Directional shift current in mirror-symmetric BC$_2$N

We present a first-principles theoretical study of the shift current in a noncentrosymmetric polytype of graphitic BC$_2$N, and find that the photoconductivity exhibits two distinctive features at the band edge. First, it ranks among the largest bulk nonlinear responses reported to date, with the peak value occurring in an energy range suitable for optical manipulation. Secondly, it is strongly anisotropic, due to the vanishing of particular tensor components not foretold by phenomenological symmetry arguments; this is a consequence of dipole selection rules imposed by mirror symmetry, which imply that the relative parities between valence and conduction bands are key for determining the directionality of the band-edge response. Our work identifies graphitic BC$_2$N as a promising candidate for next-generation photovoltaics, and opens up a broad framework for future studies.

Introduction -The conversion of light into electrical current via the photovoltaic effect is one of the most copious supplies of renewable energy on Earth. Traditional solar cells make use of p-n junctions to generate a builtin electric field that drives the photoexcited electrons. It has long been known that single-phase noncentrosymmetric crystals exhibit a different type of photovoltaic effect, called the bulk photovoltaic effect (BPVE) [1][2][3]. This is a nonlinear optical response that consists in the generation of a photovoltage (open circuit) or photocurrent (closed circuit) when light is absorbed via intrinsic or extrinsic processes. Since the BPVE occurs in homogeneous systems, it needs no sophisticated interfaceengineering techniques for its usage as sunlight harvester. As further appealing features of the BPVE, the photovoltage is not limited by the band gap of the material, nor is device performance constrained by the Shockley-Queisser limit that applies to conventional solar cells [4].
In the past few years, the study of the BPVE for solarcell applications has been reinvigorated by the search for novel materials with large photoresponsitivies [5][6][7]. There is also renewed interest on more fundamental aspects of the BPVE, particularly its sensitivity to the geometric and topological properties of the electronic wave functions [8][9][10].
Recently, considerable attention has been paid to the directionality of the BPVE, namely to the relative orientation and size of the generated photocurrent as a function of the polarization of light. As an example, the sign of the shift-current contribution to the BPVE has been suggested as a possible probe for detecting topological phase transitions [11,12]. In addition, recent experimental work on the Weyl semimetal TaAs found large and anisotropic contributions to the shift-current BPVE driven by the low-energy Weyl-node physics [13].
In this work, we report a distinctive shift-current response at the band edge of a noncentrosymmetric polytype of graphitic BC 2 N, a layered semiconductor made of alternating zigzag chains of carbon and boron nitride [14][15][16][17][18][19][20]. Our ab initio calculations reveal two unusual features of the shift-current spectrum near the fundamental band gap. First, its peak magnitude ranks amongst the largest bulk nonlinear responses reported to date. In addition, the calculated response reveals a strong anisotropy due to the vanishing of certain tensor components not foretold by phenomenological symmetry arguments. This allows, for example, to engineer particular conditions where the current flows perpendicularly to the applied field. We trace the origin of this anisotropy to the mirror symmetry of the crystal structure, which imposes selection rules on dipole transitions between the top of the valence band and the bottom of the conduction band. We capture the essential physics of this phenomenon by generalizing a two-band tight-binding model introduced in Ref. 21, thus providing a suitable framework for a broad class of materials.
Shift-current BPVE -The BPVE is a nonlinear optical response of the form [1] where a, b, and c are Cartesian indices. Since both the electric field E and and the current J are odd under inversion, the two sides of Eq. (1) pick up opposite signs under this symmetry; hence, the BPVE can only occur in systems where inversion symmetry is broken. The righthand side of Eq. (1) can be split into symmetric and antisymmetric parts under b ↔ c, known respectively as the linear and circular BPVE [1]; the former occurs in piezoelectric crystals, and the latter in gyrotropic crystals. The shift current is the intrinsic (interband) contribution to the linear BPVE. In the independent-particle picture, the shift-current contribution to the photoconductivity in Eq. (1) takes the form [22] σ abc (ω) = C d 3 k (2π) 3 n,m where C = −πg s |e| 3 /2 2 is a combination of fundamental constants (g s = 2 accounts for the spin degeneracy), f nm = f n − f m and ω nm = E n − E m are differences in band occupations and band energies, and the matrix element reads Here r b mn is the interband dipole (the off-diagonal part of the Berry connection matrix A b mn = i u mk |∂ k b u nk ) which only depends on the initial and final states m and n, and r c;a nm is a generalized derivative defined as r c;a nm = ∂ ka r c nm − i (A a nn − A a mm ) r c nm , which depends implicitly on intermediate virtual states [22]. Like the intrinsic anomalous Hall effect in ferromagnets [23], the shift current arises from purely interband velocity matrix elements and probes the intrinsic Berry-phase geometry of the Bloch bands [21,22,24,25].
Mirror symmetry and the role of band-edge parities -We next analyze the constraints imposed on the form of the shift photoconductivity tensor by the presence of mirror symmetry; without loss of generality, we take that symmetry to be M x : x → −x. If M x leaves the crystal structure invariant, then according to Eq. (1) the tensor σ abc can be nonzero only when x appears an even number of times (zero or two) in abc. This is a phenomenological constraint, which holds at any frequency and irrespective of the mechanism behind the BPVE; it also holds if the space-group operation is not a pure reflection but a glide. Now assume that the space-group operation is a pure reflection M x , and that the minimum direct gap E g = E c − E v is located on a M x -invariant plane in the BZ. Under these conditions, the shift photoconductivity at frequencies close to E g is further restricted by dipole selection rules, in much the same way as the absorptive part of the linear conductivity [26,27]. The reason is that the states |v and |c at the top of the valence band and at the bottom of the conduction band are now eigenstates of M x , with eigenvalues ±i in the spinful case and ±1 in the spinless case; introducing a "relative mirror parity" P x vc that equals +1 (−1) when |v and |c have equal (opposite) eigenvalues, the dipole matrix elements are found to satisfy [26,27] By virtue of Eqs. (2) and (3), these selection rules set some components of σ abc (ω) to zero for ω ≈ E g . Let us first consider the shift current induced by light with linear polarization along b = c. In this case, the phenomenological constraint mentioned above becomes σ xbb = 0, and so the current must flow parallel to the mirror plane (we will refer to this as "in-plane flow"). Concerning the band-edge response, one can distinguish two scenarios on the basis of Eq. (4). When P x vc = +1 the matrix element I abb mn vanishes for b = x so that σ axx = TABLE I. Selection rules for the band-edge shift photoconductivity σ abc ( ω ≈ Eg) in the crystal class mm2, when the two reflection operations are not glides. Each column lists one of the symmetry-allowed components of the photoconductivity tensor, followed by the relative Mx and Mz parities of the valence and conduction band-edge states that allow that component to be non-negligible for ω ≈ Eg. The relative parities for BC2N-A2, P x vc = −1 and P z vc = +1, are marked in bold; they imply that only σ yxx and σ xxy (also marked in bold) are non-negligible at the band edge of this material.
, and when P x vc = −1 it vanishes for b = x so that σ ayy = σ azz = 0. Thus, when P x vc = +1 (P x vc = −1) the shift current flows in response to the in-plane (outof-plane) component of the optical electric field; in both cases, it flows strictly in-plane. We emphasize that this sharp separation only occurs at the band edge.
To complete the present discussion, let us consider the possible contributions to the shift current from σ abc with b = c. Such contributions average to zero for unpolarized light (e.g., sunlight), which is why those tensor components are often ignored when discussing potential solarcell applications of the BPVE [6,21,28]. When they are present, the current is no longer constrained to flow inplane. At the band edge, P x vc = +1 does not impose any restriction at all while P x vc = −1 forces σ abc to vanish if both b = x and c = x.
Physical realization -We now show that graphitic BC 2 N provides a striking illustration of the preceding discussion. We begin by noting that while a single layer breaks inversion symmetry [see Fig. 1(a)], whether inversion is still broken in the bulk structure depends on the stacking pattern, which remains to be determined experimentally. There are two types of stackings, denoted A or B depending on whether consecutive layers have the same or opposite orientation [29]. B-type structures have a center of inversion between the layers, while those of type A break inversion symmetry. As a result, the photoconductivity vanishes in B-type structures but remains finite in those of type A.
We consider the most stable A-type bulk structure identified in Ref. 29, namely the A2 structure illustrated in Fig. 1(a). The space group is Pmm2 (No. 25), and the point group is mm2. There are two mirrors M x and M z , and a rotation C y 2 about the polar axis. Point-group symmetry allows five out of nine components of the linear BPVE tensor σ abc = σ acb to be nonzero: three involving in-plane directions only (yxx, xxy = xyx, and yyy), and two that also involve z (yzz and zzy = zyz). Since M x and M z are pure reflections, at the band edge further re- strictions emerge from dipole selection rules as detailed in Table I. The scalar-relativistic band structure of BC 2 N-A2 near the band edge is displayed in Fig. 1(b). We only show the dispersion on the k z = 0 plane because the weak interlayer coupling produces a quasi-2D band structure with virtually no k z dispersion [29]. Inspection of the figure reveals that the dispersion from S to X is also relatively weak. The minimum direct band gap of E g ≈ 1.18 eV is located approximately midway between those two time-reversal invariant momenta (TRIM), as shown in Fig. 1(c). On the S-X line, whose points remain invariant under M x , the energy eigenstates are also eigenstates of M x , with eigenvalues ±1 as depicted by the solid and dashed lines in Fig. 1(b). We have explicitly verified that the upper-valence and lower-conduction bands have opposite M x eigenvalues, i.e., P x vc = −1. Moreover, since all bands in Fig. 1(b) are derived from p z -type Wannier functions (see below), they all have the same M z eigenvalue −1 on the k z = 0 plane, so that P z vc = +1. With these two parity values in hand, we can consult Table I to find out which components of the shift photoconductivity tensor are expected to be present at the band edge.
We have computed the shift photoconductivity of BC 2 N-A2 by means of density-functional theory (DFT). The calculations were performed using the Quantum ESPRESSO code package [30], taking the structural parameters from Ref. 29. The core-valence interaction was treated using scalar-relativistic projector augmentedwave pseudopotentials constructed using the Perdew-Burke-Ernzerhof exchange-correlation functional [31]. The self-consistent calculations were carried out on a 10 × 10 × 10 k -point mesh with the plane-wave cutoff set at 60 Ry. In a post-processing step, maximally-localized Wannier functions [32,33] were generated with the Wannier90 package [34], starting from atom-centered p z trial orbitals to model the bands around the Fermi level. Finally, the photoconductivity was calculated using a recently-developed Wannier-interpolation scheme [35].
The calculated photoconductivity is shown in Fig. 2(a). As predicted in Table I, three of the five symmetryallowed components, σ yyy , σ zzy and σ yzz , have negligible values in the band-edge region indicated by the gray area. The other two, σ yxx and σ xxy , grow rapidly from the onset at E g until reaching peak values of σ yxx ∼ 50 µAV −2 and σ xxy ∼ 30 µAV −2 at E X ≈ 1.33 eV; above E X they drop gradually and then stabilize at roughly half their peak values, before peaking again near 2 eV (not shown). The three previously-negligible components become sizeable above E X due to contributions from valence and conduction bands outside the mirror-invariant S-X line [see Figs. 1(b,c)], but they remain small compared to the other two.
For light with linear polarization along a crystallographic axis [b = c in Eq. (1)], the spectrum in Fig. 2(a) can be rationalized as follows on the basis of the symmetry arguments discussed previously. At all frequencies ω > E g the shift current flows along the line of intersection between the two mirror planes (along y), and at band-edge frequencies it only flows in response to the field component along x (normal to the M x mirror with P x vc = −1, and parallel to the M z mirror with P z vc = +1). Having understood the directionality of the BPVE at the band edge of BC 2 N-A2, we now examine the origin of the strong peak in σ yxx at E X . The peak value is comparable to the largest reported photoconductivity for a gapped material, σ zzz ∼ 50 µAV −2 in ferroelectric PbTiO 3 at 6 eV [36]. In addition, it occurs at a frequency of ω ≈ 1.3 eV that is suitable for optical manipulation, and where bulk semiconductors typically have much smaller responses [6,37,38]. (In experiment the peak is likely to occur at a higher energy in the visible range, due to the typical band-gap underestimation in DFT; for the B12 polytype of BC 2 N, that underestimation is around 0.6 eV [29]). The Glass coefficient [6,39], which quantifies photocurrent generation in bulk materials taking absorption into account, peaks at the same frequency as the shift current, with a value G yxx ∼ 3 · 10 −8 cm/V that again ranks among the largest reported to date [13].
To analyze this large response, we write the photoconductivity close to the band edge as the product between the shift-current matrix element and the JDOS [21], where the coefficient C is the same as in Eq. (2). The JDOS, plotted in Fig. 2(b), exhibits a peak at E X . That peak is a 2D-like Van Hove singularity due to a saddle point in the direct band gap [see Fig. 1(c)], and it boosts the amount of electronic transitions that contribute to the shift current around that energy. Moreover, those transitions carry a sharply-enhanced matrix element for σ yxx . This can be seen in Fig. 3, which displays a heatmap plot of the matrix element I yxx mnk of Eq. (2) summed over the upper-valence and lower-conduction bands: around X, it is more than two orders of magnitude larger than almost anywhere else in the 2D BZ. Two-band model in 2D -Motivated by the quasi-2D nature of graphitic BC 2 N-A2, we now construct a minimal 2D model that captures the effect of mirror parities on the photoconductivity near the band edge. The model lies flat on the (x, y) plane, and has both M x and M z symmetry. For simplicity we assume that the band edge is located at a TRIM on a M x -invariant line in the 2D BZ, not at a generic point along such a line as in the case of BC 2 N-A2. With these constraints, the most general two-band k · p Hamiltonian that can be obtained by expanding up to second order in k= (k x , k y ) around the TRIM is Essentially the same model was considered in Ref. 21, with the following differences. (i) In Ref. 21 the coordinate system was chosen to eliminate terms linear in k x ; instead, we have oriented the axes such that the vertical mirror is M x , which requires keeping terms linear in both k x and k y . (ii) We took as basis states the energy eigenstates at the valence and conduction band edges; as a result, our Hamiltonian is diagonal at k = 0. Since our basis states are also eigenstates of M x with eigenvalues ±1, the operator M x is represented by the identity matrix when P x vc = +1 and by σ z when P x vc = −1. In a model with translational symmetry, M x invari- . When applied to Eq. (6) using the two forms for M x , this condition yields v x = α xy = β xy = 0, when P x vc = +1, (7a) The relative band-edge parity P x vc therefore defines two classes of models with very different properties. The model with P x vc = +1 was used in Ref. 21 to describe the band-edge photoconductivity of monolayer GeS, while the model with P x vc = −1 applies to BC 2 N-A2. (Regarding M z symmetry, the model in Eq. (6) has P z vc = +1 because all atomic orbitals have the same parity and lie on the same plane. In this case, M z invariance does not impose any further constraints on the model parameters.) Starting from the two-band Hamiltonian in Eq. (6), the matrix element in Eq. (5) can be evaluated as described in Refs. 21 and 40. Because the model is purely 2D, the photoconductivity vanishes when any of the three indices equals z; on the (x, y) plane, the nonzero components are Using Eq. (7) we find that when P x vc = +1 the yxx component vanishes, while for P x vc = −1 it is the yyy component that vanishes. These results are in agreement with the first three columns of Table I for the case P z vc = +1. Besides illustrating the mirror selection rules, our model reveals a simple quantitative relation, σ yxx = 2σ xxy , when P x vc = −1 and P z vc = +1, (9) between the two surviving components of the band-edge photoconductivity. The above relation is satisfied rather well by our ab initio spectrum throughout the entire band-edge region in Fig. 2(a). This can be understood from the fact that Eq. (9) is quite robust: it follows directly from Eq. (3) once we set P x vc = −1 and P z vc = +1 in Eq. (4), and use the identity r y;x cv = r x;y cv that holds for any two-band tight-binding model once off-diagonal position matrix elements are discarded [35].
Discussion -To conclude, we discuss the prospects for realizing the physics described herein. The experimental evidence for the stacking sequence in graphitic BC 2 N remains inconclusive [15,16]. According to DFT calculations, the two most stable polytypes are the A2 structure studied in this work and a B-type structure denoted B12, with a difference in formation energy of only 1.2 meV/atom favoring the latter [29]. Both are indirectgap semiconductors, and while B12 provides a slightly better qualitative match to the experimental band structure [29], neither of them fits quantitatively the measured direct and indirect gaps [15,16]. Further work is clearly needed to establish the stacking sequence in bulk BC 2 N samples, and the linear BPVE could be useful in this regard as it is only present in acentric (A-type) structures.
One intriguing possibility is that it may be possible to grow both polytypes of BC 2 N using current synthesis techniques. This has been achieved for other layered materials, such as transition metal dichalcogenides [41]. For example, bulk MoS 2 grows in two different polytypes, centrosymmetric 2H and noncentrosymmetric 3R, and the effects of inversion symmetry breaking can be clearly detected in the latter [42]. The reported energy difference between them ranges from 0.1 to 2 meV/atom depending on the calculation [43][44][45], which is comparable to that between the B12 and A2 structures of BC 2 N [29]. We hope that our work will stimulate similar progress in graphitic BC 2 N, enabling the unambiguous identification of the A2 phase via its large and highly anisotropic photogalvanic effect.