# Intersubband terahertz transitions in Landau level system of cascade GaAs/AlGaAs quantum well structures in strong tilted magnetic field

- Maksim P Telenkov
^{1, 2}Email author, - Yury A Mityagin
^{1}and - Petr F Kartsev
^{3}

**7**:491

**DOI: **10.1186/1556-276X-7-491

© Telenkov et al.; licensee Springer. 2012

**Received: **18 July 2012

**Accepted: **21 August 2012

**Published: **31 August 2012

## 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 inter-Landau 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

*ν*th) subbands is lower than the optical phonon energy (i.e., when the optical phonon scattering is suppressed), the population of zeroth LL in

*ν*th 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

*ω*

_{ c }is the cyclotron frequency. A scheme of transitions between Landau levels of subbands 1 and 2 in a quantum well structure considered in [1] is shown in Figure 1. The main problem arising is that in a magnetic field directed perpendicularly to the structure layers, the optical transition of interest, shown in Figure 1, is forbidden, i.e., the corresponding dipole matrix element is exactly equal to zero.

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 two-well periodic structure was proposed as a possible solution maximizing the optical matrix element of the terahertz transitions of interest.

## Methods

### Theoretical background

Here, ${V}_{\text{QW}}\left(z\right)$ is the quantum well potential, $m\left(z\right)=\{\begin{array}{l}{m}_{w},z\in \text{w}ell\phantom{\rule{0.1em}{0ex}}\\ {m}_{b},z\in \text{b}arrier\end{array}$ is the effective mass, ${\ell}_{\perp}=\sqrt{\frac{\hslash c}{e{B}_{\perp}}}$ and ${\ell}_{\parallel}=\sqrt{\frac{\hslash c}{e{B}_{\parallel}}}$ are the magnetic lengths for transverse (B_{⊥}) and longitudinal (B_{∥}) magnetic field components and L is the thickness of the structure.

where ${\Phi}_{n}\left(y\right)$ is the wave function of harmonic oscillator with mass ${m}_{w}$ and frequency ${\omega}_{\perp}=e{B}_{\perp}/\left({m}_{w}c\right)$, and ${\u03f5}_{\nu}$ and ${\varphi}_{\nu}\left(z\right)$ are the energy and wave function of νth subband. Here, the small effect of the effective lowering of the barrier height with the increasing of the Landau level number n [3] is neglected.

is exactly equal to zero for any polarization due to the orthogonality of subband ($\u3008{\varphi}_{{\nu}_{1}}|{\varphi}_{{\nu}_{2}}\u3009={\delta}_{{\nu}_{1},{\nu}_{2}}$) and oscillator ($\u3008{\Phi}_{{n}_{1}}|{\Phi}_{{n}_{2}}\u3009={\delta}_{{n}_{1},{n}_{2}}$) wave functions, that is, the considered (2,0) → (1,1) transition is optically forbidden.

arising in Equation 4, the variables in the Schroedinger equation are no longer separated, resulting in the mixing of in-plane and out-of-plane electron motions [4] and lifting of the above selection rule. The effect is similar to the violation of the Δ*n* = 0 selection rule for the resonant tunneling transitions between the Landau levels in the tilted magnetic field [4–10].

From this expression, one can see that the dipole matrix element becomes nonzero only if the values ${\u3008z\u3009}_{1}$ and ${\u3008z\u3009}_{2}$ are substantially different.

In symmetric well potential ${V}_{\text{QW}}\left(z\right)$, the subband wave functions ${\varphi}_{\nu}\left(z\right)$ are symmetric or antisymmetric with respect to symmetry center of the potential, and the averages $\left(z\right)$ are the same for all subbands. So, in symmetric potential, the transition matrix element continues to be close to zero even in the tilted magnetic field. Thus, to provide a nonzero dipole matrix element for transitions of interest along with the application of the tilted magnetic field, it is necessary to introduce an asymmetric potential along the direction of the structure growth.

## Results and discussion

_{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 inter-Landau 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 inter-Landau level transition matrix element was demonstrated for an asymmetric double-well structure.

## Declarations

### Acknowledgments

The work was supported by the Russian Basic Research Foundation (grants nos. 09-02-00671 and 08-02-92505-NCNIL), the RF President grant for young scientists (no. МК-916.2009.2), 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”.

## Authors’ Affiliations

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## Copyright

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