Moire metal for catalysis

Yang Zhang,1 Claudia Felser,2 and Liang Fu1

1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA2Max-Planck-Institut fur Chemische Physik fester Stoffe, 01187 Dresden, Germany

The search for highly efficient and low-cost catalysts based on earth abundant elements is one ofthe main driving forces in organic and inorganic chemistry. In this work, we introduce the conceptof moire metal for hydrogen evolution reaction (HER). Using twisted NbS2 as an example, we showthat the spatially varying interlayer coupling leads to a corresponding variation of Gibbs free energyfor hydrogen absorption at different sites on the moire superlattice. Remarkably, as a HER catalyst,twisted NbS2 is shown to cover the thermoneutral volcano peak, exceeding the efficiency of thecurrent record platinum. The richness of local chemical environment in moire structures providesan advantageous and versatile method to increase chemical reaction speed.

Heterogeneous catalysis [1, 2] plays a key role in chem-ical and medical industries by enabling large-scale pro-duction and selective product formation. Currently, theproduction of 90% of chemicals (by volume) is assistedby solid catalysts. An important chemical reaction forthe supply of “green energy” is to produce hydrogen(H2) from water using thermal, solar or electric energy.In particular, the electrochemical generation of hydro-gen molecules, known as hydrogen evolution reaction(HER), has the advantage of high reaction speed andeco-friendliness. The key component of electrochemicalreaction system is electrocatalysts with high intrinsic re-activity and long-term stability. For the industry-levelapplications, the rational design of low-cost electrocat-alysts based on earth abundant elements is in high de-mand.

The hydrogen electrocatalytic reaction speed is knownto correlate with the intrinsic properties of solid cata-lysts. Various descriptors, such as d-band center, d-bandcharge, pH values, bond length, active site density, etc.,have been introduced to predict the tendency of the elec-trocatalytic reactivities [3–5]. Beyond these empirical de-scriptors, the most quantitative intrinsic figure of meritfor HER catalyst is the Gibbs free energy (∆G(H)) ofhydrogen absorption [6, 7], at the equilibrium potential.Up to now, the most efficient electrocatalysts for HERare Pt-group metals, as Pt and Pd have ∆G(H) close tothe ideal HER volcano diagram peak ∆G(H) = 0 eV.

Metal catalysts rely on the surface dangling bonds asthe active sites for chemical absorption. In industry, solidcatalysts are often porous or dispersed on a supportingmaterial to maximize surface area and enhance catalyticreactivity. However, for bulk materials, the active cat-alytic surface area-to-mass ratio is still quite limited. Re-cently, atomic thin van der Walls (vdW) materials havebeen studied for catalysis [8–17] due to the maximizedsurface/volume ratio, potentially enabling “all surfacereaction”. The calculations of ∆G(H) of vdW materi-als within the basal plane show that the best catalyst ismonolayer TMD NbS2 with ∆G(H) =0.31 eV [10], whichstill falls far behind Pt/Pd in terms of reactivity.

In this work, we propose a new method to enhance

MX XXMM

2𝐻#

2𝐻*

2𝑒%𝑗

𝐻'

FIG. 1: Schematic of HER process at three high-symmetrystacking regions of 2H TMDs, where two protons and twoelectrons from electrode are combined to form a hydrogenmolecule. The tuning of HER catalytic properties in the bi-layer vdW metals are related to the local d-band center shift[5] and bandwidth at different stacking regions. We circleMX, MM and XX regions in moire superlattice from left toright direction, and present the the enlarged side view of threeregions at lower panel.

chemical reactivity under a broad range of environmen-tal conditions using vdW metals with moire structures.Such “moire metals” can be created by stacking two lay-ers of identical vdW metals with a twist angle [18, 19],or two different layers with a lattice mismatch. Our keyidea is that in a moire superlattice, the spatial varia-tion of stacking configuration produces a correspondingvariation of Gibbs free energy for hydrogen absorption indifferent local regions. Such variation of ∆G(H) enablesthe self-adaptive optimization of chemisorption propertyfor a wide range of reaction environments.

Using first-principles calculations, we further identifythe twisted AB stacked NbS2 as the superior catalystwith highest reactivity for HER compared to elementarymetals and topological metals. At intermediate twist an-

arX

iv:2

111.

0305

8v2

[co

nd-m

at.m

trl-

sci]

22

Nov

202

1

2

MM XX

MXML(a) (b)

(d)(c)

Energy(e

V)Energy(e

V)

FIG. 2: DFT band structure of (a) monolayer, (b) MXstacked bilayer (natural stacking), (c) MM stacked bilayer,(d) XX stacked NbS2 with anti-parallel orientation. The in-terlayer coupling has a large variation at different stackingregions as seen in the corresponding Γ pockets splitting.

gle θ = 5.08◦, an 150 meV range of Gibbs free energy forhydrogen adsorption at different locations in the basalplane is observed, covering the ∆G(H) = 0 eV HER vol-cano diagram peak for standard reaction environment.

To reveal the origin of the spatial variation of ∆G(H),we calculate stacking dependent band structures and findthat interlayer coupling varies significantly. By develop-ing a simple tight-binding model, we present a heuristicargument that the spatially varying interlayer couplingon the moire superlattice is responsible for the variationof ∆G(H) found in our large-scale DFT calculation.

Presently, semiconductor based moire materials havebeen intensively studied in the physics community [20],such as twisted bilayer graphene [18, 19] and group-VI semiconductor TMD heterostructures [21–36, 36, 37].Here, for the purpose of catalysis, we consider moire met-als formed by stacking two layers of group-V metallicTMDs MX2 in 2H structure, such as NbSe2 and NbS2.In the case of homobilayers with a small twist angle θ(relative to the natural bilayer stacking), the moire su-perlattice, shown in Fig. 3, has a superlattice constantaM = ab/

√δ2 + θ2 where δ = (ab − at)/at, with ab(at)

the lattice constant of each layer. In this superlattice,there are three high symmetry regions denoted as MM,XX, and MX, see Fig. 1. In MX region, the M atomon the bottom layer is locally aligned with the X atomon the top layer, as in natural TMD bilayers. On theother hand, in MM (or XX) region, the M (or X) atomson both layers are locally aligned. Importantly, MM andXX stacking configurations are thermodynamically un-stable and made possible only by moire engineering.

We calculate the electronic structures of bilayer NbS2

for different stacking configurations. Monolayer NbS2

MM XX MX MMΔ𝐺(𝐻)(meV

)

MoiréMX

E(eV)

DOS(a.u.)

(a)

(b)

FIG. 3: (a) Comparison of density of states for natural stack-ing MX and moire superlattice. The density of states nearFermi level is significantly modified by twisting. (b) Upperpanel is the Gibbs free energy diagram for hydrogen evolutionunder standard conditions (1 bar of H2 and pH=0 at 300 K).Energies of the intermediate states are calculated using theSCAN+rVV10 vdW functional as described in the Methods.Coverage of one hydrogen atom per moire superlattice is usedfor all calculations. Lower panel is the wavefunction plot fortwisted NbS2 between Ef −0.5 eV to Ef along MM, XX, andMX path. The wavefunction is more spread out at XX region.

features with three electron pockets near K, K ′ andΓ valleys, and the half filled conduction bands are en-tirely formed from d orbitals of Nb atom. As shown inFig. 2, the Γ valley in bilayers exhibits a large bonding-antibonding splitting on the order of 0.5eV due to vdWinterlayer coupling. Importantly, this bilayer splittingvaries significantly depending on the layer stacking: 0.6eV for MM, 0.5 eV for MX and 0.46 eV for XX. Theunderlying variation in interlayer coupling is due to dif-ferent distances between adjacent metallic atoms on thetwo layers.

In twisted NbS2 moire superlattice, the spatial varia-tion of interlayer coupling in different local regions pro-

3

vides a variety of local charging environments for thechemisorption of hydrogen atoms and molecules. Thesurface hydrogen binding energy directly determines itshydrogen production speed. The HER process involveshydrogen absorption and desorption as shown in Fig. 1.Initially, the Volmer reaction transfers one electron toa proton to form an adsorbed hydrogen atom on thesurface of catalyst (H+ + e− + ∗ → H∗, where * de-notes the active catalyst and H∗ is the intermediate).Next, the desorption of H2 can be achieved by eitherTafel reaction (2H∗ → H2 + 2∗) or Heyrovsky reaction(H+ + e− +H∗ → H2 + ∗).

Therefore, under the standard electrochemical reac-tion condition (room temperate and atmospheric pres-sure), the overall HER reaction rate is largely deter-mined by the binding energy (∆EH) of the intermedi-ate H∗ and the density of the active sites in the cat-alyst. A strong bonding between hydrogen atom andcatalyst slows down the desorption step (Heyrovsky orTafel), while a weak bonding slows down the adsorptionstep (Volmer reaction). A direct descriptor for reactionspeed is Gibbs free energy for hydrogen absorption, de-fined as ∆GH = ∆EH+∆EZPE−T∆S, with ∆EZPE and∆S as the difference in zero-point energy and entropy be-tween the adsorbed species and the gas molecule, respec-tively. Hereafter we use the value ∆EZPE − T∆S = 0.3eV for standard electrochemical reaction condition. Theideal reaction rate is achieved at thermoneutral conditionwhen ∆GH ∼ 0 eV [38].

TaAs

NbPNbS2 (2H)TaS2 (2H)

TaS2 (1T)

MoTe2 (Td/1T’)Nb

W

Os

Re

PtPdIr

Au

Cu

Ag

Zn

TwistedABstackedNbS2

FIG. 4: Predicted relative activities of various HER cata-lysts following the volcanic scheme as a function of calculatedGibbs free energy of hydrogen adsorption on the surface ofthe catalyst. Circle stands for elemental metals and topolog-ical metals, star stands for vdW metals at basal plane. TheGibbs free energy data for untwisted metal is collected fromthe previous literature [10, 39–41].

To investigate ∆G(H) for moire metal, we first cal-culate the electronic structure of twisted bilayer NbS2

with angle θ = 5.08◦. To obtain fully relaxed latticestructures, we perform large-scale density functional the-

ory calculations with the SCAN+rVV10 van der Waalsdensity functional [42], which captures the intermediate-range van der Waals interaction through its semilocalexchange term, resulting in a better estimation of layerspacing. Owing to the different local atomic arrange-ments in the MM, XX and MX regions, the layer dis-tance is found to have a strong spatial variation as shownin the supplementary material, which are 6.14A, 6.11Aand 6.39A for MM, MX and XX regions in θ = 5.08 ABstacked moire superlattices, respectively. The twistingsignificantly modified the density of states compared tothe natural MX stacked bilayer, and the density of statespeak has a shift of 100 meV. Plotting the charge densityfor states from Ef−0.5 eV to Ef ( Ef is the Fermi level),we find the out of plane distributions are quite differentat three moire regions, and XX region is more spread outthan others.

In the twisted AB stacked NbS2 with angle θ = 5.08◦,we calculate Gibbs free energy for hydrogen adsorptionat the basal plane, especially for the line connecting threehigh-symmetry stacking regions as shown in Fig. 3a.∆G(H) is obtained as ∆GH = ∆EH + ∆EZPE − T∆S.Here, ∆EH is the hydrogen binding energy which wascalculated as ∆EH = E( moire +H) − E( moire ) −1/2E (H2) for one hydrogen atom per moire superlattice.The E( moire +H), E( moire) and E (H2) are the totalenergy of respective parts.

Remarkably, the spatially dependent ∆G(H) in thebasal plane covers the ∆G(H) = 0 eV. At XX region,which is thermodynamically inaccessible without twist-ing, we find a Gibbs free energy ∆G(H) = 0.01 eV. AtMM region with largest interlayer coupling, the ∆G(H)is -0.13 eV, lower than the natural stacking region MXwith ∆G(H) = −0.09 eV. In the line plot across threehigh-symmetry stacking regions as shown in Fig. 3d, wefind half of the basal plane sites have absolute reactivitycomparable or better than platinum.

In the volcano-shaped diagram shown in Fig. 4, wecollect the Gibbs free energy for hydrogen adsorptionof the twisted vdW metal NbS2 with a variety of sin-gle metals, topological metals and 2D materials collectedfrom the literature [6, 43]. Among all studied bulk ma-terials for HER catalysis, Pt with a Gibbs free energy∆G(H) = −0.09 eV is the most efficient electrocatalystthus far, and used in industry despite the high materialcost. A recent experiment on doped bulk NbS2 demon-strates the ultrahigh-current-density for hydrogen evolu-tion, which is close to platinum [44]. Unlike the limitedtypes of catalytic sites in conventional bulk or vdW ma-terials, moire engineering provides a wide and adjustablerange of hydrogen absorption energies for the HER pro-cess in various environments with different pressure, tem-perature and hydrogen potentials. Importantly, the AB-stacked bilayer NbS2 at θ = 5.08◦ covers the volcanopeak, providing a superior HER catalyst over all knownmaterials at equilibrium potential.

4

To gain insight into the origin of the stacking-dependent hydrogen binding energy, we consider a sim-plified tight-binding model for a twisted bilayer metal,including nearest neighbor intralayer and interlayer hop-ping:

txy(d) = t1

(d · zd

)2

; tz(d) = t2

(1− d · z

d

)2

(1)

Herein t1 = t01e−(r−a)/δ1 , t2 = t02e

−(r−dc)/δ2 , and a =3.34 A, d = 6.12 A is the inplane lattice constant andlayer distance of MM stacked region, intralayer hoppingt01 = 0.184 eV is fitted to match the DFT calculateddensity of states of Γ pocket, t02 = 0.30 eV is determinedby the band splitting at Γ of MM stacking configuration,the scaling parameters δ1 = 0.3 A and δ2 = 1.1 A aredetermined from total energy of tight binding model.

To calculate the total energy for hydrogen atoms ad-sorbed at the basal plane, we introduce one hydrogenatom attached to the sulfur site with constant onsite po-tential and hopping energy. The onsite energy of hy-drogen atom is set to be VH = 0.45 eV relative to thatof Nb atoms. We fit the hopping from hydrogen to Nbatom as tH = 2.45 eV by matching the DFT calcu-lated density of states (Γ pocket Fermi surface) in MMstacked region. Although our model is greatly simpli-fied, it correctly captures the binding energy variationacross the high-symmetry stacking regions: MM, MXand XX. We plot the Gibbs free energy in the same man-ner as the large-scale DFT calculation shown in Fig. S2.The tight-binding description captures the overall energyscale quite well, and shows a good agreement for enlarged∆G(H) of XX and MM region compared to MX region,which are -96 meV, -50 meV and 1 meV, respectively.

The dependence of chemical adsorption on local stack-ing configuration in moire metals can be directly probedby scanning tunneling microscopy (STM). Previous STMstudies have directly imaged adsorbed hydrogen atoms onthe surface of conventional metals [45]. For moire metal,we expect that the spatially varying surface-hydrogenbond strength leads to an inhomogeneous hydrogen den-sity distribution that correlates with the local stackingvariation.

While we have focused on the twisted homobilayerNbS2 as a prime example of moire metals, similar stack-ing dependent chemisorption is expected in heterobilay-ers such as NbSe2/NbS2. The powerful chemical va-por deposition (CVD) [46] method enables the scalablepreparation of high quality and low cost 2D heterostruc-tures, towards the real world application of moire metalcatalysts. Besides HER, we also expect the moire engi-neering for metals can optimize the chemisorption of thealkali metal ions for the design of electrode materials,and nitrogen and oxygen atoms for the catalysis of smallorganic molecules and oxygen reduction reaction (ORR).The chiral nature of moire superlattices further provide aunique way towards heterogeneous asymmetric catalysis.

In conclusion, we have shown that the moire engineer-ing strongly enhances the HER reactivity of vdW metalsdue to the spatially varying interlayer bonding strength.We have identified twisted NbS2 as the ideal catalystfor the H2 evolution reaction, covering the volcano peak∆G(H) = 0 eV. Our work provides a guiding principle forthe design of highly efficient and low-cost HER catalystsfrom the emerging field of artificial moire materials. Ina broad sense, the moire engineering of surface chemicalproperties provides a physical way for the rational designof superior electrocatalysts for a variety of chemical reac-tions under different environmental conditions, with theaim of closing the carbon-, hydrogen-, and nitrogen-cyclewith renewable electricity.

Method We performed the density functional cal-culations using generalized gradient approximation [47]with SCAN+rVV10 van der Waals density functional[42], as implemented in the Vienna Ab initio SimulationPackage [48]. Pseudopotentials are used to describe theelectron-ion interactions. We first construct AA and ABstacked NbS2/NbS2 homobilayer with vacuum spacinglarger than 20 A to avoid artificial interaction betweenthe periodic images along the z direction. Dipole correc-tion is added to the local potential in order to correct theerrors introduced by the periodic boundary conditions inout of plane direction. The structure relaxation is per-formed with force on each atom less than 0.01 eV/A. Weuse Gamma-point sampling for structure relaxation andself-consistent calculations, due to the large moire unitcell.

Acknowledgment

We acknowledge Yan Sun, Guowei Li, Jinfeng Jia,Vidya Madhavan and Eva Andrei for valuable discussionsand comments on the manuscript.

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6

Supplemental material for “Moire metal for catalysis”

Yang Zhang,1 Claudia Felser,2 and Liang Fu1

1 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA2 Max-Planck-Institut fur Chemische Physik fester Stoffe, 01187 Dresden, Germany

GIBBS FREE ENERGY FOR HYDROGEN ABSORPTION FROM TIGHT-BINDING MODEL

We use the tight-binding model developed in main text to evaluate Gibbs free energy for hydrogen absorption. Wecalculate the total energy of moire superlattice with N unit cells before and after hydrogen absorption as follows:ETB

total(∗) =∑Ni occ εi(∗), ETB

total(∗+H) =∑N+1i occ εi(∗+H), where * stands for one moire superlattice.

The hydrogen binding energy of this toy model is obtained from ∆E(H) = ETBtotal(∗ + H) − ETB

total(∗). Assumingstandard reaction condition as in the main text, we calculate Gibbs free energy ∆G(H) and plot it along MM, XX,and MX path in Fig. S1. The overall trend matches reasonably well with DFT calculations, especially the three highsymmetry stacking regions.

Hmu=0.45eVHbond =2.45eV

(a)

(b)

MM XX MX MM

Δ𝐺(𝐻)(meV

)Energy(e

V)

Γ 𝑀 Γ𝐾

FIG. S1: (a) Band structure of tight-binding model at MM stacking region, (b) Gibbs free energy derived from hydrogenbinding energy ∆E(H) calculated in tight binding model.

LATTICE RELAXATION OF AA, AB HOMOBILAYER NBS2

We also study TMD homobilayers with a small twist angle starting from AA stacking, where every metal (M) orchalcogen (X) atom on the top layer is aligned with the same type of atom on the bottom layer. Within a local

7

(a)

(b)

Layerdistance(Å)

MMMX

XM

MM

XXMX

𝜃 = 5.08∘ 𝜃 = 4.41∘

AA

AB

FIG. S2: The interlayer distance of the twisted structure obtained from DFT is shown, demonstrating a large variationbetween different moire regions. (a) AA stacked homobilayer NbS2 with twist angle θ = 5.08◦ and θ = 4.41◦; (b) AB stackedhomobilayer NbS2 with twist angle θ = 5.08◦ and θ = 4.41◦.

region of a twisted bilayer, the atom configuration is identical to that of an untwisted bilayer, where one layer islaterally shifted relative to the other layer by a corresponding displacement vector d0. For this reason, the moire bandstructures of twisted TMD bilayers can be constructed from a family of untwisted bilayers at various d0, all having1× 1 unit cell. Our analysis thus starts from untwisted bilayers.

In particular, d0 = 0, (−a1 + a2) /3, (a1 + a2) /3, where a1,2 is the primitive lattice vector for untwisted bilayers,correspond to three high-symmetry stacking configurations of untwisted TMD bilayers, which we refer to as MM,XM, MX. In MM (MX) stacking, the M atom on the top layer is locally aligned with the M (X) atom on the bottomlayer, likewise for XM. The bilayer structure in these stacking configurations is invariant under three-fold rotationaround the z axis.

We present the relaxed lattice structure and corrugation effect in AA, AB homobilayer NbS2 in Fig. S2.

GIBBS FREE ENERGY OF AA STACKED HOMOBILAYER NBS2

When twisting from AA stacked homobilayer, there are two high symmetry stacking regions, denoted as MM, MX,and XM region. And the MX region is symmetry related to XM region via C2y operation. In the untwisted structure,we see four spin and layer splitted bands at K point in both MM and MX (XM) region, and an enlarged bonding-antibonding splitting at M and Γ point in MM region compared to MX (XM) region due to direct contact of metallicatom as shown in Fig. S3. The interlayer coupling strength at MM and MX (XM) region are 0.336 eV and 0.254 eV,respectively.

We further calculate the Gibbs free energy of hydrogen absorption in AA stacked moire superlattice. At twist angleθ = 5.08, we get -0.171 eV at MM region and -0.087 eV at MX region, with a range of 84 meV. In analogy to the ABstacking moire, we plot the wavefunction from Ef − 0.6 eV to Ef to check the real space charge variation at differentstacking regions.

TWIST ANGLE DEPENDENT GIBBS FREE ENERGY AND DENSITY OF STATES

Apart from twist angle θ = 5.08◦, we also calculated AB stacked moire superlattice at nearby angle θ = 4.4◦ andθ = 6.0◦. For θ = 6.0◦ and Lm = 3.19 nm, we got ∆G(H) = −0.114,−0.046,−0.066 eV at MM, MX, and XX region.

8

MM XM

MXML(a) (b)

(d)(c)Energy(e

V)Energy(e

V)

AA

FIG. S3: DFT band structure of (a) monolayer NbS2; (b) MX stacked bilayer, (c) MM stacked bilayer, (d) XM stacked NbS2

with parallel configuration . The interlayer coupling has a large variation at MM and MX (XM) regions as seen in the Γ pocketssplitting.

MOIRE BAND STRUCTURE

In this section, we present the band structures of moire superlattices in AB and AA stacked NbS2, and their densityof states compared to the natural stacked bilayer. Compared to the splitted density of states peak in AB stackingmoire, we have a single peak in AA stacking moire in Fig. S4a. The moire band structures shown in Fig. S4(c,d) aremassively complicated, as expected from the folding of large Fermi surface of monolayer NbS2.

9

E(eV)

DOS(a.u.)

(a) (b)

(d)(c)

Energy(e

V)

AB

ABMoiréAAMoiréMXbilayer

MM MX XM MM

FIG. S4: (a) Density of states for moire superlattice with AA (AB) stacking, compared to natural MX stacking bilayer. (b)Wavefunction plot along high symmetry line for AA stacking moire superlatttice of NbS2 from Ef − 0.6 eV to Ef . DFT bandstructure of (c) twisted AB stacked NbS2; (d) twisted AA stacked NbS2.