Relaxation of photoexcitations in polaron-induced magnetic microstructures

We investigate the evolution of a photoexcitation in correlated materials over a wide range of time scales. The system studied is a one-dimensional model of a manganite with correlated electron, spin, orbital, and lattice degrees of freedom, which we relate to the three-dimensional material ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$. The ground-state phases for the entire composition range are determined and rationalized by a coarse-grained polaron model. At half doping a pattern of antiferromagnetically coupled Zener polarons is realized. Using time-dependent density-matrix renormalization group (tDMRG), we treat the electronic quantum dynamics following the excitation. The emergence of quasiparticles is addressed, and the relaxation of the nonequilibrium quasiparticle distribution is investigated via a linearized quantum-Boltzmann equation. Our approach shows that the magnetic microstructure caused by the Zener polarons leads to an increase of the relaxation times of the excitation.

Figure 1

One-dimensional chain of corner connected ${\mathrm{MnO}}_6$ octahedra of the model manganite in the coordinate system chosen.

Figure 2

Degrees of freedom of the tight-binding model. (top) Orbital degrees of freedom of the $e_g$ electrons, which are treated explicitly, and classical spin degree of freedom related to the $t_{2g}$ electrons. (bottom) Breathing mode $Q_1$ and Jahn-Teller-active phonon modes $Q_2, Q_3$.

Figure 3

Projected density of states of the tight-binding model for the polarons in the 1D manganite chain as a function of energy in eV. Top left: Two adjacent unoccupied sites $V$; top-right: two adjacent Jahn-Teller polarons $P^{JT}$; middle left: electron polaron $P^e$; middle right: hole polaron $P^h$; bottom Zener polaron $P^Z$. The horizontal line indicates the Fermi level. Empty and filled $y$ lines indicate $d_{3z^2-r^2}$ orbitals pointing along the chain and $d_{x^2-y^2}$ orbitals orthogonal to the chain, respectively. The density of states is broadened by 0.05 eV.

Figure 4

Energy per Mn site of the model calculation (line) as a function of the electron occupation $1-x=N_e/N_s$ compared to the sum of polaron energies given in Eq. (3) (symbols). The energy $(1-x){\mu }_0$ of the particle reservoir has been included.

Figure 5

Sketch of the effective Hubbard-type many-body model derived in this section. The unit cell consists of four sites, with each site hosting a spin-up and a spin-down state. The two states are energetically separated by the Hund's splitting $\mathrm{\Delta }=2J_H$. Electrons can hop from one site to another with the hopping amplitude $t_{\mathrm{hop}}$, provided the two states have the same spin. For the sake of clarity this is only shown for the spin-$\downarrow$ direction. If two electrons are located on the same site, they feel the screened Coulomb repulsion $U$ (depicted by the curly line). Otherwise, this repulsion is screened by the positively charged atoms.

Figure 6

One-particle band structure of PCMO for different values of the Hund's splitting $\mathrm{\Delta }$, which is measured in units of $t_{\mathrm{hop}}$. $\mathrm{\Gamma }$ denotes the origin of the $k$ points and $\mathrm{X}=\pi /4a$ the zone boundary with the $\mathrm{Mn}-\mathrm{Mn}$ spacing $a$. One can see that the distance between the center of the upper two bands and the center of the lower two bands is close to $\mathrm{\Delta }$ for large values of $\mathrm{\Delta }$. Furthermore, in the same limit, the distance of the upper two bands (as well as the one of the lower two bands) is approximately $2t_{\mathrm{hop}}$.

Figure 7

Time evolution of the local density $\langle {\widehat{n}}_i\rangle$ following an excitation by applying operator Eq. (12) at the center of the system. The panels show tDMRG results for different values of $\mathrm{\Delta }/t_{\mathrm{hop}}$ for chains with $L=40$ lattice sites. Left side: $U=0$; right side: $U/t_{\mathrm{hop}}=4.3$. The solid lines indicate the maximal group velocity of the excited electrons obtained from the noninteracting band structure Eq. (10), assuming that one electron gets excited from the first to the second band. The dotted and dashed lines indicate the phase velocity at the $k$ value with the maximal group velocity, as discussed in the text.

Figure 8

Time evolution of the local density $\langle {\widehat{n}}_{R+1}\rangle$ obtained by tDMRG for a system with $L=40$ lattice sites at the right site of the Zener polaron at which operator Eq. (12) was applied to, $R=L/2+1$. Green: $U=0$; purple: $U/t_{\mathrm{hop}}=4.3$. The lines for $\mathrm{\Delta }/t_{\mathrm{hop}}=8$ show a fit using a function of the form $f\left(x\right)=\frac{1}{4}\left(cos\left(ax\right)+cos\left(bx\right)\right)+\frac{1}{2}$.

Figure 9

Time evolution of the local density $\langle {\widehat{n}}_{R+1}\rangle$ after the operator Eq. (12) was applied to $R=L/2+1$ for $\mathrm{\Delta }/t_{\mathrm{hop}}=2.3$ and $U/t_{\mathrm{hop}}=4.3$, which is close to the parameters of Table 1. The plot compares tDMRG results for systems with $L=16$ (violet boxes), $L=24$ (green circles), $L=32$ (blue up-pointing triangles), $L=40$ (red down-pointing triangles), $L=48$ (yellow diamonds), and $L=64$ (dark blue pentagons). The results displayed are obtained with MPS matrix dimension ${\chi }_{\text{MPS}}=5000$. Additionally the time evolution for $\mathrm{\Delta }=0$ and $U/t_{\mathrm{hop}}=4.3$ for a system with $L=40$ and ${\chi }_{\text{MPS}}=500$ is plotted (black pluses).

Figure 10

Momentum distribution for a system with $L=40, \mathrm{\Delta }/t_{\mathrm{hop}}=2.3$, and $U/t_{\mathrm{hop}}=4.3$ before (magenta) and just after (green) the photoexcitation by applying operator Eq. (12) at the center of the system as obtained by the DMRG.

Figure 11

Time evolution of the population ${\sum }_k{n}_{\sigma ,\nu }^{\text{el}}(k,t)$ of each band following the photoexcitation for $\mathrm{\Delta }/t_{\mathrm{hop}}=2.3$ and $U/t_{\mathrm{hop}}=4.3$ as obtained by the tDMRG. Additionally the population of the thermal state is given by the horizontal lines.

Figure 12

Time evolution of the population ${\sum }_k{n}_{\sigma ,\nu }^{\text{el}}(k,t)$ of each band following the photoexcitation as in Fig. 11, but for $\mathrm{\Delta }/t_{\mathrm{hop}}=8$, as obtained by the tDMRG. Additionally the population of the thermal state is given by the horizontal lines.

Figure 13

Comparison of the electronic and the quasiparticle momentum distributions obtained with DMRG via Eq. (19) for a system with $L=40, \mathrm{\Delta }/t_{\mathrm{hop}}=2.3$, and $U/t_{\mathrm{hop}}=4.3$ just after the photoexcitation by applying operator Eq. (12) at the center of the system.

Figure 14

Relaxation rates as obtained from the linearized Boltzmann equation approach for $\mathrm{\Delta }=1,2,4$ from left to right. The eigenvalues ${\lambda }_n$ of $\widehat{\mathcal{L}}$ are sorted by their magnitude, and we plot the lowest ones ($n=1,2,3,...$) as a function of inverse temperature $\beta$. In all of the plots, the full lines are calculations with the lowest two bands, the dashed lines indicate a calculation with the three lowest bands, and the dotted lines denote a computation with all four bands. The eigenvalues $n=1,2,3$ are zero within numerical precision. Note that, for the 3- and 4-band calculations, there is an additional zero eigenvalue, which we omit for comparability. As two examples for relaxation times, we consider $u=U/t_{\mathrm{hop}}=4.3$ at room temperature $\beta =t_{\mathrm{hop}}/(k_B\times 300\mathrm{K})\approx 23$ and at the temperature after excitation, which we estimate in Appendix pp3, $\beta =1.4$. With $\mathrm{\Delta }=2.3, t_{\mathrm{hop}}\approx 0.585\mathrm{eV}$, and $t_0:=\hslash /\pi t_{\mathrm{hop}}\approx 0.12\mathrm{fs}$ the smallest relaxation time is $1/{\lambda }_4={10}^9{t}_0/u^2\approx 6.5\mathrm{ns}$ for room temperature and $1/{\lambda }_4={10}^3{t}_0/u^2\approx 6.5\mathrm{fs}$ for the temperature after the excitation treated in Sec. 3.

Figure 15

Momentum distribution for a system with open boundary conditions, $L=40, \mathrm{\Delta }/t_{\mathrm{hop}}=2.3$, and $U/t_{\mathrm{hop}}=0$ before (magenta) and just after (green) the photoexcitation by applying operator Eq. (12) at the center of the system as obtained by the DMRG.

Figure 16

Momentum distribution as in Fig. 15 but with periodic boundary conditions.

Figure 17

Energy of systems with $L=40, \mathrm{\Delta }/t_{\mathrm{hop}}=2.3,8$, and $U/t_{\mathrm{hop}}=0,8$ as a function of the inverse temperature $\beta$. The results are obtained using DMRG by an imaginary time evolution starting from the ground state of (C1), which is a suitable state with $\beta =0$ in the physical space. The dashed horizontal lines indicate the energy after the excitation, which is obtained by computing the expectation value $E_{\mathrm{exc}}=\langle \psi \left(t\right)|\widehat{H}|\psi \left(t\right)\rangle =\mathrm{const}$, with $\widehat{H}$ the Hamiltonian (7) and $\left|\psi \right(t)\rangle$ the state after the excitation at time $t$. The intersection of the finite-temperature results and $E_{\mathrm{exc}}$ indicates the value of $\beta$, which can be associated to the energy of the excitation.