text:
\section{Introduction}

\begin{figure*
\begin{center}
\includegraphics[width=12.0cm]{GL_PGL_FET_DET_Fig1.eps}
\end{center}
\caption{Schematic views of the split-gate GL-FET (a) and PGL-FET (c) structures
and their band digrams (b) and (d),  respectively (at  unbiased p-n junction).
}
\end{figure*}
The lateral p-n-junctions in field-effect transistor (FET)  structures with a graphene layer (GL) channel  and split gates~\cite{1,2,3,4,5,6}  can be used in new electron and optoelectronic devices. For example, the tunneling electron-transit
diodes for generating terahertz radiation (THz)~\cite{7,8}, interband THz photodetectors~\cite{9,10,11,12,13,14,15,16,17,18}, and injection THz lasers~\cite{19,20,21} have been proposed and analyzed.
Recently~\cite{22}, the THz emission from split-gate GL-FET structures with forward biased p-n- junctions was observed.
Similarly, graphene nanoribbon (GNR) and graphene bilayer (GBL) FET structures with p-n junctions can be used in different devices~\cite{23,24,25,26,27,28}.
The nonlinearity of the current-voltage characteristics of the GL~\cite{1,4,26,27,28,29}
 and GBL p-n-junctions~\cite{30,31,32,33} can be used for detecting  THz radiation using the effect
 of the current rectification and for the THz generation using the tunneling negative differential conductivity.
 Since the gated regions
of p- and n-types form the plasmonic cavities~\cite{34}, the excitation of the plasmonic oscillations
(the  oscillations of the electron or hole  density and the self-consistent electric field) by incoming
THz radiation can substantially affect the detector characteristics due to the plasmonic resonances.
An idea to use the plasmonic properties of the electron (or hole)  system in the channel of a FET
for the resonant THz detection  was brought forward a long time ago~\cite{34}.
The key points of this idea are: (1) the resonant excitation of the plasmonic oscillations  and (2)
the rectification of the ac current is due to the hydrodynamics nonlinearity of the  electron system dynamics
in the channel. Later, the concept of FET-based THz detectors   was successfully realized and
such detectors based on III-V compound materials with the enhanced characteristics were manufactured
and used in applications~(see, for example,~\cite{35,36,37,38,39}).
The concepts of using the combination of plasmonic effects and other types of the current  nonlinearity
(stronger than the hydrodynamic nonlinearity) were also considered.

Due to unique electron properties of GLs, GNR, and  GBLs~\cite{40,41,42,43} and a remarkable progress
in fabrication of different transistor heterostructures based on GLs,
such FETs can be used
for  THz detectors (and frequency multipliers) with enhanced performance~\cite{44,45,46,47}.
The advantages of the GL-based heterostructures are associated with the enhanced mobility
at room   temperature (about 110,000~cm$^2$/V s in the epitaxially grown GLs on SIC substrate~\cite{43}).
This leads to  a high quality factor
of  the plasmonic oscillations at room temperature and  high frequencies of the plasma modes in relatively long channels
due to the elevated plasma-wave velocity~\cite{48}.
Besides higher room-temperature  quality factors compared to those for the FET-detectors based on the standard materials,
the symmetry of the hole and electron energy spectra provides an opportunity to
realize coherent coupled plasmonic oscillations in the p- and n-sections of the device.
The vertical GL-based structures can also be used for the resonant plasmonic THz detection~\cite{49,50}.
The GNR  transistor structures~\cite{23,51} can serve the basis for the THz detectors using different operational
principles~\cite{14}, including the resonant
absorption of THz radiation~\cite{53,54} leading to the rectification and bolometric effects.
High quality factor of the plasmonic oscillations in GL-heterostructures,
opens up the prospect of realization of the room-temperature  THz detectors
exhibiting pronounced resonant behavior due to the plasmonic effects with very high peak responsivity and
strong spectral selectivity.

In this paper, we  evaluate the proposed resonant THz detector based on lateral
split-gate GL-FET and split-gate FET with the GL perforated between the gates (PGL-FET) in which
the channel is partitioned into the p- and n-type sections.

The operation of such
detectors is enabled  by the  nonlinearity of the p-n-junction and the excitation of coupled plasmonic oscillations in the gated channel sections. We demonstrate that the responsivity of the GL-FETs under consideration as function of the incoming THz radiation frequency
can exhibit the resonant response. However,  the height  of the spectral dependence responsivity  peaks
  and their sharpness are rather moderate. This is because of relatively high  conductance of the p-n junction in the GLs associated with the interband tunneling
can be compared with the conductance of the channel. In contrast, in the PGL-FETs
the tunneling mechanism can be effectively suppressed by appropriate choice of the energy gap in the perforated portion of the channel (perforated depletion region of the p-n junction). Apart from this, the thermionic conductance of the p-n junction is also
limited by the channel nanoconstrictions. As a result,
the  net conductance of the p-n junction can be much smaller than that of the gated channel
sections that is beneficial for the achievement of pronounced spectral selectivity and elevated peak responsivity.

\section{Device model}

In  FETs with undoped  GL-channel and  with the split gates, the formation of the lateral p-n junction is  realized
by applying  voltages  $V_p$ and  $V_n$ between the source
and the neighboring gate section (source gate section)  and between the  drain and the drain gate section.
Figure~1 shows the  GL- and PGL-FETs, which can be used for the THz detection and their  band diagrams at properly chosen  applied gate voltages.

The main feature of the PGL-FETs is the energy band gap in the depletion region
caused by the GL perforation substantially affecting
the conductivity, nonlinear, and plasmonic   properties of  the p-n-junction (as in FETs based on  heterostructure made of the standard materials~\cite{55,56}). This region can be considered as an array of the nanoconstrictions or GNRs similar to the GNR channels of FETs~\cite{51}  and bolometric detectors~\cite{52}.
The source-drain voltage comprises the dc and ac signal components: $V = V_0 + \delta V_{\omega}\exp(-i\omega\,t)$ is applied between the side contacts,
where $V_0$ is the dc bias voltage in the case of  photocurrent detection and the dc signal voltage produced by the incoming radiation in the photovoltaic regime, $\delta V_{\omega}$ and $\omega$
are the amplitude and frequency of the  ac signal received by an antenna.

 %
We assume that  the steady-state hole and electron densities in the GL channel,
$\Sigma_0^{+}$ and $\Sigma_0^{-}$,
 are equal to each other: $\Sigma_0^{\pm} = \Sigma_0$.
 If, in addition,   both the gates have similar length,
 $L_g$, and due to the  symmetry of the hole and electron energy spectra,  the plasmonic properties of the p- and n-channel sections are similar.
 In this case,
 the plasmonic oscillations in the p- and n-channel sections can be synchronized.

If the length of the gates $L_g$ and  the net  length of the channel  $L$   ($L_g \leq L$)  are  markedly larger than  the gate layers
thickness $W_g$,  one can apply
 the  gradual channel
 approximation~\cite{57}.
 Assuming   that
the two-dimensional hole system (2DHS) in the p-section and two-dimensional electron system (2DES) in the n-section of the GL channel  are degenerate,
one can write down the following equations for the hole (in the p-section) and electron
(in the n-section) density in the undoped channel $\Sigma_0$ (at zero Dirac point):

\begin{equation}\label{1}
\Sigma_0 = \frac{\kappa(eV_g - \varepsilon_F)}{4\pi\,e^2W_g}.
\end{equation}
Here we set $V_p = - V_g <0$, $V_n = V_g >0$ $e = |e|$ is the electron charge, $\kappa$ is the dielectric constant of the gate layer material, and $\varepsilon_F$ is the hole and electron  Fermi energy in the pertinent channel sections.
The Fermi energy is   equal to $\varepsilon_F = \hbar\,v_W\sqrt{\pi\Sigma_0}$, where
 $v_W \simeq 10^8$~cm/s
is the characterisctic velocity of electrons and holes in GLs,  and $\hbar$ is the Planck constant.
The  term with $\varepsilon_F$ in the right-hand side of  equation~(1) is associated with
the effect of the quantum
capacitance~\cite{58}.
The hole and electron systems degeneracy implies that $\varepsilon_{F} \gg T$,
 where $T$ is the temperature (in the energy units).

Consequently, the small-signal variations of the hole density $\delta \Sigma_{\omega}^{+}$
and the electron density $\delta \Sigma_{\omega}^{+}$ caused by the ac variations of the
electric potential in the channel $\delta \varphi_{\omega}^{\pm}$ are given by

\begin{equation}\label{2}
\delta\Sigma_{\omega}^{\pm} = \mp \frac{C}{e}  \delta \varphi_{\omega}^{\pm}.
\end{equation}
Here
$C = C_{g}C_{quant}/(C_{g} + C_{quant})$  is the net gate-GL capacitance (per unit area) accounting for
the geometrical and quantum capacitances~\cite{57}, $C_{g} = (\kappa/4\pi\,W_g$),
$C_{quant} = 2e^2\sqrt{\Sigma_0}/\sqrt{\pi}\hbar\,v_W$ (in GLs~\cite{58}),
i.e., $C = (\kappa/4\pi\,W_g)[1 + (\kappa\hbar^2v_W^2/8e^2W_g\varepsilon_F)]^{-1}$.

\section{Thermionic and tunneling conductance and parameter of nonlinearity}

The ac source-drain  current is determined by the admittances  of the p- and n- sections of the channel and the p-n-junction.
The net ac  current density through the p-n-junction  $\delta j_{\omega}^{pn}$ (the current per unit length in the direction along the gate edges) at the signal frequency $\omega$, associated with the ac voltage between the source and drain contacts induced by the incoming radiation, includes  the ac  interband  tunneling
 and
thermionic components $\delta j_{\omega}^{tunn}$ and $\delta j_{\omega}^{th}$, respectively,
and the displacement component $\delta_{\omega}^{disp}$:

\begin{equation}\label{eq3}
\delta j_{\omega}^{pn} = \delta j_{\omega}^{th} + \delta j_{\omega}^{tunn} + \delta j_{\omega}^{disp}.
\end{equation}
The latter term in Eq.~(3) is given by

\begin{equation}\label{eq4}
\delta j_{\omega}^{disp} = -i \omega c^{pn}(\delta\varphi_{\omega}^{+}|_{x = -l} -
\delta\varphi_{\omega}^{-}|_{x = +l}),
\end{equation}
where $c^{pn} \simeq (\kappa/2\pi^2)\ln (2L_g/l)$ is the geometrical capacitance of the lateral p-n-junction
under consideration,  $2l$ being the length of the  depletion region (which is comparable with
the spacing between the gate edges) in the p-n-junction (see, for example,~\cite{61}),
$\delta\varphi_{\omega}^{+}|_{x = -l}$  and
$\delta\varphi_{\omega}^{-}|_{x = +l})$ are the ac electric potentials of the quasi-neutral p- and n-sections  at their edges
 $x=-l$ and $x = +l$, respectively, and the axis $x$ is directed from source to drain.

\begin{figure}[t
\centering
\includegraphics[width=8.0cm]{GL_PGL_FET_DET_Fig2.eps}
\caption{Spatial distributions of the   ac potential amplitudes  for different
 $\xi$ (schematic view) at the signal frequency $\omega = \Omega_g$. Inset shows an example of the spatial dependence
of the potential at low frequencies ($0 \leq \omega \ll \nu, \Omega_g$).
}
\end{figure}

The  ac current density associated with the tunneling and thermionic processes caused by the ac signal can be presented as

\begin{equation}\label{eq5}
\delta j_{\omega}^{th} +\delta j_{\omega}^{tunn}
 \simeq g^{pn}(\delta\varphi_{\omega}^{+}|_{x = -l} -
\delta\varphi_{\omega}^{-}|_{x = +l})\\
= g^{pn}\delta V^{pn}_{\omega}.
\end{equation}
The the variation of the dc  current density , $\overline{ \Delta j }
=   \overline{j}  - j_0 $, arising from the  averaging of the nonlinear components
of the ac current density  and the ac voltage drop across the p-n-junction over the fast oscillations (the rectified current and voltage components),  is given by

\begin{equation}\label{eq6}
\overline {\Delta j } =  \beta^{pn}
\overline{|\delta\varphi_{\omega}^{+}|_{x = -l} - \delta\varphi_{\omega}^{-}|_{x = +l}|^2} =
\beta^{pn} \overline{|\delta V^{pn}_{\omega}|^2}.
\end{equation}
In Eqs.~(6) and (7),
\begin{equation}\label{eq7}
g^{pn} =  \frac{d j_0}{d \varphi}\biggr|_{V =V_{0}},\qquad
\beta^{pn} =  \frac{1}{2}\frac{d^2 j_{0}}{d\varphi^2}\biggr|_{V = V_{0}},
\end{equation}
 where  $j_0$ is the dc current through the p-n junction at the dc source-drain voltage
$V_0 $ (bias voltage or radiation induced voltage) . The quantities $g^{pn}$ and
$\beta^{pn}$  are the p-n-junction  differential conductance (i.e., the   real part of its net conductance
$Y_{\omega}^{pn} = g^{pn}-i\omega c^{pm}$)  and  the parameter  of nonlinearity  of the p-n junction current-voltage characteristics (leading to the rectification and frequency doubling effects), respectively.
The latter formulas are valid when
 the space-charged (depletion) region of the p-n junction is sufficiently  narrow, so that the characteristic  time of electron and hole transit across this region is small in comparison with the inverse signal frequency $\omega^{-1}$,  In the opposite case, for example, at sufficiently strong the reverse bias, so that $l$ can be relatively large, $g^{pn}$ and $\beta^{pn}$ can be
 the frequency dependent  (i.e., $g^{pn} = g_{\omega}^{pn}$ and $\beta^{pn} =\beta_{\omega}^{pn}$) due to the electron and hole transit-time effects~\cite{7,8}
 (see Section VIII).
Considering both the interband tunneling and the thermionic contributions, one can obtain

 \begin{equation}\label{eq8}
g^{pn} =  g^{th}( 1 + \eta), \qquad \beta^{pn}= \beta^{th}(1 + \zeta).
\end{equation}
Here the first and the second terms in equation~(9)  describe the contributions of the tunneling and thermionic processes, respectively. Considering equations~(A5), (A6), (A9), and (A10) in the Appendix A, for the GL-FETs $g^{th} = g^{GL}$ and
$\beta^{th} = \beta^{GL}$  we arrive at

\begin{equation}\label{eq9}
g^{GL} =\biggl(\frac{e^2}{\pi^2\hbar}\biggr)\biggl(\frac{4T}{\hbar\,v_W}\biggr)
\exp\biggl(-\frac{\varepsilon_F}{T}\biggr),
\end{equation}

\begin{equation}\label{eq10}
\beta^{GL} = \biggl(\frac{e^2}{\pi^2\hbar}\biggr)\biggl(\frac{2e}{\hbar\,v_W}\biggr)
\exp\biggl(-\frac{\varepsilon_F}{T}\biggr) = \frac{eg^{GL}}{2T},
\end{equation}
where $T$ is the electron and hole temperature in the channel (which we equalize to the lattice temperature) in the energy units,  The quantities $\eta = \eta^{GL}$ and $\zeta
= \zeta^{GL}$ describe
 the relative contributions of the intersection tunneling (see the Appendix A).
 In the GL-FETs,

\begin{equation}\label{eq11}
\eta^{GL} = \sqrt{\frac{\hbar\,v_W}{8\pi\,l^*\varepsilon_F}}\biggl(\frac{\varepsilon_F}{T}\biggr)\exp\biggl(\frac{\varepsilon_F}{T}\biggr),
\end{equation}
\begin{equation}\label{eq12}
\zeta^{GL} = \sqrt{\frac{\hbar\,v_W}{32\pi\,l^*\varepsilon_F}}\exp\biggl(\frac{\varepsilon_F}{T}\biggr) = \eta^{GL}\biggl(\frac{T}{2\varepsilon_F}\biggr),
\end{equation}
where $l^*$ is the effective width of the p-n junction.
Deriving equations~(11) and  (12), we have accounted for that the GL p-n-junction built-in voltage $V_{bi} = 2\varepsilon_F/e$, set for simplicity the built-in electric field in the p-n-junction depletion region equal to $E^{bi} = V^{bi}/\pi\,l^* = \varepsilon_F/el^*$ with $l^* \geq l$ considered as a phenomenological parameter (see the Appendix A) and assumed that the dc voltage is created only by the rectification effect, i.e., it is small
 (the photo-voltaic detection).

In the PGL-FETs, the densities of the displacement, hole, and electron currents
are spatially periodic in the direction along the gate edges. Considering the PGL-FETs with  the period of the perforation of the depletion region $D \ll L_g $, we average
these current densities, conductances, and the  parameter of nonlinearity over short-range periodical spatial variations.
Taking this into account, for the PGL-FETs  the quantity $g^{th} = g^{PGL}$ averaged over the
periodic spatial variations is presented as (see the Appendix B)

\begin{multline}\label{eq13}
  g^{PGL}  = \frac{4e^2Nb}{\pi\hbar}\exp\biggl(-\frac{\varepsilon_F+\Delta_g/2}{T}\biggr)\\
 = B\exp\biggl(-\frac{\Delta_g}{2T}\biggr)g^{GL},
 \end{multline}

 \begin{multline}\label{eq14}
\beta^{PGL}\simeq  \frac{e^2J_s^{PGL}}{2T^2} \simeq \biggl(\frac{2e^3N}{\pi\hbar\,T}\biggr)
\exp\biggl(-\frac{ \varepsilon_F + \Delta_g/2}{T}\biggr)\\
=B\exp\biggl(-\frac{ \Delta_g}{2T}\biggr) \beta^{GL}.
 \end{multline}
Here the quantity
$\varepsilon_F + \Delta_g/2$ is the activation energy of the thermionic transitions between the channel sections through the channel nanoconstrictions between the perforations (GNRs),
   $\Delta_g = 2\pi\hbar\,v_W/d$ is the energy gap in these nanoconstrictions, $d$ is their  width, $N$ is the number of the nanoconstrictions per unit length, $0.5 <b < 1$ is a numerical factor, which is determined by the shape of the energy barrier in the
   nanoconstrictions ($b = 1$ and $b=0.5$ for the sharp trapezoidal and smooth barriers, respectively), and  $B =\pi\,Nb\hbar\,v_W/T$ is a small parameter describing the limitation of the current across the p-n-junction due to the nanoconstrictions. If $N = (1 -10)~\mu$m$^{-1}$ and $b= 0.75$, at room temperature
   one obtains $B \simeq 0.055 - 0.555$.

  Since the incorporation of the nanoconstrictions is aimed to
 substantially  decrease the p-n-junction conductance  in comparison with the conductances of the gated sections of the channel, we assume in the following that
 the length of GNRs $l^*$ and their energy gap $\Delta_g$ are sufficiently large, so that the hole and electron tunneling through them is suppressed. In line with this,
 for PGL-FETs we set $\eta^{PGL}, \zeta^{PGL} \ll 1$ (see the Appendix B).

\section{Low-frequency GL-FET and PGL-FET detector responsivity}

The current (in A/W units) and voltage (in V/W units)  detector responsivities,
$R_{\omega}$ and ${\cal R}_{\omega}$, to the incoming signals with the frequency $\omega$
are defined as

\begin{equation}\label{eq15}
R_{\omega} = \overline{\Delta J}/SI_{\omega}   , \qquad {\cal R}_{\omega} = \overline{\Delta V}/SI_{\omega}.
\end{equation}
Here $\overline{\Delta J} = \overline{\Delta j} D$ is the variation of the dc current
associated with the current rectified component, $D$ is the lateral size of the device in the direction along the gate edges,
$\overline{\Delta V}$ is the rectified dc voltage between the side contacts induced  by the received  signals,
which is somewhat smaller than the rectified voltage component across the p-n junction $\overline{\Delta V}^{pn}$,
hence, in the photovolataic regime,
$V_0 = \overline{\Delta V}$. The quantities  $S = \lambda_{\omega}^2G/4\pi$, $I_{\omega}$ and $\lambda_{\omega} = 2\pi\,c/\omega$
are  the wavelength
of the
incident THz  radiation  and its intensity.the intensity, where  $c$ is the speed of light  in vacuum and $G \sim 1.5$ is the antenna gain.
 The quantities $(\delta V_{\omega})^2$  and $I_{\omega}$ are related to each other as
$(\delta V_{\omega})^2 = 4\lambda_{\omega}^2I_{\omega} /\pi\,c$.

The quantities $\overline{\Delta J}$ and $\overline{\Delta V}$ are proportional to
$\overline{|\delta V^{pn}_{\omega}|^2}$, which, in turn, is determined by
the spatial distributions of the ac potentials  $\delta\varphi_{\omega}^{\pm}
= \delta\varphi_{\omega}^{\pm}(x)$.
The latter is found taking into account
that

\begin{equation}\label{eq16}
\delta \varphi_{\omega}^{\pm}|_{x = \mp L} = \mp\frac{\delta V_{\omega}}{2},
\end{equation}

At relatively low signal frequencies ($\omega \rightarrow 0$), the hole and electron densities as well as
the electric potential manage to follow the variations of the potential at the contacts.
In this case,
the ac potentials in the gated channel sections $\delta \varphi_{\omega}^{\pm} = \varphi_{\omega}^{\pm}(x)$ at $l \leq |x| \leq L$ are  linear functions of the coordinate $x$  along the channel:

\begin{equation}\label{eq17}
\delta\varphi_{0}^{\pm} = \mp\frac{\delta\,V_{0}}{2(1 + r_0)}
\biggl[
1 \mp r_0\frac{(x \pm l)}{L_g}\biggr]
\end{equation}
 with

\begin{equation}\label{eq18}
\delta \varphi_{0}^{\pm}|_{x = \mp l} =
\mp\frac{\delta V_{0}}
{2(1 + r_0)}.
\end{equation}
Here $r_0 = 2L_gg^{pn}/\sigma_0$ is the ratio of the p- and n- junction differential conductance $g^{pn}$ and
the net dc conductance of the p- and n-sections $\sigma_0/2L_g$ ($L_g \simeq L -l$, see Fig.~1)
with
\begin{equation}\label{eq19}
\sigma_{\omega} = \frac{e^2\varepsilon_F}{\pi\hbar^2}\frac{i}{(\omega + i\nu)},
\end{equation}
so that $\sigma_0 = (e^2\varepsilon_F/\pi\hbar^2\nu)$,
where $\nu$ is the electron and hole collision frequency in the pertinent section associated with the scattering on impurities and acoustic phonons.
A schematic view of the potential distribution at low frequencies is shown (for $r_0 =0.5$) in the inset in Fig.~2.

As a result,
the quantities $\overline{|\delta V^{pn}_{0}|^2}$ and $\overline{\Delta j} $
given by Eq.~(6) are  equal to

\begin{equation}\label{eq20}
\overline{|\delta V^{pn}_{\omega}|^2} = \frac{(\delta V_{\omega})^2}
{2(1 + r_0)^2}, \qquad \overline{\Delta J} =  \frac{D\beta^{pn}(\delta V_{\omega})^2}
{2(1 + r_0)^2}.
\end{equation}

As seen from Eq.~(20), it is desirable that the quantity  $r_0   \ll 1$, i.e.,
the conductance of the gated sections of the channel markedly exceeds  the differential conductance  of the p-n-junction,

Equations ~(15) and (20) yield the following formula for the low-frequency current responsivity:

\begin{equation}\label{eq21}
R_{0}^J = \frac{8\beta^{pn}}{cG(1 + r_0)^2}= \frac{4e^2J_S(1 + \zeta)}{cGT^2(1 + r_0)^2}.
\end{equation}

Considering equation~(21) and using equations~(9), (10), and (12) for $\beta^{GL}_s$ and $\zeta^{GL}$, we find for the low-frequency current responsivity of the GL-FET detectors

 \begin{equation}\label{eq22}
 R_{0}^{GL} = \frac{0.0118}{G}\frac{eD}{\hbar\,v_W}\frac{(1 + \zeta^{GL})}{(1 + r_0^{GL})^2}\exp\bigg(-\frac{\varepsilon_F}{T}\biggr),
 \end{equation}

The low-frequency current responsivity of the PGL-FET detectors is given by

 \begin{multline}\label{eq23}
 R_{0}^{PGL} = \frac{0.0372}{G}\frac{eDN}{T}\frac{(1 + \zeta^{PGL})}{(1 + r_0^{PGL})^2}\exp\bigg(-\frac{\varepsilon_F+ \Delta_g/2}{T}\biggr)\\
 \simeq \frac{0.0372}{G}\frac{eDN}{T}\exp\bigg(-\frac{\varepsilon_F+ \Delta_g/2}{T}\biggr).
 \end{multline}
 %
In equations~(21) and (22),  $r_0^{GL} = 2L_gg^{GL}(1 + \zeta^{GL})/\sigma_0 \leq 1$
and $r_0^{PGL} = 2L_gg^{PGL}(1 + \zeta^{PGL})/\sigma_0 \ll 1$, respectively,
with $\zeta^{GL} > 1$ (or $\zeta^{GL} \gg 1$) and $\zeta^{PGL} \ll 1$.
Due to a relatively large value of $\zeta^{GL}$ (because of the contribution  of the tunneling in the GL-FETs) and  large ratio of   the "thermionic" exponential  factors
in equations~(22) and (23), the current responsivity of the GL-FET detectors substantially
exceeds that of the PGL-FET detectors.

In the photo-voltaic regime, the current density $\overline{\delta j}$ is compensated by
the dc current caused by the induced dc voltage $\overline {\Delta V}$. Considering this and taking into account that the net channel dc resistance $r_{ch} = 2L_g/\sigma_0 + 1/g^{pn}  = (1 + r_0)/g^{pn}$, we obtain

\begin{equation}\label{eq24}
\frac{g^{pn}\overline{\Delta V}}{(1 + r_0)} = \frac{\beta^{pn}(\delta V_{0})^2}{2(1 + r_0)^2}.
\end{equation}
Hence, the dc voltage induced by the THz radiation between the source and drain contact is equal to

\begin{equation}\label{eq25}
\overline {\Delta V} = \frac{2\beta^{pn}}{g^{pn}(1 +r_0)}\frac{\lambda_{\omega}^2I_{\omega}}{\pi\,c}.
\end{equation}

Considering this and using equations~(20), for the voltage responsivity
we obtain following formula valid for the detectors of both types:

\begin{equation}\label{eq26}
{\cal R}_{0} \simeq  \frac{8\beta^{pn}}{g^{pn}cG(1 + r_0)}.
\end{equation}
Consequently,
\begin{equation}\label{eq27}
{\cal R}_{0}^{GL} \simeq
\overline{{\cal R}}_0^{GL}\biggl(\frac{1 + \zeta^{GL}}{1 + \eta^{GL}}\biggr)
\cdot{\cal P}_0^{GL},
\end{equation}
with $\overline{{\cal R}}_0^{GL} = \displaystyle\frac{4e(1+r_0^{GL})}{cGT}$
 and
\begin{equation}\label{eq28}
{\cal R}_{0}^{PGL} \simeq
\overline{{\cal R}}_0^{PGL}\biggl(\frac{1 + \zeta^{PGL}}{1 + \eta^{PGL}}\biggr)
\cdot{\cal P}_0^{PGL}
\end{equation}
with $\overline{{\cal R}}_0^{PGL} = \displaystyle\frac{4e(1+r_0^{PGL})}{cGT}$,
where ${\cal P}_0^{GL/PGL}= (1 + r_0^{GL/PGL})^{-2}$.

 As follows from equation~(27),
the enhancement of the tunneling probability (an increase of the quantity $ \eta^{GL}$ in the GL-FETs
results in a decrease in the  ${\cal R}^{GL}_{0}$. Thus the tunneling through the p-n junction plays a negative role.
This is because the tunneling component of the p-n junction conductance  can exceed the thermionic component at small $l^*$ providing relatively low p-n junction resistance. A rise of the tunneling conductivity $g^{tunn}$ is accompanied by a relatively slow
increase in  the tunneling  nonlinearity parameter $\beta^{tunn}$.

At the realistic parameters,
$\eta^{GL} \gg 1$, so that $\zeta^{GL} =(T/2\varepsilon_F)\eta^{GL}$.
Taking into this condition, for the low-frequency voltage responsivities we obtain

\begin{equation}\label{eq29}
{\cal R}_{0}^{GL} \simeq
\frac{4e}{cGT}\biggl(\frac{T/2\varepsilon_F}{1+r_0^{GL}}\biggr),\qquad {\cal R}_{0}^{PGL} \simeq
\frac{4e}{cGT}\frac{1}{1(+r_0^{PGL})}.
\end{equation}
The estimates for the low-frequency responsivities of the GL-FET and PGL-FET detectors
at room temperature, $\varepsilon_F = 50$~meV,  $r_0^{GL} \leq 1$,
$r_0^{PGL}, \eta^{PGl}, \zeta^{PGL} \ll 1$ (in the PGL-FETs with sufficiently long nanoconstrictions and properly chosen
their  the energy gap) yield
${\cal R}_0^{GL}  \leq 1\times 10^3$ V/W ($r_0^{GL} \leq 1$)
and
 ${\cal R}^{PGL}_0 \simeq 5\times 10^3$~V/W, respectively.

\section{Plasmonic oscillations in  GL- and PGL-FETs}

At higher frequencies, the reactive components of the two-dimensional hole system (2DHS)  and two-dimensional electron system (2DES)   ac conductivity $\sigma_{\omega}$
(which correspond to the inertia of the electron and hole motion~\cite{61}) , the p-n-junction conductance, and the gate-channel capacitances become more important
because they can subtantially affect the spatio-temporal distributions of the ac potential in the different portions of the channel.
In this situation, one needs to determine $\delta \varphi_{\omega}^{\pm}|_{\mp}
= \delta \varphi_{\omega}^{\pm}|_{\mp} (x)$
accounting for these factors.
We find the spatial
distributions of $\delta\varphi_{t}^{\pm}(x)$   in the  GL plane (along the axis $x$) and, in particular ,
$\delta \varphi_{\omega}^{\pm}|_{\mp} $,   from the
linearized kinetic~\cite{48} or  the hydrodynamic equations for the gated 2DHS and 2DES~\cite{62,63} (adopted for  the energy spectra of the electrons and holes in GLs) coupled with the Poisson equation in the gradual channel approximation, i.e., equation~(2). As a result, we arrive at the following system equations:

\begin{equation}\label{eq30}
\frac{d^2\delta\varphi_{\omega}^{\pm}}{dx^2}
+\frac{\omega(\omega + i\nu)}{s^2}\delta\varphi_{\omega}^{\pm}
=  0,
\end{equation}
Here
$s = \displaystyle \sqrt{\frac{e^2\varepsilon_F}{\pi\,C\hbar^2}} \geq
\sqrt{\frac{4e^2W_g\varepsilon_F}{\kappa\hbar^2}} \propto \Sigma_0^{1/4}$
is
the characteristic plasma-wave velocity in the gated GL structures~\cite{48,61}.
The boundary conditions for Eqs.~(29) and (30) are given by Eq.~(16) (at the side contacts $x = \mp L$ and the following conditions at $x = \mp l$:

\begin{equation}\label{eq31}
-\sigma_{\omega}\frac{d\,\delta\varphi_{\omega}^{\pm}}{dx} \biggr|_{x = -l}  = \pm\delta j_{\omega}^{pn},
\end{equation}
where  $\delta j_{\omega}^{pn}$  is given by equation~(3) and the following ones and  we $l = L - L_g \ll L_g, L$.
Writing down equation~(30), we have disregarded nonlinear terms associated with the nonlinearity
of the 2DES and 2DHS dynamics (in particular, the nonlinearity of the hydrodynamic equations for the 2DHS and 2DES - we have used their linearized versions!).
This is because,  we focus on the GL-FET operation associated with another and stronger
nonlinearity (nonlinearity of the p-n-junction current-voltage characteristics).

Equation~(30)  with the boundary conditions ~(16) and (31)  yield the following formulas for the spatial distributions
of ac potentials $\delta\varphi_{\omega}^{\pm}$:

\begin{multline}\label{32}
\delta\varphi_{\omega}^{\pm} = \mp\frac{\delta V_{\omega}}{2}\biggl\{\cos[\ae_{\omega}(x \pm L)]\\
+ \frac{\xi_{\omega}\sin(\ae_{\omega}L_g) - \cos(\ae_{\omega}L_g)}{\xi_{\omega}\cos(\ae_{\omega}L_g) + \sin(\ae_{\omega}L_g)}\sin[\ae_{\omega}(x \pm L)]\biggr\}
\end{multline}
for the range $-L \leq x \leq -l$ for $\delta\varphi_{\omega}^{+}$,  and
$l  \leq x \leq L$ for $\delta\varphi_{\omega}^{+}$. Here the wavenumber $\ae_{\omega}$
and the characteristic plasma frequency for the gated
2DES and 2DHS $\Omega_g$ are given by

\begin{equation}\label{33}
\ae_{\omega} = \frac{\pi\sqrt{\omega(\omega + i\nu)}}{2\Omega_gL_g}, \,\,\,\, \Omega_g = \frac{\pi\,s}{2L_g}= \sqrt{\frac{\pi\,e^2\varepsilon_F}{4L_g^2\,C\hbar^2}}.
\end{equation}
The parameter $\xi_{\omega} =\sigma_{\omega} \ae_{\omega}/2Y_{\omega}^{pn}
= (\sigma_{\omega} \ae_{\omega}/2g^{pn}/(1 - i\omega\tau)$, where $\tau =c^{pn}/g^{pn}$
is the p-n junction recharging time,   characterizes the ratio of the channel and p-n junction conductivities. It  is given by

\begin{equation}\label{34}
\xi_{\omega}  =
i\sqrt{\frac{\omega}{\omega + i\nu}}
 \frac{ \xi}{(1 -i\omega\tau)}.
\end{equation}
At low frequencies, equation~(32) reduces to equation~(16).

For the GBL-FETs, the quantities  $\xi = \xi^{GL}$ and $\tau = \tau^{GL}$ are equal to

\begin{equation}\label{eq35}
\xi^{GL} = \frac{\pi}{16(1 + \eta^{GL})e}\sqrt{\frac{\kappa\varepsilon_F}{W_g}}
\frac{\hbar\,v_W}{T}
\exp\biggl(\frac{\varepsilon_F}{T}\biggr),
\end{equation}
\begin{equation}\label{eq36}
\tau^{GL} = \frac{\kappa\hbar}{8(1 +\eta^{GL})T}\biggl(\frac{\hbar\,v_W}{e^2}\biggr)
\ln\biggl(\frac{2L_g}{l}\biggr)\exp\biggl(\frac{\varepsilon_F}{T}\biggr).
\end{equation}
The parameter $\eta^{GL}$ given by in equations~(11) varies in a wide rage depending on the effective widths of the depletion region $2l^*$, the Fermi energy $\varepsilon_F$, and the temperature $T$.
At $T = 300$~K, setting $l^* = 5 - 25$~nm and $\varepsilon_F = 25 - 75$~meV,
we obtain $\eta \simeq 0.54 - 15.85$. When  $\eta^{GL}~\gg~1$, from equation~(34) we arrive at
$\xi^{GL}
\simeq \displaystyle\frac{\pi^{3/2}\sqrt{\hbar\,v_W}}{2^{5/2}e}\sqrt{\frac{\kappa\,l^*}{W_g}}
 \simeq 0.648\times\sqrt{\frac{\kappa\,l^*}{W_g}}$.
 For $\kappa = 4 -20$ and $l^*/W_g = 0.5 - 5.0$, one obtains $\xi^{GL} \simeq 0.9 - 6.5$. As seen in the following, the latter parameter  essentially determines the character of the plasmonic effect.

\begin{figure}[t
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig3.eps}
\caption{Frequency dependences of plasmonic factor ${\cal P}_{\omega}^{GL}$
and normalized GL-FET detector  responsivity ${\cal R}_{\omega}^{GL}/\overline {{\cal R}}_{0}^{GL}$
for  $\nu = 0.5\times10^{12}$~s$^{-1}$ and $L_g = 325$~nm: solid lines- $l^* = 25$~nm,  $W_g/\kappa = 1$~nm,
 and $\Omega_g/2\pi = 0.86$~THz  and dashed lines -
$l^* = 10$~nm, $W_g/\kappa = 2.5$~nm,
 and $\Omega_g/2\pi = 1.17$~THz.
}
\end{figure}
\begin{figure}[h
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig4.eps}
\caption{The same as in figure~2, but for  $\nu = 1.0\times10^{12}$~s$^{-1}$.
}
\end{figure}

Due to the smallness of $\tau$, the factor $(1 - i\omega\tau)^{-1}$ in Eq.~(33) can be significant at
very high frequencies.
Indeed,
at  $L_g = 250~$nm,  $l^* =10$~nm, $W_g = 10$~nm,
$\kappa = 4 - 20$, and $\varepsilon_F = 50$~meV,
one obtains
$\tau^{GL} \simeq (0.05 -0.25)\times 10^{-13}$~s.
Hence, even for $\omega/2\pi = 5$~THz, one obtains  $\omega\tau^{GL} \simeq 0.15 - 0.75$.

For PGL-FETs, the quantities $\xi = \xi^{PGL}$
and $\tau = \tau^{PGL}$ for the parameter $\xi_{\omega}$ given by equation~(33) [compare with equations~(35) and (36)], are as follows:

\begin{equation}\label{eq37}
\xi^{PGL} = \frac{\pi}{16Be}\sqrt{\frac{\kappa\varepsilon_F}{W_g}}
\frac{\hbar\,v_W}{T}
\exp\biggl(\frac{\varepsilon_F + \Delta_g/2}{T}\biggr),
\end{equation}

\begin{equation}\label{eq38}
\tau^{PGL} = \frac{\kappa\hbar}{8BT}\biggl(\frac{\hbar\,v_W}{e^2}\biggr)
\ln\biggl(\frac{2L_g}{l}\biggr)\exp\biggl(\frac{\varepsilon_F + \Delta_g/2}{T}\biggr).
\end{equation}

Setting $\kappa = 4$,$W_g = 8$~nm,   $d = 10$~nm ($\Delta_g \simeq 200)$~meV),
$N = 10~\mu$m$^{-1}$, $b = 0.75$, $L_g/l = 10$
 $\varepsilon_F = 25 - 75$~meV,
 we obtain
 $\xi^{PGL} \simeq 116.47 - 1487$
 and $\tau^{PGL} \simeq (4.35 - 32.17)\times 10^{-12}$~s.
Thus, $\xi^{PGL}$ is fairly large. Apart from this, the recharging time, $\tau^{PGL}$,
of the perforated p-n junction in the PGL-FETs is relatively long (in comparison with $\tau^{GL}$) due to a small  p-n junction conductance.
Figure~2 schematically shows the amplitudes of the ac potential as a function of the coordinate $x$ along the channel in the devices under consideration with different values of the parameter $\xi$ ($\xi^{GL}$ or $\xi^{PGL}$). The radiation frequency
$\omega$ is chosen to be equal to $\Omega_g$ with $\nu = \pi\Omega_g/40$ and $l/L = 10$. This corresponds to the maximum span of the plasmonic oscillations. One can see from Fig.~(2) that an increase  in $\xi$
results in a substantial increase in the plasmonic oscillations, which, in turn,
provides an elevated ac potential drop across the the depletion region.
An example of the potential distribution  shown in the inset in Fig.~2 and described by  equations~(16) and (17) corresponds
to relatively low frequencies ($0 \leq \omega \ll \nu, \Omega$).

\section{Effect of plasmonic oscillations on the GL-FET and PGL-FET detectors responsivity }

Equations~(31) and~(6) yield [compare with equation~(20)]

\begin{equation}\label{39}
\overline{|\delta V_{\omega}^{pn}|^2}
= \frac{(\delta V_{\omega})^2}{2}{\cal P}_{\omega},\qquad \overline{ \Delta J } = \frac{D\beta^{pn}(\delta V_{\omega})^2}{2} {\cal P}_{\omega},
\end{equation}
where the factor
\begin{equation}\label{eq40}
 {\cal P}_{\omega} =
\biggl|\frac{\xi_{\omega}}{\sin(\ae_{\omega} L_g) +\xi_{\omega} \cos(\ae_{\omega}L_g)}\biggr|^2
\end{equation}
describes the effect of plasmonic oscillations.
In the range of low frequencies $\omega$, the plasmonic factor ${\cal P}_{\omega}$  tends to ${\cal P}_{0} = (1 + r_0)^{-2} $, and the quantities in equation~(38)  coincide with
those given by equation~(20).
As a result, for the PGL-FET detectors we can use the following formula:

\begin{equation}\label{eq41}
{\cal R}_{\omega}^{GL} \simeq \overline{{\cal R}}_{0}^{GL}\,{\cal P}_{\omega}^{GL},
\qquad {\cal R}_{\omega}^{PGL} \simeq \overline{{\cal R}}_{0}^{PGL}\, {\cal P}^{PGL}_{\omega}
.
\end{equation}

\begin{figure}[t
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig5.eps}
\caption{Spectral dependences of   normalized GL-FET detector responsivity
${\cal R}_{\omega}^{GL}/\overline {{\cal R}}_{0}^{GL}$ calculated for different hole and electron Fermi energies  (
$\nu = 0.5\times10^{12}$~s$^{-1}$,
$L_g = 325$~nm, $l^* = 25$~nm,  $W_g/\kappa = 1$~nm).
}
\end{figure}

As follows from equations~(40) and (41), the plasmonic factors  ${\cal P}_{\omega}^{GL}$
and ${\cal P}_{\omega}^{PGL}$, as well as
the pertinent  responsivities ${\cal R}_{\omega}^{GL}$
and
${\cal R}_{\omega}^{PGL}$
 are generally  oscillatory functions
of $\omega/\Omega_g$.
The oscillations of ${\cal P}_\omega{GL}$ and  ${\cal R}_{\omega}^{PGL}$  can be pronounced, i.e.,  the pertinent maximuma are high and sharp  if the
 quality factor
$Q = 4\Omega_g/\pi\,\nu$  is sufficiently large.

Figures~3 and 4 show the frequency dependences of the plasmonic factor
${\cal P}_{\omega}{GL}$ calculated using equations~(40) and (41)  with equations~(33) - (36)
for $\nu = 1.0\times10^{12}$~s$^{-1}$,
 and $\nu = 0.5\times10^{12}$~s$^{-1}$.
It is assumed that $L_g = 325$~nm, $\varepsilon_F = 50$~meV,
$l^* = 25$ and  $W_g/\kappa = 1.0$ for solid lines and   $l^* =10$~nm
and $W_g/\kappa = 25$~nm for dashed lines.
In particular, if $\kappa = 10$
(SiC or MoS$_2$ gate layer) and $\kappa = 20$ (Hf0$_2$ gate layer),
the solid lines with  pronounced maxima,
correspond to $W_g = 10$ and $W_g = 20$~nm, respectively. The solid lines in figures~2 and 3 can also be attributed to
the devices with  hBN or SiO$_2$ gate layers of the thickness
$W_g \simeq 4$~nm.

Figure~5  shows the normalized GL-FET detector responsivity as a function of the signal frequency
calculated for different  Fermi energies of holes and electrons, i.e., their different
densities for  the same other parameters corresponding to
the solid lines in figure~2.

The pronounced resonant peaks in figures~3 - 5 correspond to the frequencies
$\omega \simeq (2n-1)\Omega_g$,  where $n = 1,2,3,...$ is the plasmonic mode index.

Larger values of the quality factor $Q$ correspond to higher  peaks  of the plasmonic factor and the GL-FET detector responsivity.  In particular,  solid lines correspond from  $Q =  13.76$  (in figure~3) and $Q = 6.88$
 (in figure~4). However, the comparison of the resonant peaks
 in figures~3 and 4 shows that an increase in the quality factor (twice) leads only to a modest
increase in the peak heights.
 %
The point is that  the spectral behavior of
${\cal P}_{\omega}^{GL}$ and  ${\cal R}_{\omega}^{GL}$ is also determined by the parameter
$\xi^{GL}$ characterizing the ratio of the p-n junction and the channel resistances.
In particular, the data in Figs.~3 and 4 correspond to fairly moderate values of $\xi^{GL} \propto \sqrt{\kappa\,l^*/W_g}$:
$\xi^{GL} = 2.189$ for solid lines and $\xi^{GL} = 1.296$ for dashed lines.
This is the manifestation of the effect of relatively high conductance of the p-n junction in GL-FETs, which drops when $l^*$ decreases.
The role of this effect also  reveals in the fact that
${\cal R}_{0}^{GL}/\overline{{\cal R}}_{0}^{GL}< 1$ seen in figures~2~-~4.

\begin{figure}[t
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig6.eps}
\caption{Spectral dependences of   normalized PGL-FET detector responsivity
${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_{0}^{PGL}$ with $\Delta_g = 200$~meV
and different hole and electron Fermi energies at
 $\nu = 0.5\times10^{12}$~s$^{-1}$,
$L_g = 325$~nm,  $W_g/\kappa = 1$~nm, and $\kappa = 4$.
}
\end{figure}

\begin{figure}[t
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig7.eps}
\caption{The same as in figure~6 but for $\nu = 1.0\times 10^{12}$~s$^{-1}$
(different vertical axis scale).
}
\end{figure}

 In the PGL-FETs the parameters $\xi = \xi^{PGL}$ and $1/r_0^{PGL}$ can be very large due to relatively small the p-n junction conductance $g^{PGL}$ and negligible contribution of the tunneling mechanism ($\eta^{PGL}, \zeta^{PGL} \ll 1$).

Figures~6 and 7 show the spectral dependences of the normalized  responsivity
${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_0^{PGL}$ of PGL-FET detectors
with different hole and Fermi energy energy $\varepsilon_F$
 the energy gap in the constrictions $\Delta_g = 200$~meV ($d = 10$~nm), and their density $N = 10~\mu$m$^{-1}$
calculated using equations~(39) and (40)
with equations~(36) and (37).
It is assumed that $\nu = 0.5\times 10^{12}$~s$^{-1}$ and $\nu = 1.0\times 10^{12}$~s$^{-1}$, $L_g = 325$~nm, $L_g/l = 10$,  $W_g/\kappa =1$~nm, $\kappa = 4$, and
  $b = 0.75$ ($B = 0.555$). The lines for $\varepsilon_F = 50$~meV   in figure~5 and in figure~6
  correspond to the same parameters of the gated
sections in the PGL- and GL-FETs. Comparing figures~6 and 7, one can see
that the height of the resonant responsivity peaks markedly drops with increasing hole and electron collision frequency $\nu$.
This is also seen from figure~8, which
demonstrates how  the spectral dependence of the PGL-FET-detector normalized
 ${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_0^{PGL}$ varies when the $\nu$ changes. As seen, when $\nu$ increases threefold, the height of the first resonant peak ($n = 1$) drops nine times. Such a roll-off corresponds to
 ${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_0^{PGL} \simeq Q^2 \propto \nu^{-2}$ [see the solid and dashed lines in the inset in figure~8].

\section{Comparison of GL- and PGL-FET detectors}

The variation of $\Delta_g$in rather wide range $\Delta_g = 100 -300$~meV at fixed $\varepsilon_F$ does not lead to a marked change in
the spectral dependences of ${\cal R}_{\omega}^{PGL}/\overline {{\cal R}_0^{PGL}}$.

It is clearly seen that the plasmonic response of the PGL-FETs is much more pronounced
than that of the GL-FETs (at similar parameters of the gated channel sections).

\begin{figure}[t
\centering
\includegraphics[width=7.0cm]{GL_PGL_FET_DET_Fig8.eps}
\caption{Transformation of spectral dependence of PGL-FET-detector normalized
responsivity ${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_{0}^{PGL}$ with varying collision frequency $\nu$ and $\varepsilon_F = 75$~meV, other parameters are the same as in figures~6 and 7.
Solid  line in inset gives the   maximum (peak) value of ${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_{0}^{PGL}$ as a function of  collision frequency; dashed line corresponds to max~${\cal R}_{\omega}^{PGL}/\overline {{\cal R}}_{0}^{PGL} = Q^2 \propto \nu^{-2}$.
}
\end{figure}

Equation~(40) for  the plasmonic factor at the frequencies $\omega = (2n-1)\Omega_g$
when  $Q > 1$ yields

\begin{equation}\label{eq42}
{\cal P}_{(2n-1)\Omega_g} \simeq \frac{\xi^2Q^2}{(\xi + Q)^2 + (2n-1)^2Q^4 (\pi\nu\tau/4)^2}.
\end{equation}
At  $\omega = 2n\Omega_g$, one obtains

\begin{multline}\label{eq43}
{\cal P}_{2n\Omega_g} \simeq \frac{\xi^2Q^2}{Q^2[\xi+ 2n(\pi\nu\tau/4)]^2 + 1}\\
\simeq  \frac{\xi^2}{[\xi+ 2n(\pi\nu\tau/4)]^2}.
\end{multline}
Hence, for both GL- and PGL-FETs, ${\cal P}_{2n\Omega_g} < 1$.
Taking into account equation~(40),
for the responsivity  at $\omega = (2n-1)\Omega_g$,
one can obtain

\begin{multline}\label{eq44}
\frac{{\cal R}^{GL}_{(2n-1)\Omega_g}}{ \overline{{\cal R}}_0^{GL}}\simeq
\biggl(\frac{1 + \zeta^{GL}}{1 + \eta^{GL}}\biggr)\\
\times\frac{(\xi^{GL}Q)^2}{[(\xi^{GL} + Q)^2 + (2n-1)^2Q^4 (\pi\nu\tau^{GL}/4)^2]}
\end{multline}
for GL-FETs, and

\begin{equation}\label{eq45}
\frac{{\cal R}^{PGL}_{(2n-1)\Omega_g}}{\overline{{\cal R}}_0^{PGL}}
\simeq  \frac{(\xi^{PGL}Q)^2}{[(\xi^{PGL} + Q)^2 + (2n-1)^2Q^4 (\pi\nu\tau^{PGL}/4)^2]}
\end{equation}
for PGL-FETs.

Using equation~(44) and taking into account a long rechrging time $\tau^{GL}$ in the GL-FETs,
for their responsivity in the fange of several THz, we find

\begin{multline}\label{eq46}
\frac{{\cal R}^{GL}_{(2n-1)\Omega_g}}{ \overline{{\cal R}}_0^{GL}}\simeq
\biggl(\frac{1 + \zeta^{GL}}{1 + \eta^{GL}}\biggr)
\biggl(\frac{1}{\xi^{GL}} + \frac{1}{Q}\biggr)^{-2}\\
\simeq \biggl(\frac{T}{2\varepsilon_F}\biggr)\biggl(\frac{1}{\xi^{GL}} + \frac{1}{Q}\biggr)^{-2}.
\end{multline}
Thus, the height of the GL-FET detector responsivity maxima is determined by
min$\{\xi, Q\}$. Substituting to equation~(46) the data used for the dependences shown
in figures~3 and 4, we naturally obtain the same values of the peak height as in these plots.

Since the quantity $\xi^{PGL}$ can be much larger than $Q$,for the low-index resonances
equation~(45) can be simplified:

\begin{multline}\label{eq47}
\frac{{\cal R}^{PGL}_{(2n-1)\Omega_g}}{\overline{{\cal R}}_0^{PGL}}
\simeq  \frac{Q^2}{1 + \displaystyle(2n-1)^2Q^4\biggl(\frac{\pi\nu\tau^{PGL}}{4\xi^{PGL}}\biggr)^2}\\ \leq Q^2.
\end{multline}
The dependence ${\cal R}^{PGL}_{(2n-1)\Omega_g}/\overline{{\cal R}}_0^{PGL} \simeq
Q^2$ at the realistic parameters is also confirmed by the proximity of the solid and dashed lines in the inset in figure~8.

Comparing equations~(46) and (47), one can find that the  responsivity peaks of the GL- and PGL-FET detectors (1) exhibit rather different height and sharpness and (2) change
variously  with increasing
$\varepsilon_F$.

First, assuming that in reality  $\xi^{GL}  \leq Q \ll \xi^{PGL} $, disregarding for simplicity the "recharging" terms in the denominators
of equations~(44) and (45), taking into account equations(29),
we find
$$
\frac{{\cal R}^{PGL}_{(2n-1)\Omega_g}}{{\cal R}^{GL}_{(2n-1)\Omega_g}} \simeq
\biggl(\frac{2\varepsilon_F}{T}\biggr)^2\biggl(\frac{Q}{\xi^{GL}}\biggr)^2 \gg 1.
$$

Second,
${\cal R}^{GL}_{(2n-1)\Omega_g}$ decreases with increasing
$\varepsilon_F$, while ${\cal R}^{PGL}_{(2n-1)\Omega_g}$ increases.
Indeed, according to Eq.~(45) with Eqs.~(11) and (34) at large values of
$\varepsilon_F$ and $Q$, one obtains ${\cal R}^{GL}_{(2n-1)\Omega_g} \propto
(T/\varepsilon_F)^{3/2}$. Such a decreasing dependence on $\varepsilon_F$ is associated with both a decrease in the tunneling nonlinearity parameter and in an increase in the p-n junction conductance.
In contrast, ${\cal R}^{PGL}_{(2n-1)\Omega_g} \propto Q \propto \Omega_g \propto \sqrt{\varepsilon_F}$. Different behavior of the resonant peaks clearly  seen comparing figures~5 and (6). A  sensitivity of the responsivity maxima to the Fermi energy  implies an opportunity to optimize the PGL-FET detector characteristics.

The peaks heights can markedly roll-off with increasing resonance index $n$, especially
if $\nu\tau^{PGL}$ is not too small. One needs to mention that
$\tau^{PGL}/\xi^{PGL} \propto 1/\sqrt{\varepsilon_F} $ and
 the second term in the denominator in equation~(45) is approximately proportional to $\varepsilon_F$, whereas both these quantities are  independent of $\Delta_g$.
 At very high Fermi energies, the dependence
 ${\cal R}_{\omega}^{PGL}/\overline{{\cal R}}_0^{PGL}$  versus $\varepsilon_F$ saturates and results in the limit
 $$
 \frac{{\cal R}^{PGL}_{(2n-1)\Omega_g}}{\overline{{\cal R}}_0^V } \rightarrow \frac{1}{(2n-1)^2}
 \biggl(\frac{L_g}{2W_g}\biggr)^2.
$$

\section{Comments}

 In the above calculations we disregarded possible frequency dispersion of the p-n junction conductance, which actually contain the factor $J_0(\omega\,t_{tr}/2)$, where
 $J_0(\tau)$ is the Bessel function and $t_{tr} = l/v_W$ (in the GL-FETs) and $t_{tr} = (l/v_W)\sqrt{\Delta_g/2T}$ (in the PGL-FETs) are the transit times across the depletion region.
Such a transit-tme effect  might be important at the frequencies commensurable
with the inverse hole and electron time when  the argument of the Bessel function $J_0(\omega\,t_{tr}/2)$
is  about of 2 or more).
 At $\omega/2\pi = 1 - 3$~THz, $l=  25$~nm,
and $\Delta_g = 200$~meV, one obtains $\omega\,t_{tr} =  0.15 - 0.45$.
Hence, the assumption that Re~$g^{pn}$ is independent of the frequency used in our model is valid.

 The incident THz radiation heats the 2DHS and 2DES due to the Drude and the interband absorption. The variation of the hole and electron effective temperature  can lead to the dc source-drain current if the device is asymmetric.
The latter can be achieve by applying of the bias voltage. This effect can be used for
the so-called photothermoelectric  detection~\cite{52,54}. However, in the case of symmetric structure in the absence of the dc bias (considered above), the effective hole and electron temperatures in the pertinent sections of the channel are equal, and, hence, the  photothermoelectric effect can be disregarded (in contrast to the voltage-biased p-n junctions~\cite{52}).

Apart from using the engineered band gap (as in the PGL-FETs under consideration),
an effective suppression of the interband tunneling in the GL-FET p-n junction
can also be achieved by using the saw-shaped gate electrodes instead of  conventional bar-like gate electrodes~\cite{64,65,66,67}. In such a case, the dramatically reinforced hole and electron backscattering
from the barrier in the p-n junction can lead to a substantial drop in the tunneling current and the p-n junction net conductance. The GL-FET detectors with the saw-shaped gate electrodes arebeyond the scope of the present paper, need a separate consideration, and will be considered elsewhere.

\section{Conclusions}
In summary, we considered the proposed resonant THz detectors using FETs with the split gates, which electrically induce the lateral p-n junction in uniform  and perforated  GLs, GL-FET and PGL-FET detectors. The gated regions of the GL- and PGL-FETs play the role of plasmonic resonat cavities whereas the p-n junction depletion GL or PGL region
enable the nonlinearity of the current-voltage characteristics.
We calculated the responsivity the spatial distributions of the ac and rectified components of electric potential
thermionic and tunneling currents caused by the incident THz radiation. Relating
the rectified voltage component and the intensity of the THz radiation, we obtained
formulas for the responsivity of the GL-FETs and PGL-FETs operating as the photovoltaic THz detectors.
As shown, the spectral dependences of the responsivity can exhibit sharp resonant maxima associated with the excitation of plasmonic oscillations. The height of the responsivity peaks is determined by the hole and electron collision frequency in the channel section and by the relative conductance of the p-n junction. Due to a  suppression of the tunneling current in the PGL-FETs (with the perforated p-n-junction), their resonant response is much more pronounced and the responsivity
 is substantially higher than that in GL-FETs
The PGL-FET detectors  operating at room temperature can be of interest for different THz communication systems.

\begin{acknowledgements}
One of the authors (VR) thanks D. Svintsov for useful discussion and N. Ryabova for assistance.
The work was supported by
the Japan Society for Promotion of Science (Grant GA-SR-A $\#$16H02336)
and the Russian Scientific Foundation (Project $\#$14-29-00277).
The works at  RPI was  supported by the US Army Research Laboratory
Cooperative Research Agreement.
\end{acknowledgements}

\section*{Appendix A. Differential conductance and nonlinearity parameter in GL-FETs}
\setcounter{equation}{0}
\renewcommand{\theequation} {A\arabic{equation}}

{\bf Thermionic (activation) current.}
Both the thermionic transfer of the carriers and their tunneling between
the gate sections enable the nonlinear current voltage characteristics
of the p-n junction.
Using the  Landauer-Buttiker formula~\cite{68} for 2D channels with the
barrier transparency is given

 \begin{equation}\label{eqA1}
 {\cal T}(\varepsilon, V^{pn}) = \Theta(\varepsilon - 2\varepsilon_F +eV^{pn}),
 \end{equation}
where $\Theta(\varepsilon)$ is the unity step function, and
considering that the thermionic current is created by both the injected electron and hole components, for the thermionic current density one can obtain

\begin{equation}\label{eqA2}
j^{th} = J_s^{GL}
\biggl[\exp\biggl(\frac{eV^{pn}}{T}\biggr)  - 1\biggr],
 \end{equation}
Here $V^{pn}$
is the signal voltage across the p-n junction, which includes  the ac
and rectified components, $\delta V_{\omega}^{pn}$ and $\overline{\Delta V}^{pn}$,
respectively, and
he saturation current density $J_s$ is given by

 \begin{equation}\label{eqA3}
J_s^{GL}
 \simeq \frac{4eT^2}{\pi^2\hbar^2v_W}\exp\biggl(-\frac{\varepsilon_F}{T}\biggr)
 \end{equation}
 Here $T$ is the electron and hole temperature in the channel (in the energy units)
 and has been taken into account that the built-in voltage $V^{bi} = 2\varepsilon_F/e$.

 As a result, at $|V^{pn}| \ll T/e$, from Eqs.~(A1) and (A2) calculating $g^{th} = (d j^{th}/d V)|_{V^{pn} = 0}$ and $\beta^{th} = \frac{1}{2}(d^2 j^{th}/d V^2)|_{V^{pn} = 0}$t, we arrive at  Eqs.~(9) and (10).

{\bf Tunneling current.}
Considering that  the net electric field in the p-n junction
is equal to $E^{bi} + E^{pn}$, where $E^{bi}$ and $E^{pn}$ are the electric fields produced built-in voltage and the signal electric field component,
the tunneling current density in the GL-FETs, which is proportional to $\sqrt{( E^{bi} + E^{bi})|_{x = 0}}V$
~\cite{1,4,8}, can be presented as

\begin{equation}\label{eqA4}
j^{tunn} = g^{tunn}_0V^{pn}
\sqrt{(E^{bi} + E^{pn})|_{x=0}}.
\end{equation}
Here

\begin{equation}\label{eqA5}
g^{tunn}_0 =
\biggl(\frac{e^2}{\pi^2 \hbar}\biggr)\sqrt{\frac{eE^{bi}}{\hbar\,v_W}}.
\end{equation}

If the spacing  between the gate edges $2l \leq W_g$, the electric potential
in the region $-l < x < l$ can be presented as $\varphi \simeq [(V^{bi} + V)/2l]x$, where $V^{bi} = 2\varepsilon_F/2e$.
Hence,

\begin{equation}\label{eqA6}
E^{bi} + E \simeq  \frac{V^{bi} + V^{pn}}{2l}.
\end{equation}
When $2l > W_g$,
neglecting the electron and hole space charges in the  region $-l < x < l$
and taking into account the smallness of the screening
length $r_S = \kappa\hbar^2v_W^2/4e^2T\ln[1 + \exp(\varepsilon_F/T)] \leq 10$~nm~\cite{21,24,69},
the electric-field spatial distribution in this region is given by~\cite{8,59}:

\begin{equation}\label{eqA7}
E^{bi} + E^{pn} = \frac{V^{bi} + V^{pn}}{\pi\sqrt{(l^2 - x^2)}}.
\end{equation}
Therefore , $(E^{bi} + E^{pn})|_{x=0} = (V^{bi} + V^{pn})/\pi\,l$.
The electron and hole space charge decrease the electric field in the p-n junction center~\cite{8,21,60}, so that one can put  $(E^{bi} + E)|_{x=0} = (V^{bi} + V)/\pi\,l^* \leq (V^{bi} + V)/\pi\,l$, where $l^* \geq l$.
Considering this, we find that
$g^{tunn} = (d j_0^{tunn}/d V)|_{V^{pn} = 0}$ and $\beta^{tunn} = \frac{1}{2}(d^2 j_0^{tunn}/d V^2)|_{V^{pn} = 0}$ are equal to

\begin{equation}\label{eqA8}
g^{tunn} \simeq g^{tunn}_0 =\biggl(\frac{e^2}{\pi^2\hbar}\biggr)\sqrt{\frac{2\varepsilon_F}{\pi\hbar\,v_W\,l^*}},
\end{equation}

\begin{equation}\label{eqA9}
 \beta^{tunn} \simeq \frac{g^{tunn}_0}{2V_{bi}} = \biggl(\frac{e^2}{\pi^2\hbar}\biggr)\frac{e}{2\sqrt{2}\sqrt{\pi\hbar\,v_W\varepsilon_F\,l^*}},
\end{equation}
respectively.

From equations~(A8) and (A9) for the parameters $\eta^{GL} = g^{tunn}/g^{th}$ and $\zeta^{GL} = \beta^{tunn}/\beta^{th}$ defined by  equation~(8) in the main text, considering equations~(9) and (10), we obtain equations~(11) and (12).

\section*{Appendix B. Differential conductance and nonlinearity parameter in PGL-FETs}
 \setcounter{equation}{0}
\renewcommand{\theequation} {B\arabic{equation}}

{\bf Thermionic injection: sharp barrier.}
If the barrier in the depletion layer associated with the nanoconstrictions is sufficiently sharp, its shape can be approximated by a trapezoid.  In this case, equation~(A1) should be replaced by

\begin{equation}\label{eqB1}
j^{th} = J_s^{PGL}
\biggl[\exp\biggl(\frac{eV^{pn} - \Delta_g/2}{T}\biggr)  - 1\biggr],
 \end{equation}
where $J_s^{PGL}= 4eNb/\pi\hbar$, $N$ is the density of the nanoconstrictions, and $\Delta_g$ is the energy gap in the nanoconstrictions. Calculating $(dj^{th}/dV)|_{V^{pn} = 0} $ and $(d^2j^{th}/dV^2)|_{V^{pn} = 0} $,
from equation~(B1) we get equations~(13) and (14) which accounts for the 1D transport
in the nanoconstrictions and the exponential factor $\exp(-\Delta_g/2T)$ inherent to  the activation processes.

{\bf Tunneling current: sharp barrier.}
Assuming that the hole and electron dispersion lows in the constrictions (GNRs)
 are $\varepsilon^{\pm} = \mp\sqrt{p^2v_W^2 + (\Delta_g/2)^2}
 \simeq \mp[(\Delta_g/2) + p^2/2m]$ (where $m = \Delta_g/2v_W^2$) and using the  Landauer-Buttiker formula~\cite{68} for 1D channels with the tunneling transparency ${\cal T}(\varepsilon, V^{pn})$ of the trapezoid barrier in the PGL-FET perforated region derived in
 the WKB approximation, we find

 \begin{multline}\label{eqB2}
j^{tunn} \propto  2eN
\times \exp \biggl\{-\frac{4\sqrt{m\varepsilon_F}l}{3\hbar}\biggl[\biggl(\frac{\Delta_g}{2\varepsilon_F} + 1\biggr)^{3/2}\\
 - \biggl(\frac{\Delta_g}{2\varepsilon_F} - 1\biggr)^{3/2}
\biggr]\biggr\}
\end{multline}
(for $\Delta_g/2\varepsilon_F = Z \geq 1$). Consequently,

 \begin{equation}\label{eqB3}
 g^{tunn} \simeq \frac{ej^{tunn}}{\varepsilon_F}, \qquad \beta^{tunn}\simeq \frac{e^2j^{tunn}}{2\varepsilon_F^{2}},
 \end{equation}

Comparing    the thermionic and tunneling
contributions, one can find that in the case of the trapezoid tunneling barrier  ($Z = \Delta_g/2\varepsilon_F \geq 1$), the former contribution
 substantially exceed  the tunneling contribution
 (so that $\eta^{PGL}$ and $\zeta^{PGL}$ are much smaller than unity) when

\begin{equation}\label{eqB6}
l \gg l^{tunn}\frac{1 + Z}{\sqrt{Z}[(Z + 1)^{3/2} - (Z - 1)^{3/2}]}
\end{equation}
 $l^{tunn} = 3\hbar\,v_W/4T$. At room temperature $l^{tunn} \simeq 18$~nm.
 If $Z = 2 - 3$, the tunneling  can be disregarded in the PGL-FETs with  $l \gg 7 - 8$~nm.

Since $g^{tunn}$ and $\beta^{tunn}$ in the PGL-FETs are proportional to the tunneling current, which is weak, we can disregard their contribution to
$g^{pn}$ and $\beta^{pn}$ [see equation~(8)], keeping in mind that $\zeta^{PGL}$ and
$\eta^{PGL}$ are exponentially small.

{\bf Thermionic and tunneling currents: parabolic barrier potential model.}
In the PGL-FET device structure, the shapes of the barriers for holes and electrons  can also be approximated
by  parabolic functions~\cite{70,71}, so that in the barrier ($|x| < l$)
one has $\Delta_g(x) = \Delta_g(1 - x^2/l^2)$,
where in such a case $\Delta_g$ is the energy gap in the constriction center. Assuming the latter and using   the Landauer-Buttiker formula~\cite{72},
(adopted for  the case of 1D transport in the nanoconstrictions under consideration)
in which the barrier transparency is given by the Kemble formula~\cite{68}

 \begin{equation}\label{eqB19}
 {\cal T}(\varepsilon, V^{pn}) = \biggl[1 + \exp\biggl(\frac{\Delta_g^*/2 - \varepsilon}{\Theta}\biggr)\biggr]^{-1},
 \end{equation}
 %

where
 \begin{equation}\label{eqB10}
 \Delta_g^* = \Delta_g\biggl(1 - \frac{eV^{pn}}{2\Delta_g} \biggr)^2
 \simeq \Delta_g - eV^{pn}
 \end{equation}
 is the maximum barrier height with respect to the bottom of the conduction band in the n-section. The "tunneling" temperature is given by

 \begin{equation}\label{eqB11}
 \Theta = \frac{\hbar}{2\pi}\sqrt{\frac{\Delta_g}{l^2m}} \sim \frac{\hbar}{2\pi}\frac{v_W}{l}
 \end{equation}
 %
 characterizes the contributions of the transition below the barrier top,  where
 $m$ is an effective mass in the barrier.
 Setting $l = 25 - 100$~nm, we obtain $\Theta \sim 11.5 - 46.0$~K. It implies
 that for  room temperature  one can assume $\Theta \ll T$.
Taking the latter into account, for the PGL-FETs with not too wide constrictions
(in which $\Delta_g \gg 2\Theta$), we again arrive at equations~(13) and  (14)
with $\zeta^{PGL} < \eta^{PGL} \ll  1$.
However, if $l = 7 - 8$~nm, $\Theta \sim 390 - 450$~K. In this case, the tunneling processes can be marked. This is in line with the results of the above trapezoidal barrier model.

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text:
Frugal Entrepreneur
Run and Grow
Home / Blog / Would You Let William Nobrega Sleep On Your Sofa?
Would You Let William Nobrega Sleep On Your Sofa?
You find a bunch of college kids, and you're going to hear a lot of "living with your parents" jokes. That's the fate they'll jokingly predict for one another after graduation, when they can't find a job.
It's something that a lot of recent grads grudgingly endure, none too proud of it, probably because of the stigma.
It's also the fate of one of today's most successful entrepreneurs William Nobrega, an indignity he suffered during middle age, when he had an eleven-year-old son of his own. The chain of events leading to this began when a big fund busted on him, leaving him broke. He would then become more broke when his divorce took his house. He then took his son and moved into his mom's place.
Is necessity the mother of invention? Does living with one's mother necessitate invention? Well, in this case it was from mother to insurance. Nobrega got an idea to sell insurance in emerging markets, with Brazil and other Latin American countries as his prime target, and Southeast Asia secondary.
The model was to get customers to use a phone app that would allow Nobrega and his then-incipient company, CQS International to collect information on them and analyze it to provide them with the best possible insurance. Predictive intelligence is the key to Nobrega's approach.
The Team, The Team…
With that theoretical framework in place, it was then time for Nobrega to assemble his team and get things running. He had been a member of the Army Special Forces earlier in life, and he used some principles from that experience to choose team members. It was important to Nobrega to have a dream team, and to him this meant good communicators who could take criticism. They had to have energy and they had to have an entrepreneurial spirit. Not an easy combination to possess.
But Nobrega didn't just want off the sofa: he wanted excellence.
Nobrega is a big believer in the power of a good team, as well as in the power of belief and confidence. All indicators on him are that he is a fierce competitor with an intense drive and a no-nonsense attitude. Well, what isn't nonsense is that his company was recently valued at $170 million.
Nobrega is the kind of person who others very well might like to learn from. He has obliged the world with a big book named Riding the Indian Tiger. The animal in the title is a metaphor for India's market, which, at the 2008 first pressing, was projected to begin outpacing China's for growth. The book shows potential entrepreneurs how to tap into this emerging market.
Nobrega knows emerging markets. He also knows he knows them. He plays to his strengths, throwing intense energy into the enterprise. If that sort of courage can lead to one losing everything, it's also the only thing that can win it back.
Filed Under: Blog, Inspiring Tagged With: example entrepreneurs, inspiring, William Nobrega
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text:
Visiting Lecturer Dr. Michael Strmiska: Latvian Paganism Is Currently in a Very Interesting, Creative Phase
Researchers Up Close
Michael Strmiska is currently working at Rīga Stradiņš University (RSU) on a Fulbright Scholarship from the United States. He has a PhD in Religious Studies and is in Latvia to teach religion-oriented courses at the departments of anthropology and media studies. He became interested in European pagan movements after visiting Iceland in 1996 on another Fulbright Scholarship. Witnessing an autumn equinox ritual there and feeling its spiritual power, Dr. Strmiska's interest in paganism grew, and took over his academic life.
How did you come to be at RSU?
I've been interested in the Baltic region for a long time. I actually have partial Lithuanian ancestry. My first visit to Riga was in the early 2000s and I met members of Dievturi then, so I was already aware of them.
I have observed how the movement has changed over the years. Each new generation of people brings new ideas, so it's not the same as it was 20 years ago. Right now they are in a very interesting, creative phase.
I received a US-funded Fulbright Fellowship back in 2004 to teach at Šiauliai University in Lithuania, and now I wanted to come back to the region, so I thought "why not to Latvia". Each year, Fulbright offers specific opportunities at particular universities. A couple of years ago there was an opening within anthropology at RSU. I pursued this and came to know Klāvs Sedlenieks and Ieva Puzo.
Why did you decide to stay in Latvia during the coronavirus lockdown?
Part of my calculation was that if I were to return to the US, I would be going to the New York City area, which is clearly one of the worst places to be in the world right now in terms of the coronavirus. But honestly, I just wanted to stay here and continue my activities, even under lockdown.
What do you teach at RSU?
I teach two courses here. We had our final meeting for one of them the night before the lockdown took effect at the university. This was a course on religion and media as part of Communication and Media Studies at the Faculty of Communication. The other was a course on the anthropology of religion.
The last class on 10 March before restrictions were imposed. Photo: Linda Klusa
Could you explain why people are drawn to paganism?
If you were to spend a lot of time getting to know people in Dievturi or similar movements, you would find that maybe sometimes they're more concerned with national identity, while at other times they may be feeling an intense connection with nature. At other times they may be reflecting on the dainas, or on myths and feeling 'oh, there's a much deeper meaning here', as people do with the Latvian ornamental symbols, for example. The symbolic vocabulary of Latvia is very rich, so people go very deeply into that and come up with their own interpretations. Paganism offers many possibilities that people can pursue according to their interests and needs.
Does this cause arguments and divisions?
There certainly can be divisions, but they're just as often over personality clashes as anything else. Internal divisions do not occur so much in Baltic paganism. In Scandinavian paganism, or Scandinavian-based paganism (also called Ásatrú or heathenry, or Norse- or Nordic paganism) however, some people add a kind of racial dimension to it. This is where we sometimes get into right-wing, neo-Nazi territory. People may not go quite that far, but they are opening the door to it, and can be somewhat sympathetic to this ideology.
Have you studied the political aspects of paganism?
I actually did want to explore the politics of the American Norse pagans about ten years ago, but I found that it was disastrous for my research, because people just didn't want to talk to me anymore. They saw me as an extremist who was overly concerned with racism and such like. I really haven't been able to do anything with the American Norse pagans since then. That door has kind of shut. It is a very sensitive issue, for sure. I've increasingly come to focus on the political side, however. You can see the rise of ethnic right-wing nationalism that we observe in so many countries nowadays (from USA to Europe and beyond) also reflected in paganism. I organised a conference on politics and paganism at Masaryk University in Brno in the Czech Republic four years ago, because I had taught there for one semester in 2015 as a visiting professor. The papers from that conference were published in a journal called The Pomegranate. My keynote lecture for the conference, later published in the journal, was titled "Pagan Politics in the 21st century: Peace and Love, or Blood and Soil?" in which I lay out a lot of these issues.
How do pagan religions organise?
Regarding structure, Pagan groups like the Ásatrú Society in Iceland or Dievturība in Latvia or Romuva in Lithuania typically have a regular form of organisation with an elected leadership. They have rules that they follow as regards their meetings and their finances. Generally, you have to demonstrate certain formalities in order to be registered with the government.
As far as beliefs go, these is more loosely defined. In Dievturi, for example, there's a general understanding of being dedicated to old Latvian traditions, to the old Latvian gods and goddesses, to folklore, and of course, especially to the folk songs. The dainas are at the heart of it all.
A looser structure can actually be more appealing, because people in western countries are more individualistic these days. They don't like being told what to think or believe. If someone makes a very good case for certain ideas, while someone else has slightly different ideas, but there's an overlap between them, people actually seem quite comfortable with that.
What would you say are the main differences between Nordic, and Latvian and Lithuanian paganism?
That's interesting. The biggest differences are in the source materials they have to work with. In the Scandinavian countries, what they have from the old pagan days, from the original pagan times, is literature. They have a lot of texts that were primarily written down in Iceland (mainly by Christian monks, strangely enough). These texts give us a great deal of information about the gods and they tell stories about people who practiced the religion, but they don't have any music. Old styles of music were forbidden by the authorities, particularly by Christian authorities. In the Baltic case it's almost the opposite. Here you don't have as much rich mythological literature, or rather, you don't have it in a form that's very attractive and accessible. The written materials in Scandinavia are very attractive, enjoyable, accessible, and obviously have worldwide appeal. In the Baltic case, while you don't have that kind of rich literary foundation, what you do have here is the music, the folk songs. That tradition is obviously very, very strong and appealing here.
So when I look at the two, it's almost like each has the piece the other is missing.
Christianity came in and influenced both regions - how would you say that this influence differed between Scandinavia and the Baltic States?
Although of course Lithuania is more Catholic and Latvia ultimately more Lutheran, in both cases the old pagan traditions were able to live on. This was actually partly because of foreign domination. The pagan traditions would be better known among the Lithuanian and Latvian peasants, whereas the ruling classes - whether they were Polish, Russian, German, or Swedish - were not very interested in native traditions. That kind of gave space for them to live on and thrive.
To what extent is paganism just re-enacting a romanticised version of the past?
There is certainly some romanticisation, however, a lot of people keep up with scholarly research, so I think that many pagans actually have a pretty nuanced, accurate view of the past. There will be arguments about how to interpret this or that aspect, or whether something was really done this way, or that way. I know on some occasions, people have actually quit these movements because they were so upset at what they felt was an incorrect interpretation, or reconstruction of a kind of head covering, for example.
I think there is romanticism on an emotional level, but it's countered by an intellectual awareness that there are a lot of gaps in our knowledge of the past.
In the past, was it an actual doctrine, or religion, or was it just how you lived on a farm with the seasons?
The latter.
So where's the line between simply recreating history, and living according to a moral, spiritual structure?
Here we get to the problem of how we define religion, because if we take the Christian church as the paradigm of religion, then anything that doesn't look like the Christian church will not seem like a religion.
I always think it's more useful to define religion more broadly as worship, or a relationship to things that are considered sacred, and the beliefs, myths and organisation attached to that.
What you're saying is a very important issue, though, because it would seem to be true that in pre-Christian times people weren't really conscious of following a "religion", per se. It's only in later times that we look at what they were doing and say 'Oh, they see the river as sacred. They see the sun and the tree as sacred. They're singing songs about these things. They're dancing like the sun. Seems pretty damn religious'. That is a modern interpretation, however. I think that what people have seen is that making it into a religion today is a way to preserve the old traditions, including their spiritual content.
So, we tend to kind of put the structure of Christianity onto other religions?
Yes. What's really funny, is that some would say that Ernests Brastiņš, the man who created this organisation of Dievturi back in the 1920s and 1930s, did it in a somewhat Christian format. He created a kind of meeting structure that was not unlike a Lutheran church service, and I believe he even wanted an organ in the room. He also proposed that there be a kind of trinity of Latvian deities - Dievs, Māra, Laima. And if anyone knows anything about Christianity, you know that the trinity is central. I would not, however, take that as something negative. I think in adapting religions over time, it's natural to make use of elements you see around you that you see as working well. So you could say that the Dievturi organisation may be a bit like a church, but so what? If church organisation works well, if you want to use it for your traditions, and you want to adapt it, I see nothing wrong with that.
So in a way, pagan religions are damned if they do, damned if they don't. If they are different from Christianity, people don't think of them as religions, but if they borrow elements from Christianity, people will say they're not pagan.
Can you talk more about the Latvian symbols?
I have to confess that it's not something I have really delved into deeply. The symbol I've been most interested in is the thunder cross that has associations of the 1930s and the Pērkoņkrusts movement, the thunder cross movement, and possible right-wing fascist associations because it has similarities to the swastika. I have discussed this a lot with my students who ask me whether I don't think it's unfair that Latvia gets blamed for using the Nazi swastika, when actually the Latvian thunder cross is something quite separate. I tell them that it depends on context.
If you use the thunder cross within Latvia among Latvians where people understand these symbols, I think it's perfectly fine to use it, but the problem is when you're trying to relate to people from other countries who only know about the swastika. If you're going to use this in an international context, or international eyes are going to be on you, you have to explain yourself very carefully. Frankly I would advise against using the symbol except in very specific situations where you know that everyone has the same frame of reference, because this could so easily be demonised against Latvia. It's very unfortunate. I mean, the symbol is innocent, it's not the fault of the symbol, but you can't avoid, or ignore, the historical context.
How do you feel about missing all the spring and summer traditions, specifically Līgo?
I am a little vulnerable to the virus because of my age (I just turned 60 in isolation), and have some pre-existing health conditions. At first I thought I might have to avoid Līgo altogether, but I reached out to my colleagues and students for suggestions of where I could go to participate in a kind of 'socially distanced' Līgo. Thanks to their help, I think I have now found something I can participate in near Liepāja, at a kind of ethnographic farm museum in Vītolnieki.
I really wonder what will happen to Jāņi and Līgo this year, how people will celebrate them. My students have told me that it will probably mostly be a family affair, that people will gather in small family groups, so maybe I'll visit with a family. I don't know what bigger events will be possible around Riga.
I have a visa issue at the moment because Fulbright was shut down, but I've really enjoyed teaching at RSU and I've discussed the possibility of teaching the Anthropology of Religion course online with students in the future. Maybe the Media and Religion course too. I had great students for both and that is a major incentive for me. If I can do that, I would like to come back to Latvia from the USA for a week or two at the end of each semester to teach and meet with students and discuss their research projects as well as do a little research of my own. One way or another I do plan on coming back to Latvia.
One of the benefits of COVID-19, is that it's opened the door to more online teaching.
Something I've always tried to do when coming to Latvia, or the Baltics, is to form continuing partnerships. I really like connecting with the scholars here and, if I can, helping them get published and gain access to international opportunities, if I can, just as I have benefited from being an international scholar coming here and learning from people and receiving their help in so many ways. I really believe it should be a two-way street and that we should all help one another.
What's changed in Latvia over the years since you were first here?
I really worry about my students. Coming to Latvia this time, to Riga, before things were shut down, I noticed something very interesting; I thought I felt a different attitude among younger people here. They were much more open, much more relaxed than people were on my first visit 20 years ago, or even when I was last here in 2015. Young people today don't have to live with the memory of the Soviet times, where people in the past had a kind of fear hanging over them. What's sad now is that young people are looking at a future clouded by COVID-19 where no one knows how it's going to turn out. I'm really worried we'll go into an economic depression, and that it will badly affect travel and international contacts.
Michael is studying Asian religions in the Baltic States, for instance Krishna and Buddhist temples. He plans to start interviewing members of the growing population of Asian students in Latvia on the interplay between their experience of their religion and Westerners' interpretation of Eastern religions, as well as how Asian students feel about living in Latvia, including any problems they may have with racism or discrimination.
Michael is currently planning a chapter on Latvian and Lithuanian folk religion and seasonal traditions that will be part of the Oxford Handbook of Slavic and East European Folklore.
He is also working on a book for Springer Publications called Unchristian Eastern Europe: Pagans, Jews, Gypsies and Muslims. The idea of the book is to argue against the common idea that Europe is an exclusively Christian civilisation by highlighting how these other groups have contributed so much to the life, culture and traditions of Eastern Europe.
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Announcement About the 2021 Spring Semester Study ProcessFor Students, Crisis
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Experts from RSU Study COVID-19 Personal Protective EquipmentFor Students, For RSU Employees, Research, Crisis
15 Oct Exhibition RSU – 70
18 Jan January 2021 Pre-courses for International Students: Chemistry, Physics, Biology
25 Jan January 2021 Pre-courses for International Students: Latin, Latvian
27 Jan Orientation Week


text:
Gottlieb Giese (* 17. November 1787 in Greifswald; † 3. April 1838 ebenda; vollständiger Name Gottlieb Christian Johannes Giese, auch Christian Johann Gottlieb Giese) war ein deutscher Maler, Architekt und Zeichenlehrer.

Leben
Gottlieb Giese war Schüler von Johann Gottfried Quistorp und besuchte die Berliner Kunstakademie unter Karl Friedrich Schinkel. Nach seinem Entwurf wurde zwischen 1821 und 1826 eine achteckige Kapelle auf dem Greifswalder Alten Friedhof errichtet. Giese hatte von 1824 bis 1832 die Aufsicht bei der Renovierung des Greifswalder Doms, bei der der Kunsttischler Johann Christian Adolf Friedrich (1779–1843), ein Bruder Caspar David Friedrichs die Holzarbeiten fertigte. Er übernahm die Wiedereinrichtung der St.-Marien-Kirche, deren Hochaltar nach seinem Entwurf von Christian Friedrich 1835–1837 ausgeführt und mit der von Friedrich August von Klinkowström 1807 gemalten Kopie des Gemäldes "Heilige Nacht" von Correggio ausgestattet wurde.

Der künstlerische Nachlass Gieses wurde zum größten Teil 1871 bei einem Brand zerstört. Mehrere Gemälde befinden sich in der Galerie des Pommerschen Landesmuseums. Eine 1822 für die St.-Marien-Kirche in Barth gemalte Bilderfolge von acht Darstellungen des Lebens Jesu Christi befinden sich in der kirchengeschichtlichen Sammlung der Marienkirche.
Anlässlich der Wiedereinweihung des Greifswalder Doms nach der in den 1980er Jahren durchgeführten Restaurierung wurde 1989 eine Ausstellung zum Werk Gieses im Dom durchgeführt, auf der alle erhalten gebliebenen Gemälde und mehrere Zeichnungen gezeigt wurden.

Werke (Auswahl)
 Winterlandschaft 1814.
 Sommerlandschaft mit Viehherde 1814.
 Einsamer Wanderer vor nächtlicher Kirche 1815.
 Bildnis des Superintendenten Dr. Johann Christian Friedrich Finelius 1820.

Literatur
 Hans Reddemann: Der denkmalgeschützte Alte Friedhof in der Universitäts- und Hansestadt Greifswald. Teil III, Greifswald 2012, ISBN 978-3-00-027660-6, S. 93.
 Norbert Buske: Symposium und Ausstellung anlässlich der Wiedereinweihung des Doms St. Nikolai in Greifswald im Juni 1989 (darin Werkverzeichnis der Arbeiten von Giese). Thomas Helms Verlag Schwerin 2005
 Norbert Buske: Zwei neue Bilder von Gottlieb Giese. Vorläufiges Werkverzeichnis konnte erweitert werden. In: Pommern. Zeitschrift für Kultur und Geschichte. 43/2005, S. 44–46

Weblinks

Einzelnachweise

Architekt (Preußen)
Person (Greifswald)
Maler (Deutschland)
Geboren 1787
Gestorben 1838
Mann

redpajama_set_name: RedPajamaWikipedia


text:
Power-packed surge by Volex
By This Is Money
Updated: 19:00 EST, 7 November 2000
Electronic cable and power cords group Volex has reported a 76% jump in interim pre-tax profits to £16.1m for the six months to end September on soaring demand from the telecoms sector.
Group sales were up by more than half to £217.2m and earnings per share almost doubled to 40.7p. The interim dividend is raised 8% to 9.4p. The group has set a target of £1bn in sales within five years.


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Let's talk about Ballet barre exercises.
Barre Workout for a Ballet Booty: Envious of the perfectly toned ballerina tush? With this barre workout, you too can get that ballet booty by practicing these ballet barre workout exercises at home. Play some upbeat music (it doesn't have to be classical!) while you plié and relevé your backside into shape with this intermediate-level barre workout.
When used for promoting general fitness, barre exercises may incorporate a wide variety of activities to increase stamina, flexibility and strength, including ballet movements that require balance and stimulate the core such as and .
Barre exercises to help you lose weight and workout more often. These no-equipment barre exercises can be done at home, the gym, or the studio. Train like a ballet dancer and get that ballet barre body with these beginner barre exercises.
With a combination of ballet barre exercises we felt comfortable doing and ceiling-suspended band work we cursed for days, Barre X mimics the training routines of Cirque du Soleil acrobats. No, really: The teacher who developed it actually toured with one of the shows.
Barre Exercises for barre enthusiasts of any skill level! Browse this list of barre exercises for easy, medium, and hard barre exercises that you can do at home, or in the studio. This barre exercises list uses more traditional ballet-barre style movements, in contrast to the pilates or yoga-heavy style of barre movements that some other instructors use.
Whatever your ballet barre fitness preference, you will be sure to find some new barre exercises on this list to try out. Get that ballet body with this ballet barre exercises list!
Now, you are going to repeat the above Ballet barre exercise, but this time from 2nd position. So with feet and your arm in 2nd, and a beautifully straight back, bend your knees out to the side. As for the 1st position version, keep your knees over your feet.
Get in on the barre craze with this at-home ballet routine that will help lift, lengthen, and define your entire body.


text:
Dilar parthenopaeus is een insect uit de familie van de Dilaridae, die tot de orde netvleugeligen (Neuroptera) behoort.

Dilar parthenopaeus is voor het eerst wetenschappelijk beschreven door A. Costa in 1855.

Dilaridae


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Give it back to me.
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The above articles are a small sample of public information regarding the abortifacient nature of most so-called contraception birth control.
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Douglas eventually agreed to speak with Lincoln.
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Shirley saw a large bird.
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text:
Gardens of the Cotswolds Tour
Discover the glorious treasures of the Cotswolds and its gardens. The parkland of magnificent stately homes, quaint and picturesque villages and lovely explosions of colour in smaller gardens.
Gardens of the Cotswolds
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Slideshow: Highgrove House; Garden at Bourton House © Bourton House; Sudeley Castle.
Photo to the right of the tour: Hidcote Manor Garden © National Trust Images/ Nick Daly.
Our tour guide was absolutely wonderful! She was extremely knowledgeable and really made the tour fun. Most importantly Lana planned and coordinated a very special trip itinerary that made all the participants feel like royalty!
Garden of England and Chelsea Flower Show Tour - May
Private Gardens
Henry VIII and Tudor England Tour
Young Royals Tour


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text:
Water relations and transpiration of honey mesquite on 2 sites in west Texas.
Resource Link:
https://journals.uair.arizona.edu/index.php/jrm/article/view/8578
Wan, C.
Sosebee, R.E.
Rangeland Ecology & Management / Journal of Range Management Archives
Transpiration rates and internal water relationships of honey mesquite (Prosopis glandulosa) were investigated weekly during May through September 1986 on sandy loam and clay loam, both upland sites in west Texas. Average transpiration rates peaked at approximately 7 mmol m-2 s-1 at 1100 hr during wet periods and reached a plateau between 4 and 5 mmol m-2 s-1 between 1200 and 1400 hr. During dry periods, the average transpiration rates reached their maximum plateau of 2 mmol m-2 s-1 at 1000 hr and declined between 1200 and 1600 hr. The transpiration rates ranged from an average of 3.28 +/- 2.05 mmol m-2 s-1 for trees on a sandy loam site to an average of 3.85 +/- 1.94 mmol m-2 s-1 for those on a clay loam site. Stomatal closure in midsummer caused a substantial increase in leaf temperature. Mesquite has developed other means, such as leaf orientation, wax accumulation, and reduction in canopy development, to avoid drought. Stomatal conductance of mesquite is very responsive to soil water availability and dryness of the air, and is less responsive to internal water status. This research further substantiates that mesquite behaves like a facultative phreatophyte in west Texas.
Journal Issue/Article
Non-AGROVOC Keywords:
leaf angle
facultative phreatophytes
stress response
matric potential
clay loam soils
Prosopis glandulosa
phreatophytes
sandy loam soils
Soil water content
Rangeland Ecology & Management (REM) - formerly Journal of Range Management (JRM)
Rangeland Ecology & Management (formerly the Journal of Range Management) serves as a forum for the presentation and discussion of facts, ideas, and philosophies pertaining to the study, management, ecology, and use of rangelands and their resources. The journal is peer-reviewed and provides international exchange of scholarly research and information among persons interested in rangelands. The Global Rangelands collection includes REM content up to 5 years from the current year. More recent content is available by subscription from BioOne and the Society for Range Management, and may be available at your local university library.
(Become a SRM member)
http://www.rangelands.org/


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text:
Marie-Paule Alluard est une coloriste de bande dessinée.

Biographie

Publications

Séries réalisées
 Noël et Marie, scénario de Jean Ollivier et François Corteggiani, dessin de Jean-Yves Mitton, Éd. Messidor/La Farandole, 1989
 Névé, dessinateur Emmanuel Lepage, scénario Dieter, éd. Glénat, 1991-1997
 Trent, dessinateur Leo, scénariste Rodolphe, éd. Dargaud, 1991-2000
 Les Maîtres de l'orge, dessinateur Francis Vallès, scénario Jean Van Hamme, éd. Glénat, 1992-1999
 Largo Winch, dessinateur Philippe Francq, scénario Jean Van Hamme, éd. Dupuis, 1995-2008
 Dixie Road, dessinateur Hugues Labiano, scénariste Jean Dufaux, éd. Dargaud, 1997-2001
 Gothic, dessinateur Marcelé, scénariste Rodolphe, éd. Delcourt, 2001-2002
 Tosca, dessinateur Francis Vallès, scénario Stephen Desberg, éd. Glénat, 2001-2003
 Rafales, dessinateur Francis Vallès, scénario Stephen Desberg, éd. Le Lombard, 2005-2008
 Bunker, dessinateur Christophe Bec et Nicola Genzianella, scénario Christophe Bec et Stéphane Betbeder, éd. Dupuis, 2006-2012
 Pandemonium tome 1, Les Collines de Waverly, dessinateur Stefano Raffaele, scénario Christophe Bec, éd. Les Humanoïdes Associés, 2007
 Indicible, dessinateur Francisco Ruizgé, scénario Patrick Renault, éd. Soleil Productions, 2013

Autres ouvrages
 Conseil de pro, la couleur traditionnelle: Les couleurs traditionnelles de Marie-Paule Alluard, propos recueillis par Vincent Henry, Calliope  octobre-

Références

Annexes

Bibliographie

Liens externes

Alluard, Marie-Paule
Date de naissance non renseignée (XXe siècle)