1.

INTRODUCTION

The ultimate speed of electronic devices is the key issue for future communications and data processing systems. The 300 GHz communication links are already being actively developed. The pursuit of higher operating frequencies and shorter switching times focused attention on higher mobility semiconductors, such as Ge, InGaAs, InAs and, more recently, graphene. However, at short device features sizes typical for high speed devices, the conventional notions of the mobility determining the device speed and of the transit time dominated cutoff frequencies no longer apply. The quasi-ballistic and ballistic transport [1] starts playing the dominant role. The electron inertia and the viscosity of the two-dimensional electron fluid in the field effect transistor (FET) channels become very important. Our analytical estimates and detailed hydrodynamic simulations [2-4] reveal that the electron response becomes faster with the mobility increase only in a limited range of relatively low mobility values. With a further increase in the electron mobility, the plasmonic ringing determines the characteristic response time that becomes of the order of the momentum relaxation time. Therefore, the response time actually starts increasing proportionally to the mobility values up to the point, where this dependence saturates due to the dominant effect of the electron viscosity. The minimum response time and the maximum device modulation frequency correspond to the subpicosecond and terahertz ranges, respectively. The recent experiments of the FET switching using femtosecond optical laser pulses are in good agreement with the predicted sub picosecond switching times and demonstrate a larger sensitivity enhancement due to the constructive interference of the impinging THz pulse and the optical pulse field rectified by the device nonlinearity. We introduce two related measures of the transistor response. The first one is the characteristic time scale of the drain voltage induced by a quick step in the gate-to-source bias Figure 1 a. In our experiments, the application of a fast-varying gate-to-source bias is done by illuminating the device by a focused beam as schematically shown in Fig. 1 b. Figure 2 shows the expected response for low-mobility, high damping rate devices (the top panel) and the for high-mobility, low damping rate devices (the bottom panel.

2.

RESPONSE SPEED (DRIFT MODEL)

The drift model equations for the two-dimensional electron gas (2DEG) are given by

Here m* is the effective mass, q is the electron charge, τ_m is the momentum relaxation time, U is the channel potential, and v is the electron drift velocity. These equations are valid when 2πfτ_m ≪1 and 2πf_pτ_m ≪1, where f is frequency, f_m is the fundamental plasma frequency,

is the 2DEG concentration [5], U_gt is the gate voltage swing, k_B is the Boltzmann constant, η is the ideality factor, T is temperature, n₀ = Cηk_BT/q, C = ε₀ε/d = C_ch/WL is the gate-to-channel capacitance per unit area, C_ch is the gate-to-channel capacitance, ε₀ is the vacuum dielectric permittivity, ε is the dielectric constant. W is the gate width, L is the gate length, and d is the channel-to-gate separation.

Figure 1.

(a) Equivalent circuit diagram; (b) Space distribution of the electric field in the focused beam. Frequency is 1 THz. The size of a 10 μm wide HEMT is shown for comparison.

Figure 2.

The expected response for low-mobility, high damping rate devices (the top panel) and for the high-mobility, low damping rate devices (the bottom panel.

Figure 3.

Modulated output (a) and the definition of the maximum modulation frequency f_m (b).

The analysis of Eqs. (1-2) using the boundary conditions [6]

where j_d is the drain current density and μ = qτ_m/m* is the low field mobility, yields the following estimate for the ultimate response time for the above threshold regime (U_gt >> ηk_BT/q):

(see reference [7]). As seen, the ultimate response time is limited by the transit time of electrons filling or emptying the channel propagating with the velocity v = μU_gt/L. This estimate for τ is applicable if μU_gt/L ≪ v_s, where v_s is the electron saturation velocity. Interpolating this equation beyond the range of its applicability leads to the following estimate at large gate voltage swings:

Figure 4 compares the values of the response time for the voltage swing U_gt =0.1 V

Figure 4.

Comparison of response times τ and τ_s for U_gt = 0.1 V and v_s = 10⁵ m/s (a) and 3.5×10⁵ m/s (b).

The larger of these two response times τ and τ_s dominates. As seen, the response time could be in the sub picosecond range corresponding to the modulation frequency f_m = 1/2πτ in the sub-THz range. The major drawback of this model is that it does not account for the electron inertia and for the viscosity of the electronic fluid (traditionally referred to as 2DEG). These important effects could be accounted for in the frame of the hydrodynamic model.

3.

RESPONSE SPEED (HYDRODYNAMIC MODEL)

The equations of the hydrodynamic model, see, for example [2-4], are the continuity equation

the Naiver-Stokes equation

and the energy balance equation

Here c_v is the heat capacity, χ/c_v is the thermometric conductivity, and the electron energy is the sum of the internal energy and the drift energy m*v²/2, and γ is the viscosity of the 2DEG. The linear analysis of these equations using the unified charge control model (UCCM) leads to the following equations for the ultimate response time τ and plasma wave velocity S= (S_p² + S_ac²)^1/2:

Here s_p and s_ac are the electric and acoustic components of the plasma wave velocity. Above threshold (when the gate voltage swing U_gt >> k_B/T/q), the expression for the plasma velocity simplifies to become s = qU_gt/m*. The estimated viscosity of the 2DEG is γ ~ 15 cm²/s (comparable to that of castor oil or glycerin at room temperature) [8]. The analysis of eq. (11) reveals three distinguished regimes: (1) collision dominated transport; (2) ballistic transport, and (3) viscous transport (see Table 1). A more detailed analyses accounting on the differences between the momentum relaxation time and energy relaxation time and their dependence on energy will be presented elsewhere. Figure 5 showing the computed response time and the corresponding maximum modulation frequency illustrates the most important conclusions of the hydrodynamic model.

Figure 5.

The computed response time (a) [9] ©IEEE (2016) and the corresponding maximum modulation frequency (b) (from [2]

Table 1

Transport regimes of 2DEG

4.

SUMMARY OF EXPERIMENTAL RESULTS

The experimental study of the ultrafast response was presented in [10]. It was using a new technique based on the termination of the response by flooding the transistor by the electron hole plasma generated by the band-to-band optical pulse. These measurements confirmed the results of the hydrodynamic modeling predicting the ultra-fast transistor plasmonic response at the time scale much shorter than the electron transit time and revealed a large sensitivity enhancement (more that in 7 times) due to the constructive interference of the impinging THz pulse and the optical pulse field rectified by the transistor nonlinearity.

5.

CONCLUSIONS

Our analytical estimates and hydrodynamic simulations show that the electron response becomes faster with the mobility increase only for relatively low mobility values and/or relatively large feature sizes. For large mobility values, the plasmonic ringing determines the characteristic response time and, eventually, the dominant effect of the electron viscosity determine the response time. The minimum response time and the maximum device modulation frequency correspond to the sub picosecond and terahertz ranges, respectively. The recent experiments of the FET switching using femtosecond optical laser pulses are in good agreement with the predicted sub picosecond switching times and demonstrate a larger sensitivity enhancement due to the constructive interference of the impinging THz pulse and the optical pulse field rectified by the device nonlinearity.