Research Article Vol. 38, No. 4 / April 2021 / Journal of the Optical Society of America B 1367

Black phosphorus photoconductive terahertzantenna: 3D modeling and experimental referencecomparisonJose Santos Batista,1,* Hugh O. H. Churchill,2 AND Magda El-Shenawee1

1Department of Electrical Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA2Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA*Corresponding author: jisantos@uark.edu

Received 20 January 2021; revised 2 March 2021; accepted 9 March 2021; posted 9 March 2021 (Doc. ID 419996);published 31 March 2021

This paper presents a 3D model of a terahertz photoconductive antenna (PCA) using black phosphorus, an emerg-ing 2D anisotropic material, as the semiconductor layer. This work aims at understanding the potential of blackphosphorus (BP) to advance the signal generation and bandwidth of conventional terahertz (THz) PCAs. TheCOMSOL Multiphysics package, based on the finite element method, is utilized to model the 3D BP PCA emitterusing four modules: the frequency domain RF module to solve Maxwell’s equations, the semiconductor moduleto calculate the photocurrent, the heat transfer in solids module to calculate the temperature variations, and thetransient RF module to calculate the THz radiated electric field pulse. The proposed 3D model is computationallyintensive where the PCA device includes thin layers of thicknesses ranging from nano- to microscale. The symmetryof the configuration was exploited by applying the perfect electric and magnetic boundary conditions to reduce thecomputational domain to only one quarter of the device in the RF module. The results showed that the temperaturevariation due to the conduction of current induced by the bias voltage increased by only 0.162 K. In addition, theelectromagnetic power dissipation in the semiconductor due to the femtosecond laser source showed an increasein temperature by 0.441 K. The results show that the temperature variations caused the peak of the photocurrentto increase by ∼3.4% and ∼10%, respectively, under a maximum bias voltage of 1 V and average laser power of1 mW. While simulating the active area of the antenna provided accurate results for the optical and semiconductorresponses, simulating the thermal effect on the photocurrent requires a larger computational domain to avoid falserise in temperature. Finally, the simulated THz signal generation electric field pulse exhibits a trend in increasingthe bandwidth of the proposed BP PCA compared with the measured pulse of a reference commercial LT-GaAsPCA. Enhancing signal generation and bandwidth will improve THz imaging and spectroscopy for biomedical andmaterial characterization applications. ©2021Optical Society of America

https://doi.org/10.1364/JOSAB.419996

1. INTRODUCTION

Terahertz photoconductive antennas (PCAs) have been in theliterature for almost a decade [1,2]. The use of two-dimensional(2D) materials as the semiconductor substrate for PCAs hasgained attention in recent years [3,4]. These new materials haveexhibited tunable optoelectronic properties, and therefore theystand out as potential candidates to exceed the performanceof conventional PCAs. The 2D materials have exhibited highoptical absorption and responsivity, sub-picosecond carrier life-time, and high carrier mobility, which supersede semiconductormaterials used in conventional THz PCAs [3,4]. Black phos-phorus (BP) has received substantial research interest for THzdevices [5–7]. Furthermore, numerous works were reported inthe literature demonstrating PCA with plasmonic structures toenhance the LT-GaAs THz emitted power. The LT-GaAs is the

most commonly used semiconductor in THz PCA devices (e.g.,[8–11]).

Black phosphorus comes in bulk or is processed in thin flakesof nanoscale thickness. The material has an anisotropic crystalstructure consisting of two axes with armchair and zigzag atomconfigurations [12]. This anisotropic arrangement provideslight polarization-dependent optical properties [13], whichproduces a higher absorption of light in the armchair direc-tion compared to the zigzag direction. This is due to the largerimaginary part of the complex relative permittivity of BP inthe armchair compared with the zigzag direction (see Table 2)[13]. In addition, the effective mass of BP in the armchair islower than that of the zigzag direction [14]. This has been doc-umented, including in the reflectance measurements in ourrecent work [7], where the absorption in the armchair direction

0740-3224/21/041367-13 Journal © 2021Optical Society of America

1368 Vol. 38, No. 4 / April 2021 / Journal of the Optical Society of America B Research Article

was observed to be 20% higher than in the zigzag directionfor 780 nm laser excitation. Besides black phosphorus’s highoptical absorption, it exhibits tunable electrical properties suchas a thickness-dependent direct bandgap ranging from 2 eVfor a single layer compared to 0.3 eV for its bulk configuration[15]. This property permits its application in optoelectronicdevices in the near infrared range, where its carrier lifetime hasbeen reported to be 0.36 ps for laser excitation around 780 nm[16]. Another outstanding characteristic of black phosphorusis its high carrier mobility, which contributes to the drift of thephotocarriers under the application of bias voltage. This mate-rial has reported a carrier mobility as high as 1500 cm2/V · sfor electrons when doped with aluminum adatoms [17], and5000 cm2/V · s for holes at room temperature [18]. The highsaturation velocity, high mobility, and subpicosecond carrierlifetime make BP a potential material for emitter devices at theTHz frequency band.

BP was mostly used in PCAs as detectors of THz signals[5,6], where its short carrier lifetime and high mobility havebeen crucial for THz detection. However, to the best of ourknowledge, the first fabricated BP-based PCA emitter wasreported in [7], where the photo-response characterization ofthe device, as well as a 2D geometry computational model, werepresented. This 2D model demonstrated the performance ofthe device at two laser wavelengths (780 and 1560 nm), wherethe THz signal emitted by the BP device was ∼78.8% higherthan that of the conventional LT-GaAs PCA at a bias voltageof 1 V and∼55.6% when the LT-GaAs device bias voltage wasincreased to 30 V at 780 nm. In addition, the signal increase was∼86.5% at a bias voltage of 1 V and∼71.7% when the LT-GaAsdevice bias voltage was increased to 30 V at 1560 nm.

Here, we investigate a more comprehensive 3D model ofthe BP PCA device at 780 nm in an effort to understand thepotential enhancement of signal generation and bandwidthversus the conventional LT-GaAs device. Furthermore, weinvestigate the thermal effect of the temperature variation dueto the conduction of current induced by the bias voltage andthe electromagnetic power dissipation of the laser in the BPmaterial and its effect on the photocurrent density generated inthe device. This work modeled a realistic PCA emitter by elimi-nating several spatial assumptions made in the 2D model suchas the distribution of the laser in the photoconductor as wellas inclusion of the actual geometry of the metallic electrodes.Furthermore, a 3D structure presents a more realistic modelof the antenna to be fabricated. One of the defining factors ofthe performance of a PCA emitter lies in its capacity of trans-forming the optical pump (femtosecond laser) into a terahertzsignal. Here, we cover the study of the temperature variationof the device under working conditions and its impact on thegenerated photocurrent.

Furthermore, increasing the pump power of the laser andincreasing the bias voltage of the device could produce detri-mental effects on the emitter such as the generation of ohmiclosses in the BP layer as well as the conduction of higher currentsin the gap of the device translated to Joule heating [19]. Thesefactors contribute to the temperature increase of the device,which could affect the performance of the emitter with anadditional thermal current to the existing drift–diffusion com-ponents. In fact, the increase of the laser power and bias voltage

of the emitter could result in thermal runaway of the device witha nonlinear increase of the current leading to a breakdown [20].The capacity of the device to handle these conditions highlydepends on the heat-transfer properties of the materials as wellas the geometry of the device. For instance, the breakdownvoltage for BP was documented to be 4 V for a 50 nm layer ina 1.5 µm gap device configuration under a continuous-wavelaser [21]. This factor presents the significance of modeling thetemperature variation of the device under working conditions.

The main contribution of this work is the comprehensive3D modeling of the BP PCA THz emitter to investigate itstemperature variation, signal generation, and bandwidth ofthe device. We consider the anisotropic optical response to thefemtosecond laser excitation, the effect of its high mobility andshort carrier lifetime in the photo-response, and the impact ofthe temperature variation on the photocurrent. The computersimulations used the multiphysics finite element method com-mercial package COMSOL on a variety of platforms. However,the intensive computational work in the largest case of the simu-lations was executed on the XSEDE supercomputer, where morethan 0.313 billion unknowns were solved, requiring ∼3.2 TBRAM and∼25 h CPU time, only for the optical response of thedevice. The solutions of the electrical response, THz signal gen-eration, and temperature variations in the device were executedon local platforms (as summarized later in Table 5).

This paper is organized as follows. Section 2 presents themethodology of the work. Section 3 presents the results ofMaxwell’s solution, photocurrent, temperature variation, THzsignal generation, and time domain and spectrum measurementcomparison against a reference commercial LT-GaAs PCA.Conclusions and future work are discussed in Section 4.

2. METHODOLOGY

The proposed photoconductive THz emitter consists of abowtie-shaped antenna electrode with a BP layer placed ontop of the gap between the electrodes as shown in Fig. 1. Thefemtosecond laser illuminates the gap where the BP nano-layeris placed such that the photocarrier absorption is maximized. Alayer of the hexagonal boron nitride (hBN) covers the BP mate-rial as protection from environmental conditions. The highestquality of BP devices in terms of mobility and saturation velocityare those encapsulated in hBN because of its atomically flat,crystalline, and low-disorder surface [5,7]. This is in contrastto the atomic layer deposition metal oxides, which are typicallyamorphous and less pure with a higher density of charge trapsthat degrade transport properties.

A zoomed-in top view of the antenna geometry is shown inFig. 1(b). This sketch illustrates the hBN/BP layers covering thegap of the antenna and part of the electrodes. The crystal struc-ture of BP is shown in Fig. 1(c) with the armchair and zigzagdirections. The BP layer is positioned at the gap of the antennawith the armchair axis in the x direction, which coincides withthe laser polarization to maximize the optical power absorption.A cross-section of the geometry is presented in Fig. 1(d), whichwas obtained at the red dashed line drawn across the gap of theantenna in Fig. 1(b). The layer of hBN is shown in dark graywith a thickness of 150 nm, and the black layer represents the BPmaterial with a thickness of 40 nm. The silicon dioxide (SiO2)

Research Article Vol. 38, No. 4 / April 2021 / Journal of the Optical Society of America B 1369

Fig. 1. Proposed 3D BP PCA emitter. (a) Antenna geometry withthe hBN/BP layers in black color. (b) Top view of the antenna geom-etry zoomed-in at the gap of the antenna where the hBN/BP coversthe gap and part of the electrodes. The hBN material is placed on topof the BP for protection from oxidation. (c) BP crystal structure witharmchair and zigzag axis directions. (d) Cross-section of the proposedantenna taken along the red dashed line in (b).

layer has a thickness of 300 nm shown in light green, and thesilicon (Si) substrate is simulated as half space by positioning theabsorbing scattering boundary conditions right below a 0.5 µmthickness of the Si. The air space shown in Fig. 1(d) is of height1.5 µm. The size of the antenna gap is 4 µm in the x directionand 5 µm in the y direction in all cases considered in this workunless otherwise mentioned.

A. Optical Response

The optical response represents the modeling of the incidentfemtosecond laser excitation and its interaction with the multi-layer device shown in Fig. 1. The wave equation is solved in thefrequency domain where the electric and magnetic fields arecalculated at each point in the computational domain of Fig. 1.The femtosecond laser excitation in the x direction (armchair)was modeled as a plane wave modulated in space by a Gaussianprofile in the x and y directions given as [10]

EE inc = x̂

√4ηo Pave

πD2x

exp

(4 ln(0.5)

(x − x0)2

Dx2

)

× exp

(4 ln(0.5)

(y − y0)2

Dy2

), (1)

where ηo is the free space impedance, Pave is the average powerof the laser pulse, x0 and y0 are the center location of the laserpump, and Dx and Dy define the half power beam width in thex and y directions, respectively. The maximum amplitude ofEq. (1) is derived from the concept of the average peak power of atrain of laser pulses, with the goal of obtaining results resemblingthe average power measured by a regular power meter [22]. Theparameters of the laser are listed in Table 1. The electromagneticsolution of the optical model accounted for the anisotropicbehavior of the BP material upon considering the anisotropicproperties of the complex relative permittivity and the electricalconductivity as listed in Table 2.

The frequency-domain solution of the wave equation pro-vides the phasor components of the electric EE (Er ) and magneticEH(Er ) fields at each point (Er ) in the computational domain.

Table 1. Parameters of the Modeled FemtosecondLaser Source

Parameter Symbol Value

Laser wavelength λ 780 nmAverage power PAve 1 mWPulse x axis center location x0 0µmPulse y axis center location y0 0µmPulse center location in time t0 0.6 psHPBW–x direction Dx 2µmHPBW–y direction Dy 2µmPulse width–time Dt 100 fs

Table 2. Material Properties for Optical Response

Symbol σ ε̂ (780 nm)Description Conductivity Relative PermittivityUnits S/m 1

Au 2.892 ∗ 107 [23] −25.06−1.60i [23]Cr 7.752 ∗ 106 [24] −2.21−21.07i [25]SiO2 5 ∗ 10−14 [26] 2.38 [27]Si 4.348 ∗ 10−4 [28] 13.623−0.044i [29]hBN 1 ∗ 10−6 [30] 4.84 [31]

x 250 [32] 16.0565−1.7283i [13]BP y 92.6 [32] 14.763-0.096i [13]

z 43.5 [32] 8.3 [33]

Their time-domain representations are then calculated asEE (Er , t) and EH(Er , t). With these field components, we apply thePoynting theorem to obtain the time-dependent power densityEP (Er , t) [34] and select the maximum values over time Pmax(Er ),

which is the driving factor of the carrier generation rate G(Er , t)in Eq. (2), which in turn is the source of the electrical response ofthe device.

B. Electrical Response

The semiconductor module solves the well-known coupledPoisson’s and drift-diffusion equations to obtain the time andspatial electric potential, carrier concentration, and thermalvariation [35]. From the solution of these unknowns, we canobtain the time-dependent photocurrent density. Here, thecarrier generation formulation accounts for the femtosecondlaser temporal Gaussian envelope as [10]:

G(Er , t)=4π κ P max(Er )

hcexp

(4 ln(0.5)

(t − t0)2

Dt2

), (2)

where Pmax(Er ) is the maximum power density obtained fromthe Maxwell’s equations solution in the optical response, and κis the extinction coefficient of the BP material. Here, we assumethat the material is isotropic by using the extinction coefficient κof the BP in the x direction where the laser excitation was polar-ized in the armchair direction (x direction); further, the value ofκ in the zigzag direction (y direction) is much smaller than thatin the armchair direction by almost a factor of∼17.2 [13]. Thesymbol h is the Plank’s constant, and c is the speed of light infree space. The femtosecond pulse width Dt was applied basedon the laser parameters listed in Table 1 as well as the centerlocation in time of the laser pulse t0. The recombination of the

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carriers was accounted for based on the Shockley–Read–Hallmodel [36].

The current density provided by the semiconductor moduleis defined by the addition of the drift and diffusion componentsof the current at room temperature. However, when the tem-perature variation is considered in the model, a third thermalcomponent is included in the calculation of the current densityas [35]

EJn,p(Er , t)= EJ Driftn·p (Er , t)+ EJ Diffusion

n·p (Er , t)+ EJ Thermaln·p (Er , t).

(3)The thermal component arises from the spatial variation of

the temperature in the BP semiconductor substrate, as will bediscussed in the Section 3. Equation (4) shows the calculationsof the three current density components in the same order listedin Eq. (3) as

EJn,p(Er , t)=(µn,p∇E c ,v

)m(Er , t)±µn,p kB T F

(m(Er , t)

Nc ,v

)

×∇m(Er , t)±(

q Dth(n,p)

T∇T

)m(Er , t).

(4)

In Eq. (4), m(Er , t) stands for the carrier concentration ofelectrons n(Er , t) and holes p(Er , t) for the calculation of EJn(Er , t)and EJ p(Er , t), respectively. E c ,v represents the conduction andvalance band, kB is the Boltzmann constant, T represents thelattice temperature of the semiconductor, and F is the Fermifunction dependent on the carrier concentration and the densityof states for the conduction and valance band Nc ,v . The param-eter Dth(n,p) represents the thermal diffusion coefficient andµn,p represents the mobility for the electrons and holes [35].

In the solution of the semiconductor response of the BP PCA,the mobility was considered dependent on the bias electric fieldby the application of the Caughey–Thomas mobility model[37]. The saturation velocity of the electrons and holes for theapplication of this model is specified later in this paper, andthe model fitting parameter value was 2 for the BP [38]. Whenthe temperature variation is considered in the model, the powerlaw mobility model (µT

n,p =µin(T/Tref)−βn,p ) is added as input

to the Caughey–Thomas mobility model to account for thetemperature dependence of the mobility. The fitting parameterβn,p was used as 0.45 for electrons [17] and 2 for holes [18].

C. Terahertz Response

The volume current density EJn,p(Er , t) A/m2 in Eq. (4) wasmultiplied by the cross-sectional area of the BP layer and thendivided by the width of the electrodes to obtain surface currentdensity EJ s (Er , t) A/m in the gap. The transient RF module wasutilized to calculate the THz generation signal in Eq. (5) as [39]

ETHz(Er , t)=−µ0

4π

∂

∂t

∫EJ s(Er , t −

(∣∣Er − Er ′∣∣ /c ))|Er − Er ′|

ds 8, (5)

where µ0 is the magnetic permeability of free space, |Er − Er ′| isthe distance between the excitation source and the field points,and ds 8 is the increment of the surface area at a displacement Er ′

from the excitation source.

Table 3. Material Properties for Electrical Response

Parameter Symbol Units Black Phosphorus

Doping profile NA 1/cm3 2 ∗ 1015 [32]Electron mobility µn cm2/V · s 1,500 [17]Hole mobility µp cm2/V · s 5,000 [18]Bandgap E g V 0.3 [15]Electron affinity χ V 4.4 [15]Electron lifetime τn ps 0.360 [16]Hole lifetime τp ps 0.360 [16]Electron saturationvelocity

vn,sat cm/s 1.0 ∗ 107 [38]

Hole saturation velocity vp,sat cm/s 1.2 ∗ 107 [38]Effective density ofstates, conduction band

NC 1/m3 5.933 ∗ 1025 [40]

Effective density ofstates, valence band

NV 1/m3 1.052 ∗ 1023 [40]

3. NUMERICAL RESULTS AND EXPERIMENTALMEASUREMENTS

Several computational domains were simulated using differ-ent modules of COMSOL, as discussed earlier. The materialproperties for the RF frequency-domain and semiconductormodules are listed in Tables 2 and 3, respectively. For the elec-trical response, one must be careful about the time domainmeshing to capture all information in the carrier generationpulse; otherwise, problems with convergence will arise. For thispurpose, we carefully used a non-uniform time discretizationwith higher resolution around the pulse peak to account for theultrafast behavior of the generated carriers. The heat-transferproperties of the materials are listed in Table 4. The differentsimulation computational domains and associated platforms aresummarized in Table 5 for all cases considered in this work.

A. Maxwell’s Equations Solution

The computational domain of the active area of the device ofFig. 1 is shown in Fig. 2. The thickness of the BP and hBNlayer were selected to maximize the optical power absorptionwithin the BP layer based on the work reported in [7]. Part ofthe antenna electrodes were also considered in this model as a5/60 nm layer of chromium/gold (Cr/Au). The optical proper-ties of all materials involved are listed in Table 2, including theanisotropic optical properties of the BP.

Table 4. Heat Transfer Properties of Materials

Symbol κ ρ C p

DescriptionThermal

Conductivity DensityHeat Capacity

Constant PressureUnits W/mK kg/m3 J/kgK

Au 318 [41] 19300 [41] 129 [41]Cr 93.7 [41] 7150 [41] 450 [41]SiO2 1.4 [42] 2200 [43] 703 [44]Si 150 [42] 2329 [44] 700 [44]hBN 390 [45] 2279 [45] 818.18 [46]

x 25.29 [47] 2420 [48] 695.5 [49]BP y 48.79 [47]

z 4.00 [50]

Research Article Vol. 38, No. 4 / April 2021 / Journal of the Optical Society of America B 1371

Fig. 2. Optical computational active area. (a) Optical responsegeometry with BP shown in black color, hBN in gray color, SiO2 inlight green color, Si in dark green color, and electrodes in yellow color.(b) Meshing discretization at λ/10 in material. (c) Maximum powerdensity Pmax(Er ) at each point in the layered active area. All dimensionsare given inµm. This is case 4 in Table 5.

Since the optical wavelength involved in the femtosecondlaser excitation, compared to the size of the antenna, represents amulti-scale problem, the 3D simulation of the complete geom-etry of a photoconductive antenna becomes computationallyprohibited. To minimize the computational cost of solving theoptical response of the antenna geometry, only the active area ofthe photoconductive antenna was modeled as reported in [51].However, a large case was simulated with more than 313 millionunknowns to validate this approximation, as will be discussedhere. Moreover, with the x polarized laser excitation focusedon the center of the antenna gap, the symmetry of the compu-tational problem was exploited by applying a perfect electricconductor (PEC) boundary condition parallel to the y axisand a perfect magnetic conductor (PMC) boundary conditionalong the x axis. This application of the boundary conditionsdivides the antenna active area into four quadrants and allowsthe simulation of only one quarter of the actual geometry.

To obtain a better understanding of the computational cost ofthis problem, Fig. 2(b) shows the discretization required for thesolution. As known, the computational solution of Maxwell’sequations problem requires a high wavelength-dependent res-olution for the discretization. In fact, the mesh size for this partof the model was defined as one tenth of the wavelength in thematerial of each layer representing more than 11.8 million meshelements for the case shown in Fig. 2(b) (see Table 5).

The maximum power density calculated from the electricand magnetic field solutions at each point in the layered domainquadrant shown in red dashed lines in Fig. 2(c), is extendedby using the appropriate symmetry based on the boundaryconditions PEC and PMC discussed earlier. A custom-madeMATLAB code was developed for filling of maximum powerdensity data in all quadrants. The results of Fig. 2(c) demon-strate that a maximum power density of around 3× 108 W/m2

is shown in the air space. This value agrees with the incidentaverage laser power that was implemented in the excitationfollowing the definition of the average peak power of a train oflaser pulses. This solution is significantly important, as it servesas the driving excitation for the photocurrent generation in theelectrical response module.

B. Photocurrent Solutions

The electrical response of this model is based on the accelerationof the photocarriers due to the application of the bias electricfield. The drift-diffusion equations are solved to obtain thephotocurrent density generated in the device. For the semi-conductor solution, we only considered the BP layer whereall parameters are listed in Table 3. The boundary conditionsapplied in this part of the model correspond to ohmic contactsfor the electrode-to-BP interface and the insulating boundaryconditions for all other boundaries. The initial conditions arecritical for the convergence of the solution. In this way, theelectrical response includes two steps. The first step is to achievethe initial conditions for the second step, which is the actualtime-dependent electrical response solution of the PCA. Thefirst step represents a steady-state solution, in which the voltagewas gradually increased from 0 V with a step voltage of 0.1 Vuntil 1 V, which is the desired working condition here. Then,the solution of this stationary study was used at 1 V as initialcondition to the time-dependent solution in the second step.The spatial discretization was important in the convergence ofthe electrical response of the model. We applied a maximumspatial discretization of 0.1 µm in the x and y directions whilemaintaining the same maximum discretization that was usedin the optical response for the z direction (at λ/10 in material).Furthermore, as stated before, the time discretization is crucialfor the convergence of the solution because the femtosecondpulse represents an ultrafast switch on the generation of carriers.To overcome this convergence issue, we applied a non-uniformtime discretization with a higher time resolution around thepulse peak. The time steps were applied as 0.05 ps for the timerange of 0 to 0.5 ps, 0.01 ps for the time range of 0.51 to 1.0 ps,and 0.05 ps for the time range of 1.05 to 5.0 ps, which covers thetime duration of the photocurrent density results presented inthis work.

Based on the model description, the maximum power densityobtained from the optical response solution is the driving factorof the carrier generation rate in the semiconductor G(Er , t), asdefined in Eq. (2). Obtaining a 3D solution for the laser beaminteracting with the device represents an extensive computa-tional problem as discussed earlier. Some reported works in theliterature avoided this issue by using an approximation to cal-culate the carrier generation rate without solving the Maxwell’sequations in the semiconductor [36,52,53]. Their calculationwas based on the incident intensity of the laser and the proper-ties of the material including the quantum efficiency, calculatedfrom the absorption capacity of the material [36]. The approachwas applied in [52] to model a LT-GaAs photoconductiveantenna where their carrier generation rate Gapr(Er , t) wasapproximated as [52,53]:

Gapr(Er , t)=W0ηQE (z)h(x , y ) f (z, t), (6)

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Table 5. Summary of All Cases Presented in This Work

Solved ModelsMesh

ElementsDOF

(Unknowns)

RequiredMemory

(GB)CPU Time

(hours, minutes) Platform

OpticalResponse

Case 1–Fig. 9 6 159 517 39 188 546 288.4 5 h,39 m AHPCC–2 x Intel Xeon Gold 6126 CPU at2.60 GHz. (2 sockets–24 cores)

Case 2–Fig. 9 7 704 717 49 002 884 372.37 8 h,34 m AHPCC–2 x Intel Xeon Gold 6126 CPU at2.60 GHz. (2 sockets–24 cores)

Case 3–Fig. 9 9 329 505 59 321 684 463.07 11 h,40 m AHPCC–2 x Intel Xeon Gold 6126 CPU at2.60 GHz. (2 sockets–24 cores)

Case 4–Fig. 2, 9 11 822 474 75 151 948 617.34 18 h,52 m AHPCC–2 x Intel Xeon Gold 6126 CPU at2.60 GHz. (2 sockets–24 cores)

Case 5 (LargestCase)

49 328 365 313 252 784 3 202.98 25 h,6 m XSEDE Bridges–16 x Intel Xeon CPUE7-8880 v3 at 2.30 GHz. (16 sockets–24

cores)ElectricalResponse

Case 4–Fig. 3 311 958 5 000 711 36.16 6 h,51 m 4 x AMD Ryzen Threadripper 2990WXCPU at 3.00 GHz. (4 sockets–32 cores)

Case 5 (LargestCase)–Fig. 5

2 544 387 25 480 658 146.49 25 h,47 m AHPCC–2 x Intel Xeon Gold 6130 CPU at2.10 GHz. (2 sockets–24 cores)

THz Response 1max = 35 µm–7 114 674 800 016 6.06 3 h,47 m 4 x AMD Ryzen Threadripper 2990WXCPU at 3.00 GHz. (4 sockets–32 cores)

1max = 20 µm–7 554 407 3 713 386 18.56 15 h,16 m 4 x AMD Ryzen Threadripper 2990WXCPU at 3.00 GHz. (4 sockets–32 cores)

1max = 15 µm–7 1 293 643 8 552 746 40.64 72 h,10 m 4 x AMD Ryzen Threadripper 2990WXCPU at 3.00 GHz. (4 sockets–32 cores)

Temperature Case 5 (LargestCase) Figs. 4, 5(Joule Heating)

2 544 387 25 480 658 147.96 40 h,58 m AHPCC–2 x Intel Xeon Gold 6130 CPU at2.10 GHz. (2 sockets–24 cores)

Case 5 (LargestCase) Figs. 4, 5(Laser Heating)

2 544 387 25 480 658 148.98 24 h,14 m AHPCC–2 x Intel Xeon Gold 6130 CPU at2.10 GHz. (2 sockets–24 cores)

where W0 = I0λα/hc represents the maximum carrier gener-ation rate, which depends on the intensity of the incident laserbeam I0, laser wavelength λ, and absorption coefficient of thematerial α. In Eq. (6), ηQE represents the quantum efficiencyalong the thickness of the semiconductor, and h(x , y ) andf (z, t) define Gaussian profile distributions in both space andtime, respectively.

Here, we present a comparison between using the carriergeneration rate driven by the maximum power density obtainedfrom the solution of Maxwell’s equations G(Er , t) in Eq. (2) andthe approximation of the generated carriers based on the inten-sity of the incident laser Gapr(Er , t) in Eq. (6). It is important tonote that for the Maxwell’s equations approach, the distributionof the laser along the propagation direction is obtained from theinteraction of the incident electric field with the BP substrate.However, for the approximate calculation, the distributionof the incident laser pump along the propagation direction isdefined by the profile f (z, t), which is based on the attenuationfrom the absorption coefficient as defined in [52]. For this solu-tion, the quantum efficiency was assumed to be 1, and the spatialdistribution h(x , y ) in x and y was defined in the same way asthe incident electric field applied in the optical response of themodel [52,53].

The carrier generation rate profiles are shown in Figs. 3(a)and 3(b) at their maximum values. These plots present thecharacteristic circular distribution in the x and y directions due

to the Gaussian profile in space. However, upon comparing thetwo figures, the spatial distribution size of the carrier genera-tion rate Gapr(Er , t) in Fig. 3(b) is larger than that of G(Er , t) inFig. 3(a). This difference arises from the fact that the Maxwell’sequations approach carries the Gaussian spatial distributionfrom the maximum power density after its calculation due to theincident electric field while the approximate approach appliedthe Gaussian distribution directly to the carrier generation rate.This discrepancy creates a spatial overestimation of the carriergeneration rate, where more carriers are generated over a largerspace in the BP layer for the approximation approach. Besidesthis spatial exaggeration, the maximum carrier generation ratefor the approximate calculation is also larger than that of theMaxwell’s equations approach. The carrier generation rate isthe driving term of the drift-diffusion model, so this spatialand amplitude overestimation is transferred to the photocur-rent density calculation. In fact, the photocurrent densityresults from this study are shown in Figs. 3(c) and 3(d) for theMaxwell’s equation and the approximation approach, respec-tively. The over-estimated calculation in space and amplitudeof the photocurrent density produced by the application of theGapr(Er , t) carrier generation rate are noticeable. These plotsshow large photocurrent densities at the corners of the bowtieelectrodes due to edge effect associated with the solution of thebias electric field at those points. However, most of the pho-tocurrent density is concentrated at the center of the gap, where

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the laser beam is focused and where the carriers are generated.Figure 3(e) demonstrate the photocurrent density over time at asingle point in the center of the gap at 10 nm below the surfaceof the BP layer. We can report that an overestimation of thephotocurrent of ∼1.75 times is shown. These results highlightthe significance of solving the 3D Maxwell’s equations model inthis work.

C. Device Temperature

Here, we model the temperature variation of the BP antennaemitter under a maximum bias voltage of 1 V and a laser sourceof average power 1 mW. For this solution, the heat transferin solids module solves the heat equation and provides thetemperature over the modeling domain as [54]:

ρC p∂T∂t+ ρC p Eu · ∇T +∇ · Eq = Q + Qted, (7)

where T is the temperature, C p is the heat capacity of the mate-rial at constant pressure, ρ is the density of the material, Eu isvelocity vector related to moving objects in the domain, andEq is equal to −κ∇T, where κ is the thermal conductivity ofthe material [54]. Qted represents the thermoelastic dampingterm and Q defines a heat source or heat sink, which in thiscase represents the source of heat considered in this model.Here, we accounted for two sources of heat to calculate thetemperature distribution: the steady-state Joule heating and thetime-averaged electromagnetic power dissipation. Therefore,the temperature T in Eq. (7) is function of space and not ofinstantaneous time.

The geometry of the first case modeled in this study isthe same used for the calculation of the optical solution ofthe device without the application of the air layer. Instead ofconsidering the air layer, we applied a convective heat flux(q0 = h · (Text − T)) at the boundary of the air-to-hBN inter-face with an external temperature of 300 K and a convectiveheat transfer coefficient (h) of 3475 W/m2K for hBN [55]. Theremaining boundaries were considered as insulating boundaries.The properties of the materials used in these calculations arelisted in Table 4, where the anisotropic thermal conductivity ofBP and isotropic thermal conductivity of hBN were applied.The spatial discretization used in this part of the model is thesame as that used in the electrical response in the BP layer. Amaximum mesh size of 0.64µm was assigned to the other layerswhere the current flow heat source was not applied.

As shown in Fig. 4, the first heating source is the Joule heatingdue to the conduction of the current induced by the DC biasvoltage in the BP antenna emitter [19]. The second source inFig. 4 is the average electromagnetic power dissipation of thefemtosecond laser in the BP material [34]. Here, we simulatedthe temperature rise due to the heat produced in the BP layeronly, where the power absorption was considerably higher thanthat in the hBN, SiO2, and Si [7].

The geometry of Figs. 4(a) and 4(c) represents the compu-tation of the active area of the smaller case of Fig. 2 (case 4 inTable 5), while the geometry of Figs. 4(b) and 4(d) represents thecomputation of a larger area. The size the hBN/BP in Figs. 4(b)and 4(d) is 10 µm× 10 µm, compared with 6 µm× 6.34 µmin Figs. 4(a) and 4(c). These layers are positioned on top of

Fig. 3. Electrical response results (Dimensions in µm). (a) Carriergeneration rate from the Maxwell’s equations solution approachG(Er , t). (b) Carrier generation rate from the approximation approachG apr(Er , t) [52,53]. (c) Photocurrent density from the Maxwell’sequations solution approach. (d) Photocurrent density from theapproximation approach [52,53], (e) Photocurrent density compari-son between the Maxwell’s equations solution and approximationapproach [52,53]. This is Case 4 in Table 5.

the gap of the emitter, leading to larger modeling domain toconsider the air-to-electrodes interface. Surprisingly, the resultsof Figs. 4(a) and 4(c) show high temperature levels of more than519 and 695 K, respectively. This observation can be attributedto the fact that modeling a smaller active area where the hBNand BP layers totally cover the electrodes could be inaccurate.

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Fig. 4. Temperature calculation in the device due to Joule heat-ing and the electromagnetic power dissipation (laser heating).(a) Simulation of the active area with the hBN and BP layer cover-ing the electrodes for the Joule heating. (b) Simulation of a largeractive area with inclusion of the air-to-electrode interface for the Jouleheating. (c) Simulation of the active area as (a) for the laser heating.(d) Simulation of a larger active area as (b) for the laser heating. Cr/Aurepresents the exposed electrodes. The dashed black square representsthe simulation size of (a) and (c). All dimensions are inµm.

To better understand the reason, we increased the size of thecomputational domain in Figs. 4(b) and 4(d) to partially exposethe electrodes to the air interface. The results interestinglydemonstrate drop in temperature variation around 300 K. Thismetal-to-air interface was also modeled as a convective heatflux with the convective heat transfer of 125× 106 W/m2K,calculated from the thermal resistance of the electrodes [56]. As

such, we believe that the electrodes provide heat sink to the sys-tem, bringing the temperature down to the level close to roomtemperature due to the low thermal resistance of Cr/Au films ondioxide-on-silicon layers (Rth = 0.8× 10−8 m2K/W) [56].

To calculate the temperature variation in Fig. 4(d), wefirst obtained the average electromagnetic power dissipation( 1

2 Re[ EJ . EE ∗]) from the optical response and used it as the inputto the heat transfer in solids module. For this large case, themodel size increased to more than 313.2 million unknownswith the same discretization and domain assumptions describedabove. To solve this model, we used the 12 TB extra-large mem-ory nodes at the XSEDES Bridges cluster in the PittsburghSupercomputing Center (PSC). As summarized by Table 5, theoptical response solution of this case (Case 5) required morethan 3.2 TB of RAM memory, with a solution time of ∼25 hthat represented more than 80 service unit hours at XSEDES.The Maxwell’s equation solution of this case was comparedagainst the optical response results in Fig. 2(c). We selecteda point 10 nm below the surface of the BP layer at the centeron the gap, and the difference in maximum power flux den-sity was reported as only 0.0161% between both cases. Thisfact supports the idea of simulating only the active area of theantenna gap for the laser interaction with the photoconductorto minimize the size of the modeling domain while maintaininga considerable accuracy for the results. On the other hand, thetemperature results of Figs. 4(b) and 4(d) became more accu-rate upon modeling a larger size of the electrodes. Specifically,the results demonstrate that modeling only the active area forthe optical response and the electrical response of this deviceis sufficiently accurate, but modeling the same active area sizefor the temperature variation was not sufficient and couldprovide false rise in the device temperature. The bias voltageof 1 V applied to the device produced a steady-state currentdensity of 10.8 kA/cm2. Upon multiplying the bias voltageand the current density with the cross-section of the BP at thegap (40 nm× 5 µm), it produces electrical power of 21.6 µW.On the other hand, the average laser power of 1 mW was usedas excitation. This difference in power values can explain theobserved temperature difference between the Joule and the laserheating.

The photocurrents due to no thermal and thermal effect ofthe preceding two heat sources are shown in Fig. 5. The pho-tocurrent due to a constant room temperature of 300 K is shownby the blue curve, which is labeled as No-Thermal Effect. Thisresult represents the time-dependent photocurrent densityacquired from the drift and diffusion components without anythermal variation. This plot was also measured at a single point10 nm below the surface of the BP layer at the center of the gapof the device. To calculate the thermal effect on the photocur-rent, a time-dependent solution of the semiconductor modulewas solved at a spatially variable temperature. In this case, thetemperature distribution displayed in Figs. 4(b) and 4(d) wereused as input to the semiconductor module, which delivereda photocurrent density composed of the drift, diffusion, andthermal components. The red dashed plot represents the ther-mal effect due to Joule heating, which agrees with the resultsreported in [53]. The green plot in Fig. 5 represents the thermaleffect due to the laser heating demonstrating a larger increasein the photocurrent compared to the photocurrent rise due to

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Fig. 5. Photocurrent density without and with thermal effect ofJoule heating in Fig. 4(b) and laser heating in Fig. 4(d). This is theelectrical response of the largest case, which represents case 5 in Table 5.

the Joule heating. Upon comparing the three plots in Fig. 5, theJoule heating and the laser heating seem to cause an increase inthe photocurrent by∼3.38% and∼9.98%, respectively.

D. Simulated THz Pulse and ExperimentalMeasurements

The modeling domain of the antenna is illustrated in Fig. 6(a),consisting of a half space of air on the top layer (gray) withhalf space of Si in the bottom layer (green), with a size of500 µm× µm size in the x and y directions. PEC and PMCboundary conditions were used to take advantage of the sym-metry of the problem with the excitation of the model definedas a surface current density flowing along the gap of the antenna[11]. The profile for this excitation current is obtained fromthe photocurrent density in Fig. 3(e) for the Maxwell’s equa-tion solution approach. This volume photocurrent density(A/m2) is multiplied by the cross-sectional area of the BP layer(5 µm× 0.04 µm) and divided by the width of the antennaelectrode (5 µm) to obtain the surface current density (A/m)flowing in the gap. A top view of the overall antenna geometryis presented in Fig. 6(b), where the transmission line is shown aswell as the non-uniform discretization of the model that becamefiner towards the smaller structures in the domain such as theantenna shape and transmission line. The dimensions of theantenna are presented in a zoomed-in top view in Fig. 6(c),where we modeled only one quadrant of the real geometry basedon the implemented PEC on the x axis and PMC on the y axis.It is important to mention that the hBN, BP, and SiO2 layerswere not considered in this transient RF module because theirnanoscale thickness is negligible compared to the wavelengthat THz frequencies. Furthermore, the complex relative per-mittivity was 11.53-0.0047i for the Si substrate, which wasmeasured at the University of Arkansas Terahertz Lab using theTPS Spectra3000 system at a frequency range of 0.1–3.5 THz.These measurements agreed with those reported in [57].

The results of the simulated THz signal generation pulse ofthe proposed BP THz emitter are shown in Fig. 7(a) for the timedomain and in Fig. 7(b) for the frequency domain. The simu-lations were conducted at three discretization values of 35, 20,and 15 µm. For the simulations, the electric field componentwas calculated at a fixed point located 250 µm below the gap ofthe antenna. The goal of solving the same model with differentdiscretization values is to achieve mesh convergence and guaran-tee the accuracy for the computation. The discretization value of

Fig. 6. Antenna model geometry for the THz signal generationof the BP PCA. (a) Isometric view. (b) Top view with the discretiza-tion. (c) Zoomed-in the antenna geometry with the dimensionsa = 17.9 µm, b = 12.1 µm, c = 20.8 µm, and d = 5 µm. This geom-etry was used to produce the cases 1max = 35, 20, and 15 µm. Theircomputational details are provided in Table 5.

Fig. 7. Simulated THz signal generation pulse of the BP PCAemitter. (a) Normalized time-domain signal. (b) Normalizedfrequency-domain spectrum. The measurement was collected usingthe TeraAlign system shown in Fig. 8. The 1max represents the maxi-mum discretization size. The computational details of these plots areprovided in Table 5 as Case1max = 35, 20, and 15µm, respectively.

15 µm provided the best computational performance in termsof the solution CPU time and required RAM (see Table 5).

The simulated THz signal generation pulse of the BP PCA ofFig. 7 is compared with measurements conducted using a com-mercial photoconductive emitter based on the conventionalLT-GaAs technology. The measurements were performed using

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Fig. 8. TeraAlign bench top time-domain THz system. The dashedred line represents the laser after the fast delay line, and FP stands forfiber port.

a TeraAlign benchtop time-domain system at the University ofArkansas, as shown in Fig. 8. The TeraAlign system (TeraView,Cambridge, UK) is composed of a femtosecond laser of 780 nmwavelength, 100 fs pulse width, 80 mW average power, and100 MHz repetition rate. A group velocity compensator (GVD)adjusts the dispersion of the laser pulse. A beam splitter dividesthe laser between the emitter and detector. Three delay lines areused, one in the path of the emitter and two in the path of thedetector. The dashed square indicates two movable stages inthe x , y , and z directions, one for the emitter and one for thedetector. Two gold-coated ellipsoidal mirrors focus the emittedTHz signal onto the detector antenna. The measurementsobtained from the TeraAlign system are shown in Fig. 7(a)together with the results of the simulations of the proposed BPPCA. The normalized simulated results of the BP PCA THzsignal generation pulse agreed in principle with the LT-GaAsreference measurements in terms of pulse width and pulse shape.However, the time-domain pulses of the simulated BP PCA arenarrower compared with the LT-GaAs PCA measurements. Theresults of Fig. 7(b) show the spectrum of the pulses of Fig. 7(a),where we show the working frequencies of both the simulatedBP and the LT-GaAs reference PCAs. It is important to mentionthat the measurements performed with the commercial PCAwere developed with an averaging factor of 1800 measurementsin an unpurged environment. This explains the spikes in thefrequency-domain spectrum that represent the water signaturedue to the humidity of the air.

The simulated THz signal generation electric field pulse wascalculated at 250 µm from the gap while the measured pulsewas obtained in the far field at several centimeters from the gap.The results of Fig. 7 demonstrate a good agreement betweenthe proposed BP PCA and the conventional LT-GaAs basedon the signal generation pulse shape in the time and frequencydomains. Furthermore, the results show a trend of increasingthe bandwidth of the simulated BP PCA compared with thereference commercial LT-GaAs PCA. This observed trend inincreasing the bandwidth is based on comparing the overall BPPCA device against an existing commercial LT-GaAs devicepurchased from TeraView, Ltd. The higher bandwidth of the BPdevices holds even if a longer BP carrier lifetime is assumed (seeAppendix B).

4. CONCLUSION

This work reported the solution of Maxwell’s equations forthe application of a 780 nm femtosecond laser source againstan anisotropic BP PCA antenna. The required wavelength-dependent discretization and the multiscale nature of theantenna structure represented a computationally intensiveproblem. We reduced the computational cost of the modelthrough the simulation of only the active area of the antenna gapand implementing the PEC and PMC boundary conditions,leading to simulating one quadrant of the domain. We reporteda difference of 0.0161% upon comparing a medium sizedomain in the Maxwell’s equation solution (Case 4 in Table 5),with the solution using a larger domain (Case 5 in Table 5). Thisslight difference in the optical response solution proves the highaccuracy obtained from modeling only the active area of thedevice, which represents a huge reduction in the computationalcost of the problem.

For the semiconductor response solution, the results demon-strated overestimation of carrier generation rate between theapproximation described in [52,53] compared to the applica-tion of the 3D Maxwell’s equation solution presented in thiswork. This overestimation in the generated carriers influencedthe photocurrent density calculation, affecting its accuracy andproviding a larger photocurrent by a factor of∼1.75. This high-lights the significance of solving the optical response of the 3Dmodel. These results were also compared to a larger case (Case 5in Table 5), in which the electrical response solution exhibitedsimilar solution accuracy to the medium case of only the activearea of the antenna around the gap. In fact, these solutions werecompared at their maximum photocurrent densities, reportinga difference of only 0.651%. This means that a model using anactive area around the gap of the antenna produces results withsufficient accuracy as that of the larger domains for the opticaland electrical responses.

Finally, the conduction of current due to the applicationof the bias voltage at the gap of the antenna and the electro-magnetic power dissipation of the femtosecond laser in the BPmaterial generates a temperature increase in the device. Here, weconclude that to model the thermal effect, the air-to-electrodeinterfaces must be considered in the computational domain.The results obtained here showed that the electrodes of theantenna represent thermal sink to the device due to the lowthermal resistance of the Cr/Au metal structures. Hence, ifthe interaction of these elements with air is not considered inthe simulation domain, the results demonstrated false largetemperature levels that lead to inaccurate temperature effecton the device performance. Once we increased the size of theactive area to account for the thermal sink, the results showedonly a slight temperature variation around room temperatureof 300 K between the hottest part of the device at the antennagap and the electrodes where most of the heat is exchangedwith the environment. The temperature variation due to Jouleheating produced a difference in the photocurrent density of∼0.0136 kA/cm2, and the thermal effect of the femtosecondlaser heating produced a variation of ∼0.0399 kA/cm2 com-pared to the photocurrent density profile obtained at roomtemperature (i.e., upon neglecting the thermal effect at biased

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volte of 1 V). We should mention that the temperature varia-tion due to the phonon interaction with the photocarriers isimportant, but it is outside of the scope of this work and will beinvestigated in the future [58].

The normalized simulated THz signal generation pulse ofthe proposed BP PCA agrees with the measurements performedusing a commercial LT-GaAs PCA emitter in terms of the pulseshape in the time and frequency domains. However, the sim-ulated BP PCA THz signal generation pulse demonstrated atrend of increased bandwidth. In our previous work [7], wemodeled the LT-GaAs PCA and compared the photocurrent andTHz signal generation results with the BP device. The carrierlifetime of 0.36 ps for electrons and holes that we used for BPwas comparable to that of LT-GaAs of 0.3 ps for electrons and0.4 ps for holes [52]. Here, to demonstrate that the proposedBP PCA has a potential of THz emission, we compared thesimulated BP PCA results with measurements of a referencecommercial LT-GaAs. However, it is not possible to simulate thereference device due to the IP protection by the manufacturer(TeraView, Cambridge, UK).

The fabrication of the proposed BP PCA and its performancemeasurement is a topic of current ongoing research. This workhas established the methodology of 3D modeling of BP PCAwith a metallic bowtie antenna. In future work, the metallicelectrodes will be optimized to other designs to enhance thebandwidth, increase the emitted THz power, and improve theemitter efficiency.

APPENDIX A

To estimate the required RAM and CPU time before execut-ing cases on the supercomputers, we developed a regressionanalysis model based on the optical response solution of fourcases of different computational domain size. This estima-tion was based on a regression model given in Eq. (A1). Uponimplementing the PEC and PMC boundary conditions, wesimulated only one quadrant of the actual active area. Thesmallest case ¬ has a quadrant active area with the antennagap size of 2.0 µm× 2.0 µm and one-half electrode of size0.5 µm× 2.0 µm. For case ­, a quadrant active area was con-sidered with the antenna gap size of 2.0 µm× 2.5 µm andone-half electrode of size 0.5 µm× 2.5 µm. Case ® applied amodel size with a gap of 2.0 µm× 2.5 µm and half of the elec-trode with a size of 1.0 µm× 2.5 µm. Case ¯ with a quadrantactive area of 2.0 µm× 2.5 µm for the gap size and a trapezoidalelectrode with a width of 1.0 µm and heights of 2.5 µm and3.17 µm. This case ¯ is displayed in Fig. 9(b) for clarity. Theresults of Fig. 9(a) demonstrate the CPU solution times next toeach plot, which is also summarized in Table 5. Case ¬, witharound 39.2 million unknowns or degrees of freedom (DOF)required more than 288 gigabytes of memory compared tocase ¯, with 75.1 million of DOF that required around 617gigabytes of memory. The dashed line in Fig. 9(a) representsthe regression polynomial-based model with three coefficientsas [59]:

f (x )= 0.02739x 2+ 5.999x + 11.62, (A1)

where x is the number of DOF of the case expressed in millions.This polynomial was used to estimate the required memory

Fig. 9. Estimation of required RAM memory and iteration erroranalysis of the optical response module based on the number ofunknowns in the optical response. (a) Memory requirement anditerative solution CPU time for the four cases considered in this study.(b) Illustration of the size of the active area simulated for case ¯.(c) Convergence plot for the four cases shown in (a) with pre-assignederror threshold of 1× 10−5. These are Cases 1, 2, 3, and 4 in Table 5.

to solve models with dimensions larger and smaller than thepresented in this study.

Furthermore, based on our experience in this work, it is rec-ommended to use iterative solvers, which are known to requireless memory than the direct solvers. Here, we used the GMRESiterative solver in COMSOL with a multigrid preconditionerand a threshold error of 1× 10−5. The convergence plot of thisstudy is shown in Fig. 9(c). All cases converged after 11 iterationswith different solution CPU times. The same discretization wasused in all four cases presented in Fig. 9. The computations of allthese cases were performed using 24 cores at the High MemoryNodes (768 GB) in the Pinnacle cluster at the Arkansas HighPerformance Computing Center (AHPCC).

APPENDIX B

We observed a significant discrepancy in the literature con-cerning the values of carrier lifetime for BP, wherein they rangefrom 0.36 to 1800 ps depending on the sample preparation,measurement technique, wavelength of the laser source, and,more importantly, the curve-fitting parameter extraction fromthe measurements. Some researchers used a single exponentialcurve fitting that can lead to longer carrier lifetimes [60,61].Other researchers have used a bi-exponential curve fittingthat provided shorter carrier lifetimes [16,62,63]. Anotherdiscrepancy in the bi-exponential fit is that some researchersconsider the second-time constant as the carrier lifetime of 83 ps[62], while others consider the first-time constant as the directrecombination lifetime of ∼15 ps [63]. We believe that using

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Fig. 10. Simulation of BP device at carrier lifetime of 83 ps. (a) Photocurrent density. (b) Time-domain generated THz signal. (c) THz signalgeneration spectrum. This is case1max = 15 µm in Table 5.

two time constants in curve fitting provides results that are moreaccurate; therefore, we simulated the photocurrent and THzsignal generation using carrier lifetime of 83 ps, as shown inFig. 10. The results demonstrate that the difference in the carrierlifetime between 0.36 and 83 ps manifests itself as a longer tail inthe photocurrent in Fig. 10(a). This reflects on the signal gener-ation spectrum at low frequencies in Fig. 10(c). Therefore, thereported enhancement of the bandwidth using 83 ps still holdsat THz frequencies above 0.2 THz as shown in Fig. 10(c). Thelifetime values remain an open question for more investigationsuntil the ongoing fabrication of the BP device is completed andmeasured.

Funding. National Science Foundation (1948255).

Acknowledgment. The authors acknowledge the tremendous helpoffered by Mr. Michael Evans from TeraView Ltd. UK on setting up the open-bench TeraAlign system. This work used the Extreme Science and EngineeringDiscovery Environment (XSEDE) Bridges Large Memory at PSC throughallocation ELE200004, which is supported by National Science Foundation.This work also used the large memory nodes on the Pinnacle at the ArkansasHigh Performance Computing Center (AHPCC). The authors acknowledgethe technical support provided by Mr. Daniel Klein, at the University ofArkansas, who coordinated the licensing and installation of the COMSOLsoftware on the XSEDE and AHPCC. The authors also acknowledge the helpfrom Ms. Nagma Vohra on the experimental measurements. The authors alsoacknowledge the academic support from Menlo Systems.

Disclosures. The authors declare no conflicts of interest.

Data Availability. Data underlying the results presented in this paper arenot publicly available at this time but may be obtained from the authors uponreasonable request.

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