Partially coherent electron transport in terahertz quantum cascade lasers based on a Markovian master equation for the density matrix

Abstract

We derive a Markovian master equation for the single-electron density matrix, applicable to quantum cascade lasers (QCLs). The equation conserves the positivity of the density matrix, includes off-diagonal elements (coherences) as well as in-plane dynamics, and accounts for electron scattering with phonons and impurities. We use the model to simulate a terahertz-frequency QCL, and compare the results with both experiment and simulation via nonequilibrium Green’s functions (NEGF). We obtain very good agreement with both experiment and NEGF when the QCL is biased for optimal lasing. For the considered device, we show that the magnitude of coherences can be a significant fraction of the diagonal matrix elements, which demonstrates their importance when describing THz QCLs. We show that the in-plane energy distribution can deviate far from a heated Maxwellian distribution, which suggests that the assumption of thermalized subbands in simplified density-matrix models is inadequate. We also show that the current density and subband occupations relax toward their steady-state values on very different time scales.

Appendix: Calculation of \({\mathcal {G}}\) terms

This appendix is devoted to explicit calculation of ${\mathcal {G}}$. This task involves the evaluation of Eq. (21) for different scattering mechanisms, which is repeated here for convenience:

$$\begin{aligned} \mathcal G_g^\mathrm{em}(E_k,Q_z,E_q)\equiv \frac{1}{2\pi } \int _{0}^{2\pi }\mathrm{d}\theta \mathcal W_g^\mathrm{em}(\mathbf Q,E_k). \end{aligned}$$

(44)

The matrix element $\mathcal W_g^\mathrm{em}(\mathbf Q,E_k)$ can always be written in terms of $E_q=\hbar ^2q^2/2m^*, E_k=\hbar ^2k^2/2m^*, E_z=\hbar ^2Q_z^2/2m^*,$ and the angle $\theta$ between $\mathbf k$ and $\mathbf q$. Note that this definition of $E_z$ is only to make expressions more compact and readable, and the actual energy in the z-direction is contained in the $\varDelta _{nm}$ terms. Derivations of the various phonon matrix elements $\mathcal W_g^\mathrm{em}(\mathbf Q)$ used in this section can be found in Refs. [26, 27].

For the case of longitudinal acoustic (LA) phonons, we employ the equipartition approximation and get

$$\begin{aligned} \mathcal W_\mathrm{LA}^\mathrm{em}(\mathbf Q)\simeq \frac{D_\mathrm{ac}^2k_BT_L}{2m^*v_s^2}=\beta _{LA}, \end{aligned}$$

(45)

where $D_\mathrm{ac}$ is the deformation potential for acoustic phonons, and $v_s$ is the sound velocity in the material. In this case, the $\theta$ integration in Eq. (44) gives $\mathcal G_\mathrm{LA}^\mathrm{em}=\beta _\mathrm{LA}$. Since acoustic phonons are treated elastically, the emission and absorption terms are identical.

As with the acoustic phonons, the nonpolar optical phonon scattering is isotropic, so the phonon matrix element is constant $\mathcal W_\mathrm{op}^\mathrm{em}(\mathbf Q)=(N_\mathrm{op}+1)\beta _\mathrm{op}$. The angular integration in Eq. (44) gives $\mathcal G_\mathrm{op}^\mathrm{em}=(N_\mathrm{op}+1)\beta _\mathrm{op}$.

The phonon matrix element for electron scattering with polar optical phonons, with screening included, is given by

$$\begin{aligned} \mathcal W_\mathrm{pop}^\mathrm{em}(\mathbf Q)=(N_\mathrm{pop}+1)\beta _\mathrm{pop} \frac{Q^2}{(Q^2+Q_\mathrm{D}^2)^2} \ , \end{aligned}$$

(46a)

where $Q_\mathrm{D}$ is the Debye wave vector defined by $Q_\mathrm{D}^2=ne^2/(\varepsilon k_B T)$ and

$$\begin{aligned} \beta _\mathrm{pop}=\frac{e^2E_\mathrm{pop}}{2\varepsilon _0} \left( \frac{1}{\varepsilon _r^\infty } - \frac{1}{\varepsilon _r} \right) \ , \end{aligned}$$

(46b)

where $\varepsilon _r^\infty$ and $\varepsilon _r$ are the high-frequency and low-frequency relative permittivities of the material, respectively, and n is the average electron density. The effects of screening are quite small at the electron densities considered in this work; however, the $1/Q^2$ singularity poses problems in numerical calculations due to the high strength of the POP interaction. These problems are avoided by including screening. We can now calculate

$$\begin{aligned}&\frac{\mathcal G_\mathrm{pop}^\mathrm{em}(E_k,E_z,E_q)}{(N_\mathrm{pop}+1)}\nonumber \\&\quad =\frac{\beta _\mathrm{pop}}{2\pi } \int _0^{2\pi } \mathrm{d}\theta \frac{ |\mathbf Q-\mathbf k|^2 }{( |\mathbf Q-\mathbf k|^2+Q_\mathrm{D}^2 )^2} \nonumber \\&\quad =\frac{\beta _\mathrm{pop}}{2\pi }\int _0^{2\pi } \mathrm{d}\theta \frac{Q_z^2+q^2+k^2-2qk\cos (\theta )}{( q_z^2+q^2+k^2-2kq\cos (\theta ) +Q_\mathrm{D}^2 )^2} \nonumber \\&\quad = \frac{\beta _\mathrm{pop}}{Q_z^2+q^2+k^2} \left[ 1+\frac{Q_\mathrm{D}^2}{Q_z^2+q^2+k^2} - \frac{4k^2q^2}{(Q_z^2+q^2+k^2)^2 } \right] \nonumber \\&\qquad \times \left[ \left( 1+\frac{Q_\mathrm{D}^2}{Q_z^2+q^2+k^2}\right) ^2 -\frac{4k^2q^2}{(Q_z^2+q^2+k^2)^2} \right] ^{-\frac{3}{2}} \nonumber \\&\quad = \frac{\hbar ^2}{2m^*}\frac{\beta _\mathrm{pop}}{E_z+E_k+E_q} \nonumber \\&\qquad \times \left[ 1+\frac{E_\mathrm{D}}{E_z+E_q+E_k} - \frac{4E_kE_q}{(E_z+E_q+E_k)^2 } \right] \nonumber \\&\qquad \times \left[ \left( 1+\frac{E_\mathrm{D}}{E_z+E_q+E_k}\right) ^2 -\frac{4E_kE_q}{(E_z+E_q+E_k)^2} \right] ^{-\frac{3}{2}}, \end{aligned}$$

(47)

where $E_\mathrm{D}=\hbar ^2Q_\mathrm{D}^2/2m^*$ is the Debye energy.

The matrix element for ionized impurities is given by

$$\begin{aligned} \mathcal W_\mathrm{ii}^\mathrm{em}(\mathbf Q)=\frac{\beta _{ii}}{|\mathbf Q|^4} \end{aligned}$$

(48a)

with

$$\begin{aligned} \beta _{ii}=\frac{N_IZ^2e^4}{2\varepsilon _r^2\varepsilon _0^2}\ , \end{aligned}$$

(48b)

where $N_I$ is the impurity density, and Z is the number of unit charges per impurity. This matrix element gives

$$\begin{aligned}&\mathcal G_\mathrm{ii}^\mathrm{em}(E_k,E_z,E_q)\nonumber \\&\quad =\frac{\beta _{ii}}{2\pi } \int _0^{2\pi } d \theta \frac{1}{(Q_z^2+q^2+k^2-2qk\cos (\theta ))^2}\nonumber \\&\quad = \beta _{ii} \frac{Q_z^2+q^2+k^2}{[ (Q_z^2+q^2+k^2)^2-4q^2k^2 ]^{3/2}} \nonumber \\&\quad =\beta _{ii}\frac{\hbar ^4}{4m^2}\frac{E_z+E_k+E_q}{[(E_z+E_k+E_q)^2-4E_kE_q]^{3/2}} . \end{aligned}$$

(49)

Since ionized-impurity scattering is elastic, the absorption term is identical to the emission term.