Coherent spin dynamics of electrons and holes in CsPbBr3 perovskite crystals

The lead halide perovskites demonstrate huge potential for optoelectronic applications, high energy radiation detectors, light emitting devices and solar energy harvesting. Those materials exhibit strong spin-orbit coupling enabling efficient optical orientation of carrier spins in perovskite-based devices with performance controlled by a magnetic field. Here we show that elaborated time-resolved spectroscopy involving strong magnetic fields can be successfully used for perovskites. We perform a comprehensive study of high-quality lead halide perovskite CsPbBr3 crystals by measuring the exciton and charge carrier g-factors, spin relaxation times and hyperfine interaction of carrier and nuclear spins by means of coherent spin dynamics. Owing to their ‘inverted’ band structure, perovskites represent appealing model systems for semiconductor spintronics exploiting the valence band hole spins, while in conventional semiconductors the conduction band electrons are considered for spin functionality.

The bottom of the conduction band has R − 6 symmetry, see Refs. 1,3 . According to Ref. 1 , the valence band is mainly composed by the s-orbitals of the metal, |S 0 ⟩, with admixture of the halogen p-orbitals (a combination ∝ |X 1 ⟩ + |Y 2 ⟩ + |Z 3 where I is the nuclear spin, S e (S h ) is the electron ( where ℓ e(h) ≤ 1 is the leakage factor characterizing the losses of nuclear spin polarization due to relaxation processes other than the hyperfine coupling.
Via the hyperfine interaction, the polarized nuclear spins produce the Overhauser field where the summation is carried out over all nuclei, so that the index j includes all chemical elements, all isotopes of the element abundant in the sample, as well as all positions R j of the nuclei. Under the standard assumption of a uniform nuclear spin polarization ⟨I⟩ Eq. (5) can be written in a simple form as where the sum is carried out over the different elements and isotopes denoted by the subscript i. Indeed, the summation over unit cells assuming homogeneous nuclear polarization, can be transformed to an integral as where N iso is the number of corresponding isotopes in the unit cell. In line with the smooth envelope method, we assume that φ e(h) (R  The hyperfine interaction constants for CsPbBr 3 can be estimated as follows. In the valence band the hyperfine coupling is dominated by the contact interaction with the the lead atoms. The constant A (0) h calculated per isotope, i.e., disregarding the abundance, can be written as 9 : where S 0 (r) is the Bloch function at the nucleus position normalized per volume of the unit cell, is the Bohr magneton, µ I is the nuclear magnetic moment, I is the spin of the nucleus. It is important to note that the transformation from the electron to the hole representation results in the inversion of both the direction of spin and the energy axis, leaving the hyperfine constant sign the same. That is why we can use the electron representation for evaluation of the hyperfine coupling for holes.
For holes the relevant isotope is 207 Pb with an abundance of about 22%, the nuclear spin I = 1/2 and µ I = 0.58µ N , where µ N ≈ 7.62 MHz/T is the nuclear magneton. The hyperfine interaction can be estimated from the atomic constants 11,12 . From Ref. 11 (which uses an approach that typically overestimates the value, as known from the comparison for III-V semiconductors) we have (per nucleus): Note that inclusion of the so-called Mackey-Wood correction gives a ∼ 3.15-fold enhancement up to 336 µeV. From Ref. 12 (which approach typically underestimates the hyperfine coupling), disregarding the anisotropy factor we have : Thus, we take A Note that the dipole-dipole interaction with 79 Br and 81 Br (both I = 3/2) can be estimated is the admixture of the p-shell of Br to the s-shell of Pb in the valence band Bloch function, therefore, it can be disregarded.
For the conduction band states it is sufficient to account for the dipole-dipole interaction with the bromine nuclei only. Estimates based on Refs. 11,12 give A e ≈ 7 µeV. We use this constant for both the 79 35 Br and 81 35 Br isotopes whose spin is the identical and whose total abundance is 100%. The parameters of the hyperfine interaction are summarized in Supplementary Table 1. For complete nuclear polarization, where |⟨I⟩| = I according to our estimates after Eq. (6) the maximum Overhauser fields read for the holes and electrons, respectively. Interestingly, the maximal Overhauser field is slightly Here µ B is the Bohr magneton, g e and g h are the electron and hole g-factors, respectively, S e = S h = ±1/2 are the electron and hole spins. Note that we use the standard representation for the exciton Hamiltonian where the electron part is taken in the electron and the hole part is taken in the hole presentation, that is, why the Hamiltonian is the sum of the electron and the hole contributions. In this definition the g-factors are positive both for electrons and holes if the ground state is the state with spin projection s z = −1/2 onto the direction of the magnetic field. Note that just like for the hyperfine coupling, the value and the sign of the hole g-factor is the same, both for the electron and hole representations. The exciton g-factor is defined as In case of the perovskites the exciton g-factor is the sum of the electron and hole g-factors: g X = g e + g h . The slight discrepancy between the values of the g X inferred directly from optical spectroscopy, on the one hand, and from the sum of g e and g h , on the other hand, may be attributed to the effects of the electron-hole exchange interaction and band non-parabolicity which gives rise to a renormalization of the g-factors of the charge carriers in the exciton 19 . We consider the case of small modulation frequencies where the spin inertia is not important.
Introducing ω N as the characteristic fluctuation of the Overhauser field we obtain We recall that ω L,h is the hole spin precession frequency in the external magnetic field and τ s is the spin relaxation time unrelated to the hyperfine coupling. Equation (14)  The value of the nuclear spin fluctuation δB N ≈ 6.6 mT allows us to make a crude estimate of the hole localization length. To that end we evaluate ω N , assuming independent contributions of the Pb isotopes as ω N = √ 2 3h 2 I(I + 1)