K-Projection in Electronic Structure Analysis

• K-Projection is a method that decomposes supercell wavefunctions into primitive-cell momentum sectors using projector operators based on crystal translational symmetry.
• It employs fast Fourier transforms to efficiently extract plane-wave contributions and compute spectral weight for band unfolding and spatial localization.
• The technique provides practical insights for differentiating intrinsic electronic bands from interface or defect-induced states in heterogeneous systems.

Searching arXiv for the primary paper and closely related uses of “k-projection” / projection terminology. In electronic-structure theory, k-projection denotes a projector-based decomposition of a supercell or interface wavefunction into components that transform according to crystal momenta of a chosen primitive-cell translational symmetry. In the formulation of Chen and Weinert, the method serves two coupled purposes: it unfolds calculated electronic bands from supercells onto a primitive-cell Brillouin zone, and it yields local band structure by integrating the projected states over specified regions of space, a step that can be implemented efficiently using fast Fourier transforms (Chen et al., 2018). Its main use is in heterogeneous systems—interfaces, overlayers, slabs, and defective structures—where supercell eigenstates mix contributions from several constituents and direct comparison with angle-resolved measurements is otherwise obscured.

1. Operator-theoretic definition

The construction begins from a “primitive” crystal with translation operators $\{T_t\}$ associated with direct-lattice vectors $a_i$. Its one-dimensional irreducible representations are labelled by a crystal momentum $k$ in the first Brillouin zone, with character

$\chi_k(t)=e^{i\,k\cdot t}.$

The projector onto the $k$ subspace is

$\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$

where $h$ is the total number of translations in the supercell. Acting on a wavefunction $\psi(r)$, possibly non-periodic with respect to the primitive cell, it produces

$\psi_k(r)=\hat P_k\,\psi(r).$

Because $\hat P_k\psi_k=\psi_k$ and $a_i$0, any supercell state may be decomposed into primitive-cell $a_i$1 components (Chen et al., 2018).

In a plane-wave representation, a primitive-cell Bloch wave has the expansion

$a_i$2

Within this representation, $a_i$3 simply selects those plane-wave Fourier coefficients whose total wavevector $a_i$4 belongs to the $a_i$5 sector of the primitive Brillouin zone. This makes the method especially natural in plane-wave DFT implementations.

A central conceptual point is that the operator does not alter the underlying supercell eigenstate; it decomposes that state according to the translational symmetry chosen for analysis. This suggests that k-projection is best understood as a representation-theoretic filter for supercell wavefunctions rather than as an independent electronic-structure approximation.

2. Unfolding supercell bands and spectral weight

For band unfolding, one considers a supercell with lattice vectors

$a_i$6

A supercell Bloch state at $a_i$7 has plane-wave coefficients $a_i$8 labelled by supercell reciprocal vectors $a_i$9. Because the supercell and primitive cell are commensurate, any $k$0 can be uniquely decomposed as

$k$1

where $k$2 defines one of the folded primitive-cell momenta associated with $k$3. The corresponding primitive momentum is

$k$4

The unfolded spectral weight of supercell band $k$5 at primitive-cell momentum $k$6 is

$k$7

Equivalently, one sums the squared moduli of all supercell plane-wave coefficients that fold onto the same primitive-cell $k$8 (Chen et al., 2018).

This quantity has a precise interpretation. When the supercell truly retains the primitive translational symmetry, $k$9 for one $\chi_k(t)=e^{i\,k\cdot t}.$0 and zero otherwise, giving exact unfolding. In the presence of defects or interfaces, $\chi_k(t)=e^{i\,k\cdot t}.$1 lies between $\chi_k(t)=e^{i\,k\cdot t}.$2 and $\chi_k(t)=e^{i\,k\cdot t}.$3, indicating how “primitive-like” the supercell state is. Unfolded band structures are then plotted as $\chi_k(t)=e^{i\,k\cdot t}.$4 versus primitive $\chi_k(t)=e^{i\,k\cdot t}.$5, with point size or color proportional to $\chi_k(t)=e^{i\,k\cdot t}.$6.

A recurrent misconception is to read every line in an unfolded plot as a primitive-cell eigenband. The formalism instead assigns spectral weight to each supercell eigenstate after projection. The result is therefore a weighted representation of supercell states in primitive-cell momentum space, not a re-solution of the primitive-cell eigenproblem.

3. Local and layer-resolved k-projection

The same formalism can be made spatially selective by introducing a mask or window function $\chi_k(t)=e^{i\,k\cdot t}.$7 that equals $\chi_k(t)=e^{i\,k\cdot t}.$8 in a region of interest and $\chi_k(t)=e^{i\,k\cdot t}.$9 elsewhere. For a projected state $k$0, the local spectral weight is

$k$1

where $k$2 are the Fourier coefficients of the density $k$3 (Chen et al., 2018).

Because the window $k$4 is piecewise constant, its Fourier transform $k$5 is known analytically. One therefore computes $k$6 by a forward FFT of $k$7 on the real-space grid, multiplies by $k$8, and sums. The stated scaling is $k$9 per band. For a slab window $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$0 along $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$1,

$\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$2

This layer projection is the mechanism by which the method separates substrate, interface, and overlayer contributions. In practical interface studies, that separation is often the difference between identifying an intrinsic overlayer feature and identifying a hybridized or substrate-derived state. The formalism is also basis-agnostic at the projection stage: the same procedure may be adapted to other basis sets by first expanding the wavefunction on a real-space grid and then applying the real-space density and FFT steps.

4. Computational realization in plane-wave calculations

In a plane-wave DFT workflow, the implementation consists of a standard supercell calculation followed by a projection post-processing stage (Chen et al., 2018). The supercell calculation provides, for each band index $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$3 and supercell $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$4 point, the plane-wave coefficients $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$5. One then chooses the primitive-cell momenta of interest, namely all $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$6 generated by the commensurate folding relation.

For each pair $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$7 and each supercell reciprocal vector $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$8, one computes the fractional part of $\hat P_k=\frac{1}{h}\sum_t \chi_k^*(t)\,\hat T_t,$9 in the primitive-cell basis, identifies the corresponding $h$0, and assigns the coefficient $h$1 to the primitive channel $h$2. Summing $h$3 over all $h$4 mapped to the same $h$5 yields $h$6.

If a layer-resolved quantity is required, the workflow continues with a primitive-projected wavefunction in reciprocal space, obtained by zeroing coefficients that do not belong to the chosen $h$7. An inverse FFT gives $h$8 on the uniform grid; one forms $h$9, FFTs to $\psi(r)$0, and contracts with the analytic window transform $\psi(r)$1 to obtain $\psi(r)$2.

The algorithm is therefore modular: unfolding and local projection are separate operations coupled through the projected wavefunction. This suggests a useful practical distinction between momentum disentangling and spatial disentangling, both derived from the same projector formalism.

5. Applications to interfaces, overlayers, and surfaces

The method was illustrated on four systems: a graphene bilayer on H-saturated SiC(0001), BAs monolayer on ferromagnetic CrI$\psi(r)$3, silicene on Ag(111), and the Bi$\psi(r)$4Se$\psi(r)$5 surface (Chen et al., 2018).

System	k-projection result	Reported implication
Graphene bilayer / 6H-SiC(0001)	Si face: $\psi(r)$6 meV gap; C face: $\psi(r)$7 eV gap and $\psi(r)$8 doping	Strong termination dependence
BAs / CrI$\psi(r)$9	Spin splitting up to $\psi_k(r)=\hat P_k\,\psi(r).$0 meV in S1 and $\psi_k(r)=\hat P_k\,\psi(r).$1 meV in S3	Magnetic proximity in BAs
Silicene / Ag(111)	Linear “half-Dirac” features near Ag-BZ edge are interface states	Not true Dirac cones
Bi$\psi_k(r)=\hat P_k\,\psi(r).$2Se$\psi_k(r)=\hat P_k\,\psi(r).$3 slab	Topological Dirac state arises predominantly from $\psi_k(r)=\hat P_k\,\psi(r).$4 valence states split from bulk	Connects Shockley and topological pictures

For graphene bilayer on 6H-SiC(0001), the unfolded $\psi_k(r)=\hat P_k\,\psi(r).$5-projected bands around $\psi_k(r)=\hat P_k\,\psi(r).$6 show strong dependence on termination. On the Si-terminated face, only a $\psi_k(r)=\hat P_k\,\psi(r).$7 meV gap appears and the graphene remains essentially neutral. On the C face, a $\psi_k(r)=\hat P_k\,\psi(r).$8 eV gap opens and the bilayer becomes $\psi_k(r)=\hat P_k\,\psi(r).$9-doped, with the Fermi level $\hat P_k\psi_k=\psi_k$0 eV above the Dirac point. The layer projection localizes this effect to the graphene region.

For BAs on ferromagnetic CrI$\hat P_k\psi_k=\psi_k$1, three lateral stackings, S1, S2, and S3, were tested. Unfolding onto $\hat P_k\psi_k=\psi_k$2 BAs reveals magnetic-proximity-induced spin splittings in the BAs conduction bands up to $\hat P_k\psi_k=\psi_k$3 meV for S1 and $\hat P_k\psi_k=\psi_k$4 meV for S3.

For silicene on Ag(111), ARPES-like cuts labeled “A” and “B” were unfolded onto both the $\hat P_k\psi_k=\psi_k$5 silicene and $\hat P_k\psi_k=\psi_k$6 Ag Brillouin zones, with spatial windows $\hat P_k\psi_k=\psi_k$7 for a five-layer Ag slab and $\hat P_k\psi_k=\psi_k$8 for the first Ag layer plus silicene. The calculations reproduce the ARPES results, including linearly dispersing bands at the edge of the first Brillouin zone of Ag(111). However, the method shows that these bands originate from interface states produced by silicene-substrate interaction and are not Dirac states.

For the Bi$\hat P_k\psi_k=\psi_k$9Se$a_i$00 surface, a 10-QL slab calculation was layer-projected onto the top 3 QLs and $a_i$01-projected onto bulk-cell $a_i$02. The topological Dirac state was found to arise predominantly from $a_i$03 valence states split off from the bulk. The paper states that this ties the “normal” Shockley picture to the topological inversion.

6. Interpretation, limitations, and terminological scope

In the interface-band context, k-projection is a symmetry-analysis and visualization technique for supercell states. Its diagnostic power comes from two quantitative objects: the unfolded spectral weight $a_i$04 and the local spectral weight $a_i$05 (Chen et al., 2018). The first measures primitive-cell momentum character; the second measures where that projected character resides in real space. This separation is particularly important in heterostructures, where a linearly dispersing feature, a gap opening, or a spin splitting may be associated with an overlayer, a substrate, or a hybridized interface state.

The method also imposes an interpretive discipline. A high-weight unfolded feature indicates strong correspondence with a primitive-cell momentum sector, but not necessarily an isolated quasiparticle branch of the primitive constituent. Likewise, a layer-resolved projection can localize spectral weight to a region of space, but it does not by itself establish a fully decoupled subsystem description. The silicene/Ag(111) example is the clearest illustration: linearly dispersing bands appear in the unfolded picture, yet k-projection demonstrates that they are interface states rather than true Dirac cones.

The expression “k-projection” is not unique to this literature. In arXiv-indexed work it also appears in distinct settings, including $a_i$06th projection functions of convex bodies in integral geometry (Goodey et al., 2015), restricted projections onto $a_i$07-planes and onto $a_i$08-geodesics in Grassmannians [(Oberlin, 2011); (Gan, 2024)], Euclidean projection onto top-$a_i$09-sum sublevel sets in optimization (Roth et al., 2023), and $a_i$10-player projection games in complexity theory (Bhangale et al., 2023). In condensed-matter usage, however, the term refers specifically to the projector-based unfolding and spatial decomposition of supercell electronic states formulated for interfaces and related systems.