Two-dimensional (2D) semi-Dirac systems, such as 2D black phosphorus and arsenene, can exhibit a rich topological phase transition between insulating, semi-Dirac, and band inversion phases when subjected to an external modulation. How these phase transitions manifest within the quantum transport and shot noise signatures remains an open question thus far. Here, we show that the Fano factor converges to the universal F ≈ 0.179 at the semi-Dirac phase and transits between the sub-Poissonian ( F ≈ 1 / 3) and the Poissonian shot noise ( F ≈ 1) limit at the band inversion and the insulating phase, respectively. Furthermore, the conductance of a 2D semi-Dirac system converges to the contrasting limit of G / G 0 → 1 / d and G / G 0 → 0 at the band inversion and the insulating phases, respectively. The quantum tunneling spectra exhibits a peculiar coexistence of massless and massive Dirac quasiparticles in the band inversion regime, thus providing a versatile sandbox to study the tunneling behavior of various Dirac quasiparticles. These findings reveal the rich interplay between band topology and quantum transport signatures, which may serve as smoking gun signatures for the experimental studies of semi-Dirac systems near the topological phase transition.

Shot noise is generated by electrical current fluctuations arising from the discrete nature of charge carriers. Shot noise can be used as an indicator of the correlation between charged carriers.1–4 A well-known characterization of shot noise is the Fano factor, F, which is defined as the ratio between the actual shot noise and the Poisson shot noise with F = 1. Prominent examples of systems with F 1 include the super-Poissonian shot noise with F > 1 in zero-dimensional quantum dots5,6 and the sub-Poissonian shot noise with F = 1/3 in both disordered conductors7,8 and graphene.9–11 Furthermore, the well-celebrated maximal Fano factor value in graphene is shown to be associated with the minimal conductivity in orders of e 2 / .9 Importantly, shot noise provides a useful tool to probe the quantum transport properties of an electronic system12 and has been widely employed in the experimental studies of graphene and their heterostructures.13,14

Two-dimensional (2D) semi-Dirac material (SDM) represents an interesting system that simultaneously host linear (relativistic) energy dispersion in one direction and parabolic (nonrelativistic) energy dispersion in the orthogonal direction.15–17 SDMs have been realized in a large variety of systems, including (TiO2)m/(VO2)n nanostructure,18 strained organic salt,19 photonic crystals,20 Bi1−x Sbx,21 striped boron sheet,22 on surface states of topological insulators,23,24 or in non-centrosymmetric systems,25 such as phosphorus-based materials,26–32 monolayer arsenene,33 silicene oxide,34 and polariton lattice,35 and also in α-dice lattice36–38 with higher pseudospin. The electronic transport and shot noise of SDM have been studied extensively in previous works, which establish a Fano factor of F = 1 and F 0.179 at the band insulator phase with nonzero bandgap and at the semimetallic gapless limit, respectively.38–47 Nevertheless, 2D SDM can undergo complex topological phase transitions. Beyond the band insulating and the semimetallic regime, SDM can exhibit a band inversion phase25 in which two distinct Dirac cones emerge, thus rendering the band-inverted SDM a strong potential for valleytronic device applications.45 Nevertheless, the shot noise and conductance signatures of band-inverted SDM remain an open question thus far.

In this work, we study the quantum transport of SDM near the topological phase transitions. Focusing on the quantum transport occurring along the relativistic direction—not covered in the previous quantum transport study,45 we observe the intriguing coexistence of massless9,48 and massive Dirac fermions in the tunneling spectra at fixed transport channel at all quasiparticle energies, which is distinctive from that of bilayer graphene49 in which the massive and massless Dirac quasiparticles occur at various transversal momenta at different quasiparticle energies. We further calculate the conductance and Fano factors of 2D SDMs as the band topology changes continuously from band insulating to band inversion phases. Remarkably, we found that the Fano factor converges to sub-Poissonian shot noise with F 1 / 3 and to Poissonian shot noise with F 1 in the band inversion and insulating phase, respectively [Fig. 1(a)]. Such shot noise signatures have negligible thermal noise contributions, even at room temperature. Our findings reveal the exotic quantum transport behavior and the shot noise signatures of 2D SDM at various phases, thus uncovering shot noise as a useful tool in probing the band topology of 2D SDM.

FIG. 1.

Illustration of the insulating ( Δ > 0), semi-Dirac (Δ = 0), and band inversion ( Δ < 0) phases tuned by Δ in Eq. (1) with its corresponding Fano factors in (a). Illustration of chiral tunneling across a potential barrier with width d and transmission probability T along the linear dispersion (ky) in (b). The band inversion phase in (a) can co-exhibit both gapless/gapped band crossings along ky depending on its transverse momenta kx in (c).

FIG. 1.

Illustration of the insulating ( Δ > 0), semi-Dirac (Δ = 0), and band inversion ( Δ < 0) phases tuned by Δ in Eq. (1) with its corresponding Fano factors in (a). Illustration of chiral tunneling across a potential barrier with width d and transmission probability T along the linear dispersion (ky) in (b). The band inversion phase in (a) can co-exhibit both gapless/gapped band crossings along ky depending on its transverse momenta kx in (c).

Close modal
2D SDM can be described by a two-band effective Hamiltonian,50,51
H ̂ = ( α k x 2 + Δ ) σ x + β k y σ y ,
(1)
with α = 2 / ( 2 m * ) and β = v y, where m * and vy are the effective mass along x ̂ and Fermi velocity along y ̂, respectively. A phase transition parameter, Δ, acts as a perturbation factor that continuously changes the band topology from the band insulating phase ( Δ > 0) and the semi-Dirac phase (Δ = 0) to the band inversion phase ( Δ < 0). We can nondimensionlize Eq. (1) by defining the characteristic momentum and energy ( k 0 = 2 m * v y and ε 0 = k 0 v y) to obtain H ̂ = ( k x 2 + Δ ) σ x + k y σ y, which has the following energy dispersion:
ε k = ± ( k x 2 + Δ ) 2 + k y 2 ,
(2)
where ± indicates conduction/valence (+/–) bands. This parameter has been utilized for directional dependent transport40,52–56 and phase-dependent transport.36,44,57,58 Owing to the presence of two inequivalent valleys in the band inversion phase ( Δ < 0), band-inverted 2D SDMs have been widely studied for potential applications in valleytronics.44–47 
From Fig. 1(a), the SDM (Δ = 0) with a semi-Dirac point at ( k x , k y ) = ( 0 , 0 ) can become a band insulator ( Δ > 0) or exhibit a band inversion ( Δ < 0) phase with band crossings at ( k x , k y ) = ( ± k D , 0 ), with k D = | Δ |.15,17,25 Interestingly, Eq. (1) in the band inversion regime can be decomposed into either the 1D massless and massive Dirac Hamiltonian [Fig. 1(c)] along the ky direction at k x = ± k D and at kx = 0, respectively,
H ̂ ( Δ < 0 ) = { k y σ y , k x = ± k D ; Δ σ x + k y σ y , k x = 0.
(3)

The quantum transport along the ky direction is, thus, expected to exhibit a mixture of massive and massive Dirac quasiparticles dictated in Eq. (3).

The transmission probability T can be obtained by considering a scattering potential along ky [Fig. 1(b)] with U ( y ) = U 0 ( Θ ( y ) Θ ( y d ) ), where U 0 U 0 / ε 0 and d d 0 k 0 are the dimensionless potential height and barrier width, respectively. In the L x d limit, the Hamiltonian decouples into a 1D eigenvalue equation along k y i y, with
Ψ j = A j ( ε j k x 2 + Δ + i k j ) e i k j y + B j ( ε j k x 2 + Δ i k j ) e i k j y ,
(4)
where the index j denotes the L (left), B (barrier), and (R) right regions. The energy and wavevector with index j denote ε L / R = ε k , ε B = ε k U 0 ε q with k L / R = k y = λ ε k 2 ( k x 2 + Δ ) 2 , k B = q y = λ ε q 2 ( k x 2 + Δ ) 2 , λ = sgn ( ε k ) and λ = sgn ( ε q ). For the left incident wavefunction, AL = 1 and BR = 0 are enforced with the transmission coefficient t = AR. For only forward-moving electronic states in the region R, the wavevectors are enforced in Eq. (4) through k y > 0 and q y < 0. For the n–p–n junction, the conservation of current (non-dimensionlized), J ̂ i + J ̂ r = J ̂ t with J ̂ y = 2 Ψ σ ̂ y Ψ = 2 Ψ σ ̂ y Ψ , gives us the relation | r | 2 + | t | 2 = 1. By matching the boundary conditions at y = 0 and d, we obtain T = | t | 2 as
T = 4 ε k 2 ε q 2 k y 2 q y 2 4 ε k 2 ε q 2 k y 2 q y 2 cos 2 ( λ q y d ) + sin 2 ( λ q y d ) [ ( k x 2 + Δ ) 2 U 0 2 + ε k 2 q y 2 + ε q 2 k y 2 ] 2 ,
(5)
in agreement with a band insulator for Δ > 0 [top left panel of Fig. 2(a)] and a SDM40 for Δ = 0 by noticing that tan ϕ = ( k x 2 + Δ ) / k y and tan θ = ( k x 2 + Δ ) / q y [top right panel of Fig. 2(a)]. Remarkably, the rotational invariant of Eq. (2) around z ̂, i.e., [ H , R z ] = 0 with Rz being a rotation operator about the z ̂, allows Eq. (5) to fully capture the Klein tunneling behavior, as evident under rotation and incident angle.39,43 However, this cannot be generalized along the parabolic direction due to intervalley scattering.39–41,45
FIG. 2.

Transmission probability T for Δ = 0.3, 0, and –0.3 is shown in the top left/top right/bottom panels in (a). Pseudospin vectors from Eq. (6) are shown in (b) with red/blue arrows indicating the direction of Sy.

FIG. 2.

Transmission probability T for Δ = 0.3, 0, and –0.3 is shown in the top left/top right/bottom panels in (a). Pseudospin vectors from Eq. (6) are shown in (b) with red/blue arrows indicating the direction of Sy.

Close modal
We now focus on the band inversion phase for Δ < 0 [bottom panel of Fig. 2(a)]. The tunneling behavior37 in Eq. (5) at different transverse momenta, kx, can be understood from the pseudospin texture45 (non-dimensionalized) shown in Fig. 2(b) and is given by
S = Ψ σ Ψ = ( k x 2 + Δ ε k , λ k y ε k ) .
(6)
As T is symmetric about kx [green shading in the lower half contours], it suffices to analyze the non-positive transverse momenta with k x 0. At a Dirac point k x = k D, the conservation of pseudospin across the potential barrier [orange dotted box] enables Klein tunneling at normal incidence with T = 1 [Fig. 2(a)], which resembles the 1D tunneling of massless Dirac fermions at k x = k D in Eq. (1). Similarly, transmission resonances occur at 0 k x < k D [purple box in Fig. 2(a)], which resembles the 1D tunneling of gapped massive Dirac fermions.

As illustrated in Eq. (3), there exist two special transport channel at kx = kD in which the quasiparticles tunnel with full transmission. Such Klein tunneling behavior arises because of the 1D gapless massive Dirac Hamiltonian at kx = kD [see Eq. (3)]. At kx = 0, the effective 1D Hamiltonian [Eq. (3)] reduces to that of a gapped Dirac quasiparticle. The tunneling spectra at the transport channel of kx = 0, thus, deviates from perfect Klein tunneling and exhibits oscillations instead. The quasiparticle tunneling, thus, exhibits a peculiar coexistence of massive and massless fermions where quasiparticles with kx = 0 perform Klein tunneling (i.e., massless Dirac fermion) while those with kx = kD perform oscillatory tunneling (i.e., massive Dirac fermion). It should be noted that the coexistence of massive and massless Dirac fermions has previously been reported in twisted bilayer graphene.49 However, in twisted bilayer graphene, the massive and massless Dirac quasiparticles occur at different transport channel (i.e., transversal momenta) dependent on the energy of the quasiparticles. This aspect is in stark contrast to the case of 2D SDM studied here, in which the massless and massive Dirac quasiparticles occur at fixed transport channels of kx = kD and kx = 0, respectively. The 2D SDM, thus, provides a potentially convenient platform to study the quantum transport behaviors of massive and massless Dirac quasiparticles.

The modulation of pseudogap through U0 and Δ is shown in Fig. 3(a). While the full transmission, oscillatory, and pseudogap regions are expected from Klein tunneling,9,49 an abrupt cutoff for ε k = 0.5 at Δ = 0.5 is observed. This is attributed by the crossover of the incident wavefunction from propagating to evanescent when k y = ε k 2 Δ 2 becomes imaginary at kx = 0. This implies that the cutoff for ε k = 1.5 appears only at Δ = 1.5. Hence, the decoupling of the band inversion phase with Δ 0, in the semi-Dirac studies,37,50,51 can be attributed to the crossover of propagating to evanescent incident wavefunction.

FIG. 3.

The relationship of U0 against Δ on T is plotted in (a) with both ε k = 0.5 and 1.5. The effects of potential barrier width d on T are shown in (b) for d = 100 and in the n–p limit at Δ = 0.3.

FIG. 3.

The relationship of U0 against Δ on T is plotted in (a) with both ε k = 0.5 and 1.5. The effects of potential barrier width d on T are shown in (b) for d = 100 and in the n–p limit at Δ = 0.3.

Close modal
The retention of the Klein tunneling behavior from d = 30 [Fig. 2(a)] to d = 100 and subsequently to the n–p limit is in Fig. 3 by setting t = AB, A R = B R = 0 in Eq. (4) such that
T n p = 4 λ q y k y ε q ε k λ ( | λ k y ε q + λ ε k q y | 2 + U 0 2 ( k x 2 + Δ ) 2 ) ,
(7)
with the conservation of pseudospin dictating | r | 2 + | t | 2 λ q y ε k / ( λ k y ε q ) = 1 and by neglecting the evanescent wave contributions.
We now investigate the quantum transport signatures, i.e., conductance G and Fano factor F [Fig. 4]. In the zero-temperature limit, G and F can be expressed as
G = G 0 T d k x ; F = S I G ,
(8)
with G 0 = g e 2 / ( 2 π h ), g is the degeneracy term (spin and valley), e is the electronic charge, h is the Planck's constant, S I = T ( 1 T ) d k x is the current noise, and T is obtained from Eq. (5) in the grazing energy ( ε k / U 0 1) and tall barrier ( L x d) limit,9 such that
T sech 2 ( ( k x 2 + Δ ) d ) ,
(9)
with ε k U 0 , ε q 0 , q y 2 ( k x 2 + Δ ) 2, and k y 2 U 0 2 ( k x 2 + Δ ) 2.
FIG. 4.

The conductance G / G 0 (top) and Fano factor F (bottom) [Eq. (8)] are plotted, respectively, across Δ with d = 30 (blue), d = 60 (red), and d = 100 (green). (Top): The broken line is the SDM conductance value, i.e., G SDM / G 0 = 1.9 / d. The top inset shows the relationship between ( G G SDM ) / G 0 (blue solid line) with d = 30 against its linear approximation (orange dotted line) with 0.5 d Δ 0 in the band inversion phase. (Bottom): The broken line is the SDM Fano factor, i.e., F 0.179. The bottom inset shows the thermal noise correction up to 300 K for all Δ.

FIG. 4.

The conductance G / G 0 (top) and Fano factor F (bottom) [Eq. (8)] are plotted, respectively, across Δ with d = 30 (blue), d = 60 (red), and d = 100 (green). (Top): The broken line is the SDM conductance value, i.e., G SDM / G 0 = 1.9 / d. The top inset shows the relationship between ( G G SDM ) / G 0 (blue solid line) with d = 30 against its linear approximation (orange dotted line) with 0.5 d Δ 0 in the band inversion phase. (Bottom): The broken line is the SDM Fano factor, i.e., F 0.179. The bottom inset shows the thermal noise correction up to 300 K for all Δ.

Close modal

The conductance G / G 0 and the Fano factor F at d = 30 (blue), d = 60 (red), and d = 100 (green) are shown in Fig. 4. Away from the semi-Dirac point with ( G / G 0 1.9 / d , F 0.179),40 the values converge toward the insulating ( Δ > 0) phase with ( G / G 0 = 0 , F 1) and band inversion ( Δ < 0) phase with ( G / G 0 = 1 / d [see the inset of top figure], F 1 / 3),9 respectively. The convergence to the sub-Poissonian value of 1/3 is akin to the cases massless Dirac fermions in graphene9 and classical diffusion in disordered metals.7,8 Such sub-Poissonian shot noise arises because the quasiparticles follow conic dispersions centered around k x = ± k D in the band inversion phase.50 When Δ 0, the two conic band structures become far separated in the momentum space and becomes nearly decoupled. Such a decoupling effect gives rise to two copies of “graphene-like” Dirac cones centered at k x = ± k D. The shot noise, thus, approaches to the sub-Poissonian value of 1/3 at the decoupling limit of numerically large and negative Δ.

G in Eq. (8) can be conveniently utilized as a litmus test of the underlying band structure by linearly expanding Eq. (9) around the semi-Dirac phase (Δ = 0) with T T ( Δ = 0 ) T ̃, where T ( Δ = 0 ) is the tunneling probability at the semi-Dirac point and
T ̃ = 2 d Δ tanh ( k x 2 d ) sech 2 ( k x 2 d ) .
(10)
Correspondingly, the deviation of G away from around the semi-Dirac point with universal conductance G SDP / G 0 = 1.9 / d is
G G SDP G 0 2 d Δ ( tanh ( k x 2 d ) sech 2 ( k x 2 d ) ) d k x .
(11)
The range of validity of Eq. (11) [orange dotted line] against the actual deviation, i.e., ( G G SDP ) / G 0 [blue solid line] at d = 30 is shown in the top figure inset of Fig. 4.
Elevated temperature can affect noise measurement by broadening the current fluctuations through thermal excitation, which effectively increase the current noise, SI, and F by approximately 1% at T = 300 K [the inset of Fig. 4 in the bottom figure]. This is accounted by the measured averaged Fano factor F, which deviates from the true F with the inclusion of the Δ-invariant thermal noise61 such that
F ( coth ( U 0 2 k B T ) 2 k B T U 0 ) F ,
(12)
where U0 plays the role of eV here, where V is the applied voltage. This approximation reduces to S T = 4 k B T G = 0.0082 < 1 % if T is highly dependent on ε k [see Eq. (5)] and e V = U 0 k B T. Hence, the miniature contributions of the thermal noise in systems that govern by Eq. (1) remove the need for filtering thermal noise in electronics noise measurements as F F.

The F values for SDMs in Table I are calculated using Eq. (8) with the extracted material coefficients. Finally, we briefly comment on the experimental relevance of the model parameter, Δ, which plays the important role in determining the band topology of 2D SDMs. The Δ term can be externally tuned by a large variety of methods, such as varying the dopant distance,26 doping density,29 applied electric field,26,27 or strain.31,34,50,59 However, cautionary measures should be taken to preserve the space-time inversion symmetry62 to observe the phase transitions. Counterexamples are observed either by straining perpendicular to the valleys31 or lacking the required symmetry, such as in monolayer arsenene or non-Tb-stacked phosphorous.22,33

TABLE I.

Calculated Δ and Fano factors F using material parameters from semi-Dirac materials with α 1 / m * , β v y , k 0 m * v x, and ε 0 m * v x 2 in Eq. (1). Both the Δ / ε 0 and F values are taken from the insulating and band inversion phases and are calculated using d = 30 such that the width is approximately 2–10 nm.

Material Method m * (me) (vx,vy) (106 m / s) k0−1) ε 0 (eV) Δ / ε 0 F
Black P (Refs. 26 and 29 Doping  1.42  ( 0.86 , 0.28–0.56a 1.37  5.065  (0.036, −0.006)  (0.733, 0.154) 
5L black P (Refs. 26 and 27 E-field  1.2  (0.202,0.253)  0.436  0.580  (0.483, −0.362b (1, 0.334) 
Bilayer P (Refs. 31 and 59 Strain  1.25  (0.562, 0.75c 0.607  1.545  (0.044, −0.03)  (0.822, 0.27) 
Material Method m * (me) (vx,vy) (106 m / s) k0−1) ε 0 (eV) Δ / ε 0 F
Black P (Refs. 26 and 29 Doping  1.42  ( 0.86 , 0.28–0.56a 1.37  5.065  (0.036, −0.006)  (0.733, 0.154) 
5L black P (Refs. 26 and 27 E-field  1.2  (0.202,0.253)  0.436  0.580  (0.483, −0.362b (1, 0.334) 
Bilayer P (Refs. 31 and 59 Strain  1.25  (0.562, 0.75c 0.607  1.545  (0.044, −0.03)  (0.822, 0.27) 
a

Values from both the insulating and band inversion phases.26 

b

Linearly extrapolated from the insulating phase.27 

c

Approximated from single-layer Fermi velocities.60 

In conclusion, we study the quantum transport and shot noise signature of a 2D semi-Dirac system along the relativistic dispersion direction. The quantum tunneling spectrum exhibits a peculiar coexistence of massless and massive Dirac quasiparticles with full and oscillating transmission probabilities, respectively. 2D semi-Dirac system, thus, offers a versatile material platform to study the Klein tunneling phenomena.

We further obtain the ballistic tunneling conductance G and the Fano factor F values across different Δ, namely, ( G / G 0 1 / d , F 1 / 3), ( G / G 0 1.9 / d , F 0.179, and ( G / G 0 0 , F 1) for the band inversion, semi-Dirac, and the band insulating phases, respectively. The conductance and shot noise shall provide useful signatures to experimentally probe the phase transitions of a 2D semi-Dirac system.

W.J.C. acknowledges MOE Ph.D. RSS. Y.S.A is supported by the Singapore Ministry of Education Academic Research Fund Tier 2 (No. MOE-T2EP50221-0019). L.K.A is supported by A*STAR AME IRG (No. A2083c0057).

The authors have no conflicts to disclose.

Wei Jie Chan: Formal analysis (equal); Investigation (equal); Writing – original draft (lead); Writing – review & editing (equal). Lay Kee Ang: Funding acquisition (equal); Investigation (equal); Project administration (supporting); Supervision (supporting); Writing – review & editing (equal). Yee Sin Ang: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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