Ultrafast high harmonic generation in transition metal dichalcogenide quantum dots

We study theoretically the generation of high harmonics by transition metal dichalcogenides disk-shaped quantum dots placed in an ultrashort optical pulse. The quantum dots are described within a massive Dirac type effective model with infinite mass boundary conditions. For a linearly polarized pulse, the cutoff frequency and the intensities of high-order harmonics increase with the quantum dot radius. For the frequency of the pulse below the system band gap, the radiation spectra show strong dependence on the quantum dot size, while for the large pulse frequency, i.e., above the band gap, both the intensities of high harmonics and cutoff frequency weakly depend on the quantum dot radius. For an elliptically polarized pulse, the intensities of harmonics monotonically decrease with the pulse ellipticity and become strongly suppressed for a circularly pulse. The cutoff frequency also monotonic decreases with the pulse ellipticity and shows weak dependence on the quantum dot material.

Figure 1

Schematic illustration of a TMDC QD. The optical pulse is incident normally to the QD layer.

Figure 2

Energy spectra of TMDC QDs ($K_{\uparrow }$ valley) for different TMDC materials. The QD radius is (a), (c), (e) $R=7$ nm and (b), (d), (f) $R=12$ nm. The type of TMDC material is shown in each panel. For all cases, the spectrum is shown as a function of the magnetic quantum number, $m$, which takes integer values.

Figure 3

Profile of the electric field of the pulse as a function of time. The amplitude of the pulse is $F_o=0.25\text{V/nm}$. The frequency of the pulse is $\omega =1$ rad/fs and the pulse duration is ${\tau }_{\mathrm{pulse}}=15$ fs. The pulse is linearly polarized.

Figure 4

Radiation spectra of ${\mathrm{WS}}_2$ QDs. The frequency $\omega$ and the amplitude $F_o$ of the pulse are marked in each panel. In each panel, different lines correspond to different radii $R$ of a ${\mathrm{WS}}_2$ QD. The radiation intensity $I\left(\omega \right)$ is shown on a logarithmic scale and the frequency is shown in the units of the frequency of the pulse, ${\omega }_o$.

Figure 5

Radiation spectra of ${\mathrm{WSe}}_2$ QDs. The frequency $\omega$ and the amplitude $F_o$ of the pulse are marked in each panel. In each panel, different lines correspond to different radii $R$ of a ${\mathrm{WSe}}_2$ QD. The radiation intensity $I\left(\omega \right)$ is shown on a logarithmic scale and the frequency is shown in the units of the frequency of the pulse, ${\omega }_o$.

Figure 6

Radiation spectra of ${\mathrm{MoS}}_2$ QDs. The frequency $\omega$ and the amplitude $F_o$ of the pulse are marked in each panel. In each panel, different lines correspond to different radii $R$ of a ${\mathrm{MoS}}_2$ QD. The radiation intensity $I\left(\omega \right)$ is shown on a logarithmic scale and the frequency is shown in the units of the frequency of the pulse, ${\omega }_o$.

Figure 7

Radiation spectra of TMDC quantum dots. The frequency ${\omega }_o$ and the amplitude $F_o$ of the pulse are marked in each panel. The QD radius is $R=15$ nm. The intensity $I\left(\omega \right)$ is shown on the logarithmic scale and the frequency is shown in the units of the frequency of the pulse, ${\omega }_o$.

Figure 8

Single-particle density of states for different TMDC materials is shown for the $K^{\uparrow }$ valley. The QD radius is 15 nm. For each QD level, the Lorentzian broadening factor of 0.06 eV has been introduced.

Figure 9

Intensities of the (a) third and (b) fifth harmonics generated by TMDC QDs as a function of QD radius. The pulse frequency is $\omega =1$ rad/fs and the field amplitude is $F_o=0.25$ V/nm. In each panel, different lines correspond to different TMDC materials as marked in the figure.

Figure 10

Cutoff frequency as a function of the QD radius for different TMDC materials. Plots (a), (d), and (c) show the results. The pulse frequency is (a) ${\omega }_o=1$ rad/fs, (b) ${\omega }_o=2$ rad/fs, and (c) $\omega =2.7$ rad/fs. In each panel, different lines correspond to different TMDC materials: ${\mathrm{WS}}_2$, ${\mathrm{WSe}}_2$, and ${\mathrm{MoS}}_2$ as marked in the figure. The field amplitude is $F_o=0.25\text{V/nm}$.

Figure 11

Top-down view of a cycle of a circularly polarized pulse. The amplitude of the pulse is $F_o=0.25$ V/nm, its frequency is ${\omega }_o=1$ rad/fs, and the duration of the pulse is ${\tau }_{\mathrm{pulse}}=15$ fs.

Figure 12

Radiation spectra of a ${\mathrm{MoS}}_2$ QD as a function of the harmonic order and the polarization angle $\theta$ of an elliptically polarized pulse. The QD radius is $R=10$ nm, the pulse frequency is ${\omega }_o=1$ rad/fs, and the pulse amplitude is $F_o=0.25$ V/nm.

Figure 13

The radiation spectra of TMDC QDs for different QD radii. The pulse frequency ${\omega }_o$ and the pulse amplitude $F_o$ are marked in each panel. Different lines correspond to different ellipticities ${\alpha }_{\mathrm{incident}}$ of the incident pulse. Here ${\alpha }_{\mathrm{incident}}=0$ corresponds to a linearly polarized pulse, while ${\alpha }_{\mathrm{incident}}=1$ describes a circularly polarized pulse. The radiation intensity $I\left(\omega \right)$ is shown on the logarithmic scale and the frequency is shown in the units of the pulse frequency, ${\omega }_o$.

Figure 14

Intensities of the high-order harmonics generated by a TMDC QD with the radius of $R=7$ nm as a function of ellipticity ${\alpha }_{\mathrm{incident}}$ of the incident optical pulse. The data are shown for the first three harmonics: (a) the third harmonic, (b) the fifth harmonic, and (c) the seventh harmonic. The frequency and amplitude of the pulse are ${\omega }_o=1$ rad/fs and $F_o=0.25$ V/nm, respectively. The relaxation time is ${\tau }_{\mathrm{relax}}=25$ fs. In different panels, different lines correspond to different TMDC materials as marked in the figure.

Figure 15

Cutoff frequency as a function of the ellipticity ${\alpha }_{\mathrm{incident}}$ of the incident optical pulse. The results are shown for different QD sizes with the radius of (a) 5 nm, (b) 7 nm, and (c) 10 nm. The frequency and the amplitude of the pulse are ${\omega }_o=1$ rad/fs and $F_o=0.25$ V/nm, respectively. In each panel, different lines correspond to different TMDC materials as marked in the figure.

Figure 16

Ellipticities ${\alpha }_{\mathrm{harm}}$ of high-order harmonics generated by TMDC QDs as a function of the ellipticity ${\alpha }_{\mathrm{incident}}$ of the incident optical pulse. The data are shown for the first three harmonics: (a), (d), (g) the third harmonic; (b), (e), (h) the fifth harmonic; and (c), (f), (i) the seventh harmonic. The frequency and the amplitude of the pulse are ${\omega }_o=1$ rad/fs and $F_o=0.25$ V/nm, respectively. The QD radius is shown in each panel. The relaxation time is ${\tau }_{\mathrm{relax}}=25$ fs. In each panel, different lines correspond to different TMDC materials as marked in the figure.