Terahertz plasmonic detector controlled by phase asymmetry

I. V. Gorbenko; V. Y. Kachorovskii; Michael Shur

doi:10.1364/OE.27.004004

1. Introduction

The TeraFETs [1,2] – the devices converting the impinging terahertz (THz) radiation into a dc current by using the excitation of overdamped or resonant plasma oscillations have been implemented in Si [3,4], GaAs [5,6], GaN [7], graphene [8], and other materials systems. The TeraFETs are now being commercialized as tunable and fast detectors of sub-THz and THz radiation. The underlying physical mechanism proposed in Refs [1,2] is based on the rectification of plasmonic oscillations enabled by the device nonlinearity and the asymmetry of the setup. The latter determines the direction of the dc current response (or the sign of the induced dc voltage). One of the possible ways to introduce the asymmetry is to impose different boundary conditions on the source and drain of the device. In particular, when the source-gate input is excited by a signal with amplitude $U_{a}$ , while the drain current is fixed the generated open circuit dc voltage is

V \propto U_{a}^{2}

Such drain boundary condition implicitly implies that the external circuit is connected to the drain via an antenna or a contact pad with an infinite inductive impedance. Special “plasmonic stub” structures [13] might help implement such a boundary condition. Another option is to use identical antennas but apply signals with the different amplitudes

U_{a}

and

U_{b}

at the source and drain, respectively. For such a setup, the induced dc current is proportional to the difference of the squared voltages at the source and drain (assuming that these signals have the same phase) [9]

V \propto U_{a}^{2} - U_{b}^{2} .

Importantly, a strong asymmetry can be also induced by the phase shift between the THz signals at the source and drain. For equal amplitudes of signals on the source and drain

U_{a} = U_{b}

nonzero phase shift

θ

between signals induces the dc voltage [10–12]

V \propto U_{a}^{2} \sin θ .

In [10–12], the circularly-polarized radiation was used to induce such a phase shift. Remarkably, this shift depends very weakly on

L / λ

(here L is the channel length and

λ

is the THz radiation wavelength) and remains finite even in the limit of a homogeneous field. In this limit, the phase shift is simply given by geometrical angle

\pm θ

[see Fig. 1(a)] with the sign being determined by the clockwise or counterclockwise helicity.

Fig. 1 TeraFET Spectrometer principle of operation: (a) phase shift induced by asymmetric antennas and circularly polarized radiation (b) nonzero incident angle of incoming radiation

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Another possibility is to use the identical antennas and linearly polarized radiation but change the angle between incoming radiation and the plane of the device, see Fig. 1b. For such a setup, the phase shift $θ$ depends on $L_{e f f} / λ$ , where $L_{e f f}$ is determined by the source and drain antenna design and separation. For simplicity, we assume $L_{e f f} < < λ$ . In contrast, as we will show below, the plasmonic-related dependence of the response is governed by the parameter $L / λ_{p l}$ , where $λ_{p l}$ is the wavelength of the plasma wave. In particular, in the most interesting resonant regime, $L / λ_{p l} \sim 1$ , while the parameter $L_{e f f} / λ$ is, by assumption, much smaller than 1 and does not change essentially within the width of the plasmonic resonances. This allows one to neglect dependence of $θ$ (as well as $U_{a}$ which is usually taken as phenomenological parameters of the model [2]) on $L_{e f f} / λ$ and, consequently, on the radiation frequency, focusing on plasmonic-related effects.

Based on this physical idea, we show below that a single transistor can be used as spectrometer of the THz or sub-THz radiation. The presented results reveal the new physics of the TeraFET spectrometer and provide the design, characterization and parameter extraction tools for the new generation of the phase sensitive THz interferometers/spectrometers.

We will also discuss the homodyne detection scheme [14] enabled by the phase asymmetry. In this scheme, a strong local oscillator signal with the amplitude $U_{local}$ symmetrically excites the source and drain with respect to the gate, while the incoming radiation with the amplitude $U_{signal} ≪ U_{local}$ induces the phase-shifted (by the phase $θ$ ) signals on opposite sides of the channel. We will show that the dc response in this case reads

V \approx U_{local} U_{signal} [A (1 - \cos θ) + B s i n θ],

where coefficients A and B have qualitatively different frequency dependencies.

2. TeraFET spectrometer principle of operation

Figure 1 showing the TeraFET spectrometer structure illustrates its principle of operation. A THz radiation impinging on the FET couples to the two antennas at the opposite sides of the channel. The asymmetry is caused by the phase shift $θ$ between these antennas. This shift depends on the polarization of the radiation and the geometry of the setup. For a circular polarization, $θ$ is nonzero provided that antennas configuration is asymmetric with respect to direction from the source to drain. For a linear polarization, the finite phase shift $θ \propto Sin φ$ appears for a nonzero incidence angle $φ$ (the proportionality coefficient in this equation depends on the geometry detail). This phase shift enters the boundary conditions for the electron fluid in the transistor channel

{\begin{cases} U (0) = U_{g} + U_{a} \cos (ω t), \\ U (L) = U_{g} + U_{a} \cos (ω t + θ) . \end{cases}

Here $U (0)$ and $U (L)$ are the voltages at the source and drain of the channel, respectively, $U_{a}$ is the THz inducted voltage amplitude, $U_{g}$ is the gate-to-channel voltage swing (counted from the threshold voltage) and $ω$ is the round frequency of the impinging THz radiation. The 2D electron fluid in the device channel is described by the standard hydrodynamic equations

\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} + γ v = - \frac{e}{m} \frac{\partial U}{\partial x},

\frac{\partial U}{\partial t} + \frac{\partial (U v)}{\partial x} = 0.

Here v is the velocity of the fluid, U is the local value of the gate-to-channel voltage swing. related to the electron concentration in the channel:

n_{s} = C U / e,

_$γ$is the inverse momentum relaxation time, The solution of Eqs. (6)-(8) with the boundary conditions given by Eq. (5) is obtained using the same approach as in [2,12]

V = \frac{β ω U_{a}^{2} \sin θ}{4 U_{g} {| \sin (k L) |}^{2} ω^{2} + γ^{2}},

Here

β = 8 \sinh (\frac{Γ L}{s}) \sin (\frac{Ω L}{s})

,

k = (Ω + i Γ) / s

, and

Ω = \sqrt{\frac{\sqrt{ω^{4} + ω^{2} γ^{2}}}{2} + \frac{ω^{2}}{2}}, Γ = \sqrt{\frac{\sqrt{ω^{4} + ω^{2} γ^{2}}}{2} - \frac{ω^{2}}{2}} .

Here k is the wave vector, ω is frequency,

Ω

is the plasma frequency,

Γ

is the effective damping rate,

s = \sqrt{e U_{g} / m}

is the plasma wave velocity, e is the electron charge, and m is the effective mass. In the resonant regime

ω ≫ γ, s / L ≫ γ

these equations simplify

Ω \approx ω, Γ \approx \frac{γ}{2}, Ω ≫ Γ,

and one finds a sharply-peaked response at the resonant frequencies

ω_{N} = π N s / L :

V \approx \frac{4 δ ω γ {(- 1)}^{N}}{4 U_{g} (δ ω^{2} + γ^{2} / 4)} U_{a}^{2} \sin θ,

where

δ ω = ω - ω_{N} ≪ ω_{N} .

We notice an asymmetric form of the resonances,

V (δ ω) = - V (- δ ω)

, in contrast to the symmetric resonances obtained under the open drain boundary conditions [2].

Both in the resonant and non-resonant case, Eq. (9) shows a periodic variation with frequency. In particular, the response exactly turns to zero for $Ω = Ω_{N} = π s N / L$ due to factor $\sin (Ω L / s)$ in the coefficient $β$ . These results enable the application of a TeraFET as a spectrometer. This spectrometer operates as follows. At each frequency, the gate-to-source voltage should be adjusted till the response is zero at every incidence angle. This yields the values of the frequency satisfying the following condition

\sqrt{\frac{\sqrt{ω_{N}^{4} + ω_{N}^{2} γ^{2}}}{2} + \frac{ω_{N}^{2}}{2}} = \frac{π s}{L} N .

Here N = 1, 2, 3…. From Eq. (13), we find dependence of

ω_{N}

on the gate voltage

ω_{N} = \frac{{(\frac{π s N}{L})}^{2}}{\sqrt{{(\frac{π s N}{L})}^{2} + \frac{γ^{2}}{4}}} = \frac{\frac{e π^{2} N^{2} U_{g}}{m L^{2}}}{\sqrt{\frac{e π^{2} N^{2} U_{g}}{m L^{2}} + \frac{γ^{2}}{4}}} \propto {\begin{cases} U_{g}, for U_{g} ≪ U_{γ}^{N} \\ \sqrt{U_{g}}, for U_{g} ≫ U_{γ}^{N} \end{cases},

where

U_{γ}^{N} = m L^{2} γ^{2} / 4 e π^{2} N^{2} .

Hence, with increasing the gate voltage, all frequencies

ω_{N}

shift to higher values. The dependence of

ω_{N}

on the gate voltage qualitatively changes from the linear dependence in the non-resonant case (a small gate voltage,

ω_{N} ≪ γ

) to the square route dependence in the resonant case (a large gate voltage,

ω_{N} ≫ γ

).

For a monochromatic signal with a frequency $ω$ , one can tune $ω_{N}$ to be equal to $ω$ by changing the gate voltage. This allows to measure $ω$ . For a more general case of the radiation with a spectrum broadened around $\bar{ω}$ within a certain interval $Δ ω$ with the wave amplitude given by $U_{a} (ω)$ , the dc response can have found by replacing in Eq. (9) $U_{a} \to U_{a} (ω)$ and integrating over $ω$ . Tuning the resonant frequency $ω_{N}$ to cover the interval $\bar{ω} \pm Δ ω$ by the gate voltage allows to extract the spectral density $| U_{a} (ω) |^{2}$ .

3. Homodyne detector operation scheme

Let us now assume that. in addition to the phase-shifted signal of the incoming radiation, we apply a strong fully symmetric signal of a local oscillator. This situation can be modeled by the following boundary conditions (see Fig. 2)

Fig. 2 Homodyne detector operation scheme

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{\begin{cases} U (0) = U_{l o c a l} \cos ω t + U_{s i g n a l} \cos ω t, \\ U (L) = U_{l o c a l} \cos ω t + U_{s i g n a l} \cos (ω t + θ) . \end{cases}

These equations can be rewritten in the form used in Ref [12].

\begin{array}{l} U (0) = U_{a} \cos ω t, \\ U (L) = U_{b} \cos (ω t + \tilde{θ}), \end{array}

where parameters

U_{a}, U_{b}

and

\tilde{θ}

can be found from the following equations

U_{a} = U_{l o c a l} + U_{s i g n a l}, U_{b} e^{i \tilde{θ}} = U_{l o c a l} + U_{s i g n a l} e^{i θ}

, From these equations, we find that in the presence of the local oscillator, the phase asymmetry of the incoming signal induces an effective asymmetry of the amplitudes. For

U_{l o c a l} ≫ U_{s i g n a l},

we find

U_{a}^{2} - U_{b}^{2} \approx 2 U_{l o c a l} U_{s i g n a l} (1 - \cos θ)

. We also find

U_{a} U_{b} \sin \tilde{θ} \approx U_{l o c a l} U_{s i g n a l} \sin θ

. The calculations analogous to [2,12] yield

\begin{array}{l} {\begin{cases} A = \frac{ω [(1 + \frac{γ Ω}{Γ ω}) s i n h^{2} (\frac{Γ L}{s}) - (1 - \frac{γ Γ}{ω Ω}) s i n^{2} (\frac{Ω L}{s})]}{2 U_{g} {| s i n (k L) |}^{2} ω^{2} + γ^{2}}, \\ B = \frac{2 ω}{U_{g} {| s i n (k L) |}^{2} ω^{2} + γ^{2}} s i n h (\frac{Γ L}{s}) s i n (\frac{Ω L}{s}) . \end{cases} \end{array}

In the resonant regime

ω ≫ γ, s / L ≫ γ

, we find from Eqs. (4) and (17)

V \approx \frac{U_{l o c a l} U_{s i g n a l}}{2 U_{g}} \frac{(3 γ^{2} / 4 - δ ω^{2}) (1 - \cos θ) + 2 {(- 1)}^{N} γ δ ω \sin θ}{(δ ω^{2} + γ^{2} / 4)} .

As seen, there are two contributions to the response having qualitatively different frequency dependencies in vicinity of plasmonic resonances: the symmetric term, which is proportional to

(1 - c o s θ)

and the asymmetric term, which is proportional to

\sin θ

. For

δ ω \to \infty

response remains finite. Extracting value of response at

δ ω = \infty,

we get

V (δ ω) - V (\infty) = \frac{U_{l o c a l} U_{s i g n a l}}{2 U_{g}} F (\frac{δ ω}{γ})

where function

F (x) = \frac{1 - \cos θ + 2 {(- 1)}^{N} x s i n θ}{x^{2} + 1 / 4}

is plotted in Fig. 3.

Fig. 3 Resonant dependence of dimensionless response on the radiation frequency [Eq. (20)] for fundamental plasmonic frequency (N = 1) at different $θ$ ( $θ / π = 0.1, 0.2, 0.3, 0.4, 0.5$ ) increasing from bottom to the top at negative x.

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As seen from the above calculations, the response depends on the damping rate $γ$ . In particular, in the resonant regime, the asymmetric part of the response, taken at fixed $δ ω,$ turns to zero for $γ \to 0$ [see Eqs. (12) and (18)]. On the other hand, resonant response can be written as a dimensionless function of dimensionless parameter $δ ω / γ$ [see Eqs. (19) and (20)] This means that the amplitude of the resonant response does not depend on $γ,$ which yields additional tool for analyzing the experimental results and comparing them with the theory. In particular, extracting from static I-V curves dependence of $γ$ on temperature $T$ , one can measure the response for different temperatures and plot it as a function of $δ ω / γ (T) .$ The plotted curves for different temperatures should coincide.

4. Calculation of response for various materials

Next, we present the response calculations for different materials. Figures 4-8 show the calculation results for p-diamond FETs, Si NMOS, AlGaN/GaN and InGaAs/InP HEMTs, respectively, for the channel lengths of 25 nm, 65 nm, and 130 nm. We present in these figures the normalized response $R = V U_{g} / U_{a}^{2} \sin θ .$ Table 1 lists the parameters used in the calculation. Scattering rates in the table are taken for room temperature.

Fig. 4 Diamond TeraFET normalized response for U_g = 0.1 V and L = 130 nm (a) L = 65 nm (b) and L = 25nm (c).

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Fig. 5 Silicon TeraFET normalized response for 130 nm (a), 65 nm (b) and 25nm (c) channel lengths and U_g = 0.1 V.

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Fig. 6 AlGaN/GaN TeraFET normalized response for 130 nm (a), 65 nm (b) and 25nm (c) channel lengths and U_g = 0.1 V

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Fig. 7 InGaAs/GaAs TeraFET normalized response for 130 nm (a), 65 nm (b) and 25nm (c) channel lengths and U_g = 0.1 V.

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Fig. 8 Normalized response for different values of the gate voltage (Vg = 0.1 V, 0.2 V and 0.3 V) for 250 nm p-diamond (a) and 130 nm silicon (b) FETs.

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Table 1. Materials parameters

View Table

As seen from Fig. 4, the p-diamond FETs should enable room temperature spectroscopy in the sub-THz range. Other materials systems also offer unique capabilities for THz spectroscopy. We also see that the responses for all these materials strongly depend on frequency even for the parameters corresponding to room temperature. In Fig. 8, we show that increasing the gate voltage shifts nodes of the response to higher values in agreement with Eq. (14). Compared to a conventional TeraFET detector, the approach based on the phase induced asymmetry is much easier to control because it eliminates the issues related to the frequency dependent input, output, and load impedances, since it could be realized using identical antennas at the source and drain of a TeraFET. Even silicon TeraFETs seem to be a reasonable option, which allows to interface the TeraFET with inexpensive standard VLSI processing and data acquisition hardware. The results for p-diamond [diamond] confirm a big potential of this material for applications in the 240-320 GHz and 500 to 600 GHz atmospheric windows (the potential frequency ranges for beyond 5G).

For the resonant homodyne regime of operation the response is given by Eqs. (19) and (20) where the values of $γ$ for different materials are given in Table 1 for room temperature.

5. Conclusions

To conclude, we developed a theory of the phase-asymmetry-induced dc photoresponse in the FET subjected to THz radiation. We found that the response has sharp resonant peaks in vicinity of the plasmonic resonances. The peaks have an asymmetric shape as a function of frequency. We also discussed the homodyne operation with the response being drastically increased by using a strong local oscillator signal. In this case, the response contains two terms with symmetric and asymmetric frequency dependencies. We present detailed calculations of the response for different materials with different mobilities and find sufficiently relaxed conditions for realization of the resonant response. The results of this work can be used for developing compact, tunable spectrometers of the THz radiation.

Funding

U.S. Army Research Laboratory through the Collaborative Research Alliance for Multi-Scale Modeling of Electronic Materials; Office of Naval Research; Foundation for the Advancement of Theoretical Physics “BASIS”; and Russian Foundation for Basic Research (17-02-00217).

References

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7. A. El Fatimy, S. Boubanga Tombet, F. Teppe, W. Knap, D. B. Veksler, S. Rumyantsev, M. S. Shur, N. Pala, R. Gaska, Q. Fareed, X. Hu, D. Seliuta, G. Valusis, C. Gaquiere, D. Theron, and A. Cappy, “Terahertz detection by GaN/AlGaN transistors,” Electron. Lett. 42(23), 1342–1343 (2006). [CrossRef]

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10. C. Drexler, N. Dyakonova, P. Olbrich, J. Karch, M. Schafberger, K. Karpierz, Yu. Mityagin, M. B. Lifshits, F. Teppe, O. Klimenko, Y. M. Meziani, W. Knap, and S. D. Ganichev, “Helicity sensitive terahertz radiation detection by field effect transistors,” J. Appl. Phys. 111(12), 124504 (2012). [CrossRef]

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Material	Parameters
Material	*Effective mass, m_r*	*Mobility (m²/Vs)*	*Scattering Frequency, γ (10¹²/s)*
p-diamond	0.74	0.53	0.45
n-Si	0.19	0.10	9.3
n-GaN	0.24	0.15	4.9
n-InGaAs	0.041	0.8	5.4

Terahertz plasmonic detector controlled by phase asymmetry

Abstract

1. Introduction

2. TeraFET spectrometer principle of operation

3. Homodyne detector operation scheme

4. Calculation of response for various materials

5. Conclusions

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (20)

Optics Express