Millimeter waves from frequency generation and optical rectification in quantum dot structure

Abstract

In the form of quantum disks under applied electric field, the second-order nonlinear susceptibility (SONS) in self-assembled quantum dots (QDs) was studied. Nonlinear optical processes results in SONS are discussed, they are: second-harmonic generation (SHG), sum-frequency generation (SFG), difference frequency generation (DFG) and optical rectification (OR). Two types of structures are used the first structure uses the interband transitions in QDs while the second structure uses valence intersubband transitions. SONS relations for SHG, SFG, DFG and OR are derived. A high SONS for the first structure is obtained at long wavelength while a millimeter wavelength is obtained from the second structure which is important in optical communication applications.

Appendices

Appendix 1 (derivation of SHG in interband transition)

From Eq. (A-5) for u = 2, v = 1, u′ = 3, and n = 1

$$\begin{aligned} \frac{\partial }{\partial t}\rho_{21}^{(2)} & = ( - iw_{21} - \gamma_{21} )\rho_{21}^{(2)} \\ & \quad + \frac{i}{\hbar }\left[ {M_{21} \rho_{11}^{(1)} - \rho_{21}^{(1)} M_{11} + M_{22} \rho_{21}^{(1)} - \rho_{22}^{(1)} M_{21} + M_{23} \rho_{31}^{(1)} - \rho_{23}^{(1)} M_{31} } \right]\tilde{E}(t) \\ \end{aligned}$$

(A-1)

where M₃₁ = zero forbidden transition, and $\rho_{11}^{(1)} = \rho_{22}^{(1)} = \rho_{31}^{(1)} = 0$

$$\tilde{E}(t) = E_{s} (2w_{s} )e^{{ - i2w_{s} t}} + E_{p} (2w_{p} )e^{{ - i2w_{p} t}} + c.c.$$

(A-2)

$$\begin{aligned} \rho_{21}^{(2)} (t) & = \rho_{21}^{(2)} (2w_{s} )e^{{ - i2w_{s} t}} \\ & \quad \Rightarrow \frac{\partial }{\partial t}\rho_{21}^{(2)} (t) = \rho_{21}^{(2) \cdot } (2w_{s} )e^{{ - i2w_{s} t}} - i2w_{s} \rho_{21}^{(2)} (2w_{s} )e^{{ - i2w_{s} t}} \\ \end{aligned}$$

(A-3)

$$\begin{aligned} \rho_{21}^{(2) \cdot } (2w_{s} )e^{{ - i2w_{s} t}} - i2w_{s} \rho_{21}^{(2)} (2w_{s} )e^{{ - i2w_{s} t}} & = ( - iw_{21} - \gamma_{21} )\rho_{21}^{(2)} (2w_{s} )e^{{ - i2w_{s} t}} \\ & \quad + \frac{i}{\hbar }\left[ {\rho_{21}^{(1)} (M_{22} - M_{11} )} \right]\tilde{E}(t) \\ \end{aligned}$$

(A-4)

At the steady state

$$\left[ {i\left( {w_{21} - 2w_{s} } \right) + \gamma_{21} } \right]\rho_{21}^{(2)} (w_{s} )e^{{ - i2w_{s} t}} = \frac{i}{\hbar }\left[ {\rho_{21}^{(1)} (M_{22} - M_{11} )E_{s} e^{{ - i2w_{s} t}} } \right]$$

(A-5)

$$\rho_{uv}^{(0)} \, = \,0\,\,\,\,\,for\,\,u\, \ne \,v$$

$$\rho_{21}^{(2)} (2w_{s} ) = \frac{{\rho_{21}^{(1)} (M_{22} - M_{11} )E_{s} }}{{\hbar \left[ {\left( {w_{21} - 2w_{s} } \right) - i\gamma_{21} } \right]}}$$

(A-6)

Similarly,

$$\rho_{21}^{(1)} (2w_{s} ) = \frac{{M_{21} (\rho_{11}^{(0)} - \rho_{22}^{(0)} )E_{s} }}{{\hbar \left[ {\left( {w_{21} - 2w_{s} } \right) - i\gamma_{21} } \right]}}$$

(A-7)

Substituting Eq. (A-7) into (A-6) obtain

$$\rho_{21}^{(2)} (2w_{s} ) = \frac{{M_{21} E_{s}^{2} (\rho_{11}^{(0)} - \rho_{22}^{(0)} )\left( {M_{22} - M_{11} } \right)}}{{\hbar^{2} \left[ {\left( {w_{21} - 2w_{s} } \right) - i\gamma_{21} } \right]^{2} }}$$

(A-8)

Substituting Eq. (A-8) into Eq. (A-7) obtain

$$\varepsilon_{0} \chi^{(2)} (2w_{s} )E_{s} (2w_{s} )e^{{ - i2w_{s} t}} = \frac{N}{V}\left[ {\rho_{21}^{(2)} (2w_{s} )M_{12} e^{{ - i2w_{s} t}} } \right]$$

(A-9)

$$\chi^{(2)} (2w_{s} ) = \frac{{N\mu_{21} \mu_{12} }}{{\hbar^{2} \varepsilon_{0} }}\left\{ {\frac{{(\rho_{11}^{(0)} - \rho_{22}^{(0)} )(\mu_{22} - \mu_{11} )}}{{\left[ {(w_{21} - 2w_{s} ) - i\gamma_{21} } \right]^{2} }}} \right\}$$

(A-10)

Appendix 2: Quantum disk model under applied electric field

Dots are considered as a quantum disks with radius of a and a height of h grown on a wetting layer (WL) in a form of QW with a finite constant potential is assumed for both quantum disk and WL. The Hamiltonian in the cylindrical coordinates (ρ, ϕ, z) is given by

$$H = - \frac{{\hbar^{2} }}{{2m^{*} }}\left[ {\frac{1}{\rho }\frac{\partial }{\partial \rho }\left( {\frac{1}{\rho }\frac{\partial }{\partial \rho }} \right) + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }}} \right] + V$$

(B-1)

where the effective mass is m ^* = m ^* _d inside the disk and m ^* = m ^* _b in the barrier. Similarly, the electric potential is V = V _d inside the disk and V = V _b in the barrier. Solving the Schrodinger equation under the parabolic-band model gives the wave function of the quantum disk. Each state can be characterized by three integral quantum numbers (nml), where n, m and l correspond to ρ-ϕ (transverse) and z dependence, respectively. An approximate wave function of the quantum disk can be obtained (Kim and Chuang 2006) by solving the well-known problems of the two-dimensional circular potential well in the ρ-ϕ direction. In the ρ-ϕ direction, we have a solution of the form

$$\varPsi (\rho ,\varphi ) = \frac{{e^{im\varphi } }}{{\sqrt {2\pi } }}\left\{ \begin{aligned} C_{1} J_{m} (p\rho )\quad \rho \le a \hfill \\ C_{2} K_{m} (q\rho )\quad \rho \rangle a \hfill \\ \end{aligned} \right.$$

(B-2)

where $p = \sqrt {2m_{d}^{*} (E_{\rho } - V_{d} )} /\hbar \quad and\quad q = \sqrt {2m_{b}^{*} (V_{b} - E_{\rho } )} /\hbar$. Note that J _m (pρ) and K _m (qρ) are the Bessel function of the first kind and the modified Bessel function of the second kind, respectively. Using the boundary condition in which the wave function Ψ and its first derivative, divided by the effective mass, i.e. (1/m ^*)(dΨ/dρ) are continuous to obtain the eigen-equation. The procedure of derivation is described well in Kim and Chuang (2006). In Al-Husaini et al. (2009) the results of the modal are compared with that obtained from tight-binding calculations and are found convenient with it. If the potential in the disk is taken as V _d  = 0, the transverse Eigen-energy E _ρ is obtained by Kim and Chuang (2006)

$$E_{\rho } = \frac{{\hbar^{2} }}{{2m_{d}^{*} }}\frac{{(pa)^{2} }}{{a^{2} }}$$

(B-3)

Now, we go to treat the energy subbands under the applied electric field, where it is assumed applied along z-direction. Generally, when an electric field is applied to a QW structure as schematically illustrated in Fig. 2, the profile of the potential will be changed. The total potential is given by

$$V(z,F) = V(z,0) - ezF$$

(B-4)

where V(z,0) is the potential profile of the QW without the applied electric field, F is the applied electric field in (V/m), e is the electronic charge and z is the associated spatial coordinate. We choose the origin to be at the center of the well. The Schrodinger equation of the system in the effective-mass approximation under the applied electric field is given by

$$\begin{aligned} - \frac{{\hbar^{2} }}{{2m^{*} }}\frac{{d^{2} }}{{dz^{2} }}\psi (z) + \left| e \right|Fz\psi (z) = E\psi (z)\quad \left| z \right| \le L /2 \hfill \\ - \frac{{\hbar^{2} }}{{2m^{*} }}\frac{{d^{2} }}{{dz^{2} }}\psi (z) + \left( {V_{o} + \left| e \right|Fz} \right)\psi (z) = E\psi (z)\quad \left| z \right| \ge L /2 \hfill \\ \end{aligned}$$

(B-5)

where the potential profile of the QD in the z-direction is given by

$$V(z) = \left\{ {\begin{array}{*{20}c} 0 & {for\,\left| z \right|\, \le } & {L /2} \\ {V_{o} } & {for\,\left| z \right|\, \ge } & {L /2} \\ \end{array} } \right.$$

(B-6)

note that the potential height due to band offset between the QD and the WL is V _o  = B _eff [E _gw  − E _gd ] where B _eff is the band offset, E _gw and E _gd are the bandgaps of WL and QD, respectively. The wave function in the QD and WL regions are describes as

$$\varPsi (z) = \left\{ \begin{aligned} C_{3} Ai(\eta_{2} )\quad z > - L /2 \hfill \\ C_{4} Ai(\eta_{1} ) + D_{2} Bi(\eta_{1} )\quad \left| z \right| \le L /2 \hfill \\ C_{5} \left[ {Bi(\eta_{2} ) + iAi(\eta_{2} )} \right]\quad z < - L /2 \hfill \\ \end{aligned} \right.$$

(B-7)

where C ₃, C ₄, C ₅, D ₂ and D ₃ are constants, Ai and Bi are the homogeneous Airy function. From the properties of Airy function it is clear that Bi(η ₂) increases with increasing η ₂ and becomes infinity when η ₂ goes to infinity. In order to make the wave function well behaved in the entire region, this part is not added in the wave function in the region z > −L/2. Note that,

$$\begin{aligned} \eta_{1} = - \left[ {\frac{{2m^{*} }}{{(e\hbar F)^{2} }}} \right]^{1/3} (E - \left| e \right|Fz) \hfill \\ \eta_{2} = - \left[ {\frac{{2m^{*} }}{{(e\hbar F)^{2} }}} \right]^{1/3} (E - V_{o} - \left| e \right|Fz) \hfill \\ \end{aligned}$$

(B-8)

The required boundary conditions for the coefficients are obtained from the current continuity conditions at the heterojunction as

$$\begin{aligned} \left. {\varPsi (\eta_{1} )} \right|_{{z = z_{o} }} & = \left. {\varPsi (\eta_{2} )} \right|_{{z = z_{o} }} \\ \frac{1}{{m_{b}^{*} }}\left. {\frac{{d\varPsi (\eta_{1} )}}{dz}} \right|_{{z = z_{o} }} & = \frac{1}{{m_{d}^{*} }}\left. {\frac{{d\varPsi (\eta_{2} )}}{dz}} \right|_{{z = z_{o} }} \\ \end{aligned}$$

(B-9)

This results in the determinant,

$$\det \left| {\begin{array}{*{20}c} {Ai(\eta_{1}^{ + } )} & {Bi(\eta_{1}^{ + } )} & { - Ai(\eta_{2}^{ + } )} & 0 \\ {A^{\prime}i(\eta_{1}^{ + } )} & {B^{\prime}i(\eta_{1}^{ + } )} & { - A^{\prime}i(\eta_{2}^{ + } )} & 0 \\ {Ai(\eta_{1}^{ - } )} & {Bi(\eta_{1}^{ - } )} & 0 & { - \left[ {Bi(\eta_{2}^{ - } ) + iAi(\eta_{2}^{ - } )} \right]} \\ {A^{\prime}i(\eta_{1}^{ - } )} & {B^{\prime}i(\eta_{1}^{ - } )} & 0 & { - \left[ {B^{\prime}i(\eta_{2}^{ - } ) + iA^{\prime}i(\eta_{2}^{ - } )} \right]} \\ \end{array} } \right| = 0$$

(B-10)

where η ^± ₁ and η ^± ₂ are the values of η ₁ and η ₂ evaluated at z = L/2 and z = −L/2, respectively. Then, we obtain,

$$\left| \begin{aligned} A_{11} \quad A_{12} \hfill \\ A_{21} \quad A_{22} \hfill \\ \end{aligned} \right| = 0$$

(B-11)

where

$$\begin{aligned} A_{11} = \left\{ {\left[ {A_{i} (\eta_{1} )B^{\prime}_{i} (\eta_{2} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)A^{\prime}_{i} (\eta_{1} )B_{i} (\eta_{2} )} \right] + i\left[ {A_{i} (\eta_{1} )A^{\prime}_{i} (\eta_{2} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)A^{\prime}_{i} (\eta_{1} )A_{i} (\eta_{2} )} \right]} \right\} \hfill \\ A_{12} = \left\{ {\left[ {B_{i} (\eta_{1} )B^{\prime}_{i} (\eta_{2} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)B^{\prime}_{i} (\eta_{1} )B_{i} (\eta_{2} )} \right] + i\left[ {B_{i} (\eta_{1} )A^{\prime}_{i} (\eta_{2} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)B^{\prime}_{i} (\eta_{1} )A_{i} (\eta_{2} )} \right]} \right\} \hfill \\ A_{21} = \left\{ {\left[ {A^{\prime}_{i} (\eta_{1} )A_{i} (\eta_{2} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)A_{i} (\eta_{1} )A^{\prime}_{i} (\eta_{2} )} \right]} \right\} \hfill \\ A_{22} = - \left\{ {\left[ {A^{\prime}_{i} (\eta_{2} )B_{i} (\eta_{1} ) - \left( {\frac{{m_{b}^{*} }}{{m_{d}^{*} }}} \right)A_{i} (\eta_{2} )B^{\prime}_{i} (\eta_{1} )} \right]} \right\} \hfill \\ \end{aligned}$$

(B-12)

The eigenenergy E _z is obtained by solving Eq. (B-12), the total eigenenergy of the quantum disk E_d is approximately the summation of the transverse and longitudinal eigen-energies and is expressed as $E_{d} \, = \,E_{\rho } \,\, + \,\,E_{z}$. This gives the eigen-energy of the QD structure under applied electric field.