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Tianlin Li, Hanying Chen, Kun Wang, Yifei Hao, Le Zhang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Xia Hong

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[Transport Anisotropy in One-Dimensional Graphene Superlattice in the High Kronig-Penney Potential Limit](https://mdr.nims.go.jp/datasets/13959dbc-c165-4e2d-9dd2-c5dab3955087)

## Fulltext

1  Transport Anisotropy in One-dimensional Graphene Superlattice in the High Kronig-Penney Potential Limit Tianlin Li,1 Hanying Chen,1 Kun Wang,1 Yifei Hao,1 Le Zhang,1 Kenji Watanabe,2 Takashi Taniguchi,3 Xia Hong1,* 1  Department of Physics and Astronomy and Nebraska Center of Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA 2 Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 3 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan *  Email: xia.hong@unl.edu  Abstract One-dimensional graphene superlattice subjected to strong Kronig-Penney (KP) potential is promising for achieving the electron-lensing effect, while previous studies utilizing the modulated dielectric gates can only yield a moderate, spatially dispersed potential profile. Here, we realize high KP potential modulation of graphene via nanoscale ferroelectric domain gating. Graphene transistors are fabricated on PbZr0.2Ti0.8O3 back-gates patterned with periodic, 100-200 nm wide stripe domains. Due to band reconstruction, the h-BN top-gating induces satellite Dirac points in samples with current along the superlattice vector 𝑠̂𝑠, a feature absent in samples with current perpendicular to 𝑠̂𝑠. The satellite Dirac point position scales with the superlattice period (L) as ∝ 𝐿𝐿𝛽𝛽, with β = -1.18±0.06. These results can be well explained by the high KP potential scenario, with the Fermi velocity perpendicular to 𝑠̂𝑠 quenched to about 1% of that for pristine graphene. Our study presents a promising material platform for realizing electron supercollimation and investigating flat band phenomena.   mailto:xia.hong@unl.edu2  Two-dimensional (2D) van der Waals materials subjected to artificially designed superlattice (SL) potential modulation are versatile platforms exhibiting a rich variety of emergent phenomena [1-3], including band reconstruction and Brillouin zone folding [4-12], correlation driven Mott transitions [13], superconductivity [14], magnetism [15, 16], ferroelectricity [17, 18], and topological orders [19]. Among them, the one-dimensional (1D) graphene superlattice (GSL) has gained considerable research interests [1] as the strong anisotropy and flattening of energy bands [6, 7, 20] can lead to electron-supercollimation effect [21, 22] as well as correlated states. While 1D GSL has been intensively investigated theoretically [1, 6, 20-26], only few experimental demonstrations have been reported, exploiting a dielectric gate with either nanoscale electrode-arrays [7] or periodic thickness-modulation to generate the Kronig-Penney (KP) potential [11]. The modulated-dielectric-gating approach has two intrinsic limitations. First, the doping capacity of conventional dielectrics such as Al2O3 and SiO2 is typically <1013 cm-2, which imposes only moderate KP potential in the graphene channel. The resulting electronic structure, even though distorted from the isotropic configuration, is not viable for hosting the electron-lensing effect. Second, due to the finite gate-thickness, the locally applied gate-bias becomes spatially dispersed when mapped on the conducting channel [Fig. 1(a)]. To date, 1D GSL modulation in the high KP-potential limit has never been achieved experimentally. A promising material scheme to overcome these challenges is to exploit a ferroelectric gate with periodically patterned domain structures [27], utilizing the nonvolatile, switchable polarization to induce the SL potential modulation in graphene. The ferroelectric field effect has previously been adopted to induce nonvolatile modulation of the resistance and quantum transport in graphene [28-30]. Combining it with nanoscale domain patterning further enables the design of reconfigurable functionalities [31-35] and directional conduction paths [36] in a 2D channel. For ferroelectrics such as oxide Pb(Zr,Ti)O3 [37] and copolymer P(VDF-TrFE) [36], the remnant polarization corresponds to a high 2D carrier density well exceeding 1013 cm-2. Since the surface bound-charge of ferroelectric domains is in direct contact with the 2D channel, the induced density variation in graphene across a sharp domain wall (DW) can produce a step-like potential change [Fig. 1(a)].  In this work, we report the first realization of 1D GSL in the high KP-potential limit via nanoscale domain patterning in a ferroelectric PbZr0.2Ti0.8O3 (PZT) gate [Fig. 1(b)]. Monolayer graphene field effect transistors (FETs) are fabricated on prepatterned periodic stripe domains of 3  PZT, with the period L varying from 200 to 300 nm. The polarization reversal of PZT shifts the Fermi level (EF) of graphene by up to 0.9 eV. For the GSL samples, satellite Dirac points (DPs) emerge in the longitudinal resistance along the SL vector 𝑠̂𝑠 and evolve into multiple Landau-fan branches in high magnetic fields, a feature absent for resistance perpendicular to 𝑠̂𝑠. The average carrier density interval between consecutive DPs scales with the SL period as ∝ 𝐿𝐿𝛽𝛽, with β = -1.18±0.06, which can be well described by the band structure of 1D GSL under high KP potential. The Fermi velocity perpendicular to 𝑠̂𝑠 is quenched to about 1% of that for pristine graphene, paving the path for designing the electron-lensing effect. We work with 50 nm epitaxial (001) PZT films deposited on 10 nm La0.67Sr0.33MnO3 (LSMO) buffered SrTiO3 (STO) substrates, with the polar axis of PZT along surface normal. The PZT films have high crystallinity and surface roughness of 4-5 Å, with details in the Supplemental Material  [38]. Piezoresponse force microscopy (PFM) switching hysteresis shows robust ferroelectric switching with coercive voltages of +1.2/-2.5 V for the polarization down (Pdown) and up (Pup) states [Fig. 1(c)]. In the ambient condition, the surface bound charges of PZT would be quickly screened by charged adsorbates or trap states prior to graphene transfer and thus do not induce doping in graphene at 300 K [27, 37]. Upon cooling, the polarization field increases due to the pyroelectric effect [29], and we leverage the additional polarization to dope graphene at low temperature. We characterize the PZT pyroelectric effect-induced variation of carrier density in graphene using a Hall bar device [38]. Figure 1(d) shows the temperature dependence of the converted polarization with respect to the 300 K value, ΔP = |P-P(300 K)|, which yields a polarization increase of about ~2.5 µC cm-2 at 2 K. Switching PZT polarization (2ΔP) can thus lead to a 2D carrier density of up to 3×1013 cm-2 in graphene.  To fabricate the 1D GSL, we prepattern Cr/Au (2 nm/10 nm) electrodes on PZT in the four-point configuration. Periodic stripe domains are written between the voltage probes by conductive atomic force microscopy in two orientations, either parallel or perpendicular to the current path. The width of the Pdown (Pup) domains is 100 nm (100-200 nm), yielding a SL period of L = 200-300 nm [Fig. 1(e)] [38]. Monolayer graphene flakes with h-BN top-layers (25-40 nm) are transferred on PZT using the dry transfer approach [39], in direct contact with the prepatterned domain structures and Au electrodes [Fig. 1(f)].  Cr/Au (10 nm/50 nm) are then deposited on h-BN as the global top-gate electrode [Fig. 1(g)]. Magnetotransport measurements are performed in 4  a Quantum Design Physical Property Measurement System using the standard lock-in technique with 50 nA current. The results reported are based on six GSL samples with current along the SL vector 𝑠̂𝑠 (denoted as D1-D6) and one GSL sample with current perpendicular to 𝑠̂𝑠 (denoted as D7), as summarized in Supplemental Table 1 [38]. PFM studies show that the domain structure is robust against sample fabrication and electrical measurements [38].   Figure 2 shows the sheet resistance R□ as a function of electron doping δn induced by the top-gate at 2 K for two GSL samples, D1 with L = 205±6 nm [Fig. 2(a)] and D7 with L = 199±6 nm [Fig. 2(b)]. As 𝑠̂𝑠 is along x-axis in the laboratory coordinates, we denote R□ for these two samples as Rxx and Ryy, respectively.  Here δn = αVg, where Vg is the top-gate voltage and α = εrε0/ed  is the gating efficiency, with εr = 3.76 the dielectric constant of h-BN [40], ε0 the vacuum permittivity, e the elementary charge, and d the h-BN thickness [38]. For comparison, we also show R□(δn) for a pristine graphene sample [Fig. 2(a)]. The pristine sample exhibits a single peak at the charge neutral point (CNP), with the field effect mobility µFE of about 20,000 cm2V-1s-1 for holes and 12,000 cm2V-1s-1 for electrons. For the doping level of interest in this study (|n| = 2-8×1012 cm-2), this corresponds to a mean free path of about 200-660 nm, confirming that the SL induced band reconstruction is viable at the chosen SL period (200-300 nm). For sample D1, Rxx(δn) shows two more satellite peaks symmetrically displaced from the main Dirac point, locating at δn = -1.34×1012 cm-2 and 1.73×1012 cm-2 [Fig. 2(a)]. This carrier density level is more than one order of magnitude smaller than that expected for pyroelectric polarization doping, excluding inhomogeneous doping as the origin of the emergent peaks. For sample D7, Ryy(δn) exhibits a single peak [Fig. 2(b)], similar to that of the pristine graphene.  The transport anisotropy between Rxx and Ryy is a direct manifestation of the emergent 1D GSL [6, 7, 11] induced by the periodic ferroelectric polarization gating effect. At 2 K, the pyroelectric effect induced ΔP generates a KP-type potential with an equivalent amplitude of about 𝑉𝑉0 =2ℏ𝑣𝑣𝐹𝐹�𝜋𝜋∆𝑃𝑃(2 K)/𝑒𝑒 = 0.9 eV  [Fig. 1(a)], where 𝑣𝑣𝐹𝐹 = 108  cm s-1 is the Fermi velocity of graphene. This results in a renormalized band structure, with the Fermi velocity unchanged along 𝑠̂𝑠 (vx) and highly suppressed perpendicular to 𝑠̂𝑠 (vy) [22]. The extra peaks observed in Rxx(δn) correspond to the emergent DPs at the band crossing of the highly folded Brillouin zone [1, 6]. The positions of the satellite DPs do not vary significantly from 1.8 to 10 K (Supplemental Fig. S7) [38], consistent with the saturation of ΔP in PZT below 100 K [Fig. 1(d)]. In contrast, Ryy(δn) 5  only exhibits a single resistance peak at the original CNP due to the suppressed Klein tunneling along the stripe domains [6, 23, 41, 42]. The broadening of the peak compared with pristine graphene can be attributed to either the suppression of relaxation time associated with the high density of states of the SL or the nanoscale variation of current orientation with respect to y-direction. Similar broadening has been reported previously in GSL fabricated via modulated dielectric gates with increasing KP potential [11]. We then investigate the transport anisotropy of GSL in magnetic fields [7, 11]. Figure 3(a) shows Rxx versus δn and magnetic field B for sample D1, where we observe Shubnikov de Haas oscillations associated with three sets of Landau-fan structures [38]. The central Landau-fan branches out from the original DP, while the two satellite fans emanate from δn = -1.41×1012 cm-2 and 2.16×1012 cm-2, consistent with the emergent DPs at zero field [Fig. 2(a)]. In Fig. 3(b), we model the Landau-fan structures using 𝑛𝑛 =  𝜈𝜈𝜈𝜈𝜈𝜈/ℎ +  𝑛𝑛DP, with 𝜈𝜈 = 4𝑙𝑙 + 2 the filling factors, l the Landau-level index, h the Plank constant, and nDP the carrier density of the associated DP position. As shown in Fig. 3(a), the simulated Landau fans are in excellent agreement with the experimental Rxx data. Figure 3(c) shows the Ryy data of sample D7, which possesses a similar SL period. As expected, it exhibits only one set of Landau fan emanating from the original DP [Fig. 3(c)-(d)]. Furthermore, at high magnetic fields, Rxx is more than one order of magnitude higher than Ryy, similar as that reported in GSLs with modulated dielectric gates, which has been attributed to the highly localized electron wavefunction along 𝑠̂𝑠 in magnetic fields [11]. The emergent satellite DPs are observed in Rxx taken on all six GSL samples with current along 𝑠̂𝑠 (Supplemental Fig. S9) [38]. We adopt multiple-peak fits to Rxx(δn) to identify the order of extra DPs [38]. To minimize the error in calculating the average carrier density interval between two consecutive DPs ∆nDP, we select two most prominent peaks at high doping levels, with one from the hole branch (order Nh at nDP,h) and one from the electron branch (order Ne at nDP,e), and deduce ∆nDP = (nDP,e - nDP,h)/(Ne - Nh). Figure 4 shows ∆nDP as a function of L, which can be well described by: ∆𝑛𝑛DP = 𝐴𝐴𝐿𝐿𝛽𝛽,   with 𝛽𝛽 = −1.18 ± 0.06.  (1)  Here A is the proportionality constant. To understand the L-dependence of ∆𝑛𝑛DP, we model the band structures of 1D GSL at moderate and high KP potentials. For 1D GSL under the long wavelength approximation, the Hamiltonian can be written as 𝐻𝐻 = 𝑣𝑣𝐹𝐹𝝈𝝈��⃑ ∙ 𝒑𝒑� + 𝑉𝑉1D(𝑥𝑥), with 𝝈𝝈��⃑  the 6  Pauli matrices vector, 𝒑𝒑� the momentum operator, and 𝑉𝑉1D(𝑥𝑥) the 1D scalar KP potential [1, 6, 24]. Assuming the widths of the well and barrier regions are Ww and Wb, respectively [Fig. 1(a)], we deduce the dispersion relation 𝐸𝐸(𝑘𝑘𝑥𝑥 ,𝑘𝑘𝑦𝑦) using the transfer matrix method [6]: cos(𝑘𝑘𝑥𝑥𝐿𝐿) = cos �𝜆𝜆𝑤𝑤𝑊𝑊𝑤𝑤𝐿𝐿� cos �λ𝑏𝑏𝑊𝑊𝑏𝑏𝐿𝐿� − 𝐺𝐺 sin �𝜆𝜆𝑤𝑤𝑊𝑊𝑤𝑤𝐿𝐿� sin �λ𝑏𝑏𝑊𝑊𝑏𝑏𝐿𝐿�, (2) with 𝜆𝜆𝑤𝑤,𝑏𝑏 = �𝜖𝜖𝑤𝑤,𝑏𝑏2 − 𝑘𝑘𝑦𝑦2𝐿𝐿2�12 , 𝐺𝐺 = 𝜖𝜖𝑤𝑤𝜖𝜖𝑏𝑏−𝑘𝑘𝑦𝑦2𝐿𝐿2𝜆𝜆𝑤𝑤𝜆𝜆𝑏𝑏, 𝜖𝜖𝑤𝑤 = 𝜖𝜖 + 𝑢𝑢 𝑊𝑊𝑤𝑤𝐿𝐿, and 𝜖𝜖𝑏𝑏 = 𝜖𝜖 − 𝑢𝑢 𝑊𝑊𝑏𝑏𝐿𝐿. Here 𝜖𝜖𝑤𝑤,𝑏𝑏  can be viewed as the dimensionless effective energy of the wavefunction in the well or barrier region, which is given by the dimensionless dispersion 𝜖𝜖 = 𝐸𝐸𝐸𝐸ℏ𝑣𝑣𝐹𝐹 modified by the scaled dimensionless KP potential 𝑢𝑢 = 𝑉𝑉0𝐿𝐿ℏ𝑣𝑣𝐹𝐹.  We first consider the case for the moderate KP potential as those generated by the modulated dielectric gates [7, 11]. Figure 5(a) shows the modeled band structures for u = 9π, where new bands emerge along 𝑠̂𝑠 with periodic band crossing points locating at kx = Nπ/L for the Nth reconstructed band. The isopotential contours show very complicated features, hosting multiple satellite DPs in one band [Fig. 5(b)]. The Fermi velocity becomes anisotropic, with vx remaining unchanged and vy suppressed to about 0.1vF at the original DP. With increasing u, vy is oscillatory with a damped envelop following vy = vFsin(u/4)/(u/4) [6]. For our 1D GSL generated by ferroelectric domains, the SL period L = 200 nm corresponds to a high u ~ 90π, which quenches vy to about 0.01 vF. As shown in Fig. 5(c), the SL energy band possesses a highly flattened energy dispersion that can be approximated as 𝐸𝐸 = ±ℏ𝑣𝑣𝐹𝐹(𝑘𝑘𝑥𝑥 + 2𝑁𝑁𝑁𝑁/𝐿𝐿).  We then evaluate the relation between EF and n for the SL band structure at the high u limit. From Luttinger theorem, the carrier density is given by the surface area of the Fermi surface. As shown in Fig. 5(d), the isopotential contours at u = 90π closely resemble rectangles that extend to the entire Brillouin zone. The enclosed area of each isopotential contour can thus be approximated as 𝛼𝛼𝑘𝑘𝑥𝑥, with α on the order of the size of graphene Brillouin zone. Given that 𝑛𝑛 = 𝛼𝛼𝑘𝑘𝑥𝑥/𝜋𝜋2, we obtain 𝐸𝐸𝐹𝐹 = ℏ𝑣𝑣𝐹𝐹𝑛𝑛𝜋𝜋2/𝛼𝛼. For the satellite DPs located at the Nth band crossing, 𝐸𝐸 = ℏ𝑣𝑣𝐹𝐹𝑁𝑁𝑁𝑁𝐿𝐿 and the corresponding doping level is ∆𝑛𝑛DP = 𝑁𝑁𝑁𝑁/𝜋𝜋𝜋𝜋 ∝ 1/𝐿𝐿 . The inversely proportional relation between ∆𝑛𝑛DP  and L closely resembles our experimental results (Fig. 4 and Eq. 1). In sharp contrast, in a moderate KP potential, the extra DP position exhibits an opposite trend and increases 7  with the SL period L at a given KP potential V0 [11]. Setting the fitted proportionality constant in Eq. 1 as 𝐴𝐴 = 𝛼𝛼/𝜋𝜋, we obtain 𝛼𝛼 = 26.2 nm−1, which is in excellent agreement with the Brillouin zone size of graphene 2𝜋𝜋/𝑎𝑎0 = 25.5 nm−1, further confirming that our 1D GSL samples are in the high KP potential limit.  The extremely flattened SL band associated with the high KP potential, on the other hand, can also limit the observation of extra DPs, as the enhanced density of states broadens the resistance peaks [11]. Compared with the multi-satellite DPs observed in 1D GSL subjected to moderate KP potential, e.g., those generated by modulated dielectric gate [7, 11], our samples only exhibit a couple of extra peaks in Rxx(Vg), which reflects the flattened dispersion at the band crossings [Fig. 5(c)]. There are additional factors that can compromise the observation of the emergent DPs. First, the ferroelectric DWs are intrinsically rough [43], which leads to a randomness in the SL period L and perturbs the SL band reconstruction [44]. It has been shown theoretically that the transport signature of extra DPs can be damped by ~90% at 5% randomness in the high KP potential limit [45]. Second, the DW roughness for PZT on LSMO is about 6 nm [38]. To minimize its impact, we work with SL period L ≥ 200 nm. Even though the typical mean free path of our 1D GSL samples is larger than L, these two length scales become comparable at lower carrier density, which may blur the ideal SL band structure and lead to decoherent transport. Third, we work with finite numbers of the stripe domains (35-50) [38], and the convoluted effect of finite SL modulation also broadens the reconstructed band. It is worth noting that the limitations associated with a PZT back-gate, including the compromised KP potential due to interfacial screening charges and the non-programmable domain structure after sample fabrication, can be overcome by adopting a suspended PZT membrane top-gate [34, 35]. In conclusion, we realize 1D GSL via periodic ferroelectric stripe domains patterned in a PZT back-gate. The samples exhibit high transport anisotropy between the current directions along and perpendicular to the SL vector, with satellite DPs emerging in the former configuration due to Brillouin zone folding. The scaling behavior of the carrier density interval between consecutive DPs with the SL period can be well modeled by the reconstructed band structure subject to high KP potential. The ferroelectric-domain-controlled 1D GSL presents a promising material platform for studying electron supercollimation as well as designing collective phenomena associated with flat bands, including magnetism, superconductivity, and topological sub-bands. 8   Acknowledgement  The authors thank Wuzhang Fang, Dawei Li, Qiuchen Wu, and Yuhan Zhang for the valuable discussions, and Anandakumar Sarella for technical assistance. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award No. DE-SC0016153, NSF EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE), under Award No. OIA-2044049, and Nebraska Center for Energy Sciences Research (NCESR). K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. The research was performed, in part, in the Nebraska Nanoscale Facility: National Nanotechnology Coordinated Infrastructure, the Nebraska Center for Materials and Nanoscience, which are supported by NSF ECCS: 2025298, and the Nebraska Research Initiative.    References  [1] M. Barbier, P. Vasilopoulos, and F. M. Peeters, Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac lines, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, 5499 (2010). [2] Y. K. Ryu, R. Frisenda, and A. Castellanos-Gomez, Superlattices based on van der Waals 2D materials, Chemical Communications 55, 11498 (2019). [3] F. He, Y. Zhou, Z. Ye, S.-H. Cho, J. Jeong, X. Meng, and Y. 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Esmailpour, R. Asgari, and M. R. R. Tabar, Conductance of a disordered graphene superlattice, Physical Review B 79, 165412 (2009).       13  Figure Captions FIG. 1. (a) Schematics of 1D GSLs imposed via modulated dielectric (DE) gate (top) and ferroelectric (FE) domain structure (bottom), and the corresponding potential profiles V(x). (b) Sample schematic. (c) PFM amplitude (top) and phase (bottom) switching hysteresis of 50 nm PZT/10 nm LSMO on STO. (d) ∆P(T) for PZT. (e) PFM phase image of stripe domains with period of L = 205 nm on PZT. (f) Optical image of a h-BN/graphene stack transferred on prepatterned domain structure of PZT. The dashed (dotted) lines highlight the graphene edges (SL region). (g) Optical image of the same sample after the deposition of Au top-gate electrode.  FIG. 2. (a) R□(δn) for pristine graphene and sample D1. (b) R□(δn) for sample D7. Upper panels: schematics of GSL configurations.  FIG. 3. (a) Rxx vs. δn and B for sample D1. The dashed lines illustrate the modeled Landau fans emanating from the original and satellite DPs. (b) Modeled Landau fans in (a) with the filling factors labeled. (c) Ryy vs. δn and B for sample D7. The dashed lines illustrate the modeled Landau fan emanating from the original DP. (d) Modeled Landau fan in (c) with the filling factors labeled. Resistance in (a) and (c) is on the log scale.  FIG. 4. ∆nDP vs. L for samples D1-D6 with a fit to Eq. 1 (dashed line) on linear and (inset) semi-log scales. The error bars for ∆𝑛𝑛DP  and L are deduced from the variation in h-BN thickness (Supplemental Table S1) and DW roughness, respectively [38].  FIG. 5. (a) Calculated band structure at u = 9π and Ww = Wb, and (b) its dimensionless energy contour plot of conduction band. (c) Calculated band structure at u = 90π and Ww = Wb, and (d) its dimensionless energy contour plot of conduction band. The dimensionless energy values are labeled in (b) and (d).     Figure 1Figure 2Figure 3-14 -10 -6 -2 2 6 10-2 2 6-10 -6 -2 2 614 -6 -2 2 6 10 14 18 22 26D7(c)(a)(b)D1(d)Figure 4200 250 3001.01.21.41.61.8200 250 30011.21.41.61.8n DP (1012 cm-2)L (nm)∝ 𝐿Figure 5(c)0.60.10.20.30.40.50.30.50.70.90.1(b)(d)−202−202(a)