Evolution of the magnetic and polaronic order of ${\mathrm{Pr}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_3$ following an ultrashort light pulse

The dynamics of electrons, spins, and phonons induced by optical femtosecond pulses has been simulated for the polaronic crystal ${\mathrm{Pr}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_3$. The model used for the simulation has been derived from first-principles calculations. The simulations reproduce the experimentally observed melting of charge and orbital order with increasing fluence. The loss of charge order in the high-fluence regime induces a transition to a ferromagnetic metal. At low fluence, the dynamics is deterministic and coherent phonons are created by the repopulation of electronic orbitals, which are strongly coupled to the phonon degrees of freedom. In contrast to the low-fluence regime, the magnetic transitions occurring at higher fluence can be attributed to a quasithermal transition of a cold-plasma-like state with hot electrons and cold phonons and spins. The findings can be rationalized in a more complete picture of the electronic structure that goes beyond the simple ionic picture of charge order.

Figure 1

Left: CE-type magnetic order and orbital order in the $ab$ plane of the ground state of ${\mathrm{Pr}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_3$. Black and white symbols indicate up and down spins. The orbital-polarized central orbitals are indicated by a $d_{3z^2-r^2}$ orbital symbol in the corresponding direction. The corner sites have no orbital polarization and are indicated by circles. The gray square indicates the magnetic unit cell of the material in the $ab$ plane. Also shown are the lattice vectors $a$ and $b$ of the orthorhombic $Pbnm$ unit cell. Right: Wannier-type orbitals along a trimer of the zigzag chains of the CE-type magnetic structure. The sign of the orbital lobes are indicated by black and white. The Wannier-type orbitals are orthogonal within and between trimers. The arrows connect orbitals with dipole-allowed transitions. Transitions between all other orbital pairs within and between trimers are dipole forbidden. The orbital $|w_1\rangle$ in the majority-spin direction is filled. The optical excitation lifts electrons from $|w_1\rangle$ to $|w_2\rangle$ in the majority-spin direction.

Figure 2

Diffraction patterns for charge order (left), orbital order (middle), and spin order (right) of the CE-type low-temperature phase of ${\mathrm{Pr}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_3$. The $a$ axis points right, the $b$ axis toward the back, and the $c$ axis up. The small white spheres indicate points with integer $h,k,l$ in the $Pbnm$ setting. Reciprocal space is shown for $h,k,l\in [-1.25,1.25]$.

Figure 3

Density of states of the ground state of ${\mathrm{Pr}}_{1/2}{\mathrm{Ca}}_{1/2}{\mathrm{MnO}}_3$ projected on the Wannier-type states $|w_1\rangle$ (red), $|w_2\rangle$ (white), $|w_3\rangle$ (blue), and $|w_4\rangle$ (green). The axis of the majority-spin direction points right and that of the minority-spin direction points left.

Figure 4

Photon-absorption density $D_p$ defined in Eq. (23) as function of the photon energy $E=\hslash \omega$ for different intensities and pulse lengths. The dashed line indicates the photon energy $\hslash \omega =1.17\mathrm{eV}$ chosen for the simulations described below.

Figure 5

Photon-absorption density $D_p$ defined in Eq. (23) as function of the amplitude of the exciting field $A_0$ and a photon energy of $\hslash \omega =1.17\mathrm{eV}$. Also given are the pump fluences $F_p$ in units of $\mathrm{mJ}/{\mathrm{cm}}^2$ according to Eq. (24) for the boundaries of the four regimes discussed below in Sec. 4b.

Figure 6

Identification of the distinct regimes on the basis of the intensity of magnetic diffraction spots. The pulse lengths are 50, 75, and 100 fs from top to bottom. Open blue circles show the minimum values of the spin-correlation function $C_S(\frac{1}{2},1,1)$ characteristic for the CE-type magnetic structure during the first 2 ps following the excitation. Red-filled circles are the maximum intensities $C_s(0,0,1)$ characteristic of the A-type magnetic structure and open green squares are maximum intensities $C_S(0,1,1)$ characteristic for the G-type structure.

Figure 7

Time evolution of the electron and hole distributions induced by a femtosecond pulse of different intensities. Regimes I to IV from top to bottom, i.e., $A_0=0.20,0.45,0.53,2.50\hslash /\left(e{a}_0\right)$. The instantaneous one-particle spectrum of Born-Oppenheimer energies is shown in gray. The intensity of blue color indicates conduction electrons and that of red color indicates holes.

Figure 8

Charge-order correlation $C_Q(1,0,0)$ (black) and orbital-order correlation $C_O(0,\frac{1}{2},0)$ (red) as function of time for regime I with $A_0=0.2\hslash /\left(e{a}_0\right)$. The correlations are scaled each so that their initial value is unity. The correlations can be represented well by a superposition of two harmonic oscillations with frequency ${\nu }_1=10$ THz and ${\nu }_2=16$ THz, which are the frequencies of the two coherent phonons.

Figure 9

Short-term dynamics for polarization along the $a$ axis (top) and along the $b$ axis (bottom) in regime I with $A_0=0.20\hslash /\left(e{a}_0\right)$. Shown are the deviations $\mathrm{\Delta }q$ of the charges from a trimer of the CE-type ground state as function of time: central site (blue) and corner sites (red/green). The instantaneous amplitude (arbitrary scale) of the light pulse is shown in gray.

Figure 10

Schematic drawing of Bloch waves involved in the photoexcitation with the electric field $(\vec{E}\parallel {\vec{e}}_A)$ polarized perpendicular $(\vec{E}\parallel \vec{a})$ (1) and (2) or along $(\vec{E}\parallel \vec{b})$ (3) and (4) to the zigzag chains of the CE-type spin order. Each box shows the initial state with $|w_1\rangle$ character in the upper left and the final state with $|w_2\rangle$ character in the lower right. For each polarization, two pairs of initial and final states, one without and one with sign change from one segment to the other, are shown. The induced charge density is related in first order to the product of initial and final states.

Figure 11

Total occupancies $F_j^{\mathrm{tot}}\left(t\right)$ (full lines) of Wannier-type orbitals $|w_j\rangle$ and their spin polarization $F_j^{\text{spin}}\left(t\right)$ (dashed lines) as function of time for fluence regime I with $A_0=0.20\hslash /\left(e{a}_0\right)$. Occupancies of $|w_1\rangle$ (red), $|w_2\rangle$ (black), $|w_3\rangle$ (blue), and $|w_4\rangle$ (green).

Figure 12

Phonon modes of region I with $A_0=0.20\hslash /\left(e{a}_0\right)$. Shown are the expansions $\mathrm{\Delta }d$ of the oxygen distances along the octahedral axes. The superscript $3+$ refers to the formal charge state of the central Mn ion in a trimer of the zigzag chain of the CE-type magnetic structure, while $4+$ refers to the corner site. For the central octahedron, the expansion along the trimer axis is $\mathrm{\Delta }{d}_{\parallel }^{3+}$ and the expansion in the $ab$ plane perpendicular to the trimer axis is $\mathrm{\Delta }{d}_{\perp }^{3+}$. The planar displacements at the corner atoms along and in the $ab$ plane perpendicular to the trimer axis $\mathrm{\Delta }{d}_x^{4+}$ and $\mathrm{\Delta }{d}_y^{4+}$ are identical. The expansions in the $c$ direction, indicated by a subscript $z$, vanish. Visible is the initial reduction of the Jahn-Teller distortion, followed by the coherent breathing mode with 16 THz on the corner sites and a Jahn-Teller mode at the central site with 10 THz. The displacements are averaged over equivalent atoms.

Figure 13

Spin-correlation function of regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$. The correlation functions $C_S(\frac{1}{2},1,1)$ and $C_S(\frac{1}{2},\frac{1}{2},1)$ are characteristic for the CE-type ground state. The correlation function $C_S(0,0,1)$ is characteristic for A-type magnetic structure. The ferromagnetic (B-type) peak $C_S(0,0,0)$ is shown in orange. The CE-type structure recovers after a transient period.

Figure 14

Spin-diffraction patterns in regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$. At about 0.29 ps (left), the magnetic structure is a superposition of patterns of the CE-type and the A-type antiferromagnetic structures. At 0.5 ps (middle), a spin wave emerges, which is similar to obtained in equilibrium with $\delta =15$% excited electrons (right).

Figure 15

Spin correlations in regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$ as function of time. ${\mathrm{\Phi }}_{n,n^{\prime }}$ is the angle between the spins ${\langle \vec{S}\rangle }_n$ of neighboring zigzag chains $C_n$ and $C_{n^{\prime }}$. The mean angle of a chain is ${\langle \vec{S}\rangle }_n=\frac{1}{N}{\sum }_{j\in C_n}{\vec{S}}_j$, where $N$ is the number of Mn sites in the chain. The top four figures show the angles ${\mathrm{\Phi }}_{n,n^{\prime }}$ for neighboring chains in the same $ab$ plane. The middle four figures show the angles for neighboring chains stacked along the $c$ direction. The bottom four figures show the ferromagnetic spin correlations within a chain ${\mathrm{\Delta }}_n=\sqrt{\frac{1}{N}{\sum }_{j\in C_n}{\left[\angle \left({\vec{S}}_j,{\langle \vec{S}\rangle }_n\right)\right]}^2}$.

Figure 16

Total occupancies $F_j^{\mathrm{tot}}\left(t\right)$ (full lines) of Wannier-type orbitals $|w_j\rangle$ and their spin polarization $F_j^{\text{spin}}\left(t\right)$ (dashed lines) as function of time for fluence regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$. Occupancies of $|w_1\rangle$ (red), $|w_2\rangle$ (black), $|w_3\rangle$ (blue), and $|w_4\rangle$ (green).

Figure 17

Charge-order correlation $C_Q(1,0,0)$ (black) and orbital-order correlation $C_O(0,\frac{1}{2},0)$ (red) as function of time for regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$. The correlations are scaled each so that their initial value is unity.

Figure 18

Phonon modes of regime II with $A_0=0.45\hslash /\left(e{a}_0\right)$. For a description of the symbols, see Fig. 12.

Figure 19

Spin-diffraction intensity of regime III with $A_0=0.53\hslash /\left(e{a}_0\right)$. The peaks $C_S(0.5,1,1)$ and $C_S(0.5,0.5,1)$ shown in red, respectively green, are characteristic for the CE-type ground state. The peak $C_S(0,0,1)$ shown in blue is characteristic for A-type magnetic structure. The ferromagnetic (B-type) peak $C_S(0,0,0)$ is shown in orange. The original CE-type magnetic pattern is quickly destroyed, while an A-type magnetic pattern emerges. The latter evolves over time into a the ferromagnetic phase.

Figure 20

Magnetic diffraction patterns of region III with $A_0=0.53\hslash /\left(e{a}_0\right)$ at 0.3, 1.2, and 6.1 ps. The $a$ axis points right, the $b$-axis toward the back, and the $c$ axis up. The small white spheres indicate points with integer $h,k,l$ in the $Pbnm$ setting. Reciprocal space is shown for $h,k,l\in [-1.25,1.25]$. At 0.3 ps the diffraction pattern is dominated by an A-type pattern. At 1.2 ps the diffraction pattern exhibits spots from both A and B types, while at 6.1 ps the diffraction pattern is ferromagnetic, i.e., B type. The double spots are a sign of ferromagnetic magnetic domains, respectively a long-wavelength spin wave, rather than a pure ferromagnet.

Figure 21

Charge-order correlation $C_Q(1,0,0)$ (black) and orbital-order correlation $C_O(0,\frac{1}{2},0)$ (red) as function of time for regime III with $A_0=0.53\hslash /\left(e{a}_0\right)$. The correlations are scaled each so that their initial value is unity.

Figure 22

Phonon modes of region III with $A_0=0.53\hslash /\left(e{a}_0\right)$. For a description of the symbols, see Fig. 12.

Figure 23

Spin-diffraction intensity of regime IV with $A_0=2.50\hslash /\left(e{a}_0\right)$. The peaks $hkl=(0.5,1,1)$ and $(0.5,0.5,1)$ shown in red, respectively green, are characteristic for the CE-type ground state. The ferromagnetic (B-type) peak $C_S(0,0,0)$ is shown in orange. The peak with $hkl=(1,0,1)$ is characteristic for G-type magnetic structure. Over time, a new, noncollinear magnetic structure (see Fig. 25) emerges evidenced by the occurrence of the peak $hkl=(0.5,0.5,1)$.

Figure 24

Magnetic diffraction patterns of region IV with $A_0=2.50\hslash /\left(e{a}_0\right)$ at 0.3, 0.8, and 6.2 ps. At 0.3 and 0.8 ps the diffraction pattern is dominated by a G-type pattern. At 6.2 ps a new diffraction pattern occurs, which can be attributed to a noncollinear spin structure described in the text and in Fig. 25. Reciprocal space is shown for $h,k,l\in [-1.25,1.25]$. The $a$ axis points right, the $b$ axis toward the back, and the $c$ axis up. The small spheres indicate points with integer $hkl$ in the $Pbnm$ setting.

Figure 25

Idealized model for the spin distribution reached in regime IV beyond 2 ps. The noncollinear spin structure in the $ab$ plane of the $Pbnm$ setting is shown on the left. Shown is the grid of Mn sites with the orientation of the $t_{2g}$ spins. Planes are stacked antiferromagnetically in the $c$ direction. On the right, the corresponding magnetic diffraction pattern is shown. The diffraction pattern is nearly indistinguishable from the one shown in Fig. 24, which is obtained in regime IV at 6.2 ps after the light pulse. The $a$ axis points right, the $b$ axis toward the back, and the $c$ axis up. The small spheres indicate points with integer $hkl$ in the $Pbnm$ setting. Reciprocal space is shown for $h,k,l\in [-1.25,1.25]$.

Figure 26

Charge-order correlation $C_Q(1,0,0)$ (black) and orbital-order correlation $C_O(0,\frac{1}{2},0)$ (red) as function of time for regime IV with $A_0=2.5\hslash /\left(e{a}_0\right)$. The correlations are scaled each so that their initial value is unity.

Figure 27

Phonon modes of region IV with $A_0=2.50\hslash /\left(e{a}_0\right)$. For a description of the symbols, see Fig. 12.

Figure 28

Temperatures of the electron (blue), spin (red), and phonon (black) subsystems as function of time. The phonon temperatures are scaled by 100. From top to bottom are the representative examples from regime I to IV, i.e., $A_0=0.20,0.45,0.53$ and 2.50.

Figure 29

Distributions of electron occupations $\overline{f}$ at different times, namely, $t=0.2$ ps (open circles) and 4.8 ps (blue filled circles). From top to bottom are the representative examples from regime I to IV, i.e., $A_0=0.20,0.45,0.53$, and 2.50 $\hslash /\left(e{a}_0\right)$. The left figure shows the occupations (circles) versus Born-Oppenheimer energy ${\epsilon }_j^{\text{BO}}$. The dashed black line is the Fermi distributions at 0.2 ps obtained from the energy and particle-number sum rules. The full blue line is the Fermi distribution at 4.8 ps. On the right, the occupations $\overline{f}$ are transformed by $\mathrm{arctanh}(1-2\overline{\mathrm{f}})$ which maps a Fermi distribution to a straight line with slope $1/\left(k_B{T}_{\psi }\right)$ and zero ${\mu }_{\psi }$.

Figure 30

Nonequilibrium phase diagram with hot electrons and cold spins and phonons. Top: spin angles. ${\varphi }_c$ is the spin angle between spins in the $c$ direction of the $Pbnm$ setting. ${\varphi }_{ab}$ is the spin angle of neighboring Mn sites in the $ab$ plane. Below $T=T_1$, the angle ${\varphi }_{ab}$ is replaced by one angle ${\varphi }_a$ perpendicular to the zigzag chains of the CE-magnetic structure and ${\varphi }_{\text{chain}}$ is the angle within the zigzag chains. Bottom figure: charge-order correlation function $C_Q$ and Jahn-Teller distortion $Q^{\text{JT}}$ of the central site as function of temperature. Both are divided by their zero-temperature value.

Figure 31

Energy conservation for three intensities of the excitation. The initial rise is due to the excitation. The largest violation of energy conservation is a slow dissipation of the quantum systems.

Figure 32

Fit of the phonon modes and diffraction intensities. The top figure shows $\mathrm{\Delta }{d}_{\parallel }^{3+}$ (red) and the fit (dashed). The middle graph shows $\mathrm{\Delta }{d}_y^{4+}$ and its fit. The bottom graph shows the correlation functions $C_Q$ for charge and $C_O$ for orbital order and their fits. The derived frequencies are 9.7 and 15.7 THz. The trajectory has been performed for a light amplitude of $A_0=0.20\hslash /\left(e{a}_0\right)$.