Article pubs.acs.org/JPCC
Cite This: J. Phys. Chem. C 2019, 123, 14701−14706
Unraveling the Spin Relaxation Mechanism in Hybrid Organic− Inorganic Perovskites Zhi-Gang Yu*,†,‡ and Yan S. Li§ †
ISP/Applied Sciences Laboratory, Washington State University, Spokane, Washington 99210, United States Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, United States § Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, United States ‡
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S Supporting Information *
ABSTRACT: The hybrid organic−inorganic perovskite (HOIP) CH3NH3PbI3 exhibits a long spin coherence despite its strong spin−orbit coupling. Elucidation of the spin relaxation mechanism is a prerequisite to harness this spin coherence for spintronic applications. Here, we show that exciton spin relaxation behaviors can be quantitatively accounted for by the Rashba splitting in the exciton bands together with piezoelectric (PE) coupling, both arising from structural inversion asymmetry of HOIPs. A large exciton size and Debye screening influence piezoelectric scattering and result in the unusual temperature and photoexcitation dependences in spin relaxation. Our results suggest that the strong PE coupling helps retain spin coherence in HOIPs with strong spin−orbit coupling and Rashba splittings.
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INTRODUCTION
electrons cjez and valence-band holes v̅jhz characterized by angular momenta je(h)z = ± 1/2. In the TRFR experiment, circularly polarized light excites Γ5 excitons (c1/2v1/2 or ̅ c −1/2v−̅ 1/2 ) with spin polarization along the z-axis. The Γ5 excitons are a superposition of exciton eigenstates in a transverse magnetic field B, and a quantum-beating signal emerges.10 The spin Hamiltonian of the 1s exciton under B = Bex reads
Hybrid organic−inorganic perovskites (HOIPs) such as MAPbI3 (MA = CH3NH3), with photovoltaic efficiencies exceeding 20% in solution-processed samples, represent a revolutionary breakthrough for low-cost solar cells.1−3 For a material with heavy elements such as Pb that have an strong spin−orbit coupling (SOC),4 spin coherence is generally deemed too ephemeral to be seen. The situation is exacerbated in HOIPs by the Rashba effect (RE), resulting from the concomitant SOC and structural inversion asymmetry (SIA).5−9 The RE introduces a momentum-dependent effective magnetic field, around which the spins precess erratically. Surprisingly, recent time-resolved Faraday rotation (TRFR) measurements revealed a long spin lifetime in MAPbI3, exceeding 1 ns at 4 K.10 Here, we show that this apparent paradox can be unraveled by another consequence of SIA: pronounced piezoelectric (PE) coupling between electrons and acoustic phonons. The PE suppresses the exciton spin relaxation via the Dyakonov−Perel (DP) mechanism,11 while both Elliot−Yafet (EY)12 and Bir− Aronov−Pikus (BAP)13 mechanisms are insignificant. Thus, HOIPs provide a paradigm of achieving a long spin coherence in the presence of large SOC and RE. Moreover, we demonstrate that spin relaxation can be tuned by modifying the effective PE coupling via photoexcited carriers.
Hex = J σezσhz + J⊥(σexσhx + σeyσhy) +
(1)
where σe(h) are the Pauli matrices for electron (hole) pseudospin je(h), ge and gh are the electron and hole g-factors in the x− y plane, respectively, μB is the Bohr magneton, and J∥ (J⊥) is the exchange between electron and hole spins parallel (perpendicular) to the crystal axis. The RE is not explicitly included in eq 1 because it contributes mainly to spin relaxation (see below) but little to quantum-beating frequencies. The TRFR signal can be obtained from a time evolution of the exciton density matrix, defined as ρ̂ex = ∑mm′; nn′ρmm′; nn′|cmv̅n⟩⟨cm′v̅n′|, ∂ρ̂ex/∂t = i[ρ̂ex, Hex]/ℏ − ∂ρ̂ex/ ∂t|sr, which includes both quantum beating and irreversible spin relaxation. The latter can be written as
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METHODS AND RESULTS Exciton States under a Magnetic Field. The optical excitations in MAPbI3, as created in the TRFR experiment, are loosely bound Wannier excitons,14,15 with conduction-band © 2019 American Chemical Society
1 (g σex + gh σhx)μB B 2 e
Received: May 5, 2019 Published: May 15, 2019 14701
DOI: 10.1021/acs.jpcc.9b04261 J. Phys. Chem. C 2019, 123, 14701−14706
Article
The Journal of Physical Chemistry C ∂ρmm′; nn′ ∂t
zyz 1 jijj 1 jjρmm′; nn′ − δmm′ ∑ ρm″m″; nn′ zzz z τse j 2 m′ k { zyz 1 ijjj 1 − jjρmm′; nn′ − δnn′ ∑ ρmm′; n″n″ zzz z τsh j 2 n″ k {
Indeed, by fitting the TRFR data, the exchange in MAPbI3 is found to be around J∥( ⊥ ) ≈ 10−6 eV, which can be overcome by a field of B > 10 mT. In this weak-exchange regime, the TRFR signals, as well as exciton energies and Larmor frequencies, are displayed in Figure 1b,e. Exciton Spin Relaxation. The electronic structure of lowtemperature orthorhombic and tetragonal CH3NH3PbI3 can be well described by an effective-mass model,15 in which the valence and conduction bands are derived from Pb’s 6s and 6p orbitals. Because of Pb’s strong SOC, the 6p orbitals are more appropriately characterized by the total angular momentum j = l + s with l (l = 1) and s being orbital and spin angular momenta, respectively. The SOC is reflected in the energy splitting between the lowest conduction band (j = 1/2) and higher conduction bands (j = 3/2). In addition, the lack of inversion symmetry in orthorhombic and tetragonal CH3NH3PbI3 breaks the spin degeneracy at a given wavevector k in the conduction and valence bands, that is, the Rashba effect
=− sr
(2)
with τse(h) being the relaxation time of Se(h) = Trρ̂exσe(h). Here, we neglect exciton recombination, a much slower process than spin relaxation in HOIPs. If the exchange coupling is temporarily neglected (see the Supporting Information), dynamics of Se and Sh are decoupled, and their z-components, Sez ≡ ∑n ρ1/21/2; nn − ρ−1/2 − 1/2; nn and S hz ≡ ∑m ρmm;1/21/2 − ρmm; −1/2 − 1/2 , are described by a damped oscillation, Se(h)z ∝ cos [ωe(h)t]e−t/τse(h), with frequency ωe(h) = ge(h)μBB/ℏ. The TRFR signal, Sz ≡ ρ1/21/2;1/21/2 − ρ−1/2 − 1/2; −1/2 − 1/2 = Sez + S hz , contains two components with distinct oscillation frequencies, as shown in Figure 1a. The corresponding exciton energies and the electron’s and hole’s Larmor frequencies are displayed in Figure 1d as a function of B.
R Hc(v) = αc(v)(k yσx − kxσy)
(3)
where αc(v) is the Rashba strength in the conduction (valence) bands and directly related to the material’s inversion asymmetry parameter.15 In a semiconductor, spin relaxation can be caused by EY, DP, and BAP mechanisms. A direct spin-phonon process, which may become important in spin relaxation under strong magnetic fields (>1 T),17 is not effective at low magnetic fields ( 150 K28 because of their strong ionic nature. At low temperatures where polar optical phonons are scarce, acoustic-phonon scattering should be dominant. In this case, scattering potential Hp can be the short-range deformation potential and, in systems with SIA as in MAPbI3, the long-range piezoelectric (PE) potential29 Hp = ± ∑λq(Cλqeiq · rbλq + C*λqe−iq · rb†λq) where + and − are for electrons and holes, respectively, because of the PE coupling’s electrostatic origin. b†λq creates an acoustic phonon with eigenvector ξ(qλ) in branch λ at a wave vector q, and the coupling strength30 (see the Supporting Information)
1/2 ij e 2 yz i 2πg y ℏcλq3/2 zzp , p Cλq = jjjj λ zzzz , gλ = jjj 2 2 j ℏϵcλ zz λ̅ λ̅ k V { q + q0 k { ÄÅ ÉÑ2 Å Ñ 4π ÅÅÅ ek , ijqk qjξi(qλ) ÑÑÑ ÅÅ ÑÑ = ÑÑ qωλ ϵρ ÅÅÅÅ ÑÑÖ Ç
Figure 2b compares the momentum scattering times due to the PE and deformation coupling, showing that the PE scattering is an order of magnitude stronger for T < 200 K. Moreover, the PE scattering time decreases with temperature as T−1/2 for free carriers, as compared to the T−3/2 dependence of deformation scattering. This weak temperature dependence suggests that the PE scattering becomes more dominant at cryogenic temperatures. For T > 100 K, the temperature dependence of PE scattering levels off, as shown in Figure 2b, even though the number of acoustic phonons increases with temperature. This arises from the exciton’s form factor φe(h)(q), which limits effective scattering to long-wave length phonons q < a−1 0 . The PE scattering time τp over the entire temperature range varies from 0.1 to 1 ps, much shorter than the free induction time τf. Therefore the spin lifetime is greatly enhanced through frequent PE scattering. The enhancement factor η is shown in Figure 2c, which is about 50 for T < 10 K but reduces to 5 for T > 100 K, where spin precession becomes much faster, while scattering of the electron in an exciton remains largely unchanged. The averaged spin lifetime in eq 8, delineated in Figure 2d, quantitatively explains the experimentally measured spin lifetime over the entire temperature range. Contributions from the EY mechanism is orders of magnitude weaker than those from the DP mechanism. The dominant PE scattering in spin relaxation should be the limiting factor of carrier mobility at low temperatures in HOIPs also. Using the PE strength to estimate carrier mobility, νe = eτp/me, we find that the free electron mobility in an MAPbI3 crystal would be 102 − 103 cm2/V s for T < 100 K. Based on the results from carrier scattering and spin dynamics in HOIPs at low temperatures, the long spin lifetime in MAPbI3 can be unraveled via an auspicious combination of several factors: (1) the weak exchange coupling inhibits the BAP mechanism, (2) the minimal spin mixing due to the large energy gap makes the EY mechanism ineffective, and most importantly, (3) the strong PE coupling suppresses spin relaxation induced by the Rashba splitting via the DP mechanism. With the information of the momentum scattering time and spin lifetime at low temperatures, we can estimate the Rashba strength as αc = (M /me)/ 3τesτpMkBT . Using the calculated τp(T), we find that αc = 7 × 10−3 eV Å can explain the measured spin lifetime as a function of temperature. This value is considerably smaller than the Rashba splitting from earlier first-principles calculations reported in the literature, αc ≥ 1 eV Å,8,9 but in line with recent studies37 and consistent with values estimated from second harmonic generation experiments.38 It should be noted that when the Rashba splitting is large enough to have τf < τp, the spin lifetime would be determined by the free induction, τes ≃ τf; for T = 10 K and αc = 1 eV Å, τes = 5 × 10−14 s, which is several orders of magnitude shorter than the measured value. The effective PE coupling in eq 9 depends on the Debye screening q0, which changes the electrostatic potential from 1/ ϵr to e−q0r/ϵr. Since q0, according to the Debye−Huckel
(10)
Here, ek, ij is the PE tensor with subscripts being Cartesian component x, y, or z, q0 is the reciprocal Debye screening length, ρ is the density, and the dimensionless coupling gλ is the product of the fine structure constant for the speed of sound cλ and electromechanical coupling p̅λ, which measures the percentage of mechanical energy that is converted into electric energy. Table 1 lists the PE and mechanical properties31,32 reported in the literature as well as calculated gλ values for MAPbI3. The PE coupling strength is exceptionally large, even higher than that in CdS (gT = 3.7 and gL = 0.21),33,34 where PE scattering is known to be the dominant scattering channel at low temperatures.35 The huge gT in MAPbI3 is the result of a strong electromechanical coupling due to its ionic feature and a low speed of sound due to its elastic softness for the transverse modes, particularly in the tetragonal phase.36 Since PE and polar couplings share the same electrostatic origin, the strong PE coupling in MAPbI3 is consistent with its large polar coupling. 14704
DOI: 10.1021/acs.jpcc.9b04261 J. Phys. Chem. C 2019, 123, 14701−14706
Article
The Journal of Physical Chemistry C theory,39 can be influenced by excess electron and hole densities, ne and nh, repectively q02 =
Notes
The authors declare no competing financial interest.
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2
4πe (ne + nh) ϵkBT
ACKNOWLEDGMENTS This work was partly supported by the U.S. Army Research Office under contract no. W911NF-17-1-0511. Y.S.L. acknowledges support from the Center for Hybrid Organic Inorganic Semiconductors for Energy (CHOICE), an Energy Frontier Research Center, funded by the Office of Basic Energy Sciences, Office of Science within U. S. Department of Energy.
(11)
PE scattering, and accordingly, spin relaxation, can be tuned by varying photoexcitation intensity. Figure 3a,b describes screen-
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Figure 3. Debye screening parameter q20 (a), momentum scattering time (b), and spin lifetime (c) as a function of photoexcitation intensity. In (b), blue and red lines correspond to deformation and piezoelectric scatterings, respectively. In (c), circles and squares are experimental data on two similarly prepared MAPbI3 samples. I0 = 0.25 W/cm2.
ing q20 and momentum scattering time τp due to deformation and PE couplings as a function of photoexcitation intensity. The measured spin lifetimes of different MAPbI3 samples, as plotted in Figure 3c, follows a common intensity dependence. We see that, while photoexcitation does not affect the shortrange deformation coupling, it weakens the PE scattering and reduces the spin lifetime, which satisfactorily accounts for the observed intensity dependence of spin lifetime, as shown in Figure 3c. In summary, we have unraveled the puzzling long-exciton spin lifetime and its dependence on temperature and photoexcitation in MAPbI3. Our microscopic theory provides a solid foundation for achieving a long spin coherence in HOIPs, which, together with versatile spin manipulation afforded by strong SOC and RE, promises great potential for HOIPs in spintronics.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b04261.
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REFERENCES
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Detailed derivations and discussions on the electronic structure, exciton states, exciton spin dynamics, exciton bands, the Elliot−Yafet mechanism, and piezoelectric coupling in HOIPs (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Zhi-Gang Yu: 0000-0002-1376-9025 14705
DOI: 10.1021/acs.jpcc.9b04261 J. Phys. Chem. C 2019, 123, 14701−14706
Article
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DOI: 10.1021/acs.jpcc.9b04261 J. Phys. Chem. C 2019, 123, 14701−14706
