Resonant tunneling diodes in semiconductor microcavities: Modeling polaritonic features in the terahertz displacement current

We develop in this work a qualitative quantum electron transport model, in the strong light-matter coupling regime under dipole approximation, able to capture polaritonic signatures in the time-dependent electrical current. The effect of the quantized electromagnetic field in the displacement current of a resonant tunneling diode inside an optical cavity is analyzed. The original peaks of the bare electron transmission coefficient split into two new peaks due to the resonant electron-photon interaction, leading to coherent Rabi oscillations among the polaritonic states that are developed in the system in the strong coupling regime. This mimics known effects predicted by a Jaynes-Cummings model in closed systems and shows how a full quantum treatment of electrons and electromagnetic fields may open interesting paths for engineering new THz electron devices. The computational burden involved in the multi-time measurements of THz currents is tackled by invoking a Bohmian description of the light-matter interaction. We also show that the traditional static transmission coefficient used to characterize DC quantum electron devices has to be substituted by a new displacement current coefficient in high-frequency AC scenarios.

Figure 1

(a) 3D spatial representation of the transport through a RTD whose active region is inside a cavity, and whose transport direction size is much smaller than the lateral sizes, $L_y,L_z\gg L_x=16$ nm. (b) Zero-photon $|0\rangle$ and single-photon $|1\rangle$ states for the quantized single mode cavity field with energies $\hslash \omega /2$ and $3\hslash \omega /2$. (c) 1D view of the RTD device showing ground $|0\rangle$ and first excited $|1\rangle$ electron states with energies $E_0$ and $E_1$, so that ${\omega }_e=(E_1-E_0)/\hslash$ is in the THz range; the light-matter interaction is effective only inside the active region, while a much larger simulation box is used to deal with open boundary conditions, with $x_{\text{left},\text{right}}$ indicating the positions where wave packets are initialized. (d) $|\text{electron},\text{photon}\rangle$ states inside the RTD/cavity in the resonant strong coupling regime: state $|0,0\rangle$ almost unnafected; polaritonic states formed out of $\left(\right|0,1\rangle \pm |1,0\rangle )/\sqrt{2}$ split by $2E_r=2\hslash {\omega }_r$ in comparison to the degenerate decoupled energies (dashed line); state $|1,1\rangle$ would create another polariton subspace, in a larger basis set, with state $|0,2\rangle$. An external battery yielding a signal $V_{\text{in}}\left(t\right)$ with frequency $f$, as well as an ammeter, are also shown in panel (c). The dimensions of the cavity $L_{x,y,z}^c$ are tens of $\mu \mathrm{m}$ as to yield a frequency $\omega \approx {\omega }_e$.

Figure 2

(a) Potential profile of the simulated RTD device as function of position $x$; well (barriers) width is 16 nm (2 nm). (b) Transmission probability $T\left(E\right)$ from Eq. (64); only ground and first excited electron states are shown at energies $E_0=28$ meV and $E_1=106$ meV.

Figure 3

(a) Evolution of electron probabilities for three scenarios. $P_0(x,t)$ for no light-matter interaction $H_{xq,e}^{\left(D\right)}$ in Eq. (67). (b) $P_S(x,t)$ for semiclassical interaction $H_{xq}^{\left(\text{SC}\right)}\left(t\right)$ in Eq. (76). (c) $P_Q(x,t)$ for quantum interaction $H_{xq}^{\left(D\right)}$ in Eqs. (77) and (78).

Figure 4

Transmission coefficients as function of energy for three different scenarios: (black) no light-matter interaction $T_0\left(E\right)$, (green) semiclassical interaction $T_S\left(E\right)$ for a resonant photon $\omega ={\omega }_e$, and (red) quantum interaction $T_Q\left(E\right)$ for a resonant photon $\omega ={\omega }_e$. The symbols and arrows will be used to explain Fig. 5.

Figure 5

Dependence of $D^{\left(f\right)}(E,0)-T\left(E\right)$ on external frequency $f$ of an AC small signal ($V_A=10$ mV) in three scenarios: (a) no light-matter interaction $D_{0,\mathrm{\Theta }}^{\left(f\right)}(E,0)$; (b) semiclassical interaction $D_{S,\mathrm{\Theta }}^{\left(f\right)}(E,0$); (c) quantum interaction $D_{Q,\mathrm{\Theta }}^{\left(f\right)}(E,0$). Electrons are injected with the DC resonant energy $E=E_1=106$ meV. Two different phases of the input signal, $\mathrm{\Theta }=0$ (black) and $\mathrm{\Theta }=-\pi /2$ (red), are considered.

Figure 6

Time-evolution of the semiclassical Rabi model. (a) Ground- and excited-state probabilities; (b) wave-function distribution $|{\mathrm{\Psi }}_R{\left(t\right)|}^2$; (c) expectation values of position $\langle {\mathrm{\Psi }}_R\left(t\right)\left|x\right|{\mathrm{\Psi }}_R\left(t\right)\rangle$ and momentum $\langle {\mathrm{\Psi }}_R\left(t\right)\left|p\right|{\mathrm{\Psi }}_R\left(t\right)\rangle$ (amplified by 20); (d)“energy” $E\left(t\right)=\langle {\mathrm{\Psi }}_R\left(t\right)|H_R^e+H_R^{\omega }\left(t\right)|{\mathrm{\Psi }}_R\left(t\right)\rangle$; (e) spectral density.

Figure 7

Time-evolution of the quantum Jaynes-Cummings model. (a) Ground- and excited-state probabilities; (b) wave-function distribution $|{\mathrm{\Psi }}_J{\left(t\right)|}^2$; (c) expectation values of position $\langle {\mathrm{\Psi }}_J\left(t\right)\left|x\right|{\mathrm{\Psi }}_J\left(t\right)\rangle$ and momentum $\langle {\mathrm{\Psi }}_J\left(t\right)\left|p\right|{\mathrm{\Psi }}_J\left(t\right)\rangle$; (d) energy $E=\langle {\mathrm{\Psi }}_J\left(t\right)|H_J^e+H_J^{\omega }+H_J^i|{\mathrm{\Psi }}_J\left(t\right)\rangle$; (e) spectral density.

Figure 8

(a) Example of two Bohmian trajectories in the 2D $xq$ plane guided by the analytical evolution from the quantum scenario. (b) Schematic representation of different regions on the $xq$ configuration space in the quantum well, with white (gray) regions corresponding to the wave function (where the values $x\left[t\right]$ and $q\left[t\right]$ have higher probability) occupying the zero (one) photon and excited (ground) electron energy state.