### Theoretical background

Let us consider the electron in the quantum well structure in the tilted magnetic field

$\mathrm{B}={B}_{\parallel}{\mathrm{e}}_{y}+{B}_{\perp}{\mathrm{e}}_{z}$, where z is the growth axis. In Landau gauge

$\mathrm{A}=\left({B}_{\parallel}z-{B}_{\perp}y\right){\mathrm{e}}_{x}$, the electron envelope wave function is given by [

2]

$\psi \left(x,y,z\right)=\frac{exp\left(ikx\right)}{\sqrt{L}}\xb7f\left(y-{\ell}_{\perp}^{2}k,z\right)$

(2)

where component

$f\left(y,z\right)$ is determined by a two-dimensional Schroedinger equation

${H}_{2\text{D}}f\left(y,z\right)=Ef\left(y,z\right)$

(3)

with Hamiltonian

${H}_{2\text{D}}={H}_{\perp}+{H}_{\text{tilt}}$

(4)

where

${H}_{\perp}=-\frac{\partial}{\partial z}\frac{{\hslash}^{2}}{2m\left(z\right)}\frac{\partial}{\partial z}+{V}_{\text{QW}}\left(z\right)-\frac{{\hslash}^{2}}{2m\left(z\right)}\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\hslash}^{2}}{2m\left(z\right)}\frac{{y}^{2}}{{\ell}_{\perp}^{4}}$

(5)

is the Hamiltonian for the case of magnetic field

$\mathrm{B}={\u0412}_{\perp}{\mathrm{e}}_{z}$ normal to the structure layers, and

${H}_{\text{tilt}}=\frac{{\hslash}^{2}}{2m\left(z\right)}\xb7\frac{{z}^{2}}{{\ell}_{\parallel}^{4}}-\frac{{\hslash}^{2}}{m\left(z\right){\ell}_{\perp}^{2}{\ell}_{\parallel}^{2}}\xb7yz\text{.}$

(6)

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.

In the case of the magnetic field

$\mathrm{B}={\perp}_{}{\mathrm{e}}_{z}$ being normal to the structure layers, the variables in the Schroedinger equation are separated, and energy levels and electron wave functions are given by the expressions [

3]

$E{\left(\nu ,n,\right)}_{}={\u03f5}_{\nu}+\hslash {\omega}_{\perp}\xb7\left(n+1/2\right)$

(7)

and

$f{\left(\nu ,n,\right)}_{}\left(y,z\right)={\varphi}_{\nu}\left(z\right){\Phi}_{n}\left(y\right)\text{,}$

(8)

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.

It can be easily seen that in this case, the dipole matrix element

${\mathbf{D}}_{\left(2,0\right)\to \left(1,1\right)}=\left.\u3008\frac{exp\left(i{k}_{1}x\right)}{\sqrt{L}}{f}_{\left(2,0\right)}\left(y-{\ell}_{\perp}^{2}{k}_{1},z\right)\right|\mathbf{r}\left|\frac{\text{exp}\left(i{k}_{1}x\right)}{\sqrt{L}}{f}_{\left(1,1\right)}\left(y-{\ell}_{\perp}^{2}{k}_{1},z\right)\right.\u3009$

(9)

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.

However, the matrix element of the specified transition can be made nonzero by applying an additional component

${B}_{\u2551}$ of the magnetic field parallel to the layers, that is, by tilting the magnetic field with respect to the structure layers. Now, due to an additional term

${H}_{\text{mix}}=-\frac{{\hslash}^{2}}{m\left(z\right)}\frac{yz}{{\ell}_{\parallel}^{2}{\ell}_{\perp}^{2}}$

(10)

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].

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

$\hslash {\omega}_{\perp}<\Delta {E}_{12}$. The structure of a single-electron spectrum in the tilted magnetic field in this case does not change significantly [

4–

10]. The main effect of

${B}_{\parallel}$ is the shift of the harmonic oscillator center in Equation

8 by the value

$\frac{{\ell}_{\perp}^{2}}{{\ell}_{\parallel}^{2}}{\u3008z\u3009}_{\nu}$[

7], where

${\u3008z\u3009}_{\nu}={\displaystyle \int dz{\left|{\varphi}_{\nu}\left(z\right)\right|}^{2}z}$ is the average value of the electron coordinate along the z axis in the νth subband state:

${f}_{\left(\nu ,n\right)}\left(y,z\right)={\varphi}_{\nu}\left(z\right){\Phi}_{n}\left(y-\frac{{\ell}_{\perp}^{2}}{{\ell}_{\parallel}^{2}}{\u3008z\u3009}_{\nu}\right)\text{.}$

(11)

Substituting wave functions (Equation 11) into Equation

9, the following expression can be obtained for the squared modulus of the dipole matrix element:

${\left|{\mathbf{D}}_{\left(2,0\right)\to \left(1,1\right)}\right|}^{2}={\delta}_{{k}_{1},{k}_{2}}\xb7{\left|\left.\u3008{\varphi}_{2}\left(z\right)\right|z\left|{\varphi}_{1}\left(z\right)\right.\u3009\right|}^{2}\xb7G\left(\xi \right)\text{,}$

(12)

where

$G\left(\xi \right)={\xi}^{2}/2\xb7exp\left(-{\xi}^{2}/2\right)\text{,}$

(13)

$\xi =\left[{\u3008z\u3009}_{2}-{\u3008z\u3009}_{1}\right]\xb7\frac{{\ell}_{\perp}}{{\ell}_{\parallel}^{2}}\text{.}$

(14)

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.