Visualizing moiré ferroelectricity via plasmons and nano-photocurrent in graphene/twisted-WSe2 structures

Ferroelectricity, a spontaneous and reversible electric polarization, is found in certain classes of van der Waals (vdW) materials. The discovery of ferroelectricity in twisted vdW layers provides new opportunities to engineer spatially dependent electric and optical properties associated with the configuration of moiré superlattice domains and the network of domain walls. Here, we employ near-field infrared nano-imaging and nano-photocurrent measurements to study ferroelectricity in minimally twisted WSe2. The ferroelectric domains are visualized through the imaging of the plasmonic response in a graphene monolayer adjacent to the moiré WSe2 bilayers. Specifically, we find that the ferroelectric polarization in moiré domains is imprinted on the plasmonic response of the graphene. Complementary nano-photocurrent measurements demonstrate that the optoelectronic properties of graphene are also modulated by the proximal ferroelectric domains. Our approach represents an alternative strategy for studying moiré ferroelectricity at native length scales and opens promising prospects for (opto)electronic devices.


Supplementary note1: Two approaches to quantifying the carrier density in graphene with s-SNOM
There are two approaches to quantifying the carrier density in graphene using s-SNOM data.One is to extract the carrier density from the period of the propagating plasmon polariton 1,2 , and another is based on the near-filed amplitude or phase evolution 3 .Now we discuss these two approaches.We note that the second approach is more suitable for samples with inhomogeneous doping, such as the samples with moiré patterns in this work.
(1) Approach 1: to extract the carrier density of graphene from the plasmon polariton period In two-dimensional electron systems, the plasmon can be excited by a photon with proper momentum.With s-SNOM, whose tip can impart momentum, the plasmon excitations can be measured.Once the tip momentum is in resonance with the plasmon polariton momentum, the propagating plasmon polariton can be formed, manifesting in fringes in the real space s-SNOM image.The plasmon polariton obeys the scaling law   ∝  1/2  1/4 , where   is the plasmon energy,  is the momentum, and  is the carrier density 4 .The momentum  can be read from the plasmon propagating period  ( = 2  ).Therefore, for a given photon energy, the carrier density information can be extracted from the plasmon polariton period.
We note that for a device with multilayer materials, the scaling law would be modified by the device structure and the dielectric functions of these materials.However, the resultant scaling law can be readily obtained by solving the Fresnel reflection of p-polarized light (  calculations).
It is also noteworthy that approach 1 is only suitable for the cases with homogenous carrier doping.For a device with carrier density inhomogeneity, the boundaries of the electron/hole puddles can launch and reflect the plasmon polariton, thereby forming complex plasmon propagating patterns.The extracted plasmon period and the corresponding carrier density would be averaged ones.
(2) Approach 2: to extract the doping from near-field scattering amplitude or phase contrast As discussed above, for a sample with moiré patterns, we cannot use the propagating plasmon polariton to extract the carrier density in each domain.Now we should focus on the near-field scattering amplitude or phase.In this work, the near-field amplitude was measured.For graphene, the scattering amplitude is a function of carrier density.Therefore, we can first obtain the relationship between the near-field amplitude and the carrier density, and then use this established relationship to get the carrier density information.
In this work, we are interested in the doping difference between the graphene above AB and the graphene BA ferroelectric domains.Thus, by comparing the near-field amplitude of graphene above these two domains, an accurate carrier density can be extracted.To obtain the carrier density, we need to know the charge density point of each domain and the geometric capacitance of the device.

To get the charge density point (CNP):
We recorded the near-field amplitude, s 4 , on each type of domain when the backgate voltage,   , was swept.Namely, we obtained s 4 versus   on two domains.The CNPs of the two types of domains correspond to the peak positions of s 4 , in the s 4 versus   curves.We denote the CNPs of the AB and BA domains as  − , and  − , respectively.We emphasize that we should use the doping regimes without propagating/reflected plasmon polariton to analyze the carrier density.In the plasmon resonance regime, the measured near-field signal would be contributed to by the propagating plasmon polariton launched from nearby domains.In this case, the near-field amplitude is not exclusively from the local domain, and a carrier density error might arise.phase image of piezoresponse force microscopy (PFM) on the R-stacking WSe2/graphene device.The triangular domains are observed.

Device B
Supplementary Figure 2| Device B a,b, Optical microscope image of Device-B.The gold contacts are numbered.Contact "S" is grounded.Contact "D" is connected to a preamplifier for photocurrent measurements.c, The amplitude image of piezoresponse force microscopy (PFM) on the t-WSe2/graphene device.The domain boundaries are observed.

Analytical result of the near-field signal
Before using the light-rod model to numerically calculate the near-field signal, we first use the simplified model to provide an intuitive analytical result on the near-field signal evolution.
The tip-sample coupling G is determined by the tip momentum weighting function () and momentum-dependent Fresnel reflection coefficient   .That is,  ≈ ∫ ()  (, ) (1) The sample has three layers: the WSe2 bilayer on the top, graphene, and dielectric (h-BN, SO2 on Si) on the bottom. with Here −  2 , j=0, 1, 2, which correspond to vacuum, WSe2, and h-BN.
1) At the low doping level and the given excitation energy, the plasmon resonance momentum is much larger than the tip momentum, The real permittivity decreases as the carrier density increases, as shown in Supplementary Figure 5.It should be noted that the peak in the real permittivity at 2 =  is smeared out by the damping [5][6][7][8] .At low temperatures, this peak could appear.So (  (, )) decreases when the Fermi energy is tuned away from the charge neutrality point, thus resulting in the decrease in the near-field scattering amplitude.
2) However, when the carrier density is high enough, the plasmon momentum will shift towards the tip momentum, eventually overlapping.Then the near-field amplitude will increase, as shown in Supplementary Figure 6.
Supplementary Figure 5| The optical conductivity and permittivity of graphene with a photon energy of 860 cm -1 .Top: Conductivity of graphene with a photon energy of 860 cm -1 as a function of Fermi energy.The Drude response only considers the intraband transition, whereas the random phase approximation (RPA) includes both the intraband and interband transitions.Middle: The real permittivity of graphene as a function of Fermi energy.Bottom: The imaginary permittivity of graphene as a function of Fermi energy.The relaxation rate used here is 100 cm -1 .

Numerical simulations of the near-field signal
The near-field signals are simulated using the light rod model 9 .With doping, the near-field amplitude near  = 860  −1 first decreases and then increases, which is consistent with the experimental results.The Fresnel reflection coefficient is calculated using the transfer matrix method.We find that the plasmonic resonance, which corresponds to a peak of (  ), gradually shifts toward lower momentum with increasing doping.At a specific doping, the plasmon momentum overlaps the momentum imparted by the tip.To further verify that the observed moiré contrast originates from the ferroelectric modulated plasmon response in graphene, we performed control experiments on a sample without graphene.The sample structure is the same as that in Devices A and B, except for excluding a graphene layer.The sample structure is shown in Supplementary Figure 9a.The back gate electrode is a graphite layer, and the gating dielectric is a h-BN flake with a thickness of 40 nm.No noticeable near-field amplitude evolution was observed in WSe2 with carrier density doped up to 1.3 × 10 13  −2 (Supplementary Fig. 9 d,e)., where N,  3 , and  3 ̅ are number of pixels, near-field amplitude of each pixel, and the mean value of near-field amplitude.

General Remarks
The electrical potential immediately above a 2D plane made of 'ferroelectric' moiré superlattices can be well approximated by due to lattice relaxation, the domain walls are much thinner than the domain period.Away from the 2D plane, this potential decays with a decay length on the order of the moiré period.The periodic ferroelectric potential causes doping of graphene placed parallel to the plane of moiré superlattices.The resulting local Fermi energy () of graphene is determined by where ) is the screening electrical potential due to the doped charge on graphene,  = 1   2   2 ℏ 2 Sign[] is the local charge density,   is the gate voltage, and  is the distance of graphene to the gate.
In the simple case of a stripe moiré lattice, the potential has the analytical form: where  is the strip period, and ℎ is the distance away from the 2D plane.If the screener is a perfect metal, the screening charge is

Estimations by single Fourier component
Supplementary Figure 10| Schematics of the device.
For the device in Supplementary Fig. 10, where the moiré ferroelectric is on top, the resulting screening charge in graphene may be represented as a doping chemical potential satisfying:

𝜖
) of the device, this voltage corresponds to a doping density of  = 3.7 × 10 11 cm −2 , considering the thickness  1 = 60 nm and out-of-plane dielectric   = 3.48 for h-BN and the thickness  2 = 285 nm and dielectric  = 3.9 for SiO2.
This density corresponds to a doping level of 71 meV.To explain such a large doping level, one needs to assume that the ferroelectric potential is   ∼ 1.1 eV.

Effect of the water layer on ferroelectricity-induced doping
In addition, we also performed theoretical calculations to show the effects of potential contaminations on ferroelectricity-induced doping.We took water as an example since water is a prototype polar molecule.

Numerical Results for the nonlinear screening problem
Supplementary Figure 13| Numerical results with a grid of  × .Left: The ferroelectric potential (about 66 meV) of twisted WSe2.The moiré period (length of a lattice vector) is 340 .
The distance between the graphene and the gate is 340 nm.The dielectric of the environment is  = 5.Middle: The Fermi energy of graphene on top of WSe2.Right: Line-cut of the middle panel at  = 0.43 .

Charge transfer due to in-gap states of WSe2
We assume there are some immobile electronic states, with a density of states D, at energies inside the gap of WSe2 such that there is charge transfer between WSe2 and graphene.Experimentally, bare WSe2 is slightly n-doped.Therefore, there is probably electron transfer from WSe2 to graphene.These in-gap states have wavefunctions evenly distributed among the two WSe2 layers, and therefore they are unaffected by the ferroelectric potential.Since electrons in graphene experience the ferroelectric potential, the charge transfer between graphene and WSe2 will be inhomogeneous, alternating across AB and BA domains.The resulting local doping level  in graphene is: where   is the ferroelectric potential,  0 is the chemical potential assuming no ferroelectric potential,  = Therefore, the same ferroelectric potential leads to a large doping level on graphene.
If  is large, then considering that ℏ   ≈ 400 meV ≫   , one has  ≫   .This scenario seems to be consistent with the large alternate doping in the experiment.
Since the polarization can be approximated as a sinusoidal function, only the leading term will be kept: The electric field is (, ) = ∇V = sgn(−) This is the case that we have studied in this work.In contrast to the special potential profile with a sinusoidal form, the profile can be approximated by a square function.The step function can still be represented as a sum of multiple sinusoidal waves.

𝑃(𝑟(𝑅
At the TMD surface,  ≪ .Thus First, the neighboring domains display scattering amplitude contrast, and this contrast reverses when graphene doping is tuned via the back gate.Second, when graphene is being doped away from CNP, the scattering amplitude first decreases to a minimum and then increases.This increased scattering amplitude arises from the better coupling between the plasmon modes and the tip.When the photon energies increase, this coupling occurs at higher carrier doping.To clearly illustrate this dependence, we plot the extracted scattering amplitude line profiles in Supplementary Figure 14e.This finding further confirms that the observed scattering amplitude contrast originates from plasmon excitations as the graphene plasmon energy   , and the carrier density  obey the scaling rule,   ∝ √ 4 (Ref. 4,7,17,18).In all these data, the near-field amplitude across the domains shows a sinusoid-like profile rather than a step function profile.This spatial feature can be attributed to the carrier density gradient and the gradual transition of near-field amplitude across the domain walls (Supplementary note 9).
To probe the ferroelectric using plasmon response, in principle, the lasers from THz to the middle infrared range can be used.We should select the optimal excitation frequency, based on the plasmonic dispersion and the carrier density of the devices.However, in the spectrum measurements, we observed photoluminescence from defect states of h-BN and WSe2.The emission peak around 700 nm is from the defect states in h-BN (Ref. 21).
The emission peak around 835 nm originates from the defect excitons in WSe2 (Ref. 22).near the wrinkle.The position is denoted by an orange line with an arrow in a.The data are fitted by the blue line.

Photovoltaic effect in graphene photocurrent
Due to the ferroelectric polarization, a potential difference naturally develops across the domain wall.However, the photocurrent from this potential can be ignored.In our nano-photocurrent experiment, the decay length of photocurrent at the domain wall is hundreds of nanometers, which is much larger than the potential junction width of ~10 nm.The photocurrent originating from the potential at the domain wall is expected to show a fast spatial decay, with a scale of junction width.
Conversely, a much slower decay was observed in our nanometer-resolved photocurrent mapping.
Our nano-photocurrent results, in concert with previous far-field measurements, corroborate that the photocurrent in graphene is dominated by photothermal effect.Therefore, it is absolutely necessary to introduce the Seebeck effect.

Spatial scales in the photocurrent measurements
The laser with a wavelength of ~11 um is focused on the sample and tip using a parabolic mirror; the laser spot diameter on the sample is ~30 um.It should be noted that this incident light is locally enhanced by the sharp metallized tip.The nano-photocurrent, acquired by demodulation at the tiptapping frequency, is induced by the locally enhanced field at the apex of the tip.Therefore, to analyze the nano-photocurrent, the more relevant spatial scale is the size of the locally enhanced field, rather than the laser spot size.This local field is confined to ~ tens of nanometers underneath the tip and is much smaller than the moiré period in this work, which is around hundreds of nanometers.

2 .
To calculate the carrier density of each domain: a) The carrier density of the AB domain at backgate voltage   is   =   (  −  − ), where C is the geometric capacitance.;b) The carrier density of the BA domain at backgate voltage   is   =   (  −  − ); c) The carrier density difference between the AB and BA domains is ∆n =   ( − −  − ).This carrier density difference originates from ferroelectricity.

Supplementary
Figure 6| Numerically calculated near-field scattering signal.a,b, Near-field scattering amplitude (a) and phase (b) as a function of photon energy and chemical potential.The device geometry is the same as that of Device A, and the light rod model is used.c-f Fresnel reflection coefficient at a series of chemical potentials.The photon energy  = 850  −1 .The blue dashed lines denote the near-field coupling weight function.The different plasmonic behaviors of WSe2 and graphene originate from their distinct energy dispersions.For WSe2, the energy dispersion is parabolic, whereas the graphene shows linear energy dispersion.As a result, they exhibit different Drude weight formulas and plasmon dispersion.For WSe2, the Drude weight is  =  2   , whereas graphene's Drude weight is  =  2   √ ℏ , where  is carrier density,  is effective mass of carrier, ℏ is reduced Planck constant, and   is Fermi velocity 12 .Supplementary Figure 8| Calculated dispersion of the plasmon polariton.The dispersions are visualized using a false-color map of the imaginary part of the reflection coefficient of p-polarized light ((  )).The red dashed line indicates the momentum at which the tip can strongly couple with the polaritons.Left panel: The dispersion of doped double-layer WSe2.Right panel: The dispersion of doped double-layer WSe2/graphene.These two stacks are put on 58 nm h-BN/280 nm SiO2.Graphene and each WSe2 layer are doped with a carrier density of 5 × 10 12  −2 .
Figure 9| Plasmon response of R-stacking WSe2 bilayer.a, Schematic of the device structure.b, Optical micrograph of the device.c, Near-field amplitude images.The left side is WSe2, and the right side is a gold pad used to renormalize the near-field signal.The image was acquired using a CO2 laser with photon energy =900 cm -1 .d,e, Near-field amplitude signal at various back gate voltages for gold and WSe2.To acquire the data, we first obtained a series of near-field scattering amplitude images, like c, at various back gate voltages.Then, each data point in d and e is obtained by averaging the signals in the areas delineated by squares in c.No noticeable plasmonic response is observed.The root-mean-square-deviation is calculated by  =

2 ) 2 𝜖)
is the density of transferred charge due to the ferroelectric potential, and  are the geometric capacitance, and the distance between graphene and the central plane of WSe2, respectively,   is the charge density in graphene at zero gate voltage, and 1/  = is the inverse of the capacitance between graphene and the back gate.The charge transfer compensates for the ferroelectric potential in two ways: generating the electrostatic potential ()  and the chemical potential difference  −  0 the local dipole density between the graphene layer and the WSe2 layer and is much weaker than that generated by a charge modulation () only on graphene.

n
(,  ≪ ) maximum   (,  ≪ )( = 2 /, b=100 nm) To simplify the calculation of   , we assume that the bottom dielectric is thick h-BN and only consider the energy range without phonon resonance from h-BN.Then, the reflection structure is vacuum (labeled by 0)/WSe2 of thickness t (labelded by 1)/graphene on thick h-BN (labeled by 2).The vacuum, WSe2, and h-BN permittivities are  0 ,  1 and  2 , respectively.For WSe2 and h-BN, the in-plane and the out-of-plane permittivity are not the same, so  1 ,  2 are effective permittivity, are denoted by   = √     , where j=1 or 2,   ,   are in- 13f.13).The  top ( bottom ) is the reflection coefficient for the electrostatic potential at the top (bottom) side of the 2D graphene-ferroelectric system.Without any screening layers above WSe2, given a moiré period of 340 nm , one has   = 2.8 meV ,  eff = 3.1 and  = 0.51 .
4and the doping level is about √  eff     ≈ 16 meV .(Adding a 20 nm water layer would boost the screening to  eff = 16 and  = 1.7, and the doping level to √  eff     ≈ 65 meV.)From the experiment, we see that the voltage needed to cancel the doping in a domain is about  = 6 .Given the capacitance