Journal of the Korean Physical Society, Vol. 56, No. 3, March 2010, pp. 801∼804
Variation of Dynamic Fano Resonances in Semiconductor Superlattices Driven by Dc and THz Fields Koo Chul Je∗ Department of Physics, College of Liberal Arts and Sciences, Anyang University, Anyang 708-113 (Received October 26 2009) We describe the dynamic Fano resonances in a biased three-dimensional semiconductor superlattice driven by combined dc and THz fields. The Fano resonances on the Wannier Stark ladders are dominantly changed by the THz field via the band collapse phenomenon. The Fano coupling matrix is closely related to the separated energy distance between the two types of exciton resonances produced by the dc and the THz fields. The dynamic Fano coupling strength increases as the exciton resonance induced by the THz field approaches the Wannier exciton states. PACS numbers: 73.21.Cd, 73.23.-b, 73.50.Fq Keywords: Dynamic Fano resonance, THz fields, Miniband collapse, Dynamical localization, Dynamical Stark ladders DOI: 10.3938/jkps.56.801
ulated by the strength of the THz field via changing the localization properties of the excitons due to the band collapse phenomenon. The periodic form is directly related to the changing bandwidth with the spherical Bessel function Jn (f ). The dynamic Fano coupling strength is determined by the separated energy distance between the hh0 exciton resonance with the THz field and the hh0 exciton resonance without the field. As the energy distance increases, the Fano coupling decreases. In general, for an increasing static bias field, the axial localization is stronger, and the Wannier Stark spacing increases. Thus, the momentum mismatch between the discrete exciton and the continuum state increases. This leads to a decrease in the coupling matrix, which is a coupling of the exciton to the continuum states of the same well. Thus, the Fano resonance strength for a given Wannier Stark ladder transition generally decreases with increasing static bias field [2].
I. INTRODUCTION
The Fano effect occurs through the quantummechanical cooperation between resonance and interference [1] and is dominantly changed by the applied electric fields. The Fano interference, which has been found ubiquitous in a large variety of experiments including neutron scattering, atomic photoionization, Raman scattering, and optical absorption, is controlled by the magneticfield, the electrostatic field, and the time-dependent electric field [2–10]. The Fano interference leads to asymmetrical Fano line in the linear absorption spectra. The Fano resonances (FR) in a biased semiconductor superlattice were observed to result from the coupling of Wannier Stark ladders to the adjacent continuum [2]. Recently, the nonlinear Fano effect was observed in self-assembled quantum dots embedded in a layer [3,4]. When the bias of an intense THz field was applied to the semiconductor superlattices, the dynamical Fano resonance (DFR) predicted by the coupling between a discrete quasi-energy exciton and the continuum of quasienergy excitons associated with a neighbor sideband was investigated by using the one-dimensional tight-binding model [5]. In this paper, we investigate the FR effect due to the coupling of Wannier Stark ladders and dynamical Stark ladders with the continuum states of minibands by using realistic three-dimensional model calculations on the excitornic absorption spectra of dc- and THzbiased GaAs/AlGaAs semiconductor superlattices. We find that the dynamic Fano coupling strength is mod∗ E-mail:
II. THEORY The combined dc and THz fields are given by F (t) = Fdc +Fac cos ωL t, where Fdc and Fac are the strengths of the dc and the THz fields with frequency ~ωL = 10 meV , respectively. Under the combined external fields, the motion of a carrier in an energy band can be described by the acceleration theorem as k(t) = k0 − eF~dc t − eFT Hz ~ωL sin(ωL t + φ). The energy states change from the initial state (k0 ) without external fields to (k(t)). When the external field F (t) is applied along the growth axis of the semiconductor superlattice, the semiconduc-
je
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Journal of the Korean Physical Society, Vol. 56, No. 3, March 2010
tor Bloch equations can be expressed as [11–13] ~∂t Pij;k (t) = −iij (k(t)) Pij;k (t) +ieF (t)∂k Pij;k (t) +iµk E(t) Pij;k (t) X 0 0 V ij;i j (k, k0 ) Pij;k0 (t) +i i0 ,j 0 ;k0
−~γ2 Pij;k (t) .
(1)
Here, Pij;k ≡ is the interband polarization, where i(j) is an electron (hole) miniband index and αi,k (βj,−k ) is the electron (hole) annihilation operator for the quasi-momentum state i(j). ij (k) is the bandgap energy between the i- and the j-minibands in the presence of external fields, (ij (kz ) is obtained from the Kr¨onig-Penney equation for the superlattice), and V (k, k0 ) denotes the Coulomb matrix elements [14, 15]. E(t) is the incident optical laser field, and µk is the optical dipole matrix element. γ2 is the dephasing rate, which phenomenologically accounts for the scattering effect beyond the Hartree-Fock approximation in the optical absorption spectra. From the Fourier P transformation of the optical response P (t) ≡ ij,k µij;k Pij;k , the absorption coef-
Fig. 1. Absorption spectra in the dc- and the THz-driven superlatice for various strengths of the combined fields, (n, f ). [m + m0 ] denotes the Stark ladder index, where m and m0 are the Wannier Stark and the dynamical Stark ladder indices, respectively. hh0 is the heavy-hole exciton resonance.
P (ω) ω ficient is obtained as α(ω) = c n(ω) Im( E(ω) ), where n(ω) is the background refractive index. We introduce the numerical values of the parameters ωB = eFdc d/~, f = eFT Hz d/~ωL , and n = eFdc d/~ωL with a structural parameter d. The miniband width is proportional to Jn (f ) as ∆F = ∆F =0 Jn (f ). Dynamical localization occurs under the conditions of Jn (f ) = 0, where the minibands are completely collapsed [16–20]. The lineshapes of the absorption peaks vary drastically in the dynamical localization condition. The asymmetric excitonic peaks have the fallowing Fano lineshape [1]:
α() =
(q + ε)2 , 1 + ε2
(2)
where q is the lineshape parameter and ε(ω) = (ω−ω0 )/Γ is a normalized detuning energy from the resonance with a width Γ that corresponds to the resonance broadening due to the Fano coupling.
Fig. 2. (a) Fano coupling parameter Γ, (b) inverse of the Fano line shape parameter q obtained by fitting with hh0 exciton resonances, (c) relative oscillator strength of the hh0 exciton, and (d) changing energy of the exciton resonance as functions of THz-fields f . (α0 denoted in an unit is a peak of the hh0 exciton without external fields.)
III. RESULTS AND DISCUSSION We assume a superlattice of 111-˚ A GaAs well layers and 17-˚ A AlGaAs barrier layers. The width of combined miniband is set as ∆0 = ∆c + ∆v = 14 meV , where ∆c (∆v ) is the width of the first conduction (valence) miniband. Figure 1 shows the absorption spectra for various field strengths of (n, f ). The dc field generates Wannier Stark ladders with an energy spacing m~ωB around the heavy-hole(hh0 ) exciton resonances, and the THz field generates dynamical Stark ladders with a spacing
m0 ~ωL . Here, m and m0 are integers. The absorption spectrum consists of a series of peaks with energies of ij (k0 ) + m~ωB + m0 ~ωL , and the discrete quasienergy states can be overlapped by the Stark ladders [12]. The ij (k0 ) of the hh0 exciton resonance and its oscillator strength strongly depend on the band collapse phenomenon that determines the coupling of the resonance on the continuum states of minibands. They lead to the dynamic Fano effects. Figure 2 shows the FR of the hh0 exciton and the
Variation of Dynamic Fano Resonances in Semiconductor Superlattices Driven by Dc and THz Fields – Koo Chul Je
Fig. 3. (a) Fano coupling parameter Γ, (b) inverse of the Fano line shape parameter q obtained by fitting with hh0 exciton resonances, (c) oscillator strength of the hh0 exciton relative to the oscillator strength (α0 ) with intensity n = 1 of the dc field and (d) changing energy of the exciton resonance as functions of the THz fields f . The dotted line plots J1 (f ) as a guide for the eye.
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of bandwidth due to the band collapse phenomenon induced by the THz field can be estimated from the function ∆F = ∆F =0 Jn=1 (f ) [16]. When the THz field leads to delocalization, the bandwidth becomes wide, and the exciton resonance shifts far away from the Wannier exciton resonance generated by the dc field. When the THz field leads to localization, the bandwidth becomes narrow, and the hh0 exciton resonance approaches that of the Wannier exciton and is strongly related to J1 (f ). The 1/q and the Γ have similar behaviors with respect to the oscillator strength and the excitonic position. The Fano matrix has the same behavior for various hh0 excitonic positions, which means that the DFR is determined by the band collapse phenomenon. The Fano matrix increases, as the exciton resonance approaches the Wannier exciton state. The reverse is also true. Therefore, we can see that Fano coupling is determined by the separated energy distance between the two types of exciton resonances, which are Wannier Stark ladders and dynamical Stark ladders.
IV. CONCLUSIONS deviation of the exciton resonance peak in the presence of only THz fields. The width of the miniband, which corresponds to the continuum states in the Fano coupling matrix, is completely collapsed at f = 2.43, that is, ∆F = ∆F =0 J0 (f = 2.43) = 0. This means dynamical localization where the Fano effect is suppressed. For further strengths of the field (f > 2.43), the electrons are delocalized again. For low strengths of the THz fields (f < 0.75), the Fano coupling parameter increases with increasing THz field, but the inverse line shape parameter decreases, as in the case of a dc field. The amplitude of the electronic contribution to the exciton wave function is become to be strongly reduced in its center well, but to have high values in the neighboring wells [2]. Figure 2(c) shows that the hh0 exciton peak shifts to high energies and that its oscillator strength is smaller than the oscillator strength of the exciton without an external field. This peculiar behavior also appears in the presence of a THz field. In the region, the Γ increases; that is, homogeneous broadening of the hh0 exciton state is enhanced in spite of the blue shift of the exciton peak. For further increases in the strengths of the fields, the homogeneous broadening decreases until dynamical localization occurs with increasing oscillator strength. The inverse of the line shape parameter decreases as the hh0 exciton peak shifts in high energies. For further increasing in the strengths of the fields, the 1/q increases with increasing strength of the THz fields, because the electrons are delocalized. Figure 3 shows the FR for different strengths of the THz fields in the presence of Wannier Stark ladders. The Fano coupling and the inverse parameter are also directly related to the miniband collapse. The variation
We find that the dynamic Fano coupling strength is modulated by the strength of the THz field by changing the localization properties of the excitons due to the band collapse phenomenon. In the case of only a THz field, there is a region where homogeneous broadening of the hh0 exciton state is enhanced in spite of the increase in the energy distance, because the hh0 oscillator strength is smaller than the oscillator strength of the exciton without external fields. The dynamic Fano coupling strength is determined by the separated energy distance between the hh0 exciton resonance with the THz field and the hh0 exciton resonance without the field. As the energy distance increases, the Fano coupling decreases. Consequently, the dynamic Fano resonance can be controlled by adjusting the excitonic energy distance induced by the band collapse phenomenon. Our calculations can be used to accurately determine the degree of decoherence present in scattering devices, including metal composite quantum dots and quantum dots attached to metal.
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