## Introduction

Two-dimensional materials currently receive immense attention for application in next-generation electronic devices, particularly as channel materials in field effect transistors (FETs)1,2,3,4,5. Despite an ultrahigh carrier mobility of ~105 cm2V−1s−1, graphene cannot be used as channel material due to its zero bandgap nature. Monolayer transition metal dichalcogenides possess (direct) bandgaps of 1–2 eV6,7,8,9 and provide high Ion/Ioff ratios of ~105–10810,11, while low carrier mobilities of ~102 cm2V−1s−112 are not appropriate for high performance applications. Stacking of transition metal dichalcogenides in van der Waals heterostructures can improve the carrier mobility13. In addition, such heterostructures provide high Ion/Ioff ratios of ~103 for WSe2–MoS2 and MoTe2–SnSe214, ~104 for MoS2–CuInP2S615, ~106 for MoTe2–SnS216, and ~107 for WS2-graphene17. However, understanding of the carrier transport and realization of large-area growth are challenges that limit applications18,19. FETs based on wide bandgap oxides such as β-Ga2O3 (bandgap of 4.8 eV20; Ion/Ioff ratio of ~107–109 and electron mobility of 32–180 cm2V−1s−121) and In2O3 (bandgap of 3.1 eV22; Ion/Ioff ratio of ~106 and electron mobility of 127 cm2V−1s−123) provide excellent Ion/Ioff ratios but the carrier mobilities are low. Hole mobilities in excess of 104 cm2V−1s−1 are predicted for MXenes, a family of two-dimensional transition metal carbides and nitrides24,25. Though they can be employed in FET sensors26,27, application in logic circuits is impeded by degradation in the presence of water and air28. Remarkably, with bandgaps of 0.3–2.0 eV29, monolayer and few-layer black phosphorene achieve competitive hole mobilities of ~103 cm2V−1s−1 and Ion/Ioff ratios of ~10530,31 but also degrade in the presence of water and air32,33.

The common metal contacts used to inject charge carriers into the two-dimensional channel material cause Ohmic resistance and/or Schottky barriers that dramatically impact the efficiency of the injection. As it is a great challenge to reduce the contact resistance34, junction-free FETs (consisting of a single material) are an interesting alternative35. While the realization of metallic electrode regions turns out be difficult, it can be achieved in the case of black and blue phosphorene nanoribbon junction-free FETs by O functionalization36. However, this requires restriction of the O functionalization to the nanoribbon edges to avoid adverse effects on the material properties37,38 and only rather low Ion/Ioff ratios of ~102 (with transconductances of ~104) are reported36. In the present work, we demonstrate that these limitations can be overcome by blue phosphorene junction-free FETs. Monolayer blue phosphorene has a bandgap of 1.5–2.0 eV32,33 and bilayer blue phosphorene is metallic39. This unique combination of material properties paves the way to junction-free FETs with monolayer blue phosphorene as the semiconducting channel material and bilayer blue phosphorene as the electrode material. We show that such a design gives rise to favorable FET characteristics. The Ion/Ioff ratio reaches high values of ~104 due to the junction-free design with low contact resistance. The transconductance is as high as 603 (811) μS/μm for transport along the armchair (zigzag) direction.

## Results

### Design

Figure 1 shows the optimized structures of monolayer and bilayer blue phosphorene as well as the corresponding band structures and projected densities of states. Monolayer blue phosphorene turns out to be an indirect bandgap semiconductor in agreement with the literature32 and bilayer blue phosphorene turns out to be a metal with an interlayer distance of 1.84 Å in agreement with the literature39. The metallic nature of bilayer blue phosphorene is a result of the short interlayer distance in the energetically favorable A1B−1 stacking. The metallic states are almost equally due to the P 3s, 3px,y (the px and py orbitals are degenerate by symmetry), and 3pz orbitals. The hole mobility of blue phosphorene calculated by deformation potential theory (2.2 × 103 cm2V−1s−1) clearly exceeds the value reported for black phosphorene (~102 cm2V−1s−1, both experimental and theoretical40,41) and compares well with the value reported for few-layer black phosphorene (~103 cm2V−1s−130).

Figure 2a, e depict the proposed bilayer-monolayer-bilayer FET design with monolayer blue phosphorene serving as the channel material and bilayer blue phosphorene serving as the electrode material. The key advantages are: First, there is no intrinsic strain due to lattice mismatch between the channel and electrode materials, since the device is based on blue phosphorene only. Second, the continuous semiconductor-metal contacts reduce the resistance. To quantitatively evaluate the potential of the device, we apply a field-effect gate model42, considering transport in both the armchair and zigzag directions.

High quality monolayer blue phosphorene can be grown on Au(111) subtrate by molecular beam epitaxy33, while there is so far no report of synthesis of bilayer blue phosphorene though the material is predicted to be dynamically stable39. It is likely that bilayer blue phosphorene can be fabricated in the energetically favorable A1B−1 stacking by layer transfer techniques, because the large interlayer interaction prevents spontaneous changes in the structure39. Specifically, the wetting transfer method for preparing twisted bilayer graphene43 can be transfered to blue phosphorene due to the fact that the two materials share their hexagonal structure. The method adopts polydimethylsiloxane as substrate for the transfer, which previously was used for transferring monolayer and few-layer black phosphorene44,45,46. Once bilayer blue phosphorene is obtained, the proposed bilayer-monolayer-bilayer design can be achieved by local etching. In particular, the atomic layer etching technique enables layer-by-layer etching through alternation between chemical adsorption to modify the exposed layer and physical desorption to remove exactly this modified layer47,48. As an alternative, the methodology of layer-controlled thinning of black phosphorous by an Ar+ ion beam with narrow energy distribution49 can be transferred to blue phosphorene due to the flexible tunability of the Ar+ ion energy50,51. Chemical vapor deposition opens up another route to the bilayer-monolayer-bilayer design, as it was already used to fabricate monolayer-bilayer-monolayer lateral heterojunctions of transition metal dichalcogenides52,53 and to synthesize few-layer black phosphorene from black phosphorous as precursor54. The method is transferable to blue phosphorene due to the common hexagonal structure with the transition metal dichalcogenides and the fact that the same precursor can be used as for black phophorene.

### Properties

To address the transport, we plot the transmission spectrum obtained for the armchair and zigzag devices in Fig. 2b, f. We find a wide transmission gap that represents the semiconducting channel. As the data show hardly any influence of the channel length, studying a short channel is sufficient to capture the transport properties of the device in the ballistic regime. It turns out that a 1 × 6 × 1 k-mesh is sufficient to obtain converged results. The first eigenchannel wave function of the left-to-right transmission is plotted in Fig. 3. Due to the symmetry of the FET design, the result for right-to-left transmission (not shown here) is similar with opposite phase55. Figure 3 demonstrates delocalization of the first transmission eigenchannel wave function into the channel region for both the armchair and zigzag directions, and even into the other electrode for the zigzag direction, pointing to excellent coupling at the semiconductor-metal contacts.

Figure 2c, g depicts for the armchair and zigzag devices the channel current as a function of the gate charge at channel biases of 0.2 and 0.4 V. The gate bias is given by the modification of the Hartree potential due to the presence of the gate charge. We obtain Ion as the channel current at a gate charge of 13.6 e (giving rise to a conservative estimate of the Ion/Ioff ratio) and Ioff as the minimum of the channel current. At a channel bias of 0.2 V we obtain for transport along the armchair (zigzag) direction Ion = 9.1 × 100 (1.5 × 101) μA and Ioff = 1.3 × 10−3 (5.6 × 10−4) μA, resulting in an Ion/Ioff ratio of 7.0 × 103 (2.6 × 104), while at a channel bias of 0.4 V we obtain Ion = 1.2 × 101 (2.3 × 101) μA and Ioff = 3.3 × 10−3 (1.5 × 10−3) μA, resulting in a lower Ion/Ioff ratio of 4.2 × 103 (1.5 × 104). The obtained Ion/Ioff ratios are significantly higher than those reported for black and blue phosphorene nanoribbon junction-free FETs (2.3 × 102 and 3.5 × 102, respectively)36 and compare well with the experimental values reported for black phosphorene FETs (102–104)56. The threshold voltage estimated by the constant current method for a channel bias of 0.2 V and a channel current of 0.1 μA times the ratio of the gate width and length is −0.4 (1.9) V for the armchair (zigzag) device.

Figure 2d, h shows for the armchair and zigzag devices that the channel current generally increases with both the channel bias and gate bias. The semiconductor-metal contact of the zigzag device turns out to be Ohmic with efficient charge injection while that of the armchair device is of Schottky type. Using the transfer length method, we obtain from the interception of the linear fitting in Fig. 4 with the total resistance axis for the zigzag device a contact resistance of 3 × 10−4 MΩ. Multiplied with the width of the device this results in 0.3 Ωμm, which compares very well with experimental reports for graphene (23 Ωμm)57, Bi-MoS2 (123 Ωμm)58, and MoS2 (200–300 Ωμm)34 devices. The behavior for positive and negative channel bias is very similar for the armchair device, but not precisely the same, since the left and right electrodes are not completely symmetric, see Fig. 2a, e. To quantify the dependence of the channel current on the gate voltage Vgate, we calculate the transconductance normalized by the width W of the device (width of the supercell) as (dI/dVgate)/W59. We approximate dI/dVgate = [I(Vgate + ΔVgate) − I(Vgate − ΔVgate)]/(2ΔVgate), where ΔVgate = 0.24 (0.22) V for the armchair (zigzag) device. As expected from the IV characteristics in Fig. 2d and h, the transconductances in Fig. 5 are higher for the zigzag than the armchair device. At a channel bias of −1.0 V we obtain a transconductance of 603 μS/μm for the armchair device and at a channel bias of −0.4 V we obtain a transconductance of 811 μS/μm for the zigzag device. These values significantly exceed the experimental values reported for black phosphorene (282 μS/μm)60 and monolayer MoS2 (185 μS/μm)61.

## Discussion

We propose a junction-free FET based on blue phosphorene, which is semiconducting as a monolayer and metallic as a bilayer. The IV characteristics obtained for the bilayer-monolayer-bilayer design (does not require doping to create source and drain) demonstrate efficient electron transport between the channel material and electrodes while a decent Ion/Ioff ratio is provided. The obtained transconductances outperform competing FETs based on two-dimensional materials. It turns out that a channel length of ~30 Å is sufficient to simulate the transport properties of the device. While experimental realization of a device with such a short channel is challenging, an increased channel length will only improve the Ion/Ioff ratio of the FET62. The transport properties turn out to be slightly better for the zigzag device than the armchair device due to better monolayer-bilayer coupling. From a cost perspective the proposed FET is favorable over junction-free devices based on PdS2 (requires expensive Pd)35 and phosphorene nanoribbons (requires O functionalization limited to the nanoribbon edges)36. The performance can be further improved by using a gate oxide instead of the simulated vacuum and by optimizing the gate length and underlap (space between the end of the gate and the electrode along the transport direction)63. The proposed FET design can be realized by established fabrication methods.

## Methods

### First-principles calculations

We perform first-principles calculations using density functional theory within the generalized gradient approximation of Perdew-Burke-Ernzerhof for the exchange-correlation functional, as implemented in the Quantum-ESPRESSO code64. The total energy convergence threshold is set to 10−8 Ry and all structures are optimized until the Hellmann–Feynman forces stay below 10−5 Ry/bohr. A 80 Ry cut-off is used in the plane wave expansion and a 640 Ry cut-off for the augmentation charge. The first Brillouin zones of monolayer and bilayer blue phosphorene are sampled on Monkhorst-Pack 12 × 12 × 1 k-meshes. The densities of states are calculated by the tetrahedron method with Monkhorst-Pack 24 × 24 × 1 k-meshes. Periodic boundary conditions are used with 20 Å vacuum slabs to create two-dimensional models.

### Transport calculations

Transport calculations are performed adopting the non-equilibrium Green’s function method in combination with density functional theory65, which provides excellent agreement with experimental results63,66. A 300 Ry cut-off is used for the double-zeta polarized basis set. Monkhorst-Pack 1 × 6 × 100 k-meshes are employed for the semi-infinite electrode calculations and Monkhorst-Pack 1 × 6 × 1 k-meshes for the transport calculations. The transmission spectrum of electrons with energy E subject to a channel bias V between the electrodes (bilayer blue phosphorene) through the scattering region (monolayer blue phosphorene) is calculated as67

$$T(E,V)={{{\rm{Tr}}}}[-G(E,V)[{\Sigma }_{{{{\rm{L}}}}}(E)-{\Sigma }_{{{{\rm{L}}}}}^{{\dagger} }(E,V)]{G}^{{\dagger} }(E,V)[{\Sigma }_{{{{\rm{R}}}}}(E)-{\Sigma }_{{{{\rm{R}}}}}^{{\dagger} }(E,V)]],$$
(1)

where G(E, V) is the retarded Green’s function of the scattering region and ΣL,R(E, V) are the self-energies of the left (L) and right (R) electrodes. The current is given by67

$$I(V)=\frac{2e}{h}\int\nolimits_{-\infty }^{\infty }[f(E-({E}_{{{{\rm{F}}}}}-eV/2))-f(E-({E}_{{{{\rm{F}}}}}+eV/2))]T(E,V)\,{\rm{d}}E,$$
(2)

where f is the Fermi–Dirac distribution function, EF is the Fermi energy, and e is the elementary charge. We consider a 1 × 3 × 5 (1 × 2 × 9) supercell of monolayer blue phosphorene to model the channel material and a 1 × 3 × 2 (1 × 2 × 3) supercell of bilayer blue phosphorene to model the electrodes for transport along the armchair (zigzag) direction. The in-plane supercell dimensions given as width × length are 10.09 × 29.13 (11.65 × 30.27) Å2 for the channel material and 10.09 × 11.65 (11.65 × 10.09) Å2 for the electrodes. A uniformly charged gate is added at a distance of 20 Å in the out-of-plane direction to realize a FET geometry. The transmission eigenchannel wave functions are calculated by means of the inelastica package68.