SUPPLEMENTARY INFORMATION - Nature...Chuang Zhang 1, Dali Sun , Zhi-Gang Yu2, Z. Valy Vardeny 1, Yan...

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4145

NATURE PHYSICS | www.nature.com/naturephysics 1

Supplementary Informationfor Manuscript

Spin-polarized exciton quantum beating in hybridorganic-inorganic perovskites

Patrick Odenthal1†, William Talmadge1†, Nathan Gundlach1, Ruizhi Wang1∗,Chuang Zhang1, Dali Sun1, Zhi-Gang Yu2, Z. Valy Vardeny1, Yan S. Li1

1Department of Physics and Astronomy,

University of Utah, UT 84112, USA and

2ISP/Applied Sciences Laboratory, Washington State University, Spokane, WA 99210, USA

[1] † These authors have made equal contributions to the work.[2] ∗ Present address: School of Electronic and Optical Engineering, Nanjing University of Science and

Technology, Nanjing, Jiangsu 210094, China[3] Author to whom correspondence should be addressed; e-mail: [email protected].

http://dx.doi.org/10.1038/nphys4145

http://www.nature.com/naturephysics




1. Sample characterization

1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

0.0

0.5

1.0

1.5

Pho

tolu

min

ecen

ce (A

.U.)

Energy (eV)

Abs

orba

nce

(OD

)

10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

2θ (degree)

(110)

(220)

(312)

(211)

b c

ASEMimageshowingthetypicalmorphologyofCH3NH3PbI3-XClX.BXRD2Thetascan.CPhotoluminescence(blue)andabsorbanceredofatypicalsampleat5K.

a

Figure S 1: CH3NH3PbClxI3−x film characterization. a, Scanning Electron Microscope

image showing the typical morphology of the polycrystalline films. b, X-Ray Diffraction, 2θ scan

at room temperature. c, Photoluminescence (blue) and absorption (red) of a typical sample at 5

K.

The thickness of the films is 250 to 300 nm as determined by a profilometer. The XRDshows a polycrystalline pattern with some preferences in crystal orientation in the tetragonalphase at room temperature. Since the resolution of XRD is not sufficient to estimate thegrain size, we phenomenologically estimate it to be ∼200 nm from the SEM image. Theabsorption spectrum shows a pronounced exciton peak, and the photoluminescence displayscommonly seen dual emission peaks for polycrystalline films [1–4].






2. Setup of the spin dynamics measurements

Tsunami 90:10 beamsplitter

Chopper (f2)

PEM (f1)LP

LP

f1 Lock-in

f2 Lock-in

2λ

Wollaston

Cryostat/Magnet

Photodiode Bridge

Figure S 2: Setup of the time-resolved Faraday rotation measurement. LP: linear polar-

izer. PEM: Photo elastic modulator. λ/2: half wave plate.

MBR: 730-780 nmTunable, CW

PEM (f1)

LP

LP

f1 Lock-in

2λ

Wollaston

Cryostat/Magnet

PD Bridge

635 nm, CW

Figure S 3: Setup of the optical Hanle measurement.

Both TRFR and Hanle effect measurements use a circularly polarized “pump” for opticalorientation of spin-polarized excitons, and a linearly polarized “probe” to measure excitonspin polarization via Faraday rotation. The measurements are set up in the Voigt geometry,where the magnetic field is applied through a split coil electromagnet. The probe beam is setto be at normal incidence on the sample for accurate measurement of Faraday rotation, andpump beam has a small angle of incidence through the same focusing lens with the probebeam. The Faraday rotation is analyzed with a half wave plate and a Wollaston polarizingbeam splitter, and measured with a balanced photodiode bridge.






3. Exciton relaxation dynamics

5

4

3

2

1

0Li

fetim

e (n

s)

806040200Temperature (K)

∆T (a

.u.)

2000150010005000Time delay (ps)

80 K

50 K

20 K

4 K

a b1.0

0.8

0.6

0.4

0.2

0.0

Rel

ativ

e Am

plitu

de

806040200Temperature (K)

c

Figure S 4: Transient absorption (TA) measurement. TA curves were measured with

same wavelengths as used in TRFR. a, Transient absorption curves taken at 4, 20, 50 and 80

K. b, Temperature dependence of the two time scales seen in transient absorption. Error bars

indicate errors in the fitting. c, Relative amplitudes of the two exponential decays vs. temperature.

The relative amplitude represented by open square (triangle) symbols corresponds to the decay

component in b represented by the square (triangle) filled symbols.

With the wavelength of both pump and probe beams tuned to the exciton absorption, thetransient absorption (TA) signal should be due to photobleaching of the exciton transition[5], and the time evolution of this signal measures exciton relaxation processes. At 4 K,the TA decays bi-exponentially with a small magnitude that decays fast (∼150 ps), and adominant one that lasts much longer (> 4 ns). We speculate that the fast decay is relaxationto trap states [3], as it disappears at higher temperatures (Figure S4c), and the slow decay(> 4 ns) is due to exciton recombination or decay to bound exciton states [3, 6]. Thenonmonotonic relationships of the lifetime of the fast decay vs. T and relative amplitude vs.T we suspect to be related to more than one process. Although not measurable with the TA,the exciton lifetime should be longer if excited at much lower intensity as used for TRFR.It requires more future work to determine the exact nature of these exciton relaxationprocesses following optical excitation. Nevertheless, we identify the dominant long-livedexciton population, giving rise to the Faraday rotation signal in TRFR measured at thesame wavelength. Finally, we remark that the carrier/exciton dynamics in the orthorhombicphase CH3NH3PbI3 is different from that in the tetragonal phase, which has been the focusof a vast number of studies, and in which the long carrier recombination time is favorablefor photovoltaic applications.






4. TRFR sample statistics and intensity dependence

Figure S5 shows the high consistency of the fitting with the measured curves on sample“A” discussed in the main text. Figure S6 summarizes the TRFR data on another sample“B”, showing results similar to those of sample “A”. Table I gives the sample statistics.

0.2

0.0

Fara

day

rota

tion

(µra

d)

2000150010005000Time delay (ps)

0.2

0.0

0.2

0.0

0.2

0.0

0 mT

100 mT

400 mT

700 mT

000

Figure S 5: Fitting of the TRFR curves. The black cross denote measured data

points, and the blue curves are fitting curves by the equation θF = A1e−t/τ1 cos (2πν1t+ φ1) +

A2e−t/τ2 cos (2πν2t+ φ2).

0 250 500 750 1000

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Fara

day

Rot

atio

n (µ

Rad

)

Delay (ps)

05

10152025

0 200 400 6000

200

400

600

800

|g2|=0.44

Freq

uenc

y (G

Hz)

|g1|=2.66

T* 2 (ps

)

B (mT)

0 mT

100 mT

400 mT

700 mT

a b AMagne&cFielddependenceofTRKRtakenat4K.BMagne&cfielddependenceofthetwoprocessionfrequencieswiththecalculatedg-factors.CMagne&cfielddependenceofthespindephasing&me.Thestrongdependenceisduetogfactorinhomogeneity

Figure S 6: TRFR data on another sample at 4 K. a TRKR in different transverse magnetic

fields. b Magnetic field dependence of the two procession frequencies and spin lifetimes.






Sample T∗2 |g1| ∆g1 |g2| ∆g2

A 1170 2.63 0.07 0.33 0.15

B 830 2.66 0.07 0.44 0.09

C 850 2.65 0.08 0.45 0.07

D 425 2.59 0.07 0.38 0.12

E 445 2.64 0.05 0.38 0.10

F 1100 2.61 0.07 0.35 0.18

TABLE I: Sample Statistics

34567

100

2

34567

1000

τs (p

s)

4 6 10 2 4 6 100 2 4 6

Intensity (W/cm2)

S1 S2

8

6

4

2

0

Sig

nal (

a.u.

)

2000150010005000Time delay (ps)

TRFR (5 W/cm2)

TRFR (200 W/cm2)

TA (200 W/cm2)

a b

Figure S 7: Intensity dependence of spin lifetime. a, The data points of round symbols

were measured on sample 1 and those of triangle symbols were from sample 2. b, Comparison of

the time scales of TA and TRFR under the same pump intensity, and those of TRFR under two

different intensities.

Figure S7a shows the pump intensity dependence of the spin lifetime. Only the longerdecay time in the bi-exponential decay is plotted for simplicity. The spin lifetime decreasesmonotonically as intensity increases, consistent with the natural interpretation of spin re-laxation as a result of exciton-exciton scattering. The typical pump intensity used in TRFRis 5 W/cm2, at which it’s difficult to obtain TA data. In order to compare the decay timescales of TRFR and TA, we measured TRFR under higher intensities. The spin lifetime of∼100 ps (middle curve in Figure S7b) is much shorter than the exciton lifetime of a few nsimplied from TA (top curve in Figure S7b). It is therefore a valid assumption that TRFRmeasures spin lifetime of the long-lived excitons characterized by TA, and the contributionof exciton relaxation to spin dynamics can be ignored.






5. Exciton fine structure

The description here is borrowed from the recent effective mass model for the tetragonalphase perovskites [7], which is also a good approximation to the orthorhombic phase, asthe lattice parameters of a and c for the orthorhombic phase are very close [8]. The b-axisof the orthorhombic phase corresponds the c-axis in the tetragonal phase. In the followingderivation, we assume crystal axis notation of the tetragonal phase, and c-axis ‖ z. Thewave functions for the valence and conduction bands are

v+(−) = S ↑ (↓),

c+(−) = − 1√2[X + (−)iY ] cos ξ ↓ (↑)− (+) sin ξZ ↑ (↓),

where S, X, Y , and Z are s and three p orbital wavefunctions, the coefficient cos ξ and sin ξare related to the band structure parameters, and sin ξ = 1/

√3 for the cubic phase.

Considering the crystal symmetry, with space group of C4v for the tetragonal phase, thefour 1S exciton wavefunctions in the perovskite can be written as

ψ1 =1√2(c+v−−c−v+) = −cos ξ

2(X+iY )S ↓e↓h +

cos ξ

2(X−iY )S ↑e↑h −sin ξ√

2ZS(↑e↓h + ↓e↑h),

ψ2 =1√2(c+v−+c−v+) = −cos ξ

2(X+iY )S ↓e↓h −cos ξ

2(X−iY )S ↑e↑h −sin ξ√

2ZS(↑e↓h − ↓e↑h),

ψ+5 = c+v+ = −cos ξ√

2(X + iY )S ↓e↑h − sin ξZS ↑e↑h,

ψ−5 = c−v− = −cos ξ√

2(X − iY )S ↑e↓h +sin ξZS ↓e↓h .

The electric-dipole matrix elements for transitions between the ground state and the excitonstates are

〈ψ1|ε · p|0〉 = 0, 〈ψ2|ε · p|0〉 = i√2 sin ξεzP‖,

〈ψ+5 |ε · p|0〉 = −i cos ξε−P⊥, 〈ψ−

5 |ε · p|0〉 = i cos ξε+P⊥,

with ε± = 1√2(ex ± iey). Hence ψ2 is active with light excitation polarized along the z-

axis, and ψ±5 can be excited by absorbing right or left circularly polarized light in the

x − y plane. ψ1, however, is dark because it contains only spin triplets and is thereforedipole-forbidden. These four states split in energy because of exchange couplings, Hex =J‖σ

ezσ

hz + J⊥(σ

exσ

hx + σe

yσhy ), E1 = −J‖ − 2J⊥, E2 = −J‖ + 2J⊥, E5 = J‖.

Two orthogonal states, transverse and longitudinal states, can also be constructed out ofψ±5 states, ψ5T = 1√

2(c+v+ + c−v−) and ψ5L = 1√

2(c+v+ − c−v−).

ψ5T = −cos ξ

2(X + iY )S ↓e↑h +

cos ξ

2(X − iY )S ↑e↓h −sin ξ√

2ZS(↑e↑h + ↓e↓h),

ψ5L = −cos ξ

2(X + iY )S ↓e↑h −cos ξ

2(X − iY )S ↑e↓h −sin ξ√

2ZS(↑e↑h − ↓e↓h).






In applied magnetic field, the Hamiltonian of excitons includes both the exchange andZeeman energies,

H = Hex +HZ =J‖σezσ

hz + J⊥(σ

exσ

hx + σe

yσhy )

+1

2µB

[ge‖σ

ezHz + gh‖σ

hzHz + ge⊥(σ

exBx + σe

yBy) + gh⊥(σhxBx + σh

yBy)],

where σe/2 = se, and σh/2 = sh.Since the measurement is performed on polycrystalline samples, we consider two special

crystal orientations in the Voigt geometry.

(1) B ⊥ c-axis, and B ‖ x. TRFR measurement is along the z direction.

We can construct the four eigenstates. In this configuration, pair ψ1 and ψ5L, as well aspair ψ2 and ψ5T are coupled via the magnetic field,

H1 =

(E1 (gh⊥ − ge⊥)µBB/2

(gh⊥ − ge⊥)µBB/2 E5

).

H2 =

(E2 (gh⊥ + ge⊥)µBB/2

(gh⊥ + ge⊥)µBB/2 E5

).

And the eigenstates of H1 are

|Φ1+〉 = cos θ|ψ1〉+ sin θ|ψ5L〉, |Φ1−〉 = − sin θ|ψ1〉+ cos θ|ψ5L〉,

and corresponding energies are

λ1± = −J⊥ ± 1

2

√4(J‖ + J⊥)2 + (ge⊥ − gh⊥)2µ2

BB2.

The eigenstates of H2 are

|Φ2+〉 = cosφ|ψ2〉+ sinφ|ψ5T 〉, |Φ2−〉 = − sinφ|ψ2〉+ cosφ|ψ5T 〉.

with corresponding energies of

λ2± = J⊥ ± 1

2

√4(J‖ − J⊥)2 + (ge⊥ + gh⊥)2µ2

BB2.

θ and φ are coefficients determined by exchange couplings, g-factors, and applied magneticfield. If we define two ratios of the Zeeman energies vs. exchange couplings,

r1 =(ge⊥ − gh⊥)µBB

E5 − E1

=(ge⊥ − gh⊥)µBB

2(J‖ + J⊥),

r2 =(ge⊥ + gh⊥)µBB

E5 − E2

=(ge⊥ + gh⊥)µBB

2(J‖ − J⊥),

then

cos2 θ =r21

2(1 + r21 +√1 + r21)

, cos2 φ =r22

2(1 + r22 +√1 + r22)

.






The energy levels of four states are plotted in Figure 3 in the main text as a function ofmagnetic field.

At t = 0, circularly polarized light (E+ = 1√2E(ex+ iey)) creates exciton state ψ+

5 , so the

initial wave function is a superposition of all four states,

|Ψ(t = 0)〉 = 1√2[sin θ|Φ1+〉+ cos θ|Φ1−〉+ sinφ|Φ2+〉+ cosφ|Φ2−〉] ,

where an overall coefficient cos ξ is omitted for simplicity, but we will take it into accountlater when we compare the optical responses in different directions. The time evolution of thewave function depends on all four energies. The Faraday rotation at time t is proportionalto sz, and measures the population difference between ψ+

5 and ψ−5 .

sz ∝1

2

[|〈ψ+

5 |Ψ(t)〉|2 − |〈ψ−5 |Ψ(t)〉|2

]

=1

4

∣∣sin2 θe−iλ1+t + cos2 θe−iλ1−t + sin2 φe−iλ2+t + cos2 φe−iλ2−t∣∣2

− 1

4

∣∣− sin2 θe−iλ1+t − cos2 θe−iλ1−t + sin2 φe−iλ2+t + cos2 φe−iλ2−t∣∣2

= c1 cos(λ2− − λ1−)t+ c2 cos(λ2+ − λ1−)t+ c3 cos(λ1+ − λ2−)t+ c4 cos(λ1+ − λ2+)t.

The four coefficients ci are related to θ and φ, specifically

c1 = cos2 θ cos2 φ, c2 = cos2 θ sin2 φ, c3 = sin2 θ cos2 φ, c4 = sin2 θ sin2 φ,

which are all equal to 1/4 in high magnetic field, but strongly depend on magnetic field inthe low field regime.

The θF contains the four beating frequencies, ν1,2,3,4

hν1 = (λ2−−λ1−) = 2J⊥+1

2

√4(J‖ + J⊥)2 + (ge⊥ − gh⊥)2µ

2BB

2−1

2

√4(J‖ − J⊥)2 + (ge⊥ + gh⊥)2µ

2BB

2.

hν2 = (λ2+−λ1−) = 2J⊥+1

2

√4(J‖ + J⊥)2 + (ge⊥ − gh⊥)2µ

2BB

2+1

2

√4(J‖ − J⊥)2 + (ge⊥ + gh⊥)2µ

2BB

2.

hν3 = (λ1+−λ2−) = −2J⊥+1

2

√4(J‖ + J⊥)2 + (ge⊥ − gh⊥)2µ

2BB

2+1

2

√4(J‖ − J⊥)2 + (ge⊥ + gh⊥)2µ

2BB

2.

hν4 = (λ1+−λ2+) = −2J⊥+1

2

√4(J‖ + J⊥)2 + (ge⊥ − gh⊥)2µ

2BB

2−1

2

√4(J‖ − J⊥)2 + (ge⊥ + gh⊥)2µ

2BB

2.

The four frequencies and their amplitudes are plotted in Fig. S8 as a function of magneticfield B, with J⊥ = 1.0 µeV, J‖ = 1.5 µeV, ge⊥ = 2.65, gh⊥ = −0.32. The four frequenciescan be grouped into a fast frequency pair (ν2 and ν3), and a slow frequency pair (ν1 andν4), and both pairs are separated by 4J⊥. This calculation resembles the data if each pairmerge into one. The energy of the four states and the four frequencies will become distinctlydifferent and deviate significantly away from a linear function of B in the same magneticfield range, if the exchange couplings are larger by one order of magnitude or more [7].

We now derive linear function approximations to the four frequencies. Since the Zeemanenergy at 100 mT is still one order of magnitude larger than µeV for the exchange couplings,it is valid that 2(J‖±J⊥)/(ge∓gh)µBB 1 for the transverse magnetic field range, in whichthe Faraday rotation oscillation frequencies can be measured. Doing a Taylor’s expansionand keeping only the linear terms of this small quantity, we obtain






30

25

20

15

10

5

0

Freq

uenc

y (G

Hz)

6004002000B (mT)

ν1 ν2 ν3 ν4

1.0

0.8

0.6

0.4

0.2

0.0

Ampl

itude

6004002000B (mT)

c1 c2 c3 c4

a b

Figure S 8: Four frequencies. a, magnetic field dependence of the four frequencies. b, the

amplitudes of four frequencies in the TRFR measurement. The parameters chosen for simulation

are J⊥ = 1.0 µeV, J‖ = 1.5 µeV, ge⊥ = 2.65, gh⊥ = −0.32.

hν2 ≈ ge⊥µBB + 2J⊥, hν3 ≈ ge⊥µBB − 2J⊥

hν1 ≈ −gh⊥µBB + 2J⊥, hν4 ≈ −gh⊥µBB − 2J⊥.

(2) B ‖ c-axis, along z direction. TRFR measurement is along the x direction.

The four eigenstates are: ψ±5 with energies E5±, ψ1 and ψ2 (mixed from ψ1 and ψ2).

E5± = J‖ ±1

2(ge‖ + gh‖)µBB.

ψ1 and ψ2 are mixed in magnetic field with

H3 =

(E1 (ge‖ − gh‖)µBB/2

(ge‖ − gh‖)µBB/2 E2

),

and the new eigenstates are ψ1,2 with energies E1,2.

E1,2 = −J‖ ±1

2

√16J2

⊥ + (ge‖ − gh‖)2µ2BB

2.

When the optics measurement is along x-axis, the electric fields of RCP/LCP light canbe expressed as

E± =1√2E(ey ± iez).

The excited states are E± · p|0〉 with p being the dipole operator and |0〉 the ground state.They can be expressed in terms of the four eigenstates.

|ψR〉 = −i1

2cos ξ(ψ+

5 − ψ−5 ) + i sin ξ(− sinαψ1 + cosαψ2)

|ψL〉 = −i1

2cos ξ(ψ+

5 − ψ−5 )− i sin ξ(− sinαψ1 + cosαψ2).






Where (− sinαψ1 + cosαψ2) = ψ2, and the coefficient α can be obtained by solving H3.Assuming RCP pump, Ψ(t = 0) = ψR, and

Ψ(t) = −i1

2cos ξ(e−iE5+t/ψ+

5 − e−iE5−t/ψ−5 ) + i sin ξ(−sinαe−iE1t/ψ1 + cosαe−iE2t/ψ2).

Thus the Faraday rotation along x-axis is proportional to sx,

sx ∝1

2

[|〈ψR|Ψ(t)〉|2 − |〈ψL|Ψ(t)〉|2

]

=1

8

∣∣∣cos2 ξe−iE5+t/ + cos2 ξe−iE5−t/ + 2 sin2 ξ sin2 αe−iE1t/ + 2 sin2 ξ cos2 αe−iE2t/∣∣∣2

− 1

8

∣∣∣cos2 ξe−iE5+t/ + cos2 ξe−iE5−t/ − 2 sin2 ξ sin2 αe−iE1t/ − 2 sin2 ξ cos2 αe−iE2t/∣∣∣2

=2 cos2 ξ sin2 ξ

[1

2sin2 α cos(E5+ − E1)t/+

1

2sin2 α cos(E5− − E1)t/

+1

2cos2 α cos(E5+ − E2)t/+

1

2cos2 α cos(E5− − E2)t/

]

Similar to the other crystal orientation, the linear approximation to the four frequenciesare ge‖µBB ± 2J‖ and −gh‖µBB ± 2J‖, when the spin exchange couplings are much smallerthan the Zeeman energy.

The magnitude of sx differs from that of sz roughly by a factor of 2 sin2 ξ/ cos2 ξ if pumpedwith same RCP/LCP light. Rigorously, in configuration (1), there is a coefficient cos ξ inthe RCP light excited state, and cos4 ξ in the derived sz

2 sin2 ξ/ cos2 ξ =

1 when sin ξ = 1/

√3 in cubic phase;

0.4112/(1− 0.4112) = 0.4 when sin ξ = 0.411 is used for tetragonal phase.

Estimation of g-factors and exchange couplingsBased on the above derivations, it is straightforward to see that the Voigt geometry

TRFR measurement is sensitive to g-factors and exchange couplings along the applied fielddirection. The measured oscillation frequencies on the polycrystalline samples should beweighted averages over all crystal orientations. Average |ge| and |gh| are simply the slopesof frequencies vs. B (Figure 2c), if we consider each pair of very close frequencies mergeinto one frequency. Because of the two distinct slopes, it is easy to differentiate ge and ghby comparing them to the calculated g-values by Yu [7]. We assign g1 to ge = 2.63 and g2to gh = −0.33.

The exchange couplings can be estimated to be on the order of µeV by considering thefollowing two facts: i) The observation of two frequencies instead of four in magnetic fieldimplies that 4J‖,⊥ is smaller than the energy/frequency resolution, 4J‖,⊥ < h/T ∗

2 ∼ 5µeV(esitmated from TRFR curve in B = 700 mT). The four frequencies can be grouped intoa fast frequency pair and a slow frequency pair, and both pair differ by 4J‖,⊥. ii) Thevertical intercept of the linear fitting to frequency vs. B provides an lower bound estimateof the exchange couplings, 2J‖,⊥ ≥ 1.18 µeV (intercept of slow frequency vs. B). Thefour frequencies could all be approximated as linear functions of B using Taylor’s expansionin high field limit, and the intercepts are ±2J‖,⊥. Depending on the amplitudes of eachfrequency, the intercept should still be in the range of (−2J‖,⊥,+2J‖,⊥). When magnetic






field is high, the amplitudes of four frequencies are equal, but in low magnetic fields, thedominant one is the low frequency one with −2J⊥ intercept, therefore it is reasonable thatthe fast frequency vs. B has zero intercept, and the slow frequency vs. B has a slightlynegative intercept.

6. Discussion of spin relaxation mechanisms

Our experimental data indicate that the spin relaxation time τs has a temperature de-pendence of τs ∝ T−1.5 between 10 K and 100 K. Spin relaxation in a semiconductorlike CH3NH3PbI3 is ultimately caused by spin-orbit coupling (SOC) and carrier scatter-ing. Common spin relaxation mechanisms include the Elliott-Yafet, D’yakonov-Perel’, andBir-Aronov-Pikus mechanisms. Here we examine these mechanisms against the experimentalresult.

Elliott-Yafet mechanismIn the Elliott-Yafet (EY) mechanism, the SOC causes the mixing of wave functions with

opposite spins. Consequently, in the presence of momentum scattering, the electron spinbecomes disoriented. The rate of spin relaxation, or spin-flip scattering, can be expressed as

τ−1s ∝ τ−1

p χ2,

where χ describes the SOC-induced spin-mixing and τp is the momentum scattering time,which is related to the carrier mobility via µ = eτp/m

∗ with m∗ being the carrier effectivemass. Typically, the spin-mixing factor in a semiconductor can be written as

χ ∼ σ · (k× k′)

Eg

where σ are the Pauli’s matrices, Eg is the band gap, and k (k′) is the carrier’s wave vectorbefore and after the scattering. Thus the spin relaxation rate due to the E-Y mechanismwould be

τ−1s ∝ τ−1

p

(Ek

Eg

)2

∝ τ−1p T 2,

where the system is nondegenerate and the additional temperature dependence T 2 comesfrom the average of carrier’s momentum distribution.

In literature, the carrier momentum scattering in CH3NH3PbI3 is due mainly to phononscattering with a temperature dependence of τp ∝ T−3/2 for temperatures above 100 K. Ifwe assume that this temperature dependence remains valid between 10 K and 100 K, thespin relaxation time due to the EY mechanism would have a temperature dependence ofτs ∝ T−7/2, which is inconsistent with experiment. To explain the observed τs ∝ T−7/2

via the EY mechanism, the momentum scattering time should depend on temperature asτp ∝ T 1/2.

We would like to point out the difference between a metal [Sci. Rep. 6, 22706] and awide-gap semiconductor like CH3NH3PbI3. In the former, the momentum scattering occursat the Fermi energy, which is much larger than kBT , and consequently, the average ofkinetic energy Ek over the Maxwell distribution is virtually independent of temperature.Whereas in the latter, the carriers concentration is low and their kinetic energy is sensitiveto the temperature. This difference gives rise to distinct temperature dependences in spin






relaxation. In particular, in the metal the EY mechanism would suggest a same temperaturedependence of τs and τp.

D’yakonov-Perel’ mechanismThe D’yakonov-Perel (DP) mechanism becomes operative for systems without an inver-

sion center, where the spin degeneracy is lifted for k = 0, i.e., electrons with a same wavevector k but opposite spins have different energies. This energy difference, denoted as Ω(k)for a given k, can be regarded as the Larmor precession frequency of an effective internalmagnetic field. The momentum scattering would interrupt the spin precession and reducethe spin relaxation rate.

τ−1s ∝ τpΩ2(k).

If we assume that CH3NH3PbI3 has the C4v symmetry, which is inversion asymmetric [41],

Ωx,y kx,y.

After averaging the Maxwell distribution, the spin relaxation rate due to the DP mechanismin CH3NH3PbI3 is

τ−1s ∝ τpk

2 ∝ τ − pT.

Again, if we assume τp ∝ T−3/2 between 10 and 100 K, the spin relaxation time wouldincrease with the temperature as τs ∝ T 1/2, which is inconsistent with experiment. Toexplain the dependence of τ2 ∝ T−3/2, via the DP mechanism, the momentum scatteringshould have the temperature dependence of τp ∝ T 1/2.

Bir-Aronov-Pikus mechanismThe Bir-Aronov-Pikus (BAP) mechanism describes electron spin relaxation due to ex-

change via scattering on holes, which can be expressed as

τ−1s ∝ Npa

3B

∆2

EB

kaB,

where Np is the hole concentration, EB is the exciton binding energy, aB is exciton Bohrradius, and ∆ is the exchange coupling. In our experiment, spin relaxation due to theBAP mechanism would be negligible because the hole concentration is low and the exchangecoupling is very small, as determined by our experiment.

Based on the above analysis, the observed temperature dependence of spin relaxationtime in the hybrid perovskite may not be readily explained by existing theories of spinrelaxation in semiconductors, although it seems that the E-Y mechanism better accountsfor the experimental results. It is possible that in CH3NH3PbI3 spin relaxation is due to thecombination of various mechanisms described above and/or to some novel mechanism to beidentified. A careful mobility measurement would be very useful for developing a systematictheory to elucidate the unusually long spin lifetime in hybrid perovskites.

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SUPPLEMENTARY INFORMATION - Nature...Chuang Zhang 1, Dali Sun , Zhi-Gang Yu2, Z. Valy Vardeny 1, Yan...

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