Tunable quantum interference effect on magnetoconductivity in few-layer black phosphorus

In this study, we develop a systematic weak localization/antilocalization theory fully considering the anisotropy and Berry phase of the system, and apply it to various phases of few-layer black phosphorus (BP), which has a highly anisotropic electronic structure with an electronic gap size tunable even to a negative value. The derivation of a Cooperon ansatz for the Bethe-Salpeter equation in a general anisotropic system is presented, revealing the existence of various quantum interference effects in different phases of few-layer BP, including a crossover from weak localization to antilocalization. We also predict that the magnetoconductivity at the semi-Dirac transition point will exhibit a nontrivial power-law dependence on the magnetic field, while following the conventional logarithmic field-dependence of 2D systems in the insulator and Dirac semimetal phases. Notably, the ratio between the magnetoconductivity and Boltzmann conductivity turns out to be independent of the direction, even in strongly anisotropic systems. Finally, we discuss the tunability of the quantum corrections of few-layer BP in terms of the symmetry class of the system.

In this study, we develop a systematic weak localization/antilocalization theory fully considering the anisotropy and Berry phase of the system, and apply it to various phases of few-layer black phosphorus (BP), which has a highly anisotropic electronic structure with an electronic gap size tunable even to a negative value. The derivation of a Cooperon ansatz for the Bethe-Salpeter equation in a general anisotropic system is presented, revealing the existence of various quantum interference effects in different phases of few-layer BP, including a crossover from weak localization to antilocalization. We also predict that the magnetoconductivity at the semi-Dirac transition point will exhibit a nontrivial power-law dependence on the magnetic field, while following the conventional logarithmic field-dependence of 2D systems in the insulator and Dirac semimetal phases. Notably, the ratio between the magnetoconductivity and Boltzmann conductivity turns out to be independent of the direction, even in strongly anisotropic systems. Finally, we discuss the tunability of the quantum corrections of few-layer BP in terms of the symmetry class of the system.
Introduction. -The presence of disorder enriches the physics of mesoscopic systems [1][2][3][4]. One of the remarkable effects of disorder on transport properties is a negative correction to the dc conductivity, so called weak localization (WL). WL arises as electrons obtain an enhanced probability of backscattering due to a quantum interference effect. The theoretical description of the WL effect is accessible in a Feynman diagram formalism [5][6][7][8], and its signal has been explored extensively through experiments [9,10]. In contrast, electrons carrying π Berry phase result in a positive quantum correction to the dc conductivity, which is referred to as weak antilocalization (WAL). WAL is attributed to a suppressed probability of backscattering due to an additional minus sign by π Berry phase. The WAL effect has been studied intensely in topological states of matter with π Berry phase, such as graphene [11][12][13][14], topological semimetals [15][16][17][18][19], and surface states of topological insulators [20][21][22]. Such materials show complex quantum interference corrections due to the competition between the WL and WAL effects [11,12,19].
In this study, we develop a general framework for the quantum interference effect, which captures the anisotropy and Berry phase effect of the system. This is achieved by deriving a Cooperon ansatz which applies to a general system with an anisotropic energy dispersion. By applying this framework to the various phases of fewlayer BP, we find that the insulator phase and SDTP have the anisotropic WL corrections, while the DSM phase can host either WL or WAL arising from internode and intranode scatterings, respectively. We also estimate the effect of a magnetic field on the quantum corrections in each phase, predicting a nontrivial power-law dependence on the magnetic field at the SDTP. Notably, we show that the ratio between the magnetoconductivity and Boltzmann conductivity is independent of the direction, even in highly anisotropic systems. Finally, we discuss the tunability of the quantum interference effect in few-layer BP, with comments on the symmetry class.
Model. -The low-energy effective Hamiltonian of fewlayer BP is given by [29, 31, 33, 34] where m * is the effective mass along the zigzag (x) direction, E g is the band gap, v y is the velocity along the armchair (y) direction, and σ x,y are the Pauli matrices. The corresponding energy eigenvalues are given by E = Without band gap tuning, the Hamiltonian has a direct band gap (E g > 0) and few-layer BP is in the gapped insulator phase [ Fig. 1(a)]. At E g = 0, the Hamiltonian has a band touching point , a momentum state has time-reversed counterparts both within the node and in the opposite node. Thus, the quantum interference effect is contributed by two types of scatterings: intranode (red dashed arrow) and internode (blue dashed arrow) scatterings. As the Fermi energy increases (upper panel), the Fermi surface is distorted and the time-reversal symmetry around a node is broken. Thus, the quantum interference effect via intranode scattering is suppressed.
at (k x , k y ) = (0, 0) and few-layer BP is in the SDTP [ Fig. 1(b)]. At this point, the energy dispersion is linear in the armchair direction and quadratic in the zigzag direction. As the band gap decreases further to a negative value (E g < 0), few-layer BP becomes the DSM phase with two nodes at K ± = (± m * |Eg| 2 , 0) [ Fig. 1(c)]. At a sufficiently low Fermi energy satisfying E F |E g |, the Hamiltonian near each node can be linearized as where v x = |Eg| m * is the velocity along the zigzag direction. For later convenience, we adopt the parametriza- and v 0 κ sin φ κ = v y k y , where κ is an effective momentum measured from the nodes. In this work, we neglect the effects of spin-orbit coupling due to its negligible size in few-layer BP.
The quantum correction is determined not only by the electronic structure, but also by the type of impurity. The quantum interference effect is only induced by static and non-magnetic impurities as it is destroyed by nonstatic or magnetic impurities [3,4]. Another aspect we should consider is the range of scattering. At low Fermi energies, regardless of the range of scattering, an electron in the insulator phase and SDTP has a single timereversed counterpart on the whole Fermi surface. In contrast, for the DSM phase we consider two types of scatterings: intranode and internode scatterings [ Fig. 1(d)].
In the E F E g limit, intranode scattering may occur by long-range impurities, whereas internode scattering may arise from short-range impurities, such as lattice vacancies. The relative strengths of intranode and internode scatterings lead to competition between the WAL and WL effects in the DSM phase, as will be discussed later. Diagrammatic approach. -In a diagrammatic approach, effects of weak disorder on transport can be studied by incorporating relevant corrections into the current-current correlation function with disorderaveraged Green's functions [44,45]. Calculating the selfenergy up to the first-order Born approximation, we obtain the quasiparticle lifetime (τ qp k ) −1 = 1 V k W k ,k where V is the volume of the system and W k ,k is the transition rate from k to k . Note that for impurity scattering, the transition rate is given by W k ,k = 2πnimp |V k ,k | 2 δ(ξ k − ξ k ) where n imp is the impurity density, V k ,k = k |V |k is the matrix element of the scattering potential V , and ξ k ≡ ε k − µ is the energy of an electron with respect to the chemical potential µ. The ladder vertex correction combined with the firstorder Born approximation is the leading impurity correction to the current vertex [ Fig. 2(a)], which gives results equivalent to those of the semiclassical Boltzmann transport theory. The ladder vertex correction obeys the selfconsistent Dyson's equation [ Fig. 2(b)], and for isotropic systems, it gives a well-known result with the (1 − cos θ) factor in the inverse transport relaxation time [46], suppressing the low-angle scattering contribution to transport. In an anisotropic system, the relation is generalized to a coupled integral equation, which relates transport relaxation times at different states. In a d-dimensional anisotropic system, it reads [33, 47, 48] where τ (i) k and v (i) k = 1 ∂ε k ∂ki are the transport relaxation time and velocity, respectively, along the ith direction. The ladder vertex correction modifies the velocity vertex along the ith direction asṽ . In the zero-temperature limit, the ladder approximation yields the dc conductivity in the semiclassical Boltzmann regime as where g is the degeneracy of the system and the superscript B denotes the Boltzmann conductivity. Consideration of further diagrams, called maximally crossed diagrams, leads to the quantum interference correction [Figs. 2(c) and 2(d)]. The quantum correction can be boiled down to three leading terms referred to as a bare Hikami box and two dressed Hikami boxes [12]. The Hikami boxes can be computed by the Cooperon operator, which obeys the following Bethe-Salpeter equation [3,4] (in the following, we omit for simplicity): where C AR Q (k, k ) is the Cooperon with the momentum Q = k + k , and the superscripts A and R represent the advanced and retarded functions, respectively. Note that frequencies in the Cooperon are set to zero as we focus on the dc conductivity. The retarded and advanced Green's functions are G R,A (p, ξ) = [ξ − ξ p ± i/2τ qp p ] −1 . As the Green's functions near the Fermi surface contribute mostly to the momentum summation in the right-hand side of Eq. (5), we can perform the ξ p −integral separately as . Using the Ward identities in anisotropic systems [48], we rewrite the denominator of the right-hand side of the ξ p −integral as p . Accordingly, we have the Bethe-Salpeter equation in anisotropic systems as follows: where the Cooperon diverges as Q → 0, and thus we ignore terms of order higher than Q 2 . For the detailed derivations, see the Supplemental Material [49]. Importantly, we capture the full anisotropy of the system by introducing f Q (p), which is determined by the anisotropic velocities and transport relaxation times on the Fermi surface.
The Bethe-Salpeter equation in Eq. (7) can be solved self-consistently by collecting the most divergent contribution in the Q → 0 limit [11,15,18]. Alternatively, we directly derive the following Cooperon ansatz by computing the average of Eq. (7) on the Fermi surface [49]: where N 0 is the density of states at the Fermi surface and D ij are the diffusion coefficients defined by It is worth noting that the ansatz in Eq. (8) includes the electronic structure-dependent phase factor . The ansatz is valid as long as the composition property holds, as in the case of the various phases of few-layer BP. For the insulator phase and SDTP, F (k, k ) is unity, whereas for the DSM phase, F (κ, κ ) = e i(φκ−φ κ ) (unity) for intranode (internode) scattering. Notably, the phase factor reflects the Berry phase of the system which determines the sign of the quantum correction, as will be shown below.
WL and WAL corrections. -Using the ansatz in Eq. (8), we compute the total quantum correction ∆σ ii = ∆σ bare ii +2∆σ dressed ii for each phase [49]. For the insulator phase and SDTP, we find the following WL correction: where φ is the phase coherence length, e is the meanfree path, and D ≡ D xx D yy . We note that Eq. (10) applies not only to few-layer BP but also to a general 2D anisotropic system regardless of scattering potential [49], with the additional minus sign for the WAL correction. Remarkably, the ratio of the quantum correction to the Boltzmann conductivity is the same irrespective of direction even in strongly anisotropic systems, which also holds in 3D anisotropic systems [49].
For the DSM phase, we compute the quantum corrections due to intranode and internode scatterings, respectively. In this study, for simplicity we consider constant intranode and internode scattering potentials given by V intra and V inter , respectively. When backscattering is governed by intranode scattering, the Cooperon is given by where (τ qp intra ) −1 = nimpV 2 intra EF 2vxvy . Importantly, the phase factor F (κ, κ ) = e i(φκ−φ κ ) reflects the π Berry phase of the DSM phase, which yields the WAL correction to the dc conductivity as where intra is the mean-free path for intranode scattering. On the other hand, the Cooperon for internode scattering is obtained as where (τ qp . Thus, internode scattering gives rise to the WL correction as where inter denotes the mean-free path for internode scattering. where Vimp denotes the strength of the impurity potential. As for the DSM phase, (c) intranode scattering induces the WAL effect, (d) while internode scattering leads to the WL effect. Both corrections are plotted in units of α2 = nimpV 2 imp / 2 vxvy. As for the field-dependence, the black dashed lines denote the ± ln B−dependence, while the red dashed line and blue dashed line in (b) represent the B ν −dependence with ν ≈ 0.31 and B 2/3 −dependence, respectively. The exponent ν is not universal but depends on the system parameter.
Magnetoconductivity. -Applying an external magnetic field gives an additional phase to each backscattering path, and thus destroys the quantum interference effect [2][3][4]. In experiments, the phase coherence length can be obtained through the measurement of magnetoconductivity. The magnetoconductivity can be computed using the quantization of Landau levels [3,18]. Assuming an external magnetic field along the z direction, we obtain the ratio for each phase as follows [49]: is the magnetic length. Eq. (15c) is obtained for intranode scattering, and the field-dependence for internode scattering only differs by the sign, with intra (τ qp intra ) replaced by inter (τ qp inter ). For the insulator phase [ Fig. 3(a)] and DSM phase [Figs. 3(c) and 3(d)], we find that the fielddependence of the magnetoconductivity is well approximated by P (i) (B) ∝ ± ln B, which is in good agreement with the conventional prediction for 2D systems [5]. However, we predict that the magnetoconductivity of the SDTP will not follow the conventional prediction, but rather will show φ ) with the exponent ν, which depends on the system parameter and can be found numerically [49]. This nontrivial field-dependence may be attributed to strong anisotropy in the band dispersion [4], leading to a quantum diffusion which deviates from the 2D behavior. We note that P (i) (B), the ratio of the field-induced change in the magnetoconductivity to the Boltzmann conductivity, is the same irrespective of the direction i for each phase.
Discussion. -Our analysis shows that the sign of the quantum correction is determined by the electronic structure-dependent phase factor F (k, k ), which reflects the symmetry class of the system [11,17,51]. The unity phase factor of the insulator phase and SDTP indicates that the system has spinless time-reversal symmetry, belonging to the orthogonal class. Backscattering induced by a time-reversal operator of this kind leads to WL. As for the DSM phase, in the absence of internode scattering, the phase factor F (κ, κ ) = e i(φκ−φ κ ) indicates that the system has time-reversal symmetry around a node without spin-rotational symmetry, belonging to the symplectic class. A system possessing a time-reversal operator of this kind exhibits WAL. In contrast, the presence of internode scattering can induce a crossover from the symplectic to orthogonal class, leading to the corresponding crossover from WAL to WL [11]. Thus, the overall quantum correction in the DSM phase is determined by the dominant scattering mechanism, which depends on the separation between the two nodes [49]. In the E F |E g | limit, a large separation between the nodes will suppress the internode scattering rates, and thus the WAL effect might be dominant over the WL effect. In addition, increasing the Fermi energy leads to the distortion of the Fermi surface, suppressing WAL.
In this study, we developed a quantum interference theory for anisotropic systems by solving the Bethe-Salpeter equation for the Cooperon operator, fully considering the anisotropy and Berry phase of the system. We elaborated the Cooperon ansatz and diffusion coefficients in a compact and physically intuitive form with transport relaxation times in anisotropic systems, generalizing the previous work by P. Wölfle et al. [7]. Furthermore, we considered systems with nontrivial Berry phase, providing a systematic quantum interference theory for both WL and WAL effects, and the crossover between them. cation to phosphorene, Nano Lett. 15, 1222 (2015 In this section, we present the eigenstates for the various phases of few-layer BP, following the parametrization of Park et al. [1]. For the insulator phase and SDTP (E g ≥ 0), we can rewrite the Hamiltonian in Eq. (1) as For the DSM phase with a sufficiently low Fermi energy (E F E g ), the Hamiltonian near the positive/negative node K ± can be rewritten as where we adopt the parametrization v 0 κ cos φ κ = v x (k x − K ± x ) and v 0 κ sin φ κ = v y k y . Note that we introduce an effective momentum κ to describe momentum states near the node. The corresponding eigenstate for the upper band is given by |κ = 1 √ 2 (1, ±e ±iφκ ) T , and thus we have φ −κ = φ κ + π.

II. LADDER VERTEX CORRECTIONS
In this section, we provide a brief explanation on the ladder vertex correction for elastic impurity scattering. In a many-body diagrammatic approach, the dc conductivity can be obtained from the current-current correlation function. First, let us consider a d−dimensional isotropic system. The single bubble diagram captures the Drude conductivity σ Drude = e 2 N 0 D where D = v 2 F τ qp /d is the diffusion constant with the quasiparticle lifetime τ qp . The ladder vertex correction is the leading impurity correction to the current vertex, which obeys the self-consistent Dyson's equation [2,3] represented in Figs. 2(a) and 2(b). In a single-band isotropic system, it is equivalent to replacing τ qp in D by the transport relaxation time τ tr which satisfies [4] 1 τ tr In an anisotropic system, the relation in Eq. (S.3) is no longer valid, and a further systematic relation is required fully considering the anisotropy of the system. In a diagrammatic approach, the ladder vertex can be related to the transport relaxation time by τ where the ladder vertex Λ (i)AR (k, ξ k , ξ k ) modifies the velocity operator along the ith direction byṽ . As a result, the Dyson's equation turns out to be equivalent to the integral equation relating transport relaxation times at different states by which is the generalization of Eq. (S.3). Note that for elastic impurity scattering, the transition rate is given by Consequently, the dc conductivity incorporating the ladder vertex correction reads where the superscript B denotes the Boltzmann conductivity and S 0 (ξ) = − ∂f 0 (ξ) ∂ξ = βf 0 (ξ) 1 − f 0 (ξ) . Here, f 0 (ξ) = e βξ + 1 −1 is the Fermi-Dirac distribution function in equilibrium with β = 1 kBT . In the zero-temperature limit, S 0 (ξ) is given by the Dirac-delta function δ(ξ).

A. Bethe-Salpeter equation
This section is devoted to transforming the Bethe-Salpeter equation into a form capturing the full anisotropy of the system. Henceforth, we omit for simplicity. The Cooperon obeys the following Bethe-Salpeter equation: where V k ,k = k |V |k is the matrix element of the scattering potential V . Here, we assume that V Q−k ,Q−k ≈ V −k ,−k since the Cooperon diverges as Q → 0 and the dominant contribution comes from small Q. We compute the integral over energy with the aid of a contour integral method: We deal with the denominator of the above equation using the relation between the self-energy and quasiparticle lifetime, Using the Ward identity, v in anisotropic systems [5], we have Accordingly, the Bethe-Salpeter equation can be rewritten as p . Note that terms of order higher than Q 2 are ignored.

B. Cooperon ansatz
Now, we are in a position to solve the Bethe-Salpeter equation in Eq. (S.10). We note that there appears V k ,k V −k ,−k in Eq. (S.10). Let us assume that V −k ,−k ≡ V * k ,k F (k, k ), where the phase factor F (k, k ) is determined from the electronic structure of the system. Here, we consider systems where F (k 1 , k 3 ) = F (k 1 , k 2 )F (k 2 , k 3 ) holds, such as in few-layer BP. IntroducingC AR Q (k, k ) ≡ C AR Q (k, k )F (k , k), the Bethe-Salpeter equation reduces tõ Performing 1 V k δ(ξ k ) on both sides of Eq. (S.11), we have where we use the definition of the quasiparticle lifetime 1 . We consider the following ansatz for Eq. (S.12): where D ij denotes the diffusion coefficients. Plugging this ansatz into Eq. (S.12), we have Since the most divergent terms in the left and right-hand sides are identical as we plug this ansatz into Eq. (S.11), it is consistent with the ansatz obtained from the iterative method in previous works [6][7][8]. Finally, we obtain the general ansatz for the Bethe-Salpeter equation as

A. Insulator phase and SDTP
The quantum correction to the dc conductivity is contributed by a bare Hikami box and two dressed Hikami boxes as illustrated in Fig. 2(c). Before computing each term, we note the following identities: First, let us compute the bare Hikami box contribution: where g s = 2 is the spin degeneracy factor.
Next, let us compute the dressed Hikami box contribution. It includes two diagrams which contribute equally to where φ is the phase coherence length and e is the mean-free path. Here, we assume that the lower and upper cutoffs of the integral are given by −1 φ and −1 e , respectively. Restoring , we finally obtain the WL correction as which applies not only to few-layer BP but also to a general 2D anisotropic system. Importantly, the quantum correction is proportional to the Boltzmann conductivity, and thus the ratio of the quantum correction to the Boltzmann conductivity does not depend on the direction regardless of the anisotropy of the system. Rewriting Eq. (S.19) as we note that this direction-independent ratio generally holds even in 3D systems.

B. DSM phase with intranode scattering
Before studying the quantum interference effect in the DSM phase due to intranode scattering, we note several identities: Note that the transport relaxation time τ tr is isotropic and equals 2τ qp . Using these identities, we obtain the explicit form of the Cooperon as We note that for backscattering, the phase factor becomes e i(φ−φ ) = −1, and thus the Cooperon has a negative value. Therefore, intranode scattering in the DSM phase induces the WAL effect in the E F |E g | limit, as will be demonstrated in the following.
We are now in a position to compute the quantum correction of the DSM phase in the presence of intranode scattering. First, let us compute ∆σ bare xx : Next, one of the dressed Hikami boxes reads Accordingly, the total quantum correction to the dc conductivity is The Q−integral can be computed by introducingQ x = Dxx D Q x andQ y = Dyy D Q y with D = D xx D yy , similarly as we did for the insulator phase and SDTP: where intra is the mean-free path corresponding to intranode scattering. Restoring , we obtain the total quantum correction to the dc conductivity as or equivalently is the Boltzmann conductivity along the ith direction. Note that the ratio ∆σ ii /σ B ii is the same irrespective of the direction.

C. DSM phase with internode scattering
Using the eigenstates in the DSM phase (see Sec. I), we can obtain the matrix elements for internode scattering as where V inter is internode scattering amplitude, and the superscripts + and − denote the positive and negative nodes, respectively. Thus, V +,− κ ,κ V −,+ −κ ,−κ = |V +,− κ ,κ | 2 and we have the Cooperon ansatz as (S.31) Repeating the steps in the previous section, we obtain the quantum correction due to internode scattering as or equivalently which differs only by the sign from the quantum correction for intranode scattering in Eq. (S.29) with intra (V intra ) replaced by inter (V inter ). Note that as for the intranode scattering, the ratio ∆σ ii /σ B ii is the same irrespective of the direction.
In the presence of both intranode and internode scatterings, the quantum interference effect in the DSM phase is determined by the dominant scattering process. Thus, we expect that the WAL (WL) effect will occur when intranode (internode) scattering is dominant.

A. Insulator phase
In the following, we restore for clarity. According to J. M. Pereira, Jr et al. [9], the Landau levels in small B have a linear dependence on magnetic field, as in the case of a free electron gas. Thus, for the insulator phase, we adopt the Hamiltonian of an anisotropic free electron gas as H = Here, we define the magnetic length B = c 4eB . Following the above quantization condition, we modify the Q−integral in Eq. (S.19) by 1 x+k . Accordingly, we obtain the magnetoconductivity of the insulator phase as Since the digamma function follows the asymptotic form Ψ 1 2 + x ≈ ln x + 1 24x 2 + · · · for x → ∞, the magnetoconductivity reduces to Eq. (S.21) in the B → 0 limit. Thus, the ratio between the magnetoconductivity [Eq. (S.37)] and the dc conductivity [Eq. (S.5)] reads

(S.38)
Again, the ratio is irrespective of the direction.

B. DSM phase
Now, let us consider the DSM phase. Applying an external magnetic field perpendicular to the xy plane, say B = (0, 0, B), the crystal momentum is transformed as k = p + e c A where p is the canonical momentum and A is the vector potential. Thus, the Hamiltonian becomes where we chose the Landau gauge A = (0, Bx, 0) for the vector potential. Then we have the nth Landau level as Note that the momentum along the y axis is a good quantum number. Thus, the crystal momentum follows the The quantization for Q = k + k is obtained by doubling the magnetic field: which is equivalent to Eq. (S.34) in the insulator phase. Therefore, we compute the magnetoconductivity in the DSM phase in a similar manner as in the insulator phase: We note that in the B → 0 limit, Eqs. (S.42a) and (S.42b) reduce to Eq. (S.28a) and (S.28b), respectively. Similarly, we can compute the magnetoconductivity for internode scattering, which only differs from the result in Eq. (S.43) by the sign, with intra (τ qp intra ) replaced by inter (τ qp inter ).

C. SDTP
Following Petra Dietl et al. [10], we have the Landau quantization for the momentum Q at the SDTP as follows: . Note that we consider the region whereQ