function of $T$ for
fixed fillings, $n/n_{s}=-0.56$ and $-0.65$. Both $S$ and $dR/dT$ exhibits
prominent peaks around $T_{c}$. However, peak in $S$ enclosing region $I$ in
Fig. 3b has hardly any direct correspondence in the resistance data in Fig.
3a, as can be seen in Fig. 3d, where $S(T)$ exhibits broad peak at
$n/n_{s}=-0.74$ and $-0.80$, but $dR/dT$ does not show any such feature around
the same temperature.

Apart from the correlation between locus of the thermopower peak in the $n-T$
plane and $T_{c}(n)$ over a large part of the superconducting dome, the most
important clue for the possible origin of the unusual thermopower peak is
obtained by applying a $B_{\perp}$. As shown in Fig. 3c, the peak in $S$
completely disappears with the application of tiny $B_{\perp}=0.1$T. This is
demonstrated in Fig. 3e by plotting $S(T)$ for $n/n_{s}=-0.65$ and $-0.8$.
These observations suggest that anomalous peak in $S$ in Fig. 3b, particularly
enclosing the region $II$, directly relates to the superconductivity of MtBLG.
One cannot help but to notice the superficial resemblance of the trace of the
broad thermopower peak enclosing region $I$ with the putative pseudogap
temperature line in the doping-temperature plane of cuprates [41]. However,
the origin of the boundary enclosing region $I$ remains unclear at this moment
and will be an interesting direction for future studies, like the possible
role of nematicity and competing orders in MtBLG [18]. Nonetheless, even the
observed anomalous thermopower peak around $T_{c}$ is quite striking.
Anomalously large thermopower response around $T_{c}$, and even extending far
above $T_{c}$, have been reported in transverse thermoelectric coefficient,
i.e. the Nernst coefficient [26], for high $T_{c}$ cuprate superconductors.
However, experimental observations of peaks in the longitudinal component of
$S$ are scarce [42] and remain controversial. Such thermopower peaks have been
theoretically predicted [42, 43] to exist above, albeit close to, $T_{c}$ from
superconducting fluctuations under certain situations, like for a
superconductor in the dirty limit [42, 43]. The observed thermopower in our
MtBLG device might have similar origin, as disorder due to twist-angle
inhomogeneity [25, 44] is naturally present in the system. Effects of
superconducting fluctuations in the thermopower [42, 43] are expected to be
much more enhanced in MtBLG near half filling due to inherent PH asymmetry
from the low-energy VHS of the flat band as well as the emergent PH asymmetry
due to the proximate weak cascade transition in the hole side.

Figure 4: Cascade of Dirac revivals and thermopower peaks around integer
fillings. a, The occupation ($n_{\alpha},~{}\alpha=1,\dots,4$) of individual
flavors as a function filling $n/n_{s}$, obtained from Hartree-Fock (HF)
calculations for $T=0.005W$ and a local inter-flavor interaction $U=1.2W$. At
zero filling, the Dirac cones corresponding to the four spin-valley degrees of
freedom are degenerate. A cascade of Stoner-like transitions close to the
integer fillings lead to complete filling of one, two and three of the flavors
successively while the filling of the remaining flavor(s) resets to Dirac
point. b, The resultant HF DOS $g(\epsilon)$ at the chemical potential at
$T=0.005W$ exhibits the sawtooth feature. The effective single-particle DOS
$g(\epsilon)$ changes drastically at each integer fillings and shows strong
low-energy particle-hole asymmetry (SI-Fig. 20). c, The calculated thermopower
$S$ in HF approximation (red circles with line) shows peak-like features
around the integer fillings due to the Dirac revivals (Fig. a). The top to
bottom panels are in the order of decreasing temperatures
$T=0.27W,0.14W,0.08W,0.005W$. The thermopower $S_{0}$ obtained using ’rigid’
non-interacting single particle DOS (Fig. c, top panel, inset) is shown by the
solid blue lines. The non-interacting $S_{0}$ exhibits one sign change around
the half filling for higher temperature $T\gtrsim 0.08W$, and multiple sign
changes across the two VHSs of the non-interacting DOS at very low temperature
(the bottom panel), but these peaks depend on the details of non-interacting
DOS and are not necessarily tied to integer the fillings, unlike $S(\nu)$ in
the interacting cases. d, Non-monotonic temperature dependence of thermopower
at integer fillings from the cascaded transitions. The Dirac revived symmetry
broken state at $n/n_{s}=0.50$ only gets stabilized at finite temperature for
the particular non-interacting DOS, as indicated by a sign change in $S$
around $T\sim 0.1W$.

Emergent low-energy particle-hole asymmetry and giant thermopower peaks. As
already mentioned, the thermopower peaks suggest strong emergent low-energy PH
asymmetry of the putative correlated states at integer fillings, at least,
within the effective single-particle or Hartree-Fock descriptions of various
possible symmetry broken states [45, 46, 47]. As discussed in method and
SI-16, we use a simple minimal model [25] with four fermionic flavors,
corresponding to the spin and valley degrees of freedom, each described in
terms of a single-particle DOS [25, 1, 48], and interacting via a local
Coulomb interaction. We treat the latter via self-consistent Hartree-Fock (HF)
approximation and use the resulting HF DOS to calculate the resistivity and
thermopower as a function of filling and temperature, using Kubo formulae
(SI-16). We have used different non-interacting DOSs, obtained from both
effective continuum models [1, 48], with and without lattice relaxation
effects [48], as well as tight-binding model [49] (SI-16). The main results
are summarized in Fig. 4, where the peak value of $S$ reaches $\sim
50-100~{}\mu\mathrm{V/K}$ for $\nu\simeq 2,3$ at $T\sim 0.1W$, consistent with
our experimental observations (Fig. 2b). The temperature range $T\simeq
0.005W-0.27W$ corresponds to $\sim 200~{}\mathrm{mK}-13~{}\mathrm{K}$, for a
bandwidth $2W\simeq 10~{}\mathrm{meV}$. For comparison, in Fig. 4c (solid blue
lines), we have shown the $S_{0}$ for the non-interacting case (see Fig. 4c
caption for details). We find the thermopower peak around an integer filling
to be a robust feature whenever the Dirac revival is stabilized within the HF
approximation, and support the simultaneous presence of thermopower (Fig. 2b)
and resistance (Fig. 1b) peaks, as well as the non-monotonic temperature
dependence of $S$ (Fig. 2c) in our experiment.

Discussion. Our theory qualitatively captures the thermopower peaks, but the
$S(n)$ follows an overall ‘background’ profile dictated by the non-interacting
$S_{0}(n)$ (Fig. 4c) and its sign change around the half filling. There could
be several reasons behind the deviation of $S$ obtained from HF approximation
compared to the experimental one, e.g., effects of more complex and realistic
single-particle DOS for MtBLG than the used continuum model [25, 1, 48], twist
angle inhomogeneity [25, 44] and strong correlations in the strange metal
state [50] (see SI-16 for a detailed discussion). Moreover, we should note
that there are theoretical models [45, 46, 47] which lead to a small gap
($\Delta$) at the Dirac revivals. This will be consistent with the
simultaneous presence of thermopower, and resistance peaks at integer fillings
provided $k_{\mathrm{B}}T\gtrsim\Delta$. At very lower temperatures, $S$ is
expected to change sign across the position of resistance peak. Thus, our
thermopower results put a tighter upper bound, $\Delta\sim 0.1-0.2$ meV
(activation gap in SI-11), on the correlation-induced gap at integer fillings.
In the SI-16, we also discuss the expected thermopower from various other
kinds of ground states and their possible signatures in our measurements.

Conclusion. In summary, our experiments reveal unusual low-temperature
thermopower response and anomalously large peaks in Seebeck coefficient
originating from emergent highly particle-hole asymmetry in the flat band of
MtBLG. In the hole side of the MtBLG flat band, close to half-filling, our
measurements also reveal two interpenetrating dome-like structures, traced by
peaks in the thermopower on the doping-temperature plane. The boundary of the
dome at lower temperature coincides with the line of superconducting
transitions, presumably providing a rare glimpse of the enhanced
superconducting fluctuations in MtBLG.

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##  1 Methods

###  1.1 Device fabrication and measurement scheme:

The devices consist of hBN encapsulated twisted bilayer graphene (tBLG) on a
$Si/SiO_{2}$ substrate. The typical length and width of the devices are $\sim
6\mu m$ and $\sim 2\mu m$, respectively. The usual ‘tear and stack’ technique
[2, 3] is used to fabricate the device and is described in detail in the
supplementary information (SI-1). For the resistance measurement, we employ
the low-frequency ($\sim 13Hz$) lock-in technique (SI-3). For the thermopower
measurement, an isolated gold line, as shown in Fig.1a, is placed parallel to
one side of the tBLG at a separation of $\sim 3\mu m$. Passing a current
($I_{\omega}$) through the heater creates a temperature gradient across the
length of the tBLG as depicted by the color gradient (red to blue) in Fig. 1a.
As a result, the contact near to the heater will be hotter ($T_{h}$) compared
to the far contact ($T_{c}$). The temperature of the far contact ($T_{c}$) is
maintained at the bath temperature of the cryo-free dilution fridge by
directly anchoring it to the cold finger attached to the mixing chamber plate
, which we call a cold ground (c.g). To measure $\Delta T$, we have utilized
Johnson noise thermometry. As shown in Fig. 1a, the thermometry circuit
consists of a LC resonant ($f_{r}\sim 720kHz$) tank circuit, followed by a
cryogenic amplifier (ca). The relay sitting at the mixing chamber plate (Fig.
1a) is used to switch between the thermoelectric voltage and temperature
measurement.

###  1.2 Activation gaps, band-width and superconducting transition
temperature of MtBLG:

The value of the resistance at the full filling ($\nu=\pm 4$) continuously
decreases with increasing temperature up to much higher T $\sim 100K$. On the
other hand, the value of the resistance at $\nu=0$ and $2$ decreases with
increasing temperature up to $\sim 10K$ and then increases linearly, showing
metallic nature (SI-10). These observations are consistent with earlier
reports for MtBLG [2, 4, 5, 6, 7, 3, 8, 9]. The gap ($\Delta$) determined from
the activated plot for $\nu=0$, $\nu=2$ and $\pm 4$ are, respectively, $\sim$
$0.05meV$, $0.25meV$, $11.5meV$ and $9.25meV$ as shown in the SI-10 and SI-11.
Furthermore, it can be seen (SI-10) that there is a crossover from metallic
nature to insulating one at a higher temperature due to interband excitation
of the carriers between the flat and dispersive bands. From the crossover
temperature, $\sim 150K$ around the Dirac point, one can estimate the
bandwidth ($2W$) and found to be of the order of $\sim 10meV$ for the MtBLG.
In Fig. 1e and 1f, we have shown the differential resistance versus bias
current with temperature and perpendicular magnetic field around the
superconducting dome. In order to extract the transition temperature at a
giving filling, we compare the experimental data with the theoretically
generated critical current versus temperature using BCS theory,
$I_{c}(T)=I_{c}(0)(1-T/T_{c})^{2}$, where the $I_{c}(0)$ is the experimentally
measured critical current at $T=20mK$, and vary the $T_{c}$ such that the
theoretically generated $I_{c}(T)$ traces the experimentally measured $I_{c}$
in Fig. 1e. This was repeated for other carrier densities and shown in the
SI-12. The extracted the critical temperature was found to be $\sim 500mK$ at
$\nu\sim-2.5$ (SI-12). The measured value of $T_{c}$ and critical field
($B_{c}\sim 100mT$) of our device matches reasonably well with the available
data for MtBLG [3, 4, 7, 8, 6, 9]. Note that the Fraunhofer-like pattern in
Fig. 1f can be explained by the interference between percolating
superconducting paths separated by the normal islands, which is generic
feature in MtBLG due to twist-angle in-homogeneity [25, 44, 9]. These patterns
further establish the existence of the superconductivity in our device though
the measurement was carried out in two-probe geometry.

###  1.3 Theory:

As discussed in detail in SI-16, we compute the thermopower and resistivity as
a function of filling and temperature for the model of Ref.[25] using Hartree-
Fock (HF) approximation. The model consists of four spin-valley flavors,
interacting with local Coulomb interaction $U$. For the results reported in
the main text, we have taken $U=1.2W$, where $W$ is the band width of the
conduction (valence) band. The HF self-consistency equations depend on the
non-interacting DOS of the moire’ flat bands. We use various non-interacting
DOSs, e.g. DOSs obtained from the continuum Bistritzer-MacDonal model [1] in
Ref. [25] and the DOS generated from the continuum model of Ref. [48], which
includes lattice relaxation effects. The self-consistent HF DOS is then used
to compute thermopower and resistivity via Kubo formulae neglecting vertex
corrections. We assume a constant band velocity and use a small impurity
scattering rate $\Gamma_{0}=0.001W$ in the Kubo formulae.

##  2 Acknowledgements

A.D. thanks the Department of Science and Technology (DST), India for
financial support (DSTO-2051), the MHRD, Government of India under STARS
research funding (STARS/APR2019/PS/156/FS), and also acknowledges the
Swarnajayanti Fellowship of the DST/SJF/PSA-03/2018-19. K.W. and T.T.
acknowledge support from the Elemental Strategy Initiative conducted by the
MEXT, Japan and the CREST (JPMJCR15F3), JST.

##  3 Author contributions

S.C., A.K.P. and U.R. contributed to device fabrication. A.G. and A.K.P.
contributed to data acquisition and analysis. R.D. contributed in initial
measurements. A.D. contributed in conceiving the idea and designing the
experiment, data interpretation and analysis. K.W and T.T synthesized the hBN
single crystals. A.P., A.A., S.M. and S.B. contributed in development of
theory, data interpretation, and all the authors contributed in writing the
manuscript.

See pages - of MtBLG_Supplementary_Information_page1.pdf See pages - of
MtBLG_Supplementary_Information-pages-2-5.pdf See pages - of
MtBLG_Supplementary_Information-pages-6-11.pdf See pages - of
MtBLG_Supplementary_Information-pages-12-18.pdf See pages - of
MtBLG_Supplementary_Information-pages-19-25.pdf