Resonant plasmonic detection of terahertz radiation in field-effect transistors with the graphene channel and the black-As $$_x$$ P $$_{1-x}$$ gate layer

BlockDAG’s CEO and Team Announced: Expected to Surpass Hedera and Polygon with a Projection

BlockDAG’s CEO and Team Announced: Expected to Surpass Hedera and Polygon with a $1 Projection

30 July 2024

VeChain (VET) And Hedera (HBAR) Triple Red Charts. Rollblock (RBLK) Seems The Summer Sure Money Bet – Times Tabloid

28 July 2024

Thermionic DC and AC

At not-too-small electron densities in GCs, the characteristic time of the electron-electron collisions $\tau _{ee}$ is shorter than the pertinent times associated with the optical phonons $\tau _0$, acoustic phonons $\tau _{ac}$, and impurities $\tau _i$, respectively. This implies that the electron distribution function is close to the Fermi distribution function $f(\varepsilon ) = [\exp (\varepsilon – \mu )/T +1]^{-1}$, characterized by the effective electron temperature T generally different from the lattice (thermostat) temperature $T_0$ (in the energy units) and the electron Fermi energy $\mu $. Hence, at $\varepsilon > \mu $, $f(\varepsilon ) \simeq \exp [(\mu – \varepsilon )/T]$. However, in the energy range $\varepsilon > \Delta _C$, the electron escape over the BL can markedly decrease $f(\varepsilon )$. To account for this effect, in the range in question, one can set $f(\varepsilon ) \simeq \xi \exp [(\mu – \varepsilon )/T]$, where $\xi = \tau _{\bot }/(\tau _{ee} + \tau _{\bot })$ with $\tau _{\bot }$ being the electron try-to-escape time.

Considering that the height of the potential barrier for the electrons in the GC and in the metal gate are equal to $\Delta _C – \mu $ and $\Delta _M + e(V_G + \varphi )$, respectively, the density of the thermionic electron current can be presented as

$$\begin{aligned} j \simeq j^{max} \biggl [\exp \biggl (\frac{\mu – \Delta _C}{T}\biggl ) – \exp \biggl (-\frac{\Delta _M + e \Phi }{T_0}\biggr )\biggr ]. \end{aligned}$$

(1)

Here $j^{max}= e\Sigma /\tau _{\bot }$ is the characteristic (maximum) GC-gate DC density, $\Sigma $ is the electron density in the GC induced by the donors and gate voltage, and $e =|e|$ is the electron charge. One can assume that $\tau _{\bot }$ is determined by the momentum relaxation time, associated with the quasi-elastic scattering of the high-energy electrons, i.e., with acoustic phonons (in sufficiently perfect GCs). Due to this, it is natural to assume that $\tau _{\bot } > \tau _{ac} \gg \tau _{ee}$. The Fermi energy $\mu $ is determined by both the GC doping and the gate voltage.

Equation (1) leads to the following expressions for the thermionic DC density ${{\overline{j}}}$, corresponding to the DC temperature ${\overline{T}}$:

$$\begin{aligned} {{\overline{j}}}=j^{max}\biggl [\exp \biggl (\frac{\mu – \Delta _C}{{{\overline{T}}}}\biggl ) -\exp \biggl (\frac{\mu -\Delta _C -eV_G}{T_0}\biggr )\biggr ]. \end{aligned}$$

(2)

Due to the dependence of $\mu $ on $V_G$, Eq. (2) provides the GC-gate I-V characteristics. Since ${{\overline{T}}}$ also depends on $V_G$ (because of the electron heating in the GC by the lateral DC), the latter dependence can somewhat contribute to the GC-FET characteristics as well.

At sufficiently high GC lateral conductivity in the situations under consideration (large $\Sigma $ and $\mu $), the DC potential and the DC effective temperature nonuniformity along the GC are weak (${{\overline{T}}}\simeq const$). This implies that we disregard the possible DC crowding. A high electron thermal conductivity additionally suppresses the above nonuniformity.

The AC variation $\delta j_{\omega }$ due to the potential oscillations leading to the electron heating is given by

$$\begin{aligned} \delta j_{\omega } = j^{max}\frac{\delta T_{\omega }}{ {{\overline{T}}}} \frac{(\Delta _C -\mu )}{ {{\overline{T}}}}\exp \biggl (\frac{\mu – \Delta _C}{ {{\overline{T}}}}\biggr ) \end{aligned}$$

(3)

Here we omitted the term containing the factor $ (e\delta \varphi _{\omega }/T_0 )^2/2$ with $\delta \varphi _{\omega }$ being the GC potential ac component. In this case, the quantity $\delta j_{\omega }$, given by Eq. (3), does not depend explicitly on the AC variations of the GC potential (only via the effective temperature variation $\delta T_{\omega }$). This is due to a specific shape of the energy barrier for the electrons in the GC (see Fig. 1c).

Rectified current and effective carrier temperature

The incoming THz radiation results in variations of the potential in the GC. This leads to extra electron heating and the variation of the electron temperature $\delta T = T – {{\overline{T}}}$. According to Eq. (3), the variation of the net gate current associated with the effect of the incoming THz radiation averaged over its period (rectified photocurrent) is given by

$$\begin{aligned}<\overline{\delta J_{\omega }}>= J^{max}{{\mathscr {F}}}(V_G) \frac{<\overline{\delta T_{\omega }}>}{{{\overline{T}}}}. \end{aligned}$$

(4)

Here $J^{max} = 2LHj^{max}$, and 2L and H are the GC length and width,

$$\begin{aligned} {{\mathscr {F}}}(V_G) = \frac{(\Delta _C -\mu )}{{{\overline{T}}}}\exp \biggl (\frac{\mu – \Delta _C}{{{\overline{T}}}}\biggr ) = \frac{[\Delta _M -(\mu -\mu _D)]}{{{\overline{T}}}}\exp \biggl [\frac{(\mu -\mu _D) – \Delta _M}{{{\overline{T}}}}\biggr ] \end{aligned}$$

(5)

is the barrier factor, and the symbols $<…>$ and $\overline{<…>}$ denote the averaging over the signal period $2\pi /\omega $ and the length of the GC, respectively, with

$$\begin{aligned}<\overline{\delta T_{\omega }}>= \frac{1}{2L}\int _{-L}^Ldx<\delta T_{\omega }>. \end{aligned}$$

(6)

The dependence of the factor ${{\mathscr {F}}}(V_G)$ on the gate voltage as associated with the voltage dependence of the electron Fermi energy (see below).

The effective electron temperature T is determined by the balance of the electron energy transfer to the lattice and the energy provided by the electric field along the GL. At room temperature, the emission and absorption of the optical phonons by the electrons in GLs can be considered as a main mechanism of electron energy relaxation. In this case, the power transferring from the electrons in the GC to the optical phonons due to the intraband transitions is^30,31,32,33.

$$\begin{aligned} P_0^{intra} = \hbar \omega _0 R_0^{intra}. \end{aligned}$$

(7)

Here

$$\begin{aligned} R_0^{intra} = R_0 \frac{\hbar \omega _0\mu ^2}{T_0^3}\biggl [\biggl (1 + \frac{1}{{{\mathscr {N}}}_0}\biggr )\exp \biggl (-\frac{\hbar \omega _0}{T}\biggr )- 1\biggr ], \end{aligned}$$

(8)

$\hbar \omega _0 \sim 200$ meV is the optical phonon energy, ${{\mathscr {N}}}_0 = [\exp (\hbar \omega _0/T_0)-1]^{-1} \simeq \exp (-\hbar \omega _0/T_0)$, $R_0$ is the characteristic rate of the interband absorption of optical phonons, and $T_0$ is the lattice temperature. At moderate THz power, the effective electron temperature T is close to the optical phonon temperature $T_0$, and Eq. (8) yields for $R_0^{intra}$:

$$\begin{aligned} R_0^{intra} \simeq R_0 \frac{\hbar \omega _0\mu ^2}{T_0^3}\biggl (\frac{1}{T_0} – \frac{1}{T}\biggr ). \end{aligned}$$

(9)

Equalizing $ R_0^{intra}$ given by Eq. (9) and the Joule power associated with the AC in the GC, for the THz range of frequencies (in which one can assume $\omega \gg 1/\tau _{\varepsilon }$), we arrive at the following energy balance equation:

$$\begin{aligned} \frac{<\overline{\delta T_{\omega }}>}{\tau _{\varepsilon }} = \frac{\textrm{Re}~\sigma _{\omega }}{2\Sigma _0L} \int _{-L}^{L}dx\biggl |\frac{d\varphi _{\omega }}{dx}\biggr |^2. \end{aligned}$$

(10)

Here Re $\sigma _{\omega } = \sigma _0\nu ^2/(\nu ^2+\omega ^2)$ is the real part of the GC Drude conductivity, $\sigma _0 = e^2\mu /\pi \hbar ^2\nu $ is its DC value, $\nu $ is the frequency of the electron collisions on impurities, acoustic phonons, as well as due to the carrier viscosity (see⁴², and the references therein). Accounting for the deviation of the optical phonon temperature $T_0$ from the lattice temperature $T_l$, the carrier energy relaxation time $\tau _{\varepsilon }$ associated with the interaction with optical phonons is estimated as³² $\tau _{\varepsilon } = \tau _0 (1 + \xi _0)(T_l/\hbar \omega _0)^2\exp (\hbar \omega _0/T_l) \simeq \tau _0 (1 + \xi _0)(T_0/\hbar \omega _0)^2\exp (\hbar \omega _0/T_l)$, where $\tau _0$ is the characteristic time of the spontaneous optical phonon intraband emission by the electrons and $\xi _0 = \tau _0^{decay}/\tau _0$, and $\tau _0^{decay}$ is the decay time of optical phonons in GCs.

Plasmonic oscillations factor

The description of the spatio-temporal oscillations of the electron density and the self-consistent electric field, i.e., the plasmonic oscillations in the GLs (see, for example^{32,33,34,35,36,37}) forced by the incoming THz signals can be reduced to a differential equation for the AC potential of the gated GC filled by the electrons (followed from a hydrodynamic electron transport model equations^43,44,45 coupled with the Poisson equation), $\delta \varphi _{\omega }(x)$ :

$$\begin{aligned} \frac{d^2\delta \varphi _{\omega }}{dx^2} + \frac{\omega (\omega +i\nu )}{s^2}\delta \varphi _{\omega } =0, \end{aligned}$$

(11)

supplemented by the following boundary conditions:

$$\begin{aligned} \delta \varphi _{\omega }|_{x= \pm L}| =\pm \frac{\delta V_{\omega }}{2}\exp (-i\omega t), \qquad \delta \varphi _{\omega }^s|_{x= \pm L}| = \delta V_{\omega }\exp (-i\omega t). \end{aligned}$$

(12)

Here $s = \sqrt{4\,e^2\mu \,w/\kappa \hbar ^2}$ is the plasma-wave velocity in the gated GC.

The above equations yield the following formula for the AC potential along the GC

$$\begin{aligned} \delta \varphi _{\omega }^a = \frac{\delta V_{\omega }}{2}\frac{\sin (\gamma _{\omega }x/L)}{\sin \gamma _{\omega }}, \qquad \delta \varphi _{\omega }^s = \delta V_{\omega }\frac{\cos (\gamma _{\omega }x/L)}{\cos \gamma _{\omega }}. \end{aligned}$$

(13)

Here

$$\begin{aligned} \gamma _{\omega } = \pi \frac{\sqrt{\omega (\omega +i\nu )}}{\Omega }, \qquad \Omega = \sqrt{\frac{4\pi ^2\,e^2\mu \,W}{\kappa \hbar ^2L^2}} \end{aligned}$$

(14)

are the normalized wavenumber and the characteristic frequency of the plasmonic oscillations of the electron system in the GC-FET under consideration.

The AC electric field along the GC is equal to

$$\begin{aligned} \frac{d\varphi _{\omega }^a}{dx} = \frac{\delta V_{\omega }}{2} \frac{\gamma _{\omega }}{L}\frac{\cos (\gamma _{\omega }x/L)}{\sin \gamma _{\omega }}, \qquad \frac{d\varphi _{\omega }^s}{dx} = -\delta V_{\omega } \frac{\gamma _{\omega }}{L}\frac{\sin (\gamma _{\omega }x/L)}{\cos \gamma _{\omega }} \end{aligned}$$

(15)

that, accounting for Eq. (12), yields

$$\begin{aligned} \frac{<\overline{ \delta T_{\omega }}>^{a,s}}{\tau _{\varepsilon }} = \biggl |\frac{\delta V_{\omega }}{2}\biggr |^2 \frac{\sigma _0}{{\overline{\Sigma }}L^2}{{\mathscr {P}}}_{\omega }^{a,s}. \end{aligned}$$

(16)

Here

$$\begin{aligned} {{\mathscr {P}}}^a_{\omega } = \frac{\nu ^2}{(\nu ^2+\omega ^2)} \int _0^1d\zeta \biggl |\frac{\gamma _{\omega }\cos (\gamma _{\omega }\zeta )}{\sin \gamma _{\omega }}\biggr |^2, \qquad {{\mathscr {P}}}^S_{\omega } = \frac{\nu ^2}{(\nu ^2+\omega ^2)} \int _0^1d\zeta \biggl |\frac{\gamma _{\omega }\sin (\gamma _{\omega }\zeta )}{\cos \gamma _{\omega }}\biggr |^2 \end{aligned}$$

(17)

are the plasmonic factors, which can be also presented as

$$\begin{aligned} {{\mathscr {P}}}^a_{\omega } \simeq \biggl (\frac{\pi \nu }{\Omega }\biggr )^2 \frac{\omega }{\sqrt{(\nu ^2+\omega ^2)}}\frac{ P^a_{\omega }}{|\sin \gamma _{\omega }|^2}, \qquad {{\mathscr {P}}}^s_{\omega } \simeq \biggl (\frac{\pi \nu }{\Omega }\biggr )^2 \frac{\omega }{\sqrt{(\nu ^2+\omega ^2)}}\frac{ P^s_{\omega }}{|\cos \gamma _{\omega }|^2}, \end{aligned}$$

(18)

with $P^a_{\omega } = \int _0^1d\zeta |\cos (\gamma _{\omega }\zeta )|^2$ and $P^s_{\omega } = \int _0^1d\zeta |\sin (\gamma _{\omega }\zeta )|^2$ being functions of the order of unity oscillating with the frequency. If $\omega \ll \Omega $ ($\gamma _{\omega }$ tends to zero), Eqs. (17) and (18) yield ${{\mathscr {P}}}^a_{\omega } \simeq 1$ and ${{\mathscr {P}}}^s_{\omega } \simeq 0$.

Combining Eqs. (4), (6), and (16), we obtain

$$\begin{aligned} \frac{<\overline{\delta J_{\omega }}>^{a,s}}{J^{max}} = \biggl |\frac{\delta V_{\omega }}{2}\biggr |^2\frac{\sigma _0\tau _{\varepsilon }}{{\overline{\Sigma }}L^2} {{\mathscr {F}}}(V_G)\,{{\mathscr {P}}}^{a,s}_{\omega }. \end{aligned}$$

(19)

The detector response depends on the antenna type (see, for example,^46,47). Using an antenna specially desined for the THz range could substantially increase the collected power⁴⁷. Here we define the GC-FET detector current responsivity (in the A/W units) and its voltage responsivity (in the V/W units) as

$$\begin{aligned} {{\mathscr {R}}}_{\omega } = \frac{<\overline{\delta J_{\omega }}>^{a,s}}{ S_{\omega }},\qquad {{\mathscr {R}}}_{\omega }^V = \frac{<\overline{\delta J_{\omega }}>^{a,s}}{S_{\omega }} \rho , \end{aligned}$$

(20)

respectively. Here $S_{\omega }$ is the THz power collected by an antenna and $\rho = 2L/H\sigma _0$ is the GC DC resistance (for the case of load resistance equal to the GC resistance). This collected power is estimated as $S_{\omega } = I_{\omega }A_{\omega }$, where $I_{\omega }$ is the intensity of the impinging radiation and $A_{\omega } = \lambda _{\omega }^2g/4\pi $ is the antenna aperture⁴⁶, $\lambda _{\omega }$ is the radiation wavelength, and g is the antenna gain. Consideringm as an example, the half-wavelength dipole antenna, for which $|\delta V_{\omega }|^2 \simeq I_{\omega } (8\pi /c)(\lambda _{\omega }/\pi )^2$, where c is the speed of light in vacuum, we obtain $|\delta V_{\omega }|^2 = 32S_{\omega }/gc$.

Accounting for Eqs. (18) and (19), we obtain

$$\begin{aligned} {{\mathscr {R}}}_{\omega } = \frac{32}{g c}\frac{<\overline{\delta J_{\omega }}>^{a,s}}{|\delta V_{\omega }|^2}, \qquad {{\mathscr {R}}}_{\omega }^V = \frac{32}{gc}\frac{<\overline{\delta J_{\omega }}>^{a,s}}{|\delta V_{\omega }|^2}\,\rho . \end{aligned}$$

(21)

The latter equations yield

$$\begin{aligned} {{\mathscr {R}}}_{\omega }= {{\mathscr {R}}}_0 {{\mathscr {F}}}(V_G){{\mathscr {P}}}^{a,s}_{\omega },\qquad {{\mathscr {R}}}_{\omega }^V = {{\mathscr {R}}_0}^V {{\mathscr {F}}}(V_G)\,{{\mathscr {P}}}_{\omega }^{a,s}, \end{aligned}$$

(22)

where

$$\begin{aligned} {{\mathscr {R}}}_0 =\frac{16}{g}\frac{e\sigma _0}{{{\overline{T}}} c} \frac{\tau _{\varepsilon }}{\tau _{\bot }}\frac{H}{L},\qquad {{\mathscr {R}}}_0^V= \displaystyle \frac{32}{g}\frac{e}{{{\overline{T}}}c}\frac{\tau _{\varepsilon }}{\tau _{\bot }}. \end{aligned}$$

(23)

According to Eq. (23), the characteristic voltage responsivity ${{\mathscr {R}}}_0^V$ does not explicitly depend on the frequency of electron collisions $\nu $.

It is instructive that the responsivity at $V_G=0$ does not turn to zero because of the factor ${{\mathscr {F}}}(0) \ne 0$, so that $<\overline{\delta J_{\omega }} >0$.

Source link