Stopping dynamics of ions passing through correlated honeycomb clusters

A combined nonequilibrium Green functions–Ehrenfest dynamics approach is developed that allows for a time-dependent study of the energy loss of a charged particle penetrating a strongly correlated system at zero and finite temperatures. Numerical results are presented for finite inhomogeneous two-dimensional Fermi-Hubbard models, where the many-electron dynamics in the target are treated fully quantum mechanically and the motion of the projectile is treated classically. The simulations are based on the solution of the two-time Dyson (Keldysh-Kadanoff-Baym) equations using the second-order Born, third-order, and $\mathrm{T}$-matrix approximations of the self-energy. As application, we consider protons and helium nuclei with a kinetic energy between 1 and 500 keV/u passing through planar fragments of the two-dimensional honeycomb lattice and, in particular, examine the influence of electron-electron correlations on the energy exchange between projectile and electron system. We investigate the time dependence of the projectile's kinetic energy (stopping power), the electron density, the double occupancy, and the photoemission spectrum. Finally, we show that, for a suitable choice of the Hubbard model parameters, the results for the stopping power are in fair agreement with ab initio simulations for particle irradiation of single-layer graphene.

Figure 1

Lattice structure of circular honeycomb clusters with $L=24$ (black) and 54 sites (blue). The green point indicates the position where the projectile hits the lattice plain. For further reference, we label four sites in the center of the clusters, where we will monitor the time-dependent electron density in Sec. 4b. Furthermore, $a_0$ denotes the lattice spacing, $J$ $\left(U\right)$ is the nearest-neighbor hopping (the onsite interaction), and $W_{ii}$ is the local energy defined in Eq. (2).

Figure 2

Left panel: average double occupation ${\langle d\rangle }_{\mathrm{av}}$ on the honeycomb clusters with $L=24$, 54, and 96 sites for different interaction strengths $U/J$ in the local second Born, third-order, and $\mathrm{T}$-matrix approximations, using the initial state (A). The black triangles correspond to exact data for the extended honeycomb lattice (taken from Ref. [42]). Right panel: time evolution of ${\langle d\rangle }_{\mathrm{av}}$ for the local second Born calculations starting from the Hartree ground state [Eq. (11)], for which ${\langle d\rangle }_{\mathrm{av}}=\langle d_i\rangle =\langle n_{i\uparrow }\rangle \langle n_{i\downarrow }\rangle =0.25$.

Figure 3

(a) Photoemission signal $I_1\left(\omega \right)$ for the honeycomb cluster with $L=54$ sites in Hartree approximation $(U$ independent) and in local second Born approximation for different $U/J$, using the initial state (A) of Sec. 2c. Black dashed line: spectrum of the extended lattice at $U/J=0$ (e.g., Refs. [48, 49]), black solid line: spectrum for $U/J=5$, as obtained from a cluster-DMFT calculation [from Ref. [47] and also shown in panel (b)]. (b) Comparison of the photoemission signal $I_1\left(\omega \right)$ at $U/J=5$ for local second Born calculations with different initial states. Green line: spectrum [as in panel (a)] for the initial state (A). Red line: correlated ground state (B).

Figure 4

Energy loss $S_{\mathrm{e}}$ for protons passing through honeycomb clusters of size $L=24$ [panels (a) and (c)] and $L=54$ [panel (b)]. In all panels, the value of the onsite interaction $U/J$ is encoded in the line style, and the black lines indicate the results of the Hartree approximation. In panel (a), the red curves show the energy loss in the local second Born approximation with initial state (A). In panel (b), we present the same analysis for the larger cluster, including results of the full second Born (yellow), the third-order (blue), and the $\mathrm{T}$-matrix approximation (green). Panel (c) shows the influence of initial correlations, comparing the second Born results of panel (a) to local and full second Born, third-order, as well as $\mathrm{T}$-matrix calculations for the initial state (B). The arrows in panel (c) indicate the two situations analyzed in more detail in Fig. 5.

Figure 5

Coupled proton-electron dynamics for the honeycomb cluster with $L=24$ sites for two different proton energies, $E_{\mathrm{kin}}=0.96\mathrm{keV}/\mathrm{u}$ (left panels) and 11.82 keV/u (right panels), corresponding to the arrows shown in Fig. 4. The coupling strength is fixed, $U/J=4$, and three many-body approximations are compared (see inset). Top panels: change of the projectile's kinetic energy $\mathrm{\Delta }{E}_{\mathrm{kin}}\left(t\right)=E_{\mathrm{kin}}\left(t\right)-E_{\mathrm{kin}}(t=0)$ as function of time. Center panels: time evolution of the electron density averaged over the central sites labeled 1 to 4 in Fig. 1, i.e., $\langle N_4\rangle \left(t\right)=\frac{1}{4}{\sum }_{i=1}^4\langle n_{i\sigma }\rangle \left(t\right)$. Bottom panels: time evolution of the double occupation $\langle d_1\rangle \left(t\right)=\langle n_{1\uparrow }{n}_{1\downarrow }\rangle \left(t\right)$ on the site 1, which is closest to the impact point of the projectile.

Figure 6

Time-resolved photoemission spectrum $I(\omega ,t_{\mathrm{p}})=I_0^{-1}{\sum }_{i=1}^L{I}_i(\omega ,t_{\mathrm{p}})$ of a cluster with $L=24$ and $U/J=4$ for the stopping dynamics of a proton with energy $E_{\mathrm{kin}}=11.82\mathrm{keV}/\mathrm{u}$ [the scenario is similar to Fig. 5]. The calculations are performed in full second Born approximation with a correlated initial state [case (B)]. Red curve: initial state, black dashes: ground state of an infinite system at $U/J=0$. Blue line: spectrum when the projectile passes through the lattice plane. Green line: “final” state after the collision. To resolve the spectrum at the different stages of the dynamics, we have set the probe pulse width here to $\tau =2t_0$ (cf. Appendix pp1).

Figure 7

Energy loss for the honeycomb cluster with $L=54$ sites in second Born approximation. Comparison of the full two-time simulation and the GKBA (see text for details). Black and red solid lines are the same as in Fig. 4; red dashed and blue dotted curves: GKBA results for the initial states (A) and (B), respectively.

Figure 8

Temperature dependence of the energy loss $S_{\mathrm{e}}$ for the honeycomb cluster with $L=24$ sites and $U/J=4$. Black lines: zero-temperature Hartree calculations, as in Fig. 4. Colored lines: Hartree results for different temperatures. Symbols: local second Born results for the same temperatures $\beta J=5$, 1.5, 1, and 0.5. (a) Local second Born calculations with initial state (A). (b) GKBA calculations as in Sec. 4d with initial state (B).

Figure 9

Time-dependent energy change $\mathrm{\Delta }{E}_{\mathrm{kin}}\left(t\right)=E_{\mathrm{kin}}\left(t\right)-E_{\mathrm{kin}}(t=0)$ of the proton (upper panels) and electron density $\langle N_4\rangle \left(t\right)=\frac{1}{4}{\sum }_{i=1}^4\langle n_{i\sigma }\rangle \left(t\right)$ at the four sites around the impact point (lower panels) for four different temperatures. Same cases (Hartree dynamics, $U/J=4)$ as shown in Fig. 8.

Figure 10

Energy loss of (a) hydrogen ions $\left({\mathrm{H}}^{+}\right)$ and (b) alpha particles $\left({\mathrm{He}}^{2+}\right)$ penetrating through a single layer of graphene. In both panels, the strength of the Coulomb interaction is $U/J=1.6$, the fit parameters are $J=3.15\mathrm{eV}$ and $\gamma =0.55$, and the initial $z$ position of the projectile is $z=20a_0$. The Hartree results for different cluster sizes, ranging from $L=24$ (bottom) to $L=384$ (top), are shown by different line styles. For the cluster with $L=54$ sites, we also performed second Born calculations (green circles) showing that correlation corrections are rather small in this case. For a detailed discussion of the $U,J$, and $\gamma$ dependence, see Appendix pp2. The black symbols and lines correspond to ab initio TDDFT calculations and srimdata, respectively (taken from Refs. [7, 8]).

Figure 11

Photoemission spectrum $I_1\left(\omega \right)$ for different uncorrelated honeycomb clusters of size $L$, measured at a probe time $t_{\mathrm{p}}=8t_0$ with $\tau /t_0=4$ [cf. Eq. (A1)]. The black dashed line shows the density of states in the valence and conduction band of the extended honeycomb lattice at half-filling, e.g., Refs. [48, 49].

Figure 12

Proton stopping dynamics as in Sec. 5, but for different parameters $U,J$, and $\gamma$ in Hartree approximation. The system size is $L=54$. (a) Influence of the onsite interaction $U$ for fixed $J=3.2\mathrm{eV}$ and $\gamma =0$. (b) Influence of the hopping amplitude $J$ and the orbital overlap $\gamma$ for fixed interaction strength of $U/J=1.6$.