DC and RF Performance of an N-channel Monolayer Black Phosphorus Nanoribbon Transistor

: Two-dimensional black phosphorus is a relatively new discovery. There are numerous studies on black phosphorus two-dimensional transistors that focus on analog and RF performance. However, the RF performance of black phosphorus nanoribbon transistors is yet to be explored. We use a four-band tight binding Hamiltonian in conjunction with a non-equilibrium Green’s function quantum transport simulator to investigate both the DC and RF performance of a monolayer black phosphorus nanoribbon transistor. We found that electron intra-band tunneling is responsible for current flow in the off-state, while in the on-state, the electrons flow over the top of the channel barrier potential. With a V DD of 0.4 volt and a gate length of 5 nm, our black phosphorus nanoribbon transistor has DC performance metrics of 510 µA/µm on-state current, 10 5 on/off current ratio, and 65 mV/dec inverse subthreshold slope. The device’s RF performance characteristics are as follows: cut-off (unity current gain) frequency of 772.84 GHz, maximum oscillation (unity power gain) frequency of 1.15 THz, and open circuit voltage gain of 26.7 dB with transistor operating in the on-state. The RF performance of the device is found to be significantly impacted by the source and drain contact resistances. With source and drain resistances set to zero, the cut-off frequency increases to 995.23 GHz and the unity power gain frequency increases to 4.16 THz. The device shows unconditional stability above 893 GHz and it is conditionally stable below this frequency.


Introduction
Due to their unique electrical, mechanical, and thermal properties, two-dimensional (2D) materials have garnered considerable interest in the scientific community [1][2][3][4]. Even at the short channel limits, the 2D fieldeffect transistors (FETs) show great mobility, high on/off current ratio, and minimal leakage [5][6][7][8][9]. Graphene [5,8,9] and transition metal dichalcogenides (TMDs) [7] are two of the most investigated materials on the extensive list of 2D candidates. However, graphene has a zero band gap, and the poor mobility of TMDs restricts their usefulness [10]. Black phosphorus (BP) is a new and promising addition to the list. Its puckered shape and layer-dependent direct band gap, which varies from~0.35 eV (bulk) to~1.85 eV (monolayer) [11], make it an interesting material to study. This material also possesses notable in-plane anisotropic properties [12,13].
Because of its high mobility, controlled band gap, and anisotropic band structure, BP is an excellent option for use in electronic as well as optoelectronic applications [12]. On BP 2D FETs, a significant amount of research has been done [14][15][16][17][18][19][20][21]. It has been said that the performance of BP transistors can be higher than that of their MoS2 equivalents [16]. Applications of BP as radio-frequency (RF) transistors have also been investigated alongside nanotransistors [19,[22][23][24][25][26][27][28]. Theoretically estimated high RF figures of merit have a unity power gain frequency of fmax = 950 GHz [27]. Experiments have shown that a device with a channel length of 300 nm can achieve a fmax of 20 GHz [22].
Despite substantial research on 2D BP FETs, one-dimensional (1D) black phosphorus nanoribbons (NRs) and nanotubes (NTs) have received less attention [29][30][31][32][33][34]. Both experiments and theoretical first-principles simulations have been used to investigate the structural [29] and thermal [35,36] stability, band gap fluctuation [37], and characterization [38] of 1D BP NRs. Feng et al. [30] experimentally showed complementary BP NR top gate FETs. According to their findings, the BP NR 1D FETs outperform the BP 2D FETs by an order of magnitude in terms of on-state current, five times in terms of subthreshold slope, and four times in terms of peak transconductance. However, we realize that the RF performance of BP 1D FETs is yet to be explored.
This study uses an in-house developed quantum transport code and a four-band tight binding Hamiltonian to theoretically investigate the DC and RF performance of a monolayer 5 nm gate length BP NR transistor. The transistor exhibits a high on/off current ratio with an inverse subthreshold slope of 65 mV/dec when operated at a DC bias. In the off-state, electron intra-band tunneling regulates the current flow, and in the on-state, thermionic emission over the channel potential barrier does the same. An equivalent circuit is utilized in order to assess the radio-frequency performance. The parameters of the circuit are established by the use of numerical simulation. The RF performance parameters calculated are as follows: a unity current gain frequency of 772.84 GHz, a unity power gain frequency of 1.15 THz, and an open circuit voltage gain of 26.7 dB.

Device Structure and Simulation Approach
With gate length LG = 5 nm, effective oxide thickness EOT = 0.45 nm, and VDD = 0.4 V, which are near the specifications of 2027 low-power logic [39], we simulate a double-gate monolayer BP NR 1D FET. tox = 2.15 nm and  ox = 19 represent the physical oxide thickness and dielectric constant, respectively, that we employ in our simulation. Figure 1 depicts the cross-section of the device. The lengths of the source and drain extensions, LS and LD, are both set to 10 nm. Both the source and the drain are doped with a donor concentration of ND = 1.25 × 10 13 cm -2 , making the channel intrinsic. In order to align the source conduction band with the source Fermi level, we selected this doping density. A four-band tight binding Hamiltonian is used to model BP [11] In this equation, i and j refer to the lattice sites,  ij t is the in-layer hopping parameter between sites i and , j and  i c and j c are the operators for creating and destroying electrons at sites i and , j respectively. Note that the on-site energies have been set to zero. The model has been validated against the ab initio self-consistent 0 GW results [11]. The red rectangle in Figure 1(a) represents the four atoms that make up the BP monolayer's unit cell. Our recursive Green's function algorithm [34,40,41] uses nanoribbon unit cell depicted by the blue vertical lines in Figure 1(a). The layer (or unit cell) Hamiltonian L h and the layer-to-layer (or unit cell-to-unit cell) coupling matrices , 1 have sizes that are 4 times the number of unit cells along the y (zigzag) direction. L h and , 1 can be written as [42] and † , 1  uL  uLF and  uLB are 4 × 4 matrices, and are provided in Ref. [42]'s Eqs. (2)(3)(4)(5). There in Table 1, the hopping parameters are also provided.
We have the required three matrices, , We can now apply the recursive Green's function approach to determine the electron density from the diagonal portions of the source and drain spectral functions [34,40,41], In this equation, ,   f and  S stand for grid volume, Fermi distribution function, and Fermi energy levels at the source and drain, respectively. The spectral functions are given by † , where the broadening function is supplied by [43,44] is used to get the boundary self-energy 1,1 1,0 0,0 0,1 .   h g h The required blocks of full Green's function are determined by sequentially solving the following equations.
Poisson's solver is used to calculate the layer's potential energy, .
Potential profile is obtained by solving a three-dimensional Poisson's equation in Cartesian coordinates The potential profile and location dependent dielectric constant are represented by ( )   V U  [47]. The contact resistance used in this work is thus not too far from becoming achievable. The first step in the self-consistent loop is to make an educated guess about the potential profile. After that, we compute the charge density, Eq. 4, as well as the current [40,41]    The intrinsic component of the device potential is updated using the computed current in the following way: These newly calculated voltages serve as the boundary condition when Poisson's equation is solved using the data from the most recent computation of charge density.
The loop remains active until convergence is reached. Anderson mixing [48] is used to hasten the convergence.
After the loop has reached convergence, the results are saved, and then we move on to calculating the next bias value. The simulation software is developed internally using open-source Julia programming language [49]. Figure 2 depicts the E-k dispersions of monolayer BP nanoribbon with a width of W = 2.3 nm that we used in our simulation. The energy reference is located in the band gap's midpoint. Seven 4-atom unit cells (Figure 1), or 28 atoms, make up the nanoribbon unit cell. The transport route is in the armchair direction (x in Fig. 1), whereas the quantization direction (nanoribbon width) is along the zigzag direction (y in Figure 1). The nanoribbon has a 1.918 eV direct band gap. The top valence subbands merge as a result of a significantly heavier hole in the quantization (zigzag) direction [18], but the bottom conduction subbands are closely spaced.    [17]. It is important to note that these values were determined by conducting measurements at a low temperature of 20 0 K for a device that had a long channel and a gate bias of −4 volts. In order to acquire a better understanding of electron transport, we depict the energy profile of current, denoted by JE, overlaid on the conduction band profile in Figure 4(a) at a gate bias of 0.31 volt. The Fermi level at the source is used as the zero energy reference. The dashed line indicates that the top of the conduction band is located at 0.063 eV. This value represents the barrier energy for electrons that are injected from the source contact. Any injected electrons with energies lower than this barrier energy will be able to "tunnel" through it and reach the drain terminal. This tunneling current can be determined by integrating JE from −0.4 to 0.063 eV (barrier top), which accounts for 49.5% of the total current when the gate bias is 0.31 volt. In the remaining 50.5% of the total current, known as the thermal current, electrons pass over the barrier top. The current is broken down into its tunneling component and its thermal component across the entire voltage range in Figure  4(b). Clearly, electron tunneling dominates in low bias, whereas thermionic emission predominates at higher biases. Current in the off-state is 83% tunneling, while current in the on-state is 74% thermionic. After having determined the DC figures of merit (on-state current, inverse subthreshold slope, and on-off current ratio) and having gained an understanding of the mechanism underlying electron transport, we moved on to evaluating the device' s RF performance parameters. In order to accomplish this, we will be looking at the small signal equivalent circuit that is depicted in Figure 5. The following is how we determine the transconductance, denoted by m g , and the output conductance, denoted by 0 . g Transconductance m g is a figure of merit that determines the amplification delivered by a transistor. It increases with gate bias, and its value in on-state is 6.7 mS/ m.  In linear region 0 g is large due to uniform channel, and it decreases with drain bias due to channel length modulation. In on-state 0 0.31 mS/ m,   g that is, the output

Simulation Results and Discussions
where ch Q refers to the channel charge and  s refers to the surface potential. Figure 7 shows the computed capacitances, gs C and . gd C These capacitances play crucial roles for analyzing RF performance, such as the cutoff frequency, T f and the maximum oscillation frequency, max . f To determine the RF performance metrics, we first derive the admittance matrix of the intrinsic part of the equivalent circuit, Figure 5, as follows That is, the admittance matrix of the full equivalent circuit is From the admittance matrix, we can calculate the short circuit current gain, 21 The voltage gain at DC can be written as [52]. After substitution of m g and 0 g in onstate we get (0)  system will not show any oscillation above 893 GHz for any passive load at the input or output ports. Below 893 GHz operation, the network shows stable behavior for certain combinations of passive loads.

Conclusion
To summarize our findings, we used a four-band tight binding Hamiltonian and a non-equilibrium Green's function quantum transport code to investigate the transport mechanism and DC performance of a monolayer black phosphorus nanoribbon transistor. When in the off-state, electron transport is controlled by the intra-band tunneling, and when in the on-state, it is controlled by thermionic emission over the channel potential barrier. The device exhibits a current density of 510 µA/µm in its on state and a slope of 65 mV/dec in its subthreshold region, with an on/off current ratio of 10 5 . Next, we use the admittance matrix of a two-port network with environmental influences to assess the device's RF performance. The gate-source terminals serve as the input port, while the drain-source terminals serve as the output port. With a maximum oscillation frequency of 1.15 THz, a cut-off frequency of 772.84 GHz, and an open-circuit voltage gain of 26.7 dB, this device boasts excellent RF performance metrics. The device is completely stable above 893 GHz and partially stable below that frequency.