1. Introduction

Since the initial success of Auston et al with photoconductive switches (PC switches) in 1983, there has been an increasing demand for terahertz (THz) sources. Many approaches for generation of coherent THz radiation have been proposed so far, each having its own advantages and disadvantages [1,2]. Among alternatives, THz PC switches have received considerable attention, mostly due to wide frequency tunability range, fabrication simplicity and recent advancement of ultrafast pulse lasers [3–8]. Different types of THz photoconductive switches have been developed in recent decades mostly by the use of fast semiconductor materials such as low temperature grown GaAs (LT-GaAs), LT-InGaAs and more recently GaAsBi. These materials are used as the base material that feeds an antenna structure with the photocurrent (Fig. 1(a) ) [3–8]. THz PC switches can also work in continuous wave mode (CW mode) as THz photomixers, offering more flexibility in different applications. However, low output power (less than 10µW) for PC switches and photomixers has been a major challenge, leaving these THz sources as inefficient choices for many applications [4,7,8].

 

Fig. 1 (a) Illustration of optical excitation of a THz photoconductive switch that has a center fed dipole antenna structure. (b) A PC switch made on a thin SiO₂ oxide layer grown on a Si wafer. The gap is filled with aligned SWNTs and the antenna pattern is fabricated similar to conventional PC switches through common lithography processes (c) Block diagram of different steps of the analysis that links input laser excitation power and output THz emitted power.

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Nanomaterials —especially carbon nanotubes (CNT) and graphene— offer a set of exceptional properties that are potentially suitable for more efficient THz PC switches and photomixers [8–10]. This includes both using these materials as high impedance waveguides in the antenna structure and as a high mobility, high absorption, base material in the antenna gap (Fig. 1(b)) [8–10].

In both cases, increase in output power is anticipated, based on primary estimations [8,10]. However, there has been no design guidance for fabrication of such devices. So the effect of different parameters, such as tube alignment, carrier lifetime, exciton-exciton annihilation rate and contact resistance on the performance of such devices remains questionable. In order to choose a design path, the dynamics of a PC switch based on CNT material should be addressed thoroughly and that is the main focus of this paper.

In this paper, we present a theoretical analysis that links the input optical pump excitation power and the output THz power (Fig. 1(c)). The first step in the link deals with fast free photocarrier dynamics of the single-walled carbon nanotube (SWNT) films. The second step is to model the photoconductivity based on the photocarrier dynamics from the first step. Finally, the dynamics of the metallic antenna structure and equivalent circuit model are engaged. The purpose is to address the effect of important parameters and highlight the limits and behaviors of each one on the THz PC switching process. This procedure then can be further modified for any specific desired antenna and gap structure.

2. Conversion of input power to free photocarriers in SWNT film

The connection between input power and free photocarriers in the SWNT film is found in two steps. First, the initial absorbed photon density is calculated and then, the dynamics of the excitons and photocarriers are considered in the film. The initial absorbed photon density is expressed by:

The conversion of the input light into photocurrent in the THz spectrum range is related to fast photocarrier dynamics of the SWNT film. This is different from previously measured photocurrents for SWNT films under infrared and visible illumination [14–16]. The latter is mainly considered for solar cell applications and is described by drift-diffusion equations that are proper for lower frequency dynamics [17,18]. Experimental studies on fast photoconductivity of CNT films have confirmed that given the initial absorbed photon density$n_{abs}$, the fast carrier dynamics can be described by a set of joint continuity equations between exciton density function $n_e(t)$ and total photogenerated carrier density $n(t)$ as in Eq. (2) [19].

In this equation ${\gamma }_{EE}$ is exciton-exciton annihilation rate, ${\gamma }_{CC}$is the carrier generation rate by exciton dissociation, and${\gamma }_d$is the carrier decay rate equal to the reciprocal of the carrier life time τ [20]. In Eq. (2), n(t) itself can be written as sum of free (n_f(t)) and localized (n(t)-n_f(t)) photocarrier densities; however, the equation holds regardless of the nature of the total photocarrier density. In the boundary conditions of Eq. (2) the initial exciton density is equated to the initial absorbed photon density; therefore, it is implicitly assumed that each absorbed photon immediately generates an exciton in the material. Also it is further assumed that excitons don’t decay during the period of integration (0 to T) in Eq. (1). It is noteworthy that this mathematical assumption holds in pulsed excitation because p(t) is typically nonzero only for a very short period of time (from 0 to pulse width, which is typically less than 200fs) and the excitons will enter the carrier dynamics (Eq. (2)) immediately after the pulse have been absorbed by the material. In other words, for pulsed excitation the exciton decay is ignored in the small period of excitation itself, however, for continuous wave excitation a proper period of integration should be considered in Eq. (1) so that n_e(0) can be fairly equated to n_abs in Eq. (2). Equation (2) can be solved explicitly, and the result is in the form of a hypergeometric function ₂F₁ ;

 

Fig. 2 (a) Peak total photocarrier density as a function of excitation pulse duration and carrier decay rate, ${\gamma }_d$ (carrier life time, $\tau =1/{\gamma }_d$). The inset graph depicts the carrier density as a function of time (optical excitation pulse duration = 100fs, n_abs = 10¹⁹ cm⁻³ and decay rate = 1.23 ps⁻¹). (b) Peak total photocarrier density as a function of n_abs. The inset graph shows the behavior of the same function within allowable carrier life time of SWNT.

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As it will be seen in sections 3 and 4, two key parameters of the carrier density function are its peak value and its pulse width (bandwidth). Both of these parameters are strongly influenced by excitation profile and the other rate parameters involved in Eq. (3). A typical set of experimental values, for randomly deposited SWNT films, is mentioned in Table 1 [19–21].

Table 1. A Typical Set of Experimental Values for Rate Parameters in Eq. (2)

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Figure 2 shows the influence of $n_{abs}$ and excitation pulse width on the peak total number of photocarriers.

As it can be seen in Fig. 2(a), the peak photocarrier density curve has a knee after which the peak $n(t)$ falls with exponential behavior. The location of the knee is directly influenced by carrier lifetime. For example, if a reduction to 90% of the maximum $n(t)$ is allowed in a design, based on Fig. 2(a), the maximum of allowable input pulse width is increased from 0.26 ps to 4.42 ps with a decrease of carrier decay rate from 3.19 ps⁻¹ to 0.03 ps⁻¹. Care must be taken when using this graph in its lower limits, as the physics of the material excitation can change for pulse width less than 20fs range. The curves in Fig. 2(a) are likely to roll off on the left hand side as well [22,23]. Also, the saturation of peak $n(t)$ in higher absorbed carrier densities is captured with Eqs. (3) and (4), as shown in Fig. 2(b). This is consistent with the experimental measurement of photocarriers reported in [19]. The inset graph in Fig. 2(b) reveals that the variation of carrier lifetime has an exponential dependence for values less than 3 ps. The peak photocarrier density, however, enters a saturation region immediately after this value, and thus further increase of carrier lifetime will no longer have a significant contribution.

It should be further emphasized that unlike the case of CW THz photomixing where a short carrier lifetime (less than 2ps) is vital to deepen the THz photoconductivity modulation of the material in the gap, short carrier lifetime is not a necessity for THz PC switching. This is due to the pulsed nature of THz PC switching and the fact that the pulses are well distanced in time relative to carrier lifetime. In THz PC switching the THz components of the microantenna feed current are generated mainly by the initial sharp rise in the photocarrier density as seen in the inset graph of Fig. 2(a). Consequently, the smoother roll-off of photocarrier density that is affected by the carrier lifetime does not induce THz components in the current. In THz PC switching the carrier lifetime affects the peak photocarrier density as seen in the inset graph of Fig. 2(b) and the peak photocarrier density can affect the output THz power as it will be explained in the rest of this study.

3. Photocarriers conversion to photoconductivity in SWCNT film

The calculation of photoconductivity of a SWNT film is a challenging problem. It can be viewed from a diverse range of perspectives, varying from non-equilibrium ab initio simulations [24] and consideration of Luttinger liquid behavior for individual CNTs [25] to use of an equivalent drude model for the entire sample [26]. Other than free photocarrier density that was calculated in the previous section, there are a considerable number of other parameters that can affect the photoconductivity of SWNT film. In order to focus this study, we will determine the anticipated range based on modified Drude-Smith (DS) model. Different fabrication conditions are considered, ranging from the idealistic fabrication condition of perfectly aligned, perfectly purified to totally random and partially purified cases. The DS model has given fair results for the study of ultrafast conductivity of varieties of nanomaterials [19,27,28]. In general the photoconductivity can be calculated via the Drude-Smith model as [27]

The previously reported value ranges and dimensions of these parameters are given in Table 2 . These values are experimentally justified in previous studies [19,29–31].

Table 2. Previously Reported Values for Drude Smith Modeling of CNT Films

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For the case of a perfectly aligned, perfectly purified CNT film, values of mobility, localization constant, and percentage of free photocarriers are chosen within the range from Table 2 so that the conductivity is maximized. In a real case, however, parameters such as carrier lifetime, carrier mobility, electrical coupling and alignment with the applied field, etc., can be different for each CNT in the film. Therefore, a Monte Carlo (MC) integration method should be used for a more realistic result [32]. This integration is based on Eq. (5) and is presented in Eq. (6) for q^th iteration of calculation.

The MC integration is performed by iteratively generating the initial distribution of the photocarriers. This distribution is based on a 3D probability density function with a 2D Gaussian (normal) distribution cross section and an exponential distribution along the radiation axis. This represents an illumination with Gaussian beam, and a material with Beer-Lambert absorption behavior [33]. In each iteration, the simulation bins the carriers into J × K cells along the gap (Fig. 3(a) ), applies randomly generated parameters to each cell, and does an overall summation. The outcomes of Q iterations are then averaged to give the final result.

 

Fig. 3 (a) An illustration of binning process in MC calculation of conductivity in a partially aligned SWNT film. Each cell is a rectangular cuboid extending along the gap. (b) SEM image of a typical dip and slip deposition method results for s-SWNTs on a SiO₂ substrate.

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There have been many different reports on mobility of CNTs in semiconducting SWNT samples [24,30,31]. Based on Eq. (5), it is found that the conductivity is approximately linearly affected by the mobility at lower frequencies (f<1MHz). Here the mobility itself is assumed to be affected by alignment of CNTs with the applied field and the local carrier density of the bin ρ_jk as below [30]:

In Eq. (7), u₁ and u₂ are two independent random variables, sampled from a uniform distribution between zero and one, and mobility has units of ${\text{m}}^2\text{/Vs}$. ρ_max is the maximum local carrier density in the total volume. Also, the average effective mass is affected by local carrier density ($m_{jk}^{-1}\approx (18-5{\rho }_{jk})m_e^{-1}$) as reported in [30]. An angular deviation of almost 15° is consistent with our observation under scanning electron microscope (SEM) (Fig. 3(b)). This alignment is realizable via the slip-stick method [34]. Alignment is assumed to affect the mobility so that the conductance can be approximated with simple classical formulation without concerns of CNTs directional conductance [35] (section 4).

Figure 4 depicts the peak photoconductance, so each point is the peak of a time varying photoconductance that has the same temporal profile as n(t) in the inset graph of Fig. 2(a).

 

Fig. 4 Photoconductivity for three cases of: perfectly aligned and purified, partially aligned and purified, and randomly deposited and partially purified s-SWNT.

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The lower limit (blue dotted line) is based on the random alignment and experimental verification in [19]. The higher limit shown with dotted red line is based on the experimental semiconductor SWNT mobility measurement of 10 ${\text{m}}^2\text{/Vs}$ reported in [30,31]. The MC integration result agrees closely with the DS model with mobility of 0.55 ${\text{m}}^2\text{/Vs}$ for lower carrier concentrations. The inset graph in Fig. 4 shows a close-up of the extra photoconductivity saturation effect due to mobility reduction with carrier density increase. This saturation is only due to phonon scattering effects, although further defects can also cause such behavior [30].

4. Modeling the photoconductance effect on output power

For both PC switches and THz photomixers the last step of the analysis that connects the input power to the output THz power is the photoconductance, which appears in the equivalent circuit model (Fig. 5 ).

 

Fig. 5 Schematic of the circuit model for the PC switch with s-SWNT in the gap. V_B is the DC bias voltage, R_L is the antenna resistance,V_c is the contact voltage, R_c and cʹ are contact resistance and capacitance, and C is the capacitance of the gap.

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Unlike two previous steps, this step is not affected by the physics of the material, but it is rather affected by the design of the electrodes and fabrication of the antenna itself. The effects of antenna structure and contact fabrication are some widely studied topics both in PC switches and in CNT film research [36–39]. We address the dynamics, rather than proposing an optimized design by focusing on parameters that affect and the output power. The design of the gap can affect the conductance; assuming the gap has dimensions of x, y, z depicted in Fig. 3(a), then the conductance can be calculated by classical definition of conductance$G(t)=({\sigma }_{dark}+{\sigma }_{photo}(t))yz/x$. This is the average conductance in the gap, since the misalignment and directional conductivity of CNTs [35] have already been considered in calculation of photoconductance.

Based on the equivalent circuit model in Fig. 5, the voltage V across both the time varying conductance G(t) and the contacts is given by the solution to the following set of coupled, first-order inhomogeneous linear differential equations;

The output THz power can be calculated directly from the voltage V(t) and the Kirchoff voltage law in the main loop of the circuit as:

Since carrier density is a linear function of input power as shown in Eq. (1), the analytic link between input power and output THz power, illustrated in Fig. 1(c), is completed at this point.

As an example, consider a design with typical gap dimensions of x = 5 µm, y = 5 µm and z = 100 nm; a common set of values for the parameters in Eq. (8) and Eq. (1) are assumed as follow: V_B = 36V [7,38,39], R_L≈30Ω [6,7], C≈10fF [6,7], cʹ≈100fF [40], R_c≈20kΩ [40,41], G₀≈2.5µΩ⁻¹ [41], $\eta (\nu )=\eta$ = 0.1 [12], pulse duration of 70fs, repetition rate = 76MHz, and the illumination wavelength of 1030nm. Using these values (some estimated relative to gap dimensions) and the photoconductivity values of Fig. 4, the THz output power can be calculated as a function of excitation power. In this calculation, n_abs is converted to average power with means of Eq. (1) and consideration of repetition rate of the excitation laser. As can be seen in Fig. 6 , the average output power for CNT is higher than GaBiAs PC switches [42], LT-GaAs PC switches and LT-GaAs photomixers [42–44]. Previous work reports a few microwatts of output THz power for GaBiAs, hundreds of nanowatts for LT-GaAs PC switches and tens of nanowatts for LT-GaAs photomixers, biased with the same voltage (36V) and excited with incident power of 10 to 30 mW. At 20mW of input power, the MC results for CNT’s THz output power show 66 fold improvement relative to GaBiAs and 95 fold increase relative to LT-GaAs PC switches. Due to lower efficiency in continuous wave photomixing, the average output power for the LT-GaAs photomixer hardly shows up on the scale; it is lower than the CNT output power by almost 3 to 4 orders of magnitude [44].

 

Fig. 6 Emitted average THz power vs average input excitation power. CNT curves are based on the conductivity values in Fig. 4. The GaBiAs PC switch and GaAs based photomixer output powers are inserted in 10, 20 and 30 mW of excitation power for comparison [42–44].

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Figure 6 reveals that GaBiAs and LT-GaAs PC switches perform better than randomly aligned and partially purified CNTs; the case that we have chosen as lower limit condition (blue dotted line in Fig. 6). Improvement in alignment and purification boosts the CNT THz power to around 103µW, calculated based on MC results for photoconductivity; the output power is then limited by the saturation effect induced by high carrier density and circuit dynamics at around 561µW in 36.6mW of input power. Figure 6 shows that curves for higher conductivity are distorted by circuit dynamics, and as a result the power reaches saturation for lower carrier densities compared to the conductivity in Fig. 4. This is the direct result of the nonlinear relation between these two parameters. Although SWNT films are well known for high thermal conductance [45], it must be considered that the experimental validation of this increase in THz power is also subject to the thermal stability of the device.

This specific example shows the emitted THz power range for the full spectrum of the pulse. Further dynamics of the output power relative to different parameters, however, can only be found by sweeping each parameter in the solution of Eq. (8). This is necessary, since each parameter can affect both power level and bandwidth. Via this method, it is found that the peak power changes approximately quadratically with V_B. This is consistent with previous models for fast semiconductor cases [6,7]. Also, the dynamics of the output power with variation of each parameter in Eq. (8) is presented in Fig. 7 .

 

Fig. 7 The dynamics of output THz power with: (a) antenna resistance, R_L (b) CNT to metal plate contance resistance, R_c (c) contance capacitace, cʹ (d) gap capacitance, C (e) photoconductance, G_photo (f) dark conductance, G₀ .

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Based on Fig. 7(a), P_THz has a maximum in certain value of R_L. Increasing R_L widens the pulse and thus reduces the bandwidth. This suggests that increase of R_L might be desired up to a certain limit (here 500Ω) with trade off for bandwidth of the emitted THz pulse. Figure 7(b) shows the exponential decrease of power with increase in contact resistance. This exponential decrease is the result of voltage drop on the contacts, and also, it depends on the relative value of contact capacitance and G₀. The dependence of P_THz on contact capacitance can be of interest for cases in which a capacitive coupling mechanism is unavoidable due to difficulties in fabrication of low resistance contacts [38–41]. This can be the case when the gap is wider than the average length of CNTs in the film. It is found that with variation of contact capacitance, the power changes between two constant values with an exponential transition (Fig. 7(c)). These two lower level and higher level powers are reached when cʹ impedance is considered a short circuit or open circuit compared to R_c. Additionally, it is seen that the higher value of cʹ decreases the bandwidth; this is highlighted with dashed orange line in Fig. 7(c). Such increase of the pulse bandwidth is expected based on circuit theory, and the low-pass behavior of a capacitive contact. The gap capacitance C can also play a similar role. It is seen that by increasing the gap capacitance, the bandwidth and amplitude of the output THz power is dramatically reduced (Fig. 7(d)). This can be a significant factor in the design of the PC switch structure, since$C={\epsilon }_{film}yz/x$, where${\epsilon }_{film}$ is the average permittivity of the CNT film placed in the gap.

The photoconductance (G_photo) and dark conductance (G₀) are also affected by the geometry of the gap. The variation of these two parameters also affects the terahertz power. As can be seen in Figs. 7(e) and 7(f), the power has a non-monotonic behavior with variations of G₀, while it increases exponentially with initial increase in G_photo and then saturates at higher values. The variation of G_photo in its typical µΩ⁻¹ range is too small to have a significant effect on the output pulse bandwidth.

Finally, it is worthwhile to mention that the circuit dynamics presented here can also be applied to the photomixing case in which G(t) is changed in Eq. (8) with a sinusoidal function of time.

6. Conclusion

Based on exceptional properties of SWNTs such as high mobility (up to 10 m².v⁻¹s⁻¹) [29–31], high thermal conductance [45], and high absorption in infrared region [33], SWNTs were previously proposed as an attractive material for efficient THz PC switching and photomixing [10,46]. Through this study, we established a theoretical link between the input excitation power on a PC switch made by a SWNT film as semiconducting material and its output THz power. We addressed the effect of each parameter. Based on the fast photocarrier dynamics of the SWNT nanomaterial, it was found that the photocarrier density temporal profile can be expressed as a hypergeometric function. The effect of key parameters such as mobility, carrier lifetime, pulse width, and exciton-exciton annihilation rate were explicitly addressed in this function. The upper and lower limits of the photoconductivity were calculated, based on the Drude-Smith theory, and were compared with the average expected value, determined using Monte Carlo integration. Finally, the dependence of the emitted THz power with variation of different PC switch design parameters such as: contact impedance, dark resistance, and antenna resistance was explored. We found that higher mobility in highly purified and aligned SWNTs can increase the THz power. The power may also be increased by other desired parameters such as smaller contact resistance and smaller gap capacitance for the microantenna. Based on a numerical example with typical values for different parameters and assumption of thermal stability of the device, approximately 2 orders of magnitude increase in CNT THz power are anticipated compared to conventional GaBiAs and LT-GaAs based PC switches.