Bloch gain in dc-ac-driven semiconductor superlattices in the absence of electric domains

We theoretically study the feasibility of amplification and generation of terahertz radiation in dc-ac-driven semiconductor superlattices in the absence of electric domains. We find that if in addition to a dc bias a strong terahertz pump field is applied, a Bloch gain profile for a small terahertz signal can be achieved under the conditions of a positive static differential conductivity. Here, the positive differential conductivity arises, similarly to the case of a large-signal amplification scheme [H. Kroemer, arXiv:cond-mat/0009311 (unpublished)], due to modifications in dc current density caused by the application of a high-frequency ac field [K. Unterrainer et al., Phys. Rev. Lett. 76, 2973 (1996)]. Whereas the sign of absorption at low and zero frequencies is sensitive to the ac fields, the gain profile in the vicinity of the gain maximum is robust. We suggest to use this ac-induced effect in a starter for a terahertz Bloch oscillator. Our analysis demonstrates that the application of a short terahertz pulse to a superlattice allows the suppression of the undesirable formation of electric domains and the achievement of a sustained large-amplitude operation of the dc-biased Bloch oscillator.

Figure 1

(Color online) The Esaki–Tsu characteristic. The figure demonstrates the quantum derivative for the frequency $\omega =3{\tau }^{-1}$ taken at the operation point $E_{\mathit{dc}}=5E_{\mathit{cr}}$. Absorption at frequency $\omega$ is proportional to the difference quotient, Eq. (5), which is the slope of the inclined (red online) segment. Inset: Small-signal gain as a function of $\omega$ for $E_{\mathit{dc}}=5E_{\mathit{cr}}$. The Bloch gain exists almost up to $\omega \approx {\omega }_B=5{\tau }^{-1}$ with the maximum of gain at $\omega \approx {\omega }_B-{\tau }^{-1}=4{\tau }^{-1}$.

Figure 2

$VI$ characteristic of a superlattice modified by the strong monochromatic field with $E_{\delta }=5E_{\mathit{cr}}$ and $\omega \tau =5$. Note the appearance of the photon-assisted peak centered at $E_{\mathit{dc}}∕E_{\mathit{cr}}=6$. At this peak, the differential conductivity is positive for $4=\omega \tau -1<E_{\mathit{dc}}∕E_{\mathit{cr}}<\omega \tau +1=6$.

Figure 3

(Color online) Generated power density as a function of the probe field amplitude $E_{\delta }$ for fixed $E_{\mathit{dc}}=5.5E_{\mathit{cr}}$ and $\omega \tau =5$. The dark gray (red online) segment indicates the range of negative differential conductivity. If one assumes that the electric instability due to NDC at small probe amplitudes does not prevent the field in the resonator from growing, then the generation with a positive differential conductivity can be achieved for $E_{\delta }∊[2.5E_c,10.8E_c]$.

Figure 4

(Color online) Regions of NDC (red online) and gain (marked area bounded by the blue curve) for $\omega \tau =5$. The region of gain overlaps with the region of positive differential conductivity (blank) for $5=\omega \tau <E_{\mathit{dc}}∕E_{\mathit{cr}}<\omega \tau +1=6$ and $E_{\delta }>2E_{\mathit{cr}}$. The figure indicates the possibility of reaching a large-signal amplification in a dc-biased SL without NDC.

Figure 5

(Color online) $VI$ characteristic of SL under the action of a strong pump field ($\Omega \tau =5$, $E_{\mathit{ac}}=6E_{\mathit{cr}}$). In addition to the modified Esaki–Tsu peak, two photon-assisted peaks are visible. The operation point is chosen at the part of the second peak with a positive slope $(E_{\mathit{dc}}=9.8E_{\mathit{cr}})$. The quantum derivative [see Eq. (15)] at this operation point is negative for a wide range of $\omega$ values demonstrating the possibility of small-signal gain without NDC. Inset: Small-signal absorption as a function of the probe frequency $\omega$ at $E_{\mathit{dc}}=9.8E_{\mathit{cr}}$. There are two distinct gain resonances at frequencies $\omega \tau \approx 3.8$ and $\omega \tau \approx 8.8$.

Figure 6

(Color online) Small-signal absorption as a function of the probe frequency $\omega$ for $\Omega \tau =5$ and different values of the pump amplitude $E_{\mathit{ac}}∕E_{\mathit{cr}}≕0,1.2,2.1,3$. The gain profile near the maximum of gain only slightly changes with increase in the pump amplitude. However, the absorption at low frequencies changes from negative to positive indicating the transition from NDC (online red curves) to PDC (online black curves) at the operation point $E_{\mathit{dc}}=4.8E_{\mathit{cr}}$.

Figure 7

(Color online) Regions of NDC (red online) and gain (marked area bounded by the blue curve) for $E_{\mathit{ac}}=2E_{\mathit{cr}}$, $\Omega \tau =5.5$, and $\omega \tau =5$. The figure illustrates the feasibility of gain in conditions of suppressed electric instability for the wide ranges of signal amplitudes $E_{\delta }$ and bias values $E_{\mathit{dc}}$ under the action of a moderate ac pump.

Figure 8

(Color online) Regions of NDC (red online) and gain (marked) in the plane of pump-probe amplitudes for (a) $\Omega \tau =5$, ${\omega }_B\tau =4.8$ and $\omega \tau =4$ and (b) $\Omega \tau =5$, ${\omega }_B\tau =9.8$ and $\omega \tau =9$. Gain can exists without NDC. The arrows schematically illustrate the use of an ac pump in the terahertz starter for a Bloch oscillator (see Sec. 4 for details).

Figure 9

(Color online) Generated power density as a function of the probe amplitude for several increasing values of the pump. The black curves correspond to the generation at the first photon-assisted peak ($\Omega \tau =5$, $E_{\mathit{dc}}=4.8E_{\mathit{cr}}$, $\omega \tau =4$) induced by the ac pump with $E_{\mathit{ac}}∕E_{\mathit{cr}}=:2.1,3.5$. The gray (brown online) curves demonstrate that the generated power at the second photon-assisted peak ($\Omega \tau =5$, $E_{\mathit{dc}}=9.8E_{\mathit{cr}}$, $\omega \tau =9$) more strongly depends on the pump amplitude $(E_{\mathit{ac}}∕E_{\mathit{cr}}=:6,7)$. As can be seen from Fig. 8, the generation in both cases can be realized without the electric instability.

Figure 10

(Color online) Regions of NDC (red online) and gain (marked) for ${\omega }_B\tau =4.8$, $\omega \tau =4$, and the trichromatic pump field [Eq. (21)] with $\Omega \tau =5$, ${\Omega }_1\tau =5.5$, and ${\Omega }_2\tau =4.75$. The comparison of this figure with Fig. 8a demonstrates that the realization of gain in conditions of PDC under the polychromatic pump requires larger values of $E_{\mathit{ac}}$ than in the case of a monochromatic pump.

Figure 11

(Color online) Regions of NDC (red online) and gain (marked) in dc-biased superlattice [Eq. (2)] for the probe field with the frequency $\omega \tau =5$ and for several values of $\nu$: (a) $\nu =1$, (b) $\nu =0.4$, and (c) $\nu =0.1$. The plots illustrate the feasibility of a large-signal amplification under suppressed electric instability for various ratios of the scattering constants.

Figure 12

(Color online) Regions of NDC (red online) and gain (marked) in an ac-pumped superlattice [Eq. (10)] for $\Omega \tau =5$, $\omega \tau =4$, ${\omega }_B\tau =4.8$, and several values of $\nu$: (a) $\nu =1$, (b) $\nu =0.4$, and (c) $\nu =0.1$. The plots illustrate the feasibility of terahertz generation under suppressed electric instability for various ratios of the scattering constants.