Quantum derivatives and terahertz gain in a superlattice
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TURES AND LOW-DIMENSIONAL SYSTEMS
Quantum Derivatives and Terahertz Gain in a Superlattice A. V. Shorokhova and K. N. Alekseevb a
Ogarev Mordovian State University, Saransk, 430000 Russia b University of Oulu, Oulu, 90579 Finland e-mail: [email protected]
Abstract—Simple formulas describing terahertz absorption and gain in a semiconductor superlattice irradiated by a microwave pump field are derived for the case when the signal frequency is a half harmonic of the pump. A simple qualitative analysis provides a geometric interpretation of the derived formulas, which can be used to determine if gain is feasible. PACS numbers: 73.21.Cd, 07.57.Hm, 72.20.Ht DOI: 10.1134/S1063776107070436
Superlattices have unique properties that make it possible to amplify and detect high-frequency (0.3–10 THz) radiation [1]. Theoretically, radiation of this kind can be generated and amplified in a superlattice under a dc bias causing a negative differential conductance [2]. However, a superlattice that has a negative static differential conductance is unstable with respect to formation of high-field domains, which makes terahertz gain impossible [3]. Various solutions to this problem are currently being proposed. However, a weak high-frequency field can also be amplified when an ac field is applied to the superlattice [4]. It is important that adverse effects of negative differential conductance can be avoided if a microwave pump field is used instead of a dc bias E0 [5]. In this approach, a strong microwave field
the case of ω2 = mω1 (m ∈ ), which made it possible to determine optimum conditions for absorption from geometrical considerations entirely based on a simple qualitative analysis. In this paper, we extend our approach to the case of ω2 = mω1/2, where m is an odd number, and show that quantum derivatives can be used to determine if gain is feasible by a simple geometric analysis. Consider the response of miniband electrons to the field E ( t ) = E 0 + E 1 cos ω 1 t + E 2 cos ω 2 t.
(1)
Normalized time-averaged superlattice absorption is defined as T
2 A ( ω 2 ) = 2 〈 V ( t ) cos ω 2 t〉 t = --- dtV ( t ) cos ω 2 t, (2) T
∫
E p ( t ) = E 1 cos ω 1 t
0
interacts with electrons in the superlattice, whereas the signal wave E s = E 2 cos ω 2 t has a higher frequency (ω2 > ω1). In a real device, Es may be a cavity mode tuned to the desired terahertz frequency. The detailed analysis of this scheme presented in [5] has shown that a quasistatic pump field can be used to completely suppress domain formation in a superlattice. The objective of our research is a further study of this generation scheme. In a typical experiment [6], the interaction between a microwave field and miniband electrons is quasistatic since ω1τ 1, where τ ≈ 100 fs is a relaxation time. The signal field can or cannot be treated as quasistatic depending on whether ω2τ 1 or ω2τ ≥ 1, respectively. The latter case, when ω1τ 1 and ω2τ ≥ 1, can be referred to as semiquasistatic interaction [7]. In [7], simple expressions having the form of quantum d
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