High Harmonic Generation with Many-Particle Coulomb Interaction in Graphene Quantum Dot

Abstract

The multiphoton excitation and high harmonic generation (HHG) processes are considered using the microscopic quantum theory of nonlinear interaction of strong coherent electromagnetic (EM) radiation with rectangular graphene quantum dot (GQD) with zigzag edges and more than 80 atoms. The dynamic Hartree–Fock approximation has been used to consider the quantum dot-laser field nonlinear interaction at the nonadiabatic multiphoton excitation regime. The many-body Coulomb interaction is described in the extended Hubbard approximation. By numerical results, we show the significance of the rectangular GQD lateral size, shape, and EM wavefield orientation in rectangular GQD of the zigzag edge in the HHG process allowing for increasing the cutoff photon energy and the quantum yield of higher harmonics.

APPENDIX: INTERACTION HAMILTONIAN

Here we presented briefly the total Hamiltonian by the empirical TB model [91] in the form:

$$\hat {H} = {{\hat {H}}_{0}} + {{\hat {H}}_{{\operatorname{int} }}},$$

(5)

where

$${{\hat {H}}_{0}} = \frac{1}{2}\sum\limits_{\langle i,j\rangle }^{} {{{V}_{{ij}}}{{n}_{i}}{{n}_{j}}} + \frac{U}{2}\sum\limits_{i\sigma }^{} {{{n}_{{i\sigma }}}{{n}_{{i\bar {\sigma }}}}} - \sum\limits_{\langle i,j\rangle \sigma }^{} {{{t}_{{ij}}}c_{{i\sigma }}^{\dag }{{c}_{{j\sigma }}}} $$

(6)

is the free rectangular GQD Hamiltonian. Here $c_{{i\sigma }}^{\dag }$ is the operator of the creation of an electron with spin polarization σ = ( ↑, ↓) at site i, and 〈i, j〉 is the summation over nearest neighbor sites with the transfer energy t_ij ($\bar {\sigma }$ is the opposite to spin polarization); and n_iσ = $c_{{i\sigma }}^{\dag }$c_iσ is the electron density operator with total electron density for the site i: n_iσ = n_i↑ + n_i↓. The first and the second terms in free Hamiltonian (6) correspond to the EEI within the extended Hubbard approximation (${{\hat {H}}_{{ee}}}$) with inter-site V_ij and on-site U Coulomb repulsion energies. The inter-site Coulomb repulsion is described by the distance d_ij between the nearest-neighbour pairs varied over the system: V_ij = Vd_min/d_ij (d_min is the the minimal nearest-neighbor distance). For all calculations we have taken V = 0.3U [97, 98]. The third term in (6) is the kinetic energy part of the TB Hamiltonian with tunneling matrix element t_ij-neighboring sites. The hopping integral t_ij between the nearest-neighbor atoms of GQDs can be determined experimentally, and is usually-taken to be t_ij = 2.7 eV [57]. Note, that we neglected the lattice vibrations in the Hamiltonian.

The laser-rectangular GQD interaction is described in the length-gauge via the scalar potential:

$${{\hat {H}}_{{\operatorname{int} }}} = e\sum\limits_{i\sigma }^{} {{{{\mathbf{r}}}_{i}} \cdot {\mathbf{E}}(r)c_{{i\sigma }}^{\dag }{{c}_{{i\sigma }}},} $$

where r_i is the position vector, e is the elementary charge. We will obtain evolutionary equations for the single-particle density matrix $\rho _{{ij}}^{{(\sigma )}}$ = 〈$c_{{j\sigma }}^{\dag }$c_iσ〉 from the Heisenberg equation i$\hbar$∂$\hat {L}$/∂t = [$\hat {L}$, $\hat {H}$], where L is a Lagrangian of a quantum system. Let us the system relaxes at a rate γ to the equilibrium $\rho _{{0ij}}^{{(\sigma )}}$ distribution. The EEI will be considered under the Hartree–Fock approximation, and to describe a closed set of equations for the single-particle density matrix $\rho _{{ij}}^{{(\sigma )}}$, we will assume the Hamiltonian (6) in the form:

$$\begin{gathered} \hat {H}_{0}^{{HF}} \simeq - \sum\limits_{\langle i,j\rangle \sigma }^{} {{{t}_{{ij}}}c_{{i\sigma }}^{\dag }{{c}_{{j\sigma }}} + U\sum\limits_i^{} {({{{\bar {n}}}_{{i \uparrow }}} - {{{\bar {n}}}_{{0i \uparrow }}}){{n}_{{i \downarrow }}}} } \\ \, + U\sum\limits_{i\sigma }^{} {({{{\bar {n}}}_{{i \downarrow }}} - {{{\bar {n}}}_{{0i \downarrow }}}){{n}_{{i \uparrow }}}} + \sum\limits_{\langle i,j\rangle }^{} {{{V}_{{ij}}}({{{\bar {n}}}_{j}} - {{{\bar {n}}}_{{0j}}}){{n}_{i}}} \\ - \sum\limits_{\langle i,j\rangle \sigma }^{} {{{V}_{{ij}}}c_{{i\sigma }}^{\dag }{{c}_{{j\sigma }}}(\langle c_{{i\sigma }}^{\dag }c_{{j\sigma }}^{{}}\rangle - {{{\langle c_{{i\sigma }}^{\dag }c_{{j\sigma }}^{{}}\rangle }}_{0}}),} \\ \end{gathered} $$

(7)

with ${{\bar {n}}_{{i\sigma }}}$ = 〈$c_{{i\sigma }}^{\dag }$c_iσ〉 = $\rho _{{ii}}^{{(\sigma )}}$. Thus, the following equation for the density matrix is obtained:

$$\begin{gathered} i\hbar \frac{{\partial \rho _{{ij}}^{{(\sigma )}}}}{{\partial t}} = \sum\limits_k^{} {({{\tau }_{{kj\sigma }}}\rho _{{ik}}^{{(\sigma )}} - {{\tau }_{{ik\sigma }}}\rho _{{kj}}^{{(\sigma )}}) + ({{V}_{{i\sigma }}} - {{V}_{{j\sigma }}})} \rho _{{ij}}^{{(\sigma )}} \\ \, + e{\mathbf{E}} \cdot ({{{\mathbf{r}}}_{i}} - {{{\mathbf{r}}}_{j}})\rho _{{ij}}^{{(\sigma )}} - i\hbar \gamma (\rho _{{ij}}^{{(\sigma )}} - \rho _{{0ij}}^{{(\sigma )}}), \\ \end{gathered} $$

(8)

and the matrixes V_iσ, τ_ijσ are approximated by the density matrix ∂$\rho _{{ij}}^{{(\sigma )}}$:

$$\begin{gathered} {{V}_{{i\sigma }}} = \sum\limits_{j\alpha }^{} {{{V}_{{ij}}}(\rho _{{jj}}^{{(\alpha )}} - \rho _{{0jj}}^{{(\alpha )}}) + U(\rho _{{ii}}^{{(\bar {\sigma })}} - \rho _{{0ii}}^{{(\bar {\sigma })}})} , \\ {{\tau }_{{ij\sigma }}} = {{t}_{{ij}}} + {{V}_{{ij}}}(\rho _{{ji}}^{{(\sigma )}} - \rho _{{0ji}}^{{(\sigma )}}). \\ \end{gathered} $$

(9)

In this representation, the initial value of density matrix 〈$c_{{i\sigma }}^{\dag }{{c}_{{j\sigma }}}$〉₀ is defined via TB Hamiltonian $\hat {H}_{0}^{{TB}} = - \sum\limits_{\langle i,j\rangle \sigma }^{} {{{t}_{{ij}}}c_{{i\sigma }}^{\dag }{{c}_{{j\sigma }}}.}$ We will numerically diagonalize the TB Hamiltonian ${{\hat {H}}_{0}}$. That is, in the static limit the Hartree–Fock Hamiltonian vanishes $\hat {H}_{{ee}}^{{HF}}$ $\simeq$ 0, and the EEI in Hartree–Fock limit is included in empirical hopping integral between the nearest-neighbor t_ij which is chosen to be close to experimental data [57]. Thus, the EEI in the Hartree–Fock approximation is correspond only to quantum dynamics initiated by the pump laser field.