---
title: K-Projection in Electronic Structure Analysis
url: https://www.emergentmind.com/topics/k-projection
type: topic
---

# K-Projection in Electronic Structure Analysis

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 [1808.00112]. 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 $\sum_k \hat P_k=1$, any supercell state may be decomposed into primitive-cell $k$ components [1808.00112].

In a plane-wave representation, a primitive-cell Bloch wave has the expansion
$$
\psi_k(r)=\sum_G c_k(G)\,e^{i\,(k+G)\cdot r}.
$$
Within this representation, $\hat P_k$ simply selects those plane-wave Fourier coefficients whose total wavevector $q=k+G$ belongs to the $k$ 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
$$
A_i=\sum n_{ij}a_j.
$$
A supercell Bloch state at $k_s$ has plane-wave coefficients $\psi_{n,k_s}(G_s)$ labelled by supercell reciprocal vectors $G_s$. Because the supercell and primitive cell are commensurate, any $G_s$ can be uniquely decomposed as
$$
G_s=G+\kappa,\qquad G\in \mathrm{RLP}_{\mathrm{prim}},\ \kappa\in \mathrm{1st\,BZ}_{\mathrm{prim}},
$$
where $\kappa=G_s^0$ defines one of the folded primitive-cell momenta associated with $k_s$. The corresponding primitive momentum is
$$
k=k_s+G_s^0
\quad (\mathrm{mod\ reciprocal\ lattice\ of\ primitive\ cell}).
$$

The unfolded spectral weight of supercell band $n$ at primitive-cell momentum $k$ is
$$
w_n(k)=\langle \psi_{n,k_s}|\hat P_k|\psi_{n,k_s}\rangle
      =\sum_{G_s\to k}|\psi_{n,k_s}(G_s)|^2.
$$
Equivalently, one sums the squared moduli of all supercell plane-wave coefficients that fold onto the same primitive-cell $k$ [1808.00112].

This quantity has a precise interpretation. When the supercell truly retains the primitive translational symmetry, $w_n(k)=1$ for one $k$ and zero otherwise, giving exact unfolding. In the presence of defects or interfaces, $w_n(k)$ lies between $0$ and $1$, indicating how “primitive-like” the supercell state is. Unfolded band structures are then plotted as $E_n(k_s)$ versus primitive $k$, with point size or color proportional to $w_n(k)$.

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 $U(r)$ that equals $1$ in a region of interest and $0$ elsewhere. For a projected state $\psi_{n,k}(r)$, the local spectral weight is
$$
w_n(k;W)=\int d^3r\,U(r)\,|\psi_{n,k}(r)|^2
       =\sum_G U^*(G)\,\rho_{n,k}(G),
$$
where $\rho_{n,k}(G)$ are the Fourier coefficients of the density $|\psi_{n,k}(r)|^2$ [1808.00112].

Because the window $U(r)$ is piecewise constant, its Fourier transform $U(G)$ is known analytically. One therefore computes $\rho(G)$ by a forward FFT of $|\psi_{n,k}(r)|^2$ on the real-space grid, multiplies by $U^*(G)$, and sums. The stated scaling is $O(N\log N)$ per band. For a slab window $z_1<z<z_2$ along $a_3\parallel\hat z$,
$$
U(G)=\delta_{G_\parallel,0}\cdot
\frac{2\,e^{-i\,G_z(z_1+z_2)/2}}{G_z\,a_3}
\sin\!\biggl[\frac{G_z(z_2-z_1)}{2}\biggr].
$$

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 [1808.00112]. The supercell calculation provides, for each band index $n$ and supercell $k_s$ point, the plane-wave coefficients $\psi_{n,k_s}(G_s)$. One then chooses the primitive-cell momenta of interest, namely all $k=k_s+G_s^0$ generated by the commensurate folding relation.

For each pair $(n,k_s)$ and each supercell reciprocal vector $G_s$, one computes the fractional part of $G_s$ in the primitive-cell basis, identifies the corresponding $\kappa$, and assigns the coefficient $\psi_{n,k_s}(G_s)$ to the primitive channel $k=k_s+\kappa$. Summing $|\psi(G_s)|^2$ over all $G_s$ mapped to the same $k$ yields $w_n(k)$.

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 $k$. An inverse FFT gives $\psi_{n,k}(r)$ on the uniform grid; one forms $\rho(r)=|\psi(r)|^2$, FFTs to $\rho(G)$, and contracts with the analytic window transform $U^*(G)$ to obtain $w_n(k;W)$.

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$_3$, silicene on Ag(111), and the Bi$_2$Se$_3$ surface [1808.00112].

| System | k-projection result | Reported implication |
|---|---|---|
| Graphene bilayer / 6H-SiC(0001) | Si face: $\lesssim 10$ meV gap; C face: $\approx 0.13$ eV gap and $n$ doping | Strong termination dependence |
| BAs / CrI$_3$ | Spin splitting up to $\approx 50$ meV in S1 and $\approx 25$ 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$_2$Se$_3$ slab | Topological Dirac state arises predominantly from $k_z\approx 0$ valence states split from bulk | Connects Shockley and topological pictures |

For **graphene bilayer on 6H-SiC(0001)**, the unfolded $k$-projected bands around $K$ show strong dependence on termination. On the Si-terminated face, only a $\lesssim 10$ meV gap appears and the graphene remains essentially neutral. On the C face, a $\approx 0.13$ eV gap opens and the bilayer becomes $n$-doped, with the Fermi level $\approx +0.3$ eV above the Dirac point. The layer projection localizes this effect to the graphene region.

For **BAs on ferromagnetic CrI$_3$**, three lateral stackings, S1, S2, and S3, were tested. Unfolding onto $1\times1$ BAs reveals magnetic-proximity-induced spin splittings in the BAs conduction bands up to $\approx 50$ meV for S1 and $\approx 25$ meV for S3.

For **silicene on Ag(111)**, ARPES-like cuts labeled “A” and “B” were unfolded onto both the $1\times1$ silicene and $1\times1$ Ag Brillouin zones, with spatial windows $W_1$ for a five-layer Ag slab and $W_2$ 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$_2$Se$_3$ surface**, a 10-QL slab calculation was layer-projected onto the top 3 QLs and $k$-projected onto bulk-cell $k_\parallel+k_z$. The topological Dirac state was found to arise predominantly from $k_z\approx 0$ 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 $w_n(k)$ and the local spectral weight $w_n(k;W)$ [1808.00112]. 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 $k$th projection functions of convex bodies in integral geometry [1502.06747], restricted projections onto $k$-planes and onto $k$-geodesics in Grassmannians [1107.4913; 2404.04290], Euclidean projection onto top-$k$-sum sublevel sets in optimization [2310.07224], and $k$-player projection games in complexity theory [2312.04783]. 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.

Source: https://www.emergentmind.com/topics/k-projection