Abstract
The tunable terahertz intersubband Landau level transitions in resonant tunneling cascade quantum well structures are considered. The way of lifting the selection rule forbidding the interLandau level terahertz transitions of interest by applying a magnetic field tilted with respect to the structure layers is proposed. The importance of asymmetric structure design to achieve considerable values of transition dipole matrix elements is demonstrated.
Keywords:
Quantum well structures; Landau levels; terahertz transitions.Background
Recently, the possibility to achieve a population inversion in the system of Landau levels (LL) in cascade quantum well structures in strong magnetic field under a condition of sequential resonant tunneling, i.e., in strong transverse electric field, was shown [1]. If the spacing between the first and any upper (νth) subbands is lower than the optical phonon energy (i.e., when the optical phonon scattering is suppressed), the population of zeroth LL in νth subband can exceed that of the first LL in the first subband. So, the stimulated emission of terahertz radiation can be achieved on the transitions between these LLs, and the emission frequency may continuously be tuned in a wide range of terahertz frequencies by the variation of the magnetic field strength according to the relation
where
Figure 1. A scheme of transitions between Landau levels in a quantum well. The thick arrow indicates the (2, 0) → (1,1) radiative transition, and the wavy arrows mark the transitions due to the electron–electron scattering.
In [1], we proposed a possible way to overcome the difficulty and to provide nonzero matrix element values for transitions of interest by tilting the magnetic field with respect to the structure layers. In the present paper, we investigated the effect of magnetic field tilt on the optical matrix element of the intersubband Landau level transitions. The importance of an asymmetric structure design to achieve substantial values of transition dipole matrix elements was revealed, and an asymmetric twowell periodic structure was proposed as a possible solution maximizing the optical matrix element of the terahertz transitions of interest.
Methods
Theoretical background
Let us consider the electron in the quantum well structure in the tilted magnetic
field
where component
with Hamiltonian
where
is the Hamiltonian for the case of magnetic field
Here,
In the case of the magnetic field
and
where
It can be easily seen that in this case, the dipole matrix element
is exactly equal to zero for any polarization due to the orthogonality of subband
(
However, the matrix element of the specified transition can be made nonzero by applying
an additional component
arising in Equation 4, the variables in the Schroedinger equation are no longer separated, resulting in the mixing of inplane and outofplane electron motions [4] and lifting of the above selection rule. The effect is similar to the violation of the Δn = 0 selection rule for the resonant tunneling transitions between the Landau levels in the tilted magnetic field [410].
Here, we will consider the situation when the matrix element of the Hamiltonian (Equation
6) over the first and second subband stated (Equation 8) is much lower than the subband
spacing. This is the case in the magnetic field range when
Substituting wave functions (Equation 11) into Equation 9, the following expression can be obtained for the squared modulus of the dipole matrix element:
where
From this expression, one can see that the dipole matrix element becomes nonzero only
if the values
In symmetric well potential
Results and discussion
The simplest solution is to apply an electric field along the structure growth axis especially as the electric field is necessary to provide the resonant tunneling pumping of the LLs of the upper subband. In Figure 2, the calculated dependence of the dipole matrix element of the (2,0) → (1,1) transition in the GaAs/AlGaAs quantum well on the applied electric field is shown. It can be seen that the application of the electric field results in the nonzero dipole matrix element. Nevertheless, since the possible values of the electric field strength are determined by resonant tunneling conditions and cannot be selected independently, this way of providing a nonzero matrix element is not very effective.
Figure 2. Calculated dependencies of the dipole matrix element on the applied electric field. Squared modulus of the dipole matrix element  D_{(2,0)→(1,1)}^{2} versus the voltage drop eFa per quantum well for different values of the parallel component of the magnetic field B _{∥} = 1 to 5 T. The calculation was performed for the 250Å GaAs/Al_{0.3} Ga_{0.7}As quantum well. The transverse component of the magnetic field is B _{⊥} = 5 T.
More effective is the use of an asymmetric design of the structures themselves. One
of the possible solutions is to introduce an asymmetric double well as an active element
of the periodic cascade quantum well, consisting of two strongly coupled wells with
different widths (Figure 3a), as an active element of the periodic cascade quantum well structure. In the asymmetric
double quantum well, the first subband wave function is located mainly in the wider
well, while the second subband wave function is shifted to the narrower well (Figure
3b). As a result, a significant difference between average coordinates
Figure 3. An asymmetric double well as an active element in periodic cascade quantum well structure. ( a) Proposed design of the active element of the periodic cascade quantum well structure and the calculated wave functions of the first and second subbands. ( b) The dependence of the 1 to 2 intersubband spacing on the narrow well width a_{R}. (c) The values of the averages 〈z〉_{ν} for corresponding subbands as a function of narrow well width a_{R}.
The dipole matrix element for transitions between the zeroth LL of the second subband and the first LL of the first subband is presented in Figure 4 as a function of narrow quantum well width a_{R.}. Here, the width of wider well is fixed, while the width a_{R} of the narrow well is varied. A pronounced maximum can be seen at a_{R} = 110 Å, and the maximum achievable value of D_{(2,0)→(1,1)}^{2} is considerably higher than that in previously considered case of the single symmetric quantum well in transverse electric field (see Figure 2 for comparison).
Of course, the structure considered is an example proposed here to illustrate the general way of how the selection rule forbidding the transitions of interest can be overcome. More detailed simulations, including the direct calculations of the tunneling characteristics and optical gain, are necessary to optimize the structure design.
Conclusions
Finally, the terahertz transitions between Landau levels of different subbands in resonant tunneling quantum well structures in a tilted magnetic field were considered. An effective way was proposed to lift the selection rule forbidding the intersubband interLandau level transitions by placing the structures into the tilted magnetic field. An importance of asymmetrical structure potential was revealed, and the possibility to achieve considerable values of interLandau level transition matrix element was demonstrated for an asymmetric doublewell structure.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
MPT participated in the general task formulation and carried out the theoretical background and intersubband matrix element calculations. YAM of conceived the general task formulation, participated in the study design and coordination, and drafted the manuscript. PFK carried out the simulation program development and energy band structure calculations and participated in the sequence alignment. All authors read and approved the final manuscript.
Acknowledgments
The work was supported by the Russian Basic Research Foundation (grants nos. 090200671 and 080292505NCNIL), the RF President grant for young scientists (no. МК916.2009.2), the MISIS grant no. 3400022, and the Ministry of Science and Education of the Russian Federation program “Scientific and Pedagogical Personnel of Innovative Russia in 2009–2013”.
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