# THz Superradiance from a GaAs: ErAs Quantum Dot Array at Room Temperature

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}may be limited by the lifetime of inversion of a two-energy-level QD which is in the order of 1 ns, and the homogeneous (T

_{2}) and inhomogeneous broadening lifetime (T

_{2}*) of the array is in the range of T

_{2}≈ T

_{2}*~0.1–1 ps (i.e., the dephasing time of the polarization). If the QD array is pumped by a short and strong external pulse, the resulting spontaneous emission from the inversion can become superradiant [6]. However, experimental evidence is rare, and only in the past decade or so has there been confirmation of radiation-field-induced cooperative behavior amongst quantum dots [7]. Here we first present measurements of THz emission from an antenna-coupled QD array pumped by an ultrashort 1550 nm laser. Then we provide evidence suggesting that the THz emission is a consequence of cooperative spontaneous emission from the QD optical dipoles. This can also help explain why these devices have displayed such high efficiency, ~0.2% [8], when used as pulsed THz emitters.

## 2. Materials and Methods

^{20}cm

^{−3}such that the erbium incorporated into the GaAs in the form of ErAs quantum dots. This was proven by transmission electron microscope (TEM) imagery (Figure 1a). The most likely diameter is estimated to be d ≈ 2.0 nm, and the density of quantum dots is n

_{QD}~1 × 10

^{18}/cm

^{3}. These ErAs quantum dots contain several quasi-bound levels near the mid-gap of GaAs, allowing strong bound-to-bound, and possibly bound-to-continuum transitions as well, at around 1550 nm. This is proven by the resonant attenuation coefficient around 1550 nm, derived from the infrared transmission measurements, and plotted in Figure 1b. The peak value is α ≈ 7.4 × 10

^{3}cm

^{−1}.

## 3. Results

#### 3.1. Pulsed Terahertz (THz) Emission

_{THz}α V

^{1.97}. Figure 3b displays the power spectrum from the FFT of the interferogram obtained with a Michelson autocorrelator. It displays a maximum spectral density around 250 GHz.

_{0.53}Ga

_{0.47}As cross-bandgap-excited PC switch with the same 1550 nm setup and the same receiver (Figure 5). No such ringing was observed. Figure 4d shows the power spectrum of 4b after FFT. Similar to Figure 2b, most of the power occurs in a band-limited range between ~100 and 600 GHz. For the purpose of comparison, Figure 4c shows the power spectrum of 4a after FFT. As expected, the 780 nm power spectrum is broader than the 1550 nm power spectrum.

#### 3.2. Efficiency and Photomixing Results

^{−8}. The power dropped continuously as the difference frequency increased but was still detectable up to 1.2 THz, consistent with the THz signal coming from photomixing and not just heating. Thus the broadband pulsed-mode is ~1.5 × 10

^{5}more efficient than the narrowband photomixing mode, which is significantly greater than in cross-bandgap-driven (e.g., 780 nm) THz photoconductive devices. Since superradiance is only expected to be significant in 1550 nm pulsed mode (see below), it is the logical discriminator.

## 4. Discussion

#### 4.1. Quantum Dot (QD) Modeling

_{p}= 1476 nm (Figure 7b) which is equivalent to a photon energy of 0.84 eV, for d = 2.6 nm. Generally speaking, the absorption coefficient α(λ) is proportional to the product of N × A(λ) × g(λ), where N is the population difference between two energy levels or approximately by the number of QDs, and g(λ) is the normalized lineshape function with a fullwidth at half-maximum determined by the homogeneous and inhomogeneous broadening of the QD array [6]. Therefore, the size distribution of QD diameters can broaden the absorption spectrum around the peak wavelength λ

_{p}. Nevertheless, our calculations reveal that the optical transition between two quantum-dot energy levels can explain the resonant behavior around λ = 1550 nm. Furthermore, we estimate that the spontaneous lifetime is on the order of T

_{sp}= 1/[A(λ

_{p})] = 60 ns, which is comparable to the spontaneous lifetime for QDs reported in the literature e.g., References [5,6], and much greater than the spontaneous lifetime in typical bulk direct-bandgap semiconductors.

_{1}, the X-point upper level E

_{2}and a bound-exciton level E

_{3}just below it. The bound exciton should be very stable and have a large oscillator strength. It is analogous to the indirect Γ-X exciton in bulk silicon, but with a much higher binding energy and, therefore, temperature stability. As is well known in many-body theory, the energy levels obtained above through the Schrödinger equation and the EMA is not a complete picture. There is a correlative effect between an excited electron in a quantum dot and the hole that remains in the ground level that the electron started in levels. In semiconductors this correlation is associated with the fundamental quasiparticle called an exciton. The exciton binding energy in the limit of small diameter quantum dots is estimated by the famous Brus formula [11]. Strictly speaking, this is valid only for the ground state, (Γ-point) n = 1, l = 0, to (X-point) n = 1, l = 0 transition in the quantum dots. Evaluation for the 2.6 nm quantum dot (supporting the 1550 nm optical transition) yields E

_{ex,qd}= 0.132 eV, ≈18 times larger than the bulk binding energy. And remarkably, it is much greater than the thermal energy k

_{B}T = 0.025 eV (k

_{B}being Boltzmann’s constant) at T = 300 K, meaning that if such excitons were excited optically, they would be quite stable thermally even in the presence of electron-photon scattering. Not only can excitonic levels in quantum dots be thermally stable, but they have very large, resonant oscillator strength at 1550 nm, too.

_{1}(≈T

_{sp}= 60 ns) even at room temperature. While this sheds light on the poor performance of photomixing results, it raises question on how the lifetime of the inversion of two energy levels can be shortened sufficiently for the fast electron relaxation and then the resulted THz emission. Superradiance provides an “escape route” for this. Under ultrashort strong laser pumping, the almost immediate inversion of QDs triggers spontaneous emission and the QDs cooperate with each other through the same radiated field. This is because the QD size is in the order of ~2 nm, much shorter than the ~1550 nm wavelength, and the QDs’ distribution is dense [7]. The collective interaction of QDs with the same radiation field can reduce the spontaneous lifetime of QDs significantly, resulting in a build-up of the superradiant pulse.

#### 4.2. Comparison of THz Data with Superradiant Theory

_{1}≈ 60 ns. In addition, the modeling of Ref. [6] predicts that the FWHM is less than the dephasing time of the polarization T

_{2}(on the order of ~1 ps) because of the radiation field-QD coupling [6].

_{eff}= 2.65). It can’t be explained by the reflection of the traveling wave from each corner of the square spiral as the length difference of each turn is only ~14 µm (~0.12 ps) either. Instead, we propose that the ringing is a telltale sign of superradiance, which has been observed experimentally in several material systems, and studied theoretically. It originates from the feedback between a radiation field and a collective system of multiple optical dipoles. The radiation field swings both positive and negative, displaying a displacement current-like alternation, which cannot be explained as normal photoconductive relaxation. And the fact that the same device pumped at 780 nm displayed no ringing rules supports this proposal.

_{QD}W

^{2}L = 1.62 × 10

^{8}where n

_{QD}= 1 × 10

^{18}cm

^{−3}is the quantum dot density, W = 9 μm is the gap width (W × W = active area), and L is the quantum-dot layer thickness (2 μm). The instantaneous polarization is the summation over all available dipoles, and

**P(t)**= ${{\displaystyle \sum}}_{i}^{N}{\mathit{p}}_{i}\approx $ N

**p**when the dipoles are initially aligned by the strong ultrafast laser field. The initial tipping angle of the Bloch vector

**P**is approximated by θ

_{0}≈ 1/$\sqrt{N}$ = 7.9 × 10

^{−5}. Assuming the dipoles radiate coherently, we solved the sine-Gordon equation with this initial value and obtained the “ringing” electric field plotted in Figure 8a. Only two other quantities were required for the solution, the superradiance time constant T

_{R}of ≈201 fs, and the individual quantum-dot spontaneous lifetime T

_{sp}≈ 60 ns. The fact that, T

_{R}<< T

_{sp}is an essential aspect of superradiant systems. The comparison between experiment and theory, Figure 4b vs. Figure 8a, is good in the ringing behavior beyond the first peak. Namely, the polarization field oscillates about zero, and is damped over about four periods. As a telltale sign of sine-Gordon behavior, the period increases with time as the ringing evolves, which is why the waveform in Figure 8a is a “quasi-sinusoid” rather than a pure sinusoid.

**H**=

**J**

_{C}+ ∂

**D**/dt =

**J**

_{C}+ (ε

_{r}/χ

_{e}) ∂

**P**/dt where

**J**

_{C}is the conduction current, ∂

**D**/dt (∂

**P**/dt) is the displacement (polarization) current, χ

_{e}is the electrical susceptibility and ε

_{r}is the relative electric permittivity, and the last step follows from classical electrodynamics. In the gap region where the pulse laser is focused, the current is primarily polarization current, but automatically gets converted to a proportionate conduction current in the antenna electrodes, assuming the metal-to-gap junction is an ohmic contact, or nearly so. In fact, this is just another example of the continuity of total current which arises from conservation of total charge, and that goes on in any type of capacitor, which is how Maxwell justified his fourth and final equation.

#### 4.3. Comparison of This Research with Previous Results

^{−3}%) than the photomixing (~5 × 10

^{−7}%), no ringing was observed in its time-domain profile. The working mechanism was attributed to reduction of the photocarrier recombination time by an Auger-related process [14], not the superradiance proposed in this research.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) A TEM image of ErAs quantum dots. The image was taken by an EAG Laboratory in Raleigh, NC, USA. (

**b**) Attenuation spectrum of GaAs: ErAs sample.

**Figure 2.**(

**a**) Square spiral-antenna design, (

**b**) fabricated square spiral antenna with photoconductive gap at the center.

**Figure 3.**(

**a**) Terahertz (THz) power vs. bias voltage at a laser power of 65 mW. (

**b**) Power spectrum obtained from auto-correlation measurement.

**Figure 4.**(

**a**) The waveform in time domain when the pulsed device was pumped with 780 nm pulsed laser. (

**b**) The waveform in time domain when the pulsed device was pumped with 1550 nm pulsed laser. (

**c**) The power spectrum of the 780 nm driven radiation. (

**d**) The power spectrum of the 1550 nm driven radiation.

**Figure 7.**(

**a**) GaAs-ErAs band offsets and the most probable band-to-band transition between a Γ-pt S state (quantum numbers l = 0, n = 1) and a X-pt P state (l = 1, n = 2, m = −1, 0 or 1). (

**b**) Spontaneous emission rate for transition in (

**a**).

**Figure 8.**(

**a**) Analytic solution to sine-Gordon equation the characteristic “ringing” of superradiance. (

**b**) The polarization current and coupling to THz antenna.

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**MDPI and ACS Style**

Zhang, W.; Brown, E.R.; Mingardi, A.; Mirin, R.P.; Jahed, N.; Saeedkia, D. THz Superradiance from a GaAs: ErAs Quantum Dot Array at Room Temperature. *Appl. Sci.* **2019**, *9*, 3014.
https://doi.org/10.3390/app9153014

**AMA Style**

Zhang W, Brown ER, Mingardi A, Mirin RP, Jahed N, Saeedkia D. THz Superradiance from a GaAs: ErAs Quantum Dot Array at Room Temperature. *Applied Sciences*. 2019; 9(15):3014.
https://doi.org/10.3390/app9153014

**Chicago/Turabian Style**

Zhang, Weidong, Elliott R. Brown, Andrea Mingardi, Richard P. Mirin, Navid Jahed, and Daryoosh Saeedkia. 2019. "THz Superradiance from a GaAs: ErAs Quantum Dot Array at Room Temperature" *Applied Sciences* 9, no. 15: 3014.
https://doi.org/10.3390/app9153014